Eight Books on Mechanics, in which, by one and the same principle of the lever, the forces of machines are physically explained and geometrically demonstrated, and a method of composing machines of every kind is set forth

Creator: Paolo Casati | Date: 1684 | Notes: Original title: Mechanicorum libri octo, in quibus uno eodemque principio vectis Machinarum vires physice explicantur & geometrice demonstrantur A systematic Latin didactic treatise on mechanics in eight books, written in the mixed-mathematical tradition and printed at Lyon in 1684 under royal privilege by the Anisson–Posuel–Rigaud partnership. Distilling decades of the author's lecture courses at the Collegio Romano, it reduces all the simple machines—the lever, the balance, the pulley, the wheel and axle, the wedge, and the screw—to a single principle of the lever, whereby advantage gained in weight is paid for exactly in velocity, a proportion Casati calls a perpetua quædam justitia. The work couples Archimedean geometric demonstration with a physical account of natural and forced motion cast in the scholastic vocabulary of impetus, defends a positive levity as a real quality, and treats gravity, the centre of gravity, the inclined plane, percussion, reflection, and the law of the screw. Organized in chapters rather than theorems, it interleaves geometrical demonstration with the investigation of the physical causes of mechanical effects, drawing on tower observations at Bologna, correspondence with Grimaldi, and reports from Martini's Atlas Sinicus. 👉 <a href="https://tryleo.ai/collections/exlatinis/UID key not found">Read our introductory primer, full report, and finding guide here</a> 📜 <a href="https://archive.org/details/ita-bnc-mag-00000750-001">View the original source file</a> This text was transcribed and translated as part of the ExLatinis project—an effort by Leo to make English translations of every published text in Latin in early modern Europe (between 1450 and 1750) available to the public for free online.

Title: Eight Books on Mechanics, in which, by one and the same principle of the lever, the forces of machines are physically explained and geometrically demonstrated, and a method of composing machines of every kind is set forth
Creator: Paolo Casati
Date: 1684
Notes: Original title: Mechanicorum libri octo, in quibus uno eodemque principio vectis Machinarum vires physice explicantur & geometrice demonstrantur A systematic Latin didactic treatise on mechanics in eight books, written in the mixed-mathematical tradition and printed at Lyon in 1684 under royal privilege by the Anisson–Posuel–Rigaud partnership. Distilling decades of the author's lecture courses at the Collegio Romano, it reduces all the simple machines—the lever, the balance, the pulley, the wheel and axle, the wedge, and the screw—to a single principle of the lever, whereby advantage gained in weight is paid for exactly in velocity, a proportion Casati calls a perpetua quædam justitia. The work couples Archimedean geometric demonstration with a physical account of natural and forced motion cast in the scholastic vocabulary of impetus, defends a positive levity as a real quality, and treats gravity, the centre of gravity, the inclined plane, percussion, reflection, and the law of the screw. Organized in chapters rather than theorems, it interleaves geometrical demonstration with the investigation of the physical causes of mechanical effects, drawing on tower observations at Bologna, correspondence with Grimaldi, and reports from Martini's Atlas Sinicus. 👉 <a href="https://tryleo.ai/collections/exlatinis/UID key not found">Read our introductory primer, full report, and finding guide here</a> 📜 <a href="https://archive.org/details/ita-bnc-mag-00000750-001">View the original source file</a> This text was transcribed and translated as part of the ExLatinis project—an effort by Leo to make English translations of every published text in Latin in early modern Europe (between 1450 and 1750) available to the public for free online.

Document notes

Original title: Mechanicorum libri octo, in quibus uno eodemque principio vectis Machinarum vires physice explicantur & geometrice demonstrantur A systematic Latin didactic treatise on mechanics in eight books, written in the mixed-mathematical tradition and printed at Lyon in 1684 under royal privilege by the Anisson–Posuel–Rigaud partnership. Distilling decades of the author's lecture courses at the Collegio Romano, it reduces all the simple machines—the lever, the balance, the pulley, the wheel and axle, the wedge, and the screw—to a single principle of the lever, whereby advantage gained in weight is paid for exactly in velocity, a proportion Casati calls a perpetua quædam justitia. The work couples Archimedean geometric demonstration with a physical account of natural and forced motion cast in the scholastic vocabulary of impetus, defends a positive levity as a real quality, and treats gravity, the centre of gravity, the inclined plane, percussion, reflection, and the law of the screw. Organized in chapters rather than theorems, it interleaves geometrical demonstration with the investigation of the physical causes of mechanical effects, drawing on tower observations at Bologna, correspondence with Grimaldi, and reports from Martini's Atlas Sinicus. 👉 Read our introductory primer, full report, and finding guide here 📜 View the original source file This text was transcribed and translated as part of the ExLatinis project—an effort by Leo to make English translations of every published text in Latin in early modern Europe (between 1450 and 1750) available to the public for free online.

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CASATI MECHANICA 1 6 199

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CASATI MECHANICA 1 6 199

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1 6 199 BIBLIOTECA NAZIONALE CENTRALE - FIRENZE

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1 6 199 NATIONAL LIBRARY CENTRAL - FLORENCE

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MECHANICA CAPSALI R.P. PAULI

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MECHANICA CAPSALI R.P. PAULI

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R. P. PAULI CASATI PLACENTINI SOCIET. JESU MECHANICORUM LIBRI OCTO, IN QUIBUS UNO EODEMQUE principio Vectis vires Physicè explicantur & Geometricè demonstrantur, Atque Machinarum omnis generis componendarum methodus proponitur. LUGDUNI, Apud ANISSONIOS, IOAN. POSUEL & CLAUDIUM RIGAUD. M. DC. LXXXIV. CUM PRIVILEGIO REGIS

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Reverend Father PAULI CASATI PLACENTINI of the Society of Jesus Mechanics Eight Books, in which, by one and the same principle, the powers of the lever are physically explained and geometrically demonstrated, and the method of composing machines of every kind is proposed. Lyon, at the press of the ANISSONS, IOAN. POSUEL and CLAUDIUS RIGAUD. 1684. With the King’s Privilege

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1. 6. 199

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1. 6. 199

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CHRISTIANISSIMO GALLIARUM ET NAVARRÆ REGI LUDOVICO MAGNO. AD Majestatis Tuæ pedes, REX INVICTISSIME, me, meamque hanc de rebus Mechanicis lucubrationem, ignotus homo, vix fortasse credibili confidentiâ, isto: Sed quâ Regiâ comitate omnium animos concilias, eâdem sustentor, ne repulsam timeam. In Te Orbis universi conjecti sunt oculi, quos Tuæ Gloriæ splendor allicit: à communi felicâ

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MOST CHRISTIAN KING OF THE GAULS AND OF NAVARRE, LOUIS THE GREAT. At the feet of Your Majesty, MOST INVINCIBLE KING, I, an unknown man, and this my little work on mechanical matters, with perhaps scarcely credible confidence, place before you. But by that royal graciousness with which You win the hearts of all, I am sustained, lest I fear rejection. Upon You the eyes of the whole world are fixed, which the splendor of Your Glory attracts: from the common blessedness

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citate quid me paterer excludi? Amplissima Tua in Societatem nostram merita, quorum nullam partem, ne cogitandâ quidem gratiâ, consequi possumus, hoc saltem officij ab universo Ordine re- petunt, ut singuli, quem cordi penitissimè impressum gestamus non ingrati LUDOVICUM, in libris palàm inscriptum velimus. Me verò Natu- ræ atque Artis mutuam societatem coëuntium in Machinis, ferè dixerim, miracula contemplari assuetum rapuere admirabundum, quæ ipse patrasti, & bello, & pace, egregia atque præclara facinora non modò mirabilia, sed prodigiis similia. Neque illa quidem aut ex rerum magnitudine ac difficul- tate, aut ex multiplicato numero, aut ex dissimi- lium varietate, aut ex serie non interruptâ, me- tienda duxi, quamquam & in his admirabilitatis plurimum insit: Verùm longè omnem admirationem multumque superare mihi videtur, quòd paucis lustris vel sæcula complexus, unus pluribus Regibus par, tot, tantáque perficere valuisti. Ingentis pon- deris gravitatem vincit adhibita Machina, sed diuturno impulsu agitanda, ut proficiat aliquid: At plurima immensis munita difficultatibus exiguo tem- poris spatio expugnare, atque ad optatum exitum perducere, ita Tuum est, REX INVICTISSIME, ut quemadmodum rerum gestarum gloriâ, ac nomi- nis celebritate, nemini superiorum Regum secundus prædica

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Why should I say that I would be shut out? Your most ample merits toward our Society, of which we can obtain no part, not even by the thought of gratitude, at least demand this duty from the whole Order: that each of us, bearing in our inmost heart the non-forgotten LUDOVICUS, whom we cherish, should wish him to be openly inscribed in our books. But I, accustomed to contemplate, with wonder, the mutual alliance of Nature and Art joining together in Machines, may almost say, miracles, have been carried away in admiration by the things you yourself have achieved: both in war and in peace, your outstanding and illustrious deeds are not only wonderful, but akin to prodigies. Nor indeed did I judge those things by their greatness and difficulty, or by their multiplicity, or by the diversity of unlike matters, or by their unbroken sequence, though in these too there is very much to admire: rather, it seems to me far to surpass all admiration, and by much, that, having embraced within a few lustra, or even centuries, you alone, equal to many Kings, have been able to accomplish so many and so great things. The weight of a vast burden is overcome by a machine when applied, but it must be driven by a long-continued impulse if it is to make any progress: yet to conquer within a short space of time the greatest things fortified with immense difficulties, and to bring them to the desired outcome, is so much your own, MOST INVINCIBLE KING, that, just as in glory for your achievements and in the fame of your name, second to none of the Kings who came before you, you are to be proclaimed

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prædicaris, sic Tibi secundum, qui Tuis planè in- sistat vestigiis, ventura sæcula sperare vix audeant. Patere igitur pro summâ, quâ præditus es, huma- nitate, qualemcumque hanc rerum Mechanicarum tractationem Regio insigniri Nomine, ut, quos meas hasce commentationes legere non piguerit, vel hinc discant, aliud esse non imitabile genus Facultatis, quâ ingentia citò perficiantur, si LUDOVICI MAGNI mens accesserit. In columnem Te diu servet DEUS Catholica Fidei incremento, Regnique Tui felicitati; audiat- que bonorum omnium Largitor vota, quæ pro Ma- jestate Tuâ supplex nuncupat MAJESTATIS Tuæ Parmæ Kal. Maij 1683. Humillimus atque Obsequentissimus Servus PAULUS CASATUS è Soc. IESU.

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You are acclaimed in such a way that those who, second to You, would quite closely follow in Your footsteps may scarcely dare to hope for future ages. Therefore, out of that supreme humanity with which You are endowed, permit this treatise on Mechanical matters to be distinguished by the Royal Name, so that those who have not begrudged reading these my commentaries may learn even from this that there is another, inimitable kind of ability, by which great things are quickly accomplished, if only the mind of LUDOVICUS MAGNUS be applied. May GOD long preserve You as a pillar, for the increase of the Catholic Faith and for the happiness of Your Kingdom; and may the Giver of all good things hear the prayers which, as a suppliant, I now address for Your Majesty. Of Your Majesty, Parma, the Kalends of May 1683. Your most humble and most obedient servant, PAULUS CASATUS of the Society of JESUS.

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Facultas R. P. Provincialis Societatis Iesu in Provincia Veneta. Ego Octavius Rubeus Societatis Iesu in Provincia Veneta Præpositus Provincialis, potestate ad id mihi factâ ab Adm. R. P. N. Præposito Generali Io. Paulo Oliva, facultatem facio, ut Opus inscriptum, Mechanicorum Libri octo, Authore P. Paulo Casato Societatis Nostræ Sacerdote, ejusdem Societatis Doctorum hominum judicio approbatum, typis mandetur, si ita iis, ad quos pertinet, videbitur. Cujus rei gratiâ has litteras meâ manu subscriptas, & sigillo officij mei munitas dedi. Parmæ 23. Februarij 1681. OCTAVIUS RUBEUS. Summa Privilegij à Christianissimo Rege concessi. LUDOVICUS MAGNUS Galliarum & Navarræ Rex Christianissimus, Diplomate suo sanxit, nequis per universos Regnorum suorum fines intra decem proximos annos à die publicationis exemplarium computandos, imprimat seu typis excudendum curet & venale habeat Opus quod inscribitur, Mechanicorum Libri octo, Authore R. P. Paulo Casato Soc. Iesu; præter Anissonios Bibliopolas Lugdunenses, aut illos quibus ipsimet concesserint. Prohibuit insuper eadem auctoritate Regia omnibus suis subditis, idem Opus extra Regni sui limites imprimendum curare, & impressum divendere, vel quempiam ubicumque fuerit ad id agendum impellere; ac instigare fine consensu dictorum ANISSONIORUM; Qui secus faxit, confiscatione librorum, aliaque gravi poenâ multabitur, uti latius patet in diplomate regio. Dabatur Versaliis die vigesima prima Ianuarij anno Dom. 1684. Ex mandato Regis. JUNQUIERES. MECHA

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Permission of the Reverend Father Provincial of the Society of Jesus in the Venetian Province. I, Octavius Rubeus of the Society of Jesus in the Venetian Province, Provincial Superior, by authority granted to me for this purpose by the Most Reverend Father General, John Paul Oliva, give permission that the work entitled Mechanicorum Libri octo , by Father Paul Casato, priest of our Society, approved by the judgment of the learned men of the same Society, be put to press, if so it seem good to those to whom it pertains. For which matter I have given these letters, signed by my hand and sealed with the seal of my office. At Parma, February 23, 1681. OCTAVIUS RUBEUS. Summary of the privilege granted by the Most Christian King. LOUIS THE GREAT, Most Christian King of France and Navarre, by his diploma ordained that no one, throughout all the bounds of his kingdoms, within the next ten years, reckoned from the day of publication of the copies, shall print or cause to be printed and keep for sale the work entitled Mechanicorum Libri octo , by the Reverend Father Paul Casato of the Society of Jesus; except the Anisson booksellers of Lyons, or those to whom they themselves shall have granted it. By the same royal authority he further forbade all his subjects to cause the same work to be printed outside the limits of his kingdom, or to sell the printed book, or to urge or incite anyone wherever he may be to do so, without the consent of the said ANISSONS; whoever acts otherwise shall be punished by confiscation of the books, and by another severe penalty, as is more fully set forth in the royal diploma. Given at Versailles on the twenty-first day of January in the year of Our Lord 1684. By the King's command. JUNQUIERES. MECHA

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AD LECTOREM. ERO in lucem prodit hæc Mechanicorum tractatio, & vix fide me abduco, quam dedi, cùm Dissertationes de Terrâ Machinis motâ quasi Prodromum emisi ante plures annos: scilicet à studiis tunc abstractus, utpote alieni juris, & ad munera his non affinia translatus, multam salutem & Mathematicis disciplinis & Physicis dicere coactus sum; adeò ut demum tot elapsis annis urgente jam senio cogitationem omnem abjecerim de hujusmodi commentationibus, diffidens me posse ad hanc scriptionem satis temporis invenire, quin eam proxima mors interciperet, & susceptum alienissimo tempore laborem irritum faceret. Adde quòd (pro meâ negligentiâ, quæ calamo parcit) temporis diuturnitate deletæ ex animo pleræque imagines vix tenue vestigium reliquerant, cui novis inductis coloribus eas redintegrari posse considerem. Amicorum tamen officiosis stimulis me urgeri passus sum, ut subcisivis, quæ incurrebant, temporibus tentarem, an destinatam animo tractationem, cujus brevem Synopsis auditoribus meis in Romano Collegio, anno labentis sæculi decimi septimi quinquagesimo quarto, tradideram, redordiri, & aliquâ ratione perficere liceret. Licuit autem, præter spem, toties dimissum calamum resumere, ut tan- e

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TO THE READER. This treatise on Mechanics is now brought to light, and I can scarcely refrain from believing the promise I made when, many years ago, I issued my Dissertationes de Terrâ Machinis motâ as a sort of preface: namely, that at the time I was withdrawn from studies, inasmuch as I had been transferred to offices not connected with them, and was compelled to bid much farewell to both Mathematical and Physical disciplines; so that at length, after so many years had passed and old age was now pressing upon me, I had dismissed every thought of writings of this kind, doubting that I could find enough time for this composition, without death intercepting it near at hand, and rendering vain the labor undertaken at a most unsuitable time. Add to this that (through my negligence, which spares the pen) the images, worn away in the course of time from my mind, had left scarcely a faint trace, to which I might think they could be restored by the introduction of new colors. Yet by the dutiful urgings of friends I allowed myself to be pressed, so that in the spare moments that occurred I should try whether it might be permitted to revive, and in some way complete, the treatise I had intended in my mind, of which I had delivered a brief Synopsis to my hearers in the Roman College, in the year of the waning seventeenth century and fifty-fourth, and then to complete it in some way. And indeed it was permitted, beyond hope, to take up again the pen so often laid aside, so tha- e

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AD LECTOREM. dem de singulis Mechanicis Facultatibus aliquid me scripsisse invenerim, quod Mathematicarum disciplinarum candidatis profuturum amici censuerunt, si publici juris fieret. Quapropter alienæ utilitati serviendum potiùs fuit, quàm meæ voluntati. Verùm ne te moveat, Amice Lector, quòd Mechanici inscribantur libri, cùm tamen aliqua ad Centrobaryca, aliqua ad Statica pertineant. Cùm enim hæc ad pleniorem eorum intelligentiam, quæ de Machinis disputanda erant, referantur, nomen à scopo desumendum fuit: Nec deerat ex Aristotele (si tamen ipsi tribuenda sit illa tractatio) suffragium, qui Mechanicas Quæstiones inscripsit libellum, in quo non de solis Mechanicis facultatibus agitur. Methodum ne culpes, quòd non in Theoremata & Propositiones rem totam digesserim, sed in Capita distribuerim, & quidem aliquando longiuscula: Brevitati nimi-rum studens non amavi codicem titulis implere, ne fortè, ad ostendendam consequentium cum præcedentibus connexionem, cogerer idem sæpiùs inculcare. Facilius autem duxi ea, quæ conjuncta sunt, uno eodemque capite complecti, ut ex ipsâ verborum consecutione rerum cognatio innotescat. Præterquam quod, si formâ illâ Mathematicis familiari usus fuissem, animum fortasse induxisses, me mihi ineptè blandiri, & quasi Geometricas ratiocinationes obtrudere ea, quæ satis probabili conjecturâ stabilire conatus sum. Quamvis enim non pauca attulerim, quæ Geometricas demonstrationes recipiunt, nec mihi videar pseudographis syllogismis deceptus; quia tamen & apud Physicos & apud Mathematicos agenda erat causa, multa fuere ad Philosophicas rationes revocanda; & quidem, quoad ejus fieri potuit, à receptis in scho- lis

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TO THE READER. I have found that I had written something on each of the Mechanical Faculties, which my friends judged would be of use to candidates in the mathematical disciplines, if it were made public. Therefore I was bound rather to serve the utility of others than my own will. But let it not move you, friendly reader, that the books are entitled Mechanical, although some things pertain to Centrobarics and some to Statics. For since these are referred to a fuller understanding of those matters which had to be discussed concerning machines, the name had to be taken from the end in view: nor was there lacking the approval of Aristotle (if indeed that treatise should be attributed to him), who inscribed a little book Mechanical Questions, in which it is not only mechanical faculties that are discussed. Do not find fault with the method, because I did not arrange the whole matter into Theorems and Propositions, but divided it into Chapters, and indeed sometimes somewhat long ones. Devoted above all to brevity, I did not like to fill the volume with titles, lest perhaps, in order to show the connection of what follows with what precedes, I should be forced to repeat the same thing more than once. I found it easier to include in one and the same chapter those things which are connected, so that from the very sequence of the words the kinship of the matters may become clear. Besides, if I had used that form familiar to mathematicians, you might perhaps have believed that I was foolishly flattering myself, and as it were thrusting geometrical reasonings upon matters which I have tried to establish by sufficiently probable conjecture. For although I have brought forward not a few things that admit of geometrical demonstrations, and I do not seem to myself to have been deceived by pseudo-logical syllogisms, yet because the case had to be handled both before physicists and before mathematicians, many things had to be brought back to philosophical reasoning; and indeed, as far as it could be done, from the accepted in the schools...

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AD LECTOREM. lis opinionibus mihi non erat hîc recedendum, ne quid temerè sine argumentis proferrem, aut ne longiùs ab in- stituto recederem, si quid novi, quæsitâ veri similitudine, molirer. Hoc videlicet mihi potissimum curæ fuit, ut Phy- sicam admirandorum per Machinas motuum causam in- vestigarem: in Physicis autem modum sciendi Geome- tricum inquirens, ne ab Aristotele redarguerer, timerem. Quare alia Geometricè, alia Physicè tractata æquo animo patere. Stylum autem quid excusem? Non est, fateor, con- stans & perpetuus, siúque similis: tum quia non eadem semper subjecta materia est, tum quia, prout tempus fe- rebat, animum inæqualiter affectum ad scribendum at- tuli; nec poterat æquabiliter fluere toties intercisa oratio. Unum est inter cætera, quod fortasse desideres, nimi- rum illorum, qui de hoc eodem argumento scripserunt, sententias explicari, & quæ à me dicuntur, eorum autho- ritate muniri. Plurimum sanè mihi lucis affulsisset ex do- ctorum virorum Commentariis, neque contemnenda or- namenti accessio hujus meæ lucubrationis tenuitati fieret ex diversis Authorum opinionibus: Verùm ut nunc res se ha- bet, opportunâ librorum supellectile destitutus authorum mentionem facere plenam non potui, jejunam non debui, ne quis per contemptu prætermissus videretur. Mihi autem non ea est memoriæ firmitas, quæ, quid aliquando lege- rim, aut ubi legerim, satis explicatâ recordatione suggerat. Quòd si placuisset, corrogatis aliunde libris, magnificam hanc eruditionis pompam meæ qualicumque commenta- tioni adhibere, non satis otii ad legendum suppetebat, & nimium temporis postulasset scriptio, si exponendæ pri- mùm, dein confirmandæ aut refellendæ fuissent aliorum e ij

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TO THE READER. It was not my intention here to depart from those opinions, lest I should bring forward anything rashly and without arguments, or lest, if I attempted anything new under the appearance of probability, I should stray too far from the plan I had set myself. My chief concern, plainly, was to investigate in Physics the cause of those admirable motions accomplished by Machines: and in the Physical sciences, while seeking the geometric method of knowing, I feared lest I should be reproached by Aristotle. Therefore, accept in good part what is treated geometrically and what is treated physically. Why should I apologize for my style? I confess it is not constant and uniform, nor alike throughout; partly because the subject matter is not always the same, and partly because, as the time allowed, I brought to writing a mind affected unevenly; nor could discourse, so often interrupted, flow with equal smoothness. There is one thing among others that perhaps you may miss, namely, an explanation of the opinions of those who have written on this same subject, and support for what is said by me with their authority. Indeed, much light would have shone upon me from the Commentaries of learned men, and a not unworthy addition of ornament to the slenderness of this little work would have been made from the differing opinions of authors. But as matters now stand, being deprived of a suitable supply of books, I could not make a full mention of authors, and I ought not to have made a scant one, lest anyone should seem to have been passed over in contempt. For I do not have such firmness of memory that it can, with adequate recollection, tell me what I once read, or where I read it. And if it had pleased me to gather books from elsewhere and bring this splendid display of learning to bear upon my humble essay, I did not have enough leisure for reading, and the writing itself would have required too much time, if first the opinions of others had to be set forth, and then confirmed or refuted. e ij

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AD LECTOREM. sententiæ: propterea satius duxi, quæ animo occurrebant, pro meâ consuetudine breviter simplicitérque scribere, vix aliquando tactâ alicujus Authoris opinione, quam in adversariis jampridem notatam inveni. Nec te pluribus volo, Amice Lector. Multa habebis, quæ pro tuâ humanitate mihi condones, plura quæ amanuensi, plurima fortasse quæ Typographo, ubi præsertim de Numeris, & de Majori aut Minori Ratione sermo est; facilis enim contingit oscitanti hallucinatio, ut ab Auto- grapho aberret exemplar, & Numerus numero, verbum verbo commutetur: Non ægrè tamen ex adjunctis peti poterit correctio. In iis verò, in quibus à me per imprudentiam peccatum fuerit, à tuâ Sapientiâ facilè patiar me dedoceri. Vale. ELENCHUS

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TO THE READER. Since, as I have come to observe, opinions are diverse; therefore I judged it better to write briefly and simply, according to my usual practice, the thoughts that came to my mind, scarcely ever touching on the opinion of any author, than to seek one already long ago noted down in the notes. I do not wish to say more to you, friendly reader. You will find many things that, in your kindness, you may forgive me; more that you may attribute to the amanuensis; and perhaps very many to the printer, especially where numbers, and the greater or lesser ratio, are concerned. For it is easy for a careless person to slip into error, so that the copy departs from the autograph, and number is exchanged for number, word for word. Yet the correction may readily be sought from the context. But in those things in which I shall have erred through inadvertence, I shall easily allow myself to be corrected by your wisdom. Farewell. INDEX

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ELENCHUS CAPITUM. LIBER PRIMUS. De Centro Gravitatis. CAP.I. Qvid sit Centrum Gravium & Levium. II. An corpora prædita sint gravitate & levitate. III. Quid sit Centrum Gravitatis, & Linea Directionis. IV. An gravia centro vicina minùs gravitent. V. Qua ratione Centrum gravitatis corporum inveniatur. VI. Affertur ratio prædictarum praxeon. VII. Quomodo gravia sponte ascendentia descendant. VIII. Cur gravium in plano inclinato descendentium alia repant, alia rotentur. IX. Cur turres inclinatæ non corruant. X. An plurium structurarum capax sit Mons, quàm subjecta planities. XI. Quomodo animalium motus ordinentur ex centro gravitatis. XII. An tellus moveatur motu trepidationis. XIII. Qua ratione minuatur gravitatio in plano inclinato. XIV. Qua ratione corpus gravitet in planum inclinatum. XV. Inquiruntur Rationes gravitationis corporum suspensorum. XVI. Tractiones ac elevationes obliquæ expenduntur. LIBER SECUNDUS. De Causis Motûs Machinalis. CAP.I. QVem ad finem Machinæ instruantur. II. Impetus motum proxime efficientis naturæ explicatur. III. Qua ratione semel conceptus impetus pereat. IV. Qua ratione vis movendi cum impedimentis comparetur. V. In quo Machinarum vires sitæ sint. VI. Quid attendendum sit in Machine collocatione, atque materiæ. VII. Præstetne Machinam augere? an componere? 3

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TABLE OF CHAPTERS. BOOK ONE. On the Center of Gravity. CHAP. I. What the center of heavy and light bodies is. II. Whether bodies are endowed with heaviness and lightness. III. What the center of gravity is, and the line of direction. IV. Whether bodies nearer the center are less heavy. V. By what means the center of gravity of bodies is found. VI. The reason for the aforesaid procedures is given. VII. How bodies that ascend of themselves descend. VIII. Why among heavy bodies descending on an inclined plane some creep, others roll. IX. Why leaning towers do not fall. X. Whether a mountain is capable of supporting more structures than a plain surface beneath it. XI. How the motions of animals are ordered from the center of gravity. XII. Whether the earth moves by a motion of trembling. XIII. By what means gravitation is lessened on an inclined plane. XIV. By what means a body gravitates on an inclined plane. XV. The reasons for the gravitation of suspended bodies are investigated. XVI. Oblique traction and elevation are examined. BOOK TWO. On the Causes of Mechanical Motion. CHAP. I. To what end machines are constructed. II. Impetus, as the nature that immediately produces motion, is explained. III. By what means impetus once conceived perishes. IV. By what means the moving force is compared with impediments. V. In what the powers of machines are situated. VI. What must be considered in the placing of a machine and in the material. VII. Whether it is better to enlarge a machine or to combine it? 3

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ELENCHUS CAPITUM. VIII. Cur majores rotæ motum juvent præ minoribus. IX. Quid cylindri & Scytalæ ad faciliorem ponderis motum præstent. X. Circulorum Concentricorum motus explicatur. LIBER TERTIUS. De Libra. CAP.I. Libra forma & natura exponitur. II. Libra inæqualium brachiorum expenditur. III. Quomodo Corporum æquilibria explicentur. IV. An, & cur libra ab æquilibrio dimota ad illud redeat. V. An fieri possit libra Curva. VI. Quænam libra sint omnium exactissimæ. VII. Libra dolosæ vitia reteguntur, VIII. Stateræ Natura & Forma explicatur. IX. Antiquorum Stateræ examinatur. X. Libra & Stateræ usus extenditur. XI. Fundamenta præmittuntur ad explicandum, Cur gravia suspensa modò præponderent, modò æquilibria sint. XII. Præponderatio & Æquilibritas gravium fune suspensorum consideratur. XIII. An aliqua sit Libra Obliquæ utilitas. LIBER QUARTUS. De Vecte. CAP.I. Vectis forma & vires explicantur. II. Quid in hypomochlij collocatione sit observandum. III. Quæ ratione statuendus sit Ponderi locus in Vecte primi generis. IV. Momenta Ponderis in Vecte secundi generis considerantur. V. Quæ sit Ratio Vectis hypomochlium mobile habentis. VI. Quænam sint momenta Vectis Pondus fune connexum trahentis. VII. Quid conferat Potentiæmo ventis applicatio ad Vectem. VIII. Oneris ex Vecte pendentis momentum inquiritur. IX. An duo pondus gestantes æqualiter premantur. X. An

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TABLE OF CHAPTERS. VIII. Why larger wheels aid motion more than smaller ones. IX. What cylinders and scytalae contribute to the easier movement of a weight. X. The motion of concentric circles is explained. BOOK THREE. On the Balance. CHAP. I. The form and nature of the balance are explained. II. The balance with unequal arms is considered. III. How the equilibria of bodies are explained. IV. Whether, and why, a balance moved from equilibrium returns to it. V. Whether a curved balance can exist. VI. Which balances are the most exact of all. VII. The defects of a deceitful balance are exposed. VIII. The nature and form of the steelyard are explained. IX. The steelyard of the ancients is examined. X. The uses of the balance and steelyard are extended. XI. Foundations are laid for explaining why heavy bodies suspended by a cord are now in excess, now in equilibrium. XII. The preponderance and equilibrium of heavy bodies suspended by a cord are considered. XIII. Whether there is any use in an oblique balance. BOOK FOUR. On the Lever. CHAP. I. The form and powers of the lever are explained. II. What must be observed in the placement of the fulcrum. III. By what rule the position of the weight on a lever of the first kind is to be determined. IV. The moments of weight in a lever of the second kind are considered. V. What is the ratio of a lever having a movable fulcrum. VI. What are the moments of a lever drawing a weight connected by a cord. VII. What benefit is conferred by applying the power of winds to a lever. VIII. The moment of a load hanging from a lever is investigated. IX. Whether two men carrying a weight are equally pressed. X. Whether

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ELENCHUS CAPITUM. X. An vis Elastica ad aliquod Vectis genus pertineat. XI. Cur longiora corpora faciliùs flectantur, difficiliùs sustineantur. XII. Unde oriantur forcipum, & forficum vires. XIII. Cur Tollenones juxta puteos constituantur. XIV. Remoram vires in agenda navi expenduntur. XV. Quomodo Naves à Gubernaculo moveantur. XVI. An Malus in motu navis habeat Rationem Vectis. XVII. An ex Rationibus Vectis pendeat usus Anchora. XVIII. Plures Vectis usus exponuntur. LIBER QUINTUS. De Axe in Peritrochio. CAP.I. Axis in Peritrochio forma, & vires describuntur. II. Succulæ & Ergata usus consideratur. III. Tympani à calcante circumacti vires expenduntur. IV. An Axis in Peritrochio inveniatur etiam sinè tractione. V. Axium in suis Peritrochiis Compositione vires augentur. VI. Tympanorum dentatorum usus, & vires exponuntur. VII. Moletrinarum artificium ex Axe in Peritrochio pendet. VIII. Axis cum Vecte compositus auget Potentiæ momenta. IX. Multiplex Rotarum dentatarum usus innuitur. LIBER SEX TUS. De Trochlea. CAP.I. Trochlearum forma & vires exponuntur. II. An Trochlea ad Vectem revocanda sit. III. An Orbiculi Magnitudo quicquam conferat. IV. Qua Ratione Trochlearum vires augeantur. V. Trochlea Trochless additæ plurimum augent momenta Potentiæ. VI. Trochlearum ope moveri potest pondus velociter. VII. Quàm validum esse oporteat Trochlearum retinaculum. VIII. Aliqui Trochlearum usus indicantur. LIBER

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Table of Chapters. X. Whether Elastic Force pertains to any kind of Lever. XI. Why longer bodies are more easily bent, more difficultly sustained. XII. Whence the forces of pincers and scissors arise. XIII. Why the windlasses are set up beside wells. XIV. How by a drag the forces are expended in moving a ship. XV. How Ships are moved from the Rudder. XVI. Whether the Mast in the motion of a ship has the nature of a Lever. XVII. Whether the use of the Anchor depends on the principles of the Lever. XVIII. Several uses of the Lever are explained. BOOK FIVE. On the Axis in the Wheel and Axle. CHAP. I. The form and forces of the Axis in the Wheel and Axle are described. II. The use of the capstan and windlass is considered. III. The forces of treadwheel drums are examined. IV. Whether the Axis in the Wheel and Axle is found even without pulling. V. By the composition of Axes in their Wheel and Axles, the forces are increased. VI. The use and forces of toothed drums are explained. VII. The machinery of mills depends on the Axis in the Wheel and Axle. VIII. The Axis combined with the Lever increases the moments of Power. IX. The multiple use of toothed wheels is indicated. BOOK SIX. On the Pulley. CHAP. I. The form and forces of pulleys are explained. II. Whether the Pulley should be referred to the Lever. III. Whether the size of the wheel contributes anything. IV. By what means the forces of pulleys are increased. V. A pulley added to another pulley greatly increases the moments of Power. VI. By means of pulleys a weight may be moved quickly. VII. How strong the fastening of pulleys ought to be. VIII. Some uses of pulleys are indicated. BOOK

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ELENCHUS CAPITUM. LIBER SEPTIMUS. De Cuneo, & Percussionibus. CAP.I. Cunei forma & vires explicantur. II. Cunei inflexi vsus ad movendum. III. Cuneus Perpetuus circulo excentrico effingitur. IV. Ex Cylindro construi potest Cuneus Perpetuus. V. Cuneum Perpetuum Circulus inclinatus imitatur. VI. Unde oriatur vis Percussionis. VII. Quàm dispares ex motûs velocitate sint Percussiones. VIII. An validior sit ictus Mallei à Situ Verticali ad Horizontalem, an verò ab Horizontali ad Verticalem descendentis. IX. Quomodo Percussiones ex Mole pendeant. X. Quid conferat resistentia corporis percussi. XI. Quomodo ex Percussionibus determinentur Reflexiones, XII. Quomodo Impetus in Percussione communicetur. XIII. Cunei usus promovetur. LIBER OCTAVUS. De Cochlea. CAP.I. Cochleæ forma & virtus describitur. II. An utilis sit Cochlea duplex contraria. III. Cochlea cum Vecte, atque cum Axe componitur. IV. Cochleæ Infinitæ vires explicantur. V. Cochleæ usus aliqui indicantur. MECHA

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TABLE OF CONTENTS. BOOK SEVEN. On the Wedge and Impacts. CHAPTER I. The form and powers of the wedge are explained. II. The use of the bent wedge for moving. III. A perpetual wedge is fashioned by means of an eccentric circle. IV. A perpetual wedge can be constructed from a cylinder. V. An inclined circle imitates the perpetual wedge. VI. Whence the force of impact arises. VII. How different impacts are according to the speed of motion. VIII. Whether a blow of a hammer descending from vertical to horizontal is stronger, or rather from horizontal to vertical. IX. How impacts depend on mass. X. What the resistance of the struck body contributes. XI. How reflections are determined from impacts, XII. How impetus is communicated in impact. XIII. The use of the wedge is furthered. BOOK EIGHT. On the Screw. CHAPTER I. The form and power of the screw are described. II. Whether a double contrary screw is useful. III. The screw is combined with the lever, and with the axle. IV. The powers of the infinite screw are explained. V. Some uses of the screw are indicated. MECHA

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MECHANICORUM LIBER PRIMUS. De Centro Gravitatis. MACHINARUM vires, quibus innatæ corporum in motum aut quietem propensioni obsistimus, explo- raturus, præterire non possum gravitatem ipsam: ne scilicet ignoretur, quid arte vincendum sit. Ideò primum hunc Librum Centro gravitatis tribuen- dum censui, cùm plura ex illo pendeant examinanda in poste- rioribus. Neque tamen hîc subtilissimam illam statices partem persequar, quæ in corporibus singulis gravitatis centrum in- vestigat: id enim, & abundè ab aliis præstitum, & mihi in hac tractatione minimè necessarium; quippe cui satisfuerit cen- trum illud physicè perspectum habere, quatenus præcaven- dum est, ne alienâ ponderis ad machinam applicatione longè alia fiat momentorum ratio, quàm oporteat. Ut autem Centri gravitatis notitia clarior habeatur, non inutile ducam quæstio- nes aliquot ad illud enucleatiùs explicandum pertinentes ad- dere, ut ipsis etiam tyronibus fiat satis: quamquam enim illis machinalis scientia carere posse alicui fortasse videatur, rem tamen penitiùs introspiciens eas extrà mechanicæ considera- tionis fines positas non esse cognoscet. CAPUT I. Quid sit Centrum gravium, et levium. Quoniam hæc rerum universitas corpora diversæ inter se rationis complectitur, eorum ordo aliquis necessarius fuit A

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MECHANICORUM BOOK THE FIRST. On the Center of Gravity. In investigating the powers of machines, by which we resist the innate tendency of bodies to motion or rest, I cannot pass over gravity itself, lest it should be unknown what must be overcome by art. Therefore I have judged that this first Book should be devoted to the Center of Gravity, since many things depending on it are to be examined in what follows. Nor, however, shall I here pursue that most subtle part of statics which investigates the center of gravity in individual bodies; for that has already been done sufficiently by others, and is least necessary to me in this treatment, since it will be enough for me to have that center understood physically, insofar as it must be taken care that, through the application of weight to the machine in another place, the ratio of moments may not become very different from what it ought to be. But that a clearer knowledge of the Center of Gravity may be had, I shall not think it useless to add some questions pertaining to its more exact explanation, so that even beginners may be satisfied: for although to some it may seem that they can do without mechanical science, yet one who looks more deeply into the matter will recognize that these things are not placed outside the bounds of mechanical consideration. CHAPTER I. What the Center of the heavy and of the light is. Since this universe of things includes bodies of different kinds among themselves, some order of them was necessary

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Mechanicorum ut suo unumquodque loco disponeretur; atque adeò æquum fuit, ut singulis à natura ea tribueretur facultas, quâ & se suo in loco, hoc est, juxta insitam propensionem sibi debito, conservare possint, & ad illum se ipsa promovere, si fortè indè dimota fuerint. Quia verò æqualia non nisi æqualiter, simili- que ratione disponenda erant, nullum autem corpus præter sphæram habet perfectam in partium dispositione æqualitatem, debuerunt corpora omnia orbem unum constituere. At in sphæra punctum unum est, à quo æqualibus radiis extremæ superficies partes removentur: igitur ex ordine ad punctum hoc, quod Centrum dicitur, comparanda sunt corpora; qua- tenus cùm naturâ impellente moventur, ut in loco sibi debito, à quo per vim se juncta fuere, demum consistant, vel ad cen- trum hoc accedunt, vel ab eo recedunt. Et quidem si ad centrum accedant, gravitare dicuntur, si verò recedant, levitare: & quæ propiora centro consistunt, graviora, quæ autem remotiora, leviora quoque consentur secundùm speciem gravitatis, & levitatis: quicquid sit quod æqualia esse possint secundùm gravitatem absolutam, aut etiam sæpè contingat minus habere gravitatis absolutæ id, quod est gravius secundùm speciem. Sic libra plumbi æqualis est libræ aquæ, immò minor centum libris aquæ; quia tamen plumbum infra aquam descendens sit centro vicinius, etiam gra- vius est secundùm speciem. Quod si comparare velis duo cor- pora solida, quæ sibi sua duritie ita obsistunt, ut neutrum intra alterum moveri possit tanquam in medio; illud esse secundùm speciem gravius affirmabis, quod datâ paritate molis cum alio corpore, cum quo comparatur, staterâ expensum in eodem medio, in quo utrumque gravitat puta in aëre, plus habere ponderis deprehendes. Sic aurum est ferro gravius in specie, quia ex æqualibus molibus auri & ferri, aurea est ponderosior. Generatim autem loquendo ea sunt in specie graviora, quæ sunt densiora, ea verò in specie leviora, quæ rariora: nam & inflata vesica ob aërem constipatum gravior est, quàm flaccida; & Æolipilam candentem, aëre intus vi caloris raro, leviorem primùm, posteà, ubi refrixerit, graviorem esse experimento didicimus, aëre assumptam raritatem abjiciente. Cùm enim radij à sphæræ centro ad superficiem ducti longiùs à se invi- cem

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Mechanic things were to be arranged so that each might be set in its proper place; and therefore it was altogether just that by nature to each there should be granted that faculty by which it may both preserve itself in its own place, that is, according to the innate tendency due to it, and, if by chance it has been moved away from thence, promote itself back to it. But because equal things were to be arranged only equally, and in the same way, whereas no body except a sphere has a perfect equality in the arrangement of its parts, all bodies ought to constitute one single orb. Now in a sphere there is one point, from which the outer parts of the surface are removed by equal rays: therefore, bodies must be considered with reference to this point, which is called the Center; inasmuch as, when impelled by nature, they move so as at last to come to rest in the place due to them, from which they were joined together by force, and thus either approach this center or recede from it. And indeed if they approach the center, they are said to gravitate; if they recede, to levitate: and those which are situated nearer the center are also judged heavier, those farther away lighter, according to the species of gravity and levity; whatever may be the case that they can be equal according to absolute gravity, or even that it often happens that what is heavier according to species has less absolute gravity. Thus a pound of lead is equal to a pound of water, indeed even less than a hundred pounds of water; yet because lead, descending beneath the water, is nearer to the center, it is also heavier according to species. But if you wish to compare two solid bodies, which by their hardness resist one another so that neither can move within the other as in a medium; you will affirm that to be heavier according to species which, with an equal bulk to the other body with which it is compared, when weighed on a balance in the same medium in which both are weighed, namely in air, you will find to have greater weight. Thus gold is heavier than iron in species, because from equal masses of gold and iron, the golden one is the more weighty. Speaking generally, those things are heavier in species which are denser, and those lighter in species which are rarer: for even a distended bladder, because of the compressed air, is heavier than a flaccid one; and we have learned by experiment that an Æolipila, while glowing hot, is at first lighter, then, when it has cooled, heavier, the air having taken on rarity and then casting it off again. For since the rays drawn from the center of the sphere to the surface are farther apart from one another...

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Liber primus. CAPUT I. & II. cem recedant, æquum fuit, ut quæ plus habent materiæ atque substantiæ sub minori mole, in angustiore spatio collocarentur; ea verò, quæ sub majoribus dimensionibus continentur, am- pliora spatia occuparent, ubi radij magis distant: ut videlicet hac ratione æqua substantiæ distributio fieret in totâ sphærâ. Hinc vides, cur idem corpus, eo ipso quod rarum fit, ascendat, ut aqua in vaporem resoluta (nisi aliunde ad descendendum determinetur, ut aurum fulminans) quia materies eadem sub majoribus dimensionibus petit longiùs abesse à centro, ibiquè tantisper conquiescit, dum constipata, atque minorem in mo- lem redacta, iterum descendat. Quare centrum hoc, quod motus, vel quies corporum respi- cit, dicitur Centrum gravium, & levium; atque idem creditur esse cum centro universi: vel saltem (ne parùm utili nos dispu- tatione torqueamus) centrum eorum, quæ in hac sphærâ ele- mentari gravia, aut levia dicuntur, idem est cum centro ter- raquei hujus globi, ut quotidiana docet experientia: quicquid sit, an pars lunaris globi, si à lunâ sejungeretur, reditura esset ad lunam, ut ad centrum sui motus. Tam itaquè, quæ hujusmo- di centro proxima sunt, deorsum posita dicuntur, sursum verò, quæ ab eo longiùs collocata sunt. Hinc telluris superficiei in- sistentes caput sursum, pedes deorsum habere dicimur. Ille verò, quamvis rectus, & pedes, & caput sursum haberet, cu- jus umbilicus huic centro universi congrueret. Per quod pa- riter centrum si scala ducta intelligatur, duo possent sibi non occurrere invicem, licet alter ascenderet, alter descenderet; hic siquidem accederet ad centrum, ille inde recederet: per eam verò posset uterque ascendere, & tamen licet, æquali mo- tu moverentur, semper invicem distarent magis, quò à centro ad oppositas partes recederent. CAPUT II. An corpora prædita sint gravitate, & levitate. Inter ea, quæ planè homogenea sunt, ordo esse non potest à naturâ institutus: hinc si nulla esset corporum dissimilitudo, A 2

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Book One. CHAPTER I. & II. that they may recede from one another, it was fitting that those which have more matter and substance should be placed in a smaller bulk, in a narrower space; but those which are contained under greater dimensions should occupy larger spaces, where the rays are more distant: so that, namely, by this means an equal distribution of substance might be made throughout the whole sphere. Hence you see why the same body, by the very fact that it becomes rarefied, ascends, as water resolved into vapor does (unless it is determined from elsewhere to descend, as fulminating gold is), because the same matter, under greater dimensions, seeks to be farther from the center, and there rests meanwhile, until, being compacted and reduced to a smaller bulk, it descends again. For this center, which regards the motion or rest of bodies, is called the Center of the heavy and the light; and it is believed to be the same as the center of the universe: or at least (that we may not trouble ourselves with a dispute of little use) the center of those things which in this elementary sphere are called heavy or light is the same as the center of this earthly globe, as everyday experience teaches. Whatever may be the case, whether a part of the lunar globe, if separated from the moon, would return to the moon as to the center of its motion. So then, those things which are nearest to such a center are said to be placed below, but those farther from it are above. Hence those standing upon the surface of the earth are said to have the head above and the feet below. But he, though upright, would have both feet and head above, whose navel corresponded to this center of the universe. In the same way, if a ladder were imagined to be drawn through that center, two people could not meet each other, although one were ascending and the other descending; for this one would indeed be approaching the center, that one would be receding from it: but by that ladder both could ascend, and yet, although they were moved with equal motion, they would always be farther apart from one another, the more they receded from the center to opposite parts. CHAPTER II. Whether bodies are endowed with heaviness and lightness. Among things that are entirely homogeneous, there can be no order established by nature: hence if there were no dissimilarity among bodies, A 2

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Mechanicorum 4 sed ex omninò similibus substantiæ partibus totus hic orbis conflaretur, nulla quoque esset aut gravitas, aut levitas. Quid enim hæc potiùs pars, nulla naturæ conditione à cæteris discreta, petat abesse à centro, illa verò exigat in eo conquiescere? verùm quia multiplici corporum genere coagmentata rerum universitas inconcinna esse non potuit, suum cuique locum natura tribuit, in quo se sisteret, ut infra hæc quidem descenderet, suprà illa verò ascenderet, si quando sibi invicem contigua fierent ordine præposterò, nec ullus esset motui obex. Cum itaque corpora singula insitam habeant propensionem (ab Aristotele dicitur opum) qua petunt certum locum in universo; constat præter descentium gravitatem dari etiam positivam levitatem, quâ corpus aliquod se ipsum promovet ad superiores partes universi à centro magis distantes, neque solùm admittendam levitatem negativam, quâ corpora minùs gravia censentur levia, si eorum cum gravioribus fiat comparatio. Nam si ea, quæ levia dicuntur, eatenus dicas ascendere, quatenus à gravioribus in inferiorem locum descententibus propelluntur; mihi æquè liberum erit tollere omnem positivam gravitatem, solâ levitate admissâ; & omnia pariter solvam dicendo ea gravia censeri, quæ minùs levia sunt, atque ideò tantùm descendere, quòd extrinsecùs à levioribus ascendentibus loco pulsa detrudantur, non quòd ab internâ facultate deorsum impellantur. Quod si vel gravitas de medio tollenda sit, vel levitas, satius est levitatem relinquere; naturâ videlicet ad altiora semper, & perfectiora aspirante, nec adeò contendente de infimo loco. Quare cùm per gravitatem solam æquè ac per solam levitatem motus isti explicentur, cæteroqui autem ingenita sit unicuique corpori sui loci exigentia; utramque admittere rationi maximè consentaneum fuerit. Vitreum globum vacuum, qui in tubulum recurvum desinat, quoad fieri potest, calefactum, ut inclusus aër rarescat, Hermeticè claude: tum adjiciatur congruens plumbi gravitas, quâ infra aquam deprimatur. Sit autem globus, unà cum adjecto plumbo, connexus cum exquisitæ libræ brachio, aut lance, ejusque gravitas intrà aquam exploretur: ubi gravitas innotuerit, adhuc sub aquâ retineatur globus, sed longiore forcipe extremum tubuli caput occlusum frangatur: & animadverteres

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Mechanics 4 but if this whole world were compounded from parts of substance altogether similar, there would likewise be neither gravity nor lightness. For why should this part rather, distinguished from the others by no condition of nature, seek to be away from the center, while that other should demand to rest in it? But because the universe, made up of a manifold kind of bodies, could not be unsuitably arranged, nature assigned to each its own place, in which it might stand still, so that these should indeed descend below, but those should ascend above, if at any time they became adjacent to one another in a reversed order, and there were no obstacle to motion. Since therefore individual bodies have an innate tendency (Aristotle calls it a desire) by which they seek a certain place in the universe, it is clear that besides the gravity of descending bodies there is also a positive lightness, by which some body moves itself to the higher parts of the universe, farther from the center, and that we must admit not only negative lightness, by which bodies are accounted less heavy, if a comparison be made of them with heavier bodies. For if you say that those things which are called light ascend only insofar as they are driven by heavier things descending into a lower place, it will be equally open to me to abolish all positive gravity, admitting only lightness; and I shall equally dispose of everything by saying that those are judged heavy which are less light, and therefore descend only because, from outside, they are pushed and thrust from their place by lighter bodies ascending, not because they are driven downward by an internal faculty. But if either gravity must be removed from the middle, or lightness, it is better to leave lightness; since nature is always aspiring toward higher and more perfect things, and is not so much striving for the lowest place. Wherefore, since these motions are explained equally well by gravity alone as by lightness alone, and otherwise an exigency of its own place is innate in each body, it will have been most in accord with reason to admit both. Heat as much as possible a glass globe, empty, which ends in a recurved tube, so that the enclosed air rarefies, and seal it hermetically: then add a suitable weight of lead, by which it may be sunk below water. Let the globe, together with the added lead, be connected with the arm or pan of an accurate balance, and let its weight be examined in water: when its weight has become known, keep the globe still under water, but with a longer forceps break off the closed end of the tube: and you will observe

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Liber primus. CAPUT II. 5 vertes globi vitrei cum appenso plumbo gravitatem augeri; cu- jus incrementum indicabitur ab addito in oppositâ lance pon- dere ad constituendum æquilibrium. Cùm itaque idem maneat vitrum, idemque plumbum, & nulla facta sit alicujus gravita- tis accessio, illud unum superest, quòd aër rarus intrà globum conclusus levior, quàm idem aër, aperto tubulo, sibi restitu- tus, plus elidit gravitatis plumbi & vitri; atque moles compo- sita ex plumbo, vitro, & aëre raro, secundùm speciem levior est, quàm moles ex plumbo, vitro, & aëre non raro. Aër igi- tur intra aquam ita levis est, ut aliquid gravitatis imminuat: Nam si globum eundem ex aquâ extractum, omni aëre exclu- so, aquâ repleveris, & iterum eodem plumbo adjecto ejusdem gravitatem intrà aquam examinaveris, illam adhuc majorem deprehendes; quia scilicet nulla levitas aëris adest, quæ ali- quam deterat gravitatem, sed illa solùm perire videtur, quam infert discrimen gravitatum secundùm speciem, ut ex Hy- drostaticis constat. Neque suspiceris hæc gravitatum incre- menta oriri ex aquâ subeunte per apertum tubulum, cùm aër assumptam ex calore raritatem abjicit, se in naturalem suam molem restituens, sivè, aëre prorsus excluso, ex aquæ globum implentis gravitate. Si enim vitrum aliud aut nullius, aut mo- dicissimæ aquæ capax, sed ejusdem in aëre ponderis cum as- sumpto globo, similiter in aquâ expendas, eandem invenies gravitatem, sive multâ, sive modicâ aquâ repletum fuerit. Non igitur aqua intrà aquam gravitatem auget. Sed illud, ut reliqua fileam, non leviter suadere potest cor- pora suis nutibus non deorsum tantùm, sed etiam sursum co- nari, quod mihi haud ita pridem aliud investiganti contigit observare. Cum enim animadvertissem aliquando, quàm dis- par esset gravitas aquæ dimidiam situlam implentis, si illa in su- perficie horizontali libraret sese, ac quandò supposita ligneo globo firmiter cum superiore tigillo cohærenti altiùs ad latera assurgebat locum globo concedens, quem tamen non sustine- bat; subiit animum cupido tentandi, an bubula vesica inflata transversis virgulis infra vasis labra depressa ita, ut eam aqua circumplecteretur, vim haberet pariter augendi momenta gra- vitatis; aquam siquidem cogebat assurgere ad altitudinem ma- jorem perpendicularem, ac quandò, vesicâ liberè innatante, A 3

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Liber primus. CHAPTER II. 5 you will find the gravity of the glass globe with the lead attached increased; the amount of this increase will be shown by the added weight placed in the opposite scale to establish equilibrium. Since, therefore, the same glass remains, and the same lead as well, and no addition of weight has been made, this alone remains: that the rare air enclosed within the globe is lighter than the same air, restored to itself when the little tube is opened, and that it removes more of the weight of the lead and glass; and the composite mass made up of lead, glass, and rare air is, in appearance, lighter than the mass made up of lead, glass, and air not rare. Air, therefore, within water is so light that it diminishes some of the weight. For if you take the same globe out of the water, with all air excluded, fill it with water, and again, with the same lead attached, examine its weight within the water, you will find it still greater; namely because there is no lightness of air present to take away any part of the weight, but only that seems to disappear which is caused by the difference of specific weights, as is evident from Hydrostatics. Nor should you suspect that these increases in weight arise from water entering through the open tube, since the air, abandoning the rarity acquired from heat as it restores itself to its natural bulk, or, with the air entirely excluded, from the weight of the globe filled with water. For if you weigh in water another glass vessel, capable of holding either none or very little water, but of the same weight in air as the globe taken up, you will find the same weight, whether it has been filled with much water or with little. Therefore water within water does not increase weight. But this, to pass over the rest in silence, can strongly suggest that bodies by their own tendencies strive not only downward but also upward, which I not long ago happened to observe while investigating another matter. For when I once noticed how different was the weight of water filling half a pail, if it balanced itself on a horizontal surface, and when, placed beneath a wooden globe firmly joined to the upper crosspiece, it rose higher at the sides, yielding place to the globe, which nevertheless it did not support; there came into my mind the desire to try whether an ox bladder, inflated and pressed down by transverse rods beneath the rim of a vessel, so that the water surrounded it, might likewise have the power of increasing the moments of gravity; for it forced the water to rise to a greater perpendicular height than when, with the bladder freely floating, A 3

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Mechanicorum 6 subsidebat. Inveni tamen nullum planè observari posse in gravitate discrimen, quamvis tam ampla esset vesica, ut facilè dimidiam vasis capacitatem impleret: in utroque enim casu pon- dus fuit lib. 44 ́. Id mihi, fateor, accidit præter opinionem: Nam si ex pariete extet tigillus, cui adnectatur cylindrus P, aut vesica ritè firmata, ferè im- plens capacitatem vasis A B, vasque illi sup- ponatur ita, ut aqua deinde infusa possit libe- rè cylindro circumfundi; percipies onus lon- gè majus, quàm pro gravitate aquæ infusæ, si permitteretur subsidere: & si vas ex staterâ pendeat, adducto reductóve sacomate appa- rebunt momenta gravitatis longè majora, quàm si tota illa aqua fundum peteret, & cylindri pars, quæ priùs immerge- batur, abscissa, aut vesica innataret. Intelligebam id ex majori altitudine perpendiculari aquæ supra eandem basim oriri; nam depresso vase ita, ut paulatim cylindus emergat, & aqua sub- sidat, semper minuitur pondus: idem futurum sperabam, si vesica intra aquam non ab extrinseco obice detineretur, sed à virgulis cum vase ipso connexis; quandoquidem aqua ad can- dem pariter altitudinem assurgebat super basim eandem: at spem fefellit eventus. Nec alia mihi se obtulit probabilior ra- tio, quàm ut existimarem aquam altiorem vehementius qui- dem deorsum niti, vesicam tamen levirem altiùs depressam, conantem sursum, æqualiter contendere, ut emergeret; cùm verò nisus iste sursum oppositas virgulas, atque adeò vas cum illis connexum urgeret, elidi adversum impetum deorsum, qui à majore altitudine perpendiculari aquæ oriebatur, & so- lum remanere conatum ex ipsorum corporum substantiâ pro- manantem, quæ sicut eadem semper erat, sivè innataret vesi- ca, sivè per vim immergeretur, ita eadem obtinebat gravita- tis momenta. Quo experimento (quamquam non me lateat, quid pro se afferre hîc possent aliter sentientes) visus mihi sum deprehendere non obscurum positivæ levitatis vesti- gium. Ut autem levitatem corporibus adimendam assererent in- geniosi Academici, hoc potissimum ducti sunt experimento. Ligneum

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Mechanics 6 descended. Yet I found that no difference at all in weight could plainly be observed, although the bladder was so large that it could easily fill half the capacity of the vessel: for in either case the weight was 44 lb. This, I confess, happened to me contrary to expectation. For if from a wall there projects a little beam to which the cylinder P, or a properly fastened bladder, is attached, nearly filling the capacity of vessel A B, and the vessel is placed beneath it so that water, when afterward poured in, may freely flow around the cylinder, you will perceive a load far greater than what would correspond to the weight of the water poured in, if it were allowed to sink; and if the vessel hangs from a balance, by moving the weight to and fro there will appear moments of gravity far greater than if all that water were seeking the bottom, and the part of the cylinder that was previously submerged were cut off, or the bladder were floating. I understood that this arose from the greater perpendicular height of the water above the same base; for if the vessel is lowered so that the cylinder gradually emerges and the water sinks, the weight is always diminished: I had hoped the same would happen if the bladder within the water were not held back by an external obstacle, but by little rods connected with the vessel itself; since in that case the water likewise rose to the same height above the same base: but the event deceived my expectation. Nor did any other more probable reason present itself to me than that I should think the higher water indeed presses more strongly downward, yet the bladder, being pressed lower, strives upward in equal degree, so as to emerge; and when this upward effort presses upon the opposing rods, and thus upon the vessel connected with them, the contrary downward impulse, which arose from the greater perpendicular height of the water, is checked, and there remains only the effort proceeding from the substance of the bodies themselves, which, since it was always the same, whether the bladder floated or was forced under, thus produced the same moments of gravity. By this experiment, though I am not unaware what those who think otherwise might here advance in their own behalf, I seemed to myself to have detected no obscure trace of positive lightness. But in order that the ingenious Academics might assert that lightness is to be taken away from bodies, they were led chiefly by this experiment. Wooden

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Liber primus. CAPUT II. 7 Ligneum cylindru[m] ABC plano horizontali D, E, perpendicularem statuerunt; & ut cylindri basis subjecto plano exactè congrueret, laminas duas accuratissimè lævigatas, tùm cylindri basi, tùm subjecto plano firmiter adnexas voluerunt. Tùm ne aër facilè inter utrumque subiret, erecto supra planu[m] in orbem ex creta, aut cerâ aggerulo, argentu[m] vivum infuderunt. Cylindrum extremo libræ jugo G, alligârunt, addito in oppositâ libræ extremitate H pondere L cylindri pondus adæquante; quod utique cylindrum elevare non potest. Additum igitur est & aliud pondus M usque eò, dum cylindrus à subjecto plano avelleretur, & fuit librarum circiter trium: quam mensuram arguunt esse resistentiæ cylindri contiguo plano adhærentis metu vacui. His peractis concavum vas cylindricum NOP, æqualis aut majoris altitudinis parârunt, laminâ pariter perpolitâ vasis fundo adnexâ, cui impositus fuit cylindrus, adeoque adhæsit, ut, pleno-mercurij vase, omninò non avelleretur, ut innataret; sed tunc demum argento vivo innatavit, cùm per vim à vasis fundo avulsus est cylindrus: cui, ut iterum fundum peteret, & argento vivo immergeretur, imponendum fuit pondus Q librarum circiter quinque. Vis ergò levitatis ligni in mercurio (si qua levitas esset) æstimanda esset ut quinque, cùm vis adhæsionis metu vacui solùm inventa sit ut tria: debuisset igitur levitas ita prævalere, ut adhæsionem vinceret, & cylindrus sponte elevaretur. Non est itaque levitas, quæ ligneum cylindrum innatare cogit, sed mercurij gravitas major ipsa est, quæ lignum elevat, cum primùm locus patet, in quem descendat. Sed antequam experimentum hoc ad examen revocemus, ut innotescat, quid hinc confici possit ad levitatem excludendam, haud ægrè permiserim, cùm in abeuntis suâ sponte cor- poris

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Book One. CHAPTER II. 7 A wooden cylinder ABC was set perpendicular to the horizontal plane D, E; and, so that the base of the cylinder might exactly fit the underlying plane, they placed two plates most accurately polished, firmly attached, one to the base of the cylinder, the other to the underlying plane. Then, lest air easily enter between the two, they raised around the plane a little mound in a circle out of chalk or wax, and poured in quicksilver. They fastened the cylinder to the end hook G of a balance, adding at the opposite end of the balance the weight L, equal to the weight of the cylinder; which of course cannot lift the cylinder. Accordingly another weight M was added until the cylinder was detached from the underlying plane, and it was about three pounds: this measure they argue to be the resistance of the cylinder adhering to the adjacent plane out of fear of the vacuum. When these things had been done, they prepared a hollow cylindrical vessel NOP, of equal or greater height, with a likewise polished plate attached to the bottom of the vessel, on which the cylinder was placed, and it adhered so strongly that, with the vessel filled with mercury, it was by no means detached so as to float; but it floated on the quicksilver only when the cylinder had first been forcibly torn away from the bottom of the vessel: to make it once again seek the bottom and be immersed in the quicksilver, it was necessary to impose a weight Q of about five pounds. The force of the lightness of the wood in mercury, therefore (if there were any lightness), should be estimated as five, whereas the force of adhesion, arising from fear of the vacuum, was found to be only three: therefore lightness ought to have prevailed so much that it would overcome the adhesion, and the cylinder would rise of itself. It is not therefore lightness that compels the wooden cylinder to float, but rather the greater weight of the mercury itself, which raises the wood, when once there is room for it to descend. But before we call this experiment back to examination, so that it may become clear what may be inferred from this against the exclusion of lightness, I shall not readily allow, since in a body departing of its own accord the cor-

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8 Mechanicorum poris locum corpus aliud suapte vi, & naturâ succedit, ab hoc illud urgeri posse, ut velociùs moveatur: duo scilicet corpora diversæ secundùm speciem gravitatis si fuerint perturbatè dispositis intra medium, in quo utrumque gravitat, nil mirum, si à graviore majori nisu conante extrudatur minùs grave: id quod etiam de duobus levibus dicendum perturbatè dispositis in medio, ubi utrumque levitat: duobus enim simul currentibus, ab eo qui ponè subsequitur, si majoribus viribus polleat, priorem urgeri atque impelli palam est, quamquam motus universus impulsioni tribuendus non sit. Ita quoque ascendentem in mercurio ligneum cylindrum à descendente mercurio sursum urgeri aliquatenus posse non diffitebor, sicut & mercurium ipsum repugnare, ne sursum propellatur, atque ab eodem lignum innatans prohiberi, ne descendat: hinc tamen non sequitur ligni ascendentis motum, aut innatantis quietem, prægravis mercurij viribus omnino adscribi jure debere, nam, & sua vis ascendendi, atque consistendi, ligno ipsi tribuenda est. Quid quòd ipsæ innatantis cylindri portiones, altera quidem mercurio immersa, altera verò extans, levitatem ipsi ligno insitam declarant? Quid enim partis immersæ ad extantem (si soles spectetur) ea ratio est, quæ specificæ gravitatis ligni ad differentiam gravitatum ligni, atque mercurij? nisi quia portionis mercurio immersæ levitas, atque extantis in aëre gravitas, æquilibritatem constituent; quemadmodum in Terra machinis mota differt. 5. n. 105. explicatum est. Hanc porrò æqualitatem Algebricè sic ostendo. Ratio gravitatis ligni ad gravitatem mercurij sit ut S. ad R; differentia est R-S. Ponatur cylindri pars immersa. A. Quia igitur ut specifica gravitas corporis innatantis ad differentiam gravitatum, hoc est ut S ad R-S, ita pars cylindri immersa A, ad extantem R in A-S in A --------S--------; Si pars extans in aëre in suam gravitatem S ducatur, pars verò immersa A in differentiam gravitatum R-S, hoc est in-R+S, quia est deficiens, efficitur hinc quantitas R in A-S in A, hinc verò-R in A+S in A, quæ se invicem elidunt. Æqualia igitur sunt levitatis, & gravitatis momenta. Sit enim exempli causâ gravitas ligni ad gravitatem mercurij,

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8 Mechanics If, in the place of the body, another body by its own force and nature succeeds, then it is no wonder that the one may be pressed by the other, so that it is moved more swiftly: namely, if two bodies of different specific gravities are irregularly placed within a medium in which each gravitates, it is no surprise if the lighter is thrust out by the heavier, which strives with greater effort: the same must also be said of two light bodies irregularly placed in a medium where each has levity; for if two run together at the same time, it is plain that the one which follows behind, if it has greater force, presses upon and impels the one before it, although the whole motion is not to be attributed to the impulse. In like manner I will not deny that a wooden cylinder ascending in mercury may be pressed upward to some extent by the mercury descending, just as mercury itself resists being driven upward, and the same mercury prevents the floating wood from descending: yet from this it does not follow that the motion of the ascending wood, or the rest of the floating body, ought to be wholly ascribed by right to the forces of the heavy mercury; for the wood itself must also be credited with its own force of rising and remaining in place. What of this, that the parts of the floating cylinder itself, one indeed immersed in the mercury and the other projecting out, show the levity inherent in the wood itself? For what is the relation of the immersed part to the projecting part, if the sun be considered, other than the relation of the specific gravity of the wood to the difference between the gravities of the wood and the mercury? unless because the levity of the portion immersed in the mercury and the gravity of the portion exposed in the air will form an equilibrium; as was explained in the case of a machine moved on the Earth, 5. n. 105. I demonstrate this equality algebraically as follows. Let the ratio of the gravity of the wood to the gravity of the mercury be as S to R; the difference is R - S. Let the immersed part of the cylinder be A. Since, therefore, as the specific gravity of the floating body is to the difference of the gravities, that is, as S to R - S, so is the immersed part of the cylinder A to the part projecting, R in A - S in A --------S--------; if the projecting part in the air is multiplied by its gravity S, and the immersed part A by the difference of the gravities R - S, that is, by -R + S, because it is deficient, there results from this the quantity R in A - S in A, and from that -R in A + S in A, which cancel one another. Therefore the moments of levity and gravity are equal. For let it be, for example, that the gravity of the wood to the gravity of the mercury,

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Liber primus. CAPUT II. 9 mercurij, ut S. ad 13. differentia est 8. Est igitur cylindri pars immersa ejusdem 1/13, extans verò 8/13: at portio immersa de- ficit à gravitate mercurij secundùm speciem ut 8; igitur 1/13 in - 8 dant 40/13: item partis extantis gravitas in aëre est S; igitur 8/13 in 5 dant 40/13: confligunt itaque inter se pari conatu levitas 40/13, & gravitas 40/13, adeóque fit consistentia & innatat lignum. Sed jam ad propositi experimenti examen descendamus. Aio cylindri resistentiam ex adhæsione metu vacui non satis explo- ratam fuisse per libram; hæc enim dum ex pondere M deorsum inclinatur, extremitas G sursum elevata arcum describit, ac proinde cylindri ascendentis motus non est per lineam horizon- tali plano perpendiculariter insistentem, sed per inclinatam: Quare cùm A. versùs I libræ centrum trahatur, cylindri basis non incipit elevari parallela horizonti, sed cum inclinatione, ita ut C priùs elevetur, quàm B: ea autem, quæ sibi invicem adhæ- rescunt, multò faciliùs divelli manifestum est, si id cum inclina- tione fiat, quàm si servandus sit parallelismus. Adde in hac in- clinatione faciliùs adhuc divelli cylindrum à supposito plano, quò longior cylindrus fuerit; habet scilicet rationem vectis, cujus potentia est in A, hypomochlion in B, resistentia vin- cenda in C. Quare pondus M non aptè metitur resistentiam, quæ oritur ex corporum adhærescentiâ, metu vacui, sed hæc multò major est, si ad perpendiculum motus fieri debeat; quemadmodum & fieri oporteret, si in vase N O P mercurij pleno cylindrus fundo adhærens rectâ ascenderet. Quamvis igitur pondus Q librarum quinque admitteretur mensura levi- tatis, non continuò argui potest hujus excessus supra resisten- tiam adhæsionis. Quin immo affirmare ausim, si libræ loco adhibita fuisset amplior trochlea, & ex funiculo ejus orbitam côplectente hinc cylindrus A, hinc verò pondus M ad perpen- diculum pependissent, non satis futurum fuisse pódus librarum trium, sed multò majus adhibendum fuisse, ut cylindri resisten- tiam superaret; fuisset enim avellenda basis servato parallelismo. Quantum autem virium, ferè supra fidem, habeat vacui horror ad corpora retinenda, satis apertè declarant gravia, quæ suspenduntur. Ego sanè vidi marmoreum mortarium commu- nis magnitudinis satis vulgari artificio suspendi vitreo cyatho: B

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Liber primus. CAPUT II. 9 of mercury, so that with S. to 13 the difference is 8. Therefore the immersed part of the cylinder is 1/13 of the same, and the part extending above is 8/13: but the immersed portion falls short of the specific gravity of mercury by 8; therefore 1/13 in - 8 give 40/13: likewise the gravity of the extending part in air is S; therefore 8/13 in 5 give 40/13: thus lightness 40/13 and gravity 40/13 contend with each other with equal effort, and so consistency arises and the wood floats. But now let us descend to the examination of the proposed experiment. I say that the resistance of the cylinder from adhesion, through fear of a vacuum, was not sufficiently investigated by the balance; for when this, as it inclines downward by the weight M, lifts the end G upward and describes an arc, the motion of the ascending cylinder is not along a line standing perpendicular to the horizontal plane, but along an inclined one: wherefore, since A is drawn toward the center of the balance I, the base of the cylinder does not begin to rise parallel to the horizon, but with an inclination, so that C is lifted up before B: and it is manifest that things which adhere to one another are much more easily pulled apart if this is done with an inclination than if parallelism must be preserved. Add that in this inclination the cylinder is even more easily detached from the supporting plane, the longer the cylinder is; for it has the nature of a lever, whose power is at A, the fulcrum at B, the resistance to be overcome at C. Therefore the weight M does not suitably measure the resistance which arises from the adhesion of bodies, through fear of a vacuum, but this is much greater if the motion must be made perpendicularly; just as indeed it ought to be if, in the vessel N O P filled with mercury, the cylinder adhering to the bottom were to rise straight up. Although therefore the weight Q of five pounds was admitted as a measure of lightness, it cannot straightway be inferred that this exceeded the resistance of adhesion. Nay indeed, I would dare affirm that, if instead of the balance a larger pulley had been used, and from the rope encompassing its groove the cylinder A on this side, and the weight M on that side, had hung perpendicularly, it would not have been enough to use a weight of three pounds, but a much greater one would have had to be applied in order to overcome the resistance of the cylinder; for the base would have had to be torn away while the parallelism was preserved. How great, however, almost beyond belief, is the power of the horror of a vacuum for retaining bodies, the heavy things that are suspended sufficiently plainly declare. For my part I certainly saw a marble mortar of ordinary size suspended by common workmanship from a glass cup: B

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Mechanicorum 10 mortarij scilicet fundo exteriùs aptata fuerat massa ex farinâ ad formandos panes recens macerata, & aquâ ita subacta, ut illi tenaciter cohæreret: tum vitreo calici injecta stuppa admo- to igne exarsit, applicitusque calix massæ eam attraxit, sicut & medicorum cucurbitulæ carnem attrahunt: quare accepto ca- licis vitrei pede facile fuit mortarium elevare, & suspendere. Quod si marmoreum mortarium ex metu vacui in aëre pendu- lum hæsit, quid mirum si & ligneus cylindrus subjecto plano adhærescens in mercurio stetit? Nondum itaque ex hoc experimento, aut ex similibus, ubi metu vacui suos motus moliri corpora non possunt, satis habemus argumenti, quo levitatem, solâ gravitate retentâ, expungamus. Hujusmodi est illud, ubi in lignei vasis fundo exca- vatur scaphium, cui exquisitè congruat eburneus globus, qui superinfuso hydrargyro non ascendit. Neque enim ideò non ascendit, quia rima nulla patet argento vivo, per quam subiens extrudat eburneum globum, sed quia ita sibi exquisitè con- gruunt ebur, & lignum, ut vis ipsa ascendendi vincere non va- leat vim adhærescentiæ. Nam & eadem vis in aëre suspendit corpora gravia, ne descendant. Quamvis autem non totum hemisphærium globi eburnei, sed solùm ejus maximus circu- lus congrueret excavato ligno, & cavitas ipsa aëre repleretur, non propterea tollitur vis adhærescentiæ illius annularis; quia scilicet vis ascendendi in hydrargyro tanta non est, ut valeat inclusum ibi aërem distrahere, sicut opus esset ad incipiendum motum citra periculum vacui, & præterea superanda est re- sistentia hydrargyri dividendi; corpora enim in motu divi- dunt medium, pro cujus crassitudine resistentiam experiuntur. Adde hemisphærium inferius in aëre tanquam in loco positum gravitare non minùs, quàm hemisphærium superius levitet in hydrargyro; proinde nil mirum, si globus non ascendat. Quod si aëre excluso locum illum impleveris hydrargyro, & ebur- neum globum ita foramini aptaveris, ut illi exquisitè congruat; si in superinfuso hydrargyro globus non ascendat, indicio est ita globum esse foramini infixum, ut neque valeat elevari à sub- jecto hydrargyro in scaphij formam per vim excavato: neque enim facilè mihi persuadebis specificarum gravitatum diffe- rentiam exigere, ut hemisphærium integrum præcisè extet: præter

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Mechanics 10 namely, the bottom of a mortar had been fitted on the outside with a mass of dough, freshly kneaded for making bread, and so worked with water that it adhered firmly to it; then, when tinder was applied to a glass cup that had been placed upon it, it burst into flame, and the cup, being applied to the mass, drew it up, just as physicians’ cupping-glasses draw the flesh: wherefore, having grasped the foot of the glass cup, it was easy to lift and suspend the mortar. If, then, a marble mortar, hanging in the air from fear of a vacuum, remained stuck, what wonder is it if a wooden cylinder, adhering to the plane beneath it, stood in mercury? Thus, from this experiment, or from similar ones, in which bodies cannot set their motions in train because of fear of a vacuum, we do not yet have sufficient argument to dismiss lightness, retaining only gravity. Of this kind is that case in which, in the bottom of a wooden vessel, a shallow cavity is hollowed out to fit exactly an ivory globe, which, when mercury is poured over it, does not rise. Nor does it fail to rise because no fissure is open to the quicksilver through which, entering, it might drive out the ivory globe; rather, it is because ivory and wood fit together so exactly that the very force of rising is unable to overcome the force of adhesion. For the same force in air suspends heavy bodies so that they do not descend. And although not the whole hemisphere of the ivory globe, but only its largest circle, matched the hollowed wood, and the cavity itself was filled with air, this does not on that account remove the force of that annular adhesion; because, namely, the force of rising in mercury is not so great that it can draw apart the air enclosed there, as would be necessary to begin motion without danger of a vacuum, and moreover the resistance to dividing the mercury must be overcome; for bodies in motion divide the medium, and according to its thickness they experience resistance. Add that the lower hemisphere in air, as in a place, does not gravitate less than the upper hemisphere is lightened in mercury; therefore it is no wonder if the globe does not rise. But if, having excluded the air, you fill that place with mercury, and so fit the ivory globe to the hole that it matches it exactly, then if the globe does not rise in the mercury poured over it, it is a sign that the globe is so fixed in the hole that it cannot be lifted even by the mercury beneath it, hollowed out by force into the shape of a shallow vessel; for you will not easily persuade me that a difference of specific gravities requires that the whole hemisphere should stand out precisely; moreover

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Liber primus. CAPUT II. præter quam quod si non valebat subjectum aërem distrahere, multò minùs id in hydrargyro præstare potest, ut vacuum evitetur. At, inquis, fistulam quadricubitalem spiritu vini plenam cum globulo innatante si clauseris, & inverteris deorsum, ascendet globulus spatio 200 vibrationum perpendiculi; in eâdem verò fistulâ communis, & simplicis aquæ plenâ ascendet subduplo tempore 100 vibrationum. Cur hoc? nisi quia aqua ut pote gravior validiùs extrudit globulum, quàm spiritus vini. Nihilominus: si gravia in levibus magis gravitant, & velociùs descendunt, quò major est specificarum gravitatum differentia; vicissim levia in gravibus magis levitant, & velociùs ascendunt, quò major est secundùm speciem levitatis differentia: Atqui spiritus vini magis accedit ad specificam levitatem innatantis globuli, aqua autem magis differt; in aquâ igitur globulus magis levitat, & velociùs ascendit, sicut lapis in aëre velociùs descendit quàm in aqua, aut in melle. Addis iterum. Vitreo vasculo, cui longior fistula adhæreat, fomitem cum filo sulphurato ope fili ferrei ingere, ut vitrum tangat: totum imple hydrargyro, & converso deorsum osculo descendit hydrargyrus; atque subsistit in altitudine cubiti, & quadrantis: admotâ lucernâ vitrarij vitrum calefiat, ut fomes cum filo sulphurato accendatur: fumus descendit, nec nisi aperto superiore vasis osculo ascendit, aëre videlicet subeunte, à quo extrudatur sursum. Nego fumum ab aëre sursum extrudi, sed qui gravior spiritu raro mercurij in illo descendebat, ubi aërem tangit, ut pote levior in illo ascendit. Non ausim tamen in lapide, qui gravitatem in aquâ & aëre, levitatem in mercurio, aut plumbo liquente obtinet, duplicem statuere virtutem, quarum altera sursum, altera deorsum connitatur: Cum enim impetus motum efficiens (ut infrà constabit) ejusdem naturæ sit, in quamcunque demum orbis plagam dirigatur motus; satis video ab uno eodemque principio, pro variâ contigui corporis conditione, ascensum, descensumve prodire posse. Quandoquidem motus, qui in eadem lineâ perficitur, similes planè includit ubicationes successivè acquisitas, sivè ascensus sit, sivè descensus, ordine tantùm in earum adeptione, commutato. Quare cum ascensus à descensu hoc B 2

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Book One. CHAPTER II. apart from the fact that if it could not draw apart the subjacent air, much less can it do so in mercury, so that a vacuum may be avoided. But, you say, if you close and invert downward a four-cubit tube filled with spirit of wine and with a floating globe, the globe will rise in the space of 200 vibrations of the plumb line; but in the same tube, filled with common and simple water, it will rise in half the time, in 100 vibrations. Why is this? unless because water, being as it were heavier, more strongly drives out the globe than spirit of wine. Nevertheless: if heavy bodies gravitate more in lighter bodies, and descend more quickly, the greater the difference of specific gravities; in turn, light bodies levitate more in heavy bodies, and rise more quickly, the greater the difference in levity according to species: but spirit of wine comes nearer to the specific levity of the floating globe, whereas water differs more; therefore in water the globe levitates more and rises more quickly, just as a stone in air descends more quickly than in water, or in honey. You add again. Into a glass vessel to which a longer tube is attached, insert, by means of an iron wire, a wick with a sulphured thread, so that it touches the glass: fill the whole with mercury, and with the mouth turned downward the mercury descends; and it comes to rest at the height of a cubit, and a quarter: when a lamp is brought near, let the glass be heated so that the wick with the sulphured thread may be lit: the smoke descends, and rises only when the upper mouth of the vessel is opened, namely as air enters, by which it is driven upward. I deny that the smoke is driven upward by the air, but rather that, being heavier than the rare spirit of mercury in it, it descended there; when it touches the air, being lighter, it rises in it. Yet I would not dare, in the case of a stone which obtains heaviness in water and air, lightness in mercury, or in molten lead, to establish a double power, one of which tends upward, the other downward: for since the impulse producing motion (as will be established below) is of the same nature, in whatever quarter of the sphere the motion is directed; I clearly see that from one and the same principle, according to the various condition of the contiguous body, ascent or descent may arise. For a motion which is accomplished in the same line includes plainly similar successive positions acquired, whether it be ascent or descent, the order alone in their attainment being changed. Wherefore, since ascent and descent by this B 2

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Mechanicorum 12 uno differat, quòd quam ubicationem lapis demùm obtineret post alias propè finem motûs, si fuisset centro propior quàm mercurius, eam acquirat sub initium motûs ante alias, si in mercurij locum aër aut aqua surrogetur centro vicinior quàm lapis: ad ordinem hunc permutandum non videtur necessaria virtutis motricis dissimilitudo; nihil quippe producitur dissimile. Sed si quis sufficere dicat conditionum varietatem, nihil absonum fortè loquatur: debuit enim una virtus activa in sui effectus productione non uni tantùm conditioni alligari, sed pro earum varietate modum quoque operandi mutare posse, modò præstitutos fines, quoad substantiam, non transiliret. Neque arbitror hoc tantùm sensu negatam ab aliquibus levi- tatem positivam; potuissent enim æquè negare gravitatem, ad- missa solùm potentia motrice. Sed si vis ista se movendi deor- sum gravitas positiva dicenda est, cùm eadem sit virtus se mo- vendi sursum, cur levitas positiva non fuerit? Qui enim levita- tem à gravitate sejunctam negat, non illicò levitatem expun- git: quemadmodum Angelos intelligentiâ aut voluntate dimi- nutos non asserunt ij, qui vitalium facultatum distinctionem non agnoscunt. Nullum igitur corpus simpliciter, & absolutè grave dicendum est, nisi quod cæteris omnibus ita petat subesse, ut nequeat raritatem assumere, vi cujus evadat levius corpore simili quidem secundùm naturam, dissimilis tamen raritatis: nullum simpliciter, & absolutè leve, nisi quod ita exigat extre- mam orbis laciniam occupare, ut nunquam constipari possit, ac fieri gravius proximo corpore rariore. Reliqua omnia non nisi comparatè gravia, aut levia dici possunt: sic plumbum grave est in aëre, grave in aqua, at pariter leve in mercurio, leve si cum auro conferatur. Hinc corpus in loco sibi debito constitutum, séque ibi con- servans (extra tamen sphæræ centrum, nec in extimâ orbis ele- mentaris superficie) ob idipsum, quia obsistit non tantùm, ne infra subjectum corpus deprimatur, verùm etiam, ne in locum superioris attollatur, & levitare simul dicendum est, & gravi- tare. At si in alienum locum transferatur, quia in medio levio- re ita repugnat ascensui, ut petat descendere, solùm gravitat; quia verò in graviore ita depressioni reluctatur, ut exigat ad superiora evadere, solùm levitat. Quod si corpora hujusmodi in

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Mechanics 12 one differs in this, that the position which the stone would finally obtain after the others, near the end of motion, if it were closer to the center than Mercury, it acquires at the beginning of motion before the others, if air or water is substituted in the place of Mercury, being nearer the center than the stone: to alter this order, a difference in the motive power does not seem necessary; for nothing dissimilar is produced. But if someone says that the variety of conditions is sufficient, he will perhaps say nothing absurd: for one active power, in producing its effect, ought not to be bound to only one condition, but, according to their variety, to be able also to change its mode of operation, provided it does not overstep the preassigned limits, as to substance. Nor do I think that positive lightness has been denied by some in this sense only; for they could equally have denied gravity, admitting only motive power. But if that power of moving downward is to be called positive gravity, since the same is the power of moving upward, why should there not also be positive lightness? For he who denies lightness separate from gravity does not thereby abolish lightness, just as those who do not acknowledge the distinction of vital faculties do not assert that Angels are diminished in intelligence or will. Therefore no body ought to be called simply and absolutely heavy, except that which so seeks to be beneath all the rest that it cannot assume rarity, by virtue of which it might become lighter than a body indeed similar in nature, though of different rarity: nor simply and absolutely light, except that which so demands to occupy the outermost edge of the sphere that it can never be condensed, and become heavier than a neighboring body that is rarer. All other things can be called only comparatively heavy or light: thus lead is heavy in air, heavy in water, but likewise light in mercury, light if compared with gold. Hence a body placed in the place due to it, and preserving itself there (yet outside the center of the sphere, and not on the extreme surface of the elemental orb), for that very reason, because it resists not only being pressed downward beneath the body below it, but also being raised into the place of the one above, is to be said both to levitate and to gravitate. But if it is transferred to another place, because in the lighter medium it resists ascent in such a way that it seeks to descend, it only gravitates; but because in the heavier medium it resists depression in such a way that it seeks to rise upward, it only levitates. But if bodies of this kind in

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Liber primus. CAPUT II. 13 in actu secundo gravitare aut levitare tunc solùm dixeris, quan- do illa in locum non suum translata aut descendere expetunt, aut ascendere, vel re etiam ipsâ descendunt, aut ascendunt, non admodum repugnabo; modò conatum illum, quo se suo tutantur in loco, gravitationem, & levitationem saltem in actu primo, aut pariter asseras, aut pariter neges. Porrò motus omnis gravium, & levium sicut in vacuo exer- ceri non potest (ut in Vacuo Proscripto cap. 2. num. 9. ostendi) ita in medio fit, vel tardiùs, vel citiùs, tùm pro majori vel minori ipsius medij resistentia ad scissionem partium magis, vel minùs connexarum, tùm comparatâ gravitate seu levitate mobilis cum levitate seu gravitate medij. Hinc est gravibus minùs resistere leviora, magis verò, quæ minùs levia, cæteris pari- bus: sic aër minùs resistit lapidi cadenti, quàm si idem lapis in- ciperet moveri in aquâ, quæ minùs levis est, quàm aër. Ex opposito autem levibus graviora minùs resistunt, quæ au- tem minùs gravia, magis resistunt: sic exhalatio ex fundo aquæ, in vitreâ phialâ ad ignem expositâ, per aquam ascendit velociùs, quàm deinde extra aquam posita ascendat in aëre, ubi fumeam naturam induerit. Unde patet non adeò solidum ab aliquibus ex hoc experimento sumi argumentum negandi positivam levitatem. Quæ enim de gravibus ex comparatione cum levibus dicuntur, ea de levibus, proportione servatâ, di- cenda sunt, si cum gravibus conferantur. Cur autem gravibus leviora, levibus graviora minùs resistant, ratio est, quia mo- bile movetur in medio propter dissimilitudinem; nam si corpus contiguum esset, simile non moveretur; quando igitur major est dissimilitudo, debet velociùs moveri, segniùs autem, & len- tiùs, quò propiùs abest à similitudine, donec in simili demum quiescat. Est itaque in corporibus gravitas, & levitas, vi cujus motus ali- quos juxta naturæ propensionem perficiunt, ut certo denique in loco consistant, ejusdemque vi resistunt, ne oppositis motibus cieantur, & à suæ quietis loco avellantur. Quamvis autem eade[m] maneat gravitas aut levitas, non idem tamen est semper momentu[m] (Græcis e[ss]e[m] p[ro]p[ter]n) hoc est actualis ad motum inclinatio, dum in actio- ne est; hæc enim, ut infra patebit, ut plurimum ex positione, & situ mutatur, vel comparatè ad mediu[m], in quo perficitur motus. B 3

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Liber primus. CHAPTER II. 13 in second actuality you may say that to gravitate or levitate only when that thing, having been transferred to a place not its own, either seeks to descend, or to ascend, or in fact descends, or ascends, I shall not much object; provided only that you either equally assert, or equally deny, that tendency by which it is protected in its own place, as gravitation and levitation at least in first actuality. Moreover, every motion of heavy and light bodies, just as it cannot be exercised in a vacuum (as I showed in Vacuo Proscripto , chap. 2, no. 9), so also takes place in a medium, either more slowly or more quickly, both according to the greater or lesser resistance of the medium itself to the separation of its more or less closely joined parts, and according to the compared gravity or levity of the moving body with the levity or gravity of the medium. Hence lighter things resist heavy bodies less, but things that are less light resist them more, other things being equal: thus air resists a falling stone less than if the same stone began to move in water, which is less light than air. Conversely, heavier things resist light bodies less, but those that are less heavy resist them more: thus an exhalation from the bottom of water, in a glass vial exposed to fire, rises through the water more quickly than afterward, when placed outside the water, it rises in the air, where it has assumed a smoky nature. Hence it is evident that from this experiment some do not draw a very solid argument for denying positive levity. For what is said of heavy bodies in comparison with light bodies must, with proportion preserved, also be said of light bodies if they are compared with heavy ones. And why lighter things resist heavy bodies less, and heavier things resist light bodies less, is because the mobile body is moved in a medium on account of dissimilarity; for if a body were contiguous, what is similar would not be moved. When, therefore, the dissimilarity is greater, it ought to move more quickly; but more sluggishly and more slowly the nearer it is removed from similarity, until it finally comes to rest in what is similar. There is therefore in bodies gravity and levity, by virtue of which they accomplish certain motions according to the inclination of nature, so that at last they may remain fixed in a definite place, and by the same force they resist, lest they be set in motion by contrary motions and be torn away from the place of their rest. Although the same gravity or levity remains, nevertheless the same is not always the momentum (for the Greeks, being, as it were, an act on account of it), that is, the actual inclination toward motion while it is in action; for this, as will appear below, is for the most part changed by position and situation, or comparatively with the medium in which the motion is accomplished. B 3

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CAPUT III. Quid sit centrum gravitatis, & linea directionis. Quamvis non minùs levitate, quàm gravitate prædita sint corpora, quia tamen frequentiùs gravitatem vincere conamur, quàm levitatem; ideò illa potissimùm cadit sub contemplationem scientiæ Machinalis: vix enim aliquando contingere poterit, ut opus sit infra aquam corpus aliquod leve per vim deprimere. Hinc factum est, ut de solo gravitatis centro sermo communiter sit, levitatis autem centrum silentio obvolvatur: quia nimirùm quæ de gravitate descendente explicantur, ea de levitate ascendente, pro rata portione, dicta facile intelliguntur. Ad centrum terræ (quod & centrum gravium ac levium dicimus) properant corpora quæcumque gravia in medio leviore constituta sibi redduntur, ut motus suos perficiant. Quoniam verò natura finem propositum per media, quæ potest, brevissima prosequitur, ambages, & diverticula fugiens; moventur per lineam rectam, ut pote brevissimam, nisi externo aliquo impedimento cogantur à rectitudine deflectere: Hæc autem recta linea intelligi debet ex terræ centro ducta ad corpus ipsum, quod movetur; ac proinde tùm in sphæricam superficiem, tùm in planum Horizontis ad perpendiculum cadit. Sed quia corpus, quod deorsum contendit, plures habet partes, quibus constat, singulas suâ gravitate præditas, lineæ verò à singulis hisce partibus exeuntes in terræ centro concurrunt; fieri non potest, ut servatâ corporis figurâ, atque continuo partium nexu non dissoluto, per rectam suam lineam ad centrum ductam unaquæque pars descendat. Si enim parallelepipedum AB in aëre dimittatur, ut sponte

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CHAPTER III. What the center of gravity is, and the line of direction. Although bodies are endowed no less with levity than with gravity, yet because we more frequently try to overcome gravity than levity, therefore it is chiefly the former that comes under the consideration of the mechanical science: for it can scarcely ever happen that there is need, beneath water, to force some light body downward by strength. Hence it has come about that only the center of gravity is commonly spoken of, while the center of levity is wrapped in silence; because, indeed, what is explained concerning descending gravity is easily understood, in due proportion, of ascending levity. Bodies of whatever kind, when placed in a lighter medium, hasten toward the center of the earth—which we also call the center of heavy and light bodies—so that they may complete their motions. But since nature pursues the intended end by the shortest means it can, avoiding detours and byways, they are moved by a straight line, as the shortest one, unless by some external obstacle they are compelled to turn aside from rectitude: this straight line, however, must be understood as drawn from the center of the earth to the body itself that is moved; and therefore it falls at right angles both to the spherical surface and to the plane of the horizon. But because a body that strives downward has many parts of which it consists, each endowed with its own gravity, and because the lines proceeding from each of these parts meet at the center of the earth, it cannot happen that, while the body's shape is preserved and the continuous union of its parts is not dissolved, each part descends along its own straight line drawn to the center. For if a parallelepiped AB is let fall in the air, so that of its own accord

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Liber primus. CAPUT III. 15 te sua descendat, fieri non potest, ut A rectam A C percur- rat, quin oppositum extremum B à recta BC longissimè rece- dat, & contra: utramque verò extremitatem simul A & B rectâ in centrum C tendere non posse est manifestum: Quare cum sibi invicem obsistant æqualiter, ob gravitatis æqualita- tem, eas ex perpendicularibus A C, B C æqualiter sededere oportet ad latera, atque parallelas B E, A F descendendo des- cribere. Eadem est ratio de cæteris partibus æquali intervallo sejunctis à medio D; omnes enim à suis perpendicularis rece- dunt, præter punctum medium D, cujus perpendicularis D C parallela est lineis à reliquis partibus in motu descriptis. Ex omnibus itaque particulis datum grave componentibus, eæ solùm, quæ puncto D imminent, per rectam D C in centrum moventur; quæ tàm plano horizontis in C, quàm superficii sphæricæ in H perpendicularis est; cæteræ verò parallelæ B E, A F perpendiculares quidem in horizontem cadunt, sed sphæ- ricam superficiem obliquè secant. Iam verò si ejusdem parallelepipedi aliud planum A O hori- zonti parallelum moveri versùs C intelligas, erit in eo similiter aliud punctum unicum, quod rectam D C percurrat; & intra corporis soliditatem unica linea puncto illi imminens viâ eâdem in centrum perget non declinans à perpendiculo: cæteræ partes, tam quæ ad dextrâ, quàm quæ ad levâ, tam quæ antè, quàm quæ ponè, sibi mutuò adversantes à recto in centrû itinere deflectent æqualiter. Cum itaque, in priori positione, linea puncto D imminens, esset in communi sectione planorum, quorum alte- rum partes dextras à sinistris, alterum anteriores à posterioribus æqualiter secernebat; in secundâ autem positione linea à per- pendiculo non recedens sit quoquè in duorum planorum com- muni sectione, quibus pariter corporis gravitas in æquas tribui- tur partes; unum verò ex planis secantibus sit utrique positioni commune; unicum est punctum tribus planis commune, in quo binorum planorum sectiones se invicem secant, & sit ex. gr. punctum I; quod unicum rectâ pergit in centrum C, quemcum- que tandem situm in motu obtineat corpus datum A B, ipsum enim est duabus illis lineis commune, quæ in singulis positioni- bus ad sui perpendiculi latera non recedunt: cætera illarum li- nearum puncta, mutatâ positione corporis, lineam quoque mo- tûs mutant. Illud

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Book One. CHAPTER III. 15 if it descends of itself, it cannot happen that A passes through the straight line A C, without the opposite extremity B receding farthest from the straight line B C, and conversely: nor is it manifest that both extremities A and B can at once tend by a straight line toward the center C. Wherefore, since they equally oppose one another, because of the equality of gravity, they ought equally to move aside from the perpendiculars A C, B C toward the sides, and, descending, describe the parallels B E, A F. The same reason holds for the other parts separated by an equal interval from the middle D; for all recede from their own perpendiculars, except the middle point D, whose perpendicular D C is parallel to the lines described in motion by the remaining parts. Of all the particles therefore composing the given weight, only those which lie under point D move by the straight line D C toward the center; which is perpendicular both to the horizontal plane in C and to the spherical surface in H; but the others, parallel to B E, A F, indeed fall perpendicularly to the horizon, yet cut the spherical surface obliquely. Now if you imagine another plane A O of the same parallelepiped, parallel to the horizon, being moved toward C, there will likewise be in it another single point which traverses the straight line D C; and within the solid body a single line lying under that point will proceed by the same path into the center, without departing from the perpendicular: the other parts, both those on the right and those on the left, both those in front and those behind, opposing one another, will equally deviate from the straight course to the center. Therefore, since in the former position the line lying under point D was in the common section of planes, of which one equally separated the right parts from the left, the other the anterior from the posterior; but in the second position the line not receding from the perpendicular is likewise in the common section of the two planes, to which the weight of the body is equally distributed in equal parts; and since one of the intersecting planes is common to both positions, there is one single point common to the three planes, in which the sections of the two planes intersect one another, and let it be, for example, point I; which alone proceeds in a straight line to the center C, whatever position the given body A B may finally take in motion, for it is common to those two lines which, in each position, do not recede to either side of their perpendicular. The other points of those lines, when the body’s position is changed, also change the line of motion. That

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Mechanicorum 16 Illud itaquè punctum in quocumque corpore gravi, quod semper in motu describit lineam rectà in terræ centrum ductam, dicitur Centrum Gravitatis; & linea, quæ centrum gravitatis conjungit cum terræ centro, Linea directionis dicitur; secundùm quam videlicet dirigitur motus, & dimentienda est corporis à centro terræ distantia, si quatenus grave considere- tur. Porrò punctum I centrum gravitatis dicitur, quia centri nomen tribuitur puncto, quod est medium: & quemadmodum magnitudinis alicujus centrum vocatur punctum illud, quod æquales magnitudines circumstant, si partes, quæ ex adverso sunt, accipiantur; ita in gravibus centrum gravitatis dicitur, quod æquales gravitates, vel æqualia gravitatum momenta cir- cunstant. Quod si punctum I non haberet hinc, & hinc æqua- les gravitatum vires, ab alterutrâ parte præstante viribus pro- pelleretur in latus extra lineam directionis, à quâ nunquam re- cedit, si liberè moveatur. Cave tamen, ne partium æqualita- tem dimetiaris linearum longitudine à centro gravitatis exeun- tium, ita ut singulas lineas æqualiter dividendas putes; sed to- tum corpus debet intelligi divisum bifariam à plano per cen- trum gravitatis ipsius corporis, & per centrum gravium ac le- vium transeunte, ita ut si planum à dextrâ in sinistram ductum secernat partes anteriores à posterioribus, æqualia sint gravita- tum momenta antè, & ponè; si aliud planum per eandem di- rectionis lineam ductum partes dextras à sinistris distinguat pa- ria similiter hinc & hinc gravitatum momenta relinquat. Gravitatum, inquam, momenta, non gravitates; ne locus pateat æquivocationi; neque enim quoties æqualia sunt mo- menta, toties æquales sunt gravitates hinc & hinc centrum gra- vitatis complectentes, ut patebit ex iis, quæ de æquilibrio dice- mus. Unde fit in iis tantùm corporibus, quæ partibus unius ejus- demque naturæ, ac ductu perpetuo similiter constitutis, constât, idem esse centrum gravitatis atque magni- tudinis; reliqua certis regulis non circum- scripta, aut ex variis naturis composita, in alio puncto, molis centrum habere, in alio, gravitatis. Si enim duo solida V T, cujus centrum gravitatis, & magnitudinis R, & MN, cujus centrum S, æqualia secun- dùm

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Mechanics 16 That point in any heavy body which always, in motion, describes a line drawn straight to the center of the earth is called the Center of Gravity; and the line which joins the center of gravity with the center of the earth is called the Line of Direction, according to which, namely, the motion is directed, and by which the body's distance from the center of the earth is to be measured, insofar as it is considered as heavy. Moreover, the point I is called the center of gravity because the name of center is given to a point that is in the middle; and just as the center of some magnitude is called that point which is surrounded by equal magnitudes, if the parts that are opposite are taken, so in heavy bodies the center of gravity is said to be that which is surrounded by equal weights, or equal moments of weights. For if the point I did not have equal forces of gravity on this side and on that, it would be driven by the stronger forces from one side toward the side outside the line of direction, from which it never departs if it moves freely. But beware lest you measure the equality of the parts by the length of the lines running out from the center of gravity, as though you thought each line should be divided equally; rather, the whole body must be understood as divided in two by a plane passing through the center of gravity of the body itself and through the center of heavy and light things, so that if a plane drawn from right to left separates the front parts from the back parts, the moments of gravity before and behind are equal; if another plane drawn through the same line of direction distinguishes the right parts from the left, it likewise leaves equal moments of gravity on this side and on that. I say moments of gravity, not weights, lest there be room for ambiguity; for it is not the case that whenever the moments are equal, the weights on this side and on that enclosing the center of gravity are equal, as will be evident from what we shall say about equilibrium. Hence it happens that only in those bodies which consist of parts of one and the same nature, and are perpetually and uniformly arranged in the same way, is the center of gravity the same as the center of magnitude; the rest, not bounded by fixed rules, or composed of different natures, have in one point the center of mass, in another the center of gravity. For if two solids V T, whose center of gravity and center of magnitude is R, and MN, whose center is S, are equal according to

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Liber primus. CAPUT III. 17 dùm gravitatem coagmententur, non erit centrum gravitatis totius molis compositæ in I, ubi planum transiens per V N se- cat lineam R S jungentem centra singularum gravitatum æqua- lium, sed erit in L, ubi recta R S bifariam dividitur: planum autem per centrum terræ, & punctum L ductum non ita secat hanc molem, ut sint æquales hinc, & hinc gravitates, quamvis æqualia sint gravitatum inæqualium momenta, quæ ex figuræ positione potissimùm pendent. Quod si corporis V T gravitas ad corporis M N gravitatem, eam haberet rationem, quam S I ad I R, esset I gravitatis centrum molis compositæ, quæ à plano per terræ centrum, & punctum I ducto non in gravitates æqua- les, sed in momenta æqualia divideretur; ut in loco inferiùs ex- plicabitur. Observa autem non semper centrum gravitatis esse in ipso corpore gravi, ut patet in corporibus annularibus, aut angulos cavos habentibus, in quibus nullum est punctum per quod tran- seuntia plana quæcunque dividant in æquas partes momenta gravitatum: ita tamen est extra corporis cavi soliditatem, ut sit intra ipsam cavitatem punctum, ex quo si intelligatur annulus, vel frustum annulare suspendi, manet positionem habens hori- zonti parallelam, cum habeat æqualia hinc, & hinc gravita- tum momenta. Quod si corpus in cavos angulos sinuatum ha- beat particulam aliquam procurentem, potest contingere, ut in illius particulæ extremo sit totius molis centrum gravitatis: sic brevioris alicujus bacilli extremitati alteri si duos cultros in- fixeris, ut singuli cum bacillo hinc, & hinc angulum acutum ad easdem partes constituant, ita inclinari possunt, ut extremo ungue supposito reliquæ bacilli extremitati tota illa moles susti- neatur citrà periculum cadendi, cùm gravitatis centrum in illa extremitate, intrà cavitatem, quam inclinati cultri faciunt, æqualia habeat ex omni parte gravitatum momenta, si planum secans per illud transeat. C

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Liber primus. CAPUT III. 17 when the gravities are combined, the center of gravity of the whole composite mass will not be in I, where a plane passing through V N cuts the line R S joining the centers of the individual equal gravities, but will be in L, where the straight line R S is divided in half: but the plane drawn through the center of the earth and the point L does not cut this mass in such a way that the weights on this side and on that are equal, although the moments of the unequal gravities are equal, which depend chiefly on the position of the figure. But if the gravity of body V T had to the gravity of body M N that ratio which S I has to I R, I would be the center of gravity of the composite mass, which by a plane drawn through the center of the earth and the point I would be divided not into equal weights, but into equal moments; as will be explained below. Observe also that the center of gravity is not always in the body itself that is heavy, as is clear in ring-shaped bodies, or those having hollow angles, in which there is no point through which any planes passing divide the moments of the weights into equal parts: yet it is outside the solidity of the hollow body, so that inside the cavity itself there is a point from which, if a ring or ring-shaped segment is understood to be suspended, it remains in a position parallel to the horizon, since it has equal moments of weights on this side and on that. But if a body bent into hollow angles has some projecting part, it may happen that at the end of that part is the center of gravity of the whole mass: thus if to one end of a shorter staff you insert two knives, so that each with the staff on this side and on that forms an acute angle toward the same sides, they can be inclined in such a way that, the end claw being placed underneath, the whole mass is supported at the other end of the staff without danger of falling, since the center of gravity, at that end, within the cavity which the inclined knives make, has equal moments of the weights on every side, if a plane cutting through it should pass. C

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CAPUT IV. An gravia centro vicina minùs gravitent. Corpora non intelliguntur gravitare nisi in alieno loco; quando scilicet corpus contiguum inter illa & centrum terræ interjectum, quod medii rationem habere potest, levius est; petit enim infra illud esse: nisum autem hunc deorsùm Gravitationem dicimus. Sed quoniam nisus iste videtur idcircò à naturâ institutus, ut perturbatus corporum ordo restituatur; si ex fine ratio petenda sit, satis apparet corpora gravia centro terræ vicina minùs gravitare. Quemadmodum enim quoties- cunque aliquis à proposito fine magis distat, eò magis anxius est, atque solicitus de mediis ad illum assequendum necessariis, & animo æquiore toleratur modica, quàm multa violentia; ita natura minorem ordinis debiti perturbationem sentiens, si gra- ve parùm absit, quàm si longè abesset, à loco, ubi juxta inge- nitam propensionem exigit consistere, minùs solicita esse debet de illo restituendo, nec adeò vehementi conatu, hoc est gravi- tatione, illud urgere debet in locum suum. Ad hæc omnibus apertissimè liquet eò majore naturæ impe- tu corpora deorsùm niti, quò levius est corpus, in quo tan- quam in medio perficiendus est motus, si dimittantur. Sic à saxo in aëre pendente manum deorsùm validiùs trahi senti- mus, quàm ab eodem aquæ immerso trahatur, & multò lan- guidiùs conatur deorsum lapis in melle descendens, quàm in aqua; quia videlicet aqua levior est melle, & aër levior aquâ. Hinc est quod, si medij partes fuerint diversâ gravitate prædi- tæ, pars centro terræ propior etiam erit gravior; atque ideò corpus in parte medij graviore minùs gravitabit propè centrum terræ, quàm procul. Esse autem ejusdem medij non commoti partes graviores in imo, omnium ferè hominum sensus est: quotus enim quisque est, qui nesciat mellis optimam partem esse, quæ in vasis fundo, vini quæ in medio, olei quæ in sum- mo? id autem verum non esset, nisi liquoris ejusdem partes essent diversâ gravitate delatæ in loca à terræ centro dispari- bus

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CHAPTER IV. Whether bodies near the center gravitate less. Bodies are not understood to gravitate except in an alien place; when, that is to say, a contiguous body is interposed between them and the center of the earth, which may serve in the role of a medium, and is lighter; for it seeks to be beneath it: this impulse downward we call gravitation. But since this impulse seems therefore to have been established by nature, so that the disturbed order of bodies may be restored; if the reason is to be sought from the end, it is sufficiently apparent that heavy bodies near the center of the earth gravitate less. For just as whenever someone is farther from a proposed end, the more anxious and solicitous he is about the means necessary for attaining it, and with a calmer mind a moderate difficulty is borne than a great violence; so nature, sensing a smaller disturbance of the due order, if a heavy body be only a little distant rather than far away from the place where, according to its innate tendency, it requires to stand, ought to be less concerned with restoring it, and ought not to press it back to its place with so vehement an effort, that is, with gravitation. To these things it is most plainly clear to all that bodies are driven downward with a greater impulse of nature the lighter the body is in which, as in a medium, the motion is to be completed, if they are let go. Thus from a stone hanging in the air we feel the hand pulled downward more strongly than from the same one immersed in water, and a stone descending in honey tries to go downward far more languidly than in water; because water is lighter than honey, and air lighter than water. Hence it is that, if the parts of the medium have been endowed with different gravity, the part nearer the center of the earth will also be the heavier; and therefore a body in the heavier part of the medium will gravitate less near the center of the earth than far away. Now that the heavier parts of the same unmoved medium are below is the sense of almost all men: for how many are there who do not know that the best part of honey is that which is at the bottom of the vessels, of wine that which is in the middle, of oil that which is on the top? But this would not be true unless the parts of the same liquid had been brought, with different gravity, into places at differing distance from the center of the earth

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Liber primus. CAPUT IV. 19 bus intervallis remota: Quia enim oleum eò perfectius est, quò propiùs aëris levitatem spirituum subtilitate æmulatur, ideò quod in summo vase innatat, optimum est: At vini sua- vitas in exquisitâ sui tartari sufficienti humore diluti cum spi- ritibus permistione consistens medium locum in vase exigit, sicut media est illius gravitas inter vagantium spirituum levita- tem, & fæculenti tartari gravitatem: Mellis demùm dulcedo ex sui salis, seu sacchari, copiâ proveniens iis partibus potissi- mum inest, quæ multo sale refertæ graviores quoquè sunt, & in fundo subsidunt. Nec est iis abroganda fides, qui in altissi- mo mari adeò gravem aquam à se deprehensam alicubi testan- tur, ut supta reliquum maris fundum ambulantes ad altissi- mam fossam venerint, in quam penetrare sæpiùs irrito conatu tentârint: his enim non ægrè fidem habeo, qui aërem in imis vallibus crassiorem atquè graviorem, in summis verò montibus puriorem atque leviorem ab omnibus admitti video. Cum ita- que (si ex notis ad minùs nota progredi philosophando liceat) propè centrum gravium ac levium medij partes graviores sint, quàm procul ab illo; minor est gravitatio corporum, si centro propiora fiant, ac quando longè ab illo remota detinebantur. Hinc autem responderi potest quærentibus, cur in fodinis lon- gè faciliùs crudi metalli massa moveatur, quàm in superficie terræ: aër scilicet profundis illis cuniculis inclusus gravior mul- tò ac crassior est aëre isto, quem inspiramus, atque adeò ibi metallum minùs gravitat. Quòd si libeat minorem hanc gravitationem experimento deprehendere, sume vitream fistulam supernè clausam longio- rem pedibus tribus Romanis, eam imple argento vivo, digito- que osculum accuratè claudens inverte, ac argento vivo sub- jecti vasis immerge; tùm amoto digito descendet mercurius in fistulâ, iterúmque ascendet, & in certâ demum altitudine per- pendiculari quiescet. Observatâ igitur altitudine perpendicu- lari, quam mercurius obtinet, si in imâ valle experimentum instituatur, eâque comparatâ cum altitudine perpendiculari, in qua consistit, cùm in summo montis altissimi vertice expe- rimentum idem sumitur, animadvertes altitudinem mercurij per vim in fistulâ suspensi minorem esse in summo monte, quàm C 2

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at intervals removed: For the more perfectly oil is so much the more it emulates the lightness of air by the subtlety of its spirits; therefore that which floats on the top of the vessel is the best. But the sweetness of wine, consisting in the exquisite mixture of its own tartar sufficiently diluted with humours and blended with spirits, requires a middle place in the vessel, just as its weight is middle between the lightness of wandering spirits and the weight of the feculent tartar. Lastly, the sweetness of honey, arising from the abundance of its salt, or sugar, is chiefly found in those parts which, being filled with much salt, are also heavier and settle at the bottom. Nor is credit to be denied to those who testify that in the deepest sea they have found water so heavy somewhere that, walking beneath the remaining bottom of the sea, they came to a very deep pit, into which they often tried in vain to penetrate: for I readily believe such men, since I see that in the lowest valleys the air is thicker and heavier, but on the highest mountains purer and lighter, and this is admitted by all. Since then, if from things known it is allowed, in philosophizing, to proceed to things less known, around the center of heavy and light things the middle parts are heavier than those farther from it; the gravitation of bodies is less when they are made nearer the center than when they were held far away from it. Hence an answer may be given to those asking why in mines the mass of crude metal is moved much more easily than on the surface of the earth: namely, because the air enclosed in those deep mine-shafts is much heavier and thicker than the air which we breathe, and therefore the metal there weighs less. But if it please you to detect this lesser gravitation by experiment, take a glass tube closed at the top, three Roman feet long; fill it with quicksilver, and, carefully closing the mouth with your finger, invert it and immerse it in a vessel containing quicksilver; then, when the finger is removed, the mercury will descend in the tube and rise again, and at last will come to rest at a certain perpendicular height. Therefore, having observed the perpendicular height which the mercury obtains, if the experiment is performed in a deep valley, and that height is compared with the perpendicular height at which it stands when the same experiment is made on the summit of a very high mountain, you will notice that the height of the mercury suspended by force in the tube is smaller on the high mountain than

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Mechanicorum in valle; Quia nimirum mercurius intra fistulam detentus tan- quam in vale, est in aëre fistulam ambiente tanquam in loco; in aëre autem leviori cùm magis gravitet, in minori etiam altitudine perpendiculari consistit. Experimentum hoc in valle, & in monte sumere mihi otium non fuit, quamvis in eo sæpiùs me exercuerim: sed de illius veritate ambigere non sinunt testes in Galliâ luculentissimi, qui discrimen hoc in mercurij altitudine observârunt in altioribus montibus. Verùm, ex alio præteteà capite imminui debet gravitatio corporum in minori à centro remotione, habitâ solùm ratione sitûs. Cùm enim totius corporis gravitatio conflata sit ex singularum partium impetu, quo deorsum nituntur, manifestum est singulis partibus languidiùs deorsum conantibus, totius corporis gravitationem esse pariter languidiorem. Quoniam verò quicquid in motu cogitur à recto secundùm naturam tramite deflectere, lentiùs atque remissiùs pergit ad præstitutum motûs terminum; particulæ autem corporis solidi gravis, propiores centro factæ, magis à suo perpendicularo, sibi invicem adversantes, declinant; satis constat singulas fractis quodammo- do viribus languentes plurimum de conatu remittere. Si enim solidum A B fiat centro vicinius ita, ut A sit in K, & B in L, lineæ directionis partium extremarum sunt K C, L C: at coguntur per lineas K F, L E parallelas descendere, fiuntque anguli C K F, C L E externi majores internis C A K, C B L per 16. l. 1. magis igitur in K & L recedunt à perpendicularo, quàm recederent in A & B. Quia itaque pars in K existens magis impeditur ab oppositâ extremitate, quæ in L, ne per K C descendat (nisi enim pars, quæ in L, urgeret oppositam tentans per L C descendere, non cogeretur pars in K existens adeò recedere à suâ directionis lineâ) minori etiam impetu deorsum fertur. Est autem eadem de reliquis partibus ratio, præter

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in a valley; For indeed mercury, being confined within the tube as in a valley, is in the air surrounding the tube as in a place; but in thinner air, since it weighs more lightly, it also stands at a lesser perpendicular height. I did not have the leisure to make this experiment in a valley and on a mountain, although I had often exercised myself in it: but the most illustrious witnesses in France do not allow me to doubt its truth, who observed this difference in the height of the mercury on higher mountains. Yet, from another source, moreover, the gravitation of bodies must be diminished by a lesser remotion from the center, regard being had only to position. For since the gravitation of the whole body is composed from the impulse of the individual parts, by which they strive downward, it is manifest that if the individual parts tend downward more feebly, the gravitation of the whole body is likewise more feeble. But since whatever in motion is compelled to deviate from its straight path by nature proceeds more slowly and more languidly toward the predetermined end of the motion; and the particles of a solid heavy body, being brought nearer to the center, deviate more from their perpendicular, opposing one another; it is sufficiently clear that each, its powers in some way broken, weakens greatly in its effort. For if the solid AB be made nearer the center so that A is at K and B at L, the lines of direction of the outermost parts are KC, LC: but they are compelled to descend along the parallel lines KF, LE, and the external angles CKF, CLE are greater than the internal angles CAK, CBL, by 16. l. 1. therefore in K and L they recede more from the perpendicular than they would in A and B. Since therefore the part existing in K is more hindered by the opposite extreme part, which is in L, so that it may descend through KC (for unless the part which is in L urged the opposite part, attempting to descend through LC, the part existing in K would not be compelled to depart so much from its line of direction), it is carried downward with a lesser impulse. And the same reason applies to the remaining parts, except

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Liber primus. CAPUT IV. 21 præter eas, quæ in eâdem directionis lineâ sunt cum centro gravitatis; singulæ enim ad centrum terræ accedentes magis à suo perpendiculo recedunt, minúsque deorsum gravitant. Quî igitur fieri possit, ut debilitato singularum particularum conatu, atque impetu deorsum, non minuatur pariter totius corporis gravitatio, si fiat centro vicinius? Illud tamen non diffiteor, quod si medij levitates, aut angulorum CLE, CBL inclinationes eo tantùm discrimine secernantur, quod omnem sensum fugiat, vel saltem ex medij gravitate, & anguli magnitudine conjunctim sumptis oriri non possit varietas, quæ sub sensum cadat; neque percipietur gravitationis differentia in majori vicinitate. Sed hoc non facit, quin inter gravitationes discrimen intercedat; neque enim continuò, si quid sensum latet, id omninò non esse dicendum est: contingere si quidem potest motum aliquem ita sensim, & sine sensu fieri, ut non nisi elapso temporis spatio demùm innotescat. Sic si vinum, cujus gravitas vix minor sit gravitate aquæ arte satis notâ affuderis aquæ ita, ut innatet, & supremam vasis partem occupet, aliudque vas simili vino plenum, sed paulò altius, habeas, tum ex libra centrum motûs habente in centro gravitatis jugi pendeant æqualia pondera intrà vinum utriusque vasis; fiet utique ponderum æquilibrium, & consistent eo in situ, quem illis dederis: at si alterum libræ extremum ita deprimas, ut pondus, quod ex eo pendet, ex vino ad aquam vix graviorem transeat, reliquo pondere intra vinum manente; initio quidem non apparebit motus libræ se restituentis, quia pondus in vino non excedit gravitationem ponderis æqualis in aquâ nisi eo excessu, quo gravitas aquæ superat gravitatem vini; hic autem excessus cum minimus sit, motum quoque efficiet, quem ægrè à quiete discernas, nisi ubi post aliquod tempus deprehenderis pondus altius descendisse, depressius autem ascendisse. Haud secus philosophandum est de majore, aut minore corporum gravitatione, si disparibus intervallis à terræ centro removeantur, diutiùs enim propè centrum incumbere poterunt sustinenti, quàm procul: id quod satis erit ad minorem gravitationem patefaciendam, quæ non statim innotescat. C 3

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Liber primus. CHAPTER IV. 21 except those which are in the same line of direction with the center of gravity; for each, as it approaches the center of the earth, departs more from its perpendicular, and gravitates less downward. How then can it happen that, the effort and impulse of each particular particle downward being weakened, the gravitation of the whole body should not likewise be diminished, if it be brought nearer the center? Yet I do not deny this: that if the lightnesses of the mediums, or the inclinations of the angles CLE, CBL, are distinguished by no other difference than one that escapes all perception, or at least if no variety can arise, taken jointly from the gravity of the medium and the magnitude of the angle, that would fall under the senses; then neither will the difference of gravitation be perceived in greater nearness. But this does not mean that there is no difference between gravitations; for it does not follow at once that, if something escapes the senses, it must therefore be said not to exist at all: indeed it may happen that some motion proceeds so gradually, and without sensation, that it is not known until after the lapse of some time. Thus if you pour wine, whose gravity is scarcely less than the gravity of water, by an art well enough known upon the water in such a way that it floats upon it and occupies the upper part of the vessel, and if you have another vessel filled with similar wine, but somewhat higher, then from a balance having its center of motion at the center of gravity let equal weights hang within the wine of each vessel; there will certainly be an equilibrium of weights, and they will remain in the position you have given them: but if you depress one end of the balance so that the weight hanging from it passes from the wine into the water, which is only slightly heavier, while the remaining weight stays within the wine; at first indeed no motion of the restoring balance will appear, because the weight in the wine does not exceed the gravity of an equal weight in water except by that excess by which the gravity of water surpasses the gravity of wine; but since this excess is very small, it will produce a motion too, which you can hardly distinguish from rest, unless after some time you observe that the higher weight has descended, and the lower one risen. No otherwise must we philosophize about the greater or lesser gravitation of bodies, if they are removed at unequal distances from the center of the earth; for they will be able to rest upon the supporting body for a longer time near the center than far away: this will be enough to reveal a lesser gravitation, which does not become known immediately. C 3

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Mechanicorum Hæc autem non leviter confirmari videntur ex iis, quæ quo- tidiè ferè videmus; nam si circinus, quo circulos describere solemus, cadat, semper nodus prævertit cuspides, & prior terram ferit; nisi fortè nodus ad perpendiculum immineat cruribus: & omnia ferè corpora, quæ centrum gravitatis ex una parte habent, si ex modicâ altitudine dimittantur, videntur quidem cadere parallela; sed ex majori altitudine si descendant, pars gravior prior terram attingit. Sit enim corpus E S, cujus gravitatis centrum H, linea directionis H A; si horizonti parallelum descenderet, per rectas E I, S R parallelas lineæ directionis moveretur; id quod in modicâ tantùm altitudine contingere videtur, quia nondum facta est ea gravitationis imminutio in extremitate S, quæ percipi possit. Si enim E per E I descenderet, S verò per S R, angulus I E A æqualis alterno E A H per 29. lib. 1. minor esset angulo R S A, qui æqualis est alterno H A S; nam ex hypothesi minùs distat E, quàm S, à centro gravitatis H, & est angulus E A H minor angulo H A S; pars igitur S magis deflecteret à suo perpendiculo S A, quàm E deflecteret ab E A; cùm itaque S magis in latus propelleretur, plus etiam de conatu deorsum remitteret, quàm E; atque adeò non posset æqualiter descendere ac moveri, contra hypothesim parallelismi. Dicendum est igitur non per parallelas E I, S R fieri motum, sed intra illas paulatim partem E graviorem præcurrere: quia scilicet partes omnes extra lineam directionis A H constitutæ dum removentur à suo perpendiculo, aliquid amittunt de impetu, quo deorsum nituntur, propiores quidem minus, remotiores autem plus; pars si quidem G in principio motûs descendens parallela lineæ directionis per G M facit angulum A G M internum per 16.lib.1. minorem externo G M S, qui per 29. 1. est æqualis alterno M S R. Quia ergo A G M minor

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Mechanics This, however, seems to be confirmed not slightly by those things which we see almost daily; for if the compass, with which we are accustomed to describe circles, falls, the knob always precedes the points and strikes the ground first; unless perhaps the knob is hanging vertically over the legs. And almost all bodies which have the center of gravity on one side, if they are let fall from a moderate height, seem indeed to fall parallel; but if they descend from a greater height, the heavier part touches the ground first. For let there be the body E S, whose center of gravity is H, and the line of direction H A; if it were to descend parallel to the horizon, it would move along the straight lines E I, S R parallel to the line of direction; which seems to happen only at a moderate height, because that diminution of gravitation has not yet been made in the extremity S which can be perceived. For if E were to descend by E I, but S by S R, the angle I E A, equal to the alternate E A H by 29, lib. 1, would be smaller than the angle R S A, which is equal to the alternate H A S; for by hypothesis E is farther from the center of gravity H than S, and the angle E A H is smaller than the angle H A S; therefore the part S would deviate more from its perpendicular S A than E would from E A; since therefore S would be driven more to the side, it would also yield more of its downward effort than E; and thus it could not descend and move equally, contrary to the hypothesis of parallelism. We must therefore say that the motion is not made through the parallels E I, S R, but that within them the heavier part E gradually gets ahead: because, namely, all parts situated outside the line of direction A H, while they are removed from their perpendicular, lose something of the impulse by which they strive downward, the nearer parts less, but the more remote parts more; for indeed the part G, descending at the beginning of the motion parallel to the line of direction through G M, makes the internal angle A G M, by 16. lib. 1, smaller than the external G M S, which by 29. 1 is equal to the alternate M S R. Since therefore A G M is smaller

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Liber primus. CAPUT IV. 23 minor est angulo A S R, pars G minus de suo impetu deorsum amittit, quàm pars S; & quamvis initio discrimen hoc non percipiatur, demum fit, ut additis pluribus differentiis mani- festè appareat partem S minùs gravitare, quia tardiùs deor- sum movetur; & tandem ipsa sequitur partem E præcur- rentem, postquam minori illâ gravitatione permisit parti E, ut propiùs accederet ad lineam directionis, fieretquè quæ- dam virtualis conversio circa centrum gravitatis H, in qua extremitas E occuparet infimum locum, S autem supre- mum. Quare cùm nos doceat experientia partem H S æquiponderantem parti H E, si suspendantur ex H, in mo- tu tamen minùs gravitare, quàm oppositam, ideóque fieri illam conversionem, ut pars E fiat inferior; neque aptior assignari possit ratio, quàm quæ petitur ex recessu partium majori à suo perpendiculo: satis liquet, quantum momenti habeat hæc declinatio à perpendiculo ad minuendam gra- vitationem. Ex majori igitur declinatione à lineâ perpen- diculari, quæ consequitur corpus constitutum non adeò procul à centro terræ ut priùs, non ineptè arguitur minor corporis gravitatio in eo situ, si cætera sint paria: neque enim comparo corpus, quod per motum descendit, perse- verans in suo motu, cum corpore in loco altiori transeun- te à quiete ad motum; nam tunc ex impetu per motum concepto major est gravitatio in loco inferiore, quàm in su- periore: sed tantùm corpora invicem comparo, vel pariter quiescentia, vel æquali tempore mota, illudque, quod ter- ræ vicinius est, assero, vel minori nisu conari à quiete in loco alieno transire ad motum, vel æquali tempore, quo præ- cessit motus, minus impetus acquisuisse ac minoribus viribus motum continuare. Ex his quæ de gravibus hactenus disputata sunt, aliquis fortassè inferat levia à centro remotiora minùs levitare, si- cut gravia centro propiora minùs gravitant. Verùm res est pensiculatiùs examinanda, nec simpliciter ex oppositis gra- vium, ac levium naturis definienda, quasi ob id ipsum, quia sibi gravitas atque levitas adversantur, contraria ha- berent omnia consequentia. Et quidem quod spectat ad solam

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Book one. CHAPTER IV. 23 the angle A S R is smaller, the part G loses less of its downward impetus than part S does; and although at first this difference is not perceived, in the end it comes about that, as several differences are added together, it clearly appears that part S gravitates less, because it moves downward more slowly; and at length it follows the part E, which was moving ahead, after it has allowed by that lesser gravitation the part E to come nearer to the line of direction, and there is produced a kind of virtual turning about the center of gravity H, in which the end E occupies the lowest place, while S occupies the highest. Therefore, since experience teaches us that part H S, being equal in weight to part H E when they are suspended from H, nevertheless in motion gravitates less than the opposite part, and thus brings about that turning so that part E becomes lower; and since no more suitable reason can be assigned than that which is drawn from the greater departure of the parts from their plumb line: it is sufficiently clear how much importance this deviation from the plumb line has for diminishing gravitation. From a greater deviation, then, from the perpendicular line, which follows when a body is situated not so far from the center of the earth as before, one may not ineptly infer a lesser gravitation of the body in that position, if all other things are equal: for I am not comparing a body that descends by motion, persisting in its motion, with a body in a higher place passing from rest to motion; for then, from the impetus conceived through motion, gravitation is greater in the lower place than in the higher one. Rather, I compare only bodies with one another, either equally at rest or moved for an equal time, and I assert that that body which is nearer the earth either strives with less effort to pass from rest to motion in some foreign place, or, in the equal time during which motion has preceded, has acquired less impetus and continues motion with lesser forces. From what has so far been discussed concerning heavy bodies, someone may perhaps infer that light bodies, being farther from the center, levitate less, just as heavy bodies gravitate less when nearer the center. But the matter must be examined more carefully, and not defined simply from the opposite natures of heavy and light things, as if for that very reason, because gravity and lightness are contrary to one another, all their consequences would also be contrary. And indeed, as far as concerns lightness alone

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Mechanicorum folam corporis levioris positionem, non minuitur levitatio, sed potiùs augetur in majoribus à terræ centro intervallis; ubi minùs à suo perpendiculo declinant partes centrum levitatis circunstantes, & idcirco minùs de conatu remittunt, quò nituntur ad superiora evadere. Sit namque Globus H G, cujus centrum levitatis M, & linea discretio- nis O M N; cui parallelæ sunt H D & G F, quas des- cribunt ascendendo extremitates H & G, & motum eum- dem continuabunt, si globus in N translatus intelligatur. Quando igitur globus est in M, extremitas H recedit à per- pendiculo O I, & cum eo facit angulum I H T; quan- do autem est in N, extremitas T ascendens per T D fa- cit cum perpendiculo O R an- gulum R T D, qui per 15.lib.1. æqualis est angulo H T O ad verticem, hic autem, inter- nus cum sit, per 16.1. minor est externo I H T. Est ergo R T D minor angulo I H T, atque ideò plus habet mo- menti sursum, ubi minus à recto secundum naturam tra- mite deflectit. Discrimen hoc momentorum ab angulorum inæqualitate proveniens optimè intelligit natura, quæ ita motum perficit, ut, si duo inæqualiter levia coagmentata fuerint, le- vius præcurrat. Sic si A cortex suberis coagmentetur ligno fagino B, & intra aquam mediocriter profundam horizont- taliter collocetur solidum D C, ita per lineam directionis

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Mechanics for a lighter body placed alone, levitation is not diminished, but rather increased at greater distances from the center of the earth; where the parts surrounding the center of levity deviate less from their perpendicular, and therefore give up less of the effort by which they strive to escape upward. Let there be, then, the globe H G, whose center of levity is M, and the line of separation O M N; to which parallel are H D and G F, which the extremities H and G describe in ascending, and they will continue the same motion, if the globe be understood as transferred to N. When therefore the globe is in M, the extremity H departs from the perpendicular O I, and with it makes the angle I H T; but when it is in N, the extremity T ascending by T D makes with the perpendicular O R an angle R T D, which by 15. lib.1. is equal to the angle H T O at the vertex; but this, because it is an interior angle, by 16.1. is less than the exterior I H T. Therefore R T D is less than the angle I H T, and therefore it has more moment upward, where it deviates less from the straight course according to nature. This difference of moments arising from the inequality of the angles nature understands best, which thus completes motion, so that, if two bodies of unequal lightness are joined together, the lighter may go in advance. Thus if A, a cork, be joined to the wood of beech B, and within water of moderate depth the solid D C be placed horizontally, thus by the line of direction

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Liber primus. CAPUT IV. 25 nis T O ascendit centrum levitatis, ut demum A in loco superiore, B autem in inferiore constituatur, extremo D per rectam D O ascendente: Quo in motu natura magnum invenit compendium. Quia enim partes centro levitatis viciniores magis levitant, quòd linea parallela lineæ directionis faciat minorem angulum cum earum perpendiculo (sic si linea directionis sit FL, eique parallelæ NG, RX, angulus NGX internus per 29. 1. est æqualis externo RXY, at PGX externus per 16. 1. major est interno GXF, hoc est VXY ad verticem, ergo PGX major est angulo VXY, & si uterque auferatur ex æqualibus NGX, RXY, remanet NGP minor angulo RXV, ideoque G magis levitat, quam X) ex majore impedimento, quod initio motûs habetur ob anguli HDI magnitudinem, dum pars D minùs levitat, centrum levitatis per SO ascendens inclinat corpus DC, & extremitas D in recta DO constituitur, in qua longè citiùs minuuntur impedimenta, quàm si per parallelam DI ascenderet: vix enim ascendit in E, cum impedimenta sunt æquè diminuta, ac si ascendisset in I; quandoquidem angulus KEI per 29. 1. est æqualis alterno EID, atque adeò etiam angulo, quem in I faceret parallela DI cum perpendiculo; est igitur angulus KEI minor quocunque alio angulo, qui fieret in punctis intermediis lineæ DI; sed quoniam centrum levitatis ascendendo acquisivit majorem impetum, quàm extremitas in E existens, per vim illam rapit extra parallelam EK, trahitque per lineam EO, & perpendiculum facit angulum semper minorem cum lineâ directionis; unde fit partem inferiorem semper faciliùs trahi, quo minùs in diversa D

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Book the First. CHAPTER IV. 25 until the center of gravity rises to O, so that at last A is established in the higher place, but B in the lower, with the end D ascending along the straight line DO: in which motion nature finds a great shortcut. For since the parts nearer the center of gravity are more lightly moved, because a line parallel to the line of direction makes a smaller angle with their perpendicular (so if the line of direction be FL, and the parallels to it NG, RX, the angle NGX, internal, by 29.1, is equal to the external RXY, but the external PGX, by 16.1, is greater than the internal GXF, that is, VXY at the vertex; therefore PGX is greater than the angle VXY, and if both be subtracted from the equal angles NGX, RXY, there remains NGP smaller than the angle RXV, and therefore G is more lightly moved than X), from the greater impediment that is present at the beginning of the motion because of the size of the angle HDI, while the part D is less lightly moved, the center of gravity, rising through SO, inclines the body DC, and the end D is set in the straight line DO, in which the impediments are diminished far more quickly than if it were rising through the parallel DI: for it scarcely rises as far as E, when the impediments have been diminished as much as if it had risen to I; since the angle KEI, by 29.1, is equal to the alternate EID, and consequently also to the angle which the parallel DI would make in I with the perpendicular; therefore the angle KEI is smaller than any other angle that would be formed at the intermediate points of the line DI; but because the center of gravity, in rising, has acquired a greater impulse than the end existing at E, by that force it is carried beyond the parallel EK, and is drawn by the line EO, and makes with the line of direction an angle always smaller; whence it happens that the lower part is always more easily drawn, the less in different D

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Mechanicorum 26 abit ejus perpendicularum, cum quo semper minorem, & minorem angulum facit linea motûs D O; donec demùm totum solidum obtineat situm perpendicularem; quod initio erat in æquilibrio. Cæterum, quamvis habitâ ratione sitûs, levia altiora magis levitent, sivè parallela horizonti jaceant extrema, sivè inclinata, ratione tamen medij, quod in superioribus est levius, quàm in inferioribus, minùs levitant: experientia enim ostendit ea lentiùs ascendere, quæ propiùs accedunt ad medij naturam secundùm levitatem: nam ex tribus globulis sphæricis, quorum diameter unc. 2 ́, pedis Romani, cereus erat ponderis drachmarum 24, faginus drachm. 22, vitraëreus drachm. 7. in aëre expensi, sed eorum motus in aquâ ad altitudinem pedum 14, valdè inæqualis fuit, numeratis vibrationibus ejusdem perpendiculi; cereus siquidem ascendit lentissimè vibrationibus 88, faginus vibrationibus 37, vitraëreus vibrationibus 33: unde patet cereum, qui minimùm ab aquâ differt in pondere (aquæ etenim molis æqualis est drachm. 25 ́) minùs in eâ levitare. Sicut igitur diversa levia in eodem medio inæqualiter levitant, sic idem leve in medio dissimili inæqualiter levitabit pro majore aut minore levitatum dissimilitudine. Conveniunt itaque gravia, & levia, quod hæc procul à centro offendentia medium levius minùs levitant, illa propè centrum habentia medium gravius minùs gravitant. Differunt autem ratione positionis, quia, in loco remotiore à centro, perpendicula omnia concurrunt ad angulos magis acutos, minùsque differunt à lineâ rectâ, ideo quasi collatis viribus magis gravitant, & magis levitant; at prope centrum cum perpendicula magis in diversa abeant, & levia minùs levitant, & gravia minùsgravitant. Porrò hanc similitudinem gravitationis gravium, & levitationis levium in eodem loco, à me vocari discrimen, & differentiam, quia habita ratione oppositorum videbatur leve remotius debere minùs levitare, sicut grave propius minùs gravitat, ne te moveat; litem de verbo non faciam. CAPUT

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Mechanics 26 its perpendicular distance, with which the line of motion D O always makes a smaller and smaller angle; until at last the whole solid obtains a perpendicular position, which at first was in equilibrium. Moreover, although, regard being had to position, lighter bodies at a greater height are more buoyant, whether their ends lie parallel to the horizon or are inclined, yet in respect of the medium itself, which is lighter above than below, they are less buoyant: for experience shows that those ascend more slowly which approach more nearly to the nature of the medium in respect of lightness; for from three spherical globes, of diameter 2 inches of a Roman foot, one of wax weighed 24 drachms, one of beech 22 drachms, one of glass 7 drachms, when weighed in air; but their motion in water to the height of 14 feet was very unequal, the vibrations of the same plumb line being counted; for the wax one ascended most slowly with 88 vibrations, the beech one with 37 vibrations, the glass one with 33: whence it is clear that the wax one, which differs least from water in weight (for the mass of water is indeed equal to 25 1/2 drachms), buoyed up in it less. As therefore different light bodies buoy up unequally in the same medium, so the same light body in a different medium will buoy up unequally according to the greater or lesser difference of buoyancy. Heavy and light bodies thus agree in this, that the latter, striking far from the center, buoy up less in a lighter medium, the former, having a heavier medium near the center, gravitate less. But they differ in respect of position, because in a place farther from the center all the plumb lines converge at sharper angles and differ less from a straight line; therefore, as it were with combined forces, they gravitate more and buoy up more; but near the center, as the plumb lines depart more in different directions, the light bodies buoy up less and the heavy bodies gravitate less. Moreover, this similarity of the gravitation of heavy bodies and the buoyancy of light bodies in the same place is, by me, called a difference and distinction, because, regard being had to contraries, it seemed that a light body farther away ought to buoy up less, just as a heavy body nearer the center gravitates less; let that not move you; I shall make no dispute about the word. CHAPTER

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Liber primus. CAPUT V. 27 CAPUT V. Quâ ratione centrum gravitatis corporum inveniatur. Opus mechanicum plerunque non indiget puncto illo, quod intra corporum soliditatem latet, ac centrum gra- vitatis definivimus; sed satis est si in extimâ corporis superfi- cie innotescat punctum, aut linea imminens ipsi gravitatis centro, pro ratione sitûs, in quo corpus grave consistere cu- pimus. Ideo geometricum laborem inveniendi punctum illud intimum Centrobarycæ relinquens, mechanica tantùm inqui- sitione, & quasi tentans, pervestigo punctum illud, aut li- neam in corporis superficie, cui respondet planum per lineam directionis ductum, & secans corpus in certo situ constitu- tum. Et quidem si corpus sphæricum fuerit ex partibus ejus- dem naturæ conflatum, aut saltem ex partibus heterogeneis quidem, sed circa sphæræ centrum similiter dispositis ita, ut intima sphærula folliculis quibusdam obvolvatur; quia idem est molis atque gravitatis centrum, punctum quodcumque in sphærica superficie assumatur, aptum erit; singula enim si- milem habent positionem. Sin autem aut sphæræ segmentum, aut sphæra ex partibus heterogeneis inæqualiter dispositis fue- rit; imponatur plano horizontali accuratè levi, & maximè æqua- bili; & quod punctum tangetur à supposito plano, ubi motus omnis cessaverit, illud est, quod potissimùm quæritur, ac punctum superius, quod huic è regione est, erit pariter aptum ad propositum finem. Quod si cylindricum fuerit oblatum corpus, aut prisma quod- cunque continuo, & simili ductu productum; secetur bifariam longitudo, & punctum habebitur cylindri centro gravitatis respondens: prismatis autem singula plana parallelogramma si dividantur in æquas tum longitudinis, tum latitudinis partes, planum per inventa puncta ductum transibit per centrum gravitatis prismatis, dividet enim in partes æquales, & simi- liter positas, unde oritur momentorum gravitatis æqualitas. D 2

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Book One. CHAPTER V. 27 CHAPTER V. By what method the center of gravity of bodies is found. Mechanical work for the most part does not require that point which lies hidden within the solidity of bodies, and which we have defined as the center of gravity; it is enough if there be known on the outer surface of the body some point, or line, lying over against the center of gravity itself, according to the position in which we wish the heavy body to stand. Therefore, leaving to geometry the labor of finding that innermost point of the Centrobaryce, I investigate only by a mechanical inquiry, and as it were by trial, that point or line on the surface of the body to which corresponds a plane drawn through the line of direction, and cutting the body in the fixed position in which it has been set. And indeed, if the body be spherical and formed of parts of the same nature, or at least of heterogeneous parts, but similarly arranged about the center of the sphere so that the innermost little sphere is wrapped in certain envelopes; since the center of mass and the center of gravity are the same, any point whatever taken on the spherical surface will be suitable, for all the points have a like position. But if it be either a segment of a sphere, or a sphere made of heterogeneous parts unequally arranged, let it be placed upon a horizontal plane, accurately smooth and as level as possible; and the point that is touched by the supporting plane, when all motion has ceased, is the one chiefly sought, and the upper point which lies opposite to this will likewise be suitable for the intended purpose. But if the body presented be cylindrical, or any prism whatever produced by a continuous and similar drawing out; let it be cut lengthwise into two parts, and the point corresponding to the center of gravity of the cylinder will be obtained: but in a prism, if each parallelogrammatic face be divided into equal parts both of length and of breadth, the plane drawn through the found points will pass through the center of gravity of the prism; for it divides into equal parts, and similarly placed, whence arises the equality of the moments of gravity. D 2

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Mechanicorum Ut si parallelepiedi B C plana ita dividantur, ut habeant puncta me- dia I, & O, & per ea agatur pla- num, constat æqualia esse momenta gravitatis partium I B, & I C, cùm nullo ex capite possit oriri momento- rum inæqualitas. At si non facies parallelogrammæ prismatis dividendæ sint, sed potius basis, quæ sæpè varia est, & irre- gularis, tunc inveniendum est in ea punctum, in quo sibi oc- currunt sectiones planorum secantium datum corpus in mo- menta æqualia, illudque respondet centro gravitatis intra so- liditatem existenti. Sit autem primò basis prismatis trigona A H I; dividatur unum ex lateribus ex. gr. H I bifariam in G, planum enim transiens per A & G, atque bifariam secans pa- rallelogrammum H V transibit per centrum gravitatis prismatis trigo- ni. Nam si datum prisma secetur pluribus planis parallelis plano H V facientibus sectiones M L, B O, N S, C E, & ex harum sectionum extremis exeant alia plana secantia parallela plano A G; abscinduntur ex prismate dato pa- rallelepipeda L F, O K &c. quæ à plano A G dividuntur in partes G L, G M æquales ac similiter positas; item D O, D B, &c. Igitur singula in eodem plano A G habent gravitatis centrum, ac proinde tota moles ex iis parallelepipedis composita in eo- dem plano habet centrum gravitatis. Quoniam verò, si adhuc plana secantia frequentiora sint, plura fiunt parallelepipeda, quorum omnium moles composita adhuc minus differt à mole totius prismatis dati, ita ut toties multiplicari possit bisectio, ut demum relinquatur differentia minor quacunque minimâ mole excogitabili; hinc fit molem compositam ex parallelepi- pedis illis infinitis (sic loqui liceat, quia non est certus eorum numerus explicabilis) habere centrum gravitatis in plano A G; ac

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Mechanics If the bases of the parallelepiped B C are divided into planes in such a way that they have the middle points I and O, and a plane is drawn through them, it is clear that the moments of gravity of the parts I B and I C are equal, since no inequality of moments can arise from any source. But if it is not the faces of the prism to be divided that are parallelograms, but rather the base, which is often variable and irregular, then one must find in it the point at which the sections of the planes cutting the given body meet in equal moments; and this corresponds to the center of gravity existing within the solid. Let the base of the prism be triangular A H I; let one of the sides, e.g. H I, be divided in half at G, for the plane passing through A and G, and cutting the parallelogram H V in halves, will pass through the center of gravity of the triangular prism. For if the given prism is cut by several planes parallel to the plane H V, making the sections M L, B O, N S, C E, and from the extremities of these sections other planes parallel to the plane A G are drawn cutting it, there are cut from the given prism parallelepipeds L F, O K, etc., which are divided by the plane A G into equal and similarly situated parts G L, G M; likewise D O, D B, etc. Therefore each has its center of gravity in the same plane A G, and consequently the whole mass composed of those parallelepipeds has its center of gravity in the same plane. Now since, if the cutting planes are made more frequent, more parallelepipeds are formed, the combined mass of all of them differs less and less from the mass of the whole given prism, so that the division can be repeated so often that at last there remains a difference smaller than any smallest conceivable mass; hence it follows that the mass composed of those infinite parallelepipeds (if I may so speak, since there is no definite number of them that can be stated) has its center of gravity in the plane A G; and

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Liber primus. CAPUT V. 29 ac proinde etiam prisma trigonum ex iis conflatum parallelepi- pedis habere in eodem plano A G centrum suæ gravitatis, quandoquidem non differt ab illis nisi differentiâ minore qua- cumque minimâ excogitabili. Sunt igitur partium A G H, A G I momenta æqualia; quia si inæqualia essent haberent differentiam, qua posset dari minor (neque enim esset indivi- dua) hæc autem differentia si esset, alia non esset, quàm quæ intercedit inter prisma datum, & omnia parallelepipeda, cu- jus differentiæ inæquales partes essent in A G H, & A G I: igitur differentia partium A G H, A G I esset minor differ- entiâ prismatis, & omnium parallelepipedorum; nam esse non potest major, vel illi æqualis: sed jam ex hypothesi differentia inter molem compositam ex omnibus parallelepipedis, & pris- ma, est minor quacumque minimâ datâ, ergo si essent inæ- qualia momenta partium A G H, A G I haberent differen- tiam minorem, & non minorem eâdem differentiâ inter pris- ma & omnia parallelepipeda. Non sunt igitur inæqualia. Res autem fortassè sic breviùs explicabitur; si partes A G H, A G I non sunt æquales, sit A G H minor quàm A G I, differentiâ Y. Tot autem fiant bisectiones, ut parallelepipeda relinquant differentiam minorem quàm Y. Quia ergo parallelepipeda in A G I habent differentiam minorem quàm Y, à parte pris- matis A G I, illa sunt majora quàm pars prismatis A G H, quæ deficit à parte A G I differentiâ Y. Atqui parallelepepida in A G H sunt æqualia parallelepipedis in A G I, ergo etiam parallelepipeda in A G H majora sunt, quàm tota pars A G H, quod est manifestè falsum. Non est igitur altera pars major, altera minor. Porrò ex continua bisectione laterum A C, & C N &c. relinqui semper minorem differentiam, hoc est se- missem præcedentis differentiæ, constat, quia si A C secetur in P, & ducantur plana parallela planis A G, & H V, dividi- tur C T bisariam in Q, & est T P parallelepipedum ablatum duplum prismatis trigoni C P Q, cui æquale est prisma A P X; adeóque duobus hisce prismatis æquale est ablatum parallele- pipedum T P, quod est semissis differentiæ A T C, quæ priùs relinquebatur: & eadem est de cæteris ratio. Quare si ex datâ quantitate auferatur semissis, & iterum semissis residui, & sic in infinitum, necesse est aliquando eò devenire, ut residua D 3

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Book One. CHAPTER V. 29 and therefore also the triangular prism formed from these, and the parallelepiped, have in the same plane A G the center of their gravity, since it differs from them only by any difference however small, even the least conceivable. Therefore the moments of the parts A G H, A G I are equal; for if they were unequal, they would have a difference, by which a smaller could be assigned (for it would not be indivisible). But if there were this difference, there would be no other than that which intervenes between the given prism and all the parallelepipeds, whose unequal parts would be in A G H and A G I: thus the difference of the parts A G H, A G I would be smaller than the difference of the prism and of all the parallelepipeds; for it cannot be greater, or equal to it. But now by hypothesis the difference between the mass composed of all the parallelepipeds and the prism is smaller than any given least quantity; therefore if the moments of the parts A G H, A G I were unequal, they would have a difference smaller, and not smaller than the same difference between the prism and all the parallelepipeds. Therefore they are not unequal. The matter, however, may perhaps be explained more briefly thus: if the parts A G H, A G I are not equal, let A G H be smaller than A G I by difference Y. Then let so many bisections be made that the parallelepipeds leave a difference smaller than Y. Since therefore the parallelepipeds in A G I have a difference smaller than Y, from the part of the prism A G I they are greater than the part of the prism A G H, which falls short of the part A G I by difference Y. Yet the parallelepipeds in A G H are equal to the parallelepipeds in A G I, therefore also the parallelepipeds in A G H are greater than the whole part A G H, which is manifestly false. Therefore one part is not greater and the other smaller. Moreover, from the continuous bisection of the sides A C, and C N, etc., there is always left a smaller difference, that is, a half of the preceding difference, as is clear, because if A C is cut at P, and planes parallel to the planes A G and H V are drawn, C T is bisected at Q, and T P is the removed parallelepiped double the triangular prism C P Q, to which the prism A P X is equal; and so the removed parallelepiped T P is equal to these two prisms, which is the half of the difference A T C, which had previously been left: and the same reasoning holds for the rest. Wherefore if from a given quantity a half is taken away, and again a half of what remains, and so on to infinity, it is necessary that eventually one comes to the point where the remnants D 3

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Mechanicorum quantitas minor sit quacunque datâ quantitate, ut colligitur ex prop. 1. lib. 10. Eucl. Ideo fieri non potest, ut prismatico di- viso à plano A G, altera pars excedat momenta alterius quan- titate Y, quia tot possunt abscindi parallelepipeda, ut relin- quatur differentia illorum à prismatic minor, quàm sit Y: pla- num autem A G æqualiter dividit momenta parallelepipedo- rum, igitur cum tota residua differentia minor sit quam Y, esse omnino non potest, ut altera pars habeat excessum quan- titati Y respondentem: si enim quantitates illæ differrent, pos- set dari quantitas minor illarum differentiâ; sed non potest hu- jusmodi minor quantitas dari, nam quælibet data est major, igitur non differunt, sed sunt æquales. His ita constitutis facilè definitur punctum centro gravitatis imminens in basi prismatis: quia enim ostensum est planum ab angulo per medium latus oppositum ductum transire per centrum gravitatis, & dividere in momenta æqualia totum prisma, centrum gravitatis erit non solùm in plano A G, sed etiam in plano I N propter eandem rationem. Punctum igi- tur D, in quo occurrunt sibi communes sectiones planorum secantium, & basis, est punctum, quod quæritur, imminens centro gravitatis. Punctum D autem secare rectam N I ita, ut N D ad D I sit ut 1 ad 2, sic ostenditur. Ducatur recta N G, quæ per 2. lib. 6. est paral- lela ipsi A I; ergo ut H G ad H I, ita N G ad A I per 4. lib. 6. ergo N G ad A I est ut 1 ad 2: ergo triangula N G A, A G I sunt ut 1 ad 2, per 1. lib. 6. Cum autem ut N D ad D I, ita N D A ad D I A, & N D G ad D I G per 1. 6. erit etiam, ex 12. lib. 5. ut N D ad D I, ita N G A ad A G I, hoc est 1 ad 2. Eadem ratione ostenditur G D ad D A esse, ut 1 ad 2. Vel etiam breviùs: Quia enim N G, A I sunt pa- rallelæ, triangula N D G, A D I sunt similia propter angulo- rum æqualitatem; ergo ut N G ad A I, hoc est ut 1 ad 2, ita G D ad D A, & N D ad D I. Quare satis erit latus unum trianguli bifariam secare, & ab opposito angulo rectam duce- re; cujus tertia pars versus basum divisam dabit centrum gravi- tatis trianguli. Iam verò si basis prismatis quadrangula fuerit parallelogram- ma,

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Mechanics quantity is less than any given quantity, as is gathered from prop. 1, lib. 10 of Euclid. Therefore it cannot happen that, the prism being divided by the plane A G, one part exceeds the moments of the other by the quantity Y; because so many parallelepipeds can be cut off that there remains a difference of them from the prism smaller than Y. But the plane A G divides the moments of the parallelepipeds equally; therefore, since the whole remaining difference is less than Y, it is altogether impossible for either part to have an excess corresponding to the quantity Y. For if those quantities differed, a quantity smaller than their difference could be given; but no such smaller quantity can be given, for every given one is greater; therefore they do not differ, but are equal. These things being established, the point lying under the center of gravity on the base of the prism is easily determined: for since it has been shown that the plane drawn from the angle through the midpoint of the opposite side passes through the center of gravity, and divides the whole prism into equal moments, the center of gravity will be not only in the plane A G, but also in the plane I N for the same reason. Therefore the point D, where the common sections of the cutting planes and the base meet, is the point sought, lying under the center of gravity. But the point D cuts the straight line N I so that N D is to D I as 1 to 2, as follows. Let the straight line N G be drawn, which by 2, lib. 6 is parallel to A I itself; therefore as H G is to H I, so N G is to A I by 4, lib. 6. Therefore N G is to A I as 1 to 2; hence the triangles N G A and A G I are as 1 to 2, by 1, lib. 6. But since as N D is to D I, so N D A is to D I A, and N D G to D I G by 1, 6, it will also be, from 12, lib. 5, as N D is to D I, so N G A is to A G I, that is, 1 to 2. By the same reason it is shown that G D is to D A as 1 to 2. Or even more briefly: since N G and A I are parallel, the triangles N D G and A D I are similar because of the equality of their angles; therefore as N G is to A I, that is as 1 to 2, so G D is to D A, and N D to D I. Wherefore it will suffice to cut one side of the triangle in half, and from the opposite angle to draw a straight line; the third part of this, toward the divided base, will give the center of gravity of the triangle. Now if the base of the prism is a quadrilateral parallelogram,

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Liber primus. CAPUT V. 3I ma, ductis diametris apparebit quæsitum punctum, per quod transeunt omnia plana dividendia æqualiter corporis dati mo- menta, cum sint partes utrinque æquales, & similiter positæ. Et ob eandem rationem si basis prismatis fuerit aliqua ex figu- ris ordinatis, seu æquilateris; centrum figuræ est punctum im- minens centro gravitatis; planum si quidem per illud transiens, & per unum angulorum, dividit totum prisma in partes æquales simi- literque positas; atque adeò momenta hinc, & hinc sunt æqualia. At si basis trapezia fuerit, duc utramque diametrum EC, & BD: tum in basis trigonâ BCD prismatis partialis inveniatur punctum centro gravitatis respondens (punctum hoc deinceps, brevitatis gratiâ, dice- tur centrum gravitatis, quamvis per abusionem) & sit H; & in opposita basis trigona reliqui prismatis BDE pariter invenia- tur punctum F; & per utrumque punctum transeat planum FH; nam in hoc eodem plano est centrum gravitatis totius prismatis trapezij, quod dividitur in momenta æqualia: hoc si- quidem planum transiens per H gravitatis momenta æqualia habet hinc, & hinc in prismatic trigono BDC; similiter cum transeat per F, habet hinc, & hinc momenta æqualia gravitatis prismatis trigoni BED: si igitur æqualia æqualibus jungantur, planum idem æqualiter partitur momenta gravitatis prismatis trapezij EDCB, & in eo est centrum gravitatis illius. Eadem ratione in basis trigona EBC inveniatur punctum G, & in basis EDC punctum S, per quæ si agatur planum GS, in eo pariter erit centrum gravitatis totius prismatis trapezij. Est igitur centrum gravitatis in communi sectione planorum FH, & GS; ac proinde punctum I illud est, quod quæritur. Aliter etiam, & facillimè in basis trapezia ABCD invenitur centrum gravitatis: ductis enim diametris AC, BD, altera diameter ex. gr. AC bifariam secetur in E, ducanturque rectæ DE, BE; trianguli ADC centrum gravi- tatis est in recta DE, & quidem in F, ita ut EF sit tertia pars totius ED, ut constat ex paulò ante demonstratis. Ducatur igitur FG pa-

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ma, by drawing the diameters, the sought point will appear, through which all planes dividing equally the moments of the given body pass, since the parts on either side are equal and similarly placed. And for the same reason, if the base of the prism is any of the regular, or equilateral, figures, the center of the figure is the point corresponding to the center of gravity; for a plane passing through it and through one of the angles divides the whole prism into equal parts similarly placed; and thus the moments on this side and on that are equal. But if the base is a trapezium, draw both diameters EC and BD; then in the triangular base BCD of the partial prism let there be found the point corresponding to the center of gravity (this point hereafter, for the sake of brevity, will be called the center of gravity, though improperly), and let it be H; and in the opposite triangular base BDE of the remaining prism let the point F likewise be found; and let the plane FH pass through both points; for in this same plane is the center of gravity of the whole trapezoidal prism, which is divided into equal moments: indeed this plane, passing through H, has equal moments of gravity on this side and on that in the triangular prism BDC; likewise, when it passes through F, it has on this side and on that equal moments of gravity of the triangular prism BED. Therefore, if equal things be joined with equal things, the same plane equally divides the moments of gravity of the trapezoidal prism EDCB, and in it is the center of gravity thereof. By the same reasoning, in the triangular base EBC let the point G be found, and in the base EDC the point S, through which if the plane GS be drawn, the center of gravity of the whole trapezoidal prism will likewise be in it. Therefore the center of gravity is in the common intersection of the planes FH and GS; and consequently that point I is the one sought. Otherwise also, and most easily, in the trapezoidal base ABCD the center of gravity is found: for, the diameters AC and BD having been drawn, let one diameter, e.g. AC, be bisected at E, and let straight lines DE and BE be drawn; the center of gravity of triangle ADC is on the line DE, and indeed at F, so that EF is one third part of the whole ED, as is clear from what was demonstrated a little before. Let FG therefore be drawn pa-

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Mechanicorum 32 rallela alteri diametro BD, & erit similiter G centrum gravitatis trianguli ABC, quia per 2. lib. 6. ut EF ad FD, ita EG ad GB; Quia ergo diameter AC secatur in H, sumatur FO æqualis ipsi GH, & est O centrum gravitatis trapezij, est enim triangulum ABC ad triangulum ADC, ut FO ad OG, hoc est ut HG ad HF. Est autem HG ad HF ut BI ad ID propter parallelismum linearum GF, BD. Porrò constat triangulum ABC ad triangulum ADC esse ut BI ad ID, nam triangula ABI, ADI sunt ut bases BI, DI, item BCI, DCI sunt ut eædem bases BI, DI per 1. lib. 6; igitur, & totum triangulum ABC ad totum ADC est ut BI ad DI: igitur, & triangulum ABC ad triangulum ADC est ut FO ad OG. Hinc facilis patet via ad investigandum idem punctum in basi prismatis pentagoni BDEAC. Primùm enim ducto plano per BE, inveniatur in basi trigonâ BD E punctum R, & in basi BEAC quadrangulâ punctum P; & ducto plano per RP, in eo erit centrum gravitatis prismatis pentagoni, cum in eodem sint centra gravitatis partium. Deinde ducto per D & A plano, inveniatur in basi trigona DEA punctum L centrum gravitatis, & in basi quadrangulâ ACBD punctum M centrum gravitatis: in plano pariter ducto per ML est centrum gravitatis totius prismatis pentagoni, quod proinde est in communi planorum per PR, & LM ductorum sectione; atque adeò punctum, quod quæritur, est O. Eadem est methodus in prismate hexagono; ducto enim plano dividente in duo prismata, quorum alterum est trigonum, alterum pentagonum, inveniatur utriusque centrum gravitatis, & per inventa puncta agatur planum. Deinde iterum alio plano secetur in duo prismata, quorum alterum pariter sit trigonum, alterum pentagonum, & per inventa singularia gravitatum centra agatur planum: duo siquidem plana ducta per centra gravitatis partium, transeunt pariter per centrum gravitatis totius, quod est in communi eorum sectione. Eademque de reliquis prismatis est ratio. Sed

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Mechanics 32 parallel to the other diameter BD, and G will likewise be the center of gravity of triangle ABC, because by 2. lib. 6, as EF is to FD, so is EG to GB; since therefore diameter AC is cut in H, let FO be taken equal to GH itself, and O is the center of gravity of the trapezium; for triangle ABC is to triangle ADC as FO to OG, that is, as HG to HF. But HG is to HF as BI is to ID, because of the parallelism of the lines GF, BD. Moreover, it is clear that triangle ABC is to triangle ADC as BI is to ID, for triangles ABI, ADI are as the bases BI, DI, likewise BCI, DCI are as the same bases BI, DI by 1. lib. 6; therefore the whole triangle ABC to the whole ADC is as BI to DI: therefore triangle ABC to triangle ADC is also as FO to OG. Hence a ready way is clear for finding the same point in the base of the pentagonal prism BDEAC. First, indeed, by drawing a plane through BE, let there be found in the triangular base BDE the point R, and in the quadrangular base BEAC the point P; and by drawing a plane through RP, in it will be the center of gravity of the pentagonal prism, since in the same plane are the centers of gravity of the parts. Then, by drawing a plane through D and A, let there be found in the triangular base DEA the point L, the center of gravity, and in the quadrangular base ACBD the point M, the center of gravity: likewise, in the plane drawn through ML is the center of gravity of the whole pentagonal prism, which therefore is in the common intersection of the planes drawn through PR and LM; and thus the point sought is O. The same method applies to the hexagonal prism; for by drawing a plane dividing it into two prisms, of which one is triangular and the other pentagonal, let the center of gravity of each be found, and through the points thus found let a plane be drawn. Then again let it be cut by another plane into two prisms, of which one likewise is triangular and the other pentagonal, and through the separate centers of gravity thus found let a plane be drawn: for the two planes drawn through the centers of gravity of the parts likewise pass through the center of gravity of the whole, which lies in their common intersection. And the same reasoning applies to the remaining prisms. But

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Liber primus. CAPUT V. 33 Sed hæc indicasse sufficiat, quæ operi Mechanico satis esse possunt in omnibus ferè prismatis: Si enim basis non fuerit planè rectilinea, inscripto polygono rectilineo, quod mini- mùm differat à plano basis, quæres ejus centrum gravitatis, methodo jam traditâ; illoque usurpato tanquam vero dati pris- matis centro quæsito, minimum aberrabis; aliquando tamen aberrabis, aliquando continget, ut inventum cum quæsito conveniat. Quod si accuratiori investigatione opus fuerit: quemadmodum in cæteris corporibus, quæ continuum ductum non habent, sed inæquali crassitudine crescunt, aut decres- cunt, ut in obeliscis, aut pyramidibus truncatis, reliquisquè planè inordinatis molibus; tunc ad geometricam Centrobary- ces methodum confugiendum est; quam hic ego non perle- quor. Praxes igitur aliquæ proponendæ sunt, quibus centrum gravitatis physicè perspectum habere possimus in corporibus, quorum frequentior, vulgarisque usus esse potest. Prima praxis sit ad inveniendum gra- vitatis centrum in cingulis, quæ laminis quoque communis esse potest. Sit datum cingulum A H, quod primùm suspenda- tur ex H, & inde pendens perpendicu- lum secet oppositum latus I A in C; note- tur igitur punctum C. Deinde iterum suspendatur ex R, & perpendiculum ca- dat in punctum F, quod notetur. His cognitis ducatur filum ex R in F, ibique intentum alligetur; aliud filum similiter ex H in C ducatur, & secans in S filum R F, dabit punctum S quæsitum centrum gravitatis: ex quo si suspenderetur datum cingulum, maneret horizonti parallelum. Quod si esset corpus talis figuræ, ut spatium non clauderet, sed haberet angulum cavum, aut esset frustum annulare, eadem est methodus factâ suspensione illius ex duobus punctis, ex quibus perpendiculum cadere possit intrà corporis superficiem; in qua si notentur puncta, per quæ transit, & ducantur fila, ut priùs, eorum com- munis sectio dabit quæsitum centrum gravitatis. Hinc si vel la- mina esset perforanda, ut axi infigeretur, vel cingulum esset axi imponendum, in utrâque superficie oppositâ quærere opor- teret punctum S, ut axis per centrum gravitatis transiret, eique E

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Book One. CHAPTER V. 33 But it will be enough to have indicated these things, which for the mechanical work may suffice in almost all prisms: for if the base is not perfectly rectilinear, by inscribing a rectilinear polygon that differs as little as possible from the plane of the base, you will seek its center of gravity by the method already given; and, taking that as though it were the true center sought of the prism, you will err as little as possible; yet sometimes you will err, and sometimes it will happen that the found center agrees with the one sought. But if a more exact investigation is needed, as in other bodies that do not have a continuous outline, but increase or decrease with unequal thickness, as in obelisks, or truncated pyramids, and the rest of the completely irregular masses; then one must have recourse to the geometric method of Centrobaryces, which I do not here undertake to explain. Therefore some practical methods must be proposed, by which we may have the center of gravity, physically considered, in bodies whose more frequent and ordinary use may be this. The first practical method is for finding the center of gravity in hoops, which may also be common to plates. Let the given hoop A H be first suspended from H, and, hanging there, let a plumb line cut the opposite side I A in C; therefore mark the point C. Then let it be suspended again from R, and let the plumb line fall on the point F, which is to be marked. These being known, let a thread be drawn from R to F, and there tightly fastened; let another thread in the same way be drawn from H to C, and, cutting the thread R F at S, it will give the sought point S, the center of gravity: from which, if the given hoop were suspended, it would remain parallel to the horizon. But if the body were of such a figure that it did not enclose a space, but had a hollow angle, or were a ring-shaped segment, the same method applies, by suspending it from two points from which the plumb line can fall within the surface of the body; and if points be marked there through which it passes, and threads be drawn as before, their common intersection will give the sought center of gravity. Hence, if a plate were to be perforated so that it might be fixed to an axis, or if a hoop were to be placed on an axis, in either opposite surface one ought to seek the point S, so that the axis may pass through the center of gravity, and to it

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Mechanicorum 34 uterque polus responderet: in cingulis autem præterea habenda esset ratio transversariorum, per quæ axis infigendus esset, ea enim possunt centrum gravitatis compositæ in alio puncto constituere. Secunda praxis laminis potissimùm accommodata, in quibus punctum medium satis accuratè inquiritur, ut si lamina metallica esset in calicem excavanda, hæc esse potest. Impone laminam acutæ cuspidi cultri, aut styli, eamque ultrò citróque tantisper move, dum consistat citrà periculum cadendi punctum enim, quod à cultri aut styli cuspide notatur, centrum est quæsitum. Tertia praxis sit iis corporibus conveniens, quæ præstant longitudine, qualia sunt pseudocylindrica, conica, pyramides &c. quæ si non prædita sint multâ gravitate, imponantur funiculo brevi horizontaliter extenso, at si graviora fuerint, vel cylindrulo vel aciei prismatis trigoni imponantur, & usque dum in æquilibrio consistant, promoveantur: ubi enim quieverit corpus impositum, ex loco contactûs innotescet vel punctum, si in puncto se contingant, vel linea, si in linea, per quam si ducatur planum à centro terræ, distinguetur impositum corpus in momenta gravitatis æqualia. Inventâ autem hujusmodi lineâ facilè prodet se quæsitum punctum. Quarta praxis non multùm distat à superiore: si nimirum oblatum corpus imposueris plano alicui horizontali, quod tamen à pavimento absit mediocri aliquo intervallo, habeat autem extremum marginem exactè rectum: extra suppositi plani marginem illud paulatim promove, donec eò venerit, ut si vel minimum ulteriùs promoveretur, sponte cæderet; ibique secundùm rectitudinem marginis plani duc stylo lineam in corpore imposito. Deinde superficie eâdem planum tangente, si corpus, præter longitudinem, non modicam præterea habeat latitudinem, convertatur aliquantulum, & simili methodo invenietur linea alia secans priorem in puncto quæsito, quod scilicet respondet centro gravitatis intra corporis soliditatem delitescenti. Hæc sunt quæ Mechanices instituto sufficere possint ad centrum gravitatis inveniendum; in molibus enim majoribus, quæ plerumque vix differunt à prismatis, non indigemus communiter

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Mechanics 34 either pole would answer to it; but in the girdles there must moreover be regard had to the crosspieces, by which the axis was to be fixed in place, for these can set the center of gravity of the composite body at another point. The second method, suited chiefly to plates, in which the middle point is sought with sufficient accuracy, may be this: if a metal plate is to be hollowed into a bowl, place the plate upon the sharp point of a knife or stylus, and move it to and fro for a little while, until it rests without danger of falling; for the point marked by the knife’s or stylus’s tip is the center sought. The third method is suitable for those bodies that excel in length, such as pseudocylindrical forms, cones, pyramids, etc.; if these are not endowed with much weight, let them be placed upon a short thread stretched horizontally, but if they are heavier, let them be placed either upon a small cylinder or upon the edge of a triangular prism, and let them be advanced until they come to rest in equilibrium: for where the body placed there has settled, the place of contact will make known either a point, if they touch at a point, or a line, if at a line; and if through this a plane is drawn from the center of the earth, the placed body will be distinguished into equal moments of gravity. When such a line has been found, the sought point will readily reveal itself. The fourth method does not differ much from the preceding: namely, if you place the offered body upon some horizontal plane, which, however, is at some moderate distance from the floor, and has an exactly straight outer edge, move it gradually beyond the edge of the supporting plane until it has reached the point that, if moved ever so little farther, it would fall of itself; and there, along the straightness of the plane’s edge, draw a line upon the body with a stylus. Then, with that same surface touching the plane, if the body, besides length, also has not a little breadth, let it be turned somewhat, and by a similar method another line will be found, intersecting the first at the sought point, which indeed corresponds to the center of gravity hidden within the body’s solidity. These are the methods that may suffice for the purpose of Mechanics in finding the center of gravity; for in larger masses, which usually differ little from prisms, we ordinarily do not need

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Liber primus. CAPUT VI. 35 niter Geometricâ subtilitate. Illud restat, ut earum, quas at- tuli praxes, ratio, & causæ explicentur, ex quibus clarior ha- beatur notitia eorum, quæ ad centrum gravitatis pertinent. CAPUT VI. Affertur ratio prædictarum praxeon. UT palam fiat praxibus capite superiore allatis inveniri punctum respondens centro gravitatis, quod inquiritur, indicandi sunt fontes, ex quibus illæ deducuntur. Earum ita- que ratio petenda est ex gravium naturâ, quæ extra locum sibi debitum constituta, in medio videlicet leviore, conantur de- orsum pro viribus, nisi impediantur: quod si interpellentur quidem, non tamen prorsus descensu prohibeantur, descen- dunt, prout fert obstantium impedimentorum conditio. Sic lapis sphæricus in montis clivo positus cùm non valeat rectâ; sicut in aëre libero, deorsum ferri, per planum illud inclina- tum descendit: Sic plumbum, quod filo adnectitur laqueari, à perpendiculo remotum descendit circulariter. Porrò quæ de toto ipso corpore vera esse intelligimus, ejus quoque partibus singulis conveniunt; cùm enim singulæ suam habeant gravita- tem, nisi quid obstet, descendunt. Jam verò si contingat ita corpus grave opposito extrinsecùs obice impediri, ut cunctæ simul partes, quasi moles unà descendere nequeant; sublato partium nexu descendunt, quæcunque carent impedimento: ut si ceream candelam, aut glaciem, quam manu sustines, igni admoveas; haud dubium, quin partes extremæ igni proximæ liquescentes, solutâ unione cum cæteris, suis nutibus deorsum latæ liberè descendant. At si partes omnes colligatæ invicem permaneant, eandemque figuram servent; corpore illo suspen- so aut sustentato, fieri non potest, ut partes aliquæ descendant, quin aliæ, quæ è regione sunt trans suspensionis, aut sustenta- tionis punctum, ascendant; id autem harum gravitati re- pugnat: non igitur ascendere possunt, nisi descendentes op- positæ viribus ac momentis præstent ita, ut harum gravitati E 2

Transcription: Translated (English)

Book One. CHAPTER VI. 35 by geometric subtlety. What remains is that the rationale and causes of the practices I have set forth be explained, from which a clearer knowledge may be had of those things that pertain to the center of gravity. CHAPTER VI. The rationale of the aforesaid practices is given. THAT it may be made plain that by the practices set forth in the foregoing chapter the point corresponding to the center of gravity, which is being sought, is found, the sources from which they are drawn must be indicated. Their rationale, therefore, must be sought from the nature of heavy bodies, which, when placed outside the place due to them, namely in a lighter medium, strive downward with all their might, unless prevented: and if they are indeed obstructed, yet not altogether prevented from descending, they descend according to the condition of the resisting obstacles. Thus a spherical stone placed on a mountain slope, since it cannot move straight downward, as in free air, descends along that inclined plane. Thus lead, which is attached by a thread to the ceiling, when moved away from the perpendicular, descends in a circle. Moreover, what we understand to be true of the whole body itself is also fitting for each of its parts; for since each has its own gravity, they descend unless something stands in the way. But if it should happen that a heavy body is so hindered by some opposing external obstacle that all its parts together cannot descend as one mass, then, when the bond between the parts is removed, those parts that lack impediment descend: as if you should bring to the fire a wax candle, or ice, which you are holding in your hand; there is no doubt that the outer parts nearest the fire, melting, are released from their union with the rest, and, carried downward by their own impulse, descend freely. But if all the parts remain bound together and preserve the same shape, then, with that body suspended or supported, it cannot happen that some parts descend unless others, which are opposite across the point of suspension or support, ascend; and that is contrary to the gravity of these parts: therefore they cannot ascend unless, by descending, the opposite parts surpass them in forces and moments in such a way that to their gravity E 2

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Mechanicorum 36 vim inferre valeant. Quare si fiat corporis suspensi, aut sustentati consistentia, argumentum est æqualitatis momentorum punctum suspensionis, aut sustentationis hinc, & hinc usque- quaque circumstantium; si qua enim esset inæqualitas, alterutra pars præponderaret, & ad motum incitaretur. Sit corpus grave AB, cujus centrum gravitatis H, linea directionis HT in centrum universi producta. Si suspendatur ex puncto C, quod est in eadem lineâ directionis, necessariò consistit corpus horizonti parallelum, quia rectâ descendere non potest per HT, cum in C retineatur; neque alterutra pars potest descendere, quia momenta partis HB, quibus deorsum nititur, æqualia sunt momentis, quibus pars HA resistit, ne elevetur; & vicissim viribus gravitatis AH cæteroqui descensuræ reluctatur gravitas HB pari nisu repugnans, ne attollatur; totum ergo consistit. At si ex M puncto suspendatur, non potest quidem per MT perpendicularem descendere versùs terræ centrum, sed neque consistet horizonti parallelum; quia si planum intelligatur ex terræ centro per rectam MT ductum, non dividitur corpus in momenta æqualia, cum non transeat per H centrum gravitatis; igitur cum majora sint momenta partis MB, quàm partis MA, illa præponderabit, atque descendens circa punctum M permanens convertetur, donec centrum gravitatis H sit in perpendiculari MT, cui congruat rectæ MO: tunc autem demum consistet, quia planum transiens per MHO æqualiter dispertit momenta gravitatis; neutrâ autem parte præponderante, uttraque quiescit. Idem dicendum, si corpus ex I puncto suspenderetur; tunc enim solùm fieret consistentia, ubi in eadem directionis lineâ esset punctum I atque H centrum gravitatis. Quod si duplici funiculo suspendatur pondus, & illi paralleli non sint, quia neque horizonti perpendiculares, illi si producantur, concurrent in punctum aliquod lineæ directionis, sivè supra pondus, sivè infra, pro ratione angulorum, quos constituunt. Si

Transcription: Translated (English)

Mechanics 36 can impart force. Therefore, if there is equilibrium of a suspended or supported body, it is a proof of the equality of the moments on this side and on that side of the point of suspension or support all around it; for if there were any inequality, one side would outweigh the other, and would be moved into motion. Let there be the heavy body AB, whose center of gravity is H, the line of direction HT produced to the center of the universe. If it be suspended from point C, which is on the same line of direction, the body must necessarily remain parallel to the horizon, because it cannot descend in a straight line through HT, since it is held at C; nor can either side descend, because the moments of part HB, by which it tends downward, are equal to the moments by which part HA resists, lest it rise; and, conversely, the weight of AH otherwise about to descend is opposed by the weight of HB resisting with equal effort, lest it be raised; thus the whole remains at rest. But if it be suspended from point M, it certainly cannot descend through perpendicular MT toward the center of the earth, but neither will it remain parallel to the horizon; because if a plane be imagined drawn from the center of the earth through the straight line MT, the body is not divided into equal moments, since it does not pass through H, the center of gravity; therefore, since the moments of part MB are greater than those of part MA, that part will outweigh the other, and descending while remaining about point M will turn until the center of gravity H is in the perpendicular MT, to which the straight line MO corresponds: then at last it will come to rest, because the plane passing through MHO distributes the moments of gravity equally; and with neither part outweighing the other, each remains at rest. The same must be said if the body were suspended from point I; for then equilibrium would occur only where point I and center of gravity H were on the same line of direction. But if a weight is suspended by a double cord, and the cords are not parallel, because they are not perpendicular to the horizon, if they are extended they will meet at some point on the line of direction, either above the weight or below it, according to the ratio of the angles which they form. Si

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Liber primus. CAPUT VI. 37 Sit enim corpus A B, cujus cen- trum gravitatis O, linea directionis I O C, si ex I suspendatur per O, in eo situ manebit; ergo etiam, si funiculi sint I H, I L, manebit: ergo etiam, si sint P H, S L, funiculorum enim longitudo nihil facit; Idem etiam dicendum cum funiculi sunt D H, F L; pro- ducti enim concurrunt cum linea directionis in C, semper scilicet perinde se habet atque, si ex I suspenderetur. Quæ verò de suspensione dicta sunt, ea, analogiâ servatâ, de sustentatione quoque dicta intelligantur; tunc solùm videlicet corpus consistere, cùm ex centro gravitatis ducta directionis linea transit per punctum sustentationis, quia tunc solùm æqualia hinc, & hinc sunt momenta virtutis ad descendendum, atque resistentiæ ad ascendendum: ut quando corpus aliquod imponitur cono, vel prisma sphæræ, vel segmentum sphæricum, plano, vel cylindrus aciei prismatis trigoni in transversum; cadet enim in alterutram partem impositum corpus, nisi in eadem linea fuerint centrum terræ, punctum contactûs, & centrum gravitatis. Quod si corpus sustentans, atque sustentatum se tangant in lineâ, opus est lineam illam esse in plano per lineam directionis ducto, ut fiat æqualium momentorum consistentia. Quare si impositum corpus consistat, certissimo argumento constabit punctum, seu lineam, contactûs respondere centro gravitatis. Hinc patet ratio secundæ, & tertiæ praxis. In prima praxi quia facies extima, supra quam perpendiculum liberè movetur, est in plano verticali, perpendiculum H C est parallelum lineæ directionis corporis gravis, quæ transit etiam per punctum suspensionis H: planum igitur transiens per punctum suspensionis H, & per perpendiculum H C, transit quoque per centrum gravitatis corporis. Cum verò idem prorsus dicendum sit de plano transeunte per punctum suspensionis R, & perpendiculum R F, illud scilicet transire per centrum gravitatis corporis; apertum est centrum gravitatis esse in E 3

Transcription: Translated (English)

Book first. CHAPTER VI. 37 Let there be a body A B, whose center of gravity is O, and line of direction I O C; if it is suspended from I by O, it will remain in that position; therefore also, if the strings are I H, I L, it will remain: therefore also, if they are P H, S L, for the length of the strings makes no difference; the same must also be said when the strings are D H, F L; for their produced extensions meet the line of direction at C, since it is always exactly as if it were suspended from I. What has been said of suspension should also, by analogy, be understood of support; namely, that the body stands still only when the line of direction drawn from the center of gravity passes through the point of support, because only then are the moments of the force tending downward and of the resistance tending upward equal on this side and on that: as when a body is placed upon a cone, or upon the prism of a sphere, or a spherical segment, upon a plane, or a cylinder lying across the edge of a triangular prism; for the body placed on it will fall to one side or the other, unless the center of the earth, the point of contact, and the center of gravity are in the same line. But if the supporting body and the body supported touch each other along a line, that line must lie in the plane drawn through the line of direction, so that there may be a stable equilibrium of equal moments. Therefore, if the body placed on it remains at rest, it will be certain proof that the point, or line, of contact corresponds to the center of gravity. Hence the reason for the second and third experiments is clear. In the first experiment, because the outer face, over which the plumb line moves freely, is in a vertical plane, the plumb line H C is parallel to the line of direction of the heavy body, which also passes through the point of suspension H: therefore the plane passing through the point of suspension H and through the plumb line H C also passes through the center of gravity of the body. And since the same must be said of the plane passing through the point of suspension R and the plumb line R F, namely, that it passes through the center of gravity of the body, it is evident that the center of gravity is in E 3

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Mechanicorum 38 communi illorum planorum sectione, eique respondere punctum S inventum. Quia demum, si corpus quod sustinet, & id, quod sustine- tur, in superficie se tangant, corpus impositum in alterutram partem cadere non potest (nisi fortè suppositum planum fuerit inclinatum) quin planum per lineam directionis ductum ita sit extra superficiem, in qua sit contactus, ut neque illam con- tingat; constat ratio quartæ praxis. Si namque planum ex ter- ræ centro ductum per C cen- trum gravitatis dati corporis O S, secet subjectum planum, pars corporis extra marginem F E in aëre extans minora ha- bet momenta gravitatis, quàm reliqua pars; hæc igitur gra- vior non potest ab illa elevari: ubi verò promotum corpus eò venerit, ut planum per cen- trum gravitatis C ductum tangat extremum marginem sub- jecti plani ita, ut in eodem plano, in quo est centrum gravi- tatis C, sit etiam F E, æqualia sunt gravitatis momenta par- tis C S in aëre extantis, ac C O partis plano incumbentis; & si vel minimum ulteriùs promoveretur, pars extra planum sub- jectum extans gravior esset, adeóque descenderet. Quare si in corporis O S superficie infimâ lineam descripseris secundùm marginem F E, ea erit in plano transeunte per centrum gravi- tatis. Quia verò idem contingit, si iisdem superficiebus se con- tingentibus alium situm corpori dederis, pariterque eò usque promoveris, ut citrà cadendi periculum promoveri ulteriùs non possit; alia linea secundùm marginem F E ducta erit pari- ter in plano per gravitatis centrum transeunte, secabitque priorem lineam, punctum mutuæ linearum sectionis illud esse, quod quæritur, satis liquet. Hæc est dispar philosophandi ra- tio, si pars C O adeò longa esset, ut etiam extaret extra an- gustias subjecti plani; semper enim consistit impositum corpus, quandiù planum per lineam directionis transiens, aut tangit, aut secat subjectum planum. Quando cunque enim linea di- rectionis non transit per punctum, vel lineam, vel superficiem, in

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Mechanics 38 by the common section of those planes, and the point S corresponding to it is found. For, finally, if the body that supports and that which is supported touch each other on a surface, the body placed upon it cannot fall to either side (unless perhaps the supporting plane be inclined), unless the plane drawn through the line of direction is so outside the surface in which the contact is made that it does not even touch it; thus the reason of the fourth practice is clear. For if a plane drawn from the center of the earth through the center of gravity C of the given body OS cuts the supporting plane, the part of the body extending in the air beyond the margin FE has less weight than the remaining part; therefore this latter cannot be lifted by the former. But when the moved body has come so far that the plane drawn through the center of gravity C touches the outer edge of the supporting plane in such a way that FE is also in the same plane in which the center of gravity C is, then the moments of weight of the part CS extending in the air and of the part CO resting on the plane are equal; and if it were moved even the least further, the part extending beyond the supporting plane would be heavier, and would therefore descend. Therefore, if on the lowest surface of the body OS you have drawn a line along the edge FE, that line will be in the plane passing through the center of gravity. But since the same thing happens if, with those same surfaces touching each other, you have given the body another position, and likewise have moved it as far as it can be moved without danger of falling, another line drawn along the edge FE will likewise be in the plane passing through the center of gravity, and will intersect the former line; that the point of intersection of the mutual lines is the one sought is sufficiently clear. This is the different way of philosophizing, if the part CO were so long that it would also extend beyond the limits of the supporting plane; for the placed body always remains in balance so long as the plane passing through the line of direction either touches or cuts the supporting plane. For whenever the line of direction does not pass through a point, or a line, or a surface, in

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Liber primus. CAPUT VII. 39 in quibus corpus grave tangitur à sustentante (idem dic de suspensione) semper in alterutram partem grave inclinatur, in eam scilicet, in qua reperitur centrum gravitatis, cùm plura sint ex ea parte momenta gravitatis. CAPUT VII. Quomodo gravia spontè ascendentia descendant. Ex his, quæ proximè dicta sunt, grave sustentatum in eam partem inclinari, in qua est gravitatis centrum, oritur aliquid quando ascensus gravium, qui rerum naturalium ignaros in admirationem adducit non mediocrem, si maximè tunc corpus descendere intelligant, quando illud cernunt altiùs ab horizonte ascendere. Sit enim super planum inclinatum R N rota tantæ latitudinis, ut possit in plano verticali erecta permanere, dum convertitur; habeat autem ad P O adnexam laminam plumbeam crassiorem, adeò ut totius rotæ centrum gravitatis sit S. Iam verò ea sit plani subjecti inclinatio, ut rotâ illud tangente puncto H, linea à terræ centro per H punctum contactûs transiens non transeat per S centrum gravitatis (seu ut veriùs dicam, quia extima superficies rotæ cylindrica tangit planum in lineâ, planum ex centro terræ per lineam contactûs in H ductum non transeat per S) sed illud relinquat versus superiorem plani partem N; planum per rectam HO perpendicularem ductum distinguit rotam in momenta gravitatis inæqualia: non potest igitur rota in H consistere, sed convertitur, ita ut tangat planum in I primùm, deinde in E, demùm in P, ubi consistet, cùm

Transcription: Translated (English)

Book One. CHAPTER VII. 39 in which a heavy body, when touched by a supporter (the same may be said of suspension), always inclines to one side rather than the other, namely to that side in which the center of gravity is found, since there are greater moments of weight on that side. CHAPTER VII. How heavy bodies rising of their own accord descend. From what has just been said, that a supported heavy body inclines toward that part in which the center of gravity is, something arises concerning the ascent of heavy bodies, which brings those ignorant of natural things into no small wonder, especially if they then understand that the body is descending when they see it rising higher above the horizon. Let there be, then, upon the inclined plane R N, a wheel of such breadth that it can remain upright in a vertical plane while it turns; and let it have attached to P O a thicker lead plate, so that the center of gravity of the whole wheel be S. Now let the inclination of the underlying plane be such that, the wheel touching it at point H, the line passing from the center of the earth through the point of contact H does not pass through the center of gravity S (or, more truly speaking, because the outer cylindrical surface of the wheel touches the plane along a line, the plane drawn from the center of the earth through the line of contact at H does not pass through S), but leaves it toward the upper part of the plane, N; the plane drawn through the straight line H O, perpendicular, distinguishes the wheel into unequal moments of gravity: therefore the wheel cannot stand at H, but turns, so that it touches the plane at I first, then at E, and finally at P, where it will come to rest, when

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Mechanicorum cùm linea directionis ex gravitatis centro S ducta in terræ cen- trum transibit per P locum contactûs. In hac autem conver- sione dum rotæ partes inter H & P deinceps aptantur subjecto plano, centrum quidem molis ascendit, sed centrum gravita- tis S descendit. Lineam porrò S P minorem esse lineâ S E, & hanc minorem lineâ S I, & hanc lineâ S H, constat ex prop. 7. lib. 3. Eucl. si nimirum per S, & C centrum agatur diameter. Non est tamen censendum quamlibet ponderis additionem in O P satis esse, ut in quolibet plano inclinato rota ascendat; si enim distantia centri gravitatis à centro rotæ minor fuerit, quàm Sinus inclinationis plani, semper descendet; si eidem Sinui æqualis, non ascendet; si demum eo sinu major, poterit ascendere. Sit planum inclinatum A B, quod in H contingat circulum (hunc sumo cir- culum, qui transeat per centrum tum molis tum gravitatis rotæ) cujus cen- trum C, & ducatur recta C H, quæ cum perpendi- culari H O faciat angu- lum C H O. Quia enim O H producta cadit in ho- rizontem A D perpendicularis, & angulus O H A per 32. lib. 1. æqualis est duobus internis H F A, F A H, est autem A H C ad contingentem factus à semidiametro rectus per 18. lib. 3. sicut & H F A est rectus; reliquus C H O æqualis est angulo H A F inclinationis plani. Certum est igitur, quòd in eam partem ro- ta convertetur, in qua fuerit centrum gravitatis. Quoniam verò C I est Sinus anguli CH I, posito radio CH, est au- tem C I minima omnium, quæ ex C puncto cadant in rectam H O, manifestum est, quòd, si centrum gravitatis fuerit cen- tro rotæ vicinius, ut in R, rota semper descendet, quia cen- trum gravitatis respicit declivitatem plani: at, si fuerit in I, ascendere non potest, quia pars respiciens acclivita- tem plani non præponderat: si demum longiùs à centro distiterit, ut in S, ascendere poterit, usque dum punctum S fuerit

Transcription: Translated (English)

Mechanics when the line of direction drawn from the center of gravity S will pass into the center of the earth through the point of contact P. But in this turning, while the parts of the wheel between H and P are successively adapted to the supporting plane, the center of mass indeed rises, but the center of gravity S descends. Moreover, it is clear from prop. 7, book 3 of Euclid that line SP is less than line SE, and this less than line SI, and this than line SH, if indeed a diameter be drawn through S and center C. Yet it is not to be thought that any addition of weight at OP is sufficient for a wheel to ascend on any inclined plane; for if the distance of the center of gravity from the center of the wheel be less than the sine of the inclination of the plane, it will always descend; if equal to that sine, it will not ascend; if finally greater than that sine, it will be able to ascend. Let AB be the inclined plane, which touches the circle at H (I take this circle to be the one passing through the center both of the mass and of the gravity of the wheel), whose center is C, and let the straight line CH be drawn, which with the perpendicular HO makes the angle CHO. For since OH, produced, falls on the horizon AD, perpendicular, and the angle OHA, by 32, book 1, is equal to the two interior angles HFA, FAH; and since AHC, formed from the semidiameter to the tangent, is right by 18, book 3, just as HFA is right; the remaining angle CHO is equal to the angle HAF, the inclination of the plane. Therefore it is certain that the wheel will turn toward that side on which the center of gravity lies. But since CI is the sine of the angle CHI, with CH taken as radius, and CI is the least of all lines falling from point C to the straight line HO, it is manifest that, if the center of gravity were nearer the center of the wheel, as in R, the wheel would always descend, because the center of gravity looks toward the downward slope of the plane; but if it were in I, it cannot ascend, because the part looking toward the upward slope of the plane does not preponderate; if finally it were farther from the center, as in S, it will be able to ascend, until point S has been

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Liber primus. CAPUT VII. 41 fuerit in lineâ perpendiculari ad horizontem transeunte per punctum contactûs. Ex his apertè constat futurum, ut rota descendat, si angulus, quem in puncto contactûs faciunt lineæ ductæ ex centris mo- lis, & gravitatis (suppono molis centrum idem esse cum centro rotæ, quâ rota est) minor fuerit angulo inclinationis plani, tunc enim centrum gravitatis respicit declivitatem plani; fu- turum autem, ut rota ascendat, si angulus ille major fuerit eo- dem angulo inclinationis, quia centrum gravitatis respicit ac- clivitatem plani; futurum demùm, ut consistat, si angulus il- le fuerit æqualis eidem angulo inclinationis plani, quia nimi- rum planum perpendicularare dividit æqualiter momenta gravi- tatis, cum transeat per centrum gravitatis existens in lineâ perpendiculari. Hinc patet semper descensuram rotam, si habeat centrum gravitatis R, quia semper facit angulum, de quo dictum est, minorem angulo inclinationis, hoc est angulo CHI, nam si ducatur ad CR perpendicularis RE, & ex centro ducatur recta CE, angulus CER est maximus omnium, quos faciunt lineæ ex punctis C, & R ductæ ad idem punctum circumfe- rentiæ, ut mox ostendam; atqui CER minor est angulo CHI, (quia ob lineas RE, IH parallelas, angulus IHC internus per 29. lib. 1. est æqualis externo RLC, & RLC externus per 16. lib. 1. major est interno CER, ac proinde IHC major quàm CER) igitur quicunque angulus constitutus à rectis exeuntibus ex C, & R minor est angulo inclinationis; atque adeò semper descendet. At si centrum gravitatis fuerit S, ductâ ad CS perpendicu- lari SM, angulus omnium maximus est CMS: hic autem est æqualis externo CKI, cum IK, & SM parallelæ sint consti- tutæ; angulus verò CKI externus major est interno CHI, igitur angulus CMS major est angulo CHI, hoc est angulo inclinationis. Ascendere igitur poterit rota, quando angulus ad contractum factus à lineis ex C, & S exeuntibus major est angulo inclinationis; sin autem contactus fiat in eo puncto, ad quod fit angulus æqualis, consistet; si in iis punctis, ad quæ fit angulus minor, descendet. Porrò quamvis iis, qui in Astronomicarum Prostaphæreseon F

Transcription: Translated (English)

Book One. CHAPTER VII. 41 will be in a perpendicular line to the horizon passing through the point of contact. From these things it is plainly evident that the wheel will descend if the angle which, at the point of contact, is made by the lines drawn from the centers of the mass and of gravity (I suppose the center of the mass to be the same as the center of the wheel, by which the wheel is the wheel) is less than the angle of inclination of the plane; for then the center of gravity looks toward the slope of the plane. But it will be that the wheel ascends if that angle is greater than the same angle of inclination, because the center of gravity looks toward the rise of the plane; and finally it will stand still if that angle is equal to the same angle of inclination of the plane, because namely the perpendicular plane divides the moments of gravity equally, since it passes through the center of gravity existing in the perpendicular line. Hence it is clear that the wheel will always descend if it has the center of gravity R, because it always makes the angle just spoken of less than the angle of inclination, that is, the angle CHI; for if a perpendicular RE be drawn to CR, and from the center the straight line CE be drawn, the angle CER is the greatest of all those which the lines drawn from the points C and R to the same point of the circumference make, as I shall soon show; but CER is less than the angle CHI (because, by reason of the parallel lines RE and IH, the interior angle IHC by proposition 29 of Book 1 is equal to the exterior angle RLC, and RLC, being exterior, by proposition 16 of Book 1 is greater than the interior angle CER, and consequently IHC greater than CER) therefore whatever angle is formed by the straight lines going out from C and R is less than the angle of inclination; and thus it will always descend. But if the center of gravity were S, a perpendicular SM being drawn to CS, the greatest of all angles is CMS: now this is equal to the exterior angle CKI, since IK and SM are taken to be parallel; but the exterior angle CKI is greater than the interior angle CHI, therefore the angle CMS is greater than the angle CHI, that is, than the angle of inclination. The wheel therefore will be able to ascend, when the angle made at the contact by the lines going out from C and S is greater than the angle of inclination; but if the contact is made at that point to which an equal angle is made, it will stand still; if at those points to which a smaller angle is made, it will descend. Moreover although to those who in Astronomical Prostaphæreses F

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Mechanicorum 42 doctrinâ versati sunt, supervacaneum sit ostendere angulum ad peripheriam factum à Radio circuli, & à linea perpendiculari in diametrum, esse maximum omnium, qui fieri possint à Radio, & à lineâ ductâ ex eodem diametri puncto, in quod cadebat perpendicularis; ut omnibus tamen fiat satis, non pigebit hîc demonstrare. Sit in diametro circuli punctum R extra centrum C, & ad CR ducatur perpendicularis HR, quæ producta in G, bifariam dividitur in R: & ductis ex centro rectis CH, CG æqualibus, sunt anguli CHR, CGR æquales, per 5. vel 8. lib.1. Fiat angulus CER, ductis ex C & R rectis lineis ad idem punctum E peripheriæ. Dico angulum CER minorem esse angulo CHR. Ducatur enim recta EG; & erunt in Isoscele CEG æquales anguli CEG, CGE. Quia verò, per 7. lib.3. RE major est quàm RG, angulus RGE major est angulo REG, per 18. lib.1. & ablatis æqualibus remanet REC minor angulo RG C, hoc est RHC. Similiter ostendetur angulum RIC minorem esse angulo RHC: ductâ enim IG, anguli CIG, CGI sunt æquales: & quoniam per 7. lib.3. RG major est quàm RI, angulus RIG major est angulo RG I, per 18. lib.1. si igitur ex æqualibus auferantur inæquales anguli, remanet RIC minor, quàm RG C, hoc est quàm RHC. Eadem erit methodus demonstrandi angulos ad puncta peripheriæ propiora puncto H esse majores angulo CER. Ductâ enim RD æquali ipsi RE, ad punctum scilicet D æqualiter distans à diametro, ac distet punctum E, & ducto radio CD, est angulus CDR æqualis angulo CER. Sit autem puncto H vicinior angulus COR, quem dico esse majorem angulo CER per 7. lib.3. & 8. lib.1. Ducta lineâ OD, anguli COD, CDO sunt æquales, quia latera CO, CD æqualia sunt: at per 7. lib.3. RO minor est, quàm RE, hoc est RD, igitur angulus ROD per 18. lib.1. major est angulo RDO, & ablatis æqualibus remanet ROC major quàm RDC, hoc est quàm REC. Anguli itaque recedentes à puncto H semper fiunt minores, accedentes verò fiunt majores. Hoc

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42 Being versed in the doctrine of the Mechanicians, it would be superfluous to show that the angle made by a radius of the circle to the periphery, and by a line perpendicular to the diameter, is the greatest of all those that can be made by the radius and by a line drawn from the same point of the diameter at which the perpendicular fell; yet, to satisfy everyone, it will not be troublesome to demonstrate it here. Let there be on the diameter of the circle a point R outside the center C, and let HR be drawn perpendicular to CR, which, produced to G, is bisected in R; and if straight lines CH, CG equal to one another be drawn from the center, the angles CHR, CGR will be equal, by 5. or 8. book 1. Let the angle CER be formed by straight lines drawn from C and R to the same point E of the circumference. I say that the angle CER is less than the angle CHR. For let the straight line EG be drawn; and in the isosceles CEG the angles CEG, CGE will be equal. But since, by 7. book 3, RE is greater than RG, the angle RGE is greater than the angle REG, by 18. book 1; and when the equal parts are taken away, there remains REC less than the angle RGC, that is, RHC. In like manner it will be shown that the angle RIC is less than the angle RHC: for, when IG is drawn, the angles CIG, CGI are equal; and since, by 7. book 3, RG is greater than RI, the angle RIG is greater than RG I, by 18. book 1. If therefore unequal angles be taken away from equal ones, there remains RIC less than RGC, that is, than RHC. The same method will be used to prove that the angles at points of the circumference nearer to the point H are greater than the angle CER. For if RD be drawn equal to RE, namely to the point D equally distant from the diameter as the point E is, and the radius CD be drawn, the angle CDR is equal to the angle CER. But let the angle COR, which I say is nearer to the point H, be greater than the angle CER, by 7. book 3 and 8. book 1. If the line OD be drawn, the angles COD, CDO are equal, because the sides CO, CD are equal; but by 7. book 3, RO is less than RE, that is, RD, therefore the angle ROD is greater than the angle RDO, by 18. book 1; and after the equal parts are taken away, there remains ROC greater than RDC, that is, than REC. Therefore the angles receding from the point H always become smaller, but those approaching it become greater. This

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Liber primus. CAPUT VII. 43 Hoc probato consequens est illud, quod in rotæ peripheriâ duo sunt puncta, inter quæ quodlibet punctum contingat planum inclinatu[m], rota ascendit, si angulus maximus factus à lineis ductis ex centro rotæ, & ex centro gravitatis sit major angulo inclinationis; quia nimirum anguli à puncto H recedentes ad utramque partem semper fiunt minores; ergo ad utramque est angulus unus æqualis angulo inclinationis, & spatium inter hujusmodi angulos est quantitas peripheriæ, quæ ascendens potest coaptari plano inclinato: ac proinde ex horum punctorum distantia definietur spatium, quod potest rota ascendens percurrere. Sit igitur rota, cujus centrum C, & centrum gravitatis S: sit autem C S partium II, quarum C H Radius est 16: est igitur C S æqualis Sinui gr. 43. 26. qui erit maximus angulus C I S ad peripheriam factus à Radio, & à lineâ I S perpendiculari ad S C. Quare in quolibet plano habente minorem inclinationem poterit ascendere. Ponatur plani inclinatio gr. 15, cui æqualis sit angulus CH S. Fiat igitur ut C S II ad C H 16, ita Sinus anguli CH S 25882 ad 37646 Sinum Anguli C S H gr. 22. 7; eritque angulus S C H gr. 142. 53. Crescet ergo supra angulum H angulus ad peripheriam, si ultra punctum H fiat contactus rotæ in alio puncto viciniore puncto I, ex quo ad S C perpendicularis cadit; & ex I decrescit usque dum in P fiat angulus S P C grad. 15 æqualis angulo inclinationis. In triangulo itaque S P C invenitur ex iisdem datis angulus P S C gr. 157. 53. & angulus S C P gr. 7. 7. qui ex angulo S C H gr. 142. 53 ablatus relinquit P C H gr. 135. 46. quæ est quantitas arcûs H I P, quæ plano coaptatur in ascensu. Quoniam verò quarum partium C G Radius est 16, peripheria est 100 1/2 earum parirer est arcus H P ferè 38, si Radius rotæ fuerit unciarum pedis 16, rota ascendet in plano percurrens spatium pedum 3, & eo ampliùs. Hinc poteris aut rotæ diametrum augere, aut plani inclinationem minuere, si volveris rotam longiore spatio moveri: auctâ enim rotæ diametro augetur peri- F 2

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Book One. CHAPTER VII. 43 Having established this, it follows that in the circumference of a wheel there are two points between which, whichever point touches the inclined plane, the wheel ascends, if the greatest angle formed by the lines drawn from the center of the wheel and from the center of gravity is greater than the angle of inclination; because, namely, the angles receding from point H on either side always become smaller. Therefore on either side there is one angle equal to the angle of inclination, and the space between such angles is the quantity of the circumference which, in ascending, can be fitted to the inclined plane; and consequently, from the distance of these points will be determined the space which the ascending wheel can traverse. Let there therefore be a wheel whose center is C, and the center of gravity S; and let CS be 11 parts, of which CH, the radius, is 16. Therefore CS is equal to the sine of 43° 26', which will be the greatest angle CIS made at the circumference by the radius and by the line IS perpendicular to SC. Wherefore it can ascend on any plane having a lesser inclination. Let the inclination of the plane be 15°, to which let the angle CHS be equal. Thus let it be as CS, 11, is to CH, 16, so is the sine of angle CHS, 25882, to 37646, the sine of angle CSH, 22° 7'; and the angle SCH will be 142° 53'. Therefore the angle at the circumference will increase above angle H, if beyond point H the wheel is made to touch at another point nearer to point I, from which a perpendicular falls to SC; and from I it decreases until at P there is formed angle SPC of 15°, equal to the angle of inclination. In triangle SPC therefore, from the same given data, are found angle PSC, 157° 53', and angle SCP, 7° 7', which, subtracted from angle SCH, 142° 53', leaves PCH, 135° 46', which is the quantity of arc HIP that is fitted to the plane in ascending. Since, moreover, if CH, the radius, is 16 parts, the circumference is 100 1/2 of those parts, likewise the arc HP is about 38; if the radius of the wheel were 16 inches of the foot, the wheel would ascend on the plane, traversing a distance of 3 feet, and even more. Hence you may either increase the diameter of the wheel, or lessen the inclination of the plane, if you wish the wheel to be moved through a longer distance; for by increasing the diameter of the wheel, the circum- F 2

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Mechanicorum 44 pheria, servatâ ratione eadem distantiæ centri gravitatis. At si data fuerit rota ( oportet non ignorari distantiam centri gravi- tatis à centro rotæ, poterit autem primâ praxi cap. 5. investiga- ri) certum est illam non posse ascendere nisi per spatium mi- nus longitudine semiperipheriæ; constituto autem spatio inve- nietur inclinatio plani necessaria, hac methodo. Data spatij longitudo PH reducatur ad denominationem graduum, & erit notus angulus PCH: & quoniam anguli ad H & ad P debent esse æquales, anguli verò in R ad verticem sunt æquales, erunt pariter æquales PCH, & PSH, qui proinde notus est. Hujus semissis auferatur ex recto CSI, & innotescet angulus CSH, cum quo & duobus lateribus CS, CH invenietur per Trigo- nometriam angulus CHS æqualis angulo inclinationis plani necessariæ. Quod autem angulus HSI sit semissis totius HSP, hoc est dati PCH, sic ostendo. Quia in duobus triangulis CSP, CHS idem latus CS opponitur angulis æqualibus ad H, & ad P, æqualia autem latera CH, & CP opponuntur angulis quæsitis CSH, & CSP, constat horum duorum angulorum esse unum eundemque sinum; ergo simul sumpti sunt æquales duobus rectis; auferatur ex eorum summâ unus rectus, rema- nebunt duo anguli simul CSH, ISP æquales uni recto, hoc est angulo ISC: auferatur communis CSH, remanebit HSI æqualis angulo ISP: id quod oportuit demonstrare. Colligere possumus ex his, quæ hactenus explicata sunt, fie- ri quidem posse, ut, si rota in plano inclinato primùm consti- tuta exactè tangat in H, prorsus consistat; id tamen vix posse sperari, quia si in alio puncto remotiore ab I tangat, cadet, si in puncto viciniore, ascendet. At ubi venerit in P, si ex con- cepto impetu pergat adhuc aliquantulum ascendere; centro gravitatis S translato versùs plani declivitatem, & diminuto angulo, descendet; & ubi translierit punctum P, iterùm aucto angulo ascendet, donec omninò in P consistat. Ubi licet animadvertere non idem esse punctum contactus, in quo quiesceret in plano horizontali, ac inclinato; in plano enim horizontali quiesceret in O, ubi linea à centro rotæ C perpen- dicularis horizonti, ac transiens per S centrum gravitatis, ter- minatur: in eo autem puncto O consistere non posse supra pla- num inclinatum satis patet ex dictis. Porrò hæc, quæ de rotâ consistent

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Mechanics 44 by preserving the same ratio of the distance from the center of gravity. But if a wheel is given (the distance of the center of gravity from the center of the wheel must not be unknown, but it can be found by the first practice of chapter 5), it is certain that it cannot ascend except through a space less in length than a semiperiphery; but once the space has been established, the necessary inclination of the plane will be found by this method. Let the length of the given space PH be reduced to a denomination of degrees, and the angle PCH will be known; and since the angles at H and at P must be equal, and the angles at R are vertical angles, they will likewise be equal, PCH and PSH, which therefore is known. Let its half be subtracted from the right angle CSI, and the angle CSH will be known; with this, and the two sides CS, CH, by Trigonometry the angle CHS will be found, equal to the angle of the necessary inclination of the plane. But that angle HSI is the half of the whole HSP, that is, of the given PCH, I prove thus. Since in the two triangles CSP, CHS the same side CS is opposite the equal angles at H and at P, and the equal sides CH and CP are opposite the sought angles CSH and CSP, it is clear that these two angles have one and the same sine; therefore taken together they are equal to two right angles; let one right angle be subtracted from their sum, and there will remain the two angles CSH, ISP together equal to one right angle, that is, to the angle ISC; let the common angle CSH be subtracted, and HSI will remain equal to the angle ISP: which had to be demonstrated. We can gather from these things, which have so far been explained, that it is indeed possible, if a wheel first placed on an inclined plane exactly touches at H, for it to come completely to rest; yet this can hardly be hoped for, because if it touches at some other point farther from I, it will fall, if at a nearer point, it will ascend. But when it has come to P, if by the force already acquired it continues to ascend a little further; the center of gravity S being moved toward the declivity of the plane, and the angle diminished, it will descend; and when it has passed the point P, the angle being again increased, it will ascend, until it comes entirely to rest in P. Here it may be observed that the point of contact is not the same at which it would rest on a horizontal plane as on an inclined one; for on a horizontal plane it would rest at O, where the line from the center of the wheel C, perpendicular to the horizon and passing through S, the center of gravity, ends: but that it cannot come to rest at that point O on an inclined plane is sufficiently clear from what has been said. Moreover, these things, which concerning the wheel

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Liber primus. CAPUT VII. 45 consistente, aut cadente disputata sunt, dicenda esse de sphærâ quiescente in plano inclinato, clarius est, quàm ut oporteat pluribus explicare. Unum superesse videtur ostendendum, quî verum sit cen- trum gravitatis descendere ita, ut fiat horizonti vicinius, dum rota ascendit, & sit remotior. Id ut manifestum fiat, primò in- veniatur H S: & sit ut Sinus anguli CH S gr. 15. ad sinum an- guli S C H gr. 142. 53. hoc est ut 25882 ad 60344, ita C S partium 11 ad H S 25 1/4: quæ est altitudo centri gravitatis ante motum. Deinde inveniatur S P; & sit ut Sinus S P C gr. 15 ad Sinum S C P gr. 7. 7 hoc est, ut 25882 ad 12389, ita C S par- tium 11 ad S P 5 1/4, quæ in fine motus erit altitudo centri gravi- tatis supra planum inclinatum; huic autem addenda est altitu- do, quam supra horizontem habet punctum illud plani inclinati, in quotanget P. Quia ergo inclinatio plani est gr. 15, & H P est partium 38, tantum est spatium, quod in plano percurritur à rota ascendente, fiat ut Radius 100000 ad 25882 Sinum an- guli inclinationis, ita 38 ad 9 1/4 altitudinem supra horizontem, cui si addas S P 5 1/4, erit in fine motûs altitudo centri gravitatis supra horizontem partium 15, cùm initio distaret partibus 25 1/2. Centrum igitur gravitatis simpliciter, & absolutè descendit, dum rota in plano inclinato ascendit. Possem hîc afferre aquam vi suæ gravitatis ascendentem in cochleâ Archimedis, dum cylindrus, quem cochlea ambit, convertitur: abstineo tamen, quia non vacat hîc examinare, an motus ille compositus sit ex conversione, quâ pulsu externo agitata aqua attollatur, & ex naturali descensu, quo per tubum in spiras sinuatum descendat; an verò quemadmodum supposi- to cuneo reluctans pondus elevatur, vel etiam cochleâ trahitur in plano horizontali, ita dicendum sit aquam vi suæ gravitatis in imo persistentem à cochleâ sensim subeunte elevari simul, & trahi, quin illa sponte sua ascendat: nam aquæ facilè tribuitur aliquando motus, qui subjecto corpori, cui illa insidet, conve- nit; ut liquet si ampliorem peluim ex fune suspenderis, vel lu- brico in plano horizontali collocaveris, in qua sit non multa aqua in depressiore fundi parte quiescens; vase siquidem ex improviso vehementiùs impulso videtur aqua in oppositam par- F 3

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Liber primus. CAPUT VII. 45 Whether, with the sphere either standing still or falling, the matters discussed ought to be said of a sphere resting on an inclined plane, is clearer than that it should need a fuller explanation. One thing seems still to remain to be shown, namely, how it is true that the center of gravity descends so as to become nearer the horizon while the wheel ascends, and more remote while it descends. To make this manifest, first find H S: and let it be as the sine of angle C H S, 15 degrees, to the sine of angle S C H, 142 degrees 53 minutes; that is, as 25882 to 60344, so C S of 11 parts is to H S 25 1/4; which is the height of the center of gravity before motion. Then find S P; and let it be as the sine of S P C, 15 degrees, to the sine of S C P, 7 degrees 7 minutes; that is, as 25882 to 12389, so C S of 11 parts is to S P 5 1/4, which at the end of the motion will be the height of the center of gravity above the inclined plane; to this must however be added the height which that point of the inclined plane has above the horizon, in which the angle P lies. Since therefore the inclination of the plane is 15 degrees, and H P is 38 parts, so much is the distance traversed on the plane by the ascending wheel; let the Radius 100000 be to the sine 25882 of the angle of inclination, so 38 is to 9 1/4, the height above the horizon, to which, if you add S P 5 1/4, the height of the center of gravity above the horizon will be at the end of the motion 15 parts, whereas at the beginning it was 25 1/2 parts distant. The center of gravity therefore simply and absolutely descends, while the wheel ascends on the inclined plane. I could here mention water ascending by the force of its own gravity in Archimedes’ screw, while the cylinder which the screw surrounds is turned: I refrain, however, because there is not time here to examine whether that compound motion is made up of a rotation, by which the water, agitated by an external impulse, is raised, and of a natural descent, by which it descends through the tube bent into coils; or whether, as when a resisting weight is lifted by a wedge set beneath it, or even drawn by a screw on a horizontal plane, it should rather be said that water, by the force of its own gravity remaining at the bottom, is simultaneously raised and drawn by the screw gradually advancing, without rising of itself: for motion is sometimes easily attributed to water, when it properly belongs to the body in which it is contained; as is clear if you have suspended a larger basin by a rope, or placed it on a slippery horizontal plane, in which there is not much water resting in the lower part of the bottom; for if the vessel is suddenly driven more violently, the water seems to move into the opposite part F 3

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46 Mechanicorum tem refluere, cum tamen vas ipsum potiùs infra aquam mo- veatur, quàm aqua in vase: quanquam ratione adhæsionis aquæ ad peluim etiam ipsa motum concipiat. Quare in censu sponte ascendentium numeranda non videtur aqua tubo speciali cy- lindrum circumplexo elevata. Videatur fortasse aqua sponte ascensura in tubo non æquabi- li sed conico, in plano verticali rotæ spiraliter circumducto: dum enim aqua æquilibrium superficiei faciens in parte tubi ampliore præponderat, convertitur rota, & illa iterum æqua- liter se librans totius molis compositæ centrum gravitatis trans- fert extra lineam perpendicularem: si tamen ea cautio adhi- beatur, ut tanta sit aquæ quantitas, quæ non planam obtineat superficiem sed tubi inflexione conformetur; neque ita sit spiræ ascendentis ardua altitudo, ut aqua post superficiei libra- tionem ex ea parte ob sui paucitatem non præponderet; & præ- terea ejus figuræ sit tubus, ut aqua in parte angustiore remo- tior à perpendiculari, non ita ratione sitûs augeat momenta sui conatûs deorsum, ut repugnare valeat aquæ ampliorem tubi partem occupanti. Si hæc, inquam, observentur (an autem ita facile sit ea observare, ut quidam autumant, hic non de- finio) & centrum gravitatis transferatur extra perpendicula- rem versùs ampliorem tubi spiralis partem, futurum quidem est, ut aqua ascendat; id tamen non est opus centri gravitatis, sed potius virtutis illius, qua humor se æquabiliter librat. CAPUT VIII. Cur gravium in plano inclinato descendentium alia repant, alia rotentur. Quæ capite superiori dixi de globi aut rotæ super planum inclinatum consistentiâ in puncto, in quo linea à centro globi, aut rotæ ducta cum eâ, quæ ex centro gravitatis duci- tur, facit angulum æqualem angulo inclinationis plani, non ita intelligi velim; quasi motus omnis deorsum adimatur rotæ aut globo cujuslibet gravitatis, & in quovis plano inclinato: ibi enim consistentiæ, aut quietis nomine solam conversionem excipio,

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46 Mechanics to flow back, although the vessel itself is rather moved beneath the water than the water in the vessel; although, by reason of the adhesion of the water to the basin, it too acquires motion. Wherefore water lifted by a special tube encircling a cylinder does not seem to be to be counted among things that ascend of their own accord. Perhaps water would seem to ascend of its own accord in a tube that is not uniform but conical, spirally wound in the vertical plane of a wheel: for while the water, forming equilibrium of the surface, outweighs in the wider part of the tube, the wheel turns, and the water again, balancing itself equally, transfers the center of gravity of the whole composite mass outside the perpendicular line. Yet this requires the caution that the quantity of water be such as not to occupy a flat surface but to conform to the bending of the tube; and likewise that the height of the ascending spiral not be so steep that the water, after the leveling of the surface, does not outweigh on that side because of its small amount; and moreover that the tube be of such a shape that the water, in the narrower part, being farther from the perpendicular, does not increase the moments of its downward tendency by reason of its position so much as to be able to resist the water occupying the wider part of the tube. If these things, I say, are observed (but whether they can so easily be observed, as some suppose, I do not here determine), and if the center of gravity is transferred beyond the perpendicular toward the wider part of the spiral tube, then it will indeed happen that the water ascends; yet this is not the work of the center of gravity, but rather of that virtue by which the liquid balances itself equally. CHAPTER VIII. Why among heavy bodies descending on an inclined plane some creep, others rotate. What I said in the preceding chapter about the support of a globe or wheel standing on an inclined plane at that point in which the line drawn from the center of the globe or wheel, together with the line drawn from the center of gravity, makes an angle equal to the angle of inclination of the plane, I do not wish to be understood in such a way as if every downward motion were taken away from a wheel or globe of whatever weight, and on any inclined plane whatever; for there, under the name of support or rest, I admit only rotation,

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Liber primus. CAPUT VIII. 47 excipio, non lapsum nego. Fieri si quidem potest, ut adeò con- tinuo lævore lubricum sit planum, exactéque rotundatus globus, ut nullam ex eminulis particulis moram recipiens deorsum la- batur, volubilitate ipsâ motum nihil juvante, sed solo pondere urgente, cum in lineâ ad horizontem perpendiculari semper maneat centrum gravitatis, & punctum contactûs. Neque esset diversa ratio sphæræ centrum gravitatis haben- tis extra centrum molis, ac cæterorum corporum non sphæri- corum: Nam gravia quæcunque in plano inclinato constituta tantum habent ad descendendum momenti, ut asperitatis re- sistentiam vincant, repunt quidem, si linea directionis ab eo- rum gravitatis centro in terræ centrum ducta transeat per can- tactum subjecti plani, & impositi gravis; rotantur verò, si di- rectionis linea in plani declivitatem cadat extra contactum: sivè demùm in puncto, sivè in lineâ, sivè in superficie con- tactus fiat. Est autem animadvertendum non esse opus, ut una continua superficies sit, aut linea, secundùm quam se tangant; sed pro superficie aut linea contactûs accipitur totum illud spa- tium, quod inter extrema contingentia rectis lineis conjuncta intercipitur. Sit planum inclinatum A B, cui globus C incumbit con- tingens in puncto D. Ex cen- tro gravitatis C, quod & cen- trum molis est ex hypothesi, cadat linea directionis C E perpendicularis in horizon- tem F B; quæ necessariò ca- dit extra punctum contactûs D; alioquin eadem linea C E caderet ad angulos rectos su- pra planum inclinatum, & supra horizontale, id quod fieri non potest, cum hujusmodi plana non sint invicem parallela. Per D igitur punctum sustentationis ductâ G H parallelâ lineæ directionis, si per utramque plana parallela ducantur, planum per G H secat sphæram in partes inæqualiter graves; & id cir- co pars præponderans, in qua est centrum gravitatis globi, mo- vetur circa punctum sustentationis D, atque adeò in gyrum conversa

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Book One. CHAPTER VIII. 47 I except; I do not deny the slipping. For it may indeed happen that a plane, made very smooth by rubbing, and a globe exactly rounded, receive from no projecting particles any hindrance, and so slide downward; their own rotundity contributing nothing to the motion, but only their weight pressing them onward, while in a line perpendicular to the horizon the center of gravity and the point of contact always remain. Nor would the reasoning be different in the case of a sphere whose center of gravity lies outside the center of the mass, from that of other bodies not spherical. For all heavy bodies placed on an inclined plane have just so much tendency to descend as is enough to overcome the resistance of roughness; they creep, indeed, if the line of direction drawn from the center of their gravity to the center of the earth pass through the contact of the supporting plane and the body placed upon it; but they roll if the line of direction falls upon the declivity of the plane outside the point of contact: whether the contact be made at a point, or along a line, or upon a surface. And it is to be noted that there is no need for there to be one continuous surface or line along which they touch; but for the surface or line of contact is taken the whole space which is intercepted between the extreme points of contact joined by straight lines. Let A B be an inclined plane, upon which the sphere C rests, touching at point D. From the center of gravity C, which by the hypothesis is also the center of mass, let the line of direction C E fall perpendicular to the horizon F B; which necessarily falls outside the point of contact D; for otherwise the same line C E would fall at right angles above the inclined plane, and above the horizontal one, which cannot happen, since such planes are not parallel to one another. Therefore, through the point of support D, drawing G H parallel to the line of direction, if parallel planes be drawn through each, the plane through G H cuts the sphere into parts unequally heavy; and therefore the heavier part, in which lies the center of gravity of the globe, moves about the point of support D, and thus, turned around in a circle

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48 Mechanicorum conversa circa centrum C descendit, ac rotatur. Quod si inæqualis fuerit sphæræ substantia, & centrum gravitatis I in perpendiculari G H, non descendet sphæra in gyrum acta, sed tantùm repet, cum neutra pars præponderet. Simili ratione parallelepipedum K L, cujus centrum gravi- tatis M, non repit; quia, cùm linea directionis M N cadat ex- tra basim K O, quæ contingit subjectum planum, si per extre- mam lineam K P transeat planum P Q horizonti perpendicu- lare, dividitur parallelepipedum in duo prismata inæqualia, & non æquiponderantia: cum verò prisma trapezium Q L K P præponderet prismati trigono K O Q, quod sustinetur à basi, illud necessariò descendit, & circa lineam K P convertitur. Contrà autem quando intra basim contactûs, ut in cubo P R, cujus centrum S, cadit linea directionis S T, tunc repit, & non rotatur cubus; quia scilicet ab extrema sustentationis lineâ K P ductum planum horizonti perpendiculare dividit cubum in partes inæquales ita, ut pars illa, in qua est centrum gravitatis, & quæ à subjecto plano tota sustinetur, præponderet, nec pos- sit à reliquâ parte elevari, ut circa K P convertatur. Hinc apparet ad quantam altitudinem pertinere possit paral- lepipedum, ut in dato plano inclinato non rotetur, sed repat: nam ab extremâ sustentationis lineâ K P excitatum planum horizonti perpendiculare P Q, quod bifariam in partes æqui- ponderantes dividit parallelepipedum K Q, determinat altitu- dinem maximam X Q; in omni quippe majori altitudine non repit, sed rotatur, quia linea directionis cadit extra basina sustentationis: in omni verò minori altitudine non rotatur, sed repit, quia linea directionis cadit intra basim sustentationis. Hoc idem in corporibus cæteris, quamvis non parallelepipe- dis, observandum est, an scilicet linea directionis cadat extra basim sustentationis, nec ne. Quæ tamen de cubo repente dicta sunt, intelligi velim specta- tâ per se gravium figurâ: quia per accidens fieri potest, ut cor- pus non repat, sed rotetur, quamvis linea directionis cadat in- tra basim, quæ planum inclinatum contingit. Nam si in motu occurrat super plano inclinato offendiculum aliquod, cui de- scendens corpus illidatur, fieri potest, ut impetus ex motu con- ceptus ita promoveat centrum gravitatis in anteriora, ut linea directionis

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48. Mechanicorum turns around center C and descends, and rotates. But if the substance of the sphere were unequal, and the center of gravity I on the perpendicular G H, the sphere, when set in motion, would not descend by turning, but would only creep, since neither part preponderates. In a similar way the parallelepiped K L, whose center of gravity is M, does not creep; because, since the line of direction M N falls outside the base K O, which touches the supporting plane, if through the outer line K P there passes the plane P Q perpendicular to the horizon, the parallelepiped is divided into two unequal prisms, and not equally weighted: but since the trapezoidal prism Q L K P outweighs the triangular prism K O Q, which is supported by the base, it necessarily descends and turns about the line K P. On the contrary, when the line of direction S T falls within the base of contact, as in the cube P R, whose center is S, then the cube creeps and does not rotate; because, namely, a plane drawn from the outer line of support K P perpendicular to the horizon divides the cube into unequal parts in such a way that that part in which the center of gravity is, and which is wholly sustained by the underlying plane, preponderates, nor can it be lifted by the remaining part so as to turn about K P. Hence it appears to what height a parallelepiped may extend, so that on a given inclined plane it may not rotate, but creep: for the plane P Q, raised from the outer line of support K P and perpendicular to the horizon, which divides the parallelepiped K Q in two equal parts, determines the maximum height X Q; for in every greater height it does not creep, but rotates, because the line of direction falls outside the base of support: in every lesser height, however, it does not rotate, but creeps, because the line of direction falls within the base of support. The same is to be observed in other bodies, although they are not parallelepipeds, namely, whether the line of direction falls outside the base of support or not. What, however, has been said about the cube creeping, I would have understood with regard to the figure of heavy bodies as such: because by accident it can happen that a body does not creep, but rotates, although the line of direction falls within the base that touches the inclined plane. For if, in motion, some obstacle should occur on the inclined plane, against which the descending body may strike, it may happen that the impetus conceived from the motion so advances the center of gravity toward the front that the line of direction

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Liber primus. CAPUT IX. 49 directionis cadat extra basim ultrà punctum illud, quod proxi- mum est offendiculo, ac proinde circa illud convertatur. Hæc autem potissimùm est ratio, cur ex clivis descendentes lapides, quamquam nec orbiculares, nec admodum alti, rotentur ta- men; quia scilicet multa offendicula in clivo occurrunt, & ab impetu per motum concepto partes superiores promoventur ulteriùs, inferioribus retardatis. Sic sæpè cespitantes cadimus, quia ab offendiculo retinentur pedes, cum interim corpus re- liquum ex concepto impetu ulteriùs promoveatur, ita ut linea directionis cadat extra basim sustentationis. CAPUT IX. Cur turres inclinatæ non corruant. Observandum est, ait Vitruvius lib.6. cap.11, uti omnes structuræ perpendiculo respondeant, neque habeant in ulla parte proclinationes. Nemo est qui non intelligat præ- ceptum hoc ad ædificiorum consistentiam pertinere; sed neque defuerunt, qui rem subtiliùs, quàm par sit, perpendentes ina- ni timore se torquebant, ne fortè aliquando domus corrueret, cujus parietes inter se paralleli fuerant constituti; cùm enim perpendicula sibi demum in terræ centro occurrant, fieri non posse putabant, ut simul paralleli essent parietes. Id quod Geo- metricè quidem verum est; Physicè tamen parallelismus cum perpendiculis consentit: nam si funiculos duos longitudinis ped.100. clavo affixos ita extendas, ut extrema eorum palmi intervallo distent, angulum facient acutissimum; & si lineas duas bipedales duxeris eorum extremitatibus congruentes, vix different à parallelis, cum intervalla jungentia utrosque linea- rum terminos differant inter se solum palmi parte quinquage- sima. Longè autem majorem rationem terræ semidiameter ha- bet ad quamlibet ædificiorum altitudinem; ut proinde à paral- lelismo multo minùs recedant parietes, etiamsi fuerint turrium instar altissimi. Ponantur enim parietes duo, aut potiùs turres, distare inter se pass.300; sit autem parietum, vel turrium alti- tudo pass.60, hoc est ped.300. Constat mihi, ut aliàs ostendi, terrenam semidiametrum non esse minorem passibus Rom. G

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Book One. CHAPTER IX. 49 the line of direction falls outside the base beyond that point which is nearest to the obstacle, and therefore it turns around it. This too is chiefly the reason why stones descending from slopes, although neither spherical nor very high, nevertheless revolve; because, namely, many obstacles occur on the slope, and, from the force of motion acquired, the upper parts are advanced farther, the lower parts being retarded. Thus we often fall when stumbling, because the feet are held back by an obstacle, while in the meantime the rest of the body, from the impetus already acquired, is carried farther forward, so that the line of direction falls outside the base of support. CHAPTER IX. Why leaning towers do not fall. It is to be observed, says Vitruvius, lib. 6, cap. 11, that all structures should answer to the plumb-line, and have no inclinations in any part. There is no one who does not understand that this precept concerns the stability of buildings; but there have been some who, pondering the matter too subtly, more than was proper, tortured themselves with groundless fear, lest perhaps at some time a house should collapse whose walls had been set parallel to one another; for since plumb-lines meet only at the center of the earth, they thought it could not happen that walls should be parallel at the same time. Which is certainly true geometrically; yet physically parallelism agrees with plumb-lines: for if you stretch two cords, 100 feet long, fastened to a nail, so that their ends are separated by the distance of a palm, they will form an acute angle; and if you draw two lines, two feet long, corresponding to their ends, they will scarcely differ from parallel lines, since the intervals joining the two ends of the lines differ from one another by only the fiftieth part of a palm. But the semidiameter of the earth bears a far greater ratio to any height of buildings; so that the walls depart much less from parallelism, even if they are as high as towers. For let two walls, or rather towers, be placed 300 paces apart; let the height of the walls, or towers, be 60 paces, that is, 300 feet. It is clear to me, as I have shown elsewhere, that the semidiameter of the earth is not less than 60 Roman paces...

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50 Mechanicorum antiq. 4128635: quarè si fiat ut terræ semidiameter 4128635 ad altitudinem 60, ita distantia parietum, aut turrium in imo 300, ad aliud, proveniet differentia, qua distantia turrium in summo vertice superat earum distantiam in imo pede, & erit partium 4359/1000000 unius passus, quæ est minor quàm 2/3 digiti: quis autem parallelas non dixerit turres, quæ vix uno aut altero hordei grano distant à parallelismo? Quod si in tanta altitudine atque distantiâ discrimen hoc adeò exiguum est, satis patet, quid de columnarum parallelismo dicendum sit. Constat autem ex his ædificia in altissimis montibus constituta habere parietes minùs à parallelismo recedentes, si fuerint ad perpendicularum ædificati, quàm in locis depressioribus: atque adeò, si duæ columnæ eandem inter se positionem servantes descenderent cum subjecto plano, ita ut alterutra columnarum illarum ad perpendicularum descenderet, reliqua demùm adeò inclinaretur, ut caderet. Sed quàm inanem sibi struant solicitudinem, qui nimis exigè, & exiliter ad calculos revocant structurarum perpendiculara, satis indicant turres inclinatæ, quæ post aliquot secula consistunt citrà ullum ruinæ periculum, quamvis illam timeant imperiti. Duas habemus in Italiâ turres ob insignem inclinationem conspicuas; altera est Bononiæ quadrata opere lateritio, altera Pisis rotunda ex albo marmore affabrè expolito, & columnis 284 rite dispositis ornata. Ædificari coepit anno 1173 Germano quodam architecto, quem ab aliis Guillelemum, ab aliis Ioannem OEnipontanum dici reperio. Rotunda est forma duplici muro concludente scalas cochleæ in modum ab imo ad summum ductas: parietis crassities est cubitorum 6 1/3, turris altitudo cubitorum 7 1/3, ut ex literis ad me inde datis habeo; quamvis apud aliquos legerim tantùm cubitos 7, apud alios 6 1/3. Factâ ne fuerit illa inclinatio de industriâ, an verò subsidentibus fundamentis, incertum est. Ego non facilè eo in illorum sententiam, qui id scribunt contigisse ex artificis imperitia, cui non satis perspecta esset soli natura; tum quia fundamenta altitudinem

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50 Mechanicorum antiquity 4128635: wherefore if it happens that the semidiameter of the earth, 4128635 to the height 60, so the distance of walls, or of towers at the bottom, 300, to the other, there will result a difference, by which the distance of the towers at the top exceeds their distance at the bottom by a foot, and it will be of 4359/1000000 parts of one pace, which is less than 2/3 of a finger: and who would not call towers parallel, which scarcely differ from parallelism by one or two barleycorns? But if at so great a height and distance this difference is so very small, it is sufficiently clear what must be said concerning the parallelism of columns. It is also established from this that buildings erected on the highest mountains have walls which depart less from parallelism, if they have been built perpendicular, than those in lower places: and indeed, if two columns, preserving the same position with respect to each other, were to descend with the underlying plane, so that one of those columns would descend perpendicularly, the other would finally incline so much that it would fall. But how vain a concern those build for themselves who bring the perpendiculars of structures too closely and too finely to calculation is sufficiently shown by leaning towers, which after some centuries stand firm without any danger of ruin, although the inexperienced fear it. We have two towers in Italy notable for their remarkable inclination; one is at Bologna, square and of brickwork, the other at Pisa, round, of white marble carefully polished, and adorned with 284 columns properly arranged. It began to be built in the year 1173 by a certain German architect, whom I find called by some Guillelemus, by others John of Oenipontum. It is round in form, enclosed by a double wall containing stairs carried in a spiral manner from bottom to top: the thickness of the wall is 6 1/3 cubits, the height of the tower 7 1/3 cubits, as I have from letters sent to me from there; although I have read in some authors only 7 cubits, in others 6 1/3. Whether that inclination was made intentionally, or rather through the settling of the foundations, is uncertain. I do not readily agree with those who write that this happened from the architect’s ignorance, to whom the nature of the soil had not been sufficiently known; both because the foundations, the height

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Liber primus. CAPUT IX. 51 nèm habent, atque amplitudinem ingentem, quibus con- struendis annus solidus satis non fuit; tum quia nullam unquam egit rimam, id quod subsidente solo rarissimum est; tum quia potuit architectus excitari ad artis specimen exhibendum à tur- ri Bononiensi Garisendâ excitatâ anno 1110. Turris Bononiensis altitudinem habet pedum Bonon. 130; exteriùs inclinatur ped. 9, interiùs verò ped. 1, & paulo am- plius: muri crassities in parte infimâ est pedum 6 ́, in supre- ma ped. 4; cava turris ped. 7. quare lateris longitudo est ped. 20, & ambitus, quoniam quadrata est, ped. 80. Ex his men- suris, quas in Bononiâ Perlustratâ anno 1650 typis evulgatâ at- tulit Antonius Pauli Masini, turris spe- ciem exhibeo, & est A B latus unum ped. 20, B D inclinationis mensura ped. 9. D C altitudo perpendicularis ped. 130; E B & A F ped. 6 ́ crassities imi parietis, & C H ped. 4. crassities ejusdem parietis E C exteriùs inclinati. At quoniam inclinatio interior F I dici- tur esse ped. 1, & paulo ampliùs, erit I D paulo major ped. 21; erecta autem ex I perpendicularis dabit punctum G termi- num crassitiei muri A G in parte supre- mâ, & erit C G major ped. 21, cum sit æqualis ipsi I D. Quare fieri non potest, ut K G sit ped. 4; quemadmodum H C; alioquin esset C K saltem ped. 25, cum basis A B sit tantum ped. 20. Hinc si li- ceat conjecturas persequi (quandoqui- dem veritatem assequi non potui, cum non careat periculo ascensus per scalas ligneas à pluviis maximam partem cor- ruptas) existimo A F majorem esse quàm E B, hoc est majorem pedibus 6 ́, K G verò minorem quam H C, ut turri sua constet Eurithmia; id quod obtineretur, si I D uno, aut alte- ro pede minor esset quàm A B, differentia enim inter I D, & A B esset crassities K G. Et sanè memini aliquando me au- G 2

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Book One. CHAPTER IX. 51 do not have them, and an enormous breadth, for the construction of which a full year was not sufficient; then because it has never suffered any crack, which is most uncommon in ground that subsides; then because the architect could be prompted to exhibit a specimen of his art by the tower of Bologna, the Garisenda, erected in the year 1110. The Tower of Bologna has a height of 130 Bolognese feet; it inclines outward 9 feet, inward however 1 foot, and a little more; the thickness of the wall at the lowest part is 6 1/2 feet, at the top 4 feet; the hollow of the tower 7 feet. wherefore the length of the side is 20 feet, and the perimeter, since it is square, 80 feet. From these measurements, which Antonius Pauli Masini set forth in Bologna Surveyed , published in print in the year 1650, I present the appearance of the tower, and A B is one side 20 feet, B D the measure of the inclination 9 feet. D C the perpendicular height 130 feet; E B and A F 6 1/2 feet the thickness of the lower wall, and C H 4 feet, the thickness of the same wall E C inclined outward. But since the inward inclination F I is said to be 1 foot, and a little more, I D will be a little more than 21 feet; and the perpendicular raised from I will give the point G, the limit of the thickness of the wall A G at the top, and C G will be greater than 21 feet, since it is equal to I D itself. Therefore it cannot happen that K G is 4 feet, as H C is; otherwise C K would be at least 25 feet, since the base A B is only 20 feet. Hence if I may be allowed to pursue conjectures (since I was not able to arrive at the truth, because there is no danger-free ascent by wooden stairs, for the most part ruined by the rains) I think A F is greater than E B, that is, greater than 6 1/2 feet, and K G less than H C, so that Eurithmia may be preserved in its tower; which would be achieved if I D were one, or two feet less than A B, for the difference between I D, and A B would be the thickness K G. And indeed I remember that once I au- G 2

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divisse supremam crassitiem muri oppositi parti inclinatæ non excedere integrum pedem. Id autem valde opportunum accidebat, ut longè faciliùs paries A F G K suâ mole staret: neque enim casu inclinatam fuisse turrim dicere poteris, quam constat prope Asinellam rectissimam ideò fuisse conditam, ut multo clariùs appareret inclinatio: præterquam quod inclinatio interior minor externâ satis ostendit muros nunquam fuisse parallelos. Porrò ut constet ex hujusmodi inclinatione non magis esse de ruinâ timendum, quàm si exactè perpendicularis esset, examinemus, si placet, centrum gravitatis in turri Bononiensi; hinc enim facilis erit conjectura de cæteris. Et primò parietis maximè inclinati sectio verticalis illum bifariam secans ac transiens per centrum gravitatis sit H C B E: cujus latera parallela H C, E B bifariam secta in V & R jungantur rectâ V R, cujus longitudo investiganda est, ut in eâ definiatur punctum S centrum gravitatis, ac innotescat utrum perpendicularis S X, scilicet linea directionis cadat intra basim E B sustentantem. Et ut à fractionibus minus incommodi subeamus, liceat assumere pedem in partes centesimas divisum. Cum autem E B sit ped. 6 ́, semissis R B est ped. 3. 25 ́; & quia H C est ped. 4, V C est ped. 20 ́. Et ducatur recta B V. In triangulo B D C rectangulo datis B D, inclinatione ped. 9 ́ ́, & altitudine perpendiculari C D ped. 13 ́ ́, additis laterum quadratis fit quadratum hypothenusæ B C, quæ est ped. 130 31 ́. Ex datis autem lateribus B D, & D C invenitur angulus C B D gr. 88. 33 ́, cui æqualis est inter parallelas V C, B D alternus V C B: angulus verò C B R gr. 91. 27 ́. In triangulo V C B datis lateribus V C ped. 2. ́ ́, C B ped. 130. 31 ́, & angulo verticali V C B gr. 88. 33 ́, reperitur CVB

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The maximum thickness of the wall opposite the inclined part does not exceed one full foot. And this was very convenient, so that the wall A F G K could much more easily stand by its own mass: for you could not say that the tower had been inclined by chance, since it is known to have been built very upright near the Asinella, in order that the inclination might appear much more clearly; besides, the fact that the interior inclination is smaller than the exterior sufficiently shows that the walls were never parallel. Moreover, in order that it may be clear from such an inclination that there is no more reason to fear ruin than if it were exactly perpendicular, let us examine, if you please, the center of gravity in the Tower of Bologna; from this a conjecture about the rest will be easy. And first, let the vertical section of the wall most inclined, cutting it in two and passing through the center of gravity, be H C B E: whose parallel sides H C, E B, cut in two at V and R, are joined by the straight line V R, the length of which must be investigated, so that in it the point S, the center of gravity, may be determined, and it may be known whether the perpendicular S X, that is, the line of direction, falls within the supporting base E B. And so that we may have less trouble with fractions, let us assume a foot divided into hundredths. Since then E B is 6 feet, half of R B is 3.25 feet; and because H C is 4 feet, V C is 20. And let the straight line B V be drawn. In the right triangle B D C, given B D, the inclination of 9, and the perpendicular height C D of 13, adding the squares of the sides gives the square of the hypotenuse B C, which is 130 31 feet. But from the given sides B D and D C, the angle C B D is found to be 88.33 degrees, equal to the alternate angle V C B between the parallels V C, B D: the angle C B R, however, is 91.27 degrees. In triangle V C B, given the sides V C 2, C B 130.31, and the vertical angle V C B 88.33 degrees, there is found CVB

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Liber primus. CAPUT IX. 53 CVB gr. 90. 34'. 14", & VBC gr. 0. 52'. 46" .. Ex his autem investigatur VB ped. 130. 26". Quoniam autem angulus CBR notus erat gr. 91. 27', si dematur ex illo angulus VBC gr. 0. 52'. 46". remanet VBR gr. 90. 34', 14", æqualis angulo C VB alterno inter parallelas; & nota sunt latera illum constituentia BR ped 3. 25". & BV ped. 130. 26". Ex quibus datis invenitur angulus BRV gr. 88. 0'. 2", BV R gr. 1. 25'. 44" & basis VR ped. 130. 326". Jam verò, ex prop. 15 lib. 1. Æquipond. Archimedis, dividatur VR in S eâ ratione, ut sit VS ad SR, ut duplum EB majoris parallelarum unâ cum minore HC, ad duplum HC unâ cum majore EB, hoc est (quia EB est ped. 6 ́) & HC ped. 4.) ut 17 ad 14 ́. Igitur ut 31 ́ ad 14 ́, ita VR 130. 326", ad SR ped. 59. 99". Demum ex S ducta perpendiculari SX, quia in triangulo R XS rectangulo datur angulus SRX gr. 88. 0'. 2". atque adeò ejus complementum R SX gr. 1. 59'. 58". & latus SR ped. 59. 99". invenitur latus RX ped. 209". Est igitur RX linea minor, quàm RB posita ped. 3. 25"; & idcirco perpendicularis linea directionis SX cadit intrà basum parietis EBCH. Sed quia facturum me puto rem aliquibus gratam, si quas inij rationes hîc exhibeam, calculi totius progressum per logarithmos hîc addo, ut illum possis, si placeat examinare. In Triangulo BDC rectang BD ped. 900' — rl 7, 04575, 74906 DC ped. 130. 00' — l. 4, 11394, 33523 CBD gr. 88. 33'. m 1, 15970, 08429 In Triangulo VBR VB + BR ped. 13351' — rl 5, 87448, 62041 VB — BR ped. 12701 — l 4, 10383, 79160 Semisumma ang. gr. 44. 42'. 53", -m 9, 99567, 51920 differentia gr. 43. 17', 9 m 9, 97399, 93121 In Triangulo VCB CB + CV ped. 132. 31' — rl 5, 87840, 73306 CB — CV ped. 128. 31 — l 4, 10816, 05050 Semisumma ang. ad basum g. 45. 43' 30". m 10, 01099, 19826 differentia g. 44. 50. 44. m 999765, 98182 Angul. C VB g. 90. 34. 14 Ang. VBC g. 0. 52. 46 CBR g. 91. 27. 0 VBR g. 90. 34. 14 VBC gr. 0. 52'. 46' — rl 1, 81393, 17962 V CB gr. 88. 33. 0. — l 9, 99986, 09115 V C ped. 200'. — — — l 2, 30102, 99957 V B ped. 130. 26' — — — l 4, 11482, 27034 G 3

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Book One. CHAPTER IX. 53 CVB gr. 90. 34'. 14", & VBC gr. 0. 52'. 46" .. From these, however, VB is found, ped. 130. 26". Now since angle CBR was known, gr. 91. 27', if from it angle VBC, gr. 0. 52'. 46", be subtracted, there remains VBR gr. 90. 34', 14", equal to the alternate angle CVB between parallels; and the sides constituting it are known, BR ped. 3. 25". & BV ped. 130. 26". From these given data there is found angle BRV gr. 88. 0'. 2", BVR gr. 1. 25'. 44" & base VR ped. 130. 326". Now then, from proposition 15, book 1, of Archimedes' Equilibrium, let VR be divided at S in such a ratio that VS to SR be as twice EB, the greater of the parallels, together with the lesser HC, to twice HC together with the greater EB, that is (because EB is 6 ft.) & HC 4 ft.) as 17 to 14. Therefore as 31 to 14, so VR 130. 326", to SR ped. 59. 99". Finally, from S draw the perpendicular SX; because in the right triangle RXS the angle SRX gr. 88. 0'. 2" is given, and therefore its complement RSX gr. 1. 59'. 58". & side SR ped. 59. 99". side RX is found to be ped. 209". Therefore RX is a line smaller than RB, set at ped. 3. 25"; and for that reason the perpendicular line of direction SX falls within the base of the wall EBCH. But because I think I shall do something pleasing to some persons, if I set out here the reasons by which I arrived, I add here the progress of the whole calculation by logarithms, so that you may, if you wish, examine it. In triangle BDC, right-angled BD ped. 900' — rl 7, 04575, 74906 DC ped. 130. 00' — l. 4, 11394, 33523 CBD gr. 88. 33'. m 1, 15970, 08429 In triangle VBR VB + BR ped. 13351' — rl 5, 87448, 62041 VB — BR ped. 12701 — l 4, 10383, 79160 Semisum of angle gr. 44. 42'. 53", -m 9, 99567, 51920 difference gr. 43. 17', 9 m 9, 97399, 93121 In triangle VCB CB + CV ped. 132. 31' — rl 5, 87840, 73306 CB — CV ped. 128. 31 — l 4, 10816, 05050 Semisum of angle to the base g. 45. 43' 30". m 10, 01099, 19826 difference g. 44. 50. 44. m 999765, 98182 Angle C VB g. 90. 34. 14 Angle VBC g. 0. 52. 46 CBR g. 91. 27. 0 VBR g. 90. 34. 14 VBC gr. 0. 52'. 46' — rl 1, 81393, 17962 V CB gr. 88. 33. 0. — l 9, 99986, 09115 V C ped. 200'. — — — l 2, 30102, 99957 V B ped. 130. 26' — — — l 4, 11482, 27034 G 3

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54 Mechanicorum Quod si paries exteriùs inclinatus etiam solitarius consistere posset, modò ea esset partium connexio, ut unum quid soli- dum conflarent, quia directionis linea intra basim sustentan- tem cadit, & planum per extremam basis lineam, & terræ cen- trum transiens relinquit interiorem parietis partem præponde- rantem exteriori: quis possit de turris ruinâ dubitare, si eâdem methodo deprehendat oppositi parietis A G centrum gravita- tis esse in O, ac proinde comparatis reliquorum duorum pa- rietum centris gravitatum, totius turris centrum gravitatis esse in intimis turris partibus? Quò igitur firmiùs sibi cohærebunt partes turris, eò major erit inclinatio, quam obtinere potest ci- tra cadendi periculum. Id quod pueris ipsis notissimum est, qui turriculas inclinatas architectantur ex buxeis orbiculis, quibus in alveolo ludunt. Et ut res ista planissimè ostendatur, sit supra planum inclinatum A B, pa- rallelepipedum ligneum I D ita, ut recta C E ad horizontem perpendicu- laris transeat per centrum gravitatis: constat ex dictis cap. 8. futurum esse, ut grave I D repat, non autem rote- tur, quia pars C E D non præponderat parti C E I, siqui- dem possit descendere per planum inclinatum; quod si à lap- su impediatu, subsistet. Iam verò intellige per C planum F H horizontale, & adnecti prisma trigonum C I K pa- rallelepipedo I D; utique pars C E K præponderat parti C E D, multóque minùs dubitandum erit de solidi K D rui- nâ versus H. Quid autem aliud est solidum K D, quam tur- ris inclinata? Scripseram hæc jam tum ab anno labentis sæculi quinquage- simo sexto; cum animum subiit suspicari, an superiùs allatæ ex Masino turris Bononiensis mensuræ omninò veritati responde- rent. Quare litteris ad P. Franciscum Mariam Grimaldum da- tis rogavi, ut pro eâ, quam ad res omnes conferre solebat, di- ligentiâ, accuratè mensuras illas inquireret: hæc igitur ex ejus responsione habui, quibus superiùs dicta corrigenda sunt; quæ tamen expungere nolui, ut si lubeat, vulgarem opinionem se- qui valeas. Extimus

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54 Mechanica But if an exterior wall, inclined outward, could also stand alone, provided only that the connection of the parts were such that they formed one solid body, because the line of direction falls within the supporting base, and the plane passing through the extreme line of the base and the center of the earth leaves the inner part of the wall outweighing the outer: who could doubt the fall of the tower, if by the same method he should find that the center of gravity of the opposite wall A G is in O, and consequently, when the centers of gravity of the remaining two walls are compared, the center of gravity of the whole tower is in the innermost parts of the tower? Therefore, the more firmly the parts of the tower cohere among themselves, the greater the inclination that it can assume without danger of falling. This is something even boys know very well, when they build little leaning towers out of wooden disks with which they play in a bowl. And so that this matter may be made perfectly clear, let there be on the inclined plane A B a wooden parallelepiped I D in such a way that the straight line C E, perpendicular to the horizon, passes through the center of gravity: it is evident from what was said in chapter 8 that the heavy body I D will not move onward, but will not rotate either, because the part C E D does not outweigh the part C E I, since it may descend along the inclined plane; but if it is prevented from slipping, it will stand still. Now, however, suppose that by C there is understood the horizontal plane F H, and that a triangular prism C I K be attached to the parallelepiped I D; certainly the part C E K outweighs the part C E D, and there will be much less doubt about the fall of the solid K D toward H. But what else is the solid K D, than an inclined tower? I had written these things already in the year fifty-six of the present century; when it came into my mind to suspect whether the measurements of the Bolognese tower given above from Masino corresponded in every respect to the truth. Therefore, by letters sent to Father Francesco Maria Grimaldi, I asked that he should, with the care he was accustomed to devote to all matters, accurately investigate those measurements: from his reply I therefore obtained these things, by which what was said above should be corrected; yet I did not wish to delete them, so that, if you wish, you may be able to follow the common opinion. Extimus

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Liber primus. CAPUT IX. 55 Extimus turris ambitus tam in imâ, quam in supremâ parte æqualis est, adeò ut oppositæ facies parallelæ excurrant: singulorum autem laterum ad basim latitudo est ped. Bonon. 17. unc. 8. murorum crassities in imo æqualis est; eo tantum discrimine, quod murus, qua parte ostium patet, crassus est ped. 5. unc. 11. qui verò Septentrionem spectat, propiùs accedit ad pedes 6. Porrò in summâ turri murorum crassities pariter æqualis est, & vix deficit à pedibus 5, quantum quidem ex aspectu à superiori proximæ turris Asinellæ podio conjicere potuit singulorum murorum lateres numerans. Areæ demum vacuæ ad basim latus unum est ped. 6. alterum ped. 6. unc. 1. Cum autem pluvia per hiantem, & patulum turris verticem deciduæ scalas corruperint, nec eò veniri possit, ut demisso perpendiculo altitudo turris investigetur, subsidium petendum fuit ex Trigonometriâ, & ex proximâ turri Asinellâ, cujus mensuræ multiplici observatione innotuerant. Sit itaque turris inclinata DC, superioris autem podij Asinellæ altitudo EB ped. 234 1/2, unde observatus est angulus CEB gr. 18.40. Item in eadem turri Asinellâ patet fenestra in F, adeò ut distantia EF sit ped. 141: ibi pariter observatus est angulus EFC gr. 51.51. Quare in triangulo CEF, notum est latus EF, & duo anguli adjacentes, ex quibus datis colligitur EC distantia ped. 117 1/2. Jam verò intelligantur ex C cadere duæ perpendiculares, altera quidem CH in planum horizontale, altera verò CG in turrim Asinellam; erit enim altitudo CH æqualis altitudini GB, nam CG est parallela horizonti, cui turris EB perpendicularis insistit. Ut igitur innotescat quæsita altitudo, inveniatur in triangulo rectangulo CGE, ex datis latere CE ped. 117 1/2 & angulo observato CEG, gr. 18.40, latus EG ped. 111 1/2. Jam verò si EG ped. 111 5/12 dematur ex EB ped. 234 1/2, remanet altitudo GB, ped. 123 1/12. hoc est CH, Demum

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Liber primus. CHAPTER IX. 55 The outer circumference of the tower is equal both at the bottom and at the top, so that the opposite faces run parallel; the width of each side at the base is 17 ft. 8 in. of Bologna measure. The thickness of the walls at the bottom is equal, with this only difference, that the wall on the side where the door opens is 5 ft. 11 in. thick, while the one facing north comes closer to 6 ft. Moreover, at the top of the tower the thickness of the walls is likewise equal, and scarcely falls short of 5 ft., as far indeed as could be conjectured from the view from the upper platform of the nearby Asinelli Tower, by counting the bricks of each wall. Finally, the empty area at the base is 6 ft. on one side and 6 ft. 1 in. on the other. But since rain, entering through the open and exposed top of the tower, had damaged the descending stairs, and since one could not get there so as to determine the height of the tower by dropping a plumb line, recourse had to be had to Trigonometry and to the nearby Asinelli Tower, whose dimensions had been made known by repeated observation. Let therefore the leaning tower be DC, and let the height of the upper platform of Asinelli, EB, be 234 1/2 ft., from which the angle CEB, 18.40 degrees, was observed. Likewise, in the same Asinelli Tower there is a window open at F, so that EF is 141 ft.; there also the angle EFC, 51.51 degrees, was observed. Therefore, in triangle CEF, the side EF and the two adjacent angles are known, from which data EC is found to be 117 1/2 ft. Now let two perpendiculars fall from C, one CH to the horizontal plane, the other CG to the Asinelli Tower; then the height CH will be equal to the height GB, since CG is parallel to the horizon, to which the tower EB stands perpendicular. Thus, in order that the required height may be known, let the right triangle CGE be solved from the given side CE, 117 1/2 ft., and the observed angle CEG, 18.40 degrees, giving the side EG as 111 1/2 ft. Now if EG, 111 5/12 ft., is subtracted from EB, 234 1/2 ft., there remains the height GB, 123 1/12 ft., that is, CH. Finally

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56 Mechanicorum Demum ad investigandam turris inclinationem, applicito ad punctum I perpendiculo observatus est angulus DIL gr. 3. 10': cùm autem IL parallela sit perpendiculari CH, erit pariter angulus DCH gr. 3. 10'. Igitur in triangulo DCH rectangulo ad H notum est latus CH ped. 123 1/12, & angulus DCH gr. 3. 10', ergo & innotescit latus DH ped. 6. 10/12, quæ est mensura inclinationis quæsitæ. Ex his accuratioribus mensuris indagemus, si placet, in orientali pariete inclinato centrum gravitatis, & lineam di- rectionis methodo eâdem, qua superiùs usi sumus; eademque figura sectionis verticalis resumatur. Est igitur EB ped. 6. ac propterea RB ped. 300"; & quia HC est ped. 5, VC est ped. 2. 50". BD autem est ped. 6. unc. 10, hoc est ped. 6 10/12. In Triangulo BDC rectangulo datis BD ped. 6. 10/12, & altitudine perpendiculari CD ped. 123 1/12, additis laterum quadratis fit qua- dratum hypothenusæ BC, quæ est ped. 123. 27". Fiat igitur ut CB ped. 123. 27", ad BD ped. 6. 83". ita Radius ad sinum anguli BCD gr. 3. 10' 34". Quare angulus reliquus CBD gr. 86. 49'. 26", cui æqualis est alternus VCB inter parallelas VC, RD; angulus autem, qui est deinceps, CBR gr. 93. 10'. 34'. In triangulo VCB datis lateribus VC ped. 2. 50", CB ped. 123. 27", & angulo verticali VCB gr. 86. 49'. 26", reperitur CVB gr. 92. 0'. 36", & VBC. gr. 1. 9', 58". Ex his verò invenitur VB ped. 122. 76". Jam verò in Triangulo VBR, notus est angulus RBV æqualis alterno CVB gr. 92. 0'. 36'. & nota sunt latera RB ped. 300", & VB ped. 122. 76". Quare invenitur angulus V RB gr. 86. 35' 43". BVR gr. 1. 23'. 41", & basis VR ped. 123. 17'. Tum fiat ut 17 ad 16; hoc est duplum majoris EB cum mi- nore HC, ad duplum minoris HC cum majore EB, ita VS ad SR, & erit SR ped. 59. 72". Ductâ igitur ex S centro gra- vitatis

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56 Mechanicorum Then, in order to investigate the inclination of the tower, a plumb line having been applied to point I, the angle DIL was observed to be 3° 10'. Since IL is parallel to the perpendicular CH, the angle DCH will likewise be 3° 10'. Therefore, in the right triangle DCH at H, the side CH is known, 123 1/12 feet, and the angle DCH, 3° 10'; hence the side DH is also found, 6 10/12 feet, which is the measure of the inclination sought. From these more accurate measurements let us examine, if you please, on the eastern inclined wall the center of gravity, and the line of direction, by the same method that we used above; and let the same figure of the vertical section be resumed. Thus EB is 6 feet, and therefore RB is 300"; and since HC is 5 feet, VC is 2. 50". BD, however, is 6 feet 10 inches, that is, 6 10/12 feet. In the right triangle BDC, given BD 6 10/12 feet, and the perpendicular height CD 123 1/12 feet, by adding the squares of the sides the square of the hypotenuse BC is obtained, which is 123. 27". Let it therefore be as CB, 123. 27", to BD, 6. 83", so Radius is to the sine of angle BCD, 3. 10'. 34". Therefore the remaining angle CBD is 86. 49'. 26", equal to the alternate VCB between the parallels VC, RD; but the angle which follows is CBR, 93. 10'. 34'. In the triangle VCB, with the sides VC, 2. 50", CB, 123. 27", and the vertical angle VCB 86. 49'. 26", there is found CVB, 92. 0'. 36", and VBC, 1. 9', 58". From these, moreover, VB is found to be 122. 76". Now in the triangle VBR, the angle RBV is known, equal to the alternate CVB, 92. 0'. 36'. And the sides RB, 300", and VB, 122. 76", are known. Therefore the angle VRB is found to be 86. 35' 43", BVR, 1. 23'. 41", and the base VR is 123. 17'. Then let it be as 17 to 16; that is, the double of the greater EB together with the smaller HC, to the double of the smaller HC together with the greater EB, so VS to SR, and SR will be 59. 72". Therefore, having drawn from S, the center of gravity

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Liber primus. CAPUT IX. 57 vitatis perpendiculari lineâ directionis S X, ex datis latere S R ped. 59. 72", & angulo V R X gr. 86, 35', 43", innotescit R X ped. 3. 54". Quare R X major est quàm R B: & si paries ille solitarius esset, non utique consisteret; sed quoniam reliqui tres parietes adjecti sunt, constat ita totius molis centrum gra- vitatis esse in intima turris parte, ut linea directionis cadat in- trà turris basim sustentantem. Ex his discuties timorem eorum, qui soliciti sunt de obelis- corum consistentiâ, ex inclinatione aliquâ verticis ruinam proximam præsagientes: cum enim in hujusmodi molibus cen- trum gravitatis vicinius sit basi quàm vertici, si centrum incli- netur in alterutram partem spatio tantùm digitali, vertex in- signem acquiret inclinationem, consistet tamen, quandiu linea directionis transibit per basim sustentationis. Inclinatio enim non est spatium illud, quod inter basim, & perpendicularum à turris, vel obelisci vertice demissum intercipitur (quamvis hoc vocabulo hactenus abuti placuerit, ne à vulgo discreparem) sed est angulus, quem turris facit cum plano; & manente ea- dem inclinatione, intervallum illud mutari potest pro majore, aut minore turris longitudine. Quare quò longior est moles in- clinata, cæteris paribus, minùs est timendum, quia minor est declinatio à perpendiculari: si enim K E sit pedum 100, K C verò ped. 1. angulus K E C æqualis declinationi à perpendicularo est gr. 0. 34. 22". at si K E sit ped. 50, & K C iterum ped. 1. angulus K E C est grad. 11. 32'. 13". Hîc autem quasi præteriens satisfaciam quærenti, cur lon- giores hastas faciliùs, quàm breviores virgas digiti extremitate lustineamus, quin cadant. Quia nimirum minimus angulus declinationis à perpendicularo statim se prodit hastæ vertice ad partem unam sededente, cui statim occurrimus hastæ calcem manu transferentes, ac sub vertice collocantes: verùm quia fa- cilior hastæ consistentia innotescit etiam, quando à suppositâ manu calx ejus non movetur (nam si militarem sarissam terræ perpendiculariter insistentem constitueris, potest semel in gy- rum contorquere, & illam quasi perpendicularem recipere, id quod in breviore hastâ non obtinebis) alia est ratio petenda primum ex dictis, quia scilicet longior hasta, cæteris paribus, minùs declinat à perpendicularo, ideóque difficiliùs descendit; H

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Liber primus. CHAPTER IX. 57 By drawing the vertical line of direction SX, from the given side SR 59 ft. 72 in., and angle VRX 86° 35' 43", RX is found to be 3 ft. 54 in. Hence RX is greater than RB; and if that solitary wall were standing alone, it certainly would not remain upright; but since the other three walls are added, it is thus clear that the center of gravity of the whole mass is in the innermost part of the tower, so that the line of direction falls within the base supporting the tower. From this you may dispel the fear of those who are anxious about the stability of obelisks, foretelling a near ruin from some inclination of the summit: for in structures of this kind the center of gravity is closer to the base than to the top; so if the center inclines to either side by only a finger's breadth, the summit acquires a considerable inclination, yet it will stand so long as the line of direction passes through the base of support. For inclination is not that space which is intercepted between the base and the perpendicular let down from the top of the tower or obelisk (though I have hitherto chosen to use the word in that sense, so as not to differ from common usage), but it is the angle which the tower makes with the plane; and while that same inclination remains, that interval may vary according to the greater or lesser length of the tower. Therefore, the longer the inclined mass is, ceteris paribus, the less cause there is for fear, because the deviation from the perpendicular is smaller: for if KE be 100 feet and KC 1 foot, the angle KEC, equal to the deviation from the perpendicular, is 0° 34' 22"; but if KE be 50 feet and KC again 1 foot, the angle KEC is 11° 32' 13". Here, by the way, I shall briefly satisfy the question why we can more easily hold long spears than shorter rods at the tip of a finger, without letting them fall. The reason is that the smallest angle of deviation from the perpendicular immediately reveals itself when the spear's top begins to lean to one side, and we at once counter it by moving the spear's butt with the hand and placing it under the top; but the spear's stability is also shown when its butt, supported by the hand, does not move (for if you set a military sarissa standing perpendicularly on the ground, it can once be twisted round in a circle and, as it were, receive a perpendicular position again, which you will not achieve with a shorter spear). Another reason must first be sought from what has been said, namely, that a longer spear, ceteris paribus, deviates less from the perpendicular, and therefore descends more difficultly;

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58 Mechanicorum deinde quemadmodum longiorem hastam si in aquâ agitaveris majorem percipies resistentiam, quàm si breviorem virgam incitares; ita aërem variis semper motibus turbatum plus etiam impedire descensum longioris hastæ censendum est, præsertim si in superiore parte aër versùs unam, in inferiore autem versùs aliam partem moveatur: id quod in breviore virgâ non accidit, quam modicus aër contingit, nec potest aut adeò resistere divisioni, aut adeò diversis motibus cieri. Hinc asta longior tardiùs descensum molitur, & faciliùs sustinetur, quia major aëris dividendi quantitas, ac motus varus, magis resistit, & datâ æqualitate motûs minùs declinat à perpendiculo. CAPUT X. An plurium structurarum capax sit mons, quàm subjecta planities. Potest mons cum subjectâ planitie, cui insistit, dupliciter comparari; primùm conferendo solam planitiem in vertice montis existentem cum parte subjecti plani sibi respondente; deinde clivum montis comparando cum plano horizontali. Et sanè si planities in summo montis jugo consideretur, certum est illam esse plurium structurarum capacem, quàm subjectum planum in superficie globi terrestris: Quemadmodum enim superficies sphæræ majoris plura capit ædificia, quàm minor, ita etiam sphærarum inæqualium partes similes inæqualis sunt capacitatis: Constat autem planitiem in summo monte pertinere ad sphæram majorem, quàm pertineat similis planities illi subjecta; ac proinde & amplior est, & magis capax. Harum verò planitierum differentia ea erit, quæ est quadratorum distantiarum à centro terræ: quòd si quadratorum hujusmodi differentia exigua sit & contemnenda, eo quod ad illam quadratum semidiametri terræ habeat nimis magnam rationem; planitierum pariter differentia fugiet omnem sensum. Sit

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58 Mechanicorum then, just as if you agitate a longer spear in water you will perceive a greater resistance than if you stir a shorter rod; so too the air, when continually disturbed by various motions, must be considered to hinder even more the descent of the longer spear, especially if in the upper part the air moves toward one direction, and in the lower part toward another: which does not happen in the shorter rod, which only a moderate amount of air touches, nor can it either resist division to such a degree, or be stirred by such diverse motions. Hence the longer spear makes its descent more slowly, and is more easily supported, because a greater quantity of air must be divided, and its varied motion resists more strongly, and, given equal speed, it deviates less from the perpendicular. CAPUT X. Whether a mountain can contain more structures than the plain beneath it. A mountain may be compared with the plain beneath it in two ways; first, by comparing only the plain existing on the summit of the mountain with the corresponding part of the subjacent plain; secondly, by comparing the slope of the mountain with the horizontal plain. And indeed, if the plain at the top of the mountain is considered, it is certain that it is capable of more structures than the subjacent plain on the surface of the terrestrial globe: for just as the surface of a greater sphere contains more buildings than that of a smaller one, so also similar parts of unequal spheres are of unequal capacity: but it is clear that the plain on the summit of a mountain belongs to a greater sphere than does the similar plain beneath it; and therefore it is both wider and more capable. The difference between these plains will be that which exists between the squares of the distances from the center of the earth: but if the difference of such squares is slight and negligible, because the square of the earth’s semidiameter has in relation to it an excessively large ratio; then likewise the difference between the plains will escape all perception. Let it be

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Liber primus. CAPUT X. 59 Sit terræ semidiameter CS, altitudo autem montis SR, in cujus vertice sit planities RH, cui similis est in superficie globi terreni planities SO illi parallela: hæ autem planities similes habent, per 20. lib. 6. duplicatam Rationem laterum RI, SL, hoc est, per 4. lib. 6. duplicatam Rationis, quam habet CR ad CS. Est igitur ut quadratum distantiæ CR ad quadratum distantiæ CS, ita planities RH ad planitiem SO. Plura itaque ædificia perpendiculariter insistentia possunt in planitie RH majori excitari in montis vertice, quàm in subjectâ planitie. At si montis clivus RMOL comparetur cum subjectâ planitie SO, certum est illum esse majorem, sicuti latu R L oppositum angulo RSL, qui non est minor recto, majus est latere SL in triangulo RSL, & RM ad SF est ut RC ad SC: superficies igitur LM comprehensa sub majoribus lateribus, & angulis non minoribus, quàm superficies SO, major erit, si illa per se consideretur. Non tamen continuò major dicenda est capacitas, quæ plura aut ampliora recipiat ædificia; nisi mons ad ingentem altitudinem ascendat; tunc enim perpendiculara non sunt inter se parallela, propter insignem eorum distantiam. Nam si super clivo AB sit structura AL, cujus parietes perpendiculares, sint etiam paralleli LB, DA, illi non magis inter se distant, quàm si super plano horizontali NB fuissent excitati: quicquid sit, quod, sicut linea AB major est quàm NB, ita planum inclinatum majus sit plano horizontali. Non igitur plures aut ampliores structuras recipit clivus collis, quàm subjectum planum horizontale. Quod verò de structuris dicitur, de cæteris quoque intelligendum est, quæ perpendicularia insistunt, & spatium implent; at si ita se habeant, ut H 2

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Book One. CHAPTER X. 59 Let the semidiameter of the earth be CS, and the height of the mountain SR, on whose summit let there be a plane RH, to which, on the surface of the terrestrial globe, there is a plane SO parallel to it: these planes, being similar, have, by 20. lib. 6., the duplicate ratio of the sides RI, SL, that is, by 4. lib. 6., the duplicate ratio of the proportion which CR bears to CS. Therefore, as the square of the distance CR is to the square of the distance CS, so is the plane RH to the plane SO. Accordingly, more buildings standing perpendicularly may be erected on the larger plane RH on the summit of the mountain than on the underlying plain. But if the slope of the mountain RMOL be compared with the underlying plane SO, it is certain that the former is greater, since the side RL opposite to the angle RSL, which is not less than a right angle, is greater than the side SL in triangle RSL, and RM is to SF as RC is to SC: therefore the surface LM, enclosed under greater sides and angles no smaller than the surface SO, will be greater, if it be considered in itself. Yet it is not immediately to be said that the capacity is greater, which receives more or larger buildings; unless the mountain ascend to an enormous height; for then the perpendiculars are not parallel to one another, because of their great distance. For if on the slope AB there be a structure AL, whose walls are perpendicular, they are also parallel, LB, DA; these are no more distant from one another than if they had been erected on a horizontal plane NB: whatever the case may be, as line AB is greater than NB, so the inclined plane is greater than the horizontal plane. Therefore the slope of a hill does not receive more or larger structures than the horizontal plane beneath it. And what is said of structures is likewise to be understood of other things which stand perpendicularly and occupy space; but if they be such as to

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Mechanicorum 60 perpendicularia non insistant, certum est plures aut longiores homines jacere posse in clivo A B, quos non capit planum N B: vel si in clivo se minùs invicem impediant, tunc plura hujusmodi corpora in colle esse possunt quàm in planitie: si enim rami arboris inferioris respondeant trunco superioris, certum est quod multò viciniores esse possunt arbores, quàm in planitie, ubi rami se vicissim impedientes majorem postulant truncorum distantiam; ac proinde etiam multo plures arbores intra easdem parallelas erunt. Sic plures homines esse possunt in gradibus amphitheatri, quàm in subjecto plano, quia graciliores partes superiorum respondent crassioribus inferiorum, & se minùs invicem impedientes minus relinquunt spatij vacui: quod si non homines, sed parallelepipeda, statueres in gradibus, non plura statui in iis possent, quàm in planâ areâ gradibus subjectâ. Hæc autem ædificiorum æqualitas in clivo & in planitie, locum non habet nisi intra illud spatium, quod intercipitur à perpendicularis Physicè parallelis; statim enim ac à parallelismo recedunt perpendiculara, si ea fuerit altitudo, ad quam clivus ascendens venit, ut planities parallela plano horizontali in eâ altitudine major sit, quàm similis planities depressior, etiam plura ædificia recipiet clivus, quàm unica planities horizontalis subjecta. Ponamus enim perpendiculara GC, & OC jam non esse parallela, eamque esse altitudinem KG, ut planum per G transiens horizonti parallelum majus sit plano per O intra eadem perpendiculara intercepto, erit quidem capacitas plani inclinati GOLF æqualis capacitati subjecti plani EKOL: at ulteriùs ascendendo capacitas FGMR non erit æqualis capacitati plani SK continuati cum priore plano EO, federit major, quippe quæ æqualis est capacitati plani VG; est autem planum VG ad planum simile SK, ut quadratum GC ad quadratum KC: major igitur est totius clivi ML capacitas, quàm planitiei SO. Et ut res apertius constet, quandoquidem clivi altissimorum montium, si eandem servent inclinationem, non sunt ab imo pede ad summum jugum æquabili, & continuo ductu extensi, Sit terræ centrum H, & superficies AD;

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Mechanics 60 where the perpendiculars do not remain parallel, it is certain that more or longer men can lie on the slope A B than the level N B can contain: or if on the slope they hinder one another less, then more such bodies can be on the hill than on the plain; for if the branches of the lower tree correspond to the trunk of the upper tree, it is certain that the trees can be much closer together than on the plain, where the branches, mutually obstructing one another, require a greater distance between trunks; and therefore there will also be much more trees within the same parallels. Thus more men can be in the steps of an amphitheatre than on the level ground beneath, because the thinner parts of the upper correspond to the thicker parts of the lower, and by hindering one another less, leave less empty space: if, however, instead of men, you were to place parallelepipeds on the steps, no more could be placed in them than on the flat area lying beneath the steps. But this equality of buildings on a slope and on a plain holds only within that space which is intercepted by physically parallel perpendiculars; for as soon as perpendiculars depart from parallelism, if the height be such that to which the ascending slope comes, so that a plain parallel to the horizontal plane at that height is greater than the lower plain of the same kind, then the slope will also receive more buildings than the single horizontal plain beneath. For let the perpendiculars GC and OC now no longer be parallel, and let the height be KG, so that the plane passing through G parallel to the horizon is greater than the plane intercepted through O within the same perpendiculars, then indeed the capacity of the inclined plane GOLF will be equal to the capacity of the plane EKOL beneath: but by advancing farther upward the capacity FGMR will not be equal to the capacity of the plane SK continued with the earlier plane EO, but will be greater, since it is equal to the capacity of the plane VG; and the plane VG is to the plane SK as the square of GC to the square of KC: therefore the capacity of the whole slope ML is greater than that of the plain SO. And that the matter may be more clearly evident, since the slopes of the highest mountains, if they preserve the same inclination, are not extended from the foot at the bottom to the summit in an even and continuous line, let the center of the earth be H, and the surface AD;

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Liber primus. CAPUT X. 61 AD; cujus arcus dividatur in par- tes AB, BC, CD æquales, ita ut singuli arcus pro rectâ lineâ, & su- perficies pro plano horizontali Physicè usurpari possint; & tunc solùm intelligatur mutari horizon, quando ex A jam venerit in B, deinde in C &c. Si igitur sit pla- num inclinatum AE, ubi venerit in E punctum perpendiculi HB producti, non potest rectâ progre- di, quin mutet inclinationem supra horizontem novum, ad quem venit; quare ut servetur similis inclinatio, deflectit in EF, & est angulus HEF æqualis angulo HAE cui demum ubi ve- nerit in F, debet fieri æqualis angulus HEG. Centro autem H, intervallis HE & HF describantur arcus EI, & FK. Certum est duarum linearum angulum constituentium partem aliquam extremam esse, secundùm quam lineæ illæ non differunt, sensu judice, à parallelis; at si major pars accipiatur, jam perit paral- lelismus: Sic RA, & EB pro parallelis usurpari si possint, non poterunt similiter pro parallelis accipi RA, & LB: Sic LE, & FI sumuntur tanquam parallelæ citra errorem, at non item LB, & MC. Quare perpendiculara non solùm recedunt à parallelis- mo sensibili, quia majorem angulum in centro H constituunt, sed etiam quia major eorum pars assumitur, in qua jam apparet convergentia, quæ in parte minore latebat. Cum itaque structuræ perpendiculares in plano inclinato occupent spatium eodem modo, ac si essent in plano horizon- tali intra easdem parallelas, jam constat clivi partem EF com- parandam esse cum plano EI, non autem cum plano BC; quia in E, & I terminatur parallelismus linearum LE, FI. Est igi- tur capacitas clivi EF æqualis capacitati EI; at capacitas EI major est quàm capacitas BC, ergo capacitas clivi AF major est, quàm capacitas planitiei AC. Eademque esto de cæteris ratio. Hinc manifestum est non omninò in universum vera esse, quæ passim dicuntur de æquali capacitate collium, & planitiei subjectæ, nisi hæc certis limitibus circumscribantur; videlicet si sermo sit de iis quæ tantùm perpendiculariter insistunt, &c H 3

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Book One. CHAPTER X. 61 AD; whose arc let it be divided into the equal parts AB, BC, CD, so that the individual arcs may be used physically as a straight line, and the surfaces as a horizontal plane; and then let the horizon be understood to change only when it has already come from A to B, then to C, and so on. If therefore there be an inclined plane AE, where it shall have arrived at E, the point of the produced perpendicular HB, it cannot proceed straight on without changing its inclination above the new horizon to which it has come; wherefore, in order that a similar inclination may be preserved, it bends into EF, and the angle HEF is equal to the angle HAE, which at last, when it shall have arrived at F, must become equal to the angle HEG. With center H, and intervals HE and HF, let the arcs EI and FK be described. It is certain that in the angle formed by two lines there is some extreme part, according to which those lines, in the judgment of the senses, do not differ from parallels; but if a larger part be taken, parallelism is now lost: thus RA and EB can be used as parallels, but RA and LB cannot similarly be taken as parallels: thus LE and FI are taken as parallel, without error, but not LB and MC. Wherefore perpendiculars depart not only from sensible parallelism, because they form a larger angle at center H, but also because a larger part of them is assumed, in which convergence now appears, which lay hidden in the smaller part. Since therefore perpendicular structures on an inclined plane occupy space in the same way as if they were on a horizontal plane within the same parallels, it is now clear that the part EF of the slope must be compared with the plane EI, and not with the plane BC; because at E and I is terminated the parallelism of the lines LE, FI. Therefore the capacity of the slope EF is equal to the capacity of EI; but the capacity of EI is greater than the capacity of BC; therefore the capacity of the slope AF is greater than the capacity of the plain AC. And let the same reasoning apply to the rest. Hence it is manifest that what is commonly said about the equal capacity of hills and the plain beneath them is not altogether true in general, unless this be circumscribed within certain limits; namely, if the discussion be of those things which stand only perpendicularly, etc. H 3

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Mechanicorum intrà illud spatium, ac in eâ altitudine, ubi perpendiculorum convergentia adeò exigua est, ut evanescat. Cæterùm satis mihi videor ostendisse fieri posse, ut clivus aliquis plures structuras recipere possit, quàm superficies sphærica globi illi respondens. Si enim eadem est semper, ut supponitur, plani inclinatio, etiam latera turrium, vel domorum parietes æquè invicem remoti intercipient æquales partes plani inclinati: Si ergo structura interciens semissem plani A E transferatur in E F, æqualem partem intercipiet; at hæc minor est semisse ipsius E F, igitur duæ structuræ occupantes totum planum A E, translatæ in E F æquale spatium occupabunt, & relinquent adhuc partem spatij inanem. Esse autem E F lineam majorem linea A E patet; quia triangula A H E, E H F æquiangula sunt, & latera habent proportionalia, adeóque ut A H ad H E, ita A E ad E F; atqui H E excedit lineam H A; igitur & E F major est quàm A E: ergo multo major erit superficies ipsius E F, quàm superficies similis ipsius A E. In spatio igitur, quo superficies E F excedit superficiem A E, poterit alia præterea structura excitari. CAPUT XI. Quomodo animalium motus ordinentur ex centro gravitatis. DEi sapientiam nunquam satis admirari possumus, quæ in ordinandis naturæ motibus elucet; animalia enim solo naturæ ductu adeò accuratè se ipsa sistunt in lineâ directionis, ut nemo mathematicus Geometriæ apices perscrutatus possit tam subtiliter deprehendere, ac brevissimo temporis momento, centrum gravitatis. Quandoquidem sive consistentium quie- tem, sivè gradientium motum, sivè reclinantium se se inflexio- nem consideres, miram naturæ artem intelliges, quâ præcavit, ne corpus ingenitâ gravitate delatum præceps caderet. Id au- tem assecuta est motus ita disponendo, ut linea directionis nun- quam

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Mechanics within that space, and at that height, where the convergence of the perpendiculars is so slight that it vanishes. In any case, I seem to myself to have shown sufficiently that it is possible for some slope to receive more structures than the spherical surface corresponding to that globe. For if the inclination of the plane is always the same, as is assumed, then even the sides of towers, or the walls of houses, equally distant from one another, will enclose equal parts of the inclined plane. If therefore a structure intercepting half of plane A E is transferred to E F, it will intercept an equal part; but this is less than half of E F itself, therefore two structures occupying the whole plane A E, when moved to E F, will occupy an equal space, and will still leave part of the space empty. But that E F is a line greater than line A E is clear; because triangles A H E and E H F are equiangular, and have proportional sides, and thus as A H is to H E, so is A E to E F; but H E exceeds line H A; therefore E F also is greater than A E: therefore the surface of E F itself will be much greater than the surface similar to A E. In the space, then, in which the surface of E F exceeds the surface of A E, another structure may moreover be erected. CHAPTER XI. How the motions of animals are ordered from the center of gravity. We can never admire enough the wisdom of God, which shines forth in ordering the motions of nature; for animals, guided by nature alone, place themselves so accurately in the line of direction that no mathematician, however he has examined the heights of Geometry, could detect so subtly, even in the briefest moment of time, the center of gravity. For whether you consider the rest of those standing still, or the motion of those walking, or the bending of those reclining themselves, you will understand the marvelous art of nature, by which it has taken care that a body carried along by its innate gravity would not fall headlong. And it has achieved this by so arranging motion that the line of direction never

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Liber primus. CAPUT XI. 63 quam caderet extrà basim sustentationis, nisi fortè in cursu, in quo tamen satis consultum est animalis incolumitati, dum ab anteriore pede, ubi terram attigerit, retinetur, ne ulteriùs descendat. Basis autem sustentationis non sunt soli pedes, sed totum illud spatium interceptum à lineis pedum extremitates jungentibus; sic in quadrupedibus linea directionis debet cadere intrà spatium comprehensum lineis, quæ jungunt extrema pedum terram contingentium, ut possit animal consistere. Hinc equus in posteriores pedes se erigens flexis poplitibus reclinat se se in posteriora, & tantisper in eo situ consistit, dum centrum gravitatis imminet spatio, quod à pedibus occupatur, & ab illis intercipitur; & si extra illud spatium cadat linea directionis, vel aversus cadit, vel iterum quatuor pedibus insistit. Ubi tamen observandum est ex equo & equite fieri unam molem compositam unum habentem commune centrum gravitatis: unde fit equum magis defatigari, si eques non rectus insideat; sed inclinatus in alterutram partem, centro enim gravitatis translato motûs facilitas mutatur; & equite in anteriora inclinato ac premente caput equi in posteriores pedes erecti, centrum gravitatis in anteriora transfertur, & occurritur periculo, ne equus aversus cadat. Porrò dum spatium à pedibus occupatum voco basim sustentationis, non semper satis est lineam directionis cadere non extrà pedes; quia si pedes ipsi solùm ex parte tangant subjectum corpus, ut contingit in funambulis, debet linea directionis cadere in funem, cui insistunt pedes, & si extra illum cadat, certa est ruina, quia latitudo pedum non juvat. Cum autem difficillimum sit diutiùs consistere ita, ut centrum gravitatis semper immineat funi, ideò funambuli, vel hastam plumbeis laminis gravem in extremitatibus manu tenent, vel brachiis expansis se librant, ut hastam vel brachia extendentes in partem oppositam ei, in quam gravitas inclinat, centrum gravitatis constituatur in puncto, quod immineat funi sustentanti. Hinc oritur difficultas consistendi, quam experiuntur grallatores; cum enim grallæ exiguâ sui parte tangant terram, est quasi linea, in qua fit sustentatio, extra quam facilè cadit linea directionis: ideò tertium gestant baculum, cui innitantur,

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Book One. CHAPTER XI. 63 so that he would fall outside the base of support, unless perhaps in motion, in which case the animal’s safety is sufficiently provided for, since it is held by the forefoot, when it has touched the ground, lest it descend further. Now the base of support is not only the feet, but the whole space enclosed by the lines joining the extremities of the feet; thus in quadrupeds the line of direction ought to fall within the space enclosed by the lines which join the outermost points of the feet touching the ground, so that the animal may be able to stand. Hence a horse, when raising itself on its hind legs with bent hocks, leans backward and for a little while remains in that position, while the center of gravity hangs over the space occupied by the feet and enclosed by them; and if the line of direction falls outside that space, it either falls backward, or else stands again on all four feet. Yet here it must be observed that a horse and rider together form one composite mass, having a common center of gravity: whence it follows that the horse is more fatigued if the rider is not seated upright, but inclined to either side; for when the center of gravity is shifted, the ease of movement is altered; and when the rider leans forward and presses the horse’s head, the horse, with its hind legs raised, transfers the center of gravity forward, and the danger is averted that the horse may fall backward. Moreover, when I call the space occupied by the feet the base of support, it is not always enough that the line of direction should not fall outside the feet; because if the feet themselves only partly touch the surface beneath, as happens with tightrope walkers, the line of direction must fall on the rope on which the feet stand, and if it falls outside it, a fall is certain, because the width of the feet does not help. But since it is extremely difficult to remain standing for long in such a way that the center of gravity is always over the rope, tightrope walkers therefore either hold in their hands a staff weighted with lead plates at the ends, or balance themselves with outstretched arms, so that by extending the staff or the arms to the side opposite to that toward which gravity inclines, the center of gravity may be placed in the point over the supporting rope. Hence arises the difficulty of standing experienced by stilts-walkers; for since the stilts touch the ground only with a very small part of themselves, there is, as it were, a line on which support is made, beyond which the line of direction easily falls: therefore they carry a third staff, on which they lean,

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Mechanicorum 64 ininitantur, quoties quiescere voluerint, lineâ directionis cadente intrà spatium triangulare comprehensum à grallis, & baculo. Hîc autem maximè se prodit naturæ providentia in tam variâ pedum conformatione, ut ad sustentandum idonei essent: quadrupedibus siquidem non adeò amplos pedes tribuit, quia ex eorum inter se distantiâ plurimum spatium intercipitur, cui immineat centrum gravitatis: bipedibus verò latiores tribuit pedes, quâ parte timeri potuit casus: sic quia ex duorum crurum modicâ divaricatione non facilè periculum erat cadendi in alterutrum latus, ideò humanis pedibus minorem dedit latitudinem, quàm longitudinem; hanc verò non in æquas distribuit partes, sed minimam calci (præterquam in Scauris, quos pravis fultos male talis appellat Horatius, talis scilicet extantioribus) maximam anteriori parti concessit, ne impetu per motum concepto translatum centrum gravitatis in anteriora transiliret basim sustentationis. Aliquam tamen mediocrem latitudinem pedibus concessit, ut posset homo, si res ferret, uni tantùm pedi insistere, & esset aliqua spatij amplitudo, intrà quam quodlibet punctum opportunum esset consistentiæ centri gravitatis. Sic aves illæ, quæ uni pedi insistunt, cujusmodi sunt grues, & ciconiæ, digitos habens longiores, quos valdè explicant quasi in gyrum, ut amplior sit basis sustentationis; intrà quam ut cadat linea directionis, altero pede elevato inclinatur corpus in oppositam partem, ut centrum gravitatis immineat pedi sustentanti. Eandem ob causam anseres, & anates, quæ multâ carne abundant, & amplo sunt pectore, alternâ quadam, in dextrum, & sinistrum latus inclinatione gradiuntur, ideóque ampliores habent palmas, ut citrà cadendi periculum centrum gravitatis faciliùs vel immineat pedi sustentanti, vel minimùm ab eo declinet, ne majore, quàm par sit, impetu descendens corpus & anteriori pedi incumbens, tibiæ musculos, & tendines lædat. Aves verò, quæ subtilioribus ramusculis insident non palmipedes sunt, sed digitatæ (palmæ enim avibus amphibiiis ad natandum potissimum datæ videntur) ut ramis tenaciùs inhæreant; quæ præterquam quod exiguæ sunt gravitatis, facilè se sistunt in lineâ directionis, quæ cadat in ramusculum, cui insistunt, majore, vel minore angulo, quem faciunt

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Mechanics 64 are balanced, whenever they wish to rest, with the line of direction falling within the triangular space enclosed by the crutches and the staff. But here nature’s providence shows itself most of all in such a varied formation of the feet, so that they might be suitable for support: for she gave quadrupeds not so broad feet, because from the distance between them much space is taken up, over which the center of gravity would hang; but to bipeds she gave broader feet, in the part where a fall could be feared: thus, since from the slight spreading of the two legs there was not easily danger of falling to either side, she therefore gave human feet less breadth than length; and this she did not distribute into equal parts, but assigned the smallest part to the heel (except in the Scauri, whom Horace calls badly supported by crooked ankles, namely those with projecting ankles), and the greatest to the front part, lest, through the impetus acquired by motion, the shifted center of gravity should spring forward beyond the base of support. Yet she granted the feet a moderate breadth, so that a man, if the situation required it, could stand on only one foot, and there would be some width of space within which any point would be suitable for the center of gravity to rest. Thus those birds which stand on one foot, such as cranes and storks, having longer toes, spread them very widely, almost in a circle, so that the base of support may be broader; within it, so that the line of direction may fall, the body inclines to the opposite side when the other foot is raised, so that the center of gravity may hang over the supporting foot. For the same reason geese and ducks, which abound in much flesh and have a broad breast, walk with a certain alternating inclination to the right and left side; and therefore they have broader palms, so that, without danger of falling, the center of gravity may more easily either hang over the supporting foot, or at least deviate from it, lest the body, descending with greater force than is fitting and resting on the front foot, should injure the muscles and tendons of the shinbone. Birds, however, which perch on slenderer twigs are not web-footed, but toe-footed (for the feet of amphibious birds seem to have been given chiefly for swimming), so that they may cling more tenaciously to the branches; and besides being of little weight, they easily place themselves in the line of direction, which falls upon the twig on which they perch, at a greater or lesser angle, which they make

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Liber primus. CAPUT XI. 65 faciunt tibiæ cum coxâ; ideò ubi ramum arripuerint, subsul- tantes se librant, ramumque arctè apprehentes prohibent, ne repentino casu circumagantur à centro gravitatis nondum im- minente basi sustentationis. Verùm quoniam ad aves delapsus sum, prætereundus non est usus centri gravitatis involatu; quia enim avis dum alis aërem verberans in volatu se librat atque suspendit, ita alas debet extendere, ut centrum gravitatis existat intra illud alarum spatium, in quo exercetur sustentatio; ideò si vo- luerit ad superiora volatum dirigere, alas in anteriora ver- sus caput extendit, ut centro gravitatis in posterioribus re- licto, ac deorsum præponderante, caput sursum dirigatur: contra verò, ut motum deorsum dirigat, alas retrahit, ut caput præponderet, ac deorsum feratur. Hinc satis patet, cur ubi Pavo caudæ pompam explicuerit, erecto pectore & capite insistat pedibus, quibus immineat centrum gravita- tis: at si caput ad anteriora inclinare voluerit, & pectus inflectere, cogitur explicatam caudam demittere, ut syrma- te illo æquilibrium statuat corpori, ne proruat, ut verè pro- cumberet, si pectore inclinato expansa cauda retineretur in positione eâdem. Infinitum esset singulos animalium motus persequi, in qui- bus centri gravitatis ratio habetur; satis fuerit observasse nos ex declivi loco descendentes non insistere plantis pedum ad angulos rectos; sed paululum in posteriora inclinari; contra verò ascendentes jugum acclive curvari in anteriora; ut nimi- rum linea directionis cadat intrà spatium, cui pedes insistunt; extra quod illa si caderet, nec alteri fulcro inniteremur, quod unà cum pedibus includeret basim sustentationis, necessariò nobis cadendum esset. Quòd si quis onus habens dorso impo- situm in montosâ regione iter habeat, multò magis curvari de- bet, cum ascendit, ut pedibus immineat centrum gravitatis compositæ ex corpore, & ex onere: quare sapientissimè rustici aliqui in Alpibus, quæ Germaniam ab Italiâ disterminant, ar- culam ex levibus asserculis, & virgulis compactam habent, cui onera immittunt, basis autem arculæ, quæ gestantis corpori adhæret, imitatur Resc Hebraicum, ita ut pars quidem dor- so, pars autem capiti incumbat: unde fit, ut centrum gravita- I

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Book One. CHAPTER XI. 65 they make with the thigh; therefore, when they have seized a branch, they, by springing upward, balance themselves, and, grasping the branch firmly, prevent it from suddenly revolving them around the center of gravity, while the base of support is not yet lowered. But since I have fallen upon birds, the use of the center of gravity in flight must not be passed over; for since a bird, while beating the air with its wings in flight, balances and suspends itself, it must extend its wings in such a way that the center of gravity lies within that span of the wings in which support is exercised; therefore, if it wishes to direct its flight upward, it stretches its wings forward toward the head, so that, the center of gravity being left in the rear and weighing downward, the head may be directed upward: on the contrary, if it wishes to direct its motion downward, it draws back its wings, so that the head may preponderate and be carried downward. Hence it is sufficiently clear why, when the Peacock has displayed the pomp of its tail, it stands with breast and head raised upon its feet, on which the center of gravity lies: but if it wishes to incline its head forward and bend its breast, it is compelled to lower the spread tail, so that by that train it may establish equilibrium for the body, lest it should fall forward, as it would truly do if, the breast being inclined, the extended tail were kept in the same position. It would be endless to pursue the individual motions of animals in which account is taken of the center of gravity; it will suffice to have observed that, when we descend from a sloping place, we do not place the soles of the feet at right angles; but incline ourselves somewhat backward; whereas, when ascending a steep rise, we bend forward; namely, so that the line of direction may fall within the space on which the feet rest; for if it fell outside that space, and we did not lean upon another support, which together with the feet would enclose the base of support, we would necessarily have to fall. But if someone, carrying a burden on his back, travels in a mountainous region, he must bend forward much more when ascending, so that the center of gravity of the body together with the load may lie over the feet: wherefore some very wise peasants in the Alps, which separate Germany from Italy, have a little chest made of light planks and twigs, into which they place their burdens; and the base of the chest, which adheres to the carrier’s body, imitates the Hebrew Resc, so that one part rests on the back and another on the head: whence it follows that the center of gravita-

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68 Mechanicorum suerit, tribuendum censuissem, nisi Author ipse modicum illum excessum pedum sex cum dimidio redargueret. Quare contingere facilè potuit, ut ille, qui tunc Romæ degebat, ex aliquo manuscripto codice meam sententiam rescribens, ubi mensuram hanc pedibus definiebam, brevitatis ergo ad passus revocaverit, quam litera P notatam demùm pro pedibus sit interpretatus. Cæterùm prudens, & attentus lector me facillimè ab hoc errore vindicabit, si terræ ambitum mill.21600. dividat per mill.500; & quotientem 43 multiplicet per 15/17 unius pedis; deprehendet enim totum excessum pedum ferè 38, qui excedunt passus septem cum dimidio. Quod si ex diametro pedum 34400000, & ex diametro pedum 34400012, quas ibi Author ponit congruentes peripheriæ juxta Rationem 7 ad 22 considerentur, erit differentia circulorum pedum 38 eadem plane cum nostrâ; sed longissimè minor eâ, quam ille ibi statuit. Cæterùm quantus sit peripheriæ majoris excessus supra minorem, habebitur facillimè, si majoris Radij T F, excessum B F, statuas tanquam circuli Radium; hujus namque circuli peripheria est æqualis excessui illi. Quia enim ut minor Radius TB ad majorem Radium T F, ita minor peripheria ad majorem peripheriam, etiam convertendo & dividendo, ut TB ad B F, ita minor peripheria ad excessum peripheriæ majoris, & vicissim permutando ut Radius TB minor ad suam minorem peripheriam, ita B F excessus Radij majoris ad excessum majoris peripheriæ. Atqui excessus hic B F assumptus ut Radius circuli habet ad suam peripheriam eandem Rationem, quam TB Radius minor ad suam peripheriam; igitur est eadem Ratio B F excessûs Radij, ad excessum peripheriæ majoris, quæ est ejusdem B F ut Radij ad suam peripheriam: ergo per 9. lib. 5. hæc peripheria æqualis est illi excessui peripheriæ majoris. Cum itaque Ratio diametri ad peripheriam sit ut 7 ad 22, seu ut 113 ad 355, fiat ut Radius 7 ad peripheriam 44, seu ut 113 ad 710, ita B F altitudo ped. 6. ad ped. 37. unc.8: qui numerus consentit cùm superiore. CAPUT

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68 Mechanicorum I would have thought it should be attributed to the author’s excess of six and a half feet, unless the author himself disproved that slight discrepancy. Therefore it could easily have happened that someone then living in Rome, copying my opinion from some manuscript codex, where I defined this measure in feet, for the sake of brevity converted it to paces, which he finally interpreted as feet, because it was marked with the letter P. However, a prudent and attentive reader will very easily free me from this error, if he divide the earth’s circumference 21,600 miles by 500; and multiply the quotient 43 by 15/17 of a foot; for he will find the whole excess to be nearly 38 feet, which exceeds seven and a half paces. But if the diameters of 34,400,000 feet and of 34,400,012 feet, which the author there places as corresponding to the circumference according to the ratio 7 to 22, are considered, the difference of the circles will be 38 feet, exactly the same as ours; but far smaller than that which he there sets down. Moreover, how great the excess of the larger circumference over the smaller may be, will be obtained very easily, if you take the excess B F of the larger radius T F as though it were the radius of a circle; for the circumference of this circle is equal to that excess. For since, as the smaller radius T B is to the larger radius T F, so is the smaller circumference to the larger circumference, then also by converting and dividing, as T B is to B F, so is the smaller circumference to the excess of the larger circumference; and, reversing the terms, as the smaller radius T B is to its smaller circumference, so is the excess B F of the larger radius to the excess of the larger circumference. But this excess B F, taken as the radius of a circle, has the same ratio to its circumference as the smaller radius T B to its circumference; therefore the ratio of the excess B F of the radius to the excess of the larger circumference is the same as that of the same B F, as radius, to its circumference: hence, by 9. lib. 5., this circumference is equal to that excess of the larger circumference. Since therefore the ratio of diameter to circumference is as 7 to 22, or as 113 to 355, let it be as radius 7 is to circumference 44, or as 113 is to 710, so is B F, a height of 6 feet, to 37 feet 8 inches: which number agrees with the above. CAPUT

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Liber primus. CAPUT XI. 65 faciunt tibiæ cum coxâ; ideò ubi ramum arripuerint, subsul- tantes se librant, ramumque arctè apprehentes prohibent, ne repentino casu circumagantur à centro gravitatis nondum im- minente basi sustentationis. Verùm quoniam ad aves delapsus sum, prætereundus non est usus centri gravitatis involatu; quia enim avis dum alis aërem verberans in volatu se librat atque suspendit, ita alas debet extendere, ut centrum gravitatis existat intra illud alarum spatium, in quo exercetur sustentatio; ideò si vo- luerit ad superiora volatum dirigere, alas in anteriora ver- sus caput extendit, ut centro gravitatis in posterioribus re- licto, ac deorsum præponderante, caput sursum dirigatur: contra verò, ut motum deorsum dirigat, alas retrahit, ut caput præponderet, ac deorsum feratur. Hinc satis patet, cur ubi Pavo caudæ pompam explicuerit, erecto pectore & capite insistat pedibus, quibus immineat centrum gravita- tis: at si caput ad anteriora inclinare voluerit, & pectus inflectere, cogitur explicatam caudam demittere, ut lyrma- te illo æquilibrium statuat corpori, ne proruat, ut verè pro- cumberet, si pectore inclinato expansa cauda retineretur in positione eâdem. Infinitum esset singulos animalium motus persequi, in qui- bus centri gravitatis ratio habetur; satis fuerit observasse nos ex declivi loco descendentes non insistere plantis pedum ad angulos rectos; sed paululum in posteriora inclinari; contra verò ascendentes jugum acclive curvari in anteriora; ut nimi- rum linea directionis cadat intrà spatium, cui pedes insistunt; extra quod illa si caderet, nec alteri fulcro inniteremur, quod unà cum pedibus includeret basim sustentationis, necessariò nobis cadendum esset. Quòd si quis onus habens dorso impo- situm in montosâ regione iter habeat, multò magis curvari de- bet, cum ascendit, ut pedibus immineat centrum gravitatis compositæ ex corpore, & ex onere: quare sapientissimè rustici aliqui in Alpibus, quæ Germaniam ab Italia disterminant, ar- culam ex levibus asserculis, & virgulis compactam habent, cui onera immittunt, basis autem arculæ, quæ gestantis corpori adhæret, imitatur Resc Hebraicum, ita ut pars quidem dor- so, pars autem capiti incumbat: unde fit, ut centrum gravita- I

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Book One. Chapter XI. 65 They are made by the leg with the thigh; therefore, when they have seized a branch, springing up they balance themselves, and, grasping the branch tightly, they keep themselves from being suddenly turned round from the center of gravity, since the base of support is not yet beneath them. But since I have come to birds, the use of the center of gravity in flight must not be passed over; for since a bird, while beating the air with its wings in flight, balances and suspends itself, it must so extend its wings that the center of gravity lies within that span of the wings in which support is exercised; therefore, if it wishes to direct its flight upward, it extends its wings forward toward its head, so that, with the center of gravity left behind in the rear and weighing downward, the head is directed upward: on the other hand, in order to direct its motion downward, it draws back its wings, so that the head may outweigh and be carried downward. Hence it is sufficiently clear why, when the Peacock has spread the display of its tail, it stands with breast and head erect on its feet, beneath which the center of gravity lies: but if it wishes to incline its head forward and bend its breast, it is forced to lower its spread tail, so that by that rein it may establish equilibrium for the body, lest it topple forward, as it truly would lean down if, with the breast inclined, the spread tail were held in the same position. It would be endless to pursue the individual movements of animals in which account is taken of the center of gravity; it will be enough to observe that, when we descend from a sloping place, we do not place the soles of our feet at right angles, but incline ourselves somewhat backward; while, on the contrary, those ascending a steep slope bend forward, so that the line of direction may fall within the space on which the feet stand; for if it were to fall outside that space, and we had no other support that, together with the feet, enclosed the base of sustentation, we would necessarily have to fall. And if anyone, having a load placed on his back, travels in a mountainous region, he must bend himself still more when ascending, so that the center of gravity of body and load may lie over the feet: wherefore certain very wise peasants in the Alps, which separate Germany from Italy, have a little frame made of light planks and twigs, into which they place their burdens; and the base of the frame, which adheres to the bearer’s body, imitates the Hebrew Resc, so that one part rests on the back and another on the head: whence it comes about that the center of gravity of the body,

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Mechanicorum 68 suerit, tribuendum censuissem, nisi Author ipse modicum illum excessum pedum sex cum dimidio redargueret. Quare contingere facilè potuit, ut ille, qui tunc Romæ degebat, ex aliquo manuscripto codice meam sententiam rescribens, ubi mensuram hanc pedibus definiebam, brevitatis ergo ad passus revocaverit, quam litera P notatam demùm pro pedibus sit interpretatus. Cæterùm prudens, & attentus lector me facillimè ab hoc errore vindicabit, si terræ ambitum mill.21600. dividat per mill.500; & quotientem 43 multiplicet per 15/17 unius pedis; deprehendet enim totum excessum pedum ferè 38, qui excedunt passus septem cum dimidio. Quod si ex diametro pedum 34400000, & ex diametro pedum 34400012, quas ibi Author ponit congruentes peripheriæ juxta Rationem 7 ad 22 considerentur, erit differentia circulorum pedum 38 eadem plane cum nostrâ; sed longissimè minor eâ, quam ille ibi statuit. Cæterùm quantus sit peripheriæ majoris excessus supra minorem, habebitur facillimè, si majoris Radij T F, excessum B F, statuas tanquam circuli Radium; hujus namque circuli peripheria est æqualis excessui illi. Quia enim ut minor Radius TB ad majorem Radium T F, ita minor peripheria ad majorem peripheriam, etiam convertendo & dividendo, ut TB ad B F, ita minor peripheria ad excessum peripheriæ majoris, & vicissim permutando ut Radius TB minor ad suam minorem peripheriam, ita B F excessus Radij majoris ad excessum majoris peripheriæ. Atqui excessus hic B F assumptus ut Radius circuli habet ad suam peripheriam eandem Rationem, quam TB Radius minor ad suam peripheriam; igitur est eadem Ratio B F excessûs Radij, ad excessum peripheriæ majoris, quæ est ejusdem B F ut Radij ad suam peripheriam: ergo per 9. lib. 5. hæc peripheria æqualis est illi excessui peripheriæ majoris. Cum itaque Ratio diametri ad peripheriam sit ut 7 ad 22, seu ut 113 ad 355, fiat ut Radius 7 ad peripheriam 44, seu ut 113 ad 710, ita B F altitudo ped. 6. ad ped. 37. unc.8: qui numerus consentit cùm superiore. CAPUT

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Mechanics 68 would have been to be attributed, had not the Author himself refuted that slight excess of six and a half feet. Wherefore it could easily happen that the one who was then living at Rome, copying my opinion from some manuscript codex, where I defined this measure by feet, for the sake of brevity converted it into paces, which, being marked with the letter P, he finally interpreted as feet. However, a prudent and attentive reader will very easily rescue me from this error, if he divide the circumference of the earth, 21600 miles, by 500; and multiply the quotient, 43, by 15/17 of a foot; for he will find the whole excess to be nearly 38 feet, which exceeds seven and a half paces. But if the diameters of 34,400,000 feet and 34,400,012 feet, which the Author there presents as corresponding to the circumference according to the ratio 7 to 22, are considered, the difference of the circles will be 38 feet, exactly the same as ours; but far less than that which he there establishes. Moreover, how great the excess of the larger circumference above the smaller is, will be obtained very easily if you take the excess B F of the larger radius T F as though it were the radius of a circle; for the circumference of this circle is equal to that excess. For since, as the smaller radius T B is to the larger radius T F, so the smaller circumference is to the larger circumference, then also, converting and dividing, as T B is to B F, so the smaller circumference is to the excess of the larger circumference; and, by a reciprocal transposition, as the smaller radius T B is to its smaller circumference, so B F, the excess of the larger radius, is to the excess of the larger circumference. But this excess B F, taken as the radius of a circle, bears to its circumference the same ratio as the smaller radius T B to its circumference; therefore the ratio of the excess B F of the radius to the excess of the larger circumference is the same as that of B F itself, as radius, to its circumference; therefore, by Book 5, proposition 9, this circumference is equal to that excess of the larger circumference. Since therefore the ratio of diameter to circumference is as 7 to 22, or as 113 to 355, let it be as radius 7 is to circumference 44, or as 113 to 710, so is B F, a height of 6 feet, to 37 feet 8 inches: which number agrees with the above. CHAPTER

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CAPUT XII. An tellus moveatur motu trepidationis. Quoniam centrum gravitatis est in quolibet corpore punctum illud, quod æquales gravitates circumstant, manifestum est non permanere idem gravitatis centrum, si aliqua corpori additio fiat, aut detractio; neque enim manet eadem momentorum gravitatis æqualitas circa illud punctum; sed aliud est punctum, per quod ducta plana dividunt totius corporis gravitatem in momenta æqualia, & est novum cen- trum gravitatis. Hinc patet in telluris globo, qui plurimas mutationes subit, corporibus gravibus ex alio in alium locum translatis, tolli æqualitatem partium saltem in actu primo gra- vitantium, cum hæc quidem, quæ oppositæ parti ante erat æqualis, subtractione nunc fiat minor, illa verò, quæ pariter sibi oppositæ parti proximè fuit æqualis, additione evadat ma- jor. Ex quo necessariò colligitur mutatio centri gravitatis. Sed quia, ut tellus suis librata ponderibus in loco sibi debi- to consisteret, debuit initio ejus centrum gravitatis congrue- re centro universi, circa quod gravia & levia disponuntur; id- circò dubitari potest, utrum mutato gravitatis centro terra mo- veri debeat, ut novum gravitatis centrum collocetur in centro universi. Quoniam verò huc illuc passim translatis corpori- bus, terra nunc in hanc, nunc in illam partem moveretur, ut proinde quasi trepidaret; hinc factus est quæstioni locus, an tellus moveatur motu trepidationis; quicquid sit an motus iste sub sensum cadat, nec ne. Terram universam & singulas ejus partes suâ gravitate re- pugnare, ne sursum moveantur, certum est; at universi cen- trum occupare, toti quidem elemento gravissimo convenit, sed non partibus singulis: neque enim gravitas est appetitus sub- sistendi in centro, quem natura non satis aptè gravibus singu- lis indidisset; cui nimirùm fieri satis non potest, nisi corpora se invicem penetrent; unum autem grave in centro existens I 3

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CHAPTER XII. Whether the earth is moved by a motion of trepidation. Since the center of gravity in any body is that point which is surrounded by equal weights, it is evident that the center of gravity does not remain the same if any addition to the body is made, or any subtraction; for the equality of the moments of gravity around that point does not remain the same. But there is another point through which planes drawn divide the gravity of the whole body into equal moments, and this is a new center of gravity. Hence it is clear that in the globe of the earth, which undergoes many changes when heavy bodies are transferred from one place to another, the equality of the parts at least in the first act of gravitating is taken away, since that which was formerly equal to the opposite part becomes now smaller by subtraction, and that which was equally near to its opposite part becomes greater by addition. From this it is necessarily inferred that the center of gravity changes. But since, in order that the earth, balanced by its own weights, might stand in the place due to it, its center of gravity at the beginning ought to have corresponded with the center of the universe, around which heavy and light things are disposed, therefore a doubt may arise whether, when the center of gravity is changed, the earth ought to be moved so that the new center of gravity may be placed at the center of the universe. But since, when bodies are transferred hither and thither at random, the earth would now be moved toward this part, now toward that, so that it would seem as if it were trembling, hence a question arose whether the earth is moved by a motion of trepidation; whether this motion falls under the senses or not. That the whole earth and each of its parts resist by their own gravity, lest they move upward, is certain; but to occupy the center of the universe indeed suits the whole heaviest element, but not each individual part. For gravity is not a desire to remain in the center, which nature would not have suitably implanted in single heavy bodies; and this, namely, cannot be satisfied unless bodies penetrate one another; but one heavy body existing in the center

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72 Mechanicorum vior vicinior centro, conetur deorsum; certum est illum de- scendere non posse, quin totam reliquam terram impellat, ejus- que resistentiam superet; resistit autem primò segmentum IDEL, cujus omnes partes magis à centro removerentur; ni- si igitur mons FHG major sit segmento sphærico I DEL (vel saltem non multò minor, si quidem ob majorem à centro distantiam augerentur momenta gravitatis, ex dictis cap. 4.) non poterit subjectam terram loco dimovere. Prætereà etiam hemisphærium I AL repugnat descensui montis FHG, quia fieri non potest hic motus, nisi hemisphærij partes transiliant planum IL, atque magis à centro recedant. Quantâ igitur gravitate præditum esse montem oporteret, qui tantam re- sistentiam superare valeret? At nunquam fieri tantam partium permutationem, ut id quod transfertur, sit non minus semisse hemisphærij, ut saltem ratione habitâ distantiæ à centro pos- sit prævalere, ita omnibus est manifestum, ut probatione non indigeat. Quare neque hanc gravium translationem motus ul- lus consequitur, quo tellus trepidare dicatur. At, inquis, si in utrâque libræ lance sint unciæ 100, & al- terutri uncia una addatur, lanx illa deprimitur, & opposita elevatur; ergo exiguum pondus vim habet movendi ingens pondus; ergo pariter mons FHG producere potest impetum, qui ad movendum segmentum IDEL, quantumvis gravius, abundè sufficiat. Ego vero nego consequentiam; quia non ab unciâ illâ additâ solâ elevatur oppositum pondus, sed omnes unciæ simul in medio leviore suspensæ collatis viribus deorsum conantur, atque præponderantes oppositæ lancis pondus at- tollunt. Hoc autem nil in rem nostram facit, ubi neque mons FHG solitariè sumptus potest sursum propellere molem IDEL majorem se, neque juvari potest ab hemisphærio I AL, quod cum nihil infrà se habeat, quod & levius sit, & inter ipsum ac universi centrum intercipiatur, neque potest se ipsum versùs centrum urgere secundùm aliquas sui partes ab eo remo- tiores, cum maximè partes centro proximæ valde reluctantur, ne ab illo removeantur. Id quod in libræ lance, cui uncia fue- rit addita, reperire non poteris; totum siquidem lancis pon- dus deorsum nititur. Quod si ex librâ similitudinem ducere placeat, petenda po- tiùs

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72 Mechanics closer to the center, let it attempt to move downward; it is certain that it cannot descend without pushing the entire rest of the earth and overcoming its resistance; but first the segment IDEL resists, all of whose parts would be farther removed from the center; unless, therefore, the mountain FHG is larger than the spherical segment I DEL (or at least not much smaller, since indeed, because of the greater distance from the center, the moments of gravity would be increased, as was said in chap. 4.), it will not be able to displace the earth beneath it. Moreover, the hemisphere I AL also opposes the descent of the mountain FHG, because this motion cannot take place unless the parts of the hemisphere pass over the plane IL and recede farther from the center. With what weight, then, must the mountain be endowed that it might be able to overcome so great a resistance? But such a shifting of parts can never occur, so that what is transferred is not less than half of a hemisphere, so as at least, when account is taken of the distance from the center, to be able to prevail; this is so evident to all that it needs no proof. Wherefore no motion follows from this transfer of heavy bodies by which the earth is said to tremble. But, you say, if there are 100 ounces on each pan of a balance, and one ounce is added to either pan, that pan is depressed and the opposite one is raised; therefore a tiny weight has the power of moving a huge weight; therefore likewise the mountain FHG can produce an impulse which would abundantly suffice to move the segment IDEL, however much heavier it is. Yet I deny the inference; because it is not by that ounce added alone that the opposite weight is raised, but all the ounces, together suspended in the lighter middle, strive downward with their combined forces and lift up the heavier opposing pan. But this has nothing to do with our case, where neither the mountain FHG, taken by itself, can drive upward the mass IDEL greater than itself, nor can it be aided by the hemisphere I AL, which, since it has nothing beneath it that is both lighter and interposed between itself and the center of the universe, cannot press itself toward the center through any of its parts farther from it, since the parts nearest the center resist very strongly, so as not to be removed from it. You will not find the same thing in a balance to whose pan an ounce has been added; for the whole weight of the balance tends downward. And if it pleases you to draw the comparison from a balance, rather must it be sought

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Liber primus. CAPUT XII. 69 CAPUT XII. An tellus moveatur motu trepidationis. Quoniam centrum gravitatis est in quolibet corpore punctum illud, quod æquales gravitates circumstant, manifestum est non permanere idem gravitatis centrum, si aliqua corpori additio fiat, aut detractio; neque enim manet eadem momentorum gravitatis æqualitas circa illud punctum; sed aliud est punctum, per quod ducta plana dividunt totius corporis gravitatem in momenta æqualia, & est novum cen- trum gravitatis. Hinc patet in telluris globo, qui plurimas mutationes subit, corporibus gravibus ex alio in alium locum translatis, tolli æqualitatem partium saltem in actu primo gra- vitantium, cum hæc quidem, quæ oppositæ parti ante erat æqualis, subtractione nunc fiat minor, illa verò, quæ pariter sibi oppositæ parti proximè fuit æqualis, additione evadat ma- jor. Ex quo necessariò colligitur mutatio centri gravitatis. Sed quia, ut tellus suis librata ponderibus in loco sibi debi- to consisteret, debuit initio ejus centrum gravitatis congrue- re centro universi, circa quod gravia & levia disponuntur; id- circò dubitari potest, utrum mutato gravitatis centro terra mo- veri debeat, ut novum gravitatis centrum collocetur in centro universi. Quoniam verò huc illuc passim translatis corpori- bus, terra nunc in hanc, nunc in illam partem moveretur, ut proinde quasi trepidaret; hinc factus est quæstioni locus, an tellus moveatur motu trepidationis; quicquid sit an motus iste sub sensum cadat, nec ne. Terram universam & singulas ejus partes suâ gravitate re- pugnare, ne sursum moveantur, certum est; at universi cen- trum occupare, toti quidem elemento gravissimo convenit, sed non partibus singulis: neque enim gravitas est appetitus sub- sistendi in centro, quem natura non satis aptè gravibus singu- lis indidisset; cui nimirùm fieri satis non potest, nisi corpora se invicem penetrent; unum autem grave in centro existens I 3

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Book One. CHAPTER XII. 69 CHAPTER XII. Whether the earth is moved by a motion of trembling. Since the center of gravity in any body is that point which is surrounded by equal weights, it is evident that the same center of gravity does not remain if any addition to the body is made, or anything taken away; for the equality of the moments of gravity about that point does not remain the same; but there is another point through which planes drawn divide the gravity of the whole body into equal moments, and this is a new center of gravity. Hence it is clear that in the globe of the earth, which undergoes very many changes, with heavy bodies transferred from one place to another, the equality of the parts, at least in respect of those gravitating in the first act, is destroyed, since that part which before was equal to the opposite part is now made smaller by subtraction, while that which was likewise nearest equal to its opposite part becomes greater by addition. From this a change of the center of gravity must necessarily be inferred. But because, for the earth, balanced by its own weights, to remain in the place due to it, its center of gravity ought at the beginning to have agreed with the center of the universe, around which heavy and light things are disposed, therefore it may be doubted whether, when the center of gravity has changed, the earth ought to be moved so that the new center of gravity may be placed in the center of the universe. But because, with bodies transferred here and there in various places, the earth would now be moved in this direction, now in that, so that it would seem, as it were, to tremble, from this there arose the question whether the earth is moved by a motion of trembling; whether or not this motion falls under the senses, and so on. That the whole earth and each of its parts resist by their own weight, so as not to be moved upward, is certain; but to occupy the center of the universe is indeed fitting for the heaviest element as a whole, but not for its individual parts: for gravity is not a tendency to remain in the center, which nature would not have suitably bestowed on single heavy things; and this cannot be fulfilled unless bodies penetrate one another; but one heavy body existing in the center...

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72 Mechanicorum vior vicinior centro, conetur deorsum; certum est illum de- scendere non posse, quin totam reliquam terram impellat, ejus- que resistentiam superet; resistit autem primò segmentum IDEL, cujus omnes partes magis à centro removerentur; ni- si igitur mons FHG major sit segmento sphærico I DEL (vel saltem non multò minor, si quidem ob majorem à centro distantiam augerentur momenta gravitatis, ex dictis cap. 4.) non poterit subjectam terram loco dimovere. Præterèa etiam hemisphærium I AL repugnat descensui montis FHG, quia fieri non potest hic motus, nisi hemisphærij partes transiliant planum IL, atque magis à centro recedant. Quantâ igitur gravitate præditum esse montem oporteret, qui tantam re- sistentiam superare valeret? At nunquam fieri tantam partium permutationem, ut id quod transfertur, sit non minus semisse hemisphærij, ut saltem ratione habitâ distantiæ à centro pos- sit prævalere, ita omnibus est manifestum, ut probatione non indigeat. Quare neque hanc gravium translationem motus ul- lus consequitur, quo tellus trepidare dicatur. At, inquis, si in utrâque libræ lance sint unciæ 100, & al- terutri uncia una addatur, lanx illa deprimitur, & opposita elevatur; ergo exiguum pondus vim habet movendi ingens pondus; ergo pariter mons FHG producere potest impetum, qui ad movendum segmentum I DEL, quantumvis gravius, abundè sufficiat. Ego vero nego consequentiam; quia non ab unciâ illâ additâ solâ elevatur oppositum pondus, sed omnes unciæ simul in medio leviore suspensæ collatis viribus deorsum conantur, atque præponderantes oppositæ lancis pondus at- tollunt. Hoc autem nil in rem nostram facit, ubi neque mons FHG solitariè sumptus potest sursum propellere molem IDEL majorem se, neque juvari potest ab hemisphærio I AL, quod cum nihil infrà se habeat, quod & levius sit, & inter ipsum ac universi centrum intercipiatur, neque potest se ipsum versùs centrum urgere secundùm aliquas sui partes ab eo remo- tiores, cum maximè partes centro proximæ valde reluctantur, ne ab illo removeantur. Id quod in libræ lance, cui uncia fue- rit addita, reperire non poteris; totum siquidem lancis pon- dus deorsum nititur. Quod si ex librâ similitudinem ducere placeat, petenda po- tiùs

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72 Mechanics being nearer the center, let it try to move downward; it is certain that it cannot descend without pushing the whole remaining earth and overcoming its resistance; but first the segment IDEL resists, all whose parts would be moved farther from the center; unless therefore the mountain FHG be greater than the spherical segment IDEL (or at least not much smaller, since, because of the greater distance from the center, the moments of gravity would be increased, as was said in ch. 4), it will not be able to displace the earth lying beneath it. Moreover, the hemisphere IAL also opposes the descent of the mountain FHG, because this motion cannot occur unless the parts of the hemisphere pass across the plane IL and move farther away from the center. With what gravity, then, must a mountain be endowed that it may be able to overcome so great a resistance? But never can so great a transposition of parts take place that what is transferred is not less than half the hemisphere, so that, at least when regard is had to the distance from the center, it may prevail; this is so evident to everyone that it needs no proof. Therefore no motion follows from this transposition of heavy bodies, by which the earth is said to tremble. But, you say, if there be 100 ounces on each pan of a balance, and one ounce be added to either side, that pan is depressed and the opposite one is raised; therefore a small weight has the power of moving a great weight; therefore likewise the mountain FHG can produce an impulse which would abundantly suffice to move the segment IDEL, however much heavier it may be. But I deny the consequence; because it is not by that ounce added alone that the opposite weight is raised, but all the ounces, suspended together in the lighter middle, strive downward with their combined force, and lift up the weight of the opposite pan, which is the heavier. But this has nothing to do with our case, where neither the mountain FHG taken by itself can drive upward a mass IDEL greater than itself, nor can it be aided by the hemisphere IAL, which, since it has nothing beneath it that is both lighter and intercepted between it and the center of the universe, cannot urge itself toward the center by any of its parts farther removed from it, since the parts nearest the center most strongly resist being removed from it. This you will not find in the pan of a balance, to which an ounce has been added; for the whole weight of the pan strives downward. And if you prefer to draw the comparison from a balance, rather

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Liber primus. CAPUT XII. 73 tiùs est ex librâ, cujus lanx altera subjecto plano incumbat, al- tera in aëre libera pendeat; si enim utraque lanx plena æquali- bus ponderibus consistat in æquilibrio, & incumbenti lanci ad- datur ponderis pars, quæ à pendulâ lance detrahatur, lances non moventur, nec inter se mutuò consigunt ponderum gra- vitates, nisi quatenùs lanx gravior semper magis resistit leviori, ne ab illâ elevetur: cæterùm gravior lanx non movet leviorem, nisi ubi demum tanto pondere prægravata fuerit, ut subjecti plani resistentiam vincens illud aut frangat, aut saltem depri- mat. Sic hemisphærium I A L habet rationem lancis non tan- tùm subjecto plano incumbentis, sed, quod potius est, suo in loco quiescentis; cui quò plus addideris ponderis, auges qui- dem resistentiam ne sursum versùs H propellatur, ipsum verò non conatur deorsum versùs C; sed totus conatus imposito & adjecto monti tribuendus esset, vel (ut sim maximè liberalis) etiam excessui illi, quo hemisphærium I A L superat segmen- tum sphæricum I D E L, qui excessus est æqualis ipsi monti, hoc est segmento D E B. Quare si fuerit abscissa tertia pars hemisphærij unius, & addatur alteri hemisphærio è regione se- cundùm diametrum, tunc ad summum æqualis erit pars terræ deorsum nitens F M G H parti oppositæ repugnanti I D E L; & si velis partem F M G H remotiorem à centro magis gravitare ita, ut ratio hujus excessûs in gravitando possit vincere non so- lùm resistentiam segmenti I D E L, ne sursum propellatur, sed etiam segmenti F I L G, ne secundùm partes I L centro proxi- mas ab eo removeatur; non admodum repugnabo. Sed cum nunquam millesima, ne dum sexta, pars terreni globi ex alio in alium locum ex diametro oppositum transferatur, nulla un- quam sit gravium permutatio, vi cujus tellus trepidet. Sed unum adhuc superest, quod per dissimulantiam præ- tereundum non videtur. Esto inquis, nulla fiat in tellure gra- vium translatio, quæ tanta sit, ut novum gravitatis centrum in universi centro constituere valeat, ac proinde nulla sit centri terræ trepidatio: circa centrum saltem nutabit tellus motu conversionis, validâ ventorum vi summos montes impellente, orbemque totum, pro variâ ipsorum incursione, modò hanc, modò illam partem versante: unde fortasse ortam acû magne- ticæ eodem in loco post aliquot annos variationem suspicari K

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Book One. CHAPTER XII. 73 It is more like a balance, one scale of which rests on the underlying plane, while the other hangs free in the air; for if both scales, loaded with equal weights, are in equilibrium, and to the scale resting on the plane there is added a part of the weight which is taken away from the hanging scale, the scales do not move, nor do the weights mutually influence one another, except insofar as the heavier scale always resists the lighter more, lest it be raised by it. Moreover, the heavier scale does not move the lighter unless at last it has been loaded with so much weight that, overcoming the resistance of the underlying plane, it either breaks it or at least presses it down. Thus the hemisphere I A L has the character not only of a scale resting upon the underlying plane, but, what is more to the point, of one at rest in its place; and the more weight you add to it, the more indeed you increase the resistance lest it be driven upward toward H, but you do not thereby cause it to move downward toward C; rather, the whole effort ought to be attributed to the mound placed upon and added to it, or, if I am most liberal, even to that excess by which the hemisphere I A L exceeds the spherical segment I D E L, an excess equal to the mound itself, that is, to the segment D E B. Wherefore, if one-third part of one hemisphere were cut off and added to the other hemisphere opposite it along the diameter, then at most the part of the earth pressing downward F M G H would be equal to the opposing part resisting I D E L; and if you wish the part F M G H, being farther from the center, to weigh more so that the ratio of this excess in gravitating may be able to overcome not only the resistance of the segment I D E L, lest it be driven upward, but also that of the segment F I L G, lest it be removed from it according to the parts I L nearer the center, I shall not object greatly. But since never even a thousandth part, let alone a sixth part, of the terrestrial globe is transferred from one place to another diametrically opposite, there is never any change among heavy bodies by which the earth should tremble. But one thing still remains, which does not seem to be something that should be passed over in silence. Suppose, you say, that no transfer of heavy bodies takes place on the earth so great as to be able to establish a new center of gravity in the center of the universe, and therefore there is no tremor of the earth’s center; yet at least the earth will wobble about the center by a motion of rotation, the strong force of winds driving the highest mountains, and turning the whole globe, according to their varying impact, now this part, now that; whence perhaps one might suspect that the variation of the magnetic needle in the same place after some years has arisen

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Mechanicorum 74 quis possit. Cum enim tellus æqualibus circâ centrum nutibus librata permaneat, multo faciliùs omnem in partem converti posse videtur, quàm rota ingens suo in axe suspensa: Rota scilicet suo pondere axem premens illum, dum convertitur, terit; hancque affrictûs difficultatem vincat necesse est, quod una ex parte additur pondus, vel quæ applicatur Potentia, ut conversionem efficiat: tellus verò in orbem diffusa nec centrum premit, nec axem, cum quo ullus fiat affrictus; ac proptereà faciliorem præbet conversionis ansam Potentiæ unam aliquam in partem urgenti. Hujusmodi autem Potentia ventus est, non ad perpendiculum in terram incidens, sed obliquè in præaltos saltem montes incurrens; cujus viribus nihil obstare videtur, quin telluris globum sibi obsecundantem inclinet; quemadmodum, & ingentes naves, vela implens, impellit. Huic difficultati ut me subducam, non me in abditos magnetismi recessus recipio, asserendo tellurem ita arcanis nodis cælo connexam, ut à summo axium polorumque cælestium atque terrestrium consensu divelli ac distrahi prorsùs nequeat: neque enim hisce magnetismi latebris me satis protectum existimarem; demptâ quippe solis Australibus atque Borealibus ventis hâc facultate tellurem convertendi, ne scilicet terrestres poli à cælestibus discrepent, quid prohibeat reliquos ad Ortivum, aut Occiduum limitem pertinentes, quin suo flatu orbem hunc volvant, adhuc superesset explicandum. Hoc quidem satis esse videretur ad submovendam suspicionem illam de acûs magneticæ variatione ob telluris conversionem; manente nimirum axe terrestri ita, ut cum cælesti conveniat, aut illi saltem parallelus existat, nihil est quod, etiam tellure circa axem conversâ, magneticam declinationem commutare queat: nam quod ad syderum aspectus spectat, parum interest, tellusne? an cælum volvatur; si igitur diurna cæli conversio magnetis declinationem non mutat, neque ad illam mutandam sufficeret telluris circa suum axem conversio, vi cujus alia atque alia sydera respiceret: Præterquam quod non id temporum lapsu accideret; sed ubi ventorum impetus elanguisset, illicò variatio illa declinationis magneticæ deprehenderetur: id quod ab omni experimento longè abest. Verùm adeò à nostris sensibus sejunctæ sunt magneticorum symptomatum causæ, ut ad aliarum

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Mechanics 74 how this might be. For since the earth, balanced about the center by equal motions around it, remains poised, it seems much more easily able to be turned in every direction than a huge wheel suspended on its axle: for a wheel, pressing its axle by its own weight, wears it away while it turns; and this difficulty of friction must necessarily be overcome by that which is added on one side, namely the weight or the Power applied, so that the turning may be effected: but the earth, spread out into a globe, presses neither on a center nor on an axle, with which no friction takes place; and therefore it offers a much easier opportunity for a Power urging it in one direction or another. Now a Power of this kind is wind, not falling perpendicularly upon the earth, but striking obliquely at least upon very high mountains; and nothing seems able to prevent its force from inclining the globe of the earth to obey it, just as it drives on even huge ships by filling their sails. To free myself from this difficulty, I do not take refuge in the hidden recesses of magnetism, asserting that the earth is connected to heaven by such secret bonds that it cannot possibly be torn apart and dragged away from the complete agreement of the celestial and terrestrial axes and poles: nor indeed would I think myself sufficiently protected by these hiding-places of magnetism; for if only the southern and northern winds were deprived of this power of turning the earth, lest the terrestrial poles disagree with the celestial, what would prevent the remaining winds, those belonging to the eastern or western limit, from revolving this globe by their breath, would still have to be explained. This might indeed seem enough to remove that suspicion about the variation of the magnetic needle because of the earth’s rotation; for if the terrestrial axis remains such that it agrees with the celestial one, or at least exists parallel to it, there is nothing that can, even if the earth is turned about its axis, alter the magnetic declination: for as far as the appearance of the stars is concerned, it makes little difference whether the earth or the heaven revolves; if therefore the daily rotation of the heavens does not change the declination of the magnet, neither would the earth’s rotation about its own axis suffice to change it, by virtue of which it would look now to one set of stars, now to another: besides, this would not happen with the lapse of time; but as soon as the force of the winds had slackened, that change in magnetic declination would be detected at once: a thing which is far removed from every experiment. Yet so removed from our senses are the causes of magnetic symptoms, that to the causes of other things

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Liber primus. CAPUT XII. 75 aliarum difficultatum solutionem non facilè advocandus sit in Philosophicam scenam magnetismus. Illud potius hîc attendendum videtur, quod montis altitudo, atque magnitudo ad totius telluris molem Rationem habet satis exiguam. Cum enim terræ ambitus probabiliter statuatur, ut aliàs ostendi, milliarium Rom. antiq. 30598, ejusque propterea diameter sit proximè mill. 9738 1/4, tota superficies sphætica (ut pote quadrupla maximi circuli ex demonstratis ab Archimede) est mill. quadratorum 297. 987800 proximè. Mons statuatur altitudinis perpendicularis milliarium quinque; hæc est ad s errestrem diametrum ut 1 ad 1947: basis montis occupet millaria quadrata 500; hæc est ad sphæricam totius globi superficiem, ut 1 ad 595975. Finge jam pro monte granum hordei, quod promineat secundùm suam latitudinem ex sphærâ habente diametrum granorum 1947, hoc est passuum geometricorum sex, seu pedum Rom. antiq. 30. circuli maximi ambitus erit pedum 94 1/4: quare hujus sphæræ superficies habet pedes quadratos 2827, hoc est quadratas latitudines grani hordei paulò plures quàm 11. 579000. Igitur grani hordei jacentis altitudo ad hujus sphæræ diametrum eandem ex hypothesi habet rationem, quam prædicti montis altitudo ad telluris diametrum: & si decem grana sibi invicem attigua disponantur, ut montis basim æmulentur, eadem erit ratio ad superficiem. Quamvis itaque sphæra illa intelligatur planè inanis ac levissima solam habens superficiem papyraceam, ex qua granum ordei agglutinatum promineat, an putas à statu quantumvis valido per fistulam emisso in granum illud hordei incurrente convertendum esse globum papyraceum? Id sanè ex cæteris experimentis conjicere non licet; perinde enim est atque si nihil promineret; neque vel minimum obest Physicæ rotunditati. Quare neque montis altitudo constituta quicquam detrahet orbicularis figuræ, quod sub Physicam considerationem cadat; ac propterea nihil virium ad tellurem convertendam obtinet ventus in montem incurrens. Et quidem conversionem hanc re ipsâ non fieri manifestum est; si quidem cum nulla vincenda esset gravitas, quæ longius à centro gravium recederet, vel quæ axem tereret, facillima videretur esse globi totius conversio circa centrum, non folum K 2

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Book One. CHAPTER XII. 75 Nor can magnetism be conveniently summoned into the philosophical arena as a solution of other difficulties. Rather, what seems here to be noted is that the height and magnitude of a mountain bear a sufficiently slight relation to the mass of the whole earth. For, since the circumference of the earth may probably be taken, as I have elsewhere shown, at 30,598 ancient Roman miles, and therefore its diameter at about 9,738 1/4 miles, the entire spherical surface (being, as shown by Archimedes, four times the area of the greatest circle) is approximately 297,987,800 square miles. Let a mountain be assumed with a perpendicular height of five miles; this is to the terrestrial diameter as 1 to 1947. Let the base of the mountain occupy 500 square miles; this is to the spherical surface of the whole globe as 1 to 595975. Now imagine, in place of the mountain, a grain of barley projecting, according to its breadth, from a sphere having a diameter of 1947 grains, that is, six geometric paces, or 30 ancient Roman feet. The circumference of the greatest circle will be 94 1/4 feet; wherefore the surface of this sphere has 2827 square feet, that is, somewhat more than 11.579000 square breadths of a grain of barley. Therefore the height of a grain of barley lying upon the ground bears to the diameter of this sphere, on the same hypothesis, the same ratio as the aforesaid height of the mountain to the diameter of the earth; and if ten grains be placed adjacent to one another so as to imitate the base of a mountain, the ratio to the surface will be the same. Although therefore that sphere be understood to be altogether empty and very light, having only a paper surface, from which a grain of barley protrudes, do you think that a globe of paper struck by a blast, however powerful, issuing through a tube and falling upon that grain of barley, must be turned? Surely this cannot be inferred from the other experiments; for it is just as though nothing projected at all; nor does it in the least hinder physical roundness. Therefore neither will the determined height of a mountain detract anything from a circular figure that falls under physical consideration; and consequently the wind striking against a mountain has no force at all to turn the earth. And indeed it is manifest that this turning does not in fact occur; since, as no gravity would have to be overcome, whether that which recedes farther from the centre of gravities, or that which rubs against the axis, the conversion of the whole globe about its centre would seem to be easiest, not only K 2

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76 Mechanicorum validioribus atque incitationibus, sed temperatis etiam atque mediocribus ventis flantibus. Hi autem aliquando diuturni sunt; cujusmodi potissimum sunt Etesiæ, quibus maritimi cur- sus celeres, & certi diriguntur. Tot igitur dierum spatio, ven- to oppositos montes vehementiùs urgente, non modica fieret terreni globi inclinatio; ac propterea non eadem demum per- maneret eodem in loco Poli suprà Horizontem altitudo, quo- ties ab alterutro cardine Australi Boreali ve, aut à solstitiali Brumali-ve limite tam ortivo quàm occiduo ventus spiraret, at- que multarum ædium facies non eandem ampliùs respicerent cæli plagam; quare & scietherica Horologia quantumvis ac- curatè semel descripta post non adeò multas temporum inclina- tiones toto ferè cælo discreparent; aliis enim, atque aliis sub- inde flantibus ventis, varia oriretur orbis conversio, atque alia planorum cum circulis horariis sectio, quæ descriptis lineis non congrueret. Hujus autem mutationis nullum in toto terra- rum orbe vestigium apparet, nisi fortè fabulas liceat com- minisci. Quòd si conversionem hanc non omninò circa centrum quamcumque in partem fieri, sed tantummodo circa axem, dixeris, ut argumenti vim effugias; Quid illud est, quod ita terrestrem axem cum cælesti colligatum velit, ut tamen ter- restres meridianos à primâ mundi molitione constitutos tem- poris lapsu cum cælestibus meridianis non convenire permit- tat? Sed & aliud profectò, nec illud quidem leve, incommo- dum subeas necesse est; dum enim conversionem adstruis ab ortu in occasum, & vicissim ab occasu in ortum, fieri poterit, ut post aliquot annos non planè spernenda conversio facta fue- rit, ac proinde temporum numeratio cælo non respondeat. Nam si ab ortu in occasum ex. gr. processerit tellus, minus tem- poris numerabitur quàm pro ratione cælestium motuum; ut contigisse fertur navi cui à Victoria nomen inditum est, in ex- peditione Magellanicâ; cum scilicet post totius orbis ambitum redux in Hispalensem portum, ex quo ante tres annos solve- rat, intraret, tunc primùm observarunt se à rectâ temporis nu- meratione defecisse die uno; quippe qui cum juxta diurnam cæli conversionem ab ortu in occasum iter instituissent, justo cardiùs semper sol illis occiderat, exiguo quidem singulis die- bus,

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76 Mechanicorum with stronger and more intense, but also with tempered and moderate winds blowing. These, however, are sometimes long-lasting; such especially are the Etesian winds, by which swift and certain sea voyages are directed. So over the span of so many days, while the wind presses more strongly against the mountains opposite it, there would be no small tilting of the terrestrial globe; and therefore the altitude of the Pole above the Horizon would not in the end remain the same in the same place, whenever from either of the southern or northern cardinal points, or from the solstitial or winter limit, the wind should blow both from the east and from the west, and the faces of many houses would no longer look upon the same quarter of the sky; wherefore even scietherical clocks, however accurately once drawn, after not so many shifts of time would differ almost entirely from the heavens; for with different and different winds blowing in succession, a varied turning of the sphere would arise, and another cutting of the planes with the hour-circles, which would not agree with the lines described. Yet of this change no trace appears throughout the whole earth, unless perhaps one may be allowed to invent fables. But if you should say that this turning does not take place altogether around the center in any direction whatever, but only around an axis, so as to escape the force of the argument; what then is that which so wills the terrestrial axis to be joined to the celestial one, yet does not allow the terrestrial meridians, established from the world’s first construction, over the lapse of time to fail to agree with the celestial meridians? But there is also another inconvenience, and indeed no slight one, which you must necessarily incur; for while you assert a rotation from east to west, and conversely from west to east, it may happen that after some years a not inconsiderable turning has been made, and thus the reckoning of time does not correspond to the sky. For if the earth should have advanced, for example, from east to west, less time will be counted than the ratio of the celestial motions requires; as is said to have happened to the ship named Victoria, on the Magellanic expedition; when, after completing the circuit of the whole world and returning into the harbor of Seville, from which it had set out three years before, they then first observed that they had fallen short in the correct reckoning of time by one day; for since, following the daily rotation of the sky, they had undertaken the voyage from east to west, the sun had always set for them a little earlier than was due, each day,

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Liber primus. CAPUT XII. 77 bus, quibus procedebant, discrimine, sed quod demùm modi- cis illis accessionibus in integrum diem excreverat. Contra ve- rò accideret, si ab occasu in ortum semper navigaretur; justo enim breviores essent dies, ac propterea eorum numeru ac- cresceret. Hæc autem in temporum numeratione inconstantia, si ventorum impetu tellus modò in ortum, modò in occasum converteretur, quantam perturbationem inveheret in Astronomiam? Neque tibi quicquam suffragari existimes, si ex varia ventorum oppositas in plagas sivè simul, sivè subinde, spirantium commutatione conversiones illas compensari dixeris: id enim ad incertum revocat omnes Astronomorum calculos, ubi meridianorum circulorum sectiones stabiles non permaneant; cum ad orbem totum inclinandum, ut tu quidem autumas, satis sit, si unâ aliquâ in regione ventus montes impellat; quî verò certus sim factam ab Argeste telluris conversionem in ortum, æquatam demum fuisse à Vulturno, aut ab Euro-Austro? Verùm quàm infirmæ sint validissimorum ventorum vires ad globum hunc terraqueum inclinandum, expendamus, etiamsi montium perpendicula non quinque tantùm milliaribus definita velis, sed multò altiora. Statue in ingenti lacu compositam ex trabibus aliquot ratem, quam in littore stans facilè funiculo modereris: Tùm ratem aliam paris quidem latitudinis, sed centuplò longiorem, compone: Poteris-ne hanc funiculo eodem, ac labore non majori, trahere perinde atque priorem? Negabis utique, quamvis enim utraque lacui stagnanti innatet, nec vincenda sit alterutrius gravitas, ut à centro gravium magis recedat; licet utraque parem in motu ab aquâ dividendâ resistentiam inveniat (ejusdem quippe sunt latitudinis solâ discrepantes longitudine, & æqualis est utriusque immersio propter eandem singularum trabium molem, atque specificam gravitatem) quia tamen dispar est ratium magnitudo, & impetu extrinsecùs accepto utraque eget, ut moveatur, palàm est majore impetu opus esse, ut ratis major trahatur, ac propterea posse hanc adeò augeri, ut impetus ad illam movendam necessarius excedat vires Potentiæ ratem minorem funiculo moderantis. Ita planè est. Sed jam animum transfer ad institutam disputationem, ut dispicias, undè irrepserit dubitatio hæc de telluris K 3

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Book I. CHAPTER XII. 77 but because of the difference of the places by which they were proceeding, but because at last by those slight additions it had grown into a full day. The contrary would happen if one were always sailing from west to east; for the days would be just so much shorter, and therefore their number would increase. But if this inconsistency in the reckoning of time, when by the force of the winds the earth were sometimes turned toward the east, sometimes toward the west, what confusion would it bring into Astronomy? Nor should you think that anything favors you, if you were to say that those conversions are compensated by the varying changes of winds blowing against opposite quarters, sometimes together, sometimes in succession: for this brings all the astronomers’ calculations back to uncertainty, when the sections of the meridian circles do not remain fixed; whereas, for the whole globe to be inclined, as you indeed suppose, it is enough if in some one region the wind drives the mountains; and how indeed am I to be certain that, when the Argestes has turned the earth toward the east, it was at last balanced by the Vulturnus, or by the Euro-Auster? But let us consider how feeble are the powers of the strongest winds for inclining this terrestrial globe, even if you wish the plumb lines of the mountains to be not limited to five miles, but much higher. Suppose in a vast lake a raft made from several beams, which, standing on the shore, you can easily control with a rope: then make another raft of equal breadth, but a hundred times longer. Can you pull this one with the same rope, and with no greater labor, just as you would the former? You will certainly deny it; for although each floats on the stagnant lake, and the weight of neither has to be overcome so that it may recede farther from the center of gravity; although each finds equal resistance from the water in being moved aside (for they are of the same breadth, differing only in length, and the immersion of each is equal because of the same mass and specific gravity of the individual beams), yet because the size of the rafts is different, and each needs an external impulse in order to move, it is clear that a greater impulse is required for the larger raft to be drawn, and therefore that this may be increased so much that the force necessary to move it exceeds the power of the one controlling the smaller raft with a rope. So it is indeed. But now turn your mind to the argument we have undertaken, so that you may see whence this doubt about the earth has crept in. K 3

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78 Mechanicorum conversione ex ventorum impulsu, & quàm facilè fucum fecerit rota suo in axe suspensa, quæ levi negotio, nec valido impulsu, volvitur. Rota siquidem tota deorsum gravitat, ac proptereà axem premit; quia autem in axe suspenditur, fieri non potest, ut pars altera descendat, quin opposita ascendat. Quandiu conatus ad descendendum æqualis est resistentiæ ad ascendendum, rota quiescit; nec volvitur, nisi alterutri parti fiat accessio Potentiæ, quæ pariter descensum juvet, vel quia ipsa quoquè deorsum conatur cum parte descendente, vel quia fursum nitens partem alteram elevat, oppositamque deprimet suapte naturâ descendentem. Non tamen hujusmodi rotæ suspensæ conversio tribuenda est soli Potentiæ; sed pars rotæ descendens atque Potentia collatis viribus elevant partem rotæ ascendentem, eique impetum imprimunt. At in telluris circa fuum centrum, vel axem, conversione nihil adesset, quod Potentiam juvaret; quia nulla est pars, quæ deorsum conetur, aut fursum, ut possit oppositæ parti impetum aliquem imprimere; nulla etenim pars in hujusmodi conversione ad centrum gravium accederet, aut ab illo recederet. Totus igitur impetus à vento imprimendus esset toti telluris globo, ut à suâ, quæ secundùm naturam est, quiete dimoveretur. Atqui globi terraquei ea est moles, ut contineat milliaria cubica proximè 48670.200000 (omnis nimirum sphæra æqualis est cono, cujus altitudo par est Radio sphæræ, basis autem æqualis superficiei sphæræ, ex dictis verò paulò superiùs, & superficies & Radius globi hujus innotescit) nullus igitur adeò vehemens est ventus, qui tantæ moli impetum imprimere valeat; nullus siquidem excogitari potest ventus, qui globum marmoreum, aut etiam ex argillâ, in planitie æquissimâ constitutum, si mille passus Geometricos in diametro numeret, convolvere valeat. Adde in telluris conversione, si illa fieret, quò vehementior esset ventus in montem incurrens, validior esset resistentia aëris à reliquis montibus dividendi; sed & multorum ingentium aluminum contrariam in partem labentium impetus obsisteret, ne tellus vento flanti obsecundaret. Quod si hæc levis esse momenti dixeris ad obsistendum, levis pariter momenti esse ventorum impetum, necesse est, fatearis: neque hîc arduum esset ventorum atque aluminum vires invicem conferre, aquarum- que

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78 Mechanics from the turning caused by the impulse of the winds, and how easily it would have deceived a wheel suspended on its axle, which, with little effort and without a strong impulse, is turned. For the wheel as a whole tends downward, and therefore presses on the axle; but because it is suspended on the axle, it cannot happen that one part descends without the opposite part rising. As long as the effort to descend is equal to the resistance to rising, the wheel remains at rest; nor does it turn unless to one or the other part there is added a force which likewise helps the descent, either because it too strives downward with the descending part, or because, striving upward, it raises one part and depresses the opposite one, which by its nature descends. Nevertheless, the turning of such a suspended wheel is not to be attributed to force alone; but the descending part of the wheel and the force, acting together, raise the rising part of the wheel and impart motion to it. But in the earth’s turning about its own center, or axle, there would be nothing present to assist the force; because there is no part that strives downward, or upward, so as to be able to impart any motion to the opposite part; for in such a turning no part would approach the center of gravity, nor move away from it. Therefore the whole impulse would have to be imparted by the wind to the whole globe of the earth, so that it might be moved from its state of natural rest. But the mass of the earthly globe is such that it contains approximately 48670.200000 cubic miles (for every sphere is equal to a cone whose height is equal to the radius of the sphere, and whose base is equal to the surface of the sphere; and from what has been said a little above, both the surface and the radius of this globe are known), therefore there is no wind so violent as to be able to impart motion to such a mass; for no wind can be imagined that could set a marble globe, or even one made of clay, placed on a perfectly level plain, and measuring one thousand geometrical paces in diameter, rolling. Add that in the turning of the earth, if it were taking place, the more violent the wind striking a mountain, the stronger would be the resistance of the air being separated from the remaining mountains; and also the impetus of many huge valleys sliding in the opposite direction would resist, lest the earth yield to the blowing wind. But if you should say that these are of little moment in resisting, you must likewise admit that the impulse of the winds is of little moment; nor here would it be difficult to compare the forces of the winds and of the valleys against one another, and of the waters-

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Liber primus. CAPUT XIII. 79 que impetum multò validiorem ostendere; sed ad alia prope- randum est: satisfuerit monuisse non mediocrem intercedere analogiam inter aquarum guttas in rivulos primùm, deinde in majores rivos, ac demum in torrentem concurrentes, atque terræ expirationes in ventum congregatas, quæ multum vi- rium obtinent, si plurimæ in unum coëant, quemadmodum & aquis contingit. CAPUT XIII. Quâ ratione minuatur gravitatio in plano inclinato. Planum inclinatum dicitur planum quodcumque non tran- sit per centrum gravium & levium, hoc est per centrum universi; hujusmodi siquidem planum non cadit ad angulos æquales in sphæricam terræ superficiem. Hinc etiam planum horizonti parallelum reipsâ est inclinatum, nisi adeò exiguum sit ac breve, ut puncti vicem obtineat, si cum terreni globi su- perficie conferatur. Sit universi centrum A, plana B A, & C A sunt verticalia & perpendicularia, qui- bus si corpus aliquod grave appli- cueris, illud non impedietur, quin per suam directionis lineam descen- dat. At verò tam planum B C, quam planum C D inclinata sunt, nec cor- pus grave illis impositum potest rectâ secundùm directionis lineam descendere, sed ab illâ declinare co- gitur plano obsistente. Sunt autem anguli inclinationis A B C, A C D. Quod si planum parallelum horizonti ita exiguum sit, ut à sphæricâ superficie, quam tangit, non recedat; tunc in quacumque ejus parte constituatur corpus grave, perinde est, atque si in puncto D collocatum concipiatur. Sin autem ita à puncto

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Book One. CHAPTER XIII. 79 to show a much stronger impulse; but we must proceed to other matters: it will suffice to have noted that there is no small analogy between drops of water, which first run together into brooks, then into larger streams, and finally into a torrent, and the exhalations of the earth gathered into wind, which obtain great force if many come together in one, just as happens with waters. CHAPTER XIII. By what means gravitation is diminished on an inclined plane. An inclined plane is called any plane that does not pass through the center of the heavy and light bodies, that is, through the center of the universe; for a plane of this kind does not fall at equal angles upon the spherical surface of the earth. Hence also a plane parallel to the horizon is in fact inclined, unless it be so very small and short that it takes the place of a point, if compared with the surface of the terrestrial globe. Let the center of the universe be A; the planes BA and CA are vertical and perpendicular, and if you apply some heavy body to them, it will not be prevented from descending along its line of direction. But both the plane BC and the plane CD are inclined, nor can a heavy body placed upon them descend straight along the line of direction, but is forced to deviate from it by the opposing plane. Now the angles of inclination are ABC and ACD. And if a plane parallel to the horizon be so small that it does not recede from the spherical surface which it touches, then in whatever part of it a heavy body is placed, it is the same as if it were conceived to be set at point D. But if in such a way from point

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80 Mechanicorum puncto D distiterit, ut à sphæricâ superficie recedat, quemadmodum si esset planum DF, illud est inclinatum, & fit angulus DFA inclinationis. Ubi observandum est non eandem esse singularum plani partium inclinationem; angulus enim inclinationis A E C major est inclinatione ABC, per 16. lib. 1. & similiter A F D maior est angulo A C D. Quare statim atque ea est puncti E à puncto B distantia, ut angulus à perpendicularis in centro A factus contemni non possit, alia est etiam physicè inclinatio, & corporis ejusdem gravitatio mutatur. Quoniam verò corpus grave plano inclinato impositum ita aëre circumfunditur, ut petat infrà illum descendere, & resistat, ne sursum moveatur; ideò gravitare dicitur. Sed cavendum est, ne ex vocabulorum similitudine error subrepat: quandoquidem aliud est gravitare in plano inclinato, aliud gravitare in planum inclinatum: nam intrà aërem corpus grave, putà, lapis, gravitat in quocunque plano etiam perpendiculari, non tamen gravitat in planum perpendicularare, nullasque vires suæ gravitatis contra illud exercet, quamvis in eo existens, & resistat sursum trahenti, & conetur, ut vincat vires retinentis, ac quicquid moram infert, & impedimentum motui. In plano itaque inclinato existens corpus grave (subjectum planum supponitur optimè lævigatum, nec motui officiens partium prominularum asperitate) gravitat quidem, sed minùs quàm in plano perpendiculari, & pro variâ planorum inclinatione, varia pariter est gravitatio, ut quotidiana nos docet experientia. Quâ igitur ratione gravitatio minuatur, hîc est examinandum; capite sequenti gravitatio in Planum inclinatum explicabitur. Cognoscitur autem gravitatio ex resistentiâ, quâ corpus repugnat contra vires illud retinentis, ne deorsum feratur, aut sursum trahentis; neque enim alio nisu gravia gravitant, quàm quo resistunt impedienti motum gravitati convenientem. Et quidem experimento aliquo potest gravitationis varietas investigari; si nimirum planum BO ex ligno, aut marmore accuratè lævigetur, & extremitati B adnectatur orbiculus D facillimè circa axem versatilis, ponderi

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80 Mechanicorum recedes from point D, so that it departs from the spherical surface, just as if there were a plane DF; that plane is inclined, and the angle DFA of inclination is formed. It is to be observed here that the inclination of the several parts of a plane is not the same; for the angle of inclination AEC is greater than the inclination ABC, by 16. lib. 1. And similarly AFD is greater than angle ACD. Therefore, as soon as the distance of point E from point B is such that the angle made at the center A from the perpendicular cannot be disregarded, there is also, physically, another inclination, and the gravitation of the same body is changed. But since a heavy body placed upon an inclined plane is so surrounded by air that it seeks to descend beneath it, and resists, lest it be moved upward, therefore it is said to gravitate. But care must be taken lest error creep in from the similarity of words: for it is one thing to gravitate in an inclined plane, and another to gravitate toward an inclined plane. For within the air a heavy body, for example a stone, gravitates upon any plane even a perpendicular one, yet it does not gravitate in a perpendicular plane, nor does it exert any forces of its gravity against it, although, being in it, it resists one drawing it upward, and strives to overcome the forces retaining it, and whatever causes delay and obstruction to motion. In an inclined plane, therefore, a heavy body being present (the supporting plane is supposed to be very well smoothed, and not hindering motion by the roughness of protruding parts) does indeed gravitate, but less than in a perpendicular plane, and according to the varying inclination of the planes, the gravitation is likewise varied, as daily experience teaches us. By what ratio, then, gravitation is diminished must here be examined; in the following chapter gravitation in an inclined plane will be explained. Moreover, gravitation is known from the resistance by which a body opposes the forces retaining it, lest it be carried downward, or drawn upward; for heavy things do not gravitate by any other effort than that by which they resist whatever impedes the motion proper to gravity. And indeed by some experiment the variation of gravitation may be investigated; if, namely, the plane BO is carefully smoothed from wood or marble, and to the extremity B there is attached a little wheel D very easily turnable about its axis, the weight

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Liber primus. CAPUT XIII. 81 deri autem A subjiciantur rotulæ, & adnectatur funi- culus per D transiens, ex cujus extremo pendeat lanx E, cui pondera immitti pos- sint: pro variâ enim plani B O inclinatione etiam pon- dera in lance mutare opor- tebit, ut pondus A susti- neatur, & plura erunt, quò magis ad perpendiculare accedet planum B O. Verùm quia nunquam carere poteris suspicione, an corporum affrictus aliquid afferat impedimenti; ideò seclu- sis omnibus, quæ extrinsecùs accidere possunt, resistentiam ex solâ gravitate ortam opus est considerare. Resistentia verò omnis respondet violentiæ, quam patitur id quod resistit; minori etenim conatu minorem vim illatam propulsare studet natura, quæ validiùs obsistit majori violen- tiæ: id quod ita rationi est consonum, & obviis experimentis manifestum, ut in hoc demonstrando supervacaneum sit im- morari. Constituantur itaque duo æqualis ponderis corpora in D & in C; singulis alligetur funiculus, qui per B transeat, & sursum tra- hantur simul ita, ut æqualiter mo- veantur. Absolutâ motûs particu- lâ, corpus alterum ex D ascendit in H in plano perpendiculari; al- terum in plano inclinato ex C ve- nit in E, & C E linea æqualis est lineæ motûs D H. Non eandem tamen utrumque grave subiit vio- lentiam; nam motus D H fuit simpliciter, & absolutè violen- tus; at motus C E eatenus solùm gravitati adversatur, quate- nus ascendit; ascensum autem metitur linea D G, quam ab- scindit E G horizonti parallela. Hîc scilicet planum D C in- tellige horizontale nihil à sphæricâ superficie discrepans, ut communiter contingit: quòd si non ita se haberet; sed esset amplissimum planum, mensura violentiæ illatæ ponderi in C L

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Book One. Chapter XIII. 81 but let rollers be placed under A, and let a cord be attached, passing through D; from whose end let there hang the pan E, into which weights may be put: for according to the different inclination of plane B O, the weights in the pan must also be changed, so that the weight A may be sustained, and they will be greater the more the plane B O approaches the perpendicular. But because you can never be free from suspicion whether the friction of bodies may bring some hindrance; therefore, setting aside everything that may happen from without, it is necessary to consider the resistance arising from gravity alone. Now all resistance corresponds to the violence which that which resists suffers; for by a smaller effort nature endeavors to drive back a smaller force applied, and it resists more strongly a greater violence: this is so consonant with reason, and so evident from ordinary experiments, that it would be superfluous to dwell on proving it. Let there therefore be placed two bodies of equal weight, one in D and the other in C; let a cord be attached to each, passing through B, and let them be drawn upward together so that they move equally. When the part of the motion is completed, one body ascends from D to H on the perpendicular plane; the other on the inclined plane comes from C to E, and line C E is equal to line D H of the motion. Yet the same violence is not undergone by each heavy body; for the motion D H was simply and absolutely violent; but the motion C E opposes gravity only so far as it ascends; and the ascent is measured by line D G, which E G cuts off, parallel to the horizon. Here, indeed, understand plane D C as horizontal, differing not at all from a spherical surface, as commonly happens: but if it were not so, and were an exceedingly broad plane, the measure of the violence inflicted on the weight in C L

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82 Mechanicorum constituto, in E elevato desumenda esset ex differentiâ inter K C & O E. Est itaque gravitatio in plano perpendiculari ad gravitationem in plano inclinato, ut resistentia ad ascendendum in uno ad resistentiam ad ascendendum in alio; resistentiæ autem sunt, ut violentia, quam corpora subeunt in motu; violentia demum est ut HD ad G D, hoc est per 7. lib. 5. ut C E ad D G. Sed ut C E ad D G, ita E B ad G B, per 2. lib. 6. & ut B E, ad B G ita B C ad B D, per 4. lib. 6. igitur gravitatio in perpendiculari ad gravitationem in inclinato est ut B C ad B D, hoc est ut Secans anguli inclinationis ad Radium. Quæ autem de totis D H, & C E lineis dicta sunt, de singulis earum particulis æqualibus dicta intelligantur; ductis quippe parallelis horizonti, eadem est omnium Ratio: hîc namque supponimus planum B C non adeò magnum esse, ut singula ejus puncta cum diversis horizontibus comparanda sint, omnes siquidem perpendiculares lineæ directionis non quasi convergentes, sed physicè parallelæ accipiuntur. Quòd si tam longum esset planum, ut physicè mutatus intelligeretur angulus inclinationis, non eadem esset Ratio gravitationis in toto, ac in partibus: sed mutato angulo inclinationis mutaretur utique ejus Secans; ac proinde inæqualium Secantium Ratio ad eumdem Radium inæqualis, gravitationum pariter inæqualem rationem ostenderet. Quod si ascendentium per vim extrinsecùs illatam corporum resistentiam atque gravitationem metimur ex violentiâ, quam pro planorum varietate subeunt; eorum pariter in descendendo efficacitatem ex ipso descensu argui æquum esset, datâ motûs in diversis planis æqualitate. Sed quia descensus naturæ propensioni congruit, fieri non potest, ut in alio atque alio plano æquales sint motus isochroni; tardior enim est, qui in plano inclinato perficitur, neque, si æqualis ponderis corpora descendant ex H & E, quando illud ad D pervenit, hoc potest attingere punctum C: ideò non ex descensu gravitationem metiri oportet, cum motus æquales non habeantur: nisi fortè easdem movendi vires tribuas gravitati non impeditæ in perpendiculari, ac impeditæ in plano inclinato. Qua propter gravitationis momenta ad descendendum non aliunde meliùs æstimantur, quàm ex repugnantiâ ad ascendendum: sic enim vulgari argu- mento

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82 Mechanicorum being established, in E elevated it ought to be taken from the difference between K C & O E. It is therefore gravitation in the plane perpendicular to gravitation in the inclined plane, as resistance to ascending in one is to resistance to ascending in the other; and resistances are as the violence which bodies undergo in motion; violence finally is as H D to G D, that is, by 7. lib. 5., as C E to D G. But as C E to D G, so is E B to G B, by 2. lib. 6.; and as B E to B G, so is B C to B D, by 4. lib. 6.; therefore gravitation in the perpendicular to gravitation in the inclined is as B C to B D, that is, as the secant of the angle of inclination to the radius. But what has been said of the whole lines D H & C E is to be understood as said of each of their equal parts likewise; for by drawing parallels to the horizon, the ratio of all is the same: for here we suppose the plane B C not to be so large that its individual points are to be compared with different horizons, since all the perpendicular lines of direction are taken not as if converging, but physically parallel. But if the plane were so long that the angle of inclination would be understood to change physically, the ratio of gravitation in the whole would not be the same as in the parts: but as the angle of inclination changes, its secant would certainly change as well; and consequently the ratio of unequal secants to the same radius, unequal likewise, would show the ratio of gravitations. But if we measure the resistance and gravitation of bodies ascending by the force applied from outside, from the violence which they undergo according to the variety of the planes; then likewise in descending their efficacy ought fairly to be inferred from the descent itself, given equal motion in different planes. But because descent agrees with the inclination of nature, it cannot happen that isochronous motions are equal in one plane and another; for it is slower when accomplished on an inclined plane, and if bodies of equal weight descend from H & E, when that body reaches D, this one cannot reach point C: therefore gravitation ought not to be measured from descent, since equal motions are not available: unless perhaps you assign the same moving powers to gravity when unobstructed in the perpendicular, and when obstructed in the inclined plane. For this reason the moments of gravitation for descending are estimated nowhere better than from resistance to ascending: for thus by the common argument

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Liber primus. CAPUT XIII. 83 mento singulorum corporum gravitates librâ expendimus, tan- tumque iis ad descendendum virium tribuimus, quantum re- sistunt, ne ab oppositâ libræ lance deorsum conante eleventur. Eadem igitur est gravitationis Ratio, seu propensionis ad de- scendendum, quæ est resistentiæ ad ascendendum: Cum verò resistentiam in plano inclinato ad resistentiam in perpendicu- lari ostensum sit esse, ut Radius ad Secantem anguli inclinationis, hoc est ut BD ad BC, erit pariter vis descendendi in plano BC ad vim descendendi in plano BD, reciprocè ut BD ad BC. Eadem ratione in plano CD superficiem globi tangente, gravitatio in CD ad gravitationem in perpendiculari CA est ut CD ad CA; est enim CA Secans anguli inclinationis DCA. Si enim ducatur K F Tangens, triangula CKF, CDA sunt similia, angulus enim ad C communis est, & am- bo rectangula ad D & K; quare ut CK ad CF, ita CD ad CA; sed gravitatio in CF ad gravitationem in CK est reci- procè ut CK ad CF: igitur gravitatio in plano inclinato CD globum tangente, ad gravitationem in perpendiculari CA, est ut CD ad CA. Hinc est quod in planis horizontalibus, quæ ut plurimum habemus, corpora non descendant, aut moveantur: quia ni- mirum à puncto, in quo grave statuitur, ex. gr. F, ductæ li- neæ FA perpendicularis & FD Tangens faciunt angulum DFA inclinationis adeò magnum, ut Radius ad ejus secan- tem penè infinitam non habeat sensu perceptibilem Rationem, vel saltem non tantam, ut gravitatio, quæ ratione inclinatio- nis plani congruit corpori, non elidatur à resistentiâ, quæ ori- tur ex corporum asperitate. Quare sublatâ, aut potiùs impeditâ, gravitatione corpus quiescit in plano horizontali. Et hæc est ratio, cur violentiam determinans, quam grave ascendens patitur, assumpserim in perpendiculari BA par- tem GD, quam abscindit parallela horizonti; hæc enim mensura physicè non discrepat à verâ mensurâ, quæ assumen- da esset, si mente concipias rectam lineam DC tangere circu- lum, cujus semidiameter sit millecuplo major. Mensura si qui- dem ascensûs petenda est ex excessu, quo perpendicularis EA superat perpendicularem AC; illo enim intervallo, quo magis recessit à centro, ascendit. L 2

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Book One. CHAPTER XIII. 83 we weigh the gravities of individual bodies by the balance, and assign to them only so much force to descend as they resist, lest, by the opposite scale of the balance striving downward, they should be lifted up. The same ratio therefore belongs to gravitation, or propensity to descend, as belongs to resistance to ascend: and since it has been shown that the resistance on an inclined plane to the resistance on a perpendicular is as the Radius to the Secant of the angle of inclination, that is, as BD to BC, it will likewise be that the force of descending on plane BC to the force of descending on plane BD is reciprocally as BD to BC. For the same reason, in the plane CD touching the surface of the globe, the gravitation in CD to the gravitation in the perpendicular CA is as CD to CA; for CA is the Secant of the angle of inclination DCA. For if the tangent KF be drawn, the triangles CKF, CDA are similar; for the angle at C is common, and both are right-angled at D and K; wherefore as CK is to CF, so is CD to CA; but the gravitation in CF to the gravitation in CK is reciprocally as CK to CF: therefore the gravitation in the inclined plane CD touching the globe, to the gravitation in the perpendicular CA, is as CD to CA. Hence it is that on horizontal planes, such as for the most part we have, bodies do not descend, or move: because, namely, from the point where the heavy body is placed, for example F, the drawn lines FA perpendicular and FD Tangent make the angle of inclination DFA so great that the Radius to its secant, almost infinite, has no ratio perceptible to sense, or at least not one so great that the gravitation, which by reason of the plane’s inclination agrees with the body, is not defeated by the resistance arising from the roughness of bodies. Therefore, when gravitation is removed, or rather prevented, the body rests on the horizontal plane. And this is the reason why, determining the force which a heavy body experiences in ascending, I have assumed on the perpendicular BA the part GD, which the parallel to the horizon cuts off; for this measure does not differ physically from the true measure, which would have to be assumed if you mentally conceive the straight line DC touching a circle whose semidiameter is a thousand times greater. For the measure of ascent is indeed to be sought from the excess by which the perpendicular EA exceeds the perpendicular AC; for by that interval, by which it has receded more from the center, it ascends. L 2

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Mechanicorum Ex quo fit quod, si planum inclinatum BC cum perpendi- culari C A faceret angulum acutum A CB, corpus ex C usque in L (in quod punctum cadit perpendicularis A L) descende- ret, quia semper magis ad centrum accederet: ex L autem in E ascenderet, & ascensum metiretur excessus perpendiculari E A suprà perpendicularum L A. Quare ut ex C ascenderet, debe- ret esse planum inclinatum I C, quod cum C A faceret angu- lum I C A saltem rectum. Ubi ex occasione licet observare posse dari duos montes, qui cum valle intermedia planitiem unam constituant; si nimirum montium vertices essent E, & C, ex quibus in imam vallem L descenderetur: & aqua per mon- tium venas descendens in L posset fontem aut lacum creare. Re autem ipsâ semper contingit angulum B C A esse obtusum vel non minorem recto. Ponatur enim terræ semidiameterer D A 1000, & planum D C: (esset autem planum D C longius milliar.4) erit angulus D A C, gr. o. 3'. 26"; atque adeò D C A gr. 89. 56. 34". Iam verò sit CD ad DB ut 100 ad 87; erit angulus B C D gr. 41. 1'. 23": quare totus B C A gr. 130. 57'. 57". Nunc si libeat comparare perpendicularum E A cum perpendi- culo G A, statue G D semissem totius BD; est igitur & G E semissis ipsius D C: Quare G E est partium 50, quarum G A est 100043 1/2: addantur quadrata G E 2500 & G A 10008701892 1/4, & summæ radix quadrata 100043 102343/200086 major verâ est E A, quæ non excedit perpendicularem G A 100043 1/2 nisi particulis 2500/400172 Quoniam autem D A C angulus inventus est grad. o. 3'. 26"; ejusque Secans A C est partium 100000 5017/100000, quarum A D posita est 100000; discrimen inter A C, & A E superiùs in- ventam, est partium 43 46227/100000, quæ est proximè eadem mensu- ra, ac D G posita partium 43 1/2. Quod si in plani inclinati lon- gitudine tantâ Rationem habente ad terræ semidiametrû, quan- ta constituta est, potest citrà errorem assumi tanquam mensura ascensûs pars perpendiculari B A intercepta ab horizontali D C, & parallelâ E G, satis patet id multò magis licere in planorum longitudinibus minorem Rationem habentibus ad eandem ter- ræ semidiametrum. Manet itaque constituta regula gravitatio- nis, videlicet gravitationem in plano inclinato ad gravitationem in perpendiculari esse, ut est Radius ad secantem anguli incli- nationis. Quamvis

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Mechanics From this it follows that, if the inclined plane BC makes with the perpendicular CA an acute angle ACB, the body would descend from C to L (the point at which the perpendicular AL falls), because it would always approach the center more and more: but from L it would ascend to E, and the ascent would be measured by the excess of the perpendicular EA over the perpendicular LA. Therefore, in order for it to ascend from C, there would have to be the inclined plane IC, which with CA would make the angle ICA at least a right angle. Here, by the way, it is permissible to observe that there could be two mountains which, with the intervening valley, constitute one plain; namely, if the peaks of the mountains were E and C, from which one would descend into the lowest valley L: and water descending through the veins of the mountains into L could create a spring or lake. In fact, however, it always happens that the angle BCA is obtuse, or not less than a right angle. For let the semidiameter of the earth DA be 1000, and the plane DC: (but the plane DC would be 4 miles long) then the angle DAC will be 0° 3' 26"; and therefore DCA 89° 56' 34". Now let CD be to DB as 100 to 87; the angle BCD will be 41° 1' 23"; therefore the whole angle BCA will be 130° 57' 57". Now if one wishes to compare the perpendicular EA with the perpendicular GA, let GD be set as half of the whole BD; therefore GE is also half of DC. Thus GE is 50 parts, of which GA is 100043 1/2: add the squares GE 2500 and GA 10008701892 1/4, and the square root of the sum, 100043 102343/200086, is greater than the true EA, which does not exceed the perpendicular GA 100043 1/2 by more than 2500/400172 parts. Since the angle DAC has been found to be 0° 3' 26"; and its secant AC is 100000 5017/100000 parts, of which AD is taken as 100000; the difference between AC and AE found above is 43 46227/100000 parts, which is approximately the same measure as DG, taken as 43 1/2 parts. Therefore, if in an inclined plane whose length has such a ratio to the semidiameter of the earth as has been established, the ascent can without error be taken as the portion of the perpendicular BA intercepted by the horizontal DC and the parallel EG, it is quite clear that it is much more permissible to do so in planes whose lengths have a smaller ratio to the same semidiameter of the earth. Thus the established rule of gravitation remains, namely, that gravitation on an inclined plane to gravitation on the perpendicular is as the radius is to the secant of the angle of inclination. Although

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Liber primus. CAPUT XIII. 85 Quamvis verò in partibus inferioribus plani inclinati sit sem- per major angulus inclinationis, quàm in superioribus, & pro- inde minor sit Ratio, quam habet Radius ad secantem anguli majoris, ac ea, quam idem Radius habet ad secantem anguli minoris: non tamen ea est gravitationis differentia, cujus ratio habenda sit; cum enim adeò exiguus sit angulus B A C, ejus quantitas distribuitur per omnes inclinationis angulos, qui fiunt in punctis intermediis inter B & C; atque adeò contem- nendum est in praxi discrimen illud, quod oritur ex alio atque alio inclinationis angulo in eodem plano. Quod si insignis esset Rationum varietas, notabilis quoque esset gravitationis diver- sitas idem enim contingeret, ac si non idem esset planum. Sed hoc communiter non accidit. Ex his illud manifestâ consecutione conficitur, quod si duo plana inclinata inter se comparentur, ejusdem corporis gravita- tiones in illis sunt reciprocè ut Secantes angulorum inclinatio- nis: hoc est, si fuerint duo plana inclinata BS, BC, gravitatio in BS ad gravitationem in BC est ut BC ad BS. Quia enim gravitatio in BC ad gravitationem in BD est ut BD ad BC; & gravitatio in BD ad gravitationem in BS est ut BS ad BD, igitur ex æqualitate, per 23. lib.5. gravitatio in BC ad gravi- tationem in BS est ut BS ad BC. Hinc prætereà sit, ut, si gravia in planis constituta habeant Rationem eandem, quam secantes angulorum inclinationis ha- bent inter se vel ad Radium, eorum gravitationes sint æquales. Sit ad horizontalem, SC per- pendicularis BD, & inclina- tæ BS, BC, per quas lineas ducta intelligantur plana, & in planis gravia diversa, & ut BD ad BC ita pondus O ad pondus M, & ut BD ad BS ita pondus O ad pondus N. Dico ponderum M, O, N, gravitationes in suis planis esse æquales. Quoniâ enim duorum gravium gravitationes in eâdem perpendiculari BD sunt ut ipsoru[m] pondera, gravitatio M in per- pendiculari BD, ad gravitationem O in eadem perpendiculari, est ut M ad O, hoc est ut BC ad BD; sed gravitatio M in per- L 3

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Book One. Chapter XIII. 85 Although, indeed, in the lower parts of the inclined plane the angle of inclination is always greater than in the upper parts, and therefore the ratio which the Radius bears to the secant of the greater angle is smaller than that which the same Radius bears to the secant of the lesser angle: nevertheless this is not a difference of gravity that must be taken into account; for since the angle B A C is so very small, its quantity is distributed through all the angles of inclination that occur in the intermediate points between B and C; and therefore in practice that difference, which arises from one angle of inclination or another in the same plane, is to be disregarded. But if the variety of Ratios were considerable, the diversity of gravity would likewise be notable; for the same thing would happen as if it were not the same plane. But this does not commonly occur. From these things it is manifestly concluded that, if two inclined planes are compared with each other, the gravitations of the same body in them are reciprocally as the secants of the angles of inclination; that is, if there be two inclined planes BS, BC, the gravitation in BS to the gravitation in BC is as BC to BS. For since the gravitation in BC to the gravitation in BD is as BD to BC; and the gravitation in BD to the gravitation in BS is as BS to BD, therefore by equality, according to 23, book 5, the gravitation in BC to the gravitation in BS is as BS to BC. Hence moreover it follows, that if bodies placed on planes have the same ratio as the secants of the angles of inclination bear to each other or to the Radius, their gravitations are equal. Let BD be perpendicular to the horizontal SC, and BS, BC inclined, through which lines let the planes be understood to be drawn, and in the planes different bodies, and let the weight O be to the weight M as BD is to BC, and as BD is to BS let the weight O be to the weight N. I say that the gravitations of the weights M, O, N in their own planes are equal. For since the gravitations of two bodies in the same perpendicular BD are as their weights, the gravitation of M in the perpendicular BD, to the gravitation of O in the same perpendicular, is as M to O, that is, as BC to BD; but the gravitation of M in the per- L 3

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86 Mechanicorum pendiculari BD, ad gravitationem ejusdem M in inclinatâ BC, est pariter ut BC ad BD; igitur per 11. lib.5. gravitatio M in perpendiculari ad gravitationem O in perpendiculari est, ut gravitatio M in perpendiculari BD ad gravitationem M in inclinatâ BC; igitur per 14. lib.5. gravitatio O in perpendiculari BD æqualis est gravitationi M in inclinatâ BC. Eâdem methodo ostenditur æqualem esse gravitationem N in inclinatâ BS, gravitationi O in perpendiculari BD. Quare gravitationes M & N æquales inter se sunt, cum æquales sint gravitationi O. Constat itaque iisdem viribus retineri posse, aut sursum trahi, majus pondus in plano inclinato, quàm in perpendiculari, eadem enim est illorum gravitatio, ut ostendi; vires autem retinentis aut trahentis debent gravitationi corporis proportione respondere. Quare datis viribus, quæ possint datum pondus O sustinere in perpendiculari BD, cognosci potest gravitas ponderis quod eædem vires sustinere valebunt in dato plano BC inclinato: si nimirùm fiat ut Radius ad secantem anguli datæ inclinationis, ita datum pondus O ad pondus M quæsitum. Detur O lib.15. & angulus DBC gr.36. Fiat ut radius 10000000 ad secantem 12360680, ita lib.15. ad lib.18 1/2; quod est pondus M æquè gravitans in plano BC cum pondere O in perpendiculari. Contra verò dato pondere M sustinendo iisdem viribus, quibus sustinetur O in perpendiculari, invenietur inclinatio plani: si fiat ut pondus O lib.15. ad pondus M datum lib.50, ita Radius 10000000 ad 333.33333. secantem anguli inclinationis DBC gr.72.32'.32". Demum dato pondere & plani inclinatione nota fiet potentia, si ut Secans datæ inclinationis ad Radium, ita fiat datum pondus ad aliud pondus, quod potentia valet sustinere in perpendiculari. Sit enim DBC gr.36, & M lib.50. Erit ut Secans 12360680 ad Radium 10000000, ita M lib.50 ad pondus O ferè lib.40 1/2, quod possit à potentia in aëre libero sustineri. Quare potentia sustinens pondus in plano inclinato est ad pondus, ut Radius ad Secantem anguli inclinationis; & potentia potens movere cum sit major potentiâ sustinente, etiam majorem habet Rationem quàm habeat Radius ad Secantem. Id quod intelligitur ex vi præcisè gravitationis; quicquid inferat discriminis partium conflictus. CAPUT

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86 Mechanics perpendicular BD, to the gravitation of the same M in the inclined BC, is likewise as BC to BD; therefore, by 11. lib.5, the gravitation of M in the perpendicular to the gravitation of O in the perpendicular is as the gravitation of M in the perpendicular BD to the gravitation of M in the inclined BC; therefore, by 14. lib.5, the gravitation of O in the perpendicular BD is equal to the gravitation of M in the inclined BC. By the same method it is shown that the gravitation of N in the inclined BS is equal to the gravitation of O in the perpendicular BD. Wherefore the gravitations of M and N are equal among themselves, since they are equal to the gravitation of O. It is therefore clear that by the same forces a greater weight may be held, or drawn upward, on an inclined plane than on a perpendicular; for their gravitation is the same, as I have shown; but the forces retaining or drawing it ought to correspond proportionally to the gravitation of the body. Wherefore, given the forces which can sustain a given weight O on the perpendicular BD, the gravitation may be known of the weight which those same forces will be able to sustain on the given inclined plane BC: namely, if the Radius be taken to the secant of the angle of the given inclination, as the given weight O to the sought weight M. Let O be 15 lb., and the angle DBC 36 degrees. Let it be as the radius 10000000 to the secant 12360680, so 15 lb. to 18 1/2 lb.; which is the weight M, weighing equally on the plane BC with the weight O on the perpendicular. On the other hand, given the weight M to be sustained by the same forces by which O is sustained on the perpendicular, the inclination of the plane will be found: if the weight O, 15 lb., be to the weight M given, 50 lb., as the Radius 10000000 to 333.33333, the secant of the angle of inclination DBC, 72.32'.32". Finally, given the weight and the known inclination of the plane, the power will be found, if as the Secant of the given inclination to the Radius, so the given weight be made to another weight, which the power is able to sustain on the perpendicular. For let DBC be 36 degrees, and M 50 lb. It will be as the Secant 12360680 to the Radius 10000000, so M 50 lb. to the weight O, about 40 1/2 lb., which can be sustained by the power in free air. Therefore the power sustaining a weight on an inclined plane is to the weight as the Radius to the Secant of the angle of inclination; and the power able to move, since it is greater than the sustaining power, also has a greater ratio than the Radius has to the Secant. This is understood from the force precisely of gravitation; whatever difference the conflict of the parts may introduce. CHAPTER

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Liber primus. CAPUT XIV. 87 CAPUT XIV. Quâ ratione corpus gravitet in planum inclinatum. Constituta Ratione gravitationis in plano inclinato, deter- minatis scilicet momentis, quæ ad descendendum obtinet corpus grave existens in plano inclinato, superest explicanda gravitatio, quam idem corpus exercet in planum inclinatum illud urgendo, atque deorsum premendo. Certum est autem planum verticale seu perpendiculare nullo pacto urgeri à cor- pore gravi, quod liberè descendere potest per suam directionis lineam, quæ cum non occurrat plano verticali, nullum ab eo recipit impedimentum. Quare corporis gravitas vires totas exercet, aut descendendo, aut repugnando contra retinentem, qui non plus adhibere debet conatûs in retinendo, etiam si pla- num verticale amoveatur: atque adeò nihil omninò gravitat in planum verticale. Contra verò in planum horizontale, quam maximè gravitant corpora; eò quod directionis lineâ in illud incurrente ad angulos rectos, motus omnis impeditur, & cunctas gravitatis vires deorsum contendentes ita subjectum planum excipit, ut nihil reliquum sit virium, quas vel minimo motu exerceat. Hinc si corporis in plano horizontali jacentis ansam teneas, nihil tibi prorsus est laborandum, nec quicquam percipis ponderis; at submoto plano lacertis omnibus est con- tendendum, ut illud retineas; tota enim gravitatio cum reti- nente luctatur, quæ planum sustinens urgebat. In hoc itaque planum verticale cum horizontali comparatur, quod cum ver- ticale nihil impediat motum, corpus in plano verticali omninò gravitat, sed in illud non gravitat: cum autem horizontale prorsus impediat motum, corpus in plano horizontali nihil gra- vitat, sed in illud totam suam gravitationem exercet. Eædem igitur vires, quæ ad descendendum in plano verticali impen- derentur, in urgendo plano horizontali insumuntur. Quæ cum ita sint, satis constat corpora gravia ita in pla- no inclinato gravitare, & obtinere momenta ad descenden- dum,

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Book One. CHAPTER XIV. 87 CHAPTER XIV. On what principle a body gravitates on an inclined plane. The principle of gravitation on an inclined plane having been established, namely, the moments which a heavy body, existing on an inclined plane, possesses for descending, there remains to be explained the gravitation which the same body exerts upon that inclined plane by pressing against it and pressing downward. Now it is certain that a vertical, or perpendicular, plane is in no way pressed by a heavy body which can descend freely along its line of direction, which, since it does not meet the vertical plane, receives from it no impediment. Therefore the weight of the body exerts its whole force, either by descending, or by resisting the holder, who need not use any greater effort in retaining it, even if the vertical plane be removed: and thus it gravitates not at all to the vertical plane. On the contrary, bodies gravitate as much as possible upon a horizontal plane; because, since the line of direction falls upon it at right angles, all motion is impeded, and the subject plane receives all the forces of gravity tending downward, so that there remains no force left which it might exercise by the least motion. Hence, if you hold the handle of a body lying on a horizontal plane, you have absolutely nothing to do, nor do you perceive any weight; but if the plane be removed, all your muscles must be strained in order to hold it; for the whole gravitation struggles with the holder, which was pressing upon the plane sustaining it. In this respect, therefore, the vertical plane is compared with the horizontal, because, since the vertical does not impede motion, the body on the vertical plane gravitates not at all to it, but in it; whereas the horizontal entirely impedes motion, so the body on the horizontal plane gravitates not at all, but exerts its whole gravitation upon it. Thus the same forces which would be expended in descending on a vertical plane are spent in pressing upon a horizontal plane. Since these things are so, it is clear enough that heavy bodies gravitate thus on an inclined plane, and possess the moments for descending,

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88 Mechanicorum dum, ut etiam in illud, à quo impediuntur, gravitent, il- ludque urgeant. Id verò fieri non potest nisi pro ratione impedimenti & mo- ræ, quam subjectum planum motui infert sustinendo corpora gravia; quæ proinde sibi relicta à directionis lineâ declinant, motumque deflectunt. Porrò in plano inclinato quantum sub- sit impedimenti, statim apparet, ac innotescit, quantum reli- quum sit virium ad descendendum; vires enim, quæ reliquæ sunt, si adjiciantur viribus impeditis, totam virium omnium summam conflare debent. Atqui ex superiori capite notæ sunt vires, quibus corpus gravitat in plano inclinato; igitur quæ est differentia gravitationis in plano inclinato, à gravitatione in plano verticlai, quod & perpendiculare, ea est mensura im- pedimenti, quod à subjecto plano infertur motui; atque adeò gravitationis corporis in planum. Cum itaque ostensum fuerit gravitatione in plano B S ad gravitationem in plano B D esse reciprocè ut B D ad B S, hoc est, ut Ra- dus ad secantem anguli inclinationis cum verticali, hoc est ut B V ad B S, patet vires non impeditas ad vires impeditas esse ut B V ad V S, quandoquidem totas gravita- tis vires refert B S. In planum igitur inclinatum B S gravitatio est ut V S, quæ in planum horizontale esset secundùm totas vires ut B S. Quare gravitatio in planum horizontale ad gra- vitationem in planum inclinatum est ut Secans B S ad exces- sum Secantis supra Radium, V S; seu, quod in idem recidit, si gravitatio in plano inclinato ad gravitationem in verticali po- natur ut Sinus complementi anguli inclinationis ad Radium, ita B R Radius ad D R Sinum versum anguli inclinationis. Id autem, quod de plano B S dictum est, de plano quoque B C, & cæteris quibuscunque dictum intelligatur; cum enim gravita- tio in plano inclinato B C ad gravitationem in perpendiculari sit ut B D, hoc est B X, ad B C, erit gravitatio in planum ho- izontale ad gravitationem in inclinatum ut B C ad X C, hoc est ut B T ad D T. Quare gravitatio in planum B S ad gravi- tationem in planum B C, est ut D R Sinus versus inclinationis D B S, ad D T Sinum versum inclinationis D B C; assumptis scilicet

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88 Mechanics so that they also incline toward that in which they are hindered, and press upon it. But this cannot happen except in proportion to the impediment and delay which the plane subject to motion imposes upon the sustaining of heavy bodies; and therefore, being left to themselves, they deviate from the line of direction and deflect their motion. Moreover, in an inclined plane, how much impediment there is appears at once, and it becomes known how much force remains for descent; for the forces that remain, if added to the obstructed forces, must make up the entire sum of all the forces. But from the preceding chapter the forces by which a body gravitates on an inclined plane are known; therefore, the difference between gravitation on an inclined plane and gravitation on a vertical plane, that is, perpendicular, is the measure of the impediment introduced by the underlying plane to the motion; and thus of the body's gravitation toward the plane. Since therefore it has been shown that the gravitation on plane B S to the gravitation on plane B D is reciprocally as B D to B S, that is, as the radius to the secant of the angle of inclination with the vertical, that is as B V to B S, it is clear that the unimpeded forces to the impeded forces are as B V to V S, since B S refers to the whole forces of gravity. In the inclined plane B S therefore, gravitation is as V S, which in a horizontal plane would, according to the whole forces, be as B S. Wherefore gravitation in the horizontal plane to gravitation in the inclined plane is as the secant B S to the excess of the secant over the radius, V S; or, which comes to the same thing, if gravitation in the inclined plane to gravitation in the vertical is put as the sine of the complement of the angle of inclination to the radius, then B R, the radius, is to D R, the versed sine of the angle of inclination. And what has been said of plane B S is likewise to be understood of plane B C and of any others whatsoever; for since gravitation in the inclined plane B C to gravitation in the perpendicular is as B D, that is B X, to B C, gravitation in the horizontal plane to gravitation in the inclined one will be as B C to X C, that is as B T to D T. Wherefore gravitation in plane B S to gravitation in plane B C is as D R, the versed sine of inclination D B S, to D T, the versed sine of inclination D B C; namely, assuming

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Liber primus. CAPUT XIV. 89 scilicet numeris tabulatis ad eundem Radium relatis; nam si li- neæ spectentur, non est Ratio ut DR ad DT, sed ut OT ad DT; neque enim idem est Radius BS & BC; ac proinde OT major est, quàm DR, sicuti Radius BI major est Radio BS; vel assumpto eodem Radio BD, Ratio est ut VS ad XC, excessus secantium supra Radium. Id verò ex dictis sub finem capitis superioris videtur mani- festum: nam si in plano BC retinetur pondus lib. 50. iisdem viribus, quibus in perpendiculari suspenderentur lib. 40 ́, pa- tet à plano sustineri lib. 9 ́; ac proinde grave, quod habet gra- vitatem totam ut 100, in plano BC gravitabit ut 81, & urge- bit ut 19 subjectum planum. Ex his fieri potest satis quæ- renti, cur sustinens columnam OR plus gravitatis percipiat, quàm qui sustinet columnam SR: quia nimirum, qui sustinet, est pars plani inclinati, in quo ja- cens concipitur columna: quan- do igitur est pars plani habentis inclinationem LOR, gravitas, quæ sustinetur à subjecto plano, se habet ad totam gravitatem ut Sinus Versus anguli LOR ad Radium; Quando autem est pars plani habentis inclinationem VSR, gravitatio in sub- jectum planum sustinens est ad totam gravitationem ut Sinus Versus anguli VSR ad eundem Radium. Atqui Sinus Versus anguli VSR minoris minor est Sinu Verso anguli LOR ma- joris; igitur minor est gravitatio SR, quam OR. Verum qui- dem est illud, quod si in R aliquo obice prohibeatur, ne de- scendat; variatâ inclinatione, quo fit minor sustinentis labor, eò augetur magis conatus potentiæ in R detinentis columnam, ne juxta plani inclinationem descendat. Hinc si duo sint co- lumnam inclinatam deferentes, qui illam in R sustinet, plus subit laboris, quàm qui in O, aut S: quia præter gravitatio- nem, quam percipit tanquam pars plani inclinati SR aut OR, debet præterea retinere columnam proclivem ad descensum propter plani inclinationem; ideò cùm scalas, aut montis cli- vum conscendunt, qui in superiore loco est, minimum subit M

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namely, by the tabulated numbers referred to the same Radius; for if the lines are considered, the proportion is not as DR to DT, but as OT to DT; nor indeed is the Radius BS the same as BC; and therefore OT is greater than DR, just as the Radius BI is greater than the Radius BS; or, taking the same Radius BD, the proportion is as VS to XC, the excess of the secants above the Radius. And this indeed appears manifest from what has been said at the end of the preceding chapter: for if in the plane BC a weight of 50 lb. is retained by the same forces by which 40 lb. would be suspended in the perpendicular, it is clear that 9 lb. are sustained by the plane; and therefore a heavy body, which has the whole weight as 100, in the plane BC will weigh as 81, and will press the plane subject beneath it as 19. From these things an answer may be given to the question why the supporter of the column OR perceives more weight than he who supports the column SR: namely, because the supporter is part of the inclined plane in which the lying column is conceived. When therefore he is part of a plane having the inclination LOR, the weight sustained by the supporting plane is to the whole weight as the versed sine of the angle LOR to the Radius; but when he is part of a plane having the inclination VSR, the gravitation upon the supporting plane is to the whole gravitation as the versed sine of the angle VSR to the same Radius. But the versed sine of the smaller angle VSR is less than the versed sine of the larger angle LOR; therefore the gravitation at SR is less than at OR. Yet this is true: if at R it be hindered by some obstacle from descending, then, the inclination being altered, the less the labor of the supporter, the more the effort of the power holding the column at R, so that it may not descend according to the inclination of the plane, is increased. Hence if there are two who bear the inclined column, he who supports it at R undergoes more labor than he at O or S: because, besides the gravitation which he perceives as part of the inclined plane SR or OR, he must moreover hold back the column, which tends toward descent because of the inclination of the plane; therefore, when they ascend stairs or a mountain slope, he who is in the higher place undergoes the least

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90 Mechanicorum laboris. Huc etiam revocari posse videtur ratio, ob quam in elevando pontes illos versatiles, qui arcium portis opponuntur, initio major percipiatur difficultas, sed demùm facillimè eleventur. Verùm id ex dicendis inferiùs clariùs constabit; neque enim omnium gravium, quocunque se tandem modo habeant, eadem est ratio; cum animum diligenter advertere oporteat, ut innotescat planum inclinatum, in quo suam gravitationem exercent, & habent vires ad descendendum. Non est autem per dissimulantiam prætereunda difficultas, quæ facessere posset aliquid negotij, & gravitationis Rationem constitutam convellere videretur. Est siquidem certum apud omnes mechanicos, tam ubi de libra, quàm ubi de vecte sermo est, aliam servari Rationem quàm Sinuum Versorum in momento potentiæ, aut ponderis determinando. Sit vectis, aut libræ brachium EC, hypomochlion seu centrum C; attollatur in H, aut in D; omnes consentiunt momentum potentiæ aut ponderis in E ad momentum in H, esse ut HC ad IC, ad momentum verò in D esse ut DC ad FC. Est igitur, inquis, gravitatio in planum DC ad gravitationem in planum horizontale EC, ut FC ad DC; in planum verò HC, ut IC ad HC, hoc est ut Sinus Rectus anguli inclinationis ad Radium. Priùs verò, quàm me ab hac difficultate expediam, ostendo non satis aptè gravitationem in planum inclinatum desumi posse ex Sinu Recto anguli inclinationis. Quandoquidem vis descendendi in plano DC ad tota corporis liberi gravitatione est ut DF ad DC, igitur si gravitatio in planu[m] DC ad totam gravitatione est ut FC ad DC, tota virium summa est DF plus FC, ac tota vis gravitandi, ubi nullum est impedimentum, est DC; igitur DC, & DF plus FC, æquales sunt, contra 20. lib. 1. Eucl. Neque hic liceat ad æqualitatem potentiarum confugere, ut sicut per 47. lib. 1. Eucl. linea DC potest quadrata linearum DF, FC, ita vis totius gravitatis æqualis gravitationibus in plano inclinato & in planum inclinatum eandem servet proportionem laterum trianguli DFC, adeò ut totam gravitatem Secans

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90 Mechanical operations. This also seems able to be referred to the reason why, in raising those movable bridges which are placed before the gates of fortresses, at first a greater difficulty is perceived, but afterward they are most easily raised. But this will be made clearer from what is to be said below; for not all heavy bodies, however they may be situated, have the same rationale; since care must be taken to observe, so that it may become evident, the inclined plane in which they exert their gravitation, and have force to descend. Nor, however, must the difficulty be passed over in silence, which might create some trouble, and seem to overthrow the established rationale of gravitation. For it is indeed certain among all mechanicians, both where the lever and where the balance are discussed, that a different rule is followed than that of reversed sines in determining the moment of power, or of weight. Let there be the arm EC of a lever or balance, the fulcrum or center C; let it be raised in H, or in D; all agree that the moment of power or weight at E to the moment in H is as HC to IC, but to the moment in D as DC to FC. It is therefore, you say, gravitation on the plane DC to gravitation on the horizontal plane EC, as FC to DC; but on the plane HC, as IC to HC, that is, as the sine of the angle of inclination to the radius. But before I free myself from this difficulty, I show that gravitation on an inclined plane cannot adequately be taken from the sine of the angle of inclination. Since the force of descending on the plane DC, compared to the whole gravitation of the free body, is as DF to DC, therefore if gravitation on the plane DC compared to the whole gravitation is as FC to DC, the whole sum of forces is DF plus FC, and the whole force of gravitating, where there is no impediment, is DC; therefore DC and DF plus FC are equal, contrary to Euclid, Book 1, Proposition 20. Nor may one here take refuge in the equality of powers, so that just as by Euclid, Book 1, Proposition 47, the line DC can square the lines DF, FC, thus the force of the whole gravity, equal to the gravitations on the inclined plane and into the inclined plane, may preserve the same proportion to the sides of the triangle DFC, so that the whole gravity Secans

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Liber primus. CAPUT XIV. 91 Secans anguli inclinationis exprimat, gravitationem in plano inclinato Radius, Tangens verò gravitationem in planum inclinatum. Si enim Quadratum DC æquale est quadratis DF, & FC simul sumptis, non tamen linea DC æqualis est aggre- gato linearum DF & FC: neque eadem est inter lineas DF & DC Ratio, quæ inter earum quadrata; sed est sub duplicata quadratorum: Quare cum gravitatio in plano inclinato DC ad gravitationem in perpendiculari, non sit ut quadratum DF ad quadratum DC; sed ut linea DF ad lineam DC, frustrà ad quadrata confugimus, quorum nulla hîc habetur ratio. In eo itaque æquivocatio consistit, quod pondus in D constitutum, & applicatum brachio DC concipitur esse in plano inclinato DC, contra quàm res est: in eo siquidem plano intelligendum est, in quo ad motum determinatur; illud autem est planum DG, quod tangit circulum ED; & sic deinceps, pro ut diversa circuli puncta à diversis planis contingi possunt. Quare in D momentum ad descendendum per DG ad totam gravitationem est ut DF ad DG, hoc est ut FC ad CD, per 8. lib.6. hoc est ut FC ad EC. Est igitur brachium libræ seu vectis CD, sustinens pondus seu potentiam D, quæ cum habeat vires universas ut EC, gravitationis autem momenta habeat solùm ut FC, impeditur à sustinente ut FE; est autem EF Sinus Versus anguli FCD, hoc est anguli inclinationis FDG. Quare gravitatio ponderis contrà subjectum corpus, quod impedit motum perpendicularem, ad totam gravitationem est, ut Sinus Versus anguli inclinationis plani, per quod fieri potest motus, ad Radium. Hinc vides valdè disparem esse rationem gravitationis in sustinendo corpore inclinato, si illud liberè moveri possit, ac si circa centrum perfici debeat motus. Nam si DC sit columna, aut pons versatilis, retineaturque in C, jam punctum C vicem obtinens subjecti plani, illiusque munere fungens, sustinet ponderis partem EF, reliqua FC, quæ est mensura momenti ad descendendum, debet sustineri à potentia motum impediente per DG. Sin autem per DC planum columna moveri possit rectâ & descendere, vis descendendi ad totam gravitationem est ut DF ad DC, gravitatio autem contra sustinentem est ad totam gravitationem ut Sinus Versus anguli inclinationis M 2

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Book One. CHAPTER XIV. 91 The sine of the angle of inclination expresses the gravitation in the inclined plane; the radius, however, expresses the gravitation in the inclined plane. For if the square of DC is equal to the squares of DF and FC taken together, yet the line DC is not equal to the sum of the lines DF and FC: nor is the ratio between the lines DF and DC the same as that between their squares; but it is under the duplicate ratio of the squares. Wherefore, since the gravitation in the inclined plane DC to the gravitation in the perpendicular is not as the square of DF to the square of DC, but as the line DF to the line DC, we resort in vain to squares, since no ratio of them is here available. The equivocation therefore consists in this, that the weight placed in D, and applied to the arm DC, is conceived to be in the inclined plane DC, whereas the fact is otherwise: it ought indeed to be understood as being in that plane in which it is determined to move; and that is the plane DG, which touches the circle ED; and so likewise in subsequent cases, according as different points of the circle may be touched by different planes. Wherefore, in D, the moment for descending through DG to the whole gravitation is as DF to DG, that is, as FC to CD, by 8. lib. 6., that is, as FC to EC. Therefore the arm of the balance or lever CD, supporting the weight or power D, which, since it has universal forces as EC, but moments of gravitation only as FC, is hindered by the support as FE; and EF is the versed sine of the angle FCD, that is, of the angle of inclination FDG. Therefore the gravitation of the weight against the body beneath, which impedes the perpendicular motion, to the whole gravitation is as the versed sine of the angle of inclination of the plane by which motion can occur, to the radius. Hence you see that the ratio of gravitation in supporting an inclined body is very different, if it can move freely, from what it is if the motion must be performed about a center. For if DC be a column, or a movable bridge, and be held at C, then the point C, taking the place of the subjacent plane and performing its office, sustains the part EF of the weight; the remaining FC, which is the measure of the moment for descending, must be sustained by the power impeding motion through DG. But if the plane through DC can move upright as a column and descend, the force of descending to the whole gravitation is as DF to DC; the gravitation, however, against the supporter is to the whole gravitation as the versed sine of the angle of inclination M 2

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Mechanicorum 92 F D C ad Radium; qui enim sustinet grave, dum descendit inclinatum, habet rationem plani inclinati. Neque id mirum videri debet, quandoquidem plurimum refert, an per planum D G an verò per DC sit determinatio ad motum, & quâ ratione sustinens opponatur virtuti motivæ: quare cùm diversâ ratione opponatur motui circa centrum C, ac motui per planum D C, etiam dispar erit in sustinendo difficultas. Ex his, quæ tùm hoc, tùm superiori capite disputata sunt, habes quid funambulis respondeas volatum mentiri meditantibus, cum pectore insistentes intento funi, diductis cruribus & extensis brachiis, corpus æqualibus momentis librant, séque ex editâ turri in depressiorem locum præcipites dant; si fortè, ut noverint, quàm solidus esse debeat ac validus funis, quo iis utendum est, quærant, quantis momentis corpus urgeat subjectum funem. Datâ enim turris altitudine B D C R X S B D C R X S B D C R X S Datâ enim turris altitudine BD, & depressioris loci, in quem descendendum est, distantiâ DC, collectisque in summam harum quadratis, Radix summæ dabit BC funis longitudinem; ex quâ si auferatur BX turris altitudini BD æqualis, erit BC divisa in X juxtà Rationem momentorum, quæ corporis gravitas exercet in plano inclinato, & in planum inclinatum. Sic positâ BD ped. 150, & DC ped. 200, BC est ped. 250: ex quâ si auferatur BD, erit BX 150, & XC 100. Statue autem totius gravitatis corporis funambuli momenta 220; hæc dividantur in duas partes, quarum major sit sesquialtera minoris, sicut BX inventa est ipsius XC sesquialtera, & erunt momenta quidem ad descendendum in plano inclinato 132, momenta verò gravitationis in planum inclinatum, hoc est in subjectum funem, 88. Hæc tamen intelligenda sunt eâ factâ hypothesi, quòd funis rectâ intentus permaneret: cæterùm cum & suopte pondere, & sub impositi corporis mole subsidat, atque inflectatur, præsertim circà medium, satis apparet adhuc majorem subjecti plani inclinationem æstimandam esse, quàm quæ ex altitudine DB & distantiâ DC inferatur, quin & illam pro diversâ ab extremitatibus distantiâ subinde mutari, ac proinde validiori fune opus esse. CAPUT

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Mechanics 92 F D C to the radius; for he who supports a heavy body while it descends on an incline has the property of an inclined plane. Nor ought this to seem surprising, since it matters greatly whether the motion is determined by the plane D G or indeed by DC, and in what way the supporter opposes the motive force: wherefore, since it is opposed to motion around the center C in one way, and to motion through the plane D C in another, the difficulty in sustaining will also be different. From what has been discussed both in this and in the preceding chapter, you have what you may answer to rope-dancers who, while pretending to fly, standing with the chest pressed against the taut rope, legs spread apart and arms extended, balance the body by equal moments and cast themselves headlong from a high tower to a lower place; if perhaps, as they may wish to know how solid and strong the rope must be that they are to use, they ask by how many moments the body presses upon the rope beneath it. For given the height of the tower B D C R X S B D C R X S B D C R X S Given then the height of the tower BD, and the distance of the lower place to which one must descend, DC, and the squares of these having been collected into a sum, the root of the sum will give BC, the length of the rope; from which if BX, equal to the height of the tower BD, be subtracted, BC will be divided in X according to the ratio of the moments which the weight of the body exercises on the inclined plane and on the inclined plane. Thus, with BD set at 150 feet, and DC at 200 feet, BC is 250 feet: from which if BD be subtracted, BX will be 150, and XC 100. Now let the moments of the whole weight of the rope-dancer’s body be set at 220; let these be divided into two parts, the greater of which shall be one and a half times the lesser, just as BX found is one and a half times XC; and the moments for descending on the inclined plane will be 132, but the moments of gravitation toward the inclined plane, that is, toward the rope beneath it, 88. These, however, are to be understood on the hypothesis that the rope remained stretched in a straight line; otherwise, since it yields both by its own weight and under the mass of the body laid upon it, and bends, especially around the middle, it is quite evident that the inclination of the plane beneath must still be judged greater than that inferred from the height DB and the distance DC, and moreover that it is continually altered according to its distance from the ends, and therefore that a stronger rope is needed. CHAPTER

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Liber primus. CAPUT XV. 91 CAPUT XV. Inquiruntur Rationes gravitationis corporum suspensorum. Consideratâ corporum gravitatione tùm in plano inclinato, tùm in planum inclinatum, consequens est, ut ad eorum- dem gravitationem, si ex fune suspendantur, gradum facia- mus; hæc enim illi valdè affinis est speculatio: id quod facilè intelligat, quisquis animum advertere voluerit, remque totam penitiùs introspicere. Ex his si quidem, quæ hactenus disputa- ta sunt, lux, opinor, non modica ad hanc, quam examinandam suscipimus quæstionem, derivabitur. Pendeat ex clavo C ad perpen- diculum globus ferreus A, quem suppositum planum horizontale B D ita exactè contingat, ut nihil de funiculi C A intentione remit- tatur. Satis apparet subjecto pla- no B D non incumbere globum A, sed omnia suæ gravitationis, qua deorsum nititur, momenta exer- cere contrà clavum C, ex quo suspensus ad perpendiculum pendet. Quod si aut clavus C, nemine funem retinente, revel- leretur, aut funis C A præcideretur, jam tota vis descendendi, quæ corpori A inest, urgeret subjectum planum B D; nec ta- men in motum erumperet globus, quia planum B D; pari usque- quaque ad perpendiculum inclinatione libratur, atque adeò motui prorsus obsistit. Jam verò si globum A pariter ex perpendiculo C A penden- tem contingat planum aliud non quidem horizontale, sed in- clinatum E F, manifestum est totam pariter gravitationem exerceri contra clavum C retinentem, planumque contingens M 3

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Book One. CHAPTER XV. 91 CHAPTER XV. The causes of the gravitation of suspended bodies are investigated. Having considered the gravitation of bodies both on an inclined plane and toward an inclined plane, it follows that we should advance to their gravitation when suspended by a cord; for this is a speculation very closely akin to the former. Anyone who will give it attention and look more deeply into the whole matter will easily understand this. From what has thus far been discussed, I think, not a little light will be derived for this question, which we undertake to examine. Let an iron ball A hang by a plumb line from nail C, and let the horizontal plane B D placed beneath touch it so exactly that nothing of the tension of the cord C A is relaxed. It is sufficiently evident that the ball A does not rest on the plane B D beneath, but exerts all the forces of its gravitation, by which it tends downward, against the nail C from which it hangs suspended perpendicularly. But if either the nail C were pulled out, with no one holding the cord, or the cord C A were cut, then the whole force of descent which is in body A would press upon the plane B D beneath; yet the ball would not break into motion, because the plane B D is balanced by an inclination everywhere equal to the perpendicular, and therefore wholly resists motion. Now indeed, if the ball A, hanging likewise from the plumb line C A, should touch another plane not horizontal, but inclined, E F, it is manifest that the whole gravitation likewise is exerted against the retaining nail C, and the plane touching it

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Mechanicorum 94 omninò non urgeri, nisi præciso funiculo sibi relinquatur glo- bus, ut in inclinato plano EF ad descensum pronus contra sub- jectum planum nitatur, à quo cogitur, ut in motu à recto, quod ad universi centrum est, itinere deflectat. Quod si planum inclinatum EF ita suspenso globo A subji- ciatur, ut recta linea centrum gravitatis A, & punctum sus- pensionis H conjungens parallela sit lineæ EF, quam in plano inclinato descendens globus percurreret; momenta quidem gravitationis, quæ in eo plano obtineret globus ad descenden- dum, exercebit adversùs clavum retinentem in H, subjectum verò planum EF perinde urgebitur, atque si nullo retinente li- bera esset globo descendendi facultas: vis enim, quâ prohibe- tur globus, ne moveatur secundùm rectam lineam, ut constat, opponitur descensui in plano inclinato; ejus autem directio AH non opponitur nitenti in planum, cui parallela est. Contra verò si globus in plano inclinato constitutus retinea- tur secundùm rectam lineam, quæ ad perpendiculum cadit in subjectum planum EF, nimirum secundùm lineam LO, im- peditur quidem, ne contra planum nitatur; sed vis ista sic reti- nens nullâ ratione adversatur motui in plano inclinato, quin iisdem gravitatis momentis descendat globus in eo plano; si quidem retinentis directio LO maneat semper adversùs illud planum perpendicularis. Nam si potentia retinens secundùm eam directionem agat, ut neque congruat perpendiculari LO, neque parallelæ HA, obsistet gravitationi corporis sivè in pla- no inclinato, sivè in planum inclinatum pro ratione anguli, quem retinentis directio inter perpendicularem LO, & paral- lelam HA interjecta, constituet cum plano inclinato. Quæ enim inter LO & CA fuerit, elidet omnem corporis conatum adversùs planum, à quo illud avellit; non autem omnem eum, qui in plano inclinato deorsum rapit. Quæ verò fuerit inter CA & HA, tollet quidem descensum in plano EF inclinato; sed non omninò prohibebit, quin subjectum planum, cui aliqua- tenus nititur, urgeat. Id quod facilè intelligas, si plana subjecta BD horizontale, & EF inclinatum ex maximè flexili mate- ria, puta, papyro, concipias; in quâlibet enim suspensione inter C, & L, planum BD horizontale deflectetur ex pondere, non autem inclinatum EF: contrà verò in omni suspensione inter

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Mechanics 94 cannot be pressed upon at all, unless, the cord being cut away, the globe be left to itself, so that on the inclined plane EF, being prone to descent, it may press against the plane beneath, by which it is compelled to deviate from a motion in a straight line, which is toward the center of the universe. But if the inclined plane EF be so placed beneath the suspended globe A that the straight line joining the center of gravity A and the point of suspension H be parallel to the line EF, which the globe descending on the inclined plane would traverse, then indeed the moments of gravitation, which in that plane the globe would exert in descending, it will exert against the retaining nail in H; but the plane EF beneath will be pressed just as though the globe had free power to descend with no retainer at all: for the force by which the globe is prevented from moving in a straight line, as is clear, is opposed to descent on the inclined plane; but its direction AH is not opposed to the tendency toward the plane, to which it is parallel. On the other hand, if the globe placed on the inclined plane be retained according to the straight line which falls perpendicularly upon the plane beneath EF, namely according to the line LO, it is indeed prevented from pressing against the plane; but this retaining force in no way opposes motion in the inclined plane, so that the globe descends in that plane with the same moments of gravity; provided that the direction of the retainer LO always remains perpendicular to that plane. For if the retaining power acts in such a direction that it corresponds neither with the perpendicular LO nor with the parallel HA, it will resist the gravity of the body, either in the inclined plane or against the inclined plane, according to the angle which the direction of the retainer, placed between the perpendicular LO and the parallel HA, will make with the inclined plane. For whatever lies between LO and CA will destroy every effort of the body against the plane from which it is pulled away; but not every effort by which it is drawn downward in the inclined plane. But whatever lies between CA and HA will indeed remove descent on the inclined plane EF; yet it will not altogether prevent the subject plane, against which it presses to some degree, from being urged. This you may easily understand if you imagine the subject planes BD, horizontal, and EF, inclined, made of a very flexible material, such as paper; for in any suspension between C and L, the horizontal plane BD will bend under the weight, but not the inclined plane EF: on the contrary, in every suspension between

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Liber primus. CAPUT XV. 95 inter C & H, planum inclinatum E F rectetur; at non item ho- izontale BD, quia nimirum inclinatum E F prohibet, ne recta HA ad perpendicularum accedens verticalis fiat. Unum hîc præterea considerandum venit, quod superiori capite subindicatum fuit; si videlicet non ex flexili fune deorsum pendeat globus, sed rigido bacillo circà axem inferiùs positum versatili adnectatur superiùs. Sit rectus bacillus AB, cujus extremitas altera adnexum habeat globum B, altera sit circà axem A versatilis. Satis aperta conjectura est bacillum AB vicem subire plani, cui innitatur globus in B, qui proinde prohibetur, tùm ne ad perpendicularum cadat per BD, tùm ne per BA delabatur: linea igitur plani, per quod moliri motum poterit globus B, nulla alia congruentiùs assignari queat præter BC, quæ cum bacillo BA rectum angulum constituit. Perindè igitur in motum incitabitur, atque si in plano esset, cujus inclinatio angulum efficeret æqualem angulo elevationis bacilli supra planum horizontale GA. Cum enim recta BD producta cadens in planum horizontale, angulum BS A Rectum efficiat, reliqui duo simul SAB, AB S, Recto ABC æquales sunt; & communi AB S dempto, superest SAB elevationis angulus æqualis angulo SBC inclinationis plani. Quare ductâ Tangente DE, erit BE Secans anguli inclinationis, BD verò Radius: ac proptereà ad descendendum in hujusmodi plano BC momenta, ad totam gravitatem in perpendicularo BD, erunt ut Radius BD ad Secantem BE, juxta ea, quæ cap. 13. hujus lib. demonstravimus. Quia tamen in motu globus ex bacilli conversione circà axem A non potest percurrere rectam BC, sed ita retinetur à bacillo, cui adnectitur, ut descendat in F, jam in alio plano minorem inclinationem habente constitutus intelligitur, nimirùm in plano FG, quod cum perpendiculo FL efficit angulum inclinationis GFL æqualem angulo LAF elevationis: id quod eâdem planè methodo, ac superiùs factum est, demonstratur. Ex

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Liber primus. CHAPTER XV. 95 between C and H, let the inclined plane E F be straightened; but not the horizontal B D, because, namely, the inclined E F prevents the straight line H A, as it approaches the perpendicular, from becoming vertical. One other thing must here moreover be considered, which was hinted at in the preceding chapter: namely, if the globe does not hang downward from a flexible cord, but is attached above to a rigid rod movable around a lower axis. Let AB be a straight rod, one end of which has attached to it the globe B, while the other is movable around the axis A. It is sufficiently clear by conjecture that the rod AB takes the place of the plane on which the globe at B rests, and is therefore prevented both from falling toward the perpendicular along B D, and from slipping along B A: the line of the plane, therefore, by which the globe B can be made to move, can more fittingly be assigned to no other than B C, which with the rod B A forms a right angle. It will accordingly be set in motion just as if it were on a plane whose inclination made an angle equal to the angle of elevation of the rod above the horizontal plane G A. For when the straight line B D, extended and falling upon the horizontal plane, makes the right angle B S A, the other two together, S A B and A B S, are equal to the right angle A B C; and with the common B S A removed, there remains S A B, the angle of elevation, equal to the angle S B C of inclination of the plane. Wherefore, if the tangent D E be drawn, B E will be the secant of the angle of inclination, and B D the radius: and therefore the moments for descending on such a plane B C, as compared with the whole gravity on the perpendicular B D, will be as the radius B D to the secant B E, according to what we demonstrated in chapter 13 of this book. Since, however, in motion the globe, from the turning of the rod around the axis A, cannot traverse the straight line B C, but is so restrained by the rod to which it is attached that it descends in F, it is now understood to be placed in another plane having a smaller inclination, namely in the plane F G, which with the perpendicular F L makes the angle of inclination G F L equal to the angle of elevation L A F: this is demonstrated by exactly the same method as was done above. Ex

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Mechanicorum Ex quo fit, quemadmodum in hujusmodi conversione globus in alio atque alio plano inclinato constituitur, ita alia atque alia obtinere gravitatis momenta: in B siquidem gravitat ut BD ad BE, in F verò ut HF ad FI. Cum igitur Radius utrobiqve ex hypothesi æqualis sit, videlicet DB, & HF, major autem sit BE Secans majoris anguli DBE, quàm FI Secans minoris anguli HFI, constat ex 8. lib.5. majorem Rationem esse HF ad FI minorem, quàm DB ad BE majorem, atque adeò globum magis in F quàm in B gravitare, ut deorsum moveatur, atque adeò minùs etiam conniti contrà planum, in quo est, videlicet adversùs bacillum FA, magis verò adversùs bacillum BA. Ex his attentè perpensis facilis est transitus ad suspensorum corporum gravitationem investigandam. Sit enim jam non inferiùs, sed superiùs positus Axis A, circa quem versatilis est funiculus AB, cui globus B adnectitur. Constat sanè non ad perpendiculum BD cadere posse globum B; sed à recto deorsum tramite deflectere, funiculo scilicet AB eum retinente, quemadmodum rigidus bacillus OB eum aliquatenùs sustineret. Quia autem bacillo OB sustinente, vis descendendi ea esset, quæ per planum inclinatum BC, eadem pariter est funiculo retinente; videlicet per planum BC, in quod recta AB ad rectos angulos incidit. Momenta igitur gravitatis in eo plano inclinato, ad gravitatis momenta si corpus liberè descenderet, in eâ sunt Ratione, quæ est DB ad BE; hoc est DO ad OB per 8. lib.6. hoc est KB ad BA per 4. lib.6. Haud dispari methodo ratiocinantes ostendemus globi in F constituti momenta ad gravitandum esse perinde, atque si esset in plano inclinato FI, in quod ad rectos angulos cadit funiculus AF; ac proinde gravitatio in F, si descendendi vis præcisè spectetur, ad gravitationem globi liberi, est ut HF ad FI, hoc est, ut GF ad FA. Ex quo apertiùs liquet, quàm ut in eo explicando diutiùs immorari

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Mechanics From which it follows that, just as in this kind of turning the globe is situated in one inclined plane and then in another, so it obtains one and another moment of gravity: in B indeed it gravitates as BD is to BE, but in F as HF is to FI. Since therefore the Radius in both cases is equal by hypothesis, namely DB and HF, but BE is greater as the secant of the greater angle DBE than FI, the secant of the smaller angle HFI, it is clear from Book 5, Proposition 8 that the greater ratio of HF to FI is less than that of DB to BE, and therefore the globe gravitates more in F than in B, so that it moves downward, and therefore also presses less against the plane in which it is situated, namely against the rod FA, but more against the rod BA. Having carefully considered these things, the transition is easy to the investigation of the gravitation of suspended bodies. Let there now be an Axis A placed not below but above, around which a string AB is able to turn, to which globe B is attached. It is certainly clear that globe B cannot fall perpendicularly along BD, but deviates from the straight downward path, namely because the string AB restrains it, just as a rigid rod OB would sustain it to some extent. But since, with the rod OB sustaining it, the force of descending would be that which is through the inclined plane BC, the same is likewise through the string restraining it; namely through the plane BC, on which the straight line AB falls at right angles. The moments of gravity, therefore, in that inclined plane, compared with the moments of gravity if the body were to descend freely, are in that ratio which is DB to BE; that is, DO to OB by Book 6, Proposition 8; that is, KB to BA by Book 6, Proposition 4. By reasoning by no different method we shall show that the moments of the globe placed in F for gravitating are just as if it were in the inclined plane FI, on which the string AF falls at right angles; and therefore the gravitation in F, if the force of descending is considered exactly, is to the gravitation of the free globe as HF is to FI, that is, as GF is to FA. From which it is more clearly evident than that it need not be dwelt upon any longer in explaining it

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Liber primus. CAPUT XV. 97 immorari oporteat, alia subinde atque alia esse momenta gra- vitatis corporis suspensi, pro ut major aut minor est angulus declinationis à perpendiculo AG, haud aliter quàm si in aliis atque aliis planis inclinatis constitueretur; quo enim minor est declinationis angulus GAF, eò major est angulus inclinationis plani, quippe qui est illius complementum. Constat si qui- dem angulos GAF, GFA simul, esse æquales tùm Recto AFI, tùm Recto GFH; ac proinde dempto communi GFI, remanet HFI angulus inclinationis plani æqualis angulo GFA, qui est complementum anguli declinationis GAF. Quare quò declinationis angulus major est, eò minus est complementum, ac propterea est minor angulus inclinationis plani: in plano autem minùs inclinato majora sunt gravitatis momenta. Quò igitur corpus suspensum magis à perpendiculo removetur, eò majora percipiuntur gravitatis momenta, ma- jorque vis requiritur in eo, qui motum prohibere voluerit, ut & ipsa experientia unicuique facilè demonstrat, & ratio evin- cit; cum enim AB & AF æquales sint, major est Ratio KB ad BA, quàm GF ad FA per 8. lib. 5. est nimirum KB major, & GF minor. Quoniam verò quò major est gravitatio in plano inclinato, minor est in planum inclinatum; hoc ipso, quod facto declina- tionis angulo GAB majore, quàm GAF, major est ad descen- dendum propensio, minor est conatus adversùs axem A reti- nentem. Id quod manifesto etiam experimento deprehen- des, si observaveris minùs intentum esse funiculum AB, quàm AF. Hinc & illud satis dilucidè apparet, quod longitudinis funiculi non exigua ratio habenda est; ex eâ scilicet pen- det, quod in plano magis aut minùs inclinato constitutum censeatur corpus grave suspensum. Si enim globus F ex fu- niculo AF pendeat, declinationis angulus est GAF: at verò si funiculus, quo suspenditur, sit MF, angulum de- clinationis facit GMF, qui cum externus sit, major est interno MAF per 16. lib. 1. ac propterea minor est incli- natio plani FN facientis cum rectâ MF angulum Rectum, quàm sit inclinatio plani FI, cui perpendicularis est recta AF. Plus igitur momenti ad gravitandum habet glo- N

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Book One. CHAPTER XV. 97 it ought to remain in place; thus the moments of the weight of a suspended body are now one and now another, according as the angle of deviation from the perpendicular AG is greater or less, no otherwise than if it were placed in different inclined planes; for the smaller the angle of deviation GAF, the greater is the angle of inclination of the plane, since it is the complement of that angle. Indeed, the angles GAF and GFA together are equal, both to the right angle AFI and to the right angle GFH; and therefore, subtracting the common angle GFI, there remains the angle HFI, the inclination of the plane, equal to the angle GFA, which is the complement of the angle of deviation GAF. Wherefore, the greater the angle of deviation is, the smaller is the complement, and therefore the smaller is the angle of inclination of the plane: but on a less inclined plane the moments of gravity are greater. Thus, the more a suspended body is moved away from the perpendicular, the greater are the moments of gravity perceived, and the greater force is required in one who would prevent the motion; experience itself easily demonstrates this to anyone, and reason proves it; for since AB and AF are equal, the ratio of KB to BA is greater than that of GF to FA, by Book 5, Prop. 8; for KB is greater and GF smaller. Since, moreover, the greater the gravitation on an inclined plane, the less is it in the inclined plane itself, by the very fact that, when the angle of deviation GAB is made greater than GAF, there is a greater tendency to descend and a smaller effort against the axis A that holds it back. This you will also clearly observe by experiment, if you note that the cord AB is less taut than AF. Hence this also becomes sufficiently clear, that no small account must be taken of the length of the cord; for upon it depends whether a suspended heavy body is deemed to be placed on a plane more or less inclined. For if the globe F hangs from the cord AF, the angle of deviation is GAF: but if the cord by which it is suspended is MF, it forms the angle of deviation GMF, which, being external, is greater than the internal angle MAF by Book 1, Prop. 16; and therefore the inclination of the plane FN, which forms a right angle with the straight line MF, is less than the inclination of the plane FI, to which the straight line AF is perpendicular. Therefore the globe has more effect in gravitating...

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98 Mechanicorum bus F, si ex breviore funiculo M F pendeat, quàm si ex longiore A F. Quæ cum ita sint, haud sanè incongrua se nobis offert me- thodus pondus ex depressiore in altiorem locum transferendi; si videlicet id curemus, ut ex satis valido & longiore fune sus- pendatur; sublato etenim partium attritu, qui fieret, si per pla- num raptaretur pondus, minore virium jacturâ trahi potest. Sit corpus grave ubi A, quod at- tollere oporteat, & in superiorem locum R S transferre. Si ex C brevio- ri fune suspendatur, trahere illud po- terit usque in R, quicunque facto de- clinationis angulo A C R potest illud cum aliquo virium excessu retinere, & obsistere gravitatis momentis, quæ obtinet in R. At si ex longiore fune D A pendeat, idem corpus A trahi poterit, & retineri in S, ne deorsum labatur, & quidem mino- re conatu; facto enim declinationis angulo A D S minore, quàm A C R, in S pariter minùs gravitat quàm in R. Angu- lum autem A D S minorem esse angulo A C R constat, si rectæ A R, A S ducantur: nam C A, C R æqualia sunt latera ex hy- pothesi, item D A, D S æqualia; est scilicet idem funiculus, qui primum perpendicularis eadit, deinde à perpendicularo re- movetur: in Triangulo Ioscele C A R anguli ad basim A R æquales sunt per 5. lib. 1. item in triangulo Ioscele D A S an- guli ad basim A S æquales inter se sunt. Porrò angulus D A S major est angulo C A R; ergo & reliquus D S A major reliquo C R A. Cum itaque tres anguli utriusque trianguli sint æquales duobus Rectis per 32. lib. 1. si ex summâ duorum Rectorum au- ferantur duo majores anguli D A S, D S A, relinquitur A D S minor, quàm si ex eâdem duorum Rectorum summâ auferan- tur duo minores C A R, C R A, hoc est minor quàm A C R. Ut autem clariùs innotescat, quænam sit gravitationum Ratio pro funiculi longitudine, sit corpus grave in R: & primùm quidem ex C pendeat funiculo breviore C R, deinde ex D lon- giore funiculo D R: quisquis retineat corpus in R constitu- tum, atque descensu prohibeat, faciliùs retinebit, cum ex D, quàm

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98 Mechanics if it hang by the shorter cord M F, than if by the longer A F. Since these things are so, there presents itself to us no unfit method of transferring a weight from a lower to a higher place; namely, if we take care that it be suspended from a sufficiently strong and longer rope; for when the friction of the parts is removed, which would occur if the weight were dragged along a plane, it can be drawn with less expenditure of force. Let there be a heavy body at A, which must be lifted and transferred to the higher place R S. If it be suspended from C by a shorter rope, it will be able to draw it as far as R, whoever, the angle of inclination A C R having been made, can hold it with some excess of force, and resist the moments of gravity which prevail in R. But if it hang from the longer rope D A, the same body A will be able to be drawn and held in S, lest it slip downward, and indeed with less effort; for the angle of inclination A D S having been made smaller than A C R, it likewise weighs less in S than in R. And the angle A D S is indeed smaller than the angle A C R, as is clear if the straight lines A R, A S are drawn: for C A, C R are equal sides by hypothesis, likewise D A, D S are equal; that is to say, it is the same cord, which first falls perpendicular, and then is moved away from the perpendicular: in the isosceles triangle C A R the angles at the base A R are equal, by book 1, prop. 5; likewise in the isosceles triangle D A S the angles at the base A S are equal to each other. Moreover the angle D A S is greater than the angle C A R; therefore the remaining angle D S A is also greater than the remaining C R A. Since therefore the three angles of each triangle are equal to two right angles, by book 1, prop. 32, if from the sum of two right angles there be taken away the two larger angles D A S, D S A, there remains A D S smaller than if from the same sum of two right angles there be taken away the two smaller C A R, C R A, that is, smaller than A C R. But that it may be more clearly understood what is the ratio of gravitation according to the length of the cord, let there be a heavy body in R: and first let it hang from C by the shorter cord C R, then from D by the longer cord D R: whoever retains the body placed in R, and prevents its descent, will more easily retain it when from D, than

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Liber primus. CAPUT XV. 99 quàm cùm ex C, pendebit; quia declinationis angulus XCR major est angulo XDR per 16. lib.1. Verùm qua Ratione, in- quis, vires, quas in utroque casu retinens exerit, discriminan- tur? utique secundùm Reciprocam funiculorum Rationem co- natur obsistens corporis propensioni ad descensum; quæ enim Ratio gravitationum corporis, ea est virium gravitationibus repugnantium: comparatâ autem corporis in R constituti gra- vitatione, si ex C pendeat, cum ejusdem ibidem positi gravita- tione, si pendeat ex D, est reciprocè ut DR ad CR; igitur & vires retinentis corpus ex C pendens sunt ut DR, retinen- tis verò idem corpus ex D pendens sunt ut CR. Id quod hinc conficitur, quia corpus in suspensione, positionem habens CR, gravitat ut XR ad RC, positionem verò habens DR gravitat ut XR ad RD; duæ autem Rationes XR ad RC, & XR ad RD sunt reciprocè ut RD ad RC. Quotiescumque enim duæ sunt Rationes, quarum idem est Antecedens terminus, & di- versus Consequens, ex sunt reciprocè ut consequentes. Quòd si quis Rationes inter se comparare non assuetus de hoc ambigeret, an Rationes eumdem vel æqualem anteceden- tem terminum habentes sint reciprocè ut Consequentes, facilè intelliget, si animadvertat Rationes eumdem Consequentem terminum habentes esse inter se directè, ut antecedentes. Quemcumque enim interrogaveris, quæ sit Ratio 2 ad 6 illicò respondebit esse subtriplam, secunda scilicet ter continet pri- mam, ut constat si ter positam Rationem 2 in summam colligas; neque enim id est Rationem Rationis esse subtriplam, ac sub- triplicatam; Ratio siquidem 2 est subtriplicata Rationis 8/343. Si igitur pariter quæras, quænam sit Ratio 2 ad 7 rectè responde- bit eam esse triplam, hoc est reciprocè ut 6 ad 2: id quod ma- nifestè apparebit, si illas ad denominationem eandem, hoc est ad eumdem Consequentem terminum reduxeris, sunt nimirum ut 42/12 ad 14/12, hoc est ut 6 ad 2. Ex quibus obiter patet methodus exponendi per lineas pro- portionem duarum Rationum etiam numeris non explicabi- lium; si videlicet fiat ut Antecedens secundæ Rationis ad suum Consequentem, ita Antecedens datus primæ Rationis ad alium novum Consequentem; erit enim prima Ratio data ad secun- N 2

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Book One. CHAPTER XV. 99 than when it hangs from C; because the angle of deflection XCR is greater than the angle XDR by 16, lib. 1. But by what reason, you ask, are the forces, which it exerts while retaining it in either case, distinguished? Certainly according to the reciprocal ratio of the cords, since it strives to resist the body’s tendency toward descent; for the ratio of the body’s gravities is the same as that of the forces resisting those gravities. Now, comparing the gravity of the body situated at R, if it hang from C, with the gravity of the same body placed there, if it hang from D, the ratio is reciprocally as DR to CR; therefore the forces retaining the body hanging from C are as DR, but those retaining the same body hanging from D are as CR. This is proved from the fact that the body, in suspension with position CR, gravitates as XR to RC, but with position DR it gravitates as XR to RD; and the two ratios XR to RC and XR to RD are reciprocally as RD to RC. For whenever there are two ratios whose antecedent term is the same and whose consequent differs, they are reciprocally as the consequents. And if anyone, not accustomed to comparing ratios with one another, should doubt about this, whether ratios having the same or equal antecedent term are reciprocally as the consequents, he will easily understand if he observes that ratios having the same consequent term are directly as the antecedents. For anyone you may ask what the ratio of 2 to 6 is will immediately answer that it is subtriplicate, namely that the second contains the first three times, as is clear if you add together the ratio 2 taken three times. Nor is that to say that the ratio itself is subtriplicate and subtriplicated; for the ratio of 2 is subtriplicated to the ratio of 8/343. If then you similarly ask what the ratio of 2 to 7 is, he will rightly answer that it is triple, that is, reciprocally as 6 to 2: this will plainly appear if you reduce them to the same denomination, that is, to the same consequent term; they are namely as 42/12 to 14/12, that is, as 6 to 2. From which it is incidentally clear what method there is of expressing by lines the proportion of two ratios not even expressible by numbers; namely, if the antecedent of the second ratio be to its consequent as a given antecedent of the first ratio is to another new consequent, for then the first given ratio will be to the second N 2

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Mechanicorum dam rationem datam reciprocè ut novus Consequens terminus ad datum Consequentem primæ Rationis: aut etiam si fiat ut Consequens secundæ Rationis ad suum Antecedentem, ita con- sequens primæ Rationis ad alium novum Antecedentem; erit enim prima ratio data ad secundam Rationem datam, directè ut datus Antecedens primæ Rationis ad novum Antecedentem. Consideratâ hactenus unicâ & simplici corporis gravis sus- pensione, gradum facere oportet ad gravitationis rationes in- vestigandas, si duplex fuerit suspensio. Sit enim globus A tùm ex B, tùm ex C suspensus fu- niculis B A & C A. Haud du- bium quin tota corporis gravi- tas ex B & C pendeat; sed quâ Ratione singulæ vires eidem gravitati obsistant, de hoc po- test ambigi. Verùm nisi mea mihi nimium blanditur opi- nio, ex dictis facilis videtur explicatio. Corpus siquidem ex duplici fune suspensum ita constitutum est, ut alterutro fune præciso ex reliquo pen- deat, & descendens moveatur circà punctum, cui alligatur funis. Quare unusquisque obsistit momentis, quibus ex altero gravitat; nimirum funiculus C A retinens globum, ne descen- dat, repugnat momentis gravitatis, quibus globus A se ipse deorsùm urget circa punctum B ex fune B A: Contrà verò fu- niculus B A eundem globum retinet, ne circa punctum C ex funiculo C A moveatur descendens, atque adeò obsistit, mo- mentis gravitatis ad descendendum circà idem punctum C. At- qui momenta descendendi ex fune B A ad gravitatem in per- pendiculo sunt ut D A ad A B, & ex fune C A sunt ut E A ad A C, ex his, quæ superiùs disputata sunt. Sunt igitur duæ Ra- tiones D A ad A B, & E A ad A C. Quare fiat angulus D A F æqualis angulo E A C, & est trian- gulum D A F ob angulorum æqualitatem simile triangulo E A C; ac propterea per 4. lib. 6. ut E A ad A C, ita D A ad A F.

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Mechanics gives a ratio reciprocally, so that the new consequent term is to the given consequent of the first ratio; or also if it be as the consequent of the second ratio is to its antecedent, so is the consequent of the first ratio to another new antecedent; for then the first given ratio will be to the second given ratio directly, as the given antecedent of the first ratio is to the new antecedent. Hitherto having considered a single and simple suspension of a heavy body, we must advance to investigating the ratios of gravitation, if the suspension be double. Let the globe A therefore be suspended both from B and from C by the cords BA and CA. There is no doubt that the whole weight of the body hangs from B and C; but in what ratio the several forces resist that same weight may be doubted. Yet unless my opinion flatters me too much, an easy explanation seems to be given by what has been said. For a body suspended by a double cord is so constituted that, if either cord be cut, it hangs from the remaining one and, descending, moves about the point to which the cord is fastened. Wherefore each resists the moments by which it is pulled down by the other; namely, the cord CA, holding the globe so that it does not descend, opposes the moments of gravity by which the globe A urges itself downward about the point B from the cord BA. On the other hand, the cord BA holds the same globe, so that it does not move descending about the point C from the cord CA, and thus resists the moments of gravity tending to descend about the same point C. But the moments of descent from the cord BA to gravity in the perpendicular are as DA is to AB, and from the cord CA as EA is to AC, from those things which were discussed above. There are therefore two ratios, DA to AB, and EA to AC. Therefore let the angle DAF be made equal to the angle EAC, and the triangle DAF is, by the equality of the angles, similar to the triangle EAC; and therefore by book 6, proposition 4, as EA is to AC, so is DA to AF.

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Liber primus. CAPUT XV. 101 AF. Ergo vis descendendi ex CA est ut DA ad AF, & vis descendendi ex BA est ut DA ad AB: igitur duæ hæ Rationes sunt reciprocè ut BA ad AF; atque adeò B quidem retinens, ne descendat ex CA, exerit vires ut BA; C verò retinens, ne descendat ex BA, adhibet conatum ut FA; & quæ componitur ex BA, AF, totum gravitatis momentum, quod corpori suspenso inest, repræsentat. Momentum, inquam, gravitatis potiùs, quàm gravitatem totam; totius si quidem gravitatis nomine vires ipsas descendendi intelligimus, quas corpus grave obtinet sibi prorsùs relictum secluso quolibet impedimento, à quo certam descendendi regulam accipiat: Momenti autem vocabulo ipsas descendendi vires significamus non per se & solitariè acceptas; sed quatenus ex corporis positione, cæterorumque quæ circumstant, ad majorem aut minorem motûs velocitatem determinatur. Considerato itaque nisu corporis A ad descendendum & cùm perpendicularis est funiculus BD, & cum declinat BA, Ratio momentorum est ut BA ad AD. Similiter momentum ex perpendiculari CE ad momentum ex declinante CA est ut CA ad AE, hoc est ut FA ad AD: est igitur corporis A ex duplici funiculo BA, CA pendentis totum gravitandi momentum, quod ex lineis BA, AF componitur. Hîc autem hæsitantem videre mihi videor non neminem ex iis, quæ dicebantur, colligentem corpus A primùm ex declinante BA æquè ac ex perpendiculari BD gravitare; deinde plus ad descendendum momenti obtinere, si ex duobus funiculis, quàm si ex unico pendeat. Si enim angulus declinationis DBA sit gr. 22. 12; est DA sinus dati anguli ad radium BA ut 37784 ad 100000: & si angulus declinationis EC sit gr. 54. 35, est EA sinus dati anguli ad Radium CA ut 81496 ad 100000. At ex constructione triangulum DAF simile est triangulo EAC; igitur DA ad AF est ut 81496 ad 100000. Est autem DA in particulis Radij BA partium 37784; igitur si fiat ut 81496 ad 100000, ita 37784, ad aliud, erit AF earumdem particularum 46363, quarum BA est 100000. Quare composita BA, AF momenta sunt 146363, cum tamen momentum in perpendiculari AD sit tantum 100000. Cum verò dictum sit B clavum resistere ponderi A ut BA, C autem N 3

Transcription: Translated (English)

Book One. CHAPTER XV. 101 AF. Therefore the force of descending from CA is as DA to AF, and the force of descending from BA is as DA to AB: therefore these two ratios are reciprocally as BA to AF; and thus B indeed, holding fast, lest it descend from CA, exerts a force as BA; but C, holding fast, lest it descend from BA, applies a tendency as FA; and what is compounded from BA, AF, the whole moment of gravity, which is in the suspended body, represents. I say the moment of gravity rather than the whole gravity; for by the name of the whole gravity we understand the forces themselves of descending, which a heavy body possesses when left entirely to itself, every hindrance being excluded, from which it may receive a certain rule of descent: but by the term moment we signify the forces of descending themselves, not taken by themselves and separately, but insofar as, from the position of the body and the other things that surround it, they are determined to a greater or lesser speed of motion. Having therefore considered the endeavour of body A to descend both when the cord BD is perpendicular, and when it inclines BA, the ratio of the moments is as BA to AD. Likewise the moment from the perpendicular CE to the moment from the inclined CA is as CA to AE, that is, as FA to AD: therefore the whole gravitating moment of body A hanging from a double cord BA, CA is composed from the lines BA, AF. But here I seem to see not a few hesitating, who from what has been said gather that body A first gravitates from the inclined BA as much as from the perpendicular BD; and then that it has more momentum toward descending, if it hang from two cords than if from one only. For if the angle of declination DBA be 22°. 12′; DA is the sine of the given angle to the radius BA as 37784 to 100000: and if the angle of declination EC be 54°. 35′, EA is the sine of the given angle to the Radius CA as 81496 to 100000. But by construction the triangle DAF is similar to the triangle EAC; therefore DA to AF is as 81496 to 100000. Now DA is in the parts of the Radius BA 37784 parts; therefore if it be made as 81496 to 100000, so 37784 to another, AF will be of the same parts 46363, of which BA is 100000. Wherefore the combined momenta BA, AF are 146363, whereas the moment in the perpendicular AD is only 100000. Since however it has been said that B the nail resists the weight of A as BA, but C on the other hand N 3

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Mechanicorum 102 ut F A, manifestum est B clavum retinere ut 100000 quando declinat B A à perpendiculo: Atqui etiam in perpendiculo BD retinet ut 100000, igitur idem est ponderis tùm ex BD, tùm ex B A momentum; id quod est absurdum. Sed & illud prætereà ex dictis consequi videtur, quod ejusdem corporis majus sit momentum, si ex duobus funiculis, quàm si ex unico pendeat. Fiat enim angulus D B H æqualis angulo declinationis E C A, & assumptâ B H æquali ipsi B A, ducatur ad BD perpendicularis H I: erit utique triangulum B H I simile triangulo C A E, ac propterea ut E A ad A C, ita I H ad H B, hoc est ad A B. Sunt igitur duæ Rationes eundem Consequentem terminum habentes, atque adeò inter se in ratione Antecedentium, ac proinde cùm vis descendendi ex B A sit ut D A ad A B, & vis descendendi ex C A sit ut I H ad A B, vires descendendi invicem comparatæ sunt ut D A ad I H, totumque momentum componitur ex D A 37784, & I H 81496. Quare momentum quod in perpendiculari, si unico funiculo penderet ex BD, esset 100000, pendente corpore A ex duo- bus funiculis B A, C A, fit majus, scilicet 119280. ut quid igitur ex pluribus funiculis illud suspendere oportuit? Quibus difficultatibus ut fiat satis, & id, quod inquirimus, enucleatiùs explicetur, illud observo, quod funiculus B A si præcisè spectetur, quatenus ex eo corpus grave pendet, retinet globum A, ne rectâ descendat per lineam ipsi B D parallelam, sed cogit illum deflectere in motu: quare adversùs clavum B, globus A exercet ea momenta, quæ exerceret in planum inclinatum, cui B A ad rectos angulos insisteret. At si globus ex alio prætereà funiculo C A pendeat, idem funiculus B A resistit etiam momentis illis, quibus globus A descenderet in plano inclinato, cui C A ad rectos angulos insisteret, quæ momenta (ut summum) sunt ad B A radium ut 81496. Momenta verò quibus urgeret planum inclinatum perpendiculare ad B A, sunt, ex dictis superiori capite, ut Sinus Versus anguli inclinationis plani; inclinatio autem plani, ut paulò superiùs hoc eodem capite demonstravimus, est complementum anguli declinationis D B A. Quare differentia inter D A 37784 sinum rectum anguli declinationis, & radium B A 100000, cum sit Sinus Versus anguli inclinationis plani, sunt momenta 62216 addenda prioribus

Transcription: Translated (English)

Mechanics 102 as BA is manifest, it retains the weight as 100000 when BA inclines from the perpendicular: but also in the perpendicular BD it retains it as 100000; therefore the moment is the same, whether from BD or from BA; which is absurd. But furthermore it seems to follow from what has been said, that the moment of the same body is greater if it hangs from two cords than if it hangs from a single one. For let the angle DBH be made equal to the angle of declination ECA, and, B H being taken equal to BA, let HI be drawn perpendicular to BD: then the triangle BHI will certainly be similar to the triangle CAE, and therefore as EA is to AC, so is IH to HB, that is, to AB. There are therefore two ratios having the same consequent term, and thus among themselves in the ratio of the antecedents; and therefore since the force of descending from BA is as DA to AB, and the force of descending from CA is as IH to AB, the forces of descending are compared to one another as DA to IH, and the whole moment is composed of DA 37784 and IH 81496. Wherefore the moment which, in the perpendicular, if the body hung by a single cord from BD, would be 100000, when the body A hangs from two cords BA, CA, becomes greater, namely 119280. Why then should it have been proper to suspend it from several cords? To satisfy these difficulties, and to explain more clearly the matter we are investigating, I observe this: that the cord BA, if considered precisely as such, insofar as the heavy body hangs from it, retains the globe A so that it does not descend straight along a line parallel to BD, but compels it to deflect in motion; wherefore, against the peg B, the globe A exerts those moments which it would exert on an inclined plane, to which BA would stand at right angles. But if, in addition, the globe hangs from another cord CA, the same cord BA also resists those moments by which the globe A would descend on an inclined plane to which CA would stand at right angles; which moments (at most) are to the radius BA as 81496. But the moments by which it would press upon the inclined plane perpendicular to BA, are, from what was said in the preceding chapter, as the versine of the angle of inclination of the plane; and the inclination of the plane, as we demonstrated a little above in this same chapter, is the complement of the angle of declination DBA. Therefore the difference between DA 37784, the sine of the right angle of declination, and the radius BA 100000, since it is the versine of the angle of inclination of the plane, gives moments 62216 to be added to the former

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Liber primus. CAPUT XV. 103 prioribus 81496; adeò ut summa sit 143712 momentorum, qui- bus funiculus B A repugnat, si pondus pendeat etiam ex C A; cum tamen si ex ipso tantùm funiculo B A penderet, & aliquis esset præcisè obluctans viribus ad descendendum, idem funicu- lus B A resisteret solùm momentis 62216. Eâdem methodo deprehendes funiculum C A, si ex eo solo globus pendeat, retinere momenta 18504: at si etiam ex B A globus pendeat, additis momentis 37784, tota momentorum summa est 56288. Iam summam hanc priori 143712 adde, & erit tota momentorum summa 200000: perinde atque si corpo- ris gravitas fuisset duplicata. Id quod deprehendes, quoscum- que demùm declinationis angulos statueris sivè majores, sivè minores; semper enim eandem summam momentorum om- nium invenies 200000: & funiculus minoris declinationis plus momentorum sustinebit, tùm quia Sinus Versus majoris incli- nationis plani major est, tum quia Sinus Rectus alterius anguli declinationis majoris item major est. Hæc tamen ut veritati congruant, ita solùm accipienda sunt, ut momenta singula ex utrâque funiculorum declinatione orta particulatim sumantur: pondus scilicet ex utroque suspensum perinde hactenus consideratum est, ac si momenta ipsa descen- dendi in diversas partes abeuntia momentum quoddam ex utrisque temperatum non constituerent; re autem ipsâ quod ex iis componitur momentum, non ex ipsorum momentorum ad- ditione conflatur, sed ex ipsis temperatur. Si enim mobile sit ubi A, impetum verò cum tali directione habeat, quâ deferri possit æquabiliter per rectam A B, alio autem impetu feratur æquabiliter directum in C, no- tum omnibus est motum, qui ex A B & A C componitur, non fieri ex earum additione, sed tem- perari in lineam A D, quæ dimetiens est parallelogrammi, quod ex earumdem linearum A B, A C longitudine, ac mutuâ incli- natione formam desumit. Quâ in re plurimum interest, quam invicem habeant inclinationem directiones motuum in diversa abeuntium; quò enim acutiorem angulum constituunt, eò lon- giùs provehitur mobile, ut A B, A C in acutum angulum coëuntibus

Transcription: Translated (English)

Book One. CHAPTER XV. 103 by the former 81496; so that the total is 143712 moments, by which the cord B A resists, if the weight also hangs from C A; whereas if it hung only from the cord B A itself, and there were some force exactly opposing it as it tended to descend, the same cord B A would resist only 62216 moments. By the same method you will find that the cord C A, if the globe hangs from it alone, retains 18504 moments: but if the globe also hangs from B A, with 37784 moments added, the whole sum of moments is 56288. Now add this sum to the former 143712, and the total sum of moments will be 200000: just as if the body's gravity had been doubled. And this you will find, whatever angles of declination you set, whether greater or smaller; for you will always find the same total sum of all the moments, 200000: and the cord of lesser declination will sustain more moments, both because the sine versed of the greater inclination of the plane is greater, and because the sine rectus of the other angle of greater declination is likewise greater. These things, however, in order to agree with the truth, are to be understood only in this sense: that the several moments arising from the declination of the two cords are taken separately, each by itself; that is to say, the weight suspended from both has thus far been considered as though the moments of descent themselves, going off into different parts, did not constitute some one moment tempered from both. But in the thing itself, the momentum that is composed from them is not formed from the addition of the moments themselves, but is tempered from them. For if a movable body be at A, and yet have an impulse with such a direction that it may be carried uniformly along the straight line A B, but be moved by another impulse uniformly in the direction C, it is known to all that the motion which is composed from A B and A C is not made by their addition, but is tempered into the line A D, which is the diagonal of the parallelogram that takes shape from the length and mutual inclination of the same lines A B and A C. In this matter it makes a very great difference what inclination the directions of the motions, going off in different ways, have toward one another; for the sharper the angle they make, the farther the movable body is carried onward, as A B, A C, when meeting in an acute angle

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Mechanicorum coëuntibus mobile ex A in D venit: quò verò obtusior fuerit angulus, eò etiam brevius est iter ipsius mobilis, ut contingit, si ex B directum per rectas B A, BD ad obtusum angulum constitutas moveatur, sistitur enim in C, & brevior est diamet- ter BC quàm AD, ut ex 24. lib. 1. satis manifestum est geo- metris, & ipsa motuum natura postulat; qui nimirum sibi in- vicem magis adversantur, magisque in diversa abeunt, se ma- gis elidunt, id quod fit ex angulo obtuso DBA; qui verò mi- nùs in diversa abeunt, id quod fit ex angulo acuto C AB, se pa- riter minùs elidunt. Sint itaque, ut priùs, funiculi B A, C A, ex quibus A pon- dus suspenditur: ducatur ad B A perpendicularis AR, & est planum inclinatum, in quo descendendi momentum est ut DA; similiter ad C A perpendicularis AG ducatur referens planum inclinatum, in quo descendendi momentum est AE. Sumatur igitur AR quidem ipsi AD æqualis, AG verò ipsi AE pariter æqualis, si funiculi B A, & CA æquales fuerint; sin autem inæquales sint, fiat angulus DBH æqualis angulo declinationis ECA, & sumptâ BH æquali ipsi BA, duca- tur ad BD perpendicularis HI, eritque ut EA ad AC, ita IH ad HB, hoc est ad AB; ac propterea ipsi IH, quæ refert momentum AE, sumatur AG æqualis. Ex quo fit cor- pus A suspensum hâc ratione momenta descendendi habe- re in diversas partes abeuntia AR, AG: perfecto igitur paral- lelogrammo ARNG, ex duobus illis momentis temperatur momentum AN. Ipsius autem AN longitudinem investigare non est diffici- le; cum enim noti supponantur anguli declinationum DBA, ECA, angulus RAG conflatur ex eorum complementis, quippe qui æqualis est duobus angulis inclinationis planorum AR, & AG. Porrò ex hypothesi sunt angulus DBA gr. 22. 12, & angulus ECA gr. 54. 35: jungantur simul, & eorum summa gr. 76. 47 auferatur ex gr. 180, ut residuum gr. 103. 13 sit angulus RAG, cui æqualis est oppositus RNG; ac proinde notus est angulus G, qui est suo opposito R æqualis, uterque scilicet gr. 76. 47 quæ est summa angulorum decli- nationis. Sunt igitur in triangulo AGN nota latera AG, GN (est enim ex 34. lib. 1. GN opposito lateri AR æquale) una

Transcription: Translated (English)

When the forces are combined, the moving body travels from A to D: but the more obtuse the angle is, the shorter also is the path of the moving body, as happens if from B it is moved along the straight lines BA, BD placed at an obtuse angle, for it is stopped at C, and the chord BC is shorter than AD, as from Book 1, Prop. 24, it is sufficiently clear to geometers, and the very nature of motions requires it; for those motions that more strongly oppose one another and depart more in different directions more strongly destroy one another, which happens from the obtuse angle DBA; but those that depart less in different directions, which happens from the acute angle CAB, likewise destroy one another less. Let there be, then, as before, the cords BA, CA, from which the weight is suspended at A: let the perpendicular AR be drawn to BA, and there is the inclined plane in which the force of descending is as DA; likewise let the perpendicular AG be drawn to CA, representing the inclined plane in which the force of descending is AE. Therefore let AR indeed be taken equal to AD itself, and AG likewise equal to AE, if the cords BA and CA were equal; but if they are unequal, let the angle DBH be made equal to the angle of declination ECA, and, with BH taken equal to BA itself, let HI be drawn perpendicular to BD, and it will be as EA is to AC, so is IH to HB, that is, to AB; and therefore, for IH itself, which represents the force AE, let AG be taken equal. From this it follows that the suspended body A, by this method, has descending forces going off into different parts, AR and AG: therefore, when the parallelogram ARNG is completed, the force AN is compounded from those two forces. But to investigate the length of AN itself is not difficult; for since the angles of declination DBA and ECA are supposed known, the angle RAG is formed from their complements, since it is equal to the two angles of inclination of the planes AR and AG. Moreover, by hypothesis the angle DBA is 22° 12', and the angle ECA 54° 35': let them be added together, and let their sum, 76° 47', be subtracted from 180°, so that the remainder, 103° 13', may be the angle RAG, to which the opposite RNG is equal; and therefore the angle G is known, which is equal to its opposite R, namely 76° 47', which is the sum of the angles of declination. Therefore in triangle AGN the known sides are AG, GN (for by Book 1, Prop. 34, GN is equal to the opposite side AR)

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Liber primus. CAPUT XV. 105 unâ cum angulo G comprehenso, & ex Trigonometriâ innotescit tertium latus AN. Quare cum latus AG sit ex superiùs constitutis 81496, & GN, hoc est AR, 37784, fiat ut laterum AG, GN summa 119280 ad eorumdem differentiam 43712, ita semisummæ angulorum ad basim, hoc est gr. 51.36 ́/5 Tangens 126205 ad 46249 Tangentem gr. 24. 49 ́/5 differentiæ infra, vel supra eandem semisummam. Est igitur angulus GAN gr. 26. 47 ́/10. In triangulo itaque AGN noti sunt duo anguli A, & G, ac latus GN angulo A oppositum; igitur ut anguli A gr. 26. 47 ́/10 Sinus 45070 ad anguli G gr. 76. 47' Sinum 97351, ita latus GN 37784 ad latus AN 81613. Ex quibus apparet descendendi momentum, quod componitur ex momentis in planis inclinatis, non esse 119280 ex eorum summâ, sed ita temperari, ut longè minus sit, videlicet solùm 81613. Methodo eâdem operantes deprehendemus ponderis in H constituti, ac ex funiculis BH, CH suspensi momentum ita componi ex momento HI bis sumpto (si quidem anguli declinationum DBH, ECH & funiculi æquales sint) ut in unum ex utroque nimirum HI & HO temperatum HS coalescat. Unde constabit quò majores fuerint declinationum anguli, eò longiorem futuram lineam HS, atque adeò etiam majus momentum descendendi; plana siquidem inclinata acutiorem angulum constituunt. Quam momentorum varietatem paulò inferiùs manifesto experimento comprobabimus: ubi constabit pondus hâc ratione suspensum ex duobus funiculis plus habere aliquando momenti ad descendendum, quàm in perpendiculari suspensione. Quemadmodum verò de momentis descendendi in planis inclinatis ratiocinati sumus, ita pariter in unum coalescere dicenda sunt momenta, quibus funiculi pondus retinentes ipsum quodammodo avellere conantur à plano inclinato, ne illud urgeat; hæc enim pariter momenta in diversa abeunt secundùm ipsam funiculorum directionem. Sunt autem momenta illa Sinus Versi angulorum inclinationis planorum; qui habentur, si Sinus Recti complementorum, hoc est angulorum de- O

Transcription: Translated (English)

Book One. Chapter XV. 105 together with the angle G having been determined, and from trigonometry the third side AN is known. Therefore, since side AG, from the things established above, is 81496, and GN, that is AR, 37784, let the sum of the sides AG and GN, 119280, be taken to the difference of the same, 43712, as the semisum of the angles at the base, that is 51.36 ́/5 degrees, is to the tangent 126205 to 46249, the tangent of 24.49 ́/5 degrees below, or above, that same semisum. Therefore angle GAN is 26.47 ́/10 degrees. In triangle AGN, therefore, two angles, A and G, and side GN opposite angle A are known; therefore, as the sine of angle A, 26.47 ́/10 degrees, 45070, is to the sine of angle G, 76.47' degrees, 97351, so is side GN, 37784, to side AN, 81613. From these things it appears that the force of descending, which is composed from the forces on inclined planes, is not 119280 from their sum, but is so moderated that it is much less, namely only 81613. By operating in the same method we shall discover that the force of a weight placed at H, and suspended by the cords BH and CH, is composed in such a way from the force HI taken twice (if indeed the angles of declination DBH, ECH, and the cords are equal) that it coalesces into one, tempered from each, namely HI and HO, HS. Whence it will be clear that the greater the angles of declination are, the longer the line HS will be, and therefore also the greater the force of descending; for inclined planes make a sharper angle. We shall confirm this variation of forces a little below by a manifest experiment: where it will be clear that a weight suspended in this way from two cords sometimes has more force to descend than in perpendicular suspension. And just as we have reasoned concerning the forces of descending on inclined planes, so likewise the forces by which the cords, retaining the weight, try in some measure to tear it away from the inclined plane so that it may not press upon it, must be said to coalesce into one; for these forces likewise go off in different directions according to the direction of the cords themselves. And those forces are the versed sines of the angles of inclination of the planes; which are obtained if the sines of the complements, that is, the angles of de- O

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Mechanicorum declinationis funiculorum, de- mantur ex Radio. Itaque ex BA auferatur BF ipsi DA æqualis, & est FA Sinus Ver- sus anguli inclinationis: posita est autem declinatio DBA gr.22.12, igitur FA est parti- cularum 62216; & declinatio ECA gr.54.35; igitur factâ CG æquali ipsi AE, remanet GA particularum 18504, quarum CA est 100000. Quare ut habeantur particulæ ejusdem rationis cum particulis AF, fiat ut CA ad AG, ita BA ad AH, & est AH particularum 18504 homologarum particulis AF. Perficiatur parallelogrammum AHIF; & quia funiculus CA retrahit à plano inclinato juxta momentum ac directionem HA, funiculus verò BA retrahit à plano inclinato secundùm momentum ac directionem FA, di- rectionibus in diversa abeuntibus, temperatur ex his momentis momentum AI diameter parallelogrammi. Porrò in diametri AI investigatione methodus est eadem, quâ paulò antè utebamur: Cum enim tres anguli BAD, BAC, CAE sint duobus Rectis æquales, anguli verò BAD, CAE noti sint, quippe complementa angulorum declinationis DBA, ECA, innotescit reliquus FAH, qui æqualis est summæ an- gulorum declinationis. Est igitur FAH gr.76.47, ac proinde angulus AFI gr.103.13 notus est, unâ cum lateribus FA 62216 & FI 18504. Fiat igitur ut laterum summa 80720 ad eorum- dem differentiam 43712, ita angulorum ad basim AI semisum- mæ gr.38.23 ́. Tangens 79235 ad 42907 Tangentem dif- ferentiæ infra vel supra eandem semisummam, hoc est gr.23. 13 ́. dempta igitur hæc differentia ex semisummâ gr.38.23 ́ reliquum facit angulum FAI gr.15.10. Fiat demùm ut anguli FAI gr.15.10. Sinus 26163 ad anguli AFI gr.103.13. hoc est ad supplementi gr.76.47. Sinum 97351, ita latus FI 18504 ad basim AI 68852. Inventa itaque momenta composita tùm in planis inclinatis, tùm in plana inclinata, dividantur juxta Rationem momento- rum

Transcription: Translated (English)

The moments of the cables’ declination are taken from the radius. Thus from BA let BF be taken away, equal to DA itself, and FA is the sine versed of the angle of inclination; but the declination DBA is given as 22.12 degrees, therefore FA is of 62216 parts; and the declination ECA is 54.35 degrees; therefore, after CG has been made equal to AE itself, there remains GA of 18504 parts, of which CA is 100000. Wherefore, in order that the parts may be had of the same ratio as the parts AF, let it be as CA is to AG, so BA is to AH, and AH is 18504 parts homologous to the parts AF. Let the parallelogram AHIF be completed; and because the cable CA draws back from the inclined plane according to the moment and direction HA, but the cable BA draws back from the inclined plane according to the moment and direction FA, the directions going off in different ways, the moment AI, the diagonal of the parallelogram, is compounded from these moments. Moreover, in the investigation of the diagonal AI the method is the same as that which we used a little before: for since the three angles BAD, BAC, CAE are equal to two right angles, and the angles BAD, CAE are known, being the complements of the angles of declination DBA, ECA, the remaining FAH is known, which is equal to the sum of the angles of declination. Therefore FAH is 76.47 degrees, and consequently the angle AFI is known, 103.13 degrees, together with the sides FA 62216 and FI 18504. Let it therefore be as the sum of the sides 80720 is to their difference 43712, so is the sum of the angles to the half-sum at the base AI, 38.23 degrees. The tangent 79235 is to 42907 the tangent of the difference below or above the same half-sum, that is, 23.13 degrees. This difference being taken away from the half-sum, 38.23 degrees, there remains the angle FAI, 15.10 degrees. Finally, let it be as the sine of angle FAI, 15.10 degrees, 26163, is to the sine of angle AFI, 103.13 degrees, that is, to the sine of its supplement, 76.47 degrees, 97351, so is side FI 18504 to base AI 68852. The composite moments having thus been found, both on inclined planes and on the inclined plane, let them be divided according to the ratio of the moments

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Liber primus. CAPUT XV. 107 rum simplicium, ut innotescat, quid demum cuique funiculo tribuendum sit in pondere retinendo. Momentum descendendi compositum inventum est susperiùs 81613, simplicia sunt 81496, & 37784. Fiat ut igitur ut simplicium momentorum summa 119280 ad eorum alterutrum, puta ad 37784, ita momentum compositum 81613 ad aliud, & provenit 25852 pars illius momenti pertinens ad funiculum C A, qui retinet pondus; cujus vis descendendi est D A 37784. Reliqua autem momenti 81613 pars 55761 pertinet ad funiculum B A retinentem pondus, cujus vis descendendi est E A 81496. Pari ratione fiat ut Sinuum Versorum angulorum inclinationis simplicium 62216, atque 18504 summa 80720 ad eorum alterutrum, puta ad 18504, ita momentum compositum inventum 68852 ad aliud, & provenit pro minori 15783, pro majori verò 53069. Quare funiculus B A minorem habens declinationem, & plus sustinet in suo plano magis inclinato, cui perpendicularis est, nimirum ut 53069, & plus retinet in plano reliquo minùs inclinato, nimirum ut 55761: contra verò funiculus C A, & minus sustinet, scilicet ut 15783, & minus retinet scilicet ut 25852. Funiculus itaque B A exercet vires ut 108830, & funiculus C A ut 41635, & totum corporis suspensi momentum est 150465. Non sola autem momenta descendendi in planis inclinatis considerari oportere, sed & ea, quæ essent adversùs plana ipsa inclinata, uti dictum est, ex eo apertè conficitur, quòd ubi funiculi concurrerent ad acutissimum angulum, vix quicquam virium in retinendo pondere exercere opus esset; tenuissimum quippe, esset momentum, quod ex parvis momentis per acutissimorum angulorum Sinus Rectos definitis componeretur: si verò nihil præterea momenti addendum esset; à magnâ gravitatione, quæ in perpendiculari est, ad ferè nullam transitus esset, facta vel modicâ à perpendiculo declinatione; atque adeò vix intenti esse deberent funiculi: id quod manifesto experimento adversatur. Illud postremò hîc ostendendum superest, plus scilicet inesse posse momenti ad descendendum corpori ex duobus funiculis invicem inclinatis suspenso, quàm si ex unico ad perpendiculum pendeat. Orbiculo circà suum axem C versatili, O 2

Transcription: Translated (English)

Book One. Chapter XV. 107 For the simple forces, so that it may become clear what must finally be assigned to each cord in retaining the weight. The combined descending moment was found to be 81613; the simple ones are 81496, and 37784. Let it therefore be as the sum of the simple moments, 119280, to either one of them, say to 37784, so is the combined moment, 81613, to the other; and there results 25852, the part of that moment belonging to the cord C A, which retains the weight; whose descending force is D A 37784. The remaining part of the moment, 81613, namely 55761, belongs to the cord B A retaining the weight, whose descending force is E A 81496. In like manner let it be as the Sinuum Versorum of the angles of inclination of the simple ones, 62216 and 18504, the sum 80720, to either one of them, say to 18504, so is the combined moment found, 68852, to the other; and there results, for the lesser, 15783, for the greater, 53069. Wherefore the cord B A, having the smaller declination, both supports more in its more inclined plane, to which it is perpendicular, namely as 53069, and retains more in the remaining less inclined plane, namely as 55761: on the contrary the cord C A supports less, namely as 15783, and retains less, namely as 25852. Therefore the cord B A exerts forces as 108830, and the cord C A as 41635, and the whole moment of the suspended body is 150465. Now it is clear that not only the descending moments in inclined planes ought to be considered, but also those which would be against the inclined planes themselves, as has been said; because where the cords meet at the sharpest angle, scarcely any force would need to be exerted in retaining the weight; for the momentum would be exceedingly small, which would be composed from the small moments determined by the sines of the sharpest angles: if, however, nothing further were to be added to the momentum, there would be a passage from a great gravitation, which is in the perpendicular, to one almost none at all, with only a slight declination from the perpendicular having been made; and indeed the cords would scarcely need to be taut: which is manifestly contradicted by experiment. It remains at last to show here that more momentum may be in a body suspended by two cords inclined to one another for descending than if it hung from a single cord perpendicularly. A little wheel revolving around its axis C,

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Mechanicorum ac secundùm extremam oram excavato, inseratur funiculus AFB, ex quo æqualia hinc, & hinc pondera A, & B pendant: nullus planè sequitur motus, quia utrumque ex perpendiculo pendet, & quantâ vi alterum conatur deorsum, pari nisu alterum repugnat, ne elevetur. Quærenti igitur, quantum momenti pondus B habeat ad descendendum, utique respondebis omninò par esse momento ponderis A. Iam verò sit funiculus AFD, qui in D religetur, & ponderi A sumatur æquale pondus E, vel potiùs ipsum B transferatur in E, & funiculo AFD adnectatur in H; ut sint quasi duo funiculi DH, FH. Quæro quantum ad descendendum momenti habeat pondus E, hoc est pondus B in H translatum, quod est æquale ponderi A: si tantumdem habet momenti, quantum pondus A, planè manebit immotum, intento funiculo FD; at si E descendens cogat ascendere pondus A, utique plus momenti habet quàm A, hoc est, plusquam B perpendiculariter pendens. Id quod re ipsâ contingit; & quidem tàm certo experimento, ut non solùm pondus E prævaleat ponderi A, si sit ei æquale, verùm etiam si minus sit eodem pondere A. Non igitur hoc absurdum est, quod constitutam à nobis momentorum hypothesim consequatur, sed potiùs ipsi naturæ nostra consentit hypothesis, cui robur adjicit experientia; nec ex eo capite perperam philosophati videmur, quòd in perpendiculo minus momenti, quàm ex duplici funiculo suspensum pondus habere dicendum sit. Ex his, quæ de corpore ex binis funiculis suspenso hactenus disputata sunt, non difficilis erit conjectura eorum, quæ dicenda sint, si ex tribus aut quatuor suspendatur, sivè illi immediatè adnectantur ipsi ponderi, sivè funiculus unus demum in plura capita dividatur, ex quibus fiat suspensio: neque enim his diutius ad nauseam immorandum censeo.

Transcription: Translated (English)

Mechanics and, after the extreme edge has been excavated, let the cord AFB be inserted, from which equal weights A and B hang here and there: absolutely no motion follows, because each hangs from the plumb line, and with whatever force one attempts to go downward, the other resists with equal effort, lest it be raised. Therefore, to the question how much force weight B has for descending, you will certainly answer that it is altogether equal to the force of weight A. Now then, let there be the cord AFD, which is fastened at D, and let a weight E equal to weight A be taken, or rather let B itself be transferred to E, and attached by a cord AFD at H; so that there are as it were two cords DH and FH. I ask how much force weight E has for descending, that is, weight B transferred to H, which is equal to weight A: if it has just as much force as weight A, it will clearly remain motionless, with cord FD taut; but if E, descending, compels weight A to rise, then it certainly has more force than A, that is, more than B hanging vertically. This in fact happens; and indeed so certainly by experiment, that not only does weight E prevail over weight A if it be equal to it, but even if it be less than the same weight A. It is not, therefore, absurd that what follows from the hypothesis of moments established by us should be so; but rather our hypothesis agrees with nature herself, and is strengthened by experience; nor do we seem to have reasoned wrongly on that account, because a weight suspended by a double cord is said to have less force in the plumb line than in the plumb line. From what has thus far been discussed concerning a body suspended by two cords, it will not be difficult to conjecture what must be said if it is suspended by three or four, whether these are attached immediately to the weight itself, or whether one cord is finally divided into several branches, from which the suspension is made: nor do I think it necessary to linger over these matters any longer, to the point of nausea.

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Liber primus. CAPUT XVI. 109 CAPUT XVI. Tractiones ac elevationes obliquæ expenduntur. PRoxima est iis, quæ hactenus disputata sunt, præsens in- vestigatio gravitationis corporum, sive nisûs, quo motui resistunt, cùm obliquè in plano aliquo trahuntur, aut elevan- tur: sicut enim toto conatu repugnant elevanti ad perpendicu- lum, & abstrahenti à plano, cui insident, ita pro majori, aut minori obliquitate tractionis aut elevationis magis etiam, aut minùs, obsistere experimur. Et primùm quidem super plano inclinato AB duo pondera prorsus æqualia, & similia intelligantur posita in B & C, atque linea CE sit horizonti BE perpendicu- laris, ac pondus C filo DC ad perpendiculum suspen- datur, ita tamen, ut con- tingat planum in C, & sit recta DE. Item ex D puncto ducatur filum DB, ut sursum trahatur B pon- dus incumbens plano in- clinato, dum pariter pon- dus C sursum rectâ trahi- tur, & à plano avellitur: horum autem funiculorum trahatur ex D pars æqualis. Quando igitur C venerit in V, æquali men- surâ BP multatum intelligitur filum DB, & remanet longi- tudo DP, hoc est DO; pondus enim, cum filum in D trahe- retur, ex B venit in O. Ductâ itaque lineâ ON horizonti pa- rallelâ, erit EN altitudo perpendicularis, ad quam ascendit pondus B in plano inclinato interea, dum pondus C venit in V, aut E venit in M, est enim EM assumpta ipsi CV æqualis. Quare cum pondus B obliquè trahitur super planum inclina- O 3

Transcription: Translated (English)

Book the first. CHAPTER XVI. 109 CHAPTER XVI. Oblique tractions and elevations are considered. Next to those matters which have hitherto been discussed is the present investigation of the gravity of bodies, or the tendency by which they resist motion when they are drawn or raised obliquely on some plane: for just as they resist with their whole effort the one elevating them perpendicularly and withdrawing them from the plane on which they lie, so, according as the traction or elevation is more or less oblique, we observe that they also resist more or less. And first, indeed, upon the inclined plane AB let two weights, exactly equal and similar, be understood to be placed in B and C, and let the line CE be perpendicular to the horizon BE, and let the weight C be suspended by the thread DC perpendicularly, yet so that it touches the plane at C, and let the straight line DE be drawn. Likewise from the point D let the thread DB be drawn, so that the weight B resting on the inclined plane may be drawn upward, while at the same time the weight C is drawn upward in a straight line and is separated from the plane: but of these cords let an equal part be drawn out from D. When therefore C has come to V, the thread DB is understood to have been shortened by an equal measure BP, and there remains the length DP, that is DO; for the weight, when the thread was being pulled at D, has moved from B to O. Therefore, after the line ON has been drawn parallel to the horizon, EN will be the perpendicular height to which the weight B ascends on the inclined plane meanwhile while the weight C comes to V, or E comes to M; for EM is assumed equal to CV. Wherefore, since the weight B is drawn obliquely over the inclined plane O 3

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Mechanicorum 110 tum, minorem subit violentiam, quàm cum ab illo perpendiculari elevatione avellitur. Hoc tamen ita intelligendum est, ut observetur alia esse momenta, cùm tractionis linea parallela est ipsi plano inclinato, ac cùm in planum inclinatum cadit obliqua, ut hîc linea DB. Si enim in plano inclinato sumatur BR æqualis perpendiculari EM, gravitatio per rectam BC, seu per lineam eidem parallelam, ad gravitationem in perpendiculo CE est reciprocè ut EC ad BC, seu ut ES ad BR aut EM, ex superiùs dictis cap. 13. At verò cum tractio obliqua est, gravitatio est ut EN ad EM, sivè ut BO ad BX: punctum autem O altius est puncto R, ac proptereà in hujusmodi obliquâ tractione plus violentiæ infertur ponderi, quàm in tractione parallelâ, plus enim ascendit. Porrò lineam BO longiorem esse lineâ BR est manifestum; siquidem duo latera DO, OB per 20. lib. 1. majora sunt reliquo DB: est autem ex hypothesi DP ipsi DO æqualis, ergo reliqua BP minor est, quàm BO: sed & ipsi BP, hoc est ipsi EM, æqualis assumpta est BR; igitur BR minor est quàm BO. Id quod etiam hinc constat, quia in triangulo Isocele DOP angulus OPB infra basum major est recto, cum sit deinceps angulo DPO ad basum acuto; ergo per 19. lib. 1. latus BO majus est latere BP, hoc est BR; igitur etiam EN major est quàm ES, & plus difficultatis percipitur in obliquâ hâc tractione, quàm in tractione parallelâ. Similiter intelligatur pondus C elevatum fuisse ex D (quod punctum D concipiatur multò altius, quàm in præsenti schemate) ad perpendiculum altitudine æquali ipsi ET, pondus verò B æquali tractione funiculi venisse ex B in G, demptâ scilicet longitudine BF ipsi ET æquali, atque adeò DF, DG æquales sunt: ipsi autem ET æqualis sumatur BI; quæ simili ratione demonstratur brevior, quàm BG: ex quo pariter fit hîc etiam ad majorem altitudinem perpendicularem EH elevari, quàm si tractio parallela fuisse plano inclinato, & elevatio ad altitudinem EL. Ex his manifestum est plus virium requiri ad trahendum pondus

Transcription: Translated (English)

Mechanics 110 then it undergoes a lesser force, than when it is detached from that perpendicular elevation. This, however, must be understood in such a way that it is observed that the moments are different, when the line of traction is parallel to the inclined plane itself, and when it falls obliquely upon the inclined plane, as here line DB. For if in the inclined plane BR is taken equal to the perpendicular EM, the gravitation through straight line BC, or through a line parallel to it, to the gravitation in the perpendicular CE, is reciprocally as EC to BC, or as ES to BR or EM, from what was said above in ch. 13. But when the traction is oblique, the gravitation is as EN to EM, or as BO to BX: but point O is higher than point R, and therefore in this kind of oblique traction more force is inflicted on the weight than in parallel traction, for it ascends more. Moreover, that line BO is longer than line BR is evident; since the two sides DO, OB by 20. lib. 1. are greater than the remaining DB: but by hypothesis DP is equal to DO, therefore the remaining BP is less than BO: but BR, that is, equal to BP, that is, equal to EM, was assumed; therefore BR is less than BO. This also is clear from this, because in the isosceles triangle DOP the angle OPB below the base is greater than a right angle, since next to the angle DPO at the base it is acute; therefore by 19. lib. 1. side BO is greater than side BP, that is, BR; therefore EN also is greater than ES, and more difficulty is perceived in this oblique traction than in parallel traction. Likewise let it be understood that weight C was lifted from D (which point D is to be conceived much higher than in the present figure) perpendicularly to a height equal to ET, but weight B by equal traction of a cord came from B to G, namely, the length BF equal to ET having been taken away, and therefore DF, DG are equal: but let BI, equal to ET, be assumed; which by a similar reasoning is shown to be shorter than BG: from which it likewise follows here that it is raised to a greater perpendicular height EH than if the traction had been parallel to the inclined plane, and the elevation to the height EL. From these things it is evident that more force is required to draw the weight

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Liber primus. CAPUT XVI. III pondus idem per lineam DB, aut DO, aut DG obli- quas, quàm per lineam plani inclinati BC, aut illi paral- lelam: dum enim per obliquas illas lineas fit tractio, pon- dus quidem non omninò abstrahitur à plano, sicut in tractio. ne perpendiculari, sed nec omninò incumbit plano, si- cut in tractione parallelâ ipsi plano; ac propterea, quò ma- gis tractio ad perpendicularem accedit, eò majorem inve- nit in pondere resistentiam. Patet autem altitudinum per- pendicularium EH, EL differentiam HL majorem esse, quàm sit altitudinum perpendicularium EN, ES differen- tia NS. Comparatis enim triangulis isoscelibus DPO, DFG, anguli ad basim PO majores sunt angulis ad basim FG, quia angulus PDO minor est angulo FDG: ergo angulus BPO, qui est infra basim, minor est angulo BFG infra basim. Fiat igitur ipsi BPO æqualis angulus BFK, ac proinde K cadit inter puncta I & G. Sunt ergo triangula BPO, BFK habentia angulum ad B communem æquiangula, & similia, ac per 4. lib.6. ut PB, hoc est BR, ad BO, ita FB, hoc est BI, ad BK; & invertendo, ac dividendo, & iterùm invertendo ut BR ad RO, ita BI ad IK. Atqui IG major est quàm IK, ergo per 8. lib.5. Ratio BI ad IG minor est Ratione BI ad IK, hoc est BR ad RO. Cum itaque per 2. lib.6. ut BR ad RO, ita ES ad SN; & ut BI ad IG, ita EL ad LH, major est Ra- tio ES ad SN, quàm EL ad LH, & permutando major est Ratio ES ad EL, quàm SN ad LH; est autem ES minor quàm EL, ergo etiam SN multò minor est quàm LH; ac proinde quo magis à perpendiculari recedet obli- qua tractio, momentum ponderis magis accedit ad momen- tum ejusdem in plano inclinato per tractionem parallelam, hoc est, minore differentiâ hoc excedit. Momentum igitur perpendicularis tractionis ad momentum obliquæ tractionis minorem Rationem habet, quàm ad momentum tractionis pa- rallelæ plano inclinato. Ex his observare est aliquod paradoxum, pondus scilicet obli- quâ hâc elevatione tractum plus moveri, quàm potentiam tra- hentem; hæc enim movetur secundùm mensuram funiculi tracti, hoc est BP seu BR illi æqualis, ostensum est autem BR

Transcription: Translated (English)

Book one. CHAPTER XVI. III the same weight through the line DB, or DO, or DG, oblique lines, than through the line BC of the inclined plane, or a line parallel to it: for while the pull is made through those oblique lines, the weight is not wholly drawn away from the plane, as in a perpendicular pull, but neither does it wholly rest upon the plane, as in a pull parallel to the plane; and therefore, the more the pull approaches the perpendicular, the greater resistance it finds in the weight. Now it is evident that the difference HL of the perpendicular heights EH, EL is greater than the difference NS of the perpendicular heights EN, ES. For, comparing the isosceles triangles DPO, DFG, the angles at the base PO are greater than the angles at the base FG, because the angle PDO is less than the angle FDG: therefore the angle BPO, which is below the base, is less than the angle BFG below the base. Let therefore the angle BFK be made equal to the angle BPO, and consequently K falls between the points I and G. The triangles BPO, BFK, therefore, having the angle at B common, are equiangular and similar; and by book 6, proposition 4, as PB, that is BR, is to BO, so is FB, that is BI, to BK; and reversing, and dividing, and again reversing, as BR is to RO, so is BI to IK. But IG is greater than IK; therefore, by book 5, proposition 8, the ratio of BI to IG is less than the ratio of BI to IK, that is, of BR to RO. Since therefore, by book 6, proposition 2, as BR is to RO, so is ES to SN; and as BI is to IG, so is EL to LH, the ratio of ES to SN is greater than that of EL to LH, and by permutation the ratio of ES to EL is greater than that of SN to LH; but ES is less than EL, therefore SN is also much less than LH; and consequently, the more the oblique pull departs from the perpendicular, the more the weight’s moment approaches the moment of the same weight on an inclined plane by a parallel pull, that is, by a smaller difference does this exceed it. Therefore the moment of a perpendicular pull has a smaller ratio to the moment of an oblique pull than it does to the moment of a pull parallel to the inclined plane. From these things it is to be observed that there is a certain paradox, namely, that a weight drawn by this oblique elevation is moved more than the drawing power; for the latter is moved according to the measure of the rope drawn, that is, BP or its equal BR; but it has been shown that BR

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Mechanicorum BR minorem esse quàm BO. Id quod etiam manifestum est, tractio obliqua non abstrahat pondus à plano, sed quasi il- lud adversùs planum trahat. Sit enim planum AB, super quo globus C, & funiculus obliquus DC; ex D autem pendeat ad perpendiculum æquale pondus E. Uterque fu- niculus pariter trahatur, & cum E venerit in F, æqualis pars CG decedit funiculo DC; remanet autem longitu- do DG æqualis longitudini DH, & centrum globi C ve- nit in H. Dico CH motum globi majorem esse supra CG motum potentiæ trahentis. Ducatur enim recta GH; est Isosceles DGH, ergo angulus HG C infra basim major est recto; ergo CH per 19. lib.1. major est quàm CG. Ipsi autem CH æqualem esse distantiam contactuum RS manifestum est, quia ex centris H & C rectæ cadunt in S & R ad angu- los rectos, atque adeò sunt parallelæ: sunt æquales CR & HS, ut pote Radij ejusdem globi; igitur per 33. lib.1. CH, & RS æquales sunt & parallelæ. Quare sivè centrum spectetur, sivè puncta contactuum, perinde est; semper enim major est glo- bi motus motu potentiæ trahentis; & quia RS major est quàm CG, hoc est quàm motus, qui fieret in ipso plano inclinato tractione parallelâ, hinc est quod hujusmodi obliquâ tractio- ne ad majorem altitudinem perpendicularem pari tempore tra- hitur, majoremque proptereà violentiam subiens majoribus indiget viribus, quàm si tractione parallelâ elevaretur. Sed jam trahatur iterum funiculus ita, ut ipsi CG primæ tractioni æqualis sit secunda tractio HL; & erit centrum globi in M, & æquales DM, DL. Anguli MDH, HDI si di- cantur æquales, etiam per 3. lib. 6. ut MD ad DC ita MH ad HC; est igitur MH minor quàm HC, major tamen quàm HL, quia subtensa est angulo MLH obtuso, ut pote infra ba- sim

Transcription: Translated (English)

Mechanics BR is less than BO. This is also manifest, since an oblique pull does not draw the weight away from the plane, but, as it were, pulls it against the plane. Let AB be the plane, upon which the globe C lies, and the oblique cord DC; and let a weight E, equal [to it], hang perpendicularly from D. Let each cord be drawn equally, and when E has come to F, an equal part CG departs from the cord DC; but the length DG remains equal to the length DH, and the center of the globe C comes to H. I say that the motion CH of the globe is greater than the motion CG of the pulling force. For let the straight line GH be drawn; DGH is isosceles, therefore the angle HGC beneath the base is greater than a right angle; therefore CH, by 19, lib. 1, is greater than CG. But it is manifest that CH is equal to the distance RS of the points of contact, because from the centers H and C straight lines fall to S and R at right angles, and thus are parallel: CR and HS are equal, as being radii of the same globe; therefore by 33, lib. 1, CH and RS are equal and parallel. Wherefore, whether the center is considered or the points of contact, it makes no difference; for the motion of the globe is always greater than the motion of the pulling force; and because RS is greater than CG, that is, than the motion that would occur in the inclined plane itself by parallel pulling, this is why by such oblique pulling it is drawn to a greater perpendicular height in equal time, and consequently, undergoing greater violence, it requires greater forces than if it were raised by parallel pulling. But now let the cord be drawn again so that the second pull HL is equal to the first pull CG; and the center of the globe will be in M, and DM and DL will be equal. If the angles MDH and HDI are said to be equal, then also by 3, lib. 6, as MD is to DC, so MH is to HC; therefore MH is less than HC, yet greater than HL, because it is subtended by the obtuse angle MLH, as being beneath the base

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Liber primus. CAPUT XVI. 113 tim Isoscelis MDL. Atqui ex hypothesi anguli MDL, HDG sunt æquales; ergo Isoscelium anguli infra bases, hoc est MLH, HGC sunt æquales: angulus autem externus MHL major est interno HCD, hoc est HCG, per 16. lib. 1. igitur reliquus HML minor est reliquo CHG. Itaque in duobus triangulis, angulis CGH, HLM ex hypothesi ostensis æqualibus sub- tenditur illi quidem majus latus CH, huic verò minus HM, & angulis inæqualibus CHG majori, HML minori æquale latus CG, HL: id quod omninò absurdum esse constat ex doctrinâ & Canone Sinuum; subtensæ siquidem inæquales an- gulorum æqualium sunt in circulis inæqualibus, major in majori circulo, minor in minori, in quibus utique fieri non potest, ut angulorum inæqualium subtensæ sint æquales. Non igitur fieri potest ut factâ secundâ tractione HL æquali priori CG, angu- lus MDH æqualis sit angulo HDC; alioquin triangulum HLM (cujus basis HM ex hypothesi arguitur minor base CH, quæ tamen sunt angulis ad G & L æqualibus subtensæ) esset in circulo minore, quàm sit circulus, in quo esset triangu- lum CGH; in circulo autem minore, angulo minori HML subtensa HL esset æqualis ipsi CG subtensæ angulo majori CHG in circulo majore. Quod si dicatur angulus MDH minor, quàm HDC, ergo angulus MLH infra basim minor est angulo HGC infra ba- sim: atqui angulus MHL externus major est interno HCG; igitur reliquus angulus LMH vel est æqualis angulo GHC, vel illo minor, vel illo major. Sit æqualis: quoniam æqualibus lineis CG, HL subtenduntur, sunt in circulis æqualibus; ergo cùm angulus MHL major sit angulo HCG, etiam oppositum latus ML majus est quàm HG: ergo Isosceles MDL habens angulum minorem sub brevioribus lateribus habet majorem basim, & Isosceles HDG habens angulum majorem sub late- ribus logioribus habet brevioré basim; id quod est manifestè ab- surdu[m], ut patet ex 24. & 25. lib. 1. Fieri igitur non potest, ut anguli LMH, GHC sint æquales, si MDH minor est quàm HDC. Quandoquidem igitur LMH, GHC non sunt æquales, dica- tur angulus LMH minor quàm GHC, & quia æqualibus li- neis HL, CG subtenduntur, triangulum HLM est in circulo majore, triangulum verò CHG in minore. Cum autem angu- P

Transcription: Translated (English)

Book One. CHAPTER XVI. 113 of the isosceles MDL. But by hypothesis the angles MDL, HDG are equal; therefore the angles beneath the bases in the isosceles, that is MLH, HGC are equal: but the external angle MHL is greater than the internal HCD, that is HCG, by 16. book 1. therefore the remaining HML is less than the remaining CHG. Thus in the two triangles, the equal angles CGH, HLM being shown by hypothesis, there is subtended to the one indeed the greater side CH, to the other the lesser HM, and to the unequal angles CHG the greater side CG, to HML the lesser HL: which is clearly absurd, as is established from the doctrine and Canon of Sines; for unequal subtenses of equal angles are in unequal circles, the greater in the greater circle, the lesser in the lesser, in which indeed it cannot happen that the subtenses of unequal angles should be equal. Therefore it cannot happen that, the second line HL having been drawn equal to the earlier CG, the an- gle MDH should be equal to angle HDC; otherwise the triangle HLM (whose base HM, by hypothesis, is shown to be less than base CH, which nevertheless are subtended by the equal angles at G & L) would be in a smaller circle than the circle in which the triangle CGH would be; but in the smaller circle, the angle HML being smaller, HL would be subtended equal to CG, which is subtended by the greater angle CHG in the greater circle. But if it be said that angle MDH is smaller than HDC, then the angle MLH below the base is smaller than the angle HGC below the base: but the external angle MHL is greater than the internal HCG; therefore the remaining angle LMH is either equal to angle GHC, or less than it, or greater than it. Let it be equal: since by equal lines CG, HL they are subtended, they are in equal circles; therefore since the angle MHL is greater than angle HCG, the opposite side ML is also greater than HG: therefore the isosceles MDL, having the smaller angle under the shorter sides, has the greater base, & the isosceles HDG, having the greater angle under the longer sides, has the shorter base; which is manifestly ab- surd, as is clear from 24. & 25. book 1. Therefore it cannot happen that the angles LMH, GHC are equal, if MDH is smaller than HDC. Since therefore LMH, GHC are not equal, let it be said that the angle LMH is smaller than GHC, & because by equal lines HL, CG they are subtended, the triangle HLM is in the greater circle, the triangle however CHG in the smaller. But when the angul-

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114 Mechanicorum lus MHL, ex sæpiùs dictis, sit major quàm HCG, etiam sub- tensa illius, ut potè in circulo majori, scilicet ML major est quàm HG subtensa anguli minoris in circulo minori: atque hinc idem quod priùs, sequitur absurdum angulum verticalem. MDL, ex hypothesi minorem, & brevioribus lateribus com- prehensum basum habere majorem, quàm sit basis anguli verti- calis HDG majoris sub lateribus longioribus. Sed neque dici potest angulus HML major quàm CHG; quia, si MDL minor est quàm HDG, angulus DML ad ba- sim Isoscelis major est quàm DHG pariter ad basim; ergo si DML majori addatur major HML, & DHG minori adda- tur minor CHG, erit totus DMH major toto angulo DHC, internus scilicet major externo, contra 16. lib.1. Si igitur an- gulus HML comparatus cum angulo CHG non potest esse æqualis, neque minor, neque major, factâ hypothesi anguli MDL minoris quàm HDC, necessariâ consecutione confici- tur angulum MDL non esse minorem angulo HDG. Cum itaque angulus MDL neque æqualis, neque minor sit angulo HDG, sequitur quod sit major: igitur & angulus in- fra basim MLH major est angulo HGC; item angulus MHL major est quàm HCG; ergo HML reliquus minor est reliquo CHG: at istis æquales lineæ HL, CG subtenduntur, igitur triangulum HML est in majore circulo, ac proinde angulo MLH majori, quàm CGH, etiam majus latus subtenditur: quapropter MH, hoc est SN, illi parallela & æqualis, major est quàm CH, hoc est RS: atq[ue] adeò ad majorem altitudi- nem elevatur per SN, quàm per RS factâ æquali tractione, seu æquali motu potentiæ trahentis. Ex quo & manifestum est pro majori obliquitate & recessu tractionis à parallelismo cum pla- no inclinato etiam trahenti difficultatem augeri. Facilè ex dictis colliges, quanto laboris compendio Romæ altioribus rotis instruantur birota (antiquis Cisia dicebantur) adeò ut unicus equus temoni applicitus, illumque subjecto pla- no proximè parallelum servans, dum clivum ascendit, ingentia pondera trahat, quibus sanè par non esset, si rotarum axis mi- nùs à subjecto plano distaret, & equittractio esset obliqua sur- sum: quamvis, ut aliàs suo loco explicabitur, ipsa rotarum am- plitudo plurimum conferat. Similiter in navium tractione, quæ adverso

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114 Mechanics since MHL, as has been said more than once, is greater than HCG, its subtense also, as being in the greater circle, namely ML, is greater than HG, the subtense of the smaller angle in the smaller circle: and hence the same absurdity as before follows concerning the vertical angle MDL, which, by hypothesis, is smaller, and has the base contained by the shorter sides greater than the base of the larger vertical angle HDG contained by the longer sides. But neither can it be said that angle HML is greater than CHG; because, if MDL is smaller than HDG, the angle DML at the base of the isosceles is greater than DHG likewise at the base; therefore if to the greater DML the greater HML be added, and to the lesser DHG the lesser CHG be added, the whole DMH will be greater than the whole angle DHC, that is, an internal angle greater than an external one, contrary to Book 1, Proposition 16. If therefore the angle HML, compared with angle CHG, can be neither equal, nor less, nor greater, the hypothesis having been made that angle MDL is smaller than HDC, it is necessarily concluded that angle MDL is not smaller than angle HDG. Since therefore angle MDL is neither equal to nor smaller than angle HDG, it follows that it is greater: therefore the angle below the base MLH is also greater than angle HGC; likewise angle MHL is greater than HCG; therefore the remaining HML is less than the remaining CHG: but to these are subtended the equal lines HL and CG, therefore triangle HML is in the greater circle, and consequently, to the greater angle MLH than CGH, there is also subtended a greater side: wherefore MH, that is SN, parallel and equal to it, is greater than CH, that is RS: and so it is raised to a greater height by SN than by RS, the traction being made equal, or the power drawing moving with equal motion. From this it is also manifest that the greater the obliquity and departure of the pull from parallelism with the inclined plane, the more the difficulty for the puller increases. Easily from what has been said you will gather how much labor is saved in Rome by fitting carriages with higher wheels (in ancient times they were called Cisia), so that a single horse, harnessed to the pole and keeping it nearly parallel to the lower plane beneath, when it ascends a slope, can draw enormous weights, to which it surely would not be equal if the axle of the wheels were less distant from the plane beneath, and the horse-draught were oblique upward: although, as will be explained elsewhere in its place, the very size of the wheels contributes greatly. Similarly in the hauling of ships, which against the current

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Liber primus. CAPUT XVI. 115 adverso flumine deducuntur fune absidi mali conjuncto, aliquid juvare funis longitudinem, ut scilicet minùs obliqua sit tractio, ex dictis confirmatur: quamvis enim tractiones in plano inclinato consideraverimus, ut gravium elevationem expenderitemus, aliquid etiam facit obliquitas tractionis in plano horizontali, cujusmodi est aqua, cui navis innatat; pars siquidem demersa obstantem undam repellere debet; nec planè inutile est, secundùm quam lineam dirigatur motus potentiæ trahentis, vi cujus impedimentum superandum est. Hactenus nobis de tractione sermo fuit, quæ motum inferens non nisi spatiis, per quæ motus est, determinari potuit. Quoniam verò in obliquis tractionibus non eandem semper analogiam servari, quæ in parallelâ tractione eadem perpetuò est, deprehendimus, inquirendum superest, quæ demum Ratio momentorum sit pro singulis obliquitatibus, ut constet, quibus viribus retineri possit, ne in proclive labatur pondus, etiamsi vires ad illud ulteriùs elevandum non suppetant. Quamquam autem pondera quasi molis expertia unico puncto expressimus in plano ipso inclinato, ut in 1. fig. hujus cap. re tamen verâ centrum gravitatis attendendum est, ut in 2. schemate, quod utique distat à plano, cui corpus grave incumbit: hujus verò distantiam nulla certior mensura definit, quàm linea ex eo cadens in subjectum planum ad angulos rectos, hæc quippe omnium brevissima est. Sit igitur planum inclinatum A B, cui impositus globus centrum habet gravitatis C, & contingit planum in D; ac propterea etiam, quæ à centro ad contactum ducitur recta C D, distantiam determinat, cum sit plano perpendicularis ex 18. lib. 3. Jam recta C E parallela plano ducatur, & sit linea suspensionis, quam claritatis gratiâ parallelam vocemus: & per D punctum, in quod cadit linea distantiæ centri gravitatis transeat perpendicularis horizonti linea F D quæ in G, secat lineam C E. Constat trian- Early European Books, Copyright 2011 ProQuest LLC. Images reproduced by courtesy of the Biblioteca Nazionale Centrale di Firenze. CFMAGL. 1.6.199

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Book I. CHAPTER XVI. 115 when they are led downstream by a rope attached to the mast-head, something is gained by the length of the rope, namely that the pull is less oblique, is confirmed by what has been said: for although we have considered pulls on an inclined plane, in order to weigh the raising of heavy bodies, the obliquity of the pull on a horizontal plane also has some effect, such as water, on which a ship floats; for the submerged part must repel the opposing wave; and it is by no means useless, according to what line the motion of the power that pulls is directed, by whose force the obstacle must be overcome. Thus far we have spoken of traction, which, as producing motion, could be determined only by the spaces through which the motion takes place. But since in oblique traction we have found that the same analogy is not always preserved, which in parallel traction is always the same, the question remains to be investigated, what finally the ratio of moments is for each degree of obliquity, so that it may be known by what forces a weight may be held back from slipping downhill, even if there are no forces sufficient to raise it further. Although however we have represented weights, as if free of mass, by a single point in the inclined plane itself, as in fig. 1 of this chapter, in reality the center of gravity must be considered, as in diagram 2, which is certainly at a distance from the plane on which the heavy body rests: but no surer measure defines this distance than the line falling from it onto the underlying plane at right angles; for this is the shortest of all. Let there be then an inclined plane A B, upon which a globe is placed having its center of gravity C, and touching the plane at D; and therefore also the straight line C D, drawn from the center to the point of contact, determines the distance, since it is perpendicular to the plane, from lib. 3, prop. 18. Now let the straight line C E, parallel to the plane, be drawn, and let it be the line of suspension, which for the sake of clarity we shall call parallel; and through the point D, into which the line of the center's distance falls, let the perpendicular to the horizon be drawn, line F D, which in G, cuts the line C E. It is clear that the trian- Early European Books, Copyright 2011 ProQuest LLC. Images reproduced by courtesy of the Biblioteca Nazionale Centrale di Firenze. CFMAGL. 1.6.199

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Mechanicorum 116 gulum DGC simile esse triangulo BAS: quia enim GD par- allela est lineæ AS pariter perpendiculari ad horizontem, an- guli SAB, ADG alterni æquales sunt per 27. lib. 1. Et quo- niam angulus CDA ex constructione est rectus, complemen- tum CDG æquale est angulo complementi ABs; anguli verò DCG, BSA sunt recti, hic quidem ex hypothesi, ille autem propter linearum CE, DA parallelismum: igitur reliquus CGD reliquo BAS æqualis est; ac proptereà per 4. lib. 6. ut BA ad AS, ita DG ad GC. Quoniam itaque, si pondus in plano inclinato ad pondus in perpendiculari sit ut inclinata BA ad perpendicularem AS, eorum momenta æqualia sunt, & æquiponderant, etiam globus æqualia ad descendendum habet momenta, ac potentia habeat vires ad retinendum in parallelâ EC, si globi gravitas ad potentiam retinendum sit ut DG ad GC. Verum quidem est globum non per lineam FD, sed per CT à centro gravitatis perpendicularem horizonti deorsum ni- ti: Sed quia CT ipsi FD parallela est, triangulum CTD triangulo DGC simile est & æquale; atque adeò parùm in- terest, utrum lineis DG, GC, an verò lineis CT, TD eadem Ratio exponatur. Sed jam retineatur globus per rectam CH; utique perinde se- cundùm eam directionem se habet, atque si esset planum HCK; globus enim sustinetur per lineam DC, & retinetur ex H, ac proinde secundùm rectâ HCK conatur deorsum eo situ: quam- quam subjecti plani inclinatio obstaret, ne secundùm rectam HCK procederet, si sibi dimitteretur, & alia atque alia plana constituerentur. Planum itaque illud HC declinat à perpen- diculari, cum quâ constituit angulum CID æqualem externo KCT propter parallelismum perpendicularium FD, CT per 27. lib. 1. qui utique CID minor est externo CGD per 16. lib. 1. & quidem differentia anguli ICG per 32. lib. 1. Fiat ergo angulus BAP æqualis angulo CIG; quia BAS ostensus est æqualis ipsi CGD, remanet PAS æqualis angulo ICG. Quare BPA externus æqualis est duobus internis, scilicet recto PSA, & acuto SAP, per 32. lib. 1. igitur idem angulus BPA æqualis est toti angulo DCI. Sunt itaque æquiangula & simi- lia duo triangula BAP & DIC, atque per 4. lib. 6. ut BA ad AP, ita DI ad IC. Atqui pondera super BA & AP, quæ sint ut

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Mechanics 116 the angle DGC is similar to the triangle BAS: because GD is parallel to the line AS, which is likewise perpendicular to the horizon, the alternate angles SAB, ADG are equal by 27, book 1. And since the angle CDA is, by construction, a right angle, the complement CDG is equal to the angle of the complement ABs; and the angles DCG, BSA are right, the former indeed by hypothesis, the latter because of the parallelism of the lines CE, DA: therefore the remaining angle CGD is equal to the remaining BAS; and therefore by 4, book 6, as BA is to AS, so is DG to GC. Since therefore, if the weight on the inclined plane to the weight on the perpendicular be as the inclined BA is to the perpendicular AS, their moments are equal and balance each other, the globe also has equal moments for descending, and power has force for retaining it in the parallel EC, if the globe’s gravity to the power retaining it be as DG is to GC. But it is indeed true that the globe does not press downward by the line FD, but by CT, from the center of gravity perpendicular to the horizon: but because CT is parallel to FD itself, the triangle CTD is similar and equal to the triangle DGC; and therefore it matters little whether the same ratio is expressed by the lines DG, GC, or rather by the lines CT, TD. But now let the globe be retained by the straight line CH; it will in every respect behave according to that direction, as if there were the plane HCK; for the globe is sustained by the line DC, and is retained from H, and therefore tends downward in that position according to the straight line HCK: although the inclination of the underlying plane would prevent it from proceeding according to the straight line HCK, if it were left to itself, and if one plane after another were constructed. That plane HC therefore declines from the perpendicular, with which it makes the angle CID equal to the external KCT because of the parallelism of the perpendiculars FD, CT by 27, book 1, which is indeed less than the external CGD by 16, book 1, and indeed by the difference of the angle ICG by 32, book 1. Let therefore the angle BAP be made equal to the angle CIG; because BAS has been shown equal to CGD itself, there remains PAS equal to the angle ICG. Wherefore BPA, the external angle, is equal to the two internal, namely the right PSA and the acute SAP, by 32, book 1. Therefore the same angle BPA is equal to the whole angle DCI. Thus the two triangles BAP and DIC are equiangular and similar, and by 4, book 6, as BA is to AP, so is DI to IC. But the weights on BA and AP, which be as

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Liber primus. CAPUT XVI. 117 ut B A ad A P, æquiponderant ex dictis cap. 13. ergo etiam æqualium momentorum est globus, & potentia retinens per H C, si globus ad potentiam sit ut D I ad I C, hoc est ut C N ad N D, si ex D intelligatur exire D N parallela ipsi H C. Eâdem ratione si linea obliqua, per quam globus retinetur, sit infra parallelam C E, ut si sit C X, ostendetur globi gravitatem ad potentiam retinentem esse ut D Q ad Q C, est enim quasi planum inclinatum faciens cum perpendiculari angulum D Q C majorem interno D G C, hoc est majorem angulo B A S illi æquali. Fiat igitur angulo D Q C æqualis angulus B A Y: & quia A B Y æqualis est angulo C D Q, ut superiùs dictum est, triangula B A Y, D Q C sunt æquiangula & similia, ac per 4. lib. 6. ut B A ad A X, ita D Q ad Q C: ergo quia pondera super B A, & A Y, quæ sint in Ratione B A ad A Y, æquiponderant, etiam globi & potentiæ retinentis momenta æqualia sunt, si fuerint ut D Q ad Q C. Hîc autem tria observanda occurrunt. Primum est, quòd Rationes prædictæ momentorum potentiæ retinentis comparatæ ad pondus idem, quamvis pro diversâ obliquitate aliis atque aliis lineis explicentur D Q ad Q C, & D G ad G C, D I ad I C, omnes tamen exponuntur comparatè ad eandem B A in triangulo B A Y; in quo ipsæ quoque inter se invicem comparari possunt. Secundum est, quòd si obliquitas tàm supra, quàm infra parallelam C E æqualis sit, hoc est angulus I C G æqualis sit angulo G C Q, momenta potentiæ retinentis in H & X æqualia sunt; inter se siquidem sunt ut A P, & A Y, quæ lineæ æquales sunt; nam anguli P A S, Y A S æquales sunt ex hypothesi, & constructione, anguli autem ad S sunt recti & latus A S est utrique triangulo commune; ergo etiam per 26. lib. 1. latera A P & A Y æqualia sunt. Tertium est, quòd in lineâ C E parallelâ minus virium exigitur ad retinendum globum, quàm in cæteris: nam & linea A S vires potentiæ repræsentans omnium minima est, utpote perpendicularis. Ex his & illud colligitur, quod si linea, secundùm quam pondus retinetur in plano inclinato, sit parallela horizonti, eadem est philosophandi methodus. Si enim super plano inclinato A B sit pondus tangens in C, cujus gravitatis centrum sit D, & linea retentionis D E horizonti parallela, ducatur P 3

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Book the First. CHAPTER XVI. 117 as B A to A P are equal in weight, from what was said in chap. 13; therefore also the sphere is of equal moments, and the retaining power by H C, if the sphere to the power be as D I to I C, that is as C N to N D, if from D it is understood that D N is drawn parallel to H C. For the same reason, if the oblique line by which the sphere is retained be below the parallel C E, as if it were C X, it will be shown that the gravity of the sphere to the retaining power is as D Q to Q C; for it is as it were an inclined plane making with the perpendicular the angle D Q C greater than the internal angle D G C, that is, greater than the angle B A S equal to it. Let there therefore be made equal to the angle D Q C the angle B A Y: and because A B Y is equal to the angle C D Q, as was said above, the triangles B A Y, D Q C are equiangular and similar, and by 4. lib. 6. as B A is to A X, so is D Q to Q C: therefore because the weights upon B A, and A Y, which are in the ratio of B A to A Y, are in equilibrium, the moments of the sphere and of the retaining power are also equal, if they be as D Q to Q C. Here, however, three things present themselves to be observed. The first is, that the aforesaid ratios of the moments of the retaining power compared with the same weight, although for different obliquities they are expressed by different and different lines, D Q to Q C, and D G to G C, D I to I C, yet all are set forth in comparison with the same B A in the triangle B A Y; in which they also may be compared among themselves. The second is, that if the obliquity both above and below the parallel C E be equal, that is, if the angle I C G be equal to the angle G C Q, the moments of the retaining power in H and X are equal; for between themselves they are as A P and A Y, which lines are equal; for the angles P A S, Y A S are equal by hypothesis and construction, and the angles at S are right angles and the side A S is common to both triangles; therefore also by 26. lib. 1. the sides A P and A Y are equal. The third is, that in the line C E parallel, less force is required to retain the sphere than in the others: for the line A S, representing the forces of the power, is the smallest of all, since it is perpendicular. From these things this also is gathered, that if the line according to which a weight is retained in an inclined plane be parallel to the horizon, the same method of reasoning applies. For if upon the inclined plane A B there be a weight touching at C, whose center of gravity is D, and the line of retention D E parallel to the horizon, let there be drawn P 3

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Mechanicorum CF perpendicularis horizonti; & Ratio ponderis ad vires retinentes erunt ut CF ad FD. Fiat enim angulus B A H æqualis angulo C F D, qui utique est rectus, cum DE ex hypothesi sit horizonti parallela, FC verò perpendicularis: ergo super AB, AH æquiponderant pondera, quæ sint ut AB ad AH; paria igitur sunt momenta, si pondus ad vires potentiæ retinentis in eâdem Ratione sit ut AB ad AH, hoc est ut CF ad FD. Quia enim B AH angulus est rectus per 8. lib. 6. est ut BA ad AH, ita BG ad GA; est autem BG ad GA ut CF ad FD; quia nimirum FC perpendicularis horizonti est parallela ipsi AG, & anguli BAG, FC A alterni sunt æquales per 27. lib. 1. DCA verò est rectus ex hypothesi; igitur & DCF complementum recti æquale est angulo ABG: utrumque triangulum est rectangulum; ergo ut BG ad GA, ita CF ad FD. Hinc apparet fieri posse, ut ad retinendum pondus in tali situ aliquando plus virium requiratur, quàm ad sustinendum illud in perpendiculari; quando videlicet ex inclinatione plani AB consequitur lineam CF minorem esse quàm FD: immò crescit retinendi difficultas, si adhuc retentio fiat per lineam inferiorem horizontali DE, quæ cum perpendiculari CF constituat angulum DIC obtusum; cum enim cresceret linea DI supra DF, & IC decresceret infra FC, esset minor Ratio ponderis in perpendiculo ad potentiam obliquè retinentem, quæ proinde major esse deberet, ut fieret momentorum æqualitas. Concipe autem sublatum triangulum totum BAH, & DC esse columnam, quæ in eodem situ inclinata retineri debeat: jam satis constat ex dictis, quâ ratione disponi oporteat funes, ut qui funium extremitates tenent, minus laboris impendant. Non est tamen eadem funis retinentis, & fulcri sustentantis ratio: in supponendis enim fulcris illud potissimùm attenditur, quòd fulcrum ipsum integrum permaneat, citrà scissionis aut fractionis periculum; id quod habetur, quò magis perpendiculari ad horizontem situi proximum collocatur; parùm scilicet interest,

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Mechanics CF perpendicular to the horizon; and the ratio of the weight to the retaining forces will be as CF to FD. Let the angle B A H be equal to the angle C F D, which certainly is a right angle, since DE by hypothesis is parallel to the horizon, and FC perpendicular: therefore upon AB, AH the weights are in equilibrium, which shall be as AB is to AH; the moments are therefore equal, if the weight to the retaining force of power be in the same ratio as AB is to AH, that is, as CF is to FD. For since the angle BAH is a right angle, by 8. lib. 6. it is as BA to AH, so BG is to GA; but BG to GA is as CF to FD; because namely FC, being perpendicular to the horizon, is parallel to AG itself, and the angles BAG, FCA are alternate and equal by 27. lib. 1. But DCA is a right angle by hypothesis; therefore also DCF, the complement of a right angle, is equal to the angle ABG: both triangles are right-angled; therefore as BG is to GA, so CF is to FD. Hence it appears that it can happen that, for retaining a weight in such a position, sometimes more force is required than for supporting it in the perpendicular; namely when from the inclination of the plane AB it follows that the line CF is less than FD: indeed, the difficulty of retention increases if the holding is made by means of a line lower than the horizontal DE, which with the perpendicular CF forms the obtuse angle DIC; for as the line DI would increase above DF, and IC would decrease below FC, the ratio of the weight in the plumb line to the power obliquely retaining it would be smaller, and therefore the latter would need to be greater, so that equality of moments might be attained. Now suppose the whole triangle BAH removed, and DC to be a column, which must be retained inclined in the same position: it is now sufficiently clear from what has been said in what way the ropes ought to be arranged, so that those who hold the ends of the ropes may expend less labor. However, the relation of the rope holding and the support sustaining is not the same: for in setting supports, the chief thing considered is that the support itself may remain whole, without danger of splitting or breaking; this is achieved the more, the nearer it is placed to a position parallel to the horizon, as far as possible; that is, it makes little difference,

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Liber primus. CAPUT XVI. 119 interest, quanto conatu subjectam tellurem urgeat modò certi simus de fulcri ipsius firmitate. Cæterùm si tu ipse sustem manu tenens cogaris inclinatam columnam sustinere, punctum autem sustentationis, cui fulcrum applicatur, magis à sub- jecto plano distet, vel saltem non minùs, quàm centrum gra- vitatis columnæ, experieris minori conatu opus esse, si ful- crum axi columnæ perpendiculare sit, qui situs respondet re- tentioni parallelæ plano inclinato, majorem verò adhiben- dum esse conatum, si fulcrum cum eodem axe acutum aut ob- tusum angulum constituat; id quod obliquis elevationibus respondet. Quòd si infra centrum gravitatis applicetur fulcrum, jam constat hoc ita esse collocandum, ut ei idem centrum im- mineat, alioquin aut columna corruet, aut multis viri- bus tibi contendendum erit, ut illam sustentes à lapsu; si tamen ea sit complexio tùm inclinationis, tùm obicis co- lumnæ pedem retinentis, ne excurrat, aut elevetur, tùm po- sitionis fulcri, ut aliquatenus sustineri columna possit, ne pror- sùs ruat. Sed quoniam hîc columnæ mentio incidit, præstat ele- vationes corporum, quæ non tota elevantur, sed eorum altera extremitas subjecto alicui fulcro aut plano innititur, altera elevatur aut suspenditur, considerare: neque enim hîc reputanda sunt momenta gravitatis perinde, ac si totum cor- pus elevaretur aut suspenderetur, quemadmodum paulò an- te dicebatur; immò verè longè minora sunt pro ratione distantiæ à centro gravitatis, ut ex inferiùs dicendis, ubi de æquilibrio, atque de vecte sermo erit, constabit. Cavendum autem plurimum est ab æquivocationibus, quæ obrepere possunt, nisi animum advertas ad gravitatem, sivè per totam longitudinem, quæ movetur, aut ad motum incitari potest, diffusam, sivè quasi in unum punctum ibi collectam, ubi ele- vans applicatur, ut in vecte, aut librâ; hinc enim non mo- dica momentorum inæqualitas oritur. Nam si puncto appli- cationis respondeat centrum gravitatis, multò majores ad elevandum, aut suspendendum corpus requiruntur vires, quàm si centrum gravitatis à puncto applicationis aliquo in- tervallo fejungatur. Hinc

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Book One. CHAPTER XVI. 119 it matters how much effort is used to press upon the body laid beneath, provided only that we are certain of the firmness of the support itself. Moreover, if you yourself, holding it in your hand, are compelled to sustain a leaning column, and the point of support to which the fulcrum is applied is farther from the subjacent plane, or at least no less distant, than the center of gravity of the column, you will find that less effort is needed if the fulcrum is perpendicular to the axis of the column, a position corresponding to a support parallel to an inclined plane; but greater effort must be applied if the fulcrum makes an acute or obtuse angle with the same axis, which corresponds to oblique elevations. But if the fulcrum is applied below the center of gravity, it is now clear that it must be so placed that that same center overhangs it; otherwise either the column will fall, or you will have to contend with many efforts in order to sustain it from slipping. This, however, depends upon a combination both of the inclination, and of the obstacle retaining the foot of the column, lest it run out or be lifted, and of the position of the fulcrum, so that the column may be sustained to some extent, and not fall outright. But since mention has here been made of a column, it is better to consider the elevations of bodies which are not raised as a whole, but where one extremity rests on some underlying fulcrum or plane, while the other is lifted or suspended; for here the moments of gravity are not to be reckoned in the same way as if the whole body were being lifted or suspended, as was said a little earlier. Indeed, they are very much smaller in proportion to the distance from the center of gravity, as will be made clear from what is said below, where the discussion will be of equilibrium and of the lever. Very great caution, however, must be used against ambiguities that may creep in, unless you pay attention to gravity, whether diffused through the whole length that is moved, or that can be set in motion, or else as though gathered into a single point there where the lifter applies force, as in the lever or the balance; for from this there arises a not inconsiderable inequality of moments. For if the center of gravity corresponds to the point of application, much greater forces are required to raise or suspend the body than if the center of gravity is separated from the point of application by some interval. Hence

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Mechanicorum Hinc si sit prisma AB hor- izontaliter collocatum, ejus- que extremitas A innitatur apici pyramidis, altera verò extremitas B suspendatur per- pendiculari funiculo CB, vel sustentetur supposito ad per- pendiculu[m] fulcro DB, æqua- liter res se habet, & pares requiruntur vires tàm in suspenden- te CB, quàm in sustentante DB: hæ tamen vires non pares esse debent toti ponderi prismatis; sed quia centrum gravita- tis E ab utroque extremo æqualiter distare supponitur, se- missis tantùm gravitatis percipitur in B. Quod si in eodem horizontali situ retineatur prisma sivè à suspendente obliquo IB, sivè ab obliquo sustentante OB, utique retinentis, aut sustentantis vires æquipollere debent viribus retinentis aut sustentantis ad perpendiculum CB aut DB. Quemadmo- dum igitur pondera illa super BO & BD æquiponderant, quæ sunt ut BO ad BD, ita vires, quæ secundùm easdem lineas ac directiones æqualem effectum præstare debent; in eâdem Ratione BO ad BD esse oportet: Vires ergo retinen- tis BI obliqui ad vires retinentis CB ad perpendiculum sunt ut BO ad BD, hoc est, ductâ parallelâ CI, ut IB ad CB, propter triangulorum OBD, CBI similitudinem. Ut autem non hîc perperam nos philosophari innotescat, finge sublatam ex A pyramidem, & constitutam in G ita, ut ex B ad perpendiculum dependeat pondus aliquod æqui- librium efficiens cum prismate: quo perpendiculari pondere sublato, ut prisma horizontale permaneat, certum est super plano inclinato BO requiri pondus, quod ad pondus per- pendiculare ex BD sit ut BO ad BD: igitur si loco pon- deris applicentur secundùm eandem rectam lineam BO vires alicujus viventis, à quo retineatur prisma in eodem situ ho- izontali, satis apparet conatum debere esse ut BO ad cona- tum, qui secundùm perpendicularem requireretur ut BD. Sicut itaque conatus deorsum trahens, cum fulcrum est in G citrà centrum gravitatis E, ex inclinatione lineæ, secun- dùm quam fit, desumitur, ita etiam conatus suspendens IB, aut

Transcription: Translated (English)

Mechanics Hence if the prism AB be placed horizontally, and its end A rest on the apex of the pyramid, while the other end B is suspended by the perpendicular cord CB, or supported by the perpendicular prop DB, the case is the same, and equal forces are required both in suspending by CB and in supporting by DB: yet these forces ought not to be equal to the whole weight of the prism; rather, since the center of gravity E is assumed to be equally distant from either end, only half the weight is perceived in B. But if the prism be kept in the same horizontal position either by the oblique suspender IB or by the oblique supporter OB, certainly the force of the retainer or supporter must be equivalent to the force of the retainer or supporter at right angles CB or DB. Therefore, just as those weights on BO and BD are in equilibrium, which are as BO is to BD, so too the forces which along those same lines and directions ought to produce equal effect; in the same ratio BO to BD must they be. The force of the oblique retainer BI, therefore, to the force of the perpendicular retainer CB is as BO to BD; that is, drawing the parallel CI, as IB is to CB, by reason of the similarity of triangles OBD and CBI. But lest it be thought that we are here philosophizing incorrectly, imagine the pyramid removed from A and placed at G, so that some weight hanging perpendicularly from B produces equilibrium with the prism: this perpendicular weight being taken away, in order that the prism may remain horizontal, it is certain that upon the inclined plane BO a weight is required which, compared with the perpendicular weight from BD, is as BO is to BD. Therefore if, in place of the weight, the forces of some living agent are applied along the same straight line BO, by which the prism is retained in the same horizontal position, it is plainly seen that the effort ought to be as BO to the effort which would be required along the perpendicular BD. Just as therefore the downward-pulling effort, when the fulcrum is at G on this side of the center of gravity E, is taken from the inclination of the line along which it acts, so also the suspending effort IB, or

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Liber primus. CAPUT XVI. 121 aut sursum urgens OB, cum fulcrum est in A ultrà centrum gravitatis E, desumendus est pariter ex inclinatione lineæ, se- cundùm quam applicatur prismatici, comparatè ad conatum per- pendicularem CB, vel DB, habitâ semper ratione distantiæ fulcri à centro gravitatis. Ne quid verò dubitationis supersit, utrum OB deorsum, & IB sursum trahentium pa- res sint vires secundùm ean- dem rectam lineam OI, sint rotulæ duæ H & F circa suum axem versatiles infixæ extre- mitatibus regulæ, aut tigilli, & ex funiculo rotularum ca- vitatibus inserto dependeant æqualia pondera L & G. Hæc pondera sibi vicissim æquipon- derare manifestum est, quem- cumque tandem situm sivè perpendicularem, sivè incli- natum, habeat regula, aut ti- gillus, cui rotulæ infixæ sunt. Sit libræ jugum AB æqualiter in E divisum, circa quod punctum stabile moveri queat, & in A adnectatur funiculo HF: ex B autem dependeat pondus D æquale ponderi G, sed ita obliquè dispositum, ut linea BO parallela sit lineæ AF. Submove pondus L, remanent G & D, quorum neutrum prævalere potest; sunt enim æqualia inter se, & per lineas similiter inclinatas AF, BO agunt. Re- pone pondus L, & amove pondus G, item removeatur pon- dus D, & sursum ponatur æquale C; aio libræ jugum AB adhuc retinere eumdem situm; quia scilicet pondera C & D vicissim æquiponderabant, sicut etiam G & L: igitur quantum virium habebat pondus D ad æquiponderandum ipsi G, tan- tumdem virium habet pondus C ad æquiponderandum ponde- ri L, hoc est eidem ponderi G. Sivè igitur in superiori sche- mate considerentur vires deorsum trahentes aut sustentantes OB, sive retinentes IB, perinde est, & æqualium momento- rum censendæ sunt. Q

Transcription: Translated (English)

Liber first. CHAPTER XVI. 121 whether the force OB, pressing upward, when the fulcrum is at A beyond the center of gravity E, is to be taken likewise from the inclination of the line along which the prismatic piece is applied, compared with the perpendicular effort CB, or DB, always regard being had to the distance of the fulcrum from the center of gravity. So that no doubt may remain, whether the forces OB drawing downward, and IB drawing up- ward, are equal along the same straight line OI, let there be two pulleys H and F, turnable about their axis, fixed to the ends of a rule, or beam, and from the cord inserted into the cavities of the pulleys let equal weights L and G hang. It is manifest that these weights counterbalance one another, whatever position, whe- ther perpendicular or inclined, the rule, or beam, to which the pulleys are fixed may have. Let the beam of the balance AB be equally divided at E, about which fixed point it may move, and at A let a cord HF be attached: from B, however, let a weight D, equal to the weight G, hang, but so obliquely disposed that line BO may be parallel to line AF. Remove weight L, and G and D remain, neither of which can prevail; for they are equal to one another, and act through similarly inclined lines AF, BO. Replace weight L, and remove weight G; likewise let weight D be taken away, and in its place let an equal C be put upward; I say that the beam AB will still retain the same position; because, namely, weights C and D counterbalanced one another, as also G and L: therefore, as much force as weight D had for counterbalancing G, so much force does weight C have for counterbalancing weight L, that is, the same weight G. Whether therefore in the foregoing figure the forces drawing downward or sustaining OB, or retaining IB, be considered, it is all the same, and they are to be reckoned as moments of equal value. Q

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Mechanicorum Non jam horizontale sit prisma AB, sed inclinatum, & puncto A stabili innixum: momenta ad descendendum, ac proinde repugnantia ad ascendendum, ut superiùs innuimus cap. 14; æstimanda sunt in plano DC inclinato, quod cum AB angulos facit rectos, & cum horizonte AE concurrit in puncto E. Ducatur per B perpendicularis ad horizontem FH, & ex H ad BE perpendicularis HO. Momen- ta gravitatis prismatis in perpendiculari ad momenta ejusdem in inclinatâ sunt reciprocè ut inclinata EB ad perpendicularem BH, hoc est per 8. lib. 6. ut HB ad BO, sive (ductâ ex D super DB inclinatam perpendiculari DG secante rectam HF in F) ut BF ad BD, propter similitudinem triangulorum OBH, DBF. Vires ergo retinentes in D ad vires retinentes in F sunt ut DB ad BF. Retineatur prisma secundùm obliquam GB, quæ producæ usque ad Horizontalem concurrat in L. Iterum ex L ad DE cadat ad angulos rectos LC, quæ perpendicularem FH secabit in I: est autem IC parallela ipsi HO; ac propterea per 4. lib. 6. ut HB ad BO, ita IB ad BC, & per 11. lib. 5. ut IB ad BC, ita BF ad BD. Ad retinendum igitur prisma in eodem situ inclinationis BAE per obliquam GB, vires æquipollentes viribus retinentibus in perpendiculari FB esse oportet ut BL ad BI, quemadmodum retinentes per rectam DB sunt ut BC. Quare datâ corporis inclinatione, cujus gravitas retinenda est in eodem situ, sumatur ejusdem axis transiens per gravitatis centrum, & ad axis extremitatem mobilem ducatur ipsi axi perpendicularis DB, in quâ assumpto quolibet puncto D, ducatur prædicto axi parallela DG, quæ secans lineas quaslibet obliquas, & perpendicularem ad Horizontem, dabit omnium obliquarum suspensionum Rationem: Sic recta DG secans perpendicularem FB & obliquam GB determinat Rationem virium in utrâque suspensione, ut scilicet sunt in Ratione BF ad BG, & sic de reliquis. Quòd

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Mechanics Now let the horizontal prism AB no longer be, but inclined, and resting on the fixed point A: the moments tending to descend, and therefore the resisting forces tending to ascend, as we hinted above ch. 14, are to be estimated in the inclined plane DC, which makes right angles with AB, and meets the horizon AE at point E. Through B draw the perpendicular FH to the horizon, and from H to BE the perpendicular HO. The moments of the prism’s weight in the perpendicular are to its moments in the inclined line reciprocally as the incline EB to the perpendicular BH, that is, by 8. lib. 6, as HB to BO, or (with DG drawn from D perpendicular to the inclined DB, cutting the straight line HF at F) as BF to BD, on account of the similarity of the triangles OBH, DBF. Therefore the retaining forces at D to the retaining forces at F are as DB to BF. Let the prism be retained along the oblique GB, which being extended meet the Horizontal in L. Again, from L let LC fall at right angles to DE, which will cut the perpendicular FH at I: and IC is parallel to HO itself; and therefore, by 4. lib. 6, as HB is to BO, so IB is to BC, and by 11. lib. 5, as IB is to BC, so BF is to BD. In order therefore to retain the prism in the same state of inclination BAE by the oblique GB, the equivalent forces to the forces retaining it in the perpendicular FB ought to be as BL to BI, just as the retaining forces along the straight line DB are as BC. Wherefore, given the inclination of the body, whose weight is to be retained in the same position, let its axis passing through the center of gravity be taken, and to the movable extremity of the axis let a line perpendicular to the axis DB be drawn, and on any point D assumed in it, let DG be drawn parallel to the said axis, which, cutting any oblique lines and the perpendicular to the Horizon, will give the ratio of all oblique suspensions: thus the straight line DG, cutting the perpendicular FB and the oblique GB, determines the ratio of the forces in both suspensions, namely, as BF is to BG, and so on for the rest. That

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Liber primus. CAPUT XVI. 123 Quòd si in gradibus data sit inclinatio prismatis, & funiculi obliquè suspendentis declinatio à perpendiculo, statim ex tabulis Sinuum, aut etiam Secantium, apparebit Ratio quæsita linearum: angulus enim, quem perpendicularis ad axem facit cum perpendiculari ad Horizontem, æqualis est angulo inclinationis prismatis; angulo siquidem B A E inclinationis prismatis, æqualis est angulus E B H per 8. lib. 6. ac proptereà etiam ex 15. lib. 1. qui illi est ad verticem D B F. Hinc si inclinationis angulus sit gr. 36. D B ad B F erit ut Radius ad Secantem gr. 36. vel ut Sinus gr. 54. complementi gr. 36. ad Radium. At angulus, quem facit linea obliquæ suspensionis cum perpendiculari ad horizontem transeunte per prismatis punctum; in quo suspenditur, est æqualis angulo, quem eadem suspensionis linea facit cum perpendiculo transeunte per aliud extremum ejusdem lineæ suspensionis, cui applicatur potentia retinens: duæ enim perpendiculares prædictæ sunt inter se parallelæ, & linea suspensionis in eas incidens alternos angulos facit æquales per 27. lib. 1. Si igitur G B à suo perpendiculo, quod ex G in horizontem cadat, declinat gr. 25. etiam F B G est gr. 25. Totus igitur angulus D B G est aggregatum anguli inclinationis prismatis, & anguli declinationis funiculi suspendentis: igitur D B G est gr. 61, & positâ D B ut Radio, erit B G Secans gr. 61. Vel si comparanda sit B G cum B F, qui angulus G B B externus per 32. lib. 1. æqualis est duobus internis oppositis trianguli D B F, erit G B G gr. 126; at F B G est gr. 25, igitur F B G est gr. 29. Quare B F ad B G est ut Sinus gr. 29. ad Sinum gr. 126, hoc est supplementi gr. 54. Apparet ex his primò minimas vires exerceri, si linea retentionis cadat ad perpendiculum in axem corporis elevati cum inclinatione; quia scilicet cum in D sit angulus rectus, recta B D est omnium linearum ex B puncto exeuntium, & in rectam D G cadentium minima: quò autem major fuerit obliquitas, eò etiam majores vires requiri, quia longiores sunt Secantes angulorum majorum in B posito Radio B D. Secundò fieri potest, ut parè vires requirantur, si linea retentionis faciat cùm axe corporis elevati angulum acutum, ac si faciat cùm eodem angulum obtusum, ut si fuerit recta M B; ipsa enim pariter opponitur angulo recto B D M, ac proinde Q 2

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Book One. CHAPTER XVI. 123 If, in the graduated scales, the inclination of the prism is given, and the deviation of the cord hanging obliquely from the perpendicular, the ratio sought of the lines will immediately appear from the tables of sines, or even of secants: for the angle which the perpendicular to the axis makes with the perpendicular to the horizon is equal to the angle of inclination of the prism; for the angle B A E of inclination of the prism is equal to the angle E B H, by 8. lib. 6, and therefore also, by 15. lib. 1, to that at the vertex D B F. Hence, if the angle of inclination be 36 degrees, D B to B F will be as Radius to the secant of 36 degrees, or as the sine of 54 degrees, the complement of 36 degrees, to Radius. But the angle which the oblique line of suspension makes with the perpendicular to the horizon passing through the point of the prism at which it is suspended, is equal to the angle which the same line of suspension makes with the perpendicular passing through the other extremity of the same line of suspension, to which the retaining force is applied: for the two aforesaid perpendiculars are parallel to each other, and the line of suspension falling upon them makes equal alternate angles, by 27. lib. 1. If therefore G B deviates from its perpendicular, which from G falls to the horizon, by 25 degrees, F B G is also 25 degrees. Therefore the whole angle D B G is the sum of the angle of inclination of the prism and the angle of deviation of the suspending cord; therefore D B G is 61 degrees, and if D B be taken as Radius, B G will be the secant of 61 degrees. Or, if B G is to be compared with B F, then the angle G B B, external by 32. lib. 1, is equal to the two opposite interior angles of triangle D B F; hence G B G will be 126 degrees; but F B G is 25 degrees, therefore F B G is 29 degrees. Wherefore B F to B G is as the sine of 29 degrees to the sine of 126 degrees, that is, of the supplement 54 degrees. From these things it appears, first, that the least forces are exerted if the line of retention falls perpendicularly on the axis of the elevated body with inclination; because, namely, when the angle in D is a right angle, the straight line B D is the least of all lines issuing from point B and falling upon the straight line D G: but the greater the obliquity, the greater the forces required, because the secants of the greater angles are longer when Radius B D is placed at B. Secondly, it may happen that nearly equal forces are required, if the line of retention makes an acute angle with the axis of the elevated body, as if it makes an obtuse angle with the same, as if the straight line were M B; for it is likewise opposed to the right angle B D M, and therefore

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Mechanicorum 124 eò major est quàm recta BD, quò fuerit major angulus MBD, qui potest esse æqualis angulo DBF, vel DBG; quo casu etiam ipsa BM æqualis erit ipsi BF aut BG. Ex quo ulteriùs sequitur, si à retinente obliquè fiat tractio elevando magis ac magis prisma sic inclinatum, mutari subinde momenta: hoc tamen intercedit discrimen, quod trahentis linea initio applicata, ut angulum faciat acutum cum axe prismatis, in ipsâ tractione semper majorem facit cum ipso axe angulum, donec veniat ad angulum rectum constituendum, ut si MB traheretur, donec coincidat cùm DB, quæ pariter moveri intelligatur: contrà verò trahentis linea applicata, ut cum axe faciat angulum obtusum, in ipsâ tractione magis adhuc obtusum angulum constituit, donec tractionis linea (si tamen fieri id possit) in unam rectam lineam cum axe prismatis conveniat. Quare in primâ illâ tractione minuitur conatus, in hac secundâ augetur. MECHA

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Mechanics 124 is greater than the straight line BD, the greater the angle MBD is, which may be equal to the angle DBF, or DBG; in which case also BM itself will be equal to BF or BG. From this it follows further that, if the pull is made obliquely by the holder, while lifting the prism thus inclined more and more, the moments are successively changed: yet there is this difference, that the line of the pull, applied at first so as to make an acute angle with the axis of the prism, in the very act of pulling always makes a greater angle with that axis, until it comes to form a right angle, as if MB were being drawn until it coincides with DB, which is likewise understood to move; but, on the contrary, the line of the pull, when applied so as to make an obtuse angle with the axis, in the very act of pulling makes that obtuse angle still more obtuse, until the line of the pull (if indeed this can be done) agrees in one straight line with the axis of the prism. Therefore, in the first kind of pulling the effort is diminished; in this second it is increased. MECHA

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MECHANICORUM LIBER SECUNDUS. De causis motus Machinalis. INNOTUIT, opinor, quantum ad præsens institu- tum satis esse possit, centrum gravitatis ex iis, quæ libro superiore dicta sunt: nunc propiùs ad ipsam machinalem scientiam accedendum, quam Mecha- nicam dicimus. Hæc Geometriæ subjicitur; neque enim, ut illa, puram corporum quantitatem molisque exten- sionem abstractè considerat, sed quatenus gravitati illigatam aut levitati; nihil tamen solicita de ipsâ corporum materie, au- reáne sit, an lapidea. Quamvis autem ea quoque Statices pars, quam Hydrostaticen indigitamus, se pariter in corporum gra- vitate considerandâ exerceat, aliam tamen sibi contemplatio- nem assumit; motum siquidem corporum singulorum naturæ congruentem, pro humorum, in quos incurrunt, diversitate, potissimùm speculatur: Mechanice verò eatenus solùm ingeni- tam corporibus propensionem in motum aut quietem explorat, ut earum facultati perspectæ vim possit opportunâ instrumento- rum machinatione inferre. Quapropter ut certâ methodo ma- chinas oneribus movendis pares construere valeamus, motus machinalis causas antè cognitas habere necesse est, quàm ma- chinas ipsas aggrediamur. His porrò jactis fundamentis ope- rosum non erit inædificare, & machinarum singularum vires, sivè simplices illæ sint, sivè compositæ, exponere: adeò ut iis ritè intellectis, quæ hoc secundo libro disputabuntur, vix quic- quam in reliquo opere supersit difficultatis. Q. 3

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MECHANICORUM BOOK THE SECOND. On the causes of mechanical motion. It has now been made known, I think, so far as may suffice for the present undertaking, what the center of gravity is, from the things said in the preceding book: now we must come more closely to the science itself of mechanics, which we call Mechanica. This is subject to Geometry; for it does not, like that, consider abstractly the pure quantity of bodies and the extension of mass, but only insofar as it is bound up with gravity or levity; yet it is in no way concerned with the matter of the bodies themselves, whether they be of gold or of stone. And although that part of Statics which we call Hydrostatics likewise applies itself to considering the gravity of bodies, it nevertheless takes up a different contemplation; for it chiefly speculates on the motion proper to individual bodies, according to the diversity of the fluids into which they fall. Mechanics, however, investigates only so far the innate tendency of bodies toward motion or rest, that, once their capacity has been understood, it may be able, by a suitable contrivance of instruments, to apply force. Wherefore, in order that we may be able by a certain method to construct machines suited for moving loads, it is necessary that we should first have known the causes of mechanical motion before we undertake the machines themselves. These foundations having been laid, it will not be laborious to build upon them, and to explain the powers of the several machines, whether they be simple or compound: so that, when these are rightly understood, there will scarcely remain anything difficult in the rest of the work. Q. 3

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CAPUT I. Quem ad finem Machinæ instruantur. Finis, quò demum unaquæque actio refertur, primus animo concipitur, præstituiturque, & idonea ad agendum subsidia, quæ deligenda sunt, moderatur. Hinc ille primus nobis in hâc contemplatione occurrit; quem scilicet ad finem ma- chinæ instituantur, instruanturque, considerandum; ut ad hanc quasi regulam cæteræ causæ dirigantur, & formentur. Fortè dixerit quispiam magnificè, eo consilio machinas à no- bis excogitatas, ut naturam arte vincamus; quemadmodum enim scribit Antipho Poëta apud Aristotelem in quæst. Mechan. sub initium, πεχην πρατοῦμεν, ὑν φύσει νικώμεθα. Sed hic planis- simè philosophandi locus est, non gloriandi insolentiùs. Quare fatendum est apertè, adhiberi machinas in subsidium infirmi- tatis; ut quod virium imbecillitas onus loco movere, aut omni- nò, aut nisi ægerrimè sola nequiret, illud demum facilè, quò libuerit, aut trahat, aut impellat, aut etiam expellat quantum- vis reluctans, si machina accedat. Dupliciter autem insita corporibus gravitas obsistit moventi, si ab alio in alium locum transferenda fuerit: disparibus enim momentis mora infertur motui, si hic fluido in corpore ac se- quaci, puta in aëre aut aquâ, perficiatur, ac si suprà solidam consistentemque planitiem raptetur moles, sive Horizonti pa- rallela jaceat planities, sive molli aut arduâ inclinatione eriga- tur in clivum. Et quidem si solidum in corpus non incumbat onus, sed in aëre suspensum pendeat, ac sursum trahere opor- teat, certos ad calculos revocari gravitatis momenta poterunt, quibus machina proportione respondeat: nam quamvis aër aëri præstet tenuitate, non ea tamen est in levitatibus differentia, ut hinc in gravium corporum momentis dissimilitudo notabilis oriatur. Quare sicut laberetur turpiter, qui machinam saxo ab imo mari ad summam superficiem elevando parem instrueret, si nullâ factâ virium accessione illud in aërem extrahi posse sibi persuade

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CHAPTER I. For what purpose machines are constructed. The end, to which each action is ultimately directed, is first conceived in the mind and set beforehand, and it regulates the suitable means of action that are to be chosen. Hence this first matter occurs to us in this discussion: namely, for what end machines are established and constructed, that the other causes may be directed and formed according to this as a kind of rule. Perhaps someone might say, in a lofty way, that machines were invented by us with the purpose of conquering nature by art; for, as Antipho the poet writes in Aristotle, in the Mechanical Questions, at the beginning, πεχην πρατοῦμεν, ὑν φύσει νικώμεθα. But this is plainly a place for philosophizing, not for boasting arrogantly. Therefore it must be openly admitted that machines are employed as an aid to weakness; so that what weakness of strength could not move from its place, or could do only with great difficulty, or not at all, may at last easily drag, or push, or even expel, however much it resists, if a machine be brought to bear. Now gravity inherent in bodies hinders the mover in two ways, if it must be transferred from one place to another: for unequal moments cause delay in motion, whether this is accomplished in a fluid and yielding body, such as in air or water, or whether a mass is carried over a solid and firm plane, whether that plane lies parallel to the horizon, or is raised into a slope with a gentle or steep inclination. And indeed, if the load does not press upon a solid body, but hangs suspended in the air, and must be drawn upward, the moments of gravity can be reduced to definite calculations, to which the machine may correspond by proportion: for although air yields to air in thinness, yet the difference in lightness is not such that from this a notable dissimilarity should arise in the moments of heavy bodies. Therefore, just as he would behave foolishly who should devise a machine for lifting a stone from the bottom of the sea to the surface, if he persuaded himself that it could be drawn through the air without any addition of force had been made,

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Liber secundus. CAPUT I. 127 persuaderet; ita nimis exiguè & exiliter ad calculos revocaret aërem, qui pro dispari ejus levitate modum machinæ statueret; in materiâ etenim, ex quâ machina componitur, nullus est huic minutæ subtilitati locus, quæ aciem omnem fugit, nisi cum veritas in disputatione limatur. Id quod de eâ pariter gravitationis inæqualitate dictum velim, quæ ex inæquali à cen- tro gravium distantiâ ortum habet, ut lib.1.cap.4. disputatum est: Quia in tantulo Spatio, in quo nos labor noster exercet, illa momentorum exuperantia sub sensum non cadit. Quo cir- ca satis superque habemus, quòd moventis vires ac molis mo- vendæ pondus reputantes ita inter se conferamus, ut virium imbecillitas adhibitâ machinâ convalescat, & repugnanti one- ris gravitati non resistat modò, sed & præstare possit, nullâ aut loci aut aëris habitâ ratione. Verùm quàm facile est corporis gravitatem cùm ex mate- riæ specie, tùm ex molis magnitudine investigare; tàm mul- tis difficultatibus impedita res est, si examinandum sit, quantùm ex mutuo corporum se contingentium tritu retardetur motus: non enim quisquis pendulum in aëre majoris campanæ malleum potest à perpendiculo dimovere, earum est virium, ut illum pariter in terrâ jacentem propellere valeat: & decennis puer arrepto fune illigatam cymbam, modicè fluctuante salo, ad se trahit; quam vix, aut ne vix quidem, robustioris lacerti vir dimoveat, ubi arenoso vado insederit: cum tamen eadem aut ligneæ cymbæ aut ferreo malleo gravitas innata permaneat. Est autem tùm subjecti corporis consistentis, tùm impositi one- ris movendi superficies spectanda, quatenus se contingunt: Nam si lapideum globum pondo 100 in planitie constitutum non rotare modo, sed & rectâ urgere possis, non itidem cubum pondere parem & materiâ similem æquali facilitate urgebis; quia scilicet globus tenuissimâ sui parte suppositam planitiem contingens minus invenit impedimenti ex proximè subjecti corporis asperitate, quæ prominulas impositi globi particulas re- moretur; at cubus longè pluribus sui partibus plano adhæret, at- que adeò multiplicatâ partium hujus in illius partes incurren- tium resistentiâ, augeri quoque movendi difficultate necesse est. Quoniam verò obtineri nequit, ut corporum se contingen- tium superficies sint continuo lævore lubricæ, earum autem asperitates

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Book the Second. CHAPTER I. 127 would persuade; thus it would too narrowly and too feebly reduce to reckoning the air, which, by its differing lightness, would determine the measure of the machine; for in the material of which the machine is composed, there is no place for this minute subtlety, which escapes every eye, unless truth is sharpened in discussion. The same I would say of that unequal gravitation, which arises from an unequal distance from the center of gravity, as was discussed in book 1, chapter 4: because in so small a space, in which our labor occupies us, that excess of moments does not fall under the senses. Wherefore we have sufficiently and more than sufficiently, in weighing the forces of the mover and the weight of the thing to be moved, compared them among themselves in such a way that the weakness of the forces, with the machine applied, may recover strength, and may not only resist the gravity of the opposing burden, but even be able to overcome it, without regard either to place or to air. But how easy it is to investigate the heaviness of a body, both from the species of the matter and from the magnitude of the mass, so many difficulties hinder the matter, if one must examine how much the motion is retarded by the mutual friction of bodies touching one another: for not every man who can move from the plumb line a pendulum, or a hammer of a large bell, while it hangs in the air, has forces such that he can likewise propel it when it lies on the ground; and a ten-year-old boy, seizing a rope tied to a boat, when the sea is moderately agitated, draws it to himself; whereas scarcely, or not even scarcely, would a man of stronger arm move it, once it had settled on a sandy shoal, although the same natural heaviness remains in the wooden boat or the iron hammer. But the surfaces both of the body lying below and of the burden placed above are to be considered, insofar as they touch one another: for if you can not only roll but also push in a straight line a stone sphere weighing 100 pounds, placed on a level surface, you will not likewise push with equal ease a cube of the same weight and similar material; because, namely, the sphere, touching the supporting plane with its most slender part, finds less impediment from the roughness of the body lying nearest beneath it, which holds back the projecting particles of the sphere resting upon it; but the cube adheres to the plane in far more of its parts, and thus, by the multiplied resistance of the parts of this body striking upon the parts of that one, it is necessary that the difficulty of moving also be increased. But since it cannot be obtained that the surfaces of bodies touching one another should be continuously smooth and slippery, and their roughnesses

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128 Mechanicorum asperitates anomalæ sunt ac multiformes, resistentia indè pro- veniens sub certam legem non cadit; sed quantum conjecturâ assequi valemus, illa potius ex antiquis experimentis æstimanda videtur, quàm mathematicis ratiocinationibus indaganda. In hoc uno nimirùm facem præferre potest Geometria, ut si reli- qua prorsus paria sint, nec alia sit quàm molis aut figuræ dissi- militudo, quantum ex hoc capite movendi difficultas augea- tur, minuaturve, innotescat: cæterùm plenè atque perfectè explicare, quantum resistentiæ ex asperarum superficierum conflictione oriatur, quis nisi temerè conetur? Posteriori huic malo, quod superficierum aliqua asperitas creat, occurritur, si pingui sequacique materiâ oblitæ lubri- cæ fiant: Sic Automatis, rotarum se se mutuâ collabellatione mordentium conversione, horas indicantibus velocitas conci- liatur, si quis denticulos oleo leviter perungat: sic plaustrorum tarditatem, equorumque laborem, ut imminuant aurigæ, axes rotarumque modiolos axungiâ illinunt; & cæmentarij majora saxa attollentes, trochleæ orbiculis sapone perfricatis, quærunt laboris compendium. Hinc Amstelodami passim observatur lubricas fieri trahas ceruisiæ doliis, similîve pondere, onustas; cum enim equus non procul abest à ponte, in quem ascenden- dum est, is, qui equum agit, centonem unguine delibutum currenti trahæ substernit, ut expressus ex centone pinguis hu- mor inficiat duo illa longiora tigna, quibus traha insistit, ac proinde lubrica machina faciliùs raptetur per vias lateribus stratas. Sic Dio lib.50. de Augusto loquens. Audivi eum trie- mes ex mari exteriore per murum in sinum translulisse, & loco Pa- langum, per quos ducerentur, tergoribus animalium recens cæsorum oleo inunctis usum, Et Silius Ital. lib.13.v.444. Lubrica roboreis aderant substramina plaustris, Atque recens cæsi tergo prolapsa juvenci, Æquoream rota ducebat per gramina puppim. Verùm nec frequens esse potest, nec commodum, remedium hoc ex pingui liquore petitum; illud certius erit ad imminuen- dam moram ex tritu corporum ortam, quod ea se invicem quàm minimùm contingant. Quoniam verò deducendi one- ris superficiem amplam mutare sæpè nequimus, aut illud rap- tandum trahæ imponimus, quæ non nisi tigillis duobus læviga- tis

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128 The asperities of Mechanicorum are anomalous and multiform; the resistance arising from them does not fall under any fixed law. Yet, so far as we can reach by conjecture, it seems rather to be estimated from ancient experiments than investigated by mathematical reasoning. In this one respect, indeed, Geometry can hold up a torch: namely, if all else is exactly equal, and there is no difference except in bulk or figure, it can make known how much the difficulty of moving is increased or diminished by this cause. But to explain completely and perfectly how much resistance arises from the friction of rough surfaces—who would attempt it except rashly? This latter evil, which some roughness of surfaces creates, is met when they are made smooth by being coated with a greasy and adhesive material. Thus in Automata, in the turning of wheels that bite one another by mutual wear, velocity is gained for the clocks that indicate the hours if someone lightly oils the teeth. Thus, to lessen the slowness of carts and the labor of horses, drivers smear the axles and the wheel-hubs with axle-grease; and masons lifting large stones seek to reduce labor by rubbing the pulleys’ wheels with soap. Hence at Amsterdam it is commonly observed that sledges carrying beer barrels, or loads of similar weight, are made slippery; for when the horse is not far from the bridge up which it must ascend, the driver lays under the running sledge a blanket smeared with grease, so that the oily moisture pressed out of the blanket may infect those two longer beams on which the sledge rests, and thus the slippery machine may be more easily dragged along the streets paved with cobbles. Thus Dio, speaking of Augustus, book 50: “I heard that he had conveyed triremes from the outer sea through the wall into the harbor, and had used for the places by which they were to be drawn, skins of freshly slain animals smeared with oil.” And Silius Italicus, book 13, v. 444: “Slippery supports were there beneath the oak-built carts, And the wheel, slipping from the back of a freshly slain young bull, Was carrying the sea-like ship across the grass.” But this remedy drawn from greasy liquid can neither be frequent nor convenient. More certain, for diminishing the delay arising from the rubbing of bodies, will be that by which they touch one another as little as possible. Yet since we often cannot change the broad surface for conveying a load, or lay on that which is to be dragged a sledge which has not only two smoothed beams

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Liber secundus. CAPUT I. 129 tis subjectam planitiem tangit; aut in plaustrum injicimus, cu- jus rotæ solum calcantes dum convertuntur, axem tantum- modo terunt, compendio sanè mirabili; nam dum rotæ modio- lus axem semel terit, pedes circiter viginti provehitur onus, aut demum sublato corporum mutuo tritu cylindros, vel scy talas illi subjicimus, ut nihil noceat soli asperitas, nisi quatenus hæc cylindrorum vel scy talorum conversionem remoratur. Huc spectat id, quod non sine voluptate observare aliquan- do contigit Bononiæ. Tres erant viri nec admodum robusti, qui ut aliquot ingentes saccos farinâ plenos in domum infer- rent, paratum habuerunt axem binis rotulis circiter sesquipal- maribus instructum; axi jungebatur crassiusculus temo sacco- rum longitudinem vix superans. Erecto sacco machinulam ap- plicabant, tùm saccum pariter cum temone reclinabant, & ne temoni incumbens juxtâ longitudinem saccus in alterutram partem inclinaretur, duo hinc & hinc retinebant pariter, ac propellebant, ut tertium arrepto temone trahentem labore le- varent: Hâc ratione alium atque alium saccum tenuissimo la- bore in domum importarunt; erectoque iterum temone delap- sus est ex machinulâ saccus, stetitque erectus. Ex his itaque constat in machinâ instruendâ non solùm in- genitæ corpori movendo gravitatis rationem habendam esse; sed & plani, super quo illud deducendum est, jacens-ne sit? an erectum? læve, an asperum? amplâ, an tenui superficie contingat? hinc si quidem varia resistentiæ momenta exur- gunt. Illud tamen plerumque contingit, quod si attollendo ad perpendiculum oneri par fuerit machina, illa pariter sufficiat ad onus idem super plano horizontali, aut inclinato deducen- dum: vix enim fieri potest (nisi summa sit superficierum se contingentium asperitas) ut quantum resistentiæ demitur à plano sustinente, tantumdem addatur ex mutuo prominentium particularum conflictu. Quamquam & ipsa asperitas facit aliquod laboris compen- dium: nam licèt continens ac perpetuus non sit motus, sed al- ternâ quiete interruptus super arduo clivo, modico tamen co- natu prohibetur moles, ne prolapsa sispheum creet laborem; quia aspera superficies motui obsistens efficit ne corporis gravi- tas deorsum conetur pro plani inclinatione. Satis igitur fuerit R

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Book Two. CHAPTER I. 129 touches the level surface beneath it; or we place it on a cart, whose wheels, while they are turned by the soles that tread upon them, wear only the axle, with truly marvelous saving; for while one hub of the wheel wears the axle once, the load is carried forward about twenty feet; or finally, the mutual rubbing of bodies being removed, we apply cylinders or rollers beneath it, so that the roughness of the ground harms nothing except insofar as it hinders the turning of the cylinders or rollers. To this belongs what I once had occasion to observe in Bologna, not without pleasure. There were three men, not very strong, who, in order to carry several large sacks filled with flour into a house, had prepared an axle fitted with two rollers of about a palm and a half in diameter; to the axle was attached a somewhat thick pole, scarcely exceeding the length of the sacks. With the sack raised, they applied the little machine; then they laid the sack down together with the pole, and lest the sack, resting on the pole and lying lengthwise, should incline to one side or the other, two men on this side and that both held it in place and pushed it, so that they might relieve the third, who, seizing the pole, was doing the pulling. In this way they carried one sack after another into the house with very slight labor; and, the pole being raised again, the sack slipped down from the little machine and remained upright. From these things it is clear, therefore, that in constructing a machine one must consider not only the weight of the body to be moved, but also the surface over which it is to be conveyed: is it lying down or upright, smooth or rough, touching with a broad or a narrow surface? For from this various moments of resistance arise. Yet this generally happens: that if a machine is equal to raising a load vertically, it will also suffice to convey the same load over a horizontal or inclined plane; for it is scarcely possible, unless the roughness of the contacting surfaces be extreme, that as much resistance is taken away by the supporting plane as is added by the collision of the protruding particles. And yet roughness itself makes some compensation for labor: for although the motion is not continuous and uninterrupted, but interrupted by alternate rest on a steep slope, the mass is nevertheless kept back by moderate effort, lest, having slipped, it create a sispheum of labor; because the rough surface opposing the motion prevents the weight of the body from tending downward according to the inclination of the plane. It will therefore be sufficient R

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Mechanicorum 130 absolutæ oneris gravitati machinam ita respondere, ut illi ad perpendiculum sustollendo cæteroqui impares vires sufficiant: qui enim valuerit, adhibitâ machinâ, molem attollere, poterit illam pariter, ejusdem machinæ ope, in plano quocunque tra- here aut propellere; si maximè cylindri aut rotæ ei subji- ciantur. Hîc autem fortè nec à præsenti instituto alienum, nec lectori injucundum accidat, si quæ, aliquando comminisci placuit, subjiciam, cum narrantem quendam audirem de campanâ in- gentis ponderis facillimè agitatâ subjectis æneis rotulis, quæ demum longo ævo confectæ dissipatæ fuere; sed quonam artifi- cio, quóve ordine dispositæ fuissent, ennarrare omninò non poterat. Quare mecum ipse reputans, quî fieri id potuisset, in eam incidi sententiam, ut existimarem gravissimam campanam potuisse facilè pulsari, imminutâ resistentiâ, quæ oritur ex mu- tuo fulcri, & axis tritu. Sint enim binæ rotulæ B & C ex ære solido, quarum diameter sit in aliquâ Ratione multiplici ad diametrum axis, cui cam- pana innititur. Axis autem se- midiameter sit A E, rotulæ ve- rò B E in ratione duplâ; ergo & periphæriæ sunt in eâdem Ratione: dum igitur punctum I in H perficit quadrantem, convertit pariter rotulam; cujus pe- ripheriæ semiquadranti coæquatur. Quare si rotula infixa esset axi, cujus semidiameter B G esset æqualis semidiametro A E, fieret affrictus cum octante peripheriæ axis rotulæ B; sed quia etiam in rotulâ C fieret æqualis affrictus cum ejusdem axe, jam nihil ferè emolumenti haberetur, quia totus affrictus æquè es- set, ac si quadrans E O in fulcro stabili & cavo converteretur: & potiùs laboris in agitandâ campanâ compendium esset, si ro- tulæ fixæ hærerent, axis si quidem cylindricus cum sit, subjectas rotulas in lineâ tangeret modico scilicet tritu; rotularum autem axes concavis earum partibus congruunt in superficie, quæ te- ritur, dum rotulæ convertuntur: nisi fortè cylindrica axis B G superficies convexa paulò minor esset concavâ rotulæ superficie, exque propterea secundùm lineam se continge- rent,

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Mechanics 130 can so adapt a machine to the absolute heaviness of a load that forces otherwise inadequate for lifting it vertically will suffice: for whoever has been able, with the aid of a machine, to raise a mass, will be able equally, by means of the same machine, to draw or propel it on any level surface, especially if cylinders or wheels are placed beneath it. Here, too, perhaps it would be neither foreign to the present purpose nor unpleasant to the reader if I should add what once I was pleased to devise, when I heard someone relate how a bell of enormous weight had been moved with the greatest ease by bronze rollers placed underneath, which at length, worn out by long use, were broken up and destroyed; yet he could not by any means relate by what contrivance, or in what arrangement, they had been set. Wherefore, reflecting with myself how this could have been done, I fell upon the opinion that a very heavy bell could easily be rung, with the resistance lessened that arises from the mutual friction of the support and the axle. Let there be, then, two rollers B and C of solid bronze, whose diameter is in some multiple ratio to the diameter of the axle on which the bell rests. But let the semidiameter of the axle be A E, and that of the roller B E in a double ratio; therefore the circumferences also are in the same ratio: thus while point I completes a quadrant in H, it likewise turns the roller, whose semi-quadrant of circumference corresponds to it. Therefore, if the roller were fixed to the axle, whose semidiameter B G were equal to the semidiameter A E, there would be friction with an eighth part of the circumference of roller B’s axle; but because an equal friction would also arise in roller C with its axle, there would now be scarcely any advantage, since the whole friction would be just as great as if quadrant E O were turned in a fixed and hollow support: and rather there would be a saving of labor in moving the bell if the rollers adhered fixedly, since the cylindrical axle, as it is, would touch the rollers underneath only along a line with slight friction, whereas the axes of the rollers fit their hollow parts on the surface which is rubbed while the rollers are turning: unless perhaps the cylindrical surface B G of the axle were a little smaller than the concave surface of the roller, and therefore they would touch one another along a line,

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Liber secundus. CAPUT I. 131 rent, ut ex 13. lib. 3. facilè est demonstrare; id quod nec rarò contingit. Verùm non est necesse rotulis B & C tàm solidos axes dare; nam si axis A E toti campanæ oneri ferendo par est, bini æquales axes duplici ponderi resistunt: satis igitur esset, si axes singuli B & C, oneris semissem sustinerent. Cum verò cylindrorum resistentiæ, ne frangantur, sint in triplicatâ Ratione suarum diametrorum, sufficeret inter semidiametrum A E, & ejus semissem duas medias proportione continuâ reperire, quæ enim proximè minor esset ipsâ A E, esset sufficiens semidiameter cylindri subduplam habentis soliditatem ac resistentiam. Sed adhuc minor requiritur semidiameter, quia onus axes rotularum B & C obliquè premit; ex quo fit campanæ gravitationem in axes illos esse secundùm lineas A B, A C, non autem juxtà perpendiculum A D: igitur ut A D ad A B, ita reciprocè gravitatio super A B ad gravitationem super A D: atqui gravitatio in alterutrum axium, ut summum subdupla est totius gravitationis; ergo gravitatio super B A minor est subduplâ. Quâ autem Ratione minor sit constat. Cum enim detur tùm semidiameter A E, tùm etiam B E, nota est tota B A, & B D, pariter, ipsi B E æqualis, nota est; igitur ex 47 lib. 1. etiam A D innotescit, cujus scilicet quadratum habetur, si ex B A quadratodematur quadratum B D. Cum itaque, ex hypothesi, B A sit 3, cujus quadratum 9, & B D 2, cujus quadratum 4, remanet quadratum 5, ejusque Radix 2. 23". est recta D A: gravitatio igitur super B A ad totam campanæ super utrumque axem B, & C, gravitationem est 223 ad 600". Quoniam verò solidorum similium resistentia est in triplicatâ Ratione laterum homologorum (in cylindris autem diametrorum ratio habetur) quærantur duo medij proportionales numeri inter 600" & 223". Id quod assequeris, si cujuslibet extremi quadratum ducas in alium extremum, producti enim Radix cubica est terminus proximus illi numero, cujus quadratum assumpsisti. Primi igitur 600 quadratum 3600000 duc in 223, & producti 80280000, Radix cubica est 431 1/5 proximè: alterius verò extremi 223 quadratum 49729 ductum in 600 dat 29837400, cujus Radix cubica 310 proximè est alter medius. Sunt igitur quatuor numeri 600.431 1/5. R 2

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Book Two. CHAPTER I. 131 it is easy to demonstrate from Book 3, proposition 13; and this is not rarely the case. But it is not necessary to give the rollers B and C such solid axles; for if the axle A E is equal to bearing the whole weight of the bell, then two equal axles resist a double weight: it would therefore be enough if the individual axles B and C supported half the load. But since the resistance of cylinders, lest they break, is in the triplicate ratio of their diameters, it would suffice to find between the semidiameter A E and its half two mean proportionals in continued proportion; for that which would be nearest and less than A E itself would be a sufficient semidiameter for a cylinder having half the solidity and resistance. But an even smaller semidiameter is required, because the load presses the axles of the rollers B and C obliquely; from which it follows that the weight of the bell upon those axles is according to the lines A B, A C, and not by way of the perpendicular A D: therefore, as A D is to A B, so inversely is the gravitation upon A B to the gravitation upon A D. Yet the gravitation upon either axle, at most, is half of the total gravitation; therefore the gravitation upon B A is less than half. By what ratio it is less is evident. For since both the semidiameter A E and also B E are given, the whole B A is known, and likewise B D, equal to B E, is known; therefore from Book 1, prop. 47, A D is also made known, namely its square is obtained by subtracting from the square of B A the square of B D. Since therefore, by hypothesis, B A is 3, whose square is 9, and B D is 2, whose square is 4, there remains a square of 5, and its root, 2. 23"., is the straight line D A: therefore the gravitation upon B A to the total gravitation of the bell upon both axles B and C is 223 to 600". But since the resistance of similar solids is in the triplicate ratio of the homologous sides (in cylinders, however, the ratio of the diameters is taken), let two mean proportional numbers be sought between 600" and 223". This you will achieve by multiplying the square of either extreme by the other extreme; for the cube root of the product is the term nearest to that number whose square you have assumed. Therefore, take the square of the first extreme, 600, namely 3,600,000, and multiply by 223; the cube root of the product, 80,280,000, is approximately 431 1/5: but the square of the other extreme, 223, namely 49,729, multiplied by 600, gives 29,837,400, whose cube root is approximately 310, the other mean. Therefore there are four numbers: 600, 431 1/5. R 2

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Mechanicorum 132 310.223 continuè proportionales proximè, spretis fractiunculis. Quare si fiat ut 600 ad 431, ita semidiameter AE ad BN, erit hæc semidiameter quæsita sufficienter resistens. Quoniam itaque BE dupla est ipsius AE, & AE ad BN facta est ut 600 ad 431, erit BE ad BN ut 1200 ad 431; & secundum hanc eandem Rationem se habebunt semiquadrantes ab illis descripti. Atqui octans peripheriæ ex Radio BE æqualis est quadranti ex Radio. AE; igitur quadrans EO ad semi- quadrantem ex Radio BN est pariter ut 1200 ad 431: Qui igitur affrictus axis campanæ cum fulcro stabili & cavo esset 1200, rotulæ B cum suo axe est 431, cui æqualis est alterius rotulæ C affrictus cum suo axe; ac proinde subjectis rotulis, quarum diameter sit tantum dupla diametri axis campanæ, affrictus est ut 862, ad affrictum qui esset ut 1200. Si itaque rotularum diameter ad campanæ axem non tantùm dupla, sed vel tripla, vel quadrupla sit, multò minor erit affrictus, majorque in agitandâ campanâ facilitas. Quamvis autem istâ consimilivè diligentiâ industriâque plurimum imminui possit particularum conflictus, quæ se vicissim terentes moram atque impedimentum motui inferrent; non illa tamen ex eo propriè veréque dicitur motio machinalis, quòd instrumento atque apparatu aliquo perficiatur, nisi, spectatâ dumtaxat oneris gravitate, potentia illi movendo cæteroqui impar, subsidium sibi comparet ex machinâ. Machina autem non idem est, si plenè atque perfectè interpretari velis, ac instrumentum; licet enim machina omnis instrumentum sit, non tamen instrumentum quodlibet machinæ vocabulum continuò sortitur, si motionem aliquatenùs juvet; sed illud præterèa efficiat necesse est, quod ejus ope naturalem ac insitam vim corporis loco dimovendi superet vis minor extrinsecùs adhibita. Cum ergò onus hærere in salebrâ, non ex insitâ vi, sed ex proximi etiam atque continentis corporis asperitate proveniat, & instrumenta, quibus hoc tantummodo impedimentum tollitur, idem planè efficiant, quod pinguis humor lubricum parans iter; neque hæc machinæ magis dici possunt, quàm centones unguine delibuti, si ritè substernantur, neque motus propterea inter machinales numerandus videtur, quorum hîc causas vestigare nobis propositum est. Quamquam negandum non sit hæc pari- ter

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Mechanics 132 310.223 proportionals, as nearly as possible, neglecting fractions. Therefore if it be so that 600 is to 431, as the semidiameter AE is to BN, this semidiameter sought will be sufficiently resistant. Since therefore BE is double of AE, and AE to BN is made as 600 to 431, BE will be to BN as 1200 to 431; and according to this same Ratio the semi-quadrants described by them will be related. But indeed the eighth part of the circumference from radius BE is equal to a quadrant from radius AE; therefore quadrant EO to the semi- quadrant from radius BN is likewise as 1200 to 431: Wherefore, if the friction of the bell’s axis with the stable and hollow fulcrum were 1200, the wheel B with its axis is 431, to which equal is the friction of the other wheel C with its axis; and consequently, with the wheels placed beneath, whose diameter is only double the diameter of the bell’s axis, the friction is as 862, to the friction which would be as 1200. If therefore the wheels’ diameter to the bell’s axis be not only double, but even triple, or quadruple, the friction will be much less, and the ease in moving the bell greater. Although by such similar diligence and industry the conflict of the particles, which by rubbing against one another would bring delay and hindrance to motion, can very greatly be diminished; nevertheless it is not for that reason properly and truly called mechanical motion, because it is accomplished by some instrument and apparatus, unless, the weight of the load alone being considered, a power otherwise unequal to moving it procure for itself assistance from a machine. But a machine is not the same thing, if you would interpret the word fully and perfectly, as an instrument; for although every machine is an instrument, yet not every instrument forthwith obtains the name of machine, if it in any way aids motion; but it must moreover effect this, namely that by its means a lesser force applied from without may overcome the natural and inherent force of a body for moving it from its place. Since therefore a load’s sticking on a rough place arises not from an inherent force, but from the roughness of the neighboring and adjacent body as well, and instruments which merely remove this hindrance plainly accomplish the same thing as a greasy fluid preparing a smooth passage; nor can these more be called machines than patches smeared with grease, if properly laid beneath, nor does motion on that account seem to be numbered among mechanical motions, the causes of which it is here our purpose to investigate. Although it must not be denied that these also are equally

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Liber secundus. CAPUT I. 133 ter ad mechanicam contemplationem pertinere; quippe quæ machinis, præcipuo nimirum mechanices scopo, affinia sunt; etiamsi ad illas non velut subjectæ partes ad genus revocentur: & instrumentis hujusmodi si machinæ appellationem tribuere placuerit, non admodum de nomine disputabo; res enim hîc spectatur, non verba penduntur. Sed neque hîc disputare velim, utrum in motuum machinalium censum irrepant, an verò iis ritè annumerandi sint motus illi, quos sursum deorsum, ultrò citróque perficiendos eatenus expeditè, nec exiguo laboris compendio, molimur, quatenus eos intervallis ita distinguimus, ut nos quidem corpus deprimamus, ut adducamus, ab alio verò extollatur, aut reducatur: in his siquidem sæpè nihil est, quod nostram imminuat operam, si motiones singulæ attendantur; quamquam motui universo adjumentum importat continens illa conatûs nostri, alienique subsidij, vicissitudo. Hinc si quis ad contundendam in æneo mortario A contumacem aliquam materiam graviore pistillo ferreo opus habeat, haud dubium quin ei multâ lacertorum vi contendendum sit, ut illum extollat; cumque operosius multo sit inflexum corpus erigere, quàm erectum inclinare, multóque molestius brachia tanto pondere prægravata attollere, quàm eorum gravitati obsecundando deprimere, satis constat, quantum si bi laboris detractum eat, si superiore in loco transversum tigillum CD circa axem E versatilem statuat, paribusque intervallis hinc ex C pendeat funé suspensus pistillus B, hinc verò in D plumbea massa adnectatur, quâ ita pistillus præponderetur, ut, nemine hunc retinente aut deprimente, illa aliquanto gravior in subjectum prodeuntis è pariete tigni caput G recidens sponte subsidat. Omnis scilicet extollendi pistilli labore sublato, vel solum brachiorum pondus pistillo additum satis esse aliquando poterit ad leviusculè tundendam materiam, licebitque R 3

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Book Second. CHAPTER I. 133 to pertain to mechanical contemplation; since they are akin to machines, the principal object of mechanics, although they are not referred to them as subordinate parts of the genus: and if it should please anyone to give instruments of this kind the name of machines, I shall not greatly dispute about the name; for the thing is here considered, not the words weighed. But neither would I here dispute whether those motions creep into the number of mechanical motions, or whether they are rightly to be reckoned among them, by which we contrive, conveniently and with no small saving of labor, to perform upward and downward, to and fro, operations by so distinguishing them by intervals that we ourselves indeed depress the body, to bring it forward, while it is raised or brought back by another: for in such cases there is often nothing that diminishes our labor, if the individual motions be attended to; although to the total motion the continuous alternation of our effort and of another's aid brings assistance. Hence if anyone should have need, for pounding some stubborn material in a bronze mortar, of a heavier iron pestle, there is no doubt that he must strive with great force of his arms to lift it; and since it is much more laborious to raise a bent body than to incline one that is erect, and much more troublesome to lift arms weighed down by such a burden than, by yielding to their weight, to depress them, it is well enough known how much labor would be taken from him if he were to place a transverse beam CD in the upper position, movable about the axis E, and if at equal intervals there were suspended from C a pestle B hanging by a rope, but from D a leaden mass were attached, by which the pestle would thus be counterpoised, so that, with no one holding or pressing it down, the latter, somewhat heavier, falling upon the head G of the beam projecting from the wall below, would of its own accord sink down. Thus all the labor of lifting the pestle being removed, even the mere weight of the arms added to the pestle will at times be enough to pound the material rather lightly, and it will be possible R 3

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Mechanicorum 134 modò contento, modò remisso conatu opus urgere. Id quod pariter continget, si operâ unâ opus duplex efficere placuerit; nam si ex D plumbeæ massæ loco alius pendeat æque, ac plum- bum, gravis pistillus, pondere præpollens elevabit pistillum B, aliâmque vicissim in altero subjecto mortario conteret mate- riam sponte suâ cadens: cumque pistillorum gravitates non ad- modum inter se dispare sint, neque multum laboris eum subi- re necesse erit, cui pistillum B deprimendi munus incumbit. Quâ in re, si motus universus ita tribuatur in partes, ut tun- dentis quidem motiones singulæ seorsim spectentur, non ille profectò se juvari sentit, quippe quem, præter vires ad commi- nuendam materiam necessarias, conatum quoque adhibere oportet ad vincendam præponderantis plumbi, aut pistilli gra- vitatem. Cæterùm si totius motûs, qui Ars pariter constat ac Thesi, habeatur ratio, inficiari nemo poterit, minus multo la- boris impendi, quàm si hæc omnia sublata intelligentur. Qua- re nec incongruum prorsus videatur motûs machinalis voca- bulum, cum versatilis tigillus CD ad libræ Rationes manifestò revocetur, quam certè ex machinarum albo nemo expungit, ni- si qui solas quinque facultates, & quæ ex his componuntur, ma- chinas indigitare voluerit, & libram ad vectem referri posse pernegarit. Nec dissimilis ineunda videtur dicendi ratio, si quid alternis ciendum motibus sic disponitur, ut, cum primùm quidem mo- vetur, corpus aliud vi flectatur, quod postmodum facultate elasticâ, se restituens illud vicissim moveat; quemadmodum passim in eorum officinis videre est, qui rudes arborum, aut elephantini dentis particulas in toreumata elaborant: primùm enim artifex pede subjectum vectem premens, toreuma in gy- rum ducit, hastulâmque superiore in loco positam pariter in- flectit; quæ sibi mox suam reparans rectitudinem, funiculum- que cylindrulo versatili circumplicatum retrahens, illud iterum sua per vestigia versat, ut accuratè exquisitéque tornetur. Sic aliquid subtiliter ac delicatè secturus, ut serrulam rectâ addu- cas, reducásque, operæ tantùm semissem tibi reservans, arcum intentum ex adverso statuito, ac medio nervo serrulam alliga- to; hac enim adductâ magis flectetur arcus, qui se se mox resti- tuens illam vicissim reducet. Hæc

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Mechanicorum 134 to urge the work now with a tighter, now with a looser effort. The same thing will happen if, by the same labor, it has pleased one to accomplish a double task; for if from D, in place of the leaden mass, there hangs another heavy pestle, equal in weight to the lead, its greater weight will raise pestle B, and in turn will grind the material in the other mortar beneath, falling by its own force: and since the weights of the pestles are not greatly unequal among themselves, he who has the task of depressing pestle B will not need to expend much labor. In this matter, if the whole motion is thus divided into parts, so that the individual motions of the striker are considered separately, he certainly does not feel himself to be aided, since he must, in addition to the force needed to crush the material, also apply effort to overcome the preponderating weight of the lead, or of the pestle. But if account is taken of the whole motion, which consists alike in Art and in Thesis, no one can deny that much less labor is expended than if all these things were understood as removed. Wherefore the term mechanical motion would not seem wholly inappropriate, since the movable crosspiece CD is manifestly referred to the ratios of the balance, which certainly no one strikes from the roll of machines, unless he wishes to designate only the five faculties, and the machines composed of them, and denies that the balance can be referred to a lever. Nor does a dissimilar manner of speaking seem to be necessary if something is arranged by alternate motions so that, as soon as it is moved, another body is bent by force, which afterward, by an elastic faculty restoring itself, in turn moves that other; as is everywhere to be seen in the workshops of those who fashion rough pieces of wood, or bits of elephant ivory, into carvings: for first the craftsman, pressing with his foot on the lever beneath, turns the carving in a circle, and bends the small rod placed above; this, soon restoring its straightness and drawing back the cord wound around the rotating cylinder, turns it again along its own track, so that it may be turned with accuracy and finesse. Thus, if you are to cut something finely and delicately, so that you draw the saw forward and back, reserving for yourself only half the labor, place an already bent bow opposite, and fasten the saw to the middle string; for when the saw is drawn, the bow will bend more, and, soon restoring itself, will draw it back in turn. These

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Liber secundus. CAPUT I. 135 Hæc sanè laboris in movendo compendia ex elasmate, vel ex antisacomate petita, quemadmodum & ea, quæ mutuum cor- porum tritum atque conflictum minuunt, ut pote Mechanico artificio constituta, eumdemque in finem ac machinæ, quibus hoc nomen præcipuè tribuitur, videlicet in infirmæ potentiæ subsidium excogitata, esto illis primas deferant, non tamen omninò rejicerem, si in machinarum censu prodirent, iisque se peterent adscribi. Triplicem enim in speciem tribui posse vi- detur universum machinarum genus: Prima eas complectitur facultates, quarum ope motui facilitas conciliatur, quocum- que tandem ex capite sivè tantummodo ex insitâ in corporibus gravitate, sivè non ex eâ dumtaxat, sed ex partium asperitate movendi difficultas consurgat. Altera est, quæ mutuam qui- dem corporum se contingentium conflictionem minuit, sed ad vincendam oneris gravitatem ipsi potentiæ momenta non addit. Tertia demùm eatenus per se, quia talis est, moventem juvat, quatenus ejus operam alternam efficit, cum tamen neque gra- vitatem vincat, neque quod ex partium tritu impedimentum oritur, extenuet, nisi cum alterutra, aut utraque superiori spe- cie, amico foedere copuletur. Alternam autem operam appel- lo, cum in motu ex duplici motione composito alterutram effi- cit potentia, sivè illæ sibi invicem adversantes succedant, ut Arsis ac Thesis, Adductio atque Reductio, sivè in unam tem- perentur, ut cum premere simul oportet ac agitare: sic plana vitra expolientes in specula, inter ipsa, & lacunar bacillum in- flectunt, qui se restituere tentans vi elasticâ, speculum validè, quantum opus est, admovet atque applicat ad subjectum pla- num, adeò ut ad artificem à pressu immunem nil aliud spectet, quàm speculum urgere, retrahere, contorquere. Verùm ta- metsi de his omnibus in hac tractione passim se offeret dicendi locus, primus tamen disputationis nostræ scopus erit prima illa species, ipsæ nimirum facultates, quarum potissimum momen- ta expendimus, cum motûs machinalis causas inquirimus. CAPUT

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Book Second. CHAPTER I. 135 These certainly are expedients for lessening labor in moving things, taken from the elasmate, or from the antisacoma, as likewise are those devices which reduce the mutual rubbing and collision of bodies, being, as it were, constituted by mechanical contrivance and invented for the same end as machines, to which that name is chiefly assigned, namely, as aids to weak power; although I would give the preference to them, I would not altogether reject them if they were admitted into the class of machines and sought to be enrolled among them. For the whole kind of machines seems able to be divided into three species. The first includes those faculties by whose aid ease of motion is granted, from whatever source the difficulty of moving arises, whether only from the gravity inherent in bodies, or not from that alone, but also from the roughness of the parts. The second is that which indeed lessens the mutual collision of bodies touching one another, but does not add any force to overcome the heaviness of the load. The third, finally, helps the mover of itself, in so far as it is such, because it makes his action alternate, although it neither overcomes gravity nor diminishes the obstruction arising from the friction of the parts, unless it is joined in friendly alliance with one or other, or with both, of the superior kinds. Now I call it alternate action when, in a motion composed of a double movement, one force performs the one part, whether those movements follow one another in opposition, as in Arsis and Thesis, Adduction and Reduction, or are blended into one, as when one must both press and agitate at the same time: thus those polishing flat glasses in a mirror bend a rod between them and the ceiling, which, trying to restore itself by elastic force, strongly presses and applies the mirror, as much as is needed, to the flat surface beneath; so that, for the artificer, who is free from pressure, nothing else is required than to press, draw back, and turn the mirror. But although there will be occasion to speak of all these matters here and there in the course of this treatise, the first aim of our inquiry will nevertheless be that first species, namely the faculties themselves, whose chief powers we examine when we investigate the causes of mechanical motion. CHAPTER

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Mechanicorum CAPUT II. Impetus motum proximè efficientis natura explicatur. Quicquid movetur, qualecumque est, causam habeat mo- ventem necesse est, ut hoc quidem sponte suâ, illud ve- ro alienâ vi ex alio in alium locum migret. Suopte ingenio mo- ventur tùm corpora gravia aut levia, ut si extrà præscriptum sibi à naturâ locum constituta fuerint, suo quæque ordine dis- ponantur; tùm rara aut densa, ut si per vim hæc extenuata fue- rint, illa concreverint, naturæ statum sibi reparent; tùm ani- mantia, quibus cum à naturâ tributum sit, ut se, vitam, cor- pusque tueantur, stimulos admovet appetitus, ut ea declinent, quæ nocitura videantur, omniaque, quæ sint ad vivendum ne- cessaria, acquirant, & parent. Vi extrinsecus impressâ locum mutant, quæcumque in motu non serviunt naturæ, sed alieno reguntur arbitrio; ut iis contingit, quæ raptantur, pelluntur, in gyrum ducuntur, projiciuntur, & hujus generis motibus cientur. Quoniam verò gravium, & levium celeritatem naturâ ur- gente incitari, jaculorum autem, ac missilium, motum usque eò sensim languescerè, ut planè deficiat, observamus; etiamsi moventi naturæ, quæ ex Philosophi decretis substantia est, mo- tûs originem ultimam tribuamus, jure tamen optimo aliquid naturæ ipsi ac motui, interjectum agnoscimus (Impetum no- minamus) cujus intentionem ac remissionem velocitas ac tar- ditas consequatur. Cum enim eadem descendentis lapidis na- tura perseveret, nec illa in suâ potestate sit, aut optione delatâ, ut eligat utrum velit, motum arbitrio suo incitare, aut remit- tere valeat; quî fieri possit, ut descendens velocitatem augeat, nisi ei, quem primùm produxit, alium atque alium momentis singulis impetum adjiciat? Illud certè extrà omnem controver- siam positum videtur, naturam gravem sponte suâ non ascen- dere:

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Mechanics CHAPTER II. The nature of the impetus of the proximate efficient motion is explained. Whatever is moved, of whatever kind it may be, must necessarily have a moving cause, so that the one may indeed pass of itself, and the other by foreign force, from one place to another. Things are moved by their own nature, both heavy bodies and light ones, as if, when placed outside the position prescribed to them by nature, each were arranged in its proper order; both rare and dense bodies, as if, when the former have been thinned and the latter made to gather together by force, they restore to themselves the state of nature; and living creatures, to which, since it has been granted by nature to preserve themselves, their life, and their body, appetite applies its spur, so that they may avoid what seems likely to harm them, and acquire and prepare everything necessary for living. By force impressed from outside, whatever in motion does not serve nature, but is governed by another's will, changes place, as happens to those things which are dragged, driven, spun in a circle, thrown, and moved by motions of this kind. But since we observe that the speed of heavy and light bodies is increased by the pressure of nature, whereas the motion of darts and missiles gradually grows so faint that it entirely fails, even if we attribute the ultimate origin of motion to the moving nature, which, according to the philosophers' decree, is substance, nevertheless with the best right we recognize something intervening between nature itself and motion (we call it impetus), on whose increase and decrease velocity and slowness depend. For since the same nature of a stone descending persists, and it is not in its power, or a matter of choice, once carried down, to choose whether it wishes to increase motion or diminish it at its own will, how could it happen that a descending body should increase its speed unless to it, to which it first gave rise, it adds at every single moment another and another impetus? Surely it seems to be placed beyond all controversy that heavy nature does not ascend of its own accord:

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Liber secundus. CAPUT II. 137 dere: quid ergo illud est, quod eburneum globulum in sub- jectam rupem delapsum resilire cogit, aut sibi relictum plumbum ex fune suspensum ultrà perpendiculum, naturâ repugnante, sursum provehit, & eò quidem altiùs, quò ex altiore loco globulus aut plumbum decidserunt: nisi quia conceptus naturâ procurante impetus pergit motum efficere, ipsâ etiam naturâ quantum potest, obsistente. Quòd si corpus alienâ vi longiùs emissum moveatur, extrinsecùs impetum imprimi necesse est: quem sanè non concipit, ubi primùm à projiciente sejunctum fuerit; nihil enim prodesset ad longiorem lapidis jactum fundam iterum ac tertiò circumducere, nisi alium atque alium impetum lapis conciperet, quandù funditori adhærens unâ cum ipso movetur. Quæcumque igitur moventur, impetum habent, quo feruntur; cui satis probabili conjecturâ, proxima vis motum efficiendi tribuenda videtur. Id quod in projectis quidem, iisque omnibus, quæ naturâ repugnante moventur, ita manifestum est, ut id pluribus demonstrare non oporteat; nulla siquidem adest insita motûs causa; ab impetu igitur illo extrinsecùs impresso motum effici necesse est. At in cæteris, quibus se movendi principium inest, nemo jure negaverit aut in motu impetum acquiri, aut velocitatis incrementum ex impetus accessione oriri: quî enim fieret, ut excurrentes objectam fossam ampliore saltu transilirent faciliùs, quàm nullo præcedente cursu, si in cursu ipso conceptus impetus non augeretur? Iam verò si secundo temporis momento incitatur magis motus, quàm primo, urgente scilicet etiam impetu, quem corpus priore motu acquisivit; hic utique impetus, quem nunc gignere non potest prior motus, cum perierit, extitit pariter cum priore motu: natura igitur movens priore momento & motum effecit & impetum. Atqui impetum ex eorum saltem genere esse, quæ motum efficiant, constat ex velociore motu posterioribus momentis, naturâ prorsus immutatâ, factoque impetûs incremento: contrà verò motu, quâ motus est, impetum non augeri satis indicant missilia, quorum velocitas, dum moventur, sensum elanguescit. Igitur & priore illo temporis momento non motus impetum; sed impetus motum proximè effecit; impetum autem procreavit innata movendi vis; cui id circo motio tri- S

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Book Second. CHAPTER II. 137 Why then is it that an ivory ball, when it has fallen onto the rock beneath it, is forced to rebound; or that lead, left to itself and suspended from a rope, rises upward beyond the perpendicular, contrary to nature, and that indeed the higher, the greater the height from which the ball or the lead has fallen? It can be only because the impulse, conceived by nature’s provision, continues to produce motion, even while nature itself resists as much as it can. And if a body, sent farther by an external force, is moved, it is necessary that an impulse be impressed from outside; and surely that impulse is not conceived once it has been separated from the thrower; for it would be of no use to swing a sling around a second and a third time for a longer cast of the stone, unless the stone conceived one impulse after another, since, adhering to the slinger, it is moved together with him. Therefore whatever things are moved have an impulse by which they are carried onward; and to this impulse, by a sufficiently probable conjecture, the nearest cause of motion seems to be attributed. This is indeed so manifest in projectiles, and in all things that are moved contrary to nature, that there is no need to demonstrate it at greater length; for no innate cause of motion is present there, and therefore motion must necessarily be produced by that impulse impressed from outside. But in the rest, in which the principle of self-motion is inherent, no one would rightly deny either that impulse is acquired in motion, or that an increase of speed arises from the addition of impulse: for how would it come about that runners would more easily leap over an obstacle ditch with a longer bound than with no preceding run at all, if in the run itself the conceived impulse were not increased? Now then, if in the second moment of time motion is made more brisk than in the first, namely also under the force of the impulse which the body acquired by its earlier motion; this impulse, which now the prior motion can no longer generate, having perished, must have existed together with the prior motion as well: therefore the moving nature in the first moment produced both motion and impulse. And yet that impulse is clearly of the kind that produces motion, at least from the swifter motion in the later moments, when nature is altogether changed and there is an increase of impulse; on the other hand, from the motion as motion, it is sufficiently indicated that the impulse is not increased, since missiles, whose speed, while they are moving, grows faint to the senses. Therefore, in that first moment of time also, it was not motion that produced impulse; but impulse produced motion immediately; and innate moving force produced the impulse; for which reason the motion S

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Mechanicorum 138 buitur, quia id illa gignit, quod proximè motus conseqvitur, & ad motum efficiendum natura destinavit. Quid? quòd mo- tui per se, quia ex alio in alium locum continuata migratio est, efficientiam ægrè tribuere possumus: quippe qui, cum in fluxione consistat, ita ut locus loco, seu potius, ut scholæ lo- quuntur, Ubicatio Ubicationi, priori scilicet pereunti succedat posterior æquè fugax, inferioris notæ censendus est quàm im- petus naturâ suâ aliquandiu permanens: labentia enim stanti- bus deteriora esse, cæteris paribus, quis neget? effectum au- tem causâ præstabiliorem esse non posse ipsa originis notio sua- det, ne quid effectus habeat, quod non acceperit, aut aliquid causa dederit, quo ipsa careret. Non igitur impetum motus, sed motum impetus efficit. Porrò cum definitas ad agendum vires unaquæque causa ob- tineat, certa est impetûs mensura, quæ cum innatâ movendi facultate ita adæquatur, ut eo quasi termino circumscripta cen- senda sit potentia movens, nec unquam validiore conatu possit se ipsa urgere; si tamen omnem impetum antecedente motu as- sumptum mente secernas. Et quidem omne animal (quippe cui inest appetitio & declinatio naturalis ejus, quod naturæ ac- commodatum est, aut infensum) non semper universam illam impetûs mensuram exequitur, sed ut vult, ita utitur motu sui corporis, quem aucto aut diminuto impetu modò intendit, mo- dò remittit, pro ut interiore motu, rerumque appetitu simula- tur. Contrà verò inanimum non suo arbitrio motûs intentionem moderatur, sed naturæ juribus obsequens nihil prætermit- tit impetûs, & quantum eniti potest, opportunum in locum, si- bique à naturâ constitutum, contendit. Cave tamen existimes parem esse lapidis ejusdem, & in aëre, & in aquâ descendentis impetum: natura scilicet ex medio dividendo, in quo perficien- dus est motus, metitur impetûs modum. Sed quoniam non pauca sunt, quæ motui sæpè adversantur, hinc est non semper eandem esse corporis se moventis velocita- tem, quamvis pari impetu producto connitatur: deteritur nimi- rum tantum impetus, quantum satis est ad impedimentum sub- movendum. Sivè enim objectum corpus propellendum sit, sivè medij particulæ locum ægrè dantes divellendæ aut compri- mendæ sint, sivè connexam molem pariter rapi oporteat, sivè quid

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Mechanics 138 is bestowed, because it produces that which most nearly follows motion, and nature has destined it for producing motion. What? Since to motion itself, because it is a continuous migration from one place into another, we can scarcely attribute efficacy: for it, consisting in fluxion, so that place succeeds place, or rather, as the schools put it, Ubication to Ubication, the latter, namely, fleeting as the former perishes, ought to be deemed of lower rank than impetus, which by its nature remains for a time; for who would deny that things slipping away are worse than things standing still, ceteris paribus? Moreover, that the effect cannot be superior to the cause the very notion of origin suggests, lest the effect have anything it has not received, or the cause have given something of which it itself was lacking. Therefore it is not the impetus of motion that produces motion, but motion that produces impetus. Furthermore, since each cause possesses definite powers for acting, there is a certain measure of impetus, which is so proportioned to the innate faculty of moving that the moving power is to be regarded as circumscribed by that sort of boundary, nor can it ever urge itself on by a stronger effort; provided, however, that you mentally set aside every impetus taken up by antecedent motion. And indeed every animal—for it has within it a natural inclination and aversion toward what is suited to nature, or hostile to it—does not always exert that whole measure of impetus, but as it wills, so it uses the motion of its body, which now it intensifies, now it relaxes, as the inward motion and appetite for things are imitated. But inanimate things do not regulate the degree of motion by their own choice; rather, obeying the laws of nature, they omit nothing of the impetus, and, as far as they can strive, they press toward the appropriate place assigned to them by nature. Yet beware lest you suppose that the impetus of the same stone descending both in air and in water is equal: nature, namely, by dividing the medium in which the motion is to be accomplished, measures the amount of impetus. But since there are many things that often oppose motion, this is why the speed of a body moving itself is not always the same, even though it strives with an equal impetus applied; for so much impetus is worn down as is enough to remove the obstruction. For whether the body to be driven forward is an obstacle, or the particles of the medium, yielding with difficulty, must be torn apart or compressed, or whether some connected mass must be carried along as well, or whether some

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Liber secundus. CAPUT II. 139 quid aliud hujusmodi adsit, cui nisi vis inferatur, ut ex alio in alium locum migret præter naturam, irritus reddatur corpo- ris in motum propensi conatus; satis constat illud motu agitandum esse exterius: atque adeò quantum impetus illi imprimitur oppositæ propensioni æquale, motui tantumdem sub- trahitur. In iis sanè, quæ alienâ vi extrinsecùs moventur, quia infi- nitè progredi non licet, aliqua demum origo deprehenditur, cui naturalis sit motus: natura siquidem vis est ciens motus in corporibus necessarios; ita tamen certis tenetur legibus uni- versitatis rerum concinnitatem spectantibus, ut ne ab iis discedat, singularibus corporibus vim aliquam inferri permittat, ubi adversis propensionibus inter se confligentibus validior præstat imbecilliori. Sic quia nefas est aut corpora inanitatibus interjectis concisa hiare, aut unum in proximi corporis locum, nisi eo recedente, penetrare, aut diverticula flexionesque in motu sponte quærere; ideò & liquor in longiore siphonis, aut spirtalis diabetis, crure descendens continuum liquorem in brevior re crure ascendere cogit, totumque ex vase demum exhaurit; & rapidè lapsus torrens saxa rapit, objectasque moles disjicit; & ad perpendicularum cadens lapis subjectum vitrum comminuit, suique vestigium in terrâ validiùs pressâ relinquit. Verùm illud firmum ac perpetuum est, quòd ubi plus violentiæ opus est, parem conatum languidior motus consequitur. Id quod in siphone A B C observare in promptu est, ex cujus osculo C inæqualis aquæ copia defluit paribus temporis intervallis: quò enim magis aquæ superficies in vase deprimitur, eò lentiùs aqua ex siphone dilabitur: quamvis scilicet aquæ crus B C implentis pares sint semper ad descendendum vires, si nihil, aut saltem non inæqualiter, repugnet, aquæ tamen crus B D brevius, & B I longius, & B A adhuc longius implentis dispar est in ascensu repugnantia; ac propterea cum earumdem virium B C minor sit Ratio ad majorem resistentiam B I, quàm ad minorem B D, languidior quoque motus est descendentis aquæ ex B C, cùm graviorem aquam B I, quàm cùm minùs gravem B D sursum trahere oporter. At S 2

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Book Two. CHAPTER II. 139 whatever else of this kind may be present, which, unless force be applied to it, so that it may pass from one place into another contrary to nature, the effort of a body inclined to motion is rendered vain; it is sufficiently clear that it must be moved from without: and indeed, as much impetus as is impressed upon it in opposition to its propensity, so much is taken away from motion. In those things certainly which are moved from outside by another force, because it is not permitted to proceed infinitely, at length some origin is discovered to which motion is natural: for nature is a force initiating motion in bodies that require it; yet it is held by certain laws, looking to the harmony of the universe, so that it does not depart from them, but allows some force to be applied to individual bodies, where, opposing propensities contending among themselves, the stronger prevails over the weaker. Thus, because it is unlawful either for bodies cut asunder by intervening voids to gape open, or for one to penetrate into the place of a neighboring body unless that body withdraws, or for motion to seek by itself detours and bends; therefore also a liquid descending in the longer leg of a siphon, or of a spiral diabeta, forces the continuous liquid to rise in the shorter leg, and at last exhausts the whole from the vessel; and a torrent rushing swiftly carries away rocks and hurls aside masses laid in its path; and a stone falling perpendicularly crushes the glass beneath it and leaves its mark more strongly impressed upon the earth. But this is firm and perpetual: that where greater violence is needed, a weaker motion results in a corresponding effort. This is easy to observe in the siphon A B C, from whose mouth C an uneven quantity of water flows out at equal intervals of time; for the more the surface of the water in the vessel is depressed, the more slowly the water glides from the siphon: although, of course, the powers are always equal for the water filling the leg B C to descend, if nothing, or at least nothing unevenly, resists it, yet the resistance is unequal in ascent for the water filling the shorter leg B D, the longer B I, and still longer B A; and therefore, since the Ratio of the same forces B C to the greater resistance B I is less than to the lesser B D, the motion of the water descending from B C is also weaker, when it must draw upward the heavier water B I, than when it must draw upward the lighter water B D. But S 2

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Mechanicorum 140 si externum siphonis crus ità decurtatum sit in E, ut osculum E & aquæ in vase superficies I paribus absint ab Horizonte inter- vallis, aquam ideò hærere, nec amplius ex E fluere constat, quia aquæ B E ad descendendum propensionem, par aquæ B I repugnantia, ne ascendat, elidit. Quòd si demum aquam in vase imminuas, ut ejus superficies paulò infra I, atque adeò infra E osculum deprimatur, non jam aqua hærer in E, sed sua per vestigia in E B remeare cogitur, præponderatâ nimirum majore gravitate aquæ implentis crus paulò longiùs quàm B I, atque adeò quàm B E, quod illi ex hypothesi constituimus æquale; tantóque velociùs ab aquâ interioris cruris raperetur exterior, quantò depressior facta fuisset in vase aquæ super- ficies. Hinc itaque sit, ut pro variâ corporis motui obsistentis re- pugnantiâ modò plus, modò minus impetûs reliquum sit, quo motûs celeritas aut tarditas perficiatur. Et si tanta sit eorum omnium, quæ motui moram inferunt, obsistentia, ut ad eam vincendam plus impetûs necesse sit, quàm pro potentiæ facul- tate, tunc nullus efficitur motus, quo corpus ex loco in locum transferatur, sed aliqua ex peregrino impetu sit partium com- pressio, aut distractio; neque enim omnes corporis particulæ homogeneæ sunt, aut ita compactæ citrà omnes poros, ut nul- la tenuiorum particularum compressio aut distractio consequi possit. Quod si ea sit corporis per vim movendi natura aut posi- tio, ut nullum planè sivè lationis, sivè rotationis, sivè vibrationis, sivè constipationis, sivè dilatationis motum concipere pos- sit, aut violento in statu permanere languido illo impetu, quem vis extrinseca efficere valeret, nullum quoque impetum reci- pit; quippe qui idcircò imprimeretur, ut motum præter natu- ram efficeret, aut ut naturalem motum retunderet, aut etiam prorsus impediret. Quemadmodum enim si corporis alicujus specificam gravitatem in aquâ mutari non posse constet, infer- re continuò licet, corpus idem neque raritatem neque densita- tem in aquâ assumere posse; ex his siquidem specificæ gravita- tis mutatio oriretur: ita pariter ubi nihil haberi potest eorum, quæ impetum extrinsecùs impressum necessariò consequuntur, impetum quoque abesse non immeritò conjectamus. Si quis tamen animum diligentiùs adverrat, manifestò de- prehendet

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Mechanics 140 if the external leg of the siphon is thus shortened in E, so that the mouth E and the surface of the water in the vessel I are at equal distances from the Horizon, it is evident that the water therefore clings, and no longer flows out from E, because the tendency of the water B E to descend is counteracted by the equal resistance of the water B I, so that it does not rise. But if at length you reduce the water in the vessel, so that its surface is somewhat below I, and thus below the mouth E, it is no longer the case that the water clings in E, but it is forced to return by its own path in E B, since the greater weight of the water filling the leg, being somewhat longer than B I, and therefore than B E, which by hypothesis we have made equal to it, prevails; and the more rapidly the external water would be drawn away by the water of the inner leg, the more the surface of the water in the vessel had been depressed. Hence it comes about that, according to the varying resistance opposed to the motion of the body, sometimes more, sometimes less impetus remains, by which the speed or slowness of the motion is accomplished. And if the resistance of all those things which delay motion be so great that more impetus is required to overcome it than the power has available, then no motion is produced by which the body is transferred from place to place, but instead some compression or stretching of the parts results from the foreign impetus; for not all the parts of a body are homogeneous, nor so compacted without all pores that no compression or stretching of the more subtle parts can follow. But if the nature or disposition of the body to be moved by force is such that it can admit of no motion at all, whether of translation, rotation, vibration, condensation, or dilation, or can remain in a violent state only with that languid impetus which external force might produce, then it also receives no impetus; for such impetus would be impressed for the purpose of producing motion contrary to nature, or of checking natural motion, or even of completely hindering it. For just as, if it is established that the specific gravity of some body cannot be changed in water, one may immediately infer that the same body cannot assume either rarity or density in water, since from these the change of specific gravity would arise; so likewise, where nothing can be obtained of those things which necessarily follow an externally impressed impetus, we no less reasonably conjecture that impetus itself is absent. If, however, someone should attend to this more carefully, he will plainly discover

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Liber secundus. CAPUT II. 141 prehendet corpus idem magis repugnare motui, si celeriùs mo- vendum sit, minùs verò, si tardiùs: sic ferreæ ansæ cubiculi ostio infixæ magnetem armatum applicui, & siquidem paulò velociùs magnetem traherem, disjungebatur ab ansâ; at len- tiùs trahentem subsequebatur ostium, magnetis scilicet vim non superans, ubi lentè res peragebatur. An non oneri, quod potentia præ sui tenuitate propellere non posse videtur, motus, qui momentis singulis sensum om- nem fugiat, conciliari potest, adeò ut, si illa quidem constan- ter urgeat, elapso demùm longo temporis intervallo appareat? Sic incumbentem glebam tenerrimus nascentis frugis cauicu- lus tandem discutit; durissima marmora scindens caprificus lo- co movet; & ædificia subsedisse, ac inæquabile solum pressisse, rimæ demùm loquuntur. Tota igitur corporis, quod præter naturam movendum est, repugnantia metienda est, quâ ex principio ipso motum detrectante, quâ ex motûs celeritate, aut tarditate: adeò ut pro variâ horum connexione dispar movendi difficultas oriatur. Ex quo sit impetu eodem moveri celeriùs posse corpus, quod minorem subit violentiam, tardiùs verò, cui vis major infer- tur, & , si eadem sit reciprocè Ratio tarditatis ad velocitatem, quæ est minoris violentiæ ad majorem violentiam, parem fore utrobiqve movendi difficultatem, cùm par sit repugnantia, quæ ex motûs tùm specie, tùm intentione componitur. Si enim mo- les aliquâ tantâ vi raptetur, ut, quo tempore decies arteria pul- sum edit, passum unum conficiat; quantum virium adhiberi oporteat, ut paribus temporis momentis ad tres passus eadem moles promoveatur? utique, si cætera omnia paria sint, triplo majorem conatum adhibendum concedes, intensione exten- sionem compensante: nam quemadmodum iterùm ac tertiò re- petendus fuisset prior ille conatus ad æquale semper spatium pa- ri tarditate percurrendum; ita quamvis conatui conatus non succedat, triplici tamen conatu opus erit, ut tempore eodem motus ille triplo major perficiatur. Nonnè & agricolæ terram subigentes fossione glebarum, tam multiplices adhibent operas, quàm breviori tempore opus absolvere meditantur? Eò igitur magis resistit corpus motui, quò celeriùs agitandum est; con- trà verò minùs repugnat, quò tardiùs. S 3

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Book the Second. CHAPTER II. 141 the same body resists motion more, if it must be moved more quickly; less, however, if more slowly: thus I applied a magnet armed with iron to the iron handle fixed in the door of a room, and if I pulled the magnet somewhat more quickly, it was separated from the handle; but if I pulled it more slowly, the door followed, the force of the magnet, of course, not being overcome when the matter was carried on slowly. Can there not be imparted to a load, which from the weakness of its power seems unable to drive it forward, a motion that escapes all sensation at each moment, so that, if that force indeed urges steadily, only after a long interval of time has elapsed does it at last appear? Thus the tender stalk of the sprouting crop at length throws off the clod that lies upon it; the wild fig, breaking the hardest marbles, displaces them; and buildings having settled, and having pressed the uneven ground, are at last betrayed by cracks. Therefore the whole resistance of a body that is to be moved contrary to nature must be measured, both by that which from the very beginning rejects motion, and by that which arises from the speed or slowness of the motion: so that according to the varying combination of these there arises a differing difficulty of moving. Hence, with the same impulse, a body that undergoes less force may be able to move more quickly, but more slowly that on which greater force is applied; and, if the ratio of slowness to speed be reciprocally the same as that of lesser force to greater force, the difficulty of moving will be equal in both cases, since the resistance compounded both from the kind of motion and from its intensity is equal. For if a mass were dragged with such great force that, in the time in which the artery beats ten times, it would complete one pace, how much force ought to be applied so that, in equal moments of time, the same mass might be advanced three paces? Surely, if all else is equal, you will grant that a threefold greater effort must be applied, intensity compensating for extension: for just as that former effort would have had to be repeated a second and a third time in order to traverse always the same distance with equal slowness; so, although effort does not follow upon effort, nevertheless a triple effort will be needed if that motion, three times greater, is to be accomplished in the same time. Do not farmers too, when they break up the earth by digging the clods, employ operations as many in number as they plan to complete the work in a shorter time? Therefore the body resists motion the more, the more quickly it must be moved; conversely, it resists the less, the more slowly. S 3

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Mechanicorum 142 Quare si duo sint corpora, quorum alterum alteri præstet triplo majori gravitate, atque hæc pari celeritate attollenda sint; disparem exigunt conatum pro gravitatis Ratione: si par sit eo- rum gravitas, motus autem alterius reliquo triplo velocior esse debeat, inæqualem pariter exigunt conatum, sed pro ratione velocitatis: si demùm & dispar sit gravitas, & inæqualis velo- citas, eam esse constat repugnantiam, quæ tùm ex gravitate, tùm ex velocitate componitur; atque adeò si corpus alterum triplo gravius triplo etiam velociùs movendum esset, noncuplex esset ejus repugnantia; sin autem triplo levius triplo majori velocitate quàm corpus triplo gravius, moveretur, par esset eo- rum obsistentia, paremque conatum exigerent. Hinc satis apertè constat, datâ tum resistentiarum, tum velo- citatum Ratione, si gravitas altera nota sit, reliquam facilè inno- tescere: si nimirùm nota gravitas per suam velocitatem ducatur, & in datâ Ratione resistentiarum reperatur huic producto ter- minus homologus; quo per ignotæ gravitatis velocitatem da- tam diviso, prodibit Quotiens index quæsitæ gravitatis. Sint duo corpora inæqualia, & ad ea movenda requiratur conatus in Ratione sesquialterâ, motus autem eorum sint ut 7 ad 8, & illud quod minùs resistit, moveturque velocitate ut 7, numeret gravitatis libras 4. Reliqui corporis validiùs resistentis, cujus velocitas est ut 8, gravitas sic invenietur. Libræ 4 ducantur per numerum suæ velocitatis 7, & fit 28. Quia igitur resistentiæ sunt, ut 2 ad 3 ex hypothesi, & unius corporis resistentiâ, quæ ex gravitate & motûs velocitate com- ponitur, est 28, fiat ut 2 ad 3, ita 28 ad aliud, & erit 42 re- sistentia alterius corporis composita ex ejus velocitate & gravi- tate. Atqui velocitas nota est 8; igitur divisâ totâ resistentiâ 42 per 8; prodibit quotiens 5 1/4 index quæsitæ gravitatis. Quare ad movendas libras 5 1/4 velocitate ut 8, requiritur conatus ses- quialter conatûs necessarij ad movendas libras 4 velocitate ut 7. Eadem esto de reliquis ac similibus conjectura. Ex his præterea manifestum est corporis per vim dimovendi resistentiam ex solâ naturâ, & principio insito, quod motui re- pugnat, absolutè definiri non posse; motum si quidem ab omni prorsùs celeritatis aut tarditatis mensurâ sejungere non possu- mus; idcircò non nisi habitâ ratione celeritatis, aut tarditatis, ex

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Mechanics 142 Wherefore if there be two bodies, of which one exceeds the other in weight by a factor of three, and these are to be lifted with equal speed; they require an unequal exertion in proportion to the weight: if their weight be equal, but the motion of one must be three times swifter than the other, they likewise require an unequal exertion, but in proportion to the speed: if finally both the weight be unequal and the speed unequal, it is clear that the resistance is compounded both from weight and from speed; and therefore if one body were to be moved three times as heavy and also three times as fast, its resistance would be ninefold; but if, on the other hand, a body three times lighter were moved with a speed three times greater than that of a body three times heavier, their resistance would be equal, and they would require equal exertion. Hence it is sufficiently clear that, given the ratio both of the resistances and of the speeds, if one weight be known, the other can readily be discovered: namely, if the known weight be multiplied by its speed, and in the given ratio of resistances there be found a homologous term for this product; by dividing this by the given speed of the unknown weight, the quotient will come forth as the index of the sought weight. Let there be two unequal bodies, and for moving them let the exertion be required in the ratio of one and a half, but let their motions be as 7 to 8, and let that which resists less, and is moved with a speed as 7, have 4 pounds of weight. The weight of the remaining body, which resists more strongly and whose speed is as 8, will thus be found. Let 4 pounds be multiplied by the number of its speed, 7, and 28 results. Since therefore the resistances are as 2 to 3 by hypothesis, and the resistance of one body, which is compounded from weight and speed of motion, is 28, let it be as 2 to 3, so 28 to another, and there will be 42 as the resistance of the other body compounded from its speed and weight. But the speed is known, namely 8; therefore, if the whole resistance 42 be divided by 8, the quotient 5 1/4 will emerge, the index of the sought weight. Wherefore, to move 5 1/4 pounds with a speed as 8, an exertion one and a half times that necessary for moving 4 pounds with a speed as 7 is required. Let the same conjecture hold for the rest and for similar cases. From these things moreover it is manifest that the resistance of a body to be moved by force cannot be absolutely defined from its nature alone, and from the innate principle which opposes motion; for indeed we cannot separate motion from every measure whatsoever of swiftness or slowness; therefore only when due account has been taken of swiftness or slowness, from

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Liber secundus. CAPUT II. 143 ex quibus resistentia componitur, resistentia ipsa innotescere poterit. Quare & impetus à facultate movendi principium ha- bente productus major sit necesse est, quàm dimoti corporis repugnantia; quæ varia prorsùs cùm sit, nunc quidem majo- rem, nunc verò minorem impetum exigit, ut ab eo vincatur; nam si pares configerent vires, à neutrâ parte staret victoria. Quod autem ad ipsam motûs originem spectat, ea, quæ vi- vunt, ab iis, quæ vitâ omnino carent, secernenda sunt: hæc enim (scilicet non viventia) propterea motum expetunt, ut violentiam, quam subeunt, excutiant, nec unquam à loco, seu statu, secundùm naturam opportuno sponte recedunt; quem- admodum eunti per singula constabit. Sic gravibus & levibus suis in locis quietem natura indixit, non motum; nec deor- sum conantur aut sursum, nisi alieno in loco, hoc est, in me- dio dispari gravitate aut levitate prædito constitutâ: sic quæ- cumque elasticâ facultate pollent, motum non moliuntur, nisi cum sibi naturalem partium figuram, situmque reparare opor- tet. At motum, cujus origo vita est, natura perficit, etiamsi nulla præcesserit violentia: sic stirpes dum augentur, & cres- cunt, earum particulæ locum mutant; sic vitali facultate in- fluentibus per nervos in animaliu[m] musculos spiritibus, quos ani- males vocant, intenduntur musculi, motusque membrorum con- sequitur: quamvis ante motum nec stirpis particulæ, nec anima- lis membra vim ullâ subierint in loco minimè congruo retenta. Quæcunque igitur ob id ipsum in motum prona sunt, quia vim patiuntur, impetum illicò concipiunt, ac vis iis illata est, quo naturalem locum, seu statum, recipere valeant, licèt sæpè irrito conatu, nisi quatenùs adverso hoc impetu illatam ab ob- sistente violentiam retundunt, vim aliquam illi vicissim infe- rentes. Sic onera bajulorum humeros, quibus sustinentur, premunt, aut penduli brachij; ex quo suspenduntur, muscu- los ac ligamenta fatigant: id quod pariter in corpore inanimo cernere licet; quemadmodum enim ex diuturnâ prementis deorsum ponderis, ac musculorum sursum urgentium luctâ, dissipatis spiritibus, lassitudo in animali oritur, ita pariter sub- jectum asserem longâ temporis morâ pondus curvat, aut etiam demùm frangit, & funem, ex quo pendet, non intendit solùm, sed etiam tandem aliquando corrupto particularum nexu disjicit. Quo

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Book the second. CHAPTER II. 143 from which resistance is composed, resistance itself may be made known. Therefore an impetus produced by a principle having the power of moving must necessarily be greater than the resistance of the body being moved; since this resistance is altogether various, it now requires a greater, now a lesser impetus to be overcome by it; for if the forces were equal, victory would stand with neither side. But as regards the very origin of motion, those things which live must be distinguished from those which are wholly devoid of life: for the latter, that is, non-living things, seek motion for this reason, that they may shake off the violence which they endure, and they never of their own accord recede from the place, or state, suitable according to nature; as will be clear to the reader step by step. Thus nature has appointed rest, not motion, for heavy and light things in their own places; nor do they attempt downward or upward motion unless placed in an alien place, that is, in a medium endowed with a different gravity or levity: thus whatever things possess an elastic faculty do not produce motion unless they need to restore to themselves the natural shape and position of their parts. But motion, whose origin is life, nature brings to completion even if no violence has preceded it: thus while plants increase and grow, their particles change place; thus, in animals, the spirits flowing through the nerves into the muscles, which they call animal spirits, cause the muscles to contract, and the motion of the limbs follows: although before the motion neither the particles of the plant nor the limbs of the animal have undergone any force while being held in a place least suitable to them. Whatever things therefore are inclined to motion for that very reason, because they are subjected to force, immediately conceive an impetus, and the force inflicted upon them is so that they may be able to recover their natural place, or state, although often with a futile effort, unless insofar as, by this opposing impetus, they check the violence inflicted by the resisting body, inflicting some force upon it in return. Thus burdens press upon the shoulders of porters, by which they are supported, or upon the hanging arm from which they are suspended, fatiguing the muscles and ligaments: which may likewise be seen in an inanimate body; for just as, from the long struggle between the downward pressure of weight and the muscles urging upward, when the spirits are dissipated, weariness arises in an animal, so likewise a beam placed underneath, by the long delay of time, bends under a weight, or even at length breaks, and the rope from which it hangs not only is stretched, but also at last, the connection of the particles being corrupted, is torn asunder. Wherefore

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144 Mechanicorum Quo id autem pacto contingat, explicare operosum non fue- rit funiculi texturam consideranti; ex tenuissimis scilicet linei aut cannabini corticis longâ maceratione, & plurimâ tunstione extenuati particulis in spiram contortis filum cohæret; ex filis autem plusculis in spiram pariter contortis funiculus, & pluri- bus funiculis crassiores rudentes constantur: quod si dissolvatur omnis spira, non cohærent funiculi aut fili partes. Spira dis- solvitur factâ in contrarium revolutione; quò autem laxioribus gyris flectitur, eò faciliùs villi singuli ex cæteris, quibus im- plicantur, extrahuntur; & uno ab aliorum communione se- juncto, amplitudo spatij faciliorem exitum proximis relinquit: ex quo fit faciliùs semper ac faciliùs posse funiculum frangi; filo enim uno rupto, aut extracto, facilior est in contrarium re- volutio, & spira fit amplior, ac reliqua fila faciliùs extrahun- tur. Observamus autem non rarò appensum ex funiculo pon- dus aliquandiu in gyrum contorqueri; dum scilicet suâ gravi- tate deorsum connitens intendit funiculum, contorta fila in contrarium revolvuntur. Sed &, quamvis nulla fieret in con- trarium revolutio, satis constat ex illâ intensione funiculum distrahi, ac produci; atque adeò spiram laxiorem fieri, paula- timque unum aut alterum villum educi, locumque fieri vapo- ribus, qui proximum villum corrumpentes faciliori scissioni pa- rant, atque adeò, serpente lue, demùm non tot integri super- sunt villi, qui possint ponderis gravitati obsistere, quin dif- fringantur. Ex quo satis apparet suspensum pondus, licèt non omninò descendat, impetum tamen concipere, quo retinenti repugnat, & vim aliquam vicissim infert. Nec absimili ratione in reliquis vim patientibus contingere observabimus, ea scilicet moliri illicò naturalis statûs repara- tionem, aliquidque efficere, licèt tenuissimum, quod demum appareat, ubi temporis morâ augmentum ceperit. Sic hastam per vim inflexam si continuò dimittas, illa sese restituit, facul- tate elasticâ; at si dies aliquot, aut etiam diutiùs per vim si- nuata permanserit, sibi dimissa antiquam rectitudinem non re- parat; elanguit nimirùm facultas elastica, quæ ex violentâ par- ticularum compressione aut distractione oriebatur. Cùm enim primùm hasta flectitur, particulæ concavam curvaturæ partem respicientes comprimuntur, contra verò, quæ convexam respi- ciunt,

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144 Mechanicorum How this happens, however, it would be laborious to explain to one considering the structure of a rope: namely, from the extremely slender particles of flaxen or hempen bark, thinned by long maceration and much beating, the thread coheres when they are twisted into a spiral; from several threads likewise twisted into a spiral, the rope, and from several ropes thicker cables are made. But if every spiral is undone, the parts of the ropes or threads do not cohere. The spiral is undone by a reverse turning; and the looser the turns with which it is bent, the more easily the individual fibers are pulled out from the others with which they are intertwined; and when one has been separated from the common union of the rest, the wideness of the space leaves an easier exit for the neighboring ones. Hence it follows that a rope can always and more easily be broken; for when one thread has broken or been drawn out, reversal in the opposite direction is easier, the spiral becomes wider, and the remaining threads are more easily pulled out. We observe moreover not infrequently that a weight hung from a rope twists for a time in a circle; namely, while by its own gravity striving downward it stretches the rope, the twisted threads turn in the opposite direction. But even if no reversal in the opposite direction occurred, it is sufficiently clear that by that stretching the rope is drawn apart and lengthened; and thus the spiral becomes looser, and one fiber or another is gradually drawn out, and room is made for vapors, which by corrupting the neighboring fiber prepare it for easier splitting; and thus, as the disease spreads, at length there do not remain so many intact fibers as can withstand the weight’s heaviness without breaking. From this it is sufficiently apparent that a suspended weight, though it does not wholly descend, nevertheless acquires an impulse, by which it resists the thing holding it and in turn exerts some force. In a not dissimilar way we shall observe the same thing to happen in other bodies subjected to force, namely that they immediately strive to restore the natural state and produce something, however slight, which will finally become apparent when, with the passage of time, it has increased. Thus if you continuously release a spear bent by force, it restores itself by its elastic faculty; but if for several days, or even longer, it has remained curved by force, when left to itself it does not recover its former straightness: the elastic faculty, namely, which arose from the violent compression or stretching of the particles, has become weakened. For when the spear is first bent, the particles looking toward the concave part of the curve are compressed, but on the other hand those looking toward the convex part,

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Liber secundus. CAPUT III. 145 -tiunt, distrahuntur; quare tùm quæ raræ, tùm quæ densæ factæ sunt, dum vim illicò prorsùs excutere conantur, conspirant, ut pristinam hastæ rectitudinem moliantur: Quod si id non li- cuerit, hæ quidem aliam ex angustiis evadendi, quâ facilior patet via, rationem tentant, ita ut demùm subtilissimas in ru- gas crispentur, illæ verò sese ad angustiora spatia sensim reci- pientes mutuum nexum solvunt, tenuissimosque poros relin- quunt, aut si qui priùs interjecti fuerint, ampliùs hiare per- mittunt. Id quod ubi jam contigerit, frustrà submoves, quæ admoveras impedimenta; & spontè curvaturam hasta servat, nisi fortè particulis omnibus adhuc per tempus non licuerit vim totam excutere; tunc enim se se languidiùs restituunt, pro ratione reliquæ violentiæ. Hinc patet arcum, quò fuerit con- tentus atque adductus vehementiùs, remitti aliquando, & ma- nualium tormentorum rotas interdum laxari oportere, ne vis elastica languidior facta minùs utilis fiat. Ex his igitur paulò enucleatiùs explicatis, in quibus longio- re temporis fluxu motum aliquem tardissimum contigisse, at- que adeò etiam impetum jam tum ab initio statim fuisse pro- ductum constat, conjecturam in reliquis capio, & ab iis impe- tum concipi statuo, quæ aut loco naturali dimota, aut incon- gruam partium positionem nacta id repetunt, quod natura exi- git. Motus autem non pro impetûs tantum, sed & pro re- sistentiæ modo consequitur. CAPUT III. Quâ ratione semel conceptus impetus pereat. UT impetûs natura, quam inquirimus, explicatiùs atque distinctiùs innotescat, ex quo pariter, quæ corpora, quâ- ve ratione, impetum respuant, intelligamus, hîc nobis est vestigandum, quâ ratione conceptum semel impetum abji- ciant: hinc nimirum in uberiorem ipsius resistentiæ notitiam venientes ad explicandam motûs machinalis causam propiùs accedemus. T

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Book Two. CHAPTER III. 145 -are scattered, are drawn apart; wherefore then those which have been made rare, and those which have been made dense, while they try at once to shake off the force entirely, conspire so as to strive after the original straightness of the spear: but if this cannot be done, these indeed attempt another means of escaping from the narrowness, by which a freer way lies open; so that at last they curl themselves into very fine wrinkles, while those, on the other hand, gradually drawing back into narrower spaces, loosen their mutual connection and leave the finest pores, or if any had been previously interposed, allow them to gape more widely. And when this has once happened, you remove in vain the impediments you had brought near; and the spear of itself preserves its curvature, unless perhaps all the particles have not yet been allowed for a time to shake off the whole force; for then they restore themselves more sluggishly, according to the amount of the remaining violence. Hence it is clear that a bow, the more violently it has been stretched and drawn, must sometimes be relaxed, and that the wheels of mechanical engines must sometimes be loosened, lest the elastic force, having become feebler, become less useful. From these matters, therefore, now a little more clearly explained, in which it is evident that during a longer flow of time some very slow motion has occurred, and indeed that an impulse has already been produced at once from the very beginning, I take the conjecture in the remaining cases, and I determine that an impulse is conceived from those things which, either removed from their natural place or having obtained an unsuitable position of their parts, tend back to what nature requires. Motion, however, follows not only as a result of impulse, but also as a mode of resistance. CHAPTER III. By what means an impulse once conceived perishes. In order that the nature of impulse, which we are investigating, may become known more clearly and distinctly, and so that from this we may understand as well what bodies, and in what way, reject impulse, it is here our task to inquire by what means they cast away an impulse once conceived: from this, indeed, moving into a fuller understanding of resistance itself, we shall come more closely to explaining the cause of mechanical motion. T

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Mechanicorum Et sanè conceptum impetum, naturâ suâ, nec stabilem semper per permanere, nec ad unicum temporis punctum durare, satis constat: sivè enim spontè profluat ex naturâ debitum sibi locum quærente, sivè alienâ vi impressus suo loco corpus extrudat, perpetuus esse nequit; omnis scilicet motus terminum habeat necesse est; nam si violentus quidem est, perennis utique non est; sin autem naturalis, quem violentus præcesserit, certis definitur terminis; à loco enim, in quo quietem natura indixit, corpus infinito intervallo non abest, ac proinde ubi eum attigerit, demùm conquiescet, nec impetu perpetuo opus erit, cùm motum cessare oporteat. Sed neque temporis momento circumscribi impetum sivè in naturali motu acquisitum, sive in violento impressum, plura sunt, quæ palam faciunt: ut enim reliqua sileam nullæ essent funependulorum oscillationes, nullus emissæ sagittæ motus, si conceptus impetus illicò periret. In duo autem veluti genera tribuendus est Impetus ex naturâ dimanans; alius Innatus, seu quasi insitus, alius Acquisitus dicitur, Innatum, seu quasi insitum, voco, non quem corpus jugiter obtineat, sive suo in loco, sive in alieno quiescat; sed eum, qui facultati se movendi præcisè respondet, nullo facto per continuam adjectionem incremento: quandùm enim corpus ita simili secundùm gravitatem corpore circumfunditur, ut naturali in loco consistere dicendum sit, quare conctur motum? conatum autem hîc ab impetu non distinguo: satis igitur citrà quemlibet impetum suo sè tutatur in loco per hoc, quod eâ facultate sit præditum, quæ in contrariam partem conniti valeat illicò, ac vis inferri cæperit. Hinc nullum aquæ impetum tribuo intrà aquam consistenti; sed tunc solùm cùm situla plena è lacu extrahitur, ea aquæ pars impetum habet, quæ supra subjectam lacûs superficiem aëre circumfusa motum expetit, quo suum repetat locum repugnans sustinenti. Impetum hunc, qui naturali se movendi facultati respondet, & est ipsa gravitatio, seu naturalis ad descensum propensio, Innatum voco, & is est, cui extrinseca causa repugnat motum impediens. Quòd si suspensum corpus sibi relinquatur, ita suum in locum contendit, ut vis naturalis æquè semper ad agendum applicata, nec impedita, momentis singulis novum impetum acquirat, qui proptere Acquisitus

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Mechanics And indeed it is quite clear that a conceived impulse, by its nature, can neither always remain stable and endure continuously, nor last for a single moment of time: for whether it flows of itself from nature, seeking the place due to it, or whether, impressed by an external force, it thrusts the body from its place, it cannot be perpetual. Every motion, that is to say, must have an end; for if it is violent, it certainly is not everlasting; but if it is natural, and has been preceded by a violent motion, it is defined by fixed limits: for the body is not at an infinite distance from the place in which nature has assigned it rest, and therefore when it has reached that place, it will at last come to rest, and there will be no need of perpetual impulse, since the motion must cease. But neither can the impulse, whether acquired in natural motion or impressed in violent motion, be confined to a single instant; there are many things that make this plain: for to pass over the rest, there would be no oscillations of a hanging rope, no motion of an arrow shot forth, if the conceived impulse were to disappear at once. Now the impulse arising from nature must be assigned, as it were, to two kinds: one is called innate, or as it were inborn; the other acquired. I call innate, or as it were inborn, not that which a body continually possesses, whether it rests in its own place or in another; but that which precisely corresponds to the faculty of moving itself, with no increase produced by continual addition. For when a body is so surrounded by another body of like kind according to gravity that it is said to stand in its natural place, why should it strive for motion? Here I do not distinguish striving from impulse: it is enough, then, that without any impulse it protects itself in its place by the fact that it is endowed with that faculty which can immediately exert itself in the opposite direction as soon as force begins to be applied. Hence I assign no impulse to water remaining within water; but only when a full bucket is drawn from the lake does that part of the water have impulse which, above the surface of the lake beneath it, surrounded by air, seeks motion in order to recover its place, resisting the supporting body. I call this impulse, which corresponds to the natural faculty of moving oneself, and which is gravity itself, or the natural tendency to descend, innate; and it is that against which an external cause resists by impeding motion. But if a suspended body be left to itself, it so strives toward its own place that the natural force, always equally applied to act and not impeded, acquires at each moment a new impulse, which therefore is acquired

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Liber secundus. CAPUT III. 147 Acquisitus dicitur, & posterior priori additus intentionem ef- ficit: sapienti sanè naturæ instituto; nam si corpora per se ipsa ac suâ sponte mota non accelerarent; sed naturalis motus pla- nè æquabilis esset, tardè nimis locum suum consequerentur; atque adeò augendus continuò fuit impetus, ut & motus in- crementum acciperet: at si innatus impetus valdè intēsus esset, corpora nonnisi ægerrimè aliò transferri, aut alieno in loco re- tineri pro animalium, & hominis utilitate possent; finge scili- cet animo tibiam tanto impetu innato repugnare, ne attollatur, quanto impetu in aëre ex 200 passuum altitudine descenderet; quanto id tibi esset incommodo? Quare peropportunum acci- dit, ut vehemens non esset singularum particularum impetus innatus, qui tamen ubi motum efficeret, novâ accessione pos- set augeri. Quod ad impetum Innatum spectat, quem à gravitatione ipsâ & proximâ motus exigentiâ non sejungo, utique frustrà esset, si omni prorsus effectu careret; impetus autem motum aut efficit, aut saltem exigit: propterea illum statim perire au- tumo, ac fuerit corpus in loco suo: Id quod hoc deprehendes experimento. Scrobem defossâ humo altè excavato; situam aquæ plenam, & noti ponderis, intrà illam suspendito; tùm aquam in scrobem tantâ copiâ derivato; ut situam usquequa- que circumplectatur: illicò evanescet totius aquæ priùs in situ- lâ gravitantis pondus, quin & situla ipsa pro gravitatum secun- dùm speciem dissimilitudine levior apparebit, ut ex Hydrosta- ticis constat. Periit ergo innatus impetus, quo aqua situam replens descensum moliebatur. At impetum Acquisitum non continuò perire, ac eò ventum fuerit, ubi quiescendum esset, hinc saltem disces, quod ligneum globum aquæ cæteroqui innataturum si in sublime at- tollas, & ex illâ altitudine cadere permittas, infrà aquæ super- ficiem descendere, ac penitùs immergi videbis; quamquam postea emergat, & ubi aliquoties subsultaverit, demùm pro gravitatum aquæ, & ligni disparitate emersus quiescat. Quæ sanè immersio, nisi Acquisitus impetus adhuc duraret, omninò non contingeret. Verùm nihil rem per se satis abstrusam æquè in lucem evocat, ac funependulorum motus; plumbum enim ex filo suspensum, & à perpendiculo dimotum, ita descendens T 2

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Book Two. Chapter III. 147 An acquired impetus is called such, and when added to a prior one it produces intention: very wisely in the design of nature; for if bodies, moved by themselves and of their own accord, did not accelerate, but natural motion were altogether uniform, they would reach their place far too slowly; and therefore the impetus had to be continually increased, so that motion might also receive augmentation. But if the innate impetus were exceedingly strong, bodies could only with great difficulty be carried elsewhere, or be kept in a different place for the benefit of animals and of man. Imagine, for example, a flute resisting with so much innate force that it cannot be lifted, as the force with which it would descend through the air from a height of 200 paces. How troublesome would that be to you? Wherefore it proved very fitting that the innate impetus of individual particles should not be violent, yet should be capable, once it had produced motion, of being increased by a new addition. As regards the Innate impetus, which I do not separate either from gravity itself or from the immediate requirement of motion, it would certainly be useless if it lacked every effect whatsoever; but impetus either produces motion, or at least demands it. For this reason I think it dies away at once, as soon as the body has reached its place. You will perceive this from the following experiment. Dig a pit and hollow it out deeply in the earth; suspend within it a bucket full of water and of known weight; then pour water into the pit in such quantity that it surrounds the bucket on every side; immediately the weight of the whole water previously gravitating in the bucket will vanish, and even the bucket itself will appear lighter, according to the difference in specific gravities, as is established by hydrostatics. Therefore the innate impetus, by which the water filling the bucket was striving downward, has perished. But that the Acquired impetus does not instantly perish when the point is reached at which it ought to come to rest, you will at least learn from this: if you raise a wooden globe, which would otherwise float on water, up into the air and allow it to fall from that height, you will see it descend below the surface of the water and become completely submerged; although afterwards it rises again, and after it has bobbed up and down a few times, it finally comes up and rests according to the difference in gravity between the water and the wood. Certainly such immersion would not occur at all unless the Acquired impetus still endured. Yet nothing brings a matter in itself sufficiently obscure into clearer light so well as the motion of pendulums; for a lead weight suspended from a string, and displaced from the vertical, in descending thus T 2

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Mechanicorum arcum describit, ut ferè parem arcum, & vix (aut fortè ne vix quidem) minori tempore ascendens describat. Cui autem, repugnante plumbi gravitate à naturâ insitâ, tribuatur ascensus, nisi impetui acquisito dum descenderet, adhuc post descensum duranti? Quemadmodum verò in descensu posteriores motûs partes prioribus velociores sunt, factâ nimirum novi impetûs accessione, ita ex opposito ascensus ex celeritate in tarditatem desinit, factâ acquisiti impetûs decessione continuâ, donec ita elanguerit, ut gravitas ipsa superet, & iterum descendens alternas vibrationes efficiat. Perit igitur Acquisitus impetus non totus simul; sed sensim extenuatur; idque non aliâ ratione, quàm quâ proportione impeditur motus, quocunque tandem ex capite impedimenta oriantur. Cum enim impetus contrarium impetum non habeat, si præcisa quidem impetûs natura spectetur (quippe qui unus & idem contrariorum motuum origo est, ut ex funependulis ultrò citróque sponte vibratis & ex pilâ lusoriâ deorsum cadente, ac vi concepti impetûs sursum resiliente, constat) reliquum est, ut pereat pro ratione eorum, quæ aut motui corporis obsistunt, aut illud aliò quoquomodo dirigunt. Præstat autem hîc funependuli motum paulò attentiùs considerare. Sit plumbeus globulus B filo A B connexus clavo in A. Si globulo liceret, quâ impetus innatus urget viâ, descendere, utique rectam BC percurreret; sed funiculo retinente cogitur arcum B K describere, adeò ut semper in alio & alio plano inclinato constitutus, alia, & alia habeat gravitatis momenta, ut lib. I. cap. I 5 explicatum est; hæc autem sunt pro Ratione Sinuum angulorum declinationis à perpendiculo A K. Quare totum momentum, quod in B esset ut A B, singulis momentis in descensu libero per rectam BC paribus saltem incrementis augeretur (Quicquid sit an etiam pro Ratione duplicatâ temporum, de quo alias disputabimus) sed cum à rectitudine deflectat, cum venerit in D, non additur momentum ut EF, sed ut ED: similiter in G momentum non est

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Mechanics describes an arc, so that it describes a nearly equal arc, and ascending scarcely (or perhaps not scarcely at all) in less time. But to what is the ascent attributed, in spite of the inherent gravity of the lead, unless to the acquired impetus while descending, which still continues after the descent? Moreover, just as in descent the later parts of the motion are faster than the earlier ones, because of the addition of a new impetus, so conversely the ascent ends from speed in slowness, because the acquired impetus continually diminishes, until it has so weakened that gravity itself prevails, and, descending again, it produces alternating vibrations. Thus the acquired impetus is not lost all at once, but gradually grows weaker; and this is for no other reason than in proportion as the motion is hindered, from whatever cause the hindrances may arise. For since impetus has no contrary impetus, if one considers the exact nature of impetus itself (for it is one and the same origin of contrary motions, as is clear from a pendulum vibrating freely to and fro, and from a ball game-piece falling downward and, by the force of the conceived impetus, rebounding upward), it remains that it perishes in proportion to those things which either oppose the body’s motion or in some other way direct it elsewhere. Here it is better to consider the motion of the pendulum a little more carefully. Let the leaden globe B be connected by the string A B to the nail at A. If the globe were allowed, by the path along which its innate impetus urges it, to descend, it would certainly traverse the straight line BC; but being held by the cord it is forced to describe the arc B K, so that, always being situated in another and another inclined plane, it has one and another moment of gravity, as explained in book I, chapter I 5; and these are in proportion to the sines of the angles of declination from the perpendicular A K. Wherefore the whole momentum, which at B would be as A B, in free descent along the straight line BC would at least increase by equal increments at each moment (whatever may be said as to whether also in the ratio of doubled times, about which we shall discuss elsewhere); but when it is deflected from straightness, when it has come to D, the momentum is not added as EF, but as ED: similarly at G the momentum is not

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Liber secundus. CAPUT III. 149 est ut HI, sed ut HG. Augetur igitur impetus in descensu BK non omninò pro Ratione momentoru[m] temporis, quo motus durat, sed pro Ratione momentorum gravitatis, quæ subinde obtinet minora & minora; pars siquidem impetus ab insitâ globuli gravitate producti deteritur in intendendo filo, quo retinetur. Quapropter ubi in K venerit per arcum BK, non tantum habet impetus, quantum si per lineam perpendicularem arcui BK æqualem descendisset; in motu enim ad perpendiculum cum nihil retineat aut impediat, totus impetus ad descensum urget velociùs, quàm ubi repugnat aliquid. Ex quo fit quod, cùm arcus BK ad Radium AB, hoc est ad BC æqualem, sit proximè ut 11 ad 7, ex Cyclometricis, multò plus temporis in percurrendo arcu BK, quàm in rectâ BC, insumitur; tardiùs scilicet movetur quàm in perpendiculari, quæ ad BC esset ut 11 ad 7. manente itaque, quamdiu corpus naturâ urgente movetur, impetu acquisito, qui resistentiam excedit, in fine descensûs in K totus impetus est ut aggregatum omnium Sinuum Quadrantis: at in perpendiculari BC in fine descensûs in C esset ut aggregatum omnium parallelarum ipsi AB in Quadrato AC; ac propterea (in re Physicâ si liceat cum geometrizantibus per Indivisibilia ratiocinari) erit impetus per ar- cum BK acquisitus ad impetum per rectam BC acquisitum ut Quadrans ABK ad Quadratum AC, hoc est ut 11 ad 14, ex iis quæ in Cyclometriâ demonstrantur. Quoniam verò ubi ad perpendiculum AK globulus descendens venerit, nihil objicitur, quod motum prorsùs impediat, quin ad easdem partes pergat ferri ex præconcepti impetus directione, non sistit in perpendiculo; sed ulteriùs pergens ascendit, nec nisi per arcum circà centrum A, funiculo scilicet retinente. Sed jam repugnat ascensui gravitas plumbi, non quidem quantum in perpendiculo KA, verùm pro ratione Sinuum angulorum declinationis; qui cum semper ascendendo crescant, major est etiam momentorum gravitatis Ratio nitentium contrà impetum descendendo acquisitum. Quare tantum abest, ut novus singulis temporis punctis impetus sursum directus producatur, ut potius ex eo tantumdem dematur, quanta est ascendentis plumbi repugnantia. Hinc est ascensum initio velociorem esse, quia adhuc multus est impetus acquisitus, & pro T 3

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Book Two. CHAPTER III. 149 it is as HI, not as HG. The impetus is therefore increased in the descent BK, not in strict proportion to the moments of time during which the motion lasts, but in proportion to the moments of gravity, which continually become smaller and smaller; for part of the impetus produced by the inherent weight of the little ball is worn away in stretching the string by which it is held. Wherefore, when it has come to K by the arc BK, it does not have so much impetus as if it had descended through a perpendicular line equal to the arc BK; for in motion straight down, since nothing restrains or hinders it, the whole impetus is urged more swiftly toward the descent than where something opposes it. Whence it follows that, since the arc BK to the radius AB, that is to BC equal to it, is nearly as 11 to 7, according to cyclometric calculations, much more time is spent in traversing the arc BK than in the straight line BC; that is, it moves more slowly than in the perpendicular, which to BC would be as 11 to 7. Therefore, while the body is moved by natural force, with acquired impetus exceeding resistance, at the end of the descent in K the whole impetus is as the aggregate of all the sines of the quadrant; but in the perpendicular BC, at the end of the descent in C, it would be as the aggregate of all the parallels to AB in the square AC; and therefore (in physical matters, if it be permitted to reason with geometers by indivisibles) the impetus acquired through the arc BK will be to the impetus acquired through the straight line BC as the quadrant ABK to the square AC, that is, as 11 to 14, according to what is demonstrated in cyclometry. But since, when the descending little ball has reached the perpendicular AK, nothing is placed in its way to prevent the motion altogether, it continues to be carried toward the same parts by the direction of the impetus previously formed; it does not stop at the perpendicular, but going farther on it rises, and only by the arc around the center A, the string retaining it. But now the gravity of the lead opposes the ascent, not indeed as in the perpendicular KA, but in proportion to the sines of the angles of declination; which, since they always increase while ascending, also makes the ratio of the moments of gravity of those tending against the impetus acquired in the descent greater. Wherefore it is so far from being the case that a new impetus directed upward is produced at each point of time, that rather so much is taken away from it as is the resistance of the lead ascending. Hence it is that the ascent is faster at the beginning, because there is as yet much acquired impetus, and pro-

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Mechanicorum 150 Sinuum declinationis brevitate, exigua illius pars deteritur, atque adeò motus efficitur celerior: quia verò diminuto sensim impetu, & auctis cōtrariæ gravitatis momētis pro Sinuum declinationis incremento, minor fit ipsius impetûs ad contrariu[m] nisum Ratio, tardior sequitur motus, & plus acquisiti impetûs perit, donec demùm prorsus evanuerit, & superante gravitate globulus iterum descendat. Quamvis autem si positio sola spectetur, iisdem Reciprocè gradibus minui videatur impetus, quibus fuit auctus, totidemque momentis temporis, ita ut quantum postremo temporis puncto accessit, tantumdem primo decedat, adhuc tamen aliqua est obsistentiæ appendicula ex aëre dividendo, ac propterea paulo ampliùs extenuatur impetus acquisitus, quàm pro Ratione incrementi Sinuum declinationis: quò autem velocior est motus, magis etiam aër dividendus comprimitur, densatûsqve plus obsistit quàm rarus; quòd si medium non fuerit compressionis capax, saltem æquali tempore plures medij partes scinduntur, quàm in motu tardiori, ac propterea etiam multiplex est medij resistentia: Ex quo fit arcum ascensûs paulò minorem semper esse arcu descensûs, & cum vicissim globus remaneat ex humiliore loco ac priùs descendens, breviorum pariter secundi ascensûs arcum perfici, atque ita deinceps, ut servatâ eâ in motu semper minori reciprocando constantiâ demum quiescat in perpendiculo. At, inquis, dura magis obsistunt corpori, ejusque motum validiùs impediunt, quàm mollia, quæ dum se comprimi patiuntur, & loco paulisper cedunt, motui aliquantùm & ex parte obsecundant: si igitur pro Ratione impedimenti debilitatur acquisitus impetus, minus detrahitur impetûs corpori, quod ex alto decidens à substratis paleis excipitur, quàm si ad saxum allideretur; vehementiùs igitur à luto quàm à saxo reflecteretur, contrà quàm docet experientia. Fateor eburneum globum segniùs resilire delapsum in glebam humore perfusam, quàm in marmor; non tamen his consequens est, ut impetûs acquisiti diminutioni alius statuendus sit modus, quàm ex impedimento: ubi enim globus cadens estimam subjecti corporis superficiem attigerit, non quiescit, sed pergit moveri, aut deorsum comprimendo corpus molle, aut illicò sursum reflexum à duro. Ita autem à corpore molli excipitur,

Transcription: Translated (English)

Mechanics 150 By the briefness of the declination of the sines, a small part of it is worn away, and thus the motion becomes quicker: but because, as the impetus is gradually diminished, and the moments of the contrary gravity are increased with the increase of the declination of the sines, the ratio of the impetus itself to the contrary tendency becomes smaller, a slower motion follows, and more of the acquired impetus is lost, until at last it has entirely disappeared, and the ball, gravity prevailing, descends again. Yet although, if position alone is considered, the impetus seems to be diminished in the same reciprocal degrees in which it was increased, and in as many moments of time, so that as much as was added at the final point of time, so much departs at the first, there is nevertheless still some little obstacle from the air being divided, and therefore the acquired impetus is somewhat further weakened than in proportion to the increase of the declination of the sines: but the more rapid the motion is, the more also the medium to be divided is compressed, and being made denser it offers more resistance than when rarefied; and if the medium is not capable of compression, at least in equal time more parts of the medium are cut apart than in a slower motion, and therefore the resistance of the medium is also multiple: whence it follows that the arc of ascent must always be somewhat smaller than the arc of descent, and since in turn the ball remains from a lower place and having descended before, a shorter arc of the second ascent is likewise completed, and so on, until, preserving that constancy of motion in always diminishing reciprocation, it finally comes to rest at the perpendicular. But, you say, hard things resist a body more, and hinder its motion more strongly, than soft things, which, while they allow themselves to be compressed, and give way for a little while, in some measure and in part support the motion: if therefore the acquired impetus is weakened in proportion to the hindrance, less impetus is taken away from the body which, falling from above, is received by straw laid beneath it, than if it were to strike against a rock; therefore it would be reflected more violently from mud than from a rock, contrary to experience. I admit that an ivory ball rebounds more sluggishly when it falls onto ground soaked with moisture than onto marble; yet it does not follow from this that, for the diminution of the acquired impetus, another mode must be posited than that from impediment: for when a falling ball has touched the upper surface of the body beneath it, it does not come to rest, but continues to move, either downward by compressing a soft body, or immediately upward by rebounding from a hard one. And thus it is received by a soft body,

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Liber secundus. CAPUT III. 151 cipitur, ut licèt hoc cedat, impediat tamen & remoretur mo- tum; ac proinde quò magis cedit subjectum corpus, eò diutiùs movetur globus cum ipso, vel intrà ipsum; atque interea plus impetûs perit: quid igitur mirum, si languidiùs postea resiliat, cum exigua impetûs portio reliqua sit? Quòd si duru[m] esset sub- jectum corpus, impetu nondum debilitato reflecteretur vali- diùs. Hinc fieri potest adeò molle esse subjectum corpus, ut dum illud penetrat decidens globus, tantum impetûs deper- dat, ut, quod reliquum fit, non satis sit ad vincendam insitam globo gravitatem, qui propterea neque resilire valeat. Quam- vis itaque corpus molle minùs obsistat quàm durum, diutiùs tamen resistit; & per aliquot momenta aliquoties diminutus impetus minore mensurâ, eò decrementi venire potest, ut ma- gis imminutus demum fuerit, quàm si unico momento magis obstitisset corpus durum. Cæterùm paribus momentis plus pe- rit impetûs ex allisione ad corpus durum, quàm ad molle, quip- pe quod magis opponitur motui. Porrò huic rei explicandæ similitudo aliqua peti posset ex luce, cui sanè si contingat per medium diaphanum quidem, sed densum, pergere, languidiùs multò reflectitur à speculo, in quod incurrit, si densioris me- dij longior fuerit tractus, quàm si brevior, perinde atque eò minùs reflectitur corpus, quò molliori magisque subsidenti cor- pori occurrit. sed quoniam quæ de luce dicenda essent, fortè ob- scuriora acciderent, ab hujusmodi similitudine prudes abstineo. Sed ex illud est in durorum corporum collisione observan- dum, quod aliqua particularum compressio aliquando contin- git sivè in alterutro, sivè in utróque, quæ se facultate elasticâ reslituentes motum reflexum juvant: id autem manifesto ex- perimento constat in pilâ ex gummi, ut vocant, Indico, quæ ad terram elisa frequentissimè subsultat; at ubi in corpus molle incidit, neque hujus neque illius partes violentam compressio- nem subeunt, quam sese restituentes excutere debeant. Sic & pilâ in sphæristerio ludentes satis nôrunt eam validiùs reflecti objecto recticulo, quàm ligneo batillo; intenti scilicet nervi ex contortis siccatisque animalium intestinis reticulum constituen- tes cùm pilæ ictum excipiunt, flectuntur quidem aliquantu- lum; sed illicò sibi pristinam rectitudinem reparantes pilam ex- cutiunt (id quod ligneo bastillo non contingit) novoque hoc impetu

Transcription: Translated (English)

Book II. CHAPTER III. 151 is received, so that although it yields, it nevertheless hinders and retards the motion; and therefore, the more the subject body yields, the longer the ball is moved with it, or within it; and meanwhile more impulse is lost: what wonder then, if it afterwards rebounds more feebly, when only a small portion of the impulse remains? But if the subject body were hard, it would be reflected more strongly while the impulse was not yet weakened. Hence it may happen that the subject body is so soft that, while the falling ball penetrates it, it loses so much of its impulse that what remains is not sufficient to overcome the gravity inherent in the ball, which therefore is unable to rebound. Although, then, a soft body offers less resistance than a hard one, it nevertheless resists for a longer time; and, after the impulse has been diminished several times by several moments by a lesser measure, it can come to such a decrease that it is at last more reduced than if a hard body had resisted more at a single moment. Moreover, with equal moments, more impulse is lost from collision with a hard body than with a soft one, since it opposes the motion more strongly. Furthermore, for explaining this matter some comparison might be drawn from light, which certainly, if it happens to pass through a transparent medium indeed, but a dense one, is reflected much more feebly from the mirror on which it falls, if the passage through the denser medium has been longer, than if shorter; just as a body is reflected the less, the softer and more yielding the body it meets. But since what ought to be said about light might perhaps turn out more obscure, I prudently refrain from such a comparison. But this must be observed in the collision of hard bodies: that some compression of the particles sometimes occurs, either in one or the other, or in both, which, restoring themselves by an elastic power, aid the reflected motion: and this is manifestly shown by experiment in the ball made of Indian rubber, as it is called, which, when struck to the ground, very often rebounds; but when it falls upon a soft body, neither the parts of this nor of that undergo a violent compression which, by restoring themselves, ought to throw it back. Thus those who play with a ball in the sphairisterium know well that it is reflected more strongly from a string-net than from a wooden paddle; for the stretched nerves made of the twisted and dried intestines of animals, when they receive the blow of the ball, bend somewhat; but immediately restoring their former straightness, they throw the ball back (which does not happen with a wooden paddle) and by this new impulse

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Mechanicorum impetu auctus reliquus pilæ impetus motum quoquè efficit majorem: quòd si in reticulo flaccidi, & remissi sint nervi, lan- guidè pila reflectitur. Ad quandam autem reflexionis speciem pertinere censenda est concussio, sive vibratio, aliquarum saltem corporis partium, ubi totum ex reliquo impetu resilire nequit: sic corpus ita at- tollens, ut summis pedibus innitaris, postmodum recidens in talos, eò validiorem partium concussionem percipies, quò ve- lociùs recides. Simile quid etiam in inanimis contingere ratio suadet, neque enim ita semper solida aut prorsus homogenea tota moles est, ut nullæ omninò partes concuti valeant: quin etiam allisi corporis partes, si non adeò tenaci vinculo inter se cohæreant, ex reliquo impetu aliæ aliò distractæ desiliunt. Hinc, docente naturâ, ex alto desilientes ubi terram pedi- bus attigerint, genua antrorsum inflectunt, quasi calcaneis in- sessuri, ne conceptus ex saltu impetus superiorem corporis par- tem deorsum validiùs urgens subjectas tibias, & genua ita pre- mat, ut inde divisio aliqua membrorum, aut ossium luxatio, aut nervorum seu tendinum nimia distensio dolorem gignat: hoc autem valet illa genuum inflexio ad extenuandum impetum, quod & flexili mollitiâ subsidens terra uliginosa, si quando la- pis in eam ex alto decidert. Sic Atlas Sinicus pag. 123. in XI. Provinciâ Fokion, ubi sermo est de flumine Min, quod vio- lento cursu per saxa volvitur, ait naves, quibus ibi navigatur, ex diverbio vocari Papyraceas, eo quòd tenuibus ac minimè re- sistentibus constant asseribus, imò ne clavis quidem compaginatis; sed vimine quodam lentissimo; unde tametsi in saxa impingat na- vis, sæpè tamen minimè rumpitur, quia vix resistit. Et pag. 127. de catadupis aquarum in flumine per quod ad Jenping naviga- tur loquens ait. Cum naves transeunt, ne cum aquâ decidentes fractionis incurrant periculum, scitè præmittunt nautæ aliquot stra- minis fasces, ad quos navis levius impingat, ac transeat. Iam verò ad impetum extrinsecùs impressum mentem ocu- losque intendentes non illum semper momento perire animad- vertimus, aut illicò, ac externus agitator cessat. Unde enim fit, ut concitato navigio, cùm vela nautæ con- traxerunt, aut remiges inhibuerunt, retineat tamen ipsa navis motum & cursum suum, intermisso ventorum incursu, pulsúve remorum?

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By the force of the mechanism, the remaining impetus of the ball also makes its motion greater: for if in the net the strings are slack and loose, the ball rebounds languidly. A certain kind of concussion, or vibration, of at least some parts of the body ought also to be regarded as belonging to a species of reflection, where the whole cannot rebound from the remaining impetus: thus, if you raise the body so that you stand on the tips of your feet, then, falling back upon your heels, you will perceive a stronger concussion of the parts the more quickly you fall. Reason suggests that something similar happens in lifeless things as well, for the whole mass is not always so solid or so completely homogeneous that no parts can be shaken at all; indeed, even the struck parts of a body, if they do not cohere to one another by so tenacious a bond, leap away, some drawn off in one direction and some in another, from the remaining impetus. Hence, as nature teaches, those who leap down from a height, when they have touched the ground with their feet, bend their knees forward, as if about to sit upon their heels, lest the impulse received from the jump, pressing the upper part of the body more strongly downward, should so press the lower legs and knees that some separation of the members, or dislocation of the bones, or excessive stretching of the nerves or tendons should produce pain. But that bending of the knees serves to diminish the impulse, just as moist ground, yielding with flexible softness, does when a stone has fallen into it from a height. Thus the Chinese Atlas, p. 123, in the Eleventh Province of Fokion, where mention is made of the river Min, which rolls over rocks with violent course, says that the boats used there are called, by a borrowed term, Papyraceous, because they are made of thin planks and are scarcely resistant, indeed not even joined together with nails, but with a very pliant sort of wickerwork; whence, although the boat strikes against rocks, it is nevertheless often not broken at all, because it offers scarcely any resistance. And on p. 127, speaking of the cataracts of waters in the river by which one sails to Jenping, he says: when the boats pass through, so that they do not incur the danger of breaking by falling with the water, the sailors cleverly send ahead several bundles of straw, against which the boat may strike more gently and pass on. Now then, when we turn our attention to externally impressed impetus with respect to the mind and the eyes, we observe that it does not always perish in an instant, or immediately when the external mover ceases. For whence comes it that a ship, once driven onward, when the sailors have furled the sails, or the rowers have ceased, nevertheless retains its motion and its course, even after the onrush of the winds or the strokes of the oars have stopped?

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Liber secundus. CAPUT III. 153 remorum? nisi quia navis, etiam nullo impellente, vi impressâ urgetur. Quid rhedam cursu procedente faciliùs quàm initiò promovet, equis licèt languidius connitentibus? curve onus aliquod ingens protrudentes, aut trahentes hoc maximè ca- vent, ne contentionem illam quies interrumpat, experientiâ satis edocti incitatum semel minori labore propelli, quàm com- moveri quiescens? nisi quia reliquus ex priore motu impetus adhuc perseverans posteriorem motum juvat. Hoc tamen tria hæc differunt, quòd onus, cessantibus iis, qui protrudebant, consistit illicò (nisi fortè volubilitatem habens, aut subjectis cylindris innixum, adhuc modicum quid volvi aut progredi pergat) rheda currentes equos subitâ funium abruptione dis- junctos sequitur ad passus aliquot non adeò multos pro viæ æquabilitate præcedentisque velocitatis ratione; navigium verò submissis antennis, remisque cessatione torpentibus aliquandiu, intervallo non sanè contemnendo, provehitur. Oneris scilicet motui, cui volubilitatem neque ars, neque natura dederit, im- pedimento est ipsa extremitas aspera subjectam planitiem sale- bris quandóque non carentem contingens, gravitasque ita va- lidè premens, ut major futurus esset partium tritus, quàm pro impetûs modo, qui reliquus esset, superari posset: Id quod cur- renti rhedæ idcircò non contingere planum est, quia licèt nihilo levior sit quàm onus protrusum, minùs tamen rotarum modioli leniter cum axibus confligentes motum retardant. At navis sponte suâ innatans, ventorum incursione, remorumve pulsu diutiùs acta, vix, aut fortè ne vix quidem, mole suâ re- luctatur, nisi quatenus diffindenda est aqua; nec sinè multo fa- cilitatis compendio, prior siquidem unda, quam prora impel- lens excitat, aliam ante se urget ad easdem partes: propterea impressus navi impetus modicum nactus impedimentum diù durat, illâmque promovet. Quare idem de impetu extrinsecùs assumpto dicendum est, quod de acquisito; nimirùm minui pro Ratione eorum, quæ instituto motui obsistunt, aut etiam pror- sùs perire. Præter ea autem quæ utrique motui tùm naturali, tùm vio- lento æquè opponuntur, (cujusmodi est medium dividendum, objecti corporis occursus, aut contingentis tritus atque con- flictus, retinaculum, quod certo limite motum definiat, & alia V

Transcription: Translated (English)

Second Book. CHAPTER III. 153 of the oars? except because a ship, even with no one propelling it, is driven onward by impressed force. What makes a coach, once it has begun to move, advance more easily than at the beginning, even though the horses exert themselves more languidly? Why, when pushing or pulling some huge load, do they especially take care that rest should not interrupt that strain, being sufficiently taught by experience that what has once been set in motion is moved on with less labor than it is set in motion from rest? Except because the remaining impetus from the earlier motion, still continuing, helps the later motion. Yet these three differ in this: that a load, when those who were pushing it cease, stops at once (unless perhaps, having some rolling power or resting on cylinders beneath it, it continues to roll or advance a little farther); a coach follows the running horses, after the sudden breaking of the traces has separated it from them, for some distance, though not so great a one, according to the smoothness of the road and the speed previously attained; but a ship, with the sails lowered and the oars falling idle, is carried on for some time, over an interval not to be despised at all. For in the motion of a load, to which neither art nor nature has given rolling power, the roughness itself of the ground below, touching a surface that is level but sometimes not free of loose stones, is an obstacle, and the weight presses so strongly that the wearing of the parts would be greater than could be overcome by the remaining force of the impulse. This clearly does not happen in the case of a running coach, because, although it is no lighter than the load that is pushed, the naves of the wheels, lightly rubbing against the axles, nevertheless retard the motion less. But a ship, floating of itself, once driven for a longer time by the rush of winds or the beating of oars, scarcely, or perhaps not even scarcely, resists its own mass, except insofar as the water must be parted; nor without great saving of ease, since the wave first raised by the prow as it drives forward presses another wave before it in the same direction. Therefore the impetus impressed on a ship, having found only slight impediment, lasts a long time and moves it onward. Wherefore the same must be said of impetus taken in from outside as of acquired impetus: namely, that it is diminished in proportion to the things that oppose the motion once begun, or even altogether destroyed. Besides these things, however, there are also obstacles equally opposed to either motion, both natural and violent, such as the medium to be divided, the collision of an opposing body, or the friction and impact of something touching it, a restraint that defines motion within fixed limits, and other

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Mechanicorum id genus) illa est externo impulsui peculiaris repugnantia. quæ ex inhærente corpori gravitate oritur, sivè illi innatus impetus, sive acquisitus modum statuat. Neque id simpliciter tantùm, sed comparatè considerandum est, quam scilicet in plagam impulsus motum dirigat, & quatenus gravitatis propensioni opponatur. Quemadmodum enim qui in pilâ aroma- ta pinsunt, nihil repugnantem, quin & impulsui obsecundan- tem, experiuntur pistilli gravitatem deprimentes; contrà verò attollentes fatigat eadem gravitas directò deorsum urgens; me- dium autem quiddam tenet in obsistendo, si motio transversa contingat; sicut experiri licet, si ex funiculo pendens idem pistillus à perpendiculo dimoveatur; minore enim conatu opus est: ita quò minùs in oppositam gravitati plagam dirigitur im- pulsus, eò etiam diutiùs perseverat minus habens impedimenti. Hinc est quod gravitas æquabiliter toto corpore fusa si aut ex centro suspendatur, aut coni apici insistat, levi negotio, ac sa- ris diù, in gyrum convertitur; innatum videlicet gravitatis impetum vis ipsa suspendens aut sustentans elidit; nihil verò im- pulsum remoratur præter aut funiculi suspendentis spiras paulò spissiores, aut tritum cum subjecto cono, aërisque dividendi resistentiam; quæ tamen si tollatur in corpore orbiculari circà centrum commoto, etiam longior fit conversio. Sic ferream sagittam palmarem crassiusculam instar acûs magneticæ in æquilibrio constitutam levissimo impulsu ac diutissimè in gy- rum agi observavi; vix enim acutissimum verticem, cui innite- batur, terebat, & aëris intrà eumdem gyrum circumducti mo- dica erat resistentia. Id autem multo luculentiùs apparet in verticillo, cujus axem perpolito alveolo insistentem extremo pollice ac indice leviter comprimens, ac paulò celeriùs vertens, eò diuturniori vertigine contorqueri videbis, quò pauciores minoresque offenderit in subjectâ tabulâ asperitates, ad quas al- lisus paululùm inclinetur, aut aliò reflectatur. Quòd si magnetis polo ritè armato chalybeum axiculum congruo verticulo instructum admoveris, ut planè à magnete suspendatur, tùm summis digitis opportunè axem terentibus vertiginem ei delicatè ac molliter conciliaveris, miraculi loco tibi erit tàm diuturna conversio; quippe cui non subjecti alveoli asperitates saltitare cogentes, non gravitas ipsa premens, tritum- que

Transcription: Translated (English)

Mechanics (that kind) is that peculiar resistance to external impulse, which arises from the gravity inherent in a body, whether that be an inborn or an acquired tendency. Nor is this to be con- sidered simply, but comparatively, namely how much the impulse directs the motion toward one side, and to what extent it op- poses the tendency of gravity. For just as those who pound spices in a mortar experience the pestle’s weight as offering no resistance, and even yielding to the impulse, while depressing it; but when raising it, that same weight, pressing directly down- ward, fatigues them; and something midway is found in resist- ance if a transverse motion occurs; as can be experienced if the same pestle, hanging from a cord, is moved away from the plumb line, for less effort is then needed: so the less the impulse is directed toward the side opposite gravity, the longer also it con- tinues, having less hindrance. Hence it is that gravity, evenly diffused through the whole body, if it be either suspended from the center or placed on the point of a cone, is turned in a circle with little trouble, and for a long time; that is to say, the force itself, suspending or supporting the innate impulse of gravity, overcomes it; but nothing delays the impulse except either the somewhat thicker coils of the suspending cord, or the friction with the cone beneath, and the resistance of the air being parted; and if these be removed in a circular body moved about its center, the rota- tion becomes even longer. Thus I have observed an iron dart, a span thick, set in balance like a magnetic needle, to be driven in a circle by the slightest impulse and for a very long time; for it scarcely rubbed against the sharp point on which it rested, and the resistance of the air carried around within the same circle was slight. This is seen much more clearly in a top, whose axis, resting in a polished groove, if gently pressed by the thumb and index finger at the end and spun a little faster, you will see it twist into a longer-lasting whirl the fewer and smaller the roughnesses it encounters on the supporting board, against which, when it strikes, it is slightly inclined or deflected elsewhere. But if you should properly fit a steel pivot to the pole of a magnet, and bring near it a steel pin suitably furnished with a little turning point, so that it hangs entirely from the magnet, then if, with the fingertips duly touching the axis, you delicately and gently impart a spin to it, it will seem a miracle to you to see such a long-lasting revolution; since neither the roughnesses of the supporting groove, which force it to skip, nor gravity itself pressing down, nor the friction

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Liber secundus. CAPUT III. 155 que augens, non suspendentis funiculi violenta contortio ob- sistunt, motumve aliquatenus impedientes impressum impe- tum imminuunt; sed magnetico radio suspensus intra se perpe- tuò volvitur lævissimum chalybem magnetis polo adhærentem lenissimè terens. Illud etiam in motu, qui ab extrinseco provenit, conside- randum est, quòd contingere potest duos adesse motores, qui corporis motum in diversas partes dirigant: quare alter alteri obsistit, & motus ex duplici directione compositus is est, qui non respondeat mensuræ duplicis illius impetus, si singuli in- tegrè accipiantur. Constat enim, si æquabili & æquali cona- tu urgeant corpus, moveri aut per diametrum Quadrati, si di- rectiones sint ad angulum rectum constitutæ; aut per Diago- nalem lineam Rhombi, si directiones obliquæ sint: si verò æquabiles quidem sint, sed inæquales conatus, per diametrum Rectanguli aut Rhomboidis moveri, pro ut ad rectum aut obli- quum angulum directiones sibi invicem respondent. Semper autem minor est motus quàm pro duorum illorum impulsuum ratione; diameter siquidem brevior est aggregato duorum adjacentium laterum. Quòd si æquabiles non sint impetus, vel saltem alter æquabilis sit, alter acceleratus aut retardatus, linea curva describitur; quæ pariter minor est duabus rectis, quæ vi singulorum impetuum describerentur; ab illis si qui- dem continetur. Hîc tamen advertendus animus est, & observare oportet æquabilem impulsum (si continuus sit, nec morulis inter- ruptus) esse non posse, nisi ab animali semper æqualiter conan- te efficiatur; quia gravium descensus naturaliter acceleratur; elasmata verò dum se restituunt, semper languidiùs singulis momentis conantur, si quidem virtus elastica consideretur: quamquam posteriore momento quod est reliquum prioris im- petûs, intensionem efficit additum posteriori licèt remisso. Vix igitur contingere potest motum unum à duplici impetu extrinsecùs impresso fieri per lineam rectam nisi corpus à du- plici motore æquabiliter urgeatur. Cum itaque impetus acquisitus, aut aliundè impressus, sit qualitas propter motum instituta, quæ non nisi in motu pro- ducitur, ita pariter nisi in motu, & cum motu non conserva- V 2

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Second Book. Chapter III. 155 which, increasing it, by the violent twisting of the hanging cord do not resist, or, by somewhat impeding the motion, diminish the impressed impetus; but, suspended by a magnetic ray, it perpetually turns within itself, very gently rubbing the softest steel adhering to the pole of the magnet. It must also be considered, in motion which arises from an external cause, that it can happen that there are two movers present, directing the motion of the body in different parts: wherefore one resists the other, and the motion composed from a double direction is such as does not correspond to the measure of that double impetus, if the individual impulses are taken in their entirety. For it is clear that, if they press upon a body with equal and similar effort, it will move either through the diameter of a square, if the directions are set at a right angle; or through the diagonal line of a rhombus, if the directions are oblique: but if indeed they are equal, yet unequal in effort, it will move through the diameter of a rectangle or rhomboid, according as the directions correspond to one another at a right or oblique angle. Yet the motion is always less than would be expected from the proportion of those two impulses; for the diameter is shorter than the sum of the two adjacent sides. But if the impulses are not equal, or at least if one is equal and the other accelerated or retarded, a curved line is described; which likewise is less than the two straight lines that would be described by the force of the individual impulses; if indeed it is contained by them. Here, however, the mind must take notice, and it ought to be observed, that an equal impulse (if it be continuous and not interrupted by pauses) cannot exist unless it is produced by an animal always exerting itself equally; because the descent of heavy bodies is naturally accelerated; but springs, while they restore themselves, always strive more languidly at each moment, if indeed the elastic force is considered: although in the later moment what remains of the prior impulse produces an intensity added to the later, though relaxed. Thus it can scarcely happen that one motion caused by a double impulse impressed from outside is produced in a straight line unless the body is urged equally by a double mover. Since therefore an acquired impetus, or one impressed from elsewhere, is a quality established for the sake of motion, which is produced only in motion, so likewise unless in motion, and together with motion, it is not preserved

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Mechanicorum tur. Quare si corpus eò deveniat, ut nullo prorsus pacto agitari queat, aut interiore motu cieri, quo momento impeditur motus, ne sit, eo momento impetus perit, cessante videlicet causâ effectivâ ab ejus conservatione eo ipso quod cessat finis, propter quem impetus est. Quod si impedimentum occurrat non prorsùs motum tollens (ut si globus in plano horizontali rotatus veniat ad planum inclinatum, per quod ex concepto impetu ascendat) tunc pro ratione impedimenti extenuatur impetus, donec tandem pereat. CAPUT IV. Quâ ratione vis movendi cum impedimentis comparetur. Motus omnis nec in oppositas, nec in diversas plagas, sed per certam lineam dirigitur; unico quippe in loco, non in pluribus, eodem temporis puncto esse potest corpus. Nihil igitur motui moram & impedimentum inferre potest, nisi directò aut obliquè illi secundùm eam lineam, per quam instituendus esset, antè, ponè, ad dextram, ad lævam, sursum, deorsum opponatur. Si enim duo corpora eâdem pergerent viâ, & maximâ velocitatis, aut tarditatis conspiratione consentirent, tunc neque posterius ab eo quod antè est, traheretur, neque prius à posteriore urgeretur, neque alterum alteri impedimento esset. Hinc manifestum est non posse impedimentum superari, quin ei vis aliqua inferatur. Rem porrò universam duas in partes tribuere possumus, ut duplex Resistentiæ genus statuatur; Formalem alteram, alteram Activam scholæ vocarent. Corpus enim, quod obstat, aut retinet, si motum prorsùs nullum conetur instituto aut destinato motui adversantem, resistit quidem, sed Formaliter; nihil scilicet efficit, quo repugnet, sed suo tantùm se tutatur in loco: Sin autem & contrà nitatur, aut retrahat, jam non obsistit solùm, ne loco per vim dimoveatur; sed etiam impetum in con- trariam

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Mechanics. Therefore, if a body reaches such a state that it can in no way at all be moved, or be set in motion by an internal impulse, at the very moment in which motion is impeded, at that moment the impetus perishes; namely, because the efficient cause, by its conservation, ceases at the very point where the end for the sake of which the impetus exists ceases. But if an impediment occurs that does not altogether abolish motion (as if a ball, rolled on a horizontal plane, should come to an inclined plane, by which, by the impetus already conceived, it ascends), then in proportion to the impediment the impetus is diminished, until at last it disappears. CAPUT IV. How the moving force is compared with impediments. All motion is directed neither to opposite nor to diverse directions, but along a certain line; for a body can be in one place only, not in several, at the same moment of time. Nothing, therefore, can bring delay or impediment to motion unless it be opposed to it directly or obliquely, according to that line along which it would have to proceed, before it, behind it, to the right, to the left, above, below. For if two bodies were to proceed by the same path, and agreed with the greatest harmony of speed or slowness, then neither would the latter be drawn along by that which is in front, nor would the former be pressed by the latter, nor would either be an impediment to the other. Hence it is clear that impediment cannot be overcome unless some force is brought against it. Moreover, we can divide the whole matter into two parts, so that a twofold kind of Resistance may be established: one Formal, the other Active, as the schoolmen would call them. For a body that stands in the way or holds back, if it attempts no motion altogether contrary to the intended or directed motion, does indeed resist, but formally; that is, it does nothing by which it may oppose, but merely protects itself in its own place. But if it also strives against it, or draws back, it now not only prevents being moved by force from its place; but also brings an impetus against the contrary

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Liber secundus. CAPUT IV. 157 trariam plagam directum efficit, cujus vi motum impedit, ac proptereà Activè resistit. Huic autem verbo, cùm Resistere di- cimus, subjecta notio est, in causâ esse ne motus fiat, aut sal- tem non eâ velocitate, quæ virtuti movendi non impeditæ cæ- teroqui responderet. Sic paries, in quem incurris, tibi resistit Formaliter, ne procedas, & aqua stagnans, cui collo tenus im- mergeris, progredienti resistit Formaliter, ne velociter, sicut intrà aërem movearis pro ratione impetûs, quo conaris progre- di: qui verò occurrens te repellit, ut si coneris contra ictum fluvij, non Formaliter tantùm, sed etiam Activè resistit; non solum enim obstat, quia ejus in locum succedere non potes, nisi eum loco dimoveas, sed etiam tibi adversum impetum im- primit, ut te loco extrudat. Cum itaque impedimenta motûs externo impetu submoven- da sint, virtus autem movendi certa sit ac definita, constat vi- res omnes, quæ in corpore promovendo, si nihil obstaret, exer- cerentur, duas in partes distrahi, ad movendum scilicet cor- pus, & ad tollenda impedimenta, Concipit igitur impetum, qui motum efficiat, & obstanti corpori impetum imprimit, ut loco cedat. Quid igitur mirum, si distractis viribus languidior sequatur motus? Quia verò quò majori velocitate corpus obstans propellendum est, aut trahendum, majori quoque im- petu impresso opus habet, palàm est majorem quoque in pro- pellente, aut secum rapiente, impetum requiri, ut majorem re- sistentiam vincens se ipsum pariter moveat. Hîc autem quid monuisse oporteat vim resistendi superan- dam esse à virtute movendi? quis enim ambigat, an, si pares illæ fuerint, nullus futurus sit motus? Quòd si impedimentum prorsùs immotum adversùs conantem perstat, nullum pariter recipit impetum; qui scilicet, etiam si priùs fuisset, motu ces- sante periret. Hinc in animali defatigatio membrorum oritur, quando prorsùs in irritum conatus cadit; impetus enim, quem concipit, ut æqualem motum imprimeret impedimento, si hoc superari posset, in animali ipso motum aliquem efficit, sed quia progredi vetatur ab ostante aut retinente impedimento, impe- tus ille non totius animalis motum ulteriùs promovet; sed mem- brorum partes alias comprimit, alias distendit, unde & dolor aliquis, & lassitudo provenit. At si corpus, cui motus debetur, V 3

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Book the Second. CHAPTER IV. 157 produces a motion directed toward the opposite side, by whose force it hinders motion, and thus resists actively. But when we say “to resist,” the notion involved in this word is that of being the cause that motion should not take place, or at least not with that velocity which would otherwise correspond to an unhindered power of moving. Thus a wall against which you run resists you formally, so that you do not go forward; and stagnant water, into which you immerse yourself up to the neck, resists a person advancing formally, so that he does not move forward as quickly as he would move through the air in proportion to the impetus with which he tries to advance. But that which, meeting you, repels you, as when you try to go against the current of a river, resists not only formally, but also actively; for it not only obstructs, because you cannot take its place unless you move it from that place, but it also impresses an opposite impetus upon you, so as to drive you away from the place. Since, then, the impediments to motion are to be removed by external force, but the power of moving is fixed and definite, it is clear that all the forces which would be exerted in moving the body, if nothing stood in the way, are divided in two directions: namely, to move the body and to remove the impediments. Therefore it conceives an impetus that may produce motion, and it impresses impetus upon the body that stands in the way, so that it yields its place. What wonder, then, if motion follows more sluggishly when the forces are divided? And since the faster a body that stands in the way must be pushed on or drawn along, the greater the impetus impressed upon it is also needed, it is plain that in the one who pushes or carries along with him a greater impetus is likewise required, in order that, overcoming a greater resistance, he may move himself as well. Here, however, what should be noted is that the resistance to be overcome must be surpassed by the power of moving; for who would doubt that, if those forces were equal, there would be no motion? But if the obstacle remains altogether unmoved against one who strives, it likewise receives no impetus; and that impetus, even if it had existed before, would perish when motion ceases. Hence in an animal the fatigue of the limbs arises, when the attempt falls altogether to no effect; for the impetus which it conceives, so that it might impress an equal motion upon the obstacle if this could be overcome, produces some motion in the animal itself, but because it is prevented from advancing by the obstacle standing before it or holding it back, that impetus does not further promote the motion of the whole animal; rather, it compresses some parts of the limbs and stretches others, whence some pain and weariness arise. But if the body to which the motion is due...

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158 Mechanicorum cùm inanimum sit, nequeat impetum, quemadmodum animantes, ex arbitrio temperare, & quia solidum est ac durum, nullam pati compressionem aut distentionem partium possit, sicut & corpus obstans aut retinens compressionem omnem aut distentionem respuit; tunc nullum concipit aut imprimit impetum præter innatam gravitationem, aut levitationem, cùm per vim in loco non debito detineatur. Ex hoc conjecturam capere licet de eo, quod contingit, quando virtute movendi resistentiam vincente impedimentum submovetur; impediri videlicet, ne producatur motus, juxta resistentiæ modum atque mensuram; quæ sicuti non quâlibet minimâ vi superari potest, ita majori cedit. Verùm quonam id pacto contingat, ut explicare conemur, illud observa, quòd si corpus idem quadruplo velociùs moveri debeat, ac moveretur priùs certâ impetûs mensurâ, utique quadruplo majorem impetum exigit, ut pro impetûs intensione aut remissione velocior aut tardior sequatur motus. At si corpus aliud movendum quadruplo gravius exhibeatur, in hoc impetus ille quadruplex subquadruplam efficiet intensionem, ac propterea etiam motum habebit tardiorem, si cætera sint paria, pro impetûs intensione. Si cætera, inquam, sint paria; sæpè enimaër, aut aqua plus velociori motui resistunt, quàm tardiori, & moles major efficit, ut non omninò velocitas intensioni impetûs respondeat. Hæc tamen nunc mente secernamus, perinde atque si nihil officerent motui. Quoniam igitur motus ab omni velocitatis aut tarditatis mensurâ sejungi nequit, finge corpus per vim movendum hujusmodi esse, ut spectatâ mole seu materiâ, ac specificâ gravitate, ad percurrendum spatium passuum 100 unius horæ quadrante, indigeret impetu, cujus intensio esset particularum 4 in singulis corporis movendi partibus: molem autem, exempli gratiâ, distinctam concipe in particulas 100 minimas. Quare spectatâ tùm extensione tùm intensione impetûs, necesse est illi à motore imprimi impetûs particulas 400. Quòd si corporis per vim movendi moles ac materia esset quadruplex alterius, si nimirum ratione materiæ extensionis particulas haberet 400, jam impetus idem subquadruplam efficeret intensionem, & singulæ impetûs particulæ singulis corporis particulis inessent; atque adeò

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158 Mechanics since it is lifeless, it cannot, like living things, moderate the impulse at will; and because it is solid and hard, it cannot suffer any compression or stretching of its parts, just as a body that resists or retains gives back any compression or stretching; then it receives or impresses no impulse beyond its natural gravitation, or levitation, when it is held by force in an improper place. From this one may infer what happens when, by the power of moving, the resistance having been overcome, the obstacle is removed: namely, that the motion is prevented from being produced, according to the mode and measure of the resistance; and this, just as it cannot be overcome by any whatsoever, however small, force, so it yields to a greater one. But to try to explain by what means this happens, observe that if the same body must be moved four times more quickly than it was previously moved by a certain measure of impulse, it certainly requires four times greater impulse, so that, according to the increase or diminution of the impulse, a swifter or slower motion follows. But if another body to be moved is presented, four times heavier, in this case that quadruple impulse will produce a less than quadruple increase, and therefore it will also have a slower motion, if the other things are equal, in proportion to the increase of the impulse. If, I say, the other things are equal; for often air or water resist a swifter motion more than a slower one, and greater bulk makes it so that the speed does not correspond wholly to the increase of the impulse. Let us now set these things aside, however, as though they impeded the motion not at all. Since, then, motion cannot be separated from every measure of speed or slowness, imagine a body to be moved by force of such a kind that, considering its bulk or material and specific gravity, in order to traverse a distance of 100 paces in a quarter of an hour it would need an impulse whose intensity was 4 particles in each part of the body to be moved; but let the bulk be conceived, for example, as divided into 100 smallest particles. Therefore, considering both the extension and the intensity of the impulse, it is necessary that 400 particles of impulse be impressed upon it by the mover. But if the bulk and material of the body to be moved by force were quadruple that of another, if, namely, by reason of the extension of the material it had 400 particles, then that same impulse would produce a less than quadruple increase in intensity, and the individual particles of the impulse would be found in the individual particles of the body; and thus

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Liber secundus. CAPUT IV. 159 adeò etiam hujus velocitas esset subquadrupla prioris velocita- tis: partamen utrobiqve esset, illud quidem velociùs, hoc tar- diùs movendi difficultas, cum in utroque particulas 400 impe- tûs produci oporteret; utriusque enim impetûs extensiones & intensiones essent Reciprocè in eadem Ratione. In corpore itaque, ex quo motus originem ducit, tanta vis movendi inesse debet, ut & corpori impedienti, quod submovetur, congruen- tem motui impetum imprimat, hoc est particulas 400, & ipsum se pariter promoveat: nihil enim accepto extrinsecùs impetu agitatur à motore prorsùs immoto, ut eunti per singula patebit. Iam verò quoniam idem corpus modò remissiùs, modò con- citatùs moveri pro impetûs intensione videmus, probabilis conjectura est in iis, quæ non suo arbitrio, sed naturæ reguntur imperio, totum impetum produci, qui virtuti efficiendi respon- det: hæc autem in impedimento, cujus resistentia vincitur, impetum eâ intensionis mensurâ imprimit, quæ illi motûs ve- locitatem conciliet ipsius corporis moventis velocitati con- gruentem, adeò ut movendi facultas totas suas vires exerat partim impetum imprimens submovendo impedimento, partim motum efficiens in ipso corpore: ex quo fit quod eò remissiorem motum in se motor efficiat, quò major secundùm intensionem impetus impeditur ab impedimento. Sic plumbeus globus bili- bri, si, funiculo excavatæ volubilis orbiculi curvaturæ inserto, connectatur cum globulo subduplæ gravitatis, non eâ veloci- tate descendit, quâ descenderet sibi relictus absque ullâ appen- dice; velociùs tamen movetur, quàm si esset globuli adjuncti tantùm sesquialter; quia scilicet ut ad æqualem velocitatem temperentur motus tùm impedimenti sursum, tùm corporis mo- ventis deorsum, minor intensivè impetus impediendus est à glo- bulo subduplo quàm à subsesqualtero; ac propterea major est secundùm intensionem reliquus impetus motum efficiens con- citiatiorem. Quòd autem à globo descendente imprimatur impetus glo- bulo, quem sursum trahit, hinc constat, quod si globulus ille non sit admodum gravis, tùm demum subsilit, ubi globus ve- lociter descendens subjectum planum attigerit: quid enim il- lum subsilire cogeret quiescente jam globo, à quo trahebatur, nisi adhuc aliquid impressi impetûs remaneret? At quòd im- pressus

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Second Book. CHAPTER IV. 159 so too its speed would be four times less than the former speed: yet in both cases there would be, the one indeed a quicker, the other a slower difficulty of moving, since in each there ought to be produced 400 parts of impulse; for the extensions and intensities of each impulse would be reciprocally in the same proportion. In the body, therefore, from which motion takes its origin, there must be so great a force of moving that it may both imprint upon the body being impeded, and being displaced, an impulse corresponding to the motion, that is, 400 parts, and likewise move itself forward: for nothing is moved by a mover altogether at rest, receiving impulse from outside, as will be evident to one proceeding through each point. But now since we see the same body sometimes moved more languidly, sometimes more briskly, according to the intensity of the impulse, it is a probable conjecture that in those things which are governed not by their own choice, but by the command of nature, the whole impulse is produced which corresponds to the power of effecting; and this, indeed, in the impediment, whose resistance is overcome, impresses an impulse by that measure of intensity which may secure for that motion a speed corresponding to the speed of the moving body itself, so that the faculty of moving exerts all its strength, partly by imprinting impulse while the impediment is displaced, partly by effecting motion in the body itself: from which it follows that the mover brings about in itself a motion the more languid, the greater the impulse, according to intensity, is hindered by the impediment. Thus a leaden sphere of two pounds, if, with a cord inserted into the curvature of a hollowed rolling wheel, it be connected with a little sphere of half its weight, does not descend with the speed with which it would descend if left to itself without any addition; yet it moves more quickly than if it were only a sphere of one and a half times its weight; because, namely, in order that the motions of both the upward impediment and the downward moving body may be adjusted to an equal speed, a smaller impulse, in respect of intensity, must be impeded by the half-weighted sphere than by the one-and-a-half-weighted sphere; and therefore the remaining impulse, according to intensity, which effects the motion, is greater for the quicker-moving body. But that an impulse is impressed by the descending sphere upon the sphere which it draws upward is evident from this: if that little sphere is not very heavy, then it springs up only when the swiftly descending sphere has reached the plane beneath it. For what would compel it to spring up, when the sphere by which it was being drawn is now at rest, unless something of the impressed impulse still remained? But that impressed

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Mechanicorum 160 pressus hîc impetus non ab ipso motore, sed ab impetu, quem ille concepit, proximè efficiatur, hinc sibi suadent plures, quia ex alterâ parte impetum ab impetu produci posse manifestum videtur ex percussionibus projectorum, ut cùm globus pro- jectus in quiescentem globum impactus illum trudit; ex alterâ causam proximam effectui homogeneam congruenter naturæ statuimus; sic enim & calorem in nobis à calore potiùs quàm à substantiâ ignis proximè produci existimamus. Sed quid de percussionum impetu dicendum sit, suo loco constabit inferiùs. Motoris demùm velocitatem intensioni impetûs concepti non respondere experimur, cum valdè conantes ut onus raptemus; parùm progredimur; at si funis ex improviso abrumpatur, illicò corruimus, impetu scilicet concepto motum validiùs efficiente, ubi desierit impetum oneri, quod raptabatur, imprimere. Hinc sit quòd, si ea fuerit corporum dispositio, ut impedi- mentum tardè submovendum sit, ac proinde remissiore impetu opus habeat, qui sibi imprimatur; corpus verò, cui motus omnis tribuitur, non æquali tarditate cum impedimento ferri necesse sit, sed velociùs præ illo moveri possit, hoc sanè eò mi- nùs habet resistentiæ, quò minorem in intentione impetûs mensuram impedimento eidem imprimere debet, ut illud submoveatur. Contrà verò si ita fuerint disposita, ut impedimentum velociùs præ ipso motore moveri oporteat, multò magis resistit, quàm si pariter moverentur, plus enim impetûs imprimendum est, ut motus consequatur. Hactenùs resistentiam potissimùm Formalem, impedimento nihil in adversum conante, contemplati sumus; jam ad Acti- vam transeamus, cum scilicet duo corpora invicem aut omni- nò, aut ex parte repugnant, quia motum in diversas aut oppo- sitas plagas directum moliuntur. In medio vase aquâ pleno sta- tuatur lignea tabella crassiuscula, eique lapis imponatur: dum illa conatur ascendere, hic descendere, se invicem urgent; sed cum se vicissim permeare nequeant, si paribus quidem viribus confligant, sine motu consistunt; sin autem imparibus, aut ambo ascendunt, aut ambo descendunt, pro ut sive tabellæ le- vitas, sive lapidis gravitas oppositam vicerit. Quod si lapis ta- bellæ non impositus, sed suppositus, arctè tamen connexus fuerit,

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Mechanics 160 This impulse, pressed here, is made to arise not from the motor itself, but from the impulse which it has conceived, for this reason many are persuaded, because on the other hand it seems evident that impulse can be produced by impulse, from the blows of projectiles, as when a projected ball striking a resting ball pushes it along; on the other hand we suitably assign the proximate cause to the effect as homogeneous with nature; for thus we think that heat in us is produced more by heat than by the substance of fire. But what is to be said about the impulse of impacts will become clear in its proper place below. We also experience that the speed of the motor does not correspond to the intensity of the conceived impulse, when, striving greatly to seize a load, we make little progress; but if the rope should suddenly snap, we immediately fall down, namely because the conceived impulse produces motion more strongly when it has ceased to impress impulse upon the load that was being dragged. Hence it follows that, if the arrangement of bodies is such that the obstacle is to be removed slowly, and therefore requires a more moderate impulse to be impressed on it; but the body to which all motion is imparted need not be carried with the obstacle at an equal slowness, but may move faster than it, then indeed it has so much less resistance as it must impress a smaller measure of impulse upon the obstacle, so that it may be removed. On the contrary, if they are so arranged that the obstacle must move faster than the motor itself, it resists much more than if they were moving equally, for more impulse must be impressed in order that motion may follow. Thus far we have considered chiefly formal resistance, with the obstacle making no effort in opposition; now let us pass to active resistance, when two bodies either altogether or in part resist one another, because they endeavor to direct motion toward diverse or opposite directions. In the middle of a vessel filled with water let a somewhat thick wooden board be placed, and upon it let a stone be set: while the board attempts to rise, the stone to sink, they press one another; but since they cannot pass through one another, if indeed they contend with equal forces, they remain at rest without motion; but if unequal, either both rise or both sink, according as either the lightness of the board or the weight of the stone has overcome the opposite force. But if the stone were not placed upon the board, but beneath it, yet closely connected with it,

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Liber secundus. CAPUT IV. 161 fuerit, adhuc contrarios motus conantur, non se tamen invicem urgent, sed vicissim retrahunt, quandiu vinculum non revellatur, aut rumpatur. Hîc verò subdubitet quispiam, utrum corpora, quæ contrario nisu reluctantur, sibi vicissim impetum imprimant, nec ne, aut æqualem, si pares fuerint vires, aut, si impares, inæqualem: Quando enim ob virium æqualitatem utrumque corpus consistit, eodem pacto quies sequitur, si unumquodque suam gravitationem aut levitationem servans nihil alteri imprimat, ac si lignea tabella levitans partem impetûs sursum directi conferat imposito lapidi, à quo gravitante vicissim recipiat tantumdem impetûs deorsum directi; ex quo fiat, ut lapis habens concepti ac innati impetûs deorsum directi vires æquales viribus impetûs sursum directi consistat, idemque in ligneâ tabellâ contingat. Cùm verò inæquales fuerint vires, id quod validius est, eodem modo superat, sive nihil contrarij impetûs ab infirmiore opposito recipiat, sed minorem motum vi sui impetûs producat pro ratione virium, quibus superat; sivè partem impetûs contrarij recipiat, quæ proprij impetûs vires attenuet. Quotidianum est hujus æqualitatis aut inæqualitatis experimentum in iis, quæ innatant humori; hæc enim humori imposita, quia in aëre gravitant, descendunt; pars verò immersa levitat in humore; prægravata tamen à reliquâ parte extante deorsum adhuc urgetur, donec inter partem immersam & extantem fiat æquilibrium, & tantumdem pars immersa levitet in humore, ac extans gravitat in aëre. Sic massa plumbea argento vivo imposita descendit, donec molis plumbeæ pars 2/13 extet; est enim specifica plumbi gravitas ad specificam mercurij gravitatem ut 1 1 ad 1 3. levitat itaque plumbum in mercurio ut 2, gravitat in aëre ut 1 1; igitur plumbeæ massæ partes 1 1 levitantes singulæ ut 2 parem habent conatum sursum, ac partes 2 gravitantes singulæ ut 1 1 conantur deorsum. Quòd si ita depimeretur plumbum, ut ejus partes 1 2 immergerentur, & una extaret; jam unica pars gravitans ut 1 1 vinceretur à partibus 1 2 levitantibus singulis ut 2, ac propterea adhuc pars una emergeret: quemadmodum si quatuor partes extarent, & novem immergerentur, harum levitas 1 8 ab illarum gravitate 44. vinceretur, ideóque adhuc duæ immergerentur. X

Transcription: Translated (English)

Book Second. CHAPTER IV. 161 if it has been, still opposite motions are attempted, yet they do not press against one another, but alternately draw back, so long as the bond is not torn apart or broken. Here indeed someone may doubt whether bodies, which resist with contrary effort, do in turn impress an impulse upon one another or not; whether an equal one, if the forces be equal, or, if unequal, an unequal one. For when, by reason of the equality of forces, each body comes to rest, in the same way rest follows if each, preserving its own gravitation or levitation, impresses nothing on the other, as if a floating wooden board should impart part of an upward-directed impulse to a stone laid upon it, from which, as it gravitating in turn, it receives just as much downward-directed impulse; from which it comes about that the stone, having the force of an acquired and innate downward-directed impulse, comes to rest equal to the force of the upward-directed impulse, and the same happens in the wooden board. But when the forces are unequal, that which is the stronger prevails in the same manner, whether it receives no contrary impulse at all from the weaker opposite, but produces a smaller motion by the force of its own impulse in proportion to the forces by which it prevails; or whether it receives part of the contrary impulse, which weakens the force of its own impulse. The daily experiment of this equality or inequality is seen in those things which float upon a liquid; for these, being placed upon the liquid, because they gravitate in the air, sink; but the part immersed levitates in the liquid; yet being overweighed by the remaining part projecting above, it is still driven downward, until between the immersed and the projecting part an equilibrium is made, and the immersed part levitates in the liquid in the same amount as the projecting part gravitatess in the air. Thus a leaden mass placed upon quicksilver sinks until 2/13 of the leaden body project above; for the specific gravity of lead is to the specific gravity of mercury as 11 to 13. Thus lead levitates in mercury by 2, and weighs in the air by 11; therefore the parts of the leaden mass, 11 levitating each one as 2, have an equal tendency upward, while the parts, 11, gravitating each one as 11, tend downward. But if the lead were so divided that its 12 parts were immersed, and one projected above, then the single gravitating part, as 11, would be overcome by the 12 parts levitating, each as 2, and therefore one part would still emerge: just as if four parts projected above, and nine were immersed, their levity 18 would be overcome by the gravity of those 44, and therefore two would still be immersed. X

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Mechanicorum Iam si dixeris à partis immersæ levitantis momentis 18 impedi- diri momenta 18 partis extantis gravitantis, adeò ut supersint tantùm vires juxtà excessum gravitatis, scilicet momentorum 26, juxta quem excessum impetum imprimat parti immersæ, ut deprimatur, tunc autem cum paria fuerint levitatis atque gravitatis momenta, jam non invicem agere, sed se vicissim impedi- dire, probabilior fortasse videatur alicui philosophandi ratio hîc, ubi directè sibi invicem adversantur directiones; alteruter enim aut neuter impetus movet oppositum corpus. Verùm quoniam ubi lineæ directionum motûs non sunt in directum positæ; sed inclinationem habent, motus mixtus, qui sequitur, ex utroque impetu unum motum temperari indicat, in eam feror sententiam, ut existimem duo corpora obliquè sibi invicem repugnantia vicissim imprimere, & recipere impetum in diversas plagas directum pro modo virtutis uniuscujusque, adeò ut si paria sint momenta, medius planè inter utramque directionem sequatur motus, si disparia, sequatur pro modo excessûs. Fieri autem hanc mutuam impetûs communicationem hinc apparet, quòd si duo corpora, quorum virtus movendi ut A B & A C, in loco, ubi A, constituta moveri cœperint, alterum quidem, quod ad dexteram est, cum directione A B, alterum verò, quod ad sinistram, cum directione A C, ita se impediunt, ut quod ad lævam est, urgeat reliquum, ne per rectam A B procedat; hoc verò quod ad dextera est, illud impediat, ne per rectam A C incedat; sed propellat ita, ut ambo habeant directionem mixtam A D. Hæc autem linea A D cum major sit singulis lateribus A B, A C in rectangulo, aut rhomboide, ut quadrato, aut rhombo, cavè nè putes singulis corporibus supra proprium impetûs modum factam esse aliquam ab externo impetu virium accessionem: quî enim fieri possit, ut corpus nullo repugnante possit certo tempore percurrere lineam A B, diminutis verò impetûs viribus ex resistentiâ, pari tempore longiorem lineam A D percurrat? An quia recipiat à corpore repugnante impetum, cujus accessione augeatur proprius impetus, qui reliquus est? At si propter virium æqualitatem percur- ran

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Mechanics Now if you should say that the moment of the immersed part, by its levity, 18, is impeded by the moment 18 of the part protruding, by its gravity, so that there remain only the forces according to the excess of gravity, namely the moments 26, according to which excess it imparts an impetus to the immersed part, so that it is depressed, then, when the moments of levity and gravity are equal, they no longer act on one another, but rather mutually impede one another, perhaps a more probable way of philosophizing may seem to some here, where the directions are directly opposed to each other; for either one or neither impetus moves the opposing body. But since where the lines of the directions of motion are not placed in a straight line, but have an inclination, the mixed motion that follows indicates that one motion is tempered from both impulses, I am led to the view that I think two bodies, resisting one another obliquely, alternately impart and receive impetus directed into different quarters, according to the measure of the strength of each, so that if the moments are equal, the motion follows exactly midway between the two directions; if unequal, it follows according to the measure of the excess. That this mutual communication of impetus occurs is shown by this, that if two bodies, whose power of moving is as A B and A C, in a place where A is located, should begin to move, the one indeed, which is on the right, with direction A B, the other however, which is on the left, with direction A C, so hinder one another that what is on the left urges the other, lest it proceed along the straight line A B; but this one, which is on the right, hinders that one, lest it move along the straight line A C; but it drives it on so that both have a mixed direction A D. Now this line A D, since it is greater than each side A B, A C in the rectangle, or rhomboid, as in the square or rhombus, take care not to think that some addition of forces has been made to each body above its own mode of impetus by an external impetus: how indeed could it happen that a body, with no resistance opposing it, could in a certain time traverse the line A B, but with the forces of impetus diminished by resistance, in the same time traverse the longer line A D? Or is it because it receives an impetus from the body resisting, by whose addition its own impetus, which remains, is increased? But if, because of the equality of the forces, it should traver...

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Liber secundus. CAPUT IV. 163 rant Quadrati diametrum, utique tantumdem alterum ab alte- ro recipit impetûs, quantum tribuit: igitur non est major vis impetûs, quàm si nihil repugnaret: ex quo fit neque motum ve- lociorem esse posse, ut pari tempore diametrum percurrant, quo singula describerent latus Quadrati. Non igitur ex illâ mutuâ impetûs in diversâ directi commu- nicatione fit in singulis corporibus impetûs intensio major (si propriè loquendum sit, habent enim impetus illi, conceptus scilicet, & impressus, directionem diversam) quàm ferat pro- pria singulorum virtus: id autem potissimùm constat, quando singuloru[m] directiones valdè obtusum angulu[m] constituunt; cor- pora enim in motu breviorem Rhombi aut Rhomboidis diametru[m] describunt, quæ linea aliquando minor est singulis lateribus. Finge itaque corpus, quod percurreret A B, nullo impedi- mento prohiberi, quin moveatur eâdem velocitate per A D; utique solùm æquale spatium A I decurreret, impediret tamen, ne aliud corpus habens directionem A C, illique perpetuò adhærens, decurreret juxta suam directionem spatium æquale ipsi A C; sed tantùm E I, hoc est Sinum anguli B A D loco Tangentis ejusdem anguli, posito Radio A I. Finge iterum alterum corpus habens directionem A C eâ- dem velocitate moveri per A D; utique non nisi spatium A F, ipsi A C æquale, motu dimetiretur, prohiberetque, ne reli- quum corpus habens directionem A B, illique perpetuò adhæ- rens, progrederetur nisi in F, hoc est spatio æquali ipsi B D; sed versùs B non procederet nisi juxta mensuram A G mino- rem ipsâ A C. Atqui utrumque suam habet directionem, & non per A D, seque vicissim impediunt; igitur dum simul mo- ventur, neque subsistunt in F, neque veniunt in I; sed medio loco consistunt, puta in O. Dixeris fortasse A O æqualem ipsi A E ita, ut sit sicut D B ad B A, ita I E ad E A, hoc est ad A O, aut A O esse medio loco proportionalem inter A F & A I, hoc est inter A C & A B mensuras virium impetûs singulorum corporum. Hoc tamen secundo loco propositum non facilè admiserim, quia ubi æqua- les sunt virtutes movendi, medio loco proportionalis est æqua- lis singulis extremis, ac propterea utrumque corpus impeditum æque velociter moveretur, ac non impeditum. Primum verò, X 3

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Liber secundus. CHAPTER IV. 163 …they were drawing the diameter of the square, it certainly receives from the other just as much of the impulse as it gives; therefore the force of the impulse is no greater than if nothing resisted: from which it follows that the motion cannot be quicker, so that they would traverse the diameter in the same time in which each would describe the side of the square. Therefore it is not from that mutual communication of impulses directed differently that there arises in the individual bodies a greater intensity of impulse (if one must speak properly; for those impulses, namely the conceived and impressed ones, have a different direction) than that which the proper strength of each can bear: and this is most clearly seen when the directions of the individual bodies form a very obtuse angle; for bodies in motion describe the shorter diameter of a rhombus or rhomboid, a line which is sometimes less than the individual sides. Imagine, then, a body which would traverse A B, not being prevented by any impediment from moving with the same velocity through A D; certainly it would traverse only the equal space A I, yet it would hinder another body having the direction A C, and continually adhering to it, from traversing along its own direction a space equal to A C itself; but only E I, that is, the sine of angle B A D in place of the tangent of the same angle, the radius A I being taken. Imagine again another body having the direction A C to move with the same velocity through A D; certainly it would not measure off in motion any space except A F, equal to A C itself, and it would prevent the remaining body having the direction A B, and continually adhering to it, from advancing except to F, that is, to a space equal to B D itself; but toward B it would advance only according to the smaller measure A G, rather than A C itself. Yet each has its own direction, and not through A D, and they hinder one another reciprocally; therefore while they move together, they come to rest neither at F nor at I, but settle in the middle place, namely in O. Perhaps you would say that A O is equal to A E in such a way that, as D B is to B A, so I E is to E A, that is, to A O, or that A O is the mean proportional between A F and A I, that is, between the measures of the impulsive forces A C and A B of the individual bodies. Yet I should not easily admit the latter proposition, because where the powers of moving are equal, the mean proportional is equal to each of the extremes, and therefore each body, though impeded, would move just as quickly as when not impeded. The first, however,

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Mechanicorum 164 quod scilicet AO æqualis sit ipsi AE, gratis asseritur; neque enim potior ulla apparet ratio, cur ad instituendam analogiam assumatur potiùs IE, quàm quælibet alia minor linea cadens inter G & E. Ego autem libentiùs profiteor me nescire, quâ Ratione analogia hæc instituatur, quàm aliquid certi divinando statuere. Verùm quamvis non utrumque corpus velociùs moveatur quàm pro suâ virtute, alterum tamen quod urgetur, seu rapitur à validiori, potest, factâ impetûs accessione, plus spatij percurrere, quàm pro suis viribus: impeditur siquidem motus non absolutè, sed juxtà eam directionem. Hinc fit corpus habens directionem & velocitatem AC minorem velocitate AB promoveri ultrà punctum F in linea mixti motûs AD. At inquis: an si nautæ remis incumbant, velisque obliquis ventum excipiant, tardior erit motus, quàm si navis vel à solis remigibus, vel à solo vento impelleretur: contrarium sanè videtur experientia evincere. Verùm si rem attentiùs consideres, aliam planè esse rationem deprehendes, cum duo corpora se moventia vicissim se impediunt, aliam cùm unum à duplici extrinseco impetu in diversa directo impellitur: de illis hactenùs sermo fuit, neque ulla ratio suadere potest velocius à tardiore incitari, quamquam tardius à velociore urgeatur, ut dictum est. At si unum corpus à duobus æqualis aut inæqualis virtutis impetum recipiat, utique magis intensus, vel si intentionem propriè dictam neges, certè major est impetus, quàm si ab alterutro tantùm reciperet impetum: quare nil mirum, si ea motûs velocitas consequatur, quæ utrumque impetum singillatim sumptum vincat, quamvis utroque simul sumpto minor sit, quia habent directiones oppositas, ut alibi explicabitur. Hinc est navim velociùs agi velis remisque, quàm si aut solâ ventorum vi, aut solâ remigum ope propelleretur, & cymbam, dum secundo flumine rapitur, simulque remis ad alteram ripam impellitur, velociùs moveri, quàm aut in stagno eâdem remigum operâ, aut à flumine cessantibus remis ageretur. Quemadmodum enim neque ventus remos impellit, neque ab his ventus impellitur, ita neque se vicissim immediatè impediunt, aut sibi mutuò repugnant; atque adeò non est hîc eadem philosophandi ratio, ac cum duo corpora sibi invicem immediatè resistunt, &

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Mechanics 164 that, namely, AO is equal to AE, is asserted without proof; for no better reason appears why, for establishing the analogy, IE should rather be assumed than any other smaller line falling between G and E. For my part, I would much rather confess that I do not know by what reason this analogy is established than to determine something certain by guessing. But although neither body moves faster than is due to its own force, nevertheless the one that is pressed or drawn by the stronger may, with the addition of momentum, traverse more space than its strength would allow. For motion is hindered not absolutely, but according to that direction. Hence it comes about that a body having the direction and velocity AC, less than the velocity AB, is carried beyond the point F in the line of mixed motion AD. But you ask: if sailors bend to the oars and catch the wind with oblique sails, will the motion be slower than if the ship were driven either by oarsmen alone or by the wind alone? Experience surely seems to prove the contrary. But if you consider the matter more carefully, you will discover that the case is entirely different when two moving bodies mutually hinder one another from when one is driven by a double external impulse directed in different ways. Of the former we have spoken thus far, nor can any reasoning suggest that the slower is excited by the faster, though it is urged more slowly by the faster, as has been said. But if one body receives an impulse from two forces of equal or unequal power, then certainly the more intense one, or, if you deny intensity in the strict sense, at least the greater impulse, is greater than if it received an impulse from either one alone. Hence there is nothing surprising if a speed of motion results that surpasses each impulse taken separately, although it is less than both taken together, because their directions are opposite, as will be explained elsewhere. Hence it is that a ship is driven faster by sails and oars than if it were propelled either by the sole force of the winds or by the sole aid of the rowers, and that a boat, while being carried by a favorable current and at the same time propelled by oars toward the opposite bank, moves faster than it would either in a pond with the same exertion of the rowers, or from the river if the oars were ceased. For just as the wind does not drive the oars, nor is the wind driven by them, so they do not immediately hinder one another, nor do they mutually oppose each other; and so the manner of philosophizing here is not the same as when two bodies immediately resist one another,

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Liber secundus. CAPUT IV. 165 & alterum alterius vires extenuat impediens, ne juxtà propriæ virtutis mensuram motum concipiat. Ex his quæ hactenùs dicta sunt, illud satis constare videtur, quòd animal eatenùs in motu difficultatem ac resistentiam per- cipit, quatenùs multum impetûs concipere debet, ex quo mus- culorum contentio oritur, neque tamen ea sequitur motûs ve- locitas, quæ tanto impetui responderet, dum submovendo im- pedimento maximam virium partem impendit impetum impri- mens: unde fit plurimum influentis spiritûs animalis absumi in tàm diuturnâ, vel tàm validâ musculorum contentione, ac proinde lassitudinem sequi, atque aliquando etiam contento- rum musculorum dolorem, cum id non contingat sine aliquâ partium compressione aut distentione. Quò igitur velociùs moveri potest animal pro ratione concepti impetûs, eò mino- rem percipit in submovendo impedimento difficultatem; & quidem maximè si alternâ contentionis ac remissionis muscu- lorum vicissitudine labor mitescat. Curiosiùs autem inquirenti, quam Rationem habeat motoris impetus ad impetum corpori, quod movetur, quatenus move- tur, impressum, ut aliquatenus satisfaciam, assero ut minimum duplam esse, non quidem intensivè, aut extensivè; sed enti- tativè. Quatenùs, inquam, movetur, hoc est quatenus vinci- tur ejus resistentia: cæterùm potentia movens in se producit, & in mobili æqualem impetum; sed quemadmodum ubi calor fri- gori permiscetur illud vincens, non percipitur nisi quatenus excedit vim frigoris, ita impetus oneri impressus eatenus mo- vet, quatenùs ejusdem resistentiam superat: Hunc autem ex- cessum subduplum impetûs motoris satis probabili conjecturâ affirmo. Illud enim hoc mihi suadet, quòd motoris virtutem metitur excessus impetûs, quem ille habet suprà impedimenti resistentiam: resistentiæ autem modus, ut sæpiùs dictum est, ex velocitate motûs, quæ concilianda est gravitati corporis sub- movendi, desumitur; hoc enim ideò resistit partibus ex gr.100 impetûs, quia si solùm fuerint 100 partes impetûs, fieri non po- test ut moveatur tantâ velocitate, sed pluribus impetûs parti- bus indiget: excessus igitur virtutis motoris æqualis est ut mi- nimum resistentiæ mobilis; atque adeò tota virtus motoris, hoc est impetus ab eo conceptus, æquivalet tùm resistentiæ mobi- lis

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Book Two. CHAPTER IV. 165 and one weakens the force of the other by impeding it, lest it conceive motion beyond the measure of its own power. From what has so far been said, it seems sufficiently clear that an animal perceives difficulty and resistance in motion only insofar as it must conceive a great impetus, from which the strain of the muscles arises; nor does there follow the velocity of motion that would correspond to so great an impetus, while in removing the impediment it spends the greatest part of its strength in producing the impetus. Whence it comes that a great part of the animal spirit flowing in is consumed in so long-lasting, or so strong, a strain of the muscles, and consequently fatigue follows, and sometimes even pain in the strained muscles, since this does not happen without some compression or stretching of the parts. Therefore, the more quickly an animal can move in proportion to the impetus conceived, the less difficulty does it perceive in removing the impediment; and indeed especially if the labor is moderated by the alternate succession of muscle tension and relaxation. But to satisfy more exactly the inquirer, I assert, as to the ratio which the impetus of the mover bears to the impetus impressed upon the body that is moved, insofar as it is moved, that it is at least double—not indeed intensively or extensively, but entity-wise. Insofar, I say, as it is moved, that is, insofar as its resistance is overcome; for the moving power produces in itself, and in the movable body, an equal impetus; but just as where heat is mixed with cold, that which prevails is not perceived except insofar as it exceeds the force of the cold, so the impetus impressed on the load moves only insofar as it surpasses its resistance: and I affirm this excess to be, by a sufficiently probable conjecture, one-half of the mover’s impetus. For this consideration persuades me that the power of the mover is measured by the excess of impetus which it has over the resistance of the impediment; but the measure of the resistance, as has often been said, is taken from the velocity of the motion, which must be made to accord with the weight of the body to be moved. For this body resists for this reason, with, for example, 100 parts of impetus, because if there were only 100 parts of impetus, it could not be moved with such velocity, but it needs more parts of impetus. Therefore the excess of the moving power is at least equal to the resistance of the movable body; and thus the whole moving power, that is, the impetus conceived by it, is equivalent both to the resistance of the movable body

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166 Mechanicorum lis juxta mensuram requisitam ad motum, qui sequitur, tùm principio motûs ejusdem mobilis: atqui motus hic æqualis est motui, cui illud resistit, totus igitur impetus motoris duplus est impetûs, qui motum efficit in mobili, quatenus movetur. Hinc est eodem conatu motoris disparem effici motum, si potentia æqualiter moveatur cum mobili, ut constat: quia nimirum impetus mobili impressus inæqualem habet intensione[m] , quamvis entitativè æqualis sit. Si enim tota motoris virtus sit 2, & decem impetûs particulas resistentiam superantes mobili imprimat, in quo intensio fiat ut 1, in mobili gravitatis fesqualteræ, particulæ eædem decem impetûs intensionem efficiunt ut 1/2; quare & hujus motus erit subsesqualter, ac proinde motor, qui æqualiter cum mobili movetur, etiam tardiorem habet motum, quàm cùm motum priori mobili conciliabat. Patet igitur ex his nunquam fieri posse, ut corpus grave minoris aut æqualis virtutis alterum moveat ita, ut planè in velocitate consentiant; illud enim corpus minùs aut æquè grave concipere non potest impetum, qui & sibi ad motum sufficiat, & alteri impetum imprimat: finge scilicet animo fuisse impetum impressum corpori æquè vel magis gravi; hîc utique cum non excedat resistentiam mobilis, nullum efficere potest motum; igitur neque impressus fuit impetus, ne sit omninò inutilis. Quòd si eâ ratione disponantur ut motor velociùs moveri possit quàm mobile, jam fieri potest, ut à minore majus moveatur: nam si motor certâ quâdam velocitate movere possit pondus unius libræ motu sibi æquali, eodem conatu & eâdem velocitate se movens movebit pondus centum librarum, si hoc ita sit dispositum, ut centuplo tardiùs moveatur: quia nimirum idem entitativè impetus in hoc pondere centuplo remissior, quàm in pondere unius libræ, sufficit ad motum centuplo tardiorem. Motus siquidem centum librarum subcentuplus in velocitate, æqualis est motui unius libræ centuplo in velocitate; si enim libra percurrit centum spatij digitos sibi succedentes in longitudine, pari tempore centum libræ percurrunt quidem unicum digitum longitudinis spatij, centum tamen spatia digitalia percurrunt, singulæ scilicet libræ digitum. CAPUT

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166 Mechanicorum in proportion to the measure required for the motion that follows, both at the beginning of that same body’s motion: but this motion is equal to the motion to which it resists; therefore the whole impetus of the mover is double the impetus which produces motion in the moving body, insofar as it is moved. Hence it comes about that by the same effort of the mover a different motion is produced, if the power is moved equally with the moving body, as is clear: because indeed the impetus impressed on the body has an unequal intensity, though it is equal in entity. For if the whole force of the mover be 2, and it impresses on the moving body ten particles of impetus overcoming resistance, in which the intensity is made as 1, then in a body of twice equal weight those same ten particles produce an intensity of impetus as 1/2; wherefore the motion of this too will be sub-double, and therefore the mover, when it is moved equally with the moving body, also has a slower motion than when it imparted motion to the earlier body. It is therefore evident from these things that it can never happen that a heavy body of lesser or equal force moves another in such a way that they are wholly in agreement in speed; for that body, being less heavy or equally heavy, cannot conceive an impetus which would both suffice for its own motion and impress motion on the other: imagine, then, in the mind that an impetus had been impressed on a body equally heavy or heavier; here indeed, since it does not exceed the resistance of the moving body, it can effect no motion at all; therefore no impetus was impressed, lest it be entirely useless. But if they are arranged in such a way that the mover can be moved faster than the moving body, it can then happen that the greater is moved by the lesser: for if the mover can, with a certain speed, move a weight of one pound with a motion equal to itself, then moving itself with the same effort and at the same speed, it will move a weight of one hundred pounds, if this be so arranged that it moves one hundred times more slowly: because indeed the same impetus in entity, being one hundred times more diminished in this weight than in a weight of one pound, suffices for a motion one hundred times slower. For the motion of one hundred pounds, being one one-hundredth in speed, is equal to the motion of one pound one hundred times in speed; for if a pound traverses one hundred successive spans of digits in length, in the same time one hundred pounds indeed traverse only a single digit of the length of space, yet they traverse one hundred digit-spaces, that is, each pound a digit. CAPUT

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Liber secundus. CAPUT V. 167 CAPUT V. In quo Machinarum vires sitæ sint. Potentiam oneri movendo cæteroqui imparem præstare posse, si machina adhibeatur, quotidiano experimento discimus; adeò ut ipsa unica pluribus potentiis machinâ destitutis virtute æqualis sit, & quæ pondus solitarium ac simplex loco prorsus movere non poterat, ubi se ad machinam applicuerit, jam non ponderi tantùm, sed & machinæ motum conciliet. Quid ergo illud sit, ex quo hujusmodi virium incrementum oritur, hîc pervestigandum est; & ad illud causæ genus revocatur, quam Scholæ Formalem appellant; est scilicet ratio, per quam sit, ut sit, atque dicatur Machina: hoc autem incrementum virium, ut ex dicendis constabit, ex machinæ figurâ pendet secundùm quam potentiæ, & ponderis motus sibi invicem pro ratâ portione respondent. A machinâ quâ machina est, potentiæ moventis vires non augeri certum est; nihil enim illi interioris virtutis impertitur, & quâ machina est, ab omni innatâ gravitate sejuncta intelligitur: vectis siquidem, ferreus sit, sive lignens, machinæ rationem non immutat, si sola intercedat materiæ gravioris aut levioris disparitas. Quòd si faciliùs ferreo vecte tricubitali deorsum premens attollas saxum, quàm si ligneo vecte pariter tricubitali utaris (quia nimirum ferreus vectis habet sibi adnexam ex gravi materiâ, quâ constat, potentiam, quæ deorsum urgendo te juvat, ut saxum attollatur,) id planè esse extra vectis naturam, quâ vectis est, manifestum erit, si non deorsum, sed sursum, aut à lævâ in dextram connitendum sit, ut duo connexa disjungas; tunc enim ferrei vectis gravitas sustentanda laborem potiùs creabit, quàm ut præ simili ligneo vecte motum hunc faciliorem reddat. Quare præter Mechanicæ facultatis institutum machinis accidit, ut gravitate suâ potentiæ moventis vires adaugeant, non quidem illam immutando, facto interiore virtutis

Transcription: Translated (English)

Book Two. CHAPTER V. 167 CHAPTER V. In which the powers of Machines are shown to reside. That a power can be made equal to a burden which it would otherwise be unequal to moving, if a machine be employed, we learn from daily experience; so much so that a single power, lacking several machines, is equal in force to many, and that which by itself and simply could not move a weight from its place at all, when it has applied itself to a machine, now imparts motion not only to the weight, but also to the machine. What, then, that thing is from which such an increase of force arises must here be examined; and it is referred to that kind of cause which the Schools call Formal; that is, the reason by which it is, and is said to be, a Machine: but this increase of force, as will be clear from what is to be said, depends on the figure of the machine, according to which the motions of power and weight correspond to one another in due proportion. It is certain that from a machine, insofar as it is a machine, the force of the moving power is not increased; for nothing of an inward virtue is imparted to it, and insofar as it is a machine it is understood to be separated from all innate heaviness: for a lever, whether it be iron or wooden, does not alter the nature of the machine, if only the difference of heavier or lighter matter intervene. But if with an iron lever three cubits long you more easily lift a stone by pressing downward than if you used a wooden lever of the same length (because indeed the iron lever has annexed to it, from the heavy material of which it consists, a power which helps you by pressing downward, so that the stone may be lifted), this will plainly be shown to be outside the nature of a lever, insofar as it is a lever, if it is not downward, but upward, or from left to right, that you must strain in order to separate two things joined together; for then the weight of the iron lever will create labor to be borne, rather than make this motion easier than a like wooden lever would. Therefore, beyond the design of the mechanical art, it happens to machines that by their own weight they increase the force of the moving power, not indeed by changing it, by an inward act of virtue

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Mechanicorum tis additamento; sed aliam potentiam, quæ conjunctis cum illâ viribus agat, consociando. Sed & illud animadvertendum est, vix unquam fieri posse, ut potentia movens nihil prorsus impedimenti à machinâ recipiat: sivè enim machinæ ipsius pars aliqua gravis elevanda est; sivè membrorum, in quæ machina distribuitur, invicem confligentium, seque vicissim terentium asperitas obsistit; sive motus (ut machinæ ipsi, cui applicatur potentia, obsecundet) à suâ directione inflectitur; sivè quid hujusmodi intercedit, quod aliquid de motûs velocitate imminuat, quæ cæteroqui conceptum potentiæ ab omni machinâ absolutæ impetum consequeretur. Ex his tamen aliqua sunt, quæ ita motui potentiæ officiunt, ut ad retinendum onus juvent; hujus siquidem gravitas minùs adversùs potentiam valet, si & ipsum, quia machinæ illigatum à recto in centrum gravium tramite deflectere, vel mutuum partium se terentium conflictum vincere cogatur, ut vim potentiæ inferat. Verùm hæc, quamvis, ubi res ad praxim deducitur, per incuriam dissimulanda non sint, sub staticam considerationem hîc non cadunt, ubi machinarum vires expenduntur; harum enim figura perindè attenditur, atque si nihil adjumenti, nihil detrimenti ex materiâ accederet. Ad rem itaque propiùs accedentibus recolenda sunt ea, quæ in superioribus hujus libri capitibus disputata sunt, proximam videlicet motûs effectricem causam impetum esse sive ab interiore virtute manantem in iis, quæ sponte suâ moventur, sivè extrinsecùs aliunde impressum iis, quæ naturâ repugnante per vim cientur: ex cujus impetûs intensione, quatenùs omnem resistentiam superat, motuum velocitas oritur: nunquam autem à velocitate aut tarditate motum sejungi posse certum est, quippe qui nec sine spatio per quod decurratur, nec sine partium sibi certâ lege succedentium continuatione ac serie intelligi potest. Quare & resistentiæ momenta tùm ex corporis movendi gravitate, tùm ex velocitate componi sæpiùs innuimus, ut hinc innotescat fieri facilè posse, ut, sicut ejusdem gravitatis resistentia inæqualis est, si velocitate inæquali movenda sit, & gravitatum inæqualium disparia sunt resistentiæ momenta, si Ratio, quæ ex gravitatum & velocitatum Rationibus componitur, sit Ratio Inæqualitatis, quia gravior velo- ciùs,

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Mechanics by this addition; but by uniting another power that acts together with it. But this too must be observed: it can hardly ever happen that the moving power receives absolutely no hindrance from the machine: for either some heavy part of the machine itself must be raised; or the roughness of the members into which the machine is distributed, and which strike against one another and wear against each other in turn, impedes it; or the motion (so as to accommodate the machine itself, to which the power is applied) is deflected from its direction; or something of this kind intervenes, which diminishes somewhat the speed of the motion, which otherwise would obtain the impulse conceived by a power wholly free from any machine. Among these there are nevertheless some things which so hinder the motion of the power that they help to retain the load; for indeed its weight has less force against the power, if it is itself, because it is bound to the machine, forced to turn aside from the straight course toward the center of heavy bodies, or to overcome the mutual conflict of the parts rubbing against one another, so as to exert the force of the power. But these things, although, where the matter is brought down to practice, they must not be disregarded through carelessness, do not here fall under static consideration, where the forces of machines are weighed; for the shape of these is regarded just as though nothing of help, nothing of detriment, were added from the material. Therefore, for those approaching the matter more closely, the things must be recalled which were discussed in the earlier chapters of this book, namely that the proximate efficient cause of motion is impulse, whether flowing from an inner power in those things that move of themselves, or impressed from outside elsewhere upon those things which, though contrary to nature, are set in motion by force: from the intensity of this impulse, insofar as it overcomes all resistance, the speed of motions arises: but it is certain that motion can never be separated from speed or slowness, since it can be understood neither without the space through which it runs, nor without the continuity and series of parts succeeding one another according to a fixed law. Therefore we have often indicated that the moments of resistance are composed both from the weight of the body to be moved and from its speed, so that it may thus become known that it can easily happen that, just as the resistance of the same weight is unequal if it must be moved with unequal speed, so too the moments of resistance of unequal weights are different, if the ratio, which is composed from the ratios of weights and speeds, is a ratio of inequality, because the heavier more quickly,

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Liber secundus. CAPUT V. 169 ciùs, minùs gravis tardiùs movetur; ita gravitatum inæqualium par sit resistentia, si quæ inter gravitates intercedit Ratio, ea- dem reciprocè inter velocitates inveniatur. Quemadmodum enim quæcumque calori adversantur, vehementiorem quidem validissimè respuunt, tenuissimum verò facillimè admittunt; haud dispari ratione pondera, si velociùs incitare velis, im- pensiùs reluctantur, minimo ac tardissimo motui levissimè ob- sistunt. Quoniam igitur naturâ definitum est, quantam gravitatem, quantâque velocitate, pro certâ impressi impetûs mensurâ, mo- vere possit Potentia concepto impetu, qui pro ratâ portione respondeat impetui quem illa oneri imprimit, ut Potentia, & onus æquali velocitate moveantur; satis constat eandem impe- tûs mensuram parem esse movendo oneri graviori, si quâ Ra- tione posterior hæc gravitas priorem gravitatem vincit, eâdem Reciprocè Ratione prioris velocitas posterioris tarditatem su- peret; utrobique scilicet par est resistentia, ac proinde ab eâ- dem potentiâ vinci potest. Cùm enim ea, quæ simul æqualiter moventur, æquali impetu ferantur; si Potentia tàm tardè mo- veretur ac pondus per machinam, indigeret impetu ex. gr. sub- quintuplo ejus quo illa movetur quintuplo velociùs ac ipsum Pondus. Verùm impetus hîc subquintuplus ineptus esset ad oneris resistentiam quintuplo ferè majorem vincendam; sed so- lum superare posset ac movere ́ ponderis. Quinque igitur im- petus huic æquales possunt totam resistentiam superare. Cum itaque in motu quintuplo velociori Potentiæ sit verè impetus quintuplus, poterit etiam elevare pondus, quod est quintuplo majus, quàm sit ́ ipsius. Verùm hîc ubi de motûs velocitate sermo est, non is quidem absolutè accipiendus est; sed quâ parte gravium naturæ repugnat: si enim plumbeus globus A ex C dependeat funiculo C A, & circà versatilem or- biculum B stabili axi infixum ducatur filum connectens globos A & D, cettum quidem est globum A, si usque ad B perveniat, tantumdem spatij in arcu AB percurrere, non tamen tantumdem ascendere, quantum globus D se- cundùm rectam BD descendit; sed ascensum metitur AE, nimirum Sinus Versus arcûs AB, qui minor est eodem arcu (arcus siquidem major est rectâ AB lineâ ipsum sub- Y

Transcription: Translated (English)

Book Second. CHAPTER V. 169 the more heavily, the more slowly it moves; thus, for unequal weights, the resistance is equal, if the ratio that intervenes between the weights is found reciprocally in the velocities. For just as whatever things oppose heat, resist the more violent most strongly, but most readily admit the very gentle; so, by no dissimilar a rule, weights, if you wish to hasten them more quickly, resist more strongly, but to the least and slowest motion they oppose themselves most lightly. Since therefore it has been determined by nature how much weight, and with what velocity, in proportion to a certain measure of impressed force, Power can move, an impulse having been conceived which corresponds in due proportion to the impulse that it impresses on the load, so that Power and load are moved with equal velocity; it is sufficiently clear that the same measure of impulse is equal to moving a heavier load, if by whatever Ratio this later weight surpasses the former weight, by the same Reciprocal Ratio the velocity of the former may overcome the slowness of the latter; in both cases, namely, the resistance is equal, and therefore can be overcome by the same power. For since those things which are moved equally at the same time are borne by equal impulse; if Power were moved as slowly as the weight by means of the machine, it would need an impulse, for example, five times smaller than that by which the same Power is moved five times more quickly than the Weight itself. But here an impulse five times smaller would be unfitted for overcoming a resistance of the load nearly five times greater; but it could only overcome and move a fifth part of the weight. Therefore five such impulses equal to this can overcome the whole resistance. Since therefore in a motion five times quicker the Power truly has an impulse fivefold, it will also be able to raise a weight which is five times greater than its own. But here, where the discussion concerns the velocity of motion, that velocity is not indeed to be understood absolutely; but in so far as it is contrary to the nature of heavy things: for if a leaden globe A hang from C by the cord C A, and, around the movable little wheel B fixed in a stable axis, there be drawn the string connecting the globes A and D, it is certainly true that the globe A, if it should reach B, traverses as much space in the arc AB, yet does not ascend as much as the globe D descends along the straight line BD; but the ascent is measured by AE, namely the versed sine of the arc AB, which is less than the same arc (for the arc is greater than the straight line AB itself, which it under- Y

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Mechanicorum tendente, quæ opposita recto angulo E major est quàm trianguli basis A E) ac propterea resistentiæ momenta non ea sunt, quæ ex velocitate motûs A B, sed A E, & ipsâ globi A gravitate componuntur. Ex quo fit globum D quamvis minorem posse globo A graviori præstare, ac illum ad certam altitudinem ele- vare, ut cuilibet experiri licet, cum tamen illi ascensum suo descensui æqualem nullatenùs conciliare possit. Quòd si idem globus A ex breviore funiculo H A dependeat, experimento constat opus esse globo D gravitatem addere, ut valeat illum per arcum A F elevare ad eandem altitudinem A E: magis quippè laboriosum est breviore motu A F, quàm longiore motu A B ad eandem altitudinem ascendere; atque adeò plus virium in D requiritur, ut globo A majorem impetum imprimat, ex cujus intensione plus singulis temporis momentis ascendat in hoc posteriore motu, quàm in priore. Ne tamen motui globi D tribue mensuram arcûs A B sed rectæ A B. Sicut autem ubi potentiæ & oneris æquales esse debent motus, potentiæ vires gravitate oneris majores esse oportet, ut vim illi inferant; ita pariter ubi potentia & onus in motuum velocitate dissentiunt, & illa quidem velociùs, hoc tardiùs movetur, necesse est majorem esse Rationem Potentiæ ad Onus (licet illa minor sit onere) quàm sit Ratio tarditatis hujus ad illius velocitatem; ut scilicet ratio Potentiæ ad onus, quæ ex motuum, & virium Rationibus componitur, sit Ratio majoris inæqualitatis. Sit ex. gr. Ratio motûs Potentiæ ad motum Oneris ut 3 ad 2; si Ratio virium potentiæ absolutè sumptæ ad gravitatem oneris sit Reciprocè ut 2 ad 3, Ratio ex his Rationibus composita est Æqualitatis, scilicet 1 ad 1, & motus nullus sequitur; multò minùs si fuerit Ratio minor quàm 2 ad 3; prove- niret

Transcription: Translated (English)

For the moving power, since the line opposite the right angle, E, is greater than the base of the triangle A E, and therefore the moments of resistance are not those which are composed from the velocity of the motion A B, but from A E and the very weight of the globe A, it follows that the smaller globe D can nevertheless overcome the heavier globe A, and raise it to a certain height, as anyone may փորձience; although it cannot in any way reconcile to it an ascent equal to its descent. But if the same globe A hang from the shorter cord H A, experience shows that it is necessary to add weight to the globe D, so that it may be able to raise it through the arc A F to the same height A E: for it is more laborious to ascend by the shorter motion A F than by the longer motion A B to the same height; and therefore greater force is required in D, so that it may impart a greater impetus to the globe A, by the increase of which it rises more at each moment of time in this latter motion than in the former. Yet do not assign to the motion of globe D the measure of the arc A B, but of the straight line A B. And just as where powers and loads ought to have equal motions, the force of the powers must be greater than the weight of the load, in order that they may impart motion to it; so likewise where power and load differ in the velocity of motion, and the former indeed moves more quickly, the latter more slowly, it is necessary that the ratio of Power to Load be greater, though the former be less than the load, than is the ratio of the slowness of this to the velocity of that; that is, the ratio of Power to Load, which is composed from the ratios of the motions and the forces, may be the ratio of a greater inequality. Let, for example, the ratio of the motion of the Power to the motion of the Load be as 3 to 2; if the ratio of the force of the power, taken absolutely, to the weight of the load be reciprocally as 2 to 3, the ratio composed from these ratios is that of equality, namely 1 to 1, and no motion follows; much less if it should be a ratio less than 2 to 3; it would result...

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Liber secundus. CAPUT V. 171 niret enim Ratio minoris Inæqualitatis: debet ergo esse major Ratione 2 ad 3. Sit ex hypothesi Ratio 4 ad 5; jam Ratio com- posita ex Rationibus 3 ad 2, & 4 ad 5, est Ratio 6 ad 5 majoris Inæqualitatis. Neque hoc ita dictum intelligas, quasi motus ipse Potentiæ, ejusque velocitas, efficiendi vim haberet; sed ex ipsâ majore potentiæ velocitate innotescit impetum, qui radix est motûs, minus invenire impedimenti ex onere, quod minùs resistit, eo quòd tardiùs movendum est, quàm si æqualem velocitatis gra- dum cum potentiâ sortiri deberet. Quare licet potentia minor sit, ac pauciores entitativè particulas impetûs producere valeat, quàm potentia major, satis in aperto est fieri posse, ut potentia major majorem inveniens resistentiam nequeat impetum im- primere, ac movere onus, quod movebitur à minore potentiâ, si onus idem minùs resistat, cum sit tardiùs movendum: impe- tus enim à minore potentiâ oneri impressus satis est ad vincen- dam minorem hanc resistentiam; cum tamen potentia major non satis habeat virtutis, ut eam impetûs mensuram oneri im- primat, quæ majorem illius resistentiam superaret. In eo igitur totum Mechanices artificium consistit, ut sua instrumenta ita disponat, locisque congruis ita Potentiam, & Onus collocet, ut Potentiæ motus velocior sit præ motu Oneris: tùm horum motuum Ratione attentè perspectâ definies, quæ- nam Potentia datum Onus movere, vel quodnam Onus à datâ Potentiâ moveri queat; si nimirum Potentiæ vires ad oneris gravitatem majorem habeant Rationem, quàm sit Ratio motûs Oneris ad motum Potentiæ. Neque enim Machina aut Poten- tiæ vires auget, aut oneris gravitatem minuit, sed Ponderis re- sistentiam ad Potentiæ virtutem accommodat. Physica autem causa hæc est, quia impetus à Potentiâ pro- ductus, qui in onere minori movendo æque velociter cum po- tentiâ majore haberet intentionem, in onere majore sed tardiùs movendo minorem quidem habet intentionem, sed quæ satis est pro minore resistentiâ. Fac enim oneris particulas graves esse 20, illique à Potentiâ aliquâto graviore imprimi particulas 1 co impe- tûs, quibus vincitur Oneris resistentia: intensio in singulis par- ticulis gravitatis est particularum impetûs 5, juxta quam inten- sionis mensuram sequitur motus æque velox Potentiæ & oneris, Y 2

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Book the Second. Chapter V. 171 For the Ratio of the smaller Inequality would be less; therefore it must be greater than the Ratio 2 to 3. Let it be, by hypothesis, the Ratio 4 to 5; now the Ratio composed of the Ratios 3 to 2 and 4 to 5 is the Ratio 6 to 5, of the greater Inequality. Nor should you understand this as if the motion itself of the Power, and its velocity, had efficacious force; but from the greater velocity of the power itself it is made known that the impulse, which is the root of motion, finds less impediment from the load, because it resists less, inasmuch as it is to be moved more slowly, than if it were bound to share an equal degree of velocity with the power. Wherefore, although the power may be less, and may be able to produce fewer entitative particles of impulse than a greater power, it is sufficiently evident that it can happen that the greater power, finding greater resistance, may be unable to impress impulse and move the load, which will be moved by the lesser power, if the same load resist less, since it is to be moved more slowly: for the impulse impressed on the load by the lesser power is enough to overcome this lesser resistance; whereas the greater power does not have enough force to impress on the load that measure of impulse which would overcome its greater resistance. Therefore the whole artifice of Mechanics consists in arranging its instruments in such a way, and placing the Power and the Load in suitable positions, that the motion of the Power be swifter than the motion of the Load: then, by carefully considering the ratio of these motions, you will determine which Power can move a given Load, or what Load can be moved by a given Power; namely, if the forces of the Power have a greater ratio to the weight of the load than is the ratio of the motion of the Load to the motion of the Power. For neither does the machine increase the force of the Power, nor lessen the weight of the load, but it accommodates the resistance of the weight to the force of the Power. The physical cause, however, is this: because the impulse produced by the Power, which in moving a smaller load at the same speed with a greater power would have an intensity, in a larger load but moving more slowly, has indeed a lesser intensity, but one which is sufficient for the lesser resistance. For suppose the particles of the load to be heavy as 20, and to be impressed by some somewhat heavier Power with 1 particle of impulse, by which the resistance of the Load is overcome: the intensity in each particle of weight is 5 particles of impulse, according to which measure of intensity the motion of the Power and the load is equally swift, Y 2

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172 Mechanicorum hujus quidem per vim fursùm, illius verò juxtà naturam deorsum. Sit adhuc eadem Potentia; sed offeratur Onus, cujus particulæ gravitatis sint non jam 20; sed 50: Potentiæ virtus est eadem; quapropter non nisi resistentiam vincere potest, cui vincendæ sufficiant particulæ 100 impetûs; hæ autem in One- re graviore ut 50 efficerent solùm intentionem ut 2: Non igitur Potentia & onus æquè veloci motu, qui respondeat intentioni ut quinque, sicuti priùs, moveri poterunt; sed ut onus moveri possit, impetumque à potentiâ recipere, opus est ita illud col- locare, ut quò magis Ratione gravitatis resistit; eò minùs ra- tione tarditatis motûs resistat, séque eâ ratione temperent duæ hæ resistentiæ, ut una confletur resistentia non major illâ, quæ oriebatur ex onere gravi ut 20 æqualiter movendo: id quod fiet, si motus Potentiæ, quatenùs machinæ applicatur, ad mo- tum oneris sit ut 5 ad 2 in Reciprocâ Ratione intentionum im- petûs producti. Quare motus Potentiæ ad motum oneris est duplus sesqualter, quemadmodum posterior hæc oneris gravi- tas ut 50 est prioris gravitatis ut 20 dupla sesqualtera: atque hinc manifestum est particulas gravitatis 50 resistentes ut 2 ra- tione motûs comparati cum motu potentiæ, requirere particu- las 100 impetûs, quemadmodum particulæ gravitatis 20 re- sistentes ut 5 ratione motûs comparati cum motu ejusdem Po- tentiæ requirunt particulas 100 impetûs. Quid igitur mirum, si potentia eadem eodem conatu movet onus ut 50 velocitate ut 2, quo conatu movet onus ut 20 velocitate ut 5? Servatur itaque perpetua quædam justitia inter potentiæ vi- res, oneris gravitatem, spatia motuum, ac tempora; quò enim decrescunt potentiæ vires, aut oneris gravitas augetur, eò bre- viora sunt spatia, & longiora tempora motuum ipsius oneris; sed ampliora spatia motuum potentiæ debilioris, quæ præ one- re velociùs movetur. Hinc dato onere graviori submovendo, aut potentiam augeri, aut, si illa immutata permaneat, oneris motum imminui, seu potentiæ motum augeri necesse est: Te- nui enim potentiâ ingens pondus citò moveri non potest. Formalem igitur Machinæ Rationem, quâ Machina est, in eo sitam esse deprehendimus, quòd ea figura sit, quæ potentiæ, & oneris motibus legem ita statuat, ut Potentia velociter, Pon- dus lentè moveatur; sic enim sit, ut minor oneris resistentia vir- tuti

Transcription: Translated (English)

172 Of the mechanics indeed, in the one case by force upward, in the other by nature downward. Let the same Power still remain; but let a Load be offered, whose particles of gravity are no longer 20, but 50: the force of the Power is the same; wherefore it can overcome only a resistance, for which particles of impulse 100 would suffice; but these, in a heavier Load, would produce only an intensity as 50 to 2: therefore Power and load cannot be moved with an equally swift motion, corresponding to an intensity as five, as before; but in order that the load may be moved and receive impetus from the power, it is necessary so to place it, that the more it resists by reason of gravity, the less it resists by reason of slowness of motion; and so let these two resistances be tempered, that one resistance may be made, not greater than that which arose from moving a load of gravity 20 equally; and this will happen if the motion of the Power, insofar as it is applied to the machine, is to the mo- tion of the load as 5 to 2, in the reciprocal ratio of the intensities of the im- pulse produced. Therefore the motion of the Power to the motion of the load is two and a half, just as this latter gravity of 50 is to the prior gravity of 20 as two and a half: and from this it is manifest that particles of gravity 50 resisting as 2 in ra- tion to the motion compared with the motion of the power, require particu- les of impulse 100, just as particles of gravity 20 re- sisting as 5 in relation to the motion compared with the motion of the same Po- wer require particles of impulse 100. What wonder, then, if the same power with the same effort moves a load of 50 at a speed of 2, with which effort it moves a load of 20 at a speed of 5? Thus a certain perpetual justice is preserved between the powers of the power, the gravity of the load, the spaces of the motions, and the times; for the more the powers of the power decrease, or the gravity of the load increases, the more short are the spaces, and the longer the times of the motions of the load itself; but the spaces of the motions of the weaker power, which moves faster than the load, are greater. Hence, given a heavier load to be removed, it is necessary either that the power be increased, or, if that remains unchanged, that the motion of the load be diminished, or the motion of the power be increased: For with a weak power a great weight cannot be quickly moved. We therefore find that the formal ratio of the Machine, by which it is a Machine, consists in this: that it is a figure which so establishes a law for the motions of the power and the load, that the Power moves swiftly, while the Weight moves slowly; for thus it comes about that the lesser resistance of the load to the virtue

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Liber secundus. CAPUT V. 173 tuti vim movendi, etiamsi minorem, habenti pro ratâ portio- ne respondeat. Satis igitur erit, ubi singularum machinarum vires expendendæ erunt motuum inire rationes, qui ex machi- næ agitatione oriuntur: nam si Potentia præ Onere velociùs moveatur, operæ pretium faciet Machinator; modò non adeò tenuis sit motuum Ratio, ut quicquid utilitatis ex machinæ fi- gurâ accedit, deferatur ex partium se terentium conflictu; nam perinde esset, ac si oneri gravitas adderetur. Ex his liquet à nou paucis plus operæ laborisque consump- tum, quàm par esset, ut Aristoteli adhærerent in referendis machinarum viribus in circuli naturam planè admirandam: Quapropter inquit initio qq. Mechan. non est inconveniens ipsum miraculorum omnium esse principium. Ea igitur quæ circà libram fiunt, ad circulum referuntur; quæ verò circa vectem, ad ipsam libram; alia autem ferè omnia, quæ circa mechanicas sunt motiones, ad vectem. Nisi enim fucum veritati faciamus, quæ demum mi- racula ita circulum à reliquo figurarum vulgo secernunt, ut in eum admiratio omnis corrivata confluat, nec nisi hinc in cæte- ras derivetur? An quòd linea eadem, quâ circuli ambitus de- finitur, omnis latitudinis expers, cava pariter atque convexa amico foedere copulat, quæ sibi invicem repugnant? Cavum si quidem à convexo, quæ recto interjecto discriminantur, per- inde dissidere censemus, atque minus à majori, inter quæ sibi adversantia id, quod æquale est, intercedit. At hæc ita vulga- ria sunt, ut non Hyperbolæ solùm, ac Parabolæ, aut Nicome- dis Conchoidi, aut Archimedis Spiralibus, aut Dinostrati Quadratici, cæterisque omnibus extrà Geometricas leges cur- vis lineis communia sint; verùm etiam in angulo quocumque rectilineo facilè ab omnibus observentur; cum lineæ rectæ, qui- bus inclinatis angulus constituitur, hinc quidem sibi mutuis nutibus annuere, hinc verò abnuere videantur; quibus oppo- sitis nutibus media pariter interjacet directa positio, omni in- clinatione submotâ. An ipsâ nascentis Circuli exordia admiratione non carent, quòd æquè ex Radij ejusdem in centro subsistentis quiete, ac circumlati motu oriatur? Sed quid hæc in circulo potiùs sus- piciamus, quàm in Helice, cui genesis haud dispar contingit? Quòd si circulo primas ideò deferendas existimemus, quòd Y 3

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Book the Second. CHAPTER V. 173 the force of moving, even if smaller, having a proportionate share. It will therefore be enough, when the powers of the several machines are to be weighed, to examine the ratios of the motions that arise from the working of the machine: for if the Power move more swiftly than the Load, the contriver will make a gain; provided only that the ratio of the motions be not so slight that whatever advantage is gained from the shape of the machine is carried off by the conflict of the parts rubbing against one another; for it would be as though weight were added to the load. From these things it is clear that not a few have spent more labour and toil than was fitting in adhering to Aristotle when referring the powers of machines to the nature of the circle, which is utterly admirable: wherefore he says at the beginning of the Mechanical Questions, “it is not inappropriate that it should be the principle of all marvels.” Those things, then, which are done around the balance are referred to the circle; those which are around the lever, to the balance itself; and almost all the others, which concern mechanical motions, to the lever. For unless we do violence to the truth, what wonders are there that so set the circle apart from the rest of the figures, that all admiration is drawn together and converges upon it, and from there is derived only to the others? Is it because the same line by which the circumference of the circle is defined, being devoid of all breadth, joins hollow and convex alike in friendly union, things which are opposed to one another? For we judge the hollow to differ from the convex, which are distinguished by the interposed straight line, just as the less from the greater, between which there intervenes that which is equal, though they are contrary to one another. But these things are so common that they are shared not only by the Hyperbola and Parabola, or Nicomedes’ Conchoid, or Archimedes’ Spirals, or Dinostratus’ Quadratic curve, and all the other curved lines outside the laws of geometry; but they are also easily observed by everyone in any rectilinear angle whatever; since the straight lines by which an angle is formed when they are inclined seem on the one hand to assent to one another by mutual nods, and on the other hand to refuse assent; and between these opposed nods there likewise lies the middle direct position, all inclination removed. Does the very beginning of a circle, in its generation, lack admiration, since it arises equally from the rest of the same Radius standing in the center and from its motion around? But why should we admire this in the circle rather than in the Helix, whose origin is not unlike? But if we think the first place should therefore be given to the circle, because Y 3

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Mechanicorum 174 in se recurrens peripheria ibi sui motûs terminum inveniat, unde sumpsit exordium; & circumacta, quæ ex adverso sunt, partes oppositis cieat motibus, ita ut progredientibus supremis infimæ regrediantur, & in ima detrudantur si- nistræ, dextris in altiora provectis: Quid Ellipsis præjudi- cio repellimus? cum & hæc unico limite cavo pariter atque convexo in sese redeunte circumscripta in contrarias partes incitetur; nec à rectâ tantummodo lineâ alternis auctâ cre- mentis, imminutáque decrementis altero terminorum quies- cente, sed etiam (quod verè miraculo proximum est) utroque extremo flexilis lineæ in binis Ellipseos umbilicis defixo ab illâ in alios, atque alios angulos sinuatâ des- cribatur. At, inquis, in circulo semidiametri partes eodem im- pellente circà centrum agitatæ ita dispari velocitate ferun- tur, ut earum tarditas aut concitatio intervallo, quo sin- gulæ à centro absunt, sit analoga. Verùm & hoc Ellipsi, ac plano Helicoidi aliquatenùs pro suo modulo commune est; semidiametri enim circumactæ puncta à centro remo- tiora velociùs feruntur. Partes autem quiescenti centro pro- pios cunctabundas moveri, naturæ pro viribus opposita determinantis instituto consentaneum esse nemo non videt, qui tarditatem interjici videt quietem inter, ac motûs ve- locitatem. Quare sapientissimo consilio factum, ut eorum, quæ firmo nexu invicem solidata subsistunt, vel particu- læ omnes æquis passibus moveantur, vel si qua moræ dis- pendium subeat, finitimarum velocitas, servatâ aliquâ vi- cinitatis analogiâ minuatur: ne scilicet solutâ compage dis- siliant. Quæ verò ad explicandum, cur ea, quæ centro propiora sunt, tardiùs in gyrum contorqueantur, Author illius libri Quæst. mechan. comminiscitur de duplici motu, naturali vi- delicet, ac præter naturam, quibus feratur ea, quæ circu- lum describit linea (quasi breviorem lineam vis major à tra- hente centro illata magis à naturali motu, qui secundùm Tangentem est, deflecteret) ea sunt, quæ facillimè cor- ruant, & minimè cum Aristotelis doctrinâ cohæreant, qui lib. 1. de Cælo. summa 4. circularem motum & simplicem, & naturalem,

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Mechanics 174 the periphery, recurring into itself, may find there the limit of its motion from which it took its beginning; and, having been turned about, the parts that lie opposite may rouse opposite motions, so that as the upper parts advance, the lower recede, and the left are driven down into the lower region, the right being carried upward: why do we reject the prejudice against the Ellipsis? since this too, bounded by a single limit, hollow and likewise convex, returning into itself, is stirred toward contrary parts; and not only from a straight line increased by alternate increments and diminished by decrements, with one of the termini at rest, but also (which is truly next to a miracle) with both ends of the flexible line fixed in the two centers of the Ellipsis, it is traced, curved away from that line into other and other angles. But, you say, in a circle the parts of the semidiameter, being driven around the center by the same mover, are carried with such unequal speed that their slowness or swiftness is analogous to the distance at which each lies from the center. Yet this also is in some measure common to the Ellipsis, and to the plane helix according to its own scale; for the points of the revolving semidiameter that are farther from the center are carried more quickly. But that the parts nearer the resting center should be moved more slowly is seen by everyone to be consonant with the plan of nature, which determines things in opposition to the strength of the force acting upon them, for slowness lies between rest and the speed of motion. Wherefore it was done with most wise design that, among those things which subsist firmly bound together by mutual connection, either all the particles should move at equal pace, or, if any disparity of delay arises, the speed of neighboring parts should be reduced, while some analogy of proximity is preserved; lest, namely, the compact structure be loosened and they fly apart. As for what the author of that book, Quæst. mechan. , devises to explain why those things that are nearer the center are twisted more slowly in rotation—namely, a double motion, one natural and the other contrary to nature, by which the line describing the circle is carried (as though a greater force, acting from the center and drawing more along a shorter line, would deflect it more from the natural motion, which is according to the tangent)—these are things that most easily fall apart and are least in agreement with the doctrine of Aristotle, who in book 1 of De Cælo , question 4, calls circular motion both simple and natural,

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Liber secundus. CAPUT V. 175 naturalem, & priorem recto disertissimè pronunciat; Perfectum enim, inquit text. 12; prius naturâ est imperfecto; circulus autem perfectorum est, recta verò linea nulla, Quis ergo in circulo motus præter naturam? necessarium est, ait text. 8. esse ali- quod corpus simplex, quod natum est ferri circulari motu secundùm suam ipsius naturam. Ea certè quibus insita est in mo- tum propensio, in gyrum aguntur, ut sydera; aut saltem mo- tu in se recurrente circulum æmulantur, ut ex cerebri & cor- dis systole ac diastole spirituum ac sanguinis circuitio oritur; aut plurium circularium motuum commixtione unum tempe- rant motum, ut animalia cum progrediuntur; ossa siquidem, quibus membra subsistunt, ita à musculis commoventur, ut unumquodque sui motus centrum constituat in eâ sinitimi ossis parte, cui sivè , sive flexili com- page inseritur. At motu recto, ut potè brevissimo, nihil fertur, nisi cui ex naturæ instituto cedit quies certo in loco, à quo abstractum fuerit, eóque sibi redditum spontè remigrat. Nihil igitur præter naturam in circuli motu deprehendi potest, ex quo dispar illa intimarum atque extimarum partium velocitas petenda sit; cum vix alium natura per se expetat simplicem motum præter circularem. Cur autem qui secundùm rectam extremæ semidiametro ad perpendiculum insistentem lineam sit motus, naturalis censeatur? An quia gravia suis nutibus ad terræ centrum rectâ feruntur? Semidiametro igitur, nisi in verticali plano constituatur horizonti parallela, motus qui se- cundùm lineam circuli Tangentem est, præter naturam con- tinget, quippe qui à rectâ, quæ gravia in centrum dirigit, de- flectat: & in circulo horizonti parallelo circumacta semidiame- ter nullo naturali motu agitabitur; nulla enim recta linea cir- culi Tangens in eo plano est, quæ lineæ directionis graviùm congruat: & tamen quemcumque demum situm circulus ejus- que semidiameter obtineat, eandem semper motuum analo- giam servant partes pro ratione intervalli à centro, citrà ullam motuum naturalis, & præter naturam, commistionem. Verùm mirifica sit circuli natura; quid hæc ad explicandam Mechanicarum motionum causam? an ut hanc ignotam fatea- mur, quia admirandam prædicamus? sed unico argumento, commenta hujusmodi disjiciamus. Si minor potentia majori ponderi

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Book Second. Chapter V. 175 he most distinctly declares natural, and prior to straight line; for the perfect, says the text 12, is by nature before the imperfect; but the circle is of perfect things, whereas the straight line is of none. Why then is motion in a circle contrary to nature? It is necessary, says text 8, that there be some simple body which is born to be carried by circular motion according to its own nature. Certainly those things in which an inclination to motion is implanted are carried round, as the stars; or at least by a motion recurring into itself they imitate a circle, as from the systole and diastole of the brain and heart the circulation of spirits and blood arises; or by the blending of several circular motions they compose one tempered motion, as animals do when they advance; for the bones, on which the limbs rest, are moved by the muscles in such a way that each constitutes the center of its own motion in that part of the neighboring bone to which it is joined, whether by a firm or flexible connexion. But in straight motion, since it is very brief, nothing is carried unless that for which, by nature’s ordinance, rest in a certain place is proper, and from which, when removed, it returns of its own accord to itself. Therefore nothing contrary to nature can be detected in circular motion, from which that unequal speed of the inner and outer parts should be sought; since nature by itself scarcely seeks any simple motion other than circular. But why should motion according to a straight line standing perpendicular upon the outer semidiameter be deemed natural? Is it because heavy bodies are carried by their own impulses toward the center of the earth in a straight line? Therefore, unless the semidiameter is set in a vertical plane parallel to the horizon, a motion along the line tangent to the circle will happen contrary to nature, since it departs from the straight line which directs heavy bodies toward the center; and in a circle turned around parallel to the horizon the semidiameter will be agitated by no natural motion; for there is no straight line tangent to the circle in that plane which agrees with the direction line of heavy bodies: and yet, whatever position the circle and its semidiameter may occupy, the parts always preserve the same analogy of motions according to the proportion of their distance from the center, without any mixture of natural motion and motion contrary to nature. But however marvelous the nature of the circle may be, what is this to the explanation of the cause of mechanical motions? Are we to confess this cause unknown because we proclaim it admirable? But with a single argument let us reject such inventions. If a lesser power to a greater weight

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Mechanicorum 176 ponderi prævaleat, nullúsque intercedat circularis motus, certu[m] est hoc virtutis incremétum neque in Vectem, neque in libram neque in Circulum referri posse: adeóque principium aliud esse magis latè patens, à circulo absolutum: Atqui citrà omnem cir- cularem motu[m] minor potentia præpollet graviori ponderi: Mani- festum est igitur frustrà ex circulo peti Mechanicarum motio- num principium; sed illud esse, quod à nobis indicatum est, quippe quod, ubicumque reperitur, hoc efficit, ut minor po- tentia majori ponderi motum conciliet, nec is unquam sine illo contingit. Assumptionis veritas ut innotescat, ingensque pondus tardè movendum à tenui virtute sine circulari motu propelli posse confirmem, non ego te in suburbanum campum deducam, ut tenerrimo germini suppullulanti incumbentes glebas demùm loco cessisse observes, aut marmora Messalæ scindentem capri- ficum obtrudam, turresque longâ annorum serie labefactatas enatis fruticibus atque virgultis; ne mihi fortè herbescentes cuneos obtrudas, quos ad vectem, & circulum revocare velis. A D C B E Sed age raptandus sit in plano horizontali, aut inclinato, aut etiam elevandus sit ad perpendiculum cylindrus A. Experire primùm quanto labore id præstes illum trahens illigato fune in C, & arreptâ extremitate funis B. Tùm in B infixo firmi- ter paxillo ductarius funis alligetur; hic porrò inseratur annu- lo C optimè ferruminato, & quoad ejus fieri poterit exquisitè polito, arreptáque alterâ funis extremitate D iterum trahe cy- lindrum, & quantò minori labore id perficias, tu te ipse doce- bis. At hîc nulla circuli vides miracula; hîc libra nulla; nullus hîc vecti locus: motus enim tùm potentiæ trahentis, tùm cy- lindri, rectus est. Facilitatis autem discrimen non ex ullo cir- culari motu, qui nusquam apparet, sed ex eo oritur, quòd pri- mùm potentia & onus æqualiter moventur; postèa verò cylin- dri

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Mechanicorum 176 if the weight prevails, and no circular motion intervenes, it is certain that this increase of force can be referred neither to the pulley, nor to the balance, nor to the circle: and so there must be another principle, more widely extending, free from the circle. But without any circular motion, a lesser power prevails over a greater weight. It is therefore clear that the principle of mechanical motions is sought in vain from the circle; but that it is what we have pointed out, namely, that wherever it is found, it causes lesser power to impart motion to a greater weight, and this never happens without it. So that the truth of the assumption may be made known, and that I may confirm that a heavy mass, to be moved slowly, can be driven by a slight force without circular motion, I shall not lead you into a suburban field, to observe how the clods at last have given way beneath the tender germ sprouting forth; nor shall I thrust upon you the fig tree of Messala splitting marbles, or towers weakened through a long series of years by the shoots of bushes and little shrubs; lest you perhaps should thrust upon me greenish wedges, which you would wish to refer to the lever and the circle. A D C B E But come, let cylinder A be drawn in a horizontal plane, or an inclined one, or even raised perpendicular. First try how much labor you expend when you pull it by a rope fastened at C, and by seizing the end of the rope at B. Then, with a peg firmly fixed at B, let the guiding rope be tied; next let this be inserted into the ring C, excellently soldered and as finely polished as possible, and, taking hold of the other end of the rope at D, draw the cylinder again, and you will teach yourself how much less labor you accomplish it with. But here you see no miracles of the circle; here no balance; here no place for the lever: for the motion of both the traction and the cylinder is straight. The difference in ease, however, arises not from any circular motion, which appears nowhere, but from this, that at first power and load are moved equally; afterward, however, the cylinder

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Liber secundus. CAPUT V. 177 dri velocitas subdupla est velocitatis potentiæ; quia cum ex C cylindrus venit in B funis ultrà B extenditur juxtà longitudinem CB usque in E; ac propterea motus potentiæ duplus est, scilicet CE. Statue item in pariete puncta duo A & B (quo autem major re intervallo disjuncta fuerint, res meliùs succedet) ibique clavos rotundos nihil habentes asperitatis infige. Tùm pondera duo H & G æqualia assume, eáque funiculo nullis nodis aspero, sive serico crudo, sive crinibus equinis connexa impone claviculis A & B, ut liberè ex iis dependeant: suâ autem gravitate funiculum AB intentum Horizonti parallelum servabunt, & neutro prævalente ob gravitatis æqualitatem prorsùs immota consistent. Elige jam pondus tertium I, quod alteri datorum H & G æquale sit, aut etiam singulis aliquantò minus; illudque in E extento funiculo AB adnecte: statim pondus I secundùm rectam EF descendens videbis; pondera autem H & G per rectas HA, & GB ascendentia, quâ mensurâ funiculi inflexi partes AF, BF simul sumptæ excedunt rectam AB. Nullus igitur motus circularis hîc est; sed omnes recti ad perpendiculum, & tamen potentia I minor commovet majus pondus, quod ex H & G conflatur. Id autem ideò contingere, quia motus EF descendentis I major est motu ascendentium H & G, hinc manifestum est, quòd pondus I usque ad certum terminum descendit, ibique subsistit: quòd si illud manu apprehensum adhuc deorsum trahens eleves pondera H & G, ubi manum indè abstraxeris, pondera H & G prævalent, ac descendentia elevant pondus I ad certum illum terminum, ubi sponte substiterat: quia nimirum ultrà illum terminum non jam major est Ratio ponderis I ad pondera HG, quàm sit Ratio motuum H & G ad motum I. Hæc autem inferiùs, ubi de librâ & Æquilibrio sermo erit, paulò fusiùs & dilucidiùs explicabuntur; nunc enim satis est Z

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Book the Second. CHAPTER V. 177 the velocity of the descent is subdouble to the velocity of the power; because when from C the cylinder comes to B, the rope is extended beyond B according to the length CB as far as E; and therefore the motion of the power is double, namely CE. Likewise set on the wall two points A & B (the greater the interval by which they are separated, the better the result will be), and there drive in round nails, having nothing rough about them. Then take two equal weights H & G, and place them on the pegs A & B, connected by a cord with no rough knots, whether of raw silk or horsehair, so that they hang freely from them: by their own weight they will keep the cord AB tense, parallel to the horizon, and with neither prevailing on account of their equal weight they will remain altogether immobile. Choose now a third weight I, equal to one of the given ones H & G, or even somewhat less than each; and attach it to the cord AB stretched to E: you will immediately see the weight I descending along the straight line EF; but the weights H & G ascending along the straight lines HA & GB, by as much as the folded parts of the cord AF, BF together exceed the straight line AB. There is therefore no circular motion here; but all motions are straight toward the perpendicular, and yet the lesser power I moves the greater weight, which is made up of H & G. And this happens for this reason, because the motion of the descending I along EF is greater than the motion of the ascending H & G; hence it is clear that the weight I descends up to a certain limit, and there stops. But if, having seized it by hand, and still drawing it downward, you lift the weights H & G, when you remove your hand from there, the weights H & G prevail, and descending raise the weight I to that certain limit, where it had of itself stopped: because beyond that limit the ratio of the weight I to the weights HG is no longer greater than the ratio of the motions of H & G to the motion of I. These matters, however, below, where there will be discussion of the balance and equilibrium, will be explained somewhat more fully and clearly; for now it is enough Z

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Mechanicorum 178 pro institutâ disputatione ostendisse minorem gravitatem præ- pollere citrà omnem motum circularem. Ratum itaque esto ad nullum certum machinæ genus cætera esse revocanda; sed omnibus commune esse principium, ex quo vires desumunt; impetûs scilicet à potentiâ producti proportio ad ponderis resistentiam (quæ eò minor est, quò tardiùs mo- veri debet) ea est, quæ motûs facilitatem conciliat; nullus quippe adeò tenuis impetus reperitur, cui lentissimus aliquis motus non respondeat, si intereà à velociori motu potentia non prohibeatur. Ubi autem de potentiæ velocitate sermo est, non ea intelligatur, quæ esset, ubi præter se nihil ipsa moveret, ab- soluta ab omni resistentiâ; sed eam velocitatem intellige, quæ comparatè dicitur, ubi ejus motus cum oneris motu confertur. Semper tamen impetus, qui in Potentiâ reperitur quatenùs ex- cedit resistentiam ponderis, majorem in eâ intentionem ha- bet, quàm in pondere, quamvis pares entitativè sint impetus Potentiæ, & oneris. Hæc autem clariùs patebunt lib.4.cap.I. CAPUT VI. Quid attendendum sit in Machinæ collocatione, atque materie. Quamvis instructarum Machinarum vires ad calculos revocentur inspectâ earum figurâ, ut Potentiæ atque oneris motus invicem comparentur; quo tamen loco & situ Machina ipsa collocetur, dispiciendum est, ut innotescat, quanta illi vis inferatur tùm ab oneris gravitate, tùm à potentiæ conatu: ex hoc siquidem decernendum erit, quàm solidam construi oporteat Machinam. Quotus enim quisque est, qui ignoret longè solidiorem requiri machinam, si ex illa dependeat, aut illi in- cumbat onus, quàm si non machinæ; sed subjecto plano, inni- tatur idem pondus, aut aliunde dependeat? alia scilicet sunt gravitatis momenta contrà virtutem sustinentem etiam citrà motum, alia verò momenta, quatenus motui adversatur. Hinc

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Mechanicorum 178 had already shown by the dispute instituted that lesser gravity prevails without any circular motion. Let it therefore be taken as established that all other things are not to be referred to any fixed kind of machine; rather, there is a principle common to all, from which they draw their force, namely: the proportion of the impulse produced by the power to the resistance of the weight (which is the less, the more slowly it must be moved) is what secures the ease of motion. For there is found no impulse so slight to which some very slow motion does not correspond, if in the meantime the power is not prevented by a swifter motion. But when the speed of the power is spoken of, let not that be understood which it would have if, apart from itself, it moved nothing else, being free from all resistance; but understand that speed which is called relative, when its motion is compared with the motion of the load. Yet always the impulse found in the Power, insofar as it exceeds the resistance of the weight, has in it a greater intensity than in the weight, although the impulses of the Power and of the load are equally real in entity. These matters, however, will appear more clearly in book 4, chapter 1. CHAPTER VI. What is to be considered in the placement of a Machine, and in its material. Although the forces of constructed Machines are reduced to calculation by examining their shape, so that the motion of the Power and that of the load may be compared with one another, still the place and position in which the machine itself is to be set must be considered, so that it may become known how much force is inflicted on it both by the weight of the load and by the effort of the power: from this, indeed, it must be determined how solid a machine ought to be built. For who is there who does not know that a far more solid machine is required if a load hangs from it or rests upon it than if the same weight rests not on the machine but on the underlying plane, or hangs from somewhere else? Clearly, the moments of gravity are one thing against the sustaining force even without motion, but their moments are another thing insofar as they oppose motion. Hence

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Liber secundus. CAPUT VI. 179 Hinc operæ pretium fuerit non contemnendum, si res ita à Machinatore disponantur, ut pondus, quàm minimum fieri possit, à machinâ sustineatur: hâc enim ratione fiet, ut lon- giùs avertatur periculum luxationis aut fractionis membrorum, quibus machina distinguitur, etiamsi exilior illa fuerit; & ma- chinæ gravitas aliqua subtrahetur, dum moles ipsa minuitur, atque proinde movendi oneris difficultas non augebitur ex ma- chinâ; quæ etiam minore impendio parabitur. Sit exempli gratiâ pondus A, quod sit trochleâ attollendum in D. Poterit id duplici ratione fieri; primùm raptando illud in plano Horizontali ita, ut ex B veniat in C, tùm alligatâ tro- cleâ in I illud attollendo ad perpendicularum usque in D: cum raptatur, totum incumbit pondus subjecto plano; cum at- tollitur, totum ex trochleâ de- pendet. At si trochleâ utaris, de cujus firmitate subdubites, & loci dispositio ferat, ut pos- sit ex E & H onus suspendere, res faciliùs perficietur. Ponde- ri enim A adnecte funem O E, ex quo pendere possit in E, ac præterea tantumdem funis O S liberè vagantis; trochleam au- tem alliga in F: ubi verò ope trochleæ adduxeris pondus ex O in G, tùm funem O S liberè vagantem eleva, ac benè inten- tum adnecte in H, ut jam pondus ex H dependeat ad perpen- diculum: Ex hoc fiet, ut resoluto fune O E, liberéque vagan- te, ope trochleæ in F alligatæ adducas pondus ex G in D mul- tò minori labore, quàm si ex B in C illud raptâsses, & ex C in D sustulisses. Constat autem pondus idem minùs conniti adversùs lineas F G aut F D, quàm adversùs perpendiculares H G aut ID, ex iis quæ disputata sunt lib. 1. cap. 15, ac propterea etiam minùs dubitari potest de trochleæ firmitate. Hoc autem compendium elevandi pondera perinde, atque si per planum inclinatum attollerentur, ea scilicet suspendendo atque obliquè trahendo, ubi in praxim ritè deduxeris, appa- Z 2

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Book Two. CHAPTER VI. 179 Hence it would be well worth noting, if the matter be so arranged by the Mechanician that the least possible weight is borne by the machine: for by this means the danger of dislocation or breaking of the parts by which the machine is distinguished will be more removed, even if it be of slenderer build; and some of the machine’s heaviness will be taken away, as the bulk itself is lessened, and therefore the difficulty of moving the load will not be increased by the machine; which will also be prepared at less expense. Let there be, for example, the weight A, which is to be raised by a pulley to D. This may be done in two ways; first, by dragging it along a horizontal plane so that it comes from B into C, then, after the pulley has been fastened in I, by lifting it up to the perpendicular in D: when it is dragged, the whole weight rests upon the underlying plane; when it is lifted, the whole depends from the pulley. But if you use a pulley whose strength you distrust, and the arrangement of the place allows that the load may be suspended from E and H, the matter will be accomplished more easily. To the weight A therefore attach the rope O E, from which it may hang at E, and besides so much again of the freely moving rope O S; fasten the pulley in F. But when by means of the pulley you have drawn the weight from O to G, then raise the freely moving rope O S, and attach it well stretched in H, so that the weight now hangs from H vertically: from this it will follow that, the rope O E having been slackened and the freely moving rope released, you may, by means of the pulley fastened in F, draw the weight from G to D with far less labor than if you had dragged it from B to C and lifted it from C to D. Now it is clear that the same weight presses less against the lines F G or F D than against the perpendiculars H G or I D, from what has been discussed in book 1, chapter 15; and therefore the strength of the pulley can also be less doubted. But this contrivance for raising weights, just as if they were lifted by an inclined plane, namely by suspending them and pulling them obliquely, when you have duly reduced it to practice, appears Z 2

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Mechanicorum 180 rebit quanto labori, & quàm magnis sumptibus parcatur: cum neque vincendus sit partium tritus atque conflictus inter pondus, ac subjectum planum, neque sternendum sit multo robo- re planum ipsum, quod oneri sustinendo non impar sit. At ubi funem E O, quoad ejus fieri poterit, intenderis, aquâ largiter imbuito; hoc enim fiet, ut sele contrahens etiam paulò intentior, atque ad destinatum opus evadat aptior. Quæ cum ita sint, alia se offert methodus elevandi pondera non levi laboris compendio, si nimirùm duplex adhibeatur trochlea, altera quidem in A imminens ponderi ad perpendicularum, altera verò in B. Adhibita igitur trochlea B elevabit pondus ex C in D, ibique totum ex B pendebit: tùm vicissim trochleâ A utere, & ex D in E ascendet pondus, quod ibi totum ex A pendebit: iterum igitur adhibe trochleam B, ut ex E in F ascendat; atque vicissim, adhibita trochleâ A ascendet ex F in G; & sic deinceps. Ubi vides motum ponderis ascendentis per arcus C D E F G majorem esse quàm si rectâ ad perpendicularum elevatum fuisset ex C in G. Quia verò altitudines perpendiculares singulis arcubus respondentes subinde majores fiunt, propterea plus virium à potentia movente adhibendum est in progressu. Quâ autem Ratione altitudines illæ perpendiculares crescant, facilè innotescet, si aruum singulorum Sinus versos suis Radiis respondentes ad calculos revocaveris; arcus enim superiores & plurium esse graduum, & ex Radio minori, manifestum est: distantia autem parallelarum AC, BD perpendicularium eadem semper est; quapropter & æquales lineæ sunt Sinus Recti aruum inæqualium in circulis inæqualibus, videlicet aruum majorum in circulis minoribus. Quamquam nec omninò necesse est ità singulis tractionibus pondus attollere, ut ad perpendicularum dependeat, si maximè trochleæ invicem non modicum distarent; sed sufficeret alternis operis trochleas agitare, ut ascendens pondus modò ad hoc, modò ad illud perpendicularum accederet, ita tamen ut ultrò citróque transgrediatur perpendicularum, quod medium cadit inter extremas AC & BD; alioquin

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Mechanics 180 is a matter in which labor is spared, and at very great expense: since there is no need either to overcome the rubbing and friction between the weight and the plane beneath it, or to raise the plane itself with much strength, as it is no less able to bear the load. But when you have tightened the rope E O as much as possible, soak it thoroughly with water; for in this way it will happen that, contracting as it does, it becomes somewhat tighter, and more suitable for the intended work. Since these things are so, another method presents itself for raising weights with no small saving of labor, namely if a double pulley is used, one at A, hanging to the weight perpendicularly, and the other at B. So, with pulley B applied, the weight will be raised from C to D, and then the whole will hang from B; then in turn use pulley A, and the weight will ascend from D to E, where it will hang wholly from A; again therefore apply pulley B, so that it may ascend from E to F; and similarly, with pulley A applied, it will ascend from F to G; and so on. Here you see that the motion of the ascending weight through the arcs C D E F G is greater than if it had been raised in a straight line perpendicularly from C to G. But since the perpendicular heights corresponding to the several arcs are successively greater, therefore more force must be exerted by the moving power in the progress. By what reason, however, those perpendicular heights increase will easily become clear if you reduce the sines of the several arcs corresponding to their radii to calculation; for it is manifest that the upper arcs are of more degrees, and with a smaller radius. But the distance of the parallel perpendiculars AC, BD is always the same; wherefore the rectilinear lines are also equal, namely the sines of unequal arcs in unequal circles, that is, of the greater arcs in smaller circles. Although it is not altogether necessary, in each pull, to raise the weight so that it hangs perpendicularly, if especially the pulleys are not a little distant from one another; rather, it would suffice to set the pulleys in motion by alternate operation, so that the ascending weight might now approach this perpendicular, now that one, yet in such a way that it passes to and fro beyond the perpendicular, which lies midway between the outermost AC and BD; otherwise

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Liber secundus. CAPUT VI. 181 alioquin par non esset utriusque trahentis labor. Cæterùm satius est A & B parùm distare. Ut autem exemplo aliquo res manifesta fiat, statuamus altitudinem A C esse pedum 70, distantiam verò A B pedum 30, cui æqualis est ea, quæ ex D cadit perpendicularis in A C, scilicet D S. Quare in triangulo A S D rectangulo nota est Hypothenusa A D, quæ æqualis est ipsi A C, & nota est Basis S D. Atqui constat Perpendicularum A S esse medio loco proportionale inter summam atque differentiam Hypothenusæ ac basis, scilicet inter 100 & 40; igitur ducta prima in tertiam, videlicet ducta summa in differentiam dabit 4000 Quadratum Mediæ (hoc est perpendiculari A S) cujus Radix ped. 63 1/4 ferè est Perpendicularum A S. Igitur elevatio C S est ped. 6 3/4. Cum itaque B D æqualis sit ipsi A S ( jungunt enim parallelas æquales A B & S D) iterum in triangulo B V E rectangulo nota est Hypothenusa B E ped. 63 1/4, & Basis E V est ped. 30: Quare inter summam ped. 93 1/4, ac differentiam ped. 33 1/4 media proportionalis ped. 55. 67". est Perpendicularum B V; atque adeò elevatio D V est ped. 7. 58". major quàm C S. Et sic de reliquis. At statue distantiam A B solùm ped. 20: reperies perpendicularum A S vix excedere ped. 67; quare elevatio C S erit ped. 3 ferè; ac propterea etiam Perpendicularum B V erit paulò majus ped. 63. 94"; & elevatio D V ped. 3. 06"; & sic de cæteris. Potentiæ verò elevantis motum metitur differentia, quæ inter lineas B C & B D intercedit: quando autem distantia A B est ped. 30, linea B C est ped. 76. 1 5"; at cum est ped. 20, B C est ped. 72 2/5. Cum igitur in primo casu B D sit ped. 63 1/4, motus potentiæ est ped. 12 9/10; in secundo autem casu cum B D sit ped. 67; linea autem B C sit ped. 72 4/5, motus potentiæ est ped. 5 4/5. Quare in primo Ratio motûs Potentiæ ad motum ponderis est 12 9/10 ad 6 3/4, in secundo Ratio est 5 4/5 ad 3: & factâ reductione ad alias denominationes, prima Ratio est 86 ad 45, secunda Ratio est 29 ad 15, quæ si ad eumdem denominatorem 45 reducatur, erit 87 ad 45. Constat autem majorem esse Rationem 87 ad 45, quàm 86 ad 45. per 8. 1. 5. Majorem igitur Rationem habet motus Potentiæ ad motum ponderis, quan- Z 3

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Book Two. Chapter VI. 181 otherwise the labor of both pullers would not be equal. However, it is better for A and B to be a little distant from one another. But to make the matter clear by an example, let us suppose the height A C to be 70 feet, and the distance A B to be 30 feet, equal to that which falls perpendicularly from D upon A C, namely D S. Therefore in the right triangle A S D the hypotenuse A D is known, which is equal to A C itself, and the base S D is known. Now it is established that the perpendicular A S is the mean proportional between the sum and the difference of the hypotenuse and the base, namely between 100 and 40; therefore, by multiplying the first by the third, that is, multiplying the sum by the difference, there will be 4000, the square of the mean (that is, of the perpendicular A S), whose root is about 63 1/4 feet, which is the perpendicular A S. Therefore the elevation C S is 6 3/4 feet. Since then B D is equal to A S itself (for the equal parallels A B and S D are joined), in the right triangle B V E again the hypotenuse B E is known, 63 1/4 feet, and the base E V is 30 feet: therefore between the sum, 93 1/4 feet, and the difference, 33 1/4 feet, the mean proportional, 55.67 feet, is the perpendicular B V; and thus the elevation D V is 7.58 feet, greater than C S. And so for the rest. But suppose the distance A B to be only 20 feet: you will find the perpendicular A S scarcely exceeding 67 feet; wherefore the elevation C S will be about 3 feet; and therefore the perpendicular B V will also be a little greater than 63.94 feet; and the elevation D V 3.06 feet; and so on for the others. But the power elevating the motion is measured by the difference which lies between the lines B C and B D: now when the distance A B is 30 feet, line B C is 76.15 feet; but when it is 20 feet, B C is 72 2/5. Since therefore in the first case B D is 63 1/4 feet, the motion of the power is 12 9/10 feet; but in the second case, when B D is 67 feet and line B C is 72 4/5, the motion of the power is 5 4/5 feet. Therefore in the first case the ratio of the motion of the power to the motion of the weight is 12 9/10 to 6 3/4; in the second the ratio is 5 4/5 to 3: and, reduced to other denominations, the first ratio is 86 to 45, the second ratio is 29 to 15, which, if reduced to the same denominator 45, will be 87 to 45. Now it is clear that the ratio 87 to 45 is greater than 86 to 45, by 8.1.5. Therefore the motion of the Power has a greater ratio to the motion of the weight, when- Z 3

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Mechanicorum 182 do A & B minùs distant, quàm cum separantur intervallo ma- jore; atque adeò major est etiam movendi facilitas. Quòd si rei hujus minimè dubium experimentum sumere placeat, ipsisque oculis rem totam subjicere citrà omnem de- ludentis phantasiæ suspicio- nem, firmetur in A orbiculus circà suum axem versatilis, & ex eo æqualia pondera D & E funiculo connexa dependeant ad perpendiculum; quæ prop- ter gravitatis æqualitatem im- mota permanent. Tùm in B firmetur orbiculus circà suum axem pariter versatilis, & as- sumatur pondus C ponderi E æquale, cui adnectatur funi- culo EBC. Si manu retineas pondus C, ne gravitet, per- sistit pondus E in suo perpendiculo: jam manu retine pondus D, ne prorsus moveatur, ac dimitte pondus C, vi- debis hoc quidem descendere, pondus verò E ascendere, donec ex B dependeat, & in æquilibrio cum pondere C subsistat. Iterum retine pondus C, & dimitte pondus D, pariterque pondus D descendens videbis, E verò adhuc ascendens; & sic deinceps usque eò, dum pondus E uni- cum ambobus D & C æquipolleat, ut superiori capite in- dicatum est. Id igitur quod à ponderibus D & C præstatur, à quâlibet potentia æquali in D & C constitutâ præstari posse manifestum est. Si itaque simplicibus orbiculis fit, ut pondus æquale possit prævalere, multò magis id fiet, si trochleæ adhi- beantur. Ex his apparet, quid & in cæteris machinarum generibus, analogiâ servatâ, dicendum sit, ex quarum opportunâ col- locatione facilitas movendi augentur. Si enim, exempli gra- tiâ, cubus A marmoreus elevandus fuerit vecte BC, mul- tò faciliùs id fiet, si ille supponatur cubo, quàm si ex I ad perpendiculum elevaretur eodem vecte suspensum: ex I sci- licet totus cubus à vecte sustineretur; at subjectus vectis BC

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Mechanics 182 A and B are less distant from one another than when they are separated by a greater interval; and therefore the ease of moving is greater as well. But if one wishes to take a most certain experiment of this matter, and to submit the whole thing to the eyes themselves, without any suspicion of a deceptive fancy, let a small wheel be fixed in A, turning about its axis, and let equal weights D and E be sus- pended from it by a cord and hang perpendicularly; because of their equal weight they remain unmoved. Then let a small wheel be fixed in B, likewise turning about its axis, and let the weight C, equal to weight E, be taken, and be attached to the cord EBC. If you hold weight C by hand, so that it does not descend, weight E remains in its perpendicular. Now hold weight D by hand, so that it does not move at all, and let go of weight C; you will see this one descend, but weight E rise, until it hangs from B, and remains in equilibrium with weight C. Again hold weight C, and let go of weight D, and you will likewise see weight D descending, while E still rises; and so on, until weight E alone with both D and C is equal in force, as was indicated in the foregoing chapter. What therefore is accomplished by the weights D and C, it is clear can be accomplished by any equal force placed in D and C. If, then, with simple wheels it comes about that an equal weight can prevail, much more will this happen if pulleys are used. From these things it appears what must also be said in the other kinds of machines, analogy being preserved, from the suitable arrangement of which the ease of moving is increased. For if, for example, the marble cube A is to be raised by the lever BC, it will be done much more easily if it is placed under the cube than if it were to be raised from I by the same lever suspended perpendicularly: for from I the whole cube would be supported by the lever; but the lever BC

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Liber secundus. CAPUT VI. 183 B C ita cubum sustentat, ut etiam reliquo latere cubus idem subjecto plano incumbat. Quemadmodum autem non quemlibet vectem cuilibet oneri elevado parem esse om- nes intelligunt; sed habita ra- tione materiæ, ex quâ constat, congrua soliditas ei tribuenda est; ita pariter in cæteris omnibus, quæ hûc spectant (sive sint machinarum membra, sive paxilli sint aut tigilli, quibus machinæ adnectuntur) materiæ soliditatem attendendam esse manifestum est, ne frangantur. Et quidem quod ad materiam attinet, non omnium solidorum partes pari nexu cohærent, sed alia aliis fragiliora sunt: sic lignum quernum difficiliùs frangitur, quàm fraxineum aut populeum: neque enim in omni ligno æque operosa similisque staminum textura repe- ritur; cum etiam lignum idem quæqua versum findi non pos- sit pari facilitate; permagni quippe interest, recta ne juxta venarum ductum? an obliquè? sectio facienda sit. Id quod in ipsis quoque lapidibus, atque marmoribus observare quan- doque necesse est, ubi non æquè per omnes partes compacta materia venas habet scissioni maximè obnoxias. In metallis pariter eorum natura consideranda est, mollisne illa sit, ac flexibilis? an verò dura? ut eam, quam semel induit figu- ram, constanter retineat. Ex quo sit, ut pro materiæ dissi- militudine dispar etiam crassities requiratur: quis enim nesciat, quantum ligneum inter ac ferreum ejusdem molis vectem in- tersit? Verùm illud potiùs considerandum videtur, quod ad soli- ditatem ipsam spectat, etiamsi materies diversa non sit; pro variâ enim crassitudine mutatur frangendi difficultas; & quia in mole majori plures insunt partes divisioni resistentes, fran- gendi pariter difficultas augetur pro Ratione multitudinis par- tium, si cætera paria sint. Dubitare videlicet nemo potest à duplici partium dividendarum numero duplicem oriri resisten- tiam. Si cætera, inquam, sint paria; nam si filum serietum ut rumpatur, requirit vim ut unum, & decem fila serica paris crassitiei

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Book the Second. CHAPTER VI. 183 B C thus supports the cube, so that also on the remaining side the cube rests upon the same underlying plane. But just as everyone understands that not every lever is equal to every load to be raised; rather, account being taken of the material of which it is made, a suitable solidity must be assigned to it; so likewise in all other things which pertain hither (whether they are members of machines, or pins or beams by which the machines are attached) it is evident that the solidity of the material must be attended to, lest they break. And indeed, as far as the material is concerned, the parts of not all solids cohere with equal firmness, but some are more brittle than others: thus oak is broken with greater difficulty than ash or poplar; for in every kind of wood there is not found an equally laborious and similar texture of fibres; since even the same wood cannot be split every way with equal ease; for it makes a great difference whether the cut must be made straight, along the course of the veins, or obliquely. This must sometimes be observed also in the stones themselves and in marbles, where the material, not being equally compact in all its parts, has veins most liable to splitting. In metals likewise their nature must be considered, whether it be soft and flexible, or rather hard, so that it may steadfastly retain the shape once it has assumed. Whence it comes about that, according to the dissimilarity of the material, a different thickness is also required: for who does not know how much difference there is between a wooden and an iron lever of the same size? But that rather seems to be considered which concerns solidity itself, even if the material is not different; for with varying thickness the difficulty of breaking is altered; and because in a greater bulk there are more parts resisting division, the difficulty of breaking likewise increases in proportion to the number of the parts, if the other things are equal. Doubtless no one can doubt that from a doubled number of parts to be divided there arises a double resistance. If, I say, the other things are equal; for if a silk thread, in order that it may be broken, requires a force of one, and ten silk threads of equal thickness

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184 Mechanicorum crassitiei ac longitudinis parallela simul posita requiram vim decuplam; si in unum funiculum decem illa fila ritè contorqueantur, multò majorem vim quàm decuplam requiri, ut funiculus frangatur, manifestum est: quemadmodum & ligneis tigillus multo validiùs resistit fractioni, quàm virgarum fasciculus eidem tigillo æqualis; major est enim particularum unio, ubi in unum corpus coalescant, quàm ubi plura minora corpora constituantur. Hinc si fuerint duo parallelepipeda quadrata A & B, quorum latera sint in Ratione quadruplâ, altitudines verò A C, & B D æquales; constat ex 32. 1.11 ea esse inter se ut bases; bases autem sunt quadrata laterum; igitur parallelepipedum B est sedecuplum parallelepipedi A. Finge sexdecim parallelepipeda ipsi A æqualia in fasciculum colligata, & scissionem faciendam juxta lineam O S vi oneris in O positi: certum est faciliùs frangi posse sexdecim illa parallelepipeda, quàm parallelepipedum B illis omnibus æquale; ut enim scindatur, curvari oportet vi oneris incumbentis; illa autem sexdecim faciliùs curvantur quàm ipsum B. Id quod manifestum fiat, si virgam ex salicto decerpens, eamque leniter inflectens observes, quâ quidem parte virga curvata est, tenerum corticem in rugas assurgere atque crispari, quâ verò parte convexa est, corticem distrahi atque distendi. Ex quo facilè arguimus, quid durioribus corporibus contingat, quæ non adeò manifestè corrugari possunt; flecti scilicet nequeunt, quin aliqua fiat interiorum partium compressio, & exteriorum distractio. Hinc in parallelepipedo B, quod flecti intelligitur, ut scindatur, partes, quæ circa O, comprimuntur; quæ verò circa S, distrahuntur: huic autem motioni repugnant omnes particu- læ

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184 Mechanicorum I shall require a force ten times greater for thickness and length placed parallel together; if those ten strands are properly twisted into one cord, it is evident that much greater force than tenfold is required for the cord to be broken: just as a wooden beam resists fracture much more strongly than a bundle of rods equal to that beam; for the union of the particles is greater, when they cohere into one body, than when several smaller bodies are formed. Hence, if there are two square parallelepipeds A and B, whose sides are in a ratio of four to one, but whose heights A C and B D are equal; it is clear from 32. 1.11 that they are to one another as their bases; but the bases are squares of the sides; therefore the parallelepiped B is sixteen times the parallelepiped A. Imagine sixteen parallelepipeds equal to A, gathered together into a bundle, and a split to be made along the line O S by the force of a load placed in O: it is certain that the sixteen parallelepipeds can be broken more easily than the parallelepiped B equal to them all; for in order that it be split, it must be bent by the force of the load pressing upon it; but those sixteen bend more easily than B itself. This may be made clear if, taking a willow rod and bending it gently, you observe that on the side where the rod is bent, the tender bark rises into wrinkles and becomes crumpled, whereas on the convex side the bark is stretched and drawn out. From this we can easily infer what happens in harder bodies, which cannot be so clearly wrinkled; namely, they cannot be bent without some compression of the inner parts and stretching of the outer parts. Hence in the parallelepiped B, which is understood to be bent in order that it may be split, the parts around O are compressed; those around S are stretched: and all the partic- les resist this motion.

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Liber secundus. CAPUT VI. 185 Iæ vi nexûs, quo unaquæque cum sibi proximè cohærentibus particulis colligatur. Cum autem sexdecim illa parallelepipeda minora non sint invicem connexa, quemadmodum particulæ omnes parallelepipedi B in unam molem coaluerunt, constat pauciores nexus faciliùs, quàm plures, dissolvi. Hoc verò ut pleniùs atque apertiùs explicetur, intellige solidum longiusculum R S in plures tenues laminas plano R I parallelas divisum, sibique ita vicissim congruentes, ut earum extremitates constituant planum H I. Omnes hasce laminas secundùm extremitates fulcris impositas pondus super D C constitutum adeò premat, ut curvari aliquantulum cogantur. Observabis illicò extremitates illas non jam ampliùs in eandem planitiem H I exæquari; sed eas quidem laminas, quæ cavitatem spectant, magis curvari; minùs verò eas, quæ convexitati respondent, ac proptereà extimæ laminæ extremitatem ab extremitate intimæ laminæ, quæ ponderi imposito cohæret, magis recedere, quàm intermediarum extremitates. Constat itaque in hoc motu singularam laminarum particulas, dum curvantur, non iis respondere adhærentis laminæ particulis, quas priùs contingebant, cùm omnis curvitatis expertes erant, atque faciliùs potuisse singulas laminas moveri, quia nullo nexu invicem copulantur. Quòd si ex iis unum solidum R S planè integrum coalescat, manifestum est planitiem H I permanere, ac propterea, dum curvatur, necesse est, ut interiores particulæ invicem connexæ distrahantur, cum nequeant aliæ ab aliis secedere, quemadmodum in laminis contingere observavimus. Hinc oritur major solidi, quàm laminarum, resistentia, ne frangatur. Non negarim tamen aliquando satius esse duobus mediocribus tigillis uti, quàm crassiore tigno illis æquali; quia nimirum alterutro labem patiente rimaivè agente, alter faciliùs integer perseverat; in crassiore autem tigno, si rimam du- A a

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Book Second. CHAPTER VI. 185 Of the manner of the connections by which each particle is joined with those nearest to it. But since those sixteen smaller parallelepipeds are not connected with one another, just as all the particles of the parallelepiped B have coalesced into one mass, it is evident that fewer connections are more easily dissolved than more numerous ones. Now, that this may be explained more fully and clearly, suppose a rather long solid R S divided into several thin plates parallel to the plane R I, and so fitting one another alternately that their extremities form the plane H I. If all these plates, being supported at their ends on props, have a weight placed above D C pressing upon them so strongly that they are forced to bend somewhat, you will immediately observe that those extremities are no longer level with one another in the same plane H I; but the plates looking toward the hollow bend more, and those corresponding to the convexity less, and therefore the extremity of the outermost plate recedes more from the extremity of the innermost plate, which is joined to the imposed weight, than do the extremities of the intermediate plates. It is thus clear that in this motion the particles of individual plates, while they are bending, do not correspond to the particles of the adjoining plate which they previously touched when all were free from curvature; and that each plate could more easily be moved because they are not linked together by any bond. But if of these one solid R S is made, wholly continuous, it is plain that the plane H I remains unchanged; and therefore, when it bends, it is necessary that the interior particles, being connected with one another, be stretched apart, since they cannot separate from one another, as we observed to happen in the plates. Hence arises the greater resistance of the solid, as compared with the plates, to prevent it from breaking. Yet I would not deny that sometimes it is better to use two moderate-sized timbers than a thicker timber equal to them; because if either one suffers a defect or crack, the other more easily remains sound; but in a thicker timber, if a crack be du- A a

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Mechanicorum 186 cere occoeperit, periculum est, ne malum serpat juxta venarum aut fibrarum ductum. Cæterum sublato hujusmodi periculo, ubi reliqua paria sint, crassiora corpora difficiliùs franguntur. Quare solidorum resistentia, ne frangantur, major est quam pro Ratione sectionum; hæc siquidem Ratio sectionum servari quidem intelligitur, si limâ aut serrâ secari corpora oporteat; illæ enim tantummodo particulæ resistunt, quæ sectionem admittunt; at ubi de fractione agitur, quæ præter motum particularum, quæ dividuntur, motum etiam aliquem exigit aliarum, quas comprimi aut distrahi opus est, plus, minùs, pro Ratione vicinitatis, longè alia est Ratio, pro ut compressio illa atque distractio particularum faciliùs aut difficiliùs perfici poterit. Hoc autem ex ipsâ figurâ potissimum pendet. Solidi enim R S sectio C D E eadem quidem est, sivè illud circà D E longiorem lineam, sivè circà C D breviorum, curvari debeat, ut frangatur; sed non eadem est in fractione C D ac in fractione D E frangendi difficultas; nam cum propiores sint puncto D partes, quæ ad C, quàm quæ ad E sitæ sunt, constat has quidem magis cum circà lineam C D curvatur solidum, illas verò, cùm circà lineam D E curvatur, minùs distrahi oportere, ut fractio sequatur. Quò autem magis distrahi debent particulæ, quæ ex D versus E recedunt, magis interim comprimi necesse est eas, quæ ad D accedunt secundùm lineam R O in plano R I. Major igitur est difficultas, si circà breviorem lineam C D curvetur, & fractio secundùm longiorem lineam D E sequatur, quàm si contrà curvetur circà longiorem D E, & fractio sit juxta breviorem C D. Iam igitur si duo solida invicem comparentur, quæ ejusdem sint materiæ ejusdemque longitudinis, & in pari ab extremitatibus distantiâ frangi oporteat, itatuatur in utroque solido punctum fractionis, per quod intelligatur planum secans similiter inclinatum, faciensque in utroque solido superficies, quas vocemus Bases. Item planum per quod movetur Potentia vim frangendi habens, ita productum intelligatur, ut Basibus prædictis simili inclinatione occurrens describat sectionum lineas, quas vocemus Crassities. Ut si fuerint duo solida

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Mechanics 186 if the break should begin, there is danger lest the damage spread along the course of the veins or fibres. But this danger removed, where the rest are equal, thicker bodies are broken with greater difficulty. Therefore the resistance of solids, so that they may not be broken, is greater than in proportion to their sections; for this ratio of sections is understood to be preserved only if bodies have to be cut with a file or saw: in that case only those particles resist which admit the cut; but when fracture is in question, which, besides the motion of the particles that are divided, also requires some motion of the others, which must be compressed or drawn apart, the ratio is altogether different, more or less, according to the proximity of the parts, just as the compression and separation of the particles can be accomplished more easily or more difficultly. This depends chiefly on the figure itself. For in the solid R S the section C D E is indeed the same, whether it must be bent around D E, the longer line, or around C D, the shorter; but the difficulty of breaking is not the same in the fracture at C D as in that at D E; for since the parts situated at C are nearer to point D than those situated at E, it is clear that these latter must indeed be bent more when the solid is curved around line C D, but those must be drawn apart less when it is curved around line D E, if fracture is to follow. And the more the particles which recede from D toward E must be drawn apart, the more in the meantime it is necessary that those which approach D be compressed, along line R O in plane R I. Therefore there is greater difficulty if it be bent around the shorter line C D, and fracture follows along the longer line D E, than if, on the contrary, it be bent around the longer D E, and the fracture be near the shorter C D. Now therefore, if two solids are compared with one another, of the same material and the same length, and must be broken at an equal distance from the extremities, let in each solid a point of fracture be established, through which let a cutting plane be understood, similarly inclined, and making in each solid surfaces which we shall call Bases. Likewise let the plane through which the Power possessing the force of breaking moves be understood as so drawn that, meeting the aforesaid Bases with a similar inclination, it describes the lines of section, which we shall call Thicknesses. For example, if there are two solids

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Liber secundus. CAPUT VI. 187 da CD & EF æqualis longitudinis, parieti infixa secundùm æquales partes CI & EH, ut in punctis I & H fiat fractio, ex hypothesi. Si per ea puncta agantur plana similiter inclinata, erunt superficies IL & HM, quas vocamus hîc Bases. Iam in extremitatibus D & F æquè remotis à punctis I & H sint Potentiæ vim frangendi habentes, & per lineam motûs hujusmodi Potentiarum intelligantur plana cum simili inclinatione occurrentia basibus IL & HM, ponamusque communes horum planorum sectiones esse lineas parallelas, & æquales lineis IN & HO; quas sectiones vocamus Crassities solidorum, atque pro earum mensurâ usurpamus lineas IN & HO. Cum itaque frangendi difficultas oriatur tùm ex numero partium, quæ separandæ sunt, has autem ipsæ Bases IL & HM definiunt, tùm ex violento motu distractionis partium, qui ex ipsâ solidorum crassitie IN, & HO dignoscitur; illud consequens est, quòd Resistentiæ solidorum Ratio ea sit, quæ ex Ratione Basium, & Ratione Crassitierum componitur. Hinc est quòd si Bases fuerint similes, & quæ est Ratio laterum homologorum, ea etiam sit Crassitierum Ratio, resistentiæ ad fractionem invicem comparatæ erunt in Ratione triplicatâ laterum homologorum; ac propterea cylindrorum resistentia ad fractionem erit in Ratione triplicatâ Diametrorum, seu Crassitierum. Hanc, de quâ hactenus nobis sermo fuit, Resistentiam absolutam dicimus, quam solidum habet, ne dividatur: quò enim plures partes debent præter naturam comprimi, aut distrahi, plures sunt resistentiæ; & quò magis hoc motu debent momento eodem præter naturam moveri, eò etiam magis resistunt: quâ igitur ratione plures sunt resistentes, & quâ Ratione magis resistunt, tota resistentiæ ratio componitur; quæ ex ipsâ corporis soliditate pendet, nullâ habitâ ratione longitudinis ipsius solidi: Propterea Absoluta dicitur. Nam si longitudines frangendorum corporum comparemus, quæ suâ varietate mutant frangendi difficultatem, aut facilitatem, re- A a 2

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Book Two. Chapter VI. 187 that CD & EF of equal length, fixed in the wall according to equal parts CI & EH, so that fracture occurs at the points I & H, by hypothesis. If through those points planes similarly inclined be drawn, there will be the surfaces IL & HM, which we here call Bases. Now at the extremities D & F, equally distant from the points I & H, let there be Potencies having the power of breaking, and by the line of motion of such Potencies let there be understood planes with a similar inclination meeting the bases IL & HM, and let us suppose the common sections of these planes to be lines parallel to, and equal to, the lines IN & HO; which sections we call the Thicknesses of the solids, and we use the lines IN & HO as their measure. Since, therefore, the difficulty of breaking arises both from the number of parts that must be separated, which the Bases IL & HM themselves determine, and from the violent motion of the parts’ separation, which is recognized from the very thickness of the solids IN & HO, it follows that the Ratio of the Resistance of solids is that which is composed from the Ratio of the Bases and the Ratio of the Thicknesses. Hence it is that if the Bases are similar, and the Ratio of the homologous sides is such, then the Ratio of the Thicknesses being also such, the resistances to fracture compared with one another will be in the triplicate Ratio of the homologous sides; and therefore the resistance of cylinders to fracture will be in the triplicate Ratio of the Diameters, or of the Thicknesses. This Resistance, of which we have hitherto spoken, we call the absolute Resistance which a solid has, lest it be divided: for the more parts there are that must be compressed or drawn apart contrary to nature, the greater are the resistances; and the more, by this motion, they must at the same instant be moved contrary to nature, the more they also resist: therefore, in whatever ratio there are more resisting parts, and in whatever Ratio they resist more, the whole ratio of resistance is composed; which depends on the very solidity of the body, no regard being had to the length of the solid itself: therefore it is called Absolute. For if we compare the lengths of bodies to be broken, which by their variety change the difficulty or ease of breaking, re- A a 2

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Mechanicorum sistentia hæc dicenda erit Respectiva; quæ aliquando ea esse potest, ut corpus majore resistentiâ absolutâ præditum reddatur magis obnoxium fractioni; longitudo siquidem auget frangendi facilitatem: ideo autem Respectivam dicimus, quia comparatè ad momenta potentiæ sumitur; hæc verò momenta ex variâ longitudine, seu distantia à puncto fractionis pendere manifestum est. Sit enim solidum AB, quod ita flectatur, ut fiat fractio CD: Potentia movens in B constituta dum perficit spatium BE, distractio particularum solidi sit solùm per spatium CD (aut veriùs per CHD, nam etiam partes inter C & H distrahuntur; Sed hîc claritatis gratiâ solùm extremæ CD considerantur) quod est multo minus spatio BE secundùm Rationem HD ad HE. At si solidum frangendum sit AF, aut si sit totum AB, tamen Potentia movens sit solùm applicata in F, Potentia perficiens spatium FG (quod est minus quàm BE in Ratione HF ad HB) major esse debet quàm Potentia in B secundùm Rationem Reciprocam motuum BE & FG, ut sequatur idem motus distractionis partium CD; nam ex 8.1.5. minor est Ratio FG ad CD, quàm sit Ratio BE ad eandem CD. Constat igitur à longitudine augeri facilitatem frangendi, ac proinde Resistentiam hanc Respectivam esse secundùm Reciprocam Rationem longitudinum. Ex quo obiter apparet, cur solida Horizonti perpendicularia magis resistant fractioni, si potentiæ motus, seu conatus, sit ad perpendicularum Horizonti: quia videlicet in hujusmodi motu ad perpendicularum æqualiter moveri oportet Potentiam cum solidi particulis, quæ distrahi aut comprimi debent: ut autem Potentia superet vim restititivam, aut major esse debet Ratio motûs potentiæ ad motum corporis resistentis, quàm sit Ratio virium resistendi ad virtutem movendi, aut virtus movendi absolutè major esse debet vi resistendi: Cum itaque in motu perpendiculari intercedere non possit motuum inæqualitas, necesse est virtutem movendi vehementer augeri, ut superet vim, quâ

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The resistance in mechanics here spoken of shall be called relative ; which at times may be such that a body endowed with greater absolute resistance is made more liable to fracture, for length certainly increases the ease of breaking. We call it relative, because it is taken comparatively with the moments of force; and these moments clearly depend on the varying length, or distance from the point of fracture. Let there be a solid AB, so bent that fracture occurs at CD: the moving force applied at B, while it traverses the space BE, produces in the particles of the solid a separation only through the space CD (or rather through CHD, for even the parts between C and H are drawn apart; but here, for the sake of clarity, only the extreme points CD are considered), which is much less than the space BE in the ratio of HD to HE. But if the solid to be broken be AF, or if it be the whole AB, yet the moving force be applied only at F, then the force traversing the space FG (which is less than BE in the ratio of HF to HB) must be greater than the force at B, according to the reciprocal ratio of the motions BE and FG, so that the same motion of separation of the parts CD may follow; for from 8.1.5 the ratio of FG to CD is less than the ratio of BE to the same CD. It is therefore clear that the ease of breaking increases with length, and consequently that this resistance is relative according to the reciprocal ratio of lengths. From this it also appears incidentally why solids perpendicular to the horizon resist fracture more when the force of motion, or effort, is directed along the perpendicular to the horizon: namely, because in such motion the force must move equally with the particles of the solid, which are to be torn apart or compressed; but in order that the force may overcome the restoring force, either the ratio of the motion of the force to the motion of the resisting body must be greater than the ratio of the resisting forces to the moving power, or the moving power must be absolutely greater than the resisting force. Since, therefore, in perpendicular motion no inequality of motions can intervene, it is necessary that the moving power be greatly increased, so that it may overcome the force by which

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Liber secundus. CAPUT VI. 189 quâ particulæ solidi invicem connexæ repugnant, ne distra- hantur, aut comprimantur. Hinc ex hastâ ad perpendiculum suspensâ pendebit ingens saxum, & tigillum perpendiculariter terræ insistentem pre- met moles, penè dixerim, immensa, citrà hastæ aut ti- gilli fractionem: quia omnes hastæ atque tigilli partes & æqualiter cum onere suspenso aut incumbente moveri de- berent, & omnes æqualiter resistunt distractioni aut com- pressioni: At si ad horizontem inclinata aut parallela fue- rint hujusmodi solida (hasta videlicet atque tigillus) non est æqualis omnium partium distractio aut compressio, mi- nùs enim distrahuntur, quæ puncto H proximæ sunt, quam quæ ad D accedunt (concipe H in media crassitie) con- trà verò illæ magis, hæ minùs comprimuntur; quemad- modum neque motui distractionis aut compressionis esset æqualis motus oneris deorsùm urgentis in hastæ, vel tigil- li non perpendicularium extremitate constituti, sed multò major esset hîc oneris motus. Quoniam verò rerum natu- ra magis repugnat corporum penetrationi, ad quam quodam- modo accedere videtur compressio, quàm corporum unito- rum divisioni, ubi vacui metus absit; hinc est majorem molem faciliùs sustineri à fulcro ad perpendiculum subjecto, quàm suspendi ex solido perpendiculari citrà fractionis pe- riculum. Quamvis negandum non sit ad hujusmodi facili- tatem, quam experimur in sustinendo potiùs, quàm in re- tinendo onere, conferre plurimum, quòd tellus, cui ful- crum infigitur, demùm non subsidit; at laqueare seu for- tix ex quo solidum pendet onere prægravatum, tantam gravitatem non ita facilè ferre potest. Quare ad tollenda in superiores ædificiorum partes ingentia saxa multo cau- tiùs atque tutiùs ij operantur, qui longam trabem, aut plu- ra tigna ritè connexa, quasi navis malum rudentibus us- quequeque firmatum, ne à perpendiculo deflectat, sta- tuunt, cui superiorem trochleam adnectant; quàm qui tra- bem Horizonti parallelam parieti infigunt ad idem munus præstandum; hæc siquidem horizonti parallela magis fractio- ni obnoxia est, quàm perpendicularis; præterquam quod parietem aliquatenus labefactare potest, cum habeat ratio- A a 3

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Book Second. CHAPTER VI. 189 in which the particles of the solid, joined to one another, resist being torn apart or compressed. Hence from a rod suspended perpendicularly there will hang a huge stone, and an immense mass, one may almost say, pressing upon a beam standing upright on the earth, without the rod or beam being broken; because all the parts of the rod and beam ought equally to move with the suspended or incumbent load, and all equally resist tension or compression. But if such solids (namely, the rod and the beam) are inclined to the horizon or parallel to it, there is not an equal tension or compression of all the parts; for those near the point H are drawn apart less than those approaching D (imagine H in the middle thickness); on the contrary, those are compressed more, these less. In like manner, neither would the motion of the load pressing downward at the end of a rod or beam not set perpendicular be equal to the motion of tension or compression, but here the motion of the load would be much greater. But since the nature of things resists more the penetration of bodies, to which compression in some measure seems to approach, than the division of bodies united together, where fear of a vacuum is absent; hence it is that a greater mass is more easily sustained by a support placed underneath it perpendicularly, than suspended from a vertical solid without danger of breaking. Although it must not be denied that much contributes to this ease, which we experience in sustaining rather than in retaining a load, namely, that the ground into which the support is fixed does not in the end give way; yet the ceiling or roof from which the solid hangs, when heavily burdened with a load, cannot so easily bear so great a weight. Therefore, in order to raise huge stones to the upper parts of buildings, those work much more cautiously and safely who set up a long beam, or several beams duly joined together, like the mast of a ship firmly secured with ropes throughout, lest it deviate from the perpendicular, and to which they attach the upper pulley; than those who fix a beam parallel to the horizon into a wall for accomplishing the same task; for this, being parallel to the horizon, is more liable to fracture than a perpendicular one; besides, it can in some measure weaken the wall, since it has a relation A a 3

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Mechanicorum 190 nem vectis in superiora propellentis saxo deorsum urgente; nisi huic periculo ex arte obviam eatur. Comparatis itaque invicem solidorum frangendorum longitudinibus, hoc est intervallis inter fractionum puncta & locum, ubi potentia vim frangendi habens constituta intelligitur, quò major est longitudo, eò minor est resistentia solidi, ne frangatur. Qua propter ubi duo data solida conferantur, quæcumque demùm illa sint, non solùm eorum Resistentia Absoluta, quæ ex Rationibus Basium, & Crassitierum componitur, attendenda est, sed etiam Resistentia Respectiva, quæ ex longitudinibus pendet: atque adeò adæquata Ratio resistentiæ, ne frangantur, ea est, quæ componitur ex Rationibus Basium & Crassitierum atque ex Ratione longitudinum Reciprocè sumptarum: cùm enim longitudini majori respondeat minor resistentia, manifestum est longitudinum Rationem esse Reciprocè sumendam, ut resistentiæ, quæ ex illis oritur, Ratio habeatur. Hinc est fieri aliquando posse, ut solidum crassius minùs resistat fractioni, quàm subtilius, si hoc breve sit, illud verò valdè longum, si videlicet longitudo crassioris ad longitudinem subtilioris Rationem habeat majorem, quàm sit ea, quæ ex Rationibus Basium, & Crassitierum componitur. Sic si duo fuerint cylindri, & alter triplo crassior fuerit reliquo, sed etiam trigecuplo longior fuerit illo, minùs etiam fractioni resistet; quia resistentia absoluta majoris cylindri ad minorem est ut 27 ad 1, sed resistentia Respectiva ejusdem majoris ad minoris resistentiam pariter respectivam est ut 1 ad 30: Ratio ergo ex his Rationibus 27 ad 1, & 1 ad 30 Composita, est Ratio 27 ad 30, hoc est 9 ad 10, ac propterea major cylindrus resistit fractioni ut 9, minor verò fractioni resistit ut 10. Desine jam mirari, si quando paxillum maximis viribus resistere videris; quia nimirùm potentia, quæ motum conatur, proximè applicata est parieti aut plano, cui paxillus infigitur: quòd si remotior illa fuerit, etiam minùs hic resistet. Sic defixo in terram paxillo AB, cui funis A C alligatur, experientia docet paxillum eò resistere validiùs, quò propriùs ad A alligatur funis, debiliùs autem resistere, quò magis

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Mechanics 190 of a lever driving upward under a stone pressing downward; unless this danger is met by art. Comparing therefore the lengths of solids to be broken with one another, that is, the intervals between the points of fracture and the place where the power that has the force of breaking is understood to be applied, the greater the length, the less the resistance of the solid to being broken. Wherefore, when two given solids are compared, whatever they may be, not only their Absolute Resistance, which is composed from the Ratios of the Bases and Thicknesses, must be considered, but also the Relative Resistance, which depends on the lengths: and thus the equated Ratio of resistance, so that they are not broken, is that which is composed from the Ratios of the Bases and Thicknesses and from the Ratio of the lengths taken reciprocally: for since to a greater length a lesser resistance corresponds, it is evident that the Ratio of lengths must be taken reciprocally, in order that the Ratio of the resistance arising from them may be had. Hence it comes about that it can sometimes happen that a thicker solid resists fracture less than a more slender one, if this be short, but that one very long, if indeed the length of the thicker bears to the length of the more slender a Ratio greater than that which is composed from the Ratios of the Bases and Thicknesses. Thus if there be two cylinders, and one were three times thicker than the other, but also thirty times longer than it, it will also resist fracture less; because the absolute resistance of the larger cylinder to the smaller is as 27 to 1, but the Relative resistance of the same larger one to the relative resistance of the smaller is likewise as 1 to 30: therefore the Ratio composed from these Ratios, 27 to 1 and 1 to 30, is the Ratio 27 to 30, that is 9 to 10, and therefore the larger cylinder resists fracture as 9, while the smaller resists fracture as 10. Cease now to wonder if at times you see a peg resisting the greatest forces; because, namely, the power that attempts the motion is applied very near to the wall or plane to which the peg is fixed: but if it be farther away, this will also resist less. Thus, with a peg AB fixed in the earth, to which a cord AC is tied, experience teaches that the peg resists more strongly the more closely the cord is tied to A, but resists more weakly the more

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Liber secundus. CAPUT VI. 191 magis ad B accedit; in A nimirùm motus potentiæ trahentis vix excederet motum pa- xilli, qui ibi flectere- tur ex hypothesi; at fune in B posito, po- tentia ibi constituta, & per funem applica- ta multò velociùs mo- veretur, quàm paxilli partes propè A, quæ ibi flecterentur. Quòd si loci conditio, aut ipsa oneris movendi constitutio id exigat, ut funis propè B alligetur, & de paxilli A B firmi- tate dubitetur, paxillum alterum D E paulò remotiorem com- modo loco depange ita, ut funis primùm in D firmetur, de- inde circa B convolutus extendatur, pro ut operis faciendi ra- tio fieret. Eâdem ratione si tigillus, ex quo onus dependere debet, pa- rieti sit infixus, & sit G H, fractioni magis erit obnoxius, quò propriùs accedet pondus ad H: propterea aut ei subjicitur brevior tigillus I R omninò contiguus, aut supponitur fulcrum O S in- clinatum; quod fractionem eò va- lidiùs impediet, quò minùs dista- bunt H & S, & quò acutior fue- rit angulus, quem fulcrum S O cum pariete constituit, seu, quod eôdem recidit, quò magis ad recti anguli quantitatem acce- det angulus G S O. Quæ omnia ita ex dictis aperta sunt, ut ulte- riori explicatione non egeant. Sed & illud hîc, ubi de Resistentiâ Respectivâ sermo est, adjiciendum videtur, quòd ex solâ majori longitudine hæc non minuitur, nisi cùm longitudo solidi ad perpendiculum insistit Horizonti;

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Book Second. CHAPTER VI. 191 more nearly approaches B; for in A, indeed, the motion of the attracting power would scarcely exceed the motion of the peg, which there would be bent according to the hypothesis; but if the rope were placed at B, the power being established there, and applied through the rope, would be moved much more swiftly than the parts of the peg near A, which would there be bent. But if the condition of the place, or the very arrangement of the burden to be moved, should require it, that the rope be tied near B, and if one should doubt the firmness of the peg A B, let another peg D E, somewhat more remote, be driven in at a convenient place, so that the rope may first be fastened at D, and then, being wound around B, may be extended, as the reason for the work to be done would require. In the same way, if the small beam from which the burden is to hang be fixed in a wall, and be G H, it will be more exposed to breaking, the nearer the weight approaches H: therefore either a shorter beam I R, altogether contiguous to it, is placed beneath it, or an inclined prop O S is set under it; this will prevent break- ing the more effectively, the less H and S are distant from one another, and the more acute the angle shall be which the prop S O makes with the wall, or, which amounts to the same thing, the more nearly the angle G S O shall approach the quantity of a right angle. All these things are so clear from what has been said that they do not need further explanation. But this also, where the discussion is of Relative Resistance, seems to be added, namely, that by mere greater length this is not diminished, except when the length of the solid stands perpendicular to the horizon;

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Mechanicorum 192 Horizonti; tunc enim gravitas ipsa solidi tota incumbit subjecto plano; & tantùm Potentia oblique atque in transversum trahens applicata extremitati longioris solidi plus habet momenti, quàm applicata extremitate brevioris, quia velociùs, & faciliùs movetur secundùm Rationem longitudinum illarum. At quando solida sunt horizonti parallela, aut ad illum ita inclinata, ut centrum gravitatis partis illius, quæ erumpit ex corpore, cui solidum infigitur, non immineat basi sustentationis, non sola longitudo attendenda est, sed & ipsa gravitas, quæ etiam nullo addito extrinseco motore sua habet momenta, quibus deorsum connititur. Ex quo fit pro majori gravitate etiam frangendi facilitatem augeri, ipsa nimirum gravitas est potentia conjuncta, quæ augetur pro ratione materiæ; materia autem augetur pro ratione longitudinis (cætera siquidem paria esse hîc claritatis gratiâ, ponamus) ac propterea longius prisma comparatum cum breviori prismatic, eo quòd majorem habeat gravitatem, minùs resistit fractioni secundùm Reciprocam Rationem longitudinum. Atqui Ratio motûs hujusmodi Potentiæ conjunctæ est secundùm Rationem longitudinum, & ex dictis Ratio Resistentiæ in ordine ad hujusmodi motum est permutatim ac Reciprocè secundùm eandem longitudinum Rationem: igitur Ratio duplicatur, & resistentia longioris ad resistentiam brevioris est secundùm subduplicatam Rationem longitudinum reciprocè sumptarum. Id quod etiam hinc constat, quia cùm singula illius longirudinis puncta suam habeant gravitatem, sua omnibus insunt momenta pro Ratione distantiæ à puncto quod est veluti centrum motûs; ergo aggregata momentorum sunt ut sectores ab illis longitudinibus tanquam à Radiis descripti: sunt autem similes sectores in duplicatâ Ratione Radiorum. Quare si longitudines sint ut 3 ad 2, Resistentia respectiva longioris ad resistentiam brevioris est ut 4 ad 9. Tota igitur solidorum resistentia, ne frangantur, componitur ex Rationibus Basium, & Crassitierum, & ex subduplicatâ Ratione longitudinum permutatim ac reciprocè sumptarum. Ex his itaque, quæ de solidorum resistentiâ, ne frangantur, hactenùs disputata sunt, conjecturam facilè accipiet prudens

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Mechanics 192 Horizontally; for then the whole weight of the solid itself rests upon the supporting plane; and only the force drawn obliquely and transversely, applied to the extremity of the longer solid, has more moment than when applied to the extremity of the shorter one, because it is moved more quickly and more easily according to the ratio of those lengths. But when solids are parallel to the horizon, or inclined to it in such a way that the center of gravity of that part which projects from the body into which the solid is fixed does not overhang the base of support, length alone is not to be considered, but the weight itself also, which, even without any added external mover, has its own moments by which it tends downward. From this it follows that, as weight increases, the ease of breaking also increases; namely, weight itself is the combined force, which increases in proportion to the material; but material increases in proportion to length (for the sake of clarity, let us here suppose all other things equal), and therefore a prism of greater length, compared with a shorter prism, because it has greater weight, resists fracture less, in the reciprocal ratio of the lengths. But the ratio of the motion of this kind of combined force is according to the ratio of the lengths, and from what has been said the ratio of resistance in relation to this kind of motion is alternately and reciprocally according to the same ratio of lengths: therefore the ratio is doubled, and the resistance of the longer to the resistance of the shorter is according to the subduplicate ratio of the lengths taken reciprocally. This is also evident from the following, because since each point of that length has its own weight, in all of them there are moments according to the ratio of the distance from the point which is as it were the center of motion; therefore the aggregated moments are as sectors described from those lengths as from radii: but similar sectors are in the duplicate ratio of the radii. Wherefore if the lengths are as 3 to 2, the respective resistance of the longer to the resistance of the shorter is as 4 to 9. The total resistance, then, of solids, so that they may not break, is composed from the ratios of the bases and the thicknesses, and from the subduplicate ratio of the lengths taken alternately and reciprocally. From these things, therefore, which have thus far been discussed concerning the resistance of solids, lest they break, the prudent man will easily gather a conjecture

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Liber secundus. CAPUT VI. 193 prudens machinator, quàm solida & crassa statui debeant quæque machinarum membra, quóve loco collocanda sint, ut & materia & forma respondeant fini, in quem machinæ destinantur: neque enim satis est concinno, & eleganti dia- grammate machinam oculis repræsentasse, ejusque vires ad calculos revocâsse, quantum quidem ex machinæ figurâ col- ligitur, si demùm, instituto motu machina pondere prægra- vata luxetur. Illud tamen præterea Machinator animadvertat, oportet, quod spectat ad momenta virium, quas potentia movens exercet; neque enim sola ponderis gravitas machinam, aut corpus, cui machina alligatur, aut innititur, urget aut pre- mit, sed & ipsa potentia, dum adversùs ipsum pondus co- natur machinam movens, aliquando auget gravitatem ex oppositâ parte, adeò ut & huic & ponderi resistere debeat machina, aut id, quod machinam retinet. Si enim fuerit vectis A B in- nixus super ba- culum C D, ex B pendeat glo- bus plumbeus E, & extemi- tas A quiescat aliquo corpore retinente, ut si fuerit parieti in- fixa; solo globo E gravitante minus periculum subest fractio- nis tùm vectis, tùm baculi C D sustentantis, quàm si in A sit potentia F; cujus conatus deorsum oppositus conatui de- orsum ponderis E faciliùs curvitatem, aut etiam demùm fractionem vectis efficere potest in I, ut patet; immò & ba- culus C D sustentans vectem, non solùm momenta ponderis E, sed & momenta Potentiæ F, quæ in I uniuntur, in se recipit; atque adeò utrisque ferendis par esse debet. Simile quiddam observare est, si ex orbiculo O, in clavo M suspenso, circà suum axem versatili, dependeat pondus S, & Potentia in R deorsum conata cogat pondus S ascendere: certum est enim ab axe orbiculi, & à clavo M sustineri non B b

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Book Two. CHAPTER VI. 193 a prudent engineer must determine how solid and how thick the several members of the machines ought to be, and in what place they should be set, so that both the material and the form may correspond to the end for which the machines are intended: for it is not enough to have represented a machine to the eye in a neat and elegant diagram, and to have reduced its forces to calculation, as far indeed as can be gathered from the shape of the machine, if at last, when the motion has been undertaken, the machine, overburdened by its weight, should be strained out of shape. The Engineer ought moreover to observe this also, which concerns the moments of the forces that the moving power exerts; for it is not only the heaviness of the weight that presses or urges the machine, or the body to which the machine is fastened or on which it rests, but the power itself also, while it strives against that very weight in moving the machine, sometimes increases the gravitation from the opposite side, so that the machine, or that which holds the machine, must resist both this and the weight. For if there be a lever A B resting on a staff C D, and from B hang a leaden ball E, while the end A remains at rest by some retaining body, as if it were fixed into a wall, with only the ball E weighing down, there is less danger of breaking both the lever and the staff C D supporting it than if at A there be the power F; whose downward effort, opposed to the downward effort of the weight E, can more easily produce a bending, or even at length a breaking of the lever at I, as is evident; indeed the staff C D supporting the lever receives into itself not only the moments of the weight E, but also the moments of the Power F, which meet in I; and therefore it ought to be equal to bearing both. Something similar may be observed if from a small wheel O, suspended on a nail M and revolving around its own axis, there hang a weight S, and a Power at R, striving downward, force the weight S to ascend: for it is certain that it is held up by the axis of the wheel and by the nail M, not

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Mechanicorum solum pondus S, sed & Potentiam, quæ est in R. Contrà verò si orbiculus V sit adnexus ponderi T, funis autem orbiculo insertus alligetur clavo in N, & potentia P sursum trahat, constat ab axe quidem orbiculi sustineri solum pondus T; à clavo verò N non totum pondus T sustineri, sed ejus semissem, nam etiam Potentia P sustinet pondus. Validior igitur esse debet clavus M quàm clavus N, hic enim ponderis semissem fert, ille verò plus quàm duplum. Potentia enim R major est pondere S. Quòd si tàm pondera S & T, quàm clavi M & N, atque Potentiæ R & P non in plano Verticali, sed in Horizontali constituantur, certum est pondera S & T non suspensa sed jacentia, nihil adversùs clavos M & N; aut adversùs suorum orbiculorum O & V axes conari, immò neque adversùs Potentias R & P; quandoquidem toto nisit plano subjecto incumbunt, nullâmque exercent Activam Resistentiam; sed Formalem tantummodo, quâ repugnent Potentiis moventibus: quæ quidem resistentia, tùm ex ipâ ponderum gravitate, tùm ex attritu subjecti plani componitur. Clavorum igitur M & N ea sit, oportet, soliditas atque firmitas, quæ potentiarum R & P conatibus respondeat; ne forte clavi ipsi frangantur faciliùs, aut revellantur, quàm pondera suo loco dimoveantur. Sed hæc innuisse sat fuerit, ut singula diligenter à machinatore circumspicienda esse intelligatur; neque tamen in his ad nauseam diutiùs immorandum.

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Mechanics only the weight S, but also the power, which is in R. On the other hand, if the little wheel V is attached to the weight T, and the rope inserted in the wheel is fastened by the nail in N, and the power P pulls upward, it is clear that from the axle of the wheel only the weight T is supported; but by the nail N not the whole weight T is supported, but half of it, for the power P also sustains weight. Therefore the nail M ought to be stronger than the nail N; for this one bears half the weight, while the other bears more than double. For the power R is greater than the weight S. But if both the weights S and T, and the nails M and N, and the powers R and P are placed not in the Vertical plane, but in the Horizontal, it is certain that the weights S and T, being not suspended but lying down, attempt nothing against the nails M and N, or against the axes of their wheels O and V, indeed not even against the powers R and P; since they rest upon the whole supporting plane beneath them and exert no Active Resistance; but only Formal resistance, by which they oppose the moving powers: and this resistance is composed both of the very weight of the bodies and of the friction of the supporting plane. Therefore the nails M and N must have such solidity and firmness as will answer to the efforts of the powers R and P; lest perchance the nails themselves should break more easily, or be torn out, than the weights be displaced from their places. But it will have been enough to have suggested these things, so that it may be understood that each particular case must be carefully examined by the machinist; nor, however, should we dwell on these matters longer to the point of nausea.

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Liber secundus. CAPUT VII. 195 CAPUT VII. Præstet-ne Machinam augere? an componere. EX iis, quæ de Machinarum viribus disputata sunt satis Eliquet nullum dari finitum Pondus quod data Potentia mo- vere non possit si congruens machina adhibeatur: cum etenim data sit Ratio Ponderis ad Potentiam, eo artificio Machina disponatur, ut Ratione illâ datâ fiat major Ratio motûs Potentiæ ad motum Ponderis; & Pondus cedet Potentiæ moventi. Sic vicissim si oblata fuerit machina, examinandus primùm est lo- cus, ubi Potentia applicanda est, ubi Pondus collocandum; tùm utriusque motûs rationes ineundæ: & pronunciabis majo- rem requiri rationem Potentiæ ad Pondus, quàm sit Ratio mo- tûs Ponderis ad motum Potentiæ. Sit enim ex. gr. motuum hu- jusmodi Ratio, quæ est 3 ad 8; Potentia vim movendi habens ut 3 non movebit Pondus, cujus vis resistendi, & momentum, sit ut 8; sed opus est, ut illa major sit quàm 3. At neque Po- tentiam augere potes, ut oportet, neque Ponderi quicquam de- trahere: vide igitur utrum fieri possit, ut mutetur in machinâ motuum Ratio, aut Potentiæ motum augendo, aut ponderis motum minuendo. Hinc manifestum est machinam majorem non plus afferre facilitatis præ minore, si illæ quidem omninò similes fuerint (modò utraque satis solida sit, ne fractioni sit obnoxia) mo- tuum enim Ratio eadem est in utráque. Sic Vectis 1 co pal- morum si ita ab hypomochlio distinguatur in partes ut hinc palmos 20, hinc 80 relinquat, non majorem movendi faci- litatem præbebit, quàm vectis palmorum quinque ita divi- sus ab hypomochlio, ut hinc palmus unus, hinc verò quatuor relinquantur. Ut igitur longior ille Vectis utilior accidat, si hypomochlium quidem transferri queat, remove illud à Po- tentiâ, & admove Ponderi, motuumque Ratio augebitur; pa- tet scilicet majorem esse Rationem 85 ad 15, quam 80 ad 20; Quod si verò hypomochlium ita fixum sit ac vecti adnexum, Bb 2

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Book Two. CHAPTER VII. 195 CHAPTER VII. Whether a machine should be made larger, or adjusted. From what has been discussed concerning the powers of machines, it is sufficiently clear that no finite weight exists which a given power cannot move, if a suitable machine is applied: for since the ratio of the weight to the power is given, let the machine be so arranged by that device that, with that ratio given, there may arise a greater ratio of the motion of the power to the motion of the weight; and the weight will yield to the moving power. Likewise, if a machine is presented, the first thing to examine is the place where the power is to be applied and the place where the weight is to be placed; then the ratios of the motions of each must be considered: and you will conclude that a greater ratio of power to weight is required than is the ratio of the motion of the weight to the motion of the power. For example, let the ratio of such motions be 3 to 8; a power having moving force as 3 will not move a weight whose resisting force and momentum are as 8; but it is necessary that that be greater than 3. But neither can you increase the power as needed, nor take anything from the weight: see therefore whether it may be possible for the ratio of motions in the machine to be changed, either by increasing the motion of the power or by diminishing the motion of the weight. Hence it is plain that a larger machine offers no more facility than a smaller one, if indeed they are entirely alike (provided each is sufficiently solid, so as not to be liable to breakage); for the ratio of motions is the same in both. Thus a lever of 100 palms, if it is so divided from the fulcrum into parts that 20 palms remain on one side and 80 on the other, will not provide greater ease of movement than a lever of five palms divided from the fulcrum so that one palm remains on one side and four on the other. Therefore, if that longer lever is to prove more useful, and if the fulcrum can indeed be moved, remove it from the power and bring it nearer to the weight, and the ratio of motions will be increased; for it is clear that the ratio 85 to 15 is greater than 80 to 20. But if the fulcrum is fixed in such a way as to be attached to the lever, Bb 2

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Mechanicorum 196 ut mutari loco nequeat, abscinde palmos 5 15, adeò ut hinc sint palmi 80 ut priùs, hinc autem sint palmi 14 2/17, & eadem erit Ratio, quæ est 85 ad 15. Quare breviore vecte plus ponderis movebis, quàm longiore; vis enim, quæ longiore illo 100 palmorum movebat pondus librarum 100, breviore hoc palmorum 94 2/17 movebit libras 141 2/3: Quia quamvis in utroque Vecte hypomochlium habente post palmum octuagesimum, Potentia eodem semper motu moveatur, non tamen idem est ponderis motus, qui in minore vecte minor est, in majore major, ac proinde motûs Potentiæ ad motum Ponderis Ratio major est in minore, minor in majore vecte. Quod si demùm nec hypomochlium transferre, nec vecte mutilato uti liceat, licebit sanè fustem, vel quid simile, firmiter ad alligatum Vecti adjungere, potentiamque ab hypomochlio longiùs removere: oporteret autem additamentum hujusmodi esse palmorum 33 1/3; nam ut 15 ad 85, ita 20 ad 113 1/3; adeóque totus vectis esset palmorum 133 1/3. Porrò hîc observa, quantò facilius sit ponderis motum minuere, quàm potentiæ motum augere: in allato siquidem exemplo, manente eodem potentiæ motu, minuitur ponderis motus decurtato vecte ac diminuto palmis 5 25/17; manente autem eodem ponderis motu augetur Potentiæ motus acuto vecte palmis 33 1/3: Quia nimirum in Ratione majoris Inæqualitatis sit Consequens terminus minor minuatur, aut Antecedens terminus major augeatur, fit adhuc major Inæqualitas; ut autem eadem Ratio servetur aucto Antecedente ac diminuto Consequente, manifestum est, quæ pars Consequentis integri est consequens diminutus, eam debere esse partem Antecedentis aucti Antecedentem datum: atqui Antecedens datus est major dato Consequente; igitur plus addendum est Antecedenti, quàm dematur Consequenti. Sic data sit Ratio 8 ad 6: Consequens bifariam secetur, ejusque semissis fiat novus Consequens; erit Ratio 8 ad 3 majoris adhuc inæqualitatis; hæc enim est dupla superbipartiens tertias, illa verò erat solùm sesquitertia. Ut igitur retento priori Consequente 6 sit eadem Ratio dupla superbipartiens tertias, sicut Consequens fuit bifariam divisus, ita datus Antecedens 8 est duplicandus, ut sit Ratio 16

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Mechanics 196 so that it cannot be moved from its place, cut off 5 15 palms, so that on this side there are 80 palms as before, but on that side there are 14 2/17 palms, and the ratio will be the same, which is 85 to 15. Therefore, with the shorter lever you will move more weight than with the longer one; for the force which with that longer lever of 100 palms moved a weight of 100 pounds, with this shorter one of 94 2/17 palms will move 141 2/3 pounds: because although in either lever, having the fulcrum after the eightieth palm, the Power is always moved by the same motion, yet the motion of the weight is not the same, which is smaller in the lesser lever, greater in the greater, and therefore the ratio of the motion of the Power to the motion of the Weight is greater in the lesser, smaller in the greater lever. But if at last it is not permitted either to transfer the fulcrum or to use a mutilated lever, it will surely be permitted to attach firmly to the lever a stick, or something similar, and to remove the power farther from the fulcrum: but the addition of this kind ought to be 33 1/3 palms; for as 15 to 85, so 20 to 113 1/3; and therefore the whole lever would be 133 1/3 palms. Moreover, observe here how much easier it is to diminish the motion of the weight than to increase the motion of the power: for in the example given, the motion of the weight is reduced while the same motion of the power remains, by shortening the lever and reducing it by 5 25/17 palms; but while the same motion of the weight remains, the motion of the Power is increased by an acute lever of 33 1/3 palms: because, namely, in the Ratio of greater inequality, if the consequent term is diminished, or the antecedent term increased, there is still a greater inequality; but in order that the same Ratio may be preserved by increasing the antecedent and diminishing the consequent, it is clear that the part of the whole consequent which is the diminished consequent must be the same part of the increased antecedent as the given antecedent is; but the given antecedent is greater than the given consequent; therefore more must be added to the antecedent than is taken away from the consequent. Thus let the Ratio be 8 to 6: let the consequent be divided in half, and let its half become a new consequent; the Ratio will be 8 to 3, of still greater inequality; for this is double and superpartient to the third, whereas that one was only sesquitertian. Therefore, in order that, the former consequent 6 being retained, the same Ratio may be double and superpartient to the thirds, just as the consequent was divided in half, so the given antecedent 8 must be doubled, so that the Ratio may be 16

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Liber secundus. CAPUT VII. 197 16 ad 6: plus autem est totus antecedens major qui additur, quàm sit semissis Consequentis minoris qui demitur. In re au- tem nostrâ semper Ratio motûs Potentiæ per machinam vali- dioris factæ ad motum dati ponderis est Ratio Majoris inæqua- litatis: Quapropter satius est Ponderis motum minuere, quam potentiæ motum auctâ machinâ augere. Hæc quidem, quæ in vecte proposita facilè ac in promptu est perspicere, in cæteris pariter mechanicis Facultatibus, ut in Trochleis, Cochleâ, & reliquis intelligenda sunt, ut ex iis, quæ inferiùs dicentur, suo loco manifestum fiet. Sed quoniam ad ponderis motum extenuandum certos quosdam fines ipsa machinarum materia præscribit; neque enim quemadmodum quantitatem omnem, & corporum molem in subtiliores, ac subindè subtiliores partes mente concidimus, ita etiam id re ipsâ perficere atque in praxim deducere possumus: propterea ut plurimum cogimur Potentiæ velociorem motum conciliare, ut majorem obtineat Rationem ad motum Ponderis. Quis ete- nim non incassum uti possit Vecte, cujus hypomochlium à pondere satis gravi non ampliùs distet, quàm per digitis semis- sem? aut Cochleam adhibere, cujus spiras intervallum capilla- ceum secernat? Verùm cum id duplici methodo præstare possimus, videlicet aut Machinam ipsam, specie non mutatâ, augentes, aut illam ex pluribus membris componentes, sive ejusdem generis sint, sive diversi; operæ pretium fuerit perpendere, majus-ne in augmento? an verò in compositione? compendium inveniatur. Augmentum voco (ne ullus subsit æquivocandi locus) cum ejus- dem Facultatis species immutata permanet, factâ solum partis alicujus accessione; ut si, quia Vectis justo brevior est, Poten- tiæ ab hypomochlio distantiam longiorem facias; cum Tro- chleæ adhibeantur oneri movendo impares, amplificatis locu- lamentis orbiculorum numerum augeas; quia Cochlea ob spi- rarum raritatem minùs valida est quàm oporteat, lineam ipsam ita inclines, ut spissioribus spiris circumducatur. At verò Com- posita dicitur Machina, cum invalidæ Facultati membra alia adjiciuntur, aut generis ejusdem, ut cum Vectis Vecti, Co- chleæ Coehlea, Trochleis Throchleæ adjunguntur; aut diver- si generis, ut cum facultates ipsæ permiscentur, vecti trochleas, Bb 3

Transcription: Translated (English)

Book Two. CHAPTER VII. 197 16 to 6: but the antecedent is the greater, by as much as the entire one added is greater than the half of the lesser consequent that is taken away. But in our case, the ratio of the motion of the Power, through a machine made stronger, to the motion of the given weight is always the ratio of the greater inequality: wherefore it is better to lessen the motion of the Weight than to increase the motion of the Power by enlarging the machine. These things, which are easily and at once seen in the lever, must likewise be understood in the other mechanical Faculties, as in pulleys, the screw, and the rest, as from what shall be said below in its place it will plainly appear. But since the very material of the machines prescribes certain limits for reducing the motion of the weight; for just as we mentally divide every quantity and the mass of bodies into subtler and ever subtler parts, so also we cannot in reality carry this out and put it into practice: therefore for the most part we are forced to join to the Power a swifter motion, so that it may obtain a greater ratio to the motion of the Weight. For who could make use to no purpose of a lever whose fulcrum is not farther from a sufficiently heavy weight than by half a foot? or employ a screw whose threads are separated by a hair's breadth? But since we can accomplish this in a double method, namely either by enlarging the machine itself, without changing its form, or by composing it from several members, whether they be of the same kind or of different kinds; it will be worth considering whether the greater saving is found in enlargement, or rather in composition. I call it enlargement, lest there be any room for ambiguity, when the species of the same Faculty remains unchanged, with only the addition of some part; as if, because a lever is shorter than is proper, you make the distance of the Power from the fulcrum longer; when pulleys are used for moving a load that are unequal, you increase the number of sheaves by enlarging the blocks; because a screw, on account of the rarity of its threads, is less strong than it ought to be, you incline the line itself so that it is carried round with denser threads. But a Machine is called Composite when other members are added to an invalid Faculty, either of the same kind, as when a Lever is joined to a Lever, a Screw to a Screw, pulleys to pulleys; or of a different kind, as when the faculties themselves are mixed together, levers with pulleys, B b 3

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198 Mechanicorum Cochleæ vectem, Trochleis Cochleam, & deinceps, adjungendo. Prioris Compositionis intrà idem genus specimen aliquod exhibui in Terrâ Machinis motâ: Dissertat. 1. & inferius fuis locis de eâ redibit sermo: Posterioris autem Compositionis diversarum Facultatum, ubi de singulis disputabimus, exempla aliqua subjiciemus, ut discat Tyro Machinarum vires ritè ad calculos revocare, solertiamque machinandi acquirat. Quamvis autem quæstio hæc multò dilucidiùs explicaretur, si unamquamque Facultatem singillatim attingeremus, quàm si unâ comprehensione omnia complectamur; hîc tamen doctrinæ ratio exigit, ut dimissis rivulis fontem ipsum aperia- mus, ex quo in Machinam Compositam vis major, quàm in Amplificatam, majore compendio derivatur. Et quidem cum res tota ex potentiæ atque Ponderis motuum Ratione pendeat, quamdiu in simplici aliquâ facultate consistimus, motus Potentiæ ad motum Ponderis simplicem habet Rationem; si verò Facultas una cum aliâ quâpiam facultate conjungitur, atque connectitur, jam Potentiæ motus ad motum ponderis eam habet Rationem, quæ ex singularum facultatum rationibus componitur. Voco autem singularum Facultatum Rationem eam, quæ inter ipsos Potentiæ ac Ponderis motus intercederet, si facultas illa solitaria adhiberetur; Atqui Ratio hæc motuum in singulis Facultatibus modum recipit ex Facultatis ipsius partibus, quarum altera ad Potentiam, ad Pondus altera spectare videtur; ut per singulas Facultates eunti constabit. In Vecte enim Ponderis ab hypomochlio distantia pertinet ad Pondus, Potentiæ autem distantia ab eodem hypomochlio penes potentiam est: In Trochleis ipsarum Trochlearum distantia Pondus respicit; funis autem explicatio Potentiam: In Axe in Peritrochia crassities Axis Ponderi, Peritrochij amplitudo Potentiæ tribuitur: In Cuneo longitudo ad Potentiam spectat, crassities ad Pondus: In Cochleâ demùm spiræ circumductæ perimeter ad Potentiam attinet, extremitatum spiralis lineæ intervallum, ad Pondus. Manifestum est igitur, ubi simplex motuum Ratio in singulis Facultatibus augenda fuerit, manente eâ parte, quæ ad Pondus spectat, necessariò ita augendam esse partem reliquam, quæ Potentiæ tribuitur, ut majori illi motuum Rationi respondeat. Sic dato Vecte palmorum sex, quo potentia mo- veatur

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198 of Mechanics By adding the screw to the lever, pulleys to the screw, and so on. I have exhibited some example of the former composition within the same class in the Earth-driven Machine: Dissertation 1, and below in those places there will be further discussion of it. But of the latter composition of different faculties, where we shall dispute each in turn, we shall set forth some examples, so that the beginner in machines may learn to reduce the powers of machines rightly to calculation, and acquire skill in mechanizing. Although this question would indeed be explained much more clearly if we touched each faculty separately than if we embraced everything in one summary, here nevertheless the method of teaching requires that, leaving the rivulets, we open up the very source itself, from which, in a Composite Machine, a greater force than in an Amplified one is derived with greater saving. And indeed, since the whole matter depends on the proportion of the motions of power and weight, as long as we remain with some simple faculty, the motion of the power has a ratio to the simple motion of the weight; but if one faculty is joined and connected with another faculty, then the motion of the power to the motion of the weight now has the ratio that is composed from the ratios of the individual faculties. Now I call the ratio of individual faculties that which would intervene between the motions of power and weight themselves, if that faculty were employed alone. But this ratio of motions in the several faculties takes its form from the parts of the faculty itself, one of which seems to belong to power, the other to weight, as will be clear as we go through the several faculties. In the lever, for example, the distance of the weight from the fulcrum pertains to the weight, but the distance of the power from the same fulcrum belongs to the power. In pulleys, the distance of the pulleys themselves concerns the weight; the unfolding of the rope, however, concerns the power. In the axle in the wheel-and-axle, the thickness of the axle is assigned to the weight, and the breadth of the wheel to the power. In the wedge, length pertains to power, thickness to weight. In the screw finally, the perimeter of the spiral thread pertains to power, while the interval between the extremities of the spiral line pertains to weight. It is therefore clear that, wherever the simple ratio of motions must be increased in the several faculties, while that part which pertains to the weight remains unchanged, the remaining part, which is assigned to the power, must necessarily be increased in such a way that it corresponds to that greater ratio of motions. Thus, given a lever of six palms, by which the power may move

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Liber secundus. CAPUT VII. 199 veatur in quintuplâ Ratione ad Pondus, si maneat eadem pon- deris ab hypomochlio distantia, & motuum Ratio esse debeat vigecupla, satis constat totum vectem requiri palmorum 21, ut unus Ponderi cedat, Potentiæ autem viginti. At verò si motuum Ratio ex Rationibus componenda sit, sa- tisfuerit datæ Facultati minorem Rationem continenti, quàm oporteat, Facultatem aliam adjicere, cujus Ratio cum priori Ratione composita quæsitam Rationem constituat. Sic dato Vecti quintuplam rationem continenti adjunge aliam quamli- bet facultatem quadruplæ Rationis; ex quadruplâ enim Ratio- ne & quintuplâ componitur Ratio vigecupla quæsita. Ita au- tem secunda hæc Facultas priori Facultati adnectenda est, ut quemadmodum duorum Magnetum oppositi poli junguntur, Australis videlicet unius Aquilonari alterius, sic duarum Fa- cultatum oppositæ partes connectantur, ut scilicet quo loco ad priorem Facultatem applicanda esset Potentia, eidem admo- veatur locus Ponderi in secundâ Facultate destinatus: proinde siquidem se res habebit, atque si pondus diminutum pro Ra- tione prioris facultatis, videlicet sub quintuplum, in secun- dam hanc Facultatem transferretur, in quâ ejus motus ad mo- tum Potentiæ Rationem haberet subquadruplam: re enim ve- râ duabus hisce Facultatibus junctis, Potentiæ motus vigecu- plus est ad motum Ponderis; nam Pondus in vectis extremita- te alterâ constitutum quintuplo tardiùs movetur, quàm reli- qua vectis extremitas; hæc autem posteriori Facultati loco Ponderis adjuncta quadruplo tardiùs movetur quàm Poten- tia; igitur Ponderis motus vigecuplo tardior est motu Po- tentiæ. Statuamus exempli gratiâ secundam hanc Facultatem Vecti adjunctam esse pariter Vectem ejusdem generis quinque pal- morum ita ab hypomochlio distinctum in partes, ut hæ in qua- druplâ sint Ratione: Ecce quanto compendio rem assequamur; id enim quod simplici Vecte palmorum 21 præstandum esset, compositis vectibus duobus altero palmorum sex, altero palm. quinque perficimus, servatâ semper eâdem Ponderis ab hypo- mochlio distantiâ, nimirum palmi unius. Hæc tamen de duo- bus hisce vectibus dicta ita intelliges velim, ut ad motum sim- pliciter pertineant; non verò ad motûs quantitatem; satis enim scio

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Book Two. CHAPTER VII. 199 it would be in a quintuple ratio to the Weight, if the distance of the weight from the fulcrum remained the same, and the ratio of the motions were to be twentyfold, it is plain enough that the whole lever would be required to be 21 palms long, so that one may yield to the Weight and twenty to the Power. But if the ratio of the motions is to be made up from ratios, it will be enough to add to a given faculty, containing a smaller ratio than ought to be, another faculty, whose ratio, when compounded with the former ratio, will make up the required ratio. Thus, to a Lever containing a quintupl e ratio, add any other faculty of a quadruple ratio; for from the quadruple ratio and the quintupl e ratio the required twentyfold ratio is composed. And this second faculty is to be attached to the former in such a way that, just as the opposite poles of two magnets are joined, namely the south of the one with the north of the other, so the opposite parts of the two faculties are connected, so that, namely, the place which in the former faculty would be applied to the Power, the place assigned to the Weight in the second faculty is brought into contact with it: and so the matter will stand as if the reduced weight, in accordance with the ratio of the former faculty, namely diminished to a fifth, were transferred into this second faculty, in which its motion would have a ratio to the motion of the Power of one fourth. For in truth, with these two faculties joined together, the motion of the Power is twenty times the motion of the Weight; for the Weight, placed at one end of the lever, moves five times more slowly than the other end of the lever; but this latter, joined to the second faculty in the place of the Weight, moves four times more slowly than the Power; therefore the motion of the Weight is twenty times slower than the motion of the Power. Let us suppose, by way of example, that this second faculty is attached to a Lever of the same kind, five palms long, so distinguished into parts from the fulcrum that these are in a quadruple ratio: see with what compendiousness we achieve the matter; for what would have to be done with a simple lever of 21 palms, we accomplish with two combined levers, one of six palms, the other of five palms, while always preserving the same distance of the Weight from the fulcrum, namely one palm. Yet I would have you understand what is here said of these two levers as referring only to motion, and not to the quantity of the motion; for I know well enough

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Mechanicorum 200 scio non ad eam distantiam promoveri posse Pondus adhibito secundo hoc vecte, ad quam promoveretur Vecte palmorum 21: Verùm hîc sola movendi facilitas consideratur. Quòd si non alterum Vectem adhibeas; sed aliud facultatis genus, ut Trochleas binis orbiculis instructas, & Vecti in loco Potentiæ adnexas, multò adhuc faciliùs movebitur Pondus, cujus motus erit subvigecuplus motûs Potentiæ funem Trochlearum trahentis, & tantus erit Ponderis motus, quantus esset, si extremitati Vectis palmorum sex apponeretur Potentia quadrupla datæ Potentiæ. Idem planè de cæteris dicendum Facultatibus. Hinc manifestum est compositis tribus, quatuorve, aut pluribus Facultatibus, Rationem Compositam motus potentiæ ad motum Ponderis fieri multò majorem; cui si æqualem Rationem habere velimus unicâ atque simplici Facultate, hujus magnitudinem aliquando enormem fieri necesse esset; ut suis locis infrà declarabitur. In eo igitur elucebit Machinatoris industria, si Facultates ipsas aptè congruenterque disponat, atque permisceat, spectatâ materiæ soliditate, spatij amplitudine, Ponderis positione, Potentiæ virtute, temporis ad movendum concessi opportunitate: hæc enim omnia attentissimè perpendenda sunt; ne, dum nimis sollicitè laborem imminuere studet, motum plus æquo imminuens, tardioremque efficiens temporis jacturam faciat, aut totum spatium machina implens in eas angustias Potentiam moventem conjiciat, ut motum expeditè perficere nequeat. CAPUT VIII. Cur majores Rotæ motum juvent præ minoribus. O Nera si ex alio in alium locum deportanda fuerint, gemino labore opus est, conatu videlicet, quo sustineantur, & impetu, quo transferantur: proptereà satius est ita res disponere, ut vires omnes ad transferendum exerceantur, citrà conatum sustinendi; ut eâ ratione vel gravius onus vel idem multò

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Mechanics 200 I know that by using this second lever the weight cannot be moved to as great a distance as it would be by a lever of 21 palms: but here only the ease of moving is considered. But if you do not use another lever, but some other kind of mechanical device, such as pulleys fitted with two sheaves, and attached in the place of the power at the lever, the weight will be moved much more easily still, and its motion will be one twenty-first part of the motion of the power pulling the rope of the pulleys; and the motion of the weight will be as great as it would be if a power four times the given power were applied to the end of a lever six palms long. The same is plainly to be said of the other mechanical devices. Hence it is clear that, when three, four, or more devices are combined, the compounded ratio of the motion of the power to the motion of the weight becomes much greater; and if we wish to have an equal ratio by means of a single and simple device, its size would at times have to become enormous, as will be explained in its proper place below. Therefore the ingenuity of the engineer will shine forth if he arranges and combines the devices themselves suitably and appropriately, considering the solidity of the material, the extent of the space, the position of the weight, the force of the power, and the opportunity of the time allowed for moving: for all these things must be weighed most attentively, lest, while too anxiously seeking to diminish the labor, he diminish the motion beyond measure and make it slower, causing loss of time; or, by filling the whole space of the machine, he cast the moving power into such straits that it cannot complete the motion expeditiously. CHAPTER VIII. Why larger wheels assist motion more than smaller ones. O Nera if things are to be conveyed from one place to another, a twofold labor is required, namely the effort by which they are sustained, and the impulse by which they are transferred: therefore it is better so to arrange matters that all the forces are applied to the transfer, without the effort of sustaining; so that in this way either a heavier load or the same one much

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Liber secundus. CAPUT VIII. 201 multò faciliùs à potentia moveatur, quàm si ea illud sustinere pariter atque transferre cogeretur. Quoniam verò (cum one- ra subjecto plano imposita illud premant, atque tùm onerum tùm subjecti plani facies, quæ se invicem contingunt, non ita læves sint, ut partes omnes in rectum directæ nihil habeant asperitatis; quin immò ut plurimum, & salebris impedita via sit, & movendi corporis partes aliæ præ aliis extent atque emi- neant) ex mutuo prominentium particularum tritu atque con- flictu difficultas ad movendum oriretur; idcircò optimo consi- lio factum est, ut oneribus ipsis subjiciantur Cylindri aut Rotæ, quæ dum in gyrum aguntur, conflictum illum partium tollunt, qui vitari non posset, si onera super plano raptarentur. Hinc Ci- sia, Sarraca, Vehes, Carri & genus omne plaustrorum. Id quod etiam homines ipsi, ut terrestre iter commodiùs habeant, & minori jumentorum labore illud perficiant, quàm si iis insi- dentes veherentur, suos in usus retulerunt: Hinc Belgæ sua esseda, Galli petorita & rhedas, Hispani pilenta, Itali carpen- ta; & pro suâ quisque voluntate diversa vehiculorum genera excogitârunt, quæ subjectis rotis aguntur: dum enim Rota convertitur, ejusque curvaturæ partes aliis atque subinde aliis subjectæ planitiei partibus aptantur, adeóque currus promove- tur, solus rotæ modiolus axis ambitum axungiâ lubricum terit; ex quo tritu aut nulla aut levis mora motui infertur. Illud autem est omnibus exploratissimum, & quotidiano ex- perimento confirmatum, quo majoribus rotis instructi currus (nisi discrimen aliquod in cæteris intercedat) multò faciliùs trahuntur, passimque observatur Romæ in vulgaribus illis vehi- culis (ab antiquis Cisiis aut parum aut nihil distant) quæ cum ex celeberrimi Architecti Bonarotæ præscripto duas ingentes rotas habeant, tantis ponderibus onusta cernuntur, ut miracu- lo proximum videatur ab unico equo tam ingentia onera trahi posse: id quod alibi neutiquam fieri potest, ubi minoribus Rotis vehicula hujusmodi instructa longè minoribus oneribus defe- rendis paria sunt, si unicus equus adhibeatur. Hujus rei causam indaganti acquiescendum non est iis, qui illam ex rationibus Vectis petendam esse existimant, perinde atque si rotæ majoris semidiameter esset longior Vectis, mino- ris verò brevior; ac proptereà majore rotâ faciliùs moveretur C c

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Book the Second. CHAPTER VIII. 201 is moved much more easily by power than if that power were compelled to sustain it and carry it at the same time. But since (when burdens placed upon a flat surface press it down, and when the surfaces both of the burdens and of the flat surface beneath them, which touch one another, are not so smooth that all their parts, being directed in a straight line, have nothing rough in them; nay rather, for the most part, the road is hindered by roughness, and the parts of the body to be moved stand out and rise above others) difficulty in moving arises from the mutual rubbing and collision of the projecting particles; therefore it was most wisely contrived that Cylinders or Wheels should be placed beneath burdens themselves, which, while they are turned in a circle, remove that conflict of the parts which could not be avoided if burdens were dragged over a plane. Hence Cisia, Sarraca, Vehes, Carri, and every kind of cart. This also men themselves, in order that they might have a more convenient journey by land, and accomplish it with less labor of the beasts of burden than if they were carried seated upon them, have turned to their own use: hence the Belgians their esseda, the Gauls their petorita and rhedae, the Spaniards pilenta, the Italians carpenta; and each nation, according to its own will, devised different kinds of vehicles, which are driven by wheels beneath them: for while the Wheel turns, and the parts of its curvature are fitted now to these, now to other parts of the surface beneath, the carriage is thereby advanced; only the nave of the wheel, that is, the axle, grinds the circumference of the axle in lubricating grease; from which rubbing either no delay or but a slight one is introduced to the motion. But this is most familiar to all, and confirmed by daily experience: the greater the wheels with which carts are furnished (unless some other difference intervenes), the more easily are they drawn; and it is commonly observed at Rome in those ordinary vehicles (which differ little or nothing from the ancient Cisia) that, when they have two huge wheels according to the prescription of the most famous architect Bonarota, they are seen loaded with such great weights that it would seem almost a miracle that so huge a burden could be drawn by a single horse: which can by no means be done elsewhere, where vehicles of this sort equipped with smaller wheels are fit only for carrying much smaller loads, if a single horse is employed. In investigating the cause of this matter, one must not acquiesce in those who think that it is to be sought from the principles of the lever, as though the semidiameter of the larger wheel were a longer lever, and that of the smaller a shorter one; and therefore that the larger wheel is moved more easily C c

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Mechanicorum 202 vehiculum onustum, quàm minore, quia & longiore vecte fa- ciliùs pondera moventur, quàm breviore. Hoc, inquam, à veritate abesse palam fiet, si animadvertamus potentiam tra- hentem medio temone applicatam esse axi, cui pariter axi in- nititur onus; atque adeò tùm onus tùm Potentiam concipi quasi in Rotæ centro, cujus semidiametri altera extremitas hy- pomochlij punctum designaret. Atqui Vectis, in quo Potentia & onus ab hypomochlio eandem aut æqualem distantiam ha- bent, parùm aut nihil habet utilitatis: immò in Vecte, quâ vectis est, tria puncta diversa tribuenda sunt Potentiæ, oneri, & Hypomochlio, ut infrà, ubi de Vecte disputabitur: in Rotâ autem duo tantummodo puncta considerantur, scilicet cen- trum & semidiametri extremitas. Igitur in Rotâ ratio Vectis non invenitur, ideóque neque major Rota accipienda est qua- si longior Vectis. Aliundè itaque petendam esse causam, cur majores rotæ præ minoribus motum juvent, manifestum est. Et primùm quidem, quod ad moram illam attinet, quæ ex modioli Rotæ atque axis tritu oritur, eam minorem esse in ma- joribu Rotis, satis constàt, si attendamus axis crassitiem, non Rotæ magnitudini respondere, sed oneris gravitati, quam opus est sustinere; quapropter axi satis valido pro ratione ponderis sustinendi parùm refert, utrùm Rota, cujus radij bipalmares sint, an verò tripalmares, infigatur: manente igitur eodem axe aut major, aut minor Rota vehiculo subjici potest. Sed quo- niam Rota major, cujus diameter sesqualtera est minoris, dum conversionem unam perficit, spatium quoque sesqualterum decurrit, eumdem tamen axem, quem minor Rota, terit, hinc fit, per 8. lib. 5. eumdem axis ambitum ad majoris Rotæ peri- metrum (hoc est ad ejus motum) minorem habere rationem quàm ad perimetrum minoris Rotæ (hoc est ad minorem mo- tum) atque adeò tritus ille modioli, & axis minùs impedit ma- jorem motum quàm minorem. Deinde, ut cap. 16. lib. 1. subindicatum est superiùs, majo- res rotæ efficiunt, ut axis magis à terrâ distet; ac proinde te- mo, cui alligatus est equus, vel subjecto plano parallelus est, vel minimum à parallelismo recedit: ex quo fit tractionem aut parallelam esse, aut saltem minùs obliquam, quam si Rota mi- nor esset, & axis depressior: quò autem minor est tractionis obliquitas,

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Mechanics 202 A loaded vehicle with a larger wheel, than with a smaller one, because with a longer lever weights are moved more easily than with a shorter one. This, I say, will plainly appear to be far from the truth, if we observe that the dragging power applied to the middle of the axle is applied to the axle, on which likewise the load rests; and thus both the load and the power are conceived as if in the center of the wheel, one extremity of whose semidiameter would mark the point of the fulcrum. But in a lever, in which the power and the load have the same or equal distance from the fulcrum, there is little or no utility: indeed, in the lever, insofar as it is a lever, three distinct points must be assigned to the power, the load, and the fulcrum, as below, where the lever will be discussed; but in the wheel only two points are considered, namely the center and the end of the semidiameter. Therefore, in the wheel the ratio of the lever is not found, and for that reason a larger wheel is not to be taken as if it were a longer lever. Hence it is manifest that the reason why larger wheels aid motion more than smaller ones must be sought elsewhere. And first, as to that delay which arises from the friction of the nave of the wheel and the axle, it is sufficiently established that it is smaller in larger wheels, if we consider that the thickness of the axle does not correspond to the size of the wheel, but to the weight of the load it must support; wherefore, for an axle sufficiently strong in proportion to the weight to be borne, it matters little whether a wheel whose radii are two palms or three palms be fitted: the axle remaining the same, either a larger or a smaller wheel may be placed under the vehicle. But since a larger wheel, whose diameter is one and a half times that of a smaller one, while accomplishing one revolution, also covers a distance one and a half times as great, yet wears the same axle as the smaller wheel, it follows, by 8, lib. 5, that the same circumference of the axle has a smaller ratio to the perimeter of the larger wheel (that is, to its motion) than to the perimeter of the smaller wheel (that is, to the smaller motion); and thus that friction of the nave and axle hinders the greater motion less than the smaller. Next, as was indicated above in chap. 16, book 1, larger wheels cause the axle to be farther from the ground; and consequently the pole to which the horse is fastened is either parallel to the supporting plane, or deviates from parallelism only very little: from which it follows that the traction is either parallel, or at least less oblique, than if the wheel were smaller and the axle lower; and the less oblique the traction is,

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Liber secundus. CAPUT VIII. 203 obliquitas, minorem quoque esse trahendi difficultatem loco citato explicatum est. Ad hæc viarum asperitatem impedimento esse nemo nescit; offendicula autem, in quæ vehiculorum Rotæ incurrunt, ma- gis obsistere minori Rotæ, quàm majori, facilè ostenditur; hîc enim pariter (id quod de magnitudinibus demonstrat Eucli- des lib. 5. prop. 8.) idem majorem habet Rationem ad minus, quàm ad majus. Nam si Rotæ minoris semidiameter CB fuerit, majoris autem CD, & in planis parallelis BA, DE voluntur, ut impedimentum simile similiterque positum invenient, multò majus esse oportet illud, quod majori Rotæ objicitur, quàm quod minori. Sit enim minoris offendiculum GI; ducatur ex centro per I recta, quæ sit CIE secans majoris Rotæ peripheriam in H: erit igitur arcus IB similis arcui HD, & ille quidem minor, hic verò major, ut manifestum est. Ducatur in planum perpendicularis HF, & hoc erit impedimentum majoris Rotæ simile impedimento minoris IG, nam similem arcum à conversione circà centrum cum plani contactu impedit; necesse quippe est Rotam majorem converti circà punctum H, sicut & minorem circà punctum I, ut transgredientur obsistens offendiculum. Porro lineam HF majorem esse quàm IG sic ostenditur. Quoniam AB & ED parallelæ sunt, triangula CBA, & CDE similia sunt: ergo per 4. lib. 6. ut CB ad CD, hoc est ut CI ad CH, ita CA ad CE; & permutando ut CI ad CA, ita CH ad CE; & dividendo ut CI ad IA, ita CH ad HE: at CI minor est quàm CH; igitur per 14. lib. 5. etiam IA minor est quàm HE. Item quia AB & ED ex hypothesi parallelæ sunt, recta IE in illas incidens facit angulos IAG & HEF æquales per 29. lib. 1. sunt autem triangula IGA & HFE rectangula ad G & F ex constructione; sunt igitur similia, & Cc 2

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Book II. CHAPTER VIII. 203 It has been explained in the passage cited that obliquity, and likewise that the difficulty of drawing is less. In addition, no one is ignorant that the roughness of roads is an impediment; but that obstacles into which the wheels of vehicles run oppose a smaller wheel more than a larger one is easily shown; for here as well (as Euclid demonstrates of magnitudes in Book 5, Prop. 8) the same thing has a greater ratio to the smaller than to the greater. For if the semidiameter CB of the smaller wheel be assumed, and that of the larger CD, and if they revolve in parallel planes BA, DE, so that they encounter a similar obstacle similarly placed, that which is opposed to the larger wheel must necessarily be much greater than that which is opposed to the smaller. Let the obstacle of the smaller wheel be GI; draw from the center through I a straight line, CIE, cutting the circumference of the larger wheel at H: then the arc IB will be similar to the arc HD, and the former indeed smaller, the latter however larger, as is evident. Draw HF perpendicular to the plane, and this will be the impediment of the larger wheel similar to the impediment IG of the smaller, for it impedes a similar arc by the turning around the center with contact of the plane; indeed it is necessary that the larger wheel be turned around point H, just as the smaller around point I, in order that it may pass over the obstructing obstacle. Moreover, that line HF is greater than IG is shown thus. Since AB and ED are parallel, the triangles CBA and CDE are similar: therefore by Book 6, Prop. 4, as CB is to CD, that is, as CI is to CH, so is CA to CE; and by permutation, as CI is to CA, so is CH to CE; and by division, as CI is to IA, so is CH to HE: but CI is less than CH; therefore by Book 5, Prop. 14, IA also is less than HE. Likewise, because AB and ED, by hypothesis, are parallel, the straight line IE, cutting them, makes the angles IAG and HEF equal by Book 1, Prop. 29. Now the triangles IGA and HFE are right-angled at G and F by construction; therefore they are similar, and Cc 2

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Mechanicorum per 4. lib. 6. ut .IA ad IG, ita HE ad HF: quare cum ex dictis I A minor sit quàm HE, erit per :4. lib. 5. etiam IG mi- nor quàm HF. Cum itaque HF major sit quàm IG (assumptâ DM æqua- li ipsi IG, & ductâ perpendiculari MS, donec occurrat peri- phæriæ in S) inter Tangentem ED & arcum circuli statuatur perpendicularis SL æqualis ipsi IG; & ex centro C ducatur per S recta CO. In triangulo igitur CEO angulus internus E, per 16. lib. 1; minor est externo SOL; igitur etiam angu- lus SOL major est quàm IAG: adde utrique angulum rectum, ergo duo SLO, SOL simul majores sunt duobus IGA, IAG simul; ac propterea etiam externus LSC major est externo GIC per 32. lib. 1. Quapropter semidiameter CS obliquior incidit in offendiculum SL, quàm semidiameter CI incidat in æquale offendiculum IG: minùs igitur impeditur Rotæ majoris conversio, quàm minoris, quippe cui minus di- rectè opponatur æquale offendiculum. Præterea cum trahendi difficultas hinc oriatur, quòd Rotæ incurrens in obstantem lapidem, aut quid simile, jam non cir- cà suum centrum convoluta aptatur subjecto plano, sed, dum Rota adhæret atque insistit offendiculo, necesse est plaustrum cum imposito onere elevari pro objecti impedimenti altitudi- ne; faciliùs ab eâdem Potentia elevatur plaustrum omnium, si major fuerit Rota, quàm si minor, quia videlicet motus Poten- tiæ ad eandem elevationem majorem habet Rationem in Ro- tâ majore quàm in minore, cum illâ enim plus movetur, quàm cum istâ. Sit majoris Rotæ impedimentum LS pla- nè æquale impedimento GI minoris; producatur perpendicularis LS in T, & perpendicularis GI in V: tùm intervallo SC describatur arcus CT, & intervallo IC describatur arctus CV. Certum est in motu Rotæ majoris propter obicem LS manente puncto S transferri centrum C in T, ita ut S T sit Rotæ semi- diameter æqualis semidiametro CD, & similiter in motu Rotæ minoris propter offendiculum GI manente puncto I transferri centrum C in V, ita ut IV æqualis sit semidiametro CB. Quo- niam verò CD, VG, TL ad angulos rectos subjecto plano in- sistunt, & parallelæ sunt, anguli alterni VIC, ICB æquales sunt per 29. lib. 1, eorumque mensuræ, arcus videlicet VC & IB,

Transcription: Translated (English)

Mechanics by 4. lib. 6. as IA is to IG, so is HE to HF: wherefore, since from the things said IA is less than HE, it will be by :4. lib. 5. also IG less than HF. Since therefore HF is greater than IG (DM being assumed equal to IG, and the perpendicular MS being drawn until it meet the peri- phery at S) between the tangent ED and the arc of the circle let a perpendicular SL be set equal to IG; and from the center C let the line CO be drawn through S. In the triangle therefore CEO the internal angle E, by 16. lib. 1, is less than the external SOL; therefore also the angle SOL is greater than IAG: add to each a right angle, therefore the two SLO, SOL together are greater than the two IGA, IAG together; and therefore also the external LSC is greater than the external GIC by 32. lib. 1. Wherefore the semidiameter CS falls more obliquely upon the obstacle SL than the semidiameter CI falls upon the equal obstacle IG: therefore the turning of the greater wheel is hindered less than that of the smaller, since an equal obstacle is opposed to it less directly. Moreover, since the difficulty of drawing arises from this, that the wheel running against a resisting stone, or something similar, is no longer, as it revolves around its center, fitted to the ground beneath, but while the wheel adheres to and rests upon the obstacle, the cart with the load placed upon it must necessarily be raised by the height of the obstructing object; the cart is more easily raised by the same power, if the wheel be greater than if it be smaller, because, namely, the motion of the power toward the same elevation has a greater ratio in the greater wheel than in the smaller, for with the former it is moved more than with the latter. Let the obstacle LS of the greater wheel be exactly equal to the obstacle GI of the smaller; let the perpendicular LS be produced to T, and the perpendicular GI to V: then with interval SC let the arc CT be described, and with interval IC let the arc CV be described. It is certain in the motion of the greater wheel that, because of the obstacle LS while point S remains fixed, the center C is transferred to T, so that ST is a semidiameter of the wheel equal to semidiameter CD, and similarly in the motion of the smaller wheel, because of the obstacle GI while point I remains fixed, the center C is transferred to V, so that IV is equal to semidiameter CB. Since however CD, VG, TL stand at right angles upon the ground below, and are parallel, the alternate angles VIC, ICB are equal by 29. lib. 1, and their measures, namely the arcs VC and IB,

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Liber secundus. CAPUT VIII. 205 IB, æquales sunt; & ob eandem Rationem anguli alterni TSC, SCD, eorumque mensuræ arcus TC & SD, sunt æquales. Atqui arcus SD major est quàm IB; igitur & arcus TC major est quàm VC; hi autem arcus TC & VC respon- dent motui Potentiæ trahentis: longiore igitur ac majore mo- tu Potentiæ fit eadem elevatio, ac proinde faciliùs in Rotâ ma- jore quàm in minore. Porrò arcum SD majorem esse arcu IB, magisque distare punctum S à puncto D, quàm punctum I à puncto B, illicò manifestum fiet, si duos circulos datis duobus, æquales descripseris se intùs contingentes, & ad contactûs punctum lineam Tangentem duxeris, quocumque enim posito minoris circuli offendiculo inter Tangentem, & circulum mi- norem interjecto, illud idem offendiculum longiùs à con- tactûs puncto removeudum videbis, ut inter Tangentem eandem, & circulum majorem interjici possit: Id quod adeò manifestum est, ut non sit in eo explicando diutiùs immo- randum. Quòd si ad calculos rem hanc curiosiùs revocare libeat, sit ex gr. Rotæ minoris semidiameterer CA pedum duorum, scilicet digitorum 32, offendiculi verò DE altitudo digitorum 4. Cum igitur FD & CA parallelæ sint, sicut & FC ac DA per 34. lib. 1. FD & CA æquales sunt, remanetque EF digit. 28, & est Sinus anguli FCE, quo cognito innotescit complemen- tum, arcus scilicet quæsitus EA. Fiat itaque ut CE ad EF, hoc est ut 32 ad 28, seu ut 8 ad 7, ita 100000. Radius ad 87500 Sinum arcûs gr. 61. 2'42"; erit enim quæsi- tus arcus EA gr. 28. 57 18". Jam verò positâ semidiametro CA digitorum 32, fiat ut 113 ad 355, ita data semidiameter digit. 32 ad semiperipheriam circuli digitorum ferè 100 1/3, sci- licet 100. 53": ergo arcus EA est proximè digitorum 16. At Rotæ majoris semidiameterer BA sit sesqualtera (quic- quid sit quòd figura solùm exprimat sesquiquartam) pedum scilicet trium, hoc est digitorum 48, & offendiculum GH Cc 3

Transcription: Translated (English)

Book the Second. CHAPTER VIII. 205 IB are equal; and for the same reason the alternate angles TSC, SCD, and their measures, the arcs TC and SD, are equal. But the arc SD is greater than IB; therefore the arc TC is also greater than VC; now these arcs TC and VC correspond to the motion of the drawing Power: therefore by a longer and greater motion of the Power the same elevation is produced, and consequently more easily in a larger Wheel than in a smaller one. Moreover, that the arc SD is greater than the arc IB, and that point S is farther from point D than point I is from point B, will at once be made evident if, given two circles, you describe them equal, touching one another internally, and draw a Tangent at the point of contact; for however you may place the obstacle of the smaller circle between the Tangent and the smaller circle, you will see that the same obstacle must be set farther from the point of contact, so that it may be placed between the Tangent and the larger circle: which is so evident that there is no need to dwell longer on explaining it. But if one should wish to refer this matter more curiously to calculation, let there be, for example, the semidiameter CA of the smaller wheel, two feet, that is to say 32 inches, and the height of the obstacle DE 4 inches. Since therefore FD and CA are parallel, as also FC and DA by 34 of Book 1, FD and CA are equal, and there remains EF 28 inches; and this is the sine of the angle FCE, which being known makes known the complement, namely the sought arc EA. Let it therefore be as CE to EF, that is, as 32 to 28, or as 8 to 7, so 100000, the Radius, to 87500, the Sine of an arc of 61° 2' 42"; for the sought arc EA will be 28° 57' 18". Now if the semidiameter CA be taken as 32 inches, let it be as 113 to 355, so the given semidiameter 32 inches to the semiperiphery of the circle, nearly 100 1/3 inches, that is, 100. 53"; therefore the arc EA is nearly 16 inches. But let the semidiameter BA of the larger wheel be one and a half times as large (whatever may be the case, since the figure expresses only one and a quarter times), that is to say 3 feet, namely 48 inches, and the obstacle GH

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Mechanicorum 206 pariter digit. 4. Quare HI est digit. 44 Sinus anguli IBH, ex quo innotescet arcus complementi HA. Fiat ut BH 48 ad HI 44, seu ut 12 ad 11, ita Radius 100000 ad 91666 Sinum arcûs gr. 66. 26'. 33"; & est quæsitus arcus HA gr. 23. 33'. 27". Iam sit ut 113 ad 355, ita semidiameter 48 ad semiperipheriam digitorum 150 1/4 ferè: igitur arcus HA est proximè digitorum 20. Cum itaque dum onus elevatur ut 4, Potentia in minore Rotâ moveatur ut 16, in majore autem ut 20 (ut paulò superiùs ostensum est motum centri æqualem esse arcibus EA, & HA) facilitas movendi, quæ hinc oritur, erit ut 5 ad 4. Ex his manifestum est, in vehiculis, quæ quatuor rotis instruuntur, quarum binæ priores minores sunt, posteriores verò majores, faciliùs superari impedimenta à posterioribus rotis quàm à prioribus, ac propterea minori labore currum ab equis trahi, quàm si posteriores prioribus essent æquales. Id quod opportunè factum est, quia ut plurimum (quemadmodum in antiquioribus Rhedis viatoriis cernere est) in posteriorem potiùs, quàm in anteriorem currus partem, onus rejicitur, atque adeò posterior axis magis premitur: quærendum igitur fuit aliquod laboris compendium. Quamquam non negarim alio prorsus consilio primùm excogitatem hanc Rotarum inæqualitatem; ut nimirum onus constitutum quasi in plano trahentem versùs inclinato, faciliùs quoque illum ex impresso anterioris tractionis impetu sequeretur, si in planitie quidem tractio fieret; ubi verò superandus esset clivus, ut minùs adversùs trahentem repugnaret onus se ipsum in proclive urgendo; nam si Rotæ æquales essent, longè faciliùs vehiculum in posteriora relaberetur, pro ipsius clivi inclinatione, cui parallelum esset planum oneri subjectum insistens axibus æqualium Rotarum: at Rotis inæqualibus positis, & posterioribus quidem majoribus, planum, cui onus incumbere intelligitur à posteriori axe ad anteriorem deductum minùs inclinatur, quàm collis proclivitas ferat; ac propterea trahentibus equis minùs repugnat. Licèt autem non semper ascendendum sit in colles & clivos, quorum ascensus manifestè arduus est atque difficilis, rarò tamen, aut ferè nunquam, adeò æquata est viarum planities, quin leviter saltem inflexæ modò ascendere

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Mechanicorum 206 likewise digit. 4. Therefore HI is digit. 44, the sine of angle IBH, from which the complementary arc HA will be known. Let BH 48 be to HI 44, or as 12 to 11, so Radius 100000 is to 91666, the sine of the arc 66. 26'. 33"; and the sought arc HA is 23. 33'. 27". Now let 113 be to 355, so the semidiameter 48 is to the semiperiphery of digits 150 1/4, nearly: therefore the arc HA is approximately digits 20. Since therefore while the load is raised as 4, the Power in the smaller Wheel is moved as 16, but in the larger as 20 (as was shown a little above, the motion of the center being equal to the arcs EA and HA) the ease of moving, which arises from this, will be as 5 to 4. From these things it is clear that, in vehicles which are fitted with four wheels, of which the two front ones are smaller, but the rear ones larger, obstacles are more easily overcome by the rear wheels than by the front ones, and for that reason the carriage is drawn by the horses with less labor than if the rear wheels were equal to the front ones. This was suitably done, because for the most part (as may be seen in older traveling carriages) the load is thrown more to the rear part of the carriage than to the front, and thus the rear axle is more pressed down: therefore some saving of labor had to be sought. Although I would not deny that this inequality of the Wheels was first devised for an entirely different purpose; namely, that a load set, as it were, on a plane inclined toward the pulling side might more easily follow the impulse of the forward traction impressed on it, if the pulling were made on level ground; but where a slope had to be overcome, so that the load might less resist the pull by urging itself down the incline; for if the Wheels were equal, the vehicle would much more easily slide back toward the rear, in proportion to the inclination of the slope, to which the plane supporting the load, resting on axles with equal Wheels, would be parallel: but when the Wheels are unequal, and indeed the rear ones larger, the plane on which the load is understood to rest, drawn from the rear axle to the front, is less inclined than the slope of the hill would require; and therefore it offers less resistance to the horses pulling. Although, however, one does not always need to ascend hills and slopes, whose ascent is manifestly steep and difficult, yet rarely, or almost never, is the levelness of roads so even that they are not at least slightly bent so as to ascend

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Liber secundus. CAPUT VIII. 207 ascendere cogant, modò descendere: in quâ ascensuum atque descensuum vicissitudine non modicè utilis est illa Rotarum inæqualitas. Hinc manualia illa curricula (seu rusticæ vehes) quæ binis brachiis instructa unicum habent in anteriore parte rotam & sublevatis brachiis conversa Rotâ promoventur, faciliùs construi possent, si propè vectorem duæ essent Rotæ majores illâ anteriore Rotâ, ita ut harum diameter triplex esset diametri illiùs: hunc enim unicus homo multò majus pondus transferre potest vel impellendo, cùm in planitie est, aut clivum ascendit, vel trahendo, cùm ex declivi descendit; levatur si quidem labore sustinendi, & omnes vires exercet impellendo aut trahendo; & illa Rotarum inæqualitas in causâ est, cur faciliùs impellatur pondus versùs illam partem, in quam inclinatur. Et quoniam in Rotarum inæqualium mentionem incidi, illud hîc pariter observandum videtur, commodiùs currum moveri, cùm anteriores Rotæ à posterioribus aliquantulum distant, quàm cùm valdè vicinæ sunt (ubi tamen reliqua omnia paria fuerint, neque aliud præter Rotarum distantiam, intercedat discrimen) si in planitie quidem, & viâ minimum flexuosâ deducendus sit. Quia nimirum quo propiores fuerint axes, planum, cui onus incumbit, magis inclinatur, ac propterea anteriores Rotas premens adversùs subjectam tellurem minus obliquè conatur, ideóque pondus illam validiùs urgens majorem creat movendi difficultatem: contrà verò si axes invicem paulò remotiores fuerint, minùs inclinato plano, minor est priorum rotarum pressus in subjectam tellurem. Sic si Rotæ fuerint A & B, planum, cui onus insidet, est A B, at si Rotæ fuerint A & C, planum est A C, quod utique minùs inclinatum est, magisque accedit ad parallelismum cum Horizonte DE, atque adeò Rota B magis terram premit, quàm Rota C. Si enim in utroque plano pondus fuerit similiter positum (puta circà medium)

Transcription: Translated (English)

Book Two. CHAPTER VIII. 207 to make them ascend, now to descend: in which alternation of ascent and descent, that inequality of the wheels is not a little useful. Hence those hand-carts also (or rustic vehicles) which are furnished with two handles and have only one wheel in the front part, and which, the handles being lifted, are advanced by the turning wheel, could be constructed more easily if near the bearer there were two larger wheels than that front wheel, so that their diameter were three times the diameter of that wheel: for one man can transfer a much greater load, either by pushing, when on level ground or climbing a slope, or by pulling, when descending from an incline; he is in fact relieved from the labor of sustaining it, and exerts all his strength either in pushing or in pulling; and that inequality of wheels is the reason why a load is more easily driven toward that side to which it inclines. And since I have fallen into mention of unequal wheels, it seems here likewise to be observed, that a cart is moved more conveniently when the front wheels are somewhat distant from the rear ones than when they are very close together (provided, however, that all else be equal, and that no other difference intervene except the distance of the wheels), if it is to be drawn across level ground and a road of little winding. For, indeed, the closer the axles are, the more the plane on which the load rests is inclined; and therefore the front wheels, pressing against the ground beneath, attempt less obliquely, and thus the weight, pressing it more strongly, creates greater difficulty in moving; on the other hand, if the axles are a little more remote from one another, with the plane less inclined, there is less pressure of the forward wheels upon the ground beneath. Thus if the wheels were A and B, the plane on which the load rests is A B; but if the wheels were A and C, the plane is A C, which certainly is less inclined, and more nearly approaches parallelism with the horizon D E; and accordingly Wheel B presses the ground more than Wheel C. For if on both planes the weight were similarly placed (say about the middle)

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Mechanicorum dium) linea directionis à centro gravitatis ponderis ducta cadet ad angulos magis inæquales in planum AB magis inclinatum, quam in AC minùs inclinatum, atque momentum gravitatis ponderis magis accedet ad B quam ad C, ut infrà suo loco explicabitur, & subindicatum est superiùs lib.1. cap.14. S. Ex his fieri potest. Hinc Hamburgensia plaustra, quibus merces Hamburgo Norimbergam devehuntur, longiora sunt, quia nec altiores clivi in itinere frequentes occurrunt, nec angustæ sunt viarum flexiones, ex quibus oriatur aut ascendendi, aut plaustrum inflectendi difficultas. Quare illis & majora onera imponi possunt, & sex equi non bini & bini, sed singuli recto ordine adjunguntur; quo fit ut non in diversa trahentes, omninò simili impetu currum deducant. Quòd si viæ plus haberent difficultatis tùm ex clivis, tùm ex flexioniibus, non expediret tàm longa plaustra construere, nec equos tam longâ serie disponere, ut cuique rem vel leviter consideranti statim patebit. CAPUT IX. Quid Cylindri & Scytalæ ad faciliorem ponderis motum præstent. A Deò ingentia aliquando pondera transferenda proponuntur, ut ea carris imponere transvehenda aut nimis operosum sit, aut periculo non vacet, ne rotarum axes pondere prægravati differingantur, aut propter soli mollitudinem rotæ devorentur: propterea rationem aliquam inire oportet, quâ voti compotes simus, citrà hujusmodi pericula. Et quidem si corpus teres sit, nec viarum salebræ, aut angustiæ impedimento sint, ipsum versari in gyrum poterit simili artificio, quo ad deportandos Ephesum ex lapicidinis scapos columnarum centum viginti septem altitudine pedum sexaginta usus est Ctesiphon Gnossius (sic eum vocat Plinius lib.7. cap.37. cum Vitruvio lib.10. cap.6, quem tamen idem Plinius lib.36. cap.14. cum

Transcription: Translated (English)

Mechanics line drawn from the center of gravity of the weight will fall at more unequal angles onto the plane AB, which is more inclined, than onto AC, which is less inclined, and the force of the gravity of the weight will tend more toward B than toward C, as below in its proper place will be explained, and was indicated above in book 1, chapter 14. §. From these things this can be understood. Hence the Hamburg wagons, by which goods are conveyed from Hamburg to Nuremberg, are longer, because neither steeper hills nor frequent occur in the journey, nor are the bends of the roads narrow, from which might arise either difficulty in ascending, or in turning the wagon. Therefore greater loads can also be placed upon them, and six horses are attached not two by two, but singly in a straight line; by which it comes about that, not pulling in different directions, they draw the cart with altogether the same impulse. But if the roads had more difficulty, both from hills and from turns, it would not be expedient to construct such long wagons, nor to arrange the horses in so long a series, as will at once be clear to anyone who considers the matter even lightly. CHAPTER IX. What Cylinders and Scytalæ contribute to the easier movement of weight. Since sometimes enormous weights have to be transferred, so that to place them on carts for transport is either too laborious, or not free from danger, lest the axles of the wheels, overloaded by the weight, be broken, or because of the softness of the ground the wheels be swallowed up: therefore some method must be devised by which we may attain our desire, without such dangers. And indeed if the body be round, and neither the roughness nor the narrowness of the roads be an obstacle, it will be able to roll in a circle by a similar contrivance to that which, for carrying column shafts from the quarries to Ephesus, of a hundred and twenty-seven columns, sixty feet in height, Ctesiphon the Gnossian used (so Pliny calls him, book 7, chapter 37, with Vitruvius book 10, chapter 6, whom nevertheless the same Pliny, book 36, chapter 14, with

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Liber secundus. CAPUT IX. 209 cum Strabone vocat Chersiphronem) celeberrimo Dianæ templo construendo præfectus, & quidem felici eventu: ca- pitibus enim scaporum, ubi axis extremitates desinebant, sub- scudis in modum inseruit, atque implumbavit ferreos axes: tùm de materiâ trientali scapos (hoc est ligneos tigillos crassi- tudinis unciarum quatuor pedis, seu pollicum quatuor) duos longiores juxtà columnæ longitudinem, duosque breviores transversarios ita compegit, ut parallelogrammum constituen- tes columnam possent complecti; mediisque transversariis ferreas armillas inseruit, quibus axes ferrei infigebantur, adeò ut liberè versari possent, cum boves traherent; quem- admodum & in gyrum volvuntur cylindri marmorei aut la- pidei, quorum usus est in exæquandis ambulationibus. Est autem maximè verisimile, & probabile, ita firmiter ligneum illud parallelogrammum fuisse compactum, ut non solùm extremis transversariorum capitibus anterioribus alli- gari possent boves; sed etiam per totam anterioris scapi lon- gitudinem distribui, ut faciliùs columna transferretur. Prosperum exitum consecuta scaporum vectura animum adjecit Methageni Ctesiphontis filio, ut paternam in- dustriam æmularetur in Epistylis vehendis: cum enim ho- rum figura non ea esset, quæ perinde atque cylindrica vol- vi posset, duabus rotis pedum circiter duodenum singula epistylia firmiter inclusit; rotarumque centris ferreos axes infixit, qui in armillis similem haberent versationem, ac dictum est in scaporum vecturâ. Cum enim boves ligneo parallelogrammo alligati traherent, Rotæ volvebantur, at- que cum illis pariter epistylia Rotis cohærentia in gyrum versabantur; quippe quæ in subjectum solum non incurre- bant, cum solæ Rotæ terram attingerent. Hâc methodo corporibus, quæ non sunt ad volubilitatem rotundata, faci- lem conversionem conciliare possumus; ex Rotis nimirum & pondere moles una compingitur, cujus extremitatibus cylin- dricis tota innititur, nihilque refert, cujus demum figuræ sit pars media, scilicet pondus, modò hæc à solo aliquantulum distans motum non impediat. Quâ autem ratione aut Rotæ construantur, aut illis onus includatur, artificis seu architecti solertiæ relinquitur. D d

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Liber secundus. CHAPTER IX. 209 with Strabo calling him Chersiphron) was put in charge of building the famous temple of Diana, and with happy success: for in the ends of the beams, where the extremities of the axle terminated, he inserted them like brackets, and fitted iron axles into them; then from timber one-third in thickness he joined together beams (that is, wooden pieces four inches thick, or four fingers’ breadth), two longer ones along the length of the column, and two shorter transverse ones, so as to form a parallelogram able to embrace the column; and in the middle transverse pieces he inserted iron rings, into which the iron axles were fixed, so that they could turn freely while the oxen drew; just as marble or stone cylinders are rolled round, the use of which is in leveling walks. And it is most likely and probable that that wooden parallelogram was so firmly joined together that not only could oxen be fastened to the front ends of the transverse pieces at the extremities; but also they could be distributed along the whole length of the front beam, so that the column might be transferred more easily. The successful transport of the beams encouraged Methagenes, son of Ctesiphon, to imitate his father’s industry in carrying architraves: for since their shape was not such as could be rolled along like a cylinder, he firmly enclosed each architrave in two wheels of about twelve feet each, and fixed iron axles in the centers of the wheels, which in the rings should have a similar turning, as was said in the transport of the beams. For when the oxen, attached to the wooden parallelogram, drew, the wheels turned, and with them the architraves, attached to the wheels, turned round together; for they did not fall upon the ground beneath, since only the wheels touched the earth. By this method we can give an easy motion to bodies that are not rounded for rolling; for from the wheels and the weight a single mass is formed, on whose cylindrical extremities the whole rests, and it matters not at all what shape the middle part is, namely the weight, provided that, being somewhat raised from the ground, it does not hinder the movement. But the manner in which either the wheels are constructed, or the load is enclosed in them, is left to the skill of the craftsman or architect. D d

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Mechanicorum Methagenis artificium imitatus Paconius, teste Vitruvio lib. 10. cap. 6. lapideam basim longam pedes duodecim, la- tam pedes octo, & altam pedes sex Apollinis colosso resti- tuendam, duabus Rotis pedum circiter quindecim, simili- ter inclusit: sed aliâ ratione ac Methagenes deducere statuit. A Rotâ ad Rotam circâ lapidem fusos sextantales, hoc est crassitudinis pollicum duorum, ad circinum compegit ita, ut fusus à fuso non distaret pedem unum. Tùm circâ fusos fu- nem involvit, qui bobus trahentibus explicabatur, & con- vertebantur Rotæ. Verùm quia funis circumvoluti spiræ ad unam, aut ad alteram partem spectabant, non poterat viâ rectâ ad lineam deduci moles illa; sed modò in hanc, mo- dò in illam partem deflectebat, ut opus esset retroducere, adeò ut ducendo & reducendo pecuniam contriverit, & ope- ram luserit Paconius. Potuisset tamen huic malo occurrere, nec sui inventi laude fraudari, si circâ fusos non unicum, sed duplicem funem ita involvisset, ut funium spiris vel ab extremitatibus fusorum, vel à medio, incipientibus, funis uterque paribus semper intervallis à sibi proximâ Rotâ dista- rent; sic enim factum fuisset, ut boves æqualiter utrumque funem trahentes, æqualiterque evolventes, molem illam rectâ viâ deducerent. Quamquam autem suâ laude non careant hujusmodi arti- ficum inventa, expeditissimè tamen, & citrà impendium, one- ra ingentia traducuntur subjectis cylindris, qui pondere pressi, cùm illud trahitur, convertuntur. Palangas peculiari voca- bulo Veterès dixere frestes teretes, qui navibus subjiciuntur, cùm attrahuntur ad pelagus, vel cùm ad littora subducuntur; ut apud Nonium Marcellum legisse me memini. Neque aliud quidpiam censendus est Cæsar intellexisse, ubi lib. 3. Belli Civil. scribit Quatuor biremes subjectis scutulis (fortasse scuta- lis, hoc est scy talis, antiquis enim Romanis literam usurari solitam loco y literæ Græcæ notum est) impulsas vectibus in interiorem partem transduxit. Sunt autem scy talæ ut apud Sui- dam, rotunda & polita ligna: aliquid tamen peculiare ad- dit Aristoteles in Mechan. quæst. 11. quærens, cur super scy- talas faciliùs portantur onera quam super currus, cum tamen ij magnas habeant rotas, illæ verò pusillas? Scy talis nimirum pu- sillas

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Mechanics Paconius, imitating the device of Methagenes, as Vitruvius testifies, lib. 10. cap. 6, similarly enclosed for the restoration of the stone base of the Colossus of Apollo, twelve feet long, eight feet wide, and six feet high, with two wheels of about fifteen feet, but by another method than Methagenes had resolved to use. From wheel to wheel he joined, around the stone, rollers a sextant of a foot in thickness, that is, two inches thick, arranged in a circle, so that each roller was a foot distant from the next. Then he wound a rope around the rollers, which was unwound by oxen pulling, and the wheels were turned. But because the coils of the wound rope faced to one side or the other, that mass could not be drawn in a straight line; instead it now swerved this way, now that way, so that it was necessary to pull it back, so that Paconius, by dragging and hauling it back, wasted money and played the fool with his work. Yet he might have avoided this fault, and not been deprived of the praise of his invention, if, instead of a single rope, he had wound a double rope around the rollers in such a way that, whether the coils of the ropes began from the ends of the rollers or from the middle, each rope would always be at equal intervals from the wheel nearest to it; for thus it would have happened that the oxen, pulling both ropes equally and unwinding them equally, would have drawn that mass in a straight course. Although inventions of this kind do not lack their own praise, nevertheless very large burdens are conveyed most easily and without expense by means of rollers placed underneath, which, being pressed by the weight when the load is being drawn, turn. The ancients called palangas, by a special term, smooth round beams placed under ships when they are being hauled to sea, or when they are drawn up to the shore; as I remember reading in Nonius Marcellus. Nor should Caesar be thought to have understood anything else where, in book 3 of the Civil War, he writes: “Four biremes, set upon rollers underneath, were moved by levers and drawn into the interior part” (perhaps scutulis, that is, scytalis; for it is known that the ancient Romans were accustomed to use the letter i in place of the Greek letter y). Now scytale, as in Suidas, are round and polished logs: Aristotle in the Mechanical Problems, question 11, adds something peculiar, asking why burdens are carried more easily on scytalae than on carts, although the latter have large wheels and the former very small ones? Namely, the scytalae small

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Liber secundus. CAPUT IX. 211 sillas rotas adjectas intelligit, non eas quidem circà axem, sed cum axe ipso, cui adnectuntur, versatiles; cujusmodi essent in hoc schemate rotulæ A & B cum suo axe connexæ. Porrò duplicem hujusmodi scytalarum usum considero: si enim onus impositum incumbat Rotulis ipsis, vel quia plana sit ejus superficies, vel quia tabulato fuerit superpositum, perinde res se habet, atque si cylindrus esset, cujus diameter idem esset cum rotularum diametro: neque tunc admodum refert, cujusnam figuræ sit axis, quem onus non tangit, si- ve rotundus ille sit, sive angulatus. At si onus ipsi axi in- cumbat, promineantque hinc & hinc rotulæ, omninò ne- cesse est axem rotundum esse, ut fieri possit rotularum con- versio, atque ita longum, ut inter rotulas onus laxè interci- piatur; maximè quippe cavendum est, ne rotulæ onus con- tingant, alioquin ex mutuo conflictu mora non mediocris motui crearetur. Ideò autem excogitatæ videntur hujusmo- di scytalæ, ut minimâ sui parte secundùm extremitates tan- gerent subjectum planum, atque adeò in pauciora incurre- rent offendicula, quàm cylindri totâ suâ longitudine incum- bentes plano. Sed illæ ab usu artificum jam diù intermissæ locum simplicibus cylindris concessere, quippe qui ob con- tinentem sibique semper similem figuram solidiores sunt, & periculo carent, cui obnoxiæ sunt scytalæ, ne videlicet Ro- tulæ illæ labem aliquam faciant cum rotunditatis, atque adeò etiam motûs, detrimento. Illud verò commodum, quod ex offendiculorum evitatione oriebatur, obtinemus pariter, si duplicem planorum tigillorum seriem substernamus capitibus cylindrorum; hinc enim sit, ut viarum salebræ evitentur, & Cylindri modicâ sui parte contingant subjectos tigillos, qui viam planam & æquabilem constituentes moram nullam mo- tui injiciunt. Sed & in hoc cylindrorum usu communiter censetur ali- quid inesse facilitatis majoris ad onera deducenda, quàm si illa currui imponerentur; tùm quia currui sua inest gravitas, quæ unâ cum impositâ sarcinâ majus onus constituit, ac D d 2

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Book Two. CHAPTER IX. 211 he understands wheels attached to sledges, not indeed turning about the axle itself, but revolving together with the axle to which they are fastened; such as the wheels A and B in this diagram, connected with their axle. Moreover, I consider a twofold use of cylinders of this kind: if the load laid upon them rests on the wheels themselves, either because its surface is flat, or because it has been placed upon a platform, the matter is much the same as if it were a cylinder whose diameter were the same as that of the wheels: nor then does it greatly matter of what shape the axle be, which the load does not touch, whether it be round or angular. But if the load rests on the axle itself, and the wheels project here and there, it is altogether necessary that the axle be round, so that the turning of the wheels may be possible, and also long enough, so that the load may be loosely received between the wheels; for above all care must be taken lest the wheels touch the load, otherwise, from their mutual collision, no small delay would be created in the motion. Therefore it seems that cylinders of this kind were invented so that they might touch the supporting plane only with the least part of themselves, along their ends, and thus run into fewer obstacles than cylinders resting on a plane along their whole length. But these, long since discontinued in the use of craftsmen, have yielded place to simple cylinders, since, because of their continuous and ever-same shape, they are more solid, and free from the danger to which sledges are exposed, namely, that those wheels may suffer some damage, with loss of roundness, and therefore also of motion. That advantage, however, which arose from avoiding obstacles, we obtain equally if we lay a double row of flat beams beneath the ends of the cylinders; for by this means the roughness of the roads is avoided, and the cylinders touch the beams beneath only with a small part of themselves, the beams constituting a smooth and even way that causes no delay to motion. But also in this use of cylinders it is commonly thought that there is something of greater ease in drawing loads, than if they were placed upon a cart; both because a cart has its own weight, which together with the load placed upon it makes a greater burden, and

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Mechanicorum 212 propterea in utroque transferendo is, qui trahit, majorem impendit laborem; at subjectis oneri cylindris, horum gra- vitas nihil officit trahenti: Tùm quia currûs Rotæ, cum sint circa suum axem, cui insiguntur, mobiles, aut hûc & illuc nutant, si laxa sint capita, nec clavo exquisitè coërceantur, aut si arctiùs axi cohæreant, axem quem complectuntur, & clavum quo coërcentur, validiùs terunt; & ex utroque hoc capite movendi difficultas oritur, cùm aliquid impressi im- petûs aut in illâ inconstantiâ, aut in hoc conflictu contera- tur: nihil autem hujusmodi cylindris contingit. Tùm etiam quia Rotæ modiolus ab axe premitur, & deorsum pondere urgente, & antrorsum impetu ad anteriora trahente; ex quo quantum difficultatis in movendo oriatur, hinc manifestum est, quod nisi axungiâ aut amurcâ illinantur curruum axes, ægrè convertuntur rotæ, & denso stridore, quantus sit par- tium tritus atque conflictus, testatum faciunt. At Cylindri quantumvis ab onere premantur, nullo pingui liquore obli- nendi sunt, ut lubrici fiant; nulla enim impositi oneris aspe- ritas cylindrorum conversionem impedire potest. Nam si fue- rit ingens lapis A B cylin- dris subjectis impositus, & cylindri punctum C con- gruat puncto A lapidis, dia- metri C D altera extremitas D tangit subjectum planum; cum verò saxum ex B ver- sùs A propellitur, seu tra- hitur ex A, ita cylindrus convertitur, ut D F ar- cus sensim ad subjectum planum, contrà verò arcus C E ad impositum saxum accom- modetur, citrà omnem saxi & cylindri affrictum. Hinc tamen aliquid etiam incommodi cylindris adhæret, si eum plaustrorum rotis conferantur; hæ scilicet motum con- tinuant, cum sine fine volvantur, quippe quæ axi infixæ, im- posito oneri pariter, ut ita loquar, cohærent; illos verò, ni- mirum cylindros, onus dum promovetur, post se relinquit; ac proinde aut cylindrorum copia non exigua suppetere debet, qui

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Mechanics 212 therefore, in carrying both burdens, the one who drags expends greater labor; but with cylinders placed under the load, their weight hinders nothing in the dragger: also because the wheels of a cart, since they are movable about their own axle, on which they are fixed, either nod hither and thither if their heads are loose, and are not exactly restrained by the pin, or if they adhere more tightly to the axle, they wear down more severely the axle which they embrace and the pin by which they are restrained; and from either of these causes difficulty in moving arises, since some of the impressed impetus is consumed either in that instability or in this conflict: but nothing of this sort happens with cylinders. Then also because the hub of the wheel is pressed by the axle, both downward by the burden urging it, and forward by the impetus drawing it onward; from which how much difficulty in moving arises is here made evident, since unless the axles of carts are smeared with axunge or olive lees, the wheels are turned with difficulty, and with a harsh creaking they make manifest how great is the friction and conflict of the parts. But cylinders, however much they are pressed by the load, need not be coated with any greasy liquid in order to become slippery; for no roughness of the imposed load can impede the turning of cylinders. For if a huge stone A B be placed on cylinders underneath it, and if the point C of the cylinder coincide with the point A of the stone, the other end D of the diameter C D touches the underlying plane; but when the rock is pushed from B toward A, or pulled from A, the cylinder is thus turned, that the arc D F gradually accommodates itself to the underlying plane, while on the other hand the arc C E accommodates itself to the stone placed above, without any friction whatever between the stone and the cylinder. Hence, however, some inconvenience also attaches to cylinders if they are compared with the wheels of wagons; for these continue the motion, since they roll endlessly, because, being fixed to the axle, they, along with the imposed load, as it were adhere to it; but those, namely cylinders, the load leaves behind as it is moved onward; and therefore either a not inconsiderable supply of cylinders ought to be at hand, which

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Liber secundus. CAPUT IX. 213 qui longâ serie dispositi onus alij ex aliis excipiant, aut qui relinquuntur, subinde transferendi sunt, ut iterùm oneri subjiciantur. Verùm hæc alterna cylindrorum translatio non adeò gravis est; quin plus habeat adjumenti, quàm incom- modi; cum enim plurimùm referat, utrùm qui subjicitur cy- lindrus, reliquis posterioribus cylindris parallelus, an obli- quus statuatur, ut onus ad lineam viâ rectâ deducatur, aut motus sui vestigium inflectat; facillimum est opportunâ cylin- dri translati collocatione parallelâ, aut obliquâ, destinatum oneris motum administrare. Illud autem non immeritò hîc examinandum occurrit, utrùm majores cylindri minoribus potiores censendi sint, & an præstet subjicere oneri cylindrum G I majorem, an verò minorem G H. Et quidem si figuræ dumtaxat magnitudo atque parvi- tas spectetur, hoc unum discrimen invenio, quòd ad certam motûs mensuram perficiendam crebriùs volvi oportet cylin- drum minorem, quàm majorem; onus verò à subjecto plano distare majoris diametri G I intervallo potiùs, quàm minoris G H, non video, quid conferat ad motûs facilitatem; tantum enim promovetur onus, quantus est peripheriæ arcus, cui illud in motu aptatur, eique æqualis est arcus oppositus, qui plano pariter in motu congruit: ac propterea parum refert, utrùm eadem arcus mensura sit majoris circuli pars minor, an minoris circuli pars major. Verùm si qua inter motum occurrant offendicula, hæc minùs officere majori cylindro, quàm minori, dicendum est, quemadmodum & de rotis majoribus dictum est superiori ca- pite; siquidem majoris cylindri diameter obliquior incidit in idem offendiculum, quod minùs directè opponitur motui, & longiore motu Potentiæ sit eadem ponderis elevatio, ut ibi ex- plicatum est. Aliud est præterea, nec sanè nullius momenti, quod majo- ri cylindro incitatiorem dat volubilitatem; quòd videlicet (quemadmodum & globo majori contingit) major cylindrus, quamvis Geometricam Rotunditatem non assequatur, tamen propriùs accedit ad figuram exquisitè Rotundam, quàm mi- nor: si enim à circulo Geometricè perfecto æqualiter recedant utriusque cylindri majoris ac minoris bases, non tamen æqua- Dd 3

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Book Second. CHAPTER IX. 213 which, arranged in a long series, receive the burden one from another, or which are left behind must in turn be transferred, so that they may again be subjected to the load. But this alternate transfer of the cylinders is not so troublesome; rather, it has more advantage than inconvenience. For since it matters greatly whether the cylinder placed underneath, parallel to the remaining cylinders behind it, or oblique, is so set that the burden may be carried along in a straight line, or may bend the track of its motion, it is very easy, by a suitable parallel or oblique placement of the transferred cylinder, to regulate the intended movement of the burden. However, it here not undeservedly comes up for examination whether larger cylinders are to be judged preferable to smaller ones, and whether it is better to place under the burden the larger cylinder G I, or indeed the smaller G H. And certainly, if size and smallness alone of the figure are considered, I find only this difference, that in order to accomplish a certain measure of motion the smaller cylinder must be turned more frequently than the larger; but that the burden, by the interval of the larger diameter G I, rather than of the smaller G H, is kept away from the subject plane, I do not see what this contributes to the ease of motion. For the burden is advanced only by so much as is the arc of the circumference to which it is fitted in motion, and the opposite arc, which likewise agrees with the plane in motion, is equal to it. And therefore it matters little whether the same measure of arc be the smaller part of the greater circle or the larger part of the smaller circle. But if any obstacles arise in the motion, it must be said that these interfere less with the larger cylinder than with the smaller, just as has also been said of larger wheels in the preceding chapter; for the diameter of the larger cylinder meets the same obstacle more obliquely, which is more directly opposed to the motion, and by a longer motion of the Power the elevation of the weight is the same, as was explained there. There is also another thing, and certainly not of no importance, namely that it gives the larger cylinder a more eager rotatory movement; because, just as happens with a larger sphere, the larger cylinder, although it does not attain geometric roundness, nevertheless approaches more closely to an exquisitely round figure than the smaller one. For if the bases of both the larger and the smaller cylinder depart equally from a circle geometrically perfect, still not equally

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Mechanicorum 214 liter angulata est utraque basis, sed in majori major est angulus, in minori minor, atque adeò ille magis, quàm hic, ad rotunditatem accedit. In majori autem circulo angulum, qui peripheriam complectitur, majorem esse palam est, quia idem excessus majori Radio additus constituit secantem anguli minoris, quàm si minori Radio addatur; ac propterea angulus Complementi major est in majori, quàm in minori. Id quod, per se quidem satis clarum, dilucidiùs explicabitur, si ex minore circulo extet particula, cujus altitudo sit ON, ex majore autem circulo æqualis altitudo emineat IM. Ductis Tangentibus & Radiis, certum est Secantis excessum ON supra Radium LO minorem, habere majorem Rationem ad suum Radium, quàm habeat æqualis excessus IM ad suum Radium LI majorem ex 8. lib. 5. Est igitur MLP angulus minor angulo NLS, & Complementum LMP majus est Complemento LNS quare totus angulus VMP major est toto angulo TNS, ac proinde magis ad rotunditatem accedit. CAPUT X. Circulorum Concentricorum motus explicatur. Circuli motus, ob id ipsum quia circulus est, circa suum centrum perficitut eâ ratione, ut superiores partes progrediantur, inferiores retrocedant, anteriores descendant, posteriores ascendant, servatâ semper pari oppositorum progressûs atque regressûs, descensûs atque ascensûs mensurâ; pro ut unicuique rem vel leviter consideranti patet. Quare dum in gyrum circulus agitur, centrum quidem manet, reliquæ verò partes ita singulæ ex alio in alium locum sibi invicem

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Mechanics 214 is angular at either base, but in the greater one the angle is greater, in the lesser, lesser; and therefore that one more than this approaches roundness. In the greater circle, moreover, it is clear that the angle which contains the periphery is greater, because the same excess added to the greater radius constitutes a secant of the smaller angle, than if it be added to the smaller radius; and therefore the angle of the complement is greater in the greater than in the lesser. This, though of itself sufficiently clear, will be explained more clearly if from the lesser circle there protrudes a segment whose height is ON, and from the greater circle a like height projects IM. After drawing the tangents and radii, it is certain that the excess of the secant ON above the radius LO, being smaller, has a greater proportion to its radius than does the equal excess IM to its greater radius LI, from book 8, prop. 5. Therefore the angle MLP is less than angle NLS, and the complement LMP is greater than the complement LNS; wherefore the whole angle VMP is greater than the whole angle TNS, and therefore approaches more to roundness. CHAPTER X. The motion of concentric circles is explained. The motion of a circle, because it is a circle, is performed around its own center in such a way that the upper parts advance, the lower recede, the front descend, the rear ascend, with the measure of advance and retreat, descent and ascent, always preserved equal and opposite; as is evident to anyone who considers the matter even slightly. Wherefore while the circle is turned in a circle, the center indeed remains, but the remaining parts, however, each in turn from one place into another, to one another

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Liber secundus. CAPUT X. 215 cem succedentes commeant, ut circulus totus spatium, in quo volvitur, omninò non mutet. Quemadmodum observare est in Solis orbitâ, quam Eclipticam vocant; hæc enim diurnâ conversione circa Mundi axem Solem secum rapiens à suo lo- co non recedit, Sole ab ortu in Occasum commigrante: id multò magis in singulorum circulorum circà sua centra revo- lutione manifestum apparet. Quod si circulus aut horizonti parallelus, aut illi ad perpendiculum insistens, raptetur; mo- tus ille nihil habet circulari affine, cum circà centrum non perficiatur, sed singula circuli puncta solo motu recto unâ cum centro moveantur. Sin autem axis circulo versatili infixus trahatur, jam circu- lus & cum axe pariter movetur, & circa axem volvitur: atque adeò singularum circuli partium motus is est, qui ex recto cen- tri, & circulari ipsius orbitæ componitur. Hinc semicirculi superioris partes cum progrediantur versùs cumdem locum, ad quem centrum tendit, suum motum motui centri addunt: Contrà verò inferioris semicirculi partes retrocedentes suum motum à centri motu detrahunt. Rotæ igitur puncta omnia, dum currus trahitur, si non summatim tota revolutio, sed par- ticulatim, accipiatur, non æquali velocitate moventur. Sit explicandi gratiâ, circulus B D A E, cujus centrum C moveatur versus F, & sit tangens G A, cui in motu appli- catur ipsius circu- li orbita; in quâ accipiatur sextans hinc & hinc A D, & A E. Igitur in Conversione, dum Centrum C trahitur ad F, punctum D venit in G, & arcus D A æqualis est rectæ G A, cui in motu subinde per partes congruit: atque adeò, quarum partium semidiameter C A est 21, earum arcus A D, & recta A G est 22, & motus cen- tri illi æqualis C F est pariter 22. Quoniam verò in motu or- bitæ

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Liber secundus. CAPUT X. 215 as the points succeeding one another pass on, so that the whole circle does not alter at all the space in which it is moved. This may be observed in the orbit of the Sun, which they call the Ecliptic; for this, in its daily revolution about the axis of the World, carrying the Sun along with it, does not depart from its place, the Sun moving from East to West: this appears much more manifest in the revolution of individual circles about their own centers. But if a circle, either parallel to the horizon or standing perpendicular to it, be carried along, that motion has nothing akin to circular motion, since it is not completed about a center, but each point of the circle is moved by a purely rectilinear motion together with the center. But if the axis fixed in a movable circle be drawn along, then the circle both moves along with the axis, and revolves about the axis: and therefore the motion of the several parts of the circle is composed of the rectilinear motion of the center and the circular motion of the circle itself. Hence the parts of the upper semicircle, as they advance toward the same place toward which the center tends, add their motion to the motion of the center: whereas the parts of the lower semicircle, moving backward, subtract their motion from the motion of the center. Therefore all the points of a wheel, while a carriage is being drawn, if, instead of taking the whole revolution at once, it is considered part by part, are not moved with equal speed. Let it be, for the sake of explanation, circle B D A E, whose center C be moved toward F, and let G A be the tangent, to which in motion the path of the circle itself is applied; in which let there be taken the sixth part on this side and on that, A D, and A E. Therefore, in the revolution, while the center C is drawn to F, point D comes to G, and arc D A is equal to straight line G A, with which in motion it continually agrees by parts: and therefore, of those parts whose semidiameter C A is 21, their arc A D, and straight line A G is 22, and the motion of the center C F equal to it is likewise 22. But because in the motion of the orbit

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Mechanicorum bitæ circa suum centrum, punctum A ascendens in E retrocede- dit juxta mensuram sinûs S E (qui ad Radium C A 21 est ut 18) hinc est post conversionem, in qua D est in G, punctum A ita ascendisse, ut sit in lineâ H E parallelâ Tangenti G A, sed motui centri tantum detraxerit, quantus est sinus S E. Quia igitur Radius C D ubi congruit punctis F G, secat in H rectam H E, sumatur H I æqualis sinui S E, & puncti A totus progressus remanet S I partium 4, quarum S H, seu C F est 22. Quare A est in I, quando D est in G. Contrà verò in superiore semicirculo sumatur item ex B hinc, & hic sextans B K & B L; atque in conversione ubi cen- trum C venerit in F, & punctum orbitæ D in G, erit K in O, & diameter D K secabit parallelam K N in M. Igitur punctum B ita descendit ad parallelam N K, ut motui centri C F, hoc est B O seu R M, addiderit suum progressum juxta mensuram R L Sinum Sextantis B L, hoc est 18. Venit igitur B in N; atque additis R M 22, & M N 18, totus progressus puncti B est R N 40. Comparatis itaque invicem curvis lineis A I & B N, manifestum est puncta B & A non æque velociter mo- veri, cum eodem temporis spatio inæqualia loci spatia per- currant. Eadem erit methodus, si reliquorum orbitæ punctorum ve- locitates aut tarditates considerandæ sint: si tamen adverteris non eandem esse omnium circuli Quadrantum rationem in de- terminandâ mensura motûs addendi, aut demendi motui cen- tri. Nam in anteriori Quadrante superioris semicirculi, & in posteriori Quadrante inferioris semicirculi, mensura progres- sûs addendi in illo, & regressus demendi in isto, attendenda est ex Sinu Recto arcûs, qui describitur in motu circa cen- trum à puncto, 'cujus velocitas inquiritur, aut tarditas: Et quidem integer Sinus Rectus accipitur, si punctum à summo vertice descendens, vel ab infimo contactûs puncto ascendens movetur, ut ex B vel ex A: sin autem punctum consideretur, quod intrà eosdem Quadrantes distet ab extremitatibus diam- etri subjecto plano insistentis, puta L aut E, quæ moventur in V, aut in P, progressûs aut regressûs mensura desumitur ex dif- ferentiâ Sinuum Rectorum, qui respondent arcubus B L & B V, aut arcubus A E & A P. In posteriori verò Quadrante supe- rioris

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Mechanics of the body around its center, the point A ascending in E retreated by the measure of the sine S E (which to the radius C A 21 is as 18). Hence it follows that after the revolution, in which D is in G, the point A has ascended so far that it is in the line H E parallel to the tangent G A, but has detracted from the motion of the center only as much as is the sine S E. Since therefore the radius C D, where it coincides with the points F G, cuts in H the straight line H E, let H I be taken equal to the sine S E, and the whole progress of point A remains S I of 4 parts, of which S H, or C F, is 22. Therefore A is in I, when D is in G. Conversely, in the upper semicircle, let there likewise be taken from B this way, and here a sextant B K and B L; and in the revolution where the cen- ter C has come to F, and the point D of the orbit to G, K will be in O, and the diameter D K will cut the parallel K N in M. Therefore point B has descended thus to the parallel N K, that to the motion of the center C F, that is B O or R M, it has added its own progress according to the measure of R L, the sine of the sextant B L, that is 18. Therefore B comes to N; and with R M 22 and M N 18 added, the whole progress of point B is R N 40. Comparing thus the curved lines A I and B N with each other, it is clear that points B and A do not move equally fast, since in the same space of time they traverse unequal spaces of place. The same method will apply, if the velocities or slownesses of the remaining points of the orbit are to be considered: provided, however, that you observe that the ratio of all the quadrants of the circle is not the same in determining the measure of the motion to be added to, or subtracted from, the motion of the center. For in the anterior quadrant of the upper semicircle, and in the posterior quadrant of the lower semicircle, the measure of the progress to be added in the former, and of the retreat to be subtracted in the latter, must be attended to from the straight sine of the arc which is described in the motion around the center by the point whose velocity or slowness is being investigated. And indeed the entire straight sine is taken, if a point descending from the top vertex, or ascending from the lowest point of contact, is moved, as from B or from A: but if the point considered is one within the same quadrants, distant from the extremities of the diameter resting on the underlying plane, such as L or E, which are moved in V, or in P, the measure of progress or retreat is taken from the difference of the straight sines which correspond to the arcs B L and B V, or to the arcs A E and A P. But in the posterior quadrant of the upper

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Liber secundus. CAPUT X. 217 rioris semicirculi, & in anteriori Quadrante inferioris semicirculi, progressus addendus, aut regressus demendus, motui centri, mensuram desumit ex Sinubus Versis, aut ex eorum differentiâ, pro ut puncti motus ascendens aut descendens incipit ab extremitate Quadrantis, aut à loco medio, ut facilè cuique constat: neque enim schema multiplici linearum descriptione ad confusionem implere operæ pretium est. Cum itaque in oppositis Quadrantibus similem mensuram recipiant incrementa atque decrementa sive à sinubus Rectis, sive à Versis, addenda aut demenda motui centri, manifestum est punctum quodlibet in integrâ conversione demùm progressum fuisse pari mensurâ cum motu centri. Si enim Algebricè statuatur motus Centri Z, incrementum in superiore semicirculo addendum +A, decrementum in inferiore semicirculo tollendum - A; manifestum est totum motum, qui componitur, Z +A - A non esse nisi Z. His ita constitutis, quæ ita clara sunt, ut nihil habere videantur dubitationis, nec in controversiam vocari queant, jam eximendus est scrupulus, quem philosophantibus injecit Aristoteles Mechanic. quæst. 24. de circulorum concentricorum motu, quando alter ad alterius motum promoto communi centro movetur. Sit enim major circulus, cujus Radius CB, minor autem, cujus Radius CS; quos tangant parallelæ BF & ST, quibus item recta per centrum ducta parallela sit CO, quam videlicet percurrit centrum, dum trahitur. Negari non potest in hâc circulorum tractione & conversione peripherias tùm majoris, tùm minoris Circuli suis Tangentibus ita coaptari, ut factâ Quadrantis BD conversione, fiat pariter Quadrantis SI Ee

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Book Two. CHAPTER X. 217 in the former semicircle, and in the anterior quadrant of the lower semicircle, the progress to be added, or the regress to be subtracted, to the motion of the center, takes its measure from the versed sines, or from their difference, according as the motion of the point, ascending or descending, begins from the end of the quadrant, or from the middle place, as is easily evident to anyone: for it is not worth the trouble to fill the diagram with a confusing multiplicity of lines. Since, therefore, in opposite quadrants the increments and decrements receive a similar measure, whether from the recta sine or from the versed sine, to be added or subtracted to the motion of the center, it is manifest that any point in a full revolution has at last advanced by a measure equal to the motion of the center. For if, algebraically, the motion of the center be set as Z, the increment in the upper semicircle to be added as +A, the decrement in the lower semicircle to be removed as -A, it is manifest that the whole motion, which is composed, Z + A - A is nothing other than Z. These things being thus established, which are so clear that they seem to admit of no doubt, and cannot be brought into controversy, there must now be removed the scruple which Aristotle has raised among philosophers, Mechanic. quest. 24, concerning the motion of concentric circles, when one is moved by the motion of the other, the common center being advanced. Let there be a greater circle, whose radius is CB, and a smaller one, whose radius is CS; let the parallels BF & ST touch them, and let the straight line drawn through the center be CO, to which, as it were, the center runs, while it is being drawn. It cannot be denied that in this drawing and turning of the circles, the circumferences both of the greater and of the smaller circle are so fitted to their tangents that, when quadrant BD has been turned, quadrant SI likewise is made to turn Ee

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Mechanicorum 218 conversio, & ubi punctum D venerit in F, punctum I sit in T, & centrum C in O, atque adeò Radius C D mutato situ factus sit OF. Major igitur Quadrans percurrit spatium BF, & mi- nor spatium ST. At quia æquales rectæ OF & CB perpen- diculares sunt ad eandem rectam BF, etiam sunt parallelæ, junguntque parallelas ST & BF, quæ propterea etiam sunt æquales, ex 34. lib. 1. Igitur arcus S I minor arcu BD, coap- tatur spatio æquali ipsi arcui Quadrantis BD, cui supponitur æqualis recta BF. Quarum itaque partium 7 est Radius CB, earum est Quadrans BD, hoc est recta BF 11, estque pariter ST 11. At quarum partium 7 est Radius CB, earum sit Ra- ditus CS 4; igitur Quadrans S I est 6 ́, multo minor quàm recta ST, cui ipse Quadrans S I in motu congruit. Id enim verò tantum præ se fert difficultatis, ut mirum sit, quot Ixiones rota hæc torqueat, & quàm varias in partes se alij aliter versent; quorum sententias si examinare liberet, in lon- gum nimis sermonem me vocaret ista disputatio, nec satis sci- rem, utrum plus aliquid lucis propositæ quæstioni affunderetur. Quid igitur probabilius dicendum videatur, paucis expono. Priùs tamen observa in dictâ Quadrantis revolutione, quan- do Centrum C venerit in O, & D in F, & in I in T, tunc punctum B esse in E (est enim OE æqualis Radio CB) atque punctum S in V (est scilicet OV æqualis Radio CS) ita ut B ascendat per curvam BE, punctum autem S ascendat per curvam SV, & similiter punctum D descendat per cur- vam DF, punctum verò I descendat per curvam IT. Ex quo patet punctum S minoris circuli plus promoveri, quàm punctum B majoris circuli; hujus enim progressus est CE, il- lius autem est CV: & pari ratione constat magis ad anterio- ra promoveri punctum I minoris circuli, cujus progressûs men- sura est IO, quàm punctum D majoris circuli, cujus progres- sus est DO. Et hæc quidem, quando centri motus legem accipit à pe- ripheriâ majoris circuli; ad cujus motum minor circulus con- centricus movetur; eo quod major circulus insistit subjecto pla- no, cui orbita subinde coaptatur rectam lineam sibi æqualem designans ex hypothesi, dumque movetur, secum rapit interio- rem circulum. Quod

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Mechanicorum 218 rotation, and where point D has come to F, point I is at T, and center C at O, and so Radius C D, with its position changed, has become OF. Therefore the greater Quadrant traverses the space BF, and the smaller the space ST. But because the equal straight lines OF and CB are perpendicular to the same straight line BF, they are also parallel, and they join the parallel lines ST and BF, which for that reason are also equal, by 34 of book 1. Therefore the arc SI, smaller than arc BD, is matched to a space equal to the said arc of Quadrant BD, under which lies the equal straight line BF. Of which parts then 7 is Radius CB, of those is Quadrant BD, that is straight line BF 11, and likewise ST is 11. But of which parts 7 is Radius CB, let Radius CS be 4; therefore Quadrant SI is 6́, much smaller than straight line ST, to which the Quadrant SI itself in motion corresponds. Indeed this presents such a difficulty that it is surprising, how many Ixions this wheel may twist, and how variously different men may turn themselves in different directions; and if it were pleasing to examine their opinions, this dispute would call me into too long a discourse, nor would I sufficiently know whether any more light would be shed on the proposed question. What then seems most probable to be said, I set forth briefly. Yet first observe in the said revolution of the Quadrant, when Center C has come to O, and D to F, and I to T, then point B is in E (for OE is equal to the Radius CB), and point S in V (namely OV is equal to the Radius CS), so that B ascends through curve BE, but point S ascends through curve SV, and similarly point D descends through curve DF, while point I descends through curve IT. From which it is clear that point S of the smaller circle is moved forward more than point B of the greater circle; for the former’s progress is CE, while the latter’s is CV: and by the same reasoning it is evident that point I of the smaller circle is moved farther toward the front, its measure of progress being IO, than point D of the greater circle, whose progress is DO. And this indeed, when the motion of the center takes its law from the perimeter of the greater circle; to whose motion the smaller concentric circle is moved; because the greater circle rests upon the supporting plane, to which the orbit is then adapted, marking out a straight line equal to itself by hypothesis, and while it moves, it draws along with itself the inner circle. What

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Liber secundus. CAPUT X. 219 Quod si minor circulus insistat subjecto sibi plano, legem- que det motui centri; quia minor peripheria designat rectam sibi æqualem, res contrario modo procedit, quia dum ad mi- noris circuli motum circulus major movetur, hujus orbita de- signat in plano subjecto lineam minori peripheriæ æqualem. Hinc si arcus S I designat rectam S G sibi æqualem, ubi 1 ve- nerit in G, etiam D erit in H, atque totus Quadrans B D de- signabit solùm rectam B H æqualem rectæ S G. Erit igitur recta S G æqualis Quadranti S I 6 ́; cui pariter æqualis est B H: Ex quo fit punctum B, quia distat à centro C partibus 7, non solùm non procedere in revolutione Quadrantis; sed re- trocedere per ́ interea, dum commune centrum C promove- tur per 6 ́. Non absimili ratione punctorum B, & S jam in E & V translatorum motus per consequentes circuli Quadrantes, do- nec integra revolutio perficiatur, considerandus est: & quæ de uno puncto cujusque circuli deprehenduntur, de singulis ejusdem orbitæ punctis dicta faciliùs intelliguntur, quàm ut uberiori explicatione opus sit. Ex his apertè liquet eam lineam rectam in subjecto plano de- signari à peripheriâ tùm majoris, tùm minoris circuli, quæ æqualis sit motui centri, prout ille legem accipit à majore aut à minore orbitâ, ad cujus motum altera movetur; ac proinde modò longiori, modò breviori lineæ rectæ in motu coaptantur ambæ peripheriæ; ut enim rectè loquitur Aristoteles loc. cit. Quando hic quidem movet, ille verò movetur ab isso, quantum uti- que moverit alter, tantum alter movebitur. Cur igitur parem lineam rectam designat in plano utraque orbita major & minor? constat ex dictis: quia nimirum cu- juslibet circuli quodlibet punctum dum trahitur simul, & vol- vitur, promovetur non nisi pro ratione motûs centri: sed con- centricorum circulorum unum & idem est centrum; ergo uni- cus est centri motus, & secundùm unam eandemque mensu- ram motûs centri, omnia puncta tùm majoris, tùm minoris or- bitæ, demum absolutâ conversione, promota sunt; singulorum enim incrementa, dum superiorem semiperipheriam motu describunt, ab oppositis decrementis elisa in inferioris semipe- E e 2

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Second Book. CHAPTER X. 219 Now if the smaller circle rests upon the plane subject to it, and gives law to the motion of the center; because the smaller periphery marks out a straight line equal to itself, the matter proceeds in the opposite way, because while the greater circle is moved in accordance with the motion of the smaller circle, its orbit marks out in the subjacent plane a line equal to the smaller periphery. Hence if the arc S I denotes the straight line S G equal to itself, where 1 has arrived at G, then D will also be in H, and the whole quadrant B D will designate only the straight line B H equal to the straight line S G. Therefore the straight line S G will be equal to the quadrant S I 6 ́; to which B H is likewise equal: from which it follows that the point B, because it is distant from the center C by 7 parts, does not merely fail to advance in the revolution of the quadrant; but moves backward by ́ in the meantime, while the common center C is advanced by 6 ́. In a not dissimilar way, the motion of the points B and S, now translated to E and V through the successive quadrants of the circle, must be considered until the full revolution is completed: and what is found concerning one point of each circle is more easily understood as said of the individual points of the same orbit, so that a fuller explanation is not needed. From these things it is clearly evident that that straight line is marked out on the subjacent plane by the periphery of both the greater and the smaller circle which is equal to the motion of the center, according as that motion receives its law from the greater or from the smaller orbit, by whose motion the other is moved; and therefore sometimes the two peripheries are adapted in motion to a longer, sometimes to a shorter straight line: for, as Aristotle speaks rightly in the passage cited, when the one indeed moves, the other is moved by it, and however much the one has moved, so much the other will move. Why then does each orbit, the greater and the smaller, mark out an equal straight line in the plane? It is clear from what has been said: because, namely, every point of any circle, while it is drawn along and at the same time rolls, is advanced only in proportion to the motion of the center: but in concentric circles there is one and the same center; therefore the motion of the center is single, and according to one and the same measure of the motion of the center, all the points of both the greater and the smaller orbit, once the revolution has been completed, have been advanced; for their individual increments, while they describe the upper semicircumference in motion, are offset by opposite decrements in the lower semipe- E e 2

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Mechanicorum ripheriæ descriptione, solum centri motum relinquunt. Nil itaque mirum, si tres lineæ, quarum primam centrum percurrit, secundam orbita minor designat, tertiam orbita major, planè æquales sunt; pendent enim ab unico & communi motu centri, cui nihil additur, aut demitur ex integrâ conversione circa centrum, sivè illa latiùs excurrat in majore circulo, sivè arctiùs in minore coërceatur. At, inquis, difficile est cogitatione assequi, & oratione explicare, quî fieri possit, ut peripheriâ utráque subjectum sibi planum semper tangente, nullóque puncto manente sine motu, ita ut plana subjecta ab aliis subinde atque aliis punctis tangantur, pauciora puncta minoris peripheriæ totidem punctis rectæ lineæ coaptentur, ac plura puncta majoris peripheriæ. Sunt qui difficultatem hanc declinant adstruentes infinita puncta tùm in circulorum peripheriis, tùm in lineis rectis, negantesque inter infinitas multitudines, quæ invicem comparentur, affirmari posse totidem in unâ infinitâ multitudine, ac in aliâ pariter infinitâ unitates reperiri, nulla enim est infiniti ad infinitum Ratio, ac proinde nulla fieri potest, perinde ac in multitudinibus finitis, comparatio minoris, aut majoris, aut propriè, &, ut aiunt, positivè æqualis. Hæc tamen (quamvis quod ad infinita Ratione carentia spectat, à me ultrò admittantur, Rationem scilicet habere dicuntur inter se magnitudines, idem & de multitudinibus dicendum, quæ possunt multiplicatæ se mutuò superare, ut definit Euclides lib. 5. ubi autem nullus est terminus, ut in infinito, nullus pariter excessus intercedere potest quavis factâ multiplicatione) non facient satis comparanti omnia puncta unius lineæ cum omnibus punctis alterius lineæ, non quâ infinitæ punctorum multitudines sunt, sed quâ finitæ magnitudines ex punctis illis quantumvis infinitis constituuntur: finitas autem magnitudines comparari invicem posse, ac Rationem inter se habere nemo negaverit. Superest igitur explicandum, quomodo peripheria minor coaptetur lineæ rectæ æquali illi eidem, cui commensuratur peripheria major. Propterea, duce Galilæo Dialog. 1. de motu, observant similium polygonorum concentricorum motum ac conversionem, in quâ polygonum, ex quo centri motus legem accipit, singula

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In the description of the Mechanics of the periphery, they leave only the motion of the center. So it is no wonder if the three lines, of which the center traverses the first, the minor orbit traces the second, and the major orbit the third, are clearly equal; for they depend on the single and common motion of the center, to which nothing is added or taken away by the complete revolution around the center, whether it ranges more widely in the larger circle or is more narrowly confined in the smaller. But, you say, it is difficult to grasp in thought and explain in speech how it can happen that, with either periphery always touching the plane subject to it, and with no point remaining without motion, so that the planes beneath are touched now by some points and now by others, the fewer points of the smaller periphery are fitted to just as many points of a straight line as the more numerous points of the greater periphery. There are some who avoid this difficulty by asserting infinite points both in the circumferences of circles and in straight lines, and denying that among infinite multitudes, which are compared with one another, it can be affirmed that as many units are found in one infinite multitude as in another equally infinite one; for there is no ratio of the infinite to the infinite, and therefore no comparison can be made, just as in finite multitudes there is comparison of less, or greater, or properly, as they say, positively equal. Yet these things, although as far as concerns things without ratio among infinites I freely admit them—for it is said that magnitudes have ratio to one another, and the same must also be said of multitudes, which, being multiplied, can surpass one another, as Euclid defines in Book 5—where there is no limit, as in the infinite, likewise no excess can intervene by any multiplication whatever. But this will not suffice for one who compares all the points of one line with all the points of another line, not insofar as there are infinite multitudes of points, but insofar as finite magnitudes are constituted from those points, however infinite they may be; and finite magnitudes can undoubtedly be compared with one another and have ratio to one another. It remains, then, to explain how the smaller periphery is fitted to the straight line equal to that same line to which the greater periphery is commensurate. For this reason, guided by Galileo, Dialog. 1. de motu, they observe the motion and rotation of similar concentric polygons, in which the polygon from which it receives the law of the motion of the center, each individual

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Liber secundus. CAPUT X. 221 la latera ita æqualibus lineæ rectæ partibus accommodat, ut in integrâ conversione linea recta subjecti plani sit æqualis peri- metro polygoni: at non item partes omnes lineæ, cui alterum polygonum in motu coaptatur, si unica comprehensione su- mantur, lineam æqualem polygoni majoris perimetro consti- tuunt. Res, clarita- tis gratia, explicetur in Hexagonis, quo- rum commune cen- trum sit A, & latera BC, DE incumbant parallelis lineis BH, DK. Det primùm le- gem motui centri po- lygonum exterius, & majus, fiatque conversio circa punctum C, demùm latus CF congruet rectæ CH, & centrum A per arcum AF erit translatum in F; latus verò minoris polygoni EG congruet parti IK, intactam relinquens partem EI, ita tamen; ut tota EK æqualis sit ipsi CH. Id quod est mani- festum, quia factâ translatione centri in F, semidiameter, quæ ex F pertingit ad H, est parallela ipsi AC, cum ad similes an- gulos incidat in subjectam lineam; sunt autem parallelæ etiam AF, DK, & BH; igitur tres lineæ AF, EK, CH sunt æqua- les, ex 34. lib.1. Atqui quod uni lateri contingit, etiam reli- quis lateribus commune est; igitur factâ integrâ conversione Hexagonum majus designabit lineam sextuplicem ipsius CH æqualem toti perimetro, & Hexagonum minus percurrer li- neam similiter ipsius EK sextuplicem, quæ æqualis est perime- tro majoris Hexagoni, sumendo tàm partes lineæ DK, quas intactas relinquit, quàm quæ tangunrur. Cæterùm si ex so- lùm, quæ ab Hexagono minore tanguntur, accipiantur, patet illas simul sumptas non esse majores perimetro ejusdem mino- ris Hexagoni. Deinde polygonum interius & minus det legem motui cen- tri, & conversio fiat circa punctum E, postquam latus E G congruit lineæ EI, & centrum est in G (in hoc enim exem- plo ad vitandam in Schemate confusionem literarum assump- Ee 3

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Liber secundus. CAPUT X. 221 it so fits the sides with equal parts of straight lines, that in a full turn the straight line of the underlying plane is equal to the perimeter of the polygon: but not so all the parts of the line to which the other polygon is fitted in motion, if they are taken in a single comprehension, do they make up a line equal to the perimeter of the greater polygon. The matter, for the sake of clarity, shall be explained in Hexagons, whose common center is A, and whose sides BC, DE rest upon the parallel lines BH, DK. Let the law of motion of the center of the outer and greater polygon be first given, and let the rotation be about point C; then side CF will coincide with line CH, and center A will, by the arc AF, be transferred to F; but side EG of the smaller polygon will coincide with part IK, leaving part EI untouched, yet so that the whole EK is equal to CH itself. This is evident, because, after the translation of the center to F, the semidiameter, which extends from F to H, is parallel to AC itself, since it meets the underlying line at equal angles; and AF, DK, and BH are also parallel; therefore the three lines AF, EK, CH are equal, by prop. 34, book 1. But what happens to one side is common also to the remaining sides; therefore, after a full rotation, the greater Hexagon will trace a sixfold line equal to the whole perimeter of CH itself, and the smaller Hexagon will likewise traverse a sixfold line of EK, which is equal to the perimeter of the greater Hexagon, taking both the parts of line DK which it leaves untouched and those which it touches. Moreover, if from the smaller Hexagon only those parts which are touched are taken, it is clear that these taken together are not greater than the perimeter of the same smaller Hexagon. Then let the inner and smaller polygon give the law of motion of the center, and let the rotation be about point E, after side EG coincides with line EI, and the center is in G (for in this example, to avoid confusion of letters in the diagram, it is assumed)

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Mechanicorum 222 tum est Hexagonum minus subquadruplum majoris, latera scilicet minoris subdupla sunt laterum majoris) cum interim punctum C retrocesserit in L, & demum latus C F congruat lineæ L M. Igitur majus polygonum solùm designat in motu, quo progreditur, lineam C M æqualem lateri minoris polygoni E I; & factâ integrâ conversione, designata erit linea sextuplex ipsius C M & ipsius E I; atque adeò utrumque polygonum æqualem lineam progrediendo designat. Hæc quæ de Hexagonis concentricis exempli gratiâ dicta sunt, de omnibus similibus atque concentricis polygonis dicta intelliguntur, quotcumque sint laterum. Iam verò Authores illi concipiunt circulos tanquam polygona infinitorum laterum: & quemadmodum minus polygonum totidem spatia subjectæ lineæ intacta relinquit, totidemque tangit, quot habet latera; ita pariter in circuli minoris conversione, infinita spatia vacua non- quanta (ne scilicet si quanta essent, opus esset lineâ infinitâ) intermista spatiis, quæ tanguntur, adstruunt, adeò ut demùm ex omnibus spatiis tactis simul & intactis coalescat linea æqualis ei, quæ tangitur à majore peripheriâ majoris circuli. Mihi tamen arridere non potest illa loquendi formula, quæ circulum polygonum infinitorum (& quidem infinitorum simpliciter) laterum dicit. Polygonum enim utique regulare circulus esset; polygonum autem esse non potest illud, quod angulis caret; neque anguli esse possunt, ubi non est lineæ ad lineam inclinatio; in peripheriâ verò circuli linea nulla esse potest, essent siquidem infinitæ lineæ æquales invicem, quæ utique constituerent extensionem simpliciter infinitam. Quod si infinita dixeris puncta; non est puncti ad punctum inclinatio, quæ possit angulum constituere, ac proinde circulus non est polygonum infinitorum laterum, nisi vocabulis ad opinandi licentiam immoderatè abutamur. Adde quod omnia diametri puncta ad omnia puncta peripheriæ essent in Ratione, quam Archimedes lib. de dimensione circuli definivit contineri inter Rationem 7 ad 22, & Rationem 71 ad 223: non igitur infinita esse possunt aut diametri, aut peripheriæ, aut utriusque puncta; ab infinitis enim Rationem omnem ablegant iidem Authores. Si itaque

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Mechanics 222 then it is that the smaller Hexagon is four times less than the larger, that is to say, the sides of the smaller are half the sides of the larger) when in the meantime the point C has moved back to L, and finally the side C F coincides with the line L M. Therefore the larger polygon alone, in its motion by which it advances, marks the line C M equal to the side of the smaller polygon E I; and when the full revolution has been completed, a sixfold line of C M itself and of E I will have been marked out; and thus each polygon, by advancing an equal line, marks it out. What has here been said by way of example concerning concentric Hexagons is to be understood as said of all similar and concentric polygons, however many sides they may have. But now those Authors conceive circles as polygons of infinite sides: and just as the smaller polygon leaves untouched as many spaces as it touches on the line beneath it, namely as many as it has sides; so likewise, in the revolution of the smaller circle, they posit infinite empty spaces, not quantitative indeed (lest, if they were quantitative, an infinite line would be required), intermingled with the spaces that are touched, so that at last from all the spaces touched and untouched together there should be formed a line equal to that which is touched by the larger periphery of the larger circle. Yet I cannot approve that manner of speaking which calls a circle a polygon of infinite, and indeed simply infinite, sides. For a polygon would certainly be a regular circle; but that cannot be a polygon which lacks angles; nor can there be angles where there is no inclination of line to line; and on the circumference of a circle there can be no line at all, since there would be infinite lines equal to one another, which would certainly constitute an extension simply infinite. And if you were to say infinite points, there is no inclination of point to point that can constitute an angle, and therefore a circle is not a polygon of infinite sides, unless we abuse words in an immoderate way for the freedom of conjecture. Add that all the points of the diameter would be in Ratio to all the points of the circumference, which Archimedes, in the book on the measurement of the circle, defined as contained between the Ratio of 7 to 22 and the Ratio of 71 to 223: therefore the points either of the diameter or of the circumference, or of both, cannot be infinite; for the same Authors banish all Ratio from infinite things. If therefore

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Liber secundus. CAPUT X. 223 itaque circulus polygonus non est, adhuc indiget explicatione, quomodo ad circulos concentricos traducantur ea, quæ de po- lygonorum concentricorum conversione considerata sunt. Quòd si circulum ita in polygonum convertamus, ut nec illi fixum definitumque laterum numerum tribuamus, nec sim- pliciter infinitum; sed liceat minora semper atque minora late- ra concipere, ut laterum ipsorum numerus semper augeatur, ita ut non simpliciter infinitus, sed indefinitus dicatur, non abnuo: proposita enim difficultas satis commodè hâc ratione explicabitur. Verùm in hac laterum extenuatione, si ad mini- mam extensionem deveniamus, quæ à puncto physicè non dif- ferat; non infinitus est hujusmodi punctorum numerus, sed certus est atque definitus: Nec ipsis punctis, seu minimis Phy- sicis sua figura detrahenda est, in majori enim peripheriâ mi- nùs curvantur interiùs, minúsque convexa sunt exteriùs, pro- piúsque ad lineam rectam accedunt; in minori autem orbitâ puncta hæc circularia curvantur magis, magisque convexa sunt exteriùs, & à rectitudine magis deflectentia ita absunt à sub- jectâ rectâ lineâ, ut, dum conversio sit circuli, & trahitur, des- cribant in motu lineam curvam magis obsecundantem motui centri, quàm quæ describitur à punctis similiter positis in ma- jore peripheriâ. Cærerùm cavendum est maximè ab eo, quod quia subest æquivocationi, difficultatem in hâc quæstione auget; illud au- tem est, quod punctum peripheriæ cum puncto lineæ Tangen- tis perperam comparatur, quasi in contactu coæquarentur; id quod à veritate longè abest; se enim contingunt circulus & li- nea incommensurabiliter, si contactus præcisè spectetur: at si contactus & motus componantur, jam quædam extensio conci- pitur, quæ aliquâ ratione comparari potest cum spatio lineæ, quæ tangitur, quatenùs huic aut illi parti lineæ in motu coapta- tur circulus, aut ejus pars. Quare circuli minoris, qui ad ma- joris circuli motum movetur, singula puncta non aptè compa- rantur cum singulis subjectæ rectæ lineæ punctis, quasi circuli punctum, quod est tertium à contactu, antequam incipiat mo- tus, in conversione tangat tertium rectæ lineæ punctum; sed tanget fortasse quintum aut sextum pro ratione magnitudinis aut

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Book Two. Chapter X. 223 so then, since the polygonal circle is not yet explained, it still needs to be made clear how what has been considered concerning the conversion of concentric polygons is to be applied to concentric circles. But if we convert a circle into a polygon in such a way that we assign to it neither a fixed and definite number of sides, nor simply an infinite one; but allow ever smaller and smaller sides to be conceived, so that the number of sides itself is always increased, in such a way that it is said to be not simply infinite, but indefinite, I do not object: for the difficulty proposed will be explained quite suitably by this method. Yet in this diminution of the sides, if we come to the smallest extension, which does not differ physically from a point; the number of such points is not infinite, but certain and definite. Nor should the form of the points themselves, or of the smallest physical elements, be taken away; for in a larger circumference they curve less inwardly, and are less convex outwardly, and approach more nearly to a straight line; but in a smaller orbit these circular points curve more, are more convex outwardly, and deviate more from straightness, so that they are so far from the underlying straight line that, while the conversion is that of a circle and it is being drawn along, they describe in motion a curved line more obedient to the motion of the center than that which is described by points similarly placed on a larger circumference. Moreover, the greatest caution must be used against what, because it lies under an equivocation, increases the difficulty in this question; and that is the improper comparison of a point of the circumference with a point of the tangent line, as though they were made equal in contact; which is far from the truth. For the circle and the line touch one another in an incommensurable manner, if the contact is considered precisely; but if contact and motion are combined, then some extension is conceived, which in some way can be compared with the space of the line that is touched, insofar as the circle, or part of it, in motion is adapted to this or that part of the line. Therefore the individual points of the smaller circle, which is moved by the motion of the larger circle, are not suitably compared with the individual points of the underlying straight line, as though the point of the circle, which is third from the contact, before motion begins, in the turning should touch the third point of the straight line; but perhaps it will touch the fifth or sixth, according to the proportion of magnitude or

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Mechanicorum 224 aut parvitatis ipsius circuli; pro ut in polygonis concentricis observare est; quò enim majus est interius polygonum, eò etiam minora sunt intervalla, quæ intacta relinquuntur. Et quamvis in circuli contactu intervalla hujusmodi intacta non admittantur, non est tamen abs re puncto circuli, quod voluitur simul & trahitur cum ipso circulo, vim tribuere tangendi plus quàm unum subjectæ rectæ lineæ punctum, quemadmodum majoris peripheriæ punctum in motu contingit ex punctis subjectæ lineæ rectæ non communicantibus minus quàm unum, si ad interioris circuli motum circulus exterior moveatur: nam ad majoris, & exterioris motum minor, & interior promovetur; ad minoris verò & interioris motum major & exterior circulus retroagitur. Quapropter si interior circulus in primo casu velociùs, & exterior in secundo casu tardiùs movetur comparatè ad spatium collocatum cum eorum peripheriis, nil mirum in motu perfici ab illius puncto Physico plus spatij, quàm ferat ejus magnitudo, ab hujus autem puncto Physico minus spatij: in continuâ enim quantitate partes minores subinde ac minores vera, ut opinor, Philosophia admittit. Sed quia hæc esset infinita, concertationumque plena disputatio, satis ea sint, quæ diximus, & ad utiliora gradum faciamus. MECHA

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Mechanics 224 or from the smallness of the circle itself; as may be observed in concentric polygons: for the larger the inner polygon is, the smaller also are the intervals left untouched. And although in the contact of circles such untouched intervals are not admitted, yet it is not out of place to attribute to the point of the circle, which is rolled and drawn along with the circle itself, the power of touching more than one point of the underlying straight line, just as a point of the greater circumference, in motion, meets with points of the underlying straight line that communicate less than one, if an outer circle is moved to the motion of an inner circle: for to the motion of the greater and outer circle, the smaller and inner is advanced; but to the motion of the smaller and inner, the greater and outer circle is drawn backward. Wherefore, if in the first case the inner circle moves more quickly, and in the second case the outer more slowly, in comparison with the space laid out along their circumferences, it is no wonder that in motion more space is accomplished by the physical point of the one than its magnitude allows, but by the physical point of the other less space: for in continuous quantity smaller and ever smaller parts are, in truth, admitted by philosophy, as I think. But because this would be an infinite discussion, full of disputes, let what we have said suffice, and let us proceed to more useful matters. MECHA

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MECHANICORUM LIBER TERTIUS. De Libra. EXPLICATIS superiore Libro Causis motûs Ma- chinalis, ordinis ratio postularet, ut ad ipsas Ma- chinas, seu, ut ab Antiquioribus apud Pappum lib.8. Collect. Mathem. prop.10. vocantur, Facul- tates, ad quas Machinamenta ab artificibus exco- gitata reducuntur, aut ex quibus hæc componuntur, exami- nandas & explicandas progrederemur: Et fortè alicui videatur ab instituto nostro alienum libram hîc considerare, quippe quæ non ad motum oneribus conciliandum inventa est, ideóque nec inter Facultates enumeratur, sed usum omnem habet in motu prohibendo, ubi factum fuerit ponderibus æquilibrium. Nec eo quidem consilio libræ momenta hîc expendo, ut indè Vectis rationes explicentur (quemadmodum non paucis placet) non enim Vectis vires ad libræ Rationes revocandas existimo, cum sua cuique Facultati causa insit, communis illa quidem, sed quæ perinde in Vecte reperitur, atque si nulla prorsus existeret libra. Verùm eatenus libram Mechanicæ contem- plationi inferendam censeo, quatenus non minoris artis est ea, quæ in motum prona sunt, cohibere & sistere, quàm onera quiescentia per vim suo loco dimovere: Cum maximè ad libram pertineat Statera, in qua modicum pondus multò majori pon- deri æquipollet, æquatis in dispari gravitate gravitationum Ff

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MECHANICORUM LIBER TERTIUS. On the Balance. Having explained in the preceding Book the causes of mechanical motion, the order of the subject would require that we proceed to examine and explain the machines themselves, or, as they are called by the older writers in Pappus, lib. 8. Collect. Mathem. prop. 10, the faculties, to which the devices invented by artificers are reduced, or out of which these are composed: and perhaps it may seem to someone foreign to our purpose to consider here the balance, since it was not invented for bringing loads into motion, and therefore is not numbered among the faculties, but has all its use in preventing motion, when equilibrium has been established by weights. Nor indeed do I here expound the moments of the balance with the intention that the principles of the lever may thereby be explained (as not a few are pleased to do); for I do not think the powers of the lever should be referred to the reasons of the balance, since each faculty has its own cause in it, indeed a common one, but one found in the lever just as if no balance existed at all. Yet I judge the balance should be introduced into mechanical contemplation insofar as it is no less an art to restrain and stop things inclined to motion than to move by force loads at rest from their place: especially since the balance belongs to the scale, in which a small weight is equivalent to a much greater weight, the gravitations having been equalized in unequal heaviness. Ff

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Mechanicorum momentis, ut infra in loco ostendetur. Præterquam quod explicato æquilibrio, faciliùs declaratur in motu Machinali, quid præstet major illa Ratio momentorum agendi ad momen- ta resistendi, quàm sit reciproca Ratio gravitatum, seu vi- rium oppositarum, absolutè sumptarum extrà machinam; ex qua majore Ratione momentorum, etiam Potentiæ moventis virtus innotescit. Nihil autem officit libræ dignitati, quod Cain authorem agnoscere videatur, qui, ut Josephus lib. I. Antiq. Iud. cap. 2. loquitur, Simplicem hactenus vivendi rationem excogitatis mensuris & ponderibus immutavit, pristinamque sinceri- tatem & generositatem ignaram talium artium, in novam quan- dam versutiam depravavit. Quid enim si quis præclaro artifi- cio ex naturæ thesauris deprompto abutatur? Dolos & fallacias, aut errores, quibus infici potest libræ usus, ideò retegemus: ut nimirum quod Iustitiæ commutativæ symbolum datur, om- ni injustitiæ suspicione vacet. Cæterùm quæ nobis inest arbi- trij libertas, potissima naturæ rationis compotis prærogativa, libræ, aut stateræ jure merito comparatur, quâ iniqui abuten- tes dicuntur Psalm. 61. Mendaces filij hominum in stateris: ubi S. Basilius hom. in Psalm. 61. ait Cuilibet nostrûm intus statera quædam est à Conditore omnium apparata, per quam rerum naturam possis probè dignoscere. & infra: Tibi namque propria datur libra, quæ sufficiens discrimen boni, ac mali demonstrat. Corporea enim pondera in libræ lancibus probamus; quæ verò ad instituendam vi- tam eligenda veniunt, per liberum arbitrium discernimus: quod & stateram nominavit, quòd momentum æquale ad utrumlibet possit capere. CAPUT I. Libræ forma, & natura exponitur. E O consilio instituta est libra, ut certis, ac notis ponderi- bus, ignotæ gravitatis quantitas indagetur, quæ demùm innotescit, cum æquatis hinc & hinc ponderum libræ adnexo- rum momentis, neutro prævalente, libra consistit. In hoc instru-

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of mechanical moments, as will be shown below in its place. Besides, once the equilibrium has been explained, it is more easily shown in mechanical motion what the greater ratio of the acting moments to the resisting moments is, compared with the reciprocal ratio of the gravities, or opposite forces, taken absolutely outside the machine; and from this greater ratio of moments the power of the moving force is also made known. Nor does it at all detract from the dignity of the balance that Cain seems to be acknowledged as its inventor, who, as Josephus speaks, book I of the Antiquities of the Jews, chapter 2, changed the simple way of life hitherto followed by devising measures and weights, and perverted the former sincerity and generosity, ignorant of such arts, into a certain new cunning. For what if someone should misuse a splendid art drawn from the treasures of nature? We shall therefore uncover the deceits and fallacies, or errors, by which the use of the balance may be infected; so that, namely, what is given as the symbol of commutative justice may be free from every suspicion of injustice. Moreover, the freedom of choice that is in us, the chief prerogative of rational nature, is rightly compared with a balance, or scales, by which the wicked are said, Psalm 61, “The sons of men are deceitful in the balances”; where St. Basil, homily on Psalm 61, says, “In each of us there is within a certain balance prepared by the Creator of all things, by which you may rightly discern the nature of things.” And below: “For to you there is given a balance of your own, which shows a sufficient distinction between good and evil. For bodily weights we test in the pans of a balance; but those things which come to be chosen for the ordering of life we distinguish by free choice: and he called it a balance, because it can take an equal weight to either side.” CHAPTER I. The form and nature of the balance are explained. For this purpose the balance was established, that by certain and known weights the quantity of unknown heaviness might be investigated, which at last becomes known when, the moments of the weights attached to the balance on this side and on that being equal, neither prevailing, the balance stands still. In this instruc-

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Liber tertius. CAPUT I. 227 instrumento consideratur primùm Iugum, seu scapus, seu librile AB: hoc bifariam dividitur in C, quod, Centrum libræ dicitur, non quia sit necessariò Centrum gravitatis libræ, sed quia est Centrum, circa quod agitur, seu versatur jugum, infixo nimirum in C axiculo, qui & Agina Latinis, Græcis apud Aristotelem in quæst. Mechan. Spartum dicitur. Partes autem jugi videlicet CA, & CB. Brachia, Radij, aut etiam ab aliquibus Librilia vocantur. Ex medio jugi ad perpendiculum assurgit lingula CD, quæ inseritur ansæ EF complectenti capita axiculi, adeò ut suspensâ ex F ansâ, quæ horizonti ad perpendiculum immineat, tùm demùm intelligatur factum æquilibrium, cum lingula ansæ congruit, & jugum consistit horizonti parallelum. Utrùm autem Trutina dicenda sit ipsa lingula, an verò ansa, non conveniunt Authores: item Grammaticis dirimendam relinquo. Extremis brachiorum punctis A & B adnectitur utrumque pondus, tam notum, quod est alterius mensura, quàm ignotum, cujus gravitas examinatur. Nihil autem refert, an pondera uncinis adnexa dependeant, an verò lancibus indè pendentibus imponantur; id quod vulgare est magisque usitatum, & libræ fecit nomen Bilanci. Illud enim præcipuum est, ac maximè attendendum, quòd omnia hinc & hinc æqualia sint, nimirum pondus unius lancis cum funiculis seu catenulis æquale sit ponderi alterius lancis cum suis appendiculis (pondus, inquam, ponderi æquale sit; nil enim interest æquales ne? an inæquales fuerint utriusque lancis funiculi secundùm longitudinem, modò in æquali distantiâ à centro adnectantur) & brachium alterum majus non sit reliquo brachio non solùm quoad gravitatem, quæ materiæ jugi inest, sed potissimùm quoad ipsorum brachiorum longitudinem. Porrò hæc brachiorum longitudo non est desumenda, ut ita loquar, materialiter, à centro jugi ad extremitatem, ubi materia desinit, ex quâ constat, sivè ferrum sit, sivè lignum, sivè aliud quidpiam: sed brachiorum longitudinem definiunt F f 2

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Book Three. CHAPTER I. 227 the instrument considers first the yoke, or beam, or rule AB: this is divided in two ways in C, which is called the center of the balance, not because it is necessarily the center of gravity of the balance, but because it is the center, about which the yoke is turned, or revolves, namely with the little axle fixed in C, which in Latin is called Agina, and among the Greeks, in Aristotle’s Mechanical Questions, Spartum. The parts of the yoke, namely CA and CB, are called Arms, Radii, or also by some Librilia. From the middle of the yoke, perpendicularly, rises the tongue CD, which is inserted into the handle EF enclosing the ends of the little axle, so that, with the handle suspended from F, which hangs vertically to the horizon, then and only then is it understood that equilibrium has been achieved, when the tongue matches the handle, and the yoke stands parallel to the horizon. But whether the Trutina should be called the tongue itself, or rather the handle, the Authors do not agree: I leave this also to the Grammarians to settle. To the end points A and B of the arms are attached the two weights, one known, because it is the measure of the other, the other unknown, whose gravity is being tested. And it makes no difference whether the weights, attached by hooks, hang down, or are placed in the pans hanging therefrom; this is the common and more usual method, and from it the balance received the name Bilancis. For the chief point is, and must especially be observed, that everything on this side and that side be equal, namely that the weight of one pan with its cords or chains be equal to the weight of the other pan with its own attachments (I say, the weight be equal to the weight; for it matters not whether the cords of both pans were equal or unequal in length, provided they are attached at an equal distance from the center) and that one arm be no greater than the other arm, not only with regard to the weight which is inherent in the material of the yoke, but chiefly with regard to the length of the arms themselves. Moreover, this length of the arms is not to be taken, so to speak, materially, from the center of the yoke to the extremity, where the material ends, of which it is made, whether it be iron, or wood, or something else: but the length of the arms is defined by

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Mechanicorum puncta jugi, ex quibus pondera dependent: horum etenim distantiam à centro omnino æqualem esse oportet. Hujusmodi autem puncta non alia sunt, quàm puncta contactûs jugi & an- nulorum seu uncinorum illi infixorum, quibus deinde lances aut pondera adnectuntur. Hoc illud est, in quo maxima arti- ficis industria, atque diligentia collocanda est, ut exactissimam brachiorum æqualitatem assequatur. Data itaque hac, quàm diximus, brachiorum æqualitate, si æqualia pondera hinc & hinc addantur, manifestum est jugum libræ ex aginâ suspensum ad neutram partem inclinari, sed ma- nere horizonti parallelum; fieri namque non potest, ut extremitas altera descendat, quin opposita extremitas cum adnexo pon- dere ascendat, & quidem æquali motu propter brachiorum æqualitatem. Finge enim pondus B descendere in F, utique pondus A ascendet in E, at- que describent arcus B F & A E æquales, quippe qui æqualibus angulis ad verti- cem in C subtenduntur, & ab æqualibus radiis C B, C A describuntur. At æqualis est in B vis descendendi atque in A repugnantia ad ascendendum; illa igitur præpollere non potest. Siquidem vis descendendi componitur ex ponderis gravitate, & non impeditâ motûs naturalis velocitate; repugnantia verò ad ascendendum componitur & ex ponderis contranitentis gra- vitate, & ex velocitate motûs præter naturam: sunt autem gra- vitates ex hypothesi æquales, motus etiam per arcus B F & A E essent æquales; ac proinde vis tendendi deorsum inveniens æqualem oppositam repugnantiam ad motum sursum nequit illi imprimere impetum, quo per vim moveatur: ut enim sequa- tur motus, aut gravitates dispares esse oportet, aut motuum Po- tentiæ moventis & Ponderis moti velocitates inæquales, ut ma- jor sit Ratio hujusmodi velocitatum, quàm sit reciproca Ratio gravitatum: alioquin nulla esset virium movendi & resistentiæ inæqualitas, ubi omnia essent æqualia. Cum itaque in librâ sic constitutâ intercedat omnimoda æqualitas & brachiorum, qui- bus definitur motus, & gravitatum, quæ sibi invicem æquali- ter obsistunt, ac proinde eadem sit reciproca Ratio gravitatum &

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Mechanical points of the beam from which weights are suspended: for their distance from the center must be altogether equal. Such points are none other than the points of contact of the beam and the rings or hooks fixed to it, to which afterward the pans or weights are attached. This is the point in which the artificer’s greatest skill and diligence must be applied, so that he may achieve the most exact equality of the arms. Therefore, given this equality of the arms, as we have said, if equal weights be added on this side and on that, it is manifest that the beam of the balance, suspended from the pivot, inclines to neither side, but remains parallel to the horizon; for it cannot happen that one end descends without the opposite end, with the attached weight, ascending, and that too with equal motion on account of the equality of the arms. For imagine weight B descending to F: weight A will surely ascend to E, and they will describe the equal arcs BF and AE, since they are subtended by equal angles at vertex C and are described from equal radii CB, CA. But the power in B of descending is equal to the resistance in A to ascending; therefore the former cannot prevail. For the force of descent is composed of the heaviness of the weight and of the unimpeded velocity of natural motion; whereas the resistance to ascending is composed both of the heaviness of the opposing weight and of the velocity of motion contrary to nature. But the heavinesses are, by hypothesis, equal; the motions also along the arcs BF and AE would be equal; and therefore the force tending downward, finding an equal opposite resistance to upward motion, cannot impart to it the impulse by which it may be moved by force. For motion to follow, either the weights must be unequal, or the velocities of the moving power and of the moved weight must be unequal, so that the ratio of such velocities may be greater than the reciprocal ratio of the weights; otherwise there would be no inequality of the forces of moving and resisting, where everything would be equal. Since, therefore, in a balance thus constituted there exists a complete equality both of the arms by which the motion is determined and of the weights, which equally oppose one another, and consequently the same reciprocal ratio of the weights and

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Liber tertius. CAPUT I. 229 & motuum, jugum libræ horizonti parallelum consistere ne- cesse est; & in alteram partem si inclinetur, manifestum est in illâ lance plus ponderis fuisse impositum, quàm in reliquâ. Ut autem quàm exactissimè ponderum ignota gravitas exa- minari queat, opus est ut axiculus jugo infixus (saltem in supe- riore parte, cui scapus incumbit) exquisitè cylindricam figu- ram obtineat; hinc enim fiet, ut cum rotundo foramine scapi contactus fiat in lineâ, quamcumque tandem positionem ha- beat ipse scapus: nam quemadmodum ex prop. 1 3. lib. 3. duo circuli se intùs contingentes tangunt in puncto, ita duæ super- ficies cylindricæ, cava altera, altera convexa, se tangunt in li- neâ. Id si fiat facilè ab æquilibrio deflectet scapus, si vel modi- ca intercedat ponderum inæqualitas. At si angulatus fuerit axi- culus, vel superior foraminis pars rotunditatem non fuerit asse- cuta, jam non in unâ lineâ, sed in pluribus contactus fieret, at- que adeò iners esset ad motum scapus, etiamsi non omninò æqualia essent pondera lancibus imposita. Quare artifices illos non probo, qui axem ita efformant, ut superior pars in aciem desinat, illud sibi persuadentes, quod minore partium conflictu se tangentes axis & scapus faciliorem relinquant in alterutram partem motum libræ. Id quod ut ve- rum sit, non tamen vacat periculo, ne, dum axis capita inse- runtur ansæ, acies illa planè sursum non dirigatur, sed modi- cum in alterutram partem vergat: quæ declinatio si contingat, foramen autem exactè rotundum fuerit, miraculo proximum cense, si libra vacua æquilibrium constituat, ita ut lingula ritè collocata congruat ansæ; acies si quidem illa dividit inæquali- ter scapi longitudinem, & brachium alterum altero longius est, atque præponderat. Hoc vitium ubi libra contraxerit, inepti artifices nihil suspicati ab axe malè conformato, aut perperam disposito, ortum duxisse, vel brachium extenuant, vel lancem immutant, donec æquilibrium inveniant. Verùm libram hu- jusmodi dolosam esse inferiùs constabit propter brachiorum in- æqualitatem: quæ quidem levem infert ponderum differen- tiam in rebus exigui momenti contemnendam; sed in iis, quæ exquisitam ponderis mensuram exigunt, non leve damnum hinc potest emergere. Quod si axis non sit ansæ, sed scapo, firmiter infixus, volua- F f 3

Transcription: Translated (English)

Book Third. CHAPTER I. 229 & of motions, the beam of the balance must necessarily remain parallel to the horizon; and if it be inclined to either side, it is evident that upon that scale more weight has been placed than upon the other. But in order that the unknown gravity of weights may be examined as exactly as possible, it is necessary that the small axle fixed in the beam (at least in the upper part, upon which the stem rests) should have an exquisitely cylindrical form; for thus it will come about that contact with the round hole of the stem is made along a line, whatever position the stem itself may have: for just as, according to proposition 13 of book 3, two circles touching internally touch at a point, so two cylindrical surfaces, one hollow and the other convex, touch along a line. If this be the case, the stem will easily depart from equilibrium, even if there be only a slight inequality of the weights. But if the axle be angular, or if the upper part of the hole has not attained roundness, then contact is made not along one line but along several; and thus the stem would be sluggish in moving, even though the weights placed in the scales were not altogether equal. Wherefore I do not approve of those artificers who fashion the axle in such a way that the upper part ends in a sharp edge, persuading themselves that, because the axis and the stem touch with less conflict of parts, the balance will move more easily to either side. Even if this were true, yet there is no lack of danger lest, while the ends of the axle are inserted into the yoke, that edge should not be directed exactly upward, but should incline a little to one side or the other: if such a deviation occur, and yet the hole be exactly round, count it almost a miracle if the balance, when empty, establishes equilibrium, so that the properly placed pointer agrees with the yoke; for that edge divides the length of the stem unequally, and one arm is longer than the other, and preponderates. When a balance has contracted this defect, bungling artificers, suspecting nothing of its having arisen from a badly shaped or wrongly placed axle, either thin down one arm or alter one scale, until they find equilibrium. But it will be shown below that a balance of this sort is deceitful, because of the inequality of the arms: this indeed introduces a slight difference of weights, to be disregarded in matters of little moment; but in those things which require an exact measurement of weight, no small harm may hence arise. But if the axis be not firmly fixed to the yoke, but to the stem, Ff 3

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Mechanicorum 230 tur autem in ansæ foraminibus (id quod artificibus non paucis magis arridet) jam non superior; sed inferior axiculi pars at- tendenda est; quippe quæ inferiorem foraminum ansæ partem contingit; & eadem, quæ de superiore parte dicebantur, ob- servanda sunt. Illud tamen præterea in ansæ foraminibus ob- servandum venit, quod eorum infima pars ita sit constituta, ut axis illis incumbens parallelus sit horizonti, quando ansa sus- penditur, ut liberè pendeat, vel ita collocatur, ut ad perpen- diculum horizonti immineat: alioquin axe inclinato, jugum urgeret alteram ansæ partem, ab alterâ recederet; ex quo jugi cum ansâ conflictu aliqua motui difficultas crearetur. Iam verò quod ad pondera attinet, supervacaneum est mo- nere non omnia pondera omnibus libris convenire: quamvis enim libra, quâ libra est, nullam prorsus respuat ponderum gra- vitatem, sed omnem quorumcumque ponderum æqualitatem apta sit indicare suo æquilibrio; quia tamen ex materiâ constat, quæ definitam habet soliditatem atque partium firmitatem (ut nihil dicam de certis atque definitis viribus retinentis ansam, & cum ansâ libram, ac utrumque pondus) fieri potest, ut adeò gravia lancibus imponantur onera, quæ brachiorum rectitudi- nem inflectant, & eorum æqualitatem corrumpant: Quare te- nuioribus libris parva pondera examinantur, crassioribus ma- jora. Illud potiùs cavendum est, ne pondera, quibus tanquam mensurâ utimur, fallacia sint, quia falsa, aut excedendo legi- timam gravitatis quantitatem, aut ab illâ deficiendo. Quamvis autem tot pondera minimæ mensuræ adhibere pos- semus, quot numerare oporteret ad explorandam propositæ gravitatis ignotæ quantitatem, hoc tamen valde incommodum esset: quid enim, si lanius carnem in macello vendens grana numerare cogeretur, quæ æquilibrium cum carne constituunt? sed & inutilis esset labor, nam multa sunt, quorum quantitas non est ad vivum resecanda, & minutissimæ particulæ frustra investigantur. Subtilitas hæc relinquatur gemmaniis, aurifici- bus, aurique monetalis cursoribus, quibusdamnum esset minu- tias contemnere. Quamquam nec istis author fuerim, ut sin- gularibus granis uterentur, sed potiùs ponderibus, quæ pluri- bus granis æquivalerent, si enim singula grana à legitimo pon- dere deficiunt per centesimam grani partem, quæ facilè sensûs aciem

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Mechanics 230 But now, in the holes of the handle (which is what many craftsmen prefer), it is no longer the upper, but the lower part of the little axle that must be attended to; namely, that part which touches the lower side of the holes of the handle; and the same things that were said about the upper part must be observed. This too must moreover be noted in the holes of the handle, that their lowest part be so arranged that the axis resting upon them be parallel to the horizon, when the handle is suspended so that it hangs freely, or is so placed that it stands perpendicular to the horizon; otherwise, if the axis were inclined, the yoke would press upon one side of the handle and recede from the other; and from the collision of the yoke with the handle some difficulty in motion would arise. Now as for the weights, it is needless to warn that not all weights suit all balances: for although a balance, as a balance, rejects absolutely no heaviness of weights, but is fitted to indicate by its equilibrium the equality of any weights whatsoever; yet because it is made of material which has a definite solidity and firmness of its parts (to say nothing of the fixed and definite strength of the suspending handle, and of the balance together with the handle, and of each weight), it may happen that burdens so heavy are placed on the pans that they bend the straightness of the arms and destroy their equality. Therefore, in lighter balances small weights are tested, in heavier ones larger. Rather must this be guarded against, that the weights which we use as a measure may not be deceptive, because false, either by exceeding the lawful amount of heaviness or by falling short of it. Although we could indeed apply as many weights of the smallest measure as would need to be counted in order to discover the quantity of the unknown weight proposed, this would nevertheless be very inconvenient: for what if a butcher selling meat in the market were forced to count the grains which establish equilibrium with the meat? But the labor would also be useless, for there are many things whose quantity is not to be cut down to the quick, and the tiniest particles are sought in vain. Let this subtlety be left to jewelers, goldsmiths, and the mint workers of coined gold, for whom it would be a loss to despise minutiae. Although I would not even advise these men to use single grains, but rather weights that would equal several grains; for if each grain falls short of the lawful weight by one-hundredth part of a grain, which easily escapes the sharpness of sense...

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Liber secundus. CAPUT I. 231 aciem fugit, additis centum hujusmodi granis error est inte- gri grani deficientis; & in unciâ libræ Romanæ ponderalis ad monetam pertinentis cum grana 576 contineantur, in unciâ auri error esset granorum ferè sex deficientium, & in integrâ librâ, quæ est granorum 6912, esset error granorum 69; qui tamen error vix contingat, si assumatur integra uncia, aut li- bra: illud si quidem, quod solitarium præ suâ tenuitate in con- spectum non cadit, cum pluribus similibus conjunctum evadit demum notabile atque conspicuum. Quare ad paranda pon- dera hujusmodi subtiliora, assume laminam metallicam ponde- re unius libræ, sed æquabiliter extensam, ejusque duodecimam partem accipe; hæc erit Uncia, quam sepones. Alterius Unciæ octavam partem assumens habebis Drachmam. Drachmæ pars tertia dabit scrupulum. Scrupuli semissis est obolus. Oboli triens est siliqua. Demùm siliquæ quadrans est Granum. Ex hac minutâ divisione satis constat, quàm obnoxiæ errori sint minores particulæ præ majoribus; idemque error, qui in unciâ singularis esset, & ut nullus consideraretur, toties repetitus, quot grana in unciâ continentur, jam non esset contemnen- dus. Id autem dictum intelligatur etiam in majoribus ponde- ribus, ubi unciæ non reputantur, satius esse majora pondera habere, quàm minimam mensuram sæpiùs multiplicatam as- sumere. Sed quoniam adhuc incommodum accideret tot habere mensuras, quæ juxta seriem naturalem numerorum crescerent, ut propositæ paucitatis examinandæ quantitas indagetur, ob- servatum est non leve compendium, quod offert progressio Geometrica ab unitate incipiens, & in Ratione duplâ aut tri- plâ progrediens. Nam maximum terminum progressionis du- plæ sibimet ipsi additum si mulctaveris unitate, & in progres- sione triplâ maximo termino unitate mulctato si residui semis- sem addideris, numerum habebis gravitatum omnium, quæ paucis illis ponderibus examinari possunt. Sic dentur octo pon- dera in Ratione duplâ incipiendo ab uncia 1; octavum est unc. 128: hunc numerum duplica, & à 256 aufer unitatem, reliquus numerus 255 indicat octo illis ponderibus posse in li- brâ examinari omnes gravitates ab uncia 1 ad uncias 255. Si- mili modo in Ratione triplâ dentur quatuor pondera 1. 3. 9. 27. aufer

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Book Two. CHAPTER I. 231 if the eye is deceived, with the addition of one hundred grains of this sort the error is the deficiency of one entire grain; and in the ounce of the Roman weighing-pound belonging to coinage, since it contains 576 grains, in an ounce of gold the error would be nearly six grains lacking, and in a whole pound, which is 6912 grains, there would be an error of 69 grains; yet this error scarcely happens if an entire ounce, or pound, is taken, for that which, when solitary, does not come into view because of its smallness, when joined with many similar parts finally becomes notable and conspicuous. Therefore, for making weights of this more delicate kind, take a metal plate weighing one pound, but evenly extended, and take its twelfth part; this will be an Ounce, which you will set aside. Taking the eighth part of another Ounce, you will have a Drachm. One third of the Drachm will give a scruple. Half of the scruple is an obolus. One third of the obolus is a siliqua. Finally, one fourth of the siliqua is a Grain. From this minute division it is clear enough how much smaller particles are exposed to error than larger ones; and the same error, which in a single ounce would exist and be regarded as nothing, repeated as many times as there are grains in an ounce, would no longer be negligible. And this too should be understood as applying even in the case of larger weights, where ounces are not reckoned, that it is better to have larger weights than to take the smallest measure multiplied many times over. But since it would still be inconvenient to have so many measures increasing according to the natural series of numbers, so that the quantity to be examined in the proposed small number may be investigated, a useful shortcut has been observed, namely the Geometric progression beginning from unity, and advancing in the ratio of two or three. For if you subtract one from the maximum term of the double progression added to itself, and in the triple progression if, after subtracting one from the maximum term, you add half of the remainder, you will have the number of all the weights that can be examined by those few weights. Thus let eight weights be given in a double ratio, beginning from 1 ounce; the eighth is 128 ounces: double this number, and from 256 subtract one, the remaining number 255 indicates that by those eight weights all weights from 1 ounce to 255 ounces can be examined in a pound. In similar fashion, in a triple ratio, let four weights be given, 1, 3, 9, 27. subtract

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232 Mechanicorum aufer ab ultimo unitatem, remanet 26, cujus semissis 13 additus numero 27 dat 40: cujus igitur gravitatis est primum pondus ut 1, tot gravitates usque ad 40 examinari possunt illis solis quatuor ponderibus. Præstat autem uti ponderibus in Ratione duplâ, quia licèt plura pondera requirantur, omnia tamen seorsim in propriâ libræ lance collocantur: at si Ratio ponderum sit tripla, aliquâ commutatione uti necesse est, ut in adjecta Tabella observabis, quæ usque ad numerum 40. extenditur: Ubi etiam vides in Ratione triplâ sufficere quatuor pondera 1. 3. 9. 27, at in duplâ exigi sex videlicet 1. 2. 4. 8. 16. 32. Pondera in Ratione Dupla 1. 2. 4. 8. 16. 32. Res Pondus Res Pondus Res Pondus Res Pondus 1 1 11 8.2.1. 21 16.4.1. 31 16.8.4.2.1. 2 2 12 8.4. 22 16.4.2. 32 32. 3 2.1 13 8.4.1. 23 16.4.2.1. 33 32.1. 4 4. 14 8.4.2. 24 16.8. 34 32.2. 5 4.1 15 8.4.2.1. 25 16.8.1. 35 32.2.1. 6 4.2 16 16. 26 16.8.2. 36 32.4. 7 4.2.1. 17 16.1. 27 16.8.2.1. 37 32.4.1. 8 8. 18 16.2. 28 16.8.4. 38 32.4.2. 9 8.1. 19 16.2.1. 29 16.8.4.1. 39 32.4.2.1. 10 8.2. 20 16.4. 30 16.8.4.2. 40 32.8. Pondera

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232 Mechanicorum Subtract unity from the last, 26 remains, of which half, 13, added to the number 27 gives 40: of what gravity therefore is the first weight to 1, so many weights up to 40 can be examined with those only four weights. It is better, however, to use weights in the Ratio dupla, because although more weights are required, yet all are placed separately on their own scale pan: but if the ratio of the weights is triple, some change is necessary to use, as you will observe in the added Table, which extends up to the number 40. Where you also see that in the triple ratio four weights suffice, 1, 3, 9, 27, but in the double six are required, namely 1, 2, 4, 8, 16, 32. Weights in the Double Ratio 1, 2, 4, 8, 16, 32. Item Weight Item Weight Item Weight Item Weight 1 1 11 8.2.1. 21 16.4.1. 31 16.8.4.2.1. 2 2 12 8.4. 22 16.4.2. 32 32. 3 2.1 13 8.4.1. 23 16.4.2.1. 33 32.1. 4 4. 14 8.4.2. 24 16.8. 34 32.2. 5 4.1 15 8.4.2.1. 25 16.8.1. 35 32.2.1. 6 4.2 16 16. 26 16.8.2. 36 32.4. 7 4.2.1. 17 16.1. 27 16.8.2.1. 37 32.4.1. 8 8. 18 16.2. 28 16.8.4. 38 32.4.2. 9 8.1. 19 16.2.1. 29 16.8.4.1. 39 32.4.2.1. 10 8.2. 20 16.4. 30 16.8.4.2. 40 32.8. Weights

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Liber tertius. CAPUT I. 233 Pondera in Ratione Tripla 1.3.9.27 & 12. Res Adde Pondus Res Adde Pondus Res Adde 1 I I. 14 9.3.1. 27. 27 2 I 3. 15 9.3. 27 28 27.1. 3 3. 16 9.3. 27.1. 29 I. 27.3. 4 3.1. 17 9.1. 27. 30 27.3. 5 3.1. 9. 18 9. 27. 31 6 3. 9. 19 9. 27.1 32 3.1. 7 3. 9.1. 20 9.1. 27.3. 33 3. 8 I. 9. 21 9. 27.3. 34 3. 9 9. 22 9. 27.3.1. 35 I. 27.9. 10 9.1. 23 3.1. 27. 36 27.9. 11 I. 9.3. 24 3. 27. 37 12 9.3. 25 3. 27.1. 38 I. 27.9.3. 13 9.3.1. 26 1. 27. 39 27.9.3. At contingere potest paratis hisce ponderibus in Ratione duplâ aut triplâ aliquid abundare, & maximum terminum cæteris additum excedere quæsitum numerum, (ut hic, si opus esset provenire solum ad 40, maximus terminus 32 est abundans) proptereà retentâ cæterorum summâ adde aliud pondus, ut quæsitum numerum compleat, & est illud, quo opus est; sic 1.2.4.8.16. conficiunt summam 31; aufer 31 ex 40, residuum est 9; sit igitur sextum pondus 9, & satis erit usque ad 40; quia cum habeantur reliquis ponderibus omnes numeri infra 31, jam ex 23 & 9 fit 32, ex 24 & 9 fit 33, & sic de reliquis deinceps. Idem dic de aliâ qualibet summâ majore quàm ferant data pondera, minore tamen quàm opus sit, si adhuc unum pondus in eâdem progressione adderetur; sufficit enim residuum. Exemplum habes in superiore Tabella ponderum in Ratione triplâ, ubi quatuor conficiunt 40, sed si adderetur quintum in eadem Ratione 81, esset nimis magnum, G g

Transcription: Translated (English)

Book Three. Chapter I. 233 Weights in the Triple Ratio 1.3.9.27 & 12. Thing Add Weight Thing Add Weight Thing Add 1 I I. 14 9.3.1. 27. 27 2 I 3. 15 9.3. 27 28 27.1. 3 3. 16 9.3. 27.1. 29 I. 27.3. 4 3.1. 17 9.1. 27. 30 27.3. 5 3.1. 9. 18 9. 27. 31 6 3. 9. 19 9. 27.1 32 3.1. 7 3. 9.1. 20 9.1. 27.3. 33 3. 8 I. 9. 21 9. 27.3. 34 3. 9 9. 22 9. 27.3.1. 35 I. 27.9. 10 9.1. 23 3.1. 27. 36 27.9. 11 I. 9.3. 24 3. 27. 37 12 9.3. 25 3. 27.1. 38 I. 27.9.3. 13 9.3.1. 26 1. 27. 39 27.9.3. It can happen that, with these weights prepared in the double or triple ratio, something remains over, and the largest term, when added to the others, exceeds the number sought (as here, if it were necessary to reach only 40, the greatest term 32 is excessive). Therefore, keeping the sum of the others, add another weight so that it completes the number sought, and that will be the one needed. Thus 1.2.4.8.16 make a sum of 31; subtract 31 from 40, the remainder is 9; let therefore the sixth weight be 9, and it will suffice up to 40; for since, with the other weights, all numbers below 31 are available, now from 23 and 9 comes 32, from 24 and 9 comes 33, and so on for the rest thereafter. The same may be said of any sum greater than those the given weights yield, yet less than what is needed, if only one more weight were added in the same progression; for the remainder suffices. You have an example in the foregoing table of weights in the triple ratio, where four make 40, but if a fifth in the same ratio, 81, were added, it would be too large, G g

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234 Mechanicorum si solùm habere velimus pondera infra 121: quæratur usque ad 52, & quia inter 40 & 52 differentia est 12, quintum pondus ut 12 sufficiet. Hinc quia ad libram requiruntur solum 24 semunciæ, ad unciam 24 scrupuli, ad scrupulum 24 grana, si pondera sint in Ratione triplâ, sufficiunt tria ponderâ 1. 3. 9. quæ conficiunt 13, & quartum pondus sit 11, ut compleatur summa 24: & in Ratione duplâ sufficiunt quatuor pondera 1. 2. 4. 8. quæ conficiunt 15, & quintum pondus 9 complens summam 24 illud est, quod requiritur, ut ex adjectis Tabellis liquet. Pro 24 semunciis ad libram, aut 24 scrupulis ad unciam, aut 24 granis ad scrupulum 1. 2. 4. 8. 9. Res Pondera. 16 9. 4. 2. 1. 17 9. 8. 18 9. 8. 1. 19 9. 8. 2. 20 9. 8. 2. 1. 21 9. 8. 4. 22 9. 8. 4. 1. 23 9. 8. 4. 2. 24 9. 8. 4. 2. 1. Pro semunciis 24 1. 3. 9. 11. Res. Adde Pondera. 14 II 3. 15 II. 3. 1. 16 3. 1. II. 9. 17 3. II. 9. 18 3. II. 9. 1. 19 1. II. 9. 20 II. 9. 21 II. 9. 1. 22 1. II. 9. 3. 23 II. 9. 3. 24 II. 9. 3. 1. Unum hîc, ubi de Ponderibus sermo est, obiter moneo, libræ nomen apud Romanos æquivocum fuisse, alia enim erat libra Ponderalis aridorum, alia Mensuralis liquidorum (& potissimum olei, quod cornu librali metiebantur) quam incisis & insculptis lineis in uncias 12 partiebantur, quemadmodum & libra pondo in uncias pariter 12 distinguebatur: sed inter utramque libram, si materia ipsa ad pondus revocabatur, non exiguum erat discrimen; ut enim ex proprio experimento testatur

Transcription: Translated (English)

234 Mechanics if we wish to have weights only below 121: let us seek up to 52, and because the difference between 40 and 52 is 12, the fifth weight of 12 will suffice. Hence, since for the pound only 24 semunciae are required, for the ounce 24 scruples, for the scruple 24 grains, if the weights are in a triple Ratio, three weights suffice, 1. 3. 9. which make 13, and let the fourth weight be 11, so that the sum of 24 may be completed: and in a double Ratio four weights suffice, 1. 2. 4. 8. which make 15, and the fifth weight 9, completing the sum 24; this is what is required, as is clear from the appended tables. For 24 semunciae to the pound, or 24 scruples to the ounce, or 24 grains to the scruple 1. 2. 4. 8. 9. Things Weights. 16 9. 4. 2. 1. 17 9. 8. 18 9. 8. 1. 19 9. 8. 2. 20 9. 8. 2. 1. 21 9. 8. 4. 22 9. 8. 4. 1. 23 9. 8. 4. 2. 24 9. 8. 4. 2. 1. For 24 semunciae 1. 3. 9. 11. Things. Add Weights. 14 II 3. 15 II. 3. 1. 16 3. 1. II. 9. 17 3. II. 9. 18 3. II. 9. 1. 19 1. II. 9. 20 II. 9. 21 II. 9. 1. 22 1. II. 9. 3. 23 II. 9. 3. 24 II. 9. 3. 1. One thing I note here, in passing, where the subject is weights: the name libra among the Romans was equivocal, for there was one libra for the weight of dry goods, another mensural libra for liquids (and especially oil, which they measured with a libra cornu), which they divided into 12 unciae by incised and engraved lines, just as the libra pondo was likewise distinguished into 12 unciae: but between the two librae, if the material itself was reduced to weight, there was no small difference; for, as he testifies from his own experiment

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Liber tertius. CAPUT II. 235 tur Galenus lib. 6. cap. 8. de compositione medicam. per genera. Libra mensura solùm uncias decem continebat, quarum li- bra pondo erat duodecim: quapropter uncia mensuralis ad un- ciam ponderalem erat ut 5 ad 6 spectatâ gravitate & quantita- te materiæ. CAPUT II. Libra inæqualium brachiorum expenditur. USus libræ brachiorum inæqualium minus necessarius est, ac propterea neque communis aut vulgaris, nisi quatenus ad stateram traductus est: illam tamen hîc considerare erit operæ pretium, ut æquilibrij rationes magis innotescant. Sit libra AB, cujus centro C dividatur jugum inbrachia inæqualia CA & CB. Certum est, etiam si nul- lum addatur pondus, ju- gum ex centro C suspen- sum retinere non posse po- sitionem AB horizonti pa- rallelam; quia licet punctum C sit centrum motûs libræ, non est tamen centrum gravitatis illius; hoc enim est in puncto ju- gum (quod hîc æquabiliter ductum ponitur) bifariam dividen- te, videlicet in I, quod æquales gravitates I A & I B cir- cumstant. Verùm interim ex hypothesi fingamus lineam AB omni gravitate carentem; & in ipsis libræ extremitatibus sta- tuamus pondera eam inter se reciprocè Rationem habentia, quæ est Ratio brachiorum, & ut CA ad CB, ita sit pondus B ad pondus A. Pondera hæc, quæ in lancibus libræ vulgaris æqualium brachiorum magnam momentorum inæqualitatem haberent, quia inæqualiter gravia, hîc æquilibrium consti- tuunt, quamvis inæquales sint eorum gravitates absolutæ, quia libræ brachia reciprocè: secundùm eandem Rationem in- æqualia: quatenus enim alligantur pondera hæc extremita- G g 2

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Liber tertius. CAPUT II. 235 ... Galenus lib. 6. cap. 8. de compositione medicam. per genera. A balance of measure contained only ten ounces, whereas a pound by weight was twelve ounces; wherefore the ounce by measure was to the ounce by weight as 5 to 6, regard being had to the weight and quantity of the material. CAPUT II. A balance with unequal arms is weighed. The use of the balance with unequal arms is less necessary, and for that reason neither common nor usual, except insofar as it has been brought over into the steelyard: yet it will be worth the trouble to consider it here, so that the reasons of equilibrium may become better known. Let there be a balance AB, whose beam is divided at its center C into unequal arms CA and CB. It is certain that, even if no weight be added, a beam suspended from the center C cannot retain the position AB parallel to the horizon; because although the point C is the center of motion of the balance, it is not however its center of gravity; for this is at the point I, where the beam (which here is assumed to be drawn uniformly) is divided in two, namely in I, which is surrounded by equal weights IA and IB. But in the meantime, by hypothesis, let us suppose the line AB to be without any weight whatever; and at the very ends of the balance let us place weights having a reciprocal ratio to one another, which is the ratio of the arms, and as CA is to CB, so let the weight B be to the weight A. These weights, which on the pans of an ordinary balance with equal arms would have a great inequality of moments, because they are unequally heavy, here establish equilibrium, although their absolute weights are unequal, because the arms of the balance are reciprocally unequal in the same ratio; for inasmuch as these weights are fastened to the extremities...

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Mechanicorum 236 tribus libræ, æqualia obtinent momenta, nec jugum A B potest in alterutram partem inclinari, cum neutrum pon- dus possit ab altero assumere vim, qua sursum moveatur, majorem oppositâ virtute innatâ descendendi, qua repu- gnat, ne elevetur. Sit C A ad C B ut 1 ad 4, & vicissim pon- dus B ut 1 ad pondus A ut 4. Si gravitates dumtaxat con- siderentur, virtus ponderis A est ut 4, virtus verò ponderis B ut 1: sed quia à centro motûs C retinentur, nec liberè rectâ viâ moveri possunt, impedimentum recipiunt pro brachiorum lon- gitudine, minûsqque impeditur descensus aut ascensus rectus ponderis, quod longiori brachio adjacet, magis, quod brevio- ri. Illud igitur pondus, quod majori brachio adnectitur, si descendat, magis descendit, si ascendat, magis ascendit; quod verò breviori, si ascendat, minùs ascendit, & si descendat, minùs descendit: atque adeò si B descenderet in E, mensura descensus esset perpendicularis E G, assensum autem ponderis A in D metiretur perpendicularis D F: idem dic si A descen- deret, & B ascenderet. Porrò D F & E G sunt in Ratione brachiorum C A & C B ut patet, quia triangula rectangula C F D, & C G E, præter rectos angulos ad F & G æquales, ha- bent etiam æquales ad C angulos ad verticem, & per 32. lib. 1. sunt æquiangula; igitur per 4 lib. 6. ut C D ad C E, ita D F ad E G; at C D æqualis est ipsi C A, & C E ipsi C B (est enim eadem linea, quæ mutatâ positione A B venit in D E) igitur ut C A ad C B ita D F ad E G. Quare ratione positionis pon- dus B vim habet descendendi, & resistit ascenlui, ut 4, pon- dus autem A vim habet descendendi, ac proinde etiam re- sistendi, ne ascendat, solùm ut 1. Cum itaque momentum descendendi (idem esto judicium de momento repugnantiæ, ne ascendat) componatur tùm ex gravitate ponderis, tùm ex propensione ad motum, hoc est ex motûs, qui conseqvi posset, velocitate, manifestum est gravi- tatem ut 4, cujus motus esset ut 1, nec posse vincere gravitatem ut 1, cujus motus esset ut 4, nec vicissim posse ab illâ vinci; est siquidem inter gravitatem quadruplum semel, & gravita- tem subquadruplam quater Ratio æqualitatis; victoria autem obtineri non potest, nisi intercedat virium inæqualitas. Si enim pondera essent æqualia, ponderis A resistentia ratione notûs

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Mechanics 236 three pounds have equal moments, and the beam A B cannot incline to either side, since neither weight can receive from the other the force by which it is moved upward, with a greater innate power of descending opposing it, by which it resists being raised. Let C A be to C B as 1 to 4, and, conversely, let the weight B be as 1 to the weight A as 4. If only the gravities are considered, the force of the weight A is as 4, and the force of the weight B as 1: but because they are held from the center of motion C, and cannot freely move in a straight path, they receive an impediment according to the length of the arms, and the straight descent or ascent of the weight is the less impeded the longer the arm is, and the more the shorter. Therefore that weight which is attached to the longer arm, if it descends, descends more; if it ascends, ascends more; but the weight attached to the shorter arm, if it ascends, ascends less, and if it descends, descends less: and thus, if B should descend to E, the measure of the descent would be the perpendicular E G, but the ascent of the weight A to D would be measured by the perpendicular D F: say the same if A should descend and B ascend. Moreover, D F and E G are in the ratio of the arms C A and C B, as is evident, because the right triangles C F D and C G E, besides the right angles at F and G being equal, also have equal angles at C at the vertex, and by 32, book 1, they are equiangular; therefore by 4, book 6, as C D is to C E, so is D F to E G; but C D is equal to C A, and C E to C B (for it is the same line, which, when the position of A B is changed, comes into D E); therefore as C A is to C B, so is D F to E G. Wherefore, by reason of position, weight B has the power of descending, and resists ascent, as 4; but weight A has the power of descending, and consequently also of resisting, so that it may not ascend, only as 1. Since therefore the moment of descending (the same judgment holds for the moment of resistance, so that it may not ascend) is composed both of the weight of the body and of the tendency toward motion, that is, of the velocity of a motion which could follow, it is manifest that a gravity as 4, whose motion would be as 1, cannot overcome a gravity as 1, whose motion would be as 4, nor in turn can it be overcome by it; for between a quadruple gravity once, and a subquadruple gravity four times, there is the ratio of equality; but victory cannot be obtained unless there intervenes an inequality of forces. For if the weights were equal, the resistance of weight A on account of motion

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Liber tertius. CAPUT II. 237 motûs esset subquadrupla, sed quadruplicatur ratione gravita- tis, ergo resistentia est æqualis: item si longitudines essent æquales, resistentia ponderis B esset subquadrupla ratione gravitatis, sed quadruplicatur ratione distantiæ CB; ergo in B est æqualis. Neutrum igitur pondus potest opposito ponderi impetum imprimere, quo elevetur; quia nimirum unaquæque gravitas majorem impetum alteri communicare non potest, quàm pos- sit ipsa concipere, ac propterea impetus gravitatis B, quæ est ut C A, potens conari deorsum ut G E, si imprimeretur gravi- tati A, quæ est ut CB, deberet illam elevare ut FD: Atqui gravitas ipsius A, quæ est ut CB, conatur deorsùm ut FD, & ejus impetus si gravitati B, quæ est ut C A, imprimeretur, il- lam elevare deberet ut G E: igitur in unaquæque gravitate æqualis esset ejusdem conatus deorsùm & vis illata nitens sur- sùm, nec plus præstare posset impetus impressus, quàm innatus. Utraque igitur consistere debet, & neutra impetum acquirit, aut ab alterâ impetum accipit, quia frustra esset impetus acqui- situs aut impressus, quem nullus consequi potest motus. Quare cum eadem sit gravitatum Ratio ut C A ad CB, atque motuum reciprocè ut FD ad G E, ex 16 lib. 6. rectangulum sub extre- mis C A, hoc est pondere B, ut 1, & motu G E, ut 4, æquale est rectangulo sub mediis CB, hoc est pondere A ut 4, & mo- tu FD ut 1: sunt igitur æqualia momenta, quæ componuntur ex gravitate ut 1 & motu ut 4, atque ex gravitate ut 4 & motu ut 1. Ex his apertissimè liquet, cur superiori capite tantopere in- culcata sit brachiorum æqualitas in libræ jugo, ut ex æquili- brio innotescat propositi ponderis ignota gravitas; hæc enim æqualis censetur notæ gravitati, ubi cùm oblato pondere illa æquâ lance libratur: quia scilicet, si inæqualia essent brachia, inæquales essent propensiones ad motum, seu motuum veloci- tates, quæ ad componendam momentorum Rationem concur- runt; adeóque fieri non posset, ut æquales essent gravitates in lancibus; nam minor gravitas ex brachio longiore plus habet momenti, quàm ex breviore, pro ratione inæqualitatis brachio- rum. Verum est libram hujusmodi brachiorum inæqualium vacuam posse priùs ad æquilibritatem reduci, deinde, illâ sic G g 3

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Book Three. CHAPTER II. 237 its motion would be subquadruple, but it is quadrupled in proportion to gravity; therefore the resistance is equal: likewise if the lengths were equal, the resistance of weight B would be subquadruple in proportion to gravity, but it is quadrupled in proportion to the distance CB; therefore in B it is equal. Therefore neither weight can impart an impulse to the opposite weight, by which it may be raised; because, namely, each gravity cannot communicate to the other a greater impulse than it can itself conceive, and for that reason the impulse of weight B, which is as CA, being able to tend downward as GE, if it were imparted to gravity A, which is as CB, ought to raise it as FD: but the gravity of A itself, which is as CB, tends downward as FD, and its impulse, if it were imparted to gravity B, which is as CA, ought to raise it as GE: therefore in each gravity there would be equal in it a downward tendency and an upward force pressing against it, and the impressed impulse could perform no more than the innate one. Both therefore must stand still, and neither acquires an impulse, nor receives an impulse from the other, because the acquired or impressed impulse would be useless, which no motion can follow. Wherefore, since the same is the ratio of the gravities as CA to CB, and of the motions reciprocally as FD to GE, from book 6, proposition 16, the rectangle under the extremes CA, that is, weight B, as 1, and motion GE, as 4, is equal to the rectangle under the means CB, that is, weight A as 4, and motion FD as 1: therefore the moments are equal, composed of gravity as 1 and motion as 4, and of gravity as 4 and motion as 1. From these things it is most plainly clear why in the preceding chapter so much stress was laid on the equality of the arms in the balance-beam, so that from the equilibrium the unknown gravity of the proposed weight might be ascertained; for this is judged equal to the known gravity, when, with the weight presented, it is balanced in an equal scale: because, namely, if the arms were unequal, the propensities to motion, or the velocities of motions, which contribute to the composition of the ratio of moments, would be unequal; and so it would not be possible for the gravities in the scales to be equal; for a smaller gravity, from the longer arm, has more moment than from the shorter one, according to the ratio of the inequality of the arms. Yet it is true that a balance of this kind with unequal arms, when empty, can first be brought back to equilibrium, and then, with that so G g 3

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Mechanicorum æquilibri constitutâ posse lancibus imponi Reciprocè pondera pro Ratione inæqualium brachiorum, & ex æquilibrio argui ponderum illorum Rationem, non tamen æqualitatem: sed artificium hoc, quod peritioribus nihil officeret, ansam non modicam furacibus, & dolosis mercatoribus præberet decipiendi imperitos; quamvis enim libræ hujusmodi æquilibri impositis, hinc & hinc ponderibus adhuc fieret æquilibrium, signum quidem esset æqualibus momentis addita esse æqualia momenta gravitatis, non tamen verùm esset additas esse æquales gravitates, ut rudioribus fortasse videretur. Hinc est libram brachiorum inæqualium in usu non esse, ne locus pateat dolis. Dixi autem expressè priùs statuendam esse libræ vacuæ æquilibritatem, deinde sumenda pondera reciprocè pro Ratione longitudinis brachiorum: nisi etenim priùs æquilibritas illa statueretur, si pondera imposita essent reciprocè in Ratione longitudinis brachiorum, semper pondus minus additum brachio longiori præponderaret, quia etiam ipsa brachij longioris gravitas sua habet momenta, & quidem non modica, majora momentis brachij brevioris, quæ omninò computanda sunt: nam si ponderum in ea Ratione reciprocè positorum momenta sint æqualia, illisque adjiciantur inæqualia gravitatis brachiorum momenta, manifestum est momentorum summam, cui plus additur, majorem esse reliquâ, cui additur minus. Sed quænam sunt, & quanta utriusque brachij momenta? Ut hæc investigemus, & certâ ratione definiamus, ponamus jugum ipsum secundùm suas omnes partes uniusmodi, & gravitatem æquabiliter fusam per totam illius longitudinem. Sit igitur datum prisma AB, quod in quinque partes æquales dividatur, singulas pondo libram unam; & per singula gravitatis centra ducatur recta a u: fiatque secundùm rectam HI, à qua pars una C abscinditur à reliquis, totius prismatis suspensio, ita ut centrum motûs sit in S. Proculdubio unaquæque pars à cæteris sejuncta si appenderetur secundùm longitudinem jugi a u, quod infigeretur per centra gravitatum a, e, i, o, u, obtineret suum

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With the mechanism of an equal balance established, weights can be placed reciprocally on the pans in proportion to the inequality of the arms, and from the balance the ratio of those weights, though not their equality, can be inferred; but this device, which would do no harm to the more experienced, would offer no small opportunity to crafty and deceitful merchants for deceiving the ignorant. For although, when weights of this kind are placed on the balance, equality would still result from weights on this side and on that, it would indeed be a sign that equal moments of weight had been added to equal moments, but it would not truly be the case that equal weights had been added, as might perhaps seem to the less skilled. Hence a balance with unequal arms is not used, lest room be given for fraud. Now I said expressly that first the equilibrium of the empty balance must be established, and then the weights must be taken reciprocally in proportion to the length of the arms; for unless that equilibrium were first established, if the weights were placed reciprocally in proportion to the length of the arms, the smaller weight added to the longer arm would always preponderate, because even the weight of the longer arm itself has its own moments, and indeed not slight ones, greater than the moments of the shorter arm, which must by all means be taken into account. For if the moments of the weights, placed reciprocally in that proportion, are equal, and there are added to them unequal moments of the arms’ weight, it is manifest that the sum of the moments to which more is added is greater than the rest, to which less is added. But what are, and how great are, the moments of each arm? In order that we may investigate these and define them by a certain rule, let us suppose the beam itself to be uniform in all its parts, and its weight evenly distributed throughout its entire length. Let there then be given a prism AB, which is divided into five equal parts, each weighing one pound; and through each center of gravity let a straight line be drawn to a u; and let there be, along the straight line HI, from which one part C is cut off from the rest, the suspension of the whole prism, so that the center of motion is at S. Beyond doubt each part, if separated from the others, when hung along the beam a u, which would be fixed through the centers of gravity a, e, i, o, u, would obtain its own

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Liber tertius. CAPUT II. 239 suum momentum juxtà distantiam centri suæ gravitatis à centro motûs. Quid autem refert (quod quidem attinet ad hanc momentorum Rationem) si in unum continuum corpus unitæ illæ partes coagmententur, an verò divisæ solo contactu sibi invicem adhæreant? eadem quippe est gravitas singulis insita, eadem singularum à centro distantia. Cum itaque centra gravitatum a & e æqualiter distent ab S centro motûs, partes C & D æquiponderant: at distantia S i tripla est distantiæ S a; ergo momentum partis E triplum est momenti partis C; simi- lique ratione pars F habet momentum quintuplum, & pars G septuplum. Igitur componendo, momentum totius aggregati quatuor partium D, E, F, G, est sedecuplum momenti partis C; neque enim singulæ partes ex hoc quod cum cæteris pen- deant, illisque cohæreant, suum amittunt momentum. Hinc sit momenta brachiorum esse inter se ut Quadrata longitudinum eorumdem brachiorum: siquidem ostenditur singularum partium momentum crescere secundùm Rationem numero- rum imparium, prout secundùm eandem Rationem crescunt distantiæ centrorum gravitatis illarum. Sic brachiorum longitudines si essent in Ratione 2 ad 7, illorum momenta ratione suæ gravitatis innatæ & ratione positionis essent ut 4 ad 49. Hæc Ratio momentorum in Ratione Quadratorum longi- tudinis, si res attentè perpendatur, omnibus est manifesta: Nam singulorum brachiorum gravitates juxta hypothesim æquabiliter fusæ per totum libræ jugum Rationem inter se habent, quam illorum longitudinis propensiones ad motum, seu, quod eòdem recidit, distantiæ à centro motûs eandem pariter Rationem habent, quam brachiorum longitudines: Quoniam igitur (ut sæpiùs dictum est, sæpiùsqque iterùm inculcandum) momenta componuntur ex gravitatibus ratio- ne materiæ, & ex propensionibus ad motum ratione sitûs seu positionis, componuntur duæ Rationes longitudinum; atque adeó momentum unius brachij ad momentum alterius bra- chij est in duplicata Ratione suarum longitudinum, hoc est, ut ipsarum longitudinum Quadrata. Id quod adhuc ul- teriùs sic explicari posse videtur. Sit libræ jugum M. N, & motûs centrum O: intelligatur moveri, ut obtineat positio- nem

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Book the Third. CHAPTER II. 239 its moment according to the distance of the center of its gravity from the center of motion. But what does it matter (so far as this ratio of moments is concerned) whether those united parts are joined together into one continuous body, or whether, being separated, they adhere to one another only by contact? For the gravity inherent in each is the same, and the distance of each from the center is the same. Since therefore the centers of gravity a and e are equally distant from the center of motion S, the parts C and D are of equal weight; but the distance S i is triple the distance S a; therefore the moment of part E is triple the moment of part C; and by a similar reasoning the part F has a moment five times as great, and the part G seven times as great. Therefore, by combining these, the moment of the whole aggregate of the four parts D, E, F, G is sixteen times the moment of part C; for the several parts do not lose their moment from the fact that they hang with the others and adhere to them. Hence let it be that the moments of the arms are among themselves as the squares of the lengths of those arms: since it is shown that the moment of each part increases according to the ratio of odd numbers, just as according to the same ratio the distances of their centers of gravity increase. Thus if the lengths of the arms were in the ratio of 2 to 7, their moments, by reason of their innate gravity and by reason of position, would be as 4 to 49. This ratio of moments in the ratio of the squares of length, if the matter be carefully considered, is evident to all: for the gravities of the several arms, being uniformly diffused through the whole beam of the balance according to the hypothesis, have among themselves the same ratio which their propensities to motion, or, which comes to the same thing, their distances from the center of motion, likewise have; and the distances from the center of motion have the same ratio as the lengths of the arms. Since therefore (as has been said often enough, and must again and again be impressed) moments are composed of gravities in the ratio of matter, and of propensities to motion in the ratio of situation or position, two ratios of lengths are combined; and thus the moment of one arm to the moment of another arm is in the doubled ratio of their lengths, that is, as the squares of the lengths themselves. Which may still seem capable of being explained further as follows. Let the beam of a balance be M. N, and the center of motion O: let it be understood to be moved so as to obtain the position

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Mechanicorum 240 nem PR. Momentum brachij minoris OM referre videtur sector MOP, momentum verò brachij majoris ON referre videtur sector NOR; singularum quippe partium motus ab arcu descriptus illarum momentum ob oculos ponit, & totius brachij momentum illius motus, scilicet sector in motu descriptus. At ob æqualitatem angulorum ad verticem in O, sectores MOP, NOR sunt similes, & quia uterque sector est similis pars sui circuli, eam inter se habent sectores Rationem, quæ est circulorum, per 15. lib.5. circuli autem sunt in duplicatâ Ratione diametrorum, ex 2. lib.12. seu Radiorum OM & ON; igitur & sectores sunt in duplicatâ Ratione OM ad ON, hoc est quadrati OM ad quadratum ON. At quæris. In proposito prismatic A B, momentum brachij S A ad momentum brachij S B est ut 1 ad 16: An, ut habeatur æquilibrium in S, addendum erit in A pondus librarum 15: quandoquidem pars C est libræ unius, reliquum autem brachium lib.4, & longitudo S B est quadrupla longitudinis S A. Hoc sanè non est iis, quæ dicta sunt, consequens, nec ex illis efficitur: aliud quippe est momenta brachiorum esse ut 1 ad 16, aliud verò perinde se habere, atque si ex brachiorum gravitate carentium extremitatibus penderent libræ 1 & 16, ut ad æquilibrium constituendum opus sit breviori brachio addere libras 15. Primum illud verum est, etiam si extremitatibus adnecti intelligamus hinc quidem libræ semissem; hinc verò libras octo, manet scilicet eadem Ratio 1 ad 16. Alterum à formà veritatis prorsùs alienum videtur, nam licet libræ 4 in extremitate B positæ æquivaleant libræ unciæ simul cum pondere lib.15. in extremitate A; non est tamen eadem ratio librarum 4 secundùm longitudinem brachij S B distributarum; quo enim propiores sunt partes centro motûs, eò minus habent momenti: non igitur libræ 4 sic distributæ æquivalent libris 16, nec addendum erit pondus librarum 15 in oppositâ extremitate ad æquilibrium constituendum, quandoquidem nec ipsa unica

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Mechanics 240 ... The sector MOP seems to represent the momentum of the shorter arm OM, and the sector NOR the momentum of the longer arm ON; for the motion of each of the parts, described by the arc, places its momentum before our eyes, and the momentum of the whole arm is that of its motion, namely, the sector described in the motion. But because the angles at the vertex in O are equal, the sectors MOP and NOR are similar; and since each sector is a similar part of its circle, the sectors have between themselves the same ratio that circles have, by Prop. 15, book 5. But circles are in the duplicate ratio of their diameters, from Prop. 2, book 12, or of the radii OM and ON; therefore the sectors are also in the duplicate ratio of OM to ON, that is, of the square of OM to the square of ON. But you ask: In the proposed prism AB, the momentum of the arm SA to the momentum of the arm SB is as 1 to 16. Will it then be necessary, in order to obtain equilibrium at S, to add a weight of 15 pounds at A, since part C is one pound, but the remainder of the arm is 4 pounds, and the length SB is four times the length SA? This certainly does not follow from what has been said, nor is it established by it: for it is one thing for the momenta of the arms to be as 1 to 16, and another for them to behave as if from the ends of arms without weight there hung weights of 1 and 16, so that, in order to establish equilibrium, it is necessary to add 15 pounds to the shorter arm. The first is true, even if we understand that half a pound is attached to one end and eight pounds to the other; the same ratio of 1 to 16 remains. The other seems wholly alien to the form of the truth, for although 4 pounds placed at the end B may be equivalent to 1 ounce together with a weight of 15 pounds at the end A, nevertheless the same is not true of 4 pounds distributed according to the length of arm SB; for the closer the parts are to the center of motion, the less momentum they have: therefore 4 pounds so distributed are not equivalent to 16 pounds, nor will it be necessary to add a weight of 15 pounds at the opposite end in order to establish equilibrium, since not even the single...

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Liber tertius. CAPUT II. 241 unica libra partis C tantumdem habet momenti, quantum ha- beret si totâ ex A penderet. Equidem ex his, quæ paulò ante dicebam de sectoribus re- ferentibus momenta brachiorum, aliquando eò deveni, ut sus- picarer totam gravitatem brachij ON (idem dic de reliquo OM) intelligendam esse ibi exercere totum momentum, ubi est quasi centrum omnium suorum momentorum, hoc est, ubi momenta bifariam dividuntur. Si autem sector NOR refert totum momentum brachij ON; non est intelligendum cen- trum hoc momentorum esse punctum L, ubi est semissis bra- chij ON; quia Sector LOQ ad Sectorem NOR est in Ra- tione Quadrati OL ad Quadratum ON, quod est illius qua- druplum. Quod si inter OL & ON sumatur media propor- tionalis OV, jam sector VOT est ad Sectorem NOR in du- plicatâ Ratione Radiorum OV, & ON, hoc est ut OL ad ON, hoc est ut 1 ad 2; ac propterea Sector VOT æqualis est Trapezio NVTR; proinde in V videbantur divisa æqualiter momenta. Hinc arguebam vel totam brachij gravitatem cen- sendam esse sua exercere momenta in puncto distantiæ à centro motûs mediæ proportionalis inter semissem brachij & totam brachij longitudinem, vel in extremitate brachij censen- dam esse pendere gravitatem, quæ medio loco proportiona- lis sit inter totam brachij ejusdem gravitatem & ejus se- missem. Verùm, ut quod res est sincerè eloquar, quamvis in Secto- ribus illis, quos paulò ante commemorabam, imaginem quandam momentorum gravitatis secundùm brachiorum longitudinem distributæ agnoscerem, non tamen in re Physicâ satis fidebam Geometricæ illi commentationi: quip- pe qui observabam à Sectoribus quidem poni ob oculos Ra- tionem momentorum singulorum brachiorum ex motu, qui idem est, sivè multa, sivè modica sit gravitas, sivè in uno, sivè in alio puncto constituta intelligatur, non tamen defi- niri ipsius gravitatis momenta. Quare satius duxi ad experi- menta potiùs confugere, ut hinc lux aliqua suboriretur, qua gravitatis quæsita momenta innotescerent. Primùm igitur assumptus est ligneus cylindrus, cujus dia- meter CE unc. 1. 06" pedis Romani antiqui, & addito in A Hh

Transcription: Translated (English)

Liber tertius. CAPUT II. 241 a single pound of the part C has precisely the same moment as it would have if it hung entirely from A. Indeed, from what I was saying a little earlier about sectors representing the moments of the arms, I was at times led to suspect that the whole weight of the arm ON (the same may be said of the rest OM) should be understood to exert its full moment there where it is, as it were, the center of all its moments, that is, where the moments are divided into two equal parts. But if the sector NOR represents the whole moment of the arm ON, it is not to be understood that this center of moments is the point L, where the midpoint of the arm ON lies; because sector LOQ is to sector NOR in the ratio of the square of OL to the square of ON, which is four times as much. If, however, between OL and ON a mean proportional OV is taken, then sector VOT is to sector NOR in the duplicate ratio of the radii OV and ON, that is, as OL is to ON, that is, as 1 to 2; and therefore sector VOT is equal to trapezium NVTR; thus at V the moments seemed to be equally divided. From this I argued either that the whole weight of the arm should be thought to exert its moments at a point distant from the center of motion by a mean proportional between half the arm and the whole length of the arm, or that at the extremity of the arm the weight should be thought to hang, which is midway proportionally between the whole weight of that arm and half of it. But, to speak sincerely about the matter as it is, although in those sectors I was mentioning a little earlier I recognized some image of the moments of gravity distributed according to the length of the arms, still in physical matters I did not sufficiently trust that geometrical conjecture: for I observed that the sectors indeed present to the eye the ratio of the moments of each arm arising from motion, which is the same whether the gravity is great or small, whether it is understood to be placed in one point or another; yet they do not determine the moments of gravity itself. Therefore I thought it better to have recourse to experiments, so that some light might arise from them, by which the sought moments of gravity might become known. First, then, a wooden cylinder was taken, whose diameter CE was 1.06 inches of the old Roman foot, and with the addition in A Hh

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Mechanicorum 242 pondere D unciarum 40 ́ collocatus est in æquilibrio, quod factum est in B puncto. Fuit autem longitudo B A unciarum pedis Romani 7 ́ BC verò unc. 42 ́. Resecto demùm subtilissimè cylindro, repertum est pondus A B unciarum 2 ́, pondus autem B C unc. 13 ́. His observatis cum nullus dubitarem, quin momenta brachiorum essent ut quadrata longitudinum, ipsas longitudines A B unc. 7 ́, & B C unc. 42 ́ ad unicam denominatione reduxi, videlicet ́ & ́ : & assumptis numeratorum Quadratis 136900 atque 4481689 hanc posui Rationem momentorum. Tùm sic ratiocinatus sum Algebricè; ut 136900 ad 4481689, ita momentum B A 1 R[ecipe] ad 32.73" R[ecipe] momentum B C. Cum igitur æqualitas esset inter momentum brachij B C, & momentum brachij B A plus ipso pondere D; hæc enim constituebant æquilibrium, æquatio Algebricè est inter momentum B C 32.73" R[ecipe] & B A + D, hoc est 1 R[ecipe] + unc. 40 ́: & per Antithesim demptâ utrinque 1 R[ecipe], æquatio est inter 31.73" R[ecipe] & unc. 40. ́. Factâ itaque numeri absoluti 40 ́ divisione per numerum Radicum prodit pretium 1 R[ecipe] pondo unc. 1.27", quod est momentum brachij B A; ac proinde momentum brachij B C: est pondo unc. 41. 57". Quare perinde est atque si gravitas unc. 1.27" poneretur in extremitate Alineæ Mathematicæ, ac in extremitate C poneretur gravitas unc. 41. 57". At in A fuit additum pondus unc. 40 ́: ergo momentum brachij B C æquivalet ponderi D, & præterea unc. 1.07", qui est semissis gravitatis brachij A B observatæ unc. 2 ́, hoc est in centesimis paulò ultra 2. 12". Si verò momentis brachij B A pondo unc. 1.27" addatur gravitas D pondo unc. 40. 50", fit aggregatum 41. 77", quod excedit inventum momentum brachij B C unc. 41. 57". excessu ́/100 unciæ: quæ discrepantia facillimè potuit oriri ex aliquâ exili, ac minime notabili differentiâ vel in dimetiendis brachiorum longitudinibus, vel in ponderandis eorum gravitatibus; cum maximè resegmina illa, & scobs, non computarentur in gravitate. Quod si fiat ut longitudo B C 2117 ad longitudi

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Mechanicorum 242 was placed in equilibrium by the weight D of 40 ounces, which was effected at point B. Now the length B A was 7 inches of a Roman foot, and B C 42 inches. After the cylinder was finally cut away with the greatest nicety, the weight of A B was found to be 2 ounces, but the weight of B C 13 ounces. Having observed this, since I did not doubt at all that the moments of the arms were as the squares of the lengths, I reduced the lengths A B 7 inches, and B C 42 inches, to a single denomination, namely ́ and ́ : and having taken the squares of the numerators, 136900 and 4481689, I set down this Ratio of moments. Then I reasoned thus algebraically: as 136900 is to 4481689, so the moment B A 1 R[ecipe] is to 32.73" R[ecipe] the moment B C. Since therefore there was equality between the moment of the arm B C, and the moment of the arm B A plus the weight D itself; for these constituted the equilibrium, the algebraic equation is between the moment B C 32.73" R[ecipe] and B A + D, that is 1 R[ecipe] + 40 ounces: and by Antithesis, subtracting 1 R[ecipe] on each side, the equation is between 31.73" R[ecipe] and 40 ounces. Thus, when the absolute number 40 ounces is divided by the number of Roots, there results the value 1 R[ecipe] weighing 1.27 ounces, which is the moment of the arm B A; and therefore the moment of the arm B C is 41.57 ounces. Wherefore it is as if a weight of 1.27 ounces were placed at the extremity A of the Mathematical line, and a weight of 41.57 ounces were placed at the extremity C. But at A there was added a weight of 40 ounces: therefore the moment of the arm B C equals the weight D, and moreover 1.07 ounces, which is half the weight of the arm A B observed, of 2 ounces, that is, in hundredths a little over 2.12 ounces. But if to the moments of arm B A of 1.27 ounces there be added the weight D of 40.50 ounces, the total becomes 41.77 ounces, which exceeds the found moment of the arm B C, 41.57 ounces, by 2/100 of an ounce: which discrepancy could very easily have arisen from some slight and hardly noticeable difference either in measuring the lengths of the arms, or in weighing their weights; since especially those shavings and filings were not reckoned in the weight. If it were made so that the length B C 2117 to the lengthi

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Liber tertius. CAPUT II. 243 longitudinem AB 370, ita pondus in A unc. 41. 77" ad pon- dus in B unc. 7. 30", constat esse ferè semissem gravitatis unc. 13 1/2: sed est excessus semunciæ ob minùs accuratam ob- servationem. Qua propter aliud experimentum quàm accuratissimè insti- tui ligneo parallelepipedo, cujus longitudo palmorum Roma- norum 7. unc. 6. 566" , ejus verò pondus lib. 1. unc. 1 1/4. Alte- ri extremitati additus est plumbeus cylindrus ad per- pendiculum pendens, cujus pondus unc. 20. Impositum est parallelepipedum rotun- do claviculo ferreo, qui horizonti parallelus erat, & factum est æquilibrium in puncto, ubi tota longitudo in duas partes dividebatur, quarum minor ponderi adhærens fuit mensurâ unc. 18 1/5, partes verò major fuit mensurâ palm. 6. unc. 2/5. Cum itaque longitudo CB observata fuerit unciarum mensuralium 72. 40", & AC unciarum mensuralium 18. 16", in eadem pariter Ratione ponuntur brachiorum gravitates absolutæ. Quare CB pondo unc. 1059", AC verò pondo unc. 2. 66". Igitur ut longitudinis BC quadratum 52417600 ad longitudi- nis AC quadratum 3297856, ita momentum BC 1 Rx ad 3297856/52417600 Rx momentum brachij AC: cui additur cylindrus D unc. 20: Est ergo æquatio inter AC + D, hoc est 3297856/52417600 Rx + unc. 20. 00" & 1 Rx; & factâ Antithesi est æquatio inter unc. 20. 00" & 49119744/52417600 Rx: demum institutâ divisione consurgit pretium 1 Rx, hoc est momentum BC, unc. 21. 342" & paulo amplius: atque momentum brachij AC est pondo unc. 1. 343", cui additâ gravitate cylindri fit summa unc. 21. 343" planè æqualis momento brachij BC. Et ut hanc operandi methodum confirmarem, iterum insti- tui argumentationem assumendo quadrata gravitatum utrius- que brachij, sunt enim ex hypothesi gravitates in Ratione lon- gitudinum. Cum igitur sit CB pondo unc. 10. 59"; & AC pondo unc. 2. 66." fiat ut quadratum CB 1121481 ad quadra- tum AC 70756, ita ipsius CB momentum 1 Rx ad 70756/1121481 Rx momentu[m] ipsius AC. Quoniam verò AC + D hoc est 70756/1121481 Rx Hh 2

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Liber tertius. CHAPTER II. 243 length AB 370, so the weight at A is 41.77 oz. compared with the weight at B, 7.30 oz., it is found to be nearly half the gravity, 13 1/2 oz.; but there is an excess of half an ounce because of a less accurate observation. For this reason I carried out another experiment as accurately as possible with a wooden parallelepiped, whose length was 7 Roman palms, 6.566 in.; and its weight was 1 lb. 1 1/4 oz. To one end there was added a lead cylinder hanging vertically, whose weight was 20 oz. The parallelepiped was placed on a round iron pin, which was parallel to the horizon, and equilibrium was achieved at the point where the whole length was divided into two parts, of which the smaller part attached to the weight measured 18 1/5 in., while the larger part measured 6 palms, 2/5 in. Therefore, since the observed length CB was 72.40 mensural inches, and AC 18.16 mensural inches, the absolute weights of the arms are likewise put in the same ratio. Hence CB is 1059 oz. by weight, while AC is 2.66 oz. by weight. Therefore, as the square of length BC, 52417600, is to the square of length AC, 3297856, so the moment BC 1 Rx is to 3297856/52417600 Rx the moment of arm AC: to this is added cylinder D, 20 oz. Therefore there is an equation between AC + D, that is 3297856/52417600 Rx + 20.00 oz. and 1 Rx; and by anti-thesis an equation results between 20.00 oz. and 49119744/52417600 Rx: finally, by division, the price of 1 Rx is obtained, that is, the moment of BC, 21.342 oz. and a little more; and the moment of arm AC is 1.343 oz. by weight, to which, when the weight of the cylinder is added, the sum becomes 21.343 oz., exactly equal to the moment of arm BC. And to confirm this method of procedure, I again undertook the argument by taking the squares of the weights of each arm, for by hypothesis the weights are in the ratio of the lengths. Since therefore CB is 10.59 oz. by weight, and AC 2.66 oz. by weight, let the square of CB, 1121481, be to the square of AC, 70756, as the moment of CB itself, 1 Rx, is to 70756/1121481 Rx the moment of AC itself. But since AC + D, that is 70756/1121481 Rx Hh 2

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Mechanicorum 244 tunc. 20. 00" æquatur momento BC hoc est 1 Rx, factâ per Antithesin communi subtractione 70756/1121481 Rx, remanet æquatio inter pondus unc. 20. 00" & 1050735/1121481 Rx, & factâ divisione emergit pretium 1 Rx, hoc est momentum BC pondo unc. 21. 347" atque adeò momentum ipsius AC est pondo unc. 1. 347"; cui si addatur cylindri D gravitas unc. 20, totum momentum in A est unc. 21. 347", omnino æquale momento ipsius B: id quod ab initio vix sperare audebam, cum hæc operatio à superiore differat solùm per 1000. Hîc pariter brachij AC gravitas absoluta pondo unc. 2. 66". habet momentum unc. 1. 347", cum ejus semissis sit unc. 1. 330", quæ est minima atque prorsùs contemnenda differentia: quî enim fieri potuit, ut, quantalibet adhiberetur diligentia in metiendo, & ponderando, ne pilum quidem à verò aberrarem? aut quis omninò certus sit omnes parallelepipedi partes æquali prorsùs fuisse præditas gravitate, itaut quæ pars ad arboris radicem vergebat, non fuerit paulò densior, aut interiùs nodulum aliquem latentem habuerit, quo factum fuerit, ut vera gravitas instituto calculo non exactissimè responderet? simili ratione semissis gravitatis brachij BC intelligitur in extremitate B: nam fiat ut longitudo BC 72. 40" ad longitudinem AC 18. 16", ita reciprocè pondus in A unc. 21. 347" ad pondus in B unc. 5. 354": erat autem brachij BC gravitas absoluta unc. 10. 59" cujus semissis 5. 295". differt ab invento pondere solùm per 10/1000 unciæ, hoc est ferè sesquiscrupulum, seu grana 34. Ex his quidem satis apparebat brachij gravitatem in libræ jugo intelligendam esse, quasi ejus semissis in ipsâ extremitate constitueretur, seu, quod idem est, tota gravitas brachij ad mediam longitudinem applicaretur (eadem siquidem esse momenta totius gravitatis in dimidiatâ distantiâ, ac dimidiæ gravitatis in totâ distantiâ, ex sæpiùs dictis est manifestum) mihi tamen satisfactum non existimabam, nisi ulteriore experimento veritatis vestigia persequerer. Quare eundem plumbeum cylindrum, cujus longitudo erat palmi 1. unc. 1. 9/10, ita in extremitate A collocavi, ut super AI jaceret, & factum est æquilibrium in E, eratque E A longitudo unc. 22 4/10. Tùm divîso bifariam in O spatio AI, quod cylindrus jacens occupabat, ex puncto

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Mechanics 244 then. 20. 00" is equal to the moment BC, that is 1 Rx; after the common subtraction by antithesis of 70756/1121481 Rx, there remains the equation between the weight of 20. 00" and 1050735/1121481 Rx; and after division there emerges the price of 1 Rx, that is the moment BC, weighing 21. 347" ounces, and thus the moment of AC itself is 1. 347" ounces; to which if the weight of cylinder D, 20 ounces, be added, the total moment at A is 21. 347" ounces, wholly equal to the moment at B: something which at the start I could hardly have hoped, since this operation differs from the former only by 1000. Here likewise the absolute weight of arm AC, weighing 2. 66" ounces, has a moment of 1. 347" ounces, when its half is 1. 330", which is the smallest and altogether negligible difference: for how could it have happened that, however great care was used in measuring and weighing, I should not have erred by even a hair’s breadth from the truth? Or who can be altogether certain that all parts of the parallelepiped were endowed with exactly equal weight, so that the part tending toward the root of the tree was not a little denser, or had some small hidden knot inside it, whereby it came about that the true weight did not correspond exactly with the calculated result? In a similar way the half of the weight of arm BC is understood at the extremity B: for let the length BC 72. 40" be to the length AC 18. 16", so inversely the weight at A, 21. 347" ounces, is to the weight at B, 5. 354" ounces: but the absolute weight of arm BC was 10. 59" ounces, of which the half was 5. 295". It differs from the found weight by only 10/1000 of an ounce, that is, nearly a scruple and a half, or 34 grains. From these things it was indeed sufficiently evident that the weight of the arm in the balance beam must be understood as if its half were placed at the very end, or, what is the same, the whole weight of the arm were applied at the midpoint (since, as has been said often, the moment of the whole weight at half the distance is the same as that of half the weight at the whole distance); nevertheless I did not think myself satisfied unless I pursued the traces of the truth by further experiment. Therefore I placed the same lead cylinder, whose length was 1 palm 1. 9/10 inches, at the extremity A in such a way that it lay upon AI, and equilibrium was established at E, and the length EA was 22 4/10 inches. Then, the space AI occupied by the lying cylinder having been divided in half at O, from the point

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Liber tertius. CAPUT II. 245 puncto O suspendi cylindrum, & factum est pariter æquilibrium exactissimè in E, sicut priùs, cum jacebat super A I. Deinde cylindrum eumdem iterum parallelepipedo imposui jacentem, sed ea ratione illum ultrò citróque promovebam, ut omnino propè fulcrum consisteret, donec demùm factum est æquilibrium in H, & fuit H A palm.2. unc.10 2/10: Factâ verò suspensione cylindri ex L, ita ut HL esset dimidiata cylindri jacentis longitudo, æquilibrium pariter in H factum est. Relictâ igitur illâ sectorum analogiâ, deprehendi per illas quidem ob oculos poni motum, non verò momentum, seu propensionem ad motum, quæ ex distantiâ à centro motûs in ipsâ longitudine definienda est: & quod ad gravitatem attinet, nullus mihi relictus est dubitandi locus ita computandam esse totius brachij gravitatem per ipsum æquabiliter diffusam, quasi tota in dimidiatâ distantiâ à centro motûs collocaretur: quamvis enim particularum gravium, quæ ultrâ semissem longitudinis magis à centro removentur, momentum crescat pro Ratione distantiæ, reliquarum tamen numero æqualium citrà longitudinis semissem centro propiorum momentum similiter pro Ratione minoris distantiæ minuitur; ac proptereà tantùm ista momenta simul sumpta decrescunt, quantum illa simul sumpta augentur. Ex quo oritur quædam quasi æqualitas, perinde atque si momenta omnia majora & minora in illam particulam confluerent, quæ media est Arithmeticè inter extrema (momenta si quidem ratione distantiæ Arithmeticè crescent, prout Arithmeticè ipsa distantia crescit) hæc autem est in semisse longitudinis brachij. Ex quo iterum confirmatur momenta brachiorum esse ut quadrata longitudinum; sunt enim in duplicatâ Ratione illarum; semisses quippè sunt in Ratione integrarum longitudinum, gravitates sunt in Ratione earumdem longitudinum, ergo Ratio composita est duplicata ejusdem Rationis longitudinum. Hinc datâ jugi æquabilis, & uniformis gravitate absolutâ, & datâ Ratione longitudinum brachiorum inæqualium libræ, dividatur data gravitas secundùm datam Rationem brachiorum: tùm fiat ut longitudo minor ad longitudinem majorem, ita dimidia gravitas majoris brachij ad aliud, ex quo quarto ter- H h 3

Transcription: Translated (English)

Book III. CHAPTER II. 245 at point O I suspended the cylinder, and there resulted likewise a balance most exactly in E, just as before, when it lay upon A I. Then I again placed the same cylinder lying upon the parallelepiped, but in such a way that I moved it back and forth, until at length it stood almost entirely near the fulcrum, until finally there came to be equilibrium in H, and H A was 2 palms, 10 2/10 inches: but when the suspension of the cylinder was made from L, so that HL was half the length of the lying cylinder, equilibrium likewise occurred in H. Having therefore set aside that analogy of sectors, I found from them that motion is indeed shown before the eyes, but not momentum, or the tendency toward motion, which must be defined from the distance from the center of motion in the very length itself: and as for weight, no place of doubt was left to me that the weight of the whole arm must be calculated as being evenly diffused through it, as if the whole were placed at half the distance from the center of motion: for although the momentum of the heavier particles, which are removed farther from the center beyond the half-length, increases in proportion to the distance, yet that of the remaining equal in number particles nearer the center, this side of the half-length, likewise diminishes in proportion to the smaller distance; and therefore only so much do these momenta, taken together, decrease as those, taken together, increase. From this there arises a certain kind of equality, just as if all the greater and smaller momenta flowed together into that particle which is the arithmetical mean between the extremes (for the momenta, insofar as they increase arithmetically according to distance, as the distance itself increases arithmetically) and this is at the half-length of the arm. From this it is again confirmed that the momenta of arms are as the squares of the lengths; for they are in a doubled ratio of those lengths; for the halves are in the ratio of the whole lengths, the weights are in the ratio of those same lengths, therefore the composite ratio is a doubled ratio of the same lengths. Hence, given the absolute weight of an equal and uniform balance, and given the ratio of the lengths of unequal arms of the balance, let the given weight be divided according to the given ratio of the arms: then let it be as the shorter length is to the longer length, so let half the weight of the greater arm be to the other, from which by the fourth ter- H h 3

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Mechanicorum 246 mino invento si auferatur dimidia gravitas brachij minoris, re- siduum indicabit pondus addendum extremitati brachij mino- ris, ut fiat æquilibrium cum solâ gravitate brachij longioris. Vel potiùs fiat ut quadratum longitudinis brachij minoris ad differentiam inter quadrata brachiorum, ita semissis gravitatis brachij minoris ad pondus ipsi addendum. CAPUT III. Quomodò corporum æquilibria explicentur. Quamvis libro primo plura de Gravitatis centro, prout hujus operis instituto congruebat, disputata sint, eorum tamen plenior explicatio ex his, quæ duobus præcedentibus capitibus dicta sunt, petenda est, si quidem Physicam æquilibrij causam nosse velimus. Neque enim Gravitatis centrum illud est, quod æquales gravitates, sed quod æquales gravitationes, aut æqualia gravitatis momenta, hoc est æquales ad descendendum propensiones ac vires circumstant. Nam gravitas eâ Ratione per universum corpus grave distribuitur, quâ Ratione materia ipsa, cui illa inest, diffusa intelligitur; quæ si uniusmodi sit & homogenea, ibi centrum habet, ubi est molis ipsius centrum; ubi siquidem bifariam moles & materia, ibi pariter gravitas illi insita bifariam dividitur. Quoniam verò fieri potest, ac sæpiùs contingit, materiam quidem corporis & molem invariantam permanere, figuram autem mutari; ex quo nunc in hanc, nunc in illam partem migrat gravitatis centrum, quia alia atque alia fiunt gravitatis momenta pro variâ corporis secundùm suas partes positiones; proptereà hujusmodi momentorum æqualitas ex libræ Rationibus desumenda est, sivè æqualium, sivè inæqualium brachiorum libra intelligatur, prout varia corporis gravis suspensio aut sustentatio contingit. Sed quia in communi usu non adeò frequens est illa suspensio, qua corpus pendeat quasi ex puncto lineæ directionis transeuntis per centrum gravitatis, & ad universi centrum deductæ, aut illa sustentatio, qua corpus grave acutissimo apici incumbat,

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Mechanics 246 If the lesser arm is found, and half the weight of the lesser arm is taken away, the remainder will indicate the weight to be added to the end of the lesser arm, so that equilibrium may be produced with the sole weight of the longer arm. Or rather, let the square of the length of the lesser arm be to the difference between the squares of the arms as half the weight of the lesser arm is to the weight to be added to it. CHAPTER III. How the equilibria of bodies are explained. Although in the first book several things concerning the center of gravity were discussed, as was fitting to the purpose of this work, their fuller explanation is nevertheless to be sought from what has been said in the two preceding chapters, if indeed we wish to know the physical cause of equilibrium. For the center of gravity is not that which is surrounded by equal weights, but that which is surrounded by equal gravitations, or equal moments of gravity, that is, equal tendencies and forces to descend. For gravity is distributed through the whole heavy body in the same way as the matter itself, in which it resides, is understood to be diffused; and if that matter be of one kind and homogeneous, it has its center where the center of the mass itself is; and if the mass and matter are divided in two, the gravity inherent in it is likewise divided in two. But since it can happen, and often does happen, that the matter and mass of a body remain unchanged, while its shape is altered; whereby now to this, now to that part the center of gravity moves, because the moments of gravity become different and different according to the positions of the body in its parts; therefore the equality of such moments must be deduced from the principles of the balance, whether a balance with equal or unequal arms be understood, according as the various suspension or support of the heavy body occurs. But because in common use that suspension is not very frequent, by which a body hangs as if from a point of the line of direction passing through the center of gravity and carried down to the center of the universe, or that support by which a heavy body rests upon the sharpest apex,

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Liber tertius. CAPUT III. 247 incumbat, cui immineat idem gravitatis centrum; quinimmo ita plerumque suspenditur, aut sustinetur corpus, ut ductâ per Gravitatis centrum lineâ, aut ex hujus extremitatibus tanquam polis illud suspendatur, aut subjecto fulcro lineæ huic parallelo illud sustineatur; ideò hujusmodi lineam per centrum gravitatis ductam liceat appellare Diametrum Gravitatis; quæ diameter quasi in librâ locum Axis seu Aginæ obtinet, corporis verò partes hinc & hinc positæ rationem habent brachiorum libræ, atque pro distantiarum seu longitudinum Ratione sua habent momenta. Sit propositum Trapezium, cujus gravitatis centrum C puncto respondeat, & sustineatur secundùm rectam lineam ACN (similis esset philosophandi ratio, si assumeretur recta R C S) quæ proptereà Diameter Gravitatis à me dicitur, quia sicut circuli diameter per centrum ducta illum in semicirculos æquales distinguit, ita hæc per gravitatis centrum transiens dividit Trapezium in momenta æqualia, itaut in neutram partem inclinetur, juxta dicta de centro Gravitatis. Sed cur fiat æquilibrium intelliges ex Rationibus libræ Brachiorum inæqualium: ducatur enim ad rectam AN per C perpendicularis DCE, & fiunt brachia CD, CE inæqualia; sunt igitur momenta CE longioris majora momentis CD brevioris. Ductis verò ipsi DE parallelis BF & ML, secatur diameter gravitatis AN in punctis H & I: quare inæqualia sunt brachia HB longius, & HF brevius, & vicissim IM est brevius, & IL longius: Ex quo sit momenta in L & E majora esse momentis in M & D, at momentum in F minus esse momento in B; atque adeò componendo majora cum minoribus ex eâdem parte, fieri compositum momentum unius partis æquale toti momento oppositæ partis. Vel si non placeat particulatim Trapezium distinguere quasi in tot libras, quot ductæ intelliguntur parallelæ, dic totius gravitatis ADN semissem intelligi in D, & totius gravitatis AEN semissem intelligi in E; & quamvis pars ADN absolutè & seorsim accepta major sit & gravior parte AEN absolutè sumptâ, quia tamen sunt reciprocè in Ratione distantiarum

Transcription: Translated (English)

Liber tertius. CAPUT III. 247 rests upon it, over which the same center of gravity may hang; indeed it is usually so suspended or supported that, a line having been drawn through the center of gravity, either it is suspended from that line, as from poles at its extremities, or supported by a prop placed under it parallel to that line; therefore such a line drawn through the center of gravity may rightly be called the Diameter of Gravity; this diameter, as it were in a balance, occupies the place of the Axis or Agina, while the parts of the body placed here and there have the relation of the arms of the balance, and according to the ratio of the distances or lengths they have their moments. Let a trapezium be proposed, whose center of gravity C corresponds to the point and is supported along the straight line ACN (a similar mode of reasoning would apply if the straight line R C S were assumed), which for that reason I call the Diameter of Gravity, because just as the diameter of a circle drawn through the center distinguishes it into equal semicircles, so this line passing through the center of gravity divides the trapezium into equal moments, so that it inclines to neither side, according to what has been said about the center of gravity. But why equilibrium occurs you will understand from the reasons of a balance with unequal arms: for let a perpendicular DCE be drawn to the straight line AN through C, and the arms CD, CE become unequal; therefore the moments of CE, the longer arm, are greater than the moments of CD, the shorter one. But if BF and ML are drawn parallel to DE, the diameter of gravity AN is cut at the points H and I: wherefore the arms HB, longer, and HF, shorter, are unequal, and in turn IM is shorter, and IL longer: from which it follows that the moments at L and E are greater than the moments at M and D, but the moment at F is less than the moment at B; and thus by combining the greater with the lesser on the same side, the compounded moment of one part becomes equal to the whole moment of the opposite part. Or if it does not please you to distinguish the trapezium piece by piece as though into as many balances as the parallel lines drawn are understood to be, say that the semisum of the whole gravity ADN is to be understood in D, and that the semisum of the whole gravity AEN is to be understood in E; and although the part ADN, taken absolutely and separately, is greater and heavier than the part AEN, taken absolutely

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Mechanicorum 248 rum C E & C D, propterea æquilibrium constituere; pars enim minùs gravis ex positione majorem habet propensionem ad mo- tum, qui esset velocior; partis verò gravioris minor est propen- sio ad motum, qui esset tardior; atque adeò hæc minùs resistit ratione motûs, magis autem ratione gravitatis; at illa ex adver- so magis resistit ratione motûs, sed minùs ratione gravitatis, servatâ reciprocè eâdem Ratione inter gravitates & motus. Nil igitur mirum si æquatis hinc & hinc viribus agendi, & resisten- di sequatur consistentia. Hinc manifestum est, cur mutatâ figurâ centrum gravitatis ad eam partem transferatur, quæ longiùs à sustentationis vel suspensionis loco recedit; quia nimirum crescunt ex illâ parte comparatè ad oppositam momenta ratione distantiæ majoris, ac proinde, ut fiat momentorum æqualitas, centrum ad illam par- tem secedit. Sic cespitantes à naturâ docentur in partem op- positam illi, in quam inclinantur, brachium illicò extendere, ut brachij gravitas longiùs à corpore translata plus habeat mo- menti, quàm cùm reliquo corpori adhæret, atque hinc sequa- tur centri gravitatis in illam partem translatio. Veritas hæc sa- tis nota est ipsis funambulis, cùm corpus universum super ex- tento fune librant; neque enim temerè crura & brachia exten- dunt aut contrahunt, sed certâ lege, ut centrum momento- rum gravitatis totius corporis hac vel illâ ratione dispositi im- mineat, & incumbat funi. Sic plumbeæ virgæ rectæ ex medio suspensæ, & in æquilibrio manentis, si brachium alterum in- flexeris, fieri non potest, ut reliquum brachium rectum servet positionem horizonti parallelam, sed deorsum inclinabitur, quia cum longius sit brachio inflexo, majora habet momenta ac prævalet. Quod si ob inæqualem virgæ crassitiem non planè ad mediam illius longitudinem facta sit suspensio, sed æquili- britas contingat in puncto, quod propius est crassiori extremiti- tati virgæ, factâ alterutrius brachij inflexione tollitur æquili- brium, quia non jam ampliùs eadem est reciprocè Ratio longi- tudinum, quæ & gravitatum. Ex his pariter consequens est aliquando minimam virtutem satis esse ad dimovenda ab æquilibrio ingentia corpora, si ita sustineantur, ut fulcrum vel in puncto, vel in lineâ contingant: quoniam si corpus grave insistat apici coni, aut pyramidis, aut angulo

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Mechanicorum 248 C E and C D are therefore made into an equilibrium; for the less heavy part, by its position, has a greater tendency to motion, which would be swifter; but the heavier part has a smaller tendency to motion, which would be slower; and thus it resists less in respect of motion, but more in respect of gravity; whereas the other, on the contrary, resists more in respect of motion, but less in respect of gravity, the same ratio between weights and motions being reciprocally preserved. Therefore it is no wonder if, when the forces of acting and resisting are equal on this side and on that, stability follows. Hence it is manifest why, when the shape is changed, the center of gravity is transferred to that part which departs farther from the place of support or suspension; namely because, on that side, the moments increase in comparison with the opposite side on account of the greater distance, and therefore, in order that the moments may be equal, the center recedes toward that side. Thus those who stumble are taught by nature to extend at once the arm toward the side opposite to that toward which they are leaning, so that the weight of the arm, carried farther from the body, may have more moment than when it adheres to the rest of the body, and that hence the center of gravity may be transferred to that side. This truth is well known even to tightrope walkers, when they balance the whole body upon the stretched rope; for they do not rashly extend or contract their legs and arms, but according to a definite rule, so that the center of the moments of the whole body, thus or otherwise disposed, may hang over and rest upon the rope. Thus a straight leaden rod suspended from the middle and remaining in equilibrium, if one arm be bent, it is impossible for the other arm to preserve a straight position parallel to the horizon, but it will incline downward, because since it is longer than the bent arm it has greater moments and prevails. But if, because of the unequal thickness of the rod, the suspension is not made exactly at the middle of its length, but equilibrium occurs at a point nearer the thicker extremity of the rod, then, when one or the other arm is bent, the equilibrium is destroyed, because the same ratio of lengths, and also of weights, no longer remains reciprocally the same. From these things it likewise follows that sometimes the smallest force is sufficient to displace from equilibrium enormous bodies, if they are so supported that the fulcrum touches them either at a point or along a line: for if a heavy body rests on the apex of a cone or pyramid, or on an angle

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Liber secundus. CAPUT III. 249 angulo solido, aut portioni sphæricæ, quam contingat idem corpus sive planâ, sive sphæricè cavâ, sive sphæricam æmulante superficie, contactus in puncto efficitur, ac propterea quacunque in extremitate corporis addatur vis movendi, æquilibrium tollitur, & quidem eò faciliùs, quo magis à puncto contactûs extremitas illa removetur; in illâ quippe distantiâ vis movendi apta velociorem motum efficere, quàm si propior esset, plus habet momenti: Id quod adhuc faciliùs accidit, si ab extremitate, ubi vis movendi applicatur, ductâ per contingentis fulcri punctum rectâ lineâ ad oppositam extremitatem, inæqualiter divisa sit in puncto contactûs, & vis ipsa movendi in magis distante extremitate constituta fuerit; tunc enim non sua tantùm momenta addit, sed illa multiplicat pro Ratione excessûs suæ distantiæ; quemadmodum de inæqualibus libræ brachiis dictum est. Sin autem fulcrum sustinens, quod horizonti parallelum ponitur, sit acies prismatis, aut latus pyramidis jacentis, aut portio cylindrica seu conica jacens; tunc in lineâ fit contactus, si vel plana sit, vel circulariter concava corporis insistentis superficies: sed si vis movendi, quantacumque sit, addatur secundùm rectam lineam, quæ efficit Gravitatis diametrum, puta in A vel N, non mutat æquilibritatem, si fulcrum congruit toti diametro A N: si verò fulcrum brevius est quàm A N, & ex. gr. congruit solùm ipsi A I, jam centrum motûs est I, & oportet vim movendi tantam esse in N, ut aggregatum ex parte MLN ac virtute additâ in N habeat ad partem MAL reliquam majorem Rationem, quàm sit Ratio distantiæ I A ad distantiam I N. Quare in hujusmodi contactu lineari vis movendi, æquilibrium facilè tollens, esse debet ad latus diametri gravitatis, & pro ratione distantiæ majus erit momentum; maximum autem erit momentum in E distantiâ maximâ. Non igitur facilè inter fabulas rejicienda sunt, quæ Atlas Sinicus pag. 32. de Montibus circa urbem Peking loquens ait, Púon mons altissimus ac præruptus varios attollens vertices, in cujus summitate ingens est lapis, qui minimo contactu movetur ac titubat: fieri siquidem potuit, ut lapis ille in infimâ parte excavatus innitatur subjecto saxo, à quo vel in puncto, vel in lineâ tangatur, sicuti dictum est; & cum sit perfectè libratus, modico impulsu tangentis, quâ saltem parte ad illum patet accessus, po- I i

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Book the second. CHAPTER III. 249 in a solid angle, or in a spherical portion, which the same body touches, whether with a plane, or with a surface hollowed spherical-wise, or emulating a sphere, contact is effected in a point; and therefore, in whatever way a moving force be added at one extremity of the body, equilibrium is destroyed, and indeed the more easily, the farther that extremity is removed from the point of contact; for at that distance the moving force, suited to produce a swifter motion than if it were nearer, has greater moment: which happens still more easily if, from the extremity where the moving force is applied, a straight line be drawn through the point of contact of the support to the opposite extremity, and it be unequally divided at the point of contact, and the force itself of motion be placed in the more distant extremity; for then it not only adds its own moments, but multiplies them according to the ratio of the excess of its distance; just as was said of the unequal arms of a balance. But if the supporting fulcrum, placed parallel to the horizon, be the edge of a prism, or the side of a lying pyramid, or a cylindrical or conical portion lying down; then contact is made in a line, if the surface of the body resting upon it be either plane or circularly concave: but if the moving force, whatever it be, be added along the straight line which forms the diameter of gravity, say at A or N, it does not alter the equilibrium, if the fulcrum agrees with the whole diameter A N: but if the fulcrum is shorter than A N, and, for example, agrees only with A I itself, then the center of motion is I, and the moving force must be so great at N that the aggregate from the part M L N and the added virtue at N may have, in relation to the remaining part M A L, a greater ratio than is the ratio of the distance I A to the distance I N. Therefore, in contact of this kind, linear, the moving force, easily destroying equilibrium, ought to be at the side of the diameter of gravity; and according to the ratio of the distance the moment will be greater; but greatest of all will be the moment at the farthest distance E. Therefore those things are not easily to be rejected among fables, which the Chinese Atlas, p. 32, speaking of the mountains around the city of Peking, says: Púon, a very lofty and precipitous mountain, raising up various summits, on whose top there is a huge stone, which is moved and totters at the slightest touch: for it could well have happened that that stone, hollowed out in its lower part, rests upon a subjacent rock, by which it is touched either at a point or in a line, as was said; and since it is perfectly balanced, by a slight impulse of one touching it, from whatever side access to it lies open, po- I i

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Mechanicorum 250 test ab æquilibrio dimoveri: quòd si usquequaque circum- obeundo lapidem quâcumque in parte tangatur, sequitur illius trepidatio, signum est contactum subjecti fulcri esse in puncto. Simili ratione explicanda sunt, quæ idem Atlas Sinicus in XI Provincia Fokien habet pag. 125, ubi ait, Versùs Urbis Changcheu Orientalem partem mons est Cio dictus, in quo lapidem esse scribunt altum perticas quinque, crassum decem & octo, qui quo- ties tempestas imminet, titubat omninò, ac movetur: hic enim la- pis in perfecto æquilibrio constitutus suprà fulcrum, à quo in puncto, vel in lineâ tangatur, & fortasse etiam ab eodem fulcro distinctus in longitudines inæquales, violento impulsu hali- tuum aut infernè subeuntium, aut ex superiore nubium parte obliquè reflexorum, facilè moveri potest ac titubare, si extre- mitas à fulcro remotior impellatur. Et quoniam de Sinensibus mentio incidit, non injucundum fuerit hîc aliud addere pertinens ad eorum industriam in ser- vando æquilibrio. Idem Atlas Sinicus, cum sermo est de Pro- vincia Peking, ubi solum esse arenosum atque planissimum testatur, hæc habet pag. 28. Modus itineris faciendi hisce locis non infrequens, nec incommodus est. Plaustrum adhibent cum unâ rotâ ita constitutum, ut uni illius medium occupandi, & quasi equo insidendi sit locus, aliis duobus ab utroque latere adsidentibus; auri- ga plaustrum retro ligneis vectibus urget ac promovet non securè mi- nùs, quàm velociter. Si rem conjecturis indagare liceat, ego ro- tam concipio ita inclusam ligneo loculamento majoris segmen- ti circuli figuram habente, ut huic insitus sit rotæ axis, ad dex- tram autem & ad lævam extantia tabulata tantæ latitudinis, ut quis modò propè rotam, modò longiùs adsidere queat ad æquilibrium constituendum inter duos viatores inæqualiter graves: Aurigæ locus est in suprema parte loculamenti, cui quasi equitans insidet, binosque contos, seu vectes concinnè locatos, ut manubrium ante se habeat, extremitas altera (for- tassè in acumen desinens, ut leviter solo infigatur) post se ter- ram respiciat, utrâque manu apprehendens solum obliquè pre- mit, & currum in anteriora velociter promovet. Id quod nemi- ni difficile videatur, qui sæpiùs observaverit à puero fabri lignarij aut ferrarij rotam curulem identidem impulsam per urbis vias velociter deduci; quæ dum impresso impetu veloci- ter

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Mechanics 250 be displaced from its equilibrium test: but if, by going all around it, the stone is touched in whatever part, and its trembling follows, it is a sign that the contact of the supporting point is at a single point. In a similar way must be explained what the same Chinese Atlas says in the 11th Province of Fokien, p. 125, where he says: Toward the eastern part of the city of Changcheu there is a mountain called Cio, in which they write that there is a stone five poles high and eighteen thick, which, whenever a storm threatens, trembles greatly and moves: for this stone, being set in perfect equilibrium upon a support, may easily be moved and made to tremble by violent impulses of air either rising from below or obliquely reflected from the upper part of the clouds, if the end farther from the support be struck. And since mention has fallen upon the Chinese, it will not be unpleasant here to add another thing concerning their skill in preserving equilibrium. The same Chinese Atlas, when speaking of the Province of Peking, where he states the ground to be sandy and very level, has the following on p. 28: The mode of traveling in these regions is not uncommon, nor inconvenient. They use a cart with one wheel so arranged that there is a place for one person to occupy the middle of it, and as it were to sit astride, with two others seated on either side; the driver pushes and propels the cart backward with wooden poles, not less safely than quickly. If I may infer the matter by conjecture, I imagine the wheel enclosed in a wooden box having the shape of a larger segment of a circle, so that the axle of the wheel is set in this box, while on the right and left there are projecting platforms of such breadth that one may sit now near the wheel, now farther away, in order to establish equilibrium between two travelers of unequal weight. The driver's place is on the upper part of the box, on which he sits as if riding, holding neatly arranged two poles or shafts, so that he has the handle before him; the other end, perhaps ending in a point so that it may be lightly fixed in the ground, faces backward toward the earth. Seizing both with his hands, he presses obliquely upon the ground and quickly drives the cart forward. This would seem difficult to no one who has often observed how, from boyhood, the smith’s or carpenter’s wheelbarrow, repeatedly pushed along the streets of a city, is swiftly brought along; which, while driven by the force imparted to it, swiftly

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Liber secundus. CAPUT III. 251 tet conversa in anteriora promovetur, licet huc atque illuc nutabunda inclinetur, ob velocem conversionem immunis est à casu: quemadmodum etiam stanneum aut argenteum orbem apici cultri impositum, si in gyrum velociter agatur, à casu im- munem videmus, etiamsi punctum sustentationis non exactissi- mè centro respondeat. Sic aliquis suppositam sphærulam altero pede, etiam summis digitis premens, celeriter in gyrum totum corpus contorquet, qui non ita facilè citrà cadendi periculum eidem sphærulæ insistens quietus consisteret; ipsâ nimirum conversionis celeritate gravitatis propensionem eludente. Non absimili igitur ratione in hujusmodi rotæ Sinici plaustri conver- sione veloci deteritur, quicquid in alterutram partem inclinatio- nis oriretur vel ex modicâ viæ inæqualitate, vel ex æquilibrio non adeò exactè servato, ut etiam consistente plaustro insiden- tes viatores consisterent æqualiter librati absque alicujus artifi- cij subsidio: Quod artificium in promptu esse non dubito; ne- que enim Sinenses ita sibi præsidentes existimo, ut aliquâ ratio- ne sibi non præcaveant à periculo casûs, si fortè rotâ in obicem incurrente plaustrum seu loculamentum in anteriorem, aut in posteriorem partem improvisâ inclinatione convertatur. Sed singula persequi nec otium est, nec operæ pretium: quapropter generatim dicendum corporis æquilibrium ibi fieri, ubi in duas partes ita distinguitur, ut illarum gravitates sint reciprocè in Ratione longitudinum seu distantiarum à puncto suspensionis seu sustentationis, quemadmodum in librâ dictum est. Quare si tota moles proposita eâdem gravitatis specie prædita fuerit, nec facile sit in illâ centrum gravitatis invenire, quia nimis irregu- laris est, distingue illam in duas partes, & singularum inventa centra gravitatis junge rectâ lineâ, quæ quasi libræ jugum divi- datur in reciprocâ Ratione illarum partium; est enim punctum illud, in quod cadit divisio, punctum æquilibrij, & centrum gra- vitatis totius. Sic Trapezij, NPMQ in- venies punctum æquilibrij, si duorum triangulorum NQM, NPM, in quæ di- viditur, singularia centra gravitatis inve- nias O & B: hæc jungantur rectâ OB; tum fiat ut triangulum NQM ad trian- gulum NPM, ita reciprocè BD ad DO,

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Book second. CHAPTER III. 251 then, turned forward, it is carried on, although it inclines now this way now that and wavers, it is, by reason of its rapid turning, immune from falling: just as we see a pewter or silver disc placed on the point of a knife, if it be swiftly whirled in a circle, immune from falling, even though the point of support does not correspond exactly to the center. Thus someone, placing a small sphere beneath one foot and even pressing it with the tips of the toes, quickly twists the whole body around in a circle; he would not so easily stand still upon that same sphere without danger of falling, since the very speed of the turning eludes the tendency of gravity. Not unlike this, therefore, in the turning of this sort of Chinese cart’s wheel is worn away whatever would arise as an inclination to either side, whether from a slight inequality of the road, or from the balance not being preserved with sufficient exactness, so that even when the cart is standing still, the passengers seated in it would remain evenly poised, without the aid of any device: a device which, I do not doubt, is ready at hand; for I do not think the Chinese are so confident in themselves that they would not in some way take precaution against the danger of falling, if by chance the wheel, striking an obstruction, the cart or compartment should be turned by an unexpected inclination toward the front or toward the rear. But to go through each point in detail there is neither leisure nor is it worth the trouble: wherefore it must be said in general that the equilibrium of a body takes place there where it is divided into two parts in such a way that their weights are reciprocally in the ratio of the lengths or distances from the point of suspension or support, as was said in the balance. Therefore, if the whole mass proposed is endowed with the same kind of gravity, and it is not easy to find in it the center of gravity, because it is too irregular, divide it into two parts, and join the centers of gravity of each, once found, by a straight line, which, like the beam of a balance, is divided in the reciprocal ratio of those parts; for that point into which the division falls is the point of equilibrium, and the center of gravity of the whole. Thus in the trapezium NPMQ you will find the point of equilibrium, if you find the individual centers of gravity O and B of the two triangles NQM and NPM, into which it is divided: let these be joined by the straight line OB; then let it be as the triangle NQM to the triangle NPM, so reciprocally BD to DO,

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Mechanicorum & est D punctum æquilibrij, seu centrum gravitatis Trapezij quæsitum. At si Trapezio addatur triangulum NLP ejusdem specificæ gravitatis, emergit Pentagonum irregulare LPMQN: inveniatur additi trianguli centrum singulare gravitatis A, & jungatur recta AD; tùm fiat ut Trapezium ad triangulum ad- ditum, ita reciprocè AS ad SD, & est punctum S centrum commune gravitatis totius Pentagoni, in quo fit æquilibrium; perinde enim est ac si in jugo libræ AD inæqualiter distributæ appenderetur ex A quidem triangulum NLP; ex D verò Tra- pezium NQMP, quæ in illis distantiis à centro motûs æqualia haberent momenta. Quòd si tota moles proposita constet partibus non ejusdem specificæ gravitatis, non jam satis est invenisse singularia cen- tra, ut ducatur jugum libræ illa connectens, & notam esse Ra- tionem molis ad molem; sed præterè opus est notam habere Rationem gravitatis specificæ ad gravitatem specificam; quia Ratio gravitatum absolutarum componitur ex Rationibus quantitatum, & gravitatum secundùm speciem. Quamobrem si additum triangulum habeat specificam gravitatem majorem gravitate specificâ Trapezij, quia hoc lignetum est, illud fer- reum, non cadet in S punctum æquilibrij, sed accedet ad punctum A, quia factâ hujusmodi Rationum compositione, minor est inæqualitas gravitatum absolutarum; si enim Trape- zium excedit mole Triangulum, cedit illi specificâ gravitate. Ponamus namque Rationem molis Trapezij ad molem Trian- guli esse ut 7 ad 2; specificæ verò gravitatis Rationem ut 5 ad 42, gravitas absoluta Trapezij lignei est ut 35, gravitas Trian- guli ferrei ut 84: sunt igitur gravitates in Ratione 5 ad 12: di- vidatur itaque jugum AD in I reciprocè, ut sit AI 5, ID 12, & erit I centrum gravitatis compositæ, ac punctum æquilibrij, quia ab illo inæquales gravitates habent suas distantias in Ra- tione reciprocâ ipsarum gravitatum. Eadem est in corporibus omnibus Ratio, & methodus deprehendendi punctum æqui- librij, seu centrum gravitatis, per quod deinde duci potest dia- meter gravitatis, ut fiat opportuna suspensio. Quia tamen aliquando evenit suspensum corpus aut susten- tatum, dum positionem horizonti parallelam servare contendit, aliquod incommodum subire in motu corporis, cui innititur; proptereà

Transcription: Translated (English)

Mechanics and D is the point of equilibrium, or the center of gravity of the trapezium sought. But if to the trapezium there is added the triangle NLP of the same specific gravity, there arises the irregular pentagon LPMQN: let the single center of gravity A of the added triangle be found, and let the straight line AD be joined; then as the trapezium is to the added triangle, so inversely is AS to SD, and S is the common center of gravity of the whole pentagon, in which equilibrium is established; for it is just as if on the balance beam AD, unevenly distributed, there were hung from A the triangle NLP, and from D the trapezium NQMP, which at those distances from the center of motion would have equal moments. But if the whole proposed body consists of parts not of the same specific gravity, it is no longer enough to have found the individual centers, in order to draw the balance beam connecting them, and to know the ratio of mass to mass; but in addition it is necessary to know the ratio of specific gravity to specific gravity; because the ratio of absolute weights is compounded from the ratios of quantities and of specific gravities. Wherefore if the added triangle have a specific gravity greater than the specific gravity of the trapezium, because this is wooden and that iron, it will not fall at the point S of equilibrium, but will move toward the point A, because, when such a composition of ratios is made, the inequality of the absolute weights is less; for if the trapezium exceeds the triangle in mass, it yields to it in specific gravity. Let us suppose, namely, the ratio of the mass of the trapezium to the mass of the triangle to be as 7 to 2; but the ratio of specific gravity as 5 to 4; the absolute weight of the wooden trapezium is as 35, the weight of the iron triangle as 84: therefore the weights are in the ratio of 5 to 12. Let the beam AD therefore be divided at I inversely, so that AI is 5, ID 12, and I will be the center of gravity of the composite body, and the point of equilibrium, because from that point the unequal weights have their distances in the reciprocal ratio of those same weights. The same ratio and method holds for all bodies for detecting the point of equilibrium, or center of gravity, through which afterward the line of gravity may be drawn, so that a suitable suspension may be made. Because, however, it sometimes happens that a suspended or supported body, while striving to maintain a position parallel to the horizon, undergoes some inconvenience in the motion of the body on which it rests; therefore

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Liber tertius. CAPUT III. 253 proptereà huic occurrendum est artificio, quo situm eumdem perpetuò servet. Rem exemplo declaro. In pyxide nauticâ in- sistit cuspidi acus magnetica æqualibus momentis librata, ut horizonti parallela jaceat, quamcumque in partem dirigatur. Si alicui navis plano pyxis ipsa adhæreret ita, ut infimâ sui par- te illi congrueret, quamcumque in partem navis inclinaretur, ipsum pariter pyxidis fundum inclinari manifestum est, & alte- ri acûs magneticæ positionem horizonti parallelam servantis extremitati occurrens illius motum impediret, aut saltem retar- daret. Ut igitur semper pyxis tùm acui magneticæ, tùm hori- zonti parallela consistat, suspendenda fuit, non quidem funi- culo, ne incertis motibus jactaretur, sed duobus polis, super quibus opportunè versaretur æqualiter librata. Verùm duobus hisce polis non tollitur omne incommodum; si etenim poli respiciant navis latera, elevatâ aut depressâ prorâ juvant, sed navi in dextrum aut in sinistrum latus inclinatâ, alter deprime- retur, alter elevaretur, nisi & ipsi infigerentur circulo super alios polos proram & puppim respicientes versatili. Sit pyxis ipsa ABCD, in qua venti des- cripti sint, & in centro O acus magnetica volubilis insistat: py- xidem circulus E I F H com- plectatur, cui poli D & B facilè versatiles infigantur, ut inclinatâ navi in A vel in C pyxis horizon- ti parallela maneat; & ut eumdem parallelismum servet, etiam si na- vis in B aut D inclinetur, circu- lus ille E I F H duos pariter polos facilè versatiles habeat in E & F externæ pyxidi immobili infixos: hac enim ratione fiet, ut in quacumque navis inclinatione pyxis nautica à suo parallelismo & æquilibrio non recedat. Hoc eodem artificio construitur lucerna ferreo aut æneo globo inclusa multipliciter perforato, ut fumo exitus pateat, quæ citrà effusionem olei in solo rotata non extinguitur; est si- quidem vasculum plumbeum, ut sua gravitate securiùs deor- sum vergat, polis versatilibus suspensum in circulo, qui pariter 1 i 3

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Book Three. CHAPTER III. 253 therefore this difficulty must be met by an artifice, by which it may constantly preserve the same position. I explain the matter by an example. In a nautical box there is placed upon the point of a magnetic needle, balanced by equal moments, so that it lies parallel to the horizon, however it may be directed. If the box itself were attached to the flat deck of a ship in such a way that its lower part corresponded with it, it is evident that, in whatever direction the ship might incline, the bottom of the box itself would likewise incline; and, encountering the end of the magnetic needle which preserves a position parallel to the horizon, it would hinder, or at least retard, its motion. Therefore, in order that the box may always stand parallel both to the magnetic needle and to the horizon, it had to be suspended, not indeed by a cord, lest it should be tossed about by uncertain motions, but by two poles, upon which, being equally balanced, it might conveniently turn. Yet even by these two poles every inconvenience is not removed; for if the poles face the sides of the ship, when the prow is raised or lowered they assist, but if the ship inclines to the right or to the left, one would be depressed and the other raised, unless they too were fixed in a ring turning upon other poles that face the prow and the stern. Let the box itself be ABCD, in which the winds are marked, and in the center O let the revolving magnetic needle stand: let the ring E I F H encompass the box, to which poles D and B are easily fixed so as to turn, in order that when the ship inclines toward A or toward C the box may remain parallel to the horizon; and in order that it may preserve the same parallelism even if the ship inclines toward B or D, let that ring E I F H likewise have two easily turning poles in E and F, fixed into the immovable outer box: for by this means it will come about that, in whatever inclination of the ship, the nautical box does not depart from its parallelism and equilibrium. By this same contrivance is constructed a lamp enclosed in an iron or bronze globe, perforated in many places so that smoke may find an exit, which, when turned about without spilling the oil on the floor, is not extinguished; for it is a small leaden vessel, that it may more securely lean downward by its own weight, suspended by movable poles in a ring, which likewise

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Mechanicorum 254 polos inferit secundo circulo, secundus similiter tertio, tertius demum scaphio, seu inferiori hemisphærio globi, cui includitur, eâ dispositione, ut quemadmodum pyxidis nauticæ hîc descriptæ ambitus in quatuor partes distinguitur à polis, ita lucernæ hujus ambitus in octo partes à polis distribuatur, atque proinde facilior sit globi in omnem partem volutatio citrà periculum inclinationis vasculi oleum cum ellychnio continentis. Nec pluribus opus est hîc explicare, quàm proclive sit artificium hoc ad plura traducere, quorum usus est in plano horizontali, ne libellâ semper & normâ indigeamus, ut illa ritè collocentur: ut si horologium horizontale statuendum sit quo cumque in plano, sit illud pyxidi inclusum cum circulo, quemadmodum de pyxide nauticâ dictum est: si lectulum viatorium in rhedâ sternere oporteat, in quo citrà jactationem, etiam viâ salebrosâ, quiescere liceat, ferreo parallelogrammo complectere lectulum ex polis suspensum circâ medium eo loco, ut corpus in lectulo jacens sit horizonti parallelum, ipsum verò parallelogrammum polis rhedæ infixis & versatilibus ad caput & ad pedes suspendatur: & alia hujusmodi, quæ facilè pro rerum opportunitate excogitari possunt. Verùm quàm facilè est super polos in æquilibrio constituere corpora gravitatis centrum habentia vel in ipsâ sustentationis lineâ, vel infrà illam, tam multis difficultatibus implicitum opus est in æquilibrio statuere corpus, cujus gravitatis centrum in parte superiori reperitur, & quidem maximè si multùm inde removeatur; tunc enim sufficit vel minima inclinatio, ut totum corpus revolvatur, cum ex alterâ parte sint plura gravitatis momenta, quàm in oppositâ. Nam si corpus BC, cujus centrum gravitatis sit A, suspendatur super polis in I, quando axi sustentanti ad perpendiculum respondet centrum gravitatis A, manet æquilibrium, sed factâ corporis inclinatione, ut A recedat à perpendiculo, jam versùs C plures sunt partes gravitatis descendentes, quàm versùs B sint partes ascendentes, & illæ velociùs moventur deorsum, quàm hæ sursum; quapropter illæ majora

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Mechanics 254 places the poles of the second circle, the second likewise into the third, the third at last into the lower hemisphere of the globe, in which it is enclosed, with this arrangement, that just as the circumference of the nautical box here described is divided by the poles into four parts, so the circumference of this lamp is distributed by the poles into eight parts, and therefore the rolling of the globe in every direction is more easy, without danger of tilting the vessel containing the oil with the wick. Nor is it necessary here to explain at greater length how readily this device may be extended to many other uses on a horizontal plane, so that we may not always need a plumb line and a level, in order that they may be properly placed: as if a horizontal clock were to be set up on any plane whatsoever, let it be enclosed in a box with a circle, just as has been said of the nautical box: if a travelling bed is to be spread in a carriage, in which one may rest without jolting, even on a rough road, enclose the bed, suspended from poles about the middle, in an iron parallelogram, in such a place that the body lying in the bed may be parallel to the horizon, and suspend the parallelogram itself, with poles fixed and movable in the carriage, at the head and at the feet: and other things of this kind, which can easily be devised according to the convenience of circumstances. But just as easy as it is to place in equilibrium upon poles bodies that have their center of gravity either on the very line of support or below it, so much the more difficult is the task of placing in equilibrium a body whose center of gravity is found in the upper part, and especially if it is removed a great distance from that line; for then even the slightest inclination is enough for the whole body to overturn, since on one side there are more moments of gravity than on the opposite side. For if a body BC, whose center of gravity is A, be suspended on poles at I, when the supporting axis corresponds perpendicularly to the center of gravity A, equilibrium is maintained; but when the body is inclined, so that A moves away from the perpendicular, then toward C there are more descending parts of gravity than toward B there are ascending parts, and those move downward more quickly than these upward; wherefore those exert greater

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Liber tertius. CAPUT III. 255 majora habent momenta, quibus deorsum urgentibus corpus revolvitur. Id quod multò magis contingit in Acrobarycis, quæ nimirum gravitatem in summitate habent, ut si corpori BC in superiori parte adnexa esset pyramis D; cum enim totius com- positæ molis ex solido BC, & pyramide D, centrum commu- ne gravitatis non esset in A, sed adhuc superius procul à polo I, qui est centrum motûs, factâ levi inclinatione multo plus gravitatis esset ex parte C, quàm ex oppositâ B, ut constat: nam quò altius & remotius est centrum gravitatis, eò faciliùs linea directionis cadit extra punctum vel lineam sustentationis, factâ pari inclinatione. Liceat autem hîc obiter, quasi corollarij loco, attingere æquilibria corporum humido insidentium, & Acrobarycorum fluitantium, in quibus pariter Rationes libræ agnoscentur, si rectè perpendatur, ubi fiat sustentatio. In omni igitur corpo- re fluitante duplex pars consideranda est, & quæ intrâ humi- dum mergitur, & quæ in aëre extat: illa quidem utpote secun- dùm speciem minùs gravis, quàm humor, levitat, hæc verò aëre gravior gravitat: Quare & illa suum habet centrum levi- tatis, & hæc centrum gravitatis; nec posset corpus datam posi- tionem servare, nisi in eâdem lineâ perpendiculari ad universi centrum tendente esset utrumque centrum & levitatis & gra- vitatis; cumque par sit virtus ascendendi virtuti descendendi, neutrâ prævalente, & sibi vicissim utrâque obsistente, consistit corpus. Quòd si non in eodem perpendiculo sit utrumque centrum, utrumque suâ viâ pergere potest, illud ascendendo, hoc descendendo. Sic baculum rectum in aquam immittens, manûque retinens, ne in alterutram partem inclinetur, mergi quidem illum videbis pro Ratione specificæ suæ gravitatis, quæ minor est specificâ gravitate aquæ, sed erectus non manebit, nisi quandiu retinueris; nam ubi illum dimiseris, statim cen- trum gravitatis descendet, & levitatis centrum ascendet, quia vel exiguus aquæ motus partem immersam inclinans satis est, ut centra illa non eidem perpendiculo respondeant; ac prop- terea demùm baculus jacens innatabit. Quiescente igitur corpore in humoris superficie, mani- festum est centrum gravitatis partis extantis in eodem perpen- diculo esse cum centro levitatis partis demersæ. Quare si ligneum

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Book three. CHAPTER III. 255 have greater force, by which, when pressing downward, the body is turned over. This occurs much more in Acrobarycs, which, namely, have their heaviness at the top, as if a pyramid D were attached to the upper part of body BC; for since the common center of gravity of the whole composite mass, made up of the solid BC and the pyramid D, would not be in A, but still higher, far from the pole I, which is the center of motion, then, with a slight inclination made, there would be much more weight on the side C than on the opposite B, as is evident: for the higher and more remote the center of gravity is, the more easily does the line of direction fall outside the point or line of support, when an equal inclination is made. Here, however, let us be allowed incidentally, as it were by way of a corollary, to touch upon the equilibria of bodies resting in a liquid, and of floating Acrobarycs, in which likewise the reasons of the balance will be recognized, if it is rightly considered where the support is made. In every floating body, therefore, a twofold part is to be considered: both that which is immersed within the liquid, and that which is above the air. The former, indeed, as being by nature less heavy than the liquid, has levity; the latter, however, as heavier than the air, has gravity. Wherefore that part has its center of levity, and this its center of gravity; nor could the body preserve the given position unless both centers, both of levity and of gravity, were on the same perpendicular line tending toward the center of the universe; and since the force of ascending is equal to the force of descending, neither prevailing, and each resisting the other in turn, the body remains at rest. But if both centers are not on the same perpendicular, each can proceed by its own path, the one ascending, the other descending. Thus, if you immerse a straight stick in water and hold it with your hand so that it does not incline to either side, you will indeed see it sink in proportion to its specific gravity, which is less than the specific gravity of water, but it will not remain upright unless you keep holding it; for when you release it, the center of gravity immediately descends and the center of levity ascends, because even a slight motion of the water inclining the immersed part is enough for those centers no longer to correspond to the same perpendicular; and therefore at length the stick, lying down, will float. Therefore, when a body is at rest on the surface of a liquid, it is manifest that the center of gravity of the part projecting above is on the same perpendicular as the center of levity of the submerged part. Wherefore if a wooden

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Mechanicorum 256 ligneum prisma A C aquæ imponatur, & immergatur ita, ut pars demersa & levitans sit E C, pars verò extans in aëre & gravitans sit A F, centrum gravi- tatis est G, centrum levitatis est H, quæ sibi directè adversantia in oppositas partes conantur æqualibus viribus, atque prop- terea nullus sequitur motus. Quòd si aut H recederet versùs D, aut G versùs B, & hoc posset descendere, & illud ascendere neutro contranitente. Iam verò quiescenti prismati imponatur aliquod pon- dus, certum est partem in aëre extantem, conflatam ex parte prismatis & ex addito pondere, graviorem esse, ac proinde prævalere viribus partis in aquâ levitantis, illam- que deprimere, quoadusque fiat æqualitas inter levitatem & gravitatem. Sed multùm interest, utrum additi pon- deris centrum gravitatis in eodem perpendiculo sit cum cen- tro gravitatis G, ut rectâ deprimatur prisma infrà superfi- ciem aquæ; an verò sit extrà illud perpendiculum; id quod si accidat, commune centrum gravitatis transfertur ver- sus A, aut B. Sit ex. gr. ad partes A propè S; cumque non immineat puncto H centro levitatis, descendit prisma ad partes A, & opposita pars ascendit, ita ut E deprimatur infrà superfi- ciem aquæ, F verò emergat. Sed dum ad partes C F prisma emergit ex aquâ, ad partes autem D E deprimitur, centrum levi- tatis non manet in H, sed ad majorem partem depressam secedit, donec fiat V, atque in eodem perpèdiculo sit cum centro gravi- tatis S; & tunc quiescit prisma, nec amplius demergitur in E, aut emergit ex F. Sustinetur itaque centrum gravitatis S à cen- tro levitatis V, & vicissim centrum levitatis V retinetur à cen- tro gravitatis S; & fit tùm inter gravitates, tùm inter levitates æquilibrium, quia gravitas in A major minùs distat à puncto, vel potius à lineâ sustentationis factâ à plano transeunte per V, & gravitas in B minor magis distat; ideóque neutra prævalet: & similiter levitas in D E major minùs distat à lineâ detentio- nis facta à plano transeunte per S, ac levitas minor in C magis distat;

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Mechanicorum 256 Let the wooden prism AC be placed upon the water and immersed in such a way that the part submerged and floating is EC, while the part extending into the air and weighing is AF; the center of gravity is G, the center of buoyancy is H, which, being directly opposed to one another, strive toward opposite parts with equal forces, and therefore no motion follows. For if either H were to move away toward D, or G toward B, and this one could descend and that one ascend, neither resisting the other. Now indeed, upon the prism at rest let some weight be placed; it is certain that the part extending into the air, composed of the prism itself and the added weight, is heavier, and therefore prevails over the force of the part floating in the water and presses it down, until equality between buoyancy and weight is achieved. But it makes a great difference whether the center of gravity of the added weight lies on the same perpendicular as the center of gravity G, so that the prism is pressed straight down beneath the surface of the water; or whether it lies outside that perpendicular; which if it happen, the common center of gravity is shifted toward A or B. Let it be, for example, toward A near S; and since it does not hang over the point H, the center of buoyancy, the prism descends toward A, and the opposite part rises, so that E is pressed beneath the surface of the water, while F emerges. But while the prism emerges from the water toward the parts CF, and is pressed down toward the parts DE, the center of buoyancy does not remain in H, but moves away to the greater depressed part, until it becomes V, and is on the same perpendicular as the center of gravity S; and then the prism comes to rest, nor is it submerged any further in E, nor does it emerge from F. Thus the center of gravity S is sustained by the center of buoyancy V, and conversely the center of buoyancy V is retained by the center of gravity S; and an equilibrium is established both between weights and between buoyancies, because the weight in A, being greater, is less distant from the point, or rather from the line of support made by the plane passing through V, and the weight in B, being smaller, is more distant; and therefore neither prevails: and likewise the buoyancy in DE, being greater, is less distant from the line of retention made by the plane passing through S, and the smaller buoyancy in C is more distant;

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Liber tertius. CAPUT III. 257 distat; quare vis tardiùs ascendendi major prævalere non po- test minori virtuti repugnanti ad descendendum velociùs. Quemadmodum verò si tantum ponderis adderetur in A, ut centrum commune gravitatis non posset imminere centro levi- tatis partis demersæ, nemo non intelligit futuram omnimodam depressionem partis A infrà superficiem aquæ, & omnimodam emersionem oppositæ partis C; ita in Acrobarycis fluitantibus manifestum est, quò altiùs attollitur gravitas, eò faciliùs factâ inclinatione transferri commune centrum gravitatis ultrà per- pendiculum, in quo est centrum levitatis partis demersæ. Sic si justo longior sit in navi malus, factâ ex fluctibus inclinatione in latus, aut saltem impulsu venti suprema carbasa implentis, facilis erit navis submersio, quia plus momentorum gravitatis est ex alterâ parte, quàm ex oppositâ, translato in navis latus, aut ultra illud, centro gravitatis totius partis extantis in aëre. Sed de his, Deo dante, pleniùs in Hydrostaticis differendum erit, ubi ostendetur ad navium stabilitatem necessariam esse eam centrorum dispositionem, ut centrum gravitatis totius na- vis cum omnibus impositis sit infrà centrum levitatis partis de- mersæ in eodem perpendiculo, in quo pariter erit centrum gra- vitatis partis extantis. CAPUT IV. An, & cur libra ab æquilibrio dimota ad illud redeat. Nemini dubium esse potest æquilibrium tolli ob momento- rum gravitatis inæqualitatem, vel quia in unâ libræ æqui- libris lance additum est pondus, vel quia altera jugi extremitas, alicujus elevantis aut deprimentis vi, recedit à positione horizonti parallelâ. Illud in quæstionem revocari potest, an sublato ponderis excessu, aut cessante impulsu extrinseco, li- bra redeat ad æquilibrium, & positionem horizonti parallelam sibi ipsa restituat. Certè Keplerus in Astronomiâ Opticâ cap. 1. K k

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Liber tertius. CAPUT III. 257 distant; wherefore the force of rising more slowly cannot prevail against the lesser force tending to descend more quickly. For just as if so much weight were added in A that the common center of gravity could not fall below the center of buoyancy of the submerged part, everyone understands that the part A would then be entirely depressed below the surface of the water, and the opposite part C entirely raised out; so in floating Acrobarycs it is evident that, the higher the weight is raised, the more easily, once an inclination has been made, can the common center of gravity be transferred beyond the perpendicular, in which is the center of buoyancy of the submerged part. Thus, if a mast in a ship be longer than is proper, then, when the ship has been inclined to one side by waves, or at least by the force of the wind filling the sails at the top, the ship will sink easily, because there are more moments of gravity on one side than on the opposite, after the center of gravity of the whole part projecting into the air has been moved to the side of the ship, or beyond it. But concerning these matters, God willing, a fuller discussion must be made in Hydrostatics, where it will be shown that for the stability of ships there is needed such a disposition of centers, that the center of gravity of the whole ship with all its load is below the center of buoyancy of the submerged part in the same perpendicular, in which likewise will be the center of gravity of the part projecting above water. CAPUT IV. Whether, and why, a balance, once moved from equilibrium, returns to it. No one can doubt that equilibrium is destroyed because of the inequality of the moments of gravity, either because a weight has been added to one of the equal pans of the balance, or because one end of the beam, by some force lifting or depressing it, departs from a position parallel to the horizon. What may be questioned is whether, once the excess of weight has been removed, or the external impulse has ceased, the balance returns to equilibrium and restores of itself the position parallel to the horizon. Certainly Kepler, in Optics of Astronomy, chapter 1. K k

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Mechanicorum prop. 20. asserit eum, qui negat libram brachiorum æqualium ad horizontis æquilibrium redituram, non antiquitati tantùm, sed rerum naturæ, sed utilitati generis humani bellum indicere. At ex adverso Authores ferè omnes, qui de his accuratiùs scripserunt, triplicem libræ speciem distinguentes unam tantummodo agnoscunt, quæ se restituat horizonti parallelam. Hoc si- quidem tanquam certum assumunt, corpus quodcumque grave, quod suspensum, aut sustentatum liberè in aëre pendeat, in eò tantum situ quiescere, in quo gravitatis centrum cum suspensionis aut sustentationis puncto in eâdem directionis lineâ reperiatur; descendit enim quantum potest, neque ei opponitur punctum suspensionis aut sustentationis, nisi in eodem perpendiculo ad universi centrum ducto utrumque sit. Cum itaque libra sit corpus grave suspensum, & suum habeat centrum gravitatis, tunc demùm quiescet, ubi eam positionem obtinuerit, in quâ suspensionis punctum, & gravitatis centrum in eâdem sint directionis lineâ. Punctum verò supensionis libræ non illud hîc intelligitur, ex quo pendet ansa, cui libra inseritur, sed ipsa Agina, seu spartum, ut Aristotelico vocabulo utar, est suspensionis punctum; ex illo enim proximè libra suspenditur. Hinc oritur triplex libræ species, quia tripliciter componi possunt centrum motûs, & centrum gravitatis; primò scilicet possunt in uno eodemque puncto convenire, deinde centrum motûs potest esse superius, demum inferius centro gravitatis. Et quidem si unum idemque punctum sit motûs & gravitatis centrum A, & æqualibus brachiis AB, A C æqualia sint adnexa pondera B & C, utique æquilibrium horizontale manet, propter momentorum æqualitatem tùm ratione gravitatum æqualium, tùm ratione æqualium propensionum ad motum. Si igitur applicatâ manu in B deprimatur libra, ut sit DE; amotâ manu, cur redeat libra ad priorem positionem BC? adhuc enim momenta utrinque sunt æqualia, & tantumdem ascendere deberet D, quantum descenderet E: par igitur est resistentia

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Mechanicorum prop. 20 asserts that anyone who denies that a balance with equal arms will return to horizontal equilibrium is waging war not only on antiquity, but on the nature of things, and on the usefulness of the human race. But, on the other hand, almost all authors who have written more accurately about these matters, distinguishing a threefold species of balance, acknowledge only one that restores itself parallel to the horizon. Indeed, they assume as certain that any heavy body, when suspended or supported, hangs freely in the air, and can rest only in that position in which the center of gravity and the point of suspension or support are found on the same line of direction. For it descends as far as it can, and the point of suspension or support opposes it only if both are on the same perpendicular drawn to the center of the universe. Since, therefore, a balance is a heavy body suspended, and has its own center of gravity, it will then at last be at rest when it has obtained that position in which the point of suspension and the center of gravity are on the same line of direction. Now by the point of suspension of the balance is not here understood that from which hangs the loop into which the balance is inserted, but the Agina itself, or the beam, to use the Aristotelian term, is the point of suspension; for the balance is suspended from that part immediately. Hence arises a threefold species of balance, because the center of motion and the center of gravity can be combined in three ways: first, they can meet in one and the same point; then the center of motion can be above the center of gravity; and finally, below it. And indeed, if one and the same point of motion and gravity be A, and if, with equal arms AB, AC, equal weights B and C are attached, horizontal equilibrium certainly remains, because of the equality of the moments, both by reason of the equal weights and by reason of the equal tendencies to motion. If therefore, with the hand applied at B, the balance is depressed so that it becomes DE; when the hand is removed, why does the balance return to its former position BC? For the moments are still equal on both sides, and D ought to rise as much as E descends; therefore the resistance is equal.

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Liber tertius. CAPUT IV. 259 resistentia ipsius D propensioni ad motum ipsius E: neutro ita- que prævalente fiet in eo situ DE consistentia. Attamen huic argumentationi, quamvis legitimæ, non ac- quiescunt nonnulli, qui libram hujusmodi in quâcumque posi- tione quiescentem se visuros desperant, quia nunquam vide- runt: quare potiùs causam inquirunt, cur ad æquilibrium re- deat libra æqualium brachiorum, quamvis ex medio jugo sus- pendatur. Existimant aliqui posse vim argumenti eludi, si con- cedant quidem in uno eodemque puncto convenire centrum motûs & centrum gravitatis jugi, non tamen libræ: nam si præter jugum assumantur etiam uncini aut lances, quibus ad- nectuntur aut imponuntur pondera, multò magis si eadem pon- dera assumantur, centrum gravitatis hujusce molis compositæ reperiri asserunt infrà ipsum jugum, ac propterea nullam esse hujusmodi primam speciem libræ. Sit libræ jugum A B; centrum motûs & gravitatis jugi sit C: pendeant lances D & E, singularumque cum suis appendiculis gravitas sit æqualis gra- vitati jugi, ut facere con- sueverunt accuratiores monetarij. Lancium igi- tur simul sumptarum commune gravitatis cen- trum est in F: jungantur centra gravitatum C & F; & erit demum totius libræ vacuæ D A B E commune gravitatis cen- trum in G. Quod si lan- cibus D & E imponan- tur æqualia pondera, commune centrum gravitatis erit inter G & F, atque quò gra- viora erunt pondera, eò propiùs accedet ad F. Est igitur ma- nifestum centra motûs & gravitatis totius libræ non in eodem puncto convenire, sed gravitatis centrum esse infrà centrum motûs, seu spartum C. Verum effugium hoc nullum esse censeo: inclinetur enim libra, & acquirat positionem H I, jam HM & IN lineæ di- K k 2

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Book three. CHAPTER IV. 259 resistance of D itself to the motion of E: thus, neither prevailing, there will result in that position DE a state of rest. Nevertheless, some do not acquiesce in this argument, although legitimate, because they despair of ever seeing such a balance at rest in any position whatever, since they have never seen it; and therefore they rather inquire into the cause why a balance with equal arms returns to equilibrium, although suspended from the middle of the beam. Some think that the force of the argument can be evaded, if they grant indeed that in one and the same point the center of motion and the center of gravity of the beam coincide, but not of the balance: for if, in addition to the beam, the hooks or pans be also assumed, to which the weights are attached or placed, and much more if the same weights be included, they assert that the center of gravity of this composite body is found below the beam itself, and therefore that there is no such first species of balance. Let AB be the beam of the balance; let C be the center of motion and gravity of the beam: let the pans D and E hang, and let the weight of each together with its attachments be equal to the weight of the beam, as more accurate moneyers are accustomed to do. Therefore the common center of gravity of the pans taken together is at F: let the centers of gravity C and F be joined; and finally the common center of gravity of the whole empty balance D A B E will be at G. But if equal weights are placed on the pans D and E, the common center of gravity will be between G and F, and the heavier the weights are, the nearer it will approach F. It is therefore manifest that the centers of motion and gravity of the whole balance do not coincide in the same point, but that the center of gravity is below the center of motion, or point C. Yet I think there is no escape here: for let the balance be inclined, and let it acquire the position HI; now HM and IN the lines of the K k 2

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Mechanicorum 260 rectionis lancium sunt æquales, quia eædem cum AD & BE, & sunt parallelæ, quia ambæ perpendiculares ad horizontem, ac propterea ex 33. lib. 1. æquales sunt ac parallelæ HI & MN. Cumque CF linea directionis centri gravitatis jugi sit iisdem HM & IN parallela, & exeat ex C medio rectæ HI, cadet pariter in medium rectæ MN ex 34 lib. 1. & idem punctum F est commune centrum gravitatum M & N; atque proinde libræ MHIN commune centrum gravitatis erit in eâdem rectâ lineâ CF. Si itaque quiescit corpus grave suspensum, quando in eâdem directionis lineâ est punctum suspensionis, & gravitatis centrum, etiam in positione HI deberet libra quiescere, esto in C non conveniant contra motûs & gravitatis totius libræ. Nicolaus Tartalea lib. 8. quæsito 32. ideo libram ad parallellismum horizontis redire existimat, quia in inclinatione jugi putat majora esse momenta brachij elevati, quàm depressi. Id quod hâc methodo conatur ostendere. Si ex C æqualiter dissent pondera æqualia A & B, fuerintque ab æquilibrio remota, describunt circulum, in quo sumptis partibus æqualibus, dum A descendit ex F in A, vis descendendi est NO, at ex A in G vis descendendi est OP major, quàm NO, ut constat ex doctrinâ Sinuum. Similiter vis descendendi ipsius B ex I in B est KL major, quàm LM vis descendendi ex B in H. Est autem KL ipsi OP, & LM ipsi ON æqualis; igitur OP est etiam major, quàm LM. Cum itaque in situ A CB pondus B gravitet solùm ut LM, & pondus A gravitet ut OP, major est potentia ipsius A, quàm ipsius B: igitur ad æquilibrium descendere oportet pondus A. Sed peccat hæc Tartaleæ argumentatio, quia in pondere B non est consideranda vis descendendi in H, sed repugnantia ad ascendendum in I, secundùm quàm obsistit opposito ponderi A; hujus autem resistentiæ mensura est LK æqualis ipsi OP potentiæ seu propensioni ipsius A ad descendendum: æquatur

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Mechanics 260 the directions of the strings are equal, because they are the same as AD and BE, and they are parallel, because both are perpendicular to the horizon, and therefore, from Book 1, proposition 33, they are equal and parallel to HI and MN. And since CF is the line of direction of the center of gravity of the yoke, parallel to the same HM and IN, and issues from C, the middle of the straight line HI, it will likewise fall into the middle of the straight line MN, by Book 1, proposition 34; and the same point F is the common center of gravity of M and N; and therefore the common center of gravity of the balance MHIN will be in the same line CF. If therefore a suspended heavy body is at rest when the point of suspension and the center of gravity are in the same line of direction, then even in the position HI the balance ought to be at rest, although in C they do not coincide, contrary to the motion and gravity of the whole balance. Nicolaus Tartalea, book 8, question 32, therefore thinks that the balance returns to the parallelism of the horizon, because in the inclination of the yoke he judges that the moments of the raised arm are greater than those of the depressed arm. This he tries to show by this method. If from C two equal weights A and B are equally removed, and are moved away from equilibrium, they describe a circle, in which, taking equal parts, while A descends from F to A, the force of descent is NO, but from A to G the force of descent is OP, greater than NO, as is clear from the doctrine of sines. Similarly the force of descent of B itself from I to B is KL, greater than the force of descent LM from B to H. But KL is equal to OP, and LM to ON; therefore OP is also greater than LM. Since therefore in the position ACB the weight B gravitates only as LM, and the weight A gravitates as OP, the power of A is greater than that of B: therefore, for equilibrium, the weight A ought to descend. But this argument of Tartalea errs, because in the weight B one must not consider the force of descent in H, but the resistance to ascent in I, according to which it opposes the opposite weight A; and the measure of this resistance is LK, equal to the power or tendency of A itself to descend: it is equal

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Liber tertius. CAPUT IV. 261 æquatur ergo potentia resistentiæ, nec ullus fieri potest motus, quamdiu hæc æqualitas permanet. Ioannes Keplerus Astronomiæ Opticæ loco citato, cur libræ brachia revolvantur ad æquilibrium, infert ex eo, quòd altero brachiorum prægravato additione ponderis, ita jugum libræ consistit, ut quod est gravius non planè imum locum petat, & quod est levius, non planè in apicem attollatur. Cujus rei causam inquirens statuit libræ jugum C D bifariam in A divisum; & centro A descripto circulo ducit perpendiculum B A F: ex quo manifestum est neutrum pondus posse deprimi infra F, aut attolli supra B. Sed quia pondus D ponitur gravius, quàm pondus C, & utrumque naturâ suâ ad imum tendit, contenduntque invicem, partiuntur inter se descensum B F in proportione, quâ ipsa sunt: adeò ut B H descensus ponderis C sit ad B G descensum ponderis D, ut pondus C ad pondus D. Est autem F G linea æqualis lineæ B H, quia ex æqualibus A B & A F auferuntur æqualia latera A H & A G, cum enim triangula C H A, D G A rectangula sint, & angulos ad verticem A æquales habeant, & latera A C, A D æqualia; etiam per 26. lib. 1. latus A H est æquale lateri A G. Igitur ut pondus C ad pondus D, ita F G ad G B. Ducatur ex F ad A D perpendicularis F K: similiter triangula A G D, A K F rectangula, & cõmunem angulum in A habentia, cum latere A F æquali lateri A D, per eandem 26. lib. 1. habēt latera A G & A K æqualia: ergo & residua F G, D R æqualia sunt. Igitur propter æqualitate diametroru[m] F B & D C, erit etiam G B linea æqualis lineæ K C. Quare ut pôdus D ad pondus C, ita G B ad G F, hoc est ita K C ad K D: ac propterea factâ jugi suspensione in K pondera C & D inæqualia secundùm Rationem brachiorum reciprocè posita æquiponderabunt & consistent. Cum igitur in hac eâdem Ratione sit descensus B H & B G, ut est pondus C ad pondus D, fiet consistentia in situ C A D. Ergo per subsumptionem patet, subdit Keplerus, cujus superiorem doctrinam conatus sum paulo clariùs exponere, cur libræ brachia K k 3

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Book the third. CHAPTER IV. 261 Thus the power is equal to the resistance, and no motion whatsoever can take place so long as this equality remains. Ioannes Keplerus, in the place cited in Astronomiæ Opticæ, infers the reason why the arms of the balance turn back toward equilibrium from this: that, if one of the arms be made heavier by the addition of weight, the beam of the balance is so situated that that which is heavier does not descend all the way to the lowest point, and that which is lighter is not raised all the way to the top. Seeking the cause of this matter, he sets down the beam of the balance C D divided in two at A; and, with center A, he describes a circle and draws the perpendicular B A F: from which it is manifest that neither weight can be depressed below F, or raised above B. But since the weight D is placed heavier than the weight C, and both by their nature tend downward, and contend with one another, they divide between them the descent B F in the proportion in which they themselves are; so that the descent B H of weight C is to the descent B G of weight D as the weight C to the weight D. Now F G is a line equal to line B H, because from the equal lines A B and A F are taken the equal sides A H and A G; for since the triangles C H A, D G A are right-angled, and have equal angles at the vertex A, and equal sides A C, A D; also, by proposition 26, book 1, side A H is equal to side A G. Therefore as weight C is to weight D, so is F G to G B. Let a perpendicular F K be drawn from F to A D: likewise triangles A G D, A K F are right-angled, and having a common angle at A, and side A F equal to side A D, by the same proposition 26, book 1, they have sides A G and A K equal: therefore the remaining lines F G, D R are equal also. Therefore, because of the equality of the diameters F B and D C, G B also will be a line equal to line K C. Wherefore, as weight D is to weight C, so G B is to G F, that is, so K C is to K D: and therefore, if the suspension of the beam be made in K, the unequal weights C and D, placed reciprocally according to the ratio of the arms, will balance and remain at rest. Since therefore in this same ratio are the descents B H and B G, as weight C is to weight D, equilibrium will result in the position C A D. Therefore, by subsumption it is clear, Kepler adds, whose earlier doctrine I have tried to explain a little more clearly, why the arms of the balance K k 3

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Mechanicorum revolvuntur ad æquilibrium; cum enim æque ponderent, æquales etiam in circulo fieri descensus par est. Meam hebetudinem dissimulare non possum, qui hujusce Keplerianæ argumentationis vim satis assequi non valeo: quid enim, si fieret æquilibrium horizontale ponderum, facta in K suspensione? an propterea consequens est fieri æquilibrium etiam in situ CAD, nisi aliunde probetur? sed quod ad rem nostram attinet, pondera alligata, & adnexa libræ non ita con- sideranda sunt, ut ambo descendant, si comparatè sumantur, sed alterius propensio ad motum deorsum comparanda est cum alterius repugnantiâ ad motum sursum, & vicissim hujus pro- pensio ad descendendum cum illius resistentiâ, ne ascendat. Quapropter si ex D pondere majore auferatur excessus supra pondus C, & fiant æqualia pondera, non possunt ad æquili- brium horizontale redire, nisi C descendat, D verò ascendat: Cum autem hujus ascensus GA sit æqualis descensui HA, nul- la est ratio, cur propensio ponderis C vincere debeat æqualem ponderis D resistentiam. Deinde quid intelligendum est, cum dicitur ipsius C descen- sus esse BH, ipsius verò D descensus esse BG? ex B enim non utrumque descendit, sed alterutrum: & si pondus D descendis- set ex B, ex adverso pondus C ascendisset ex F; cùmque illius descensus esset BG, hujus ascensus esset FH; sunt autem BG & FH æquales. Quòd si non motus præcedens, sed sola pro- pensio ad descendendum & repugnantia ad ascendendum con- sideretur pro ratione positionis, pondus D habet mensuram propensionis ad descendendum, non motum (qui fortasse tran- sait) ex B in D, sed quem in eo situ posset perficere ex D in F: atque adeò ipsius D descensus est GF, ejusque resistentia, ne ascendat usque ad summum est GB, & vicissim ponderis C pro- pensio ad descendendum non est ex B in C, sed ex C in F, si usque ad imum descendat, habens mensuram HF, ejus verò repugnantiam ad ascendendum metitur HB. Est igitur mani- festum uniuscujusque ponderis propensionem habere opposi- tam resistentiam æqualem (est enim propensio GF æqualis re- sistentiæ HB, & propensioni HF æqualis est resistentia GB) æc proinde nullum sequi posse motum ponderum æqualium à centro A æqualiter distantium. At, inquis, quid causæ est, cur

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Mechanics are brought back to equilibrium; for when they weigh equally, equal descents in a circle are likewise proper. I cannot conceal my dullness, since I am not able sufficiently to grasp the force of this Keplerian argument: for what if there were an equilibrium of weights in the horizontal position, a suspension having been made at K? Is it therefore consequent that an equilibrium should also be made in the position CAD, unless it be proved from elsewhere? But as far as our matter is concerned, weights attached and joined to the balance are not to be so considered, as though both should descend, if they are taken comparatively; rather, the inclination of one toward downward motion is to be compared with the other's resistance to upward motion, and vice versa, this one's inclination to descend with that one's resistance, lest it ascend. Wherefore, if from the greater weight D the excess above weight C is taken away, and the weights are made equal, they cannot return to horizontal equilibrium unless C descends and D ascends: but since the ascent of this one, GA, is equal to the descent HA, there is no reason why the inclination of weight C should overcome the equal resistance of weight D. Next, what is to be understood when it is said that the descent of C itself is BH, but the descent of D is BG? For from B neither descends both, but one or the other: and if weight D had descended from B, on the opposite side weight C would have ascended from F; and since the descent of that one would be BG, the ascent of this one would be FH; but BG and FH are equal. But if not the preceding motion, but only the inclination to descend and the resistance to ascend, is considered according to the position, weight D has the measure of an inclination to descend, not the motion (which perhaps passes away) from B to D, but that which in that position it could complete from D to F: and indeed the descent of D itself is GF, and its resistance, lest it ascend all the way to the top, is GB; and conversely the inclination of weight C to descend is not from B to C, but from C to F, if it descends all the way to the bottom, having the measure HF, and its resistance to ascending is measured by HB. It is therefore evident that the inclination of each weight has an opposite equal resistance (for the inclination GF is equal to the resistance HB, and to the inclination HF is equal the resistance GB), and therefore no motion of equal weights equally distant from the center A can follow. But, you say, what is the cause why

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Liber tertius. CAPUT IV. 263 cur similem libram in quâcumque positione quiescentem non habemus? sed omnis libra ea est, ut vel ad æquilibrium redeat, vel omninò quantum potest descendat, quâ parte habet bra- chium inclinatum? Responsio in promptu est; quia scilicet dif- ficillimum est duo illa puncta exquisitè convenire, hoc est cen- trum motûs & centrum gravitatis, nimirùm punctum illud, quod brachiorum longitudinem discriminat. Quòd si vel mi- nimum duo illa centra discrepent, natura omnes sui juris api- ces exactissimè persequitur, & est spartum non in medio, sed aut in superiore, aut in inferiore parte jugi (si quidem brachia sint æqualia; nam si ad latus esset in eâdem rectâ lineâ, libra es- set inæqualium brachiorum, & tunc non adnexorum ponderum æqualitas esset consideranda, sed eorum Ratio, sumptâ recipro- cè brachiorum Ratione) ex quo sequitur aut reditus ad æquili- brium, aut ulterior descensus brachij inclinati. Hinc est de illâ duplici tantummodo libræ specie locutum fuisse Aristotelem in Mechan. q. 2. omisâ priore hac, quæ vi- detur speculantis intellectûs terminis coërceri, nunquam in praxim nisi fortuito deducenda. Non enim satis est accuratis- simè inquirere centrum gravitatis jugi, ut illud sit pariter cen- trum motûs, sed necesse est punctum hoc in eâdem rectâ lineâ esse, quæ jungit puncta contactuum jugi & annulorum, ex quibus lances dependent: nam nisi hoc contingat, centrum il- lud gravitatis assumptum non est punctum, à quo brachiorum longitudines discriminantur, ut inferiùs constabit dilucidiùs ex iis, quæ de librâ curvâ dicentur. Quærendum est itaque, cur libra aginam habens in supe- riore loco, si ab æquilibrio horizontali dimoveatur, ad illud re- deat. Et ne locus æquivocationi pateat, dum ad hoc de- monstrandum assumuntur puncta notabili intervallo inter se distantia (ne videlicet linearum brevitas confusionem aut ob- scuritatem pariat) observa lingulæ nomine non eam solùm par- tem intelligi, quæ supra libræ jugum intrà ansam excurrens extat; sed lingulæ, seu, ut aliis placet, trutinæ pars est etiam linea, quæ in ipsa jugi crassitie descripta intelligitur perpendi- cularis ad lineam longitudinis brachiorum, & transiens per centrum motûs. Quare hujus lineæ pars intercepta inter cen- trum motûs, & lineam longitudinis brachiorum, sivè exigua sit,

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Book third. CHAPTER IV. 263 why do we not have a similar balance resting in whatever position? but every balance is such that either it returns to equilibrium, or else it descends as much as it can, on that side where it has the inclined arm? The answer is at hand: namely, because it is extremely difficult for those two points to coincide exactly, that is, the center of motion and the center of gravity, namely that point which marks off the length of the arms. And if the two centers differ even in the least, nature follows with the utmost exactness all the extremities that are within her power, and the fulcrum is not in the middle, but either in the upper or in the lower part of the beam (if indeed the arms are equal; for if it were on the side in the same straight line, the balance would be of unequal arms, and then the equality of the attached weights would not be to be considered, but their ratio, the ratio of the arms having been taken reciprocally) from which it follows either a return to equilibrium, or a further descent of the inclined arm. Hence it is that Aristotle in the Mechan. q. 2. spoke only of that twofold kind of balance, omitting this earlier one, which seems to be confined within the bounds of the speculative intellect, and never to be brought into practice except by chance. For it is not enough to inquire most accurately into the center of gravity of the beam, so that it may likewise be the center of motion; rather, it is necessary for this point to be on the same straight line that joins the points of contact of the beam and the rings, from which the pans hang: for unless this happens, that assumed center of gravity is not the point from which the lengths of the arms are distinguished, as will be made clearer below from what will be said about the curved balance. It must therefore be asked why a balance having the fulcrum in the upper place, if it is moved away from horizontal equilibrium, returns to it. And lest there be room for ambiguity, while points separated from one another by a notable interval are taken for this demonstration (so that the brevity of the lines may not produce confusion or obscurity), observe that by the name of tongue is understood not only that part which projects above the beam of the balance within the handle; but also the part of the tongue, or, as others prefer, of the trutina, is likewise the line which is understood to be drawn in the very thickness of the beam, perpendicular to the line of the length of the arms, and passing through the center of motion. Therefore, the part of this line intercepted between the center of motion and the line of the length of the arms, whether it is small,

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Mechanicorum sit, sivè valde notabilis (quod quidem ad præsentem considerationem attinet) nihil interest, nam eadem planè semper est ratio, atque demonstratio. Sit libra æqualium brachiorum A B, cujus puncto medio C insistat perpendicularis CD, & sit in ipsâ jugi crassitie centrum motûs punctum D, impositisque æqualibus ponderibus in A & B, maneat in æquilibrio horizontali A B. Deprimatur extremitas A, ut veniat in E, reliqua extremitas B ascendit in F, & C venit in G. Non potest igitur manere libra in positione E F sublato deprimente in E, sed manentibus æqualibus ponderibus redit ad æquilibrium, séque restituit in A B; tùm quia centrum gravitatis non est in lineâ directionis transeunte per D punctum suspensionis, tùm potissimum quia momenta ipsius F majora sunt momentis ipsius E ratione positionis & propensionis ad motum; potest enim F descendere juxta mensuram F H, dum E ascendit juxta mensuram E I; est autem major Ratio motûs F H ad motum E I, quàm sit Ratio ponderum, quæ est Ratio æqualitatis, nimirum ut F G ad G E. Nam per 8 lib. 5. FO ad G E majorem habet Rationem quàm F G ad G E, & FO ad O E majorem habet Rationem quàm FO ad G E; ergo multo major est Ratio FO ad O E, quàm FG ad G E. At similia sunt triangula FHO, EIO, quia æquiangula (nam propter parallelismum linearum directionis F H & IE, alterni E & F, & alterni I & H, qui etiam recti ponuntur, & qui ad verticem O, æquales sunt) igitur per 4. lib. 6. ut FO ad OE, ita FH ad EI. Est igitur major Ratio descensûs F H ad ascensum EI, quàm sit Ratio ponderum, quæ est ut FG ad G E. Hinc patet clara solutio quæstionis à Keplero propositæ: quia si pondus E majus sit pondere F, illud non ad imum locum descendet, sed ibi libra obliquè subsistet, ubi pondera erunt in Ratione reciprocâ motuum; quando scilicet ratione positionis ita propensio ad descendendum ponderis F erit ad resistentiam ponderis E, ne ascendat, ut est vicissim pondus E ad pondus F: & tunc perpendicularis linea directionis ex D puncto

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Whether the mechanics be slight, or even very notable, as indeed concerns the present consideration, it makes no difference, for the ratio and demonstration are always exactly the same. Let there be a balance with equal arms A B, at whose midpoint C let the perpendicular C D stand, and let the center of motion be at the very thickness of the beam in the point D; and with equal weights placed at A and B, let it remain in horizontal equilibrium A B. Let the end A be depressed, so that it comes to E; the other end B rises to F, and C comes to G. Therefore the balance cannot remain in the position E F, the depressing force having been removed from E, but, the equal weights remaining, it returns to equilibrium and restores itself to A B; both because the center of gravity is not in the line of direction passing through the point D of suspension, and especially because the moments of F itself are greater than the moments of E itself by reason of position and inclination to motion; for F can descend by the measure F H, while E rises by the measure E I. Now the ratio of the motion F H to the motion E I is greater than the ratio of the weights, which is the ratio of equality, namely as F G to G E. For by Book 5, Proposition 8, F O has a greater ratio to G E than F G to G E, and F O to O E has a greater ratio than F O to G E; therefore the ratio of F O to O E is much greater than F G to G E. But the triangles F H O and E I O are similar, because they are equiangular (for, because of the parallelism of the lines of direction F H and I E, the alternate angles E and F, and I and H, which are also taken as right angles, and which are equal at the vertex O), therefore by Book 6, Proposition 4, as F O is to O E, so is F H to E I. The ratio of the descent F H to the ascent E I is therefore greater than the ratio of the weights, which is as F G to G E. Hence the clear solution to the question proposed by Kepler is evident: because if the weight E be greater than the weight F, it will not descend to the lowest place, but there the balance will stand obliquely, where the weights will be in the reciprocal ratio of the motions; namely, when by reason of position the inclination of the weight F to descend is so related to the resistance of the weight E, so that it does not ascend, as the weight E is to the weight F in the opposite direction: and then the perpendicular line of direction from the point D

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Liber tertius. CAPUT IV. 265 puncto suspensionis demissa cadet in centrum gravitatis compositæ libræ & ponderum. Cujus rei argumentum est manifestum, quod libra quiescens in positione EF si moveatur ab aliquo deprimente ulteriùs aut elevante, sibi relicta non minùs redit ad eumdem situm obliquum, quàm redeat ad æquilibrium horizontale, si pondera sint æqualia. Quæ omnia ex dictis plana sunt & aperta; sed an hoc idem ritè probaverit Keplerus, viderint alij. Eadem philosophandi ratio erit in librâ brachiorum inæqualium LM, in qua sint pondera L & M (computatis ipsorum brachiorum gravitatibus juxta momenta, quæ habent in illâ eâdem longitudine, ut dictum cap. 2. hujus libri) reciprocè in Ratione brachiorum NM & NL. Deprimatur L in P, & elevabitur M in Q, & N in V. Dico libram summoto deprimente, ad æquilibrium LM redituram. Ducantur perpendiculares PT & QR, productâ LM horizontali, si opus fuerit. Triangula SQR, SPT sunt similia; igitur per 4 lib. 6. ut QS ad SP, ita ponderis Q propensio ad descendendum QR, ad ponderis P resistentiam, ne ascendat, PT. Est autem major Ratio QR ad PT, quàm sit ponderis P ad pondus Q; igitur pondus Q prævalebit. Majorem autem esse Rationem sic ostenditur. Pondus P ad pondus Q est ut NM ad NL ex hypothesi, hoc est ut QV ad VP: sed per 8. lib. 5. major est Ratio QS ad VP, quàm QV ad VP, & major Ratio QS ad SP, quàm QS ad VP: igitur major est Ratio QS ad SP, quàm QV ad VP, hoc est quàm pondus P ad pondus Q. Est autem demonstratum ita esse QS ad SP, ut QR ad PT; igitur major est Ratio descensûs QR ad ascensum PT, quàm sit Ratio ponderis P ad pondus Q: Ergo vis descendendi major est, quàm opposita resistentia, ac proptereà restituet se libra in æquilibrio horizontali. Ex his manifestum est rem contrario modo se habere, quando spartum est in crassitie jugi ita collocatum, ut sit infra lineam, quæ constituit longitudinem brachiorum; tunc enim al- L1

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Book three. CHAPTER IV. 265 let down from the point of suspension will fall to the center of gravity of the composed balance and weights. The proof of this is manifest, for a balance at rest in the position EF, if moved by some further lowering or raising, when left to itself returns no less to the same oblique position than it returns to horizontal equilibrium, if the weights are equal. All these things are plain and evident from what has been said; but whether Kepler has rightly proved this same thing, let others judge. The same way of philosophizing will apply to a balance with unequal arms LM, in which let the weights be L and M (the weights of their arms being computed together with them according to the moments they have at that same length, as was said in chap. 2 of this book) in reciprocal ratio to the arms NM and NL. Let L be lowered to P, and M will be raised to Q, and N to V. I say that, when the lowering force is removed, the balance will return from LM to equilibrium. Let the perpendiculars PT and QR be drawn, the horizontal LM having been produced, if need be. The triangles SQR and SPT are similar; therefore by 4, book 6, as QS is to SP, so is the inclination of weight Q to descend QR, to the resistance of weight P, lest it ascend, PT. But the ratio QR to PT is greater than the ratio of weight P to weight Q; therefore weight Q will prevail. Now that the ratio is greater is shown thus. The weight P to the weight Q is as NM to NL by hypothesis, that is, as QV to VP: but by 8, book 5, the ratio QS to VP is greater than the ratio QV to VP, and the ratio QS to SP is greater than the ratio QS to VP: therefore the ratio QS to SP is greater than the ratio QV to VP, that is, than the weight P to the weight Q. And it has been demonstrated that QS is to SP as QR is to PT; therefore the ratio of the descent QR to the ascent PT is greater than the ratio of weight P to the weight Q: therefore the force of descending is greater than the opposite resistance, and therefore the balance will restore itself to horizontal equilibrium. From these things it is evident that the matter is otherwise when the spar is placed in the thickness of the yoke so that it lies below the line which constitutes the length of the arms; for then al-

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Mechanicorum tero brachiorum inclinato, tantum abest, ut libra revertatur ad priorem parallelismum cum horizonte, ut potiùs, nullo ulteriùs deprimente, brachium inclinatum descendat omninò, donec impediatu ab ansâ, in quam incurrit alterum brachium elevatum: quod si superiori aut inferiori brachio nullum occurreret impedimentum, ita fieret totius libræ conversio & revolutio, ut spartum esset in loco superiore, & tunc demùm in æquilibrio horizontali jugum quiesceret. Quæ omnia licet perspicua sint, si superiores duæ figuræ invertantur, clarioris tamen ex- plicationis gratiâ, sit iterum jugum AB æqualiter divisum in C, & in perpendiculari CD sit axis, & centrum motûs inferiùs in D: positis æqualibus ponderibus A & B sit æquilibrium horizontale: & quoniam æqualia sunt pondera, atque æquales ad motum propensiones, centrumque gravitatis est in eâdem perpendiculari lineâ directionis cum puncto sustentationis D, manent in æquilibrio. Deprimatur A in E, elevatur pariter B in F, & C deprimitur in G. Dico libram, si sibi ipsa dimittatur, non redituram ad positionem AB supra punctum D; sed pondus E ulteriùs descensurum. Ductis enim perpendicularibus EI & FH, propensio ponderis F ad motum deorsum, ut se restituat in priore æquilibrio, est FH, resistentia ponderis E ad motum sursum est EI. Est autem major Ratio resistentiæ EI ad propensionem deorsum FH, quàm sit Ratio ponderis F ad pondus E, aut vicissim; hæc enim æqualia sunt ex hypothesi, & est eorum Ratio ut AC ad CB, hoc est ut EG ad GF: Non igitur potest à pondere F, cujus momenta minora sunt elevari pondus E, cujus momenta sunt majora ex dispositione ad motum. Constat verò major Ratio resistentiæ EI ad propensionem FH, quàm ponderis F ad pondus E, quia in triangulis OIE, & OHF similibus eâdem est Ratio EI ad FH, quæ est EO ad OF; sed ex 8 lib.5. EO ad OF majorem habet Rationem quam EG ad GF: igitur major est Ratio EI ad FH, quam EG ad GF, hoc est ponderis ad pondus. Descendet itaque E, & nullo occurrente obice ea fiet totius libræ revolutio circà centrum D, ut demum

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Mechanics with the axis of the arms inclined, it is so far from the balance’s returning to its former parallelism with the horizon, that rather, no longer being pressed down, the inclined arm descends altogether, until it is stopped by the loop into which the other raised arm falls: but if the upper or lower arm encountered no obstacle, there would thus take place a turning and revolution of the whole balance, so that the cord would be in the upper place, and then at last the beam would rest in horizontal equilibrium. Although all these things are plain, if the two upper figures are inverted, yet for the sake of a clearer ex- planation, let the beam AB once again be equally divided at C, and in the perpendicular CD let there be the axis, and the center of motion below at D: with equal weights A and B placed, there is horizontal equilibrium: and since the weights are equal, and the tendencies to motion equal, and the center of gravity is on the same perpendicular line of direction with the point of support D, they remain in equilibrium. Let A be lowered to E, B likewise raised to F, and C lowered to G. I say that the balance, if left to itself, will not return to the position AB above the point D; but the weight E will descend further. For if perpendiculars EI and FH are drawn, the tendency of the weight F to motion downward, by which it may restore itself to its former equilibrium, is FH; the resistance of the weight E to motion upward is EI. But the ratio of the resistance EI to the downward tendency FH is greater than the ratio of the weight F to the weight E, or conversely; for these are equal by hypothesis, and their ratio is as AC to CB, that is, as EG to GF: therefore the weight F, whose moments are smaller, cannot raise the weight E, whose moments are greater by reason of its disposition to motion. It is clear, moreover, that the ratio of the resistance EI to the tendency FH is greater than that of the weight F to the weight E, because in the similar triangles OIE and OHF the ratio EI to FH is the same as EO to OF; but from book 5, proposition 8, EO to OF has a greater ratio than EG to GF: therefore the ratio of EI to FH is greater than EG to GF, that is, of weight to weight. Therefore E will descend, and if no obstacle occur the whole balance will then revolve around center D, so that at length

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Liber tertius. CAPUT IV. 267 demum jugum EF sit infrà punctum D, & quod initio fuit punctum sustentationis, fiat punctum suspensionis libræ. Ea- dem dicta intelligentur de librâ brachiorum inæqualium, quæ supervacaneum est iterum inculcare. Oblatâ itaque librâ facilè dignosces, cujus speciei illa sit, quamvis ob punctorum propinquitatem, scilicet centri mo- tûs, & puncti brachiorum longitudinem discriminantis, non valeat oculus dijudicare: impositis enim æqualibus ponderi- bus, ut habeat æquilibrium horizontale, aliquantulum depri- me alterutrum brachiorum, & sublato deprimente, si quidem manserit obliqua (id quod rarissimè continget) pronunciabis centrum motûs convenire cum puncto brachiorum longitudi- nem discriminante: sin autem ad æquilibrium redierit, cen- trum motûs erit in superiore loco; si ulteriùs descenderit, cen- trum motûs erit infra lineam longitudinis brachiorum. Vel etiam facto æquilibrio horizontali, adde pondus alteri lanci; si descendat ita, ut jugum obliquè consistat aut magis aut mi- nùs, prout major aut minor factus est excessus ponderis, pro- nunciabis centrum motûs esse in superiore loco: at si factâ ponderum inæqualitate lanx gravior usque ad imum deprima- tur, quantum potest, indicabit centrum motûs esse in inferio- re loco, aut convenire cum puncto brachia discriminante: sed hoc ultimum temerè non affirmabis, nisi restitutâ ponderum æqualitate, sequatur quies in quacumque positione, aut con- versâ deorsum ansâ non contingat obliqua jugi consistentia: si enim factâ ansæ suspensione centrum illud fuisset in inferio- re loco, factâ conversione esset in superiore loco, & continge- ret æquilibrium in positione obliquâ. CAPUT V. An fieri possit libra Curva. Quamvis ad ponderum examen instituendum rarô contin- gere possit, ut librâ Curvâ uti cogamur, quia tamen in machinamentis aliquibus ita aut loci angustiæ, aut opportuna Ll 2

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Liber third. CHAPTER IV. 267 so that at last the yoke EF is below point D, and what at first was the point of support becomes the point of suspension of the balance. The same things will be understood of a balance with unequal arms, which it is superfluous to repeat again. A balance having thus been presented, you will easily discern of what species it is, although because of the proximity of the points, namely the center of motion and the point distinguishing the length of the arms, the eye may not be able to judge: for if equal weights are placed upon it, so that it has a horizontal equilibrium, depress slightly either arm, and when the depressing weight is removed, if it remains oblique (which will happen very rarely) you will pronounce that the center of motion agrees with the point distinguishing the length of the arms: but if it returns to equilibrium, the center of motion will be in the upper position; if it descends further, the center of motion will be below the line of the length of the arms. Or also, when horizontal equilibrium has been established, add weight to one scale-pan; if it descends so that the yoke stands obliquely, more or less, according as the excess of weight has become greater or smaller, you will pronounce the center of motion to be in the upper position: but if, when the weights are unequal, the heavier pan is depressed all the way down as far as it can, this will indicate that the center of motion is in the lower position, or agrees with the point distinguishing the arms: but you will not rashly affirm this last, unless, when the equality of the weights is restored, there follows rest in any position whatever, or, when the handle is turned downward, an oblique standing of the yoke does not occur: for if, when the handle was suspended, that center had been in the lower position, when turned it would be in the upper position, and equilibrium would occur in an oblique position. CHAPTER V. Whether a Curved balance can exist. Although for the examination of weights it can rarely happen that we are compelled to use a Curved balance, nevertheless because in certain machines either by reason of the narrowness of the place, or by reason of a suitable Ll 2

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Mechanicorum 268 corporum movendorum dispositio, exigunt collocari ponde- ra, ut & libræ Rationes serventur, & tamen jugi rectitudo nul- la appareat; non erit hîc inutile libram curvam examinare, ut, si quando eâ uti contigerit, innotescat, quænam sint brachiorum, & motuum Rationes. Libram autem curvam voco, quæ à communi formâ deflectens latera habet non in directum posi- ta, sed in angulum concurrentia, aut in arcum sinuata, quo- rum extremitates sivè sursum, sivè deorsum respiciunt: factâ enim suspensione sivè ubi angulum latera constituunt, sivè in aliquo arcûs puncto, ea fieri potest hinc & hinc ponderum ad- ditio, quam horizontale æquilibrium consequatur. Sed quia imperitis fucum facere posset apparens hæc laterum longitudo, caveant, ne ex illis jugum libræ deductum intelligant: contin- gere scilicet potest, ut planè varia sit hujusmodi libræ forma, & magnitudo, idem tamen sit semper libræ jugum, in quo brachia desumenda sunt. Sint enim in angulum compacta duo latera recta AB & AC; non est tota jugi magnitudo computanda ex horum late- rum longitudinibus; sed ex ipsâ extre- mitatum B & C distantiâ BC; quæ sem- per eadem est, sivè sit arcus BEFC, sivè alia sint latera DB & DC, aut GB & GC, atque suspensio fiat sivè in A, sivè in D, sivè in G, sivè in quo- cumque alio puncto, quod sit intra spa- tium à lineis AB, AC, BC comprehensum. Est igitur idem jugum BC, quia in B & C adnexa intelliguntur pondera, eo- rumque distantia, prout libræ adnectuntur, ea est, quæ jugi longitudinem determinat. Verùm an libra æqualium sit po- tiùs, quàm inæqualium brachiorum, definiendum est ex puncto suspensionis, à quo ad extremitates B & C deducen- dæ sunt rectæ lineæ; quæ si æquales fuerint, libra est æqualium brachiorum; sin autem inæquales, inæqualium. Hinc si late- ra AB & AC jungantur transversario HI, in eóque sumatur punctum suspensionis D, nil refert æqualia-ne, an inæqualia sint latera AB & AC; sed attendenda est æqualitas aut in- æqualitas linearum ex D ductarum ad extremitates B & C. Neque me arguas, quòd dixerim jugum esse BC, & attenden- dam

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Mechanics 268 The arrangement of bodies to be moved requires weights to be placed so that the ratios of the lever are preserved, and yet no straightness of the yoke appears; it will not be useless here to examine a curved balance, so that, if it should ever happen to be used, it may become known what the ratios of the arms and motions are. By a curved balance I mean one which, departing from the common form, has sides not placed in a straight line, but meeting at an angle, or bent into an arc, whose ends look either upward or downward: for when the suspension is made, either where the sides form an angle, or at some point of the arc, addition of weights may be made on both sides, whereby horizontal equilibrium is attained. But because this apparent length of the sides might deceive the inexperienced, let them beware not to understand from them the yoke of the balance as determined: for it may happen that the form and size of such a balance are quite various, and yet the yoke of the balance is always the same, in which the arms are to be taken. For let two straight sides AB and AC be joined together at an angle; the whole size of the yoke is not to be computed from the lengths of these sides, but from the distance BC of the ends themselves B and C; which is always the same, whether there be an arc BEFC, or whether the sides DB and DC, or GB and GC, are otherwise, and the suspension is made either in A, or in D, or in G, or in any other point whatever that lies within the space enclosed by the lines AB, AC, BC. Therefore the same yoke is BC, because the weights are understood to be attached at B and C, and their distance, according as the balance is attached, is what determines the length of the yoke. But whether the balance be of equal rather than unequal arms is to be determined from the point of suspension, from which straight lines are to be drawn to the ends B and C; if these are equal, the balance is one of equal arms; if unequal, of unequal arms. Hence, if the sides AB and AC are joined by a crossbar HI, and in it the point of suspension D is taken, it matters not whether the sides AB and AC be equal or unequal; rather, one must attend to the equality or inequality of the lines drawn from D to the ends B and C. Nor should you object to me because I said that the yoke is BC, and that it should be attended to

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Liber secundus. CAPUT V. 269 dam æqualitatem aut inæqualitate linearu[m] ex puncto suspensionis ductarum, puta DB & DC; brachia siquidem in ipso jugo consideranda sunt; illæ aute[m] lineæ nihil habent cum jugo commune præter puncta extrema B & C. Quamvis enim lineæ hujusmodi brachia libræ non sint, si res propriè consideretur, inferunt tamen æqualitatem aut inæqualitatem brachiorum, quatenus ex puncto suspensionis D ducta intelligitur ad BC jugum perpendicularis DM, quæ jugum dividit in partes BM & CM æquales aut inæquales. Nam quia triangula BMD & CMD sunt rectangula, quadrato BD, ex 47. lib.1. æqualia sunt duo quadrata DM & MB, & quadrato DC æqualia sunt duo quadrata DM & MC. Si igitur lineæ DB & DC æquales sunt, earum pariter quadrata sunt æqualia; ex quibus dempto communi quadrato DM, remanent quadrata BM & CM æqualia, ac proinde lineæ MB & MC æquales. Si verò lineæ BD & CD sunt inæquales, quadrata earum sunt inæqualia; ex quibus dempto communi quadrato DM, residua sunt quadrata BM & CM inæqualia, eorumque latera (scilicet lineæ MB & MC) inæqualia erunt pronuncianda. Brachia itaque hujus libræ curvæ propriè sumpta non illa sunt, quæ apparent, & quia ex illis libræ curvæ moles constat, vulgariter hoc vocabulo donantur; sed sunt segmenta lineæ jungentis extremitates, quibus pondera adnectuntur; in quæ segmenta dividitur à perpendiculo, quod ad illam ducitur ex puncto, quod est motûs centrum. Cum igitur punctum hoc, quod tanquam centrum legem dat motui, sit extrà lineam extremitates illas jungentem, aut in superiore, aut in inferiore loco erit; ac proptereà altera erit ex duabus illis speciebus libræ, de quibus capite superiore sermo fuit, habentibus spartum aut suprà, aut infrà; & huic curvæ ea omnia convenient, quæ ibi dicta sunt, ut fiat æquilibrium horizontale, aut obliquum. Si enim sit libræ scapus rectus AB bifariam divisus, centrum motûs habens in C & pondera adnexa in D & E æqualia, habet æquilibrium horizontale, ad quod redit, si ab illo dimoveatur; & si pondera D & E sint inæqualia, habet æquilibrium obliquum pro Ratione discriminis ponderum; L1 3

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Book Second. CHAPTER V. 269 by the equality or inequality of the lines drawn from the point of suspension, namely DB and DC; for the arms are to be considered in the beam itself; but those lines have nothing in common with the beam except the extreme points B and C. For although lines of this sort are not the arms of the balance, if the matter be properly considered, they nevertheless imply the equality or inequality of the arms, inasmuch as from the point of suspension D there is understood to be drawn to the beam BC the perpendicular DM, which divides the beam into the equal or unequal parts BM and CM. For since the triangles BMD and CMD are right-angled, by proposition 47 of Book I, the two squares DM and MB are equal to the square BD, and the two squares DM and MC are equal to the square DC. If therefore the lines DB and DC are equal, their squares likewise are equal; from which, if the common square DM be taken away, there remain the squares BM and CM equal, and consequently the lines MB and MC equal. But if the lines BD and CD are unequal, their squares are unequal; from which, after the common square DM is taken away, the remaining squares BM and CM are unequal, and their sides (that is, the lines MB and MC) will have to be pronounced unequal. The arms therefore of this curved balance, properly speaking, are not those which appear, and because the mass of the curved balance consists of them, they are commonly given this name; but they are the segments of the line joining the extremities to which the weights are attached; into which segments it is divided by the perpendicular, drawn to that line from the point which is the center of motion. Since therefore this point, which as it were gives law to the motion as a center, lies outside the line joining those extremities, it will be either in the upper or in the lower position; and therefore this balance will be one of the two kinds spoken of in the preceding chapter, having the fulcrum either above or below; and to this curved balance all those things will apply which were there said, so that there may be a horizontal or an oblique equilibrium. For if the stem of the balance be a straight AB, divided into two equal parts, having the center of motion in C and the weights attached in D and E, equal, it has a horizontal equilibrium, to which it returns if moved away from it; and if the weights D and E are unequal, it has an oblique equilibrium in proportion to the difference of the weights. L1 3

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Mechanicorum 270 quia scilicet centrum motûs C est supra lineam DE jungentem puncta contactuum, quibus pondera adnectuntur. Factâ autem figuræ conversione, ut C sit in inferiore loco, & linea DE in superiore, in solo æquilibrio horizontali manet, à quo si removeatur, ad illud non redit, neque ullum habet æquilibrium in positione obliquâ, ut dictum est. Iam ex jugo AB omnia superflua resecentur, & remaneant virgulæ CD & CE connexæ in C centro motûs: manifestum est non esse immutata ponderum momenta, & eundem esse motum libræ curvæ DCE ac rectæ AB; sivè C intelligatur in parte superiori, sivè in inferiori. Quare & de hac curvâ, quod ad æquilibrium spectat, eadem dicenda sunt, quæ de librâ spartum superiùs aut inferiùs habente sunt dicta. Et quidem si latera illa, quibus libra curva constat, secundùm longitudinem æqualia sint, & paris gravitatis, additis hinc & hinc æqualibus ponderibus fiet æquilibrium horizontale; quia vera linea jugi in segmenta æqualia dividitur, sunt autem omnes Rationes Æqualitatis, omninò similes. At si latera illa sint inæqualia, non erunt addenda reciprocè pondera (etiam computatâ ipsorum laterum gravitate) in Ratione illarum longitudinum; sed in Ratione segmentorum jugi, ut fiat æquilibrium: quia ex laterum illorum inæqualitate statim quidem infertur etiam veram lineam jugi dividi in segmenta inæqualia; sed non illico consequens est similem esse Rationem Inæqualitatis: Immò si inæqualia sint illa latera, fieri omnino non potest, ut segmenta, quæ fiunt à perpendiculari cadente in basim, videlicet in lineam jugi, sint in eâdem Ratione; alioquin si basis segmenta essent in Ratione laterum adjacentium, angulus, ex quo perpendicularis demittitur, esset bifariam sectus, per 3 lib.6. atque adeò duo triangula haberent duos angulos duobus angulis æquales, nimirum rectum & acutum, atque latus haberent commune; ergo per 26.lib.1. & reliqua latera essent æqualia, contra hypothesim. Sit enim libra curva laterum inæqualium BAC, linea recta BC est vera linea jugi, in quam cadens perpendicularum AD definit brachiorum

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Mechanics 270 because, namely, the center of motion C is above the line DE joining the points of contact to which the weights are attached. But when the figure is turned so that C is in the lower position and the line DE in the upper, it remains in horizontal equilibrium alone; if removed from this, it does not return to it, nor does it have any equilibrium in an oblique position, as has been said. Now from the yoke AB let all superfluous parts be cut away, and let there remain the little rods CD and CE connected at the center of motion C: it is evident that the moments of the weights have not been changed, and that the motion of the curved balance DCE is the same as that of the straight one AB, whether C be understood in the upper part or in the lower. Therefore, with respect to equilibrium, the same things must be said of this curved one that were said of the balance having the cord above or below. And indeed, if those sides of which the curved balance is composed are equal according to their length, and of equal weight, then by adding equal weights here and there a horizontal equilibrium will result; because the true line of the yoke is divided into equal segments, and all the relations of equality are altogether similar. But if those sides are unequal, the weights are not to be added reciprocally (even when the weight of the sides themselves has been taken into account) in the relation of those lengths; but in the relation of the segments of the yoke, so that equilibrium may be made: because from the inequality of those sides it is indeed immediately inferred that the true line of the yoke is also divided into unequal segments; but it does not at once follow that the relation of inequality is similar. Indeed, if those sides are unequal, it is altogether impossible for the segments which are made by the perpendicular falling to the base, namely to the line of the yoke, to be in the same relation; otherwise if the segments of the base were in the relation of the adjacent sides, the angle from which the perpendicular is let down would be bisected, by 3 lib.6. and thus the two triangles would have two angles equal to two angles, namely a right angle and an acute one, and would have a side in common; therefore, by 26 lib.1., the remaining sides would be equal, contrary to the hypothesis. Let therefore there be a curved balance of unequal sides BAC, the straight line BC is the true line of the yoke, into which falling the perpendicular AD determines the arms

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Liber tertius. CAPUT V. 271 rum DB & DC longitudinem. Non est autem DB ad DC ut BA ad AC, alioquin angulus BAC esset bifariam sectus, & duo triangula DAB, DAC haberent præter rectos ad D, etiam acutos ad A æquales, atque latus AD commune, ac proinde essent etiam latera BA & AC æqualia contra hypo- thesim. Sunt igitur anguli ad A inæquales, & minor est, qui adja- cet minori lateri AC, quàm qui adjacet majori lateri AB: quia in triangulo BAC major est angulus C oppositus majori lateri BA, quàm angulus B oppositus minori lateri AC, ex 18. lib.1. igitur in triangulis BDA, CDA rectangulis ad D, comple- mentum CAD minus est complemento BAD. Qua propter si angulus BAC sit bifariam dividendus, recta AE auferet ali- quid ex majore angulo BAD, & constituens angulum BAE. cadet in basim inter B & D. Est itaque, per 3. lib.6. ut BA ad AC, ita BE ad EC: sed minor est Ratio BE ad EC quàm BD ad EC, & multo minor quàm BD ad DC. per 8. lib.5. igitur minor est Ratio BA ad AC, quàm sit Ratio brachij BD ad brachium DC. Si igitur pondera in C & B essent reciprocè ut BA ad AC, haberent minorem Rationem, quàm BD ad DC, ac propterea non essent apta ad constituendum æquilibrium horizontale. Retento igitur pondere B, augendum esset pon- dus C, vel retento pondere C, minuendum esset pondus B, ut essent in reciprocâ Ratione brachiorum BD & DC. Hinc etiam constat retentis eodem latere AB eademque li- neâ horizontali BC cum eodem angulo B, si velis uti minori pondere, quod cum pondere B faciat æquilibrium, addendum esse in A latus majus latere AC, puta latus AF, itaut tota BF sit jugi longitudo, & brachia sint BD & DF. Manifestum est autem ex 8. lib.5. majorem Rationem esse ejusdem BD ad DC minorem, quàm ad DF majorem; ad pondera debent esse in F & B ut BD ad DF; igitur minus pondus in F æquivalet eidem ponderi B, cui in C æquivalet pondus majus. Porrò nemini dubium esse potest, an latus AF majus sit latere AC, quippe quod in triangulo CAF opponitur angulo obtuso ACF, per 19. lib.1. Sed si res fuerit in praxim deducenda, indicare oportet, quâ methodo utendum sit, ut quæsitam ponderum Rationem, hoc est

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Book three. CHAPTER V. 271 the length DB and DC. But DB is not to DC as BA is to AC; otherwise the angle BAC would be bisected, and the two triangles DAB, DAC would have not only the right angles at D, but also equal acute angles at A, together with the common side AD; and therefore the sides BA and AC would also be equal, contrary to the hypothesis. Therefore the angles at A are unequal, and the smaller is that which is adjacent to the smaller side AC, than that which is adjacent to the greater side AB: because in triangle BAC the angle C, opposite the greater side BA, is greater than angle B, opposite the smaller side AC, by 18. Book 1. Therefore in the right triangles BDA, CDA at D, the complement of CAD is less than the complement of BAD. Wherefore if the angle BAC is to be divided into two equal parts, the straight line AE will take away something from the greater angle BAD, and, forming angle BAE, will fall on the base between B and D. It is therefore, by 3. Book 6, as BA is to AC, so is BE to EC: but the ratio BE to EC is smaller than BD to EC, and much smaller than BD to DC, by 8. Book 5. Therefore the ratio BA to AC is smaller than the ratio of the arm BD to the arm DC. If therefore the weights at C and B were reciprocally as BA to AC, they would have a smaller ratio than BD to DC, and therefore would not be suitable for establishing a horizontal equilibrium. Retaining therefore the weight B, the weight C would have to be increased; or, retaining the weight C, the weight B would have to be diminished, so that they might be in the reciprocal ratio of the arms BD and DC. Hence it is also evident that, the same side AB and the same horizontal line BC being retained, with the same angle B, if you wish to use a smaller weight which, together with weight B, may make equilibrium, there must be added at A a side greater than the side AC, namely the side AF, so that the whole BF may be the length of the beam, and the arms may be BD and DF. But it is manifest from 8. Book 5 that the ratio of the same BD to DC is greater when DC is smaller than when it is DF, greater; and the weights ought to be at F and B as BD is to DF; therefore a smaller weight at F balances the same weight B, to which a greater weight balances at C. Moreover, no one can doubt whether the side AF is greater than the side AC, since in triangle CAF it is opposite the obtuse angle ACF, by 19. Book 1. But if the matter is to be carried into practice, it ought to be indicated by what method it should be used, so that the required ratio of the weights, that is

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Mechanicorum est ipsa jugi segmenta inveniamus, quippe quod solâ mente concipitur ad laterum extremitates jungendas deductum. Hæc autem esse poterit praxis. Laterum A B & A C longitudines metire, tùm ex B ad C extentum funiculum ad similem mensuram revoca. His paratis certum est hanc jugi longitudinem communiter majorem esse longitudine singulorum laterum, semper tamen saltem alterius, tanto excessu, ut possit ab eâ auferri pars, de quâ mox dicetur; debet scilicet excedere medium proportionalem inter aggregatum laterum, & eorum differentiam. Cum enim linea jugi à perpendicularo cadente ex angulo verticali dividenda sit, utrumque latus cum jugo facit angulos acutos; alioquin si alteruter angulorum rectus esset, aut linea jugi non esset parallela horizonti, aut latus esset idem perpendicularum; & si obtusus esset, perpendicularum caderet extra lineam extremitates jungentem. Debet igitur tanta esse jugi longitudo, ut differentia partium, in quas dividitur ad differentiam laterum sit ut summa laterum ad totum jugum. Quare fiat ut jugi longitudo funiculo deprehensa ad laterum summam, ita laterum differentia ad partem auferendam ex longitudine jugi; cujus residuum bifariam divisum dabit minoris brachij longitudinem. Hujus operationis ratio manifesta est ex corollario primo prop. 36. lib. 3, & ex 3. ejusdem lib. 3. Sit exempli gratia latus A B partium 20, latus A C partium 9, distantia B C partium 23. Fiat ut 23 ad 29 summam laterum, ita laterum differentia 11 ad 13 20/23 partem auferendam ex jugi longitudine 23: Residuum partium 9 3, bifariam dividatur, & ejus semissis 4 13/23 est longitudo brachij minoris D C; quod reliquum est jugi partium 18 10/23 dat longitudinem alterius brachij majoris B D. Est igitur brachiorum (atque adeò etiam ponderum reciprocè) Ratio ut 424 ad 105. Quod si his cognitis investigare oporteat, quanta sit hujus lineæ horizontalis B C distantia à puncto suspensionis A, nimirum quanta sit perpendicularis A D, statim ex 47. lib. 1. innotescet, si ex quadrato lateris A C 81 auferas brachij D C quadratum 20 445/529; nam residuum 60 84/529 est quadratum perpendiculari A D, quod proinde est partium 7 17/23 proximè. At si pro ratione tui instituti nimia sit hujus perpendiculari longitudo,

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In mechanics, let us find the segments of the beam itself, since it is conceived only by the mind as extended to join the extremities of the sides. This practice may be as follows. Measure the lengths of the sides AB and AC, then bring back to a like measure the cord stretched from B to C. When these are prepared, it is certain that this length of the beam is commonly greater than the length of either side singly, yet always at least greater than one of them by so much that a part may be taken away from it, of which more will be said presently; indeed, it must exceed the mean proportional between the sum of the sides and their difference. For since the line of the beam, to be divided by the perpendicular falling from the vertical angle, each side makes acute angles with the beam; otherwise, if either angle were right, either the line of the beam would not be parallel to the horizon, or the side would be the same as the perpendicular; and if it were obtuse, the perpendicular would fall outside the line joining the extremities. Therefore the length of the beam must be such that the difference of the parts into which it is divided to the difference of the sides is as the sum of the sides to the whole beam. Wherefore let the length of the beam, found by the cord, be to the sum of the sides as the difference of the sides is to the part to be removed from the length of the beam; the remainder, being divided in half, will give the length of the smaller arm. The reason for this operation is clear from corollary 1 of proposition 36, book 3, and from proposition 3 of the same book 3. Let, for example, side AB be 20 parts, side AC 9 parts, and distance BC 23 parts. Then as 23 is to 29, the sum of the sides, so the difference of the sides, 11, is to 13 20/23, the part to be removed from the length of the beam 23: the remainder, 9 3 parts, being divided in half, its half, 4 13/23, is the length of the smaller arm DC; and what remains of the beam, 18 10/23 parts, gives the length of the other, larger arm BD. Therefore the ratio of the arms (and consequently also of the weights reciprocally) is as 424 to 105. But if, these things being known, it is necessary to investigate how great is the distance of this horizontal line BC from the point of suspension A, namely how great the perpendicular AD is, this will be known at once from book 1, proposition 47, if from the square of side AC, 81, you subtract the square of arm DC, 20 445/529; for the remainder, 60 84/529, is the square of the perpendicular AD, which therefore is about 7 17/23 parts. But if, according to the purpose of your design, the length of this perpendicular should be too great,

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Liber tertius. CAPUT V. 273 longitudo, & opportuniùs accidat jugum BC horizontale mi- nùs distare à puncto suspensionis A, jam constat latera AB & AC explicanda in majorem angulum; quapropter etiam major erit jugi longitudo, ex 24. lib.1. Sit ergo definita per- pendiculi AD altitudo partium 4: hujus quadratum 16 aufer ex 81 quadrato lateris AC, & residuum 65 est quadratum bra- chij minoris DC, quod idcircò est partium 8 1/16 ferè. Simili- ter ipsius AD quadratum 16 aufer ex 400 quadrato lateris AB, & residuum 384 est quadratum brachij majoris BD, quod est partium 19 3/9 proximè; & totum jugum BC est partium 27 25/30. Quare brachij BD ad brachium DC Ratio esset ut 764 ad 314, quæ reciprocè esset & ponderum. Ex quibus perspicuum est, positis iisdem libræ curvæ late- ribus, disparem esse ponderum Rationem: in priore enim posi- tione Ratio est 424 ad 105, hoc est proximè ut 4 ad 1. in poste- riore positione, ubi in majorem angulum laterâ explicantur, Ratio est 764 ad 314, hoc est ut 2. 43 ad 1; quæ minor est Ratio, quàm prior ut 4 ad 1. Si autem latera eadem essent in directum constituta, esset ponderum Ratio ut 20 ad 9, hoc est ut 2. 22 ad 1; quæ est minima Ratio omnium, quæ intercede- re possunt inter pondera æquilibrium horizontale constituen- tia ex illorum laterum extremitatibus: quæ extremitates quo- minùs distabunt, inflexis subinde latcribus, eo majus pondus requiretur in extremitate lateris brevioris, ut æquè ponderet cum uno eodemque pondere collocato in extremitate lateris longioris. Porrò ubi de ponderum Ratione sermo est, cave ne ipsorum laterum inæqualium libræ curvæ gravitatem contemnas; si enim æqualia illa essent, æqualia quoque essent eorum mo- menta tùm ratione gravitatis, tum tatione positionis, nam per- pendiculum caderet in medium jugum, & latera essent simi- liter inclinata, ac proinde sola ponderum æqualitas spectaretur: at laterum hujusmodi inæqualium momenta sunt ex utroque capite inæqualia, videlicet & ratione gravitatis insitæ, quæ ex hypothesi singulis lateribus inest pro Ratione molis inæqualis, & ratione positionis, quæ valde diversa est, cùm non sint late- ra illa simili angulo ad perpendiculum inclinata; sed magis in- M

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Book Three. CHAPTER V. 273 length, and the horizontal yoke BC more conveniently falls at a lesser distance from the point of suspension A, it is now clear that the sides AB and AC are to be opened to a greater angle; wherefore the length of the yoke will also be greater, from lib. 1, prop. 24. Let therefore the height of the perpendicular AD be defined as 4 parts: subtract its square, 16, from 81, the square of side AC, and the remainder, 65, is the square of the shorter arm DC, which is therefore about 8 1/16 parts. Likewise subtract the square of AD, 16, from 400, the square of side AB, and the remainder, 384, is the square of the longer arm BD, which is nearly 19 3/9 parts; and the whole yoke BC is 27 25/30 parts. Therefore the ratio of arm BD to arm DC would be as 764 to 314, which reciprocally would be that of the weights. From which it is evident that, with the same sides of the curved balance being set, the ratio of the weights is different: for in the former position the ratio is 424 to 105, that is, nearly as 4 to 1; in the latter position, where the sides are opened to a greater angle, the ratio is 764 to 314, that is, as 2.43 to 1; which is a smaller ratio than the former, as 4 to 1. But if the same sides were set in a straight line, the ratio of the weights would be as 20 to 9, that is, as 2.22 to 1; which is the least ratio of all that can intervene between the weights constituting a horizontal equilibrium from the extremities of those sides: the less those extremities are separated, as the sides are bent more and more, the greater a weight will be required at the extremity of the shorter side, so that it may weigh equally with one and the same weight placed at the extremity of the longer side. Moreover, when the ratio of the weights is under discussion, beware of neglecting the weight of the unequal sides of the curved balance themselves; for if those were equal, their moments would also be equal, both in respect of gravity and in respect of position, since the perpendicular would fall in the middle of the yoke, and the sides would be similarly inclined, and thus only the equality of the weights would be considered: but the moments of such unequal sides are unequal in both respects, namely both in respect of the inherent gravity, which by hypothesis belongs to each side in the ratio of its unequal mass, and in respect of position, which is very different, since those sides are not inclined to the perpendicular at a similar angle; but more in- M

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Mechanicorum 274 clinatur latus longius faciens cum perpendiculo majorem angulum: pro varia autem inclinatione ipsam ejusdem lateris gravitatem varia obtinere momenta manifestum videtur. Pona- mus laminam metallicam A B clavo infixam in A, circa quem quasi cen- trum describat semicirculum B D C. Si obtineat perpendicularem positionem A B, tota gravitas innititur clavo A sustinenti, & nullam vim habet de- scendendi; similiter in perpendiculari positione A C tota gravitas retinetur à clavo A, nec potest descendere. At si positionem habeat A D horizonti pa- rallelam, omnino nec sustinetur, nec retinetur à clavo, sed toto conatu suas descendendi vires exerit. In locis igitur intermediis partim sustinetur aut retinetur à clavo A, partim conatum deorsum exercet: sic ex B veniens in E sustinetur juxta mensuram F E, & deorsum tendit juxta mensuram G E; at ex B veniens in H sustinetur juxta mensuram I H, & deorsum tendit juxta mensuram K H. Simili modo contingit in quadrante in- feriore; nam in positione A L retinetur juxta mensuram I L, nec descensum potest habere nisi ut L M; atque in O impedimentum à retinente est ut F O, conatum deorsum metitur O N. Quia scilicet si ab aliquo sustineatur in L, perinde se habet ac si esset in plano habente inclinationis angulum C A L; in quo plano gravitatio est ad gravitationem in perpendiculo ut Radius ad secantem, seu ut Sinus Complementi ad Radium, hoc est ut IL ad AL: ac propterea vires clavi retinentis in eâ inclinatione ad vires retinentis in perpendiculo debent esse ut IL ad AC, hoc est ad AL: At gravitatio, quâ urgetur planum inclinatum, est ut PC Sinus Versus anguli inclinationis, qui planè æqualis est ipsi LM. Cùm autem hîc nullum habeatur subjectum planum, quod prematur à gravitante laminâ metallicâ, exerit hunc conatum deorsum adversùs aliud oppositum pondus, quod elevare conatur, vel cui conanti resistit, ne ab eo elevetur. Si igitur in lineâ AC perpendiculari lamina AC contra

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Mechanics 274 the farther side is inclined, making a greater angle with the perpendicular: but according to the different inclination, it is evident that the gravity of that same side obtains different moments. Let us suppose a metal plate A B fixed by a nail in A, around which, as a center, it may describe the semicircle B D C. If it holds the perpendicular position A B, the whole weight rests upon the supporting nail A and has no force to descend; similarly, in the perpendicular position A C the whole weight is retained by the nail A, nor can it descend. But if it has the position A D parallel to the horizon, it is neither supported nor retained by the nail at all, but exerts its descending force with its whole effort. Therefore, in the intermediate places it is partly supported or retained by the nail A, and partly exerts a downward tendency: thus, coming from B to E, it is supported according to the measure F E, and tends downward according to the measure G E; but coming from B to H, it is supported according to the measure I H, and tends downward according to the measure K H. In like manner this happens in the lower quadrant; for in the position A L it is retained according to the measure I L, and can have no descent except as L M; and in O the impediment from the retainer is as F O, and it measures the downward tendency O N. For if it be supported by something at L, it is the same as if it were in a plane having the angle of inclination C A L; in which plane the gravitation is to the gravitation in the perpendicular as radius to secant, or as the sine of the complement to the radius, that is, as I L to A L: and therefore the forces of the retaining nail in that inclination to the forces of the retaining nail in the perpendicular must be as I L to A C, that is, to A L. But the gravitation by which an inclined plane is urged is as P C, the versed sine of the angle of inclination, which is plainly equal to L M itself. But since here there is no underlying plane, which is pressed by the gravitating metal plate, it exerts this downward tendency against another opposite weight, which it tries to raise, or against which, when trying to rise, it resists, so that it may not be lifted by it. If therefore, on the perpendicular line A C, the plate A C against

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Liber tertius. CAPUT V. 275 contra clavum A exercet momenta totius gravitatis deorsum nitentis, & in AL impeditur, ac retinetur secundùm mensuram IL, fiat ut AC ad IL, ita tota gravitas laminæ ad aliud, & prodibit quantitas gravitationis contra retinentem, residuumque LM erit illa gravitatio, quæ consideranda est in eâ positione inclinata AL. Sed quoniam AL à centro motûs A distantiam habet AI, comparanda erit hæc distantia cum distantia oppositi lateris libræ, ut habeantur momenta invicem comparata. Observandum tamen est non rem perinde se habere, ac si tota gravitatio laminæ inclinatæ AL posita esset in L, atque adeò in distantiâ AI; sed quia distribuitur secundùm totam ipsam longitudinem AL, & partes remotiores plus habent momenti, quàm propiores centro, juxtà Rationem distantiarum, proptereà vel tota gravitas lateris AL, quæ est LM, intelligenda est in mediâ distantiâ inter A & I, vel semissis gravitationis AL, hoc est semissis ipsius LM, intelligendus est in I, quemadmodum hujus libri 3. cap. 2. dictum est totam gravitatem AD intelligendam in mediâ distantiâ inter A & D, aut ejus semissem in extremitate D. Quamvis autem ex inclinatione CAL oriatur distantia AI, hæc tamen venire pariter in computationem debet, quia comparari debent hæc momenta cum momentis distantiæ oppositæ, quæ momenta orta ex Ratione distantiarum eadem sunt, sive AL sit lamina, sive trabs; quamquam valde dispares sint gravitates, quæ assumendæ sunt ex eâdem inclinatione; ac propterea & LM indicans gravitationem comparatè ad totam gravitatem absolutam, & AI definiens momentum ex distantiâ, considerari debent. Hoc pacto habetur totum momentum lateris AL; similiterque habebitur momentum lateris oppositi. Ex quo patet laterum inclinatorum in librâ curvâ momenta componi & ex Ratione distantiarum, & ex Ratione momenti, quod habent singula latera ex inclinatione ad perpendiculum. At subdubitas, utrum ista, quæ hîc dicuntur, cum iis aptè cohæreant, quæ lib. 1. cap. 15. dicta sunt, ubi ponderis in L constituti vires ad descendendum definiri diximus à Sinu anguli declinationis à perpendiculo CAL, qui æqualis est ipsi AI: hîc verò laminæ AL gravitationem constituimus ex Mm 2

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Book three. CHAPTER V. 275 against the pin A it exerts the moments of the whole gravity tending downward, and in AL it is impeded and retained according to the measure IL, so that AC is to IL as the whole gravity of the plate is to the other, and there will come forth the quantity of gravitation against the retainer, and the remainder LM will be that gravitation which is to be considered in that inclined position AL. But since AL has from the center of motion A the distance AI, this distance must be compared with the distance of the opposite side of the balance, so that the moments compared with one another may be had. It must nevertheless be observed that the matter is not as if the whole gravitation of the inclined plate AL were placed in L, and therefore at the distance AI; but because it is distributed through the whole length of AL itself, and the more remote parts have more moment than the nearer parts to the center, according to the ratio of distances, therefore either the whole gravity of the side AL, which is LM, is to be understood at the mean distance between A and I, or half of the gravitation of AL, that is, half of LM itself, is to be understood in I, just as in the 3rd chapter of this book, chap. 2, it was said that the whole gravity AD is to be understood at the mean distance between A and D, or its half at the extremity D. Although, however, from the inclination CAL the distance AI arises, this must nevertheless also come into the calculation, because these moments must be compared with the moments of the opposite distance, which moments arising from the ratio of distances are the same, whether AL be a plate or a beam; although the gravities to be assumed from the same inclination are very different. And therefore both LM, indicating the gravitation relatively to the whole absolute gravity, and AI, defining the moment from the distance, must be considered. In this way the whole moment of the side AL is obtained; likewise the moment of the opposite side will be obtained. From this it is clear that the moments of the inclined sides in a curved balance are composed both from the ratio of distances and from the ratio of the moment which the individual sides have from the inclination to the perpendicular. But you doubt whether the things here said cohere suitably with those which were said in book 1, chap. 15, where we said that the powers of a weight placed in L for descending are defined by the sine of the angle of declination from the perpendicular CAL, which is equal to AI: here, however, we establish the gravitation of the plate AL from the sine of the angle ... Mm 2

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Mechanicorum 276 Sinu complementi ejusdem anguli CAL, nimirum ex li- neâ IL. Quapropter observa non eandem esse rationem gravitationis lateris AL libræ, atque ponderis adnexi in extremitate L; hu- jus enim momenta perinde computantur, ac si esset in I; quia scilicet AI æqualis est brachio libræ PL, & planum inclina- tum, in quo pondus L constitutum intelligitur, non est AL, sed Tangens in L ad angulos rectos, ut loco citato explicatum est. At libræ latus AL suam habens gravitatem aliter se habet: nam quemadmodum si inniteretur clavo in A, non tamen illi infigeretur, atque ab aliquo sustineretur in puncto L, certum est planum inclinatum, in quo moveretur, esse AL, contra quæ momenta descendendi in plano inclinato reluctatur clavus in A positus, & retinens; ita sublato sustinente in L, & posito contranitente reliquo latere libræ, non tollitur munus clavi A retinentis, sed substituitur latus illud oppositum loco sustinen- tis in L: igitur contra illud latus hoc latus AL exercet eadem momenta gravitationis, quæ exerceret adversùs sustinentem in L, hoc est in planum inclinatum; quæ momenta ea sunt, quæ remanent demptis IL momentis gravitationis in plano in- clinato, nimirum residuum LM. Quia verò qui sustineret la- tus AL in L, non esset unicum sustinens, sed planum inclina- tum est AL, & ita latus retinetur in clavo A, ut etiam ab eo aliquatenus sustineatur, atque adeò lamina inclinata sustinea- tur à duobus in A & L, retineaturque solùm ab A; propterea non totum momentum LM, sed ejus semissem accipiendum diximus, ut habeantur momenta, quibus contranititur oppo- situm latus, si addantur momenta, quæ oriuntur ex distantiâ à centro motûs, ut dictum est. Hæc autem ut exemplo claria fiant, sint eadem, quæ priùs in præcedente figurâ posita sunt, latera libræ curvæ BAC, lon- gius BA partium 20, brevius CA partium 9, & quidem in eâ positione, ut perpendiculum AD cadens in jugum sit partium 7 17/23, & brachium jugi DC adjacens minori lateri sit partium 4 13/23, reliquum verò jugi brachium DB partium 18 10/23. Primùm quære momenta laterum ex eorum inclinatione: Cumque per- pendiculum AD sit æquale Sinui Complementi anguli incli- nationis

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Mechanics 276 by the sine of the complement of the same angle CAL, namely from the line IL. Therefore observe that the ratio of the weight of the side AL of the balance is not the same as that of the weight attached at the end L; for the moments of the latter are computed just as if it were at I, because, namely, AI is equal to the arm PL of the balance, and the inclined plane, on which the weight L is understood to be placed, is not AL, but the tangent at L at right angles, as was explained in the cited place. But the side AL of the balance, having its own weight, is otherwise situated: for just as if it rested upon a nail at A, though not driven into it, and were supported by something at the point L, it is certain that the inclined plane on which it would move is AL, against which the nail placed at A, holding it fast, resists the moments of descent along the inclined plane; so, when the support at L is removed and the opposite side of the balance is placed to resist, the function of the nail holding at A is not taken away, but that opposite side is substituted in the place of the support at L. Therefore this side AL exercises against that side the same moments of gravitation that it would exercise against a supporter at L, that is, against an inclined plane; and these moments are those which remain after the moments IL of gravitation on the inclined plane have been subtracted, namely the remainder LM. But because the one who would support the side AL at L would not be the only support, since the inclined plane is AL, and thus the side is retained at the nail A, so that it is also in some measure supported by it, and accordingly the inclined plate is supported by two supports at A and L, but is held fast only by A, for that reason we have said that not the whole moment LM, but half of it, is to be taken, so that there may be obtained the moments by which the opposite side resists, if there are added the moments which arise from the distance from the center of motion, as was said. Now, to make these things clearer by example, let there be the same sides of the curved balance BAC as were placed before in the preceding figure, the longer BA of 20 parts, the shorter CA of 9 parts, and indeed in such a position that the perpendicular AD falling on the yoke is of 7 17/23 parts, and the arm DC of the yoke adjacent to the smaller side is of 4 13/23 parts, while the remaining arm DB of the yoke is of 18 10/23 parts. First seek the moments of the sides from their inclination: and since the perpendicular AD is equal to the sine of the complement of the angle of inclination

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Liber tertius. CAPUT V. 277 nationis DAC, posito Radio AC, notus est Sinus Versus ejusdem anguli inclinationis, scilicet differentia inter AD & AC, quæ est partium 1 6/23: & simili methodo Sinus Versus anguli inclinationis DAB est partium 12 6/23. Ratio igitur gravitationis lateris AB ad gravitationem lateris AC ex inclinatione est ut 282 ad 29; Ratio momentorum ex distantiâ à centro, ut supra diximus, est ut 424 ad 105. Compositis igitur duabus hisce Rationibus, est totius momenti lateris AB ad totum momentum lateris AC Ratio ut 119568 ad 3045, hoc est in minimis terminis ut 39.267" ad 1. Sit igitur gravitas absoluta lateris AB unciarum 20; gravitatio respondens semissi Sinus Versi anguli inclinationis est unciarum 6 3/23. Item gravitas absoluta lateris AC sit unc. 9: gravitatio respondens semissi Sinus Versi anguli inclinationis est unc. 29/46. Hæc gravitatio 29/46 ducatur in distantiam à perpendicularo partium 4 13/23, & est momentum 2.878". Similiter gravitatio unc. 6 3/23 ducatur in distantiam à perpendicularo partium 18 10/23, & est momentum 113.013". Diviso itaque majore numero 113013 per minorem 2878, in minimis terminis Ratio est ut 39.268" ad 1: quæ minimum differt à priore illa Ratione propter neglectas fractiunculas in divisionibus. Nunc inquiramus, quantum ponderis addendum sit lateri minori, ut fiat æquilibrium cum solâ majoris lateris gravitate. Statuatur pondus addendum Algebricè 1 Rx, cujus distantia à perpendicularo cum sit partium 4 13/23, ponderis additi momentum est 105/23 Rx addendum momento lateris minoris invento. Quare 2.878" + 105/23 Rx æquantur momento 113.013" lateris majoris: & utrinque demptis 2.878", remanet æquatio inter 105/23 Rx & 110.135". Demum institutâ divisione prodit pretiu[m] 1 Rx, hoc est ponderis addendi, unciarum 24 1/8. Huic itaque ponderi additâ gravitatione lateris minoris AC unc. 29/46 hoc est in millesimis 630", erit in C totum pondus unc. 24.755"; & in B intelligitur gravitas unc. 6 3/23, hoc est in millesimis unc. 6.130" ferè. Vides igitur hæc pondera esse reciprocè posita in Ratione distantiarum DB & DC: & quamvis demum in his Rationibus non sibi exactissimè respondeant numeri, satis pa- Mm 3

Transcription: Translated (English)

Book third. CHAPTER V. 277 In triangle DAC, with radius AC laid down, the Versed Sine of the same angle of inclination is known, namely the difference between AD & AC, which is 1 6/23 parts: & by a similar method the Versed Sine of the angle of inclination DAB is 12 6/23 parts. Therefore the ratio of the gravitation of side AB to the gravitation of side AC from inclination is as 282 to 29; the ratio of the moments from distance from the center, as we said above, is as 424 to 105. Combining therefore these two ratios, the ratio of the whole moment of side AB to the whole moment of side AC is 119568 to 3045, that is in the least terms as 39.267" to 1. Let therefore the absolute gravity of side AB be 20 ounces; the gravitation corresponding to half the Versed Sine of the angle of inclination is 6 3/23 ounces. Likewise let the absolute gravity of side AC be 9 oz.; the gravitation corresponding to half the Versed Sine of the angle of inclination is 29/46 oz. Let this gravitation 29/46 be multiplied by the distance from the perpendicular of 4 13/23 parts, & the moment is 2.878". Likewise let the gravitation of 6 3/23 oz. be multiplied by the distance from the perpendicular of 18 10/23 parts, & the moment is 113.013". Dividing therefore the greater number 113013 by the lesser 2878, in the least terms the Ratio is as 39.268" to 1: which differs only a little from the former Ratio because of neglected little fractions in the divisions. Now let us inquire how much weight must be added to the smaller side, that an equilibrium may be made with the gravity of the greater side alone. Let the weight to be added be set down algebraically as 1 Rx, whose distance from the perpendicular being 4 13/23 parts, the moment of the added weight is 105/23 Rx to be added to the moment of the smaller side already found. Wherefore 2.878" + 105/23 Rx are equal to the moment 113.013" of the greater side: & subtracting 2.878" from both sides, there remains an equation between 105/23 Rx & 110.135". At length, by division, there comes out the price of 1 Rx, that is the weight to be added, of 24 1/8 ounces. To this weight therefore, adding the gravitation of the smaller side AC, 29/46 oz., that is in thousandths 630", the total weight in C will be 24.755 oz.; & in B is understood the gravity of 6 3/23 oz., that is in thousandths about 6.130". You see therefore that these weights are reciprocally placed in the Ratio of the distances DB & DC: & although at length in these Ratios they do not respond exactly to one another, the numbers sufficiently agree. Mm 3

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Mechanicorum 278 tet exiguum hoc discrimen oriri ex neglectis fractiunculis. Cæterùm hæc tam minutè persequi in librâ curvâ, cujus latera non adeò notabili gravitate sunt prædita, labor quidem videtur inutilis: sed quoniam hujusmodi libræ præcipuus usus esse potest in machinationibus, ubi latera libræ sunt tigilli crassiores non mediocris gravitatis, operæ pretium fuit indicare, quâ methodo ipsorum laterum gravitates & momenta computari oporteat, ut non casu, sed ex certâ ratione pondera collocentur, & æquipondia statuantur. CAPUT VI. Quænam libræ sint omnium exactissimæ. Instrumenti cujusque bonitas æstimatur ex fine, ad quem fuit institutum, prout ad illum assequendum aptum fuerit, aut ineptum, eóque melius censetur instrumentum, quò certiùs per illud propositus finis obtinetur; quemadmodum per singula eunti facilè constabit. Ut igitur exactissimum libræ genus innotescat, satis patet inquirendum esse, quænam libra facillimè ab æquilibrio recedat; quo recessu indicans vel minimam ponderum inæqualitatem, etiam suo æquilibrio exquisitam ponderum inæqualitatem ostendit; id quod per libram vestigamus. Hîc autem de librâ inæqualium brachiorum sermo est, quâ communiter uti solemus: quamquam aliqua etiam ad libram inæqualium brachiorum proportione traduci queant. Ex duplici capite libram, quâ libra est, ponderum gravitates præ aliis libris exquisitè examinare contingit, videlicet aut ex brachiorum longitudine, aut ex sparti, seu centri motûs, positione; reliqua enim impedimenta, aut adjumenta materiam potiùs sequuntur, quàm libræ formam. Et quidem quod ad brachiorum longitudinem spectat, adeò certum Aristoteli videtur majoribus libris, majori scilicet brachiorum longitudine præditis, accuratiùs examinari ponderum æqualitatem, ut in Mechanicis quæstionibus hoc primum ab

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Mechanics 278 this small difference arises from neglected little fractions. But to pursue these things so minutely in a curved balance, whose sides are not endowed with weight of any notable amount, seems indeed a useless labor: yet since the chief use of a balance of this kind may be in machines, where the sides of the balance are thicker beams of no small weight, it was worth while to indicate by what method the weights and moments of those sides themselves are to be computed, so that the weights may be placed, and the counterweights set, not by chance, but by a certain rule. CHAPTER VI. Which balances are the most exact of all. The goodness of any instrument is estimated from the end for which it was instituted, according as it is fit or unfit for attaining that end; and the instrument is judged the better, the more certainly the intended end is achieved by it, as will easily be clear to anyone proceeding step by step. Therefore, in order that the most exact kind of balance may be known, it is sufficiently clear that inquiry must be made as to which balance most easily departs from equilibrium; for by that departure, indicating even the smallest inequality of weights, it also shows, by its own equilibrium, the most exact inequality of weights; this is what we investigate by means of the balance. Here, however, the discussion is of the balance with unequal arms, which we commonly use: although some things may also be transferred to a balance of unequal arms by proportion. From a double cause it happens that a balance, insofar as it is a balance, examines with exactness the weights of bodies above others balances, namely either from the length of the arms, or from the position of the spatium, or center of motion; for the remaining impediments or aids follow rather the material than the form of the balance. And indeed, as regards the length of the arms, it seemed so certain to Aristotle that in larger balances, that is, those furnished with greater length of arms, the equality of weights is examined more accurately, that in the Mechanical Questions this is first of all by

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Liber tertius. CAPUT VI. 279 ab eo quæratur, Cur majores libra exactiores sunt minoribus? Causam autem ex eo desumendam putat, quòd spartum sit centrum, brachia verò quasi lineæ à centro exeuntes; & quia Radij longiores ab eodem centro cum brevioribus exeuntes si pariter moveantur, majorem arcum describunt, propterea etiam citius moveri necesse est extremitatem librae, quò plus à sparto discesserit. Hinc est in minore libra posse aliquando ex tenui inæqualitate ponderum fieri motum non conspicuum, atque adeò illam occultè discedere ab æquilibrio; id quod in majore libra contingere non potest, quia longioris brachij extremitas notabili motu inclinatur. Sit enim libra longior AB, cujus spartum sit C; moveatur, & describat arcus BG, & AF, qui sunt multò magis conspicui & majores, quàm qui à libra minore DE habente idem motûs centrum C, describantur arcus EI & DH. Constat igitur motum puncti E prorsus fugere omnem oculorum aciem, si motus extremitatis B vix sit conspicuus. Ex quo illud etiam consequens est, quod major libra clariùs indicat æquilibrium. Verùm si hæc ita accipiantur, prout communi huic interpretationi subest Aristoteles, vix aliquid habent momenti: quis enim pondera vix inæqualia bilance subtiliter examinans jugi extremitates respicit, ut videat, an lineæ horizonti parallelæ congruat jugum? & non potiùs lingulam CO considerat, an cum ansâ perpendiculari illa conveniat? Quod si lingula attendatur, idem est ejus motus sive longior sit libra AB, sive brevior DE; factâ enim inclinatione aut majore motu BG, aut minore motu EI, eadem est lingulæ positio CS. Hoc tantùm habent emolumenti brachia longiora, quod faciliùs dividuntur bifariam æqualiter quàm breviora: & si minimum aliquod discrimen intercedat, hoc minorem habet Rationem ad brachium longiùs, quàm ad brevius. Quare aliâ ratione accipienda est libra: nam si in uno eodemque puncto C conveniant spartum & jugi divisio, aut spartum sit inferius, sive longiora, sive breviora sint brachia, ponderum inæqualitas illicò innotescit, quia extremitas præponderans, ad imum locum, quan- tum

Transcription: Translated (English)

Liber tertius. CHAPTER VI. 279 It is asked of it, Why are larger balances more exact than smaller ones? He thinks, however, that the cause is to be drawn from the fact that the fulcrum is the center, while the arms are as it were lines running out from the center; and because rays longer, issuing from the same center together with shorter ones, if moved equally, describe a larger arc, for that reason also the extremity of the balance must move more quickly, the farther it has departed from the fulcrum. Hence it is that in a smaller balance, from a slight inequality of weights, a motion not perceptible may sometimes be produced, and so it may secretly depart from equilibrium; which cannot happen in a larger balance, because the extremity of the longer arm is inclined by a noticeable motion. For let the longer balance be AB, whose fulcrum be C; let it be moved and describe the arcs BG and AF, which are much more conspicuous and larger than those which are described by the smaller balance DE, having the same center of motion C, namely the arcs EI and DH. It is therefore clear that the motion of point E completely escapes all sight, if the motion of extremity B is scarcely perceptible. From this it also follows that the larger balance more clearly indicates equilibrium. But if these things are taken in the way Aristotle is understood in this common interpretation, they have scarcely any weight: for who, when subtly testing with a balance weights that are scarcely unequal, looks to the ends of the arms in order to see whether the beam agrees with the line parallel to the horizon? And does he not rather consider the pointer CO, whether it agrees with the perpendicular handle? But if the pointer is attended to, its motion is the same whether the balance AB is longer or DE shorter; for once inclination has been made, either with a greater motion BG or with a smaller motion EI, the position of the pointer CS is the same. The longer arms have only this advantage, that they can more easily be divided exactly into two equal parts than shorter ones: and if some very small difference intervene, this has less relation to a longer arm than to a shorter one. Therefore the balance must be understood in another way: for if the fulcrum and the division of the beam meet in one and the same point C, or if the fulcrum is lower, whether the arms are longer or shorter, the inequality of the weights is immediately made known, because the preponderating extremity, toward the lower place, as much as

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Mechanicorum tum potest, descendit. Locutus igitur videtur Aristoteles de librâ spartum habente in superiore jugi loco extrà lineam, quæ jugi longitudinem definit. Sit iterum libra longior AB, & brevior DE, utraque bifariam divisa in C; & sit linea lingulæ perpendicularis CK, in quâ sumatur spartum, seu motûs centrum O, & residuum OK sit lingula, ex cujus declinatione à perpendiculo ansæ, dignoscitur sublatum æquilibrium. Sit pondus A ad pondus B ut 5 ad 3: centrum gravitatis jugi & ponderum commune non potest esse C, quod brachia CA & CB æqualia constituit; sed erit ut pondus A ad pondus B, ita reciprocè longitudo BG ad longitudinem GA, eritque punctum G centrum gravitatis, nec libra consistet, nisi recta GOH fiat perpendicularis horizonti: lingula igitur OK declinabit à perpendiculo ansæ juxta angulum HOK. Eadem pondera transferantur in minorem libram DE; & si fiat ut pondus D5 ad pondus E3, ita EF ad FD, erit F centrum gravitatis libræ DE & ponderum: quare libra non consistet, nisi recta FOI sit horizonti perpendicularis, & tunc à perpendiculo declinabit lingula OK juxta angulum IOK. Quoniam verò est ut 4 ad 1, ita AC ad CG, ita DC ad CF, & AC major est quàm DC, erit etiam ex 14 lib.5. GC major quàm FC; igitur angulus COF minor est angulo COG, pars minor toto; ac proinde ad verticem angulus KOI minor est angulo KOH. Positis igitur ponderibus iisdem in libræ longioris AB extremitatibus, declinabit lingula à perpendiculo, cum eo constituens angulum majorem, quàm sit angulus ab eadem lingulâ constitutus cum perpendiculo, quando pondera illa inæqualia adnectuntur libræ breviori DE. Hinc est quòd si inæqualitas ponderum exigua sit, centrum gravitatis in utrâque librâ non multùm recedat à puncto C, parùm in majore, minimùm in minore, ac proinde lingulæ deflexio fortasse inobservabilis erit in minore librâ, quæ in majore evadet notabilis atque conspicua. Hinc etiam patet, cur extremitas A descendens magis moveatur, quàm extremitas D minoris libræ; quia scilicet angulus OGA, per 16. lib.1. major est quàm angulus

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When it can, it descends. Aristotle therefore seems to have spoken of a balance having a plumb line in the upper place of the beam, outside the line which determines the length of the beam. Let there again be a longer balance AB, and a shorter one DE, each divided equally at C; and let CK be the perpendicular line of the tongue, on which let the plumb line, or center of motion, O, be taken, and let the remainder OK be the tongue, from whose deviation from the perpendicular of the handle the lifting of the equilibrium is recognized. Let the weight A be to the weight B as 5 to 3: the common center of gravity of the beam and weights cannot be at C, since that would make the arms CA and CB equal; but as the weight A is to the weight B, so reciprocally is the length BG to the length GA, and the point G will be the center of gravity, and the balance will not stand unless the straight line GOH is perpendicular to the horizon: therefore the tongue OK will deviate from the perpendicular of the handle according to the angle HOK. Let the same weights be transferred to the smaller balance DE; and if, as the weight D5 is to the weight E3, so EF is to FD, F will be the center of gravity of the balance and the weights: wherefore the balance will not stand unless the straight line FOI is perpendicular to the horizon, and then the tongue OK will deviate from the perpendicular according to the angle IOK. But since, as 4 is to 1, so are AC to CG and DC to CF, and AC is greater than DC, it will also be, from lib. 5, prop. 14, that GC is greater than FC; therefore the angle COF is smaller than the angle COG, the smaller part being less than the whole; and consequently at the vertex the angle KOI is smaller than the angle KOH. Therefore, if the same weights are placed at the ends of the longer balance AB, the tongue will deviate from the perpendicular, forming with it a greater angle than the angle formed by the same tongue with the perpendicular, when those unequal weights are attached to the shorter balance DE. Hence it is that if the inequality of the weights be slight, the center of gravity in either balance will not recede much from point C, little in the larger one, least in the smaller, and therefore the deflection of the tongue may perhaps be imperceptible in the smaller balance, while in the larger it becomes noticeable and conspicuous. Hence also it is clear why the end A, descending, moves more than the end D of the smaller balance; because, namely, the angle OGA, by lib. 1, prop. 16, is greater than the angle

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Liber tertius. CAPUT VI. 281 angulus OFD, ac propterea ubi O G facta sit perpendicularis, linea AG cum illâ faciens obtusiorem angulum, magis depri- metur infrà lineam AB horizontalem. Sed jam inquirendum est, utrum expediat centrum motûs magis distare à lineâ jugi, an verò illi propiùs admoveri, ut clariùs innotescat recessûs jugi ab æquilibrio horizontali: illa quippe sparti positio eligenda est, quæ etiam minimum mo- tum indicet notabili lingulæ declinatione. Dico itaque spar- tum lineæ jugi proximum utilis esse, quàm remotum. Sit enim libra AB bifariam in C di- visa, & ex hoc puncto exeat per- pendicularis CI; in quâ pro cen- tro motûs eligatur punctum S; ponantur verò pondera A & B ita esse inæqualia, ut centrum gravi- tatis commune sit D. Igitur DSR est linea, quæ facta perpendicularis constituit cum lingulâ SI angulum ISR. Deinde reliquis omnibus manentibus, sit cen- trum motûs O remotius à lineâ jugi, & linea DOV facta per- pendicularis declinabit à lingulâ OI juxta angulum IOV, quem constat esse minorem angulo ISR; nam angulus DSC externus major est interno DOS, per 16. lib. 1. est autem huic ad verticem IOV, & illi ad verticem ISR; igitur ISR angu- lus est major angulo IOV. Quòd si centrum motûs adhuc propiùs admoveatur medio jugi puncto C, adhuc majorem angulum constituet cum lin- gulâ, ac proptereà adhuc multò notabilior erit deflexio lingu- læ à perpendicularo, etiam si exiguus sit motus ex eo, quod cen- trum gravitatis D proximè accedat ad punctum C: est siqui- dem extrà controversiam, quò minor est ponderum inæquali- tas, eò etiam minorem esse puncti D à puncto C distantiam. Ex quo manifestum evadit exiguam ponderum differentiam non dignosci, si spartum notabili intervallo recesserit à lineâ ju- gi; hæc enim sparti distantia habet rationem Radij, distantia centri gravitatis à medio jugi locum obtinet Tangentis; igitur si fiat major sparti distantia, eadem Tangens ad majorem Ra- dium minorem Rationem habebit, atque adeò subtendet mul- tò acutiorem angulum, qui proptereà minùs observari poterit. Nn

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Liber tertius. CAPUT VI. 281 angle OFD, and therefore, where O G has been made perpendicular, the line AG, making with it an obtuse angle, will be depressed more below the horizontal line AB. But now it must be inquired whether it is advantageous for the center of motion to be placed farther from the line of the beam, or rather to be brought nearer to it, so that the recession of the beam from horizontal equilibrium may be made clearer: for that position of the index is to be chosen which shows even the slightest motion by a noticeable declination. I say, therefore, that a beam near the line of the beam is more useful than one farther away. For let the balance AB be divided equally at C, and from this point let the perpendicular CI arise; on this let point S be chosen as the center of motion; but let the weights A and B be such unequal that their common center of gravity is D. Therefore DSR is the line which, when made perpendicular, forms with the index SI the angle ISR. Next, all other things remaining the same, let the center of motion O be farther from the line of the beam, and the line DOV, when made perpendicular, will decline from the index OI according to the angle IOV, which it is clear is smaller than the angle ISR; for the angle DSC, external, is greater than the internal DOS, by 16, book 1; but this angle is opposite at the vertex to IOV, and that one to ISR; therefore the angle ISR is greater than the angle IOV. But if the center of motion be brought still closer to the middle point C of the beam, it will make still a greater angle with the index, and therefore the deflection of the index from the perpendicular will be still much more noticeable, even if the motion be slight, because the center of gravity D approaches point C closely: for beyond controversy, the smaller the inequality of the weights, the smaller also is the distance of point D from point C. From this it becomes clear that a slight difference of weights cannot be discerned if the beam has receded from the line of the beam by a noticeable interval; for this distance of the beam has the relation of a radius, the distance of the center of gravity from the middle of the beam takes the place of a tangent; therefore if the distance of the beam be made greater, the same tangent, to a greater radius, will have a smaller ratio, and so it will subtend a much sharper angle, which therefore will be less observable. Nn

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Mechanicorum 282 Quare pro eâdem ponderum inæqualitate dignoscendâ, si concurrant minima sparti à jugo distantia, & ob longitudinem majorem brachiorum libræ major centri gravitatis distantia à medio jugi puncto, patet multò faciliùs dignosci inæqualia esse pondera, quia majore angulo linea deflectit à perpendiculo; & posito minimo Radio Tangens major angulo majori opponitur. Hæc quidem de librâ spartum habente suprà lineam jugi dicta accommodari possunt libræ spartum habenti infrà jugi lineam, si eadem schemata inverso situ posita intelligantur: quò enim majore angulo deflectit à perpendiculo linea jungens gravitatis centrum, & centrum motûs, eò faciliùs brachium, in quo est gravitatis centrum, inclinatur. Verùm si duplex hæc libræ species, quæ suprà, & quæ infrà jugi lineam spartum habet, invicem comparetur, satis apertum est multò faciliùs à posteriore hâc specie indicari ponderum inæqualitatem; quia videlicet si centrum gravitatis in alterutram partem vel minimum recedat à medio jugi, non ampliùs imminet sparto in eodem perpendiculo, neque potest sustineri, sed illicò, quantum potest ad imum locum descendit. At in priore illâ specie libræ spartum in superiore loco habentis, recedente in alterutram partem centro gravitatis, descendit illud quidem; sed non nisi pro ratione excessûs ponderis; qui descensus inobservabilis erit, si exigua sit ponderum differentia. Hinc non semel animadverteri accuratissimas bilances, quibus aurearum monetarum pondera examinantur, eas esse, quæ spartum in inferiore loco habent; lanx enim, quæ pondere prægravatur, ad imum, quantum potest descendit: factâ autem libræ conversione ita, ut ansa inferiùs sustentata libram sustineat, iisdemque ponderibus impositis, lanx prægravata non descendit ad imum locum; sed manet libra in obliquâ positione, quæ ponderum inæqualitati congruè respondet; &, si ea sit ponderum inæqualitas, quæ omnem observantis subtilitatem effugiat, videtur libra in æquilibrio horizontali posita, cum tamen in priore situ, antequam libra inverteretur, non posset in ullo æquilibrio consistere. Non ita tamen hæc dicta intelligi velim, ut nulla sit habenda ratio materiæ, ex qua libra constat; hæc siquidem tantæ gravitatis esse potest, ut axem vehementiùs premens motum aliqua- tenus impediat, ac propterea levis illa virtus effectiva motûs, qui

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Mechanics 282 Therefore, in detecting the same inequality of weights, if the smallest distance of the beam from the fulcrum, and, because of the greater length of the arms, the greater distance of the center of gravity from the middle point of the fulcrum, are involved, it is clear that the unequal weights are much more easily detected, because the line deviates from the perpendicular at a greater angle; and, the smallest radius being posited, the tangent opposes the greater angle. These things indeed, said of a balance having the beam above the line of the fulcrum, can be adapted to a balance having the beam below the line of the fulcrum, if the same diagrams are understood as placed in the opposite position: for the greater the angle at which the line joining the center of gravity and the center of motion departs from the perpendicular, the more easily is the arm in which the center of gravity lies inclined. But if this double kind of balance, namely that which has the beam above, and that which has the beam below the line of the fulcrum, be compared with each other, it is sufficiently clear that the inequality of weights is much more easily indicated by this latter kind; because, namely, if the center of gravity recedes ever so little from the middle of the fulcrum to either side, it no longer hangs over the beam in the same perpendicular, nor can it be sustained, but immediately descends, as far as it can, to the lowest position. But in the former kind of balance, having the beam in the upper position, when the center of gravity departs to either side, it does indeed descend; but only in proportion to the excess of the weight; and this descent will be imperceptible if the difference of the weights be slight. Hence it is not once observed that the most accurate balances, by which the weights of gold coins are examined, are those which have the beam in the lower position; for the scale that is overburdened by weight descends to the bottom as far as it can: but when the balance is inverted so that the handle, supported below, sustains the balance, and the same weights are placed upon it, the overburdened scale does not descend to the lowest position; but the balance remains in an oblique position, which appropriately corresponds to the inequality of the weights; and, if the inequality of the weights be such as to escape all the observer’s exactness, the balance seems to be placed in horizontal equilibrium, though in the former position, before the balance was inverted, it could not have stood in any equilibrium at all. I would not, however, have these things understood in such a way that no account is to be taken of the material of which the balance is made; for this may indeed be of such weight that, pressing more strongly upon the axis, it somewhat hinders the motion, and therefore that slight effective force of motion, which

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Liber tertius. CAPUT VI. 283 qui ponderum adnexorum inæqualitatem cæteroqui conseque- retur, ex hâc pressione, & prominularum particularum se vi- cissim contingentium conflictu elidatur, atque jugi æquili- brium horizontale permaneat. Gravitatem autem motui im- pedimento esse ex eo constat, quòd faciliùs quando sine pondere est, movetur libra, quàm cum pondus habet, ut observavit Aristoteles 9. 19. Mechan. Cui tamen in assignandâ hujus difficultatis causâ non aquiesco, licet ultrò concedam in con- trarium ei, ad quod vergit onus, movere difficile esse; si enim libræ vacuæ lances minùs graves sunt; imposito autem pondere fiunt graviores, & proptereà lanx elevanda facta gravior difficiliùs movetur contra insitam gravitati propensionem, etiam vicissim lanx deprimenda facta gravior ex adnexo pondere faciliùs ob- secundat naturali gravium propensioni, atque adeò augere de- beret movendi facilitatem, vel saltem hanc imminui non per- mitteret. Non aliunde igitur ortum ducere videtur hujusmo- di difficultas movendi libram onustam, quàm ex majore pre- mentis gravitatis conatu: pressione autem motum impediri quis neget, si super planam superficiem continuo lævore lubricam ducat regulam metallicam exquisitè politam, quam nunc te- nui, nunc validiori conatu premat? utique percipiet pro vario prementis conatu aliam atque aliam esse trahendæ regulæ me- tallicæ difficultatem. Adde graviori libræ crassiorem axem, ut ei proportione respondeat, necessariò adjungi; hic autem si non sit exquisitè cylindricus, quâ parte fit contactus, sed aliquatenùs angulatus duobus in locis contingat, satis manifestè apparet magis impe- diri motum libræ, quàm si axis tenuior esset, atque subtilior; licet enim hic pariter similique ratione angulatus esset, quia tamen anguli minùs distarent invicem, quàm in axe crassiore, minùs etiam libræ conversionem impedirent. Idem accidit, si axis quidem cylindricus, foramen autem, cui axis inseritur, non exquisitè rotundum sed angulatum fuerit. Cur autem libræ conversio impediatur, si fiat contactus in duobus punctis, pa- làm est; quia nimirum quamdiu centrum gravitatis compositæ interjicitur inter duos illos contactus (vel saltem linea directio- nis per illud centrum ducta transit per intervallum illud duo- rum contactuum) non potest fieri libræ in alterutram partem N n 2

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Book Three. CHAPTER VI. 283 which would otherwise follow the inequality of the attached weights, is, by this pressure and the conflict of the small projecting particles touching one another in turn, overcome, and the horizontal equilibrium of the beam is maintained. But that gravity is an impediment to motion is clear from the fact that a balance moves more easily when it is without a weight than when it has a weight, as Aristotle observed, 9. 19. Mechan. Yet I do not assent to the explanation assigned for this difficulty, although I readily concede that it is difficult to move a load in the direction toward which it inclines; for if the pans of an empty balance are lighter, but when a weight is placed on them they become heavier, then the pan that must be raised, becoming heavier from the attached weight, is moved with greater difficulty against the natural tendency of gravity; and likewise the pan that must be depressed, becoming heavier from the attached weight, more readily yields to the natural tendency of heavy bodies, and thus ought to increase the ease of movement, or at least not allow it to be diminished. Therefore this sort of difficulty in moving a loaded balance does not seem to arise from any other source than from the greater effort of the pressing weight: and who would deny that motion is impeded by pressure, if he were to draw over a plane surface a polished metal ruler, exquisitely smooth, sometimes pressing it lightly, sometimes with greater force? He will surely perceive that, according to the varying force of the presser, the difficulty of drawing the metal ruler is different and different. Add also that a thicker axle must necessarily be attached to a heavier balance, so that it may correspond to it in proportion; but if this axle is not exquisitely cylindrical where contact is made, but is somewhat angular and touches in two places, it clearly appears that the motion of the balance is more impeded than if the axle were thinner and more slender; for although this too would be angular in the same way and for the same reason, because nevertheless the angles would be less distant from one another than in a thicker axle, they would also less hinder the turning of the balance. The same happens if the axle indeed is cylindrical, but the hole into which the axle is inserted is not exquisitely round but angular. But why the turning of the balance is impeded if contact is made at two points is plain; because, namely, as long as the center of gravity of the composite body lies between those two contacts (or at least the line of direction drawn through that center passes through the interval between those two contacts), the balance cannot be moved to either side. N n 2

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Mechanicorum 284 couverso; quæ proinde ut convertatur, tantum ponderis alte- ri lanci addi necesse est, ut centrum gravitatis omninò cadat extrà illud spatium, quod à contactibus comprehenditur. Hinc patet, cur libræ crassiores, & majores ingentibus sar- cinis onustæ inertes fiant ad motum, etiam si adnexus ponderi- bus insit aliquot unciarum, aliquando fortasse etiam librarum, disparitas. Contrà verò aurificibus, & gemmariis, quibus mi- nutias contemnere damno esset, valdè exiguæ libræ in usu sunt; quippè quæ subtilissimo axe contentæ sunt, & levi jugo constant, cujus gravitati æqualis est singularum lancium gra- vitas: quare cum nec vehemens pressio contingat, nec axis adeò tenuis facilè angulos admittat, exilioribus hujusmodi li- bris etiam minima ponderum inæqualitas exploratur, si cæte- roqui fuerint ritè constructæ. At quærat hîc quispiam. Proponitur libra, quæ vacua æqui- librium ostendit, nec ita gravis est, ut de validiore axis pres- sione dubitetur: ut inquiratur, quàm facilè mobilis illa sit, alte- ri lanci singula subinde grana delicatè imponuntur, quot satis sint ad primò tollendum æquilibrium, tùm aliâ librâ tenuiori examinatum granorum omnium pondus (rejecto ultimo grano, cujus additione primò facta est libræ inclinatio) deprehendi- tur unciæ unius, exempli gratiâ. Quæritur, an, si eidem lanci imponantur merces, & oppositæ lanci legitima pondera, sit semper numeranda uncia una amplius, ut verum mercis pon- dus habeatur; quandoquidem deprehensum est non mutari æquilibrium, nisi uncia addatur. Ut quæstioni satisfaciam, tanquam certum statuamus hanc libræ inertiam non oriri ex multâ jugi & lancium gravitate axem premente; si enim ex hujusmodi pressione oriretur, ad- ditis hinc & hinc ponderibus multò major fieret pressio, ex quâ movendi difficultas major crearetur; & si minorem pressio- nem vix unius unciæ excessus vincit, utique majorem pressio- nem non nisi plurium unciarum excessus vincere poterit. De- finire autem hujusmodi pressionum vires motum libræ retar- dantes, meæ tenuitatis non est; quippè qui nec divinare au- deo, nec certam rationem pressiones illas dimetiendi invenio. Illud igitur reliquum est, seclusâ pressione, quòd axis con- tactus non omninò in unico puncto, sed in pluribus fiat, ac propterea

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Mechanics 284 covered; therefore, so that it may turn, so much weight must be added to the other pan that the center of gravity falls altogether outside that space which is enclosed by the points of contact. Hence it is clear why thicker, larger balances, loaded with immense burdens, become inert to motion, even though there may be in the attached weights a difference of some ounces, perhaps even at times of pounds. On the other hand, for goldsmiths and jewellers, for whom it would be a loss to disregard minutiae, very small balances are in use; indeed, they are provided with the most delicate axis, and consist of a light beam, whose weight is equal to the weight of each of the pans: wherefore, since there is neither any violent pressure, nor does an axis so slender easily admit angles, even the smallest inequality of weights is detected with balances of this sort, provided they have been properly constructed in all other respects. But someone may ask here: a balance is proposed which, when empty, shows equilibrium, and is not so heavy that one may doubt the stronger pressure of the axis; in order to inquire how readily movable it is, single grains are successively and delicately placed in one pan, as many as are sufficient first to disturb the equilibrium; then, with another, more delicate balance, the total weight of all the grains is ascertained (the last grain, by whose addition the balance first began to incline, being rejected), and found to be, for example, one ounce. The question is whether, if merchandise be placed in that same pan, and lawful weights in the opposite pan, one extra ounce must always be counted in order to obtain the true weight of the goods, since it has been discovered that the equilibrium is not altered unless an ounce is added. To satisfy the question, let us assume as certain that this inertness of the balance does not arise from the great weight of the beam and pans pressing upon the axis; for if it arose from such pressure, then by adding weights here and there the pressure would become much greater, from which a greater difficulty of movement would be produced; and if a smaller pressure is scarcely overcome by an excess of one ounce, certainly a greater pressure can be overcome only by an excess of several ounces. But to define the forces of such pressures, which retard the motion of the balance, is not for my small ability; for I neither dare to divine, nor do I find any sure method of measuring those pressures. What remains, then, setting pressure aside, is that the contact of the axis does not take place altogether at a single point, but at several, and therefore

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Liber tertius. CAPUT VI. 285 propterea alterutri vacuæ libræ lanci imponendam unciam, ut primò disposita sit libra ad recedendum ab æquilibrio. Hoc au- tem indicat, libræ prorsus vacuæ centrum gravitatis esse inter extrema puncta contactûs axis; sed additâ unciâ compositæ gra- vitatis centrum convenire cum extremo puncto contactûs axis. Quærendum est igitur, quo intervallo extremum hoc punctum, quod etiam est gravitatis centrum, distet à medio jugi puncto. Id quod ut innotescat, observetur jugi & lan- cium gravitas; tùm in extremitatibus jugi intelligatur semissis singularum brachiorum, & addatur singularum lancium gra- vitas: sint autem hinc & hinc ex. gr. unciæ duodecim tota gra- vitas: alteri addatur uncia, & erunt hinc quidem unciæ 12; hinc verò unciæ 13. Quare jugum reciprocè distinguatur in duas partes, quarum altera lit 13, altera 12: igitur punctum hoc divisionis jugi distat à medio jugi puncto parte unâ quin- quagesimâ totius longitudinis ejusdem jugi: hæc siquidem lon- gitudo distincta intelligitur in partes 25 æquales; punctum medium ab extremitate distat partibus 12 ́, centrum gravita- tis compositæ distat partibus 12; igitur punctorum istorum in- tervallum est ́. Iam imponatur alteri lanci merx, quæ cum pondere le- gitimo lib. 2. faciat æquilibrium: aio non posse pronuncia- ri mercem esse unc. 25: nam si ponatur merx unc. 25: ad- ditâ gravitate lancis & brachij unc. 12 ex hypothesi, hinc quidem essent unciæ 37, hinc verò unciæ 36; igitur divi- so jugo in partes 73, centrum gravitatis distaret à medio jugi puncto parte ́. At punctum extremum contactûs axis & jugi distat parte ́, igitur multo majus pondus supra unciam adden- dum est merci, ut æquilibrium exquisitè faciat cum pondere legitimo lib. 2. Nimirum instituenda est analogia ut 12 ad 13, ita unciæ 36 ad uncias 39; dempto igitur pondere lancis & bra- chij libræ, quantitas mercis est unc. 27. Ex quo liquet, quò majora pondera lancibus imponuntur, eò majorem esse diffe- rentiam à pondere legitimo. Hinc ulteriùs patet hujusmodi librâ satius esse multam mercem simul ponderare, quàm per partes: pone enim esse uncias 12 legitimi ponderis, cum quo Nn 3

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Liber tertius. CHAPTER VI. 285 therefore an ounce must be placed on one of the two empty pans, so that the balance may first be set in a position to move away from equilibrium. This indicates that the center of gravity of the balance when entirely empty is between the two extreme points of contact of the axis; but when an ounce is added, the center of gravity of the compound body coincides with the extreme point of contact of the axis. It must therefore be asked at what interval this extreme point, which is also the center of gravity, is distant from the midpoint of the beam. In order that this may be known, let the weight of the beam and pans be observed; then at the extremities of the beam let half of each arm be understood, and let the weight of each pan be added: let the total weight on each side, for example, be twelve ounces; to one side add one ounce, and there will be, on the one hand, 12 ounces; on the other, 13 ounces. Therefore let the beam be reciprocally divided into two parts, one of which shall be 13, the other 12: thus this point of division of the beam is distant from the midpoint of the beam by one fiftieth part of the whole length of that beam; for if that length is understood to be divided into 25 equal parts, the midpoint is 12 1 &frasl; 2 parts from the extremity; the center of gravity of the compound body is 12 parts away; therefore the interval between those points is 1 &frasl; 50 . Now let merchandise be placed on the other pan, which together with the lawful weight of 2 lb. may produce equilibrium: I say that it cannot be declared that the merchandise is 25 oz.; for if the merchandise is supposed to be 25 oz., then, with the added weight of the pan and arm, 12 oz. according to the hypothesis, there would be 37 oz. on one side and 36 oz. on the other; therefore, if the beam is divided into 73 parts, the center of gravity would be distant from the midpoint of the beam by 1 &frasl; 146 . But the extreme point of contact of the axis and beam is distant by 1 &frasl; 50 ; therefore a much greater weight than an ounce must be added to the merchandise in order that it may make the equilibrium exactly with the lawful weight of 2 lb. Namely, an analogy must be established thus: as 12 is to 13, so are 36 ounces to 39 ounces; therefore, after subtracting the weight of the pan and arm of the balance, the quantity of merchandise is 27 oz. From this it is clear that the greater the weights placed on the pans, the greater is the difference from the lawful weight. Hence it further appears that with a balance of this sort it is better to weigh a large quantity of merchandise all at once than in parts: for suppose the lawful weight to be 12 ounces, with which Nn 3

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Mechanicorum æquilibrium constituitur, merx erit unicarum 14, quia ut 12 ad 13, ita unc. 24 ad 26, & demptis unciis 12 ad brachium & lancem spectantibus, remanent mercis unciæ 14: quare bis factâ ponderatione erit differentia unc. 4; unica autem ponde- ratio dabat tantùm uncias 3: quia videlicet singulis vicibus ad- ditur id, quod respondet gravitati lancis oppositæ; atque adeò differentia sæpiùs repetita major est, quàm simplex: sic qua- tuor libris ponderis legitimi responderent in alterâ lance mer- cis lib. 4. unc. 5; quòd si quatuor vicibus operando singulas libras expendisses, differentia demùm esset unciarum 8. Unum adhuc superesse videtur hîc observandum, quoniam longioribus brachiis exquisitiùs indicari æquilibrium diximus: cavendum scilicet, ne in aliud incommodum incidamus, quo illud idem pereat, quod persequimur. Si enim longiora fiant brachia, additur gravitas, quæ magis axem premens motui ali- quam difficultatem creat: quod si retentâ eâdem brachiorum gravitate illorum crassities extenuetur, & in longitudinem ex- tendantur, vide ne nimis exilia evadant ita, ut flexioni obnoxia sint, vel suâ ipsorum, vel expendendorum ponderum gravita- te. Præterquam quod longiora brachia plus habere videntur momenti ad premendum axem, etiam si par sit longiorum at- que breviorum libræ brachiorum gravitas absoluta; cujus se- missis in extremitate brachij longioris plus habet momenti ad descendendum, quàm in extremitate brevioris. Et si longior hasta ex medio suspensa faciliùs sponte suâ flectitur circa me- dium (id quod breviori non accidit) indicio est obicem reti- nentem magis premi; idem igitur & axi libræ contingere po- test, cujus pressio major esse videtur ex longioribus brachiis, etiamsi in cæteris nullum intercedat discrimen. Sic Aristote- les quærit quæst. 17. Mechan. Cur si valde procerum fuerit idem pondus, difficiliùs super humeros gestatur, etiam si medium quispiam illud ferat, quam si brevius sit? cujus difficultatis causam ille tri- buit validiori vibrationi extremitatum magis distantium ab hu- mero sustinente: sed hoc non nisi in motu contingit, & cùm flexile est pondus, cujusmodi esset longior hasta aut bractea ferrea mediocris crassitiei. Certè longiori columnæ marmoreæ jacenti, cujus medio recens fulcrum subjectum fuit, jam pu- trescentibus extremis fulcris, sua longitudo obfuit, ut frange- retur:

Transcription: Translated (English)

Mechanics balance is established, the merchandise will be 14 units, because as 12 is to 13, so 24 ounces are to 26, and with 12 ounces subtracted, which pertain to the beam and the pan, 14 ounces of merchandise remain: therefore, after the weighing has been done twice, the difference will be 4 ounces; but a single weighing yielded only 3 ounces: because, namely, at each time there is added what corresponds to the weight of the opposite pan; and thus the difference, repeated several times, is greater than the simple one: thus with four pounds of lawful weight, in the other pan there would correspond merchandise of 4 lb. 5 oz.; but if, working four times, you had weighed the individual pounds, the difference would finally be 8 ounces. One more point seems here to remain to be observed, since we have said that with longer arms the balance is indicated more precisely: care must be taken, namely, lest we fall into another inconvenience, by which that same thing which we are seeking should be lost. For if the arms are made longer, weight is added, which, pressing more strongly on the axis, creates some difficulty of motion: but if, the same weight of the arms being retained, their thickness is reduced, and they are extended in length, see to it that they do not become too slender, so that they may be liable to bending, either from their own weight or from the weight of the things to be weighed. Moreover, not only do longer arms seem to have more force in pressing the axis, even if the absolute weight of the scale-beams, long and short, be equal; the half of which at the end of the longer arm has more force to descend than at the end of the shorter one. And if a longer rod, suspended from the middle, bends more easily of itself around the middle (which does not happen with a shorter one), it is a sign that the obstacle holding it back is pressed more strongly; therefore the same can also happen to the axis of a balance, whose pressure seems greater with longer arms, even if in the other respects there is no difference. Thus Aristotle asks in question 17 of the Mechanics: Why, if the same weight be very tall, is it carried with greater difficulty on the shoulders, even if someone carries it at the middle, than if it is shorter? He attributes the cause of this difficulty to the stronger vibration of the extremities, which are farther from the shoulder that supports it: but this happens only in motion, and when the weight is flexible, such as would be a longer rod or an iron plate of moderate thickness. Certainly, in the case of a longer marble column lying down, beneath whose middle a support had recently been placed, while the end supports had already decayed, its length was a hindrance, so that it broke:

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Liber tertius. CAPUT VI. 287 retur: id quod æqualis ponderis columnæ breviori ex graviore secundùm speciem marmore non ita facilè accidisset: non nisi quia gravitas magis à fulcro distans plùs habet momenti, etiamsi non contingat vibratio corporis, quemadmodum in motu. Illud postremò non omittendum, quod ad lingulam pertinet, hanc enim longiusculam esse præstat, quàm brevem, ut vel levi inclinatione libræ, apex lingulæ magis conspicuo motu extra ansam ad latus secedat, & sublatum æquilibrium indicet. Dum tamen lingulæ longitudinem affectas, cavendum, ne illa momentum addat suâ gravitate brachio, quod inclinatur; quamvis enim hoc nihil referat, ubi sublatum horizontale æquilibrium indicatur; in librâ tamen, quæ in æquilibrio obliquo potest consistere, videretur indicare majorem ponderum inæqualitatem, quàm revera sit. Cæterùm communiter libræ hoc periculo vacant; sola enim ponderum æqualitas horizontali æquilibrio inquiritur, non ponderum Ratio obliquo æquilibrio investiganda proponitur: quare communiter nil de lingulæ gravitate timendum est, quod nos solicitos habeat. Quare præter exquisitam brachiorum æqualitatem, & accuratam lingulæ cum ipso jugo positionem ad angulos rectos, ad libram exactissimam constituendam concurrunt brachiorum & lingulæ longitudo, jugi & lancium modica gravitas, axis subtilitas, sparti & jugi quàm maxima propinquitas, & ipsius sparti infrà jugi lineam positio. Quæ tamen omnia cum rectâ ratione sunt administranda, ut ponderibus examinandis proportione respondeant libræ partes; majoribus enim sarcinis validior axis, & crassiora libræ brachia conveniunt; & sic de reliquis. CAPUT VII. Libræ dolosæ vitia reteguntur. Libram dolosam voco, quæ solitariè accepta sinè ponderibus justa apparet, & æquilibrium ostentat, re tamen verâ injusta est, quia adnexis ponderibus suo æquilibrio non tribuit æqualita

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Book Three. Chapter VI. 287 would be transferred: something that in a column of equal weight, shorter by reason of a greater mass according to species, would not so easily happen: only because a weight farther removed from the fulcrum has more force, even if the body does not undergo vibration, as in motion. Finally, it must not be omitted what concerns the tongue: for it is better that this be somewhat longer than short, so that even with a slight inclination of the balance, the tip of the tongue may depart more visibly by motion outside the handle to the side, and indicate the lifting of equilibrium. Yet while you seek the length of the tongue, care must be taken lest it add momentum by its weight to the arm that is inclined; for although this matters nothing where the lifting of horizontal equilibrium is indicated, in a balance, however, that can stand in oblique equilibrium, it would seem to indicate a greater inequality of weights than there really is. Moreover, balances generally are free from this danger; for only the equality of weights is sought in horizontal equilibrium, not the ratio of weights to be investigated in oblique equilibrium is proposed: therefore generally nothing need be feared from the weight of the tongue that should make us anxious. Therefore, besides the exact equality of the arms, and the careful position of the tongue with the yoke itself at right angles, to constitute the most accurate balance there contribute the length of the arms and of the tongue, the moderate weight of the yoke and pans, the slenderness of the axis, the greatest possible closeness of the stirrup and the yoke, and the position of the stirrup itself below the line of the yoke. Yet all these things must be managed according to right reason, so that the parts of the balance correspond in proportion to the weights to be weighed; for with larger loads a stronger axis, and thicker arms of the balance, are suitable; and so with the rest. Chapter VII. The faults of the deceitful balance are exposed. I call a balance deceitful which, taken by itself without weights, appears to be correct and displays equilibrium, but in reality is unjust, because with weights attached it does not grant equality to its own equilibrium

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Mechanicorum æqualitatem, vel quia ponderum æqualitatem non indicat ve- ro æquilibrio. Quare nullus mihi sermo de iniquorum vendi- torum sycophantiis, quibus, justam licèt libram adhibentes, rudem ac simplicem emptorem circumveniunt, aut imprimen- do impetum sursum brachio, cui legitimum pondus adnectitur, ut merx præponderare videatur, aut ponderibus iniquis & justo minoribus utendo, aut subjectam mensam, cui lanx mercis in- cumbit, materiâ aliquatenus tenaci illinendo, ut sublatâ in aërem librâ priùs attollatur lanx ponderis quàm mercis, quæ omninò præponderans apparet, si libra spartum habeat infrà jugum, aut similes imposturas excogitando: sed de illis tantum deceptionibus agendum, quæ ex ipsius libræ constructione, aut positione ortum habere possunt. Et primò quidem se offert dolus, cujus meminit Aristoteles quæst. 1. Mechan. familiaris eo tempore vendentibus purpuram, & ea, quorum modica quantitas pretium exigebat non contem- nendum: hi enim librâ utebantur, quæ brachiis non omninò paribus constabat, ita tamen, ut hæc inæqualitas non se oculis statim proderet. Ut autem lateret dolus, scapum seu jugum libræ ex ligno construebant, cujus partes omnes non eandem specificam gravitatem obtinerent, quamvis nulla secundùm molem diversitas intuenti occurreret: quia enim nodi, & partes radici propiores, ut potè magis densæ, graviores sunt, quàm reliquæ partes à radice remotiores & nodis carentes, partem il- lam graviorem breviori brachio tribuebant, vel si materia pla- nè uniusmodi esset, & æquabili gravitate prædita, breviori brachio aliquid plumbi infundebant, ut materiæ gravitate mo- mentum, quod ratione positionis deerat, supplente, appareret æquilibrium lancium in vacuâ librâ. Sed ubi demum merx lanci longioris brachij imponebatur, hæc erat justo minor, quamvis cum opposito pondere esset æquilibris; non enim erat illi æqualis, sed in Ratione reciprocâ longitudinis brachij mi- noris ad longitudinem majoris. Hîc spectat inæqualitas bra- chiorum orta ex eo, quòd jugi ferrei pars altera ex validiore, & diuturniore percussione mallei facta densior, etiam gravior est; nam puncto longitudinem jugi bifariam dividendi non respon- det centrum gravitatis; sed recedit à medio versùs extremita- tem densiorem, atque graviorem; ac proptereà, ut æquili- brium

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Mechanic equality, or because it does not indicate equality of weights by true balancing. So I say nothing here of the frauds of dishonest sellers, by which, though they employ a fair balance, they deceive the rude and simple buyer, either by thrusting upward with the arm to which the lawful weight is attached, so that the merchandise seems to outweigh the other side, or by using unjust weights smaller than what is right, or by smearing the surface of the table beneath which the scale-pan of the goods rests with some somewhat adhesive material, so that when the balance is lifted into the air the weight-pan is raised before the merchandise-pan, which then appears altogether to outweigh it, if the balance has a cord beneath the beam, or by devising similar tricks: but only those deceptions are to be discussed which can arise from the construction of the balance itself, or from its position. And first there presents itself the fraud mentioned by Aristotle, question 1 of the Mechanics, familiar in his time to those who sold purple dye, and to those goods for which a small quantity demanded no contemptible price: these men used a balance whose arms were not altogether equal, yet in such a way that the inequality did not immediately reveal itself to the eye. So that the fraud might be concealed, they made the beam or yoke of the balance from wood, all the parts of which did not have the same specific gravity, although no difference in bulk would be apparent to the observer. For since the knots and the parts nearer the root, being more dense, are heavier than the other parts farther from the root and lacking knots, they assigned that heavier part to the shorter arm; or, if the material was plainly of one kind and possessed uniform weight, they poured some lead into the shorter arm, so that the weight of the material, supplying the effect that was lacking because of its position, made the equilibrium of the pans appear in an empty balance. But when at last the merchandise was placed in the pan of the longer arm, it was less than it ought to be, although it was in equilibrium with the opposite weight; for it was not equal to it, but in the reciprocal ratio of the length of the shorter arm to that of the longer. Here belongs the inequality of the arms arising from the fact that one part of the iron yoke, being made denser by the stronger and longer-continued blows of the hammer, is also heavier; for the center of gravity does not correspond to the point dividing the yoke into two equal parts in length, but departs from the middle toward the denser and heavier end; and therefore, so that the equilibrium

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Liber tertius. CAPUT VII. 289 brium appareat, centrum motûs inæqualiter dividit longitudinem jugi. Similiter si jugi quidem materia æquabiliter sit gravis, sed brachiorum inæqualitatem suppleat lancium gravitas reciprocè inæqualis; æquilibris erit libra vacua; sed damno emptoris merx longiori brachij adnectitur. Quare ut pateat dolus, facto æquilibrio inter mercem ac pondus, statim commuta lances, & pondus majus ex longiore brachio multò plus habebit momenti, quàm merx ex brachio breviore: idcircò, si ex pondere dematur, quantum satis sit ad æquilibrium cum merce iterum statuendum, plus mercis habebit emptor, quàm pro oppositi ponderis mensurâ. Secundò sit jugi materia planè æquabilis, & ab axe jugum dividatur omnino bifariam: sed puncta contactuum annulorum, ex quibus pendent lances, non æqualiter distent à medio: etiamsi lancis propioris gravitas suppleat momentum, quod deest ratione sitûs, & æquilibris appareat libra vacua, non tamen æqualia pondera lancibus imposita constituent æquilibrium, sed illud gravius apparebit, quod ex distantia majore appendetur: & si pondera æquilibrium faciant, inæqualia erunt reciprocè juxtà Rationem inæqualitatis distantiarum à medio. Similiter igitur facto ponderum æquilibrio, lances commuta, & quidem si post commutationem iterum æquilibrium fiat, justa est libra, secùs verò si alterum gravius appareat, quod priùs æquale videbatur. At quæris, quâ methodo possis deprehendere, quanta sit brachiorum inæqualitas, quando quidem non habetur æquilibrium post factam lancium commutationem, & planè ignoratur, quanta sit mercis gravitas. Ut quæstioni satisfaciam, accipio legitima pondera, & primùm facto æquilibrio observo legitimi ponderis quantitatem: Commuto deinde lances, & cum non fiat æquilibrium cum eâdem merce, tantum accipio legitimi ponderis, quantum requiritur ad æquilibrium. Demum inter hæc duo pondera legitima invenio terminum medio loco proportionalem, & hoc est mercis pondus, quod collatum cum alterutro ex legitimis ponderibus dat reciprocè longitudinis brachiorum Rationem. Hanc methodum esse certam patet, quia cum bis fiat æquilibrium, bis inter pondera est eadem Ratio reciproca brachiorum. Sint brachia, quæ brevitatis gratia Oo

Transcription: Translated (English)

Book three. CHAPTER VII. 289 If the center of motion is not apparent, it divides the length of the beam unequally. Likewise, if the material of the beam itself is indeed of equal weight, but the inequality of the arms is compensated by the weights of the pans being reciprocally unequal, the empty balance will be in equilibrium; but to the buyer’s detriment the merchandise is attached to the longer arm. Wherefore, that the fraud may be made clear, once equilibrium has been established between the merchandise and the weight, immediately exchange the pans, and the greater weight, from the longer arm, will have much more force than the merchandise from the shorter arm: therefore, if from the weight there is taken away as much as is sufficient to restore equilibrium with the merchandise, the buyer will have more merchandise than the measure of the opposite weight. Secondly, let the material of the beam be perfectly uniform, and let the beam be divided by the axis exactly into two equal parts; but let the points of contact of the rings, from which the pans hang, not be equally distant from the middle: even if the weight of the nearer pan make up the momentum that is lacking by reason of its position, and the empty balance appear to be in equilibrium, nevertheless equal weights placed in the pans will not constitute equilibrium, but that weight will appear the heavier which is suspended from the greater distance; and if the weights make equilibrium, they will be reciprocally unequal according to the ratio of the inequality of the distances from the middle. Likewise, therefore, once the weights have been balanced, exchange the pans, and indeed if after the exchange equilibrium again occurs, the balance is just; otherwise, if the other side appears heavier, that which formerly seemed equal. But you ask by what method you may discover how great the inequality of the arms is, since equilibrium is not obtained after the pans have been exchanged, and the weight of the merchandise is entirely unknown. To satisfy the question, I take lawful weights, and first, having established equilibrium, I observe the amount of the lawful weight. Then I exchange the pans, and since equilibrium does not occur with the same merchandise, I take as much of the lawful weight as is required for equilibrium. Finally, between these two lawful weights I find the term proportional midway, and this is the weight of the merchandise, which, when compared with either of the lawful weights, gives the reciprocal ratio of the lengths of the arms. That this method is certain is clear, because equilibrium is established twice, and twice the same reciprocal ratio of the arms exists between the weights. Let the arms, for brevity’s sake, Oo

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Mechanicorum 290 vocemus R & S; igitur ut R ad S ita primum pondus legiti- mum in S ad mercem in R: & factâ commutatione ponitur merx in S, & iterum fit ut R ad S, ita reciprocè merx eadem in S ad secundum pondus legitimum in R: igitur, per 1 1. lib. 5. ut primum pondus ad mercem, ita merx ad secundum pondus: sunt autem nota duo pondera legitima; igitur & innotescit mer- cis gravitas: quæ si comparetur ut conseqvens terminus cum primo pondere, aut ut Antecedens cum secundo pondere, ha- bebitur Ratio R ad S. Sit itaque ex. gr. in primo æquilibrio primum pondus legitimum unc. 72, in secundo æquilibrio se- cundum pondus legitimum sit unc. 69 18/100. Est ergo merx me- dio loco proportionalis unc. 70 576/1000; ac propterea R ad S est ut 72 ad 70 576/1000, aut ut 70 576/1000 ad 69 18/100, hoc est ut 4500 ad 4411. Sit demum totius jugi longitudo distincta in partes 200: addantur termini Rationis inventæ, & fiat ut 8911 ad 4411 ita 200 ad 99, & hæc est longitudo brachij brevioris, erit au- tem longioris brachij longitudo partium 101: distat ergo spar- tum à puncto medio per unam ducentesimam partem totius ju- gi. Quòd si res subtilissimè ad calculos revocanda esset, hujus ducentesimæ partis gravitas, quæ est semissis gravitatis diffe- rentiæ brachiorum esset computanda, atque subducenda, vel addenda, ut mercis pondus exquisitè innotescat. Tertiò. Accidere potest lingulam ex medio libræ scapo as- surgere ad angulos rectos, lineamque lingulæ transeuntem per centrum motûs ita occurrere lineæ jungenti puncta, ex quibus lances pendent, ut eam bifariam æqualiter dividat, in eam ta- men ad angulos inæquales cadat. Aio nec brachia esse verè æqualia, nec lingulam, quamvis ansæ congruens videatur, in- dicare æquilibrium horizontale, esse veram lingulam, etiamsi pondera in eo æquilibrio consistentia sint æqualia, & non in Ratione brachiorum. Sit scapus libræ A B, ex quo perpendicularis assurgat lingula CD, & ex D per O centrum mo- tûs ducta recta linea occurrat li- neæ S V jungenti extrema puncta, ex quibus lances pendent, eam- que bifariam dividat in 1: sed quo- niam punctum S est paulò altiùs quam

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Mechanics 290 let us call them R and S; therefore, as R is to S, so is the first legitimate weight in S to the merchandise in R: and, the exchange having been made, the merchandise is placed in S, and again it happens that, as R is to S, so reciprocally is the same merchandise in S to the second legitimate weight in R: therefore, by 1 1. lib. 5., as the first weight is to the merchandise, so is the merchandise to the second weight: but two legitimate weights are known; therefore the gravity of the merchandise is also known: if this be compared, as the consequent term with the first weight, or as the antecedent with the second weight, the Ratio R to S will be had. Let it then be, for example, in the first equilibrium the first legitimate weight 72 oz., in the second equilibrium let the second legitimate weight be 69 18/100 oz. The merchandise is therefore the mean proportional, 70 576/1000 oz.; and accordingly R to S is as 72 to 70 576/1000, or as 70 576/1000 to 69 18/100, that is, as 4500 to 4411. Let at last the whole length of the beam be divided into 200 parts: add the terms of the ratio found, and let it be as 8911 to 4411 so 200 to 99, and this is the length of the shorter arm; the length of the longer arm will be 101 parts: therefore the suspended load is distant from the midpoint by one two-hundredth part of the whole beam. But if the matter were to be reduced to calculations with the greatest exactness, the weight of this two-hundredth part, which is half the weight of the difference of the arms, would have to be computed and then subtracted, or added, so that the weight of the merchandise may be precisely known. Thirdly. It may happen that the tongue rises from the middle of the balance beam at right angles, and that the line of the tongue passing through the center of motion so meets the line joining the points from which the pans hang as to divide it equally in two, yet falls upon it at unequal angles. I say that neither are the arms truly equal, nor is the tongue, although it may seem to agree with the handle, indicating a horizontal equilibrium, the true tongue; and even if the weights standing in that equilibrium are equal, they are not in the ratio of the arms. Let A B be the beam of the balance, from which the tongue C D rises perpendicularly, and from D through O, the center of motion, let a straight line be drawn meeting the line S V joining the extreme points from which the pans hang, and dividing it in two: but since the point S is somewhat higher than than

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Liber tertius. CAPUT VII. 291 quàm punctum V, fiat angulus SIO minor, & VIO major. Dico lineam SV esse quidem jugum, sed brachia non esse æqualia, non enim sunt IS & IV: quandoquidem ductis rectis O S & OV, est libra curva SOV latera habens inæqualia, SO minus, & VO majus. Nam in triangulis SIO, VIO latus IS ex hypothesi est æquale lateri IV, latus IO commune est, angulus SIO est ex hypothesi minor, quàm angulus VIO; ergo per 24. lib.1. basis SO minor est basi VO. Igitur ex O perpendicularis linea cadens in jugum SV dividit illud in bra- chia inæqualia, & perpendicularum ex O cadit inter S & I, pu- ta in H, quia ex hypothesi angulus SIO est acutus. Vera igitur lingula non est ID, sed linea, quæ ad angulos rectos insistens jugo SV ex H per O ducitur. Quare si CD con- gruit an[n]æ perpendicularis horizonti, jugum SV non est ho- rizonti parallelum, non est igitur æquilibrium horizontale, sed obliquum: quia tamen est I centrum commune gravitatis pon- derum æqualium in S & V, ac per illud transit perpendicularum ex O cadens in horizontem, proptereà possunt esse ponde- ra æqualia, & æquilibrium ostendere, quod modicâ obliquita- te inclinatum mentiatur æquilibrium horizontale. At si alia fieret hypothesis, scilicet lineam jugi SV non dividi æqualiter, pondera non essent æqualia, sed essent reciprocè in Ratione motuum, quos perficere possent extremitates S & V, juxta su- periùs dicta cap.4 hujus lib.3. Vitium igitur hujus libræ non in eo consistit, quòd ponde- ra non sint æqualia, sed quòd indicet æquilibrium horizontale, cum sit obliquum, & pondera æqualia nunquam possint ad æquilibrium horizontale devenire; ut enim hoc fieret, ponde- ra esse oporteret inæqualia reciprocè in Ratione brachiorum SH & HV. Quòd si contingat punctum O centrum motûs, esse idem cum puncto I, pondera æqualia verè habebunt æqui- librium horizontale; sed lingula CD declinabit ab ansâ, quasi æquilibrium non esset. Libræ hujusmodi vitium deprehendi non potest ponderum commutatione in lancibus; quia cùm æqualia ex hypothesi sint pondera, eadem utrobiqque habent momenta, servant quippè eamdem distantiam, & æqualiter sunt ad motum disposita. Rarò tamen continget jugum SV planè æqualiter dividi à lineâ lingulæ ad angulos obliquos in- Oo 2

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Third Book. CHAP. VII. 291 than point V, let the angle SIO be smaller, and VIO greater. I say that the line SV is indeed the beam, but that the arms are not equal; for they are not IS and IV: since, lines OS and OV being drawn, SOV is a curved balance having unequal sides, SO smaller and VO greater. For in the triangles SIO, VIO the side IS, by hypothesis, is equal to side IV, the side IO is common, and the angle SIO is, by hypothesis, smaller than the angle VIO; therefore, by 24 of book 1, the base SO is smaller than the base VO. Hence the perpendicular line falling from O upon the beam SV divides it into un- equal arms, and the perpendicular from O falls between S and I, namely at H, because by hypothesis the angle SIO is acute. The true pointer, therefore, is not ID, but the line which, standing at right angles on the beam SV, is drawn from H through O. Wherefore if CD con- forms to the vertical line of the plumb line, the beam SV is not parallel to the horizon; it is therefore not a horizontal balance, but an oblique one: yet since I is the common center of gravity of the equal weights at S and V, and the perpendicular falling from O through it passes through the horizon, there can therefore be equal weights, and they may show equilibrium, which with a slight obliquity feigns a horizontal equilibrium. But if another hypothesis were made, namely that the line of the beam SV were not divided equally, the weights would not be equal, but would be reciprocally in the ratio of the motions which the extremities S and V could perform, according to what was said above in chap. 4 of this book 3. The defect, therefore, of this balance does not consist in this, that the weights are not equal, but in this, that it indicates a horizontal equilibrium, when it is oblique, and equal weights can never attain to horizontal equilibrium; for in order that this might happen, the weights ought to be unequal reciprocally in the ratio of the arms SH and HV. But if it should happen that the point O, the center of motion, is the same as point I, then equal weights will truly have a horizontal equilibrium; but the little pointer CD will deviate from the handle, as though there were no equilibrium. The defect of this kind of balance cannot be detected by exchanging the weights in the pans; because, since by hypothesis the weights are equal, they have the same moments in either pan, for they keep the same distance, and are equally disposed to motion. Yet it will rarely happen that the beam SV is divided exactly equally by the line of the pointer at oblique angles,

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Mechanicorum cidente, quæ tamen ad scapum perpendicularis appareat: proptereà factâ ponderum in lancibus commutatione prodet se momentorum inæqualitas. Quartò. Libra, quam diutissimè justam expertus es, potest momento à suâ justitiâ deficere, si vel modicum inflectatur alterutrum brachiorum, vel si utrumque non æqualiter flectatur; hinc enim oritur brachiorum inæqualitas; quam deprehendes commutatis ponderibus in utrâque lance; quæ scilicet æquilibrium constituebant propter reciprocam Rationem brachiorum, quibus adnectebantur, non ampliùs eandem servant in aliâ positione Rationem. Quintò. Axis, qui duobus in punctis contingat (scio contactum fieri in lineâ; sed puncta assumo in ipsis lineis, per quæ transit planum perpendicularare ad horizontem, in quo est linea jugi) vel quia ipse est angulatus, vel quia foramen, cui inseritur, non exquisitè rotundum, quâ saltem parte fit contactus, libram constituit dolosam: quia videlicet duo illa puncta axis perinde se habent, ac si duo essent centra motûs. Manifestum est autem eandem jugi lineam non posse in duobus punctis æqualiter dividi. Tripliciter potest hoc fieri. Primò unum ex his punctis potest exactè respondere medio jugi; secundò potest utrumque hoc punctum æqualiter à medio jugi distare; Tertiò possunt ab eodem medio hinc & hinc inæqualiter distare. Sit linea jugi AB, cujus medium C: puncta contactuum axis, ex quibus ad jugum ducitur perpendicularis, ea sint primò, ut respondeant in jugo punctis C & D. Si lanci in B imponatur legitimum pondus, tùm in A ponatur merx usque ad æquilibrium, à quo proximè recederet, si aliquid amplius mercis adderetur, fiet æqualitas, quia ex C puncto æqualiter ab extremitatibus distante fit suspensio libræ. At si positâ primùm merce in A, deinde legitima pondera addantur in B, utique plura pondera, quàm par sit, addentur: quia videlicet non inclinabitur libra infrà B, nisi ponderum ad mercem Ratio excedat Rationem reciprocam brachiorum AD ad DB; est enim D quasi centrum motûs. Deinde

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Mechanics, when the center falls, which nevertheless appears perpendicular to the shaft: therefore, when the weights in the pans are changed, the inequality of the moments will reveal itself. Fourth. A balance which you have long found to be just may at once cease to be just, if either arm is bent even a little, or if the two are not bent equally; for from this arises an inequality of the arms, which you will detect by exchanging the weights in either pan; for they, of course, constituted equilibrium on account of the reciprocal ratio of the arms to which they were attached, and in another position no longer preserve the same ratio. Fifth. A pivot that touches at two points (I know that contact occurs in a line; but I take the points in the lines themselves through which the plane perpendicular to the horizon passes, in which the line of the beam lies), either because it is angled, or because the hole into which it is inserted is not perfectly round, at least where contact is made, makes a balance deceptive: because those two points of the pivot behave just as if there were two centers of motion. But it is clear that the same line of the beam cannot be equally divided at two points. This can happen in three ways. First, one of these points may correspond exactly to the middle of the beam; second, both of these points may be equally distant from the middle of the beam; third, they may be unequally distant from the same middle, on this side and on that. Let AB be the line of the beam, with C its middle: let the points of contact of the pivot, from which the perpendicular is drawn to the beam, be such that, first, they correspond in the beam to the points C and D. If the lawful weight is placed in B, then merchandise is placed in A until equilibrium is reached, from which it would depart most nearly if something more were added to the merchandise; equality will be achieved, because the suspension of the balance is made from the point C, which is equally distant from the ends. But if, merchandise having first been placed in A, the lawful weights are then added in B, certainly more weights than is proper will be added: because the balance will not incline below B unless the ratio of the weights to the merchandise exceeds the reciprocal ratio of the arms AD to DB; for D is, as it were, the center of motion. Then

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Liber tertius. CAPUT VII. 293 Deinde puncta illa contactuum axis possunt respondere jugi punctis E & D æqualiter à medio C distantibus: & tunc, ut tollatur æquilibrium, necesse est tantum ponderis uni lanci ad- dere, ut pondera sint in majori Ratione, quàm sit Ratio reci- proca brachiorum; erit si quidem extremitas A proxime dispo- sita, ut facto additamento gravitatis inclinetur, si fuerit ut B E ad E A, ita pondus in A ad pondus in B; & vicissim extremitas B erit proximè disposita, ut auctâ gravitate inclinetur, si ut A D ad D B ita pondus in B ad pondus in A. Quia autem ex hypo- thesi D C & E C æquales sunt, etiam residua E A & D B æqua- lia sunt, item A D & B E: quapropter ut A D ad D B, ita B E ad E A; ex quo consequens est ex folâ lancium commutatione (si centrum motûs modò sit D, modò sit E) non posse dignosci hoc libræ vitium, sicut dignosceretur in primo casu, si ut A D ad D B, ita pondus in B ad pondus in A; factâ enim lancium commutatione, pondus ex B in A translatum præponderaret ex centro motûs C, cum tamen in priori positione circa cen- trum motûs D non tolleret æquilibrium. Similiter in tertio casu, quando puncta contactuum axis es- sent F & D à medio C inæqualiter distantia, & ut A F ad F B, ita pondus in B ad pondus in A daret æquilibrium; factâ pon- derum in lancibus commutatione non maneret æquilibrium, quia pondus translatum in B ad pondus translatum in A post hanc commutationem adhuc esset ut B F ad F A; sed ad æqui- librium circa D centrum motûs deberet esse ut A D ad D B, est autem B F prima major, quàm A D tertia, & F A secunda minor est, quàm D B quarta; igitur est major Ratio B F ad F A, quàm A D ad D B: igitur pondus, quod priùs erat in B, transla- tum in A impar est ad æquilibrium constituendum. Ad dignoscendum, an libra hoc vitio laboret, uti poteris hac methodo. Lancibus impone pondera, ut fiat æquilibrium: tùm lances commuta; & siquidem iterum fiat æquilibrium, adde alteri lanci aliquid ponderis, à quo si libra inclinetur, aufer ad- ditum pondus, & oppositæ lanci impone; quæ si persistat non inclinata, adde adhuc pondus, quantum ferre potest citrà in- clinationem: iterum commutatis lancibus, nullo pacto manere æquilibrium videbis, & indicio erit contactum axis fieri in duobus punctis, quorum alterum respondet medio jugi siqui- Oo 3

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Book the Third. CHAPTER VII. 293 Then those points of contact of the axis may correspond to the points E and D of the beam, equally distant from the middle C; and then, in order that the equilibrium may be destroyed, it is necessary only to add so much weight to one scale, that the weights may be in a greater ratio than is the reciprocal ratio of the arms; for the extremity A will be found to be so situated that, when weight is added, it inclines, if it be as B E to E A, so is the weight in A to the weight in B; and conversely the extremity B will be found to incline, when the weight is increased, if as A D to D B, so is the weight in B to the weight in A. But since, by hypothesis, D C and E C are equal, the remaining E A and D B are also equal, likewise A D and B E: wherefore, as A D is to D B, so is B E to E A; from which it follows that by the mere interchange of the scales (if the center of motion be now D, now E) this fault of the balance cannot be detected, as it would be detected in the first case, if as A D is to D B, so were the weight in B to the weight in A; for, the scales being interchanged, the weight transferred from B to A would preponderate from the center of motion C, although in the former position about the center of motion D it did not destroy the equilibrium. Likewise in the third case, when the points of contact of the axis were F and D, unequally distant from the middle C, and if, as A F is to F B, so the weight in B to the weight in A would produce equilibrium; after the interchange of the weights in the scales, the equilibrium would not remain, because the weight transferred in B to the weight transferred in A after this interchange would still be as B F to F A; but for equilibrium about the center of motion D it ought to be as A D to D B; but B F, the first, is greater than A D, the third, and F A, the second, is less than D B, the fourth; therefore the ratio of B F to F A is greater than that of A D to D B: therefore the weight, which before was in B, transferred to A, is unequal for establishing equilibrium. To detect whether the balance labors under this fault, you may use this method. Place weights on the scales so that equilibrium is established: then interchange the scales; and if equilibrium is again established, add something of weight to the other scale, and if the balance inclines from this, remove the added weight and place it on the opposite scale; if this remains uninclined, add still more weight, as much as it can bear without inclining: then, after the scales are again interchanged, you will see that equilibrium cannot by any means remain, and it will be a sign that the contact of the axis occurs in two points, one of which corresponds to the middle of the beam if indeed

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Mechanicorum 294 dem in primâ lancium commutatione mansit æquilibrium; & est primus casus. Quòd si facto æquilibrio, alterutri lancium addas pondus, & æquilibrium maneat, adde quantum satis est, ut libra sit proximè inclinanda in eam partem, si adhuc pondus adderetur, tum oppositæ lanci similiter additum pondus si non tollat æquilibrium, indicat inter puncta contactuum axis esse medium punctum C, quod bifariam dividit jugum: & videbis posse sine sine alternis additamentis augeri pondera singularum lancium, quia commune centrum gravitatis modò migrat ad unum punctum contactûs, modò ad aliud extremum. Sed ad internoscendum, utrum puncta hæc æqualiter, an inæqualiter à puncto C medio distent, observa additamenta illa, æqualia ne sint? an inæqualia? Nam ut centrum gravitatis migret ex D in E, & iterum ex E in D, æqualia addenda sunt primùm in B, deinde in A, pondera. At ut migret gravitatis centrum ex D in F, plus addendum est ponderis in A, quàm addatur in B, ut migret ex F in D; quia scilicet B magis distat à D centro mo- tûs, quàm A distet ab F centro motûs: igitur plus ponderis ad- dendum est in A, ut habeat momentum æquale momento pon- deris additi in B. Hoc vitium minoribus libris, quarum exilis est axis, non facilè inerit; majores libræ, quæ crassiori axe in- digen, illi obnoxiæ esse possunt, nisi artificis industria in eo ex poliendo solicita fuerit. Sed quid si axis, quâ parte contingit, in angulum simplicem desinat, non tamen in eum cadat per- pendicularis linea lingulæ, quæ jugum bifariam dividit? Iam constat à centro motûs dividi jugum in brachia inæqualia, ac proptereà æquilibrium horizontale esse non posse, inter pon- dera verè æqualia. Sextò. Si libra exactissimè habens brachia æqualia, & lin- gulam perpendicularem, & lances æquales, & funiculorum pondera æqualia, habeat tamen funiculum alterum altero lon- giorem, incumbatque plano horizontali, impositis æqualibus ponderibus non apparebit æquilibrium, si centrum motûs fue- rit in medio jugi puncto, vel infrà illud; sed ad illam partem inclinabitur, quæ breviorem funiculum habuerit. Hoc ideò accidit, quia libram attollens extendit breviorem funiculum longiori adhuc languescente, ac proinde pondus huic lanci im- positum non resistit sursum trahenti, nisi cum funiculus iste fuerit

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Mechanics 294 the equilibrium remained in the first exchange of the pans; and this is the first case. But if, after equilibrium has been established, you add a weight to either of the pans, and equilibrium remains, add as much as is enough for the balance to be inclined almost toward that side, if more weight were still added; then, if a similar weight added to the opposite pan does not overthrow the equilibrium, it indicates that between the points of contact of the axis there is the middle point C, which divides the beam in two: and you will see that, without alternating additions, the weights of the individual pans can be increased, because the common center of gravity now moves to one point of contact, now to the other end. But in order to distinguish whether these points are equally, or unequally, distant from the middle point C, observe those additions: are they equal, or unequal? For as the center of gravity moves from D into E, and again from E into D, equal additions must first be made in B, then in A, to the weights. But for the center of gravity to move from D into F, more must be added to the weight in A than is added in B, in order for it to move from F into D; because, namely, B is farther distant from the center D of motion than A is distant from F, the center of motion: therefore more weight must be added in A, so that it may have a moment equal to the moment of the weight added in B. This defect will not easily be present in smaller balances, whose axis is slender; larger balances, which need a thicker axis, may be liable to it, unless the craftsman’s care has been diligent in polishing it. But what if the axis, on the side where it touches, ends in a simple angle, and yet the perpendicular line of the tongue, which divides the beam in two, does not fall upon it? It is now clear that the beam is divided by the center of motion into unequal arms, and therefore horizontal equilibrium cannot exist between truly equal weights. Sixth. If a balance, having arms exactly equal, and a perpendicular tongue, and equal pans, and equal weights of the cords, should nevertheless have one cord longer than the other, and rest upon a horizontal plane, then when equal weights are placed on it equilibrium will not appear, if the center of motion lies in the middle point of the beam, or below it; but it will incline toward that side which has the shorter cord. This happens because the one lifting the balance stretches the shorter cord while the longer one still slackens, and therefore the weight placed on this pan does not resist the upward pull, except when this cord has been

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Liber tertius. CAPUT VII. 295 fuerit extentus: quare libræ jugum ex hâc parte ascendit sine resistentiâ, dum ex alterâ, quæ funiculum habet breviorem, invenit resistentiam; atque alterâ extremitate manente, alterâ ascendente, jugum inclinatur, extento demùm utroque funi- culo lanx utraque attollitur. Sed quia ex hypothesi omnia sunt æqualia, vel remanet jugum in eâdem positione inclinatum, si punctum libræ brachia disterminans congruat centro motûs, vel pars inclinata ulteriùs descendit, si partum sit inferiùs po- situm. Hinc pondera apparent inæqualia, quamvis verè æqualia sint; & non rarò accidit monetas aliquas aureas tanquam le- ves rejici, quamvis reverâ sint justi & legitimi ponderis; quia lancis, cui imponuntur, funiculus longior est, & libra ad hanc partem, in quâ est pondus, inclinatur; ideóque tribuitur mo- netæ levitas, quia libra vacua in aëre suspensa justissima appa- ret. Vicissim igitur potest fieri, ut moneta levis appareat præ- ponderans, in librâ spartum inferiùs habentè, si moneta levis fuerit imposita lanci, cujus funiculus brevior est; factâ scilicet jam jugi ad hanc partem inclinatione, cum postea lanx utra- que à plano separatur, legitimum pondus, quod gravius qui- dem est, non potest descendere, nisi attollat oppositam lan- cem, cujus ascendentis motus major esse deberet motu legitimi ponderis descendentis; ac proptereà nisi sit major Ratio pon- deris ad monetam, quàm motûs monetæ ascendentis ad motum ponderis descendentis, moneta videbitur præponderans: & tantisper latebit dolus, dum facta fuerit in lancibus ponderis, & monetæ commutatio: apparebit siquidem levius id, quod in lance pendet ex funiculo longiore. Quòd si libra hujusmodi funiculis inæqualibus instructa spartum haberet in loco supe- riore, initio quidem imposita æqualia pondera apparerent in- æqualia, quia non viderentur æquilibria, sed demùm se libra in æquilibrio constitueret, si verè omnia æqualia sint, ut fert hy- pothesis. At si, ut non paucis venditoribus vulgare est, ita li- bra sit constituta, ut lanx altera, cui legitimum pondus impo- nitur juxtà quæsitam mercis quantitatem, subjecto plano in- sistat, altera merci destinata in aëre pendeat, lingulâ ansæ congruente, quæ æquilibrium ostendit; sit verò funiculus lan- cis plano incumbentis fortassè non satis extentus (quia ita con- textus,

Transcription: Translated (English)

Book the Third. CHAPTER VII. 295 has been stretched: wherefore the beam of the balance ascends on this side without resistance, while on the other, which has the shorter cord, it encounters resistance; and, the one end remaining still while the other ascends, the beam inclines, and at last, when both cords are stretched, both pans are raised. But because, by hypothesis, all things are equal, either the beam remains in the same inclined position, if the point dividing the arms of the balance coincide with the center of motion, or the inclined part descends further, if it be placed lower. Hence weights appear unequal, although they are truly equal; and it not infrequently happens that some gold coins are rejected as light, although in reality they are of just and lawful weight; because the cord of the pan on which they are placed is longer, and the balance inclines to that side in which the weight is, and therefore the coin is judged light, because a balance suspended empty in the air appears most just. Conversely, therefore, it may happen that a light coin appears to preponderate in a balance having the beam lower on one side, if a light coin has been placed on the pan whose cord is shorter; for the beam having already been inclined to that side, when afterward both pans are separated from the plane, the lawful weight, which is indeed the heavier, cannot descend unless it raise the opposite pan, whose ascending motion would need to be greater than the motion of the lawful weight descending; and therefore, unless the ratio of the weight to the coin is greater than the motion of the coin ascending to the motion of the descending weight, the coin will seem to preponderate; and the deceit will remain hidden so long as the exchange of the weights and the coin has been made in the pans: for that will appear lighter which hangs in the pan from the longer cord. But if a balance of this kind, furnished with unequal cords, had the beam in the upper place, then at first indeed the equal weights placed in it would appear unequal, because they would not seem to be in equilibrium, but at last the balance would place itself in equilibrium, if indeed all things are equal, as the hypothesis states. But if, as is common among not a few sellers, the balance be so set up that one pan, on which the lawful weight is placed according to the quantity of merchandise sought, rests upon the ground beneath it, while the other, destined for the goods, hangs in the air, the tongue of the handle coinciding, which shows the equilibrium; but if the cord of the pan resting on the ground should perhaps not be sufficiently stretched (because so it is woven,

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Mechanicorum textus, ut majore vi extendatur, quâ cessante se iterum contrahat) merx videbitur præponderans, etiamsi non sit major legitimo pondere; quia deorsum suâ gravitate connitens, dum pondus ex alterâ parte resistit, inclinat lingulam, & oppositæ lancis funiculum extendit. Septimò. Ex ipso plano, cui libra incumbit, antequam attollatur, oriri potest fallacia æqualibus ponderibus inæqualitatem tribuens, etiamsi nullum libræ insit vitium aut ratione inæqualitatis brachiorum, aut ratione lingulæ perperam inclinatæ ad jugum, aut ratione axis angulati, aut ratione funiculo-rum inæqualium. Nam si planum ab horizonte deflectat, & ad illum inclinetur; cùm ad perpendiculum ansa attollitur, funiculi pariter horizonti perpendiculares intelliguntur, & quia æquales sunt, jugum libræ est parallelum plano, ac proptereà perpendiculum ansæ ad angulos inæquales incidit tùm in jugum libræ, tùm in planum inclinatum; lingula igitur, quæ jugo insistit ad angulos rectos, declinat ab ansâ, & sublatâ in aërem librâ, inclinatur lingula ad depressiorem plani paftem, manetque inclinata, quamvis pondera æqualia sint, si centrum motûs & punctum brachia disterminans in eodem puncto conveniant; si verò spartum inferius sit, adhuc magis inclinatur, videturque lanx illa omninò præponderans: at si spartum in superiore loco fuerit, libra primùm inclinata, demùm in aëre suspensa ad æquilibrium horizontale veniet. Octavò. Si contingat ita pondus in lance collocari, ut ipsius ponderis singulare centrum gravitatis non omninò in eodem perpendiculo sit cum puncto jugi, ex quo lanx illa dependet, æquilibrium non indicabit æqualitatem ponderum in utráque lance positorum: Nam si linea directionis per hujusmodi centrum gravitatis transiens incurrat in jugi punctum, quod sit centro motûs vicinius, quàm punctum extremum brachij, oppositæ lancis pondus erit minus; sin autem occurrat lineæ jugi (quæ produc[ti]a intelligitur) remotiùs à centro motûs, oppositæ lancis pondus erit majus; quia scilicet hæc centri gravitatis ponderis collocatio perinde se habet, atque si brachium illud aut imminutum sit, aut auctum: quapropter etiam pondera æquilibria sunt in Ratione reciprocâ brachiorum, ut ex sæpius dictis liquet. Hinc si pondus præter opinionem gravius aut le- vius

Transcription: Translated (English)

Mechanicorum textus, so that it may be extended with greater force (and when that ceases, contract itself again), the weight will seem to be the greater, even though it be not greater than the lawful weight; because, pressing downward by its own gravity, while the weight on the other side resists, it inclines the tongue, and extends the cord of the opposite scale-pan. Seventh. From the plane itself on which the balance rests, before it is raised, a fallacy may arise, assigning inequality to equal weights, even though there be no defect in the balance either because of inequality of the arms, or because of the tongue being wrongly inclined to the yoke, or because of a bent axis, or because of unequal cords. For if the plane declines from the horizon and inclines toward it, when the handle is lifted to the perpendicular, the cords are likewise understood to be perpendicular to the horizon, and because they are equal, the yoke of the balance is parallel to the plane; and therefore the perpendicular from the handle falls at unequal angles both upon the yoke of the balance and upon the inclined plane. The tongue therefore, which rests on the yoke at right angles, deviates from the handle, and when the balance is lifted into the air, the tongue inclines toward the lower part of the plane, and remains inclined, although the weights are equal, if the center of motion and the point dividing the arms coincide in the same point. But if the lower support be below, it inclines still more, and that scale-pan appears altogether to outweigh the other; but if the support be in a higher place, the balance, first inclined, will at last, when suspended in the air, come to a horizontal equilibrium. Eighth. If it should happen that the weight is placed in the scale-pan in such a way that the singular center of gravity of that weight is not altogether in the same perpendicular with the point of the yoke from which that scale-pan hangs, equilibrium will not indicate equality of the weights placed in the two pans. For if the line of direction passing through such a center of gravity falls upon the point of the yoke, which is nearer to the center of motion than the end point of the arm, the weight of the opposite scale-pan will be less; but if it meets the line of the yoke (which is understood as extended) farther from the center of motion, the weight of the opposite scale-pan will be greater; because, namely, this placement of the center of gravity of the weight acts just as if that arm had been diminished or increased. Wherefore also weights are in equilibrium in the reciprocal ratio of the arms, as is clear from what has been said more than once. Hence if the weight unexpectedly be heavier or lighter

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Liber tertius. CAPUT VIII. 297 vius appareat, ejusque pars maxima extrà lancem extet, illud aliter in lance dispone, ut centro gravitatis ponderis facilè im- mineat punctum jugi, ex quo lanx illa suspenditur; & tunc certior fies, an verè gravitas illa ponderi insit, an verò irrep- serit fallacia ex ineptâ ipsius ponderis positione priori. Hoc tamen intellige, quando ex hujusmodi positione sequeretur in- æqualis velocitas motuum oppositorum ponderum. CAPUT VIII. Stateræ natura & forma explicatur. Hactenùs de librâ sermo fuit, in quâ, cum brachia æqua- lia sint, legitimum pondus est æquale gravitati rei, cujus quantitatem ex gravitate investigamus: & quidem quando exi- gua, vel etiam mediocria sunt pondera, res commodè hujus- modi bilance perficitur; at ubi ingentium sarcinarum quanti- tas examinanda est, prorsùs incommodum esset opportunas bi- lances aut habere, aut adhibere: quot enim & quanta pondera parare oporteret, ut centenas aliquot fæni libras, seu mercato- rios fasces, seu saccos farinæ plenos expendereamus? & ex alio in alium locum si transferenda esset libra cum legitimis ponde- ribus tantæ gravitatis, nonne opus esset plaustro, ut tàm in- gens onus in destinatum locum transveheretur? Quare Statera excogitata est tanquam libra brachiorum inæqualium, in quâ pondus minus longiori brachio adnexum æqualia habet mo- menta cum majori pondere, quod ex breviore brachio suspen- ditur. Sed ne varia pondera in promptu habere cogeremur, quæ longioris brachij extremitati adnecterentur, pro variâ oneris gravitate explorandâ, sapientissimè à majoribus sta- tera constructa est quæ eodem æquipondio modò in majo- re, modò in minore distantiâ à centro motûs, æquilibrium constitueret. Ex quo fit stateram eandem vires subire plu- rium librarum, prout plura longioris brachij puncta percur- rit æquipondium; mutantur siquidem Rationes distantiarum ponderum, manente eâdem mercium à sparto distantiâ, ac P p

Transcription: Translated (English)

…so that it may appear more clearly, and its greatest part extend beyond the scale-pan, place it otherwise in the balance, so that the point from which that pan is suspended may easily lie beneath the center of gravity of the weight; and then you will be more certain whether that weight truly has gravity in it, or whether a fallacy has crept in from an unsuitable prior placing of the weight itself. This, however, you must understand when from such a position there would follow unequal speed of the opposite motions of the weights. CAPUT VIII. The nature and form of the steelyard are explained. Hitherto we have spoken of the balance, in which, since the arms are equal, the legitimate weight is equal to the gravity of the thing whose quantity we investigate by its weight: and indeed when the weights are small, or even moderate, the matter is conveniently carried out by a balance of this kind; but when the quantity of very large loads must be examined, it would be altogether inconvenient either to have or to use suitable balances: for how many and how great weights would it be necessary to prepare, if we were to weigh several hundred pounds of hay, or merchants’ bundles, or sacks full of flour? And if from one place to another the balance were to be transferred with legitimate weights of such great heaviness, would it not be necessary to use a wagon, so that so enormous a burden might be conveyed to the intended place? Wherefore the steelyard was devised, as a balance of unequal arms, in which the lesser weight attached to the longer arm has equal moments with the greater weight which is suspended from the shorter arm. But lest we should be forced to have various weights at hand to be attached to the extremity of the longer arm, for investigating the varying heaviness of the load, the steelyard was most wisely constructed by our forebears, so that the same counterpoise would establish equilibrium, now at a greater, now at a lesser distance from the center of motion. From this it follows that the same steelyard bears the force of several pounds, according as the counterpoise traverses more points of the longer arm; for the ratios of the distances of the weights are indeed changed, while the distance of the goods from the scale end remains the same, and P p

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Mechanicorum 298 proinde etiam idem æquipondium variam habet Rationem ad merces inæquales. Sunt autem stateræ partes Jugum, Ansa, Uncus aut lanx, Æquipondium, quod aliis Sacoma, aliis Cursorium dicitur. Jugum est, quod in partes inæquales divisum ab axe, qui An- sæ inseritur, definit Rationem ponderum, quæ momentis æqualibus librantur. Ansa est, ex quâ suspenditur statera, ut liberè utramque in partem versetur. Uncus, aut lanx, oneri sustinendo destinatur; quæ enim facilè molem unam efficiunt, possunt ex Unco suspendi; sed quæ ex pluribus non facilè in unam molem coëuntibus constant, lance subjectâ recipi oporter Æquipondium est certæ gravitatis pondus, ex quo oppositæ gravitatis Ratio innotescit. Sit AB jugum ab axe inæqualiter in C divisum, sitque CA brachium minùs, cujus extremitati A catena aut funis adnecti- tur cum unco aut lance E, & CB brachium majus, cujus longitu- dinem pro opportunitate percurrit æquipondium F. Ansa respondens lingulæ CD, ipsius axis extremit- tates recipit, ut facilè convolvi possit. In minoribus & mediocri- bus stateris lingula crassiuscula ad- ditur, quæ ansæ intercapedinem ita impleat, eique congruat, ut tamen nullo partium conflictu impediatur motus; in majori- bus & longioribus stateris aliquando lingula omittitur, vel quia spartum est infrà rectam lineam jugi, quod non nisi horizonta- liter consistit, vel quia si spartum est in superiore loco, non multùm à vero pondere aberrare permittit ipsâ brachij longitu- do, quæ facilè prodit parallelismum aut inclinationem ad ho- izontem; mediocris autem error in mercibus, quæ hujusmodi magnis stateris expenduntur, neque emptori, neque venditori incommodo est; quapropter in iis subtilitatem scrupulosè per- sequi inutile est, & ineptum. Quæ in librâ circâ Axem, lin- gulam, Ansam observanda monuimus, stateræ pariter commu- nia sunt, neque hîc iterum inculcanda. Potissimum, quod in staterâ observandum est, pertinet ad divisionem longioris brachij in minutiores particulas, ut exqui- sitiùs

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Mechanics 298 therefore this same balance also has a different relation to unequal goods. Now the parts of the balance are the beam, the handle, the hook or pan, and the counterweight, which is called by some the Sacoma, by others the Cursorium. The beam is that which, divided into unequal parts by the axis inserted into the handle, determines the relation of weights, which are balanced by equal moments. The handle is that from which the balance is suspended, so that it may turn freely to either side. The hook, or pan, is intended for supporting the load; for those things which easily make up one mass may be suspended from a hook; but those which consist of several parts not easily coalescing into one mass ought to be received upon a pan placed beneath them. The counterweight is a weight of a certain heaviness, from which the ratio of the opposite heaviness becomes known. Let AB be the beam, unequally divided at C by the axis, and let CA be the shorter arm, to the extremity A of which a chain or cord is attached together with the hook or pan E, and CB the longer arm, along whose length, as convenience requires, the counterweight F moves. The handle, corresponding to the little plate CD, receives the extremities of the axis itself, so that it may easily be turned round. In smaller and medium-sized balances a rather thick little plate is added, which fills the space of the handle in such a way and fits it so well that nevertheless the motion is impeded by no conflict of the parts; in larger and longer balances sometimes the little plate is omitted, either because the beam is set below the straight line, which consists only horizontally, or because, if it is set in the upper position, the length of the arm itself allows it to deviate little from the true weight, since it readily reveals the parallelism or inclination to the horizon; but a moderate error in goods weighed on such large balances is inconvenient neither to buyer nor seller; wherefore in these it is useless and improper to pursue exactness scrupulously. What we have noted as to be observed in the scale about the axis, the little plate, and the handle, is equally common to balances, and need not here be repeated again. What chiefly must be observed in the balance concerns the division of the longer arm into smaller parts, so that more accurately

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Liber tertius. CAPUT VIII. 299 sitiùs innotescat Ratio mercis ad æquipondium, quæ denota- tur ab incisis in brachio notis indicantibus Rationem brachij longioris ad brevius; est scilicet minoris brachij longitudo transferenda in alterum brachium, quoties fieri potest; & quia hoc longius produci potest infinitè, proptereà statera vocari potest libra quasi infinita brachiorum inæqualium. Sic distan- tia AC translata in brachium CB ex. gr. quater, facit ut pon- dus in E possit esse quadruplum æquipondij F, si æquipondium sit in extremitate B: quia, ut dictum est de librâ brachiorum inæqualium, ut AC ad CB, ita pondus in B ad pondus in A: & si æquilibrium contingat sacomate existente in G, erit ut AC ad CG ita Sacoma in G ad pondus in E. Hîc animadvertendum est distantiam AC, si sit valdè nota- bilis, capacem esse multiplicis divisionis, ac proptereà æqua- lem partem HG posse subtiliùs dividi, ut non solùm uncias, sed & unciæ quadrantes, aut etiam drachmas ostendat, si tran- situs ex H in G sit nota unius libræ. Verum est in brachio CB hujusmodi majores partes minori brachio æquales non multas esse posse: sed huic malo occurritur in adversâ parte jugi; con- versâ enim statera aliam habet ansam, puta SV, quæ minùs distat ab extremitate A; hæc autem distantia sæpiùs iterata plu- res exhibet partes, & factâ suspensione VS, æquipondium in extremitate B positum æquilibratur cum majori pondere, quàm cùm ex DC statera suspenditur; est scilicet major Ratio BS ad SA, quàm BC ad CA; nam ad eandem CA, majorem Ratio- nem habet BS major, quàm BC minor, & eadem BS majo- rem Rationem habet ad SA minorem, quàm ad CA majorem ex 8 lib. 5, manifestum est igitur majorem esse Rationem BS ad SA, quàm BC ad CA. Si igitur pondera sunt reciprocè ut brachiorum longitudines, idem æquipondium in extremitate B positum minorem habet Rationem ad pondus in A, quando brachia sunt BS & SA, quàm cùm brachia sunt BC & CA: ac propterea tunc pondus in A est majus. Verùm hactenùs de staterâ perinde locutus sum, ac si nulla illi inesset gravitas; quæ tamen omninò contemnenda non est, quantumvis minuta sit ipsa statera atque exilis, hac enim mi- norum ponderum gravitatem scrupulosiùs exploramus: ideò autem gravitatem à materiâ mente præcidere satius duxi, ut P p 2

Transcription: Translated (English)

Book third. CHAPTER VIII. 299 so that the ratio of the weight to the balance may more clearly be made known, which is indicated by the marks cut on the arm showing the ratio of the longer arm to the shorter; namely, the length of the shorter arm is to be transferred to the other arm, as often as can be done; and because this can be prolonged infinitely, therefore the balance may be called a scale of, as it were, infinitely unequal arms. Thus the distance AC, transferred to the arm CB, for example four times, makes it so that the weight at E may be quadruple the equilibrium F, if the equilibrium be at the extremity B: because, as has been said of the balance of unequal arms, as AC is to CB, so is the weight at B to the weight at A; and if equilibrium occurs with the counterpoise being at G, then as AC is to CG, so is the counterpoise at G to the weight at E. Here it is to be noted that the distance AC, if it be very noticeable, is capable of being divided in many ways, and therefore the equal part HG can be more subtly divided, so that it may show not only ounces, but also quarter-ounces, or even drachms, if the passage from H to G be the indication of one pound. It is true that in the arm CB, parts larger than the smaller arm can scarcely be many: but this inconvenience is met on the opposite side of the beam; for when the balance is turned over, it has another handle, namely SV, which is less distant from the extremity A; and this distance, repeated more often, exhibits more parts, and when suspension is made by VS, the counterpoise placed at the extremity B is balanced with a greater weight than when the balance is suspended from DC; that is, the ratio of BS to SA is greater than that of BC to CA; for to the same CA, the larger BS has a greater ratio than the smaller BC, and the same BS has a greater ratio to the smaller SA than to the larger CA; from Book 5, proposition 8, it is manifest, therefore, that the ratio of BS to SA is greater than that of BC to CA. If therefore the weights are reciprocally as the lengths of the arms, the same counterpoise placed at the extremity B has a smaller ratio to the weight at A when the arms are BS and SA than when the arms are BC and CA: and therefore then the weight at A is greater. However, up to this point I have spoken of the balance as if no weight were in it; yet that is not to be wholly disregarded, however minute and slender the balance itself may be, for by this means we more carefully examine the weight of lesser weights: therefore I thought it better to set aside the weight itself from the material in my mind, so that P p 2

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Mechanicorum statim appareat vis momentorum, quæ pro variâ distantiâ obtinet æquipondium; prout ad majorem, aut ad minorem motum comparatè cum motu ponderis in A, est dispositum. Cæterùm pondus in A, quod æquilibrium facit cum sacomate F, majus est quàm pro Ratione distantiarum reciprocè sumptâ; quia videlicet ipsius brachij longioris gravitas sua habet momenta majora momentis brachij brevioris, ac propterea præter pondus, quod Sacomati respondet, addendum est etiam pondus, quod respondeat excessui momentorum brachij majoris suprà momenta brachij minoris. Cùm itaque ex dictis cap. 1. hujus lib. momenta brachiorum singulorum perinde se habeant, atque si semissis gravitatis singulorum esset in extremitatibus, posito jugo æquabilis crassitiei, si nota sit totius jugi gravitas, & brachiorum Ratio, singulorum quoque gravitas innotescit; cujus semissis per sibi congruum terminum Rationis ductus exhibet singulorum momenta. Sit AB jugum lib. 5. unc. 10, hoc est omninò unc. 70: Ratio AC ad CB sit ut 2 ad 5; igitur gravitas AC est unc. 20, & CB unc. 50: semissis AC unc. 10 ductus per 2 (qui est terminus Rationis illi congruens) dat momentum 20: semissis CB unc. 25 ductus per 5, dat momentum 125: differentia momentorum est 105 dividenda per terminum Rationis congruum distantiæ AC, videlicet per 2: Quare ut fiat æquilibrium cum solâ gravitate brachij longioris, addendæ sunt extremitati A unciæ 52 ́: igitur adddito semisse gravitatis AC, intelliguntur in A unciæ 62 ́; & in B unciæ 25: sunt autem 62 ́ ad 25, ut 5 ad 2, quæ est Ratio reciproca brachiorum. Quare si jugum AB æquabile sit, ut fert hypothesis, & in extremitate B sit Sacoma lib. 2, pondus in A (computatâ etiam gravitate catenæ & unci AE) non erit solùm lib. 5. ut exigit Ratio longitudinis brachiorum, sed præterea unc. 52 ́, hoc est omnino lib. 9. unc. 4 ́. Quia verò aliquando accidit properatâ ad subitum usum staterâ uti, videlicet crassiore tigillo, cujus gravitas non est planè contemnenda, sed valdè notabilis; proptereà hîc brevem praxim adjicere placet, quæ etiam minùs peritis usui esse possit, ut statim inveniant gravitatis quantitatem, quæ soli gravitati brachij longioris respondet. Sit tigillus AB, in quo intelliga- tur

Transcription: Translated (English)

Mechanics let the force of the moments, which is obtained for a varying distance, of the balance-beam immediately appear, according as it is arranged for a greater or a smaller motion compared with the motion of the weight in A. Moreover, the weight in A, which produces equilibrium with the counterpoise F, is greater than would be required by the ratio of the distances taken reciprocally; because, namely, the weight of the longer arm itself has greater moments than the moments of the shorter arm, and therefore, besides the weight which corresponds to the counterpoise, there must also be added the weight which corresponds to the excess of the moments of the greater arm over the moments of the lesser arm. Since therefore, from what was said in chap. 1 of this book, the moments of the several arms are the same as if half of the weight of each were at the extremities, if a beam of uniform thickness be given, and if the weight of the whole beam and the ratio of the arms are known, the weight of each is also made known; and half of this, multiplied by its proper term of the ratio, exhibits the moments of each. Let AB be a beam of 5 lb. 10 oz., that is, in all 70 oz.; let the ratio of AC to CB be as 2 to 5; therefore the weight of AC is 20 oz., and of CB 50 oz.: half of AC, 10 oz., multiplied by 2 (which is the term proper to that ratio), gives moment 20; half of CB, 25 oz., multiplied by 5, gives 125: the difference of the moments is 105, to be divided by the term proper to the ratio of the distance AC, namely by 2: wherefore, in order that equilibrium may be made with the weight of the longer arm alone, 52 1/2 oz. must be added to the end A; therefore, adding half the weight of AC, there are understood at A 62 1/2 oz., and at B 25 oz.: and 62 1/2 to 25 are as 5 to 2, which is the reciprocal ratio of the arms. Wherefore, if the beam AB is uniform, as the hypothesis states, and at the end B there is a counterpoise of 2 lb., the weight at A (the weight of the chain and the point AE being computed also) will not be merely 5 lb., as the ratio of the length of the arms requires, but in addition 52 1/2 oz., that is, in all 9 lb. 4 1/2 oz. But because sometimes it happens that a balance is used hurriedly for immediate use, namely with a thicker beam, whose weight is not at all to be neglected, but is highly considerable; for that reason it seems fitting here to add a brief practical method, which may also be useful to those less skilled, so that they may at once find the quantity of weight which corresponds to the weight of the longer arm alone. Let AB be a beam, in which let it be understood

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Liber tertius. CAPUT VIII. 301 tur ipsi A C brachio minori æqualis pars CH; est igitur bra- chiorum differentia HB. Ponamus totam jugi longitudinem esse distinctam in partes 22, quarum A C sit 4, C B 18, ac dif- ferentia H B 14. Sit verò tigilli pondus lib. 84, cujus semissem lib. 42 accipio. Tum fiat ut longitudo brachij minoris 4 ad dif- ferentiam brachiorum 14, ita semissis gravitatis jugi lib. 42 ad aliud, & provenient lib. 147 addendæ brachio minori, ut fiat æquilibrium cum solâ gravitate longioris. Sic in superiore exemplo, ubi brachia erant ut 2 ad 5, differentia 3, pondus ju- gi unc. 70, cujus semissis unc. 35; fiat ut 2 ad 3, ita unc. 35 ad uncias 52 ́, quod est pondus ibi inventum pluribus calculus. Ex his infertur jugum æquabilis crassitiei si suspendatur ex quartâ parte suæ longitudinis, sustinere sinè æquipondio pon- dus additum minori brachio, cujus gravitas æqualis sit gravita- ti totius jugi. Si ex sextâ parte suspendatur, sustinet pondus duplex gravitatis ipsius jugi: si ex octavâ parte, sustinet pon- dus triplex gravitatis jugi; si ex decima parte, sustinet pondus quadruplex; si ex duodecimâ, sustinet pondus quintuplex, & sic deinceps. Ut igitur ex ratione & certâ methodo construeretur statera exquisitè distincta in suas particulas, oporteret brachium mi- nus cum adnexus appendiculis, catenâ, unco, seu lance, tantæ gravitatis esse, ut cum solâ longioris brachij gravitate æquili- brium constitueretur: tùm distantia inter punctum, ex quo onus suspenditur, & centrum motûs transferenda esset ex eo- dem centro motûs in brachium longius, quoties fieri posset, & singula intervalla in certas partes minores dividenda, vel pro libito vel (quod magis rationi congruum est) in partes pro- prias mensuræ, quæ adhibetur, ut si libra sit in uncias, si un- cia, in drachmas. Hoc autem pendet ex gravitate sacomatis, quod eligitur: nam si libram unam pendat unà cum suo annu- lo æquipondium, tot erunt ponderis libræ, quot partes minori brachio æquales intercipiuntur inter spartum & ipsum æqui- pondium: at si bilibre sit sacoma, jam partes illæ assumptæ æquales minori brachio sunt bifariam dividendæ, ut singula- rum librarum notæ in jugo habeantur. Quod si constructâ jam hoc modo staterâ, & majoribus partibus distinctis in particulas ex libito assumptas, velis apponere æquipondium majus, quàm P p 3

Transcription: Translated (English)

Liber tertius. CAPUT VIII. 301 thus the part CH is equal to the smaller arm AC; therefore the difference of the arms is HB. Let the whole length of the beam be divided into 22 parts, of which AC is 4, CB 18, and the difference HB 14. Suppose also that the weight of the beam is 84 lb., of which I take the half, 42 lb. Then let it be as the length of the smaller arm, 4, to the difference of the arms, 14, so the half of the weight of the beam, 42 lb., is to another quantity; and there will result 147 lb. to be added to the smaller arm, so that equilibrium may be obtained with the weight of the longer arm alone. Thus in the preceding example, where the arms were as 2 to 5, the difference 3, and the weight of the beam 70 oz., of which the half is 35 oz.; let it be as 2 to 3, so 35 oz. to 52 ́ oz., which is the weight found there by the former calculation. From these things it is inferred that a beam of equal thickness, if suspended from the fourth part of its length, will support, without counterpoise, an added weight on the smaller arm whose heaviness is equal to the heaviness of the whole beam. If it is suspended from the sixth part, it supports a weight double the beam’s own heaviness; if from the eighth part, it supports a weight triple the beam’s heaviness; if from the tenth part, it supports a quadruple weight; if from the twelfth, a quintuple weight; and so on thereafter. Therefore, in order that a balance finely divided into its parts might be constructed by reason and a fixed method, it would be necessary for the shorter arm, together with the attached appendages, chain, hook, or pan, to be of such weight that equilibrium would be established with the weight of the longer arm alone; then the distance between the point from which the load is suspended and the center of motion would need to be transferred from that center of motion to the longer arm, as often as possible, and each interval divided into smaller determinate parts, either at pleasure or, what is more in accordance with reason, into the proper units of the measure that is used, as if a pound were divided into ounces, or an ounce into drachms. But this depends on the weight of the counterpoise that is chosen: for if one pound hangs together with its ring as the counterweight, there will be as many pounds of weight as there are parts equal to the smaller arm intercepted between the beam and the counterpoise itself; but if the counterweight is two pounds, then those parts assumed as equal to the smaller arm must already be divided in two, so that the marks of individual pounds may be obtained on the beam. But if, a balance having been thus constructed, and the larger parts divided into particles assumed at will, you wish to add a larger counterpoise than the one previously mentioned, then P p 3

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Mechanicorum 302 fortè ab artifice destinaretur, licebit; modò memineris reciprocam esse distantiarum Rationem & ponderum, quæ in æquilibrio sunt. At si contigerit ea omnia, quæ breviori brachio adhærent, non constituere æquilibrium cum brachio longiore seorsim sumpto absque sacomate, vel quia graviora sunt, vel quia minùs gravia; satis apparet æquipondium in distantiâ à sparto duplâ brachij minoris non habere duplum momentum, sed inveniendum esse aliud punctum, à quo distantiæ mensura desumatur. Sit statera A C B, quæ in C suspendatur: gravitas brachiorum ita se habet, ac si illius semissis in sua cujusque brachij extremitate poneretur. Hujusmodi semisses gravitatum repræsententur à lineis B D & A E, quæ sunt utique invicem in Ratione brachiorum (quoniam jugum æquabile & uniforme ponitur) & ut A C ad C B, ita A E ad B D. Sed ut fiat æquilibrium debet esse vicissim ut A C ad C B, ita B D gravitas in B ad A F gravitatem in A: Est igitur A E ad A F in duplicatâ Ratione brachiorum A C ad C B, hoc est ut Quadratum A C ad Quadratum C B: Ergo etiam dividendo, per 17. lib. 5. ut Quadratum C B minus Quadrato A C ad Quadratum A C, ita A F minùs A E ad A E; hoc est ut, differentia Quadratorum utriusque brachij ad Quadratum brachij minoris, ita F E pondus addendum, ad A E semissem gravitatis brachij minoris, ut fiat æquilibrium cum semisse gravitatis, & momento brachij C B longioris. Id si factum fuerit, assumantur in C B, incipiendo à puncto C, partes æquales ipsi C A, & tunc ad mercem additam in F habebit gravitas sacomatis H eam Rationem, quam habuerit A C ad distantiam ejusdem sacomatis à puncto C, ut superiùs dicebatur. Verùm si præter A E gravitatem respondentem minori brachio A C, pendere intelligatur ex A non solùm gravitas E F, quæ sufficiat ad æquilibrium cum longiore brachio C B, sed præterea sit etiam gravitas F G, ita ut tota gravitas addita sit E G;

Transcription: Translated (English)

Mechanics 302 perhaps might be assigned by the craftsman, it will be permissible; only remember that the ratio of distances and of weights which are in equilibrium is reciprocal. But if it should happen that all those things which adhere to the shorter arm do not establish equilibrium with the longer arm taken separately without the scale, either because they are heavier or because they are less heavy, it is clear enough that a counterweight placed at a distance double that of the shorter arm does not have double the moment; rather, another point must be found, from which the measure of the distance is to be taken. Let there be the balance A C B, suspended at C: the weight of the arms is such as if half of it were placed at the extremity of each arm. Let such halves of the weights be represented by the lines B D and A E, which are certainly in reciprocal ratio to one another, since an equal and uniform beam is assumed, and as A C is to C B, so A E is to B D. But for equilibrium to occur, it must reciprocally be as A C is to C B, so must the weight B D at B be to the weight A F at A. Therefore A E is to A F in the doubled ratio of the arms A C to C B, that is, as the square of A C to the square of C B. Therefore also, by division, by 17, lib. 5, as the square of C B minus the square of A C to the square of A C, so A F minus A E to A E; that is, as the difference of the squares of each arm to the square of the shorter arm, so the weight F E to be added to the half of the weight A E of the shorter arm, so that equilibrium may be obtained with the half of the weight, and the moment of the longer arm C B. If this be done, let equal parts to A C be taken on C B, beginning from point C, and then, in addition to the merchandise placed at F, the weight of the scale H will have that ratio to it which A C will have to the distance of the same scale from point C, as was said above. But if, besides the weight A E corresponding to the shorter arm A C, it is understood that from A there hangs not only the weight E F, which suffices for equilibrium with the longer arm C B, but furthermore also a weight F G, so that the total added weight is E G;

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Liber tertius. CAPUT VIII. 303 EG; tunc assumpto æquipondio H notæ gravitatis, debet fieri ut pondus H ad pondus FG excessum suprà id, quod requiritur ad æquilibrium, ita distantia AC ad aliud ex. gr. CI: & ex I initium sumere debet divisio transferendo in longius brachium, & iterando distantiam CA ita, ut AC æqualis sit ipsi IN: si enim in G addatur tantum mercis, cujus gravitas GM sit ad æquipondium H, ut IN ad AC, fiet in N æquilibrium. Quia scilicet ut FG gravitas ad gravitatem H, ita IC distantia ad distantiam CA ex constructione; & ut gravitas H ad gravitatem GM, ita CA distantia ad distantiam IN; erit ex æqualitate per 22. lib.5. ut gravitas FG ad gravitatem GM, ita distantia CI ad distantiam IN; Ergo componendo, per 18. lib.5. ut FM ad GM, ita CN ad IN; sed ut GM ad H, ita IN ad CA ex hypothesi; igitur ex æqualitate ut FM gravitas ad gravitatem H, ita CN distantia ad distantiam CA. Cùm itaque pondera addita ultrà æquilibrium, quod additâ gravitate EF sit in C puncto suspensionis, sint in Ratione reciprocâ distantiarum à sparto C, necessariò sequitur æquilibrium in N. Idem dicendum de cæteris deinceps punctis iterando distantiam IN, prout brachij longitudo ferre potest, nam duplicatâ distantiâ IN, poterit in G addi gravitas dupla gravitatis æquipondij H. Quod si demùm partes minori brachio CA adjacentes non essent tantæ gravitatis, ut fieret cum longiore brachio CB æquilibrium, quemadmodum si essent ut OE ad EA semissem gravitatis brachij minoris; primò observa, quantum desit gravitatis, ut fiat æquilibrium, scilicet sit quantitas OF, quæ ponatur minor gravitate æquipondij H: intelligatur itaque gravitas æqualis gravitati æquipondij H, & sit excessus FG. Quare sicuti paulò antè dicebatur, fiat ut pondus H ad gravitatem FG, ita AC ad CI, & erit I punctum à quo incipienda est divisio jugi, ita tamen ut facto æquilibrio in I intelligatur addita merx æqualis gravitatis cum æquipondio H, & erit ex. gr. prima libra. At verò si OE tam modica gravitas esset, ut etiam addita gravitas æqualis gravitati sacomatis H, nondum adæquaret gravitatem EF, addatur duplex, triplex, quadruplex gravitas sacomatis H ita, ut demum excedat gravitatem EF necessariam ad æquilibrium cum solo brachio longiore; tum fiat sicuti

Transcription: Translated (English)

Book Three. CHAPTER VIII. 303 Thus, taking the weight H of the notation as the equivalent, it must be arranged that the weight H to the weight FG, the excess over that which is required for equilibrium, is as the distance AC to another, e.g. CI; and from I the division must begin, transferring it to the longer arm, and repeating the distance CA so that AC is equal to IN: for if in G there be added only such a weight of merchandise, whose gravity GM is to the equivalent H as IN is to AC, there will be equilibrium at N. For, namely, as the weight FG is to the weight H, so the distance IC is to the distance CA, by construction; and as the weight H is to the weight GM, so the distance CA is to the distance IN; therefore, from equality, by 22. lib. 5, as the weight FG is to the weight GM, so is the distance CI to the distance IN; therefore, by composition, by 18. lib. 5, as FM is to GM, so is CN to IN; but as GM is to H, so IN is to CA, by hypothesis; therefore, by equality, as the weight FM is to the weight H, so is the distance CN to the distance CA. Since therefore the added weights beyond the equilibrium, which with the added weight EF is at the point C of suspension, are in the reciprocal ratio of the distances from the fulcrum C, it necessarily follows that equilibrium is at N. The same must be said of the other following points, by repeating the distance IN, as far as the length of the arm can bear; for with the distance IN doubled, a double weight of the equivalent H can be added in G. But if at length the parts adjacent to the smaller arm CA were not of such weight that equilibrium would occur with the longer arm CB, as if they were as OE to EA, a half of the weight of the smaller arm; first observe how much weight is lacking for equilibrium, namely let the quantity be OF, which is to be set less than the weight of the equivalent H: let a weight therefore be understood equal to the weight of the equivalent H, and let FG be the excess. Wherefore just as was said a little before, let the weight H be to the gravity FG, so AC is to CI, and I will be the point from which the division of the yoke must begin, however so that, when equilibrium is made at I, there is understood to be added a load equal in weight to the equivalent H, and it will be, for example, the first pound. But if the weight OE were so slight that even the added weight equal to the weight of the balance H would still not equal the weight EF, let there be added double, triple, quadruple weight of the balance H, so that at last it exceeds the weight EF necessary for equilibrium with the longer arm alone; then let it be thus

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Mechanicorum 304 sicuti priùs, ut pondus H ad excessum illum, scilicet ad FG, ita AC ad CI, & est I punctum quæsitum, ex quo incipit divi- sio, & in quo si fiat æquilibrium mercis cum sacomate, indicat mercis gravitatem esse duplam, triplam, quadruplam gravita- tis sacomatis H, prout hanc duplicare oportuit, aut triplicare. Sed quas habemus communes stateras ab hâc sedulitate pro- cul remotas esse omnibus constabit, si observaverint amplitu- dines priorum divisionum non omninò respondere brachij mi- noris longitudini, hoc est, intervallo, quo pondus distat à spar- to; neque id solùm, quia artifices tantam adhibere diligentiam recusant pro tenui mercede; verùm etiam ne adeò graves existant majores stateræ, si minori brachio tanta esset addita gravitas, quæ longioris brachij momenta æquaret. Propterea jugum construunt, uncum seu lancem cum suis catenulis ad- nectunt, ex ansâ suspendunt, sacoma non certi ponderis sed ex arbitrio eligunt, quod tamen additæ lanci, aut unco aliquate- nus respondeat juxta minoris brachij longitudinem; nam si hoc valde breve sit, augent lancis pondus, & minuunt æquipon- dium; & ex adverso, si illud longiusculum sit, minuunt lancem, augent sacoma; quia nimirum in illâ brevitate brachij minoris majora sunt momenta brachij longioris, & minus æquipon- dium plus habet momenti; contrà verò auctâ minoris brachij longitudine decrescunt momenta tùm longioris brachij tùm æquipondij. His paratis statuunt in lance legitimum aliquod pondus jux- tà denominationem mensuræ, quam assumunt tribuendam sta- teræ, puta libram (idem dic de majoribus ponderibus in aversâ stateræ parte inscribendis, ut lib. 25 aut 100 juxtà regionis mo- rem) deinde tantisper sacoma adducunt vel reducunt, dum fiat exquisitè æquilibrium; & punctum adnotant, in quo sacoma quiescit. Tùm aliam adhuc libram, aut, primâ sublatâ, bilibre pondus, lanci imponunt, & sacoma retrahunt, ut magis à mo- tûs centro distet; iterumque facto æquilibrio punctum notant. Demum intervallum inter hæc duo notata puncta in jugo ite- rant, quoties possunt; & ut uncias habeant, singula intervalla in duodecim æquales particulas distinguunt, quæ in minuscu- lis stateris adhuc minores divisiones recipiunt. Quod si adhuc pondera infrà libram unam, hoc est infra un- cias

Transcription: Translated (English)

Mechanicorum 304 as before, so that the weight H to that excess, namely to FG, is as AC to CI; and I is the point sought, from which the division begins, and in which, if the goods and the counterweight be made to balance, it indicates that the weight of the goods is double, triple, or quadruple the weight of the counterweight H, according as this latter ought to be doubled or tripled. But that the common scales we have are far removed from this exactness will be evident to everyone, if they observe that the extents of the earlier divisions do not correspond entirely to the length of the shorter arm, that is, to the distance by which the weight is removed from the pivot; and not only because craftsmen refuse to apply so much diligence for so small a wage; but also lest the larger scales should become too heavy, if so much weight were added to the shorter arm as would equal the moments of the longer arm. For this reason they construct the beam, attach the hook or pan with its chains, suspend it from the handle, and choose a counterweight not of a fixed weight but at their discretion, such however as may in some measure correspond to the added pan or hook according to the length of the shorter arm; for if this be very short, they increase the weight of the pan and diminish the counterpoise; and on the other hand, if it be somewhat longer, they lessen the pan and increase the counterweight; because, naturally, with that shortness of the shorter arm the moments of the longer arm are greater, and the counterpoise has more moment; but conversely, when the length of the shorter arm is increased, the moments both of the longer arm and of the counterpoise decrease. These things being prepared, they place in the pan some lawful weight according to the denomination of the measure which they choose to assign to the scale, for example a pound (the same may be said of larger weights to be inscribed on the reverse side of the scale, such as lib. 25 or 100 according to the custom of the region); then they bring the counterweight up or back until exact balance is achieved; and they note the point at which the counterweight rests. Then they place yet another pound, or, the first being removed, a two-pound weight, in the pan, and draw back the counterweight so that it may be farther from the center of motion; and again, balance being obtained, they note the point. Finally they repeat, as often as they can, the interval between these two marked points on the beam; and in order to have inches, they divide each interval into twelve equal parts, which in smaller scales receive still smaller divisions. And if weights still below one pound, that is below ounces

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Liber tertius. CAPUT VIII. 305 cias 1 2, hac staterâ examinare libeat, inter punctum primò no- tatum atque spartum minusculas illas divisiones transferunt, incipiendo ab illo puncto. Quid autem hîc meminerim puncta hujusmodi omnia in ju- gi acie, seu angulo solido superiore notari, majores autem di- visiones certis lineis ad latus ductis significari? hæc enim vul- garia sunt. Illud potius notandum est, quod in unâ eâdemque staterâ trium regionum stateras habere possumus: quia enim stateræ scapus communiter quadrangularis est, & in superiore angulo libras hujus regionis insculpsit artifex, in duobus angu- lis hinc, & hinc libras duabus regionibus, cum quibus com- mercia miscentur, peculiares inscribere licebit (nam pondera simili nomine in pluribus regionibus donata, non esse inter se æqualia docemur experientiâ, quæ libras Parisiensem, Ro- manam, Venetam inæquales esse ostendit) & æquipondij an- nulus unâ eâdemque operâ in tribus angulis diversarum regio- num pondus ejusdem mercis indicabit. Hîc verò curiosiùs inquirenti, præstantiorne dicenda sit sta- tera? an libra? vix poterit quisquam absolutè respondere: nam minoribus ponderibus, ut gemmis, aureis monetis, & simili- bus examinandis parùm opportuna est statera; at ingentibus oneribus hæc aptissima est, libra autem incommoda. Compen- dium habet statera unico sacomate contenta; pluribus ponderi- bus eget libra. Vicissim in librâ securiùs artifices laborem im- pendunt, quia faciliùs æqualitatem assequuntur brachiorum, quàm proportionem justo æquilibrio necessariam; & in librâ quidem si æqualitatem perfectam semel statuant, nil est quæ- rendum ampliùs; sed in staterâ singula divisionum puncta suam habent Rationem, suamque exposcunt diligentiam; in pluribus verò aliquando peccare proclivius est, quàm in uno. Quòd si libræ perfecta æqualitas desit, saltem lancium & ponderum commutatione, ut superiùs monuimus, deprehenditur error; at si falsa sit statera, non aliter innotescet, quàm si pondus idem iterùm librâ examinemus, ut appareat, an sibi constet eadem gravitas: quis enim aliter iniqui venditoris imposturam rete- gat, qui, ut major appareat mercis gravitas, ex æquipondio, aut ex capite longioris brachij, quasi nitidiùs illa expoliens, notabilem aliquam gravitatis particulam limâ abrasit? cum ta- Q9

Transcription: Translated (English)

Book third. CHAPTER VIII. 305 If it pleases us to examine 1, 2, with this steelyard, they transfer those little divisions between the first marked point and the spar, beginning from that point. What need I here remind the reader that all such points are marked on the continuous edge, or upper solid angle, while the larger divisions are indicated by certain lines drawn to the side? These things are indeed common enough. What should rather be noted is that on one and the same steelyard we may have the steelyards of three regions: for since the beam of the steelyard is commonly quadrangular, and the craftsman has inscribed the pounds of this region on the upper angle, it will be allowable to inscribe in the two angles on either side the pounds peculiar to the two regions with which trade is conducted (for experience teaches us that weights granted the same name in several regions are not equal among themselves; Paris, Roman, and Venetian pounds have been shown to be unequal) and thus the equilibration ring will, by one and the same operation, indicate in the three angles the weight of the same merchandise according to different regions. But here, to the more inquisitive, whether the steelyard should be called the better instrument, or the balance, hardly anyone will be able to answer decisively: for for weighing smaller weights, such as gems, gold coins, and the like, the steelyard is of little use; but for very heavy burdens this is most suitable, whereas the balance is inconvenient. The steelyard has the advantage of using only a single weight; the balance requires several weights. On the other hand, in the balance artisans work with greater security, because they more easily achieve equality of the arms than the proportion needed for just equilibrium; and in the balance, indeed, once perfect equality has been established, nothing further need be sought; but in the steelyard each point of division has its own ratio and requires its own care; and in several such points it is sometimes easier to err than in one. But if the perfect equality of the balance is lacking, at least the error is detected by exchanging the pans and weights, as we noted above; yet if the steelyard is false, it will not become known except by examining the same weight again with the balance, so that it may appear whether the same heaviness is consistent with itself: for how else will one unmask the fraud of an unfair seller, who, so that the merchandise may seem heavier, has filed away with a file some notable portion of weight from the equilibrium point, or from the end of the longer arm, as though polishing it more brightly? when the...

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Mechanicorum men à minore brachio expoliendo manum abstinuerit; quippe qui satis norat id fieri non posse citrà ipsius venditoris damnum: constitutâ siquidem staterâ, nihil ex hac aut ex illâ parte de- mendum, nihil addendum, ne mutetur Ratio, quæ intercedit inter ipsorum brachiorum momenta, aut ne æquipondium di- minutis momentis magis removendum sit à sparto, quàm pro gravitate mercis. Siverò hoc acciderit, occultum manet state- ræ vitium, nec ipsa se prodit. Et quoniam de stateræ vitio sermo incidit, cavendum vendi- tori est, ne illâ utatur, si facta fuerit curva; cùm enim recta fuerit ab artifice suas in partes ritè distincta, & quidem juxta Rationem brachiorum, curva non eandem servat Rationem, ut ostensum est hîc cap. 5. & venditoris damno plus mercis ad- dendum esset lanci, ut haberetur æquilibrium; ut ex ibi dictis constat. CAPUT IX. Antiquorum Statera examinatur. Dubitatur à non paucis, utrum nostræ, quâ nunc utimur, stateræ similis esset Antiquorum, saltem Græcorum, sta- tera. Dubitationi locum fecit Aristoteles in quæst. 20. Mechan. quærens, Cur statera, quâ carnes ponderantur, pauco appendiculo magna ponderat onera? quæstioni autem satisfaciens plurium spartorum mentionem fecit. Quemadmodum autem si una li- bra multæ sint libræ; sic talia insunt sparta multa in ejusmodi li- brâ; quorum uniuscujusque quod intrinsecùs est ad appendicu- lum, stateræ est dimidium. & post pauca. Hujusmodi autem existens multæ sunt libræ, totque, quot fuerint sparta. Semper au- tem quod lanci propinquius est spartum appensoque oneri, majus trahit pondus. Plura hæc sparta, quorum Aristoteles meminit, Blancano in locis Mathem. Arist. occasionem præbuerunt stateram quan- dam comminiscendi, quasi illa fuerit Antiquorum statera: cu- jus sententiam probare non potui, cum Mechanicam doctri- nam

Transcription: Translated (English)

Mechanicorum men had refrained from stripping the hand from the smaller arm; for he well knew that this could not be done without injury to the seller: since, the balance having been set, nothing is to be taken away from this side or from that, nothing added, lest the Ratio which exists between the weights of the two arms be altered, or lest the counterpoise, the weights having been diminished, be moved farther away from the spartan, than is proper to the weight of the merchandise. But if this should happen, the fault of the balance remains hidden, and it does not betray itself. And since mention has fallen upon a defect of the balance, the seller must be careful not to use it if it has become bent; for when it has been straightly divided into its parts by the craftsman, and indeed in accordance with the Ratio of the arms, a bent one does not preserve the same Ratio, as has been shown here in chap. 5, and at the seller’s loss more merchandise would have to be added to the scale in order to obtain equilibrium; as is clear from what was said there. CAPUT IX. Examination of the balance of the Ancients. A doubt has arisen among not a few whether the balance of the Ancients, at least of the Greeks, was similar to our balance, which we use now. Aristotle gave rise to the doubt in question 20 of the Mechanics, asking, Why does the balance by which meats are weighed weigh great loads with a small attachment? In answering the question he made mention of several sparta. For just as if one pound there were many pounds; so in such a pound there are many such sparta; of each of which that part which is inward toward the attachment is half of the balance. And a little later: Such a thing, however, being many pounds, is as many as there are sparta. But always the spartum that is nearer the scale and the suspended load draws the greater weight. These several sparta, which Aristotle mentions, gave Blancanus, in the mathematical places of Aristotle, the opportunity to invent a certain balance, as though that were the balance of the Ancients: whose opinion I could not approve, since the doctrine of Mechanics

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Liber tertius. CAPUT IX. 307 nam anno labentis sæculi 34 in Collegio Romano explicans, publici juris facerem hæc eadem, quæ nunc post annos viginti scribo. Quoniam verò quæ tunc Blancano opposui, video placuisse Authori Magiæ Naturalis P. Gaspari Schoto tunc ibi degenti (eaque cum aliis quibusdam in suam Magiam staticam transtulit, me identidem suprà meritum, pro suâ humanitate, laudato) hîc iterum proferre non gravabor, ut meliùs stateræ natura innotescat. Statuit itaque Blacanus stateram illam fuisse hastam oblongam A B in certas partes distributam inter se æquales, puta 12, ex quibus exirent trutinæ diversæ, ut modò ex hâc, modò ex illâ suspenderetur statera, prout carnis vendendæ quantitas postulabat, singulisque trutinis insculptam fuisse notâ ponderis mercis. In extremitate A pedebat lanx capax mercis, in oppositâ extremitate B æquipondium, quod ut ille ait, debet habere tantum pondus, quantum est in lance nudâ, ut sic tota statera sit per se solam æquilibralis; & præterea debet habere pondus statum ac legitimum, ex. gr. unius libræ, aut duarum, aut trium, prout magis trutinandæ merci idoneum erit, & hoc erit proprium æquipondij pondus. Ponamus æquipondium esse librarum 12. Dico quod trutina C dabit in lance pondus mercis 12 lib. si ex eâ fiat æquilibrium; est enim ut A C ad C B, ita permutatim æquipondium 12 ad mercem; sed A C ipsi C B est æqualis; ergo etiam æquipondium 12 erit merci æquale, hoc est utrinque erit 12 lib. Similiter si fieret æquilibrium ex trutinâ D, esset ut A D 3 ad D B 9, ita 12 ad 36. Tandem trutinâ E æquilibrante, esset ut A E 9 ad E B 3, ita 12 ad 4. Si igitur trutina C notetur 12 numero, trutina D numero, 36, trutina E numero 4, & idem de cæteris, statim facile erit quodlibet pondus per hujusmodi stateram exhibere. Unde videas contrario ab illis modo in nostris stateris æquipondium totam hastam percurrere, in illis verò manente æquipondio trutinam quodammodo per hastam moveri. Hæc ille. Plures hasce trutinas sic expositas, quasi solidas ansas hastæ infixas, quæ pro opportunitate apprehenderentur, nunquam Q9 2

Transcription: Translated (English)

Book the Third. CHAPTER IX. 307 for in the year 34 of the waning century, while explaining it in the Roman College, I would have made these same things public, which now after twenty years I am writing. But since what I then opposed to Blancanus I see pleased the Author of Magia Naturalis , Father Gaspar Schott, then residing there (and who transferred those points together with certain others into his own Magia statica , praising me again and again, beyond my merits, in his kindness), I shall not hesitate to put them forth here again, so that the balance of nature may be better understood. Thus Blancanus held that that balance was a long beam A B divided into certain equal parts, say 12, from which different poises would hang, so that now from this, now from that side the balance would be suspended, as the quantity of meat to be sold required, and that on each poise there was engraved the mark of the weight of the merchandise. At the extremity A there hung the pan capable of holding the merchandise, and at the opposite extremity B a counterweight, which, as he says, ought to have just as much weight as there is in the empty pan, so that the whole balance may of itself alone be in equilibrium; and besides, it ought to have a fixed and lawful weight, for example of one pound, or two, or three, as shall be more suitable for weighing the merchandise, and this will be the proper weight of the counterweight. Let us suppose the counterweight to be 12 pounds. I say that the poise C will give in the pan a weight of the merchandise of 12 lb. if equilibrium is produced by it; for it is as A C is to C B, so, by permutation, is the counterweight 12 to the merchandise; but A C is equal to C B; therefore the counterweight 12 will also be equal to the merchandise, that is, on both sides there will be 12 lb. Likewise if equilibrium were produced by the poise D, it would be as A D 3 is to D B 9, so is 12 to 36. Finally, with poise E balancing, it would be as A E 9 is to E B 3, so is 12 to 4. If therefore poise C is marked with the number 12, poise D with the number 36, poise E with the number 4, and the same for the others, it will immediately be easy to show any weight by means of such a balance. Whence you may see, contrariwise, how in our balances the counterweight in that manner traverses the whole beam, whereas in theirs, the counterweight remaining fixed, the poise is in a manner moved along the beam. This he says. Many have explained these poises thus, as if they were solid handles fixed into the beam, which would be seized as occasion required, never

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Mechanicorum potui in animum inducere, ut mihi persuaderem fuisse anti- quis in usu; cùm enim non possent summis digitis suspendi ob nimiam mercis gravitatem, puta lib. 36 (& multò plurium, si ex F statera penderet) manu fuissent validè apprehendendæ; quis autem non videt, quibus dolis obnoxia fuisset statera ex levissimâ manûs inclinatione æquilibrium mentiente? Neque plicatiles fuisse hujusmodi trutinas, videlicet funiculos forami- nibus insitos in divisionum locis, existimo, quia vel nimis fre- quentes esse debuissent, vel, nisi æquipondium fuisset levissi- mum, non potuissent, citrà venditoris, aut emptorum incom- modum non leve, exhibere quæsitum pondus. Si enim (ut in- sistam ratiocinantis Blancani vestigiis) in D exhibentur libræ 36 mercis, in G exhiberentur libræ 60, quia ut A G 2 ad GB 10, ita æquipondium 12 ad mercem 60: quâ igitur ratio- ne innotescere poterat pondus mercis, si deprehendebatur esse majus quidem libris 36, sed minus libris 60? Et si æquilibrium fuisset inter F & G, pondus fuisset majus libris 60, minus li- bris 132: quàm latè igitur patuisset campus erroribus in tantâ ponderum differentiâ? Quare si hoc stateræ genere utendum esset, in quâ manen- te æquipondio spartum percurreret jugi longitudinem, inse- renda potius esset hasta annulo solidè firmato, intrà quem hasta ipsa ultrò citróque promoveretur, donec haberetur æquili- brium; eâ enim ratione in minutiores particulas posset hasta distingui; & plurima essent sparta, seu centra motûs. Aut etiam jugum parari posset crassioris lam- inæ in speciem, cujus- modi esset MO, per cujus longitudinem ductâ incisurâ seu cre- nâ SI excurrere posset axis exquisitè cylin- dricus infixus ansæ DE cujus ansæ extremitas in apicem E desinens indicaret par- ticulas in lineâ MO notatas. Verùm quia adversùs hasce state- ras faciunt pleræque rationes mox contrà Blancani stateram afferendæ, proptereà illas ut parùm aptas rejicio. Et

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Mechanicorum I have not been able to persuade myself that they were in use among the ancients; for since they could not be suspended by the tips of the fingers because of the excessive heaviness of the goods, say 36 pounds (& much more, if they were to hang from the F balance), they would have had to be firmly grasped by hand; but who does not see to what tricks such a balance would have been liable, from the slightest inclination of the hand falsely maintaining equilibrium? Nor do I think that balances of this sort were made foldable, namely with cords inserted through holes at the points of division, because either they would have had to be far too numerous, or, unless the counterpoise had been extremely light, they could not have shown the required weight without some not inconsiderable inconvenience to the seller or the buyer. For if (to follow the reasoning of Blancanus) at D there are displayed 36 pounds of goods, at G there would be displayed 60 pounds, because as AG 2 is to GB 10, so is the counterpoise 12 to the 60-pound load: by what means then could the weight of the goods be known, if it were found to be greater than 36 pounds, but less than 60? And if the equilibrium were between F and G, the weight would be greater than 60 pounds, less than 132 pounds: how wide then would the field for error have been opened in so great a difference of weights? Therefore, if this kind of balance had to be used, in which, the counterpoise remaining fixed, the cord would run along the whole length of the beam, it would rather be necessary to insert a rod with a ring firmly fastened within it, inside which the rod itself could be moved to and fro until equilibrium was obtained; for by that method the rod could be divided into smaller parts, and there would be many cords, or centers of motion. Or else a beam could be made in the form of a thicker plate, such as MO, through whose length, with a slit or groove SI cut along it, there could pass an exquisitely cylindrical axle fixed in the handle DE, the end of which handle, terminating in the point E, would indicate the particles marked on the line MO. But because against these balances most of the arguments soon to be brought forward against Blancanus’ balance can be turned, for that reason I reject them as not very suitable. And

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Liber tertius. CAPUT IX. 309 Et primùm quidem difficile videatur, quâ ratione fieri pos- set, ut in C puncto medio indicetur mercis pondus lib. 12, si ex illo statera ipsa est per se solam æquilibralis, ut Blancanus loqui- tur, positâ lance æqualis gravitatis cum æquipondio: Assumen- da fuisset trutina quarta H, quia ut A H 4 ad H B 8, ita 12 ad 24, & subductâ gravitate lancis 12, reliquæ fuissent lib. 12 mercis. Hinc patet neque in D indicari pondus mercis lib. 36; hoc enim est pondus mercis & lancis simul sumptarum; quare merx solum esset lib. 24; & ut haberentur mercis lib. 36, opor- teret spartum accipere, quod hastam divideret in partes, qua- rum proxima lanci esset 1, reliqua 4, quia ut 1 ad 4, ita 12 ad 48, & demptâ lancis gravitate lib. 12 remanerent mercis lib. 36. Sed illud à veritate longissimè abest, quod à Blancano additur, ex trutinâ E indicari mercem lib. 4. Immò addo nullum po- tuisse ibi fieri æquilibrium, & maximam partem illarum truti- narum futuram fuisse prorsus inutilem; nam si lanx A æquè gravis est ac æquipondium B, lanx cum merce gravior est æqui- pondio; igitur lanx cum merce in distantiâ majore, quàm sit æquipondij distantia majora habet momenta quàm æquipon- dium, cum quo nunquam poterit æquilibrium constituere. Quare omnes trutinæ inter B & C, & ipsa trutina C inutiles sunt, si lanx æqualis gravitatis sit cum æquipondio B: proptereà lancem multò levirem esse oporteret, ut cum impositâ merce posset habere ad æquipondium Rationem reciprocam distantia- rum à sparto. Sed si lanx levior sit æquipondio, ut inter C & B haberi possit æquilibrium; jam non omnes quidem; sed aliquæ tantum trutinæ inter B & C inutiles evadent; ubi enim hasta dividitur reciprocè in Ratione gravitatu lancis, & æquipondij, ibi esset statera per se solam æquilibralis, juxtà Blancani ratio- cinium: igitur nulla trutina inter illud punctum, & B esset uti- lis; quia diminutâ æquipondij à sparto distantiâ, ejus momenta decrescunt, & auctâ lancis ab eodem sparto distantiâ, ipsius lan- cis momenta augentur; igitur multò magis augentur facto pon- deris in lance additamento; ac proinde fieri non poterit æqui- librium. Verùm fortasse Author ille, cùm stateram dixit per se solam æquilibralem ex lancis, & æquipondij gravitatibus æqualibus, hoc tantùmmodo voluit (& ex ejusdem verbis inferendum vi- Qq 3

Transcription: Translated (English)

Liber tertius. CAPUT IX. 309 And first indeed it may seem difficult in what way it could happen that at the midpoint C the weight of the merchandise, 12 lb., is indicated, if from that point the balance itself is, by itself alone, in equilibrium, as Blancanus speaks, with the pan of equal weight with the counterpoise placed there: the fourth division H ought to have been taken, because as A H 4 is to H B 8, so is 12 to 24, and, after subtracting the weight of the pan, 12, there would have remained 12 lb. of merchandise. Hence it is clear that at D too the weight of the merchandise, 36 lb., is not indicated; for this is the weight of the merchandise and the pan taken together; therefore the merchandise alone would be 24 lb.; and in order that 36 lb. of merchandise might be obtained, it would be necessary to take the spar, which would divide the beam into parts, of which the one nearest the pan would be 1, the other 4, because as 1 is to 4, so is 12 to 48, and, after the weight of the pan, 12 lb., had been removed, 36 lb. of merchandise would remain. But what Blancanus adds is very far from the truth, namely, that from the balance E 4 lb. of merchandise are indicated. Indeed I add that no equilibrium could be established there, and that the greater part of those balances would have been altogether useless; for if pan A is equally heavy as counterpoise B, the pan with the merchandise is heavier than the counterpoise; therefore the pan with the merchandise at a greater distance has greater moments than the counterpoise, with which it could never establish equilibrium. Wherefore all the balances between B and C, and the balance C itself, are useless, if the pan be of equal weight with counterpoise B: therefore the pan ought to be much lighter, so that with the merchandise placed upon it it might have, in relation to the counterpoise, the reciprocal ratio of the distances from the spar. But if the pan be lighter than the counterpoise, so that equilibrium may be had between C and B, then not indeed all; but only some of the balances between B and C become useless; for where the beam is divided reciprocally in the ratio of the weights of the pan and counterpoise, there would the balance itself be, by itself alone, in equilibrium, according to Blancanus’ reasoning: therefore no balance between that point and B would be useful; because, with the distance of the counterpoise from the spar diminished, its moments decrease, and, with the distance of the pan from the same spar increased, the moments of the pan itself increase; therefore they are much more increased by the addition of weight placed on the pan; and consequently equilibrium cannot be brought about. But perhaps that Author, when he said that the balance is of itself alone in equilibrium from the equal weights of the pan and the counterpoise, intended only this (and from his own words it must be inferred Qq 3

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Mechanicorum detur) ut æquipondium ultrà libras 12 sibi peculiares, tantam prætereà haberet gravitatem, quæ si solitariè assumeretur, pos- set cum lance vacuâ æquilibrium facere in C: quo pacto lanx non esset lib. 12; sed levior. Per hæc tamen non omne incom- modum sublatum esset, neque Blancani dicta consisterent; quia sit lanx unius libræ, & item æquipondium ultrà libras 12 habeat libram unam; in C quidem esset æquilibrium cum merce lib. 12; quia merx cum lance, item æquipondium totum sunt lib. 13. At facto æquilibrio in D, distantiæ essent ut 3 ad 9, igi- tur æquipondium ad mercem cum lance ut 13 ad 39; & sub- ductâ lancis gravitate lib. 1, esset merx lib. 38, non verò 36. Sic in E facto æquilibrio, distantiæ essent ut 9 ad 3, igitur æquipon- dium ad mercem cum lance ut 13 ad 4 1/3, & lancis gravitate lib. 1. demptâ, esset merx lib. 3 1/3 non autem lib. 4. Et in ultima trutinâ prope B esset ut 11 ad 1, ita 13 ad 1 2/11, & lance sublatâ lib. 1, esset merx lib. 2/11, cum juxta Blancani ratiocinium debe- ret esse solum lib. 1/11. Deinde iugi brachia sua habent gravitatis momenta, quæ pro variâ longitudine inæqualitatem subirent; & hæc in hujusmo- di staterâ modò majora, modò minora essent, aliquando adden- da lanci, aliquando æquipondio. Nam si spartum sit in D, ab- scindens quartam iugi partem, sola brachij DB gravitas susti- net in A pondus æquale gravitati totius iugi; ac proinde facto in D æquilibrio, pondus totum additum in A est non solùm tri- plum æquipondij, ut fert reciproca distantiarum Ratio; sed est præterea æquale gravitati iugi. At si spartum in F abscindat ju- gi partem duodecimam, non solùm pondus unâ cum lance est æquipondij undecuplum, sed etiam quintuplum gravitatis iugi: & sic de cæteris. Contra verò si quando æquilibrium fieret in- ter C & B, ex æquipondio demenda esset gravitas respondens momento brachij oppositi; tum ex residuo colligeretur gravitas lancis cum merce, & subductâ demùm lance, gravitas mercis innotesceret. Sic in E facto æquilibrio, quia E B est quarta pars iugi, ex æquipondio B lib. 12 auferenda est gravitas iugi ex. gr. lib. 4, remanent lib. 8: igitur ut A E 3 ad E B 1, ita lib. 8 ad lib. 2 2/3: si demas pondus lancis, quæ utique valde levis esse de- bet, vide quanta gravitas sit demùm tribuenda merci. At si lanx adeò

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Mechanics given) so that a counterweight beyond its own 12 pounds might likewise have such a weight that, if taken alone, it could make equilibrium in C with the empty scale-pan; in which case the scale-pan would not be 12 lb., but lighter. Yet by these means not every inconvenience would be removed, nor would Blancani’s statements hold good; because if the scale-pan be of one pound, and likewise the counterweight beyond 12 pounds have one pound, then indeed in C there would be equilibrium with merchandise of 12 lb.; because the merchandise together with the pan, and likewise the whole counterweight, are 13 lb. But equilibrium being made in D, the distances would be as 3 to 9, therefore the counterweight to the merchandise with the pan as 13 to 39; and, the weight of the pan, 1 lb., being subtracted, the merchandise would be 38 lb., not 36. So, equilibrium being made in E, the distances would be as 9 to 3, therefore the counterweight to the merchandise with the pan as 13 to 4 1/3, and the weight of the pan, 1 lb. being deducted, the merchandise would be 3 1/3 lb., not 4. And in the last balance near B it would be as 11 to 1, so 13 to 1 2/11, and the pan being removed, 1 lb., the merchandise would be 2/11 lb., whereas according to Blancani’s reasoning it ought to be only 1/11 lb. Next, the arms of the beam have moments of gravity, which, according to the various length, would undergo inequality; and in a balance of this kind these would sometimes be greater, sometimes less, sometimes to be added to the pan, sometimes to the counterweight. For if the cord be at D, cutting off the fourth part of the beam, the weight of the single arm DB alone sustains in A a weight equal to the weight of the whole beam; and therefore, equilibrium being made at D, the whole weight added in A is not only triple the counterweight, as the reciprocal ratio of the distances demands; but is also equal to the weight of the beam. But if the cord at F cut off the twelfth part of the beam, not only is the weight together with the pan eleven times the counterweight, but also five times the weight of the beam: and so on in the rest. On the contrary, if at any time equilibrium were made between C and B, the weight corresponding to the moment of the opposite arm would have to be deducted from the counterweight; then from the remainder the weight of the pan with the merchandise would be inferred, and, finally the pan being subtracted, the weight of the merchandise would become known. Thus, equilibrium being made in E, because EB is the fourth part of the beam, from the counterweight B of 12 lb. must be taken away the weight of the beam, for example 4 lb., there remain 8 lb.: therefore, as AE 3 to EB 1, so 8 lb. to 2 2/3 lb.: if you subtract the weight of the pan, which indeed must be very light, see how much weight must at last be assigned to the merchandise. But if the pan is so

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Liber tertius. CAPUT IX. 311 adeò levis sit, manifestum est, quantò plus mercis apponendum sit, quando spartum à medio secedit versus lancem A. Quare patet genus hoc stateræ, ut pote parùm utile, reji- ciendum, nec potuisse Antiquis usitatum esse, quin facilè de- prehenderetur erroribus non levibus obnoxium; cum præsertim oblongam fuisse hastam (non utique levissimam) commi- niscatur Blancanus, & qui eum ducem sequuti sunt. Non ne- gârim quidem posse à perito mathematico ita iniri rationes, ut certis mercium ponderibus sua puncta in jugo inscriberentur, in quibus æquilibrium fieret cum æquipondio manente in extre- mitate jugi: sed hunc laborem subiisse antiquos Mathematicos, ut stateras carnem in macello vendentibus pararent, suaderi non potest; artificibus autem tantum fuisse industriæ, omnem fidem superat. Ex his mihi certissimum videtur aliam prorsùs adhibendam esse Aristotelicis verbis interpretationem: Nam ponamus stateram illam, de quâ Aristoteles loquitur, planè si- milem fuisse nostræ stateræ, quis neget unam libram brachiorum inæqualium esse multas libras, hoc ipso quod æquipon- dium in multis distantiis ab eodem puncto varias brachiorum Rationes constituit? sunt autem plura sparta, quia punctum idem determinans brachia varias Rationes habentia æquivolet multis, & quàm multas Rationes brachiorum definire potest, tàm multas constituit libras. Demùm quamvis lancis à sparto eadem materialiter sit distantia, non est tamen eadem formali- ter, neque enim solitariè accipienda est, sed comparatè cum distantiâ æquipondij à sparto; ac propterea cum major æqui- pondij distantia ad eandem lancis & oneris distantiam majo- rem habeat Rationem, potest etiam dici tunc spartum esse lan- ci & oneri propinquius; nam si in unâ æquipondij distantiâ bra- chia sint, ut 2 ad 5, & remoto æquipondio Ratio distantiarum sit ut 2 ad 6, patet comparatè ad æquipondij distantiam, esse minorem priore posteriorem hanc lancis à sparto distantiam. Cùm itaque nulla hîc intercedat violenta interpretatio, nil pro- hibet existimare Aristotelem de staterâ nostris non dissimili lo- cutum fuisse. CAPUT

Transcription: Translated (English)

The third book. CHAPTER IX. 311 that it is so light is manifest, inasmuch as a greater weight of merchandise must be placed on it when the sparum departs from the middle toward the pan A. Wherefore it is clear that this kind of balance, being of little use, must be rejected, and could not have been commonly used by the Ancients without its error, prone as it was to no small mistakes, being easily detected; especially since Blancanus, and those who followed him as their guide, imagines the beam to have been oblong (not by any means very light). I do not deny that a mathematical expert could arrange the calculations in such a way that, for certain weights of merchandise, their points might be marked on the yoke, at which equilibrium would occur while the counterpoise remained at the end of the yoke: but that ancient Mathematicians underwent this labor in order to prepare balances for those selling meat in the market cannot be believed; and that craftsmen had so much industry is beyond all credence. From these considerations it seems to me absolutely certain that a wholly different interpretation must be given to Aristotle’s words. For let us suppose that the balance of which Aristotle speaks was altogether like our own balance: who would deny that one pound of unequal arms is many pounds, simply because the counterpoise, placed at many distances from the same point, establishes various ratios of the arms? There are, moreover, several sparums, because the same point determining arms having various ratios is equivocal to many; and however many ratios of arms it can determine, so many pounds it constitutes. Finally, although the distance of the pan from the sparum is materially the same, it is not formally the same; for it must not be taken in isolation, but compared with the distance of the counterpoise from the sparum. And therefore, since a greater distance of the counterpoise has, for the same distance of pan and load, a greater ratio, it may also then be said that the sparum is nearer to the pan and the load; for if, at one distance of the counterpoise, the arms are as 2 to 5, and with the counterpoise removed the ratio of the distances is as 2 to 6, it is clear that, in comparison with the distance of the counterpoise, this later distance of the pan from the sparum is smaller than the former. Since therefore no violent interpretation is involved here, nothing prevents us from thinking that Aristotle was speaking of a balance not unlike our own. CHAPTER

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Mechanicorum CAPUT X. Libræ & stateræ usus extenditur. Quæ semel aliquem in finem excogitata sunt, non ea sunt, ut illis tantùm terminis coërceantur, sed ad plura extendi possunt; & fundamentis positis alia superstrui licet, modò non desit artificis industria atque solertia. Quos in usus libra & sta- tera à vulgo destinentur, omnes nôrunt; sed ad quos alios tra- duci possint, iis manifestum est, qui illarum naturam diligen- tiùs scrutati sunt. Qua propter ut aliquâ ratione industriis arti- ficibus præeam, qui similia, & multò meliora comminisci po- terunt, pauca quædam hoc capite innuam, quibus libræ & sta- teræ usus extenditur. Distinctionis autem atque claritatis gratiâ, in plures propo- sitiones caput hoc tribuere commodum accidet. PROPOSITIO I. Libram construere, quâ innatantium solidorum in humido speci- ficam levitatem, & ipsorum humidorum specificam gravita- tem investigare possumus. E Rigatur tigillus A B fulcro ritè instructus in B, ut firmiter constitui possit horizonti perpendicularis: transversa juga duo CD, & EF bifariam æqua- liter divisa, & circà suos axes versatilia inserantur tigillo, pro- ut opportunius fuerit, ita tamen, ut in eâdem perpendiculari li- neâ VS sint axes, & inferiori jugo addatur exteriùs axis capi- ti insertus index GI, qui ubi convenerit cum perpendiculari lineâ VS in facie tigilli descrip- tâ,

Transcription: Translated (English)

Mechanics CHAPTER X. The use of scales and balances is extended. Things which have once been devised for some end are not so confined to those limits, but may be extended to more uses; and once the foundations are laid, other things may be built upon them, provided the skill and ingenuity of the artificer are not lacking. For what purposes scales and balances are intended by common usage, all know; but to what other uses they may be applied is evident to those who have more diligently examined their nature. Wherefore, that I may in some measure go before industrious craftsmen, who can devise similar things, and much better, I shall indicate in this chapter a few points by which the use of scales and balances is extended. Now, for the sake of distinction and clarity, it will be convenient to divide this chapter into several propositions. PROPOSITION I. To construct a balance by which we may investigate the specific lightness of solid bodies floating in a liquid, and the specific gravity of the liquids themselves. Let the beam A B, properly fitted with a support at B, be so placed that it may be firmly set perpendicular to the horizon: let two cross-yokes, CD and EF, each equally divided into two parts, and movable around their axes, be inserted into the beam, as shall be more convenient, provided however that the axes be in the same perpendicular line VS, and let to the lower yoke be added externally a pointer GI, with its axis inserted into the head, which, when it has agreed with the perpendicular line VS described on the face of the beam,

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Liber tertius. CAPUT X. 313 tâ, æquilibrium horizontale jugorum CD & EF indicet. Tum extremitates C & E vel solido, vel plicatili vinculo CE connectantur, & in D quidem addatur lanx; in C verò mo- mentum plumbi, ut æquilibrium suâ gravitate constituant. Postquam in F adnexus fuerit stylus in triplicem cuspidem de- sinens, ut faciliùs deprimatur corpus solidum H infrà humo- rem, in quo levitat, addatur pariter in E aliquid plumbi, ut jugum EF in æquilibrio maneat; nisi fortè tanta sit ipsius vin- culi CE gravitas, ut plumbum addere non sit opus. Demùm habeatur vas humore implendum, quod subjici possit extremitati F, unà cum solido H innatante. Primò quæritur levitas solidi H in aquâ. Expendatur soli- dum H exactè in aëre librâ communi & consuetâ; ejusque pon- dus adnotetur: deinde imponatur vasi aquæ pleno, ita ut soli- dum totum immergatur; id quod tunc solùm fiet, cùm lanci in D fuerit impositum pondus congruum, nam descenden- te D, ascendit C, & secum trahit E sursum, ac proin- de F deprimit solidum H infrà aquam. Ubi lingula G I in- dicaverit æquilibrium solido H aquæ prorsus immerso, obser- va pondus lanci D impositum: hoc adde ponderi priùs in- vento ejusdem solidi H in aëre; & pronunciabis, ut hæc sum- ma ponderum ad pondus solidi in aëre, ita esse gravitatem specificam aquæ ad gravitatem specificam propositi solidi. Fuerit pondus in aëre unc. 20; additæ sint in lance D unciæ 5; igitur ut 25 ad 20, hoc est ut 5 ad 4, ita gravitas specifica aquæ ad gravitatem specificam solidi. Veritas ostenditur ex iis, quæ in Hydrostaticis certa sunt. Si enim ponamus aquæ gravitatem ad solidi H gravitatem se- cundùm speciem esse ut 5 ad 4, emergit ex aquâ pars quinta solidi gravitans ut 4; reliquæ quatuor infrà aquam levitant sin- gulæ ut 1, quæ est differentia specificarum gravitatum: igitur pars quinta solidi extans est unc. 4; quia totum in aëre est unc. 20; & pars immersa levitat tanto nisu, ut æqualis sit contrario conatui unc. 4. Igitur si quinque partes demergan- tur, resistent unciis quinque, quæ solido superimponeren- tur; idem autem est, si unciæ quinque imponantur lanci D; eandem enim deprimendi vim habent. Si igitur solidum gra- ve in aëre ut 20, levitat in aquâ ut 5, aquæ moles æqualis Rr

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Liber tertius. CAPUT X. 313 that the horizontal equilibrium of the beams CD & EF may be indicated. Then let the extremities C & E be connected by a solid or flexible link CE, and let a pan be added at D; but at C let a weight of lead be added, so that they may establish equilibrium by their own weight. After a stylus ending in a threefold point has been attached at F, so that the solid body H, floating below the fluid, may be more easily depressed, let some lead be likewise added at E, so that the beam EF may remain in equilibrium; unless perhaps the weight of the link CE itself be so great that there is no need to add lead. Finally, let there be a vessel to be filled with fluid, which may be placed beneath the extremity F, together with the solid H floating upon it. First, the lightness of the solid H in water is sought. Let the solid H be weighed exactly in air with the common and usual balance; and let its weight be noted: then let it be placed in a vessel full of water, so that the solid may be entirely immersed; which will then be done only when a suitable weight has been placed in the pan at D, for as D descends, C ascends, and with it draws E upward, and therefore F depresses the solid H below the water. When the little tongue G I has indicated equilibrium with the solid H fully immersed in water, observe the weight placed in pan D: add this to the weight previously found of the same solid H in air; and you will declare that, as this sum of weights is to the weight of the solid in air, so is the specific gravity of the water to the specific gravity of the proposed solid. Let the weight in air be 20 ounces; let 5 ounces be added in pan D; therefore as 25 to 20, that is as 5 to 4, so is the specific gravity of the water to the specific gravity of the solid. The truth is shown from those things which are certain in Hydrostatics. For if we suppose the gravity of water to the gravity of the solid H according to species to be as 5 to 4, one-fifth of the solid emerges from the water, weighing as 4; the remaining four parts beneath the water each float with a force of 1, which is the difference of the specific gravities: therefore one-fifth of the solid standing out is 4 ounces; because the whole in air is 20 ounces; and the part immersed floats with so great a force that it is equal to the contrary effort of 4 ounces. Therefore if five parts are sunk, they will be resisted by five ounces, which would be superimposed on the solid; but the same is true if five ounces are placed in pan D; for they have the same power of depressing. If therefore a heavy solid in air is as 20, but floats in water as 5, a mass of water equal

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Mechanicorum 314 est 25, atque adeò aqua ad solidum est ut 25 ad 20 secundùm gravitatis speciem. Secundò comparandi sint humores, uter gravior sit. Idem solidum H notæ gravitatis in aëre unc. 20, quod priori aquæ immersum requirebat in lance D uncias 5, immergatur eodem modo alteri aquæ, ita, ut in lance sint unc. 4. drachmæ 5: igitur solidi gravitati in aëre unc. 20. addantur unc. 4. drach. 5. & erit aquæ secundùm molem æqualis specifica gravitas unc. 24 5/8; hæc ergo posterior aqua ad priorem aquam est ut 197 ad 200. Tertiò. Notâ solidi secundùm speciem gravitate comparatâ cum gravitate specificâ humoris, cognoscere possumus alterius molis ejusdem speciei gravitatem in aëre. Sit cognita Ratio gravitatum secundùm speciem ut 4 ad 5. Requiratur in lance D pondus unc. 8, ut infrà aquam deprimatur solidum. Fiat ut differentia specificarum gravitatum 1, ad specificam gravitatem solidi 4, ita unciæ 8, ad unc. 32: Est ergo solidum in aëre unciarum 32, & aquæ moles æqualis unc. 40. Placeat fortasse alicui rem hanc aliter perficere. Libræ jugum EF ita firmetur in G, ut alteri extremitati E adnexus funiculus ascendat orbiculo X circumvolutus, & appositâ lance D, atque in F stylo tricuspide, omnia sint æquilibrata, addito, si opus fuerit, in F plumbi momento: Pondus etiam lanci impositum sursum trahens E deprimit F, & pariter solidum subjecto humori innatans à stylo deprimitur, & immergitur. Proposi

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Mechanicorum 314 is 25, and so water to the solid is as 25 to 20 according to specific gravity. Second, the fluids are to be compared, to see which is heavier. Let the same solid H, of a gravity in air of 20 ounces, which when immersed in the former water required 5 ounces in pan D, be immersed in the other water in the same way, so that there are in the pan 4 ounces 5 drachms: therefore to the gravity of the solid in air, 20 ounces, add 4 ounces 5 drachms, and the specific gravity of a volume equal to water will be 24 5/8 ounces; this later water therefore to the former water is as 197 to 200. Third. When the specific gravity of the solid has been compared with the specific gravity of the fluid, we can know the gravity in air of another body of the same kind. Let the ratio of the specific gravities be known as 4 to 5. Let there be required in pan D a weight of 8 ounces, so that the solid may be depressed beneath the water. Let it be as the difference of the specific gravities, 1, to the specific gravity of the solid, 4, so 8 ounces to 32 ounces: therefore the solid in air is 32 ounces, and the volume of water equal to it is 40 ounces. Perhaps someone may prefer to carry out this matter otherwise. Let the beam EF of the scales be so fixed in G that the cord attached to the other end E may ascend, wound around the wheel X, and, with pan D placed thereon, and with the three-pronged stylus at F, let all things be in equilibrium, adding, if necessary, at F a weight of lead: a weight also placed in the pan, pulling upward, depresses E and F, and likewise the solid floating on the fluid beneath is depressed by the stylus and is immersed. Proposi

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Liber tertius. CAPUT X. 315 PROPOSITIO II. Horologium arenarium ex librâ construere, quod horæ minu- ta indicet. Jugum libræ æqualium brachiorum A B paretur, spartum O in superiore loco habens: huic enim tantummodo libræ spe- ciei convenire potest æqui- librium obliquum. Lingu- lam O I habeat longiuscu- lam, quæ indicis munere fungi possit, & quam levissima sit. Tum assumpta lanx, quæ figuram conicam æmuletur, in imâ parte, quâ apex desinit, foramen ha- beat exiguum, ex quo pos- sit sensim arena fluere; cu- jusmodi ea est, quâ in vul- garibus horologiis arenariis utimur. Suspendatur lanx seorsim à jugo, & impleatur arenâ, quæ in subjectum vas defluat spatio horæ unius: horâ elapsâ servetur arena, quam vas excepit, reliqua, quæ in lan- ce, rejiciatur. Sed quoniam ubi multum erat arenæ in lance, plus defluxit, quàm par est, iterum arena hæc vasis subjecti in lancem infun- datur, & toties experimentum repetatur rejiciendo reliquam, quoties opus fuerit, ut certi simus arenæ defluxum exquisitè metiri unius horæ longitudinem. Habitâ jam congruâ arenæ quantitas diligenter servetur, ne pereat aliquid illius, & novum laborem subire cogamur. Hujus arenæ gravitas examinetur librâ exactissimâ: item lancis cum suis appendiculis pondus inquiratur: quibus cognitis inter gra- vitatem solius lancis C vacuæ, & gravitatem lancis congruâ arenâ plenæ inveniatur terminus medio loco proportionalis, qui dabit gravitatem ponderis D ex opposito libræ brachio appen- R r 2

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Book Three. Chapter X. 315 PROPOSITION II. To construct out of a scale a sand-hourglass that indicates the hour by minutes. Let the yoke of a scale with equal arms A B be prepared, having the cord O in the upper part; for to this kind of balance alone can an oblique equilibrium be fitted. Let it have a somewhat longer tongue O I, which may serve the office of an index, and be as light as possible. Then take a pan, which imitates the form of a cone, and in its lower part, where the apex ends, let it have a small hole, through which the sand may flow gradually; of such a kind is that with which we use in ordinary sand-hourglasses. Let the pan be suspended separately from the yoke, and filled with sand, which shall run into the vessel beneath in the space of one hour: when the hour has elapsed, let the sand which the vessel has received be kept, and the remainder, which is in the pan, be thrown back. But since, when there was much sand in the pan, more flowed out than was fitting, let this sand be again poured from the vessel beneath into the pan, and let the experiment be repeated so many times as may be needed, throwing away the remainder, until we are certain that the outflow of the sand most exactly measures the length of one hour. The proper quantity of sand having now been obtained, let it be carefully preserved, lest any part of it be lost and we be compelled to undergo the labor anew. Let the weight of this sand be determined with a most accurate balance; likewise let the weight of the pan with its little appendages be ascertained: these being known, between the weight of the empty pan C and the weight of the pan filled with the proper sand, let there be found the middle proportional term, which will give the weight of the weight D to be hung from the opposite arm of the balance. R r 2

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Mechanicorum Demùm intervallo O I longitudinis lingulæ, quæ scilicet à sparto incipit, describatur vel in lamellâ, vel in crassiore papyro sextans circularis limbi E I F, qui divisus in partes 60 ita ap- tandus est, ut lingula suo apice notata puncta percurrens me- dio puncto I congruat, ubi libræ jugum A B horizontale fuerit. Quare cum lingulæ apex I erit in E, declinabit lingula à per- pendiculo angulo gr. 30: id quod pariter in oppositâ parte con- tinget, quando lingulæ apex venerit in F. Cum igitur jugum similiter inclinari debeat, ut æquilibrium similiter obliquum fiat hinc lancis C arenâ plenæ depressæ cum pondere D elevato, hinc ponderis D depressi cum lance vacuâ elevatâ; constat eandem esse oportere Rationem gravitatis lan- cis C arenâ plenæ ad pondus D, quæ est ponderis D ad gravi- tatem lancis vacuæ: Est igitur ponderis D gravitas medio loco proportionalis inter gravitates lancis vacuæ, & lancis plenæ. Sit deprehensa gravitas lancis vacuæ pondo unc. 5 ́, lancis au- tem cum arenâ unc. 18: igitur pondus D requiritur unc. 10. Sed quærendum est, quantum distare oporteat spartum à li- neâ jugi, ut fiat hujusmodi æquilibrium obliquum gr. 30. Sit CD libra, & in C pondus unc. 18. in D unc. 10; & fiat æquilibrium ita, ut O I lingula faciat cum perj en- diculo HS angulum HOI gr. 30. Ergo in S est cen- trum gravitatis, & est reci- procè ut pondus C ad pon- dus D, ita longitudo D S ad longitudinem S C: igi- tur quarum partium tota CD est 28, & CG 14, earum partium est GS 4. In triangulo igitur O GS rectan- gulo, GS est Sinus gr. 30, & GO est Sinus gr. 60; ac propterea si GS est 4, GO est 6.928''': tanta itaque debet esse distantia sparti O à lineâ jugi. Hîc autem observabis lineam jugi inclinatam, cum lineâ ho- zontali, quam secat, constituere angulum æqualem angulo declinationis lingulæ à perpendiculo; nam angulo lingulæ cum perpen-

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Mechanics Finally, with the interval O I of the length of the tongue, which indeed begins at the sparteum, let there be drawn, either on a thin plate or on thicker paper, a circular sextant of the rim E I F, which, being divided into 60 parts, must be so adjusted that the tongue, with its tip marked, passes through the points and agrees with the middle point I, where the beam A B of the balance is horizontal. Wherefore, when the tip I of the tongue is at E, the tongue will deviate from the perpendicular by an angle of 30 degrees; and the same will happen on the opposite side, when the tip of the tongue comes to F. Since therefore the beam must likewise be inclined, so that the equilibrium may likewise become oblique, with the pan C filled with sand depressed and the weight D raised, and with the weight D depressed and the empty pan raised, it is clear that the ratio of the weight of the pan C filled with sand to the weight D must be the same as that of the weight D to the weight of the empty pan: therefore the weight of D is to be taken as proportionally intermediate between the weights of the empty pan and of the full pan. Let the weight of the empty pan be found to be 5 ounces, and that of the pan with sand 18 ounces: therefore the weight D is required to be 10 ounces. But it must be asked how far the sparteum ought to be from the line of the beam, so that such an oblique equilibrium of 30 degrees may occur. Let CD be one pound, and in C the weight of 18 ounces, in D 10 ounces; and let the equilibrium be made so that the tongue O I makes with the perpendicular HS the angle HOI of 30 degrees. Therefore in S is the center of gravity, and reciprocally, as the weight C is to the weight D, so is the length D S to the length S C: therefore, if the whole CD is 28 parts, and CG 14, GS is 4 of those parts. In the right triangle OGS, GS is the sine of 30 degrees, and GO is the sine of 60 degrees; and therefore if GS is 4, GO is 6.928: such therefore must be the distance of the sparteum O from the line of the beam. Here however you will observe that the line of the beam, inclined with the horizontal line which it intersects, makes an angle equal to the angle of the tongue’s deviation from the perpendicular; for the angle of the tongue with the perpendicular...

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Liber tertius. CAPUT X. 317 perpendiculari HOI æqualis est ad verticem angulus SOG: & quia horizontalis VR secat perpendicularum HS ad angulos rectos in B, duo triangula OGS, & EBS rectangula, & com- munem angulum ad S habentia, sunt æquiangula, atque adeò angulò SOG, æqualis est angulus SEB, cui ad verticem æqualis est angulus DER, qui proptereà æqualis est ipsi HOI. Sed quoniam GS est 4, & GO est 6.928", per 47. lib.1. innotescit OS partium 7.999" ex quâ aufertur OB æqualis ipsi GO (est enim distantia sparti ab horizontali æqualis distantiæ ejusdem sparti à jugo) remanet BS partium 1.071". In triangulis igitur SGO, SBE similibus ut GS 4 ad SO 7.999", ita BS 1.071" ad SE partium 2.142": remanet igitur EG partium 1.858". Quare tota DE est partium 15. 858", angulus E in triangulo EMD rectangulo est gr.30, ut ostensum est; igitur DM altitudo, ad quam elevatur pondus est partium 7.929". Et similiter quia EC est partium 12.142", depressio NC est partium 6.071". Ex quo habetur sub- jectum vas, quod cadentem arenam excipit, hoc saltem inter- vallo depressum esse infrà lancem pendentem ex jugo horizon- tali posito. Et ut subjecti vasis longitudinem invenias, quâ possit caden- tem arenam excipere, invenienda est distantia lancis à per- pendiculo HS, & cùm in summâ depressione est, & cùm est maximè elevata: Cùm depressa est, distat intervallo BN, cùm horizontalis est, distat intervallo BV, cùm demùm est elevata, distat intervallo æquali ipsi BM. Sunt investigandæ distantiæ BN & BM: Et quia in triangulo EMD rectangulo angulus est gr.30, & Radius ED est partium 15.858"; Sinus Com- plementi EM est partium 13.733". Et in simili triangulo ENC, quia EC Radius est partium 12.142", Sinus Comple- menti EN est partium 10.515". Et iterum in simili triangu- lo EBS, quia ES Radius inventus est partium 2.142", Sinus Complementi EB est partium 1.855". Itaque ex EN aufer EB, remanet BN 8.660", ipsi verò EM adde EB, est BM partium 15.588". Demum ex BM aufer BN, & residuum partium 6.928" est longitudo, quam percurrit lanx ascenden- do, & est æqualis distantiæ sparti à lineâ jugi; ac propterea vas Rr 3

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Liber tertius. CHAPTER X. 317 the perpendicular HOI is equal to the vertical angle SOG: and because the horizontal VR cuts the perpendicular HS at right angles in B, the two triangles OGS and EBS, being right-angled and having a common angle at S, are equiangular; and therefore the angle SOG is equal to the angle SEB, to which, at the vertex, is equal the angle DER, which therefore is equal to HOI itself. But since GS is 4, and GO is 6.928", by 47. lib.1. it is known that OS is 7.999" parts, from which is subtracted OB, equal to GO itself (for the distance of the spoon from the horizontal is equal to the distance of the same spoon from the yoke), there remains BS of 1.071" parts. Therefore, in the similar triangles SGO, SBE, as GS is to SO 7.999" in the ratio of 4 to 7.999", so BS 1.071" to SE 2.142" parts: therefore there remains EG 1.858" parts. Wherefore the whole DE is 15. 858" parts; the angle E in the right triangle EMD is 30 degrees, as has been shown; therefore DM, the height to which the weight is raised, is 7.929" parts. And similarly, because EC is 12.142" parts, the depression NC is 6.071" parts. From this it is concluded that the vessel set below, which receives the falling sand, is depressed by this at least to the extent of the interval below the scale hanging from the yoke placed horizontally. And in order that you may find the length of the vessel below, by which it may receive the falling sand, the distance of the scale from the perpendicular HS must be found, both when it is in its greatest depression and when it is most highly raised: when depressed, it is distant by the interval BN; when horizontal, by the interval BV; when finally raised, it is distant by an interval equal to BM itself. The distances BN and BM are to be sought: and because in the right triangle EMD the angle is 30 degrees, and the Radius ED is 15.858" parts, the Sine of the Complement EM is 13.733" parts. And in the similar triangle ENC, since EC is the Radius of 12.142" parts, the Sine of the Complement EN is 10.515" parts. And again in the similar triangle EBS, since ES, the Radius found, is 2.142" parts, the Sine of the Complement EB is 1.855" parts. Therefore subtract EB from EN, there remains BN 8.660"; but to EM itself add EB, BM is 15.588" parts. Finally subtract BN from BM, and the remainder, 6.928" parts, is the length traversed by the scale in rising, and is equal to the distance of the spoon from the line of the yoke; and therefore the vessel Rr 3

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Mechanicorum 318 excipiendæ arenæ destinatum longitudinem habeat necesse est, quæ saltem sit quarta pars longitudinis totius jugi, quæ ex da- tis est partium 28. Hæc quæ hactenus dicta sunt, eo consilio attuli, ut si quis velit rem ex certâ ratione peragere, intelligat, quâ sit illi uten- dum methodo: Cæterùm nemini author fuerim, ut hæc omnia calculis indagare eligat, cùm possit citrà laborem citissimè as- sequi propositum finem. Statutis enim ponderibus, scilicet lance, arenâ, & æquipondio (quod, ut dixi, medio loco pro- portionale esse oporter inter vacuam lancem, & lancem ean- dem cum arenâ) assumatur libræ jugum quodcumque, modò sit æqualium brachiorum, & spartum in superiore loco habeat, tùm adnexus hinc lance cum arenâ, hinc æquipondio, libra consistat obliqua; & in plano Verticali libræ proximo notetur punctum, cui lingulæ apex congruit: deinde extractâ arenâ vacuam lancem relinquat, & librâ consistente notetur pariter punctum in plano, quod apici lingulæ respondet; & hæc sunt extrema puncta arcûs, qui à circumductâ lingulâ describi po- test in eodem plano verticali, & dividi in quæsitas partes 60, ut horæ minuta indicentur. Quò autem propius ad jugi lineam accedet spartum, & longior fuerit lingula, major quoque erit hujusmodi arcus, & faciliùs in partes 60 dividetur. Vasis de- mum longitudinem ipsa libræ positio duplex & cum arenâ, & sine arenâ statim ostendet. Hîc verò ubi de arcûs divisione in partes 60 sermo est, liceat mihi dissimulare partes illas, si res subtilissimè examinetur, non esse omninò inter se æquales; sed in re Physicâ subtilitatem hanc persequi inutile est. PROPOSITIO. III. Ex Libræ Rationibus aliquod Motûs perpetui rudimentum proponere. Hoc saxum jamdiu multi versant; sua cuique cogitata pla- cent; quem corporibus tribuere nondum potuerunt arti- fices perpetuum motum, hunc sibi vendicant Philosophorum mentes inquietâ vertigine illius vestigiis insistentes; sed nimis fugacem

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Mechanics 318 the vessel intended for receiving the sand must have a length, which must be at least the fourth part of the whole length of the beam, which, from the given data, is 28 parts. What has been said thus far I have brought forward in order that, if anyone wishes to carry out the matter from a definite method, he may understand what procedure he should use. However, I would advise no one to choose to investigate all these things by calculation, since he can attain the intended end very quickly without effort. For, with the weights established, namely the pan, the sand, and the counterpoise (which, as I said, ought to be proportionally placed in the middle between the empty pan and the same pan with sand), let any beam of a balance be taken, provided it have equal arms and a loop in the upper part; then, with the pan containing sand attached on one side and the counterpoise on the other, let the balance stand inclined; and in the vertical plane nearest the balance let the point be marked to which the tip of the pointer corresponds: then, after the sand has been removed and the empty pan left behind, let the balance again stand, and let the point likewise be marked in the plane that corresponds to the tip of the pointer; and these are the extreme points of the arc which can be described in the same vertical plane by the rotating pointer, and divided into the sought 60 parts, so that the minutes of the hour may be indicated. But the more the loop approaches the line of the beam, and the longer the pointer is, the larger also will be an arc of this kind, and the more easily will it be divided into 60 parts. Finally, the length of the vessel will be shown immediately by the position of the balance itself, both with sand and without sand. Here, however, where mention is made of dividing the arc into 60 parts, let me be allowed to say nothing of those parts, if the matter is examined most carefully, not being altogether equal to one another; but in a physical matter it is useless to pursue this subtlety. PROPOSITION III. To propose some rudiment of perpetual motion from the relations of the balance. This stone has now long been turned over by many; each man is pleased with his own thoughts; what the craftsmen have not yet been able to attribute to bodies, namely perpetual motion, the minds of philosophers claim for themselves, following its traces with restless vertigo; but too elusive

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Liber tertius. CAPUT X. 319 fugacem nunquam assequentes. Liceat & mihi hîc aliquid proponere quasi rudimentum naturæ motum perpetuum efficere condiscentis. Videtur autem omninò certum, ut motus semel institutus sine fine perseveret (seclusâ materiæ corruptione, quæ ævo confecta tabescit) opus esse alterno quodam virium incremento atque decremento, ut idem viribus auctis prævaleat, viribus diminutis minùs resistat: propterea simplicissimam machinulam, quasi duplicem libram æqualium brachiorum ad angulos rectos compactam aliquando excogitavi, in quâ alterna hæc vicissitudo contingere posse videtur. Scapi duo A B & D E ad angulos rectos in C compingantur, & sit in C axis, circa quem facilè versari possint: quia verò opposita brachia ex hypothesi æqualia sunt, & centrum motûs planè in medio congruens centro gravitatis ponitur, in quâ cumque positione æqualibus momentis librata quiescunt. Sint autem singula brachia tubi in morem excavata ab extremitate usque ad decussationis locum æqualiter, ita tamen, ut ex uno brachio in aliud brachium sivè oppositum, sivè proximum nullus pateat exitus: extremum autem tubi osculum congruâ cochleâ possit exquisitè claudi. Hæc, inquam, omnia ea sint, quæ æquilibrium in quâcumque positione constituant: id quod improbus labor accurati artificis assequi se posse non desperat. Duplici hac librâ sic paratâ, singulis brachiis certa & omninò æqualis quantitas Argenti Vivi infundatur, aut major aut minor pro ratione magnitudinis & gravitatis tuborum, ita tamen ut non sit immodica quantitas. Occlusis diligentissimè tuborum osculis, erigatur D E ad perpendiculum. Utique hydrargyrus in superiore brachio D C totus quiescit propè C, in inferiore brachio C E totus est in extremitate E: in brachiis autem C A & C B horizonti parallelis se æqualiter librat juxta brachiorum longitudinem; quare libra tota manet immota, cum sint hinc & hinc æqualia momenta, tùm ratione brachio

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Book Three. CHAPTER X. 319 fugitives never overtaken. Let me here propose something, as it were, for a rudiment of Nature learning to produce perpetual motion. But it seems altogether certain that, once motion has been initiated, it would continue without end—apart from the corruption of the material, which, worn out by age, decays—that there must be some alternate increase and decrease of forces, so that the same body, when its forces are increased, may prevail, and when its forces are diminished, may resist less. For that reason I once devised a very simple little machine, as it were a double balance of equal arms joined at right angles, in which this alternating change appears able to occur. Let two shafts, A B and D E, be joined together at right angles at C, and let there be at C an axis around which they may easily turn. But because the opposite arms, by hypothesis, are equal, and the center of motion is placed exactly in the middle, coinciding with the center of gravity, in whatever position they are balanced by equal moments they come to rest. Let each arm, moreover, be hollowed out like a tube from the end to the point of intersection equally, but in such a way that no passage opens from one arm into the other arm, whether the opposite one or the adjacent one: and let the mouth of the tube at the end be capable of being exquisitely closed with a fitting screw. All these things, I say, are to be such as establish equilibrium in whatever position; and an assiduous labor of a skilled artificer does not despair of being able to achieve this. When this double balance has thus been prepared, let a certain and altogether equal quantity of quicksilver be poured into each arm, either more or less according to the size and weight of the tubes, but in such a way that the quantity not be excessive. After the mouths of the tubes have been most carefully closed, let D E be raised perpendicular. Certainly the mercury in the upper arm D C rests entirely near C, while in the lower arm C E it is entirely at the end E; but in the arms C A and C B, parallel to the horizon, it balances itself equally according to the length of the arms; wherefore the whole balance remains unmoved, since on this side and on that the moments are equal, both in respect of the arm

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Mechanicorum 320 brachiorum æqualium, tùm ratione argenti vivi æqualis, & æqualiter ad motum dispositi: illud verò quod est propè C, & propè E, non potest mutare æquilibrium, ut patet. Incline- tur extremitas B aliquantulum deorsum; illicò totus hydrargy- rus brachij CB confluit ad extremitatem B, contrà verò qui est in brachio C A, totus confluit propè centrum C: Facta est igitur libra inæqualium brachiorum, & æqualia argenti vivi pondera inæqualiter distant à centro motûs; ac proinde juxtà naturam libræ spartum in ipsâ jugi lineâ habentis extremitas B descendit quantum potest. Cum autem grave quodcumque sponte sua descendens acquirat impetum non statim pereun- tem, sed qui adhuc juxta priorem directionem ad easdem par- tes ferat corpus grave etiam contra naturæ propensionem, ut in perpendiculo ascendente est manifestum, quid prohibeat ex- tremitatem B hydrargyro prægravatam, ex concepto impetu dum descendit, vel, modicum quid translire perpendicularem positionem ultrà punctum E: Id quod si accidat, extremitas D, dum tota libra convertitur, inclinata infrà horizontalem A B totum hydrargyrum habet non jam in C; sed in D, quare & illa simili modo descendit, nam hydrargyrus, qui erat in E, elevato brachio C E supra horizontalem A B, totus confluit prope C: neque difficilis est descensus; quia B, ubi translierit perpendiculum D E, ulteriùs ex concepto impetu sponte ascen- deret; sed multò magis ascendit ex impetu impresso brachij descendentis, à quo urgetur. Fateor equidem in primâ conversione post quietem, hydrar- gyrum E reluctari, nec juvare quicquam ad motum; quia sci- licet, cùm debeat ascendere ex solo impetu impresso brachij C B descendentis, nihil confert ad motum, nisi quatenus E initio sui ascensus modicum ascendit, B verò initio sui descen- sûs multum descendit, ac propterea plus imprimi potest impe- tûs, ratione cujus, crescente quamvis ascensuum mensurâ, ha- betur aliquid facilitatis ex prævio impulsu. Hinc est in primis conversionibus opus esse manûs adjumento, quæ sursum pellat insimum brachium C E: concepto autem jam impetu, nondum video, cur motus cessaturus sit. Nam si nullâ factâ ponderum alternâ translatione (quæ semper novum motûs principium af- fert) sed ponderibus semper in extremitate brachiorum manen- tibus,

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of equal arms, then by reason of the equal quantity of mercury, and equally disposed for motion: but that which is near C and near E cannot change the equilibrium, as is evident. Let the end B be slightly inclined downward; immediately all the mercury of arm CB flows to the end B, while all that is in arm C A flows toward the center C: thus a balance of unequal arms is made, and equal weights of mercury are unequally distant from the center of motion; and therefore, according to the nature of a balance having the beam itself in the horizontal line, the end B descends as far as it can. But since any heavy body, descending of its own accord, acquires an impetus that does not immediately perish, but still carries the heavy body, according to its former direction, toward the same parts, even against the tendency of nature, as is manifest in the case of a rising plumb line, what prevents the mercury-loaded end B, by the impetus it has acquired as it descends, from passing a little beyond the perpendicular position toward the point E? If this should happen, the end D, while the whole balance turns, being inclined below the horizontal A B, will have all the mercury no longer in C, but in D; wherefore that side also descends in like manner, for the mercury which was in E, when the arm C E is raised above the horizontal A B, flows entirely near C: nor is the descent difficult; because B, when it has passed the perpendicular D E, would further rise of itself from the impetus already acquired; but it rises much more from the impressed impetus of the descending arm, by which it is urged. I freely admit that in the first turning after rest, the mercury at E resists, and contributes nothing to the motion; because, namely, since it ought to rise by the sole impressed impetus of the descending arm C B, it contributes nothing to the motion except insofar as at the beginning of its rise E rises a little, whereas B at the beginning of its descent descends much, and therefore a greater impetus can be impressed; by reason of which, although the measure of the rises increases, there is nevertheless some ease from the previous impulse. Hence, in the first turns, the help of the hand is needed, to push upward the lower arm C E: but once impetus has been conceived, I do not yet see why the motion should cease. For if no alternate transference of the weights is made (which always brings a new beginning of motion), but the weights always remain at the ends of the arms,

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Liber tertius. CAPUT X. 321 tibus, post aliquot conversiones externo impulsu factas sponte sua diu convertitur rota, aut etiam simplex scapus, non nisi ex impresso impetu tamdiu permanente, quidni perseveret in mo- tu, si in singulis conversionibus novum impetum concipiat? Sed hæc indicasse sufficiat, ut saltem longiorem motum, si non per- petuum, quis assequi possit suo instituto atque proposito op- portunum: mihi enim satis est rationes libræ hujusmodi com- mentatione aliquantò uberiùs explicare. Unum tamen hîc ad- dere fuerit operæ pretium, videlicet, si non placuerit scapos A B & D E invicem ad angulum rectum compactos excavare, sed solidos retinere volueris, posse singulis brachiis æquales tubulos hydrargyri quantitate æquali impletos adalligari, ita ta- men, ut similem brachij faciem contingant, ex quo fiet, ut sint ipsi tubuli alternatim dispositi, qui sibi ex adverso respondent, nimirum alter superior, alter inferior, alter ad dexteram, alter ad sinistram. PROPOSITIO IV. Dato unico pondere legitimo examinare bilance gravitatem multiplicem materiæ dividua. Res est facilis, non tamen omittenda, ne fortè quis sibi per- suadeat non nisi longissimâ operâ id perfici posse. Datum sit unicum pondus legitimum, ex. gr. uncia, & oblata sit ma- teria dividua, quæ particulatim examinari possit, ut sal, & cæ- tera minuta. Non sunt singulæ unciæ ponderandæ; sed pri- mò quidem fiat cum unciâ æquilibrium salis; deinde in lancem eandem cum pondere legitimo transferatur sal; iterum cum alio sale fiat æquilibrium, & hic in lancem ponderis refundatur, totiesque simili methodo repetatur ponderatio, donec oblatæ materiæ plus quàm semissem exhauseris; & adnota, quoties operam illam repetieris; tot enim termini in Ratione duplâ in- cipiendo ab unitate assumpti, & in summam redacti, dabunt gravitatem salis jam examinati. Sint ex. gr. decem termini; postremus est 512, cujus duplum demptâ unitate est summa omnium; sunt igitur unciæ 1023, hoc est libræ 85 1/4. Quod re- S s

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Liber tertius. CHAPTER X. 321 tubes, after several turns made by an external impulse, the wheel, or even a simple axle, continues to revolve of its own accord for a long time, only as long as the impressed impulse lasts; why should it not continue in motion if at each revolution it receives a new impulse? But it will suffice to have indicated these things, so that at least a longer motion, if not a perpetual one, may be attained by anyone according to his design and purpose as something suitable: for my part, it is enough to explain the reasons for this kind of balance somewhat more fully by this commentary. However, it will be worthwhile to add one thing here, namely, that if it should not please you to hollow out the axles A B and D E, joined to one another at a right angle, but you wish to keep them solid, equal tubes of mercury, filled with an equal quantity, can be fastened to each of the arms, provided however that they touch a similar face of the arm, from which it will follow that the tubes themselves are disposed alternately, answering one another opposite each other, namely one upper, one lower, one to the right, one to the left. PROPOSITION IV. Given a single lawful weight, to examine on a balance the multiple gravity of divisible matter. The thing is easy, though not to be omitted, lest perhaps someone should persuade himself that it can be accomplished only with very long labor. Let there be given a single lawful weight, for example an ounce, and let divisible matter be presented, which can be examined part by part, such as salt and other small things. The individual ounces are not to be weighed; but first let equilibrium be made with the ounce of salt; then let the salt be transferred into the same pan with the lawful weight; again let equilibrium be made with another portion of salt, and let this be poured into the pan of weight, and let the weighing be repeated in this same manner as often as necessary, until you have exhausted more than half of the presented matter; and note how many times you have repeated that operation; for so many terms in the double ratio, beginning from unity, and reduced to a sum, will give the gravity of the salt already examined. Let there be, for example, ten terms; the last is 512, whose double minus unity is the sum of all the others; therefore there are 1023 ounces, that is, 85 1/4 pounds. Which re-

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Mechanicorum siduum est salis, iterum simili ratione examinetur, donec habeas plus quàm semissem illius residui, acceptisque ite- rum tot terminis progressionis duplæ habebis ejus quantita- tem: & sic deinceps, donec totius propositæ molis pondus innotescat. Quòd si certam salis mensuram extrahere ex totâ illa mole desideras, ex. gr. libras tres, hoc est uncias 36, observa quot terminis progressionis duplæ proximè accedas ad propositam quantitatem, & erunt quinque termini, quorum postremus est 16, & tota summa 31. Quare operatio, ut supra, quinquies repetenda est, & habentur unciæ 31: quibus sepositis inqui- rantur unciæ 5 addendæ, nam duplici operatione singulas un- cias accipiens in eandem lancem cum unciâ legitimâ repones, & facto demum æquilibrio reliquas tres uncias habebis, ut summa conficiatur 36. unc. PROPOSITIO V. Libram æqualium brachiorum construere ad plura pondera tùm multiplicia tùm submultiplicia ejusdem æquipondij examinanda. Illud, in quo præstat libræ statera, est, quòd uno eo- demque stateræ æquipondio plura pondera examinamus. Non dissimile compendium invenire possumus in librâ æqua- lium brachiorum, quæ tamen spartum in superiore loco habeat; hæc enim pro diversâ ponderum inæqualitate va- riam habet inclinationem, in quâ quiescat obliquè posita. Expedit autem spartum à lineâ jugi aliquanto intervallo distare. Sit scapus planus, in quo linea jugi recta AB bifariam dividatur in C; ex quo ad angulos rectos assur- gat firmiter adnexa quasi lingula CD, ita tamen ut in D statuatur Axis ansæ insertus, circâ quem versanda est libra; & ex axe pendeat perpendiculum DE, cujus lon- gitudo tanta esse debet, ut non sit minor intervallo DA aut DB. Tum ex A sumatur totius lineæ jugi AB tertia pars A,

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The remainder of the salt should be examined again in a similar manner, until you have more than half of that remainder; and, having again taken so many terms of the doubled progression, you will have its quantity. And so on, until the weight of the whole proposed mass is known. But if you desire to extract from that whole mass a certain measure of salt, for example three pounds, that is, 36 ounces, observe how many terms of the doubled progression you approach most nearly to the proposed quantity; there will be five terms, the last of which is 16, and the whole sum 31. Therefore the operation, as above, is to be repeated five times, and 31 ounces are obtained; these being set aside, inquire for the 5 ounces to be added, for by the double operation, taking the individual ounces, you will place them in the same scale with the lawful ounce, and when equilibrium is finally achieved you will have the remaining three ounces, so that the sum may be made 36 oz. PROPOSITION V. To construct a balance with equal arms for examining several weights, both multiples and submultiples, of the same equivalence. That in which the balance scale excels is that by one and the same counterpoise of the balance we examine many weights. A not dissimilar convenience may be found in a balance of equal arms, which, however, has the spur in the upper part; for this, according to the different inequality of the weights, has a varying inclination, in which it rests when placed obliquely. But it is advantageous for the spur to be at some distance from the line of the beam. Let there be a flat shaft, in which the straight line AB of the beam is divided into two equal parts at C; from this point, at right angles, there rises firmly attached, as it were, a little tongue CD, so that at D the pivot-axis is set, around which the balance is to be turned; and from the axis let there hang the plumb line DE, the length of which ought to be such that it is not less than the distance DA or DB. Then from A let there be taken one third part of the whole line AB, A,

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Liber tertius. CAPUT X. 323 A 2, quarta A 3, quinta A 4, sexta A 5, & sic deinceps, quatenus commodè fieri poterit: quæ eædem partes ex B in alterum brachium transferatur quàm accuratissimè. Demum ex A & B æquales lances pendeant, quæ æquilibrium constituant. Hujus libræ usus est ad multiplicia vel submultiplicia pondera cum uno eodemque æquipondio comparata invenienda: Nam ubi æquipondium legitimum statueris in lance B, mercem verò in lance A, si æqualitas intercedat, ita jugum manet, ut perpendicularum DE congruat puncto C: si merx major sit æquipondio, inclinatur deorsum lanx A, & perpendicularum DE ad angulos inæquales secans lineam jugi congruit alicui ex punctis notatis inter C & A, scilicet in 2. si fuerit dupla, in 3 si tripla, & sic de reliquis: si demum merx fuerit minor æquipondio, lanx B inclinabitur, & perpendicularum DE congruet alicui ex punctis inter C & B notatis, indicabitque mercem esse æquipondij aut semissem, aut trientem, aut quadrantem, &c. Hinc si volueris plures uncias, aut unciæ partem aliquotam habere, statue in B legitimum unciæ pondus; si verò plures libras, aut libræ partem aliquotam quæsieris, statue in B libram legitimam. Verùm potissima hujus libræ utilitas se prodet, ubi dati ponderis, cujus gravitas secundùm legitimas mensuras ignota est, quæritur pars aliquota, aut illius multiplex pondus. Hujus autem libræ constructio innititur superiùs dictis, & manifesta est ratio, quia ex ponderum inæqualitate centrum commune gravitatis respondet jugi puncto, quod congruit perpendiculo pendenti ex eodem puncto suspensionis libræ. S s 2

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Book the Third. CHAPTER X. 323 A 2, the fourth A 3, the fifth A 4, the sixth A 5, and so on, as far as may conveniently be done: and let the same parts be transferred from B to the other arm as exactly as possible. Finally, let equal pans hang from A and B, which establish the equilibrium. The use of this balance is to find multiple or submultiple weights compared with one and the same counterpoise: for when you have placed a lawful counterweight in pan B, and the commodity in pan A, if equality exists, the beam remains so placed that the perpendicular DE corresponds to point C: if the commodity be greater than the counterweight, pan A inclines downward, and the perpendicular DE, cutting the line of the beam at unequal angles, corresponds to some one of the marked points between C and A, namely at 2 if it be double, at 3 if triple, and so on for the rest: if finally the commodity be less than the counterweight, pan B will incline, and the perpendicular DE will correspond to some one of the marked points between C and B, and will indicate that the commodity is half, or a third, or a fourth, etc., of the counterweight. Hence if you wish to have several ounces, or some aliquot part of an ounce, set in B the lawful weight of an ounce; but if you seek several pounds, or some aliquot part of a pound, set in B a lawful pound. However, the chief usefulness of this balance will appear when, from a given weight whose heaviness according to lawful measures is unknown, one seeks an aliquot part, or a multiple of that weight. The construction of this balance rests on the principles stated above, and the reason is clear, because from the inequality of the weights the common center of gravity corresponds to the point of the beam which agrees with the plumb line hanging from the same point of suspension of the balance. S s 2

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Mechanicorum Propositio VI. Staterâ examinare pondus majus, quàm ipsa communiter ferat. Certum esse pondus, quod unaquæque statera ferat pro ra- tione suæ magnitudinis, & gravitatis æquipondij, omni- bus manifestum est: & quidem si oblatum pondus dividuum sit, explorari potest per partes ejus gravitas, ut tota demùm in- notescat; sed si moles quædam solida sit, quæ se dividi non pa- tiatur, statera autem sit impar tanto oneri, artificium aliquod adhiberi potest, quo gravitatem illam majorem hâc eâdem sta- terâ investigemus. Et primò quidem ponamus stateram ita fuisse constructam, ut lancis gravitas suis momentis æquet mo- menta brachij longioris, adeò ut, dempto æquipondio stateræ jugum consistat in æquilibrio horizontaliter. Tunc certum est æquipondium ad onus esse reciprocè in Ratione distantia- rum oneris, & æquipondij à centro motûs. Quare eadem sta- tera poterit quodammodò multiplex fieri, si nimirum æquipon- dium duplicetur, aut triplicetur; poterit enim duplex aut tri- plex pondus staterâ examinari; ut, si proprium stateræ æqui- pondium sit unius libræ, & brachium longius sit brevioris bra- chij quindecuplex, examinari poterit pondus ut summum li- brarum quindecim; assumptum verò æquipondium novum bi- libre habebit momentum æquale libris 30; si trilibre sit no- vum æquipondium, momentum erit æquale libris 45; & sic de reliquis, etiam si æquipondium hoc novum non esset ad anti- quum omninò in Ratione multiplici; sed in quâcumque alia Ratione etiam super particulari, aut superpartiente; ducto enim pondere novi æquipondij per numerum notatum in sta- teræ brachio, habebitur quantitas ponderis, quod potest exa- minari; sic si æquipondium novum sit ad antiquum ut unc. 20. ad unc. 12. ducto 20 per 15, sit pondus unc. 300, hoc est lib. 25, quibus novum æquipondium in extremitate stateræ positum æquivolet. Verùm illud est incommodum, quòd hujusmodi æquipon- dio majori non possumus exploratam habere gravitatem pon- deris, si fortè gravitas æquipondij non sit illius pars aliquota: nam

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Mechanics Proposition VI. To weigh by a balance a weight greater than it commonly bears. It is evident to all that there is a certain weight which every balance bears in proportion to its size and the gravity of the counterpoise; and indeed, if the weight presented is divisible, its heaviness can be tested in parts, so that the whole is at last ascertained. But if there is some solid mass that does not allow itself to be divided, while the balance is unequal to so great a burden, some device may be used by which we may investigate that greater heaviness with the same balance. And first let us suppose the balance to have been so constructed that the weight of the pan, by its moments, equals the moments of the longer arm, so that, the counterpoise being removed, the beam stands in horizontal equilibrium. Then it is certain that the counterpoise to the load is reciprocally in the ratio of the distances of the load and the counterpoise from the center of motion. Wherefore the same balance may in a certain way be made multiple, namely if the counterpoise be doubled or tripled; for a double or triple weight can then be weighed by the balance. Thus, if the proper counterpoise of the balance be one pound, and the longer arm be fifteen times the shorter arm, a weight can be weighed up to fifteen pounds at most; but a new two-pound counterpoise will have a moment equal to 30 pounds; if the new counterpoise be three pounds, the moment will be equal to 45 pounds; and so on for the rest, even if this new counterpoise were not altogether in a multiple ratio to the old one, but in any other ratio as well, whether superparticular or superpartient. For by multiplying the weight of the new counterpoise by the number marked on the arm of the balance, there will be obtained the quantity of weight that can be weighed. Thus if the new counterpoise is to the old as 20 ounces to 12 ounces, multiplying 20 by 15 gives 300 ounces of weight, that is, 25 pounds, with which the new counterpoise placed at the end of the balance will be equivalent. But the inconvenience is this: with a counterpoise of this kind, larger than the ordinary one, we cannot have the weight of the load ascertained, if perchance the weight of the counterpoise is not an aliquot part of it: for

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Liber tertius. CAPUT X. 325 nam si novum æquipondium sit bilibre, non indicabit nume- rum disparem librarum ponderis in punctis libras denotantibus ( sed solummodo in punctis selibrarum) vel saltem singulas un- cias non indicabit, quia omnes numeri in staterâ notati dupli- candi essent: similiter dicendum de æquipondio triplici, quo adhibito omnes numeri triplicandi essent. Propterea, ut huic incommodo occurratur, retineatur anti- quum æquipondium in jugo stateræ, sed simul novum æqui- pondium in jugi extremitate apponatur duplum, vel triplum, vel quadruplum antiqui æquipondij, prout proximè requiri- tur ad explorandam dati oneris gravitatem; tùm antiquum æquipondium in jugo stateræ admoveatur vel removeatur, quatenus opus fuerit ad æquilibrium constituendum. Nam si numerus librarum novi æquipondij ducatur per numerum om- nium librarum, quas ferre potest statera, huicque addatur nu- merus ab antiquo æquipondio indicatus, habebitur ipsa pon- deris gravitas, quæ inquiritur. Proponatur pondus aliquod gravitatis ignotæ, quod stateræ lanci imponatur, & æquipon- dium antiquum ac proprium stateræ in extremitate positum non valeat pondus elevare ad æquilibrium; addatur æquipon- dium duplum, hoc adhuc impar est; addatur triplum, neque hoc satis est; addatur quadruplum, & hoc unà cum antiquo æquipondio in extremitate brachij posito præponderans illud est, quod requiritur; manente enim novo hoc æquipondio qua- druplo in extremitate, antiquum æquipondium admoveatur versùs spartum, & fiat æquilibrium in puncto lib. 7. unc. 9: quia stateræ numerus extremus est ex hypothesi lib. 15, & æqui- pondium novum est lib. 4, jam sunt lib. 60; adde lib. 7. unc. 9. tota gravitas ponderis quæsita est lib. 67. unc. 9. At quæris, an eodem hoc artificio uti liceat in communibus stateris, quas nostratibus artificibus construere solemne est; in quibus nec statera est per se solam æquilibris, nec æquipondij stateræ jugo ita innexi, ut inde pro libito auferri nequeat, gra- vitatem indagare possumus, ut æquipondium illius multiplex eligamus. Opportunè utique dubitas; nam pondus & æquipon- dium in vulgaribus stateris non sunt omninò in reciprocâ Ra- tione distantiarum à sparto, ut superiùs suo loco dictum est. Propterea uti quidem possumus eodem artificio, sed certâ ratio- S s 3

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Book the Third. CHAPTER X. 325 for if the new balance weight be of two pounds, it will not indicate the unequal number of pounds of weight in the points marking pounds ( but only in the points marking half-pounds) or at least it will not indicate the single ounces, because all the numbers marked on the balance would have to be doubled: similarly, the same must be said of a threefold balance weight, by the use of which all the numbers would have to be tripled. Therefore, in order to remedy this inconvenience, let the old balance weight be retained on the fulcrum of the scale, but at the same time let a new balance weight be applied at the end of the beam, double, or triple, or quadruple the old balance weight, as shall be nearest required for discovering the heaviness of the given load; then let the old balance weight be moved toward or removed from the fulcrum of the scale, as far as may be necessary in order to establish equilibrium. For if the number of pounds of the new balance weight be multiplied by the number of all the pounds which the scale can carry, and to this be added the number indicated by the old balance weight, the very weight sought will be obtained. Let some weight of unknown heaviness be proposed, to be placed on the scale pan, and let the old and proper balance weight of the scale, placed at the end, be unable to raise the weight to equilibrium; add a double balance weight, and this is still insufficient; add a triple one, and this too is not enough; add a quadruple one, and this, together with the old balance weight placed at the end of the arm, overweighs it, which is what is required; for while this new quadruple balance weight remains at the end, let the old balance weight be moved toward the spar, and let equilibrium be established at the point lib. 7. unc. 9: because the extreme number of the scale is, by hypothesis, lib. 15, and the new balance weight is lib. 4, there are now lib. 60; add lib. 7. unc. 9, the whole heaviness of the weight sought is lib. 67. unc. 9. But you ask whether the same contrivance may be used in common scales, which it is customary for our artisans to construct; in which neither is the scale itself, by itself, in equilibrium, nor are the balance weights so attached to the beam of the scale that they cannot be removed at will, can we discover the heaviness by choosing its multiple balance weight. You doubt rightly enough; for the weight and balance weight in ordinary scales are not at all in reciprocal proportion to the distances from the spar, as was said above in its place. Therefore indeed we may use the same contrivance, but with a certain ratio S s 3

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Mechanicorum ne: quia enim antiquum æquipondium cum stateræ notis lon- gè aliter se habet, ac in staterâ superiùs assumptâ, hoc retinea- tur, quod antiquum æquipondium indicabit gravitatem ponde- ris juxta notas, stateræ impressas; sed æquipondium novum as- sumatur certæ ac notæ gravitatis proximè tàm submultiplicis ponderis examinandi, quàm submultiplex brachij longioris est brachium minus stateræ; & hoc æquipondium adnectatur non planè in stateræ extremitate, sed in puncto, in quod cadit lon- gitudo multiplex brachij minoris. Sit ex. gr. statera communis, quæ elevet pondus lib. 15; sed comparato breviore brachio cum longiore, hoc non est illius omnino quindecuplum; assumo, quoties assumi potest brachium minus, ex. gr. quaterdecies; & in illo puncto statuendum erit novum æquipondium notæ gra- vitatis; & quoniam suspicor propositam gravitatem non mul- tum abesse à lib. 50, assumo æquipondium lib. 3. quæ in notato puncto æquivaleat libris 42 (nam ter 14 dant 42) & promoto versùs spartum antiquo æquipondio, fit æquilibrium in puncto lib. 5. unc. 3: erit igitur proposita gravitas lib. 47. unc. 3. Id quod est manifestum, quia antiquum æquipondium cum notis state- ræ impressis indicat gravitatem ponderis habitâ ratione mo- mentorum brachij stateræ & cæterarum illius partium, quas semel attendere opus est; reliquæ gravitatis momenta non nisi ratione distantiarum consideranda sunt. Quòd si plurium æquipondiorum supellectile careas, & ur- geat necessitas statim explorandi gravitatem illam majorem, ob- vium aliquod pondus, puta lapidem, vel quid ejusmodi, state- râ tuâ expende, ut ejus gravitas innotescat: hoc suspende ex opportuno stateræ puncto, de quo dictum est, & ejus gravita- tem duc per 14 (vel alium quemlibet numerum minorem aut majorem, prout opportuna ejus suspendio, aut stateræ longitu- do feret) ut habeas gravitatem huic novo æquipondio respon- dentem; Cætera ut priùs absolve. Non videtur autem necessa- riò monendus hîc lector posse plura nova æquipondia vel di- versæ, vel paris gravitatis, addi in diversis distantiis à sparto; ut si æquipondium lib. 3. in distantia 14, & aliud lib. 2 in distan- tia 11 simul apponantur, æquivalebunt lib. 42 & 22, hoc est li- bris 64; hæc enim clariora sunt, quàm indigeant uberiori ex- plicatione. PROPOSI

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Mechanics No: because the old counterweight with the marks of the steelyard is very different from what it would be in the steelyard assumed above, let this be retained, namely, that the old counterweight will indicate the heaviness of the weight according to the marks impressed on the steelyard; but let a new counterweight be taken, of a certain and known heaviness, as nearly as the submultiple of the weight to be examined is the submultiple of the longer arm to the shorter arm of the steelyard; and let this counterweight be attached not exactly at the end of the steelyard, but at the point where the multiple length of the shorter arm falls. Let there be, for example, an ordinary steelyard that will raise a weight of 15 lb.; but, comparing the shorter arm with the longer, this is not exactly fifteen times that arm; I take the shorter arm, as many times as it can be taken, for example fourteen times; and at that point the new counterweight of known heaviness must be placed; and since I suspect that the proposed weight is not far from 50 lb., I take a counterweight of 3 lb., which at the marked point is equivalent to 42 lb. (for three times 14 gives 42), and, by moving toward the spar with the old counterweight, equilibrium is reached at the point of 5 lb. 3 oz.; therefore the proposed weight will be 47 lb. 3 oz. This is evident, because the old counterweight, together with the marks impressed on the steelyard, indicates the heaviness of the weight, regard being had to the moments of the arm of the steelyard and of its other parts, which need only be considered once; the remaining moments of the heaviness are to be considered only in relation to the distances. But if you lack a supply of several counterweights, and there is urgent need immediately to discover that greater weight, weigh some object at hand, such as a stone, or something of that sort, with your steelyard, so that its weight may be known: suspend this from a suitable point of the steelyard, of which mention has been made, and multiply its weight by 14 (or by any other lesser or greater number, as the convenience of its suspension, or the length of the steelyard, may allow) so that you may have the weight corresponding to this new counterweight; then finish the rest as before. Nor does it seem necessary here to warn the reader that several new counterweights, either of different or equal weight, may be added at different distances from the spar; so that if a counterweight of 3 lb. at a distance of 14, and another of 2 lb. at a distance of 11, are placed at once, they will be equivalent to 42 and 22 lb., that is, 64 lb.; for these things are clearer than to need a fuller explanation. PROPOSI

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Liber tertius. CAPUT X. 327 PROPOSITIO VII. Stateram parare ad minusculas gravitates expendendas. S tateræ hujus jugum non differt à vulgaribus; sed æquipon- dij & ponderis est contraria positio; pondus enim longiori brachio, breviori æquipondium adnectitur, & quò levius fue- rit pondus, eò magis à sparto removetur. Paretur jugum cum lance adnexâ, quæ suâ gravitate æquet momenta brachij lon- gioris, & in perfecto æquilibrio consistat. Tum brevioris bra- chij longitudo accuratè transferatur in brachium majus, quod minoris saltem decuplum vellem, & singulas partes iterum in decem minores particulas tribuerem, ut totum longius bra- chium in centum particulas distingueretur. Sit stateræ jugum AB ita in C à sparto divisum, A I F C G E D A B ut CB sit de- cuplex ipsius CA: ex A au- tem pendeat lanx D suâ gravitate æquè librans momenta bra- chij CB longioris; quod distinctum in longitudines decem æquales brachio minori CA, in singulis divisionibus indicabit Rationem ponderis ad æquipondium. Collocetur enim æqui- pondium in lance D, pondus examinandum si leviusculum sit ita, ut serico crudo suspendi possit, jugo CB imponatur, & à sparto removeatur, donec fiat æquilibrium: nam si in primo puncto divisionis consistat, erit æqualis gravitatis cum æqui- pondio; si in secundo puncto, erit semissis gravitatis æquipon- dij; si in tertio, erit triens, si in quarto, quadrans, & sic de cæ- teris. At singulis divisionibus minori brachio æqualibus ite- rum in decem particulas distinctis, indicabitur gravitas à fractione, cujus numerator est 10, denominator est numerus particularum omnium, quæ inter spartum C & locum ponde- ris æquèlibrati intercipiuntur: ut si ex. gr. æquilibrium fiat in F, hoc est in tertiâ particulâ post duas integras divisiones priores, jam sunt particulæ 23; igitur pondus est 10/23; ipsius æquipondij in D positi; ut constat ex eo, quòd ut distantia CF 23 ad distantiam

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Book three. Chapter X. 327 Proposition VII. To prepare a balance for weighing minute weights. The beam of this balance does not differ from ordinary ones; but the position of the counterpoise and the weight is opposite: for the weight is attached to the longer arm, the counterweight to the shorter, and the lighter the weight is, the more it is removed from the pivot. Let a beam be prepared with a pan attached, which by its own weight shall equal the moments of the longer arm, and stand in perfect equilibrium. Then let the length of the shorter arm be accurately transferred to the longer arm, which I would wish to be at least ten times the smaller, and let each part again be divided into ten smaller particles, so that the whole longer arm may be distinguished into one hundred particles. Let the beam AB be thus divided at C by the pivot, A I F C G E D A B so that CB may be ten times CA: from A moreover let the pan D hang, balancing by its own weight the moments of the longer arm CB; this, divided into ten equal lengths corresponding to the shorter arm CA, will in each division indicate the ratio of the weight to the counterweight. For let the counterweight be placed in pan D, and if the weight to be examined is somewhat light, so that it can be suspended by raw silk, let it be placed on beam CB and moved away from the pivot until equilibrium is reached: for if it rests at the first point of division, it will be equal in weight to the counterweight; if at the second point, it will be one half the weight of the counterweight; if at the third, one third; if at the fourth, one fourth, and so on for the rest. But if each division, equal to the shorter arm, is again divided into ten parts, the weight will be indicated by a fraction whose numerator is 10, and whose denominator is the number of all the particles that lie between the pivot C and the place of the balanced weight: thus if, for example, equilibrium is achieved at F, that is, in the third particle after the first two entire divisions, there are now 23 particles; therefore the weight is 10/23 of the counterweight placed at D; as is clear from the fact that as the distance CF is 23 to the distance

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Mechanicorum 328 distantiam CA 10, ita æquipondium in D ad pondus in F 10/13. Hujus stateræ utilitas satis latè patet, quia non alligatur certo æquipondio, sed in lance D statui potest sivè drachma, si- vè uncia, sivè libra, & ponderis minoris gravitas examinabitur; quæ quidem habebitur secundùm Rationem partis ad assem, sed deinde ad certam ponderis mensuram, sivè scrupula, sivè grana revocabitur. Quòd si pondus examinandum non facilè suspendi possit serico crudo, ut dictum est, paratam habeto lancem minusculam, cui imponi possit pondus; & demùm facto æquilibrio, gravitate ponderis inventâ, atque ad homogeneam cum æquipondio mensuram redactâ, subducenda est hujus lancis cum suo funiculo gravitas, ut sola ponderis impositi gravitas habeatur. Ex quo patet adhibitâ hujusmodi lance, quæ percurrat stateræ jugum, posse expendi gravitatem multò minorem: propterea lancis hujus gravitas minor esse deberet, quàm subdecupla gravitatis æquipondij impositi lanci D, ut in extremo stateræ puncto B fieri posset æquilibrium: verùm si æquipondium in D sit uncia, aut aliquid unciâ minus, majus tamen decimâ ejus parte, satius fuerit lancem illam excipiendo ponderi destinatam esse decimam unciæ parrem. Ponatur enim æquipondium uncia, lanx ponderis cursoria 1/10 unciæ: impositum pondus faciat æquilibrium in F puncto particulæ 23: est igitur pondus cum suâ lance 10/23 unciæ, aufer ratione lancis 1/10 unciæ, residuum 27/230 unciæ est gravitas ponderis; hoc est scrupulorum 8. Similiter fiat æquilibrium in puncto 99; ergo pondus cum lance est 10/99 unciæ; aufer 1/10, residuum est 1/990 unciæ, quod est levissimum pondus paulò majus semisse grani. Si in parte 98, pondus erit 1/1 unciæ, hoc est grani 1/6, si in puncto 97, pondus erit 3/970 hoc est ferè grani; 1/5: & sic de cæteris. PROPOSI

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Mechanics 328 distance CA 10, thus the counterpoise in D to the weight in F is 10/13. The usefulness of this balance is sufficiently evident, because it is not tied to any fixed counterpoise, but in pan D there may be placed either a drachm, or an ounce, or a pound, and the weight of a smaller body will be examined; which indeed will be obtained according to the ratio of the part to the whole, and then reduced to a certain measure of weight, whether scruples or grains. But if the weight to be tested cannot easily be suspended by raw silk, as has been said, have ready a small pan on which the weight may be placed; and finally, equilibrium having been achieved, and the gravity of the weight found and reduced to a homogeneous measure with the counterpoise, the weight of this pan together with its thread must be subtracted, so that only the gravity of the weight placed upon it may be had. From this it is clear that, by using such a pan which traverses the beam of the balance, a much smaller weight can be weighed: therefore the weight of this pan ought to be less than one-tenth of the weight of the counterpoise placed in pan D, so that equilibrium could be made at the extreme point B of the balance; but if the counterpoise in D be an ounce, or something less than an ounce but greater than one-tenth of it, it would be better for the pan intended to receive the weight to be one-tenth part of an ounce. For let the counterpoise be one ounce, and the movable pan one-tenth of an ounce: let the placed weight produce equilibrium at point F, division 23; therefore the weight together with its pan is 10/23 of an ounce; subtract, by the ratio of the pan, one-tenth of an ounce; the remainder, 27/230 of an ounce, is the gravity of the weight; that is, 8 scruples. Likewise let equilibrium be made at division 99; therefore the weight together with the pan is 10/99 of an ounce; subtract 1/10, the remainder is 1/990 of an ounce, which is a very light weight a little greater than half a grain. If at division 98, the weight will be 1/1 of an ounce, that is 1/6 of a grain; if at point 97, the weight will be 3/970, that is nearly 1/5 of a grain; and so on for the rest. PROPOSI

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Liber tertius. CAPUT X. 329 PROPOSITIO VIII. Ad ingentia onera examinanda stateras communes componere. Si opportunas stateras parare oporteret ingentibus oneribus sexaminandis pares, cujusmodi esset æs campanum, aut bellicum tormentum majus, eas esse debere aut longissimas, aut immani æquipondio instructas, manifestum est. Fac enim tormentum esse lib. 17000 circiter, & stateram habere uncialem distantiam sparti ab extremitate, cui pondus adnectitur, æquipondium verò esse lib. 25; utique ut 25 ad 17000, ita uncia pedis ad uncias 680, hoc est pedes 56. unc. 8: atque adeò tota statera esset ped. 56 3/4 ut minimum: cui longitudini si congrua crassities respondeat, an non machineâ opus est, ut sola statera transferatur? præterquam quod ipsa longioris brachij gravitas momenta non exigua haberet. Quòd si, ut non paucis solemne est, ita trabem ex mediâ longitudine suspendas, ut æquilibris maneat, tùm alteri extremitati propositum onus adnectas, oppositæ autem extremitati plura minora pondera adjicias, donec æquilibrium fiat, quorum singulæ gravitates in summam redactæ propositi oneris gravitatem manifestam reddant, non solùm methodus hæc artificio caret, sed & falsitatis periculo non vacat, incertum quippe est an trabis centrum gravitatis planè in mediâ longitudine sit, cùm pars radici proxima gravior sit reliquâ, ac proinde libra sit inæqualium brachiorum, quæ censetur æqualium. Satius igitur fuerit stateras plures minores componere, ut indicatum est lib. 2. cap. 7, quàm ingentem stateram construere. Assumantur tres stateræ AB, DE, GH, quarum brachium minus sit majoris subdecuplum, & ita omnes ex superiore loco suspendantur, ut orbiculi M & N facilè veriatiles inferiùs firmati excipere possint funiculos BMD, & ENG, quibus extremitates junguntur: ex quo fiet, ut dum H vi æquipondij deprimitur, extremitas E, atque extremitas B pariter deprimantur, pondus verò in A ad- T

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Book three. CHAPTER X. 329 PROPOSITION VIII. To devise common balances for weighing very heavy loads. If suitable balances were needed for weighing very heavy loads, such as a campana bell, or a large cannon, it is evident that they must be either extremely long or furnished with an enormous counterpoise. For example, suppose a cannon to weigh about 17,000 lb., and a balance to have an uncial distance of a span from the end to which the weight is attached, while the counterpoise is 25 lb.; then, as 25 is to 17,000, so one inch of a foot is to 680 inches, that is, 56 feet, 8 inches: and so the whole balance would be at least 56 3/4 feet long. If a suitable thickness corresponded to such a length, would not machinery be needed merely to move the balance itself? besides, the weight of the longer arm itself would create no small moment. But if, as is customary with not a few, you suspend a beam from its midpoint so that it remains in equilibrium, and then attach the proposed load to one end and add several smaller weights to the opposite end until equilibrium is achieved, the individual weights of which, reduced to a sum, make the weight of the proposed load clearly known, this method not only lacks ingenuity, but is also not free from the risk of error; for it is uncertain whether the center of gravity of the beam is exactly at the midpoint, since the part nearest the root is heavier than the rest, and therefore the balance has unequal arms, though it is taken to be one of equal arms. It would therefore be better to compose several smaller balances, as was indicated in Book 2, chapter 7, than to construct one enormous balance. Let three balances AB, DE, GH be taken, of which the shorter arm is one-tenth of the longer; and let them all be suspended from above in such a way that the little wheels M and N, easily turning and fixed below, may receive the cords BMD and ENG, by which the ends are joined: from which it will follow that, while H is depressed by the force of the counterpoise, the end E, and likewise the end B, are also depressed, but the weight in A to-

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Mechanicorum nexus elevetur. Motus autem staterarum non sunt æquales: nam sicut depressio ipsius H est decupla elevationis ipsius G, cui elevationi æqualis est depressio extremitatis E, ita hæc ejusdem E depressio decupla est elevationis ipsius D: quare depressio H est centupla elevationis D; ac propterea quia depressio B æqualis elevationi D est decupla elevationis A, depressio æquipondij in H est millecupla elevationis ponderis in A constituti. Ex quo sequitur æquipondium in H æquivalebit ponderi lib. 17000. Quod autem hactenus de stateris æqualibus dictum est, etiam de inæqualibus dictum intelligatur, componendo Rationes, quas singularum staterarum brachia habent. Hinc si Ratio A C ad C B sit 1 ad 10, Ratio D F ad F E sit 1 ad 8, Ratio G I ad I H sit 1 ad 12, Ratio composita est 1 ad 960, quæ potest intercedere inter æquipondium & onus. Hinc manifestum est plures addi posse stateras, quot opus fuerit, quocumque tandem ordine collocentur, sive secundùm rectam lineam, sive invicem parallelæ, prout commodius accidet, & loci opportunitas feret. Si stateræ istæ fuerint ita constructæ, ut jugum dempto æquipondio æquilibre sit, quia extremitas brachij minoris gravitate tantâ prædita est, ut gravitati longioris brachij æquipollat, res planissima est, quia sola brachiorum longitudinis Ratio attendenda est; & præterea æquipondium in H augeri posset, aut minui. Immò hîc etiam adhiberi posset artificium, de quo prop. 6. dicebatur, addendo novum æquipondium certæ gravitatis, ut si præter æquipondium H etiam esset L lib. 25, quod in puncto jugi septimo æquivaleret ponderi septingenties majori, hoc est lib. 1400. Quâ methodo addi possunt etiam plura æquipondia in punctis jugi diversis: quod sanè esset egre- gium

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The nexus of the machine be raised. But the motions of the balances are not equal: for just as the depression of H itself is ten times the elevation of G, to which elevation the depression of the end E is equal, so the depression of this same E is ten times the elevation of D; therefore the depression of H is a hundred times the elevation of D; and consequently, because the depression of B, equal to the elevation of D, is ten times the elevation of A, the depression of the counterweight in H is one thousand times the elevation of the weight placed in A. From this it follows that the counterweight in H will be equivalent to a weight of 17,000 lb. What has so far been said about equal balances is to be understood also of unequal ones, by combining the ratios which the arms of each balance have. Hence, if the ratio of AC to CB be 1 to 10, the ratio of DF to FE be 1 to 8, and the ratio of GI to IH be 1 to 12, the compounded ratio is 1 to 960, which may intervene between the counterweight and the load. Hence it is clear that more balances may be added as many as are needed, in whatever order they are arranged, whether in a straight line or parallel to one another, as shall be more convenient and as the opportunity of the place shall permit. If these balances are so constructed that the beam, with the counterweight removed, is in equilibrium, since the end of the shorter arm is endowed with such a weight as to balance the weight of the longer arm, the matter is perfectly clear, because only the ratio of the lengths of the arms must be considered; and moreover the counterweight in H could be increased or diminished. Indeed, here also that contrivance could be used which was mentioned in prop. 6, by adding a new counterweight of a certain weight, so that if, besides the counterweight H, there were also L, weighing 25 lb., it would, at the seventh point of the beam, be equivalent to a weight seven hundred times greater, that is, 1,400 lb. By this method it is possible also to add several counterweights at different points of the beam: which certainly would be excellent

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Liber tertius. CAPUT X. 331 gium compendium, & ut plurimum duabus stateris ponderatio ipsa perficeretur. At si stateræ cujusque jugum non fuerit æquilibre, contemnenda non est brachij longioris gravitas, ut dati ponderis gravitas ritè examinetur: Nam semissis gravitatis brachij IH in extremitate H constitutus æquivolet ponderi decuplo in G minus gravitate semissis brachij IG. Igitur perinde est, atque si hujusmodi pondus additum fuisset in E, ubi habet momentum decuplum æqualis ponderis in D, & centuplum æqualis ponderis in A. Quare momentum brachij IH est ut 50, & momentum IG ut 1/2, atque adeò momentum ut 49 1/2 intelligitur additum in E, quod propterea comparatum cum extremitate A habet momentum ut 4950. Sic momentum gravitatis FE comparatum cum extremitate A est ut 495; & momentum brachij CB est ut 49 1/2. Tota igitur momentorum, quæ ex brachiorum gravitate oriuntur (si illa æqualiter ducta intelligantur) summa est 5494 1/2, sive sint unciae, sive libræ, prout staterarum moles requirit. Id quod quia ægrè innotescit, si jugum non fuerit æquabiliter ductum, idcircò expeditius fuerit stateris ritè dispositis, ac dempto æquipondio, addere in A tantum gravitatis, ut juga sint horizonti parallela (cujus parallelismi indicium potissimum dabit extremæ stateræ lingula, quæ plus cæteris movetur) quæ gravitas ubi innotuerit, addenda erit gravitati, quam deinde æquipondium indicabit, cùm onus ipsum in A additum fuerit. Sic pone momenta illa 5494 1/2 esse uncias, hoc est lib. 457. unc. 10 1/2, & expendendo onus additum in A, æquipondium librale H indicet æquilibrium in puncto septimo, hoc est lib. 700, addantur lib. 457. unc. 10 1/2, erit tota oneris gravitas lib. 1157. unc. 10 1/2. Verùm quia vulgares stateræ, quibus communiter utimur ad majorum ponderum gravitatem examinandam, non ita sunt fabrefactæ, ut brachium longius in partes aliquotas minori brachio æquales distinguatur, propterea minoris brachij longitudo, quoties fieri id poterit, transferatur in brachium longius, ut inveniatur punctum, cui adnectendus est funiculus, quo cum alterius proximæ stateræ extremitate connectitur. Sed antequam opus aggrediaris, amoto æquipondio secundæ & ter- T t 2

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Book the third. CHAPTER X. 331 an abridged compendium, and, for the most part, the weighing was performed with two balances. But if the beam of either balance has not been made level, the weight of the longer arm is not to be disregarded, so that the weight of the given load may be duly examined: for half the weight of the arm IH, placed at the extremity H, is equivalent to ten times the weight in G, less the weight of half the arm IG. Therefore it is as though a weight of this kind had been added at E, where it has a moment ten times that of an equal weight in D, and one hundred times that of an equal weight in A. Wherefore the moment of the arm IH is as 50, and the moment of IG as 1/2, and thus a moment as 49 1/2 is understood to have been added at E, which therefore, compared with the extremity A, has a moment as 4950. In this way the moment of the weight FE, compared with the extremity A, is as 495; and the moment of the arm CB is as 49 1/2. The whole sum therefore of the moments arising from the weight of the arms (if they are understood as drawn out equally) is 5494 1/2, whether they be ounces or pounds, as the size of the balances requires. Since this is difficult to ascertain, if the beam has not been equally drawn out, it will therefore be easier, the balances being rightly arranged and the counterpoise removed, to add in A only so much weight as is needed for the beams to be parallel to the horizon (of which parallelism the indicator of the outermost balance will chiefly give evidence, namely the tongue, which moves more than the others); when this weight has been ascertained, it must be added to the weight which the counterpoise will then indicate, when the load itself has been added in A. Thus suppose those moments 5494 1/2 to be ounces, that is, lb. 457, oz. 10 1/2, and by weighing the load added in A, let the pound counterpoise H indicate equilibrium at the seventh point, that is, lb. 700; if lb. 457, oz. 10 1/2 be added, the total weight of the load will be lb. 1157, oz. 10 1/2. But because ordinary balances, which we commonly use for examining the weight of larger loads, are not so made that the longer arm is divided into equal aliquot parts corresponding to the shorter arm, therefore the length of the shorter arm, whenever this can be done, should be transferred to the longer arm, so that the point may be found to which the string is to be attached, by which it is connected with the extremity of the other nearest balance. But before you begin the work, the counterpoise of the second and third having been removed,

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Mechanicorum tiæ stateræ, vide quantum ponderis singulæ, quantum con- nexæ requirant ad æquilibrium cum longiore brachio, ut inno- tescat, quantum adhuc gravitatis oneri tribuendum sit, præter illam, quæ ab æquipondio indicatur. Sit ex. gr. secunda statera CD, cujus brachium longius SD æquivaleat lib. 42, & tertiæ stateræ FG longius brachium VG æquivaleat libris 37. Ponamus tertiæ stateræ (cui onus erit adnectendum) brachium minus F V duodecies contineri in longiore brachio usque ad I, ubi funiculus connectit illud cum extremitate C secundæ stateræ. Item secundæ stateræ CD brachium minus C S tredecies sumi possit in brachio lon- giore usque ad punctum E, ubi illam funiculus connectit cum primæ stateræ extremitate A. Igitur quia momentum brachij SD æquivalet libris 42 ex hypothesi, & intelligitur translatum in I, ubi duodecuplo velociùs movetur quàm punctum F, du- cantur lib. 42 per 12, & æquivalet libris 504, quibus addenda sunt momenta brachij VG lib. 37, & ponderi invento ex æqui- pondio demum addendæ erunt lib. 541. Iam statuamus æqui- pondium H primæ stateræ AB constituere æquilibrium in puncto indicante libras 14: perinde igitur est, atque si libræ 14 ponerentur in E; & quia E S ad S C est ut 13 ad 1, libræ 14 in E æquivalent ponderi in C librarum 182, quæ in I positæ (quia IV ad V F est ut 12 ad 1) æquivalent ponderi in F librarum 2184. Quod si punctum illud, in quo æquipondium H con- sistit, non esset nota librarum simplicium 14, sed ponderum, quæ singula libras 25 continent (ut nobis Italis præsertim in Galliâ Cisalpinâ solemne est) utique onus in F adnexum esset lib. 54600, quibus adhuc addendæ essent libræ 541, propter momenta brachiorum secundæ & tertiæ stateræ, & esset tota gravitas lib. 55141. At si stateras communes habeas, nec possis æquipondia jugo inserta amovere, ut inquirere possis momenta gravitatis brachij, longioris,

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of the mechanics of the balance, see how much weight each one, and how much when connected, requires for equilibrium with the longer arm, so that it may become known how much of the weight still remains to be attributed to the load, besides that which is indicated by the counterpoise. Let, for example, the second balance CD, whose longer arm SD is equal to 42 lb., and the longer arm VG of the third balance FG equal 37 lb. Let us suppose that the shorter arm FV of the third balance (to which the load is to be attached) is contained twelve times in the longer arm, up to I, where the cord connects it with the extremity C of the second balance. Likewise, let the shorter arm CS of the second balance CD be able to be taken thirteen times in the longer arm, up to the point E, where the cord connects it with the extremity A of the first balance. Therefore, since the moment of the arm SD is equal by hypothesis to 42 lb., and is understood to be transferred to I, where it moves twelve times faster than point F, multiply 42 by 12, and it is equal to 504 lb., to which must be added the moments of the arm VG, 37 lb., and to the weight finally found from the counterpoise there must be added 541 lb. Now let us suppose the counterpoise H of the first balance AB to establish equilibrium at a point indicating 14 lb.: it is therefore the same as if 14 lb. were placed at E; and because ES is to SC as 13 to 1, 14 lb. at E are equivalent to a weight at C of 182 lb., which, placed in I (because IV is to VF as 12 to 1), are equivalent to a weight at F of 2,184 lb. But if that point, in which the counterpoise H is situated, were not an indication of simple 14 lb., but of weights which each contain 25 lb. (as is customary among us Italians, especially in Cisalpine Gaul), certainly the load attached at F would be 54,600 lb., to which still 541 lb. must be added on account of the moments of the arms of the second and third balance, and the total weight would be 55,141 lb. But if you have ordinary balances, and cannot remove the counterweights inserted in the yoke, so that you may inquire into the moments of the weight of the longer arm,

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Liber tertius. CAPUT X. 333 longioris, hoc unum in secundâ, & in tertiâ staterâ, aut etiam pluribus, si opus fuerit, observa, quoties nimirum brachium minus in longiore contineatur, ut punctum I & E innotescat, quod cum proximæ stateræ extremitate C & A connectendum est: in singulis autem stateris sua æquipondia admoveantur, vel removeantur, donec fiat æquilibrium. Non est autem ne- cesse singulas stateras insculptas esse notis homogeneis gravi- tatum; prima enim AB potest habere notas indicantes quar- tam partem Centenarij, hoc est lib. 25, secunda verò & ter- tia possunt indicare tantum singulas libras cum suis unciis. Fac enim constituto æquilibrio, æquipondium H esse in puncto pond. 9. lib. 7. duc 9 per 25, & sunt lib. 225, & additis lib. 7, sunt lib. 232; quæ ducuntur primò per Rationem secundæ sta- teræ 13 ad 1, & fiunt 3016, quæ ductæ per Rationem tertiæ stateræ 12 ad 1, dant demum lib. 36192. Deinde æquipon- dium secundæ stateræ CD sit in puncto lib. 7. unc. 8: hæ du- cendæ sunt per Rationem tertiæ stateræ 12 ad 1, & fiunt lib. 92. Demum æquipondium tertiæ stateræ indicet lib. 5. unc. 6, addantur hi tres numeri 36192, 92, & 5. unc. 6; tota gravitas oneris in F adnexi erit lib. 36289. unc. 6. Ideò autem inquirenda dixi puncta I & E, ut longitudines VI & SE sint multiplices longitudinum brachiorum minorum FV & CE, atque fractionum molestia evitetur. Cæterùm si volueris extremitates ipsas G & D cum extremitatibus C & A connectere, omnino licebit, ubi innotuerit, quota pars brachij minoris sit IG & ED. Nam si Ratio DS ad SC deprehen- datur ut 13 2/5 ad 1, Ratio autem GV ad VE ut 12 3/4 ad 1, gra- vitas indicata ab æquipondio H ducenda primùm erit per 13 2/5, deinde numerus productus per 12 3/4 ductus dabit quæsitam oneris gravitatem respondentem æquipondio H, quod, ex hy- pothesi superiùs constitutâ, indicans pond. 9. lib. 7, hoc est lib. 232, monet ducendas libras 232 per 13 2/5, & fit 3108 4/5, qui numerus ducatur per 12 3/4, & fiunt lib. 39637 1/5. Deinde æqui- pondium secundæ stateræ positum in puncto lib. 7. unc. 8 indi- cat has ducendas per 12 3/4, & erunt lib. 97 3/4: quibus si adda- tur numerus primæ stateræ, & numerus quem dat tertia stateræ lib. 5. unc. 6, summa erit omnino lib. 39740. unc. 5. Tl 3

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Book Three. Chapter X. 333 of the longer; observe this one thing in the second, and in the third balance, or even in more, if need be, namely, how many times the shorter arm is contained in the longer, so that the point I and E may be known, which is to be connected with the nearest end C and A of the balance: in each balance, however, let their own counterweights be applied, or removed, until equilibrium is achieved. But it is not necessary for the individual balances to be engraved with homogeneous marks of weights; for the first AB may have marks indicating a fourth part of a hundredweight, that is, 25 lb.; while the second and third may indicate only single pounds with their ounces. For suppose, with equilibrium established, that the counterweight H is at the point of 9 pounds 7 ounces; multiply 9 by 25, and there are 225 lb., and adding 7 lb., there are 232 lb.; these are first multiplied by the ratio of the second balance, 13 to 1, and there result 3016, which, multiplied by the ratio of the third balance, 12 to 1, finally give 36192 lb. Next, let the counterweight of the second balance CD be at the point of 7 lb. 8 oz.; these must be multiplied by the ratio of the third balance, 12 to 1, and there result 92 lb. Finally, let the counterweight of the third balance indicate 5 lb. 6 oz.; add these three numbers, 36192, 92, and 5 lb. 6 oz., and the total weight of the load attached at F will be 36289 lb. 6 oz. I said that the points I and E should therefore be sought, so that the lengths VI and SE may be multiples of the lengths of the shorter arms FV and CE, and the trouble of fractions may be avoided. Otherwise, if you wish to connect the extremities G and D themselves with the extremities C and A, this will be wholly permissible, once it is known what part of the shorter arm IG and ED are. For if the ratio of DS to SC is found to be as 13 2/5 to 1, and the ratio of GV to VE as 12 3/4 to 1, the weight indicated by the counterweight H must first be multiplied by 13 2/5; then the resulting number, multiplied by 12 3/4, will give the required weight of the load corresponding to the counterweight H, which, from the hypothesis established above, indicating 9 lb. 7 oz., that is, 232 lb., gives the instruction to multiply 232 lb. by 13 2/5, and the result is 3108 4/5; this number, multiplied by 12 3/4, gives 39637 1/5 lb. Next, the counterweight of the second balance placed at the point 7 lb. 8 oz. indicates that these are to be multiplied by 12 3/4, and there will be 97 3/4 lb.; if to these the number of the first balance, and the number given by the third balance, 5 lb. 6 oz., be added, the sum will be altogether 39740 lb. 5 oz. Tl 3

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Mechanicorum PROPOSITIO IX. In librâ brachiorum æqualium posse non æqualia esse ponderum æqualium momenta. SIt libra AB, cujus centrum C, prorsus in medio, jugum in brachia dividat æqualia: sint autem in brachiorum extre- mitatibus annuli vel unci, quibus adnectenda sunt pondera, quæ as- sumantur gravi- tatis exquisitè æ- qualis, computatâ etiam funiculo- rum gravitate. Sed alterum quidem pondus D unco adnectatur unâ cum suo funicu- lo; alterius verò ponderis E funiculus suâ extremitate inferiùs in F paxillo alligetur, & transiens per annulum, vel uncum suspendat connexum pondus E. Experimento disces pondus E semper prævalere æquali ponderi D, si per annulum vel uncum funiculus liberè valeat excurrere, descendente ipso pon- dere E. Sed rei primâ facie admiratione dignæ causam inquirenti illa se statim offert, quæ Machinalium motionum causa à nobis af- fertur; quia videlicet pondus E descendens duplo velociùs de- scendit, quàm pondus D ascendat; ubi enim pondus E vene- rit in F, extremitas libræ B ibi consistet, ubi duplicatus est fu- niculus, mediâ nimirum viâ; atqui extremitas A non nisi tan- tumdem ascendit, & cum eâ pondus D; igitur pondus E velo- ciùs descendens potiora habet momenta, nec erit æquilibrium, nisi pondus E sit ponderis D subduplum. Cave tamen existimes semper esse motuum Rationem du- plam; id enim tunc solùm accidit, cum funiculus extentus est horizont

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Mechanics PROPOSITION IX. In a balance with equal arms, the moments of unequal equal weights may be unequal. Let AB be a balance, whose center C, exactly in the middle, divides the beam into equal arms: and let there be at the extremities of the arms rings or hooks, to which weights are to be attached, and let these be assumed to be of exactly equal gravity, the weight of the cords also being reckoned in. But let one weight D be attached to the hook together with its cord; while the cord of the other weight E is fastened by its end below to peg F, and, passing through the ring or hook, suspends the connected weight E. By experiment you will learn that weight E always prevails over equal weight D, if the cord can freely run through the ring or hook, while the weight E itself descends. But to one inquiring into the cause of this thing, which on the first sight deserves admiration, there immediately presents itself that which we have assigned as the cause of mechanical motions; namely, because the descending weight E moves down twice as fast as weight D rises. For when weight E has come to F, the end of the balance B will be at the place where the cord has been doubled, that is, at the middle of the way; but the end A rises only by the same amount, and with it weight D; therefore, weight E, descending more quickly, has greater moments, and there will be no equilibrium unless weight E be one-half the weight of D. But do not think, however, that the ratio of motions is always double; for that occurs only when the cord is stretched horizontally

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Liber tertius. CAPUT X. 335 horizonti perpendicularis, cujusmodi est FE: at si fuerit in- clinatus, non est eadem motuum Ratio, sed ut duplex funicu- li GE longitudo ad altitudinem perpendicularem EF, ita se habet motus ponderis E ad differentiam, quâ excedit motum ponderis D, seu depressionis libræ B. Sit funiculus GE, alti- tudo perpendicularis, per quam descendit pondus E, sit EF; distantia GF: descendente pondere E, ubi hoc attigerit pla- num horizontale in F, funiculus, qui erat GE, factus est G I F; igitur libra deprimitur usque in I, & est I F differentia motuum EF & EI. Quare cùm GI sit GE minùs IF, quadratum GI æquale est quadrato GE plus quadrato IF, minùs rectangulo sub GE & IF bis comprehenso. At eidem quadrato GI æqualia sunt quadrata IF & GF simul sumpta ex 47. lib. 1: propterea ause- ratur utrinque quadratum IF, & remanet quadratum GE, mi- nùs rectangulo bis sub GE & IF comprehenso æquale quadra- to GF: Addatur utrinque rectangulum sub GE & IF bis, & utrinque dematur quadratum GF, & est quadratum GE mi- nus quadrato GF (hoc est quadratum EF ex 47. lib. 1.) æqua- le rectangulo bis sub GE & IF. Igitur ex 17 lib 6. ut bis GE ad EF, ita EF ad IF. Ponderis itaque motus deorsum EF comparatus cum ascensu ponderis D, est ad differentiam mo- tuum IF, ut duplex longitudo funiculi GE ad altitudinem perpendicularem EF, per quam descendit pondus E. Ex quo ulteriùs colligitur, quò obliquior est funiculus, eò minorem esse differentiam IF, ac propterea minorem esse Ra- tionem descensûs EF ad ascensum ponderis oppositi, ideóque etiam minus habere virium ad prævalendum. Hinc ex diver- sâ funiculi longitudine & obliquitate, si æquilibrium fiat, lice- bit arguere ipsam ponderum inæqualitatem, ratione habitâ mo- tuum reciprocè sumptorum; qui motus cum habere non possint Rationem multiplicem majorem duplâ, ut constat funiculi ipsius flexionem consideranti, neque pondus D potest esse minus pon- dere E, neque eodem majus quàm duplum, si fiat æquilibrium; minus autem erit quàm duplum, si funiculus sit obliquus, & ex motuum differentiâ, quæ singulas funiculi obliquitates conse- queretur, etiam ipsa ponderum inæqualium differentia in- fertur. PROPOSIT

Transcription: Translated (English)

Book Third. CHAPTER X. 335 horizontal, of which kind is FE: but if it be inclined, the ratio of the motions is not the same, but as the double length of the cord GE is to the perpendicular height EF, so is the motion of weight E to the difference by which it exceeds the motion of weight D, or the depression of balance B. Let the cord GE be the perpendicular height through which weight E descends, let it be EF; the distance GF: when weight E descends, when this has reached the horizontal plane at F, the cord, which was GE, has become G I F; therefore the balance is depressed as far as I, and I F is the difference of the motions EF and EI. Wherefore, since GI is GE minus IF, the square of GI is equal to the square of GE plus the square of IF, minus the rectangle twice comprehended under GE and IF. But to the same square GI are equal the squares IF and GF taken together, by 47, Book 1: therefore the square IF is removed from both sides, and there remains the square GE, minus the rectangle twice comprehended under GE and IF, equal to the square GF: add on both sides the rectangle under GE and IF twice, and subtract on both sides the square GF, and the square GE minus the square GF (that is, the square EF, by 47, Book 1) is equal to the rectangle twice under GE and IF. Therefore, by 17, Book 6, as twice GE is to EF, so is EF to IF. Therefore the downward motion EF of the weight, compared with the ascent of the opposite weight D, is to the difference of the motions IF, as the double length of the cord GE is to the perpendicular height EF, through which the weight E descends. From which it is further inferred that the more oblique the cord is, the smaller is the difference IF, and therefore the smaller is the ratio of descent EF to the ascent of the opposite weight, and consequently the less has it power to prevail. Hence, from the different length and obliquity of the cord, if equilibrium be made, it will be allowable to infer the inequality itself of the weights, regard being had to the motions taken reciprocally; which motions, since they cannot have a multiple ratio greater than the double, as is clear to one considering the bending of the cord itself, neither can weight D be less than weight E, nor greater than it by more than double, if equilibrium is established; but it will be less than double if the cord be oblique, and from the difference of the motions, which would follow from each obliquity of the cord, the difference of the unequal weights itself is also inferred. PROPOSIT

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Mechanicorum Propositio X. Æqualia pondera similis figuræ, sed diversæ substantiæ, similibus & æqualibus pyxidibus inclusa discernere. SInt duo globi, alter ferreus H, alter argenteus S, inclusi æqualibus & similibus pyxidibus AB & CD ita æqualis ponderis, ut pyxides vacuæ librâ examinatæ æquiponderent, & adjecti globi pariter sint æquales ratione ponderis, quamvis moles inæquales sint, major enim est ferreus, minor argenteus. Oporteat igitur discernere, utra pyxis argenteum globum contineat. Singularum pyxidum longitudo bifariam dividatur in F & E, ex quibus punctis fiat suspensio; quâ factâ utique descendent extremitates B & D. Addantur tum in A, tum in C pondera, ut fiat æquilibrium. Pondus majus indicabit ibi esse globum argenteum. Vel si unico æquipondio uti placeat, invento æquilibrio unius pyxidis, idem æquipondium ad alteram pyxidem transferatur: si enim apposita extremitas præponderet, ibi est argentum, si sursum attollatur, ibi est ferrum. Manifesta autem est ratio, quia majoris globi centrum gravitatis propius est medio pyxidis, ex quo fit suspensio, ac propterea minus habet momenti, quàm minor globus, cujus centrum magis distat. Quamvis verò suspensio facta fuerit ex medio, nihil refert, etiamsi ad alterutram extremitatem accedat ut in K, dummodo æqualis assumatur distantia in L; eadem enim semper ratio pro inæqualitate momentorum militat, inæqualis scilicet distantia centrorum gravitatis. At si non ea esset pyxidum longitudo, ut extremitatibus A & C facilè adnectatur æquipondium, assume regulam BZ longiorem ipsâ pyxide, eamque alliga funiculo per K transeunte, & in Z æquipondium statuatur: deinde regulam eandem similiter alliga alteri pyxidi, ut sit DX, & funiculus per L transeat: nam

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Mechanics Proposition X. To distinguish equal weights of similar shape, but of different substance, enclosed in similar & equal boxes. Let there be two spheres, one iron H, the other silver S, enclosed in equal & similar boxes AB & CD, of such equal weight that the empty boxes, weighed on the scale, balance one another, & the added spheres likewise are equal in respect of weight, although their masses are unequal, for the iron one is larger, the silver one smaller. It is therefore necessary to distinguish which box contains the silver sphere. Let the length of each box be divided in two at F & E, from which points let suspension be made; when this is done, the extremities B & D will certainly descend. Let weights then be added at A and at C, until equilibrium is produced. The greater weight will indicate that the silver sphere is there. Or if it should be preferred to use a single counterpoise, when the equilibrium of one box has been found, let the same counterpoise be transferred to the other box: for if the end placed against it should preponderate, silver is there; if it is lifted upward, iron is there. The reason is evident, because the center of gravity of the larger sphere is nearer to the middle of the box, from which the suspension is made, and therefore has less effect than the smaller sphere, whose center is farther away. But although the suspension has been made from the middle, it makes no difference, even if it is moved toward either end as in K, provided an equal distance is taken in L; for the same reason always argues for the inequality of the moments, namely the unequal distance of the centers of gravity. But if the length of the boxes were not such that a counterpoise could easily be attached to the extremities A & C, take a rod BZ longer than the box itself, and fasten it by a cord passing through K, and place the counterpoise at Z: then fasten the same rod in the same way to the other box, so that it is DX, & the cord passes through L: for

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Liber tertius. CAPUT XI. 337 nam idem æquipondium in X si nimis leve sit, indicat ibi ar- gentum esse; id quod pariter indicabit æquipondium majus faciens æquilibrium. Quòd si pondus idem utrobiq[ue] faceret æquilibrium, indicio esset aut inclusa corpora non esse secundùm molem similia, aut si similia fuerint non esse in pyxidibus similiter posita in extre- mitate, contrà hypothesim. Id quod ut deprehendas, ita pyxi- des converte, ut ad latus constituatur pars, quæ priùs erat infi- ma; tunc enim ponderis aliqua diversitas apparebit. Si autem adhuc æquilibrium constituatur, minorem molem ita ex arte collocatam fuisse, ut centrum gravitatis æqualem distantiam habeat à puncto suspensionis, ac moles major in alterâ pyxide, manifestum est. Tunc igitur utraque pyxis intrà aquam pon- deranda est; quæ enim minùs gravis apparebit, continet ar- gentum; hoc quippe minus spatij occupans quàm ferrum, ma- jori aëris moli in pyxide locum relinquit: major autem aëris moles plus deterit ponderis pyxidi intra aquam: pyxidum sci- licet moles ponuntur æquales. CAPUT XI. Fundamenta præmittuntur ad explicandum, cur gravia suspensa modò præponderent, modò æquilibria sint. Locus hic est obstrictam non semel in superioribus fidem liberandi, cùm me ostensurum suscepi in corporibus sus- pensis aliquando minùs gravia gravioribus prævalere, nec ta- men ullum libræ aut Vectis vestigium deprehendi, neque mo- tum propriè circularem tribui posse potentiæ moventi, quæ vi suæ gravitatis juxtà directionis lineam deorsum conatur, atque movetur motu recto, sursum ascendente rectà corpore gravio- re, quod per vim elevatur. Sed ut res tota capite sequenti cla- riùs & breviùs explicari valeat, propositiones aliquot hîc lem- matum loco præmittendæ videntur, & problemata, quibus cer- V u

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Liber tertius. CAPUT XI. 337 for the same counterpoise, if it be too light in the X, it indicates that there is silver there; this will likewise be indicated by making the counterpoise greater and producing equilibrium. But if the same weight should produce equilibrium in both cases, it would be an indication either that the enclosed bodies are not similar in bulk, or, if they are similar, that they are not placed in the boxes in like manner at the extremity, contrary to the hypothesis. To detect this, turn the boxes so that the part which was formerly at the bottom is placed at the side; then some difference of weight will appear. But if equilibrium is still established, it is clear that the smaller bulk was so placed by art that its center of gravity had an equal distance from the point of suspension, as the larger bulk in the other box. Then, therefore, both boxes are to be weighed in water; for the one that appears less heavy contains silver. For since it occupies less space than iron, it leaves room in the box for a greater mass of air: but a greater mass of air lessens the weight of the box more when it is in water; the bulk of the boxes, of course, being equal. CAPUT XI. Foundations are laid down for explaining why suspended heavy bodies sometimes outweigh, and sometimes are in equilibrium. This is the place to discharge the pledge not once given in the preceding pages, when I undertook to show that in suspended bodies the lighter sometimes prevail over the heavier, and yet no trace of the balance or lever is to be detected, nor can a properly circular motion be attributed to the moving power, which by virtue of its own gravity strives downward along the line of direction, and is moved in a straight motion, while the heavier body rises upward in a straight line, being raised by force. But in order that the whole matter may be explained more clearly and briefly in the following chapter, it seems necessary that some propositions should here be set forth in the place of lemmas, and problems, by which cer- V u

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Mechanicorum ta methodus præscribatur, ut pro instituto corpora ipsa graviâ eligantur, atque suis quæque locis disponantur. PROPOSITIO I. Excessus secantis cujuscumque anguli supra Radium, minor est Tangente ejusdem anguli. Sit datus angulus quilibet DBC, ejus Tangens DC, secans BID, & excessus secantis supra Radium DE. Dico DE minorem esse Tangente DC. Ducatur recta CE dato angulo subtensa faciens angulos ad basim æquales ex 5. lib. I. ac proinde acutos: igitur angulus DEC complementum ad duos rectos est obtusus, & maximus in triangulo DEC, ac propterea ex 19. lib. I. maximum latus est, quod illi opponitur, nimirum Tangens DC. PROPOSITIO II. Cujuslibet anguli Tangens est media proportionalis inter excessum secantis supra Radium, & aggregatum ex Radio & secante ejusdem anguli. Datus sit idem angulus DBC, Tangens DC, excessus secantis DE: producatur recta DB usque in A, & est recta DA aggregatum ex Radio BA & secante BD. Dico Tangentem DC esse mediam proportionalem inter ED & DA. Cùm enim ex 36. lib. 3. rectangulum sub ED & DA æquale sit quadrato, quod à Tangente CD describitur, per 17. lib. 6. sunt tres continuè proportionales ED, DC, DA. Hinc sequitur excessum secantis supra Radium ad aggregatum ex Radio & secante habere Rationem duplicatam Rationis,

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The method of mechanics is to be prescribed in such a way that, according to the purpose, the bodies themselves are chosen with gravity, and each is arranged in its proper place. PROPOSITION I. The excess of the secant of any angle above the radius is less than the tangent of the same angle. Let there be given any angle DBC, its tangent DC, secant BID, and the excess of the secant above the radius DE. I say that DE is less than the tangent DC. Draw the straight line CE subtending the given angle, making the angles at the base equal, from Book I, Prop. 5, and therefore acute: thus the angle DEC, being the supplement to two right angles, is obtuse, and the greatest in triangle DEC, and therefore, from Book I, Prop. 19, the greatest side is that which is opposite to it, namely the tangent DC. PROPOSITION II. The tangent of any angle is a mean proportional between the excess of the secant above the radius and the sum of the radius and the secant of the same angle. Let the same angle DBC be given, with tangent DC and excess of the secant DE: produce the straight line DB to A, and then the straight line DA is the sum of the radius BA and the secant BD. I say that the tangent DC is a mean proportional between ED and DA. For since, from Book 3, Prop. 36, the rectangle contained by ED and DA is equal to the square described on the tangent CD, by Book 6, Prop. 17, ED, DC, DA are three continuously proportional quantities. Hence it follows that the excess of the secant above the radius bears to the sum of the radius and the secant the ratio duplicated of the ratio,

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Liber tertius. CAPUT IX. 339 nis, quam idem excessus habet ad Tangentem, hoc est, se ha- bere ut quadratum ED ad quadratum DC, ita ED ad DA; igitur & dividendo ut quadratum ED ad differentiam quadra- torum ED & DC, ita excessus ED ad Radij duplum EA, dif- ferentiam inter ED & DA. PROPOSITIO III. Dato angulo, ad cujus secantis excessum supra Radium suæ Tangens habet Rationem datam, cujuscumque anguli mino- ris Tangens ad excessum suæ secantis habet Rationem majo- rem datâ; cujuscumque autem anguli majoris Tangens ad excessum suæ secantis habet Rationem minorem datâ Ratione. Angulus DBC sit datus, & illius Tangens DC ad DE ex- cessum suæ secantis habeat datam aliquam Rationem. Pri- mò sit minor angulus FBC. Dico ejus Tangentem FC ad suæ secantis excessum FZ habere majorem Rationem quàm DC ad DE. Quia angulus CFB exterior major est interno CDB ex 16. lib.1. fiat huic æqualis angulus CFG, eruntque ex 28.lib.1. parallelæ lineæ DB & FG, & ex 29.lib.1. DBF & GFH al- terni æquales: sunt autem BHE & FHG æquales per 15. lib.1. ut pote ad verticem; ergo & reliquus angulus BEH est reliquo angulo FGH æqualis. Similia itaque sunt triangula, & per 4.lib.6. ut EB ad BH, ita GF ad FH: est autem EB major quàm BH (nam BH minor est Radio BZ, cui æqualis est Radius BE) igitur & GF major est quàm FH; ergo & mul- tò major quàm FZ. Sed quoniam GF & ED sunt parallelæ, & triangula CFG, CDE sunt æquiangula, ex 4.lib.6. eadem est Ratio CF ad FG, quæ est CD ad DE: CF autem ad FG majorem ex 8.lib.5. habet minorem Rationem quàm ad FZ minorem; ergo CF Tangens anguli minoris habet ad FZ excessum suæ secantis supra Radium, Rationem majorem quàm CD ad DE. Secundò sit angulus IBC major dato angulo DBC: Dico illius Tangentem CI ad suæ secantis excessum KI habere mi- V u 2

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Book third. CHAPTER IX. 339 nis, than the same excess has to the Tangent, that is, to be as the square of ED is to the square of DC, so is ED to DA; therefore, by dividing, as the square of ED is to the difference of the squares of ED and DC, so is the excess ED to the double Radius EA, the difference between ED and DA. PROPOSITION III. Given an angle, whose secant’s excess above the Radius has a given ratio, the Tangent of any lesser angle has a greater ratio to the excess of its secant than the given ratio; but the Tangent of any greater angle has a smaller ratio to the excess of its secant than the given ratio. Let the angle DBC be given, and let its Tangent DC to the excess DE of its secant have some given ratio. First let FBC be a lesser angle. I say its Tangent FC to the excess FZ of its secant has a greater ratio than DC to DE. Because the exterior angle CFB is greater than the interior CDB by 16. lib.1., let angle CFG be made equal to this; and by 28. lib.1. the lines DB and FG will be parallel, and by 29. lib.1. DBF and GFH are alternate equal angles: but BHE and FHG are equal by 15. lib.1., as being at the vertex; therefore the remaining angle BEH is equal to the remaining angle FGH. The triangles are therefore similar, and by 4. lib.6. as EB is to BH, so is GF to FH: but EB is greater than BH (for BH is less than the Radius BZ, to which the Radius BE is equal), therefore GF is also greater than FH; therefore much more than FZ. But since GF and ED are parallel, and the triangles CFG, CDE are equiangular, by 4. lib.6. the ratio of CF to FG is the same as that of CD to DE: but CF to FG has, by 8. lib.5., a smaller ratio than to the smaller FZ; therefore the Tangent CF of the lesser angle has, to FZ the excess of its secant above the Radius, a greater ratio than CD has to DE. Secondly, let the angle IBC be greater than the given angle DBC: I say its Tangent CI to the excess KI of its secant has a smaller ratio than the ratio given.

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Mechanicorum 34o norem Rationem, quàm CD ad DE. Quoniam externus angulus CDB major est interno CIB, fiat illi æqualis angulus CIO, & lineæ C E productæ occurrat in O, linea IO, quæ parallela est lineæ BD; & sunt anguli OIL & EBL alterni æquales, quemadmodum & anguli ad verticem in L æquales sunt. Quapropter in triangulis IOL & EBL æquiangulis per 4. lib.6. ut LB ad BE, ita LI ad IO: est autem LB major quàm BE (nam LB major est Radio BK) ergo etiam LI, & multo magis KI major est quàm IO. Sed ut CD ad DE ita CI ad IO; ergo minor est Ratio CI ad IK majorem, quàm sit CI ad IO minorem; ergo est minor Ratio Tangentis CI ad excessum suæ secantis KI, quàm sit Ratio CD ad DE. PROPOSITIO IV. Differentia inter Tangentes duorum quorumlibet angulorum major est, quàm differentia inter eorum secantes. SInt anguli BAC, BAD, eorum Tangentes BC & BD, Squarum differentia CD: angulorum secantes AC & AD, secantium differentia (assumptâ AG æquali ipsi AC) est DG. Dico CD esse majorem quàm DG. Ducatur recta CG, & est triangulum CAG isosceles, ideóque angulus CGA acutus, & qui est illi deinceps, CGD obtusus, & maximus in triangulo CGD: quare per 18. lib. 1. major est CD Tangentium differentia quàm DG secantium differentia. PROPOSITIO V. Ratio differentiæ Tangentium ad differentiam secantium fit semper minor. Esto anguli BAC Tangens BC, anguli BAK Tangens BK; descripto arcu COG, differentia secantium est KO, &

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Mechanicorum 34o a greater Ratio than CD to DE. Since the external angle CDB is greater than the internal CIB, let the angle CIO be made equal to it, and let the line IO, produced from C E, meet in O, a line parallel to BD; and the angles OIL and EBL are alternate equal, just as the angles at the vertex in L are equal. Therefore, in the triangles IOL and EBL, being equiangular, by 4. book 6. as LB to BE, so LI to IO: but LB is greater than BE (for LB is greater than the radius BK), therefore LI also, and much more KI, is greater than IO. But as CD to DE so CI to IO; therefore the Ratio CI to IK, the greater, is less than the Ratio CI to IO, the lesser; therefore the Ratio of the Tangent CI to the excess of its secant KI is less than the Ratio CD to DE. PROPOSITION IV. The difference between the tangents of any two angles whatever is greater than the difference between their secants. Let BAC, BAD be angles, their tangents BC and BD, whose difference is CD: the secants AC and AD of the angles, the difference of the secants (AG assumed equal to AC) is DG. I say that CD is greater than DG. Let the straight line CG be drawn, and the triangle CAG is isosceles, and therefore the angle CGA is acute, and the one next to it, CGD, obtuse, and the greatest in the triangle CGD: wherefore by 18. book 1. the difference of the Tangents CD is greater than the difference of the secants DG. PROPOSITION V. The ratio of the difference of the Tangents to the difference of the secants is always less. Let the angle BAC have Tangent BC, the angle BAK have Tangent BK; the arc COG being described, the difference of the secants is KO, and

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Liber tertius. CAPUT XI. 341 & Tangentium differentia est CK. Item anguli BAD Tangens BD, & descripto arcu KH, differentia Tangentium BK & BD est KD, atque secantium AK & AD differentia est HD. Dico majorem Rationem esse CK ad KO, quàm KD ad DH. Ducantur rectæ CG & KH. In triangulis iso- scelibus CAG & KAH, anguli ad basim CG minores sunt angulis ad basim KH, quia angulus CAG major est angulo KAH: quapropter angulo CGA fiat æqualis angulus IHA. Cum itaque ex 28. lib. 1. I H' & CG sint parallelæ, per 2. lib. 6. ut CI ad HG, hoc est ad KO, ita ID ad DH: atqui CK major est quàm CI; ergo major est Ratio CK ad KO quàm CI ad KO ex 8. lib. 5. hoc est quàm ID ad DH. Sed ID est major quàm KD; ergo per 8. lib. 5. major est Ratio ID ad DH, quàm KD ad DH; ergo multò major est Ratio CK ad KO, quàm KD ad DH. Idem de cæteris consequentibus angulis nec dissimili methodo demonstrari poterit, minorem scilicet fieri Rationem differentiæ Tangentium ad differentiam secantium. PROPOSITIO VI. Dato Radio, & datâ Ratione Tangentis ad excessum secantis, invenire Tangentem & secantem, earumque angulum. DAtus Radius sit B, data Ratio Tangentis ad excessum secantis suprà Radium sit R ad S. Oportet Tangentem ipsam atque secantem invenire. Tangens esto A: ut R ad S ita A ad A in S R excessum secantis supra Radium; igitur secans integra est B + A in S R; hujus quadratum est B quad. + 2B in A in S + A quad. in S quad. quod R R quadr. ex 47. lib. 1. æquale est quadratis Radij & Tangentis simul, hoc est B quad. + A quad. Utrinque dempto B quad. tum omnibus per A divisis, deinde omnibus ductis per R quad. V u 3

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Book Three. Chapter XI. 341 & the difference of the tangents is CK. Likewise, the tangent of angle BAD, BD, and an arc KH being described, the difference of the tangents BK & BD is KD, and the difference of the secants AK & AD is HD. I say that the greater ratio is CK to KO, than KD to DH. Let the straight lines CG & KH be drawn. In the isosceles triangles CAG & KAH, the angles at the base of CG are less than the angles at the base of KH, because angle CAG is greater than angle KAH: wherefore let angle IHA be made equal to angle CGA. Since therefore, by 28. of book 1, I H' & CG are parallel, by 2. of book 6, as CI is to HG, that is to KO, so is ID to DH: but CK is greater than CI; therefore the ratio of CK to KO is greater than CI to KO by 8. of book 5, that is, than ID to DH. But ID is greater than KD; therefore by 8. of book 5, the ratio of ID to DH is greater than KD to DH; therefore much greater is the ratio of CK to KO than KD to DH. The same, concerning the other consequent angles, and by a similar method, can be demonstrated, namely, that the ratio of the difference of the tangents to the difference of the secants becomes smaller. PROPOSITION VI. Given the radius, and the ratio of the tangent to the excess of the secant, to find the tangent and secant, and their angle. Let the given radius be B, and the given ratio of the tangent to the excess of the secant above the radius be R to S. It is necessary to find the tangent itself and the secant. Let the tangent be A: as R is to S, so let A be to A in S R, the excess of the secant above the radius; therefore the whole secant is B + A in S R; its square is B quad. + 2B in A in S + A quad. in S quad. which R R quadr. by 47. book 1, is equal to the squares of the radius & tangent together, that is B quad. + A quad. After B quad. is taken away from both sides, then after all are divided by A, then after all are multiplied by R quad. V u 3

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Mechanicorum 342 demum factâ Antithesi A in R quad. — A in S quad. æquatur 2 S in B in R. Quare revocatâ ad Analogiam æquatione, est ut R quad. — S quad. ad 2 S in R, ita B Radius ad A Tangentem quæsitam. Tum fiat ut R ad S ita A inventa ad aliud, & erit excessus secantis, qui additus Radio B dabit quæsitam secantem. Sit R 3, S 2: horum quadratorum 9 & 4 differentia est 5; duplum rectangulum sub R & S est 12. Igitur ut 5 ad 12, ita B Radius 100000 ad 240000 Tangentem gr. 67. 22. 48". Iterum ut 3 ad 2 ita 240000 ad 160000 excessum secantis; Igitur ad- dito Radio, Secans quæsita est 260000; quæ etiam in Canone responderet eidem angulo. Itaque generatim loquendo, fiat ut differentia inter quadra- ta terminorum datæ Rationis ad rectangulum bis sub iisdem terminis comprehensum, ita datus Radius ad aliud, & prove- niet Tangens quæsita; quæ habita facilè dabit secantis exces- sum in Ratione datâ. Quòd si rem Geometricè perficere velis, circâ majorem Ra- tionis datæ terminum R describe semicirculum, & in eo ac- commoda minorem Rationis terminum S; nam linea T dabit quadratum, quod est differentia quadratorum ex R & ex S, ut est manifestum ex eo, quod angulus in semicirculo est rectus per 31.lib.3.& ex 47 lib.1. quadratum unius lateris circa rectum est differentia quadratorum hypothenusæ & reliqui lateris. Deinde inter alterutrum terminorum duplicatum, & reliquum terminum quære mediam proportionalem, & sit V potens qua- dratum æquale duplo rectangulo sub terminis datis. Quoniam verò ex 20.lib.6. quadrata sunt in duplicatâ Ratione laterum, & T quadratum ad V quadratum est in duplicatâ Ratione T ad V; inveniatur tertia proportionalis X. Demum ut T ad X ita fiat B ad Z, quæ est quæsita Tangens, & ad angulum rectum constituta cum Radio B dabit hypothenusam secantem quæsi- tam, quæ cum Radio constituet quæsitum angulum. Vel etiam ex corollario prop.2. fiat ut differentia quadrato- rum ex R & ex S ad S quadratum, ita duplum Radij B ad exces- sum secantis: deinde hic excessus inventus ad Tangentem quæsitam fiat ut S ad R; & summa ex dato Radio atque exces- su invento dabit quæsitam secantem. P R O P O S I

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Mechanics 342 finally, after the antithesis has been made, A in R squared — A in S squared is equal to 2 S in B in R. Wherefore, the equation being brought back to the analogy, it is as R squared — S squared to 2 S in R, so is B the Radius to A the tangent sought. Then let it be as R to S, so A found to something else, and there will be the excess of the secant, which, added to the Radius B, will give the secant sought. Let R be 3, S 2: the difference of their squares, 9 and 4, is 5; the double rectangle under R and S is 12. Therefore as 5 to 12, so B Radius 100000 to 240000, tangent 67. 22. 48". Again as 3 to 2, so 240000 to 160000 the excess of the secant; therefore, with the Radius added, the secant sought is 260000; which would also correspond in the Canon to the same angle. Thus, speaking generally, let the difference between the squares of the terms of the given ratio be to the rectangle twice comprehended under the same terms, so let the given Radius be to something else, and the tangent sought will result; and this, when found, will easily give the excess of the secant in the given ratio. But if you wish to complete the matter geometrically, about the greater term R of the given ratio describe a semicircle, and in it fit the lesser term S of the ratio; for the line T will give the square, which is the difference of the squares from R and from S, as is manifest from the fact that the angle in a semicircle is right, by 31, lib. 3, and from 47, lib. 1, the square of one side about the right angle is the difference of the squares of the hypotenuse and the remaining side. Then, between either of the terms doubled and the remaining term, seek a mean proportional, and let V be the square power equal to the double rectangle under the given terms. But since from 20, lib. 6, squares are in the doubled ratio of the sides, and the square T to the square V is in the doubled ratio of T to V, let a third proportional X be found. Finally, as T to X, so let B be to Z, which is the tangent sought; and, placed at a right angle with the Radius B, it will give the sought secant, the hypotenuse, which with the Radius will constitute the sought angle. Or also, from corollary prop. 2, let the difference of the squares from R and from S be to the square of S, so let double the Radius B be to the excess of the secant: then this excess found, to the tangent sought, let it be as S to R; and the sum of the given Radius and the excess found will give the secant sought. P R O P O S I

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Liber tertius. CAPUT XI. 343 PROPOSITIO VII. Datâ Tangente communi duorum circulorum inæqualium, & datis Rationibus excessum Secantium ad eandem Tangentem, invenire Circulorum Radios. S It super lineam CD indefinitam erecta ad perpendiculum Srecta AB, quam in B oporteat tangere duos circulos inæquales, ita ut sit Tangens duorum angulorum inæqualium, excessus autem secantis unius sit ad datam Tangentem ut E ad G, alterius verò secantis excessus sit ad eandem ut F ad G: & hujusmodi circulorum semidiametros invenire oporteat. Fiat ut G ad E ita AB data ad H; & ut H ad AB ita AB ad MS, ex quâ dematur MO ipsi H æqualis, reliquæ OS semissi RS æqualis sumatur BD pro Radio circuli BL. Item fiat ut G ad F ita AB data ad I; & ut I ad AB ita AB ad NT, ex quâ dematur NP æqualis ipsi I, & reliquæ PT semissi VT æqualis statuatur BC semidiameter circuli BK. Junctis CA, & DA erunt excessus secantium suprà suos Radios ad Tangentem, videlicet KA & LA ad AB in datis Rationibus. Quia enim recta TP secta est bifariam in V, & adjecta est illi PN, per 6. lib. 2. quadratum NV est æquale quadrato VT (hoc est quadrato CB) unà cum rectangulo TNP: huic autem rectangulo, ex 17. lib. 6. æquale est quadratum AB, quæ ex constructione est media proportionalis inter PN, hoc est I, & NT. At iisdem quadratis CB & BA simul sumptis æquale est

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Liber tertius. CAPUT XI. 343 PROPOSITION VII. Given a common tangent of two unequal circles, and the ratios of the excesses of the secants to the same tangent, to find the radii of the circles. Let there be erected upon the indefinite line CD, at right angles, the straight line AB, which in B ought to touch two unequal circles, so that it may be the tangent of two unequal angles; and let the excess of one secant above the given tangent be as E to G, and the excess of the other secant above the same as F to G: and it is required to find the semidiameters of such circles. Let it be as G to E so AB, given, to H; and as H to AB so AB to MS, from which let MO, equal to H itself, be taken away; let the remaining OS, equal to the half RS, be taken for BD, the radius of circle BL. Likewise let it be as G to F so AB, given, to I; and as I to AB so AB to NT, from which let NP, equal to I itself, be taken away, and let the remaining PT, equal to the half VT, be set down for BC, the semidiameter of circle BK. Then, joining CA and DA, KA and LA will be the excesses of the secants above their respective radii to the tangent, namely in the given ratios as KA and LA to AB. For since the straight line TP is cut in half at V, and PN is added to it, by 6. lib. 2. the square of NV is equal to the square of VT (that is, the square of CB) together with the rectangle TNP: but to this rectangle, by 17. lib. 6., the square of AB is equal, which, by the construction, is the mean proportional between PN, that is I, and NT. But to the same squares CB and BA taken together is equal

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Mechanicorum 344 est quadratum CA ex 47. lib. 1. igitur quadratum CA æquatur quadrato NV, & linea CA æqualis est lineæ NV. Sunt autem VP & CK æquales (nam & æquales sunt lineis VT & CB) ergo etiam KA reliqua æqualis est reliquæ PN, hoc est I. Cum itaque I ad AB sit ut F ad G ex constructione, etiam KA ad AB est in eâdem datâ Ratione F ad G. Nec dissimili methodo utendum erit ad ostendendum LA ad AB esse in datâ Ratione E ad G: id quod indicasse sufficiat, nec pluribus est opus. Quare CB & DB sunt quæsitorum circulorum semidiametri. PROPOSITIO VIII. Datis duobus inæqualibus circulis se contingentibus in B, datisque eorum Radiis CB & DB, invenire Tangentem communem BA, ad quam secantium excessus habeant datas Rationes E ad G, & F ad G. O Portet secantis excessum, qui ad Tangentem habet majorem Rationem, quàm alter excessus; pertinere ad minorem circulum; qui verò minorem Rationem habet, pertinere ad majorem circulum. Cum enim rectangula sub excessibus & aggregatis suarum secantium suorumque Radiorum sint inter se æqualia, ut pote ex 36. lib. 3. eidem Tangentis quadrato æqualia, erit per 16. lib. 6. ut excessus secantis majoris circuli ad excessum minoris, ita aggregatum ex secante & Radio minoris ad aggregatum ex secante & Radio majoris. Sicut ergo eadem Tangens habet majorem Rationem ad Radium minoris circuli quàm ad Radium majoris, subtenditque majorem angulum in circulo minori quàm in majori; ita suæ secantis excessus habet majorem Rationem ad eandem Tangentem, quàm excessus secantis minoris anguli in circulo majori. Sit itaque major Ratio F ad G quàm E ad G, & pertinebit ad circulum minorem. Fiat ut F ad G ita G ad QX, ex quâ dematur QZ æqualis ipsi F. Tum fiat ut XZ ad ZQ, ita minoris Radij duplum TP ad PN: & inter PN & NT inveniatur media proportionalis BA, quam ex B ad perpendiculum erectam

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Mechanics 344 is the square CA from 47. book 1; therefore the square CA is equal to the square NV, and line CA is equal to line NV. But VP and CK are equal (for they are also equal to the lines VT and CB); therefore also KA, the remainder, is equal to PN, the remainder, that is, I. Since therefore I to AB is as F to G by construction, KA also to AB is in the same given ratio of F to G. Nor will a dissimilar method need to be used to show LA to AB to be in the given ratio E to G: it is enough to have indicated this, and there is no need for more. Wherefore CB and DB are the semidiameters of the sought circles. PROPOSITION VIII. Given two unequal circles touching at B, and given their radii CB and DB, to find the common tangent BA, to which the excesses of the secants have the given ratios E to G and F to G. It is necessary that the excess of the secant which has the greater ratio to the tangent should belong to the smaller circle; but the one which has the smaller ratio should belong to the greater circle. For since the rectangles under the excesses and the sums of their secants and their radii are equal to one another, as being equal to the same square of the tangent from 36th of book 3, it will be, by 16th of book 6, that as the excess of the secant of the greater circle is to the excess of the smaller, so is the sum of the secant and radius of the smaller to the sum of the secant and radius of the greater. Therefore, just as the same tangent has a greater ratio to the radius of the smaller circle than to the radius of the greater, and subtends a greater angle in the smaller circle than in the greater; so the excess of its secant has a greater ratio to the same tangent than the excess of the secant of the smaller angle in the greater circle. Let the greater ratio F to G therefore be greater than E to G, and it will belong to the smaller circle. Let it be as F to G, so G to QX; from which subtract QZ equal to F itself. Then let it be as XZ to ZQ, so the double TP of the smaller radius to PN: and between PN and NT let there be found a mean proportional BA, which from B erected perpendicularly

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Liber tertius. CAPUT XI. 345 crectam jungat cum centro C rectâ CA : nam KA ad Tan- gentem AB habet datam Rationem F ad G. Cùm enim ea- dem AB, quæ ex constructione est media inter PN & NT, sit etiam ex 36. lib.3. & 17. lib.6. Media inter KA & ACB, & extremarum NT & ACB excessus supra sibi respondentes extremas PN & KA sint ex constructione æquales (sunt scili- cet PT & KCB duplum Radij CB) etiam ipsæ extremæ sunt æquales, nimirum NT æqualis ipsi ACB, & PN, æqualis KA. Atqui ut XZ ad ZQ, ita ex constructione TP ad PN, & componendo atque convertendo ut ZQ ad QX ita PN ad NT; ergo etiam ut ZQ ad QX ita KA ad ACB. Quare si- cuti ZQ ad QX est duplicata Rationis F ad G ex constructio- ne, etiam KA ad ACB est ejusdem Rationis F ad G duplica- ta; ergo KA ad mediam AB, hoc est Excessus secantis ad Tan- gentem, est ut F ad G. Eâdem methodo fiat ut E ad G ita G ad Ya, ex quâ dema- tur Yb æqualis ipsi E : & fiat ut ab ad bY, ita Radij majoris BD duplum SO ad OM; atque inter OM & MS erit media proportionalis eadem AB: similique ratiocinatione ostendetur excessum LA ad Tangentem AB esse in datâ Ratione E ad G. Ut in praxim res faciliùs deduci queat, exemplo illustretur. Sit Radius minor CD 12, F ad G ut 16 ad 35: inveniatur his tertia proportionalis QX 76 2/16. Dematur F 16, remanet XZ 60 2/16. Fiat ut 60 2/16 ad 16, ita Radij duplum TP 24 ad PN 6 2/5 proximè. Est ergo NT 30 2/5. Inter 6 2/5 & 30 2/5 me- dia est 14. Item sit Radius major BD 18, E ad G ut 12 ad 35: inve- niatur his tertia proportionalis Ya 102 1/1, & auferatur E 12, remanet ab 90 1/5. Fiat ut 90 1/5 ad 12, ita Radij duplum SO 36 ad OM 4 2/5. Est ergo MS 40 2/5. Inter 4 2/5 & 40 2/5 est media proportionalis 14: in his autem exemplis neglectæ sunt fractiunculæ. xx

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Liber tertius. CAPUT XI. 345 Erecting it, join with the center C the straight line CA: for KA has, to the tangent AB, the given ratio F to G. For since the same AB, which by construction is the mean between PN and NT, is also, by 36. lib. 3. and 17. lib. 6., the mean between KA and ACB, and since the excesses of the extremes NT and ACB above the corresponding extremes PN and KA are by construction equal (namely PT and KCB are the double of the radius CB), the extremes themselves are also equal, namely NT equal to ACB, and PN equal to KA. But as XZ is to ZQ, so by construction TP is to PN, and by composition and inversion, as ZQ is to QX, so is PN to NT; therefore also as ZQ is to QX, so is KA to ACB. Wherefore, as ZQ is to QX is the duplicate of the ratio F to G by construction, so also KA to ACB is the duplicate of the same ratio F to G; therefore KA to the mean AB, that is, the excess of the secant over the tangent, is as F to G. By the same method let it be made that E is to G as G is to Ya, from which let Yb be taken away equal to E: and let it be made as ab is to bY, so is the double of the greater radius BD, SO, to OM; and between OM and MS there will be the same mean proportional AB: and by a like reasoning it will be shown that the excess LA to the tangent AB is in the given ratio E to G. That the matter may be more easily reduced to practice, let it be illustrated by example. Let the lesser radius CD be 12, and F to G as 16 to 35: find the third proportional to these, QX 76 2/16. Let F 16 be subtracted, there remains XZ 60 2/16. Let it be as 60 2/16 to 16, so is the double of the radius TP 24 to PN 6 2/5 approximately. Therefore NT is 30 2/5. Between 6 2/5 and 30 2/5 the mean proportional is 14. Likewise let the greater radius BD be 18, and E to G as 12 to 35: find the third proportional to these, Ya 102 1/1, and let E 12 be taken away, there remains ab 90 1/5. Let it be as 90 1/5 to 12, so is the double of the radius SO 36 to OM 4 2/5. Therefore MS is 40 2/5. Between 4 2/5 and 40 2/5 the mean proportional is 14: in these examples the fractional parts have been neglected. xx

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Mechanicorum PROPOSITIO IX. Si duorum circulorum se exterius contingentium centra jungat recta linea, & ab unius centro ad alterius convexam peripheriam rectæ ducantur, subtensa arcus abscissi major est quam differentia linearum angulum in illo centro constituentium. Duorum circulorum centra sint A & B, qui se tangant in C, & jungat centra recta AB. Ex centro A in alterius convexam peripheriam ducatur recta AD abscindens arcum CD. Dico linearum AD & AC angulum in centro A constituentium differentiam ED minorem esse subtensâ CD. Quia ex 20. lib. 1. duæ lineæ AC & CD simul majores sunt rectâ AD; auferantur AC & AE æquales, remanet CD major quam ED. Simili ratione CI major est quam IF. & si sumatur angulus IAD, etiam ID major est quam DH differentia inter AI & AD, quia in triangulo AID duo latera AI & ID majora sunt reliquo DA, demptisque æqualibus AI & AH remanet ID major quam DH. PROPOSITIO X. Si duo circuli se exterius contingant, & in uno æquales arcus sumantur, ad quorum extremitates ducantur rectæ à centro alterius circuli; differentia sinuum arcûs simpli & dupli ad differentiam Excessuum harum rectarum supra suum Radium habet minorem Rationem, quam sinus arcûs simpli ad Excessum lineæ ad ipsum ductæ. SInt duo circuli, quorum centra A & B, se contingentes in C, sumantur æquales arcus CI & ID, ad quos ex centro B ducantur

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Mechanics PROPOSITION IX. If the centers of two circles touching each other externally are joined by a straight line, and straight lines are drawn from the center of one to the convex periphery of the other, the chord of the cut-off arc is greater than the difference of the lines making the angle at that center. Let the centers of two circles be A and B, which touch at C, and let the centers be joined by the straight line AB. From center A, let straight line AD be drawn to the convex periphery of the other circle, cutting off arc CD. I say that the difference ED of the lines AD and AC forming the angle at center A is less than the chord CD. For, by Book 1, Proposition 20, the two lines AC and CD together are greater than the straight line AD; let AC and AE, equal, be taken away, there remains CD greater than ED. By a similar reason CI is greater than IF. And if angle IAD be taken, ID is also greater than DH, the difference between AI and AD, because in triangle AID the two sides AI and ID are greater than the remaining side DA, and if the equal parts AI and AH are taken away, ID remains greater than DH. PROPOSITION X. If two circles touch each other externally, and equal arcs are taken in one of them, to whose extremities straight lines are drawn from the center of the other circle; the difference between the sines of the simple arc and the double arc has a smaller ratio to the difference of the excesses of these lines above their radius than the sine of the simple arc has to the excess of the line drawn to it. Let there be two circles, whose centers are A and B, touching each other at C; let equal arcs CI and ID be taken, to whose extremities let straight lines be drawn from center B

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Liber tertius. CAPUT XI. 347 ducantur rectæ BI & BD secantes circulum CE in F & E: Radium B excedunt excessibus FI & ED, qui ex 8. lib.3. in- æquales sunt, & ma- jor est ED quàm FI differentiâ KD. Ar- cium subtenæ CI & ID æquales sunt, sinuum IH & DG differentia est LD. Dico majorem Ra- tionem esse HI ad IF, quàm LD ad DK. Primò ducantur rectæ EF, KI: EF autem producatur ita, ut occurrat rectæ DI productæ in O. Quia triangula BEE & BIK sunt isoscelia, & angulus BEF æqualis est angulo BKI, rectæ EO & KI ex 28. lib.1. sunt parallelæ: igitur ex 2 lib.6. in triangulo DOE ut DI ad IO, ita DK ad KE: Atqui DI major est quàm IO, ergo etiam DK major quàm KE. Proba- tur autem DI majorem esse quàm IO, quia DI æqualis est ipsi CI ex hypothesi; punctum verò O est extra circulum CE, quem linea EFO secat: ergo linea EF producta occurrit li- neæ IC citrà punctum C in S. Sed quoniam angulus BEF est acutus, qui est illi deinceps DEO est obtusus; ergo per 16. lib.1. externus DOS multo magis est obtusus: ergo per 19 lib.1. major est IS quàm IO, ergo multò major est IC quàm IO, hoc est ID major est quàm IO. Deinde angulus MCI major est angulo NID, majori enim arcui MDI ille insistit, hic autem minori ND ex 33. lib:6: triangula verò HIC & LDI rectangula æquales habent hy- pothenusas, hoc est Radios CI & ID, ergo majoris anguli HCI major est sinus HI; minoris verò anguli LID minor est sinus LD. Igitur ex 8. lib.5. HI major ad KE, hoc est ad IF, habet majorem Rationem quàm ad eandem KE habeat LD minor: & eadem LD habet minorem Rationem ad DK ma- jorem quàm ad KE minorem: Ergo HI ad IF majorem habet Rationem, quàm LD ad DK. xx 2

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Liber tertius. CHAPTER XI. 347 Let the straight lines BI and BD be drawn, cutting the circle CE in F and E: they exceed the radius B by the excesses FI and ED, which from 8, book 3, are unequal, and ED is greater than FI by the difference KD. The chords CI and ID are equal, and the difference of the sines IH and DG is LD. I say that the ratio HI to IF is greater than LD to DK. First, let the straight lines EF and KI be drawn: and let EF be produced so that it may meet the straight line DI produced in O. Because the triangles BEE and BIK are isosceles, and the angle BEF is equal to the angle BKI, the straight lines EO and KI are parallel, by 28, book 1: therefore, from 2, book 6, in triangle DOE, as DI is to IO, so is DK to KE: but DI is greater than IO, therefore DK is also greater than KE. Now DI is proved to be greater than IO, because DI is equal to CI itself by the hypothesis; but the point O is outside the circle CE, which the line EFO cuts: therefore the produced line EF meets the line IC on this side of the point C, at S. But since the angle BEF is acute, that which is next to it, DEO, is obtuse; therefore by 16, book 1, the external angle DOS is much more obtuse: therefore by 19, book 1, IS is greater than IO, therefore much more IC is greater than IO, that is, ID is greater than IO. Next, the angle MCI is greater than the angle NID, for the former rests on the greater arc MDI, but the latter on the smaller ND, by 33, book 6: and the right-angled triangles HIC and LDI have equal hypotenuses, that is, the radii CI and ID; therefore, for the greater angle HCI the sine HI is greater; and for the lesser angle LID the sine LD is lesser. Thus, from 8, book 5, HI has a greater ratio to KE, that is to IF, than LD lesser has to the same KE; and the same LD has a lesser ratio to the greater DK than to the lesser KE: therefore HI has a greater ratio to IF than LD has to DK. xx 2

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Mechanicorum CAPUT XII. Præponderatio & Æquilibritas gravium fune suspensorum consideratur. Propositum est lib. 1. capit. 5. Experimentum, cujus hîc symptomata explicanda, causam afferendo omninò conso- nam iis, quæ sæpiùs inculcata sunt. Funiculi extremitatibus al- ligantur pondera prorsùs æqualia; tùm claviculis duobus à se invicem aliquo intervallo disjunctis, sed in eâdem horizontali lineâ constitutis (exquisitè tamen, quoad ejus fieri poterit ro- tundis atque politis, ne suâ asperitate motui impedimento sint) funiculus imponitur. Deinde tertium pondus assumitur duo- bus illis simul acceptis levius, aut singulis illis æquale, aut etiam illis minus, & funiculo inter utrumque claviculum adnectitur: hoc sibi dimissum ita duobus illis ponderibus, quæ ob gravita- tis æqualitatem sibi mutuo nisu obsistebant, ne moverentur, prævalet, ut ipsum descendens vi suæ gravitatis cogat utrum- que illud ascendere. Id quod admiratione carere non potest, cum duo majora pondera, suum æqualem conatum singula vi- cissim elidentia, conjunctis viribus minori gravitati præstare non valeant. Funiculo C A B D jungantur æquales gravitates C & D ex claviculis A & B pendentes, quæ æqualiter deorsum conniten- tes, sibique æqualiter repugnantes ne ascendant, quiescunt. Adnecta- tur in E pondus: huic etiamsi mi- nori illæ gravitates C & D omnino obsistere non possunt, quin ex E descendat in F ex. gr. & funicu- lum trahens cogat illas ascendere C quidem in I, D verò in K. Qua- propter funiculo EBD æqualis est funiculus FBK, & funicu- lo EAC æqualis est funiculus FAI: cum autem rectæ BE & BG æquales sint (nam centro B, intervallo BE descriptus est arcus)

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Mechanics CHAPTER XII. The preponderance and equilibrium of heavy bodies suspended by a rope are considered. The experiment proposed in book 1, chapter 5, whose symptoms are here to be explained, I account for by giving a cause altogether in agreement with those things which have been more often repeated. Equal weights are attached to the ends of a rope; then the rope is placed over two pegs separated from one another by some interval, but situated in the same horizontal line (yet the pegs should be made as nearly round and smooth as possible, so that their roughness may not hinder the motion). Next a third weight is taken, lighter than the two together, or equal to each of them, or even less than them, and is attached to the rope between the two pegs. Left to itself, this weight prevails over the two others, which, because of the equality of their gravity, resist one another with equal effort so that they do not move; so that, descending by the force of its own gravity, it compels both of them to ascend. This cannot fail to arouse wonder, since the two greater weights, each in turn destroying the equal effort of the other, are not able by their combined forces to overcome the lesser gravity. Let equal weights C and D, hanging from pegs A and B, be joined by the rope C A B D; since they are pulling equally downward and resisting one another equally, they remain at rest, not ascending. Let a weight be attached at E: although these lesser gravities C and D cannot entirely resist it, they cannot prevent it from descending from E to F, for example, and, pulling the rope, force them to ascend, C indeed to I, and D to K. Therefore the rope EBD is equal to the rope FBK, and the rope EAC is equal to the rope FAI: now since the straight lines BE and BG are equal (for the arc is described with center B and radius BE)

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Liber tertius. CAPUT XII. 349 arcus) his ablatis, BD æquatur ipsi FG plus BK; & demptâ communi BK, remanet GF æqualis ipsi DK. Eâdem ratione HF ostenditur æqualis ipsi CI. Est igitur mensura motûs pon- derum C & D ascendentium HFG, ponderis verò intermedij descendentis EF. At ex prop. 1. capitis superioris Tangens EF major est secantis BF excessu GF, item secantis A F excessu HF: contingit autem aliquam Tangentem majorem esse utro- que excessu simul sumpto: potest igitur gravitas minor velociùs descendens præstare utrique ponderi tardiùs ascendenti. Quamdiu itaque spatium descendentis per Tangentem ma- jus est spatio ascendentium, quod metitur excessus secantium, ita ut Ratio motûs descendentis ad motum ascendentium major esse possit Ratione, quam habent pondera extrema ad pondus, intermedium; hoc minore illa majora præponderantur. Ubi ve- rò eò ventum sit, ut jam neutra Ratio alteri præstet, tunc pon- dera subsistunt, & quies est. Si demùm ponderi intermedio pondus addatur, vel vis aliqua inferatur ponderis vicem subiens, utique adhuc descendit, quia Ratio ponderum extremorum ad pondus intermedium auctum facta est minor; sed sublato hoc ponderis additamento, illa extrema majorem habent Rationem ad pondus intermedium, quàm possit esse motuum reciprocè sumptorum Ratio; ac proinde, illa descendentia hoc tantisper elevant, dum fiat Rationum æqualitas. Non est autem hîc opus ea, quæ uberiùs superiore libro ex- plicata sunt, replicare, videlicet, gravium resistentiam, ne mo- veantur, non esse attendendam penè ipsam gravitatem dum- taxat, verùm etiam motûs, qui situm ipsum atque positionem consequeretur, velocitate aut tarditate dimetiendam; hanc ve- rò unius tarditatem cum alterius velocitate comparari non pos- se nisi ex longitudine spatiorum, quæ utrumque eodem tempo- ris intervallo percurreret. Ex quo manifestâ consequutione con- ficitur satis esse, si spatiorum inæqualitas aut æqualitas ostenda- tur; ut præponderatio aut æquilibritas innotescat: ac propterea satis est hîc secantium excessus cum Tangente comparare; hæc enim ponderis intermedij, illi ponderum extremorum motum definiunt. Quapropter animum in rem ipsam attentiùs intendentes ob- servamus descendentis ponderis intermedij funiculum B.F.A. X x 3

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Book Third. CHAPTER XII. 349 (arcs) these being removed, BD is equal to FG itself plus BK; and BK being subtracted in common, there remains GF equal to DK. By the same reasoning HF is shown equal to CI. Therefore the measure of the motion of the ascending weights C and D is HFG, and of the intermediate descending weight EF. But from Proposition 1 of the preceding chapter, the tangent EF is greater than the secant BF by the excess GF, likewise the secant AF by the excess HF: and yet it may happen that some tangent is greater than both excesses taken together; therefore a smaller weight descending more swiftly may counterbalance each of the two weights ascending more slowly. So long therefore as the space of the descending weight through the tangent is greater than the space of the ascending weights, which is measured by the excess of the secants, so that the ratio of the motion of the descending weight to the motion of the ascending weights may be greater than the ratio which the extreme weights bear to the intermediate weight, the greater are outweighed by the lesser. But when it has come to this point, that now neither ratio exceeds the other, then the weights come to rest, and there is equilibrium. If finally some weight be added to the intermediate weight, or some force be applied taking the place of a weight, it certainly still descends, because the ratio of the extreme weights to the augmented intermediate weight has become less; but when this addition to the weight is removed, those extremes have a greater ratio to the intermediate weight than can be the ratio of the motions taken reciprocally; and therefore those descending weights raise this one for a little while, until an equality of ratios is reached. Nor is it here necessary to repeat those things which were explained more fully in the preceding book, namely that, in the resistance of heavy bodies against being moved, not only the weight itself is to be considered, but also the motion which would follow the very situation and position, to be measured by swiftness or slowness; and that the slowness of one cannot be compared with the swiftness of another except from the length of the spaces which each would run through in the same interval of time. From which it is clearly inferred that it is enough if the inequality or equality of the spaces be shown, so that preponderance or equilibrium may be known; and therefore it is enough here to compare the excesses of the secants with the tangent, for these define the motion of the intermediate weight, and those the motion of the extreme weights. Wherefore, directing our attention more closely to the matter itself, we observe the cord B.F.A. of the descending intermediate weight

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Mechanicorum cum horizontali lineâ B A angulos constituere ad B & A pri mùm quidem acutissimos, deinde majores & majores; ac propterea Tangentis ad Excessum secantis Rationem semper minui ex propos. 3. ideóque tandem ad eam deveniri Rationem, quæ non sit major Ratione ponderum reciprocè sumptorum. Quid igitur mirum, si tandem fiat quies, ubi non est Ratio- num inæqualitas? Vicissim autem quia ponderum certa est Ra- tio; certa est etiam Ratio Tangentis ad Excessum secantis certi cujusdam anguli; igitur ex eâdem prop. 3. minoris anguli Tan- gens ad Excessum suæ secantis majorem habet Rationem, quàm sit Ratio ponderum reciprocè: ideóque pondus in E constitutum positionem habens, ex quâ aliquis major motus deorsum consequi potest, quàm ascendant extrema pondera, descendit, & superat eorum resistentiam. Sed quoniam suppositâ extre- mis ponderibus manu ita elevare ea possumus, ut pondus inter- medium descendens funiculumque intendens constituat ad B & A angulos, quorum communis Tangens E F habeat ad Ex- cessum secantium H F G Rationem minorem, quàm sit reci- procè Ratio ponderum extremorum ad pondus intermedium, fatis constat, cur illa extrema præponderent, cùm & plus gra- vitatis & majora momenta, hoc est propensionem ad majorem motum, obtineant. Quamvis enim ex prop. 4. differentia inter Tangentes duorum in eodem circulo arcuum inæqualium ma- jor semper sit differentia, quæ inter eorumdem secantes inter- cedit; quia tamen ex prop. 5. Ratio hæc semper sit minor, quò anguli augentur, idcircò si Tangens sit duobus circulis com- munis, fieri potest, ut utriusque circuli secantium differentiæ simul sumptæ majores sint ipsâ Tangente, vel saltem Tangens ad illas simul sumptas eam habeat Rationem, quæ minor sit Ra- tione ponderum reciprocè. Et ut veritas exemplis ante omnium oculos posita nullum du- bitationi locum relinquat, data sit Ratio extremorum ponde- rum ad pondus intermedium, & inquiratur Tangens similem Rationem habens ad utriusque secantis Excessum: intelligatur autem hîc facilitatis gratiâ punctum E omninò æqualiter distans ab A & B ita, ut æquales etiam sint secantium excessus H F & G F. Et primò quidem ponatur pondus medium æqua- le singulis extremis. Est igitur quæsita Ratio dupla Tangentis E F

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Mechanics when they make angles with the horizontal line B A at B and A, first indeed very acute, then greater and greater; and therefore, by proposition 3, the ratio of the tangent to the excess of the secant is always diminished, and so at length it comes to that ratio which is not greater than the ratio of the weights reciprocally taken. Why then should it be surprising if rest finally occurs, where there is no inequality of ratios? Conversely, since the ratio of the weights is fixed, the ratio also of the tangent to the excess of the secant of a certain angle is fixed; therefore, by the same proposition 3, the tangent of the smaller angle has to the excess of its secant a greater ratio than is the ratio of the weights taken reciprocally: and so the weight placed at E, having a position from which some greater downward motion can result than the extreme weights can ascend, descends and overcomes their resistance. But since, with the extreme weights assumed, we can raise them by hand so that the intermediate weight, descending and stretching the cord, forms with B and A angles whose common tangent E F has to the excess of the secants H F, G R a ratio smaller than the reciprocal ratio of the extreme weights to the intermediate weight, it is sufficiently clear why those extremes outweigh it, since they both possess more gravity and greater moments, that is, a tendency toward greater motion. For although, by proposition 4, the difference between the tangents of two unequal arcs in the same circle is always greater than the difference which lies between their secants; yet because, by proposition 5, this ratio is always smaller the greater the angles become, therefore if the tangent is common to two circles, it may happen that the differences of the secants of both circles taken together are greater than the tangent itself, or at least that the tangent has to those taken together the ratio which is less than the reciprocal ratio of the weights. And so that the truth, set before the eyes of all by examples, may leave no room for doubt, let the ratio of the extreme weights to the intermediate weight be given, and let there be sought a tangent having a similar ratio to the excess of each secant: let it also here, for the sake of convenience, be understood that the point E is altogether equally distant from A and B, so that the excesses of the secants H F and G F are also equal. And first, let the middle weight be set equal to each of the extremes. The ratio sought is therefore the double of the tangent E F

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Liber tertius. CAPUT XII. 351 EF ad Excessuum summam HFG, cujus summæ semissis est GF, atque adeò Ratio Tangentis EF ad GF est quadrupla, hoc est ut 4 ad 1. Ergo ex corollar. prop. 2. ut quadratum Excessûs ad differentiam inter quadrata Excessûs & Tangentis (sunt autem quadrata 1 & 16) hoc est ut 1 ad 15, ita excessus secantis ad duplum Radij BE. Quare Excessus secantis ad Radium BE est ut 1 ad 7 ́. Posito igitur Radio BE 100000, Excessus secantis GF est 13333 ́, & ejus quadrupla Tangens EF 53333 ́ dat angulum EBF gr. 28. 4. 21, cujus secans BF est 113333 ́. Distantia AB statuatur pedum quatuor, hoc est digitorum 64: est BE dig. 32. Igitur ut BE 100000 ad GF 13333, ita BE dig. 32. ad GF dig. 4 ́. & Tangens hujus Excessûs quadrupla erit descensus EF dig. 17, ascensus verò DK aut CI dig. 4 ́ singuli, & ambo simul 8 ́. In omnibus igitur angulis minoribus angulo gr. 28. 4. 21. Ratio Tangentis ad Excessuum secantium summam major est Ratione duplâ, quæ est ponderum Ratio, in angulis verò majoribus minor est Ratione duplâ: ac propterea ibi pondus intermedium superat extrema, hîc superatur ab illis, & quiescunt in invento angulo gr. 28. 4. 21. Generaliter autem ut invenias, quantum ascendere possint extrema pondera vi ponderis medij descendentis, sit nota Ratio ponderum: tùm minoris termini Rationis datæ semissem accipe (quia unicus Excessus hîc sumitur, & pondus medium æquali intervallo distat ab A & B) & hujus semissis quadratum deme ex quadrato termini majoris: Deinde fiat ut hæc quadrorum differentia ad quadratum illius semissis, ita duplum Radij, hoc est tota claviculorum distantia AB ad aliud, & erit Excessus unius secantis, quæ est mensura ascensûs æqualis ponderum DK aut CI. Ponderum extremorum Ratio simul sumptorum ad intermedium sit ex. gr. ut 7 ad 6: termini minoris 6 semissis est 3, cujus quadratum 9 ex 49 quadrato termini majoris 7 deme, & est differentia 40. Distantia claviculorum A & B sit digitorum 80; fiat igitur ut 40 ad 9 ita 80 ad 18, & vi ponderis illius intermedij poterunt extrema pondera ascendere dig. 18. Ut verò innotescat, quantum descendat pondus medium, inter Excessum secantis

Transcription: Translated (English)

Liber tertius. CHAPTER XII. 351 EF to the sum of the excesses HFG, of which sum half is GF, and therefore the ratio of the tangent EF to GF is quadruple, that is, as 4 to 1. Therefore, from corollary 2 of proposition 2, as the square of the Excess to the difference between the squares of the Excess and the Tangent (now the squares are 1 and 16), that is, as 1 to 15, so is the excess of the secant to twice the Radius BE. Hence the excess of the secant to the Radius BE is as 1 to 7 ́. Therefore, taking the Radius BE as 100000, the excess of the secant GF is 13333 ́, and its quadruple, the Tangent EF, 53333 ́, gives the angle EBF of 28. 4. 21 degrees, whose secant BF is 113333 ́. Let the distance AB be set at 4 feet, that is, 64 inches: BE is 32 inches. Therefore, as BE 100000 to GF 13333, so BE 32 inches to GF 4 ́ inches. And the tangent of this Excess, quadruple, will be the descent EF 17 inches; the ascents, however, DK or CI, 4 ́ inches each, and both together 8 ́. In all angles therefore smaller than the angle 28. 4. 21 degrees, the ratio of the Tangent to the sum of the secants of the Excesses is greater than the double ratio, which is the ratio of the weights; but in angles greater than that, it is less than the double ratio: and for that reason there the intermediate weight exceeds the extremes, here it is exceeded by them, and they rest in the found angle 28. 4. 21 degrees. Generally, however, in order to find how much the extreme weights may ascend by the force of the descending middle weight, let the ratio of the weights be known: then take half of the lesser term of the given ratio (because only a single Excess is taken here, and the middle weight is at equal interval from A and B) and subtract the square of this half from the square of the greater term: then let it be as this difference of the squares to the square of that half, so is twice the Radius, that is, the whole distance AB of the little handles, to another, and there will be the Excess of one secant, which is the measure of the equal ascent of the weights DK or CI. Let the ratio of the extreme weights taken together to the middle be, for example, as 7 to 6: the lesser term 6, half of it is 3, whose square 9 subtract from the square 49 of the greater term 7, and the difference is 40. Let the distance of the handles A and B be 80 inches; therefore let it be as 40 to 9 so 80 to 18, and by the force of that intermediate weight the extreme weights will be able to ascend 18 inches. But in order that it may be known how much the middle weight descends, between the Excess of the secant

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Mechanicorum 352 secantis 18, & 98 summam secantis & Radij, quære mediam proportionalem, & ex prop. 2. hæc est Tangens dig. 42: dupli- catus autem 18 pro utroque excessu secantis dat 36, atque mo- tuum Ratio 42 ad 36 eadem est cum reciprocâ Ratione ponde- rum 7 ad 6. Quòd si angulum E B F tantummodo quæris, quem funiculus FB constituit cum horizontali AB, fiat similiter ut 40 ad 9 ita Radij duplum 200000 ad 45000 Excessum Radio addendum, ut habeatur secans 145000 gr. 46. 24. Ex his facilè intelligitur cur pro majore claviculorum A & B intervallo pondus medium magis descendat, quia scilicet atten- denda est anguli magnitudo, ex quâ pendet Tangentis & se- cantis Ratio; ubi verò major est Radius, majorem quoque esse similis anguli Tangentem atque secantem manifestum est. Quare si exiguum sit pondus medium, & vix appareat, an ab illo extrema pondera eleventur, atque dubitetur, an ideò so- lùm illud descendat, quia funiculum magis intendit; adhibe longiorem funiculum, cui eadem pondera adnectas, & augea- tur, quantum opus fuerit, claviculorum A & B intervallum; demum enim apparebit extremorum ponderum ascendentium motus: acutissimus scilicet angulus in majore circulo habet se- cantis Excessum suprà Radium faciliùs notabilem quàm in mi- nore. Sic vides posito Radio habente unitatem cum septem cyphris, non inveniri Excessum secantis nisi gr. 0. 1. 10. uni- tatem: at posito Radio cum quindecim cyphris, habetur ejus- dem anguli secantis Excessus supra Radium partium 57585857: immò habetur etiam unius secundi secans, cujus Excessus su- pra Radium est 11752. Hinc etiam desines mirari, cur longiores funes aut catenæ nullâ vi ita intendi possint, ut in lineâ horizonti parallelâ rectam positionem habentes consistant, sed aliquantulum sal- tem inflectantur; quia nimirum insitum funi aut catenæ pon- dus idem præstat, quod in hoc experimento pondus in medio appensum. Id quod nautæ non ignorantes sæpius malunt uni anchoræ funem duplo longiorem adnectere, quàm duabus an- choris simplici & subduplo fune instructis navem firmare: nô- runt siquidem longè majore vi opus esse ut funis longitudinem habens ducentorum cubitorum intendatur, quàm si centum tantummodo cubitorum longitudo esset; ac proinde undarum impetum

Transcription: Translated (English)

Mechanics 352 of the secant 18, and 98, seek the mean proportional of the sum of the secant and the radius, and by proposition 2 this is the tangent, 42. But the doubled 18 for each excess of the secant gives 36, and the ratio of the motions, 42 to 36, is the same as the reciprocal ratio of the weights, 7 to 6. But if you are seeking only the angle E B F, which the cord FB makes with the horizontal AB, let it be similarly as 40 to 9 so the double of the radius, 200000, to 45000, the excess to be added to the radius, so that the secant 145000 may be obtained; 46. 24. From these things it is easily understood why, for a greater interval of the pins A and B, the middle weight descends more, because, namely, one must attend to the magnitude of the angle, on which the ratio of the tangent and secant depends; but where the radius is greater, it is clear that the tangent and secant of a similar angle are also greater. Therefore, if the middle weight be small and it scarcely appear whether the extreme weights are lifted by it, and it be doubted whether it descends only for this reason, because it stretches the cord more, use a longer cord, to which you attach the same weights, and increase, as much as shall be necessary, the interval of the pins A and B; for then at last the motion of the rising extreme weights will appear: namely, the sharpest angle in a larger circle has the excess of the secant above the radius more easily noticeable than in a smaller one. Thus you see that, the radius being taken as one with seven zeros, the excess of the secant is not found unless it be 0. 1. 10. of the unit: but, the radius being taken with fifteen zeros, the excess of the secant of the same angle above the radius is obtained as 57585857 parts: indeed even the secant of one second is obtained, whose excess above the radius is 11752. Hence also you will cease to wonder why longer ropes or chains cannot by any force be so tightened that, when placed in a straight position parallel to the horizon, they remain so, but are bent at least a little; because, namely, the weight inherent in the rope or chain does the same thing that, in this experiment, the weight suspended in the middle does. Not ignorant of this, sailors often prefer to attach a rope twice as long to a single anchor, rather than to secure the ship with two anchors equipped with a single and shorter rope: for they know that far greater force is needed to stretch a rope having a length of two hundred cubits than if the length were only one hundred cubits; and therefore the force of the waves

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Liber tertius. CAPUT XII. 353 impetum longior funis faciliùs eludit, eóque minùs timendum est, ne dirumpatur, quò difficiliùs intendi potest. Simile quiddam dicendum videtur, cùm longiorum prisma- tum aut cylindrorum extremitates subjectis fulcris totam longi- tudinem horizonti parallelam in aëre quasi suspensam sustinent; suo enim pondere si non franguntur, saltem curvantur; id quod brevioribus cylindris aut prismatis non contingit. Quia vide- licet ex ipsâ positione partes, quæ in mediâ longitudine locum obtinent, & quæ his proximæ sunt, aptæ sunt velociùs moveri quàm remotiores: & quemadmodum pondus in medio positum descendens vincit resistentiam extremorum ponderum ascen- dentium, ita vis harum partium mediarum superat vim, quâ partes invicem nectuntur, ac proinde distractæ flectuntur sal- tem, & demum separantur. Sed antequam planè ex animo effluat, unum hîc observan- dum (de quo fortasse maluisses initio præmoneri) aliud esse quod ex naturæ instituto, aliud quod ex iis, quæ accidunt, con- tingit. Quæ hactenus diximus de Ratione motuum spectatis ponderum gravitatibus, intelligenda sunt, nisi quid interveniat, quod legem hanc infringat; cujusmodi est aliqua funiculi re- missio, vel minor intensio, ita ut hic faciliùs à medio pondere descendente adhuc intendatur, quàm extrema pondera eleven- tur; ubi enim eò devenerit pondus medium, ut intentus funi- culus cum lineâ horizonti parallelâ angulum faciat, cujus Tan- gens ad secantium Excessus Rationem habet reciprocam pon- derum, ibi subsistit, etiamsi extrema pondera elevata non fue- rint nisi juxtâ mensuram differentiæ secantium duorum angu- lorum, ejus videlicet quem demum funiculus constituit, & ejus qui funiculi remissionem ipso motûs initio consequitur: quia ulterior descensus ad ulteriorem ascensum non haberet majo- rem Rationem, sed minorem Ratione ponderum reciprocè sumptorum. Quòd si valde inæqualia fuerint pondera, eveni- re potest totam vim descendendi, quam pondus medium habet, absumi in funiculo intendendo, nec quicquam virium superes- se ad extrema pondera attollenda. Húc etiam spectat impedimentum, quod ex funiculi clavi- culos terentis conflictu oritur; cùm enim descendentis ponde- ris medij momentum semper decrescat, ut ex prop. 5. constat, Y y

Transcription: Translated (English)

Book three. CHAPTER XII. 353 a longer rope more easily eludes the shock, and therefore there is less fear that it will break, the more difficult it is to stretch it. Something similar seems to be said when the ends of longer prisms or cylinders, supported by props placed underneath, sustain the whole length, as it were suspended in the air, parallel to the horizon; for by their own weight, if they are not broken, at least they bend; and this does not happen with shorter cylinders or prisms. For, plainly, from the very position, the parts which occupy the middle of the length, and those nearest to them, are fit to move more quickly than the more distant parts: and just as a weight placed in the middle, descending, overcomes the resistance of the weights at the ends as they rise, so the force of these middle parts exceeds the force by which the parts are bound together, and therefore when pulled apart they bend at least, and finally separate. But before it quite slips from memory, one thing must here be observed (which perhaps you would have preferred to be forewarned of at the beginning): that one thing happens according to nature’s arrangement, and another according to what occurs by accident. What we have so far said about the ratio of motions, with the weights and their heaviness considered, must be understood, unless something intervene to break this law; of which kind is some loosening of the rope, or lesser tension, so that here it may more easily be further tightened by the middle weight descending, than the end weights be raised; for when the middle weight has reached the point where the rope, being stretched tight, makes an angle with the line parallel to the horizon, whose tangent has to the excess of the secants an inverse ratio of the weights, there it comes to rest, even if the end weights have not been raised except in proportion to the difference between the secants of the two angles, namely that which the rope finally makes, and that which follows the loosening of the rope at the very beginning of the motion: because any further descent would not have a greater ratio to the further ascent, but a smaller one, in the ratio of the weights taken reciprocally. But if the weights are very unequal, it may happen that the whole force of descent which the middle weight has is spent in tightening the rope, and no force remains for raising the end weights. This also relates to the impediment that arises from the rope’s collision with the pegs that wear it down; for since the moment of the descending middle weight always diminishes, as is clear from prop. 5, Y y

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Mechanicorum adeò extenuari potest, ut jam superare non valeat extremorum ponderum ascendentium momenta aucta momento, quod ex partium conflictu oritur; qui conflictus si non adesset, pergeret illud adhuc descendendo. Propterea si claviculos ipsos con- gruentibus rotulis inseras, adeò ut funiculus excavatæ absidi insideat, longè majorem motum faciliusque perfici videbis; minùs enim rotula cum suo axe confligit, quàm funiculus cum claviculo, si illum terat; & quidem quò major fuerit ro- tula, circa eundem axem faciliùs volvitur, minor siquidem partium tritus fit, si cætera omnia sint paria. Simili modo si pondus medium plus æquo per vim deprimas, faciliùs suum in locum redibit adhibitis rotulis, quàm si funiculus clavicu- lis insisteret: quia pondera extrema superare non valent & gra- vitatem ponderis medij & impedimentum, quod oritur ex ma- jori tritu funiculi & claviculorum, quàm rotularum & axium. Observabis etiam adhibitis rotulis pondus medium sibi re- lictum tanto impetu à lineâ horizonti parallelâ descendere, ut ex concepto impetu fines suos transiliat, ac idcirco desinente impetu, quem in motu acquisivit, iterum sursum trahi ab ex- tremis ponderibus, quæ sicut minorem Rationem habebant ad gravitatem ponderis medij auctam impetu acquisito, ita ma- jorem Rationem habent ad eandem spoliatam illo impetu. Porrò hæc quæ hactenus de pondere in mediâ planè distan- tiâ inter claviculos aut rotulas constituto dicta sunt, intelli- genda sunt pariter de pondere claviculorum intervallum inæ- qualiter dividente, quod quidem spectat ad æquilibrium aut præponderationem propter Rationum æqualitatem aut inæqua- litatem. Peculiare tamen aliquid observandum est, videlicet aliquando contingere, ut hoc pondere medio descendente pondus proximum ascendat, remotum verò descendat, utró- que autem pondere extremo ascendente magis ascendere quod proximum est, minùs quod remotum. Hujus inæqualis ascen- sûs (si pondus medium rectâ ad perpendiculum descendat) causa in promptu est ex iis, quæ prop. 8. indicata sunt, nam e usdem Tangentis quadrato æqualia sunt, atque adeò & in- ter se æqualia, rectangula, quæ fiunt sub Excessu secantis & aggregato secantis & Radij: sunt igitur ex 14. lib. 6. Excessus secantium reciprocè in Ratione aggregatorum secantis & Ra- dij:

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The force of the machines can be so reduced that it is no longer able to overcome the moments of the descending extreme weights, increased by the moment arising from the conflict of the parts; and if that conflict were not present, it would still continue descending. Therefore, if you insert the little pins themselves into matching pulleys, so that the cord lies in the hollowed groove, you will see a much greater motion accomplished more easily; for the pulley conflicts less with its axle than the cord with the little pin, if it rubs against it. And indeed, the larger the pulley is, the more easily it turns about the same axle, since there is less friction of the parts, all other things being equal. In like manner, if the middle weight is pressed down beyond measure by force, it will return to its place more easily with the help of pulleys than if the cord rested on little pins; because the extreme weights are not able to overcome both the heaviness of the middle weight and the hindrance arising from the greater friction of the cord and the little pins than of the pulleys and axles. You will also observe that, when pulleys are used, the middle weight, being left to itself, descends with so great an impulse from a line parallel to the horizon that, from the impulse it has acquired, it passes beyond its limits, and therefore, when the impulse acquired in motion ceases, it is again drawn upward by the extreme weights, which, just as they had a smaller ratio to the heaviness of the middle weight increased by the impulse acquired, so now have a greater ratio to the same weight deprived of that impulse. Moreover, what has been said thus far about a weight placed exactly midway between the little pins or pulleys must likewise be understood of a weight unequally dividing the interval between the little pins, which indeed concerns equilibrium or preponderance on account of the equality or inequality of the ratios. Yet something special is to be observed, namely, that it sometimes happens that, while this middle weight descends, the nearer weight ascends and the more distant one descends; but when both extreme weights ascend, that which is nearer ascends more, and that which is farther away ascends less. The cause of this unequal ascent, if the middle weight descends in a straight line perpendicularly, is readily found from what was indicated in Proposition 8, for the rectangles formed by the excess of the secant and the aggregate of the secant and radius are equal to one another, and therefore equal among themselves when they arise from the same tangent squared; hence, from Book 6, Proposition 14, the excesses of the secants are reciprocally in the ratio of the aggregates of the secant and the radius:

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Liber tertius. CAPUT XII. 355 dij: quapropter ubi major est Radius & secans, ibi minor est secantis Excessus, hoc est remoti ponderis ascensus, & contra ubi minor est Radius & secans, ibi major est secantis Excessus, hoc est ponderis proximi ascensus. Cur autem aliquando proximum pondus ascendat, atque remotum descendat, quando nimirum valde inæquales sunt ponderis medij à claviculis distantiæ, hinc fit, quod idem pondus ex longiore funiculo majorem habet vim descendendi, quàm ex breviore; cui majori momento cum resistere debeat pondus proximum, faciliùs cedit descendenti, atque adeò non rectâ deorsum tendit pondus medium, sed obliquè, accedendo ad pondus remotum, quod propterea descendit. Sic positum pondus in E valde inæqualia habet momenta comparatum cum extremis ponderibus D & C, quæ in punctis B & A exercent suas vires adversùs pondus medium; quod ubi infra horizontalem AB descéderit, illico inæquales angulos cum horizontali lineâ AB constituit inflexus funiculus; ut si intelligatur pondus ex E venisse in F, angulus FBA major est angulo FAB ex 18. lib.1. quia latus AF est majus latere FB. Igitur angulus FB D, quem funiculus inflexus FB facit cum perpendiculari BD minor est angulo FAC; ergo ex dictis lib.1. cap.15. pondus in F minora habet momenta ad descendendum versus perpendicularum BD, quàm ad descendendum versus AC, & quidem duplici titulo, scilicet anguli FB D minoris, & funiculi FB brevioris. Cum itaque pondus illicò ac ex E descendit magis pronum sit ad descendendum versùs perpendicularum AC, non per rectam EF perpendicularé descendit; sed obliquè per lineam EG, ita ut funiculus GA brevior sit funiculo EA, ac propterea cedit ponderi C deorsum trahenti. Et quia funiculus GB longior est funiculo FB, & multo magis funiculo EB, propterea aliquando contingere potest pondus D magis ascendere, quàm ascenderet, si E fuisset planè in mediâ distantiâ inter A & B. Ex quo etiam sit descensum perpendicularem ponderis medij minorem esse; nam punctum G Y y 2

Transcription: Translated (English)

Book the Third. CHAPTER XII. 355 therefore: where the Radius and secant are greater, there is the lesser Excess of the secant, that is, the ascent of the more distant weight; and on the contrary, where the Radius and secant are lesser, there is the greater Excess of the secant, that is, the ascent of the nearer weight. But why at times the nearer weight ascends, and the more distant descends, when, namely, the distances of the middle weight from the pins are very unequal, this comes about because the same weight from a longer cord has a greater force of descending than from a shorter one; to which greater moment the nearer weight must resist, it yields more easily to the descending one, and so the middle weight tends not straight downwards, but obliquely, by approaching the more distant weight, which therefore descends. Thus the weight placed in E has very unequal moments when compared with the extreme weights D and C, which at the points B and A exert their forces against the middle weight; which, when it has descended below the horizontal AB, at once the bent cord forms unequal angles with the horizontal line AB; so that if it is understood that the weight has come from E to F, the angle FBA is greater than the angle FAB, from lib. 1, cap. 18, because the side AF is greater than the side FB. Therefore the angle FBD, which the bent cord FB makes with the perpendicular BD, is less than the angle FAC; therefore, from what was said in lib. 1, cap. 15, the weight in F has lesser moments for descending toward the perpendicular BD than for descending toward AC, and indeed for a double reason, namely because the angle FBD is smaller, and because the cord FB is shorter. Since therefore the weight, as soon as it descends from E, is more inclined to descend toward the perpendicular AC, it does not descend perpendicularly along the straight line EF; but obliquely along the line EG, so that the cord GA is shorter than the cord EA, and therefore yields to the weight C pulling downward. And because the cord GB is longer than the cord FB, and much more than the cord EB, therefore it can happen at times that the weight D ascends more than it would ascend if E had been exactly in the middle distance between A and B. From which it also follows that the perpendicular descent of the middle weight is less; for the point G

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Mechanicorum minus distat ab horizontali AB, quam punctum F, & tamen major est differentia inter EB & GB; ideò minor est Ratio IG ad Excessum GL, quam EF ad Excessum FO. Hanc momentorum inæqualitatem perspicies, si pondus me- dium singulis extremis æquale inter claviculos æqualiter consti- tutum descendere permittas, suóque in loco co[n]sistere; cùm enim æqualis sit funiculorum illud sustinentium longitudo, & æqua- les faciat angulos tùm cum horizontali, tùm cum perpendicularibus, contra utrumque extremum æqualibus momentis pugnat, ac rectâ ad perpendicularum descendit. Tum alteri extremorum aliquid adde ponderis; hoc utique descendens secum rapit & ponderis medij & reliqui extremi gravitates, quas cogit ascen- dere, donec ea fiat funiculorum inæqualitas, ut momenta, quæ pondus medium habet ad descendendum ratione distantiæ à cla- viculo remotiori, jam superari non valeant à pondere illo ex- tremo cum suo additamento. Nec dispar est philosophandi methodus, cum funiculi extre- mitas alterutri claviculo alligatur, unico pondere in alterâ extre- mitate pendente ex altero claviculo: pondus enim inter clavi- culos funiculo adnexum, quia velociùs movetur descendendo, quam reliquum pondus ascendendo, superare potest illius gra- vitatem. Sit enim funiculus alligatus in A, & pendeat pondus D ex clavicu- lo B: pondus (utrùm æquale sit, an ma- jus, an minus, parum refert) adnectatur in C: utique descendens describit ar- cum CI circa centrum A; est autem funiculus IB longior quam CB ex 8. lib. 3. Sed quoniam duo latera BC & CI simul majora sunt reliquo latere IB ex 20. lib. 1. major est recta CI, & multo magis arcus CI spatium quod percurrit pon- dus medium descendens) quam IE Excessus lateris IB supra CB, hoc est mensura motûs ponderis D ascendentis. Quia verò ponderis medij descendentis circa centrum A momenta decres- cunt ex dictis lib. 1. cap. 15. circa centrum autem B decrescunt quidem, quia minor sit angulus declinationis à perpendicular GBD, sed decrementum hoc temperatur, quia momenta cres- cunt ratione longitudinis funiculi, quæ semper augetur ex 8. lib. 3.

Transcription: Translated (English)

Mechanics is farther removed from the horizontal AB than point F, and yet the difference between EB and GB is greater; therefore the ratio of IG to the excess GL is smaller than EF to the excess FO. You will perceive this inequality of moments if you let a middle weight, equal to the two ends, equally placed between the pins, descend and remain in its place; for since the length of the cords supporting it is equal, and makes equal angles both with the horizontal and with the perpendiculars, it resists both ends with equal moments, and descends straight down toward the perpendiculars. Then add some weight to one of the ends; this, as it descends, will certainly draw along with it both the gravity of the middle weight and that of the remaining end, which it compels to rise, until such inequality of the cords is produced that the moments which the middle weight has for descending by reason of its distance from the farther pin can no longer be overcome by that end weight with its added load. Nor is the method of reasoning dissimilar when one end of the cord is fastened to either pin, with a single weight hanging at the other end from the other pin: for the weight attached to the cord between the pins, because it moves downward more quickly than the remaining weight rises, can overcome the gravity of the latter. Let the cord therefore be fastened at A, and let weight D hang from the pin B: let a weight (whether equal, greater, or less, it matters little) be attached at C. As it descends, it certainly describes arc CI about center A; but cord IB is longer than CB, by 8. lib. 3. Yet since the two sides BC and CI together are greater than the remaining side IB, by 20. lib. 1, the straight line CI is greater, and much more so the arc CI (the space traversed by the middle weight descending) than IE, the excess of side IB over CB, that is, the measure of the motion of the rising weight D. But because the moments of the descending middle weight about center A decrease, as has been said in lib. 1, cap. 15, and about center B they do indeed decrease, because the angle of declination from the perpendicular GBD is smaller, yet this decrease is moderated because the moments increase in proportion to the length of the cord, which is always augmented, as in 8. lib. 3.

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Liber tertius. CAPUT XII. 357 lib. 3. propterea ad momentorum æqualitatem venit, ubi demùm quiescit. Quantum autem descendat, pendet ex ipsius ponderis gravitate absolutâ sive majori, sive minori, sive æquali compara- tâ cum pondere D, & ex distantiâ à centro A: si enim valde pro- pinquium sit centro, parùm descendit, etiamsi cæteroqui gravius sit; & si per vim adhuc deprimatur, ut veniat in G, cessante vi extrinsecùs illatâ pondus D descendens illud iterum attollit. Cave tamen ponderis medij descendentis momenta metiaris ex arcu, quem describit, sed potiùs illa definienda sunt ex ipso descensu perpendiculari, cum moveatur vi suæ gravitatis. Quo- niam verò æqualibus arcubus descriptis non respondent paria perpendicularium linearum incrementa ex prop. 10. sed semper minora fiunt; contra verò incrementa secantium augentur, hinc est deveniri ad momentorum æqualitatem, ita ut pondus me- dium gravius pondere extremo aptum sit minùs descendere quàm illud ascenderet secundùm reciprocâ Ratione gravitatum. Hinc elici potest compendium aliquod in attollendo ponde- re cæteroqui valde gravi; sit enim pondus P attollendum fune circumducto rotulæ A: quò longior funis potest alligari in B, eò faciliùs sequetur motus, si ad servandam in mediâ distantiâ positionem poten- tiæ moventis simplicem trochleam aut annulum in C addideris, cui in- seratur funis B A: nam applicata po- tentia in D deorsum trahens multo faciliùs attollet pondus P, quàm si arreptâ funis extremitate B idem onus elevare conaretur ad eam al- titudinem, ad quam attolleretur à pondere in C adnexo, quod æqualibus viribus præditum esset cum potentiâ in D trahente. Ubi jam sit attollendi difficultas, suppone aliquid ponderi P, cui illud incumbat, nec contra funem conetur: tùm iterum funem intende, & alliga in B, ut sit A B horizonti parallelus, & ite- rum in D deorsum trahens priorem facilitatem experieris: id quod toties iterari poterit, quoties opus fuerit. Ex his omnibus, quæ toto hoc capite disputata sunt, mani- festum est non referendas esse machinarum vires ad Rationes Y y 3

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Book the Third. CHAPTER XII. 357 book 3. therefore it comes to equality of moments, where at last it rests. But how far it descends depends on the absolute heaviness of the weight itself, whether greater, smaller, or equal, compared with the weight D, and on the distance from the center A: for if it be very near the center, it descends little, even though otherwise it be heavier; and if by force it is pressed down further, so as to come to G, when the externally applied force ceases, the weight D descending raises it again. But beware that you do not measure the moments of the descending middle weight from the arc which it describes, but rather they must be determined from its very perpendicular descent, since it moves by the force of its own gravity. For since equal arcs being described do not correspond to equal increments of perpendicular lines, according to prop. 10, but are always smaller; on the other hand the increments of secants increase, hence it comes about that moments are brought to equality, so that the middle weight, being heavier than the weight at the end, is fitted to descend less than that one would ascend according to the reciprocal ratio of the gravities. From this some saving may be drawn in lifting a weight otherwise very heavy; let there be a weight P to be raised by a rope passed around the pulley A: the longer the rope can be tied at B, the more easily the motion will follow, if, to maintain the position at the middle distance, you add a simple pulley or ring at C, into which the rope B A is inserted: for the applied power at D, pulling downward, will raise the weight P much more easily than if, having seized the end of the rope at B, it were trying to lift the same load to that height to which it would be raised by the weight attached at C, which would be endowed with equal forces with the power pulling at D. When the difficulty of lifting is now present, suppose something be placed upon the weight P, upon which it may rest, and that it not press against the rope: then again stretch the rope, and tie it at B, so that A B is parallel to the horizon, and again, pulling downward at D, you will experience the former ease: and this may be repeated as many times as is needed. From all these things, which have been discussed throughout this whole chapter, it is manifest that the powers of machines are not to be referred to Ratios

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Mechanicorum circuli aut Vectis, quandoquidem hic videmus minori pondere majus pondus moveri absque ullo motu circulari. CAPUT XIII. An aliqua sit Libræ obliquæ utilitas. L ibram obliquam vocat Simon Stevinus Static. lib.3. prop.6. L rotulam L funiculi in excavatâ apside capacem pondus cum æquipondio jungentis, & in suo lo- culamento facillimè versatilem, cujus par- ticula extans E possit pro re natâ eximi, at- que iterum inseri foraminibus, quibus exactè congruat, tigilli P firmè infixi pedi satis gravi, ne valeat à ponderis examinan- di gravitate rapi & inclinari. Hanc ille ad ponderum obliquorum momenta investi- ganda utilem existimavit, eamque sæpiùs ingerit Static. lib.1. prop.19. & seqq quam- vis semper illam cum elevante directo conjunctam adhibeat. Propterea, an aliquid ex illâ emolumenti, si solitaria adhibeatur, capere possimus in ponderum momentis investigandis sivè sus- pensorum, sivè in plano inclinato jacentium, hîc examinare ope- ræ pretium fuerit; nam & à superioris capitis argumento non aliena videtur esse præsens disputatio. Antequam verò rem aggrediar, monendum te censeo, Ami- ce Lector, opportunius accidere, si tigilli perforati loco cylin- drum in cochleam efformatum statueris, cui congruat in simi- lem helicem excavatum foramen S in rotulæ L loculamento: sic enim faciliùs elevabitur aut deprimetur rotula, prout exiget ipsius ponderis positio. Dupliciter itaque contingere potest ponderis obliquitas, seu quia suspensum non in eodem perpendiculo, in quo est punctum suspensionis, habet centrum suæ gravitatis, seu quia plano incli- nato incumbit; utroque enim in casu momenta habet ad descen- dendum, quæ communi librâ aut staterâ vestigare utique non possumus: an libræ obliquæ ope id assequemur? Et primò qui- dem si pondus examinandum è funiculo suspensum fuerit, ejus- que momenta pro variâ declinatione à suo perpendiculo inqui- rantur

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of the mechanic’s circle or lever, since here we see a greater weight moved by a smaller weight without any circular motion. CHAPTER XIII. Whether there is any use for the oblique balance. Simon Stevinus calls an oblique balance Static. lib. 3, prop. 6. He calls it a small wheel L of cord, in a hollowed recess, joining the weight with the counterweight, and easily turnable in its socket, whose projecting part E can, as occasion requires, be taken out and inserted again into the holes, with which it exactly fits, the peg P being firmly fixed in a foot sufficiently heavy, lest it should be dragged and tilted by the weight to be examined. He thought this useful for investigating the moments of oblique weights, and he frequently introduces it, Static. lib. 1, prop. 19 and following, although he always uses it together with the direct raising apparatus. Therefore, whether by using it alone we may derive any advantage in investigating the moments of weights, whether suspended or lying on an inclined plane, it will be worth examining here; for this present discussion does not seem alien to the argument of the preceding chapter. But before I enter upon the matter, I think it necessary to warn you, dear Reader, that it would be more convenient if, instead of the perforated peg, you were to place a cylinder fashioned into a screw, to which a hole S , hollowed in a similar helix, should correspond in the socket of the wheel L ; for in this way the wheel may be raised or lowered more easily, according as the position of the weight itself requires. Thus the obliquity of a weight can happen in two ways: either because, when suspended, its center of gravity is not in the same vertical line as the point of suspension, or because it rests upon an inclined plane; for in either case it has moments tending downward, which we certainly cannot investigate with an ordinary balance or steelyard. Shall we achieve this by means of an oblique balance? And first, indeed, if the weight to be examined is suspended from a cord, and its moments are sought according to its varying deviation from the vertical,

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Liber tertius. CAPUT XIII. 359 rantur, res manet incerta, si in praxim deducatur, quia plurimum interest, quâ obliquitate inclinetur, atque à suo perpendiculo de- fectat funiculus libræ obliquæ, si maximè cum diversâ obliqui- tate jungatur dispar funiculi illius longitudo. Nam ex A sus- pendatur pondus B habens B A C angulu[m] declinationis à suo perpendiculo A C; & primùm sit libra obliqua D, itaut æquipo[n]dium E retineat pondus B in eodem situ: deinde transferatur libra obliqua ex D in F, & æquipondium G retineat pariter in eodem situ pondus B cum declinationis angulo B A C. Si in eâdem rectâ lineâ sint B D F, nulla est momentorum inæqualitas, quamvis disparitas intercedat inter funi- culi longitudines B D, & B F. Sin autem F paulo superior fuerit aut paulo inferior, jam B D & B F angulum in B constituunt, & momenta mutantur. Quoniam enim I E & H G perpediculares sunt parallelæ, in easque incidit recta B D F producta, anguli B I E, & B H G sunt æquales per 29.lib.1.at verò si libra obliqua F non planè in eâdem rectâ lineâ, sed superiore loco collocaretur, angulum constitueret cum perpendiculo H G acutiorem, & inferiùs posita angulum efficeret minùs acutum. Quare pondus B, quò acutior est angulus, & magis accedit ad perpendiculu[m] F G, eò etiam magis conatur contra F, & ad æqui- librium exigit majorem gravitatem in G, quàm cum angulus est minùs acutus. Id quod experimento allato superiori capite ma- nifestum sit; si enim funiculi extremitates jungant pondera inæ- qualia, pondus intermedium magis accedit ad perpendiculum, in quo est major gravitas. Hinc quia valde incertum est in praxi, utrùm B, D, & F in eâdem sint rectâ lineâ, propterea etia[m] incertu[m] erit ex gravitate ponderis G inferre, quanta sint ponderis B mo- mēta cum declinatione B A C: Nisi fortè duplicè instituas libræ obliquæ positionem in D, & in F atque eode[m] semper pòdere tam in E quàm in G retineatur pondus B in positione eâde[m]. Ita tam[m]e collocanda est libra obliqua, ut angulus A B D sit rectus; ex illo quippe æstimatur planu[m] inclinatu[m], in quo pondus B conatur des- cendere, ut dietu[m] est lib.1.cap.15.alioquin si acutus fuerit aut ob- tusus ille angulus, quamvis in eâdem declinatione B A C reti- neatur, valde inæqualia apparebunt momenta. Quis autem de anguli

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Book Three. Chapter XIII. 359 it remains uncertain in practice, because it matters very much at what obliquity the oblique balance is inclined, and how much its cord departs from its own perpendicular, especially when a cord of unequal length is joined with a different obliquity. For let weight B be suspended from A, having the angle BAC of deviation from its perpendicular AC; and first let the oblique balance D be so placed that the equal weight E holds weight B in the same position. Then let the oblique balance be transferred from D to F, and let the equal weight G likewise hold weight B in the same position with the angle of declination BAC. If BDF lie on the same straight line, there is no inequality of moments, although there is a disparity between the lengths of the cords BD and BF. But if F is a little higher or a little lower, then BD and BF form an angle at B, and the moments are changed. For since IE and HG are perpendiculars and parallel, and the straight line BDF extended falls upon them, the angles BIE and BHG are equal by Book 1, Proposition 29. But if the oblique balance F were placed not exactly on the same straight line, but in a higher position, it would form with the perpendicular HG an acute angle; and if placed lower, it would make a less acute angle. Therefore, the more acute the angle is, and the more weight B approaches the perpendicular FG, the more it also tends toward F, and to produce equilibrium it requires a greater gravity in G than when the angle is less acute. This is made clear by the experiment given in the preceding chapter; for if the ends of the cords join unequal weights, the intermediate weight approaches more nearly to the perpendicular, where the greater gravity lies. Hence, because in practice it is very uncertain whether B, D, and F are on the same straight line, it will also be uncertain, from the weight of G, to infer how great are the moments of weight B with the declination BAC, unless perhaps you establish the position of the oblique balance twice, at D and at F, and with the same always weight B is held in the same position, both in E and in G. Yet the oblique balance must be placed so that the angle ABD is a right angle; for from this the inclined plane is estimated, on which weight B seeks to descend, as was said in Book 1, Chapter 15. Otherwise, if that angle is acute or obtuse, although it is retained in the same declination BAC, the moments will appear very unequal. But who would of the angle

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360 Mechanicorum anguli illius rectitudine certus fuerit? cùm maximè rectam DB oporteat ad perpendiculum insistere lineæ jungenti pu nctum A suspensionis cum centro gravitatis ponderis B. Ex pon dere itaque, quod est in E, aut in G, nemo potest certò d e fini re momenta ponderis B suspensi. At si dato quopiam plano inclinato jaceat pondus, velisque librâ hujusmodi obliquâ explorare, quanta habeat pro eâ plani inclinatione ad descendendum momenta, ego sanè nihil certi affirmare auderem; quippè qui semper incertus hærere, an æquipondium libræ obliquæ indicaret ipsa momēta ponderis in plano inclinato pro ratione inclinationis; nam plani subjecti non omninò lubrica superficies, & ponderis illi incumbentis asperitas impedientes motum, non nihil detrahunt momenti ad descendendu[m]. Cum verò pro diversâ inclinatione planu[m] inæqualiter prematur ab insistent p[ro]dere, adhuc eadem superficieru[m] se contingentiu[m] asperitas magis obsistit motui, quò major est plani inclinatio declinans à perpendiculo. Quare adhuc magis incerta essent momenta, quæ ab æquipondio libræ obliquæ indicarentur. Nihil aliud itaque commodi hinc sperari potest præter notitiâ momenti, quod planorum asperitas detrahit momēto descendendi. Si enim nota sit ponderis dati gravitas absoluta, & plani inclinatio innotuerit, videlicet angulus, quem planum inclinatu[m] cum plano horizontali constituit, fiat ut Radius ad Sinum noti anguli inclinationis, ita gravitas absoluta dati ponderis ad momēta, quæ habet in plano inclinato: Tum librâ obliquâ exploretur, quanto æquipondio opus sit ad retinēdum pondus in plano inclinato, ne deorsum labatur: nam differentia inter gravitatem æquipondij, & momenta inventa pro tali inclinatione indicabit, quantum impedimenti oriatur ex planoru[m] se contingentium asperitate, si æquipondij gravitas minor sit momentis, quæ ab hujusmodi inclinatione exiguntur. Sic ex. gr. sit ponderis dati absoluta gravitas unciarum 30, inclinationis angulus dati plani cum plano horiz [n]o[n]tali sit gr. 60. fiat ut 100000 Radius ad 86603 Sinum gr. 60. ita 30 ad 25.98". Si applicata libra obliqua æquipondium habeat solùm unc. 24, manifestum est à planorum asperitate detrahi momenti partem ferè decimam tertiam, cùm desint justo æquipondio ferè unciæ 2. Verùm & hîc observandum, opus esse funiculi, à quo pondus retinetur, parallelismum cum plano inclinato, prout ex iis, quæ de obliquis tractionibus lib. 1. cap. 16. dicta sunt, satis constat. MECHA

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360 Mechanics Would anyone be certain of the uprightness of that angle? since the straight DB ought most exactly to stand perpendicular to the line joining point A of suspension with the center of gravity of the weight B. From the weight therefore, which is in E or in G, no one can certainly determine the moments of the suspended weight B. But if a weight lie on some given inclined plane, and you wish to examine with such an oblique balance how much moment it has, for descending, according to that plane’s inclination, I would certainly dare to affirm nothing definite; rather, I would always remain uncertain whether the counterpoise of the oblique balance itself indicated the moments of the weight on the inclined plane in proportion to the inclination; for the not altogether smooth surface of the plane beneath, and the roughness of the weight resting upon it, which impede motion, take away some part of the moment for descending. And since, for different inclinations, the plane is pressed unequally by the body standing upon it, the same roughness of the surfaces in contact still more resists motion, the greater the inclination of the plane declining from the perpendicular. Therefore the moments indicated by the counterpoise of the oblique balance would still be more uncertain. Nothing else of advantage can therefore be hoped from this, except the knowledge of the moment which the roughness of the planes takes away from the moment of descent. For if the absolute gravity of a given weight be known, and the inclination of the plane be known, namely the angle which the inclined plane makes with the horizontal plane, let it be as Radius to the Sine of the known angle of inclination, so the absolute gravity of the given weight is to the moments which it has on the inclined plane. Then let the oblique balance be used to examine how much counterpoise is needed to keep the weight on the inclined plane, lest it slip downward: for the difference between the gravity of the counterpoise and the moments found for such an inclination will show how much hindrance arises from the roughness of the planes touching one another, if the gravity of the counterpoise be less than the moments required by such an inclination. Thus, for example, let the absolute gravity of a given weight be 30 ounces, and let the angle of inclination of the given plane with the horizontal plane be 60 degrees. Let it be as 100000 Radius to 86603 Sine of 60 degrees, so 30 to 25.98. If, when the oblique balance is applied, the counterpoise has only 24 ounces, it is clear that about one thirteenth part of the moment is taken away by the roughness of the planes, since nearly 2 ounces are lacking to the proper counterpoise. But here also it must be observed that the cord by which the weight is held must be parallel with the inclined plane, as is sufficiently clear from what was said in Book 1, chapter 16, concerning oblique pulls. MECHA

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MECHANICORUM LIBER QUARTUS. De Vecte. ACTENUS de instrumentis ad movenda pondera idoneis nihil, nisi fortassè obiter, dictum est: jam ad illa explicanda accedimus, quibus veteres facul- tatibus nomen indiderunt. Quamvis autem in quinque facultatibus enumerandis primum locum Vecti Pappus lib. 8. Collect. Math. non tribuat, placuit tamen de Vecte ante cæteras facultates differere, est siquidem paratu facillimus, & ad subitum usum promptissimus, atque censeri potest, ut idem Pappus loquitur, fortasse præmeditatio motus cir- ca excedentia pondera: statuentes enim quidam magna pondera mo- vere [quoniam primùm à terrâ attollere oportet, ansas autem non habebant) quòd omnes partes basis ipsius ponderis solo incumberent, paulum suffodientes, & ligni longi extremitatem subjicientes sub onus, adducebant ex alterâ extremitate, supponentes ligno propè ipsum onus lapidem, qui Hypomochlium appellatur. Cùmque illis vi- sus esset hic motus valde facilis, exi, imaverunt fieri posse, ut hoc paclo magna pondera moveretur. Vocatur autem tale lignum Vectis, sive quadratum sit, sive rotundum, & quanto propinquius oneri poni- tur hypomochlium, tanto faciliùs pondus movetur. Hæc ille vectis ortum & procreationem quodammodo indigitans. Contingere quidem potest, ut Vecte aliquando utamur ad sustinendum ingens pondus, non autem ad movendum, adeò ut potentia exigua sustinens, in alterâ vectis extremitate posita, ZZ

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MECHANICORUM BOOK FOUR. On the Lever. Up to this point nothing has been said, except perhaps incidentally, about instruments suitable for moving weights: now we come to explain those to which the ancients gave the name of powers. Although, however, Pappus in book 8 of the Collection of Mathematics does not assign the first place among the five powers to the Lever, it has nevertheless seemed fitting to discuss the Lever before the other powers, since it is very easy to prepare and most ready for immediate use, and it may be regarded, as the same Pappus says, as perhaps a premeditation of motion around weights that exceed us. For some, when proposing to move large weights (since one must first raise them from the ground, and they had no handles), made the whole base of the weight rest on the ground a little dug away, and inserting the end of a long piece of wood beneath the load, then, from the other end, placing under the wood near the load a stone called the hypomochlion, they drew upon it. And when this motion seemed to them very easy, they judged that it could be done in this way to move great weights. Such a piece of wood is called a lever, whether it be square or round; and the nearer the hypomochlion is placed to the load, the more easily the weight is moved. Thus he, indicating, as it were, the origin and generation of the lever. Indeed, it can happen that we use a Lever sometimes to support an immense weight, not however to move it, so that a small force, sustaining it, placed at the other end of the lever,

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Mechanicorum 362 habeat rationem æquipondij retinentis pondus in oppositâ ex- tremitate collocatum: & tunc locum habet Aristotelis senten- tia Mechan. quæst. 3. dicentis, Ipse vectis est in causâ libra existens, sartum infernè habens, in inæqualia divisa; hypomochion enim est sartum, ambo namque sunt ut centrum. Verùm cùm propriè, & pressè tunc facultas esse non videatur, neque exerceat munus vectis, quia non movet, sed sit quasi jugum stateræ; frustra Vectis quâ vectis est, ad libram revocatur: præsertim cùm ali- quod vectis genus sit, in quo nullum libræ vestigium depre- hendi potest, etiamsi pondus cæteroqui ruiturum sustineat; si nimirum pondus ipsum inter vectis extremitates constitutum sustineatur, aut potentia ipsa sustentans medium locum occu- pet inter pondus & hypomochlium, ut infra dicetur. Quid enim pariter non revocetur libra aut statera ad Vectem, si ex altera jugi extremitate pondus addatur, quod ad oppositum pondus majorem habeat Rationem, quàm libræ, aut stateræ brachia reciprocè sumpta? tunc enim (quasi stateræ aut libræ centrum motûs esset hypomochlium) sequitur motus prout ex vecte. Quemadmodum igitur libra aut statera ad ponderum æquilibrium institutæ, non verò ad eorum motum, libræ aut stateræ munus non exercent in motu, quâ motus est; ita pari- ter vectis hypomochlium inter extremitates habens non exer- cet munus vectis in quiete: alioquin & vectis ad libram, & vi- cissim libra ad vectem absurdo circulo revocaretur. Adde verò genus hoc vectis hypomochlium inter extremitates habentis, si adhibeatur ad onus in plano horizontali movendum, non verò ad illud sustentandum, nihil habere commercij cum librâ, onus si quidem nullam exercet vim suæ gravitatis adversùs ipsum vectem, nam cessante potentiâ onus illicò quiescit; at in libra sublato æquipondio pondus descendit. Quid si vecte utamur ad corpus leve infra aquam deprimendum? an erit illa libra in- versa? Non igitur me frustra conficiam labore enitens rationes libræ in vecte recognoscere, sed ipsum per se considerans, quæ opportuniora censuero, disputabo. CAPUT

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Mechanics 362 take account of a balance, retaining the weight placed at the opposite extremity: and then Aristotle’s statement in the Mechanical Questions, question 3, is applicable, where he says, “The lever itself is in the case of a balance, having a support beneath, divided into unequal parts; for the hypomochlion is the support, since both are as a center.” But since, properly and strictly speaking, it does not then seem to have the faculty, nor perform the office, of a lever, because it does not move but is as it were the beam of a scale, it is in vain to refer the lever, insofar as it is a lever, to the balance; especially since there is a certain kind of lever in which no trace of a balance can be perceived, although it sustains a weight that would otherwise fall; namely, if the weight itself is sustained while placed between the extremities of the lever, or if the sustaining power itself occupies the middle place between the weight and the hypomochlion, as will be said below. For why should not the balance or scales likewise be referred to a lever, if to one extremity of the beam a weight is added which, in relation to the opposite weight, has a greater ratio than the arms of the balance or scales taken reciprocally? For then, as though the hypomochlion were the center of motion of the scales or balance, the motion follows as from a lever. Just as therefore the balance or scales, instituted for the equilibrium of weights, and not for their motion, do not perform the office of a balance or scales in motion, as motion is such; so likewise a lever having the hypomochlion between its extremities does not perform the office of a lever in rest: otherwise both the lever would be referred to the balance and, in turn, the balance to the lever, in an absurd circle. Moreover, add that this kind of lever having the hypomochlion between its extremities, if it is employed to move a load on a horizontal plane, and not to sustain it, has nothing in common with a balance, since the load exercises no force of its gravity against the lever itself, for when the power ceases the load immediately comes to rest; but in a balance, when the counterweight is removed, the weight descends. What if we use a lever to force a light body downward beneath water? Will that be an inverted balance? Therefore I shall not labor in vain trying to recognize the reasons of the balance in the lever, but, considering the lever by itself, I shall discuss whatever I judge to be more appropriate. CHAPTER

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Liber quartus. CAPUT I. 363 CAPUT I. Vectis forma, & vires explicantur. Vectis ob id ipsum quia Vectis est & Facultas mechanica, longitudo quædam est, in qua tria puncta assignantur, pri- mum Potentiæ moventi, alterum Ponderi movendo, tertium Fulcro, seu Hypomochlio, cui innixus vectis tanquam ex centro duos arcus describens duplicem motum definit, Poten- tiæ videlicet & Ponderis, pro variâ illorum ab eodem fulcro distantiâ. Hinc quia tripliciter in hac longitudine tria hæc puncta disponi possunt, tria oriuntur vectis genera. Primum est vectis genus, cùm extremitates occupantur à Potentia A & Pondere B, medius locus Hypomochlio C cedit. Secundum genus est, cum extremitati alteri F innititur vectis, alteri Potentia D adjungitur, & inter utramque extremitatem collocatur Pondus E. Tertium genus est, cum Potentia & Pondus loca secundi generis invicem permutant, Potentia G videlicet in medio, Pondus H in extremitate constituitur, manente alterâ extremitate I tanquam motuum centro. Cum itaque nulla alia fieri possit trium hujusmodi punctorum diversa dispositio, patet tria solùm Vectis genera excogitari potuisse: quod enim quartum Vectis genus, scilicet inflexum R S V comminisci quibusdam placuit, omnino ineptum est, quippe quod à primo genere nihil differt, nisi quia, loco subjecti fulcri, adnexum habet hypomochlium inter extremitates constitutum in S, ubi sinuatur in angulum, cui in motu innititur. Quemadmodum autem inter hæc tria Vectis genera dissimilitudo, ita non modica inter eorum vires discrepantia interce- Zz 2

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Book Four. CHAPTER I. 363 CHAPTER I. The form of the lever and its powers are explained. A lever is so called for this very reason, because it is a lever and a mechanical faculty; it is a certain length in which three points are assigned: the first, to the moving power; the second, to the weight to be moved; the third, to the fulcrum, or hypomochlion, upon which the lever rests, and, as if from a center, describes two arcs, thereby determining a double motion, namely that of the power and that of the weight, according to their varying distance from the same fulcrum. Hence, because these three points may be arranged in this length in three ways, three kinds of lever arise. The first is the kind of lever when the ends are occupied by the power A and the weight B, and the middle place is given to the hypomochlion C. The second kind is when the lever rests on one end F, the power D is applied to the other, and the weight E is placed between the two ends. The third kind is when the power and the weight exchange the places of the second kind, namely the power G is placed in the middle, the weight H at the end, while the other end I remains as the center of motion. Since therefore no other different arrangement of these three points can be made, it is clear that only three kinds of lever could have been devised; for that fourth kind of lever, namely the bent R S V, which some have pleased themselves to invent, is altogether inept, since it differs from the first kind in nothing except that, instead of a supporting fulcrum, it has attached to it a hypomochlion set between the ends at S, where it bends into an angle, upon which it rests in motion. And just as there is dissimilarity among these three kinds of lever, so there is also no slight difference in their powers between them— Zz 2

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Mechanicorum 364 dit. Primum enim genus, si ab hypomochlio inæqualiter di- vidatur longitudo vectis, ut ab eo plus distet Potentia, quàm Pondus, juvat Potentiam; secus verò, si Potentia & Pondus æqualibus intervallis ab hypomochlio absint, aut propior sit Potentia quàm Pondus; Potentiæ etenim tunc vectis vel nihil affert adjumenti, vel plurimum detrimenti. Secundum genus Potentiæ laborem semper minuit, Tertium semper auget. Quo- nam id pacto contingat, manifestum fiet, si vectis vires unde ortum habeant, aperiamus. Certum est fieri non posse, ut pondus aliquod per vim mo- veatur, nisi potentiæ moventis virtus superet ponderis resisten- tiam; si enim pari conatu confligerent, anceps esset victoria, & nullus esset motus; multo minùs à potentiâ infirmiore, quàm par sit, vinci poterit innata ponderis propensio. Hoc igitur ipso quod motus efficitur, argumento est potentiæ virtutem re- sistentiâ ponderis esse majorem: Quod verò pondus eodem temporis intervallo plus spatij aut minus decurrat, pro ratione excessûs virium potentiæ supra ponderis resistentiam definitur; nam si perexiguus fuerit excessus, movebitur quidem pondus, sed tardè; sin autem potentiæ virtus longè excedat ponderis vires, eam celerior motus consequetur. Et hæc quidem intel- ligi hactenus velim, quando potentia & pondus juxta æqualem spatij longitudinem pari velocitate promoventur, ut ipsa expe- rientia omnibus manifestum facit; nemo siquidem dubitat, an currus à validioribus equis celerius quàm à debilibus canthe- ris trahatur; & à robustiore bajulo citiùs quàm ab imbecillio- re onus in destinatum locum transferri quotidie videmus. Ut igitur vecte pondus moveri valeat, lex hæc eadem stabi- lis & firma permaneat, necesse est, ut ponderis resistentia mi- nor sit virtute potentiæ moventis. Quia verò resistentia com- ponitur ex innatâ ponderis gravitate, & ex motûs violenti tar- ditate aut velocitate, hoc est ex motûs hujusmodi quantitate intra datam temporis mensuram; propterea ita duo hæc tempe- rari oportet, ut quod alteri additur, alteri dematur; ne adeò resistentia augeatur, ut jam minor non sit virtute potentiæ. Quare in vecte, cujus extremitati A potentia applicatur certæ virtutis, ita statuendus est hypomochlio C locus, ut compara- to motu potentiæ in A cum motu ponderis in B, ea sit motûs B tarditas;

Transcription: Translated (English)

Mechanicorum 364 says. For in the first kind, if the length of the lever is divided unevenly from the fulcrum, so that Power is farther from it than Weight is, it assists Power; otherwise, if Power and Weight are at equal intervals from the fulcrum, or if Power is nearer than Weight, the lever then brings Power either no help at all, or the greatest detriment. The second kind always diminishes the labor of Power; the third always increases it. By what means this happens will be made clear if we explain whence the force of the lever arises. It is certain that it is impossible for a weight to be moved by force unless the power of the moving agent surpasses the resistance of the weight; for if they were to contend with equal effort, the victory would be uncertain, and there would be no motion; much less could a weaker power than is equal overcome the innate tendency of the weight. Therefore, the very fact that motion is produced is proof that the force of the power is greater than the resistance of the weight. But the fact that the weight travels more or less distance in the same interval of time is determined according to the ratio of the excess of the power’s force over the weight’s resistance; for if the excess be very small, the weight will indeed move, but slowly; but if the force of the power greatly exceeds the force of the weight, a swifter motion will follow. And this, indeed, I would have understood thus far: when power and weight are moved side by side with equal speed over an equal length of space, as experience itself makes clear to everyone; for no one doubts whether a carriage is drawn more quickly by stronger horses than by weak ones, and we daily see a burden carried to its appointed place more quickly by a sturdy porter than by a feeble one. Therefore, in order that a weight may be able to be moved by a lever, this same law must remain fixed and firm: it is necessary that the resistance of the weight be less than the force of the moving power. But since resistance is made up of the innate heaviness of the weight, and of the slowness or speed of violent motion, that is, of the quantity of such motion within a given measure of time, these two things must therefore be so balanced that what is added to the one is taken away from the other; lest resistance increase so much that it is no longer less than the force of the power. Wherefore, in a lever to whose extremity A a power of certain strength is applied, the place of the fulcrum C must be so set that, when the motion of the power at A is compared with the motion of the weight at B, there is such slowness of the motion of B;

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Liber quartus. CAPUT I. 365 tarditas; quæ addita gravitati ponderis B resistentiam compo- nat minorem virtute movendi potentiæ A. Quoniam enim, manente puncto C tanquam centro motûs potentiæ descenden- tis & ponderis ascendentis, manifestum est eam esse motuum Rationem, quæ est Radiorum CA & CB idcirco quò major erit hujusmodi Radiorum inæqualitas, eò etiam major erit Ra- tio motûs potentiæ ad motum ponderis, cujus tarditas gravi- tatem compensans minuet resistentiam, ut virtuti potentiæ, pro- portione respondeat. Hîc verò, si rem paulò attentiùs introspicias, deprehendes tamdiu solùm admirationi esse machinarum vires, quamdiu causâ occulta manet; quæ si in medium proferatur, admiratio- ni nobis est ipsa nostra admiratio. Aio igitur potentiam tan- tumdem planè motûs in pondere efficere cum vecte conjunctam (idem de cæteri pariter Facultatibus intelligatur, ne idem sæ- piùs ad nauseam inculcare oporteat) ac si solitaria eodem cona- tu pondus aliquod secum pari velocitate adduceret, aut eleva- ret. Sit potentia A æqualiter, ac pondus B, distans à fulcro C; & quo conatu movetur potentia descendens spatio digitorum decem, dum arteria bis pulsat; cogat oppositum pondus libræ unius ascendere pariter eodem tempore per digitos decem; esse enim æquales oppositos hujusmodi motus, qui ex æqualibus Radiis arcus æquales describunt, certum est. Iam manente Ra- dio CA, singe Radium CB mutilum atque decurtatum adeò, ut sola ejus pars decima reliqua sit, & CB ponderis distantia ab hypomochlio sit subdecupla distantiæ CA potentiæ ab eo- dem hypomochlio: erit igitur motus in B subdecuplus motûs in A. Quare pondus unius libræ in hac subdecuplâ distantiâ cùm subdecuplo tardius moveatur (percurrit enim tempore eo- dem spatium subdecuplum) indiget solùm subdecuplo impetu ejus, quem prius exigebat, ut æqualiter cum potentiâ move- retur. Totus igitur impetus ille, quem potentia ponderi unius libræ imprimebat, ut æquali velocitate pariter moverentur, illa descendendo, hoc ascendendo, si decem ponderibus similibus distribuatur, satis est, ut omnia illa moveantur subdecuplâ ve- locitate. Quia autem duorum arteriæ pulsuum spatio singula ascendunt digitum unum, & sunt decem ascensus digitales, dum potentia descendit digitos decem, & dum potentia primo Z z 3

Transcription: Translated (English)

Book Four. Chapter I. 365 slowness; which, added to the gravity of the weight B, may make the resistance less than the power of motion of the power A. For since, the point C remaining as the center of the descending power’s and ascending weight’s motion, it is manifest that there is that ratio of motions which is that of the radii CA and CB; and therefore the greater the inequality of such radii shall be, the greater also will be the ratio of the motion of the power to the motion of the weight, whose slowness, compensating for gravity, will lessen the resistance, so that it may correspond proportionally to the power’s force. Here, however, if you examine the matter a little more attentively, you will discover that the powers of machines are admired only so long as the cause remains hidden; but if it is brought forth into the open, our very admiration becomes an object of admiration to us. I say, therefore, that the power, when joined with the lever, produces exactly as much motion in the weight (the same should be understood of the other faculties as well, lest the same thing need to be repeated too often to the point of annoyance) as if, acting alone, with the same exertion, it were to draw along or lift some weight with equal speed. Let the power A and the weight B be equal, and let them be distant from the fulcrum C; and with what exertion the descending power is moved through a space of ten digits, while the artery beats twice, let it compel the opposite weight of one pound to ascend likewise through ten digits in the same time; for it is certain that such opposite motions are equal, since they describe equal arcs from equal radii. Now, with radius CA remaining, shorten radius CB so much, and so diminish it, that only one-tenth of it remains, and let the distance of the weight CB from the hypomochlion be one-tenth of the distance of the power CA from the same hypomochlion: therefore the motion in B will be one-tenth of the motion in A. Wherefore a weight of one pound in this one-tenth distance, since it moves one-tenth more slowly (for in the same time it traverses a one-tenth space), requires only one-tenth of the impetus it previously needed in order to move equally with the power. Therefore the whole impetus which the power imparted to the weight of one pound, so that they might move together with equal speed, this descending and that ascending, if distributed among ten similar weights, is sufficient for all of them to be moved with one-tenth the speed. But since in the time of two pulsations of the artery each rises one digit, and there are ten digital ascents, while the power descends ten digits, and while the power first Z z 3

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Mechanicorum arteriæ pulsu decurrit digitos quinque, decem illa pondera motum quinque digitorum perficiunt, singula videlicet per se- midigitum (id quod pariter observari facilè poterit in singulis minutioribus temporis particulis) tantumdem motus perficit potentia ac pondus, sive toto impetu uni libræ impresso libra una habeat motum decem digitorum, sive decimâ impetûs par- te singulis libris impressâ, singulæ habeant motum digitalem: utrobique scilicet sunt decem motus digitales, sive unius pon- deris, sive decem ponderum eodem tempore. Quis verò mi- retur, si ille idem, qui decem aureis nobili hospiti splendidio- res epulas parare posset, decem hominibus frugalem mensam instrueret singulis aureis in singulos homines tributis? Desinat igitur pariter mirari, si potentia eadem, quæ decem impetûs particulis libram unam secum pari velocitate movet, singulis particulis in singulas libras tributis moveat decem libras, sin- gulas subdecupla velocitate; neque enim hic plus conatûs, quàm ibi, requiritur. In hoc itaque Vectis vires sitæ sunt, quod ex Potentiæ & Ponderis positione ita temperantur motus, ut impetûs quem potentia ponderi imprimere valet, aut re ipsa imprimit, inten- sio respondeat tarditati aut velocitati motûs ipsius ponderis. Hinc si Potentia, & Pondus æqualibus intervalliis ab hypomo- chlio distent; motus æquales sunt; & perinde ac si potentia so- litaria sine vecte (si illa quidem vivens sit) attolleret pondus, vectis nihil juvat potentiam, quia pondus hoc recipit totam impetûs intentionem, quam illa efficere potest. Sin autem Potentia quidem magis, Pondus verò minùs à fulcro absit, tar- dior ponderis motus minorem exigit impetûs intentionem; ac proinde entitas eadem impetûs, quæ est intensivè minor, po- test fieri extensivè major, & communicari ponderi majori, ac priùs. Quare pro Ratione tarditatis motûs extenuatur impetûs intensio, atque ideò pro eadem Ratione augeri potest ponderis extensio, hoc est gravitas; ut quæ Ratio est velocitatis motûs in pondere æqualis velocitati motûs in potentiâ, ad tarditatem motûs in pondere minoris motu in potentiâ, eadem sit directè Ratio intentionis impetûs in pondere æquè veloci ad intensio- nem impetûs in pondere tardiori, & reciprocè eadem sit Ratio ponderis tardioris majoris ad pondus illud minus, quod æquè velociter

Transcription: Translated (English)

In mechanics, the pulse of the arteries runs through five fingers; these ten weights accomplish a motion of five fingers, each one, that is, by itself a semi-finger’s worth (which can likewise be easily observed in smaller parts of time), so much motion is accomplished by power as by weight: whether, with the whole impulse impressed on one pound, one pound have a motion of ten fingers, or, with a tenth part of the impulse impressed on each of ten pounds, each have a finger’s motion, in both cases there are ten finger-motions, whether of one weight or of ten weights in the same time. Who, then, would wonder if the same man who could prepare a more splendid banquet for a noble guest with ten gold coins would set a frugal table for ten men, one gold coin being assigned to each man? Let him likewise cease to wonder if the same power, which moves one pound with equal speed through ten parts of impulse, moves ten pounds, each assigned to the individual parts, with a tenfold slower speed; for no greater effort is required here than there. In this, therefore, lies the force of the Lever: that from the position of the Power and the Weight the motions are so adjusted that the degree of impulse which the power can impress on the weight, or actually does impress, corresponds to the slowness or speed of the motion of the weight itself. Hence, if Power and Weight are at equal distances from the fulcrum, the motions are equal; and it is as though the power alone, without a lever (if indeed it be living), were lifting the weight, the lever then helping the power not at all, because the weight receives the whole force of impulse that the power can produce. But if the Power is farther from the fulcrum and the Weight nearer, the slower motion of the weight requires a smaller degree of impulse; and therefore the same amount of impulse, which is intensively smaller, can become extensively greater, and be communicated to a greater weight than before. Wherefore, in proportion to the slowness of the motion, the intensity of the impulse is diminished, and therefore in the same proportion the extension of the weight, that is, its heaviness, can be increased; so that, as the ratio of the speed of motion in the weight is equal to the speed of motion in the power, and the slowness of motion in the weight is less than the motion in the power, the same ratio is directly that of the intensity of the impulse in the weight moving equally fast to the intensity of the impulse in the slower weight, and reciprocally the same ratio is that of the slower and greater weight to that lesser weight which moves equally fast

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Liber quarius. CAPUT I. 367 velociter cum potentia moveri potest. Quòd si potentia pro- pior fuerit hypomochlio, quàm pondus, potentia tardiùs, pon- dus movetur velocius: plus igitur intentionis impetûs requiri- tur in pondere quàm in potentiâ, adeò ut impetus, qui in po- tentiâ non vivente est extensivè major, intensivè minor, con- tra in pondere sit extensivè minor, intensivè major: ac propterea pondus tantò levius esse oportet pondere, quod æquè velociter cum potentia moveretur, quantò velociùs movetur præ illo æquè veloci. Non igitur vectis juvat potentiam, ut faciliùs moveat, sed movendi difficultatem auget. Id quod in Tertio vectis genere semper contingit, in quo potentia G mi- nus ab hypomochlio I distat, quam pondus H, & tardiùs mo- vetur. Accidit autem hoc idem etiam in Primo genere, cum vectis inæqualiter ab hypomochlio distinguitur in partes, si lo- ca permutentur, ut potentia propior sit, quàm pondus. His tamen uti possumus, quoties quidem viribus abundamus, sed spatium, in quo potentia moveatur, angustum est, oportet au- tem ponderi velocem motum conciliare. Contra verò in vecte Secundi generis potentia à fulero semper remotior est, quàm pondus; idcirco semper juvat potentiam; quia quo tardior est ponderis motus, eò minorem ponderis pars, quæ æqualis sit ponderi æquè veloci, exigit impetûs intentionem; ac propterea quod reliquum est impetûs à potentia producendi, pluribus aliis similibus ponderis partibus impertiri potest; atque adeò absolutè majus est pondus, quàm quod æquè velociter mo- veretur. Hæc eadem, quæ de ponderibus vecte movendis dicta sunt, intelligi pariter oportet de ponderibus vecte sustentandis citra motum; eo tantùm observato discrimine, quod ad motum ma- jor requiritur potentiæ virtus, quàm sit ponderis resistentia, in sustentatione verò par resistentiæ ponderis est virtus potentiæ. Resistentia autem in sustentatione non ex motûs tarditate aut velocitate, quæ re ipsa sit, sed ex eâ, quæ esset, si motus fieret, quatenus pondus est vecti connexum, definienda est; & pro lujusmodi momentorum Ratione, quibus pondus deorsum co- natur, etiam impetûs contraritentis intentionem dimetiri ne- cessè est. Quia igitur pondus cum vecte connexum quo propiùs ad hypomochlium accedit, eo tardiùs sibi relictum descenderet; propterea

Transcription: Translated (English)

Liber quarius. CAPUT I. 367 can be moved rapidly with power. But if the power is nearer to the fulcrum than the weight is, the power moves more slowly, the weight more quickly: therefore a greater intensity of impetus is required in the weight than in the power, so that the impetus, which in the non-living power is extensively greater, is intensively smaller, while, on the contrary, in the weight it is extensively smaller and intensively greater. And for this reason the weight ought to be so much lighter than a weight that would be moved as quickly as the power, as it is moved more quickly than that one would be moved equally quickly. The lever therefore does not aid the power so that it may move more easily, but rather increases the difficulty of moving. This always happens in the Third kind of lever, in which the power G is farther from the fulcrum I than the weight H, and is moved more slowly. But the same thing also occurs in the First kind, when the lever is unequally divided by the fulcrum into parts, if the positions are exchanged so that the power is nearer than the weight. Yet we can use these whenever indeed we have abundance of strength, but the space in which the power is moved is narrow, and it is necessary to impart rapid motion to the weight. On the contrary, in the Second kind of lever the power is always farther from the fulcrum than the weight; therefore it always aids the power, because the slower the motion of the weight is, the smaller is the part of the weight that, being equal to a weight moving equally quickly, requires a degree of intensity of impetus; and for this reason the remainder of the impetus to be produced by the power can be imparted to many other similar parts of the weight; and thus absolutely the weight is greater than one that would be moved equally quickly. These same points, which have been said concerning weights to be moved by a lever, must likewise be understood of weights to be sustained by a lever without motion; with only this difference observed, that for motion a greater force of the power is required than is the resistance of the weight, but in sustaining, the force of the power is equal to the resistance of the weight. The resistance, however, in sustaining must be determined not from the slowness or swiftness of the motion, which in fact exists, but from that which would exist if motion were taking place, insofar as the weight is connected to the lever; and according to such a ratio of moments, by which the weight tends downward, it is also necessary to measure the intensity of the opposing impetus. Since therefore a weight connected with the lever, the closer it comes to the fulcrum, would descend more slowly if left to itself, therefore

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Mechanicorum 368 propterea etiam minorem contranitentis impetus intentionem requirit: Ex quo fit eodem potentiæ conatu, quo illa pondus si- ne vecte sustineret, posse majorem ponderis gravitatem sustineri adhibito vecte, eóque majorem, quo major est Ratio distantiæ potentiæ ad distantiâ ponderis à fulcro; & vicissim potentia mi- nore conatu idem pondus sustinebit, si hoc propius admoveatur ad hypomochlium, quàm priùs, cum opus erat majore conatu. Porrò conatum potentiæ de industria dixi, ut vocabulo ute- rer, quo tum potentia vivens, tùm inanimata æquè comprehen- deretur; quia aliquando quidem potentia conatum adhibet in- natâ suâ gravitate, aliquando autem præter, aut contra gravitatis propensionem. Gravitate utitur, quæ inanima est, & vires suas exerit totas, quodcunque demum pondus vecte movendum aut sustentandum proponatur. Potentia verò vivens suo consulens commodo, ne se inani conficiat labore, non plus operæ confert, quàm opus fuerit, sed vires ex opportunitate administrat, modò majores, modò minores impendens, quippe quæ musculorum contentione voluntarios motus perficit, & non solùm deorsum premendo, sed etiam sursum connitendo, aut in transversum ur- gendo, vecte uti potest: At inanimata potentia non nisi descen- dendo vi suæ gravitatis cogere potest adversum pondus ad ascen- dendum; atque si primum vectis genus demas, cui potest illa proximè admoveri, in cæteris generibus, si attollendum sit pon- dus, artificium aliquod excogitandum est, quo interjecto, aut potentiæ virtus, aut ipsum pondus ad vectem applicetur, ut propositum finem assequamur; conatus enim potentiæ & pon- deris, licet inæquales, non tamen oppositi sunt, sed ad eandem partem sua gravitate contendunt. Sic vecte R S, cujus fulcrum sit in extremitate R, non potest pondus V attolli à po- tentia inanimata P, si proximè illi adjungatur in S; ac propterea rotu- la in T figenda est ver- satilis, & funiculo S T P jungenda potentia P, quæ deorsum connitens elevat vectem in S, at- que

Transcription: Translated (English)

Mechanics 368 therefore also requires a lesser intention of the counter-resisting impulse: from which it follows that, with the same exertion of power by which it would sustain that weight without a lever, a greater heaviness of weight can be sustained by employing a lever, and a still greater one the greater the ratio is of the distance of the power from the fulcrum to the distance of the weight from the fulcrum; and conversely, the power will sustain the same weight with less effort if this is moved closer to the hypomochlion than before, when a greater effort was required. Moreover, I have purposely said the exertion of the power, in order to use a term by which both living and inanimate power might equally be comprehended; because sometimes indeed power employs exertion by its innate gravity, but sometimes apart from, or against, the tendency of gravity. The inanimate uses gravity, and exerts all its strength, whatever weight is proposed to be moved or sustained by the lever. But living power, considering its own convenience, lest it wear itself out by useless labor, contributes no more effort than shall be necessary, but administers its strength as occasion requires, now expending more, now less; since it accomplishes voluntary motions by muscular effort, and can use a lever not only by pressing downward, but also by straining upward, or by urging sideways. But inanimate power can compel an opposing weight to rise only by descending through the force of its gravity; and if you take away the first kind of lever, to which it can be brought nearest, in the other kinds, if a weight is to be lifted, some contrivance must be devised by which, being interposed, either the force of the power or the weight itself may be applied to the lever, so that we may attain the proposed end; for the efforts of power and weight, though unequal, are nevertheless not opposed, but by their gravity strive toward the same side. Thus by a lever R S, whose fulcrum is at the end R, the weight V cannot be raised by the inanimate power P, if it be joined to it nearest at S; and therefore a wheel must be fixed at T, turning freely, and the power P joined by the cord S T P, which, straining downward, raises the lever at S, and

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Liber quartus. CAPUT I. 369 que adeò etiam pondus V. Simili ratione sit vectis Secundi generis MN, & hypomochlium in M, locus autem ponderis in H: si potentia N inanimata vecti proximè adnectatur, uti- que elevare non poterit pondus in H collocatum: quare sta- tuatur in loco superiore rotula K, & funiculo HKL jungatur pondus L cum puncto H; nam potentia N sua gravitate descen- dens deprimendo punctum H vectis elevabit pondus L. Idem continget, si vectis MN sit tertij generis, & N sit pondus at- tollendum, potentia verò inanimata collocanda sit in H. Nihil utique præstabit descendendo in H; ut igitur punctum H ascendat, rotula K adhibeatur, & à potentia L descendente elevabitur idem punctum H, ac proinde etiam pondus N. Vel si in vecte R S tertij generis statuatur potentia V, illa descen- dens deprimet velociter extremitatem S, & pari velocitate ascendet pondus P. Quid hoc simplex artificium aliquando in scenicis motionibus præstare possit emolumenti, facilè prudens machinator intelligit. Ex his, quæ de Vectis viribus explicata sunt, apertè liquet omnino veritati consentanea esse ea, quæ lib. 2. cap. 8. diximus, in rotis curruum inveniri non posse rationem vectis, quia duo tantummodo sunt puncta, scilicet extremitas Radij subjectam tellurem tangentis, & rotæ centrum, cui & innititur pondus, & medio temone applicatur potentia. Cum igitur potentia & pondus eandem habeant positionem, & æquali velocitate mo- veantur, nullum habetur ex Vectis rationibus compendium. Eatenus enim Vectis in Mechanicarum Facultatum censu nu- meratur, quoad potentia & pondus dispari celeritate moventur, vel quia potentia se velociter movens exiguo conatu tardè mo- vet pondus, ut in primo & secundo genere vectis, vel quia po- tentia se tardè movens multo conatu celeriter movet pondus, ut in tertio genere. Quare semper in motu ponderis per vectem aliquid lucri habetur, nimirum aut major ponderis gravitas, quæ movetur, aut saltem major velocitas, qua movetur. A a a

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Book Four. CHAPTER I. 369 so that likewise the weight V. In a similar way, let there be a lever of the second kind MN, with the fulcrum in M, and the place of the weight in H: if the power N, being lifeless, be attached very close to the lever, it will by no means be able to raise the weight placed in H; therefore let a pulley K be placed above, and let the weight L be joined by the rope HKL to the point H; for the power N, descending by its own weight, by depressing the point H will raise the weight L. The same will happen if the lever MN be of the third kind, and N be the weight to be lifted, while the lifeless power is to be placed in H. In that case, descending in H, it will accomplish nothing; so that the point H may rise, let the pulley K be employed, and by the descending power L the same point H will be raised, and consequently also the weight N. Or if in the lever R S of the third kind the power V be placed, that power descending will quickly depress the extremity S, and with equal speed the weight P will rise. What advantage this simple device may sometimes produce in theatrical motions, a prudent mechanician can easily understand. From what has been explained concerning the forces of levers, it is plainly evident that what we said in book 2, chapter 8, is altogether in agreement with the truth: namely, that in the wheels of carts no ratio of the lever can be found, because there are only two points, namely the extremity of the radius touching the ground beneath, and the center of the wheel, on which the weight rests, while the power is applied to the middle of the shaft. Since therefore the power and the weight have the same position and move with equal speed, no advantage is obtained from the laws of the lever. For the lever is numbered among the mechanical powers only insofar as power and weight move with unequal speed, either because the power, moving swiftly, moves the weight slowly with little effort, as in the first and second kind of lever, or because the power, moving slowly, moves the weight swiftly with much effort, as in the third kind. Therefore, in the motion of a weight by means of a lever, there is always some gain, namely either the greater heaviness of the weight that is moved, or at least the greater speed with which it is moved. A a a

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Mechanicorum CAPUT II. Quid in hypomochlij collocatione sit observandum. Tria in Vecte, ut dictum est, puncta constituuntur & de- signantur duo quæ moventur, tertium illorum motuum centrum, quod alicui corpori innititur, ut vectis consistat, nec à ponderis gravitate, aut à potentiæ vi abripiatur: huic cor- pori Hypomochlio nomen inditum est à Græcis, quasi (si verbum è verbo volumus) subvectis, nam ut plurimum vecti subjicitur, nos Fulcrum dicimus, quia vectem sibi incumbentem fulcit. Cæterùm non est hæc constans, & perpetua hujus corporis positio, ut sub vecte sit, quamvis semper Hypomochlij aut Ful- cri nomine donetur; quandoquidem in vecte tertij generis, ubi pondus in extremitate est, potentia medium locum obtinet, si infra alteram vectis extremitatem esset corpus hujusmodi, uti- que à potentia nequiret attolli pondus, ut patet: in superiore igitur parte sit oportet, ut potentiâ sursum conante, pondere deorsum contranitente, impediatur altera vectis extremitas, ne frat totius vectis conversio obsecundans aut potentiæ conatui, aut gravitati ponderis, quod esset attollendum. Quod si hoc vecte tertij generis deprimendum esset infra aquam per vim corpus aliquod leve, tunc sub vecte constitueretur hypomo- chlium: contrà vectis primi & secundi generis si ad premen- dum aut deprimendum adhibeatur, exigit hypomochlium in superiori parte. Similiter non est sub vecte, sed ad latus adja- cet, quoties pondus est movendum in plano horizontali, sive in eodem plano sit vectis, sive in plano verticali, ut cùm duo mar- mora non elevanda sunt, sed immisso inter illa vecte invicem disjungenda. Quemadmodum igitur lapis à lædendo pedem vocabulum habet, etiamsi non lapides omnes pedem lædant; ita corpus illud, cui punctum vectis quiescens innititur, hypo- mochlij & fulcri nomen retinet, quamvis non semper sub vecte sit, illumque suffulciat. Quid autem profuerit immutare vo- cabula,

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Mechanics CHAPTER II. What is to be observed in the placement of the fulcrum. As has been said, three points are established and marked in a lever: two that move, and the third, the center of those motions, which rests upon some body, so that the lever may remain steady and be neither carried away by the weight of the load nor by the force of the power. From the Greeks this body has been given the name hypomochlion, as it were, if we want a word-for-word rendering, “supporting-under,” for for the most part it lies beneath the lever; we call it fulcrum, because it sustains the lever resting upon it. However, this position of the body is not fixed and perpetual, namely, that it should be under the lever, although it is always called by the name hypomochlion or fulcrum; since in a lever of the third kind, where the weight is at the end and the power occupies the middle place, if a body of this sort were beneath the other end of the lever, the weight could not indeed be lifted by the power, as is evident. Therefore it ought to be in the upper part, so that, while the power strives upward and the weight resists downward, the other end of the lever may be held back, lest the turning of the whole lever, corresponding either to the effort of the power or to the gravity of the weight to be lifted, take place. But if in this third kind of lever a light body were to be depressed below water by force, then the hypomochlion would be placed under the lever; on the contrary, in a lever of the first and second kind, if it is employed to press or depress, the hypomochlion is required in the upper part. Likewise it is not beneath the lever, but lies at the side, whenever a weight is to be moved in a horizontal plane, whether the lever itself is in the same plane or in a vertical plane, as when two marbles are not to be lifted, but are to be separated from one another by inserting a lever between them. Thus, just as a stone has its name from harming the foot, although not all stones harm the foot, so that body upon which the resting point of the lever rests retains the name hypomochlion and fulcrum, even though it is not always under the lever and supporting it. But what use would there be in changing the words,

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Liber quartus. CAPUT II. 371 cabula, ubi rem ipsam tenemus? Immò punctum ipsum vectis quiescens, quod hypomochlio respondet, non rarò ab iis hy- pomochlium dicitur, aut fulcrum, qui verborum compendio claritati consultum volunt; mihique hanc loquendi facultatem, ubi res tulerit, resero. Quiescens autem voco punctum vectis, quod est centrum motuum potentiæ, & ponderis; non quia semper omnino quiescat, sed quia si aliquo motu moveatur, tardissimum certè est omnium punctorum; cætera quippe vectis puncta circa hoc tanquam circa centrum describunt lineam inflexam ac recur- vam: alioquin si punctum hoc plus moveretur quàm pondus, mutatæ fuissent vices, & quod pondus dicitur, esset reipsa hy- pomochlium, corpus verò, quod hypomochlium dicitur, esset pondus, quod à potentiâ potissimum moveretur. Observandum enim est non pondus solum, verùm etiam hypomochlium acci- pere vim externam potentiæ vectem agitantis, resistente vide- licet pondere, ex quo fit illud premi; quod si inæqualiter re- sistant, licet utrumque moveatur, in illud potiùs exercet vir- tutem suam potentia, quod languidiùs resistit, altero validiore hypomochlij rationem habente. Sic vecti ad attollendum mar- mor applicato si glebam, hypomochlij loco, supposueris, non marmor attolles, sed glebam vecte conteres: marmor igitur est hypomochlium vecti superpositum, & glebæ est pondus contri- tum vecte secundi generis: At si pro gleba lignum subjicias, quod non frangatur, sed aliquantulum cedens comprimatur, & vectis vestigium recipiat, ita tamen, ut marmor moveatur, du- plex vectis genus hîc intercedit, prout duplex effectus poten- tiæ conatum consequitur; ad comprimendum scilicet lignum vectis est secundi generis hypomochlium habens impositum marmor, ad elevandum autem marmor vectis est primi generis, cujus hypomochlium est subjectum lignum. Cujusmodi sit hy- pomochlium, sive sit funis vectem retinens, sive axis infixus, circa quem volvatur vectis, sive quodcumque aliud corpus, cui ille incumbat, aut innitatur, modò absit incommodi periculum ex ejusfragilitate, parum refert: satis est, si par fuerit ferendo oneri, quod vecte elevatur. Ex ponderis autem gravitate hy- pomochlij soliditas atque materies definienda est; ex motûs A a a 2

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Book Four. CHAPTER II. 371 Why do we seek the thing itself? Nay, even the resting point of the lever, which corresponds to the hypomochlion, is not infrequently called by them the hypomochlion, or fulcrum, by those who wish to provide for clarity by means of brevity of expression; and I reserve for myself this way of speaking, when the matter shall require it. I call the resting point of the lever that which is the center of the motions of power and weight; not because it always rests entirely, but because if it is moved by some motion, it is certainly the slowest of all points; for the other points of the lever, around this one as around a center, describe a bent and curved line: otherwise, if this point were moved more than the weight, the roles would have been changed, and what is called the weight would in fact be the hypomochlion, while the body that is called the hypomochlion would be the weight, which would be moved chiefly by the power. For it must be observed that not only the weight, but also the hypomochlion, receives the external force of the power moving the lever, namely the weight resisting it, from which it comes about that it is pressed; and if they resist unequally, although both are moved, the power exerts its force rather on that which resists more feebly, the other having the stronger role of hypomochlion. Thus, if a lever applied to lift marble, with a clod placed under it in the place of the hypomochlion, you will not lift the marble, but will crush the clod with the lever: therefore the marble is the hypomochlion placed upon the lever, and the clod is the weight crushed by the lever of the second kind. But if instead of the clod you place beneath it wood, which is not broken, but, yielding somewhat, is compressed and receives the impression of the lever, yet so that the marble is moved, here a double kind of lever intervenes, insofar as the double effect follows the effort of the power; namely, for compressing the wood the lever is of the second kind, having the marble placed upon it as hypomochlion, but for elevating the marble the lever is of the first kind, whose hypomochlion is the wood placed beneath it. What sort of thing the hypomochlion may be, whether it be a rope holding the lever, or an axis fixed in place around which the lever turns, or any other body on which it rests or leans, provided there is no danger of inconvenience from its fragility, matters little: it is enough if it is equal to bearing the burden that is lifted by the lever. But from the weight of the burden the solidity and material of the hypomochlion must be determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; from the motion of the burden, the solidity and material of the hypomochlion are determined; A a a 2

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Mechanicorum 372 qualitate (spectatâ loci, in quo perficiendus est, positione) forma hypomochlij statuatur. Illud examinandum videtur, quandónam præstet uti vecte primi generis, quando vecte Secundi generis, hoc est an plus commodi afferat fulcrum in vectis extremitate collocatum, ut in secundo genere, an verò inter pondus atque potentiam interjectum, ut in primo genere. Proposita sit vectis longitudo decem palmorum, quo oporteat pondus ita attollere, ut ejus motus sit respondens arcui descripto ex Radio duorum palmorum. Si vectis sit primi generis, pondus & potentia sunt in vectis extremitatibus, hypomochlium dividit totam longitudinem in partes duas, quarum major ad potentiam spectans est quadrupla minoris spectantis ad pondus; est scilicet illa octo, hæc duorum palmorum. At si vectis fuerit secundi generis, hypomochlium & potentia illius extremitates occupant, pondus ab hypomochlio distat palmos duos: quare potentiæ distantia ab hypomochlio cum sit tota vectis longitudo, est quintupla distantiæ ponderis. Cum igitur ponderis motus cum potentiæ motu comparatus hic quintuplo tardior sit, ibi verò solum quadruplo tardior, minore impetu indiget, ut moveatur vecte secundi generis. Cæterùm considerato hoc duplici vectis genere, observandum est in secundo generê à potentiâ elevandum non solum pondus sed etiam vectem ipsum, qui si valde gravis sit (ut aliquando contingere potest trabem fungi vectis munere) auget potentiæ movendi difficultatem: Contra verò in vecte primi generis ipsa vectis gravitas juvat potentiam; & quidem si homo sit, qui vectem premat, ipsa corporis gravitas accessionem facit, ad impetum, qui à vitali conatu oritur: præterquam quod hic liberè & facillimè potentiam inanimatam adhibere possumus, & aliam atque aliam adjicere prout opus fuerit; at non item in vecte secundi generis, nisi adhibito artificio, de quo superiori capite dictum est. Datâ igitur ponderis movendi gravitate, & datâ potentiæ virtute (quæ videlicet tanto conatu adhibito potest certam gravitatem sola sine vecte movere in simili plano sive horizontali, sive inclinato, sive verticali) distinguatur vectis in duas partes ita, ut vel pars ad partem, si sit primi generis, vel totus ad par- tem,

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Mechanics 372 the form of the hypomochlion should be determined according to the quality of the place, with regard to the position in which it is to be completed. It seems worth examining when it is preferable to use a lever of the first kind, and when a lever of the second kind; that is, whether it brings more advantage for the fulcrum to be placed at the end of the lever, as in the second kind, or between the load and the power, as in the first kind. Let the length of the lever be proposed as ten palms, and let it be necessary to raise the load in such a way that its motion corresponds to an arc described from a radius of two palms. If the lever is of the first kind, the load and the power are at the extremities of the lever; the hypomochlion divides the whole length into two parts, of which the greater, relating to the power, is four times the lesser, relating to the load; namely, that one is eight palms, this one two palms. But if the lever is of the second kind, the hypomochlion and the power occupy its extremities, and the load is distant from the hypomochlion by two palms: wherefore, since the distance of the power from the hypomochlion is the whole length of the lever, it is five times the distance of the load. Therefore, since the motion of the load compared with the motion of the power is here five times slower, and there only four times slower, a smaller impulse is required for it to be moved by a lever of the second kind. Moreover, having considered this double kind of lever, it must be observed that in the second kind not only the load but also the lever itself is to be lifted by the power; and if this is very heavy, as at times can happen when a beam performs the office of a lever, it increases the difficulty of moving on the part of the power. On the contrary, in the first kind of lever, the very weight of the lever assists the power; and indeed, if it is a man who presses the lever, the weight of his body itself adds something to the impulse arising from vital effort. Moreover, here we can freely and easily apply an inanimate power, and add one after another as needed; but this is not so in a lever of the second kind, unless by the use of some contrivance, of which mention was made in the preceding chapter. Therefore, given the heaviness of the load to be moved, and given the strength of the power (which, namely, by employing so much effort, can move a certain weight by itself without a lever, on a similar plane, whether horizontal, inclined, or vertical), let the lever be divided into two parts so that either part to part, if it is of the first kind, or the whole to part,

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Liber quartus. CAPUT II. 373 tem, si sit secundi generis, eandem Rationem habeat, quæ est dati ponderis ad pondus, quod à potentiâ solâ sine vecte po- test moveri. Sic data Potentia virtutem habeat movendi pon- dus lib. 6. certo conatu, oporteat autem hoc eodem conatu movere lib. 30: quia virtus potentiæ est subquintupla pon- deris dati, propositus vectis intelligatur primùm distinctus in partes sex, quarum una tribuatur distantiæ ponderis ab hypomochlio, reliquæ quinque tribuantur distantiæ poten- tiæ, ita ut reciprocè sit distantia potentiæ ad distantiam ponderis, ut pondus datum ad virtutem potentiæ: & hic est vectis primi generis. Deinde ut habeatur vectis secun- di generis, distinguatur totus vectis in partes quinque, & una ex illis sit distantia ponderis ab hypomochlio in vectis extremitate constituto. In utroque enim casu motus poten- tiæ est quintuplus motûs ponderis, atque adeò potentia poterit vecte movere pondus quintuplum ponderis, quod so- la potest movere. Potentiæ virtutem dixi, non potentiæ gravitatem, tùm quia non omnis potentia vim movendi habet ex gravitate, tum quia potentiæ gravitas movere non potest gravitatem æqualem, sed minorem, nam cum æquali facit æquili- brium, & solùm potest illam suspendere. Quare si poten- tia vi suæ gravitatis moveat, non satis erit, si fiat ut po- tentiæ gravitas ad ponderis gravitatem, ita reciprocè pon- deris distantia à centro motûs ad distantiam potentiæ ab eo- dem centro; sed distantia ponderis ad distantiam potentiæ exigit habere minorem Rationem. Hinc si potentia sit pon- deris subquintupla ratione suarum gravitatum, pondus ab hypomochlio distare debet minus quàm parte quinta distan- tiæ potentiæ ab eodem hypomochlio. Quod si vectis is esset, cujus gravitas notabile momentum adderet potentiæ, tunc distantia ponderis, quæ esset subquintupla distantiæ potentiæ, sufficeret, minor enim esset Ratione potentiæ adæquatè ac- ceptæ ad Pondus. Ubi verò ponderis gravitatem considerare oportet, non satis est illam notam habere, ac si staterâ expenderetur, sed considerandum est planum, in quo illud movendum est; neque enim eadem habet momenta, si sursum elevandum sit Aaa 3

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Book Four. CHAPTER II. 373 then, if it be of the second kind, it have the same ratio which is that of the given weight to the weight which can be moved by the power alone without the lever. Thus, let the given power have the ability to move a weight of 6 lbs. with a certain effort, and it ought to move 30 lbs. with the same effort: because the force of the power is one-fifth of the given weight. Let the proposed lever be first understood to be divided into six parts, of which one is assigned to the distance of the weight from the fulcrum, the remaining five to the distance of the power, so that reciprocally the distance of the power to the distance of the weight is as the given weight to the force of the power: and this is a lever of the first kind. Then, in order to have a lever of the second kind, let the whole lever be divided into five parts, and let one of them be the distance of the weight from the fulcrum, with the fulcrum placed at the end of the lever. For in either case the motion of the power is five times the motion of the weight, and therefore the power will be able by means of the lever to move a weight five times greater than the weight which it alone can move. I said the force of the power, not the heaviness of the power, both because not every power has the force of moving by reason of its heaviness, and because the heaviness of the power cannot move a heaviness equal to it, but only a lesser one; for with an equal one it makes equilibrium, and can only suspend it. Wherefore if the power, by the force of its own heaviness, move, it will not be enough if the ratio be made such that the heaviness of the power is to the heaviness of the weight, so reciprocally is the distance of the weight from the center of motion to the distance of the power from the same center; but the distance of the weight to the distance of the power requires a smaller ratio. Hence, if the power be in the one-fifth ratio of the weight in respect of their heavinesses, the weight ought to be distant from the fulcrum by less than one-fifth part of the distance of the power from the same fulcrum. But if the lever were such that its own weight would add a notable moment to the power, then the distance of the weight, which would be one-fifth less than the distance of the power, would suffice, for it would be less by the ratio of the power adequately taken to the weight. But where the heaviness of the weight must be considered, it is not enough to have it known merely as if it were weighed on a balance, but the plane in which it is to be moved must be considered; for it does not have the same moments if it must be lifted upward Aaa 3

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Mechanicorum in plano Verticali, ac si urgendum sit in plano inclinato, aut propellendum in horizontali: propterea in Ratione assignandâ partibus vectis non est attendenda gravitas absoluta ponderis, sed quatenus in proposito plano. Idem est de gravitate potentiæ dicendum. Ex dictis patet non quamcumque vectis longitudinem semper opportunam esse, quamvis verum sit quemlibet vectem posse secundùm quamcumque Rationem in partes distingui, atque proinde quodcumque pondus à quacumque datâ potentiâ posse moveri, si ritè applicari posset. Unum enim est incommodum, quod, quo propiùs ad centrum motuum admovetur pondus, eo minor est illius motus: & contingere potest adeò exiguam esse ponderis ab hypomochlio distantiam, ut motus adeò tenuis nulli futurus sit usui. Quapropter longiori vecte utendum erit, ut, servatâ eâdem distantiarum Ratione, intervallum inter pondus & centrum motuum sit notabile & conspicuum, ex quo motus sufficiens obtineri possit. Quid enim juvaret, si vecte palmorum 25 tentares attollere pondus centuplum virtutis potentiæ? an ut pondus ab hypomochlio distans per digitum (sumo digitos quatuor pro singulis palmis) elevaretur ad altitudinem unius aut alterius grani hordei? Præterquam quod tam ingens pondus ægrè posset in tantillo spatio ad vectem opportunè applicari. Quod autem ad hypomochlium attinet, curandum maximè est, ut qua parte vectem contingit, minimum sit, &, si fieri potest, proximè in aciem desinat; ut scilicet eandem semper in motu vectis partem contingat; si enim alia atque alia vectis pars hypomochlio insistat, mutantur ponderis atque potentiæ momenta, ideoque augeri potest movendi difficultas. Sit vectis secundi generis AB innixus saxo, quod contingit in C, & centri gravitatis ponderis locus sit D: utique quia DC minore est quàm DB, major est Ratio AB ad DC minorem, quàm ejusdem AB ad DB majorem, per 8. lib. 5. At elevato vecte, ut habeat positionem FE, si- cut

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In mechanics, in the vertical plane, as if it must be driven in an inclined plane, or propelled in a horizontal one: therefore, in assigning the ratio among the parts of the lever, one must not regard the absolute weight of the weight, but only insofar as it lies in the proposed plane. The same must be said of the gravity of the power. From what has been said it is clear that not every length of lever is always suitable, although it is true that any lever can be divided into parts according to whatever ratio, and consequently that any weight can be moved by any given power, if it could be rightly applied. For there is one inconvenience: the closer the weight is brought to the center of motion, the smaller its motion is; and it can happen that the distance of the weight from the fulcrum is so small that the motion, being so slight, will be of no use. Therefore a longer lever must be used, so that, while preserving the same ratio of distances, the interval between the weight and the center of motion may be notable and conspicuous, from which a sufficient motion can be obtained. For what would it profit, if with a lever of 25 palms you tried to lift a weight a hundred times the force of the power? Would it be so that the weight, being distant from the fulcrum by a finger's breadth (I take four fingers for each palm), would be raised to the height of one or another grain of barley? Moreover, so huge a weight could hardly be suitably applied to the lever in so small a space. As for the fulcrum itself, care must be taken especially that where the lever touches it, the contact be as small as possible, and, if possible, end nearly in an edge; so that it may always touch the same part of the lever in motion. For if now one part and now another of the lever rests upon the fulcrum, the moments of the weight and of the power are changed, and therefore the difficulty of moving may be increased. Let there be a lever of the second kind AB, resting on a stone, which it touches at C, and let the location of the center of gravity of the weight be D: certainly, because DC is less than DB, the ratio of AB to the lesser DC is greater than that of the same AB to the greater DB, by 8. book 5. But when the lever is raised so that it has the position FE, as

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Liber quartus. CAPUT II. 375 cut A venit in F, ita B venit in E, ubi saxo innititur, & pon- dus D venit in G. Est igitur FE ad GE, ut AB ad DB; ergo etiam FE ad GE habet minorem Rationem quàm AB ad DC. Quo autem minor est motuum Ratio, eò etiam minus est po- tentiæ momentum ad momentum ponderis; igitur si Ratio AB ad DB minor sit, quàm AC ad DC, etiam Ratio FE ad GE minor erit quàm Ratio AC ad DC. Quare tunc solùm ea- dem movendi facilitas manebit (quod quidem spectat ad ra- tionem hypomochlij quicquid sit an ex alio capite mutetur, ut infra) quando CB pars extrema vectis, quæ innititur hypo- mochlio, ea est, ut eadem sit Ratio AB ad DB, quæ est AC ad DC: Hoc autem fieri omnino non potest, quia AB & DB sunt idem ac AC, atque DC, si his utrisque addatur eadem pars CB. Si ergo ut AC plus CB ad DC plus CB esset ut AC ad DC, etiam permutando, & dividendo, & iterum per- mutando, per 16. & 17. lib. 5. esset ut AC ad DC ita CB ad CB, ac propterea AC totum æquale esset parti DC. Non igitur fieri potest, ut maneat in motu eadem facilitas ratione hypomochlij, si accidat, ut vectis positiones in motu se de- cussent; id quod evenit, si alia atque alia pars vectis hypomo- chlium tangat. Et quia major est Ratio totius AB ad totam DB, quàm sit ablatæ CB ad ablatam CB, erit etiam, per 33. lib. 5. reliquæ AC ad reliquam DC major Ratio quàm totius AB ad totam DB, hoc est major Ratio quàm FE ad GE. Similiter in vecte primi generis, si fulcrum sit cylindricum, tangit quidem in puncto, sed dum vectis deorsum urgetur, aliud atque aliud ejus punctum aliis cylindri punctis congruit: nam si fuerit potentia in C, & pondus in E, vectis autem tangat in F, in con- versione cum E venerit in I, & C in L, jam contactus sit in H ita, ut HL minor sit quàm FC, contrà verò HI major sit quàm FE. Decrescunt ergo potentia momenta, cujus distantia à motûs centro minuitur, augentur autem ponderis momenta, cujus distan- tiæ à motûs centro aliquid semper accedit. Et quidem quò crassior fuerit cylindrus, factâ pari vectis inclinatione, major etiam oritur distantiarum differentia; ut facilè demonstratur, si

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Book Four. Chapter II. 375 but A comes into F, so B comes into E, where it rests on the stone, and the weight D comes into G. Therefore FE is to GE as AB is to DB; therefore also FE to GE has a smaller ratio than AB to DC. But the smaller the ratio of the motions is, the smaller also is the moment of the power to the moment of the weight; therefore if the ratio AB to DB is smaller than AC to DC, then the ratio FE to GE will also be smaller than the ratio AC to DC. Wherefore then only will the same ease of moving remain, as far indeed as concerns the ratio of the hypomochlion, whatever may be changed from another source, as below, when the outer part CB of the lever, which rests upon the hypomochlion, is such that the ratio AB to DB is the same as AC to DC. But this cannot happen at all, because AB and DB are the same as AC and DC, if to both of these the same part CB be added. If therefore, as AC plus CB would be to DC plus CB, so were AC to DC, then also, by permutation and division and again permutation, by 16 and 17 of Book 5, it would be as AC is to DC, so is CB to CB, and therefore the whole AC would be equal to the part DC. It is therefore not possible that the same ease remain in motion by reason of the hypomochlion, if it should happen that the positions of the lever in motion cross one another; which occurs if now one and now another part of the lever touches the hypomochlion. And because the ratio of the whole AB to the whole DB is greater than that of the removed CB to the removed CB, by 33 of Book 5 the remaining AC to the remaining DC will also have a greater ratio than the whole AB to the whole DB, that is, a greater ratio than FE to GE. Similarly in a lever of the first kind, if the fulcrum be cylindrical, it does indeed touch at a point, but while the lever is pressed downward, one and another point of it coincides with different points of the cylinder: for if the power be at C, and the weight at E, but the lever touch at F, in the turning, when E comes to I and C to L, now the contact is at H, so that HL is smaller than FC, but on the contrary HI is greater than FE. Therefore the moments of the power decrease, since its distance from the center of motion is lessened, but the moments of the weight increase, since something is always added to the distances from the center of motion. And indeed, the thicker the cylinder is, once the lever has been given an equal inclination, the greater also is the difference of the distances; as is easily demonstrated, if

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Mechanicorum si duo circuli se intus contingant in O, ubi vectem sustinent, & deinde vectis inclinetur, ut faciat angulum O I G tangens cylindrum minorem in G, aut faciat angulum O H S illi æqualem tangens cylindrum majorem in S: duo si quidem triangula I R G & H M S sunt æquiangula, quia vectes C K & B D sunt paralleli ex hypothesi, lineæ verò à centris R & M ad puncta contactuum G & S ductæ cadunt ad angulos rectos, ex 18. lib. 3. quapropter & anguli ad centra R & M sunt æquales: igitur etiam arcus O G & O S sunt similes in Ratione suarum semidiametrorum O R & O M: major ergo est arcus O S quàm arcus O G, ac propterea illi major quàm huic vectis pars in conversione aptatur, adeóque distantia ponderis ab hypomochlio minùs augeatur ab O in G, quàm ab O in S, factâ æquali vectis inclinatione. Illud tamen habetur compendij, si crassior cylindrus vecti supponatur, quod non adeò inclinandus sit vectis, ut ad certam altitudinem attollatur pondus, ac illum inclinare oporteret, si exilior cylindrus fulcri munere fungeretur. Quæ de cylindro dicta sunt, manifesta quoque apparent, si hypomochlium planum sit, ut O S: est nimirum longè alia Ratio V O ad O R atque X S ad S T; nam additur ipsi O R longitudo O S, ut habeatur S T. Cum ergo minor sit potentiæ distantia X S, quàm V O, minora sunt potentiæ momenta: contra verò cum major sit ponderis distantia T S, quàm R O, majora pariter sunt ponderis momenta. Ut itaque in vectis motu momentorum Ratio stabilis ac firma perseveret, satius est hypomochlium vecti objicere aciem anguli, in quem duæ subjecti corporis facies concurrunt, aut vecti axem infigi, circa quem ille convolvatur. CAPUT

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Mechanics if two circles touch each other internally at O, where they support a lever, and then the lever is inclined, so as to make the angle O I G, touching the smaller cylinder at G, or so as to make the angle O H S equal to it, touching the larger cylinder at S: for the two triangles I R G and H M S are equiangular, because the levers C K and B D are parallel by hypothesis, and the lines drawn from the centers R and M to the points of contact G and S fall at right angles, from book 3, proposition 18; wherefore the angles at the centers R and M are equal: therefore also the arcs O G and O S are similar in the ratio of their semidiameters O R and O M: therefore the arc O S is greater than the arc O G, and consequently for the former a greater part of the lever is adapted in turning than for the latter; and thus the distance of the weight from the fulcrum is less increased from O to G than from O to S, when an equal inclination of the lever has been made. Yet there is some convenience in this, if a thicker cylinder be placed under the lever, namely, that the lever need not be inclined so much as to raise the weight to a certain height, as it would have to be inclined if a slimmer cylinder were serving the function of a fulcrum. What has been said about the cylinder appears plainly also if the fulcrum be flat, as O S: for the ratio of V O to O R is very different from that of X S to S T; for the length O S is added to O R, so that S T may be obtained. Since therefore the distance of the power X S is less than V O, the moments of the power are less: on the other hand, since the distance of the weight T S is greater than R O, the moments of the weight are likewise greater. Thus, therefore, in the motion of a lever, that the ratio of moments may remain stable and firm, it is better to place against the lever the edge of the angle into which the two faces of the body beneath converge, or to insert the axis of the lever, around which it turns. CHAPTER

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Liber quartus. CAPUT III. 377 CAPUT III. Qua Ratione statuendus sit ponderi locus in Vecte primi generis. Quoniam pondus vecte movendum non est corpus aliquod planè individuum, sed partes habet, quarum aliæ sunt puncto fulcri, hoc est, centro motûs, propiores, aliæ remotiores; animum diligenter advertere opus est, cuinam vectis puncto intelligendum sit adjunctum onus, ut ex eo ad fulcrum distantia determinetur. Et quidem vix cuiquam dubium esse potest, an inter omnia ponderis puncta illud unum eligendum sit, in quo gravitas vires suas omnes exercere intelligitur, videlicet circa quod paribus momentis deorsum nititur, si ipsa sibi relinquatur: hoc autem est Gravitatis centrum ipsi ponderi insitum, in quod singularum partium conatus confluere, & secundum quod per directionis lineam deorsum vectem urgeri concipimus. Sit enim pondus P, quod vecti AB infixum, & longitudini AC congruens, suo gravitatis centro I deorsum nititur per lineam directionis IH. Dico vectem perinde à toto pondere urgeri, atque si tota ejus gravitas esset in puncto I, atque ideò distantiam ponderis ab hypomochlio D esse, neque AD maximam, neque CD minimam, sed ID mediam: quia, etsi partibus singulis sua insit gravitas, & singula pro suâ à puncto D distantia sua habeant momenta, ita majora momenta remotiorum particularum à minoribus vicinarum compensantur, ut intelligenda sit vel tota gravitas in media distantia ID vel semissis gravitatis in extrema distantia AD, prout lib. 3. cap. 2. de momentis brachiorum inæqualium libræ ostensum est. Hoc autem, quod de pondere secundùm molem & gravitatem æquabili dicitur, etiam de ponderibus, quorum anomala est figura, vel ex diver- B b b

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Liber quartus. CAPUT III. 377 CAPUT III. In what way the place of a weight in a lever of the first kind is to be determined. Since a weight to be moved by a lever is not some utterly indivisible body, but has parts, of which some are nearer to the point of support, that is, the center of motion, others farther away; one must carefully consider to which point of the lever the load is to be understood as attached, so that its distance from the fulcrum may be determined from that point. And indeed hardly anyone can doubt that, among all the points of the weight, that one ought to be chosen in which gravity is understood to exert all its force, namely the point about which, if left to itself, it tends downward with equal moments; and this is the center of gravity inherent in the weight itself, into which the efforts of the individual parts are understood to converge, and according to which we conceive the lever to be pressed downward along the line of direction. Let there be a weight P, fixed in the lever AB, and corresponding to the length AC, tending downward by its center of gravity I along the line of direction IH. I say that the lever is pressed by the whole weight just as if its entire gravity were in point I, and therefore that the distance of the weight from the fulcrum D is neither the greatest AD nor the smallest CD, but the middle one ID: because, although each individual part has its own gravity, and the individual parts have their own moments according to their distance from point D, so the greater moments of the more remote parts are compensated by the smaller ones of the nearer parts, that it may be understood either that the whole weight is at the middle distance ID or that half the gravity is at the extreme distance AD, as was shown in book 3, chapter 2, concerning the moments of unequal arms of the balance. But this, which is said of a weight uniform in bulk and gravity, is also true of weights whose figure is irregular, or from diver- B b b

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Mechanicorum sis secundùm speciem gravitatibus composita, intelligendum est, si eorum centro gravitatis congruat vectis longitudo; nam ponderis distantia non est Arithmeticè media inter maximam & minimam, sed est intervallum, quod inter fulcrum & centrum gravitatis interjicitur. Sed quia non rarò pondus aut vecti totum incumbit, aut plu- ribus funiculis firmiter alligatum ex illo suspenditur, propterea observandum est, in quod vectis punctum incidat Directionis linea ex centro gravitatis ponderis ducta; hæc enim definiet distantiam ponderis ab hypomochlio, & innotescent momenta, quibus illud resistit potentiæ elevanti. Id quod per libram æqualium brachiorum (ne illorum inæqualitas aliquam pariat difficultatem) instituto æquilibrio facillimè experiri poteris, si laminas ligneas, aut metallicas, in varias figuras conformave- ris, in quibus centrum gravitatis inventum fuerit, & ita singu- las secundùm unum latus immobiliter uni brachio aptaveris, ut illi congruant, atque in oppositâ jugi extremitate æquipon- dium addideris; facto enim æquilibrio, & demisso perpendicu- lo per centrum gravitatis notatum transeunte, apparebit, cui- nam libræ puncto respondeat; atque inter hoc punctum, & cen- trum motûs libræ, distantia erit ad reliqui brachij totam lon- gitudinem, ut æquipondij gravitas ad ponderis examinati gra- vitatem. Quod si pondus ex unico fune pendulum adnectatur vecti, satis constat, ex quo vectis puncto desumatur ejus distantia, ni- mirum ex puncto suspensionis; intentus enim funis à pendente gravitate lineam Directionis ostendit. Quamvis autem si hujus puncti tantummodo ratio habeatur, eadem videantur futura ponderis momenta, quæcumque tandem fuerit vectis positio sive horizonti parallela, sive obliqua, examinandum tamen erit inferius cap.8. utrum ratione anguli, secundùm quem pon- dus deorsum trahere conatur vectem, ejus momenta mutentur. Nunc autem pondus firmiter vecti adnexum, non verò ex unico fune pendulum, consideremus, sive vecti incumbat, sive infra vectem collocetur; hoc nimirum est illud, in quo, propo- sitis majoribus ponderibus, non videtur connivendum; neque enim nihil refert, utrum infra, an supra vectem sit movendæ gravitatis centrum, quantóque intervallo hoc ab illo absit, ibi si

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Mechanics in weights composed according to their kind, it must be understood if the length of the lever agrees with their center of gravity; for the distance of the weight is not an arithmetic mean between the greatest and the least, but is the interval that is inserted between the fulcrum and the center of gravity. But because not rarely a weight either rests wholly on the lever, or is firmly tied to it by several cords and hangs from it, therefore it is to be observed at what point of the lever the line of direction drawn from the center of gravity of the weight falls; for this will determine the distance of the weight from the fulcrum, and the moments will be made known, by which it resists the elevating power. This you can very easily test by means of a balance with equal arms (so that the inequality of those arms may cause no difficulty), if you have formed wooden or metal plates into various shapes, in which the center of gravity has been found, and thus have fitted each one immovably to one arm on one side so that they agree, and have added an equal weight at the opposite end of the beam; for when equilibrium has been achieved, and a plumb line passed through the marked center of gravity is let down, it will appear to which point of the balance it corresponds; and between this point and the center of motion of the balance, the distance will be to the total length of the remaining arm as the weight of the equal weight is to the weight of the weight under examination. But if the weight is attached hanging from a single cord to the lever, it is sufficiently clear from what point of the lever its distance is to be taken, namely from the point of suspension; for the taut cord from the hanging weight indicates the line of direction. Although, however, if only this point is taken into account, the moments of the weight seem to be the same, whichever the position of the lever may be, whether parallel to the horizon or inclined, nevertheless it will have to be examined below in chapter 8 whether, because of the angle according to which the weight attempts to pull the lever downward, its moments are changed. Now however let us consider a weight firmly attached to the lever, and not hanging from a single cord, whether it rests on the lever or is placed below the lever; this is indeed the point in which, larger weights having been proposed, one must not seem to be careless; for it is by no means irrelevant whether the center of the movable weight is below or above the lever, and how great an interval this is removed from that, there if

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Liber quartus. CAPUT III. 379 si quidem gravitas collocata intelligitur, ubi suas omnes vires omnium partium conspiratione exercet. Quapropter, ut pon- deris momenta innotescant, centri gravitatis motum perpen- dere, ac dimetiri oportet. Hinc est pondus firmiter adnexum vecti perinde se habere, atque si vectis quidam curvus in an- gulum inflexus ad punctum hypomochlij, si sit vectis pruni ge- neris, extremitatem alteram in centro gravitatis ponderis, al- teram in potentiâ haberet. Sit Vectis rectus AB horizonti parallelus, hypomochlium habens in C, & in parte inferiore stabili nexu adjungatur pon- dus, cujus gravitatis centrum I. Ex I in vectem horizontalem cadat perpendicularis linea directionis IE; hoc enim perpendicularum de- finit distantiam gravitatis à vecte. Est igitur potentia in A, & pondus in I perinde, atque si esset vectis ACI; & ut pondus atque potentia in eâdem linea horizontali consistant, non est attendenda vectis positio AB, sed rectæ lineæ AI jungentis centrum po- tentiæ A cum centro gravitatis ponderis I; quæ linea AI simul ut æquè ab horizonte distabit, & linea CH ad angulos rectos cadens in eandem lineam AI congruens erit rectæ lineæ jun- genti punctum hypomochlij C cum centro terræ, æquilibrium indicabit; eademque definiet Rationem ponderis ad potentiam sustinentem horizontaliter, juxta reciprocam eorumdem distantiam à puncto H; pro ut lib.3.cap.5. de librâ curvâ ex- plicitatum est. In positione autem obliqua AI, quando recta ex C ad centrum terræ ducta est CG cadens super AI ad angu- los inæquales, potentia sustinens est ad pondus, ut IG ad GA. Cum igitur sit IG minor quàm IH, contrà verò GA sit ma- jor quàm HA, erit minor Ratio IG ad GA, quàm IH ad HA. Quoniam verò linea directionis ponderis IE perpendicula- ris est ad vectem AB horizontalem ex hypothesi, & parallela lineæ CG, est ut AG ad GI, ita AC ad CE, per 2.lib.6. ac propterea, in situ vectis parallelo horizonti, locus ponderis est in vecte determinatus à lineâ directionis ponderis occurrente B b b 2

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Liber quartus. CAPUT III. 379 if indeed a load is understood to be placed where it exercises all its force by the union of all its parts. Wherefore, in order that the moments of weights may be made known, it is necessary to consider and measure the motion of the center of gravity. Hence a weight firmly attached to a lever behaves as if a certain curved lever, bent into an angle at the point of the hypomochlion, if it is a lever of the prun- kind, had one end in the center of gravity of the weight, and the other in the power. Let the straight lever AB be parallel to the horizon, having its hypomochlion in C, and let a weight be joined by a fixed attachment to the lower part, the center of whose gravity is I. From I let the perpendicular line of direction IE fall upon the horizontal lever; for this perpendicular determines the distance of the gravity from the lever. Thus the power at A and the weight at I are just as if there were the lever ACI; and in order that the weight and the power may lie on the same horizontal line, one must not attend to the position of the lever AB, but to the straight line AI joining the center of power A with the center of gravity of the weight I; which line AI, as soon as it is equally distant from the horizon, and the line CH falling at right angles coincides with the same line AI, will be congruent with the straight line joining the point of the hypomochlion C with the center of the earth, it will indicate equilibrium; and the same will define the ratio of the weight to the sustaining power horizontally, according to their reciprocal distances from point H; as was explained in book 3, chapter 5, on the curved lever. But in an oblique position of AI, when the straight line drawn from C to the center of the earth is CG, falling upon AI at unequal angles, the sustaining power is to the weight as IG is to GA. Since therefore IG is smaller than IH, but on the other hand GA is greater than HA, the ratio IG to GA will be smaller than IH to HA. But since the line of direction of the weight IE is perpendicular to the horizontal lever AB by hypothesis, and parallel to the line CG, as AG is to GI, so is AC to CE, according to 2. book 6; and therefore, in the position of the lever parallel to the horizon, the place of the weight on the lever is determined by the line of direction of the weight meeting the B b b 2

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Mechanicorum 380 ipsi vecti. Et quia major est Ratio A G ad G I, quàm sit A H ad H I, etiam major est Ratio A C ad C E, quàm sit A H ad H I: Ergo convertendo E C ad C A minorem habet Rationem, quàm I H ad H A, per 26. lib. 5. Atqui potentia sustinens pondus datum, quando recta A I æquè distat ab horizonte, est ad pondus ut I H ad H A; quando autem pondus est infra lineam B A illud cum potentiâ jungentem horizonti parallelam est ut E C ad C A. Igitur potentia sustinens in horizontali pondus habet majorem Rationem ad illud, quàm ad idem pondus habeat potentia sustinens illud infra horizontalem. Ergo, ex 8. lib. 5. potentiâ sustinens pondus infra horizontalem minor est potentiâ illud sustinente in horizontali. Finge enim esse libram curvam A C I habentem spartum in C: utique si in A esset æquipondium, quod ad pondus I esset ut I H ad H A, non maneret in eadem positione obliqua, sed A descenderet ad positionem horizontalem, ut dictum est lib. 3. cap. 4. ut igitur obliqua maneat, æquipondium A debet esse minus. Ad sustinendum autem pondus, hîc in vecte idem à Potentiâ præstatur, ac ab æquipondio in librâ brachiorum inæqualium. Simili omnino methodo ostendetur pondus idem vecti A B horizontali impositum, cujus centrum gravitatis sit D, linea directionis D I occurens vecti in E, esse ad potentiam A, ut est A C ad C E; at si recta A D jungens potentiam cum centro gravitatis D esset horizonti parallela, pondus ad potentiam esset ut A L ad L D, quàm Rationem determinat C L cadens ad angulos rectos in rectam A D. Quia enim D E & I E sunt æquales ex hypothesi, cum sit idem pondus, & latus E A est commune, anguli verò ad E sunt recti, etiam, per 4. lib. 1. lineæ A D & A I, item anguli E A D & E A I sunt æquales. Præterea in triangulis CH A, CL A rectangulis ad H & L, latus C A est commune, & anguli ad A sunt æquales; igitur, per 26. lib. 1. lineæ A L & A H sunt æquales, igitur & residuæ L D & H I sunt æquales. Quapropter ut A H ad H I, ita A L ad L D: quia igitur Ratio A H ad H I ostensa est superiùs minor Ratione A C ad C E, etiam minor est Ratio A L ad L D, quàm A C ad C E. Sed ut A C ad C E, ita A O ad O D, per 2. lib. 6. propter parallelismum linearum C O & E D; ergo minor est Ratio A L ad L D, quàm A O ad O D. Atqui cùm A D parallela

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Mechanics 380 carried by themselves. And because the ratio of A G to G I is greater than that of A H to H I, the ratio of A C to C E is also greater than that of A H to H I: therefore, by conversion, E C has a smaller ratio to C A than I H has to H A, by book 5, prop. 26. But the force sustaining a given weight, when the straight line A I is equally distant from the horizon, is to the weight as I H is to H A; but when the weight is below the line B A joining it with the force, parallel to the horizon, it is as E C is to C A. Therefore the sustaining force in the horizontal position has a greater ratio to that weight than the sustaining force has to the same weight when it is below the horizontal. Therefore, from book 5, prop. 8, the force sustaining the weight below the horizontal is less than the force sustaining it in the horizontal position. For suppose a curved balance A C I having a pivot at C: certainly if at A there were an equal weight, which to the weight I would be as I H is to H A, it would not remain in the same oblique position, but A would descend to the horizontal position, as was said in book 3, chapter 4; therefore, if it is to remain oblique, the equal weight at A must be smaller. But for sustaining the weight, here on the lever, the same is performed by the force as by the equal weight in a balance of unequal arms. By a wholly similar method it will be shown that the same weight placed on the horizontal lever A B, whose center of gravity is D, with the line of direction D I meeting the lever at E, is to the force A as A C is to C E; but if the straight line A D joining the force with the center of gravity D were parallel to the horizon, the weight to the force would be as A L is to L D, a ratio determined by C L falling at right angles upon the straight line A D. For since D E and I E are equal by hypothesis, since it is the same weight, and the side E A is common, and the angles at E are right angles, also, by book 1, prop. 4, the lines A D and A I, likewise the angles E A D and E A I, are equal. Moreover, in the right triangles C H A and C L A at H and L, the side C A is common, and the angles at A are equal; therefore, by book 1, prop. 26, the lines A L and A H are equal, and therefore the remaining parts L D and H I are equal. Wherefore, as A H is to H I, so is A L to L D: since therefore the ratio of A H to H I was shown above to be less than the ratio of A C to C E, the ratio of A L to L D is also less than that of A C to C E. But as A C is to C E, so is A O to O D, by book 6, prop. 2, because of the parallelism of the lines C O and E D; therefore the ratio of A L to L D is less than that of A O to O D. But since A D parallel

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Liber quartus. CAPUT III. 381 parallela est horizonti, pondus D impositum vecti ad potentiam A sustinentem est ut A L ad L D, in positione verò obliquâ AD est idem pondus ad potentiam sustinentem ut A O ad OD; ergo in priori positione horizontali pondus ad potentiam habet minorem Rationem, quàm in posteriori positione obliqua: ergo per 8. lib. 5. in priori est major potentia, quàm in posteriori. Quamvis autem, cùm vectis est horizonti parallelus, pondus sive illi impositum, sive suppositum fuerit, iisdem momentis reductetur potentiæ sustinenti, non ita tamen se res habet, si idem vectis AB, fulcrum habens in C, elevetur supra lineam horizontalem RT: plurimu[m] enim interest, utru[m] ponderi subjectus sit vectis, an vecti pondus. Sint, ut prius, gravitatis ponderis centra D superius, & I inferius, ex quibus in vectem perpendiculares cadunt DE & IE, quæ, ex 14. lib. 1. sunt una recta linea DI. Jungantur centra potentiæ & ponderis rectâ AD, quæ secat rectam transeuntem per fulcrum C & terræ centrum in puncto M. Quare ex dictis de librâ curva, si sint æqualia momenta ponderis atque potentiæ, erit ut AM ad MD, ita pondus D ad potentiam A. Ducatur ex D linea directionis DN parallela perpendiculari MC; & per 2. lib. 6; est ut AM ad MD, ita AC ad CN: est autem CN minor quàm CE, ergo, ex 8. lib. 5. major est Ratio AC ad CN, quàm AC ad CE. Atqui in vecte horizontali potentia ad pondus est ut EC ad CA; hic autem ut NC ad CA; igitur minor est potentia sustinens pondus impositum vecti obliquo supra horizontem, quàm potentia sustinens pondus idem vecte parallelo horizonti. At si pondus vecti subjiciatur, & sit ejus gravitatis centrum I, ducatur recta AI secans perpendiculum ex C ductum ad Bbb 3

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Book Four. CHAPTER III. 381 is parallel to the horizon, the weight D placed upon the lever is to the supporting power A as A L is to L D; but in the oblique position AD the same weight is to the supporting power as A O is to O D. Therefore, in the former horizontal position the weight has a smaller ratio to the power than in the latter oblique position; therefore, by 8. book 5, in the former there is greater power than in the latter. Although, however, when the lever is parallel to the horizon, a weight, whether placed upon it or beneath it, is reduced by the same moments to the supporting power, yet the matter is not the same if the same lever AB, having its fulcrum at C, is raised above the horizontal line RT: for it matters very much whether the lever be beneath the weight, or the weight beneath the lever. Let, as before, the centers of gravity of the weight be D above, and I below, from which perpendiculars DE and IE fall upon the lever, which, by 14. book 1, are one straight line DI. Join the centers of power and weight by the straight line AD, which cuts the straight line passing through the fulcrum C and the center of the earth at the point M. Therefore, from what has been said about the curved balance, if the moments of the weight and the power are equal, it will be as AM is to MD, so is the weight D to the power A. From D draw the line of direction DN parallel to the perpendicular MC; and by 2. book 6, it is as AM is to MD, so is AC to CN: but CN is less than CE, therefore, by 8. book 5, the ratio of AC to CN is greater than that of AC to CE. But in a horizontal lever the power to the weight is as EC to CA; here, however, as NC to CA; therefore the supporting power is less when it sustains a weight placed on an oblique lever above the horizon than when it sustains the same weight on a lever parallel to the horizon. But if the weight be placed beneath the lever, and its center of gravity be I, draw the straight line AI cutting the perpendicular drawn from C to Bbb 3

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Mechanicorum 382 centrum terræ in V. Igitur si æqualia sunt momenta ponde- ris I & potentiæ A, est pondus ad potentiam ut AV ad VI. Ex I centro gravitatis linea directionis IB parallela lineæ CV occurrat vecti in B; igitur, ex 2. lib.6. ut AV ad VI, ita AC ad CB: est autem CB major quàm CE; ergo AC ad CB ha- bet, ex 8. lib.5. minorem Rationem, quàm AC ad CE. Cum itaque in vecte horizontali potentia ad pondus esset ut EC ad CA, hîc autem in vecte obliquo sit ut BC ad eandem CA, major potentia sustinens hîc requiritur. Quare tantumdem crescit sustinendi difficultas in pondere infra vectem adjuncto, quantum decrescit in sustinendo pondere supra vectem posito. Cum enim triangula BEI, DEN sint æquiangula (quia BI & DN, per 30. lib.1. sunt parallelæ, adeóque per 29. lib.1. alterni anguli ad B & N, & alterni ad I & D sunt æquales, & reliquus reliquo, per 32. lib.1.) est, per 4. lib.6. ut IE ad ED, ita BE ad EN: sunt autem ex hypothesi DE & IE æquales, igitur & BE æqualis est ipsi EN, illa refert incrementum po- tentiæ, hæc decrementum; ergo æqualiter ibi crescit, hîc de- crescit difficultas sustinendi pondus. Contraria sunt momenta, quæ ponderibus accidunt, vecte cum pondere infra horizontalem lineam inclinato: concipe enim hoc idem schema ita conversum, ut potentia A sit in su- periore loco, pondera autem I & D sint infra horizontalem RT. Iam pondus I incumbit vecti, pondus verò D illi sub- jectum adnectitur. Igitur pondus I vecti impositum majora mo- menta habet vecte cum pondere infra horizontem inclinato, quàm vecte horizonti parallelo: in hoc autem eodem situ in- clinato pondus subjectum D minora habet momenta, nam pon- dus I ad potentiam A sustinentem est ut AC ad CB majorem, quæ est minor Ratio quàm AC ad CE minorem, ex 8. lib.5: è contrario D pondus ad potentiam A sustinentem est ut AC ad CN minorem, quæ est major Ratio, quàm AC ad CE ma- jorem. Hinc est momenta ponderis vecti ex primo genere im- positi infra horizontem majora esse, supra horizontem minora; contrà autem ponderis vecti subjecti infra horizontem minora esse, supra horizontem majora. Et hæc quid e[st] catenus dicta intelligantur, quatenus concipitur Potentia vi suæ gravitatis rectâ deorsum connitens, adeò ut Di- rectione

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Mechanics 382 the center of the earth in V. Therefore, if the moments of the weight I and of the power A are equal, the weight is to the power as AV is to VI. From I, let the line of direction IB through the center of gravity, parallel to the line CV, meet the lever at B; therefore, from Book 6, Prop. 2, as AV is to VI, so is AC to CB. But CB is greater than CE; therefore AC to CB has, from Book 5, Prop. 8, a smaller ratio than AC to CE. Since therefore in a horizontal lever the power to the weight would be as EC is to CA, but here in an oblique lever it is as BC is to the same CA, a greater sustaining power is required here. Wherefore the difficulty of sustaining increases in the same amount when the weight is placed below the lever as it decreases when the weight is placed above the lever. For since the triangles BEI and DEN are equiangular (because BI and DN, by Book 1, Prop. 30, are parallel, and therefore by Book 1, Prop. 29, the alternate angles at B and N, and the alternate angles at I and D, are equal, and the remaining angle equals the remaining, by Book 1, Prop. 32), it follows, by Book 6, Prop. 4, that as IE is to ED, so is BE to EN. But by hypothesis DE and IE are equal; therefore BE also is equal to EN. The one represents an increase of power, the other a decrease; therefore the difficulty of sustaining the weight increases there and decreases here equally. The moments that occur to the weights are opposite when the lever, with the weight, is inclined below the horizontal line: for imagine this same figure turned so that the power A is in the upper position, and the weights I and D are below the horizontal RT. Now the weight I rests upon the lever, but the weight D is attached below it. Therefore the weight I placed on the lever has greater moments when the lever is inclined below the horizon than when the lever is parallel to the horizon: but in this same inclined position the subordinate weight D has smaller moments, for the weight I to the sustaining power A is as AC to the greater CB, which is a smaller ratio than AC to the smaller CE, by Book 5, Prop. 8; on the contrary, the weight D to the sustaining power A is as AC to the smaller CN, which is a greater ratio than AC to the greater CE. Hence the moments of a weight placed on the lever of the first kind are greater below the horizon and smaller above the horizon; but, on the other hand, for a weight placed below the lever, they are smaller below the horizon and greater above the horizon. And let what has here been said be understood insofar as it is conceived that the power, by virtue of its own gravity, tends straight downward, so that by Directione

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Liber quartus. CAPUT III. 383 rectiones Potetiæ atque Ponderis sint parallelæ; propterea enim considerata est linea per centrû motûs, hoc est punctum fulcri, ducta ad centrû terræ utrique Directioni parallela. At si linea Di- rectionis Potentiæ non esset parallela Directioni gravitatis Pon- deris (si res scrupulosiùs agatur) paulo aliter consideranda vide- tur linea per punctum fulcri transiens, quæ determinet partes li- neæ jungentis Potentiam & Centrum gravitatis ponderis, linea videlicet per fulcrum ducta ex puncto, in quo concurrunt di- rectiones Potentiæ atque Ponderis. Sit Vectis A B insistens fulcro C depres- sus in A infra horizontem, ut sustineat pondus D in- cumbens vecti, à quo distat per lineam D E. Directio gravitatis ponderis est per- pendicularis DR, at di- rectio Potentiæ non sit per- pendicularis AT, verùm obliqua AR faciens cum vecte angulum B A R. Concurrunt itaque di- rectiones Ponderis, & Potentiæ in R. Quare sicuti quando sunt directiones DR & AT parallelæ, premunt fulcrum C juxta perpendicularem CV, quæ rectam AD secat in M, ita directiones DR & AR videntur premere fulcrum C juxta rectam CR, quæ producta secat rectam AD in S: ac propterea Ratio Potentiæ sustinentis ad Pondus non est ut DM ad MA, sed ut DS ad SA. Hinc est lineam directionis Potentiæ, quò majorem angu- lum constituit cum vecte in A, eò minorem angulum efficere cum perpendiculari lineâ directionis ponderis DR productâ, atque proinde cum illa concurrere multo remotiùs quàm in R, & lineam ex puncto concursûs directionum ductam ad C, & ulterius productam secare lineam AD inter M & S, adeò ut aliquando facilè citra notabilem errorem assumi possit punctum M: Cum enim DR & MV sint parallelæ, angulus DR C in- ternus æqualis est externo MCS, ex 29. lib. 1. idemque di- cendum

Transcription: Translated (English)

Liber quartus. CHAPTER III. 383 Since the directions of Power and Weight are parallel; for that reason the line through the center of motion, that is, the point of the fulcrum, drawn to the center of the earth, was considered parallel to both directions. But if the line of the direction of Power were not parallel to the direction of the gravity of the Weight (if the matter be examined more exactly), the line passing through the point of the fulcrum appears to be considered somewhat differently, namely the line determining the parts of the line joining the Power and the center of gravity of the weight, that is, the line drawn through the fulcrum from the point at which the directions of Power and Weight meet. Let the Lever AB rest upon the fulcrum C, depressed at A below the horizon, so as to support the weight D resting on the lever, from which it is distant by the line DE. The direction of the gravity of the weight is the perpendicular DR, but the di- rection of the Power may not be per- pendicular AT, but rather an oblique AR making with the lever the angle BAR. Thus the directions of the Weight and Power meet in R. Wherefore, just as when the directions DR and AT are parallel, they press upon the fulcrum C according to the perpendicular CV, which cuts the straight line AD in M, so the directions DR and AR seem to press upon the fulcrum C according to the straight line CR, which, when produced, cuts the straight line AD in S: and therefore the ratio of the sustaining Power to the Weight is not as DM to MA, but as DS to SA. Hence it is that the line of direction of the Power, the greater the angle it makes with the lever at A, the smaller an angle it forms with the perpendicular line of direction of the weight DR produced, and therefore meets that line much farther away than in R, and the line drawn from the point of meeting of the directions to C, when produced farther, cuts the line AD between M and S, so that sometimes the point M may easily be assumed without noticeable error: For since DR and MV are parallel, the internal angle DRC is equal to the external MCS, from 29 of book 1, and the same is to be said

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Mechanicorum cendum de quolibet angulo constituto cum perpendiculari DR à lineâ ex puncto concurfus directionum ducta per C punctum fulcri: ideò quo minor sit angulus ad B, minor quo- que est ad C, & punctum in lineâ AD notatum magis acce- dit ad M. Hinc pro determinanda Ratione momentorum potentiæ ad momenta ponderis pro diversâ vectis inclinatione duplici me- thodo uti poteris. Prima est, si ex centro gravitatis ponderis lineam directionis ducas, punctum enim, in quo hæc occurrit vecti, illud est, quod definit locum ponderis, in quo sua exer- cet momenta. Secunda est, si tam ex Potentiæ quàm ex Pon- deris centro gravitatis lineam ducas ad perpendiculum in li- neam horizontalem, quæ transit per C punctum fulcri; nam partes hujus lineæ horizontalis interceptæ inter puncta, in quæ cadunt perpendiculares, & punctum C, illæ sunt, quæ reci- procè sumptæ ostendunt Rationem ponderis ad potentiam. In situ namque horizontali vectis punctum E congruit puncto S, & potentia A congruit puncto X: est igitur ut AC ad CE ita XC ad CS: in positione autem obliquâ ex A in horizontalem perpendicularis cadit in Z, ex D cadit in K, ex I verò in O. Quia igitur triangula AZC & NKC sunt æquiangula, vide- licet rectangula ad Z & K, angulos ad verticem C, ex 15.lib.1; æquales habent, &, ex 32 lib.1. reliquum reliquo, est per 4. lib.6. ut AC ad CN ita ZC ad CK. Similiter triangula BOC & AZC rectangula ad O & Z angulos ad verticem C æquales habent, & reliquum reliquo, adeóque sunt similia, & ut AC ad CB, ita ZC ad CO. Quare in hac obliquâ vectis positione momentum ponderis D ad momentum potentiæ susti- nentis est ut ZC ad CK, & momentum ponderis I ad momen- tum potentiæ sustinentis est ut ZC ad CO. Ex his, quæ de potentia sustentante dicta sunt, satis apparet potentiam paulo validiorem satis esse ad pondus movendum. Verùm licèt in vecte primi generis ad pondus sustentandum opportunè animum adverterimus ad libram curvam, hæc ta- men in vecte secundi generis locum habere non possunt; propterea ad aliam explicandi rationem confugiendum est, quæ utrique generi communis sit; nec difficile erit ea, quæ sta- tim capite sequenti subjiciam pro secundo vectis genere ad pri- mum

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Mechanics. For any given angle, reckoned with the perpendicular DR, from the line drawn from the point of concurrence of the directions through the point C of the fulcrum: therefore, the smaller the angle at B is, the smaller also is that at C, and the point marked on line AD approaches M more closely. Hence, in order to determine the ratio of the moments of the power to the moments of the weight for different inclinations of the lever, you can use a twofold method. The first is this: if from the center of gravity of the weight you draw the line of direction, for the point at which this meets the lever is that which determines the place of the weight, where it exerts its moments. The second is this: if from the center of gravity both of the Power and of the Weight you draw a line perpendicular to the horizontal line which passes through the point C of the fulcrum; for the parts of this horizontal line intercepted between the points where the perpendiculars fall and the point C are those which, taken reciprocally, show the ratio of the weight to the power. For in the horizontal position of the lever, point E corresponds to point S, and power A corresponds to point X: therefore, as AC is to CE, so XC is to CS. But in the oblique position, the perpendicular from A falls on Z, from D on K, and from I on O. Since therefore triangles AZC and NKC are equiangular, namely right-angled at Z and K, and having the angles at the vertex C equal, from Book 1, proposition 15; and, from Book 1, proposition 32, the remaining angle equals the remaining angle, it follows by Book 6, proposition 4, that as AC is to CN, so ZC is to CK. Similarly, triangles BOC and AZC, right-angled at O and Z, have equal angles at the vertex C, and the remaining angle equals the remaining angle, and therefore they are similar; and as AC is to CB, so ZC is to CO. Wherefore, in this oblique position of the lever, the moment of the weight D to the moment of the sustaining power is as ZC is to CK, and the moment of the weight I to the moment of the sustaining power is as ZC is to CO. From what has been said concerning the sustaining power, it is clear enough that a power somewhat stronger is sufficient to move the weight. However, although in the lever of the first kind we have suitably paid attention, for the purpose of supporting the weight, to the curved balance, this cannot be applied in the lever of the second kind; therefore we must have recourse to another method of explanation, one common to both kinds; nor will it be difficult to apply to the first kind of lever, from the second, what I shall immediately set forth in the following chapter.

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Liber quartus. CAPUT IV. 385 num traducere. Consideratur nimirum motus ponderis com- paratus cum eodem motu potentiæ: si enim potentia sit suâ gravitate descendens, ejus descensum metitur Z A: pondus vecti impositum ascendit, ut sit supra horizontalem altitudine KD; sed ex hac demenda est centri gravitatis distantia DE, qua eminebat supra horizontalem, ut habeatur ejus motus DK minùs DE, hoc est GK. Contra verò pondus vecti sub- jectum erat infra horizontalem distantiâ IE, quæ si addatur al- titudini OI, dabit OH motum ipsius ponderis. Major est au- tem motus OI plus IE, hoc est plus DE, quàm sit motus KD minùs DE; nam posita obliquitate lineæ DI, facto centro D, intervallo DE circulus descriptus transit per G punctum de- pressius quàm E, & ex I intervallo IE descriptus transit per H punctum altius quàm E: ergo motus Z A ad minorem motum habet majorem Rationem, quàm ad majorem motum, atque adeò major est movendi facilitas. CAPUT IV. Momenta ponderis in Vecte secundi generis considerantur. IN Vecte secundi generis circa extremitatem, ubi est ful- crum, describuntur à pondere proximo & à potentiâ remotâ duo circulorum arcus tanquam circa commune centrum. Et quidem si in eadem rectâ lineâ sint punctum fulcri, centrum gravitatis ponderis, & ipsa virtus potentiæ sursum ascendentis, motus potentiæ & ponderis sunt in eadem Ratione, in qua sunt distantiæ ab hypomochlio, sive pondus supra horizontalem transeuntem per fulcrum, sive à loco inferiore ad horizontalem elevetur; quia videlicet tam pondus quàm potentia per simi- les arcus ab horizontali æqualiter remotos moventur; ac pro- inde eorum arcuum Sinus, qui metiuntur elevationem, ha- bent inter se Rationem eandem, quæ est radiorum, sive di- stantiarum. Ccc

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Liber quartus. CAPUT IV. 385 to translate. The motion of the weight is considered, indeed, compared with the same motion of the power: for if the power is descending by its own gravity, its descent is measured by Z A; the weight placed upon the vehicle rises so that it is above the horizontal by the height KD; but from this must be subtracted the distance DE of the center of gravity, by which it stood above the horizontal, in order to obtain its motion, DK minus DE, that is, GK. On the other hand, the weight placed beneath the vehicle was below the horizontal by the distance IE, which, if added to the height OI, will give OH, the motion of the weight itself. But the motion OI plus IE, that is plus DE, is greater than the motion KD minus DE; for, since the line DI is taken obliquely, with center D and radius DE, the circle described passes through the point G, lower than E, and, with interval IE from I, it passes through the point H, higher than E: therefore the motion Z A has a greater ratio to the smaller motion than to the greater motion, and so the facility of moving is greater. CAPUT IV. The moments of weight in a lever of the second kind are considered. IN a lever of the second kind, around the extremity where the fulcrum is, two arcs of circles are described by the nearby weight and the distant power, as though around a common center. And indeed, if the point of support, the center of gravity of the weight, and the very force of the power ascending upward are on the same straight line, the motion of the power and of the weight are in the same ratio as the distances from the hypomochlion, whether the weight is carried above the horizontal passing through the fulcrum, or raised from a lower place to the horizontal; because both the weight and the power are moved through similar arcs equally distant from the horizontal; and therefore the sines of those arcs, which measure the elevation, have the same ratio among themselves as the radii, or distances. Ccc

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386 Mechanicorum At verò si centrum gravitatis ponderis sit extra lineam rectam jungentem punctum fulcri cum puncto virtutis potentiæ existentis in alterâ vectis extremitate, sive supra vectem, sive infra illum sit, non manet eadem Ratio motuum, quæ est distantiarum potentiæ & ponderis (quatenus ponderis distantia sumitur à puncto, in quod à centro gravitatis cadit in vectem perpendicularis) quia ascensus & elevationes non servant eandem Rationem; ex eo quod, licèt in vectis conversione tam centrum gravitatis ponderis quàm centrum potentiæ describant in motu arcus similes, hi tamen arcus non sunt similiter positi, hoc est simili modo ab horizontali distantes: ac propterea (ut patet ex doctrina Sinuum) differentiæ Sinuum, qui conveniunt arcubus supra vel infra horizontem, ubi incipit quadrans circuli, æqualiter crescentibus, non sunt æquales: hæc autem differentiæ metiuntur motum elevationis, qui maximè attenditur, quatenus opponitur innatæ propensioni gravitatis. Sit in C fulcrum vectis CA, & in A sit potentia movens. Si centrum gravitatis ponderis sit in eadem rectâ CBA, semper motus ponderis & potentiæ sunt omnino similes, & ut CB ad ad CA; illud enim describit arcum BG, hæc verò arcum AS, & elevatio ponderis ex B in G est BR, ascensus potentiæ est AP; & propter triangulorum rectangulorum CRB & CPA similitudinem est ut CB ad CA, ita BR ad AP. Et quamvis, diviso arcu BG in partes aliquot æquales, & in totidem æquales partes diviso arcu simili AS, non sint in singulis ejusdem arcûs partibus æquales ascensus) nam BH minor est quàm HI, hic minor quàm IK, & hic minor quàm KR, similiterque AL minor quàm LM, hic minor quàm MN, & hic minor quàm NP) comparatis tamen singulis ascensibus in minore arcu BG, cum singulis ascensibus in arcu majore AS sibi invicem respondentibus, manet eadem Ratio, & ut BH ad AL, ita HI ad LM, & sic de reliquis

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386 Mechanicorum But if the center of gravity of the weight lies outside the straight line joining the point of support with the point of the power existing at the other end of the lever, whether it be above the lever or below it, the same ratio of motions does not remain, which is that of the distances of the power and the weight (in so far as the distance of the weight is taken from the point upon which the perpendicular from the center of gravity falls onto the lever), because the ascents and elevations do not preserve the same ratio; for although, in the turning of the lever, both the center of gravity of the weight and the center of the power describe similar arcs in motion, yet these arcs are not similarly situated, that is, not equally distant from the horizontal: and therefore (as is clear from the doctrine of sines) the differences of the sines corresponding to arcs above or below the horizon, where the quadrant of the circle begins, increasing equally, are not equal; but these differences measure the motion of elevation, which is chiefly considered, in so far as it is opposed to the innate tendency of gravity. Let the fulcrum of the lever CA be at C, and let the moving power be at A. If the center of gravity of the weight be on the same line CBA, the motions of the weight and the power are always altogether similar, and as CB is to CA; for the former describes the arc BG, the latter the arc AS, and the elevation of the weight from B to G is BR, the ascent of the power is AP; and by the similarity of the right-angled triangles CRB and CPA, as CB is to CA, so is BR to AP. And although, if the arc BG be divided into some equal parts, and the similar arc AS into the same number of equal parts, the ascents in the several parts of each arc are not equal (for BH is less than HI, this less than IK, and this less than KR, and similarly AL less than LM, this less than MN, and this less than NP), nevertheless, when the several ascents in the smaller arc BG are compared with the several corresponding ascents in the larger arc AS, the same ratio remains; and as BH is to AL, so is HI to LM, and so of the rest.

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Liber quartus. CAPUT IV. 387 reliquis (ut ex Sinuum doctrinâ manifestum est, nec opus est hîc ostendere) sunt enim omnes in Ratione Radij CB ad Ra- dium C A. Longè aliter se res habet, quando extra rectam lineam jun- gentem punctum fulcri cum potentiâ est centrum gravitatis ponderis. Nam si Vecti CBA impositum sit pondus, cujus centrum gravitatis sit D, potentiâ A describente arcum A Q centrum gravitatis ponderis describit arcum DE, qui licèt æqualis sit arcui BD; habet tamen ascensum HI majorem quàm BH: igitur ascensus AL ad HI majorem, habet mino- rem Rationem quàm ad BH minorem, ex 8.lib.5. igitur in hoc motu Potentia ad Ponderis motum habet minorem Rationem, quàm si centrum gravitatis ponderis esset in B; ergo majorem experitur in movendo difficultatem. Contrà verò si pondus sit vecti CBA subjectum, ejusque centrum gravitatis sit O; dum potentia A describit arcum A Q, centrum gravitatis O describit arcum OB, ejusque ascensus est OV; atqui OV minor est quàm BH; ergo AL ascensus potentiæ ad OV minorem est in majori Ratione quàm ad BH majorem; est autem HI major quàm BH; ergo AL ad OV multo majorem Rationem habet quàm ad HI. Ergo datâ eâ- dem vectis positione, eodemque motu, major facilitas erit in elevando pondere habente centrum gravitatis infra vectem in O, quàm si illud habeat supra vectem in D. Eadem erit demonstrandi methodus in cæteris ascensitùs: nam potentia percurrens arcum AT habet ascensum AM, centrum D percurrit arcum DF, cujus ascensûs mensura est HK; centrum autem O percurrens arcum OD habet ascen- sum OX: cùm igitur OX minor sit quàm BI, & hic minor quàm HK, etiam AM ad OX minorem est in majore Ratione quàm ad HK majorem. Et hæc quidem hactenus dicta intelliguntur de vecte infra lineam horizonti parallelam depresso; nam vecte supra hori- zontalem lineam elevato, contraria prorsus accidere ex dictis demonstratur. Concipe vectem AC elevatum supra horizon- tem, pondus OB est illi impositum, pondus DB est subjectum: quando potentia ascendens per arcum QA habet ascensum LA, centrum gravitatis O describit arcum BO, & ascensûs CCC 2

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Liber quartus. Chapter IV. 387 the rest (as is evident from the doctrine of the sines, and there is no need here to show it) for they are all in the ratio of the Radius CB to the Radius CA. Much differently does the matter stand when the center of gravity of the weight is outside the straight line joining the fulcrum with the power. For if a weight be placed upon the lever CBA, whose center of gravity is D, while the power at A describes the arc AQ, the center of gravity of the weight describes the arc DE, which, although it may be equal to the arc BD, nevertheless has a rise HI greater than BH: therefore the ascent AL to HI has a smaller ratio than to the smaller BH, from 8. lib. 5. therefore in this motion the Power, in relation to the motion of the Weight, has a smaller ratio than if the center of gravity of the weight were at B; therefore a greater difficulty is experienced in moving it. On the contrary, if the weight be subject to the lever CBA, and its center of gravity be O; while the power at A describes the arc AQ, the center of gravity O describes the arc OB, and its ascent is OV; but OV is less than BH; therefore the ascent AL of the power to OV has a smaller ratio in the greater one than to the greater BH; but HI is greater than BH; therefore AL to OV has a much greater ratio than to HI. Therefore, given the same position of the lever, and the same motion, there will be greater ease in raising a weight whose center of gravity lies below the lever in O than if it have that center above the lever in D. The same method of demonstration will apply in the other ascents: for the power traversing the arc AT has the ascent AM, the center D traverses the arc DF, whose measure of ascent is HK; but the center O traversing the arc OD has the ascent OX: since therefore OX is less than BI, and this less than HK, even AM to OX is in a greater ratio than to the greater HK. And these things thus far said are understood of a lever depressed below the line parallel to the horizon; for with a lever raised above the horizontal line, the contrary is altogether demonstrated to happen from the things said. Conceive a lever AC raised above the horizon, the weight OB is imposed on it, the weight DB is subject to it: when the power ascending through the arc QA has the ascent LA, the center of gravity O describes the arc BO, and the ascent CCC 2

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Mechanicorum 388 mensura est VO, at centrum gravitatis D describens arcum ED habet ascensum IH. Cum igitur ostensum sit majorem Rationem esse LA ad VO, quàm ad IH, etiam supra horizontem elevato vecte major erit facilitas in movendo pondere vecti imposito, quàm in elevando pondus habens centrum gravitatis infra vectem. Ut autem innotescat, qua Ratione in progressu motûs crescat difficultas, aut minuatur, observa ex Canone in arcubus æqualiter crescentibus Sinuum differentias ab initio quadrantis progrediendo usque ad finem Quadrantis semper decrescere, harum verò differentiarum differentias, hoc est differentias secundas, semper augeri. Hinc est ita RK Sinum arcûs GF majorem esse quàm differentiam KI, & KI majorem quàm IH, & IH majorem quàm HB, ut differentia inter Sinum RK & differentiam KI minor sit quàm differentia inter KI & IH, hæc verò differentia minor sit quàm differentia inter IH & HB. Idem dicendum de similibus differentiis inter Sinum PN, & differentias NM, & ML, & LA. In iisdem lineis PA & RB particulas assumptas donavi vocabulo Sinuum aut differentiarum, non quasi ignorans illas particulas non esse Sinus aut differentias Sinuum arcubus æqualiter crescentibus respondentium, sed claritatis gratia abutens vocabulo; quandoquidem illis æquales sunt, cum assumantur per lineas Radio CS parallelas. His positis intelligatur vectis totus CA cum pondere B intrà aquam, potentia verò sit cortex suberis, aut uter inflatus, seu vesica, aut quid hujusmodi levitans. Potentiæ motum metiri oportet ex naturalibus ascensibus AL, LM, & reliquis. Quia autem est ut AL ad LM, ita BH ad HI; etiam vicissim, per 16. lib. 5. ut AL ad BH, ita LM ad HI, & sic de cæteris, sive infra, sive supra horizontalem: propterea eadem semper manet facilitas aut difficultas elevandi pondus in aquâ gravitans, cujus gravitatis centrum congruat vecti CA. Idem dic si Potentia Sin aqua gravitans deprimeret per vim pondus G, quod in aquâ levitaret: nam PN descensus naturalis potentiæ ad RK depressionem ponderis, eandem Rationem haberet, quàm descensus NM ad depressionem KI. Si vectis sit CA, cui pondus incumbat habens centrum gravitatis

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Mechanicorum 388 is the measure VO; but the center of gravity D, describing the arc ED, has the ascent IH. Since therefore it has been shown that the ratio LA to VO is greater than to IH, even when the lever is raised above the horizon there will be greater ease in moving a weight placed on the lever than in lifting a weight whose center of gravity lies below the lever. Now, in order that it may be made clear by what ratio in the progress of motion the difficulty increases, or diminishes, observe from the Canon that in arcs increasing equally, the differences of the sines, proceeding from the beginning of the quadrant to the end of the quadrant, always decrease, but the differences of these differences, that is, the second differences, always increase. Hence it is that RK, the sine of the arc GF, is greater than the difference KI, and KI greater than IH, and IH greater than HB, so that the difference between the sine RK and the difference KI is less than the difference between KI and IH; but this difference is less than the difference between IH and HB. The same must be said of similar differences between the sine PN and the differences NM, and ML, and LA. In the same lines PA and RB, the parts assumed I have given the name of sines or differences, not as though unaware that those parts are not sines or differences of sines corresponding to arcs equally increasing, but using the term loosely for the sake of clarity; since, in fact, they are equal to them, when they are taken through lines parallel to Radius CS. These things being laid down, let the whole lever CA with the weight B in water be understood, but let the power be a cork float, or an inflated bladder, or a vesicle, or something of that kind buoying up. The motion of the power must be measured from the natural ascents AL, LM, and the rest. But since it is as AL to LM, so BH to HI; also conversely, by 16. lib. 5. as AL to BH, so LM to HI, and so in the rest, whether below or above the horizontal: therefore the same ease or difficulty always remains in lifting a weight gravitating in water, whose center of gravity coincides with the lever CA. The same may be said if a power in water were to press down by force the weight G, which would float in the water: for the natural descent PN of the power to the depression RK of the weight would have the same ratio as the descent NM to the depression KI. If the lever be CA, upon which there rests a weight having the center of gravity

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Liber quartus. CAPUT IV. 389 vitatis D, atque tam pondus quàm potentia sint in medio, in quo alterum levitet, alterum gravitet, utriusque motum qua- tenus naturalis est aut violentus, metitur linea perpendicularis in horizontalem cadens: & ut particulæ ipsæ invicem compa- rentur, Sinuum differentiæ AL, LM &c. BH, HI &c. con- siderandæ sunt. Cum itaque differentia inter BH, & HI ma- jor sit quàm differentia inter HI, & IK, utique BH magis de- ficit ab æqualitate cum HI, quàm HI cum IK; ideóque mi- nor est Ratio BH ad HI, quàm HI ad IK: Atqui eadem est Ratio BH ad HI, quæ est AL ad LM; igitur minor est etiam Ratio AL ad LM, quàm HI ad IK, & vicissim, per 27. lib. 5. minor est Ratio AL ad HI, quàm sit LM ad IK. Igitur si po- tentia A levitet, & pondus, cujus centrum gravitatis D, gravi- tet, ascendendo ad horizontalem, quæ per fulcrum C transit, acquirit movendi facilitatem. Iam figuram inverte, ut vectis moveatur supra horizontalem: vecte congruente lineæ horizontali CS, ponderis impositi cen- trum gravitatis erit in F, & ascendet juxta mensuram KI & IH, cum potentiæ ascensus erit PN & NM. Quia igitur differentia inter Sinum RK & differentiam KI minor est, quàm differentia inter KI & IH, utique RK minùs excedit æqualitatem cum KI, quàm KI cum IH: ideóque minor est Ratio RK ad KI, quàm KI ad IH. Est autem eadem Ratio RK ad KI, quæ est PN ad NM; igitur minor est Ratio PN ad NM, quàm KI ad IH, & vicissim minor est Ratio PN ad KI, quàm NM ad IH: Igitur ascendendo magis & recedendo ab horizontali crescit movendi facilitas. Demum si vecti CA subjectum sit pondus, cujus centrum gravitatis O, & potentiæ motum metiatur perpendicularis AP ascendendo versus horizontalem; quia differentia inter OV, & VX major est quàm differentia inter VX & HI, adeóque OV magis deficit ab æqualitate cum VX, quàm VX cum HI, prop- terea OV ad VX habet minorem Rationem quàm VX ad HI: sed ut VX, hoc est BH, ad HI, ita AL ad LM; ergo minor est Ratio OV ad VX quàm AL ad LM; & vicissim minor est Ra- tio OV ad AL quàm VX ad LM; ideóque faciliùs elevatur ex O in B, quàm ex B in D. Factâ autem figuræ conversione, ut ascensus Potentiæ sit PA, & ascensus Ponderis sit RB, si poten- Ccc 3

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Book Four. CHAPTER IV. 389 if D be vitatis, and as both weight and power are in the middle, in which the one lightens, the other weights, the motion of each, insofar as it is natural or violent, is measured by the perpendicular line falling into the horizontal; and, in order that the particles themselves may be compared with one another, the differences of the sines AL, LM, etc. BH, HI, etc. are to be considered. Since therefore the difference between BH and HI is greater than the difference between HI and IK, assuredly BH departs more from equality with HI than HI with IK; and therefore the ratio of BH to HI is less than that of HI to IK. But the same ratio of BH to HI is that of AL to LM; therefore the ratio of AL to LM is also less than that of HI to IK, and conversely, by 27. lib. 5, the ratio of AL to HI is less than that of LM to IK. Therefore if the power A lightens, and the weight, whose center of gravity is D, weights, in ascending to the horizontal which passes through the fulcrum C, it acquires facility of motion. Now invert the figure, so that the lever may move above the horizontal: the lever coinciding with the horizontal line CS, the center of gravity of the imposed weight will be in F, and it will ascend according to the measure KI and IH, while the ascent of the power will be PN and NM. Since therefore the difference between the sine RK and the difference KI is less than the difference between KI and IH, assuredly RK exceeds equality with KI by less than KI does with IH; and therefore the ratio of RK to KI is less than that of KI to IH. But the same ratio of RK to KI is that of PN to NM; therefore the ratio of PN to NM is less than that of KI to IH, and conversely the ratio of PN to KI is less than that of NM to IH. Therefore, by ascending more and receding from the horizontal, the facility of moving increases. Lastly, if under the lever CA there be a weight whose center of gravity is O, and the power measure the motion by the perpendicular AP, ascending toward the horizontal; since the difference between OV and VX is greater than the difference between VX and HI, and therefore OV departs more from equality with VX than VX with HI, for that reason OV has a smaller ratio to VX than VX to HI: but as VX, that is BH, is to HI, so is AL to LM; therefore the ratio of OV to VX is less than that of AL to LM; and conversely the ratio of OV to AL is less than that of VX to LM; and therefore it is more easily raised from O to B than from B to D. But after the figure has been turned, so that the ascent of the Power is PA and the ascent of the Weight is RB, if the power

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Mechanicorum 390 tia sit in Z, centrum gravitatis ponderis subjecti est in G, & dum potentia ascendit per NM & ML describens arcum ZQ, pondus ascendit per R K & K I. Atqui R K ad K I habet mi- norem Rationem quàm K I ad I H, ut superiùs ostensum est, & ut K I ad I H, ita NM ad ML; ergo minor est Ratio R K ad K I, quàm NM ad ML, & vicissim minor est Ratio R K ad NM quàm K I ad ML; ergo faciliùs movetur per R K ascen- dendo, quàm per K I, adeóque crescit difficultas elevandi pondus subjectum vecti suprà horizontalem, si comparentur inter se partes elevationis. Quare, ut in summam ea, quæ dicta sunt, referantur, si pon- dus sit infra vectem secundi generis, faciliùs elevatur eodem vectis motu versùs horizontalem, quàm si fuerit supra vectem: Contrà verò supra horizontalem faciliùs eodem vectis motu elevatur pondus vecti impositum, quàm vecti subjectum. Con- sideratis autem particulatim singulis elevationibus, diviso scili- cet in æquales particulas universo motu ejusdem ponderis, si pondus sit in eâdem rectâ lineâ cum fulcro & potentia, eadem semper est movendi facilitas aut difficultas: Si pondus sit supra vectem, & motus infra horizontalem incipiat, semper crescit movendi facilitas non solùm usque ad horizontalem, verùm etiam supra illam: At si pondus sit infra vectem, motusque in- fra horizontalem incipiat, augetur semper difficultas movendi tùm usque ad horizontalem, tùm supra illam. Hæc omnia confirmari possunt, si lineam directionis per cen- trum gravitatis ponderis ductam produci intelligamus usque ad horizontalem lineam, quæ per fulcrum transit; Secabit enim vectem, & in sectionis puncto quodammodo constitutum pon- dus concipere possumus. Sit enim infra horizontalem CR, vectis CA, & ad punctum B illi insistat perpendiculariter linea à centro gravitatis ducta, scilicet DB su- pra, & OB infra. Quando vectis CA congruet lineæ CR, & erit horizonti parallelus, pondus con- cipietur niti in B contra vectem: at infra horizontalem centrum D nititur in S, & centrum O in

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Mechanicorum 390 that there be in Z, the center of gravity of the weight subject to it is in G, and while the power ascends through NM and ML describing the arc ZQ, the weight ascends through RK and KI. But RK has a smaller ratio to KI than KI has to IH, as was shown above, and as KI is to IH, so is NM to ML; therefore the ratio of RK to KI is smaller than that of NM to ML, and conversely the ratio of RK to NM is smaller than that of KI to ML; therefore it is moved more easily through RK in ascending than through KI, and accordingly the difficulty of raising the weight subject to the lever above the horizontal increases, if the parts of the elevation are compared with one another. Wherefore, to refer in summary to what has been said: if the weight be below the lever of the second kind, it is more easily raised by the same motion of the lever toward the horizontal than if it were above the lever: on the contrary, above the horizontal, the weight placed upon the lever is more easily raised by the same motion of the lever than the weight subject to the lever. But if the several elevations are considered separately, namely if the whole motion of the same weight be divided into equal parts, if the weight be in the same straight line with the fulcrum and the power, the ease or difficulty of moving is always the same: if the weight be above the lever, and the motion begins below the horizontal, the ease of moving always increases, not only up to the horizontal, but also above it: but if the weight be below the lever, and the motion begins below the horizontal, the difficulty of moving always increases, both up to the horizontal and above it. All these things can be confirmed if we understand the line of direction drawn through the center of gravity of the weight to be produced until it reaches the horizontal line that passes through the fulcrum; for it will cut the lever, and in the point of intersection the weight may be conceived as in some way situated. For let CR be below the horizontal, CA the lever, and at point B let there stand perpendicularly upon it the line drawn from the center of gravity, namely DB above, and OB below. When the lever CA shall coincide with line CR, and be parallel to the horizon, the weight will be conceived as pressing at B against the lever: but below the horizontal, center D presses in S, and center O in

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Liber quartus. CAPUT IV. 39. in T, juxta lineas directionis D S & O T. Quia igitur punctum S magis distat à fulcro C quàm punctum T, pondus infra vectem faciliùs sustinetur sub horizontali, quàm pondus supra vectem. Contra autem supra horizontalem centrum O nititur in I remotiùs à fulcro C, & centrum D in H propiùs; ergo supra horizontalem faciliùs sustinetur pondus vecti impositum, quàm illi subjectum. Quoniam verò triangula rectangula CNT, & OBT, an- gulos ad verticem T æquales habent, & reliquum reliquo æqualem, erit, ex 4. lib. 6. ut CT ad TN, ita OT ad TB. Igitur prout ex elevatione vectis minuitur angulus ACN, etiam minuitur angulus TOB, ac propterea T recedit à ful- cro C versus B, & augetur sustinendi atque movendi difficul- tas. Isti autem accessus versus B sunt inæquales, etiam si æqua- lia sint anguli TOB decrementa, prout decrescunt angulo- rum ad O factorum Tangentes, posito Radio OB. Porrò ex Canone Tangentium constat illarum differentias semper ma- jores fieri, si augeatur angulus, minores fieri, si minuatur an- gulus. Igitur recedente lineâ directionis Centri gravitatis O à fulcro C, augetur difficultas sustinendi & elevandi pondus vecti subjectum: & quia supra horizontalem semper magis re- cedit ab eodem fulcro C ultrà punctum B versus A potentiam, puta, ut sit OI, multo adhuc major est sustinendi atque mo- vendi difficultas. Consideratis autem particulatim motibus, quia infra horizontalem differentiæ recessuum à puncto C fiunt semper minores; propterea crescit quidem difficultas, sed inæ- qualibus & minoribus incrementis; quia verò supra horizon- talem differentiæ recessuum à fulcro C fiunt semper majores, crescit adhuc difficultas, & quidem semper majoribus incre- mentis. At si pondus sit D vecti impositum, linea directionis DS accedit versus B usque ad horizontalem, supra quam re- cedit à B versus C, ut sit ex. gr. DH: semper igitur faciliùs movetur, quamquam non æqualibus facilitatis incrementis; fiunt enim incrementa infra horizontalem sensim minora, su- pra autem fiunt semper majora. Sed hic unum explicandum est, quod fortasse alicui animum minùs attentè advertenti dif- ficultatem pariat adversùs ea, quæ superiùs dicta sunt: videlicet ostensum est pondus vecti impositum, si motus incipiat infra horizon

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Book four. Chapter IV. 39. in T, according to the lines of direction D S & O T. Since therefore the point S is farther from the fulcrum C than the point T, a weight below the lever is more easily sustained than a weight above the lever under the horizontal. On the other hand, above the horizontal, the center O leans in I farther from the fulcrum C, and the center D in H is nearer; therefore, above the horizontal, a weight placed on the lever is more easily sustained than one placed beneath it. Now since the right triangles CNT and OBT have equal angles at the vertex T, and the remainder equal to the remainder, it will follow, from Book VI, Prop. 4, that as CT is to TN, so is OT to TB. Therefore, as the lever is raised, the angle ACN is diminished; the angle TOB is also diminished, and consequently T moves away from the fulcrum C toward B, and the difficulty of sustaining and moving increases. But these approaches toward B are unequal, even though the decrements of the angle TOB are equal, inasmuch as they decrease according to the tangents of the angles formed at O, the radius OB being assumed. Moreover, from the Canon of Tangents it is clear that their differences always become greater if the angle is increased, and smaller if the angle is diminished. Therefore, as the line of direction of the center of gravity O recedes from the fulcrum C, the difficulty of sustaining and raising the weight placed below the lever increases; and because above the horizontal it continually recedes from the same fulcrum C beyond the point B toward the force A, that is, such that it be OI, the difficulty of sustaining and moving is much greater still. But if we consider the motions in detail, because below the horizontal the differences of the recedings from point C become continually smaller, the difficulty indeed increases, but by unequal and smaller increments; whereas above the horizontal the differences of the recedings from the fulcrum C become continually greater, so the difficulty increases further, and indeed by ever greater increments. But if the weight D is placed on the lever, the line of direction DS approaches toward B up to the horizontal, above which it recedes from B toward C, as for example DH; it is therefore always moved more easily, although not with equal increments of ease: for the increments below the horizontal become gradually smaller, but above it they always become greater. But here one thing must be explained, which perhaps may cause difficulty to someone who does not carefully attend to what was said above: namely, it has been shown that a weight placed on the lever, if the motion begin below the horizon

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Mechanicorum horizontalem, majori difficultate moveri, quàm pondus vecti subjectum. Si enim, inquis, linea Directionis D S magis ac magis accedit ad B, utique crescit movendi facilitas; contra verò lineâ directionis O T accedente ad B crescit movendi difficultas. Ut nodum hunc solvas, observa triangula SBD, & TBO rectangula ad B, quia D S & T O sunt parallelæ, esse æquian- gula & similia, immò æqualia, quia ut D B ad O B sibi ex hy- pothesi æqualem, ita S B ad T B. Igitur qua Ratione minuitur angulus ACR, etiam minuitur angulus SDB, & angulus TOB: igitur Tangentium differentiæ fiunt semper minores. Quare in primo motu tam linea directionis D S, quàm linea directionis O T, magis accedit ad B quàm in secundo motu, & magis in secundo, quàm in tertio; accessus tamen utrius- que lineæ directionis ex eodem vectis motu sunt æquales; & qua mensurâ augetur recessus ponderis D vecti impositi, à Po- tentia A, eâdem pariter mensurâ augetur recessus ponderis O vecti subjecti, à fulcro C. Itaque crescit quidem illius facili- tas, hujus difficultas, si ponderum singulorum motus particu- latim accipiantur, ejusdemque ponderis motûs pars cum par- te conferatur: at verò si utriusque ponderis motus invicem comparentur, utique pondus D difficiliùs movetur, cùm ejus linea directionis est citra punctum B versus potentiam, quàm moveatur pondus O, quamdiu ejus linea directionis est ultra idem punctum B. Ex his, quæ de vecte secundi generis dicta sunt, quid de vecte tertij generis dicendum sit, faciliùs innotescit, quàm ut illud pluribus explicari oporteat; potentia si quidem & pondus invicem loca commutant, sed motuum Ratio eadem est, & quæ in vecte secundi generis est Ratio motûs Potentiæ ad motum Ponderis, vice versâ in vecte tertij generis est Ratio motûs Ponderis ad motum Potentiæ. Hoc te monitum velim, Amice Lector, consideratum hacte- nus vectem ad movenda sursum pondera gravia, aut deprimen- da deorsum levia, & quidem à Potentia, quæ vi suæ gravitatis aut levitatis moveatur, cujus propterea ascensum aut descen- sum consideravimus. Nam si in plano horizontali à Potentia vivente movendum sit pondus, utique Potentiæ motus circu- laris

Transcription: Translated (English)

Mechanics: to be moved horizontally, with greater difficulty, than the weight placed upon the lever. For, you ask, if the line of direction DS comes more and more near to B, surely the ease of moving increases; on the other hand, when the line of direction OT approaches B, the difficulty of moving increases. To solve this knot, observe that the right triangles SBD and TBO, because DS and TO are parallel, are equiangular and similar, indeed equal, because, as DB is to OB, which by hypothesis is equal to it, so is SB to TB. Therefore, in whatever ratio the angle ACR is diminished, the angle SDB is also diminished, and the angle TOB: thus the differences of the tangents are always made smaller. Wherefore, in the first motion, both the line of direction DS and the line of direction OT come nearer to B than in the second motion, and more in the second than in the third; yet the approach of each line of direction, from the same motion of the lever, is equal; and by whatever measure the recession of the weight D placed on the lever from the Power A is increased, by the same measure likewise the recession of the weight O placed under the lever from the fulcrum C is increased. Thus indeed the ease of the one increases, the difficulty of the other; if the motions of the individual weights are taken separately, and a part of the motion of one and the same weight is compared with a part: but if, on the other hand, the motions of both weights are compared with each other, then the weight D is moved more difficultly, when its line of direction is on this side of point B toward the power, than the weight O is moved, so long as its line of direction is beyond the same point B. From these things, which have been said about the second-class lever, what must be said about the third-class lever becomes more easily evident than that it should be explained at greater length; for indeed the power and the weight change places with each other, but the ratio of the motions is the same, and what in the second-class lever is the ratio of the motion of the Power to the motion of the Weight, in the third-class lever, conversely, is the ratio of the motion of the Weight to the motion of the Power. I would have you note this, dear Reader: hitherto we have considered the lever for moving heavy weights upward, or light weights downward, and indeed by a Power moved by virtue of its own gravity or lightness, whose ascent or descent we therefore considered. For if, on a horizontal plane, a weight is to be moved by a living Power, then certainly the circular motion of the Power

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Liber quartus. CAPUT IV. 393 laris observatur, & attendendum est vectis punctum, in quod cadit linea, quæ à centro gravitatis ponderis in vectem per- pendicularis ducitur, ut ponderis locus statuatur, & momen- ta definiantur. Naturâ quippe comparatum est, ut si vectis non occurrat huic perpendiculari, non moveatur totum pondus, sed fiat ponderis conversio vel circa gravitatis centrum, vel circa aliud punctum quod maneat immotum, aut saltem mino- re motu moveatur. CAPUT V. Quæ sit Ratio Vectis Hypomochlium mobile habentis. Non hîc hypomochlium mobile illud intelligo, quod simul cum pondere à potentiâ sustentato ad easdem partes pro- movetur; cujusmodi sunt manualia bajulorum vehicula, quæ unicâ rotâ instruuntur, & habentia rationem vectis secundi ge- neris; nam fulcrum habent in axe rotæ, & potentiam in extre- mitate manubriorum, quibus illa sustinet pondus transferen- dum: cui propterea addita est rota illa versatilis, ut etiam hy- pomochlium citra difficultatem, quin atterat subjectam plani- tiem, simul cum pondere jam elevato, atque sustentato pro- moveatur. Hujusmodi pariter est novitium vehiculi genus, cui Sellæ Rotatæ nomen fecerunt, hoc uno à lecticâ viatoriâ discrepans, quòd loco posterioris jumenti sustinentis additus est axis dua- bus rotis infixus, cui innituntur vectes ab anteriore equo sustentanti unâ cum pondere intermedio. Hic est vectis secun- di generis, cujus hypomochlium sequitur potentiam trahen- tem pariter ac sustentantem impositum pondus, non mutatâ Ratione momenti potentiæ sustinentis, sive hypomochlium moveatur, sive stabile sit ac fixum. Cæterùm quò pondus ma- gis à rotis distat, magis equum gravat, minùs autem subsilit, cùm rotæ in offendiculum incurrunt. Nomine igitur hypomochlij mobilis illud intelligo, quod D d d

Transcription: Translated (English)

Book Four. Chapter IV. 393 the lever is observed, and attention must be given to the point of the lever on which the line falls, which is drawn perpendicular from the center of gravity of the weight into the lever, so that the place of the weight may be determined, and the moments may be defined. For nature has so arranged it, that if the lever does not meet this perpendicular, the whole weight is not moved, but a turning of the weight is produced, either about the center of gravity, or about some other point which remains unmoved, or at least is moved by a lesser motion. Chapter V. What is the Ratio of a Lever having a Mobile Hypomochlion. Here I do not mean that mobile hypomochlion which, together with the weight sustained by the power, is advanced toward the same parts; of which kind are the hand-carts of porters, which are equipped with a single wheel, and have the ratio of a lever of the second kind; for they have the fulcrum in the axle of the wheel, and the power at the end of the handles, by which they sustain the weight to be carried: for this reason that turning wheel has been added, so that even the hypomochlion, without difficulty, and without rubbing the surface beneath, may be moved along together with the weight now raised and sustained. Likewise of this sort is the novel kind of vehicle, to which they have given the name Sellæ Rotatæ, differing from the travelling litter in this one respect, that in place of the rear supporting beast an axle has been added, fixed with two wheels, on which the levers rest, supported by the front horse together with the intermediate weight. This is a lever of the second kind, whose hypomochlion follows the power drawing and at the same time supporting the load placed upon it, the ratio of the moment of the supporting power being unchanged, whether the hypomochlion be moved or remain stable and fixed. Moreover, the more the weight is distant from the wheels, the more it burdens the horse, but the less it bounds upward, when the wheels run into an obstacle. By the name therefore of a mobile hypomochlion I understand that which D d d

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Mechanicorum 394 movente potentiâ atque conante adversùs pondus, resistit quidem vecti, sed & simul loco cedit ita, ut pondus & hypomochlium in oppositas partes immisso inter illa vecte moveantur. Sic contingere potest fulcrum deprimi, dum pondus elevatur, aut fulcrum elevari, dum pondus deprimitur, aut si utrumque in plano horizontali moveatur, in oppositas plagas recedere. Loquor autem de vecte primi & secundi generis, quibus communiter utimur; nam in vecte tertij generis, si hypomochlium cedat, movetur ad easdem partes cum pondere & potentia, sed tardiùs. Hinc si vecte inter duo pondera non immodicè inæqualia interjecto alterutrum movere coneris, reliquum etiam movetur; ita tamen ut neutrum tantum motûs perficiat, quantum haberent singula, si solitariè moverentur, reliquo manente immoto. Sit vectis A B inter duos lapides C & D interjectus, qui lapidem C non dimovebit, nisi eum tangat in puncto cui occurrit linea ex C gravitatis centroducta (aut potiùs planum per idem gravitatis centrum C transiens) ad perpendiculum in vectem, & sit linea C E; nisi enim in E lapis à vecte tanga- tur, movebitur quidem lapis circa centrum C, donec congruat vecti, sed non propelletur totus lapis. Idem dic de lapide D, nisi tangatur in F occurrente lineæ perpendiculari D F. Quare pondera intelligentur in E & F: & quoniam F resistit vecti, ut E propellatur versùs C, & vicissim E resistit vecti, ut F propellatur versùs D, propterea ad movendum pondus C, vectis A E est primi generis, & ad movendum pondus D, vectis A E est secundi generis; atque pondera illa vicissim habent rationem hypomochlij, quia vectis alteri innititur, ut alterum moveat. Cæterùm singulorum lapidum absoluta & simpliciter sumpta resistentia tum ex eorum ingenitâ gravitate, tum ex superficierum se tangentium asperitate atque conflictu definitur: Comparatè verò ad vectem non sic accipienda est singulorum resistentia, quasi motûs centra essent E aut F: experimento enim manifesto deprehenditur motum potentiæ A ad motum ponderis

Transcription: Translated (English)

Mechanics 394 while force is applied and acts against the weight, the lever resists indeed, but at the same time yields its place, so that the weight and the fulcrum, with the lever inserted between them, are moved to opposite sides. Thus it may happen that the fulcrum is depressed while the weight is raised, or that the fulcrum is raised while the weight is depressed, or, if both are moved on the horizontal plane, that they recede to opposite sides. But I am speaking of the lever of the first and second kind, which we commonly use; for in a lever of the third kind, if the fulcrum yields, it is moved toward the same side as the weight and the power, but more slowly. Hence, if a lever inserted between two weights not excessively unequal is used to move one of them, the other is also moved; yet so that neither accomplishes as much motion as each would have if it were moved alone while the other remained motionless. Let the lever A B be inserted between two stones C and D, which will not move stone C unless it touches it at the point where the line drawn from the center of gravity of C (or rather the plane passing through that same center of gravity) meets the lever at right angles, and let that line be C E; for unless the stone is touched by the lever at E, the stone will indeed move about the center C until it comes into line with the lever, but the whole stone will not be propelled. The same is to be said of stone D, unless it is touched at F where the perpendicular line D F meets it. Therefore the weights are to be understood at E and F: and since F resists the lever so that E may be driven toward C, and conversely E resists the lever so that F may be driven toward D, for that reason, in order to move weight C, the lever A E is of the first kind, and in order to move weight D, the lever A E is of the second kind; and those weights in turn have the relation of a fulcrum, because the lever rests upon one in order to move the other. Moreover, the absolute resistance of each stone, taken simply, is determined both by its innate heaviness and by the roughness and collision of the surfaces touching one another: but comparatively, with respect to the lever, the resistance of each is not to be understood in this way, as though the centers of motion were E or F: for by clear experiment it is found that the motion of the power A toward the motion of the weight

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Liber quartus. CAPUT V. 395 ponderis C non esse ut A F ad F E, neque ejusdem potentiæ A æqualem motum esse ad motum ponderis D ut A E ad F E. Nam si punctum E vectis, fixum esset, & potentiæ motus esset A L, motus ponderis F esset F H: Si verò punctum F maneret immotum, & potentiæ motus esset A I æqualis ipsi A L, motus ponderis esset E G. Tunc autem motus A I æqualis est motui A L, quando ut A F ad A E, ita vicissim angulus A E L ad angulum A F I: æqualium si quidem angulorum in circulis inæqualibus arcus sunt ut Radij; ergo si fuerint anguli reciprocè ut Radij, scilicet minor in majore circulo, & major angulus in minore, erunt æquales arcus illis oppositi: Sic anguli A F R æqualis angulo A E L arcus A R est ad A L, ut Radius F A ad Radium E A; sed ut F A ad E A, ita arcus A R ad arcum A I ex constructione; ergo ut A R ad A L ita A R ad A I: ergo per 9.lib.5. A I & A L sunt æquales. Quoniam igitur tam E quàm F ex hypothesi in oppositas partes moventur circumacto vecte, punctum aliquod est inter E & F, quod est veluti centrum motuum tam potentiæ quàm ponderum, in quo centro quodammodo divisa intelligitur resistentia, quæ componitur tùm ex eorum innatâ gravitate, tùm ex eorum motu, spectatâ positione ad vectem. Hinc manifestum est singula pondera minùs moveri, quàm si singula moverentur reliquo manente immoto; quia videlicet singula minùs distant à centro, circa quod moventur. Sic ponderum E & F gravitas ponatur æqualis: si intelligatur centrum motûs ab utroque æqualiter distare, ut sit K E æqualis ipsi K F, motus potentiæ factus intervallo A K æqualem habet Rationem ad motum, qui fit à singulis ponderibus. Quare potentiæ momentum perinde se habet, atque si utrumque pondus esset in E, aut utrumque in F, hypomochlium verò in K. Ponamus enim E F esse partium 6, quarum partium 7 est F A: igitur E K est 3, & K A 10; & potentia sine vecte movens lib.3, vecte A K E movebit lib.10 in E. Similiter K F est 3, & D d d 2

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Book Four. CHAPTER V. 395 of the weight C is not to A F to F E, nor is the motion of the same power A equal to the motion of the weight D as A E is to F E. For if the point E of the lever were fixed, and the motion of the power were A L, the motion of the weight F would be F H: but if the point F remained motionless, and the motion of the power were A I, equal to the same A L, the motion of the weight would be E G. Now the motion A I is equal to the motion A L when, as A F is to A E, so conversely is the angle A E L to the angle A F I: for in unequal circles the arcs of equal angles are as the radii; therefore if the angles are reciprocally as the radii, namely the smaller angle in the greater circle, and the greater angle in the smaller, the arcs opposite to them will be equal: thus, since the angle A F R is equal to the angle A E L, the arc A R is to A L as the radius F A is to the radius E A; but as F A is to E A, so, by construction, is the arc A R to the arc A I; therefore as A R is to A L, so is A R to A I: therefore by 9. book 5, A I and A L are equal. Since therefore both E and F, by hypothesis, are moved in opposite directions when the lever is turned, there is some point between E and F which is as it were the center of the motions both of the power and of the weights, in which center the resistance is in some way understood to be divided, which is composed both of their innate gravity, and of their motion, regard being had to their position with respect to the lever. Hence it is manifest that each weight is moved less than if each were moved the other remaining motionless; because each is at a lesser distance from the center about which they move. Thus let the gravity of the weights E and F be equal: if the center of motion be understood to be equally distant from both, so that K E is equal to K F itself, the motion of the power produced at the interval A K has an equal ratio to the motion which is produced by the individual weights. Wherefore the moment of the power is the same as if each weight were in E, or both in F, but the hypomochlion in K. For let E F be of 6 parts, of which F A is 7 parts: therefore E K is 3, and K A 10; and the power moving without a lever, of 3 pounds, with the lever A K E will move 10 pounds in E. Similarly K F is 3, and D d d 2

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Mechanicorum 396 K A est 10; igitur potentia ut 3 in A, movebit in F pondus ut 10: igitur etiam in A potentia ut 6, facto motûs centro K, mo- vebit vel utrumque pondus ut 10 in E & F, vel unicum pondus ut 20 sive in E, sive in F. Constituatur itaque potentiæ virtus ut 6, si hypomochlium esset F immotum, non moveret nisi pon- dus grave ut 7 positum in E; & facto hypomochlio stabili E moveret pondus grave 13 positum in F; adeóque universum pondus esset librarum 20. Quare idem pondus lib. 20 movetur ab eâdem potentia, sed non eodem motu: Nam hîc amborum simul ponderum motus circa centrum K est ut 6; at si potentiæ motus A I sit 10 (quemadmodum motus potentiæ circa cen- trum K est 10) circa F centrum, motus E G est 8 4; & si po- tentiæ motus A L sit pariter 10 circa centrum E motus F H est 4 8/13. Hinc patet singulorum ponderum motum, quando utrum- que simul movetur, minorem esse, quàm si singula solitariè moverentur, adeóque totum motum, qui ex duobus motibus coalescit, minorem esse summâ, quæ conflatur ex motu E G & motu F H. Præterea manifestum est cæteris paribus move- ri faciliùs pondus F, quod est Potentiæ A proximum, quàm pondus E ab eâdem remotum; minor enim differentia est in- ter 4 2/13 & 3, quàm inter 8 4/13 & 3. Quod si duorum ponderum E & F absoluta resistentia, quæ ex gravitate oritur, inæqualis fuerit, inæqualem pariter esse oportet resistentiam ex motûs velocitate, quæ unicuique pon- deri conveniat, sed reciprocè, ut fiat totius resistentiæ æquali- tas. Cum enim utrumque pondus movendum sit, par est ita re- sistentiam dividi, ut æqualibus momentis adversentur poten- tiæ contranitenti; quod scilicet gravius est, difficiliùs movetur, quod minus grave, faciliùs: igitur illius motus minor est, hu- jus major. Proptetea centrum motuum iis intervallis ab utro- que pondere aberit, ut quæ Ratio est gravioris ponderis ad mi- nus grave, ea sit Ratio distantiæ centri motûs à minùs gravi ad distantiam ejusdem centri à graviore. Sit ex. gr. pondus E lib. 8. & pondus F lib. 12; distantia E F eadem quæ priùs, hoc est, 6; & F A 7. Cum igitur pondera sint ut 2 ad 3, dividatur E F in quin- que partes, & propè gravius F assumantur duæ F M, reliquæ tres

Transcription: Translated (English)

Mechanics 396 K A is 10; therefore the power, as 3 at A, will move in F a weight as 10: thus also at A a power as 6, the center of motion being K, will move either both weights as 10 in E and F, or a single weight as 20, either in E or in F. Let the power therefore be taken as having the force 6; if the hypomochlion were fixed at F, it would move only a heavy weight as 7 placed in E; and with the hypomochlion fixed at E it would move the heavy weight 13 placed in F; and thus the whole weight would be 20 pounds. Wherefore the same weight of 20 lb. is moved by the same power, but not by the same motion: for here the motion of both weights together about the center K is as 6; but if the motion of the power A I be 10 (just as the motion of the power about the center K is 10) about the center F, the motion E G is 8 4/13; and if the motion of the power A L be likewise 10 about the center E, the motion F H is 4 8/13. Hence it is clear that the motion of individual weights, when both are moved together, is less than if they were moved separately, and thus the whole motion, composed of the two motions, is less than the sum made up from the motion E G and the motion F H. Moreover it is manifest that, other things being equal, the weight F, which is nearer to the power A, is moved more easily than the weight E, which is farther from it; for the difference between 4 2/13 and 3 is less than that between 8 4/13 and 3. But if the absolute resistance of the two weights E and F, which arises from gravity, is unequal, then the resistance arising from the speed of motion, which belongs to each weight, must likewise be unequal, but reciprocally, so that equality of the whole resistance may result. For since both weights are to be moved, the resistance ought to be divided in such a way that the counterstriving powers oppose equal moments; that which is heavier is moved more difficultly, that which is less heavy more easily: therefore the motion of the former is smaller, of the latter greater. Therefore the center of motions will be distant from the two weights by such intervals that, as the ratio of the heavier weight to the lighter is, so also is the ratio of the distance of the center of motion from the lighter to the distance of the same center from the heavier. Let, for example, weight E be 8 lb. and weight F 12 lb.; let the distance E F be the same as before, namely 6; and F A 7. Since therefore the weights are as 2 to 3, let E F be divided into five parts, and near the heavier F let two parts F M be taken, the remaining three

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Liber quartus. CAPUT V. 397 tres ME spectent ad minus grave E. Si itaque circa centrum M moveantur pondera E & F, habent æqualia resistentiæ mo- menta; nam lib. 12 moventur ut 2, & lib. 8 moventur ut 3. Quare AM est ad ME ut 9 2/5 ad 3 3/5, & AM ad MF est ut 9 2/5 ad 2 2/5. Fiat igitur ut AM ad ME, ita reciprocè pondus E lib. 8 ad virtutem potentiæ A movendi sine vecte libras 3 3/47: & ut AM ad MF, ita reciprocè pondus F lib. 12 ad ejusdem po- tentiæ A virtutem movendi sine vecte libras 3 3/47. In hac ita- que ponderum inæqualium dispositione paulo plus virium re- quiritur in potentia (hoc est vis movendi lib. 6 6/47) quàm si es- sent æqualia, eandemque gravitatis summam lib. 20 consti- tuerent. At si vice versâ pondus E esset lib. 12, & F lib. 8, centrum motuum esset N, atque AN esset 10 3/5: ac propterea ut AN 10 3/5 ad NE 2 2/5, ita pondus E lib. 12 ad virtutem potentiæ si- ne vecte moventis libras 2 3/53; & ut AN 10 3/5 ad NF 3 2/5, ita pondus F lib. 8 ad virtutem potentiæ A moventis sine vecte li- bras 2 3/53. Tota igitur virtus potentiæ in hac eorumdem pon- derum inæqualium collocatione sufficiet, si fuerit vis movendi lib. 5 2/53, quæ minor est eâ, quæ requiritur; quando pondera sunt æqualia, & differt à virtute, quæ requiritur, quando F gravius est quàm E, vi movendi ferè uncias 8 1/5. Simili argumentatione ratiocinando deprehendes, quo mi- nus fuerit intervallum inter E & F, etiam faciliùs duo illa pon- dera eodem vecte moveri. Nam si idem vectis AE 13 adhi- beatur, atque pondera E & F æqualia fuerint, intervallum ve- rò EF sit 4, centrum motuum distabit ab A intervallo 11, & à singulis ponderibus intervallo 2: Quare potentia ut 4 move- bit pondera singula ut 11: vel si ponantur ut prius singula lib. 10, fiat ut 11 ad 2, ita lib. 10 ad potentiam sine vecte mo- ventem lib. 1 2/11; atque ideò tota potentia sufficiens ad movenda duo pondera æqualia simul sumpta lib. 20, erit vis movendi sine vecte lib. 3 2/5. Quod si E fuerit lib. 8, & F lib. 12, E distabit à à centro motuum partibus 2 2/5, F verò part. 1 3/5, & potentia A distabit part. 10 2/5: Ex quo sit singula moveri posse à potentia D d d 3

Transcription: Translated (English)

Book Four. Chapter V. 397 let the three look toward the lighter E at least. If, therefore, around the center M the weights E and F are moved, they have equal moments of resistance; for, when weighted 12, they are moved as 2, and when weighted 8, as 3. Wherefore AM is to ME as 9 2/5 to 3 3/5, and AM to MF is as 9 2/5 to 2 2/5. Let it therefore be as AM is to ME, so, reciprocally, let weight E of 8 lb. be to the power A of moving 3 3/47 lb. without a lever: and as AM is to MF, so, reciprocally, let weight F of 12 lb. be to the power of the same A of moving 3 3/47 lb. without a lever. In this disposition, therefore, of unequal weights, somewhat more force is required in the power (that is, a moving force of 6 6/47 lb.) than if they were equal, and together constituted the same sum of weight of 20 lb. But if, conversely, weight E were 12 lb. and F 8 lb., the center of motions would be N, and AN would be 10 3/5: and therefore as AN 10 3/5 is to NE 2 2/5, so weight E of 12 lb. is to the power of moving 2 3/53 lb. without a lever; and as AN 10 3/5 is to NF 3 2/5, so weight F of 8 lb. is to the power A of moving 2 3/53 lb. without a lever. Therefore the whole power in this arrangement of the same unequal weights will suffice if it be a moving force of 5 2/53 lb., which is less than that which is required when the weights are equal, and differs from the force required when F is heavier than E, by a moving force of about 8 1/5 ounces. By similar reasoning you will find that the smaller the interval between E and F is, the more easily will those two weights be moved by the same lever. For if the same lever AE of 13 be employed, and the weights E and F be equal, but the interval EF be 4, the center of motions will be at a distance of 11 from A, and 2 from each weight: wherefore the power as 4 will move each weight as 11: or, if each be set, as before, at 10 lb., let it be as 11 to 2, so 10 lb. is to the power moving without a lever 1 2/11 lb.; and therefore the whole power sufficient to move two equal weights taken together, of 20 lb., will be a moving force without a lever of 3 2/5 lb. But if E were 8 lb. and F 12 lb., E will be distant from the center of motions by 2 2/5 parts, but F by 1 3/5 parts, and the power A will be distant by 10 2/5 parts: from which it follows that each may be moved by a power D d d 3

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Mechanicorum 398 habente virtutem movendi sine vecte lib. 1 43/53, & ambo simul à potentia habente vim movendi lib. 3 33/53. At verò si vicissim E fuerit lib. 1 2, & F lib. 8, distantia potentiæ à centro motuum erit part. 1 1 2/51, ac propterea singula pondera exigent virtutem movendi lib. 1 12/51, & tota potentia ad utrumque simul movendum sufficiens erit vis movendi sine vecte lib. 3 21/57, quæ deficit à vi movendi lib. 3 33/53, ea virtute, quæ requireretur ad movendum uncias 3 10/50, atque à vi movendi lib. 3 2/51 deficit per uncias 3 1/5 ferè. Que de corpore gravi dimovendo dicta sunt, intelligantur pariter, si vectis inter duo corpora flectenda, aut divellenda, interjiceretur; quemadmodum objectos caveæ si quæras frangere clathros: quod enim hîc gravitas, ibi ferreæ virgæ aut lignei tigilli soliditas impedimentum affert motui. Porrò in vecte tertij generis, quando potentia inter utrumque pondus mobile constituitur, aliter res se habet: adhoc scilicet, ut aliquam vectis Rationem habeat, requiritur aut inæqualitas ponderum, aut saltem inæqualitas distantiarum potentiæ à ponderibus in utrâque extremitate constitutis, ita tamen ut hæ distantiæ non sint in reciprocâ Ratione ponderum: nam si planè æqualiter distaret potentia ab æqualibus ponderibus, aut inæquales distantiæ essent in reciprocâ Ratione inæqualium ponderum, ita utrumque traheretur, aut impellere tur, ut pondera singula æquè moverentur ac potentia: ad Rationes autem vectis spectat inæqualiter moveri potentiam ac pondus, si vectis quidem obtineat vim Facultatis Mechanicæ. Quoniam igitur in hujusmodi vecte tertij generis oportet utrumque pondus opponi motui potentiæ, vel quia utrumque impellitur, vel quia alterum trahitur, alterum impellitur, sit vectis A B, in cujus extremitatibus pondera respondeant punctis A & B: Si potentia fuerit in C æquè distans ab A & B, pondera autem fuerint æqualia; potentia ex C versus D mota nullum haberet sui motûs centrum, sed pariter traheret aut impelleret ad partes D utrumque pondus; nam æquè utrumque resisteret tùm

Transcription: Translated (English)

Mechanics 398 having the force of moving without the lever lib. 1 43/53, and both together from the power having force to move lib. 3 33/53. But if, on the other hand, E were lib. 1 2, and F lib. 8, the distance of the power from the center of motion will be part. 1 1 2/51, and therefore each weight will require the force of moving lib. 1 12/51, and the whole power sufficient to move both together will be the force of moving without the lever lib. 3 21/57, which falls short of the force of moving lib. 3 33/53, by that force which would be required to move 3 10/50 ounces, and from the force of moving lib. 3 2/51 it falls short by ounces 3 1/5 nearly. What has been said about moving a heavy body should be understood in the same way, if a lever were interposed between two bodies to be bent or pulled apart; as when, if you seek to break the bars of a cage, an object is opposed: for what here is weight, there the hardness of the iron rod or wooden stick presents an obstacle to motion. Moreover, in a lever of the third kind, when the power is placed between the two movable weights, the matter is otherwise: namely, for the lever to have some ratio, either inequality of the weights is required, or at least inequality of the distances of the power from the weights placed at both ends, provided however that these distances are not in reciprocal ratio to the weights: for if the power were exactly equally distant from equal weights, or unequal distances were in reciprocal ratio to unequal weights, then both would be drawn, or pushed, so that the individual weights would be moved as much as the power: but to the ratios of the lever it pertains that the power and the weight be moved unequally, if indeed the lever possesses the force of Mechanical Faculty. Since therefore in a lever of this kind of the third kind it is necessary that each weight oppose the motion of the power, either because both are pushed, or because one is pulled and the other pushed, let there be the lever A B, at whose extremities the weights correspond to the points A and B: if the power were at C equally distant from A and B, but the weights were equal; the power moved from C toward D would have no center of its motion, but would equally draw or push toward the parts D both weights; for each would equally resist then

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CAPUT V. tùm ratione gravitatis, tùm ratione positionis & distantiæ, quæ legem daret motui, ac proinde utrumque æqualiter cederet virtuti potentiæ. At si pondus A minus fuerit, quàm pondus B, sed reciprocam Rationem habeant distantiæ potentiæ existentis in E, ut sit EB ad EA, in Ratione ponderis A ad pondus B; adhuc æquales sunt resistentiæ; sicut enim in plano Verticali potentiæ in E sustineret utrumque pondus in æquilibrio, ita in plano horizontali trahens aut impellens utrumque æqualiter moveret. Sint igitur pondera A & B sive æqualis gravitatis, sive inæ- qualis, & ita potentia sit in E, ut EB ad EA non sit in Ratio- ne ponderis A ad pondus B: utique si B moveri non posset, potentia E circa B, tanquam circa centrum, describeret ar- cum EI, & pondus A arcum AF: similiter si pondus A immo- tum maneret, potentia circa A, tanquam circa centrum, des- criberet arcum EH, & pondus B arcum BG, ex hypothesi æqualem arcui AF. Potentia igitur in E faciliùs cæteris pari- bus moveret pondus B sibi proximum, quàm pondus A remo- tum, si singula singillatim movenda essent; quia, cum arcus EH major sit arcu EI, arcus autem BG, & AF sint æquales, major est Ratio EH ad EI quàm BG ad AF; & per 27. lib.5. vicissim EH ad BG habet majorem Rationem quàm EI ad AF. Cum itaque neutra extremitas immota maneat, sed ambo pondera moveantur, minùs movetur A, quod difficiliùs, ma- gis B, quod faciliùs: ac propterea A simpliciter fungitur mu- nere hypomochlij ad motum ponderis B: hoc verò vicissim ad ponderis A motum, quamvis minorem, subit vicem fulcri: Neque enim hic unum tribus motibus, potentiæ videlicet & duorum ponderum, commune centrum reperire est, quia ad eandem partem omnium motus dirigitur. Hinc si fune alligato in E trahas vectem cum ponderibus, punctum E neque omni- no versùs I, neque omnino versùs H dirigetur, quamquam ad H potiùs, quàm ad 1 inclinabitur; quia faciliùs A vectis punctum respondens ponderi convertitur circa centrum gravi- tatis ponderis, quàm propellat aut trahat totum pondus, pro ut ferunt, & ipsius gravitas, & ejusdem distantia ab E, quæ il- lius resistentiam componunt. CAPUT

Transcription: Translated (English)

CHAPTER V. both on account of gravity, and on account of position and distance, which would determine the law of motion, and therefore each would equally yield to the power. But if the weight A be less than the weight B, yet the distances of the power existing at E be in reciprocal proportion, so that EB to EA is in the ratio of the weight A to the weight B, still the resistances are equal; for just as in the vertical plane the power at E would sustain both weights in equilibrium, so in the horizontal plane, by drawing or impelling, it would move both equally. Let therefore the weights A and B be either of equal gravity or of unequal, and let the power be at E in such a way that EB to EA is not in the ratio of the weight A to the weight B: certainly if B could not be moved, the power at E, about B as a center, would describe the arc EI, and the weight A the arc AF; similarly if the weight A were to remain at rest, the power, about A as a center, would describe the arc EH, and the weight B the arc BG, by hypothesis equal to the arc AF. The power therefore at E would more easily, other things being equal, move the weight B near to itself than the weight A more remote, if each had to be moved separately; because, since the arc EH is greater than the arc EI, while the arcs BG and AF are equal, the ratio of EH to EI is greater than that of BG to AF; and by 27 of book 5, conversely EH has a greater ratio to BG than EI has to AF. Since therefore neither extremity remains unmoved, but both weights are moved, A is moved less, since it is more difficult, B more, since it is easier: and therefore A simply serves the office of hypomochlion for the motion of weight B; but this in turn, for the motion of weight A, though lesser, takes the role of fulcrum. For here one common center for the three motions, namely of the power and of the two weights, is not to be found, because the motion of all is directed to the same side. Hence if you pull the lever with a rope tied at E, the point E will be directed neither altogether toward I nor altogether toward H, although it will incline more toward H than toward I; because the point of the lever at A, corresponding to the weight, is more easily turned about the center of gravity of the weight than it propels or draws the entire weight, according as both its gravity and its distance from E, which together make up its resistance, are understood. CHAPTER

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Mechanicorum CAPUT VI. Quanam sint momenta Vectis pondus fune connexum trahentis. Contingere potest oblato ponderi super planum horizontale, aut inclinatum, trahendo non esse parem Potentiam: hujus imbecillitati opem ferre licebit Vecte potissimùm secundi generis, cujus extremitas altera fixa & stabilis maneat in plano, in quo pondus jacet, alteram extremitatem Potentia moveat, & loco intermedio alligetur funis cum pondere connexus, qui dum vecte movetur, secum rapiat & pondus. Verùm non leviter hallucinaretur, quisquis momenta vectis ex alligati funis loco simpliciter & absolutè definiret; cum potiùs ponderis resistentiam ex ipsius motu computare oporteat. Quoniam verò duplex esse potest vectis motus, nimirum aut in plano Verticali, aut in Horizontali, propterea uterque seorsim considerandus est; diversas enim lineas in plano, in quo jacet, pondus percurrit; rectam scilicet, si vectis in plano Verticali agitatur; curvam verò, si in plano horizontali aut inclinato eodem, cui pondus incumbit etiam vectis moveatur. Sit in plano, in quo pondus jacet, linea AB, cui vectis congruere intelligatur, & concipiatur pondus in puncto C; vecti autem in D alligetur funis ita connectens pondus cum vecte, ut parti vectis DA æqualis sit funis DC. Attollatur in plano Verticali vectis, ut sit AE describens arcum BE; etiam punctum D ascendit in F, ac propterea funis est FG, & ponderis motus est CG. Iterum attollatur æqualiter vectis ex E in H; funis caput venit in I, & pondus in K. Similiter vecte

Transcription: Translated (English)

Mechanics CHAPTER VI. What are the moments of a lever drawing a weight connected by a rope. It can happen, when a weight is placed upon a horizontal or inclined plane, that by pulling there is not an equal power: to remedy this weakness one may chiefly use a lever of the second kind, one end of which remains fixed and stable in the plane in which the weight lies, while the power moves the other end, and at an intermediate point the rope connected with the weight is attached, which, while the lever is moved, may draw the weight along with it. But one would err not a little who should define the moments of the lever simply and absolutely from the place of the tied rope; since rather the resistance of the weight must be computed from its own motion. And since the motion of the lever may be twofold, namely either in a vertical plane or in a horizontal one, therefore each must be considered separately; for the weight traverses different lines in the plane in which it lies: namely a straight line, if the lever is moved in a vertical plane; but a curved line, if the lever itself is moved in the horizontal plane, or in the same inclined plane on which the weight also rests. Let there be in the plane in which the weight lies the line AB, to which the lever is understood to correspond, and let the weight be conceived at the point C; but let a rope be tied to the lever at D, so connecting the weight with the lever that the part DA of the lever is equal to the rope DC. Let the lever be raised in a vertical plane, so that it be AE describing the arc BE; the point D also rises to F, and therefore the rope is FG, and the motion of the weight is CG. Again let the lever be raised equally from E to H; the end of the rope comes to I, and the weight to K. Likewise the lever

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Liber quartus. CAPUT VI. 401 vecte in L sublato, funis venit in M, & pondus in N. Sunt igitur tres ponderis motus, CG, GK, KN, inter se inæquales, qui semper majores fiunt; motus autem potentiæ BE, EH, HL ex hypothesi sunt æquales; igitur major est Ratio motûs BE ad motum CG, quàm motûs EH ad motum GK, & hæc Ratio major est Ratione motûs HL ad motum KN. Cum itaque motibus BE, EH, HL similes sint motus DF, FI, IM, manifestum est motum ponderis non servare Rationem secundùm quam dividitur vectis ab alligati funis capite, eadem quippe semper est Ratio EA ad AF, & HA ad AI, & LA ad AM. Motus autem illos ponderis CG, GK & KN semper esse majores hinc constat, quia pars vectis inter funem alligatum atque hypomochlium A ex hypothesi est æqualis ipsi funi connectenti pondus: sunt igitur triangula Isoscelia æqualium semper laterum, sed quæ majores & majores angulos ad basim habent, ideóque minorem & minorem angulum verticalem continent. Atqui angulorum ad centrum in circulis æqualibus, vel in eodem circulo, semper æqualiter decrescentium subtensæ minores fiunt eâ lege, ut decrementa illa, hoc est, subtensarum diminutarum differentiæ augeantur, ut ex Canone Sinuum constat. Cum itaque AG sit subtensa anguli AFG, & AK sit subtensa anguli AIK minoris, & AN sit subtensa anguli AMN adhuc minoris; harum subtensarum differentiæ, videlicet CG (differentia inter diametrum circuli AC & subtensam AG) GK & KN motus ponderis semper augentur. Id quod ut manifestum fiat, triangula ipsa ad calculos revocemus singulorum basim inquirentes: ponamus verò ex. gr. arcus BE, EH, HL singulos grad. 15, & latera singula AF & GF esse partium 100. Igitur angulus AFG est grad. 150, & basis AG deprehenditur partium 193 18/100. Est autem AC ex hypothesi 200, adeóque CG part. 6 8/100. In triangulo AIK latera sunt eadem, anguli ad basim singuli grad. 30, angulus verticalis grad. 120; ergo basis AK part. 173 20/100: & inter AK atque AG differentia GK est 19 98/100. Deinde in triangulo AMN anguli ad basim singuli sunt grad. 45; igitur angulus verticalis grad. 90, & basis AN part. 141 42/100, & inter AN & AK dif- Ecc

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Liber quartus. CAPUT VI. 401 the lever being raised at L, the rope comes to M, and the weight to N. There are therefore three motions of the weight, CG, GK, KN, unequal among themselves, which are always increasing; but the motions of the power BE, EH, HL, by hypothesis, are equal; therefore the ratio of the motion BE to the motion CG is greater than that of the motion EH to the motion GK, and this ratio is greater than the ratio of the motion HL to the motion KN. Since therefore the motions DF, FI, IM are similar to the motions BE, EH, HL, it is manifest that the motion of the weight does not preserve the ratio according to which the lever is divided by the head of the fastened rope, for the ratio of EA to AF is always the same, and of HA to AI, and of LA to AM. But that those motions of the weight, CG, GK, and KN, are always greater, is made clear from this: because the part of the lever between the fastened rope and the fulcrum A, by hypothesis, is equal to the rope itself connecting the weight. Therefore the triangles are isosceles, with sides always equal, but which have larger and larger angles at the base, and therefore contain a smaller and smaller vertical angle. Yet the chords of angles at the center in equal circles, or in the same circle, which are always decreasing equally, become smaller in such a way that those decreases, that is, the differences of the diminished chords, increase, as is known from the Canon of Sines. Since therefore AG is the chord of angle AFG, and AK is the chord of the smaller angle AIK, and AN is the chord of the still smaller angle AMN; the differences of these chords, namely CG (the difference between the diameter of the circle AC and the chord AG), GK, and KN, the motions of the weight, always increase. So that this may be made clear, let us reduce the triangles themselves to calculation, inquiring into the base of each: let us suppose, for example, the arcs BE, EH, HL each 15 degrees, and the sides AF and GF each to be 100 parts. Therefore the angle AFG is 150 degrees, and the base AG is found to be 193 18/100 parts. But AC, by hypothesis, is 200, and therefore CG is 6 8/100 parts. In triangle AIK the sides are the same, the angles at the base each 30 degrees, the vertical angle 120 degrees; therefore the base AK is 173 20/100 parts: and the difference between AK and AG, GK, is 19 98/100. Then in triangle AMN the angles at the base are each 45 degrees; therefore the vertical angle is 90 degrees, and the base AN is 141 42/100 parts, and between AN and AK dif- Ecc

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Mechanicorum 402 ferentia K N est 31 78/100. Et si vectem elevando pergas, idem in consequentibus triangulis deprehendes, augeri scilicet differentias usque ad A. Hinc patet eò faciliorem esse, cæteris paribus, motum, quò majorem angulum funis cum vecte constituit, nam ab æquali potentiæ motu minor ponderis motus efficitur, quàm si major esset angulus elevationis vectis. Quare faciliùs promovebitur ad destinatum locum pondus quod trahitur, si post aliqualem vectis elevationem, iterum inclinato, quàm maximè fieri poterit, vecte, extremitatem A, hoc est hypomochlium subinde promoveas, quantum feret funis longitudo: tunc enim fune maximè inclinato tractio ponderis minùs obliqua juvabit motum, qui etiam minor est, quàm si pergeres vectem elevando. Non est tamen necesse servari hanc, quam claritatis gratiâ proposui, æqualitatem funis F G, & partis vectis F A; sed assumi potest vel longior, vel brevior funis, adeò ut ex vectis parte, ex fune, & ex distantiâ ponderis ab hypomochlio fiat triangulum scalenum: in quo si funis fuerit longior parte vectis, eodem potentiæ motu minùs accedet pondus ad hypomochlium, quàm si funis fuerit brevior eâdem vectis parte; atque quo longior fuerit funis, etiam minor erit, adeóque facilior, ponderis motus, cæteris paribus, nam & tractio minùs obliqua erit. Statue igitur ex. gr. A D esse partium 73, & D C part. 100: quare vecte jacente, distantia A C est 173. Sit angulus F A G iterum gr. 15; invenitur angulus A F G gr. 154. m. 7, & basis A G part. 168 66/100; igitur C G 4 34/100. In triangulo A I K latera A I 73, I K 100 ut priùs, angulus I A K gr. 30: invenitur angulus verticalis A I K gr. 128. m. 36, & basis A K part. 156 30/100: igitur G K est 12 36/100. Demum in triangulo A M N latera sunt eadem ut priùs, angulus M A N gr. 45: ex quibus datis invenitur angulus A M N gr. 103. m. 55, & basis A N part. 137 27/100: igitur K N 19 3/100, & totus motus C N est part. 35 73/100. Sed vicissim statue A D partium 100, D C verò funent part. 73; quibus æqualia sunt trianguli A F G latera A F 100 & F G 73; angulus autem F A G est gr. 15: invenitur angulus F G A gr. 20. m. 46, & angulus A F G gr. 144. m. 46: quare A G

Transcription: Translated (English)

Mechanics 402 The difference K N is 31 78/100. And if you continue raising the lever, you will discover the same thing in the following triangles, namely that the differences increase up to A. Hence it is clear that the motion is made easier, ceteris paribus, the greater the angle which the rope makes with the lever; for, from an equal motion of the power, a smaller motion of the weight is produced than if the angle of elevation of the lever were greater. Wherefore the weight that is being drawn will be advanced more easily to its appointed place, if after the lever has been raised to some extent, the lever is again inclined as much as possible, and you then move forward the end A, that is, the hypomochlion, as far as the length of the rope allows: for then, with the rope most inclined, the pull on the weight, being less oblique, will aid the motion, which is also smaller than if you were to continue raising the lever. It is not necessary, however, to preserve this equality of the rope F G and of the part of the lever F A, which I proposed for the sake of clarity; rather, a longer or a shorter rope may be assumed, so that from the part of the lever, the rope, and the distance of the weight from the hypomochlion, a scalene triangle is formed: in which, if the rope is longer than the part of the lever, then with the same motion of the power the weight will approach the hypomochlion less than if the rope were shorter than that same part of the lever; and the longer the rope is, the smaller, and therefore the easier, will be the motion of the weight, ceteris paribus, since the pull will also be less oblique. Set, for example, A D equal to 73 parts, and D C equal to 100 parts: thus, when the lever lies down, the distance A C is 173. Let the angle F A G again be 15 degrees; the angle A F G is found to be 154 degrees 7 minutes, and the base A G 168 66/100 parts; therefore C G is 4 34/100. In the triangle A I K, the sides A I 73 and I K 100, as before, and the angle I A K 30 degrees: the vertical angle A I K is found to be 128 degrees 36 minutes, and the base A K 156 30/100 parts; therefore G K is 12 36/100. Finally, in the triangle A M N the sides are the same as before, and the angle M A N is 45 degrees: from these given data the angle A M N is found to be 103 degrees 55 minutes, and the base A N 137 27/100 parts; therefore K N is 19 3/100, and the total motion C N is 35 73/100 parts. But conversely, suppose A D to be 100 parts, and D C rope 73 parts; equal to these are the sides of triangle A F G, namely A F 100 and F G 73; and the angle F A G is 15 degrees: the angle F G A is found to be 20 degrees 46 minutes, and the angle A F G 144 degrees 46 minutes; wherefore A G

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Liber quartus. CAPUT VI. 403 A G est part. 164 85/100, & motus C G part. 8 15/100, qui tamen su- periùs, quando D C major erat, quàm A D, deprehensus est solum 4 34/100. In Triangulo A I K similiter datur A I 100, I K 73, angulus I A K gr. 30: invenitur angulus I K A gr. 43. m. 14, ac proinde angulus verticalis A I K gr. 106. m. 46, & basis A K part. 139 79/100: igitur G K part. 25 6/100, quæ tamen priùs erat 12 36/100. Deinde in triangulo A M N latera sint eadem, & an- gulus M A N gr. 45: invenitur angulus M N A gr. 75. m. 37, & verticalis A M N gr. 59. m. 23: quare basis A N est par. 88 81/100, & motus K N part. 50 94/100, qui priùs erat 19 3/100. Verùm ele- vari vectis poterit solùm, ut funis fiat perpendicularis horizon- ti, scilicet facto angulo ad A gr. 46. m. 53; & basis erit distan- tia ab A part. 68 35/100. Ut autem innotescat, quid contingat fune adhuc longiore quàm part. 100, positâ eâdem vectis parte A D part. 73. non pigeat iterum examinare triangula. Sit ergo funis D C part. 200, quarum A D est 73; anguli elevationis vectis sint iidem, qui superiùs. In Triangulo A F G, angulus F A G est gr. 15, latus A F 73, latus F G 200: invenitur angulus F G A gr. 5. m. 25, & angulus A F G gr. 159. m. 35: ac proinde basis A G part. 269 56/100, & motus C G part. 3 44/100, qui, posito fune F G 100, erat 4 34/100. In triangulo A I K latera sunt eadem, angulus I A K est gr. 30; ergo angulus I K A gr. 10. m. 31, & verticalis A I K gr. 139. m. 29; atque basis A K part. 259 87/100; ac propterea G K part. 9 69/100, quæ priùs fuit 12 36/100. Denique in Triangulo A M N eadem latera 73 & 200 cum angulo M A N gr. 45, dant angulum M N A gr. 14. m. 57, & verticalem A M N gr. 120. m. 3; atque basim A N part. 244 82/100: quare motus K N est 15 5/100, qui in priore hypothesi erat 19 3/100. Lon- gior itaque funis dat minorem & faciliorem motum pon- deris. Quemadmodum verò elevando vectem à positione hori- zontali usque ad perpendicularum difficultas trahendi augetur, quia pondus velociùs movetur, ita ex adverso, si vectis hori- zonti perpendicularis inclinetur ad partem oppositam ponderi E e e 2

Transcription: Translated (English)

Book Four. Chapter VI. 403 A G is part 164 85/100, and motion C G part 8 15/100, which however above, when D C was greater than A D, was found to be only 4 34/100. In Triangle A I K likewise there is given A I 100, I K 73, angle I A K gr. 30: angle I K A is found to be gr. 43. m. 14, and therefore the vertical angle A I K gr. 106. m. 46, and base A K part. 139 79/100: therefore G K part. 25 6/100, which however was formerly 12 36/100. Then in triangle A M N let the sides be the same, and angle M A N gr. 45: angle M N A is found to be gr. 75. m. 37, and vertical A M N gr. 59. m. 23: wherefore base A N is par. 88 81/100, and motion K N part. 50 94/100, which was formerly 19 3/100. But the lever can be raised only so far that the cord becomes perpendicular to the horizon, namely when the angle at A is gr. 46. m. 53; and the base will be the distance from A part. 68 35/100. Now in order that it may be made known what happens with a cord even longer than part. 100, the same part of the lever A D being set at part. 73, it is not troublesome to examine the triangles again. Let the cord D C then be part. 200, of which A D is 73; let the angles of elevation of the lever be the same as above. In Triangle A F G, angle F A G is gr. 15, side A F 73, side F G 200: angle F G A is found to be gr. 5. m. 25, and angle A F G gr. 159. m. 35: and therefore base A G part. 269 56/100, and motion C G part. 3 44/100, which, with cord F G set at 100, was 4 34/100. In triangle A I K the sides are the same, angle I A K is gr. 30; therefore angle I K A gr. 10. m. 31, and vertical A I K gr. 139. m. 29; and thus base A K part. 259 87/100; and therefore G K part. 9 69/100, which formerly was 12 36/100. Finally in Triangle A M N, the same sides 73 and 200 with angle M A N gr. 45, give angle M N A gr. 14. m. 57, and vertical A M N gr. 120. m. 3; and thus base A N part. 244 82/100: wherefore motion K N is 15 5/100, which in the former hypothesis was 19 3/100. A longer cord therefore gives a smaller and easier motion of the weight. But just as by raising the lever from a horizontal position up to a perpendicular one the difficulty of drawing increases, because the weight moves more quickly, so on the other hand, if the lever, perpendicular to the horizon, is inclined toward the side opposite the weight E e e 2

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Mechanicorum 404 (adeò ut vectis sit inter potentiam & pondus) crescit trahendi facilitas, quia pondus semper tardiùs movetur, quo magis vectis ad horizontem de- primitur. Sit enim pon- dus in P, vectis perpendi- cularis CB, funis OP utique longior parte vectis OC. Inclinetur vectis per Quadrantis trientem, ut O veniat in S; funis erit ST, & pondus veniet ex P in T. Æquali inclinatione deprimatur vectis, ut S veniat in M; funis erit MV, & ponderis motus TV. Demum vectis horizonti congruat, ut M veniat in N; funis erit NI, motusque ponderis VI. Cum igitur semper tar- diùs moveatur pondus, quia spatia PT, TV, VI semper de- crescunt, æquales autem potentiæ motus illis respondeant, etiam crescit trahendi facilitas. Illa autem basium CP, CT, CV decrementa in triangulis COP, CST, CMV semper minui constabit ex Trigonome- tria; dantur enim in omnibus eadem duo latera, scilicet funis longitudo, & pars vectis, datúrque in singulis æqualiter crescens angulus funi oppositus; quare inveniuntur & bases, quarum differentiæ semper minores fiunt. Sic in triangulo OCP rectangulo sit perpendicularum OC partium 73, & hypo- thenusa OP part. 100; igitur CP basis est part. 68 34/100. Deinde quia CS est part. 73, ST part. 100, & angulus SCT gr. 120, invenitur CT part. 40 97/100: ergo PT est part. 27 37/100. Similiter MC est 73, MV 100, angulus MCV gr. 150; igitur CV in- venitur part. 29 91/100; ac proinde TV est part. 11 6/100. Demum quia NC est 73, & NI est 100, remanet CI part. 27: & ablatâ CI ex CV, relinquitur IV part. 2 91/100. Totus itaque motus PI est part. 41 34/100. Fac autem OC 73 esse quartam to- tius vectis partem, qui proinde erit part. 292. Et quia Radius ad Quadrantem peripheriæ circuli est ut 7 ad 11, si fiat ut 7 ad 11, ita 292 ad 458 5/7, potentia in vectis extremitate posita motum

Transcription: Translated (English)

Mechanics 404 (so that the lever lies between the power and the weight) the ease of drawing increases, because the weight is always moved more slowly as the lever is depressed more toward the horizon. For let the weight be at P, the perpendicular lever CB, and the cord OP, certainly longer than the part OC of the lever. Let the lever be inclined by a third of a quadrant, so that O comes to S; the cord will be ST, and the weight will move from P to T. With an equal inclination let the lever be lowered, so that S comes to M; the cord will be MV, and the motion of the weight TV. Finally let the lever coincide with the horizon, so that M comes to N; the cord will be NI, and the motion of the weight VI. Since therefore the weight always moves more slowly, because the spaces PT, TV, VI always diminish, while the motions of equal powers correspond to these, the ease of drawing also increases. Now the decrements of the bases CP, CT, CV in the triangles COP, CST, CMV will always be seen to diminish from Trigonometry; for in all of them the same two sides are given, namely the length of the cord and the part of the lever, and in each case the angle opposite the cord increases equally; wherefore the bases are found likewise, whose differences always become smaller. Thus in the right triangle OCP, let the perpendicular OC be 73 parts, and the hypotenuse OP 100 parts; therefore the base CP is 68 34/100 parts. Then because CS is 73 parts, ST 100 parts, and angle SCT 120 degrees, CT is found to be 40 97/100 parts: therefore PT is 27 37/100 parts. Similarly MC is 73, MV 100, angle MCV 150 degrees; therefore CV is found to be 29 91/100 parts; and accordingly TV is 11 6/100 parts. Finally because NC is 73, and NI is 100, there remains CI 27: and after subtracting CI from CV, IV remains 2 91/100 parts. Thus the whole motion PI is 41 34/100 parts. Let now OC = 73 be one quarter of the whole lever, which accordingly will be 292 parts. And since the radius to the quadrant of the circumference of a circle is as 7 to 11, if as 7 is to 11, so 292 is to 458 5/7, the power placed at the end of the lever moves

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Liber quartus. CAPUT VI. 405 motum habet part. 458, dum pondus movetur solum per spatium part. 41. Iam verò finge omnia eadem, præter funis longitudinem, quam statuamus O P ex. gr. partium 200, quarum O C est 73. igitur C P est 186 ́; & quia N C est 73, atque N I ex hypothesi est 200, remanet C I part. 127; atque adeò totus motus P I est part. 59 ́: ad quem motum idem potentiæ motus 458 habet minorem Rationem quàm ad motum part. 41, quem dat minor funis longitudo. Superest adhuc tertia quædam ponderis positio, vecte agitato in plano verticali; quando nimirum initio motûs statuitur pondus proximum hypomochlio, à quo in motu recedat: hujusmodi scilicet vecte uti possumus, cùm aliquid modicè quidem movendum in plano horizontali proponitur, sed multa est difficultas. Similiter si longiuscula ferrea bractea esset suis extremitatibus validè connexa cum aliquo corpore, & circa medium eam flecti oporteret, ut cuneus vel aliquid simile inter corpus & bracteam inseri posset; funi adnectoretur uncus bracteam apprehendens, qui elevato vecte, sive depresso illam aliquantulum flecteret. Sit vectis hypomochlium in R, funis in K alligatus, & funis longitudo K S 73; pars verò vectis R K 100: quare R S distantia ponderis S ab hypomochlio R est 27. Moveatur vectis sursum, & faciat angulum L R K gr. 15 funis est L I part. 73, & L R part. 100; igitur ex his datis invenitur angulus R I L gr. 159. m. 14, & R L I gr. 5. m. 46; adeóque basis R I part. 28 34/100; igitur motus ex S in I est 1 34/100. Quod si ponatur R L esse quarta pars vectis, totus Radius est part. 400, & arcus ab extremitate vectis descriptus gr. 15, est part. 104 32/100: Ex quo vides motum potentiæ ad motum ponderis esse proximè ut 78 ad 1. At verò si funis L I longior ponatur, ut sit part. 90, & reliqua sint ut priùs, invenitur angulus R I L gr. 163. Eec. 3

Transcription: Translated (English)

Book Four. Chapter VI. 405 has motion of 458 parts, while the weight is moved only through a space of 41 parts. Now moreover imagine everything the same, except the length of the cord, which let us suppose O P to be, for example, of 200 parts, of which O C is 73. therefore C P is 186′; and because N C is 73, and N I by hypothesis is 200, there remains C I part. 127; and thus the whole motion P I is part. 59′: to which motion the same motion of the power, 458, has a smaller ratio than to the motion of part. 41, which a shorter length of cord gives. There remains still a third position of the weight, with the lever moved in a vertical plane; namely when at the beginning of the motion the weight is placed near the fulcrum, from which in the motion it departs: we can use a lever of this kind, when something is proposed to be moved moderately indeed in a horizontal plane, but there is great difficulty. Likewise if a somewhat long iron plate were strongly joined at its extremities to some body, and it were necessary to bend it about the middle, so that a wedge or something similar could be inserted between the body and the plate; a hook attached to the cord, grasping the plate, would, the lever being raised or depressed, bend it a little. Let the fulcrum of the lever be at R, the cord tied at K, and let the length of the cord K S be 73; but let the part of the lever R K be 100: wherefore the distance R S of the weight S from the fulcrum R is 27. Let the lever be moved upward, and let it make the angle L R K of 15 degrees; the cord is L I part. 73, and L R part. 100; therefore from these given quantities the angle R I L is found to be 159 degrees 14′, and R L I 5 degrees 46′; and therefore the base R I is part. 28 34/100; therefore the motion from S into I is 1 34/100. But if R L be taken as the fourth part of the lever, the whole Radius is part. 400, and the arc described by the extremity of the lever, of 15 degrees, is part. 104 32/100: from which you see that the motion of the power to the motion of the weight is approximately as 78 to 1. But if the cord L I be taken longer, so that it is part. 90, and the rest remain as before, the angle R I L is found to be 163. Eec. 3

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Mechanicorum 406 m. 17, & angulus RLI gr. 1. m. 43: quare basis RI est part. 10 42/100: & quia RS ex hypothesi est solum part. 10, motus S I est 42/100 multo minor quàm cum funis brevior ponitur, ac prop- terea etiam facilior est motus, quippe qui minorem Rationem habet ad motum potentiæ. Pergendo autem in elevatione vectis adhuc per gr. 15, ita ut angulus PRO sit gr. 30, PR est 100, PO est 73: invenitur angulus ROP gr. 136. m. 46, & angulus RPO gr. 13. m. 14, atque basis RO part. 33 42/100: quare motus IO est part. 5 8/100. Et iterum elevando vectem per gr. 15, ita ut angulus QRT sit gr. 45, invenitur angulus QTR gr. 104. m. 23, & angulus RQT gr. 30. m. 37, basis autem RT part. 52 57/100: ex quo fit motum OT esse partium 19 15/100. Hinc patet æqualibus potentiæ motibus inæquales, semperque majores ponderis motus respondere, ac proinde crescere movendi difficultatem; cum enim pondus suâ gravitate insistat subjecto plano, in quo trahitur, quò magis elevatur vectis, etiam funis magis obliquus est, minusque valida fit tractio, quæ magis obliqua est. Quas hactenus recensuimus tractiones, fieri per lineam rectam vel accedendo ad punctum hypomochlij, vel ab eo recedendo, satis constat; quia, dum vectis in plano Verticali movetur, pondus non recedit ab illo eodem plano Verticali semper suâ gravitate insistens plano horizontali, atque idcirco motus illius est in communi horum planorum sectione, hoc est, in lineâ rectâ. Sin autem motus vectis fuerit in eodem plano horizontali, in quo est pondus fune trahendum, quia vectis circulariter movetur, illum sequitur pondus per lineam curvam, sed quo ad ejus fieri possit, brevissimam, ut quàm minimam patiatur violentiam. Certum quippe est oportere funem vecti congruentem citra quamlibet anguli inclinationem, esse breviorem parte illâ vectis, quæ inter hypomochlium, & locum alligati funis, intercipitur; si enim æqualis esset, circumducto vecte pondus fulcro proximum non moveretur; multo minùs, si longior esset funis. Cum itaque brevior sit, necesse est pondus quoque circumduci, sed non eâ ratione, qua moveretur, si funis eundem semper angulum cum vecte constitueret; quemadmodum contingueret, si vecti loco funis flexilis

Transcription: Translated (English)

Mechanics 406 m. 17, and angle RLI gr. 1. m. 43: therefore the basis RI is part. 10 42/100: and because RS by hypothesis is only part. 10, the motion S I is 42/100 much smaller than when the rope is placed shorter, and therefore the motion is also easier, since it has a smaller ratio to the motion of the power. But proceeding in the elevation of the lever by another gr. 15, so that angle PRO is gr. 30, PR is 100, PO is 73: there is found angle ROP gr. 136. m. 46, and angle RPO gr. 13. m. 14, and the basis RO part. 33 42/100: whence the motion IO is part. 5 8/100. And again raising the lever by gr. 15, so that angle QRT is gr. 45, there is found angle QTR gr. 104. m. 23, and angle RQT gr. 30. m. 37, but the basis RT part. 52 57/100: from which it follows that the motion OT is part 19 15/100. Hence it is clear that, for equal motions of the power, the motions of the weight are unequal, and that greater motions of the weight always correspond; and therefore the difficulty of moving increases. For since the weight, by its own gravity, presses upon the underlying plane on which it is drawn, the more the lever is raised, the more oblique also is the rope, and the less effective becomes the traction, which is more oblique. The pullings which we have so far recounted are sufficiently known to take place in a straight line, either by approaching the point of the hypomochlion or by receding from it; because, while the lever is moved in a vertical plane, the weight does not depart from that same vertical plane, always pressing by its own gravity upon the horizontal plane, and therefore its motion is in the common intersection of these planes, that is, in a straight line. But if the motion of the lever were in the same horizontal plane in which the weight to be drawn by a rope is situated, because the lever is moved circularly, the weight follows it by a curved line, but, as far as it can be, by the shortest one, so that it suffers the least violence. For it is certain that the rope corresponding to the lever, short of any inclination of angle, must be shorter than that part of the lever which lies between the hypomochlion and the place where the rope is tied; for if it were equal, then when the lever was turned about, the weight nearest the fulcrum would not move; much less if the rope were longer. Since therefore it is shorter, it is necessary that the weight also be carried around, but not in that manner in which it would move if the rope always formed the same angle with the lever; as would happen if, in place of a flexible rope to the lever

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Liber quartus. CAPUT VI. 407 flexilis rigidum brachium adjaceret, cui pondus adnecteretur. Verùm quia pondus suâ gravitate resistit, dum vectis movetur, retinetur aliquantulum funis à pondere, & angulum subinde majorem cum vecte efficit; trahitur tamen pondus, sed ita, ut violentiam subeat quàm minimam pro ratione positionis; ac propterea lineam curvam helici similem describit, quo ad fu- nis certum angulum acutum (pro ut funis, aut pars vectis lon- giores fuerint, sive breviores) cum vecte constituat; quo de- inde angulo manente pondus in gyrum ducitur per circuli am- bitum. Observabis enim positâ eadem vectis parte, quò bre- vior fuerit funis, eò majorem esse angulum illum, ad quem de- venitur, & in quo consistitur nec illum augendo, nec minuendo. Quod si in funem eâdem vectis parte longiorem ita disponas, ut non vecti congruat, sed cum illo angulum efficiat, circumducto vecte ita pondus per helicem moveri videbis, ut diminuto sub- inde angulo, demum funis vecti in eadem rectâ lineâ congruat, & ponderis ultra hypomochlium manentis tractio desinat: quia videlicet ponderis gravitas resistens licèt trahatur, retinet ta- men funem, & minuitur angulus, usque dum omnis angulus pereat. Hujusmodi motûs causam deprehendes, si attentè inspicias pondus, cum vectis in gyrum moveri incipit, ita trahi, ut etiam aliquantulum circa suum centrum gravitatis, aut circa aliud punctum (neque enim hic locus est punctum illud definiendi) volvatur; ex qua conversione fit minore motu opus esse, ut pondus consequatur trahentem: ubi verò tanta facta fuerit ponderis circa suum centrum conversio, ut si in hanc, vel in illam partem adhuc converteretur, majorem subiret in tractio- ne violentiam, hoc est, cogeretur majorem motum perficere, quàm sit motus ejusdem nullâ factâ circa suum illud centrum conversione, tunc manet angulus funis cum vecte, nec jam au- getur aut minuitur. Quia autem, cæteris paribus, quò bre- vior est funis, eò major fit ponderis circa suum centrum con- versio; propterea ad majorem angulum demum inclinatur fu- nis in vectem. Sed in hoc non est diutiùs immorandum; rarus quippe est hujusmodi tractionis usus. CAPUT

Transcription: Translated (English)

Liber quartus. CAPUT VI. 407 If the flexible part lay beside the rigid arm, to which the weight was attached. But because the weight resists by its own gravity, while the lever is being moved, the rope is held back a little by the weight, and so continually makes a greater angle with the lever; nevertheless the weight is drawn, but in such a way that it undergoes the least violence possible in proportion to its position; and therefore it describes a curved line similar to a helix, until the rope forms with the lever a certain acute angle (according as the rope, or the part of the lever, is longer or shorter); and when that angle afterwards remains fixed, the weight is carried round along the circumference of the circle. For you will observe that, with the same part of the lever positioned, the shorter the rope is, the greater is that angle at which it finally arrives and remains, without either increasing or diminishing it. But if, with the same part of the lever, you arrange a longer rope so that it does not agree with the lever, but forms an angle with it, you will see the weight, the lever being turned, moved along the helix in such a way that, as the angle is gradually lessened, at last the rope comes into the same straight line with the lever, and the pulling of the weight remaining beyond the hypomochlion ceases: because, namely, although the resisting weight is drawn, it still holds back the rope, and the angle is diminished until every angle disappears. You will detect the cause of this kind of motion if you carefully observe the weight, when the lever begins to move in a circle, as being drawn in such a way that it also turns somewhat about its own center of gravity, or about some other point (for this is not the place to define that point); from which turning it follows that less motion is needed in order for the weight to follow the one pulling. But when so great a turning of the weight about its own center has been made that, if it were still to turn in this direction or that, it would undergo greater violence in being drawn—that is, it would be compelled to perform a greater motion than would be the motion of the same weight when no turning has been made about that center—then the angle of the rope with the lever remains, and is no longer increased or diminished. And since, other things being equal, the shorter the rope is, the greater is the weight's turning about its own center; therefore the rope is at last inclined toward a greater angle in the lever. But on this we need not dwell further; for the use of this kind of traction is rare. CAPUT

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Mechanicorum CAPUT VII. Quid conferat Potentiæ moventis applicatio ad vectem. Quoniam duplex est Potentiæ genus, alia siquidem inani- ma est, cujus conatus secundùm rectam lineam in cen- trum, vel à centro, dirigitur, prout gravis est aut levis, alia est vivens, quæ pro variâ musculorum intentione, ac mem- brorum inflexione in aliam atque aliam partem dirigi potest; idcirco, cujusmodi potentiâ uti liceat, considerandum est, ut opportunum vectis genus eligatur. Nam si vectis primi gene- ris depressione attollendum sit pondus, & potentia sit vivens, ut homo, major est movendi facilitas, tum quia ipsum vectis pondus potentiam juvat suâ gravitate, tum quia in hujusmodi depressione vectis non solùm brachiorum, sed etiam quando- que totius humani corporis vecti incumbentis, vel ex vecte pendentis gravitas momentum non leve addit contentioni, qua virtus movendi impetum vecti imprimens connititur. Sin au- tem vecte secundi, aut tertij generis elevandum sit pondus idem, ipsa vectis gravitas officit, quam pariter cum ipso pon- dere attollere oportet, & majore virium contentione opus est, ut experientiâ docemur. Verùm illud, quod hîc potissimum examinandum proponi- tur, est ipsa potentiæ, quæcumque illa sit, applicatio ad vectem: neque enim satis est, si illa extremitati vectis adjun- gatur, aut certo quodam loco in vecte tertij generis collocata intelligatur; sed maximè attendendum est, secundùm quam lineam potentiæ motus dirigatur; diversa quippe sunt poten- tiæ momenta pro alia atque aliâ hujusmodi motûs directione, quatenus cum vecte comparatur. Quemadmodum enim si po- tentia vectem urgeat, aut trahat, juxta ejusdem vectis in hy- pomochlij puncto firmati longitudinem, nihil prorsùs in pon- dere efficit; ita quoquè si in vectis longitudinem obliquè inci- dat

Transcription: Translated (English)

Mechanics CHAPTER VII. What is conferred by the application of moving power to a lever. Since there are two kinds of power, one indeed is inanimate, whose effort is directed in a straight line toward the center or away from the center, according as it is heavy or light; the other is living, which, according to the varying tension of the muscles and the bending of the limbs, can be directed now in one way, now in another. Therefore, it must be considered what kind of power may be used, so that a suitable kind of lever may be chosen. For if, by lowering a first-kind lever, a weight is to be raised, and the power is living, as in a man, the ease of movement is greater, both because the weight of the lever itself helps the power by its own gravity, and because in such lowering the weight not only of the arms, but sometimes also of the whole human body leaning upon the lever, or hanging from it, adds no slight force to the effort by which the power, imparting an impulse to the lever, exerts itself. But if the same weight is to be raised by a lever of the second or third kind, the weight of the lever itself is a hindrance, since it must be raised together with the weight itself, and a greater exertion of strength is required, as experience teaches us. But what is here chiefly proposed for examination is the application of the power itself, whatever it may be, to the lever: for it is not enough if it is attached to the end of the lever, or understood to be placed at a certain point on a lever of the third kind; rather, it must especially be observed along what line the motion of the power is directed, for the moments of the power are different according to this or that direction of motion, insofar as it is compared with the lever. For just as, if the power press upon or draw the lever along the length of that same lever fixed at the point of support, it produces absolutely nothing in the weight; so also if it falls obliquely along the length of the lever

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Liber quartus. CAPUT VII. 409 dat impetûs à potentiâ concepti directio, pro ratione obliqui- tatis minuitur potentiæ momentum; quod integrum manet, si ad angulos rectos vecti occurrat linea motus, quam init po- tentia. Sit vectis primi generis A B habens hypomochlium in C, si- ve secundi aut tertij generis D E habens hypomochlium in D. Si potentia constituta in B, aut E aut F, motum suum dirigat secundùm eandem rectam li- neam B A aut E D, sive urgen- do vectem versus C aut D, sive illum inde retrahendo, mani- festum est, puncto hypomo- chlij C aut D manente, pondus in A, aut in F, aut in E consti- tutum nihil prorsus moveri, nam totus potentiæ conatus irri- tus est, nec vectem movet. Oportet igitur lineam, secundùm quam dirigitur motus potentiæ, constituere cum vectis longi- tudine angulum aut rectum, aut recto minorem, aut majorem. Si angulum acutum H B A efficiat, movetur quidem potentia & vectis, sed cùm urgeatur vectis versus hypomochlium C, impeditur potentia, nec movet vectem pro ratione impetûs, quem illa concipit. Similiter si directio motûs potentiæ sit se- cundùm lineam B G, & fiat angulus obtusus G B A, quamvis vectem moveat, minùs tamen illum flectit aut deprimit, quàm requirat impetûs concepti intensio, quia conatur vectem re- trahere ab hypomochlio C, & ab illo retinetur. Cùm autem, quò acutior aut obtusior est angulus, eò etiam majus sit impe- dimentum, hinc est pariter plus laboris à potentiâ impendi. Quare, cùm nullum sit hujusmodi impedimentum, quando ad rectos cum vecte angulos potentiæ motus dirigitur, ut I B A, propterea tunc solùm potentia obtinet omnia momenta, quæ concepto impetui respondent: nihil enim impetûs deteritur ab impedimento, quod vectis inferat, quippe qui nec versus hy- pomochlium urgetur, nec ab illo retrahitur. Porrò observa longè aliam esse lineam motûs potentiæ, à li- neâ secundùm quam ejusdem potentiæ motus dirigitur; nam potentia in B applicata movetur describendo arcum circa C F f

Transcription: Translated (English)

Book fourth. CHAPTER VII. 409 the direction of the impulse conceived from power is diminished, according to the ratio of the obliquity, the moment of the power; which remains entire, if the line of motion, which the power enters upon, meets the beam at right angles. Let there be a lever of the first kind A B having its fulcrum in C, or of the second or third kind D E having its fulcrum in D. If the power placed in B, or E or F, direct its motion along the same straight line B A or E D, whether by urging the lever toward C or D, or by drawing it back from there, it is evident that, the point of the fulcrum C or D remaining fixed, the weight placed in A, or in F, or in E is moved not at all; for the whole effort of the power is vain, nor does it move the lever. Therefore the line along which the motion of the power is directed must make with the length of the lever either a right angle, or one less than a right angle, or greater than a right angle. If it make the acute angle H B A, the power and the lever are indeed moved, but when the lever is urged toward the fulcrum C, the power is hindered, and does not move the lever according to the proportion of the impulse which it conceives. Likewise if the direction of the motion of the power be according to the line B G, and the obtuse angle G B A be formed, although it moves the lever, it bends or depresses it less than the intensity of the conceived impulse requires, because it strives to draw the lever back from the fulcrum C, and is held back by it. But since, the sharper or more obtuse the angle is, the greater also is the impediment, hence likewise more labor is expended by the power. Wherefore, since there is no such impediment when the motion of the power is directed at right angles with the lever, as I B A, for that reason then alone does the power obtain all the moments that correspond to the conceived impulse: for none of the impulse is wasted by any impediment that the lever may introduce, since it is neither urged toward the fulcrum nor drawn back from it. Moreover, observe that the line of the motion of the power is very different from the line according to which the motion of the same power is directed; for the power applied at B is moved by describing an arc around C F f

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Mechanicorum 410 punctum hypomochlij, sed pro variâ directione modò majorem, modò minorem arcum describit eodem tempore ex vi ejusdem impetûs concepti. Hinc est, nisi potentia suum motum in gyrum dirigat, fieri non posse, ut in motu eadem servet virium momenta: Nam licèt eandem directionem servaret, quatenus horizontem respicit, aut certum aliquod punctum, non esset tamen eadem directio comparata cum vecte; alium quippe atque alium cum vecte angulum constitueret illa eadem directionis linea: id quod manifestò constat, cùm vectis à potentiâ gravi deprimitur; linea enim directionis in centrum gravium directa semper obliquior incidit in vectem, qui deprimitur. Voco autem Directionem motûs lineam illam, quam potentia ex vi concepti impetûs sponte percurreret, nisi ab illâ deflectere cogeretur, quia cum vecte connectitur. Sic potentia B lineam BH ex. gr. percurreret, nisi vectis in C firmati soliditas obstaret, cogerétque arcum BR describere: idem de cæteris lineis dicendum. Hinc si longitudo BH concipiatur spatium, quod à potentiâ liberâ vi sui impetûs certo tempore perficere- tur, illa utique non recederet à lineâ AB nisi pro ratione Sinûs Recti angulo HBA convenientis posito Radio BH, scilicet per HO. Similiter si directio motûs sit BG angulum obtusum GBA constituens, potentia non recederet ab eâdem lineâ AB nisi pro ratione GK sinus Recti ejusdem anguli GBA obtusi posito Radio BG, qui ex hypothesi æqualis est Radio BH, ponitur enim utrobiqve æqualis impetus potentiæ. Quare cùm idem sit Sinus Rectus anguli acuti, atque obtusi, quorum summa æquatur duobus rectis, eadem pariter momenta virium exercet potentia, sive ad acutum, sive ad obtusum cum vecte angulum dirigatur. Hoc tamen intercedit discrimen, quando potentia eandem servat ad horizontem directionem, quod acutus angulus procedente motu fit major accedens ad Rectum, augeturquo ejus Sinus; obtusus verò angulus fit obtusior magis recedens à Recto, minuiturque ejus Sinus; ac proinde ibi augetur, hîc minuitur movendi facilitas. Potentia itaque motum suum dirigens ad acutum angulum per lineam BH, vi sui impetûs describit circa centrum C arcum BR; ad angulum rectum per lineam BI describit arcum BN;

Transcription: Translated (English)

Mechanics 410 the point of the fulcrum; but according to the different direction it describes now a larger, now a smaller arc in the same time, by the force of the same impressed impulse. Hence it is that, unless the power direct its motion in a circle, it cannot happen that in motion it preserves the same moments of force: for although it preserved the same direction, as regards the horizon, or some fixed point, it would not nevertheless be the same direction as compared with the lever; for that same line of direction would form one angle with the lever and another; which is manifestly evident when the lever is depressed by a heavy power: for the line of direction, being directed to the center of gravity, always falls more obliquely upon the lever, which is being depressed. Now I call the direction of motion that line which the power, by virtue of the impressed impulse, would of itself traverse, unless it were compelled to deflect from it because it is connected with the lever. Thus power B would traverse line BH, for example, unless the solidity of the lever fixed at C should oppose it, and compel it to describe arc BR: the same must be said of the other lines. Hence, if the length BH be conceived as the space which a free power would accomplish by the force of its impulse in a certain time, it would certainly not depart from line AB except in proportion to the sine of the right angle corresponding to angle HBA, the radius BH being posited, namely by HO. Similarly, if the direction of motion be BG, constituting the obtuse angle GBA, the power would not depart from the same line AB except in proportion to GK, the sine of the right angle of the same obtuse angle GBA, the radius BG being posited, which by hypothesis is equal to radius BH, for the impulse of the power is posited as equal in both cases. Therefore, since the sine of the acute angle is the same as that of the obtuse, the sum of which is equal to two right angles, the power likewise exerts the same moments of force, whether it be directed to an acute or to an obtuse angle with the lever. Yet there is this difference: when the power preserves the same direction relative to the horizon, the acute angle, as the motion proceeds, becomes larger, approaching the right angle, and its sine increases; but the obtuse angle becomes more obtuse, receding further from the right angle, and its sine is diminished; and therefore in the former case the ease of moving is increased, in the latter diminished. Accordingly, when the power directs its motion toward an acute angle by line BH, by the force of its impulse it describes about center C arc BR; toward a right angle by line BI it describes arc BN;

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Liber quartus. CAPUT VII. 411 BN; ad angulum demum obtusum per lineam BG describit arcum BL, qui est æqualis arcui BR, si angulus obtusus GB A sit supplementum ad duos rectos anguli acuti HBA, major autem, aut minor eodem arcu BR, si angulus obtusus sit mi- nor aut major eodem Supplemento ad duos rectos. Hæc tamen ita dicta intelligas velim, ut hujusmodi arcus toti atque integri non motum ipsum exprimant, qui revera fiat, sed virium Ra- tionem pro diversa potentiæ applicatione in minimâ arcûs des- cripti particulâ; neque enim singulis temporis momentis æqua- lis pars arcûs eidem impetui respondet, singulis nimirum mo- mentis mutatur vectis inclinatio, & manet eadem motûs di- rectio, atque varia est potentiæ ad vectem applicatio, nisi il- la impetum concipiat, quo sua sponte in gyrum ageretur, etiamsi ad motum circularem non determinaretur à vecte. Sed quoniam arcus eodem Radio CB à potentiâ descriptus est similis arcui eodem Radio CA descripto à pondere, proinde non mutatur Ratio motuum, sive potentia descri- bat eodem impetu minorem, sive majorem arcum ejusdem circuli: propterea non mutatur quidem momentum poten- tiæ cum pondere absolutè comparatæ, mutatur tamen subinde momentum potentiæ, quatenus secum ipsa comparatur, faci- liusque movere pondus tunc dicitur, quando eodem conatu majorem motum ponderi æquali tempore conciliat; id quod fit, cùm ad angulum rectum vecti applicatur. Hæc eadem, quæ in vecte primi generis explicata sunt, in reliquis pariter duobus generibus locum habent, nec opus est illa iterum inculcare. Unum in his observandum vide- tur, quando potentia movens est à vecte sejuncta, illum- que trahendo movet certo in loco firmiter constituta, vectem in motu propiùs accedere ad potentiam trahentem, ac proin- de diligenter attendendam esse ipsius potentiæ positionem, ut innotescat, utrum angulus, quem subinde cum vecte funicu- lus efficit, accedat magis ad rectum, an verò recedat à recto, quia in motu acutior aut obtusior evadat. Id quod satis fuerit subindicâsse; præstat siquidem laborem in motu minui, quàm augeri. Sic dato vecte CB secundi generis habente hypomo- chlium in C, statue quantum moveri debeat, ex. gr. per F f 2

Transcription: Translated (English)

Liber quartus. CHAPTER VII. 411 BN; at last toward the obtuse angle by the line BG it describes the arc BL, which is equal to the arc BR, if the obtuse angle GBA is the supplement to two right angles of the acute angle HBA, but greater or less than the same arc BR, if the obtuse angle be less or greater than that same supplement to two right angles. Yet I wish this to be understood as thus stated: that such arcs, whole and complete as they are, do not express the motion itself which in fact takes place, but the ratio of forces according to the different application of the power in the smallest particle of the described arc; for not in each instant of time does an equal part of the arc answer to the same impetus; namely, in each moment the inclination of the lever is changed, and the direction of the motion remains the same, while the application of the power to the lever is varied, unless it should conceive that impetus by which it would of its own accord be carried in a circle, even though it were not determined to circular motion by the lever. But since the arc described by the power with the same radius CB is similar to the arc described with the same radius CA by the weight, therefore the ratio of the motions is not changed, whether the power describe with the same impetus a smaller or a larger arc of the same circle: for that reason indeed the momentum of the power, absolutely compared with the weight, is not changed; yet the momentum of the power is nevertheless continually changed, insofar as it is compared with itself, and the power is said to move the weight more easily then, when with the same effort it imparts a greater motion to an equal weight in equal time; and this happens when it is applied at right angles to the lever. These same things, which have been explained in the lever of the first kind, have a place likewise in the other two kinds, and there is no need to repeat them again. One thing seems to be observed in these cases, namely, when the moving power is separate from the lever, and moves it by pulling, firmly fixed in a certain place, the lever in motion comes nearer to the pulling power, and therefore the position of the power itself must be carefully considered, so that it may become known whether the angle which the cord then makes with the lever comes nearer to the right angle, or rather departs from the right angle, because in motion it becomes more acute or more obtuse. This has been sufficiently indicated; for it is better that the labor in motion be lessened than increased. Thus, given a lever CB of the second kind having the fulcrum in C, determine how much it ought to move, for example, by F f 2

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Mechanicorum arcum BD. Erit igitur vectis positio CD. Excitetur ex D perpendicularis DE, & in alio quo rectæ lineæ DE puncto, puta in E, statuatur potentia, quæ funiculo EB trahens vectem ita vecti intelligatur applicata, ut majorem subinde angulum efficiat, donec ad rectum CDE deveniat. Sic facilior erit motus, & labor minuetur. Quod si potentia movendo pergeret adhuc trahens vectem, jam augeretur labor, quia applicaretur ad angulum obtusum. Porrò in recta DE eligendum esse punctum, quoad fieri poterit, proximum puncto D, ut funiculus EB minùs acutum angulum cum vecte CB constituat, apertius est, quàm ut oporteat id pluribus hîc ostendere. Eximendus tamen est omnis scrupulus, ostendendo angulum semper majorem fieri, quando distantia potentiæ E ab hypomochlio C major est longitudine dati vectis CB. Intelligatur descriptus integer circulus BGIL à vecte CB circumducto, & producatur BC in I, atque funiculus EB secet peripheriam in G. Tum vectis positio fiat CO, & funiculus EO secet peripheriam in H. Manifestum ex 33. lib. 6. angulum COE majorem esse angulo CBEnam, productâ OC in L, angulus CBE insistit arcui GI, angulus autem COE insistit arcui HL, qui major est arcu GI: omnes autem acuti sunt, quia insistunt periphe- riæ minori, quàm sit semicirculus (sunt enim, ex 20. lib. 3, subduplici suorum angulorum ad centrum iisdem peripheriis insistentium) donec funiculus ED sit Tangens circuli, & ex 18. lib. 3. angulum rectum constituat in D. Quòd si vectis adhuc trahatur à potentia E, & veniat in H & in G, constat ex 21. lib. 1. angulum

Transcription: Translated (English)

Mechanics arc BD. Therefore let the position of the lever be CD. Let the perpendicular DE be raised from D, and at another point of the straight line DE, namely at E, let the power be placed, which, pulling the lever by the cord EB, is to be understood as applied to the lever in such a way that it makes a greater and greater angle, until it reaches the right angle CDE. Thus the motion will be easier, and the labor diminished. But if the power should continue moving, still pulling the lever, the labor would now increase, because it would be applied at an obtuse angle. Moreover, it is clear enough, without our needing to show it here at length, that in the straight line DE the point chosen should be as near as possible to the point D, so that the cord EB may form with the lever CB a less acute angle. Yet every doubt must be removed by showing that the angle is always made greater when the distance of the power E from the fulcrum C is greater than the length of the given lever CB. Let the whole circle BGIL be described by the lever CB being turned about, and let BC be produced to I, and let the cord EB cut the circumference at G. Then let the position of the lever be CO, and let the cord EO cut the circumference at H. It is manifest from Book 6, proposition 33, that the angle COE is greater than the angle CBE; for, if OC be produced to L, the angle CBE stands on the arc GI, while the angle COE stands on the arc HL, which is greater than arc GI. Moreover, they are all acute, because they stand on a circumference smaller than a semicircle (for, from Book 3, proposition 20, they are the submultiple of their angles at the center standing on the same circumferences), until the cord ED is a tangent to the circle and makes a right angle at D, from Book 3, proposition 18. But if the lever is still pulled by the power E, and comes to H and to G, it is evident from Book 1, proposition 21, that the angle

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Liber quartus. CAPUT VII. 413 lum CDE minorem esse angulo CHE, hunc verò minorem angulo CGE, atque ita deinceps. Idem contingit, si distantia potentiæ R ab hypomochlio C omnino æquali sit longitudini vectis CB; nimirum trahendo vectem ex B in S, angulus RSC major est angulo RBC, & sic deinceps trahendo ex S versus R: quamvis enim semper sit an- gulus acutus, major tamen subinde fit & major; quia manente eâdem distantiâ RC æquali longitudini vectis, tam triangulum CBR quàm CSR, & reliqua omnia sunt Isoscelia; quò ergo minor sit angulus ad C, eò major sit angulus ad basin in R, cui, per 5. lib. 1. æqualis est reliquus angulus ad eandem ba- sim. At si distantia potentiæ ab hypomochlio minor fuerit lon- gitudine vectis, utique locus potentiæ est intra circulum à vecte circumducto descriptum. Consideranda est igitur varia funiculi ad vectem inclinatio: pro qua explicanda hæc præ- mitto lemmata. LEMMA I. Si intra circulum assumptum fuerit punctum E, in quo duæ rectæ lineæ BF & GH æqualiter à cen- tro distantes, ideóque ex 14. lib. 3. æquales, se invicem se- cent; & à centro C ducantur Radij CB & CG; anguli CBF & CGH sunt æquales. Producatur BC in M, & GC in N, ducanturque rectæ FM & HN. Quia MB & NG sunt diametri, anguli in semicir- culo BFM & GHN, ex 31. lib. 3. sunt recti: igitur quadra- ta BF & FM simul sumpta sunt æqualia quadratis GH & HN simul sumptis, cum, ex 47. lib. 1. æqualia sint qua- drato diametri. Est autem qua- dratum BF æquale quadrato GH, nam rectæ BF & GH ex hypothesi sunt æquales; igitur quadrata FM & HN sunt æqualia, ideoque rectæ Fff 3

Transcription: Translated (English)

Liber quartus. CHAPTER VII. 413 the angle CDE is less than the angle CHE, this again less than the angle CGE, and so on. The same happens if the distance of the power R from the fulcrum C is exactly equal to the length of the lever CB; namely, by drawing the lever from B to S, the angle RSC is greater than the angle RBC, and so on by drawing from S toward R: for although it is always an acute angle, yet it becomes greater and greater by degrees; because, the same distance RC remaining equal to the length of the lever, both triangle CBR and CSR, and all the rest, are isosceles; therefore, the smaller the angle at C may be, the greater is the angle at the base in R, to which, by Book 1, Prop. 5, the remaining angle at the same base is equal. But if the distance of the power from the fulcrum is less than the length of the lever, then plainly the place of the power is within the circle described by the turning of the lever. It is therefore necessary to consider the varying inclination of the cord to the lever: to explain this I set down the following lemmas. LEMMA I. If within the assumed circle there be a point E, in which two straight lines BF and GH, equally distant from the center, and therefore equal, by Book 3, Prop. 14, intersect one another; and from center C the radii CB and CG be drawn; the angles CBF and CGH are equal. Let BC be produced to M, and GC to N, and let the straight lines FM and HN be drawn. Because MB and NG are diameters, the angles in the semicircles BFM and GHN, by Book 3, Prop. 31, are right angles: therefore the squares BF and FM taken together are equal to the squares GH and HN taken together, since, by Book 1, Prop. 47, they are equal to the square of the diameter. But the square BF is equal to the square GH, for the straight lines BF and GH are equal by hypothesis; therefore the squares FM and HN are equal, and so the straight lines Fff 3

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Mechanicorum 414 FM & HN sunt æquales; ergo ex 28. lib. 3. subtendunt æquales peripherias, FHM & HMN; ergo ex 27. lib. 3. anguli FBM & HGN æqualibus peripheriis insistentes æquales sunt. Invenitur autem recta linea transiens per E, quæ æqualis sit rectæ BF, si facto centro E, intervallo EF, describatur circulus FRG secans datum circulum in G; nam ex G per E ducitur recta GH quæsita: est enim, per 35. lib. 3, rectangulum GEH æquale rectangulo FE B; sunt autem GE & FE æquales Radij ejusdem circuli ex constructione; igitur per 1. lib. 6, etiam EH & EB sunt æquales; ergo tota GH toti FB est æqualis. LEMMA II. Si in puncto E intra circulum assumpto secent se invicem duæ rectæ BF & IO inæquales, ac proinde ut colligitur ex 15. lib. 3. inæqualiter à circuli centro distantes, ducanturque ex centro Radij CB, & CI, angulus factus à Radio cum lineâ remotiore major est angulo facto à Radio cum lineâ propinquiore. Persiciantur triangula BFM & 'IOS rectangula ad F & O ex 31. lib. 3, quia MB & SI sunt diametri. Quadrata BF & FM simul sumpta, ex 47. lib. 1, sunt æqualia quadratis IO & OS simul sumptis: Quia autem ex hypothesi recta IO remotior est à centro quàm BF, est etiam minor, ut constat ex 15. lib. 3: igitur quadratum IO minus est quadrato BF, adeóque quadratum reliquum OS majus est reliquo quadrato FM, & linea OS major est linea FM. Quapropter etiam OS subtendit majorem arcum OMS, & FM subtendit minorem arcum FOM, & angulus SIO factus à Radio cum lineâ remotiore major est angulo MBF facto à Radio cum lineâ propinquiore. LEMMA III. Si in circulo ab extremitate diametri B exeat recta linea BC circulum secans, in qua assumatur punctum D eam bifariam æqualiter dividens, & per punctum

Transcription: Translated (English)

Mechanics 414 FM and HN are equal; therefore, by 28, book 3, they subtend equal circumferences, FHM and HMN; therefore, by 27, book 3, the angles FBM and HGN, standing on equal circumferences, are equal. But a straight line passing through E is found, which is equal to the straight line BF, if, having taken E as center and EF as radius, the circle FRG be described, cutting the given circle at G; for through G the straight line GH sought is drawn through E: for, by 35, book 3, the rectangle GEH is equal to the rectangle F E B; but GE and FE are equal radii of the same circle by construction; therefore, by 1, book 6, EH and EB are also equal; therefore the whole GH is equal to the whole FB. LEMMA II. If, a point E being taken inside the circle, two unequal straight lines BF and IO intersect one another, and therefore, as is gathered from 15, book 3, are unequally distant from the center of the circle, and from the center the radii CB and CI are drawn, the angle formed by the radius with the more distant line is greater than the angle formed by the radius with the nearer line. Let the right triangles BFM and IOS be completed at F and O, by 31, book 3, because MB and SI are diameters. The squares BF and FM taken together, by 47, book 1, are equal to the squares IO and OS taken together: but because, by the hypothesis, the straight line IO is farther from the center than BF, it is also smaller, as is clear from 15, book 3: therefore the square IO is less than the square BF, and so the remaining square OS is greater than the remaining square FM, and the line OS is greater than the line FM. Wherefore OS also subtends the greater arc OMS, and FM subtends the lesser arc FOM, and the angle SIO, formed by the radius with the more distant line, is greater than the angle MBF, formed by the radius with the nearer line. LEMMA III. If in a circle from the extremity of the diameter B a straight line BC should go out, cutting the circle, in which let a point D be taken dividing it equally in two, and through the point

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Liber quartus. CAPUT VII. 415 punctum D alia recta circulum secans ducatur, hæc vicinior est centro, & major. Ducatur ex centro S recta S D, quæ per 3.lib.3. facit angulum S D C rectum: Tum per D alia quædam linea E F transeat, quæ utique cum recta S D facit angulum S D F minorem recto, & S D E majorem recto: nam si angulos faceret rectos, esset S D utrique lineæ B C, & E F perpendicularis, secaret E F bifariam in D per 3.lib.3. adeóque duæ rectæ B C & E F se mutuo bifariam secarent, contra 4.lib.3. Igitur in rectam E F perpendicularis ducta ex centro S erit S G cadens ad partes anguli acuti. Quapropter in triangulo S G D rectangulo ad G major est hypothenusa S D, quàm perpendicularum S G. Magis ergo distat linea B C quàm linea E F à centro, ac proinde per 15.lib.3. illa est minor, hæc major. LEMMA IV. Si in eâdem rectâ B C assumatur punctum I inter extremitatem B & punctum medium D, atque, ex centro directâ rectâ S I V, inter V & B alia quæpiam per I transeat recta H L circulum secans, quæ & secet perpendicularem S D, ex. gr. in puncto K; hæc pariter H L centro propinquior est quàm B C, ac proinde major. Angulus K D I est rectus, angulus D K I, & qui est illi ad verticem, S K L est acutus; igitur perpendicularis ex centro S in rectam H L ducta cadit inter K & L, puta in M. In triangulo igitur rectangulo S M K major est S K quàm S M, ac propterea ex 15.lib.3. H L vicinior est centro, & major quàm B C. LEMMA

Transcription: Translated (English)

Book Four. Chapter VII. 415 let a straight line be drawn through point D, cutting the circle; this is nearer the center, and greater. Let the straight line S D be drawn from the center S, which, by 3. lib. 3., makes the right angle S D C: Then through D let another line E F pass, which certainly with the straight line S D makes the angle S D F less than a right angle, and S D E greater than a right angle: for if it made right angles, S D would be perpendicular to both lines B C and E F, and would bisect E F at D by 3. lib. 3., and so the two straight lines B C and E F would mutually bisect one another, contrary to 4. lib. 3. Therefore the perpendicular drawn to the straight line E F from the center S will be S G, falling toward the side of the acute angle. Wherefore in the right triangle S G D, the hypotenuse S D is greater than the perpendicular S G. Therefore the line B C is farther from the center than the line E F, and consequently, by 15. lib. 3., that is the lesser, this the greater. LEMMA IV. If on the same straight line B C there be taken a point I between the end B and the midpoint D, and, from the center, by the straight line S I V, between V and B let some other straight line H L pass through I, cutting the circle, which also cuts the perpendicular S D, for example at the point K; this likewise H L is nearer the center than B C, and therefore greater. The angle K D I is right, the angle D K I, and its vertical angle, S K L, is acute; therefore the perpendicular drawn from the center S to the straight line H L falls between K and L, namely at M. In the right triangle S M K, therefore, S K is greater than S M, and for that reason by 15. lib. 3., H L is nearer the center, and greater than B C. LEMMA

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Mechanicorum LEMMA V. Si in rectâ BC ductâ ab extremitate diametri assumatur punctum N ultra punctum medium D, atque ex S centro ductâ per N lineâ rectâ SO, productâque perpendiculari SD in P, transeat per N alia quæpiam recta QR inter P & B circulum secans in Q; hæc pariter secat in T perpendicularem productam, & est à centro S remotior quàm recta BC atque proinde minor. Quia in Triangulo NDT rectangulo ad D, angulus DTN est acutus, utique in lineâ TS assumpto puncto S, ex hoc cadet in lineam QR perpendicularis inter puncta T, & R; quam dico majorem esse perpendiculari SD. Nam si ipsa recta SN perpendicularis fuerit ad RQ, est triangulum SDN rectangulum, adeóque hypothenusa SN major est quàm latus SD: Sin autem perpendicularis ad RQ cadat in V secans rectam BC in Z, utique S Z subtendens angulum rectum SDZ major est quàm SD; est autem S V major quàm S Z, ergo & multo major, quàm SD: ergo linea QR remotior est quàm BC, & minor. Deinde si linea per N transiens, & circulum secans, extremitatem alteram habeat non inter punctum P terminum perpendicularis SD productæ, atque B terminum rectæ BC; Vel dividitur in N bifariam, & linea SN per 3.lib.3. est perpendicularis ad illam, quæ major est quàm SD, ut pote opposita angulo recto SDN: Vel dividitur inæqualiter. Si segmentum majus sit in parte superiori, hoc inter N & arcum OP, utique perpendicularis ex S centro ducta in illam lineam cadens seabit lineam BC inter puncta N & D, ac propterea ostendetur major quàm SD, ut supra ostensum est de linea RQ. At si in parte superiori, hoc est inter N & arcum OP sit segmentum minus, perpendicularis ex S in lineam ducta cadet infra punctum

Transcription: Translated (English)

Mechanics LEMMA V. If, in the straight line BC drawn from the extremity of the diameter, a point N be taken beyond the midpoint D, and from S as center, through N, the straight line SO be drawn, and the perpendicular SD be produced to P, and through N some other straight line QR, cutting the circle between P and B, pass in Q; this likewise cuts in T the produced perpendicular, and is farther from the center S than the straight line BC, and therefore less. Because in the right triangle NDT at D, the angle DTN is acute, therefore, a point S being taken on the line TS, from this there will fall on the line QR a perpendicular between the points T and R; which I say to be greater than the perpendicular SD. For if the straight line SN itself be perpendicular to RQ, then the triangle SDN is right-angled, and therefore the hypotenuse SN is greater than the side SD: But if the perpendicular to RQ falls at V, cutting the straight line BC at Z, then certainly SZ, subtending the right angle SDZ, is greater than SD; but SV is greater than SZ, therefore also much greater than SD: therefore the line QR is farther than BC, and less. Then if the line passing through N, and cutting the circle, have its other extremity not between the point P, the end of the produced perpendicular SD, and B, the end of the straight line BC; either it is bisected at N, and the line SN by Book 3, Proposition 3 is perpendicular to it, which is greater than SD, as being opposite to the right angle SDN: or it is divided unequally. If the greater segment be in the upper part, that is, between N and the arc OP, then certainly the perpendicular drawn from the center S to that line, falling upon it, will cut the line BC between the points N and D, and consequently will be shown to be greater than SD, as above was shown of the line RQ. But if in the upper part, that is, between N and the arc OP, the smaller segment be situated, the perpendicular drawn from S to the line will fall below the point

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Liber quartus. CAPUT VII. 417 punctum N, & à segmento majore abseindet particulam in- ter N & punctum perpendiculi interceptam. Hæc particula si fuerit æqualis particulæ ND, linea BC & linea ducta sunt æqualiter à centro remotæ; sin illa particula minor fuerit quàm ND, linea ducta remotior erit quàm BC; si demùm major fue- rit quàm ND, linea ducta propinquior centro erit quàm BC. Finge scilicet ductam esse rectam PNX, & segmentum majus esse NX; utique perpendicularis ex S bifariam secans totam PX cadit inter N & X, puta in Y. Est igitur SYN triangu- lum rectangulum in Y, & per 47. lib.1. quadratum SN æqua- le est quadratis NY & YS; atqui etiam triangulum SDN est rectangulum ex hypothesi, eandemque habet hypothenusam SN; igitur quadrata ND & DS æqualia sunt quadratis NY & YS. Quare si particulæ NY & ND æquales sunt, æqualia sunt & earum quadrata, ac idcircò etiam æqualia sunt quadra- ta YS & DS, atque eorum latera æqualia sunt, & lineæ BC atque PX sunt æqualiter remotæ. Quod si particula NY mi- nor est quàm ND, etiam illius quadratum minus est quadrato hujus; ergo reliquum quadratum YS majus est reliquo qua- drato DS, atque adeò linea SY major est quàm linea SD, & linea ducta PX remotior est atque minor quàm BC. Si demum NY major est quàm ND, etiam illius quadratum majus est hu- jus quadrato, & reliquum quadratum YS minus est reliquo quadrato DS: igitur linea YS minor est quàm linea DS, ac propterea linea ducta PX propinquior est centro, & major quàm BC. His præmissis facilis est solutio propositæ difficultatis, ut in- notescat, utrum in tractione minuatur labor, an augeatur, quando potentiæ trahentis distantia ab hypomochlio est minor longitudine vectis. Dato si quidem loco potentiæ datur ejus- dem distantia tùm ab hypomochlio, tum ab extremitate vectis, cum qua funiculus connectitur; sed & datur ipsius vectis longi- tudo: quare per Trigonometriam innotescit quantitas anguli, cui opponitur vectis. Nam si ille rectus est, ut SDB, per lem- ma 3. in tractione funiculus sit pars lineæ centro propinquo- ris, quàm primò assumpta DB: igitur in tractione anguli fu- niculi cum vecte fit sensim acutior ex lem. 2. augeturque difficultas trahendi. Si angulus vecti oppositus sit obtusus, ut Ggg

Transcription: Translated (English)

Book Four. Chapter VII. 417 the point N, and cuts off from the larger segment the little part intercepted between N and the perpendicular point. If this little part is equal to the little part ND, the line BC and the drawn line are equally distant from the center; but if that little part be smaller than ND, the drawn line will be farther away than BC; if, finally, it be greater than ND, the drawn line will be nearer the center than BC. Suppose, then, that the straight line PNX has been drawn, and that the larger segment is NX; certainly the perpendicular from S, bisecting the whole PX, falls between N and X, namely at Y. Therefore SYN is a right triangle at Y, and by Book 1, Proposition 47, the square SN is equal to the squares NY and YS; but the triangle SDN is also right-angled by hypothesis, and has the same hypotenuse SN; therefore the squares ND and DS are equal to the squares NY and YS. Hence if the little parts NY and ND are equal, their squares are equal too, and therefore the squares YS and DS are also equal, and their sides are equal, and the lines BC and PX are equally distant. But if the little part NY is less than ND, its square is also less than the square of the latter; therefore the remaining square YS is greater than the remaining square DS, and thus the line SY is greater than the line SD, and the drawn line PX is farther away and smaller than BC. If, finally, NY is greater than ND, its square is also greater than the square of the latter, and the remaining square YS is less than the remaining square DS: therefore the line YS is less than the line DS, and consequently the drawn line PX is nearer the center, and greater than BC. These things being premised, the solution of the proposed difficulty is easy, so that it may become known whether, in traction, the effort is diminished or increased when the distance of the tractive power from the hypomochlion is less than the length of the lever. For given the position of the power, its distance is given both from the hypomochlion and from the end of the lever to which the cord is attached; but the length of the lever itself is also given: wherefore, by trigonometry, the size of the angle opposite the lever is made known. For if that angle is right, as SDB, then, by lemma 3, in traction the cord is a part of the line closer to the center than the line DB first assumed; therefore, in traction, the angle of the cord with the lever becomes gradually sharper, by lemma 2, and the difficulty of pulling is increased. If the angle opposite the lever is obtuse, as

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Mechanicorum 418 SIB, in tractione funiculus evadit pars lineæ propinquioris centro, quàm prima IB ex lemm. 4. & similiter ex lemm. 2. sit angulus magis acutus, atque trahentis labor augetur. Si de- mum angulus vecti oppositus sit acutus, ut SNB, ex lemm. 5. minuitur labor trahentis usque ad certum terminum, quandiu scilicet vectis non secat perpendiculariter primam funiculi po- sitionem NB, hoc est vectis circumductus nondum est SP; tandiu enim funiculus est pars lineæ à centro remotioris, & fa- cit per lemm. 2. cum vecte angulum majorem. Ubi autem vectis fuerit SP, tunc observandum est, utrum angulus SNP rectus sit, an obtusus, an acutus; & eâdem methodo proce- dendum est, quasi prima funiculi positio esset NP, ut innotes- tescat, utrum funiculus in ulteriori tractione fiat pars lineæ re- motioris, an verò propinquioris, ac proinde fiat angulus sub- inde major, an verò minor. Quæ de Vecte in alterâ extremitate hypomochlium, in alte- râ potentiam habente hactenus exempli gratia explicata sunt, facilè referuntur ad vectem, quando hypomochlium, aut po- tentia inter extremitates collocantur; semper enim attendenda est hypomochlij distantia à potentia trahente, ut potentiæ obli- què trahentis momenta innotescant; angulus scilicet funiculi cum vecte pendet ab hypomochlij puncto, circa quod sit vectis conversio. Quoniam autem hujus capitis initio momentorum Rationem juxta diversam potentiæ applicationem ex arcubus vi ejusdem impetus descriptis æstimandam esse dictum est, & quis fortasse suspicetur arduum esse hujusmodi arcus inter se comparare; animadvertat ex Tabulis Trigonometricis ejusdem arcûs Si- num & Tangentem iisdem planè numeris definiri, quando ar- cus valde exiguus est. Quapropter cum quilibet arcus minor sit suâ Tangente, & major Sinu, arcuum minorum Rationem citra ullum erroris periculum explicare possumus per eorum Sinus. Cum verò hîc, ubi de potentiæ ad vectem secundùm diversos angulos applicatæ momentis sermo est, non nisi mini- mi arcus assumendi sint, eorum Ratio eadem assumitur, quæ Sinuum. Quare si vectis sit AB, hypomochlium C, vis potentiæ & directio motûs potentiæ BH: loco arcûs BK, qui in motu vi talis

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Mechanics 418 If, in SIB, in the pulling of the cord it becomes the part of the line nearer the center than the former IB, from lemma 4, and similarly from lemma 2, the angle becomes more acute, and the labor of the puller increases. But if at last the opposite angle of the lever be acute, as SNB, from lemma 5 the labor of the puller decreases up to a certain limit, namely so long as the lever does not cut perpendicularly the first position NB of the cord, that is, the lever as turned about has not yet reached SP; for as long as this happens, the cord is the part of the line farther from the center, and by lemma 2 it makes a greater angle with the lever. But when the lever has become SP, then it must be observed whether the angle SNP is right, or obtuse, or acute; and the same method must be followed, as if the first position of the cord were NP, so that it may be made known whether in the further pulling the cord becomes the part of the farther line, or indeed the nearer one, and consequently whether the angle becomes then greater, or indeed smaller. What has hitherto been explained by way of example concerning the lever when the hypomochlion is at one extremity and the power at the other, is easily transferred to the lever when either the hypomochlion or the power is placed between the extremities; for the distance of the hypomochlion from the pulling power must always be considered, so that the moments of the obliquely pulling power may be known; namely, the angle of the cord with the lever depends on the point of the hypomochlion, around which the lever is turned. And since at the beginning of this chapter it was said that the ratio of moments, according to the different application of the power, is to be estimated from the arcs described by the force of the same impulse, and someone perhaps may think it difficult to compare arcs of this kind with one another, let him observe that from the Trigonometric Tables the sine and tangent of the same arc are defined by exactly the same numbers when the arc is very small. Therefore, since any arc is smaller than its tangent and greater than its sine, we can explain the ratio of smaller arcs without any danger of error by means of their sines. But since here, where the discussion is of the moments of a power applied to a lever according to different angles, only the smallest arcs are to be assumed, their ratio is taken to be the same as that of the sines. Therefore, if the lever be AB, the hypomochlion C, the force of the power and the direction of the motion of the power BH: in place of the arc BK, which in the motion by the force of the such

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Liber quartus. CAPUT VII. 419 talis impetûs cum hac directione describitur; assumi potest an- guli HBO, Radio BH, Sinus HO, qui est æqualis Sinui arcûs BK Ra- dio CB. Est autem minimus ar- cus longè minor quàm arcus BK, sed claritatis gratia arcum notabi- lem & conspicuum assumere opor- tuit. Est igitur potentiæ ad an- gulum rectum in B applicatæ mo- mentum, ad ejusdem potentiæ ad angulum HBO acutum applicatæ momentum, ut Radius BH ad acuti anguli Sinum HO. Iam intellige vectem AB converti, & lineam BH produci, donec in D ad angulos rectos occurrat vecti habenti positio- nem EF. Dico potentiæ ad perpendicular & obliquè appli- catæ momenta invicem comparata ita esse, ac si eadem poten- tia tam in F quàm in D ad angulum rectum applicaretur, quia ut BH ad HO, ita est FC ad CD. Ducatur enim ex D ad CB perpendicularis DG, quæ est parallela ipsi HO: quare per 4. lib. 6. ut BH ad HO, ita BD ad DG, & per 8. lib. 6. ut BD ad DG, ita BC, hoc est FC, ad CD: igitur per 11. lib. 5. ut BH ad HO, ita FC ad CD; hoc est ut Radius ad Sinum angu- li, secundùm quem potentia dirigitur, ita momentum poten- tiæ perpendiculariter applicatæ ad momentum ejusdem obli- què ad angulum acutum, vel obtusum applicatæ. Nam si po- tentia in B dirigat suum motum secundùm lineam BI, utique posito Radio BI, Sinus anguli ABI obtusi est IL, & Ratio mo- menti potentiæ in B applicatæ secundùm angulum rectum, ad momentum ejusdem potentiæ in B applicatæ secundùm angu- lum obtusum ABI, est ut BI ad IL. Producatur IB, donec in D perpendicularis cadat supra CF rectam æqualem ipsi CB. Quia triâgula BIL & BCD rectangula ad L & D, & æquales angulos ad verticem B habentia, similia sunt, est ut BI ad IL, ita BC ad CD per 4. lib. 6. Perinde igitur in extremitate B ad angulum obtusum ABI applicata potentia operatur, atque si ad angulum rectum applicaretur in D puncto vectis EF, qui idem ponitur esse ac vectis AB: & momenta potentiæ sunt ut FC ad CD. Ggg 2

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Liber quartus. CAPUT VII. 419 such an impulse is described with this direction; the angle HBO, the radius BH, the sine HO, which is equal to the sine of the arc BK to the radius CB. But the smallest arc is far less than the arc BK; yet for the sake of clarity it was necessary to assume a notable and conspicuous arc. Therefore the ratio of the moment of the power applied at the right angle in B to the moment of the same power applied at the acute angle HBO is as the radius BH to the sine HO of the acute angle. Now understand the lever AB to be turned, and the line BH to be extended, until it meets at D at right angles the lever having the position EF. I say that the moments of the power applied perpendicularly and obliquely are compared with one another in such a way as if the same power were applied both in F and in D at a right angle, because as BH is to HO, so is FC to CD. For let the perpendicular DG be drawn from D to CB, which is parallel to HO itself: wherefore, by 4. book 6, as BH is to HO, so is BD to DG, and by 8. book 6, as BD is to DG, so is BC, that is FC, to CD: therefore, by 11. book 5, as BH is to HO, so is FC to CD; that is, as the Radius to the Sine of the angle according to which the power is directed, so is the moment of the power applied perpendicularly to the moment of the same power applied obliquely at an acute or obtuse angle. For if the power at B directs its motion according to the line BI, then certainly, with radius BI assumed, the sine of the obtuse angle ABI is IL, and the ratio of the moment of the power applied at B according to the right angle to the moment of the same power applied at B according to the obtuse angle ABI, is as BI to IL. Let IB be extended, until at D the perpendicular falls upon the straight line CF equal to CB itself. Because the triangles BIL and BCD, right-angled at L and D, and having equal angles at the vertex B, are similar, it is as BI to IL, so is BC to CD, by 4. book 6. In like manner therefore, at the extremity B, the power applied to the obtuse angle ABI operates just as if it were applied at a right angle at the point D of the lever EF, which is taken to be the same as the lever AB: and the moments of the power are as FC to CD. Ggg 2

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Mechanicorum Dato itaque angulo, secundum quem potentia applicatur ad vectem, si angulus sit Rectus, momentum est ut Radius; sin autem angulus acutus sit vel obtusus, momentum est ut Sinus ejusdem anguli; atque adeò comparando inter se hujusmodi angulos, Ratio illorum erit eadem, quæ est Sinuum. Hinc datus vectis A B hypomochlium habens in C, si fuerit ita inclinatus, ut positionem habeat E F, potentia in E deorsum premens per rectam E G perpendicularem ad CB, momentum habet ut CG; & si positione habeat I H, potentia in I deorsum premens aut trahens juxta rectam IK, quæ producta incidat perpendicularis ad rectam AB in L, momentum habet ut CL. Quare ex E in B augentur prementis aut trahentis momenta, quæ ex B in I minuuntur. Id quod iis etiam, qui campanas pulsant, manifestum est: si enim intelligatur vecti CB adhærere campanam, cujus centrum gravitatis sit O, utique dum B deprimitur, O elevatur, sed elevandi difficultas crescit, tum quia centrum gravitatis O arcum describens circa punctum C, æqualibus temporibus inquales, atque semper majores habet ascensus juxta incremen- ta Sinuum Versorum, tum quia ex depressione vectis ex B in I facto angulo funis & vectis semper obtusiore, momenta potentiæ minuuntur: & licet in reditu ex I in B crescerent, si quis vectem sursum traheret, hoc nihil juvat potentiam deorsum trahentem ad elevandam campanam, quæ sponte sua descendens elevat vectis caput, cui funis adnectitur. Propterea majoribus gravioribusque campanis non simplicem vectem CB sed rotam, aut rotæ segmentum adjungunt, cujus excavatæ perimetro funis inseritur; qui dum trahitur, semper est Tangens circuli; atque ideo ad Radium circuli, quasi esset novuus atque novus vectis, applicatur potentia trahens ad angulum rectum.

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Mechanics Given an angle, according to which force is applied to a lever, if the angle is a right angle, the moment is as the radius; but if the angle is acute or obtuse, the moment is as the sine of that same angle; and therefore, comparing such angles with one another, their ratio will be the same as that of the sines. Hence, given a lever AB having its fulcrum in C, if it be so inclined that it has the position EF, the force pressing downward at E along the straight line EG perpendicular to CB has a moment as CG; and if it have the position IH, the force pressing downward or pulling at I along the straight line IK, which when extended meets perpendicularly the straight line AB at L, has a moment as CL. Wherefore, the moments of the pressing or pulling force increase from E to B, and diminish from B to I. This is also evident to those who ring bells: for if a bell be understood as attached to the lever CB, whose center of gravity is O, then indeed while B is depressed, O is raised; but the difficulty of raising increases, both because the center of gravity O, describing an arc about the point C, in equal times has unequal and always greater ascents according to the increases of the versed sines, and because, by the depression of the lever from B to I, the angle made by the rope and the lever becoming ever more obtuse, the moments of the force are diminished: and although on the return from I to B they would increase, if one were to pull the lever upward, this helps nothing for the force pulling downward to raise the bell, which, descending of itself, raises the head of the lever to which the rope is attached. For this reason, for larger and heavier bells they attach not a simple lever CB but a wheel, or a segment of a wheel, into whose hollow perimeter the rope is inserted; and when this is pulled, it is always a tangent to the circle; and thus the pulling force is applied to the radius of the circle, as if it were a new and fresh lever, at a right angle.

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Liber quartus. CAPUT VIII. 421 CAPUT VIII. Oneris ex Vecte pendentis momentum inquiritur. Contingit aliquando pondus vecte elevandum fune con- necti, & pendulum ex vecte suspendi. Nemo dubitat, an gravitas ponderis ibi sua exerceat momenta, ubi cum vecte connectitur; funis si quidem intentus congruit lineæ directionis, qua pondus ipsum nititur in centrum gravium: verùm non eandem percipi in elevando difficultatem experientia testatur pro varia vectis inclinatione. Si enim ex vecte AB horizontali hypomochlium in extremitate B habente, pondus D suspendum ex C pendeat ad angulos rectos, omnia sua momenta exercet pro ratione distantiæ CB ab hypomochlio. At si elevatus vectis positionem habeat EB, & C venerit in F, pondus verò pendulum D venerit in G, ita ut linea Directionis in centrum gravium congruat funiculo suspendenti FG; etiamsi FB æqualis sit ipsi CB, non eadem tamen momenta habet pondus adversùs eandem potetiam ex A translata[m] in E; quia scilicet angulus GFB est acutus, DCB autem rectus. Id explicare ex iis, quæ superiori capite disputata sunt, non erit difficile, si animadvertamus in vectibus secundi & tertij generis utrumque genus conjungi: quemadmodum enim potentia conatur adversùs gravitatem ponderis, ita pondus conatur adversùs vim potentiæ: & in hoc conatu vicissim exercent munus potentiæ & ponderis. Finge siquidem duos homines applicari vecti AB, alterum quidem in A, alterum verò in C, sed in adversa conantes; uterque est potentia, uterque est pondus, dum sibi reluctantur. Anne ita hæc vocabula intra certos fines coëreeri existimas, ut potentiæ nomine illum solum donandum putes, qui reliquum vincit? sed quid, si horum hominum co- Ggg 3

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Book fourth. Chapter VIII. 421 Chapter VIII. The moment of a weight hanging from a lever is investigated. It sometimes happens that a weight to be raised is attached by a rope to a lever, and hangs suspended from the lever. No one doubts whether the gravity of the weight exerts its moment there where it is joined to the lever; for indeed the rope, being stretched tight, agrees with the line of direction by which the weight itself tends toward the center of gravity. Yet experience shows that the same difficulty is perceived in lifting, according to the varying inclination of the lever. For if from the horizontal lever AB, having the fulcrum at the end B, the weight D is suspended from C at right angles, it exerts all its moment in proportion to the distance CB from the fulcrum. But if the lever, raised, has the position EB, and C has come to F, and the hanging weight D has come to G, so that the line of direction toward the center of gravity agrees with the suspending cord FG, even if FB is equal to CB, the weight nevertheless does not have the same moment against the same power transferred from A to E; because, namely, the angle GFB is acute, whereas DCB is right. This will not be difficult to explain from what was discussed in the previous chapter, if we observe that in levers of the second and third kind both kinds are joined together: for just as the power strives against the gravity of the weight, so the weight strives against the force of the power; and in this struggle they alternately perform the office of power and weight. Suppose, then, that two men are applied to the lever AB, one indeed at A, the other at C, but striving in opposite directions; each is a power, each is a weight, while they contend with one another. Do you think that these terms are thus confined within certain limits, so that you judge that only the one who overcomes the other ought to be called by the name of power? But what if these men’s co-

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Mechanicorum natus sint reciprocè ut eorum distantiæ ab hypomochlio B, & A quidem conetur ut CB, C autem conetur ut AB; utique neuter superat; nec tamen negari potest ideò contingere mo- mentorum æqualitatem inter inæquales conatus, quia vecti ap- plicantur: sunt igitur sibi vicissim potentia & pondus. Si ita- que A est potentia, & C pondus; vectis est secundi generis: Si verò C est potentia, & A pondus, vectis est tertij generis. Illud igitur quod de hominibus dicitur, de reliquis omnibus vim movendi habentibus dictum intelligitur: nihil si quidem interest, utrum animata sint, an inanima, quæ vecti applican- tur, & in oppositas partes conantur. Et quamvis non interce- dat inter ipsos conatus momentorum æqualitas, quam conse- quatur quies, sed efficiatur motus; ita tamen id quod prævalet, est potentia ad motum efficiendum, ut id quod vincitur, & re- sistit, sit potentia ad motum retardandum. In omni itaque vecte sive secundi, sive tertij generis sit, utrumque genus ami- co, nec solubili foedere copulantur. In vecte autem primi ge- neris idem genus manet, licèt vicissim habeant rationem po- tentiæ & ponderis ad movendum & retardandum, similiter enim potentiæ & ponderi, licet inæqualibus intervallis, inter- jacet hypomochlium. Hîc itaque, ubi oneris ex vecte pendentis momentum inqui- ritur, considerandus est vectis tertij generis, in quo gravitas in C, aut in F posita exercet munus potentiæ conantis deprimere vim sursum connitentem in A, aut in E. Quare in positione vectis horizontali, cum sit angulus rectus D CB, neque gravitas illa vectem versus hypomochlium B urgeat, aut eum ab illo re- trahat, omnia sua momenta obtinet, quæ in hac à fulcro distantiâ gravitati huic convenire possunt. At elevato vecte ita, ut fiat an- gulus acutus GFB, licèt eadem maneat gravitas, eademque ab hypomochlio distantia, non tamen eadem manent momenta, sed decrescunt pro ratione Sinûs anguli, ut superiori capite dictum est. Producta igitur intelligatur linea directionis FG usque ad horizontalem in H: posito Radio BF, hoc est BC, est BH Si- nus anguli GFB; ac proinde ut BC ad BH, ita momentum oneris pendentis ex vecte horizontali, ad momentum ejusdem oneris pendentis ex eodem vecte inclinato. Hinc est, inclinato vecte EB, tantumdem conatûs adhibendum esse in E ad susti- nendum

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In mechanics, therefore, the efforts are reciprocally as their distances from the fulcrum B; and A indeed tends as CB, while C tends as AB; in any case neither prevails; nor can it be denied that the equality of moments between unequal efforts thus comes about, because the weights are applied to the lever: they are therefore to one another alternatively power and weight. If therefore A is the power, and C the weight, the lever is of the second kind: if, however, C is the power, and A the weight, the lever is of the third kind. That which is said of men is therefore understood to be said of all other things having a moving force: for it makes no difference at all whether the things applied to the lever, and striving in opposite directions, are animate or inanimate. And although there does not intervene between the efforts themselves an equality of moments, from which rest would follow, but motion is produced; yet that which prevails is thus a power for producing motion, just as that which is overcome and resists is a power for retarding motion. In every lever, therefore, whether it be of the second or of the third kind, each kind is a friend, nor are they joined by a dissoluble bond. But in the lever of the first kind the same kind remains, although they reciprocally bear the relation of power and weight for moving and retarding; for likewise, with unequal intervals, the fulcrum lies between the power and the weight. Here therefore, where the moment of a load hanging from a lever is investigated, the lever of the third kind must be considered, in which the weight placed at C, or at F, performs the office of a power striving to depress the force resisting upward at A, or at E. Therefore, when the lever is in a horizontal position, since the right angle DCB is formed, and that weight neither urges the lever toward the fulcrum B nor draws it away from it, it retains all its own moments, which can belong to this weight in this distance from the fulcrum. But when the lever is raised so that the acute angle GFB is formed, although the same weight remains and the same distance from the fulcrum, the moments do not remain the same, but decrease in proportion to the sine of the angle, as was said in the preceding chapter. Let the line of direction FG therefore be produced until it meets the horizontal at H: the radius BF being posited, that is BC, BH is the sine of the angle GFB; and therefore, as BC is to BH, so is the moment of the load hanging from the horizontal lever to the moment of the same load hanging from the same inclined lever. Hence, when the lever EB is inclined, just so much effort must be applied at E to sustain

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Liber quartus. CAPUT VIII. 423 nendum onus G, quanto conatu opus esset in vecte horizontali AB ad sustinendum idem onus, si penderet ex H. Quoniam igitur distantia BH minor est quàm BC, major est Ratio AB ad BH, quàm ejusdem AB ad BC, ex 8.lib.5. ideóque faciliùs sustinetur idem onus vecte inclinato, quàm vecte horizontali. Quod si ex G centro gravitatis oneris ductam intelligas ad vectem EB rectam perpendicularem GI, habes similiter momentorum differentiam, quæ scilicet intercedit inter FG, & GI, si FG repræsentet omnia momenta in vecte horizontali: sunt enim triangula FIG & FHB rectangula, communem angulum ad F habentia, adeóque similia, & ut FB ad BH, ita FG ad GI. Cave autem ne putes (ut non pauci hallucinantur) ita ex I termino rectæ GI perpendicularis desumendam esse mensuram decrementi momentorum, ut perinde se habeat, quasi pondus esset in I: hoc enim à veritate longissimè abesse deprehendes, si manente eâdem vectis inclinatione, & eadem oneris gravitate, funiculo longiore onus suspenderis; quandoquidem etiam punctum I magis accedet ad hypomochlium B, nec tamen adhibito longiore funiculo adeò minuuntur momenta; alioquin tam longo funiculo suspendere posses onus, ut recta ex oneris centro ducta ad vectem EB perpendicularis caderet in B, atque ideo nullum esset gravitatis momentum, quasi onus esset in B: id autem omnino falsum est. Quando autem dicitur faciliùs à potentiâ sustineri idem onus suspensum vecte inclinato, quàm vecte horizontali, ita intelligendum est, ut linea directionis motûs potentiæ sustinentis eundem semper faciat cum vecte angulum: nam si hæc linea alium atque alium efficiat angulum, etiam potentiæ momenta variantur, quæ cum oneris momentis comparanda sunt. Hinc est in vecte primi generis CD, cujus hypomochlium O, si potentia & pondus sint gravia M & N, licet inclinato vecte, ut habeat positionem RS, recedentibus angulis à rectitudine, singulorum momenta minora fiant, non tamen mutari momentorum potentiæ & ponderis invicem compara

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Book Four. Chapter VIII. 423 the load G being supported, how much effort would be needed in the horizontal lever AB to sustain the same load, if it were hanging from H. Since therefore the distance BH is less than BC, the ratio of AB to BH is greater than that of the same AB to BC, from Book 8, Prop. 5; and therefore the same load is more easily sustained by an inclined lever than by a horizontal lever. If, however, from G, the center of gravity of the load, a line is understood to be drawn to the lever EB, the perpendicular GI, you similarly have the difference of moments, namely that which lies between FG and GI, if FG represents all the moments in the horizontal lever; for the triangles FIG and FHB are right-angled, having the common angle at F, and therefore similar, and as FB is to BH, so is FG to GI. But take care not to suppose (as not a few are mistaken) that the measure of the diminution of the moments is to be taken from the end I of the perpendicular line GI in such a way that it would be as though the weight were at I; for you will find this very far from the truth, if, while the same inclination of the lever and the same heaviness of the load remain, you suspend the load by a longer cord. For in that case the point I will also come nearer to the fulcrum B, and yet, with a longer cord employed, the moments are not diminished to such an extent; otherwise you could suspend the load by so long a cord that the straight line drawn from the center of the load perpendicular to the lever EB would fall at B, and therefore there would be no moment of gravity, as though the load were at B: but that is altogether false. But when it is said that the same load suspended by an inclined lever is more easily sustained by the power than by a horizontal lever, this is to be understood in such a way that the line of direction of the motion of the sustaining power always makes the same angle with the lever; for if this line makes one angle and then another, the moments of the power also vary, and these must be compared with the moments of the load. Hence in the first-kind lever CD, whose fulcrum is O, if the power and the weight are the heavy bodies M and N, although, with the lever inclined, so that it has the position RS, as the angles move away from rectitude, the moments of each become smaller, nevertheless the moments of the power and of the weight are not altered in relation to one another, as compared

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Mechanicorum 424 comparatorum Rationem; quia scilicet singulorum momenta proportionaliter minuuntur. Cum enim gravia semper nitantur juxta suas lineas directionis in centrum gravium, hujusmodi lineæ parallelæ consentur, & cum vecte duos angulos efficiunt duobus rectis æquales, ac proinde si alter acutus fuerit, alter est obtusus supplementum acuti ad duos rectos. Sicut autem in eodem circulo idem est Sinus anguli acuti, atque obtusi, qui compleat duos rectos; ita in diversis circulis hujusmodi angulorum Sinus proportionales sunt suis Radiis. Quapropter inclinato vecte, ut sit RS, gravia nituntur deorsum juxta lineas directionis ST & RV parallelas, quæ occurrunt perpendiculares horizontali in Z & V. Momentum igitur gravis T ad momentum æqualis, seu ejusdem gravis N est ut OZ ad OD, & momentum gravis V ad momentum æqualis, seu ejusdem gravis M est ut OV ad OC. Quare in vecte RS inclinato momenta gravior pendentium sunt ut OZ ad OV. Quia verò triangula RVO, SZO rectangula, & angulos ad verticem O æquales habentia, sunt similia, per 4. lib. 6. ut OS ad OR, hoc est ut OD ad OC, ita OZ ad OV. Manet itaque eadem momentorum Ratio invicem comparatorum, sive integra in vecte horizontali, sive diminuta in vecte inclinato sint singulorum momenta. At in vecte secundi aut tertij generis, si potentia non fuerit vivens, fieri non potest ut eadem servetur momentorum Ratio inter potentiam & pondus, nisi fortè in eodem medio horum alterutrum grave esset, alterum leve; ut si vectis AK intra aquam constitutus adnexum haberet in K inflatum utrem V, in L verò pendulus esset lapis: tunc enim, si uter ascendens trahat vectem in B, elevabit lapidem pendulum, ut sit angulus ICA acutus, & angulus ABF obtusus; qui cum æqualis sit alterno BCI (sunt enim FBE & CI parallelæ, quia utraque ad horizontem perpendicularis est) similem habet Sinum Sinui acuti ICA secundùm Rationem Radiorum BA & CA, hoc est KA & LA; atque

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Mechanics 424 the comparison of ratios; because, namely, the moments of individual weights are diminished proportionally. For since heavy bodies always tend along their lines of direction toward the center of gravity, such parallel lines are considered, and with the lever they make two angles equal to two right angles; and therefore, if one is acute, the other is obtuse, the supplement of the acute to two right angles. But just as in the same circle the sine of an acute angle is the same as that of the obtuse angle which completes two right angles, so in different circles the sines of such angles are proportional to their radii. Wherefore, when the lever is inclined, as RS, the heavy bodies tend downward along the lines of direction ST and RV, parallel, which meet the horizontal at right angles in Z and V. Therefore the moment of the weight T relative to the moment of an equal, or the same, weight N is as OZ to OD, and the moment of the weight V relative to the moment of an equal, or the same, weight M is as OV to OC. Hence, in the inclined lever RS, the moments of the hanging weights are as OZ to OV. But since the right triangles RVO and SZO, having equal angles at the vertex O, are similar, by 4. lib. 6, as OS to OR, that is, as OD to OC, so is OZ to OV. Thus the same ratio of moments compared with one another remains, whether their moments are entire in the horizontal lever, or diminished in the inclined lever. But in a lever of the second or third kind, if the power were not living, it cannot happen that the same ratio of moments between power and weight be preserved, unless perhaps in the same medium one of them were heavy and the other light; as if the lever AK were placed within water and had attached at K an inflated bladder V, while at L there hung a stone: then, if the rising bladder should draw the lever toward B, it would raise the hanging stone, so that the angle ICA is acute and the angle ABF obtuse; and since this is equal to the alternate angle BCI (for FBE and CI are parallel, because each is perpendicular to the horizon), it has a sine similar to the sine of the acute angle ICA according to the ratio of the radii BA and CA, that is, KA and LA; and

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Liber quartus. CAPUT VIII. 425 atque ut B A ad C A, ita est E A ad I A; similia quippe sunt triangula A B E & A C I. Cæterùm si rotulæ X insistens funi- culus jungeret vectis extremitatem K & pondus aliquod ad- nexum in S fungens munere potentiæ elevantis; hoc descen- dens ex S in M elevaret vectem in B, & lapidem, qui penderet ex C: sed angulus A B X esset multò obtusior quàm Supple- mentum acuti I C A ad duos rectos, ac proinde Sinus anguli A B X esset multo minor quàm E A Sinus anguli obtusi A B F: igitur multo minor esset Ratio momenti potentiæ applicatæ in B ad momentum ponderis in C, quàm sit Ratio B A ad C A, hoc est K A ad L A. Sola igitur potentia vivens potest ita sui motûs directionem inflectere, ut eundem faciat cum vecte an- gulum, ideóque elevans vectem acquirat majorem sustinendi facilitatem. His consequens est, quantò altiùs supra horizontem eleva- tur vectis cum pondere pendulo, tantò validiùs à pondere pre- mi aut urgeri hypomochlium A. Nam quemadmodum in vecte horizontali A K pondus in L suspensum magis premit hypomo- chlium A vicinum quàm potentiam K remotam ex hypothesi, ita elevato vecte multo magis premitur hypomochlium, quia quodammodo propiùs illi admovetur pondus in I quàm in L, suâque innatâ gravitate in vecte elevato conatur versus hypo- mochlium quasi secedens à potentia; ut nihil dicam de vecte ipso, cujus gravitas, maximam partem, innititur fulcro. CAPUT IX. An duo pondus gestantes æqualiter premantur. Hactenus disputatis proximè affinis est præsens quæstio, qua inquirimus, utrum æqualis sit labor duorum in eodem pon- dere gestando consentientium. Et quidem si movendum sit pondus atque trahendum, cur duo simul faciliùs illud mo- veant, quàm singuli, omnes intelligunt; quia plus impetus à duobus producitur, quàm à singulis; & quem impetum mo- H h h

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Book Four. Chapter VIII. 425 and as B A is to C A, so is E A to I A; for the triangles A B E and A C I are similar. But if a cord resting on the pulley X were to join the end K of the lever and a certain weight attached at S, serving as the power that raises; this, descending from S to M, would raise the lever at B and the stone hanging from C. But the angle A B X would be much more obtuse than the supplement of the acute angle I C A to two right angles, and therefore the sine of the angle A B X would be much less than E A times the sine of the obtuse angle A B F. Therefore the ratio of the moment of the applied power at B to the moment of the weight at C would be much less than the ratio of B A to C A, that is, K A to L A. Only living force, then, can so bend the direction of its motion that it makes the same angle with the lever, and thus, by raising the lever, acquires greater ease in sustaining it. From this it follows that the higher the lever with the hanging weight is raised above the horizon, the more strongly is the fulcrum A pressed or urged by the weight. For just as in the horizontal lever A K, the weight suspended at L presses more upon the nearby fulcrum A than upon the distant power K, as assumed, so when the lever is raised the fulcrum is pressed much more, because the weight at I is in a manner brought nearer to it than at L, and by its natural heaviness, in the raised lever, tends toward the fulcrum as if withdrawing from the power; not to mention the lever itself, whose weight, for the greater part, rests upon the support. Chapter IX. Whether two men carrying a weight are equally burdened. Closely related to the questions already discussed is the present one, in which we inquire whether the labor of two men acting together in carrying the same weight is equal. And indeed, if a weight must be moved and dragged, why two together move it more easily than each separately is clear to everyone; because greater impetus is produced by two than by one, and the impetus that is mo H h h

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Mechanicorum 426 vendo oneri parem singuli multo conatu producerent, singulis in parte impetûs efficiendâ minùs conantibus, totus producitur, totique oneri imprimitur. At in pondere sustentando, cujus gravitas in partes non dividitur, quomodo hæc singulis levior accidat, plures in sustentando conspirent, quàm si singulis imponeretur, tunc maximè cùm nullum impetum sursum gravitatis conatui adversantem producant, non ita explicatu facile existimant aliqui. Verum ex rationibus vectis satis manifesta solutio eruitur. Claritatis autem gratiâ, observandum est, an onus palangâ (ut cum bajuli dolium ex funibus suspensum transferunt) an verò subjectis humeris sustineatur. Et primo sit palanga A B, cujus extremitates à gestatoribus sustineantur; sit autem onus in C. Duplex effectus hîc considerandus est, videlicet oneris sustentatio, & gestatorum pressio; Si primum respicias, gestatores A & B rationem habent potentiæ efficientis sustentationem, atque impedientis motum oneris suâ gravitate deorsum conantis: Si secundum, idem onus C minus potentiæ pressionem efficientis exercet, dum secum palangam deorsum trahens, oppositos gestatorum humeros comprimit, aut si manibus palanga gestatur contentos contractosque brachiorum musculos, quantum potest, distrahit, atque relaxat. Sunt enim duo conatus, gestatorum scilicet & oneris, motum in oppositas partes efficere valentes, nisi sibi mutuo impedimento essent: hinc si gestatores conari cessent, onus descendit; Si ex improviso abruptis funibus onus à palangâ sejungatur, gestatores palangam sursum attollunt, sive æqualiter, sive inæqualiter, pro ut æquales aut inæquales sunt eorum conatus. Quare æstimanda res est ex motu, quem singuli conantes efficerent tum in se, tum in opposito conante, nisi prohiberentur momentorum æqualitate. Sic potentia in A suo conatu elevaret pondus in C positum, & circa centrum B arcum describerent; similiter potentia in B suo conatu elevaret pondus idem in C positum, & circa centrum A suos motus perficerent. Quod itaque ad sustentationem spectat, gestatores A & B vicissim habent rationem potentiæ & fulcri; nam si A est potentia, fulcrum est B; atque vicissim si B sit potentia, fulcrum est A; & est

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Mechanics 426 If, in order to produce a load equal to the load to be carried, each were to exert much effort, and each contributed less to the production of that part of the motion, the whole is produced, and is impressed upon the whole load. But in sustaining a weight, whose heaviness is not divided into parts, how this comes to be lighter for each, and how several combine in sustaining it more than if it were imposed on each individually, especially when they produce no upward impulse opposing the effort of gravity, is not so easily explained, some think. Yet from the reasons of the lever a sufficiently clear solution may be drawn. For the sake of clarity, however, it must be observed whether the load is supported by a palanga (as when porters carry a cask suspended by ropes) or whether it is borne on the shoulders beneath. And first let there be a palanga A B, whose ends are supported by the bearers; let the load be at C. A twofold effect must here be considered, namely the support of the load and the pressure on the bearers. If you regard the first, the bearers A and B have the relation of the power effecting support, and of the power impeding the motion of the load, which by its weight tends downward. If the second, the same load C exerts less force in producing pressure, while, drawing the palanga downward with it, it compresses the bearers’ opposed shoulders, or, if the palanga is carried by the hands, it stretches and relaxes, as much as it can, the contracted muscles of the arms. For there are two efforts, namely those of the bearers and of the load, able to produce motion in opposite directions, unless they were mutual impediments to each other: hence if the bearers cease to strive, the load descends; if, the ropes being suddenly broken, the load is separated from the palanga, the bearers raise the palanga upward, either equally or unequally, according as their efforts are equal or unequal. Therefore the matter is to be estimated from the motion which each, by striving, would produce both in itself and in the opposing body, if they were not prevented by equality of moments. Thus the power in A, by its effort, would raise the weight placed at C, and they would describe an arc about center B; similarly the power in B, by its effort, would raise the same weight placed at C, and they would perform their motions about center A. Therefore, as regards support, the bearers A and B alternately have the relation of power and fulcrum; for if A is the power, B is the fulcrum; and conversely, if B is the power, A is the fulcrum; and it is

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Liber quartus. CAPUT IX. 427 est duplex vectis secundi generis, scilicet AB & BA. Quod verò ad pressionem attinet, in qua onus C est potentia premens, gestatores vicissim habent rationem fulcri atque ponderis pres- si; & est duplex vectis tertij generis, quo utitur unica poten- tia, sicut in duplici vecte secundi generis unicum est pondus, quod sustinent duæ potentiæ. In vecte igitur secundi generis posito fulcro B, momentum potentiæ A sursum nitentis, ad momentum ponderis C deor- sum conantis, est ut AB ad CB; ac propterea potentia sustinere valens sine vecte pondus C, ad potentiam vecte AB sustinentem idem pondus C, est ut AB ad CB; quanto igitur CB minor est quàm AB, tanto minor potentia requiritur in A, quàm require- retur in C, si in C pondus sine vecte sustineretur. Idem quod de potentiâ A, posito fulcro B, dictum est, dic vicissim de poten- tiâ B; posito fulcro A; Requiritur enim in B potentia ut CA, ad potentiam, quæ esset ut BA, si sine vecte pondus in C sustinere- tur. Hinc est vires sustinendi requiri reciprocè tantas, quanta est Ratio distantiarum à pondere ipsorum sustinentium: vires si quidem in A requiruntur ut CB, & vires in B ut CA. Si itaque æquali intervallo pondus medium distet à gestatoribus, æqua- liter eos conari oportet, ut illud sustineant in C: at si inæquali- ter ab iis remotum sit, ut in D, requiruntur in A vires tantò ma- jores quàm in B, quantò major est distantia DB quàm DA. Quapropter datâ virium inæqualitate; statim innotescet, in quo palangæ puncto adnectendum sit onus; si nimirum palangæ lon- gitudo dividatur secundùm Rationem virium, & gestatores re- ciprocè collocentur. Sint enim ex. gr. duo, quorum alter vires habeat ut 3, alter ut 2: concipe totam longitudinem AB in quinque partes distinctam, & hinc accipe duas AD, hinc verò tres BD: locus ponderi debitus est punctum D, in quod cadit divisio in duas partes juxta datam Rationem: locus debilioris gestatoris est in palangæ extremitate B, ad quam spectat major distantia ab onere in C posito. Similiter virium inæqualitatem deprehendes, si pondere in medio puncto C posito, alter se præ- gravari sentiat: palanga enim ita promota, ut pondere in D constituto neuter se ultra vires prægravatum experiatur, indi- cabit vires gestatoris A esse ut DB, ad vires gestatoris B, quæ sunt ut DA. H h h 2

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Liber quarto. CHAPTER IX. 427 is a double lever of the second kind, namely AB & BA. But as regards pressure, in which the load C is the pressing power, the bearers in turn take the place of fulcrum and of the weight pressed upon; and it is a double lever of the third kind, which is used by a single power, just as in the double lever of the second kind there is a single weight, which is sustained by two powers. In the lever, therefore, of the second kind, the fulcrum B being placed, the momentum of the power A tending upward, to the momentum of the weight C tending downward, is as AB to CB; and therefore the power able to sustain the weight C without a lever, to the power sustaining the same weight C by means of the lever AB, is as AB to CB; the smaller therefore CB is than AB, the smaller the power required at A, than would be required at C, if the weight were sustained at C without a lever. What has been said of the power A, the fulcrum B being placed, say likewise of the power B; the fulcrum A being placed; for there is required at B a power as CA, to the power which would be as BA, if the weight at C were sustained without a lever. Hence it is that the forces sustaining are reciprocally required as great as the ratio of the distances of the sustainers themselves from the weight: for the forces at A are required as CB, and the forces at B as CA. If therefore the middle weight is at an equal interval from the bearers, they must exert themselves equally in order to sustain it at C: but if it is unequally distant from them, as in D, the forces required at A are so much greater than at B, as the distance DB is greater than DA. Wherefore, given an inequality of forces, it will immediately be known at what point of the palanga the load is to be attached; that is, if the length of the palanga be divided according to the ratio of the forces, and the bearers be reciprocally placed. For let there be, for example, two men, one of whom has forces as 3, the other as 2: conceive the whole length AB divided into five parts, and thence take two AD, and thence three BD: the proper place for the weight is the point D, into which the division falls in two parts according to the given ratio: the place of the weaker bearer is at the extremity B of the palanga, to which belongs the greater distance from the load placed at C. Likewise you will detect an inequality of forces, if, the weight being placed at the middle point C, one feels himself over- burdened: for the palanga being so moved that, with the weight placed at D, neither experiences himself overburdened beyond his strength, will indicate the forces of bearer A to be as DB, and the forces of bearer B to be as DA. H h h 2

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Mechanicorum At si vectem tertij generis, quatenus gestatorum pressio ab onere efficitur, consideremus; posito fulcro in A, potentia in C aut in D existens non premit gestatorem B perinde, atque si nullo intercedente vecte gestator esset pariter in C aut in D, sed tanto minùs, quanto minor est C A aut D A, quàm B A: idem- que de gestatore A, posito fulcro in B, dicendum est. Quare re- ciproca sunt pressiones distantiis gestatorum ab onere, & A premitur ut CB aut DB, B autem premitur ut C A aut D A. Cùm enim vis ipsa gravitatis oneris deorsum conantis apta sit circa centrum A moveri pro ratione distantiæ C A aut D A, uti- que in B motum efficere debet pro ratione distantiæ B A: est autem C A major quàm D A ex hypothesi, igitur, ex 8. lib. 5. ma- jor est Ratio momenti C A quàm momenti D A ad idem mo- mentum B A, ac proinde major pressio gestatoris B efficitur ab onere in C, quàm in D, collocato. Contra verò gestatorem A magis premit onus in D quàm in C positum, quia motus respi- cit centrum B, atque adeò ad eandem distantiam A B major est Ratio distantiæ D B majoris, quàm C B minoris distantiæ: eadem autem est distantiarum, & motuum Ratio, ac proinde momentorum. Observa autem mihi ideò de gestatoribus oneris sermonem fuisse, ut duplicem effectum sustentationis atque pressionis ex- pressiùs recognoscerem; qui enim onus gestando sustentant, musculorum contentione conantes elidunt impetum oneris de- orsum nitentis, & aliquid efficientes, dum Activè resistunt, no- men Potentiæ merentur. At si onus palangæ connexum susti- neretur à duobus fulcris in extremitate positis, hæc utique cùm oneris gravitati nullo conatu adversarentur, solam resistentiam Formalem suâ soliditate exercerent, impediendo ne onus cum palangâ descenderet, sed nullam haberent Resistentia[m] Activam, quæ illis Potentiæ vocabulum tribueret. In his unicus pressionis effectus attendendus est, & validiori fulcro propiùs admoven- dum est onus, ne fortè fulcrum infirmius nimiâ pressione coga- tur succumbere. Unum adhuc in palangæ gestatoribus attendendum est, si vi- rium inæqualium fuerint, & onus non ita sit palangæ applica- tum, ut ejus distantiæ à gestatoribus sint permutatim ut eorum- dem vires; nimirum contingere posse, ut validior gestator dum juxta

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Mechanics But if we consider a lever of the third kind, insofar as the pressure of the bearers is brought about by the load, then, a fulcrum being placed at A, the power located at C or at D does not press on bearer B as though there were no intervening lever and the bearer were equally at C or at D, but by so much less as C A or D A is less than B A. And the same must be said of bearer A, the fulcrum being placed at B. Wherefore the pressures are reciprocal according to the distances of the bearers from the load, and A is pressed as CB or DB, while B is pressed as C A or D A. For since the force itself of the weight of the load, tending downward, is suited to move about center A in proportion to the distance C A or D A, certainly it ought to produce motion in B in proportion to the distance B A. But C A is greater than D A by hypothesis; therefore, from book 5, proposition 8, the ratio of the moment of C A is greater than that of the moment of D A to the same moment B A, and consequently a greater pressure on bearer B is produced by the load placed at C than when it is placed at D. On the other hand, bearer A is pressed more by the load placed at D than by that placed at C, because the motion refers to center B, and therefore, at the same distance A B, the ratio of the greater distance D B is greater than that of the lesser distance C B. But the ratio of distances and motions is the same, and therefore of moments. But observe that I have therefore spoken of bearers of a load, in order that I might more clearly recognize the twofold effect of support and pressure; for those who, by bearing the load, support it, strive by muscular tension to neutralize the downward impulse of the load, and, doing something while actively resisting, deserve the name of Power. But if a load attached to a plank were supported by two fulcrums placed at the ends, these, since they would in no way oppose themselves to the weight of the load, would exercise only formal resistance by their solidity, preventing the load from descending with the plank, but would have no active resistance, which would give them the title of Power. In such cases only one effect of pressure is to be considered, and the load must be brought nearer to the stronger fulcrum, lest perhaps the weaker fulcrum be compelled to yield under too great a pressure. One more thing must still be considered in the bearers of a plank: if they are of unequal strength, and the load is not so applied to the plank that its distances from the bearers are inversely as their strengths, it may happen, namely, that the stronger bearer, while near

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Liber quartus. CAPUT IX. 429 juxta suas vires conatur adversùs onus, magis premat infirmiorem gestatorem, quam premeretur fulcrum infirmius, si unâ cum validiore fulcro eandem gravitatem sustineret. Quia vide. licet in eâdem palanga A B vectem primi generis considerare possumus, in quo onus C deorsum nitens contra vim gestatoris habeat rationem hypomochlij, & validior gestator A sit potentia repellens gestatorem infirmiorem B conantem adversus onus, ac proinde illum premat: quemadmodum si funi deorsum firmiter alligato insereretur palanga, cui humeros subjicerent duo inæqualibus viribus sursum conantes; constat enim infirmiorem à validiore premi, & esse vectem primi generis. Ex quo vides, cur bajuli dato invicem signo curent, ne alter alterum præveniat in elevandâ palanga; ne scilicet qui segnior fuerit, pressionem, non ab onere adhuc jacente & nondum elevato, sed à socio diligentiùs suam palangæ extremitatem elevante, recipiat. Hæc eadem, quæ de onere sublevando sunt dicta, de eodem trahendo pariter intelligantur, si vecti illigatum sit onus, & vectis extremitatibus jungantur trahentes: horum enim conatus esse oportet permutatim in Ratione distantiarum ab onere, hoc est à vectis puncto, cui onus adnectitur. Id non sine jucundâ quadam animi titillatione vidi aliquando observatum à rustico, qui alterius equorum currum trahentium defatigati laborem miseratus, transversarium, cui ambo adjungebantur, ita transutulit, ut in partes inæquales à temone distingueretur, & longior transversarij pars ad debiliorem equum spectaret. Inerant siquidem transversario tria foramina, per quæ temoni necdebatur ferreo clavo; unum quidem planè æqualiter ab extremitatibus aberat, reliqua duo hinc & hinc à medio distabant modico quidem sed congruo intervallo, ut si equus dexter defatigaretur, clavus immitteretur sinistro foramini, aut contra dextro, si sinister equus languidiùs traheret. Verùm cautè modica intervalla definierat, ne nimia fieret momentorum inæqualitas; quod enim alteri equorum laboris demebatur, addebatur reliquo. Quando autem non palangâ defertur onus, sed ipsum immediatè à duobus sustinetur, eadem prorsus est philosophandi ratio; quandoquidem est quodammodo onus vecti conjunctum, atque juxta vectis longitudinem distributum. Quamvis vero H h h 3

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Book Four. Chapter IX. 429 according to their strength, it tries against the load in such a way that it presses the weaker carrier more, than the weaker support would be pressed if it were sustaining the same weight together with a stronger support. For although in the same palanga we may consider a lever of the first kind, in which the load C, pressing downward against the force of the bearer, has the place of the hypomochlion , and the stronger bearer A is the repelling power that drives back the weaker bearer B, who is trying against the load, and therefore presses him: just as if a palanga were inserted into a rope firmly tied downward, to which were attached two men straining upward with unequal forces; for it is clear that the weaker is pressed by the stronger, and that it is a lever of the first kind. From this you see why the porters, having given one another a signal, take care that one does not anticipate the other in raising the palanga ; lest, namely, the one who is slower should receive the pressure not from the load still lying there and not yet lifted, but from his companion, who is more carefully raising his end of the palanga . The same things that have been said about lifting a load are to be understood equally about drawing it, if the load is attached to the lever, and the pullers are joined to the ends of the lever: for their efforts ought to be inversely proportional to the distances from the load, that is, from the point of the lever to which the load is attached. I once saw this observed, not without a certain pleasing stirring of the mind, by a rustic who, pitying the labor of one of two horses drawing a cart when worn out, shifted the crossbar to which both were harnessed so that it was divided into unequal parts from the pole, and the longer part of the crossbar looked toward the weaker horse. For there were in the crossbar three holes, through which it was fastened to the pole by an iron pin; one was exactly equally distant from the ends, while the other two were on this side and that at a slight but suitable distance from the middle, so that if the horse on the right grew tired, the pin would be inserted into the left hole, or conversely into the right, if the left horse should pull more weakly. But he had wisely assigned only small intervals, lest there should be too great an inequality of forces; for what was taken away from the labor of one of the horses was added to the other. But when the load is not carried by a palanga , but is supported immediately by the two, the reasoning is exactly the same; since in a certain sense the load is joined to the lever, and distributed along the length of the lever. Although, however, H h h 3

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Mechanicorum 430 singulis partibus sua gravitas insit, quia tamen in unam coalescunt gravitatem, ideò totius molis gravitas ibi intelligenda est, ubi est centrum gravitatis; vectis autem longitudo æstimanda est in lineâ jungente puncta, quibus gestatores aut sustinentes applicantur: Ex quibus punctis si ponamus exire lineas parallelas lineæ directionis exeunti ex centro gravitatis, cadent omnes ad perpendiculum in lineam horizontalem transeuntem per centrum gravitatis, aut illi parallelam. Harum igitur parallelarum, quæ directionem conatûs oppositi gravitationi ponderis referunt, distantia à lineâ directionis centri gravitatis, ipsorum deferentium conatum in sustinendo, permutatim sumpta definiet; quæcumque demùm sit oneris figura. Sit onus deferendum, cujus centrum gravitatis E; linea per gestatores transiens, habensque rationem vectis, sit B C, quæ intelligatur horizonti parallela. In hanc igitur ad angulos rectos cadit linea directionis E F; & gestatores in quocumque puncto lineæ B C fuerint, permutatim habent momenta sustinendi, aut recipiunt momenta pressionis pro Ratione distantiarum à puncto F, in quod cadit linea directionis: cùm enim linea B C ex hypothesi sit horizonti parallela, omnium ipsi F E parallelarum

Transcription: Translated (English)

Mechanics 430 although each part has its own weight, yet because they coalesce into one weight, the weight of the whole mass is therefore to be understood to be there where the center of gravity is; but the length of the lever is to be estimated in the line joining the points at which the carriers or supporters are applied: from which points if we suppose lines to be drawn parallel to the line of direction issuing from the center of gravity, all will fall perpendicularly into the horizontal line passing through the center of gravity, or into a line parallel to it. Therefore the distance of these parallels, which refer the direction of the effort opposed to the weight’s gravitation, from the line of direction of the center of gravity will, taken reciprocally, define the effort of the carriers in sustaining, whatever the shape of the load may be. Let there be a load to be carried, whose center of gravity is E; the line passing through the carriers, and having the character of a lever, be B C, which is understood to be parallel to the horizon. Upon this, therefore, at right angles, falls the line of direction E F; and the carriers, wherever they may be in the line B C, have reciprocally moments of sustaining, or receive moments of pressure, according to the ratio of their distances from the point F, upon which the line of direction falls: for since the line B C, by hypothesis, is parallel to the horizon, of all the parallels to it, F E

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Liber quartus. CAPUT IX. 431 parallelarum distantia ab eâdem F E, desumenda est ex inter- vallis gestatorum & puncti F. At verò si recta B C non fuerit horizonti parallela, vel quia deferentes onus non sunt æquè alti, vel quia in clivo consistunt, utique linea directionis centri gravitatis E non cadit ampliùs in rectam B C ad angulos rectos in F, sed obliquè incidit in H. Concipiatur itaque per H transiens linea G I horizonti paral- lela, & ipsi E H sint parallelæ C D & B K: sunt igitur distan- tiæ H D & H K, quæ eandem inter se habent Rationem, quæ reperitur inter H C & HB; sunt enim triangula H D C & H K B rectangula ad D & K, æquales angulos ad verticem H habentia, ac proinde similia, & per 4. lib.6 ut HD ad HK, ita HC ad HB. Quo igitur magis ab horizonte removetur punctum B præ puncto C, etiam linea directionis ex E propior cadit puncto C; atque adeò qui inferior est, magis gravatur ab onere. Id quod ex iis, quæ hujus libri cap.4. dicta sunt, confirma- tur: Nam vectis CB habens hypomochlium B, & pondus E vecti impositum, est infra horizontem inclinatus; igitur plus la- boris potentia impendit, quàm in horizontali positione vectis. Similiter vectis BC habens hypomochlium C & pondus E vecti impositum, est elevatus supra horizontalem; igitur mi- nus laborat potentia quàm in positione horizontali. Itaque si positâ lineâ BC horizonti parallelâ æqualiter premebantur gestatores in B & in C, factâ inclinatione ad horizontem, mi- nùs premitur B superior quàm C loco inferior. Quòd si gestatores non sustineant onus subjectis humeris, sed illud manibus arreptum quasi suspensum retineant in M & N; simili ratione attendenda est distantia illorum à puncto, in quod cadit linea directionis centri gravitatis E; quæ utique ad angu- los rectos incidit in rectam MN, si hæc fuerit horizonti pa- rallela, & labor gestatorum est permutatim ut eorum distantia à puncto S. At verò si linea MN fuerit ad horizontem inclinata, & linea directionis sit EO; utique minor est distantia à superiore M, quàm ab inferiore N, ideóque plus laborabit su- perior retinendo, quàm inferior. Id quod pariter ex dictis cap.4. confirmatur; nam pondus est vecti subjectum, & vectis MN habens hypomochlium N est supra horizontalem lineam, ac

Transcription: Translated (English)

Liber quartus. Chapter IX. 431 the distance of the parallels from the same F E is to be taken from the intervals of the bearers and of point F. But if the straight line B C is not parallel to the horizon, either because those carrying the load are not of equal height, or because they stand on a slope, then the line of direction of the center of gravity E no longer falls upon the straight line B C at right angles in F, but strikes obliquely in H. Let there therefore be conceived a line G I passing through H and parallel to the horizon, and let C D and B K be parallel to the line E H: therefore the distances H D and H K are in the same ratio to one another as is found between H C and H B; for the triangles H D C and H K B are right-angled at D and K, have equal angles at the vertex H, and are therefore similar; and by Book 6, Proposition 4, as H D is to H K, so is H C to H B. Therefore, the more point B is removed from the horizon beyond point C, the more nearly does the line of direction from E fall toward point C; and thus the lower one is, the more is he burdened by the load. This is confirmed by what was said in chapter 4 of this book: for the lever C B, having the fulcrum B and the weight E placed on the lever, is inclined below the horizon; therefore power expends more labor than in the horizontal position of the lever. Likewise the lever B C, having the fulcrum C and the weight E placed on the lever, is raised above the horizontal; therefore power labors less than in the horizontal position. Thus, if the line B C is set parallel to the horizon, the bearers at B and at C were pressed equally; but when the inclination to the horizon is made, the one at B, being higher, is pressed less than the one at the lower place C. But if the bearers do not support the load on their shoulders, but hold it suspended, grasped in their hands at M and N, then the same reasoning applies to the distance of each from the point on which the line of direction of the center of gravity E falls; this certainly falls at right angles upon the straight line M N if it be parallel to the horizon, and the labor of the bearers is reciprocally as their distance from point S. But if the line M N is inclined to the horizon, and the line of direction is E O, then the distance from the upper M is certainly less than from the lower N, and therefore the upper one will labor more in holding it than the lower. This is likewise confirmed by what was said in chapter 4; for the weight lies upon the lever, and the lever M N, having the fulcrum N, is above the horizontal line, and

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Mechanicorum ac propterea potentia plus laborat quàm in horizontali: contra autem vectis NM habens hypomochlium M est infra horizontalem lineam depressus, ideoque minùs potentia laborat quàm in horizontali. Quæ omnia tam aperte respondent quotidiano experimento, ut mirum videatur potuisse aliquos authores idem planè opinari, sive gestatores sustineant impositum onus, sive illud suspensum retineant in positione vectis declivi; Si enim ductâ per O lineâ horizonti parallelâ, ducantur ex M & N rectæ MT, & NV parallelæ lineæ directionis centri gravitatis EO, utique distantiæ sunt TO & VO: atqui TO ad VO est ut MO ad NO propter triangulorum O TM & O VN similitudinem; & MO ad ON habet minorem Rationem quàm MS ad SN ex 8. lib. 5. igitur etiam TO ad OV habet minorem Rationem quàm MS ad SN: igitur in positione vectis declivi, M superior laborabit ut ON, atque N inferior laborabit ut OM. Ex his unusquisque intelligit non ad duos tantùm gestatores, sed etiam ad plures referenda esse, quæ hactenus diximus, habitâ scilicet distantiarum ratione, quibus singuli absunt à pondere, adeò ut qui æqualibus intervallis à pondere distant, æqualem conatum impendant in eo sustinendo. Sic si à pon- dere P æqualiter distent A & B, æqualiter premuntur: item C & D æqualiter distantes ab eodem pondere P æqualem pressionem recipiunt: Et si comparentur invicem D & B, aut C & A, manifestum est propinquiores premi præ remotioribus; ac propterea, si solùm positionis ratio haberetur, qui robustiores sunt, collocandi essent in C & D, infirmiores verò in A & B: sed quoniam contingit inter plures sodales aliquem aliquando connivere, ideò ut plurimum extremi A & B validiores sunt, ut si fortè mediorum aliquis languidiùs conetur sustinendo, illi faciliùs muneri suo satisfaciant.

Transcription: Translated (English)

Mechanics, and therefore power, labors more than in the horizontal position; on the other hand, when the lever NM, having the fulcrum M, is depressed below the horizontal line, power labors less than in the horizontal position. All these things correspond so clearly with everyday experience that it seems surprising that some authors could have thought exactly the same thing, whether the bearers support the load placed upon them, or retain it suspended in the position of an inclined lever. For if, a line parallel to the horizon having been drawn through O, straight lines MT and NV be drawn from M and N parallel to the direction line of the center of gravity EO, then certainly the distances are TO and VO; but TO is to VO as MO is to NO, because of the similarity of triangles OTM and OVN; and MO has a smaller ratio to ON than MS has to SN, from Book 5, proposition 8; therefore TO also has a smaller ratio to OV than MS has to SN: therefore, in the position of the inclined lever, the upper M will labor as ON, and the lower N will labor as OM. From these things everyone understands that what we have said up to now must be referred not only to two bearers, but also to more, namely with due regard to the distances by which each is removed from the weight, so that those who are at equal intervals from the weight expend equal effort in supporting it. Thus if A and B are equally distant from the weight P, they are equally pressed; likewise C and D, equally distant from the same weight P, receive equal pressure. And if D and B, or C and A, are compared with one another, it is clear that the nearer are pressed more than the more distant; and therefore, if only the position were considered, the stronger should be placed in C and D, and the weaker in A and B. But since it happens among several companions that someone may sometimes slacken, therefore usually the outermost A and B are stronger, so that if by chance one of the middle men should strive more feebly in supporting the load, they may more easily fulfill their task.

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Liber quartus. CAPUT X. 433 CAPUT X. An vis Elastica ad aliquod Vectis genus pertineat. Quoniam Græcis ἐλασμᾶ tùm laminam, tùm plicam seu flexum significat, atque ἐλασμὸν est id, quod impellit; sæpiùs autem chalybeas laminas in machinulis ita disponimus, ut primùm flexæ, deinde sibi dimissæ, dum sese restituunt, aliud corpus impellant, cui motum concilient; propterea Elasmata, seu Elasmos, hujusmodi laminas dicimus, quas Itali Susse aut molle vocamus; & facultatem illam, qua sibi congruentem fi- guram atque positionem hæ laminæ reparant, Vim Elasticam appellamus. Quamquam non solis laminis, sed cæteris quoque corporibus per vim inflexis, & ad sibi debitam rectitudinem redeuntibus, facultas hæc Elastica tribuenda est, quemadmodum flexilibus virgulorum ramis, à quibus secundus in sylvâ sibi cavere debet, & perticæ, quâ toreutæ utuntur in toreumate elaborando, dum tornum circumagunt circumducto funiculo, qui depresso suppedaneo perticam flectit; hæc enim, cessante pedis pressione, funiculum retrahens suam sibi reparat rectitudinem. An verò ignis atque aër sive externâ compressione, sive alieno frigore concretus, & in exigua spatia contractus, ubi cessante vi, aut abeunte frigore, extenuatus ampliorem locum occupat, proximumque corpus pellens à suâ sede removet, facultate Elasticâ præditus dicendus sit; quæstio Grammaticis dirimenda relinquatur: hæc enim fluida corpora, nullam partium texturam habentia, nec certis figuræ, quam expetant, terminis suapte naturâ circumscripta, vix quicquam cum Elasmate commune habere videntur. Cum itaque inæquales deprehendantur elasmatis ejusdem vires pro diversâ suarum partium positione juxta longitudinem, animum subiit cupido examinandi, an fortè in eo aliqua vectis species reperiatur, ut propterea & vectis rationibus illa virium i i i

Transcription: Translated (English)

Book Four. Chapter X. 433 Chapter X. Whether Elastic Force belongs to any kind of Lever. Since among the Greeks ἐλασμᾶ signifies both a plate and a fold or bending, and ἐλασμὸς is that which impels; and since more often we arrange steel plates in small machines in such a way that, first bent and then released, while they spring back, they impel some other body and impart motion to it; therefore we call such plates Elasmata, or Elasmos, which the Italians call Susse or molle; and that faculty by which these plates restore to themselves a fitting shape and position we call Elastic Force. Although this faculty must be attributed not only to plates, but also to other bodies bent by force and returning to their proper straightness, as with the pliant branches of rods, against which one must beware in the woods, and with the staff used by engravers in making toreumata, while they turn the lathe by means of a cord passed around it, which, pressing down the foot-rest, bends the staff; for when the pressure of the foot ceases, the staff, drawing back the cord, restores its own straightness. But whether fire and air, whether thickened by external compression or congealed by foreign cold, and contracted into small spaces, when the force ceases or the cold departs, becoming expanded and occupying a larger place, and pushing aside the neighboring body from its position, should be said to be endowed with Elastic faculty, let the question be left to Grammarians to decide: for these fluid bodies, having no texture of parts and not being by nature bounded by fixed limits of the shape they seek, seem scarcely to have anything in common with Elasma. Since therefore unequal powers of the same elasmata are found according to the different position of their parts along the length, the desire came upon me to examine whether perhaps some species of lever might be found in it, so that for that reason also its powers would be explained by the principles of the lever. i i i

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Mechanicorum inæqualitas definienda sit. Et quidem manifestum est aliquam elasmatis partem fixam esse atque manentem, sive illa extrema sit, ut in perticâ toreutæ, sive media, ut in arcu balistæ, sive utraque extremitate manente pars media flectatur in sinum, ut citharæ nervis contingit. Qui enim fieri posset, ut per vim la- mina flecteretur, si partes omnes æqualiter moverentur? Ut igi- tur externam vim recipiat, & flectatur, aliquam ejus partem oportet aut omnino immotam manere, aut saltem languidiùs moveri. Hinc est elasmatis motum, dum inflectitur, circa partem manentem perfici, ac proinde particulas, quæ ad cavam qui- dem superficiem spectant, per vim comprimi, quæ verò ad con- vexam, intendi. Quòd si particulæ illæ non ita tenaci nexu in- ter se invicem cohærerent, ut facilè distraherentur contentæ, & exprimerentur compressæ, quemadmodum plumbeæ laminæ, quæ in figuram quamlibet conformatur, accidit, amissam recti- tudinem non recuperarent. Sed quoniam arctissimo vinculo conjunguntur, quod nisi validioribus viribus revelli non potest, ut in chalybeâ laminâ observamus; cessante externâ vi, quæ contentæ fuerant, se contrahunt, quæ compressæ, se latiùs ex- plicant; atque adeo his debitam positionem sibi reparantibus, lamina ad pristinam formam eò vehementiùs redit, quò majo- rem violentiam patiebatur. Quare Potentia movens sunt ipsæ particulæ illatam vim excutientes, & ad sibi debitam positio- nem redeuntes. Licet igitur in arcu balistæ intento duplex elasina hinc atque hinc considerare; media si quidem pars arcûs balistæ manubrio infixa manet, & singula cornua sinuantur, sed eò difficiliùs, quò breviora sunt, cæteris paribus, attentâ eorum crassitie, & ferri temperatione, pari enim flexione paucioribus minoris ar- cûs particulis major violentia inferenda est, quippe quas ma- gis comprimi, magisque intendi oportet, quàm in longiore ar- cu, ubi minore plurium partium compressione & intentione flexio eadem habetur. Præterquam quod in ipsa flexione adhi- betur quodammodo vectis secundi generis, quem ipsa longitu- do repræsentat, pars manens vicem hypomochlij subit, & par- tes intermediæ, quas per vim coarctari aut dilatari oportet, lo- cum obtinent ponderis: nihil igitur mirum, si Potentia extre- mita

Transcription: Translated (English)

The inequality of mechanics must be defined. And indeed it is manifest that some part of the elasma is fixed and remains, whether that be one extremity, as in the bar of a sculptor, or the middle, as in the bow of a crossbow, or, with both extremities remaining, the middle part is bent into a hollow, as happens with the strings of a cithara. For how could it happen that a plate would be bent by force if all its parts moved equally? Therefore, so that it may receive external force and be bent, some part of it must either remain wholly unmoved, or at least move more sluggishly. Hence it is that the motion of the elasma , while it is bent, is accomplished around the part that remains; and therefore the particles that face the concave surface are compressed by force, while those that face the convex are stretched. But if those particles did not cohere with one another by so firm a bond that, when stretched, they could easily be drawn apart, and when compressed, be pressed together, as happens with leaden plates that are shaped into any form, they would not recover their straightness when it was lost. But since they are joined by a very close bond, which cannot be torn apart except by stronger forces, as we observe in a steel plate, when the external force ceases, those that had been stretched contract themselves, those that had been compressed expand more widely; and thus, with these restoring their due position, the plate returns to its original form the more forcefully, the greater the violence it had suffered. Therefore the moving power consists in the particles themselves, shaking off the force applied to them and returning to the position due to them. Although, therefore, in a crossbow bow under tension, a double elasma may be considered here and there; for the middle part of the crossbow bow remains fixed in the stock, and each horn bends, but the more difficultly the shorter they are, all other things being equal, regard being had to their thickness and the tempering of the iron. For with the same bending, greater violence must be inflicted on fewer particles of a smaller bow, since they must be compressed and stretched more than in a longer bow, where the same bending is obtained by less compression and tension of more parts. Besides, in the bending itself a lever of the second kind is, in a certain way, employed, which its own length represents; the fixed part takes the place of the fulcrum, and the intermediate parts, which must be narrowed or expanded by force, occupy the place of the weight: therefore it is no wonder if the moving power of the extremit...

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Liber quartus. CAPUT X. 415 mitatem arcûs ad se nervo adnexo trahens faciliùs moveat par- ticulas easdem, quò, longiùs absens ab hypomochlio, faciliùs movetur. Hîc autem ubi arcûs mentio incidit, in ipso nervo illud elas- matis genus occurrit, quod utramque extremitatem habet ma- nentem; curvato enim arcu nervus inflexus intenditur; postea cùm dimittitur, pars media, cui sagitta aut globus excutiendus aptatur, plus movetur quàm ejus extremitates arcûs cornibus cohærentes. Universa autem violentia, quam nervus conten- tus subit, consistit in suarum particularum intentione, quæ, dum se contrahentes aliquid juvant ad nervum ipsum juxta rectam lineam extendendum, aliquid etiam impetûs sagittæ ex- cussæ imprimunt. Quod verò ad ipsa arcûs cornua attinet, satis liquet illa si- milis crassitiei, paris longitudinis, æqualisque temperamentis esse debere, ut æqualis fiat hinc & hinc compressio atque in- tentio partium, ex qua æquales oriantur vires sese in pristinam formam restituendi. Si enim alterutra pars arcûs majorem violentiam passa velociùs atque validiùs præ reliquâ se move- ret, à destinato scopo sagitta aberraret in dexteram aut in si- nistram declinans. Ut igitur hisce prænotatis ad propositam quæstionem acce- damus, non est hîc sermo de laminâ in spiram multipheem in- flexâ, atque spisè per vim contorta, quæ amoto repagulo sese in ampliores gyros explicans secum rapit aliud corpus extremitati mobili adnexum; cujusmodi est Elasma in Automatis ho- ras indicantibus, cujus extremitati adnectitur tympanum spi- ram illam includens; dum enim ex dilatatione Elasmatis in am- pliorem spiram, circumagitur tympanum, adnexam catenu- lam conum circumplexam trahit, totique machinulæ motum conciliat. Hîc siquidem, uti nulla longitudo in consideratio- nem cadere potest, nullam vectis speciem habere possu- mus; nam facultas movendi non ratione positionis exte- nuatur, ut in vecte, sed vires initio validæ sensim lan- guescunt, quia elasmatis partes compressæ atque conten- tæ, pro ratione violentiæ, quam subeunt, excutiendæ, ve- hementiùs primùm, deinde remissiùs conantur. Quare con- troversia in illo est, utrum in elasmate, cujus aliqua 1 i i 2

Transcription: Translated (English)

Book Four. Chapter X. 415 …by drawing the middle part of the bow toward itself by the attached string, more easily moving the same particles, because, the farther it is removed from the fulcrum, the more easily it is moved. Here, however, since mention has fallen upon the bow, there occurs in the string itself that kind of elastic body which has both ends fixed; for when the bow is bent, the string is stretched by being drawn inward; afterward, when it is released, the middle part, to which the arrow or ball is to be discharged is fitted, moves more than its ends, which cling to the bow’s horns. But the whole force which the tightened string undergoes consists in the tension of its particles, which, while by contracting they help in some measure to stretch the string itself in a straight line, also impart some impetus to the discharged arrow. But as for the horns of the bow themselves, it is clear enough that they ought to be of like thickness, equal length, and equal temper, so that equal compression and tension of the parts may be produced here and there, from which equal powers of restoring themselves to their former shape arise. For if either side of the bow should suffer a greater violence and move more quickly and strongly than the other, the arrow would swerve from the intended mark, veering to the right or to the left. Therefore, since these things have been set forth in advance, let us come to the proposed question. The discussion here is not about a plate bent into a multifold spiral and tightly twisted by force, which, when the stop is removed, expands into wider circles and carries along with itself another body attached to its moving end; such as is the Elasma in automata showing the hours, to whose end is attached the drum enclosing that spiral. For when, by the expansion of the Elasma into a wider spiral, the drum turns around, it draws the attached chain winding about the cone, and gives motion to the whole little machine. Here, indeed, since no length can come into consideration, we cannot have any kind of lever; for the faculty of moving is not weakened by reason of position, as in a lever, but the forces, strong at the beginning, gradually grow faint, because the parts of the elasma, being compressed and strained, according to the degree of the violence they undergo in being discharged, strive first more vehemently, then more faintly. Wherefore the controversy is whether in the elasma, of which some… 1 i i 2

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Mechanicorum 436 longitudo designari potest, aliqua vectis species reperiatur. Et ut majori in luce quæstio versetur, perticam toreutæ oculis subjiciamus, quæ sit A B, & in A fixa atque immota perseveret, quamvis extremitas B deprimatur, ut veniat in C. In hac perticæ flexione partes, quæ circa D ex. gr. intelliguntur, maximam violentiam patiuntur, nam inter eas, quæ ad cavitatem spectantes compressione coarctantur, illæ præ cæteris hinc atque hinc cohærentibus urgentur magis; inter eas verò, quæ convexitatem respicientes distentur explicantur, quæ ibi sunt, præ reliquis à summo flexu paulò remotioribus vehementiùs tenduntur. Hinc licèt particulæ omnes in hac flexione vim passæ, dum nituntur singulæ pristinum statum sibi reparare, conatus suos exerant, majores aut minores pro ratione majoris aut minoris violentiæ; potissima tamen vis elastica ibi consideranda est, ubi summa inflexio summam vim particulis infert; ibi enim majore conatu quàm alibi violentiam excutit natura. Quamvis igitur vis elastica per universam elasmatis longitudinem, quàm particulæ compressæ atque contentæ obtinent, extendatur, ibi tamen potissimùm collocata intelligitur, ubi in summo flexu puta in D, validiùs conatur. Iam verò quis ignorat in tornando plurimum interesse, utrum funiculus in ipsâ extremitate B, an verò in E adnectatur? Siquidem, sicut ex E difficiliùs flectitur pertica, quàm ex B, æquali flexione, ita cæteris paribus in E validiùs retrahitur funiculus, & minor motus perficitur quàm in B. Est igitur hîc ratio Vectis tertij generis, in quo hypomochlium est A pars fixa & immota; Potentia movens (scilicet particulæ vim illatam excutientes) est potissimùm in D; pondus, quod movetur, est ultra D, sive in extremitate B, sive in aliqua ex partibus intermediis, ut in E. Quare in collocatione corporis, quod ope elasmatis movendum est, attendere oportet, quanto motu opus sit, ut in majore seu minore distantiâ à puncto elasmatis manente, & immoto applicetur: quo enim minor est distantia, minus spatium percurrit. Quamvis

Transcription: Translated (English)

Mechanics 436 the length may be determined, provided some kind of lever be found. And that the question may be considered in a clearer light, let us place before the eyes a bent rod, which be A B, and let it remain fixed and immovable at A, although the end B is pressed down, so that it comes to C. In this bending of the rod, the parts which are understood to be around D suffer the greatest violence; for among those that face the concavity and are narrowed by compression, those are pressed most of all which lie here and there among the contiguous parts; among those again which, facing the convexity, are stretched and expanded, those which are there, farther than the rest from the summit of the bend, are drawn with greater force. Hence, although all the particles in this bending have suffered force, and while each strives to restore itself to its former state exerts its efforts, greater or smaller according to the degree of greater or lesser violence, the chief elastic force is nevertheless to be considered there where the greatest inflection inflicts the greatest force on the particles; for there nature throws off violence with greater effort than elsewhere. Although therefore the elastic force throughout the whole length of the elasticity, which the compressed and constrained particles possess, is extended, it is nevertheless understood to be placed chiefly there where, in the greatest bend, as at D, it strives more strongly. Now who does not know that in turning it makes a great difference whether the cord is attached at the very end B, or indeed at E? For, as from E the rod is bent more difficultly than from B, though with equal bending, so, other things being equal, the cord is pulled back more strongly at E, and a smaller motion is accomplished than at B. There is therefore here the ratio of a lever of the third kind, in which the fulcrum is A, the fixed and immovable part; the moving power (namely the particles throwing off the force impressed upon them) is chiefly in D; the weight, which is moved, is beyond D, whether at the end B or at some of the intermediate parts, as at E. Wherefore in placing the body which is to be moved by means of the elasticity, it must be considered how much motion is needed, so that it may be applied at a greater or lesser distance from the point where the elasticity remains and is immovable: for the smaller the distance, the less space it traverses. Although

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Liber quartus. CAPUT X. 437 Quamvis autem elasmatis vires ad impellendum vel trahendum corpus ex hujusmodi distantiâ pendeant, & comparatis inter se duabus positionibus E atque B, validiùs operetur in E quàm B, non tamen eâdem vi motus (quicunque demum ille sit sive major, sive minor) inchoatur, atque procedit, ut supra innuimus; natura quippe remissiore nisu reluctatur, ubi minorem patitur violentiam, ac proinde sensim attenuatur conatus, quatenus particularum violenta compressio atque contentio diminuitur. Hæc quæ de elasmate prorsus recto explicata sunt, etiam de incurvo intelliguntur; cujusmodi esset lamina chalybea R T inflexa in S, cujus manens & immota extremitas esset R: dum enim pars S T propellitur versùs R, particulæ, potissimùm quæ in S, comprimuntur atque intenduntur. Quod si pars R S paulo longior fuerit, contingere potest, ut facilius sit illam inflecti saltem leviter, quàm particulas in S ulteriùs comprimi, aut intendi. Quare particulæ ipsius R S sese restituentes impellunt S, particulæ autem ipsius S T impellunt T. Semper autem Potentiam minùs moveri, quàm corpus, quod impellitur, constat, quemadmodum ratio vectis tertij generis exigit. Neque his, quæ dicta sunt, adversantur percussiones, quæ in extremitate longioris elasmatis per vim inflexi, statimque dimissi, validiores fiunt, quam in partibus mediis; sicut ipse te docere potes, si longiusculi virgulti inflexi atque dimissi primùm parti mediæ deinde extremitati manum in eodem plano verticali constitutam opponas, quam percutiat, magis enim ex secundâ quàm ex primâ percussione dolebis. Quia scilicet non impetus solùm primo productus, sed & velocitas percutientis cum impetu acquisito ex motu ante percussionem (ut suo loco dicetur) attenditur, ut validior sit ictus: majorem autem esse partis extremæ quàm mediarum velocitatem constat, quamvis initio illæ impetu eodem, aut æquali moveantur. Quando verò elasmatis vires prope partem manentem majores esse, quam procul ab illâ, dictum est, non est habita ratio percussionis, quæ prævium percutientis motum requirit, sed tractionis aut impulsionis, quæ nullum trahentis aut impellentis prævium I i i 3

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Book Four. Chapter X. 437 Although, however, the force of the elasma for pushing or drawing the body depends on such a distance, and, when two positions, E and B, are compared with each other, it acts more strongly in E than in B, nevertheless the motion is begun and proceeds not by the same force, whatever that motion may be, whether greater or less, as we have noted above; for nature resists with a gentler effort where it suffers less violence, and therefore the endeavor is gradually weakened, insofar as the violent compression and tension of the particles diminishes. These things, which have been explained concerning the perfectly straight elasma, are also understood of the curved kind; such as would be the steel plate R T bent at S, whose remaining and unmoved end would be R: for while the part S T is driven toward R, the particles, especially those at S, are compressed and stretched. But if the part R S should be a little longer, it may happen that it is easier to bend it at least slightly than to compress or stretch the particles in S any further. Wherefore the particles of R S itself, restoring themselves, drive S, but the particles of S T drive T. And it is always evident that the Power is moved less than the body which is driven, as the rule of the third kind of lever requires. Nor are the impacts, which in the extremity of a longer elasma, bent by force and immediately released, become stronger than in the middle parts, contrary to what has been said; as you may convince yourself, if you first oppose your hand, placed in the same vertical plane, to the middle part and then to the extremity of a somewhat long bent and released twig, which it strikes, for you will be hurt more by the second blow than by the first. Because not only the impulse first produced, but also the velocity of the striking body together with the impulse acquired from the motion before the impact (as will be said in its place), is taken into account, so that the blow is stronger: and it is clear that the velocity of the extremity is greater than that of the middle parts, although at the beginning they are moved with the same, or equal, impulse. But when it was said that the forces of the elasma are greater near the remaining part than farther from it, no account was taken of percussion, which requires the prior motion of the striker, but of traction or propulsion, which requires no prior motion of the one drawing or pushing I i i 3

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Mechanicorum 438 motum exigunt, eóque faciliores accidunt, quò tardiores sunt; minùs enim resistit corpus, quod tardè movetur, ac proinde validiùs trahitur aut impellitur, quo minorem potentia invenit resistentiam: Contrà quàm accidat in percussione, quæ validiorem facit ictum, quo majorem invenit resistentiam; hæc autem major est, quò velociùs moveri deberet corpus percussum, ut percutientis motui obsecundaret, cui magis resistens majorem ictum recipit; cum tamen hic languidior esset atque infirmior, si manum sensim subduceres virgulto percutienti. Quare pars clasmatis extrema validiùs percutit, quia majorem invenit resistentiam, pars media validiùs trahit aut impellit, quia minùs illi resistitur. CAPUT XI. Cur longiora corpora faciliùs flectantur, difficiliùs sustineantur. PRæsens disputatio non distat ab iis, quæ ab Aristotele inquiruntur in Mechanicis quæst. 14. Cur ejusdem magnitudinis lignum faciliùs genu frangitur, si quispiam æquè diductis manibus extrema comprehendens fregerit, quàm si juxta genu: & si terræ illud applicans pede superimposito manu longè diductâ confragerit, quàm propè. & quæst. 16. Cur quanto longiora sunt ligna, tanto imbecilliora fiunt: & si tollantur, inflectuntur magis; tametsi quod breve quidem est, ceu bicubitum, fuerit tenue; quod verò cubicorum centum, crassum? & quæst. 26. Cur difficilius est longa ligna ab extremo super humeros ferre, quàm secundum medium, æquali existente pondere? Et quidem quod ad primum, scilicet ad flexionem spectat, quam demum consequitur fractio, quemadmodum flectendi corpus aliquod longum aut frangendi difficultas oritur ex complexione atque copulatione particularum, quibus constat, ægrè dissolubili, ita illud inflectitur, atque frangitur, cum earundem particularum coagmentatio vehementi impulsioni labefactatur,

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Mechanics 438 produce motion, and the slower they are, the more easily they occur; for a body that moves slowly offers less resistance, and therefore is drawn or impelled more strongly, since the force finds less resistance. The contrary is the case in percussion, which makes the blow stronger the greater resistance it finds; and this resistance is greater the more quickly the body struck ought to move in order to yield to the motion of the striker, to whose motion, the more it resists, the greater a blow it receives; whereas the blow would be weaker and feebler if you were to withdraw your hand gradually from the twig that strikes it. Wherefore the outer part of a splinter strikes more strongly, because it encounters greater resistance; the middle part draws or impels more strongly, because it is resisted less. CAPUT XI. Why longer bodies are bent more easily, and supported with more difficulty. This discussion is not far from those questions which Aristotle investigates in the Mechanics, question 14: Why a piece of wood of the same size is more easily broken at the knee if someone, holding the ends with hands spread equally apart, breaks it than if he does so near the knee; and if, placing it on the ground and setting his foot upon it, he breaks it with his hands spread far apart rather than close by. And question 16: Why the longer pieces of wood are, the weaker they become; and if lifted, they bend more; although what is short, as two cubits long, may be slender, while what is a hundred cubits long may be thick? And question 26: Why is it more difficult to carry long pieces of wood on the shoulders from one end than by the middle, even when the weight is the same? And indeed, as regards the first, namely what concerns bending, which is followed at last by breaking, just as the difficulty of bending some long body or breaking it arises from the constitution and conjunction of the particles of which it consists, which are not easily separable, so it is bent and broken when the cohesion of those same particles is weakened by a violent impulse,

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Liber quartus. CAPUT XI. 439 factatur, his quidem per vim compressis, his verò validè in di- versa distractis. Quo igitur faciliùs compressio hæc atque di- tractio perficitur, eò etiam faciliùs flectitur corpus, aut fran- gitur. Hanc autem particularum compressionem atque di- tractionem faciliùs contingere longiori corpori quàm breviori manifestum est; quia videlicet motus ille particularum ad fle- xionem aut fractionem necessarius minorem Rationem habet ad motum Potentiæ longiùs applicatæ, quàm ad motum Po- tentiæ propioris. Potentia movens bifariam considerari potest, sivè in ipso cor- pore inclusa, cujusmodi est illi insita atque ingenita gravitas, vi cujus sponte suâ flectitur; sivè extrinsecùs adhibita, ut si onus aliquod grave deorsum premens adjiciatur, aut potentia vivens ad motum in quamcumque positionis differentiam apta: utrobique tamen est eadem ratio, ubi scilicet assumpta atque adventitia potentia applicatur, ibi operatur; atque ibi innata gravitas intelligitur sua exercere momenta, ubi partis ultra sub- jectum fulcrum extantis centrum gravitatis reperitur; illâque est à fulcro distantia potentiæ flectentis, aut etiam frangentis. Sit enim prisma AB, cujus pars AC infixa sit parieti, extra quem emineat horizonti parallela pars CB suâ gravi- tate deorsum connitens; quæ sanè non est intelligenda in B, sed quasi tota constituta esset in D, ubi est centrum gravitatis non totius cor- poris AB, sed partis extantis CB. Quod si brevius esset pris- ma AE, partis CE minor esset gravitas, quàm partis CB, & prætereà minus abesse à fulcro C intelligeretur, quippe cujus centrum gravitatis esset F multo propius quàm D. Plura igi- tur momenta habet CB quàm CE ad flectendum prisma parie- ti infixum, juxta ea, quæ uberiùs dicta sunt lib.2. cap.6. ubi so- lidorum Resistentiam respectivam consideravimus, nec vacat hîc iterum inculcare. Unum hîc considerandum est, quod ad rationes vectis atti- net, videlicet, si superiori prismatis parti, quæ respondet ipsi AC, incumberet onus, quod faciliùs loco moveri possit, quàm particularum complexio labefactari, aut omnino dissolvi, pris- ma neque frangi, aut fortasse ne flecti quidem contingeret; sed

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Liber quartus. CHAPTER XI. 439 is effected, these indeed being compressed by force, and those truly drawn apart with great violence in opposite directions. Therefore, the more easily this compression and separation are accomplished, the more easily also is the body bent or broken. Now it is manifest that this compression and separation of the particles more easily occurs in a longer body than in a shorter one; because, namely, that motion of the particles necessary for bending or breaking has a smaller proportion to the motion of the Power applied farther away than to the motion of the Power applied closer at hand. A moving Power may be considered in two ways: either as included in the body itself, such as its inherent and inborn gravity, by virtue of which it bends of its own accord; or as applied from without, as if some heavy load pressing downward were added, or a living power adapted to motion in whatever difference of position: yet in either case the reason is the same, namely, where the assumed and adventitious power is applied, there it acts; and there innate gravity is understood to exert its force, where the center of gravity of the part projecting beyond the supporting fulcrum is found; and the distance from the fulcrum is that of the bending or even breaking power. Let AB be a prism, whose part AC be fixed in a wall, beyond which the part CB projects, parallel to the horizon, striving downward by its own weight; and this surely is not to be understood at B, but as if the whole were placed at D, where is the center of gravity not of the whole body AB, but of the projecting part CB. But if the prism AE were shorter, the weight of part CE would be less than that of part CB, and moreover it would be understood to be farther removed less from the fulcrum C, since its center of gravity would be F, much nearer than D. Therefore CB has more moments than CE for bending a prism fixed in a wall, according to what has been more fully said in book 2, chapter 6, where we considered the relative Resistance of solids, and it is not necessary to insist on that again here. One thing here must be considered, which pertains to the reasons of the lever, namely, if upon the upper part of the prism, which corresponds to AC itself, there were laid a weight which might be more easily moved from its place than the cohesion of the particles could be weakened, or altogether dissolved, the prism would neither break, nor perhaps even bend;

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Mechanicorum sed rationem vectis primi generis haberet, cujus fulcrum esset in C, pondus in A, potentia in D. At si onus impositum nul- latenus dimoveri queat, quemadmodum cum prisma parieti in- figitur, si CB ejus sit longitudinis, ut vis gravitatis ad descen- dendum tali intervallo CD sejuncta à fulcro C plus habeat momenti, quàm particularum coagmentatio, ne compriman- tur, aut distrahantur; tunc vectis est secundi generis, fulcrum quidem habens in C, quatenus totum segmentum A C retine- tur prorsus immotum, & potentia in D, pondus verò, cujus vires vincuntur, eo loco, ubi maxima sit particularum com- pressio atque distractio. Hinc factum videtur satis Aristoteli quærenti, cur faciliùs flectatur lignum crassum cubitorum centum, quàm tenue bi- cubitum; quia nimirum in crasso ligno cubitorum centum è pariete extantium, si ponatur similem atque æquabilem crassi- tiem juxta totam longitudinem habere, gravitatis centrum distat à fulcro cubitis quinquaginta, tenue verò atque exile lignum similis figuræ atque materiæ centrum habet uno tantùm cubito distans à fulcro, & gravitas illius ad hujus gravitatem in eâ est Ratione, quam habent inter se ipsorum lignorum moles, quæ scilicet ex Rationibus basium atque longitudinum compo- nitur. Cum itaque longitudo ad longitudinem sit ut 50 ad 1, si basium similium latera homologa sint ut 10 ad 1, basium Ratio est ut 100 ad 1: quare cùm longioris ligni gravtia- tis Ratio ad gravitatem brevioris componatur ex Ratione ba- sium ut 100 ad 1, & ex ratione longitudinum ut 50 ad 1, gravi- tas longioris ad gravitatem brevioris est ut 5000 ad 1. Atqui momenta ad descendendum componuntur ex gravitate & di- stantia à fulcro; igitur momenta longioris ad momenta brevio- ris sunt ut 250000 ad 1. At verò resistentia absoluta, ne flectan- tur, aut frangantur hujusmodi ligna, est in Ratione compositâ ex Rationibus basium atque crassitierum; ac proinde si bases sint similes, & similiter positæ, Ratio est triplicata Rationis la- terum homologorum, hoc est Rationis 10 ad 1; atque adeò re- sistentia longioris ad resistentiam brevioris est ut 1000 ad 1. Patet igitur momenta ut 250.000 ad resistentiam ut 1000 ma- jorem habere Rationem, quàm momenta ut 1 ad resistentiam ut 1: faciliùs ergo illa quàm hæc momenta resistentiam sibi con- gruentem

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Mechanics, but let it have the ratio of a lever of the first kind, whose fulcrum is at C, the weight at A, and the power at D. But if the load placed there can by no means be moved, as when a prism is fixed into a wall, if CB is of such length that the force of gravity, to descend through such a distance CD separated from the fulcrum C, has more moment than the cohesion of the particles, lest they be compressed or stretched; then it is a lever of the second kind, having indeed the fulcrum at C, inasmuch as the whole segment AC remains completely motionless, and the power at D, while the weight, whose forces are overcome, is in that place where the compression and stretching of the particles are greatest. Hence it seems sufficient answer to Aristotle, who asks why a thick piece of wood a hundred cubits long bends more easily than a thin piece two cubits long; because, namely, in a thick piece of wood a hundred cubits long projecting from a wall, if it be supposed to have a similar and uniform thickness throughout its whole length, the center of gravity is distant from the fulcrum by fifty cubits, whereas a thin and slender piece of wood of similar figure and material has its center only one cubit distant from the fulcrum, and the gravity of the former is to the gravity of the latter in the ratio that their masses bear to one another, which is composed from the ratios of the bases and the lengths. Since therefore length is to length as 50 to 1, if the homologous sides of similar bases are as 10 to 1, the ratio of the bases is as 100 to 1: wherefore, since the ratio of the gravity of the longer piece of wood to the gravity of the shorter is composed from the ratio of the bases as 100 to 1, and from the ratio of the lengths as 50 to 1, the gravity of the longer to the gravity of the shorter is as 5000 to 1. But the moments tending downward are composed of gravity and distance from the fulcrum; therefore the moments of the longer to those of the shorter are as 250000 to 1. But the absolute resistance, lest such pieces of wood bend or break, is in a compounded ratio from the ratios of the bases and thicknesses; and therefore if the bases be similar and similarly placed, the ratio is the triplicate of the ratio of the homologous sides, that is, of the ratio 10 to 1; and thus the resistance of the longer to the resistance of the shorter is as 1000 to 1. It is therefore clear that moments as 250,000 to resistance as 1000 have a greater ratio than moments as 1 to resistance as 1: therefore the former moments more easily than the latter have a resistance corresponding to them.

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Liber quartus. CAPUT XI. 44. gruentem superant, atque faciliùs lignum crassum longius flectitur, aut frangitur, quàm brevius. Quæ autem de ligno parieti secundum alteram extremitatem infixo dicta sunt, servatâ analogiâ de eodem dicantur, si circa medium fulcro alicui insistat ita, ut hinc atque hinc habeat gravitatis momenta composita ex ipsarum partium gravitate & ex distantia centrorum gravitatis à fulcro, cui innititur: eâdem enim ratiocinatione colligitur in longiore ligno majorem esse Rationem momentorum gravitatis ad resistentiam ortam ex partium complexione, ne flectatur, quàm in breviore. Quod verò spectat ad longioris ligni faciliorem flexionem, quando utraque extremitas innixa est subjecto fulcro, non videtur pro- priè hujus loci, sed de eâ dictum est superiùs lib.3. cap.12. Ex his, quæ de prismatica extra parietem extante, quod faci- liùs flectitur, hactenus diximus, ulteriùs patet, cur ex contra- rio longius lignum ut A B, etiamsi parem cum breviore A E crassitiem habeat, alterâ extremitate æqualiter in A C ap- prehensum difficiliùs sustineatur. Nam quod longius est ad il- lud, quod brevius est, secundùm gravitatem, quæ deorsum ni- titur, eam habet Rationem, quæ est longitudinis majoris C B ad longitudinem minorem C E: & præterea momenta, quæ ex distantia oriuntur, sunt ut C D ad C F, hoc est ut C B ad C E, si quidem ex hypothesi centrum gravitatis intelligatur in me- diâ longitudine; secus autem, universaliter juxta distantias cen- tri gravitatis à fulcro. Quare tota momentorum Ratio ea est quæ componitur ex Rationibus gravitatum respondentium moli ultrà fulcrum protensæ, & distantiarum centri gravitatis. Cum itaque in longiore ligno plus inveniatur gravitatis, & ma- gis à fulcro distet centrum gravitatis, quàm in breviore ligno, nil mirum, si vis in A posita, ut contranitatur momentis lon- gioris ligni innixi fulcro C, major esse debeat, quàm ut resiste- ret momentis ligni brevioris. Desinant igitur mirari, qui sarissam decem cubitorum per- pendicularem extremo digiti apice sustineri, eandem verò ho- izontaliter jacentem non nisi valido conatu elevari vident. Res enim ex dictis perspicua est; quia dum hasta perpendicu- laris digito incumbit, centrum gravitatis rectâ deorsum urgens digito motum sibi æqualem præscribit, ac proinde vicissim K k k

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Book Four. CHAPTER XI. 44. surpass what is fitting, and thicker wood bends more easily, or breaks, than shorter wood. What has been said, however, of wood fixed into a wall at one end, let it be said of the same thing, with the analogy preserved, if it rests with its middle on some support, so that on this side and on that it has moments of weight composed from the weight of the parts themselves and from the distance of the centers of gravity from the support on which it rests: for by the same reasoning it is gathered that in longer wood there is a greater ratio of the moments of weight to the resistance arising from the combination of the parts, so that it may not bend, than in shorter wood. But as for the easier bending of a longer piece of wood, when both ends are supported on a support beneath it, this does not seem properly to belong to this place, but it was spoken of above in lib. 3, cap. 12. From these things, which up to now we have said about a prismatic beam projecting beyond a wall, which bends more easily, it is further clear why, conversely, a longer beam such as A B, even though it has the same thickness as the shorter A E, is more difficult to support when it is grasped at one end in A C. For that which is longer, compared with that which is shorter, according to the weight pressing downward, has the ratio which the greater length C B bears to the lesser length C E; and besides, the moments arising from distance are as C D to C F, that is, as C B to C E, if indeed by hypothesis the center of gravity is understood to be in the middle length; otherwise, universally, according to the distances of the center of gravity from the support. Wherefore the whole ratio of the moments is that which is composed from the ratios of the corresponding weights of the mass extending beyond the support and of the distances of the center of gravity. Since, therefore, in a longer beam more weight is found, and the center of gravity is farther from the support than in a shorter beam, it is no wonder if the force applied at A, so as to counteract the moments of the longer beam resting on support C, must be greater than that which would resist the moments of the shorter beam. Let those therefore cease to wonder who see a sarissa ten cubits long held upright by the tip of a finger, but lying horizontally lifted only with a strong effort. For the matter is clear from what has been said; because while the spear rests upon the finger vertically, the center of gravity, pressing straight downward, prescribes to the finger a motion equal to itself, and therefore in turn K k k

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Mechanicorum æqualis est digiti & hastæ motus sursum, si digitus sursum conetur: hinc est solam Rationem gravitatis comparatæ ad vires sustinendi attendendam esse, ideóque si sarissæ pondus sit ex. gr. lib. 10, solo nisu opus est, quo libræ 10 sustineantur. Cum verò hasta obliqua est, & horizonti parallela, sive ad illum inclinata, jam non idem seu æqualis convenit motus manui hastam elevanti, atque centro gravitatis, sed hoc ad motum multo majorem incitatur; ac propterea momentorum Ratio non ex solâ gravitate pendet, verùm etiam ex motuum Ratio- ne componitur. Sit hasta horizontaliter jacens A I cubitorum 10; pars manu apprehensa sit I C quinta fermè pars cubiti adeò ut I C ad C A sit ut 1 ad 49: punctum I respondet extremæ parti metacarpij, quâ carpo adhæret articulatio minimi digiti: punctum autem C respondet secundo indicis articulo; motúsfque elevationis hastæ perficitur deprimendo I & elevando C, ac motûs centrum est in juncturâ manûs cum osse cubiti; quod centrum propterea intelligitur respondere sarissæ ex. gr. in O inter C & I. Quapropter si facultas in I deprimens considere- tur, vectis est primi generis, sin autem vis in C elevans attendatur, vectis est terti generis; pondus verò movendum est si- ve tota gravitas longitudinis O A in centro gravitatis E, sive semissis gravitatis in extremitate A, ut constat ex iis, quæ disputata sunt lib. 3. cap. 2. de brachiis libræ. Intelligatur itaque, facilioris explicationis gratiâ, centrum motûs in O planè medium inter C & I; eritque tam A O ad O C, quàm A O ad O I, ut 99 ad 1. Gravitas igitur partis O A est lib. 9 2/10 ex hypothesi; illius semissis est lib. 4 10/10; cujus momentum in A ad momentum, quod haberet illa eadem in C aut in I, est ut 99 ad 1. Cùm autem potentia in I deprimens æquivalent potentiæ elevanti in C, quippe illarum distantia ab O centro motûs ex hypothesi est æqualis, perinde est atque si in C unica potentia totum pondus elevans posita esset æquivalens duplici illi potentiæ in I & in C. Quare potentia in C elevans pondus

Transcription: Translated (English)

In mechanics, the upward motion of the finger and the staff is equal, if the finger attempts to move upward; hence it follows that only the ratio of the gravity compared with the forces of support is to be considered, and therefore, if the weight of the sarissa be, for example, 10 lb., only the effort by which 10 lb. are sustained is required. But when the staff is oblique, and parallel to the horizon, or inclined to it, the same or equal motion no longer belongs to the hand raising the staff and to the center of gravity, but the latter is urged to a much greater motion; and therefore the ratio of moments depends not on gravity alone, but is also composed from the ratio of motions. Let a staff lying horizontally, A I, be 10 cubits long; let the part grasped by the hand, I C, be almost the fifth part of a cubit, so that I C is to C A as 1 to 49: point I corresponds to the extreme part of the metacarpus, where the wrist-joint adheres to the little finger; but point C corresponds to the second joint of the index finger; and the motion of raising the staff is effected by depressing I and raising C, and the center of motion is at the junction of the hand with the bone of the elbow; this center is therefore understood to correspond to the sarissa, for example in O between C and I. Wherefore, if the force depressing in I be considered, it is a lever of the first kind; but if the force raising in C be regarded, it is a lever of the third kind; but the weight to be moved is either the whole gravity of the length O A, in the center of gravity E, or half the gravity at the extremity A, as is clear from what was discussed in book 3, chapter 2, on the arms of the balance. Let it therefore be understood, for the sake of easier explanation, that the center of motion at O lies exactly midway between C and I; and then both A O to O C, and A O to O I, will be as 99 to 1. The gravity therefore of the part O A is 9 2/10 lb. by hypothesis; its half is 4 10/10 lb.; the moment of which at A, compared with the moment which that same weight would have in C or in I, is as 99 to 1. But since the power depressing in I is equivalent to the power raising in C, because by hypothesis their distance from the center O of motion is equal, it is just as though in C a single power raising the whole weight were placed, equivalent to that double power in I and in C. Therefore the power raising the weight in C

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Liber quartus. CAPUT XI. 443 pondus perpendiculare lib. 9 2/10 ad potentiam in C pariter consti- tutam elevantem lib. 4 12/20 in distantia, quæ exigat motum unde- centuplum erit ut 9 2/10 ad 490, hoc est, tàm valida esse debet, ut posset perpendiculariter elevare libras 490. Porrò elevatâ hastâ ita ut A veniat in F, jam non intelligi- tur semissis gravitatis in A, sed in G puncto, quod definitur à perpendiculari cadente ex F in horizontalem: & idcirco gra- vitas 4 19/20 ducenda est in distantiam GO minorem quàm A O; atque ita deinceps minuitur, usque dum hasta fiat in O hori- zonti perpendicularis, & facillimè sustineatur, aut attollatur. Si autem in hac ratiocinatione tibi, Lector, placuerit non negli- gere momentum illud exiguum, quod potentiæ elevanti addi- tur à gravitate particulæ OI, non abnuo, si operæ pretium te facturum existimes. Quòd si punctum I concipiatur omnino immotum, illud est centrum motûs, & vis elevans in C aliam habet Rationem; nam potentiæ motus ad motum semissis ponderis hastæ in A est ut 1 ad 50; sunt igitur lib. 5 ex hypothesi, quæ moventur motu quinquagecuplo; ac propterea vis elevandidatam hastam posi- ta in C, quando hasta est horizonti parallela, ea esse debet, quæ possit elevare libras 250 perpendiculares. Hinc est quod, si hastam eandem lib. 10. humero ita imponas in C, ut apprehen- sum calcem in I manus retineat, & C I sit pars decima totius longitudinis hastæ parallelæ horizonti, semissis (scilicet lib. 4 1/2) reliquæ hastæ ultra humerum intelligitur in A, & ut IC ad CA, hoc est ut 1 ad 9, ita lib. 4 1/2 ad lib. 40 1/2, quibus æquiva- lere debet partis C I momentum & vis manûs deorsum urgen- tis, atque in I retinentis hastam horizonti parallelam. Perinde itaque humerus in C premitur ab hastâ sic positâ, & à manu deorsum urgente, atque si ponderis librarum 81 centrum gra- vitatis immineret humero; nam si loco manûs deorsum trahen- tis adderes in I pondus faciens æquilibrium, esse oporteret lib. 40; siquidem partis C I momentum est lib. 1/2 in I. At si ex- tremitas I retineatur quidem, sed nemine deorsum urgente (quemadmodum si in parietis foramen inferatur, & à superio- re foraminis saxo impediatur, ne possit elevari) in C verò susti- neatur ab humero; tunc humeri pressio soli gravitati hastæ tri- K k k 2

Transcription: Translated (English)

Liber quartus. CHAPTER XI. 443 a perpendicular weight of 9 2/10 lb. to a power placed equally in C, lifting 4 12/20 lb. at a distance which requires a motion eleven hundred times greater, will be as 9 2/10 to 490; that is, it must be so strong as to be able to raise 490 pounds perpendicularly. Moreover, when the rod is raised so that A comes to F, the half of the weight is now not understood to be in A, but in the point G, which is defined by the perpendicular falling from F to the horizontal; and therefore the weight of 4 19/20 is to be multiplied by the distance GO, less than AO; and so on thereafter it diminishes, until the rod becomes perpendicular to the horizon in O, and is very easily sustained or lifted. But if in this reasoning you, Reader, should wish not to neglect that small moment which is added to the lifting power by the weight of the particle OI, I do not object, if you think that it would be worth the trouble. But if point I be conceived to be wholly immovable, that is the center of motion, and the lifting force in C has another relation; for the motion of the power to the motion of the half-weight of the rod at A is as 1 to 50; therefore there are, by hypothesis, 5 lb. which are moved with a fiftyfold motion; and therefore the force lifting the rod placed in C, when the rod is parallel to the horizon, must be such as can raise 250 pounds perpendicular. Hence it is that, if you place the same 10-lb. rod upon the shoulder in C in such a way that the foot grasped in I is held by the hand, and if C I is the tenth part of the whole length of the rod parallel to the horizon, the half, namely 4 1/2 lb., of the remaining rod beyond the shoulder is understood to be in A; and as IC is to CA, that is as 1 to 9, so are 4 1/2 lb. to 40 1/2 lb., to which the moment of the part C I and the force of the hand pressing downward and holding the rod parallel to the horizon in I ought to be equivalent. Thus the shoulder in C is pressed by a rod thus placed, and by the hand pressing downward, just as if the center of gravity of 81 pounds were hanging over the shoulder; for if in place of the hand pulling downward you added in I a weight making equilibrium, it ought to be 40 lb.; since indeed the moment of the part C I is 1/2 lb. in I. But if the extremity I be indeed held, yet without anyone pressing downward (as if it were inserted into a hole in a wall, and prevented by a stone from the upper side of the hole so that it cannot be lifted), while in C it is sustained by the shoulder; then the pressure on the shoulder is due only to the weight of the rod tri- K k k 2

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Mechanicorum buenda est; hasta quippe est vectis secundi generis hypomo- chlium in I habens, pondus movendum, hoc est, humerum premendum in C, potentiam verò, hoc est lib.5. semissem gra- vitatis hastæ, in A, ita ut A I distantia sit decupla distantiæ C I: premitur ergo humerus, quasi sustineat libras 50. Demum, ne intacta relinquatur Aristotelis quæstio 14. de ligno, quod terræ applicatum pede imposito faciliùs frangitur manu longè diducta quàm prope, dic ligni partem, quæ inter pedem impositum, & terram subjectam interjicitur, esse pror- sus similem parti prismatis infixi parieti, ne moveatur, manum verò esse potentiam, quæ longiùs applicata majora habet mo- menta ad vincendum nexum particularum ligni; est enim lon- gior vectis. Similiter applicato ad genu ligno, & æquè di- ductis mauibus; duo sunt vectes hinc atque hinc, fulcrum ad genu, scilicet ad duo puncta contactuum, habentes, eoque longiores, quò magis diductæ fuerint manus, ac proinde faci- liùs distrahentes particulas extimas ligni, quod circa genu curvatur, faciliùsqque comprimentes particulas ejusdem ligni ad cavam faciem pertinentes; quæ dum sibi vicissim obsistunt, uberiorem reliquarum distractionem juvant: longiorem autem vectem præ breviori eligendum esse quis nesciat? ac propterea si ad genu propiùs admoverentur manus ligno, cùm minor es- set illarum motûs Ratio ad motum particularum ligni distrahen- darum, quàm sit Ratio motûs illarum longiùs diductarum, uti- que difficiliùs frangeretur lignum; ideóque longiùs diducun- tur manus, ut longiores sint vectes. CAPUT XII. Unde oriantur forcipum & forficum vires. Forcipum duplex est usus; primus quidem ad corpus ali- quod firmiter apprehendendum, secundus verò ad evel- lendum illud faciliùs, vel suâ è sede dimovendum; id quod adhibito hujusmodi instrumento faciliùs perficitur, quàm nudâ manu.

Transcription: Translated (English)

Mechanics ought to be understood in this way: indeed, the spear is a lever of the second kind, a hypomochlion having in I the thing to be moved, that is, the shoulder to be pressed in C, and the power, namely 5 1/2 pounds of the weight of the spear, in A, so that the distance AI is ten times the distance CI: thus the shoulder is pressed as if it were bearing 50 pounds. Finally, lest Aristotle’s question 14 about wood be left untouched—that when wood applied to the ground is more easily broken with the foot placed upon it than with the hand held far apart—say that the part of the wood which lies between the foot placed upon it and the ground beneath is altogether like the part of a prism fixed into a wall so that it may not move, whereas the hand is the power which, when applied farther away, has greater moments for overcoming the cohesion of the particles of the wood; for it is a longer lever. Similarly, when wood is applied to the knee and the hands are equally separated, there are two levers on this side and that, having the knee as the fulcrum, namely the two points of contact; and they are the longer, the farther the hands are drawn apart, and therefore they more easily pull apart the outer particles of the wood, which bends around the knee, and more easily compress the particles of the same wood belonging to the concave surface; and while these resist one another in turn, they help the more abundant separation of the rest. But who does not know that a longer lever should be preferred to a shorter one? And therefore, if the hands were brought closer to the wood at the knee, since the ratio of their motion to the motion of the particles of the wood to be separated would be smaller than the ratio of the motion of those more widely separated, the wood would certainly be broken with greater difficulty; and therefore the hands are drawn farther apart so that the levers may be longer. CHAPTER XII. Whence the powers of pincers and scissors arise. Pincers have a twofold use: first, indeed, for firmly grasping some body; second, for plucking it out more easily, or moving it from its place; and this is accomplished more easily with an instrument of this kind than with the bare hand.

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Liber quartus. CAPUT XII. 445 manu. Hinc Aristoteles mechan. quæst. 21. quærit, Cur medi- ci faciliùs dentes extrahunt denti forcipis onere adjecto, quàm si so- lâ utantur manu? Quia nimirum infixum mandibulæ dentem extrahendum vix summis duobus digitis, quibus non multa vis inest, arripere valent, & ob carnis mollitudinem facilè è tra- hentibus digitis elabitur lubricus dens: at forcipulam in os im- mittere potiùs, quàm digitos, sæpe facilius est, validiùsqque trahit manus in pugnum constricta forcipi dentem per vim educenti applicata, quàm digitorum extremitates dentem adhuc in gingivâ hærentem evellere valeant. Præterquam quod in dentiforcipe, cujuscumque tandem figuræ sit, ratio vectis intercedit ad dentem firmiùs apprehendendum, dum presso manubrio arctiùs constringitur: nec facilè Chirurgus operam ludit, ubi dens forcipem subterfugere nullatenus po- test. Totam igitur vectis vim in dentiforcipe agnosco ad strin- gendum dentem, ut medica manus illum faciliùs evellat: ne- que enim eâdem ratione à medicis (nisi fortè veterinariis) ex- trahuntur dentes, quâ fabri lignarij revellunt infixos tabulæ clavos, de quibus mox erit sermo. Similiter quia ad stringendam exilem aliquam materiam inepta esset digitorum crassitudo, minutorum opusculorum fa- bricatores forcipulis utuntur, quibus illam apprehendentes fir- miter, aut limæ subjiciunt, aut opportunè collocant. Et quia candens ferrum manu tractari nequit, ut in quamcumque par- tem versetur, incudique impositum nisi retineretur, sæpè se- cundis aut tertiis malleorum ictibus se subduceret, propterea fabri ferrarij forcipes adhibent, quarum author & inventor Cinyra Cyprius Agriopæ filius scribitur à Plinio lib. 7 cap. 56; ideóque forcipes, quasi forvicapes, dictæ sunt, quòd iis for- va; id est calida, capiantur. Vis autem forcipum in eo sita est, quòd duo vectes primi generis AB, & CD in E connexi commune hypomochlium E habent; potentia verò in B & D longiorum brachiorum extremitates adducens eò validiùs stringit ferrum brevioribus brachiis EA & EC ap- prehensum, quo major fuerit Ratio BE ad EA: tamque firma retentio esse potest, ut modico pueri conatu extremitates B & D K k k 3

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Book Four. CHAPTER XII. 445 by hand. Hence Aristotle in the Mechanical Questions, 21, asks, Why do doctors extract teeth more easily with the addition of a dental forceps than if they use only the hand? Because, namely, a tooth fixed in the jaw can scarcely be seized for extraction even by the tips of the two fingers, in which there is not much force, and because of the softness of the flesh the slippery tooth easily slips from the fingers that are pulling it; but to insert a forceps into the mouth is often easier than using the fingers, and the hand, clenched into a fist, pulls more strongly when a forceps applied to it is driving the tooth out by force than the fingertips can manage to pluck out the tooth still clinging to the gum. Besides, in a dental forceps, whatever its shape may be, the principle of the lever comes into play for the firmer grasping of the tooth, as the handle is pressed and it is drawn more tightly together: nor does the surgeon labor in vain when the tooth can in no way escape the forceps. I therefore recognize the whole power of the lever in the dental forceps for tightening the tooth, so that the medical hand may more easily pull it out; for doctors do not extract teeth by the same method as woodworkers tear out nails fixed in a board (unless perhaps veterinarians do), about which more will soon be said. Similarly, because the thickness of the fingers would be unsuitable for gripping some slender material, craftsmen of small works use forceps, with which, after firmly seizing it, they either apply it to the file or place it in the proper position. And because glowing iron cannot be handled by hand, and in order that, whatever way it is turned, if placed on the anvil and not held, it would often slip away under the second or third blows of the hammers, for that reason ironworkers use tongs, whose author and inventor is said by Pliny, book 7, chapter 56, to be Cinyra the Cyprian, son of Agriopas; and therefore tongs are called, as it were, forvicapes, because by them fora, that is, hot things, are taken. Now the force of tongs lies in this, that two levers of the first kind, AB and CD, connected at E, have a common fulcrum E; but the power drawing the ends at B and D of the longer arms thus more strongly grips the iron held by the shorter arms EA and EC, the greater the ratio of BE to EA may be: and the grip can be so firm that by a slight effort of a boy the ends B and D...

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446 Mechanicorum vicissim comprimentis, robustissimi cujusque vires elidantur, ne arreptum ferrum ex AC possit eximere. Quòd si forcipibus BA & DC utatur aliquis veterinarius vice Postomidis (seu, ut aliquibus Grammaticis placet Pastomidis) equi nares, ad frænandam ejus tenaciam, ut loquitur Festus, inter longiora brachia BE & DE contingens; jam BE & DE vectes sunt secundi generis, cum illud, quod ponderis vicem subit, inter hypomochlium & potentiam interjiciatur. Hujusmodi forcipibus vectis in EC & EA non absimile fuisse existimo instrumentum antiquioribus Græcis ad frangendas absque ictu percutientis mallei nuces familiare, ut ex Aristotele Mechan. quæst. 22. colligitur: quod fortasse vel in alterutrâ, vel in utraque interiori facie breviorum brachiorum modicè excavatâ frangendæ nuci locum designabat; adducto enim in oppositas partes utroque vecte BA & DC, quo propior erat nux communi hypomochlio, puncto scilicet connexionis E, eò faciliùs frangebatur, quia eò major erat Ratio motûs potentiæ ad motum particularum nucis ex compressione dividendarum, quàm esset Ratio resistentiæ ex earumdem particularum complexione ortæ, ad vim motivam potentiæ. Et quoniam in nucum mentionem incidi, ne levitati mihi tribuas, quòd hîc puerile inventum à me puero, & tunc quidem admiratione obstufacto, observatum commemorare non erubescam. Videbam pueros clandestinis jentaculis indulgentes, ut citra multiplicis percussionis strepitum nuces confringerent, eas inter postium angulos & fores collocare; tum adductis foribus levissimo negotio unâ operâ confringere. Erat scilicet vectis primi generis, cujus majorem longitudinem definiebat foris latitudo, minorem ipsius foris crassitudo, ita ut vectis esset in angulum infexus, cujus hypomochlium cardinibus respondebant. Usque adeò natura ipsa Mechanicen, usumque vectis, vel pueros docet. His adde acutas forcipulas, quibus catenularum fabricatores extremitatem fili ferrei inflectunt: ratio enim vectis potissimùm consistit in validâ & firmâ ipsius fili ferrei apprehensione; nam quo ad ejusdem inflexionem spectat, non est, cur nos torqueamus, ut aliquam demum vectis umbram venemur: satis est, si manubrij amplitudinem considerantes, eâmque cum tenui

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446. By alternately compressing the work of mechanics, the forces of the strongest man are brought to naught, so that the seized iron cannot be drawn out of AC. But if a veterinarian uses the forceps BA and DC in place of Postomides (or, as some grammarians prefer, Pastomides) to seize a horse’s nostrils, in order to curb its obstinacy, as Festus puts it, by touching the longer arms BE and DE; then BE and DE are levers of the second kind, since that which takes the place of the weight is interposed between the fulcrum and the power. I think that an instrument of this kind, like the lever in EC and EA, was not unlike one familiar to the older Greeks for breaking nuts without the blow of a striking hammer, as is gathered from Aristotle, Mechan. quæst. 22: this perhaps indicated, in either one or in both of the inner faces of the shorter arms, a moderately hollowed place for the nut to be broken; for when both levers BA and DC were drawn toward opposite sides, the nearer the nut was to the common fulcrum, that is, to the point of connection E, the more easily it was broken, because the ratio of the motion of the power to the motion of the particles of the nut, to be divided by compression, was greater than would be the ratio of the resistance arising from the cohesion of those same particles, to the motive force of the power. And since I have come to mention nuts, I do not think I should be accused of triviality if I do not blush to recall here a childish device observed by me as a boy, and at that time, indeed, struck with wonder. I saw boys, indulging in clandestine breakfasts, place nuts between the angles of a doorpost and the door, so that they might crack them without the noise of repeated blows; then, by pulling the door in, they would break them all at once with the slightest effort. This was, of course, a lever of the first kind, whose greater length was defined by the breadth of the door, and its lesser by the thickness of the door itself, so that the lever was inserted in the corner, whose fulcrum corresponded to the hinges. So far does nature itself teach mechanics and the use of the lever, even to boys. To these add the sharp forceps with which the makers of chains bend the end of the iron wire: for the reason of the lever chiefly consists in a strong and firm grasp of the iron wire itself; for as regards its bending, there is no need for us to strain ourselves in order at last to hunt some mere shadow of a lever: it is enough if, considering the size of the handles, and comparing it with the slender...

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Liber quartus. CAPUT XII. 447 tenui apice forcipulæ, circa quem filum ferreum contorquetur, comparantes motum potentiæ manubrio applicatæ longè ma- jorem motu particularum fili ferrei, quod flectitur, deprehen- damus; hinc quippe aucta potentiæ momenta cognoscimus. Aliud forcipum genus frequentiùs usurpatur, quarum potis- simus usus est in eximendis clavis, & minora brachia A E & C E non recta sunt, sed curva; non solùm ut clavus tenaciùs apprehendatur ex- cepto ejus capite intra forcipum si- num, verùm etiam ut forcipes aliam exerceant vectis curvi rationem: cùm enim arrepto inter A & C clauo inclinantur forcipes, ut punctum H tangat subjectum planum, sive paries sit, sive tabula, jam hypomochlium est in H, & momenta potentiæ in B ad resistentiam clavi evellen- di, sunt ut B H ad H A, cùm circa punctum H perficiatur motus. Quare ad constringendum clavum momentorum Ra- tio est ut B E ad E A (perinde atque si ab E ad A ducta esset recta linea) ad revellendum verò momentorum Ratio est ut recta ex B ad H ducta ad rectam, quæ ex H ad A ducitur; ne- que enim curva linea ex H ad B, aut ex H ad A, sed recta le- gem constituit motibus potentiæ in B, & ponderis in A. Id quod pariter contingit cùm aversam mallei partem subti- liorem clavo submittimus, & in oppositam partem manu- brium retrahimus, ut clavus extrahatur: est siquidem curvus quidam vectis fulcrum habens in E, circa quod punctum manens uterque motus perfi- citur; & motus potentiæ in H ad motum clavi in I habet Rationem rectæ H E ad rectam E I. Ex quo patet pro majori manu- brij longitudine augeri etiam potentiæ mo- menta. Quoniam verò aliquando forcipes hujus- modi curvæ aciem habent in A & C, ut id, quod constringitur vehementiùs, etiam scin- datur, non est alia philosophandi ratio, quod quidem spectat ad momenta potentiæ duplici illi vecti applicatæ, hoc uno dif- ferunt, quod vis scindendi orta ex acie ferri pertinet ad ratio- nes

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Book Four. Chapter XII. 447 at the thin tip of the pincers, around which the iron wire is twisted, by comparing the motion of the power applied to the handle with the much greater motion of the particles of the iron wire, which is bent, we perceive; from this indeed we recognize the increased moments of the power. Another kind of pincers is used more frequently, whose most common use is in extracting nails, and the smaller arms AE and CE are not straight, but curved; not only so that the nail may be more securely grasped, with its head received within the bosom of the pincers, but also so that the pincers may exercise another form of curved lever: for when, with the nail seized between A and C, the pincers are inclined so that point H touches the surface beneath, whether it be a wall or a board, then H is the fulcrum, and the moments of the power at B, for the resistance of drawing out the nail, are as BH to HA, since the motion is completed around point H. Therefore, for tightening the nail, the ratio of the moments is as BE to EA (just as if a straight line had been drawn from E to A); but for pulling it out, the ratio of the moments is as the straight line drawn from B to H to the straight line drawn from H to A; for neither the curved line from H to B, nor from H to A, but the straight line establishes the law for the motions of the power at B and of the weight at A. The same thing likewise happens when we place the thinner, reverse part of the hammer upon the nail and draw back the handle toward the opposite side, so that the nail may be extracted: for it is indeed a certain curved lever having its fulcrum in E, around which point, remaining fixed, each motion is performed; and the motion of the power at H to the motion of the nail at I has the ratio of the straight line HE to the straight line EI. From this it is clear that, with a greater length of handle, the moments of the power are also increased. But since sometimes pincers of this kind have a curved edge at A and C, so that what is compressed may also be cut, and cut more forcibly, there is no other reason of philosophizing, so far as concerns the moments of the power applied to that double lever, except that in this one respect they differ, that the force of cutting arising from the edge of the iron pertains to the ratios

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Mechanicorum nes Cunei, de quo inferiùs suo loco. Idem dicendum de forficibus, quarum acies pariter ex rationibus Cunei vim scindendi habent; majora autem momenta potentiæ, quæ faciliùs scindat, petenda sunt ex rationibus vectis; sunt enim hîc pariter duo vectes in oppositas partes commoti, commune hypomochlium in puncto connexionis habentes; & quo majorem Rationem manubriorum longitudo habet ad distantiam rei scindendæ à puncto connexionis, eò etiam facilior contingit scissio. Idcirco quæ duriora sunt, prope connexionis punctum applicantur, quia eadem manubriorum longitudo ad minorem distantiam habet majorem Rationem quàm ad distantiam majorem; & quæ ad hæc duriora scindenda institutæ sunt forfices, breviora habent brachia, quæ ad scindendum exacuuntur, longiora verò ea, quibus potentia movens applicatur; cujusmodi sunt forfices, quibus fabri ferrarij ad æreas aut ferreas laminas scindendas utuntur. In harum usu illud etiam observare poteris, satis esse, si duorum vectium communi fulcro connexorum, ita ut decussati existant, alterum moveatur manente altero: hoc enim potissimum attenditur, quo pacto potentia validiùs applicetur, ubi multâ opus est virtute; cum autem unicum hujusmodi forficum brachium movetur, tota illi manus applicatur, & reliquo deorsum connitente corpore validè premit. CAPUT XIII. Cur Tollenones juxta puteos constituantur. Qui Tollenones Latinis (Ciconias aliqui vocant) Græcis Kelovia dicuntur, familiaria rusticis & olitoribus instrumenta ad hauriendas ex puteis non admodum altis aquas, aliqua habent explicatu digna, quæ ex Vectis doctrinâ petenda sunt, nec visum est Aristoteli quæstione 28. hanc eandem disputationem instituere indecorum, aut homini Philosopho minùs

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of mechanics, the wedge, of which more will be said below in its proper place. The same must be said of scissors, whose blades likewise have the power of cutting from the principles of the wedge; but the greater grounds of power, by which they cut more easily, are to be sought from the principles of the lever; for there are here likewise two levers moved in opposite directions, having a common hypomochlion at the point of connection; and the greater the ratio which the length of the handles bears to the distance of the thing to be cut from the point of connection, the more easily does the cutting take place. Therefore, what is harder is brought near the point of connection, because the same length of the handles has a greater ratio at a smaller distance than at a greater distance; and scissors intended for cutting such harder things have shorter arms, which are sharpened for cutting, but longer ones, by which the moving power is applied; such are the scissors used by blacksmiths for cutting copper or iron plates. In the use of these you can also observe this, that it is enough if, of two levers connected by a common fulcrum, so that they stand crossed, one be moved while the other remains still: for this is chiefly considered, in what manner the power may be more strongly applied, where much force is required; but when only one arm of such a pair of scissors is moved, the whole hand is applied to it, and by pressing firmly with the rest of the body that tends downward. CHAPTER XIII. Why shadoofs are set up beside wells. Those shadoofs, which the Latins call Ciconiae (some call them thus), and the Greeks Kelovia, are familiar instruments for rustics and gardeners for drawing water from wells not too deep, and they have something worth explaining, which must be sought from the doctrine of the lever, nor did it seem to Aristotle, in question 28, improper to set forth this same discussion, nor unbecoming to a philosopher less

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Liber quartus. CAPUT XIII. 449 nùs conveniens. Et primùm quidem ipsa Tollenonis con- structio pendet ex rationibus vectis primi generis, habet si qui- dem fulcrum medium inter potentiam moventem & pondus elevatum. Erecto enim tigno DA imponitur transversa hasta CE, infigitúrque axi in A, circa quem liberè converti possit. Tum extremitati E puteo appositæ alligatur plumbum, aut sa- xum, sive grave aliud quodpiam; ab extremitate autem C, quæ puteo respondet, funis CF pendet (seu hasta fune con- vexa in C, sed tamen facilè mobilis) cui in F situla adnectitur. Iam verò duplex motus in hauriendâ aqua considerandus est, alter, quo hydria vacua in puteum demittitur, alter, quo eadem hydria aquæ plena è puteo extrahitur. Priori motui utique non favet tolleno, faciliùs quippe hydria descende- ret, si nullum esset onus in E, quod depressâ hydria esset elevandum; hujus enim gravitas major est hydriæ gravita- te; ac propterea præter ejusdem hydriæ gravitatem alia po- tentia deprimens requiritur in F, ut major sit Ratio gra- vitatis & potentiæ in F ad gravitatem ponderis in E, quàm sit reciprocè Ratio distantiæ AE ad distantiam AC. Quare si AC longitudo multo major sit longitudine AE, facilor LII

Transcription: Translated (English)

Book Four. CHAPTER XIII. 449 suitable mechanism. And first indeed the very construction of the Tolleno depends on the ratios of a lever of the first kind, since it has the fulcrum in the middle between the moving power and the lifted weight. For upon the erected beam DA is placed the transverse rod CE, and it is fixed to an axle at A, around which it may freely turn. Then to the end E, placed by the well, is tied a lead weight, or a stone, or some other heavy thing; but from the end C, which corresponds to the well, hangs the rope CF (or the rod bent by the rope at C, but nevertheless easily movable), to which the bucket is attached at F. Now, however, a twofold motion must be considered in drawing water: one, by which the empty vessel is lowered into the well; the other, by which the same vessel, full of water, is drawn out of the well. The Tolleno certainly does not favor the former motion, for the vessel would descend more easily if there were no weight at E, which, with the vessel depressed, would have to be raised; for the gravity of this weight is greater than the gravity of the vessel; and therefore, besides the gravity of the vessel itself, another depressing power is required at F, so that the ratio of the gravity and power at F to the gravity of the weight at E may be greater than, reciprocally, the ratio of the distance AE to the distance AC. Therefore, if the length AC be much greater than the length AE, easier...

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Mechanicorum 450 erit hydriæ vacuæ depressio; contra verò deprimendi difficul- tas augebitur, quo magis pondus E distabit à fulcro A. Sed hæc eadem, quæ deprimendi difficultatem augent, ju- vant ad extrahendam faciliùs hydriam: pondus enim E quò longiùs aberit à fulcro A, eò plura habebit momenta adversùs gravitatem aquæ & hydriam pendentes ex C. Hinc est poten- tiæ atque ponderis vices permutari; in depressione nimirum pondus in E existens attollitur, & potentia in C descendit; at in elevatione vicissim pondus elevatur in C, & potentia in E descendit. Prudenter itaque providere oportet, ut & hastæ C E longitudo opportunè distinguatur in partes C A, A E, & pondus in E neque ita leve sit, ut parum adjumenti afferat in extrahendâ aquâ, neque ita grave, ut detrimento sit in depri- mendâ hydriâ. Præstat tamen plus aliquid laboris suscipere in deprimenda hydriâ, ut ea deinde elevetur majore compen- dio: nemo quippe dubitat, quin longè faciliùs sit homini funem F C deorsum trahenti attollere pondus E, quàm pa- rium momentorum aquam è puteo extrahere. Porrò non abs re fuerit monere hîc aliquem, ne se rusticis ridendum præbeat, ubi pro altitudine putei assumpto fune C F, hastam C E æquo longiorem constituerit præter rationem inter- valli inter tigillum D A & puteum; contingeret enim, ut hasta in putei labra incurrens necessariam funis longitudinem minueret. Quapropter tria hæc necesse est sibi invicem pro- portione respondere, videlicet hastæ C E longitudinem, tigilli A D altitudinem, ejusque à puteo distantiam; ut erecta ferè ad perpendiculum hasta eam admittat funis longitudinem, quæ & facile hydriæ jungi possit, & putei altitudinem exæquet. Cur autem tùm in deprimendo Tollenone, ut hydria im- mergatur, tùm in attollendo, ut aqua è puteo eximatur, non parem semper & æquabilem experiamur facilitatem, ratio in promptu est; quia scilicet varia est potentiæ medio fune F C tollenonem agitantis applicatio; quo enim acutior fuerit angu- lus F C A, eò minora sunt potentiæ trahentis momenta, quæ crescente angulo pariter augentur, ut tunc maxima sint, cùm funis F C, & hasta C A angulum rectum constituerint. Et qui- dem licèt, ubi funis ab angulo recto ad obtusum desciverit, ite- rum momenta potentiæ decrescant, si applicationis potentiæ ejusdem

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Mechanicorum 450 there will be the depression of the empty bucket; on the other hand, the difficulty of depressing it will increase the more the weight E is distant from the fulcrum A. But these same things, which increase the difficulty of depressing it, help to draw out the bucket more easily; for the weight E, the farther it is from the fulcrum A, the greater will be its moments against the gravity of the water and the bucket hanging from C. Hence the powers and weights exchange functions: in depressing, namely, the weight in E is raised, and the power in C descends; but in raising, conversely, the weight is raised in C, and the power in E descends. Therefore care must prudently be taken that both the length of the rod C E be suitably divided into the parts C A, A E, and that the weight in E be neither so light as to afford little help in drawing up the water, nor so heavy as to be a disadvantage in depressing the bucket. It is, however, preferable to undertake a little more labor in depressing the bucket, so that it may afterward be raised with greater ease; for no one doubts that it is much easier for a man pulling the rope F C downward to lift the weight E than to draw water out of a well by equal moments. Moreover, it will not be out of place here to warn someone not to make himself ridiculous to country people, if, taking the height of the well by the rope C F, he should set the rod C E longer than is right, beyond the measure of the interval between the crosspiece D A and the well; for it would happen that the rod, striking against the edge of the well, would reduce the necessary length of the rope. Therefore these three things must necessarily correspond to one another in proportion: namely, the length of the rod C E, the height of the crosspiece A D, and its distance from the well; so that a rod erected almost perpendicular will admit that length of rope which both can easily be attached to the bucket and will equal the height of the well. But why then, both in depressing the Tollenon so that the bucket may be immersed, and in lifting it so that the water may be drawn from the well, do we not always experience equal and uniform ease? The reason is at hand: namely, because the application of the power moving the Tollenon through the intermediate rope F C is varied; for the more acute the angle F C A is, the smaller are the moments of the pulling power, which increase likewise as the angle grows, so that they are then greatest when the rope F C and the rod C A have formed a right angle. And indeed, although, when the rope departs from a right angle toward an obtuse one, the moments of the same power decrease again, if the application of that power

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Liber quartus. CAPUT XIII. 451 ejusdem tantummodo habeatur ratio; fieri tamen potest, ut ponderis in E momenta minuantur, quo altiùs attollitur, si il- lud fuerit hastæ impositum, cum ejusdem linea directionis ca- dat in hastæ punctum, quod magis ad fulcrum A accedat, jux- ta ea, quæ hujus libri cap. 3. dicta sunt; atque adeò deprimendi facilitas, quæ hinc sumit incrementum, diminutâ ponderis E re- sistentiâ suppleat decrementum, quod obliquam potentiæ ap- plicationem consequitur. Nec absimilis momentorum varietas contingit ex disparili angulorum amplitudine, quos lineæ directionis gravitatum tum aquæ attollendæ, tum ponderis E, cum hastâ C E consti- tuunt. Nam depressâ hastâ, & pondere maximè elevato, hu- jus momenta initio minora sunt, & subinde augentur receden- te à fulcro A lineâ directionis centri gravitatis, si illud quidem hastæ incumbat: Pondere igitur E minùs conante adversùs aquam cum hydriâ attollendam, plus laborandum est homini funem sursum trahenti; cujus deinde labor minuitur auctis gra- vitatis E momentis; & tunc potissimùm præstare videntur, cum angulus F C A ex recto in acutum transit; tunc enim aquæ de- orsum connitentis ac opposito ponderi resistentis momenta de- crescere incipiunt, ac infirmiora fieri. Ex his non parum lucis affulget scenicis machinationibus, in quibus non planè ad perpendiculum, sed obliquè ascenden- dum est aut descendendum, si enim statuatur vectis Z X ha- bens in P fulcrum, & fune Z N pendeat corpus demit- tendum, utique obliquus erit descensus ex N in M, & vicis- sim obliquus ascensus ex M in N: momenta autem ponderis X, aut S, pro variâ positione, ut dictum est, dissimilia atque disparia sunt: Quapropter temperanda sunt pro motûs instituendi opportunitate; atque si pondus X levius sit corpore demittendo ex N, hoc sponte descendet; si verò in S augea- tur pondus, ut corporis in M gravitatem superet, hoc ex M in N elevabitur. Quod autem de scenicâ machinatione hîc LII 2

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Book Four. CHAPTER XIII. 451 only its own effect is to be taken into account; yet it can happen that the moments of the weight E are diminished, the higher it is raised, if it has been placed upon the staff, when its line of direction falls upon that point of the staff which is nearer to the fulcrum A, in ac- cordance with what was said in chapter 3 of this book; and thus the ease of lowering, which hence takes an increase, may make up, by the reduced resistance of the weight E, for the decrease that follows from an oblique application of the power. Nor does a not dissimilar variety of moments occur from the unequal size of the angles which the lines of direction of the weights, both of the water to be raised and of the weight E, make with the staff C E. For when the staff is depressed and the weight raised highest, the mo- ments of this weight are at first smaller, and then gradually increase as the line of direction of the center of gravity recedes from the fulcrum A, if indeed it rests upon the staff: therefore, with weight E making less effort against the water to be raised with the bucket, the man pulling the rope upward has to labor more; and afterward his labor is lessened as the moments of weight E increase; and then they seem especially to be effective, when the angle F C A passes from a right angle to an acute one; for then the moments of the water, striving downward and resist- ing the opposite weight, begin to decrease and to become weaker. From these things no small light is shed upon theatrical machines, in which one must not ascend or descend quite perpendicularly, but obliquely; for if the lever Z X be assumed, having the fulcrum at P, and from the rope Z N there hangs the body to be let down, the de- scent from N to M will certainly be oblique, and likewise the ascent from M to N will be oblique: but the moments of the weight X, or S, according to the different position, as has been said, are unlike and unequal. Wherefore they must be adjusted according to the suitability of the motion to be carried out; and if the weight X be lighter than the body to be lowered from N, this will descend of itself; but if in S the weight be increased so that it exceeds the gravity of the body in M, this will be raised from M to N. But what has been said here about a theatrical machine LII 2

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Mechanicorum innui, ad alias motiones corporum elevandorum (ut si ex navi in altiorem fluminis ripam onus transferendum esset) facilè traduci posse ita manifestum est, ut pluribus non sit opus, si accuratè examinetur altitudo, ad quam deducendum est, & amplitudo seu distantia parallelarum, intra quas obliquus motus perficiendus est, ut vecti congrua longitudo statuatur, & opportuno loco collocetur, ubi eam anguli R P Z inclinationem habeat, cui Sinus Versus R N respondeat. CAPUT XIV. Remorum vires in agendâ navi expenduntur. Remum, quo naves aguntur, Copensibus, & Platænsibus debemus, ut Plinius lib. 7 cap. 56. scribens ait, Remum Copæ, latitudinem ejus Platææ (utraque est Boëotiae urbs) invenerunt; & nomen ipsum ab inventoribus inditum videtur, nam Græcis κατων Remus, πλάτη Palmula, latior scilicet remi pars dicitur. Ratem siquidem conto propellere rudis adhuc ars nautica noverat, ubi fluminis non admodum alti fundum perticâ pertentare licebat; at ubi uberior unda prohibet, ne fundum attingatur, operam luderet, qui navim conto AB impellere se posse sibi persuaderet, nisi fortè extremitati B ligneâ tabellam adjungeret, ita levem, ut sponte suâ innataret; illam enim per vim velociter immergenti obliquam, aqua resisteret, & navis aliquantulum promoveretur: cæterum ingens esset labor in conto retrahendo, & tabellâ ex aquis extrahendâ, etiamsi scalmo tigillus longiusculus EF ad perpendiculum infigeretur, ex cujus summo vertice funis E I penderet, fune autem contus medius in I suspenderetur. Quare opus fuit instrumentum moliri,

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It is evident enough that, for other motions of bodies to be lifted, it can readily be applied elsewhere as well (as if a load were to be transferred from a ship to a higher bank of a river), so that no further explanation is needed, if one carefully examines the height to which it must be raised and the breadth, or distance between the parallels within which the oblique motion is to be carried out, so that the appropriate length of the beam may be fixed and placed in the proper position, where it may have the inclination of angle R P Z, to which the versed sine R N corresponds. CHAPTER XIV. The powers of oars are expended in propelling a ship. We owe the oar by which ships are driven to the people of Copae and Plataeae, as Pliny says in book 7, chapter 56: “The people of Copae invented the oar, and those of Plataeae its breadth” (both are cities of Boeotia); and the name itself seems to have been given by the inventors, for among the Greeks κατων denotes the oar, and πλάτη the blade, namely the broader part of the oar. Indeed, rude nautical skill had already known how to propel a raft with a pole, where the riverbed was not too deep and it was possible to probe the bottom with a pole; but where the greater depth of the water prevents one from reaching the bottom, he would be wasting his effort who thought he could drive a ship with a pole AB, unless perhaps he attached to its end B a wooden board, so light that it would float of itself. For if, when thrust with force, this board were set obliquely under water, the water would resist it, and the ship would be moved forward somewhat; but there would be great labor in drawing the pole back and in pulling the board out of the water, even if a somewhat longer piece of wood EF were fixed upright in the scalmus, from the top of which a rope E I would hang, and the middle of the pole would be suspended by the rope at I. Therefore it was necessary to devise an instrument,

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Liber quartus. CAPUT XIV. 453 moliri, quo & facile uteremur, & aliquod laboris compendium inveniremus. Ratione suæ longitudinis ad unum aliquod vectis genus referendus est remus, ad cujus caput applicatur potentia, videlicet remex; extrema palmula immergitur aquæ, & circa medium innititur scalmo: sed aquæ ne? an scalmo? ratio fulcri conveniat, disputatur. Si Aristoteles audiendus esset mechan. quæst. 4. hypomochlion fit scalmus, stat enim ille, pondus verò mare est, quod propellit remus; vectem autem movens ipse est remex. Id quidem verum esset, si quis anchoris nondum solutis, & stante navi, adumbratâ ad speciem remigatione se exerceret; nil enim præstaret præter aquarum impulsionem. Cæterùm nautæ remorum pulsu non aquam verberare, sed navim impellere contendunt. Igitur aqua, cui remi palmula immergitur, divisioni resistens, atque impediens motum palmulæ, hypomochlij, cui vectis, hoc est remus, innititur, rationem habet; navis verò ipsa, quæ promovetur, quatenus est scalmo conjuncta, utique est pondus, ex cujus movendi, non ex aquæ repellendæ difficultate æstimandus est nautarum labor: alioquin eodem remo, qui scalmo similiter insisteret, æqualis labor esset, sive actuarium, sive corbitam impellere oporteat; pari siquidem aquæ occurrit utrobique palmula. Manifestum est igitur pondus vecte promovendum navim esse, non aquam, ac propterea hypomochlij vices aquam subire, adeóque remum censendum esse vectem secundi generis, cujus extremitates potentia & fulcrum occupant. Hinc est aliquod semper haberi laboris compendium, ponderis enim motus, qui vecte perficitur, minor est motu potentiæ remi capiti applicatæ, illud enim minùs, hæc magis ab hypomochlio distat. Motus, inquam, qui vecte perficitur, minor est motu potentiæ; fieri enim contingit, ut vi impressi impetus, etiam cessante remigis impulsion, navis promoveatur, adeò ut pro ratione impetus multo major sit navis motus, quàm potentiæ impellentis. Verum hoc non ex vecte ob idipsum, quia vectis est, oritur, sed quia navis innatans aquæ non eam invenit à corpore fluido resistentiam, quam cæteroqui ex mutuo tritu inveniunt pondera corpori solido insistentia, etiamsi vecte horizontaliter moveantur; ac proinde impressus impetus, LII 3

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Book four. CHAPTER XIV. 453 to set in motion, so that we might use it easily, and find some saving of labor. By reason of its length, the oar is to be referred to some one kind of lever, to the head of which is applied the power, namely the rower; the outer blade is immersed in the water, and about the middle it rests on the thole; but whether the water, or the thole, corresponds to the function of the fulcrum, is disputed. If Aristotle in Mechanical Questions 4 were to be followed, the thole is the hypomochlion; for it stands fixed, while the weight is the sea, which the oar propels; but he who moves the lever is the rower himself. This would indeed be true if someone, with the anchors not yet cast off and the ship standing still, were to exercise himself in a mere simulation of rowing; for he would accomplish nothing except the striking of the water. But sailors maintain that by the stroke of the oars they do not beat the water, but propel the ship. Therefore the water, against which the blade of the oar is immersed, resisting division and hindering the motion of the blade, has the role of the hypomochlion, on which the lever, that is, the oar, rests; but the ship itself, which is moved forward, insofar as it is joined to the thole, is certainly the weight, and from the moving of this, not from the difficulty of repelling the water, must the labor of the sailors be estimated. Otherwise, with the same oar, which similarly rested on the thole, there would be equal labor whether one had to propel an actuary or a corbita; for in both cases the blade meets the water equally. It is therefore clear that the ship is the weight to be advanced by the lever, not the water, and for that reason the water must take the place of the hypomochlion, and thus the oar is to be judged a lever of the second kind, whose extremities are occupied by the power and the fulcrum. Hence some saving of labor is always had; for the motion of the weight, which is effected by the lever, is less than the motion of the power applied to the head of the oar, for the former is farther, the latter nearer, from the hypomochlion. I say, the motion which is effected by the lever is less than the motion of the power; for it may happen that, by the force of the impressed impulse, even after the impulse of the rower has ceased, the ship continues to move forward, so that, in proportion to the impulse, the motion of the ship is much greater than that of the impelling power. But this does not arise from the lever, precisely because it is a lever, but because the ship, floating on the water, does not encounter that resistance from the fluid body that weights resting upon a solid body otherwise find through mutual friction, even if they are moved horizontally by a lever; and therefore the impressed impulse, LII 3

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Mechanicorum 454 ceillante vi externâ, non statim perit. Id autem intelligendum est, cum in lacu vel tranquillo mari navigatur, cum scilicet aqua suo cursu non adversatur motui navis: nam si adverso flu- mine promovendum sit navigium, contrarius aquæ impulsus impetum à remige impressum elidit, fierique potest, ut utilius accidat navim trahere, quàm remigando impellere, ne subla- tis ex aquâ remis navigium vi aquæ fluentis retro-actum eò re- deat, unde discessit, & alternâ remorum immersione atque ex- tractione opera ludatur: præterquam quod quò magis immersa remi palmula ab adverso flumine repellitur, eò amplius detrahi- tur motui navis. Ideò quamvis navim trahens plus laboris sim- pliciter impendat, quàm remigans, facit tamen operæ pretium, qui enim navim adversùs profluentem trahit, etiam retinet, ne retrorsum agatur; at qui remo impellit, sublatâ ex undis pal- mulâ, recessum impedire non valet. Cum itaque remus vectis sit secundi generis, remigis vires æstimandæ sunt ex Ratione, quam longitudo remi habet ad il- lam ejusdem remi partem, quæ aquæ & scalmo interjecta est; hæc siquidem Ratio est motuum, ac proinde & momentorum, ut sæpiùs dictum est. Remi autem longitudinem non absolutam intelligas; sed primùm ea demenda est palmulæ particula, quæ aquæ immergitur; quippe quæ aquam repellens quasi hypomo- chlio incumbit. Deinde attendendum est, quam remi partem remex apprehendat; si enim plures eundem remum agitent, ut in triremibus, non sunt æqualia momenta singulorum, sed ejus, qui scalmo propior est, minora sunt (perinde atque si remo adeò brevi uteretur) ejus, qui remi caput apprehendit, maxi- ma sunt momenta; medij autem medio modo se habent. Quare longitudo vectis in remo definitur intervallo, quod in- ter aquam, & remigis manum interjectum est; ponderis distantiam ab hypomochlio metitur intervallum, quo scalmus ab aquâ palmulam excipiente disjungitur. Si igitur intervallum illud est hujus intervalli duplum aut sesquialterum, momenta Po- tentiæ ad momenta ponderis Rationem habent duplam aut ses- quialteram, & quatuor remiges ad promovendam navim tan- tumdem ferè valent ac sex aut octo, qui pari conatu navim eandem sine remis propellerent, aut traheerent. Dixi, ferè, quia cum motus cujuslibet vectis sit circularis circa punctum hypo- mochlij,

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Mechanics 454 ...when protected by an external force, it does not immediately perish. This, however, must be understood when one is sailing in a lake or on a calm sea, that is, when the water, by its own course, does not oppose the motion of the ship; for if the vessel is to be advanced against an opposing current, the contrary impulse of the water cancels the thrust imparted by the rower, and it may happen that it is more useful to tow the ship than to drive it forward by rowing, lest, with the oars lifted out of the water, the vessel be carried backward by the force of the flowing water to the place from which it departed, and the labor be wasted by the alternate immersion and withdrawal of the oars. Besides, the more the blade of the oar, when immersed, is repelled by the opposing current, the more it is drawn away from the motion of the ship. Therefore, although the person towing the ship simply expends more labor than the rower, he nevertheless performs a worthwhile service; for he who tows a ship against the current also holds it back from being driven astern, whereas he who propels it with an oar, once the blade is lifted from the waves, cannot prevent its retreat. Since therefore the oar is a lever of the second kind, the strength of rowers must be estimated from the ratio which the length of the oar bears to that part of the same oar which lies between the water and the oarlock; for this ratio is indeed the ratio of motions, and consequently also of moments, as has often been said. But by the length of the oar you should not understand its absolute length; first, that portion of the blade which is immersed in the water must be subtracted, since, repelling the water, it rests, as it were, on a fulcrum. Next it must be noted what part of the oar the rower grasps; for if several move the same oar, as in triremes, the moments of each are not equal: those of the one nearer the oarlock are smaller, as if he were using an oar that was very short, while those of the one who grasps the end of the oar are greatest; the middle positions have intermediate moments. Therefore the length of the lever in the oar is determined by the interval between the water and the rower’s hand; the distance of the weight from the fulcrum is measured by the interval by which the oarlock is separated from the blade receiving the water. If, then, that interval is double or one and a half times that interval, the moments of Power have the ratio of two to one or three to two to the moments of the weight, and four rowers to advance a ship are almost as effective as six or eight men who, with equal effort, would drive forward or tow the same ship without oars. I said “almost,” because since the motion of any lever is circular about the point of the fulcrum,

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Liber quartus. CAPUT XIV. 455 mochlij, remex, qui in dextero navis latere remigat, secundùm vectis naturam arcum describit ad dexteram inclinatum, id quod pariter contingit sinistro remigi arcum sinistrorum describenti: Cum autem navis non nisi unico motu moveri possit, ex his duobus circularibus sibi adversantibus resultat tertius medius, scilicet rectus, qui proinde tantus esse non potest, quantus esset, si sex aut octo homines æquali nisu navim sine remis impellerent aut traheerent; quia contrariæ illæ directiones ad dexteram & ad sinistram nequeant in tertiam mixtam directionem coalescere, sine aliquo impetûs detrimento. Quod si remiges omnes non consentirent in deprimendo, impellendo, atque extrahendo remo, sed alij alios præverterent, non solùm id incommodi accideret, quod ab instituto itinere deflecteret navis in alterutram partem, nisi æqualis utrinque esset impulsus, verùm etiam retardaretur motus, tùm quia minor impulsus à paucioribus navi imprimitur, tum quia remi tardiores, reliquis elevatis, adhuc immersi dum communi navis motu moventur, minùs impellunt aquam post se fugientem, & palmulæ latitudo occurrenti aquæ obversa moram infert, ut eam dividat; ex quo fit, ut aliquid impetûs ab aliis remigibus impressi deteratur, qui citra hoc impedimentum adhuc perseveraret. Sunt scilicet plures remi plures vectes, quibus idem pondus movetur; & nisi remiges omnes conspiraverint, aut navis tardiùs movetur, aut aliquorum labor augetur: haud secus ac si plures homines uni vecti ad pondus aliquod elevandum applicarentur, uno aut altero cessante reliquorum nisus augendus esset, supplementum desidiosorum. Ex rationibus igitur vectis satisfit quæstioni ab Aristotele propositæ, Cur ij, qui in navis medio sunt remiges, maxime navim movent? Allatam à Philosopho responsionem intactam relinquo; an satis commoda sit, alij examinent. Remigum alij in puppi constituuntur, qui Thranitæ dicebantur, ut est apud Suidam, alij in prorâ, qui Thalamij seu Thalamitæ, alij in navis medio, qui Zygitæ: & quamvis omnes ad promovendam navim suum conatum conferant, non tamen omnium æqualis est labor, aut par in movendo efficacitas; quia non secundùm eandem Rationem singulorum remorum longitudo in partes à scalmo distinguitur; sed quia puppis altior est, & spatium angustum, major remi

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Book Four. Chapter XIV. 455 the mochlij , the rower who rows on the right side of the ship, according to the nature of the lever describes an arc inclined to the right, and the same happens to the left rower describing the arc of those on the left: but since a ship can be moved only by a single motion, from these two circular motions, opposing one another, there results a third, intermediate one, namely a straight motion, which therefore cannot be as great as it would be if six or eight men with equal effort were to drive or draw the ship without oars; because those contrary directions to the right and to the left cannot coalesce into a third mixed direction without some loss of force. But if all the rowers did not agree in pressing down, pushing, and drawing the oar, but some acted before others, not only would that inconvenience occur, that the ship would deviate from its intended course to either side, unless the impulse were equal on both sides, but the motion would also be slowed, both because a smaller impulse is imparted to the ship by fewer men, and because the oars, moving more slowly, when the rest are raised, while still immersed as they move with the common motion of the ship, strike less against the water fleeing behind them, and the breadth of the blade, turned toward the oncoming water, causes delay, in order to divide it; from which it follows that some of the force impressed by the other rowers is wasted, a force which, without this hindrance, would otherwise continue to persist. There are, indeed, several oars, several levers, by which the same weight is moved; and unless all the rowers are in harmony, either the ship moves more slowly, or the labor of some is increased: just as if several men were applied to a single lever to raise some weight, if one or two were to stop, the effort of the others would have to be increased, as compensation for the idle men. From the reasons of the lever, then, the question proposed by Aristotle is answered: Why do those rowers who are in the middle of the ship move it most? I leave the answer given by the Philosopher untouched; whether it is sufficiently apt, let others examine. Some rowers are stationed in the stern, who were called Thranitae, as is found in Suidas; others in the prow, who are Thalamii or Thalamitae; others in the middle of the ship, who are Zygitae: and although all contribute their effort to propel the ship, nevertheless the labor of all is not equal, nor is their effectiveness in moving the same; because the length of each oar is not divided according to the same proportion in the parts from the oarlock; but because the stern is higher, and the space narrow, the greater oars

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456 Mechanicorum remi pars extra navim est, parúmque à scalmo distat remex; ideò motus potentiæ ad motum ponderis minorem habet rationem, quàm si brevior esset inter palmulam, & scalmum, longiorque inter scalmum & remigem distantia, ut contingit in medio, ubi navis depressior est, & maximam habet latitudinem; pondere enim minùs distante ab hypomochlio, majora sunt potentiæ momenta, cum eadem ponatur utrobiique vectis longitudo. Quæ autem de puppi dicta sunt, saltem quo ad spatij angustias, etiam de prorâ intelligenda sunt, quæ quia depressior est puppi, & aliquanto altior quàm circa medium, propterea Thalamiorum labor medius est inter Thranitarum & Zygitarum laborem. Dicuntur autem remiges, qui in navis medio sunt, maximè movere vim, non quia navis motus, qui circa hypomochlium tanquam circa centrum sit, ibi sit major motu, qui fit in puppi, si remiges parem arcum describant, nam potiùs oppositum contingit; sed quia remex in medio minorem inveniens ponderis movendi resistentiam plus navim impellit, quàm si in puppi pariter conaretur, ubi eodem nisu non potest eodem temporis spatio tam amplum arcum describere. Propterea fortissimi remiges ad puppim statuuntur, ut majore impetu producto vincant majorem resistentiam; ideóque Thranitis præter publicum stipendum etiam extraordinarium datum commemorat Thucydides lib.6. Hæc verò, quæ de Antiquorum navibus magis propriè dicuntur, quarum forma à nostris dissidebat, nostris tamen celocibus aut triremibus servatâ analogiâ accommodari possunt; nam etiam apud nos scalmus ad proram & ad puppim ascendit, & in medio major est navigij amplitudo, ita ut, licèt remorum capita in eâdem rectâ lineâ juxta navigij longitudinem constituantur, dispari tamen Ratione à scalmo distinguantur in partes. Sed præstat ipsum navis motum paulo attentiùs considerare; quandoquidem si hypomochlium esset prorsus immobile, & aqua locum non daret palmulæ urgenti, utique motus navis ad motum capitis remi in eâ esset Ratione, quæ intercedit inter distantias scalmi, & capitis remi ab aquâ. Nam si palmula B immota maneret, & scalmus esset in C, motus remigis AD ad motum navis CE esset ut AB ad CB. Contra verò si aqua nihil prorsus obsisteret remo (sicuti contingere,

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456 Mechanics the rower’s oar extends outside the ship, and the rower is at only a short distance from the thole; therefore the motion of the power has a smaller ratio to the motion of the weight, than if the distance between the blade and the thole were shorter, and the distance between the thole and the rower longer, as happens in the middle, where the ship is lower and has its greatest breadth; for when the weight is at a lesser distance from the fulcrum, the moments of the power are greater, the same lever length being assumed on both sides. What has been said about the stern, at least as regards the narrowness of the space, must also be understood of the bow, which because it is lower than the stern and somewhat higher than around the middle, therefore the labor of the Thalamites is midway between that of the Thranites and the Zygites. But the rowers who are in the middle of the ship are said to exert the greatest force, not because the motion of the ship, which is around the fulcrum as around a center, is greater there than the motion which takes place in the stern, if the rowers describe an equal arc, for rather the opposite happens; but because the rower in the middle, finding less resistance to the moving of the weight, propels the ship more, than if he were to try equally in the stern, where with the same effort he cannot describe so wide an arc in the same span of time. For this reason the strongest rowers are stationed at the stern, so that, having produced greater momentum, they may overcome greater resistance; and therefore Thucydides, book 6, mentions in addition to the public pay also an extraordinary allowance for the Thranites. But these things, which are said more properly of the ships of the Ancients, whose form differed from ours, can nevertheless be adapted to our light boats or triremes by preserving the analogy; for even among us the thole rises toward the bow and toward the stern, and in the middle the ship’s breadth is greater, so that, although the oar-blades are placed in the same straight line along the length of the ship, they are nevertheless distinguished into parts by a different ratio from the thole. But it is better to consider the motion of the ship itself a little more carefully; since if the fulcrum were entirely immovable, and the water gave no room to the pushing blade, then surely the motion of the ship would be to the motion of the oar’s head in it in the ratio that intervenes between the distances of the thole and of the oar’s head from the water. For if the blade B remained unmoved, and the thole were at C, the motion of the rower AD to the motion of the ship CE would be as AB to CB. On the other hand if the water opposed the oar in no way at all (as would happen,

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Liber quartus. CAPUT XIV. 457 ret, si ille admodum lentè moveretur, aut palmula nimis obliqua aquam finderet) tantumque palmula retrogrederetur per BH, quantum remex per AD progreditur, immota maneret navis in C: id quod etiam contingeret, si regressus BH ad progressum AD esset in Ratione CB ad CA. Quod si palmula à profluente rapta ex B in L majus spatium conficeret, quàm remex ex A in D (aut saltem BL ad AD esset in majore Ratione quàm CB ad CA) utique navis ipsa retrocederet, & scalmus ex C veniret in F. Cum igitur promoveatur navis, & aqua palmulæ obsistens sit hypomochlium mobile, necesse est progressu remigis AD minorem esse palmulæ regressum BI, ut scalmus ex C propellatur in E. Quare quo magis aqua resistit, minusque palmula movetur in oppositam remigis motui partem, magis promovetur navis, quia majorem impulsum recipit. Majorem autem aquæ resistentiam efficere potest aut velocior remi motio, aut major palmulæ immersio: constat si quidem, si baculo aquam lentè dividas, vix percipi in illa scindendâ laborem; at si velociter baculum immersum agitare libeat, multò validiùs illam resistere: similiter quò major palmulæ immersæ pars plus aquæ propulsat, eò majorem invenit resistentiam, difficiliùs enim multa, quàm modica aqua dividitur. Verùm cum festinato opus est, satius est velociter remum movere, & parum immergere palmulam, ut frequentiori remorum percussione plus impetûs navi imprimatur. Quod demum spectat ad remi motum, unum superest observandum, videlicet, non eum tantummodo motum capiti remi tribuendum, qui respondet partibus navis, quatenus ex remigis musculorum contentione atque membrorum inclinatione pendet, cujus mensuram definiret perpendiculum à capite remi in subjectum navis planum descendens, & in eo remi iter describens; sed præterea addendus est motus navis, qui omnibus in navi existentibus communis est, adeò ut navis vi remorum acta M m m

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Book Four. Chapter XIV. 457 if it moved very slowly, or if the blade cut the water too obliquely) and if only the blade moved backward through BH as much as the rower advances through AD, the boat would remain motionless in C: which would also happen if the retreat BH were to the advance AD in the ratio of CB to CA. But if the blade, seized by the current, from B to L were to make a greater distance than the rower from A to D (or at least if BL were in relation to AD greater than CB to CA) then certainly the boat itself would move backward, and the oarlock would go from C to F. Since therefore the boat is driven forward, and the water resisting the blade is the movable fulcrum, it is necessary that in the rower’s advance AD the blade’s retreat BI be less, so that the oarlock from C is propelled into E. Wherefore the more the water resists, and the less the blade moves in the opposite direction to the rower’s motion, the more the boat is propelled, because it receives a greater impulse. Now a greater resistance of the water can be produced either by a quicker motion of the oar, or by a greater immersion of the blade: indeed it is clear that if, with a stick, you slowly part the water, there is scarcely any labor perceived in splitting it; but if you choose to move a stick immersed in the water briskly, it resists much more strongly: similarly, the greater the part of the immersed blade that drives more water forward, the greater resistance it finds; for much water is divided with more difficulty than a little. But when haste is required, it is better to move the oar quickly, and to immerse the blade only slightly, so that by the more frequent striking of the oars more impetus may be imparted to the boat. What finally concerns the motion of the oar, one thing remains to be observed, namely, that not only that motion should be attributed to the head of the oar which corresponds to the parts of the boat insofar as it depends on the exertion of the rower’s muscles and the inclination of his limbs, the measure of which would be defined by a plumb line descending from the head of the oar onto the plane beneath the boat, and tracing on it the oar’s path; but in addition must be added the motion of the boat, which is common to all those existing in the boat, so that the boat, driven by the force of the oars M m m

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Mechanicorum 458 moveatur à motore translato. Quapropter si AD est universus capitis remi, seu manûs remigis motus, demendus ex illo est navis progressus CE, & residuus motus à remigis conatu, qua- tenus remum impellit, pendet. Sed antequàm ab hac remorum contemplatione animum avertamus, placet innuere, quæ de Sinensium remis attigit Atlas Sinicus in Præfatione pag. 10, ubi de Præsectorum navi- bus, quæ nostris triremibus æquales sunt, hæc habet. Dum cessant ventorum flatus, adsunt destinati, qui remulco trahant, aut remis moles tota impellitur motis ad modum caudæ piscium, methodo facili, & compendiosâ; quippe sine ulla aquæ percussione, extractio- neve remi, vel remo unico propellitur & dirigitur navis; adeòque unus hic sex aut octo nostraribus nautis æquivalet. Postremum hoc de uno remige sex aut octo nostraribus nautis æquivalente, adeò magnificè dictum videtur, sed & adeò jejunè expositum, ut verba mihi dari non facilè patiar, nec me libenter præbeam credulum: fundamentum constituendæ fidei fuisset remigan- di ordo descriptus, remorum forma atque positio verbis aut iconismo proposita, ut, quanta sint remigis Sinici momenta, innotescerent; aliam enim utique à nostraribus remorum for- mam esse necesse est, quippe quos flexiles esse oporteat, ut ad- modum caudæ piscium moveantur; hi scilicet postremam cor- poris sui partem flectunt priùs atque contorquent, ut caudam postmodum velociter porrigentes aquam verberent, qua re- sistente conceptus impetus totum corpus promoveat, quandiu ille perseverat. Ubi animadvertendum est, quàm sapienti na- turæ instituto factum sit, ut pisces caudam lentiùs inflectant, sed velociùs explicent, inflectant obliquam, explicent erectam; si enim erectam caudam velociter flecterent, ita aqua resiste- ret, ut potiùs retrocederent, quemadmodum Astaci fluviati- les (cammaros, alij cancros fluviatiles vocant, rectè ne ? an perperam? non est hujus loci examinare) quando timent, cau- dâ aquam validè percutientes, ac quasi ad se velociter trahen- tes non procedunt, sed retrorsum curvatâ caudâ secedunt. Sic etiam contingeret cymbæ, si quis in puppi stans ligneam tabel- lam extremæ perticæ infixam aquæ à tergo positæ immitteret, perticâmque ad se velociter traheret, nam cymba retrorsum agi videretur. Cùm autem pisces caudam & obliquè & lentiùs inflectant

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Mechanics 458 may be moved by a transferred mover. Therefore, if AD is the whole motion of the helm, or of the rower’s hand, the ship’s progress CE must be subtracted from it, and the remaining motion depends on the rower’s effort, insofar as he impels the oar. But before we turn our minds away from this contemplation of oars, it is pleasing to mention what Atlas Sinicus notes concerning Chinese oars in the Preface, p. 10, where, speaking of the vessels of the Præsecti, which are equal to our triremes, he has this to say. When the blasts of the winds cease, there are those appointed who tow by rope, or the whole mass is driven by oars moved in the manner of a fish’s tail, by an easy and concise method; for without any striking of the water, or extraction of the oar, the ship is propelled and guided by a single oar; and thus one man here is equal to six or eight of our sailors. This last statement, that one rower is equal to six or eight of our sailors, seems to have been said most magnificently, but explained in such a meager way that I cannot easily allow myself to be fooled by words, nor willingly present myself as credulous: the foundation for establishing belief would have been a description of the order of rowing, and the shape and position of the oars set forth in words or in an image, so that the force of the Chinese rower might be made known; for it must certainly be that their oars are of a different form from ours, since they must be flexible, so that they may be moved like a fish’s tail; for these first bend and twist the rear part of their body, so as afterward, swiftly extending the tail, to strike the water, and the resistance of this, while the conceived impetus endures, moves the whole body forward. Here it must be observed how wisely nature has arranged that fish bend their tails more slowly, but extend them more quickly, bend them slantwise, and extend them uprightly; for if they were to bend an upright tail quickly, the water would resist so strongly that they would rather move backward, just as river crayfish (some call them freshwater crabs; correctly? or incorrectly? it is not the place to examine here), when they are frightened, striking the water strongly with their tail and, as it were, drawing it swiftly toward themselves, do not advance, but retreat with the tail curved backward. The same would happen to a boat if someone, standing in the stern, were to thrust into the water behind him a wooden board fixed to the end of a pole, and were to pull the pole toward himself quickly, for the boat would seem to be driven backward. But since fish bend their tail both obliquely and more slowly

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Liber quartus. CAPUT XIV. 459 inflectant, minorem aquæ resistentiam percipiunt. Quare, ut remus suo motu imitetur motum caudæ piscium, opus est erectam palmulam (hoc est, in plano verticali longitudinem navis obliquè, aut ad rectos angulos, secante existentem) aquæ occurrere, ut aquâ resistente propellatur navis, eandem verò palmulam posteà obliquam fieri, ne dum, intra aquam retrahi- tur ad iterandum impulsum, tantam inveniat resistentiam, sed aquam faciliùs findat. Hinc conjecturâ aliquâ ducebar ali- quando ad suspicandum, an ita remi palmula reliquæ remi lon- gitudini adnexa esset fibulâ plicatili, ut, cùm remi caput pup- pim versùs, palmula verò in oppositam partem impelleretur, hæc occurrenti aquæ cederet, eâmque obliquè finderet modi- câ manûs remigis deflexione remum interim contorquentis. Sed, ut quod res est eloquar, vereor, ne argutum nimis, vix- que aliquid habens compendij, artificium hoc videatur: nam & nostrates cymbularij communi remo cymbam ex puppi agentes eam propellunt, & dirigunt aquam non percutientes, nec remum extrahentes, cujus varia inclinatione, loco guber- naculi, cymbæ motum temperant remigando. Ut quid ergo remum in duas partes, quæ fibulâ jungantur, divisum adhibe- re! quippe qui noceat potiùs; nam remigis motus in proram directus nullum impulsum imprimit navi, nisi quando iterum rectus factus fuerit remus. Sed flectatur & dirigatur remus in morem caudæ piscium; quid hoc, ut unus remex sex aut octo nostratibus nautis æquivaleat? Hac autem oblatâ occasione cùm varias excogitaverim rationes utendi remis aquæ semper immersis, liceat mihi per lectoris patientiam unum proponere, quod fortasse nec incommodum, nec inutile accideret, si in usum deduceretur, tunc maximè, cum plures hinc & hinc re- miges adhibentur, qui navis æquilibrio non officerent: neque enim facilè author essem, ut levioribus cymbis methodus hæc communis esset: quippe quæ deficiente ponderis hinc & hinc æqualitate in alterutram partem nimis inclinarentur, nec citra casûs nautæ, aut eversionis naviculæ periculum. Remiges statuo hinc & hinc scalmo insistentes; id quod incommo- dum non erit; quandoquidem additis extrinsecùs opportu- nis fulcris crassiorem satisfie firmam tabulam impono me- diocris latitudinis à scalmo distantem tanto intervallo, quan- M m m 2

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Book Four. Chapter XIV. 459 they incline, they experience less resistance from the water. Therefore, in order that the oar may imitate by its own motion the motion of the tail of fishes, it is necessary that the raised blade of the oar (that is, the part standing in a vertical plane, crossing the length of the ship obliquely or at right angles) should encounter the water, so that the ship may be propelled by the water’s resistance; but afterwards the same blade should be made oblique, so that, when it is drawn back through the water to repeat the stroke, it may meet with no such great resistance, but may more easily cleave the water. From this I was once led by some conjecture to suspect whether the blade of the oar might be attached to the rest of the oar’s length by a hinged clasp, so that, when the head of the oar is turned toward the stern and the blade, on the contrary, is driven in the opposite direction, the blade would yield to the oncoming water and cut it obliquely, by a moderate deflection of the rower’s hand while he meanwhile twists the oar. But, to speak plainly, I fear that this device may seem too subtle and scarcely of any advantage; for even our boatmen, driving a boat from the stern with an ordinary oar, propel and guide it not by striking the water, nor by drawing out the oar, but by varying its inclination, and by rowing temper the motion of the boat in place of a rudder. Why then should we employ an oar divided into two parts joined by a hinge? It would rather do harm; for the rower’s motion directed toward the prow produces no impulse on the ship except when the oar has again been made straight. But let the oar be bent and directed in the manner of a fish’s tail; what then, if one rower should equal six or eight of our sailors? Yet on this occasion, since I have devised various ways of using oars always immersed in the water, let me, by the reader’s patience, propose one method which perhaps would prove neither inconvenient nor useless if it were put into practice, especially when several rowers are employed here and there, so that they do not interfere with the ship’s balance. For I would not easily approve that this method be common for lighter boats, since, the equality of weight on both sides being absent, they would incline too much to one side or the other, and not without danger of the sailors’ falling, or of the boat overturning. I place rowers standing on both sides on thole-pins; this will not be inconvenient, since, with suitable supports added from outside, I place a thicker, sufficiently firm plank of moderate width at such a distance from the thole-pin as is proper, Mmm 2

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Mechanicorum to opus est, ut interjici valeat remus, commodéque agitari: externam autem additæ tabulæ oram ambiat limbus, ne facilè pes labatur; alterum enim pedem tabulæ, alterum scalmo com- modè imponere poterit remex. Remi longitudinem definit al- titudo scalmi ferè supra navis fundum, addita mediocri hominis altitudine; ipsique remi capiti cylindrulus trans- versarius injicitur, ita tamen, ut in eo- dem plano inveniatur cum remi pal- mulâ: apprehenso si quidem utrâque remigis manu hinc & hinc cylindrulo, palmulæ planities aquæ obvertitur, eâmque impellit; apprehensâ autem alterâ tantum extremitate, sive A, si- ve B, quando retrahitur remus, pal- mula DE aquam findit, & est quo- dammodo parallela carinæ. Duplicem igitur motum remo conciliare oportet, alterum quidem à puppi ad proram, & vicissim, cùm scilicet im- pellitur, & retrahitur, alterum verò circa suum axem longitu- dinis, ut convertatur nunc ad impellendam, nunc ad findendam aquam. Primus motus perficitur, si ferreo circulo H I remus in- feratur duos polos habenti; quorum alter scalmum, alter additam tabulam ingrediatur (sivè potiùs excavatæ congruæ crenæ in- cumbant, ut extrahi pro libito possint) sintque facilè versatiles: remus enim in eodem plano verticali semper existens deprimi potest, ut horizontem versùs inclinetur, atque iterum elevari ac- etiam in oppositam partem inclinari. Secundus autem motus fa- cillimè habetur, si remo ferreus rotundus claviculus F adjicia- tur circuli H I superiorem partem contingens; impedit enim, ne remus deorsum prolabatur, adeóque nullo labore converti- tur circa axem suæ longitudinis remus, dimissâ alterutrâ cylin- druli A B extremitate, quando retrahitur. Inventum hoc rudi- ter propositum expolire, atque accuratiùs excolere poteris, in- geniose Lector, qui fortasse tuâ industriâ consequeris artem mihi ignotam remos ita disponendi, ut remex unus pluribus nautis æquivaleat, quemadmodum de Sinensibus narratur. CAPUT

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A mechanical work is required, so that the oar may be inserted and be moved conveniently: and let a rim around the outer edge of the added plank prevent the foot from slipping easily; for the rower will be able to place one foot on the plank, the other on the thole with convenience. The length of the oar is determined by the height of the thole, almost above the bottom of the ship, added to a moderate human height; and to the head of the oar itself a transverse little cylinder is fixed, however, so that it may be found in the same plane with the oar’s blade: for if the rower, grasping the cylinder on both sides with both hands, turns the flat surface of the blade toward the water, and drives it; but if only one end is grasped, whether A or B, when the oar is drawn back, the blade DE splits the water, and is in some way parallel to the keel. Therefore the oar ought to be given a twofold motion, one indeed from stern to bow, and back again, namely when it is pushed and pulled back, the other about its own longitudinal axis, so that it may now be turned to drive, now to split the water. The first motion is accomplished if the oar is placed with the iron ring H I on two pivots; one of which enters the thole, the other the added plank (or rather rests upon suitable hollow notches cut into them, so that they may be taken out at will), and let them be easily turnable: for the oar, always remaining in the same vertical plane, can be lowered, so as to incline toward the horizon, and then raised again and also inclined to the opposite side. But the second motion is most easily obtained if to the oar there be added a round iron knob F, touching the upper part of the ring H I; for it prevents the oar from slipping downward, and thus the oar is turned about the axis of its length without any effort, when one end of the little cylinder A B is released, as it is drawn back. This invention, roughly proposed, you may refine and more accurately develop, ingenious Reader, who perhaps by your own industry may achieve an art unknown to me, of arranging oars so that one rower may equal several sailors, as is said of the Chinese. CHAPTER

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CAPUT XV. Quomodo Naves à gubernaculo moveantur. Res est, cui assiduus usus admirationem detraxisse videtur, non tamen propterea minus habet admirabilitatis, motus scilicet navium, quæ à gubernaculo reguntur, cùm magnum pondus temporis momento moveatur. Nam & Apostolus S. Ia- cobus in Canonicâ Epistola cap. 3. ait, Ecce & naves cum magna sint, & à ventis validis minentur, circumferuntur à modico guber- naculo, ubi impetus dirigentis voluerit. Et Aristoteles Mechan. quæst. 5. inquirit. Cur parvum existens gubernaculum, & in ex- tremo navigio, tantas habet vires, ut ab exiguo temone, & ab homi- nis unius viribus alioquin modicè utentis, magna navigiorum movean- tur moles? Partes duas in gubernaculo invenimus; alteram ex- trinsecùs navi adjectam, ligneam videlicet alam, sive cardini- bus, circa quos converti potest, postremæ puppis parti affixam, sive ad latus puppi adjacentem, tignóque, quod ex scalmo as- surgit, adalligatam, prout maritimo vel fluviatili itineri desti- nata sunt navigia; alteram intra navim, temonis in morem; ex cujus conversione aut pars illa externa in oppositam plagam convertitur, ita ut deductâ temonis extremitate ad dexteram, gubernaculi ala extremæ puppi adjacens in sinistram circa suos cardines deflectat, & vicissim hæc ad dexteram, temone in si- nistram converso: aut in navigiis, quorum in fluminibus usus est, gubernaculum ad puppis latus habentibus, depresso temo- mone superior extremitas alæ in triangulum subjecto cylindro infixum conformatae propiùs accedit ad navim, temone autem elevato ab illa recedit: contra verò si ala infra cylindrum consti- tuatur, ejus extremitas inferior ad navim accedit temone ele- vato, à navi recedit temone depresso. Quemadmodum autem ad propellendam navim instituti sunt remi, ita ad ejusdem cursum dirigendum, atque pro opportu- nitate in dexteram aut in sinistram inclinandum, clavus ad- Mmm 3

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CAPUT XV. How Ships are moved by the rudder. This is a thing to which continual use seems to have taken away our wonder, yet for that reason it is no less admirable, namely the motion of ships which are governed by the rudder, when a great weight is moved in an instant of time. For the Apostle St. James also says in the Canonical Epistle, chapter 3: Behold also the ships, though they are great, and though they are driven by strong winds, are turned about by a very small rudder, whithersoever the impulse of him that directs them will. And Aristotle, Mechan. question 5, inquires, Why does a rudder, though small and placed at the extreme part of the vessel, have such great force, that by a slight helm, and by the strength of one man otherwise using only moderate power, the great mass of the ships is moved? We find two parts in the rudder; one attached externally to the ship, namely the wooden blade, or the hinges around which it can turn, fastened to the aftermost part of the stern, or lying alongside the stern, and tied to the timber which rises from the stern-post, as is appropriate for ships intended for sea or river travel; the other inside the ship, like a helm; by the turning of which either that outer part is turned to the opposite side, so that when the end of the helm is moved to the right, the blade of the rudder adjoining the stern at the far end bends to the left around its hinges, and vice versa, this to the right when the helm is turned to the left: or, in ships which are used on rivers and have the rudder at the side of the stern, when the tiller is depressed, the upper end of the blade, formed into a triangle fixed in the cylinder beneath, comes closer to the ship; but when the tiller is raised it moves away from it; on the contrary, if the blade is placed below the cylinder, its lower end comes closer to the ship when the tiller is raised, and moves away from the ship when the tiller is depressed. Now just as oars have been instituted to propel a ship, so also to direct its course, and as occasion requires to incline it to the right or to the left, the helm is added

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Mechanicorum 462 jectus est. Quamquam enim remorum pulsu navis acta proram in dexteram obvertere possit, si remiges sinistri cessent, atque è contrario in sinistram cessantibus dexteris; aut etiam vento navim impellente fieri possit hæc in alterutram partem declinatio modò passo, modò contracto velo, ut me observasse memini, cum ex insula Seelandia per fretum Oresunticum in Scaniam (Elsingorâ scilicet Elsemburgum) navigiolo transfretarem: id tamen longè faciliùs, atque ad unius gubernatoris arbitrium perficitur converso opportunè clavo, ut quotidiano experimento docemur. Porrò dupliciter gubernaculi motum considerare possumus; neque enim eadem est ratio, cùm navis quiescit, nullusque est aquæ motus, atque cùm navis vento seu remis agitur, aut aqua ipsa movetur. Et quidem si navigium in aquâ immotâ quiescat, qui gubernaculi temonem movet, est potentia applicata vecti, cujus hypomochlium est aqua, si navis non sit tantæ gravitatis, ut faciliùs ipsa moveatur, quam tota aqua propellatur ab alâ gubernaculi; & tunc est vectis secundi generis, nam puppis, aquâ resistente, secedit ad dexteram aut ad sinistram sequens temonis conversionem. At si tanta sit navis gravitas, ut multo faciliùs tota aqua propellatur, quàm navis loco moveatur, vectis est primi generis habens hypomochlium in cardinibus, circa quos gubernaculi ala convertitur, pondus autem, quod movetur, est aqua, quæ eò faciliùs, minori scilicet labore, propellitur, quò longior est temo; tunc enim potentia plus habet momenti. Hinc duplex vectis ratio invenitur, cùm aliquâ ex parte aqua, aliquâ ex parte puppis movetur; quo in motu satis constat neque motum puppis fieri circa aquam extremæ gubernaculi alæ respondentem, neque motum aquæ respondentis extremo gubernaculo fieri circa cardines puppi inhærentes; sed conversionem fieri circa punctum aliquod intermedium reciprocè acceptum pro Ratione resistentiarum aquæ & navis, juxta dicta cap. 5. hujus libri. Cùm autem resistentia aquæ æstimanda sit ex magnitudine & figura alæ gubernaculi aquam ipsam impellentis, & resistentia navis pariter definienda sit tùm ex ejus gravitate, tum ex aquæ propellendæ quantitate, dum navis in dexteram aut in sinistram convertitur, patet nullum certum punctum navibus omnibus commune statui posse; sunt

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Mechanics 462 is moved. For although the ship, driven by the thrust of the oars, can turn its prow to the right if the rowers on the left cease, and conversely to the left if those on the right cease; or although, with the wind driving the ship, this deviation to either side may also occur by the sail being now slackened, now tightened, as I remember observing when I crossed by small boat from the island of Zealand through the Øresund Strait into Scania (that is, from Helsingør to Elsinore): yet this is accomplished far more easily, and at the will of a single helmsman, by suitably turning the tiller, as we are taught by daily experience. Moreover, we may consider the motion of the rudder in two ways; for the same account does not apply when the ship is at rest and there is no motion of the water, as when the ship is driven by wind or by oars, or when the water itself is moving. And indeed if a vessel rests in motionless water, he who moves the tiller of the rudder is applying the power to a lever whose fulcrum is the water, if the ship is not of such great weight that it is moved more easily than the whole water is pushed by the rudder blade; and then it is a lever of the second kind, for the stern, the water resisting, turns to the right or to the left following the turning of the tiller. But if the ship be of such weight that the whole water is much more easily pushed than the ship is moved from its place, the lever is of the first kind, having its fulcrum in the hinges around which the rudder blade turns, while the weight that is moved is the water, which is propelled more easily, that is, with less labor, the longer the tiller is; for then the power has greater effect. Hence a double account of the lever is found, when in some part the water is moved and in some part the stern; in this motion it is sufficiently clear that neither is the motion of the stern made about the water corresponding to the far end of the rudder blade, nor is the motion of the water corresponding to the end of the rudder made about the hinges attached to the stern; rather, the turning is made about some intermediate point, taken reciprocally according to the ratio of the resistances of the water and the ship, as stated in chapter 5 of this book. But since the resistance of the water must be estimated from the magnitude and shape of the rudder blade itself driving the water, and the resistance of the ship likewise must be defined both from its weight and from the amount of water to be displaced, while the ship is turned to the right or to the left, it is clear that no fixed point common to all ships can be assigned; there are

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Liber quartus. CAPUT XV. 463 sunt siquidem hujusmodi resistentiæ multiplici varietati ob- noxiæ. At verò cùm navis in motu est, & vento impellente seu re- mis agitur, potentia quidem pro temonis longitudine sua ha- bet momenta, & ad navis conversionem juvat, magis tamen accipiendo vim externam & ferendo, quàm agendo & facien- do, hoc est retinendo gubernaculum in illa obliquâ positione adversùs vim aquæ in contrarium nitentis, aut resistentis. Quò enim velociùs fertur navis, obviam aquam prorâ scindens illam ita dividit, ut ad navis latera hinc atque hinc velociùs refluat in puppim; ubi si gubernaculi alam inveniat rectam, pergit na- vis recto itinere; Sed si aqua refluens obli- quum gubernaculum offendat, ut si existen- te carina AB fuerit gubernaculum obli- quum CD, aqua in alam AD incurrens dum illam urget, puppim cogit declinare ex A in E, & prora obvertitur versùs F. Hæc tamen, quæ de aqua ad navis late- ra resluente dicta sunt, non ita accipi velim, ut non nisi ab ejus impetu flecti navis cur- sum existimes; sed hæc deflexio præcipuè tribuenda est resistentiæ ipsius aquæ, in quam incurrit gubernaculum obliquum, dum navis tota impellitur; eò autem major est aquæ resistentia, quò velociùs illam scindi oportet, ut sæ- piùs dictum est. Ideò quo validiore venti aut remorum impul- su agitur navis, faciliùs flectitur ope gubernaculi, majorem quippe invenit resistentiam. Cum verò resistentia hæc ex al- terutra tantùm parte inveniatur, necesse est proram in eandem obverti plagam. Cujus rei obvium experimentum sumere quis- que potest, si corpus aliquod angulatum (cujusmodi esset nor- ma, qua ad angulum rectum describendum utimur) in plano inclinato æquabili ac polito descendere permittat: nam si, quod ponè sequitur, brachium in offendiculum aliquod incurrat, il- lico reliquum brachium ad eam partem inclinari videbit, im- petu scilicet promovente corpus, atque objectum impedimen- tum declinante. Quare id, quod navim maximè movet in dexteram aut si- nistram,

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Book Four. Chapter XV. 463 they are indeed subject to a manifold variety of such resistances. But when a ship is in motion, and is driven by wind or by oars, the power certainly has its force in proportion to the length of the tiller, and helps the turning of the ship; yet it does so more by receiving an external force and bearing it than by acting and doing, that is, by keeping the rudder in that oblique position against the force of the water striving or resisting in the opposite direction. For the more swiftly the ship is carried forward, cutting the water before it with the prow, the more it divides that water, so that it flows back more quickly on both sides to the stern; and if it find the rudder blade straight, the ship continues on a straight course. But if the returning water strikes an oblique rudder, as if, the keel AB being present, the rudder CD were oblique, the water running against the blade AD, while pressing upon it, forces the stern to turn aside from A to E, and the prow is turned toward F. Yet I would not have the things here said about the water flowing back to the sides of the ship understood as though you supposed that the ship’s course is bent only by its force; rather, this deflection is to be attributed chiefly to the resistance of the water itself, against which the oblique rudder runs when the whole ship is being driven onward; and the greater that resistance of the water is, the more swiftly it must be cut, as has been said often enough. Therefore, the more strongly the ship is driven by the impulse of wind or oars, the more easily it is turned by means of the rudder, for it finds greater resistance. But since this resistance is found only on one side or the other, it is necessary for the prow to turn to the same side. Of this matter anyone can take a clear experiment, if he lets some angular body (such as the square rule with which we use to describe a right angle) descend on an even and polished inclined plane: for if, as follows from that, one arm should strike some obstacle, immediately he will see the rest of the arm incline to that side, the impulse moving the body forward, and the impediment met with causing it to turn aside. Wherefore that which moves a ship most to the right or left,

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Mechanicorum 464 nistram, est impetus ab ipso vento aut à remigibus navi impres- sus; gubernaculum autem infert moram & impedimentum, ne motus omnino fiat juxta directionem impetûs ab impellente im- pressi: quamdiu verò impedimentum perseverat, navis magis aut minùs obliquè fertur, pro ut modificata impetûs directio exigit. Juvat autem, ut dictum est, aqua, quæ à prorâ dividi- tur, & ad latera refluit, maximè si adverso flumine, aut contra marini æstus cursum navigatio instituatur; aucto enim impedi- mento faciliùs flectitur instituta progressio; sed idcircò etiam navis motus retardatur magis. Quod quidem spectat ad gubernaculum extremæ puppis pla- næ faciei adhærens, ut in majoribus navigiis maritimo itineri destinatis, satis jam explicatum est: unum addendum videtur, quod in navigiis ad devehendas merces fabricatis in fossâ qua- dam manufactâ aliquando observasse me memini; ex puppi vi- delicet extremâ in acutum assurgente, quasi caudæ in morem, clavus longiùs protendebatur apici puppis insistens remo absi- milis tantùm, quatenus palmula paulò latior, nec juxta scapi longitudinem directa, sed inflexa intra aquam immergebatur; caput autem temonis fune jungebatur navigij plano ita, ut gu- bernaculi pars externa suo pondere recidere nequiret, ac fun- dum alvei non peteret, sed palmula paulò infra aquæ superfi- ciem consisteret. Hinc enim fiebat, ut temonis capite in alteru- tram partem adducto in eandem puppis recederet aquâ resisten- te palmulæ, ac proinde prora in oppositam partem obvertere- tur: perinde atque in cymbis gubernaculo destitutis, cùm remi ad latus extremæ puppis directè immersi caput ad se retrahit nauta, puppim ipsam impellit, ac proram in oppositam partem convertit. Huc spectare possunt, quæ habet Atlas Sinicus pag. 123 in XI Provincia Fokien loquens de flumine Min, quod ex Puching ad oppidum usque Xuiken per valles & saxa ingenti impetu ac violentiâ volvitur, inde placidissimum flumen est; & quantumcumque violentum enavigatur tamen à Sinis con- suetâ illorum industriâ, ac parvarum navicularum artificio: hæ enim naves clavum, ut aliæ non habent, sed duos longissimè porrectos, ad puppim unum, ad proram alterum: his per saxa ac scopulos prominentes facillime ac velocissimè naves, ac si fræno equos continerent, dirigunt. Hæc ibi. Sed

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Mechanics 464 The impulse is from the wind itself or impressed on the ship by the rowers; but the rudder introduces delay and obstruction, so that the motion does not proceed altogether according to the direction of the impulse impressed by the propeller. So long, however, as the obstruction lasts, the ship is carried more or less obliquely, as the modified direction of the impulse requires. Moreover, as has been said, the water which is divided by the prow and flows back to the sides helps, especially if navigation is undertaken against the current of a river or against the course of the sea tide; for as the obstruction is increased, the intended movement is more easily deflected, but for that reason the motion of the ship is also more greatly retarded. What concerns the rudder attached to the flat end of the stern has already been sufficiently explained, as in larger ships intended for sea travel: one thing seems still to be added, namely what I remember observing in certain ships built for carrying merchandise in a man-made canal; namely, with the extreme stern rising into a point, almost in the manner of a tail, a long tiller extended farther, resting on the tip of the stern, resembling an oar only in so far as the blade was somewhat broader and, not directed along the length of the shaft, but bent, it was immersed in the water; but the head of the tiller was joined by a rope to the plane of the vessel in such a way that the outer part of the rudder could not, by its own weight, fall back and seek the bottom of the channel, but the blade remained a little below the surface of the water. Hence it came about that, when the head of the tiller was pulled to one side or the other, the stern itself also moved back in that direction, the water resisting the blade, and therefore the prow was turned to the opposite side: just as in boats without a rudder, when the oars inserted straight at the side of the extreme stern draw the sailor’s hand back toward them, they propel the stern itself and turn the prow to the opposite side. To this may be referred what the Chinese Atlas has on page 123, in the Eleventh Province, Fokien, when speaking of the river Min, which from Puching as far as the town of Xuiken rushes through valleys and rocks with great force and violence, and then is a most placid river; and however violent it may be, it is nevertheless navigated by the Chinese with their accustomed industry and the skill of their small boats: for these boats have no rudder, as others do, but two long projecting ones, one at the stern, the other at the prow; by these, through projecting rocks and shoals, they steer their boats most easily and most quickly, as if holding horses with a bridle. Thus far there. But

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Liber quartus. CAPUT XV. 465 Sed ut aliquid etiam de gubernaculo ad puppis latus consti- tuto in navigiis, quorum potissimus usus est in fluviis, dicatur, animadvertendum est hujusmodi navigia non solùm proram, sed & puppim habere, quæ obliquè assurgentes in acutum de- sinunt, gubernaculum autem constare ex cylindro obliquè descendente juxta puppis longitudinem, atque ex alâ triangu- lari ut plurimùm sursum respiciente, cujus latus unum cylin- dro congruit, cui infixum est. Quandiu ala sursum respicit, nihil impedit navis motum, æqualiter enim aqua hinc & hinc fluit, ac proinde navis fertur juxta impetûs à vento aut à remi- gibus impressi directionem (idem dic, cùm navis trahitur) quam sequitur, nisi aliquid fortuito interveniat, à quo turbe- tur motus, & præter nautarum voluntatem aliò flectatur. Quod si convoluto circà suum axem cylindro, ala in hanc aut illam partem vertatur, jam occurrit aquæ, ex cujus resistentiâ impe- dimentum objicitur navi, ne rectâ feratur, sed in alteram par- tem detorquetur: nam si depresso temone, qui priùs erat hori- zonti parallelus, ala versùs navim inclinetur, aqua inter guber- naculum & navim intercepta resistit, atque interfluens conatur alam gubernaculi in directum restituere: quapropter puppim in dexteram trahens, illique ad dexteram resistens (clavus scilicet dextero puppis lateri adjacet) proram obvertit ad sinistram. At si gubernaculi ala in oppositam navi partem extrorsum verta- tur, obviam habet aquam externam, qua resistente repellitur puppis in sinistram, & prora in dexteram convertitur. Quod si alam triangularem placeat potiùs cylindro subjicere, elevato te- mone ala accedit ad navim, & depresso temone ala recedit à na- vi: quapropter ibi puppis repellitur in sinistram, hîc ab aquâ in- tercurrente trahitur in dexteram, motûsque oppositi proræ conveniunt. Ex his facilè innotescit, quid præstet gubernaculum inter puppes duorum pontonum, quos impositus pons jungit, vali- dûsque rudens congruæ longitudinis retinet, ne secundo flu- mine rapiantur; prout enim in hanc vel illam partem guberna- culi ala vertitur, obvium habet interjectarum aquarum impe- tum, quo propellitur in adversam partem, eâque ratione traji- citur flumen, ut in Pado & aliis Galliæ Cisalpinæ fluviis passim videre est. N n n

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Book Four. CHAPTER XV. 465 But that something also may be said about the rudder placed at the side of the stern in those vessels whose principal use is on rivers, it must be noted that vessels of this kind have not only a prow but also a stern, which, rising obliquely, taper to a point; and that the rudder consists of a cylinder descending obliquely along the length of the stern, and of a triangular blade, for the most part facing upward, one side of which fits the cylinder, to which it is fixed. As long as the blade faces upward, nothing hinders the motion of the vessel; for the water flows equally on both sides, and therefore the vessel is carried in the direction of the impulse impressed by the wind or by the rowers (the same may be said when the vessel is towed), which it follows unless something should intervene by chance, by which its motion is disturbed and turned aside, contrary to the sailors’ intention. But if, the cylinder having been rotated around its axis, the blade is turned to this side or that, it then meets the water, and by its resistance an obstacle is offered to the vessel, so that it is not carried straight ahead, but is deflected to the other side. For if, the tiller being depressed and previously parallel to the horizon, the blade is inclined toward the vessel, the water intercepted between the rudder and the vessel resists, and, flowing past, seeks to restore the blade of the rudder to its straight position; wherefore, drawing the stern to the right and resisting it to the right also (the tiller, namely, lies beside the right side of the stern), it turns the prow to the left. But if the blade of the rudder is turned outward toward the opposite side of the vessel, it has the external water before it, and, as this resists, the stern is driven to the left, and the prow is turned to the right. But if one prefers to place the triangular blade beneath the cylinder, then, the tiller being raised, the blade approaches the vessel, and, the tiller being depressed, the blade recedes from the vessel; wherefore there the stern is driven to the left, here it is drawn by the water flowing between to the right, and the opposite motions of the prow correspond. From these things it readily becomes clear what service the rudder performs between the sterns of two barges, which a bridge laid over them joins together, and a strong rope of suitable length holds, lest they be swept away by the downward current; for as the blade of the rudder is turned to this side or that, it meets the force of the water between them, by which it is propelled to the opposite side, and in this way the river is crossed, as may be seen everywhere on the Po and on other rivers of Cisalpine Gaul. N n n

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Mechanicorum 466 Illud postremò consideratione dignum est, quod ad ipsius navis conversionem attinet nimirùm quodnam sit punctum circa quod convertitur: Manifestum est enim neque circa pup- pim tanquam circa centrum describi arcum à prorâ, neque vi- cissim circa proram quasi centrum arcum à puppi describi, quia aquæ quantitas respondens longitudini carinæ plurimum re- sistit, ne circulariter moveatur tota ad eandem partem, coge- retur scilicet nimis amplum arcum describere, nimisque veloci- ter moveri in latus, ut per destinatum navigationis Rumbum nova loxodromia institueretur: faciliùs igitur convertitur na- vis, si dum pars anterior proræ aquam in dexteram propellit, reliqua pars posterior puppi proxima aquam repellat in si- nistram, utraque enim extremitas minore arcu descripto ad ma- jorem angulum carinam inclinat atque deflectit à lineâ prioris cursûs, & minore velocitate aquam urgens minorem invenit resistentiam. Fit igitur conversio circa punctum aliquod me- dium inter proram & puppim; illud autem est, circa quod na- tura faciliùs assequitur propositum, & minore motu removetur impedimentum, quod ab aquâ occurrente infertur, quæ cùm dividatur à prorâ, resluátque juxta navis latera, æqualiter qui- dem à prorâ dispertitur, sed ubi navis ventrem, hoc est amplis- simam navigij partem prætergressa est, offendens ex alterâ parte gubernaculi alam fluere non potest, qua velocitate flue- ret nullo objecto offendiculo; propterea aquæ refluenti ex ad- verso navis latere objiciendum est obliquè puppis latus, ut illa pariter lentiùs fluat, divisóque impedimento æquales aquæ por- tiones ex utroque puppis latere fluant. Quare probabili con- jecturâ existimo conversionem fieri circa illud carinæ punctum, quod respondet maximæ navigij amplitudini; pars quippe na- vigij anterior juxta suam latitudinem occurrens aquæ invenit resistentiam; aqua igitur incurrens in gubernaculum movet partem posteriorem in latus, ubi non est tam valida aquæ re- sistentia. Cum autem in majoribus navigiis præcipuus malus statuatur in maxima navigij amplitudine, hoc est, ubi carinæ longitudo bessem relinquit puppim versus, & trientem versus proram, carina ad proram spectans minùs movetur quàm quæ ad puppim; sed propter notabilem proræ projecturam si pars na- vis suprema inspiciatur, malus ille est circa mediam totius na- vis

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Mechanics 466 It is also worthy of consideration, concerning the turning of the ship itself, namely what point it turns about. For it is manifest that an arc is described by the bow, not around the stern as around a center, nor, conversely, is an arc described by the stern around the bow as though around a center, because the quantity of water corresponding to the length of the keel offers great resistance, so that the whole ship would have to move circularly to the same side, namely, it would be forced to describe too large an arc and to move too quickly sideways, in order that a new loxodromy might be established along the intended course of navigation. Therefore the ship is turned more easily if, while the forward part of the bow pushes the water to the right, the remaining after part near the stern repels the water to the left; for both extremities, by describing a smaller arc, incline and deflect the keel through a greater angle from the line of the former course, and by pressing the water with less velocity find less resistance. Thus the turning takes place about some point midway between bow and stern; and this is the point about which nature more easily achieves its purpose, and with less motion the impediment introduced by the oncoming water is removed. For when the water is divided by the bow and flows back along the sides of the ship, it is indeed divided equally from the bow, but after the ship has passed beyond its belly, that is, the broadest part of the vessel, the water striking the other side of the rudder blade cannot flow as quickly as it would if there were no obstacle in the way. Therefore the stern side must be presented obliquely to the backflowing water on the opposite side of the ship, so that it may flow more slowly as well, and, the impediment having been divided, equal portions of water may flow from each side of the stern. Wherefore I think it probable, by conjecture, that the turning is made about that point of the keel which corresponds to the greatest breadth of the vessel; for the forward part of the vessel, meeting the water according to its own breadth, encounters resistance; the water therefore falling upon the rudder moves the after part sideways, where the resistance of the water is not so strong. But when, in larger ships, the principal mast is placed at the greatest breadth of the vessel, that is, where the keel leaves two-thirds toward the stern and one-third toward the bow, the part of the keel looking toward the bow is moved less than that toward the stern; yet because of the notable projection of the bow, if the upper part of the ship is considered, that mast is about the middle of the whole ship

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Liber quartus. CAPUT XV. 467 vis longitudinem, ibique fit conversio. Cæterùm quicumque navis formam, tormentorumque bellicorum dispositionem ac numerum observet, utique centrum gravitatis inter puppim & malum præcipuum interjectum esse affirmabit; præsertim cum id necesse sit, ne ventorum vi prora nimis deprimatur; id quod multo manifestiùs innotescit in minoribus navigiis, si fortè ve- lo uti contingat, malus enim maximè ad proram accedit. Sit igitur carina A B, maxima navis latitudo H I, malus pri- marius in G, centrum gravitatis navigij ex. gr. in K. Duo sunt principia moventia; unum est ventus in G, alterum est aqua resluens in A D: duo pariter sunt hypomochlia, seu impedimenta; vento resistit gubernaculum A D, propterea navim mo- tione transversâ promovens transfert cen- trum gravitatis K versus H: aquæ resluenti resistit vis venti in G, ita ut non valeat na- vim retrorsum agere, propterea puppim ex A transfert in E, & centro gravitatis K im- petum imprimit dirigentem versùs I, cui tamen prævalente impetu venti dirigente versùs H, obliquus navis motus efficitur. Quare duplex est vectis secundi generis; aqua in A D ad resistentiam centri gra- vitatis K habet momentum ut A G, seu D G, ad K G: Ventus in G ad ejusdem centri gravitatis K resistentiam habet momen- tum ut G A ad K A. Ex quo constat majorem quidem esse Ra- tionem A G ad K G minorem, quàm ad K A majorem; sed multo validiorem potentiam esse ventum, quàm aquam re- fluentem, nisi fortè addatur naturalis fluxus aquæ, qui aliquan- do prævalere dignoscitur ex occultis Maris Currentibus, quæ navim aliquando retrorsum agunt contrà vim venti. Sed quo- niam tam varia & multiplex est navigiorum forma, nec in iis construendis omnes artifices eandem servant partium membro- rumque Rationem, nulla assignari potest certa Ratio, quæ in- tercedat inter distantiam centri gravitatis ab extremitate pup- pis, atque distantiam puncti, circa quod fit conversio, ab ea- dem extremitate. Hic autem (ne quis facilè similiter labatur) fateor me ali- N n n 2

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Book four. CHAPTER XV. 467 the length, and there the turning takes place. Moreover, whoever observes the shape of a ship and the arrangement and number of the artillery, will certainly affirm that the center of gravity is interposed between the stern and the main mast; especially since this is necessary, lest the bow be too much depressed by the force of the winds; which becomes much more clearly evident in smaller vessels, if it should happen that they use sail, for the mast ap- proaches the bow the most. Let the keel A B therefore be given, the greatest breadth of the ship H I, the main mast in G, the center of gravity of the ship, for example, in K. There are two moving principles; one is the wind in G, the other is the water flowing back in A D: there are likewise two hypomochlia, or impediments; the rudder A D resists the wind, and therefore, by advancing the ship by a transverse motion, it transfers the center of gravity K toward H: the force of the wind in G resists the water flowing back, so that it cannot drive the ship backward; therefore it transfers the stern from A to E, and impresses upon the center of gravity K an impulse directing it toward I, to which however the prevailing impulse of the wind, directing toward H, produces the oblique motion of the ship. Wherefore there is a double second-class lever; the water in A D, in resisting the center of gravity K, has a moment as A G, or D G, is to K G: the wind in G, in resisting the same center of gravity K, has a moment as G A is to K A. From which it is clear that the ratio A G to K G is indeed smaller than to K A, the greater; but that the power of the wind is much stronger than that of the returning water, unless perhaps the natural flow of the water be added, which is sometimes known to prevail from the hidden currents of the sea, which sometimes drive the ship backward against the force of the wind. But since the shape of vessels is so varied and manifold, and in constructing them not all artisans observe the same ratio of the parts and members, no certain ratio can be assigned that intervenes between the distance of the center of gravity from the stern extremity and the distance of the point about which the turning takes place from the same extremity. Here however (lest anyone should easily fall into a similar error) I confess that I have been somewhat ... N n n 2

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Mechanicorum 468 quando veri quadam specie deceptum existimasse intervallum inter extremam proram & punctum conversionis ad quartam totius longitudinis partem proximè statuendum esse; ducebar scilicet quadam analogiâ desumpta ex cylindro ligneo innatante, cujus quiescentis extremitatem si tanto impetu percusseris, quo certum spatium percurrat, videbar mihi ritè inferre punctum, circa quod convertitur, distare ab extremitate per- cussâ ad totius longitudinis dodrantem: satis enim ipso usu in- notescebat, nec punctum medium, scilicet centrum gravitatis, nec oppositam extremitatem esse centrum conversionis. Vide- batur autem naturæ sua jura tueri conanti valde consentaneum, si corpus amans quietis externo impulsui ita obsecundet, ut quam minimo totius corporis motu impressus impetus partem percussam pro suæ intensionis modo transferat. Sit enim Cylindrus A B, cujus medium atque centrum gravitatis C: A E verò sit totius longitudinis dodrans: percutiatur extremitas A tanto impetu, quanto illa ferri posset per spatium A D, si moveretur circa centrum C. Ducatur igitur per C recta D O æqualis toti cylindro; qui si movere- tur circa punctum C, utique suo motu describeret duos Sectores, A C D, & B C O. Item per E ducatur ipsi D O parallela F I ita, ut ipsi E A æqualis sit E F, & ipsi E B æqualis sit E I. Cylindrus igitur A B conversione factâ circa punctum E describeret Sectores A E F & B E I duobus prioribus similes. Sunt autem Sectores similes, ut quadrata Radiorum; quemadmodum facilè colligitur ex 2 lib. 12: atque ideò, cum Radius A C ad Radium A E sit ut 2 ad 3, Sector A C D ad Sectorem A E F est ut 4 ad 9: & quia Radius B C ad Radium B E est ut 2 ad 1, Sector B C O ad Sectorem B E I est ut 4 ad 1. Motus igitur cylindri circa centrum C ad motum circa centrum E est ut 8 ad 10, si Sectores similes describantur. Atqui impetus impressus solum potest extremitatem A transferre per spatium æquale ipsi

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Mechanics 468 when, deceived by a certain appearance of truth, I had thought that the interval between the extreme prow and the point of conversion ought to be placed nearest to the fourth part of the whole length; I was led, namely, by a certain analogy taken from a floating wooden cylinder, whose extremity, if you strike it while it is at rest with such force that it traverses a certain distance, I seemed rightly to infer that the point about which it turns is distant from the struck extremity by three quarters of the whole length: for from actual experience it became sufficiently clear that neither the middle point, that is, the center of gravity, nor the opposite extremity is the center of rotation. And it seemed very consistent with nature’s striving to preserve its own laws, if a body that loves rest should so yield to an external impulse that, with the least possible motion of the whole body, it should transfer the struck part according to the measure of the force impressed upon it. Let there be a cylinder A B, whose middle and center of gravity is C: and let A E be three quarters of the whole length. Let the extremity A be struck with as much force as would be able to carry it through the space A D, if it were moved around the center C. Therefore draw through C the straight line D O equal to the whole cylinder; and if it were moved around the point C, it would certainly describe by its motion two sectors, A C D, and B C O. Likewise through E draw F I parallel to D O, so that E F may be equal to E A, and E I equal to E B. The cylinder A B, therefore, after a rotation made around the point E, would describe the sectors A E F and B E I, similar to the former two. But similar sectors are as the squares of the radii; as is easily gathered from Book 12, Proposition 2: and therefore, since the radius A C is to the radius A E as 2 to 3, sector A C D is to sector A E F as 4 to 9; and because the radius B C is to the radius B E as 2 to 1, sector B C O is to sector B E I as 4 to 1. Thus the motion of the cylinder around the center C is to the motion around the center E as 8 to 10, if similar sectors are described. Yet the impressed impulse can transfer only the extremity A through a space equal to it

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Liber quartus. CAPUT XV. 469 ipsi AD (est autem arcus AD ad arcum AF similem, ut Ra- dius AC ad Radium AE, hoc est ut 2 ad 3) igitur eandem transfert solùm per AG bessem arcûs AF, ac proinde motus est per Sectores AEG & BEH, qui ex ult. lib. 6. sunt bes duo- rum Sectorum AEF & BEI, quorum summa est 10; ipsius au- tem 10 bes est 6 ́. Motus igitur circa centrum E minor est motu circa centrum C, & impetus impressus æqualiter transfert extremitatem A. Fateor potuisse statui AE mediam proportionalem inter to- tam longitudinem AB & ejus semissem AC, & motus fuisset paulo minor. Ponatur enim tota AB 200, AC 100, est AE 141 ́ proximè: igitur ut quadratum AC ad quadratum AE mediæ proportionalis, hoc est ut 10000 ad 19994, ita Sector ACD ad sectorem AEF similem; & ut ipsius CB 100 qua- dratum 10000, ad ipsius EB 59 proximè quadratum 3481, ita Sector BCO ad Sectorem similem BEI. Quare summa Secto- rum ACD, BCO est 20000, Sectorum verò AEF, BEI est 23475. Sed quia ut AC ad AE, ita arcus AD ad arcum AF, quarum partium AD est 100, AF est 141 proximè: & assump- to arcu AG 100, summa Sectorum AEG & BEH, ad sum- mam Sectorum AEF & BEI erit ut 100 ad 141, hoc est, ut 16649 ad 23475: minor igitur est quàm summa Sectorum ACD & BCO, quæ est 20000. At si AE sit 150, & EB 50, summa Sectorum AEF, BEI ut 25000, bes autem 16666 ́, qui excedit numerum superiùs inventum 16649 adeò modico intervallo, ut contemnendum sit; cùm maximè impetus per ar- cum AF aliquantulo majorem motum efficiat quàm per cir- cumferentiam circuli minoris, ac propterea censendus sit arcus AG aliquantulum major quàm arcus AD; idcircò vero pro- pior est AE dodrans totius longitudinis AB, quàm AE media proportionalis inter AC & AB. Verùm quæ de cylindro in aquâ quiescente dicuntur satis probabiliter, non omninò congruere possunt motui navis, quæ præter motum aquæ percutientis gubernaculum promovetur à vento aut à remigibus, & præterea non habet æquabili ductu constitutam figuram, quemadmodum cylindrus: propterea huic analogiæ non acquiescendum duxi. Sed & illud adden- Nun 3

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Book Four. CHAPTER XV. 469 to it (for the arc AD is to the similar arc AF, as the radius AC to the radius AE, that is, as 2 to 3); therefore it transfers the same thing only through AG, a third part of the arc AF, and consequently the motion is through the sectors AEG and BEH, which, from the last proposition of Book 6, are a third of the two sectors AEF and BEI, whose sum is 10; but a third part of 10 is 6⅔. Therefore the motion about center E is less than the motion about center C, and the impressed impulse equally carries the extremity A. I confess that AE could have been taken as the mean proportional between the whole length AB and its half AC, and the motion would have been a little less. For let the whole AB be 200, AC 100, then AE is about 141; therefore as the square of AC is to the square of AE the mean proportional, that is, as 10000 to 19994, so is sector ACD to the similar sector AEF; and as the square of its CB, 100, which is 10000, is to the square of its EB, 59 approximately, which is 3481, so is sector BCO to the similar sector BEI. Therefore the sum of sectors ACD, BCO is 20000, but the sum of sectors AEF, BEI is 23475. But because as AC is to AE, so is arc AD to arc AF, of which parts AD is 100 and AF about 141; and taking arc AG as 100, the sum of sectors AEG and BEH, to the sum of sectors AEF and BEI, will be as 100 to 141, that is, as 16649 to 23475: therefore it is less than the sum of sectors ACD and BCO, which is 20000. But if AE be 150, and EB 50, the sum of sectors AEF, BEI is as 25000, but the third part is 16666⅔, which exceeds the number found above, 16649, by so small an interval that it may be disregarded; since at the very least the impulse through arc AF produces somewhat greater motion than through the circumference of the smaller circle, and therefore arc AG must be judged somewhat greater than arc AD; hence AE is closer to three quarters of the whole length AB than AE is the mean proportional between AC and AB. But the things said concerning the cylinder at rest in water, although sufficiently probable, cannot by any means agree with the motion of a ship, which, besides the motion of the water striking the rudder, is driven forward by the wind or by oars, and besides does not have a form established by a uniform curve, as a cylinder does: for this reason I judged that this analogy should not be accepted. But also that is added- Nun 3

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Mechanicorum 470 dum, quod neque de cylindro satis certus esse possum; nam si alia fiat hypothesis, & ad totius longitudinis bessem statuatur punctum conversionis ita ut semissis AC sit 3, AE verò sit 4, & EB sit 2; Sector ACD ad Sectorem AEF est ut 9 ad 16, & Sector BCO ad Sectorem BEI est ut 9 ad 4; igitur summa priorum ad summam posteriorum est ut 18 ad 20. Atqui Sector AEG ad Sectorem AEF est ut 3 ad 4 (id quod de simili Sectore BEH ad Sectorem BEI intelligendum est) igitur, cum A E F sit 16, AEG est 12, & cum BEI sit 4, BEH est 3, ac propterea summa Sectorum AEG & BEH est ut 15 ad sum- mam Sectorum ACD & BCO ut 18. In prima autem hypo- thesis quando erat AC ut 2 & AE ut 3, erat motuum Ratio ut 8 ad 6 ́, quæ est planè eadem cum Ratione 18 ad 15. Cum itaque eadem motuum Ratio sequatur, sive AE sit bes, sive dodrans totius longitudinis AB, cur dodrantem potiùs quàm bessem pronunciemus, nisi aliunde doceamur? CAPUT XVI. An malus in motu navis habeat Rationem vectis. NAvim impelli ventorum vi certum est, qui velum implent ex antennâ suspensum atque expassum, funibúsque, quos Propedes vocant, posteriori navis parti alligatum. Quoniam verò, ut quotidiano usu didicimus, quò altiùs evecta fuerit an- tenna, eò validiùs, cæteris paribus, navis à vento impellitur, quæritur ab Aristotele quæst. 6. Cur quando antenna sublimior fuerit, iisdem velis, & eodem vento, celeriùs feruntur navigia? Causam ille ex vectis rationibus petendam opinatur, quasi ma- lus sit vectis habens hypomochlium in ea carinæ parte, cui in- figitur; potentia movens sit ventus velum implens supremæ mali parti applicatus, ubi antenna cum malo connectitur; pon- dus verò sit navigium: quò igitur potentia magis ab hypomo- chlio abest, plus habere momenti manifestum est. Verùm non ego hîc inani labore suscepto, ut Philosophi dicto aliquam veri similitu

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Mechanics 470 so, because I cannot be sufficiently certain about the cylinder; for if another hypothesis be made, and the point of conversion be fixed at the eight-ninths of the whole length, so that the half AC is 3, AE is 4, and EB is 2; the Sector ACD to the Sector AEF is as 9 to 16, and the Sector BCO to the Sector BEI is as 9 to 4; therefore the sum of the former to the sum of the latter is as 18 to 20. But the Sector AEG to the Sector AEF is as 3 to 4 (which is to be understood also of the similar Sector BEH to the Sector BEI); therefore, since AEF is 16, AEG is 12, and since BEI is 4, BEH is 3, and accordingly the sum of the Sectors AEG and BEH is as 15 to the sum of the Sectors ACD and BCO as 18. But in the first hypothesis, when AC was as 2 and AE as 3, the ratio of the motions was as 8 to 6, which is plainly the same as the ratio of 18 to 15. Since, therefore, the same ratio of motions follows, whether AE be the eighth part, or the three-quarters part, of the whole length AB, why should we pronounce the three-quarters part rather than the eighth part, unless we are taught otherwise? CAPUT XVI. Whether the mast in the motion of a ship has the ratio of a lever. It is certain that a ship is driven by the force of the winds, which fill the sail hanging and spread out from the yard, and fastened by ropes, which they call Propedes, to the after part of the ship. But since, as we have learned from daily experience, the higher the yard has been raised, the more strongly, other things being equal, the ship is driven by the wind, Aristotle asks in question 6, Why, when the yard is higher, are ships carried more swiftly with the same sails and the same wind? He thinks the cause must be sought from the ratios of the lever, as if the mast were a lever having its hypomochlion in that part of the keel into which it is fixed; the moving power being the wind filling the sail applied to the upper part of the mast, where the yard is connected with the mast; and the weight being the ship. Therefore the farther the power is from the hypomochlion, the greater its efficacy is manifestly. But I here, not undertaking vain labor, in order to give some appearance of truth to the Philosopher's statement...

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Liber quartus. CAPUT XVI. 471 similitudinem adjiciam, tempus conteram: Cui otium est, Authores legat. Si quæstio esset, Cur longiores mali sint magis obnoxij periculo fractionis, facilè invenirem rationem vectis, quia ponderis vicem subeunt particulæ ipsæ, quarum nexus per vim solvendus est in fractione; quò autem longior est malus, ad motum partium, quæ dividuntur, majorem Rationem habet motus venti applicati longiori malo, quàm motus ejusdem breviori malo applicati. Sed hîc, ubi de navis motu quæstio est, sive altè assurgat malus, sive brevis sit, semper eadem est Ratio motûs venti velo excepti, atque navis. Quomodo enim pterna, id est ima mali calx, esse potest extremus vectis in carinâ, cui inferitur, velut in hypomochlio quiescens, navis verò tota simul mota æquali motu, rationem habet ponderis à vecte impulsì nonne hypomochlium, pondus, & potentia æquali planè motu moventur? neque enim velociùs movetur ventus velo exceptus, quàm ipsum velum, nec velum velociùs quàm navis; & cum ipsa navi planè æqualiter movetur carinæ punctum, cui malus infigitur. Quis autem motus per Vectem, qua Vectis est Facultas mechanica, hujusmodi æqualitatem admittit? Non igitur malus in motu, quo navis progreditur, Rationem vectis habere dicendus est. Sit malus CD cujus pterna C inferatur carinæ AB, carchesis autem D applicetur antenna cum velo pendente, cujus imæ extremitates navis lateribus opportunè ad ventum excipiendum jungantur. Certum est malum CD moveri semper sibi parallelum (nisi fortè aliquanto plus extremitas D moveatur, sicut in homine plus caput moveatur quàm pedes supra sphæricam terræ vel aquæ superficiem, sed hoc nihil refert) neque posse obtinere rationem vectis nisi comparatè ad eum motum, quo circa C tanquam circa centrum fieret conversio; quemadmodum si deprimenda esset prora & elevanda puppis, ut carina AB non esset horizonti parallela, sed

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Book Four. Chapter XVI. 471 I shall add a comparison, and waste time: Let him who has leisure read the authors. If the question were, Why longer masts are more liable to the danger of breaking, I could easily find an explanation from the lever, because the particles themselves take the place of the weight, whose connection must be broken by force in breaking; and the longer the mast is, the greater, in the motion of the parts that are divided, is the ratio of the motion of the wind applied to the longer mast, than of the motion of the same wind applied to the shorter mast. But here, where the question is about the motion of the ship, whether the mast rises high or is short, the ratio of the motion of the wind caught by the sail and that of the ship is always the same. For how can the foot of the mast, that is, the lower heel of the mast, be the end of a lever in the keel, in which it is fixed, as though resting on a fulcrum, while the whole ship is moved at the same time with equal motion? And does not the fulcrum, the weight, and the power, moved by one and the same motion, move equally? For the wind caught by the sail is not moved more quickly than the sail itself, nor the sail more quickly than the ship; and with the ship itself equally is moved the point of the keel in which the mast is fixed. But what motion through a lever, since it is a lever, that is, the mechanical faculty of a lever, admits of such equality? Therefore the mast, in the motion by which the ship advances, must not be said to have the ratio of a lever. Let the mast be CD, whose foot C is set into the keel AB, while the masthead D is attached to the yard with the hanging sail, whose lower extremities are conveniently joined to the sides of the ship for receiving the wind. It is certain that the mast CD is always moved parallel to itself (unless perhaps the extremity D is moved somewhat more, as in a man the head is moved more than the feet above the spherical surface of the earth or water, but that makes no difference) nor can it obtain the ratio of a lever except in comparison with that motion by which there would be a turning about C as about the center; just as if the prow were to be lowered and the stern raised, so that the keel AB would not be parallel to the horizon, but

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Mechanicorum 472 sed B deprimeretur infra planum horizontale, & A supra illud elevaretur. Verùm (præterquam quod non hic est navis mo- tus, de quo disputatur) observandum est in navigiis minoribus, quibus movendis unicus malus adhibetur, hunc statui non in medio navigio, sed magis accedere ad proram, in majoribus autem navibus, quæ plures malos habent, maximum quidem malum, cujus validissimæ sunt vires, aliquanto quidem magis ad puppim quàm ad proram accedere (si longitudo in superiori parte attendatur) ut in Oceano videre est Anglicas, Gallicas, Hollandicas naves, comparatè tamen ad carinam, majorem carinæ partem puppim, minorem proram respicere. Id autem eo consilio factum est, ne malus centro gravitatis navis respon- deat, neque exercere possit munus vectis deprimendo proram, puppimque elevando. Quando enim magis ad proram accedit malus, major pars navigij inter malum & puppim interjecta re- nititur sua gravitate, ne elevetur, quando verò à medio pup- pim versùs recedit, major pars navis, quæ deprimenda esset, majorem aquæ resistentiam invenit; ac proinde servatâ carinæ positione horizontali faciliùs navis movetur. Hinc tardiorem fieri navigij cursum contingit, vel quia perperam collocatus est malus, vel quia pondera in navi non sunt ritè distributa, adeò ut à malo vix absit centrum gravitatis navigij onusti; tunc enim depressâ prorâ & carinâ ad horizontem inclinatâ major vis ob- viæ aquæ resistit. Quare tantum abest malus à ratione vectis, vi cujus progrediatur navigium, ut potius caveatur, ne vectis munus ille exerceat, motum aliquem efficiendo, qui celeritati non parum officeret. In motu autem majoris navigij pluribus malis instructi non solus malus, qui præcipuus est & maximus, attenditur, sed etiam reliqui: potior tamen ad provehendam navim est malus, qui à medio ad proram accedit, quippe qui navim trahit; nam qui à centro gravitatis puppim versùs recedit, navim impellit potiùs, quàm trahat: quamquam ille, qui ad puppim proximè spectat, & velum habet triangulare, maximè juvat, ut gubernatoris pro- posito, qui clavum regit, obsecundet ad navis cursum in alteru- tram partem dirigendum. Verum quicumque malus conside- retur, in nullo rationem vectis reperies, sive ad impellendam, sive ad trahendam navim. At,

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Mechanics 472 but if B were lowered below the horizontal plane, and A raised above it. However (apart from the fact that this is not here the motion of a ship about which the discussion is concerned), it must be observed that in smaller vessels, which are moved by means of a single mast, this mast is placed not in the middle of the vessel, but rather nearer the prow; in larger ships, however, which have several masts, the principal mast, whose powers are strongest, is placed somewhat nearer the stern than the prow, if the length in the upper part is considered, as may be seen on the ocean in English, French, and Dutch ships; yet, in relation to the keel, the greater part of the keel faces the stern and the lesser the prow. This has been done for this purpose, that the mast may not correspond to the ship’s center of gravity, nor be able to perform the office of a lever by depressing the prow and raising the stern. For when the mast approaches more toward the prow, the larger part of the ship between the mast and the stern resists with its weight, lest it be raised; but when it recedes from the middle toward the stern, the larger part of the ship, which would have to be depressed, encounters greater resistance from the water; and therefore, the keel being kept horizontal, the ship is moved more easily. Hence it happens that the ship’s course becomes slower, either because the mast has been wrongly placed, or because the weights in the ship are not properly distributed, so that the center of gravity of the laden vessel is scarcely far from the mast; for then, with the prow depressed and the keel inclined to the horizon, the greater force of the opposing water resists. Therefore the mast is so far from the function of a lever by whose force the ship advances, that rather care is taken lest it should exercise the function of a lever, producing some motion that would not a little hinder speed. In the motion of a larger ship fitted with several masts, not only the principal and largest mast is considered, but the others as well; yet the more effective in propelling the ship is the mast that approaches from the middle toward the prow, since it draws the ship; for the one that recedes from the center of gravity toward the stern pushes the ship rather than draws it. Although the one that stands nearest the stern, and has a triangular sail, helps most, so as to comply with the purpose of the helmsman, who governs the tiller, in directing the ship’s course to one side or the other. Yet whichever mast is considered, in none will you find the principle of the lever, whether for driving or for drawing the ship. But,

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Liber quartus. CAPUT XVI. 473 At, inquis, si adverso flumine deducendum sit navigium si- ve à nautica turbâ sive ab equis trahentibus, cur funis ma- lo, non autem proræ, alligatur, si nihil confert facilitatis appli- catio potentiæ trahentis medio fune ad majorem altitudinem à carinâ? Ego verò ex te, quisquis hæc objicis, quæro, cur iidem nautæ si remulco navim trahere aggrediantur, funem navi non tam altè alligant; si ex vectis rationibus illa altitudo aliquod af- fert compendium laboris in trahendo. Sed satis utrique quæstio- ni factum videbis, si observes non planè æqualem esse in uni- verso alveo aquæ altitudinem, ac proinde neque posse navim æquè semper abesse à fluminis ripâ, in qua trahentes progre- diuntur; idcircò longiore fune opus est, qui suo pondere spon- te curvatus aquam secaret, & trahentium laborem augeret, aut in occultum aliquem sub aquis latentem obicem incurreret non sine gravi incommodo, si funis extremitas depressiori loco navi- gij alligaretur; propterea malo altiùs adnectitur, eo quoque consilio, ut si quæ virgulta aut arbusculæ secundùm fluminis ripam occurrant, minori impedimento sint funi obliquè incli- nato, quàm si horizonti esset ferè parallelus. Qui verò navim remulco trahunt, non adeò longè ab illa abesse coguntur, nec hujusmodi impedimentis obnoxij sunt; ideò breviore fune utuntur, quem proræ alligant. Cæterùm nullæ vectis vires exercentur; non enim prora infra aquam deprimi, & puppis elevari potest: id quod si contingeret, prora magis demersa plus inveniret resistentiæ ab aquâ dividendâ. Quid igitur, ais, causæ est, quòd antennâ usque ad carche- sium D elevatâ, magis promovetur navis, quàm si tantummodo usque ad E attolleretur? quandoquidem nulla vectis ratio hîc habetur. Eos, qui cum Aristotele sentiunt, æquivocatione la- borare facilè ostenditur: quid enim refert, utrum antenna ma- gis an minùs elevetur, si potentia, videlicet ventus velum im- plens, illi mali parti applicata intelligeretur, cui antenna ad- nectitur? hæc autem funibus, quos vocant, sursum trahitur, semperque, sive altior, sive depressior sit, adnectitur carchesio in D: quemadmodum nauta fune in D alligato na- vim trahens, semper in D applicatus intelligitur, quamvis hu- miliore in loco, quàm D, constituatur. Verùm non ibi vis venti præcisè intelligenda est, ubi antenna est, sed toti malo O o o

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Book Four. CHAPTER XVI. 473 But, you say, if a vessel must be hauled down a river, whether by a crew of sailors or by horses drawing it, why is the rope fastened to the mast and not to the prow, if attaching the power of the puller to the middle of the rope at a greater height from the keel contributes nothing to ease of movement? For my part, I ask you, whoever you are that raise this objection, why the same sailors, if they begin to tow a ship with a towing line, do not fasten the rope to the ship at so great a height, if that height, according to the principles of leverage, brings some saving of labor in towing. But you will see that enough has been said to answer both questions if you observe that the height of the water is not exactly equal throughout the whole channel, and therefore the ship cannot be kept at the same distance from the riverbank along which the haulers proceed; for that reason a longer rope is needed, which by its own weight, bending of itself, would cut through the water and increase the labor of those pulling, or would strike some hidden obstacle lying beneath the water, not without serious inconvenience, if the end of the rope were fastened to a lower part of the vessel; therefore it is attached higher up, also with this intention, that if any twigs or little trees should occur along the riverbank, they may be less of an impediment to a rope slanting obliquely than if it were nearly parallel to the horizon. But those who tow a ship with a towing line are not forced to keep so far from it, nor are they subject to such impediments; therefore they use a shorter rope, which they fasten to the prow. Moreover, no force of leverage is employed; for the prow cannot be pressed down below the water and the stern raised up: if that were to happen, the more submerged prow would find greater resistance from the water it must divide. What then, you say, is the reason that, with the antenna raised as far as the carchesium D, the ship is advanced more than if it were raised only as far as E? since no leverage is involved here. Those who agree with Aristotle are easily shown to be confused by ambiguity: for what does it matter whether the antenna is raised more or less, if the power, namely the wind filling the sail, is understood to be applied to that part of the mast to which the antenna is attached? But this is drawn upward by ropes, as they are called, and whether it is higher or lower, it is always attached to the carchesium at D; just as a sailor, when pulling a ship with a rope fastened at D, is always understood to be applying force at D, even though he is positioned in a lower place than D. But here the force of the wind is not to be understood precisely where the antenna is, but to the whole mast O o o

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Mechanicorum 474 aut ejus parti applicatur, quæ respondet velo non solùm antennæ cornibus, sed etiam navis lateribus alligato: velum autem in humiliore loco minus recipit venti, quia alta majorum navium puppis (nisi ventus ex latere spiret) vento opposita illum subtrahit velo, & præterea ventus, qui in navis puppim & latera illiditur, reflectitur, & proximas venti partes turbat, atque aliorsum dirigit, vel saltem illarum impetum imminuit; ex quo oritur minori vi impelli velum. At partes venti sublimiores ab his inferioribus reflexis, vel nihil, vel mitiùs turbantur, atque adeò plures ad implendum velum majore vi accurrunt. Adde (his etiam mente seclusis) ventum in sublimiore loco multo validiorem esse, quàm in inferiore, ac propterea quò altius attollitur velum, non solùm majorem, sed etiam validiorem ventum excipit, quo fit, ut incitator sit navigij motus. Neque de hoc venti discrimine dubitare poterit cui contingat iter habere in ampla planitie arboribus & ædificiis vacuâ vento flante; si enim ex equo desiliat, & humi sedeat, manifestè percipiet, quanto minore vi impetatur à vento. Id quod pariter ex ipsâ veli figurâ arguitur; sive enim velum triangulare fuerit, & obliquâ antennâ erigatur ita ut quasi aurem leporis imitetur, altiori vento, utpote vehementiori, pars veli strictior objicitur; sive pluribus quadrangularibus velis instruatur navigium ita, ut alia superiora, scilicet dolones, alia inferiora sint, videlicet Acatia; quæ supra Corbem statuuntur, non solùm minora sunt inferiore velo, sed etiam eorum suprema pars longè strictior est basi, ut nimirum minus recipiat venti validioris: propterea ingruente tempestate primùm superiora vela deprimuntur, ut majori ventorum vi subducantur; eriguntur autem celeritatis causa, ut si quando effusè fugere opus sit. Ecce igitur citra omnem vectis rationem, Car quando antenna sublimior fuerit, iisdem velis, & vento eodem, celeriùs feruntur navigia: quia scilicet velum altiùs sublatum & plus venti, & validiorem ventum recipit. Quod si ad vectis rationes confugiendum esset, non quæreretur, cur celeriùs ferantur navigia, sed, cur faciliùs? Nam vectis longitudo (nisi fortè in vecte tertij generis, cujus nullum vestigium deprehenditur in malo navis) non celeritatem motûs ponderi conciliat, sed facilitatem, ita

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Mechanics 474 is applied to it, or to that part which corresponds to the sail, not only fastened to the yardarms, but also to the sides of the ship: but a sail placed in a lower position receives less wind, because the high stern of larger ships (unless the wind blows from the side), standing opposite the wind, withdraws it from the sail; and besides, the wind, which strikes the stern and sides of the ship, is reflected, disturbs the nearby parts of the wind, and directs them elsewhere, or at least lessens their force; from which it follows that the sail is driven with less power. But the higher parts of the wind are by these lower reflected parts either not disturbed at all, or only more gently, and therefore more of them come together with greater force to fill the sail. Add to this (even setting these considerations aside) that the wind in a higher place is much stronger than in a lower one, and therefore the higher the sail is raised, the greater and also the stronger wind it receives, so that it becomes an incentive to the motion of the ship. Nor can anyone doubt this difference in the wind who happens to travel across an open plain, empty of trees and buildings, with the wind blowing; for if he should leap down from a horse and sit on the ground, he will clearly perceive how much less forcefully he is assailed by the wind. This is likewise inferred from the very shape of the sail; for whether the sail be triangular and raised on an oblique yard so as to imitate, as it were, a hare’s ear, the narrower part of the sail is exposed to the higher wind, since it is the more violent; or whether the ship be equipped with several quadrangular sails, so that some are upper, namely dolones, others lower, namely Acatia; those which are placed above the Corbis are not only smaller than the lower sail, but also their upper part is much narrower than the base, namely so that it may receive less of the stronger wind: therefore, when a storm arises, the upper sails are first lowered, so that they may be withdrawn from the greater force of the winds; but they are raised for the sake of speed, if at any time it is necessary to flee at full sail. Behold then, apart from any consideration of the lever, that when the yard is higher, with the same sails and the same wind, ships are driven more swiftly: namely because a sail raised higher receives both more wind and a stronger wind. But if recourse had to be had to lever principles, the question would not be why ships are driven more swiftly, but why more easily? For the length of a lever (unless perhaps in a lever of the third kind, of which no trace is found in the mast of a ship) does not impart swiftness of motion to the weight, but ease, ita

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Liber quartus. CAPUT XVII. 475 ita ut posito longiore vecte potentia servans eandem sui motûs velocitatem faciliûs quidem moveat propositum pondus, sed tardiùs quàm breviore vecte, positâ eâdem ponderis ab hypo- mochlio distantiâ: Ac propterea, si in hoc navis motu, de quo quæstio est, intercederet ratio vectis, idem ventus eadem vela altiùs sublata implens eâdem quidem velocitate moveretur, sed tardiùs navim moveret, quamquam faciliùs, hoc est magis onustam. Id autem à vero longissimè abesse testatur experien- tia; quæ idcirco confirmat navigij malo nihil esse cum vecte commercij ad navim promovendam. CAPUT XVII. An ex vectis rationibus pendeat usus anchoræ. Quandoquidem nauticas aliquot quæstiones cum Aristote- le superioribus capitibus examinare placuit, liceat & hanc addere, quæ ad usum anchoræ spectat in firmanda navi, ne à fluctibus, aut à vento abripiatur: tranquillo enim mari, aut in lacu quiescente, sua sponte subsistit navis, nec anchoræ ope in- diget, ut sua in statione permaneat. Et quidem ipsa navis gra- vitas cum suis instrumentis, & onus quod illa ferre potest (cu- jus gravitas æquat navigij gravitatem) satis per se resistunt, nec facilè cuilibet auræ aut fluctui cedunt. Quare major esse debet vis venti, aut fluctuum, aut profluentis, quàm ut illi ob- sistere valeat universum navigij pondus, ad hoc ut sit opus an- chorâ, qua navigium firmetur. In anchorâ autem spectanda est & gravitas, & figura; utra- que enim juvat; aliquando si quidem solum anchoræ pondus sufficit, ne placidiores fluctus, aut fluminis impetus, aut lenis flatus navim secum rapiant. Sic legisse me memini navim à naufragio anchoris omnibus destitutam in statione totam noctem quievisse securam firmatam facculo, quo mille trecen- ti Hispani Crucigeri (octo Reales singulis Crucigeris tribuun- tur) continebantur, rudentis autem munus supplebat evolutus Ooo 2

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Book Four. Chapter XVII. 475 so that, if by placing a longer lever the power, while preserving the same speed of its motion, were to move the intended weight more easily, it would do so more slowly than with a shorter lever, the same distance of the weight from the fulcrum being assumed. And therefore, if in this motion of a ship, of which the question is being asked, there were any relation to a lever, the same wind, filling the same sails raised higher, would indeed move with the same speed, but would move the ship more slowly, though more easily, that is, with a greater burden. But experience testifies that this is very far from the truth; and thus confirms that the ship’s mast has no dealings with the lever in promoting the ship’s movement. Chapter XVII. Whether the use of the anchor depends on the principles of the lever. Since in the preceding chapters it has pleased us to examine some nautical questions with Aristotle, let us also add this one, which concerns the use of the anchor in securing the ship, lest it be carried off by the waves or by the wind: for in calm sea, or in a still lake, the ship of itself comes to rest and needs no help from an anchor in order to remain at its station. Indeed, the weight of the ship itself with its equipment, and the load which it can bear (whose weight equals that of the vessel), sufficiently resist on their own, and do not easily yield to any breeze or wave. Therefore the force of the wind, or of the waves, or of the current, must be greater than the entire weight of the vessel can oppose, for there to be need of an anchor, by which the ship may be secured. In an anchor, however, both weight and shape are to be considered; for both are helpful. Sometimes indeed the anchor’s mere weight is sufficient, so that gentler waves, or the force of a river, or a mild breeze do not carry the ship away with them. Thus I remember reading of a ship, deprived of all anchors after shipwreck, which rested safely in its mooring throughout the whole night, secured by a bundle containing one thousand three hundred Spanish Cruzigeri (eight Reales are assigned to each Cruziger), while the office of rope was supplied by a unrolled

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Mechanicorum telæ scapus: qui enim fluctus navim aliò propellere potuissent, non satis habebant virium, ut etiam illud argenti pondus maris fundo incumbens & navi connexum pariter trahere possent. Simili igitur ratione anchora, licèt duriori solo dentem infigere non valeat, aliquando suo pondere navim firmabit. Respondet autem anchoræ gravitas oneri, quod ferre potest navis, ea Ratione, ut pro oneris libris 40000 (hoc est 20 Amphoris aut dolis, singulorum quippe doliorum gravitas statuitur librarum bis mille, & singulis libris unciæ sexdecim tribuuntur) ferri libras centum & decem habeat primaria & maxima anchora, secunda habeat primæ dodrantem, tertia bessem, quarta semissem. Rudentis vero, cui anchora adnectitur, pondus ferè ponitur duplum sesquiquartum gravitatis suæ anchoræ. Quamquam non omnino servetur hæc Ratio ponderis anchoræ in ingentibus navigiis, quæ nimirum suâ gravitate maximè resistunt fluctuum impulsioni, ac proinde minore anchorâ opus habent. Primariæ anchoræ potissimus usus communiter est, cùm validior tempestas navim aggreditur; secundæ, ut navis in statione quiescat; tertiam adhibent nautæ, ut duabus anchoris ad diversas plagas constitutis (puta, alterâ ad Subsolanum, alterâ ad Boream aut ad Borrhapeliotem,) vento & fluctibus navis resistat validiùs, nec abrepta à fluctibus anchoram pariter secum rapiat, sed tantum alternis motibus quasi circa centrum agitetur: Quartam demum lintre transferunt procul à navi juxta longitudinem funis adnexi ducentorum circiter cubitorum, quem machinâ ad id destinatâ colligentes accedunt ad anchoram, & stationem commutant, aut portum intrant, seu ab illo exeunt, ubi cessat ventus, aut adversus spirat. Ad firmandam verò navim plurimum habet momenti longitudo ipsa rudentis; satis enim manifestum est, quantâ vi opus fit, ut longior funis intendatur, qui cessante externâ vi illico sanuatur, ac propterea vehementi conatu ventorum ac fluctuum navis impellenda est, ut rudens intentus anchoram trahat. Varia est autem Rudentis longitudo pro anchorarum Ratione; longitudo si quidem rudentis anchoræ primariæ cubitos habet centum viginti, secundæ cubitos centum, tertiæ cubitos octoginta: quò enim adversùs validiorem impetum repugnandum est,

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Mechanicorum the shaft of the loom: for the waves could indeed have driven the ship elsewhere, but they did not have enough force to be able also to draw along that weight of silver, resting on the sea bottom and connected to the ship. In a similar way, therefore, the anchor, although it cannot embed its tooth in harder ground, will sometimes steady the ship by its own weight. The weight of the anchor corresponds to the burden that the ship can carry, in this way: for a load of 40,000 pounds (that is, 20 Amphorae or casks, since the weight of each cask is fixed at 2,000 pounds, and 16 ounces are assigned to each pound) the primary and largest anchor should weigh 110 pounds, the second should be three quarters of the first, the third two thirds, the fourth one half. The weight of the cable, to which the anchor is attached, is usually taken to be two and a quarter times the weight of the anchor itself. Although this ratio of anchor weight is not entirely observed in very large ships, which of course by their own weight most greatly resist the force of the waves, and therefore need a smaller anchor. The chief use of the primary anchor is commonly when a stronger storm attacks the ship; the second, so that the ship may remain at rest at anchor; the third sailors use so that, with two anchors set in different directions (for example, one toward Subsolanus, the other toward Boreas or toward Borrhapeliotes), the ship may resist wind and waves more strongly, and not be carried off by the waves together with the anchor, but rather be tossed only by alternate motions, as it were around a center: the fourth at last they transfer by boat far from the ship, along the length of the attached rope of about two hundred cubits, and by drawing it in with the machine intended for that purpose they approach the anchor, and change their station, or enter the harbor, or leave it, when the wind ceases, or blows against them. But for securing a ship the length of the cable itself is of great importance; for it is quite evident how much force is needed to stretch a longer rope, which, when the external force ceases, immediately slackens again; and therefore the ship must be driven by a violent onslaught of winds and waves, so that the tightened cable may pull the anchor. Now the length of the cable varies according to the ratio of the anchors; for the length of the cable of the primary anchor is 120 cubits, of the second 100 cubits, of the third 80 cubits: for the stronger the force to be resisted,

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Liber quartus. CAPUT XVII. 477 est, eò longior adhibetur rudens, ut difficiliùs intendatur, ac idcirco fractionis periculo minùs obnoxius sit, & venti fluctuumve impetus in rudente intendendo elisus minus vi- rium habeat ad rapiendam simul cum navi anchoram. Hinc ingentes bellicæ naves in Oceano ferè semper primariam an- choram demittunt, & tres aut quatuor rudentes capitibus in- vicem firmiter colligatis in unam longitudinem productos adji- ciunt; vix enim tanta esse potest fluctuum aut venti vis, quæ valeat tam longum rudentem intendere atque dirumpere, nisi fortè ad navis latus aut ad scopulum collisus atteratur. Id quod aliis quoque nautis placet tùm propter eandem causam, tùm ut longiùs à littore consistere possit navis, & anchora arenæ infi- gi, etiamsi altior sit aqua. Mihi sanè contingit nautarum incu- riam experiret in Albi fluvio; cùm enim anchoram breviore ru- dente demisissent, nocturno maris æstu intumescentibus undis ita elevatum est navigium, ut ex prorâ penderet suspensa an- chora, nôsqe dormientes æstus abriperet; quos demum exci- tavit fragor ex collisione cum altero navigio, in quod tanto im- petu impacti fuimus, ut abrupto fune scapham amiserimus. Sed quod ad anchoræ formam attinet, non eadem omnibus est figura; navigia enim, quorum in majoribus fluminibus usus est, ut noctu in medio alveo subsistant, anchoram habent qua- tuor aduncis brachiis instructam; cujusmodi pariter sunt trie- mium anchoræ. At in Oceano navium anchoræ non nisi duo habent brachia ad angulum acutum inflexa cum scapo; ne ve- rò demissa anchora prorsus jaceat in maris fundo, scapo prope annulum adnectitur ligneum transversarium ( cujus gravitas est ferè subquintupla gravitatis anchoræ, si tamen etiam fer- reos clavos, quibus firmatur, in computationem admittas) ejus- dem cum Scapo longitudinis, adeò ut jacente utroque brachio Scapus transversario secundum extremitatem innixus obliquè inclinetur. Ex quo etiam sit, ut extremæ brachiorum palmulæ obliquè occurrentes maris fundo faciliùs in subjectum solum penetrent. Quando igitur vehementior est fluctuum impetus, aut venti impulsus validior, quàm ut illi resistere possit ipsa an- choræ gravitas, intento rudente tantisper abripit cum navi an- choram, quæ maris fundum sulcans, ubi brachiorum palmu- læ arenis aliquantulum immerfæ inæqualem invenerint subjecti O o o 3

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Book Four. Chapter XVII. 477 the longer the cable used, so that it is stretched with more difficulty, and therefore is less exposed to the risk of breaking, and the force of the wind or waves, being spent in tightening the cable, has less power to carry off the anchor together with the ship. Hence the great warships in the Ocean almost always let down the main anchor, and add three or four cables, with their ends firmly tied together into one length. For scarcely can there be so great a force of waves or wind as to stretch and snap such a long cable, unless perhaps it be worn away by striking against the side of the ship or against a rock. This is pleasing to other sailors too, both for the same reason, and so that the ship may stand farther from the shore, and the anchor be fixed in the sand, even if the water be deeper. Indeed, I myself had occasion to experience the negligence of sailors in the river Albis; for when they had let down the anchor with too short a cable, by night, as the sea tide swelled and the waves rose, the vessel was so lifted up that the anchor hung suspended from the bow, and the tide swept us away as we slept; at last the crash of a collision with another ship woke us, into which we were driven with such force that, the rope having snapped, we lost the boat. But as for the form of the anchor, it is not the same for all; for vessels used on larger rivers, so that they may remain at night in mid-channel, have an anchor equipped with four hooked arms; such also are the anchors of triremes. But the anchors of ships in the Ocean have only two arms bent at an acute angle with the shank; and lest the anchor, when dropped, should lie flat on the bottom of the sea, a wooden crosspiece is attached to the shank near the ring, the weight of which is nearly five times the weight of the anchor itself, if indeed you also admit into the calculation the iron nails by which it is secured; it is of the same length as the shank, so that, when both arms lie down, the shank, resting on the crosspiece by its farther end, inclines obliquely. From this it also comes about that the outer flukes of the arms, meeting obliquely, more easily penetrate into the ground beneath when they encounter the bottom of the sea. Therefore, when the force of the waves is more violent, or the thrust of the wind stronger than the weight of the anchor itself can withstand, the tightened cable drags the anchor along with the ship for a time, and the anchor, furrowing the seabed, when the flukes of the arms, half-buried in the sand, have found the unevenness of the underlying

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Mechanicorum 478 soli resistentiam (quocumque tandem ex capite oriatur hæc re- sistentiæ inæqualitas) positionem mutat, nec ampliùs jacet utrumque brachium, sed illud, cui minùs obsistitur, elevatur, adnitente etiam ligneo transversario, cui naturalis est in aquâ positio horizonti parallela, quam acquirens ita anchoram con- vertit, ut Dens maris fundo inhærens magis in illud infigatur tùm urgente deorsum ipsius anchoræ gravitate, tùm trahente ipsa navi, quam fluctus aut ventus impellit; cùm etenim bra- chium cum scapo acutum angulum constituat, non ad perpen- diculum, sed obliquè fundum ingreditur, & idcirco in illud profundiùs penetrat. Cum itaque anchoræ scapo alij duplicem, alij triplicem tri- buant alterius brachij longitudinem, hæc utique major est, quàm distantia inter extremos anchoræ dentes, non enim bra- chia cum scapo rectum sed acutum angulum, ut dictum est, constituunt. Ad hanc igitur extremorum dentium distantiam major transversarij longitudo majorem habet Rationem, quàm minor; est autem longitudini scapi par transversarij longitudo; quare longioris anchoræ transversarium longius est, ejusque conversio, ut se horizonti parallelum statuat, magis juvat an- choræ conversionem, ut dens inferior profundiùs in arenam infigatur. Sit primùm anchoræ scapus A B duplex longitudinis bra- chij A C, & prope an- nulum in B æquale trans- versarium E F adjiciatur ad angulos rectos, ea ta- men conditione, ut ja- centibus brachiis A C & A D in plano hori- zontali, transversarium sit in plano verticali, ejus- que altera extremitas, ex. gr. F. maris fundum con- tingat, altera E sublimis sit, ac proinde scapus A B inclinetur ad horizon- tem grad. 30. Vento, aut fluctu, navim impellente intenditur rudens,

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Mechanics 478 the resistance of the ground (from whatever source this inequality of resistance may finally arise) changes the position; and no longer do both arms lie down, but the one to which less opposition is offered is raised, the wooden crosspiece also helping, whose natural position in water is parallel to the horizon; by acquiring this position it turns the anchor so that the tooth of the sea, adhering to the bottom, is more firmly fixed in it, both while the weight of the anchor itself presses downward and while the ship itself, driven by waves or wind, pulls. For since the arm with the shank forms an acute angle, it enters the bottom not perpendicularly but obliquely, and therefore penetrates it more deeply. Since, then, some give the shank of the anchor a double, others a triple length of the other arm, this length is certainly greater than the distance between the two extreme teeth of the anchor; for, as has been said, the arms with the shank do not form a right angle but an acute one. Therefore, with respect to this distance between the extreme teeth, a greater length of crosspiece bears a greater relation than a smaller one; but the length of the crosspiece is equal to the length of the shank; hence the crosspiece of the longer anchor is longer, and its turning, so as to place itself parallel to the horizon, more helps the turning of the anchor, so that the lower tooth may be fixed more deeply in the sand. Let first the shank A B of the anchor be twice the length of the arm A C, and near the ring at B let an equal crosspiece E F be attached at right angles, on condition, however, that when the arms A C and A D lie in the horizontal plane, the crosspiece shall be in the vertical plane, and that one of its ends, for example F, shall touch the sea bottom, while the other, E, shall be raised; and consequently the shank A B shall incline to the horizon by 30 degrees. When wind or wave drives the ship, the rope is strained,

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Liber quartus. CAPUT XVII. 479 rudens, & scapi extremitas B annulo proxima elevatur, nec ampliùs transversarium in F incumbit arenæ; propterea bra- chiorum palmulæ C & D in triangulum conformatæ, dum si- mul cum navi trahuntur, se se profundiùs in arenam insinuant: sed si inæqualem offendant resistentiam, aut altera, ex. gr. C, profundiùs infigatur præ reliquâ (sive ex subjecti soli diversi- tate, sive quia navis in transversum acta trahit anchoræ caput B in latus, & brachij alterius extremitas describens circa A in solo arcum versus navim profundiùs infigitur, atque adeò re- liqua extremitas oppositi brachij in contrarium mota circa A, vix terram mordet) vis in B trahens, neque valens pariter utrumque brachium trahere, cogitur circa C, tanquam circa centrum, seu potiùs tanquam circa polum, moveri. Et quia punctum B sublimius est puncto C, necesse est ita hujusmodi conversionem fieri, ut opposita extremitas D elevetur, atque ex fundo extrahatur. Cumque jam transversarium non æqua- liter hinc & hinc retineatur per vim in plano verticali, sed ejus superior pars B E versus C inclinetur, conatur positionem ho- rizontalem acquirere, ejusque inferior pars B F ad latus decli- nans ascendit, juvátque ipsius brachij A D ascensum; ex quo fit demum centrum gravitatis totius anchoræ imminere palmu- læ C, quæ propterea etiam urgente gravitate profundiùs in- figitur. In hac lignei transversarij conversione observandum est par- tem alteram sublimiorem B E per vim in aquâ deprimi, partem autem inferiorem B F in aquâ sponte ascendere, ac proinde, propter intermediam gravitatem in B, illam resistere huic sur- sum conanti, atque ideò illam habere rationem hypomochlij, hanc potentiæ, pondus verò esse in B, quod & convertitur: non quidem quia totum pondus sit in B, sed quia totius ancho- ræ centrum gravitatis est in scapo A B, adeóque intelligitur applicatum puncto B, quamvis ipsius centri gravitatis conver- sio fiat circa extremitatem C manentem. At verò si scapus A K fuerit triplex brachij A C, etiam trans- versarium H I scapo æquale est ejusdem brachij triplex: hinc fit ipsius longioris transversarij H I vim, qua se horizontale statuat in aquâ, majorem esse, quàm brevioris E F; lignum enim longiùs difficiliùs in aquâ erectum retinetur. Quamvis autem

Transcription: Translated (English)

Book Four. Chapter XVII. 479 When the shank, and the end of the stock B nearest the ring, is raised, the crosspiece no longer rests upon the sand in F; therefore the palms C and D of the arms, formed into a triangle, as they are dragged together with the ship, insinuate themselves more deeply into the sand. But if they encounter unequal resistance, or if one, for example C, is driven deeper than the other, either because of the difference in the ground beneath, or because the ship, being forced sideways, draws the head B of the anchor to one side, and the end of the other arm, describing around A in the ground an arc toward the ship, is driven deeper, while the remaining end of the opposite arm, moved in the contrary direction around A, scarcely bites the earth, then the force pulling at B, being unable to draw both arms equally, is compelled to move around C, as if around a center, or rather as around a pole. And because the point B is higher than the point C, it is necessary that this rotation take place in such a way that the opposite end D is raised and drawn out from the bottom. And since now the crosspiece is no longer held equally on this side and on that by force in the vertical plane, but its upper part B E inclines toward C, it strives to acquire a horizontal position; and its lower part B F, declining to the side, rises and assists the ascent of the arm A D itself. From this it finally comes about that the center of gravity of the whole anchor hangs over the palm C, which therefore is also, under the pressure of gravity, driven more deeply in. In this turning of the wooden crosspiece it is to be observed that the one higher part B E is forced down in the water, while the lower part B F rises of itself in the water, and therefore, because of the intermediate weight at B, the one resists the other striving upward, and thus has the role of the fulcrum, the other of the power, while the weight itself is in B, which is also what is turned: not indeed because the whole weight is in B, but because the center of gravity of the whole anchor is in the shank A B, and is therefore understood to be applied at the point B, although the turning of that center of gravity takes place around the remaining fixed end C. But if the shank A K were triple the arm A C, then the crosspiece H I is also equal to the shank, being triple that same arm. Hence it follows that the force of the longer crosspiece H I, by which it sets itself horizontal in the water, is greater than that of the shorter E F; for longer wood is more difficult to keep upright in the water. However, although

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480 Mechanicorum autem eadem sit Ratio FE ad BE, quæ est IH ad KH, ta- men major est Ratio motûs ipsius K ad motum centri gravitatis circa extremitatem C manentem, quàm sit Ratio motûs ipsius B ad motum centri gravitatis circa idem punctum C: in illa enim conversione centrum gravitatis existens in aliquo puncto lon- gitudinis A K elevari vix potest ad majorem altitudinem, quàm sit CL; quia in longiore anchorâ A K centrum gravitatis ma- gis recedens ab extremitate A, quàm in anchorâ breviore AB, magis imminet palmulæ C, eâmque profundius in arenam in- figit; ideóque si fortè sit inter L & K, atque ex inclinatione scapi ad horizontem paulò altius existeret quàm CL, si C ma- neret in superficie fundi maris, ipsa depressio puncti C infra il- lam superficiem demit aliquid ex altitudine. Nam quod spectat ad centrum gravitatis anchoræ longioris, certum est illud non removeri ab extremitate Scapi A secun- dùm eandem Rationem, secundùm quam ejus longitudo pro- ducitur: si enim scapus esset longitudo pari & æquabili crassitie ducta, utique sicut A K est ipsius AB sesqualtera, etiam cen- tri gravitatis distantia ab A in scapo longiore esset sesqualtera distantiæ centri gravitatis ab A in Scapo breviore. Quoniam verò & pars B K aliquanto decremento deficit à crassitie reli- quæ partis BA, & pro centro gravitatis totius anchoræ atten- denda est non solius scapi gravitas, sed & brachiorum, mani- festum est centrum gravitatis anchoræ longioris removeri ab A minùs, quàm in Ratione sesqualtera. Atqui circa punctum A ( quando jacent brachia, & elevari incipit extremitas altera sca- pi) moventur B & K pro Ratione distantiarum, hoc est in Ra- tione sesqualtera; igitur motus ipsius K ad motum sui centri gravitatis est in majore Ratione, quàm motus puncti B ad mo- tum sui centri gravitatis. Hinc est intento rudente faciliùs pro rata portione elevari extremitatem K longioris scapi, quam B brevioris, & centrum gravitatis inter A & K, hoc est inter hypomochlium & poten- tiam, habere rationem ponderis, quod elevatur vecte secundi generis AK. Quia autem facta elevatione puncti K jacentibus adhuc brachiis, postea fieri debet conversio circa palmulam C manentem, tunc punctum C habet rationem hypomochlij, & pondus intelligitur esse centrum gravitatis interjectum inter K &

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480 Mechanica however the ratio of FE to BE is the same as that of IH to KH, yet the ratio of the motion of K itself to the motion of the center of gravity about the fixed end C is greater than the ratio of the motion of B itself to the motion of the center of gravity about the same point C: for in that turning, the center of gravity, existing at some point along the length AK, can scarcely be raised to a greater height than CL; because in the longer anchor AK, the center of gravity, receding farther from the end A than in the shorter anchor AB, presses more upon the palmula C, and drives it more deeply into the sand; and therefore, if it should perhaps lie between L and K, and from the inclination of the shank to the horizon stand somewhat higher than CL, if C remained upon the surface of the sea bottom, the depression itself of point C below that surface takes away something from that height. For as regards the center of gravity of the longer anchor, it is certain that it is not removed from the end A of the shank according to the same ratio according to which its length is increased: for if the shank were drawn out with equal and uniform thickness, then indeed, just as AK is one and a half times AB, so also the distance of the center of gravity from A in the longer shank would be one and a half times the distance of the center of gravity from A in the shorter shank. But since both part BK falls somewhat short in thickness of the remaining part BA, and since for the center of gravity of the whole anchor account must be taken not only of the weight of the shank, but also of the arms, it is manifest that the center of gravity of the longer anchor is removed from A by less than in the one-and-a-half ratio. Yet around point A, when the arms lie down and the other end of the shank begins to be raised, B and K are moved in proportion to their distances, that is, in the one-and-a-half ratio; therefore the motion of K to the motion of its own center of gravity is in a greater ratio than the motion of point B to the motion of its own center of gravity. Hence, by means of the rope being strained, the end K of the longer shank is more easily raised in due proportion than B of the shorter one, and the center of gravity between A and K, that is, between the fulcrum and the power, has the function of the weight which is raised by the second-kind lever AK. But because, once point K has been raised while the arms still lie down, a turning must afterwards be made about the palmula C remaining fixed, then point C has the function of the fulcrum, and the weight is understood to be the center of gravity lying between K and

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Liber quartus. CAPUT XVIII. 481 & C, si minus sit intervallum inter K & centrum gravitatis, quàm inter K & hypomochlium C, cujusmodi esset, si cen- trum gravitatis esset citra L versùs K, & esset vectis curvus se- cundi generis. Quòd si magis distat centrum gravitatis à puncto K, quàm ab eodem puncto K distet punctum C, vectis est curvus primi generis. Quid autem, inquis, si pari interval- lo distet punctum K à puncto C, atque à centro gravitatis? cu- jusmodi generis vectis erit? primi-ne? an secundi? Respondeo in vecte hoc curvo, cujus altera extremitas ma- net, & pondus non ad perpendiculum, neque motu recto in plano verticali, sed conversione elevatur, attendenda esse pla- na, in quibus tùm potentia, tùm pondus propriam conversio- nem perficiunt; his autem planis parallelum concipe aliud pla- num, quod per extremitatem C manentem transeat, quod pla- num si interjectum fuerit inter illa plana conversionum, vectis erit primi generis, quia hypomochlium est inter potentiam & pondus; sin autem hoc extremum fuerit, & medium locum ob- tineat planum, in quo convertitur centrum gravitatis, vectis erit secundi generis. Facta demùm conversione ita, ut transversarium ligneum positionem habeat horizontalem, & utrumque brachium in eodem sit plano verticali; quia faciliùs elevatur K quàm B, & transversarium H I longius majorem habet vim sustinendi, quàm transversarium E F brevius, hinc est brachium A C ma- gis inclinari ad subjectum maris planum horizontale, ac prop- terea etiam validiùs in arenam infigi, quando à navi trahitur anchora. CAPUT XVIII. Plures Vectis usus exponuntur. Quod superiore libro præstitimus libræ atque stateræ usum extendentes, & hîc præstare operæ pretium fuerit, tum ut vectis natura ex uberiori utilitate innotescat, tum ut fax ali- P p P

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Book Four. CHAPTER XVIII. 481 ...and C, if there is a smaller interval between K and the center of gravity than between K and the hypomochlium C, such as it would be if the center of gravity were on this side of L toward K, and the lever were a curved lever of the second kind. But if the center of gravity is farther from the point K than the point C is from the same point K, the lever is a curved lever of the first kind. But what, you ask, if the point K is at an equal distance from the point C and from the center of gravity? Of what kind will the lever then be? of the first? or of the second? I reply that in this curved lever, one end of which remains fixed, and the weight is raised not perpendicularly, nor by straight motion in a vertical plane, but by rotation, the planes must be considered in which both the power and the weight perform their proper rotation; and along with these planes imagine another plane parallel to them, which passes through the fixed end C; and if that plane lies between those planes of rotation, the lever will be of the first kind, because the hypomochlium lies between the power and the weight; but if this be the terminal plane, and occupy the middle position, in which the center of gravity turns, the lever will be of the second kind. Finally, after the rotation has been made in such a way that the wooden crosspiece has a horizontal position, and both arms are in the same vertical plane, since K is raised more easily than B, and since the crosspiece H I, being longer, has greater power of support than the shorter crosspiece E F, hence the arm A C is inclined more toward the horizontal plane of the underlying sea, and for that reason also is driven more firmly into the sand when the anchor is dragged by the ship. CHAPTER XVIII. Various uses of the lever are set forth. What we accomplished in the preceding book by extending the use of the balance and scales, it would here also be worthwhile to accomplish, both so that the nature of the lever may become known from its more abundant usefulness, and so that it may be a means of illumination...

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Mechanicorum 482 qua tyronibus præferatur viam commonstrando, qua similes usus possint pro opportunitate excogitare. PROPOSITIO I. Duplex Vectis genus in uno vecte conjungere. Sæpissime contingit unico quidem vecte nos uti, re tamen Svera duplicem esse vectem; quemadmodum cum ingentis alicujus faxi extremitati vectem subjicimus, quo extremitatem illam attollimus. Sit enim faxum, cujus centrum gravitatis S, & subjecto vecte AB habente hypomochlium in C attollatur extremitas F, manente extremitate E: utique vectis primi generis est AB; sed si rem attentius perpendamus, etia[m] longitudo FE, aut potiùs BE vectis est secundi generis habens impositum pondus S, & fulcrum in E; atque quò magis supra horizontem elevatur, linea Directionis SD magis accedit versùs E, ex quo oritur movendi facilitas; quam juvat Potentiæ A depressio, ex qua fit ut B magis accedens ad F, magis etiam recedat ab hypomochlio E. Manifestum est autem pondere accedente ad hypomochlium, & potentiâ ab eodem recedente, majorem fieri Rationem motûs Potentiæ ad motum Ponderis, atque adeò augeri movendi facilitatem. Quare momenta potentiæ in A sustinentis faxum ea sunt, quæ componuntur ex Ratione AC ad CB, & Ratione BE ad EI. Sed de hoc nullus mihi hîc sermo; quia vel duo vectes sunt, ut explicatum est, alter quidem ab ipso pondere non sejunctus FE, alter verò ab eo distinctus AB; vel si unicus intelligatur vectis, qui ponderi applicatur, hic sanè ad unum pertinet genus non ad duo, ut hæc propositio exigit. Sit igitur dati vectis longitudo CD, in cujus medio hypomochlium O bifariam æqualiter dividat totam longitudinem, &

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Mechanics 482 by pointing out a path to novices, by which they may contrive similar uses as opportunity offers. PROPOSITION I. To combine two kinds of lever in one lever. It very often happens that we use indeed a single lever, while in reality it is a double lever; as when we apply a lever to the extremity of some very large beam, in order to raise that extremity. Let there be a beam, whose center of gravity is S, and with a lever AB placed under it, having its fulcrum at C, let the extremity F be raised, the extremity E remaining fixed: certainly AB is a lever of the first kind; but if we examine the matter more carefully, the length FE, or rather BE, is also a lever of the second kind, having the weight S placed upon it, and the fulcrum at E; and the more it is lifted above the horizon, the more the line of direction SD approaches toward E, whence arises the ease of moving it; which is helped by the depression of the power at A, from which it follows that B, approaching more toward F, also recedes more from the fulcrum E. Now it is manifest that, as the weight approaches the fulcrum and the power recedes from the same, the ratio of the motion of the power to the motion of the weight becomes greater, and therefore the ease of moving is increased. Wherefore the moments of the power sustaining the beam at A are those which are compounded from the ratio of AC to CB, and the ratio of BE to EI. But of this I say nothing here; because either there are two levers, as has been explained, one indeed FE not separated from the weight itself, the other AB distinct from it; or, if a single lever applied to the weight be understood, this certainly belongs to one kind only, not to two, as this proposition requires. Let therefore the length of the given lever be CD, in the middle of which the fulcrum O may divide the whole length equally into two parts, &

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Liber quartus. CAPUT XVIII. 481 & sit pondus in P. Erit, per, 7. lib. 5. eadem Ratio C O ad O P, atque D O ad O P; & si in C sit potentia deprimens, in D autem potentia elevans, æqualia habent momenta ad elevandum pondus in P. Est ergo C P vectis primi generis, & D O vectis secundi generis, cui cum primo commune est hypomochlium O, & communis pars O P. Quòd si Potentiæ inæquales fuerint, utraque autem valeat sive deprimere, sive elevare, dividatur longitudo C D in duas partes, quarum Ratio eadem sit ac Potentiarum, & in puncto divisionis statuatur fulcrum: tum in extremitatibus reciprocè collocentur Potentiæ, validior scilicet propior sit fulcro, debilior verò remotior, ut æqualium sint momentorum. Datæ Potentiæ sint ut 5 ad 3. Dividatur C D partium octo ita in P (ubi statuendum est fulcrum) ut C P sit 5, P D sit 3; & Potentia robustior, quæ est ut 5 sit in D; infirmior verò, quæ est ut 3, sit in C; & pondus sit in R, quoniam C P ad P R est ut 5 ad 1, & D P ad P R est ut 3 ad 1. Igitur si pondus R sit lib. 30, attolletur à Potentia C potente sine vecte attollere lib. 3, & à Potentia D potente sine vecte elevare lib. 5: utriusque enim momenta singillatim accepta sunt 15 composita ex virtute movendi & motûs velocitate. At si pondus P sit lib. 30, & fulcrum in O, sit autem C O ad O P, atque D O ad O P ut 4 ad 1, satis est si singulæ Potentiæ æquales C & D possint sine vecte attollere lib. 3. unc. 9. Porrò si inæqualium potentiarum altera possit solùm deprimendo vectem elevare pondus, manifestum est ad illam pertinere vectem primi generis: ac propterea si illa sit potentia validior, eidem tribuetur minor distantia ab hypomochlio; sin autem illa sit imbecillior, ipsi tribuetur distantia major, atque illam inter ac pondus statuetur fulcrum. Hinc facilè poterit potentia vivens uti ope potentiæ inanimatæ, quæ vi suæ gravitatis deorsum premat oppositam extremitatem propositi vectis. Huc spectare videtur facillimum genus antliæ simplicis, P P P 2

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Book Four. CHAPTER XVIII. 481 & let the weight be at P. It will be, by 7, lb. 5; the same ratio C O to O P, and D O to O P; & if in C there is a power depressing, but in D a power raising, they will have equal moments for raising the weight at P. Therefore C P is a lever of the first kind, & D O a lever of the second kind, with which the first has in common the fulcrum O, & the common part O P. But if the powers are unequal, and both are able either to depress or to raise, let the length C D be divided into two parts, whose ratio is the same as that of the powers, & let the fulcrum be placed at the point of division: then at the ends let the powers be set reciprocally, namely the stronger being nearer to the fulcrum, the weaker farther away, so that their moments may be equal. Let the given powers be as 5 to 3. Let C D, consisting of eight parts, be divided at P (where the fulcrum is to be placed) so that C P is 5, P D is 3; & let the stronger power, which is as 5, be in D; the weaker, which is as 3, in C; & let the weight be at R, since C P is to P R as 5 to 1, & D P is to P R as 3 to 1. Therefore if the weight R is 30 lb., it will be raised by the power C, which is able to raise 3 lb. without a lever, & by the power D, which is able to raise 5 lb. without a lever: for the moment of each, taken separately, is 15, composed of the power of moving & the velocity of the motion. But if the weight P is 30 lb., & the fulcrum is in O, and C O to O P, and D O to O P are as 4 to 1, it is enough if the individual equal powers C & D can without a lever raise 3 lb. 9 oz. Moreover, if of unequal powers one can raise a weight only by depressing the lever, it is clear that a lever of the first kind belongs to that power: and therefore if that power is the stronger, a smaller distance from the hypomochlion will be assigned to it; but if it is the weaker, a greater distance will be assigned to it, and between that power and the weight the fulcrum will be placed. Hence a living power can readily make use of the help of an inanimate power, which by the force of its gravity presses downward the opposite extremity of the proposed lever. To this seems to refer the easiest kind of simple pump, P P P 2

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Mechanicorum 484 qua ex depressiore loco in altiorem aquas attollimus. Sit enim modiolus B, cui aptè inseratur congruens embolus medio hastili CD connexus cum transversario EF versatili circa axem infixum in I: cujus transversarij extremitatem E occupet massa plumbea opportunæ gravitatis ad deprimendum embolum intra modiolum, postquam elevatus fuerit à potentia funem FG trahente adnexum in altera extremitate F. Vectis FE est primi generis duplex habens pondus, alterum in E, alterum in D, utrumque enim per vim elevatur. At vectis EI est secundi generis, in quo E est potentia deprimens embolum, & quo magis distabit ab hypomochlio I, minor massa plumbea eadem obtinebit momenta. Quòd si IF constet materiâ satis gravi, jam habet rationem ponderis, ac propterea distantia centri gravitatis illius H à puncto I determinabit ejus momenta. Quare potentia E vecte EI secundi generis deprimet embolum, & vecte EH primi generis attollet pondus brachij IF. Hinc est commodius accidere, si longitudo EK ferrea sit, in K verò

Transcription: Translated (English)

Mechanics 484 by which, from a lower place, we raise waters to a higher one. Let there be a cylinder B, into which is suitably inserted a matching piston connected by the middle rod CD with the crossbar EF, movable around an axis fixed at I: let the end E of this crossbar be occupied by a leaden mass of suitable weight for pressing down the piston within the cylinder, after it has been raised by the power pulling the rope FG attached at the other end F. The lever FE is a double lever of the first kind, having two weights, one at E and the other at D; for both are raised by force. But the lever EI is of the second kind, in which E is the power that depresses the piston, and the farther it is from the fulcrum I, the smaller a leaden mass will suffice to produce the same moments. And if IF be made of sufficiently heavy material, it already has the role of a weight, and therefore the distance of its center of gravity H from the point I will determine its moments. Therefore the power at E, by the lever EI of the second kind, will depress the piston, and by the lever EH of the first kind it will raise the weight on the arm IF. Hence it is more convenient if the length EK is of iron, and in K indeed

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Liber quartus. CAPUT XVIII. 485 verò inseratur, ut firmiter cohæreat, satis validus baculus ligneus KF; poterit enim longior esse, & faciliorem efficere antliæ agitationem, quin gravitas nimia indigeat multo plum- bo in E, ut præponderetur. Quòd si non placeret addere plumbum in E, & solo vecte primi generis FD uti velles, recurrendum esset ad vim elasti- cam, qua vel arcûs X in superiore loco firmati nervo, vel extremæ perticæ AM longiuculæ (ut Toreuticen exercen- tibus solemne est) adnecteretur funis pertingens ad F, ut ex tractione Potentiæ GF curvatus arcus, vel inflexa per- tica, cessante potentiâ, iterum se suum in statum restitueret, sursúmque traheret extremitatem F, ac proinde embolum intra modiolum B deprimeret. Tunc enim esset FD vectis primi generis, cujus extremitati F applicarentur duæ Poten- tiæ, altera deorsum, altera vicissim alterno conatu sursum trahens. At si fortè duplicem antliam velis simul componere, dupli- cémque potentiam viventem alternis operis conantem adhi- bere, jugo R S versatili cir- ca axem X adde duo, le- viora quidem, sed satis fir- ma manubria R O & S M, quorum extremitates aut premi, aut adjecto fune trahi deorsum valeant: nam depressâ extremitate O deprimitur pariter hasti- le infixum in R, & est O X vectis secundi generis, at- que attollitur hastile ad- nexum in S, & est O S vectis primi generis. Simi- liter M X vectis est secundi generis, movens pondus positum in S, atque M R est vectis primi generis attollens pondus po- situm in R. Propterea autem leviora dixi adjecta manubria R O & S M, ne suâ gravitate movendi difficultatem augeant. Verùm si solus volueris antliam utramque agitare, unus sit continuus funis ex O per rotulas P & Q transiens, atque in M P p p 3

Transcription: Translated (English)

Book Four. Chapter XVIII. 485 if it is inserted there, so as to cohere firmly, a sufficiently strong wooden rod KF may be used; for it can indeed be longer, and make the working of the pump easier, without requiring excessive weight of much lead in E to counterbalance it. But if it should not please you to add lead in E, and you wished to use only the first-kind lever FD, recourse would have to be had to an elastic force, to which either a bow X fixed in the upper place with a string, or the end of the rather long pole AM (as is customary among those who practice Toreutic) would be attached by a rope extending to F, so that, by the traction of the power GF, the curved bow, or the bent pole, when the force ceases, may restore itself again to its former state, and draw the end F upward, and thus depress the piston within the cylinder B. Then FD would be a first-kind lever, at whose end F two forces would be applied, one downward, the other in turn drawing upward by alternate effort. But if perhaps you wish to construct a double pump at the same time, and to employ a double living power striving in alternate operations, add to the movable yoke RS about the axis X two handles RO and SM, lighter indeed, but sufficiently strong, whose ends may either be pressed down or drawn downward by a rope attached: for when the end O is depressed, the rod fixed in R is likewise depressed, and OX is a lever of the second kind, while the rod attached in S is raised, and OS is a lever of the first kind. Similarly MX is a lever of the second kind, moving the weight placed in S, and MR is a lever of the first kind, raising the weight placed in R. For that reason I called the added handles RO and SM lighter, lest by their own weight they increase the difficulty of movement. But if you should wish to work both pumps by one continuous rope passing from O through the pulleys P and Q, and into M P p p 3

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Mechanicorum 486 connexus: quacumque enim in parte constitutus fueris, tantumdem funis sequitur ascendentem extremitatem, quantum trahitur deprimendo alteram extremitatem: sic trahendo funem PO deprimitur extremitas O, deinde trahendo funem PQ deprimitur extremitas M. Aut etiam sit unicum manubrium SM, & erit RM: atque premens in M attollet hastulam R, elevans aut in M attollet hastulam S. Ut autem faciliùs attollatur M, sit in superiore loco orbiculus, per quem transeat funis connexus in M, alteram enim extremitatem deorsum trahens attollet manubrium M. Quod si ab oculis remotam volveris antliam, fac per parietis foramen in proximum conclave exire funem MI orbiculi H excavatæ absidi insertum, & per orbiculos P, Q, transire funem OPQL; connexis enim funium extremitatibus I & L modò hunc modò illum funem trahendo utramque antliam agitabis. PROPOSITIO II. Antliam opportuno vecte instruere. UT aliquam speciem vectis curvi oneri movendo destina- ti exhibeam, placet in antlia, qua ad hauriendas aquas utimur, exemplum ponere, quod facilè in reliquis pro re nata imitari possimus. Est in antliâ loco ponderis aqua, quæ adducto embolo attrahitur in modiolum, eóque reducto exprimitur, & prætereà conflictus ipse emboli cum modiolo; superanda quippe est difficultas, quæ ex mutuo horum contactu oritur, & aqua per vim elevanda est, sive solùm attrahatur, ut ex modiolo per emboli reducti foramen subinde erumpens effluat, sive in modiolo compressa ab embolo, cùm reducitur, exprimatur in tubum, ut adhuc altiùs ascendat, juxta ea, quæ in Hydrotechnicis fusiùs dicuntur. Id quidem fieret si hastili, quod embolo infigitur, ipsa potentia proximè applicaretur; sed ut minus laboris illa subeat, additur vectis, ut multo major sit potentiæ motus, quàm emboli. Sit

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Mechanics 486 connected: for in whatever part you may be placed, the ascending end of the rope follows by just as much as the other end is drawn down by lowering; thus by pulling the rope PO, the end O is lowered, then by pulling the rope PQ, the end M is lowered. Or let there be a single handle SM, and there will be RM: and pressing at M will raise the little rod R, and lifting at M will raise the little rod S. But so that M may be raised more easily, let there be a small wheel in the upper place, through which the rope connected at M passes; for by drawing the other end downward it will raise the handle M. And if you wish to turn the pump away from sight, cause the rope MI, inserted into the hollow arch of wheel H, to pass through a hole in the wall into the next room, and let the rope OPQL pass through the small wheels P, Q; for, with the ends of the ropes I and L connected, by pulling now this rope, now that, you will operate both pumps. PROPOSITION II. To equip the pump with a suitable lever. THAT I may exhibit some kind of curved lever intended for moving a load, I choose, in the pump that we use for drawing water, to set an example, which we may easily imitate in the rest as occasion requires. In the pump, water is in the place of the weight, which is drawn in by the drawn-down piston into the barrel, and with it brought back is squeezed out; and moreover the resistance itself of the piston against the barrel must be overcome. For the difficulty that arises from the mutual contact of these parts must be overcome, and the water must be raised by force, whether it is merely drawn up, so that, bursting out through the opening of the returned piston, it flows away from the barrel, or, being compressed in the barrel by the piston as it is returned, it is squeezed into the tube, so that it may rise still higher, in accordance with what is explained more fully in the Hydrotechnics. This indeed would happen if the power itself were applied directly to the rod fixed into the piston; but so that it may undergo less labor, a lever is added, so that the motion of the power is much greater than that of the piston. Let there be

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Liber quartus. CAPUT XVIII. 487 Sit enim embolus A congruens modiolo B, illique in- fixum hastile C A; quo elevato aqua per subjectum modiolo tubum F attrahitur in modiolum ipsum B, quo depresso aqua cogitur ex eodem modiolo exire. Sed ut minore operâ id totum perficiatur, additur in C vectis curvus C D E versatilis circa axem D infixum parieti interjecto inter antliam & potentiam moventem. Nam extremitatem E arripiens potentia si vectem urgeat versus parietem intermedium, elevatur embolus, & aqua modiolum implet, si verò vectis extremitatem à pariete removeat, deprimitur embolus, & compressa aqua exprimitur. Hîc vectem primi generis agnoscis habentem hypomochlium in D, scilicet in axe, circa quem versatur vectis; & pro Ratione longitudinis D E ad longitudinem D C est Ratio momentorum potentiæ ad resistentiam ponderis, hoc est tantò magis augentur potentiæ vires, quò major est Ratio D E ad D C: sumitur autem D E recta linea non computato flexu D G E, qui eatenus adstruitur, quatenus parietis crassities obstaret, ne commodè uteremur vecte E D C inflexo in D. Quia verò faciliùs ab homine urgetur vectis in E, quàm ipsa extremitas E retrahatur, ideò in antliâ solùm attrahente utendo hoc vecte primi generis curvo minus est laboris, nam in deprimendo embolo minus est difficultatis quàm in elevando. At si aqua altiùs elevanda esset supra antliam non attrahentem solùm, sed etiam expellentem, faciliùs attolleretur embolus, quàm deprimeretur, propter majorem aquæ resistentiam, cùm exprimitur, juxtà altitudinem perpendicularem, ad quam expellitur: propterea tunc mutanda esset positio, ut esset vectis secundi generis; hypomochlium enim statuendum esset in C, & hastile emboli adnectendum in D. Quod si potentia viribus abundet, poterit duplicem antliam agitare, cujusmodi esset si jugum R S bifariam divisum in

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Book four. CHAPTER XVIII. 487 Let there be an embolus A fitting into the cylinder B, and a rod C A fixed in it; by raising this, water is drawn through the tube F placed under the cylinder into the cylinder B itself, into which, when depressed, the water is forced to go out of the same cylinder. But so that this may be accomplished with less labor, there is added at C a curved lever C D E, turning about the axis D fixed in the wall between the pump and the power that moves it. For if the power, seizing the extremity E, urges the lever toward the intermediate wall, the embolus is raised and the water fills the cylinder; but if the lever’s extremity is moved away from the wall, the embolus is depressed and the compressed water is expelled. Here you recognize a lever of the first kind, having its fulcrum in D, namely in the axis about which the lever turns; and in proportion as the length D E bears to the length D C, so does the ratio of the moments of the power to the resistance of the weight. That is to say, the power’s force is increased all the more as the ratio of D E to D C is greater. Now D E is taken as a straight line, not counting the bend D G E, which is admitted only so far as the thickness of the wall would prevent our conveniently using the bent lever E D C at D. But since the lever is more easily pressed at E by a man than the very end E is drawn back, therefore in a pump that only draws, using this curved lever of the first kind, there is less labor; for in depressing the embolus there is less difficulty than in raising it. But if the water were to be raised higher, above a pump not only drawing but also expelling, the embolus would be more easily lifted than depressed, because of the greater resistance of the water when it is expelled, according to the perpendicular height to which it is expelled; for that reason the position would then have to be changed so that it would be a lever of the second kind; for the fulcrum would have to be placed in C, and the rod of the embolus attached at D. But if the power should abound in strength, it could move a double pump, such as would be the case if the yoke R S, divided into two parts in

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Mechanicorum in X jungeretur in R & S duplici hastili, centrum autem motûs responderet puncto X, cui firmiter adnecteretur manubrium XZ, quod agitaretur parallelum plano, per quod transit axis jungens jugum R S cum manubrio ipso: dum enim Z versùs P movetur, deprimitur R & attollitur S, atque vicissim reman in Q deprimit embolum respondentem jugi extremitati S, & oppositum attollit. Sunt autem duo vectes curvi Z X R & Z X S primi generis partem unam, videlicet manubrium XZ, habentes communem. Sed quoniam posito longiore manubrio Z X, vel DE, faciliùs quidem attollitur aqua, quàm si illud brevius esset, major tamen corporis agitatio requiritur, & multâ membrorum inclinatione laboriosa exercitatio suscipienda est, propterea satius est uti vecte recto, ut prop. 1. dictum est, quem etiam sedens modico labore commovere poteris adnexum extremitati funem deorsum trahendo. PROPOSITIO III. Rotam in profluente positam, quæ aquam faciliùs elevet ex vectis Rationibus, constituere. A Quam ex depressiore loco in altiorem provehi vasculis ab- sidi rotæ circum circa alligatis, quæ in infimâ rotæ parte subjectam aquam immersa hauriunt, & circumactâ rotâ, ubi circuli semissem ascendendo perfecerint, descendendo effundunt, quibus per Helvetios iter facere contigit, perspectum est; si in Tigurinâ Urbe, quam lacus Limagum fluvium excipiens interluit, observârunt ab utrâque ripâ ductum ex palis confertim densatis obicem obliquum usque ad medium alveum, ut

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Mechanics if joined at X in R & S by a double rod, but the center of motion would correspond to point X, to which the handle XZ would be firmly attached, which would be moved parallel to the plane through which passes the axis joining yoke R S with the handle itself: for while Z moves toward P, R is depressed and S is raised, and vice versa when the ram in Q depresses the piston corresponding to the end of the yoke S, & raises the opposite one. There are therefore two curved levers Z X R & Z X S, sharing one common part of the first kind, namely the handle XZ. But since, when a longer handle Z X, or DE, is set in place, water is indeed raised more easily than if it were shorter, yet a greater movement of the body is required, and with much bending of the limbs a laborious exercise must be undertaken; therefore it is better to use a straight lever, as was said in Prop. 1, which even while seated you will be able to move with moderate effort by pulling downward a rope attached to the end. PROPOSITION III. To construct a wheel placed in flowing water that may more easily raise water by lever principles. A This was observed by those who had to transport water from a lower place to a higher one by means of little vessels fastened all around the circumference of a wheel, which, immersed in the water beneath the lowest part of the wheel, draw it up, and when the wheel is turned, after they have completed half the circle in ascending, they descend and pour it out; this was seen by those who traveled through Helvetia. In the city of Tigurum, which is watered by the river Limagus flowing out of the lake, they observed a sloping barrier driven from both banks by closely packed stakes, extending to the middle of the channel, so that

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Liber quartus. CAPUT XVIII. 489 ut ex angustiis erumpens aqua cæteroqui leniter defluens, ve- lociùs fluere cogatur, & validiùs in prominentes rotæ palmulas incurrens ingentem illam rotam cum adjunctis vasculis aquâ plenis faciliùs circumagat, atque adeò in subjectum vas ponti impositum effusa aqua per urbem universam dividatur. Verùm quia pondus, scilicet aqua vasculis contenta, semper à centro rotæ intervallo eodem abest, aliud rotæ genus excogitari po- test, quod aquam facilius elevet, nec adnexa, sed congenita habeat vascula. In plano ex asseribus rite conjunctis compacto, centro A, in- tervallo A B, intelli- gatur descriptus cir- culus, cujus femidia- meter aliquanto ma- jor sit altitudine, ad quam aqua evehen- da est, dividaturque descripti circuli peri- pheria in quotlibet æquales partes, ex. gr. duodecim, aut plures. Tum assump- tâ palmulæ congruâ altitudine BD, alius interior circulus co- dem centro A, in- tervallo AD describatur, qui à ductis per centrum A dia- metris similiter in totidem æquales partes dividitur. Assump- tâ itaque CF æquali ipsi DB, statuatur CE intervallum op- portunæ amplitudinis, ut aqua facilè ingredi possit. Et ductâ rectâ lineâ BE, resecetur particula exterior, ut sit BECF: idemque de cæteris partibus intelligatur, prout adjectum schema refert. Duo hujusmodi plana parentur omnino æqualia, similitérque denticulata, quæ cylindro (sive prismati similem basim haben- ti cum polygono ab initio descripto) hoc est axi inserantur in A, & parallela sint. Planorum autem intervallum definiant af- feres æquè lati, qui perpendiculares insistant lineis GBE, & Q99

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Book four. CHAPTER XVIII. 489 that, as water, flowing otherwise gently, bursts forth from a narrow opening, it may be compelled to flow more swiftly, and, striking more vigorously against the projecting vanes of the wheel, may more easily turn that great wheel together with the attached vessels filled with water, and thus, the water being poured into the vessel set upon the bridge, it may be distributed throughout the whole city. But because the weight, namely the water contained in the vessels, is always at the same distance from the center of the wheel, another kind of wheel can be devised, which may raise the water more easily, and may have vessels not attached, but integral. On a plane formed of boards properly joined together, with center A and radius AB, let a circle be described, whose semi-diameter is somewhat greater than the height to which the water is to be raised; and let the circumference of the described circle be divided into as many equal parts as desired, for example twelve, or more. Then, with the suitable height of the vane BD assumed, let another inner circle, with the same center A and radius AD, be described, which, by lines drawn through the center A, is likewise divided into the same number of equal parts. Then, CF being taken equal to DB, let the interval CE be set with a convenient width, so that the water may easily enter. And, a straight line BE having been drawn, let the exterior segment be cut off, so that it is BECF; and the same is to be understood for the remaining parts, as the appended figure shows. Two planes of this kind are to be prepared, perfectly equal and likewise toothed, which are to be inserted into the cylinder (or prism having a similar base with the polygon first described), that is, into the axis at A, and are to be parallel. The spacing of the planes, however, is to be determined by equally wide strips, which stand perpendicular to the lines GBE, and Q99

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Mechanicorum similibus; quorum asserum latitudo palmulis quoque DB, CF, & reliquis latitudinem statuet. Omnibus ritè firmatis, ac ob- structis accuratè rimulis, rota super polos axi infixos collocetur in profluente, ità ut palmula tota in aquam immergatur, quæ per apertum osculum CE ingrediens impleat spatium EBD. Impetu igitur profluentis dum rota convertitur, aqua inclusa paulatim versùs rotæ centrum secedit, donec quadrantem cir- culi ascendendo transgressa proxima fiat axi: cùm enim B vene- rit in H, aqua erit in I, cùm verò ex H in S venerit, jam aquæ in subjectum vas effluet. Quare, licèt æqualium conversionum non sint æquales ascensus in eâdem circuli peripheriâ, sed ab imo puncto usque ad finem Quadrantis crescant, quia tamen centrum gravitatis aquæ se in æquilibrio statuentis sensim cen- trum versùs recedit, ejus ascensus minor est, quàm si eodem semper intervallo abesset à centro rotæ. Est itaque vectis curvus primi generis, cujus hypomochlium respondet centro A, Potentia movens duplex est, scilicet vis profluentis applicata in B, atque vis aquæ descendentis existens in S: pro variâ autem centri gravitatis aquæ elevatæ distantiâ ab hypomochlio A, diversa etiam est motuum Ratio & momen- torum. Aqua enim in superiore semicirculo supra RS in singu- lis loculamentis sibi invicem hinc atque hinc respondentibus æqualiter disposita obtinet æqualia gravitatis momenta. Qua- propter totus profluentis conatus impenditur in elevandâ aquâ, quæ loculamentis inter B & R interceptis continetur. Quare si multæ sint profluentis vires, crassior rota statui potest, ut, planis magis distantibus, major aquæ copia singulis loculamen- tis hauriatur: quo fiet, ut palmula latior majorem incurrentis aquæ impetum recipiat. Quòd si placuerit palmulas addere la- tiores, quàm sit rotæ crassitudo, non abnuo: hæc enim, & cæ- tera, quæ constructionis facilitatem juvent, prudentis machi- natoris arbitrio relinquuntur: mihi satis est innuisse, quid com- pendij ex vectis rationibus peti possit. PROPOSI

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Mechanics similar; the width of whose boards, also the vanes DB, CF, and the rest, will determine the width. All being properly fastened, and the narrow gaps carefully blocked up, let the wheel be placed on the axles fixed to the poles in the stream, so that the whole vane is immersed in the water, which, entering through the open mouth CE, fills the space EBD. Therefore, by the force of the current, as the wheel turns, the water enclosed within it gradually moves away toward the center of the wheel, until, after rising and passing the quadrant of a circle, it comes nearest to the axle: for when B has come to H, the water will be in I; but when it has come from H to S, the water will already flow into the vessel below. Hence, although in equal rotations the ascents along the same circumference are not equal, but increase from the lowest point to the end of the quadrant, yet because the center of gravity of the water, as it establishes itself in equilibrium, gradually recedes toward the center, its ascent is less than it would be if it were always at the same distance from the center of the wheel. It is therefore a curved lever of the first kind, whose fulcrum corresponds to the center A. The moving power is twofold, namely the force of the current applied at B, and the force of the descending water existing at S; and according to the different distance of the center of gravity of the raised water from the fulcrum A, the ratio of the motions and of the moments is also different. For the water in the upper semicircle above RS, being evenly distributed in the individual compartments corresponding to one another on either side, has equal moments of weight. Wherefore the whole effort of the current is expended in lifting the water contained in the compartments intercepted between B and R. Therefore, if the powers of the current are great, a thicker wheel may be constructed, so that, with the planes being farther apart, a greater quantity of water may be drawn into each compartment: by which means a wider vane will receive a greater impulse of incoming water. But if it should please one to add vanes wider than the thickness of the wheel, I do not object: for these and the other things that assist the ease of construction are left to the judgment of the prudent machinist. It is enough for me to have indicated what advantage may be sought from the principles of the lever. PROPOSI

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Liber quartus. CAPUT XVIII. 491 PROPOSITIO IV. A pluribus hominibus ingens pondus transferri posse ita, ut omnes æqualiter ferant. ONus ingens palangâ transferri pluribus hinc atque hinc longo ordine succollantibus, notum est: sed quoniam non omnium æqualis est distantia à pondere (nisi fortè bini & bini æquè distarent à centro gravitatis) non sunt æqualia momenta; sed qui propiores sunt magis premuntur, cæteris paribus, quàm remotiores, maximè si quis sensim se subducat oneri adeò, ut inæqualis fiat oneris distantia ab iis, qui illud sustentant. Prop- terea methodus aliqua excogitanda est, qua fiat ut singuli pa- rem experiantur in deferendo onere difficultatem. Sit ponderi dato alligatus vectis AB, & gravitatis centro respondeat punctum C, atque æqualia sint in- tervalla AC & BC. Si centrum gravitatis pon- deris respondens puncto C vectis non fuerit pla- nè in mediâ ejusdem ponderis longitudine, neque fuerit vectis val- dè longior ipso ponde- re, non poterunt plu- res ita æqualiter dispo- ni, ut ad ferendum æqualiter pondus sin- gulis anterioribus sin- guli posteriores respon- deant æquè à puncto C distantes, impediente videlicet ipsâ ponderis longitudine.

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Book Four. CHAPTER XVIII. 491 PROPOSITION IV. That a very great weight can be carried by several men, so that all bear it equally. It is well known that a heavy load may be carried by a pole by several men supporting it along both sides in a long row; but since the distance of each from the weight is not equal (unless perhaps they were arranged two and two at equal distances from the center of gravity), the moments are not equal; rather, those who are nearer are pressed more, other things being equal, than those farther away, especially if anyone should gradually slip out from under the load, so that the distance of the load from those who support it becomes unequal. Therefore some method must be devised by which each may experience an equal difficulty in carrying the load. Let a lever AB be attached to the given weight, and let the point C correspond to the center of gravity, and let the intervals AC and BC be equal. If the center of gravity of the weight corresponding to point C of the lever were not exactly in the middle of the same weight’s length, nor were the lever very much longer than the weight itself, then several could not be so arranged equally that, in order to bear the weight equally, the several men behind should correspond to the several men in front, each equally distant from point C, the length of the weight itself, namely, preventing this.

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Mechanicorum Quare tam in A quàm in B duo alij vectes DE, & FG bifariam æqualiter divisi sustineant vectem AB: atque adeò quemadmodum in A (idem die de B) sustinetur semissis totius gravitatis, in D sustinetur tantùm quadrans, sicut & in E. Sustineatur similiter extremitas D alio vecte HI (id quod præstitum intellige pariter in extremitatibus E, F, G) & in H sustinetur octava pars: item extremitas H sustineatur vecte KL, & extremitas K vecte MN; percipitur in K gravitatis pars decima sexta, & in M pars trigesima secunda. Poterunt igitur hac ratione disponi homines 32, qui si possint singuli deferre lib. 100, transferent pondus lib. 3200, & erit æqualiter inter illos distributa gravitas. Quòd si spatium non prohibeat adhuc vectem singulis extremitatibus adjungere, numerus hominum deferentium duplicabitur, & vel singulorum labor dimidiatus erit, vel duplicatum pondus transferre poterunt. Porrò vectem vecti esse firmo vinculo connectendum, ne fortè in motu, vecte aliquo se subducente, luxetur machina, non opus est monere, cum per se res ipsa loquatur. Illud observa, quod vectium inter se æqualitatem, sive longitudo, sive crassis spectetur, non opus est studiosè accurare, dummodo singuli vectes æqualiter bifariam dividantur: immò postremi, & breviores esse possunt, ut minus spatij requiratur, & graciliores, minùs quippe urgentur à pondere. In Atlante Sinico hæc lego pag. 125. In ferendis oneribus scitissimi sunt sinæ, ac rustici illic non parvum sanè staticis nostris speculatoribus facesserent negotium ad causas inveniendas ac rationes, si viderent illos tormenta etiam majora, ac similia pondera, ita vectibus utrinque suspendentes, ut per arctissimas etiam montium fauces facillimè transferant; ac licèt præcedant alij, alij subsequantur, multisque passibus à pondere suspensò distent, ita tamen illud vectibus ac funibus ex æquo nôrunt dividere, ut quilibet æquale ferè sentiat onus, seu paulò remotior sit, sive vicinior. Hoc pacto ingentia marmora, atque integras etiam arbores facilè videas humeris gestare Sinas. Hæc ibi. Sed quonam id artificio in praxim deducatur, nullum planè apparet vestigium. Si

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Mechanics Why, both in A and in B, do two other levers DE and FG, divided into two equal parts, support the lever AB: and so, just as in A (and the same is said of B) half of the whole weight is supported, in D only a quarter is supported, as also in E. Let the extremity D be supported in the same way by another lever HI (and understand this likewise to be done at the extremities E, F, G), and at H one eighth part is supported: likewise let the extremity H be supported by lever KL, and the extremity K by lever MN; it is perceived that at K one sixteenth part of the weight is borne, and at M one thirty-second part. Thus, by this method, 32 men can be arranged, who, if each can carry 100 lb., will transfer a weight of 3200 lb., and the burden will be equally distributed among them. But if space does not prevent still adding a lever to each extremity, the number of men carrying will be doubled, and either the labor of each will be halved, or they will be able to transfer twice the weight. Moreover, that the lever should be fastened to the lever by a firm binding, lest perhaps, in motion, if some lever should slip away, the machine be thrown out of order, need not be pointed out, since the thing itself speaks for itself. Observe this: that the equality of the levers among themselves, whether length or thickness be considered, need not be carefully attended to, provided only that each lever be equally divided into two parts: indeed, the last ones may even be shorter, so that less space is required, and thinner, for they are burdened less by the weight. In the Chinese Atlas I read this on page 125. The Chinese are very skilled in carrying loads, and the peasants there would truly cause no small trouble for our engineers in finding causes and reasons, if they saw them even suspending larger cannon and similar weights by means of levers on both sides, so that they may be transported with the greatest ease even through the narrowest mountain passes; and although some go before, others follow after, and are many paces distant from the suspended weight, yet by levers and ropes they know how to divide it so evenly that each feels an almost equal load, whether he is somewhat farther away or nearer. In this way you can easily see the Chinese carrying huge marbles and even whole trees on their shoulders. So it says there. But by what art this may be put into practice, absolutely no trace appears.

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Liber quartus. CAPUT XVIII. 493 Si igitur funibus suspenditur pondus, & deferentes alij propiores sunt, alij remotiores, duo observanda sunt. Pri- mum est, quòd suspensio non est perpendicularis sed obli- qua, ac proinde plus virium requiritur, ut constat ex iis, quæ dicta sunt tum lib. 1. cap. 16. de elevationibus obli- quis, tum lib. 3. cap. 12, de præponderatione, & æquili- britate gravium fune suspensorum. Verùm hoc momento- rum augmentum in elevatione & suspensione obliquâ, ubi operis abundamus, non consideratur; videtur quippe satis leve incommodum, quod facilitate transferendi onus com- pensatur. Secundum est, quòd si omnes ferè æqualiter la- borant, non dissimiles esse oportet, sed proximè easdem obliquitates funium, ex quibus onus suspensum defer- tur: manifestum enim est in minori obliquitate suspen- sionis minus virium requiri, quàm in majori obliqui- tate. Quare si hanc Sinarum industriam æmulari conarer, pri- mùm oneris transferendi extremitatibus (vel saltem in pa- ri distantiâ à centro gravitatis, quantum conjecturâ asse- qui possem) vectes transversos firmissimè alligarem, ut vectium horum capitibus jungerem funes, quibus suspen- sum onus deferatur. Horum autem transversorum vectium longitudinem ita definirem, ut in lineâ vectibus parallelâ, & æquali quatuor saltem homines commodè collocari queant, quin sibi ullum impedimentum progredientes in- ferant. Deinde satis validos funes utrique vectium extre- mitati adnexos tantæ longitudinis statuerem, quantâ opus sit, ut (tribus hominibus ante onus sibi ordine recto suc- cedentibus ac mediocriter distantibus, quin posterior prio- ris calcem progrediendo feriat) ad tertij humerum pertin- gere possit; hæc enim videtur minima obliquitas suspensio- nis, & quæ proximè accedat ad suspensionem perpendicu- larem: Si verò major fuerit funium longitudo, majori labo- re deferetur onus, si maximè ita elevetur, ut multum distet à subjecto solo, major enim erit obliquitas suspensionis. Tum extremitati funis alius vectis alligetur, qui vecti- bus aliis sustentetur eâ methodo, quam paulo superiùs in- dicavi. Q99 3

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Book Four. CHAPTER XVIII. 493 If, then, a load is suspended by ropes, and some of those carrying it are nearer, others farther away, two things must be observed. The first is that the suspension is not perpendicular but oblique, and therefore more force is required, as is clear from what was said both in book 1, chapter 16, on oblique elevations, and in book 3, chapter 12, on the overbalancing and equilibrium of weights suspended by rope. But this increase of effort in oblique elevation and suspension, where we have an abundance of laborers, is not considered; for it seems a sufficiently slight inconvenience, since it is compensated by the ease of moving the load. The second is that, if nearly all are laboring equally, they ought not to be dissimilar, but the obliquities of the ropes from which the suspended load is carried should be nearly the same; for it is evident that in a lesser obliquity of suspension less force is required than in a greater obliquity. Therefore, if I were trying to emulate this Chinese contrivance, I would first most firmly bind transverse poles to the ends of the load to be moved, or at least to points at equal distance from the center of gravity, as far as I could judge by conjecture, so that I might join ropes to the heads of these poles, by which the suspended load is carried. Then I would determine the length of these transverse poles so that in a line parallel to the poles, and at equal spacing, at least four men could be conveniently positioned, without causing any obstacle to one another as they advance. Next I would attach to each end of the poles sufficiently strong ropes of such length as would be needed so that, with three men before the load following one another in a straight order and at moderate intervals, so that the one behind does not strike the heel of the one before as he advances, it might reach the shoulder of the third man; for this seems to be the least obliquity of suspension, and the one that comes nearest to perpendicular suspension. But if the length of the ropes is greater, the load will be carried with greater labor; if it is lifted so high that it is much distant from the ground beneath, the obliquity of the suspension will be greater. Then another pole should be tied to the end of the rope, and should be supported by the other poles in the method I indicated a little above. Q99 3

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Mechanicorum Sit onus transferendum P; extremitati anteriori (omnia eadem in alterâ extremitate posita intelligantur) adnectatur vectis AB, cui in A & B jungantur funes AD & BC sufficientis longitudinis, quibus in D & C alligetur vectis ab aliis vectibus, ut paulo superiùs indicatum est, sustentatus, adeò ut quarto vecti duo homines facilè humeros supponere valeant, & singulorum funium extremitates D & C à sexdecim hominibus sustineantur. Quare si totidem funes atque homines posteriori ponderis parti simili ratione applicentur, totum pondus ab hominibus 64 æqualiter laborantibus sustinetur. Ex quo fit non adeò difficile esse in exercitu, ubi non est hominum succollantium inopia, bombardas ex loco in locum transferre, si nimis arduum sit iter, nec equis trahi possint: Nam majoribus bombardis pro singulis globi ferrei libris metalli libræ 150 aut 160 dimidiatis Cartois, ut vocant, in singulas globi libras, metalli libræ 180 aut 190, campestribus & minoribus bombardis metalli libræ 238 usque ad 266 in singulas globi libras communiter tribuuntur. Quòd si ex sint viarum angustiæ, quæ octo homines pariter incedentes non capiant, adhibeatur longior funis, duos, aut etiam tres, aut plures vectes connectens ita invicem distantes, ut intentus funis rectus sit, & propiores quidem suum vectem aut manu apprehensum sustentent, aut fune suspensum alio

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Mechanics Let the load P be transferred; to the front end (the same things are to be understood as placed at the other end) attach the bar AB, to which at A and B are joined ropes AD and BC of sufficient length, by which at D and C there shall be tied a bar supported by other bars, as was indicated a little above, so that under the fourth bar two men may easily place their shoulders, and the ends D and C of each rope be supported by sixteen men. Therefore if an equal number of ropes and men be applied in a similar way to the rear part of the weight, the whole weight is sustained by 64 men laboring equally. From this it follows that it is not so difficult in an army, where there is no lack of men to shoulder burdens, to transfer cannon from place to place, if the route is too steep and they cannot be drawn by horses: For larger cannon, for each pound of iron ball, 150 or 160 pounds of metal, with half-sized carts, as they are called, for each pound of ball, 180 or 190 pounds of metal, and for field and smaller cannon 238 up to 266 pounds of metal are commonly allotted for each pound of ball. But if there are narrownesses of roads, which do not admit eight men walking together, let a longer rope be used, connecting two, or even three, or more bars, so spaced apart from one another that the tightened rope is straight, and the nearer ones either support their bar by hand, or suspended by a rope another

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Liber quartus. CAPUT XVIII. 495 alio vecte parallelo humeris gestent, remotissimi verò humeros suo vecti subjiciant. Sic disponatur funis A H, ut intentus pertingat ad humeros eorum, qui in E & F sustentant vectes D E & F I. Quoniam verò vectis M N longè depressior est, quàm humeri eorum, qui tam propè absunt à pondere; propterea vel solis manibus apprehensum vectem sustentent, vel, quod satius est, alium præterea vectem humeris gestent parallelum vecti M N, ita ut ex illo funibus ad perpendiculum intentis suspendantur extremitates M & N. Id quod etiam de reliquis, atque de consequentibus vectibus dictum intelligatur. Omnes autem æqualiter conari palàm est, quia intento fune A H eadem est obliqua suspensio ponderis, & paria sunt momenta adversùs singulos vectes, quos funis connectit. Illud tamen negari non potest, quod pro majore funis A H longitudine major est suspensionis obliquitas, ac proinde, & major sustentandi labor.

Transcription: Translated (English)

Book Four. Chapter XVIII. 495 Let them carry, by another parallel pole, on their shoulders; but the farthest should place their shoulders under their own pole. Let the rope A H be arranged so that, when stretched tight, it reaches the shoulders of those who at E & F support the poles D E & F I. But since the pole M N is much lower than the shoulders of those who are so far from the weight, for that reason either let them support the pole grasped only with their hands, or, what is better, let them also carry on their shoulders another pole parallel to the pole M N, so that from it, with ropes stretched vertically, the ends M & N may be suspended. The same is to be understood as said also of the remaining, and of the following, poles. Now it is clear that all should exert themselves equally, because with the rope A H stretched tight the oblique suspension of the weight is the same, and the moments against the individual poles which the rope connects are equal. Yet this cannot be denied: the greater the length of the rope A H, the greater the obliquity of the suspension, and consequently the greater the labor of supporting it.

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Mechanicorum 496 Unum adhuc hîc addere (ne quid intactum relinquatur) fuerit operæ pretium, videlicet, si ponderis transferendi crassities seu altitudo mediocris saltem fuerit, ita ut non solùm infimo plano subjici vectes possint, sed etiam supremæ aut mediae parti adnecti, posse eidem lateri duos aut etiam tres funes, non quidem omninò, sed proximè parallelos alligari, quibus duæ, aut tres, ferè similes obliquæ suspensiones fiant, & deferentes pondus alij aliis remotiores sint, ferè tamen æqualiter conantes. Sic ingentis saxi altitudo sit FG, & al- ligatus in F funis connectantur cum vecte in S aliis vectibus sustentato, ut supra. Item in E & in G alij funes paralleli similer jungantur cum vectibus in T & V, ut homines ibi succollantes vectibusque subjecti sibi invicem impedimento non sint. Si igitur singulis lateribus ad B, C, D tres funes hac ratione addantur, erunt 12 funes, & si homines 16 singulis funibus applicentur methodo superiùs indicatâ, pondus gestabitur à viris 192: constat igitur quàm ingens onus facilè transferri vectibus queat. PROPOSITIO V. Multiplici vecte moventis vires augere. Pro vectis longitudine majori, in eâdem ab hypomochlio distantiâ ponderis, potentiæ momenta augeri, quia Ratio motûs

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Mechanics 496 One thing still to add here, lest anything be left untouched, would be worth the effort: namely, if the thickness, or average height, of the load to be transferred is at least such that not only can levers be placed beneath its lowest plane, but also attached to its upper or middle part, then to the same side two or even three ropes may be fastened, not exactly but nearly parallel, by which two or three almost similar oblique suspensions are made, and the ropes carrying the load are farther apart from one another, yet still exerting force in nearly equal fashion. Thus let the height of a huge stone be FG, and let the rope tied at F be connected with the lever at S, supported by other levers, as above. Likewise at E and at G let other parallel ropes be similarly joined with levers at T and V, so that the men there, lifting on their shoulders and placed beneath the levers, may not hinder one another. If therefore on each side at B, C, D three ropes are added in this manner, there will be 12 ropes, and if 16 men are assigned to each rope by the method indicated above, the load will be borne by 192 men: it is therefore evident how vast a burden can easily be transferred by levers. PROPOSITION V. To increase the moving force by means of a multiple lever. Since, with a greater length of lever, at the same distance from the fulcrum of the weight, the moments of force are increased, because the ratio of motion

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Liber quartus. CAPUT XVIII. 497 motûs potentiæ ad motum ponderis augetur, satis manifestum est ex dictis. Verùm quia non rarò tam longus vectis, quanto opus esset, in promptu non est, aut ipsa longitudo illum redde- ret fractioni, aut saltem flexioni, magis obnoxium, aut, si peri- culo huic occurratur, tam immanis est vectis moles, ut non levi incommodo sit eo utentibus: propterea ars aliqua excogitanda est, qua oblati vectis brevitatem compensatione aliquâ sup- pleamus. Et primò quidem si oblatus sit vectis A B, habens hypomo- chlium in C, & pondus in B tam grave, ut unica potentia in A non satis sit ad vincendam oneris resi- stentiam, utique si altero, aut ter- tio movente opus sit, non omnes in extremitate A vectem apprehen- dere valent, sed alter in D, tertius in E; qui propterea, licèt singuli æquali robore polleant, non tamen æqualia habent momenta, sed pri- mus ut A C, secundus ut D C, ter- tius ut E C. Quapropter alter vectis G H adnectatur extremitati A ad angulos rectos, ut huic applicati motores plus ha- beant momenti. Si enim A C ad C B fuerit ut 10 ad 1, perinde est, atque si decima ponderis pars à duabus in G & H æquali- ter ab A distantibus movenda esset, ac propterea singuli semis- sem decimæ partis resistentiæ percipiunt, hoc est, habent simul sumpti momentum ut 20 ad 1: qui autem in A esset solus, habe- ret momentum ut 10, & qui in D haberet momentum ex. gr. ut 9; qui idcircò simul sumpti minùs possunt quàm G & H. At si volueris tres homines in extremitatibus vectis G H di- stribuere in potentias æqualiter conantes, distingue G H in tres partes, & sit A H triens totius longitudinis G H: tum duo ap- plicentur extremitati H, tertius verò extremitati G: ut enim potentia duplex in H ad potentiam in G, ita reciprocè duplex distantia G A ad distantiam A H. Nam quemadmodum de sustinentibus pondus vecte sive æqualiter, sive inæqualiter di- viso dictum est, ita hîc pariter de Prementibus dicendum, qui in H & in G sunt vicissim Potentia & Hypomochlium: ex eo R r r

Transcription: Translated (English)

Book four. Chapter XVIII. 497 That the power of motion is increased by the motion of the weight, is sufficiently clear from what has been said. But because not infrequently such a long lever as would be needed is not at hand, or else its very length would make it more liable to breakage, or at least to bending, or, if this danger be avoided, the bulk of so huge a lever would be no slight inconvenience to those using it: therefore some device must be invented by which we may make up for the shortness of the lever offered by some compensation. And first, indeed, if a lever AB be offered, having its fulcrum in C, and the weight at B so great that a single power at A is not enough to overcome the resistance of the load, then if the work requires a second or a third mover, not all can grasp the lever at the extremity A, but one at D and a third at E; who therefore, although each may have equal strength, do not yet have equal moments, but the first as AC, the second as DC, the third as EC. Wherefore let another lever GH be attached at right angles to the extremity A, so that the movers applied to this may have more moment. For if AC be to CB as 10 to 1, it is just as though the tenth part of the weight were to be moved by two points at G and H equally distant from A; and therefore each receives half of the resistance of that tenth part, that is, together they have a moment as 20 to 1: but the one who stood alone at A would have a moment as 10, and the one at D would have a moment, for example, as 9; and therefore those taken together are less able than G and H. But if you should wish to distribute three men at the extremities of the lever GH, in powers equally striving, divide GH into three parts, and let AH be one third of the whole length GH: then let two be applied to the extremity H, and the third to the extremity G; for as the double power at H is to the power at G, so conversely is the double distance GA to the distance AH. For just as has been said concerning those supporting a weight by a lever, whether divided equally or unequally, so here likewise it must be said of those pressing, who in H and in G are, as it were, alternately Power and Fulcrum: from this

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Mechanicorum 498 scilicet quòd potentia in H premit, habet rationem hypomo- chlij, dum resistit, ne potentia in G premens elevet ipsam ex- tremitatem H, ac propterea deprimat pondus in A existens; & vicissim potentia in G premens habet rationem hypomochlij resistendo, ne elevetur à potentiâ premente in H; quæ prop- terea deprimit pondus in A. Est itaque veluti duplex vectis se- cundi generis; & A G ad A H si fuerit ut 2 ad 1, momentum potentiæ in G ad resistentiam ponderis in A est ut 3 ad 1, hoc est ut G H ad A H: & momentum unius potentiæ in H ad re- sistentiam ponderis in A est ut 3 ad 2, hoc est, ut H G ad A G. Sed quia in H ex hypothesi sunt duæ potentiæ, duplicatæ po- tentiæ in H momentum erit ut 6 ad 2, hoc est, singularum momentum ut 3 ad 1. Igitur qui est in G habet momentum at- que conatum, quasi sine vecte moveret trigesimam partem pon- deris in B existentis; & duo, qui in H, singuli habent momen- tum æquale atque conatum similem. Ponatur pondus B lib. 60; in A percipitur ponderis 1/10, hoc est, lib. 6. Igitur potentia in G percipit resistentiam lib. 2, & duæ potentiæ in H simul lib 4, hoc est singulæ lib. 2. Quod si non placuerit longitudinem G H habere tanquam vectem, qui non alternâ quadam motione & quiete extremita- tum perficitur motus, sed G, & A, & H omnino simul & æquali motu moventur, non admodum contendo: perinde erit, atque si tres potentiæ in A essent constitutæ, quarum singulæ tertiam partem ponderis moveant conatu subdecuplo illius co- natus, quo tertia illa pars sine vecte movenda esset. Hinc saltem constat, quo virium atque conatûs compendio valeat unicus homo oblati vectis momenta augere: Nam si idem sit primi generis vectis A B, & A C ad C B sit ut 10 ad 1, adhi- be vectem secundi generis G H, & alterâ extremitate fixâ, ut ibi sit hypomochlium, idem augebit momenta juxta Rationem totius longitudinis G H ad distantiam ipsius A ab hypomo- chlio: Quare si Ratio sit dupla, aut tripla, æquivalebit duobus aut tribus, qui in A moventes haberent momentum decuplum; nam A movetur decuplo velociùs quàm B, & posito hypomo- chlio H, movetur potentia G duplo aut triplo velociùs quàm A, hoc est vigecuplo aut trigecuplo velociùs quàm B. Id quod usum habet non solùm, quando vectis A B movendus est in pla- no

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Mechanics 498 namely, because the force at H presses, it has the role of a hypomochlion, while it resists the force pressing at G, so that it may not lift the end H itself, and thereby depress the weight located at A; and conversely, the force pressing at G has the role of a hypomochlion in resisting, lest it be lifted by the force pressing at H; which therefore depresses the weight at A. Thus it is as though a double lever of the second kind; and if A G to A H be as 2 to 1, the moment of the force at G to the resistance of the weight at A is as 3 to 1, that is, as G H to A H: and the moment of one force at H to the resistance of the weight at A is as 3 to 2, that is, as H G to A G. But because in H by hypothesis there are two forces, the doubled force in H will have a moment as 6 to 2, that is, the moment of each as 3 to 1. Therefore the one which is in G has a moment and an effort as though it moved, without a lever, one thirtieth part of the weight existing in B; and the two which are in H each have equal moment and a similar effort. Let the weight B be 60 lb.; in A there is perceived one-tenth of the weight, that is, 6 lb. Therefore the force in G perceives a resistance of 2 lb., and the two forces in H together 4 lb., that is, each 2 lb. If it should not please one to have the length G H as a lever, in which the motion is accomplished not by some alternate movement and rest of the extremities, but G, and A, and H are moved altogether and with equal motion, I do not insist on that very strongly: it will be all the same, as though there were three forces placed at A, each of which moves a third part of the weight by an effort ten times smaller than that effort by which that third part would have to be moved without a lever. From this at least it is evident by what saving of force and effort a single man can increase the moments of the proposed lever: for if the lever A B of the first kind be the same, and A C to C B be as 10 to 1, use the lever of the second kind G H, and with one extremity fixed, so that there it may be the hypomochlion, it will increase the moments accordingly to the ratio of the whole length G H to the distance of A itself from the hypomochlion: wherefore if the ratio be double or triple, it will be equivalent to two or three who, moving at A, would have a tenfold moment; for A is moved ten times more quickly than B, and, the hypomochlion H being posited, the force G is moved two or three times more quickly than A, that is, twenty times or thirty times more quickly than B. This is useful not only when the lever A B is to be moved in a plane

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Liber quartus. CAPUT XVIII. 499 no Verticali, sed etiam in plano horizontali, ut si duo marmora disjungenda essent, aut clathri dissipandi. Juxta autem loci opportunitatem adjungendus est secundus vectis GH aut proximè ipsi primo vecti, aut remotè medio fune extremitatem A connectente cum secundo vecte. Sic inter duo marmora immissus ferreus clavus SR jungitur vecti TV fune SO, & potentia in T habet momentum compositum ex Rationibus TV ad V O, & SX ad XR. Neque duos tantummodo, verùm etiam plures vectes adhibere possumus, tunc maximè, cùm ingenti oneri exiguus motus tribuendus est. Sit enim marmor P attollendum subjecto vecte AB secundi generis habente hypomochlium in B, ac pondere incumbente illi in C: & AB ad CB sit ut 7 ad 1. Quia vectis attollendus est, subjice illi in A vectem alterum DE, ut ED ad AD sit in Ratione 3 ad 1. Item extremitati E subjice tertium vectem FG, & sit GF ad EF ut 8 ad 1. Igitur A movetur septuplò velociùs quàm C, & E triplo velociùs quàm A, atque G octuplo velociùs quàm E. Quare motus potentiæ in G ad motum ponderis in C est ut 168 ad 1. Quàm difficile autem accideret, si tam longum vectem parare oporteret, cujus longituduo esset ad CB ut 168 ad 1! Adde non solùm vectibus rectis hoc momentorum incrementum acquiri posse, sed etiam pro loci opportunitate vectibus curvis aut angulatis. Si enim in superiore loco fuerit vectis secundi generis MN oneri subjectus, aut oneri inferiùs posito junctus fune

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Book Four. Chapter XVIII. 499 not only in the vertical direction, but also in the horizontal plane, as if two marbles were to be separated, or grates to be pulled apart. And according to the convenience of the place, a second lever GH must be added either close to the first lever itself, or at a distance by means of a rope connecting the end A with the second lever. Thus, when an iron pin is inserted between two marbles, SR is joined to the lever TV by the rope SO, and the power at T has a composite momentum from the ratios of TV to VO, and of SX to XR. Nor can we use only two levers, but even more, especially when a small motion must be given to a great load. Let marble P, for example, be to be raised by means of the lever AB beneath it, a lever of the second kind, having its fulcrum at B, and the weight resting on it at C; and let AB be to CB as 7 to 1. Since the lever is to be raised, place beneath it at A another lever DE, so that ED to AD be in the ratio 3 to 1. Likewise place a third lever FG at the end E, and let GF be to EF as 8 to 1. Therefore A is moved seven times faster than C, and E three times faster than A, and G eight times faster than E. Hence the motion of the power at G to the motion of the weight at C is as 168 to 1. How difficult, however, it would be if such a long lever had to be prepared, whose length would be to CB as 168 to 1! Add that this increase of momentum can be obtained not only by straight levers, but also, according to the convenience of the place, by curved or angled levers. For if in the upper position there should be a lever of the second kind MN subject to the load, or joined to the load placed below by a rope

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Mechanicorum in O, non solùm possumus extremitatem M fune connectere cum vecte recto superiùs posito, sed etiam subjicere illi possumus vectem curvum K L fixum in I, & extremitas L fune L R trahi potest deorsum, ut I K elevetur, atque illo motu attollat extremitatem M, quantum ferre potest flexus I K. Non est autem opus monere inæqualia sensim fieri momenta, prout subjectus vectis curvus K I L in alio atque alio puncto contingit vectem M N, pro variâ scilicet distantiâ ab hypomochlio. In vecte tertij generis majorem esse ponderis motum motu potentiæ, ac proinde majores requiri potentiæ vires ad attollendum onus, si illa conjuncta ac sociata sit cum hujusmodi vecte, quàm si ipsa solitaria manum admoveret ponderi sublevando, manifestum est; propterea infirmiori potentiæ subsidium aliquod industriâ comparare possumus, & propositum vectem in aliam vectis speciem quasi convertere, etia[m] si spatij angustiis coarctemur, modò liceat proximum parietem perfodere. Sit parieti AB innixus vectis CD, cujus extremitati D adnectendum sit pondus ex. gr. lib. 200: potentia autem applicari nequeat nisi in E, ita ut E C sit quarta pars totius vectis CD. Igitur, cum motus in D sit quadruplus motûs in E, ut potentia sublevet onus D, tanta sit, oportet, ut ipsa se sola valeat quadruplum onus, scilicet lib. 800 attollere: id quod valdè incommodum accideret, si adeò validam potentiam invenire opus esset. Perfode igitur in superiore parte B parietem, illique immitte vectem F G facilè in B hypomochlio versatilem, ita ut B F pars imminens subjecto vecti sit æqualis parti E C, hoc est, distantiæ potentiæ E ab hypomochlio C, & fune F E connectantur: pars verò ultra parietem in proximum conclave extans B G ad partem B F sit in quacumque Ratione. Tum in inferiore loco, prout opportunius acciderit, vectem alium statue HI, cui junge superioris Vectis extremitatem G fune GM: nam Ratio composita ex Rationibus IH

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On mechanics in O, not only can we connect the extremity M by a rope with a straight lever placed above, but we can also place beneath it the curved lever K L fixed at I, and the extremity L can be pulled down by the rope L R, so that I K is raised, and by that motion lifts the extremity M, as far as the bend of I K can bear. It is not necessary, however, to point out that the moments become unequal by degrees, according as the curved lever K I L placed beneath touches the lever M N at one point or another, namely, according to the different distance from the fulcrum. In a third-class lever, the motion of the weight is greater than the motion of the power, and therefore greater force is required from the power to lift the load, if it is joined and associated with such a lever, than if it alone were to apply the hand to the load being raised, is manifest; therefore, by contrivance we can obtain some aid for a weaker power, and in effect turn the proposed lever into another kind of lever, even if we are confined by narrow space, provided we are allowed to pierce the nearby wall. Let a lever CD rest against the wall AB, to whose extremity D there is to be attached a weight, for example of 200 lb.; but the power cannot be applied except at E, so that E C is the fourth part of the whole lever CD. Therefore, since the motion at D is four times the motion at E, in order that the power may raise the load D, it must be such that it alone can lift a quadruple load, namely 800 lb.; which would be very inconvenient, if it were necessary to find a power so strong. Bore therefore in the upper part B of the wall, and insert into it a lever F G easily turning at the fulcrum B, so that the part B F projecting over the supported lever be equal to the part E C, that is, to the distance of the power E from the fulcrum C, and connect them by the rope F E: but let the part beyond the wall extending into the adjacent room B G be in any ratio to the part B F. Then in a lower place, as shall seem more convenient, set another lever HI, to which join the extremity G of the upper lever by the rope GM: for the ratio composed of the ratios IH

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Liber quartus. CAPUT XVIII. 501 IH ad MH, & GB ad BF dabit momentum potentiæ in I po- sitæ ad attollendum pondus in D constitutum per vectem da- tum CD habentem potentiam in E. Hic habes tria vectis genera; nam IH est secundi generis, quia pondus intelligitur in M inter potentiam I & hypomochlium H; GF est primi generis, quia hypomochlium B est inter potentiam G & pondus in F; CD est tertij generis, quemadmodum ab ini- tio constitutum est. Si itaque in E requireretur vis attollendi lib.800, & sit GB dupla ipsius BF, requiritur in G vis attollendi lib.400. Si verò IH ad MH sit quadrupla, requiritur in I vis elevandi lib.100. Quare & uteris vecte tertij generis CD, quo satis notabiliter movetur pondus D; & potentiæ momenta auxisti adeò, ut non solùm non requiratur potentia major pondere attollendo, sed sufficiat potentia minor, habet quippe motum duplo majorem, quàm sit motus ponderis D; nam mo- tus extremitatis F & puncti E sunt æquales; motus potentiæ I est quadruplus motus ipsius M; hoc est extremitatis G; hæc ve- rò motum habet duplum motus ipsius F: igitur motus poten- tiæ I est octuplus motûs puncti E, quod movetur motu subqua- druplo extremitatis D: Motus igitur potentiæ I ad motum ponderis D est ut 8 ad 4, hoc est ut 2 ad 1. PROPOSITIO VI. Stateræ vires addito Vecte augere. Paretur hasta AB, atque extremitati A addatur annulus, cui inseri valeat stateræ CD uncus, & extremitas B ita confor- metur, ut notabile sit & conspicuum punctum, quod hypomo- chlio respondeat; sitque certa nota, qua dignoscatur vectis pa- rallelusne sit horizonti, an inclinatus. Tum distantia B A di- R r r 3

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Book Four. CHAPTER XVIII. 501 IH to MH, & GB to BF will give the force of the power placed at I for raising the weight set at D by the given lever CD, having the power at E. Here you have three kinds of lever; for IH is of the second kind, because the weight is understood at M between the power at I and the fulcrum H; GF is of the first kind, because the fulcrum B is between the power at G and the weight at F; CD is of the third kind, as was established from the beginning. If therefore, in E, there were required a force for lifting 800 lb., and GB be double BF, there is required at G a force for lifting 400 lb. If, however, IH be quadruple MH, there is required at I a force for lifting 100 lb. Wherefore if you use also the lever CD of the third kind, by which the weight D is moved sufficiently noticeably; and you have so increased the moments of the power that not only is a power greater than the weight to be lifted not required, but a lesser power suffices, since it has, in fact, a motion twice as great as the motion of the weight D; for the motion of the extremity F and of the point E are equal; the motion of the power I is four times the motion of M itself, that is, of the extremity G; and this again has a motion twice the motion of F: therefore the motion of the power I is eight times the motion of point E, which is moved with a motion one-quarter that of the extremity D: therefore the motion of the power I to the motion of the weight D is as 8 to 4, that is, as 2 to 1. PROPOSITION VI. To increase the force of scales by adding a lever. Let a rod AB be prepared, and to the extremity A let there be added a ring, into which the hook of the scales CD may be inserted, and let the extremity B be so shaped that a notable and conspicuous point may be present, corresponding to the fulcrum; and let there be a fixed mark by which it may be recognized whether the lever is parallel to the horizon or inclined. Then the distance BA di- R r r 3

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Mechanicorum 502 vidatur primùm in duas partes, deinde in tres, & sic deinceps, quatenus commodè fieri id poterit citra confusionem, quando opus fuerit huic aut illi puncto adnectere onus expendendum, adeò ut certi scimus, quotuplex sit totius vectis A B longitudo comparata cum distantiâ ponderis ab hypomochlio B. Hoc vecte ad usum parato, examinetur staterâ communi, quantum ille gravitet parallelus horizonti: & sit æquipondium stateræ ex. gr. in H indicans lib. 2: id quod memoriâ retinen- dum est, ut, cùm ponderis gravitas explorabitur, ex numero, qui in stateræ jugo indicabitur ab æquipondio, dematur ipsa vectis gravitas deprehensa, scilicet lib. 2. Proposita igitur gravitate majori, quàm ut expendi valeat communi staterâ CD, adnecte onus vecti in aliquo ex adno- tatis punctis, ex. gr. in puncto 6, prout commodius acciderit: tum reduc tantisper æquipondium stateræ, dum ejusdem sta- teræ jugum & vectis æquè ab horizonte distent, & consistat æquipondium, puta, in puncto I indicante lib. 12. unc. 8. de- me lib. 2. gravitatem vectis, remanent lib. 10. unc. 8. Quia autem onus ex hypothesi adnexum est in puncto 6, multiplica per 6 lib. 10. unc. 8, & habebis lib. 64 gravitatem oneris quæ- sitam. Quod si plane in medio puncto 2 constitutum fuisset pondus, duplicanda esset gravitas indicata à staterâ. Mani- festa est hujus operationis ratio; siquidem æquipondium sta- teræ in puncto I sustinet lib. 12. unc. 8 adnexas extremitati D. At vis sustinendi in vecte secundi generis posita in A sustinet vectem, cujus momentum est lib. 2, & præterea sustinet pon- dus in puncto 6 positum, quod ad reliquam potentiæ virtutem in A, hoc est lib. 10. unc. 8, habet Rationem, quæ sit A B ad B 6. igitur convertendo ut 1 ad 6, ita lib. 10. unc. 8. ad lib. 64. Quoniam verò accidere potest, ut oblatum pondus exce- dat quidem datæ stateræ vires, sed ejus gravitas minor sit quàm dupla ejus, cui æquipondium in extremo stateræ jugo respon- det; propterea divisiones eædem, quæ ex 2 ad B adnotatæ sunt, transferantur ex 2 versùs A, ut habeamus diversa puncta in vecte, quibus applicari possit onus ponderandum. Ex numero igitur adnotato, cui adnectitur pondus, fiat numerator fractionis, cujus Denominator sit unitate minor ipso numeratore; & per hanc fractionem multiplicetur numerus à staterâ indicatus (demptâ

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Mechanics 502 First let it be divided into two parts, then into three, and so on, as far as this can conveniently be done without confusion, whenever it is necessary to attach to this or that point a weight to be weighed, so that we know with certainty how many times the whole length of the lever AB is as compared with the distance of the weight from the fulcrum B. This lever being prepared for use, let it be examined with an ordinary balance how much it weighs parallel to the horizon; and let there be a counterpoise on the balance, for example at H, indicating 2 lb. This must be kept in mind, so that when the weight of the lever is investigated, from the number indicated on the balance-beam by the counterpoise, the weight of the lever itself, already found, namely 2 lb., may be deducted. Therefore, if a weight greater than can be weighed by the ordinary balance CD is proposed, attach the load to the lever at one of the marked points, for example at point 6, as may be more convenient; then lower the counterpoise of the balance until the beam of the same balance and the lever are equally distant from the horizon, and let the counterpoise stand, say, at point I, indicating 12 lb. 8 oz. Subtract 2 lb., the weight of the lever, and 10 lb. 8 oz. remain. But since, by hypothesis, the load is attached at point 6, multiply 10 lb. 8 oz. by 6, and you will obtain 64 lb. as the sought weight of the load. If the weight had plainly been placed at the midpoint, point 2, the weight indicated by the balance would have to be doubled. The reason for this operation is clear; for the counterpoise of the balance, standing at point I, supports 12 lb. 8 oz. attached to the extremity D. But the sustaining force in a lever of the second kind placed at A supports the lever, whose moment is 2 lb., and moreover supports the weight placed at point 6, which, in relation to the remaining power of force at A, that is 10 lb. 8 oz., has the ratio that AB is to B6. Therefore, by converting as 1 is to 6, so is 10 lb. 8 oz. to 64 lb. Now because it may happen that the weight presented does indeed exceed the power of the given balance, yet its weight is less than twice that weight to which the counterpoise corresponds at the end of the balance-beam, therefore the same divisions, which were marked from 2 toward B, are transferred from 2 toward A, so that we may have different points on the lever to which the weight to be weighed may be applied. From the number therefore marked, to which the weight is attached, make the numerator of a fraction whose denominator is one unit less than the numerator itself; and by this fraction multiply the number indicated by the balance, after the

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Liber quartus. CAPUT XVIII. 503 (demptâ priùs vectis gravitate, ut superiùs dictum est) & habe- bitur oneris gravitas. Sit igitur inter A & 2 adnexum pondus in puncto 7, & statera indicet lib. I 3. unc. 7: demo vectis gravita- tem, quæ est lib. 2 ex hypothesi, remanent lib. I 1. unc. 7. mul- tiplicandæ per 2/6, & dabitur oneris gravitas lib. I 3. unc. 6 1/6. Quare in hoc casu fortasse nullum habetur ex vecte adjecto compendium, potuisset enim ex ipsâ statera immediatè cognosci eadem gravitas. Quod si eundem numerum indicasset sta- tera, sed onus adjunctum fuisset in puncto 3, per 1/2 multiplica- tis lib. I 1. unc. 7, provenisset gravitas oneris quæsita lib. I 7 unc. 4 1/2, quæ ex hypothesi major est, quàm ut solâ staterâ oblatâ expendi possit. Vel si rem breviùs expedire placuerit, numeri staterâ inventi accipe partem denominatam à numero vectis unitate minore, eâmque illi numero invento adde, & idem obtinebis. Sic quia in puncto 3 appensum fuit onus, ac- cipe librarum I 1. unc. 7. partem denominatam à 2, scilicet lib. 5. unc. 9 1/2, eâmque adde libris I 1. unc. 7 inventis, & habebis, ut priùs, lib. 17 unc. 4 1/2. Cur hac methodo operandum sit, manifestò constat ex ipsa vectis divisione; nam A B ad A 3 est ut 3 ad I ex constructione, atque ideò A B ad 3 B est ut 3 ad 2: igitur ut 2 ad 3, ita numerus à statera indicatus (demptâ vectis gravitate) ad numerum quæsitum, quo ponderis gravitas innotescit. Generatim itaque atque universè oblato quocumque vecte ad subitum usum properato utere, etiamsi nullæ in eo divisio- nes adnotatæ fuerint, examinato tamen priùs ipsius vectis ho- rizonti paralleli gravitatis momento, quatenus ad stateram comparatur: Tum datum pondus ibi alliga, ubi commodè à staterâ extremo vecti applicatâ elevari possit. Facto demum æquilibrio, stateræ numerum (dempto priùs vectis momen- to) multiplica per Rationem, quam habet vectis longitudo ad distantiam ponderis ab hypomochlio; & propositum obtine- bis. Hîc habes maximum compendium ad ingentium ponde- rum gravitatem explorandam: etiamsi enim vectis non sit adeò crassus, quia tamen non procul ab extremitate illius, ubi est hypomochlium, alligatur onus, validè resistit fractioni; & quo major est Ratio longitudinis vectis ad distantiam pon- deris

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Book Four. CHAPTER XVIII. 503 (after first subtracting the weight of the lever, as said above) and the weight of the load will be obtained. Let therefore the attached weight be at point 7 between A and 2, and let the balance indicate 1 lb. 3 oz. 7; subtract the weight of the lever, which by hypothesis is 2 lb., and there remain 1 lb. 1 oz. 7, to be multiplied by 2/6, and there will be given the weight of the load, 1 lb. 3 oz. 6 1/6. Therefore in this case perhaps no saving is obtained from the added lever, for the same weight could have been known immediately from the balance itself. But if the balance had indicated the same number, and the load had been attached at point 3, by multiplying 1 lb. 1 oz. 7 by 1/2 there would have resulted the desired weight of the load, 1 lb. 7 oz. 4 1/2, which by hypothesis is greater than can be weighed by the balance alone. Or if it should be preferred to finish the matter more briefly, take the denominator-part of the number found by the balance from the number one less than the lever, and add it to that found number, and you will obtain the same result. Thus, because the load was hung at point 3, take from 1 lb. 1 oz. 7 the denominator-part of 2, namely 5 lb. 9 1/2 oz., and add this to the 1 lb. 1 oz. 7 found, and you will have, as before, 1 lb. 7 oz. 4 1/2. Why this method should be used is clearly evident from the division of the lever itself; for AB is to A3 as 3 to 1 by construction, and therefore AB is to 3B as 3 to 2: therefore as 2 to 3, so is the number indicated by the balance (after subtracting the weight of the lever) to the number sought, by which the weight of the load is made known. In general, then, whenever you are provided with any lever for immediate use, make use of it even if no divisions have been marked on it, but first having examined the moment of the lever’s weight as parallel to the horizon, in so far as it is compared with the balance: then fasten the given weight there where it can conveniently be lifted by the balance applied at the end of the lever. Finally, when equilibrium has been achieved, multiply the number of the balance (after first subtracting the moment of the lever) by the ratio which the length of the lever bears to the distance of the weight from the fulcrum; and you will obtain the proposed result. Here you have the greatest saving of labor for determining the weight of enormous loads: for although the lever may not be very thick, yet because the load is fastened not far from the end of it, where the fulcrum is, it strongly resists breaking; and the greater the ratio of the length of the lever to the distance of the weight

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Mechanicorum 504 deris ab hypomochlio, tanto majore incremento augentur stateræ vires. Quod si fortè unicus vectis satis non fuerit, nihil prohibet plures adhiberi vectes multo majore compendio, quàm si unicum longiorem adhiberes. Nam si vectis AB non ita stateræ vires multiplicet, ut tormentum æneum in C alligatú elevari possit ab æquipondio stateræ, alium vectem EF statue ipsi AB parallelum, habeatque in E hypomochlium, & statera in F adnectatur, qua primùm ipsorum vectium fune HI conjunctorum & positionem horizonti parallelam habentium gravitatis momentum expendatur. Deinde facto æquilibrio dematur vectium momentum, & reliquus librarum numerus à staterâ indicatus multiplicetur primò per Rationem FE ad HE, & quod ex hac multiplicatione consurgit, secundò multiplicetur per Rationem IB ad CB; habebitur enim demum tormenti ænei gravitas quæsita. Sit ex. gr. IB ad CB ut 10 ad 1, & FE ad HE ut 12 ad 1, atque statera, dempto vectium momento, indicet libras 100: igitur 100 per 12 dat 1200, & 1200 per 10 dat lib. 12000 gravitatem ænei tormenti. His autem indicatis statim occurit animo non duos tantummodo sed plures vectes posse ita disponi, ut semper fiat major Ratio, quæ ex illorum Rationibus componitur: si nimirum inter duas trabes in solo ad perpendiculum firmatas, & æquali intervallo à se invicem dissitas interjiciantur vectes alterna hypomochlia habentes in axibus, circa quos facilè converti possint, & simili ratione jungantur, ac de duobus vectibus AB & EF dictum est: Ex singulorum enim vectium Rationibus una Ratio componitur, per quam multiplicandus est numerus à staterâ indicatus, dempto priùs vectium momento. Id quod paulo latiùs explicatum est in Terra machinis mota. dissert. -1- n. 16. nec opus est hìc transcribere. MECHA

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Mechanics 504 from the hypomochlion, the forces of the balance are increased by so much greater a ratio. But if by chance a single lever should not be sufficient, there is nothing to prevent several levers from being used, with far greater advantage than if you were to use a single longer one. For if the lever AB does not multiply the forces of the balance so that the bronze cannon tied at C can be raised by the counterweight of the balance, place another lever EF parallel to AB itself, let it have its hypomochlion at E, and let the balance be attached at F, by which first the weight moment of the joined levers themselves, having the cord HI and a position parallel to the horizon, is weighed. Then, after equilibrium has been established, remove the weight of the levers, and let the remaining number of pounds indicated by the balance be multiplied first by the ratio of FE to HE, and what arises from this multiplication be multiplied secondly by the ratio of IB to CB; for thus at last the desired weight of the bronze cannon will be obtained. Suppose, for example, IB to CB is as 10 to 1, and FE to HE as 12 to 1, and the balance, the weight of the levers having been removed, indicates 100 pounds: therefore 100 times 12 gives 1200, and 1200 times 10 gives 12,000 pounds, the weight of the bronze cannon. Having stated these things, it immediately occurs to the mind that not only two but many levers can be so arranged that a greater ratio is always obtained, composed from their ratios: namely, if between two beams firmly set upright in the ground, and at equal intervals from one another, levers are inserted having alternate hypomochlia in the axes around which they can easily be turned, and are joined in a similar manner as was said of the two levers AB and EF: for from the ratios of the individual levers one ratio is composed, by which the number indicated by the balance must be multiplied, after the weight of the levers has first been removed. This has been explained somewhat more fully in Terra machinis mota. dissert. -1- n. 16, and there is no need to transcribe it here. MECHA

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MECHANICORUM LIBER QUINTUS. De Axe in Peritrochio. UÆ de Vecte ejusque viribus superiore libro disputata sunt, illa quidem vera sunt, & ad- mirabilia, sed, nisi vectis admodum longus sit, exiguus motus conciliatur ponderi, adeò ut, si ad notabilem aliquam altitudinem attol- lendum illud sit, oporteat subinde & ipsi ponderi fulcrum supponere, ne recidat, & ipsi vecti hypomochlium altiùs subjicere, ut congruo loco statuatur. Præterquam quod pro variâ ipsius vectis inclinatione, onerisque illi impositi, aut subjecti positione, varia quoquè sunt momenta potentiæ vectem urgentis. Hinc alia Facultas excogitata est, quæ, ut pluribus placet, vectis quidam sit perpetuus, citra in- commoda, quæ in simplici Vecte, ut innuebam, occur- runt. Vectem autem appellant, quia ad vectis Rationes il- lius vim revocant; perpetuum verò, quia nullâ opus est hypomochlij mutatione: proprio tamen, tritóque jam ve- tustate vocabulo, communiter dicitur Axis in Peritrochio, quasi Axis in Rota, ut quidam interpretantur; sed fortassè clariùs, pleniúsque vocabuli vim assequeremur, si Axem Convolutum vocaremus; neque enim semper adest Rota, cum tamen semper intersit Convolutio, simul quippe vol- vitur, & Axis ipse, & id, cum quo Axis conjungitur. S s s

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MECHANICS BOOK FIVE. On the Wheel and Axle. What was discussed in the foregoing book concerning the Lever and its powers is indeed true and admirable; but unless the lever be very long, only a small motion is communicated to the weight, so that, if it is to be raised to any notable height, it is necessary to keep placing a fulcrum under the weight itself, lest it fall back, and to set the fulcrum higher under the lever itself, so that it may be positioned at a suitable place. Besides this, according to the various inclination of the lever itself, and the position of the burden placed upon or beneath it, the moments of the force pressing the lever are likewise various. Hence another faculty was devised, which, as many prefer, is a kind of perpetual lever, without the inconveniences which, as I hinted, occur in the simple lever. It is called a lever because its force is referred back to the principles of the lever; perpetual, however, because no change of fulcrum is needed. Yet, by its proper and now by long use familiar name, it is commonly called the Wheel and Axle, almost as if an axle in a wheel, as some interpret it; but perhaps we should more clearly and fully grasp the force of the term if we called it a Wound Axle; for a wheel is not always present, whereas winding is always involved, since both the axle itself and that with which the axle is joined are turned together.

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506 Mechanicorum Neque hic sumitur Axis quemadmodum in Cono, Cylin- dro, atque Sphærâ, pro linea rectâ, circa quam immo- tam corpora illa in gyrum aguntur; sed est corpus suâ præditum crassitie, cui Axis nomen inditum est, quia ro- tarum axem imitatur, non tamen circa illum fit convolu- tio, sed ipse circa idem centrum volvitur minore motu, circa quod potentia motu majore rotatur, quatenus illi ap- plicatur, ut ex his, quæ dicentur, manifestum fiet. CAPUT I. Axis in Peritrochio forma, & vires describuntur. Axis in Peritrochio forma à Pappo Alexandrino circa finem lib. 8. Collect. Mathem. describitur, quadratum scilicet lignum tympano quadra- to foramen A B eidem ligno con- gruens circa suum centrum ha- benti inseritur, ut simul verti possint: ligni autem partes è tympano prominentes in cylin- dricam rotunditatem conforman- tur; & lignum horizonti parallelum super polos æreos, aut ferreos (choinicidas Pappus vocat) congruis fulcris insisten- tes statuitur. Extremæ verò tympani orbitæ infiguntur Ra- dij CD, EF, &c, quos Pappus Scytalas, Aristoteles Col- lops nominat, longiores scilicet paxillii, quibus arreptis versatur tympanum, & cum eo Axis, quem ductarius fu- nis HI in convolutione circumpectens attollit adnexam in I sarcinam; atque hæc tantumdem attollitur, quantus funis Axem circumplicat ex convolutione. Ut Axis hujusmodi vires explicentur, communiter in eo agnoscunt Vectis Rationes: cum enim CB sit semidiame- ter cylindri, quem funis complectitur, & CE semidiame- ter

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506 Mechanics Here the axle is not taken, as in the cone, cylinder, and sphere, for a straight line around which those bodies are moved in a circle while remaining fixed; but it is a body endowed with its own thickness, to which the name axle has been given, because it imitates the axle of wheels, though a winding is not made around it, but it itself turns around the same center with a lesser motion, around which the power rotates with a greater motion, insofar as it is applied to it, as will be made clear from what will be said. CHAPTER I. The axle in the form of the peritrochium, and its powers, are described. The axle in the form of the peritrochium is described by Pappus of Alexandria near the end of book 8 of the Mathematical Collections, namely a square piece of wood is inserted into a square drum, having an opening AB fitting the same piece of wood at its center, so that they may turn together: but the parts of the wood projecting from the drum are formed into a cylindrical roundness; and the wood is placed parallel to the horizon upon bronze or iron pivots (which Pappus calls choinicidas) standing on suitable supports. On the extreme rim of the drum are fixed radii CD, EF, etc., which Pappus calls Scytalas, Aristotle Collops, namely longer pegs, by which, when grasped, the drum is turned, and with it the axle, which the carrying rope HI, in winding about it, lifts the burden attached at I; and this is lifted by as much as the rope, by winding, wraps around the axle. In order that the powers of this kind of axle may be explained, the relations of the lever are commonly recognized in it: for since CB is the semidiameter of the cylinder which the rope embraces, and CE the semidiameter

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Liber quintus. CAPUT I. 507 ter tympani circumpositi, E A verò longitudo Radij, concipiunt A B quasi Vectem primi generis habentem hypomochlium in C, adeò ut ex Ratione A C ad C B momentum potentiæ in A applicatæ computetur. Quæ quidem vera esse non negaverim, si hoc unum intelligatur, quòd Ratio A C ad C B similis sit Rationi, quam haberet æqualis Vectis similem habens positionem Potentiæ, Hypomochlij, & Ponderis. Verùm cur primi potiùs quàm secundi generis vectis dicatur Axis in Peritrochio, cùm æquè attollatur pondus P, si Radij extremitas D elevetur sursum, ac si extremitas Radij A deprimatur deorsum ? Esto facilior sit depressio, quàm elevatio. Quid, si Axis statueretur horizonti perpendicularis, tympanum autem horizonti parallelum, non ad attollendum, sed ad trahendum pondus ? Utique par esset trahendi facilitas, sive impellatur D versus H, sive A versus I : adeóque nulla esset ratio, cur primi potiùs quàm secundi generis Vectis diceretur : an utrique generi ascribendus est ? Sed quid Axem ad Vectem revocare opus est ? cùm eodem ex fonte ita utriusque vires emanent, ut etiamsi Vectem extra omnem Naturæ facultatem positum, atque inter adverata recensendum esse fingeremus, adhuc Axi sua permanerent momenta : Est nimirum, si secundum velocitatem comparentur, motûs potentiæ ad motum Ponderis Ratio major, quàm gravitatis ponderis ad virtutem potentiæ : dum enim funis ductarius semel cylindrum circumplectitur, potentia semel percurrit spatium æquale peripheriæ circuli ab extremo Radio descripti ; cùm autem sint peripheriæ circulorum in Ratione semidiametrorum, motus potentiæ A ad motum ponderis P est ut A C ad S s s 2

Transcription: Translated (English)

Book the Fifth. CHAPTER I. 507 ...the drums being placed around it, and E indeed the length of the radius, conceive A B as a lever of the first kind, having its fulcrum in C, so that from the ratio of A C to C B the moment of the power applied at A is calculated. Which indeed I should not deny to be true, if this one thing be understood, namely, that the ratio of A C to C B is similar to the ratio which an equal lever would have, having a similar position of Power, Fulcrum, & Weight. But why rather than a lever of the second kind should the Axis in the Wheel and Axle be called a lever of the first kind, when the weight P is raised equally, if the end of the radius D be lifted upward, as if the end of the radius A were pressed downward? Granted, it may be easier to depress than to raise. What if the axis were set perpendicular to the horizon, and the drum however parallel to the horizon, not for lifting, but for drawing a weight? Certainly the ease of drawing would be equal, whether D be impelled toward H, or A toward I: and thus there would be no reason why it should be called rather a lever of the first than of the second kind: or must it be assigned to either kind? But why is it necessary to refer the axis to the lever? since from the same source the strength of both so flows forth, that even if we imagined the lever placed outside all the power of Nature, and to be reckoned among things contrary to nature, still the moments would remain to the axis: for plainly, if the velocities be compared, the ratio of the motion of the power to the motion of the weight is greater than the ratio of the weight’s heaviness to the force of the power: for while the hauling rope once wraps the cylinder around, the power once traverses the space equal to the circumference of the circle described by the end of the radius; and since the circumferences of circles are in the ratio of their semidiameters, the motion of the power A to the motion of the weight P is as A C to S s s 2

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508 Mechanicorum CB. Quare potentiæ Peritrochium versantis conatus, ad conatum potentiæ sine machinâ attollentis pondus P, erit in Ratione CB ad AC; quò enim minor secundùm velocitatem est motus ponderis comparatus cum motu potentiæ, eo minor est ejusdem resistentia; minorem autem resistentiam minor conatus superat. Quæ ita ex dictis tùm lib.2.cap.5. tum lib.4.cap.1. clara sunt, ut uberior explicatio supervacanea censenda sit. Hinc apparet, quid juvet ipsius rotæ adjunctæ magnitudo, aut infixarum scyntalarum longitudo; quò enim fuerit major Potentiæ distantia à centro motûs, eò pariter major erit movendi facilitas. Quo circa si eidem Peritrochio placuerit duplicem applicare potentiam, atque ideò scy talas non exteriori ab sidi tympani infigas, sed potiùs extremam tympani oram scy talis ad ejus planum perpendicularibus transfigas; tunc ad augenda Potentiæ momenta nequicquam prodest scy talæ longitudo, sed à foramine, cui illa infigitur, usque ad centrum desumenda est potentiæ distantia, quæ ut major fiat, tympani diameter augenda est. Id quod pariter dicendum est, quando manubrium (unicus scilicet paxillus tympani plano infixus) apponitur, quod moventis manu perpetuò in conversione retinetur; ejus enim distantia à centro perinde consideratur, atque si potentia illi tympani parti fuisset proximè applicata, cui manubrium infigitur. Cavendum tamen hîc videtur, ne quis majorem aliquam rotam ultrà manubrium excurrentem cylindro circumpositam considerans, quæ aliquando plus habere videtur momenti, quàm si rota non major esset, quàm ferat manubrij à centro distantia, existimet non ex hac distantiâ computandum esse potentiæ manubrio applicatæ momentum. Observet, oportet, hoc non contingere in immanibus & colossicoteris ponderibus, immò neque in mediocribus movendis, sed in iis tantummodo, quæ leviore negotio & velociter moveri possunt: Rota enim, cujus semidiameter major est, quàm manubrij à centro distantia, impressum à movente potentiâ impetum concipit, qui levem nactus resistentiam non statim perit, sed aliquantisper perseverans motum rotæ unà cum novo potentiæ conatu efficit majorem, quàm pro solitariis potentiæ viribus: immò tanta fieri potest impetus impressi accessio, ut post aliquod tempus, etiam

Transcription: Translated (English)

508 Mechanicorum CB. Therefore the effort of the power turning the peritrochium, compared with the effort of the power lifting the weight P without a machine, will be in the ratio of CB to AC; for the smaller the motion of the weight, compared in velocity with the motion of the power, the smaller is its resistance; and the smaller resistance is overcome by the smaller effort. These things are so clear from what has been said, both in book 2, chap. 5 and in book 4, chap. 1, that a fuller explanation is to be deemed superfluous. Hence it appears what advantage is gained by the size of the wheel itself, or by the length of the fixed handles; for the greater the distance of the power from the center of motion, the greater likewise will be the ease of moving. Wherefore, if it should please one to apply a double power to the same peritrochium, and therefore not fix the handles on the outside of the drum, but rather pierce the outer edge of the drum with handles perpendicular to its plane, then, to increase the moments of the power, the length of the handles is of no use; but the distance of the hole in which it is fixed from the center must be taken as the distance of the power, and, so that this may be made greater, the diameter of the drum must be increased. The same must also be said when a handle is added, that is, a single peg fixed in the plane of the drum, which is continually held by the hand of the mover in turning; for its distance from the center is to be considered just as if the power had been applied near to that part of the drum to which the handle is fixed. However, care must here be taken lest someone, considering a larger wheel extending beyond the handle and set around the cylinder, which sometimes seems to have more effect than if the wheel were no larger than the distance of the handle from the center allows, should think that the moment of the power applied to the handle is not to be computed from this distance. One must observe that this does not happen in immense and gigantic weights, indeed nor even in those of moderate size to be moved, but only in those things which can be moved with greater ease and quickly: for a wheel whose semidiameter is greater than the distance of the handle from the center receives the impulse impressed by the moving power, which, having encountered slight resistance, does not immediately perish, but, persisting for a little while, produces the motion of the wheel together with the new effort of the power greater than from the solitary forces of the power alone; indeed the addition of the impressed impulse can become so great that after some time, even

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Liber quintus. CAPUT I. 509 etiam dimisso à potentiâ manubrio, vi ejusdem impressi impe- tûs adhuc se rota in gyrum contorqueat. Hinc est aliquando ejusdem rotæ diametrorum extremitatibus addi plumbeas mas- sas, quæ plus impetûs concipientes, atque diutiùs retinentes, rotæ conversionem validiùs promoveant, etiam cessante poten- tiâ. Sed hîc non unica est potentia, quæ manubrio applicatur, cujus momenta ex distantiâ manubrij à centro definimus; sed præterea impetus ille perseverans rationem habet alterius po- tentiæ applicatæ illis rotæ partibus, quibus inest; & pro variâ à centro distantiâ, alia pariter atque alia sunt particularum ip- sius impetûs impressi momenta ad rotam convertendam. Quo- niam verò rotæ semidiameter ex hypothesi major est, quàm manubrij à centro distantia, nil mirum, si particulæ impetûs ex- tremæ rotæ impressi multum habeant momenti, quippe quæ magis distant, & velociorem motum efficiunt. Quod verò ad cylindrum spectat, quem funis ductarius cir- cumplicat, non est necesse illum esse exactè & Geometricè ro- tundum, sed satis est si cylindricam figuram æmuletur: eatenus siquidem rotundum axem construimus, quatenus eadem volu- mus in convolutione servari momenta: si verò angulatus esset axis, perpendicularum, in quo esset pondus, modò vicinum cen- tro esset, modò ab eo remotum, ac propterea ejusdem remoti majora essent momenta, quàm vicini. Sit enim ex. gr. qua- dratus Axis B D H G: utique per- pendiculum, in quo est funis reti- nens pondus quod attollitur, va- riam habet à centro C distantiam; nam quando latus B D congruit fu- ni perpendiculari, distantiâ à cen- tro C æqualis est semissi lateris G B, & est C I; cum verò latus B D in conversione fit obliquum, distan- tia perpendiculari fit major, & est C E, ita ut demùm distantiâ maxi- ma sit æqualis ipsi C B; quæ iterum decrescit, donec funis congruat lateri B G. Potentiæ autem à centro distantia eadem semper manet A C, ideòque momentorum potentiæ ad mo- menta ponderis Ratio subinde mutatur. Quòd si non quadra- S s s 3

Transcription: Translated (English)

Book Five. CHAPTER I. 509 even when the handle has been let go by the power, by the force of the same impressed impetus the wheel still continues to turn in a circle. Hence it is that sometimes leaden masses are added to the extreme diameters of the same wheel, which, conceiving more impetus and retaining it longer, promote the rotation of the wheel more strongly, even when the power has ceased. But here there is not only the power which is applied to the handle, whose moments we define from the distance of the handle from the center; but besides this, that persevering impetus has the character of another power applied to those parts of the wheel in which it resides; and according to its varying distance from the center, the moments likewise of the particles of the impressed impetus are one thing and another for turning the wheel. But since by hypothesis the semidiameter of the wheel is greater than the distance of the handle from the center, it is no wonder if the extreme particles of the impressed impetus of the wheel have much moment, since they are farther away and produce a swifter motion. As for the cylinder, which the guiding rope winds around, it is not necessary that it be exactly and geometrically round, but it is enough if it imitates a cylindrical figure: for we construct a round axis only so far as we wish the same moments to be preserved in the winding. If, however, the axis were angular, the perpendicular in which the weight would be placed would now be near the center, now farther from it, and therefore the moments of the more distant ones would be greater than those of the nearer. Let, for example, the axis B D H G be square: certainly the perpendicular in which is the rope holding the weight that is being raised has a varying distance from the center C; for when side B D coincides with the perpendicular of the rope, the distance from center C is equal to half of side G B, and is C I; but when in the turning side B D becomes oblique, the distance of the perpendicular becomes greater, and is C E, so that at length the greatest distance is equal to C B itself; which again diminishes until the rope coincides with side B G. But the distance of the power from the center A C always remains the same, and therefore the ratio of the moments of the power to the moments of the weight is continually changing. But if not quadra- S s s 3

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Mechanicorum 510 tus sit Axis, sed plurium angulorum, ita ut latera brevissima sint, sicuti vix distat à rotunditate cylindri, ita vix momentorum disparitatem infert. Illud quidem animadversione dignum est, quòd non temerè statuenda sit Axi crassitudo, sed adeò validus esse debet ac firmus, ut ponderis gravitati obsistere possit, quin flectatur, aut dissiliat; si enim incurvesceret, augeretur movendi difficultas, quia nimirum in conversione majorem ambitum describeret, quàm pro ejus soliditate. Sed neque idcircò præter modum crassus Axis eligi debet, quia quò major ille est atque crassior, eò major etiam est potentiæ moventis labor, nisi pariter majus illi addatur Peritrochium. Hinc sit contingere posse, ut in attollendo ponde- re augeatur labor potentiæ circa finem motûs; quia vide- licet, si ductarij funis spiræ jam universam cylindri faciem circumplectantur, & sequentes spiræ non cylindro cohæ- reant, sed subjecto funi, jam intelligitur semidiameter axis aucta crassitudine funis subjecti, ac proinde secundus hic spirarum ordo majorem funis longitudinem exigit, adeóque etiam infert majorem ponderis motum, quo tempore poten- tia motum non majorem perficit: quare diminutâ Ratione mo- tûs potentiæ ad motum ponderis, minora fiunt illius momenta ad attollendum pondus. Porrò non est omnino necesse, ut ad pondus attollendum Axis statuatur in superiore loco, sed fieri potest, ut longè al- tiùs elevetur pondus supra locum Axis; si nimirum funis ductarius transeat per orbiculum superiùs firmatum: Ve- rum ita firmiter stabilienda est machina, ut hæc à nimiâ ponderis gravitate non rapiatur sursum. Cæterùm cùm fu- nis immediatè nectitur ponderi inferiùs posito, ipsa ponde- ris gravitas stabilit machinam suis fulcris insistentem solo. Hactenus quidem Axem rectum, prout magis communiter usurpatur, statuimus; pro opportunitate tamen adhiberi etiam potest curvatus. Quemadmodum si ex profluenta aquam sur- sum antliâ propellere velimus, rotæ BC congruis pinnis instructæ, in quas aqua incurrens vim suam exerceat, addi- tur crassior ferreus stylus centro A infixus, curvatusque ADEFGHIK (si IK sit alter polus, cui machina incum- bit,

Transcription: Translated (English)

Mechanicorum 510 the Axis should be, not of a single angle, but of several angles, so that the sides are very short; just as it scarcely differs from the roundness of a cylinder, so it scarcely introduces any difference in the moments. Indeed, it is worthy of notice that the thickness of the Axis should not be determined carelessly, but it ought to be so strong and firm that it can resist the weight of the burden without bending or breaking apart; for if it were to bend, the difficulty of moving would increase, because in turning it would describe a greater circuit than its solidity allows. But neither should the Axis therefore be chosen excessively thick, because the larger and thicker it is, the greater also is the labor of the moving power, unless a larger Peritrochium be added to it at the same time. Hence it can happen that, in raising a weight, the labor of the power increases near the end of the motion; because, namely, if the turns of the driving rope now completely encircle the whole face of the cylinder, and the following turns do not adhere to the cylinder but to the rope beneath, it is now understood that the semidiameter of the axis, increased by the thickness of the rope beneath, and therefore this second order of turns requires a greater length of rope, and consequently also produces a greater motion of the weight, at a time when the power does not accomplish a greater motion: wherefore, as the ratio of the motion of the power to the motion of the weight is diminished, its moments for raising the weight become smaller. Furthermore, it is not absolutely necessary, in order to raise a weight, that the Axis be placed in an upper position; rather, it can happen that the weight is lifted much higher above the place of the Axis, namely if the driving rope passes through a pulley fastened above: yet the machine must be fixed so firmly that it is not drawn upward by the excessive heaviness of the weight. Moreover, when the rope is immediately attached to the weight placed below, the very heaviness of the weight stabilizes the machine, resting on its supports upon the ground. Thus far we have set down the straight Axis, as it is more commonly used; however, as occasion requires, a curved one may also be employed. Just as if we wished to drive water upward from a stream by a pump, the wheels BC, furnished with suitable vanes, against which the water striking may exert its force, there is added a thicker iron rod fixed in the center A and curved ADEFGHIK (if IK be the other pole, on which the machine rests,

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Liber quintus. CAPUT I. 511 bit, nam si fulcrum sit propè A inter A & D, sufficit si in H, terminetur) ita ut ipsi D E æqualis sit particula HI, utriusque autem dupla FG, atque inter EF & GH annulo inseritur hasta adnexa embolo, ita ut dum alter embolus attollitur, alter deprimatur. Hîc attendenda est Ratio semidiametri rotæ, seu distantiæ potentiæ à centro, ad D E, quæ est semidiameter cylindri, qui ex ejus convolutione gignitur; perinde atque si esset cylindrus, cujus tota diameter esset FG: atque ideo non ex ipsius ferrei styli crassitudine, sed ex flexu æstimanda est Axis semidiameter; eatenus quippe crassior, aut exilis ferreus stylus eligitur, quatenus majore aut minore vi opus est in attollendo atque deprimendo embolo. CAPUT II. Succulæ & Ergatæ usus consideratur. Peritrochij usus quidem frequens est, sed & sæpiùs sine rotâ idem præstatur, vel addito Aximanubrio, vel Radiis Axi infixis; & machina Latinis succula dicitur; parvulus autem paxillus, cui funis ductarij caput adnectitur, Porculus nominatur: Si tamen paxilli loco annulum cylindro adnectas, cui funis caput insertum firmetur, perinde est. Hujusmodi est cylindrus AB suis polis insistens congruo pegmati, infixosque habens Radios CD, EF; quibus manu arreptis cylindrus volvitur, & funis adnexus paxillo I circumducitur cylindro, atque connexum in H onus attollitur. Dupliciter autem succula

Transcription: Translated (English)

Book Five. Chapter I. 511 bit, for if the fulcrum be placed near A, between A and D, it is enough if it be terminated at H, so that the particle HI is equal to DE itself, and FG twice either of them, and between EF and GH an annulus is inserted with a rod attached to the piston, so that while one piston is lifted, the other is depressed. Here attention must be paid to the ratio of the semidiameter of the wheel, or of the distance of the power from the center, to DE, which is the semidiameter of the cylinder that is generated by its turning; just as if it were a cylinder, whose whole diameter were FG: and therefore the semidiameter of the axis is to be estimated not from the thickness of the iron rod itself, but from its bend; for an iron rod is chosen thicker or thinner according as greater or lesser force is needed in lifting and lowering the piston. Chapter II. The use of the succula and ergata is considered. Indeed the use of the peritrochium is common, but it is also often accomplished without a wheel, either by adding a handle to the axis, or by spokes fixed into the axis; and the machine is called by the Latins a succula; but the little peg to which the end of the hauling rope is attached is called a porculus: if, however, in place of the peg you attach to the cylinder an annulus, into which the end of the rope is secured, it is the same. Such a device is the cylinder AB standing on its pivots in a suitable frame, and having fixed to it the spokes CD, EF; grasping these by hand, the cylinder is turned, and the rope attached to the peg I is wound around the cylinder, and the load connected at H is lifted. But the succula is of two kinds

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Mechanicorum 512 succulâ utilicet, aut Radiis perpetuò infixis, aut qui cylindri foraminibus dum versatur, subinde inserantur: Si perpetuò in- fixi maneant, unus potest Axem convertere alio post alium Ra- dio arrepto: at verò si idem Radius in aliud atque aliud fora- men immittendus sit, duo sint, oportet, qui alternâ operâ suum Radium deprimentes Axem convolvant; alioquin, nisi artificio aliquo retineatur, dum ex uno foramine extrahitur Radius, ut in aliud immittatur, pondus suâ gravitate deorsum relaberetur. Licet tamen hanc duplicis potentiæ necessitatem utilitate aliâ compensare; ubi enim duo sint, quorum vicissitudine circum- agatur Axis immissitio hujusmodi Radio, hic potest esse multo longior, quàm si eidem Axi infixus maneret; oporteret siqui- dem plures Radios perpetuò manentes infigere; id quod, si lon- giores essent, non careret incommodo. Quo autem longior Radius fuerit, eò pariter faciliùs potentia movebit, quippe quæ motum multò velociorem motu ponderis habebit, pro Ratione longitudinis Radij plus semidiametro cylindri, ad eandem cy- lindri semidiametrum. Ad hoc fortasse genus revocari possunt Scytalæ oneribus promovendis subjectæ, de quibus dictum est lib. 2. cap. 9; cùm harum capitibus aptè perforatis immittuntur ferrei aut lignei vectes, quorum ope scytalæ ipsæ convertuntur, atque incum- bens onus dum ad aliam atque aliam orbitæ partem accommo- datur, promovetur. Quo enim longioribus vectibus utimur, potentia circa cylindri centrum multò velociùs movetur quàm impositum saxum, cujus motus æqualis est conversioni peripheriæ. Nam quod motus absolutè sumptus sit aliquantulo major, quia centrum ipsum promovetur, nihil refert, quia motus hic & cylindro subjecto, & oneri, & Potentiæ commu- nis est. Præter Succulam Radiis infixis instructam, cujusmodi ea est, quæ ad hauriendas è puteis aquas vulgò usurpatur (quam- quam ob radiorum brevitatem & ipsius Axis crassitudinem non admodum potentiæ momenta augeantur) forma alia cæmenta- riis maximè familiaris est ad attollenda saxa, lateres, & calcem, duplici manubrio in oppositas partes disposito, ut quædam co- natuum constans similitudo servetur, dum altero suum manu- brium deprimente, suum alter elevat: cùm enim vi brachio- rum

Transcription: Translated (English)

Mechanics 512 with a useful tackle, either with arms fixed permanently, or which, while the cylinder is turning, are inserted into its holes from time to time: if they remain fixed permanently, one man can turn the axle by taking hold of one arm after another; but if the same arm must be put now into one hole and now into another, there must needs be two men, who, by alternate labor, while one depresses his arm, wind the axle round; otherwise, unless it were held back by some contrivance, while the arm is withdrawn from one hole in order to be inserted into another, the weight would slip back downward by its own gravity. Yet this need for a double power may be compensated by another advantage; for where there are two men, by whose alternation the axle is driven round by such insertion of the arm, that arm can be much longer than if it remained fixed in the same axle; for it would be necessary to fix several arms permanently, which, if they were longer, would not be without inconvenience. And the longer the arm is, the more easily will the power move, since it will have a motion much quicker than the motion of the weight, in the proportion of the length of the arm beyond half the cylinder’s radius to the same radius of the cylinder. To this class perhaps may be referred the scytalae employed for moving loads, of which mention was made in Book 2, chapter 9; since into the suitably perforated heads of these are inserted iron or wooden levers, by means of which the scytalae themselves are turned, and the load resting upon them is moved onward as it is adjusted now to one and now to another part of the orbit. For the longer the levers we use, the power around the center of the cylinder moves much more quickly than the stone laid upon it, whose motion is equal to the rotation of the circumference. For that the motion, taken absolutely, is somewhat greater, because the center itself is advanced, matters not, because this motion is common both to the cylinder beneath, and to the load, and to the power.

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Liber quintus. CAPUT II. 513 deorsum connitentium facilior contingat depressio, quàm ele- vatio, si manubriorum inflexio ad eandem partem collocaretur, uterque simul deprimendo faciliùs axem converteret, at uterque simul elevans aliquid amplius laboris subiret; alternis autem elevationibus atque depressionibus labor temperatur. Cæterum quod ad potentiæ momenta attinet, parum interest, quam positionem manubria habeant vicissim comparata; spectatur videlicet singulorum longitudo & cujusmodi motum potentia manubrio applicata describat: Sic manubrij longitu- do GH, hoc est potentiæ apprehen- dentis HO distantia perpendicula- ris ab axe cylindri, qui convolvitur, attendenda est, & cùm ipsius cylin- dri, semidiametro comparanda, ut Ratio motûs Potentiæ ad ponderis motum innotescat, ac proinde Po- tentiæ momentum definiatur. Hinc apparet pro ipsius GH longitudine ad cylindri axem productum perpendiculari augeri momenta potentiæ; perinde namque se habet, ac si infixus esset cylindro Radius KI ipsi GH æqualis; quia ut KI ad KE semidiametrum, ita GH ad KE, & ambitus à potentia in H descriptus ad ejusdem cylin- dri ambitum. Quare non leviter allucinantur, qui manubrij longitudinem GH non rectam, sed in hemicyclum curvatam volunt quasi hinc plus aliquid momenti potentiæ conferretur; quamvis enim circuli semiperipheria sit salte diametri sesqual- tera, potentiæ applicatæ motui non semiperipheria, sed diam- ter legem statuit: alioquin si ex ipsa manubrij inflexione mo- menta augerentur, satius esset non tantùm semiperipheriæ, sed majori circuli segmento simile esse manubriu; id quod si expe- riri voluerint, tantum abest, ut movendi facilitatem acquirant, ut potiùs momenta minui sentiant; nam in circulo maximam linea esse diametrum, & quò majoru segmentoru arcus majores fiunt, minui subtensas chordas, ex 15. lib. 3. nôrunt ipsi Elementarij. Hac igitur manubrij longitudine perpensâ, non solùm non est eorum æqualitas religiosè servanda, verùm author essem cætementariis, ut manubriorum alterum paulò longius consti- tuerent; cùm enim ut plurimum inæquales sint operarum vi- Ttt

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Book Five. CHAPTER II. 513 the depression of those striving downward would be more easy than the elevation, if the bend of the handles were placed on the same side; for then, both depressing together, they would more easily turn the axis, but both elevating together would undergo somewhat more labor; whereas by alternate elevations and depressions the labor is moderated. Moreover, as to the moments of the powers, it matters little what position the handles have in comparison with one another; for regard is had, namely, to the length of each, and to what kind of motion the power applied to the handle describes: thus the length of the handle GH, that is, the perpendicular distance of the grasping power HO from the axis of the cylinder which is wound, is to be considered, and compared with the semidiameter of the cylinder itself, so that the ratio of the motion of the Power to the motion of the weight may be known, and consequently the moment of the Power may be defined. Hence it appears that, for the perpendicular drawn to the axis of the cylinder, the moments of the power are increased according to the length of GH; for it is the same as if there were fixed in the cylinder the radius KI equal to GH; because as KI is to the semidiameter KE, so is GH to KE, and the circumference described by the power at H to the circumference of the same cylinder. Therefore they are not a little mistaken who wish the length GH of the handle to be not straight, but curved into a hemicycle, as though by this means something more of moment were imparted to the power; for although the half-circumference of a circle be at least one and a half times the diameter, yet to the motion of the applied power not the half-circumference, but the diameter sets the law: otherwise, if moments were increased by the bending of the handle itself, it would be better not only for it to be like a half-circumference, but even like a larger segment of a circle; which, if they should wish to try, so far will it be from their acquiring ease in moving, that rather they will feel the moments diminished; for they know from the Elementary Books, in Book 3, proposition 15, that in a circle the greatest line is the diameter, and that the greater the arcs of the segments, the smaller the subtended chords become. This length of the handle therefore being duly considered, not only is it not necessary that their equality be scrupulously observed, but I would advise craftsmen to make one of the handles somewhat longer; since, as for the most part, the labors of the workers are unequal, Ttt

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Mechanicorum 514 vires, si æqualia sint manubria, qui infirmior est, plus subit laboris, quàm ferre possit: at si alterum paulo longius sit, debiliorem illi applicari oportebit, ut minore incommodo præscriptum opus perficiat. Quòd si contingat ab unico homine convertendam esse succulam, non erit contemnendum laboris compendium, si possit longiore manubrio uti. Quamvis autem nullus statuatur finis conversioni, quia funis ductarius succulam non circumplectatur, eadem manet Ratio. Si enim axi polygono insistens catena singulis palmaribus, aut majoribus intervallis adnexos globulos aut discos habeat tubo, per quem transeunt, congruentes, qui intra tubum aquam intercipiètes dum ex succulæ conversione attolluntur, aquam pariter elevant, secumque rapiunt, perpetua fieri potest conversio; pondus autem, quod movetur, est aqua tubum implens. Ubi aliquorum imperitiam castigare oporteret, qui manubrij longitudinem (quæ ipse non sine inscitiæ admiratione vidi, narrow) minorem semidiametro axis, cui catena insistit, constituint, & operarum laborem frustra augent, dum minor est potentiæ motus, quàm ponderis. Quid enim paulo majorem longitudinem manubrio non tribuunt? minùs scilicet laborantes operæ concitatiùs axem volverent, & globuli celeriùs elevati minus aquæ elabi sinerent. Jam verò ad Ergatam, quæ modicum à succulâ differt, transeamus, cujus usus potissimùm est in trahendis oneribus, quamquam illâ etiam, adhibitâ videlicet trochleâ, ad onera attollenda uti possimus, & frequenter utamur. Quemadmodum autem in succulæ positione est cylindrus ut plurimùm horizonti parallelus, ita in Ergatâ statuitur horizonti perpendicularis. Cylindro enim DC ita firmato, ut vel circa extremos polos, vel in loculamento congruo converti possit, additur vectis GEF (aut etiam plures vectes eidem cylindro infiguntur) cui applicata Potentia dum cylindrum circa suum axem versat, fu- nemque

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Mechanics 514 powers, if the handles are equal, the one who is the weaker undergoes more labor than he can bear; but if one of them be somewhat longer, it ought to be applied to the weaker man, so that he may perform the prescribed work with less inconvenience. But if it should happen that the screw is to be turned by a single man, the saving of labor is not to be disregarded, if he can use a longer handle. Although, however, no limit is assigned to the turning, because the guiding rope does not embrace the screw, the same reasoning remains. For if, a chain fixed upon a polygonal axis having at every palm's length, or at greater intervals, small balls or disks attached to it, corresponding to the tube through which they pass, which, while intercepting the water within the tube as they are raised by the turning of the screw, likewise lift the water and carry it along with them, a perpetual rotation can be made; and the weight that is moved is the water filling the tube. Here one should rebuke the ignorance of some, who set the length of the handle (which I myself have seen, not without astonishment at their ignorance, narrow) less than the semidiameter of the axis on which the chain rests, and thus vainly increase the labor of the workers, while the motion of the power is less than that of the weight. Why do they not assign a somewhat greater length to the handle? for then, the laborers being less burdened, they would turn the axis more swiftly, and the balls, being raised more quickly, would allow less water to escape. Now then let us pass to the Ergata, which differs little from the screw, whose chief use is in drawing loads, although even with it, by using a pulley, we may also employ it for raising loads, and often do so. But just as in the position of the screw the cylinder is for the most part parallel to the horizon, so in the Ergata it is set perpendicular to the horizon. The cylinder DC being so fixed that it can rotate either about its extreme poles, or in a suitable socket, the lever GEF is added (or even several levers are fixed to the same cylinder) to which, when Power is applied, the cylinder turning about its axis, and the rope

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Liber quintus. CAPUT II. 515 nemque convolvit, adnexam sarcinam adducit. Æstimatur autem potentiæ momentum ex ejusdem Potentiæ distantiâ ab axe cylindri, comparata cum ipsius cylindri semidiametro: quantus nimirum funis cylindrum circumpectitur, tantus est oneris adducti motus, qui ad potentiæ motum eam habet Rationem, quæ inter cylindri ambitum circularem, & peripheriam Radio EF, aut EG, descriptam intercedit. Cum verò hujusmodi vecti EF tanta tribui possit longitudo, quantam ferre possit spatium; in quo Potentia movetur, patet longiore vecte momentum potentiæ pro arbitratu augeri posse. Verum quidem est plures potentias eidem vecti EE applicatas inæqualia habere momenta pro Ratione inæqualium distantiarum ab axe cylindri. Sed illud maximè commodum accidit in Ergatâ, quod hîc jumentorum ope hominum laborem minuere licet, dum illa extremo vecti alligata, & in gyrum acta cylindrum convolvunt; à quibus tamen subsidium petere in succulæ convolutione non possumus; nisi fortè cylindrum horizontalem Verticali peritrochio inseramus, & extremam crassioris peritrochij orbitam funis circumpectatur; qui dum jumento trahente evolvitur, cogat cylindrum converti, funemque, cui sarcina adnectitur, circa cylindri orbitam convolutum attollere pondus. Id quod etiam præstare valemus, si trahendum sit onnus, neque in locum inducere liceat jumentum: nam perpendiculari cylindro HI peritrochium, seu tympanum LM horizonti parallelum circumponitur, & pluribus spiristympano circumducitur funis, quem in O jumentum trahens quamvis procul positum explicat, atque cylindrum convertit, ac propterea onus in N adnexum adducit. Ttt 2

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Book Five. CHAPTER II. 515 and so winds it up, drawing along the attached load. The power’s effectiveness, however, is estimated from the distance of that same Power from the axis of the cylinder, compared with the semidiameter of the cylinder itself: namely, as much as the rope winds about the cylinder, so much is the motion of the load drawn in, which has to the motion of the power the same ratio as exists between the circular circumference of the cylinder and the circumference described by the radius EF, or EG. But since such a lever EF may be given a length as great as the space can bear in which the Power is moved, it is clear that by a longer lever the effectiveness of the power may be increased at will. It is indeed true that several powers applied to the same lever EE have unequal moments in proportion to their unequal distances from the axis of the cylinder. But the great convenience here lies in the capstan, that by its aid the labor of men may be reduced by means of beasts of burden, since, when attached to the end of the lever and turned round, they wind the cylinder; yet we cannot seek such assistance in winding up a winding device, unless perhaps we insert a horizontal cylinder into a vertical peritrochium, and the rope be wound about the outer circumference of the thicker peritrochium; which, as it is unwound by the beast pulling, compels the cylinder to turn, and to lift the weight by winding the rope, to which the load is attached, around the cylinder’s circumference. This too we can accomplish if the load must be drawn and it is not permissible to place the beast in the spot: for then around the perpendicular cylinder HI a peritrochium, or drum LM, parallel to the horizon, is placed, and several turns of rope are wound around the drum, which a beast pulling at O, though placed far away, unwinds, and turns the cylinder, and thereby draws the load attached at N. Ttt 2

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Mechanicorum Porrò in funis ductarij circumvolutione circa succulæ aut Ergatæ cylindrum observandum est, non esse necesse totum funem circumvolvi, illique adnecti; nimis enim multus ali- quando esset, & non leve afferret incommodum; ut satis constat, cùm solvendæ sunt anchoræ, si crassum illum ruden- tem totum cylindro circumduci opus esset, ut anchora è maris fundo extrahatur. Satis igitur est, si funis duplici aut triplici spirâ cylindrum circumplectatur, quando ingentia pondera movenda sunt; hæc siquidem valdè resistunt, & ita funis circa ipsum cylindrum constringitur, ut illum validè premat, nec fa- cilè possit excurrere, maximè si cylindrus non fuerit exquisitè tornatus; nimius scilicet partium se se mutuo contingentium affrictus, qui cum cylindri superficie fieri deberet, perinde re- sistit, atque si funis paxillo aut annulo esset idem cylindro ad- nexus. Quare satis fuerit, si puer funem in conversione expli- catum colligat. Ex dictis tùm hoc, tùm superiori capite, satis constat, quæ- nam longitudo statuenda sit Radio, cui potentia data applican- da est, si pariter cylindri semidiameter, & oneris gravitas detur. Nam si fiat ut data Potentia ad datam ponderis gravitatem, ita data cylindri semidiameter ad quæsitam Radij longitudinem, habetur longitudo sufficiens ad sustinendum pondus in aëre suspensum. Quare pro arbitratu augeatur longitudo Radij, &, cùm facta jam sit major Ratio motûs potentiæ ad motum pon- deris, quàm sit Ratio gravitatis ponderis ad virtutem potentiæ sustinentis, illa poterit propositum pondus movere. Sic quo- niam in navibus ad proram jacet horizonti parallelus versatilis cylindrus (aut potiùs hexagonum seu octogonum prisma) cu- jus extremitas decrescentibus crenis denticulata incumbentem ligneam regulam singulis subinde crenis excipit, ne ponderis vi in contrariam partem retroagi valeat, & cylindro circumdu- citur rudens (Pisma ab aliquibus dicitur) ex quo anchora pen- det; nec habere potest plures Radios perpetuò adnexos, quos videlicet spatij angustiæ ferre non possent, ideò foramina quæ- dam habet, quibus, ubi opus fuerit, inseruntur vectes. Ut vectium longitudo statuatur, anchoræ gravitas cum adjecto ligneo transversario consideranda est, quæ est ferè sub trecen- tupla gravitatis navis vacuæ, ut constat ex iis, quæ lib.4.cap.17. innummus.

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Mechanics Moreover, in the winding of the hauling rope around the drum of the capstan or windlass, it is to be observed that it is not necessary for the whole rope to be wound around it and attached to it; for otherwise sometimes too much would be required, and it would bring no slight inconvenience, as is sufficiently evident when anchors are to be let go, if that thick rough rope had to be wound entirely around the drum in order that the anchor may be drawn up from the bottom of the sea. It is therefore enough if the rope embraces the drum with a double or triple turn, when very great weights are to be moved; for these turnings resist strongly, and the rope is thus tightened around the drum itself that it presses it firmly, nor can it easily slip out, especially if the drum has not been exquisitely turned; namely, the excessive friction of the parts touching one another, which ought to take place against the surface of the drum, resists just as much as if the rope had been attached to the drum by a peg or ring. Wherefore it will be enough if the boy gathers up the rope as it unwinds in turning. From what has been said, both here and in the preceding chapter, it is sufficiently clear what length must be assigned to the Radius to which the applied power is to be applied, if the semidiameter of the drum and the weight of the load are likewise given. For if the given Power be to the given weight, as the given semidiameter of the drum is to the required length of the Radius, there is obtained a length sufficient to sustain a weight suspended in the air. Therefore let the length of the Radius be increased at pleasure, and, since the ratio of the motion of the power to the motion of the weight has now become greater than the ratio of the gravity of the weight to the force of the supporting power, it will be able to move the proposed weight. Thus, since in ships there lies near the prow a horizontal rotating cylinder (or rather a hexagonal or octagonal prism), the end of which, being furnished with diminishing notches, receives at each notch the wooden bar laid upon it, so that it may not be driven back in the opposite direction by the force of the weight, and around which is wound the rope (called by some Pisma) from which the anchor hangs; and since it cannot have several Radii permanently attached, which the narrowness of the space could not bear, therefore it has certain holes into which levers are inserted when necessary. In order to determine the length of the levers, the weight of the anchor together with the added wooden crosspiece must be considered, which is nearly three hundred times the weight of an empty ship, as is evident from what is set forth in book 4, chapter 17.

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Liber quintus. CAPUT III. 517 innuimus. Navis autem capacitas (hoc est pondus, quod navis gestare valet, & æquale est gravitati navis in aëre) vel per dolia, seu amphoras aquæ, quam sine incommodo ferre potest, numeratur, ut solent Galli & Angli singulis do- liis navalibus libras bis mille tribuentes, vel per pondera, quæ Hollandis atque Germanis Last dicuntur, singula li- brarum saltem quatuor millibus definita (nam Last Ham- burgi continet libras 4554, Amstelodami, si sit triticum habet lib. 4800, sin autem siligo lib. 4200, Stevinus verò lib. 3. staticæ pop. 10. singulos modios definit lib. 360) & singulis libris unciæ sexdecim, seu Lotones 32, hoc est se- munciæ tribuendæ sunt. Quare data navis capacitas ex. gr. doliorum 400, multiplicetur per lib. 2000; & sunt. lib. 800000, quarum pars trecentesima lib. 2666 est ferè pon- dus anchoræ cum ligneo transversario. Possunt autem non plures applicari vectes quàm quatuor, ideóque singuli quar- tam ponderis partem elevare debent, hoc est lib. 666. Si fuerit igitur cylindri semidiameter 3/4 pedis, & vis potentiæ (quia ipsa corporis gravitas vectem premit) sit elevandi lib. 100, fiat ut 100 ad 666, ita 3/4 pedis ad pedes ferè quinque; & hæc erit quæsita longitudo Radij, cui poten- tia applicanda est. CAPUT III. Tympani à calcante circumacti vires expenduntur. Tympana, quæ Græcis , Latinis retentâ vocabuli in- terpretatione Grues dicuntur, hoc differunt à Succulâ, quòd ab hominibus non brachiorum contentione, sed cor- poris calcantis gravitate moventur. Horum autem frequen- tissimus est usus tum in Hollandiâ tùm in Germaniâ juxta fluvios navigabiles, ut ingentia pondera è navibus extra- hant, & in ripâ deponant: quamquam & ad alios usus fa- T t 3

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Book five. CHAPTER III. 517 is indicated. But the carrying capacity of a ship (that is, the weight which the ship is able to bear, and is equal to the weight of the ship in the air) is reckoned either by barrels, or amphorae of water, which it can carry without inconvenience, as the French and English are accustomed to assign to each naval barrel two thousand pounds; or by weights which are called Last by the Dutch and Germans, each of them defined at no less than four thousand pounds (for a Last at Ham- burg contains 4554 pounds, at Amsterdam, if it be wheat, it has 4800 lb., but if rye 4200 lb.; Stevinus however in lib. 3, staticæ, pop. 10, defines each modius at 360 lb.) and to each pound sixteen ounces, or 32 lotones, that is to say, half-ounces, are to be assigned. Therefore, if the carrying capacity of a ship be given, e.g. 400 barrels, multiply by 2000 pounds; and there are 800000 lb., of which the three-hundredth part, 2666, is about the weight of the anchor with the wooden crossbar. But not more than four levers can be applied, and therefore each must raise a quarter of the weight, that is, 666 lb. If therefore the semidiameter of the cylinder were 3/4 of a foot, and the force of the power (since the gravity of the body itself presses on the lever) were sufficient to raise 100 lb., let it be as 100 to 666, so 3/4 of a foot is to nearly five feet; and this will be the sought length of the radius, to which the power is to be applied. CHAPTER III. The powers of drums turned by treading are considered. Drums, which by the Greeks, with the Latin interpretation of the word retained, are called Cranes, differ from a winch in this, that they are moved by men not by the exertion of the arms, but by the weight of the foot treading upon them. Their most frequent use is both in Holland and in Germany along navigable rivers, to draw huge weights out of ships and set them down on the bank: although also for other uses they are employed T t 3

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Mechanicorum cilè traduci possint, si apto loco collocentur. Cylindro AB versatili, & horizonti parallelo; ac ritè firmato, ampliorem rotæ peripheriam CDE circumponimus ex latioribus asscribus compactam, ut unus saltem homo ingredi valeat; qui dum ex C in D ascendere conatur, sua gravitate deprimens tympanum, cylindrum pariter convertit, ductariumque funem convolvit, qui per orbiculos F & G superiori trabi exporrectæ connexos transiens, cum onere in P connectitur, atque adeò ex Cylindri conversione attollitur pondus, etiamsi à machinâ ipsa absit, quantum trabs exporrigitur. Si placuerit eidem cylindro duplex tympanum, aut unicum valdè amplum apponere, licebit, ut plurium hominum operâ in elevando onere uti possimus. Machinæ hujus vires eò majores esse, quò major est semidiametri rotæ ad cylindri semidiametrum Ratio, satis manifestum est ex iis, quæ sæpissimè dicta sunt; major est enim potentiæ motus, quò amplior est rota. Cavendum tamen ne, quemadmodum in succulâ atque Ergatâ, ita etiam hîc omnino ex ipsa semidiametrorum rotæ & Cylindri Ratione definiantur potentiæ momenta: hîc scilicet potentia tympanum movens est insita homini gravitas deorsum connitens; in succulâ autem atque in Ergatâ potentia movens est impulsus ab animali facultate impressus, ac in gyrum directis. Quapropter in succulâ, atque in Ergatâ cùm eadem sit potentiæ directio similiter applicatæ in quocumque situ, eadem manent in conversione potentiæ momenta: at hominis tympanum calcantis non eædem semper sunt vires, sed quo magis ascendit versus D, augentur ejus momenta; quia videlicet perinde est, atque si à centro ad punctum orbitæ, in quo est gravitas calcans, ducta esset linea; ibi enim momentum descendendi est ut Sinus declinationis à perpendiculo, juxta dicta lib.1.cap.15. Sit

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How the parts of mechanics may be translated, if they are placed in a suitable position. With the movable cylinder AB, parallel to the horizon, and duly secured, we place around it a broader wheel-periphery CDE, made of wider planks, so that at least one man may be able to enter it; who, while trying to ascend from C to D, by his weight depressing the drum, turns the cylinder as well, and winds up the guiding rope, which, passing through the pulleys F and G connected to the upper stretched beam, is attached to the load in P; and thus, by the turning of the cylinder, the weight is lifted, even though it be far from the machine itself, as much as the beam extends. If it should please us to fit to the same cylinder a double drum, or a single one that is very broad, it will be allowed, so that we may be able to use the labor of several men in lifting the load. That the powers of this machine are greater, the greater the ratio of the semidiameter of the wheel to the semidiameter of the cylinder, is sufficiently clear from what has often been said; for the power of motion is greater, the broader the wheel. Yet care must be taken lest, as in the succula and the ergata, so also here, the moments of power be defined altogether from the ratio of the semidiameters of the wheel and the cylinder: here, namely, the power moving the drum is the weight innate in man striving downward; in the succula however and in the ergata, the moving power is the impulse impressed by animal faculty, and directed in a circle. Wherefore in the succula, and in the ergata, since the direction of the power similarly applied is the same in whatever position, the moments of power remain the same in the turning: but in the case of the man treading the drum the forces are not always the same, but the more he ascends toward D, the greater are his moments; because, namely, it is as if a line had been drawn from the center to the point of the wheel, in which is the treading weight; for there the moment of descending is as the sine of the declination from the perpendicular, according to what was said in book 1, chapter 15. Let it be

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Liber quintus. CAPUT III. 519 Sit enim rotæ CHG semidiameter EB, cylindri verò se- midiameter EO. Si homo tympanum ingressus consistat in infimo loco H, in quem scilicet cadit perpendiculum EH, utique machinam non movet, à qua ipse sustinetur: Similiter si funis OC per superioris trabis orbiculos transiens fuerit intentus, etiam. si ex H ascendat in D, in quod punctum cadit recta CO cylindrum tangens, non movetur pondus, quod ex hypothesi excedit gravitatem hominis tympanum calcantis: quia nimirum in D homo ratione positionis non habet descendendi momenta majora, quàm sit semidiameter OE, quæ pariter sunt momenta oneris funi OC adnexi: Cùm autem ratione positionis momenta ponderis atque potentiæ æqualia sint, sed ratione gravitatis potentia infirmior sit ex hypothesi quàm pondus, illa utique elevare pondus non valebit. Procedet igitur ascendendo ex. gr. usque ad I, ubi obtinebit momenta ut VE (hoc est IK sinus anguli declinationis IEH) ad momenta oneris ut OE, adeò ut quæ Ratio est VE ad OE, eadem sit Ratio gravitatis oneris ad gravitatem hominis; ac proinde ascendentis ex I in G momenta augebuntur, & in G erunt ut SE. Cùm ergo SE ad OE Ratio major sit quàm VE ad OE, hoc est gravitatis oneris ad gravitatem hominis, jam prævalet potentia, & tympanum convertitur, descenditque illius punctum G in locum, ubi erat punctum I, in quo fit consistentia & quoddam æquilibrium sustentando pondus, quod ut porrò moveatur, pergendum est in percurrendâ tympani orbitâ. Numquam igitur ratione positionis potentiæ momentum est ut semidiameter rotæ, nisi homo ita ascenderet, ut ejus centrum gravitatis responderet puncto B; ex momento enim, quod potentia obtineret in B, demendum est, quantum ab ipso centro retrahitur: in G autem

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Book Five. Chapter III. 519 Let the semidiameter of the wheel CHG be EB, and the semidiameter of the cylinder EO. If a man entering the drum stands in the lowest place H, in which, namely, the plumb line EH falls, assuredly he does not move the machine, by which he himself is sustained. Likewise, if the cord OC, passing through the pulleys of the upper beam, has been stretched, even if from H he ascends to D, to which point the straight line CO, touching the cylinder, falls, the weight is not moved, since by hypothesis it exceeds the weight of the man treading the drum; because, namely, in D the man, by reason of position, does not have greater moments of descent than the semidiameter OE, which are likewise the moments of the load attached to the cord OC. But since, by reason of position, the moments of the weight and of the power are equal, yet by reason of gravity the power is, by hypothesis, weaker than the weight, it certainly will not be able to raise the weight. He will therefore proceed ascending, for example, as far as I, where he will obtain moments as VE (that is, IK, the sine of the angle of declination IEH) to the moments of the load as OE, so that the ratio of VE to OE is the same as the ratio of the gravity of the load to the gravity of the man; and therefore the moments of the ascending man from I to G will be increased, and in G they will be as SE. Since therefore the ratio of SE to OE is greater than the ratio of VE to OE, that is, of the gravity of the load to the gravity of the man, the power now prevails, and the drum is turned, and its point G descends into the place where point I had been, in which there is a standing-still and a certain equilibrium in sustaining the weight; and in order that it may furthermore be moved, one must continue through the orbit of the drum. Therefore never by reason of position is the moment of the power equal to the semidiameter of the wheel, unless the man were to ascend in such a way that his center of gravity corresponded to point B; for from the moment by which the power prevails in B, there must be subtracted whatever is drawn back from its center itself: in G however

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520 Mechanicorum autem retrahitur juxta mensuram BS, & in I juxta mensuram BV, ac propterea ibi momentum remanet ut SE, hîc ut VE. Perinde autem se habere momentum in G ad pondus, atque si esset libra curva GEF, & ab alterâ extremitate F diametri cylindri, duceretur recta FG secans in R perpendicularem EH, manifestum est, quia ex 2. lib. 6. ut SE ad EF, hoc est OE, ita GR ad RF. Quòd si tympani orbitam limbus hinc & hinc ambiat, cui teretes paxilli inserti veluti gradus scalas constituant, quibus homo non solùm insistat pedibus, sed quos etiam manibus apprehendat; observare oportet, pedibusne tantum premat subjectum tympanum, an ex manibus quasi suspensus pendeat. Nam si in eodem perpendiculo non sint paxillus, cui insistit, & is ex quo dependet, valde disparia sunt momenta. Si verò non planè rectum sit corpus, sed quasi procumbens inclinetur, tunc potentiæ locus definitur à perpendiculo transeunte per centrum gravitatis ipsius hominis. Id quod dicendum pariter, quando tympano includitur canis (nam & à cane ingens tympanum versari vidi, quo in Solarium attollebatur non mediocris cista linteis recens ablutis plena) cujus gravitatis centrum spectandum est, ejusque distantia à perpendiculo transeunte per tympani centrum. Hinc si ex navi aliæ atque aliæ sarcinæ hac machinâ extrahantur, is qui tympanum versat, etiamsi omnino non videat onus extra machinæ domunculam positum, facilè pronuntiabit majorne? an minor sit secundæ sarcinæ gravitas comparata cum priorre: quò enim magis ascendere cogitur in tympano, eò majore est oneris gravitas; quærenda nimirum sunt momenta majora ex positione, ut majore intervallo absit à perpendiculo EH transeunte per E centrum. Simili ratione, si inter duos homines quæstio oriatur uter illorum gravior sit, facilè litem dirimes, si alter post alterum ingrediatur tympanum, ut idem onus attollat; qui enim magis ascendere cogitur, minus habet gravitatis, ideóque majora momenta quærit ex positione. Quanta autem sit oneris gravitas, dignoscetur ex artificio statim indicando. Unum hîc quasi per anticipationem addendum, quod ad funem duclarium spectat; præstat scilicet ejus extremitatem unco extremæ trabi infixo adnecti, & per orbiculum cum onere conjunctum transire, atque hinc per orbiculos G & F ad cylindrum deduci: hac enim

Transcription: Translated (English)

520 Mechanics however, is drawn back according to the measure BS, and in I according to the measure BV, and therefore there the moment remains as SE, here as VE. And it is manifest that the moment in G bears to the weight just as if there were a curved balance GEF, and from the other extremity F of the cylinder’s diameter a straight line FG were drawn, cutting the perpendicular EH at R, because from Book 2, proposition 6, as SE is to EF, that is, OE, so is GR to RF. But if the rim of the drum’s circumference encircles it on this side and on that, into which smooth pegs are inserted, forming as it were the steps of a ladder, on which a man not only stands with his feet, but which he also grasps with his hands; it must be observed whether he presses the drum beneath him only with his feet, or whether he hangs as it were suspended from his hands. For if the peg on which he stands and the one from which he hangs are not on the same perpendicular, the moments are very unequal. If his body is not perfectly upright, but inclines as if leaning forward, then the place of the force is determined by the perpendicular passing through the center of gravity of the man himself. The same must be said likewise when a dog is enclosed in the drum (for I have seen a large drum turned by a dog, by which a not inconsiderable chest full of freshly washed linens was raised up to the upper level), whose center of gravity must be considered, and its distance from the perpendicular passing through the center of the drum. Hence, if from a ship various loads are drawn out by this machine, the person turning the drum, even though he sees not at all the burden placed outside the machine’s little house, will easily declare whether the second load is greater or less in weight compared with the first; for the more it is forced to ascend in the drum, the greater is the weight of the load; namely, the greater moments must be sought from position, the farther it is from the perpendicular EH passing through E the center. In a similar manner, if a question arises between two men as to which of them is heavier, you will easily settle the dispute if one enters the drum after the other, so that he may raise the same load; for the one who is compelled to ascend more has less weight, and therefore seeks greater moments from position. But how great the weight of the load is will be recognized from the method to be indicated immediately. One thing here must be added, as it were by anticipation, concerning the hoisting rope; namely, it is preferable that its end be attached to a hook fixed in the outer beam, and pass through a pulley connected with the load, and thence be led through the pulleys G and F to the cylinder: for in this way

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Liber quintus. CAPUT III. 521 enim ratione attollendi facilitas geminatur, ut clariùs patebit ex iis, quæ sequenti libro de Trochleâ dicentur. Ut igitur innotescat, quanta sit proximè oneris gravitas ob- servandus est in tympano locus, ubi homo illud calcans facit cum pondere æquilibrium: quando scilicet eò venerit, ut paulo altiùs ascendens incipiat attollere pondus. Quoniam verò hu- jusmodi pondera ea sunt, ut in iis exiguæ differentiæ contem- nantur, exquisita quædam accuratio omnino supervacanea es- set, si singulas, aut pauculas libras ad calculos revocandas esse censeremus, cum sæpè non nisi per centenas libras eorum gra- vitas definiatur. Primùm ex centro E in ipsâ limbi crassitudine describatur circuli peripheria B C H: id quod facilè fiet funi- culo extento, & axem F R O complectente; quo funiculo cir- cumducto stylus in extremitate colligatus describet peripheriâ. Deinde si nota non sit accurata semidiametri mensura, quæ peripheriæ sextantem accipiat, punctum unum, quod placuerit, statue, ex quo peripheriam in partes aliquotas (quascumque tandem opportunitas dederit) dividere incipias: nam per nu- merum partium divisis gradibus 360, statim patebit, quot gra- dus singulæ partes contineant, quas aliquotas assumpsisti. Par- tem igitur unam in gradus sibi congruentes tribue; eorum enim mensura in consequentem arcum translata, quoties oportuerit, demùm integrum Quadrantem in gradus 90 divisum dabit. Po- namus commodam accidisse divisionem peripheriæ in partes 15: divisis gr. 360 per 15, quotiens 24 indicat numerum graduum parti decimæ quintæ tribuendorum. Quare partem unam bifa- riam divide, & intervallum gr. 12 inter punctum divisionis & assumptum punctum, ex quo divisio incipit, iterum divide bifa- riam, ut parti uni cedant gradus 6: his iterum bifariam divisis, habetur partis aliquotæ primò assumptæ pars octava gr. 3. hanc in tres æquales partes distribue, & singulorum graduum men- sura manifesta est. Acceptis itaque tribus partibus decimis quintis addantur gradus 18, & habebitur integer peripheriæ Quadrans in gradus 90 distributus, qui adeò notabiles erunt, ut etiam gradûs partes, cujusmodi est semissis, triens, & qua- drans satis clarè dignosci queant. Tertiò. Quia non arcus H G, sed semidiametri pars E S con- sideratur, ut dictum est, concipe semidiametrum E B in partes V u u

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Book five. CHAPTER III. 521 for indeed, by reason of raising, the ease is doubled, as will be shown more clearly from what will be said in the following book about the Pulley. Therefore, in order that it may be made known how great the heaviness of the load is at the nearest point, the place on the drum must be observed where a man, treading upon it, brings it into equilibrium with the weight: namely, when he has come so far that, rising a little higher, he begins to lift the weight. But since weights of this kind are such that slight differences in them are disregarded, a certain exact precision would be altogether unnecessary if we were to think that individual pounds, or a few pounds, must be reduced to calculation, since often their heaviness is determined only by hundreds of pounds. First, from the center E, on the very thickness of the rim, let the circumference B C H of a circle be described: this will easily be done with a cord stretched and encompassing the axis F R O; and with the cord drawn around, the stylus fastened at the end will describe the circumference. Then, if an accurate measure of the semidiameter is not known, one which takes the sixth part of the circumference, set down any one point that pleases you, from which you may begin to divide the circumference into aliquot parts (whatever opportunity may in the end provide): for when the degrees 360 are divided by the number of parts, it will immediately be clear how many degrees each of the parts you have assumed contains. Therefore assign one part to the degrees belonging to it; for when their measure is transferred to the following arc, repeated as often as necessary, it will finally give the whole Quadrant divided into 90 degrees. Let us suppose that a convenient division of the circumference into 15 parts has occurred: dividing 360 degrees by 15 shows 24 as the number of degrees to be assigned to the fifteenth part. Wherefore divide one part in two, and again divide in two the interval of 12 degrees between the point of division and the assumed point from which the division begins, so that 6 degrees may fall to one part: when these are again divided in half, there is obtained the eighth part of the aliquot part first assumed, namely 3 degrees. Divide this into three equal parts, and the measure of the individual degrees is clear. Thus, with the three fifteenth parts taken, add 18 degrees, and there will be obtained the whole Quadrant of the circumference distributed into 90 degrees, which will be so notable that even the parts of a degree, such as a half, a third, and a fourth, may be recognized sufficiently clearly. Third. Since it is not the arc H G, but the part of the semidiameter E S that is considered, as has been said, conceive the semidiameter E B in parts V u u

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522 Mechanicorum aliquotas distributam, primùm in duas, deinde in tres, in quatuor, & deinceps, prout opportunum accidet, ita tamen, ut non venias ad partem aliquotam minorem semidiametro Axis: Id quod deprehendes, si assumptam chordam subtensam gradibus 60, in illud genus partium aliquotarum, de quo dubitas, diviseris, & in semidiametro B E incipiendo ab extremitate B illas acceperis; si enim postrema pars aliquota residua major sit semidiametro Axis, aut illi æqualis, non est justo minor. Igitur ex Canone Sinuum exquire arcum singulis partibus, incipiendo à centro tympani, congruentem, & in peripheriâ descriptâ atque in gradus distributa arcum inventum ex Canone designa notâ partis aliquotæ 1/2, 1/3, 1/4 &c. ut statim appareat, quo loco intelligatur posita potentia sive in semisse, sive in triente, sive in quadrante semidiametri, sive in ejusdem besse aut dodrante &c. Factâ siquidem comparatione inter distantiam potentiæ à centro, & Axis semidiametrum, innotescet Ratio ponderis ad potentiam in tympano calcantem. Ponamus itaque tympani semidiametrum distinctam in partes 10, ita ut Axis semidiameter E O ad E B sit ut 1 ad 10: possunt commodè omnes partes intra decimas reperiri, pro ut in adjectâ tabellâ oculis subjicio, in qualicèt minuta secunda exprimantur, ut innotescat etiam alios in usus, quibus Sinubus quinam arcus respondeant: in præsenti tamen

Transcription: Translated (English)

522 Mechanics distributed into aliquot parts, first into two, then into three, into four, and so on, as shall be convenient, provided nevertheless that you do not come to an aliquot part smaller than the semidiameter of the Axis: this you will detect if you divide the assumed chord subtending 60 degrees into that kind of aliquot parts about which you are in doubt, and receive them beginning from the end B on the semidiameter B E; for if the last remaining aliquot part is greater than, or equal to, the semidiameter of the Axis, it is not properly smaller. Therefore from the Canon of Sines seek the arc corresponding to each part, beginning from the center of the drum, and in the described periphery and the divisions into degrees mark the found arc from the Canon with the sign of the aliquot part 1/2, 1/3, 1/4, etc., so that it may immediately appear at what place the power is understood to be placed, whether in the half, or in the third, or in the quarter of the semidiameter, or in its two-thirds or three-quarters, etc. For once the comparison has been made between the distance of the power from the center and the semidiameter of the Axis, the ratio of the weight to the power pressing on the drum will be made known. Let us therefore suppose the semidiameter of the drum divided into 10 parts, so that the semidiameter E O of the Axis is to E B as 1 to 10: all the parts within the tenths can be conveniently found, as I set before the eyes in the appended little table, in which even the minutes and seconds are expressed, so that it may also be made clear for other uses to which arcs correspond to which sines: for the present, however

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Liber quintus. CAPUT III. 523 tamen opere prorsus inutilis accideret tam exquisita accuratio: satis quippe est circiter illum gradum ejusque minuta prima rotam appingere, indicem partis, vel partium semidiametri tympani. Quartò. Funiculum Axi insistentem, & facilè excurrentem ita dispone, ut plumbeus globus in ejus extremitate pendulus intendat funiculum ipsum, qui in tympani limbo designet punctum, per quod transit linea perpendicularis ab Axis cen- tro in horizontem descendens. Tum ab hoc puncto usque ad punctum, ubi fit æquilibrium, sumatur intervallum, atque transferatur in Quadrantem gradibus distinctum: Nam punctum, in quod ab initio Quadrantis cadit altera observati intervalli extremitas, indicabit notâ in limbo prænotatâ, quotâ semidiametri parte distet à tympani centro gravitas calcans ipsum tympanum, ex. gr. 3/5 aut 4/7. Cum igitur jam innotuerit Ratio semidiametri Axis ad semidiametrum tympani, scilicet ex hypothesi 1/10, fiat ut fractio index Rationis semidiametro- rum, ad fractionem in limbo notatam, ita gravitas calcantis tympanum ad gravitatem ponderis, cum quo facit æquilibrium, videlicet ut 1/10 ad 3/5, ita gravitas hominis, puta lib. 250, ad gra- vitatem oneris lib. 1500. Hinc patet in puncto D, per quod transit linea OD tangens Axem & parallela perpendiculari EH ex centro demissæ, æquilibrium esse inter gravitates om- nino æquales, ac proinde minimum pondus esse æquale gravi- tati hominis calcantis: Nam si inter H & D fieret æquilibrium, pondus levius esset quàm homo, & communi staterâ facilè po- tes assequi illius gravitatem. Maximum autem pondus est il- lud, quod indicat semidiameter tympani ad semidiametrum Axis, homine nimirum suæ gravitatis vires exercente in B, ac propterea gravitas ponderis ad gravitatem hominis in B esset ex. gr. in Ratione decuplâ. Illud tamen hîc perpende, quòd, si homo calcans in B, aut indè pendens, non volvit Axem, atque adeò non attollit pon- dus adnexum, non constat, an sit æquilibrium, idem enim ac- cideret etiam, si pondus esset multò majus; ac præinde neque constat de ejusdem ponderis gravitate nisi hoc, quod sit ut mi- nimum decupla gravitatis hominis; quia nimirum nunquam il- V u u 2

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Book five. CHAPTER III. 523 yet such exactness would be wholly useless in the work: for it is enough to place the wheel, around that degree and its first minutes, the pointer of the part, or of half the parts of the drum. Fourth. Arrange the cord resting on the Axle, and running easily, so that the leaden globe hanging from its end may stretch the cord itself, which, on the rim of the drum, may mark the point through which passes the perpendicular line descending from the center of the Axle to the horizon. Then from this point to the point where equilibrium occurs, let the interval be taken, and transferred to a Quadrant divided into degrees: for the point into which one extremity of the observed interval falls from the beginning of the Quadrant will indicate, by the mark previously noted on the rim, by what part of the radius the weight pressing down the drum is distant from the center of the drum, e.g. 3/5 or 4/7. Since therefore the ratio of the radius of the Axle to the radius of the drum is now known, namely, by hypothesis, 1/10, let the fraction expressing the ratio of the radii be to the fraction marked on the rim, as the weight pressing the drum is to the weight with which it is in equilibrium; namely, as 1/10 to 3/5, so let the weight of the man, say 250 lb., be to the weight of the load, 1500 lb. Hence it is clear that at point D, through which passes the line OD, touching the Axle and parallel to the perpendicular EH drawn down from the center, there is equilibrium between weights altogether equal, and therefore the least weight is equal to the weight of the man pressing down: for if equilibrium were achieved between H and D, the weight would be lighter than the man, and by a common balance you could easily determine its weight. But the maximum weight is that which the ratio of the radius of the drum to the radius of the Axle indicates, namely with the man exerting the force of his own weight at B, and therefore the weight of the load to the weight of the man at B would be, for example, in a tenfold ratio. Yet consider here that if the man pressing at B, or hanging from it, does not turn the Axle, and thus does not lift the attached weight, it is not certain whether there is equilibrium; the same thing would happen even if the weight were much greater; and accordingly the weight of that same load is not certain except in this, that it is at least ten times the weight of the man; because namely it never...

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Mechanicorum 524 lud movebit; nam ascendens homo ex B versus C minora semper obtinet momenta, quàm in B. Hoc autem ubi contigerit, & velis exploratam habere oneris gravitatem, assume pondus aliquod notæ gravitatis, quod adnectere valeas oræ tympani aut in B, aut eo loco, ut deinde homo calcans infra B, attollat pondus: ubi enim demum fiat æquilibrium, duplex institue ratiocinium, alterum quidem ratione hominis, alterum verò ratione gravitatis additæ: inventi siquidem termini simul additi indicabunt quæsitam oneris gravitatem. Sic si assumptum pondus sit lib. 36, & homo calcans sit lib. 250, fiat autem æquilibrium homine existente in X, pondere verò in G; sumptis intervallis XH & GH, atque translatis in Quadrantem inveniatur pro homine 2/10, & pro pondere 1/5. Fiat primò ut 1/10 ad 2/10, ita lib. 250 ad lib. 2250: deinde ut 1/10 ad 1/5, ita lib. 36 ad lib. 288: igitur summa lib. 2538 indicat ponderis gravitatem. Quòd si funis ductarij extremitas sit adnexa extremæ trabi, ut indicatum est, atque transeat per orbiculum ponderi conjunctum, inventus numerus lib. 2538 duplicandus est & sunt lib. 5076. Ex his fortasse alicui placeat stateræ LM vires augere addito tympano hujusmodi EB ut supra prænotato. Nam firmatâ in superiore loco staterâ LM, cujus ansa sit in N, primùm observetur, quantum ponderis requiratur in M, ut fiat æquilibrium cum solo brachio NL; hæc enim gravitas semper addenda erit gravitati, quæ invenietur ex ratiocinatione, qua componuntur Rationes stateræ & tympani. Deinde in tympani limbo notetur punctum C, cui congruit funis OL, quando statera est horizonti parallela; ut hinc dignoscatur, quo in loco tympani, dum convertitur, contingat æquilibrium. Demum cùm nota sit Ratio MN ad NL, componatur cum Ratione semidiametri Axis ad partem semidia

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Mechanics 524 will move; for a man ascending from B toward C always obtains smaller moments than in B. But when this has occurred, and you wish to have the weight of the load determined, take some weight of known heaviness, which you can attach to the rim of the drum either at B or at some other point, so that then the man stepping below B may raise the weight: where at last equilibrium is reached, set up a double calculation, one indeed in terms of the man, the other in terms of the added weight; for the terms found, when added together, will indicate the desired weight of the load. Thus, if the assumed weight be 36 lb., and the stepping man be 250 lb., and equilibrium is established with the man at X and the weight at G, then, taking the intervals XH and GH and transferring them to the quadrant, one finds for the man 2/10 and for the weight 1/5. First let 1/10 be to 2/10 as 250 lb. is to 2250 lb.; then let 1/10 be to 1/5 as 36 lb. is to 288 lb.: therefore the total, 2538 lb., indicates the heaviness of the weight. But if the end of the driving rope be attached to the outer beam, as has been indicated, and pass through the little pulley joined to the weight, the number found, 2538 lb., must be doubled, and the result is 5076 lb. From these things perhaps someone may wish to increase the power of the balance LM by adding a drum of this kind EB, as noted above. For, with the balance LM fixed in the upper position, whose handle be at N, first observe how much weight is required at M, so that equilibrium may be established with the single arm NL; for this heaviness must always be added to the heaviness which will be found by the calculation by which the ratios of the balance and the drum are combined. Next, on the rim of the drum mark the point C, to which the rope OL corresponds when the balance is parallel to the horizon, so that it may thereby be known at what place on the drum equilibrium occurs while it is turning. Finally, since the ratio of MN to NL is known, let it be combined with the ratio of the semidiameter of the axis to the part of the semidia

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Liber quintus. CAPUT III. 525 semidiametri tympani, ex. gr. EO ad ES; & habetur Ratio gravitatis hominis tympanum calcantis ad gravitatem oneris, quod in M expenditur. Sit EO ad ES ut 1/10 ad 4/5, & MN ad NL ut 2 ad 25; Ratio composita est ut 1/5 ad 20, hoc est ut 1 ad 100. Igitur pondus in M expensum est, ut minimum, centuplum gravitatis hominis; nam addenda præterea est gravitas respondens gravitati brachij LN longioris ipsius stateræ. Si verò neque tympano, quod ab homine intùs calcante premitur, neque adeò incerto sacomate, cujusmodi est varia hominum gravitas, uti volueris, aut potius non ingentes sarcinas, sed onera mediocria expendere placuerit, paetur CBH discus ligneus parvulum axem habens ad centrum E, & in eo descripta sit peripheria circuli CBH, atque adnotatum punctum C, per quod funiculus OL transit, quando stateræ jugum LM est horizonti parallelum. Tum converso disco ita, ut LO transeat per C, dimisso perpendiculo insistente Axi, notetur in peripheria punctum H, per quod transit perpendicularis à centro E. Deinde ex H versùs B ascendendo accipiantur gradus juxta superiorem tabellam, affixis notis indicibus partium, similiter ac de tympano dictum est. Demum sacoma certæ gravitatis, puta unius aut alterius libræ, ita disponatur, ut per disci ambitum ex H versùs B excurrere possit, & cochleâ firmari, ubi æquilibrium contigerit: Aut potiùs singulis Quadrantis gradibus, aut saltem punctis partium notatis, claviculos infige, qui inseri possint annulo sacomatis. Nota enim Ratio semidiametri axis EO ad disci semidiametrum EB indicabit, quid faciendum sit juxta dicta, ut gravitas ponderis in M suspensi innotescat. At si volueris Quadrantem HB in suos 90 gradus distribuere, & uti Canone Sinuum, priùs innotescat Ratio semidiametri axis ad Radium in partibus Radij: deinde fiat ut partes Radij axi congruentes, ad Sinum Rectum graduum, ubi fit æquilibrium, ita Sacoma appensum ad gravitatem ponderis, quod expenditur. Vuu 3

Transcription: Translated (English)

Book five. Chapter III. 525 semi-diameter of the drum, for example EO to ES; and the Ratio is obtained of the weight of the man treading the drum to the weight of the load, which is weighed in M. Let EO to ES be as 1/10 to 4/5, and MN to NL as 2 to 25; the compound Ratio is as 1/5 to 20, that is, as 1 to 100. Therefore the weight weighed in M is, at the least, one hundred times the weight of the man; for in addition there must be added the weight corresponding to the weight of the longer arm LN of the balance itself. But if you do not wish to use either the drum, which is pressed by a man treading inside it, or that uncertain sacomum, such as the varying weight of men, or rather if it is preferred not to weigh very large burdens, but moderate loads, let there be a wooden disk CBH with a small axis at the center E, and on it let the circumference of the circle CBH be drawn, and let the point C be marked, through which the cord OL passes, when the beam LM of the balance is parallel to the horizon. Then, with the disk turned so that LO passes through C, and the plumb line being let down to rest on the axis, let the point H be marked on the circumference, through which the perpendicular from the center E passes. Then, ascending from H toward B, let the degrees be taken according to the upper table, with marks of the indices of the parts affixed, in the same way as was said about the drum. Finally, let a sacomum of a certain weight, say of one pound or two, be so arranged that it may run along the rim of the disk from H toward B, and be secured by a screw where equilibrium occurs: Or rather, at every degree of the Quadrant, or at least at the marked points of the parts, insert little pins which may be fitted into the ring of the sacomum. For the noted Ratio of the semi-diameter of the axis EO to the semi-diameter of the disk EB will indicate what is to be done according to the above, so that the weight of the burden suspended in M may be ascertained. But if you wish to divide the Quadrant HB into its 90 degrees, and use the Canon of Sines, first let the Ratio be known of the semi-diameter of the axis to the Radius in parts of the Radius: then let it be as the parts of the Radius corresponding to the axis are to the Sine Rectus of the degrees where equilibrium occurs, so is the suspended Sacomum to the weight of the burden being weighed. Vuu 3

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Mechanicorum CAPUT IV. An Axis in Peritrochio inveniatur etiam sine tractione. Hactenus ductarij funis conversionem circa Axem convolutum consideravimus, ex quo oritur ponderis fune connexi tractio: sed numquid non etiam ad hoc genus machinæ aliquæ revocari possunt, quibus non quidem trahitur pondus, sed aliqua resistentia superatur? Occurrit autem primo loco antiquus servorum metus Pistrinum, in quod detrusi frumentum tundere cogebantur molâ versatili, sive in nostrarum Moletrinarum speciem ac similitudinem metam congruo Catillo impositam manu truderent, ac circumagerent, sive ingentem lapideum discum perpendiculari cylindro coagmentatum versarent asellorum vicarij laborioso muneri succedentes, cum Vectis cylindro ad angulos rectos infixi extremitatem aut traheerent, aut propellerent: Cujusmodi machinæ genere nos quoque utimur in frenendis leguminibus, & in contundendis seminibus, ex quibus demùm oleum prælo exprimitur. Et hîc quidem non ipsius molaris lapidis gravitatem movendam attendimus, quippe qui ipsius machinæ pars est; sed potissimum corporis à molâ compressi resistentia consideranda est, quæ nimirum vincenda proponitur. Oritur autem hæc resistentia ex corporis obterendi aut contundendi duritiæ, in quod incurrit scabra molæ circumactæ superficies, cum verò illud incumbenti lapidi se subducere non possit, à lapidis gravitate & potentiæ impetu cogitur dissilire in partes. Quia igitur potentiæ cum machinâ connexæ motum impedit illa granorum aut nucleorum frangendorum durities, comparanda est distantia potentiæ moventis à centro motûs, cum distantiâ corporis comminuendi; & quò major est hujusmodi intervallorum Ratio, majora pariter sunt potentiæ momenta. Hinc vides, cur in trusatili mola (quam mediocrem esse oportet,

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Mechanics CHAPTER IV. An Axis in a Wheel and Axle may be found even without traction. Thus far we have considered the turning of a rope wound around an Axis, from which arises the traction of a weight connected to the rope: but may there not also be referred to this kind of machine some by which a weight is not indeed drawn, but some resistance is overcome? There first comes to mind the ancient terror of slaves, the mill, into which those thrust were compelled to crush grain by a turning millstone, whether they drove by hand and revolved the conical stone set upon the corresponding hopper in the manner and likeness of our hand mills, or whether they turned the great stone disk joined to a vertical cylinder, with asses taking over the laborious task, with levers fixed to the cylinder at right angles, pulling or pushing the end. We likewise use a machine of this kind in crushing legumes, and in grinding seeds, from which at last oil is expressed by the press. And here indeed we do not attend to moving the weight of the millstone itself, since it is part of the machine; but rather the resistance of the body compressed by the mill must chiefly be considered, which is, as it were, the thing proposed to be overcome. This resistance arises from the hardness of the body to be ground or crushed, against which the rough surface of the revolving mill strikes; and since it cannot withdraw from the stone pressing upon it, it is forced by the weight of the stone and the impulse of the power to burst apart into pieces. Since, therefore, the hardness of the grains or kernels to be broken impedes the motion of the power connected with the machine, the distance of the moving power from the center of motion is to be compared with the distance of the body to be comminuted; and the greater the ratio of such intervals, the greater likewise are the moments of the power. Hence you see why in the push-mill (which ought to be of moderate size,

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Liber quintus. CAPUT IV. 527 oportet, ne nimio labore frangatur molitor in immani saxo versando, catillus quidem planus est, meta verò, quâ catillum reipicit, non omnino subjecto plano congruit, sed cavam obtusissimi coni superficiem æmulatur: ut scilicet integra grana per medium foramen immissa inter utrumque lapidem intercipiantur non procul à centro, à quo potentia abest, comminuta autem peripheriam versùs accedant ad angustiora spatia, quò magis obterantur: cùm enim integra grana magis fractioni obsistant, quàm comminuta, integris frangendis majora debentur potentiæ momenta, comminutis in minusculas particulas redigendis minores vires sufficiunt. In molâ autem Asinariâ ubi lapideus discus in plano Verticali constitutus subjectum catillum modicè excavatum vix extremo ambitu contingit, eadem ferè est semper distantia à centro motûs, nisi quatenus ipsius molæ crassitudo partem aliam centro motûs propiorem, aliam remotiorem habet: porrò grana illa, quæ lapidum contactui, vel quasi contactui, propiora sunt, validiùs teruntur, quàm quæ magis ab eodem contactu recedunt: sed hoc nihil ad præsentem disputationem attinet, nisi quatenus lapidis partes remotiores subjecta grana agitantes, atque tundentes crassiùs, majorem Rationem ad potentiæ motum habent in suâ convolutione, quàm partes ejusdem minùs à centro remotæ. Haud dissimili ratione, si ex chalybe ellipticum sphæroides obliquis striis modicè asperum, quasi in limæ speciem, congruo loculamento interiùs pariter asperato includatur, ita tamen, ut spatium, quo sphæroides à loculamento distat, paulatim à latitudine in angustias se se contrahat; axi verò sphæroidis superiùs producto addatur manubrium, quo arrepto converti possit in gyrum; grana piperis, aut similia superiùs immissa levissimo negotio comminuentur: quorum scilicet durities si cum potentiæ viribus conferatur, resistentiam habet maximam pro Ratione semidiametri transversæ Ellipsis ad manubrij longitudinem: initio autem, quia grana minùs ab axe distant, minùs resistunt, si cætera fuerint paria, hoc est, si modicè comminutorum durities, integrorum duritiei omnino respondeat; nam minor distantia à centro motûs minorem habet Rationem, quàm distantia major ad eandem manubrij longitudinem. Par

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Book Five. Chapter IV. 527 it is necessary, lest the miller be worn out by excessive labor in turning the huge stone, for the lower stone is indeed flat, but the upper stone, with which it faces the lower one, does not entirely correspond to a flat surface beneath it, but imitates the hollow surface of an exceedingly blunt cone: so that, namely, whole grains, admitted through the central hole, may be caught between the two stones not far from the center, from which power is absent, but, once broken up, may move toward the circumference and into narrower spaces, the more they are crushed: for since whole grains resist fracture more than broken ones, greater forces are required for crushing whole grains, while for reducing broken ones into tiny particles smaller forces suffice. But in the Assinarian mill where a stone disc placed in a vertical plane barely touches the underlying slightly hollowed lower stone only at its extreme edge, the distance from the center of motion is almost always the same, unless insofar as the thickness of the mill itself has one part nearer to the center of motion, another farther away: moreover, those grains which are nearer to the contact, or as it were the contact, of the stones are ground more strongly than those that recede farther from the same contact: but this has nothing to do with the present discussion, except insofar as the parts of the stone that are farther away, driving and striking the grains beneath them, being thicker, have a greater relation to the motion of power in their revolution than the parts of the same stone that are less distant from the center. Not unlike this, if from steel an elliptical spheroid, moderately roughened with oblique striations, as if in the manner of a file, is enclosed in a suitable receptacle likewise roughened on the inside, yet so that the space by which the spheroid is separated from the receptacle gradually narrows from width into a constriction; and if to the upper part of the spheroid’s axis a handle is added, by grasping which it may be turned in a circle; grains of pepper, or similar things, introduced from above will be crushed with the slightest effort: whose hardness, namely, if compared with the forces of power, has the greatest resistance in proportion to the ratio of the transverse semidiameter of the ellipse to the length of the handle: at the beginning, however, because the grains are less distant from the axis, they resist less, if the other conditions are equal, that is, if the hardness of the moderately crushed grains altogether corresponds to the hardness of the whole grains; for a smaller distance from the center of motion has a smaller ratio than a greater distance with the same length of handle. Part

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Mechanicorum Par erit philosophandi ratio, si tympanis non idem centrum habentibus inclusa aqua ex interioris tympani conversione ad angustias redigatur, atque compressa exprimatur ex tubo; cujusmodi fortasse fuit veterum Hydracontisterium; de cujus formâ non est hîc disputandi locus; nam manubrij à potentiâ commoti longitudo comparanda est cum distantiâ peripheriæ tympani aquam comprimentis à centro, circa quod perficitur motus, ut momentorum Ratio perspecta sit; aqua scilicet dum impellitur, atque exprimitur, resistit. Ad hæc porrò inversus quidam Axis in peritrochio usus considerandus est, quando videlicet potentia non peritrochio, sed ipsi Axi, applicatur; id quod tunc potissimum contingit, cùm potentia viribus abundat, motui autem, qui efficiendus est, non admodum resistit corpus, quod vel modicè impellendum est, vel in orbem circumducendum. Certum quippe est potentiam Axi applicatam tardiùs multò moveri, quàm peritrochij peripheriam, pro Ratione semidiametrorum Axis & Peritrochij, ac proinde licèt impetus amplioris peripheriæ partibus impressus imbecillior quodammodo sit, ut pote distractus, satis tamen esse ad vincendam levem resistentiam. Hinc quoniam ferrum cotis tritu extenuatur, eóque faciliùs, quò celeriùs cos movetur, qui restituunt obtusas cultorum aut novacularum acies, lapidem ex cotariâ eductum in discum rotundant, ut circa axem centro infixum versatilis circumagi possit. Quamvis autem non rarò eidem axi cohæreat manubrium, quo circumducto rotatur lapis, ut tamen minori labore adhuc etiam velociùs rotetur, sapienter instituerunt amplioris rotæ absidi excavatæ funem insistere, qui rotulam eundem cum lapide axem habentem circumplectatur, ut minor hæc rotula amplioris rotæ ductum sequens secum pariter rapiat cotem; cujus peripheria, cùm adnexam rotulam valdè excedat, velociùs quoquè movetur. Quantum verò motus hic, celeritate suâ, potentiæ motum superet, facilè constabit, si motuum singulis membris convenientium ratio ineatur. Sit ex. gr. manubrij longitudo ad amplioris rotæ semidiametrum subquadrupla, hæc autem semidiameter ad rotulæ semidiametrum sit octupla: demum rotulæ eundem cum cote axem habentis semidiameter sit subtripla semidiametri ipsius cotis. Igitur puncti in cotis peripheriâ notati

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Mechanics The method of reasoning will be similar, if water enclosed in drums that do not have the same center is reduced to a narrow space by the turning of the inner drum, and, being compressed, is forced out through a tube; a device of which perhaps the ancient Hydracontisterium was a kind; but there is no place here to dispute its form; for the length of the handle moved by power must be compared with the distance of the circumference of the drum compressing the water from the center around which the motion is performed, so that the ratio of moments may be understood; for water, while it is driven and squeezed, resists. Moreover, an inverted axis used in the peritrochium is to be considered, namely when power is applied not to the peritrochium, but to the axis itself; this happens especially when power abounds in strength, while the body which is to be moved does not greatly resist the motion, whether it is to be pushed only a little, or turned around in a circle. For it is certain that power applied to the axis is moved much more slowly than the circumference of the peritrochium, in proportion to the semidiameters of the axis and of the peritrochium, and therefore although the impulse impressed upon the parts of the larger circumference is in some way weaker, since it is dispersed, yet it is sufficient to overcome slight resistance. Hence, because iron is worn thin by the friction of the whetstone, and the more easily the more quickly the whetstones are moved, those who restore the dull edges of knives or razors make the stone taken from the grindstone into a disk, so that it may be turned around an axis fixed in its center. Although a handle is often attached to the same axis, by which the stone is turned when it is revolved, nevertheless, that it may be rotated with less effort and even more quickly, they wisely arranged for a cord to be fastened to the hollow of a larger wheel, which encircles a little wheel having the same axis as the stone, so that this smaller wheel, following the motion of the larger wheel, may drag the grindstone along with it; and the circumference of the stone, since it far exceeds the attached little wheel, is moved even more quickly. But how much this motion exceeds the motion of power by its speed will easily be evident, if the ratio of the motions appropriate to the individual parts is calculated. Let, for example, the length of the handle be less than four times the semidiameter of the larger wheel; let this semidiameter, however, be eight times the semidiameter of the little wheel: lastly, let the semidiameter of the little wheel that has the same axis as the grindstone be less than three times the semidiameter of the grindstone itself. Therefore the point marked on the circumference of the grindstone

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Liber quintus. CAPUT IV. 529 notati motus triplo velocior est motu similis puncti in rotulæ peripheriâ: rotulæ motum definit funis, qui in convolutione ex- plicatur, hic enim pariter majoris rotæ motum metitur: octies ergo rotula, & cum eâ lapis rotatur, dum amplior rota semel in gyrum agitur. Quoniam verò rotæ semidiameter est ad cotis semidiametrum ut 8 ad 3 ex hypothesi, dum punctum in rotæ peripheriâ notatum movetur velocitate ut 8, simile punctum cotis movetur velocitate ut 24. Atqui motus rotæ cum motu potentiæ manubrio applicatæ comparatus est ut 4 ad 1, ex hy- pothesi; igitur si duæ Rationes 1 ad 4, & 8 ad 24 componan- tur, erit Ratio motus potentiæ ad motum cotis ut 1 ad 1 2. Sunt hîc itaque duo Axes in Peritrochiis suis compositi, & Potentia Axibus applicata intelligitur; cùm enim in Verticali plano lapis ipse versetur super polos læves atque politos, non admodum re- pugnat motui; impressus autem impetus aliquandiu manens potentiam ipsam juvat. Simile quiddam observandum occurrit in horologioru[m] motu, quæ in turribus statuuntur: nam cylindrum horizonti paralle- lum circumplicat funis, quo vi ponderis descendentis explica- to, circumagitur rota eidem axi infixa: ex hac in consequentes rotas derivatur motus semper velocior, qui demum temperatur ex quadam motûs retardati & brevissimæ morulæ vicissitudine, cum postremæ rotæ dentes in serræ modum conformati fusum, cui Tempus adnectitur alternis motibus agunt. Primum enim dens rotæ superior in pin- nulam A incurrens eam impellit, axemque HL convertit unà cum trans- versario CD & adjunctis globulis plumbeis E & F, qui similem arcum descri- bunt, ac pinnula A, sed longè majorem; propterea pro ratione gravitatis glo- bulorum, eorumque di- stantiæ ab axe, HL, etiam major vis requiritur, adeóque impeditur, ac retardatur motus rotæ dentatæ, & cum eâ reliquarum rotarum, atque ipsius pon- XXX

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Book five. CHAPTER IV. 529 the motion marked is three times swifter than the motion of the similar point on the circumference of the wheel: the rope, which is unwound in the winding, determines the motion of the wheel; for this also measures the motion of the larger wheel: therefore the little wheel, and with it the stone, rotates eight times while the larger wheel is turned once in a circle. But since the semidiameter of the wheel is to the semidiameter of the grindstone as 8 to 3 by hypothesis, while the point marked on the circumference of the wheel moves with a velocity as 8, the similar point on the grindstone moves with a velocity as 24. Yet the motion of the wheel compared with the motion of the power applied by the handle is as 4 to 1, by hypothesis; therefore if the two ratios 1 to 4 and 8 to 24 are compounded, the ratio of the motion of the power to the motion of the grindstone will be as 1 to 1 2. Here therefore there are two axes combined in their peritrochs, and the power is understood to be applied to the axes; for since the stone itself is turned in a vertical plane upon smooth and polished pivots, it does not greatly resist the motion; and the impressed impulse, remaining for some time, aids the power itself. A similar thing is found to be observed in the motion of the clocks which are set in towers: for a cord is wound around a cylinder parallel to the horizon, and when this is unwound by the force of the descending weight, the wheel fixed to the same axis is turned; from this motion is derived to the subsequent wheels a motion ever swifter, which is finally moderated by a certain alternation of retarded motion and the briefest pause, when the teeth of the last wheel, shaped in the manner of a saw, drive the spindle to which Time is attached by alternate motions. For first the upper tooth of the wheel, meeting the pin A, pushes it, and turns the axis HL together with the crossbar CD and the attached leaden balls E and F, which describe a similar arc as the pin A, but a far larger one; therefore, in proportion to the weight of the balls and their distance from the axis HL, a greater force is also required, and so the motion of the toothed wheel, and with it of the other wheels, is impeded and retarded, and of the very weigh-

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Mechanicorum deris, à quo totius machinæ motus initium sumit: qui si fuerit justo velocior, globuli E & F removentur ab L, sin autem justo tardior, admoventur, ut modò major, modò minor sit re- sistentia. Deinde quia globuli E & F ex impulsu pinnulæ A impetum conceperunt ad certam plagam directum, pergerent illorum moveri, quamdiu impressus impetus perseveraret, nisi in eâ conversione inferior pinnula B occurreret inferiori denti rotæ serratæ: hinc fit vi hujus impetus brevissimam morulam in- ferri conversioni rotæ, quæ eandem pinnulam urgens ipsos quo- què globulos in contrariam plagam reflectit. Posse autem adeò exilibus viribus morulam inferri tanto ponderi descendenti, paulò inferiùs manifestum fiet, ubi de Rotis dentatis in unam machinam compactis disseretur; illud saltem palam est, si mo- rulam nullam admittas, resistentiam esse non solùm pro gravi- tate globulorum, eorumque distantiâ, verùm etiam pro ratione impetûs impressi in antecedenti impulsion. Quare globuli iidem quando moventur impulsâ pinnulâ, rationem habent ponderis peritrochio adnexi, & potentia impellens pinnulæ, hoc est axi, applicatur: Contrà verò ad retardandum, aut tan- tisper coërcendum motum ponderis, quod pinnulæ applicatum intelligitur, iidem globuli vi impetûs sibi impressi rationem ha- bent potentiæ peritrochio applicatæ. Quoniam verò hîc horologiorum mentio incidit, cur in illis, quæ secum quisque ad perpetuum usum ferre potest, catenula seu nervus cono, non cylindro, circumducatur, manifestum est: quia scilicet chalybea lamella, à qua motûs origo est, initio in spissiorem spiram contracta suam vim elasticam exerens va- lidiùs conatur se restituere, trahênsque catenulam, seu ner- vum FE, totum conum DEC eique con- nexam rotulam dentatam AB in gyrum agit; cum verò illa fuerit in paulò laxiores spiras explicata, languidiùs conatur, atque catenu- lam, seu nervum, trahens jam non propè verti- cem coni, sed magis ad basim, eandem rotam AB circumagit. Cùm igitur motus potentiæ propè verticem coni, ad motum rotæ AB mi- norem habeat Rationem, quàm ad eundem rotæ motum mo- tus potentiæ in latiore coni parte (ibi enim breviorem, hîc ma- jorem

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of the mechanism, from which the motion of the whole machine takes its beginning: if this should be a little too swift, the globules E & F are moved away from L; if, however, it should be a little too slow, they are brought nearer, so that the resistance is now greater, now less. Then, because the globules E & F, from the impulse of the vane A, have acquired a force directed toward a certain side, they would continue to move, as long as the impressed impulse lasted, unless, in that rotation, the lower vane B encountered the lower tooth of the serrated wheel: hence it comes about that, by virtue of this impulse, a very brief delay is introduced into the rotation of the wheel, which, pressing on the same vane, also throws the globules themselves back to the opposite side. That so slight a force can introduce a delay into so great a descending weight will be made plain a little further on, when the toothed wheels compacted into one machine are discussed; at least this is clear, that if you allow no delay, the resistance is determined not only by the heaviness of the globules and their distance from one another, but also by the proportion of the impressed impulse in the preceding impulse. Therefore the same globules, when they are moved by the driven vane, have the role of a weight attached to the peritrocheum, and the impelling power of the vane—that is, of the axle—is applied to them: on the other hand, for retarding, or for restraining for a time the motion of a weight which is understood to be applied to the vane, the same globules, by the force of the impulse impressed upon them, have the role of a power applied to the peritrocheum. But since mention has here fallen on clocks, it is clear why in those which one can carry about for perpetual use, the little chain or cord is wound around a cone, not a cylinder: namely because the steel spring, from which the motion begins, being at first contracted into a tighter spiral and exerting its elastic force, strives more vigorously to restore itself, and, drawing the chain or cord FE, drives the whole cone DEC and the toothed little wheel AB connected with it around; but when it has been unwound into somewhat looser spirals, it strives more feebly, and, drawing the chain or cord, now not near the vertex of the cone but more toward the base, it turns the same wheel AB. Therefore, since the motion of the power near the vertex of the cone has a smaller relation to the motion of wheel AB than the motion of the power in the broader part of the cone has to the same motion of the wheel (for there the shorter, here the greater...

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Liber quintus. CAPUT IV. 531 jorem gyrum perficit) ut quædam motûs æquabilitas in horologio servetur, opportunum fuit potentiæ validiùs conanti majorem opponi resistentiam, minorem verò languidiùs conanti: nam si catenula non conum, sed cylindrum circumplecteretur, eadem semper esset motuum mensura atque Ratio, sed inæquales vires elasticæ motum inæqualiter velocem ef- ficerent. Huc pariter revocandas esse Terebrarum vires vix cui- quam dubium esse potest, quarum quò ampliora sunt manubria, majores pariter esse vires constat; quandoquidem potentia ampliorem circulum describit, dum terebræ acies minimo motu ligni aut metalli particulas, in quas incurrit, abscindit. Quæcumque demum sit terebræ forma, sive ejus apex in cochleam striatam exacutam desinat, ut AB manubrium habens CD rectum, sive in aciem obliquam aut planam, aut modicè excavatam exeat ut EF, manubrium autem LHI inflexum habeat circa GI versatile (quam Zerebram Gallicam aliqui Itali appellant) sive ferrea lamina in orbem convoluta, & inferiùs denticulata, ut MN, manubrio transverso OP coaptetur: Similiter semper est momentorum Ratio desumenda aut ex transversarij CD longitudine ad crassitiem cochleæ striatæ B, aut ex distantiâ puncti H à lineâ transeunte per GIEF ad integram seu dimidiatam aciei F latitudinem (prout extremum punctum F in mediâ aut in extremâ latitudine positionem habet) aut ex manubrij OP longitudine ad diametrum circularis serræ NM: partes autem subjecti corporis abscindendæ, ut illud perforetur, habent rationem ponderis movendi eò difficiliùs, quò validiore nexu illæ invicem conjunguntur. Eadem erit philosophandi methodus in eo terebræ gene- re, cui nos Itali proximè ad Græcum vocabulum accedentes nomen fecimus. Teretis baculi BC extremitati C XXX 2

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the larger circle, so that a certain equality of motion in the clock may be preserved, it was fitting that to greater force making a stronger effort a greater resistance should be opposed, and to lesser force making a weaker effort a lesser one: for if the little chain should clasp, not a cone, but a cylinder, the measure of motions and the ratio would always be the same, but unequal elastic forces would make the motion unequally swift. To this likewise the forces of augers can scarcely, to anyone, be doubted to belong; for the more ample their handles are, the greater also the forces are known to be, since the power describes a larger circle, while the cutting edge of the auger, with the least motion, cuts off the particles of wood or metal upon which it falls. Whatever, then, be the form of the auger, whether its point ends in a sharpened threaded screw, as one having a handle AB and a straight CD, or whether it passes into an oblique edge, or a flat one, or a moderately hollowed one, as EF, but has an inflected handle LHI about a movable GI (which some Italians call a Gallic auger), or whether a strip of iron, rolled into a circle and toothed underneath, as MN, is fitted with a transverse handle OP: in like manner the ratio of the moments must always be taken either from the length of the transverse part CD to the thickness of the threaded screw B, or from the distance of point H from the line passing through GIEF to the full or half width of edge F (according as the extreme point F has its position in the middle or at the end of the width), or from the length of handle OP to the diameter of the circular saw NM: but the parts of the subject body to be cut away, so that it may be pierced, are related as the weight to be moved, all the more difficultly in proportion as they are joined to one another by a stronger bond. The same method of philosophizing will apply in that kind of auger to which we Italians, approaching most closely the Greek word, have given a name. To the extremity C of the round staff BC XXX 2

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Mechanicorum additur chalybea cuspis CD ita in punctum desinens, ut ad aliquam latitudinem obliquè ascendat pro ratione semidiametri foraminis, quod max- imum artificis animus destinavit. Inferior baculi pars infigitur sphæroidi H, & trans- versarium GI in medio E ita perforatum est, ut facillimè per insertum baculum ex- currens sursum deorsum agitari possit: cum enim extremitates G & I adnexum funiculum habeant pertingentem usque in B, hoc circa baculum contorto transversarium non procul abest à B, quod si deprimatur, explicatur funiculus, & bacu- lus in gyrum agitur pariter cum infixa cuspide; qua sensim ac leniter minimas subjectæ laminæ metallicæ particulas abraden- te, demùm sæpiùs repetitâ transversarij sursum deorsum agita- tione, atque adeò celeri terebellæ conversione, foramen patet. Quamvis autem artificis manus applicetur medio transversario in E, quod deprimit; potentia tamen intelligitur applicata su- perficiei baculi medio funiculo illum circullexo, perindeat- que si inter utramque palmam alternis motibus adductam atque reductam idem baculus convolveretur: tantóque major est po- tentiæ sic applicatæ motus, quanto excessu baculi ambitus su- perat terebellæ subjectam laminam abradentis gyrum. Quo- niam verò potentia, hoc est manus, movetur descendendo, ejus motus comparandus est cum multiplici convolutione ba- culi, quæ fit, dum explicatur funis. Sed & alia potentia hîc consideranda occurrit: adjunctum enim sphæroides H, quod mediocriter grave statuitur, non so- lùm suo pondere juvat, ut cuspis paulò pressiùs adhæreat sub- jectæ laminæ, verùm etiam concepto in convolutione impetu dum explicatur funis, pergit ad easdem partes moveri, expli- catumque funem iterum circa baculum contorquens cogit transversarium GI ascendere versùs B, quo vicissim ab artificis manu depresso in contrarias partes volvitur. Impetus igitur sphæroidi H impressus, dum illud movet, rationem habet po- tentiæ cuspidem in gyrum contorquentis, cujus momenta ex distantiâ ab axe, circa quem efficitur motus, definienda sunt. CAPUT

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To the mechanic’s device a steel point CD is added, ending thus in a point, so that it rises obliquely to some width according to the radius of the hole, which the artisan’s mind has determined as the maximum. The lower part of the rod is fixed into the spheroid H, and the crosspiece GI is bored in the middle E in such a way that, with the inserted rod passing through it, it can most easily be moved up and down: for since the extremities G and I have an attached cord reaching as far as B, this, when twisted around the rod, makes the crosspiece not far from B, and if this is pressed down, the cord is unwound, and the rod is turned in a circle together with the fixed point; and as this, gently and little by little, abrades the smallest particles of the underlying metal plate, at length, after the up-and-down motion of the crosspiece is repeated more often, and thus by the rapid turning of the drill, the hole is opened. Although, however, the artisan’s hand is applied to the middle of the crosspiece at E, which presses it down, nevertheless the force is understood to be applied to the surface of the rod by the middle cord encircling it, just as though, between either palm, brought alternately together and drawn back, the same rod were turned round; and the greater is the motion of the force thus applied, by however much the circumference of the rod exceeds the circle of the drill abrading the plate placed beneath. But since the force, that is, the hand, is moved by descending, its motion must be compared with the multiple turning of the rod, which takes place while the cord is unwound. But another force also here presents itself for consideration: for the attached spheroid H, being set as moderately heavy, not only by its own weight helps so that the point adheres somewhat more firmly to the underlying plate, but also, having acquired impetus in the turning motion while the cord is being unwound, continues to move toward the same parts, and by twisting the unwound cord again around the rod it compels the crosspiece GI to rise toward B, which in turn, being pressed down by the artisan’s hand, is turned in contrary directions. The impetus therefore impressed upon the spheroid H, while it moves it, has the character of the force that turns the point in a circle, whose moments must be determined from the distance from the axis around which the motion is effected. CHAPTER

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Liber quintus. CAPUT V. 533 CAPUT V. Axium in suis peritrochiis compositione vires augentur. Contingere potest, & quidem non rarò, ad servandam pe- ritrochij & axis cum pondere & potentiâ analogiam, tam ingens tympanum aut manubrium Axi coaptandum esse, ut aut loci angustiæ commodè illud non patiantur, aut non nisi majore dispendio, quàm sit operæ pretium, tam ampla ma- china parari, aut congruè disponi queat. Quid enim, si specta- tâ potentiæ validioris virtute centuplum onus sustollemadum proponatur? an crassiori Axi, qui satis firmus sit, rotam aut tympanum cujus diameter centupla sit diametri Axis, adjun- gemus? quanto id incommodo futurum esset, quantâque sub- sidia comparare oporteret, ut tam immanis machina citrà luxa- tionem subsisteret, & congruo pegmati inniteretur, nemo non videt. Satius itaque fuerit, insistendo iis, quæ lib.2 cap.7. dicta sunt, machinam, quam ad centuplam altitudinem augere in- commodum accideret, componere, pluribus Axibus cum suis peritrochiis invicem ritè coaptatis. Statuendus primùm est Axis, cujus soliditas oneris gravi- tati sustinendæ respondeat, longitudo circumflexas ducta- rij funis spiras commodè capiat. Deinde tympanum eligatur, cujus diameter ad constituti cylindri diametrum eam habeat, quæ placuerit, Rationem, modò illa sit majoris inæqualitatis, ut manifestum est: sit ex. gr. quintupla, & cylindri crassities palmaris ponatur. Sed quoniam proposito ponderi attollendo impar est potentia applicata machinæ resistentiam gravitatis in Ratione solùm quintuplâ extenuanti, alium adhibe Axem suo Peritrochio infixum (vel quem fors tulerit ante in alios usus paratum, vel secundùm destinatam Rationem elaboratum) cu- jus ductarius funis ipsi quidem Axi congruo loco conjungatur, sed tympanum prioris Axis circumplectatur, ut in convolutio- XXX 3

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Book five. CHAPTER V. 533 CHAPTER V. The powers of axes are increased by their peritrochia by composition. It may happen, and indeed not rarely, that in order to preserve the proportion of the peritrochium and the axis to the weight and the power, so huge a drum or handle must be fitted to the Axis, that either the narrowness of the place does not conveniently allow it, or such a large machine cannot be prepared or suitably arranged except at greater expense than the work is worth. For what if, regard being had to the force of greater power, there were proposed the raising of a burden a hundredfold? Shall we attach to a thicker Axis, which is sufficiently strong, a wheel or drum whose diameter is a hundred times the diameter of the Axis? Everyone sees how inconvenient that would be, and how many supports would have to be provided, so that so enormous a machine might stand without dislocation and rest upon a suitable frame. It will therefore be better, adhering to what was said in Book 2, chapter 7, to construct, by joining together several Axes with their peritrochia in due order, a machine which it would be troublesome to make increase to a hundredfold height. First, an Axis must be set up whose solidity corresponds to the heaviness of the burden to be sustained, and whose length conveniently receives the coils of the circumvolving carrying rope. Then let a drum be chosen whose diameter bears to the diameter of the established cylinder whatever ratio may be desired, provided that ratio be one of greater inequality, as is clear; let it be, for example, fivefold, and let the thickness of the cylinder be set at a palm’s breadth. But since the applied power is unequal to the weight proposed to be raised, the machine’s resistance of gravity being diminished only in the fivefold ratio, add another Axis fixed with its own Peritrochium (either one perhaps already prepared beforehand for other uses, or worked up according to the intended ratio), whose carrying rope is indeed joined to that Axis at a suitable place, but is wound about the drum of the former Axis, so that in the winding-up... XXX 3

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Mechanicorum 534 ne secundi Axis evolutus tympano illi motum conciliet, adeô- que etiam ponderi. Duas igitur Rationes, quas singula Peritro- chia ad suos Axes habent, compone, ut potentiæ secundo Peri- trochio applicatæ momenta innotescant. Sit Ratio hæc poste- rior ex. gr. quadrupla; & Ratio, quæ ex quadrupla & quintupla componitur, est vigecupla, quæ adhuc minor est, quàm opor- teat. Quare, cùm ex Ratione centupla Ratio vigecupla subducta relinquat Rationem quintuplam, tertium Axem cum manubrio quintuplae longitudinis ad ejusdem Axis semidiametrum simili- ter appone, & erit ex his tribus Rationibus composita Ratio centupla quæsita. Quia enim potentia manubrio huic applicata movetur quintuplo velociùs, quàm punctum, cui illa in secundi Axis tympano applicaretur, hoc verò quadruplo velociùs, quàm simile punctum in tympano prioris Axis, potentia movetur vi- gecuplo velociùs, quàm si applicaretur tympano prioris Axis: huic autem tympano applicata moveretur quintuplo velociùs quàm pondus: igitur manubrium illud versans potentia move- tur centuplo velociùs quàm pondus: id quod fieri oportebat, ut proposita gravitas in altum attolleretur. Placeat jam triplicem hunc Axem cum unico illo comparare, qui solus adhiberetur, si machina simplex esset, & non compo- sita: ille siquidem si palmaris diametri esset, adjunctum tympa- num haberet altitudinis palmorum centum; in quo construen- do quàm multâ materiâ opus esset, quantoque artificio compin- genda, ne suâ mole labefactata dissolveretur? Triplex autem hic Axis cum suis duobus tympanis, & manubrio (præterquam quod multipliciter disponi potest pro loci opportunitate, & potentiæ moventis commodo) non solùm ad altitudinem palmoru[m] viginti nô assurgeret, sed longè infra illâ subsisteret, à quocu[m]que artifice nullo negotio construeretur, ab alio in aliu[m] locu[m] facillimè trâsfer- retur, levique dispedio pararetur, ut cuique cõsiderati obviu[m] est. Hoc autem, quod in tribus Axibus explicatum est, de pluribus etiam dictu[m] intelligatur: nam si Rationes singulæ peritrochij ad suum axem considerentur, & simul componantur multiplicando invicem omnes Rationum terminos Antecedentes, item omnes Consequentes, ut habeatur novus Antecedens & novus Conse- quens, apparebit Ratio motûs potentiæ ad motu[m] ponderis, adeó- que gravitatis ponderis ad virtutem potentiæ. Ex quo patet quos- cumque

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Mechanics 534 so that the development of the second axis may impart motion to that drum, and thus also to the weight. Therefore combine the two Ratios which the individual wheel-and-axles have with their axes, so that the moments of the power applied to the second wheel-and-axle may be known. Let this latter Ratio, for example, be quadruple; and the Ratio composed of the quadruple and the quintupl e is twentyfold, which is still less than is required. Wherefore, since from the hundredfold Ratio the twentyfold Ratio being subtracted leaves the quintupl e Ratio, place a third axis with a handle of quintupl e length relative to the semidiameter of that axis in like manner, and there will be, from these three Ratios combined, the hundredfold Ratio sought. For since the power applied to this handle is moved five times faster than the point to which it would be applied on the drum of the second axis, and this again four times faster than a similar point on the drum of the first axis, the power is moved twenty times faster than if it were applied to the drum of the first axis: but applied to this drum it would be moved five times faster than the weight; therefore that power turning the handle is moved one hundred times faster than the weight: which is what had to happen, in order that the proposed load might be raised upward. Let us now compare this triple axis with that single one which alone would be used if the machine were simple and not compound: for if that single axis had a palm's diameter, it would have an attached drum one hundred palms in height; in constructing which how much material would be needed, and with how much skill would it have to be joined together, lest, weakened by its own bulk, it should come apart? But this triple axis with its two drums and handle (besides the fact that it can be arranged in many ways according to the convenience of the place and the advantage of the moving power) would not only not rise to a height of twenty palms, but would remain far below that, and could be built without difficulty by any craftsman, could very easily be transferred from one place to another by another, and could be prepared with little expense, as is evident to anyone who considers it. But let what has been explained in the case of three axes also be understood of more: for if the individual Ratios of the wheel-and-axles are considered with their own axes, and then combined together by multiplying all the antecedent terms of the Ratios by one another, and likewise all the consequent terms, so that a new Antecedent and a new Consequent are obtained, there will appear the Ratio of the motion of the power to the motion of the weight, and thus the gravitating force of the weight to the power of the power. From this it is evident whichsoever

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Liber quintus. CAPUT V. 535 cumque Axes oblatos utiles esse posse, modò Peritrochiorum ad suos Axes Ratio innotescat, sive similes sint, sive dissimiles Rationes, sive multiplices, sive superparticulares, sive superpartientes: demum enim, si quid desit ad quæsitam Rationem, addi potest certus Axis cum manubrio ita, ut Ratio quæsita expleatur. Sint quinque Axes in suis Peritrochiis omnino similes, & singuli contineant Ratione decupla: quinque Rationes 10 ad 1 invicem ducantur, & erit motus Potetiæ ad motu ponderis, ut 100000 ad 1; ac propterea quo conatu Potentia solitaria moveret talentum, ac machinâ compositâ movebit centum millia talentorum. Sint item quinque Rationes, 10 ad 1, 20 ad 7, 8 ad 3, 9 ad 2, 4 ad 1 (quocumque ordine inter se disponantur) omnes Antecedentes invicem ducti faciunt novum Antecedentem 57600, omnes autem Consequentes invicem ducti dant novum Consequente 42; quare Ratio Composita est 57600 ad 42, hoc est 9600 ad 7: & potentia valens attollere libras 7, hac machinâ compositâ attollet libras 9600. Quod si oporteret moveri libras decies mille ab hac eade Potentiâ, auferatur Ratio 9600 ad 7 ex Ratione 10000 ad 7, & relinquitur Ratio 700 ad 672, hoc est 25 ad 24: quare addendus esset sextus Axis cum manubrio, cujus longitudo ad Axis semidiametrum esset ut 25 ad 24, & potentia eadem hujusmodi manubrio applicata attollere posset libras 10000. Illud habere videtur incommodi hæc Axium compositio, quod magnam vim funium tympana circumplectentium exigit, qui scilicet singulorum tympanorum motui respondeant. Cum enim tympani A diameter sit quintupla Axis B C ex hypothesi, ejus motus est quintuplo major motu ponderis P, ac proinde funis, qui tympani limbum complectitur, quintuplo longior esse debet fune P D, hoc est motu ponderis; cujus funis tympano A circumducti caput cum Axe E F connectitur, circa

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Book Five. Chapter V. 535 and indeed the axes offered may be useful, provided that the ratio of the peritrochia to their axes is known, whether the ratios are similar or dissimilar, whether multiple, submultiple, or part ratios: for finally, if anything is lacking to the desired ratio, a certain axis with a handle can be added, so that the desired ratio may be completed. Let there be five axes, all entirely similar in their peritrochia, and let each contain a ratio of ten to one: if five ratios of 10 to 1 are multiplied together, there will be a motion of power to the motion of the weight as 100000 to 1; and therefore, by the effort with which a solitary power would move one talent, with the compounded machine it will move one hundred thousand talents. Let there also be five ratios, 10 to 1, 20 to 7, 8 to 3, 9 to 2, 4 to 1 (in whatever order they are arranged among themselves); all the antecedents multiplied together make a new antecedent, 57600, and all the consequents multiplied together give a new consequent, 42; wherefore the compounded ratio is 57600 to 42, that is, 9600 to 7: and a power capable of lifting 7 pounds, by this compounded machine will lift 9600 pounds. But if it were necessary to move ten thousand pounds by this same power, let the ratio 9600 to 7 be subtracted from the ratio 10000 to 7, and there remains the ratio 700 to 672, that is, 25 to 24: wherefore a sixth axis with a handle would need to be added, whose length to the semidiameter of the axis would be as 25 to 24, and the same power applied to such a handle could lift 10000 pounds. This composition of axes seems to have the inconvenience that it requires a great supply of ropes encircling the drums, namely, ropes that correspond to the motion of each of the drums. For since the diameter of drum A is five times that of axis B C, by hypothesis, its motion is five times greater than the motion of the weight P, and therefore the rope that encompasses the rim of the drum must be five times longer than the rope P D, that is, by the motion of the weight; the end of which rope, drawn around the drum A, is connected with the axis E F, around

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Mechanicorum 536 quem in motu convolvitur. Quoniam verò tympanum O ex hypothesi diametrum habet quadruplam diametri sui Axis E F ejusque motus est ad motum sui Axis quadruplus, funis circum- plicatus tympano O quadruplus est funis, qui circa Axem E F convolvitur, ac propterea etiam vigecuplus funis D P; adeo ut, ubi totus funis evolutus fuerit, atque circa axem H G convolu- tus, pondus sublatum usque in D intelligatur. Ex quo fit poten- tiam manubrio I L applicatam, quia I G longitudo est quintu- pla semidiametri Axis G H, adhuc quintuplo velociùs moveri quàm tympanum O, cujus motum metitur evolutio funis illud circumplectentis; atque idcirco Potentia in I movetur centu- plo velociùs quàm pondus P. Quare si adhuc quartum Axem ad- dere oporteret, & loco manubrij G I tympanum suo fune in- structum apponeretur, funis ille esset ipsius P D centuplus; at- que ita deinceps pro tympanorum & Axium multiplicatione juxta singulorum Rationem augeretur funium longitudo. Verùm pro tantâ funium longitudine non est tympanorum limbo enormis amplitudo tribuenda, ut eos capiat; quia scilicet quò longiores exiguntur hujusmodi funes, eò etiam tenuiores atque exiliores esse possunt: Si enim funis D P oneri attollendo par constat funiculis contortis invicem ex. gr. centum, funis qui tympanum A complectitur, non nisi quintam ponderis par- tem resistentem habet, hoc est ipsum pondus P subquintuplo minore resistentiâ repugnans potentiæ per tympanum A reti- nenti: ac proinde si funium firmitatem funiculorum numerus metitur, satis validus erit funis constans ex funiculis viginti. Si- militer funis complectens tympanum O, quia pondus resisten- tiam subvigecuplo minorem habet, satis firmus censebitur, si ex quinque funiculis invicem contortis conflctur. Quod si justo tenuiores timeas hujusmodi funes, licebit adhuc paulò crassio- res adhibere. Illud certè manifestum est multo minores suffi- cere, quàm sit funis D P. Ne autem tympanis limbi amplitudinem funis circumducti capacem temerè constituas, singulorum funium crassitudo con- sideranda est, ut eorum diameter innotescat, & longitudinis ra- tione habitâ spirarum numerus inveniatur, per quem ducta fu- nis diameter dabit necessariam limbi amplitudinem; quæ si justo minor esset primum spirarum ordinem secundus ordo cir- cumplectere

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Mechanics 536 which turns in motion. Since, however, the drum O by hypothesis has a diameter four times the diameter of its axis E F, and its motion is four times the motion of its axis, the rope wound around drum O is four times the rope that is wound around axis E F, and therefore also twenty times the rope D P; so that, when the whole rope has been unwound and is understood to have been wound around the axis H G, the weight is raised as far as D. From this it follows that the power applied to the handle I L, because I G is five times the semidiameter of axis G H, is still moved five times faster than drum O, whose motion is measured by the unwinding of the rope encircling it; and therefore the Power at I is moved a hundred times faster than the weight P. Therefore, if a fourth axis were still to be added, and in place of the handle G I a drum furnished with its rope were set there, that rope would be a hundred times that of P D; and thus thereafter, in proportion to the multiplication of the drums and axes according to the ratio of each, the length of the ropes would be increased. But for such a length of rope, an excessive breadth must not be assigned to the rim of the drums in order to contain them; because, namely, the longer such ropes are required to be, the thinner and finer they may also be: for if the rope D P is equal to lifting a load and is made up of, for example, one hundred twisted cords, the rope that encircles drum A has resisting power equal to only a fifth part of the weight, that is, the very weight P is opposed to the retaining power by the drum A with a resistance five times smaller; and therefore, if the strength of ropes is measured by the number of cords, a rope consisting of twenty cords will be sufficiently strong. Similarly, the rope encircling drum O, because the weight has a resistance twenty times smaller, will be judged sufficiently strong if it is made from five cords twisted together. But if you fear that such ropes are too thin, it will still be permissible to use somewhat thicker ones. It is certainly clear that much smaller ones suffice than the rope D P. Lest you rashly set the rim breadth of the drums capable of the wound rope, the thickness of each rope must be considered, so that its diameter may be known, and, regard being had to the length, the number of coils may be found, by which, when multiplied by the rope’s diameter, the necessary breadth of the rim will be obtained; and if this were just a little too small, the second row would encompass the first row of coils

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Liber quintus. CAPUT V. 537 cumplecteretur: & quamvis initio hinc major aliqua movendi facilitas oriretur (auctâ scilicet peritrochij diametro) tamen evoluto secundo hoc spirarum ordine diameter peritrochij di- minuta majorem crearet movendi difficultatem; maxime si cir- ca Axem convolutus funis spirarum ordinem pariter geminaret, atque adeò Axis diametrum augeret. Funes itaque quasi cylin- dri considerandi sunt, quorum crassitudinis diametri sunt in subduplicatâ basium Ratione; ac propterea inter ipsas crassitu- dines numeris definitas inveniendus est numerus medio loco proportionalis, & hic indicabit tenuioris funis diametrum, quemadmodum primus numerus major ponitur pro diametro crassioris funis. Sic quoniam funis DP est ex hypothesi ut 100, & funis circa tympanum A subquintuplæ crassitudinis est ut 20, inveniatur inter 100 & 20 medius 44 72/100 proximè, & dia- metri funium erunt proximè in Ratione 100 ad 45. Quare quam amplitudinem requirunt 45 spiræ maximi funis DP, eandem exigunt 100 spiræ minoris funis attributi tympano A, si uterque funis circa eundem cylindrum convolvatur. Sed quia perimeter tympani A quinquies continet perimetrum Axis BC, unica tympani spira quinque Axis spiris æquatur se- cundùm longitudinem, & centum spiræ tympani quingentis Axis spiris respondent, si lineæ longitudo spectetur: satis au- tem est, si longitudinem spirarum 225 circa Axem, ille funis tympani obtineat, quia longitudo illa est ad funis DP longitu- dinem 45 quintupla. Propterea tympani limbus minorem exi- git amplitudinem, quàm sit spatium, quod in Axe BC occupa- tur à convoluto fune DP: nimirum à limbo contineri oportet sui funis circumplicati spiras 45; satis igitur fuerit dimidiata amplitudo. Simili methodo tympano O limbi amplitudinem definies: quoniam enim funis crassitudo ad crassitudinem funis DP est subvigecupla, inter 100 & 5 medio loco proportionalis 22 36/100 proximè inveniatur; & est diameter funis tympani O ad dia- metrum funis DP ut 22 36/100 ad 100. Quare si circa eundem cy- lindrum uterque funis convolveretur, quod spatium spiras fu- nis DP 45 contineret, tenuioris hujus funis spiras saltem 201 comprehenderet. Ponamus tympani O perimetrum esse ad pe- Y yy

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Book Five. Chapter V. 537 embracing it: and although at first some greater ease of motion would arise from this (namely from the increased diameter of the peritrochium), yet when this second order of spirals is unwound, the diminished diameter of the peritrochium would create greater difficulty of motion; especially if the rope, wound about the Axis, were to double the order of spirals as well, and thus increase the diameter of the Axis. Ropes, therefore, are to be considered as cylinders, whose diameters of thickness are in the subduplicate ratio of the bases; and therefore between the thicknesses themselves, defined by numbers, the number proportional in the middle place must be found, and this will indicate the diameter of the thinner rope, just as the first, larger number is taken as the diameter of the thicker rope. Thus, since rope DP is by hypothesis as 100, and the rope around drum A is of one-fifth the thickness, as 20, let the mean proportional between 100 and 20 be found, namely 44 72/100 approximately, and the diameters of the ropes will be approximately in the ratio of 100 to 45. Wherefore whatever extent 45 coils of the largest rope DP require, the same is required by 100 coils of the smaller rope assigned to drum A, if both ropes are wound around the same cylinder. But since the perimeter of drum A contains the perimeter of Axis BC five times, one coil of the drum is equal in length to five coils of the Axis, and 100 coils of the drum correspond to 500 coils of the Axis, if the length of the line be considered: yet it is enough if the length of the coils around the Axis, namely 225, be obtained by that rope of the drum, because that length is to the length 45 of rope DP as five to one. Therefore the rim of the drum requires a smaller extent than the space occupied on Axis BC by the rope DP wound about it: namely, the rim ought to contain 45 coils of its wound rope; thus half the extent will be sufficient. By a similar method you will determine the extent of the rim for drum O: for since the thickness of its rope is to the thickness of rope DP as one-twentieth, let the mean proportional between 100 and 5 be found, namely 22 36/100 approximately; and the diameter of the rope of drum O is to the diameter of rope DP as 22 36/100 to 100. Wherefore if both ropes were wound around the same cylinder, the space that would contain 45 coils of rope DP would contain at least 201 coils of this thinner rope. Let us suppose the perimeter of drum O to be to pe- Y yy

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Mechanicorum 538 rimetrum Axis BC ut 4 ad 1: igitur limbus tympani O si eandem habeat amplitudinem, quam funis DP occupat in suo Axe BC, capiet sui funis spiras 201, quæ in unam longitudinem extensæ constituunt longitudinem, quæ ad longitudinem spirarum 45 funis DP est ut 804 ad 45. Sed quia longitudo illius funis est vigecupla longitudinis funis DP, debet esse ut 900 ad 45; ideò adhuc majorem exigit amplitudinem, ut adhuc spiras 24 aut 25 supra ducentas obtineat. Quod si Axis E F subtilior sit quàm Axis BC, & tympani O diameter ad sui axis E F diametrum quadrupla sit, jam tympani perimeter ad perimetrum Axis BC habebit minorem Rationem quàm 4 ad 1, ac proinde ejus limbum adhuc ampliorem constitui necesse est. Quare si Axis BC diameter sit palmaris, spiræ 45 funis DP convoluti elevabunt pondus P ad altitudinem palmorum circiter 141, quanta nimirum esset ipsius funis convoluti longitudo: funis circa tympanum A longitudo esset palmorum saltem 705, & funis circa tympanum O longitudo palmorum 2820. Hinc quamvis præter primum Axem BC oneri sustinendo parem, reliqui consequentes Axes E F, & GH, & si qui alij adhuc sint, possint in minorem soliditatem extenuari; si ponderis resistentia attendatur, quia tamen, quò subtiliores sunt, frequentioribus etiam spiris circumplicantur, ex quo fit ut plures spirarum ordines fiant, adeoque Axis diameter aucta minuat momentorum Rationem; præstat exiles Axes non studiosè quærere, nisi fortè necessitas aliqua iis uti suadeat. Quamquam & huic incommodo occurri potest, si, quemadmodum in Ergatæ usu funem paucis aliquot spiris circumductum, dum in conversione evolvitur, puer agglomerat, ita etiam hîc funem tympanorum A & O quatuor aut quinque spiris circa Axes E & H convolutum puer colligeret: hoc enim pacto non contingeret, ut primum spirarum ordinem alter spirarum ordo superinductus circumplecteretur. Porrò ne tantam funium vim comparare cogamur, & ampliore limbo tympana circumscribere, haud sanè ineptum censerem, si pro eorum more, qui novaculas obtusas acuunt, ut aliàs innui, funis in sese rediens unâ aut altera (aut etiam triplici, si opus fuerit) spirâ tùm Axem, tum subjectum tympanum arctè complecteretur: sic enim fieret, ut Axe convoluto etiam sub- jectum

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Mechanics 538 the ratio of Axis BC as 4 to 1: therefore if the rim of the drum O has the same width as the rope DP occupies on its Axis BC, it will take up 201 coils of its rope, which, stretched out into one length, make a length that is to the length of the 45 coils of the rope DP as 804 to 45. But because the length of that rope is twenty times the length of the rope DP, it ought to be as 900 to 45; therefore it requires an even greater width, so as to obtain still 24 or 25 coils above two hundred. But if Axis E F be thinner than Axis BC, and the diameter of drum O to the diameter of its axis E F be quadruple, then already the perimeter of the drum in relation to the perimeter of Axis BC will have a smaller ratio than 4 to 1, and therefore it is necessary that its rim be made still broader. Wherefore if the diameter of Axis BC be one palm, the 45 coils of the wound rope DP will raise the weight P to a height of about 141 palms, that is, about the length the wound rope itself would have: the length of rope around drum A would be at least 705 palms, and the length of rope around drum O 2820 palms. Hence although, besides the first Axis BC, which is equal to sustaining the load, the remaining subsequent Axes E F, and GH, and if there are any others still, may be made thinner in solid substance; yet if the resistance of the weight be considered, since the thinner they are, the more frequently are they also wound with coils, from which it follows that more ranks of coils are formed, and thus the increased diameter of the Axis lessens the ratio of moments; it is better not to seek slender Axes too eagerly, unless perhaps some necessity suggests using them. Although this inconvenience too can be met, if, as in the use of the Ergata, a rope wound with a few coils, while it is unwound in turning, is gathered up by a boy, so here also a boy would gather the rope of drums A and O wound with four or five coils around Axes E and H: for in this way it would not happen that a second order of coils, superimposed, would wrap around the first order of coils. Moreover, lest we be compelled to procure so great a quantity of ropes, and to mark out the drums with a broader rim, I should certainly not think it improper, if, in the manner of those who sharpen blunt razors, as I have elsewhere indicated, the rope, returning into itself, with one or two coils (or even three, if need be), should tightly clasp both the Axis and the drum beneath it: for in this way it would come about that, the Axis being wound, the subject drum also

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Liber quintus. CAPUT V. 539 jectum tympanum volveretur; idemque funis perpetuo ordine à tympano in proximum Axem & ab Axe in tympanum suc- cedens quantolibet motui perficiendo sufficeret. Et ut omne periculum submoveatur, ne funis excurrat, satius est tùm Axis, tùm tympani ambitum non in cylindricam superficiem expoli- re, sed angulis asperum esse. Quod si aliquando languidior fu- nis non adeò pressè complecteretur Axem & tympanum, spongiam aquâ imbutam ipsi funi admove, & intentus fiet. Demùm in hujusmodi Axium compositione non sine animadversione prætereundæ videntur mutua Axium positio, atque distantia, qua secundus Axis à primo tympano abest. Sit Axis A in Peritrochio CDE, at- que ex fune perpendiculari BG dependeat onus, & GB Tangens cum Radio AB constituat angu- lum rectum ABG. Producatur recta AB usque ad tympani peri- pheriam in C, & sit ad angulum rectum Tangens CH, cui ad- nexa intelligatur potentia per axem S trahens, atque tympa- num CDE convertens; ex cu- jus conversione convolvitur Axis AB versus I, & pondus ascendit. Verùm secundus Axis S cum pri- mo tympano comparatus non hanc solùm positionem obtinere potest, ut superior sit, sed etiam constitui potest ad latus ita, ut funis ductarius cadens in hori- zontem ad perpendiculum sit KL, Tangens verò HC sit ho- rizonti parallela; aut ita disponi possunt, ut Axis A superiore loco, Axis S inferiore loco statuatur, & funis pondus retinens sit MN, cui parallelus sit funis CH. Quamcumque ex his tribus positionem habeat Axis secundus S, sivè superior, sivè ad latus, sive inferior sit (modò linea CH vel parallela sit li- neis ductarij funis BG aut MN, vel parallela lineæ AK jun- genti centrum Axis cum puncto cont. tûs perpendiculi KL) Y yy 2

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Liber quintus. CHAPTER V. 539 the suspended drum should be turned; and likewise that the same rope, proceeding in perpetual order from the drum to the next axle, and from the axle to the drum, should suffice for accomplishing whatever motion is desired. And that all danger may be removed, lest the rope slip off, it is better both for the axle and for the circumference of the drum not to be polished into a cylindrical surface, but to be rough with angles. But if at any time the slack rope does not clasp the axle and the drum closely enough, apply a sponge soaked with water to the rope itself, and it will become tight. Finally, in the composition of axles of this kind, the mutual position of the axles, and the distance by which the second axle is removed from the first drum, do not seem to be passed over without attention. Let axle A be in the peritrochium CDE, and let a weight depend from the vertical rope BG, and let GB, as tangent with radius AB, form a right angle ABG. Let the straight line AB be produced as far as C on the periphery of the drum, and let CH, tangent at right angles, be taken, to which the power drawing through axle S, and turning the drum CDE, is understood to be attached; from whose rotation the axle AB is wound toward I, and the weight rises. But the second axle S, compared with the first drum, can not only have this position, namely that it be above, but it can also be placed at the side, so that the guide rope falling to the horizon is KL, while the tangent HC is parallel to the horizon; or they can be arranged so that the axle A is set in a higher place, the axle S in a lower place, and the rope retaining the weight is MN, to which the rope CH is parallel. Whichever of these three positions the second axle S has, whether it be above, at the side, or below (provided that line CH is either parallel to the guide lines BG or MN, or parallel to line AK joining the center of the axle with the point of contact of the perpendicular KL) Y yy 2

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540 Mechanicorum eadem habere videtur momenta; quia punctum C, cui appli- cata intelligitur potentia, juxta potentiæ directionem simili Ratione accedit versus potentiam comparatè ad ascensum puncti Axis, cui applicatur pondus, ac est Ratio semidiametri tympani ad semidiametrum Axis. Concipiamus enim in con- volutione punctum C venire in F, punctum verò B in I: punctum igitur C sequens potentiæ directionem accedit versus potentiam juxta mensuram Sinûs arcûs C F, hoc est OF, quemadmodum punctum B contrà directionem gravitatis pon- deris ascendit juxta mensuram Sinûs arcûs B I: sunt autem hi sinus arcuum similium similiter positorum in Ratione Radio- rum A C ad A B. Atqui sive in K, sive in M intelligatur pon- dus, ascensus illius est æqualis ascensui B I; ergo ad illos, ut po- te huic æquales, accessus puncti C ad potentiam, qui est OF, eandem habet Rationem, quæ est Radij A C ad Radium A B. At verò si Axis secundus sit T, potentia non intelligitur ap- plicata tympano in C, sed in F, ubi circulum tangit recta TF; nec ejus directio FT est parallela directioni ponderis BG, sed obliqua, adeò ut quamvis F veniat in Q per arcum æqualem arcui CF, quia tamen non est similiter positus, punctum F se- quens directionem potentiæ accedit versus potentiam accessu, quem metitur RP; est autem RP minor quàm AR, hoc est OF, ut ex doctrinâ Sinuum constat; igitur accessus RP ad ascensum BI habet minorem Rationem, quàm accessus OF ad eundem ascensum BI. Potentia igitur volvens Axem T in li- neâ TF obliquâ minora habet momenta, quàm in parallelâ HC. Similiter si Axis fuerit V propior quàm Axis T; linea VD tangit circulum in D puncto remotiore quàm F, à puncto C; ac propterea datâ arcûs æqualitate adhuc minor est accessus in D quàm in F, multóque minor quàm in C, & idcircò mi- norem habet trahendi facilitatem. Quare quò propior est Axis secundus, si tractio sit obliqua, ut TF & V D, plus laboris re- quiritur in movendo. Neque hoc mihi inconsiderantiæ tribuas, quod assumpserim arcus CF & BI perinde atque si idem esset motus, ac quando funis HC esset firmiter alligatus in C, & ejus caput veniret ex C in F; cum tamen alia semper atque alia pars funis aliis sub- inde peripheriæ tympani partibus respondeat, in quibus sit ad angulum

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540 Mechanics seems to have the same moments; because point C, to which the power is understood to be applied, in the direction of the power advances similarly toward the power, relatively to the ascent of point Axis, to which the weight is applied, and this is in the ratio of the semidiameter of the drum to the semidiameter of the Axis. For let us conceive that in the winding point C comes to F, and point B to I: therefore point C, following the direction of the power, advances toward the power by the measure of the sine of arc C F, that is, OF, just as point B, contrary to the direction of the weight’s gravity, ascends by the measure of the sine of arc B I: but these sines of similarly placed similar arcs are in the ratio of the radii A C to A B. Moreover, whether the weight be understood to be in K or in M, its ascent is equal to the ascent B I; therefore, to those, as to things equal to this, the advance of point C toward the power, which is OF, has the same ratio as the radius A C has to radius A B. But if the second Axis be T, the power is not understood to be applied to the drum at C, but at F, where the straight line T F touches the circle; nor is its direction F T parallel to the direction of the weight B G, but oblique, so that although F comes to Q by an arc equal to arc C F, yet because it is not similarly placed, point F, following the direction of the power, advances toward the power by the advance measured by R P; but R P is less than A R, that is, OF, as is clear from the doctrine of sines; therefore the advance R P to the ascent B I has a smaller ratio than the advance OF to the same ascent B I. The power, therefore, turning Axis T along the oblique line T F, has smaller moments than in the parallel H C. Similarly, if the Axis were V, closer than Axis T; line V D touches the circle at point D, farther from point C than F is; and therefore, given equal arcs, the advance at D is still less than at F, and much less than at C, and for that reason it has less facility of drawing. Wherefore, the nearer the second Axis is, if the traction be oblique, as T F and V D, the more labor is required in moving. Nor should you attribute this to inconsiderateness in me, because I have assumed arcs C F and B I as though it were the same motion, as when the cord H C were firmly tied at C, and its end came from C to F; although in fact another and another part of the cord always correspond to the successive parts of the drum’s circumference, in which there be at an angle

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Liber quintus. CAPUT V. 541 angulum rectum cum diametro contactus, dum ille evolvitur. Eatenus enim notabilem arcum assumpsi, quatenus ob oculos ponenda erat momentorum Ratio: Cæterum satis scio non adeò notabiles arcus, ut CF & BI, considerandos esse, sed eorum particulam minimam, sive centesimam dicas, sive millesimam aut decies millesimam: eadem scilicet erit Ratio Sinuum, qui respondent minimis arcubus similibus ac similiter positis, qui nimirum incipiunt à C & B, atque Sinuum respondentium majoribus arcubus similibus ab iisdem punctis C & B incipientibus. Id quod pariter de punctis F & D comparatis cum puncto B, aut K, aut M dicendum: Nam quæ inito semel mo- tu intercedit momentorum Ratio inter potentiam & pondus ratione positionis, eadem in toto motu perseverat. At si funis non evolveretur, sed puncto C esset firmiter colligatus, in tractione ex C in F subinde mutarentur Potentiæ momenta, fieréntque semper minora, adeò ut demum perirent, & nulla essent, ubi in rectam lineam coalescererent Radius A C & fu- nis H C. Hæc quæ de secundo Axe funem primo tympano circum- ductum evolvente dicta sunt, facilè traduci possunt etiam ad funem in sese redeuntem, cujusmodi esset funis F T E F, aut D V X D: nisi enim funis contingat tympanum in puncto se- midiametri transeuntis per contactum Axis & funis ductarij (hoc est in C extremitate semidiametri A C transeuntis per B, aut M) ita ut sit funi ductario parallelus, aut in puncto semi- diametri parallelæ funi ductario KL, consultius erit, cæteris paribus, axem secundum esse remotum ut T, quàm proximum ut V: in proximo enim lineæ D V & X V productæ coirent in angulum majorem, quàm lineæ F T & E T, ac propterea comprehensus arcus DX minor est arcu FE. Cæteris, in- quam, paribus; si videlicet in eadem rectâ lineâ intelligan- tur trium Axium centra A, V, T; nam si in lineâ eadem jun- gente centra AT non esset V, sed recederet ita, ut funis tym- panum contingens minùs obliquus esset, sed magis accederet ad parallelismum cum lineâ BG, aut cum Radio Axis AK, quamvis Axis V propior esset, quàm Axis T, plus tamen haberet momenti ratione directionis potentiæ minùs obliquè trahentis. Y yy 3

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Book five. CHAPTER V. 541 the right angle with the diameter of contact, while it is unwound. For I have assumed a notable arc only so far as the ratio of the moments had to be placed before the eyes; however, I know well enough that arcs not so notable, such as CF and BI, are to be considered, but rather their smallest part, whether you call it a hundredth, or a thousandth, or a ten-thousandth: the ratio of the sines will indeed be the same, corresponding to the smallest similar arcs similarly placed, which begin, namely, at C and B, as the ratio of the sines corresponding to the larger similar arcs beginning from the same points C and B. The same must likewise be said of the points F and D compared with the point B, or K, or M: for whatever ratio of moments, once the motion has begun, intervenes between power and weight by reason of position, the same continues throughout the whole motion. But if the rope were not unwound, but were firmly tied at the point C, then in the pull from C to F the moments of the power would continually change, and would always grow smaller, so that at last they would vanish, and there would be none, when the radius AC and the rope HC coalesced into a straight line. What has been said concerning the second axis, unwinding the rope wound around the first drum, can easily be transferred also to a rope running back upon itself, such as would be the rope FT EF, or DVXD: unless, indeed, the rope touches the drum at the point of the semidiameter passing through the contact of the axis and the guide-rope (that is, at C, the end of the semidiameter AC passing through B, or M), so that it is parallel to the guide-rope, or at the point of the semidiameter parallel to the guide-rope KL, it will be more advisable, other things being equal, for the second axis to be farther away, as T, than near, as V: for in the near case the lines DV and XV produced would meet at a larger angle than the lines FT and ET, and therefore the arc DX enclosed is smaller than the arc FE. Other things, I say, being equal; that is, if the centers A, V, T of the three axes are understood to be in the same straight line; for if in the same line joining the centers AT there were not V, but it were moved away so that the rope touching the drum would be less oblique, but would come more nearly into parallelism with the line BG, or with the radius of the axis AK, then, although the axis V were nearer than the axis T, it would nevertheless have more moment by reason of the direction of the power pulling less obliquely. Y yy 3

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542 Mechanicorum Cave autem, ne hîc in latentem quendam æquivocationis scopulum incurras, si fortè permixtim accipias ponderis eleva- tionem atque ejusdem suspensi retentionem, ne recidat; hæc enim duo opposito modo contingunt, & quæ minor funis obli- quitas causa est facilioris elevationis, eadem difficiliorem ef- ficit retentionem: nam pondus B retinetur à potentiâ C, absque eo quod potentia ullo conatu urgeat polos, quibus axis & peritrochium incumbit, ideò pondus totas suas vi- res exerit adversùs potentiam sursum directè trahentem: at verò in F aut in D potentia sursum obliquè trahens tym- panum versus centrum quodammodo urget, & quidem eò magis, quò magis obliqua est tractio; ac proinde pondus non solùm vincere debet potentiæ vires, sed etiam resisten- tiam ex illâ pressione ortam; quæ quò major est pro majo- re declinatione à parallelismo cum lineâ BG, aut Radio A K, majorem quoque potentiæ tribuit retinendi facilitatem. Hinc quando secundus Axis est in inferiore loco, & potentiæ tra- hentis directio deorsum tendit, magis premuntur poli, quia & à potentia deorsum conante, & à ponderis gravitate ur- gentur, & quidem eò magis, quò magis potentiæ deorsum trahentis directio accedit ad lineam directioni ponderis MN parallelam, aut ultra illam excurrit se quodammodo invicem decussando. An non exesa publicorum puteorum marmorea labra aliquando observasti, quæ diuturno atque frequentis- simo usu à funibus, quibus aqua hauritur, detrita sunt? Utique aquam in situâ sursum trahentis labor minor esset, cæteris paribus, si solùm situæ & aquæ gravitatem vince- re oporteret, quàm si præter hanc etiam superanda sit re- sistentia, quæ ex funis conflictu cum marmore oritur. Sed quia deinde hoc eodem conflictu efficitur, ut quando tractio alternis morulis interciditur, retentio minorem potentiæ co- natum exigat; propterea etiam trahentes facilè patiuntur resistentiam augeri, ut aliquantulo laboris compendio gau- deant, quoties placuerit quietem aliquam captare. Quamquam non negaverim rudes foeminas atque pueros hoc in opere, naturâ duce, quærere etiam in trahendâ sursum si- tulâ non leve laboris compendium: si enim rectâ, intacto pu- tei labro, funis sursum trahendus esset, id utique solâ brachio- rum

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542 Mechanics But beware lest here you fall upon some hidden reef of equivocation, if perhaps you take together the raising of a weight and the holding of the same when suspended, so that it does not fall; for these two things happen in opposite ways, and the lesser obliquity of the rope, which is the cause of easier lifting, makes the holding more difficult. For the weight B is held by the power C, without the power pressing in any attempt upon the pivots on which the axis and the peritrochium rest; therefore the weight exerts all its force against the power pulling directly upward. But in F or in D the power, pulling obliquely upward, presses the drum toward the center in a certain manner, and indeed the more so the more oblique the pull is; and therefore the weight must overcome not only the force of the power, but also the resistance arising from that pressure; and the greater this is, because of the greater deviation from parallelism with line BG, or radius A K, the greater also is the power’s ease in holding. Hence, when the second axis is in a lower position and the direction of the pulling force tends downward, the pivots are more pressed, because they are urged both by the power trying downward and by the weight of the load, and indeed the more so the more the direction of the power pulling downward approaches the line parallel to the direction of the weight MN, or runs beyond it, crossing it, as it were, one against the other. Have you not sometimes observed the worn marble rims of public wells, which have been worn away by ropes, through which water is drawn, by long and very frequent use? Certainly the labor of pulling water upward in the bucket would be less, other things being equal, if one had only to overcome the gravity of the bucket and the water, than if, in addition to this, one must also overcome the resistance arising from the rope’s contact with the marble. But since by this same contact it comes about that, when the pulling is interrupted at alternating intervals, the holding requires a smaller effort of the power, for that reason those who pull also readily tolerate the resistance being increased, so that they may enjoy a small saving of labor whenever they wish to take some rest. Although I would not deny that rustic women and boys, guided by nature, also seek in this task, when drawing a bucket upward, no slight saving of labor: for if the rope had to be pulled straight up, with the lip of the well untouched, that would surely be possible only by means of the arms

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Liber quintus. CAPUT VI. 543 rum contentione perfici posset; sed ubi funis labro innititur, non solùm contentis brachiorum musculis trahunt, sed etiam inclinato retrorsum corpore hoc efficiunt, ut ipsa corporis gravitas nonnihil conferat, quo potentiæ animalis viribus fiat additamentum. Ex quo manifestum est resistentiam il- lam ex pressione ortam & difficiliorem efficere tractionem, & faciliorem retentionem: ac proinde lapsum putarem, qui tra- hentis potentiæ momenta æstimaret ex majore retinendi fa- cultate. In his, quæ in posteriore hujus capitis parte disputata sunt de hac inæqualitate momentorum pro diversa positione axis secundi, mihi videor satis probabiliter philosophatus: verùm si ad Rationes Vectis (ut pluribus placet) revocanda esset vis Axis in Peritrochio, quamvis aliqua satis commodè explica- ri possent, ubi Vectis est rectus, non omnia tamen, ubi Vectis curvus intelligendus est, congruam patiuntur explicationem, ut cuilibet rem attentè consideranti manifestum fiet; mihi enim hîc non videtur operæ pretium in re parùm utili tempus conterere; placuit tamen id obiter innuere, ut ipse tibi persua- deas inanem esse laborem, quo quis singularum Facultatum vires ad Vectem revocare conatur. CAPUT VI. Tympanorum dentatorum usus & vires exponuntur. Quæ hactenus tympana consideravimus, fune circum- ducto atque evoluto versantur; nunc genus aliud, cujus amplissimus usus est, contemplari oportet, tympana videlicet dentata, seu Rotas dentatas, in quibus sivè fuerint simplices, sivè compositæ, aut nullo prorsus fune indigemus aut illo tan- tummodo, quo pondus proximè trahitur, aut attollitur: den- tes enim majoris atque minoris tympani, ubi plura componun- tur, se mutuâ collabellatione mordentes se vicissim urgent, pro

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could be accomplished by tension; but where the rope rests on the lip, they not only pull with the muscles of the arms strained, but also, by leaning the body backward, bring it about that the weight of the body itself contributes something, so that it becomes an addition to the strength of the animal power. From this it is clear that that resistance arising from pressure makes the pulling more difficult, and the holding easier; and therefore I would think mistaken anyone who would estimate the moments of the pulling power from the greater ease of holding. In what has been discussed in the latter part of this chapter about this inequality of moments according to the different position of the second axis, I seem to myself to have philosophized with sufficient probability; but if the force of the axis in the peritrochium were to be referred to the reasons of the lever (as is preferred by many), although some things could be explained quite conveniently where the lever is straight, nevertheless not everything, where the lever must be understood as curved, admits a suitable explanation, as will be manifest to anyone who considers the matter attentively; for here I do not think it worthwhile to waste time on a matter of little utility. Yet I thought it proper to hint at this in passing, so that you yourself may be persuaded that the labor is vain by which someone tries to refer the powers of individual faculties to the lever. CHAPTER VI. The use and force of toothed drums are explained. The drums we have considered up to this point are turned by a rope carried around and unwound from them; now another kind, of the widest use, must be contemplated, namely toothed drums, or toothed wheels, in which, whether they are simple or compound, we need either no rope at all, or only that by which the weight is drawn near or lifted; for the teeth of the larger and smaller drum, when several are combined, biting one another by mutual engagement, press upon one another, pro

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Mechanicorum 544 pro ut hoc aut illud tympanum habet originem motûs. Sit chalybea lamina AB satis solida, in alte- râ extremitate, quæ pondus respicit, mo- dicè sinuata, ut in A, & in validum un- cum recurva, ut in C; latus autem DE quasi serræ in morem sit dentibus aspe- rum. Tum rotula I paris saltem cum la- minâ crassitudinis paretur dentes habens ita in orbem dispositos, ut hi in rotulæ circa suum centrum conversione denti- bus laminæ subinde congruant: colloca- tis enim in apto loculamento rotulâ, at- que laminâ ( cujus tamen pars DC extet) adeò, ut hæc ex illius conversione liberè promoveri, illa circa suum axem, cui fir- miter infixa sit, facilè versari valeat, circumducto axis manu- brio ad latus extra loculamentum extante, urgeri poterit pon- dus, aut trahi: Nimirum si rotulæ conversio fiat ex H in I, propellitur extra loculamentum lamina, ejusque extremitas A recedens à rotulâ urget pondus obvium: contrà verò si rotula convertatur ex I in H, laminam ad se intra loculamentum re- trahit, & pondus unco C connexus ad se rapit. Hinc clariùs vides, quàm ut monendus sis, oportere in attollendo, aut pro- pellendo pondere loculamentum aut ponderi suppositum firmo solo insistere, aut ponderi objectum solido repagulo inniti; in trahendo autem pondere, quod uncus C apprehendit, opposi- tam loculamenti extremitatem valido fune retineri. Illud potiùs attentè perpendendum, quod in statuendis tùm laminæ, tùm rotulæ dentibus plurimum refert, utrum rari, an spissiores sint laminæ dentes, ac proinde utrum pauci, an plures insint ipsi rotulæ, cujus peripheria in conversione aptatur lami- næ; hæc enim juxta numerum dentium rotulæ, quibus subinde coaptatur, promovetur, & cum ipsâ pondus pari velocitate aut tarditate movetur. Præstare autem pondus tardè, potentiam velociter moveri, quid opus est iterùm inculcare? Igitur quò minor erit rotula & paucioribus dentibus instructa, eodem ma- nente manubrio, faciliùs movebitur pondus; quia ut semidia- meter rotulæ ad manubrij longitudinem, ita motus ponderis, ad

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Mechanics 544 according as this or that drum has the origin of motion. Let a steel plate AB be sufficiently solid, slightly bent at the other end, which faces the weight, as at A, and recurved into a strong hook, as at C; but let the side DE be rough with teeth, after the manner of a saw. Then let a wheel I, of at least equal thickness with the plate, be made with teeth arranged in a circle, so that, as the wheel turns about its center, these may from time to time mesh with the teeth of the plate: for when the wheel and the plate have been placed in a suitable recess (the part DC of the plate however remaining outside) so that the plate can be freely advanced by the wheel’s turning, while the wheel itself can readily turn about its axis, to which it is firmly fixed, by turning the crank of the axis projecting at the side outside the recess, the weight may be pushed or pulled: namely, if the wheel be turned from H to I, the plate is driven out of the recess, and its end A, moving away from the wheel, presses against the weight in front of it; but if the wheel be turned from I to H, it draws the plate toward itself within the recess, and carries along with it the weight connected by the hook C. Hence you see more clearly, than that you should need to be warned, that in lifting or pushing a weight, the recess or the weight placed beneath it ought to rest on firm ground, or the weight opposed to it must lean against a solid barrier; but in pulling a weight which the hook C has seized, the opposite end of the recess must be held back by a strong rope. That is rather to be carefully considered, that in determining the teeth of both the plate and the wheel, it makes a great difference whether the plate’s teeth are sparse or more closely set, and consequently whether the wheel itself contains few or many teeth, to whose circumference the plate is adapted in its turning; for the plate is advanced according to the number of the wheel’s teeth with which it is from time to time engaged, and with it the weight is moved at the same speed or slowness. But what need is there to insist again that it is better for the weight to move slowly, the power to move swiftly? Therefore, the smaller the wheel and the fewer the teeth with which it is equipped, the more easily the weight will be moved, the handle remaining the same; because as the radius of the wheel is to the length of the handle, so is the motion of the weight, to

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Liber quintus. CAPUT VI. 545 ad motum potentiæ, & reciprocè ita potentiæ vis movendi, ad pondus. Quare ulteriùs manifestum est, si majore potentiæ vir- tute opus fuerit, spectatâ ponderis movendi difficultate, posse augeri manubrium, ut majora sint potentiæ momenta. Quo- niam verò non semper in promptu est opportunum manubrium, suaderem extremum axis caput, quod manubrio inseritur, qua- dratum fieri, & longiusculum esse: loco autem vulgaris manu- brij habeatur crassioris cylindri frustulum MN, in cujus imâ basi circa centrum excavatum sit quadratum foramen S ad exci- piendum caput axis, & ipsius cylindri scapum penetrent fora- mina rotunda R, T, quibus pro opportunitate inseri possint ba- culi sive longiores, sive breviores. Porrò cylindri crassitiem nihil obesse apertè constat, si quidem sola rotulæ semidiameter attenditur ad definiendum ponderis motum comparata cum ba- culi longitudine, quatenus potentia ab axe rotulæ distat. Quod si uno eodemque tempore duo pondera in oppositas partes dispellere, aut sibi invicem propiora fieri oporteat, simi- lem alteram laminam priori parallelam in eodem plano sed con- trario modo positam (ut scilicet extremitas similis ipsi A CD respiciat prioris laminæ extremitatem B) in oppositâ rotulæ parte colloca ad I, ut pariter laminæ dentes rotulæ dentibus im- plicantur: Quia enim circuli circa suum centrum circumacti partes adversæ oppositis motibus cientur, etiam laminarum ex- tremitates, quæ pondus propellunt aut trahunt, in contrarias partes à rotulâ circumactâ moventur, ita ut vel à se invicem re- cedant, vel ad se mutuò accedant. Hoc idem quod laminæ rectæ dentatæ tympano similiter den- tato implicitæ contingit, accideret pariter, si illius in circulum inflexæ extremam oram dentes ambirent: quemadmodum enim recta lamina AB, tympani HI conversi ductum sequitur, ita il- la in circulum conformata circa suum centrum moveretur à tympani dentibus impulsa; eâ tamen ratione, ut duarum hujus- modi rotarum se invicem mordentium conversiones in opposi- tas plagas tenderent; si enim prioris rotæ pars superior Occa- sum versùs converteretur, posterioris rotæ pars item superior Ortum versùs convolveretur; & si adhuc tertia rota dentata ad- deretur, hæc iterum proximæ adversata ad occasum pergeret; atque ita deinceps alternis conversionibus sibi vicissim respon- dentibus. Zzz

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Book Five. CHAPTER VI. 545 to the motion of the power, and reciprocally so is the moving force of the power to the weight. Wherefore it is further evident that, if greater force of the power be needed, considering the difficulty of moving the weight, the handle may be enlarged, so that the moments of the power may be greater. But since a suitable handle is not always at hand, I would suggest that the end of the axle which is inserted into the handle be made square, and somewhat long: and in place of the ordinary handle let there be a piece of thicker cylinder MN, in whose lower base, around the center, a square hole S be hollowed out for receiving the end of the axle, and through the body of the cylinder let round holes R, T pass, into which rods, either longer or shorter, may be inserted as needed. Moreover, that the thickness of the cylinder is no obstacle is plainly clear, since only the semidiameter of the wheel is considered in determining the motion of the weight, compared with the length of the rod, inasmuch as the power is distant from the axle of the wheel. If at one and the same time it is necessary to drive two weights in opposite directions, or to bring them closer to one another, place another similar plate parallel to the first, in the same plane but set in the contrary manner (so that, namely, its similar end ACD faces the end B of the first plate) on the opposite side of the wheel at I, so that likewise the teeth of the plate mesh with the teeth of the wheel: for since the parts of circles turned about their center are moved by opposite motions toward opposite sides, even the ends of the plates, which propel or draw the weight, are moved by the turned wheel toward contrary directions, so that they either recede from one another or mutually approach one another. The same thing that happens when a straight toothed plate is engaged with a similarly toothed drum would likewise occur if the outer edge of that plate, bent into a circle, were surrounded with teeth: for just as the straight plate AB follows the motion of the turned drum HI, so it, formed into a circle, would be moved around its own center, driven by the teeth of the drum; yet in such a way that the rotations of two wheels of this kind, biting one another, would tend toward opposite sides. For if the upper part of the first wheel were turned toward the west, the upper part of the second wheel would likewise be turned toward the east; and if a third toothed wheel were added, this would in turn proceed toward the west, opposite to the next; and so on, the alternating rotations corresponding to one another in turn. Zzz

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Mechanicorum Observandum est autem hujusmodi tympanorum, quæ dentata vocamus, multiplicem esse posse formam, eamque eligendam, quæ præstituto motui magis congruere videbitur: non solum enim pro majoribus tympanis assumi potest discus extremum limbum habens serræ in morem denticulatim incisum, verùm etiam in orbem insigi possunt paxilli dentium loco prominentes, sive peripheriam ipsam quasi radij exeuntes ambiant, sive suprà disci planum erigantur ad perpendiculum certis intervallis distributi. Hoc autem dentium insitorum genus non parum habet utilitatis prædentibus illis quasi connatis: nam si longo usu dens aliquis atteratur, aut excutiatur, facilè restitui potest novo paxillo in prioris locum immisso; at non ita facilè reparatur pars illa limbi denticulata, quæ facta est inutilis. Similiter pro minoribus tympanis non solùm dentatas rotulas adhibere possumus suis axibus infixas, sed etiam uti licet aut paulò crassioribus axibus striatis, quorum excavatæ striæ majoris tympani dentibus congruant, aut vertebris pariter striatis subtiliori axi infixis. Quando verò majus tympanum paxillos habet prodentibus, tympanum minus illi respondens est Curriculus (quê alij ex Italico idiomate Rocchetum dicunt) aliquot virgulis, ut plurimum ferreis ad firmitatem, capita duobus parallelis planis infixa habentibus constans, ita ut in majoris tympani conversione singulos paxillos excipiant singula virgularum intervalla, quibus propulsis curriculus convolvitur, & cum eo aut pondus ipsum, aut aliud tympanum movetur. Sic contingere potest ut Axe A B attollendum sit pondus, & expediat uti jumento, quod tamen non nisi in plano horizontali moveri potest; tympanum autem C D, in quo est Axis A B horizontalis, est in plano Verticali. Ad tympani C D planam faciem aversam perpendiculares paxillos in ambitu statue, & Curriculum E F circa suum axem superiùs atque inferiùs firmatum versatilem adjice, cujus virgulæ congruis intervallis distinctæ tympani dentibus respondeant. Aliud item tympanum H G horizonti paralle

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Mechanics But it should be observed that drums of this kind, which we call toothed, may have a variety of forms, and that the one should be chosen which seems most suited to the motion previously determined: for not only may a disk be used for larger drums, having its outer rim cut with small teeth like a saw, but also pegs may be set around the circle, projecting in place of teeth, either surrounding the very circumference as if rays issuing forth, or standing above the plane of the disk, set upright at fixed intervals. This kind of inserted teeth has not a little advantage over those which are, as it were, integral with the drum: for if, through long use, some tooth is worn away or knocked out, it can easily be replaced by inserting a new peg in the former place; but that toothed part of the rim is not so easily repaired once it has become useless. Likewise, for smaller drums we may use not only toothed wheels fixed upon their axles, but it is also permissible to use either somewhat thicker axles with grooves, whose hollow grooves correspond to the teeth of the larger drum, or similarly grooved cylinders fixed upon a finer axle. But when the larger drum has projecting pegs, the smaller drum corresponding to it is the Curriculus (which others, from the Italian tongue, call Rocchetum ), consisting of certain rods, for the most part iron for strength, having their heads fixed into two parallel planes, so that in the turning of the larger drum the intervals between the rods receive the individual pegs, which, being driven onward, cause the curriculus to wind up, and with it either the weight itself or another drum is moved. Thus it may happen that by the axle A B a weight must be raised, and it is convenient to use a beast of burden, which however can move only on a horizontal plane; but the drum C D, in which is the horizontal axle A B, is in a vertical plane. On the face of the drum C D turned away from you, set perpendicular pegs on the circumference, and add the curriculus E F, made movable around its axle and secured above and below, whose rods, marked off at suitable intervals, correspond to the teeth of the drum. Also another drum H G parallel to the horizon

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Liber quintus. CAPUT VII. 547 parallelum dentes habens ex peripheriâ extantes, & Curriculi E F virgulis aptè congruentes, infigatur axi perpendiculari I K, cui opportuno loco addatur vectis L M, ita ut in M commodè jungi possit jumentum: hoc enim progrediente, & tympanum ex G versùs O convolvente, dens H incurrens in virgulam cur- riculi E F illum convertit versùs tympanum C D, cujus pariter denti occurrens alia virgula, atque impellens cogit infimam tympani partem D ascendere, simulque Axem AB converti, & convoluto fune ductario R S attolli pondus. Hæc tympanorum duorum & curriculi intermedij complexio si attentè perpendatur, non auget potentiæ momenta præter ea, quæ obtineret proximè applicata tympano C D ad convolven- dum Axem A B: Nam si ponatur vectis L M non longior semi- diametro tympani dentati H G, perinde est, ac si potentia in M posita existeret in G, æquali scilicet motu cum tympani H G peripheriâ movetur. Curriculi autem E F motus æquè velox est atque motus tympani H G; licèt enim hoc sit majus, ille mi- nor, tamen dum illud semel, hic sæpiùs convolvitur pro ratione diametrorum; adeò ut si tympanum H G habeat dentes viginti, curriculum strias quinque, hic quater volvatur ex unicâ tympani conversione: quapropter quatuor curriculi subquadrupli con- volutiones uni conversioni tympani H G æquantur. Similiter & de tympano C D dicendum, cujus tantummodo dentes quin- que respondentes quinque striis aut virgulis curriculi E F ur- gentur unicâ conversione curriculi ejusdem, & idcircò æqualis est utriusque motus, ac proinde etiam duo tympana C D & H G æqualiter moventur, & potentiæ in M applicatæ momenta ea- dem sunt, quæ forent, si tympano C D proximè applicaretur. Quamobrem, ut aliqua fiat momentorum accessio in potentiâ, oportet vectem L M statuere longiorem semidiametro tympani H G: tunc enim ex Ratione longitudinis L M ad semidiam- trum tympani, & Ratione diametri tympani C D ad diametrum Axis A B, componitur Ratio, quæ definit momenta potentiæ; est scilicet Ratio motûs potentiæ ad motum ponderis. Ex quo satis vides eatenus addi tympanum H G, quatenus quærendus est jumento locus, ut in gyrum circumagi valeat: cæterum si tympanum C D cum suo Axe A B ita in superiore aut inferiore loco collocari atque firmari possit, ut nulli impedi- ZZZ 2

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Book the fifth. CHAPTER VII. 547 with parallel teeth projecting from the circumference, and the drum E F suitably fitting the grooves, let it be fixed to the perpendicular axle I K, to which, in a convenient place, let the lever L M be added, so that at M there may conveniently be attached a draft animal: for this animal, advancing and winding the drum from G toward O, the tooth H striking against the groove of the drum E F turns it toward the drum C D, whose tooth likewise meeting another groove and pushing it compels the lower part D of the drum to ascend, and at the same time the axle A B to turn, and the load to be lifted by the winding of the hauling rope R S. If this arrangement of the two drums and the intermediate wheel is carefully considered, it does not increase the force of the power beyond what would be obtained by the drum C D applied directly for winding the axle A B: for if the lever L M be supposed not longer than the semi- diameter of the toothed drum H G, it is the same as if the power placed at M existed at G, moving with the same motion as the circumference of the drum H G. But the motion of the drum E F is just as swift as the motion of the drum H G; for although this is larger and that smaller, yet while the one is wound once, the other is wound several times in proportion to the diameters; so that if the drum H G have twenty teeth, the wheel five grooves, this latter will revolve four times for a single revolution of the drum: wherefore four revolutions of the wheel, being fourfold, are equal to one revolution of the drum H G. Likewise the same must be said of the drum C D, whose only five corresponding teeth are urged by five grooves or strips of the wheel E F in a single revolution of that same wheel, and therefore the motion of each is equal; and consequently the two drums C D and H G are likewise moved equally, and the moments of the power applied at M are the same as they would be if it were applied directly to the drum C D. Wherefore, if some increase of moment is to be made in the power, the lever L M must be set longer than the semidiameter of the drum H G: then indeed, from the ratio of the length of L M to the semidiameter of the drum, and the ratio of the diameter of the drum C D to the diameter of the axle A B, there is composed a ratio which defines the moment of the power; that is, the ratio of the motion of the power to the motion of the weight. From this you clearly see that the drum H G is added only so far as a place for the animal is to be sought, so that it may be able to be turned round in a circle: otherwise, if the drum C D with its axle A B can be so placed and fixed in the upper or lower position that no obstacle may impede ZZZ 2

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548 Mechanicorum mento sit jumento in inferiore aut superiore plano existenti & circumacto, satius est labori & sumptibus parcere omisso tym- pano HG, & circa crassiorem axem construere curriculum EF, cui axi opportunè adjungatur vectis LM, ut potentia ipsum curriculum immediatè convertat; erit siquidem major Ratio ipsius vectis LM ad curriculi semidiametrum, quàm ad semi- diametrum tympani HG; ac proinde minore conatu indige- bit potentia, ut Curriculum cum adjacente tympano CD con- vertat. Non alio quàm hujusmodi artificio videtur usus Anonymus, qui de Rebus Bellicis scripsit ad Theodosium Augustum ejus- que filios Honorium, & Arcadium Cæsares, ubi liburnam pro- ponit navalibus idoneam bellis, quam pro magnitudine sui virorum exerceri manibus quodammodo imbecillitas humana prohibeat, & quocumque utilitas vocet, ad facilitatem cursûs ingenij ope subnixa animalium virtus impellit. In cujus alveo, vel capacitate bini boves machinis adjuncti, adhærentes rotas navis lateribus volvunt; qua- rum supra ambitum vel rotunditatem extantes radij currentibus his- dem rotis in modum remoram aquam conatibus elidentes miro quodam artis effectu operantur, impetu parturiente discursum. Hæc eadem tamen liburna pro mole sui, proque machinis in semet operantibus tanto virium fremitu pugnam capessit, ut omnes adversarias libur- nas cominus venientes facili attritu comminuat. Quamvis, quæ de- mum machinæ essent, quibus boves adjungebantur, Author non exponat, facile tamen est opinari boves in superiore tabu- lato circumacto versâsse axem carinæ perpendiculariter in- sistentem, cui infra tabulatum rota dentata horizonti parallela infixa esset dentes habens in alterutra tympani facie, quibus subinde apprehenderet virgulas curriculi in plano Verticali convoluti, & infixi axi horizontali, qui utrumque navis latus permearet, & in extantibus extremitatibus rotas haberet cum palmulis prominentibus, quæ aquam in conversione verbera- rent. Potuerunt autem hujusmodi machinæ juxta liburnæ lon- gitudinem multiplicari, prout ipsum schema ab Authore pro- positum exhibet. An verò hujus liburnæ, quam in Præfatione dicit velocissimum liburnæ genus, decem navibus ingenij magisterio prævalere, tantus impetus, tantâque velocitas esset, ut adversa- riæ liburnæ venientes facilè comminuerentur, dispiciat lector, cui

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548 Mechanicorum it is better, if a draft animal is standing on a lower or upper plane and being turned around, to spare labor and expense by omitting the drum HG, and to construct around the thicker axle the carriage EF, to which axle the lever LM is suitably attached, so that the power may turn the carriage itself directly; for there will indeed be a greater ratio of the lever LM itself to the semidiameter of the carriage than to the semidiameter of the drum HG; and therefore the power will require less effort in order to turn the carriage together with the adjoining drum CD. The Anonymous author who wrote On Military Affairs to the Emperor Theodosius and his sons, the Caesars Honorius and Arcadius, seems to have used no other device than one of this kind, where he proposes a liburna suitable for naval warfare, which, by reason of its size, human weakness prevents from being worked by men’s hands, and wherever utility calls, animal strength, supported by the ingenuity of invention, impels it toward the ease of movement. In the hold or capacity of this ship, two oxen attached to machines turn the ship’s wheels fixed to its sides; from which wheels the radii projecting above the circumference or roundness, running with these same wheels, strike the water like a drag or oar with their efforts, producing by a marvelous effect of art a motion born of impulse. This same liburna, however, by reason of its bulk and the machinery operating within it with such a clamor of force, engages in battle in such a way that it crushes all opposing liburnae approaching at close range by easy friction. Although the author does not explain what the machines were by which the oxen were attached, it is nevertheless easy to suppose that the oxen, on the upper deck, turned an axle set vertically in the keel, to which below the deck a horizontal toothed wheel parallel to the horizon was fixed, having teeth on one or the other face of the drum, by which it would in turn catch the little bars of a carriage wound in the vertical plane and fixed to a horizontal axle passing through both sides of the ship and having at its projecting ends wheels with protruding paddles, which would beat the water in turning. Such machines could moreover be multiplied along the length of the liburna, as the scheme itself proposed by the Author shows. But whether this liburna, which in the Preface he says is the swiftest kind of liburna, surpasses ten ships by the mastery of ingenuity, with such force and such speed that the opposing liburnae coming against it would easily be crushed, let the reader consider, to whom

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Liber quintus. CAPUT VI. 549 cui otium fuerit, quemadmodum an nostris usibus navalibus artificium hoc aliquid utilitatis afferre possit. Hinc manifesta fit illarum machinarum vis, quæ Pancratia Glossocoma, Charistia, & siquod est aliud vocabuli genus, di- cuntur, ex plurium tympanorum complexione minorum & ma- jorum, ita ut à minore tympano, cui manubrium additur, inci- piat motus, & deinceps minora majoribus consequentibus mo- tum communicent. Quandoquidem rotula minor dentata si comparetur cum rotâ majore, cum qua communem habet axem, utique tardius movetur, quàm peripheria rotæ majoris, cum qua connectitur: At verò si cum rotâ majore consequente comparetur, cujus dentes apprehendit, utique æqualis est ipsa- rum motûs velocitas, nam plures minoris conversiones æquales sunt uni conversioni majoris, quam efficiunt. Sit rota dentata A B S D C R F E I H L G O A B, cujus axi firmiter infixo additum sit manubrium ejusdem rotæ semidiametri ex. gr. quintuplum, ac propterea potentia manubrio applicata quintuplo velociùs movetur, quàm punctum in rotæ AB peripheriâ notatum. Addatur rota ma- jor BC, cujus dentes implicantur dentibus rotulæ BA: ex hu- jus conversione illa pariter convolvitur; sed si diameter BA sit diametri BC subtripla, ter rotula BA volvitur, ut rota BC compleat integram conversionem, ac proinde potentia quin- tuplo velociùs movetur, quàm rota BC. Rota hæc major sibi connexam habeat in eodem axe minorem SD, quæ apprehen- dat dentes secundæ majoris rotæ DE; quæ similiter in eodem axe conjunctam habeat minorem FR: hujus dentes mor- Z z z 3

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Liber quintus. CHAPTER VI. 549 whoever has leisure, to consider how far this art may bring any benefit to our nautical uses. From this the force of those machines becomes clear, which are called Pancratia, Glossocoma, Charistia, and, if there is any other kind of the term, from the combination of several small and large drums, so that the motion begins from the smaller drum, to which a handle is added, and then the smaller ones communicate motion to the larger ones that follow. For a smaller toothed wheel, if compared with a larger wheel with which it has a common axis, certainly moves more slowly than the periphery of the larger wheel with which it is connected: but if it is compared with the larger wheel that follows, whose teeth it catches, then the speed of their motion is certainly equal, for several revolutions of the smaller equal one revolution of the larger, which they produce. Let there be a toothed wheel A B S D C R F E I H L G O A B, to whose firmly fixed axis is attached a handle equal to, for example, five times the semidiameter of that wheel, and therefore the power applied to the handle moves five times faster than the point marked on the periphery of wheel AB. Let a larger wheel BC be added, whose teeth engage with the teeth of the small wheel BA: by the turning of this one, that also revolves; but if the diameter of BA is one-third the diameter of BC, then the small wheel BA turns three times, so that wheel BC may complete one entire revolution, and therefore the power moves five times faster than wheel BC. Let this larger wheel have connected to it, on the same axis, a smaller one SD, which catches the teeth of the second larger wheel DE; which likewise has joined to it on the same axis a smaller one FR: the teeth of this one

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550 Mechanicorum deant peripheriam tertiæ majoris rotæ FG, in qua est Axis I H, ex cujus convolutione ductarius funis L O trahit pondus. Ut potentiæ momenta habeantur, ejus motum cum ponderis motu collatum ad calculos revoca componendo Rationes, quas singulæ majores rotæ ad suas minores habent quo ad diametrum. Quare si manubrium ad semidiametrum rotulæ AB habeat Rationem quintuplam, diameter BC ad diametrum SD triplam, DE ad RF item triplam, & FG ad IH similiter triplam, compositis tribus Rationibus triplis cum Ratione quintuplâ, oritur Ratio 135 ad 1: atque adeò ut postrema rota FG & cum eâ Axis IH semel volvatur, prima rotula AB facit 27 conversiones; potentia autem manubrio applicata movetur quintuplo velociùs quàm suæ rotæ AB peripheria; igitur ponderis motus ad motum potentiæ est ut 1 ad 135. Porrò fieri 27 conversiones manubrij apertè constat, quia hoc ter volvi ponitur, ut rota BC, atque adeò etiam SD illi connexa, semel convolvatur: & quia ex hypothesi rotula SD tres conversiones habet, ut toti peripheriæ rotæ DE congruat, manubrium novies in gyrum agitur, ut rota major DE & minor RF semel convertatur: demum quia pariter ex hypothesi rota minor RF triplici convolutione indiget, ut toti peripheriæ rotæ majoris FG respondeat, ut hæc unicum circuitionem perficiat, viginti septem manubrij conversionibus opus est. Quare momentum Potentiæ manubrio applicatæ comparatæ cum rotulâ AB est ut 5, cum sequenti rotulâ SD ut 15, cum rotulâ RF ut 45, cum Axe IH ut 135. Ne verò artifex hujusmodi rotas dentatas majores atque minores parare jussus inutili demùm labore se torqueat, monendus est, ut animum diligenter advertat, utrum omnes majores rotas, item omnes minores, inter se æquales statuere velit, an inæquales; ex hoc enim ipsarum rotarum collocationem definiet, ne sibi vicissim impedimento sint. Finge enim rotulas DS & FR æquales esse, item majores BC & DE, atque alterno ordine positas, ita ut si rotula DS fuerit in parte anteriore suæ rotæ BC, vicissim rotula FR sit in parte aversâ rotæ DE: quemadmodum dentes minoris DS implicantur dentibus majoris DE, ita pariter dentes majoris BC implicantur dentibus

Transcription: Translated (English)

550 Mechanicorum the circumference of the third larger wheel FG, in which is the axis I H, by the convolution of which the carrying rope L O draws the weight. In order that the moments of the powers may be obtained, reduce its motion, compared with the motion of the weight, to calculations by composing the ratios which the several larger wheels have to their smaller ones in respect of diameter. Therefore if the handle to the semidiameter of the little wheel AB has a quintuplæ ratio, the diameter BC to the diameter SD a triple, DE to RF likewise a triple, and FG to IH likewise a triple, then, the three triple ratios being combined with the quintuplæ ratio, there arises the ratio of 135 to 1: and thus, as often as the last wheel FG and with it the axis IH is turned once, the first little wheel AB makes 27 revolutions; but the power applied to the handle is moved five times more quickly than the circumference of its wheel AB; therefore the motion of the weight to the motion of the power is as 1 to 135. Moreover, it is plainly evident that 27 revolutions of the handle are made, because this is supposed to be turned three times, so that the wheel BC, and therefore also the SD connected with it, may be turned once: and because, by hypothesis, the little wheel SD has three revolutions, so that it corresponds to the whole circumference of the wheel DE, the handle is turned nine times in a circle, so that the larger wheel DE and the smaller RF may be turned once: finally, because likewise by hypothesis the smaller wheel RF needs a triple convolution, so that it may correspond to the whole circumference of the larger wheel FG, in order that this may complete a single circuit, twenty-seven revolutions of the handle are required. Wherefore the moment of the power applied to the handle compared with the little wheel AB is as 5, with the next little wheel SD as 15, with the little wheel RF as 45, with the axis IH as 135. Lest, however, the craftsman, ordered to make larger and smaller toothed wheels of this sort, should at length torment himself with useless labor, he must be warned to pay careful attention whether he wishes all the larger wheels, and likewise all the smaller ones, to be set equal to one another, or unequal; for from this he will determine the arrangement of the wheels themselves, lest they be an impediment to one another. For suppose the little wheels DS and FR to be equal, and likewise the larger BC and DE, and placed in alternate order, so that if the little wheel DS is on the front side of its wheel BC, conversely the little wheel FR is on the back side of the wheel DE: just as the teeth of the smaller DS engage the teeth of the larger DE, so likewise the teeth of the larger BC engage the teeth of

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Liber quintus. CAPUT VI. 551 dentibus minores FR: igitur unica conversio majoris rotæ BC ter convolveret minorem rotam FR: atqui cum minore rotâ FR simul converteretur major DE in eodem axe; igitur dum semel converteretur major rotam BC, ter convolveretur rotam major DE: hoc autem omnino fieri nequit, quia unica ro- tæ BC conversio est etiam unica conversio rotulæ minoris DS, hujus autem unica conversio respondet solùm tertiæ parti con- versionis majoris rotæ DE: plurimum igitur abest à trinâ con- versione. Quare si hujusmodi æqualitas intercederet tùm inter majores, tùm inter minores rotas dentatas, oporteret minores rotas ad eandem partem respicere, ne rota major consequenti minori rotæ motum ullum communicare possit. Verùm hoc fortasse alicui videatur incommodum, quod non ita aptè in suo loculamento hujusmodi rotæ collocari valeant, si rotarum ma- jorum consequentium plana superimponantur planis antece- dentium; id quod exigit ipsa minorum rotularum positio, si omnes partem eandem respiciant. Propterea inæquales fiant rotæ ita, ut alternatim positæ minores rotulæ occurrant qui- dem singulæ peripheriæ consequentis majoris, non attingantur autem à dentibus majoris rotæ antecedentis: hoc enim pacto in suo loculamento pressiùs firmantur, & sunt quasi duo plana parallela, in quibus hinc rota minor inter duas majores, hinc verò rota major inter duas minores conspicitur. Porrò minores rotæ in eodem axe cum majoribus dupliciter disponi possunt: primùm ut major rota minori proxima sit, & earum plana se contingant; deinde ut aliquo inter se absint in- tervallo. Si minor majori cohæreat, suaderem minores rotas ex lamina paulò crassiore fieri quàm majores; sic enim in locula- mento ita disponuntur, ut plana majorum se omninò non con- tingant, ac proinde nullus sit partium se vicissim terentium con- flictus, qui moram inferat motui. Sin autem quæ in eodem axe sunt rotæ, major & minor invicem distent, nullum quidem sub- est periculum ex mutuo affrictu majorum, verùm cavendum est, ne axis longior, quàm par fuerit, etiam sit infirmior; si ni- mirum axis longioris extremitates loculamento infixæ volvan- tur. Propterea aliter etiam disponi possunt, ita ut loculamentum validissimum sit, nec fractioni obnoxium, & facilè ex alio in alium locum transferri valeat. Parentur axes rotundi, sed utra- que

Transcription: Translated (English)

Book Five. Chapter VI. 551 teeth smaller than FR: therefore a single revolution of the larger wheel BC would turn the smaller wheel FR three times; but with the smaller wheel FR the larger DE, on the same axle, would be turned at the same time; therefore, while the larger wheel BC was turned once, the larger wheel DE would be turned three times: but this is altogether impossible, because a single revolution of wheel BC is also a single revolution of the smaller wheel DS, and this single revolution corresponds only to the third part of the revolution of the larger wheel DE: therefore it falls far short of a threefold revolution. Wherefore, if such equality existed both between the larger and between the smaller toothed wheels, it would be necessary for the smaller wheels to face the same side, so that the larger wheel might not be able to communicate any motion to the following smaller wheel. But perhaps this may seem inconvenient to someone, because wheels of this kind cannot be suitably placed in their housing if the planes of the following larger wheels are superimposed upon the planes of the preceding ones; and this is required by the very position of the smaller rollers, if all face the same side. Therefore let the wheels be made unequal, so that, being placed alternately, the smaller rollers meet indeed each individual circumference of the following larger wheel, but are not touched by the teeth of the preceding larger wheel: for by this arrangement they are more firmly held in their housing, and there are as it were two parallel planes, in which here a smaller wheel is seen between two larger ones, and there a larger wheel between two smaller ones. Moreover, the smaller wheels on the same axle can be arranged in two ways with the larger ones: first, so that the larger wheel is near the smaller one, and their planes touch; then, so that they are separated by some interval. If the smaller adheres to the larger, I would advise that the smaller wheels be made from a plate somewhat thicker than the larger ones; for thus they are arranged in the housing so that the planes of the larger ones do not touch at all, and accordingly there is no collision of the parts rubbing against one another, which would delay the motion. But if the wheels which are on the same axle, the larger and the smaller, are separated from one another, there is indeed no danger from the mutual friction of the larger ones, yet care must be taken lest the axle, longer than would be fitting, should also be weaker; namely, if the ends of a longer axle, fixed in the housing, are set in motion. Therefore they may also be arranged otherwise, so that the housing is very strong, and not liable to breakage, and can easily be transferred from one place to another. Let round axles be prepared, but each of them...

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Mechanicorum 552 que extremitas quadrata sit, ut inseratur quadrato foramini, quod rotarum centro inest: tum tigni pars accipiatur crassitudinis tantæ, ut congruis foraminibus rotundis excipiat axium rotunditatem, extantibus hinc atque hinc extremitatibus quadratis. Deinde quadratas axis extremitates excipiant rotæ dentatæ alterno ordine, ut qua parte prior axis habet rotam majorem, secundus axis habet rotam minorem, & vicissim ille in oppositâ tigni parte habeat rotam minorem, hic majorem; illud semper præcavendo, ne rota minor secundi axis contingat peripheriam rotæ majoris primi axis; id quod fiet, si posteriores rotæ majores etiam paulò majorem semidiametrum habeant. Relinquitur autem artificis industriæ ita foraminum extremitates munire, ut nec axes ultrò citroque commeare valeant, rotis ipsis illos coërcentibus, nec nimio affrictu tigni faciem rotæ circumactæ terant, interjecto inter tignum & rotam exiguo circulo, cum quo tritus omnis atque conflictus exerceatur. Quamvis autem tria tantummodo tympana dentata præter primam rotulam manubrio affixam, brevitatis gratiâ, examinanda proposuerim, plura, & plura similia addi posse est manifestum, adeò ut omni arrogantiæ notâ vacent magnificæ illæ Mechanicorum propositiones, quibus se quodcumque etiam immane pondus moturos spondent, immò tellurem ipsam, si locus daretur statuendæ machinæ idoneus. Illud tamen incommodum vitari nullatenus potest, quod ex ponderis tarditate oritur: quî enim fieri possit, ut gravitatis resistentia ex motûs tarditate minuatur, quin multo tempore opus sit ad pondus movendum? Idcirco unâ eademque operâ, qua potentiæ momenta inquiris componendo Rationes, quas majora tympana habent ad minora sibi adjuncta, etiam motûs tarditatem notam facis; ac proinde constituto intra certam temporis mensuram potentiæ motu, innotescit ponderis motus, quem eodem tempore perficit. Fac esse decem Rationes quintuplas, quæ componendæ sunt, si manubrium ad suæ rotulæ semidiametrum habeat Rationem quintuplam, & similis sit Ratio majorum ad sua minora tympana. Motus potentiæ manubrio applicatæ est ad motum ponderis ut 9.765625 ad 1: tot igitur spatij pedes iteratis revolutionibus confici à potentiâ necesse est, ut pondus pedem unum percurrat. Quod si potentiam tantâ velocitate veri

Transcription: Translated (English)

Mechanics 552 that the end be square, so that it may be inserted into the square hole which is in the center of the wheels; then let the beam be taken of such thickness that, through suitable round holes, it may receive the roundness of the axles, with square extremities projecting on this side and on that. Then let toothed wheels receive the square extremities of the axles in alternating order, so that on the side where the first axle has the larger wheel, the second axle has the smaller wheel, and conversely, on the opposite side of the beam it has the smaller wheel, this one the larger; always taking care that the smaller wheel of the second axle does not touch the circumference of the larger wheel of the first axle; which will happen if the later larger wheels also have a somewhat greater semidiameter. It remains for the ingenuity of the artificer so to secure the ends of the holes that neither axle may be able to move back and forth, the wheels themselves restraining them, nor may too much friction wear the face of the beam by the wheel turned round; an exceedingly small circle being interposed between the beam and the wheel, with which all wearing and rubbing may be borne. And although, for the sake of brevity, I have proposed for examination only three toothed drums besides the first little wheel fixed to the handle, it is manifest that more and more similar ones can be added, so that those magnificent propositions of Mechanics may be free from every mark of arrogance, by which they promise that they will move whatever enormous weight may be, indeed even the earth itself, if there should be room and a suitable place for setting up the machine. Yet that inconvenience, which arises from the slowness of the weight, can by no means be avoided: for how can it happen that the resistance of gravity be diminished by the slowness of motion, without a very long time being needed to move the weight? Therefore, in the very same labor, by which you investigate the moments of power by composing the ratios which the larger drums have to the smaller ones joined to them, you also make known the slowness of the motion; and thus, with the motion of the power established within a certain measure of time, the motion of the weight becomes known, which it accomplishes in the same time. Suppose there to be ten quintuple ratios that are to be composed, if the handle have to the semidiameter of its little wheel a quintuple ratio, and the ratio of the larger to their smaller drums be similar. The motion of the power applied to the handle is to the motion of the weight as 9.765625 to 1: so many feet of distance therefore must be traversed by the power in repeated revolutions, in order that the weight may go through one foot. If, however, the power with such speed truth

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Liber quintus. CAPUT VI. 553 veri ponamus, ut horis singulis pedum quindecim millia per- currat, indigebit horis 651 1/24, hoc est diebus 27, horis 3. min. 2 1/2, ut pondus à loco in locum pedis unius intervallo distan- tem transferat: atque ideò quis tantâ oculorum acie polleat, ut ponderis motum dignoscat, nisi post aliquot horas? quando- quidem unius horæ spatio vix unius unciæ partem quinquage- simam quartam perficit, scilicet 1/2 pedis. Verùm tam immane pondus, quod ad gravitatem respondentem potentiæ machinâ destitutæ sit ut 9. 765625 ad 1, movere, licet tardissimè, satius est, quàm nullo pacto movere. Ex his liquet, quid contingat, si potentiæ & ponderis loca ita commutentur, ut potentia extremo tympano applicetur, pondus verò movendum primæ rotulæ axi aut manubrio res- pondeat; exiguus enim validioris potentiæ motus velocissimè movet pondus, motumque diu continuat; ut palam est in au- tomatis horas indicantibus, sive potentia movens sit vis elasti- ca laminæ chalybeæ inflexæ, sivè gravitas ponderis axem ma- ximæ rotæ volvens; nisi enim Tempus alternis motibus objice- ret rotæ serratæ dentibus sui fusi pinnulas, quæ moram infer- rent, rota ipsa serrata velocissimè volveretur. Sed quoniam ra- rò contingit validissimam potentiam adhibere, ut leve pondus moveatur, propterea non est frequens hujusmodi locorum commutatio inter pondus & potentiam: usum tamen aliquan- do habere posset in rebus scenicis, maximè si æquabilis esse de- beat motus; gravitas enim, quæ per unius aut alterius palmi spatium descendat, non acquirit in motu notabile aliquod velo- citatis incrementum, atque idcirco æquabilis apparet motus tam ipsius gravitatis descendentis, quàm ponderis illius virtu- te ascendentis: hoc si non rectâ sursum trahatur, sed circum- agatur, fortasse impressus impetus velociorem circumvolutio- nem efficere possit. Quod demum ad ipsos rotarum dentes attinet, singulæ qui- dem rotæ à suis dentibus in partes æquales tribuuntur; sed hal- lucinati videntur non pauci frustra requiretes in omnibus ro- tis invicem comparatis dentium æqualitatem, & ex dentium numero potentiæ momenta metientes; quasi servari nequiret eadem momentorum Ratio, etiamsi minoris tympani dentes A A a a

Transcription: Translated (English)

Liber quintus. CAPUT VI. 553 if we suppose it true, so that in each hour it may traverse fifteen thousand feet, it will require 651 1/24 hours, that is, 27 days, 3 hours, min. 2 1/2, to transfer the weight from one place to another, one foot distant: and therefore who can possess so keen an eye as to discern the motion of the weight, unless after some hours? since in the space of one hour it scarcely accomplishes the fifty-fourth part of one inch, namely 1/2 foot. But to move so huge a weight, which, in proportion to the corresponding gravity, is as 9. 765625 to 1, though most slowly, is better than not to move it at all. From these things it is clear what happens if the places of power and weight are so exchanged that the power is applied to the outer drum, and the weight to be moved corresponds to the axle or handle of the first wheel; for a slight motion of a stronger power moves the weight very quickly, and continues the motion for a long time; as is evident in automata that indicate the hours, whether the moving power be the elastic force of a bent steel spring, or the weight’s gravity turning the axle of the largest wheel; for unless Time by alternating motions were to oppose the little teeth of the serrated wheel to the vanes of its spindle, which would cause delay, the serrated wheel itself would revolve very quickly. But since it rarely happens that a very strong power is applied so that a light weight may be moved, therefore such an exchange of places between weight and power is not frequent: yet it might sometimes be useful in theatrical matters, especially if the motion ought to be uniform; for a weight descending through the space of one or another palm does not acquire any noticeable increase of velocity in its motion, and therefore the motion appears uniform both of the descending weight itself and of that weight ascending by virtue of it: if this is not drawn straight upward, but is revolved around, perhaps the impressed impetus may be able to produce a swifter revolution. As for the teeth of the wheels themselves, each wheel is indeed divided by its teeth into equal parts; but not a few seem to be mistaken in vain when they require in all wheels, compared with one another, an equality of teeth, and measure the moments of power from the number of teeth, as though the same ratio of moments could not be preserved, even if the teeth of the smaller drum

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Mechanicorum 354 non omninò similes essent, aut ut veriùs loquar, singuli non essent æquales singulis dentibus majoris tympani in eodem axe existentis. Dentium æqualitas in iis tantummodo rotis requiritur, quæ sibi mutuâ collabellatione cohærentes in convolutione dentem dentibus implicant; nisi enim ab unius rotæ dentium intervallis alterius dentes subinde reciperentur, fieri non posset utriusque rotæ conversio. Cæterùm nil prohibet, quominùs in plurium majorum tympanorum complexione alia rariores, alia spissiores dentes habeant, dummodo singulis majoribus singula minora, à quibus illa motum recipiunt, respondeant similibus dentibus instructa, etiamsi hi dissimiles sint dentibus tympani in eodem axe connexi. Si enim prioris majoris tympani peripheria sit in dentes 24 distributa, minoris autem tympani eidem axi infixi peripheria sex dentes habeat, sed eorum diametri sint ut 3 ad 2, utique eorum motus non aliam habent Rationem quàm sesqualteram, licèt dentium numeri sint in Ratione quadruplâ. Idem planè dicendum de secundo tympano majore, quod ad prioris motum convolvitur; hujus enim motus pariter comparandus est cum motu tympani minoris sibi con uncti spectatâ diametrorum Ratione, non dentium multitudine, ut momenta innotescant. Quare illæ dentium multitudines invicem comparatæ satis quidem faciunt quærenti, quoties potentia manubrium circumagat, ut semel convertatur Axis, quem ductarius funis complectitur; sed quibus momentis id perficiat ipsa potentia, sola diametrorum Ratio spectata indicabit. Sed hìc ubi diametrorum incidit mentio (quamquam res Mechanicæ in praxim deductæ tantâ subtilitate non indigeant) non est dissimulandum aliquam necessariò intercedere momentorum inæqualitatem in ipso motu, quando tympanorum ambitus est dentium incisuris asperatus: cum enim extremi dentium apices à centro magis absint, quàm anguli, in quibus sibi dentes occurrunt, non est utrobiqve eadem movendi facultas, quippe quæ in majore à centro distantiâ validior est, cæteris paribus. Eatenus scilicet rota rotam urget, quatenus rota movens sui dentis apice contingit faciem dentis rotæ, quæ movetur: hic autem contactus primùm fit propè angulum, hoc est minùs procul à centro, & sen- sim

Transcription: Translated (English)

Mechanics 354 would not be altogether similar, or, to speak more truly, the individual teeth would not be equal to the individual teeth of the larger drum existing on the same axle. Equality of teeth is required only in those wheels which, being mutually coupled together by their engagement, interlock tooth with tooth in their rotation; for unless the teeth of one wheel were continually received into the intervals of the teeth of the other, the rotation of either wheel would not be possible. Otherwise, nothing prevents, in the construction of several larger drums, some from having teeth more widely spaced, others more closely set, provided that each larger drum has corresponding smaller ones, from which they receive motion, furnished with similar teeth, even though these may differ from the teeth of the drum connected on the same axle. For if the circumference of the first larger drum be divided into 24 teeth, while the smaller drum fixed to the same axle has a circumference with six teeth, but their diameters are as 3 to 2, clearly their motions have no other ratio than one and a half to one, although the numbers of teeth are in the ratio of four to one. The same must plainly be said of the second larger drum, which is wound by the motion of the first; for the motion of this also must be compared with the motion of the smaller drum joined to it, regard being had to the ratio of the diameters, not to the number of teeth, so that the moments may be understood. Wherefore those numbers of teeth, compared with one another, are indeed sufficient for one who inquires how many times the power turns the handle, so that the axle enclosed by the driving rope may be turned once; but by what moments the power itself accomplishes this, only the ratio of the diameters, when considered alone, will indicate. But here, when mention falls upon diameters (although matters reduced to practice in Mechanics do not need such subtlety), it must not be concealed that some inequality of moments necessarily intervenes in the very motion, when the rims of the drums are roughened by the cuttings of the teeth: for since the outermost points of the teeth are farther from the center than the corners at which the teeth meet one another, the moving force is not the same in both places, since that force is stronger at a greater distance from the center, other things being equal. In this sense, therefore, one wheel pushes another only insofar as the moving wheel touches with the point of its tooth the face of the tooth of the wheel that is being moved: but this contact occurs first near the corner, that is, less far from the center, and gradually

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Liber quintus. CAPUT VI. sim dens rotæ moventis suo apice excurrents versus extre- mitatem dentis rotæ, quæ movetur, magis recedit à cen- tro. Cum igitur rota movens suam vim exerceat apice den- tis, integra semper illius diameter aut semidiameter consi- deranda est; at rota, quæ urgetur, cum non in eodem puncto recipiat moventis impulsionem, non est absolu- tè attendenda integra illius diameter aut semidiameter, sed potiùs mediocris quædam inter maximam & mini- mam à centro distantiam eligenda est, ut alter Rationis terminus habeatur. Ex quo vides (si res subtiliter elime- tur) non parum interesse, utrum minor rota majorem ur- geat, an è contrario major minorem propellat. Concipe enim majoris rotæ integram semidiametrum esse particula- rum 100, & talem esse dentium incisuram, ut angulus, in quo ipsi dentes coëunt, distet à centro particulis 94: rotæ autem minoris, quæ suos dentes illius dentibus impli- cat, semidiameter integra sit similium particularum 20, & angulus concursûs dentium distet à centro particulis 14. Uti- que si minor majorem urgeat illius Radius est ut 20, hujus verò est ut 97: contra autem si major pellat minorem illius Radius est ut 100, hujus ut 17. Quare singulæ com- paratæ cum iis, quæ secum communem habent axem, di- versam constituunt Rationem: si enim major rota urgea- tur à minore sibi proximâ, adeò ut secunda minor movea- tur ad motum majoris in eodem axe, & Ratio sit ut 20 ad 97, si majore proximâ urgente minorem moveretur major ad motum minoris in eodem axe, hujus minoris motus ad motum suæ majoris non esset pariter ut 20 ad 97, sed ut 17 ad 100, quæ est minor Ratio.

Transcription: Translated (English)

Book Five. CHAPTER VI. if a tooth of the moving wheel, advancing with its tip toward the extremity of the tooth of the wheel that is moved, recedes more from the center. Therefore, when the moving wheel exerts its force at the tip of the tooth, the whole diameter or semidiameter of that wheel must always be considered; but when the wheel that is driven does not receive the impulse of the moving wheel at the same point, its whole diameter or semidiameter is not to be regarded absolutely, but rather some mean distance between the greatest and the least distance from the center is to be chosen, so that it may be taken as the other term of the Ratio. From this you see (if the matter is examined carefully) that it makes not a little difference whether the smaller wheel drives the larger, or, conversely, the larger propels the smaller. For suppose the whole semidiameter of the larger wheel to be 100 parts, and the cut of the teeth to be such that the angle at which the teeth themselves meet is 94 parts distant from the center; and suppose the semidiameter of the smaller wheel, whose teeth engage with those of the other, to be 20 similar parts, and the angle of the meeting of the teeth to be 14 parts distant from the center. Certainly, if the smaller drives the larger, its Radius is as 20, but that of the larger is as 97; on the other hand, if the larger drives the smaller, its Radius is as 100, and that of the smaller as 17. Therefore, when each is compared with those that share a common axis with it, it forms a different Ratio: for if the larger wheel be driven by the smaller one next to it, so that the smaller second wheel moves in accordance with the motion of the larger on the same axis, and the Ratio is as 20 to 97, then if the larger, with the neighboring one driving, moved the smaller in accordance with the motion of the smaller on the same axis, the motion of this smaller wheel in relation to the motion of its larger one would not likewise be as 20 to 97, but as 17 to 100, which is the lesser Ratio.

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Mechanicorum CAPUT VII. Molestrinarum artificium ex Axe in Peritrochio pendet. Artificia omnia, quæ ex Axe in Peritrochio pendent, re- censere res esset non quidem injucunda, sed penè infi- niti laboris, historiam potiùs redolens, quam theoriam, cui potissimùm inservio Machinarum fontes indicans, ex quibus ingeniosus quisque Machinas suo instituto opportunas moliri queat. Placuit tamen in Molendinorum artificio paulisper im- morari, ut quam uberem ab Axe in Peritrochio utilitatem ad vitæ commoda percipiamus, innotescat. Quamvis autem po- tissimùm instituta sint molendina ad comminuendum triticum & alia semina, ut ex farinâ panis conficiatur, ad alios tamen usus pars eorum aliqua destinatur: omnibus quippe communis est rota exterior, quam aqua incurrens versat, & Axis, qui convolvitur. Si enim tundenda sit lana, aut Cannabis; si in pollinem redigenda elementa pulveris pyrij carbo, sulphur, nitrum; si antiqua linteorum resegmina conterenda, & in mi- nimas particulas dissipanda ad conficiendam chartam, Axi in- fixæ sunt pinnulæ, quæ in conversione occurrentes aliis pistil- lorum illos elevant, atque dimittunt, & eorum gravitate reci- dente subjecta materia aut contunditur, aut conteritur. Nec dissimili methodo disponi possent pistilli suis embolis congruentes, qui à pinnulis Axis elevati aquam in embolum attraherent, aut sponte irruentem admitterent per assarium, tùm dimissi vi suæ gravitatis aquam exprimerent per tubum, & in altiorem locum ascendere cogerent. Vel si non adeò gra- ves pistillos parare placuerit, velisque certiùs aquam in altio- rem locum pellere, dispone binos pistillos fune, aut catenâ, per excavatum rotæ superiùs positæ ambitum transeunte con- nexos, aut potiùs transversario, quasi libræ jugo conjunctos, ita ut altero depresso alter elevetur, pinnula autem Axis de- primat

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Mechanics CHAPTER VII. The operation of devices depends on the axle in the peritrochium. To review all the devices that depend on the axle in the peritrochium would be a task not indeed unpleasant, but almost of infinite labor, smacking rather of history than of theory, to which I chiefly devote myself, indicating the sources of machines, from which any ingenious person may devise machines suited to his purpose. Yet it has seemed good to linger a little over the art of mills, so that it may become known how great a usefulness we receive from the axle in the peritrochium for the conveniences of life. Although mills are established chiefly for grinding wheat and other grains, so that bread may be made from flour, some part of them is nevertheless assigned to other uses: for the outer wheel, which is driven round by incoming water, is common to all, as is the axle which is turned. For if wool or hemp is to be beaten; if the elements of gunpowder, charcoal, sulphur, and nitre are to be reduced to powder; if old scraps of linen are to be ground and scattered into very small particles for making paper, little vanes are fixed to the axle, which, as they revolve, striking against the pestles of others, lift them and let them fall, and, when their weight brings them down again, the material beneath is either pounded or ground. In a similar method, pestles could be arranged to suit their cylinders, which, lifted by the vanes of the axle, would draw water into the cylinder, or admit water rushing in of itself through the opening; then, being released, by the force of their own weight they would squeeze the water out through a tube, and force it to ascend to a higher place. Or, if it should be preferred not to make pestles so heavy, and you wish more certainly to drive water to a higher place, arrange two pestles connected by a rope or chain passing over the hollow circumference of a wheel placed above, or rather joined by a transverse bar, as if with the yoke of a balance, so that as one is depressed the other is raised, while the vane of the axle depresses

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Liber quintus. CAPUT VII. 557 primat pistillum, vi cujus aqua in tubum ascendentem exprimatur, & alter pistillus attollatur aquam inferiùs positam attrahens, qui pariter ab Axis pinnulâ ejus alæ respondente subinde deprimatur. Hinc fit posse longiorem Axem addi rotæ, & plura hujusmodi pistillorum paria disponi pinnulis in ambitu Axis ita distributis, ut non plures simul pistillos, sed singulos unum post alium premant, si non adeò valida fuerit potentia rotam versans; Sin autem validior illa fuerit, plures simul deprimant, iisque conjugatos attollant. Nisi fortè magis arri- serit duobus tantummodo pistillis conjugatis uti, tot pinnulis in Axe dispositis, ut in unâ ejusdem Axis conversione bis aut ter pistillus idem deprimatur. Huc pariter spectant, quæ passim videre est in officinis mal- leatorum cupri aut ferri, ubi & rota exterior vi aquæ labentis circumacta interiùs in conclavi quasi manubrium convolvit, quod superiori Axi horizonti parallelo infixum Radium, sibique regulâ in juncturis plicatili connexum, dum attollit, at- que deprimit, in alterâ ejusdem Axis extremitate transversa- rium hinc pariter attollens atque hinc deprimens follicus alter- num motum conciliat: Et rota alia validiorem aquæ deciden- tis impetum recipiens, suumque Axem convolvens, pinnulis axi infixis extremitatem alteram deprimit tigilli, cujus opposi- tæ extremitati elevatæ cohæret ingens ferreus malleus, qui præ- terlapsâ Axis pinnulâ sponte recidens tundit subjectum cuprum aut ferrum ignitum. In his omnibus rotæ quidem semidiameter attendenda est, in cujus extantes palmulas aqua incurrens vim potentiæ mo- ventis obtinet; sed Axis semidiameter non solitariè accipienda est, verùm & addenda prominentis pinnulæ longitudo, ita ut ex utrâque conficiatur unica semidiameter motûs, qui commu- nicatur pistillo, aut depressæ extremitati mallei. Depressæ, in- quam, extremitati mallei, nam mallei elevatio aliquanto major est, quam illa depressio, ut validior ictus sequatur; neque enim tigillus à suo axe, cui innititur, omnino æqualiter dividitur, sed ab eo aliquantulo remotior est malleus, quàm opposita ex- tremitas, quæ deprimitur: ac proinde vis illam deprimens ma- jor est, quàm si tigillus in partes æquales distingueretur. Simi- liter in follicum motu primùm comparanda est rotæ semidiame- A A a a 3

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Book Five. CHAPTER VII. 557 the first piston is pressed down, by means of which the water is forced into the ascending tube, and the other piston is raised, drawing up the water placed below, which in turn is then depressed by a vane of the Axis corresponding to that wing. Hence it comes about that a longer Axis may be added to the wheel, and several pairs of pistons of this kind may be arranged with vanes around the circumference of the Axis so distributed that they do not all press the pistons at once, but one after another, if the power turning the wheel is not sufficiently strong; but if it is stronger, they may depress several at once, and raise those joined to them. Unless perhaps it should seem better to use only two pistons joined together, with so many vanes disposed on the Axis that in one revolution of the same Axis the same piston is depressed twice or three times. To this likewise belong the things that are everywhere to be seen in the workshops of copper- or iron-smiths, where also an outer wheel, turned by the force of flowing water, inside in the chamber, as it were, winds a handle, which, being fixed to the upper Axis parallel to the horizon, and connected to itself by a rule jointed with flexible hinges, while it raises and lowers, at the other end of the same Axis a crosswise bellows similarly raising on one side and lowering on the other imparts an alternate motion: and another wheel, receiving the more powerful force of falling water and winding its own Axis, with vanes fixed to the Axis, depresses one end of a bar, to the opposite raised end of which is attached a huge iron hammer, which, as the vane of the Axis passes by, of its own accord falling back, strikes the copper or iron placed beneath, heated. In all these cases the semidiameter of the wheel must indeed be considered, for the water striking the projecting paddles acquires the force of the moving power; but the semidiameter of the Axis is not to be taken by itself, but the length of the projecting vane must also be added, so that from the two there is formed one single semidiameter of the motion that is communicated to the piston, or to the depressed end of the hammer. I say the depressed end of the hammer, for the raising of the hammer is somewhat greater than that depression, so that a stronger blow may follow; for the rod is not divided equally through the axis on which it rests, but the hammer is a little farther from that axis than the opposite end, which is depressed: and therefore the force depressing it is greater than if the rod were divided into equal parts. Likewise in the motion of the bellows, first of all the semidiameter of the wheel must be compared to the semidiameter of the wheel. A A a a 3

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Mechanicorum ter cum adhærente manubrio, deinde Radius Axi superiori infixus comparandus est cum semisse transversarij, cui folles junguntur; & ex his duabus Rationibus componitur Ratio momentorum potentiæ ad momenta ponderis movendi. At verò in molendinis, quibus mola frumentaria plano horizontali parallela circumagenda est, & quidem velociter, ut granum in farinam dissolvatur, non satis est exterior rota aquæ impetum recipiens & Axem sibi infixum volvens, sed etiam interior rota denticulata in eodem Axe requiritur; & ne Machinæ membra frustrà multiplicentur, ita molares lapides communiter disponuntur, ut ferreus axis metam sustinens, & curriculo instructus, inferiorem locum obtineat, ac proinde curriculus ipse proximè attingat superiorem partem interioris rotæ in suo plano denticulatæ eundem cum exteriore rotâ axem habentis. Quod si moleares lapides collocari non possint in plano, infra vel supra quod volvatur rota interior denticulata, sed solùm paulò infra, aut supra axem ejusdem rotæ; quia Vertebra striata proximè molari lapidi cohærens, adeóque lapidem ipsum volvens, distat, à rotâ denticulatâ, hæc autem commodè non admittit tam longos dentes, qui ejusdem Vertebrae aut curriculi virgulis aptè commisceri valeant, propterea exigitur alius Axis horizonti perpendicularis curriculo & rotæ infixus, quem convertat rota interior curriculi hujus virgulas suis dentibus impellens; simul enim rota dentata horizonti parallela, eidem Axi perpendicularis infixa volvitur, & curriculum molæ conjunctum circumagit. Hîc quoque plures Rationes componendæ sunt; prima est Ratio diametri rotæ exterioris ad diametrum rotæ interioris in eodem axe; deinde Ratio diametri curriculi molæ adhærentis ad ipsius molæ circumactæ diametrum (sive integra diameter accipienda sit, sive illa tantum pars, quæ est diameter circuli in rotatione molæ descripti à puncto inter centrum & peripheriam intermedio) & si, ut in secundo casu, interjectus fuerit Axis perpendicularis, prætereà in compositionem venit Ratio diametri curriculi ad diametrum rotæ denticulatæ in eodem Axe perpendiculari. Ex quibus apparet præstare rotæ interioris diametrum minorem esse diametro rotæ exterioris, ut aquæ hanc impellentis momenta validiora sint: sed & cavendum, ne illa

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In mechanics, the third with the handle attached; then the radius fixed to the upper axis is to be compared with half the transverse beam, to which the bellows are joined; and from these two ratios is composed the ratio of the moments of the power to the moments of the weight to be moved. But in mills, in which a millstone parallel to a horizontal plane must be turned, and indeed quickly, so that the grain may be dissolved into flour, it is not enough that the outer wheel receive the impulse of the water and revolve the axis fixed to it, but an inner toothed wheel is also required on the same axis; and lest the parts of the machine be multiplied in vain, the millstones are commonly so arranged that the iron axis, supporting the millstone and furnished with a carriage, occupies the lower position, and therefore the carriage itself nearly touches the upper part of the inner wheel, which is toothed in its plane and has the same axis as the outer wheel. But if the millstones cannot be placed in the plane beneath or above which the toothed inner wheel revolves, but only somewhat below or above the axis of that same wheel; because the grooved spindle, being attached very near to the millstone and thus turning the stone itself, is at a distance from the toothed wheel, and this wheel does not conveniently admit teeth so long that they can fitly engage the rods of the same spindle or carriage, therefore another axis perpendicular to the horizon is required, fixed to the carriage and wheel, which the inner wheel turns by driving the rods of this carriage with its teeth; for at the same time the toothed wheel fixed parallel to the horizon, perpendicular to the same axis, is turned, and drives around the carriage joined to the millstone. Here also several ratios must be composed: the first is the ratio of the diameter of the outer wheel to the diameter of the inner wheel on the same axis; then the ratio of the diameter of the carriage attached to the millstone to the diameter of the millstone itself when turned (whether the whole diameter is to be taken, or only that part which is the diameter of the circle described in the rotation of the millstone from the point midway between the center and the periphery); and if, as in the second case, an intervening perpendicular axis has been inserted, besides this there enters into the composition the ratio of the diameter of the carriage to the diameter of the toothed wheel on the same perpendicular axis. From which it appears that it is better for the diameter of the inner wheel to be smaller than the diameter of the outer wheel, so that the moments of the water impelling this wheel may be stronger: but care must also be taken lest that

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Liber quintus. CAPUT VII. 559 illa ita minor statuatur, ut ejus dentium numerus vix excedat numerum virgularum curriculi molæ adhærentis, hæc enim nimis tardè moveretur; & si intermedius fuerit Axis perpendicularis, positâ hac dentium æqualitate & virgularum curriculi, unica rotæ exterioris conversio semel tantùm convolveret rotam denticulatam horizonti parallelam, atque idcirco eodem tempore mola toties solùm converteretur, quoties numerus virgularum ejus curriculi contineretur in numero dentium rotæ denticulatæ infixæ Axi perpendiculari. Ut autem convolutionem molæ numerum augeas, cave ne movendi difficultas pariter plus justo augeatur, si nimirum in axe perpendiculari diameter curriculi sit immodicè minor diametro rotæ denticulatæ in eodem axe: potentia si quidem curriculo applicata multo tardiùs moveretur, quàm pondus extremis rotæ dentibus applicatum, ac proinde movendi difficultas augeretur. Quare omnia prudenter administranda, ut neque potentiæ moventis vires frustra conterantur, neque mola tardiùs aut velociùs, quàm par sit, moveatur. Quod si non placuerit, aut loci dispositio non tulerit, axem illum intermedium statui perpendicularem, sed horizonti parallelus commodior accidat, tunc rotæ interioris eundem cum exteriore rotâ axem habentis dentes non plano infixi, sed in extreto ambitu defixi requiruntur, ut superioris axis curriculum (sive majorem, sive minorem, prout opus fuerit) convertant, & cum eo rotam non in ambitu, sed in plano, denticulatam, à qua molæ curriculum convolvatur. Neque aliter, ac priùs, momentorum Ratio componitur, ex Rationibus videlicet tympanorum, quæ communem Axem habent, ut satis constat ex dictis. Hinc quoniam potentia movens est aqua, observamus non omnino eandem esse formam rotæ aquam excipientis; quæ enim in profluente collocantur rotæ, nimis incommodæ essent, si valdè amplam diametrum haberent; aut modico aquæ labentis impetu pellerentur, si palmulis exiguis instruerentur: propterea rotæ hujusmodi mediocrem quidem habent diametrum,

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Book five. CHAPTER VII. 559 that it be made so much smaller that the number of its teeth scarcely exceeds the number of the staves of the drum attached to the millstone; for otherwise this would be moved too slowly. And if the intermediate axis were perpendicular, with this equality of the teeth and the staves of the drum established, a single revolution of the outer wheel would turn only once the toothed wheel parallel to the horizon, and therefore at the same time the millstone would be turned only as many times as the number of the staves of its drum would be contained in the number of teeth of the toothed wheel fixed to the perpendicular axis. But in order to increase the number of revolutions of the millstone, take care lest the difficulty of movement likewise be increased beyond measure, if indeed on the perpendicular axis the diameter of the drum be excessively smaller than the diameter of the toothed wheel on the same axis: for the power applied to the drum would be moved much more slowly than the weight applied to the outer teeth of the wheel, and consequently the difficulty of movement would be increased. Therefore all things must be prudently managed, so that neither the forces of the moving power be wasted in vain, nor the millstone be moved more slowly or more quickly than is fitting. But if this does not please, or if the disposition of the place does not allow it, and that intermediate axis should not be set perpendicular, but rather it should happen that a horizontal one is more convenient, then the teeth of the inner wheel having the same axis as the outer wheel are required, not fixed on a flat surface, but fastened on the outer rim, so that they turn the drum of the upper axis (whether the larger or the smaller, as the case may require), and with it the wheel not on the rim, but in the plane, toothed, from which the drum of the millstone is turned. And in no other way than before is the ratio of the moments composed, namely from the ratios of the drums which have a common axis, as is sufficiently clear from what has been said. Hence, since the moving power is water, we observe that the form of the wheel receiving the water is not altogether the same; for wheels placed in a stream would be too inconvenient if they had a very large diameter; or they would be driven by the slight impulse of the flowing water if they were equipped with small vanes: therefore wheels of this kind have indeed a moderate diameter,

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560 Mechanicorum trum, sed valdè notabilem axis partem occupant palmulis adeò juxtà axis longitudinem expansis, ut à multâ aquâ in illas incurrente validiore impulsu circumagantur. Sic in Pa- do communiter Rotæ hujus longitudo est cubitorum 10, diameter tota cubitorum 6; interior rota diametrum habet cubit. 5 ́, dentes 108 plano infixos, & molæ curriculus in fusos 9 distinguitur; lapis autem molaris in crassitudine nu- merat uncias 6 aut 7, in diametro cubitos 2 ́. Quia ve- rò aquæ ex alto cadentis motus major est quàm profluen- tis, propterea rotarum diameter amplior statui potest, si opus fuerit, & palmularum latitudo valde mediocris suffi- cit, quippe inclusa canali, per quem aqua decidens labi- tur: modica scilicet aqua per planum magis elevatum pro- lapsa majora habet momenta, quàm per planum ferè ho- rizontale: & præterea rota amplioris diametri faciliùs vol- vitur etiam à minore aquâ, nam ad interiorem rotam, cæteris paribus, habet majorem Rationem. Porrò palmu- læ communiter quidem planæ sunt, aut non nisi mo- dicè sinuatæ, ita ut aqua hinc atque hinc diffuat; ali- quando tamen limbo ex utraque parte concluduntur, & quasi vascula aquam aliquandiu continent, ut ipsius aquæ inclusæ gravitas conversionem juvet deorsum urgendo. Ad- de in ipso canali inclinato majores esse vires aquæ in parte inferiore, quàm in superiore propè initium casûs; quia vide- licet aqua naturaliter descendens motum habet acceleratum, & ex antecedente descensu acquisivit impetum. Hactenus Molendina, quæ aquarum vi aguntur conside- ravimus, nihil addentes de iis, quæ ab hominibus, aut ab animalibus volvuntur, nihil enim hæc habent peculiare præ- terquàm quod axis primæ rotæ, quæ cæteris consequentibus membris motum conciliat, est horizonti perpendicularis, quia potentia faciliùs in plano horizontali movetur, quàm in tym- pano Verticali, quod calcaretur, & loco exterioris rotæ ab aquâ propulsæ vectis axi infigitur, quem aut jumenta trahunt, aut homines urgent. Aliquid tamen innuendum de Molendinis, quæ vento aguntur, sive ad comminuendas fruges, sive etiam ad agi- tandas

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560 Mechanics but they occupy a very notable part of the axis with their little arms spread out so close to the length of the axis that they are turned by the stronger impulse of much water running against them. Thus in the Po commonly the length of this wheel is 10 cubits, the total diameter 6 cubits; the inner wheel has a diameter of 5 cubits, with 108 teeth fixed in a plane, and the mill-run is divided into 9 spindles; but the millstone in thickness measures 6 or 7 inches, and in diameter 2 cubits. But because the motion of water falling from a height is greater than that of water flowing out, for that reason the diameter of the wheels may be made larger, if necessary, and a very moderate width of the paddles is sufficient, since they are enclosed in the channel through which the falling water glides: indeed a small quantity of water, having fallen over a more elevated plane, has greater force than through a nearly horizontal plane; and moreover a wheel of larger diameter turns more easily even with less water, for toward the inner wheel, other things being equal, it has a greater ratio. Furthermore, the paddles are commonly flat, or only moderately curved, so that the water flows off on this side and that; sometimes however they are enclosed by a rim on both sides, and like little vessels they retain the water for a while, so that the weight of the enclosed water helps the turning by pressing downward. Add that in the inclined channel itself the forces of the water are greater in the lower part than in the upper near the beginning of the fall; because, namely, water naturally descending has accelerated motion, and from the preceding descent has acquired impetus. So far we have considered mills that are driven by the force of waters, adding nothing about those that are turned by men or by animals, for these have nothing peculiar except that the axis of the first wheel, which imparts motion to the following members, is perpendicular to the horizon, because power is more easily moved in a horizontal plane than in a vertical drum, which would be treaded upon, and in place of the outer wheel driven by water a lever is fixed to the axis, which either draft animals pull or men urge. Something must however be indicated about mills that are driven by wind, whether for crushing grain, or also for driving

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Liber quintus, CAPUT VII. 561 tandas antlias, quibus aquæ depressioribus campis insiden- tes exhauriuntur. Quod enim attinet ad interius artificium rotarum & curriculorum, simillimum est iis, quæ in nostra- tibus molendinis aquâ urgente commotis reperiuntur, nisi quod in illis, ut pote à subjectâ planitie remotis ( locus si- quidem amplo ventilabro opportunus tribuendus est, & cap- tandus ventus) per scalas ascenditur, & in superiorem lo- cum comportandæ sunt fruges, quas commolere oportet, atque farina inde transferenda: quo labore levari potest molitor, si operâ eâdem, qua ventus axem primarium cum rotis versat, saccos tritico aut farinâ plenos attollat, aut de- ponat, fune ductario circa ipsum Axem convoluto, aut evo- luto. Illud potissimum in hoc molendinorum genere atten- dendum est, quod ad ipsa flabella, quibus ventus excipi- tur, spectat; neque enim quemadmodum juxta aquæ cur- sum rotæ planum dirigitur, etiam ventilabrum flabella habet ita disposita, ut venti ductum sequantur: sed superior do- munculæ pars, qua Axis cum rotâ denticulatâ continetur, usque adeò convertitur, ut ventilabrum flanti vento adver- sum statuatur. Sunt autem flabella quasi quatuor scalæ in primarij Axis extremitate conjunctæ, quibus obducitur singulis linteum, ut vento resistat; qui si justo validior fuerit, lintei pars complicata aliquem vento exitum præbet. Non tamen fla- bella hæc ita ex æquo collocantur, ut in uno eodemque pla- no Verticali constituantur, sed singulorum flabellorum pla- num modicè obliquum statuitur latere altero se paulatim sub- ducente à vento. Ex quo fit ventum inter quatuor flabel- lorum intervalla intercurrentem repellere in latus, & quasi cubito percutere ipsa flabella, atque adeò Axem converti juxta flabellorum inclinationem. Nam si nulla esset flabel- lorum obliquitas, & omnia quasi unicum planum efficerent, in quod Axis esset perpendicularis, incertum esset, quam in partem fieret conversio. Quod ad latitudinem aut longi- tudinem hujusmodi flabellorum obliquè positorum attinet, non dubitatur, quin eorum latitudo maximè juvet motum; quia eâdem obliquitate positâ, major aëris pars incurrit in BBbb

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Liber quintus, Chapter VII. 561 ...these lifting machines, by which water lying in lower fields is drawn off. For as regards the internal mechanism of the wheels and the carriages, it is very similar to those found in our mills driven by the pressure of water, except that in the latter, since they are removed from the level ground beneath (for a place suited to a large sail must certainly be provided, and the wind caught), one ascends by stairs, and the crops that must be ground, and the flour to be carried away from them, are brought to the upper place. By this labor the miller may be relieved if, by the same force with which the wind turns the main axis together with the wheels, he lifts or lowers sacks filled with grain or flour, by a carrying rope wound around, or unwound from, the Axis itself. The chief thing to be observed in this kind of mill is what concerns the very vanes by which the wind is received; for the wheel is not directed, as is the case beside the course of water, so that the plane follows it, nor are the vanes of the ventilator so arranged that they follow the direction of the wind: rather, the upper part of the little house, in which the Axis with the toothed wheel is contained, is turned so far that the ventilator is set opposite the blowing wind. These vanes are, as it were, four ladders joined to the end of the principal Axis, each of which is covered with linen, so that it may resist the wind; and if the wind is somewhat stronger than is proper, the folded part of the linen gives the wind some outlet. Yet these vanes are not placed so evenly that they lie in one and the same vertical plane, but the plane of each vane is set moderately oblique, with one side gradually withdrawing from the wind. From this it comes about that the wind, running between the intervals of the four vanes, is driven aside and, as it were, strikes the vanes with its elbow, and thus the Axis is turned according to the inclination of the vanes. For if there were no obliquity of the vanes, and all together formed, as it were, a single plane to which the Axis was perpendicular, it would be uncertain to which side the turning would occur. As for the width or length of such obliquely placed vanes, there is no doubt that their width most greatly aids the motion; because, with the same obliquity, a greater part of the air strikes in... BBbb

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Mechanicorum 562 amplius quàm in strictius linteum; & in vehementiori ven- to, ne nimia sit machinæ velocitas, experimur aliquando non nisi dimidium velum expandi. An verò fuerit operæ pretium horum longitudinem augere, incertum est: quam- vis enim potentia magis à centro motûs distans plus habeat momenti, tamen quia longiorum flabellorum extremitates valde inter se distarent, ventus ampliora spatia nactus mi- nus haberet virium; sicut & aqua fluens, velociùs atque majore conatu per angustias, quàm per patentem alveum currit. Propterea in hujusmodi flabellis non auderem omni- no definire, quo loco potentiæ moventis vires statuendæ sint quasi in centro virtutis; nam prope Axem, cui infixa sunt, modica est distantia, & ventus quasi eorum objectu compressus velociùs spirat, procul autem ab Axe in majo- re intervallo faciliùs elabens minùs incitat cursum. Cum verò non sit temerè statuendum venti compressionem om- nino respondere mutuis flabellorum distantiis, quæ in eâ- dem Ratione sunt ac distantiæ ab Axe; neque facilè asseri potest eâdem Ratione decrescere vim venti ex compressio- ne, qua ejusdem momenta crescunt ex distantiâ ab Axe: Ex quo fieret momenta composita ex distantia ab Axe, & ex vi compressionis, esse per totam flabelli longitudi- nem æqualiter diffusa, ac proinde in mediâ longitudine es- se Centrum virtutis moventis. Omnibus tamen ritè per- pensis, existimarem centrum hoc virtutis, cui applicata potentia intelligitur, haud procul abesse à mediâ flabelli lon- gitudine: Nisi fortè flabella ipsa talia essent, ut eorum la- titudo ab Axe recedens augeretur; sic enim diminutâ in extremitatibus flabellorum distantiâ, etiam venti compressio augeretur. Quod si occurrendum putares incommodo, quod subire necessè est ædiculam polo innixam ita convertendo, ut fla- bella adversum ventum excipiant, haud abs re esse duce- rem, si quis in supremo domûs fastigio, loco patente & ventis omnibus exposito, crassum satisfque validum axem ho- rizonti perpendicularem statueret, quem rota denticulata horizonti parallela complecteretur, ex cujus conversione de- mum

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Mechanics 562 more than into a tighter cloth; and in a stronger wind, lest the speed of the machine be too great, we sometimes find that not even half the sail is spread out. But whether it would be worth while to increase the length of these is uncertain: for although power farther from the center of motion has greater effect, yet because the ends of longer vanes would be very far apart from one another, the wind, having obtained larger spaces, would have less force; just as flowing water runs more quickly and with greater effort through narrow places than through an open channel. Therefore in vanes of this kind I would not at all dare to determine exactly where the forces of the moving power ought to be placed, as though in the center of the force; for near the axis to which they are fixed there is a small distance, and the wind, as it were compressed by their obstruction, blows more swiftly, but farther from the axis, at a greater interval, escaping more easily, it stimulates the course less. But since it should not rashly be assumed that the compression of the wind altogether corresponds to the mutual distances of the vanes, which are in the same ratio as the distances from the axis; nor can it easily be asserted that the force of the wind decreases by compression in the same ratio as its moments increase by distance from the axis: from which it would follow that the moments, composed of the distance from the axis and of the force of compression, are equally diffused throughout the whole length of the vane, and therefore the center of the moving force lies in the middle of the length. Yet all things duly considered, I would think this center of force, to which the applied power is understood to act, is not far from the middle length of the vane: unless perhaps the vanes themselves were such that their breadth increased as it receded from the axis; for thus, with the distance between the vanes diminished at the extremities, the compression of the wind would also increase. If however you were to think that objection should be met, which one must necessarily encounter by turning a little building supported on a pole so that the vanes receive the wind head-on, I would not think it out of place if someone were to set on the highest ridge of the house, in an open place and exposed to all winds, a thick and sufficiently strong axis perpendicular to the horizon, which a toothed wheel parallel to the horizon would encompass, from whose turning then

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Liber quintus. CAPUT VII. 563 mum mola circumageretur. At flabellorum latitudo juxta Axis longitudinem in ejusdem supremo capite extra tectum collocanda esset, ut incurrentis venti impulsum exciperent, perindè atque fluentis aquæ impetum recipiunt palmulæ ro- tarum. Sed quoniam plana flabella parùm apta videntur ad conversionem continuandam, quia, quæ sunt à diametro opposita, demùm venti viribus exponerentur æqualiter, nec dexterum potiùs quàm sinistrum impellendum esset, adeó- que cessaret conversio; propterea flabella construenda es- sent modicè incurva; hac enim ratione fieret, ut opposita inæqualiter urgerentur, & dextri quidem convexam, si- nistri verò cavam faciem ventus impeteret inæqualibus vi- ribus, illud scilicet quasi se subducit vento, nec admo- dum ejus impulsui opponitur extremitas juxta venti directionem inflexa; hoc autem cavo sinu ventum excipiens to- tum ejus impulsum recipit. Adde quod venti particula in duo proxima flabella incurrens à convexâ unius facie in ca- vam proximi faciem reflectitur, & auget impulsionem. Quod si placuerit non quatuor, sed quinque flabella sta- tuere, ne unquam duo ex diametro opponantur, non ab- nuo. Illud certum est hujusmodi flabellorum tùm longi- tudinem, tùm latitudinem plurimùm juvare, quo enim ampliora sunt, plus venti excipiunt, & quò longiora, ut pote à motûs centro magis sejuncta, plus habent momen- ti. Quomodo autem sistenda sit machina, explicanda aut complicanda vela, ne præter molitoris voluntatem agitentur flabella, nil refert hîc pluribus disputare, ubi tantummodo vis movendi consideratur. Neque solum hujus molendini usus esset in comminuendis tritici aut leguminum granis, sed etiam in attollendis atque alio derivandis aquis, ut palus exsiccetur, & cæteris hujusmodi, quæ præsente semper cor- pore movendo, non certo tempori alligantur, quemadmodum, opus molendi, quod non perpetuò exercetur. B B bb 2

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Book Five. CHAPTER VII. 563 The mill was to be turned about by means of a wheel. But the breadth of the vanes, in proportion to the length of the axis, ought to be placed at the uppermost end of the same, outside the roof, so that they might receive the force of the incoming wind, just as the paddles of wheels receive the force of flowing water. But since flat vanes seem somewhat ill-suited to maintaining continuous rotation, because those opposite each other on the diameter would at length be exposed equally to the force of the wind, and neither side would be driven more than the other, so that the turning would cease; for that reason the vanes ought to be made moderately curved. For by this means it would come about that the opposite ones were pressed unequally, and that the right-hand vanes would indeed be struck on their convex face, the left-hand ones on their hollow face, by unequal forces: the one, as it were, drawing itself away from the wind, and not greatly opposing the impact with its edge bent in the direction of the wind; the other, receiving the wind in its hollow hollow, takes in its whole impulse. Add to this that a portion of wind, striking two adjacent vanes, is reflected from the convex face of one to the hollow face of the next, and increases the impulse. And if it should please one to set up not four but five vanes, so that no two may ever stand opposite each other, I do not object. This much is certain: the length as well as the breadth of such vanes is of great advantage; for the broader they are, the more wind they receive, and the longer they are, as being farther removed from the center of motion, the greater is their leverage. But how the machine is to be stopped, or the sails opened or folded, so that the vanes may not move contrary to the miller’s will, there is no need to discuss further here, where only the moving force is under consideration. Nor would the use of this mill be only for grinding wheat or legume grains, but also for raising and diverting waters elsewhere, so that a marsh may be drained, and for other uses of this kind, which always require motion present in the body, and are not bound to any fixed time, as is the work of milling, which is not carried on continuously. B B bb 2

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Mechanicorum CAPUT VIII. Axis cum Vecte compositus auget Potentia momenta. Tanta est aliquando ponderis gravitas, ut datæ potentiæ vires illi movendo impares sint, aut de oblatæ machinæ soliditate ac firmitate dubitetur: propterea opportunum accidet Vectem cum Axe in Peritrochio componere. Primum dato Vecte AB secundi generis, cujus hypomochlium sit B, & pondus constitutum in C, Potentia, quæ in extremitate A applicanda est, minor sit, quàm pro gravitate ponderis, datâ vectis Ratione CB ad AB. Adhibeatur succula EF opportunè collata, ut funis ductarius in A alligatus Vectem attollat: momenta enim potentiæ componuntur ex Rationibus radiorum succulæ ad semidiametrum Axis, & distantiæ AB ad distantiam CB in vecte. Hinc si Ratio AB ad CB sit ut 3 ad 1, Ratio autem radiorum ad Axis semidiametrum sit ut 4 ad 1, unicus homo Succulam vertens momentum habet æquale momentis quatuor hominum in A vecti applicatorum, quorum singuli æquiparantur tribus, qui pondus idem sine vecte attollere conarentur: atque adeò unicus homo succulam convertens æquat vires duodecim hominum ponderi ipsi proximè applicatorum citra quodlibet machinæ subsidium. Deinde

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Mechanics CHAPTER VIII. An Axis combined with a Lever increases the moments of force. Sometimes the weight is so great that the strength of the given power is insufficient to move it, or one may doubt the solidity and firmness of the proposed machine: for this reason it will be fitting to combine the Lever with the Axis in the wheel-and-axle. First, let there be given a Lever AB of the second kind, whose fulcrum be at B, and the weight placed at C, so that the Power, which is to be applied at the extremity A, may be less than would be required for the weight, according to the ratio of the lever CB to AB. Let the windlass EF be suitably placed, so that the rope, attached at A, may raise the Lever: for the moments of the power are composed from the ratios of the radii of the windlass to the semidiameter of the Axis, and of the distance AB to the distance CB on the lever. Hence, if the ratio of AB to CB be as 3 to 1, and the ratio of the radii to the semidiameter of the Axis be as 4 to 1, a single man turning the windlass has a moment equal to the moments of four men applied at A on the lever, each of whom is equivalent to three who would try to raise the same weight without the lever: and thus a single man turning the windlass balances the strength of twelve men applied directly to the weight itself, apart from every assistance of the machine. Then

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Liber quintus. CAPUT VIII. 565 Deinde in vecte primi generis, quando movendo pon- deri velocitas aliqua concilianda est, validiore potentiâ opus est, & tamen adjecto Axe infirmæ potentiæ adjumentum comparare in promptu est. Sit enim vectis I G, & hypo- mochlium in H. Uti- que potentia in I tan- to major requiritur, quanto major esse de- bet ponderis motus su- pra motum potentiæ, hoc est in Ratione H G ad H I. Statua- tur Axis R S, & fu- nis ductarius Vectem apprehendat in I. Tum axi infigatur Radius V T; nam pro Ratione longitudinis V T ad Axis semidiametrum ita augeri possunt potentiæ momenta, ut non solùm ponderis gravitati paria sint, sed & illam excedant. Fac enim I H ad H G esse ut 1 ad 4, pondus verò in G esse lib. 200, certè requireretur in I potentia major libris 800, ut suâ virtute gravitati ponderis præstaret: At si Radius V T ad Axis R S semidiametrum sit ut 10 ad 1, jam potentia in T motum habet ad motum ponderis in G ut 10 ad 4: igitur reciprocè potentia in T ad pondus in G esset ut 4 ad 10, ac proinde potentia habens vires attollendi absque machinâ libras 80, applicata in T attollet libras 200. Hæc quæ de attollendo pondere dicta sunt, intellige pariter si in plano horizontali aut inclinato movendum esset; collocato scilicet Axe non parallelo horizonti, sed vel perpendiculari, vel in- clinato, pro ut loci opportunitas feret: hîc siquidem sola mo- mentorum incrementa considerantur ex harum duarum Fa- cultatum compositione. Quid autem opus est monere idem virium compendium haberi posse in Vecte pariter primi generis, quando pon- dus tardè movendum est? res enim per se clara est, hypo- mochlio scilicet magis ad extremitatem G accedente, quàm ad extremitatem I, quæ potentiæ locus est, ut si esset in L: BBbb 3

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Book five. CHAPTER VIII. 565 Next, in the first kind of lever, when some velocity must be imparted to the weight being moved, a stronger power is required; and yet, by adding an axle, it is easy to obtain the assistance of a weaker power. For let there be the lever I G, and the fulcrum at H. Certainly a greater power is required at I, the greater the motion of the weight must be above the motion of the power, that is, in the ratio of H G to H I. Let the axle R S be set up, and let the driving rope grasp the lever at I. Then let the radius V T be fixed to the axle; for according to the ratio of the length of V T to the semidiameter of the axle, the moments of the powers can be increased so that they may not only equal the weight of the load, but even exceed it. Suppose indeed that I H to H G is as 1 to 4, and that the weight at G is 200 lb.; certainly a power greater than 800 lb. would be required at I, in order that by its own force it might overcome the weight’s gravity. But if the radius V T to the semidiameter of the axle R S be as 10 to 1, then the power at T has to the motion of the weight at G the ratio of 10 to 4; therefore reciprocally the power at T to the weight at G would be as 4 to 10, and consequently a power capable of lifting 80 lb. without machinery, applied at T, will lift 200 lb. What has been said here about lifting a weight, understand likewise if it were to be moved on a horizontal or inclined plane; that is, with the axle placed not parallel to the horizon, but either perpendicular or inclined, as the convenience of the place shall require: for here only the increase of moments is considered, arising from the combination of these two faculties. But why is it necessary to note that the same saving of force may likewise be obtained in a lever of the first kind, when the weight is to be moved slowly? The matter is clear of itself, namely, the fulcrum approaching more toward the extremity G than toward the extremity I, which is the place of the power, as if it were at L: BBbb 3

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Mechanicorum id quod tunc potissimùm usurpari potest, cùm elevatio pon- deris ad aliquam non minimam altitudinem requiritur; opor- tet enim hypomochlium à pondere intervallo notabili abesse, unde & major movendi difficultas oritur, atque idcircò addi- tâ succulâ potentiam juvari necesse est. Succulam verò po- tiùs adhibendam proponere censui, quippe quæ & parabilior est, & commodior, nec multis impensis construitur: Cæte- rum nec Ergatam, nec tympana seu Grues, nec rotas denta- tas, si placuerint, excludo. Ex his satis liquet, quid de Vecte tertij generis dicendum sit, in quo Potentia media inter pondus & hypomochlium collocatur: Succula scilicet in superiore loco statuenda est, ita ut funis ductarius vectem apprehendat, ubi potentiæ lo- cus assignatur: sed quoniam minor est potentiæ, quàm pon- deris motus, & augenda sunt potentiæ momenta, ut ponde- ris gravitati elevandæ par sit, Axi addendus est Radius tantæ longitudinis, ut potentia non jam Vecti, sed Radio applicata velociùs moveatur, quàm pondus. Hactenus Axem in Peritrochio additum Vecti conside- ravimus, quatenus Vectem solitarium infirmior potentia movere nequit: Nunc Vectem addere opportet Axi in Peritrochio, ut hujus usus illo addito facilior accidat. Ma- chinulam secum deferunt communiter aurigæ in Germania, qua rotam currûs, si fortè limo profundiùs infixa inhæse- rit, sublevant, ac proinde recte Pancratium aurigarum dici potest. Lamina est chalybea denticulata, cui rotula pariter dentata congruit, cujusmodi initio capitis 6. descripsimus: parvula tamen est rotula illa, sed centrum habens commu- ne cum rotâ majore similiter dentatâ, ex cujus conversione minor convolvitur, & laminam sursum propellit. Majoris rotæ dentes apprehendit Axis striatus, cujus motûs princi- pium ducitur à manubrio extra loculamentum ad latus ex- tante. Quare duplex est Ratio, videlicet manubrij ad semi- diametrum axis striati, atque diametri rotæ majoris ad dia- metrum rotulæ minoris concentricæ; ex quibus componi- tur Ratio motûs Potentiæ manubrium versantis, ad motum ponderis sublevati. Quia autem fieri potest, ut aut de lami- næ

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Mechanics that which can then especially be used, when the raising of a weight to some not insignificant height is required; for the hypomochlion must be at a notable distance from the weight, whence a greater difficulty of moving also arises, and therefore, with the addition of a windlass, it is necessary that the power be assisted. But I judged that the windlass should rather be employed, since it is both more readily available and more convenient, and is constructed at little expense. Moreover, I do not exclude the ergate, nor drums or cranes, nor toothed wheels, if they should be preferred. From these things it is sufficiently clear what must be said about the lever of the third kind, in which the Power is placed between the weight and the hypomochlion: namely, the windlass is to be set in the upper position, so that the hauling rope seizes the lever where the place of the power is assigned; but since the power is less than the movement of the weight, and the moments of the power must be increased so that it may be equal to lifting the weight’s heaviness, a Radius of such length must be added to the axis that the power, applied no longer to the Lever but to the Radius, may move more quickly than the weight. Thus far we have considered the Axis added to the Peritrochium as it supports the solitary Lever, which an inferior power cannot move: now it is necessary to add a Lever to the Axis in the Peritrochium, so that the use of the latter with this addition may be easier. Coachmen in Germany commonly carry with them a little machine by which, if a wheel of the cart should by chance be stuck deeper in mud, they lift it; and therefore it can rightly be called the pancratium of coachmen. It is a toothed steel plate, with which a likewise toothed little wheel corresponds, of the sort we described at the beginning of chapter 6: that little wheel is indeed small, but it has a common center with a similarly toothed larger wheel, from whose turning the smaller is wound around, and drives the plate upward. The teeth of the larger wheel are engaged by a grooved axis, the principle of whose motion is derived from a handle extending outside the housing at the side. Therefore there is a double ratio, namely of the handle to the radius of the grooved axis, and of the diameter of the larger wheel to the diameter of the smaller concentric wheel; from these is composed the ratio of the motion of the power turning the handle to the motion of the lifted weight. But since it may happen that either from the plate's

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Liber quintus. CAPUT VIII. 567 næ soliditate dubitetur, aut subjectum rotæ solum non ad- mittat congruam machinulæ positionem; tunc rotæ elevan- dæ capiti subjiciatur validus fustis alterâ extremitate incumbens telluri, alterâ innixus dentatæ laminæ; quæ eò minùs à plaustri onere gravabitur, quò major erit Ratio totius lon- gitudinis fustis ad ejus partem inter rotæ caput, & solum, cui innititur, interjectam. Hinc si Ratio vectis sit ut 2 ad 1, machinæ lamina non nisi à ponderis semisse gravatur; & Potentiæ manubrio Pan- cratij applicatæ momenta geminantur. Nam si manubrij longitudo ad Axis striati semidiametrum sit ut 8 ad 1, ro- tæ autem majoris diameter ad rotulæ concentricæ diametrum sit ut 4 ad 1, potentiæ motus ad motum laminæ dentatæ est ut 32 ad 1: sed apposito vecte, cujus Ratio datur ut 2 ad 1, jam motus potentiæ ad motum ponderis elevati est ut 64 ad 1, & potentiæ conatus, qui satis esset ad attollendas sine machinâ libras 20, hoc Pancratio unâ cum Vecte attol- tolleret libras 1280. Similiter si Ergatâ AB raptandum esset onus, & poten- tia infirmior es- set, quam ut in extremitate Ra- dij CD valeret superare oneris resistentiam, ad- hibe Vectem EF, & extre- mitate E inni- tente subjecto solo, potentia applicetur ex- tremitati F; nam ejus momenta componu[n]tur ex Rationibus CD Radij ad semidiametrum Axis AB, & vectis FE ad DE. Potest autem post aliquantulum motum subinde promoveri extremitas

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Book five. Chapter VIII. 567 if there is doubt about the solidity of the axle, or if the ground beneath the wheel does not allow a suitable position for the machine; then, with the wheels to be raised, let a strong stick be placed under the head, resting at one end on the ground and at the other against the toothed plate; this will be the less burdened by the load of the cart, the greater the ratio of the whole length of the stick to that part of it which is inserted between the wheel head and the ground on which it rests. Hence, if the ratio of the lever is as 2 to 1, the plate of the machine is burdened by no more than half the weight; and the moments of the Power applied to the handle of the Pancratium are doubled. For if the length of the handle to the semidiameter of the grooved axle is as 8 to 1, but the diameter of the larger wheel to the diameter of the concentric small wheel is as 4 to 1, the motion of the power to the motion of the toothed plate is as 32 to 1: but with a lever added, whose ratio is given as 2 to 1, now the motion of the power to the motion of the raised weight is as 64 to 1, and the effort of the power, which would be sufficient to raise 20 pounds without a machine, with this Pancratium together with the lever would raise 1280 pounds. Similarly, if a load were to be dragged by means of the windlass AB, and the power were weaker than to be able, at the end of the radius CD, to overcome the resistance of the load, let the lever EF be applied, and with its end E resting on the ground beneath, let the power be applied at the end F; for its moments are compounded from the ratios of the radius CD to the semidiameter of the axle AB, and of the lever FE to DE. But after a little motion, the end may then be progressively advanced

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568 Mechanicorum extremitas E vectis, ut manifestum est. Quod si Ergatâ ipsâ uteremur ad sensim demittendum in plano inclinato onus quoddam ingens, & timeretur, ne vis gravitatis vinceret co- natum hominum in D reluctantium, ne præceps delabatur onus; adhibeatur vectis E F, quo sensum dimisso certiùs reti- netur onus, & lentiùs descendit. CAPUT IX. Multiplex rotarum dentatarum usus innuitur. Quanquam ea, quæ ad Mechanicam scientiam spectant circa tympana dentata, satis in superioribus explicata sint, quatenus ex iis subsidium petitur ad virium supplemen- tum, & fontes indicati sint, ex quibus unusquisque variam hujusmodi tympanorum complexionem pro opportunitate ex- cogitare possit; placuit tamen auctarium adjicere multiplicis usûs, etiam aliquando citra momentorum potentiæ moven- tis incrementum. Illud autem generatim observandum est, ne pluribus membris distinguatur machina, si pauciora suf- ficiant: fieri siquidem non potest, quin motui mora aliqua inferatur, ubi plurium membrorum multiplex conflictus at- que tritus contingit, etiamsi omnia ritè disponantur, & sibi invicem proportione respondeant. PROPOSITIO I. Anemoscopium, Ventorum flantium indicem describere. Si quis in conclavi manens cognoscere cupiat, quo vento impellatur aër externus, & Anemoscopium construen- dum curet, si hoc quidem in fornice, aut in laqueari des- cribendum sit, nullo opus est artificio; sed satis est intra laminæ V K foramen erecto axi perpendiculari A B, qui nodo

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568 Mechanics the end of the lever E, as is evident. But if we were to use the very Ergata for slowly lowering some huge load on an inclined plane, and there were fear lest the force of gravity should overcome the effort of the men in D resisting, so that the load might slip down headlong, let the lever E F be applied, by which, the handle being released, the load is more surely retained, and descends more slowly. CHAPTER IX. The multiple use of toothed wheels is indicated. Although those things which pertain to the science of Mechanics concerning toothed drums have been sufficiently explained above, insofar as from them aid is sought for the supplement of forces, and the sources indicated from which each person may devise, as opportunity requires, a varied construction of such drums; nevertheless it has seemed good to add something by way of supplement concerning their multiple use, even sometimes apart from any increase in the power moving the moments. But this should be observed in general: that the machine should not be divided into many parts if fewer are sufficient; for it cannot happen that some delay is not introduced into the motion, where a manifold collision and rubbing of many parts occurs, even if all things are arranged properly and correspond to one another in due proportion. PROPOSITION I. To describe an Anemoscopium, an indicator of blowing winds. If someone, while remaining in a room, should wish to know by what wind the external air is driven, and should take care to construct an Anemoscopium, if this is indeed to be described in a vault or in the ceiling, no artifice is needed; but it is enough, within the opening of the plate V K, for a perpendicular axis A B to be erected, which node

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Liber quintus. CAPUT IX. 569 node C laminæ insistat facilè versatilis, adjicere flabellum AD supra tecti fastigium loco apto ita eminens, ut directè, citra reflexionum suspicionem, cujuslibet auræ flantis impulsum excipiens, & venti ductum sequens convertatur, atque ejus extremitas D coeli plagam vento oppositam respiciat. In alterâ verò axis extremitate B infra laquearis aut fornicis faciem, in qua ritè juxta horizontis positionem descripti sint ventorum cardines, adnectatur index B F, ea lege, ut ex diametro contrariam flabello A D positionem B F obtineat: hinc enim fiet, ut quoniam ventus ex A in D directus spirat, index F eam horizontis partem, unde flat, respiciat. Sin autem in plano Verticali (auto etiam inclinato) describendum sit Anemoscopium, sit axis A H cum flabello A D transiens per C foramen, & acutâ cuspide insistens plano H, ut facillimè converti queat, vertebram stria- tam E G habens in octo æquales strias distinctam, quibus subinde exactè congruere possint rotæ M N dentes octo, in quos ferrea lamina distributa est æqualiter, antequàm in circulum inflecteretur. Ex hujus rotæ centro infixus exeat axis R parietem pervadens, & in extremitate adnexum indicé convolvens ad indicandos ventos in interiori, aut exteriori parietis facie descriptos. Verùm in ventorum descriptione cavendum, ne, quemadmodum in Mappis Geographicis supremus locus Septentrioni, infimus Austro, dexter (qui scilicet est ad dexteram aspicientis) Subsolano, sinister Favonio tribuitur, ita hìc ordinem eundem serves: quia enim vento flabellum impellente si vertebra CCcc

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Book Five. CHAPTER IX. 569 At node C, where the easily turning plate rests, add the vane AD, projecting above the ridge of the roof in a suitable place, so prominent that, directly, without any suspicion of reflection, receiving the impulse of any blowing breeze, and following the course of the wind, it may turn, and its extremity D may face that quarter of the sky opposite to the wind. But at the other extremity B of the axis, beneath the face of the ceiling or vault, in which the wind-points have been duly marked according to the position of the horizon, attach the index BF, on this condition, that it should have, diametrically opposite, the position BF opposite to the vane AD: for thus it will come about that, since the wind blows directly from A toward D, the index F will face that part of the horizon from which it blows. But if the Anemoscope is to be described on a vertical plane, or even on an inclined one, let there be an axis AH with the vane AD passing through the hole C, and resting with a sharp point on the plane H, so that it may be turned most easily; having a notched spine EG divided into eight equal notches, with which the teeth of the wheel MN can in turn correspond exactly, into which the iron plate has been evenly distributed before it was bent into a circle. From the center of this wheel let the fixed axis R emerge, passing through the wall, and at its extremity let it carry the attached pointer, in order to indicate the winds marked on the inner or outer face of the wall. But in the description of the winds it must be observed not to do as in geographical maps, where the upper place is assigned to the North, the lower to the South, the right (that is, the right of the observer) to the East, and the left to the West; do not here keep the same order: because, when the vane is driven by the wind, if the spine...

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570 Mechanicorum striata convertatur ex G in I, rota dentata ascendit ex N in M, & similiter index convolvitur ex T in L; propterea si ventus ab Arcto spirans in supremâ parte descriptus sit in T, & in infimâ qui à meridie in O, is, qui ab ortu flat, describendus est ad si- nistram in L, & qui ab Occasu, ad dexteram in P. Quare mani- festum est, quo ordine reliquos intermedios describere oporteat. PROPOSITIO II. Currûs motum metiri. Quæ Vitruvius lib. 10. cap. 14. scripsit methodum innuens, qua Veteres navi aut rhedâ vecti peractum iter dimetie- bantur, plurium ingenia excitarunt (quandoquidem non paucis Vitruvij verba obscuritate admodum laborare videbantur, quam tamen notam illi inurere nô ausim) ad varias rationes excogitan- das, quibus hoc idem assequi se posse confidant. Maneat sua cui- que Machinatori laus; neminis inventa improbo, aut aspernor: Mihi planissimâ inire viam semper placuit, qua putaverim ad id, quod volumus, perveniri posse: quapropter nec certam rhedæ formam, nec versatilem cum affixis rotis axem præscribo, sed ali- quid vulgaribus rhedis aut curribus commune comminisci pla- cuit, modò liceat alterius posteriorum rotarum (quippe anterio- ribus altiores sunt) modiolo ad partem interiorem insigere bre- viorem paxillum, quo rota ipsa, dum convertitur, motum ma- chinulæ currui alligatæ conciliet. Unum moneo, quod ad Vitruvium spectat (in quo nullam re- perio obscuritatem) non arguendum esse oscitantiæ, quòd rotæ diametrum statuerit pedum quaternum & sextantis, deinde verò totam rotæ versationem definiat pedibus duodecim; cùm tamen ex Rationibus Cyclicis sint ut minimum tredecim; atque adeò quadringentæ versationes perficiant pedes 5200, hoc est passus geometricos 40 supra milliare; ex quo integrâ die, qua milliaria 30 computarentur, error esset passum 1200, qui milliaribus 30 ad- dendi essent. Contra verò si rotæ ambitus solùm peragat pedes 12; quadringentæ versationes dant pedes 4800, & pedes 200 de- sunt ad milliaris complementum: quare & hîc in milliarium 30 computatione deessent passus 1200, qui milliaribus 30 demen- di essent. Ipse tamen Vitruvius quadringentis versationibus tri- buit spatia pedum 5000, hoc est integri milliaris. Non

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570. Mechanics is turned from G into I, the toothed wheel rises from N into M, and likewise the index is wound from T into L; therefore if the wind blowing from the North is depicted in the upper part in T, and in the lower part that from the south in O, that which blows from the east is to be drawn to the left in L, and that from the West, to the right in P. Wherefore it is manifest in what order the remaining intermediate ones ought to be drawn. PROPOSITION II. To measure the motion of a carriage. What Vitruvius wrote in book 10, chapter 14, indicating a method by which the Ancients, riding by ship or by carriage, measured the journey completed, has stirred up the ingenuity of many (since not a few seemed to think that Vitruvius’s words labored under considerable obscurity, though I would not dare to brand them with that fault), to devise various methods by which they trust that they can attain the same end. Let each mechanician keep his own praise; I blame or slight no man’s inventions: it has always pleased me to take the plainest road, which I have thought might lead to that which we desire; wherefore I prescribe neither a fixed form of carriage, nor a revolving axle with attached wheels, but I have thought fit to devise something common to ordinary carriages or carts, provided it be permitted to insert on the inner side of one of the rear wheels (for these are higher than the front wheels) a short peg in the nave, by which the wheel itself, while turning, may impart motion to a little machine attached to the carriage. I give one warning, as regards Vitruvius (in whom I find no obscurity), that he is not to be accused of carelessness because he set the diameter of the wheel at four and one-sixth feet, and then defines the entire revolution of the wheel as twelve feet; when nevertheless, according to the Cyclic Ratios, they are at least thirteen; and indeed four hundred revolutions make 5200 feet, that is, 40 geometrical paces above a mile; from which, over a whole day, if 30 miles were reckoned, the error would be 1200 paces, which would have to be added to 30 miles. On the contrary, if the circumference of the wheel should complete only 12 feet; four hundred revolutions give 4800 feet, and 200 feet are lacking for the completion of a mile: therefore here too, in the reckoning of 30 miles, 1200 paces would be lacking, which would have to be deducted from 30 miles. Yet Vitruvius himself assigns to four hundred revolutions a distance of 5000 feet, that is, an entire mile. Not

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Liber quintus. CAPUT IX. 571 Non est, inquam, oscitantiæ arguendus Vitruvius, quem ista latere non potuerunt, cùm sint admodum obvia cuique vel levi- ter Geometricis asperso; sed eo consilio rotæ diametrum supra quatuor pedes sextante auxit, ut quod deteritur ex aliquâ rotæ depressiore in solo, cui impressa vestigia relinquit, hoc augmento aliquâ ex parte restituatur, adeóque tota versatio consistat intra pedes 12 & 13; addere autem certam fractiunculam pedibus 12, temerarium fuisset, illa quippe valdè inconstans & incerta est: satis fuit demum in summâ 400 versationum medium eligere inter 5200, & 4800: neque enim error, qui notabilis esset, obre- pere poterat. Non tamen placet Vitruvianum tympanum, cujus orbita in quadringentos æquales denticulos esset distributa, nimia quippe & incommoda mihi videtur hujusmodi tympani magnitudo: si enim ligneum fuerit tympanum, singulorum détium pedem, quo orbitæ cohærent, vix puto minorem esse posse latitudine digitali, hoc est quatuor granorum hordei, si quidé satis validi, & ad pe- rennitatem constructi intelligentur: sin autem ferreum fuerit tympanum, latitudo singulorum saltem æquabit duo grana hor- dei. Quare orbita tympani in 400 hujusmodi dentes distributa, erit digitorum 400, aut 200, hoc est, palmorum 100, aut 50; ac proinde diameter erit palmorum ferè 32, aut 16. Commodius igitur acciderit minora tympana componere, quàm adeò in- gens tympanum construere in tot dentes divisum. Sit itaque primùm rota denticulata A, cujus denti insistens hastula C I axiculo in I jun- gatur laminæ H I ita fixæ in H, ut elateris, non tamen admodùm validi, vice fun- gatur. Tum hastulæ C I subjiciatur elasma D ali- quanto validius, quantum satis fuerit ad efficiendum motum, quem statim indi- cabo. Alia pariter hastula F G cum suo elasmate E ita disponatur, ut denti G occurrens non permittat rotam retroagi ex G versus B, sed solùm converti posse ex G in C, atque à sin- CCC 2

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Book Five. CHAPTER IX. 571 I do not, I say, accuse Vitruvius of negligence, since these things could not have escaped him, as they are quite obvious to anyone even slightly versed in Geometry; but with this intention he increased the diameter of the wheel by one-sixth over four feet, so that what is worn away from the lower part of the wheel, when it is depressed in the ground and leaves impressed tracks, may in some measure be restored by this addition, and so the whole revolution may remain within 12 and 13 feet; but to add some fixed little fraction to 12 feet would have been rash, since that is very inconstant and uncertain: in the end, out of 400 revolutions, it was enough to choose the mean between 5200 and 4800; for no notable error could creep in. I do not, however, approve of the Vitruvian drum, whose circumference would be divided into four hundred equal teeth, for the size of such a drum seems to me too great and inconvenient: for if the drum were made of wood, I think the breadth of each tooth, by which they adhere to the circumference, could scarcely be less than the width of a digit, that is, four barleycorns, if indeed they are understood to be sufficiently strong and built for durability; but if the drum were of iron, the breadth of each would amount at least to two barleycorns. Wherefore the circumference of a drum divided into 400 such teeth will be 400 digits, or 200, that is, 100 palms, or 50; and therefore the diameter will be about 32 palms, or 16. It will therefore be more convenient to make smaller drums than to construct so enormous a drum divided into so many teeth. Let there be first a toothed wheel A, on whose tooth the little rod C I, joined by a small axle in I, is attached to the plate H I fixed at H in such a way as to serve as a spring, though not a very strong one. Then let a somewhat stronger spring D be placed beneath the rod C I, sufficient to produce the motion which I shall indicate immediately. In like manner let another rod F G with its spring E be so arranged that, when meeting the tooth G, it does not allow the wheel to be turned back from G toward B, but only to be able to turn from G to C, and from the lef- CCC 2

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572 Mechanicorum gulis dentibus elevata statim vi elasmatis E recidat, séque illis objiciat, ne retrocedant. Additus igitur funiculus C S si trahatur, dentem rotæ convertit hastula C I impellens subjectum elasma D, eademque operâ dens unus transgreditur hastulam F G, quæ vi elasmatis E recidens prohibet, ne in contrarium fieri possit ro- tæ conversio. Quia verò hastula C I dum trahitur, dentem quo- què secum rapit, & ab eo inclinato demum liberatur, dimisso fu- niculo, vi elasmatis D sursum validè propellitur, & per obliquum dentis latus excurrans extremitas C, obliquè pariter desinens, repellit in I elaterem HI, donec hastula ipsa dentis apicem transgressa ab elatere H I sese restituente coaptetur lateri supe- riori dentis. Quo pacto singuli rotæ dentes subinde convertun- tur; atquè tandiu hujusmodi convolutio perseverat, quandiu trahitur, & dimittitur funiculus. Deinde rota altera pariter denticulata paretur, suóque axi in- fixa ita disponatur priori rotæ parallela (sed citra planorum con- tactum) ut in ejus dentes incurrat paxillus L in rotæ A plano ad perpendicularum erectus, quo post integram prioris rotæ conver- sionem dens unus secundæ rotæ promoveatur. Ex quo fiet tot prioris rotæ conversiones requiri ad posteriorem semel convol- vendam, quot in posteriore rotâ dentes numerantur. Simili ra- tione tertia, aut etiam, si opus fuerit, quarta rota denticulata pa- retur, & ita pariter parallelæ disponantur, ut paxillus secundæ rotæ tertiam, & tertiæ quartam convertat, paxillo videlicet den- tium intervalla subeunte post integram suæ rotæ conversionem. Hinc ut innotescat, quoties trahendus, atque dimittendus sit funiculus, ut rotæ convertantur, attendendus est in singulis rotis dentium numerus: tùm numerus primæ per numerum secundæ ducendus; & qui producitur indicans numerum tractionum fu- niculi, ut secunda rota semel convertatur, per numerum dentium tertiæ rotæ est multiplicandus, ut sciamus, quot funiculi tractio- nibus tertia rota gyrum integrum perficiat. Quod si hæc postre- ma non fuerit, sed & quarta rota adjiciatur, productus ex secun- dâ illâ multiplicatione numerus per numerum dentium quartæ hujus rotæ multiplicabitur: ac demum innotescet, quoties funi- culum trahere oporteat, ut quarta hæc rota totam circuli peri- pheriam percurrat. Quod si paxillis, de quibus dictum est, uti non placuerit, sed po- tiùs

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572 Mechanicorum so that the lifted teeth immediately fall back by the force of the elasma E and present themselves against them, lest they move backward. If therefore the cord C S is pulled, it turns the tooth of the wheel, the little lever C I impelling the elasma D beneath it; and by the same operation one tooth passes over the little lever F G, which, falling by the force of the elasma E, prevents the wheel from being turned in the opposite direction. But because the little lever C I, while it is being pulled, also carries the tooth along with it, and is at last freed from the tooth when it has inclined, when the cord is released it is strongly driven upward by the force of the elasma D, and, running along the oblique side of the tooth, its end C, likewise ending obliquely, pushes back the spring HI at I, until the lever itself, having passed beyond the point of the tooth, is by the spring HI restoring itself fitted to the upper side of the tooth. In this way the individual teeth of the wheel are successively turned; and this rotation continues as long as the cord is pulled and released. Then let another similarly toothed wheel be made, and fixed on its own axle, and arranged parallel to the first wheel (but without contact of the planes), so that the peg L, erected perpendicular in the plane of wheel A, strikes against its teeth, whereby, after a full rotation of the first wheel, one tooth of the second wheel is advanced. From this it will follow that as many rotations of the first wheel are required to wind the second once as there are teeth in the second wheel. In like manner a third, or even, if necessary, a fourth toothed wheel may be made, and arranged similarly in parallel, so that the peg of the second wheel turns the third, and the peg of the third the fourth, namely the peg passing through the spaces between the teeth after a full rotation of its own wheel. Hence, in order to know how many times the cord must be pulled and released for the wheels to turn, one must consider in each wheel the number of teeth: then the number of the first is to be multiplied by the number of the second; and the product, indicating the number of pulls of the cord by which the second wheel is turned once, is to be multiplied by the number of teeth of the third wheel, so that we may know by how many pulls of the cord the third wheel completes a full revolution. But if this last one is not present, and a fourth wheel is added, the number produced by that second multiplication is to be multiplied by the number of teeth of this fourth wheel; and thus it will at last become known how many times the cord must be pulled for this fourth wheel to traverse the whole circumference of the circle. But if it does not please to use the pegs mentioned above, but rather

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Liber quintus. CAPUT IX. 573 tiùs libeat singulis rotis crassiusculos axes inserere, ex quibus dēs unus promineat, qui post integram suæ rotæ conversionem den- tibus sequentis rotæ implicetur; omnino licebit, & fortasse suo commodo non carebit. Illud in rotarum collocatione intra suum loculamentum est diligenter animadvertendum, quod prioris ro- tæ paxillus (aut axis dens) non nisi post integram suæ rotæ con- versionem incurrat in dentes posterioris; alioquin in errorem non sanè levem inducere nos posset index, qui extremitati axis adnexus in exteriore loculamenti facie indicat singularum ro- tarum convolutions. Affigatur itaque posteriori rhedæ parti opportuno loco regula circa axem versatilis, cujus superior extremitas conjunctum ha- beat funiculi C S trahendi caput S, inferior auté extremitas oc- currat paxillo, quem ab initio rotæ modiolo ad partem interioré infixisti: sic enim fiet, ut paxillo regulam impellente funiculus trahatur, atque ad singulas rotæ currûs conversiones, singuli den- tes rotulæ A funiculum trahentem sequantur: ac propterea in loculaméti facie index cum axe A convolutus indicabit, quoties rota currûs cõversa fuerit; & absolutâ integrâ rotæ A conversio- ne index sequentis secundæ rotulæ ostendet integras convolu- tiones primæ; atque ita deinceps index tertiæ numerabit convo- lutiones secundæ, & index quartæ convolutions tertiæ. Hinc si rotulæ singulæ sint in dentes decé distributæ, numero, quem in- dicat secunda rotula, adde unicum cyphram o, numero tertiæ ro- tulæ adde duas cyphras ooo, & numero à quarta rotula indicato adde tres cyphras ooo; statúmque manifestus fiet numerus con- versionum rotæ currûs. Quare si posteriores currûs rotæ habeât diametrum quinque pedum, rotæ ambitus est trium passu[m] Geo- metricorum (quod est super, negligitur, nam sæpè rota solu[m] mol- liusculum penetrans extenuat diametrum) atque adeò, ut semel prima rotula cõvertatur, currûs rotâ decies cõversa percurrit spa- tium passuum 30; ut secunda unicum conversione[m] perficiat, rota currûs centies volvitur, & conficit passus 300; ut tertia gyrum ab- solvat, rota currûs millies vertitur, & tria Italica milliaria percur- rit. Ideò numerus ab indice quartæ rotulæ significatus, indicans tertiæ rotulæ integras convolutions, triplicandus est, ut peracti itineris mensura Italicis milliaribus definiatur. Ex quo fit quar- tam rotulam in dentes decem distributam sufficere ad numeran- CCC 3

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Book Five. Chapter IX. 573 it may rather be preferred to insert somewhat thicker axles into each wheel, from which one tooth shall project, which, after the complete revolution of its wheel, shall engage the teeth of the following wheel; this will certainly be permissible, and perhaps will not lack its own advantage. In the arrangement of the wheels within their receptacle, this must be carefully observed: that the peg of the preceding wheel (or the tooth of its axle) shall not engage the teeth of the following wheel until after the complete revolution of its own wheel; otherwise the index, which is attached to the end of the axle and indicates on the outer face of the receptacle the revolutions of each wheel, might lead us into no slight error. Let there be fixed, therefore, to the rear part of the carriage, in a suitable place, a rod movable around an axle, whose upper extremity shall have joined to it the end S of the cord C S to be drawn, while the lower extremity shall encounter the peg which you originally fixed in the hub of the wheel on the inner side: for thus it will happen that, when the peg drives the rod, the cord is drawn, and at each revolution of the carriage wheel the single teeth of the small wheel A follow the pulling cord; and therefore on the face of the receptacle the index, turning with axle A, will indicate how many times the carriage wheel has turned; and after the complete revolution of wheel A, the index of the next small wheel will show the complete revolutions of the first; and thus in turn the index of the third will count the revolutions of the second, and the index of the fourth the revolutions of the third. Hence, if the individual small wheels are divided into ten teeth, to the number indicated by the second small wheel add one cipher 0, to the number of the third wheel add two ciphers 00, and to the number indicated by the fourth wheel add three ciphers 000; and immediately the number of revolutions of the carriage wheel will become clear. Therefore, if the rear wheels of the carriage have a diameter of five feet, the circumference of the wheel is three geometric paces (which is in excess, but this is disregarded, for often a wheel sinking only into softer ground reduces the diameter); and thus, in order that the first small wheel may make one revolution, the carriage wheel, having turned ten times, travels a distance of 30 paces; in order that the second may complete a single revolution, the carriage wheel turns one hundred times and makes 300 paces; in order that the third may complete a circuit, the carriage wheel turns a thousand times and travels three Italian miles. Therefore the number indicated by the index of the fourth wheel, showing the complete revolutions of the third wheel, must be tripled, in order that the measure of the journey accomplished may be determined in Italian miles. From this it follows that the fourth wheel, divided into ten teeth, is sufficient for counting CCC 3

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Mechanicorum 574 da milliaria Italica 30: quod si plura velis numerare unicâ hujus rotæ conversione, in plures dentes, quàm decem, quartam rotulam distingue: sed non est opus, quia unâ convolutione absolutâ, milliaribus indicatis addi possunt milliaria 30. Si ad navis cursum dimetiendum machinulam hanc eandem traducere placeat, adjiciéda est ad navis latus rota, ex cujus conversione integrâ observatum fuerit, quâtùm navis promoveaur: nam similiter impellendo regulam, qua funiculus trahitur, rotæ conversionu[m] numerus innotescet, atque adeò etia[m] itineris spatiu[m]. Cave tamen, ne in errore incidas, qui facilè obrepere posset; cum enim navis non semper æquè mergatur in aquâ (seu quia illa non est semper æquè onusta, seu quia hæc no[n] est semper æquè crassa, aut tenuis) etiam rota inæqualiter mergitur, ac proinde una rotæ hujus conversio non semper æquali itineris spatio respondet. Nec dissimili ratione pedestria itinera metiri licebit, si parvulam hujusmodi machinulam ita corpori alligaveris, ut funiculi extremitas sub poplite adnectatur: nam ad singulos passus denticulus unus convertetur, & demum passuum numerus innotescet. PROPOSITIO III. Objecti procul visi pseudographam speciem deformare. Ontingit aliquando minùs attentos recti specie decipi: propterea hac propositione non inutile fuerit abusum quendâ rotæ dentatæ, quæ facilè fucum faciat imperitis, indicare: ne fortè sibi quasi de præclaro invento inaniter gratulentur. Quadri- laterum Prisma AB eligatur, cujus extremitas in tenuiorem cylindrum CD desinat inserendum forami- ni subjecti plani crassioris, ita ut plano ad perpendiculum insistat prisma, & servatâ positione perpendiculari, facilè converti possit in dexteram, & in sinistram. Tum rotæ dentatæ semissis M F N prismati secundùm longitudem excavato inseratur, atque circa axem A ductum per prisma

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Mechanics 574 to 30 Italian miles: but if you wish to count more by a single revolution of this wheel, divide the fourth little wheel into more than ten teeth; but there is no need, because, when one revolution has been completed, 30 miles may be added to the miles indicated. If it should please you to apply this same little machine to measuring the course of a ship, a wheel must be attached to the side of the ship, from whose complete revolution it may be observed how much the ship advances; for by similarly pushing the rod by which the cord is drawn, the number of the wheel’s revolutions will be made known, and consequently also the distance of the journey. However, beware lest you fall into an error, which might easily creep in; for since the ship is not always equally submerged in the water (either because it is not always equally loaded, or because the water is not always equally dense, or thin), the wheel is also submerged unequally, and therefore one revolution of this wheel does not always correspond to an equal distance traveled. Nor by a different method will it be possible to measure pedestrian journeys, if you shall have tied a little machine of this kind to the body in such a way that the end of the cord is fastened below the knee: for at each step one tooth will turn, and finally the number of steps will be made known. PROPOSITION III. An object seen from afar distorts the appearance of a pseudograph. It sometimes happens that those who are not attentive are deceived by the appearance of something straight; therefore in this proposition it will not be useless to indicate a certain abuse of a toothed wheel, which may easily deceive the uninstructed, lest they should perhaps vainly congratulate themselves on what seems to them a splendid invention. Let a quadrilateral prism AB be chosen, whose end terminates in a thinner cylinder CD, to be inserted into the hole of the thicker underlying plane, so that the prism stands upon the plane at a right angle, and, the perpendicular position having been preserved, may easily be turned to the right and to the left. Then let a half toothed wheel M F N be inserted into the prism hollowed out lengthwise, and around the axis A drawn through the prism

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Liber quintus. CAPUT IX. 575 prisma & rotæ dentatæ centrum circumagi possit. In eandem au- tem prismatis fissuram infra rotæ dentatæ segmentum immitra- tur regula H I superiùs exasperata in crenas dentibus rotæ tan- gentis congruentes, adeò ut ex rotæ conversione regula H I ad- ducatur, & reducatur: quæ in I calamum scriptorium, aut lapi- dem plumbarium habens (aut saltem acutum stylum, quo certa puncta lineis deinde jungenda notari valeant) in subjectâ char- tâ lineas describit sequens ductum radij optici per dioptram M N excepti. Quare quot lineas in objecto procul viso percurrit radius opticus, totidem lineæ à stylo I describuntur in chartâ. Quando igitur magis altum, aut longiùs positum objecti punctum per dioptram aspicitur, dioptræ extremitas oculo pro- xima deprimitur, atque adeò rotæ dentatæ portio ita conver- titur, ut versùs objectum promoveat regulam: contrà verò de- pressius, aut propius objecti punctum aspiciens, proximam ocu- lo extremitatem dioptræ elevat, & regulam ab objecto removet; cuicumque tandem extremitati M, aut N oculum admoveas: Si enim ex M aspicias, deprimendo M propellis stylum I versùs pris- ma, hoc est versùs objectum; atque similiter ex N aspiciés, depri- mendo N removes stylum I à prismatic, & versùs objectum im- pellis. At verò ubi transversum objecti latus aspiciendum est, factâ circa cylindrulum C D conversione, plurimum interest, utrùm ex M, an ex N aspicias: Nam si oculus sit in N, & radio optico percurrat objecti latus à sinistrâ in dexteram, etia[m] stylus I à sinistrâ in dextram movetur unâ cum extremitate M objectum respiciente. Sin autem oculus sit in M, atque stylus I inter oculu[m] & prisma, aut oculus inter stylum & prisma interjectus sit, con- trariam positionem habent puncta à stylo descripta, & sinistra mi- grant in dexteram, atque dextera in sinistram; stylus quippe ocu- lum sequitur, qui motum habet oppositum motui alterius extre- mitatis N objectum respicientis. Quamobrem expedit oculum dioptræ in N admovere, & in objectu[m] stylu[m] I obvertere, ut dextra dextris, & sinistra sinistris respondeat, prout sub aspectu[m] cadunt. Verùm, licèt objecti visi speciem aliquam hoc artificio adum- brare liceat, cavendu[m] tamen, ne ipsi nobis assentates quasi exacta[m] Ichnographiam, & subtilem, servatis corporis partiu[m] Rationibus, descriptionem nos comparasse existimemus: cuique scilicet rem accuratè perpendenti manifestum est, quandiu semicirculus in eodem

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Book Five. CHAPTER IX. 575 so that the prism and the center of the toothed wheel may be turned around. Into the same slit of the prism, beneath the segment of the toothed wheel, is inserted the rule H I, roughened above with notches matching the teeth of the touching wheel, so that by the turning of the wheel the rule H I is drawn in and pushed back; on this rule, at I, is a writer’s pen, or a lead pencil, or at least a sharp stylus, by which certain points may be marked to be joined afterward by lines; and it traces on the paper beneath lines following the path of the optical ray admitted through the diopter M N. Therefore, however many lines the optical ray traverses on the object seen at a distance, so many lines are drawn on the paper by the stylus I. When therefore a higher, or more distant point of the object is viewed through the diopter, the end of the diopter nearest the eye is depressed, and thus so much of the toothed wheel is turned that it advances the rule toward the object; but conversely, when a lower, or nearer point of the object is viewed, it raises the end of the diopter nearest the eye, and moves the rule away from the object; whichever end, M or N, you bring the eye to: for if you look from M, by depressing M you push the stylus I toward the prism, that is, toward the object; and similarly, if you look from N, by depressing N you move the stylus I away from the prism and drive it toward the object. But when the crosswise side of the object is to be viewed, after a rotation has been made around the little cylinder C D, it matters greatly whether you look from M or from N: for if the eye is in N, and the optical ray traverses the side of the object from left to right, then the stylus I also moves from left to right together with the end M looking toward the object. But if the eye is in M, and the stylus I lies between the eye and the prism, or the eye is interposed between the stylus and the prism, then the points described by the stylus have the opposite position, and move from left to right and from right to left; for the stylus follows the eye, which has a motion opposite to the motion of the other end N looking toward the object. Therefore it is advantageous to place the eye of the diopter at N, and to turn the stylus I toward the object, so that right corresponds to right and left to left, according as they fall under view. But although by this device some likeness of the object seen may be sketched, we must nevertheless be careful not to imagine that we have thereby obtained for ourselves an exact ichnography and a subtle description, preserving the proportions of the parts of the body; for to anyone who examines the matter accurately it is manifest, so long as the semicircle remains in the same

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Mechanicorum 576 eodem plano Verticali co[n]sistit, & dioptra elevatur, sive deprimi- tur, lineam objecti, quam radius opticus percurrit in plano hori- zontali, respondere differentiæ Tangentium angulorum, quos cu[m] perpendiculo A B constituit radius opticus: At linea, quam sty- lus I describit, respondet quidem (saltem proximè, & quatenus sensu in tantâ parvitate percipi potest) differentiæ Tangentium angulorum æquè differentium, quos cum perpendiculo eodem A B constituere intelligitur linea à centro A ad stylum I ducta. Non tamen fieri potest, ut deinde in omnibus positionibus muta- to Verticali eadem Ratio servetur; quia linea à centro A ad sty- lum I ducta, non est parallela radio optico, sed angulum multo minorem constituit cum perpendiculo; ac proinde angulorum minorum differentia, etiamsi æqualis differentiæ angulorum ma- jorum, non infert proportionalem differentia Tangentium. Sta- tuatur ex. gr. differentia angulorum duobus gradibus definita, & in uno Verticali majores anguli à dioptrâ constituti sint gr. 88. & 86, minores autem gr. 58. & 56: in altero Verticali majores anguli à dioptrâ constituti sint gr. 73. & 71, minores verò gr. 43, & 41. Quia idem est Radius A B, quarum partium 1000 est Radius, in primo Verticali differentia majorum Tangentium est 14336, & differentia Tangentium minorum est 118: in secundo Vertica- li differentiæ Tangentium sunt 367 majorum, & 63 minorum angulorum: inter hos autem terminos non intercedere propor- tionem manifestum est. Quando verò, factâ circa cylindrum C D conversione, fit tran- situs ab uno plano Verticali ad aliud planum Verticale, linea, quam radius opticus percurrit, & linea, quam stylus I describit, subtendunt quidem similes arcus, opponuntur enim eidem an- gulo Verticalium, sed sunt in Ratione distantiarum objecti visi, atque styli à cylindrulo tanquam centro motûs. Porrò hasce li- neas differentiis illis Tangentium non esse analogas perspicuum est. Quapropter descriptum schema non servans objecti Ratio- nes, censendum est pseudographum. Oporteret plano immobili, cui infigitur prisma, adnectere con- gruis cardinibus aut fibulis, tabellam, quæ semper parallela diop- træ cum hac pariter elevaretur & deprimeretur (non tamen cum eâ convolveretur) ut in chartâ tabellæ affixâ species magis cum objecto conveniens describeretur: Qua autem methodo? inge- niosus lector dispiciat. MECHA

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Mechanics 576 lies in the same vertical plane, and the dioptra is raised, or lowered, the line of the object, which the optical ray traverses in the horizontal plane, corresponds to the difference of the tangents of the angles which the optical ray makes with the perpendicular A B: but the line described by the stylus I does indeed correspond (at least approximately, and insofar as this can be perceived by the senses in so small a quantity) to the difference of the tangents of equally differing angles, which line drawn from center A to stylus I is understood to make with the same perpendicular A B. Yet it cannot happen that afterward, in all positions, with the vertical changed, the same ratio is preserved; because the line drawn from center A to stylus I is not parallel to the optical ray, but makes a much smaller angle with the perpendicular; and therefore a difference of smaller angles, even if equal to the difference of larger angles, does not produce a proportional difference of tangents. Let it be assumed, for example, that the difference of angles is defined by two degrees, and in one vertical the larger angles made by the dioptra are 88° and 86°, but the smaller 58° and 56°: in another vertical the larger angles made by the dioptra are 73° and 71°, but the smaller 43° and 41°. Since AB is the same radius, of which 1000 parts make the radius, in the first vertical the difference of the larger tangents is 14336, and the difference of the smaller tangents is 118: in the second vertical the differences of tangents are 367 of the larger, and 63 of the smaller angles: but it is manifest that there is no proportion between these terms. But when, after a rotation about the cylinder CD, a transition is made from one vertical plane to another vertical plane, the line traversed by the optical ray, and the line described by stylus I, do indeed subtend similar arcs, for they are opposed to the same angle of the verticals, but they are in the ratio of the distances of the object seen, and of the stylus, from the little cylinder as the center of motion. Moreover, it is clear that these lines are not analogous to those differences of tangents. Wherefore the described scheme, not preserving the proportions of the object, must be judged a pseudograph. It would be necessary to attach to the immovable plane in which the prism is fixed, by suitable hinges or clamps, a small board which would always remain parallel to the dioptra and be raised and lowered together with it (but not revolve with it), so that on the paper affixed to the board a more faithful representation of the object might be drawn: by what method? let the ingenious reader consider. MECHA

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MECHANICORUM LIBER SEXTUS. De Trochlea. ON semper commodum accidit Ergatâ, aut suc- culâ, aut Tympano uti ad pondus aliquod moven- dum: ut enim ex iis, quæ superiore libro disputa- ta sunt, manifestum est, si in altiorem locum eve- hendum sit pondus, ibi construere oporteret peg- ma, cui machina insisteret: sæpè autem id fieri non posset sine magna impensa, aut citrà incommodum sive propter loci an- gustias, sive propter temporis brevitatem pegmati construendo imparem. Hinc alia Facultas excogitata est, cui Trochleæ no- men inditum est; quippè quæ communiter ex rotulis circà axem in suo loculamento versatilibus coagmentatur, iisque cir- cumducitur funis ductarius, quo trahitur pondus trochleæ ad- nexum. Trochleam autem, ut Vitruvius lib. 10 cap. 2. testatur nonnulli Rechamum dicunt. Ex orbiculorum numero nomen ducit machina; nam si uni- cus sit orbiculus, Trochlea simplex, aut Monospatos vocatur; si duo fuerint orbiculi, Dispastos; si tres Trispastos; atque ita deinceps. In hac tamen nomenclaturâ observandum est, non eodem omnes vocabulo uti: aliqui enim cunctos orbiculos utriusque loculamenti in unam summam referunt, & ex eorum numero vocabulum statuunt; ut si alterius loculamenti duo sint orbiculi, alterius verò unicus, Trispaston appellant: Alij ta- men nomen indunt ex orbiculis singulorum loculamentorum; nam si binos orbiculos singula contineant, non Tetraspaston, sed Dispaston vocant, quia communiter ambo loculamenta æquali orbiculorum numero instruuntur, & ex alterius numero reliqui, pariter numerus innotescit. Neque omnino abs re alte- D D d d

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MECHANICORUM BOOK SIX. On the Pulley. It does not always happen to be convenient to use a ergata , or a winch, or a drum for moving some weight; for, as is made clear from what has been discussed in the preceding book, if a weight is to be lifted to a higher place, it would be necessary there to construct a scaffold on which the machine might stand. Often, however, this could not be done without great expense, or without inconvenience, either because of the narrowness of the place or because the shortness of the time made the construction of a scaffold impossible. Hence another device has been devised, to which the name Trochlea has been given; namely, that which is commonly made up of wheels mounted on an axle in its frame and turning about it, with a carrying rope passed around them, by which the weight attached to the pulley is drawn. And the pulley, as Vitruvius bears witness in book 10, chapter 2, is called by some Rechamus . The machine takes its name from the number of wheels; for if there is a single wheel, it is called a simple pulley, or Monospastos ; if there are two wheels, Dispastos ; if three, Trispastos ; and so on. In this nomenclature, however, it should be observed that not all use the same terminology: some refer all the wheels of both blocks to one total, and establish the name from their number; thus, if one block has two wheels and the other one wheel, they call it a Trispastos . Others, however, give the name from the wheels of each block; for if each contains two wheels, they call it not a Tetraspastos but a Dispastos , because ordinarily both blocks are furnished with an equal number of wheels, and from the number of one the number of the other is likewise known. Nor is it altogether without reason that furth- D D d d

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578 Mechanicorum rius tantummodo loculamenti orbiculos numerant, quia hujus facultatis vires potissimùm habentur ex solis orbiculis locula- menti, cui pondus trahendum adnectitur; reliquum scilicet lo- culamentum cum suis rotulis proptereà adjicitur, ut funis ducta- rius singulos illius orbiculos complecti possit. Ex quo fit, posito inæquali orbiculorum numero, modò Monospaston, modò Dis- paston dici, prout pondus adnectitur loculamento unum, aut duos orbiculos habenti. Cæterùm in vocabulis non est hærendum: Ego Trochleam voco loculamentum unum cum suis or- biculis; & quando opus est duplici loculamento uti, duplicem Trochleam dico, atque orbiculos numero, ne ullus subesse possit æquivocationi locus. Quantum autem Facultas hæc sit Axe, aut Vecte utilior, hinc saltem constat, quod etiamsi plures potentiæ diversis funis ductarij partibus applicentur, æqualia tamen obtinent momen- ta; id quod non contingit pluribus eundem Succulæ Radium, aut eumdem Vectem urgentibus; neque enim æqualibus à mo- tûs centro intervallis absunt. CAPUT I. Trochlearum forma, & vires exponuntur. Aliquando simplicem orbiculum, cujus excavatæ orbitæ funis ductarius insistit, adhibemus, ut onera sursum attol- lamus: & quidem communiter in superiore loco firmatur locu- lamentum cum orbiculo versatili, & alteram funis extremita- tem apprehendit Potentia, alteri adnectitur pondus sublevan- dum, quod ascendendo spatium percurrit æquale spatio, per quod Potentia descendendo movetur. Id quod eatenus excogi- tatum est, quatenus brachia deprimentibus in ponderis eleva- tione insita brachiorum gravitas vires addit, & minore lacerto- rum contentione opus est, quàm si pondus ipsum sursum trahe- remus brachia elevantes. Factus est autem orbiculus circa suum axem versatilis, ut vitetur difficultas, quæ cæteroqui conseque- retur mutuum tritum funis cum subjecto corpore, cui insisteret, si illud non versaretur. Quantus enim sit hujusmodi funis cum subjecto corpore (si illud non convolvatur) conflictus, manifestu[m] est

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578 Mechanics count only the pulleys in the block, because the power of this device is derived chiefly from the pulleys of the block to which the weight to be drawn is attached; the rest of the block, namely, together with its sheaves, is added for this reason, so that the carrying rope may be able to encompass each of its pulleys. From this it follows that, when the number of pulleys is unequal, the device is called now a Monospaston, now a Dispaston, according as the weight is attached to a block having one or two pulleys. However, one should not get caught up in the terms: I call a Pulley a single block with its pulleys; and when it is necessary to use a double block, I call it a double Pulley, and I count the pulleys, lest any occasion for ambiguity may remain. How much more useful this device is than the axle or the lever is at least clear from this: even if several forces are applied to different parts of the carrying rope, they nevertheless produce equal moments; which does not happen when several forces press upon the same radius of a windlass, or the same lever; for they are not at equal distances from the center of motion. CHAPTER I. The form and power of pulleys are explained. Sometimes we use a simple sheave, in whose hollow groove the carrying rope rests, in order to raise loads upward; and indeed, commonly the block with the movable sheave is fixed in the upper position, and the Power seizes the other end of the rope, while the weight to be lifted is attached to the other end, which in ascending travels a distance equal to the distance through which the Power is moved downward. This has been devised insofar as the arms, in lifting a weight by pressing down, have strength added to them by the weight of the arms themselves, and the work is accomplished with less strain on the muscles than if we were drawing the weight itself upward by raising the arms. Moreover, the sheave is made movable about its own axle in order to avoid the difficulty that would otherwise arise from the mutual rubbing of the rope against the body beneath it on which it would rest, if that body did not turn. For how great the conflict of a rope with the body beneath it is of this kind (if it does not revolve) is evident

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Liber sextus. CAPUT I. 579 est in puteis, quibus ad hauriendam aquam non est girgillus, hoc est, orbiculus versatilis, adjectus, sed funis transverso fusti cylindrico, verùm immobili, insistit; excavatur siquidem cylin- der ille diuturno, & frequenti tritu funium. Cæterum si non ad perpendiculum attollendum sit pondus, sed in plano horizonta- li, aut inclinato (non tamen lubrico) raptandum, vix, aut ne vix quidem, ullum compendium consequeris, si funem per or- biculum transeuntem trahas in plagam oppositam plagæ, ver- sùs quam pondus dirigitur, ac si pondus idem arrepto fune ad te directè rapias: eadem quippe est brachiorum contentio, quo- rum insita gravitas non juvat potentiam, nisi quando hæc deor- sum tendit. Adhiberi tamen hujusmodi orbiculus in planitie poterit, si commodiùs Potentia consistat in loco, ubi jacet pon- dus, quàm ibi, quò illud adducendum est. Quamquam verò orbiculus stabili loculamento infixus non sit aptus ad augendas Potentiæ vires, prout ad Machinæ ratio- nem pertinet; si tamen loculamentum ip- sum adnectatur ponderi, quod cum illo mo- veatur, geminantur Potentiæ momenta, non enim æqualis est Potentiæ & Ponderis mo- tus, sed illa duplo velociùs movetur. Sit pondus attollendum sivè raptandum A, cui adnectatur loculamentum orbiculi B; funis autem ductarius firmetur in C, & funis extremitatem reliquam apprehendat Po- tentia in D: utique Potentia ut adducat orbiculum usque in C, tantumdem pro- gredi debet ultra C, quantum orbiculus B distat à puncto C; oportet siquidem totum funem DBC explicari. Igitur po- tentia ex D venit primùm in E, deinde in F: est autem distantia DE æqualis in- tervallo BC; sed tunc, cùm illa est in E, orbiculus solùm est in I, & demum hic est in C, quando potentia est in F. Motus itaque potentiæ DF est duplus mo- tus orbiculi BC. Porrò cum orbiculo pa- riter trahitur pondus A adnexum; igitur DDdd 2

Transcription: Translated (English)

Book Six. CHAPTER I. 579 is in wells, in which there is no windlass for drawing up water, that is, no revolving pulley attached, but the rope rests upon a transverse cylindrical shaft, yet immovable; for that cylinder is hollowed out in time by the long and frequent rubbing of the ropes. Otherwise, if the weight is not to be raised perpendicularly, but to be dragged in a horizontal plane, or on an inclined one (though not a slippery one), you will scarcely, or not even scarcely, obtain any saving, if you pull the rope passing through the pulley in the opposite direction to that in which the weight is being carried, as if you were dragging the same weight directly toward you by taking hold of the rope; for the exertion of the arms is the same, whose inherent heaviness does not aid the power, except when this tends downward. Nevertheless, such a pulley can be used on a level surface, if it is more convenient for the Power to stand in the place where the weight lies than there to which it is to be brought. But although a pulley fixed in a stationary block is not suitable for increasing the force of the Power, as far as the design of the Machine is concerned; yet if the block itself is attached to the weight, so that it moves with it, the moments of the Power are doubled, for the motion of the Power and of the Weight is not equal, but the former moves twice as fast. Let the weight to be raised or dragged be A, to which the block of the pulley B is attached; and let the guiding rope be fixed at C, and let the Power grasp the remaining end of the rope at D: then indeed, in order that the Power may draw the pulley as far as C, it must advance beyond C by as much as the pulley B is distant from point C; for the whole rope DBC must be paid out. Therefore the power comes from D first to E, then to F: and the distance DE is equal to the interval BC; but then, when it is at E, the pulley is only at I, and at length it is here at C, when the power is at F. Thus the motion DF of the power is double the motion BC of the pulley. Moreover, the weight A attached is drawn along with the pulley as well; therefore DDdd 2

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Mechanicorum 580 duplo velocior est potentiæ motus præ motu ponderis. Qua- re potentia valens trahere motu sibi æquali pondus aliquod sine orbiculo, hoc addito valebit trahere pondus duplo ma- jore gravitate præditum. Ex quibus manifestum est, quantum intersit, utrum ex- tremitati funis adnectatur pondus, & orbiculi loculamen- tum stabile sit, an verò, funis extremitate manente atque immotâ, ponderi adnectatur loculamentum, quod cum ipso pondere moveatur, immò veriùs, cujus motum consequatur motus ponderis: nam in secundo hoc casu potentiæ motus duplus est ad motum ponderis; in primâ autem positione mo- tus utriusque sunt planè æquales. Hinc ulteriùs constat, quando duæ Trochleæ simplici orbiculo instructæ adhibentur, ita ut altera fixa maneat, al- tera cum pondere moveatur, nihil addi momenti Potentiæ si funis extremitas alligetur trochleæ stabili, aut loco alicui extra trochleas. Nam si in G posita sit Trochlea manens immota H, & altera funis extremitas illi jungatur in O, seu extra illam clavo, aut paxillo in C, Potentia in L ap- plicata æqualiter movetur cum puncto D: at punctum D movetur duplo velocius, quàm Trochlea B; igitur Poten- tia L movetur solum duplo velociùs quàm pondus, perinde atque si non fuisset addita trochlea H. Eatenus igitur additur Trochlea H, quatenus Potentiam & Pondus in oppositas pla- gas moveri oportet, aut potentia deorsum conari debet, ut pondus ascendat. Sin autem extremitas funis alligetur Trochleæ mobili, cui pariter adnectitur pondus, & primùm funis ab unco trochleæ mobilis deducatur ad orbiculum trochleæ immotæ, deinde ad orbiculum ejusdem Trochleæ mobilis, jam Po- tentia triplo velociùs movetur quàm Pondus; quia videli- cet etiam ipsa funis extremitas movetur trahentem sequens unâ cum pondere. Concipe enim pondus A sejunctum à Trochleâ B, quæ ita firmetur, ut immota maneat, pondus verò intelligatur translatum in G, atque Trochlea H jam sit mobilis: utique Potentia funem in L arreptum trahens in motu progreditur ultra B, quanta est longitudo funis ex- plicati OBDH, quæ longitudo dupla est intervalli OB: igitur

Transcription: Translated (English)

Mechanics 580 is twice as fast as the motion of the power compared with the motion of the weight. Therefore a force capable of drawing, with a motion equal to itself, some weight without a pulley will, with this added, be able to draw a weight endowed with twice the heaviness. From these things it is manifest how much difference it makes whether a weight is attached to the end of the rope, and the receptacle of the pulley is fixed, or whether, on the other hand, while the end of the rope remains and is unmoved, the receptacle is attached to the weight, so that it moves together with the weight itself, or rather, more truly, so that the motion of the weight follows its motion: for in this second case the motion of the power is double the motion of the weight; but in the first arrangement the motions of each are altogether equal. Hence it is further clear that, when two pulleys equipped with a single wheel are used, so that one remains fixed and the other moves with the weight, nothing is added to the advantage of the Power if the end of the rope is tied to the fixed pulley, or to some place outside the pulleys. For if the pulley H is placed in G, remaining unmoved, and the other end of the rope is joined to it at O, or outside it by a nail or peg at C, the Power applied in L moves equally with the point D: but point D moves twice as fast as pulley B; therefore the Power L moves only twice as fast as the weight, just as if pulley H had not been added. Accordingly, pulley H is added only insofar as the Power and the Weight ought to move in opposite directions, or the power ought to strive downward, so that the weight may rise. But if the end of the rope is tied to the movable pulley, to which the weight is likewise attached, and the rope is first led from the hook of the movable pulley to the wheel of the fixed pulley, and then to the wheel of the same movable pulley, now the Power moves three times as fast as the Weight; because, namely, even the end of the rope itself is moved, following the puller, together with the weight. For imagine the weight A separated from pulley B, which is so fixed that it remains unmoved, but let the weight be understood to have been transferred to G, and pulley H now to be movable: certainly the Power, pulling the rope seized in L, advances in motion beyond B by as much as is the length of the rope unrolled OBDH, and this length is twice the interval OB: therefore

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Liber sextus. CAPUT I. 581 igitur potentia L accedens ad B semel percurrit interval- lum O B, & præterea adhuc duplum spatium ultrà B, dum punctum O venit ad B simul cum pondere ad- nexo in G: triplo igitur velociùs movetur Potentia quàm Pondus. Simili omnino ratione ac de Trochleis simplicibus phi- losophamur, etiam ratiocinari oportet in Trochleis plures orbiculos habentibus; si enim singulæ duos habeant orbi- culos, attendendum est, an funis extremitas adnectatur Trochleæ immotæ, an verò mobili: si immotæ, potentia movetur quadruplo velociùs quàm pondus; sin autem mo- bili, movetur quintuplo velociùs. Generatim igitur nume- ra orbiculos trochleæ mobilis, cui scilicet jungitur pondus, & pro singulis orbiculis duplica potentiæ momenta. Hinc si tres fuerint orbiculi, momentum Potentiæ est sextuplum; si quatuor, octuplum; & sic deinceps. At si eidem Trochleæ mobili adnectatur extremitas funis, adhuc adde unitatem, & momentum erit septuplum, aut noncuplum. Funis siqui- dem uni trochleæ alligatus primùm insistit orbiculo primo reliquæ trochleæ; inde flectitur ad orbiculum primum tro- chleæ, cui adnectitur: postmodum ad secundum orbicu- lum alterius trochleæ transit, & rediens ad priorem tro- chleam insistit orbiculo ejus secundo; atque ita deinceps, al- terno ex trochleâ in trochleam excursu, donec orbiculis om- nibus insistat. Quod si duabus Trochleis non insit æqualis orbiculorum numerus, sed altera alteram unitate superet, necesse est funem alligari trochleæ pauciorum orbiculorum. Quare attendendus pariter est numerus orbiculorum trochleæ mobilis, quæ si pauciores habeat orbiculos, utique illi ad- nectitur extremitas funis; atque adeò duplicato ejus orbi- culorum numero addenda est unitas: ut, si duos habeat or- biculos, motus Potentiæ est quintuplus motus Ponderis. At si trochlea mobilis plures habeat orbiculos quàm trochlea im- mota, duplicandus solùm est illorum numerus, ut habeatur denominatio momenti; ut, si tres fuerint orbiculi, motus po- tentiæ ad ponderis motum est sextuplus. In hujusmodi Trochleis plures rotulas habentibus obser- vandum est interiores rotulas minores statui, exteriores verò DDdd 3

Transcription: Translated (English)

Book six. CHAPTER I. 581 thus the power L, approaching B, passes once through the interval OB, and moreover still a double space beyond B, while the point O comes to B together with the weight attached at G: therefore the Power moves three times as fast as the Weight. By the very same reasoning as we philosophize concerning simple pulleys, we must also reason in pulleys having more sheaves; for if each has two sheaves, it must be observed whether the end of the rope is attached to the fixed pulley or to the movable one: if to the fixed, the power moves four times as fast as the weight; but if to the movable, it moves five times as fast. Generally, therefore, multiply the number of sheaves of the movable pulley, to which the weight is joined, and for each sheave double the moments of the power. Hence, if there are three sheaves, the moment of the Power is sixfold; if four, eightfold; and so on. But if the end of the rope is attached to the same movable pulley, add one more still, and the moment will be sevenfold, or ninefold. For the rope, when fastened to one pulley, first rests on the first sheave of the other pulley; thence it is bent to the first sheave of the pulley to which it is attached; afterward it passes to the second sheave of the other pulley, and returning to the former pulley rests on its second sheave; and so on thereafter, moving alternately from pulley to pulley, until it rests on all the sheaves. But if the two pulleys do not have an equal number of sheaves, but one exceeds the other by one, it is necessary to fasten the rope to the pulley with fewer sheaves. Wherefore the number of sheaves of the movable pulley is likewise to be considered, which, if it have fewer sheaves, the end of the rope is attached to it; and thus, after doubling its number of sheaves, one must add one: so that, if it has two sheaves, the motion of the Power is five times the motion of the Weight. But if the movable pulley has more sheaves than the fixed pulley, its number alone is to be doubled, in order that the denomination of the moment may be obtained; thus, if there are three sheaves, the motion of the power to the motion of the weight is sixfold. In pulleys of this kind, having many wheels, it is to be observed that the inner wheels should be made smaller, the outer ones however

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Mechanicorum majores: nam A & C minores sunt, B & D majores, ne fu- nium ductus se invicem intercipiant, ac mo- tum mutuo tritu retardent, nisi etiam sese vicis- sim atterentes funes disrumpantur. Quare pro- bare non possum Trochleas, quæ plures orbi- culos parallelos uni & eidem axi infixos intra congruum loculamentum habent; quamvis enim Trochleis hujusmodi valde inter se distan- tibus non adeò appareat incommodum funium sese perfricantium, ubi tamen illæ propiores factæ fuerint, hoc manifestò apparet: præ- terquam quod funis obliquè insistens extremæ ipsarum rotularum orbitæ, quam contingit, non adeò facilè movetur, ac si illis exactè con- grueret, ut fit, quando singulæ rotulæ suos ha- bent axes. Et quidem quod ad axes rotularum spectat, quamvis nec admodum longi sint, & rotula suo loculamento proximè adhæreat, atque adeò non sint facilè obnoxij fractionis periculo, ca- vendum tamen est, ne nimis exiles sint, aut ex materia non satis solidâ; ne fortè ponderis attollendi gravitas illos labefactet. Verum qui- dem est non esse necesse singulos axes statuere sustinendo oneri pares; cum enim plures sint, adversus singu- los minor conatus ponderis exercetur. Si verò illi exquisitè læves atque politi fuerint, faciliorem fore rotularum iis infixa- rum revolutionem apertiùs constat, quàm ut moneri artificem oporteat. Prætereà rotularum facies optimè lævigatas velim, & lo- culamentum ipsum non placet ita amplum, ut maximam ro- tularum partem includat: satis est, si ità firmum ac solidum sit, ut axes contineat, & in extremitatibus validos uncos ha- beat, quibus & funis, & onus alligari queant: Quò scilicet minorem rotularum partem tangit, minus cum illis confligit, adeóque facilior est motus: neque enim leviora hæc compen- diam omninò contemnenda sunt. Demum funis ductarij crassitudo statuenda est, quæ reti- nendo

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Mechanics: for A & C are smaller, B & D larger, lest the courses of the ropes intercept one another and, by rubbing mutually, retard the motion, unless, by wearing against one another as well, the ropes are broken. For this reason I cannot approve of pulleys that have several parallel grooved wheels fixed to one and the same axis within a suitable housing; for although with pulleys of this kind, when they are set at a great distance from one another, the inconvenience of the ropes rubbing against each other does not seem very evident, yet when they are made closer together this plainly appears: besides, a rope running obliquely against the outer groove of the wheel it touches does not move so easily as it would if it fitted exactly, as happens when each wheel has its own axis. And indeed, as regards the axes of the wheels, although they are not very long and the wheel adheres closely to its housing, and therefore are not easily exposed to the danger of breaking, care must nevertheless be taken that they are not too slender or made of material not sufficiently solid, lest perhaps the weight of the load being lifted should weaken them. But it is indeed true that it is not necessary to make each axis as strong as would be required to bear the load; for since there are several of them, a smaller effort is exerted against each one. If, however, they are exquisitely smooth and polished, it is clear enough that the rotation of the wheels fixed in them will be easier, than that the craftsman needs to be reminded of it. Moreover, I would have the faces of the wheels as smooth as possible, and I do not like the housing itself to be so wide that it encloses the greater part of the wheels: it is enough if it is so firm and solid as to contain the axes, and if at the ends it has strong hooks, to which both the rope and the load can be fastened. For inasmuch as it touches a smaller part of the wheels, it conflicts less with them, and therefore the motion is easier; for these advantages are by no means to be entirely disregarded. Finally, the thickness of the guiding rope must be determined, in order to keep it

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Liber sextus. CAPUT I. 583 nendo ponderi respondeat: sed quia plures sunt funis à trochleâ in trochleam ductus, ideò quasi plures funes reputantur, inter quos quodammodo distribuitur sustentatio ponderis, perinde ferè, atque si ex pluribus illis ductibus funis unicus compone- retur. Hinc si pondus fuerit adnexum trochleæ I, sustinetur à quatuor funibus; sin autem trochlea I in superiore loco firmata fuerit, & pondus trochleæ H alligatum dependeat, sustinetur à quinque funibus, nam etiam Potentia in O sustinet fune R O. Ex funis autem crassitudine definitur rotularum altitudo, ut ni- mirum orbitæ excavatæ insistere possit funis, quin interiorem loculamenti faciem contingat, ne perpetuo affrictu atteratur cum disruptionis periculo, & non levi celeritatis detrimento, auctâ trahendi difficultate. Porrò cùm excavatam dico rotu- larum orbitam, nolim intelligas quasi crenam perimetro pro- fundiùs incisam; sed satius fuerit orbitam ipsam esse modicè sinuatam; hoc enim pacto facilius excurrit funis, etiamsi paulò crassior aliquando adhibendus sit, qui cæteroqui inter crenæ incisæ labra depressus non sine labore ex illis angustiis eximere- tur in rotulæ conversione. Cum itaque ea sit Trochlearum dispositio, ut pondus tardiùs moveatur, potentia velocius (si videlicet alteri Trochlearum non Potentia, sed Pondus adnectatur, alioquin si loca permu- tarent, res contrario prorsùs modo se haberet) manifestum est resistentiam ponderis minui ex tarditate; poterit igitur augeri ex gravitate: sæpiùs quippe dictum est adæquatum resistentiæ momentum componi ex insita gravitate, & ex dispositione ad motûs velocitatem, aut tarditatem. Potentia igitur valens su- perare resistentiam ponderis alicujus certæ gravitatis, si cum illa æqualiter movendum sit, poterit eodem impetu, atque co- natu superare resistentiam majoris ponderis, si ex collocatione, quatenus cum Potentiâ connectitur, ita minus velociter movea- tur, ut quæ Ratio est æqualis illius velocitatis ad minorem ve- locitatem, eadem sit Ratio majoris ponderis ad pondus illud æquè velox cum potentiâ; est enim omnino par resistentia; quia quantum addit major velocitas minori ponderi, tantum- dem addit majus pondus minori velocitati. Quamvis autem ponderis motus non sit æquè velox ac motus potentiæ, tamen ponderis motus entitativè acceptus æqualis est motui

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Book Six. CHAPTER I. 583 correspond to the load: but because there are several ropes running from pulley to pulley, they are therefore reckoned as so many ropes, among which the support of the load is in a way distributed, almost as if a single rope were made up from those several runs. Hence, if a weight is attached to pulley I, it is sustained by four ropes; but if pulley I is fixed in the upper position, and the weight attached to pulley H is allowed to hang down, it is sustained by five ropes, for the Power at O also supports it with the rope R O. From the thickness of the rope, however, the height of the pulleys is determined, namely, that the rope may rest upon the hollowed groove without touching the inner face of the housing, lest it be worn away by continual rubbing, with danger of rupture, and no small loss of speed, together with increased difficulty of pulling. Moreover, when I say the groove of the pulleys is hollowed, I do not mean that you should understand it as a notch cut deeper around the perimeter; rather it is better for the groove itself to be moderately curved inward; for in this way the rope runs out more easily, even if at times a somewhat thicker rope must be used, which otherwise, pressed down between the lips of the cut notch, could not without effort be drawn out of those narrow openings when the pulley is turned. Since, then, such is the arrangement of pulleys, that the weight is moved more slowly, the power more quickly (if, namely, to one of the pulleys Power, and not Weight, is attached; otherwise, if the places were exchanged, the matter would stand in the opposite way altogether), it is clear that the resistance of the weight is diminished by slowness; it may therefore be increased by heaviness: for it has often been said that the adequate moment of resistance is composed of inherent heaviness and of the disposition toward the velocity or slowness of motion. Therefore a Power capable of overcoming the resistance of a certain weight, if it must move equally with that weight, will be able by the same impulse and effort to overcome the resistance of a greater weight, if, by reason of its placement, insofar as it is connected with the Power, it is moved more slowly, so that the Ratio of that velocity to the lesser velocity is the same as the Ratio of the greater weight to that weight moving with equal speed as the power; for the resistance is altogether equal; because whatever the greater velocity adds to the lesser weight, the greater weight adds just as much to the lesser velocity. Although, however, the motion of the weight is not as swift as the motion of the power, still the motion of the weight, taken entitatively, is equal to the motion

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Mechanicorum 584 motui potentiæ, ac proindè mirum non est, si potentia eadem impetu eodem æqualem motum producat, atque efficiat. Pone enim in O gravitatem paulò majorem libris 100; utique si in S statuerentur libræ 100 gravitas O prævaleret, & gravitatem S elevaret: igitur illa eadem gravitas O elevabit libras 400 in I adnexas Trochleæ, nam I movetur quadruplo tardiùs quam O, ex dictis, S autem movetur æqualiter ac O; ergo ratione motûs tardioris quadruplo minùs resistit pondus lib. 400 in I, licet ratione gravitatis quadruplo magis resistat. Si itaque li- bræ 100 in S intra certum tempus percurrant unà cum Poten- tia O spatij pedes 40, eodem tempore singulæ libræ 100 gra- vitatis in I adnexæ percurrunt pedes 10: at sunt libræ 400; igitur sunt quatuor motus pedum 10, & illarum omnium mo- tus est pedum 40. Quare potentia O idem planè efficit, ac si moveret in S libras 100: id quod præstare potest absque ulla machina. Et quidem si res attentè perpendatur, nec vulga- ribus vocabulis notionem minus propriam subjiciamus, non est dicendum manente eodem conatu, & eadem velocitate Po- tentiæ augeri per Machinam potentiæ momenta, aut vires, semper enim Potentia vincit æqualem resistentiam sive adhibi- tâ machinâ, sive absque illâ, quamvis non semper vincat ean- dem gravitatem. Quemadmodum in libra nil refert, utrum corpus expendendum habeat majorem gravitatem secundum speciem, sed molem minorem, an verò minorem gravitatem specificam sub mole majori, modò reciprocè sit ut gravitas specifica ad specificam gravitatem, ita moles ad molem; est si- quidem par gravitas absoluta, quæ componitur ex gravitate spe- cificâ & mole. Ita pariter æqualis est absoluta ponderis re- sistentia, quæ ex gravitate, & velocitate componitur, si fuerit inter eas reciproca Ratio. CAPUT

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Mechanics 584 to the motion of power, and therefore it is no wonder if the same power, with the same impulse, produces and brings about an equal motion. Suppose, then, in O a weight a little greater than 100 pounds; surely if 100 pounds were placed in S, the weight in O would prevail and raise the weight in S: therefore that same weight in O will raise 400 pounds in I attached to the pulley, for I moves four times more slowly than O, as stated above, while S moves equally with O; therefore, by reason of the slower motion, the 400-pound weight in I resists four times less, although by reason of its weight it resists four times more. If, then, 100 pounds in S, within a certain time, together with the power O, travel 40 feet of space, in the same time each 100-pound weight attached in I travels 10 feet: but there are 400 pounds; therefore there are four motions of 10 feet, and the motion of all of them is 40 feet. Wherefore power O accomplishes exactly the same thing as if it moved 100 pounds in S: something it can do without any machine at all. And indeed, if the matter is considered carefully, and if we do not attach an inaccurate notion to common words, it must not be said that, while the same effort and the same speed remain, the moments or forces of power are increased by a machine; for power always overcomes an equal resistance, whether a machine is used or not, although it does not always overcome the same weight. Just as in a balance it makes no difference whether the body to be weighed has a greater specific weight but smaller bulk, or a smaller specific weight under greater bulk, provided there is reciprocally the same proportion of specific weight to specific weight as bulk to bulk; for it is the equal absolute weight, composed of specific weight and bulk. In the same way, the absolute resistance of weight is equally made up of weight and speed, if there is between them a reciprocal ratio. CAPUT

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Liber sextus. CAPUT II. 585 CAPUT II. An Trochlea ad Vectem revocanda sit. UT Machinalis motûs causa meliùs innotescat, neque opus esse Facultates omnes ad Vectem revocare, ut non pauci hactenus conati sunt, & adhuc conantur, hîc potissimùm quæstionem hujusmodi examinare placuit in Trochleâ. Aiunt siquidem in simplici orbiculo, quando ejus centrum immotum manet, & alteram funis extremitatem potentia apprehendit, ex alterâ dependet pondus, Vectem esse primi generis, cujus hypomochlium est in centro orbiculi, potentia & pondus in extremitatibus diametri; quæ cùm à centro æqualibus intervallis absint, vectis ille nil juvat potentiam. Quando verò ponderi adnectitur theca, cui orbiculus includitur, adeóque ejus centrum unà cum pondere movetur, jam pondus respondet orbiculi centro, & extremitatem alteram diametri obtinet potentia trahens funem; quapropter hypomochlium censendum est in opposita diametri extremitate. Quapropter cùm pondus sit inter potentiam, & hypomochlium, vectis est secundi generis: & quia pondus est in vectis medio, potentiæ momentum duplum est momenti ponderis, si positio ipsa spectetur. Sit orbiculus, cujus centrum C, ejusque loculamento adnexum pondus respondeat lineæ CB: funis R S D T V sit alligatus in R, & Potentia sit in V, quæ funem trahens intelligitur constituta in T, & oppositum diametri punctum S censetur hypomochlium; atque adeò momentum Potentiæ ad momentum Ponderis est ut T S ad C S. Ex quo fit, si reciprocè vis potentiæ ad gravitatem ponderis sit ut C S ad T S, ab hujusmodi potentiâ sustineri pondus, & potentia si augeatur, etiam moveri, orbiculo circà suum centrum revoluto, & versus potentiæ E E e e

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Book Six. CHAPTER II. 585 CHAPTER II. Whether the Pulley is to be referred to the Lever. So that the cause of mechanical motion may be more clearly understood, and it may not seem necessary to reduce all faculties to the Lever, as not a few have hitherto tried, and still try, it has seemed best here to examine especially such a question in the Pulley. For they say that in a simple wheel, when its center remains unmoved, and the power takes hold of one end of the rope, while the weight hangs from the other, it is a Lever of the first kind, whose fulcrum is at the center of the wheel, the power and the weight at the ends of the diameter; and since these are equally distant from the center, that lever helps the power not at all. But when a case is attached to the weight, in which the wheel is enclosed, so that its center is moved together with the weight, then the weight corresponds to the center of the wheel, and the power pulling the rope occupies the other end of the diameter; wherefore the fulcrum must be deemed to be at the opposite end of the diameter. Therefore, since the weight lies between the power and the fulcrum, the lever is of the second kind; and because the weight is in the middle of the lever, the moment of the power is double the moment of the weight, if the position itself be considered. Let there be a wheel, whose center is C, and let its attached weight in the socket correspond to the line CB: let the rope R S D T V be fastened at R, and let the Power be at V, which, pulling the rope, is understood to be placed at T, and the opposite point of the diameter S is considered the fulcrum; and thus the moment of the Power to the moment of the Weight is as T S to C S. From which it follows that, if reciprocally the force of the power to the heaviness of the weight is as C S to T S, the weight is sustained by such a power, and if the power be increased, it is also moved, the wheel being revolved about its center, and toward the power's E E e e

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586 Mechanicorum tiam attracto. In conversione autem orbiculi, prout aliæ atque aliæ sunt diametri, quas contingunt funis ductus R S, & V T, alios subinde, atque alios vectes esse comminiscuntur. Verùm hujusmodi ratiocinationi nunquam aquiescere po- tui; mihi enim perspectum est, si orbiculus non fuerit versatilis, sed omnino fixus in suo loculamento, adhuc potentiam V faci- liùs attollere pondus, quod in B intelligitur suspensum, quàm illud directè, & immediatè attolleret; & tamen diameter ea- dem T S semper maneret horizonti parallela (nam C B semper est in perpendiculo) nullumque haberet motum conversionis circà punctum, quod vocant, hypomochlij S, quo referret mo- tum Vectis proprium. Adde orbiculum in suo loculamento fixum perinde esse, atque si annulus ponderi adnectatur, & fu- nis alligatus in R inseratur annulo, atque potentia in V funem trahat; potentia enim duplo velociùs movetur, quàm annulus & pondus: hîc autem in annulo, quem nullatenus convolvi cer- tum est, quomodo Vectis vestigium deprehendes? Illud qui- dem incommodi in annulo, & in orbiculo non versatili, accide- ret, quod funis ob suam asperitatem cum orbiculi orbitâ, & cum annulo confligeret; ex quo tritu non levis movendi difficultas oriretur: propterea, ad vitandum hujusmodi incommodum adhibentur orbiculi circa suum axem versatiles; axis enim poli- tus, aut etiam addito unguine lubricus, fere nullam creat orbi- culi rotationi difficultatem, funis verò non atterit ejusdem or- biculi orbitam, quâ revolutâ ille explicatur. Cæterum quod ad Rationem motuum potentiæ & ponderis spectat, eadem est Ra- tio dupla, sive orbiculus versatilis sit, sive fixus, sive annulus ponderi adnectatur, sive etiam ponderi inseratur funis, ità ut pondus ipsum excurrere queat. Hoc scilicet unicè pendet ex ipsâ funis inflexione: nam si funis A B ita flectatur, ut ad extremitatem extremitas accedat, & B veniat in C propè A; utique non nisi media pars B E mo- vetur; adeò ut, si annulus inseratur funi in B, & per longitudinem funis, qui complicatur, excurrat, ve- niat ex B in E interea, dum extremitas B, & poten- tiam illam adducens, venit in C: quo in motu sin- gulæ funis particulæ inter B & E percurrunt spa- tium duplum distantiæ singularum à medio, ante- quam

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586 Mechanicorum thus attracted. But in the turning of the small wheel, since there are one and another diameters which the cord passes over, R S and V T, they imagine now one sort of lever, now another. But I could never assent to reasoning of this kind; for it is clear to me that if the little wheel were not movable, but entirely fixed in its socket, still the power V would more easily lift the weight understood to be suspended in B than would it lift it directly and immediately; and yet the diameter T S would always remain parallel to the horizon (for C B is always perpendicular), and it would have no motion of rotation around the point, as they call it, the hypomochlion S, by which the proper motion of the lever would be referred. Add that a little wheel fixed in its socket is the same as if a ring were attached to the weight, and a cord fastened in R were inserted through the ring, and the power in V drew the cord; for the power moves twice as fast as the ring and the weight. But here, in the ring, which by no means is certain to be wound up, how will you detect the trace of a lever? Indeed, that inconvenience would occur in the ring, and in the non-rotating little wheel, that the cord, because of its roughness, would rub against the groove of the wheel and against the ring; from which friction no slight difficulty of motion would arise. Therefore, to avoid such inconvenience, little wheels movable about their axis are used; for a polished axis, or even one made slippery by the addition of grease, creates almost no difficulty for the rotation of the wheel, and the cord does not wear away the groove of the same wheel, by whose turning it is unwound. Moreover, as far as the ratio of the motions of the power and the weight is concerned, the ratio is the same, namely double, whether the little wheel is movable or fixed, whether a ring is attached to the weight, or whether a cord is also inserted into the weight so that the weight itself can run along. This, of course, depends solely on the bending of the cord itself: for if the cord A B is bent in such a way that end approaches end, and B comes to C near A, certainly not more than the middle part B E moves; so that, if a ring is inserted into the cord at B, and runs along the length of the cord that is being folded, it will pass from B to E in the meantime, while the end B, bringing along that power, comes to C: in that motion each of the particles of the cord between B and E travels a space double the distance of each from the middle, before

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Liber sextus. CAPUT II. 587 quam complicarentur: & si potentia ex C ulteriùs progrediatur, singulæ funis particulæ inter medium & caput A inter- ceptæ perficiunt spatium duplum distantiæ singularum à capi- te A, ubi funis religatur. Iam verò statue ampliorem aliquem, & satis gravem cylin- drum D E F G, qui rotatu pro- movendus sit, aut in plano hori- zontali, aut in superiorem plani inclinati locum: applicentur au- tem homines in D & G, & quot- quot necessarij fuerint juxta cylin- dri longitudinem, qui illum im- pellant. Quæro, an ibi ulla Vectis ratio intercedat, ita ut sit quasi vectis D E, hypomochlium in E, & pondus in puncto I, quod respondet centro gravitatis, at- què adeò in cylindri conversione subinde mutetur vectis, & locus tùm potentiæ, tùm hypomochlij, prout aliis atque aliis perimetri punctis applicatur potentia im- pellens, quibus ex diametro opponuntur alia atque alia puncta, in quibus à subjecto plano cylinder tangitur. Vix, puto, au- debis Vectem ibi agnoscere, ubi demum Potentiam impellen- tem, & Pondus, quod in centro gravitatis, scilicet in Axe cy- lindri, constitutum intelligitur, æqualem motus lineam per- currisse deprehenderis, ut manifestum est in hujusmodi rotun- dorum corporum revolutione, in qua æqualem lineam percur- runt centrum, & punctum in peripheriâ notatum. Igitur duo- rum funium capita firmiter alliga in M & H, ipsósqque funes cylindro subjice, & in superiorem partem reductos ita dispo- ne, ut cylindrum complectantur, atque à duabus potentiis, quæ priùs in D & G impellebant, trahantur capita L & P. Certissi- mo constat experimento longè faciliùs cylindrum hujusmodi funibus convolvi, quàm impulsion potentiarum illi proximè applicitarum. Si nulla Vectis Ratio agnoscenda est in diame- tro D E, utique facilitas illa movendi non habetur à vecte, qu nullus est: Sin autem Vectem ibi esse constanter affirmes, igi E E e e 2

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Book Six. CHAPTER II. 587 when they are wound together: and if the force from C advances farther, the individual strands of the rope, intercepted between the middle and the head A, accomplish a space double the distance of each from the head A, where the rope is fastened. Now then, suppose some larger, and sufficiently heavy, cylinder D E F G, to be moved by rotation, either on a horizontal plane, or to a higher place on an inclined plane: let men be applied at D and G, and as many as are necessary along the length of the cylinder, to push it along. I ask whether there is here any reason for a lever, so that it is as though the lever were D E, the fulcrum at E, and the weight at the point I, which corresponds to the center of gravity; and so that in the turning of the cylinder the lever, and the place of both the force and the fulcrum, are continually changed, as the driving force is applied to different points of the perimeter, to which, diametrically opposite, are other and other points at which the cylinder touches the supporting plane. I think you will hardly dare to recognize a lever there, when at last you find that the driving force and the weight, which is understood to be placed in the center of gravity, that is, in the axis of the cylinder, have traversed an equal line of motion, as is evident in the revolution of such round bodies, in which the center and a point marked on the circumference traverse an equal line. Therefore tie the ends of two ropes firmly at M and H, and place the ropes under the cylinder, and, drawn up over the top, arrange them so that they encircle the cylinder, and let the ends L and P be pulled by the two forces which previously pushed at D and G. By very certain experiment it is found that a cylinder of this kind is wound up much more easily by ropes than by the push of forces applied near it. If no reason of a lever is to be acknowledged in the diameter D E, then surely that ease of moving is not obtained from a lever, which does not exist: but if you nevertheless firmly assert that there is a lever there, then the

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Mechanicorum tur perindè est si Potentia proximè, & immediatè applicetur puncto D, aut H, ad impellendum, atque si medio fune M H L applicetur puncto H trahens funis caput L: atqui longè majo- ra momenta habet funem L H trahens, quàm impellens in H; cum igitur utrobiqque idem Vectis; eadem scilicet cylindri dia- meter, habeatur, sed non idem momentum, non ex rationibus Vectis, sed aliundè petenda est hæc momenti accessio: Quia videlicet fune sic disposito, potentia duplo velociùs movetur quàm pondus, nullâ habitâ vectis ratione. Finge jam funem laxiorem circumplecti cylindrum, & in nodum colligi in X: utique si in X adderetur pondus aliquod raptandum unà cum cylindro promoto; facilius raptaretur cylindro hujusmodi fu- nibus revoluto, quàm si cylindrus impulsione potentiæ proxi- mè applicatæ promoveretur; & tamen major hæc facilitas ex nullo vecte addito oriretur. An non ergo cylindrus trochleæ orbiculum refert, & funis X orbiculi loculamentum, cui pon- dus adnectitur: manifesto igitur experimento habetur non ex Vectis rationibus ducendam esse majorem movendi facilitatem, quæ ex simplici trochleâ habetur, quando illi adnectitur pondus. Sed præstat examinare, quæ præterea dicuntur, quando ei- dem simplici Trochleæ, cui pondus M adnectitur, etiam funis caput alligatur; tunc enim potentiæ momentum triplex est, adeò ut ad attollendum pondus M sufficiat potentia subtripla illius potentiæ, quæ absque machinâ attol- leret idem pondus. Sic igitur ratiocinantur apud P. Schott in Magia mechanica Syn- tagm. 4. cap. 2. prop. 5. Si fuerit Vectis D E, in cujus medio C sit pondus, fuerit autem quædam potentia in C sustinens, & alia po- tentia illi æqualis sustinens in E, hypomo- chlium verò in D, unaquæque potentia est subtripla ponderis sustentati. Quia enim potentia C distat ab hypomochlio D æqua- liter ac pondus in C constitutum, sustinet pondus æquale suis viribus; potentia autem E, quia est in duplo majore distantia quàm Pondus C, sustinet pondus duplum suarum virium.

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It is in mechanics the same whether the power is applied very near and immediately to the point D or H, in order to push, or whether, by means of the rope M H L, it is applied to point H, pulling the end L of the rope: yet the rope pulling at L H has far greater force than pushing at H. Since therefore in both cases the same lever is used, namely the same diameter of the cylinder, but not the same force, this increase of force must be sought not from the reasons of the lever, but from elsewhere: namely because, with the rope so arranged, the power moves twice as fast as the weight, without any consideration of the lever. Suppose now that a looser rope is wrapped around the cylinder and gathered into a knot at X: certainly if some weight were added at X to be dragged along together with the moving cylinder, it would be dragged more easily by a cylinder revolved with such ropes than if the cylinder were moved by the impulse of a power applied very near; and yet this greater ease would arise from no lever added. Does not the cylinder therefore represent the wheel of a pulley, and the rope X the groove of the wheel to which the weight is attached? By clear experiment therefore it is shown that the greater ease of moving, which is obtained from a simple pulley when a weight is attached to it, is not to be derived from the reasons of the lever. But it is better to examine what is said besides this: when to the same simple pulley, to which the weight M is attached, the end of the rope is also tied, then the force has a triple effect, so that for lifting the weight M a force one-third of that force suffices which without the machine would lift the same weight. Thus they reason in P. Schott, Magia mechanica, Syntagm. 4, cap. 2, prop. 5. If there be a lever D E, in the middle of which C the weight is placed, and there be one power in C sustaining it, and another power equal to it sustaining at E, with the fulcrum at D, each power is one-third of the supported weight. For since the power at C is equally distant from the fulcrum D as the weight placed at C, it sustains a weight equal to its own force; but the power at E, because it is at twice the distance as the weight C, sustains a weight double its own force.

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Liber sextus. CAPUT II. 589 virium. Quoniam ergo Potentiæ ex hypothesi sunt æquales, & totius ponderis duæ partes sustinentur à Potentia E, & una à Potentia C, illa autem est subdupla ponderis à se sustentati, unaquæque est ejusdem totius ponderis subtripla quo ad vires sustentandi. Cum igitur in propositis Trochleis sit potentia F sustinens in medio, & potentia G in alterâ extremitate sustinens, unaquæque est subtripla ponderis M sustinendi, ac propterea Potentia G si sit paulo major quàm subtripla, erit etiam apta ad movendum pondus. His pariter assentiri nequeo, quæ de ponderis sustentatione dicuntur; nec satis video, an Vecti secundi generis congruant; neque enim solùm Potentiæ in medio atque in altera extremitate applicatæ, verùm etiam hypomochlium ipsum exercet vim sustinendi: ex hoc siquidem quod addatur potentia in medio, ubi est pondus, non tollitur omnino pressio, quâ hypomochlium à pondere urgetur. Quare non tota vis sustentandi dividenda est inter duas illas potentias, sed etiam admittendum est hypomochlij consortium. Dic autem, quænam est potentia in F retinens pondus? nonne statim ac potentia in G remissiorem conatum adhibet, etiam F cum pondere descendit? ipsa quippe Potentia G dum intentum funem F I retinet, sustinet etiam pondus; atque adeò non duæ sunt potentiæ sustinentes, sed unica. Et quidem, si res sincerè exponatur, pondus sustinetur & à potentiâ G sursum conante, & à clavo S, cui superior trochlea adnectitur, mediis funibus H D, I F retinente, ità ut centrum gravitatis ponderis sit in lineâ Directionis transeunte per ipsum clavum S, si funis G E sit ad perpendiculum, nec in latus retrahat trochleam C: eo autem ipso, quòd Potentia G suo conatu prohibet, ne funis excurrat, retinet pondus ex eodem clavo S suspensum. Quapropter ejusdem potentiæ G est vis illa, quæ & in F, hoc est in C & in E retinet. Quando verò sursum attolli- tur pondus, eadem est potentia G, quæ sursum trahit F, cui non minùs applicatur medio fune I F, quàm applicetur ipsi E medio fune G E; neque enim in F est alia potentia sponte sursum ascendens, & secum rapiens pondus. Sed quid frustrà confugiamus ad vim sustentandi pondus ex trochleis dependens? si pondus fuerit in plano horizontali tra- E E e e 3

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Book Six. CHAPTER II. 589 of forces. Since therefore the Potencies, by hypothesis, are equal, and two parts of the whole weight are sustained by the Potency E, and one by the Potency C, but that one is subdouble of the weight sustained by it, each is subtriple of the same whole weight in respect to the sustaining forces. Since therefore in the proposed Pulleys the potency F is sustaining in the middle, and the potency G at the other extremity is sustaining, each is subtriple of the weight M to be sustained, and therefore the Potency G, if it be a little greater than subtriple, will also be fit for moving the weight. I likewise cannot agree with these things that are said concerning the sustaining of the weight; nor do I clearly see whether the Levers of the second kind are suitable; for not only do the Potencies applied in the middle and at the other extremity act, but the hypomochlion itself also exerts a sustaining force: indeed, from this that a potency is added in the middle, where the weight is, the pressure is not wholly removed, by which the hypomochlion is pressed by the weight. Wherefore the whole force of sustaining is not to be divided between those two potencies, but the partnership of the hypomochlion must also be admitted. But tell me, what is the potency in F retaining the weight? Does not the moment the potency in G applies a more relaxed effort, F also descend with the weight? for the Potency G itself, while it holds the strained rope F I, also sustains the weight; and thus there are not two sustaining potencies, but one. And indeed, if the matter be stated sincerely, the weight is sustained both by the potency G striving upward and by the nail S, to which the upper pulley is attached, through the intermediate ropes H D, I F, so that the center of gravity of the weight is in the line of direction passing through the very nail S, if the rope G E be perpendicular and does not draw the pulley C sideways: but by that very fact that the Potency G by its effort prevents the rope from running out, it retains the weight suspended from the same nail S. Wherefore to the same potency G belongs that force which also retains in F, that is, in C and in E. But when the weight is being lifted upward, it is the same potency G which draws F upward, and to which it is applied no less through the middle rope I F than it is applied to E itself through the middle rope G E; nor, indeed, is there in F any other potency naturally ascending upward and carrying the weight along with it. But why should we in vain have recourse to a force sustaining a weight hanging from pulleys? if the weight were on a horizontal plane E E e e 3

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590 Mechanicorum hendum, nihil in trochleis reperitur, à quo sustineatur pondus omnino incumbens subjecto plano, & tamen potentia G est subtripla potentiæ, quæ sine machinâ in eodem plano trahe- ret idem pondus: ratione vectis E D solùm esse potest subdu- pla; in F nulla est potentia trahens; unde ergo ratione vectis potentia ad trahendum pondus habet momenti incrementum? Quod si dixeris eandem potentiam, quæ in G trahit, etiam tra- here in F; igitur conatum non adhibet subtriplum, sed subses- quialterum; nam conatur & in extremitate E, & in vectis me- dio C, ut tu quidem ais, ita ut utrobiq[ue] sit subtripla vis mo- vendi: fatendum est ergo potentiam trahentem conari ut 2/3, cum tamen reipsa adhibeat solum conatum ut 1/3. Consideremus demum Trochleas pluribus instructas orbicu- lis, & videamus, quid ex Vecte sperari possit. Statuunt Autho- res cum eodem P. Schott ibid. prop. 7. si fuerint duo vectes B A, & D C, ex quorum me- dio E & F dependeat pondus G, duas potentias æquales in B & D constitutas, simulque æqua- liter in sustinendo pondere la- borantes, singulas esse subqua- druplas ponderis. Nam si sola potentia D sustineret, esset pon- deris subdupla, scilicet ut F C ad D C; & si sola potentia B sustineret, esset ipsa pariter sub- dupla, nimirum ut E A ad B A. Cum igitur ambæ æquales sint, & æqualiter conentur, unicui- que respondebit subduplum subdupli, hoc est quarta pars ponderis. Atqui in Trochleis binos orbiculos habentibus sunt duo vectes H I, & P O in me- dio sustinentes pondus, hypomochlia in I & O, atque Po- tentiæ in H & P. Igitur potentia sustinens est ponderis sub- quadrupla, & movens paulò major subquadruplâ. Quæ de duobus Vectibus D C & B A dicuntur, illa quidem catenus

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590 Mechanics hendum, nothing is found in the pulleys by which the weight entirely resting on the subjacent plane is sustained, and yet the power G is one-third of the power which, without a machine, would draw the same weight on the same plane: by reason of the lever E D it can only be one-half; in F there is no drawing power at all; from what, then, does the power for drawing the weight derive an increase of force by reason of the lever? If you should say that the same power which draws at G also draws at F, then it does not exert a one-third effort, but a one-and-a-half-th part; for it exerts itself both at the extremity E and at the middle C of the lever, as you indeed say, so that in both places there is the one-third force of moving: therefore it must be admitted that the drawing power strives as 2/3, although in reality it employs only an effort of 1/3. Let us finally consider pulleys equipped with several wheels, and see what can be expected from the lever. The authors, together with the same P. Schott, ibid. prop. 7, establish that if there are two levers B A and D C, from the middle of which E and F the weight G is suspended, and two equal powers are placed at B and D, and likewise labor equally in sustaining the weight, each is one-fourth of the weight. For if only the power at D were sustaining it, it would be one-half of the weight, namely as F C is to D C; and if only the power at B were sustaining it, it too would likewise be one-half, namely as E A is to B A. Since therefore both are equal, and strive equally, to each there will correspond a half of a half, that is, the fourth part of the weight. But in pulleys having two wheels there are two levers H I and P O sustaining the weight in the middle, the fulcra at I and O, and the powers at H and P. Therefore the sustaining power is one-fourth of the weight, and the moving power somewhat greater than one-fourth. What is said about the two levers D C and B A, indeed so far

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Liber sextus. CAPUT II. 591 eatenus admitto, quatenus singulas potentias D & B sustinen- tes subquadruplas esse ponderis definiunt; nam perinde se ha- bent, atque si utraque potentia in unius, ejusdemque vectis ex- tremitate simul sustinerent, unicamque potentiam, constitue- rent, quæ subdupla est ponderis: & quia singulæ potentiae sunt ad totam & integram potentiam subduplae, singulæ sunt ponde- ris subquadruplae. Cæterum ex hoc quod ambæ potentiae æqua- les sint, & singulæ solitariæ essent subduplae arguere, quod uni- cuique respondeat subduplum subdupli, materialiter quidem verum est, non autem formaliter ex modo argumentandi; alio- quin si addatur tertius vectis, servatâ eadem argumentandi for- mâ, tres essent potentiæ, & unicuique respondebit subduplum subdupli, hoc est octava pars ponderis; id quod est falsum. Neque enim ex hoc quod potentia D sustineat pondus in F, facit illud esse minùs grave, quasi transferatur in E factum gravitatis subduplae, & potentia B subduplam gravitatem pon- deris sustineret subduplo conatu, hoc est subquadruplo ejus, qui requiritur ad sustinendum totum pondus; alioquin addito tertio vecte in illius medium transferretur gravitas subquadru- pla ponderis, quæ sustineretur à potentiâ illius subduplâ, ac proinde suboctupla totius ponderis; cum tamen in tribus vecti- bus sic dispositis tres potentiæ sustinentes singulæ sint solum subsextuplae. Quod si pondus alligetur medio primi vectis in F, tum extremitas D alligetur medio secundi vectis in E, & dein- ceps extremitas B alligetur medio tertij vectis, optimè con- cluditur potentiam in F sustinere subduplum subdupli, & po- tentiam applicatam tertio vecti sustinere subduplum subdupli subdupli, ac proinde illam esse subquadruplam, hanc verò sub- octuplam. Sed hæc dispositio nil juvaret ad explicandum Tro- chlearum momentum. Verùm in Trochleâ duas illas potentias in H & P non vi- deo; nam unica potentia in X medio fune XSH applicatur quidem puncto H, suoque conatu prohibet ne pondus suâ gra- vitate deorsum trahat ipsam Trochleam: at in P quænam alia Potentia hoc idem efficit? An non eadem Potentia X medio fune XSHILMP applicatur vecti PO in P? igitur eadem potentia exhibet conatum duarum potentiarum subquadrupla- rum: igitur Potentia non est subquadrupla, sed solum subdu- pla;

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Book Six. CHAPTER II. 591 I admit so far as they define the individual powers D and B, supporting them, to be subquadruple of the weight; for they are related just as if both powers together at the same end of one and the same lever supported a single power, which would be subduple of the weight: and because each power is subduple to the whole and entire power, each is subquadruple of the weight. However, from the fact that both powers are equal, and that each separately would be subduple, to argue that to each there corresponds the subduple of the subduple is materially true indeed, but not formally, given the mode of argument; otherwise, if a third lever be added, keeping the same form of argument, there would be three powers, and to each there would correspond the subduple of the subduple, that is, the eighth part of the weight; which is false. Nor indeed, because power D supports the weight at F, does it make it less heavy, as though the effect of subduple gravity were transferred to E, and power B supported the subduple gravity of the weight by a subduple effort, that is, by a force subquadruple of that required to support the whole weight; otherwise, if a third lever were added, the subquadruple gravity of the weight would be transferred to the middle of that lever, and would be supported by the subduple power of that lever, and consequently would be suboctuple of the whole weight; whereas in three levers thus arranged the three supporting powers are each only subsextuple. But if the weight be attached to the middle of the first lever at F, then the end D be attached to the middle of the second lever at E, and then the end B be attached to the middle of the third lever, it is quite rightly concluded that the power at F supports the subduple of the subduple, and that the power applied to the third lever supports the subduple of the subduple of the subduple, and therefore that the former is subquadruple and the latter suboctuple. But this arrangement would not help at all in explaining the force of the pulleys. But in the pulley system I do not see those two powers in H and P; for a single power in X, through the middle rope XSH, is indeed applied at the point H, and by its effort prevents the weight from pulling the pulley itself downward by its gravity: but in P, what other power accomplishes this same thing? Does not the same power X, through the middle rope XSHILMP, apply itself to the lever PO at P? Therefore the same power exhibits the effort of two subquadruple powers: therefore the power is not subquadruple, but only subduple;

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592 Mechanicorum pla; quemadmodum si duos simul vectes in D & B idem sustineret, utique tantumdem virium impenderet in utroque simul sustinendo, quantum si unicus esset vectis. Neque dixeris sustineri pondus à funibus inferiores orbiculos complectentibus: Hoc enim ad propositam quæstionem nihil est, tum quia nulla est sustentatio, si pondus raptandum sit in plano horizontali, & tamen vis Trochleæ exercetur in motu; tum quia ad pondus retinendum funes vim eandem exercerent, si tam ampla esset unius orbiculi orbita, ut funem utrumque caperet, vel unicus esset funis tam validus, ut utrique illi funi, quibus duo inferiores orbiculi insistunt, æquivaleret; tum quia vero propius est dicere, pondus sustineri à clavo, ex quo superior trochlea pendet, quàm à funibus, quemadmodum ipsa potentia sustinet; non autem vis sustinendi tribuitur funi illi, quem potentia arripit, & quo medio sustinet: Clavus autem in hujusmodi trochleis, quando potentia trahens proximè applicatur trochleæ clavo adnexæ, perinde sustinet totam atque integram ponderis gravitatem, si plures fuerint orbiculi, ac si unicus esset orbiculus; quamquam potentia minùs reluctans in pluribus orbiculis, minore impetu conetur adversùs pondus, ac proinde illa clavum minùs premat: quando verò potentia proximè applicatur trochleæ inferiori, atque sursum trahit, clavus nec urgetur ab impetu potentiæ, quem nullum recipit, nec ipse sustinet totum pondus. Quod si pondus trahatur in plano horizontali, sola potentia est, quæ adversùs clavum suam vim exercet superando resistentiam ponderis, quod nihil agit adversùs clavum, sed suâ gravitate urget subjectum planum. Ut autem manifestè deprehendas nihil esse Trochleis cum Vecte commercij, duo ligna accipe, cujuscumque tandem figuræ: singulis tria insint foramina, quoad ejus fieri poterit, exquisitè polita, ut minore conflictu funis excurrere possit: deinde funis alterno ab uno in alterum lignum ductu per foramina trajiciatur: Nam si alterum lignorum hujusmodi certo in loco firmetur, alteri adnectatur pondus, tum funis extremitatem arripiens trahas, idem planè præstabis, quod adhibitis orbiculis in communibus Trochleis: & tamen nullum hîc vectis vestigium apparet. Certè in majoribus navigiis malus hinc & hinc navis

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592 Mechanica place; just as if one were to sustain two levers at once, in D and B, he would certainly expend as much force in sustaining both together as if there were only a single lever. Nor should you say that the weight is sustained by the ropes embracing the lower pulleys: for this has nothing to do with the question proposed, both because there is no support at all if the weight is to be drawn in a horizontal plane, and yet the force of the pulley is exercised in motion; and because, to hold the weight, the ropes would exert the same force if the track of a single wheel were so broad as to contain both ropes, or if there were only one rope, so strong as to be equal to both those ropes on which the two lower pulleys rest; and also because it is more proper to say that the weight is sustained by the nail from which the upper pulley hangs than by the ropes, just as the power itself sustains it; but the force of sustaining is not attributed to that rope which the power seizes and by means of which it sustains: Now the nail in pulleys of this sort, when the pulling power is applied close to the pulley attached to the nail, supports the whole and entire weight no less, if there be several wheels, than if there were only one wheel; although the power, being less resisted in several wheels, endeavors against the weight with a smaller impulse, and therefore presses the nail less: but when the power is applied close to the lower pulley and pulls upward, the nail is neither urged by the impulse of the power, which it receives not at all, nor does it itself sustain the whole weight. But if the weight is drawn in a horizontal plane, it is the power alone that exerts its force against the nail by overcoming the resistance of the weight, which does nothing against the nail, but by its own gravity presses on the plane beneath it. But that you may clearly perceive that there is no commerce between pulleys and the lever, take two beams, of whatever shape you please: let there be in each three holes, as exactly polished as possible, so that the rope may run out with less friction: then let the rope be passed through the holes, led alternately from one beam to the other. For if one of such beams be fixed in a certain place, and a weight be attached to the other, and then you seize the end of the rope and pull, you will perform exactly the same thing as when the wheels are used in common pulleys; and yet here no trace of a lever appears. Certainly in larger ships the mast on this side and on that side of the ship...

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Liber sextus. CAPUT II. 595 navis lateribus alligatur, ut rectam positionem servet: quia au- tem rudentes aliquando remittuntur, ut in majore æstu, illóf- que intendi oportet, propterea duo hujusmodi ligna in Ellipsis ferè deformata (vel potiùs in sphæroides Hyperbolium factâ conversione non circa Axem, sed circa ordinatim Applicatam) alterum navis lateri, alterum rudenti adnectunt nautæ, & fu- nem non adeò crassum per foramina alterno ductu trajiciunt, quem etiam axungiâ, aut aliâ pinguedine inficiunt, ut faciliùs excurrat. Cum autem remissior factus fuerit rudens, funis illius caput solvunt, & trahentes cogunt ligna illa fieri propiora, ex quo rudens intenditur exiguo trahentis conatu, si animadver- tas quàm operosum & incommodum esset alio artificio ruden- tem remissum intendere. Argumentum hoc, quod olim ante annos vigintiquinque in Collegio Romano meis Auditoribus insinuavi, conatus est P. Schott ubi supra cap.3. eludere dicens ligna illa nullo modo habere rationem trochlearum, quia malus, qui est resistitivum, & debet trahi versùs latera navis, est appensus uni extremo illorum mediante fune, & potentia trahens est applicata alteri extremo eorumdem, & nihil dependet intermedium. Mirum ergo non est, si non habeat Vectis rationem. Verùm, tanti viri pa- ce dixerim, ligna illa ita habent rationem Trochlearum, ut si illorum loco communes Trochleas substituas, idem planè & eodem modo efficias, trochleâ alterâ adnexa navis alteri, alterâ rudenti intendendo: Neque enim malus est resistitivum, quod ponderis loco succedit, neque ille ad navis latus trahendus est, aut inclinandus, sed rudentis caput trahendum est, ut malo immoto ad navim accedat, adeóque intendatur: Quare rudens ipse intendendus vicem subit ponderis, quatenus intentioni repugnat, & potentia est applicata funi per lignorum foramina trajecto, sicut applicaretur funi ductario trochlearum orbiculos complexo. Quod si ligna illa non habent rationem Trochlea- rum, & tamen trahendi facilitatem præstant, ad quam Facul- tatem Mechanicam spectant? Non ad Vectem, ut ille quoquè admittit; non ad Axem, neque ad Cuneum, neque ad Co- chleam, ut manifestum est, pertinent: igitur vel novam Facul- tatem constituunt, vel omnino Trochleæ sunt. FFff

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Book six. CHAPTER II. 595 the ship is tied by the sides, so that it may keep a straight position: but because the stays are sometimes slackened, as in a higher swell, and then must be tightened, for that reason sailors attach two such pieces of wood, shaped almost into an Ellipse, or rather into a spheroid made by the conversion of a Hyperbolium not around the Axis, but around the ordinately Applied line; one to the side of the ship, the other to the stay, and they pass a not very thick rope alternately through the holes, which they also impregnate with tallow or some other grease, so that it may run more easily. But when the stay has become looser, they loosen the end of that rope, and by pulling cause those pieces of wood to come closer together, by which the stay is tightened with a small effort on the part of the puller, if you consider how laborious and inconvenient it would be to tighten a slack stay by some other contrivance. This argument, which long ago, twenty-five years ago, I communicated to my auditors at the Roman College, P. Schott, in the place cited above, chapter 3, tried to evade, saying that those pieces of wood in no way have the nature of pulleys, because the mast, which is the resistive element and ought to be drawn toward the sides of the ship, is suspended at one end of them by means of a rope, and the pulling power is applied to the other end of the same, and nothing depends in the middle. It is therefore no wonder if it does not have the nature of a lever. But, with due respect to so great a man, I would say that those pieces of wood have the nature of pulleys in such a way that, if you substitute ordinary pulleys in their place, you would accomplish exactly the same thing, and in the same manner, with one pulley attached to the ship and the other tightening the stay. For the mast is not the resistive element, which takes the place of a weight, nor is it to be drawn toward the side of the ship or inclined, but the end of the stay is to be drawn, so that the mast, remaining unmoved, may come toward the ship, and thus be tightened: wherefore the stay itself is to be tightened, inasmuch as it resists the tightening, and the power is applied to the rope passed through the holes of the pieces of wood, just as it would be applied to the hauling rope embracing the wheels of pulleys. But if those pieces of wood do not have the nature of pulleys, and yet provide ease in pulling, to what Mechanical Faculty do they belong? Not to the Lever, as he himself also admits; not to the Axis, nor to the Wedge, nor to the Screw, as is manifest: therefore either they constitute a new Faculty, or they are altogether pulleys. FFff

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Mechanicorum CAPUT III. An orbiculi magnitudo quicquam conferat. Quamquam Trochleæ Vires haberi etiam sinè orbiculis su- periùs dictum sit, communiter tamen rotulas suis thecis inclusas, & versatiles adhibemus. Quæritur autem, an rotu- larum hujusmodi magnitudo quicquam conferat ad faciliorem motum: an verò indiscriminatim rotulis sive majoribus, sive minoribus uti possimus, citra virium notabile dispendium. Quæstioni huic locum fecit Aristoteles Mechan. quæst. 9. ubi inquirit, Cur ea, quæ per majores circulos tolluntur, & trahuntur, faciliùs & citiùs moveri contingit, veluti majoribus trochleis, quam minoribus? & respondet, An quoniam quanto major fuerit illa, quæ à centro est, in æquali tempore majus movetur spatium? Quamobrem æquali inexistente onere idem faciet, quemadmodum diximus, & majores libras minoribus exactiores esse; spartum enim in illis cen- trum est. Non desunt, qui negent faciliùs attolli pondus, ex. gr. situlam aquâ plenam è puteo, si funis insistat orbiculo majo- ri, quàm si minorem complectatur, ac proptereà ab Aristotele frustrà quæri causam facilitatis, quæ nulla sit. Si enim diame- ter orbiculi sumatur ut Vectis primi generis hypomochlium habens in centro, potentia & pondus in diametri extremitati- bus æqualiter distant ab hypomochlio, ac proinde sive major sit, sive minor diameter, eadem semper manet Ratio æqualita- tis momentorum, quatenus ex positione pendent; adeóque nullum est facilitatis in movendo discrimen. Sin autem nullus agnoscatur Vectis, sed potentiæ motus cum motu ponderis comparetur, hos semper æquales esse manifestum est, sive ma- jor, sive minor rotula adhibeatur: atque hinc nullum infert mo- mentorum discrimen magnitudo, aut parvitas rotulæ. Ego tamen, Aristotelem omnino temerè majorem hanc mo- vendi facilitatem per majores orbiculos assumpsisse, affirmare non

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Mechanics CHAPTER III. Whether the size of a wheel contributes anything. Although it has been said above that the powers of pulleys may also be obtained without wheels, nevertheless we commonly employ small wheels enclosed in their sockets and movable. The question, however, is whether the size of wheels of this kind contributes anything to easier motion; that is, whether we may indiscriminately use wheels, whether larger or smaller, without any notable loss of force. Aristotle raised this question in Mechan. quæst. 9, where he asks why things that are lifted and drawn by larger circles happen to move more easily and more quickly, as with larger pulleys rather than smaller ones? And he replies: Is it because the greater that which is from the center, the greater the space it moves in equal time? For this reason, with the load remaining equal, it will do the same, as we have said, and larger balances are more exact than smaller ones; for the rope is at their center. There are not wanting those who deny that a weight is more easily lifted—for example, a bucket full of water from a well—if the rope runs over a larger wheel rather than a smaller one; and therefore they think Aristotle asked in vain for the cause of a facility that does not exist. For if the diameter of a wheel is taken as the fulcrum of a first-class lever, having its support in the center, the power and the weight are equally distant from the fulcrum at the ends of the diameter, and therefore, whether the diameter be larger or smaller, the same ratio of equality of moments always remains, insofar as it depends on position; and thus there is no difference in ease of movement. But if no lever is recognized, and rather the motion of the power is compared with the motion of the weight, it is clear that these are always equal, whether a larger or a smaller wheel is used: and from this the size or smallness of the wheel introduces no difference in the moments. I myself, however, would not dare affirm that Aristotle altogether rashly assumed this greater ease of motion by means of larger wheels, not

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Liber sextus. CAPUT III. 595 non ausim; neque enim carere potuit experimento aliquo, quo id suaderetur. Difficultas potiùs suboriri potest, an veram ille afferat causam majoris hujusce facilitatis: Nam quod innuit de libris majoribus, quæ exquisitiores sunt minoribus, quo pacto intelligendum sit, dictum est lib. 3. cap. 6: illud autem hîc lo- cum non habere manifestum est. Nemo negat in majoribus circulis, quorum major est Radius, ab extremitate Radij ma- jorem arcum describi, quàm à Radio minore, si tempore eodem similem arcum describant; sunt scilicet arcus similes in Ratione Radiorum; sed quando rotulæ inæquales commune centrum non habent, neque Radij omnino simul moventur, quasi minor sit pars majoris, quid prohibet eodem tempore arcus quidem æquales, sed dissimiles, describi? Nam si potentia trahens de- scendat per spatium palmare, sive rotula major sit, sive minor, pondus ascendit per palmum, & punctum in orbitâ rotulæ tam majoris quàm minoris notatum describit arcum palmarem: hoc autem tantummodo differunt, quod in universo ponderis ele- vati motu rotula minor sæpiùs convertitur quàm major, & con- versionum numeri sunt reciprocè in Ratione Radiorum: sic si Radius minor ad majorem sit ut 4 ad 9, novem conversiones minoris eodem tempore fiunt, ac quatuor conversiones majoris rotulæ, si à Potentiâ æqualiter moveantur. Quare æquali tem- pore major Radius non movetur per majus spatium; movetur siquidem æqualiter cùm potentiâ trahente & pondere ascen- dente, quemadmodum & minor Radius. Ut igitur Aristotelis dicto veritatem aliquam conciliemus, quæ tamen experimentis respondeat, illud observandum est, quod superiùs innui, videlicet eo consilio excogitatos esse orbi- culos, ut impedimentum ex funis attritu submoveatur, qui sanè tantus esset cum corpore, cui funis insistit, quanta est funis lon- gitudo æqualis motui ponderis, quod trahitur. At in orbiculo versatili solus axis teritur à cavâ foraminis superficie axis super- ficiei congruente, quæ eò minor est, quò minor est axis diam- ter diametro ipsius orbiculi: perimetri enim sunt in Ratione diametrorum. Quare si Axis diameter ad orbiculi diametrum sit ex. gr. subquadrupla, conflictus axis cum orbiculo est subqua- druplus ejus, qui esset orbitæ orbiculi stabilis cum fune mobili; immò adhuc minor est quàm subquadruplus, funis enim multò FF ff 2

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Book Six. CHAPTER III. 595 I would not dare; for it could not have been without some experiment by which it was suggested to him. Rather, difficulty may arise as to whether he states the true cause of this greater ease: for what he intimates about the larger wheels, which are more refined than the smaller ones, in what way this is to be understood has been said in book 3, chapter 6; but that it does not apply here is evident. No one denies that in larger circles, whose radius is greater, a larger arc is described from the end of the radius than from a smaller radius, if in the same time they describe a similar arc; there are, of course, similar arcs in the ratio of the radii. But when unequal wheels do not have a common center, and their radii are not moved altogether at the same time, as if the smaller were a part of the larger, what prevents equal arcs indeed, but dissimilar ones, from being described in the same time? For if the moving power descend through a span of one palm, whether the larger wheel be used or the smaller, the weight rises by one palm, and the point marked on the orbit of either the larger or the smaller wheel describes a palm-long arc: they differ only in this, that in the whole motion of the lifted weight the smaller wheel turns more often than the larger, and the numbers of revolutions are reciprocally in the ratio of the radii: thus if the smaller radius is to the larger as 4 to 9, nine revolutions of the smaller are made in the same time as four revolutions of the larger wheel, if they are moved equally by the power. Therefore in equal time the larger radius does not move through a greater distance; for it moves equally with the pulling power and the rising weight, just as the smaller radius does. In order then to reconcile some truth with Aristotle’s statement, one must observe what I hinted above, namely, that these small pulleys were devised with the intention that the impediment arising from the friction of the rope might be removed, which would certainly be as great on the body against which the rope bears as the length of rope equal to the motion of the weight being drawn. But in the movable pulley only the axle is worn by the hollow surface of the hole, the surface of the axle fitting it, and this is the less the smaller the axle’s diameter is in comparison with the diameter of the pulley itself: for circumferences are in the ratio of their diameters. Therefore, if the diameter of the axle is, for example, one-fourth that of the pulley, the friction of the axle with the pulley is one-fourth of that which there would be between the orbit of a fixed pulley and a moving rope; indeed it is still less than one-fourth, for the rope is much more...

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Mechanicorum 596 asperior est quàm superficies axis & foraminis sibi congruentes. Quoniam verò axis soliditas definitur ex pondere, quod ab eo sustinendum est, idem esse potest axis cum majore, & cum mi- nore orbiculo. Si ergo eidem axi major orbiculus inseratur, manifestum est minorem fieri attritionem datâ motûs æqualita- te. Nam orbiculorum orbitæ ex hypothesi sint in Ratione du- plâ, minoris autem orbiculi peripheria ad axis ambitum sit in Ratione quadrupla, jam orbita majoris orbiculi ad ambitum axis est in Ratione octupla: ponamus ambitum axis esse digitorum 4, orbita minor est digitorum 16, orbita major digit. 32: igitur si adhibeatur minor orbiculus, dum potentia & pondus pariter moventur per digitos 16, tritus cum axe est per digitos 4 (pono scilicet axem & foramen se invicem terere in puncto, in quo exercetur sustentatio) adhibito autem majore orbiculo, dum potentia & pondus per digitos 16 moventur, tritus cum axe est solum per digitos 2, semissem ambitûs foraminis. Ubi autem est minus movendi impedimentum, facilior est motus; igitur ma- jore orbiculo faciliùs movetur pondus. Sed ut rem ipsam penitiùs introspiciamus, animadvertendum est conflictum orbiculi cum Axe non fieri in centro motûs, quod idem est cum centro Axis, sed in ipsius axis super- ficie: quapropter hinc pon- dus repugnans, hinc poten- tia contranitens suas exer- cent vires in axem non per lineam ad ejus centrum ductam, sed per lineam à punctis potentiæ & ponde- ris contingentem ejusdem axis superficiem. Sic posito Axe, cujus centrum C, se- midiameter ad perpendicu- lum C A, si in extremitatibus diametri orbiculi D B sit in B po- tentia, in D pondus, illa suas vires in Axem exercet per lineam contingentem B E, hoc verò per lineam D F. Similiter si po- tentia sit in G, & pondus in I extremitatibus diametri orbiculi majoris circa eundem Axem, illa vires exercet per contingen- tem

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Mechanics 596 is rougher than the surfaces of the axle and the hole corresponding to it. But since the solidity of the axle is defined from the weight that must be borne by it, the same axle may be used with a larger or a smaller wheel. If, therefore, a larger wheel is inserted into the same axle, it is clear that there is less friction, the motion being equal. For, if the paths of the wheels are, by hypothesis, in a double ratio, while the circumference of the smaller wheel is to the circumference of the axle in a quadruple ratio, then the path of the larger wheel is to the circumference of the axle in an eightfold ratio: let us suppose the circumference of the axle to be 4 digits, the path of the smaller wheel is 16 digits, the path of the larger wheel 32 digits: therefore if the smaller wheel is used, while the power and the weight are moved equally through 16 digits, the rubbing with the axle is through 4 digits (I am, of course, assuming that the axle and the hole rub against one another at the point where the support is exercised); but if the larger wheel is used, while the power and the weight are moved through 16 digits, the rubbing with the axle is only through 2 digits, that is, half the circumference of the hole. But where there is less hindrance to motion, the motion is easier; therefore the weight is moved more easily with a larger wheel. But in order to examine the matter itself more thoroughly, it must be noted that the contact of the wheel with the axle does not occur at the center of motion, which is the same as the center of the axle, but on the surface of the axle itself: wherefore the resisting weight on one side, and the opposing power on the other, exert their forces on the axle not through the line drawn to its center, but through the line from the points of the power and the weight touching the surface of the same axle. Thus, with the axle being set, whose center is C, and semidiameter C A at right angles, if at the extremities of the diameter of the wheel D B there is power at B, and weight at D, the former exerts its force on the axle through the tangent line B E, and the latter through the line D F. Similarly, if the power be at G and the weight at I, at the extremities of the diameter of the larger wheel around the same axle, it exerts its force through the tangent

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Liber sextus. CAPUT III. 597 tem GH, hoc per IL. Potentia igitur B trahens, punctum F orbiculi minoris cogit ascendere in A, & potentia G cogit punctum L orbiculi majoris ascendere pariter in A. Porrò punctum L propius esse puncto A, quàm punctum F, est ma- nifestum, quia minor Secans CD & minor Tangens DF com- prehendunt arcum SF minorem, quàm sit arcus SL compre- hensus à majore Secante CI & majore Tangente IL. Hinc ex Doctrinâ Sinuum constat in Radio CA minorem particulam respondere arcui LA, quàm sit particula respondens æquali ar- cui incipienti ab F versus A. Igitur datâ motûs æqualitate, po- tentiæ scilicet trahentis tantum funem, quantus est arcus LA, minùs resistit ascensui punctum L, quàm punctum F, & citiùs L venit in A per breviorem arcum LA, quàm veniat F per lon- giorem arcum FA. Potentia itaque in G faciliùs, hoc est mi- nore labore, cæteris paribus movebit pondus in I positum, quàm potentia eadem minori orbiculo in B applicata moveat idem pondus in D. Neque dixeris ex æquali funis tractione pondus in suâ perpen- diculari lineâ Directionis æqualiter ascendere, sive fuerit in D, sive in I, ac propterea nullum inveniri facilitatis discrimen in illo attollendo. Quia adhuc considerandum est pondus, quate- nus est applicatu[m] Axi medio orbiculo, in quo axe dum ascendit, ascendit pariter in suâ perpediculari lineâ Directionis; & quam- vis in hac æqualiter se habeat, non tamen est æqualiter applicatu[m] axi: lineæ autem GH, IL productæ concurrerent in angulum magis obtusum, quàm lineæ BE, DF; quapropter sibi invicem minùs adversantur, quò propiùs accedunt ad rectitudinem. Verùm facilitas ista non est cum additamento momenti, quod à machinâ efficitur; Machina enim tribuens movendi facilita- tem est pariter causa tarditatis motûs; at hìc faciliùs & citiùs per majores circulos moveri pondus docet Aristoteles; quatenus vi- delicet sublatâ impedimenti particulâ, quæ ex tritu oriretur, potentia faciliùs & citiùs movetur, cum qua pariter æquali pla- nè motu etiam pondus movetur, quod tamen per machinam tardiùs moveretur, quàm potentia. Quod si cylindricam axis superficiem non admittas omnino congruere cavæ superficiei foraminis, jam contactus Axis est so- lùm ad punctum A utriusque orbiculi tam majoris, quàm mino- FFff 3

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Book Six. CHAPTER III. 597 through GH, this through IL. Therefore the power B, drawing, compels the point F of the smaller wheel to ascend to A, and the power G compels the point L of the larger wheel likewise to ascend to A. Moreover, that point L is nearer to point A than point F is is evident, because the smaller Secant CD and the smaller Tangent DF include the smaller arc SF, than is the arc SL included by the larger Secant CI and the larger Tangent IL. Hence, from the Doctrine of Sines, it is clear that on the radius CA a smaller portion corresponds to the arc LA than the portion corresponding to an equal arc beginning from F toward A. Therefore, given equality of motion, that is, with the drawing power extending only so much rope as is the arc LA, point L offers less resistance to ascent than point F, and L reaches A sooner by the shorter arc LA than F does by the longer arc FA. Thus the power in G will more easily, that is, with less labor, all else being equal, move the weight placed in I than the same power applied to the smaller wheel in B will move the same weight in D. Nor should you say that, from an equal pulling of the rope, the weight ascends equally along its perpendicular line of direction, whether it be in D or in I, and that therefore no difference of ease is found in lifting it. For it must still be considered that the weight, insofar as it is applied to the middle of the wheel, in which axle, while it ascends, it likewise ascends in its perpendicular line of direction; and although in this respect it behaves equally, it is not equally applied to the axle: for the lines GH, IL, when produced, would meet at a more obtuse angle than the lines BE, DF; wherefore they oppose one another the less, the nearer they come to straightness. But this ease is not accompanied by any addition of power produced by the machine; for a machine, while granting ease of movement, is also the cause of slowness of motion. Yet here Aristotle teaches that the weight is moved more easily and more quickly by larger circles, namely insofar as, once the impediment arising from friction has been removed, the power is moved more easily and more quickly, and with it, by the same equal motion, the weight is also moved; nevertheless, the weight would be moved more slowly by the machine than the power. But if you do not admit that the cylindrical surface of the axle perfectly agrees with the hollow surface of the hole, then the contact of the axle is only at the point A of each wheel, both of the larger and of the smaller FFff 3

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598 Mechanicorum ris, & tunc refert quandam libræ similitudinem, cujus jugum sit aut G I, aut B D, & partum in loco superiore A. Sed non eadem hîc militat ratio, quæ in libra: nam in brevioribus libræ brachiis, quando pondera sunt inæqualis gravitatis, extremitas brachij de- scendentis in suo motu deflectens à lineâ rectâ, etiam deflectit à perpendiculo eò magis, quò minor est semidiameter circuli, cu- jus arcum describit; at in longioribus brachiis majorem arcum describentibus similem minori, minùs deflectit à perpendiculo; ac proinde in descensu pauciora deteruntur gravitatis momenta, cum magis obsecundet naturali gravitatis propensioni, quæ niti- tur ad perpendiculum. At hîc in orbiculis, si Potentia movens sit gravitas aliqua major pondere attollendo, non cogitur deflectere à perpendiculo, sive major, sive minor fuerit orbiculus. Quapro- pter non ex Rationibus libræ philosophandum est, sed conside- randa est pressio superanda, quæ fit in A, tùm pondere, tùm po- tentiâ deorsum, ex hypothesi, conantibus; vel si pondus in pla- no, cui insistit, raptandum sit, pressio fit vi potentiæ trahentis pondus resistens. Avellenda est igitur ab Axe pars orbiculi illum tangens in A: sed posito æquali motu potentiæ tùm in B, tùm in G, minor motus & tardior particulæ A efficitur, si potentia mo- veat in G, quàm si moveat in B: facilius igitur illa movet præ istâ. Minorem autem & tardiorem esse motum in A, ubi vincenda est vis pressionis, cōstat, quia, ut semel foramen orbiculi minoris per- currat axem in A, totus ille convertendus est; at orbiculi majo- ris punctum in orbitâ designatum si moveatur æquali motu ac punctum minoris orbitæ, non absolvit integram revolutionem; atque adeò orbiculus major æquale habens foramen cum orbi- culo minore, sed multo majorem orbitam, motu æquali non per- currit Axem in A, nisi juxta partem, quæ respondeat revolutio- ni orbitæ, quam constat non esse integram. Hinc conjicerelicet posse orbiculo con- strui satis exactam libram. Fiat ex ligno aut ex materiâ metallicâ discus R S T, cujus centrum V, ejusque orbita ad tornum mo- dicè excavetur, ut illi insistere possint fu- niculi lancium. Tum in V centro fiat fora- men exquisitè rotundum atque politum, cui indatur Axis pariter politus & lævis: axis

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598 Mechanics ...and then it refers to a certain likeness to a balance, whose beam is either G I or B D, and the weight is in the upper place A. But here the same reason does not hold as in the balance: for in the shorter arms of a balance, when the weights are of unequal heaviness, the extremity of the descending arm, in its motion deviating from a straight line, also deviates from the perpendicular all the more, the smaller the semidiameter of the circle whose arc it describes; but in longer arms, describing a larger arc similar to a smaller one, it deviates less from the perpendicular; and therefore in the descent fewer moments of gravity are worn away, since it conforms more to the natural tendency of gravity, which strives toward the perpendicular. But here in the little wheels, if the moving Power be some gravity greater than the weight to be lifted, it is not forced to deviate from the perpendicular, whether the little wheel be larger or smaller. Wherefore one must not philosophize from the reasons of the balance, but must consider the pressure to be overcome, which occurs at A, as both the weight and the power, downward, according to the hypothesis, are making an effort; or if the weight is to be dragged along the plane on which it rests, the pressure is caused by the force of the power drawing the resisting weight. Therefore the part of the little wheel touching the axis at A must be drawn away from the axis: but if the motion of the power at B and at G be equal, a smaller and slower motion of particle A is produced if the power act at G than if it act at B: therefore it moves more easily in the former case than in the latter. But that the motion in A is smaller and slower, where the force of pressure must be overcome, is evident, because, in order that the hole of the smaller little wheel may once traverse the axis at A, the whole of it must be turned round; but if the point of the greater little wheel marked in the orbit is moved with a motion equal to that of the point of the smaller orbit, it does not complete a full revolution; and thus the larger little wheel, having a hole equal to that of the smaller little wheel but a much greater orbit, does not with equal motion traverse the axis at A except by that part which corresponds to the revolution of the orbit, which is known not to be complete. Hence it may be conjectured that a very accurate balance can be constructed by means of a little wheel. Let a disk R S T be made of wood or of metallic material, whose center is V, and let its orbit be slightly hollowed out on the lathe, so that the cords of the pans may rest upon it. Then in the center V let a hole be made, exquisitely round and polished, into which may be inserted an axis likewise polished and smooth: axis

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Liber sextus. CAPUT III. 599 axis aute[m] extremitatibus hinc atque hinc eminentibus alligen- tur fila, inter quæ interceptus discus possit suspendi. Si gravitas fuerit per universam laminam æquabiliter diffusa, c[on]sistet discus in quacumque positione; sin autem partes fuerint secundum gravitatem inæquales, ita sponte convertetur discus, ut pars gravior inferiorem occupatura locum usque eò descendat, dum centrum gravitatis sit in lineâ directionis perpendiculari tran- seunte per punctum suspensionis, & punctum contactus orbi- culi cum axe. Hanc perpendicularem lineam refert, atque de- signat filum, ex quo suspenditur. Notato igitur diligentissimè puncto S, in quo fila suspendentia tangunt extremam orbitam, ibi est locus apponendæ lingulæ, atque ibi firmandus est uter- que funiculus S R & S T. Amotis igitur funiculis, seu filis, ex quibus prius suspendebatur orbiculus, atque adjectâ opportu- nâ lingulâ, apponatur ansa V M, quæ includet lingulam, si hæc fuerit ritè collocata. Demum pendentibus funiculis adnectan- tur lances ita, ut æquilibrium constituant, quod à lingula indi- cabitur. Sic parata erit, ut opinor, exactissima libra, de qua du- bitari non possit, an centrum motûs verè respondeat lineæ, in qua est centrum gravitatis: æqualitas brachiorum V R, V T est manifesta propter faciliorem circuli constructionem, quàm bra- chiorum rectorum æqualitatem æquabili & æquali gravitate præditam: pondera autem si inæqualia lancibus imponantur, semper in eodem perpendiculo consistunt, sive descendant, sive ascendant: lingula verò quia satis longa est, quippe quæ incipit ab V, quamvis additamentum factum sit in S, vel modicissimam inclinationem in alterutram partem indicabit. Cum itaque negari non possit in simplici orbiculo aliquam demum movendi facilitatem aquiri, si ille major fuerit, quàm si minor, hoc pariter in Trochleis contingere posse non nega- rem adhibitis majoribus orbiculis potiùs quàm minoribus. Ve- rum attendendum est, an sit operæ pretium tam ingentes tro- chleas movendis ponderibus adhibere; illa enim & majore dis- pendio construerentur, & essent valde graves, & ægrè trans- ferri possent, si notabili aliquâ magnitudine præditæ essent. Quare nemini author essem, ut rejectis minoribus orbiculis ma- jores quæreret; communiter enim valde mediocribus trochleis utuntur artifices, & satis commodè perficitur motus, si orbicu- li

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Book Six. Chapter III. 599 To the extremities of the axis, projecting on this side and on that, threads are fastened, between which the disk, being held in suspension, can hang. If the weight is evenly distributed throughout the whole plate, the disk will remain in whatever position it is placed; but if the parts are unequal according to gravity, the disk will of itself turn so that the heavier part may descend until it occupies the lower place, until the center of gravity is in the direction of the perpendicular line passing through the point of suspension and the point of contact of the little wheel with the axis. This perpendicular line is marked and indicated by the thread from which it is suspended. Therefore, after the point S, where the suspending threads touch the outer circumference, has been very carefully noted, there the place for the little tongue is to be set, and there both cords S R and S T are to be fixed. Then, when the cords, or threads, from which the little wheel was formerly suspended have been removed, and a suitable little tongue has been added, let the loop V M be applied, which will enclose the tongue, if it has been properly placed. Finally, when the cords are hanging, let the pans be attached in such a way that they produce equilibrium, which will be indicated by the little tongue. Thus, as I think, the most exact balance will be prepared, concerning which there can be no doubt whether the center of motion truly corresponds to the line in which the center of gravity lies: the equality of the arms V R, V T is manifest because of the easier construction of the circle than the equality of straight arms endowed with uniform and equal gravity: but if unequal weights are placed on the pans, they always remain in the same perpendicular, whether they descend or ascend: the little tongue, however, because it is sufficiently long, since it begins at V, although an addition has been made at S, will indicate even the slightest inclination to either side. Since, therefore, it cannot be denied that in a simple little wheel some facility of motion is at length acquired if it is larger than if it is smaller, I would likewise not deny that this may happen in pulleys when larger wheels are employed rather than smaller ones. But it must be considered whether it is worth the trouble to employ such huge pulleys for moving weights; for these would be constructed at greater expense, would be very heavy, and could scarcely be transported if they were endowed with any notable size. For this reason I would advise no one, after rejecting smaller little wheels, to seek larger ones; for artisans commonly use very moderate-sized pulleys, and the motion is performed sufficiently conveniently, if the little wheels

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600 Mechanicorum li facilè convolvantur: commodum verò, quod accederet ex aliquatenus diminuto orbiculorum cum suis axibus conflictu, non tantum est, ut majore incommodo parandum sit. CAPUT IV. Qua Ratione Trochlearum vires augeantur. Ex dictis cap. 1. satis notum est Trochlearum vires augeri pro multitudine orbiculorum: sed quoniam non præstat ingentes Trochleas construere, proptereà satius est Trochleas cum aliâ quapiam Facultate componere, & potissimum cum Axe in Peritrochio, sive Sucula sit, sive Ergata, sive Tympanum, quorum Axi dum in conversione circumducitur funis ductarius, Trochleæ evadunt propiores, & adducitur pondus. Ejus rei meminit Lucret. lib. 4: Multaque per Trochleas & Tympana pondere magno commovet, atque levi sustollit machina nisu. Et quidem superiore loco, uti de Tympano agebatur, indicata est methodus geminandi vires Tympani ABCS, fi

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600 Mechanics so that they may easily be wound together: but the advantage that would arise from somewhat reducing the collision of the pulleys with their axes is not so great that it should be sought at greater inconvenience. CHAPTER IV. By what means the powers of pulleys are increased. From what was said in chapter 1, it is sufficiently known that the powers of pulleys are increased in proportion to the number of wheels: but since it is not practical to construct very large pulleys, it is therefore better to combine pulleys with some other faculty, and especially with an Axis in Peritrochium, whether it be a Winch, or a Capstan, or a Drum, by whose axis, as it is turned in rotation, the hauling rope is drawn around, the pulleys become closer together, and the weight is brought along. Lucretius mentions this in book 4: “And with pulleys and drums he moves many things with great weight, and with a light machine lifts them by effort.” And indeed in the preceding place, as the Drum was being discussed, the method of doubling the powers of the Drum ABCS was indicated, fi

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Liber sextus. CAPUT IV. 601 si nimirum trabis exporrectæ extremitati D alligetur funis ductarius, qui primùm transeat per orbiculum E ponderi attol- tellendo adnexum, deinde per orbiculos F & G trabi adhæ- rentes, per quos demum venit ad Axem H, cui circumducendus est. Quia enim potentia in F duplo velociùs movetur quàm E, & Potentia in S premens Tympanum movetur velociùs quam F, in Ratione partis semidiametri tympani ad semidia- metrum Axis, hoc est in Ratione A V ad A H, manifestum est geminari momenta tympani solitariè accepti. Quod si tam ex- tremitati D, quàm Ponderi adnecterentur Trochleæ, adhuc major esset vis Tympani aucta per Trochleas, & vicissim ma- jor Trochlearum vis aucta per Tympanum. Hinc si essent duæ trochleæ binis orbiculis instructæ, & funis caput inferiori tro- chleæ adjungeretur, quintuplex fieret tympani momentum, & vicissim trochlearum momentum acciperet incrementum in ra- tione A H ad A V. Distinguenda sunt autem onera, quorum alia sunt mediocria (nam minora facilè solis trochleis attolluntur, arreptâ ab ho- minibus funis ductarij extremitate) alia majora, & ingentia, quæ à Vitruvio, ut aliàs innui, Colossicotera dicuntur. Pro mediocribus ponderibus ad operarum numerum minuendum Trochleis adjungi potest Sucula, cui circumducatur funis ductarius: compositis enim Rationibus Suculæ & Trochlea- rum, habetur Ratio momenti potentiæ ad Pondus. Si non ad multam altitudinem attollendum sit Pondus, neque proximo parieti trabem infigi expediat, cui Trochlea adjungatur, & ex qua onus dependeat, ex Vetruvij præscripto lib. 10. cap. 2. tigna tria parantur longitudine & soliditate respondentia oneris magnitudini & gravitati; hæc à capite fibulâ aut funibus con- juncta, in imo divaricata eriguntur, quasi in pyramidis trian- gularis speciem. Quod si timeatur, ne in hanc aut illam par- tem machina inclinetur, funibus in capitibus collocatis, & circa dispositis in adversas plagas, atque firmatis, erecta reti- netur. In summo, ubi tigna coëunt, alligatur Trochlea; & inferiùs, ubi commodè applicari possit Potentia, exteriori duo- rum tignorum divaricatorum faciei firmiter affiguntur Chelo- nia, hoc est fulcra quædam rotundum foramen habentia, in quæ conjiciuntur Suculæ capita; ut Axis facilè versetur. Sucu- GGgg

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Book Six. Chapter IV. 601 If, to be sure, a haulage rope is attached to the extremity D of an extended beam, which first passes through the pulley E attached to the weight to be raised, then through the pulleys F and G attached to the beam, and finally comes to the Axis H, around which it is to be wound. For since the power at F is moved twice as quickly as at E, and the power at S pressing the drum is moved more quickly than at F, in the ratio of the semidiameter of the drum to the semidiameter of the Axis, that is, in the ratio of A V to A H, it is evident that the moments of the drum, taken by itself, are doubled. And if pulleys were attached both to the extremity D and to the weight, the force of the drum would still be greater, increased by the pulleys, and conversely the force of the pulleys would be increased by the drum. Hence, if there were two pulleys furnished with two sheaves, and the end of the rope were attached to the lower pulley, the moment of the drum would become fivefold, and conversely the moment of the pulleys would receive an increase in the ratio of A H to A V. Now loads must be distinguished: some are moderate—for smaller ones are easily lifted by pulleys alone, the end of the haulage rope being seized by men—others are greater and enormous, which, as I have indicated elsewhere, Vitruvius calls Colossicotera. For moderate weights, in order to reduce the number of laborers, a winch may be added to the pulleys, around which the haulage rope is wound; for, by combining the ratios of the winch and the pulleys, one obtains the ratio of the moment of the power to the weight. If the weight is not to be raised to a great height, and it is not convenient to fix a beam to the nearby wall, to which a pulley may be attached and from which the load may hang, then, according to Vitruvius’ prescription in book 10, chapter 2, three timbers are prepared, corresponding in length and solidity to the size and heaviness of the load; these, joined at the top by a clamp or by ropes and spread apart at the bottom, are erected, as it were, in the form of a triangular pyramid. And if it is feared that the machine may incline to one side or the other, by means of ropes placed at the heads and disposed around it toward opposite directions, and fastened, it is held upright. At the top, where the timbers meet, a pulley is attached; and lower down, where the power can conveniently be applied, to the outer face of the two diverging timbers are firmly fixed chelonia, that is, certain supports having a round opening, into which the heads of the winches are inserted, so that the axis may turn easily. Winch-

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Mechanicorum 602 la autem proximè capita aut habet infixos Radios, aut saltem bina foramina ita temperata & disposita, ut vectes in ea immiti possint variæ longitudinis pro opportunitate atque necessitate, habitâ ratione loci & ponderis. Altera Trochlea adnectitur ponderi, prout commodius acciderit, & funis ductarij extremitas superiori trochleæ adnectitur, ejusque per Trochlearum orbiculos trajecti caput ad Suculam religatur, cujus conversione attollitur pondus. Machinam hanc aliqui artifices Capram vocant. Ex his, si data fuerit ponderis gravitas, & nota Potentiæ virtus, definies trochlearum orbiculos, aut saltem vectium longitudinem, qui faciliùs parari possunt, & commutari pro re natâ, quàm aliæ trochleæ inveniri. Sint itaque trochleæ binis orbiculis instructæ; harum forma & positio non nisi quadruplum potentiæ motum determinat, si cum motu ponderis comparetur. At potentia universa sint duo homines, singuli valentes attollere libras 25; atque adeò potentia est lib. 50; quæ si proximè applicetur funi ductario trochlearum, poterit solùm attollere gravitatem quadruplam, hoc est lib. 200. Quoniam verò oblatum pondus est ex hypothesi lib. 1000, hoc est quintuplum librarum 200, addenda est trochleis Ratio quintupla Succulæ, cujus Radij aut Vectes sint quintupli semidiametri Axis ejusdem Suculæ. Nam Potentia Vectibus aut Radiis applicata quintuplo velociùs movetur, quàm extremitas funis ductarij Axem complexi; hæc autem quadruplo velociùs quàm pondus; atque idcircò potentia vigecuplo velociùs movetur quàm pondus, poteritque movere pondus vigecuplum librarum 50, hoc est lib. 1000. Quod si ad insignem aliquam altitudinem evehendum sit pondus, non est opus tria hujusmodi tigna compingere, sed ut sumptibus & labori parcatur, satis est non procul à pondere longiorem trabem, etiam ex pluribus aptè & firmiter conjunctis compositam, erigere, atque funibus in oppositas ventorum plagas dispositis ita ejusdem caput firmare, ut nullam in partem vi suspensi ponderis inclinetur. Verùm quidem est trabem hujusmodi (Antennam aliqui dicunt) non omnino ad perpendicularum erigi, sed modicè inclinatam statui, ut à summo vertice pendens ad perpendicularum sarcina, quæ attollitur, non

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Mechanics 602 …or it has inserted near the head either spokes, or at least two holes so adjusted and arranged that bars of various lengths can be inserted into them, as occasion and necessity require, having regard to the place and the weight. Another pulley is attached to the load, as may be more convenient, and the end of the carrying rope is attached to the upper pulley, and after being passed through the sheaves of the pulleys, its head is fastened to the windlass, by whose turning the weight is raised. Some craftsmen call this machine a Goat. From these, if the weight’s heaviness is given, and the power of the force is known, you will determine the sheaves of the pulleys, or at least the length of the bars, which can more easily be prepared and changed as the case requires, than other pulleys can be found. Let there therefore be pulleys fitted with two sheaves; their form and position determine nothing but a quadruple motion of the force, if compared with the motion of the weight. But let the whole force be two men, each able to lift 25 pounds; and thus the force is 50 lb.; which, if applied close to the carrying rope of the pulleys, will be able to lift only four times the weight, that is, 200 lb. But since the weight presented, by hypothesis, is 1000 lb., that is five times the 200 pounds, a quintupling ratio must be added to the pulleys of the windlass, whose spokes or bars should be five times the semi-diameter of the axis of that same windlass. For the force applied to the bars or spokes moves five times faster than the end of the carrying rope embracing the axis; but this latter moves four times faster than the weight; and therefore the force moves twenty times faster than the weight, and will be able to move a weight twenty times 50 pounds, that is, 1000 pounds. But if the weight is to be raised to some notable height, there is no need to bind together three such beams, but, so that expense and labor may be spared, it is enough to erect not far from the weight a longer beam, even one made of several pieces suitably and firmly joined together, and to secure its head with ropes arranged to opposite quarters of the winds, so that it inclines in no direction under the force of the suspended weight. However, such a beam (Some call it an Antenna) should not be erected perfectly perpendicular, but set somewhat inclined, so that from the topmost summit a load hanging perpendicular to the lifting beam, which is raised, does not

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Liber sextus. CAPUT IV. 603 non incurrat in trabem. Modicè, inquam, inclinata statuitur trabs ista (nisi fortè illa altiùs defodiatur, & circùm fistucatio- ne solidetur, tunc enim poterit magis inclinata statui) quia propter notabilem longitudinem ita potest inclinari, ut linea directionis ab illius centro gravitatis ducta cadat intrà (vel cer- tè non admodum ultrà) basim sustentationis, atque perpen- diculum à summo vertice descendens & illi lineæ parallelum tanto absit intervallo, quod satis sit ad elevandum pondus citra periculum collisionis cum trabe: cui periculo occurri non po- test in tigno breviore, quo valde inclinato ad vitandum hujus- modi periculum collisionis, linea directionis ab ejus centro gravitatis cadens multo notabiliùs recederet à basi sustentatio- nis: propterea ubi brevioribus trabibus fuerit utendum, tres modo superiùs dicto compinguntur, ut se invicem fulcientes sponte consistant, & pondus non contingant. Capiti igitur erectæ trabis longioris altera trochlea alligatur, altera oneri; sed ad trabis pedem orbiculus unus firmiter adnectitur, per quem funis ductarius juxta trabis longitudinem descendens trajicitur, & ad Ergatæ axem adducitur, ut ex ejus revolutio- ne funis trahatur: orbiculum hunc Græci ἐπαγοῦla Latini Ar- temonem vocant, ex Vitruvio lib. 10 cap. 5. Hic tamen infi- mus orbiculus cum nihil immutet aut potentiæ velocitatem, aut ponderis tarditatem, nihil addit momenti ipsi potentiæ ad onus attollendum, sed ideò potissimùm adhibetur, ut funis commodiùs Ergatæ circumducatur. Quare Potentiæ momenta componuntur ex momentis Tro- chlearum & Ergatæ; quæ si innotescant, & data sit potentiæ virtus movendi, manifestum erit pondus, quod illa Ergatæ ap- plicata movere poterit. Sic si Trochleæ binos habeant orbicu- los, qui dant Rationem quadruplum, Vectis autem Ergatæ sit ad ejusdem Axis semidiametrum ut 20 ad 1, Ratio, quæ ex quadruplâ, & vigecuplâ componitur, est octuagecupla; ac proindè potentia extremo Vecti applicata poterit movere pon- dus octuagecuplum ejus, quod sine machinâ movere potest. Hinc si potentia movere valeat libras 50, huic machinæ ap- plicata movebit pondus lib. 4000. Illud autem commodi ha- bet Ergata, quod in illâ convolvendâ uti possumus jumentis extremo vecti applicatis: & experimento didicimus trochleis GGgg 2

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Book Six. CHAPTER IV. 603 does not strike against the beam. I say that this beam is set with a moderate inclination (unless perhaps it be buried deeper and solidly rammed around, for then it can be set more inclined), because on account of its notable length it can be inclined in such a way that the line of direction drawn from its center of gravity falls within, or certainly not much beyond, the base of support; and the plumb line descending from the highest vertex, and parallel to that line, is so far removed by a distance sufficient to raise the weight without danger of collision with the beam. This danger cannot be avoided in a shorter timber, for if it were greatly inclined in order to avoid this kind of danger of collision, the line of direction falling from its center of gravity would depart much more notably from the base of support. Therefore, where shorter beams are to be used, three are joined together in the manner stated above, so that, supporting one another, they stand of themselves and do not touch the weight. To the head of the upright longer beam, then, one pulley is attached, and another to the load; but at the foot of the beam one small wheel is firmly fastened, through which the carrying rope, descending along the length of the beam, is passed, and brought to the axle of the winch, so that by its turning the rope may be drawn in: this small wheel the Greeks call ἐπαγούλα, the Latins Artemonem, from Vitruvius, book 10, chapter 5. This lowest pulley, however, since it changes neither the velocity of the power nor the slowness of the weight, adds nothing of moment to the power itself for lifting the load, but is used chiefly so that the rope may be more conveniently drawn around the winch. Therefore the moments of the power are composed of the moments of the pulleys and of the winch; and if these are known, and the force of the power to move is given, it will be clear what weight that applied to the winch it will be able to move. Thus if the pulleys have two wheels, which give a ratio of fourfold, but the lever of the winch is to the semidiameter of the same axis as 20 to 1, the ratio, composed of the fourfold and the twentyfold, is eightyfold; and therefore the power applied to the end of the lever will be able to move a weight eighty times greater than that which it can move without the machine. Hence, if the power can move 50 pounds, applied to this machine it will move a weight of 4000 lb. But the winch has this advantage, that in turning it we may use draft animals applied to the end of the lever: and by experiment we have learned pulleys GGgg 2

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Mechanicorum 604 binorum orbiculorum, & Ergatâ attolli à duobus equis pondus librarum non minùs quàm triginta millium. Cum enim sint duo equi, unusquisque movet libras 15000; sed quia Trochleæ dant Rationem quadruplam, accipe librarum 15000 quadrantem 3750, & ope trochearum, si solæ essent & ab Ergatâ sejunctæ, unicuique equo adhibendus esset nisus subquadruplus, videlicet conatus sufficiens ad movédas absque trochleis libras 3750: quoniam demum Ergatæ Vectis ad semidiametrum axis est ex. gr. decuplus, singuli equi adhibent conatum adhuc subdecuplum, quo scilicet moverent libras 375: est nimirum, ex hypothesi harum trochlearum, & hujus Ergatæ, motus potentiæ ad motum ponderis quadragecuplus; ac proindè potentia adhibet conatum, quo moveret absque machina gravitatem dati ponderis subquadragecuplam. At verò si pondera attollenda sint omnino ingentia & colossicotera, non satis fuerit trabem erigere, sed ex pluribus trabibus invicem compactis sive funibus, sive ferreis retinaculis, & clavis quasi crassiores columnas erigere, eásque transversis aliis trabibus inter se colligare, aut etiam obliquis fulcire oportet, & circa pondus componere validissimum castellum, quod nullam in partem inclinari queat: ut deinde pluribus Trochlearum paribus cum suis Ergatis ritè collocatis machinator tutò aggredi possit opus. Hîc autem multo commodius accidit plures communes Trochleas & Ergatas adhibere, quàm pauciores trochleas plurimorum orbiculorum construere, quæ longissimum funem ductarium exigerent, aut ingentes Ergatas statuere, quarum vectis valde longus non nisi in amplo spatio circumagi posset. Illud Machinatoris solertiæ relinquitur, quod Ergatas singulas atque Trochleas tam aptè disponat, ut sibi invicem impedimento non sint. Quod si, dum moles ipsa elevatur, fulcra subinde opportuno loco subjicias, quibus illa innitatur, multo certiùs, nec sine laboris compendio, rem totam perficies. Quod demum ad funes attinet, in ponderum ingentium elevatione duplex periculum præcavendum est; alterum, quod plures Trochleæ diversis ponderis partibus applicantur, ne scilicet funes aliquarum Trochlearum vi ponderis plus justo distendantur, & longiores fiant, quàm par sit, ut pondus usque in

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Mechanics 604 with two pairs of pulleys, and by a winch, a weight of not less than thirty thousand pounds can be raised by two horses. For since there are two horses, each moves 15,000 pounds; but because the pulleys give a quadruple ratio, take the quarter of 15,000 pounds, namely 3,750, and by means of the pulleys, if they alone existed and were separated from the winch, there would be assigned to each horse a subquadruple effort, namely an effort sufficient to move, without pulleys, 3,750 pounds: since finally the lever of the winch, to the semidiameter of the axis, is, for example, tenfold, each horse applies a yet subdecuple effort, by which it would move 375 pounds; thus, on the hypothesis of these pulleys and of this winch, the movement of power to the movement of the weight is fortyfold; and therefore the power applies an effort by which, without the machine, it would move the heaviness of the given weight, sub-fortyfold. But if the weights to be lifted are altogether enormous and colossal, it will not be enough to erect a beam, but from several beams compacted together, whether by ropes or by iron fastenings and nails, as it were thicker columns must be set up, and these joined together by other transverse beams, or even strengthened by oblique supports, and around the weight there must be built a very strong framework, which can incline in no direction: so that afterward the engineer, with several pairs of pulleys and their winches properly placed, may safely undertake the work. Here it is much more convenient to employ more common pulleys and winches than to construct fewer pulleys with many wheels, which would require a very long guide-rope, or to set up huge winches, whose very long lever could not be turned except in a wide space. It is left to the engineer's skill so to arrange each winch and pulley that they do not hinder one another. And if, while the mass itself is being raised, you from time to time place supports in a suitable spot beneath it, on which it may rest, you will accomplish the whole task much more certainly, and not without saving labor. As for the ropes finally, in the lifting of enormous weights a double danger must be guarded against; the one, that when several pulleys are applied to different parts of the weight, the ropes of some pulleys may be stretched by the force of the weight more than is proper, and become longer than is fitting, so that the weight may continue up to in

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Liber sextus. CAPUT IV. 605 in destinatum evehatur locum, & aptè collocetur; una enim parte jam ferè suum in locum deductâ, reliqua pars adhuc distaret, nec potentia trochleis illis applicata sola ad perficiendum motum sufficeret. Alterum est, ne ex motu, & vehementi funium cum orbiculis, aut orbiculorum cum axibus tritu, nimis incalescant, atque ignem concipiant. Sed utrique periculo occurritur, si aquam in promptu habeas, qua funes aut trochleæ madesiant; illa enim non solum incensionis periculum submovet, verùm etiam funes contrahit. In plano autem horizontali aut inclinato longè facilior est motus, potentia quippe caret labore retinendi onus, quod innitur plano; & quamvis hoc sit inclinatum (non tamen lubricum, neque pondus incumbat scy talis, seu cylindris) ita pondus suâ gravitate premit subjectum planum, ut etiam dimissum non facilè prolabatur: ut tamen in hujusmodi planis faciliùs trahatur, expedit cylindros supponere, aut rotas addere, aut illud trahæ imponere. Hic pariter ad trahendum juvari potest potentia, si funis ductarij per Trochleas trajecti caput ad Axem Ergatæ, aut Suculæ, aut Tympani referatur; prout majora aut mediocria fuerint pondera. In minoribus autem ponderibus raptandis, etiam simplici Vecte, & quidem expeditissimè, augeri possunt momenta Trochlearum; si nimirum vecti circa medium alligetur caput funis ductarij, & inclinati vectis caput subinde transferatur. Sit enim ductarij funis extremitas A; hæc in A religetur vecti B C, qui terram premat in C, quod est hypomochlium, & potentia movens sit in B; quæ, manente extremitate C, dum promovetur in D, funis caput venit ex A in E: tunc iterum inclinetur vectis C D, ut habeat positionem F G, & potentia similiter circa punctum G manens moveatur ex F versus D, atque ulteriùs adducatur funis caput E, & sic deinceps. Quod si progredi nolueris, sed eodem in loco consistere, ubi vectis positionem C D nactus fuerit, & A venerit in E, retrahe D ite- GGgg 3

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Book Six. CHAPTER IV. 605 be raised to the intended place and suitably set; for when one part has already been brought almost into its place, the remaining part is still at a distance, and the force applied through those pulleys alone would not suffice to complete the motion. The other danger is that, from the motion and the violent friction of the ropes against the pulleys, or of the pulleys against the axles, they may become too hot and catch fire. But both dangers are avoided if you have water at hand, with which the ropes or pulleys may be moistened; for this not only removes the danger of ignition, but also contracts the ropes. In a horizontal or inclined plane, however, the motion is much easier, because the force has no labor of holding back the load resting upon the plane; and although this be inclined also (yet not slippery, nor does the weight bear upon rollers or cylinders), the weight nevertheless presses the supporting plane by its own gravity so that, even when let go, it does not easily slip away: yet, so that it may be drawn more easily on such planes, it is useful to place cylinders underneath, or add wheels, or set it upon a sled. Here likewise the force may be helped in drawing, if the end of the leading rope, passed through pulleys, be brought to the axle of a windlass, capstan, or drum, according as the weights are larger or moderate. But in drawing smaller weights, the moments of the pulleys may be increased even by a simple lever, and indeed very readily; namely, if the end of the leading rope be tied about the middle of the lever, and the end of the inclined lever be shifted from time to time. Let A be the end of the leading rope; this be tied at A to the lever B C, which presses the ground at C, which is the fulcrum, and let the moving force be at B; which, while C remains fixed, as it is advanced to D, brings the end of the rope from A to E: then let the lever C D again be inclined, so that it may have the position F G, and likewise let the moving force, remaining about the point G, move from F toward D, and thus the end of the rope E is drawn farther, and so on. But if you do not wish to proceed, but to remain in the same place where the lever has obtained the position C D, and A has come to E, draw back D ite- GGgg 3

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606 Mechanicorum rum in B, atque particulam funis A E ita vecti convolve, ut excurrere nequeat; nam iterato vectis motu trahetur funis, & cum trochleâ pondus; motúque hujusmodi continuato destina- tu min locum adducetur pondus. Quantum verò sit hoc compen- dium, illicò innotescet, si observaveris ut minimum geminari mo- menta potentiæ, si videlicet punctum A præcisè medium fuerit æqualiter ab extremitatibus B & C distans: quod si A C sit triens totius B C, momentum potentiæ triplicatur, & est Ratio com- posita ex Ratione Trochlearum, & Ratione Vectis. Ut autem manifesto experimento deprehendas, quàm tenuis Potentia Trochleis cum Axe in Peritrochio compositis non leve pondus trahat, in extremitate tabulæ non ruditer dolatæ duos perpendiculares tigillulos erige intervallo digitorum quatuor, illósqque junge transversario similis crassitiei, non tamen à plano absit nisi quatuor digitos: Deinde supra transversarium interval- lo saltem digitorum octo statue axem in tigillis facillimè versati- lem, cujus diameter vix digitalis sit: alteri autem hujus axis capiti rotam circumpone, cujus diametrum digiti duodecim metiâtur: quæ rota ut levis sit, bino radiorum ordine constet aptè colliga- torum, & circa perimetrum emineant palmulæ, ut in molendino- rum aquaticorum rotis, ex levi materiâ, cujusmodi esset crassior charta, aut membrana, aut quid simile valens flatum excipere. Tum parvulæ trochleæ duæ binis orbiculis instructæ firmentur, altera quidem in transversario tigillorum, altera in extremitate asserculi, cui onus aliquod est imponendum, eique aut rotæ, aut cylindruli subjiciantur, ut facilè mobilis sit; & tro- chleæ mobili adnectatur extremitas funiculi serici, qui per or- biculos trochlearum trajectus demum ad Axem referatur. Nam si in rotæ palmulas vehementiùs insuffles, rota convol- vetur, & cum illâ Axis, atque adeò funiculum involutum se- queretur trochlea cum pondere ferè sexagecuplo ejus, quod flatu eodem exsufflare posses. Aut potius Æolipilam aptè col- loca, ut flatus ex illâ exiens in palmulas incurrat, & ex ponde- ris, quod asserculo impositum movetur, gravitate cognosces impetum, quo flatus ex Æolipilâ erumpit, si Ratio Trochlea- rum, quæ est quintupla, componatur cum Ratione diametri ro- tæ ad diametrum axis, quæ, ex constructione, est duodecupla: cum enim sit Ratio, sexagecupla, fiat ut 60 ad 1, ita gravitas ponderis,

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606 Mechanics in B, and wind the small part of the rope AE on the axle in such a way that it cannot run out; for by the repeated motion of the lever the rope will be drawn, and with it the weight by the pulley; and if such motion be continued, the intended weight will be brought to the desired place. But how great this saving is will at once become clear if you observe that the power is at the very least doubled, if, namely, the point A is exactly in the middle, equally distant from the extremities B and C: but if AC is one-third of the whole BC, the force is tripled, and it is a ratio composed of the ratio of the pulleys and the ratio of the lever. But that you may discover by plain experiment how slight a force, when pulleys combined with an axle in a windlass, draws a not inconsiderable weight, at the end of a board not roughly hewn erect two small perpendicular posts four digits apart, and join them with a crosspiece of similar thickness, yet so that it does not stand more than four digits above the plane. Then above the crosspiece, at an interval of at least eight digits, place an axle, very easily turnable in the posts, whose diameter is scarcely a digit; and on one end of this axle place a wheel, whose diameter measures twelve digits. Let this wheel be light, made of two rows of spokes suitably connected, and let small vanes project around the rim, as in the wheels of watermills, made of a light material, such as thicker paper, or parchment, or something similar capable of catching the wind. Then let two small pulleys, each fitted with two discs, be fixed, one indeed on the crosspiece of the posts, the other at the end of a small beam on which some load is to be placed, and let it rest upon either a wheel or a cylinder so that it may move easily; and to the movable pulley let the end of a silk cord be attached, which, having been passed through the discs of the pulleys, is finally brought to the axle. For if you blow strongly onto the vanes of the wheel, the wheel will revolve, and with it the axle, and thus the pulley together with the weight would follow the wound rope by almost sixty times the force that you could blow out with that same breath. Or rather, suitably arrange an aeolipile so that the wind issuing from it strikes the vanes, and from the weight moved upon the small beam you will learn the force of the blast by which the wind bursts forth from the aeolipile, if the ratio of the pulleys, which is fivefold, be combined with the ratio of the diameter of the wheel to the diameter of the axle, which by construction is twelvefold: for since the ratio is sixtyfold, let it be as 60 to 1, so the weight of the load,

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Liber sextus. CAPUT V. 607 ponderis, quod per hujusmodi machinulam trahitur, ad pon- dus, quod flatu illo impelli posset sine machinâ. CAPUT V. Trochleæ Trochleis additæ plurimùm augent momenta Potentiæ. Quodlibet oblatum pondus datâ Potentiæ virtute movere adhibitis Trochleis omnes norunt, si binas trochleas tot instruant orbiculis, quot exigit Ratio ponderis ad potentiam, aut plura communium trochlearum paria cum pluribus Ergatis adhibeant: quo in opere quàm immanes trochleas esse oporte- ret, si centenos aliquot, aut millenos orbiculos singulæ conti- nerent, aut quot Ergatæ, quantóque dispendio statuendæ es- sent, ex methodo superiori capite traditâ, nemo non videt. Hîc ergo solis Trochleis rem facillimè perfici posse me demonstratu- rum confido, prout in Terrâ machinis motâ dissert. 1. indicavi. Et primò quidem simplici orbiculo stabili elevari potest pon- dus cum incremento momentorum Potentiæ deorsum trahentis (id quod multo facilius accidit, quàm sur- sum trahere) si extremitati funis loco Po- tentiæ adnectatur alius orbiculus, cujus fu- nis in loco inferiore alligatus fuerit. Sit Pondus P, orbiculo stabili A elevandum: utique Potentia funi ductario in B appli- cata non attollet pondus, nisi ejus gravitas, aut virtus movendi major fuerit gravitate ponderis P. Adnectatur in B orbiculus ver- satilis E: & paxillo in C firmato alligetur caput funis per orbiculum E transeuntis: Nam potentia in F funem trahens duplo velociùs movetur quàm orbiculus E, hoc est B extremitas funis ductarij, quæ cum pondere P æqualiter movetur: ac proinde potentia,

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Book six. CHAPTER V. 607 of weight, which is drawn by such a little machine, to the weight which could be driven by that blast without a machine. CHAPTER V. Pulleys added to pulleys greatly increase the moments of Power. Everyone knows that any given weight can be moved with the given force of Power by using pulleys, if they set up two pulleys with as many sheaves as the ratio of weight to power requires, or if they employ several pairs of common pulleys with several tackles: in such a work, how enormous the pulleys would have to be if each contained several hundreds, or thousands, of sheaves, or how many tackles, and at what expense, would have to be set up, no one fails to see from the method set forth in the preceding chapter. Here, then, I trust that I shall demonstrate that the matter can be accomplished most easily by pulleys alone, as I indicated in the dissertation on machines moved on Earth, dissertation 1. And first, indeed, with a simple fixed sheave, a weight can be raised by an increase in the moments of a Power pulling downward (which happens much more easily than pulling upward), if to the end of the rope, in place of the Power, another sheave is attached, whose rope has been fastened in a lower place. Let there be a weight P to be raised by the fixed sheave A: certainly the Power applied to the directive rope at B will not raise the weight unless its gravity, or its moving force, is greater than the gravity of the weight P. Let a movable sheave E be attached at B, and let the head of the rope passing through the sheave E be fastened by a peg fixed at C: for the power pulling the rope at F moves twice as fast as the sheave E, that is, the end B of the directive rope, which moves equally with the weight P: and therefore the power,

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608 Mechanicorum potentia, quæ duplo velociùs movetur quàm pondus, satis est si fuerit paulo major, quàm subdupla ponderis. Ex hoc quasi rudimento continuò se se offert methodus componendi trochleas conjugatas; si nimirum duabus trochleis aptè dispositis ad trahendum pondus, funis ductarij extremitati non applicetur potentia, sed alia trochlea adnectatur, quasi ibi esset pondus, & harum trochlearum secundò positarum funi applicetur potentia, cujus momenta ex Trochlearum rationibus componuntur. Sint duæ Trochleæ A & B binis orbiculis instructæ; pondus in M adnexum sit trochleæ A; & trochlea B firmetur in C: funis autem ductarius in D alligetur trochleæ A mobili: utique Potentia in F habet momentum quintuplum, quia quintuplo velociùs movetur, quàm pondus in M adnexum. Sed iterum duæ aliæ Trochleæ H & I binos orbiculos habentes parentur, & trochlea H mobilis jungatur extremitati funis F, trochlea verò I stabilis sit. Trochlearum H, I funis ductarius alligetur in G trochleæ mobili; & potentia in K applicata quinquies velociùs movetur quàm F; ergo & vigesies quinquies velociùs quàm pondus adnexum in M. Illud igitur pondus, quod quinquies homines applicati in F traheerent, ab unico homine in K applicato atque trahente adducitur, qui unicus æquivalet vigintiquinque hominibus pondus absque trochleis trahere conantibus. Cum itaque communes Trochleæ in promptu sint, manifestum est, quàm facilè multiplicari valeant momenta potentiæ quæ cæteroqui in exemplo proposito duas trochleas singulas duodenum orbiculorum exigeret, ut unus homo præstaret idem, quod viginti quinque. Quod si adhuc duas similes trochleas adhiberes, & alteram similiter in K adnecteres, unicus homo æquivaleret hominibus 125 trahentibus: hinc si unicus ille homo tanto conatu trahat, quanto traheeret libras 50, omninò solus tribus his trochlearum paribus movebit pondus librarum 6250. Sed

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608 Mechanics power, which moves twice as fast as the weight, is sufficient if it is a little greater than half the weight. From this as it were first principle there immediately presents itself the method of combining compound pulleys; namely, if, with two pulleys suitably arranged for drawing a weight, power is not applied to the end of the rope of traction, but another pulley is attached there, as though a weight were there, and to the rope of these second-set pulleys power is applied, whose moments are compounded from the ratios of the pulleys. Let there be two pulleys A and B, furnished with two wheels each; let the weight attached at M be on pulley A; and let pulley B be fixed at C: but let the rope of traction be tied at D to the movable pulley A: certainly the power at F has a quintuple moment, because it moves five times faster than the weight attached at M. But again let two other pulleys H and I, having two wheels each, be prepared, and let movable pulley H be joined to the end of the rope F, but pulley I let it be fixed. To the pulleys H, I let the rope of traction be tied at G to the movable pulley; and the power applied at K moves five times faster than F; therefore also twenty-five times faster than the weight attached at M. Therefore that weight, which five men applied at F would draw, is brought by a single man applied at K and pulling, who alone is equal to twenty-five men trying to draw the weight without pulleys. Since therefore common pulleys are at hand, it is evident how easily the moments of power can be multiplied, which otherwise in the proposed example would require two single pulleys of twelve wheels each, so that one man would do the same as twenty-five. But if you were to employ still two similar pulleys, and similarly attach the other in K, a single man would be equal to 125 men pulling: hence if that single man pulls with such force as he would draw 50 pounds, he will entirely alone, with these three pairs of pulleys, move a weight of 6250 pounds. But

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Liber sextus. CAPUT V. 609 Sed si quem admiratio capiat duodecim orbiculis in sex tro- chleas distributis tantum pondus moveri, admiretur adhuc am- pliùs iisdè duodecim orbiculis in duodecim simplices trochleas distinctis, quæ binæ & binæ conjungentur, longè majus pondus posse trahi. Nam si trochleæ mobili, cui alligatur pondus, etiam funis extremitas adnectatur, jam singulæ Trochlearum conjuga- tiones dant rationem triplam; sunt igitur sex Rationes triplæ compositæ; ac propterea prima conjugatio dat Rationem 3 ad 1; secunda 9 ad 1; tertia 27 ad 1; quarta 81 ad 1; quinta 243 ad 1; sexta 729 ad 1: & hæc est Ratio motûs po- tentiæ ad motum ponderis. Quare si po- tentia conetur ut 50, ducatur 729 per 50, & potentia trahere valebit pondus libra- rum 36450. At verò si duodecim illæ sim- plices trochleæ nô fuerint conjugatæ, sed singulæ seorsim suos habeant funes, ita ut primæ adnectatur pôdus, & secundæ jun- gatur extremitas funis ductarij primæ, at- que ita deinceps, jam multo majus erit momentum potentiæ; erunt scilicet duo- decim Rationes duplæ cöpositæ. Sit enim trochleæ X adnexum pondus, funis illius alligatus in V, & ejusdem capiti adnexa sit secunda Trochlea T; cujus pariter funis alligatus in S reliquâ extremitate conjun- gatur cum tertia Trochleâ R, ejusque fu- nis similiter firmatus in Q veniat ad P, cui deinceps quarta trochlea adjungatur, & sic de cæteris consequentibus. Certum est T moveri duplo velociùs quàm X, & R duplo velociùs quàm T, & P duplo velo- ciùs quàm R; ac proinde P moveri octu- plo velociùs quàm pondus in X. Si igitur duodecim rationes duplæ componantur, erit demum Ratio 4096 ad 1. Quapro- ter potentia in extremitate funis trochleæ duodecimæ similiter conata ut 50, move- bit pondus librarum 204800. HHhh

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Liber sextus. CHAPTER V. 609 But if anyone marvels that so great a weight is moved by twelve pulleys distributed into six blocks, let him marvel still more that by the same twelve pulleys, divided into twelve single pulleys, which are joined two and two, a much greater weight can be drawn. For if to the movable pulley, to which the weight is attached, the end of the rope is also fastened, then each pair of pulleys gives a triple ratio; there are therefore six compounded triple ratios; and consequently the first pair gives the ratio of 3 to 1; the second 9 to 1; the third 27 to 1; the fourth 81 to 1; the fifth 243 to 1; the sixth 729 to 1: and this is the ratio of the motion of the power to the motion of the weight. Therefore if the power is exerted as 50, multiply 729 by 50, and the power will be able to draw a weight of 36,450 pounds. But if those twelve single pulleys are not joined in pairs, but each separately has its own rope, so that to the first the weight is attached, and to the second is joined the end of the driving rope of the first, and so on, then the force of the power will be much greater; namely, there will be twelve compounded double ratios. For let a weight be attached to pulley X, its rope fastened at V, and to its head let the second pulley T be attached; to the rope of this pulley likewise fastened at S let the other end be joined to the third pulley R, and its rope similarly fixed at Q let it come to P, to which afterward the fourth pulley is attached, and so on for the rest in succession. It is certain that T moves twice as fast as X, and R twice as fast as T, and P twice as fast as R; and therefore P moves eight times as fast as the weight at X. If therefore twelve double ratios are combined, the final ratio will be 4096 to 1. Wherefore, if the power at the end of the rope of the twelfth pulley likewise acts as 50, it will move a weight of 204,800 pounds. HHhh

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Mechanicorum Ex his vides posterioribus trochleis minùs repugnare pondus quàm prioribus, atque propterea funes ductarios posteriorum trochlearum posse exiliores esse, quamvis longiores; eóque de- veniri posse, ut potentia subtilissimo funiculo applicetur, & se- curè trahat valde magnum pondus. Semper autem tractionis mentionem feci, non elevationis, quia in illa faciliùs quàm in hac uti possumus hujusmodi trochlearum complexione: quam- quam etiam in elevatione ad mediocrem altitudinem, dispositis duabus trochleis, quasi illas tantum adhibere oporteret, possumus extremitati funis ductarij adjicere trochleam, cujus com- parem paxillo in terram firmiter depacto alligemus; aut etiam, si altitudo suppetat longè major eâ, ad quam attollendum est pondus, in supremo loco statuere possumus trochleam stabilem secundæ conjugationis, & mobili trochleæ adnectere extremitatem funis ductarij priorum trochlearum, in quibus propterea caput funis adnectendum est trochleæ mobili, cui adhæret pondus evehendum. Non est autem dissimulandum incommodum, quod ex hac trochlearum dispositione atque complexione oritur, scilicet magnam funium longitudinem requiri, nec non ingens spatium, in quo disponantur duo illa Trochlearum paria, quibus vigequintupla fiunt Potentiæ momenta. Quia enim in Tro- chleis adnexam sarcinam adducentibus sunt quatuor funis ductus æquales trochlearum intervallo, utique, si eidem trochleæ pondus ac funis alligatur, totus explicatur ultra terminum, cui trochlea stabilis adnectitur: quare trochleam mobilem secundæ conjugationis adnexam extremitati funis priorum trochlearum constituere oportet distantem à suâ trochleâ stabili non minùs quàm intervallo quintuplo distantiæ priorum: ac propterea harum posteriorum funis explicatus excurrit ultra terminum, cui affigitur compar trochlea stabilis spatio illius quintupli intervalli quadruplo, hoc est vigecuplo intervalli priorum trochlearum; cui si addatur distantia posteriorum quintupla distantiæ priorum, Potentia trochleæ secundæ mobilem applicata funem trahens movetur vigequintuplo velociùs quàm pondus, & exigit spatium vigequintuplum distantiæ priorum trochlearum, si illa velit progredi, quantum fert longitudo funis explicati; id quod necesse est, si funis à jumentis trahatur, nec

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Mechanics From these you see that the later pulleys resist the weight less than the earlier ones, and therefore the driving ropes of the later pulleys can be thinner, though longer; and thus it can come about that force is applied by the thinnest cord, and safely draws a very great weight. I have always mentioned pulling, not lifting, because in the former we can more easily use this kind of pulley arrangement than in the latter; although even in lifting to a moderate height, if two pulleys are arranged, as though only those should be employed, we can add a pulley to the end of the driving rope, and bind its counterpart firmly to a peg driven into the ground; or also, if the height available is much greater than that to which the weight is to be raised, we can place a fixed pulley of the second combination at the highest point, and attach to the movable pulley the end of the driving rope of the earlier pulleys, in which case the head of the rope must therefore be attached to the movable pulley to which the weight to be raised is fastened. But it must not be concealed what inconvenience arises from this arrangement and combination of pulleys, namely that a great length of rope is required, as well as a vast space in which those two pairs of pulleys are arranged, by which twenty-fivefold force is produced. For since in pulleys drawing the attached load there are four equal lengths of rope, equal to the interval between the pulleys, if the weight and rope are attached to the same pulley, the whole is paid out beyond the limit to which the fixed pulley is attached; wherefore the movable pulley of the second combination must be placed attached to the end of the rope of the earlier pulleys at a distance from its fixed pulley not less than five times the interval of the earlier ones. And therefore the rope of these later pulleys, when paid out, runs beyond the limit to which the corresponding fixed pulley is attached by a space four times that fivefold interval, that is, twenty times the interval of the earlier pulleys; and if to this be added the fivefold distance of the later from the earlier, the force applied to the movable pulley of the second combination, drawing the rope, moves twenty-five times faster than the weight, and requires a space twenty-five times the interval of the earlier pulleys if it is to advance as far as the length of the paid-out rope allows; which is necessary if the rope is drawn by draft animals, nec

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Liber sextus. CAPUT V. 611 nec circumducatur Ergatæ; tunc enim non tantum spatij re- quiritur, & momentum Ratione Ergatæ augetur. At si Poten- tia trahens sint homines, satis est si propè secundam trochleam stabilem consistant. Quare si quis voluerit hujusmodi quatuor trochlearum complexione uti, ut potentia obtineat momentum vigequintuplum, requiritur spatij longitudo quintupla spatij, per quod deducendum est pondus. Quod igitur ad funium longitudinem spectat, longitudo funis priorum trochlearum est quadrupla spatij percurrendi à pondere, & longitudo funis posteriorum est ejusdem spatij vigecupla; hic tamen posterior funis potest esse priore tenuior atque exilior, ut dictum est. Dixerit fortasse aliquis, rem minùs attentè considerans, posse posteriores trochleas habere funem non longiorem fune prio- rum; sed quia, ubi ille totus explicatus fuerit, pondus non est adductum nisi ad quintam partem spatij, posse trochleas illas posteriores ita invicem disjungi, ut ea, quæ est mobilis, adjun- gatur funi ductario propè trochleam priorem mobilem; nam potentia iterum trahens adducet pondus: id quod sæpius ite- rari potest. Verùm hoc fieri omnino non posse deprehendes, si observa- veris, nunquam hoc pacto adduci pondus nisi per quintam par- tem reliqui spatij; quare aliquid semper relinquitur, quin ad destinatum locum pondus perveniat. Si placuerit tamen hunc laborem assumere in disjungendis posterioribus trochleis, prio- res trochleas ita invicem disjunctas initio colloca, ut earum in- tervallum sit saltem sesquialterum spatij, per quod pondus mo- veri oportet; sic enim repetito quinquies trahendi labore obti- nebis propositum motum: primâ videlicet tractione deducitur pondus per totius intervalli 1/5; in secunda per ejusdem inter- valli 4/25; in tertiâ per 16/125; in quartâ per 64/625; in quintâ per 256/3125; quæ partes si in summam redigantur, dant 2101/3125, hoc est paulo amplius quàm 2/5, propositi intervalli, quantum satis est ad perfi- ciendum destinatum spatium. Ubi vides; si intervallum as- sumptum fuisset paulo majus quàm duplum destinati spatij, ter- tiâ tractione absolvi propositum motum; nam 1/5, 1/25, 16/125 si colli- gantur in summam, dant 61/125, hoc est ferè 1/5. At si duobus sim- plicibus orbiculis utaris, quibus compositis potentia habet mo- HHhh 2

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Book Six. Chapter V. 611 nor let it be carried around the Ergatæ; for then not only is more space required, but the force is increased in proportion to the Ergatæ. But if the pulling power consists of men, it is enough if they stand near the second fixed pulley. Therefore, if someone should wish to use a system of four such pulleys, so that the power may obtain a twenty-fivefold force, there is required a length of space five times the space through which the weight is to be drawn. Thus, as regards the length of the ropes, the length of the rope of the first pulleys is four times the space traversed by the weight, and the length of the rope of the latter is twenty times that same space; yet this latter rope may be thinner and more slender than the former, as has been said. Someone may perhaps say, considering the matter less attentively, that the later pulleys can have a rope no longer than that of the former; but since, when it has all been paid out, the weight is drawn only to one fifth of the space, those later pulleys can be separated from one another in such a way that the movable one is attached to the carrying rope near the first movable pulley; for the pulling power, acting again, will draw the weight on: and this can be repeated many times. But you will discover that this can by no means be done, if you observe that in this way the weight is never drawn on except by one fifth of the remaining space; wherefore something always remains, so that the weight does not reach the appointed place. If, however, it should please you to undertake this labor of separating the later pulleys, then place the first pulleys, separated from one another in this way at the outset, so that their interval is at least one and a half times the space through which the weight must be moved; for in this way, by repeating the labor of pulling five times, you will obtain the proposed motion: by the first pull the weight is drawn through 1/5 of the whole interval; by the second through 4/25 of the same interval; by the third through 16/125; by the fourth through 64/625; by the fifth through 256/3125. If these parts are reduced to a sum, they give 2101/3125, that is, somewhat more than 2/5 of the proposed interval, which is enough to accomplish the intended distance. Thus you see that if the interval assumed had been somewhat more than twice the intended space, the proposed motion would be completed by the third pull; for if 1/5, 1/25, and 16/125 are collected into a sum, they give 61/125, that is, nearly 1/5. But if you use two simple pulleys, by which, when combined, the power has mo HHhh 2

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Mechanicorum 612 mentum quadruplum, etiamsi secundi orbiculi funem statuas æqualem funi prioris orbiculi, cui adnectitur pondus, facilli- mum est orbiculum secundum retrahere ad orbiculum primum, postquam hic primâ tractione absolvit semissem spatij inter pon- dus & paxillum, cui alligatur funis; & secundâ tractione ab- solvit quadrantem totius intervalli initio constituti: Quare sa- tis fuerit funem prioris orbiculi æquari intervallo sesquitertio longitudinis spatij, per quod deducendum est pondus. Et quoniam hîc mentio incidit orbiculorum simplicium, ob- serva, quanto faciliùs duobus orbiculis perficiamus id, quod duabus trochleis binos orbiculos habentibus præstaremus in trahendo pondere, quando funis ductarius est alligatus tro- chleæ stabili; tunc enim potentia solùm habet momentum quadruplum, quod pariter obtinet duobus orbiculis. Sit enim A B distantia sesquitertia spatij A I, per quod trahendum est pondus in P adnexum orbicu- lo A: funis in B alligetur, & ejus caput C con- nectatur cum orbiculo E, cujus pariter funis in B alligetur, atque illius extremitas à Poten- tiâ F trahatur. Quando potentia F adduxerit orbiculum E propè B, erit orbiculus A in H: retrahatur orbiculus E ex B, & propè H ad- nectatur funi orbiculi A; factâ enim secundâ tractione, quando orbiculus E fuerit iterum prope B, orbiculus A erit in I; est autem ex hypothesi distantia A I æqualis spatio, per quod trahendum erat pondus, subsesquitertio intervalli A B. Ecce igitur Potentia habet momentum quadruplum, & duorum funium longitudines simul sumptæ non dant longitu- dinem triplam spatij, per quod deducendum est pondus. At si essent duæ Trochleæ cum binis orbiculis, exigerent unicum funem qua- druplum longitudinis spatij, per quod insti- tuendus est motus. Sed & illud addendum videtur, quod duobus simplicibus or- biculis etiam ad longiora spatia adduci potest pondus, ita ut quilibet trahentium habeat momentum quadruplum. Expe- dit

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Mechanics 612 a quadruple advantage, although you set the cord of the second pulley equal to the cord of the first pulley, to which the weight is attached, it is very easy to draw the second pulley back to the first pulley, after the first pull has carried through half the distance between the weight and the peg to which the cord is tied; and after the second pull it has carried through a quarter of the whole interval originally set. Therefore it will be sufficient for the cord of the first pulley to be equal to one and a half times the length of the space through which the weight must be lowered. And since mention has here fallen on simple pulleys, observe how much more easily, with two pulleys, we accomplish what with two tackle-blocks having two pulleys each we would perform in drawing a weight, when the hauling rope is tied to the fixed tackle-block; for then the power has only a quadruple advantage, which equally holds with two pulleys. Let A B be the distance one and a half times the interval A I, through which the weight attached in P to pulley A is to be drawn: let the cord be tied at B, and its end C connected with pulley E, whose cord likewise is tied at B, and let its end be drawn by the power F. When the power F has brought pulley E near B, pulley A will be at H: let pulley E be drawn back from B, and near H let it be attached to the cord of pulley A; for when the second pull has been made, when pulley E is again near B, pulley A will be at I; and by hypothesis the distance A I is equal to the space through which the weight had to be drawn, being somewhat less than one and a half times the interval A B. Behold, then, the power has a quadruple advantage, and the lengths of the two cords taken together do not make three times the length of the space through which the weight is to be lowered. But if there were two tackle-blocks with two pulleys each, they would require a single cord four times the length of the space through which the motion is to be effected. But it also seems that this should be added, that with two simple pulleys the weight may likewise be brought through greater distances, so that each of those pulling has a quadruple advantage. It is advantageous

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Liber sextus. CAPUT V. 613 dit autem trahentium numerum geminari, ut alternâ quiete faciliùs & citiùs onus trahant. Sit orbiculus M adnectendus ponderi, & sit datus funis du- ctarius S R, cu- jus extremitati- bus R & S re- plicatis quasi in laqueum, seu ansam facillimè immitti possint & paxillus R, & alterius tro- chleæ uncus S. Funis alius paretur N T prioris duplus extre- mitates similiter replicatas habens, ut in V immitti possit tra- hentis manus, & in T paxillus. Quare tantumdem paxillus R distat à Trochleâ M, quantum à paxillo T, & hic tantumdem à paxillo X. Cum igitur toto fune V N T explicato orbiculus N fuerit in T, orbiculus M erit in R, & funis extremitas S erit in T. Itaque ex paxillo T auferatur funis explicatus, & ejus loco injiciatur extremitas S. Eximatur tunc ex paxillo R extremitas funis, & adnectatur alteri Trochleæ funem habenti æqualem funi V N T, cujus extremitas alligata fuerit paxillo X, & ad T adducetur trochlea M unà cum pondere. Atque ita alternâ ope- râ adducetur pondus ad quancumque distantiam; interea enim, dum orbiculus M ex R trahitur ad T, is qui traxerat funem V, alligat illum paxillo, ad quem progrediendo pervenitur, & extre- mitatem T exemptam è paxillo trahet, ubi trochleam N eò jam deductam iterum junxerit extremitati S in T existenti. Sunt itaque pangendi in terram paxilli æqualibus intervallis. Monendus est autem Lector ad hoc caput non pertinere illam Trochlearum additionem, quæ non facit rationum Compositio- nem; quando scilicet plures trochleæ uno loculamento ita in- cluduntur, ut singulæ trochleæ tam superior, quam inferior plu- res habeant orbiculorum ordines in latitudinem collocatos, at- que adeo tot funes ductarios, quot sunt ordines illi orbiculorum, exigunt; perinde enim est atque si duæ aut tres trochleæ diver- sis loculamentis distinctæ adhiberentur. Cum autem plures sint funes ductarij, qui uno eodemque tempore adducendi sunt, di- ligenter animum advertere oportet, ut operæ omnes æqualiter trahant. HHhh

Transcription: Translated (English)

Book Six. CHAPTER V. 613 it is shown that the number of those pulling is to be doubled, so that by alternate rest they may draw the load more easily and more quickly. Let the small wheel M be attached to the weight, and let there be given the guiding rope S R, the extremities of which R and S, being doubled back as into a noose, or loop, can most easily receive both the pin R and the hook S of the other pulley. Let another rope N T be prepared, double the former, having its extremities similarly doubled back, so that the hand of the puller may be put in at V, and the pin in T. Wherefore the pin R is as far from pulley M as it is from pin T, and this again as far from pin X. When therefore, the whole rope V N T being let out, the small wheel N shall have been in T, the small wheel M will be in R, and the end S of the rope will be in T. So from pin T let the let-out rope be removed, and in its place let end S be inserted. Then let the end of the rope be taken from pin R, and let it be attached to the other pulley, having a rope equal to V N T, whose end had been tied to pin X, and pulley M together with the weight will be drawn to T. And thus by alternate operation the weight will be drawn to whatever distance; meanwhile, for indeed while the small wheel M is being pulled from R to T, the man who had pulled the rope V binds it to the pin which is reached in advancing, and will draw out the end T released from the pin, when he shall again have joined the pulley N, now brought to that place, to the end S existing in T. Therefore pins must be driven into the ground at equal intervals. However, the Reader is to be warned that the addition of pulleys does not belong to this chapter, when it does not make a composition of ratios; namely, when several pulleys are enclosed in one housing in such a way that each pulley, both upper and lower, has several rows of small wheels arranged in breadth, and therefore requires as many guiding ropes as there are those rows of small wheels; for it is just as if two or three pulleys, distinguished by different housings, were employed. But when there are several guiding ropes which are to be drawn at one and the same time, care must be taken that all the workmen pull equally. HHhh

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Mechanicorum CAPUT VI. Trochlearum ope moveri potest pondus velociter. Hactenus Trochlearum in facilè movendis oneribus vires expendimus, ubi quò majora momenta ope hujus Faculta- tis adduntur Potentiæ, eò etiam tardior est motus ponderis, po- tentiæ autem velocior: quod si velociter movendum sit pondus, necessariò augeri debet potentia. Verùm quia non rarò contin- gere potest, ut potentia quidem ipsa per se viribus abundet, il- lam tamen tardè moveri oporteat, aut contra in trahendo onere festinato sit opus, propterea hìc indicandum est, qua methodo uti possimus, ut hinc plenior hujus Facultatis notitia habeatur. Opus sit in turrim, vel in urbis moenia commeatum transfer- re velociter: operarum suppetat satis, at non item temporis. Statuatur in summa turri, aut certè in loco opportu- no, Sucula B C cum manu- briis C E F, & B D I, qui- bus plures operæ applicari possint pro gravitate oneris attollendi; immò etiam ha- beat infixos radios, ut adhuc plures recipiat, qui illam versare possint. Circà Axem involutus sit funis paulo longior semisse altitudinis, & extremitati sit adnexus girgillus G, cui insertus sit funis ductarius H G L æqualis altitudini, ad quam evehendum est onus; alte- ri hujus funis extremitati cohæreat in H validus un- cus, quo onus suspendatur, alteram verò extremitate[m] L firme

Transcription: Translated (English)

Mechanics CHAPTER VI. A weight can be moved quickly by means of pulleys. Thus far we have examined the powers of pulleys in easily moving loads, where, the greater the forces added to the Power by means of this Faculty, the slower also is the motion of the weight, and the faster that of the power: but if a weight is to be moved quickly, the power must necessarily be increased. However, because it can often happen that the power itself, considered in itself, is abundantly strong, yet must be moved slowly, or, on the contrary, there is need of haste in drawing a load, therefore it must here be indicated by what method we may use this, so that a fuller knowledge of this Faculty may be obtained. Suppose it is necessary to transfer provisions quickly into a tower, or into the walls of a city: let there be enough laborers available, but not time as well. Let there be set up on the top of the tower, or certainly in a suitable place, a Sucula B C with handles C E F and B D I, to which several laborers may be applied according to the heaviness of the load to be lifted; indeed, let it even have fixed spokes, so that it may receive still more, who can turn it. Around the axis let a rope be wound, a little longer than half the height, and to its end let there be attached the pulley G, into which let the guiding rope H G L be inserted, equal to the height to which the load is to be raised; to the other end of this rope let there be joined at H a strong hook, by which the load may be suspended, and the other end at L firmly

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Liber sextus. CAPUT VI. 615 firmet clavus, aut quid simile ad pedem turris. Nam si conver- tatur Sucula, devenient pariter in A tum girgillus G, tum onus unco H suspensum; quod sanè duplo velociùs movetur, quàm si adnexum funi ductario A K traheretur sursum ope simplicis suculæ. Quare momenta suculæ non nisi dimidiata computanda sunt, adeò ut si duo homines suculam B C cir- cumagentes valerent attollere libras 400, eodem conatu, & la- bore possint solum libras 200 attollere: at quia facilè multipli- cari possunt homines suculam versantes, geminetur eorum nu- merus, & attollent libras 400, sed breviori tempore. In con- trarium autem revoluta sucula demittet girgillum G, & suo pon- dere duplo velociùs descendet uncus H. Ex quo habetur quæ- situm temporis compendium. At si duabus trochleis simplicibus singulos orbiculos haben- tibus res perficienda esset, ita ut uni trochleæ adnecteretur Po- tentia, alteri Pondus, funis autem extremitas alicubi clavo re- ligata esset, attentè dispiciendum est, utri trochleæ adnecta- tur reliqua funis trochleas jungentis extremitas. Nam si tro- chleæ A, quam trahit potentia N, adnectatur in C funis per orbicu- los trajectus, trochleæ verò B pondus M, & funis religatus fue- rit in D, intelligitur motus inci- pere, quando trochleæ adhuc in- vicem absunt, ità ut in motu tro- chlea ponderis ad trochleam po- tentiæ accedat, cessare autem, cùm illæ proximæ factæ fuerint in maximâ distantiâ à clavo D, ubi funis extremitas alligatur. Contrà verò accidit trochleis G & H, si trochleæ H adnectatur pondus B, atque in I funis ductarij caput: nam trahente potentia S, quæ ini- tio propiores erant trochleæ, à se invicem recedunt, trochleâ po- tentiæ sedente à trochleâ pon- deris; & demum absolvitur mo- tus,

Transcription: Translated (English)

Book Six. CHAPTER VI. 615 firm the pin, or something similar, at the foot of the tower. For if the winch be turned, both the wheel G and the weight suspended from the hook H will come down into A; and this indeed moves twice as fast as if it were drawn upward by means of the simple winch through the attached haul-rope A K. Therefore the effects of the winch are to be counted as only one-half, so that if two men turning the winch B C were able to raise 400 pounds, with the same effort and labor they could raise only 200 pounds: but because the number of men turning the winch can easily be increased, let their number be doubled, and they will raise 400 pounds, but in a shorter time. On the contrary, if the winch is turned the other way, it will lower the wheel G, and by its own weight the hook H will descend twice as fast. From this is obtained the desired saving of time. But if the work is to be accomplished by two simple pulleys, each having one wheel, so that Power is attached to one pulley, Weight to the other, and the end of the rope is fastened somewhere to a nail, it must be carefully considered to which pulley the remaining end of the rope joining the pulleys is to be attached. For if to pulley A, which is drawn by power N, the rope passed through the wheels is attached at C, but to pulley B the weight M is attached, and the rope has been fastened at D, it is understood that the motion begins when the pulleys are still apart from one another, so that in motion the pulley of the weight approaches the pulley of the power, but ceases when they have become near one another at the greatest distance from the nail D, where the end of the rope is tied. The opposite happens with pulleys G and H, if to pulley H the weight B is attached, and at I the end of the haul-rope: for when power S draws, the pulleys, which were at first nearer together, separate from one another, the pulley of the power remaining away from the pulley of the weight; and finally the motion is completed,

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Mechanicorum 616 tus, cùm trochlea H ponderis accesserit ad R extremitatem funis religati. Cum itaque in utroque casu & potentia, & pondus versùs eandem partem moveantur, in primo tamen pondus, quod à potentiâ distabat, ad illam accedat, & in secundo potentia vicina ponderi ab illo recedat, manifesto indicio est in primo casu pondus, in secundo potentiam ve- lociùs moveri: quare ibi potentia augenda est, ut valeat mo- vere pondus, hîc fieri potest additamentum ponderi, ut po- tentiæ virtuti respondeat. Est autem motuum Ratio sesqui- altera, ut palàm faciunt funium ductus, eorumque explica- tio: Nam in primo casu maxima trochlearum distantia est, quando trochlea A est clavo D proxima; igitur potentia movetur per spatium, cujus longitudinem metitur funis ex- plicatus, qui est duplus distantiæ trochlearum, & pondus accedens ad potentiam insuper percurrit spatium, quo tro- chleæ distabant; igitur motus ponderis est ut 3, & poten- tia ut 2. In secundo verò casu, Trochleæ G & H cùm proximæ sunt, distant à clavo R juxta longitudinem funis explicati, cùm autem maximè invicem absunt, & potentia transgressa est clavum R, totus funis distributus est in duos ductus, & trochlearum intervallum est medietas longitudi- nis funis; quare ponderis motus est ut 1, & motus poten- tiæ ut 1 ́. Simili ratione philosophandum erit, si trochleæ inæquales proponantur, ut si altera sit duorum orbiculorum, altera unius orbiculi: Utique funis per orbiculos trajectus adnecten- dus est simplici trochleæ, ejusque altera extremitas alicubi firmanda. Non igitur indiscriminatim sivè huic, sivè illi trochleæ adjungenda est potentia, sed priùs statuendum ti- bi est, utrum velis pondus movere facilè, an velociter; si facilè, tardior sit ponderis motus, quàm potentiæ; si velo- citer, tardior sit potentia. Facilè movebis pondus, si potentia trahat simplicem orbiculum, & pondus cohæreat trochleæ duorum orbiculorum: Velociter autem movebitur pondus, si illud adnectatur simplici orbiculo, potentia verò trahat tro- chleam duorum orbiculorum. Nam in primo casu funis expli- catus replicatur, & potentia recedit à pondere; in secundo fu- nis replicatus explicatur, & pondus accedit ad potentiam. Sit

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Mechanics 616 when pulley H has approached the end R of the tied rope. Since, therefore, in either case both the power and the weight move toward the same side, but in the first case the weight, which was distant from the power, approaches it, and in the second the power, being near the weight, recedes from it, this is clear proof that in the first case the weight, and in the second the power, moves more quickly: wherefore in the former the power must be increased, so as to be able to move the weight; in the latter an addition may be made to the weight, so that it may correspond to the force of the power. Now the ratio of the motions is one and a half, as the course of the ropes and their explanation plainly show. For in the first case the greatest distance of the pulleys is when pulley A is nearest the peg D; therefore the power moves through a space whose length is measured by the extended rope, which is twice the distance between the pulleys, and the weight, approaching the power, moreover traverses the space by which the pulleys were separated; therefore the motion of the weight is as 3, and that of the power as 2. But in the second case, when pulleys G and H are nearest, they are at a distance from the peg R equal to the length of the extended rope; but when they are farthest apart from one another, and the power has passed the peg R, the whole rope is distributed into two courses, and the interval between the pulleys is half the length of the rope; therefore the motion of the weight is as 1, and the motion of the power as 1 1/2. In a similar way we must reason if unequal pulleys are proposed, as if one were of two wheels and another of one wheel: indeed, the rope passed around the wheels must be attached to the simple pulley, and its other end fastened somewhere. Therefore the power is not to be attached indiscriminately to this pulley or to that, but you must first decide whether you wish to move the weight easily or quickly; if easily, let the motion of the weight be slower than that of the power; if quickly, let the power be the slower. You will move the weight easily if the power draws the simple wheel, and the weight is attached to the pulley of two wheels. But the weight will be moved quickly if it is attached to the simple wheel, while the power draws the pulley of two wheels. For in the first case the extended rope is wound back, and the power recedes from the weight; in the second the wound rope is unwound, and the weight approaches the power. Let it be

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Liber sextus. CAPUT VI. 617 Sit Trochlea MO, & orbiculus I; huic in L adnectitur fu- nis, cujus altera extremitas religatur in A, quò demum devenire potest trochlea MO cum pondere T adjecto. Vice versâ Tro- chleæ GC adhibeatur potentia, & pon- dus S adjiciatur orbiculo E: huic in B ad- nectitur funis, qui per orbiculos trajectus desinit in F, ubi ille religatur, & trochlea GO maximè distat ab orbiculo E. Insti- tuto motu, Potentia D semper magis re- cedit à pondere T; at pondus S semper magis accedit ad Potentiam H: ibi ergo potentia celerior est pondere, hîc pondus velocius est Potentiâ; motuum autem Ra- tio est sesquitertia: Nam explicato fune toto, qui religatur in A, potentia proxi- ma est ponderi, & distant ab A pro funis longitudine; potentiâ trahente accedunt ad A, sed potentia ulteriùs progreditur, atque absoluto motu replicatus est funis in tres ductus, & Potentia distat à pondere tertiâ parte ipsius funis, ita ut pondus quidem sit clavo A proxi- mum, potentia verò transgressa sit clavum A intervallo OL: igitur motus ponderis, quem longitudo funis metitur, est ut I, potentiæ ut I ́. Ex adverso Potentia applicata trochleæ GC proxima est clavo F, cum ab illâ pondus maximè abest inter- vallo tertiæ partis ipsius funis in tres ductus replicati: inito mo- tu pondus accedit ad Potentiam, cui demum proximum est, quando jam totus funis est explicatus; igitur motum potentiæ metitur funis explicatus, motum autem ponderis adhuc tertia pars, scilicet intervallum BC: adeóque ponderis motus ad mo- tum potentiæ est ut I ́ ad I. Quæ autem de his trochleis dicta sunt, si attentè conside- rentur, etiam cæteris trochleis conjugatis, sed dispari orbicu- lorum numero instructis, conveniunt. Non posse verò orbicu- lorum numeros differre nisi unitate, satis manifestum est: nam si different binario aut ternario, eorum aliquis aut plures pla- nè otiosi essent, quippe qui recipere nequirent funem ducta- II i i

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Book Six. Chapter VI. 617 Let the pulley MO and the small wheel I be given; to the latter, at L, a rope is attached, one end of which is fastened at A, to which point the pulley MO can finally be brought when the weight T is added. Conversely, to the pulley GC let a power be applied, and let the weight S be added to the small wheel E: to this, at B, is attached a rope, which, passing through the small wheels, ends at F, where it is fastened, and the pulley GO is at the greatest distance from the small wheel E. When the motion is set in progress, the power D always recedes more and more from the weight T; but the weight S always approaches more and more toward the power H: there, therefore, the power is swifter than the weight; here the weight is swifter than the power; but the ratio of the motions is one and a half times as great. For when the whole rope, which is fastened at A, is fully unwound, the power is nearest the weight, and they are distant from A by the length of the rope; as the power draws, they approach A, but the power advances farther, and, the motion being completed, the rope is folded back into three runs, and the power is distant from the weight by the third part of the rope itself, so that the weight indeed is near the peg A, but the power has passed beyond peg A by the interval OL: therefore the motion of the weight, which the length of the rope measures, is as 1, that of the power as 1½. On the other hand, the power applied to the pulley GC is nearest to peg F, while the weight is farthest from it by the interval of the third part of the rope itself folded into three runs: when the motion begins, the weight approaches the power, to which it is at last nearest when now the whole rope has been unwound; therefore the motion of the power is measured by the unwound rope, but the motion of the weight still by the third part, namely the interval BC: and thus the motion of the weight to the motion of the power is as 1½ to 1. What has been said about these pulleys, if carefully considered, also applies to other coupled pulleys fitted with a different number of small wheels. But that the numbers of the small wheels cannot differ except by one is sufficiently clear: for if they differed by two or three, some of them would be entirely idle, since they would be unable to receive the rope drawn through the

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Mechanicorum 618 rium jam per reliquos orbiculos trajectum. Quare si altera trochlea minor duos habeat orbiculos, altera major non nisi tres habere potest, aut si minor tres habeat, major non nisi quatuor habere poterit. Attendendum est igitur, utri trochlearum trochlearum potentia applicetur; si enim illa trahendam arripiat trochleam plures habentem orbiculos, tardiùs movetur, quàm pondus in Ratione subsuperparticulari denominatâ à numero omnium simul orbiculoru[m]: ut si potentia trochleæ trium, pondus verò trochleæ duorum orbiculorum applicetur, motus potentiæ est ad motum ponderis in Ratione subsesquiquintâ, quia illa movetur ut 5, pondus ut 6: & si potentia trochleæ quatuor orbiculorum applicetur, pondus autem trochleæ trium, Ratio est subsesquiseptima, quia illa movetur ut 7, hoc ut 8. Quare augetur motus ponderis, & in eadem Ratione difficultas potentiæ. Contrà autem si pondus alligetur majori trochleæ, etiam potentiæ motus major, est motu ponderis in Ratione superparticulari denominatâ à numero omnium simul orbiculorum: sic erit Ratio sesquiquinta, si potentia duobus, pondus tribus orbiculis alligetur; nam motus potentiæ est ut 6, & motus ponderis ut 5: similiter erit Ratio sesquiseptima, quando pondus alligatum trochleæ quatuor orbiculorum movetur ut 7, dum potentia applicata trochleæ trium orbiculorum movetur ut 8. Quod si pari orbiculorum numero constet utraque trochlea, & utraque moveatur, similiter motus erunt in Ratione superparticulari denominatâ à numero omnium orbiculorum simul: hoc tamen erit discrimen, quod illud tardius movebitur, quod applicabitur trochleæ, cui extremitas funis per orbiculos trajecti adnectitur. Sic si trochleæ ambæ binos habeant orbiculos, Ratio est sesquiquarta, si ternos sesquisexta: si trochleam, cui funis ductarij extremitas adnectitur, potentia trahat, illa movetur ut 4, aut ut 6, pondus verò movetur ut 5, aut ut 7: sed si trochleæ, cui funis adnectitur, alligetur pondus, potentia movetur ut 5, aut ut 7, pondus autem ipsum ut 4, aut ut 6. Ex his itaque duplex trochlearum usus innotescit, alter communis quo potentia applicatur extremitati funis ductarij (alterâ trochlearum manente stabili) quem trahens attrahit pariter pondus, & motus potentiæ est in Ratione aliqua multiplici ad motum ponderis. Alter verò est, quando extremitas funis ductarij

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Mechanics 618 is now drawn through the remaining small pulleys. Therefore, if one smaller pulley has two small pulleys, the other, larger, can have no more than three; or if the smaller has three, the larger can have no more than four. It must therefore be noted to which of the two pulleys the power is applied; for if that power takes hold of the pulley having more small wheels, it moves more slowly than the weight, in a submultiple ratio named from the number of all the small wheels taken together: thus, if the power is applied to a pulley of three small wheels and the weight to a pulley of two small wheels, the motion of the power is to the motion of the weight in the ratio of sesquiquinta, because that one moves as 5, the weight as 6; and if the power is applied to a pulley of four small wheels, but the weight to one of three, the ratio is sesquisexta, because that one moves as 7, this as 8. Therefore the motion of the weight is increased, and in the same ratio the difficulty for the power is increased. On the other hand, if the weight be tied to the larger pulley, then the motion of the power is also greater, and is to the motion of the weight in a superparticular ratio named from the number of all the small wheels taken together: thus the ratio will be sesquiquinta, if the power is attached to two small wheels and the weight to three; for the motion of the power is as 6, and the motion of the weight as 5. Similarly the ratio will be sesquiseptima, when the weight tied to a pulley of four small wheels moves as 7, while the power applied to a pulley of three small wheels moves as 8. If both pulleys consist of the same number of small wheels, and both are moved, then likewise the motions will be in a superparticular ratio named from the total number of the small wheels taken together. Yet there will be this difference: that will move more slowly which is applied to the pulley to which the end of the rope passed through the small wheels is attached. Thus if both pulleys have two small wheels each, the ratio is sesquiquarta; if three, sesquisexta: if the power pulls the pulley to which the end of the driving rope is attached, it moves as 4, or as 6, while the weight moves as 5, or as 7. But if the weight is attached to the pulley to which the rope is fastened, the power moves as 5, or as 7, while the weight itself moves as 4, or as 6. From these things, therefore, a twofold use of pulleys becomes clear: one common use, in which the power is applied to the end of the driving rope (the other pulleys remaining fixed), and the puller draws the weight along with it, and the motion of the power is in some multiple ratio to the motion of the weight. The other, however, is when the end of the driving rope

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Liber sextus. CAPUT VII. 619 ductarij non trahitur, sed alicubi firmatur, potentia autem trahit alteram trochleam, ad cujus motum etiam reliqua tro- chlea cum pondere illorum movetur, quorsum potentia tendit: si in hoc motu trochleæ disjunguntur, & potentia recedit à Pondere, Ratio motûs potentiæ ad motum ponderis est super- particularis, & potentia consequitur aliquam movendi facilita- tem: sin autem pondus ad potentiam accedit, & trochleæ, quæ disjunctæ erat, fiunt proximæ, Ratio motûs potentiæ ad motum ponderis est subsuperparticularis, & potentiam plus adhibere conatûs oportet, quàm si illud absque trochleis tra[n]heret; quia pondus velociùs movetur quàm potentia. CAPUT VII. Quàm validum esse oporteat trochlearum retinaculum. IN Trochlearum usu communi alteram stabilem esse ac fir- mam, alteram mobilem (si enim plures essent omnino stabi- les, quantumvis multæ, non augerent motum potentiæ) illam autem ab aliquo corpore, cui alligata est, retineri, satis per se patet; propterea corpus hoc adeò validum esse oportet, ut ne- que gravitati ponderis, neque conatui potentiæ cedat, sed ita immotum persistat, ut universus potentiæ impetus ad vincen- dam ponderis resistentiam referatur. Hinc à veritate non ad- modum recessisse videntur, qui in Mechanicis motionibus qua- si duplex munus distinguunt, alterum, quo pondus retinetur, ne vi suæ gravitatis labatur, alterum, quo gravitas ipsa superatur, & cogitur inire motum suæ propensioni adversantem: poste- rius hoc soli potentiæ tribuendum, prius illud non uni poten- tiæ, sed etiam corpori, cui machina innititur, adscribendum censent, & in illud maximam oneris partem rejici asserunt. Et sanè quid prodesset trochleam superiorem aut fune, qui laxari nequiret, aut ferreo unco, quem revellere nulla gravitas pos- set, connecti cum tigno parieti infixo, si timendum esset, ne IIii 2

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Book six. Chapter VII. 619 the ductary is not pulled, but is fixed somewhere; however, the power draws the other pulley, and by its motion the remaining pulley, together with the weight, is also moved in whatever direction the power tends. If, in this motion, the pulleys separate and the power recedes from the weight, the ratio of the motion of the power to the motion of the weight is superparticular, and the power obtains some ease in moving. But if the weight approaches the power, and the pulleys, which were separated, become closer, the ratio of the motion of the power to the motion of the weight is subsuperparticular, and one must apply more effort to the power than if it were drawing that load without pulleys; because the weight moves more quickly than the power. Chapter VII. How strong the fastening of pulleys ought to be. IN the common use of pulleys, one is fixed and firm, the other movable (for if there were several that were wholly fixed, however many, they would not increase the motion of the power); but it is evident enough of itself that that fixed one is held by some body to which it is attached. Therefore this body must be so strong that it yields neither to the weight of the load nor to the effort of the power, but remains so unmoved that the whole impulse of the power is directed to overcoming the resistance of the load. Hence those do not seem far from the truth who, in mechanical motions, distinguish as it were a double function: one, by which the load is held so that it does not slip by its own weight; the other, by which weight itself is overcome and compelled to enter upon a motion contrary to its inclination. They judge that the latter should be attributed to the power alone, while the former should be ascribed not to the power alone, but also to the body on which the machine rests, and they assert that the greatest part of the burden falls upon that body. And indeed, what good would it do for the upper pulley to be connected either with a rope that could not be slackened, or with an iron hook that no weight could tear away, if it were to be feared that IIii 2

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Mechanicorum 620 tignum ipsum imbecillum, vimque gravitatis suspensæ ferre non valens, frangeretur? Quare ne magnum in discrimen res adducatur, & ad periculum omne submovendum, ne institutus motus repentina retinaculi abruptione intercidatur, atque ut certiùs eligi possit, cuinam potissimùm corpori (tigno ne parieti infixo? an antennæ erectæ?) concedenda sit oneris sustentatio, machinatori attentè dispiciendum est, quantam vim tùm oneris gravitas, tùm potentiæ conatus exerceat adversùs hujusmodi retinaculum. Propterea vim istam placuit hoc capite examinare, ut cætera securè definiri valeant. Ut verò brevitati & perspicuitati consulatur, retinaculum hoc ponamus esse clavum, ex quo trochleæ cum onere suspenso dependeant; quæ enim de hujusmodi clavo dicentur, facilè ad cætera traduci poterunt. Et primò si trochlearum funis per orbiculos ritè trajectus demum suâ extremitate in nodum colligatur, ne excurrere valeat, totam atque integram oneris gravitatem (trochleas & funem à suâ insitâ gravitate nunc quidem mente secernamus) à clavo, ex quo trochleæ suspenduntur, retineri dubium esse non potest; nihil aliud quippe adest, adversùs quod ponderis gravitas deorsum se ipsa urgens connitatur. Deinde si funis ductarij caput, quod potentia trahere solita est, alligetur solo, aut ingenti saxo longissimè graviori, quàm pondus suspensum, utique neque saxum illud subjectæ telluri incumbens, neque tellus ipsa, quippiam virium exercent adversùs pondus, cui solùm suâ longè majori gravitate resistunt Formaliter, non verò Activè; quia nimirum nullum efficiunt impetum, quo descensum moliantur; ac proinde à clavo solo pondus trochleis adnexum sustinetur, & solum pondus clavum deorsum trahere conatur. At verò si funis ductarij extremitati adnectatur alia gravitas pro trochlearum Ratione respondens ponderis gravitati, ita ut æqualibus momentis certantes ambæ suspensæ consistant, utraque gravitas collatis viribus clavum trahere conatur, utraque enim deorsum connititur: & ideò tam validum statui clavum oportet, ut utriusque gravitatis conatum ferre valeat. Id quod multo magis observandum est, quando gravitas adnexa præponderans vim infert oneri, illudque sursum trahit; ipsa scilicet gravitas plus conatur in motu, quàm in æquilibrio; ac propterea & potentiæ

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Mechanics 620 the beam itself, being weak and unable to bear the force of the suspended weight, should break? Therefore, lest the matter be brought into great danger, and to remove all risk, lest the intended motion be interrupted by the sudden breaking of the fastening, and so that it may more surely be decided to which body chiefly the support of the load should be granted, whether to a beam fixed into the wall or to an upright mast, the mechanician must carefully consider what force there is exerted both by the weight of the load and by the effort of the power against such a fastening. For this reason it has seemed proper to examine this force in this chapter, so that the rest may be safely determined. But in order to provide for brevity and clarity, let us suppose this fastening to be a nail, from which pulleys with a suspended load hang; for whatever will be said of such a nail can easily be applied to the rest. And first, if the rope of the pulleys has been duly passed through the blocks and then tied at its end in a knot, so that it cannot run out, there can be no doubt that the whole and entire weight of the load—setting aside now in thought the pulleys and the rope by reason of their own inherent weight—is held by the nail from which the pulleys are suspended; for nothing else is present against which the weight, pressing downward by its own force, may exert itself. Then, if the end of the running rope, which the power is accustomed to pull, is tied to the ground, or to a huge rock far heavier than the suspended weight, certainly neither that rock resting upon the earth beneath it, nor the earth itself, exerts any force against the weight, except by their own far greater weight; formally, however, not actively, because they produce no impulse by which they might attempt a descent. And therefore the load attached to the pulleys is supported by the nail alone, and the load alone tries to pull the nail downward. But if to the end of the running rope another weight is attached, corresponding in respect of the pulleys to the weight of the load, so that, contending with equal moments, both remain suspended in equilibrium, each weight, by joined forces, tries to pull the nail; for each strives downward. And therefore the nail must be set so strong that it may be able to bear the combined effort of both weights. This must be observed all the more when the attached weight exceeds it and exerts force upon the load, drawing it upward; for weight itself strives more in motion than in equilibrium; and therefore also the power of

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Liber sextus. CAPUT VII. 621 potentiæ deorsum connitentis in motu impetum, & oneris mo- tui sursum repugnantis gravitatem fert clavus utrique resistens suâ soliditate. Sicut igitur gravitas inanimata ex trochlearum fune pendens suspendit pondus, aut attollit; ita potentia vivens funem retinendo suo impetu virtutem ejusdem gravitatis æquat, ac similem vim exercet in clavum; funem verò trahendo virtu- tem illam gravitatis superat, atque impresso impetu quodam- modo attenuat, ita tamen, ut quod videtur gravitati demptum, intelligatur additum conatui potentiæ prævalentis. Mihi autem (quid frustra dissimulem?) non levis injicitur scrupulus & dubitatio, an vis illata clavo, ex quo trochleæ cum onere dependent, mensuram præcisè recipiat ex absolutâ gra- vitate oneris, quando abest conatus potentiæ illud attollentis aut suspendentis. Dubitandi ansam offert quædam munerum commutatio inter Potentiam, Pondus, & Clavum, si ad effectio- nes diversas referantur. Si enim oneris suspensio aut elevatio vi potentiæ ex adverso nitentis consideretur, Clavus exercet mu- nus Retinaculi: at si vim clavo illatam, ejusque inflexionem, aut revulsionem intueamur, efficientia vim hujusmodi inferens tri- buenda est aut gravitati oneris, aut impetui potentiæ trahentis: quapropter soliditas clavi inflexionem respuentis, aut ejus firma cohæsio cum pariete aut ligno, cui infixus est, vicem subit Pon- deris ope trochlearum movendi cum alterâ trochleâ connexi; Sarcina autem ex reliquâ trochleâ dependens aut retinaculi munus obtinet, si attollatur, aut Potentiæ vices subit, si deor- sum moveatur. Sit clavo A adnexa simplex Trochlea B, ejusque funis ductarius C D E: adnectatur in C sa- xum P, & à Potentia G elevatum sus- pendatur religato funis capite in E. Si saxum P accipiatur, quatenus elevatur, ipsum est Pondus, Clavus A est Retina- culum, & Potentia est G, sive illa sit inanima sua majore gravitate contrani- tens, sive sit vivens suo impetu sursum trahens, & postmodum remissiore impe- tu, & nervorum contentione impediens, ne saxum elevatum relabatur. At si ipsius Illi 3

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Book Six. CHAPTER VII. 621 the nail, resisting both by its solidity, bears the force of the power pressing downward in motion, and the weight opposing the upward motion. Just as therefore inanimate gravity, hanging by the rope of pulleys, suspends a burden or raises it, so living power, by holding the rope and by its impulse, makes equal the force of that same gravity, and exerts a similar force upon the nail; but by pulling the rope it overcomes that force of gravity, and by the force impressed upon it somehow diminishes it, yet in such a way that what seems to be taken away from gravity is understood to be added to the striving of the prevailing power. But for my part, why should I conceal it in vain? a not slight scruple and doubt is raised in me, whether the force imparted to the nail, from which the pulleys with the load are suspended, receives its measure precisely from the absolute weight of the load, when the effort of the power raising or suspending it is absent. An occasion for doubt is offered by a certain exchange of functions between Power, Weight, and Nail, if they are referred to different effects. For if the suspension or raising of the burden is considered with respect to the force of the power striving from the opposite side, the Nail performs the office of a Retainer; but if we look to the force imparted to the nail and to its bending or wrenching away, the efficient cause inflicting such force must be attributed either to the gravity of the burden or to the impetus of the pulling power. Wherefore the solidity of the nail, resisting bending, or its firm cohesion with the wall or wood into which it is fixed, takes the place of the Weight to be moved by means of pulleys when joined to another pulley; but the burden hanging from the remaining pulley performs either the office of the retainer, if it is raised, or takes the place of the Power, if it is moved downward. Let there be attached to the nail A a simple Pulley B, and its guiding rope C D E: let a stone P be attached at C, and let it be suspended, raised by the Power G, with the end of the rope tied at E. If the stone P is considered insofar as it is raised, it is the Weight; the Nail A is the Retainer; and the Power is G, whether that power be inanimate, resisting with its greater gravity, or living, drawing upward by its impetus, and afterward, with its gentler impetus and tension of the sinews, preventing the stone once raised from falling back. But if its

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Mechanicorum 622 clavi A pressio, sive inflexio consideretur, jam vis efficiendi pressionem hanc, seu inflexionem, tota tribuenda est saxo P, quod propterea inducit rationem Potentiæ, & retinaculum est paxillus E in terram firmiter depactus, qui nihil agit, sed funem duntaxat retinet. Verùm in hac positione momentum saxi adversùs clavum non est simplicis gravitatis absolutè acceptæ, perinde atque si funis F G infra orbiculum reflexus colligeretur in nodum cum fune DC: tunc enim, collecto in nodum fune, orbiculus esset planè otiosus, & nihil conferret ad momentorum varietatem, sed idem accideret, ac si funis simplex proximè & immediatè clavo adnecteretur sepositâ quacumque trochleâ. Sed fune in E religato (quasi duplex pondus ex simplici fune penderet) geminatur saxi P momentum adversùs clavum, qui nequit vel minimum flecti, quin duplo motu saxum ipsum moveatur: neque enim, quod ad geminandum momentum spectat, differt saxum à potentiâ vivente, quæ utique in C applicata funt, & trochleam trahens, adversùs pondus trochleæ adnexum habet momentum duplum ejus, quod obtineret, si funem simplicem traheret: est autem trochleæ adnexus clavus. Quod si in G contra saxum P aut gravitas inanimata, aut potentia vivens nitatur, si quidem æqualibus conatibus hinc & hinc certetur, atque suspensum consistat saxum, aut clavus similiter premitur atque libræ agina, cùm jugum à duobus æqualibus ponderibus in æquilibrio retinetur, aut alterutri munus Potentiæ, & alteri Retinaculi adscribendum est, & Potentia similiter geminato momento clavum trahit deorsum. Sin autem aut saxum P, aut virtus movendi in G, superat, huic Potentiæ ratio tribuatur, opposito munus retinaculi; sed Potentiæ absolutè acceptæ momenta non geminantur, quia retinaculum stabile non est, sed cedit; adeóque impetus à Potentia productus duos motus efficit, alterum trahendo retinaculum, alterum inflectendo clavum, qui propterea minùs flectitur, quò magis oppositum retinaculum movetur. Neque hæc quicquam habent admirationis: Nam si Vectis sit C D, habens in C hypomochlium; in medio autem puncto

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Mechanics 622 If the pressure, or bending, of the nail A is considered, the whole force producing this pressure, or bending, must be attributed to the stone P, which therefore supplies the role of Power, and the peg E firmly driven into the ground is the Retainer, which does nothing but merely holds the rope. But in this position the moment of the stone against the nail is not simply that of weight absolutely taken, just as if the rope F G, bent under the pulley, were gathered into a knot with the rope DC: for then, the rope being gathered into a knot, the pulley would be altogether idle, and would contribute nothing to the variation of the moments, but the same thing would happen as if the simple rope were tied to the nail immediately and directly, with any pulley set aside. But when the rope is fastened at E (as though a double weight hung from a single rope), the moment of the stone P against the nail is doubled; for the nail cannot bend even the least little bit without the stone itself being moved by a double motion: and indeed, so far as the doubling of the moment is concerned, the stone does not differ from the living power, which is certainly applied at C and, drawing the pulley, has against the weight attached to the pulley a moment double that which it would have if it drew the simple rope: but the nail is attached to the pulley. And if in G either an inert weight or a living power presses against the stone P, if indeed the contest on this side and that is equal, and the stone remains suspended, then either the nail is pressed similarly as the beam of a balance, when the beam is kept in equilibrium by two equal weights, or to one side the role of Power, and to the other that of Retainer, must be assigned, and Power likewise draws the nail downward with a doubled moment. But if either the stone P, or the force moving in G, prevails, let the role of Power be assigned to this one, and to the opposite side the office of retainer; but the moments of Power, absolutely taken, are not doubled, because the retainer is not stable, but yields; and thus the impulse produced by the Power causes two motions, one by drawing the retainer, the other by bending the nail, which therefore is bent less the more the opposite retainer is moved. Nor do these things involve anything to be wondered at: for if the Lever be C D, having its fulcrum at C; but in the middle point

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Liber sextus. CAPUT VII. 623 puncto E adnexus sit funiculus, qui incumbens orbiculo F versatili adnexum habeat pondus S innixum plano subjecto; utique in ex- tremitate D pondus V paulo majus quàm subduplum ponderis S illud ele- vabit, atque præcisè subduplum non elevabit quidem illud, sed adversùs orbiculum F conatur momento duplo ejus, quod obtineret, si ex E pende- ret ipsum pondus V, cui reluctaretur pondus S gravius innixum plano. Ve- rùm ad vectem retinendum in positio- ne horizontali C D nihil interest, utrùm in C aliquid superius sit prohibens, ne illa extremitas vi ponderis V attollatur, an verò inferiùs funiculo connectatur cum tellure, aut ex C pendeat onus H (sed plano subjecto in- nixum) vel æquale ipsi V, vel eo majus; semper enim pondus V eadem obtinet momenta. Quare si, amoto orbiculo F & pon- dere S, manu retineas funiculum IE, percipies ad servandum vectem horizontalem, quantâ virium accessione tibi opus sit, supra quàm exigeret simplex gravitas ponderis V, si ex E pen- deret, ubi nulla Vectis ratio intercederet. Cum itaque hæc in Vecte pariter ratione positionis pon- deris contingant, quæ trochleæ accidere diximus ratione connexionis ponderis vel cum trochleâ, vel cum paxillo telluri infixo, nil mirum si alia atque alia sint ejusdem pon- deris momenta adversùs clavum. Sicut autem quando tam ab hypomochlio quàm à potentiâ sustinetur onus in medio vecte suspensum, hypomochlium à pondere non premitur nisi juxta semissem gravitatis ponderis; ita quoque cum funis ductarius alterâ extremitate adnexus est clavo, alterâ retinetur à poten- tia, pondus ex trochleâ simplici pendens partim à Potentiâ, partim à clavo sustinetur, adversùs quem minus virium exercet ejus gravitas, ut constabit, si clavo orbiculum versatilem in- figas, & funiculo per orbiculi orbitam excavatam transeunti aliud pondus adnectas, quod satis erit, si fuerit subduplum ponderis ex trochleâ pendentis; hoc enim sustinebitur à duplici virtute subduplâ gravitatis illius. Non igitur plus resistentiæ

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Book Six. CHAPTER VII. 623 to point E let a cord be attached, which, resting on the movable pulley F, shall have attached to it the weight S resting on the subjacent plane; and thus at any rate at the extremity D the weight V, a little greater than the subduple of the weight S, will lift it, and the exact subduple certainly will not lift it, but will indeed strive against the pulley F with a moment double that which it would have if the weight V itself hung from E, to which the heavier weight S, resting on the plane, would oppose resistance. But for keeping the lever CD in a horizontal position, it makes no difference whether at C something above is placed to prevent that extremity from being raised by the force of the weight V, or whether below it is connected by a cord with the earth, or a load H hangs from C, resting on the subjacent plane, either equal to V itself or greater than it; for the weight V always obtains the same moments. Wherefore, if, the pulley F and the weight S having been removed, you hold the cord IE by hand, you will perceive how much additional force is needed to keep the lever horizontal, beyond what the simple weight of V would require if it hung from E, where no leverage would intervene. Since, then, these things happen in the lever as well by reason of the position of the weights as we have said occur in the pulley by reason of the connection of the weight either with the pulley or with the peg fixed in the earth, it is no wonder if the same weight has different moments against the nail. And just as when a load suspended in the middle of a lever is supported both by the fulcrum and by the power, the fulcrum is pressed by the weight only to the extent of half the weight; so also when the guiding cord is attached at one end to a nail and at the other is held by a power, a weight hanging from a simple pulley is supported partly by the power and partly by the nail, against which it exerts less force, as will be clear if you insert a movable pulley into the nail and attach to the cord passing through the hollow groove of the pulley another weight, which will suffice if it be subduple the weight hanging from the pulley; for this will be supported by a double virtue subduple to that weight's heaviness. Therefore there is not more resistance

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Mechanicorum 624 resistentiæ requiritur in clavo, quàm in pondere illo subduplo. His ita in unicâ simplici trochleâ constitutis, examinandæ sunt trochleæ conjugatæ; nec difficile erit ex dictis superiore capite investigare momenta ponderis adversùs clavum, cui altera trochlea adnectitur. Ibi enim alteri trochleæ potentiam sursum trahentem, alteri pondus dependens adnecti posuimus, funis verò extremitatem clavo alligari: Hîc loco clavi illius retinentis extremitatem funis intelligendum est retinaculum, quodcumque tandem illud sit, sive manus hominis, sive etiam alius clavus: sed loco Potentiæ superiorem trochleam sursum trahentis sit clavus, ex quo trochleæ fune ductario connexæ unà cum pondere dependent; gravitas autem illa suspensa ex inferiore trochleâ exercet munus potentiæ adversùs clavum, qui subit vicem ponderis movendi, quatenus aliquantulum flectitur, aut inflexioni repugnat. Sicut ergo ibi ostensum est in duabus simplicibus trochleis singulos orbiculos habentibus, si funis ductarij caput alligatum sit superiori trochleæ, motum trochleæ superioris ad motum inferioris esse ut 2 ad 3; si verò funis caput alligatum fuerit inferiori trochleæ, motum superioris ad motum inferioris esse ut 3 ad 2: Ita hîc dicendum est (ponamus clavum flecti aliquantulum) in primo casu motum flexionis clavi ad motum descensûs ponderis esse ut 2 ad 3, in secundo autem casu ut 3 ad 2. Ex quo fit in primo casu pondus habere adversùs clavum majus momentum quàm in secundo casu; & in primo casu validiùs deorsum trahere, quàm si simplici funiculo dependeret, & motus essent æquales; major siquidem est Ratio 3 ad 2, quàm 2 ad 2; in secundo verò casu debiliùs deorsum trahere, quàm si nullæ essent trochleæ, adeóque motus æquales essent; minor quippe est Ratio 2 ad 3, quàm 1 ad 1, aut 3 ad 3. Simili planè methodo philosophandum est in reliquis trochleis conjugatis: si enim duabus trochleis dispar insit orbiculorum numerus, ut altera major sit, altera minor, observandum est, an major trochlea alligetur clavo, an verò minor: Si trochlea plures habens orbiculos clavo adnectatur, motus flexionis clavi minor est motu descensûs ponderis in Ratione subsuperparticulari denominatâ à numero omnium simul orbiculorum; ac

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Mechanics 624 the resistance required in the nail is greater than in that half-sized weight. These things being thus established in the single simple pulley, the combined pulleys must be examined; and it will not be difficult, from what has been said in the preceding chapter, to investigate the moments of the weight against the nail to which the other pulley is attached. For there we set the power, drawing upward, attached to one pulley, the suspended weight attached to the other, and the end of the rope tied to the nail: here, in place of that nail retaining the end of the rope, the retainer must be understood, whatever it may be, whether a man’s hand, or even another nail: but in place of the Power drawing the upper pulley upward, there let there be the nail, from which the pulleys connected by the guiding rope hang together with the weight; and that suspended gravity, hanging from the lower pulley, performs the office of the power against the nail, which takes the place of the moving weight, insofar as it is somewhat bent, or resists bending. Therefore, as it was shown there in two simple pulleys, each having a single groove, if the end of the guiding rope is tied to the upper pulley, the motion of the upper pulley is to the motion of the lower as 2 to 3; but if the end of the rope is tied to the lower pulley, the motion of the upper to the motion of the lower is as 3 to 2: so here it must be said, let us suppose the nail to be bent somewhat, that in the first case the motion of the bending of the nail is to the motion of the descent of the weight as 2 to 3, but in the second case as 3 to 2. Whence it follows that in the first case the weight has against the nail a greater moment than in the second case; and in the first case it pulls downward more strongly than if it were hanging from a simple cord, and the motions were equal; for the ratio 3 to 2 is greater than 2 to 2. But in the second case it pulls downward more weakly than if there were no pulleys at all, and therefore the motions were equal; for the ratio 2 to 3 is less than 1 to 1, or 3 to 3. In a similar and indeed identical method one must philosophize about the remaining combined pulleys: for if in two pulleys there is an unequal number of grooves, so that one is larger and the other smaller, it must be observed whether the larger pulley is attached to the nail, or rather the smaller. If the pulley having more grooves is attached to the nail, the motion of the bending of the nail is less than the motion of the descent of the weight in the sub-superparticular ratio named from the total number of grooves taken together; and

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Liber sextus. CAPUT VII. 625 ac proinde pondus habet momentum majus, quàm si nullæ in- tercederent trochleæ: contra verò si clavo adnectatur trochlea minor, motus flexionis clavi major est motu descensûs ponde- ris in Ratione superparticulari denominatâ à numero omnium simul orbiculorum; atque ideo pondus adversùs clavum minus habet momenti, quàm si ex illo simplici fune penderet. At si utriusque trochleæ par sit orbiculorum numerus, & pariter ra- tio superparticularis, aut subsuperparticularis denominata à numero omnium simul orbiculorum; & si quidem trochleæ su- periori adnectatur funis caput, pondus adversùs clavum habet momentum majus, quàm si amotis trochleis ex simplici fune penderet; sin autem inferiori trochleæ alligetur extremitas fu- nis ductarij, ponderis momentum adversùs clavum minùs est, quàm si idem pondus ex eodem clavo simplici fune sus- penderetur. Ex his satis apparet clavo eandem vim inferri, si pondus de- pendens ex trochleis suspensum maneat, sive quia funis extre- mitas religetur paxillo, sive quia ex eâdem funis extremitate dependeat onus submultiplex ponderis ex inferiore trochleâ pendentis, secundùm Rationem, quam inferunt ipsi orbiculi. Sic ex trochleis binos orbiculos habentibus dependeat pondus, & funis extremitas religetur paxillo: ex dictis, superioris tro- chleæ clavo adnexæ, & funis ductarij caput habentis, motus, ad motum inferioris trochleæ & ponderis est subsesquiquartus; ac proinde pondus trochleis connexum cum clavo ad vim illi inferendam perinde se habet, atque si ex eodem clavo absque trochleis simplici fune appenderetur pondus aliud dati ponde- ris Sesquiquartum. At si extremitati funis adderetur pondus va- lens suspendere onus adnexum trochleæ inferiori, esset ex dictis cap. 1. dati oneris subquadruplum. Igitur duorum horum pon- derum summa ad datum pondus esset ut 5 ad 4, cujusmodi erat Ratio motuum, ex quibus momentum desumitur. An non si onus in plano horizontali raptandum simplici trochleæ adne- xum proponatur, & duo homines pariter utramque funis ex- tremitatem arripiant, atque trahant, singuli medietatem ne- cessarij conatûs adhibent? si verò alter trahentium deficiat, & illa funis extremitas alligetur paxillo, nonne qui reliquus est eodem conatu trahens solus adducet idem pondus? non nisi K K k k

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Book six. CHAPTER VII. 625 and therefore it has a greater moment of weight than if no pulleys intervened: on the other hand, if a smaller pulley be attached to the nail, the motion of the nail’s bending is greater than the motion of the descent of the weight, in the superparticular ratio named from the number of all the little wheels together; and therefore the weight against the nail has less moment than if it hung from that simple rope. But if the number of little wheels in each pulley be equal, and likewise the superparticular, or subsuperparticular, ratio named from the number of all the little wheels together; and if indeed the end of the rope be attached to the upper pulley, the weight against the nail has a greater moment than if, the pulleys being removed, it hung from a simple rope; but if the end of the guiding rope be tied to the lower pulley, the moment of the weight against the nail is less than if the same weight were suspended from the same nail by a simple rope. From these things it is sufficiently clear that the same force is brought to bear on the nail, if the weight hanging from the pulleys remain suspended, whether because the end of the rope is tied to a peg, or because from the same end of the rope there hangs a load submultiple of the weight hanging from the lower pulley, according to the ratio which the little wheels themselves introduce. Thus, if a weight hang from pulleys having two little wheels each, and the end of the rope be tied to a peg: from what has been said, the motion of the upper pulley attached to the nail, and having the end of the guiding rope, is to the motion of the lower pulley and weight as sub-sesquiquartal; and therefore the weight connected with the pulleys, insofar as it is joined to the nail in order to exert force on it, behaves just as if another weight, the sesquiquartal of the given weight, were hung from the same nail by a simple rope without pulleys. But if there were added to the end of the rope a weight sufficient to suspend the load attached to the lower pulley, it would, from what has been said in chapter 1, be the subquadruple of the given load. Therefore the sum of these two weights to the given weight would be as 5 to 4, which was the ratio of the motions, from which the moment is derived. Is it not the case that if a load to be dragged along a horizontal plane be proposed, attached to a simple pulley, and two men alike seize each end of the rope and pull, each applies half of the necessary effort? but if one of the pullers fail, and that end of the rope be tied to a peg, will not the one remaining, pulling with the same effort, draw the same weight alone? not unless K K k k

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626 Mechanicorum quia, cùm ambo trahebant, pondus & potentia æqualiter mo- vebantur; cùm alter tantùm trahit, ille movetur duplo velo- cius, quàm pondus, quod ad subduplam velocitatem satis ha- bet impetum subduplum impetûs necessarij ad velocitatem æqualem. Eadem igitur militat ratio in clavo, cui vis infertur à duobus ponderibus suspensis ex trochleis in æquilibrio, quæ simul deorsum trahentia motum habent æqualem cum motu clavi, qui flectitur, aut revellitur; sed funis capite religato fir- miter ad paxillum, pondus inferiori trochleæ adnexum motum habet velociorem comparatum cum ejusdem clavi motu, ac propterea majus momentum habet. Hinc præterea inferendum est non satis utiliter eos opera- ri, qui pondus ex superiore loco fune suspensum, sive orbicu- lus intercedat, sive non, putant firmius sustineri, si funis ca- put in inferiore loco religetur: si enim funis excurrere ne- queat, inferiùs hoc retinaculum prorsus inutile accidit, sin au- tem excurrere valeat, superius illud retinaculum geminatam vim suscipit, quasi duplex pondus ab illo sustineretur. CAPUT VIII. Aliqui Trochlearum usus indicantur. PRo more in superioribus libris servato, hîc pariter indican- di sunt aliqui Trochlearum usus, qui facilè ad similia traduci poterunt, spectato motu, qui exhibendus proponitur, ut ei trochleæ respondeant, & aptè collocentur, neque plu- ribus, quàm opus sit, orbiculis instruantur; ne dum poten- tiæ facilitatem consectaris, nimis tardè moveas pondus, aut ex adverso, dum ponderi velocitatem concilias, nimio labore potentiam opprimas. PROPOSI

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626 Mechanics because, when both were pulling, the weight and the force were moved equally; when only one pulls, that one is moved twice as fast as the weight, which at a subdouble speed has enough impetus, namely half the impetus needed for equal speed. The same reasoning therefore applies to the nail, on which force is exerted by two weights suspended from pulleys in equilibrium, which, while pulling downward together, have motion equal to the motion of the nail, which bends or is drawn out; but if the rope, with its end firmly tied to a peg, is attached to the lower pulley, the weight has a faster motion compared with the motion of that same nail, and therefore has greater momentum. Hence it must also be inferred that those do not act very usefully who think that a weight suspended by a rope from a higher place, whether a pulley intervenes or not, is more firmly supported if the end of the rope is tied in a lower place: for if the rope cannot run out, this lower fastening is wholly useless; but if it can run out, that upper fastening takes on a doubled force, as though a double weight were being supported by it. CHAPTER VIII. Some uses of pulleys are indicated. As was the custom preserved in the preceding books, here too some uses of pulleys must be indicated, which can easily be adapted to similar cases, regard being had to the motion that is proposed to be produced, so that the pulleys may correspond to it and be suitably placed, and not furnished with more wheels than is necessary; lest, while you pursue ease of power, you move the weight too slowly, or, on the other hand, while you give speed to the weight, you oppress the power with excessive labor. PROPOSI

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Liber sextus. CAPUT VIII. 627 PROPOSITIO I. Auram in Conclavi excitare. Quæritur sæpè æstivo tempore aliquod ex aëris motu re- frigerium; sed manuali flabello auram excitare aliquan- do incommodum est, si aliud agendo distinearis: propterea ventilabrum in conclavis angulo statuere possumus, quod ali- quandiu moveatur, aëremque agitet: ideóque illud ad angu- lum statuendum proposui, ut commotus aër in proximos hinc atque hinc parietes impactus reflectatur, & faciliùs reliquum conclavis aërem exagitet. Excitetur angulo congruens turricula haud absimilis iis, quibus horologia reconduntur; in supremâ turriculæ parte ab angulo ad oppositum ex diametro angulum Axis horizon- ti parallelus statuatur facilè versatilis, cujus tamen pars ex- tra turriculam promineat tantæ longitudinis, quanta flabel- lis latitudo destinatur. Pars tamen hæc Axis extima nul- lam exigit certam figuram, nihilque refert sive cylindrica sit, sive quadrata, sive quæcumque alia; modò ea sit, ut illi facilè flabella firmiter infigi, atque eximi pro opportunitate possint, iisque exemptis aptari valeat manubrium, quo faciliùs & citiùs ab homine convolvatur Axis. Parentur duæ trochleæ ternis orbiculis instructæ, altera in superiore turriculæ loco firmetur, altera ad turriculæ pe- dem constituatur adnexam habens plumbeam massam motui perficiendo congruentem: huic eidem trochleæ adnectatur extremitas funis ductarij, qui per omnes trochlearum orbicu- los trajectus demum ad Axem referatur, ibique alligetur. Tum apposito manubrio convolutum Axem circumplectetur funis ductarius, & plumbea massa in supremam turriculæ partem Axi proximam deducetur. Insigantur Axi ventilabra, & amo- to manubrio plumbea massa sibi relicta lentissimo motu descen- det, convolvensque axem cum flabellis tandiu aërem commo- vebit, quandiu illa descendet. Hic commutatas vices inter potentiam & pondus observare quilibet potest; potentia siquidem movens est plumbea mas- KKkk 2

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Book Six. CHAPTER VIII. 627 PROPOSITION I. To stir up a breeze in a chamber. It is often desired in summer time to obtain some refreshment from the movement of the air; but to stir up a breeze with a hand fan is sometimes inconvenient if one is engaged in something else. Therefore we may place a ventilator in the corner of a room, which may be kept in motion for some time and agitate the air; and for this reason I proposed that it should be set in the corner, so that the moved air, striking against the nearby walls on either side, may be reflected, and more easily stir up the rest of the air in the room. Let there be made in the corner a suitable little tower, not unlike those in which clocks are enclosed; in the upper part of the little tower, from the corner to the opposite corner across the diameter, let a horizontal axis be set, easily turnable, a part of which should project outside the tower to a length equal to the intended width of the fans. This outer part of the axis, however, does not require any definite shape; it matters not whether it be cylindrical, square, or any other form, provided only that the fans may easily be firmly fixed into it and removed as needed, and that when they are removed a handle may be fitted to it, by which the axis may be more easily and quickly turned by a person. Let two pulleys be prepared, each fitted with three wheels; let one be fixed in the upper part of the tower, and the other be set at the foot of the tower, having attached to it a leaden weight suitable for accomplishing the motion. To this same pulley let the end of a driving cord be attached, which, after passing through all the wheels of the pulleys, is finally brought back to the axis and tied there. Then, a handle being applied, the driving cord will wind around the axis, and the leaden weight will be drawn up to the top of the tower near the axis. Let the ventilators be fastened to the axis, and, the handle being removed, the leaden weight, left to itself, will descend with a very slow motion, and, winding the axis with the fans, will stir the air for as long as it descends. Here anyone may observe the exchanged turns between power and weight; for the moving power is the leaden mas-

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Mechanicorum sa, quæ septuplo tardiùs movetur quàm extremitas funis ductarij, quem aliàs trahere solita est potentia. Loco autem ponderis est aër, qui à flabellis impellitur; ac proinde quò ampliora sunt flabella, eò major est resistentia aëris com- moti, ratione cujus etiam retardatur motus potentiæ. Ubi quoquè attendenda est Ratio longitudinis flabellorum ad se- midiametrum axis convoluti: nam si hæc Ratio componatur cum Ratione septuplâ, quam Trochleæ inferunt, habebitur Ratio motûs extremi flabelli ad motum massæ plumbeæ: quamquam non ita computanda est aëris resistentia, quasi to- ta in flabelli extremitate exerceretur; hæc scilicet per univer- sam flabelli longitudinem diffunditur inæqualiter distributa pro Ratione distantiæ à centro motûs, aër quippe pro diversâ impellentis velocitate inæqualiter resistit. Quod si magis arrideret non continua convolutione flabel- la circumagi, sed alternâ quadam modò in dextram, modò in sinistram inflexione agitari; Axi, quem funis ductarius complectitur, infige rotam dentatam, cujus dentes incurrant in pinnulas fusi perpendicularis flabella sustinentes, quemad- modum in Tempore horologij: simili enim ratione, ac Tem- pus, ultrò citróque remeabunt flabella, & aërem in oppositas partes commovebunt. Cùm verò antè motum apposito manu- brio convolvendus erit Axis, ut funem ductarium recipiat, at- que trochlea inferior cum pondere attollatur, ita fusum pau- lisper elevare oportebit, ut ejus pinnulæ non occurrant denti- bus rotæ, nisi cùm iterum fusus suum in locum restituetur. Hac alternatione diuturnior erit motus. PROPOSITIO II. Corpus aliquod in gyrum celeriter volvere. IN rebus scenicis locum habere non infrequentem potest hæc propositio: aliquando scilicet solis discum in scenam producimus, quem licet auro obductum, ac multis facibus illustratum, quas spectatorum oculis ex arte subducimus, non tamen radios ejaculantem mentimur, nisi ille circa suum

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Mechanics sa, which moves seven times more slowly than the extremity of the rope that draws it, which by another means it is accustomed to pull by its power. In place of weight there is air, which is driven by the vanes; and therefore the larger the vanes are, the greater is the resistance of the air set in motion, by reason of which the motion of the power is also retarded. Here also the ratio of the length of the vanes to the semi-diameter of the wound axis must be considered: for if this ratio be compounded with the sevenfold ratio which the pulleys produce, there will be had the ratio of the motion of the outermost vane to the motion of the leaden weight: although the resistance of the air is not to be computed as though it were exerted entirely at the extremity of the vane; rather, it is diffused through the whole length of the vane, unevenly distributed in proportion to the distance from the center of motion, for air resists unevenly according to the different velocity of the impeller. But if it should seem preferable not that the vanes be turned by continuous winding, but that they be driven by an alternate bending, now to the right, now to the left, fix upon the axis which the driving rope embraces a toothed wheel, whose teeth may engage the little pins of the perpendicular spindle supporting the vanes, as in a clock. For in a similar way, like Time, the vanes will go back and forth, and will move the air in opposite directions. But when before the motion the axis must be wound by the attached handle, so that it may receive the driving rope, and the lower pulley be raised with the weight, then the spindle must be lifted a little, so that its pins do not meet the teeth of the wheel except when the spindle is again restored to its place. By this alternation the motion will be longer sustained. PROPOSITION II. To cause some body to revolve rapidly in a circle. In stage works this proposition may have frequent use: namely, sometimes we bring onto the stage the disk of the sun, which, though covered with gold and illuminated by many torches, which we artfully conceal from the spectators’ eyes, we nevertheless do not falsely represent as darting rays, unless it revolves around its own

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Liber sextus. CAPUT VIII. 629 suum centrum velociter circumagatur. Id quod variâ qui- dem methodo præstari potest infixum disci centro cylindrum convolvendo, sive ope rotæ dentatæ Vertebram striatam cy- lindro circumpositam moventis; sive fune cylindrum bis aut ter arctè complexo, & in sese redeunte, ubi majoris alicu- jus tympani orbitam pariter complexus fuerit; sive pondere funem cylindro involutum explicante: sed postremus hic modus non nisi breve temporis spatium exigit; duo priores, si paulo longior futurus sit motus, non nisi à potentiâ vi- vente commodè exhiberi possunt. Quare satius fuerit tro- chleas, ut in superiore propositione, dispositas adhibere, at- que loco flabellorum solis discum Axi adnectere; sic enim fiet, ut & celeriter in gyrum agatur, & diu perseveret motus. Similiter ad fingendum mare, & undarum motum vehe- mentiorem, statuuntur horizonti & invicem paralleli ali- quot axes, quos ambiunt spiræ profundiùs excavatæ colore marinam undam imitantes: dum enim hujusmodi axes con- volvuntur, marini æstûs cursum spectatoribus repræsentant. Ut autem axes illi citra cujusquam laborem volvantur tro- chleas duas binis, aut ternis orbiculis instructas (prout diu- turnior motus requiritur) compone, & proximas statue, al- teram firmans in superiore loco: Tum funis ductarius per omnes Trochlearum orbiculos trajectus singulorum axium ca- pita ex ordine ambiat unâ saltem aut alterâ spirâ, & demum ad peculiarem alium axem deveniat, quem totus plures in spiras complicatus circumplectatur, ita tamen, ut facilè evolvi queat. Ubi igitur tempus advenerit, inferiori tro- chleæ congruum pondus adnecte; hoc enim licèt lentè de- scendat, velociter tamen axes convolvit funem evolvens. Procellam verò mitescere aut exasperari mentieris, factâ ponderis aliquâ detractione aut accessione, id quod difficile non fuerit. KKkk 3

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Book Six. CHAPTER VIII. 629 its center may be rapidly turned around. This may indeed be accomplished by various methods: by winding a cylinder fixed in the center of the disk, either by means of a toothed wheel moving a ridged spindle placed around the cylinder; or by a cord tightly wrapped twice or thrice around the cylinder and returning upon itself, after it has likewise encircled the circumference of some larger drum; or by a weight unwinding a cord wound around the cylinder. But this last method requires only a short space of time; the first two, if the motion is to continue somewhat longer, can conveniently be effected only by living force. Wherefore it will be better to employ pulleys, as in the preceding proposition, arranged for the purpose, and in place of fans to attach the disk of the Sun to the axle; for thus it will be made to revolve quickly and the motion will continue for a long time. In like manner, to represent the sea and the more violent motion of the waves, several axes are set up parallel both to the horizon and to one another, around which there pass spirals more deeply cut and colored to imitate the marine wave: for when such axes are wound up, they represent to the spectators the course of the sea tide. But that those axes may be turned without anyone’s labor, arrange two pulleys furnished with two or three little wheels each, as a longer-lasting motion requires, and place them close together, fixing one in the upper position. Then let the guiding cord, passed through all the pulleys’ little wheels, embrace in order the ends of the several axes with at least one or two turns, and finally come to some other special axis, which, being wound about with many turns, it may nevertheless easily unwind. When therefore the time shall come, attach a suitable weight to the lower pulley; for although this descends slowly, it nevertheless quickly winds up the axes while paying out the cord. And you will feign that the storm is calming or growing more violent by making some subtraction from, or addition to, the weight, which will not be difficult. KKkk 3

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Mechanicorum PROPOSITIO III. Se ipsum ope trochlearum in altum evehere, aut promovere. Sella paretur hinc & hinc habens fulcra, quibus brachia Sinnituntur, & in hujusmodi fulcrorum extremitate ante- riore aptetur Sucula manubriata, quam sedens commodè ver- fare valeat: sella autem quatuor funibus in nodum cum an- nulo coëuntibus suspendatur ita, ut inferioris trochleæ uncus annulo indatur, & funis ductarius per cunctos trochlearum orbiculos trajectus demum suculæ alligetur. Nam in sellâ se- dens, & suculæ manubria convertens, funem ductarium trahit, atque ipse se in altum evehit eâ facilitate, quam in- fert Ratio composita ex Rationibus trochlearum, & Suculæ: est siquidem Potentia ipsa virtus animalis musculorum con- tentione versans manubria, pondus autem est insita corpori gravitas, quæ eò minor apparet, quo majores sunt, hoc est pluribus instructæ orbiculis, trochleæ, & major est Ratio manubriorum ad semidiametrum Axis, qui fune obvolvitur. Sit enim ex. gr. inferior trochlea, cui pondus movendum ad- nectitur, & funis ductarij extremitas alligatur, orbiculorum duorum; superior autem trochlea, quæ stabilis manet, tres habeat orbiculos: utique Ratio motûs potentiæ ad motum ponderis est quintupla: manubria autem Suculæ sint quadru- pla semidiametri Axis: Ratio composita ex quadruplâ & quin- tuplâ est vigecuplâ; igitur conatus Potentiæ manubria versan- tis satis est, si respondeat vigesimæ parti ponderis. Similiter si cymba adverso flumine non procul à ripâ dedu- cenda sit, & qui in eâ sunt nautæ, ita pauci sint, ut non va- leant eam adversùs vim profluentis remo agere, aut è ripâ fu- ne nautico trahere; subsidium ex trochleis petere poterunt; exscensu scilicet in terram facto, atque defixo in ripâ paxil- lo alligatur trochlea una, altera adnectitur proræ cymbæ, in qua nautæ duo funem ductarium trahentes illam adverso flumine promovent perinde, atque si essent octo aut duode- cim homines, si trochleæ binos aut ternos habuerint orbicu- los.

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Mechanics PROPOSITION III. To raise oneself upward, or advance oneself, by means of pulleys. Let a seat be prepared, having supports on this side and that, on which the arms may rest; and at the forward end of such supports let a hand-cranked winch be fitted, which the seated person may conveniently turn. The seat, however, is to be suspended by four ropes meeting in a knot with a ring, so that the hook of the lower pulley may be inserted into the ring, and the hauling rope, passed through all the sheaves of the pulleys, may at last be attached to the winch. For, while seated in the chair and turning the handles of the winch, one draws the hauling rope, and thus lifts oneself upward with that ease which arises from the ratio composed of the ratios of the pulleys and the winch; for the Power is the animal force itself, turning the handles by muscular effort, while the weight is the heaviness inherent in the body, which appears the smaller, the greater are the pulleys, that is, the more sheaves they are furnished with; and the greater is the ratio of the handles to the semidiameter of the axis around which the rope is wound. Let, for example, the lower pulley, to which the weight to be moved is attached and to which the end of the hauling rope is tied, have two sheaves; and let the upper pulley, which remains fixed, have three sheaves: plainly the ratio of the motion of the power to the motion of the weight is fivefold; and if the handles of the winch are four times the semidiameter of the axis, the ratio composed of four and five is twentyfold; therefore the effort of the Power turning the handles is sufficient if it equals the twentieth part of the weight. Likewise, if a boat must be drawn upstream near the bank, and if the sailors in it are so few that they cannot move it against the force of the current with oars, nor pull it from the bank by a rope attached to the boat; they may seek help from pulleys. For after going ashore, and fixing a post in the bank, one pulley is tied to it, another is attached to the prow of the boat, and by pulling the hauling rope two sailors in the boat move it upstream just as if there were eight or twelve men, if the pulleys had two or three sheaves.

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Liber sextus. CAPUT VIII. 631 los. Quod si trochleis illi careant, utantur artificio sequentis propositionis, unà cum iis, quæ cap. 5. dicta sunt. PROPOSITIO IV. Trochlearum defectum supplere. EX his, quæ cap. 2. hujus libri indicata sunt, satis constat etiam sinè orbiculis haberi posse momentum Trochlearum: quare his deficientibus annulos sufficere facile erit. Et primo quidem singulis annulis uti possumus: nam cùm cymba communiter adnexum proræ annulum habeat, ut medio fune, aut catenâ ad ripam religetur, funis ductarius unus A B adnectatur paxillo A in ripâ defixo, & per cymbæ annulum B trajiciatur; illius alteri extremitati C annulus alius adnectatur, per quem alter ductarius funis D E F trajectus & paxillo D alligatus si à Potentiâ in F constitutâ trahatur, illa habebit momentum quadruplum, perinde atque de orbiculis superiùs dictum est cap. 5. At si consistentes in cymbâ trahere illam velint nautæ, adjiciatur paxillo D annulus G stabilis, per quem productus funis E F transeat, & veniat in H ad nautarum manus in cymbâ; nam illum trahendo cymbæ prora ex B accedet ad A. Porrò annuli nomine notatum volo quicquid ejusmodi est, ut funis per illud trajici possit, & liberè excurrere, sive sit ligni frustum foramen habens politum & satis amplum, ut per illud funis facilè moveri valeat, sive etiam sit flexilis bacilli

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Book Six. CHAPTER VIII. 631 If they lack pulleys, let them use the method of the following proposition, together with those things that were said in chapter 5. PROPOSITION IV. To make up for the lack of pulleys. From what has been indicated in chapter 2 of this book, it is sufficiently clear that the force of pulleys can also be obtained without sheaves; therefore, if these are lacking, it will be easy to substitute rings. And first, indeed, we may use single rings: for since a boat commonly has attached to its prow a ring, in order that it may be tied by a middle rope, or chain, to the bank, let one guide rope A B be attached to peg A fixed in the bank, and passed through the boat’s ring B; to the other end C of this, let another ring be attached, through which let the other guide rope D E F be passed and tied to peg D. If this is pulled by the Power placed at F, it will have quadruple force, just as was said above in chapter 5 about pulleys. But if the sailors remaining in the boat wish to pull it, let a fixed ring G be added to peg D, through which let the extended rope E F pass, and let it come to H in the hands of the sailors in the boat; for by pulling it the prow of the boat will move from B toward A. Moreover, by the name ring I wish to designate whatever is of such a kind that a rope can be passed through it and run freely, whether it be a piece of wood with a hole, polished and sufficiently wide so that a rope may easily move through it, or even a flexible little rod

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Mechanicorum 632 bacilli particula in arcum vel modicè sinuata; modò illa non sit fractioni obnoxia. Illud autem in annulis observandum est, quod faciliùs excurrit funis, si illi crassiores fuerint & po- liti, quàm si exiles & asperi. Quod si annulis Trochleas propiùs æmulari placuerit, duos annulos R & S alliga paxillo V, duósqe alios H & G ad- necte in M ponderi trahendo: Tum funem ductarium eidem paxillo V alligatum trajice primùm per annulum G, deinde per annulum S, hinc per annulum H, demum per annulum R. Nam si extremitati I potentia trahens applicetur, movebitur quadruplo velociùs, quàm pondus in M, adeóque etiam ha- bebit momentum quadruplum. Ne autem funes ob nimiam propinquitatem sibi invicem impedimento sint se mutuo con- flictu atterentes, annulos transversis bacillis ON, & LP dis- junge. PROPOSITIO V. Resistentiam ex axium cum orbiculis conflictu in Trochleis examinare. Quoniam variarum Trochlearum usum indicavimus, mo- dò trochleâ alterâ manente atque stabili, modò utra- que commotâ, placet hìc examinare propositas duas tro- chleas, an aliquid impedimenti afferant ex conflictu axium cum orbiculis, aut etiam trochleas comparare cum annulis earum

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Mechanics 632 a small piece of a rod bent into an arc, or moderately curved, provided it is not liable to break. But this must be observed in rings, that the cord runs more easily if they are thicker and polished, than if they are slender and rough. But if it should please one to imitate pulleys more closely by means of rings, tie two rings R and S to the peg V, and attach two others H and G at M to the weight to be drawn. Then pass the guiding cord, tied to the same peg V, first through the ring G, then through the ring S, thence through the ring H, and finally through the ring R. For if the moving force is applied to the end I, it will move four times as fast as the weight at M, and thus will also have fourfold momentum. But lest the cords, by their too great proximity, should be an impediment to one another, rubbing together by mutual collision, separate the rings by transverse rods ON and LP. PROPOSITION V. To examine the resistance arising from the collision of the axles with the wheels in pulleys. Since we have indicated the use of various pulleys, now with one pulley remaining and fixed, now with both in motion, it is pleasing here to examine the proposed two pulleys, whether they bring any hindrance from the collision of the axles with the wheels, or even to compare pulleys with their rings

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Liber Sextus. CAPUT VIII. 633 earum loco adhibitis, quantum videlicet præ trochleis afferat impedimenti conflictus funis ductarij cum annulis. Sit libræ jugum A B æqualium brachiorum aginam cum examine habens in C; adnectatur in A trochlea superior, ex qua cum inferiore trochleâ pendeat saxum F notæ gravitatis, & funis ductarij extremitas religetur clavo in E. Innotescat tum trochlearum singularum, tum funis ductarij gravitas, ut congruum pondus parari possit in B appendendum. Clavus igitur E sustinet inferioris trochleæ & adnexi saxi gravitatis partem quintam, reliquas quatuor quintas partes, & præterea trochleæ superioris, atque quatuor ductuum funis gravitatem sustinet brachium libræ in A. Quare in B tantum ponderis apponendum est, quantum sufficiat ad æquilibrium; proinde sensim augendum est pondus in D, donec examen in C æqualitatem momentorum indicet. Hoc peracto adde adhuc ponderi D aliam atque aliam gravitatem, usque dum brachium B deorsum inclinetur: hujusmodi enim additamentum indicabit resistentiam ortam ex conflictu axium cum orbiculis. Jam si trochlearum loco annulos substituas, eademque methodo invento primùm æquilibrio, deinde factâ in D ponderis accessione præponderantiam quæras, deprehendes, quanto major resistentia ex funis ductarij cum annulis affrictu oriatur, quam ex axium cum suis orbiculis conflictu in trochleis. LLII

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Book Six. Chapter VIII. 633 using rings in their place, namely to what extent the conflict of the guide rope with the rings adds to the impediment in comparison with pulleys. Let the beam A B of equal arms be balanced with the scale-beam in C; let an upper pulley be attached at A, from which with the lower pulley there hang a stone F of known weight, and let the end of the guide rope be fastened by a nail at E. Let there be known both the weight of each pulley and the weight of the guide rope, so that a suitable weight may be prepared to be hung at B. Therefore the nail E supports one-fifth of the lower pulley and of the attached stone’s weight, the remaining four-fifths, and in addition the weight of the upper pulley and of four lengths of the rope, the beam arm at A sustains. Wherefore at B only so much weight must be added as is sufficient for equilibrium; accordingly the weight at D must be gradually increased until the scale-beam at C indicates equality of moments. This done, add still another and another weight to D, until the arm B inclines downward: for such an addition will indicate the resistance arising from the conflict of the axles with the wheels. Now if you substitute rings in place of the pulleys, and by the same method first find the equilibrium, then after adding weight at D seek the preponderance, you will discover how much greater resistance arises from the friction of the guide rope with the rings than from the conflict of the axles with their wheels in pulleys. LLII

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Mechanicorum 634 Simili ratione si in A sit clavus, cui superior trochlea stabili- lis permanens affigatur; extremitas verò funis ductarij M ad- nectatur brachio libræ ad æquilibrium constituendum sufficit in N quinta pars gravitatis trochleæ inferioris unà cum saxo F, & ductu funis M I. Facto igitur in H additamento gravitatis, ut tollatur æquilibrium, indicabitur quanta resistentiæ accessio fiat ponderi F ex axium cum orbiculis conflictu: atque simili- ter repositis loco trochlearum annulis, post æquilibrium aucto pondere N donec deprimatur, innotescet resistentia orta ex fu- nis cum annulis affricitu. Hinc apparet primò satius esse hoc posteriore modo ope- rari, quia longè minus pondus requiritur in N, quàm in D. Secundò ad tollendum pondus F cum trochleâ inferiore, si superior fixa maneat, tantam vim in potentiâ requiri, quan- tâ opus esset ad attollendum absque ullâ machinâ pondus N præponderans; ad attollendum verò idem saxum F cum utrâque trochleâ, trahendo scilicet sursum trochleam A, tantam vim exigi in potentia, quanta requiritur ad attol- lendum pondus D præponderans. Tertiò, retentis iisdem tro- chleis, sed mutato pondere F, examinari posse, an, & quan- to major resistentia oriatur ex majore pressione axium, quando pondus est majus. Quartò. mutatis trochleis, & pon- dere eodem retento, disparitatem aliquam inveniri, quia non omnium trochlearum axes sunt æquè teretes, ac politi, & suorum orbiculorum foramini congruentes. Quod si, exa- mine hujusmodi semel instituto, orbiculos manu paulisper convertas, & iterum idem examen instituas, neque æqua- lis inveniatur resistentia, indicium erit foramen orbiculi, aut fortasse etiam axem, non esse, exquisitè rotundum. Quintò. simili examine in annulis inito deprehendi posse, an faciliùs succedat tractio fune crassiore, an verò te- nuiore. Non ita tamen necesse est indicatâ methodo uti, ut, si non placeat jugum libræ æqualium brachiorum adhibere, nequeas loco libræ stateram applicare ut trochleæ A, aut funis extremitati M: primùm enim indicabitur æquili- brium: deinde longiùs reducto sacomate, usque dum appa- rere incipiat præponderatio, innotescet quantitas impedimen- ti,

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Mechanics 634 In a similar way, if in A there is a nail to which the fixed upper pulley is attached; and the end of the driving rope M is joined to the arm of the balance so as to establish equilibrium, it will suffice for N that one-fifth of the weight of the lower pulley together with the stone F, and the path of the rope M I. Therefore, by adding weight in H so that equilibrium is destroyed, it will be indicated how much additional resistance is added to the weight F from the conflict of the axles with the little wheels; and similarly, the pulleys being replaced by rings, after equilibrium is again exceeded by increasing the weight N until it is depressed, the resistance arising from the friction of the rope with the rings will become known. From this it appears first that it is better to work by this latter method, because a far smaller weight is required in N than in D. Second, to lift the weight F with the lower pulley, if the upper remains fixed, so much force is required in the power as would be needed to raise, without any machine, the preponderant weight N; but to lift the same stone F with both pulleys, that is, by pulling the pulley A upward, so much force is required in the power as is needed to raise the preponderant weight D. Third, the same pulleys being retained, but the weight F changed, it can be examined whether, and by how much, greater resistance arises from greater pressure on the axles when the weight is greater. Fourth, by changing the pulleys while retaining the same weight, some disparity may be found, because not all pulley axles are equally smooth and polished, and congruent with the hole of their little wheels. If, after such an examination has once been made, you turn the little wheels by hand for a short while and then make the same examination again, and equal resistance is not found, it will be an indication that the hole of the little wheel, or perhaps even the axle, is not exquisitely round. Fifth, by a similar examination begun in the rings, it can be discovered whether pulling succeeds more easily with a thicker rope or with a thinner one. It is not, however, necessary to use the method indicated in such a way that, if one does not care to employ the yoke of a balance with equal arms, one cannot instead apply a steelyard in place of the balance to the pulley A or to the end of the rope M: for first the equilibrium will be indicated; then, with the counterweight drawn farther back until preponderance begins to appear, the amount of the impediment will become known,

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Liber sextus. CAPUT VIII. 635 ti, quin opus sit gravitatem aliam atque aliam addero, ut in librâ. PROPOSITIO VI. Vim Retinaculi Trochlearum augere. Sæpè contingit infixo parieti tigillo alligari superiorem trochleam; & nisi paries valdè firmus ac solidus fuerit, cujusmodi sunt antiqui parietes, non leve periculum imminet, ne ponderis vi labefactetur ipse paries, maximè si recens fuerit, & tigillus non admodum procul à summitate insigatur; ut si recentis parietis B C foramini immittatur tigillus brevior A H, ex quo in A dependet trochlea, & ex illâ pondus cum reliquâ trochleâ: fieri enim potest, ut tigillus ipse quasi Vectis à pondere adnexo depressus attollat lateres impositos, & superioris parietis compagem dissolvat. Foramen igitur ita fiat, ut paries pervius sit, illumque pervadat longior tigillus A F, cujus caput F fune F I connectatur cum annulo in I parieti infixo: sic enim ponderis gravitas nullam inferre poterit parieti labem quamvis recenti; & quò longior fuerit tigilli pars H F supra partem H A, eò validius retinebitur trochlea in A, tigillo rationem Vectis habente. LLII 2

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Liber sextus. CHAPTER VIII. 635 to, unless I add one gravity after another, as in the scale. PROPOSITION VI. To increase the force of the pulley-retaining beam. It often happens that the upper pulley is attached to a small beam fixed into a wall; and unless the wall is very firm and solid, such as the ancient walls are, there is no small danger lest the wall itself be shaken by the force of the weight, especially if it is newly built, and the beam is inserted not far from the top; as if into the hole B C of a recent wall there is inserted the shorter beam A H, from which at A the pulley hangs, and from it the weight together with the rest of the pulley. For it may happen that the beam itself, pressed down as it were like a lever by the attached weight, raises the bricks placed above, and loosens the fabric of the upper wall. Let the hole therefore be made so that the wall may be pierced through, and let the longer beam A F pass through it, whose head F is connected by the cord F I with the ring fixed in the wall at I: for thus the heaviness of the weight will be able to inflict no damage upon the wall, even if it is new; and the longer the part H F of the beam is above the part H A, the more securely will the pulley in A be held, the beam having the nature of a lever. LLII 2

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Mechanicorum Propositio VII. Trochleis vim Vectis augere. Quamvis ex dictis obvium sit Trochleas cum aliis Faculta- tibus componere, placet tamen hîc eas cum Vecte com- ponendas indicare. Sit prælum in torculari, sed fortè dissipa- ta fuerit Cochlea, qua illud deorsum trahebatur, aut certè ad subitum usum properato prælo utendum sit: trabem statue transversam, quæ altero capite retineatur objecto repagulo, ne fursum attollatur. Vectis est secundi generis in medio habens pondus premendum. Alteri trabis extremitati adnectatur tro- chlea, ejusque compar in inferiore loco firmetur: nam funem ductarium trahentes momentum habebunt, quod ex Ratione Vectis, & ex Ratione Trochlearum componitur. Simili methodo utendum est, si Vecte secundi generis at- tollendum sit pondus: loco enim potentiæ destinato, hoc est Vectis extremitati attollendæ, adnectatur Trochlea, ejusque compar in superiore loco firmetur: hîc enim pariter Vectis at- que Trochlearum Rationes componuntur. Quòd si funem Su- culâ traxeris, aut Ergatâ, tres erunt Rationes compositæ; dua- bus quippe illis addenda est Ratio Suculæ aut Ergatæ. MECHA

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Mechanics Proposition VII. To increase the force of a lever by pulleys. Although from what has been said it is evident that pulleys may be combined with the other mechanical powers, it nevertheless seems proper here to indicate that they may be combined with the lever. Let a press be in a press-frame, but perhaps the screw by which it was drawn downward has been removed, or certainly let the press be to be used quickly for an urgent purpose: set a cross-beam, one end of which is held by a fixed stop, so that it may not be lifted upward. The lever is of the second kind, having the weight to be pressed in the middle. To one end of the beam let a pulley be attached, and let its fellow be fixed in a lower position: for those pulling the guiding rope will have a force compounded from the ratio of the lever and from the ratio of the pulleys. The same method is to be used if a weight is to be raised by a lever of the second kind: for at the place assigned to the force, that is, at the end of the lever to be lifted, let a pulley be attached, and let its fellow be fixed in a higher place: for here likewise the ratios of the lever and of the pulleys are combined. But if you draw the rope by a windlass or by a capstan, there will be three compounded ratios; for to those two must be added the ratio of the windlass or capstan. MECHA

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MECHANICORUM LIBER SEPTIMUS. De Cuneo & Percussionibus. MECHANICARUM Facultatum Quarta species, Cuneus, præsentem disputationem exigit: neque enim solam corporum gravitatem, ob id ipsum quia gravitas est, vincere oportet, ac loco dimo- vere, quemadmodum Vecte, Axe, & Trochleis, sed etiam sæpè conjunctas corporum partes, aut cohærentia proximè corpora sejungere atque divellere: id quod Cuneo potissimùm perficimus, & iis, quæ ad Cunei rationem specta- re videntur. Quoniam verò in iis, in quibus præcipuè Cu- nei vis elucet, percussione utimur, quæ sanè in paulò lon- giorem sermonem nos vocat, non erit abs re aliquanto latiùs Percussionis naturam explicare, ut potentiæ Cuneo applicatæ virtus manifesta fiat. Quamquam non semper percussione indigeat Cuneus, sed non rarò impulsioni contentus sit, ne- que semper ad divellenda ea, quæ conjuncta sunt, illo uta- mur, sed aliquando etiam ad deprimendum, aut attollendum corpus aliquod, ut ex sequentibus patebit. Ea verò, quæ Cunei figuram imitantur, quia in apicem desinunt, ut trian- gula, ideóque Cunei nomine sunt indicata sæpiùs à Vitruvio in Architecturæ libris, ad præsentem disputationem non atti- nent; quemadmodum neque subscudes, seu securiclæ, quibus arcte duæ tabulæ compinguntur; licèt enim cunei formam imitentur, non tamen similem, sed cuneo oppositam effectio- nem habent, & inter retinacula connumerandæ sunt. LLII

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MECHANICORUM BOOK THE SEVENTH. On the Wedge and Impulses. The fourth kind of mechanical powers, the Wedge, demands the present discussion: for it must not only overcome the mere heaviness of bodies, for the very reason that they are heavy, and move them from their place, as is done by the Lever, the Axis, and Pulleys, but also often separate and tear asunder the joined parts of bodies, or bodies cohering closely together: this is what we accomplish chiefly by the Wedge, and by those things which seem to belong to the nature of the Wedge. But since in those cases in which the force of the Wedge especially shines forth, we use percussion, which indeed calls us to a some- what longer discourse, it will not be out of place to explain somewhat more fully the nature of Percussion, so that the power of the force applied to the Wedge may become evident. Although the Wedge does not always require percussion, but is not rarely content with mere impulsion, nor do we always use it to tear apart things that are joined, but sometimes also to depress or raise some body, as will be clear from what follows. But those things which imitate the shape of a Wedge, because they end in a point, as triangles do, and for that reason are more often designated by Vitruvius in the books of Architecture by the name of wedge, do not concern the present discussion; just as neither do the subscudes, or securiclæ, by which two planks are tightly joined together; for although they imitate the form of a wedge, they do not have a similar, but an opposite, effect, and are to be counted among the fastenings. LLII

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Mechanicorum CAPUT I. Cunei forma, & vires explicantur. Cuneus, quo ad ligna findenda communiter utimur, satis notum est instrumentum, quod ex ampliori base fastigia- tur in acumen desinens; duas scilicet facies quadrangulas in li- neam coëuntes, & duobus triangulis hinc atque hinc connexas super quadrilateram basim erigit, adeò ut scindendo corpori acies ipsa applicetur; percussionem excipiat basis. Nihil tamen prohibet aliam cuneo figuram tribui: nam cum aliquando com- planandus esset colliculus, cujus creta arenæ permixta in lapi- deam quandam materiem concreverat, & ita obduruerat, ut li- gonibus ægerrimè cederet, parari jussi longiusculos ferreos cy- lindrulos digitorum duum crassitudine extremitate alterâ capi- tatos ad excipiendum mallei ictum, altera in planas duas super- ficies compressos atque exacutos, qui juxta lapidis intervenia ap- plicati, ac tudite adacti lapidem penetrabant; qui demum rimas agens dissiliebat in frusta satis conspicua non poenitendo labore. Sed quæcumque demum figura cuneo statuatur, illud omnibus commune est, quod ex minori latitudine in majorem procedant, ut quibus corporibus Cuneus inseritur, magis atque magis alte- rum ab altero sededat, dum ille penitiùs adigitur. Nunc verò scissionem tantisper seponamus, & solum motum corporis gravis ex cunei impulsione consideremus, sicuti si duro plano incumbentem marmoreum cubum, cui vectis subjici ne- quiret, ut attolleretur, addacto cuneo disjungeremus à sub- jecto plano: id quod faciliùs acutiore cuneo præstatur, ut omnes nôrunt, quàm si ille in mi- nus acutum angulum de- sineret. Sit enim cubus H marmoreus plano A B in- cumbens; & applicatus cuneus D E, atque tudite in D validè percussus ita cubo subjiciatur, ut hic elevetur

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Mechanics CHAPTER I. The form and forces of the wedge are explained. The wedge, which we commonly use for splitting wood, is a well-known instrument, which rises from a broader base and ends in a point; that is, it brings together two quadrangular faces into a line, and, joined on both sides by two triangles, it is raised upon a quadrilateral base, so that in splitting a body the edge itself is applied to it; the base receives the удар. Yet nothing prevents another shape from being assigned to the wedge: for when at one time a little hill had to be leveled, whose chalk mixed with sand had solidified into a kind of stony material, and had become so hardened that it yielded only with difficulty to spades, I ordered longish iron cylinders, two fingers in thickness, to be made, headed at one end to receive the blow of the hammer, and compressed into two flat surfaces at the other and sharpened, which, applied along the interstices of the stone and driven in with a mallet, penetrated the stone; which at last, making cracks, split apart into quite visible fragments, not without considerable labor. But whatever shape is assigned to the wedge, this is common to them all: that they proceed from a lesser breadth to a greater, so that into whatever bodies the wedge is inserted, it more and more separates one from another, the deeper it is driven in. Now, however, let us set aside splitting for a little while, and consider only the motion of a heavy body from the impulse of the wedge, as if, by inserting a wedge beneath a marble cube resting on a hard plane, which a lever could not be placed under in order to lift it, we separated it from the plane beneath: this is accomplished more easily by a sharper wedge, as everyone knows, than if it ended in a less acute angle. Let there be, then, a marble cube H resting on the plane A B; and let the applied wedge D E, when struck hard at D with a mallet, be so inserted beneath the cube that the cube is raised.

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Liber septimus. CAPUT I. 639 elevetur ad altitudinem F I. Cùm itaque citrà omnem dubi- tationem is, qui tuditem movet, eóque cuneum percutit, illo eodem conatu nequeat cubum attollere sinè cuneo, quæritur, unde vis tanta illi accedat, spectatâ præcisè cunei figurâ, nihil interim ad percussionem respiciendo. Qui Aristoteli Mechan. quæst. 17. in lateribus cunei ligno scindendo interjecti duplicem Vectem agnoscenti adhærent, illicò ad Vectis Rationes confugient, quas in latere D E re- cognoscere conabuntur. Verùm quærenti, primi ne ? an se- cundi generis Vectis sit D E ? vix suppetet, quid respondeant. Si primi generis, ut voluisse videtur Aristoteles; cum tria sint puncta D, & I, & E, atque primum D procul dubio potentiæ ascribatur; reliquum est medium punctum I esse hypomochlij, extremum E ponderis. Atqui nihil prorsus apparet, quod cu- bum H applicet puncto E, à quo neque sustinetur, neque tan- gitur: igitur vectis non est primi generis. Quod si in E pon- dus esse ultrò concesserim, illud certè ex hoc efficitur, atque consequens est, quod cuneo novâ percussione ulteriùs adacto, & Potentia ad hypomochlium, scilicet D ad I, accedat, & pon- dus ab hypomochlio, scilicet E ab I, remotius fiat; igitur & potentiæ momenta decrescerent, & ponderis momenta auge- rentur; ac proinde hanc momentorum decessionem, & acces- sionem, major movendi difficultas, & quidem notabilis atque conspicua, consequeretur; quæ tamen cum experimentis non consentit. Adde in Vecte primi generis potentiam & pondus oppositis motibus circa hypomochlium, quasi circà centrum, circulariter moveri; at hîc neque ullus intercedit circa punctum I motus circularis, neque potentia descendit pondere ascendente. At fortasse, quod aliis magis placet, vectem ais esse secundi generis; pondus quippe in I sustinetur, & est potentiæ in D existenti proximum; quare hypomochlio relinquitur extre- mum punctum E. Id quidem aliquantò magis appositè dictum videretur, si, quæ vectis Rationibus conveniunt, hîc quoquè in Cuneo locum habere possent; Vectis siquidem quò longior est, & potentia magis abest ab hypomochlio, cæteris paribus, plus momenti tribuit potentiæ: at Cunei D E longitudo si au- geatur, manente eodem angulo I E F, eádemque distantiâ I E, nullam

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Book Seven. Chapter I. 639 be raised to the height F I. Since therefore, beyond all doubt, the one who moves the wedge, and strikes it with the hammer, cannot by that same effort raise the cube without the wedge, the question is, whence so great a force comes to it, the precise figure of the wedge being considered, while nothing in the meantime is regarded as to the percussion. Those who cling to Aristotle, Mechan. quest. 17, acknowledging in the sides of the wedge inserted for splitting wood a double Lever, will at once have recourse to the ratios of the Lever, which they will try to recognize in the side D E. But when asked whether D E is of the first or of the second kind of Lever, they are scarcely able to reply. If of the first kind, as Aristotle seems to have intended; since there are three points, D, I, and E, and the first, D, is beyond doubt to be assigned to the power; it remains that the middle point I is the fulcrum, the extremity E the weight. Yet nothing whatever appears that applies the cube H to the point E, by which it is neither supported nor touched: therefore it is not a Lever of the first kind. But if I freely grant that the weight is at E, it certainly follows from this, and it is a consequence, that as the wedge is driven further in by a new blow, the Power approaches the fulcrum, namely D to I, and the Weight recedes from the fulcrum, namely E from I; therefore the moments of the power would decrease, and the moments of the weight increase; and consequently a greater difficulty of moving, indeed a notable and conspicuous one, would follow from this recession and approach of moments; yet this does not agree with the experiments. Add that in a Lever of the first kind the power and the weight move circularly around the fulcrum, as around a center, by opposite motions; but here there intervenes neither any circular motion around the point I, nor does the power descend while the weight ascends. But perhaps, as is more pleasing to others, you say that the lever is of the second kind; for the weight is supported at I, and is nearest to the power existing at D; wherefore the fulcrum is left at the extreme point E. This indeed would seem somewhat more suitably said, if what agrees with the ratios of the Lever could here also have a place in the Wedge; for the longer a Lever is, and the farther the power is from the fulcrum, other things being equal, the more it gives to the power: but if the length of the wedge D E is increased, while the same angle I E F remains, and the same distance I E, no

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Mechanicorum 640 nullam facit momentorum accessionem: nulla igitur ibi Vectis Ratio intercedit. Contrà verò manente eadem cunei DE longitudine, eademque distantiâ IE, diminuto autem, sive aucto angulo ad E eædem perseverarent vectis Rationes; ergo & eadem momenta movendi: id tamen longissimè à vero abesse manifestis docemur experimentis, nam major angulus ad E movendi difficultatem auget, minor minuit. Cùm verò Cuneus, ex hypothesi, semper subjecto plano incumbat, utique potentia in D ab eo semper æqualiter distat, & nunquam altiùs assurgit, pondere tamen altiùs sublato, quo magis illi cuneus subjicitur: in Vecte autem secundi generis potentia ascendit, si pondus attollitur. Non igitur cuneus habet rationem vectis secundi generis; si maximè nullum hîc haberi circa hypomochlium, tanquam circa centrum, motum circularem potentiæ animadvertas. Cùm itaque à Cuneo absint Rationes Vectis, illius vires petendæ sunt ex eo, quod olim constitutum est, Facultatibus omnibus Mechanicis communi principio: videlicet, quia Cunei forma ea est, ut majore motu moveatur Potentia Cuneum impellens, quàm pondus à Cuneo repulsum ad latus; hoc minùs resistit, quàm si æquali motu cum potentiâ moveretur; atque adeò impetus motum in potentiâ efficiens tantæ velocitatis, & valens pari velocitate movere certum pondus, cui inesset æquè intensus ac in potentiâ, poterit in majori pondere entitativè æqualis, sed minùs intensus efficere motum tardiorem pro Ratione minoris intensionis, ita tamen, ut, quæ Ratio esset intensionis majoris in minori pondere, ad intensionem minorem in majori pondere, ea pariter sit Ratio majoris gravitatis ad minorem gravitatem; sic enim contingit æqualem esse entitativè motum tardiorem majoris ponderis, atque motum velociorem minoris ponderis; quemadmodum aliàs in Vecte & in Trochleâ explicatum est. Quoniam igitur cuneus, vi impetûs impressi à potentia, dum promovetur sub pondus juxta lineam EF, repellit pondus juxta lineam FI, linea EF motum potentiæ metitur, linea autem FI motum ponderis. Atqui Cunei conformatio hoc habet, ut in triangulo EF I minimus angulus sit ad apicem E; igitur per 19. lib. 1 minimum latus est FI, atque proinde minùs movetur pondus per FI, quàm potentia per EF. Quanto

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of Mechanics 640 produces no addition of moments: therefore there is no lever ratio there. On the other hand, while the same length DE of the wedge, and the same distance IE, remain, but the angle at E is diminished or increased, the same lever ratios would remain; and therefore the same moving moments. Yet experience clearly teaches us that this is very far from the truth, for a larger angle at E increases the difficulty of moving, a smaller one diminishes it. But since the wedge, by hypothesis, always rests on the plane beneath it, the power at D is always equally distant from it, and never rises higher; yet the load is lifted higher, the more the wedge is inserted beneath it. In a lever of the second kind, however, the power rises if the load is raised. Therefore the wedge does not have the ratio of a lever of the second kind; even if, at most, you observe that here there is no circular motion of the power around the fulcrum as around a center. Since, then, lever ratios are absent from the wedge, its forces must be sought from what was once established as the common principle of all mechanical powers: namely, because the form of the wedge is such that the power driving the wedge is moved with greater motion than the load repelled sideways by the wedge; the latter resists less than if it were moved with equal motion with the power. And thus the impulse producing motion in the power, having such speed and being able with equal speed to move a certain load in which there was an intensity equal to that in the power, can, in a greater load, bring about a motion that is entitatively equal but less intense, slower in proportion to the lesser intensity; so however that, as the ratio of greater intensity in the lesser load is to lesser intensity in the greater load, so too is the ratio of greater weight to lesser weight. For in this way it happens that the slower motion of the greater load is entitatively equal to the swifter motion of the lesser load, as was elsewhere explained in the lever and in the pulley. Since therefore the wedge, by the force of the impulse impressed by the power, while it is advanced beneath the load along line EF, repels the load along line FI, line EF measures the motion of the power, while line FI measures the motion of the load. But the structure of the wedge is such that in triangle EFI the smallest angle is at the apex E; therefore, by Book 1, proposition 19, the smallest side is FI, and consequently the load is moved less through FI than the power through EF. To what extent

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Liber septimus. CAPUT I. 641 Quanto igitur major est EF quàm FI, tanto majus esse potest gravitatis momentum in I vincendum, quàm esset momentum gravitatis propellendæ in E juxta directionem FE motûs po- tentiæ. Hinc planissimè constat, cur acutiores cunei majora habeant movendi momenta, cæteris paribus. Fac enim angulum E esse adhuc minorem, utique oppositum latus minus erit quàm FI; eadem igitur linea EF ad lineam breviorem, quàm FI, habet majorem Rationem, quàm ad eandem lineam FI; ac propterea pondus adhuc multo tardiùs movetur quàm potentia, & pote- rit esse majus; vel, si majus non fuerit, potentia indigebit mino- re conatu, & faciliùs movebit. Observa autem (quantum quidem ex Cunei Ratione est) ut se initium dederit, eandem semper esse facilitatem in processu motûs, quia eadem permanet Ratio motuum potentiæ cuneo applicatæ, & ponderis: nam ex 4. lib. 6. ut EF ad FI, ita EC ad CO, & IS, hoc est FC, ad SO, propter triangulorum si- militudinem, cùm sit IS parallela ipsi EC. Quantum, inquam, est ex Cunei Ratione; quandoquidem cubi H, dum manente extremitate A elevatur ex I, momenta subinde variari, suo lo- co, superiùs indicatum est. In scindendis autem corporibus, prout variè contingit scissio, aliquando peculiaris intercedere potest causa faciliorem vel difficiliorem in processu scissionem reddens. Et quidem in scissione corporum vi cunei faciendâ non est ita proclivè Geometricas leges persequi, ad explicandam eo- rum resistentiam: neque enim sicut gravitas loco dimovenda facilè innotescit, certâmque sub mensuram cadit, ita corpo- rum resistentia, ne findantur, fieri potest manifesta: Est siqui- dem scissio partium conjunctarum separatio; earum autem con- junctionem adeò variam esse contingit, ut certam legem subi- re nequeat. Nam quemadmodum inter lapides, ut monet Vi- truvius lib. 2. cap. 7. alij ita molles sunt, ut etiam serrâ dentatâ, quasi ligna, secentur, immò secundùm oras maritimas ab sal- sugine exesa diffluant, & in locis patentibus atque apertis, prui- nâ & gelu frientur, ac dissolvantur; alij duriores, sed qui inter- veniorum vacuitates habeant, quapropter ab igne tuti non sint, quin rarescente aëre vacuitatibus illis interjecto dissiliant & dis- M M m m

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Book Seven. Chapter I. 641 So much greater, therefore, as EF is than FI, so much greater can be the momentum of gravity to be overcome in I, than would be the momentum of gravity to be propelled in E along the direction FE of the power’s motion. Hence it is quite clear why sharper wedges, other things being equal, have greater moving force. For suppose the angle at E to be still smaller; the opposite side will certainly be smaller than FI; therefore the same line EF has a greater ratio to the shorter line than FI, than it has to the same line FI; and for that reason the weight is moved much more slowly than the power, and may be greater; or, if it is not greater, the power will require less effort, and will move more easily. Observe, however, that so far as the nature of the wedge is concerned, once it has been given a beginning, the same ease always remains in the progress of the motion, because the ratio of the motions of the power applied to the wedge and of the weight remains the same: for from book 4, proposition 6, as EF is to FI, so EC is to CO, and IS, that is FC, to SO, by reason of the similarity of the triangles, since IS is parallel to EC. I say, so far as it depends on the nature of the wedge; for since, while the end A remains fixed, the cube H is raised from I, the moments are subsequently varied, as has been indicated above in its proper place. But in the splitting of bodies, as the splitting occurs in various ways, there may sometimes intervene a special cause making the splitting in progress easier or more difficult. And indeed, in splitting bodies by means of a wedge, it is not so easy to pursue geometrical laws in explaining their resistance; for just as gravity, to be moved from its place, is readily perceived and falls under exact measurement, so the resistance of bodies, so that they may be split, cannot be made manifest: for splitting is the separation of parts joined together; but their conjunction happens to be so various that it cannot be brought under a fixed law. For just as among stones, as Vitruvius warns in book 2, chapter 7, some are so soft that even with a serrated saw they are cut, like wood; indeed, along the seacoasts, eaten away by salt spray, they crumble, and in open and exposed places they are reduced to powder by frost and cold, and dissolve; others are harder, but have interstices and voids, and therefore are not safe from fire, since, the air rarefying, they burst apart and dis- M M m m

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Mechanicorum 642 sipentur; alii ita spissis compactionibus solidati, ut neque ab tempestatibus, neque ab ignis vehementiâ timeant: Ita pariter inter ligna alia aliis solidiora sunt, & unum præ alio faciliùs est fissile, prout particulæ componentes crassiores, aut tenuiores, sunt magis aut minus exquisitè permistæ, atque nimio, sive modico, sive temperato humore concretæ, & prout juxta sta- minum ductum, aut illa obliquè secando, instituitur scissio. Sunt autem corpora illa (quantum quidem ad præsentem tractionem spectat) partium separationi difficiliùs obnoxia, quorum materia ita probè subacta est, ut eorum elementa in minutissimas particulas concisa, & quasi individua corpuscula in unam naturam inobservabili permistione temperata coalue- rint eâ tantùm humoris copiâ, quæ satis fuerit ad illa firmiter agglutinanda. Ex quo fit, ut hujusmodi corpora solidiora sint, minúsque conspicuas inanitates admittant, atque proinde, si expoliantur, superficiem induant lævem & undique æquabi- lem: id quod cæteris non accidit, quorum particulæ frequenti- bus hiatibus intercisæ, cùm aliæ emineant, aliæ superentur, semper aliquid habeant asperitatis; quemadmodum animadver- tere poterit, quisquis lapides cum marmoribus comparaverit. Si igitur ex solido corpore avellenda est particula aliqua, hæc istaque disjungenda est à circumstantibus particulis, quibus conjungitur, neque fieri potest, ut illa moveatur, quin proxi- marum particularum aliæ motui oppositæ impellantur, aliæ distrahantur; omnes autem ægrè à statu sibi secundùm natu- ram debito recedentes repugnant: quò verò plures particulæ vim subire coguntur, eò major est resistentia plurium quasi col- laris viribus simul repugnantium. Hinc si duriora ligna secan- da offerantur, potior est usus subtilioris serræ minutos denticu- los habentis; quia videlicet, quò exilior atque subtilior est den- ticulus, minorem particulam obviam habet, quam impellat, & pauciores particulæ, à quibus separetur, illam circumstant; ideóque ab iis faciliùs avellitur, quàm particula major, quæ à pluribus disjungenda esset. Contra verò quorum particulæ le- vi impulsu divelluntur, quia non adeò dura sunt, crassiore ser- râ facilè secantur, quæ in durioribus majorem resistentiam in- veniens parùm utilis accideret. Hæc autem in limâ pariter ob- servari possunt; quàm enim dissipari asperitate opus est in limâ, qua

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Mechanics 642 some are so consolidated by dense compaction that they fear neither tempests nor the violence of fire. Thus, among woods also, some are firmer than others, and one is more easily split than another, according as the component particles, thicker or thinner, are more or less exactly mixed together, and solidified by too much, or too little, or moderate moisture, and according as the splitting is made along the grain, or by cutting obliquely across it. Now those bodies are, in so far as the present discussion is concerned, more difficultly subject to the separation of their parts, whose matter has been so well worked that its elements, cut into the tiniest particles and, as it were, individual corpuscles, have been tempered into one nature by an imperceptible mixture, being bound together by only so much moisture as was enough to make them firmly adhere. From this it follows that bodies of this kind are more solid, and admit fewer visible voids; and therefore, if they are polished, they assume a surface that is smooth and even on every side. This does not happen with others, whose particles, broken up by frequent gaps, as some parts project and others are left below, always retain something of roughness; as anyone can observe who compares stones with marbles. If, then, some particle is to be torn away from a solid body, it must be separated from the surrounding particles with which it is joined; and it cannot happen that it be moved without some of the nearest particles being driven in opposition to the motion, and others being dragged apart. But all, as they are with difficulty drawn away from the state due to them according to nature, resist. And the more particles are compelled to undergo force, the greater is the resistance of many, as it were, collars resisting together. Hence, if harder woods are offered to be cut, the better use is made of a finer saw with tiny teeth; because, namely, the smaller and finer the tooth, the smaller particle it encounters in front of it to push, and the fewer particles surround it from which it must be separated; and therefore it is more easily torn away from them than a larger particle, which would have to be detached from more. On the other hand, those whose particles are pulled apart by a slight impulse, because they are not so hard, are easily cut with a coarser saw, which, finding greater resistance in harder materials, would be of little use. The same observations can likewise be made concerning the file; for just as roughness must be worn away with a file, so too ...

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Liber septimus. CAPUT I. 643 qua chalybs, aut qua lignum terendo expolitur? Sed & in mar- morum sectione mirantur aliqui serras adhiberi nullis dentibus, saltem conspicuis, asperas, non satis animum adverteres ad arenas aquâ aspersas, quæ hujusmodi in opere interveniunt; ut enim loquitur Plinius lib. 36. cap. 6. arenâ hoc fit, & ferro vide- tur fieri, serrâ in prætenui lineâ premente arenas, versandoque tractu ipso secante. Expedit autem subtiliore arenâ uti, crassior enim are- na laxioribus segmentis terit, & plus erodit marmoris, majusque opus scabritiâ polituræ relinquit: Sunt scilicet arenæ granula tam quæ premuntur, quàm quæ serræ adhærent, quasi denticuli mobiles mordacis limæ eodem ductu tùm crustarum faciem le- viter expolientis, tùm subjectum marmor secantis. Est autem manifestum non eâdem vi, qua super lignum ser- ram adducentes & reducentes scissionem inchoamus, idem pa- riter obtineri, si cuneo lignum premamus citrà percussionem: quia nimirum cuneo prementes urgemus in directum subjectas ligni partes, quæ conjunctim resistunt, ne comprimantur, at- que à lateribus cohærentes particulæ repugnant, ne distrahan- tur: serram verò ducentes obliquè urgemus ligni particulas dentibus respondentes, ac proinde pauculæ illæ tantùm, quæ urgentur, resistunt compressioni, & illis attiguæ distractioni. Hinc fit cultro faciliùs aliquid scindi, si illius aciem quamvis hebetem & obtusam adversùs corpus scissile urgeas simul, atque transversam agas; quia particulæ à cultro pressæ minorem in- veniunt resistentiam in transverso motu, ubi anteriores eâdem cum posterioribus directione moventur, nec sibi adversantur, quàm si solo pressu extimæ urgerent interiores, quæ comprimi renuunt. Huc pariter referenda est causa, cur adeò valida con- tingat scissio, si Harpe ictus infligatur: sic hujusmodi genere ensis in summitate falcati, & in exteriore latere exacuti usum Perseum in amputando Medusæ capite, & Mercurium in occi- dendo Argo centoculo refert Ovidius lib. 5. Metam. Vertit in hunc Harpem madefactam cæde Medusæ: id enim non ex solâ Harpes gravitate, sed ex ipso potissimùm flexu oritur, qui ef- ficit, ut dum vi impetùs descendit, acies etiam transverso mo- tu ducatur supra partes corporis scissilis; ex quo & facilior scis- sio. Sic quidam vulgari ense, quo equitantes viatores non in speciem, sed ad usum, præcini solent, vituli caput uno ictu M M m m 2

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Book Seven. CHAPTER I. 643 by what means steel, or wood, is polished by rubbing? But even in the cutting of marble some marvel that saws are used with no teeth, or at least with none that are conspicuous, rough as they are, unless you pay close attention to the sand sprinkled with water, which intervenes in operations of this kind; for, as Pliny says, book 36, chap. 6, “it is done by sand, yet seems to be done by iron,” the saw pressing the sand in a very thin line and, by its turning motion, cutting through. It is expedient, however, to use finer sand, for coarser sand, with its looser particles, grinds and erodes more of the marble, and leaves the greater work with a roughness of polish: for there are grains of sand both those that are pressed and those that adhere to the saw, like mobile little teeth of a biting file, which by the same stroke both lightly polish the surface of the crusts and cut the marble beneath. Now it is clear that by no other force than that by which, when we draw a saw over wood back and forth, we begin the split, is the same result likewise obtained if we press a wedge into the wood without striking: because, when we press with a wedge, we force straight upon the parts of the wood beneath, which resist together so as not to be compressed, and the particles cohering at the sides resist so as not to be pulled apart; but when we draw a saw, we obliquely press the particles of the wood corresponding to the teeth, and therefore only those few that are pressed resist compression, while those adjacent to them resist separation. Hence it happens that something is more easily cut with a knife, if you press the edge, however dull and blunt, against the object to be cut while at the same time moving it across; because the particles pressed by the knife find less resistance in the transverse motion, where the parts in front move in the same direction as those behind and do not oppose one another, than if by pressure alone you were to press the outer parts upon the inner, which refuse compression. To this also must be referred the cause why such a powerful cutting occurs if a Harpe-blow is inflicted: thus Ovid, book 5 of the Metamorphoses, relates the use of this kind of sword, hooked at the summit and sharpened on the outer side, in Perseus when he cut off Medusa’s head, and in Mercury when he killed the many-eyed Argus. “He turns this Harpe, made wet with the gore of Medusa”: for this arises not from the weight of the Harpe alone, but chiefly from its very curve, which causes that, while it descends with the force of the blow, the edge is also drawn in a transverse motion over the parts of the body to be cut; from which cutting is easier. Thus some man with the common sword, which riders are accustomed to wear at their side, not for show but for use, one blow

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Mechanicorum 644 amputabat; averso scilicet ictu percutiens gladius circulum describebat, adeóque non solùm premendo secabat, verùm etiam motu transverso: quo in negotio dexteritate potiùs, quàm viribus opus est. Quòd autem alij resimum Harpes lætus in canaliculum excavant, eique aliquid argenti vivi indunt, quod à capulo ad cuspidem excurrat, id faciunt ad percussionem augendam, quia translatâ ad cuspidem gravitate Mercurij, etiam percussionis centrum transfertur longiùs à capulo, ideóque ictus sit validior, accedente præsertim impetu, quem Mercurius descendens concipit. Non est itaque comparandus gladius motu transverso scindens cum serrâ secante; hæc enim obvias particulas in scobem abeuntes sensim à ligno divellens illud demum sublatis omnibus intermediis particulis bifariam divisum relinquit: ille verò suâ acie premens atque penetrans, sed nihil abradens, interponitur partibus, quæ invicem separantur. Motus serræ motui particularum abscissarum planè æqualis est; dens quippe particulam, in quam incurrit, tangens impellit, suóque impulsu vincens nexum, quo particula sibi cohærentibus jungebatur, illam avellit. Transversus verò gladij motus si comparetur cum motu particularum compressarum, atque invicem divulsarum, multò major est illo; nam culter manûs moventis motui obsecundat; & particulæ compressæ in latus recedunt; ac proinde multo velociùs movetur potentia gladium adducens aut reducens, quàm id, cui hoc motu vis infertur: atque idcircò gladij vis scindendi hoc motu refertur ad cuneum. Quòd si gladius non motu transverso ducatur super id, quod scinditur, sed omnino motu recto pressionis, sit autem ferri crassities sensim extenuata in aciem, quemadmodum cuneis omnibus commune est, idem planè dicendum erit, quod de vulgari cuneo, cui nomen hoc præcipuè inditum est. Quare Cuneus in corpus scindendum adactus considerandus est ratione habitâ ipsius corporis, quod tenerum ac molle esse potest, atque ita flexibile, ut sequatur quocunque torqueas, aut etiam durum, & minimè tractabile. Si molle illud sit, immissum cuneum recipit, séque illi accommodat, & comprimuntur tùm subjectæ particulæ cunei aciem tangentes, tùm quæ à lateribus cunei faciei congruunt: illæ omnino æqualiter pro- moventur,

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Mechanicorum 644 he cut off; that is, striking with the edge turned aside, the sword describes a circle, and so not only cuts by pressing, but also by a transverse motion: in this business dexterity rather than strength is needed. And whereas others, when they are pleased with a back-edge Harpes, hollow out a little groove in it and put into it some quicksilver, so that it may run from the hilt to the point, they do this to increase the blow, because, with the weight of Mercury transferred to the point, the center of percussion is also carried farther from the hilt, and so the blow is stronger, especially with the added impetus which descending Mercury acquires. Therefore a sword cutting with transverse motion is not to be compared with a saw cutting; for the saw, gradually tearing away the opposed particles as they go off into sawdust from the wood, finally leaves it divided in two, all the intermediate particles having been removed: but the sword, pressing and penetrating with its edge, yet abrading nothing, is interposed between the parts which are separated from one another. The motion of the saw is quite equal to the motion of the particles cut off; for the tooth, touching the particle into which it runs, drives it on, and by its own impulse overcoming the bond by which the particle was joined to those adhering to it, tears it away. But if the transverse motion of the sword be compared with the motion of the particles compressed and torn apart from one another, it is much greater than that; for the blade yields to the motion of the hand moving it, and the compressed particles recede sideways; and therefore the power bringing or drawing back the sword moves much more quickly than that on which force is inflicted by this motion: and for that reason the cutting power of the sword in this motion is referred to the wedge. But if the sword is not drawn over what is being split by a transverse motion, but altogether by a straight motion of pressure, while the thickness of the iron is gradually thinned down into an edge, as is common to all wedges, the same thing will plainly have to be said as of the ordinary wedge, to which this name has been especially given. Wherefore the Wedge, when driven into a body to be split, must be considered with regard to the body itself, which may be tender and soft, and thus so flexible that it follows wherever you turn it, or else hard and not at all tractable. If it be soft, it receives the inserted wedge and accommodates itself to it, and the particles beneath that touch the edge of the wedge are compressed, both those that meet the sides of the wedge and those that coincide with its face: these are all moved forward equally,

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Liber septimus. CAPUT I. 645 moventur, ac adigitur cuneus, neque motus earum à cuneo pendet, quâ cuneus est, sed quâ corpus est suâ mole objectam molem trudens: hæ verò ad latus secedentes magis & magis, prout cunei crassitudo excrescit, minori motu moventur, quàm promoveatur immissus Cuneus; quandoquidem major est cunei longitudo ejusdem motum metiens, quàm crassities dextras at- que sinistras partes impellens. Sin autem durum sit corpus, cui cuneus inseritur, illud quidem vix excogitari potest ex adeò constipatis partibus inter se quàm aptissimè cohærentibus con- stare, ut nullius prorsus compressionis sit capax, neque vel te- nuissimum cunei apicem admittat. Comprimuntur igitur ini- tio partes proximè cuneo subjectæ multò magis, quàm quæ la- teri adjacent; sed propter duritiem certum compressionis mo- dum natura sinivit, extra quem subjectas partes diffindi potiùs patiatur, atque à se mutuò divelli. Qua in fissione ipsæ etiam superiores partes majori compressioni semper validiùs re- pugnantes, quò penitiùs adigitur cuneus, plurimum habent momenti, ut cunei vis, quâ cuneus est, exerceatur: Quia vide- licet superiores partes cum inferioribus connexæ, neque flexi- biles, dum ad latera cuneo urgente secedunt, cogunt pariter in- feriores ad dexteram & ad sinistram recedere; atque propterea quæ adhuc connexæ erant, distrahuntur ita, ut demum di- vellantur. Sit cuneus H I in subjectum lignum immissus inter B & C; dum percussione urgetur introrsum, partes B versùs A, & partes C versus D re- cedunt, & cum illis pariter inferiores B M, atque C M: ex quo fit partes in M con- nexas distrahi atque invi- cem divelli, & scissionem longiùs promoveri. Cum igitur hoc sit propositum cuneo scindere subjectum corpus, non attendendus est simpliciter motus in B & C, sed etiam qui in M efficitur, ibi quippe scissio contingit, semperque longiùs distat à punctis B, & C, locus scissionis, quò magis introrsum urge- tur cuneus. Ex quo fit attentè distinguendam esse facilitatem M M m m 3

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Book Seventh. CHAPTER I. 645 are moved, and the wedge is driven in; nor does the motion of them depend on the wedge, as it is a wedge, but as it is a body thrusting the opposing mass by its own bulk: but these parts, receding more and more to the side as the thickness of the wedge increases, are moved by a smaller motion than the inserted wedge is advanced; since the length of the wedge, measuring its motion, is greater than its thickness, which drives the parts on the right and on the left. But if the body into which the wedge is inserted be hard, it can scarcely be conceived to consist of parts so closely compacted and most aptly cohering among themselves that it is capable of no compression at all, nor admits even the very sharp point of the wedge. Therefore at first the parts nearest under the wedge are compressed much more than those adjacent to the side; but on account of its hardness nature has allowed a certain limit of compression, beyond which she rather permits the parts subject to it to be split apart, and to be torn away from one another. In this splitting, the upper parts themselves also, always resisting the greater compression more strongly as the wedge is driven deeper, have very great influence in causing the force of the wedge, by which it is a wedge, to be exerted: because, namely, the upper parts, being connected with the lower and not flexible, while under the pressure of the wedge they separate to the sides, compel the lower parts likewise to recede to the right and to the left; and therefore those that were still connected are so pulled apart that at length they are torn asunder. Let the wedge H I be inserted into the wood below between B and C; while, being driven inward by blows, the parts B recede toward A, and the parts C toward D, and with them likewise the lower parts B M and C M: from which it happens that the parts connected at M are drawn apart and separated from one another, and the splitting is carried farther on. Since, therefore, the purpose of the wedge is to split the body beneath it, attention is not to be paid simply to the motion in B and C, but also to that produced at M; for there the split occurs, and the place of splitting is always farther distant from the points B and C as the wedge is forced more inward. From this it follows that the ease of splitting must be carefully distinguished from the

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adigendi cunei, à facilitate scindendi: quandiu enim partes faciem cunei tangentes non admodum repugnant compressio- ni, facilè cedunt adacto cuneo; ubi verò ulteriùs comprimi renuunt, tota vis exercenda est in distractione partium sejun- gendarum; quam distractionem quò majorem esse contingit, augetur sanè & cunei adigendi, & scindendi difficultas. Hæc tamen alio ex capite minuitur, quia, quò magis punctum M, in quo distrahendæ sunt partes conjunctæ, abest à punctis B & C, faciliùs consequitur scissio, nam, cæteris paribus, motus particu- larum, quæ sejunguntur, minorem habet Rationem ad motum punctorum B & C. Sic sustem crassiorem ab alterâ extremitate fissum juxta notabilem longitudinem ulteriùs findimus etiam solis manibus eò faciliùs, quò longior fuerit prior scissio: plu- rimum siquidem interest in lignis, quorum textura certum quendam & rectum staminum ordinem habet, utrum juxta eo- rumdem staminum ductum instituatur scissio, an hæc obliquè secentur; quemadmodum & in lapidibus præstat cuneum in- terveniis applicare, ut faciliùs scindantur: propterea nodosis arborum partibus applicatus Cuneus ægrè illas findit, quia no- dorum stamina non recto tramite, sed per anfractus & tortuosè procedunt. In hac autem ligni scissione si placeat tum particu- las, quæ in M distrahuntur, tum illis subjectas atque adhuc im- motas, puta in N considerare, atque vestigium aliquod dupli- cis Vectis secundi generis recognoscere, itaut commune hypo- mochlium sit in N, longitudines vectium B N, & C N, poten- tia medio cuneo applicata in B & C, atque resistentia vincen- da in M; non me difficilem præbebo: sed & illud statim ad- dam, non esse hunc duplicem illum Vectem, quem alij in Cu- neo quærunt; cum potiùs sint duo vectes in diversa impulsi ab interjecto cuneo. Quapropter ex his, quæ latiùs explicare placuit, illud confi- citur, quod Cuneo aliquando resistit gravitas, ut cùm ille cor- pori gravi elevando supponitur, aut cùm disjunctorum quidem corporum gravium, sed proximorum, saltem alterum remove- tur; aliquando resistit partium nexus in M, qui nisi solvatur, propelli nequeunt partes B & C cunei faciem tangentes: vis si- quidem cunei proximè exercetur adversùs B & C, & propter partium connexionem etiam adversùs M, quamvis hoc postre- mum

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the driving of the wedge depends on the ease of splitting: for so long as the parts touching the face of the wedge do not resist compression very much, they readily yield to the wedge when it is driven in; but when they refuse to be compressed further, the whole force must be exerted in pulling apart the parts to be separated; and the greater this pulling apart turns out to be, the greater, indeed, becomes both the difficulty of driving in the wedge and of splitting. This difficulty is nevertheless lessened from another side, because the more the point M, at which the joined parts are to be pulled apart, is distant from the points B and C, the more easily the splitting takes place; for, other things being equal, the motion of the particles that are separated has a smaller relation to the motion of the points B and C. Thus a thicker stake, split from one end along a noticeable length, we can further split even by hand the more easily the longer the first split has been: in woods whose texture has a certain fixed and straight order of fibers, it makes a great difference whether the split is made along the direction of those fibers or whether these are cut obliquely; just as also in stones it is better to apply the wedge to the joints, so that they may be split more easily: for this reason a wedge applied to the knotty parts of trees splits them with difficulty, because the fibers of the knots do not run straight, but proceed by bends and in a winding course. But in this splitting of wood, if one wishes to consider both the particles that are pulled apart at M and those beneath them, still unmoved, say at N, and to recognize some trace of a double lever of the second kind, so that the common fulcrum is at N, the lengths of the levers BN and CN, the power applied at the middle wedge at B and C, and the resistance to be overcome at M, I shall not be hard to persuade: but I shall also add at once that this is not that double lever which others seek in the wedge; rather, there are two levers driven in opposite directions by the wedge inserted between them. Therefore, from what has been more fully explained, this is concluded: that sometimes the weight resists the wedge, as when it is placed under a heavy body to be raised, or when, of two heavy bodies that are to be separated and are close together, at least one is removed; sometimes the bond of the parts at M resists, which, unless it is loosened, the parts B and C touching the face of the wedge cannot be driven forward; for the force of the wedge is exercised most directly against B and C, and because of the connection of the parts also against M, although this last

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Liber septimus. CAPUT I. 647 mum sit scopus scindentis: quò autem validiùs partes in M conjunguntur, etiam difficiliùs urgentur partes B & C: ligna verò adhuc viridia, & lento humore plena, quia particulæ ma- jorem distractionem ferunt, nec facilè dissiliunt, difficiliùs scinduntur, quàm ligna arida. Quæ itaque de Cuneo vulgari dicta sunt, facilè innotescit ea pariter convenire forficibus, cultris ensibus, novaculis, scal- pris, dentibus hominis anterioribus, & similibus, quibus ad scindendum utimur; sunt enim cunei diversimodè juxta varios usûs conformati; neque egent percussione, quia res scindendæ non admodum resistunt, & solus impulsus sæpè sufficit. Forfi- ces autem sunt quidem cunei, sed vectem conjunctum haben- tes; adeò ut potentia momenti augmentum acquirat ex Ratio- nibus vectis juxta distantias tùm potentiæ, tùm corporis scin- dendi, à clavo, ubi decussantur. An verò etiam scalpra, qui- bus Marmorarij assulas ex operibus dejiciunt; Cuneorum ratio- nem habeant, non admodum curo; videntur siquidem non in- cidere marmor, sed partes superfluos decutere: quod si per- cussi scalpri mucro penetrat, & dividit marmor, cuneus perin- de est atque scalpra, quibus lignum cælatur. Similiter acus, subulæ, aculei, clavi ad cuneum referun- tur: & quidem hujusmodi corpora quò subtiliora sunt, eò fa- ciliùs penetrant, quia immissa, & juxta suam longitudinem progredientia valdè moventur interea, dum corporis perforan- di particulæ distrahendæ atque comprimendæ exiguo motu in latera secedunt. Hinc constat, cur terebellâ paulo minore ape- riendum sit foramen, cui immittatur clavus crassiusculus, si præsertim lignum tenue sit; ne videlicet immisso clavo tot par- tes adeò invicem comprimantur, ut ulteriorem compressionem recusantes cogant alias distrahi, ac demum rimâ factâ lignum dissiliat: sublatis autem terebrâ particulis aliquot, reliquæ com- primendæ ut clavum arctè complectantur, cum pauciores sint, faciliùs compressionem ferunt citrà periculum fractionis aut scissionis ligni. Demum securis, & gladius cæsim feriens, cuneus est, cui quo- dammodo junctus est tudes; illo siquidem percutimus: quid enim interest, quod cuneum manentem tudite percutiamus, sive cu- neo velociter moto percutiatur corpus scindendum? CAPUT

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Book Seven. CHAPTER I. 647 whose object is the wedge of the splitter; for the more strongly the parts are joined together in M, the more difficultly also are the parts B & C pressed apart: but green wood still, and full of a sluggish moisture, because its particles bear a greater separation and do not easily split asunder, is more difficult to cleave than dry wood. What has therefore been said concerning the common wedge, it is easy to see that the same applies equally to shears, knives, swords, razors, chisels, the front teeth of man, and the like, which we use for cutting; for they are wedges fashioned in different ways according to various uses; nor do they require striking, because the things to be split do not resist very much, and a mere impulse is often sufficient. Shears, however, are indeed wedges, but having a lever joined to them; so that the power of the moment gains increase from the ratios of the lever according to the distances both of the power and of the body to be split, from the pivot where they cross. Whether chisels also, with which marble workers chip off flakes from the work, have the nature of wedges, I do not much care; for they seem not so much to cut into the marble as to strike off superfluous parts: but if, when struck, the point of the chisel penetrates and divides the marble, then the wedge is just the same as chisels, with which wood is carved. In like manner needles, awls, prickles, nails are referred to the wedge: and indeed the more slender bodies of this kind are, the more easily they penetrate, because when introduced and advancing along their length, they are greatly moved meanwhile, while the particles of the body to be pierced, being drawn apart and compressed, retreat sideways with slight motion. Hence it is clear why a hole should be opened with a somewhat smaller auger into which a rather thick nail is to be inserted, especially if the wood is thin; lest, namely, when the nail is inserted, so many parts should be compressed against one another that, refusing further compression, they force others apart, and at length, a fissure being made, the wood should split asunder: but when some particles are removed by the auger, the remaining ones, to be compressed so that they closely embrace the nail, since they are fewer in number, more easily bear the compression without danger of the wood breaking or splitting. Finally, a hatchet and a sword striking by cutting are a wedge, to which in some way a striking-mallet is joined; for by that we strike: for what difference does it make whether we strike the standing wedge with the mallet, or whether the body to be split is struck by the wedge moved swiftly? CHAPTER

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Mechanicorum CAPUT II. Cunei inflexi usus ad movendum. PRæter vulgarem Cunei formam, quæ planis extremitatibus circumscribitur, si non ad scindendum, sed ad movendum adhibeatur, utilis esse potest Cuneus inflexus, ita ut, qua saltem parte movendo corpori applicatur, faciem habeat non planam, sed inflexam, moveatur autem non circa ejusdem circularis curvitatis centrum. In crassiore tabulâ assumpto A puncto tanquam centro describatur placito intervallo A B pars arcûs circuli B C, sive Quadrans, sive Quadrante minor, sive major fuerit. Tum centro alio assumpto, & majore aliquo intervallo, alia circuli pars describatur D C occurrens priori arcui B C in puncto C, sive ibi se contingant, sive secant arcus, prout tibi commodius acciderit. Resecatis igitur supervacuis tabulæ partibus, & retentâ parte curvilineâ, habetur cuneus B C D inflexus, qui suam vim exerceat non motu recto, ut cæteri cunei, sed curvo: propterea illi ad firmitatem addantur transversaria ab extremitatibus cunei exeuntia, & in unum punctum coëuntia, circa quod, tanquam centrum, moveri possit cuneus. Ad firmitatem, inquam, quia, ad motum, satis esset crassiori extremitati B D addere appendicem B M, in qua assumi possit punctum, circa quod moveatur, quodcumque illud sit, modò non sit centrum arcûs D C, si arcus ille impellat corpus movendum, neque punctum D minùs distet ab hujusmodi centro motûs, quàm punctum aliud extremum C. Contra verò si cunei conatus exercendus sit trahendo, & corpori applicetur arcus B C, oportet centrum motûs minùs abesse ab extremitate B, quàm

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Mechanics CHAPTER II. The use of the bent wedge for moving. Besides the ordinary form of the wedge, which is bounded by flat ends, if it is employed not for splitting, but for moving, a bent wedge can be useful, so that, at least on the side by which it is applied to the body to be moved, it has not a flat face, but a curved one; and it is moved not about the center of the same circular curvature. In a thicker board, taking point A as a center, let part of the arc of a circle B C be described at a chosen interval A B, whether it be a quadrant, less than a quadrant, or greater. Then, taking another center, and with some greater interval, let another part of the circle D C be described, meeting the former arc B C at point C, whether they touch there or cut each other, as may be more convenient. Therefore, after the unnecessary parts of the board have been cut away, and the curved part retained, there is obtained the bent wedge B C D, which exerts its force not by straight motion, as other wedges do, but by curved motion; for that reason, to strengthen it, there should be added crosspieces extending from the ends of the wedge and meeting at one point, around which, as around a center, the wedge may move. I say, to strengthen it, because for motion it would be enough to add to the thicker end B D a projection B M, in which a point may be taken around which it may move, whatever that point may be, provided it is not the center of the arc D C, if that arc is to push the body to be moved, nor should the point D be less distant from such a center of motion than the other end point C. On the other hand, if the wedge is to be used by pulling, and the arc B C is applied to the body, then the center of motion ought to be less distant from the end B than

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Liber septimus. CAPUT II. 649 quàm ab apice cunei C. Hùc scilicet spectare videntur ferrei uncini, quibus duo corpora fibulantur, aut resibulantur, ut cum armaria, fenestræ, aut capsulæ clauduntur & recluduntur. Quare intellectâ rectâ DM tanquam parte diametri circuli, cujus arcus exerceat vim cunei, si hic sit arcus DC, oportet ejus centrum inter extremitatem D, & punctum M centrum motûs, interjacere; sin autem sit arcus BC, inter A centrum circuli, & extremitatem B, oportet interjici centrum motûs H: ab illo quippe centro M in primo casu removeri oportet corpus movendum, & ad hoc centrum H accedere oportet corpus trahendum. Quod si recta DM non fuerit pars diametri transeuntis per centra arcûs & motûs, saltem oportet illam hujusmodi diametro propiorem esse, quàm sit recta ex centro motûs ad C ducta; id quod ex 7.lib.3. manifestum est. Firmato itaque pro loci opportunitate centro motûs, & applicatum ad impellendum pondus cunenum BCD urgens potentia in D, describit circa M centrum motûs arcum circularem DE, sed pondus non propellitur nisi juxta differentiam linearum à punctis arcûs DC ad M centrum motûs ductarum. Ex quo sit movendi facilitatem subinde augeri, cæteris paribus. Hoc autem ut planiùs explicetur, sit arcus LQ in quinque æquales partes divisus, & singulæ sint gr. 3. linea HL transeat per centrum I, & ex H ducantur rectæ HM, HN, HO &c; Certum est lineas hasce omnes ex Q ad L semper majores esse, & maximam esse HL ex 7.lib.3. atque assumptâ HF æquali ipsi HQ, differentiam totam esse FL. Quare si LQ sit latus cunei inflexi, potentia describens arcum æqualem arcui LQ haberet motum, qui ad motum ponderis esset ut arcus LQ ad rectam FL, seu QS. Sed quoniam centrum motûs est H, potentia circa illud describit arcum LS, cujus quantitas innotescit, si dato IL, hoc est IQ, Radio, atque distantiâ IH, NNnn

Transcription: Translated (English)

Book Seven. Chapter II. 649 than from the apex of the wedge C. These iron hooks, by which the two bodies are fastened together, or fastened back again, plainly seem to look to this; as when cabinets, windows, or boxes are closed and reopened. Therefore, once DM is rightly understood as a part of the diameter of the circle whose arc exerts the force of the wedge, if this be the arc DC, its center must lie between the extremity D and the point M, the center of motion; but if it be the arc BC, between A, the center of the circle, and the extremity B, the center of motion H must be interposed. For from that center M, in the first case, the body to be moved ought to be withdrawn; and to this center H the body to be drawn ought to approach. But if the straight line DM were not a part of the diameter passing through the centers of the arc and of motion, still it must be nearer to such a diameter than is the straight line drawn from the center of motion to C; which is clear from book 3, proposition 7. Therefore, the center of motion being fixed in a convenient place, and the power applied at D to impel the wedge BCD, the force, describing about M the center of motion the circular arc DE, does not push the weight except according to the difference of the lines drawn from the points of the arc DC to the center M of motion. From this it follows that the ease of moving is thereby increased from time to time, other things being equal. Now, to explain this more clearly, let the arc LQ be divided into five equal parts, and let each be 3 degrees. Let the line HL pass through the center I, and from H let the straight lines HM, HN, HO, etc. be drawn. It is certain that all these lines are always greater than the line from Q to L, and that HL is the greatest, from book 3, proposition 7; and if HF be taken equal to HQ, the whole difference will be FL. Therefore, if LQ were the side of the bent wedge, the force describing an arc equal to the arc LQ would have a motion which would be to the motion of the weight as the arc LQ is to the straight line FL, or QS. But since the center of motion is H, the force describes around it the arc LS, whose magnitude is known if IL, that is IQ, the radius, and the distance IH, are given. NNnn

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Mechanicorum cum ex hypothesi notus sit angulus LIQ, investigetur angulus IHQ, quem metitur arcus LS; hujus autem quantitas prodit ex datis partibus Radij HL. Quare angulus LIQ sit gr. 15. IL partium 10000, IH partium 5000. Igitur in triangulo HIQ angulus IHQ est gr. 10. 0. 45: atque si ex Cyclometricis ineatur Ratio mensuræ arcûs LS, invenietur in partibus Radij IL 10000 ferè par arcui LQ; hic est partium 2618, ille 2621. Quapropter in tam exiguâ circuli portione perinde est arcum LQ, atque arcum LS considerare. Singulæ itaque partes quintæ arcûs descripti sunt particularum 524. Jam verò per Trigonometriam, ex datis lateribus HI 5000, & IM 10000, atque angulo comprehenso HIM, ut pote supplemento ad duos rectos noti anguli LIM ex hypothesi gr. 3. (similiter in reliquis triangulis eadem sunt latera, & angulus comprehensus sensim per gr. 3. minuitur) inveniatur linearum longitudo; & est in iisdem Radij IL partibus 10000 linea HQ 14885, HP 14927, HO 14959, HN 14981, HM 14995, atque demum HL 15000. Sunt igitur linearum ex Q ad L incrementa inæqualia, videlicet 42, 32, 22, 14, 5, quibus respondet motus ponderis cuneo propulsi, qui semper decrescit, dum potentia motus æquales perficit. Ratio proinde motûs potentiæ ad motum ponderis initio, dum cunei pars QP subinde ponderi applicatur, atque Potentia venit ex L in M, est ut 524 ad 42, deinde in PO ut 524 ad 32, in ON ut 524 ad 22, in NM ut 524 ad 14, in ML ut 524 ad 5. Cum itaque semper major fiat Ratio motuum, augetur movendi facilitas; atque perinde fit, ac si acutior semper atque acutior cuneus adhiberetur. Hîc tamen observandum est ita temperandum esse movendi facilitatem cum ipso ponderis motu, ut illam consectando hoc minùs moveri non contingat, quàm par fuerit: quò enim punctum, quod est centrum motûs, minùs abest ab I centro arcûs LQ, eò quidem faciliùs movetur pondus, quia ad hujus motum potentiæ motus majorem habet Rationem, sed à pondere minus spatium percurritur. Nam si centrum motûs sit G, & IG partium 2500, quarum IQ est 10000, linea GQ est 12432, GP 12456, GO 12475, GN 12489, GM 12497, GL 12500: atque adeò linearum incrementa sunt 24, 19, 14, 8, 3; cum tamen quinta pars arcûs intervallo GL descripti à potentiâ

Transcription: Translated (English)

Mechanics. Since, by hypothesis, the angle LIQ is known, let the angle IHQ be investigated, which is measured by the arc LS; but the quantity of this arises from the given parts of the radius HL. Therefore, let the angle LIQ be 15 degrees, IL parts 10000, IH parts 5000. Hence in triangle HIQ the angle IHQ is 10 degrees 0. 45: and if, from the Cyclometrica, the ratio of the measure of arc LS be determined, it will be found in parts of radius IL 10000 that the arc LQ is nearly equal; this is of 2618 parts, that 2621. Wherefore in so small a portion of the circle it is much the same to consider the arc LQ and the arc LS. Therefore the single parts of the fifth part of the described arc are 524 parts. Now by Trigonometry, from the given sides HI 5000, and IM 10000, and the included angle HIM, being the supplement to two right angles of the known angle LIM by hypothesis, 3 degrees, (similarly in the remaining triangles the same sides are given, and the included angle gradually diminishes by 3 degrees) let the length of the lines be found; and in the same parts of radius IL 10000 there is line HQ 14885, HP 14927, HO 14959, HN 14981, HM 14995, and finally HL 15000. Therefore the increments of the lines from Q to L are unequal, namely 42, 32, 22, 14, 5, to which corresponds the motion of the weight driven by the wedge, which always decreases while the force of motion performs equal intervals. The ratio, therefore, of the motion of the force to the motion of the weight at the beginning, while the part QP of the wedge is being applied successively to the weight, and the Force comes from L to M, is as 524 to 42, then in PO as 524 to 32, in ON as 524 to 22, in NM as 524 to 14, in ML as 524 to 5. Since, therefore, the ratio of the motions becomes ever greater, the ease of moving increases; and it is as though a sharper and yet sharper wedge were continually employed. Here, however, it is to be observed that the ease of moving must be so adjusted to the motion itself of the weight, that in following it the weight may not happen to be moved less than is proper: for the nearer the point, which is the center of motion, is to the center I of the arc LQ, the more easily is the weight moved, because the motion of the force has a greater ratio to the motion of the weight, but by the weight a smaller space is traversed. For if the center of motion be G, and IG parts 2500, of which IQ is 10000, the line GQ is 12432, GP 12456, GO 12475, GN 12489, GM 12497, GL 12500: and so the increments of the lines are 24, 19, 14, 8, 3; although nevertheless the fifth part of the arc described by the interval GL by the force

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Liber septimus. CAPUT II. 651 tiâ sit proximè 524; Major est autem Ratio 524 ad singula hæc linearum incrementa, quàm cum motûs centrum est H. Cum hac tamen movendi facilitate connectitur exiguus ponderis mo- tus; nam inter 12432 & 12500, quæ sunt extremæ lineæ GQ & GL, differentia 68 minor est quàm differentia 115 inter HQ 14885 & HL 15000, quæ differentia inter extreamas lineas me- titur ponderis motum: est siquidem differentia inter aggrega- tum laterum & basim trianguli HIQ, aut GIQ, mensura, juxta quam pondus promovetur impulsu cunei. Cum verò IQ & IL, utpote semidiametri, æquales sint, basis autem HQ major sit basi GQ (nam in triangulo HGQ amblygonio basis HQ opponi- tur majori angulo) fieri non potest, ut eadem sit motûs ponderis mēsura æqualis ipsis FL aut QS, nisi ab assumpto motûs centro descriptus arcus (intervallo usque ad Q punctum illi centro pro- ximum) transeat per extremitates easdem Q & F, per quas tran- siret arcus ex H intervallo HQ descriptus. Quare duo circuli se in duobus punctis secantes communem haberent rectam li- neam QF, ad quam bifariam sectam in V perpendicularis VH transiret per utriusque circuli centrum, ex 3. lib. 3. ac proinde, cum ex. 5. lib. 3. non habeant idem centrum H, alterius circuli centrum esset extra rectam HI, puta in T. Utrùm autem dato eodem motûs centro, & datâ pari arcûs portione, præstet arcum esse majoris, an minoris, circuli partem, vix est dubitandi locus. Quando enim duo circuli idem planum in eodem puncto contingunt, peripheria majoris interjicitur in- ter planum datum & peripheriam minoris circuli; atque adeò ab eodem motûs centro lineæ ad illam majoris circuli periphe- riam ductæ omnes secant peripheriam minoris, ac propterea, ut- pote longiores, majorem efficiunt pressionem, longiúsque pro- pellunt corpus, quod impellitur. Hinc si viribus potentia abun- det, & ad majus spatium protrudere oporteat pondus, adhiben- da est peripheria majoris circuli; contra verò minore utendum est, si parum movendum sit, & potentia imbecillior. Porrò hîc exerceri cunei vires, quis ambigat? neque enim admodum interest, plana ne? an inflexa? sit ejus facies, modò ex ejus interjectu duo disjuncta corpora magis invicem remo- veantur, sive utrumque simul in diversas partes abeant, sive al- tero manente, alterum tantummodo moveatur. Hîc autem im- NNnn 2

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Book Seven. CHAPTER II. 651 thus the ratio is almost 524; but the ratio 524 is greater in each of these incremental lines than when the center of motion is H. Yet with this ease of motion is joined a slight movement of the weight; for between 12432 and 12500, which are the extreme lines GQ and GL, the difference 68 is less than the difference 115 between HQ 14885 and HL 15000, which difference measures the movement of the weight between the extreme lines: for it is indeed the difference between the sum of the sides and the base of triangle HIQ, or GIQ, a measure according to which the weight is advanced by the impulse of the wedge. But since IQ and IL, as semidiameters, are equal, while the base HQ is greater than the base GQ (for in the obtuse triangle HGQ the base HQ is opposed to the greater angle) it cannot happen that the same measure of the movement of the weight should be equal to FL itself or QS, unless the arc described from the assumed center of motion (at the interval up to point Q, nearest to that center) pass through the same extremities Q and F, through which the arc described from H at interval HQ would pass. Wherefore the two circles, intersecting each other at two points, would have in common the straight line QF, which, being bisected at V, the perpendicular VH would pass through the center of each circle, by book 3, proposition 3; and therefore, since by book 3, proposition 5, they do not have the same center H, the center of one circle would be outside the line HI, namely in T. But whether, given the same center of motion and an equal portion of arc, it is better that the arc be a part of the greater or of the lesser circle, there is scarcely room for doubt. For when two circles touch the same plane at the same point, the circumference of the greater lies between the given plane and the circumference of the lesser circle; and thus from the same center of motion, the lines drawn to the circumference of that greater circle all cut the circumference of the lesser, and therefore, being longer, produce greater pressure and propel the body farther. Hence, if power abounds in the forces, and the weight must be driven farther, the circumference of the greater circle should be used; on the other hand, the lesser should be used if only a little movement is required and the power is weaker. Moreover, who would doubt that here the force of the wedge is exercised? For indeed it makes little difference whether its face be flat or curved, provided that by its insertion two separated bodies are moved more apart from one another, whether both depart at once in different directions, or one remaining, only the other is moved. Here however im- NNnn 2

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Mechanicorum 652 motum manet centrum, circa quod vertitur portio circuli excentrici, quæ sive simplici impulsion, sive etiam percussione, adacta urget corpus, quod contingit, neque aliter quàm si inter validum stipitem humi defixum, atque pondus interjiceretur vulgaris cuneus planus. Verùm quamvis hactenus potentiam in ipsa cunei inflexi extremitate posuerimus ad explicandum ejus motum, nihil tamen refert: nam si etiam circa medium cuneum fuerit ansa, qua arreptâ ille valeat circumduci, perinde est; motus siquidem potentiæ ad ponderis motum eandem servat Rationem. Ex quo manifestò deprehenditur nullam esse in Cuneo Vectis umbram; in Vecte siquidem certus est potentiæ locus, quo mutato etiam momenta variantur: at in hujusmodi Cuneo non contingit momentorum mutatio, cujuscumque tandem cunei parti applicetur potentia, dummodo ea sit dispositio, ut vires suas æquè exercere valeat, sive in hac, sive in illâ arcûs extremitate, hoc est ad L aut Q, sive circa medium ad O, aut N, constituatur. Cave tamen putes æquè liberum esse in majore aut minore distantiâ à centro motûs H aut G potentiam collocare: id enim sanè perperam fieret; pro Ratione siquidem distantiæ à centro motûs majorem aut minorem arcum potentia suo motu describeret: esto nihil intersit, cuinam parti applicetur, servatâ eâdem à prædicto motûs centro distantiâ. Propterea si ad Q applicetur potentia cuneum trahens, ansa ejusmodi apponenda est sursum recurva, cui applicata potentia non minorem arcum describat, quàm si illa applicaretur puncto L impellens cuneum: non est scilicet par potentiæ motus, qui fit intervallo H Q, ac intervallo H L. Similiter autem Cuneo plano uti licebit, cujus latera si ferreo paxillo hinc atque hinc extante trajeceris, ut arreptis utrâque manu paxilli extremitatibus cuneum adducere valeas, aut impellere, duo corpora, quibus cuneus interjicitur, disjunges: immò si fissili ligno bicubitali juxta staminum ductum cuneum eumdem ita per vim immiseris, ut cuneum elevatum sequatur pariter & lignum, tùm ligni calce saxum percusseris, cuneus scissionem promovebit, quocumque tandem in loco sive juxta ipsius cunei apicem, sive juxta basim immissus fuerit paxillus ille, cui potentia applicatur. CAPUT

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Mechanics 652 the center remains fixed, around which turns a portion of the eccentric circle, which, whether by a simple push or even by a blow, drives the body that it touches, and does so in no other way than if a common flat wedge were inserted between a strong stake fixed in the ground and a weight. But although up to now we have placed the power at the very end of the bent wedge in order to explain its motion, it makes no difference; for if there were also a handle near the middle of the wedge, by seizing which it could be turned around, the effect is the same. For the motion of the power keeps the same relation to the motion of the weight. From this it is clearly understood that there is no shadow of a lever in the wedge. In a lever, indeed, there is a definite place for the power, which when changed also changes the moments; but in a wedge of this kind no change of moments occurs, whichever part of the wedge the power is applied to, provided the arrangement be such that it can exert its force equally, whether it be placed at this or that end of the arc, that is, at L or Q, or around the middle at O or N. But be careful not to think that it is equally free to place the power at a greater or lesser distance from the center of motion H or G: for that would surely be done incorrectly; since, in proportion to the distance from the center of motion, the power would describe by its motion a greater or smaller arc. Thus it makes no difference to which part it is applied, provided the same distance from the said center of motion is preserved. Therefore, if the power pulling the wedge is applied at Q, a handle of this kind must be attached, curved upward, so that the power applied to it may describe no smaller an arc than if it were applied at point L, pushing the wedge; that is, the motion of the power is not the same when it occurs through the interval HQ as through the interval HL. In the same way, however, it will be possible to use a flat wedge, whose sides, if you pass through them an iron pin projecting here and there, so that by grasping the ends of the pin with both hands you may be able to draw the wedge or push it, you will separate the two bodies between which the wedge is inserted; indeed, if into split wood two cubits long, along the line of the grain, you forcibly drive in the same wedge so that the raised wedge follows the wood as well, then, if you strike the wood with the heel of your shoe, the wedge will promote the splitting, wherever that pin to which the power is applied has been inserted, whether near the point of the wedge or near its base. CHAPTER

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Liber septimus. CAPUT III. 653 CAPUT III. Cuneus perpetuus circulo excentrico effingitur. C Uneum Perpetuum voco non eum, qui perpetuâ, hoc est majore semper atque majore impulsionem corpus propellat longiùs atque longiùs; id enim, ut satis clarum est, infinitam exigeret cunei longitudinem: sed eatenus dico Perpetuum, qua- tenus potentia illi semel applicata institutum motum juxta eandem directionem perpetuare potest: id quod neutiquam contingit secundùm rectam lineam, quæ spatium requireret infinitum perpetuo illo motu percurrendum; sed potentia in orbem progrediens, & cuneum contorquens, motus alternos perficit. Et primùm quidem id fieri potest circulo, cujus motûs centrum absit ab ejusdem circuli centro; vis enim cunei erit ad propellendum corpus intervallo duplo intervalli centrorum; momentum verò potentiæ desumetur ex semiperipheriâ circuli, cujus Radius æqualis sit dati circuli semidiametro auctæ centrorum illorum intervallo; si tamen extremitati à centro motûs maximè distanti ipsa potentia applicetur. Sit datus circulus B E D, cujus centrum A: fiat motûs centrum C, circa quod in gyrum agatur constitutus circulus. In hoc motu duo circuli concentrici describuntur; alter quidem à puncto D, Radio C D, alter verò à puncto B, Radio C B. Quare dum potentia ex B per G venit in H, punctum D per K venit in L, & corpus, quod puncto D applicitum erat, à peripheriâ circuli B E D sensim propellitur, donec veniat ex D in H. Est autem D H æqualis ipsi B L, quia ex æqualibus C H & C B auferuntur æquales C D NNnn 3

Transcription: Translated (English)

Book Seven. CHAPTER III. 653 CHAPTER III. A perpetual wedge is formed by an eccentric circle. I call a wedge perpetual, not one which by a perpetual, that is, always greater and greater impulse drives a body farther and farther; for that, as is clear enough, would require an infinite length of wedge: but I call it perpetual insofar as the power once applied to it can perpetuate the motion begun in the same direction; which by no means happens in a straight line, which would require an infinite space to be traversed by that perpetual motion; but a power advancing in a circle, and turning the wedge, produces alternating motions. And first of all, indeed, this can be done by a circle whose center of motion is distant from the center of the circle itself; for the force of the wedge will be for propelling the body at a distance double the distance between the centers; but the momentum of the power will be taken from the semicircumference of the circle, whose radius shall be equal to the semidiameter of the given circle, increased by the distance between those centers; provided, however, that the power itself be applied to the extremity farthest from the center of motion. Let the given circle B E D be assumed, whose center is A; let the center of motion be C, about which the established circle is carried around in a revolution. In this motion two concentric circles are described; one indeed from the point D, with radius C D, the other from the point B, with radius C B. Therefore while the power passes from B through G to H, the point D through K comes to L, and the body, which was applied at point D, is gradually propelled from the circumference of the circle B E D, until it comes from D to H. Now D H is equal to B L itself, because equal C H and C B have equal C D and NNnn 3 taken away from them.

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Mechanicorum & CL: inter diametros verò BD & LD differentia est BL: igitur quia diametrorum differentia dupla est differentiæ semi- diametrorum, BL est dupla ipsius AC differentiæ semidiametrorum AD & CD. Quapropter etiam DH spatium, quod à pondere propulso percurritur, duplum est intervalli centrorum AC. Demùm potentia in B, ex hypothesi, applicata momentum habet juxta Rationem semiperipheriæ BGH ad spatium DH duplum intervalli centrorum AC: hæc siquidem est Ratio motuum potentiæ & ponderis. Hinc si ponatur dati circuli Radius AB 100, & centrorum distantia AC 13, erit DH 26: At semiperipheria BGH ad suum Radium BC est ut 355 ad 113; igitur BGH ad DH est ut 355 ad 26. Quare, cæteris paribus, quò majus est centrorum intervallum, eò majores requiruntur in potentiâ vires; quia hujus intervalli duplum est spatium, per quod impellitur pondus, manente eodem potentiæ motu. Est aute attentè considerandu[m], utrum præstet, cæteris paribus, majore circulo uti: Cæteris, inquam, paribus, ut scilicet idem sit centrorum intervallum, & eadem potentiæ à cétro motûs distantia. Et primò observandu[m] est cuneum esse LBD, cujus vertex est angulus cōtingentiæ factus à peripheriâ dati circuli, & à peripheriâ circuli, quem circa centrum C in motu describit extremitas D. Deinde semiperipheria BGH à potentiâ descripta in motu (& est ex hypothesi partiu[m] 355, quaru[m] Radius CB est 113) dividatur in partes æquales duodecim, ita ut singulæ respondeant gradibus 15, & singulis competant partes 29 1/2. Similiter semiperipheria DOL in 12 æquales partes singulas gr.15. dividatur: adeò ut cùm linea BD circa punctum C circumacta angulum gr.15 descripserit, potentia sit progressa per partes 29 1/2. Examinandum est, quanto spatio interim propellatur pondus, quod erat in D, versùs H. Sit angulus DCO, hoc est arcus DO, gr.15: ducta intelligatur recta CO usque in N peripheriam dati circuli; est igitur ON motus ponderis ex D versus H. Quapropter investiganda est ipsius CN longitudo, ut appareat ejusdem excessus supra CD. Ducatur dati circuli Radius AN notus partium 100; datur item intervallum AC partium 13; notus est angulus ACN gr.165: ergo per Trigonometriam innotescit primò angulus CNA gr.1.55.41; atque ex eo reliquus angulus NAC gr.

Transcription: Translated (English)

Mechanics & CL: but the difference between the diameters BD & LD is BL: therefore, since the difference of the diameters is double the difference of the semi- diameters, BL is double the difference AC of the semidiameters AD & CD. Wherefore also the space DH, which is traversed by the driven weight, is double the interval AC of the centres. Finally, the power applied at B, by hypothesis, has a momentum according to the ratio of the semicircumference BGH to the space DH double the interval AC of the centres: for this is the ratio of the motions of the power & the weight. Hence if, for a given circle, the radius AB be taken as 100, and the distance AC of the centres as 13, then DH will be 26: but the semicircumference BGH to its radius BC is as 355 to 113; therefore BGH to DH is as 355 to 26. Wherefore, other things being equal, the greater the interval of the centres, the greater the forces required in the power; because double this interval is the space through which the weight is impelled, the motion of the power remaining the same. But it must be carefully considered whether, other things being equal, it is better to use a larger circle: other things, I say, being equal, namely that the interval of the centres be the same, and the same distance of the power from the centre of motion. And first it must be observed that the wedge is LBD, whose vertex is the angle of contact made by the periphery of the given circle and by the periphery of the circle which the extremity D describes in motion around the centre C. Then the semicircumference BGH, described by the power in motion (& which, by hypothesis, is of parts 355, whose radius CB is 113), is divided into twelve equal parts, so that each corresponds to 15 degrees, and each part is 29 1/2. Likewise let the semicircumference DOL be divided into 12 equal parts of 15 degrees each: so that when the line BD, being turned around the point C, shall have described an angle of 15 degrees, the power will have advanced through 29 1/2 parts. It must be examined how far in the meantime the weight, which was in D, is impelled towards H. Let the angle DCO be, that is the arc DO, 15 degrees: let the straight line CO be drawn, as far as the periphery of the given circle, to N; thus ON is the motion of the weight from D towards H. Wherefore the length of CN itself must be investigated, in order that its excess above CD may appear. Let the radius AN of the given circle be drawn, known as 100 parts; the interval AC of 13 parts is also given; the angle ACN of 165 degrees is known: therefore by Trigonometry there is first made known the angle CNA of 1.55.41 degrees; and from it the remaining angle NAC of gr.

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Liber septimus. CAPUT III. 655 gr. 13. 4'. 19", hoc est arcus ND: deinde habetur longitudo CN partium 87 38/100, quarum CO, hoc est CD est 87: igitur ON est 38/100. Quod si arcus DO ponatur gr. 30, in triangulo ACN dantur eade latera AN 100, & AC 13, & angulus ACN gr. 150: igitur invenitur CNA gr. 3. 43'. 37"; atque angulus NAC, hoc est arcus ND gr. 26. 16'. 23", & linea CN par- tium 88 52/100: igitur ON est part. 1 52/100. At arcus DO sit gr. 45; est angulus ACN gr. 135: datis iisdem lateribus AN 100, & AC 13, invenitur angulus CNA gr. 5. 16'. 27", atque angu- lus NAC, hoc est arcus ND gr. 39. 43'. 33", & linea CN part. 90 38/100: igitur ON part. 3 38/100. Similiter si DO sit gr. 60. invenitur ON part. 5 86/100; si verò fuerit gr. 75, est ON 8 84/100: si gr. 90, est ON 12 15/100. Mutetur jam hypothesis, & circuli dati Radius sit duplex, scilicet AD, hoc est AN, partium 200, quarum AC est 13. Sit arcus DO gr. 15: invenitur angulus CNA gr. 0. 57'. 51": atque angulus NAC gr. 14. 2'. 9": ac proinde linea GN partium 187 40/100, quarum CD, hoc est CO, est 187; quare ON est 40/100. Sit deinde arcus DO gr. 30: in- venitur angulus CNA gr. 1. 5 1'. 45", & angulus NAC, hoc est arcus DN, gr. 28. 8'. 15", atque demum linea CN part. 188 63/100: igitur ON part. 1 63/100. Denique arcus DO sit gr. 45: deprehenditur angulus CNA gr. 2. 38'. 4". angulus NAC, hoc est ar- cus DN, gr. 42, 21'. 56"; & linea CN part. 190 59/100; atque adeò ON part. 3 59/100. Si DO sit gr. 60, ON est part. 6 18/100; si DO sit gr. 75, ON est 9 35/100. Si sit gr. 90, ON est 12 57/100. Ex his manifestò constat initio motûs in primo quadrante à circulo majore paulò ampliùs propelli pondus ex D versùs H, quàm à circulo minore, datâ angulorum motûs ad centrum C paritate. Verùm in circulo majoris diametri non solùm pari graduum numero respondet longior arcus pro Ratione diametrorum, sed etiam, ut ex superioribus calculus constat, major circulus plures gradus ponderi coaptat, quàm minor. Sic in motu ad centrum C gr. 15, circulo minori, cujus Radius 100, competunt gr. 13. 4'. 19"; at circulo majori, cujus Radius 200, competunt gr. 14. 2'. 9". Quare præterquam quod duplex est longitudo

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Book Seven. Chapter III. 655 at 13 gr. 4'. 19", that is, arc ND: then the length CN is found to be 87 38/100 parts, of which CO, that is CD, is 87: therefore ON is 38/100. But if arc DO be taken as 30 gr., in triangle ACN the same sides are given, AN 100, and AC 13, and angle ACN gr. 150: therefore angle CNA is found to be 3. 43'. 37"; and angle NAC, that is arc ND gr. 26. 16'. 23", and line CN 88 52/100 parts: therefore ON is 1 52/100 parts. But let arc DO be 45 gr.; angle ACN is 135 gr.: with the same sides AN 100, and AC 13 given, angle CNA is found to be 5. 16'. 27", and angle NAC, that is arc ND gr. 39. 43'. 33", and line CN 90 38/100 parts: therefore ON 3 38/100 parts. Similarly if DO be 60 gr., ON is found to be 5 86/100 parts; if it be 75 gr., ON is 8 84/100: if 90 gr., ON is 12 15/100. Now let the hypothesis be changed, and let the Radius of the given circle be double, namely AD, that is AN, 200 parts, of which AC is 13. Let arc DO be 15 gr.: angle CNA is found to be 0. 57'. 51": and angle NAC gr. 14. 2'. 9": and therefore line GN is 187 40/100 parts, of which CD, that is CO, is 187; wherefore ON is 40/100. Let arc DO then be 30 gr.: angle CNA is found to be 1. 5 1'. 45", and angle NAC, that is arc DN, gr. 28. 8'. 15", and finally line CN 188 63/100 parts: therefore ON 1 63/100 parts. Lastly let arc DO be 45 gr.: angle CNA is found to be 2. 38'. 4". angle NAC, that is arc DN, 42, 21'. 56"; and line CN 190 59/100 parts; and thus ON 3 59/100 parts. If DO be 60 gr., ON is 6 18/100 parts; if DO be 75 gr., ON is 9 35/100. If it be 90 gr., ON is 12 57/100. From these things it is manifestly evident that at the beginning of motion in the first quadrant, the weight is propelled somewhat more strongly from D toward H by the larger circle than by the smaller circle, given the equality of the angles of motion at center C. But in the circle of greater diameter not only does a longer arc correspond to the same number of degrees in proportion to the diameters, but also, as the calculation above shows, the larger circle adapts more degrees to the weight than the smaller. Thus, in motion toward center C of 15 gr., to the smaller circle, whose Radius is 100, there belong 13. 4'. 19" gr.; but to the larger circle, whose Radius is 200, there belong 14. 2'. 9". Wherefore, besides the fact that the length is double

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656 Mechanicorum longitudo arcus majoris, quia duplex est Radius, adhuc superest longitudo gr. 0. 57'. 50": cum tamen motus ponderis in minore sit 38/100, in majore 40/100; quod discrimen 2/100 longè minus est illo excessu arcûs. Quapropter si Potentia peripheriæ dati circuli partibus subinde applicetur (ut si extarent ad orbitam paxilli perpendiculares) patet in majore circulo haberi majora momenta; multo magis, si applicetur juxta maximam à motûs centro distantiam; id quod fieri expedit, si nihil obsit: At si Potentia à centro motûs æquè absit in majore atque in minore circulo, non habetur hoc momentorum compendium, quod ex distantia facilè obtineri posset. Si itaque corpus ex D in H impulsum aut vi elasticâ restituere se possit ex H in D, aut illud sublevatum (si circulus fuerit in plano Verticali) suâ gravitate descendere valeat ex H in D, paulatim in priorem locum redibit, cum potentia transgressa punctum H per I se restituet in B: potentia igitur perpetuò in gyrum circumactâ, circulum similiter versando, corpus illud in motu reciprocando servabit constantiam. Quod si virtute elasticâ præditum sit corpus impulsum ex D in H, illa pariter, præter insitam corpori gravitatem, movendi difficultatem augebit, quippe cui vis inferenda est, quam deinde excutere valeat. Quare satius fuerit omnem virtutem elasticam amovere (si id quidem fieri possit) ut sola gravitatis resistentia superanda sit. Ut autem corpus ultro citróque remeare possit ex D in H, & vicissim ex H in D, regula statuatur in Verticali plano erecta, sed versatilis circa axem, aut annulum alteri ejusdem regulæ extremitati infixum, & reliqua regulæ extremitas mobilis occurrat circulo in B: Tum funiculo longitudine diametrum BD æquante connectatur regula cum pondere movendo; sic enim fiet ut regulæ extremitas devenerit in L, quando pondus fuerit in H; atque propellendo regulam ex L in B, pondus ex H trahetur in D, & perpetua vicissitudine tum regula, tùm pondus à circumacto cuneo impellentur: semper verò resistentia orientur ex ponderis modò impulsi, modò attracti gravitate; regula siquidem per se nihil obsistit, sed quatenus cum pondere trahendo conjungitur. Verùm qua positione collocandus sit fu- niculus

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656 Mechanicorum the length of the greater arc, because the radius is double, there still remains the length of 0° 57' 50": although the motion of the weight in the smaller circle is 38/100, in the greater 40/100; and this difference of 2/100 is far less than that excess of the arc. For this reason, if the Power is applied successively to parts of the circumference of a given circle (as if there were pegs set perpendicular to the orbit), it is clear that in the greater circle there are greater moments; and much more so, if it is applied near the greatest distance from the center of motion; and this it is expedient to do, if nothing stands in the way. But if the Power is equally distant from the center of motion in the greater as in the smaller circle, this saving of moments is not obtained, which could easily be gained from the distance. Therefore, if a body, impelled from D to H, can either restore itself by elastic force from H to D, or, being lifted (if the circle were in a vertical plane) can descend by its own gravity from H to D, it will gradually return to its former place, since the power, having passed the point H through I, will restore itself to B: the power, then, being perpetually turned round in a circle, will likewise, by turning the circle, preserve that body in reciprocating motion. But if the body, impelled from D to H, is endowed with elastic force, that force likewise, in addition to the body’s inherent gravity, will increase the difficulty of moving it, since force must be applied to it, which it may afterward be able to throw off. Therefore it will be better to remove all elastic force (if this can indeed be done), so that only the resistance of gravity need be overcome. And so that the body may be able to go back and forth of its own accord from D to H, and vice versa from H to D, let a rule be set upright in a vertical plane, but movable around an axis, or an annulus fixed to one end of the same rule, while the other end of the rule, being movable, meets the circle at B. Then let the rule be connected by a cord whose length equals the diameter BD with the weight to be moved; thus it will come about that the end of the rule arrives at L when the weight is at H; and by pushing the rule from L to B, the weight will be drawn from H to D, and by perpetual alternation both the rule and the weight will be driven by the revolving wedge: but resistance will always arise from the gravity of the weight, now impelled, now drawn; for the rule by itself offers no resistance, except insofar as it is joined with the weight to be pulled. But in what position the cord should be placed

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Liber septimus. CAPUT IV. 657 niculus circuli diametro BD respondens, an supra, an infra cir- culum BED, quid opus est explicare? satis enim cuique mani- festum est attendendum esse, qua ratione ipsi circulo applicetur potentia movens; nam si illum Potentia agitet paxillo in supe- riori aut inferiori facie extremæ orbitæ infixo, patet funiculum adversæ faciei respondere, ne in illum paxillus incurrat. Sin autem potentia circulo non proximè adhæreat, nec illum tan- gat, quia ex centro C exeunti axi additum est manubrium cir- culo parallelum, aut Vectis, aut circulus alius eidem parallelus, liberum erit funiculum alterutri circuli faciei respondentem collocare, ex neutrâ scilicet parte impedimento esse potest potentiæ se in gyrum contorquenti. Ne me verò carpendum puta, quòd integrum circulum FBED proposuerim, cum satis esse possit segmentum paulo majus semicirculo BED (ut scilicet sit locus axi infigendo in C motûs centro) cui tota vis impellendi sive pondus, sive regu- lam, tribuenda est. Eo consilio integrum circulum FBED proposui, ut liberum potentiæ sit sive in dexteram, sive in si- nistram motum instituere, atque promiscuè uti modò cuneo inflexo BED, modò BFD, prout commodius acciderit. Dein- de si semicirculo tantùm BED utamur, & vis elastica interve- niat, aut gravitas sublevata recidat, ubi potentia venerit in H, & semicirculi BED sit facta positio HML, fieri non potest, ut potentia versùs I procedat, quin illicò & quasi momento pon- dus redeat ad D; hujusmodi verò motus adeò velox vix contin- gere sæpiùs potest citra aliquod detrimentum; cui periculo occurritur, si integer fuerit circulus FBED, sensim enim sit regressus ex H in D. CAPUT IV. Ex Cylindro construi potest Cuneus perpetuus. A ltera species Cunei perpetui desumi poterit ex Cylindro Recto obliquè secto, erit siquidem sectio Ellipsis; & potis- OOO

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Book Seven. CHAPTER IV. 657 What need is there to explain whether the little circle corresponding to the diameter BD is above or below the circle BED? For it is sufficiently evident to anyone that attention must be paid to the manner in which the moving power is applied to the circle itself; for if the power drives it by a peg fixed in the upper or lower face of the outer rim, it is clear that the cord must correspond to the opposite face, lest the peg strike against it. But if the power does not adhere closely to the circle, nor touch it, because to the axle emerging from the center C there has been added a handle parallel to the circle, or a Lever, or some other circle parallel to it, it will be free to place the cord corresponding to either face of the circle, since from neither side can there be any obstacle to the power twisting itself into a revolution. Do not think, however, that I am to be blamed because I have proposed the complete circle FBED, when it may be enough to use a segment a little larger than the semicircle BED (so that there may be room for the axle to be fixed at C, the center of motion), to which the whole force of the driving, whether of a weight or of a ruler, is to be attributed. I proposed the complete circle FBED with this purpose, that it may be free for the power to set the motion either to the right or to the left, and to use alternately now the bent wedge BED, now BFD, as may prove more convenient. Then, if we use only the semicircle BED, and an elastic force intervene, or a raised weight fall back, when the power has come to H, and the position HML of the semicircle BED has been made, it is not possible for the power to advance toward I without at once, and as if in an instant, the weight returning to D; but a movement of this sort, being so swift, can scarcely happen often without some loss; this danger is avoided if the circle FBED is complete, for then the return from H to D is gradual. CHAPTER IV. From a Cylinder a Perpetual Wedge can be constructed. Another kind of perpetual Wedge may be taken from a right Cylinder cut obliquely; for the section will be an Ellipse; and the possibilitOOO

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Mechanicorum simùm inserviet ad deprimendum, si cylindrus fuerit horizonti perpendicularis, atque addito vecte circumagatur Ergatæ in morem: Sin autem cylindrus sit horizonti parallelus, inserviet ad propellendum pondus, & Radios admittet quemadmodum Sucula. Sit cylindrus BC Rectus, cujus scilicet Axis est ad basim perpendicularis, & obliquè sece- tur plano per DC; sit enim sectio DECF ellipsis. Quod si hujus- modi Ellipsis planities officere pos- sit motui, interiores partes aliquan- tulùm excavari oportebit, quando- quidem sufficit limbus perimetri, modò sit solidus, & satis validus; quem & ferreâ laminâ non ruditer politâ munire operæ pretium fue- rit. Ut autem hujusmodi sectionis obliquitas major aut minor opportunè fiat, statuenda est primùm mensura depressionis aut impulsionis, qua movendum est pondus, & sit ex. gr. C A, cui æqualis sumatur ID: tùm ex D in C fiat sectio, & erit consti- tutus cuneus A D F C, cujus latus unum est cylindri semipe- ripheria A D, aliud D F C semiperimeter Ellipsis, cujus partes ponderi in D constituto subinde applicantur ex convolutione cylindri, & quatenus ab A D recedunt, pondus deprimunt, aut impellunt, donec demum à puncto C attingatur pondus propul- sum ex D in I. Quoniam verò Ellipticum limbum ferreâ laminâ muniendum dixi, non omninò abs re fuerit indicare, qua methodo illius perimetrum indagare possimus, ut laminæ in ellipticam figu- ram inflectendæ longitudo innotescat. In circulo quidem nota est aliqua Ratio diametri ad peripheriam, sive ut 7 ad 22, si- ve ut 71 ad 223, sive ut 113 ad 355, sive quæcumque alia ma- gis arrideat majoribus numeris explicata: at in Ellipsi nulla hujusmodi Ratio perimetri ad alterutrum Axem (quod quidem sciam) deprehensa adhuc est: quapropter ad investigandam ejus perimetrum ex datis Axibus variæ tentatæ sunt viæ, quas inire nunc non est operæ pretium. Mihi hanc, utpote perbre- vem, nec à veritatis formâ, quantum res Physica patitur, re- cedentem

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will serve for depressing in the same way, if the cylinder be perpendicular to the horizon, and if, with an added lever, it be turned in the manner of a winch. But if the cylinder be parallel to the horizon, it will serve for propelling a weight, and will admit spokes, in the manner of a drum. Let the cylinder BC be straight, whose axis is, of course, perpendicular to the base, and let it be cut obliquely by the plane through DC; for let the section DECF be an ellipse. But if the flat surface of such an ellipse should hinder the motion, the inner parts ought to be somewhat hollowed out, since it is enough for the rim of the perimeter to remain, provided it be solid and sufficiently strong; and it will be worth the trouble to secure it also with an iron plate not roughly polished. But in order that the obliquity of such a section may be made greater or smaller as needed, the measure of the depression or impulse by which the weight is to be moved must first be determined, and let it be, for example, CA, equal to which let ID be taken: then from D to C let the cut be made, and there will be formed the wedge ADFC, one side of which is the cylinder’s semi-periphery AD, the other DFC the semi-perimeter of the ellipse, whose parts are then applied to the weight placed at D by the revolution of the cylinder, and, as they recede from AD, they depress or impel the weight, until at last from the point C the weight propelled from D to I is reached. And since I have said that the elliptical rim is to be secured with an iron plate, it will not be entirely out of place to indicate by what method we may determine its perimeter, so that the length of the plate to be bent into an elliptical shape may be known. In a circle, indeed, some ratio of the diameter to the circumference is known, whether as 7 to 22, or as 71 to 223, or as 113 to 355, or any other more pleasing ratio expressed by larger numbers: but in an ellipse no such ratio of the perimeter to either axis (so far as I know) has yet been discovered; wherefore, for investigating its perimeter from the given axes, various methods have been tried, which it is not worth the trouble to go into now. I, however, this one, as being very brief and not departing from the form of truth, so far as physical matter allows

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Liber septimus. CAPUT IV. 659 cedentem ineundam suscepi. Illud verò tanquam certum & demonstratum pono, quod Ellipsis, quæ ex Coni sectione, ab ea quæ ex Cylindri sectione oritur, non differt, ut quidam minùs attenti perperam existimârunt; proinde quælibet oblata Ellipsis ad aliquem Cylindrum spectare potest. Ad quem autem cylindrum illa pertineat, facile est determinare ex ipsius Ellipsis Axe minori, qui est æqualis diametro Cylindri. Datis igitur, Axibus, Majore, & Minore, invenire oportet, quanta sit in hujusmodi Cylindro obliquitas sectionis Ellipsis constituens: id quod obtinetur, si ex quadrato Axis Majoris auferatur quadratum Axis Minoris; residui enim Radix quadrata dabit in Cylindri latere longitudinem, qua distant inter se duo plana basi parallela, inter quæ intercipitur sectio obliqua. Sit data Ellipsis DECF, cujus major Axis DC 25, Minor FE 20. Axi FE æqualis est cylindri diameter CI. Igitur planum per axem cylindri ductum habet cum plano obliquè secante communem sectionem DC Axem Ellipsis, & cum cylindri base sectionem facit CI, atque in superficie dat latus DI. Quare est triangulum rectangulum CID, cujus datur hypothenusa DC 25, & basis CI 20: ex ipsius DC quadrato 625 ablatum quadratum ex CI 400, relinquit 225, quadratum perpendiculi DI, quod propterea est 15. Plana igitur CI, & AD parallela distant intervallo CI 15. Cum itaque ex basis diametro CI 20 innotescat ejusdem cylindricæ basis peripheria 62 84/100 proximè, superficies cylindrica ACID manifesta est 942 60/100, cujus semissem 471 30/100 dividit bifariam semiperimeter Ellipsis DEC. Est igitur triangulum rectangulum, cujus latera circa rectum sunt latus DI 15, & cylindricæ basis semiperipheria CHI 31 42/100: horum quadrata 225, & 987 2164/10000 in summam colligantur, & quadrati 1212 2164/10000 Radix 34 82/100 ferè, est Ellipsis semiperimeter DEC, integra verò DECF erit 69 64/100. Quapropter cum plana per AD & CI ex hypothesi sint parallela, etiam DI, & AC æqualia sunt latera. Igitur cum cylindri dati nota sit diameter 20, atque adeò semiperipheria AD 31 42/100, sit data obliquitas, quam metitur AC 15, ex summâ quadratorum rectæ AC, & semiperipheriæ AD, eruatur OOOO 2

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Book Seven. Chapter IV. 659 I undertook the following, which was to be brought to a conclusion. I state this as certain and demonstrated: an Ellipse which arises from the section of a Cone does not differ from one which arises from the section of a Cylinder, as some less attentive persons have wrongly supposed; therefore any Ellipse that is given may be referred to some Cylinder. To which cylinder it belongs is easily determined from the Ellipse’s minor axis, which is equal to the diameter of the Cylinder. Therefore, given the axes, major and minor, it is necessary to find how much obliquity of the section in such a Cylinder constitutes the Ellipse: this is obtained if the square of the major axis is subtracted from the square of the minor axis; for the square root of the remainder will give, in the side of the Cylinder, the length by which the two planes parallel to the base are distant from one another, between which the oblique section is intercepted. Let the Ellipse DECF be given, whose major axis DC is 25, minor FE 20. The axis FE is equal to the diameter CI of the cylinder. Therefore the plane drawn through the axis of the cylinder has in common with the obliquely cutting plane the section DC, the axis of the Ellipse, and with the base of the cylinder it makes the section CI, and on the surface gives the side DI. Wherefore there is a right triangle CID, whose hypotenuse DC 25 and base CI 20 are given: from the square of DC itself, 625, subtracting the square of CI, 400, leaves 225, the square of the perpendicular DI, which therefore is 15. Therefore the planes CI and AD are parallel and are separated by the interval CI 15. Since, then, from the diameter of the base CI 20, the circumference of the same cylindrical base is known approximately as 62 84/100, the cylindrical surface ACID is manifestly 942 60/100, of which half, 471 30/100, is cut in two parts by the semiperimeter of the Ellipse DEC. There is therefore a right triangle, whose sides about the right angle are the side DI 15 and the semiperimeter of the cylindrical base CHI 31 42/100: the squares of these, 225 and 987 2164/10000, are gathered into the sum, and the square root of the square 1212 2164/10000, about 34 82/100, is the semiperimeter of the Ellipse DEC; but the whole DECF will be 69 64/100. Therefore, since by hypothesis the planes through AD and CI are parallel, DI and AC are also equal sides. Thus, since the diameter of the given cylinder is known to be 20, and consequently the semiperimeter AD 31 42/100, the obliquity is given, which AC 15 measures; from the sum of the squares of the straight line AC and of the semiperimeter AD, extract the root. OOOO 2

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Mechanicorum 660 Radix quadrata, & dabit semiperimetrum Ellipsis D F C proximam veræ, quæ ex jam datis est illa eadem, quam paulo antè invenimus 34 82/100. Hinc igitur innotescunt momenta hujusmodi cunei, comparatis inter se lineis, quæ definiunt motum ponderis atque potentiæ; pondus enim movetur juxta lineam A C, potentia autem juxta semiperipheriam cylindri, quatenus videlicet præcisè atque simpliciter ratione ipsius cunei motus illi convenit. Verùm quia non facilè potentia applicatur proximè superficiei cylindri, & sæpius expedit potentiæ momenta augere; propterea Cylindro infigitur Vectis M N, cujus longitudo desumitur à puncto, ubi ille concurrit cum Axe Cylindri, usque ad extremitatem N, cui potentia applicatur. Hæc autem longitudo, manente eodem cuneo, varia omnino esse potest, atque adeò potentiæ momenta repræsentabit semiperipheria circuli ab extremitate N descripti, quæ comparanda erit cum motu ipsius ponderis ab obliquitate sectionis definito, ut dictum est. At subdubitare contingit, utrum crassiore, an graciliore cylindro uti expediat, manente eâdem obliquitatis mensurâ, atque eâdem vectis longitudine; manet siquidem eadem motuum Ratio; sed augeri videtur corporum conflictus ex mutuo tritu, nam in crassiore cylindro major est elliptica semiperimeter, quàm in tenuiore, ut manifestum est, si methodo paulò antè indicatâ res ad calculos revocetur: quamobrem ex majore hoc tritu augeri videtur difficultas movendi, cum maneat eadem corporis gravitas, eadem potentiæ virtus, eadem motuum Ratio. Longè tamen aliter se res habet; quandoquidem duorum corporum se se invicem in motu contingentium conflictus, qui ex superficiei asperitate oritur (hîc corporis unius conatum adversùs aliud vi suæ gravitatis mente secernimus à conatu, quo illud repellit præcisè vi suæ molis tanquam objectum impedimentum, etiamsi adversùs illud non gravitet) considerandus est, quatenus corpus impulsum adversatur directioni motûs corporis impellentis. Hinc est minimum esse conflictum, si ambæ facies se in plano Verticali contingant, & alterutrum corpus in eodem plano Verticali moveatur; nam reliquum corpus non repellitur, quia in illud non incurrit linea directionis motûs alterius corporis, sed solùm prominulæ utriusque corporis

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Mechanics 660 The square root, and it will give the semiperimeter of the Ellipse D F C, nearly the true one, which from the data already given is the same as that which we found a little before, 34 82/100. Hence, therefore, the moments of this kind of wedge become known, by comparing with one another the lines which define the motion of the weight and of the power; for the weight moves along the line A C, but the power along the semiperiphery of the cylinder, insofar as, namely, precisely and simply by reason of the motion of the wedge itself, those motions correspond. But since the power is not easily applied close to the surface of the cylinder, and it is often expedient to increase the moments of the power, therefore a Lever M N is inserted in the Cylinder, whose length is taken from the point where it meets the Axis of the Cylinder to the extremity N, to which the power is applied. This length, however, the same wedge remaining, can be altogether varied, and thus will represent the moments of the power by the semiperiphery of the circle described from the extremity N, which must be compared with the motion of the weight itself defined by the obliquity of the section, as was said. But a doubt may arise whether it is better to use a thicker or a thinner cylinder, the measure of the obliquity remaining the same, and also the length of the lever; for the Ratio of the motions remains the same; but the conflict of the bodies seems to increase from their mutual friction, for in a thicker cylinder the elliptical semiperimeter is greater than in a thinner one, as is manifest if the matter is reduced to calculations by the method indicated a little before: wherefore, from this greater friction, the difficulty of moving seems to increase, while the same weight of the body remains, the same force of the power, the same Ratio of the motions. Yet the matter stands very differently; since the conflict of two bodies contingently touching one another in motion, which arises from the roughness of the surface (here we mentally separate the attempt of one body against another by the force of its gravity from the attempt by which it repels it precisely by the force of its bulk as an obstructing object, even though it does not gravitate against it) must be considered insofar as the struck body opposes the direction of the motion of the striking body. Hence the conflict is least if both faces touch one another in a Vertical plane, and one or the other body moves in the same Vertical plane; for the remaining body is not repelled, because the line of direction of the motion of the other body does not fall upon it, but only the projections of each body

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Liber septimus. CAPUT IV. 661 ris particulæ, quatenus aliæ in alias incurrunt, impediunt mo- tum pro earum magnitudine & numero: quoad impedimentum maximâ ex parte tollitur, si pingui aliquo humore delibutæ fa- cies lubricæ fiant; replentur scilicet inanitates inter prominulas particulas interjectæ, quas intercapedines subire non tam faci- lè possunt corporis proximi particulæ. Sic si integer esset cylin- drus, suâ basi aut limbo CHI contingens subjectum corpus, minimo tritu cum illo confligeret in motu circà suum Axem, quia hujusmodi motui non opponitur corpus illud in I positum. At verò major est conflictus, quando directioni motûs illud ad- versatur, ut cùm prope D esse intelligitur aliquâ sui parte sub- jectum cylindro, qui obliquè sectus circumagi non potest, quin urgeat illud ex D versùs I. Quò autem majore angulo planum Ellipticum CEDF inclinatur ad basis planum CHI, eò magis conversioni cylindri adversatur objectum corpus, adeóque ma- jor invenitur difficultas. Cùm itaque in majore cylindro, datâ æquali obliquitatis mensurâ AC (æqualem obliquitatem non dico) planum obli- què secans minorem angulum cum plano basis cylindri consti- tuat, magisque ad ipsam basim accedat, minus habet resisten- tiæ ab objecto corpore, si particulæ singulæ considerentur, quamvis cunctæ resistentiæ simul collectæ demùm in æqualem summam à mensura AC definitam coëant. Sit enim minoris cy- lindri semiperipheria RS, men- sura obliquitatis RO, semiperi- meter Ellipsis SO: Manente au- tem eâdem RO, sit majoris cy- lindri semiperipheria RT, & el- lipsis semiperimeter TO; utique angulus RSO, utpote externus, major est interno opposito RTO; ac proinde alternus SOX major est alterno TOX, quibus angulis repræsentatur plani obliquè secantis inclinatio ad basim cylindri. Quamvis igitur ex cylindri convolutione semiperimeter Ellipsis SO impellat pondus juxta mensuram RO, & juxta eandem mensuram RO impellatur pondus à semiperimetro Ellipsis TO; item ab illius quadrante SP, atque hujus quadrante TQ, æqualiter impella- tur juxta mensuram MP, & NQ, quæ æquales sunt (utraque OOOO 3

Transcription: Translated (English)

Book Seven. CHAPTER IV. 661 The particles, insofar as some collide with others, hinder motion according to their size and number; but this hindrance is for the most part removed if the surfaces are smeared with some greasy moisture and thus made slippery; that is, the empty spaces lying between the projecting particles are filled up, through which the particles of the neighboring body cannot so easily enter. Thus, if a cylinder were entire, touching the body beneath it with its base or rim CHI, it would conflict with that body in its motion about its axis with the least friction, because that body placed at I does not oppose such motion. But the conflict is greater when that body opposes the direction of the motion, as when it is understood to lie near D with some part of itself subject to the cylinder, which, being cut obliquely, cannot be turned around without pressing that body from D toward I. And the greater the angle at which the elliptic plane CEDF is inclined to the plane of the base CHI, the more the body opposed to it resists the turning of the cylinder, and therefore the greater the difficulty is found to be. Therefore, when in a larger cylinder, given an equal measure of obliquity AC (I do not mean equal obliquity), the plane cutting obliquely forms a smaller angle with the plane of the cylinder’s base and comes closer to the base itself, it has less resistance from the body opposed to it, if the individual particles are considered, although all the resistances taken together finally amount to an equal total determined by the measure AC. For let the semiperiphery of the smaller cylinder be RS, the measure of obliquity RO, and the semiperimeter of the ellipse SO: but RO remaining the same, let the semiperiphery of the larger cylinder be RT, and the semiperimeter of the ellipse TO; certainly the angle RSO, being an exterior angle, is greater than the opposite interior angle RTO; and therefore the alternate angle SOX is greater than TOX, by which angles the inclination of the plane cutting obliquely to the base of the cylinder is represented. Thus, although by the revolution of the cylinder the semiperimeter of the ellipse SO impels the weight according to the measure RO, and according to the same measure RO the weight is impelled by the semiperimeter of the ellipse TO; likewise by the quadrant SP of the one, and by the quadrant TQ of the other, the weight is equally impelled according to the measure MP and NQ, which are equal (both OOOO 3

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Mechanicorum 662 scilicet est quadrans ipsius R O, siquidem propter triangulorum similitudinem, ut O S ad P S, ita R O ad M P, & ut O T ad Q T, ita R O ad N Q; sed ex hypothesi O S est quadrupla ipsius P S, sicut O T est quadrupla ipsius Q T; igitur R O est quadrupla ipsius M P, & ipsius N Q, quæ propterea sunt æquales) quia tamen Q T major est quàm P S, qua Ratione R T major est quàm R S, & O T major quàm O S; idcircò eadem resistentia distributa per plures particulas longioris Q T, seu O T, minor est in singulis particulis, quàm cùm distribuitur per pauciores particulas brevioris P S, seu O S. Minùs igitur T O suis particulis subinde contingens corpus, quod impellit, cum eo confligit, quàm confligat S O pertinens ad minorem cylindrum. Hujusmodi Cuneum inflexum ex cylindro obliquè secto perpetuum esse in reciprocando motu, satis manifestum est, si postquam pondus ex D propulsum est in I, iterum redeat ad D; absolutâ enim cylindri conversione iterum pondus ex D ad I propellitur. Id autem ut fiat, statuatur jugum K L circa axem in V versatile, ita tamen ut V respondeat axi cylindri: tùm in L adnectatur pondus, quod ex D impellitur in I, dum Cylindri dimidia revolutio ex D per F in C perficietur: quia autem in reliquâ dimidiâ cylindri revolutione jugi extremitas K jam repulsa sursum versùs A, impelletur iterum ad C, pondus restituetur ex I in D, atque ita deinceps reciprocando impulsionem tum ponderis adnexi in L, tùm extremitatis K. Hinc si ex laqueari pendeat statua ventum referens, & in speciem volantis ingentes alas expandens junctas jugo K L, atque in superiore conclavi adsit qui cylindrum circumagat, alis reciprocantibus commovebitur aër, & aura excitabitur ad refrigerandum. Pro variis demum usibus statuetur cylindrus modò horizonti, tanquam Ergata, perpendicularis, modò velut Sucula, parallelus: Eritque expeditissima ejus conversio, si centrum Ellipsis nulli polo innitatur, sed cylindrus ipse congruo loculamento ita inseratur, ut in eo sit versatilis, &, quam primò dederis, positionem

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Mechanics 662 is, namely, one quarter of RO itself, since, by reason of the similarity of the triangles, as OS is to PS, so is RO to MP; and as OT is to QT, so is RO to NQ. But by hypothesis OS is four times PS, just as OT is four times QT; therefore RO is four times MP and NQ, which therefore are equal. Yet because QT is greater than PS, for which reason RT is greater than RS, and OT greater than OS, for that reason the same resistance, distributed through more particles of the longer QT, or OT, is smaller in each particle than when it is distributed through fewer particles of the shorter PS, or OS. Therefore TO, striking the body that impels it as it successively comes against its particles, conflicts with it less than SO, which belongs to the smaller cylinder, conflicts. That a wedge of this kind, bent from a cylinder cut obliquely, is perpetually to continue in reciprocating motion is sufficiently evident if, after the weight has been driven from D to I, it again return to D; for, after the cylinder has completed its revolution, the weight is again driven from D to I. In order that this may happen, let the yoke KL be set to turn about the axis in V, however, so that V correspond to the axis of the cylinder; then let a weight be attached at L, which is driven from D to I while the cylinder completes half a revolution from D through F to C. But because in the remaining half-revolution of the cylinder the end K of the yoke, now repelled upward toward A, will again be driven to C, the weight will be restored from I to D, and so thereafter, by reciprocation, the impulse both of the attached weight at L and of the end K. Hence, if from the ceiling there hang a statue representing the wind, and spreading great wings, like a flying thing, joined to the yoke KL, and if in the upper room there be one who turns the cylinder, the air will be moved by the reciprocating wings, and a breeze will be stirred up for cooling. Finally, for various uses, the cylinder will be placed now perpendicular to the horizon, like an ergata, now parallel to it, like a sucula: and its turning will be most easy if the center of the ellipse rests on no pole, but the cylinder itself is inserted into a suitable socket in such a way that it is movable therein and can retain the position first given it.

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Liber septimus. CAPUT V. 66 sitionem deinceps servet, ac neutram in partem nutet. Id qui- dem paulò longiorem cylindrum exigit; non tamen est neces- se unius perpetuæ crassitiei esse cylindrum; sæpè enim nimis crassum atque incommodum esse contingeret; sed frusto cy- lindrico crassiori obliquè secto firmiter inter poterit gracilior cylindrus, ita ut axis axi conveniat, & rectam lineam consti- tuant, atque hic in foramen immissus, in quo versatilis est, dum contorquetur, crassiorem cylindrum pariter convolvit. CAPUT V. Cuneum perpetuum Circulus inclinatus imitatur. Tertiam hanc cunei perpetui speciem duabus superioribus adjicere non inutile fuerit, ut legenti hujus libri cap. ult. prop. 6. constabit, quanquam fortassis alicui à superiore parùm distare videatur; ibi enim Ellipsis ex cylindri recti sectione obliquâ, hîc circulum suo quidem centro insistentem, sed in- clinatum proponimus. Ut autem res clariùs exponatur, concipiamus circu- lum à plano horizontali R S sectum per centrum A, ita ut eorum commu- nis sectio sit diameter B C, semicircu- lus autem superior ad horizontem in- clinatus sit B D C; qui per A D bifa- fariam secetur plano Verticali ad sub- jectum planum R S horizontale recto; sitque horum planorum communis sectio recta A E. Tum ex D Quadran- tis extremitate demittatur per 11. lib. 11. perpendicularis ad sub- jectum planum recta D E; quæ propterea ex defin. 3. lib. 11. fa- cit cum rectâ A E angulum rectum. Accipiatur arcus D F, & per F in circuli plano ductâ F G parallelâ ipsi A D, per eam ductum intelligatur planum parallelum plano transeunti per A D. Igitur planum horizontale plana illa parallela secans, per 16. lib. 11. facit sectiones A E & G H parallels, ac proinde, cum duæ rectæ A D & A E duabus rectis G F & G H sint pa- rallelæ,

Transcription: Translated (English)

Book Seven. Chapter V. 66 and afterward preserve the same position, and incline to neither side. This indeed requires a somewhat longer cylinder; yet it is not necessary for the cylinder to be of one continuous thickness: for often it would happen that it is too thick and inconvenient; but a thicker cylindrical piece cut obliquely may firmly receive within it a thinner cylinder, so that the axis may agree with the axis and they may form a straight line; and this, inserted into the hole in which it turns, while it is twisted, likewise winds up the thicker cylinder. CHAPTER V. The perpetual wedge imitates an inclined circle. It would not be useless to add this third species of the perpetual wedge to the two preceding ones, as will be clear to the reader from the last chapter, prop. 6 of this book, although perhaps it may seem to someone to differ little from the preceding; for there we propose an Ellipse from an oblique section of a right cylinder, but here a circle standing indeed upon its own center, yet inclined. But that the matter may be explained more clearly, let us conceive a circle cut by the horizontal plane R S through the center A, so that their common section is the diameter B C, while the upper semicircle is inclined to the horizon, B D C; let it be divided in two by the vertical plane through A D, perpendicular to the underlying horizontal plane R S; and let the common section of these planes be the straight line A E. Then from the extremity D of the quadrant let the perpendicular D E be drawn down to the underlying plane, by 11. lib. 11; and it therefore, by definition 3 of lib. 11, makes a right angle with the straight line A E. Let the arc D F be taken, and through F, in the plane of the circle, let F G be drawn parallel to A D; let there be understood through it a plane parallel to the plane passing through A D. Therefore the horizontal plane, cutting those parallel planes, by 16. lib. 11, makes the sections A E and G H parallel, and consequently, since the two straight lines A D and A E are parallel to the two straight lines G F and G H,

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Mechanicorum rallelæ, etiam per 10. lib. 11 anguli D A E, & F G H (cum illæ sint similiter positæ) sunt æquales. Iam ex F demittatur in sub- jectum planum perpendicularis F H, quæ cum rectâ G H constituit angulum rectum. Cum itaque duo triangula A E D & G H F rectangula habeant angulum D A E angulo F G H æqualem, & reliquus reliquo æqualis est, atque similia sunt triangula: Quapropter ut A D ad D E, ita G F ad F H, & per- mutando ut A D ad G F, ita D E ad F H. Eadem erit ratiocinatio in triangulo K L I similiter facto, quod erit reliquis simile, & ut A D ad K I, ita D E ad I L: atque ita deinceps de cæteris omnibus triangulis, quæ efformari possunt à lineis parallelis Radio A D, tanquam hypothenusis, & à perpendicularibus cadentibus in subjectum planum ex peripheriâ circuli inclinati, & à rectis, quæ jungunt punctum, in quod cadit perpendiculum, cum extremitate altera hypothenusæ. Quoniam verò omnes lineæ parallelæ Radio A D sunt Sinus arcuum à puncto C incipientium (sic I K est Sinus arcûs I C, F G est Sinus arcûs F C) omnium illarum Ratio manifesta est ex Canone Sinuum, si arcuum quantitas in gradibus data sit, vel nota; quare etiam nota est Ratio perpendicularum D E, F H, I L. Hæc autem, quæ de Quadrante D A C dicta sunt, etiam de reliquo Quadrante D A B intelliguntur; & quæ de hoc superiore semicirculo demonstrata sunt, etiam de inferiore semicirculo vera sunt, quatenus ille ad hoc idem planum horizontale R S refertur, à quo circulus bifariam secatur. Iam verò integer circulus cum alio plano horizontali non Secante, sed Tangente circulum inclinatum in puncto infimo, comparetur: sunt autem duo hæc plana horizontalia invicem parallelæ; & perpendicularum à puncto D cadens in planum horizontale Tangens, est duplum perpendiculari D E, quemadmedum totius circuli diameter est dupla Radij A D. Quapropter cum nota sit dati circuli diameter secundùm certam mensuram, & data sit circuli inclinatio, sive perpendiculari longitudo, qui est Sinus anguli inclinationis, facile est invenire singularum perpendicularium quantitatem. Nam si datur angulus inclinationis circuli ad planum Tangens (cùm hoc sit parallelum plano Secanti) angulus ille æqualis est angulo D A E: quare sicut in triangulo D A E rectangulo datur hypothenusa A D Radius circuli,

Transcription: Translated (English)

Mechanics parallel lines, also by lib. 10, 11, the angles D A E and F G H (since they are similarly placed) are equal. Now let from F be let fall upon the subjacent plane the perpendicular F H, which with the straight line G H makes a right angle. Since therefore the two right triangles A E D and G H F have the angle D A E equal to the angle F G H, and the remaining angle is equal to the remaining one, the triangles are similar. Wherefore as A D is to D E, so is G F to F H; and by permutation, as A D is to G F, so is D E to F H. The same reasoning will hold in the similarly constructed triangle K L I, which will be similar to the rest, and as A D is to K I, so is D E to I L; and so on for all the other triangles that can be formed from lines parallel to the Radius A D, as the hypotenuse, and from perpendiculars falling upon the subjacent plane from the circumference of the inclined circle, and from straight lines which join the point on which the perpendicular falls with the other extremity of the hypotenuse. Since however, all lines parallel to the Radius A D are sines of arcs beginning from the point C (thus I K is the sine of the arc I C, F G is the sine of the arc F C), the ratio of all these is manifest from the Canon of Sines, if the quantity of the arcs be given or known in degrees; therefore the ratio of the perpendiculars D E, F H, I L is also known. And what has been said concerning the quadrant D A C is to be understood also of the remaining quadrant D A B; and what has been demonstrated concerning this upper semicircle is also true of the lower semicircle, insofar as it is referred to the same horizontal plane R S, by which the circle is cut in two. Now, then, let the entire circle be compared with another horizontal plane not cutting, but tangent to the inclined circle at the lowest point: but these two horizontal planes are parallel to one another; and the perpendicular from the point D falling on the tangent horizontal plane is double the perpendicular D E, just as the diameter of the whole circle is double the radius A D. Wherefore, since the diameter of the given circle is known according to a certain measure, and the inclination of the circle is given, that is, the length of the perpendicular, which is the sine of the angle of inclination, it is easy to find the magnitude of each perpendicular. For if the angle of inclination of the circle to the tangent plane is given (since this is parallel to the cutting plane), that angle is equal to the angle D A E: wherefore, as in the right triangle D A E the hypotenuse A D, Radius of the circle, is given,

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Liber septimus. CAPUT V. 665 circuli, & angulus acutus adjacens D A E, ex quibus invenitur latus D E, ita manifestum fit perpendicularum à summo circuli inclinati puncto D in planum horizontale Tangens, quod est duplum lateris D E inventi. Sivè igitur detur puncti D à plano horizontali Tangente distantia, sivè inveniatur, distantia hæc bipartitò dividatur, ejusque medietas tribuatur perpendiculari D E. Tum Quadrans D C in quotlibet partes æquales divisus intelligatur, puta in decem, & ex Canone accipiantur singulorum arcuum Sinus gr. 81.72.63.54.45.36.27.18.9: deinde fiat ut Radius ad singulos Sinus, ita notum perpendicularum D E ad aliud, & proveniet singulorum perpendicularorum in planum Secans cadentium mensura: quibus singillatim addenda est quantitas ipsius D E, hoc est dimidia altitudo summa, ut habeatur singulorum altitudo suprà planum horizontale Tangens. Ponatur summa circuli elevatio à positione horizontali, palmi unius: igitur D E est semipalmus, qui intelligatur distinctus in particulas 100.000, adeóque totus palmus in part. 200.000. Ideò in superiori Quadrante singulis perpendicularis addito semipalmo perpendicularum altitudo ea est, quam adjecta tabella exhibet. Perpendicular. Superioris Quadr. Inferioris Quadr. Gr. Gr. 90 200000 90 81 198769 81 72 195106 72 63 189101 63 54 180902 54 45 170711 45 36 158778 36 27 145399 27 18 130902 18 9 115643 9 0 100000 0 Pro inferiori autem Quadrante ponendo gr. 90. in puncto contactûs circuli cum plano Tangente, lineæ perpendicularares ad PPpp

Transcription: Translated (English)

Book Seven. CHAPTER V. 665 From the circle, and the acute angle adjacent D A E, from which the side D E is found, it thus becomes evident that the perpendicular from the highest point of the inclined circle, D, to the horizontal tangent plane is double the side D E found. Whether therefore the distance of point D from the horizontal tangent plane is given, or is to be found, let this distance be divided into two equal parts, and let its half be assigned to the perpendicular D E. Then let the quadrant D C be understood to be divided into any number of equal parts, say into ten, and from the Canon let the sines of the individual arcs be taken, gr. 81.72.63.54.45.36.27.18.9: then let it be as Radius to each sine, so the known perpendicular D E to another, and there will result the measure of each perpendicular falling in the secant plane: to each of which separately must be added the quantity of D E itself, that is, half the total height, so that the height of each above the horizontal tangent plane may be obtained. Let the total elevation of the circle from the horizontal position be one palm: therefore D E is half a palm, which is understood to be divided into 100,000 parts, and thus the whole palm into 200,000 parts. Therefore in the upper quadrant, by adding half a palm to each perpendicular, the height of the perpendicular is that which the appended table shows. Perpendicular. Superior Quadrant Inferior Quadrant Gr. Gr. 90 200000 90 81 198769 81 72 195106 72 63 189101 63 54 180902 54 45 170711 45 36 158778 36 27 145399 27 18 130902 18 9 115643 9 0 100000 0 But for the lower quadrant, placing 90 degrees at the point of contact of the circle with the tangent plane, the perpendicular lines to PPpp

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Mechanicorum 666 planum Secans auferendæ sunt à semisse datæ elevationis, hoc est ex semipalmo part. 100000, & residuum est distantia perpendicularis à subjecto plano Tangente singulis partium punctis respondens; quemadmodum adjectæ tabellæ pars altera ostendit. His fundamentis positis innititur species hæc cunei petita ex circulo inclinato, qui eandem servans inclinationem circa suum centrum convertitur. Sit enim circulus B F E D, cujus centrum C, ad horizontem, sive ad planum Verticale inclinatus, & sit in D pondus impellendum. Potentia in B existens, circulumque retinens in eâdem inclinatione, si illum circa suum centrum C circumagat, paulatim pondus impellit, prout illud tangitur modò à puncto E, modò à puncto F, donec demum à puncto B infimo ad extremum motûs terminum deducatur. Quare potentiæ motus definitur à semicirculi peripheriâ B F E D, motus verò ponderis à rectâ D H. Ut autem circulus in conversione eandem semper inclinationem servet, frustum ligni G obliquè in parte superiori sectum, inferiùs affigatur circulo (aut contrà in parte inferiori sectum superiùs affigatur, prout commodius acciderit) atque in ligno foramen fiat respondens circuli centro C, per quod foramen transeat polus, cui centrum insistit. Cum enim foramen illud perpendiculare maneat ad horizontem, circulus eandem retinet in conversione inclinationem. Verùm quia priùs statuendum est spatium D H, per quod ponderi commeandum est, quàm circuli amplitudo definiatur, non solùm ut Potentiæ motus ad ponderis motum Rationem habeat majorem pro circuli semiperipheriæ longitudine, sed etiam ut quàm minimum fieri possit, inclinatio ipsa recedat à parallelismo cum plano, ad quod inclinari dicitur, sive illud horizontale sit, sive Verticale, quò enim minùs directioni motûs potentiæ opponitur pondus, eò minùs resistit: Propterea data linea D H statuatur ut Sinus anguli inclinationis, & Radio respondebit diameter circuli opportuni. Sic si recta D H sit linea palmaris, & angulus inclinationis ponatur gr. 10: fiat ut gr.

Transcription: Translated (English)

Mechanics 666 The cutting plane is to be removed from half of the given elevation, that is, from the half-palm part of 100000, and the remainder is the perpendicular distance from the subject plane tangent at the individual corresponding points; as the second part of the appended table shows. Having laid down these foundations, this kind of wedge is based on a circle inclined, which, keeping the same inclination, is turned about its center. Let there be the circle B F E D, whose center C, inclined to the horizon, or to the vertical plane, and let there be at D the weight to be moved. The power at B, which holds the circle in the same inclination, if it turns it about its center C, gradually pushes the weight, as it is touched now at point E, now at point F, until at last it is brought from the lowest point B to the extreme limit of motion. Therefore the motion of the power is defined by the semicircular perimeter B F E D, but the motion of the weight by the straight line D H. Now, in order that the circle in its rotation may always preserve the same inclination, a block of wood G, cut obliquely at the upper part, should be attached below to the circle (or conversely, if cut at the lower part it should be attached above, as may be more convenient), and in the wood a hole should be made corresponding to the center C of the circle, through which hole there should pass the pole on which the center rests. For since that hole remains perpendicular to the horizon, the circle retains the same inclination in rotation. But because the space D H, through which the weight is to travel, must first be determined before the size of the circle is defined, not only so that the motion of the Power may have a ratio to the motion of the weight greater according to the length of the semicircumference of the circle, but also so that, as much as possible, the inclination itself may depart from parallelism with the plane to which it is said to incline, whether that plane be horizontal or vertical; for the less the weight opposes the direction of the motion of the power, the less it resists: therefore, given the line D H, let it be taken as the sine of the angle of inclination, and to the radius there will correspond the diameter of the suitable circle. Thus if the straight line D H be a palm line, and the angle of inclination be set at 10 degrees: let it be as though deg.

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Liber septimus. CAPUT V. 667 gr. 10 Sinus 17365 ad Radium 100000, ita 1 palmus ad pal- mos 5 76/100 ferè, quæ esset diameter circuli eam inclinationem habentis; atque adeò motus potentiæ cum circulo in gyrum actæ esset ad motum ponderis saltem noncuplus. Ex quibus sa- tis apertum est ampliorem circulum præ minore utiliorem esse, cæteris paribus. At circulum ipsum convolvere aut non placet, aut non licet, quia fortasse pondus illius peripheriæ adnexum est, atque id- circò non nisi in gyrum pariter cum circulo ageretur. Idem planè assequemur, si circulum horizonti, aut plano Verticali, constitutum parallelum potentia urgeat in B; tùm puncto D applicetur pondus; deinde potentia pergens in F & E percurrat circuli ambitum illum deprimendo & inclinando; quandoqui- dem utroque modo mutatur distantia ponderis ab infimo puncto circuli. Ponatur enim punctum F æquè distans à puncto B, atque punctum E distat à puncto D. Si potentia manens ap- plicata eidem puncto B convertat circulum ita, ut ipsa poten- tia distet à pondere arcu B E, pondus non adnexum circulo impellitur pro Ratione, quam arcus ille exigit: at verò si ma- nente pondere applicato ad punctum D, cui adnectitur, poten- tia pergat ex B in F, similiter distat à pondere arcu F D, qui est æqualis arci B E; atque proinde æqualiter deprimitur, pro- ut idem arcus exigit, juxta superius explicata, & in tabellâ ex- posita; atque ita deinceps, donec potentia veniat in D: singulas autem impulsiones metitur differentia perpendicularium. Hîc igitur ubi pondus circulo adnexum ponitur, manifesta est motûs reciprocatio: potentia siquidem ubi per F & E vene- rit in circuli punctum D, & impulerit pondus usque in H, percurrendo reliquum semicirculum D I B iterum retrahit pon- dus ex H in D. At quando pondus non connectitur cum circu- lo, & circulus ipse convertitur, tunc opus est aliquo artificio, ut pondus ex H remeet in D, quemadmodum indicatum est capite superiori. Porrò circulus iste non convolutus, sed à potentiâ ejus am- bitum percurrente secundùm alias atque alias partes inclinatus, non est à Ratione Cunei excludendus; quandoquidem parùm interest utrùm simili motu potentia atque organum moveantur, an verò dissimili motu. Quando potentia in eodem puncto B P P p p 2

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Book Seven. CHAPTER V. 667 gr. 10 A sine of 17,365 to radius 100,000; thus 1 palm to 5 76/100 palms, nearly, which would be the diameter of a circle having that inclination; and therefore the power’s motion, when the circle is driven round, would be to the motion of the weight at least as nine to one. From which it is sufficiently clear that a larger circle is more useful than a smaller one, ceteris paribus. But to revolve the circle itself either is not agreeable, or is not allowed, because perhaps the weight is attached to its circumference, and for that reason would not be moved except around in like manner with the circle. We shall plainly achieve the same result if the power urges the circle, set parallel to the horizon or to a vertical plane, at B; then the weight is applied at point D; afterwards the power, continuing through F and E, traverses that circumference, depressing and inclining it; since in either way the distance of the weight from the lowest point of the circle is changed. For let point F be equally distant from point B as point E is from point D. If the power, remaining applied to the same point B, turns the circle so that the power itself is distant from the weight by arc B E, the weight not attached to the circle is impelled according to the ratio which that arc requires: but if, while the weight remains applied at point D, to which it is attached, the power proceeds from B to F, it is likewise distant from the weight by arc F D, which is equal to arc B E; and therefore it is equally depressed, just as that same arc requires, according to what was explained above and set out in the table; and so on thereafter, until the power comes to D: each impulse, however, is measured by the difference of the perpendiculars. Here, then, where the weight is placed attached to the circle, the reciprocity of motion is clear: for when the power, having come through F and E to point D of the circle, and having driven the weight as far as H, by traversing the remaining semicircle D I B it again draws the weight back from H to D. But when the weight is not connected with the circle, and the circle itself is turned, then some contrivance is needed so that the weight may return from H to D, as was indicated in the preceding chapter. Moreover, this circle, not wound round but inclined by the power traversing its circumference through one part after another, is not to be excluded from the Ratio of the Wedge; since it matters little whether the power and the organ are moved by a similar motion, or indeed by a dissimilar motion. When the power in the same point B P P p p 2

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Mechanicorum semper applicata in gyrum pergit, simili motu cum circulo in gyrum acto movetur: quando verò potentia quidem circulari- ter movetur, sed non secum rapit circulum, quem solummodò inclinat, est quidem diversus potentiæ motus à motu organi, sed ponderis motus idem planè efficitur in utroque casu, & æqualis est potentiæ ipsius motus determinatus à Rationibus Cunei, quamvis hic non promoveatur, sed solùm impellatur. CAPUT VI. Unde oriatur vis Percussionis. Cunei vires, quatenus ex ejus formâ proveniunt, hactenus consideravimus; nunc ad id, quod potissimum in hac tractatione videtur, transeundum est, videlicet ad percussionem, qua dum adigitur Cuneus, multo faciliùs consequitur motus (sive scissio sit, sive simplex impulsio, citrà corporis divisionem) quàm si onere imposito prægravaretur, aut Vecte seu aliâ qualibet Facultate augerentur Potentiæ momenta. Certè Aristoteles Mechan. quæst. 19. quærit, Cur si quis super lignum magnam imponat securim, desuperque illi magnum adjiciat pondus, ligni quippiam, quod curandum sit, non dividit: Si vero securim extollens percutiat, illud scindit; cum alioquin multo minus habeat ponderis id, quod percutit, quàm id quod superjacet, & premit? Id quod in cæteris quoquè percussionibus, ubi nulla intervenit Cunei Ratio, manifestum est; quemadmodum in simplici compressione, ut cùm lamella aurea in subtilissimam bracteolam diducitur repetitâ mallei percussione; quod enim, licèt immensum, pondus vi suæ gravitatis tantumdem præstare posset? Percussionis igitur natura investiganda est, ut ejus vires in Cuneo innotescant. Certum autem esse debet, & extra omnem controversiam positum nihil esse in hac rerum universitate, quod vacet corpore, sed corporibus omnem obsideri locum, nullumque esse inane, in quod se recipere valeant, ac propterea corpora om- nia

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Mechanics always applied continues in a circular motion, moving with a motion similar to a circle turned round: but when, indeed, the power moves circularly, yet does not draw the circle along with it, but only inclines it, the motion of the power is indeed different from the motion of the organ, but the motion of the weight is made exactly the same in both cases, and the motion of the power itself, determined by the reasons of the wedge, is equal, although here it is not advanced, but only impelled. CHAPTER VI. Whence the force of percussion arises. We have so far considered the powers of the wedge, insofar as they arise from its form; now we must proceed to that which seems of chief importance in this treatise, namely percussion, by which, when the wedge is driven, motion is obtained much more easily (whether it be a splitting, or a simple impulse, without division of the body) than if it were weighted down by a load imposed, or if the moments of the powers were increased by a lever or some other faculty. Certainly Aristotle, Mechan. quæst. 19, asks: Why, if someone place a large axe upon wood, and from above add a great weight to it, does it not divide any part of the wood that ought to be cut? But if, lifting the axe, he strikes it, it splits it; although otherwise that which strikes has much less weight than that which lies on top and presses down? This is evident also in the other kinds of percussion, where no ratio of the wedge intervenes; as in simple compression, as when a gold leaf is drawn out into the thinnest foil by repeated blows of the hammer. For what, though immense, weight could by the force of its own gravity accomplish as much? The nature of percussion must therefore be investigated, so that its force in the wedge may be made known. Now it must certainly be held, and placed beyond all controversy, that there is nothing in this universe of things which is empty of body, but that all places are occupied by bodies, and that there is no void into which they can retreat, and therefore all bodies

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Liber septimus. CAPUT VI. 669 nia ita sibi vicissim suâ mole obsistere, ut nullum moveri va- leat, quin alterius in locum succedat; quod proinde loco pelli necesse est, quantum satis fuerit, ut subeunti corpori spatium concedat; sive id contingat, quia obsistens corpus inter an- gustias deprehensum se comprimi patiatur, sive quia divisum in latera secedat, sive quia circumfusa corpora circumpellat, quæ abeuntis vestigia sequantur. Cum itaque nec omnia pla- nè corpora perpetuò quiescant, nec omnia æquali prorsus agi- tatione commoveantur, fieri non potest, quin aliquibus vis aliqua saltem aliquando inferatur, seu quia non licet diu juxta naturæ institutum quieta consistere, seu quia externo pulsu ad velociorem motum incitantur. Quare nullius corporis ex loco in locum migratio excogitari potest, cui nullum aliud corpus adversetur & repugnet, vel ut suo se tutetur in loco juxta præ- scriptum à natura ordinem, vel ut partium nexum, & natura- lem earum positionem servet citra divisionem, aut compressio- nem, aut distractionem. Ex quo & illud consequens est, quod nullum reipsa (quicquid animo finxeris) quiescit corpus ad omnem omninò motum adeò indifferens, ut nihil prorsus re- tundat impetus ab alio corpore commoto sponte concepti, aut extrinsecùs impressi: nullum quippe est, quod neque quicquam habeat proni, neque sursum subvolare contendat, si disparis se- cundùm speciem gravitatis corpori permeabili proximum consistat, ubi fortè ordinem perturbari contigerit: ac propterea ad motum indifferens censendum non est, nisi ut exquisitam circuli peripheriam circa centrum gravium percurrat externâ vi impellente: id quod animo fingere facile est, opere exequi, ut mitissimè loquar, difficillimum; certè semper incertum. Hoc verò discrimen est inter corpora (quantum quidem ad præsentem disputationem attinet) quod aliqua ita liberè fluunt, ut nusquam adhærescere videantur, quemadmodum aër, & ex- tenuatus vapor: Alia liquida & fusa manant, atque labuntur, ut aqua cæterique humores, per quos transire & permeare li- cet, dirempti enim iterùm coëunt. Alia partibus constant, quæ junctione aliqua tenentur, & sub certâ quidem conformatione, atque figurâ consistunt, quandiu nullo impellente urgentur; quia tamen facilè comprimi queunt, in aliam figuram transfe- runtur; cujusmodi sunt lutum, cera, & reliqua mollia ac tene- PPPP 3

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Book Seven. Chapter VI. 669 Thus all things mutually oppose one another by their own bulk, so that no body can be moved without another succeeding into its place; wherefore it is necessary that something be driven from its place, so far as is enough to give room to the body entering in; whether this happens because the resisting body, caught in narrow confines, suffers itself to be compressed, or because it divides and withdraws to the sides, or because it drives on the surrounding bodies, which follow in the track of the departing body. Since therefore not all bodies are altogether at rest perpetually, nor are all moved with an absolutely equal agitation, it cannot happen that force is not at some time applied to some of them, either because it is not allowed them to remain long at rest according to nature’s ordinance, or because they are urged by an external blow to a swifter motion. Therefore no migration of any body from place to place can be conceived to which no other body opposes and resists, either to protect itself in its own place according to the order prescribed by nature, or to preserve the connection of its parts and their natural position without division, compression, or separation. From this it also follows that no body in reality rests, however much you may imagine in your mind one that is so indifferent to every motion that it offers no resistance at all to an impetus spontaneously conceived by another moved body, or impressed from without: for there is none that has nothing inclined downward, nor, if a body of a different specific gravity be set near it as a permeable body where perhaps the order has been disturbed, tends to fly upward; and therefore it is not to be reckoned indifferent to motion unless it traverses the exact circumference of a circle about the center of gravitating bodies, propelled by an external force: this it is easy to imagine in the mind, but to carry out in practice, to speak very mildly, is extremely difficult; certainly always uncertain. But this is the difference among bodies, as far indeed as the present discussion is concerned: some flow so freely that they seem nowhere to adhere, as air and rarefied vapor. Others run and glide as liquids and fluids do, like water and the other humors, through which one may pass and permeate; for when parted, they reunite again. Others consist of parts held together by some junction, and stand in a certain shape and figure so long as they are not pressed by any forcing agent; yet since they can easily be compressed, they are transferred into another shape; such are clay, wax, and the remaining soft and tena-

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Mechanicorum 670 ra, quæ aut ita tractabilia sunt, ut quamcumque in formam singantur, aut ita flexibilia, ut sequantur quocumque torqueas: Alia demum solida & dura sunt, quæ figuræ terminos, quibus circumscribuntur, non facilè mutant, & si fortè se aliquatenus comprimi patiantur, pristinam formam sibi reparant. Ex hisce quatuor corporum generibus priora rationem medij subire possunt, in quo reliquorum corporum motus exerceantur, ut ex alio in alium locum commigrent; posteriora, si unum in aliud incurrat, aut si sibi invicem occurrant, ea sunt, per quæ transitus non pateat, sed aliorum corporum motui tantisper mole suâ unumquodque obluctatur, dum pulsu externo removeatur. Porrò Impulsionem à Percussione distinguere opus est, nisi vocabulis abuti velimus; quamvis enim utraque objecti corporis resistentiam inveniat, nemo tamen dixerit idem esse, apprehensum manu Vectem impellendo, atque illum percutiendo deprimere, innatans aquæ lignum conto propellere, atque inflicto ictu illud à ripâ longiùs abstrahere, etiamsi æquè & Vectis deprimatur, & lignum promoveatur. Simplex nimirum Impulsio nullum per se antecedentem corporis impellentis motum exigit: at Percussio ob idipsum, quia Percussio est, corporis percutientis motum requirit, qui ipsorum corporum collisionem præcedat. Quare in Percussione intervenit instituti jam & inchoati motûs interruptio ex novâ objecti corporis resistentiâ. Hinc Aristoteles lib. 4. Meteor. summa 3. cap. 2. ait, Est autem Pulsio, motus à movente, qui fit à tactu; Percussio autem, cum à latione. Hæc omnia conjunctim Percussio postulat: Primò inchoatum esse jam & institutum motum oportet: quamvis etenim impulsio omnis vincat corporis urgendi aut scindendi resistentiam etiam primo motûs momento; quia tamen præcedens corporis impellentis motus, quo antè accessit ad corporis impulsis contactum, quàm illud urgere incipiat, omnino præter Impulsionis naturam accidit (hæc si quidem eadem sequetur, etiam si priùs in mutuo contactu diutissimè quiescant) propterea non satis est resistentiam invenire, sed hanc instituto jam motui intervenire necesse est, ut sit Percussio. Deinde, licèt præcesserit motus, atque adhuc continuatus novam inveniat resistentiam,

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Mechanicorum 670 Some bodies are such as are tractable, that they can be shaped into whatever form; or so flexible, that they follow wherever you twist them. Others finally are solid and hard, and do not easily change the boundaries of the figure by which they are circumscribed; and if they should perhaps allow themselves to be compressed to some extent, they restore to themselves their former shape. From these four kinds of bodies, the former can take the role of a medium, in which the motions of the remaining bodies are exercised, so that they may pass from one place into another; the latter, if one strikes against another, or if they meet one another, are those through which no passage lies open, but each, by its own mass, resists the motion of the other bodies for a time, until it is removed by an external blow. Moreover, it is necessary to distinguish Impulse from Percussion, unless we wish to abuse words; for although each encounters the resistance of the object body, no one would nevertheless say that it is the same thing to depress a lever held in the hand by pushing it, and to depress it by striking it; or, with a pole, to drive a floating piece of wood upon the water, and by a given blow to draw it farther away from the bank, even if the lever is equally depressed and the wood advanced. Simple Impulse, to be sure, does not of itself require any prior motion of the impelling body: but Percussion, precisely because it is Percussion, requires the motion of the striking body, which should precede the collision of the bodies themselves. Therefore in Percussion there intervenes the interruption of a motion already undertaken and begun, arising from the new resistance of the object body. Hence Aristotle, in book 4 of the Meteorologica, section 3, chapter 2, says: “Pushing is motion from a mover, which occurs by touch; but percussion, by striking.” All these things together Percussion requires: first, a motion must already have been begun and established; for although every impulse overcomes the resistance of the body being urged or cut, even in the first moment of motion, yet because the preceding motion of the impelling body, by which it approached the contact of the body being impelled before it begins to urge it, occurs altogether outside the nature of Impulse (for this indeed will follow in the same way, even if they should first remain at rest in mutual contact for a very long time), therefore it is not enough to encounter resistance, but this resistance must intervene in a motion already established, if there is to be Percussion. Then, although the motion has preceded and, while still continuing, encounters a new resistance,

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Liber septimus. CAPUT VI. 671 tiam, quia alias medij ejusdem scindendi partes offendit; si ta- men æquabilis perseveret pristinam velocitatem aut tarditatem nulla ex parte imminutam continenter servans, non est censenda nova resistentia; sed quemadmodum continuus est idem motus aliis atque aliis partibus sibi succedentibus, ita continuatur eadem resistentia, nec posteriores medi partes percuti dicuntur, sed, ut priùs, præcisè impelli, aut scindi: quia vide- licet præcedens motus nihil confert ad novam hanc impulsionem prioribus omnino similem. Quod si corpus in motu ob id- ipsum quia movetur, majorem atque ma, orem adhiberet celeritatem, adeò ut in medio, hoc est aëre ipso, resistentiæ modus augeretur, facilè acquiescam contendenti aërem verberari & percuti: sed quia nimis facilem se præbet aër ad hoc, ut scindatur, non de hujusmodi percussione medij, in quo sit motus, mihi hîc est sermo, sed potiùs de percussione corporis, ad quod per medium accedit corpus percutiens. Hinc aquam percuti non negaverim, quando ensis bonitatem examinaturi, utrum scilicet ritè & æquabiliter in chalybem temperatum sit ferrum, horizontalem aquæ stagnantis superficiem plano gladio vehementer percutimus; per aërem scilicet, tanquam per medium antecedentis motûs, ad aquam devenit gladius; quicquid sit, quod & ipsa aqua ad ulteriorem motum, quo ensis profundiùs immergatur, medij rationem habere possit. Similiter aërem ipsum percuti à corpore, quod ex aquâ emergit, haud ægrè concesserim, si id quidem ex vi præcedentis motûs contingat: esto, minùs obsistat aër, quàm aqua, obsistit tamen, si à quiete dimoveatur, aut velociùs moveri cogatur, quàm moveretur, si hujusmodi nova impulsio vi præcedentis motûs non accideret: Neque aër, cum primùm emergens corpus in eum incurrit, habet rationem medij, sed perinde se habet primo illo momento, atque si tabella suspensa horizonti parallela faciem aquæ proximè contingeret, & in eam ex aquâ emergens corpus incurreret; quanquam ab hac majorem, quàm ab aëre, resistentiam subiret, atque adeò validiorem huic ictum infi- geret. Sed quid vocabula in quæstionem frustra vocamus? De his loquere, ut libet: per me sanè licebit, quando corpus ab uno fluido, per quod inchoatus est motus, ad aliud fluidum transit (five

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Book Seven. Chapter VI. 671 then, because it encounters other parts of the same thing to be divided; if, however, it continues evenly, preserving continuously its former speed or slowness, diminished in no respect, then no new resistance is to be supposed; but just as the same motion is continuous, with one part after another succeeding to it, so the same resistance is continued, and the later middle parts are not said to be struck, but, as before, to be precisely driven or cut through: because, namely, the preceding motion contributes nothing to this new impulse, wholly similar to the former one. But if a body in motion were on that account because it is moving to apply greater and greater speed, so that in the medium, that is, in the air itself, the measure of resistance should increase, I should easily agree with one who contends that the air is beaten and struck; but because the air offers itself too readily for this purpose, namely, to be cut through, I am not here speaking of this sort of striking of the medium in which the motion takes place, but rather of the striking of the body to which the body that strikes approaches through the medium. Hence I should not deny that water is struck, when, in order to test the quality of a sword, namely whether the iron is duly and evenly tempered into steel, we strike violently with a flat blade the horizontal surface of stagnant water; through the air, namely as through the medium of the preceding motion, the sword comes to the water; whatever may be the case, since even the water itself may have the character of a medium for a further motion, by which the sword may be immersed more deeply. Similarly, I should not find it hard to concede that the air itself is struck by a body emerging from water, if indeed this happens by force of the preceding motion: granted, the air offers less resistance than water, yet it does resist, if it is moved from rest, or forced to move faster than it would move if this new impulse did not occur by force of the preceding motion. Nor, when a body emerging first of all rushes upon the air, does the air have the character of a medium, but it behaves at that first moment just as if a board suspended parallel to the horizon were to touch closely the face of the water, and a body emerging from the water were to rush against it; although from this it would receive greater resistance than from the air, and would therefore inflict a stronger blow upon it. But why do we vainly summon words into dispute? Speak of these matters as you wish: by me, certainly, it shall be permitted, when a body passes from one fluid through which the motion has begun to another fluid (whether

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Mechanicorum 672 (sive hoc magis, sive minus crassum atque concretum fuerit) hujus posterioris fluidi primum contactum cum impulsione Percussionem æquè appellare, atque si non fluidum esset, sed duum; validiùs scilicet impetitur vi antecedentis motûs, quàm si corpus incurrens tunc primùm à quiete recederet. Hîc soli- dorum atque consistentium corporum percussiones persequi- mur, quarum vim inquirimus, & modum recipiunt à resistentiâ corporis percussi, quæ quò major est, validior quoquè cæteris paribus efficitur percussio. Sic si quis velit alteri alapam in- fligere, nullus erit ictus, si æquè velociter ad easdem partes moveantur tum percutientis manus, tum is, cui destinata est alapa; quia nulla est resistentia motum manûs impediens, aut retardans: erit verò ictus genere ipso validissimus, si sibi oc- currant, & quò majore impetu atque velocitate occurrent, eò validior; quia nullum est majus resistentiæ genus, quàm si duo oppositi motus se invicem retundant. Quod si demum percu- tientis manus moveatur velociùs, quàm is, qui percutitur, quamvis ad easdem partes moveantur, ictus infligetur validus pro Ratione excessûs velocitatis, cui motus tardior resistit, qua- tenus corpus tardum tandem à velociore deprehenditur, atque urgetur: antè ictum verò si corpus percussum quiescat, quo ve- locior erit percutientis motus, validior quoquè erit ictus; ad eandem enim resistentiam major motus habet majorem Ratio- nem, quàm minor. Vim igitur percussionis ex antecedenti motu originem duce- re manifestum videtur; non quidem quâ motus est ex loco in locum transitus, hic enim ante corporum contactum ictum nullum infligere potest, in ictu autem ipso motus omnis præce- dens evanuit, nec jam extinctus quicquam efficere potest, etiamsi motui præsenti vis aliqua efficiendi tribueretur. Sed quia cum motu illo antecedente acquisitus est impetus, qui adhuc durans ipso percussionis momento longè plus habet vi- rium, quàm si tunc omnino inciperet motus cum impulsione; augetur siquidem in motu impetus ab eâdem causa movente productus singulis momentis, semper enim ad agendum causa necessaria applicata est, atque, si maneat, utilitate non caret impetus, quem subsequi potest motus. Quid nimirum causæ est, quare ligneus globus leniter aquæ impositus innataret, si verò

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Mechanics 672 (whether this be more or less thick and condensed) the first contact of this later fluid, together with the impulse, may equally be called percussion, as if it were not a fluid, but a solid body; that is to say, it is struck more strongly by virtue of the antecedent motion than if the body coming in were then first departing from rest. Here we are pursuing only the percussions of solid and coherent bodies, whose force we inquire into, and they receive their manner from the resistance of the body struck, which the greater it is, the stronger also, all other things being equal, is the percussion. Thus if someone should wish to strike another with a slap, there will be no blow, if both the striking hand and the one for whom the slap is intended should be moving equally swiftly toward the same parts; because there is no resistance hindering or retarding the motion of the hand: but the blow will be by its very nature most forceful, if they run against one another, and the greater the violence and speed with which they meet, the stronger it will be; because there is no greater kind of resistance than when two opposing motions repel one another. But if at last the hand of the striker move more swiftly than the one who is struck, although they move toward the same parts, the blow will be inflicted strongly in proportion to the excess of velocity, against which the slower motion resists, inasmuch as the slow body is at length overtaken by the swifter and pressed upon: but before the blow, if the struck body be at rest, the swifter the motion of the striker, the stronger also will be the blow; for to the same resistance a greater motion has a greater ratio than a smaller one. Therefore it seems manifest that the force of percussion derives its origin from antecedent motion; not indeed as motion is a passage from place to place, for this before the contact of bodies can inflict no blow; but in the blow itself all antecedent motion has vanished, nor can what is now extinguished effect anything, even if some power of producing effects were attributed to the present motion. But because with that antecedent motion impetus has been acquired, which still continuing at the very moment of percussion has far more force than if then motion were to begin altogether together with the impulse; for impetus is indeed increased in motion by the same moving cause produced at each moment, since the necessary cause for acting is always applied, and, if it remain, the impetus, which motion can follow, is not without usefulness. For what reason indeed is there, why a wooden globe gently placed upon water should float, if indeed

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Liber septimus. CAPUT VI. 673 verò ex editâ turri in subjectam fossam dimittatur, aquam al- tiùs penetrat? nisi quia impetum in motu globus acquisivit, quo perseverante terminos suæ gravitati à Naturâ præscriptos transilit, eóque demum languescente, aut illum aqua sursum extrudet, aut vi suæ levitatis sponte ascendet. Sic citrà nota- bilem doloris sensum sustinemus capiti impositum lapidem for- tè bipedalem, at non item scrupuli duorum digitorum ex alti- tudine centum cubitorum decidentis ictum ferre possumus ci- trà incommodum non sanè leve: id quod ex acquisito impetu contingere palàm est, nulla quippe alia præter impetum in promptu est causa, cui vis hæc efficiendi commodè, atque pro- babili conjecturâ, tribuenda sit. Hunc impetum in motu acquisitum Gravitatis nomine indi- gitare placuit Aristoteli, cùm propositæ quæstioni 19. satisfa- cere contendens ait, An quia omnia cum motu fiunt, & grave ipsum gravitatis magis assumit motum, dum movetur, quàm dum quiescit? Incumbens igitur connatam gravi motionem non movetur; motum verò & secundùm hanc movetur, & secundùm eam, quæ est percutientis. Neque enim adeò in rebus Physicis Aristotelem, ejusque peritiores asseclas cæcutiisse dixerim, ut gravitatem corpori insitam, quæ prima radix atque origo est, cui motus debeatur, in ipso motu revera augeri existimaverint (quamvis nullâ factâ naturæ, saltem constipatis partibus, mutatione) haud secus, ac calori calor addatur. Sed idcirco plus gravitatis assumi dicitur à corpore gravi dum movetur, quàm dum quies- cit, quia in motu vi ac potestate se movendi æquiparat corpora graviora, atque adeò plus habet gravitatis non Formaliter, sed Virtualiter & Æquivalenter, ut ipsorum Peripateticorum vo- cabulis utar. Cæterùm gravitatis nomine non ipsum pondus intelligi ab Aristotele suadet ipsa loquendi formula, qua gravi- tatem assumptam dum movetur, confert cum gravitate assump- tâ dum quiescit, ut hæc illâ minor censeatur: videtur enim Aristoteles in corpore gravi ad motum prono agnoscere as- sumptum impetum, quo fieret connata ipsi corpori gravi mo- tio, nisi impediretur, dum quiescit, & præter hunc impetum, alium in motu acquisitum, adeò ut demum utroque impetu moveatur, ac proinde dicatur in motu plus assumere gravitatis. Quòd si hæc philosophandi ratio placeat, qua corpori gravi QQ99

Transcription: Translated (English)

But if from a lofty tower it be let down into the pit below, why does it penetrate the water more deeply? Unless because the ball acquired an impetus in motion, by the continuance of which it passes beyond the bounds prescribed by Nature to its heaviness; and when that impetus at last grows faint, either the water will drive it upward, or by the force of its own lightness it will rise of itself. So too, without any notable sense of pain, we sustain a stone perhaps two feet long placed upon the head; but a pebble two fingers broad falling from a height of a hundred cubits we cannot bear striking us without no slight inconvenience: and this plainly happens from the acquired impetus, since there is in readiness no other cause besides impetus to which this power of producing that effect can, with convenience and probably conjecture, be attributed. It pleased Aristotle to call this impetus acquired in motion by the name of gravity, when, striving to answer the proposed question 19, he says: “Is it because all things take place with motion, and the very heavy thing itself assumes the motion of gravity more when it is moving than when it is at rest? The thing weighed down, therefore, by its connate motion is not moved; but by motion both according to this and according to that of the striker it is moved.” Nor indeed would I say that Aristotle, and his more skilled followers, were so blind in physical matters as to suppose that the heaviness inherent in a body, which is the first root and origin to which motion is owed, is in truth increased in the very motion itself (although no change in nature having been made, at least in the compacted parts), no otherwise than as heat is added to heat. But for this reason it is said that more gravity is assumed by a heavy body when it is moving than when it is at rest, because in motion by the force and power of moving itself it is equal to heavier bodies, and thus has more gravity not formally, but virtually and equivalently, to use the vocabulary of the Peripatetics themselves. Moreover, that by the name of gravity Aristotle did not understand the weight itself is shown by the very form of speech, by which he compares gravity assumed while moving with gravity assumed while at rest, so that the latter is judged less than the former. For Aristotle seems to acknowledge in a heavy body inclined to motion an assumed impetus, by which there would arise in the heavy body itself a connate motion, unless it were hindered while at rest; and beyond this impetus, another acquired in motion, so that at last it is moved by both impulses, and therefore is said in motion to assume more gravity. But if this manner of philosophizing should please, whereby to a heavy body QQ99

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Mechanicorum 674 idcircò saltem ad speciem quiescenti, quòd removere non va- leat ea, quæ obstant, & motum, qui sub sensum cadat, impe- diunt, conceditur impetus Innatus, qui sit ipsa actualis gravi- tatio insitæ gravitati addita, reipsa connitens aut adversùs sub- jectum corpus, aut contra vim suspendentem: Cùm gravitas in motu alium atque alium adhibeat novum conatum ad descendendum, perinde videtur contingere, ac si toties mul- tiplicata fuisset eadem gravitas, quoties multiplicatus fuit co- natus priori illi æqualis. Hac autem ratione non ineptè dixit Aristoteles grave ipsum assumere plus gravitatis in motu, quia sublato motûs impedimento & impetus Innatus suas omnes vi- res exerit, & augetur Acquisito: ac propterea minor gravitas sic æquivalenter multiplicata longè plus efficit, quàm si major gravitas incumberet, cujus impetus Innatus impediretur, ne motum efficeret ullum neque compressionis corporis subjecti, neque distentionis corporis suspendentis. Quando autem nul- lus omnino motus, sivè qui sub sensum cadat, sive qui aciem omnem fugiat, tribuitur corpori gravi, nullus quoquè superad- ditus Impetus ipsi connatæ gravitati respondens concedendus est; neque enim deorsum reipsa connititur; quamvis in se ha- beat principium & originem gravitandi, si impedimentum sal- tem ex parte removeatur. Cùm itaque omne id, quod percussionis ictum consequitur, ab impetu oriatur, neque impetum gravitas, aut ulla moven- di facultas, concipere valeat, quin aliquo saltem motu ipsa mo- veatur; nil mirum, si ingens moles prohibita, ne prorsùs mo- veatur, nullam labem inferat subjecto lapidi, quem minor gra- vitas cadens, atque percutiens in frusta comminuit: minor sci- licet gravitas liberè descendens multum concipit impetum, quem lapidi percusso communicans cogit eum in frusta dissili- re, si vis impetûs superet partium nexum, aut saltem eum con- cutit. Nullum autem effectum impetûs ab ingenti mole prorsùs quiescente expectare possumus, quippe quæ nullum imprime- re potest impetum subjecto corpori. Hinc mirari cessent, qui plumbeum globulum primo mallei ictu certam compressionem pati observant, secundo verò ictu priori omninò æquali adhuc magis comprimi, quamvis minore compressione, quia particulæ jam per vim constipatæ validiùs rejiciunt

Transcription: Translated (English)

Mechanics 674 therefore, at least in appearance, to a body at rest, because it is unable to remove the obstacles that stand in the way and prevent motion perceptible to the senses, there is granted an innate impetus, which is the actual gravitation itself added to the indwelling weight, really tending either against the body subjected to it or against the suspending force. Since weight in motion applies a new and different effort to descend, it seems to happen just as though the same weight had been multiplied as many times as the effort equal to that first one was multiplied. And for this reason Aristotle not unfittingly said that a heavy body takes on more heaviness in motion, because when the impediment to motion is removed, the innate impetus exerts all its forces and is increased by what is acquired; and therefore a lesser weight thus equivalently multiplied accomplishes far more than if a greater weight were pressing down, whose innate impetus would be prevented from producing any motion, whether by the compression of the body below or by the stretching of the suspending body. But when no motion at all, whether perceptible to the senses or escaping every eye, is attributed to a heavy body, no added impetus corresponding to its inborn weight is to be granted either; for it does not really tend downward, although it has within itself the principle and source of gravitating, if at least the impediment be removed in part. Since therefore everything that follows upon the blow of percussion arises from impetus, and weight, or any faculty of moving, cannot conceive impetus without itself being moved by some motion at least, it is no wonder if a huge mass, being prevented from moving at all, inflicts no damage on the stone beneath it, whereas a lesser weight, falling and striking it, shatters it into fragments: namely, a lesser weight descending freely conceives great impetus, and, communicating it to the stone struck, forces it to burst apart if the force of the impetus overcomes the cohesion of the parts, or at least shakes it. But from a huge mass entirely at rest we can expect no effect of impetus, since it cannot impress any impetus upon the body beneath it. Hence let those cease to wonder who observe that a leaden ball at the first stroke of the hammer suffers a certain compression, but at the second stroke, though entirely equal to the first, is compressed even more, although with less compression, because the particles already packed together by force repel more strongly

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Liber septimus. CAPUT VI. 675 rejiciunt majorem violentiam. At si globulum similem subjiciant ponderi, quod illum æquè comprimat, ac prior ictus mallei, addito adhuc æquali pondere non sequitur compressio globuli tanta, quæ respondeat secundo ictui mallei: ex quo satis constat duplicis percussionis vires non æquari à duplici gravitate. Si enim animum attentè advertant, videbunt mallei motus tam in primâ quàm in secunda percussione planè æquales esse, tùm ratione velocitatis, tùm ratione spatij, ac proinde æquali impetu malleum percutere: At factâ jam primâ subjecti globuli compressione, in qua gravitas incumbens motum habuit illi compressioni respondentem, manifestum est, propter majorem globuli jam compressi resistentiam, non posse secundam gravitatem priori æqualem additam æquali motu, nec æquali velocitate moveri, atque propterea neque posse æqualem impetum concipere, quo possit effectum secundo mallei ictui similem producere. Adde quod secundum pondus additum priori, atque illi impositum, suum habet gravitatis centrum, & commune totius ponderis globulo incumbentis centrum gravitatis transfertur in aliud molis compositæ punctum; ideóque linea directionis non similiter incurrit in subjectum globulum, adversùm quem similes exhibeat vires. Neque mihi facilè persuadebis tam accuratè secundum pondus adjectum priori, ut posterius gravitatis centrum in eâdem sit lineâ directionis, nec ab illâ quicquam deflectat. Quare vel posterior hæc gravitas addita priori, jam quiescenti, ubi facta est vis comprimendi par virtuti resistendi, omni prorsus motu caret, & nihil impetûs potest concipere, aut imprimere; vel solùm tenuissimum, & qui vix post multum tempus conspicuus fiat, motum habet, & non nisi levem impetum imprimit, quo subjectus globulus demum aliquantulo compressior appareat: ideò, ut compressio similis illi, quæ fit à secundo mallei ictu, habeatur, necesse est gravitatem additam esse adhuc majorem, ut gravitas tota composita impetum efficere valeat, quem consequatur motus æqualis secundæ illi compressioni à malleo factæ. Cur itaque securis ligno incumbens, quamvis ingenti prægravata pondere, vix levem fissionem inferat subjecto ligno, quod tamen altiùs penetratur ab eâdem securi cadente & percutiente, in promptu causa est: quia videlicet compressio, quæ QQqq

Transcription: Translated (English)

Book Seven. CHAPTER VI. 675 reject a greater violence. But if they apply a ball similar to the weight, which compresses it equally as much as the former blow of the hammer, with an equal weight added still further, the compression of the ball does not follow so great as would correspond to the second blow of the hammer: from which it is sufficiently clear that the forces of a double percussion are not equal to a double heaviness. For if they pay close attention, they will see the motion of the hammer both in the first and in the second blow to be plainly equal, both with respect to speed and with respect to space, and therefore that it strikes the hammer with equal force: But once the first compression of the ball beneath it has been made, in which the weight pressing down had a motion corresponding to that compression, it is manifest that, because of the greater resistance of the ball now compressed, the second weight, added equal to the former, cannot be moved with equal motion, nor with equal speed, and therefore can neither conceive an equal impulse, by which it may be able to produce an effect similar to the second blow of the hammer. Add also that the second weight added to the former and placed upon it has its own center of gravity, and the common center of gravity of the whole weight pressing on the ball is transferred to another point of the composite mass; and therefore the line of direction does not strike the subject ball in the same manner, against which it would display similar forces. Nor will you easily persuade me that the second weight is so accurately added to the first that the later center of gravity is on the same line of direction, and does not deviate in any way from it. Wherefore either this later weight, added to the former now at rest, when the force of compressing has been made equal to the resisting force, is wholly without motion, and can conceive or impart no impulse at all; or else it has only the slightest, and one which can scarcely be noticed until after a long time, motion, and imparts only a slight impulse, by which the ball underneath may at last appear somewhat more compressed: therefore, in order that a compression similar to that which is produced by the second blow of the hammer may be had, it is necessary that the added weight be still greater, so that the total combined weight may be able to produce the impulse which is followed by a motion equal to that compression made by the hammer. Why then does the axe resting on the wood, although burdened with a great weight, hardly inflict a slight split upon the wood beneath it, while the same axe, when falling and striking, penetrates the wood more deeply, the cause is at hand: namely because the compression, which QQqq

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Mechanicorum & impulsio est, cum motu quidem fit, sed ipso statim initio & in progressu adest resistentia, ne producatur totus impetus, quem vis motiva posset efficere, & motus non est, nisi quan- tum impellitur objectum corpus; ideóque securis vi ponderis incumbentis non valet in hujusmodi motu alium impetum con- cipere præter illum, quem fert præsens motus, qui valde exi- guus est: At percussio ea est, ut cùm primùm securis cadens applicatur ligno, jam multum habeat concepti imptus in aëre libero, & nihil adhuc resistente ligno, ac propterea possit ve- lociùs moveri comprimendo & dividendo subjectum lignum. Ex quo fit onus securi impositum tantæ gravitatis esse oportere, ut quæ Ratio est spatij à securi cadente decursi ad spatium, quo illa penetrat lignum, ea saltem sit Ratio gravitatis conflatæ ex securi & addito pondere ad gravitate simplicis securis, ut fieret æqualis scissio ab eâdem securi: ut videlicet tantumdem im- petûs concipiatur à magnâ gravitate in exiguo motu præsente resistentiâ, quantum impetûs concipitur à securi in anteceden- te motu longiore absque resistentiâ ullâ, præterquam medij. Similiter nullum adhiberi posse pondus, quo aureæ lamellæ imposito hæc diduci possit in subtilissimam bracteolam, quem- admodum vi mallei percutientis, ex iisdem principiis constat. Attende enim, quanto motu moveri possit illud pondus com- primens; utique non nisi quantum est altitudinis discrimen in- ter lamellam & bracteolam: at tantillum spatium, in quo exer- cendus esset motus, quam Rationem habet ad toties multipli- catum spatium, in quo iteratis sæpiùs ictibus liberè movetur malleus? Cùm itaque minimus motus, aut etiam fortassè nul- lus, post tenuissimam auri compressionem ingenti illi oneri con- veniat, nil mirum si exiguo impetu ferè nihil efficiat, cùm ta- men malleus novo semper impetu singulis ictibus concepto ali- quam, licèt semper minorem atque minorem, compressionem efficiat. Ut autem res hæc pleniùs innotescat, observa impulsionem, qua corpus urgetur, opponi tractioni, & compressionem par- tium distractioni, atque sicut corporis, quod urgetur, particu- læ aliquando comprimuntur, ita corporis, quod trahitur, par- ticulas aliquando distrahi, aut divelli, neque dissimilem esse resistentiam corporum vi suæ gravitatis, ne impellantur, aut ratione

Transcription: Translated (English)

Mechanics and impulsion is such that, although motion takes place, resistance is present right from the very beginning and in the progress of the motion, so that the whole impetus that the moving force could produce is not developed, and the motion is only so far as the object body is impelled; and therefore an axe, by the force of the hanging weight, is not able in this kind of motion to conceive any other impetus besides that which its present motion carries, and that is very slight: but the striking blow is such that, as soon as the falling axe is applied to the wood, it already has much of the conceived impetus in free air, and the wood as yet resists nothing, and therefore it can move more swiftly by compressing and dividing the wood beneath it. From this it follows that the load placed upon the axe ought to be of such weight that, as the ratio is of the space traversed by the falling axe to the space into which it penetrates the wood, so at least may be the ratio of the weight made up from the axe and the added load to the weight of the simple axe, so that an equal splitting might be made by the same axe: namely, that as much impetus is conceived from the great weight in a small present motion with resistance as is conceived from the axe in a longer preceding motion with no resistance at all, except that of the medium. Similarly, it is clear from the same principles that no weight can be applied by which a gold plate, when loaded, can be drawn out into the thinnest leaf, as is done by the force of a hammer striking. For consider how much motion that compressing weight can have; assuredly no more than the difference of height between the plate and the leaf: but what proportion has so small a space, in which the motion would have to be exercised, to the many times multiplied space in which, by repeated blows, the hammer moves freely? Since therefore the slightest motion, or perhaps even none at all, suits that great burden after the very thin compression of the gold, it is no wonder if by a slight impetus almost nothing is effected, whereas the hammer, each time conceiving a new impetus from each blow, effects some compression, though always less and less. But that this matter may be more fully understood, observe that impulsion, by which a body is pressed, is opposed to traction, and compression of parts to disjunction; and just as in the case of a body that is pressed, the particles are sometimes compressed, so in the case of a body that is drawn, the particles are sometimes drawn apart or torn asunder; and the resistance of bodies by virtue of their own gravity, lest they be impelled, is not dissimilar, or by reason

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Liber septimus. CAPUT VI. 677 ratione positionis partium, ne comprimantur, ac ne trahantur, aut particularum nexus dissolvatur. Quapropter ubi primùm incipit impulsio aut tractio, sive compressio aut distractio, in- cipit etiam resistentia, quæ eò major evadit, quò majorem vio- lentiam subit corpus. Hinc est potentiam impellentem aut tra- hentem semper minore impetu ferri, quàm si liberè moveretur, dum nulla adesset resistentia. Sic si quis funiculum, quem re- tinet clavus parieti infixus, arripiat, atque jam extentum trahat, illum quidem multo nisu intendit, sed nec illum dis- rumpere valet, nec clavum revellere: sed si eodem conatu fu- niculum languidum nec dum extentum trahat, celeriter mo- vetur manus, antequam funiculus extendatur, & facilè aut hic abrumpitur, aut ille revellitur. Quia nimirum extenti jam fu- niculi resistentia, ne intendatur, impedit, ne potentia pro Ra- tione sui conatûs moveatur, multo impetu absumpto in vincen- dâ illâ resistentia; neque movetur potentia nisi cunctabunda, & per brevissimum spatium, quantum vi intentionis funiculus magis extenditur: At ubi languidus est funiculus, potentia absque ullo retinente per aliquantum spatij liberè movetur, & totum impetum suo conatui respondentem in efficiendo celeri motu impendit, quem jam notabiliter auctum invenit funicu- lus, cum primùm est extentus, & adhuc magis augetur perse- verante eodem conatu. Quare cùm multò major sit impetus, satis esse potest non solùm ad intendendum funiculum, verùm etiam ad illum disrumpendum, aut, si, hujus particulæ validio- re nexu jungantur, ad revellendum clavum. Ex his habes, quid respondeas doctissimis viris vim percus- sionis investigantibus. Ut apparet, quantâ vi plumbeus globu- lus unciarum duarum ex cubitali altitudine cadens percuteret subjectum corpus, existimârunt satis innotescere, si globulus ille funiculo cubitali adnecteretur chordæ arcûs medio loco in- ter extremitates. Tum sublatus globulus usque ad chordam ipsam, dimissus est, atque observatum est punctum, ad quod adducta est chorda: proclive enim erat arguere, globulum tantâ vi percussurum subjectum corpus, quantâ vi inflectebat balistæ arcum. Quare tentando varia pondera addiderunt chordæ arcûs, donec demum pondus decem librarum chordam ad idem punctum adduxit, ad quod adducta fuerat à globo ca- QQqq 3

Transcription: Translated (English)

Book Seven. CHAPTER VI. 677 by reason of the position of the parts, lest they be compressed and lest they be drawn apart, or the cohesion of the particles be dissolved. Wherefore, as soon as impulse or traction, whether compression or separation, begins, resistance also begins; and this becomes the greater the greater violence the body undergoes. Hence it is that the impelling or drawing power is always carried with less impetus than if it were moved freely, when no resistance were present. Thus, if someone should seize a cord which a nail fixed in the wall holds fast, and now, being stretched, should pull it, he indeed strains it greatly, but can neither break it nor wrench out the nail; but if with the same effort he pulls a slack cord not yet stretched, the hand is moved quickly before the cord is extended, and easily either the cord snaps or the nail is wrenched out. For the resistance of the cord already stretched, so that it may not be stretched further, hinders the power from moving according to the measure of its effort, much of its force being spent in overcoming that resistance; nor is the power moved except slowly, and for the shortest space, so far as the cord is more extended by the force of the strain. But when the cord is slack, the power moves freely for some distance without anything restraining it, and spends the whole impetus corresponding to its effort in producing a rapid motion, which the cord, now stretched, finds notably increased, and which is increased still more if the same effort continues. Therefore, since the impetus is much greater, it may be sufficient not only to stretch the cord, but also to break it, or, if its particles are joined by a stronger bond, to wrench out the nail. From these remarks you have what to answer to the most learned men who investigate the force of percussion. As appears, how strongly a leaden ball of two ounces, falling from a height of one cubit, would strike the body beneath was thought to be sufficiently made known if that ball were attached by a cubit-long string to the middle point of the bowstring between its ends. Then the ball was lifted up to the string itself and released, and the point to which the string was drawn was observed: for it was easy to argue that the ball would strike the body beneath with the same force with which it bent the arbalest’s bow. Wherefore, by trial, they added various weights to the bowstring, until at last a weight of ten pounds drew the string to the same point to which it had been drawn by the falling ball of the ca- QQqq 3

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Mechanicorum dente, atque in eodem flexionis statu chordam & arcum detinuit. Arguebant igitur percussionem globi plumbei duarum unciarum ex cubitali altitudine cadentis æquiparari pressioni decem librarum. Ulteriùs autem progrediendo, adhibita est balista alia validior, cujus arcus ob duriorem ferri temperationem minùs erat flexibilis: quapropter cum ejusdem potentiæ eadem sit vis, ejusdem globuli ex eâdem altitudine similiter cadentis non nisi eædem esse poterant vires ad vincendam æqualem resistentiam: atque adeò durioris arcûs minor flexio æquè resistens, ac major flexio arcûs mollioris, breviore termino definivit descensum globuli plumbei, & ad propius punctum adducta est chorda. Verùm, ut in eodem flexionis statu fortior hic arcus retineretur, non satis fuit decem libras appendere, sed viginti librarum pondere opus fuit. Hinc inferebant eandem ejusdem globuli duarum unciarum percussionem æquare non solùm vires librarum decem, sed & viginti: atque usque eò argumentationem deducebant, ut assumpto robustiore aliquo arcu concluderent, ne pondus quidem librarum mille satis esse ad arcum illum in eâ positione retinendum, ad quam fuisset adductus à globo duarum unciarum cadente: id quod vim quandam percussionis infinitam indicare videbatur. Verùm quamvis hos ingeniosorum hominum conatus non modò non improbem, sed multâ commendatione dignos existimem, liceat tamen mihi argumentationis infirmitatem exponere; tam enim non est vis percussionis duarum unciarum infinita, quàm infinita non est vis pressionis decem librarum. Quando enim vi ponderis adnexi flectitur arcus, utique pondus descendit, & suâ gravitate superat rigidi chalybis vires, donec demum æqualitas quædam intercedat inter vim arcûs elasticam, & gravitatis conatum ad descendendum; tunc scilicet fit consistentia. Prout igitur robustiores sunt arcus, minùs permittunt descendere pondus chordæ appensum, si omnia sint paria: Nam si brevior sit arcus mollis & languidus, longior verò arcus durioris temperationis, fieri potest, ut idem pondus æqualiter adducat longiorem chordam atque breviorem, simili planè ratione ac de ponderibus fune suspensis præponderantibus atque æquilibribus dictum est lib. 3. cap. 12: ideò ponendi sunt arcus ita similes & æquales, ut solâ ferri temperatione discrepent.

Transcription: Translated (English)

the dente of the mechanics, and held the string and bow in the same state of flexion. They therefore argued that the impact of a two-ounce lead ball falling from a cubit’s height was equivalent to the pressure of ten pounds. Going further, another, stronger crossbow was used, whose bow, because of the harder tempering of the steel, was less flexible: wherefore, since the power is the same, the force of the same ball falling likewise from the same height could be only the same in order to overcome equal resistance; and thus a smaller flexure of the harder bow, being equally resistant, and a greater flexure of the softer bow, determined the descent of the lead ball within a shorter range, and the string was drawn to a nearer point. But in order to keep this stronger bow in the same state of flexion, it was not enough to hang ten pounds, but a weight of twenty pounds was required. Hence they inferred that the same impact of the two-ounce ball was equal not only to the force of ten pounds, but also of twenty: and they carried the argument so far that, taking some even more robust bow, they concluded that even a weight of a thousand pounds would not be enough to keep that bow in the position to which it had been drawn by the falling two-ounce ball: which seemed to indicate an infinite force of impact. But although I do not merely disapprove of these efforts of ingenious men, but consider them worthy of much commendation, yet I may be allowed to set forth the weakness of the argument; for no more is the force of the impact of two ounces infinite than the force of the pressure of ten pounds is infinite. For when a bow is bent by the force of a suspended weight, the weight certainly descends, and by its own gravity overcomes the strength of the rigid steel, until at last some equality intervenes between the bow’s elastic force and the tendency of gravity to descend; then consistency is achieved. Accordingly, the stronger the bows are, the less do they permit the weight hanging from the string to descend, if all else is equal: for if the soft and weak bow be shorter, but the bow of harder temper longer, it may happen that the same weight draws the longer string and the shorter one equally, in precisely the same way as was said in book 3, chapter 12, concerning weights suspended from a rope, both preponderating and in balance: therefore the bows are to be assumed so similar and equal that they differ only in the tempering of the steel.

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Liber septimus. CAPUT VI. 679 crepent. Si igitur validioris arcus repugnantia, ut flectatur ad duos digitos, tanta est, quanta repugnantia mollioris arcûs, ut flectatur ad sex digitos, patet non esse eumdem impetum de- cem librarum descentium solùm per duos priores digitos, atque per sex: ac proinde cùm decem libræ applicatæ arcui va- lidiori solùm possunt per duos digitos (& quidem lentiùs propter majorem resistentiam) moveri, minus possent, quàm per impetum conceptum in motu sex digitorum; & propterea neque possent illius robustioris arcûs chordam adducere ad duos digitos; sed neque adductam ab aliâ potentiâ possent reti- nere in eo statu ac positione: quia etiam si vis elastica arcûs ro- bustioris inflexi ad duos digitos par esset virtuti elasticæ arcûs imbecillioris inflexi ad sex digitos, cui reluctantur decem li- bræ; hæ minùs repugnant, ne ad duos digitos, quàm ne ad sex attollantur; igitur decem libræ minùs resistunt virtuti elasticæ arcûs fortioris, adeóque nec possunt in eo flexionis statu reti- nere arcum fortiorem & chordam: si enim pares sunt vires elasticæ arcûs inflexi ad duos digitos, & arcûs inflexi ad sex di- gitos, pari impetu se restituunt, ut parem violentiam excu- tiant; at pondus par utrique chordæ adnexum non pari veloci- tate movetur, si ad duos ac si ad sex digitos attollatur; igitur minùs resistunt decem libræ motui duorum, quàm motui sex digitorum. Porrò vis globi cadentis non est comparanda cum pondere quatenus retinente chordam in eadem flexione, sed quatenus illam adducente & flectente, ut motus cum motu, non verò motus cum quiete comparetur. In eo autem motu ponderis ad- ducentis chordam, & arcum inflectentis, quò major est re- sistentia, eò minor est impetus & velocitas, qua pondus illud movetur: igitur idem pondus non parem vim habere potest, ubi dispari impetu & velocitate movetur. At globus cadens antequam incipiat trahere chordam, nullum prorsus habet im- pedimentum, sed sivè fortior, sivè mollior sit arcus, eodem im- petu & velocitate movetur; ubi verò resistentiam invenit, so- lùm descendit ulteriùs pro ratione repugnantiæ; & factâ de- mum æqualitate inter vim descendendi à globulo acquisitam, & vim elasticam in arcu, cessat descensus, atque extincto im- petu acquisito, vi elasticâ vincente globuli gravitatem, hic sur- sum

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Book seven. CHAPTER VI. 679 creak. If therefore the resistance of the stronger bow, so that it is bent to two digits, is as great as the resistance of the softer bow, so that it is bent to six digits, it is clear that the same impulse of ten pounds descending only through the first two digits is not the same as through six: and therefore, since ten pounds applied to the stronger bow can be moved only through two digits (and indeed more slowly because of the greater resistance), they could move less than by the impulse conceived in the motion of six digits; and for that reason they could not even draw the cord of that more robust bow to two digits; but neither could they, once drawn by another force, retain it in that state and position: for even if the elastic force of the stronger bow bent to two digits were equal to the elastic virtue of the weaker bow bent to six digits, against which ten pounds resist, these resist less, not that they may be raised to two digits, than that they may be raised to six; therefore ten pounds resist less to the elastic force of the stronger bow, and accordingly cannot even retain the stronger bow and the string in that state of flexion: for if the elastic forces of the bow bent to two digits and of the bow bent to six digits are equal, they restore themselves with equal impulse, so as to repel equal violence; but an equal weight attached to each string is not moved with equal speed, whether it be raised to two digits or to six digits; therefore ten pounds resist the motion of two digits less than the motion of six digits. Moreover, the force of a falling globe is not to be compared with the weight insofar as it holds the string in the same flexion, but insofar as it draws it and bends it, so that motion is compared with motion, not motion with rest. But in that motion of the weight drawing the string and bending the bow, the greater the resistance, the smaller the impulse and velocity with which that weight is moved: therefore the same weight cannot have the same force where it is moved with different impulse and velocity. But the falling globe, before it begins to pull the string, has absolutely no impediment; but whether the bow be stronger or softer, it is moved with the same impulse and velocity; but where it finds resistance, it descends further only in proportion to the repugnance; and when at last equality has been established between the descending force acquired by the little globe and the elastic force in the bow, the descent ceases, and, the acquired impulse having been extinguished, while the elastic force prevails over the globe’s gravity, this rises upward

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680 Mechanicorum sum trahitur. Cùm itaque quicquid vi extrinsecùs assumptâ movetur, moveatur juxta excessum virtutis motivæ supra re- sistentiam; si æqualis resistentiæ mensura, quæ ex dissimilium arcuum majori aut minori flexione desumitur, eumdem exces- sum virtutis motivæ exigat, ut vincatur, & hunc excessum ha- beat globulus cadens, nil mirum, si idem globulus cadens id præstare possit, quod superat vires alicujus ponderis, cujus vis movendi non eumdem semper excessum habet supra illam re- sistentiam priori resistentiæ æqualem; quia videlicet non æqua- li impetûs intensione aggreditur motum, ubi ipso statim initio major invenitur difficultas, & tardior est motus. Non est igitur vis infinita globuli duarum unciarum nullo impedimento prohi- biti, quin ad trahendam cujuscumque arcûs chordam semper afferat, exempli gratiâ, centum gradus impetûs in motu acqui- sitos, quando pondera majora & majora tractionem incipientia à quiete non parem habent impetûs excessum, sed minorem & minorem pro duriore arcûs temperatione. An infinitam dixe- ris equi virtutem, qui solus in liberâ planitie currum trahat, ad quem trahendum in eâdem planitie altioribus atque altioribus nivibus obsitâ requiruntur plures & plures equi igitur nec in- finita est vis decem librarum, qua flectitur arcus mollis, quia ad flectendos arcus fortiores majus & majus pondus requiritur: huic autem virtuti decem librarum æqualis est vis globuli ca- dentis; hæc igitur & ipsa finita est. Nimirum aucta resistentia quodammodo imminuit virtutem agendi; ac propterea non sa- tis aptè comparantur decem libræ cum viginti libris perinde, atque si utræque essent omnino liberæ; sed unumquodque pon- dus componi debet cum suâ resistentiâ, ut demum habeatur excessus virtutis motivæ supra resistentiam. At, inquis, arcus fortior retinetur à libris viginti, & infir- mior à libris decem. Ita planè est: sed hîc pondera propriè non habent rationem efficientis, sed potiùs resistentis, quatenus impediunt arcuum vim elasticam, ne se restituant: cùm verò virtutes elasticæ ex genere suo propter disparem temperatio- nem inæquales sint, nil mirum, si ab inæqualibus resistentiis impediendæ sint, ne agant. Hinc autem non est desumenda ulla comparatio cum virtute globuli cadentis, quippe qui ac- quisitum impetum amittens non habet vim retinendi arcum in co

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680 of mechanics is drawn. Since, therefore, whatever is moved by force applied from outside is moved according to the excess of the motive power over the resistance, if the measure of equal resistance, which is taken from the greater or lesser bending of unlike arcs, requires the same excess of motive power in order to be overcome, and if the falling body has this excess, it is no wonder if that same falling body can do what surpasses the powers of some weight, whose moving force does not always have the same excess over that resistance equal to the former resistance; namely because it does not approach the motion with the same intensity of impulse, where at the very beginning there is greater difficulty, and the motion is slower. Therefore the power of a two-ounce body, hindered by no impediment, is not infinite, though in drawing the cord of any arc it always brings, for example, one hundred degrees of impulse acquired in motion, whereas greater and greater weights, beginning the pull from rest, do not have an equal excess of impulse, but a lesser and lesser one as the arc is made stiffer. Would you call the power of a horse infinite, if, alone, he were to draw a cart across a free plain, when to draw it over the same plain covered with higher and higher snowbanks more and more horses are required? Therefore the force of ten pounds by which a soft arc is bent is not infinite either, because to bend stiffer arcs a greater and greater weight is required; but equal to this force of ten pounds is the force of the falling body; therefore this also is finite. Indeed, when resistance is increased, the power of acting is somehow diminished; and for that reason ten pounds and twenty pounds are not suitably compared as though both were completely free; rather each weight must be compared with its own resistance, so that in the end one may have the excess of motive power over resistance. But, you say, a stronger bow is held by twenty pounds, and a weaker one by ten. That is indeed so: but here the weights do not properly have the role of an efficient cause, but rather of a resisting cause, insofar as they prevent the elastic force of the bows from restoring themselves; and since elastic forces, by their nature, are unequal because of differing temperaments, it is no wonder if they are to be hindered by unequal resistances so that they do not act. From this, however, no comparison should be drawn with the force of a falling body, since, as it loses the acquired impulse, it does not have the power of holding the bow in its ...

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Liber septimus. CAPUT VII. 68. eo statu; ad quem illum adduxit: at ponderis adnexi gravitas manet, & ibi retinet arcum, quò eum adduxit; nisi fortè aliquid impetum acquisierit in descensu, quo pereunte, aliquantulum præpolleat vis elastica, & sursum retrahat appensum pondus. Licet igitur globulo cadenti æqualiter resistere dicantur arcus fortior qui minùs flectitur, & mollior qui magis flectitur; postquam tamen jam per vim inflexi sunt arcus, naturaliter partes minùs flexibiles validiùs conantur se restituere, quàm flexibiliores: quemadmodum gravitas ut quatuor, & gravitas ut duo, si moveantur per vim motu reciprocè subduplo, æqualiter resistunt moventi; sed si utraque suspendatur, inæqualiter conantur suos motus naturales. CAPUT VII. Quàm dispare ex motûs velocitate sint percussiones. Percussionem ex ea parte, quatenus à simplici Impulsione distinguitur, motum exigere antecedentem, quo impetus acquiratur, superiori capite definitum est. Nunc verò, quia pro motuum velocitate diversâ dispare sunt percussionum vires, quærendum est, unde dissimilitudo ista procreetur, & quænam servari Ratio videatur, sivè inæquales ejusdem corporis, sive diversorum corporum percussiones inter se comparentur. Est autem considerandum in Impetu, qui est proximè efficiens motum, aliud esse ejus quantitatem, sivè entitatem accipere, aliud in ejusdem Intensione consistere: intensionem consequitur velocitas motûs, at ex entitate ipsâ magis extensâ, quamvis minùs intensâ, ac proinde ex motu tardiore, oriri potest validior ictus, de quo in sequentibus. Ex motûs autem velocitate, quæ corpori vi præcedentis motûs congrueret, si nihil obstaret, percussionem fieri majorem tam certis experimentis constat, ut vel cæci, si quando præfidentes concitatiùs ambulando caput ad objectum parietem allidunt, id abundè testari valeant; corpus R R r r

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Book Seven. CHAPTER VII. 68. in that state to which it has brought it; but the weight of the attached burden remains, and there it keeps the bow to which it has brought it; unless perhaps something has acquired momentum in the descent, through the loss of which the elastic force may somewhat prevail, and draw the suspended weight upward. Therefore, although those bows are said equally to resist a falling pellet, the stronger bow, which bends less, and the softer one, which bends more; yet after the bows have now been bent by force, the less flexible parts naturally strive to restore themselves more strongly than the more flexible ones: just as weight of four, and weight of two, if they are moved by force in reciprocally half-speed motion, equally resist the mover; but if both are suspended, they strive unequally toward their natural motions. CHAPTER VII. How greatly percussion differs according to the velocity of motion. That percussion, in so far as it is distinguished from simple impulsion, requires a preceding motion, by which momentum is acquired, was defined in the foregoing chapter. But now, since according to the different velocities of motions the powers of percussions are different, it must be asked whence this dissimilarity arises, and what rule seems to be observed, whether the percussions of the same body, or of different bodies, are compared with one another. It is to be considered, however, in momentum, which is the nearest efficient cause of motion, that one thing is its quantity, or entitiy, to be received, and another to consist in its intensity: velocity of motion follows intensity, but from the entity itself, if more extended though less intense, and therefore from a slower motion, a stronger blow may arise, as will be shown in what follows. And from the velocity of motion, which would correspond to the body by the force of a preceding motion, if nothing hindered, that the percussion becomes greater is established by certain experiments so clearly that even the blind, if ever, walking more hastily, they dash their head against an object or a wall, would abundantly be able to testify it; the body R R r r

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Mechanicorum 682 siquidem, quod motui resistit, majori & velociori motui magis resistit, quare & percussio fit validior. Ubi verò de motûs velocitate sermo est, non videtur dissimulanda medij scindendi resistentia; hoc quippe tardiori motui minùs, velociori magis obstat. Si enim ex lento & flexili virgulo abstractam virgam per aërem molli brachio huc, illuc, sursum, deorsum duxeris, hæc, aëre tenuissimam aut ferè nul- lam compressionem subeunte, vix, aut ne vix quidem, tantu- lum à directâ suarum partium positione deflectet: at si eam ve- hementiùs agitaveris, aëre tantam particularum compressionem renuente, manifestè inflexam videbis, & illatam sibi vim aër acuto sibilo prodet. Sic baculo aquam sensim ac leniter di- videns non admodum repugnantem experiris; at velociùs con- citanti illa validè resistit, eóque validiùs, quò crassior fuerit baculus. Ex his liquidò conficitur de percussione philosophan- tem frustra medij resistentiam mente abstrahere: Nam si nulla est sine motu percussio, nullus motus nisi per medium, neque sinè certa velocitatis aut tarditatis mensurâ, cui medium inæ- qualiter resistit; utique & motum à percussione ita mente pari- ter se jungere poteris, ut nihil prorsus de motu cogites, si nul- lam cum medio rationem habendam existimas: At motum, ejús- que velocitatem attendendam esse in percussione nemo negat; igitur neque medij resistentiam, quæ velocitati modum ali- quem statuit, omnino contemnere oportet. Hinc duæ ferè ex diametro oppositæ sententiæ cavendæ sunt, quarum altera gravium inæqualium motum statuit ipso- rum gravitatibus analogum, ut decuplò velociùs moveatur il- lud, quod est decuplò gravius: altera æqualem omnibus velo- citatem tribuit. Utramque manifesta experimenta falsitatis re- darguunt, si ex congruâ altitudine instituantur: Si enim ex val- dè editâ turri inæqualia corpora, aut ejusdem, aut diversæ se- cundùm speciem gravitatis dimittas, illud, quod gravius est, terram citiùs attingere observabis, sed tam brevi momentorum discrimine, ut nulla subesse possit suspicio servatæ velocitatum cum gravitatibus analogiæ, neque tamen de velocitatum inæ- qualitati dubitari queat. Meis scilicet auribus & oculis fidem abrogare nequeo, quicquid obtrudant aliqui in contrarium sua aut aliorum experimenta afferentes ex nimis brevi altitudine. Nam

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Mechanics 682 since that which resists motion resists more the greater and swifter motion, wherefore the impact also becomes stronger. But where the speed of motion is under discussion, the resistance of the medium through which it is cut should not seem to be overlooked; for this hinders the slower motion less, and the swifter motion more. For if from a soft and pliant switch you were to draw a rod through the air with a gentle hand this way and that, upward and downward, since the air undergoes a very slight or almost no compression, the rod would scarcely, or hardly at all, deviate from the straight position of its parts; but if you were to move it more violently, the air refusing so great a compression of its particles, you would clearly see it bent, and the air would betray the force inflicted upon it with a sharp whistle. Thus, if by a stick you divide water slowly and gently, you find it does not resist much; but if you drive it more quickly, it resists strongly, and the more strongly the thicker the stick is. From these things it is clearly concluded that, in philosophizing about impact, it is pointless to abstract the resistance of the medium in thought: for if there is no impact without motion, no motion except through a medium, and not without some fixed measure of speed or slowness, to which the medium resists unequally; then indeed you will be able to join motion to impact in thought in such a way that you think of nothing at all about motion, if you suppose no account is to be taken of the medium. But no one denies that motion, and its speed, must be attended to in impact; therefore the resistance of the medium, which sets some limit to speed, ought not to be wholly despised. Hence two almost diametrically opposite opinions must be avoided, one of which assigns the motion of unequal heavy bodies to their heaviness itself by analogy, so that that which is ten times heavier moves ten times faster; the other attributes equal speed to all. Clear experiments refute both as false, if made from a suitable height: for if from a very high tower you let fall unequal bodies, whether of the same species or of different specific gravity, you will observe that the heavier one reaches the earth sooner, but with so small a difference of moments that no suspicion can arise of a preserved analogy between speeds and weights, nor yet can the inequality of the speeds be doubted. I certainly cannot discredit my ears and eyes, whatever some may urge to the contrary by bringing forward their own or others’ experiments from too small a height. For

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Liber septimus. CAPUT VII. 683 Nam & sæpiùs in profundissimum puteum inæquales lapides dimisi simul, & ictuum sonitum alium alio priorem semper au- divi; id quod satis est ad illam velocitatum omnimodam æqua- litatem rejiciendam, quamvis uter prior aquam attigerit, certò dignoscere non valeret auris; quòd si alter in subjectam peluim æneam, alter in vas ligneum decidisset, potuisset auris dijudi- care ex sonitu: Et ex altissimâ turri Bononiensi dimissa pondera inæqualia observavi initio quasi æqualiter descendere ita, ut oculus nullam velocitatum dissimilitudinem adhuc dignosce- ret; deinde procedente descensu paulatim gravitas major præ- currere notabiliter incipiebat, semperque magis augebatur ve- locitas, adeò ut aliquando gravitas major terram attigerit, quando minor adhuc aberat intervallo pedum quadraginta, quemadmodum ex notâ in turris latere dimetiri licuit. Verum quidem est brevissimâ temporis mensurâ & hanc minorem in terram decidisse. Id quod fortasse fucum fecit non animadver- tentibus magnæ velocitati multum respondere spatij, quod quasi momento percurritur; ac propterea æqualitatem veloci- tatum utrique gravitati tribuendam censuerunt, quia exiguum erat temporum discrimen. An vellus, quantum pugno com- prehenditur, æquè velociter ac prægrande saxum descensurum existimas? Figuræ dices tribuendum plurimum, non enim ab omnibus corporibus æquè facilè dividitur aër nunquam non fluctuans. Ita sane: igitur si aër inæqualiter resistit, inæqua- liter moveri possunt corpora cadentia. Adde non dissimiliter gravia & levia ad suos motus à naturâ incitari; atque adeò si, ubi plus est levitatis, velociorem motum sursum observamus, etiam, ubi plus est gravitatis concitatiorem motum deorsum arguere debemus. Aquam in longiore fistulâ vitreâ aliquan- diu agita, ut aër fistulæ inclusus aquæ admisceatur: ubi ab agi- tatione cessatum fuerit, majores aëris particulas citò ascenden- tes videbis, dum minores cunctabundæ paulatim moventur: id quod clariùs constabit, si aquæ loco hydrargyrum in fistulam admiseris. Quidni igitur gravia pariter deorsum dispari velo- cite moveantur, si inæqualia fuerint? Non tamen servandam esse gravitatum analogiam hinc apertè constat, quod in corporibus ejusdem speciei Ratio gra- vitatum eadem est ac magnitudinum: magnitudines autem RRrr 2

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Book Seven. Chapter VII. 683 For, more than once, I let down unequal stones together into a very deep well, and I always heard the sound of their impacts one before the other; which is enough to reject that complete equality of speeds, although the ear could not certainly determine which reached the water first. But if one had fallen into a brass basin beneath, and the other into a wooden vessel, the ear could have distinguished it from the sound. And from the highest tower of Bologna I observed unequal weights let fall, and at the beginning they seemed to descend as if equally, so that the eye as yet perceived no difference in their speeds; then, as the descent proceeded, the greater weight began little by little to get ahead noticeably, and its speed was always increased more and more, so that at times the greater weight had touched the ground when the smaller was still forty feet away, as I was able to measure from a mark on the side of the tower. It is indeed true that in the briefest measure of time this smaller one also fell to the ground. This perhaps deceived those who did not notice that a great speed corresponds to a great distance, which is traversed, as it were, in an instant; and therefore they judged that equality of speeds ought to be attributed to both weights, because the difference in times was slight. Do you think that a tuft of wool, as much as can be grasped in the fist, will descend as quickly as a very large stone? You will say that this must be attributed mostly to shape, for air, never at rest, is not equally easily divided by all bodies. So indeed it is; therefore, if air resists unequally, falling bodies may move unequally. Add to this that heavy and light bodies are likewise impelled by nature toward their motions; and therefore, if where there is more lightness we observe a swifter upward motion, then also where there is more heaviness we ought to infer a more rapid downward motion. Stir water for some time in a long glass tube, so that the air enclosed in the tube may mix with the water: when the stirring has ceased, you will see larger particles of air rising quickly, while the smaller move slowly and little by little; which will be more clearly evident if, instead of water, you admit quicksilver into the tube. Why then should heavy bodies not likewise move downward at different speeds, if they are unequal? Nevertheless, that the analogy of weights must not be maintained is here plainly evident from the fact that in bodies of the same species the ratio of weights is the same as that of magnitudes; but magnitudes ...

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Mechanicorum 684 sunt in triplicatâ Ratione homologorum laterum: At impedi- mentum, quod ex medio scindendo oritur, & velocitati mo- dum statuit, non est ipsis magnitudinibus analogum, sed ad summum ea esse potest Ratio, quæ inter corporum superficies intercedit; hæ autem tantùm sunt in duplicatâ Ratione late- rum homologorum. Non igitur velocitates, quatenus ab im- pedimento temperantur, sunt directè gravitatibus analogæ. Ubi autem corpora non ejusdem secundùm speciem gravitatis, habuerint gravitates magnitudinibus reciprocè analogas, atque adeò æquali gravitate absolutâ, seu pondere, prædita fuerint, adhuc inæquales esse aëris resistentias, si figuræ similes sint, satis probabiliter concedimus, plus siquidem majori repugnat, quàm minori: si verò & dissimiles figuræ, & inæquales gravi- tates ponantur, ex omnibus simul compositis quodammodo conflari resistentiarum Rationem facile est opinari; sed Ratio- nis terminos temerè definire non ausim. Porrò certam legem, qua in resistendo aër contineatur, om- ninò afferre non possumus, si quemadmodum ille resistat, per- pendamus. Non eadem est aquæ & aëris resistendi Ratio, ne dividatur; aqua enim nondum in vaporem extenuata, & fusa, constipari se, & in angustiora spatia coarctari non sinit; sed ubi locum descendenti ex aëre corpori concedere cogitur, super- ficiem intrà vas, quo continetur, attollens tantumdem aëri, quem impellit, surripit spatij, quantum immerso corpori per- mittit. Aut si non descendat corpus, sed obliquè feratur (ut si baculum partim aquæ immissum, partim extantem transversum agas, aut navis prora illam findat) tunc quæ motui opponitur, aqua crispatur, & baculus sivè navis foveam ponè relinquit, in quam deinde aqua resluat; quò autem crassior baculus aut am- plior prora, & vehementior atque concitatior fuerit motus, aqua impulsa altiùs assurgit, & magis depressa fovea apparet. Ex quo satis apertè constat à corpore, quod movetur, proximas aquæ oppositæ particulas impelli, & per has interjectas etiam reliquas in aëris locum protrudi. At verò aër, quem facilè comprimi & dilatari tam multis ex- perimentis novimus, dum locum corpori commoto concedit, neque opus est, ut supremi ætheris regionem invadat locum sibi quærens, neque in foveam excavatus vel ad momentum hiat;

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Mechanics 684 are in the triplicate ratio of homologous sides. But the impediment which arises from the division of the medium, and sets a limit to velocity, is not analogous to the magnitudes themselves, but at most can be in that ratio which intervenes between the surfaces of bodies; and these are only in the duplicate ratio of homologous sides. Therefore velocities, in so far as they are moderated by impediment, are not directly analogous to gravities. Where, however, bodies of the same kind of gravity shall have gravities reciprocally analogous to magnitudes, and thus be endowed with equal absolute gravity, or weight, we still readily allow that the resistances of air are unequal, if the figures are similar; for indeed the greater is opposed more than the less: but if both dissimilar figures and unequal gravities are posited, it is easy to think that the ratio of the resistances is somehow compounded from all these together; but I would not dare rashly define the bounds of that ratio. Moreover, we cannot at all lay down a certain law by which air is kept within resistance, if we consider how it resists. The ratio of water and air in resisting is not the same, nor is it divided; for water, not yet attenuated and dissolved into vapor, does not permit itself to be compressed and narrowed into smaller spaces; but when it is compelled to yield place to a body descending from the air, by raising the surface within the vessel in which it is contained, it takes away from the air which it drives the same amount of space that it allows to the immersed body. Or if the body does not descend, but is carried obliquely, as if you should drive a stick partly immersed in water and partly projecting across, or a ship’s prow should cleave it, then that which opposes the motion is the water stirred up, and the stick or ship leaves behind a hollow, into which the water afterward flows back; and the thicker the stick or the broader the prow, and the more violent and impetuous the motion, the higher the water driven up rises, and the more depressed does the hollow appear. From this it is sufficiently clear that, by the body which moves, the nearest particles of the water opposed to it are driven away, and through these interposed ones the remaining particles also are pushed into the place of the air. But air, which we know from so many experiments can easily be compressed and expanded, when it yields place to a moving body, has no need to seek a place for itself by invading the region of the upper ether, nor does it open up even for a moment as a hollow excavated into a pit;

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Liber septimus. CAPUT VII. 685 hiat; sed tantisper dum ejus particulæ circumpulsæ in abeun- tis corporis relictum spatium succedant, quæ antè sunt, com- primuntur, quæ ponè, dilatantur; compressæ autem se expli- cantes aërem lateribus adhærentem repellunt, quem dilatatæ attrahunt se contrahentes. Si tardus sit motus, exiguâ aëris constipatione aut distractione opus est; at si velocior, oppositæ aëris particulæ magis comprimuntur, sequentes magis dilatan- tur; quæ proinde se restituere vehementiùs conantes, etiam velociorem efficiunt reliquarum particularum circumpulsio- nem. Verùm quia sapientissimo Naturæ instituto ita compara- tum est, ut quàm minimum ejus ordo perturbetur, & mini- mam, quoad fieri possit, corpora singula patiantur violentiam, hanc pluribus potiùs dispertiendam censuit, quàm uni subeun- dam: propterea si digitale spatium multo aëri surripiendum est, exigua contingit singulis particulis naturalis spatij jactura, quam dissimulanter ferunt, nec admodum repugnant; contra verò si modicus sit aër, & tantumdem de ejus spatio demendum sit, reluctatur acriùs, ut pro viribus naturæ jura tueatur. Hinc si corpus, quod movetur, brevi intervallo absit à corpore soli- do & duro, quod ejus motum obsistendo compescet, atque adeò etiam aërem impulsum remoratur, hunc inter angustias deprehensum magis constipari necesse est, magisque resistere. Non est tamen aëri denegandum, quod cæteris corporibus ul- tro concedimus; nam & ipse jam commotus ex concepto per impulsionem externam, aut ex vi suâ elasticâ, impetu faciliùs pergit institutum iter conficere, quàm si tunc primùm à quie- te recederet: Ex quo fit in motu corporis accelerato, licèt ra- tione habitâ velocitatis augenda esset resistentia aëris, hanc ta- men non augeri nisi pro excessu velocitatis illius supra motum, quo aër moveretur ad easdem partes, nisi acriùs ab ipso corpo- re urgeretur. Hanc aëris resistentiam paulò explicatiùs commemorare pla- cuit eo consilio, ut mihi ipse persuadeam non modò ipsum nihil omnino non officere motui, verùm etiam tam fieri non posse, ut percussionibus certissimam legem statuamus, quàm evidens est adeò inconstantem & variam esse aëris resisten- tiam, ut ad calculos subtiliter & exquisitè revocari nequeat, quippe quæ ex tam variis causis pendet: quemadmodum enim RRrr 3

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Book Seven. Chapter VII. 685 opens; but only for so long as its particles, driven around, make room in the departing body’s vacant space, those that are before are compressed, those behind are expanded; and the particles that are compressed, unfolding themselves, drive back the air adhering to their sides, which the expanded particles draw in as they contract. If the motion be slow, there is need of only a slight condensation or rarefaction of the air; but if it be swifter, the opposite particles of the air are more compressed, the following ones more expanded; and these, therefore, striving more vehemently to return to themselves, also make the circulation of the remaining particles more rapid. But since, by the most wise ordinance of Nature, the arrangement is so made that its order is disturbed as little as possible, and that each body, as far as may be, suffers the least violence, it judged that this should rather be distributed among many than borne by one: therefore, if a large space is to be withdrawn from the air, only a slight loss of natural space befalls each particle individually, which it bears without complaint, nor does it resist much; on the other hand, if the air be scanty, and the same amount must be taken away from its space, it resists more sharply, so as to defend the rights of nature to the full extent of its powers. Hence, if a body that is moving is at a short distance from a solid and hard body, which by resisting will check its motion, and thus also hold back the air that is driven against it, this air, caught between narrow limits, must be more compressed and resist more strongly. Yet the air is not to be denied what we freely concede to other bodies; for it too, once set in motion by the impulse of an external force, or by its own elastic power, continues more easily with impetus the journey it has begun than if it were then first to depart from rest: from which it follows that, in the accelerated motion of a body, although, if velocity alone were considered, the resistance of the air ought to increase, nevertheless it does not increase except in proportion to the excess of that velocity over the motion by which the air itself would be moved in the same directions, unless it were more sharply pressed by the body itself. I thought it worthwhile to mention this resistance of the air somewhat more fully, with the aim of persuading myself not only that it in no way whatever fails to impede motion, but also that it is altogether impossible for us to establish a certain law for collisions, since it is evident that the resistance of the air is so inconstant and variable that it cannot be reduced to calculations with subtlety and exactness, being as it is dependent on so many different causes: for just as RRrr 3

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686 Mechanicorum aquæ, cæterorumque liquorum dissimilium resistentia inæqualis conspicua est, ita putum ac tenuem aërem non æquè resiste- re atque crassum & concretum ratio suadet: quis autem syncerum aërem ab aëre cum terræ expirationibus permisto discernat? Quid si corpus in motu aërem aliò directum, aut in con- trarias partes reflexum, aut turbine aliquo perversum atque adhuc agitatum offendat? an non aliquis velocitatis gradus im- minuitur? Sed quis certum habeat, utrum quiescat aër, neque corporis impetum frangat, aut reprimat alienâ impressione ad- versus, an verò ad easdem partes delatus motui obsecundet, & velocitati faveat? Quare laudandi quidem quicunque percus- sionum naturam vestigantes, & ea, quibus in ejus notitiam deduci possent, conjecturâ prospicientes, in instituendis expe- rimentis sedulò se exercuerunt; parùm tamen mihi de veritate blandiri me posse arbitrarer, si hæc quasi Apodixes temerè reci- perem; sed neque ore tam duro fuerim, ut ea prorsùs rejiciam. Confirmatis igitur experimentis me duci sinam, quatenus ad veritatis similitudinem me proximè accessurum spero. Percussio itaque, si motum naturalem ex gravitate ortum sub- sequatur, certam aliquam Rationem ob idipsum sortiri videtur, quia cum velocitate consentit impetus: velocitas autem ex spa- tio deprehenditur æqualibus temporibus respondente; spatia verò cum temporibus comparata certis Rationibus definita vi- deri, iterata experimenta docuerunt, quæ vix quisquam sanus neget; in iis siquidem tot doctissimi viri post Galilæum versati sunt pari exitu, & summo consensu, ut in his omnibus insit quidam, sine ullo fuco veritatis color. Hujus rei specimen exhibeamus in globo argillaceo unciarum octo, qui spatio unius scrupuli secundi (quantus ferè est pulsus arteriæ hominis sani) observatus est percurrere pedes Romanos 15; duplo autem tempore incipiendo à quiete, hoc est scrupulis secundis duo- bus, pedes 60: quare si priori scrupulo secundo respondent pe- des 15, posteriori tribuendi sunt pedes 45: igitur motus est ce- lerior, cùm majus spatium pari tempore confecerit. Plura hujusmodi experimenta (si te à tentando absterreat labor) suppe- ditabit Ricciolius tom.1. Almag. lib.9. sect.4. cap.16. ex quibus demum infertur velocitatis incrementa fieri juxta incremen- tum progressionis Arithmeticæ numerorum imparium ab unita- te

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686 Mechanics The unequal resistance of water and other dissimilar liquids is evident, so reason suggests that thin and rare air does not resist equally with dense and compact air: but who can distinguish pure air from air mixed with exhalations from the earth? What if a body in motion should encounter air directed elsewhere, or reflected in opposite directions, or turned awry by some whirl and still agitated? Is not some degree of speed diminished? But who can be certain whether the air is at rest, and does not break the body’s impulse, or restrain it by an opposing alien impression, or whether, being carried along to the same parts, it accommodates itself to the motion and favors the speed? Therefore those are indeed to be praised who, tracing the nature of impacts and foreseeing by conjecture those things by which they might be brought to its knowledge, have diligently applied themselves to performing experiments; yet I should think that I could flatter myself too little with the truth if I were rashly to accept these things as if they were demonstrations; but neither should I be so hard of tongue as to reject them altogether. So, with the experiments confirmed, I shall allow myself to be guided by them, insofar as I hope to have come as near as possible to the likeness of truth. An impact therefore, if it follows natural motion arising from gravity, seems for that very reason to acquire some certain ratio, because impulse agrees with velocity: and velocity is ascertained from distance, corresponding to equal times; but repeated experiments have shown that distances, when compared with times, seem determined by certain ratios, which hardly anyone sane would deny; for so many most learned men have worked on these matters after Galileo, with the same result and with the highest agreement, that in all these things there is a certain color of truth without any disguise. Let us present a specimen of this matter in an eight-ounce clay ball, which was observed to travel 15 Roman feet in the space of one second part of a minute, which is about the time of the pulse of a healthy man; but in double the time, beginning from rest, that is, in two second parts of a minute, 60 feet: therefore, if 15 feet correspond to the first second part of a minute, 45 feet must be assigned to the latter; thus the motion is swifter, since it has covered the greater distance in equal time. Riccioli in vol. 1 of the Almagest, book 9, section 4, chapter 16, will supply more experiments of this kind, if labor should deter you from testing them; from which it is finally inferred that increments of velocity occur according to the increase in the arithmetic progression of odd numbers from unity

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Liber septimus. CAPUT VII. 687 te incipientis 1.3.5.7.9.11.13, &c. adeò ut, si quod spatium pri- mo momento percurritur, statuatur ut 1, triplo velociùs movea- tur corpus grave descendens in secundo momento, quintuplo velocius in tertio, septuplo velociùs in quarto, atque ita dein- ceps. Quoniam verò numerorum imparium series ab unitate incipiens hoc habet, quòd, si colligantur in summam, nume- ros quadratos constituant; hinc est, quòd collectis in summam omnibus incrementis velocitatis (hoc est omnibus spatiis, ex his quippe dignoscitur velocitas) habeatur numerus quadratus temporis, quod duravit motus. Collatis igitur invicem duobus motibus naturalibus ejusdem corporis gravis, sed non isochro- nis, erunt ut quadrata temporum ita & spatia, atque è conver- so ut spatia inter se, ita & temporum quadrata. Hinc cognito spatio, quod à dato corpore gravi percurritur dato tempore, statim innotescet, quantum spatij conficere va- leat alio tempore dato, vel quanto tempore aliud datum spa- tium. Quæratur enim, quantum spatium descendendo percur- ret uno horæ quadrante globus idem argillaceus, qui uno mi- nuto secundo Romanos pedes 15 percurrit? Datum tempus, scilicet horæ quadrans, scrupula Secunda 900 continet, cujus numeri quadratum est 810000. Fiat igitur ut 1 ad 810000, ita pedes 15 ad 12150000: qui pedum numerus in milliaria Itali- ca resolutus dat milliaria 2430, quæ uno horæ quadrante con- ficeret. Vicissim quæratur quantum temporis idem globus in- sumeret in primo milliari percurrendo, hoc est ped. 5000. Fiat ut 15 ad 5000, ita 1 quadratum dati temporis, scilicet unius scrupuli secundi, ad 333 1/4 quadratum quæsiti temporis; cujus quadrati Radix investiganda est, & demum invenitur Scrup. sec. 18 1/4 & paulo ampliùs; nam huic tempori præcisè respon- dent solùm pedes 4995 15/16. Incrementa hæc velocitatis ex concepti impetûs incremento desumenda esse nullus dubito; sed operosum videri posset au- gescentis impetûs causam exponere. Cùm junior Aristotelem interpretarer, & primas curas hujusmodi rerum contemplatio- ni impenderem, hanc excogitavi hypothesim; videlicet impe- tûs producti diuturnitatem maximam duobus tantùm momen- tis circumscriberebam, ita ut primo momento oriretur, secundo æqualem

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Book seven. CHAPTER VII. 687 the incipient 1, 3, 5, 7, 9, 11, 13, etc.; so that, if the distance traversed in the first moment be taken as 1, a falling heavy body is moved three times as fast in the second moment, five times as fast in the third, seven times as fast in the fourth, and so on. But since the series of odd numbers beginning from unity has this property, that, if they are collected into a sum, they constitute square numbers, hence it is that, when all the increments of velocity are collected into a sum (that is, all the distances, for velocity is discerned from these), there is obtained the square number of the time during which the motion lasted. Therefore, when two natural motions of the same heavy body, but not isochronous, are compared with one another, the squares of the times will be as the spaces, and conversely, as the spaces are to one another, so also are the squares of the times. Hence, once the distance traversed by a given heavy body in a given time is known, it will at once become known how much distance it can cover in another given time, or in what time it will cover another given distance. For let it be asked how much distance the same clay globe, which in one second of a minute travels 15 Roman feet, will traverse in descending in one quarter of an hour? The given time, namely a quarter of an hour, contains 900 seconds, the square of which number is 810000. Let it therefore be as 1 to 810000, so 15 feet to 12150000: which number of feet, resolved into Italian miles, gives 2430 miles, which it would accomplish in one quarter of an hour. Conversely, let it be asked how much time the same globe would consume in traversing the first mile, that is, 5000 feet. Let it be as 15 to 5000, so the square of 1, the given time, namely of one second, to 333 1/4, the square of the sought time; the square root of which must be investigated, and in the end it is found to be 18 1/4 seconds and a little more; for only 4995 15/16 feet correspond exactly to this time. These increments of velocity, I have no doubt, are to be derived from the increment of conceived impetus; but it might seem laborious to explain the cause of the increasing impetus. When I was a younger interpreter of Aristotle, and devoted my first efforts to the contemplation of such matters, I devised this hypothesis; namely, I limited the greatest duration of the produced impetus to only two moments, so that it would arise in the first moment, in the second equal

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Mechanicorum æqualem sibi impetum gigneret, in quo superstes esset, tertio periret: gravitati autem singulis momentis vim producendi certum impetûs gradum sibi congruentem tribuebam. Hinc corpus descendens primo momento primum habebat impetûs gradum à gravitate productum; secundo momento primus ille gradus alium gradum gignebat præter eum, qui à gravitate tunc oriebatur; quare duo novi gradus cum uno antiquo tres gradus constituebant. Tertio momento primus gradus peribat, duo secundi gradus duos pariter producebant, & gravitas suum tertium gradum; quare quinque gradus erant. Quarto mo- mento duobus secundis gradibus pereuntibus, tres gradus ter- tio momento producti reliqui erant, & sibi tres alios gradus addebant, quos producebant, atque gravitas suum quartum gradum efficiebat, ut in universum essent septem gradus. Ex his septem quinto momento peribant tres tertij gradus; qua- tuor reliqui item alios quatuor adjiciebant quinto gradui à gra- vitate proficiscenti, & erant novem. Atque ita deinceps, tot pereuntibus gradibus, quotum erat momentum uno interjecto præcedens, & tot productis, quotum erat ipsius motûs momen- tum. Sic momento vigesimo nono peribant gradus 27 pro- ducti momento vigesimo septimo, remanentibus gradibus 28 productis momento vigesimo octavo, à quibus totidem produ- cebantur unâ cum gradu proprio gravitatis, hoc est gradus 29, & tunc erat impetûs intensio graduum 57 momento vigesimo nono. Quemlibet verò terminum in serie numerorum impa- rium facilè invenies, si illi duplicato demas unitatem: sic quæ- rens octavum terminum ex denominator termini duplicato, scilicet bis 8, hoc est 16, deme unitatem, & 15 est octavus ter- minus: sic terminus septuagesimus habetur demptâ unitate ex 140, & est 139. Huic hypothesi cum Phenomeno optimè conveniebat, & ea statuebatur impetûs intensio, quæ velocitati efficiendæ par es- set, servatâ incrementorum Ratione, quæ ex iteratis experimen- tis innotuerat. Verùm commentitia, & fabulæ proxima vide- batur tàm brevis impetûs vita, quam non nisi duo momenta metirentur: in iis sanè, quæ vi externâ moventur, & longiùs projiciuntur, aut in gyrum aguntur, licet extinctâ effectrice causâ impressus impetus diutius permanet; quidni & impetus sponte

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Mechanics would generate for itself an equal impetus, in which it would survive the third moment and perish: but I attributed to gravity at each moment the power of producing a certain degree of impetus corresponding to itself. Hence a descending body at the first moment had the first degree of impetus produced by gravity; at the second moment that first degree generated another degree besides that which was then arising from gravity; wherefore two new degrees, together with one old one, made three degrees. At the third moment the first degree perished, the two second degrees produced two more likewise, and gravity its third degree; wherefore there were five degrees. At the fourth moment, when the two second degrees perished, there remained three degrees produced at the third moment, and these added to themselves three others, which they produced, and gravity effected its fourth degree, so that in all there were seven degrees. From these seven, at the fifth moment, three of the third degree perished; the four remaining likewise added four others to the fifth degree arising from gravity, and there were nine. And so on thereafter, as many degrees perishing as the moment was one removed from the preceding one, and as many produced as the moment of the motion itself. Thus at the twenty-ninth moment the degrees 27 perished, produced at the twenty-seventh moment, there remaining 28 degrees produced at the twenty-eighth moment, from which the same number were produced together with gravity's own degree, that is, degree 29, and then the intensity of impetus at the twenty-ninth moment was 57 degrees. But you will easily find any term in the series of odd numbers, if from its double you subtract unity: thus, seeking the eighth term from the doubled denominator of the term, namely twice 8, that is 16, subtract unity, and 15 is the eighth term: thus the seventieth term is obtained by subtracting unity from 140, and it is 139. This hypothesis agreed very well with the Phenomenon, and by it was established the intensity of impetus, which would be equal to that required for producing velocity, preserving the Ratio of increments that had been learned from repeated experiments. Yet it seemed fictitious and close to a fable that the life of impetus should be so brief as to be measured by only two moments: in those things indeed which are moved by an external force and are projected farther, or are set in a circular motion, although the effective cause has ceased, the impressed impetus lasts longer; why not also the impetus of its own accord

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Liber septimus. CAPUT VII. 689 sponte suâ conceptus, suæque origini cohærens aliquandiu per- severet? quippe qui aut ejusdem, aut saltem non deterioris na- turæ censendus est. Adde nimis incertum esse, an impetus im- petum producere valeat in eodem corpore, cui inest, quamvis impetum in alienis corporibus percussis efficiendi vis illi conce- datur: nam & calor, & cæteræ qualitates effectrices, quas de- perditas sibi forma substantialis reparare incipit, non alios simi- les gradus sibi addunt, licèt eos in proximo corpore efficere va- leant. Præterquam quod, cur illo ipso momento, quò primùm existit impetus, similem gradum non producit? nihil scilicet il- li deest, nullo impedimento prohibetur, neque causam ætate præcedere effectum, sed origine, necesse est. Si autem primo momento & oritur impetus, & impetum efficit, hic pariter suam vim primo eodem momento exerens alium impetum producit, & infinita gradum impetûs æqualium multitudo consurgit; cujus ne vestigium quidem apparere potest, cùm in causarum & effectuum serie semper ab infinitate natura discedat. Quare impetum à gravitate descendente productum, ex tam expedito interitu vendicandum, & virtute sè novo impetu au- gendi spoliandum, longè probabiliore conjecturâ censui. Im- petum igitur certâ quadam mensurâ gravitati corporis con- gruente, statim ac in motum erumpere, potest, produci, existimo, & quandiu motus perseverat, permanere; eadem enim gravi- tas, quæ primo momento illum effecit, reliquis consequentibus momentis conservare valet; finis, quò refertur, & cujus causâ productus est, adhuc obtineri potest, videlicet motus; liberè descendenti corpori nullum objicitur impedimentum; nihil adest, quod ipsius concepti impetûs interitum exigat: ergo im- petum à gravibus descentibus conceptum non perire in mo- tu si dixerimus, similitudinem veri nos consecutos arbitror. Quoniam verò gravitas inter eas causas enumeratur, in quibus inest efficiendi necessitas, & quandiu opus est juxta naturæ pro- positum, quantum possunt, efficiunt; singulis momentis, qui- bus potest descendere, singulos impetûs gradus æquales priori- bus adjicit, adeò ut, quot momenta motum metiuntur, tot gra- dus impetûs postremo momento intentionem constituant, cui motûs velocitas respondeat. Velocitatum igitur incrementa fiunt juxta naturalem numerorum progressionem 1.2.3.4.5, &c. S S s s

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Book seventh. CHAPTER VII. 689 How could it, being conceived of itself and cohering with its own origin, continue for some time? for it must be judged to be either of the same nature, or at least of no worse. Add that it is too uncertain whether impetus can produce impetus in the same body in which it exists, though power be granted to it of producing impetus in other bodies struck by it: for heat also, and the other efficient qualities, which the substantial form begins to restore to itself when they have been lost, do not add to themselves other similar degrees, although they may be able to produce them in the neighboring body. Besides, why, at that very moment when impetus first exists, does it not produce a similar degree? plainly nothing is lacking to it, it is hindered by no impediment, nor must the cause precede the effect in time, but only in origin. But if at the first moment impetus both arises and produces impetus, this in like manner, exercising its force at the same first moment, produces another impetus; and an infinite multitude of equal degrees of impetus arises, of which not even a trace can appear, since in the series of causes and effects nature always departs from infinity. Wherefore I judged, by a much more plausible conjecture, that impetus produced by descending gravity should be rescued from so quick a destruction, and should be stripped of the power of increasing itself by a new impetus. I therefore think that impetus, with a certain measure corresponding to the gravity of the body, can immediately break out into motion and be produced, and can remain as long as the motion continues; for the same gravity which produced it at the first moment is able to preserve it in the following moments. The end to which it is referred, and for whose sake it was produced, can still be attained, namely motion; no impediment is opposed to the body descending freely; nothing is present that would require the destruction of the conceived impetus. Therefore, if we say that impetus conceived by descending bodies does not perish in motion, I think we have attained a likeness of the truth. But since gravity is numbered among those causes in which the necessity of producing is inherent, and which, so long as there is need and according to nature’s plan, produce whatever they can, each moment in which it is able to descend adds to it a single equal degree of impetus, prior degrees being thereby increased, so that, as many moments as measure the motion, so many degrees of impetus at the final moment make up the intensification to which the speed of the motion corresponds. The increments of velocities therefore occur according to the natural progression of numbers 1, 2, 3, 4, 5, and so on. S S s s

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Mechanicorum 690 nam juxta hanc eandem seriem impetus, velocitatis causa, augetur, singulis gradibus in singula momenta additis. Ab omni tamen infinitatis suspicione recedendum est hîc, ubi momentorum vocabulum usurpo, quasi infinita puncta temporis agnoscerem, & ad vim percussionis infinitam adstruendam, infinitis momentis singulis impetus gradum tribuerem. Quemadmodum enim corpora punctis prorsùs individuis non constare suadetur multiplici argumento præsertim ex Asymptotis lineis desumpto, ita motui atque tempori puncta omnibus omnino partibus carentia nunquam concedenda censui. Sed huic verbo, cum momentum dico, subjecta notio est, minima temporis particula Physica, quæ licèt particulas alias adhuc minores contineat sibi ordine succedentes, ex quibus illa constituitur, tota tamen ad primi impetus effectionem ita requiritur, ut juxta naturæ leges nihil effici posset motûs, nisi integra illa temporis particula suppeteret. Hujusmodi autem non individuas particulas minimas certas atque æquales in tempore, aut motu, finito non esse nisi certo numero definitas manifestum est: Quapropter sicut momentorum, ita & graduum impetus æqualium multitudo finita est. Verùm nemo temerè hanc momentorum multitudinem ad calculos revocare instituat; res enim planè incerta est. Utique tardissimos reperiri & languidissimos motus aliquos novimus, qui diu latent, nec nisi post tempus benè conspicuum demum innotescunt: Ex quo deprehendimus in tempore aut motu, qui sensibus percipi possit, multas numerari hujusmodi momentorum myriadas: si enim in uno aliquo motu exiguo particulæ illius sibi ex ordine succedentes respondent motui longissimè majori (cujusmodi est cælorum motus) ex quo definitur tempus, & in hoc plurimæ partes notabiles, & sub Physicam mensuram eadentes numerantur, utique & in illo plurimæ particulæ omnem sensus aciem fugientes inveniuntur. Et quidem si cum illis Astronomis philosophemur, qui cælestium graduum minuta usque eò in sexagesimas partiuntur, ut demum in scrupulis Decimis consistant, cùm Æquatoris gradus quindecim in Primo mobili uni horæ respondeant, satis constat, quantus sit hujusmodi scrupulorum Decimorum numerus, in quorum fluxum unica hora resolvatur: ac proinde in uno horæ minuto Secundo,

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Mechanics 690 for according to this same series, impetus, for the sake of velocity, is increased, with successive steps added in successive moments. Yet one must here depart from any suspicion of infinity, where I use the word moments, as if I were acknowledging infinite points of time, and, in order to establish an infinite force of impact, were assigning to each of infinite moments a degree of impetus. For just as bodies are shown by many arguments, especially one drawn from asymptotic lines, not to consist of absolutely indivisible points, so I have judged that points entirely devoid of parts are never to be granted to motion or to time. But by this word, when I say moment, the notion implied is a smallest physical part of time, which, though it contains other still smaller parts succeeding one another in order, out of which it is constituted, is nevertheless wholly required for the production of the first impetus, so that according to the laws of nature nothing of motion could be effected unless that entire part of time were available. Now it is evident that such minimal, fixed, equal, non-individual parts in time or in motion are not to be found in a finite number only: wherefore, just as the multitude of moments, so too the number of equal degrees of impetus is finite. But let no one rashly set about reducing this multitude of moments to calculation; for the matter is plainly uncertain. Certainly we know that some of the slowest and most languid motions can be found, which lie hidden for a long time and are not finally known until after a quite perceptible span of time: from this we understand that in time or motion perceptible to the senses many myriads of such moments may be counted; for if in some small motion the successive parts of that thing correspond in order to a motion very much greater indeed (such as the motion of the heavens), from which time is determined, and in this motion many notable parts are counted, and are subject to physical measure, then surely in that one many parts are found escaping every reach of the senses. And indeed if, along with those astronomers, we philosophize, who divide the minutes of celestial degrees down to the sixtieth parts, so that they finally consist in tenths of scruples, since fifteen degrees of the equator in the primum mobile correspond to one hour, it is sufficiently clear how great is the number of such tenths of scruples, into whose flux one hour is resolved: and therefore in one second minute of an hour,

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Liber septimus. CAPUT VII. 691 do, hoc est in pulsu arteriæ, continentur plusquam decies mil- lies millena millia myriadum hujusmodi scrupulorum Decimo- rum, quæ momenta appellari possunt. Quapropter illud uni- cum generatim statuere possumus, in quolibet tempore Physi- cè notabili plurima esse momenta, quamvis eorum certum nu- merum explicare nequeamus: ideóque cùm incrementa velo- citatum mensuram desumant ex momentorum numero, qui semper unitatis additione augetur, intensio autem impetûs ha- beat graduum numerum parem numero momentorum; quàm difficiles explicatus habet momentorum multitudo, tam obscu- ra est impetûs intensio; si minutam subtilitatem persequamur. Sed si datum tempus in aliquot particulas nostro arbitratu distinguamus, quo plures fuerint hujusmodi particulæ, eò pro- piùs accedemus ad id, quod experimentis deprehensum est; vi- delicet, etiam si spatia in motu decursa juxta seriem naturalem numerorum augeantur in motu, demum eorum collectiones in- cipiendo à quiete habere inter se duplicatam Rationem tempo- rum inveniemus. Comparatis igitur invicem motibus alicujus corporis gravis descendentis, cujus motus unus jam innotuerit, quantum scili- cet spatij dato tempore confecerit, innotescet, quantus futurus sit alio tempore motus, si fiant ut quadrata datorum temporum, ita & spatia; vel quanto tempore percurrendum sit spatium de- finitum, si fiant ut Radices quadratæ datorum spatiorum, ita & tempora. Quia nimirum posita illa incrementa impetûs, & ve- locitatum, atque spatiorum juxta seriem naturalem numero- rum ab unitate incipientem constituunt collectiones habentes inter se proximè Rationem duplicatam temporum. Habemus experimento globum argillaceum unciarum octo percurrere uno minuto Secundo horæ pedes 15, & duobus Secundis pe- des 60, hoc est spatium quadruplum, & quia tempora sunt ut 1 ad 2, spatia sunt ut quadrata, scilicet ut 1 ad 4. Ponamus in uno Secundo esse momenta 10000; sunt igitur ultimo momento 10000 gradus velocitatis similes & æquales primo gradui primi momenti, & spatium ultimo hoc momento decursum, ad spa- tium primi momenti est ut 10000 ad 1. Coge igitur in sum- mam omnia spatia incipiendo ab unitate usque ad 10000, vi- delicet ultimi termini dimidiato quadrato adde ejusdem ultimi SSss 2

Transcription: Translated (English)

Book Seventh. Chapter VII. 691 that is, in the pulse of the artery, there are contained more than ten thousand times ten thousand myriads of such scruples of tenths, which may be called moments. Wherefore we can establish only this general principle: that in any physically notable interval of time there are more moments than we can determine by any fixed number; and therefore, since the increments of velocities take their measure from the number of moments, which is always increased by the addition of one unit, while the intensity of the impulse has a number of degrees equal to the number of moments, the multitude of moments is as difficult to explain as the intensity of the impulse is obscure, if we pursue minute exactness. But if we divide a given time into several parts at our discretion, the more numerous those parts are, the more nearly shall we approach what has been discovered by experiment; namely, even if the spaces traversed in motion increase according to the natural series of numbers in the motion, we shall at length find that their collections, beginning from rest, have among themselves a doubled ratio of times. Therefore, if we compare with one another the motions of some heavy descending body, one motion of which is already known, namely how much space it has completed in a given time, it will become known how great the motion will be at another time, if the times be as the squares of the given times, so also the spaces; or how long a time will be required to traverse a definite space, if the times are as the square roots of the given spaces, so also the times. For indeed, since those increments of impulse and of velocities, and of spaces, established according to the natural series of numbers beginning from unity, make collections having among themselves a nearly doubled ratio of times. By experiment we have that an eight-ounce clay ball traverses in one second of an hour 15 feet, and in two seconds 60 feet, that is, a quadruple space; and because the times are as 1 to 2, the spaces are as the squares, namely as 1 to 4. Let us suppose that in one second there are 10,000 moments; there are therefore in the last moment 10,000 degrees of velocity, similar and equal to the first degree of the first moment, and the space traversed in this last moment is to the space of the first moment as 10,000 to 1. Therefore gather into a sum all the spaces, beginning from unity up to 10,000, namely add to half the square of the last term the same last term itself. SSss 2

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Mechanicorum 692 termini semissem, & prodibit omnium spatiorum summa. Ultimi termini 10000 quadratum est 100000000, cui adde ipsum ultimum terminum; & hujus summæ medietas 50005000 est summa minimorum spatiorum, quibus constantur pedes 15. In duobus Secundis erunt momenta 20000, & similiter invenitur summa 200010000 minimorum spatiorum, quibus constant pedes 60. Non sunt quidem duæ hujusmodi summæ hîc inventæ 50005000 & 200010000, omnino ut 1 ad 4, sed ut 1 ad 3 49995/50005: verùm tantula differentia 10/50005 quid officit allato experimento? an potuit observari? Si 4 sunt pedes 60, quid sunt 3 49995/50005 utique pedes 59 49855/50005; deest igitur pedis particula 150/50005, hoc est unciæ quasi pars vigesima septima. Quis autem tam minutæ subtilitati locus sit in observando motu? Ut autem perspicuè appareat hanc hypothesim incrementi juxta seriem naturalem numerorum consentire cum experi- mentis, & spatia se habere ut quadrata temporum, statuamus eadem spatia, ut primum sit ad secundum in Ratione 50005000 ad 200010000. Radix primi spatij est 7071 5959/14142, Radix autem secundi spatij est 14142 6918/14142; quæ sunt ut 1 ad 2, si fractiones contemnantur; nec repugnat experimentum; nam tantula differentia temporum, ne sit Ratio præcisè dupla, discerni non potuit: quarum enim partium 7071 5959/14142 est unum minu- tum Secundum horæ, deest unius partis 5000/14142, ut sint duo mi- nuta Secunda, hoc est unius pulsûs alteriæ pars una vicies mil- lesima desideratur, ut sint planè duo Secunda. Quære argu- menta, si qua potes; an experimento revinces esse planissimè duo minuta Secunda, nec vel unicum momentum defuisse? Hoc idem, quod exempli causâ in Ratione duplâ temporum & quadruplâ spatiorum explicatum est, in cæteris pariter de- prehendes. Fac enim esse tempus quadruplum, hoc est Secun- dorum 4, hoc est minimorum temporis 40000. Tota collectio spatiorum erit 800020000. Quare 50005000 ad 800020000 est ut 1 ad 15 49945/50005, quasi ut 1 ad 16; est autem defectus 60/50005. Spatium igitur uno Secundo decursum cum sit ped. 15, qua- tuor Secundis erit ped. 240 minùs una ferè sexagesima nona particulâ unciæ. Vicissim ut tempora invenias in subduplicatâ Ratione

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Mechanics 692 If you take half of the terms, the sum of all the spaces will result. The last term 10000 squared is 100000000; add to it the last term itself, and half of this sum, 50005000, is the sum of the least spaces, by which 15 feet are made up. In two seconds there will be 20000 moments, and likewise there is found the sum 200010000 of the least spaces, by which 60 feet are made up. Indeed, these two sums found here, 50005000 and 200010000, are not exactly as 1 to 4, but as 1 to 3 49995/50005; yet what does so slight a difference, 10/50005, matter for the experiment cited? Could it have been observed? If 4 are 60 feet, what are 3 49995/50005? namely 59 feet 49855/50005; therefore a portion of a foot is lacking, 150/50005, that is, nearly the twenty-seventh part of an inch. But where is there room for such minute subtlety in observing motion? Now, that it may be made clearly apparent that this hypothesis of increase according to the natural series of numbers agrees with the experiments, and that the spaces behave as the squares of the times, let us suppose the same spaces, so that the first is to the second in the ratio of 50005000 to 200010000. The square root of the first space is 7071 5959/14142, but the square root of the second space is 14142 6918/14142; these are as 1 to 2 if the fractions are disregarded; nor is the experiment contrary to this, for so slight a difference in the times, so that the ratio may not be exactly double, could not be distinguished: for of those parts of 7071 5959/14142, one minute Second of an hour, there is lacking 5000/14142 of one part, so that there are two mi- nute Seconds, that is, one twenty-thousandth part of another pulse is lacking, so that there may be exactly two Seconds. Seek arguments, if you can; or will you refute by experiment that there are plainly two minute Seconds, and that not even a single moment was lacking? The same thing, which, by way of example, has been explained in the ratio of double times and quadruple spaces, you will find likewise in the others. Suppose, then, that the time be fourfold, that is, 4 Seconds, that is, 40000 minimal units of time. The whole sum of the spaces will be 800020000. Therefore 50005000 to 800020000 is as 1 to 15 49945/50005, almost as 1 to 16; and the deficiency is 60/50005. The space therefore traversed in one Second, since it is 15 feet, in four Seconds will be 240 feet, less by nearly one sixty-ninth part of an inch. Conversely, in order to find the times in the subdouble ratio

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Liber septimus. CAPUT VII. 693 Ratione spatiorum, quære illorum tanquam quadratorum Ra- dices; & primi quidem Radix est, ut priùs fuit inventa 7071 5959 /14142, secundi Radix est 28284 8836 /14142, quarum Ratio est quadrupla, si fractiones spernantur; at aliquid deest, ut sint integra quatuor Secunda minuta horæ; qui defectus demum vix major est quàm 1/1 unius pulsûs arteriæ. Cùm itaque constituta hypothesis incrementi spatiorum, velocitatis, atque impetûs juxta seriem naturalem numerorum sit naturæ consentanea, nequè Physicè repugnet experimentis, non debemus esse solliciti, ut aliam quæramus hypothesim ad statuenda incrementa exquisitè juxta numeros impares; cùm maximè in aquâ ob majorem resistentiam, quam in illâ divi- denda inveniunt corpora gravia descendentia, non exactè ser- vari eandem Rationem incrementorum, quæ in aëre apparet, experimenta iterata declarent, quamvis ad illam Rationem proximè accedant, ut apud Ricciolium tom. 1. Almag. lib. 9. sect. 4. cap. 16. n. 16. varia experimenta afferentem legi potest. Præterquam quod si tam in aëre quàm in aquâ adhibeantur in experimentum corpora secundùm gravitatem specificam ab il- lis minimum discrepantia, statim apparebit non servari illam temporum atque spatiorum Analogiam: id quod pariter obser- vabitur, si in diversis liquoribus eadem gravia corpora dimit- tantur: varia scilicet est resistentia; hæc autem latet, quando grave dimissum valde differt à gravitate, aut levitate medij. His ita constitutis, percussionis vires, quatenus ex velocita- te oriuntur, proximè definire poterimus, si innotescant spatia, per quæ idem corpus grave liberè descendit; nam hinc inno- tescet Ratio intensionum impetûs ultimo descensûs momento, quo contingit percussio. Est siquidem numerus graduum im- petus in motu naturali libero concepti par numero momento- rum motûs; at momenta, quibus constant tempora motuum inæqualium, sunt in Ratione subduplicatâ Rationis spatiorum: igitur sicut Radices quadratæ spatiorum indicât Rationem tem- porum, ita pariter eædem indicant Rationem intensionum im- petûs. Quare si alicujus gravis ex datâ altitudine cadentis per- cussio manifesta fuerit, facilè inferemus, quanta proximè sit futura ejusdem percussio ex majori, aut minori altitudine, si fiat SSss 3

Transcription: Translated (English)

Book Seven. CHAPTER VII. 693 By reason of the spaces, seek their roots as if of squares; and the root of the first is, as was previously found, 7071 5959 /14142, the root of the second is 28284 8836 /14142, of which the ratio is quadruple, if the fractions are disregarded; yet something is lacking, so that they may make exactly four whole seconds of an hour; which defect at length is scarcely greater than 1/1 of a single beat of the pulse. Since therefore the established hypothesis of the increments of spaces, velocity, and impetus according to the natural series of numbers is consonant with nature, and does not conflict physically with experiments, we ought not to be concerned to seek another hypothesis for establishing the increments exactly according to odd numbers; since especially in water, on account of the greater resistance than in air, in which falling heavy bodies find themselves, the same ratio of increments that appears in air is not exactly preserved, repeated experiments declare, although they approach that ratio closely, as may be read in Riccioli, vol. 1, Almagest, book 9, sect. 4, chap. 16, no. 16, where he presents various experiments. Moreover, if in both air and water bodies are used in the experiment that differ least from them in specific gravity, it will immediately appear that that analogy of times and spaces is not preserved: and the same will likewise be observed if the same heavy bodies are released into different liquids; for the resistance is various, namely, and this lies hidden when the heavy body released differs greatly from the gravity or lightness of the medium. These things being thus established, we shall be able to define approximately the force of impacts, insofar as they arise from velocity, if the spaces through which the same heavy body descends freely are known; for from this the ratio of the intensities of the impetus at the final moment of the descent, at which the impact occurs, will be known. For the number of degrees of impetus conceived in free natural motion is equal to the number of moments of the motion; but the moments of which the times of unequal motions consist are in the subduplicate ratio of the ratio of the spaces: therefore, just as the square roots of the spaces indicate the ratio of the times, so likewise they indicate the ratio of the intensities of the impetus. Wherefore, if the impact of some heavy body falling from a given height is manifest, we shall easily infer how great approximately its impact will be from a greater or lesser height, if it be done SSss 3

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Mechanicorum ut Radix datæ altitudinis prioris ad Radicem posterioris altitudinis, ita nota percussio ad quæsitam percussionem. Neque hîc conabor dicta confirmare experimentis tùm à Ricciolio loc. cit. cap. 16. n. 12, tùm à Mersenne tom. 3. in Re- flexionibus Physico-Mathemat. cap. 8. allatis, ex quibus proximè infertur hæc Ratio subduplicata spatiorum. Nam cùm adhibita sit libra, ut in alteram lancem pondus ex diversis altitudinibus dimissum elevaret pondera inæqualia oppositæ lanci imposita, res est anceps & incetta. Quandoquidem, ut observavi jam tum ab anno 54 labentis sæculi scriptis Romæ de hoc eodem argumento publicè traditis, & funiculi, ex quibus lanx percussa pendet, distrahuntur, & libræ jugum flectitur, immò & lanx ipsa ictum cadentis ponderis excipiens flexilis est; ac propterea impetûs vim retundunt, cùm maximè vi elasticâ se restituentes conantur sursum. Præterquam quod, si pondus cadens non exactè incidat in lancis centrum respondens extremitati jugi, plurimum interest ad varianda momenta, prout libræ brachium aut decurtatum aut productum intelligitur. Quò autem majus est pondus in oppositâ lance attollendum ex vi depressionis lancis percussæ, magis resistit, ac proinde locus est flexioni majori ipsius libræ, aut funiculorum distractioni, præsertim si lanx fuerit concava, & cadens pondus illam tangens prolabatur in depressiorem lancis locum. Ex quo accidit, ut docuit experientia, aucto pondere elevando non satis esse dimittere pondus cadens ex altitudine, quæ sit ad priorem altitudinem ut quadratum ponderis majoris ad quadratum ponderis minoris initio elevati; percussio enim contingit in lance, cujus resistentiam ad hoc, ut vi ponderis cadentis deprimatur, metitur resistentia ponderis elevandi in oppositâ lance: at hæc si fuerit major quàm resistentia jugi, aut lancis, ne flectatur, aut funiculorum ne distrahantur, in hac flexione aut distractione insumitur vis percussionis, quin opposita lanx attollatur. Quapropter ex altitudine adhuc majori dimittendum est pondus cadens; nam adhuc majore impetu concepto tam validè percutiet lancem, ut resistentia jugi & lancis ad flexionem ulteriorem, atque funiculorum ad longiorem distractionem, major sit quàm resistentia ponderis oppositæ lancis: atque adeò lanx percussa non deprimetur solùm, quantum funiculorum distractio

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Mechanically: as the root of the given height of the former is to the root of the height of the latter, so is the known impact to the sought impact. Nor here shall I attempt to confirm these statements by the experiments cited both by Riccioli, loc. cit. cap. 16, n. 12, and by Mersenne, tom. 3, in the Physico-Mathematical Reflections, cap. 8, from which it is next inferred that this ratio is the subduplicate of the spaces. For since a balance was used, so that on the other pan a weight released from different heights might raise unequal weights placed on the opposite pan, the matter is uncertain and doubtful. Since, as I have already observed from the year 54 of the last century in writings publicly delivered at Rome on this same subject, both the cords from which the struck pan hangs are stretched, and the beam of the balance bends, indeed the pan itself, receiving the blow of the falling weight, is flexible; and therefore they blunt the force of the impulse, since, especially as they try to restore themselves by elastic force, they spring upward. Besides, if the falling weight does not strike exactly at the center of the pan corresponding to the end of the beam, it matters greatly for varying the moments, according as the arm of the balance is understood to be either shortened or lengthened. And the greater the weight is that must be raised in the opposite pan from the force of the depression of the struck pan, the more it resists, and therefore there is greater bending of the balance itself, or stretching of the cords, especially if the pan be concave, and the falling weight, touching it, slip into a lower part of the pan. From this it happens, as experience has taught, that, when the weight to be raised is increased, it is not enough to let the falling weight drop from a height which is to the former height as the square of the greater weight to the square of the lesser weight first raised; for the impact occurs on the pan, whose resistance to being depressed by the force of the falling weight is measured by the resistance of the weight being raised in the opposite pan: but if this shall be greater than the resistance of the beam, or of the pan, lest it bend, or of the cords lest they be stretched, in this bending or stretching the force of the impact is spent, without the opposite pan being raised. Wherefore the falling weight must be let fall from still greater height; for, having conceived still greater impetus, it will strike the pan so vigorously that the resistance of the beam and of the pan to further bending, and of the cords to longer stretching, shall be greater than the resistance of the weight in the opposite pan: and thus the struck pan will not only be depressed, in proportion as the stretching of the cords

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Liber septimus. CAPUT VII. 695 distractio & ipsa flexio permittit, sed adhuc ulteriùs; atque op- posita lanx attolletur, quatenus impetûs vires excedunt oppo- sitæ gravitatis resistentiam. Ex his, quæ de Percussionibus corporum gravium naturali- ter descendentium hactenus dicta sunt, conjectura sumenda est de reliquis percussionibus, quæ motûs originem non à solâ gra- vitate ducunt: nam in his pariter ex motûs velocitate, qua in- dicatur intensio impetûs, oritur validior percussio: nam sive conceptus à potentiâ vivente impetus, sive extrinsecùs im- pressus (nisi novam offendat resistentiam, quæ illum retundat ac minuat) augetur novo impetu, quem potentia similiter ap- plicata, iisdémque viribus prædita, nec obstaculo ullo aut reti- naculo impedita, sequentibus momentis efficere potest: ideó- que quò major est potentiæ percutientis motus quoad spatium, cæteris paribus, validiùs percutit. Cæteris, inquam, paribus; nam si posterioribus momentis motûs, minor impetus addatur à potentiâ movente, quàm deperdatur ex priore impetu impres- so, quem natura repugnans excutit, languescit motus, & mi- nuitur impetus: aut si tantumdem acquiratur impetus, quan- tum deperditur, motus est æquabilis, nec ad validiorem per- cussionem quicquam confert diuturna potentiæ moventis appli- catio, cum eadem sit impetus intensio in fine horæ, atque in primo momento. Quia tamen frequentiùs (etiam si fortè ali- quid antiquioris impetûs ex novâ resistentiâ deteratur) plus ad- ditur, velocitas incrementum sumit, si potentia maneat diutius applicata. Hinc patet, cur, cum quis hostem sarissâ confodere tentat, aut postes crassiore fuste arietare, brachium retrahat quantum potest; ut nimirum potentia diutiùs applicata maneat in motu, semperque novum impetum gignens sarissæ aut fusti imprimat. Sic duobus digladiantibus, si alter alteri sinistrum latus obver- tat, & tam longo ense utatur, ut protecto corpore possit manum valde retrahere, hic validissimum ictum infliget extento dex- tro brachio, tùm quia ex celerrimâ corporis totius conversione impetus aliquis brachio communicatur præter impetum, quem conferunt musculi movendo brachio destinati, tùm quia diu- tiùs movetur gladius à manu per majus spatium. Quod si eo ictu, quem Itali Quartam vocamus, hostem impetat, adhuc va- lidior

Transcription: Translated (English)

Book Seven. Chapter VII. 695 The very deflection, and even the bending itself, allows it, but still farther; and the opposite scale will be raised insofar as the force of the impulse exceeds the resistance of the opposed weight. From what has so far been said concerning the impacts of bodies naturally descending under gravity, conjecture must be drawn concerning the remaining impacts, which do not derive the origin of motion from gravity alone: for in these also, from the speed of motion, by which the intensity of the impulse is indicated, a stronger impact arises. For whether the impulse conceived from a living power, or impressed from without, unless it encounters a new resistance to blunt and diminish it, it is increased by a new impulse, which a similarly applied power, endowed with the same forces and hindered by no obstacle or restraint, can produce in the following moments. And therefore, the greater the motion of the striking power with respect to distance, all else being equal, the more strongly it strikes. All else being equal, I say; for if in the later moments of motion a smaller impulse is added by the moving power than is lost from the earlier impressed impulse, which resisting nature casts off, the motion weakens and the impulse diminishes; or if as much impulse is acquired as is lost, the motion is uniform, and the prolonged application of the moving power contributes nothing to a stronger impact, since the intensity of the impulse is the same at the end of the hour as in the first moment. Since, however, more is more frequently added, even if perhaps some of the older impulse is worn away by a new resistance, speed takes on an increase if the power remains applied longer. Hence it is clear why, when someone tries to thrust through an enemy with a sarissa, or to batter doorposts with a heavier club, he draws back his arm as much as he can; namely, so that the power may remain longer applied in motion, and may always generate a new impulse and impress it upon the sarissa or club. Thus, in two men fighting with swords, if one turns his left side toward the other, and uses so long a sword that, with his body protected, he can draw his hand far back, he will deliver a very powerful blow with his right arm extended, both because some impulse is communicated to the arm from the very rapid turning of the whole body, in addition to the impulse contributed by the muscles appointed to move the arm, and because the sword is moved by the hand over a greater distance for a longer time. If with that stroke, which the Italians call the Quarta, he attacks the enemy, even more powerful

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Mechanicorum 696 dior erit ictus, quia longior motus; nam in conversione corpo- ris obvertitur hosti dextrum latus præcisè ita, ut brachium to- tum extendi queat; & præterea pars exterior manûs deorsum convertitur, ex quo propter conformationem juncturarum cubi- ti & manus ictus evadit duos ferè digitos longior, quàm si non fieret hujusmodi manûs conversio: demum cùm sinistrum bra- chium in posteriora projiciatur extentum, fit, ut corpori major impetus in anteriora possit imprimi citrà periculum cadendi; brachium siquidem eo pacto in posteriora projectum, translato gravitatis, ut gravitationis, centro servat totius corporis æqui- librium. Nec dissimili ratione manifestum fit, cur pugnum validiùs impingant, qui brachium magis contrahunt, ideóque validio- res ictus ab iis proveniant, qui longiora habentes brachia ea magis contrahunt; sicut calce fortiùs impetunt, qui longiora habent crura: quia videlicet diutiùs moventur, magisque im- petum augent & velocitatem, antè ictum. Sic vides ab equis calcitronibus pedem in anteriora retrahi, & ab irato tauro col- lum depressum in posteriora inflecti, corpore pariter curvato, & quasi in posteriora retracto, ut longiore motu validiùs impe- tant: hinc qui propior est equo calcitranti, minùs læditur, quia nondum tantum impetum concepit, quantum longiore motu concepisset. Hujusmodi percussionibus à potentiâ vivente, quæ suos mo- tus ex arbitrio temperat, provenientibus certam legem statui non posse nemo non videt, sive corpus percutiens impetu ex- trinsecùs assumpto feratur naturâ omnino repugnante, sive im- petus impressus cum impetu vi interiore acquisito in eumdem motum conspirent. Hoc unum tanquam manifestò comper- tum atque deprehensum tenemus, quod in longiore motu factâ novi impetûs accessione velocitas augetur, & vis percussionis est major. Hinc sicut quando fistucâ cadente pali in terram adiguntur, initio illa modicum attollitur, quia exigua superan- da est resistentia, hac autem crescente quò altiùs adacti fuerint, magis illa attollitur; sic lignarios fabros clavum in tabulam in- figentes, initio quidem breviore mallei motu uti videmus, quem deinceps augent, donec demum totâ brachij extensione con- nitantur, prout resistentiæ incrementa validiore percussione vinci

Transcription: Translated (English)

Mechanicorum 696 The blow will be stronger, because the motion is longer; for in the turning of the body the right side is turned toward the opponent precisely so that the whole arm may be extended; and besides, the outer part of the hand is turned downward, from which, because of the structure of the joints of the elbow and hand, the blow becomes almost two fingers’ breadth longer than if such turning of the hand did not take place. Finally, when the left arm is thrown backward while extended, it comes about that a greater impulse can be given to the body forward without danger of falling; for the arm, thus thrown backward, by the shifting of the center of gravity, or of gravitation, preserves the balance of the whole body. In no dissimilar way it becomes evident why those who draw back the arm more strike with greater force, and therefore why stronger blows come from those who, having longer arms, contract them more; just as those who have longer legs strike more strongly with kicks: because, namely, they move longer, and before the blow they increase momentum and velocity more. Thus you see that in kicking horses the foot is drawn back forward, and in an angry bull the lowered neck is bent backward, while the body likewise is curved and as it were drawn back, so that they may attack with a longer motion more strongly; hence the one who is nearer to a kicking horse is less injured, because it has not yet acquired as much impetus as it would have acquired with a longer motion. It is plain to everyone that with blows of this kind proceeding from a living power, which regulates its motions at will, no fixed law can be established, whether the striking body is moved by an externally taken impetus entirely contrary to nature, or whether the impressed impetus, together with the impetus acquired by internal force, concur toward the same motion. This one thing we hold as manifestly ascertained and established: that in a longer motion, when a fresh impetus is added, velocity increases, and the force of the blow is greater. Hence, just as when piles are driven into the ground by a falling rammer, at first it is lifted only a little, because only a small resistance must be overcome, but as this resistance increases the higher the piles are driven, it is lifted more; so we see carpenters, when driving a nail into a plank, use at first a shorter motion of the hammer, which they then increase, until at last they exert themselves with the full extension of the arm, as the increases of resistance are to be overcome by a stronger blow

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Liber septimus. CAPUT VIII. 697 vinci oportet. Sic rusticos ligna findentes altiùs elevare securim, aut tuditem, quo cuneum percutiant, observamus, quò major est scindendi difficultas, ut auctus impetus velociorem motum efficiat, quem percussio consequitur. Cum itaque duæ velocitates inter se comparatæ conferri possint vel ratione temporis, vel ratione spatij, ita ut vel æqualia spatia inæqualibus temporibus, vel inæqualia spatia tempore eodem conficiant; illa utique erit major velocitas, quando in motu temporis brevitas, & spatij amplitudo consenserint. Hinc percussio contingit validior à corpore, quod multum spatij brevi tempore decurrat, quàm à corpore conficiente minus spatij longiore tempore. Propterea quæ de velocitate dicta sunt, & percussionum viribus, spectatâ diuturnitate motûs, ita intelligenda sunt, ut corpus percutiens vel eodem tempore plus spatij, vel breviore tempore æquale spatium, vel breviore tempore plus spatij decurrat: hoc enim ex majore impetûs intensione oritur. CAPUT VIII. An validior sit ictus Mallei à situ Verticali ad Horizontalem, an verò ab Horizontali ad Verticalem descendentis. Certum est percussiones fieri validiores, quando cum impetu ab extraneâ potentiâ impresso consentit vis intrinseca ipsius corporis percutientis motu naturali descendentis; ipsum enim suum pariter impetum concipit, quem addit impresso: sic saxum, quod ex editâ turri deorsum rectâ projicis, validiùs percutit, quàm si illud dimitteres sponte sua casurum. Hinc qui malleo deorsum percutit aliquod corpus, ad motum eundem, cum percutientis impulsu conspirantem invenit mallei gravitatem, & citrà omnem controversiam majorem ictum infligit, quàm si sursum, aut in latus urgeret malleum, gravitate aut repugnante, aut saltem nihil juvante. TTtt

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Book Seven. CHAPTER VIII. 697 must be overcome. Thus we observe woodcutters, when splitting wood, raise the axe or mallet higher, with which they strike the wedge, the greater the difficulty of splitting, so that the increased impetus may produce a swifter motion, which the blow follows. Since therefore two velocities compared with each other can be measured either by time or by space, so that they accomplish either equal spaces in unequal times, or unequal spaces in the same time; that velocity will certainly be greater, when in the motion the brevity of time and the largeness of space agree. Hence a blow from a body that travels a great distance in a short time is stronger than from a body covering less space in a longer time. Therefore what has been said about velocity and the force of blows, considering the duration of the motion, must be understood thus: that the striking body either in the same time traverses more space, or in a shorter time an equal space, or in a shorter time more space; for this arises from a greater intensity of impetus. CHAPTER VIII. Whether the blow of a Hammer descending from the Vertical to the Horizontal, or rather from the Horizontal to the Vertical, is the stronger. It is certain that blows become stronger when the force intrinsic to the body itself, descending by natural motion, agrees with the impetus impressed by an external power; for it likewise conceives its own impetus, which it adds to that impressed: thus a stone, which you throw straight down from a high tower, strikes more strongly than if you were to let it fall of itself. Hence he who strikes a body downward with a hammer finds the weight of the hammer conspiring to the same motion with the striker’s impulse, and beyond all controversy deals a greater blow than if he were to drive the hammer upward, or to the side, with gravity either resisting or at least not helping at all. TTtt

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Mechanicorum Porrò cùm circa juncturam brachij cum humero, tanquam circa centrum, describatur semicirculus descendens, in quo sunt duo Quadrantes, alter cùm brachium summè elevatum in perpendiculo existens descendit, ut fiat horizonti parallelum, alterùm brachium à positione horizonti parallelâ ad imum de- primitur, ut iterum in perpendiculo statuatur; dubitari potest, an mallei ictus juxta duas hasce positiones sint omnino æquales, an verò inæquales; & hoc quidem non ex vi impulsionis exter- næ, quam æquabilem ponimus, homine æqualiter connitente, brachio æqualiter extento, & datâ eâdem manubrij longitudi- ne, sed ratione ipsius gravitatis mallei naturaliter descendentis. Prioris motus, quo in circulo Verticali malleus deprimitur us- que ad planum horizonti parallelum, in quo fit percussio, exem- plum præbent tùm fabri ferrarij incudem tundentes, tùm ligna- rij clavos asseribus in plano horizontali constitutis infigentes. Posterioris autem motûs, quo malleum horizonti parallelum usque eò deprimimus, ut fiat perpendicularis, specimen habemus, cum corpus in pavimento jacens, aut non procul ab illo, ita percutimus, ut percussum moveatur horizonti ferè paralle- lum; cujusmodi esset, quando cuneum jacenti saxo subijcere conamur, aut ligneam pilam ludentes malleo excutimus. Ut autem dilucidè proposita quæstio exponatur, secernendus est mallei motus naturalis ab ea parte, quam externa im- pulsio addit; & perinde con- siderandus est malleus, atque si manubrij extremitas axi in- fixa esset circa eum versatilis, adeò ut sibi relictus malleus arcum descendendo descri- beret. Sit malleus A B; & manubrij extremitas A sit circa axem in A versatilis; centrum autem gravitatis mallei intelligatur in B: quod quandiu in perpendiculo im- minet axi A, totam suam vim in illum exerens sustinetur, nec

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Mechanics Moreover, since around the junction of the arm with the shoulder, as around a center, a descending semicircle is described, in which there are two quadrants: one, when the arm, being highly raised and standing upright, descends until it becomes parallel to the horizon; the other, when the arm is lowered from the position parallel to the horizon down to the bottom, until it is again set upright; it may be doubted whether the blows of the hammer in these two positions are altogether equal, or rather unequal; and this indeed not on account of the force of the external impulse, which we suppose to be uniform, the man exerting himself equally, the arm equally extended, and the same length of handle being given, but on account of the weight of the hammer itself naturally descending. Of the former motion, by which the hammer is lowered in a vertical circle until the plane parallel to the horizon, in which the striking takes place, examples are provided both by smiths beating the anvil and by carpenters driving nails into boards placed on a horizontal plane. Of the latter motion, however, by which we lower the hammer from the position parallel to the horizon down so far that it becomes perpendicular, we have an example when we strike a body lying on the floor, or not far from it, in such a way that the thing struck moves almost parallel to the horizon; of this sort would be when we try to drive a wedge under a stone lying on the ground, or when, playing with a wooden ball, we knock it away with a hammer. But in order that the question proposed may be explained clearly, the natural motion of the hammer must be separated from that part which external impulse adds; and the hammer should likewise be considered as if the end of the handle were fixed to an axle, movable around it, so that, left to itself, the hammer would describe an arc in descending. Let the hammer be A B; and let the end A of the handle be movable around the axle in A; but let the center of gravity of the hammer be understood to be in B: so long as this hangs over the axle A in a perpendicular, supporting with all its force upon it, it is sustained, nor

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Liber septimus. CAPUT VIII. 699 nec motum inchoat, nisi à perpendiculo B A removeatur; hoc verò ubi transgressus fuerit malleus, descensum molitur: sed quia rigido manubrio connectitur cum Axe A, cogitur in latus secedere, & describere arcum B C, cui motui respondet solùm descensus B D Sinus Versus anguli B A C; & descripto integro Quadrante B E, descensum metitur Radius B A. Verùm quamvis motus hujusmodi per arcum B E sit consentaneus propensioni gravitatis, quæ proinde singulis momentis novam impetûs particulam concipiens motum, quantum potest, accelerat, non tamen illa Ratio servatur, de qua superiori Capi- te dictum est, ut Ratio spatiorum sit in duplicatâ Ratione tem- porum; neque enim hîc liberè descendit malleus, sed in singu- lis punctis Quadrantis alia atque alia habet momenta descen- dendi, singula minora momento, quod haberet idem malleus nullo impedimento prohibitus; quod momentum integrum ille obtinet tantummodo in Quadrantis extremitate E, ubi nullâ ratione sustinetur aut retinetur ab axe A manubrium sustinen- te, aut retinente. Est autem momentum gravitatis in unoquo- que arcûs puncto, quasi illa esset in plano ibi circulum tangen- te, ac propterea inclinato: idcirco, ut lib.1.cap.13. dictum est, ejusdem gravitatis momentum in plano inclinato, ad momen- tum in lineâ perpendiculari eam Rationem habet, quam in triangulo rectangulo, cujus angulus Verticalis sit æqualis an- gulo inclinationis plani, habet Perpendicularum ad Hypothe- nusam. Quare in puncto C malleus habet momentum, ac si es- set in plano inclinato F C, & ad momentum liberum ita se ha- bet, ut D F ad F C, hoc est per 8.lib.6. ut D C ad C A, & si- militer in puncto G ut I G ad G A. Ex quo patet momentorum incrementa analoga esse incrementis Sinuum Rectorum arcu- bus subinde majoribus convenientium. At hîc, ubi de gravitatis momentis sermo instituitur, caven- dum est, ne quem fortè in errorem inducat ambiguitas nomi- nis. Nam quando in C momentum dicimus esse ut D C, & in G momentum esse ut I G, hoc intelligendum est præcisè ra- tione positionis, quatenus in hoc aut illo puncto constituta gra- vitas concipitur, nullâ habitâ ratione antecedentis motûs aut quietis: & sub voce momenti Gravitatis hæc subjecta est sen- tentia, ut gravitas mallei, quæ non impedita singulis punctis TTtt 2

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Book Seven. CHAPTER VIII. 699 nor does it begin its motion unless it be removed from the plumb line B A; but when the hammer has passed beyond this, it seeks a descent: yet because it is connected by a rigid handle with the Axis A, it is compelled to deviate sideways, and to describe the arc B C, to which motion there corresponds only the descent B D, the sine of the angle B A C; and, the whole quadrant B E being described, the Radius B A measures the descent. But although a motion of this kind through the arc B E is in accordance with the tendency of gravity, which therefore at every instant, conceiving a new particle of impulse, accelerates the motion as much as it can, nevertheless that Ratio is not preserved, of which mention was made in the foregoing chapter, namely, that the Ratio of spaces should be in the duplicated Ratio of times; for here the hammer does not descend freely, but at each point of the Quadrant it has one and another momentum of descent, each smaller than the momentum which the same hammer would have if not prevented by any obstacle; which full momentum it obtains only at the end of the Quadrant E, where it is supported or retained in no way by the axis A sustaining, or retaining, the handle. Now the momentum of gravity at each and every point of the arc is as though it were in the plane there tangent to the circle, and therefore inclined: wherefore, as was said in book 1. chap. 13, the momentum of the same gravity in an inclined plane bears to the momentum in a perpendicular line that ratio which, in a right triangle whose vertical angle is equal to the angle of inclination of the plane, the perpendicular sides bear to the hypotenuse. Therefore at the point C the hammer has a momentum as if it were in the inclined plane F C, and to the free momentum it is related as D F is to F C, that is, by book 6, lib. 8, as D C is to C A, and similarly at the point G as I G is to G A. From this it is clear that the increments of the momenta are analogous to the increments of the right sines corresponding to progressively larger arcs. But here, where discussion is begun concerning the moments of gravity, care must be taken lest the ambiguity of the term perhaps lead anyone into error. For when we say that in C the momentum is as D C, and that in G the momentum is as I G, this must be understood precisely with respect to position, insofar as gravity is conceived as being placed in this or that point, without any regard being had to previous motion or rest: and under the term momentum of Gravity the following proposition is to be understood, namely, that the gravity of the hammer, which not impeded at each point TTtt 2

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Mechanicorum 700 temporis conciperet novum impetum ut A C, A G &c. quia à rigido manubrio modò magis, modò minùs sustinetur, quando est in C, impeditur, ne concipiat impetum nisi ut D C, & in G ut I G, atque ita de cæteris, donec in E concipiat impetum ut A E. Cæterùm quia in præcedentibus temporis punctis acqui- sitæ sunt particulæ impetûs respondentes Sinubus Rectis præ- cedentium arcuum, ea fit impetûs intensio, ac proinde motûs velocitas, quæ omnium illorum Sinuum aggregato ferè respon- deat: Et in fine Quadrantis in E vis est complectens omnes impetus, quibus additio facta est semper non tamen æqualis pri- mo impetui, qui valdè languidus fuit, sed semper major atque major, prout Sinus Recti excreverunt. Et hæc quidem de Superiore Quadrante. Iam inferior Qua- drans considerandus est, in quo Axis A retinet malleum, ne ex E recto tramite descendat, sed eum cogit deflectere, & arcum E S describere, in cujus singulis punctis momentum perinde est atque in plano inclinato. Quare in L intelligitur descendens in plano inclinato K L, & ibi ejus momentum ad momentum liberum est ut R K ad K L, hoc est ut M L, ad L K, hoc est per 8. lib. 6. ut M A ad A L, hoc est ut Sinus Complementi arcûs E L ad Radium. Similiter in N est ut P A ad A N; & sic de cæteris. Est autem manifestum hujusmodi Sinus Complementorum eosdem planè esse cum Sinubus Rectis Superioris Qua- drantis, sed ordine præpostero acceptis, atque adeò horum ag- gregatum esse illorum summæ æquale. Non tamen hinc statim conficitur eandem esse in S vim mal- lei descendentis ex E, atque est in E vis ejusdem descendentis ex B. Non inficior æqualem in utroque Quadrante produci impetûs entitatem, si in summam referantur omnes impetûs particulæ, quæ momentis singulis efficiuntur; sed an æqualem demum conflent intentionem, ex qua vis percussionis oritur, non omninò temerè, ut mihi quidem videor, subdubito. Cùm enim acquisitus impetus interveniente resistentiâ imminuatur, ac debilitetur, & quidem eò magis, si à rectâ secundùm natu- ram lineâ magis declinare cogitur; utique institutâ Superioris cum Inferiori Quadrante comparatio ostendit in illo quidem resistentiam semper decrescere, in hoc semper augeri, ac proin- de impetum prioribus momentis acquisitum, licèt aliquid in consequem

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of the mechanics 700 it might conceive a new impulse as AC, AG, etc.; because from the rigid handle it is now sustained more, now less; when it is at C, it is hindered from conceiving an impulse except as DC, and at G as IG, and so on through the rest, until at E it conceives an impulse as AE. Moreover, because in the preceding points of time small portions of impulse corresponding to the sines of the straight lines of the preceding arcs have been acquired, there results an intensification of the impulse, and consequently a velocity of motion which corresponds almost to the aggregate of all those sines: and at the end of the quadrant at E the force is one comprising all the impulses, to which addition is made that is not, however, equal from the beginning; for the first impulse was very languid, but always greater and greater, as the straight sines have increased. And thus much for the upper quadrant. Now the lower quadrant must be considered, in which the axis A holds back the hammer, lest it descend straight from E, but forces it to bend aside and describe the arc ES, in each of whose points the momentum is the same as in an inclined plane. Therefore at L it is understood to descend in the inclined plane KL, and there its momentum to free momentum is as RK to KL, that is, as ML to LK; that is, by 8, book 6, as MA to AL, that is, as the sine of the complement of the arc EL to the radius. Similarly at N it is as PA to AN; and so on for the rest. Now it is manifest that such sines of the complements are exactly the same as the straight sines of the upper quadrant, but taken in reverse order, and therefore the aggregate of these is equal to the sum of those. Yet from this it does not immediately follow that the force of the hammer descending from E in S is the same as the force in E of the same body descending from B. I do not deny that an equal quantity of impulse is produced in each quadrant, if all the small impulses that are effected at each moment are brought into the total sum; but whether they ultimately form an equal intensity, from which the force of percussion arises, I for one do not at all rashly, as it seems to me, doubt. For since acquired impulse is diminished and weakened by intervening resistance, and indeed the more so if it is compelled to deviate further from the straight line according to nature; certainly, if the upper quadrant is compared with the lower, it shows that in the former resistance is always diminishing, in the latter always increasing, and therefore the impulse acquired in the earlier moments, although something in the subsequen

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Liber septimus. CAPUT VIII. 701 consequentibus amittat, tamen hujus decrementi mensurâ ma- gis ac magis extenuatâ, non adeò in Superiore Quadrante lan- guescere, sicut in inferiore, ubi resistentia semper augetur, & impetus magis ac magis deteritur. Adde impetum de novo productum in posterioribus momentis, in superiore quidem Quadrante esse majorem, in inferiore verò minorem. Quare cùm in postremis motûs momentis in superiore Quadrante multus producatur impetus, & ferè nulla sit resistentia, in in- feriore autem Quadrante multa inveniatur resistentia, & valdè exiguus impetus producatur, satis probabili conjecturâ ali- quam statuemus ictuum inæqualitatem, ita ut aliquanto vali- dior sit ex B in E, quàm ex E in S. Neque obstat, quod in E maximus producatur impetus, qui deinde in L & N, atque consequentibus punctis additamen- tum accipiat, licèt semper minus ac minus, quo ita augetur, ut semper incitetur motus. Hoc enim non facit, quin impetus in E conceptus majoribus semper decrementis imminuatur usque in S, & impetus in L conceptus similiter magis languescat, at- que ita de cæteris. Finge scilicet nullum impetum novum con- cipi in L, aut nullum in N, adhuc impetus in E conceptus deor- sum tenderet, sed retinaculo illo debilitatus languidiùs adduce- ret malleum in S: idem dic de quolibet impetu singulis mo- mentis concepto, qui crescente resistentiâ majoribus decre- mentis imminueretur, & languidè veniret in S. At quoniam plurima sunt momenta in brevissimi temporis particulâ, tot sunt reliqui impetus, ut simul constituant notabilem inten- sionem. Quod autem resistentia in superiore Quadrante minuatur, argumento non est opus; manifestum quippe est gravitatem in arcu B E descendentem subinde transferri à plano magis inclinato in minùs inclinatum, & magis accedens ad perpendicu- lare: quis autem neget descendenti gravitati eò minùs obstare planum subjectum, quò fuerit minùs inclinatum? Atqui an- gulus C A F minor est angulo G A H, anguli autem ad C & G sunt recti; igitur angulus A F C major est angulo A H G; at- que propterea planum F C est magis inclinatum, quàm pla- num H G, & cætera plana consequentia usque ad planum per- pendiculare in E. Contra verò in inferiore Quadrante resisten- TTtt 3

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Book Seven. CHAPTER VIII. 701 although it lose by the ensuing ones, yet, this measure of diminution being more and more reduced, it does not languish so much in the Upper Quadrant as in the lower, where the resistance always increases, and the impulse is more and more worn away. Add that the new impulse produced in the later moments is indeed greater in the upper Quadrant, but smaller in the lower. Wherefore, since in the last moments of motion in the upper Quadrant much impulse is produced, and there is almost no resistance, whereas in the lower Quadrant much resistance is found, and a very slight impulse is produced, we shall, by a sufficiently probable conjecture, posit some inequality of the blows, so that it may be somewhat stronger from B to E than from E to S. Nor does it hinder that in E the greatest impulse is produced, which then in L and N, and in the following points, receives an addition, though always less and less, by which it is so increased that the motion is always excited. For this does not prevent the impulse conceived in E from being diminished by ever greater decrements until S, and the impulse conceived in L from languishing similarly more, and so of the rest. Imagine, namely, that no new impulse is conceived in L, or none in N; still the impulse conceived in E would tend downward, but weakened by that restraint it would more languidly carry the hammer toward S. The same may be said of every impulse conceived at each moment, which, as the resistance increases, would be diminished by greater decrements, and would come languidly to S. But since there are very many moments in the smallest part of time, there are so many remaining impulses that together they constitute a notable intensity. Now, that the resistance in the upper Quadrant diminishes, no argument is needed; for it is manifest that gravity descending in the arc B E is successively transferred from a more inclined plane to a less inclined one, and one more approaching the perpendicular: and who would deny that the less inclined the underlying plane is, the less it obstructs the descending gravity? But the angle C A F is less than the angle G A H, while the angles at C and G are right angles; therefore the angle A F C is greater than the angle A H G; and therefore the plane F C is more inclined than the plane H G, and the other successive planes up to the perpendicular plane at E. On the other hand, in the lower Quadrant the resistan-

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Mechanicorum 702 tiam semper augeri ex eo constat, quòd à plano perpendicula- ri ad inclinatum, immò ad semper magis atque magis inclina- tum, fit transitus, donec demum descendens gravitas veniat ad planum horizontale. Angulus videlicet inclinationis plani est æqualis angulo, quem denotat arcus in inferiore Quadrante decursus. Nam in triangulo ALK, cujus in basim AK cadit perpendicularis LM, ex 8. lib.6. angulo KAL æqualis est an- gulus KLM, & propter parallelismum linearum ML & KR, angulo KLM æqualis est alternus LKR angulus inclinationis plani KL; igitur hic angulus inclinationis est æqualis angulo, quem denotat arcus EL. Idem dic de angulo EAN & cæte- ris, qui semper majores indicant gravitatem transire ad plana magis & magis inclinata. CAPUT IX. Quomodo percussiones ex mole pendeant. Quamvis ad validiorem ictum infligendum corporis percu- tientis velocitas, impetûs intentionem indicans, pluri- mum conferat, ut dictum est; non ad hanc tamen velut ad uni- cam causam referenda est vis percussionis; sed & corporis ejus- dem percutientis moles attendenda est: videmus scilicet ex mole ipsâ percussiones augeri, cæteris paribus; perinde enim est, atque si tot corpora percutientia essent, quàm multiplex est moles major collata cum minore. Nam si nota est vis per- cutiendi, quæ inest globulo duarum unciarum cadenti ex cer- tâ quadam altitudine utique probabilis conjectura & ratio sua- det sextuplam esse vim globi ex simili materiâ unciarum duo- decim ex altitudine eâdem cadentis: gravitas siquidem sexies multiplicata etiam impetum efficere potest sextuplam, non qui- dem intensivè, sed entitativè; neque enim pro Ratione molis augetur velocitas, quippe quæ requireret sextuplam intensio- nem. Hanc tamen majorem vim Ratione molis ita intelligi velim, ut medij resistentia dissimuletur: nam eo ipso, quod mo- les

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Mechanics 702 It is clear from this that the inclination always increases, because there is a passage from a perpendicular plane to an inclined one, indeed to one ever more and more inclined, until at last the descending weight comes to the horizontal plane. For the angle of inclination of the plane is equal to the angle denoted by the arc in the lower quadrant of the descent. For in triangle ALK, on whose base AK the perpendicular LM falls, by 8. lib.6. angle KAL is equal to angle KLM, and, because of the parallelism of the lines ML and KR, angle KLM is equal to the alternate angle LKR, the angle of inclination of plane KL; therefore this angle of inclination is equal to the angle denoted by arc EL. The same may be said of angle EAN and the rest, which always indicate that the weight passes to planes more and more inclined. CHAPTER IX. How impacts depend on mass. Although, in inflicting a stronger blow, the velocity of the striking body, which indicates the intensity of the impetus, contributes very much, as has been said; nevertheless the force of impact is not to be referred to this alone as to its single cause; the mass of the striking body itself must also be considered: for we see that, all else being equal, impacts are increased by mass itself; indeed it is as though there were as many striking bodies as the larger mass is multiple compared with the smaller. For if the force of striking is known, which belongs to a two-ounce ball falling from a certain height, then certainly a probable conjecture and reason suggests that the force of a twelve-ounce ball of similar material falling from the same height is sixfold: for gravity, multiplied six times, can also produce an impetus six times as great, not indeed intensively, but entitatively; nor is velocity increased in proportion to mass, since that would require a sixfold intensity. Yet I would have this greater force in respect of mass understood in such a way that the resistance of the medium be disregarded: for by the very fact that the mass

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Liber septimus. CAPUT IX. 703 les ejusdem secundùm speciem gravitatis augetur, etiam super- ficiem augeri necesse est, quæ non eâdem facilitate medium di- vidit. Sed quoniam ita augeri potest moles, ut non similem servet figuram, sed alias atque alias induat figuras manente æquali gravitate, ac propterea valde incerta est resistentiæ mensura, quæ ex medij divisione oritur, prout hanc aut illam faciem medio dividendo obvertit ipsa moles; hinc est, quod il- lam resistentiam tantisper dissimulare licet, dum reliquas per- cussionis causas vestigamus. Cæterùm illius quoque ratio est habenda, ut aliquid virium detractum intelligatur percussioni, ubi & moles & spatium consideratur; quatenus velocitas, auctâ mole, hoc est aucto ex medij resistentiâ impedimento, aliquan- tulum imminuitur, ita tamen ut pariter plures majoris molis partes, collatis viribus medium urgentes, aliquid afferant faci- litatis in dividendo medio, adeóque etiam velocitatis. Ex his habetur percussionis vires componi ex mole & ex ve- locitate corporis percutientis: Moles siquidem determinat en- titativè mensuram impetûs singulis momentis producti, veloci- tas indicat intentionem, hoc est summam impetuum in motu acquisitorum, hoc est ejusdem impetûs gravitati, aut potentiæ virtuti, primo momento respondentis multiplicationem. Qua- re si duorum corporum ictus comparentur, percussionum Ratio erit composita ex Rationibus velocitatum, & gravitatum, seu potentiarum, quibus vis movendi tribuitur. Velocitatem au- tem illam intelligo, quæ ratione impetus acquisiti conveniret corpori eo momento, quo percutit, nisi inveniret resistentiam: Hujusmodi verò impetûs eo momento intentionem indicat spa- tium in antecedenti motu decursum, ex quo, prout dictum est cap.7. cognoscitur Ratio temporum, quibus est analoga inten- sio impetûs. Hinc si cadat ex altitudine 100 palmorum globus unciarum duarum, deinde ex altitudine decem palmorum glo- bus unciarum 12, sunt duæ Rationes, altera velocitatum, hoc est intensionum impetûs in subduplicatâ Ratione altitudinum, vi- delicet ut 10 ad 3 16/100, altera gravitatum ut 1 ad 6; quæ comosi- tæ dant Rationem ut 10 ad 18 96/100; ac proinde percussio globi minoris estimari poterit proximè ut 10, majoris ut 18 95/100: Nam duæ unciæ majoris descendendo per decem palmos haberent impetum

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Book Seventh. Chapter IX. 703 Since its force increases in proportion to the species of gravity, the surface also must necessarily increase, which does not divide the medium with the same ease. But since the mass can be increased in such a way that it does not retain a similar figure, but assumes one shape after another while its weight remains equal, and therefore the measure of resistance arising from the division of the medium is very uncertain, as the mass itself presents now this and now that face to the medium in dividing it; hence it is that that resistance may for the time being be left out of account, while we investigate the other causes of percussion. Nevertheless, account must also be taken of it, so that some force may be understood to be taken away from the percussion, where both mass and space are considered; insofar as velocity, with the mass increased—that is, with the hindrance increased by the resistance of the medium—is somewhat diminished, though in such a way that likewise a larger number of parts of a greater mass, pressing upon the medium with their combined forces, contribute something to the ease of dividing the medium, and therefore also to the velocity. From these things it is evident that the forces of percussion are composed of mass and of the velocity of the striking body: for mass determines quantitatively the measure of impetus produced at each moment, while velocity indicates the intention, that is, the sum of the impetuses acquired in motion, that is, the multiplication corresponding at the first moment to the gravity, or power, of the same impetus. Wherefore, if the blows of two bodies are compared, the ratio of the percussions will be composed from the ratios of the velocities and of the gravities, or powers, to which the force of moving is attributed. By that velocity, however, I mean that which would belong to the body, by reason of the acquired impetus, at the moment when it strikes, if it were not to encounter resistance: now the intention of such an impetus at that moment indicates the space covered in the preceding motion, from which, as was said in chapter 7, the ratio of the times is known, to which the intention of the impetus is analogous. Hence, if a ball of two ounces falls from a height of 100 palms, and then a ball of 12 ounces from a height of ten palms, there are two ratios: one of the velocities, that is, of the intentions of the impetus, in the subduplicate ratio of the heights, namely as 10 to 3 16/100; the other of the gravities, as 1 to 6; which, combined, give the ratio as 10 to 18 96/100; and therefore the percussion of the smaller ball may be estimated approximately as 10, and that of the larger as 18 95/100: for the two ounces of the larger body, descending through ten palms, would have an impetus

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Mechanicorum 704 impetum ut 3 16/100: ergo sexies duæ unciæ habent entitativè im- petum ut 18 96/100. Quare si gravitates fuerint reciprocè ut velo- citates, hoc est impetûs intensiones, erunt æquales percussio- nes, ut est manifestum. Hoc autem, quod de gravitate motu naturali descendente dictum est, de potentiis pariter, servatâ analogiâ, est intelligen- dum (assumendo scilicet loco gravitatis vim ipsam potentiæ, quæ similiter conata perseveret in motu antecedente percussio- nem) si innotescat, quantum potentiæ inæqualiter conentur, & per inæqualia spatia moveant idem corpus, quo ictum infligunt; nam ex Rationibus conatuum, & velocitatum componi- tur Ratio percussionum. At si potentiæ duæ inæqualiter co- nantes per inæqualia spatia moveant inæqualia corpora, qui- bus alteri corpori ictus infligatur, attendendum est, an solùm tam diuturnus sit motus ictum præcedens, ut nihil impressi im- petûs deteratur: nam si alternis quibusdam incrementis & de- crementis modò augeatur, modò minuatur, non est habenda ratio totius temporis, aut spatij, in quo factus est motus: quis enim existimet aptè computari posse, utrùm navis percurrerit sex, aut octo milliaria, ut ejus ictus, quo cymbam percutit, cognoscatur? Quare satius erit in hujusmodi motibus ab impe- tu extrinsecùs impresso provenientibus, qui ut plurimum re- pugnantem habet ipsius corporis naturam, nec totus permanet quemadmodum impetus acquisitus, ipsam velocitatem consi- derare, quatenus apparet non multo tempore antè ictum: tunc enim, quia potentia movens eum impetum imprimit, qui satis sit ad molem illam movendam tantâ velocitate, ut impetus hu- jusmodi innotescat, & molis & velocitatis ratio habenda est; atque idcircò ad comparandas invicem percussiones compo- nenda est Ratio ex Rationibus molium, & velocitatum. Hinc si navis oneraria lentè moveatur velocitate ut duo, & navis alia sextuplo minor moveatur velocitate ut decem (quia videlicet paulò antè ictum observatum est, quo tempore illa procedebat duos passus, hanc percurrisse decem passus) ictus majoris ad ictum minoris erit ut 12 ad 10, compositis scilicet Ratione mo- lium 6 ad 1, & Ratione velocitatum 2 ad 10. Hic autem ubi Molis nomen usurpamus, cavendus est in vocabuli

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Mechanics 704 the impact is as 3 16/100: therefore six times two ounces have, entitatively, an impact as 18 96/100. Wherefore, if weights are reciprocally as velocities, that is, as intensities of impulse, the impacts will be equal, as is manifest. But what has been said of gravity descending by natural motion must likewise, with due analogy preserved, be understood of powers, namely, by taking in place of gravity the very force of the power, which similarly persists in its effort in the motion preceding the impact, if it be known how much the powers strive unequally, and move the same body through unequal spaces, by which they inflict the blow; for from the ratios of the efforts and the velocities is composed the ratio of the impacts. But if two powers, striving unequally, move unequal bodies through unequal spaces, upon which the blow is inflicted on one body, it must be considered whether the motion preceding the blow is of such duration that none of the impressed impulse is worn away: for if it is now increased, now diminished, by certain alternate increments and decrements, one must not take account of the whole time or space in which the motion was made; for who would think it could be suitably computed whether a ship had run six or eight miles, so that its blow, by which it strikes a boat, might be known? Therefore it will be better in motions of this kind, arising from impulse impressed from without, which for the most part has the resisting nature of the body itself and does not wholly remain, as does acquired impulse, to consider the velocity itself, insofar as it appears shortly before the blow: for then, because the moving power impresses that impulse which is sufficient to move that mass with such velocity, so that the impulse of this kind may be known, account must be taken of both the mass and the velocity; and therefore, in order to compare blows with one another, a ratio must be composed from the ratios of masses and velocities. Hence, if a loaded ship move slowly with a velocity of two, and another ship, six times smaller, move with a velocity of ten (because, namely, shortly before the blow it was observed that, at the time when that ship was proceeding two paces, this one had covered ten paces), the blow of the larger will be to the blow of the smaller as 12 to 10, the ratio of the masses being composed, namely, 6 to 1, and the ratio of the velocities, 2 to 10. Here however, where we use the name of Mass, caution is needed in the word

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Liber septimus. CAPUT IX. 705 vocabuli ambiguitate lapsus: neque enim corporis tantummodo amplitudinem, quatenus sub Geometricam dimensionem cadens spatium occupat, intelligere oportet, verùm etiam naturam ipsam atque substantiam: ea scilicet, quæ minore secundùm speciem gravitate prædita sunt, sub magnis dimensionibus parum habent substantiæ atque materiæ, ideóque & tenuem movendi ac impetum producendi virtutem; neque propterea quod molem magnam præseferant, validiora in percutiendo censenda sunt, quasi à globo ligneo librarum duarum, quia ferè decuplo major est globo plumbeo ejusdem ponderis, expectari posset validior ictus, si ex eâdem altitudine dimittantur: nam vis producendi impetum connata est substantiæ, quâ substantia talis est, non quantitati, prout extensio est. Quare ubi molis habendam esse rationem diximus, ut fiat Rationum Compositio, ex qua percussionum incrementa aut decrementa innotescant, ipsam potissimùm substantiam intelligimus, quam, non nisi intra idem genus corporis, extensio major aut minor conseqvi solet: propterea si ferrei cylindri ictus, atque lignei, conferre invicem volueris, non ipsos cylindros, quatenus cylindri sunt sub tantâ basi & altitudine, dimetiri oportet, sed potiùs eorum gravitatem, ut quanta sit moles virtutis ipsi naturæ atque substantiæ respondens, ex gravitate inferatur. Non tamen idcircò extensionis atque figuræ animadversio otiosa est, aut contemnenda, in percussionibus; quinimmò non oscitanter consideranda, ut deprehendatur, qua sui corporis parte validissimum ictum infligat instrumentum percutiens. Hoc autem tripliciter potissimùm movetur, videlicet, primò ad perpendiculum descendendo motu naturali; deinde horizontaliter, seu obliquè, cùm à dextrâ in sinistram, aut vicissim à lævâ in dexteram, aut in anteriora extrinsecùs motu recto impellitur; demum in orbem, circuli arcum describendo. Et quidem corpus sponte suâ descendens, quodcumque tandem illud sit, suam habet Directionis lineam, per quam in motu Centrum gravitatis progreditur. In insimâ igitur corporis parte ipsa Directionis linea definit punctum, in quo si fiat corporis percutientis contactus, ille erit validissimus ictus, quem hujusmodi corpus ex datâ altitudine descendens infligere potest, ibi quippe maximam reperit resistentiam, cum æquales VVuu

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Book Seven. CHAPTER IX. 705 taken in by the ambiguity of the term: for by size one must understand not only the amplitude of the body, insofar as it occupies space falling under geometric dimension, but also its very nature and substance: namely, those things which are endowed with less weight according to species have, under large dimensions, little substance and matter, and therefore a slight power of moving and of producing impulse; nor, because they present a great bulk, are they on that account to be judged stronger in striking, as though from a wooden ball weighing two pounds, because it is nearly ten times larger than a leaden ball of the same weight, a more forceful blow could be expected if they are let fall from the same height: for the force of producing impulse is inherent in substance, inasmuch as substance is such, not in quantity, as extension is. Wherefore, when we said that account must be taken of bulk, so that a Composition of Ratios may be formed, from which the increases or decreases of blows may be made known, we understand chiefly the substance itself, which, not otherwise than within the same kind of body, a greater or lesser extension is accustomed to follow: therefore, if you wish to compare the blow of an iron cylinder with that of a wooden one, you ought not to measure the cylinders themselves, insofar as they are cylinders under such a base and height, but rather their weight, so that from the weight it may be inferred how great is the bulk of power corresponding to nature and substance itself. Nevertheless, consideration of extension and shape is not therefore idle, or to be despised, in blows; indeed it must be carefully considered, so that it may be discovered by which part of its body the striking instrument delivers the strongest blow. Now this is moved in three principal ways: namely, first, by descending vertically with a natural motion; then horizontally, or obliquely, when it is driven from right to left, or conversely from left to right, or straight onward toward the front; finally, in a circle, by describing the arc of a circle. And indeed a body descending of its own accord, whatever it may be, has its own line of Direction, along which the Center of gravity advances in motion. Therefore, in the lowest part of the body, the very line of Direction defines the point at which, if contact is made by the striking body, that will be the strongest blow which such a body descending from a given height can inflict; for there it finds the greatest resistance, when equal

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Mechanicorum 706 vires hinc atque hinc consistentes ibi conspirent, & obicem motui directè oppositum offendant, adeò ut neque in hanc, neque in illam partem Centrum gravitatis dirigatur. Quod si punctum contactûs corporum collisorum non sit in lineâ Di- rectionis corporis cadentis, sed à latere; eò validior erit ictus, quo minore intervallo punctum contactûs ab hujusmodi lineâ Directionis aberit; magis videlicet opponitur motui directo, quàm si ab eâ longiùs abesset: quando enim contactus procul est à lineâ Directionis, ab hac minùs deflectere cogitur cen- trum gravitatis, quod multò magis repellendum esset à contactu propiore. Sic globus, cujus centrum gra- vitatis sit C, descendens per lineam Di- rectionis CD, si percutiat puncto D, om- nium validissimum ictum infligit, quia corpus percussum omninò opponitur mo- tui CD, nec centro C relinquit locum saltem obliquè descendendi: at verò si contactus fiat in E, impeditur quidem descensus globi per rectam CD ulteriùs productam, potest tamen centrum gravitatis descendere descri- bendo circa punctum E manens arcum CF; quapropter in E minorem invenit resistentiam quàm in D, ubi nihil descende- re potest, si subjectum corpus loco non cedat. Similiter si con- tactus fiat in G, adhuc impeditur motus directus per CD, atta- men centrum gravitatis C potest obliquè descendere descri- bendo arcum CH. Sed quoniam per arcum CF magis declinat à perpendiculo, & minùs descendit, quàm per arcum CH (quamvis arcus illi æquales ponantur paribus Radiis EC, & GC descripti) propterea magis impeditur motus in contactu E propiori lineæ Directionis, quàm in G remotiori. Cum itaque eò validiorem ictum infligant corpora percutientia, quò ma- jorem inchoato motui resistentiam offendunt, manifestum est in corporibus naturali motu descendentibus validissimum esse ictum in puncto, quod lineæ directionis motûs respondet, sem- pèrque imbecilliores esse ictus, quò magis puncta contactûs ab- sunt à lineâ Directionis. Hoc idem, quod de lineâ Directionis gravium sponte suâ descendentium dictum est, analogiâ servatâ, traducendum est ad

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Mechanicorum 706 the forces standing here and there may conspire together, and encounter the obstacle directly opposed to motion, so that the center of gravity is directed neither toward this side nor toward that. And if the point of contact of colliding bodies is not in the line of direction of the falling body, but to the side, the blow will be the stronger the smaller the interval by which the point of contact is distant from such line of direction; that is to say, it is opposed to direct motion more than if it were farther away from it. For when the contact is far from the line of direction, the center of gravity is compelled to deviate less from that line, whereas it ought to be repelled much more by a nearer contact. Thus a ball, whose center of gravity is C, descending along the line of direction CD, if it strike at point D, inflicts the strongest possible blow, because the body struck is wholly opposed to the motion CD, and leaves center C no place even for an oblique descent; but if the contact is made at E, the descent of the ball along the straight line CD produced farther is indeed hindered, yet the center of gravity can still descend by describing, while remaining about point E, the arc CF; wherefore at E it finds less resistance than at D, where nothing can descend if the body beneath does not yield its place. Similarly, if the contact is made at G, direct motion along CD is still impeded; nevertheless the center of gravity C can descend obliquely by describing the arc CH. But since by the arc CF it departs more from the perpendicular, and descends less, than by the arc CH (although those arcs are assumed equal, described by equal radii EC and GC), for that reason motion is more hindered in the contact E, closer to the line of direction, than in G, farther away. Since, therefore, striking bodies inflict the stronger blow the greater resistance to the motion begun they encounter, it is manifest that in bodies descending by natural motion the strongest blow is at the point which corresponds to the line of direction of the motion, and that blows are always weaker the farther the points of contact are from the line of direction. The same thing, which has here been said of the line of direction of heavy bodies descending of their own accord, must, with analogy preserved, be transferred to

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Liber septimus. CAPUT IX. 707 ad ea corpora, quæ externo impulsu agitata motu recto sivè Horizonti parallelo, sivè ad Horizontem aut obliquè, aut ad perpendiculum, inclinato moventur. Cum enim, ex hypo- thesi, partes omnes hujusmodi corporis impulsi æquali veloci- tate per æqualia spatia moveantur, æqualem impetum singulæ recipiunt, à movente impressum. Similiter igitur in corpore illo concipiendum est punctum, quod Centrum Impetûs vocari potest, quia illud æquales hinc & hinc Impetus circumstant, quemadmodum Centrum Gravitatis dicitur, circa quod æqua- lia gravitatis momenta disposita intelliguntur. Hinc si corpo- ris particulæ fuerint omnino homogeneæ, adeóque æquè capa- ces impetûs recipiendi, illud idem erit Centrum Impetûs, quod est centrum molis, seu magnitudinis; nam eadem plana, quæ molem æqualiter dividunt, etiam æqualiter dividunt Impetum per singulas particulas æquabiliter diffusum. At si non ejusdem generis fuerint partes corpus illud componentes, sed raræ aliæ, aliæ densæ, hoc est ex materiâ partim tenui, partim constipatâ, sicut non esset idem Centrum Gravitatis, atque Centrum Magnitudinis, ita neque idem est cum Molis centro Centrum Impetûs impressi; quia, ut ex Projectis constat, ea quæ secun- dùm speciem leviora sunt, cæteris paribus, minorem impetum concipiunt (& globuli ex argillâ efficti, quos balistæ evibrant, majorem ictum infligunt, quàm pares globuli lignei, qui sunt argillâ leviores) ac proinde Centrum Impetûs impressi assumi potest idem, ac punctum illud, quod in motu naturali esset Centrum gravitatis ipsi corpori inexistens. Quare in motibus corporum externâ vi impulsorum atten- denda est pariter linea, secundùm quàm dirigitur motus hujus- modi Centri Impetûs: & punctum illud in corporis percutien- tis superficie, quod linea directionis motûs à Centro Impetûs ducta designat, ipsum est, in quo corpus percutiens vim suam validissimè exercet. Cum enim omnia plana per hanc Di- rectionis motûs lineam transeuntia (quorum illa est communis sectio) dividant universum Impetum in partes hinc & hinc æquales, quippe quæ etiam per Centrum Impetûs transeunt, ita ex percussione in puncto illo impeditur motus, ut neque ad hanc, neque ad illam partem deflectere possit corpus impactum in obicem, qui resistit. Quod si punctum contactûs fuerit ex- VVuu 2

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Book Seven. Chapter IX. 707 to those bodies which, set in motion by an external impulse, move in a straight line, whether parallel to the horizon, or inclined to the horizon, either obliquely or at a right angle. For since, by hypothesis, all the parts of such a body that has been impelled move with equal velocity through equal spaces, each receives an equal impulse impressed by the mover. In like manner therefore in that body there must be conceived a point, which may be called the Center of Impetus, because equal impulses on this side and on that surround it, just as the Center of Gravity is said to be that around which equal moments of gravity are understood to be disposed. Hence, if the particles of the body were entirely homogeneous, and therefore equally capable of receiving impulse, that same point would be the Center of Impetus which is the center of mass, or of magnitude; for the same planes which divide the mass equally also divide equally the Impetus diffused uniformly through the individual particles. But if the parts composing that body are not of the same kind, but some rare, others dense, that is, made partly of subtle matter and partly of compacted matter, then just as the Center of Gravity would not be the same as the Center of Magnitude, so neither is the Center of the impressed Impetus the same as the center of mass; because, as is clear from projectiles, things that are lighter by nature, other things being equal, receive a smaller impulse (and balls made of clay, which are discharged by catapults, inflict a greater blow than equal wooden balls, which are lighter than clay), and therefore the Center of the impressed Impetus may be taken to be the same as that point which in natural motion would be the Center of gravity inherent in the body itself. Wherefore, in the motions of bodies propelled by external force, there must likewise be considered the line along which the motion of this Center of Impetus is directed: and that point on the surface of the striking body which is indicated by the line of direction of the motion drawn from the Center of Impetus is the very point at which the striking body most powerfully exerts its force. For since all planes passing through this line of direction of motion (of which it is the common section) divide the whole Impetus into equal parts on this side and on that, since they also pass through the Center of Impetus, thus by the blow at that point the motion is so hindered that the body struck against the obstacle cannot deflect either to this side or to that. But if the point of contact has been ex- VVuu 2

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708 Mechanicorum tra lineam Directionis motûs, inæquales sunt impetus, & majo- re præpollente, corpus pergit in motu, quamvis ad latus in- flectatur sivè magis, sivè minùs, prout majus aut minus fuerit intervallum inter punctum contactûs, & lineam Directionis motûs. Sit corpus A B, quod translatum à potentiâ impellente ha- beat Centrum Impetûs C, & linea, per quam dirigitur motus, sit C D, cui parallelæ sunt lineæ à singulis par- tibus in motu descriptæ. Si ergo in obicem incur- rat punctum D, ita im- peditur motus, ut ulte- riùs promoveri nequeat corpus, nisi obex loco cedat; quia nimirum impetus in D A æqualis est impetui in D B, ideò neutra pars æquali impetu af- fecta promoveri potest: est igitur maxima resistentia, & ictus validissimus. Sin autem non puncto D, sed puncto E fiat per- cussio secundùm eandem directionem G E, jam impetus sunt inæquales, & minor impetus est in E A, quàm in E B; proin- de pars E B validior pergens in motu inflectitur circa obicem in puncto E, tanquam circa centrum, & resistentia est minor, quàm ad punctum D. Simile quid contingit, si fiat percussio in puncto F, multo enim major impetuum inæqualitas interce- dit inter F A, & F B, quàm inter E A, & E B, atque faciliùs sit conversio & inflexio motûs circa obicem in puncto F, quàm in puncto E: arcus siquidem majore Radio F D descriptus mi- nùs desescit à rectitudine lineæ, per quam dirigitur motus, quàm arcus minore Radio E D descriptus. Quò igitur magis punctum contactûs in percussione abest à puncto D, eò infir- mior est ictus, minorem quippe invenit resistentiam. At si corpus idem A B ita impellatur, ut linea directionis motûs ducta ex C centro impetûs sit C A, similiter constat va- lidissimum ictum fieri in A, imbecilliorem verò in extremis an- gulis ejusdem superficiei. Hinc vides, cur ex vetere disciplinâ Poliorceticâ ad murorum, aut postium expugnationem, arie- tes, quibus concutiebantur, non planâ facie, sed convexâ com- muniter, aut acutâ construerentur: quia scilicet trabem ferro in

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708 Mechanicorum when, with the line of direction of the motion, the impulses are unequal, and the greater prevailing, the body continues in motion, although it is deflected to the side, either more or less, according as the interval between the point of contact and the line of direction of the motion has been greater or less. Let there be a body A B, which, being moved by the impelling force, has a center of impulse C, and let the line along which the motion is directed be C D, to which are parallel the lines described by the several parts in motion. If therefore it strikes an obstacle at point D, the motion is so checked that the body cannot be carried farther unless the obstacle yields its place; because, namely, the impulse in D A is equal to the impulse in D B, therefore neither part, being affected by an equal impulse, can be advanced: there is thus the greatest resistance, and the strongest blow. But if the blow is made not at point D, but at point E, along the same direction G E, the impulses are now unequal, and the lesser impulse is in E A than in E B; therefore the stronger part E B, continuing in motion, is bent around the obstacle at point E as around a center, and the resistance is less than at point D. Something similar happens if the blow is made at point F, for then there is a much greater inequality of impulses between F A and F B than between E A and E B, and the conversion and bending of the motion around the obstacle at point F is easier than at point E: indeed, the arc described with the greater radius F D departs less from the straightness of the line along which the motion is directed than the arc described with the smaller radius E D. Therefore, the farther the point of contact in the blow is from point D, the weaker is the blow, since it meets less resistance. But if the same body A B is impelled in such a way that the line of direction of the motion drawn from the center of impulse C is C A, it is likewise clear that the strongest blow occurs at A, but a weaker one at the extreme angles of the same surface. Hence you see why, according to the old discipline of Poliorcetica, for the breaching of walls or gates, the rams with which they were battered were commonly constructed not with a flat face, but with a convex one, or else with a pointed one: namely because the beam in iron

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Liber septimus. CAPUT IX. 709 In capite armatam funibus suspensam (ne sustinendi laborem subirent, sed vires omnes in motu impenderent) retro ducentes, ac deinde propellentes, non planè horizontaliter, sed quasi circulariter movebant; planum autem si fuisset trabis caput, ictus infictus fuisset ab extremo illius superficiei latere, non verò à partibus circa medium existentibus, à quibus multò validior ictus expectari potuisset; quemadmodum certiùs contingit facie convexâ, aut in apicem desinente. Si demum linea directionis motûs esset CF, utique in F esset validissimus ictus, quia planum FH bifariam divideret æqualiter universum impetum, & impetus FAH æqualis esset impetui FBH. Esset autem infirmior ictus, quem infligeret punctum D, cujus directio DI parallela directioni Centri FH inæqualiter divideret impetum, & pars impetûs DBI minor esset parte DAHI; quapropter hæc circa obicem in D moveri posset, & minorem inveniret resistentiam quàm in F. Ubi observandum est non aptè quæri, quonam in puncto validissimus fiat ictus, nisi pariter statuatur, quænam sit linea Directionis motûs: Nam in eodem puncto D validissimus est ictus, si directio fuerit CD, quia tunc est maxima resistentia; nullus est ictus in directione CA, quia nihil illi opponitur; imbecillis est ictus in Directione CF, quia mediocrem offendit resistentiam. Præterea comparatis invicem Directionibus CD & CA, validior est ictus in A quàm in D; plures siquidem partes in eandem longitudinis lineam directè conspirantes plus obtinent virium, quàm pauciores in lineâ latitudinis: præterquam quod partium ad latera adjacentium lineæ, quæ propiores sunt lineæ Directionis Centri Impetûs, quasi in unam Physicè coalescunt; id quod non contingit partibus notabili intervallo disjunctis ab illâ Directionis lineâ: quæ eatenus solùm in percussionem consentiunt, quatenus cum intermediis conjunctæ nexu non facilè dissolubili eas pariter juvant; nam si esset corpus percutiens in plures partes, ceu virgulas, dissectum, iis, quæ obicem contingerent, manentibus, reliquæ sine ictu excurrerent: propterea etiam conjunctæ faciliùs à directâ positione deflectentes corporis longitudinem inspectunt: ut in tenui & gracili ligno accedere potest extremis partibus A & B, quæ ex impulsu, quo promoventur, possunt circa punctum D inflecti; ex quo infirmior VVuu 3

Transcription: Translated (English)

Book Seventh. CHAPTER IX. 709 At the head, the weapon being suspended by ropes so that they would not undergo the labor of supporting it, but would expend all their strength in the motion, they drew it back and then drove it forward, moving it not quite horizontally, but as it were circularly; but if the head of the beam had been flat, the blow would have been delivered from the extreme side of that surface, and not from the parts lying about the middle, from which a much stronger blow could have been expected; as more certainly happens with a convex face, or one ending in a point. If at last the line of the direction of motion were CF, the strongest blow would certainly be at F, because the plane FH would equally divide the whole impetus into two parts, and the impulse FAH would be equal to the impulse FBH. But the blow struck by point D would be weaker, since its direction DI, parallel to the direction of the center FH, would divide the impetus unequally, and the part of the impetus DBI would be smaller than the part DAHI; wherefore this could move about the obstacle at D, and would find less resistance than at F. Here it must be observed that it is not properly asked at what point the strongest blow occurs, unless at the same time it is also determined what the line of the direction of motion is: for at the same point D the blow is strongest if the direction be CD, because then the resistance is greatest; there is no blow in the direction CA, because nothing is opposed to it; the blow is weak in the direction CF, because it encounters only moderate resistance. Moreover, comparing the directions CD and CA with one another, the blow at A is stronger than at D; for more parts directly converging on the same line of length have greater force than fewer in the line of breadth: besides, the parts adjacent to the sides of the line, which are nearer to the line of the direction of the impetus’s center, almost coalesce physically into one; which does not happen to parts separated from that line of direction by a notable interval: these consent to the percussion only insofar as, being joined with the intermediate parts by a bond not easily dissolved, they likewise assist them; for if the striking body were divided into many parts, as into little rods, then, the parts that touched the obstacle remaining, the rest would run on without a blow. For this reason also, when joined together, they more easily depart from a direct position, regarding the length of the body: as in a thin and slender piece of wood, the extreme parts A and B can be brought together, which, from the impulse by which they are advanced, can be bent about point D; from which a weaker

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Mechanicorum 710 percussio, quàm cùm tanta est corporis crassities, ut pro longi- tudine breviore non valeat flecti. At si cum his directionibus CD & CA, ad perpendiculum incidentibus in faciem corpo- ris percutientis, comparetur Directionis linea CF obliquè in- cidens, licèt longior sit linea CF, quàm CD, non idcirco va- lidior est in F ictus Directionis CF, quàm in D ictus Directionis CD: id quod oritur ex minore resistentiâ ratione obliqui- tatis; faciliùs quippe potest ulteriùs excurrere corpus obliquè percutiens, quàm si directè percuteret. Porrò non levis error obreperet minùs accuratè perpenden- tibus ea, quæ hactenus de Centro Impetûs disputata sunt, si hoc Centrum absolutè in eo instrumento, quo ad percutiendum utimur, quærendum esse existimarent: non enim rarò etiam ejus, à quo instrumentum impellitur, considerandus est impe- tus & motus. Sic quando duo lanceis concurrunt, non est æsti- manda percussio ex solo impetu lanceæ impresso, verùm etiam ex eo, quem militis corpori imprimit equus, cui currenti insi- det, immò & ipsius equi impetus, quem virtute suâ animali concipit: universum quippe hunc impetum retundi oportet ab eo, qui ictum recipit: hinc si miles minùs robustus fuerit, infirmior est ictus, quia ipso ictûs momento ille cedit, & perin- de est, atque si lancea ipsa cederet, aut flecteretur. Non est igi- tur Centrum impetûs in lanceâ ipsâ, sed potiùs in corpore mili- tis non procul ab equo; ac proinde inclinata lancea, ut, quam minùmùm fieri possit, recedat à positione parallelâ lineæ Di- rectionis motûs, & ab hac lineâ non longè absit, validissimum ictum infliget: hoc autem quia faciliùs obtinetur longiore lan- ceâ, quàm breviore, ideò, cæteris paribus, præstat longiore lanceâ uti. Res autem aliter se habet, quando percussio contingit instru- mento non ampliùs cohærente ipsi causæ, à qua impetum re- cipit, sed jam ab eâ disjuncto; in eo enim præcisè est Cen- trum Impetûs, & attendenda est linea Directionis motûs ab hujusmodi centro ducta, ut vis percussionis maxima innotescat. At hîc quæris; si hastam manu stringentes impetum illi im- primimus, & brachium pariter impetum concipit, atque ex utroque impetu æstimandus est ictus; cur validiùs hastam ean- dem intorquemus jaculantes, quàm manu tenentes? Sic anti- quis,

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Mechanics 710 a blow, rather than when the thickness of the body is so great that it cannot be bent by a shorter length. But if, with these directions CD and CA falling perpendicularly on the face of the striking body, the line of Direction CF, falling obliquely, is compared, although the line CF is longer than CD, the blow delivered at F by the Direction CF is not for that reason more powerful than the blow delivered at D by the Direction CD: this arises from the lesser resistance due to obliquity; for a body striking obliquely can more easily travel farther than if it struck directly. Moreover, a not insignificant error would creep in among those who consider less accurately the matters hitherto discussed concerning the Center of Impetus, if they were to think that this Center must be sought absolutely in that instrument which we use for striking: for not infrequently the impetus and motion of the one by whom the instrument is propelled must also be taken into account. Thus, when two men with lances charge one another, the blow is not to be estimated from the impulse impressed on the lance alone, but also from that which the horse, upon which the soldier rides while moving, impresses upon his body, and even from the impulse of the horse itself, which it conceives by its own animal power: for this whole impulse must be resisted by the one who receives the blow. Hence, if the soldier is less robust, the blow is weaker, because at the very moment of the blow he yields, and it is the same as if the lance itself were yielding or bending. Therefore the Center of impetus is not in the lance itself, but rather in the body of the soldier not far from the horse; and accordingly, a lance held inclined, so that it departs as little as possible from the position parallel to the line of the Direction of motion, and does not lie far from that line, will inflict the strongest blow: and since this is more easily achieved with a longer lance than with a shorter one, therefore, other things being equal, it is preferable to use a longer lance. But the matter is different when the blow occurs with an instrument no longer joined to the very cause from which it receives the impetus, but already separated from it; in that case the Center of Impetus is precisely there, and the line of Direction of motion drawn from such a center is to be considered, so that the greatest force of the blow may be known. But here you ask: if, gripping the spear by hand, we impart impetus to it, and the arm likewise conceives impetus, and the blow is to be estimated from both impulses; why do we hurl the same spear more forcefully when throwing it than when holding it in the hand? Thus to the ancients,

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Liber septimus. CAPUT IX. 711 quis, ad acriùs feriendum, placuit hastis amentatis uti, ut post ejaculationem hastam loris ligatam retraherent, iterumque evibrarent: Esse autem validiorem ictum hinc cognosces, quòd hastæ evibratæ muero altiùs infigitur objectæ tabulæ, quàm cùm illam manu retinentes similiter tabulam cuspide percutimus. Ex multiplici causâ id petendum videtur. Et primò quidem, quia cùm hastam manu stringimus, caro, quæ est in volâ manûs, illicò cedit, ac obicem hastæ resistentem offendit, ex qua cessione minuitur impetûs; qui & multo magis debilitatur, si brachium pariter in posteriora modicè revocemus timentes, ne ex præconcepto impetu, & corporis percussi resistentiâ oriatur nimia aliqua partium convulsio, aut dolor: hoc autem incommodum vitatur in hastâ jam emissâ. Deinde quando aliquid jaculamur, ultimo momento, quo illud tenemus, brachium validissimo conatu in anteriora movemus, statimque retrahimus dimittentes missile, cui propterea plurimus impetus imprimitur: constat autem non posse à nobis hastam retinentibus (alias scilicet est musculorum contentio & motio) moveri brachium motu adeò concitato. Demum impetus brevissimo illo motu tantâ vi productus in missili suam retinet directionem (quicquid sit, an gravitas insita aliquid officiat) quæ in longiore motu brachij si non dimittatur, aliquantulum labefactatur, eo quod plures motus circa diversa centra, videlicet circa os humeri, & os cubiti, misceantur; atque ex diversa illâ directione vis impetûs minuitur. Cùm itaque ex omnibus hisce causis major inveniatur impetus in hastâ evibratâ, quo momento illa percutit, majorem quoquè ictum ab eâ infligi consequens est. Ad hoc percussionum horizontalium genus spectat illa percussio, qua in ludo minoris tudiculæ globus unus tudiculâ impellente emissus alium globulum percutit. Si enim in eâdem directionis motûs lineâ reperiantur centra utriusque globi, percutientis scilicet & percussi, maximus ictus infligitur, quia maximam invenit resistentiam, cum totus globulus percussus toti percutienti opponatur, cujus singularum partium lineæ directionis motûs si producantur, occurrunt globulo percussio (aequales sunt globuli ex hypothesi) quamvis sola linea directionis Centri illum contingat. Sin autem globus emissus ita alium quiescem

Transcription: Translated (English)

Book Seven. Chapter IX. 711 who, for striking more sharply, thought it best to use spears fitted with thongs, so that after hurling them they might draw the spear back by the leather straps and hurl it again: and that the blow is stronger by this means you will recognize from the fact that a spear, when thus thrown, drives its point deeper into the target board than when, holding it in the hand, we strike the board in the same way with the point. This seems to be demanded for several reasons. First indeed, because when we grasp the spear with the hand, the flesh in the palm immediately gives way and strikes against the obstacle resisting the spear; by this yielding the impetus is diminished. And it is weakened all the more if we also draw the arm slightly backward, fearing that from the previously formed impetus and the resistance of the body struck some excessive convulsion of the parts, or pain, may arise: but this inconvenience is avoided in the spear once it has been let fly. Then, when we throw something, at the last moment, while still holding it, we move the arm forward with the strongest effort, and then at once, releasing the missile, we draw back; wherefore very great impetus is imparted to it. But it is clear that, when we are holding back the spear, the arm cannot be moved with such a rapid motion (for otherwise there is a straining and movement of the muscles). Finally, the impetus, produced by that very brief motion with such force, preserves in the missile its direction, whatever the case may be, whether innate gravity impedes it somewhat; but in a longer motion of the arm, if it is not released, it is somewhat impaired, because several motions around different centers, namely around the bone of the shoulder and the bone of the elbow, are combined; and from that different direction the force of the impetus is diminished. Since, therefore, from all these causes greater impetus is found in the spear when hurled, it follows that, at the moment when it strikes, a greater blow is also inflicted by it. To this class of horizontal blows belongs that strike by which, in the game of the smaller cudgel, one ball, sent forth by the striking cudgel, strikes another ball. For if the centers of both balls, the striking one and the struck one, are found on the same line of direction of motion, the greatest blow is inflicted, because it encounters the greatest resistance, since the whole struck ball opposes the whole striking ball; and if the lines of direction of motion of each of its individual parts are extended, they meet the struck ball (the balls being equal by hypothesis), although only the line of direction of the center touches it. But if the ball sent forth so strikes another resting one

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Mechanicorum 712 quiescentem tangat, ut recta linea per contactûs punctum ducta sit utriusque globi Tangens; & lineæ directionis motûs parallela, nullus est ictus, quia nullum motui impedimentum infertur. Demum si linea, per quam dirigitur centrum globuli emissi, non occurrat centro globi percussi, ita tamen se habeat, ut lineæ utrumque globulum Tangenti occurrat extra punctum contactûs, tunc major aut minor erit ictus pro ratione impedimenti & resistentiæ, prout majori aut minori parti globuli percutientis opponitur globulus percussus: Ex quo fit eò majorem esse resistentiam, quò linea directionis motûs Centri percutientis propriùs ad punctum contactûs occurrit lineæ Tangenti. Sit globus B emissus adversùs globum A quiescentem, & linea, per quam dirigitur motus centri B, sit B C occurrens puncto contactûs C, atque adeò, ut colligitur ex 12. lib. 3. etiam centro A, si producta intelligatur. Hìc invenit maximam resistentiam globus percutiens, neque ad hanc, neque ad illam partem deflectere potest à priori directione; linea siquidem Directionis BC ad angulos rectos incidit in Tangentem DE, & omnes singularum partium directiones occurrerent globulo A, si productæ intelligentur extra globulum B. Quòd si linea directionis motûs fuisset B F parallela Tangenti DE, nullum planè inferretur impedimentum motui à globo quiescente A; nam partium globi impulsì Directiones parallelæ lineæ B F, ex essent, ut earum nulla incurreret in globum A, ideóque nullus esset ictus, ubi nulla est resistentia. At si globus B habeat Directionem B E, aut B G, & quiescentem globum tangeret in C, etiamsi neque B E, neque B G directiones centri incurrerent in globum A, tamen partium aliquarum ejusdem globi B Directiones parallelæ Directioni B E, aut B G, si productæ intelligentur, incurrerent in globum A, atque invenirent ex eo impedimentum. Sit enim Directio B E, & in globo B linea L O ipsi B E paralla, quæ producta contingeret in K globum A: utique omnes partes segmenti

Transcription: Translated (English)

Mechanicorum 712 touching the resting sphere, if a straight line drawn through the point of contact is the tangent of both spheres; and parallel to the direction of motion, there is no impact, because no impediment is offered to the motion. Finally, if the line along which the center of the emitted sphere is directed does not meet the center of the struck sphere, yet is so situated that the line tangent to both spheres meets beyond the point of contact, then the impact will be greater or lesser according to the ratio of the impediment and resistance, as the struck sphere is opposed to a greater or lesser part of the striking sphere: from which it follows that the resistance is the greater, the more closely the line of direction of the motion of the striking center approaches the point of contact of the tangent line. Let sphere B be sent against the resting sphere A, and let the line along which the motion of the center of B is directed be B C, meeting the point of contact C, and thus, as is gathered from 12. lib. 3. also the center of A, if it be understood as extended. Here the striking sphere encounters the greatest resistance, and it can deflect neither to this nor to that side from its prior direction; for the line of direction BC falls at right angles upon the tangent DE, and all the directions of the individual parts would meet sphere A, if they were understood as extended beyond sphere B. But if the line of direction of the motion had been B F parallel to the tangent DE, there would be absolutely no impediment introduced to the motion by the resting sphere A; for the directions of the parts of the impelled sphere, parallel to the line B F, would be such that none of them would strike into sphere A, and therefore there would be no impact, where there is no resistance. But if sphere B have direction B E, or B G, and should touch the resting sphere at C, even if neither the direction B E nor B G of the center should meet sphere A, nevertheless the directions of some parts of the same sphere B, parallel to the direction B E or B G, if understood as extended, would meet sphere A, and would find impediment from it. For let the direction be B E, and in sphere B let the line L O parallel to B E, which when extended would touch sphere A at K: indeed all the parts of the segment

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Liber septimus. CAPUT IX. 713 menti OLS habentes Directionem parallelam Directioni BE, quæ est directio Centri, inveniunt resistentiam, cum earum di- rectiones incurrant in oppositum globum A. Similiter si Di- rectio Centri sit BG, parallela Directio HI producta tangit in P globum A, qui proinde opponitur Directionibus omnium partium segmenti IHS. Cùm autem segmentum IHS majus sit segmento OLS, etiam major est ictus, quando Directio BG ea est, ut Tangenti lineæ DE occurrat in puncto G non ita procul à contactu C, quàm si directio BE ea esset, quæ in puncto E remotiore à contactu C occurreret eidem Tangenti DE: Maximam siquidem habet veritatis speciem, in hujusmo- di ictibus impetum in globo percusso eâ intensione imprimi, quæ proportione respondeat partibus, quæ directè secundùm illam Directionis lineam impediuntur. Quoniam verò impe- tus globo percusso impressus illum afficit æquabiliter, hinc est, quod ille non potest à percussione determinari, nisi ut moveatur per lineam Directionis, quæ conjungat punctum contactûs C cum centro A: quandoquidem æquales sunt globi partes circa centrum, adeóque & æquales impetus. Observa hîc à me assumptos esse circulos pro globis, & lineas vice planorum secantium globos, ut res faciliùs explicaretur: cæterùm quæ de lineis dicta sunt, si de planis per lineas illas transeuntibus intelligentur, rem similiter ob oculos ponent, si ipsa parallela fuerint, aut inclinata, prout de lineis constituta est hypothesis. Superest tertius percutientis motus, videlicet in arcûs circularis speciem ductus, quando, altera extremitate manente, corpus in gyrum movetur. Experimentis autem do- cemur validissimum ictum non semper fieri ab extremitate, quamvis hæc velocissimè moveatur præ cæteris punctis alte- ri extremitati manenti propioribus. Ut igitur inveniatur punctum, in quo corpus percutiens maximam habeat re- sistentiam, ponendum est illud esse æquabiliter ductum, & ex materiâ homogeneâ æqualiter capaci impetûs; atque ibi sanè maxima erit resistentia, ubi impetûs momenta æqualiter dividuntur: id quod contingit in puncto ita re- moto à motûs centro, ut cadat inter bessem, & dodrantem totius longitudinis, quæ habet rationem Radij, quo arcus describitur. XXX

Transcription: Translated (English)

Book Seven. Chapter IX. 713 having the direction OLS parallel to the direction BE, which is the direction of the center, find resistance when their directions run against the opposite globe A. Likewise, if the direction of the center is BG, the parallel direction HI, extended, touches the globe A at P, and therefore opposes the directions of all the parts of the segment IHS. But since the segment IHS is greater than the segment OLS, the blow is also greater when the direction BG is such that it meets the tangent line DE at the point G, not so far from the contact C, than if the direction BE were such as to meet the same tangent DE at the more distant point E: for it very much has the appearance of truth that, in blows of this kind, the impulse is impressed on the struck globe with that intensity which answers in proportion to the parts that are directly impeded along that line of direction. But since the impulse impressed on the struck globe affects it uniformly, it follows that it cannot be determined by the blow except insofar as it is to move along the line of direction joining the point of contact C with the center A: since the parts of the globe around the center are equal, and therefore the impulses are also equal. Observe here that I have assumed circles for globes, and lines in place of planes cutting the globes, so that the matter might be explained more easily; otherwise, what has been said of lines, if understood of planes passing through those lines, will likewise place the matter before the eyes, if they are parallel or inclined, according as the hypothesis has been set forth regarding the lines. There remains the third motion of the striking body, namely, one drawn in the form of a circular arc, when, one extremity remaining fixed, the body is moved in a circle. But by experiments we are taught that the strongest blow is not always delivered by the extremity, although this moves most swiftly of all the points nearest the other extremity that remains fixed. Therefore, in order to find the point in which the striking body has the greatest resistance, it must be taken to be moved uniformly, and to be of homogeneous matter equally capable of impulse; and assuredly there will be the greatest resistance where the moments of the impulse are equally divided: which occurs at a point so remote from the center of motion that it falls between the two-thirds and three-quarters of the whole length, which has the ratio of the radius by which the arc is described. XXX

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Mechanicorum 714 Sit corporis percutientis longitudo AB. Si motu naturali sponte suâ descenderet, & in motu positionem horizonti parallelam servaret, utique validissimus esset ictus in D puncto, quod respondet centro gravitatis C; essent enim hinc atque hinc æquales gravitates, & æqualia impetus momenta, ut superiùs dictum est. At manente extremitate A, tanquam centro motûs, & corpore ipso vi suæ gravitatis descendente, licèt singulæ particulæ, utpote naturæ ejusdem, paribus viribus sint præditæ, non tamen æquali momento feruntur; sed cum in A retineantur, quæ puncto A propiores sunt, magis detorquentur à directione naturalis gravitatis, adeoque plus momenti habent partes inter DB, quàm inter AD constitutæ. Porrò momenta sunt in Ratione Distantiarum: Momentum siquidem est Excessus virtutis moventis supra resistentiam, qua impedimentum prohibet, ne sequatur motus juxta naturalem propensionem: quare singularum partium momenta ex earum motu dignoscuntur: moventur autem per circulorum arcus similes, quorum etiam similes sunt Sinus descensum metientes, qui sunt in Ratione Radiorum, hoc est distantiarum ab A communi centro. Sic momentum puncti D est ut AD, puncti G ut AG, puncti H ut AH, atque ita de cæteris. Hinc est omnium momentorum summam constari ex illorum aggregato, quasi ex aggregato arcuum quos describunt, aut Sinuum arcubus similibus respondentium, quorum Ratio eadem est cum aggregato Radiorum, ex quibus describuntur arcus. Cum autem universa longitudo AB in particulas æquales divisa intelligatur, manifestum est distantias à centro A constituere Progressionem Arithmeticam juxta seriem naturalem numerorum, ac proinde punctum, quod vocari potest Centrum Momentorum Impetus, illud esse, in quo momenta illa bifariam æqualiter dividuntur. Hoc verò punctum esse ultra bessem totius longitudinis hinc apparet, quòd, si longitudo AB in tres æquales partes distincta intelligatur, prima centro A proxima habet momentum ut 1, secunda ut 2, tertia ut 3: igitur post finem secundæ, hoc est in G, videtur

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Mechanics 714 Let AB be the length of the body that strikes. If, by natural motion, it were to descend of its own accord, and in motion were to preserve a position parallel to the horizon, certainly the blow would be strongest at point D, which corresponds to the center of gravity C; for on either side there would be equal weights and equal impulses of force, as was said above. But with the extremity A remaining fixed, as the center of motion, and the body itself descending by the force of its own weight, although the individual parts, being of the same nature, are endowed with equal forces, nevertheless they are not carried with equal moment; but since they are retained at A, those parts which are nearer point A are more deflected from the direction of natural gravity, and therefore the parts situated between DB have greater moment than those situated between AD. Moreover, moments are in the ratio of distances: for a moment is the excess of the moving force over the resistance by which the obstacle prevents motion from following its natural tendency. Therefore the moments of the individual parts are recognized from their motion; but they are moved through arcs of similar circles, the sines measuring the descent of which are also similar, and are in the ratio of the radii, that is, of the distances from A as a common center. Thus the moment of point D is as AD, of point G as AG, of point H as AH, and so on for the rest. Hence it follows that the sum of all the moments is made up of their aggregate, as it were from the aggregate of the arcs which they describe, or of the sines corresponding to similar arcs, whose ratio is the same as the aggregate of the radii from which the arcs are described. But when the whole length AB is understood to be divided into equal parts, it is manifest that the distances from center A form an arithmetic progression according to the natural series of numbers, and therefore the point which may be called the center of moments of impulse is that in which those moments are equally divided in two. Now that this point is beyond the midpoint of the whole length is shown as follows: if the length AB is understood to be divided into three equal parts, the first, nearest center A, has a moment as 1, the second as 2, the third as 3; therefore after the end of the second, that is, at G, it appears

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Liber septimus. CAPUT IX. 715 videtur esse æqualitas momentorum; nam A G habet ut 3, & GB item ut 3. Sed non esse in G momentorum æqualitatem constat, si adhuc plures in partes A B distincta intelligatur, & eadem sit Ratio dupla A G ad GB. Sit A B partium 6; A G est 4, GB 2: Momenta A G sunt 10, GB 11; sunt scilicet ipsius A G momenta quatuor distantiarum 1. 2. 3. 4. hoc est 10, GB verò momenta quintæ & sextæ distantiæ 5 & 6, hoc est 11. Quod si ponatur A B partium 9, & A G 6, GB 3; momenta A G sunt 21, GB sunt 24. Similiter statuamus longitudinem A B partium 12, scilicet A G 8, GB 4, momenta A G sunt 36, GB 42. Item A B sit partium 15; A G 10, GB 5: momenta A G sunt 55, GB 65. Demum A B sit partium 18; A G 12, GB 6: momenta A G sunt 78, GB 93. Non igitur momen- torum æqualitas est præcisè in G ad bessem longitudinis A B, sed est ultra G versus B. Verùm, si A H sit dodrans longitudinis A B, non in H, sed citra H, inter G & H est quæsitum punctum, in quo momen- torum æqualitas invenitur. Nam si A B sit partium 4, atque A H sit 3, HB verò sit 1; momenta A H sunt 6, & HB 4: Si A B sit part. 8: & A H 6, HB 2; momenta A H sunt 21, HB 15; & sic de cæteris servatâ eâdem Ratione triplâ A H ad HB. Cum itaque momenta A G minora sint quàm momen- ta GB, contrà verò momenta A H majora sint momentis HB, constat æqualitatem momentorum esse inter G & H, hoc est inter bessem & dodrantem. Ubi autem proximè sit hujusmodi punctum, deprehendes, si totam A B statuas partium 576, & A I partium 407: Cum enim momenta omnia totius A B sint 166176, & momenta A I sint 83028, remanent momen- ta I B 83148, proximè æqualia momentis A I. Est autem Ra- tio 407 ad 576 minor Ratione 3 ad 4: & Ratio A I ad I B mi- nor est Ratione A H ad HB, hoc est minor Ratione Dodran- tis ad Assem: item Ratio 407 ad 576 major est Ratione 2 ad 3, hoc est Ratione Bessis ad Assem. Quæ de lineâ A B hactenus dicta sunt, in reliquis pariter illi parallelis vera esse deprehen- duntur simili ratiocinatione; ac propterea à lineâ IL omnes in eâdem Ratione secantur. Maxima igitur percussio à corpore A E circa lineam A F in gyrum acto fiet in lineâ IL; in qua medium punctum K denotat locum validissimi ictûs; in eo sci- XXX 2

Transcription: Translated (English)

Book Seven. CHAPTER IX. 715 it appears to be equality of moments; for A G has as 3, and G B likewise as 3. But that there is not in G an equality of moments is evident, if the part A B be understood as further divided into more parts, and the same be the double Ratio of A G to G B. Let A B consist of 6 parts; A G is 4, G B 2: the moments of A G are 10, of G B 11; namely, the moments of A G itself are of the four distances 1, 2, 3, 4, that is 10, but the moments of G B are of the fifth and sixth distances 5 and 6, that is 11. If A B is supposed to be of 9 parts, and A G 6, G B 3; the moments of A G are 21, of G B 24. Likewise let us set the length A B at 12 parts, namely A G 8, G B 4; the moments of A G are 36, of G B 42. Again let A B be of 15 parts; A G 10, G B 5: the moments of A G are 55, of G B 65. Finally let A B be of 18 parts; A G 12, G B 6: the moments of A G are 78, of G B 93. Therefore the equality of moments is not precisely at G at the two-thirds point of the length A B, but lies beyond G toward B. But if A H be the three-quarters point of the length A B, not at H, but this side of H, between G and H is the sought point, in which the equality of moments is found. For if A B be of 4 parts, and A H be 3, but H B 1; the moments of A H are 6, and of H B 4: If A B be of 8 parts, and A H 6, H B 2; the moments of A H are 21, H B 15; and so of the rest, keeping the same triple Ratio of A H to H B. Since therefore the moments of A G are smaller than the moments of G B, but on the contrary the moments of A H are greater than the moments of H B, it is clear that the equality of moments lies between G and H, that is between the two-thirds point and the three-quarters point. Where however such a point lies most nearly, you will discover if you set the whole A B at 576 parts, and A I at 407 parts: for since the moments of the whole A B are 166176, and the moments of A I are 83028, there remain the moments of I B 83148, very nearly equal to the moments of A I. Now the Ratio of 407 to 576 is less than the Ratio of 3 to 4; and the Ratio of A I to I B is less than the Ratio of A H to H B, that is, less than the Ratio of the Three-quarters point to the Whole: likewise the Ratio of 407 to 576 is greater than the Ratio of 2 to 3, that is, than the Ratio of the Two-thirds point to the Whole. What has thus far been said concerning the line A B is found to be true in the rest as well, in the lines parallel to it, by similar reasoning; and for that reason all are cut in the same Ratio by line I L. Therefore the greatest percussion from body A E, moving around the line A F in a circle, will be in line I L; in which the midpoint K denotes the place of the strongest blow; in it XXX 2

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Mechanicorum 716 scilicet omnia momenta impetûs æqualiter dividuntur tam juxta longitudinem, quàm juxta latitudinem. Sed quoniam rarò contingit corpus, quo percutimus, ita æquabili ductu partes omnes dispositas habere, sicut hactenus hypothesim in cylindro aut prismatico constituimus, idcircò frequentissimè centrum hoc momentorum Impetûs, ex quo ictus vehementia pendet, aut magis ad centrum motûs accedit, aut ab hoc magis recedit, prout ad hanc aut illam extremitatem plures sunt partes majoris impetûs, aut majorum momentorum capaces: fieri siquidem potest, ut plures partes centro motûs proximæ tenuioribus momentis sint præditæ, pauciores autem partes ab hujusmodi motûs centro remotæ majora obtineant momenta, adeò ut inæqualitas partium reciprocâ quadam inæqualitate momentorum compensetur; & fieri potest, ut plures partes cum majore distantiâ componantur, adeò ut centrum momentorum impetûs proximum sit extremitati, quæ velocissimè movetur. Hinc quia malleo & securi infligendus est ictus, in illorum manubriis statuendis cavendum est, ne nimis gravia sint, ne fortè extra malleum aut securim, quibus sit percussio, sit centrum momentorum impetûs. Centrum hoc momentorum si appellare libeat Centrum Percussionis, per me licet; neque enim hæreo in vocabulis. Ut autem oblatò quocumque corpore ad percutiendum apto, quo utendum sit motu circulari, cujusmodi est malleus, clava, securis, & similia, ejus centrum Momentorum impetûs Physicè & Mechanicè habeamus, hæc methodus fortasse non inutilis accidat. Extremam illam partem, quæ manu apprehendi solet, ex clavo immobili ita suspende, ut circa illum liberè moveri valeat: tum suspensam clavam à perpendiculo remove, & in hanc atque illam partem vibrari permitte. Interim ex subtilissimo filo æreus, aut plumbeus, globulus pendeat, qui pariter vibretur: & hujus perpendiculi vibrationes cum clavæ suspensæ vibrationibus compara, an videlicet singulæ singulis isochronæ sint, hoc est æqualis durationis, an verò inæqualis; si una perpendiculi vibratio diuturnior sit, quàm una clavæ vibratio, decurtandum est filum, si brevior, producendum usque eò, dum perpendiculi vibrationes singulæ singulis clavæ vibrationibus isochronæ fuerint. Hoc ubi consecutus fueris, haud temerè pronun

Transcription: Translated (English)

Mechanics 716 namely, all the moments of the impulse are equally divided both according to length and according to breadth. But since it rarely happens that the body with which we strike has all its parts arranged with such uniform distribution, as we have hitherto supposed in the cylinder or prism, therefore very often this center of the moments of impulse, on which the force of the blow depends, either approaches more nearly the center of motion, or recedes further from it, according as toward this or that extremity there are more parts capable of greater impulse, or of greater moments: for it may happen that more parts near the center of motion are endowed with smaller moments, while fewer parts farther from this center of motion obtain greater moments, so that the inequality of the parts is compensated by a certain reciprocal inequality of moments; and it may happen that more parts are combined with greater distance, so that the center of the moments of impulse is near the extremity which moves most swiftly. Hence, because the blow is to be delivered with a hammer and an axe, care must be taken in setting their handles, lest they be too heavy, lest perhaps outside the hammer or axe, with which the blow is made, the center of the moments of impulse should be. If it please one to call this center of moments the Center of Percussion, it is permitted by me; for I am not bound by terms. But in order that, whatever body suitable for striking may be offered, which is to be used with circular motion, such as a hammer, club, axe, and the like, we may physically and mechanically have its center of the moments of impulse, this method perhaps may prove not useless. Suspend from a fixed nail that end which is usually grasped by the hand, so that it may be able to move freely around it; then remove the suspended club from the vertical, and let it swing to this and that side. Meanwhile from the finest thread let a small brass or leaden ball hang, which may likewise be swung: and compare the vibrations of this pendulum with the vibrations of the suspended club, namely whether each isochronous with each, that is, of equal duration, or rather unequal; if one vibration of the pendulum is longer than one vibration of the club, the thread must be shortened; if shorter, lengthened until the vibrations of the pendulum are each isochronous with each of the vibrations of the club. When you have achieved this, you will not rashly pronun

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Liber septimus. CAPUT IX. 717 pronunciabis quæsitum Centrum momentorum Impetûs clavæ tanto intervallo abesse à puncto suspensionis, quanta est per- pendiculi longitudo, non quidem exactissimè & Geometricè, sed quantum satis est ad Physicum opus. Cur ita argumentari liceat, si rationem reposcas, hæc satis probabilis afferri potest; quia scilicet Centrum momentorum Impetûs est punctum illud, in quo cum sit æqualitas momentorum, omnes ipsius clavæ par- tes suam vim exercent ad motum illius oscillationis; quemad- modum in centro globuli ex filo pendentis (filum ex hypothe- si nullum habet notabile momentum ad motum adnexi globuli variandum) est æqualitas momentorum ejusdem descendentis, & ad positionem perpendicularem se restituentis. Est igitur cla- va quasi perpendiculum tantæ longitudinis, quantum est inter- vallum inter Centrum motûs atque Centrum momentorum Im- petûs. At perpendicula æqualis longitudinis sunt isochrona: igitur invento perpendiculo isochrono cum oscillationibus cla- væ, nota erit ex hujus longitudine etiam longitudo rigidi illius perpendiculi, quod concipitur in clavâ, videlicet distantia Cen- tri momentorum Impetûs à Centro motûs. Hîc tamen animadvertas velim ex hac methodo non haberi exactè punctum Centri momentorum in clavâ, sed illud adhuc paulò longiùs abesse; quia nimirum perpendiculorum omnino æqualium, præterquam in gravitate ponderis appensi, illud, quod gravius est, plures vibrationes eodem tempore perficit; atque perpendiculorum omnino æqualium, præterquam in lon- gitudine, illud, quod longius est, paucioribus vibrationibus eodem tempore agitatur. Cum autem clava sit perpendiculum gravius globulo, qui ex filo pendet, positâ æquali longitudine, clava velociùs moveretur: Si igitur motus clavæ est isochronus cum motu globuli ex filo suspensi, necesse est longitudine gra- viori inferente vibrationum raritatem compensari ejus gravita- tem, quæ crebriores efficeret vibrationes. Quare hoc certum habebis, quæsitum Centrum Momentorum esse ultra punctum illud inventum ex longitudine perpendiculi adhibiti. Sed & illud præterea observandum est, motus istos circula- res corporum percutientium communiter non habere pro sui motûs Centro alteram extremitatem, nisi fortè, quando ad solius manûs motum moventur, cubito ac brachio immotis: cæterùm XXXX 3

Transcription: Translated (English)

Book Seven. CHAPTER IX. 717 you will declare the sought Center of the moments of the hammer’s Impetus to be distant from the point of suspension by as much as the length of the plumb line, not indeed with the utmost exactness and Geometrical precision, but as much as is sufficient for a Physical work. If you ask why such reasoning may be allowed, this sufficient and probable argument may be given: namely, because the Center of the moments of the Impetus is that point in which, since there is an equality of moments, all the parts of the hammer exert their force toward the motion of that oscillation; just as in the center of a little ball hanging from a thread (the thread, by hypothesis, has no notable momentum in altering the motion of the attached ball) there is an equality of moments of the same descending body and of that which restores itself to the perpendicular position. The hammer is therefore as it were a plumb line of such length as is the distance between the Center of motion and the Center of the moments of the Impetus. But perpendiculars of equal length are isochronous: therefore, when a perpendicular is found isochronous with the oscillations of the hammer, from its length also will be known the length of that rigid perpendicular which is conceived in the hammer, namely, the distance of the Center of the moments of the Impetus from the Center of motion. Here, however, I would have you note that by this method the exact point of the Center of the moments in the hammer is not obtained, but is still a little farther off; because, namely, of perpendiculars altogether equal, except in the heaviness of the weight attached, that which is heavier performs more vibrations in the same time; and of perpendiculars altogether equal, except in length, that which is longer is moved by fewer vibrations in the same time. But since the hammer is a heavier perpendicular than the little ball hanging from a thread, with equal length assumed, the hammer would move more quickly: if therefore the motion of the hammer is isochronous with the motion of the ball suspended from a thread, it is necessary that the rarity of the vibrations brought on by greater heaviness be compensated by its heaviness, which would make the vibrations more frequent. Wherefore this you will hold as certain: that the sought Center of Moments is beyond that point found from the length of the applied perpendicular. But it must also be observed further that these circular motions of striking bodies commonly do not have the other extremity for the center of their motion, unless perhaps when they are moved by the motion of the hand alone, with the elbow and arm unmoved: otherwise XXXX 3

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Mechanicorum 718 pro centro motûs habent aut cubiti, aut humeri juncturam, prout cubitus, vel totum brachium movetur: & tunc Centrum momentorum transferri contingit, nec opus est adeò longa esse manubria; ut vides malleos, quibus ad contundendos libros utuntur bibliopægi, brevioris esse manubrij, quia extento bra- chio percutiunt, quod fungitur vice valde longi manubrij: contra verò fabrorum ferrariorum mallei, bipennes, & cætera instrumenta, quæ utrâque manu apprehensa tractamus, lon- giora habent manubria, tunc enim brachium adeò extendere nequimus. CAPUT X. Quid conferat resistentia corporis percussi. Eatenus ictum corpori percusso infligi, quatenus hoc motui corporis percutientis opponitur, illique obsistit, dictum est cap.6. eóque vehementiorem esse percussionem, quò major est resistentia. Hæc autem resistentia originem ducit ex ipsâ cor- porum naturâ; omne siquidem corpus, quâ corpus est, nulli corpori penetrabile est, neque fieri potest, citrà Divinam vir- tutem longiùs, quàm naturæ termini postulant, excurrentem, ut uno eodemque in spatio duo corpora collocentur, quemad- modum duæ substantiæ ab omni prorsus sensu disjunctæ, sed quæ solâ ratione, & intelligentiâ comprehenduntur, se vicissim eodem in loco facilè patiuntur. Corpus igitur percussum tùm ex suâ constitutione, & temperie, tum ex recedendi difficulta- te, tùm ex positione, secundùm quam ictum excipit, habet, ut modum percussioni statuat; ex triplici enim hoc capite resisten- tiarum varietas petenda est, qua corpus percussum reluctatur, ne loco cedat. Et ad primum quidem quod attinet, corpora dura magis re- sistere, quàm mollia, manifestum est ex ipsâ Duri & Mollis notione. Est autem Durum, ut ait Aristoteles lib.4. Meteor. summa 2. cap.1. quod non cedit in seipsum secundùm superficiem: Molle

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Mechanics 718 have, as the center of motion, either the joint of the elbow or of the shoulder, according as the elbow or the whole arm is moved; and then the center of the moments happens to be transferred, nor is it necessary for the handles to be so long; as you see that the hammers used by bookbinders for crushing books have shorter handles, because they strike with the arm extended, which serves in place of a very long handle. On the contrary, the hammers of blacksmiths, axes, and other tools which we handle with both hands have longer handles, for then we cannot extend the arm so far. CHAPTER X. What the resistance of the body struck contributes. It has been said in chapter 6 that a blow is inflicted on the body struck insofar as it opposes the motion of the striking body and resists it, and that the more the resistance, the more violent the striking. This resistance, however, takes its origin from the very nature of bodies; for every body, insofar as it is a body, is impenetrable to any other body, nor can it happen, without Divine power extending beyond the limits of nature, that two bodies be placed in one and the same space, just as two substances, separated from all sensation whatsoever but apprehended only by reason and intelligence, easily suffer one another to be in the same place. The body struck therefore, both from its constitution and temper, and from the difficulty of yielding, and also from the position according to which it receives the blow, has the means to set a limit to the striking; for from these three heads must be sought the variety of resistances by which the body struck makes resistance, lest it give way from the place. And as to the first point, it is evident from the very notion of Hard and Soft that hard bodies resist more than soft ones. Now Hard, as Aristotle says in Metaphysics book 4, text 2, chapter 1, is that which does not yield into itself according to the surface: Soft

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Liber septimus. CAPUT X. 719 Molle autem, quod cedit, non circumobsistendo, aqua enim non Mol- lis, non enim cedit compressione superficies in profundum, sed circum- obsistit: aqua scilicet, & humores urgenti quidem cedunt, at non induendo superficiem, quæ maneat, (ut ceræ ac luto accidit) sed ita secedendo, ut, urgente remoto, ad superficiem antiquæ superficiei similem confluant particulæ, quæ secesserant. Qua- propter ad mollium genus, in rem præsentem spectare cenlenda sunt, quæcumque se premi patiuntur, hoc est, externo impulsu in se ipsa coëunt, cùm in profundum superficies permutatur, nec dividitur; sivè imprimi & formari possint, pulsu tantùm, ut cera & argilla, aut percussione, ut plumbum; sivè impressionem & formam rejiciant, ut lana, & spongiiæ. Ea autem, quæ dura sunt, sed ductilia, quia eâdem percussione possunt simul in latus, & in profundum secundum superficiem transferri secundùm partem, ut ferro candenti, aliisque metallis sub fabri malleo contingit, aliquatenus ad mollia pertinere videntur, saltem comparatè, quia videlicet cedunt percutienti, quod propterea durius censetur. Sic in arce Antuerpiensi memini me vidisse ænea aliquot ingentia tormenta bellica, olim ex Sckenckianâ munitione, cum in Hispanorum potestatem venit, asportata, in quorum tubis non mediocres contusionum notæ ab hostilibus globis impressæ apparebant. Quæ verò corpora dura sunt, neque se ita comprimi patiuntur, ut superficies depressa crassitiem minuat, sed solùm, servatâ longitudine, flexibilia sunt eâ ratione, ut à rectitudine ad curvitatem, aut vicissim à curvitate ad rectitudinem torqueantur, cedunt quidem, sed inter mollia, ex hoc quidem capite, recensenda non sunt. Quod si vehementiore percussione non solùm flectantur, sed etiam frangantur (quemadmodum contingit crassiusculo baculo, cujus extremitates in acumen desinentes innituntur duobus vitreis cyathis; qui circa medium valido fuste percussus flectitur, & inflexione declinans vitra, iis integris frangitur) in magnas partes dividuntur, & separantur: at si in partes plures dissiliant ex unicâ percussione, friantur, ut vitrum, lapis, fictile; id quod ex duritie oritur. Hæc eadem corporis habitudo, quæ particularum componentium complexionem respicit, æquè in percutiente, ac in percusso attendenda est; quandoquidem si dispar fuerit eorum durities,

Transcription: Translated (English)

Book Seven. Chapter X. 719 But what is soft, in that it yields, does so not by resisting all around; for water is not soft, since compression does not drive the surface inward, but rather encircles it: water, and liquids indeed yield to pressure, but not by taking in the surface, which remains, as happens with wax and mud, but by drawing back in such a way that, when the pressure is removed, the particles that had withdrawn flow together again to the surface similar to the former surface. Therefore, among the class of soft things, and with regard to the present discussion, those things ought to be reckoned which suffer themselves to be pressed, that is, which under external force are compressed into themselves, when the surface is changed into depth and is not divided; whether they can be impressed and shaped merely by pressure, as wax and clay, or by striking, as lead; or whether they reject impression and shape, as wool and sponges. But those things which are hard, yet ductile, because by the same blow they can at once be transferred sideways and into depth according to the surface in part, as happens with glowing iron and other metals under the smith’s hammer, seem in some degree to belong to the soft, at least by comparison, because they yield to the striker, who for that reason is judged the harder. Thus, in the citadel of Antwerp, I remember seeing some huge bronze cannon, once carried off from the Schenck fortification when it came into Spanish power, in the tubes of which there were visible no small marks of dents impressed by hostile shot. But those bodies which are hard and do not suffer themselves to be compressed so that the depressed surface diminishes in thickness, but are merely flexible, with their length preserved, in such a way that they are bent from straightness to curvature, or conversely from curvature to straightness, do indeed yield, but on this ground they are not to be counted among soft things. And if by a more violent blow they are not only bent but also broken, as happens with a rather thick stick whose ends taper to points and rest on two glass cups, and which, when struck near the middle with a strong staff, bends and, as it gives way in bending, causes the glasses to break while remaining intact itself, they are divided into large parts and separated: but if, from a single blow, they fly into many pieces, they are shattered, as glass, stone, and pottery; and this arises from hardness. This same condition of the body, which regards the composition of the particles making it up, is to be considered equally in the striker and in the thing struck; since if their hardness be unequal,

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Mechanicorum 720 durities, fieri potest, ut ex ictu labefactetur potiùs percutiens, quàm percussum: sic globus plumbeus ex editâ turri decidens in subjectum silicem ex ictu contunditur, rotundâ superficie in planam mutatâ, qua parte fuit contactus; & vitrum ad saxa allisum friatur; & follis lusorius in parietem impactus compri- mitur. Hinc tamen non fit, quò minus corpus illud, in quod plumbeus globus, aut vitrum, aut follis incurrit, percussum di- catur; ex ictu enim saltem concutitur, & contremiscit. Neque tremorem hujusmodi temerè confictum suspicabitur, quisquis longissimæ trabis extremitati aurem admoverit, ut alterâ extre- mitate quamvis levissimè digito percussâ sonitum audiat, aut noctu scrobiculo in terrâ facto aurem immiserit, ut adventan- tis alicujus adhuc procul positi passus percipiat: nullum verò sonum fieri sine tremore & motu, extra controversiam posuit experientia. Porrò spectatâ corporum temperie, percussionum vehemen- tiâ æstimatur ex iis, quæ consequuntur resistentiam ortam ex corporum collisorum duritie seu mollitudine majori au minori, tùm absolutè, tùm comparatè. Cum Absolutè dico, alterutrius solùm duritiem seu mollitudinem considero ita, ut aut corpora percussa inter se, aut corpora percutientia similiter inter se conferantur: Comparatè autem, quando percutiens cum per- cusso comparatur, prout duritie se excedunt. Si corpus percu- tiens valdè durum ponatur, & corpus percussum molle fuerit, hoc cedendo retundit ictum; ex levi enim illâ resistentiâ tan- diu durante, quandiu fit partium compressio, minuitur in per- cutiente impetus, & quod corpori molli subjectum est corpus, levissimam impressionem ex ictu recipit. Sic apud Sinas, ut in Atlante Sinico pag. 127. In flumine, per quod ad lenping navi- gatur, Catadupæ aquarum multæ sunt, & periculosissima Syrtibus loca, duo præsertim propè Cinglieu, unus Kieulung, alter Changcung dictus. Cum naves transeunt, ne cum aquâ decidentes fractionis in- currant periculum, scitè præmittunt aliquot straminis fasces, ad quos navis levius impingat, ac transeat. Sic ferreis tormentorum glo- bis objecti sacci lanâ aut terrâ repleti illorum vim elidunt, ne diruant muros hujusmodi saccis protectos: sic farti gossypio thoraces non levi munimento sunt digladiantibus. Quò autem mollius fuerit corpus percussum, quia magis cedit, minùs læ- ditur

Transcription: Translated (English)

Mechanics 720 It is possible, from the hardness of bodies, that by a blow the striker rather than the struck body may be damaged: thus a leaden ball falling from a high tower upon a flint below is crushed by the impact, its round surface being changed into a flat one at the place of contact; and glass dashed against rocks is shattered; and a blown bladder, striking a wall, is compressed. Yet from this it does not follow that the body against which the leaden ball, or the glass, or the bladder comes in contact should not be called the struck body; for it is at least shaken and made to tremble by the blow. Nor will anyone suspect a tremor of this kind to be falsely imagined who has placed his ear against the end of a very long beam, so as to hear the sound when the other end is struck ever so lightly with a finger, or who at night has put his ear into a little hole made in the ground, in order to perceive the steps of someone approaching while still far away: but that no sound is produced without tremor and motion experience has placed beyond dispute. Furthermore, in view of the temperament of bodies, the force of blows is estimated from those things which follow the resistance arising from the greater or lesser hardness or softness of the colliding bodies, both absolutely and comparatively. When I say absolutely, I consider only the hardness or softness of one side alone, so that either the struck bodies are compared with each other, or the striking bodies likewise with each other: comparatively, however, when the striker is compared with the struck body, according as they exceed each other in hardness. If the striking body be very hard, and the struck body soft, the latter, by yielding, deadens the blow; for by that slight resistance, lasting so long as the compression of the parts continues, the impetus in the striker is diminished, and the body lying beneath a soft body receives only the slightest impression from the blow. Thus among the Chinese, as in the Chinese Atlas, page 127, in the river by which one sails to Lenping, there are many cataracts and places most dangerous like quicksands, especially two near Cinglieu, one called Kieulung, the other Changcung. When ships pass by, lest by falling into the water they incur the risk of wreck, they cleverly send ahead a few bundles of straw, against which the ship may strike more gently and pass on. In like manner, sacks filled with wool or earth, when placed before iron cannon-balls, blunt their force, so that they do not destroy walls protected by such sacks: thus cuirasses stuffed with cotton are no slight defense for swordsmen. But the softer the struck body is, because it yields more, the less it is injured

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Liber septimus. CAPUT X. 72 I ditur à percutiente; & vicissim quò durius illud fuerit, magis ab eodem percutiente læditur, cujus impulsum excipit. Hinc vides cur ferreos militum thoraces, & galeas nostro hoc ævo aliter temperare oporteat, ac antiquis temporibus, quando gladiorum, hastarum, sagittarum ictus tantummodo repellere opus erat; tunc enim durâ temperatione solidandum erat ferrum, ne prorsus cederet, hujusmodi armorum mucronem admittendo: nunc verò ut innoxiè excipientur ictus glo- borum à Sclopis emissorum, ferrum molle esse expedit, ut con- tusum flectatur, & aliquantulum cedens ita imminuat globi ejaculati vires, ut penetrare ulteriùs non valeat. Quod si in chalybem temperatus esset thorax militaris, nec admodum crassus esset, ne gravitate nimiâ incommodus, aut inutilis ac- cideret, facilè chalybs ex globi ictu dissiliret, & vulneri locum aperiret. Sed quia globi plumbei sunt, & se comprimi patiun- tur, ex hujusmodi percussione compressio quasi distribuitur in- ter plumbeum globum explosum, atque ferreum thoracem, qui multo magis contunderetur (aut fortè etiam perforaretur à glo- bulo ferreo; globulus autem plumbeus, si thorax aut ipsa galea nihil cederet, magis comprimeretur, quemadmodum cùm in marmor exploditur. At ubi corpus percussum non cedit in seipsum secundùm su- perficiem, flectitur tamen, adhuc minùs resistit, quàm corpo- ra rigida, nec flexioni notabili obnoxia. Notabili, inquam, ne in quæstionem vocemus, utrum flecti dicenda sint illa corpora, quæ ex ictu tremorem concipiunt, ut æri campano, cum pulsa- tur, accidit: nam vix excogitari potest corpus aliquod, cui ex vi percussionis accidere nequeat tremor; cum & terram ipsam licèt altiùs defossam in cuniculis concuti & contremiscere ostendant lapilli, & fabæ in tympani militaris planâ facie subsi- lientes ex profundo illo ligonis ictu. Certè, si Atlanti Sinico pag. 57. credimus, ubi in I V Provinciâ Xantung mentionem facit de monte, cui nomen Mingxe, hoc est Sonorum lapis; in hujus montis vertice cippus erectus stat centum altus perticas (Pertica apud Sinas est decem cubitorum) qui vel leviter digito percussus ad tympani modum sonum edere dicitur, à quo monti nomen; nullus autem sonus absque corporis sonori tremore efficitur. Quod si non nisi levissimè flecti queat corpus percussum, sed ci- Y Y y y

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Book Seven. Chapter X. 72 is produced by the striking object; and, conversely, the harder it is, the more it is injured by the same striker, whose impulse it receives. Hence you see why the iron breastplates and helmets of soldiers in our age must be tempered differently than in ancient times, when it was only necessary to repel the blows of swords, spears, and arrows; for then the iron had to be hardened with a hard tempering, so that it would not yield at all, while admitting the edge of such weapons: now, however, if blows from bullets discharged by firearms are to be received harmlessly, it is advantageous for the iron to be soft, so that, being struck, it bends and, yielding somewhat, thereby diminishes the force of the bullet shot out, so that it cannot penetrate further. For if a military breastplate were tempered to steel, and were not also rather thick, lest it become inconvenient or useless through excessive weight, the steel would readily split apart from the impact of the bullet and open a way for the wound. But since bullets are made of lead and allow themselves to be compressed, by such a blow the compression is, as it were, divided between the leaden bullet that has been fired and the iron breastplate, which would be much more bruised, or perhaps even pierced by an iron ball; but the leaden pellet, if the breastplate or the helmet itself yielded not at all, would be compressed more, as when a shot is fired into marble. But when the body struck does not yield inward upon itself at the surface, yet is bent, it still resists less than rigid bodies and is not subject to notable flexure. Not notable, I say, lest we raise the question whether those bodies are to be called bent which, when struck, receive a vibration, as happens with a bell when it is struck: for scarcely can any body be imagined to which the force of percussion cannot impart vibration; since even the earth itself, though buried deeper in mines, is shown by little stones and beans bouncing on the flat face of a military drum, from that deep blow of the drumstick, to be shaken and tremble. Certainly, if we believe the Chinese Atlas, page 57, where, in the fourth province, Xantung, he mentions a mountain called Mingxe, that is, the Sounding Stone; on the summit of this mountain stands a monument erected one hundred fathoms high (a fathom among the Chinese is ten cubits), which, it is said, when lightly struck with a finger, gives forth a sound like a drum, from which the mountain takes its name; and no sound is produced without a vibration of the sounding body. And if the struck body can be bent only very slightly, but ci- Y Y y y

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Mechanicorum 722 tra tremorem frangatur, aut frietur, indicium est majoris re- sistentiæ, ac proinde, cæteris paribus, vehementiorem futu- ram percussionem, quàm si conspicuam flexionem admitteret. Cæterùm resistentia ferè maxima eorum corporum est, quæ & partes nexu ægrè dissolubili copulatas habent, & congruâ cras- situdine prædita non nisi creberrimo & minutissimo tremore concuti possunt, si percutiantur. Nam omnium resistentiarum absolutè maxima est, cùm prorsus immotum à percussione ma- net corpus. Hinc si percussi corporis durities major fuerit, quàm percu- tientis, fieri potest, ut impetus qui corpori percusso imprimi non potest, disjiciat ipsius percutientis partes, aut in latus im- pellat ita, ut vel contundatur, vel frangatur, vel frietur, sitque percutientis conditio deterior, quàm percussi. Hujusmodi esset apud nos conditio gladij, quo marmor percuteremus; neque enim nostrates enses comparandi sunt cum illo, de quo Atlas Sinicus pag. 159. in XV Provincia Junnam ad urbem Chinkiang, ubi hæc habet. Ad urbis Borealem partem ad hæc usque tempora in- gens conspicitur lapis, ubi Mung Rex Sinulo alterius Regis legatos excipiens, cum illi minimè satisfacerent, extra[n]to gladio lapidem ita percussit, ut ictus ad tres cubitos penetraret, verbis insuper mi- nacibus legatos alloquens; Ite, & Regi vestro renunciare, quales apud me gladij sint. Altera resistentiæ origo habetur ex difficultate recedendi; quando videlicet corpus percussum sivè ratione figuræ, sivè ra- tione molis & gravitatis, sivè ratione obstaculi alicujus, aut retinaculi, sivè ratione motûs oppositi, nequit obsecundare motui percutientis, sed potiùs illum aut cohibet, aut retardat, aut reflectit; hæc enim tria accidere possunt motui percutientis ex percussi resistentiâ. Primum siquidem si corpus percussum voluble non fuerit, & in orbem incitari nequeat, sed planâ fa- cie incumbat solo, præsertim salebroso, quò ampliori facie fit contactus, eò difficiliùs impelli potest. Deinde etiamsi rotun- dum fuerit corpus, & facilis motionis principium habeat specta- tâ figurâ, si tamen ingens fuerit globus marmoreus, aut æreus, tanta esse potest gravitas, ut vix, aut ne vix quidem, loco dimo- veri queat. Non tamen semper faciliùs moventur ex percussio- ne, quæ leviora sunt; nam si quis ex cupressu galbulum, aut ex

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Mechanicorum 722 If, under a tremor, it is broken or rubbed to pieces, this is an indication of greater resistance, and therefore, all else being equal, of a more violent blow to come than if it were to admit a visible bending. Moreover, the resistance is almost greatest in those bodies which have parts joined by a bond not easily dissolved, and which, being of suitable thickness, can only be shaken by a very frequent and very minute tremor if they are struck. For the absolutely greatest of all resistances is when a body remains altogether unmoved by the blow. Hence, if the hardness of the body struck be greater than that of the striker, it may happen that the force which cannot be impressed upon the struck body may shatter the parts of the striker itself, or drive them sideways so that it is either crushed, or broken, or rubbed to pieces; and thus the condition of the striker is worse than that of the struck body. Such would be among us the condition of a sword with which we should strike marble; for our native blades are not to be compared with that one, of which Atlas Sinicus, p. 159, in the XV Province Junnam near the city of Chinkiang, has this account. On the northern side of the city, even to this day, a huge stone is seen, where King Mung of Sin, receiving the envoys of another king, and when they gave him no satisfaction, drew his sword and struck the stone so that the blow penetrated three cubits; moreover, addressing the envoys with threatening words, he said: Go, and report to your king what swords I have. Another source of resistance is found in the difficulty of giving way; namely, when the body struck, whether by reason of its shape, or by reason of its mass and weight, or by reason of some obstacle or restraint, or by reason of an opposite motion, cannot yield to the motion of the striker, but rather either checks it, or delays it, or reflects it; for these three things can happen to the motion of the striker because of the resistance of the struck body. First, indeed, if the body struck is not movable, and cannot be set in a circular motion, but rests on a flat surface of the ground, especially a rough one, the more extensive the surface of contact, the more difficult it can be driven. Next, even if the body were round, and by reason of its shape had an easy beginning of motion, yet if it were a huge marble or bronze sphere, such weight may exist that it can scarcely, or hardly at all, be moved from its place. Nor, however, are those things which are lighter always more easily moved by a blow; for if someone from a cypress tree a galbulus, or from

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Liber septimus. CAPUT X. 723 ex quercu gallam decerpat, & malleo percutiat, ob nimiam galbuli, & gallæ levitatem multo infirmior erit ictus, quàm si æqualem globulum eburneum percuteret. Ad hæc, forma quidem apta esse potest, nec gravitas aut moles nimia, sed quia corpus percussum nequit percutienti cedere, nisi corpus aliud proximum repellatur, propterea augeri potest resistentia: sic sublicas acuminatas in terram adigimus fistucâ sivè directas ad perpendicularum, sivè pronas; sed ea potest esse telluris densitas compressionem respuens, ut sæpiùs cadens fistuca parum proficiat. Demum si corpus percussum antè ictum non quiescat, sed opposito motu occurrat percutienti, quò velociùs movetur, & directione magis oppositâ, etiam magis resistit, & utrumque vicissim est percutiens & percussum; nisi quod percutientis vocabulum validiori conceditur. Contra verò languidior accidit percussio, si corpus percutiens assequatur aliud, quod ad easdem partes tardiùs movetur; eóque minor est resistentia, quò minor est in velocitate motuum differentia. Sic decidentis ex altitudine non modicâ lapidis ictum manu citrâ læsionem excipimus, si illius motui, ubi manum attigerit, exiguo minoris velocitatis discrimine obsecundemus: hæc siquidem exigua resistentia modicum quid impetûs deterit, & quia aliquot momenta durat, ita sensim extenuatur impetus, ut demum qui reliquus est nocere non valeat. Hoc artificio procul dubio utebatur quidam, qui ante aliquot annos, ut ex viro fide digno tanquam rem notissimam accepi, Mutinæ ensem eâ dexteritate in altum projiciebat, ut perpendicularis recideret mucrone deorsum converso, quem cadentem nuda manûs vola innoxiè excipiebat; sed cum aliquando invitus cogeretur, ut id noctu experiretur in conclavi multis facibus illustrato, cùm (deficiente constanti & clarissimâ diurnâ luce, quam æmulari non potest tremula & inconstans facium, quamvis multarum, flamma) non ita exactè assequeretur descendentis gladij motum & velocitatem, finem fecit ludo manum trajectam referens. Postremum caput, ex quo resistentiæ modus desumitur in percussionibus, est ipsa positio corporis percussi, prout directè, aut obliquè, ictum excipit, hoc est quatenus linea directionis motûs, quo fertur corpus percutiens, incurrit in corporis per- Y Y y y 2

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Book Seven. Chapter X. 723 from the oak he plucks the gall, and strikes it with a mallet, the blow will be much weaker because of the excessive lightness of the gallnut and the gall, than if he were striking an equal globule of ivory. Moreover, although the shape may indeed be suitable, and there be no excessive weight or bulk, yet because the body struck cannot yield to the striker unless another neighboring body is repelled, for that reason the resistance may be increased: thus we drive pointed piles into the ground with a pile-driver, whether they are set upright perpendicular, or inclined; but the density of the earth may be such, repelling compression, that the repeated falling of the driver accomplishes little. Finally, if the body struck does not rest before the blow, but meets the striker with an opposite motion, the faster it moves, and the more opposite its direction, the more it resists also; and each in turn is both striker and struck, except that the term striker is granted to the stronger. On the contrary, the blow is weaker if the striking body overtakes another body moving more slowly in the same direction; and the smaller the difference in the speed of their motions, the less the resistance. Thus, in receiving with the bare hand the impact of a stone falling from no small height, we avoid injury if we accommodate ourselves to its motion when it has reached the hand, by yielding a slight difference of lesser speed: for this small resistance wears away a moderate amount of impetus, and because it lasts for a few moments, the impetus is gradually diminished, so that at last what remains is no longer able to harm. Without doubt a certain man used this artifice, who some years ago, as I learned from a man worthy of trust, as from a thing well known, would throw the sword of Modena into the air with such dexterity that it would fall straight down with the point turned downward, and he would receive the falling blade unharmed in the naked palm of his hand; but when he was once compelled against his will to try this at night in a room illuminated with many torches, since (with the constant and brightest daylight lacking, which cannot be imitated by the trembling and inconstant flame of torches, though they be many) he did not so precisely match the motion and speed of the descending sword, he ended the game by reporting a wounded hand. The final chapter, from which the mode of resistance is drawn in blows, is the very position of the struck body, whether it receives the blow directly or obliquely, that is, insofar as the line of direction of the motion by which the striking body is carried impinges upon the body of the- Y Y y y 2

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Mechanicorum 724 cussi superficiem ad angulos æquales, aut inæquales. Si enim ad angulos æquales opponatur Directioni motûs, cum ad neutram partem corpus percutiens declinare possit, tota vis ictus excipitur à corpore percusso. Sin autem obliquè, & ad angulos inæquales, à corpore percusso excipiatur percutientis ictus, quò major erit angulorum inæqualitas, eò languidior erit percussio, minùs quippe motûs Directioni opponitur superficies percussa, quò fuerit angulus Incidentiæ magis acutus. Ut autem horum ictuum Ratio aliqua innotescat, nulla mihi congruentior methodus occurrit, quàm si philosophemur simili planè ratiocinatione, ac cùm lib. 1. cap. 14. expendimus gravitationem corporis in planum inclinatum; sicut enim ibi gravitatem cum suâ directione deorsum ad centrum gravium consideravimus, ita hîc in percussione impetum corporis percutientis, & ejus directionem accipere oportet: & quemadmodum in plano inclinato gravia obtinent momenta descendendi majora, aut minora, prout angulus inclinationis plani cum perpendiculo minor est, aut major; similiter in percussione momentum progrediendi juxta conceptam aut impressam directionem motûs iisdem tenetur legibus, juxta plani percussi obliquitatem; ac proinde minor invenitur resistentia, ubi majus est progrediendi momentum. Quare hîc satis erit recolere, quæ dicta sunt lib. 1. cap. 13. & 14. de gravitatione in plano inclinato, & in planum inclinatum, eáque percussionibus servatâ analogiâ applicare. Cum itaque in percutiente consideranda sit & moles, & motûs velocitas, & Directio motûs, & durities; in corpore autem percusso & naturæ temperatio, & recedendi difficultas, & positio, secundùm quam excipitur ictus, spectanda sit; manifestum est ex his omnibus ictuum vim temperari; atque adeò si duo ictus comparandi sint, assumendæ sunt in corporibus percutientibus invicem comparatis Rationes omnes & molis ad molem (hoc est gravitatis ad gravitatem, aut virtutis moventis ad virtutem moventem) & velocitatis ad velocitatem, & directionis ad directionem, & duritiei ad duritiem; & similiter in corporibus percussis Rationes eorum, quæ in illis considerantur: atque demum facta Rationum compositio indicabit Rationem ictuum. Hinc vides quàm multæ fieri possint hujusmodi Rationum

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Mechanics 724 the struck surface at equal or unequal angles. For if it is opposed to the direction of motion at equal angles, since the body struck can deviate to neither side, the whole force of the blow is received by the body struck. But if it is received obliquely, and at unequal angles, by the body struck, the greater the inequality of the angles, the weaker will be the percussion; for the struck surface is opposed to the direction of motion less, the more acute the angle of incidence is. But that some account of these blows may become known, no method more suitable occurs to me than if we philosophize with reasoning entirely similar to that which, in Book 1, chapter 14, we used in considering the gravitation of a body on an inclined plane; for just as there we considered gravity with its direction downward toward the center of heavy bodies, so here in percussion we must take up the impulse of the striking body and its direction: and just as on an inclined plane heavy bodies obtain greater or lesser tendencies to descend, according as the angle of inclination of the plane with the perpendicular is smaller or greater; so likewise in percussion the tendency to advance along the conceived or impressed direction of motion is governed by the same laws, according to the obliquity of the struck plane; and therefore the resistance is found to be less where the tendency to advance is greater. Wherefore it will be enough here to recall what was said in Book 1, chapters 13 and 14, concerning gravitation on an inclined plane and into an inclined plane, and, preserving the same analogy, to apply it to blows. Since therefore in the striking body there must be considered both mass and velocity of motion, and the direction of motion, and hardness; but in the body struck, the constitution of nature, the difficulty of moving away, and the position according to which the blow is received, must be examined; it is evident from all these that the force of blows is moderated; and thus, if two blows are to be compared, there must be assumed in the striking bodies, compared with one another, all the ratios, both of mass to mass (that is, of weight to weight, or of moving power to moving power), and of velocity to velocity, and of direction to direction, and of hardness to hardness; and similarly in the bodies struck, the ratios of those things that are considered in them: and finally the composition of the ratios made will indicate the ratio of the blows. Hence you see how many such ratios may be made

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Liber septimus. CAPUT XI. 725 Rationum complexiones; quas si juxta earum varietatem in Propositiones digerere otium esset, in molem non exiguam hæc scriptio excresceret, sed non majore fructu, quàm si tu ipse Ra- tiones, ut indicatum est, componas. CAPUT XI. Quomodo ex Percussionibus determinantur Reflexiones. UT Percussionis natura plenè perfectéque innotescat, dis- piciendum superest, quomodo ex illâ determinetur Re- flexio. Motus siquidem, qui propriè est reflexus, percussionem consequitur, quatenus id, quod motu directo ferebatur, inve- nit obicem, ne ulteriùs juxta eandem Directionem progredia- tur: sed quia adhuc acquisitus, seu impressus, impetus superest, aliam inire viam cogitur; eóque magis reflectitur, quò majo- rem invenit resistentiam ortam ex utriusque corporis impene- trabilitate, atque duritie. Quòd si utrique corpori, percutien- ti videlicet atque percusso, summa durities inesse poneretur, ita ut in neutro ex vi percussionis ulla sequeretur partium com- pressio, aut depressio, aut attritio seu divisio, perfecta quoquè intelligeretur reflexio, in qua corpus percutiens non nisi in transitu, citrà omnem vel brevissimam morulam, contingeret corpus, à quo reflectitur; & nulla fieret impetûs acquisiti, sive impressi, diminutio præter eam, quam secum trahit nova re- flectentis determinatio opposita lineæ directionis, secundùm quam priùs movebatur. Nemini autem dubium esse debet, an corpus reflexum pergat moveri ex vi impetûs adhuc residui post motum directum: nam corpus reflectens prorsus immo- tum & quiescens non potest impetum illi communicare; cum perpetuis experimentis doceamur nihil moveri ab alio quiescente. Sæpiùs tamen contingit (si quis dixerit semper, quibus argu- mentis eum coarguerem manifestæ falsitatis, me non habere YY yy 3

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Book Seven. CHAPTER XI. 725 Complexions of ratios; which, if it were worth the trouble to arrange, according to their variety, into Propositions, this writing would grow to no small bulk, but with no greater profit than if you yourself were to compose the Ratios, as has been indicated. CHAPTER XI. How Reflections are determined from Percussions. In order that the nature of Percussion may be fully and perfectly understood, there remains to be considered how Reflection is determined from it. For the motion which is properly called reflected follows percussion, in so far as that which was being carried forward by direct motion finds an obstacle, so that it cannot proceed farther along the same direction: but because the acquired, or impressed, impetus still remains, it is forced to take another path; and it is reflected the more, the greater resistance it finds arising from the impenetrability and hardness of both bodies. But if to both bodies, namely the striking and the struck, supreme hardness were supposed to belong, so that in neither would any compression, depression, or attrition or division of parts follow from the force of percussion, a perfect reflection would likewise be understood, in which the striking body would touch the body from which it is reflected only in passing, without any delay, however brief; and there would be no diminution of the acquired, or impressed, impetus except that which its new determination, opposite to the line of direction along which it was previously moving, carries with it. Nor ought it to be doubted by anyone whether the reflected body continues to move by the force of the impetus still remaining after the direct motion: for a reflecting body, being entirely motionless and at rest, cannot communicate impetus to it; since by continual experiments we are taught that nothing is moved by another thing at rest. Yet it happens more often (if anyone should say always, by what arguments I should convict him of manifest falsehood, I do not have

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Mechanicorum candidè profiteor) non fieri puram reflexionem ex merâ re- sistentiâ; sed in alterutro saltem corporum collisorum, ex per- cussione sequitur aliqua partium violenta compressio, aut distractio; hanc autem naturæ repugnantem partium positio- nem excutere dum nititur, séque in pristinum statum restitue- re, novum impetum concipit, quem & potest reflectens re- flexo imprimere, atque in eo diminuti ex resistentiâ impetus jacturam, aliqua saltem ex parte, resarcire. Hinc, in rem præ- sentem distinguere oportet, quid inter Compressionem & De- pressionem intersit: quæ enim deprimuntur, ut plumbum, cera, argilla, non resiliunt secundùm superficiem, ut pristinam figu- ram induant; ideóque quando hujusmodi corpora in aliud im- pinguntur, vel aliud in illa impingitur, valdè debilitatur re- flexio, si modò aliqua contingere potest. Quæ verò compri- muntur, externâ vi deficiente se in pristinam figuram absque cunctatione restituunt concepto novo impetu. Exemplum ex folle pugillatorio peti potest, ut res in apertum deducatur. Ca- dens in subjectum pavimentum follis lusorius, ritè inflatus, im- peditur, ne ulteriùs procedat; sed quia inclusi aëris particulæ eæ sunt, quæ per vim constipari ampliùs possint, ideò ex illis anteriores hinc urgentur à posterioribus, quæ vi acquisiti im- petûs inchoatum iter prosequuntur, hinc tellure resistente, hinc alutâ continente, inter angustias deprehensæ comprimun- tur: id quod cùm motum exigat, certam aliquam brevissimi temporis mensuram requirit, quo fluente, terra à folle tangi- tur, motûsqque aliquatenus impeditur (nunquam tamen ita, ut cesset omnino motus illarum saltem partium, à quibus anterio- res urgentur ac premuntur) & quò diutiùs hujusmodi com- pressio durat, eò magis impeditur motus totius follis, atque adeò plus impetûs deperditur. Sed quoniam status ille majoris com- pressionis aëri intrà follem constipato contra naturam accidit, ubi primùm, debilitato impetu urgente, restituere se potest aër, impetum sibi imprimit, quo moveatur ad ampliorem lo- cum occupandum, si facta fuerit condensatio, vel certè ad par- tes in pristino & naturali statu constituendas (quemadmodum alutæ contingit, cujus partes aliæ compressæ, aliæ distractæ se- se restituunt) cùmque id præstare nequeat motu ad terram di- recto, quippe quæ resistit, in oppositam partem motum dirigit; novóque

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I clearly profess in Mechanics that pure reflection does not arise from mere resistance; but in at least one of the colliding bodies, from the удар there follows some violent compression or stretching of the parts; and as this position of the parts, repugnant to nature, strives to shake itself off and restore itself to its former state, it conceives a new impulse, which, reflecting, it can impart to the reflected body, and thereby make up, at least in part, the loss of impulse diminished by resistance. Hence, for the present purpose, one must distinguish what difference there is between Compression and Depression: for things that are depressed, like lead, wax, clay, do not rebound by their surface so as to assume their former figure; and therefore when bodies of this kind strike against something else, or something else strikes against them, reflection is greatly weakened, if indeed any can occur at all. But things that are compressed, when the external force ceases, return without delay to their former shape by the newly conceived impulse. An example may be taken from the boxing-bladder, to make the matter clear. A properly inflated football, falling onto a hard floor beneath it, is prevented from proceeding further; but because the particles of the enclosed air are such that they can be pressed together further by force, the forward ones are thereby driven by the rear ones, which, by the force of the acquired impulse, continue the begun motion; thus, meeting resistance from the ground on the one hand and from the leather casing on the other, they are caught in narrow confines and compressed: and since this requires motion, it needs some definite measure of the shortest time, during which, as it passes, the ground is struck by the ball and the motion is somewhat impeded (yet never so that the motion of those parts at least by which the forward ones are driven and pressed upon ceases altogether); and the longer such compression lasts, the more the motion of the whole ball is hindered, and thus the more impulse is lost. But since that state of greater compression in the air packed within the ball occurs contrary to nature, as soon as the air can restore itself, weakened by the urging impulse, it impresses an impulse upon itself by which it may move to occupy a larger space, if condensation has taken place, or certainly to bring its parts back into their former and natural condition (as happens with the leather casing, whose parts, some compressed and others stretched, restore themselves); and since it cannot accomplish this by motion directly toward the earth, since the earth resists, it directs the motion toward the opposite side; and a new

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Liber septimus. CAPUT XI. 727 novóque hoc impetu si non æquatur, qui resistentiâ diminutus fuerat, saltem incrementi alicujus compensatione lenitur in- commodum detrimenti, & major sit motus, quàm pro Ratione residui impetûs ante percussionem & compressionem concepti. Hæc eadem proportione dicenda sunt, quando non corpus percutiens, ut follis in terram decidens, sed percussum com- primitur aut distrahitur, & virtute elasticâ se restituit; impe- tum enim concipiens, quo amissam figuram recuperet, etiam percutienti impetum imprimit, quo repellitur. Sic in sphæ- risterio immissæ pilæ si reticulum ex contortis animalium in- testinis in plagas distinctum objeceris, validiùs reflectitur pila, quàm objecto batillo ligneo, intenti enim nervi illi, ex impetu pilæ inflexi, validissimè se restituunt, id quod ligno non con- tingit, quippe quod vix elasticam hanc virtutem exercet, si ta- men à pilâ impactâ quicquam inflexionis recipit, quæ compres- sio sit potiùs, quàm depressio. Quæ scilicet corpora eam par- tium texturam habent, ut minùs ferant se à priore positione & figurâ dimoveri, illa sese majore impetu restituunt. Ex his constat, cur partium depressio officiat reflexioni cor- poris percutientis: quia nimirum à posterioribus illius partibus urgentur anteriores contactui proximæ, quæ interim vel quies- cunt, vel multò tardiùs moventur, vel ad latus secedunt, & idcircò vel totum, vel ferè totum, suum impetum deperdunt: posteriores verò dum urgent ac premunt, moventur quidem, sed reperiunt resistentiam subsidentium partium anteriorum, atque adeò in illis pariter minuitur impetus; sæpiúsque tanta fit impetûs diminutio, ut, depressione absolutâ, partes illæ poste- riores reliquum non habeant tantum impetûs, qui vincere va- leat gravitatem, & reflexionem efficere; neque enim aliquid amissi impetûs compensatur ab impetu novo partium se resti- tuentium, quemadmodum fieri diximus in compressione. Quan- do autem depressio partium accidit corpori, ad quod alliditur corpus percutiens, ut cùm in arenam siccam ac pulverulentam, aut in limosam terram decidit globus, tunc multum impetûs deperditur, ut dictum est superiùs de ictu, qui eò infirmior est, quò mollius est corpus percussum; reflexio autem eò major est, quò validiore ictu percutitur corpus reflectens. Quòd si utrumque corpus, tam percutiens, quàm percussum, patiatur compressio

Transcription: Translated (English)

Book Seven. Chapter XI. 727 and if by this new impulse it is not made equal to what had been diminished by resistance, at least the inconvenience of the loss is alleviated by some compensation of increase, and the motion becomes greater than would accord with the remaining impulse conceived before the impact and compression. The same must be said in proportion when not the striking body, as a bellows falling to the ground, but the body struck is compressed or stretched and restores itself by its elastic force; for, conceiving an impulse by which it may recover its lost shape, it also impresses upon the striker an impulse by which it is repelled. Thus in ball play, if you set against the ball thrown in a net made of twisted animal intestines divided into meshes, the ball is reflected more strongly than when a wooden bat is placed against it; for those tense cords, bent by the impulse of the ball, restore themselves with the greatest force, which does not happen with wood, since it scarcely exercises this elastic power, though, if it receives any bending from the ball that strikes it, it is rather a compression than a depression. For bodies of this kind have such a texture of parts that they are less able to be moved from their former position and shape, and they restore themselves with greater impulse. From these things it is clear why the depression of parts impedes the reflection of the striking body: namely because the anterior parts nearest the contact are pressed by the posterior parts, while in the meantime they either rest, or move much more slowly, or withdraw to the side, and therefore lose either all, or almost all, of their impulse; but the posterior parts, while they urge and press, do indeed move, yet find resistance from the sinking anterior parts, and thus the impulse in them is likewise diminished; and often the diminution of impulse becomes so great that, once the depression is complete, those posterior parts have not enough impulse left to overcome gravity and effect reflection. Nor is any part of the impulse lost made up by the new impulse of the parts restoring themselves, as we said happens in compression. But when depression of the parts occurs in the body against which the striking body is dashed, as when a ball falls into dry and dusty sand or into muddy earth, then much impulse is lost, as was said above concerning a blow, which is weaker the softer the struck body is; but reflection is greater the stronger the blow with which the reflecting body is struck. But if both bodies, both the striking and the struck, suffer compression

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728 Mechanicorum compressionem aut depressionem, aut partium attritum, tunc multò minor est reflexio, quia dum invicem cedunt, aliquo tempore durat resistentia, multóque magis minuitur impetus: id quod adhuc magis contingit, si se invicem conterant, & particulæ aliquæ majores resiliant. Quapropter cum incerta semper, & varia sit complexio hujusmodi resistentiarum & cessionum, juxta variam corporum temperationem; ut reflexionis certæ regulæ statuantur, semotis iis, quæ percussionis accidunt, consideranda & assumenda est resistentia absque ullâ cessione, perinde atque si durissimorum corporum collisio fieret. Cum itaque in reflexione, corporis duri in aliud decidentis, aut impacti, motus ad novam lineam dirigatur, nova hæc directio oritur ex lineâ directionis prioris motûs quatenus comparatâ cum plano reflectente, videlicet quatenus ad illud inclinatur, & cum eo angulum constituit in puncto contactûs. Quando autem superficies corporis reflectentis eo loco, ubi percutitur, plana non est, sed convexa (simile quid dicendum, si cava fuerit) sivè sphærica sit, sivè Elliptica, sivè Conica, reverâ nullum ibi est planum reflectens (nisi fortè hujusmodi convexas superficies ex plurimis planis minimis constitui fingas, quemadmodum circuli peripheriam ex infinitis lineolis rectis, quarum rectitudo sensum omnem fugiat, componi opinantur aliqui) sed communiter mente concipiunt planum, quod in puncto percussionis tangeret superficiem convexam; & ex illo angulos tùm Incidentiæ, tùm Reflexionis definiunt. Porrò planum reflectens (quod quidem spectat ad novam directionem motûs statuendam corpori percutienti, quem ponamus esse globum) ita se habere videtur, ac si in globum quiescentem motu parallelo impingeretur ipsum planum tanto impetu, quanto impetu fertur globus adversùs planum: si enim ex duobus collisis alterum quiescit, alterum movetur, ad rationem ictûs nil refert, utrum illorum quiescat, aut moveatur, modò cætera omnia paria fuerint; ad rationem verò reflexionis, quà reflexio est, attenditur potissimum ordinatio novæ lineæ motûs, quæ ex obstaculi positione desumitur, adeò ut nova linea directionis, quatenus à plano reflectente pendet, & à centro

Transcription: Translated (English)

728 Mechanics compression or depression, or the rubbing together of parts, then reflection is much less, because while they yield to one another, resistance lasts for some time, and the impulse is much more diminished: which happens still more, if they grind against one another, and some larger particles rebound. Therefore, since the composition of resistances and yieldings of this kind is always uncertain and variable, according to the varying constitution of bodies; so that certain rules of reflection may be established, setting aside those things which occur in impact, resistance without any yielding must be considered and assumed, just as if a collision of the hardest bodies were taking place. Since therefore in reflection, the motion of a hard body falling upon another, or striking it, is directed to a new line, this new direction arises from the line of the prior direction of motion, insofar as it is compared with the reflecting plane, namely insofar as it inclines toward it, and forms with it an angle at the point of contact. But when the surface of the reflecting body, at the place where it is struck, is not plane, but convex (something similar must be said if it be hollow), whether spherical, or Elliptical, or Conical, in truth there is no reflecting plane there (unless perhaps you imagine such convex surfaces to be composed of very many tiny planes, just as some suppose the circumference of a circle to be composed of infinitely many straight little lines, whose straightness escapes all perception) but they commonly conceive in the mind a plane, which at the point of percussion would touch the convex surface; and from that they determine both the angle of incidence and the angle of reflection. Moreover, the reflecting plane (which indeed concerns the determination of the new direction of motion for the striking body, which let us suppose to be a sphere) would seem to behave as if the plane itself were driven against a sphere at rest with a parallel motion, with as much impetus as the sphere moves against the plane: for if of two bodies colliding one is at rest and the other moves, as regards the nature of the blow it makes no difference whether one of them is at rest or is moving, provided all the rest be equal; but as regards reflection, insofar as it is reflection, the ordering of the new line of motion is chiefly considered, and this is taken from the position of the obstacle, so that the new line of direction, insofar as it depends on the reflecting plane, and on the center

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Liber septimus. CAPUT XI. 729 centro gravitatis descendentis, aut à centro Impetûs corpo- ris impacti, certâ Ratione respiciat priorem lineam directionis. Eo igitur ipso quod concipimus planum reflectens mo- veri motu parallelo, hoc est servatâ positione priori posi- tioni parallelâ, adversùs globum quiescentem, manifestum est novam determinationem ex illo ortam esse versùs lineam plano perpendicularem, ex puncto contactûs erectam: nam impetus, qui ex illo plani motu imprimeretur globo quiescenti, hunc deferret per lineam jungentem punctum contactûs cum centro globi: hæc autem linea ex centro sphæræ ducta ad punctum contactûs plani est ipsi plano per- pendicularis, ut ex Sphæricis constat. Quamvis igitur res contrario modo se habeat, scilicet planum quiescat, & glo- bus moveatur, directio tamen, quatenus orta ex resistentiâ plani, eodem modo se habet, & est versùs perpendicularem ex puncto contactûs. Sed quia cum impetu globi ut plurimùm manet adhuc prior directio, ex his duabus mo- tuum ordinationibus oritur tertia mixta; ita ut neque ad perpendicularum reflectatur, nisi incidentiæ linea perpendi- cularis fuerit, neque recta institutum iter prosequatur. Quoniam igitur ex puncto contactûs innumeræ lineæ exi- re possunt cùm variâ inclinatione ad planum reflectens, nec ulla peculiaris est causa, cur ad hos potiùs, quàm ad illos angulos, reflectatur corpus percutiens, qui majores sint aut minores angulo incidentiæ, quem linea directionis motûs constituit cum eodem plano reflectente; reli- quum est, ut angulo incidentiæ æqualis sit angulus re- flexionis; hæc siquidem linea ad angulum priori æqualem re- flexa unica est, quæ inter innumeras alias lineas magis aut minùs inclinatas potiori quodam jure exigitur à naturâ prioris directionis leges, quoad fieri potest, retinente. Non est autem necesse tyronem monere, duas lineas, di- rectam & reflexam in puncto reflexionis concurrentes esse in uno & eodem plano, ut constat ex. 2. lib. II. ab hoc autem plano secari planum reflectens, ac proinde ad lineam, quæ est duorum planorum communis sectio, referendam esse linearum illarum inclinationem. Quare sit plani reflectentis, & plani, in quo fit motus, ZZzz

Transcription: Translated (English)

Book seventh. CHAPTER XI. 729 whether it is from the center of gravity of the descending body, or from the center of the impulse of the body that is struck, it is to be regarded in a certain ratio as looking toward the prior line of direction. Therefore, by the very fact that we conceive the reflecting plane to be moved with a parallel motion, that is, with its former position preserved parallel to the prior position, against a body at rest, it is manifest that a new determination arising from this is toward a line perpendicular to the plane, raised from the point of contact: for the impulse, which would be impressed upon the body at rest from that motion of the plane, would carry it along the line joining the point of contact with the center of the body; but this line, drawn from the center of the sphere to the point of contact with the plane, is perpendicular to the plane itself, as is evident from the Sphere. Although therefore the matter is otherwise, namely, the plane is at rest and the body moves, nevertheless the direction, insofar as it arises from the resistance of the plane, is the same in kind, and is toward the perpendicular from the point of contact. But because, along with the impulse of the body, the prior direction for the most part still remains, from these two arrangements of motions a third, mixed one arises; so that it is not reflected to the perpendicular unless the line of incidence were perpendicular, nor does it continue straight on the path it has begun. Since therefore from the point of contact innumerable lines can go forth, with varying inclination to the reflecting plane, and there is no particular cause why the striking body should be reflected to these angles rather than to those, whether they be greater or less than the angle of incidence, which the line of motion makes with the same reflecting plane; it remains that the angle of reflection be equal to the angle of incidence; for indeed this line, reflected at an angle equal to the former one, is unique, and among innumerable other lines more or less inclined it is by a certain better right demanded by nature, which, retaining as far as possible the laws of the prior direction. Nor is it necessary to warn the beginner that the two lines, the direct and the reflected, meeting at the point of reflection, are in one and the same plane, as is clear from book II, proposition 2; and that the reflecting plane is cut by this plane, and therefore the inclination of those lines must be referred to the line which is the common section of the two planes. Therefore let there be of the reflecting plane, and of the plane in which the motion takes place, ZZzz

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Mechanicorum 730 communis sectio linea AB, & super planum ad rectos angulos cadat linea directionis prioris DC, per quam movetur globus tanto impetu, ut nisi planum obstaret, ulteriùs procederet rectà versus E: Verùm quoniam à plano obsisten- te repellitur per lineam perpendi- cularem CD, nova hæc determinatio ad motum est omninò & adæquatè opposita priori directioni DC, ideóque ictus est va- lidissimus propter maximam resistentiam. Hinc quia ex re- sistentiâ oritur reflexio, maxima est reflexio, quæ fit per li- neam perpendicularem, nihil enim remanet de priori directio- ne: in hoc quippe comparantur invicem reflexiones, ut illa major dicatur, in qua nova motûs ordinatio magis minuit prio- rem directionem, ut scilicet minùs pergat ad eam partem, ad quam ferebatur motu directo corpus percutiens. In reflexione autem perpendiculari ita tollitur prior directio, ut nullo pacto globus, qui ex D per DC movebatur, ampliùs versus E ten- dat. Cùm ergo nova ordinatio sit per perpendicularem CD ad angulos rectos, manifestò constat, angulum reflexionis esse æqualem angulo incidentiæ; nam omnes anguli recti sunt æquales. At moveatur corpus per lineam FC, & fiat incidentiæ an- gulus FCB acutus: nisi planum resisteret, progrederetur cor- pus juxta eandem directionem ultra C in G; quo motu rece- dens à puncto C partim tenderet à C versus A, partim à C ver- sùs E, ita ut à lineâ CA distaret intervallo AG, à lineâ au- tem CE intervallo EG; esset enim directio CG æquivalens directioni mixtæ ex CA, & CE. Verùm nova motûs ordina- tio à plano reflectente, quatenus opponitur ulteriori motui, est per lineam perpendicularem CD; hæc autem priori directio- ni FCG adversatur solùm, prout æquivalet Directioni CE (nam quatenus æquivalet directioni CA, non illi opponitur; globo scilicet, qui per CA moveretur, planum non resisteret, nec illum reflecteret) ac propterea dat oppositam directionem CD, cujus longitudinem ponamus æqualem ipsi CE. Manen- te igitur directione per CA, & directione CE mutatâ in CD, est

Transcription: Translated (English)

Mechanics 730 the common section, line AB, and let the line of direction of the prior motion DC fall upon the plane at right angles, by which the ball is moved with so great an impulse that, unless the plane were to obstruct it, it would proceed farther in a straight line toward E: But since it is repelled by the plane resisting through the perpendicular line CD, this new determination for motion is altogether and adequately opposite to the former direction DC, and therefore the blow is the strongest possible because of the greatest resistance. Hence, since reflection arises from resistance, the greatest reflection is that which is made through the perpendicular line, for nothing remains of the prior direction: in this matter reflections are compared with one another, so that that is called the greater in which the new ordering of motion diminishes the prior direction more, namely, so that the striking body moves less toward that part to which it was being carried by direct motion. But in perpendicular reflection the prior direction is so abolished that by no means does the ball, which was moving from D through DC, tend farther toward E. Since therefore the new ordering is through the perpendicular CD at right angles, it is clearly evident that the angle of reflection is equal to the angle of incidence; for all right angles are equal. But let the body be moved along the line FC, and let the angle of incidence FCB be acute: unless the plane resisted, the body would proceed along the same direction beyond C to G; by this motion, receding from point C, it would tend partly from C toward A, partly from C toward E, so that it would be distant from line CA by the interval AG, and from line CE by the interval EG; for the direction CG would be equivalent to the mixed direction from CA and CE. But the new ordering of motion from the reflecting plane, insofar as it opposes the further motion, is through the perpendicular line CD; and this line opposes the prior direction FCG only insofar as it is equivalent to the direction CE (for insofar as it is equivalent to the direction CA, it does not oppose it; namely, a plane would not resist a ball moving through CA, nor would it reflect it) and therefore it gives the opposite direction CD, whose length let us suppose equal to CE itself. Therefore, the direction through CA remaining, and the direction CE being changed into CD, it is

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Liber septimus. CAPUT XI. 731 est ex utrâque mixta directio CH, secundùm quam movetur corpus reflexum. Quoniam itaque directionum singularum mensuræ sunt CA, & CE, per A ducatur parallela ipsi DE; & per E, atque per D, ducantur EG & DH ipsi CA parallelae. Est ergo rectangulum HE; & quia CD assumpta est æqualis ipsi CE, etiam AH & AG sunt illis æquales. Quapropter cum in triangulis CAH, CAG rectangulis, latera AC & AG æqualia sint lateribus AC & AH, atque angulus comprehensus ad A sit rectus, per 4. lib. 1. angulus ACH (qui est angulus Re- flexionis) est æqualis angulo ACG: at angulo ACG æqualis est ad verticem angulus incidentiæ FCB, per 15. lib. 1: ergo angulo FCB incidentiæ æqualis est ACH angulus re- flexionis. Eâdem methodo, si angulus incidentiæ fuerit ICB, ostendemus angulum reflexionis KCA esse illi æqualem; quando- quidem directio CM mutatur in CN, & manet directio CA; ac propterea directio mixta ex CN, & CA, est CK. Atque ita de cæteris. Ex quibus observabis, quò acutior fuerit angulus incidentiæ, in reflexione ita misceri novam directionem cum anti- quâ, ut magis prævaleat antiqua; nova siquidem ad anti- quam, secundùm id, quod de illâ remanet, se habet ut Sinus Rectus anguli incidentiæ ad Sinum Complementi. Nam si incidentiæ angulus sit FCB, & illi æqualis HCA, nova directio CD, hoc est AH ad antiquæ residuum CA, se habet ut HA ad AC: Sin autem in- cidentiæ angulus fuerit ICB, hoc est illi æqualis angu- lus reflexionis KCA, nova directio ad id, quod de anti- quâ remanet, est ut KA ad CA. Est autem major Ratio AC ad AK minorem, quàm ejusdem AC ad AH majo- rem per 8. lib. 5. Quare quandiu angulus incidentiæ mi- nor est semirecto, majus est residuum antiquæ directionis (attentè observa me de solâ directione loqui) quàm nova ordinatio: ubi fuerit angulus semirectus, sunt æquales; si angulus incidentiæ fuerit semirecto major, nova ordina- tio major est eo, quod remanet de antiquâ directione: ubi demum fuerit angulus rectus in perpendiculari inciden- tiâ, nova directio ad priorem se habet ut Radius ad ZZZZZ

Transcription: Translated (English)

Book Seven. CHAPTER XI. 731 there is formed a direction CH mixed from both, according to which the reflected body is moved. Since therefore the measures of the several directions are CA and CE, let through A be drawn a line parallel to DE; and through E, and through D, let EG and DH be drawn parallel to CA. Therefore HE is a rectangle; and because CD is taken equal to CE, AH and AG are also equal to them. Wherefore, since in the right triangles CAH and CAG, the sides AC and AG are equal to the sides AC and AH, and the included angle at A is right, by 4. book 1. the angle ACH (which is the angle of re- flection) is equal to the angle ACG: but the angle ACG is equal at the vertex to the angle of incidence FCB, by 15. book 1: therefore the angle ACH of reflection is equal to the angle FCB of in- cidence. By the same method, if the angle of incidence were ICB, we shall show the angle of reflection KCA to be equal to it; since the direction CM is changed into CN, and the direction CA remains; and therefore the mixed direction composed of CN and CA is CK. And so on for the rest. From these things you will observe that the more acute the angle of incidence is, the more in reflection a new direction is mixed with the an- cient one, so that the ancient predominates more; for the new direction, compared to the ancient, in respect of that which remains of it, is as the sine of the right angle of the angle of incidence to the sine of the complement. For if the angle of incidence be FCB, and equal to it HCA, the new direction CD, that is AH, compared to the remainder of the ancient direction CA, is as HA to AC: but if the angle of incidence be ICB, that is, the angle of reflection KCA equal to it, the new direction compared to that which remains of the an- cient is as KA to CA. But the ratio of AC to the smaller AK is greater than that of the same AC to the larger AH, by 8. book 5. Therefore so long as the angle of incidence is less than a right angle, the remainder of the ancient direction is greater (consider carefully that I am speaking only of direction) than the new disposition: where the angle is a right angle, they are equal; if the angle of incidence be greater than a right angle, the new dis- position is greater than that which remains of the ancient direction: where finally there is a right angle in perpendicular inci- dence, the new direction is to the former as Radius to ZZZZZ

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Mechanicorum 732 nihil, motus enim reflexus nihil retinet de priori directione. Antè tamen quàm in hac disputatione procedamus, mentis oculos tantisper in globum C, à quo percutitur planum AB, convertamus; hactenus enim universa contemplatio in meris lineis versata est. Et quidem si directionis linea sit RS perpendicularis transiens per globi centrum C, & punctum contactûs S, nulla esse potest difficultas, quin per eandem lineam SR resiliat ad angulos rectos. Sed si in planum obliquè incidat linea directionis globi per centrum C deducta, & sit MN; certum est in plano AB punctum N, in quod directionis linea MC producta incurrit, non esse punctum contactûs; alioquin linea à globi centro ducta ad punctum contactûs, caderet ad angulos inæquales ex hypothesi, cum tamen angulos rectos constituere demonstretur in Sphæricis. Est igitur contactus in puncto S, extra lineam directionis centri, ideóque angulus reflexionis non est BNQ æqualis angulo ANM. Propterea in globo attendendum est punctum S, à quo reipsâ percutitur planum; & sicuti in circulo globum bifariam dividente punctum I delatum est per lineam MI, ita punctum S per lineam OS ipsi MI parallelam (pono hîc globum non rotari dum movetur, sed recto itinere deduci) venit ad contactum & percussionem plani. Cum igitur OS & MN sint parallelæ, anguli OS A, & MNA sunt æquales: & sicuti si punctum I solitarium esset, atque juxta suam directionem veniret in N, reflecteretur per NQ, ut reflexionis angulus QNB esset æqualis angulo Incidentiæ MNA; ita punctum S globi reflectitur per SP, & angulus reflexionis PSB æqualis est incidentiæ angulo OS A; ac proinde anguli PSB, & QNB sunt æquales inter se. Centrum igitur

Transcription: Translated (English)

Mechanics 732 nothing, for reflected motion retains nothing of the prior direction. But before we proceed in this discussion, let us for a little while turn the eyes of the mind to the sphere C, by which the plane AB is struck; for up to now the whole consideration has been concerned with mere lines. And indeed, if the line of direction RS be perpendicular, passing through the center C of the sphere and the point of contact S, there can be no difficulty that it rebounds through the same line SR at right angles. But if the line of direction, drawn through the center C, MN, should strike the plane obliquely, it is certain that in the plane AB the point N, at which the produced line MC meets, is not the point of contact; otherwise the line drawn from the center of the sphere to the point of contact would, contrary to the hypothesis, fall at unequal angles, whereas in Spherics it is shown to form right angles. The contact is therefore at the point S, outside the line of direction from the center; and therefore the angle of reflection is not equal to angle BNQ, equal to angle ANM. For this reason, in the sphere attention must be given to the point S, by which the plane is in fact struck; and just as in the circle dividing the sphere into two halves the point I was carried by the line MI, so the point S, by the line OS parallel to MI itself (I assume here that the sphere does not rotate while it is moving, but is carried along in a straight path), comes to the contact and impact of the plane. Since therefore OS and MN are parallel, the angles OSA and MNA are equal; and just as if the point I were single, and were to come along its own direction into N, it would be reflected through NQ, so that the angle of reflection QNB would be equal to the angle of incidence MNA; in like manner the point S of the sphere is reflected through SP, and the angle of reflection PSB is equal to the angle of incidence OSA; and therefore the angles PSB and QNB are equal to each other. Therefore the center

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Liber septimus. CAPUT XI. 733 igitur C cùm in directione MN haberet directionem mixtam ex directione CS versus planum, & directione SB, cum im- pediatur à globi soliditate ne ad planum ulteriùs accedat, mu- tatâ directione CS in CR, atque retentâ priore directione SB, habet directionem mixtam CH omnino similem directioni puncti S; atque propterea CH est parallela ipsi SP. Quod si globus rotari intelligatur, loco linearum, de quibus hactenus fuit sermo, concipe plana, in quibus puncta illa suas periodos describerent in motu rotationis; & plana illa essent ad planum reflectens similiter inclinata, ut de lineis dictum est. In cæteris verò corporibus non rotundis idem de eorum re- flexione dicendum est, servatâ analogiâ, quantum ferre potest anomala eorum figura, & dispar partium positio circa centrum gravitatis aut magnitudinis: in multis enim hujusmodi æqua- litas angulorum incidentiæ & reflexionis non exactè servatur. Sic hastam si obliquè contorqueas in rupem, non modò inæ- qualitatem angulorum deprehendes, sed vix reflexionem fieri admittes; quia videlicet extremo hastæ calce rupem tangente, reliquæ partes habentes circa centrum gravitatis inæqualia mo- menta, valdè turbant motum: solùm autem quando hasta in planum impingitur, aut cadit, ad perpendiculum, servatâ re- flexionis regulâ ad angulos rectos resilit; quia tunc partes om- nes circa centrum gravitatis paria habent momenta. Hæc au- tem momentorum diversitas in globo non reperitur, nisi fortè aut deficiat à perfectâ rotunditate, aut centrum magnitudinis non sit idem cum centro gravitatis, adeò ut linea punctum contactûs cum centro gravitatis, jungens non sit plano re- flectenti perpendicularis; tunc enim perturbaretur globi reflexio. Ex dictis satis apertè constat reflexionem non ex impetu de- sumendam esse, sed ex directione motus, cui opponitur corpus reflectens, juxta hujus positionem perpendicularem aut obli- quam: multus enim impetus aliquando officere potest æquali- tati angulorum, si ex collisione corporis impacti cum corpore reflectente, aut alterutrum, aut utrumque notabiliter cedat, adeò ut non contingat sincera reflexio. Cæterùm cum semper in reflexione sit nova directio priori directioni opposita, ali- quid impetus perit pro Ratione oppositionis. Ex quo fit irre- ZZzz 3

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Book Seven. CHAPTER XI. 733 therefore C, when in the direction MN it had a mixed direction from the direction CS toward the plane, and from direction SB, since it is prevented by the solidity of the globe from approaching the plane any further, having changed the direction CS into CR, and retaining the former direction SB, it has a mixed direction CH altogether similar to the direction of point S; and therefore CH is parallel to SP itself. But if the globe is understood to rotate, then, in place of the lines concerning which up to now the discussion has been, conceive planes, in which those points would describe their periods in rotational motion; and those planes would be similarly inclined to the reflecting plane, as has been said of the lines. But in other non-round bodies the same must be said of their re- flection, with the analogy preserved as far as their irregular figure and the unequal position of the parts around the center of gravity or magnitude can allow: for in many bodies of this sort the equal- ity of the angles of incidence and reflection is not exactly preserved. Thus if you strike a spear obliquely against a rock, you will not only detect inequality of the angles, but you will scarcely admit that a reflection takes place; because, namely, when the heel of the spear touches the rock, the remaining parts, having unequal moments around the center of gravity, greatly disturb the motion: only when the spear strikes a plane, or falls, perpendicularly, the rule of reflection being preserved, it rebounds at right angles; because then all the parts around the center of gravity have equal moments. But this diversity of moments is not found in a globe, unless perhaps it departs from perfect roundness, or the center of magnitude is not the same as the center of gravity, so that the line joining the point of contact with the center of gravity is not perpendicular to the reflecting plane; for then the globe’s reflection would be disturbed. From what has been said it is sufficiently clear that reflection should not be taken from impetus, but from the direction of the motion to which the reflecting body is opposed, according to its perpendicular or oblique position: for much impetus can sometimes interfere with the equal- ity of the angles, if from the collision of the striking body with the reflecting body, either one or the other, or both, yield notably, so that a genuine reflection does not occur. Moreover, since in reflection there is always a new direction opposed to the previous direction, some impetus is lost in proportion to the opposition. From this it follows irre- ZZzz 3

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Mechanicorum 734 flexione ad angulos magis acutos impetum minori decremento minui, quia nova directio minùs opponitur antiquæ, & minùs impeditur motus; idcirco globus ad angulum valde acutum re- flexus, si offendat in motu reflexo aliquem obicem, multò va- lidiùs illum percutit, quàm si ad angulum minùs acutum re- flecteretur, quia, cæteris paribus, majore impetu superstite ictum infligit. Quapropter obvium est cuique rationem reddere omnium, quæ in pilæ ludo contingunt circa saltus in pavimento, in quod pila emissa decidit, & reflexiones ad parietem, in quem illa impingitur. Duo tamen potissimùm observare placet. Primùm, quando pila cadit obliqua in pavimentum non procul à pariete, sæpè fit duplex reflexio, altera scilicet à pavimento, altera à pariete: ex quo fit, ut pila aliquando longè altiorem saltum edat, si multum habeat impetus; quia videlicet à pavimento resiliens, si in parietem non incurreret, lineam curvam in re- flexione describens vi suæ gravitatis impetum extrinsecùs im- pressum temperantis, citiùs deprimeretur, & magis à recto tra- mite deorsum deflecteret: at quia proximus ponitur esse paries, linea primò reflexa nondum differt notabiliter à lineâ rectâ; atque proinde in secundâ reflexione altiùs pila assurgit, quàm à pavimento distaret apex lineæ curvæ, quæ ex primâ reflexio- ne describeretur; nam directio illa secunda magis elevata supra horizontem minùs permittit pilam à rectâ lineâ declinare; ut in balistarum & bombardarum globis cum majori elevatione emissis constat. Deinde quando reticulis luditur, non rarò re- ticulum movetur in plano aliquo horizontali, aut valde inclinato (nos Itali dicimus Tagliare, è Trinciare una palla) ita ut, dum pilam rectâ expellit, illi etiam motum quendam imprimat, quo ipsa circa suum centrum movetur: unde fit, ut, nisi pilam excipias, repellásque antè, quàm pavimentum attingat, frustra deinde saltum illius expectes juxta regulas reflexionis, quia ni- mirum pila terram tangens, dum pergit moveri circa suum centrum motu orbiculari, nequit à plano impediente recipere directionem illam, cujus esset capax, si solùm simplici motu centri mota fuisset; motus enim peripheriæ globi contrarius est motui centri. Idem accidit quando pila leviore affrictu funem perstringit; tunc scilicet concipit motum circularem, adeóque saltus

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Mechanics 734 By reflection at sharper angles, the impact is diminished by a smaller decrease, because the new direction opposes the old one less, and the motion is less hindered; therefore a ball reflected at a very acute angle, if in its reflected motion it strikes some obstacle, hits it much more strongly than if it were reflected at a less acute angle, because, other things being equal, with a greater impulse remaining it delivers the blow. For this reason it is plain to anyone to give an account of everything that occurs in the game of the ball with regard to the rebounds on the pavement on which the thrown ball falls, and the reflections from the wall against which it strikes. Two things, however, it is especially worth observing. First, when the ball falls obliquely upon the pavement not far from the wall, a double reflection often occurs, namely one from the pavement, the other from the wall: from which it happens that the ball sometimes makes a much higher rebound, if it has much force; because, to wit, after rebounding from the pavement, if it did not run into the wall, describing a curved line in its reflection while the force of its gravity moderates the impulse impressed from without, it would be lowered more quickly and would deflect more from the straight path downward: but because the wall is assumed to be near, the first reflected line does not yet differ notably from a straight line; and therefore in the second reflection the ball rises higher than the apex of the curved line would be from the pavement, which would be described from the first reflection; for that second direction, being more elevated above the horizon, allows the ball less to deviate from the straight line; as is evident in the balls of ballistae and cannon shot projected with greater elevation. Then, when playing with nets, the net is often moved in some horizontal plane, or a very inclined one (we Italians say Tagliare, è Trinciare una palla), so that, while it drives the ball straight on, it also imparts to it a certain motion by which it moves around its own center: whence it happens that, unless you catch the ball and strike it back before it reaches the pavement, afterwards you will look in vain for its rebound according to the rules of reflection, because, namely, when the ball touches the ground, while it continues to move around its own center with circular motion, it cannot receive from the impeding plane that direction of which it would be capable if it had been moved only by the simple motion of its center; for the motion of the globe’s periphery is contrary to the motion of the center. The same happens when the ball brushes a rope with a lighter graze; then indeed it takes on a circular motion, and thus the rebound

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Liber septimus. CAPUT XI. 735 saltus fallit. Quantum autem in motu valeat directiones com- miscere, alteram centri rectam, alteram peripheriæ circularem sed oppositam, satis norûnt, qui minoribus orbiculis ludentes globum quasi pendentem ex manu tenent, dumque illum pro- jiciunt, manu ei motum circularem communicant; unde oritur, quod, ubi terram globus attigerit, vel sistit se, si directio peri- pheriæ ad motum circularem est æqualis directioni centri ad motum rectum; vel tardiùs promovetur, quàm si solam centri directionem haberet, prout directio centri major est directione peripheriæ, quæ cum primùm terram attingit, apta est suâ con- versione retrahere centrum versùs projicientem. Quandoquidem verò in ludicris philosophamur, liceat hîc vulgarem errorem retegere; quando scilicet rotundum verticil- lum impressus impetus in gyrum agit, si verticillus corruat, mo- vetur, quoad impetus extinguatur, sed ita ut videatur omnino in contrarias partes agi, ac priùs: id quod communiter tribuunt reflexioni, quia in pavimentum recidit. Nullam hîc reflexio- nem intercedere, & eandem permanere motûs directionem, memini me aliquando asserentem visum fuisse pluribus, qui aderant, paradoxum loqui: sed ubi inter nos convenit eundem motum esse, quandiu circà axem ita fit convolutio, ut quæ par- tes peripheriæ verticilli præcedebant axe insistent, eædem axe inclinato & procumbente præcedant; jussi aliquas notas peri- pheriæ imprimi, ut priores à posterioribus discerni possent; de- inde verticillo corruente, & procumbente observatum est par- tes, quæ priores erant in circuitione, easdem subinde altiùs at- tolli à pavimento, atque circa axem eandem fieri conversio- nem: & quia verticilli pes quasi centrum retinet inclinatam peripheriam, illa eadem conversio circa axem facit, ut peri- pheria secundùm posteriores partes subinde attingat subjectum alveolum, adeóque ratione habitâ alveoli videatur in contrarias partes ferri ac priùs. Quare cùm nulla sit nova motûs directio ex plani oppositione, nulla quoque est reflexio. At si duo corpora sibi invicem occurrant, sibi mutuo ob- sistunt, & diminuto ex resistentiâ impetu, si quid adhuc resi- duum fuerit impetus, qui excedat insitam repugnantiam ex gravitate ortam, fit reflexio, aut alterius tantùm, si in reliquo impetus obtundatur, aut utriusque, si fuerint sibi invicem per- cutiens

Transcription: Translated (English)

Book Seven. Chapter XI. 735 slips past. But how much in motion it can avail to mix directions, one straight from the center, the other circular from the circumference but opposite, those know well enough who, playing with small hoops, hold the ball as if suspended from the hand, and while they throw it, communicate circular motion to it by the hand; whence it comes that, when the ball has touched the ground, it either comes to a stop, if the direction of the circumference toward circular motion is equal to the direction of the center toward straight motion; or it advances more slowly than if it had only the direction of the center, according as the direction of the center is greater than the direction of the circumference, which, as soon as it touches the ground, is fitted by its turning to draw the center back toward the thrower. And since, however, we are philosophizing in playful matters, let it be permitted here to expose a common error; namely, when a round top, driven by an impressed impulse, is set spinning, if the top falls, it continues moving until the impulse is extinguished, but in such a way that it seems to be driven altogether in the opposite direction from before: this is commonly attributed to reflection, because it falls back onto the floor. I remember that I once asserted that no reflection intervenes here, and that the same direction of motion remains; several who were present thought me to be speaking paradoxically: but when it was agreed among us that the motion is the same, so long as the revolution about the axis is such that those parts of the circumference of the top which were leading stand upon the axis, and, the axis being tilted and falling, the same parts continue to lead; I had certain marks made on the circumference, so that the former parts could be distinguished from the latter; then, when the top fell and lay prone, it was observed that the parts which had been first in the circular motion were thereafter lifted higher from the floor, and that the same revolution was being made about the axis: and because the foot of the top, as it were, keeps the center in the inclined circumference, that same revolution about the axis makes the circumference, by way of the later parts, touch the supporting basin in turn, and thus, if regard is had to the basin, it seems to move in the opposite direction from before. Therefore, since no new direction of motion arises from the opposition of the plane, there is likewise no reflection. But if two bodies meet one another, they resist each other, and, the impetus having been diminished by the resistance, if any residue of the impetus remains that exceeds the innate repugnance arising from gravity, reflection takes place, either of only one body, if the impetus is weakened in the rest, or of both, if they shall have been striking one another

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Mechanicorum 736 cutiens & percussum: ut cùm duo globi sibi in motu occur- runt aut æquali, aut non immodicè inæquali impetu acti. Æquali inquam, impetu, non æquali velocitate; si enim inæqua- les fuerint globi, fieri potest, ut eorum velocitates sint in Re- ciproca Ratione gravitatum; tunc scilicet impetus æquales sunt; contingere siquidem potest majorem globum tardè quidem moveri, sed multo impetu respondente ejus moli, adeò ut ex- cedat minoris globi impetum, qui proptereà non præcisè re- flectatur, sed à majore globo & impetum recipiat, & directionem non ex solâ resistentiâ definitam, sed etiam ex ipsius ma- joris globi motu. Id quod si contingat, minor quidem reflecti- tur, sed qui majore impetu ferebatur, modicam inveniens re- sistentiam non reflectitur, quia dum minor globus cedit, plu- rimum impetûs deperditur à majore; & ubi resistentia minor est cessione, esse nequit reflexio. Ponamus itaque globos duos tanto impetu actos, ut possit uterque reflecti. Non placet inter illos ad punctum contactûs interjicere planum, ut ex angulis determinetur reflexio; hoc enim planum cogitatione nobis ipsi fingimus; sed, licèt in idem res recidat, tamen ad veritatem sinceriùs me accessurum spero, si rem ex ipsis motuum directionibus & resistentiis defi- niero. Quare occurrant sibi globi in puncto A; & illo- rum directiones primò ex sint, quæ sibi maximè ad- versantes in rectam lineam BC coëant: haud dubium quin globus V per rectam AB, & globus R per rectam AC resiliat, uterque per li- neam, quâ venit, jungentem cum puncto contactûs Centrum impetûs: simplici enim di- rectione alter adversùs alterum fertur, & sibi toto conatu re- pugnant. Deinde obliquus sit morus, & globi R directio sit DE: quapropter RE directio quædam est mixta ex lineâ maximæ re- sistentiæ RA, & ex lineâ nullius resistentiæ RI, ita ut partim versùs A, partim versùs I tendat: priorem directionem versùs A metitur linea RF, posteriorem versùs I metitur linea FE. Cum igitur

Transcription: Translated (English)

Mechanics 736 struck and struck back: as when two bodies in motion collide with one another, driven with either equal, or not greatly unequal, force. Equal, I say, in force, not equal in velocity; for if the bodies be unequal, it may happen that their velocities are in the reciprocal ratio of their weights; then indeed the forces are equal. For it may happen that a larger body is moved slowly, yet with a force corresponding to its mass so great that it exceeds the force of the smaller body; wherefore the smaller is not reflected exactly, but receives force from the larger body, and a direction not determined by resistance alone, but also by the motion of the larger body itself. If this happen, the smaller indeed is reflected; but that which was carried with greater force, finding only a slight resistance, is not reflected, because while the smaller body gives way, much of the force is lost from the larger; and where the resistance is less than the yielding, reflection cannot take place. Let us therefore suppose two bodies driven with such force that each may be reflected. It does not please me to interpose between them, at the point of contact, a plane, so that the reflection may be determined from angles; for we ourselves imagine this plane by thought; but, though the matter comes to the same thing, still I hope to come more sincerely to the truth if I define the matter from the directions of the motions themselves and from the resistances. Wherefore let the bodies meet at point A; and let their directions at first be such that they converge most directly into the straight line BC: there is no doubt that body V will rebound along AB, and body R along AC, each along the line by which it came, joined to the point of contact, the center of force: for with a simple direction each is borne against the other, and they resist one another with their whole effort. Next let the motion be oblique, and let the direction of body R be DE: therefore the direction RE is a certain mixed line, made up of the line of greatest resistance RA and the line of no resistance RI, so that it tends partly toward A, partly toward I: the former direction toward A is measured by line RF, the latter toward I is measured by line FE. Since therefore

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Liber septimus. CAPUT XII. 737 igitur globus V solùm priori directioni R F opponatur, hæc mutatur in oppositam R G, manet autem directio versùs I æqualis ipsi F E, & est G H: propterea motus centri R est R H parallelus motui puncti A, quod per lineam K A incidens in Tangentem M N, reflecteretur ad angulos æquales K A N & L A M percurrendo lineam A L. Simili ratione, si globi V di- rectio sit P O, à globo R occurrente in A reflectitur centrum per rectam V S. CAPUT XII. Quomodo impetus in percussione communicetur. Antè satisfaciendum est Physicis, quàm percussionum con- templationem dimittamus. Quoniam percussio omnis mo- tum antecedentem exigit; motus non habetur absque impetu concepto aut impresso; ex impetu pendet ictus, quo corporis percussi resistentia aliqua vincitur, sivè illud totum impellatur, sivè expellatur, sivè concutiatur, sivè flectatur, sivè compri- matur, sivè deprimatur, sivè dissiliat in partes earum unione solutâ, sivè quamcumque aliam vim subeat; corporis percussi partes, vel omnes, vel aliquæ saltem, moveantur, & impetum recipient necesse est, à quo motus ipse efficiatur impressi impe- tûs intensioni respondens. Quærat autem Physicus, cuinam tribuenda sit virtus efficiendi impetum corpori percusso im- pressum. Existimabit fortasse non nemo à virtute eâdem, quæ in cor- pore percutiente insidet, ut seipsum moveat, effici novum im- petum, quo corpus percussum impellatur, aut agitetur. Sed quid? si percutiens neque animans sit, cujus in potestate posita sit motio, neque juxta insitæ gravitatis directionem seipsum agat. Huic certè inhærens facultas se movendi planè otiosa est, quippe quæ prorsus immota consisteret, nisi impetum extra- neum reciperet. Aliunde igitur quàm ex hac se movendi facul- tate originem ducit impetus corpori percusso impressus. Dein- A A a a a

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Book Seven. CHAPTER XII. 737 therefore let the globe V alone oppose the former direction R F; this is changed into the opposite R G, but the direction toward I remains equal to itself F E, and is G H: therefore the motion of the center R is R H, parallel to the motion of point A, which, passing through line K A into the tangent M N, would be reflected with equal angles K A N and L A M, traversing line A L. In a similar way, if the direction of globe V be P O, the center, the globe R meeting it in A, is reflected through the straight line V S. CHAPTER XII. How impulse is communicated in percussion. First the Physicists must be satisfied before we dismiss the consideration of percussion. Since every percussion requires a preceding motion; motion is not had without a conceived or impressed impulse; upon impulse depends the blow, by which some resistance of the struck body is overcome, whether it be wholly driven on, or expelled, or shaken, or bent, or compressed, or depressed, or burst asunder into parts with their union dissolved, or undergo whatever other force; the parts of the struck body, whether all, or at least some, must be moved, and will receive an impulse, from which the motion itself is produced corresponding to the intensity of the impressed impulse. But let the Physicist inquire to whom the power is to be attributed of producing the impulse impressed upon the struck body. Perhaps someone will think that from the same power which dwells in the striking body to move itself, a new impulse is produced, by which the struck body is driven on, or agitated. But what if the striker be not animate, with motion placed in its power, nor move itself according to the direction of the inherent gravity. In such a case the faculty of moving itself adhering to it is certainly idle, since it would remain altogether unmoved unless it received an extraneous impulse. Therefore the impulse impressed upon the struck body derives its origin from somewhere else than from this faculty of moving itself. Then A A a a a

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Mechanicorum 738 de certum est corporis percutientis naturam non priùs imprime- re posse percusso impetum; quàm illud attingat: at in ipso per- cutientis appulsu ea est percussi resistentia, ut ejusdem percu- tientis motum ex ipsâ naturâ provenientem imminuat: cùm igitur natura percutientis vix seipsa movere valeat, quàm te- nues habet vires ad vincendam obicis resistentiam? Præterea, nisi facta fuerit notabilis in longiore motu naturali acquisiti im- petûs accessio, manifestò apparet valdè languida & enervata percussio; &, quamvis sivè longior, sivè exiguus motus præ- cesserit, eadem manens virtus movendi, nec sibi dissimilis, va- rietatem in se habet nullam: cum tamen ex disparibus incre- mentis impetûs in motu acquisiti dissimiles fiant percussiones: Non igitur à solâ insitâ vi movendi producitur in percusso im- petus. Propterea, ut una atque eadem in percussionibus omnibus assignetur produciti impetûs causa, sivè percutiens sponte suâ, sivè per vim sibi illatam moveatur, percutientis impetum plu- res consent dicendum esse principium & causam effectricem impetûs percusso impressi; ab illo enim, prout major fuerit, aut minor, hujus menfuram pendere satis innotuisse videtur ex quotidianis experimentis. Verùm, ne raptim in hanc sententiam pedarius Philosophus curram, illud me remoratur, quod, sicuti eam esse constat im- petûs naturam, ut illico prorsus pereat, ac motus cessat omni- no illius corporis, in quo priùs inerat motum efficiens, ita pari- ter eodem momento impetum minui necesse est, eáque Ratio- ne, quo momento, & qua Ratione illius ejusdem corporis mo- tus ex parte impeditur. Quò igitur magis impeditur percutien- tis motus, eò magis ejusdem impetum minui consequens est: propterea, quo momento à percutiente attingitur corpus per- cussum, extenuatur in illo impetus, quia tunc illius motus im- peditur; eóque minor evadit in percutiente impetus, quò ma- jus invenit impedimentum motûs. Cùm autem effectui tenui- tatem importet causæ imbecillitas, exiguum utique impetum in corpore percusso efficere valeret attenuatus percutientis im- petus, quo momento accidit appulsus atque allisio; eóque mi- norem impetum reciperet corpus percussum, quò magis re- sistens plus inferret impedimenti motui percutientis, quippe cujus

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Mechanics 738 it is certain that the nature of the striking body cannot first impart an impulse to the body struck before it reaches it; but at the very moment of impact there is in the struck body such resistance that it diminishes the motion of the striking body itself, arising from its own nature. Since, therefore, the nature of the striker can scarcely move itself, how slight are its powers for overcoming the resistance of the obstacle? Moreover, unless there has been a notable addition of acquired impetus in a longer natural motion, it clearly appears that the impact is very weak and enervated; and, whether a longer or a slight motion has preceded, the same power of moving remaining, and not unlike itself, has no variety in it: whereas from unequal increases of impetus in acquired motion dissimilar impacts arise. Therefore the impetus impressed upon the struck body is not produced by the innate power of moving alone. For this reason, in order that one and the same cause of the produced impetus may be assigned in all impacts, whether the striker moves of its own accord or by force applied to it, the impetus of the striker must be said to be the principal cause and efficient source of the impetus impressed on the struck body; for from it, according as it is greater or less, the measure of this seems sufficiently known from daily experience. But, lest I should too hastily run to this opinion like a mere pedestrian philosopher, one thing holds me back: namely, that, just as it is known to be the nature of impetus to perish altogether at once, as soon as the motion of that body in which it previously existed, producing motion, ceases entirely, so likewise at the same moment the impetus must be diminished, and for the same reason, at the moment and in the same way that the motion of that body is partially impeded. Therefore, the more the motion of the striker is impeded, the more its impetus necessarily diminishes; hence, at the very moment when the struck body is reached by the striker, the impetus in it is weakened, because then its motion is impeded; and the impetus in the striker becomes the smaller, the greater the obstruction it encounters to its motion. But since weakness of cause implies thinness or slightness in the effect, an attenuated impetus of the striker would indeed be able to produce a slight impetus in the struck body at the moment when the approach and collision occur; and the struck body would receive a smaller impetus the more the resisting body brought greater obstruction to the motion of the striker, since it would

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Liber septimus. CAPUT XII. 739 cujus impetus fieret languidior; neque enim quicquam juvat antiqua virtus, si nunc est effoeta. Quò igitur magis resistit corpus percussum, languidiorem ictum exciperet, cum levior insirmiórque impetus in eo efficeretur à tenuiore & languidio- re percutientis impetu. Sed cum manifesta refragetur expe- rientia validiores ictus à majore resistentia ortos demonstrans, quæso à Philosophis, ut in hac causâ mihi dent hanc veniam, ut patiantur me ab eorum placitis aliquantulum discedere, nec percutientis impetui tribuere facultatem effectricem impetûs in corpore percusso, lyceo quamvis reclamante; cui silentium si tantisper indicere possem, dum me audiret postulantem id, quod æquissimum est, ut ne quid huc præjudicati afferat, meam fortasse in sententiam volens deduceretur. Cùm itaque nec à virtute movendi, quæ corpori percutien- ti inhæret, nec ab impetu ejusdem percutientis effici novum impetum in corpore percusso, satis probabili conjecturâ dicen- dum videatur, quænam demum erit causa impetûs, & eorum, quæ impetum consequuntur, in corpore percusso? Ut quæstio- nibus satisfiat, quas percussiones excitant, nihil se mihi offert vero propius, quàm si dicamus ex percutiente in corpus per- cussum migrare impetum, aut totum, aut ex parte, prout alicu- jus motûs capax fuerit corpus, quod motui percutientis resistit. Si totus impetus à percutiente recedat, hoc neque reflectitur ab obice percusso, neque quicquam procedit in motu: Si quid impetus in percutiente remaneat, hoc aut juxta institutam di- rectionem pergit moveri unà cum corpore percusso, sive lentiùs illud sequitur, aut aliò reflectitur, pro residui impetûs inten- sione, aut vibratur, & concutitur. Hinc quia gravissima simul & durissima corpora tantum im- petûs obtinere à percutiente nequeunt, quanto opus esset, ut motum aliquem conspicuum ex percussione reciperent, pro- pterea validissimè resistunt, & reflectunt, cùm universus ferè impetus in percutiente remaneat: in corpus enim percussum non migrat nisi impetus, qui respondeat motui, cujus illud tunc est capax. Contra verò à corporibus, quæ leviter re- sistunt, & facilè moventur aliquo motu, aut nihil, aut langui- dè reflectitur percutiens; quia illa plurimum impetûs reci- piunt, & exiguus impetus in percutiente reliquus est. Hinc A A a a a 2

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Book Seven. Chapter XII. 739 whose force would become weaker; for ancient strength is of no help if it is now spent. Why then should the body struck, the more it resists, receive a weaker blow, when a lighter and weaker force would be produced in it from the lighter and weaker force of the striker? But since clear experience opposes this, showing that stronger blows arise from greater resistance, I ask the Philosophers, that in this matter they grant me this indulgence, that they allow me to depart somewhat from their doctrines, and not to assign to the force of the striker the power that produces force in the body struck, though the lycée may protest; if I could impose silence on it for a little while, while it listened to me asking what is most just, namely that it should bring in no prejudice here beforehand, it might perhaps be led willingly to my view. Since therefore it does not seem sufficiently probable to say that the power of moving, which inheres in the striking body, nor the force of the same striker, produces a new force in the body struck, what, after all, will be the cause of the force, and of the things that follow from force, in the body struck? In order to satisfy the questions that blows raise, nothing seems to me closer to the truth than if we say that the force passes from the striker into the struck body, either wholly or in part, according to how much motion the body that resists the motion of the striker is capable of. If the whole force departs from the striker, it is neither reflected by the struck obstacle nor does anything proceed in motion: if some force remains in the striker, this either continues to move together with the struck body in the direction already established, following it more slowly, or is reflected elsewhere, or vibrates and is shaken, according to the intensity of the remaining force. Hence, because the heaviest and hardest bodies cannot receive from the striker as much force as would be needed for them to receive any visible motion from the blow, for that reason they resist and reflect most strongly, since almost the whole force remains in the striker: for into the struck body there passes only so much force as corresponds to the motion of which it is then capable. On the other hand, in bodies that resist only lightly and are easily moved in some motion, the striker is reflected either not at all or only weakly; because those bodies receive a great part of the force, and only a small force remains in the striker. Hence A A a a a 2

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74o Mechanicorum pariter globus æqualem in mole & gravitate globum percutiens eâ directione, quæ per utriusque globi centra transeat, consistit in loco, ubi percutit, & percussum globum vehementer excutit; quia videlicet globus æqualis satis resistit, & capax est totius impetûs eum æquali intensione afficientis, hic destituens globum percutientem æquè velocem motum percusso conciliat, & percutiens omni destitutus impetu consistit. Sin autem percutiatur globus major & gravior; hic quidem (nisi nimia sit gravitatis aut molis differentia) loco cedit; sed quia ad motum æquè velocem plus requirit impetus, quàm illi imprimere valeat globus minor, propterea minore intensione affectus tardiùs movetur, & minorem globum aliquando reflectit. Si demum globus major minorem & leviorem percutiat, hic languidiùs resistens impetum recipit velociori motui congruum; & quia in globo majore adhuc aliquid superest impetus, ille pariter pergit moveri, sed tardiùs. At, inquis, impetus ex eo genere est, quod Accidentia tanquam partes complectitur: Accidentia autem ex subjecto in subjectum non transire, ipsi scholarum parietes clamant. Multa istiusmodi, non diffiteor, dicuntur in scholis: verùm an satis examinata, momentóque suo ponderata fuerint, ignoro: non pauca quippe habemus de manu, ut aiunt, in manum tradita, non ad aurificis stateram revocata, sed populari trutinæ permissa. In illis certè Accidentium generibus, quæ postremis novem Categoriis comprehenduntur, si sex demas, Relationem, Actionem, Passionem, Ubi, Quando, Situm, quos alij (liberaliter ne ? dicam, an prodigè ?) Modos certæ naturæ, à qua avelli nequeunt, affixos appellant, alij minimo contenti, & parciùs philosophantes, nihil esse præter mera nomina, aut abstractas à rebus inter se comparatis intelligentias existimant; vix tria reliqua genera Quantitas, Qualitas, Habitus constituere controversiam possunt. Et quidem de Habitu nullus videtur relictus ambigendi locus; quis enim neget potuisse Thersitem eâdem Achillis galeâ, eodemque thorace armari, & regiâ chlamyde servum indui ? mutatâ scilicet armorum aut indumentorum Ubicatione comparatâ cum hominis, qui armatus dicitur, aut vestitus, Ubicatione & positione. Quantitatem verò, qua locus obsidetur (nam de Numero, qui præter individua cogita- tioni

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74th Of Mechanics Likewise, when a globular body striking an equal globular body in mass and weight, in that direction which passes through the centers of both globes, comes to rest in the place where it strikes, and violently drives the struck globe away; because, namely, the equal globe resists sufficiently and is capable of the whole impulse affecting it with equal intensity, thereby leaving the striking globe, it imparts to the struck globe an equally swift motion, and the striking globe, deprived of all impulse, comes to rest. But if a greater and heavier globe be struck, this indeed yields its place (unless the difference in gravity or bulk be too great); yet because a greater impulse is required for equally swift motion than the smaller globe can impress on it, therefore, being affected with lesser intensity, it moves more slowly, and sometimes reflects the smaller globe. Finally, if a greater globe strikes a smaller and lighter one, this, resisting more languidly, receives an impulse fitting a swifter motion; and because there still remains some impulse in the greater globe, it likewise continues to move, but more slowly. But, you say, impulse is of that kind which includes accidents as parts: but accidents, the very walls of the schools cry out, do not pass from subject to subject. Many things of this sort, I do not deny, are said in the schools; yet whether they have been examined enough and weighed with due consideration, I know not: for we have many things, as they say, handed from hand to hand, not brought back to the goldsmith’s scales, but left to the common balance. Certainly, in those kinds of accidents which are comprehended in the last nine Categories, if you remove six—Relation, Action, Passion, Where, When, Position—which others call (shall I say liberally? or prodigally?) modes affixed to a certain nature from which they cannot be detached, while others, being content with the least, and philosophizing more sparingly, think them nothing more than mere names or abstractions from things compared with one another—scarcely can the remaining three kinds, Quantity, Quality, and Habit, make up a controversy. And indeed, as to Habit, no room seems left for doubting; for who would deny that Thersites could have been equipped with the same helmet as Achilles and with the same breastplate, and that a servant could be clothed in a royal cloak? namely, by changing the placing of the arms or garments, compared with the placing and position of the man who is said to be armed or clothed. As for Quantity, however, by which space is occupied (for concerning Number, which beyond individuals belongs to thought...

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Liber septimus. CAPUT XII. 741 tioni ea complectenti subjecta nihil est, non attinet dicere) quam multi à materiâ non dividunt? quot Philosophi suam sin- gulis corporeis rebus tribuunt quantitatem? De solâ igitur Qualitate oriri potest quæstio: cujus tamen species aliæ me- ram membrorum aut terminorum corporis collocationem & conformationem dicunt, ut Forma & Figura; aliæ particula- rum in extimâ superficie positionem, ut asperitas & lævor; aliæ earumdem toto corpore diffusarum complexionem, ut molli- tudo & durities, raritas & densitas; aliæ non nisi intelligentiâ secretæ accidere dicuntur Naturæ, cujusmodi non paucæ Na- turales Potentiæ & Impotentiæ; aliæ Patibiles Qualitates aut Passiones immissione corpusculorum effluentium communican- tur, quemadmodum Odores & Sapores, & fortè etiam quas Pri- mas Qualitates vocant. Sed quicquid tandem de hujusmodi Accidentibus asserere placeat (neque enim hîc de iis philosophandi est locus) ultrò demus ea esse, quæ licèt à substantiâ distinguantur, per se ta- men stare nequeant, & necessariò subjectam aliquam naturam afficiant, in qua inhærereant: verùm Qualitates omnes (nisi ex earum genere sint, quos Modos appellant, quia Actuales De- terminationes, cujusmodi sunt cogitationes, appetitiones, & motus, quibus actio vitæ continetur) quid prohibet nunc huic, mox illi subjecto inhærere, quemadmodum in locum pereun- tis Causæ Effectricis, cujus virtute hactenus conservabantur, aliam substitui causam, cujus vi adhuc permaneant, omnes fa- temur? Nonne causâ effectrice magis indigent Accidentia, quàm Materiali & Subjectivâ? Divinâ siquidem vi accidentia à Subjecto avulsa permanere posse docemur ex Mysteriis Eu- charisticis; at sinè ullâ causâ effectrice consistere nullatenus possunt: hanc subinde permutant citrà Naturæ incommodum; quidni & subjectum? Nihil igitur extra modum absonum & ab- surdum loquatur, qui impetum migrantem ex percutiente in percussum ita subjectum mutare dixerit, quemadmodum om- nes novum impetum à percutiente malleo produci in percusso & excusso globo opinantes, aliam ejusdem impetûs, quandiu durat, causam, à qua conservetur, ultrò admittunt. Quamvis autem hoc cæteris qualitatibus ratum ac firmum esset, quod ita subjecto, cui semel inhæserint, affigantur, ut A A a a a 3

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Book Seven. CHAPTER XII. 741 (to say nothing here of that to which that operation is subject, as comprehending it) how many do not distinguish it from matter? how many philosophers assign their quantity to each corporeal thing? Therefore the question can arise only about Quality: some species of which say that it is merely the arrangement and conformation of the members or boundaries of the body, as Form and Figure; others, the position of the same parts on the outer surface, as roughness and smoothness; others, the complexion of the same parts diffused through the whole body, as softness and hardness, rarity and density; others are said to occur only through an intelligible hidden nature, such as many Natural Powers and Impotencies; others, Patible Qualities or Passions, are communicated by the influx of effluent tiny bodies, as Odors and Tastes, and perhaps even those which they call Primary Qualities. But whatever may finally please us to assert about accidents of this kind (for this is not the place to philosophize about them), let us readily grant that they are things which, although distinguished from substance, nevertheless cannot stand by themselves, and necessarily affect some underlying nature in which they inhere: yet what prevents all qualities (unless they are of the kind they call Modes, because they are actual determinations, such as thoughts, appetites, and motions, by which the action of life is sustained) from adhering now to this subject, now to that, just as, in place of a perishing efficient cause, by whose power they were hitherto preserved, another cause is substituted, by whose force they still remain, as we all admit? Do not accidents need an efficient cause rather than a material and subjective one? Indeed, by divine power we are taught from the mysteries of the Eucharist that accidents can remain separated from the subject; but without any efficient cause they can in no way subsist. This they then change without inconvenience to nature; why not the subject as well? Therefore, he says nothing at all exceptionally strange or absurd who says that an impulse migrating from the striking body to the body struck changes the subject in this way, just as all those who think that a new impulse is produced in the struck and rebounding sphere by the hammer striking it also readily admit another cause of the same impulse, so long as it lasts, by which it is preserved. And although this were established and firm with regard to the other qualities, namely that they are so attached to the subject to which once they have adhered, that A A a a a 3

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742 Mechanicorum aut in illo insidere necesse sit, aut interire; impetui tamen privatam legem à Naturâ irrogatam fuisse non est incongruum, quippe qui motui efficiendo, & locorum commutationi, tamquam proxima causa, destinatus est; si enim illi corporum translatio tribuenda est, quidni & ipse à corpore, quod jam commovere nequit ob resistentiam, in aliud corpus proximum faciliùs mobile transmittat, ut submoveatur impedimentum: Neque mihi videor temerè in hanc sententiam discessisse: observavi scilicet quàm multum intersit in vehementi brachij projectione, si verè lapidem manu longiùs excutias, ac si tantummodo, eâdem quidem contentione, sed manu vacuâ, te lapidem jactare mentiaris: hoc enim postremum sinè dolore non accidit, quia impetus à brachio in lapidem jactandum transferendus si in brachio permaneat, hoc secum rapit, & nexum distrahit, quo tenetur cum humero colligatum. At ne fortè me potiùs opinionis commento, quàm re ductum suspiceris (quamquam & alij hunc eundem brachij dolorem experientes non semel probârunt) balistæ arcum chalybeum intento nervo inflecte, ac sæpiùs, nullo adjecto globo aut telo, quod explodat & ejiciat, submoto nervi adducti retinaculo dimitte: an diutiùs inani hoc ludo uti licebit? sexcenties utique & millies balistâ hac globos argillaceos ejaculaberis citrà arcûs detrimentum; sed non item sine incommodo sæpiùs vacuum nervum dimittes, quin arcus ipse in periculum ac discrimen vocetur, ne facilè disrumpatur: impetus siquidem, quem missili imprimere oportuit, in arcu, dum sese vi elasticâ restituit, permanens illum validiùs concutit, ac sæpiùs labefactans demum diffindit. Quapropter, cùm ex projectionibus satis habeamus argumenti, posse impetum ex projiciente migrare in projectum, quo momento projicitur; cur non item poterit impetus ex percutiente in percussum transire, quo momento percutitur, prout hoc motum aliquem concipere potest pro impetûs Ratione? Neque ut percussi impetum à percutientis virtute tunc primò productum adstruas, conferenda est Percussio cum Impulsione; non enim par est in Percussione aut Projectione, atque in Simplici Impulsione aut Tractione philosophandi ratio: Potentia enim corpori impulso aut raptato applicata quandiu cum illo nectitur, & se, & illud movet quasi corpus unum ex utro- que

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742 Mechanics either it must be contained within it, or perish; nevertheless it is not improper to say that an impulse has had a private law imposed on it by Nature, since it is destined, as the proximate cause, to effect motion and change of place. For if the transference of bodies is to be attributed to it, why should not impulse itself, when the body it has set in motion can no longer move because of resistance, be passed on from that body to another neighboring body that is more easily movable, so that the obstacle may be removed? Nor do I think that I have rashly departed to this opinion. I have observed, namely, how much difference there is, in a vigorous throwing of the arm, if you truly fling a stone farther with your hand, and if you merely pretend to throw the stone with the same effort, but with an empty hand: for the latter does not happen without pain, because the impulse that ought to be transferred from the arm to the stone to be thrown, if it remains in the arm, drags the arm along with it and strains the connection by which it is bound to the shoulder. But lest perhaps you should suspect that I have been led rather by a contrivance of opinion than by fact (although others too, who have experienced this same pain in the arm, have more than once confirmed it), bend the steel bow of a crossbow with the string drawn tight, and repeatedly, with no bolt or missile added to be discharged and sent forth, release the string after removing the catch that holds it drawn: will it be possible to continue this idle play for long? Certainly you may discharge clay pellets from such a crossbow six hundred times and a thousand times without injury to the bow; but not likewise, without inconvenience, may you repeatedly release the empty string, without the bow itself being brought into danger and peril of easily breaking: for the impulse which ought to have been imparted to the missile, remaining in the bow while it restores itself by elastic force, shakes it more violently, and, after often weakening it, at last splits it apart. Wherefore, since from projections we have sufficient evidence that impulse can pass from the projector into the projected object at the moment when it is projected, why likewise should not impulse be able to pass from the striker into the thing struck, at the moment when it is struck, since this can conceive some motion in proportion to the impulse? Nor, if you would establish that the impulse of the struck body is first produced by the force of the striker, is Percussion to be compared with Impulsion; for the manner of philosophizing in Percussion or Projection is not the same as in Simple Impulsion or Traction: for the power applied to a body that is pushed or dragged, so long as it is joined with that body, moves both itself and that body, as though they were one body made out of both

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Liber septimus. CAPUT XII. 743 que conflatum: propterea sicut musculi in animante ossa sibi cohærentia attollentes & se movent, & ossa; ita potentia Vecti applicata & se movet, & vectem, & pondus, atque equi cur- rui adjuncti non modò seipsi, sed & currum, trahentes movent. At Percussio sæpè corpus percussum procul à percutiente ejicit, quemadmodum & Projectio. Quod si cum Percussione junga- tur Impulsio (quæ semper Projectionem præcedit) impetus in Impulsione producitur à potentiâ impellente; sed sicut momen- to Projectionis qui erat in projiciente impetus, migrat in pro- jectum, quod discedit; ita in Percussione primo Percussionis mo- mento transit impetus in corpus percussum pro ejus capacitate: quod si præterea impellatur à corpore percutiente, cujus motus juxta suam directionem procedat, & urgeat partes corporis per- cussi (ut in iis, quæ deprimuntur, aut comprimuntur contin- git & cùm sublicas, dum panguntur, fistuca ex casu non resi- liens impellit) impetum aliquem habet ab impellente pro- ductum præter impetum ab eodem tamquam percutiente, ipso percussionis momento communicatum: sed qui ab impellente efficitur, non admodum multus est, si cum eo componatur, qui ex percussione habetur. Simile quid Impulsioni, quæ Percussionem sequitur, habe- tur in Tractione, quam Excursus præcessit, in quo acquisitus est impetus: quo enim momento Excursus cessat, & incipit Tractio, transit impetus, & minuitur in trahente; ut si lapis in pavimento jacens fune jungatur alteri lapidi paulò minori, fu- nis autem orbiculo versatili insideat, & lapis ille minor cadens, donec funem intendat, impetum ex motu acquirat; statim ac intentus est funis, & lapis jacens descendentis lapidis motui re- sistit impetus acquisitus migrat ad vincendam jacentis lapidis resistentiam, atque acceptâ à trahentis motu directione cogi- tur ascendere, quandiu alter descendit, & hunc aliquantulum trahit; sed impetu impresso languescente in lapide graviore hic descendit, & sursum vicissim rapit eum, à quo vim passus fue- rat. Sic potentia velociter languidum funem intendens mul- tum concipit impetum, quem ponderi adnexo imprimit, dum illo destituitur, cum primùm resistentiam patitur, sed & aliam impetûs particulam trahendo producit atque efficit in pondere. Cum

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Book Seven. CHAPTER XII. 743 which is formed together: therefore, just as muscles in an animal, lifting the bones cohering to them and moving themselves, move both themselves and the bones; so also the power of the lever, when applied, moves both itself and the lever and the weight, and the horses joined to a carriage not only move themselves, but also draw and move the carriage. But percussion often throws the body struck away from the striker, just as projection does. If, however, propulsion is joined with percussion, as always precedes projection, the impetus in propulsion is produced by the propelling power; but just as at the very moment of projection the impetus that was in the projector passes into the projected body, which departs, so in percussion, at the first moment of the blow, the impetus passes into the body struck according to its capacity. If, moreover, it is driven by the striking body, whose motion proceeds in accordance with its direction and presses the parts of the body struck (as happens in things that are depressed or compressed, and also when piles, while being driven in, are urged by the ram falling and not rebounding), it has some impetus produced by the propeller in addition to the impetus communicated by the same body as striker at the very moment of percussion; but that which is produced by the propeller is not very great, if it is compared with that which is obtained from percussion. Something similar to propulsion, which follows percussion, is found in traction, which is preceded by motion, in which impetus has been acquired; for at the moment when motion ceases and traction begins, the impetus passes and is diminished in the one pulling, as if a stone lying on the floor were joined by a rope to another somewhat smaller stone, and the rope were placed over a movable pulley, and that smaller stone, falling until it stretches the rope, should acquire impetus from its motion; as soon as the rope is stretched and the stone lying there resists the motion of the descending stone, the acquired impetus passes to overcome the resistance of the stone lying there, and, having received direction from the motion of the one pulling, it is forced to ascend while the other descends, and it draws this one somewhat; but when the impressed impetus grows weak in the heavier stone, it descends, and in turn drags upward the one from which it had suffered force. Thus a power stretching a quickly slackened rope conceives a great impetus, which it impresses on the attached weight, while being itself deprived of it as soon as it meets resistance; but it also produces and effects another part of the impetus in the weight by pulling.

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Mechanicorum 744 Cum igitur duplex sit in motu submovendorum impedimentorum genus, alia, videlicet, quæ inchoatum motum abrumpunt, alia quæ obsistunt, ne fiat motus; illa tollenda sunt per impetum, quo motus continuandus fuisset, nisi impedimentum occurrisset; hæc verò superat pars impetus producta à potentiâ, quæ se tardiùs movet, quia vires dividit, partem impetus sibi reservans, partem impertiens obstaculo, quod removet impellendo aut trahendo. Quare nil mirum, si impetus, qui periturus esset in percutiente, cujus motus impeditur, transeat in obicem percussum, quem submovendo locum relinquit ulteriori motui, si facultas se movendi suppetat corpori percutienti. CAPUT XIII. Cunei usus promovetur. NE quis fortè Cuneum solis rusticis ad findenda ligna usui Nesse sibi persuadeat, fontes aliquos indicare placet ex quibus non levis utilitas derivatur. Ad Machinarum scilicet Rationem pertinet potissimùm motus corporis, cujus resistentia superatur, sivè illa demum ex gravitate oriatur, sivè ex nexu, quo colligatur cum proximo corpore; id quod iis contingit, quæ in corpus unum coalescunt, & fissione sejunguntur. PROPOSITIO I. Vectis vires Cuneo augere. Contingit aliquando potentiam incommodè applicari vecti, ut cum hominem valde curvari oportet ad vectem secundi generis ferè in solo jacentem attollendum; tunc subsidium à Cuneo non incongruè peti potest. Sit Vectis A B subjectus foribus D C suis è cardinibus avellen- dis,

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Mechanics 744 Since therefore there are two kinds of impediments to be removed in motion: some, namely, which break off a motion already begun; others which stand in the way of motion’s occurring; the former must be removed by the impulse by which the motion would have been continued, had not the impediment intervened; the latter, however, are overcome by that part of the impulse which is produced by the power that moves more slowly, because it divides its forces, keeping part of the impulse for itself and imparting part to the obstacle, which it removes by pushing or pulling. Therefore it is no wonder if the impulse, which would be lost in the striking body whose motion is hindered, passes into the struck obstacle, which, being moved aside, leaves room for further motion, if the striking body has the ability to move itself. CHAPTER XIII. The use of the wedge is furthered. Lest anyone perhaps should persuade himself that the Wedge is useful only to peasants for splitting wood, it is worth pointing out certain sources from which no slight advantage is derived. For the science of machines is concerned chiefly with the motion of a body whose resistance is overcome, whether that resistance arises finally from weight, or from the bond by which it is joined to the neighboring body; as happens to those things which cohere into one body and are separated by splitting. PROPOSITION I. To increase the force of the lever by means of the wedge. It sometimes happens that power is inconveniently applied to a lever, as when a man must bend down greatly in order to raise a second-class lever lying nearly on the ground; then help may not improperly be sought from a wedge. Let the lever AB be placed beneath the doors DC, to be torn away from their hinges,

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Liber septimus. CAPUT XIII. 745 dis, ut reficiantur: hypomochlium est in A, & pondus in C. At si potentiam adeò inclinari atque curvari oporteat, ut arripiat extremum vectem B, satis manifestum est, quanto id incommodo fiat. Subjiciatur vecti in B (siquidem solum æquo mollius fuerit) asseris aut lapidis pars, quæ compressioni resistat, atque inter illam & vectem apex Cunei E immittatur. Nam si tudite cuneum percutias, vectem facilè attollet, ac proinde etiam valvas in C incumbentes. Quod si vectis secundi generis F G habens hypomochlium in G ita fuerit altiùs collocatus, ut ægrè brachiorum contentione attolle- re valeas pondus in K ad- nexum, utere cuneo in- flexo F H, quem solo in I incumbentem, & vecti in F subjectum, si propellas lateri H I, arrepto manu- brio L M, applicatus, prout commodiùs acciderit, vectem cum pondere eatenus elevabis, quoad latus I H longius, solo ad pendiculum insistat. Vectem autem, qua parte cuneum hujusmodi contingit, ita extenuatum esse oportere, ut cunei orbitæ ex- cavatæ congruat, ne elabatur, res per se ipsa loquitur. Maximè verò opportunum duxerim hujusmodi cuneo in- flexo uti, ubi tertij generis Vectis adhibendus fuerit R S, & pondus adnexum ex S in V sustollendum: Nam si loco Potentiæ destinato in T subjicias cuneum inflexum T P, solo in X incumbentem, & hunc urgeas ex latere X P, aut trahas ex latere T X, ubi P venerit in Q, pondus ex S erit in V. BBbbb

Transcription: Translated (English)

Book Seven. CHAPTER XIII. 745 so that they may be lifted up: the fulcrum is at A, and the weight at C. But if the power must be so inclined and bent, that it may seize the end of the lever B, it is sufficiently clear how great inconvenience this involves. Let there be placed under the lever at B (if the ground itself should be somewhat softer than level) a board or a part of a stone that resists compression, and between that and the lever let the point of the wedge E be inserted. For if you strike the wedge with a mallet, it will easily raise the lever, and consequently also the valves resting on C. But if a second-class lever F G, having the fulcrum at G, has been placed so much higher, that you can with difficulty raise the weight attached at K by the effort of the arms, use a curved wedge F H, which, resting on the ground in I, and placed under the lever at F, if you push it at the side H I, taking hold of the handle L M, and applying it as may be more convenient, you will lift the lever with the weight only so far as the side I H, longer, stands perpendicular to the ground. But the lever, on the side by which it touches such a wedge, ought to be so thinned that it fits the hollowed track of the wedge, lest it slip out; the thing speaks for itself. Indeed I would judge it especially convenient to use a wedge of this kind, when a third-class lever R S is to be employed, and the attached weight is to be raised from S to V: for if in the place assigned to the Power at T you place the curved wedge T P, resting on the ground at X, and drive it from the side X P, or pull it from the side T X, when P has come to Q, the weight from S will be at V. BBbbb

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Mechanicorum PROPOSITIO II. Vecte, aut Trochleâ, aut Succulâ, Cunei inflexi vires augere. Instrumenta pressoria varij generis excogitantur; sed ad subi- tum usum, & non sinè compendio, uti aliquando possumus Cuneo inflexo, cui, si validiùs premendum sit, vectem adjice- re licebit. Sit crassior tabula A B, cui supponatur id, quod premen- dum proponitur. Paretur cuneus inflexus D E, & pro illius motûs centro statuatur punctum C, cui axis infigatur: Nam urgendo lon- gius latus C E, aut trahendo bre- vius C D, subjectam tabulam A B premes. Quod si validiore pressu opus fuerit, lateri longiori C E Vectem F G adhibe: potentia si- quidem in G majorem arcum des- cribens circa centrum motûs C, majora obtinebit momenta, quàm si proximè illa applicaretur Cuneo: illa tamen momenta poten- tiæ in G sensim minuuntur, prout cunei partes tabulam con- tingentes propiores sunt extremo puncto E. Ne verò ipsa ea- dem tabula A B impedimento sit, si lateri C E proximè vectis adhæreret, affigatur cuneo unum aut alterum Chelonion H F, I K, in quæ conjectus vectis F G distet à Cuneo citrà pericu- lum incurrendi in subjectam tabulam A B. Vel si Vectem cuneo affigere non placuerit, ipsius vectis ca- put hypomochlio respondens ita collocetur, ut vectis horizonti ferè paralleli longitudo transversa cadat in latus cunei D E, séque non procul ab E decussent: hac enim ratione vecti sua constabunt momenta, quibus momenta cunei augeantur. At si fortè loci dispositio non ferat, ut vectis adhibeatur im- pellendo cuneo, trahatur ille ex D, ubi aut annulus infigatur, aut

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Mechanics PROPOSITION II. To increase the force of a bent wedge by means of a lever, a pulley, or a windlass. Various kinds of pressing instruments are devised; but for immediate use, and not without advantage, we may sometimes employ a bent wedge, to which, if greater pressure is required, a lever may be added. Let there be a thick board A B, under which let that which is to be pressed be placed. Let the bent wedge D E be prepared, and let point C be set as the center of its motion, into which the axle is to be fixed. For by pressing the longer side C E, or by drawing the shorter side C D, you will press the board A B beneath it. But if stronger pressure is needed, apply the lever F G to the longer side C E: for the power, indeed, in G describing a larger arc about the center of motion C, will obtain greater moments than if it were applied very near to the wedge; yet those moments of the power in G gradually diminish, as the parts of the wedge touching the board are nearer to the end point E. But lest the board A B itself should be an obstruction, if the lever were attached very near to the side C E, let one or two cheeks H F, I K be fixed to the wedge, in which, when the lever F G is inserted, it may be kept at a distance from the wedge so as to avoid danger of striking the board A B beneath it. Or, if it is not pleasing to attach the lever to the wedge, let the head of the lever corresponding to the hypomochlion be so placed that the transverse length of the lever, being nearly parallel to the horizon, may fall upon the side D E of the wedge, and may meet it not far from E: for in this way the lever will preserve its own moments, by which the moments of the wedge will be increased. But if perhaps the arrangement of the place does not allow the lever to be used by pushing upon the wedge, let it be drawn from D, where either a ring is fixed, or

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Liber septimus. CAPUT XIII. 747 aut foramini inseratur funis, cui deinde trochlea sive simplex, sive multiplex adnectatur, prout opus fuerit. Immò & Succula addi poterit, ad quam funis caput religetur: eruntque momenta potentiæ, quæ componuntur ex Rationibus Succulæ, Trochleæ, & Cunei. PROPOSITIO III. Cuneum inflexum validissimum construerè ad Vectem tùm trahendum, tùm repellendum. Asumatur planum aliquod circulare circa axem per centrum ductum versatile, ita crassum & validum, ut in eo insculpi possit profundiùs spira, quæ Vectis caput ferreo clavo capitato, & in globum rotundato, armatum contineat, ne elabatur. Hinc enim fiet, ut in spiræ cavitatem immissum Vectis caput aut propellatur, aut attrahatur, prout plani illius motus in hanc aut illam partem dirigitur: tantus scilicet erit Vectis motus, quanta erit Radiorum à centro ad spiræ ambitum ductorum differentia. Ex quo orietur tractio, aut impulsio; Radiis enim decrescentibus trahitur Vectis ad centrum, illis crescentibus propellitur à centro. Potentia igitur certæ plani illius circularis parti applicata integrum circulum describit, dum vectis caput per unum spiræ flexum excurrit, & tot circulos potentia describit, quot spiræ flexus vectis caput subinde complectuntur. Quare comparanda est distantia à centro plani circumacti, quam in motu caput Vectis mutavit, cum universis circulis, quos potentia interim descripsit, & statim innotescet Ratiomomentorum. Hinc si plano hujusmodi, in quo excavata est spira, addideris circa extremam orbitam Radios, quemadmodum Axi in Peritrochio, motus potentiæ satis amplos circulos describet. Statuamus, exempli gratiâ, plani circularis assumpti diametrum cubitalem, hoc est sesquipedalem, seu digitorum 24; sit autem inter spiræ excavatæ flexum & flexum intercapedo digitorum duum, adeò ut, peractâ circulatione unâ, vectis caput per spiram excurrents digitos duos à primâ suâ sede dimotum B B b b b 2

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Book Seven. CHAPTER XIII. 747 or a rope may be inserted through the opening, to which then a pulley, either simple or compound, may be attached, as need may be. Moreover, a winch may be added, to which the end of the rope shall be tied; and the moments of the power will be composed from the ratios of the winch, the pulley, and the wedge. PROPOSITION III. To construct a very strong bent wedge for both drawing and pushing a lever. Let some circular plate be taken, capable of turning about an axis drawn through the center, so thick and strong that in it there may be carved more deeply a spiral, which shall contain the head of the lever, armed with an iron-headed nail, rounded into a ball, so that it does not slip out. For thus it will come about that the head of the lever, inserted into the cavity of the spiral, is either driven on or drawn in, as the motion of that plate is directed to this side or to that: that is, the motion of the lever will be as great as the difference between the radii drawn from the center to the circumference of the spiral. From this there will arise traction or thrust; for as the radii decrease, the lever is drawn toward the center; as they increase, it is driven from the center. Therefore the power applied to a certain part of that circular plate describes a whole circle, while the head of the lever runs through one turn of the spiral; and the power describes as many circles as the turns of the spiral successively embrace the head of the lever. Wherefore the distance from the center of the rotated plate, which the head of the lever has changed in its motion, must be compared with the total circles which the power meanwhile has described, and the ratio of the moments will immediately be known. Hence if to a plate of this kind, in which a spiral has been hollowed out, you were to add radii around the outer circumference, as in the axle in the peritrochium, the motion of the power will describe circles sufficiently large. Let us suppose, by way of example, that the diameter of the circular plate taken is a cubit, that is, a foot and a half, or 24 digits; let there be, moreover, a distance of two digits between one turn of the hollowed spiral and the next, so that, after one complete revolution, the head of the lever, running through the spiral, is moved two digits from its first seat. B B b b b 2

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Mechanicorum 748 fuerit: at potentia in extremâ orbitâ plani circularis constituta suo motu integram peripheriam circuli, cujus diameter digi- torum 24, descripserit, hoc est digitorum 75. Motus igitur potentiæ ad motum capitis vectis est ut 75 ad 2: cui si addatur Ratio ad ipsum Vectem spectans, quatenus cum pondere com- paratur, fiet Ratio Composita indicans Rationem motûs poten- tiæ ad motum ponderis. Quapropter etiamsi ad conciliandum ponderi motum paulò velociorem, uteremur Vecte primi generis sed inverso, ita ut ab hypomochlio plus distaret pondus, quàm potentia capiti vectis applicata, adhuc haberetur non modicum momentorum compendium. Sit enim distantia potentiæ in capite vectis ab hypomochlio ut 1, ponderis verò ut 5; atque adeò, dum Vectis caput deprimitur digitos duos, pondus attollatur digitos de- cem: Componantur duæ Rationes 75 ad 2, & 1 ad 5; erit Ra- tio 15 ad 2, & potentia spiræ applicata movebit pondus hujus- modi vecti adnexum lib. 150, quo conatu absque ullâ machinâ moveret pondus lib. 20. Hanc propositionem hîc potiùs afferre placuit, quàm in se- quentem librum de Cochleâ reservare, quia hîc caput vectis excurrit per ipsam spiram, & proximè pertinere videtur hîc motus ad motum super faciem Cunei inflexi: in Cochleâ verò, prout communiter illa usurpatur, pondus movetur ad motum cylindri, cui insculpta est Cochlea. Dixi, prout communiter usurpatur, quia aliquid simile contingit Cochleæ infinitæ, ut videbimus. PROPOSITIO IV. Flatum vehementem non interruptum excitare folllibus adhibitis. GLebam metallicam ex fodinis erutam valido igne exco- quere oportet, ut metallum fluat, atque id, quod utile est, ab inutili secernatur. Ignis autem ut ex carbonibus excitetur eâ vehementiâ, qua opus est, etiam vehementem flatum, qui ex folllibus exprimatur adhibendum manifestum est omnibus: neque

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Mechanics 748 there has been: but if a power placed at the extreme orbit of a circular plane should by its motion describe the whole circumference of a circle whose diameter is 24 digits, that is, 75 digits. Therefore the motion of the power to the motion of the head of the lever is as 75 to 2: to which if there be added the ratio referring to the lever itself, insofar as it is compared with the weight, there will be formed a compound ratio indicating the ratio of the motion of the power to the motion of the weight. For this reason, even if, in order to impart to the weight a somewhat quicker motion, we were to use a lever of the first kind but inverted, so that the weight would be farther from the fulcrum than the power applied to the head of the lever, there would still be no small saving of force. Let the distance of the power at the head of the lever from the fulcrum be as 1, and that of the weight as 5; and so, while the head of the lever is depressed 2 digits, let the weight be raised 10 digits: combine the two ratios, 75 to 2 and 1 to 5; the ratio will be 15 to 2, and the power applied to the spiral will move a weight of this kind attached to the lever, 150 pounds, whereas by that effort, without any machine, it would move a weight of 20 pounds. I have preferred to present this proposition here rather than reserve it for the following book on the Screw, because here the head of the lever passes through the spiral itself, and the motion here seems to pertain more closely to the motion upon the face of the bent Wedge: but in the Screw, as it is commonly used, the weight is moved in relation to the motion of the cylinder on which the Screw is engraved. I said, as it is commonly used, because something similar happens in the infinite Screw, as we shall see. PROPOSITION IV. To produce a strong, uninterrupted blast by means of bellows. The metallic lump brought from the mines must be smelted with a strong fire, so that the metal may flow and what is useful may be separated from what is useless. But in order that the fire may be kindled from the coals with that vehemence which is needed, it is clearly necessary to employ also a vehement blast, such as is forced out by bellows: and not

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Liber septimus. CAPUT XIII. 749 neque enim ubique commodum reperiri potest conclave hy- pogæum, in quod præceps delapsa aqua aërem vapori mistum per tubum in camini focum impellat. Quare præter vulgarem & notissimam methodum folles alterno motu agitandi, ex his, quæ hujus lib. cap. 3. dicta sunt, rationem aliquam inire pos- sumus, qua plurimum flatus in carbones accensos immit- tamus. Quod ad folles ipsos spectat, non illos simplices vellem, sed singulos duplices, ita videlicet conformatos ut singuli ex binis asseribus constent invicem secundùm alteram extremitatem in- clinatis, quasi in angulum coituri essent, qui omnino stabiles permaneant. In ipso autem tigillo, cui firmiter infixa manent asserum illorum capita, excavatus sit congruè ductus, per quem flatus exprimatur in tubum adnexum, quo ad focum defertur: atque opportuno loco in singulis asseribus, ut moris est, fora- men excipiendo aëri destinatum assario muniatur. Hos inter immotos, planum aliud simile, ad extremitatem fibulâ versatili connexum cum tigillo illo communi, adjiciatur, & cum extre- mis asseribus corio plicatili jungatur, adeò ut duo sint conjuncti folles, quorum alter clauditur, alter recluditur, cùm ex medio hoc plano mobili exiens ansa adducitur & reducitur: hoc enim mobile planum est diaphragma sejungens folles, ne ex altero in alterum compressus aër effugiat, sed per infimum ductum in tubum erumpat, per quem ad focum devehatur. Quatuor pa- rentur hujusmodi folles duplices, quorum bini sibi ex diametro oppositi ita statuantur, ut inter illos discus circularis congruæ magnitudinis interjectus eorum ansas subinde propellere va- leat: bini autem oppositi funiculo jungantur aut loro, aut ca- tenulâ, ansas connectente longitudinis æqualis diametro cir- culi: Ex quo fiet, ut operâ eâdem follis unius ansa propellatur, oppositi verò ansa trahatur. Porrò attendendum est, quantum spatij percurrat singulo- rum follium ansa ultro citróque remeando, quo loco illa tangi- tur à circulo: hujus enim spatij semisse definietur intervallum, quo circuli centrum abesse oportet à centro, quod statuendum est, ut circa illud fiat ejusdem circuli convolutio. Huic motûs centro infigendus est firmiter axis, sivè ille sit communis exte- riori rotæ ab aquâ fluente convolutæ, sivè cui vectis opportu- BBbbb 3

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Book Seven. Chapter XIII. 749 For indeed it is not everywhere possible to find a convenient hypogeum chamber, into which water, rushing down headlong, may drive through a tube into the fire of a hearth the air mixed with vapor. Wherefore, besides the common and very familiar method of working bellows by an alternating motion, from what was said in this book, chapter 3, we may devise some plan by which we may force a great blast into the burning coals. As to the bellows themselves, I would not have them simple, but each double, so fashioned namely that each consists of two boards inclined toward one another at one extremity, as though they were about to come together in an angle, and remain altogether steady. But in the very little beam to which the heads of those boards are firmly fixed, let there be cut a fitting passage, through which the blast may be squeezed into the attached tube, by which it is carried to the hearth: and at a suitable place in each board, as is customary, let a hole be made and fitted with a board intended to receive the air. Between these immovable parts, let another similar flat piece be added, joined at one extremity by a movable clasp to that common little beam, and let it be joined to the outer boards by pliant leather, so that there are two connected bellows, of which one is closed while the other is opened, when the handle issuing from this middle moving plate is moved to and fro: for this moving plate is the diaphragm separating the bellows, lest the air compressed from one pass into the other and escape, but rather burst through the lower passage into the tube, by which it is conveyed to the hearth. Let four such double bellows be prepared, of which two, set opposite each other directly, be so placed that a circular disc of suitable size interposed between them may be able continually to drive their handles; but let the two opposite ones be joined by a cord or strap or little chain, connecting handles of equal length to the diameter of the circle: from this it will happen that by the same operation the handle of one bellows is pushed and the handle of the opposite one is drawn. Moreover, it must be observed how far the handle of each bellows travels to and fro, at the point where it is touched by the circle; for half of this distance will determine the interval by which the center of the circle ought to be distant from the center which must be established, so that the rotation of that same circle may be made around it. To this center of motion an axis must be firmly fixed, whether it be the common one of the outer wheel turned by flowing water, or one to which a lever suitably... BBbbb 3

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Mechanicorum 750 næ longitudinis adjiciatur, ut ab homine, aut à jumento con- torqueatur. Sic ansæ motus universus. Sit ex. gr. A B circuli centrum sit C: acci- piatur intervallum C D sub- duplum ipsius A B; & erit in D infigendus axis, ex cu- jus convolutione circulus pa- riter circumagatur, & fol- lium ansas in quatuor oppo- sis punctis A, E, G, F subinde tangat, easque vicissim propel- lat, & trahat: Cùm scilicet incipit propelli follis ansa, quæ est in E, propellitur pariter ea, quæ est in G (si quidem conver- sio fiat ex E in F) atque ex adverso tantumdem trahitur quæ est in A, quantum propellitur quæ est in E; atque similiter tractio ejus, quæ est in F, est æqualis impulsioni ansæ, quæ est in G. Si igitur circulus non sit in plano Verticali, & axem in D in- fixum non habeat communem cum rotâ, quæ ab aquâ volva- tur, sed sit in plano horizontali, axi infixo in D addatur vectis D H, ut potentia in H vectem impellens aut trahens circum- agat circulum. Quo autem loco statuendus sit vectis, pendet ex loci positione, prout vel in superiori, vel in inferiori, vel in eodem conclavi folles collocantur; axis siquidem certam non exigit longitudinem, sed ea illi tribuenda est, quæ commodior acciderit. Vectis tamen longitudinem ita temperare oportet, ut, dum potentiæ movendi facilitatem affectas, nimiam tardi- tatem compressionis follium effugias. Ex his satis apparet, quantum aëris impellatur in prunas à quatuor follibus, qui clauduntur, dum quatuor reliqui reclu- duntur, perpetuúsque est flatus nunquam interruptus. Quòd si potentia movens viribus abundet, & circulus fieri possit am- plior ita, ut non quatuor solùm follibus duplicibus, sed etiam sex aut octo similibus in gyrum disponendis commodus locus suppetat, satis vides, quantus excitari possit flatus. Quoniam verò ex dictis infertur folles esse erigendos, ut in eorum ansas circulus horizonti parallelus incurrat, observa pos- se illos etiam jacentes (modo assarium superioris asseris foramen claudens

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Mechanics 750 should be added to the length, so that it may be turned by a man or by a beast of burden. Thus the whole movement of the handles. Let, for example, A B be the center of the circle: let C be taken at a distance C D half the length of A B; and there shall the axis be fixed, by whose revolution the circle likewise shall be turned round, and the bellows shall in turn touch the handles at four opposite points A, E, G, F, and alternately drive them and draw them back: for when the handle of the bellows which is in E begins to be driven, that which is in G is driven likewise (if indeed the turning be from E to F), and on the opposite side there is drawn back by as much that which is in A as that which is in E is driven; and similarly the drawing of that which is in F is equal to the driving of the handle which is in G. If therefore the circle is not in a vertical plane, and does not have the axis fixed in D in common with the wheel which is turned by the water, but is in a horizontal plane, let there be added to the axis fixed in D a lever D H, so that the power, impelling or drawing the lever in H, may turn the circle. But in what place the lever is to be set depends on the position of the place, according as the bellows are placed either in the upper room, or in the lower, or in the same room; for the axis requires no fixed length, but that must be assigned to it which shall prove most convenient. The length of the lever, however, must be so adjusted that, while you seek ease in moving the power, you may avoid excessive slowness in compressing the bellows. From these things it sufficiently appears how much air is driven into the coals by four bellows, which are closed while the other four are opened, and there is a perpetual blast, never interrupted. But if the moving power abound in strength, and the circle can be made larger, so that not only for four double bellows, but also for six or eight similar ones to be arranged in a circle, there may be sufficient space, you see well enough what a great blast may be produced. And since from what has been said it is inferred that the bellows are to be raised, so that a circle parallel to the horizon may strike their handles, observe that they can also be laid down (provided the board closing the opening of the upper plank

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Liber septimus. CAPUT XIII. 751 claudes exacte fungi possit suo munere) usui esse posse, si artificiu[m] aliquod adhibeatur, quo ansa attollatur, atque deprimatur. Fiat inflexus vectis, seu quasi vectis R S G, cujus angulo S addatur axis, circa quem facilè converti possit, & ad extremitatem G sit regula GH follis ansæ adnexa; & similia omnia ex adverso parentur, atque funiculo MP conjungantur. Nam circulus impellens in R attollet extremitatem G, ac proinde ansam illi adnexam, trahendo autem funiculum MP deprimet extremitatem O, & cum illâ ansam oppositi follis: atque ita vicissim impellendo N, attolletur O, & deprimetur G. Hinc colliges hoc eodem artificio, si hastulæ GH non follis ansam, sed antliæ embolum adjeceris, fieri posse machinam, qua, multiplicatis antliis, plurimum aquæ sursum impellere valeas. An autem disci circularis exteriorem orbitam ferreo annulo polito & lævi munire, atque ferream laminam pariter politam ansæ follis, aut vecti inflexo, apponere præstet, ut quàm minimo tritu inter se confligant, non est opus monere, si fuerit operæ pretium machinæ diuturnitati consulere, & faciliorem motum exhibere. Quòd demum spectat ad ipsius circuli collocationem, quamquam cylindrus illi infixus possit inniti polis, circa quos versetur; ut tamen subter circulum liberrimè trahi possint funiculi, placeret potiùs illum omnino suspensum pendere ex crassiore (sive simplici, sive ex duobus compacto) tigno, cujus foramini inseratur cylindrus, ferreo annulo munitus tam in superiori, quàm in inferiori parte, qua foramini respondet, ita ut neque sursum agi, neque deorsum descendere valeat, sed intrà foramen illud convertatur, quod pariter utrinque circulis ferreis muniatur respondentibus superiori & inferiori annulo cylindri. Sit circularis discus AB, in quo motûs centrum sit C, cui insigatur cylindrus CD convertendus à vecte, sivè in I, sivè in H immittendo. Horizonti parallelum tignum

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Book Seven. Chapter XIII. 751 claudes exacte fungi possit suo munere ) can be useful, if some device be employed by which the handle may be lifted and depressed. Let a bent lever, or, as it were, a lever R S G be made, to whose angle S let there be added an axis, around which it may easily turn; and at the extremity G let there be a rule GH attached to the bellows handle; and let all similar parts be prepared opposite it, and joined together by the cord MP. For the impelling circle, in R, will lift the extremity G, and therefore the handle attached to it; but by drawing the cord MP it will depress the extremity O, and with it the handle of the opposite bellows: and thus, by impelling N in turn, O will be lifted and G depressed. From this you will gather that by this same device, if you attach not the bellows handle but the plunger of a pump to the little rod GH, a machine can be made by which, with pumps multiplied, you may be able to force a great quantity of water upward. As to whether it be better to protect the outer rim of the circular disk with a polished and smooth iron ring, and likewise to fit to the bellows handle, or bent lever, an equally polished iron plate, so that they may collide with as little friction as possible, there is no need to advise, if it be worth the labor to provide for the machine’s durability and to secure a smoother motion. As for the placement of the circle itself, although the cylinder fixed in it may rest on pivots around which it turns, still, so that the cords may be drawn most freely beneath the circle, it would rather please me that it hang entirely suspended from a thicker beam (whether a single beam or one made of two joined together), into whose hole the cylinder is inserted, being furnished with an iron ring both in the upper and in the lower part, where it corresponds to the hole, so that it can neither be forced upward nor descend downward, but may turn within that hole, which likewise on both sides is to be fitted with iron rings corresponding to the upper and lower ring of the cylinder. Let there be a circular disk AB, in which the center of motion is C, to which let the cylinder CD be fastened, to be turned by a lever inserted either at I or at H. A beam parallel to the horizon

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Mechanicorum 752 tignum KL secundùm suas extremitates in pariete, aut aliter, firmetur, in eóque sit foramen F capax cylindri: foramen ferreo circulo, quoad fieri possit, lævi atque polito muniatur, cui æqualis annulus cylindrum vestiens, illique affixus, respondeat. Manebit ex F suspensus cylindrus DC unâ cum adjuncto circulari disco AB. In inferiore tigni facie similiter sit circulus ferreus, & annulus, ne sursum excurrere queat cylindrus. Si tignum quidem valde distet à circulo AB, immitti poterit vectis in H; at si exiguum fuerit intervallum inter tignum & circulum AB, atque tignum existat infra planum, in quo homo aut jumentum vectem impellens aut trahens movetur, vectis in I immittatur. PROPOSITIO V. Plures antlias duplices perpetuo ductu agitare. Ubi jumentorum operâ uti oportet ad agitandas antlias, quibus aqua in superiorem locum aut attrahitur, aut impellitur, illa in gyrum agere necesse est; id quod multo tempore eget; neque enim circuitus illos currendo efficere possunt: quapropter rotarum dentatarum complexionem construere solemus, ut, dum semel jumentum suam conversionem absolvit, sæpiùs antliæ agitentur. Verùm minore impendio absque rotis idem fortasse assequemur, si potissimùm aqua in altum propellenda sit. Ad

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Mechanics 752 the beam KL, according to its ends, be fixed in the wall, or otherwise, and in it let there be a hole F capable of receiving the cylinder: let the hole be furnished with an iron ring, as smooth and polished as may be, to which let an equal annulus, clothing the cylinder and attached to it, correspond. There will remain hanging from F the cylinder DC together with the attached circular disk AB. In the lower face of the beam let there likewise be an iron ring and annulus, lest the cylinder be able to run upward. If the beam indeed be very distant from the circle AB, a lever can be inserted at H; but if the interval between the beam and the circle AB be small, and the beam lie below the plane in which the man or beast, pushing or pulling the lever, moves, let the lever be inserted at I. PROPOSITION V. To drive several double pumps by a continuous motion. Where it is necessary to use beasts of burden for driving pumps, by which water is either drawn up or forced to a higher place, it is necessary to make them go round in a circle; and this requires much time; for they cannot accomplish those circuits by running: wherefore we are accustomed to construct a combination of toothed wheels, so that, while the beast once completes its turn, the pumps are driven more often. Yet with less expense, without wheels, we may perhaps attain the same result, especially if water is to be driven upward. Ad

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Liber septimus. CAPUT XIII. 753 Ad perpendiculum erigatur tignum, quod imo puteo in- nitatur, sive solidiori tigno A B putei lateribus infixo insistat brevius tignum C D, ita tamen obliquè constitutum, ut hujus an- guli latera illius respiciant, quatenus hastulæ ex hoc exeuntes, medio jugo, nullum recipiant à sub- jecto tigno A B impedi- mentum. Suprema pars tigni C D ita secetur, ut circa axem in E infixum liberè versari possit ju- gum, cujus extremitatibus adnexæ sunt hastulæ an- tliarum embolos attollen- tes atque deprimentes. Paulò infra axem E ape- riatur foramen F, cui pa- riter immitti queat jugum alterum versatile circa axem H infixum paulò infrà crenam superiori ju- go subservientem. Porrò utriusque jugi non eadem est forma: Nam supe- rius jugum axi E infixum rectum est G I, additamente ad K auctum, ut paulo depressius sit foramen K ad reci- piendum axem, quàm sint axes ad G & I, quibus jun- guntur cum jugo hastulæ ad embolorum motum perficien- dum destinatæ. At verò jugum inferius non nisi extremit- tates L M & N O rectas habet, cætera inflexum est, & ad mediam curvaturam habet in P foramen, quo innita- tur axi in H infixo. Quantam autem esse oporteat hu- jusmodi inflexionem M P N, ex hoc definies, quod ubi jugum G I in suo axe consistens horizonti parallelum fue- rit, etiam inferioris jugi in suo axe H consistentis extre- CCccc

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Book Seven. Chapter XIII. 753 A beam should be erected perpendicularly, resting at the bottom of the well, or let a shorter beam C D be fixed into the sides of the well upon a stronger beam A B; however, it should be set obliquely, so that the sides of this angle may face those of the other, in order that the rods coming out from this, by means of the middle yoke, may receive no obstruction from the underlying beam A B. The upper part of the beam C D should be cut in such a way that the yoke, fixed around an axle set in E, may freely turn, to whose extremities are attached the rods of the pumps, raising and lowering the pistons. A little below the axle E, let an opening F be made, into which another yoke may likewise be inserted, turning about an axle H fixed a little below the notch that serves the upper yoke. Moreover, the shape of the two yokes is not the same: for the upper yoke, fixed to the axle E, is straight at G I, and is additionally enlarged at K, so that the hole K for receiving the axle may be a little lower than are the axles at G and I, to which are joined the rods attached to the yoke for accomplishing the movement of the pistons. But the lower yoke has only the extremities L M and N O straight; the rest is bent, and at the middle of its curve has a hole in P, on which it rests upon the axle fixed in H. How great, however, this kind of bending M P N ought to be, you will determine from this: that when the yoke G I, standing upon its axle, has been parallel to the horizon, then also the extremities of the lower yoke, standing upon its axle H, will have been... CCccc

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Mechanicorum 754 mitates LM & NO in eodem horizontali plano cum GI conveniant. His paratis frustum cylindricum diametri (si id quidem commodè fieri possit) non multo minoris, quàm sit jugi GI intercapedo inter hastularum axes, construatur. Quod si tantæ crassitudinis lignum præsto non fuerit, plura aptè compinge, atque ferreo circulo constringe, ne dissilire va- leant. Tum destinatæ embolorum depressioni atque eleva- tioni æqualis saltem pars QS toreutæ operâ rotundetur; deinde serrâ obliquè secetur, ut fiat ellipsis QT: cujus limbus ferreâ lamellâ exactè planâ & politâ muniatur tùm ad perpetuitatem, ne lignum atteratur, tùm ad faciliorem motum, ut minor sit cum subjectis ligneis jugis conflictus: interiores autem ellipsis partes scalpro eximi possunt, ut factâ cavitate nullum motui impedimentum afferant tigni CD su- premi anguli. Frusto huic cylindrico ad axem firmiter inseratur minor cylindrus VR, cui immitti possit vectis XY à jumento in X circumducendus. Minor hujusmodi cylindrus methodo su- periùs indicatâ sub finem prop. 4. suspendatur eâ lege, ut jugo GI maximè inclinato conveniat ellipsis diameter QT, minori autem ellipsis Axi conveniant extremitates LM & NO inferioris jugi, quæ erunt horizonti parallelæ. Hinc fiet, ut converso cylindro subinde deveniant ad maximam depressionem, atque vicissim ad maximam elevationem, sin- guli antliarum emboli. Quod si liceret in puteo, ubi aqua scaturit, aut in vase, in quod aqua influit, in altum elevanda vi antliæ propellentis, non quadratum tantùm, sed hexagonum aut octogonum pris- ma erigere ad perpendiculum, & in oppositis faciebus fora- mina excavare, quibus immitterentur inflexa juga, eâ in- flexione, quæ satis esset, ut demum omnium extremitates in eodem horizontali plano convenirent, satis manifestum est tribus aut quatuor jugis posse deinceps sex aut octo antlias agitari. PROPONI

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Mechanics 754 the extremities LM & NO may coincide in the same horizontal plane with GI. When these preparations have been made, let a cylindrical piece, of a diameter (if indeed it can conveniently be done) not much smaller than the interval between the axes of the rods, namely the spacing of the ridge GI, be made. But if wood of so great a thickness is not available, fit several pieces together appropriately, and bind them with an iron hoop, lest they be able to split apart. Then let at least an equal portion QS, intended for the depression and elevation of the pistons, be rounded by the work of the turner; afterwards let it be cut obliquely with a saw, so that an ellipse QT is formed: whose rim shall be furnished with an iron strip exactly flat and polished, both for durability, lest the wood wear away, and for easier motion, so that there may be less friction with the underlying wooden ridges; but the inner parts of the ellipse may be cut out with a chisel, so that, a cavity having been made, they may present no obstacle to the motion of the upper angle of the beam CD. Into this cylindrical piece let the smaller cylinder VR be firmly inserted at the axis, into which there may be fitted the lever XY, to be turned around at X by a beast. A smaller cylinder of this kind shall be suspended by the method indicated above at the end of proposition 4, under this condition: that when the ridge GI is most inclined, the diameter QT of the ellipse may correspond, but the extremities LM & NO of the lower ridge, which will be parallel to the horizon, may correspond to the minor axis of the ellipse. Hence it will come about that, as the cylinder is turned, each of the pump pistons shall in turn descend to the greatest depression and, conversely, rise to the greatest elevation. And if in a well where water gushes forth, or in a vessel into which water flows, it were permissible to erect vertically, for the purpose of raising water by the force of the propelling pump, not merely a square, but a hexagonal or octagonal prism, and to hollow out openings in opposite faces into which the bent ridges might be inserted, with such a bending as would be sufficient so that at length the extremities of all of them might meet in the same horizontal plane, it is quite clear that by means of three or four ridges six or eight pumps could afterward be operated. PROPOSED

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Liber septimus. CAPUT XIII. 755 PROPOSITIO VI. Alia ratione plures antlias componere. EX iis, quæ hujus libri cap. 5. dicta sunt, genus aliud ad Cuneum pertinens excogitare possumus, quo simul plures antlias agitare possit potentia, cui maximè virium copia suppetat, & valde simplex machina construenda proponatur. Ex solidis asseribus compingatur circulus: hic in octo partes distribuatur, & duæ proximæ confixum habeant tigillum A B, cujus extremitates ita extra circulum promineant, ut incisis crenis hastulæ embolo adnexæ circa suum axem versatiles sinè impedimento moveri queant. Tres similes tigilli transversarij affigantur CD, EF, GH extremitatibus similiter prominenti- bus extra circuli ambitum, & excavatis in crenas hastularum capaces. Hastularum verò formam suaderem, quæ prope embolum essent plicatiles in dextram atque sinistram, quemadmodum in supremâ parte, ubi transversariis cohærent, sunt circa axem flexiles in anteriorem atque in posteriorem partem: ex hac enim flexibilitate in omnem partem facilior oritur motus. Duos autem tigillos CD & EF existimo apponendos esse transversarios, ad majorem circuli firmitatem: quamquam sufficeret ad propositum finem breviores apponere ad CE & DF, omnino similes & æquales ipsis A B & GH. His paratis alius æqualis circulus superponatur, firmiterque CCccc 2

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Book Seven. Chapter XIII. 755 PROPOSITION VI. To construct several pumps in another way. From the things said in chapter 5 of this book, we can devise another kind, belonging to the Wedge, by which one power, possessing a great abundance of force, may at the same time drive several pumps, and a very simple machine be proposed for construction. Let a circle be framed from solid planks: let this be divided into eight parts, and let two adjacent parts have fixed in them a small beam AB, whose ends project so far beyond the circle that, the notches having been cut, the little rods attached to the plungers, being movable about their own axis, may be able to move without hindrance. Three similar transverse little beams CD, EF, GH should be fastened, with their ends likewise projecting beyond the circumference of the circle, and hollowed out into notches capable of receiving the little rods. As for the form of the rods, I would recommend that near the plunger they should be flexible to the right and to the left, just as in the upper part, where they are joined to the transverse pieces, they are flexible about the axis forward and backward: for from this flexibility there arises in every direction a more easy motion. Moreover, I think that two transverse little beams, CD and EF, ought to be added, for greater firmness of the circle: although for the proposed end it would suffice to add shorter ones at CE and DF, entirely similar and equal to AB and GH. When these things have been prepared, another equal circle shall be placed above, and firmly joined together. CCccc 2

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Mechanicorum cum inferiore cohæreat. Tum validus stylus ferreus R T figuræ primùm cylindricæ, deinde ad S sphæricæ, demum in T desinens in conum construatur, & columnæ, cui universa machina inniti debet, ad perpendiculum insigatur. Ad centrum verò circuli inferioris foramen fiat, per quod facilè globus S immitti possit, & in centro circuli superioris aliud pariter foramen aperiatur, sed tantùm capax coni S T, adeò ut machina sustineatur à globo S, & in quancumque partem facilè inclinari queat: id quod etiam faciliùs continget, si foramen illud superioris circuli, qua parte globum S contingit, annulo, seu limbo ferreo muniatur. Quod si circulus ille superior crassior fuerit, quàm ut facilè inclinari possit, ne superior ora foraminis incurrat in conum, abradi poterit, quantum satis fuerit, in calathoidem, ut magis pateat, atque liberam inclinationem permittat. Circulari hac compage impositâ stylo R S, hastulæ embolorum suis axibus adnectantur extremitatibus tigillorum prominentibus. Tum ad conciliandum motum machinæ, cylindrus I K suo centro K innitatur apici styli T, & in superiore loco, axis I congruo foramini immissus servet cylindri positionem perpendicularem. Sit autem in cylindri latere profundiùs excavata crena, cui inseri possit triangulum OPN obtusangulum ad P, quod validum sit, & cum cylindro firmissimè cohæreat: sic enim fiet, ut trianguli extremitas O tangens circulum, illum à positione horizonti parallelâ removeat, & in eam partem inclinet, atque ex adversâ elevet. Potentia verò vecti VX applicata, & cylindrum volvens, alam pariter NOP circumducet; quæ aliis atque aliis subjecti circuli partibus subinde applicata illas deprimet, & ex diametro oppositas elevabit: intermediæ autem aliæ deprimentur, ad quas scilicet extremitas O accedit, aliæ elevabuntur, à quibus eadem extremitas O recedit. Quantum autem extremitas O infra basim cylindri descendere

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Mechanicorum should be joined with the lower one. Then let a strong iron style, of the shape R T, be made, first cylindrical, then spherical at S, and finally ending at T in a cone; and let the column on which the whole machine must rest be fixed upright by the plumb line. At the center of the lower circle, however, let an opening be made, through which the ball S may easily be inserted; and in the center of the upper circle let another opening be made similarly, but only large enough for the cone S T, so that the machine may be supported by the ball S, and may easily be inclined in any direction: this will also happen more easily if that opening of the upper circle, on the side where it touches the ball S, is provided with an iron ring, or rim. But if that upper circle should be thicker than to be easily inclined, lest the upper edge of the opening strike against the cone, it may be shaved off, as much as is sufficient, into a calathoid, so that it may be more open and permit free inclination. With this circular frame placed on the style R S, the rod pieces of the pistons shall be attached to their axles by the projecting ends of the little beams. Then, in order to produce the motion of the machine, let the cylinder I K rest with its center K upon the apex of the style T, and in the upper part, let the axis I, inserted through a suitable opening, preserve the perpendicular position of the cylinder. Now let there be on the side of the cylinder a deeper notch, into which the triangle OPN may be inserted, obtuse-angled at P, which may be strong and most firmly joined to the cylinder: for in this way it will happen that the extremity O of the triangle, touching the circle, removes it from the position parallel to the horizon, and inclines it toward that side, and on the opposite side raises it. Now the force applied to the lever VX, turning the cylinder, will likewise turn the wing NOP; and this, being successively applied to various parts of the circle beneath it, will depress those parts, and raise those opposite them by a diameter: the intermediate parts will meanwhile be depressed when the extremity O approaches them, and others will be raised when the same extremity O recedes from them. But how far the extremity O descends below the base of the cylinder

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Liber septimus. CAPUT XIII. 757 dere oporteat, definiendum est primò ex motu, quem embolus elevatus atque depressus perficit; cujus motûs medietas accipienda est: deinde attendenda est distantia basis cylindri à plano circuli, si hoc constitueretur horizonti parallelum; hæc verò distantia addenda est semissi motûs emboli, ut innotescat, quantum oporteat extremitatem O, deprimi infra basim cylindri. Neque cuiquam dubium esse potest, an sic definienda sit hujusmodi depressio extremitatis O; siquidem inclinato circulo tantum extremitas altera diametri deprimitur infra planum horizontale, quantum altera attollitur; hæc autem duplicata differentia dat universum motum emboli; igitur hujus moûs semisse definitur circuli depressio & inclinatio. Quia autem ad faciliorem motum, tùm ne cylindri crassities plano circuli inclinato occurrat, tùm ne latus P O circulum tangat præterquam extremitate O, ad vitandum tritum atque conflictum partium, præstat cylindrum non proximè adhærere circulo; propterea distantia basis cylindri à centro subjecti circuli computanda est. Porrò expedire extremitatem O munitam ferreâ laminâ percurrere in subjecto circulo laminam pariter ferream exquisitè politam, non opus est monere: satis quippe per se patet. Illud cavendum est, ut modum serves in alæ N O P amplitudine; nam si nimis exigua sit, paulò difficiliùs movet, quia nimis distat ab hastulis embolorum: sin autem æquo amplior fuerit, cùm maximam resistentiæ partem illa sustineat, subit periculum luxationis. Cæterùm hoc pendebit ex circuli amplitudine, cujus diametrum constituendam esse habitâ ratione motûs embolo antliæ communicandi, nemo ignorat; quemadmodum & in simplici antliâ ex hoc eodem definitur distantia hastulæ à centro motûs. Quoniam enim motus ille depressionis & elevationis emboli connectitur cum motu circulari semidiametri circuli, cui hastulæ adnectuntur, eum Radium circulo tribuere oportet, ut arcus ab extremo puncto descriptus quàm minimum differat à lineâ rectâ; sic enim faciliùs movetur embolus. Quare arcus ejusmodi describendus est, ut illius medietas Sinum Versum habeat, quoad fieri poterit, minimum. Ponamus universum emboli motum esse unciarum 4, ejus semissem CCC 3

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Book Seven. CHAPTER XIII. 757 It must first be determined, from the motion which the raised and lowered piston performs, what amount of depression ought to be made; half of that motion is to be taken. Next, attention must be given to the distance of the base of the cylinder from the plane of the circle, if this were taken as parallel to the horizon; and this distance must in fact be added to the half-motion of the piston, so that it may be made known how far the extremity O ought to be depressed below the base of the cylinder. Nor can anyone doubt whether such a depression of the extremity O should be defined in this way; for, when the circle is inclined, only one extremity of the diameter is depressed below the horizontal plane by as much as the other is raised; but this doubled difference gives the whole motion of the piston; therefore the depression and inclination of the circle are determined by half of this motion. But since, for easier motion, both lest the thickness of the cylinder should come in contact with the inclined plane of the circle, and lest the side P O should touch the circle except at the extremity O, in order to avoid rubbing and collision of the parts, it is better that the cylinder should not adhere closely to the circle; for this reason the distance of the base of the cylinder from the center of the subjacent circle must be computed. Moreover, it is not necessary to warn that it is advantageous for the extremity O, protected with an iron plate, to run upon the subjacent circle, which likewise should be of iron and exquisitely polished; for this is sufficiently clear of itself. One must take care to observe moderation in the breadth of the wing N O P; for if it be too narrow, it moves somewhat more difficultly, because it is too far from the rods of the pistons; but if it be wider than is fitting, since it bears the greater part of the resistance, there is danger of dislocation. Moreover, this will depend on the breadth of the circle, whose diameter must be determined with due regard to the motion to be communicated to the pump piston; as indeed in a simple pump the distance of the rod from the center of motion is determined from the same consideration. For since that motion of depression and elevation of the piston is connected with the circular motion of the semidiameter of the circle to which the rods are attached, that Radius must be assigned to the circle, so that the arc described from the extreme point may differ as little as possible from a straight line; for in this way the piston is moved more easily. Therefore such an arc must be described that its middle point, as far as may be, has a versed sine as small as possible. Let us suppose the whole motion of the piston to be 4 inches, its half 3 CCC 3

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Mechanicorum unciarum 2: Sit circuli Radius BD unciarum 8. Inveniatur in Canone Sinuum arcus, cujus Si- nus ad Radium sit ut 2 ad 8, & est proximè gr. 14. 28' 40". Est igitur arcus ab extremâ semidiametro D describendus C E gr. 28. 57. 20": quo bifariam diviso in D est arcus CD gr. 14. 28'. 40"; cujus Sinus CI; & Sinus Versus I D est totius Radij BD 1/100, hoc est unius unciæ 4/25; quæ deflexio arcûs C E à rectitudine non admodum nocet. Satis igitur fuerit, si circuli diameter sit unc. 14. & ti- gilli hinc atque hinc aliquantulum præter unam unciam pro- mineant, ubi illis hastulæ embolorum adnectuntur; sic enim fiet, ut hastulæ satis commodè moveantur, maximè si lon- giores fuerint. Quod si ligneis tigillis uti nolueris, sed potius ferreis pris- matibus inter utrumque ligneum circulum aptè conferendis, adeò ut circuli plana sibi vicissim adhæreant, non dubium, quin multò firmior futura sit machina: hoc te monitum volo, quod circulos crassiusculos esse oportet, ut in illis opportunum foramen excavetur, quo commodè machina insistat styli glo- bulo, & , prout oportet, inclinetur. MECHA

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Mechanics of 2 inches: Let the radius BD of the circle be 8 inches. In the canon of sines let there be found the arc whose sine is to the radius as 2 to 8, and it is approximately 14° 28' 40". Therefore the arc CE to be described from the outer end of the semidiameter D is 28° 57' 20": when this is divided in two at D, there is the arc CD of 14° 28' 40"; whose sine is CI; and the versed sine ID is 1/100 of the whole radius BD, that is, 4/25 of an inch; which deviation of the arc CE from straightness does not greatly matter. It will therefore be sufficient if the diameter of the circle be 14 inches, and the rods project a little beyond an inch on this side and that, where the piston-rods are attached to them; for thus it will come about that the rods move quite conveniently, especially if they are longer. But if you do not wish to use wooden rods, but rather iron prisms suitably fitted between the two wooden circles, so that the flat faces of the circles adhere to one another in turn, there is no doubt that the machine will be much stronger: I wish to warn you of this, that the circles ought to be somewhat thick, so that in them an appropriate hole may be hollowed out, by which the machine may conveniently rest on the globule of the pen and incline as required. MECHA

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MECHANICORUM LIBER OCTAVUS. De Cochlea. Postremo loco inter Mechanicas Facultates numeratur Cochlea, non tamen postremo loco habenda, si ejus vires perpendantur; immò si cum cæteris Facultatibus comparetur, omnium efficacissima censenda erit, cæteris paribus, ut ex iis, quæ hoc libro disputabuntur, mani- festum fiet. Cur de Cochleâ postremus habeatur sermo, si quis inquirat, non pauci ex iis, qui inter Mechanicas faculta- tes cognationis nexus quosdam pervestigant, ideò post Cu- neum numerari Cochleam autumabunt, quia Cochlea longior quidam Cuneus cylindro convolutus censeri potest, cujus propterea vires ad Cuneum revocare contendunt. Mihi tamen, qui Facultates singulas ita à reliquis absolutas agnosco, ut nul- lo alio vinculo invicem copulentur, nisi quatenus omnes ab uno eodemque principio ortum ducunt, ea tantummodo esse videtur causa, quod reliquæ Facultates simplices sint, ac faci- liùs parabiles, quàm Cochlea, atque hæc si solitaria adhibea- tur, nec cum ullâ reliquarum Facultatum componatur, licèt validè urgeat, aut trahat, eâ tamen communiter non utamur ad majores motus efficiendos, quos unâ aliquâ reliquarum Fa- cultatum, minore operâ, consequimur. Hûc autem non spectat Archimedea Cochlea ad aquam in altum evehendam instituta: est enim tubus in spiram convolu- tus circa superficiem conicam aut cylindricam, seu in cono ipso aut cylindro ita excavatus, ut aquam continere valeat, quam extremum tubi osculum ex subjectâ profluente hausit: dum sci- licet

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MECHANICS BOOK EIGHT. Of the Screw. Last among the Mechanical Powers is reckoned the Screw, yet it should not be placed last if its force be considered; indeed, if compared with the other Powers, it will be judged the most effective of all, other things being equal, as will be made clear from what is discussed in this book. If anyone should inquire why the Screw is treated last, many of those who investigate certain ties of kinship among the Mechanical Powers will think that it is therefore numbered after the Wedge, because the Screw may be regarded as a sort of longer Wedge wound around a cylinder, and on that account they contend that its force should be referred back to the Wedge. For my part, however, since I recognize each Power as so distinct from the others that they are joined together by no other bond except insofar as they all derive their origin from one and the same principle, the only cause that seems to me is this: the other Powers are simple and more easily made than the Screw; and if this is used by itself, and is not combined with any of the remaining Powers, although it presses or draws strongly, nevertheless we do not commonly use it for producing greater motions, which we attain by some one of the other Powers with less labor. But here is not meant the Archimedean Screw devised for raising water into the height: for it is a tube wound into a spiral around a conical or cylindrical surface, or excavated in the cone itself or in the cylinder in such a way that it can contain water, which the mouth of the end of the tube has drawn up from the stream below: while, namely

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Mechanicorum 760 licet circa suum axem Conus aut Cylinder ad horizontem inclinatus convertitur, quæ ingressa fuerat aqua, per spiras ascendens ad alteram tubi extremitatem superiorem demum effunditur; atque hac ratione ad tantam altitudinem illa attollitur, quantus est Sinus anguli, quo ad horizontem inclinatur axis coni aut cylindri, posito eodem axe tanquam Radio. Hic, inquam, motus aquæ in tubo hujusmodi spirali ascendentis, non est præsentis disputationis, aqua siquidem non trahitur sursum, sed semel ingressa in tubo spirali convoluto sponte descendit, donec ad supremum osculum provehatur; haud secus ac plumbeus globulus in eundem tubum immissus, si volvatur cylindera, non valens consistere in eâ spiræ parte, quæ priùs infima & horizonti proxima, modò in conversione removetur ab horizonte & attollitur, suâ autem gravitate repugnans ascensui, sponte descendit per tubum tanquam per planum inclinatum, atque ita deinceps, quoad ex supremo tubi osculo erumpat. Idem planè contingit aquæ in hujusmodi tubo spirali vi suæ gravitatis subinde fluenti ac descendenti in singulis spiris statim, ac modicum quid elevata est in conversione. Cochlea igitur, de qua hîc disputabitur, ea est, quæ ad vim gravitati inferendam, si repugnet, instituta est, adeò ut corporis vim passi motus impulsui à Potentiâ per Cochleam communicato adæquatè tribuendus sit; & si quid gravitas ipsa conferat, id planè contingens reputetur. Nomen autem Cochleæ inditum est ex simili quadam convolutione in testâ limacis, quæ in spiras contorquetur, sicut & Cochledes dicuntur scalæ, per quas in gyrum ascenditur. CAPUT I. Cochleæ forma & virtus describitur. Cochlea, quam explicandam suscipimus, ex limacis testâ Ceatenus solùm similitudinem ducit, quatenus in spiras ducitur, cæterùm animalis illius spiræ inæquales sunt, & major spira

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Mechanics 760 Although a Cone or Cylinder inclined to the horizon, turning about its axis, causes the water which has entered it, ascending through the coils, at length to be poured out at the upper end of the tube; and by this means it is raised to so great a height as the sine of the angle by which the axis of the cone or cylinder is inclined to the horizon, taking that same axis as the radius. This motion, I say, of water ascending in a tube of this kind in a spiral, is not the subject of the present discussion, since the water is not drawn upward, but, once admitted into the coiled spiral tube, it descends of itself until it is carried to the highest opening; just as a leaden ball, let into the same tube, if the cylinder be turned, being unable to remain in that part of the spiral which was formerly lowest and nearest the horizon, but is now, by the turning, removed from the horizon and raised, and opposing ascent by its own weight, descends of itself through the tube as through an inclined plane, and so onward until it bursts forth from the top opening of the tube. The same thing plainly happens with water in such a spiral tube, flowing and descending by the force of its own weight in the several coils as soon as it has been lifted a little in the turning. The screw, therefore, of which we shall here speak, is one instituted for imparting force against gravity, if gravity resists, so that the motion undergone by the body must be attributed adequately to the impulse communicated by the Power through the Screw; and if gravity itself contributes anything, that is plainly to be regarded as incidental. Now the name Screw has been given from a certain similar coiling in the shell of a slug, which is twisted into spirals; just as staircases, by which one ascends in a circle, are also called Cochleae. CHAPTER I. The form and power of the Screw are described. The Screw, which we undertake to explain, derives its resemblance from the shell of a slug only in so far as it is led into spirals; otherwise the spirals of that animal are unequal, and the larger spiral

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Liber octavus. CAPUT I. 761 spira minorem quasi complectitur, non quemadmodum helix in plano descripta, sed ferè sicut spira in coni aut globi superficie deformata. Spira autem conicè ducta, aut sphæricè, parùm utilis accideret Machinatoris instituto; cum enim, ut firmetur, inerenda sit foramini similiter in spiram excavato, majores coni, aut globi, spiræ non congruerent minoribus spiris foraminis conici aut sphærici in modum scaphij, nec per eas promoveri possent; atque minores coni, aut globi, spiræ in amplioribus spiris foraminis firmari nequirent. Oportet igitur spiram omnino similibus ductibus, atque æqualibus constare; id quod non nisi in cylindro obtinetur. Quapropter Cochlea, de qua hìc agimus, est solida spira in superficie excavati cylindri efformata; quæ vitium Capreolos arboris ramum complexos imitata vulgari vocabulo Vitis (& fortasse aptiùs) nominatur. Receptaculum verò concavum, cui cylindrus in helicem deformatus immittitur, habétque spirales cavitates solidæ cylindri spiræ congruentes, Matrix dicitur, alij Tylum, Cochlidium alij, vocabulo ad hanc significationem detorto, vulgus Matrem Vitis nuncupat. Ut autem spiram cylindro æqualibus atque similibus gyris circumductam intelligas, concipe triangulum rectangulum, cujus perpendiculum æquale sit dato lateri aut Axi cylindri Recti, basis verò trianguli toties contineat perimetrum basis cylindri, quoties spira cylindrum ipsum complecti debet; nam hujusmodi trianguli hypothenusa lineam spiralem omnino similiter ductam in cylindri superficie describet, si triangulum cylindro circumvolvatur. Sit cylindri altitudo A B, ejusque basis circulari peripheriæ sit æqualis recta BC ad rectum angulum C B A constituta. Oporteat autem spiram quatuor gyris complecti cylindrum; idcirco recta BC producatur, ut tota BF sit ipsius BC quadrupla: ducta enim hypothenusa FA, si triangulum cylindro circumplicetur quadruplici convolutione, designabit in cylindri superficie quatuor spiras omnino similes & æquales. Spirarum æqualitatem & similitudinem fa- DDddd

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Book Eight. CHAPTER I. 761 the smaller spiral as it were encloses, not as a helix described in a plane, but almost as a spiral distorted on the surface of a cone or sphere. But a spiral drawn conically or spherically would be of little use for the Mechanician’s purpose; for since, in order to be fixed, it must be inserted into a hole likewise excavated in the form of a spiral, the larger cones or spheres would not agree with the smaller spirals of a conical or spherical hole in the manner of a scaphoid, nor could they be advanced through them; and smaller cones or spheres could not be fixed in the larger spirals of the hole. Therefore the spiral ought altogether to consist of similar courses and equal ones; and this is obtained only in a cylinder. Wherefore the Cochlea of which we here speak is a solid spiral formed on the surface of a hollowed cylinder; which, imitating a vine branch enclosing the tendrils of a vine, is in common speech called Vitis (and perhaps more aptly). But the hollow receptacle into which the cylinder, distorted into a helix, is inserted, and which has spiral cavities corresponding to the solid spirals of the cylinder, is called Matrix, by others Tylum, Cochlidium by others, a word twisted to this meaning; the common people now call it Matre Vitis. And that you may understand the spiral wound around the cylinder with equal and similar turns, conceive a right triangle, whose perpendicular be equal to the given side or axis of the right cylinder, and whose base contain as many times the perimeter of the cylinder’s base as the spiral ought to embrace the cylinder itself; for the hypotenuse of such a triangle will describe on the surface of the cylinder a spiral line drawn altogether similarly, if the triangle be wound around the cylinder. Let the height of the cylinder be AB, and let its base be equal to the circular periphery BC set at a right angle CBA. Now suppose that the spiral is to embrace the cylinder with four turns; therefore let the straight line BC be prolonged, so that the whole BF may be four times BC: for if the hypotenuse FA be drawn, then, if the triangle is wrapped around the cylinder with a quadruple winding, it will mark on the surface of the cylinder four spirals altogether similar and equal. The equality and similarity of the spirals fa-

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Mechanicorum 762 cilè demonstrabis, si trianguli basim B F, & altitudinem B A, utramque in quatuor æquales partes distinxeris; deinde ex singulis divisionum punctis rectas C M, D L, E K altitudini B A parallelas, & rectas G K, H L, I M parallelas basi B F excita- veris; sibi enim occurrentes in punctis K, L, M, divident hypothenusam in quatuor æquales partes, ut patet ex 2. lib. 6: Nimirum ut F E ad E D, ita F K ad K L; & ut F D ad D C, ita F L ad L M; & ut F C ad C B, ita F M ad M A: sunt autem F E & E D ex hypothesi æquales, igitur etiam F K & K L æquales: F D posita est ipsius D C dupla, ergo F L ipsius L M dupla; ergo L M æqualis est ipsi K L, aut F K: Demum F C ex con- structione est ipsius C B tripla; igitur etiam F M est tripla ipsius M A; quare M A æqualis est singulis reliquis partibus F K, K L, L M; & tota hypothenusa divisa est in quatuor æquales partes. Item in parallelogrammo K D, per 34. lib. 1. æqualia sunt opposita latera K N & E D, atque in parallelogrammo L C æqualia sunt L O & D C, quemadmodum & in parallelogrammo M B æqualia sunt M I & C B: Sicut igitur rectæ F E, E D, D C, C B ex hypothesi sunt æquales, etiam F E, K N, L O, M I sunt inter se æquales. Similiter ostendes sicut æquales sunt ex constructione B G, G H, H I, I A, ita æquales inter se esse E K, N L, O M, I A. Cum itaque triangula F E K, K N L, L O M, M I A habeant tria latera singula singulis æqualia, & similiter posita, ipsa sunt quoque æquian-gula, ac proinde similiter inclinatæ sunt singulæ spiræ F K, K L, L M, M A, quæ pariter demonstratæ sunt æquales. Quam si-milem inclinationem ostendit æqualitas angulorum ad F, K, L, M, propter linearum parallelismum. Triangulum igitur A B F suâ hypothenusâ F A designat in cylindri superficie qua-tuor similes & æquales spiras. Verùm quid juvaret in exteriore cylindri superficie spiralem lineam exquisitè descripsisse, nisi corpus ipsum cylindricum in solidam spiram deformaretur? Quapropter necessariò cylindrum circumplectuntur duæ spiræ, cava altera & depressa, al- tera convexa & prominens, quibus similiter atque æqualiter depressæ & prominentes duæ spiræ in receptaculi seu Matricis foramine cylindricè excavato requiruntur ita illis responden- tes, ut depressam receptaculi spiram subeat prominens cylindri spira,

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Mechanics 762 you will demonstrate it thus: if you divide the base B F of the triangle and the altitude B A into four equal parts each; then from each point of division draw the straight lines C M, D L, E K parallel to the altitude B A, and raise the straight lines G K, H L, I M parallel to the base B F; for these, meeting one another at the points K, L, M, will divide the hypotenuse into four equal parts, as is clear from 2. book 6: namely, as F E is to E D, so is F K to K L; and as F D is to D C, so is F L to L M; and as F C is to C B, so is F M to M A. But F E and E D are equal by hypothesis; therefore F K and K L are also equal: F D is set as double of D C, therefore F L is double of L M; therefore L M is equal to K L, or to F K: finally, F C is by construction triple of C B; therefore F M is also triple of M A; wherefore M A is equal to each of the remaining parts F K, K L, L M; and the whole hypotenuse is divided into four equal parts. Likewise, in the parallelogram K D, by proposition 34 of book 1, the opposite sides K N and E D are equal, and in the parallelogram L C the sides L O and D C are equal, just as in the parallelogram M B the sides M I and C B are equal: therefore, as the straight lines F E, E D, D C, C B are equal by hypothesis, so also F E, K N, L O, M I are equal among themselves. In the same way you will show that, just as B G, G H, H I, I A are equal by construction, so too E K, N L, O M, I A are equal among themselves. Since therefore the triangles F E K, K N L, L O M, M I A have three sides each equal to three sides, and likewise similarly situated, they are also equiangular, and consequently the spirals F K, K L, L M, M A are similarly inclined, and have been shown equally so. The equality of the angles at F, K, L, M demonstrates this same similar inclination, because of the parallelism of the lines. Therefore the triangle A B F, by its hypotenuse F A, marks on the surface of the cylinder four similar and equal spirals. But what would it avail to have drawn with exactness a spiral line on the outer surface of the cylinder, unless the cylindrical body itself were fashioned into a solid spiral? Wherefore the cylinder is necessarily embraced by two spirals, one hollow and depressed, the other convex and projecting; and two such spirals, equally depressed and projecting, are likewise required in the cavity of the receptacle or Matrix, cylindrically hollowed out, answering to them in such a way that the projecting spiral of the cylinder enters the depressed spiral of the receptacle,

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Liber octavus. CAPUT I. 763 spira, & vicissim prominentem receptaculi spiram excipiat de- pressa cylindri spira. Ex quo fit, ut convolutus circa suum axem cylindrus attollatur aut deprimatur, adducatur aut redu- catur, prout opus fuerit, atque cum eo corpus basi illius proxi- mum, seu adnexum urgeatur, aut trahatur, elevetur, aut pre- matur. Vulgatissimus autem & frequentissimus est hujus Facultatis usus, ubi potissimùm opus est valida pressione, ut in prælis vi- nariis ad exprimendum ex uvæ jam pressæ reliquiis tortivum mustum, apud typographos ad imprimendos subjectæ chartæ ex typis characteres, apud bibliopægos ad comprimendos li- bros, jam compactos, apud fabros ferrarios ad firmandas fer- reas laminas limâ expoliendas, atque apud alios artifices. Quamquam & sæpissimè clavorum loco, quibus ligna, aut me- tallicæ laminæ configuntur citrâ mallei percussionem, cochleis utimur, & quidem ad validiorem atque perennem firmitatem; neque enim revelli potest cochlea, aut excuti, quemadmodum clavus. Sed tunc hujusmodi cochleæ non exercent vim facul- tatis Mechanicæ; eatenus scilicet validiùs, quàm clavi, duo corpora, quæ compinguntur, connectunt, quatenus multipli- ces in cylindruli facie solidarum spirarum ductus pluribus cavis foraminum spiris implicantur ex cylindruli convolutione; qui propterea eximi non potest, nisi in contrarium revolvatur; quandiu quidem incorruptum permanet lignum neque ex hu- more putrescens, neque vermiculo erodente cariosum, neque calore nimio ita discedens atque dehiscens, ut laxato foramine jam non ampliùs solida cylindruli spira congruentibus striis coërceatur. Hinc est in sustentando pondere ex cochleâ suspenso pro- propriè non exerceri vim Mechanicam; nihil enim ampliùs co- nante Potentiâ (quemadmodum in Vecte, aut Axe in Peritro- chio, aut fune Trochlearum retinendo opus est, quæ pondus elevavit convoluto cylindro in cochleam deformato, sola spira- rum cavæ atque convexæ complexio efficit, ut cylindrus cum adnexo pondere retineatur, ne recidat, quatenus à subjectâ lo- culamenti spirâ solidâ sustinetur: quemadmodum & subscudi- bus compagem cohibentibus accidit, quatenus securicla ex mi- nore in majorem amplitudinem explicata decrescentis recepta- D D d d d 2

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Book eight. CHAPTER I. 763 the spiral, and in turn let the projecting spiral of the receptacle receive the depressed spiral of the cylinder. From this it follows that the cylinder, wound about its own axis, may be raised or lowered, advanced or drawn back, as need may be, and with it the body adjacent to, or attached to, its base may be pressed or pulled, lifted or depressed. But the most common and frequent use of this faculty is where strong pressure is especially required, as in wine presses, for expressing the pressed residue of the grapes into new wine; among printers, for impressing the characters from type upon the paper placed beneath; among bookbinders, for compressing books already bound; among smiths, for fastening iron plates to be smoothed with a file; and among other craftsmen. Although very often, in the place of nails, with which wood or metal plates are fastened together without the blow of a hammer, we use screws as well, and indeed for a stronger and more lasting firmness; for a screw cannot be torn out or knocked away, as a nail can. But then screws of this sort do not exercise the force of the Mechanical Faculty; rather, they connect two bodies joined together more strongly than nails do, in so far as the multiple turns of solid spirals on the surface of the little cylinder are intertwined with the many spiral cavities of the holes by the winding of the cylinder, which therefore cannot be removed unless it is turned back in the opposite direction; provided, of course, that the wood remains sound, neither rotting from moisture, nor worm-eaten and decayed, nor so weakened by excessive heat as to split and gape apart, so that once the hole has become loose, the solid spiral of the cylinder is no longer restrained by the matching grooves. Hence, in supporting a weight suspended from a screw, the Mechanical force is properly speaking not exercised; for nothing more is done by the Power, striving as it does only in one direction (as in a Lever, or in an Axis in a wheel and axle, or in a rope of Pulleys while keeping in place that which has lifted the weight transformed by the coiled cylinder into a screw), than that the mere combination of the hollow and convex spirals causes the cylinder, together with the attached weight, to be held fast, so that it does not fall back, inasmuch as it is sustained by the solid spiral of the underlying socket: just as also happens with the fastenings that hold the fittings together, inasmuch as the small wedge, expanded from a lesser into a greater breadth, diminishes the receptacle's descending...

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Mechanicorum 764 culi angustiis coërcetur, ne excurrat; adeóque confixum hujusmodi subscude corpus grave inferius retinetur, ne à superiore disjungatur, & cadat. Tota igitur vis Machinalis à Cochleâ exercetur in motu, quem à potentiâ illam circumagente recipit. Et sanè si potentiæ cylindrum versantis motum comparemus cum motu ponderis, quod à cochleâ urgetur, aut trahitur; statim apparebit potentiam quidem circulum describere circa convoluti cylindri axem, pondus verò rectâ moveri, prout promovetur, aut retrahitur cylindrus. Cum itaque in singulis cylindri conversionibus ejus motum definiat spiræ à spirâ intervallum; si hoc cum circulari peripheriâ conferatur, innotescet motuum Ratio, & Potentiæ momentum, quæ eò minorem in pondere resistentiam invenit, quò tardiùs hoc movetur. Hinc si cylindri altitudo ad ejusdem diametrum sit ut 20 ad 1, numeratásque spiras cylindrum complectentes inveneris esse 35, rectè definies convolutionibus 35 respondere totum cylindri motum, atque adeò spiræ à spirâ intervallum esse ad cylindri diametrum ut 4 ad 7: ex quo infertur circuli peripheriam ad spirarum distantiam, hoc est potentiæ motum ad motum ponderis, esse proximè ut 22 ad 4, atque potentiæ conatum ut 4 vincere posse quamlibet resistentiam minorem quàm ut 22, spectatâ Ratione, quam infert cylindri crassities, & spirarum obliquitas. Verùm quia non nisi parvulis cochleis, aut ubi levis conatus requiritur, ita applicatur potentia, ut cylindri superficiei applicata intelligatur, complanatâ scilicet ejusdem cylindri extremitate, quam summis digitis apprehendere valeas, communter adhuc majus est momentum Potentiæ, quàm ut ex circuli peripheriâ basim cylindri ambiente circumscribatur; additur enim aut Radius cylindri Capiti quadrato infixus, aut aliquid manubrij rationem habens, adeò ut potentia longè majorem circulum describat, quàm sit cylindri in spiram deformati basis: ac proinde non ex cylindri crassitie, sed ex distantiâ potentiæ ab axe cylindri definiendus est ejusdem potentiæ circulum perficientis motus, atque cum spirarum intervallo motum ponderis metiente comparandus. Hinc ad imprimendas metallicæ laminæ ex argento, aut auro, aut cupro imagines citrà percussionem, super solido pla- no

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Mechanics 764 is restrained by the narrowness of the eye, so that it does not run out; and thus the body fastened with this kind of wedge is held firm below, lest it be separated from the upper part and fall. Therefore the whole mechanical force is exercised by the screw in the motion which it receives from the power turning it. And indeed if we compare the motion of the turning cylinder with the motion of the weight which is pressed or drawn by the screw, it will immediately appear that the power indeed describes a circle around the axis of the wrapped cylinder, but the weight moves in a straight line, as the cylinder is advanced or drawn back. Since therefore in each revolution of the cylinder the interval from one thread to the next defines its motion; if this be compared with the circular circumference, the ratio of the motions will become known, and the force of the power, which finds less resistance in the weight the more slowly this is moved. Hence if the height of the cylinder be to its diameter as 20 to 1, and if, when the turns encircling the cylinder are counted, you find them to be 35, you will rightly determine that 35 revolutions correspond to the whole motion of the cylinder, and therefore that the interval from thread to thread is to the diameter of the cylinder as 4 to 7: from which it is inferred that the circumference of the circle to the distance of the threads, that is, the motion of the power to the motion of the weight, is approximately as 22 to 4, and that the effort of the power can overcome any resistance less than as 22, regard being had to the ratio introduced by the thickness of the cylinder and the slant of the threads. But because in no other case than with very small screws, or where only slight effort is required, is the power applied in such a way that it is understood to be applied to the surface of the cylinder, namely with the end of the same cylinder flattened, which you can grasp with the tips of the fingers, it is commonly still a greater force of the Power than can be bounded by the circumference of the circle surrounding the base of the cylinder; for either a Radius fixed into the square head of the cylinder is added, or something serving the purpose of a handle, so that the power describes a far larger circle than is the base of the cylinder deformed into a spiral: and therefore the motion of the power completing the circle is to be defined not from the thickness of the cylinder, but from the distance of the power from the axis of the cylinder, and compared with the interval of the threads measuring the motion of the weight. Hence, for impressing images on a metallic sheet of silver, or gold, or copper without percussion, upon a solid plane

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Liber octavus. CAPUT I. 765 no erectis atque infixis ad perpendiculum duobus ferreis pedibus ferreo pariter transversario firmatis, in quo excavata cochleæ congruens Matrix, typus inter laminam & cylindrum interjectus validè urgetur ex cylindri convolutione, & imaginem exprimit: quia videlicet superiori cylindri Capiti quadrato inferitur longior ferreus vectis hinc atque hinc productus, ut duplici ejus extremitati duplex potentia, si opus fuerit, applicetur. Quapropter ab axe cylindri ad vectis hujusmodi extremitatem ducta linea est Radius circuli potentiæ motum determinantis; atque si hujusmodi Radij longitudo ad spirarum intervallum fuerit ut 50 ad 1, circuli diameter est 100, ejusque peripheria major quàm 314; & potentiæ motus ad motum typi laminam prementis est ut 314 ad 1: idcirco si in vectis extremitatibus sint singuli homines perinde conantes, ac si libras 50 singuli moverent, premitur typus vi hujus cochleæ quasi à pondere librarum 31400. Quod autem de pressione dicitur, simili ratione intelligendum est de ponderis elevatione, si fortè aut inferiori cylindri basi adnexum fuerit, aut ejus capiti impositum; sicut enim corpus prementi resistit ratione particularum constipatarum, ita elevanti repugnat ratione suæ gravitatis: utrobiqque igitur similem virtutem habet potentia ad vincendam resistentiam, quando utrobiqque eadem invenitur Ratio motuum atque momentorum. Propterea in hujusmodi cochleis, quæ infixo Radio convolvuntur, non est admodum anxiè procuranda cylindri crassities, modò satis solidus sit, nec fragilis: eadem quippe est circuli à potentiæ motu descripti peripheria, sivè major sit, sivè minor cylindri crassitudo, quando eadem est potentiæ distantia à cylindri axe, ac proinde idem est momentum. Illud quidem ad rem facit maximè, quam obliquè inclinatus sit spirarum ductus; hinc enim oritur intervalli Ratio inter proximos spirarum circuitus, qui frequentissimi sunt, ac brevi intervallo disjuncti, si linea spiralis sit maximè inclinata, rari autem atque notabiliter sejuncti, si illa fuerit ad majorem angulum (acutum tamen) erecta: Est nimirum hujusmodi intervallum æquale Tangenti anguli inclinationis, posito Radio ambitu basis cylindri; ipsa autem spira est ejusdem anguli Secans. Quare datâ cylindri diametro, invenitur peripheria basis; & D D d d d 3

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Book eight. CHAPTER I. 765 with two iron feet erected and fixed perpendicularly, equally secured by an iron crosspiece, in which the nut, hollowed to fit the screw, is inserted between the plate and the cylinder and is powerfully pressed by the winding of the cylinder, and thus produces the impression: because, namely, into the square upper head of the cylinder is fitted a longer iron lever projecting on this side and on that, so that to its two extremities, if need be, a double force may be applied. Wherefore, from the axis of the cylinder to the extremity of such a lever, a line is drawn, which is the radius of the circle determining the movement of the power; and if the length of such a radius be to the interval of the spirals as 50 to 1, the diameter of the circle is 100, and its circumference greater than 314; and the motion of the power to the motion of the plate pressing the type is as 314 to 1: therefore, if at the extremities of the lever there be several men, each exerting himself as though he were moving 50 pounds apiece, the type is pressed by the force of this screw as if by a weight of 31,400 pounds. What is said, however, of pressure is to be understood in a similar way of the raising of weight, if perhaps it shall have been attached either to the lower base of the cylinder or placed upon its head; for just as a body resists the one pressing by reason of its compacted particles, so it resists the one lifting by reason of its own gravity: in both cases, therefore, the power has a similar force for overcoming resistance, when in both cases the same ratio of motions and moments is found. For this reason, in screws of this kind, which are wound around a fixed radius, the thickness of the cylinder is not to be anxiously provided for, so long as it is sufficiently solid and not fragile: for the circumference of the circle described by the motion of the power is the same, whether the cylinder's thickness be greater or smaller, when the distance of the power from the cylinder's axis is the same, and consequently the moment is the same. What matters most in the matter is how obliquely inclined the course of the spirals is; from this arises the ratio of the interval between the neighboring turns of the spirals, which are very frequent and separated by a short interval if the spiral line is very much inclined, but sparse and notably separated if it has been raised at a greater angle (yet an acute one): such an interval is, namely, equal to the tangent of the angle of inclination, the radius being taken as the circumference of the cylinder's base; but the spiral itself is the secant of the same angle. Therefore, given the diameter of the cylinder, the circumference of the base is found; & D D d d d 3

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Mechanicorum 766 dato spirarum intervallo, invenitur angulus huic intervallo tan- quam Tangenti oppositus, scilicet inclinatio spiræ, & hypo- thenusa tanquam ejusdem anguli Secans dat ipsius lineæ spira- lis longitudinem. Quod si totius lineæ spiralis universum cylindrum com- plectentis lineam desideras, toties peripheriam basis multipli- ca, quot sunt spirarum circuitus, & habebis Radium; cylindri altitudo dabit Tangentem, cui respondens Secans indicabit to- tius spiræ integram longitudinem. Sit ex. gr. cylindri altitudo ped. 3. hoc est unciarum 36. ejus diameter unciarum 7; ergo basis perimeter unc. 22: Spirarum circuitus sint 25: igitur ductâ perimetro 22 in 25, habetur 550 tanquam Radius, & 36 tanquam Tangens: igitur ut unciæ 550 ad uncias 36, ita Ra- dium 100000 ad 6545 Tangentem gr. 3. m. 45. cui respondet Secans 100214: Quare ut 100000 ad 100214, ita unciæ 550 ad uncias 551 177/1000 longitudinem totius lineæ spiralis; quam, si careas Canone Trigonometrico, etiam habebis ex 47. lib. 1. addendo quadrata numerorum 550 & 36, erit enim horum summa quadratum, cujus Radix dabit eandem quæsitam spiræ longitudinem. Cum autem hæc spiræ longitudo, sive universa, sive parti- culatim assumatur, semper longior sit, sivè multiplici, sivè sin- gulari perimetro circuli, qui est basis cylindri, utique motus potentiæ, ejusque momentum, non ex hac spirali lineâ desu- mendum est; neque enim ipsa esse potest mensura motûs po- tentiæ cylindro applicatæ ad ejus diametri extremitatem. Hinc est mihi non arridere eorum sententiam, qui cochleæ vires re- ferunt ad planum inclinatum, quod ab ipsâ lineâ spirali repræ- sentetur. In plano siquidem inclinato momentum gravitatis, ad ejusdem gravitatis momentum in perpendiculo, se habet re- ciprocè ut perpendiculum ad ipsam lineam inclinatam; ac pro- pterea eandem Rationem servant conatus Potentiæ moventis pondus aut in perpendiculo, aut in plano inclinato. At hîc po- tentia non movetur juxta lineæ inclinatæ longitudinem, sed breviore motu juxta basim trianguli rectanguli, cujus hypothe- nusa est ipsa linea inclinata: Igitur potentiæ momentum ali- quanto minus censendum est, quam pro Ratione plani inclina- ti. Adde momenta gravitatis ponderis alicujus tunc solùm fieri

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Mechanics 766 Given the interval of the spirals, the angle opposite this interval as a tangent is found, namely the inclination of the spiral, and the hypotenuse, as the secant of the same angle, gives the length of the spiral line itself. But if you desire the whole length of the spiral line embracing the entire cylinder, multiply the circumference of the base as many times as there are circuits of the spiral, and you will have the radius; the height of the cylinder will give the tangent, and the corresponding secant will indicate the complete length of the whole spiral. Let, for example, the height of the cylinder be 3 feet, that is, 36 inches; its diameter 7 inches; therefore the circumference of the base is 22 inches. Let the circuits of the spiral be 25; therefore, multiplying the circumference 22 by 25, there is obtained 550 as the radius, and 36 as the tangent. Thus, as 550 inches are to 36 inches, so the radius 100000 is to 6545, the tangent of 3 degrees 45 minutes, to which there corresponds a secant of 100214. Therefore, as 100000 is to 100214, so 550 inches are to 551 177/1000 inches, the length of the entire spiral line; which, if you lack the Trigonometric Canon, you will also obtain from book 1, proposition 47, by adding the squares of the numbers 550 and 36, for their sum will be the square, whose root will yield the same sought length of the spiral. But since this length of the spiral, whether taken as a whole or in part, is always longer than the single or multiple circumference of the circle, which is the base of the cylinder, plainly the force of the power and its moment are not to be derived from this spiral line; for it cannot itself be the measure of the motion of the power applied to the cylinder at the end of its diameter. Hence I do not approve of the opinion of those who refer the force of the screw to an inclined plane, represented by the spiral line itself. For on an inclined plane the moment of gravity, compared with the moment of the same gravity on a perpendicular, is reciprocally as the perpendicular is to the inclined line itself; and therefore the efforts of the moving power, whether it moves a weight in a perpendicular or in an inclined plane, preserve the same ratio. But here the power is not moved along the length of the inclined line, but by a shorter motion along the base of the right triangle whose hypotenuse is the inclined line itself; therefore the moment of the power must be judged somewhat less than according to the ratio of the inclined plane. Add that the moments of gravity of a weight are then only made

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Liber octavus. CAPUT I. 767 fieri minora in plano inclinato, quando illi insistit, & deorsum nititur premendo ipsum planum: at si pondus idem incumbat plano horizontali, verum quidem est planum verticale, quod adversùs pondus moveatur non secundùm directionem, quæ recta occurrat centro gravitatis ejusdem ponderis, sed obliquè, minus invenire resistentiæ: sed pondus illud propriè non mo- vetur super plano, licèt ab eo obliquè repellatur; & potiùs pla- num movetur juxta pondus: hìc verò si pondus in plano hori- zontali jacens sit adnexum cochleæ trahenti, aut oppositum cochleæ repellenti & urgenti, non movetur obliquè, sed motu directo: non igitur movetur super planum inclinatum. Porrò unum est in Cochleâ quodammodo singulare, quod in nullam aliam Facultatem æquè convenire deprehenditur: Cum enim requiratur & cylindrus in helicem inflexus, & Ma- trix illi congruens, ita ut alteri quies, alteri motus debeatur; perinde est si matrice immotâ cylindrus convertatur, atque si manente cylindro matrix ipsa convolvatur, modò Potentia æquali Radio utatur, sivè cylindri capiti, sivè Matrici infixo: eadem siquidem sunt potentiæ momenta, & æqualis motus pon- deris; æqualiter enim promovetur matrix in cylindro stabili, at- que cylindrus in Matrice immotâ. Id quod maxime locum habet, ubi opus est compressione, & vulgatissimus est apud va- rios artifices usus. Iam verò quod ad diuturnitatem spectat, diffitendum non est cochleam frequenti usu atteri, est enim perpetuus illius cum suâ Matrice conflictus, quamvis plurimùm juvet, si smegmate, aut pingui aliquo humore inungatur, quo lubrica fiat, ut faci- liùs convolvatur, minúsque atteratur. Deinde quamvis unica spira matrici sufficiat, ut vel ipsa, vel cylindrus promoveatur, aut retrahatur, nihilominus faciliùs labem patitur, quàm si plu- res in spiras fuerit excavata: cum enim aut à gravitate ponde- ris sustollendi, aut à partium constipatione repugnantium com- pressioni, ipsa unica resistentiam inveniat, utique vel ponderis gravitas ipsi uni innititur, vel potentiæ conatus, reluctante cor- pore comprimendo aut trahendo, in illam solam effunditur. Propterea expedit alteram saltem, aut tertiam spiram addere, ut diviso in plures conatu firmitati consulatur. Eandem ob causam aliquando cylindrum complectitur non unica

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Book Eight. CHAPTER I. 767 to make smaller things on an inclined plane, when it rests upon it and presses downward upon the plane itself: but if the same weight lie upon a horizontal plane, it is indeed true of a vertical plane that it is moved against the weight not in a direction that falls directly upon the center of gravity of that weight, but obliquely, so as to meet less resistance: but that weight is not properly moved over the plane, although it is repelled obliquely by it; rather the plane is moved along with the weight: here, however, if a weight lying on a horizontal plane be attached to a pulling screw, or opposed to a repelling and urging screw, it is not moved obliquely, but by a direct motion: therefore it is not moved over an inclined plane. Moreover, there is something in the Screw that is in a way singular, and is found to belong equally well to no other faculty: for since both a cylinder twisted into a helix and a corresponding Matrix are required, so that motion must belong to the one and rest to the other; it is the same whether, with the matrix unmoved, the cylinder be turned, or whether, the cylinder remaining, the matrix itself be wound around, provided the Power uses an equal Radius, whether fixed to the head of the cylinder or to the Matrix: for the moments of the power are the same, and the motion of the weight is equal; for the matrix is advanced equally in a fixed cylinder, and the cylinder in an unmoved Matrix. This is especially the case where compression is required, and its use among various artificers is very common. Now, as regards durability, it must not be denied that the screw is worn down by frequent use, for there is a perpetual conflict between it and its Matrix; though it helps greatly if it be anointed with grease or some oily moisture, so that it may become slippery, be wound more easily, and be worn less. Then, although a single spiral is enough for the matrix, whether it is to move forward or draw back the matrix itself or the cylinder, nevertheless it suffers damage more easily than if it had been hollowed out into several spirals: for since that single spiral encounters resistance either from the weight to be lifted, or from the compression of resisting parts opposing the pressure, surely either the weight’s heaviness presses upon that one spiral alone, or the effort of the power, while a body resists being compressed or drawn, is poured out upon that one alone. Therefore it is advantageous to add at least a second, or a third spiral, so that by dividing the effort among several, firmness may be better provided for. For the same reason at times the cylinder encompasses not a single

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Mechanicorum 768 unica spirarum series, sed & alia illi parallela additur ( nec quicquam prohibet, quin & plures duabus sint hujusmodi parallelarum spirarum series) ut multo validior ac firmior sit cochlea, ne facilè spira aliqua dissipetur, aut si qua labefactetur, nullum sequatur incommodum, alterâ spirâ parallelâ ejus vices supplente. Sic spiræ ABCDEF parallela statuitur altera ab I incipiens, & per KLMNOP simili lapsu serpens. Ex hac tamen multiplici spirâ non augetur momentum potentiæ applicatæ Radio VS; neque enim circulus à potentiâ in S applicatâ descriptus comparandus est cum A K, sed cum A C; unicâ siquidem cylindri convolutione promovetur cylindrus non ex A in K, sed ex A in C. Propterea oblato cylindro in Cochleam deformato diligenter attendendum est, utrum plures sint spirarum series, an unica; ne fortè ex brevi inter proximas spiras intervallo perperam conjicias lineam spiralem magis inclinatam, quàm reipsa sit; prius enim dijudicandum est, an illæ proximæ spiræ ad eandem Helicem spectent; attenditur scilicet intervallum spirarum ad eandem seriem continuo ductu pertinentium. CAPUT II. An utilis sit Cochlea duplex contraria. Quamvis ad superandam modico labore resistentiam non modicam corporis, quod Cochlea urget, aut trahit, hujusmodi Facultas sit potissimùm excogitata, sæpissimè tamen cochleam adhibemus non ad vincendam resistentiam, quæ aliquando tenuissima est, sed unicè ad motum ita temperandum, ut pro opportunitate exiguus sit; neque enim musculorum motum ita attenuare pro arbitrio potest homo, ut semper quàm minimus

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Mechanics 768 a single series of spirals, but another parallel to it is added (nor does anything prevent even more than two such series of parallel spirals), so that the screw may be much stronger and more firm, lest any spiral should easily be broken up, or if any should be weakened, no inconvenience should follow, another parallel spiral supplying its place. Thus, for the spiral ABCDEF, another is set parallel, beginning from I, and winding in a similar course through KLMNOP. Yet from this multiple spiral the moment of the applied power is not increased by the radius VS; for the circle described by the power applied at S is not to be compared with AK, but with AC; for by a single convolution of the cylinder the cylinder is advanced not from A to K, but from A to C. Therefore, when the cylinder transformed into a screw is presented, one must carefully attend to whether there are several series of spirals, or only one; lest perhaps from the short interval between the neighboring spirals you wrongly conjecture the spiral line to be more inclined than it really is; for first it must be judged whether those neighboring spirals belong to the same helix; namely, the interval between spirals continuously drawn belonging to the same series is to be considered. CHAPTER II. Whether a double contrary screw is useful. Although this sort of device was chiefly devised for overcoming, with slight labor, the not slight resistance of a body, which the screw presses or draws, nevertheless very often we employ the screw not to overcome resistance, which is sometimes very slight, but solely to regulate motion so that, as occasion requires, it may be small; for a man can no more so moderate the motion of his muscles at will, that it is always as small as possible

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Liber octavus. CAPUT II. 769 minimus contingat: propterea deformato in cochleam cylindro utimur, ut majori potentiæ motui minor motus in corpore movendo respondeat. Sic ad excipiendas objectorum corporum species opticas aut lumen, cum non eadem semper tubospecilli longitudo opportuna sit pro variâ tum objecti distantiâ, tum oculi conformatione, prudenter ab aliquibus extremus tubulus, cui lens ocularis inseritur, in spiram contorque-tur, ut faciliùs & citiùs justam longitudinem assequantur: id quod ægrè obtinerent, si rectâ tubulum illum adducerent, aut reducerent, ut satis experientiâ constat. Hinc aliquando contingit oppositos motus conciliandos esse duobus corporibus ita, ut aut ad se mutuò accedant, aut magis invicem disjungantur, sive illa motui valde repugnent, sive sola motûs tarditas requiratur. Propterea ejusdem cylindri longitudo in duas helices distinguitur, quæ simili quidem ductu cylindrum circumpectuntur, sed illis in diversâ abeuntibus, unius spiræ non sunt alterius spiris parallelæ; quæ eatenus contrariæ vocari possunt, quatenus oppositos motus efficiunt, ne- que eæ sunt, quæ in unam continuam spiram coalescere queant. Ex cylindri A B medio puncto C exeant duæ spiræ ad easdem partes inclinatæ, hinc C D versùs extremitatem A procedens, hinc verò C E versùs extremitatem B; utraque enim suam matricem habens, cui inseratur, dum convolvitur cylindrus, matricem longiùs à medio promovet, aut ad medium attrahit; atque cum matrice adnexa corpora simili & æquali motu moventur. Hinc si utraque matrix proxima sit medio puncto C, ex primâ convolutione cylindri altera per CD removetur usque in F, altera per C E in G; atque adeò sicut matrices moventur per C F, & C G, ita eadem mensura corporum adnexorum E E e e e

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Liber octavus. CHAPTER II. 769 the smallest may be effected: for this reason we use a cylinder shaped into a screw, so that a greater motion in the moving power may correspond to a lesser motion in the body being moved. Thus, for receiving the optical species of objects, or light, since the proper length of the tube of the spyglass is not always the same, both on account of the varying distance of the object and of the conformation of the eye, some have prudently twisted the outermost tube, into which the ocular lens is inserted, into a spiral, so that they may more easily and more quickly attain the proper length; something they would scarcely obtain if they drew out or pushed back that tube in a straight line, as experience sufficiently proves. Hence it sometimes happens that opposite motions must be reconciled in two bodies, so that either they move toward one another or separate more from one another, whether they strongly resist the motion or whether only slowness of motion is required. For this reason the length of the same cylinder is divided into two helices, which, indeed, encircle the cylinder with a similar course, but, diverging in different directions, are not parallel to one another’s turns; these may be called opposite insofar as they produce opposite motions, and they are not such as can merge into one continuous spiral. From the middle point C of cylinder A B let two spirals proceed inclined toward the same sides, here from C D advancing toward the extremity A, there from C E toward the extremity B; for each, having its own matrix into which it is inserted, as the cylinder is wound up, carries the matrix farther from the middle, or draws it toward the middle; and the bodies attached to the matrix are moved with a similar and equal motion. Hence if each matrix be near the middle point C, from the first winding of the cylinder one is moved through C D as far as F, the other through C E to G; and thus, just as the matrices are moved through C F and C G, so by the same measure the attached bodies E E e e e

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Mechanicorum 470 motum metitur, quæ invicem removentur intervallo FG; & ita deinceps in cæteris cylindri convolutionibus. Quod si movendorum in oppositas partes corporum resistentia exigua sit, satis fuerit extremitatibus cylindri ansulas apponere, quibus circumactis cylindrus ipse in cochleas deformatus convertatur. Sic antè annos ferè quadraginta (cum non arrideret vulgaris tunc apud artifices circinorum forma, qui interjecto elatere crura divaricant; sed inflexam in arcum cochleam alteri crurum infixam, & per alterum trajectam, decurrente matrice exteriùs appositâ, dilatationem moderatur artifex) jussi mihi parari absque ullo elatere circinum, quem ipse dilatarem atque contraherem pro arbitrio, cochleam hujusmodi duplicem in hanc atque in illam partem convertens. Ad trientem totius longitudinis à nodo, singula crura cylindricum foramen habent, ut singulis inferantur cylindruli congruentes exquisite politi, quorum superiori extremitati sunt adnexæ cochlearum matrices, inferior extremitas extra circini soliditatem exiens in helicem desinit, ut appositâ matrice cylindrulus intra foramen contineatur. Hujusmodi est cylindrulus I S crassitiei circini respondens, superior pars est matrix R, infima extra circini soliditatem in helicem deformata est S V, cui addita matrix X continet cylindrulum intrà foramen, cui inditus est, ita tamen, ut cylindruli ipsius opportunam convolutionem non impediat. Duæ igitur matrices, cujusmodi est R, coaptantur duplici cochleæ cujus deinde extremitatibus, ad facilem conversionem, ansulæ adduntur, adeò ut illæ matrices non sint exemptiles. Quare utrique crurum circini foramini, utriusque matricis cylindrulus I S inseratur, & inferius matrice X firmetur: Nam convertendo cylindrum in duplicem cochleam deformatum, circini crura divaricabis, aut adduces, ut libuerit. Neque quicquam officiet cylindri rectitudo, quia matricum cylindruli I S pro opportunitate volvuntur. Hinc circino eodem uti poteris absque cochleâ, deduci enim hæc potest, exemptis matricibus è foramine, cui inseruntur. At verò si validiore conatu opus fuerit, ad medium cylindrum, ubi cochlearum spiræ incipiunt, oportebit foramina excavare, quibus immitti queat vectis, ut potentiæ motus

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Mechanicorum 470 it measures the motion, which are in turn separated by the interval FG; and so on in the other convolutions of the cylinder. If the resistance of the bodies to be moved in opposite directions is slight, it will be sufficient to attach small rings to the ends of the cylinder, by turning which the cylinder itself, deformed into screws, may be moved. Thus about forty years ago (when the common form of compasses then used by craftsmen did not please me, in which the legs are spread apart by an inserted spring; but the screw bent into an arch, fixed to one leg and passing through the other, the outer matrix being applied, regulates the opening as it runs down), I was ordered to have made for me, without any spring, a compass which I myself might open and close at will, by turning a double screw of this kind in one direction or the other. At a third of the whole length from the joint, each leg has a cylindrical hole, so that into each may be inserted a matching little cylinder, exquisitely polished, to the upper end of which screw-matrices are attached; the lower end, passing out beyond the solidity of the compass, ends in a helix, so that, with the matrix applied, the little cylinder is held within the hole. Such is the little cylinder IS, corresponding in thickness to the compass; its upper part is the matrix R, its lower part, beyond the solidity of the compass, is deformed into a helix SV, to which the added matrix X holds the little cylinder within the hole into which it has been inserted, yet in such a way that it does not hinder the proper turning of the little cylinder itself. Two matrices, therefore, such as R, are fitted to the double screw, to the ends of which handles are then added for easy turning, so that those matrices are not removable. Wherefore into the hole of each leg of the compass, the little cylinder IS of each matrix is inserted, and fixed below by matrix X: for by turning the cylinder deformed into a double screw, you will spread the legs of the compass apart, or draw them together, as you wish. Nor will the straightness of the cylinder cause any difficulty, because the little cylinders IS of the matrices are turned as needed. Hence you will be able to use the same compass without the screw, for this can be withdrawn, the matrices being removed from the hole into which they are inserted. But if a stronger effort should be needed, it will be necessary to hollow out holes in the middle of the cylinder, where the spirals of the screws begin, into which a lever may be inserted, so that the motion of power

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Liber octavus. CAPUT II. 771 tus ad ponderum motum habeat majorem Rationem. Sic duplici cochleâ cylindro circumductâ ad medium E sint foramina, quibus vectis B C subinde inferri possit: duo autem membra F D, & M N ex materiâ satis solidâ, qua extremitate respiciunt vectem, matricem habeant cochleæ congruentem, ut ex vectis & cochleæ conversione aut ad se invicem accedant, aut sejungantur: reliqua extremitas exterior D & N cava sit, ut corpus repellendum comprehendatur, reflectatur verò quasi in uncos K & R, ut si duo corpora attrahenda fuerint, iis apprehendantur, sivè proximè & immediatè, sivè funibus adnexa. Quanta sit hujus instrumenti vis, etiam ad frangenda aut dilatanda ferrea clathra, hinc patet, quòd, longiore vecte addito, potentiæ momenta notabiliter augentur; quia potentia percurrit peripheriam circuli, cujus Radius à cylindri centro ad vectis extremitatem producitur, pondera verò non nisi pro spirarum intervallo moventur. Ubi tamen advertendum est, utrum cylindrus ita sit alicui loculamento insertus, ut ejus medium D E nec ad dexteram, nec ad sinistram declinare queat, an verò liber omnino sit. Si enim interjectum movendis corporibus instrumentum omnino liberum sit, pondera verò movenda inæqualiter resistant comparatis, aut eorum gravitatibus, aut momentis ratione planorum non uno modo inclinatorum, aut ex disparili superficierum asperitate, non sequitur æqualis eorum motus, sed qua parte major invenitur resistentia, minor quoque est motus; quamvis utrumque æqualiter distet à medio cylindri, quod repellitur quodammodo ad eam partem, ubi levior est resistentia. Si enim ad N sit aliquid obstans motui, ut paries, aut firmi- E E e e 2

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Book Eight. Chapter II. 771 thus it has a greater ratio to the motion of the weights. Thus, with a double screw turned around the cylinder, there should be holes at the middle E, through which the lever B C may be inserted from time to time: and the two members F D and M N, made of material sufficiently solid, at the end facing the lever, should have a female thread matching the screw, so that by the turning of the lever and the screw they may either approach one another or be separated: the outer remaining end D and N should be hollow, so that the body to be repelled may be grasped, but turned back as it were into hooks K and R, so that if two bodies are to be attracted, they may be seized by them, whether directly and immediately, or attached by ropes. How great the force of this instrument is, even for breaking or widening iron bars, is clear from this: that, if a longer lever be added, the moments of the power are notably increased; because the power traverses the circumference of a circle whose radius, from the center of the cylinder, is extended to the end of the lever, whereas the weights are moved only according to the interval of the turns. Yet it must be noted whether the cylinder is inserted in some kind of housing, so that its middle D E can turn neither to the right nor to the left, or whether it is entirely free. For if the instrument interposed for moving bodies is entirely free, but the bodies to be moved resist unequally in comparison, either by their weights, or by their moments with respect to planes inclined in not one way, or from the unequal roughness of their surfaces, then an equal motion does not follow, but on the side where greater resistance is found, there is also less motion; although both are equally distant from the middle of the cylinder, which is repelled, as it were, toward that part where the resistance is lighter. For if at N there is something hindering the motion, such as a wall, or a firm- E E e e 2

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Mechanicorum 472 ter infixus paxillus, ad D verò corpus aliquod repellendum; utique ex vectis B C conversione etiam cochlea convolvitur, & corpus in D positum tantumdem promovetur, quanto intervallo absunt F & M ex cochleæ conversione; nam propter impedimentum in N existens, nullo pacto ipsum M movetur. Sin autem corpus in N non omnino reluctetur motui, sed tamen resistat magis, quàm corpus in D, illud quidem aliquantulum repellitur, sed multò magis repellitur corpus, quod est in D; & in hoc motu medium cylindri punctum E ad eas partes accedit, ad quas movetur corpus in D repulsum. Quod si medium E ita esset loculamento aliquo conclusum, ut positionem mutare nequeat, sed solùm convolvi possit, tunc utrumque corpus æqualiter repellitur, quia spirarum intervalla in utráque cochleâ æqualia sunt. Hinc patet posse fieri motus inæquales, si spirarum inclinationes non fuerint æquales; minùs enim movetur illud, quod spiris spissioribus urgetur. Quamvis autem hujusmodi duplex cochlea ad duo corpora disjungenda aut attrahenda sæpè utilis sit, ubi tamen exigus motus requiritur, sivè ad augenda potentiæ momenta, sive ad affectandam tarditatem, præstabit simplicem cochleam adhibere. Nam in duplicis cochleæ conversione disjunguntur, aut ad se invicem accedunt matrices (ac proinde & corpora, quæ moventur) quantum est duplex intervallum, quo spira abest à spira; singulis nimirum cochleis suum respondet intervallum: at in simplicis cochleæ conversione motus respondet simplici duarum proximarum spirarum intervallo; ad quod idem potentiæ motus majorem habet Rationem, quàm ad duplex intervallum. Quare satis est, si alterutrum membrum cochleam includens longius sit, & matricem habeat; reliquum membrum, ut M N, brevius esse potest, quantum opus fuerit ad recipiendum cylindri caput extenuatum in minorem cylindrum, intrà foramen cylindricum exquisitè politum, ut facillimè converti possit: ita verò caput illud muniatur, ut ex loculamento extrahi nequeat, quando utendum fuerit instrumento ad corpora attrahenda: nam ad illa disjungenda cùm addhibetur, satis reluctatur major diameter cylindri in cochleam

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Mechanicorum 472 the peg inserted three times, toward D, indeed, for repelling some body; for from the turning of the lever B C, the screw is also wound up, and the body placed at D is advanced by as much as the distance by which F and M are separated through the turning of the screw; for because of the obstacle existing in N, in no way is M itself moved. But if the body in N does not wholly resist motion, but nevertheless resists more than the body in D, then indeed it is repelled somewhat, but the body that is in D is repelled much more; and in this motion the middle point E of the cylinder approaches those parts toward which the body repelled in D is moved. But if the middle E were enclosed in some socket in such a way that it cannot change position, but can only be wound around, then each body is repelled equally, because the intervals of the spirals in each screw are equal. Hence it is clear that unequal motions can be produced if the inclinations of the spirals are not equal; for that is moved less which is pressed by closer-set spirals. Although, however, a double screw of this kind is often useful for separating or drawing together two bodies, where a small motion is required, whether for increasing the force of power, or for producing slowness, it will be better to use a simple screw. For in the turning of the double screw the nuts are separated, or approach one another, and consequently the bodies that are moved, by as much as the double interval by which one spiral is distant from another; namely, each screw corresponds to its own interval: but in the turning of a simple screw the motion corresponds to the simple interval of two neighboring spirals; and to the same motion of power it has a greater relation than to the double interval. Therefore it is enough, if either member enclosing the screw is longer and has a nut; the remaining member, as M N, may be shorter, as much as is needed for receiving the tapered head of the cylinder into a smaller cylinder, within the cylindrical hole finely polished, so that it may be turned most easily: thus indeed let that head be secured so that it cannot be drawn out from the socket, when the instrument is to be used for drawing bodies together; for when it is applied to separate them, the larger diameter of the cylinder in the screw sufficiently resists

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Liber octavus. CAPUT II. 773 chleam deformati, ne intrà foramen ulteriùs excurrat. Simili planè ratione ad divaricanda aut contrahenda cir- cini crura, uti poteram unicâ & simplici cochleâ R S: cylindri il- lius extremitas extenuetur in mi- norem cylindrum exquisitè lævigatum, qui prominentis capitis O foramini cylindrico pariter polito inseratur, & exteriore ansula V converti pro arbitrio possit. Alterius clavi T X caput T matricem habeat cochleæ congruentem: nam conversa an- sula V adducet clavum T, & cum eo crus circini, ad O, aut ab hoc illum removebit, & crura divaricabit: & qui- dem faciliùs licebit minutam in accipiendis punctorum di- stantiis subtilitatem persequi; quandoquidem uni integræ conversioni cylindri respondet unicum spirarum intervallum, non autem duo intervalla hujusmodi, quemadmodum cùm duplex est cochlea. Quæ verò hîc dicta sunt, in pluribus aliis locum habe- re possunt, in quibus pro opportunitate modò simplicem, modò duplicem cochleam prudens Machinator adhibebit: Et quidem si duplex futura sit cochlea, neque æquali mo- tu movenda sint in oppositas partes corpora, cochleas ipsas non simili, sed inæquali, spirarum inclinatione formari ju- bebit. Cochleam autem ipsam opportuno loco statuet: & si fortè corporum ipsorum motus paulò velocior aut major requiratur, quàm ferat ipsa cochleæ convolutio, duobus vectibus decussatis, & circa axem in decussatione versati- libus uti poterit, atque cochleam cum suis clavis & matri- cibus (ut superiùs de circino dictum est) non procul à de- cussatione collocabit; nam modica illius convolutio non exiguum motum tribuet corporibus in vectium illorum ex- tremitate positis, quippe quæ à decussatione magis distant, quàm cochlea. At, inquies, hoc idem Ergatâ præstari poterit: si enim funes breviorum brachiorum extremitatibus C & E adnexi E E e e e 3

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Book Eighth. CHAPTER II. 773 the screw being shaped, so that it may not run farther inward than the hole. In a quite similar way, for spreading apart or drawing together the legs of a compass, I may use a single and simple screw R S: the end of that cylinder is reduced into a smaller cylinder, exquisitely polished, which is inserted into the cylindric hole of the projecting head O, likewise polished, and may be turned at will by the outer handle V. The head T of the other pin T X has a socket corresponding to the screw: for by turning the handle V the pin T will be drawn toward it, and with it the leg of the compass toward O, or it will be removed from it and the legs spread apart: and indeed it will be easier to pursue minute precision in taking the distances of points, since to one full turn of the cylinder there corresponds a single interval of the threads, and not two such intervals, as when the screw is double. But what has here been said may have application in many other cases, in which, as convenience requires, an intelligent Mechanician will use now a simple, now a double screw. And indeed, if the screw is to be double, and bodies are to be moved in opposite directions without equal motion, he will direct that the screws themselves be formed not with similar but with unequal inclination of the threads. He will place the screw itself in a suitable position; and if perhaps a somewhat quicker or greater motion of the bodies themselves is required than the winding of the screw will allow, he may use two crossed levers, movable around the axis at the crossing, and will place the screw with its pins and sockets (as was said above of the compass) not far from the crossing; for its slight winding will impart no small motion to the bodies placed at the ends of those levers, since those ends are farther from the crossing than from the screw. But, you will say, this same thing can be accomplished by the Ergoata: for if the ropes attached to the ends of the shorter arms C and E ...

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Mechanicorum 774 connectantur cum Ergatæ cylindro, ex hujus conversione ac- cedent ad se invicem ex- tremitates C & E, ac pro- pterea etiam velociùs cor- pora in F & D movebun- tur. Ita planè: non diffi- teor: sed si extremitates C & E proximas disjungere oporteat, adeóne promptus erit Ergatæ usus, quin alio artificio opus sit, ut hujus ope disjungantur? Præterquam, quod sæpè multum spatij ad col- locandam Ergatam requiritur; si maximè opus sit illi pegma construere. Quid verò si vectes ipsi promovendi sint, non retrahendi, ut in rebus scenicis contingere potest? Quid si in sublimiore loco res perficienda sit? quàm incommodè opportu- na Ergata satis longo vecte instructa ibi parabitur? Sed illud potissimum attendendum est, quod vis cochleæ longè major est; nam in Ergatâ Ratio motûs potentiæ ad motum ponderis est eadem cum Ratione peripheriæ ab extremitate vectis de- scriptæ ad cylindri ambitum, hoc est longitudinis vectis usque ad axem cylindri, ad ipsius cylindri semidiametrum: at in co- chleâ peripheria ab extremo vecte descripta non comparatur cum ipsius cylindri perimetro, sed cum proximarum spirarum intervallo, quod sæpissimè minus est (saltem potest esse minus, si magis inclinatæ & spissæ sint spiræ) quàm cylindri diameter, aut ejus perimeter: ac proinde major est Ratio motûs potentiæ ad motum ponderis. CAPUT III. Cochlea cum Vecte, atque cum Axe componitur. Contingere potest aliquando onus Vecte elevandum esse, tantam verò illius gravitatem deprehendi, ut sufficiens Vectis longitudo non suppetat pro ratione operarum, quas adhibere

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Mechanicorum 774 If they are connected with the cylinder of the winch, then by its turning the ends C and E will approach one another, and therefore the bodies in F and D will also be moved more quickly. Just so: I do not deny it. But if the adjacent ends C and E must be separated, will the use of the winch be so ready that no other contrivance is needed, by means of which they may be separated? Besides, very often a great deal of space is required for placing the winch; and if need be, a staging must be built for it. But what if the levers themselves are to be advanced, not drawn back, as may happen in theatrical machinery? What if the work is to be done in a higher place? How inconveniently shall a suitable winch, furnished with a sufficiently long lever, be prepared there? But this above all must be observed, that the force of the screw is far greater; for in the winch the ratio of the motion of the moving power to the motion of the weight is the same as the ratio of the circumference described by the end of the lever to the circumference of the cylinder, that is, of the length of the lever up to the axis of the cylinder, to the semidiameter of the cylinder itself: but in the screw the circumference described by the end of the lever is not compared with the circumference of the cylinder itself, but with the interval between the adjacent spirals, which very often is less (at least it can be less, if the spirals are more inclined and closer together) than the diameter of the cylinder, or its circumference; and therefore the ratio of the motion of the moving power to the motion of the weight is greater. CAPUT III. The screw is combined with the lever, and with the axle. It may happen at times that a load is to be raised by a lever, but its weight is found to be so great that there is not sufficient length of lever available, in proportion to the number of workmen who are to be employed, to...

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Liber octavus. CAPUT III. 775 adhibere possumus, aut saltem ex sint loci angustiæ, ut neque hujusmodi Vectis longitudinem, neque operarum multitudinem capiat: plures autem Vectes componere omnino incommodum sit, quia, ut in loco dictum est, minimus fieret oneris elevandi motus. Præstabit igitur Cochleam Vecti addere, ubi maximè frequens futura sit hujusmodi ponderum elevatio, quemadmodum propè telonia, ubi ingentes mercium sarcinæ attollendæ sunt, ut plaustris avehendæ imponantur, aut advectæ ex iis deponantur. Erigatur tignum A B ad perpendiculum firmiter infixum plano subjecto, tanta verò sit crassities tigni, ut in eo excavari possit crena CD, per quam tignum aliud EF immitti facilè valeat adeò solidum, ut Vectis munere fungatur, ubi extremo unco G funibus adnexum fuerit onus. Ut igitur facillimè oneris elevatio perficiatur, cochlea H I prismati HL infixa (satiùs fuerit, si ejusdem ligni pars in prisma, pars in cochleam deformetur) ad perpendiculum erigatur inserta matrici NM: sit autem ita solida matrix, ut illi adnecti queat tignum EF clavo E, circa quem facilè versari possit tignum ipsum, quando attollitur aut deprimitur. Demum subjecto prismati HL non

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Book Eight. Chapter III. 775 we can apply it, or at least from there the place may be narrow, so that it will hold neither the length of this kind of Lever nor the number of workers: but to combine several Levers would be altogether inconvenient, because, as has been said in the place, there would be the smallest possible movement in lifting the load. Therefore it will be preferable to add a Screw to the Lever, wherever such lifting of weights is to be most frequent, as near tollhouses, where great burdens of merchandise have to be raised, so that they may be placed on wagons to be carried away, or, having been brought by them, may be set down from them. Let the beam A B be erected upright, firmly fixed in the ground beneath; and let the thickness of the beam be such that a notch CD can be cut in it, through which another beam EF may easily be inserted, so solid as to serve the office of a Lever, when a weight attached by ropes to the hook G at its end has been fastened. Therefore, in order that the lifting of the load may be accomplished as easily as possible, let the screw H I, fixed to the prism HL (it would be better if part of the same piece of wood were fashioned into the prism and part into the screw), be erected upright, inserted into the socket NM: and let that socket be so solid that the beam EF may be attached to it by the pin E, around which the beam itself may easily turn when it is raised or lowered. Finally, with the prism HL placed beneath, not

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Mechanicorum non solùm in quatuor faciebus insint foramina, quibus immittatur Radius K O, verùm etiam in infimâ basis parte, qua respondet axi cylindri in cochleam deformati, sit polus, circa quem convolvi possit: Hic tamen (ut satis manifestum est) intrà ferream laminam ritè applumbatam lapidi in terrâ firmissimè defixo, aut certè adeò gravi, ut longè omnem elevandarum sarcinarum gravitatem vincat, ita contineri debet, ut indè nullo pacto eximi valeat, neque à prismate avelli. Quod si non fuerit inter pavimentum & laqueare intervallum enorme, faciliùs erit congruam trabis partem in cochleam deformare, illâmque basi imponere ( cujus altitudo commodam Radij K O conversionem præstet) atque circa duos polos, alterum eidem subjectæ basi, alterum lacunari infixum, convolvere. Aut saltem proximo parieti infigatur tignum horizontale, quod prominens excipiat superiorem polum, atque prohibeat, ne vi ponderis ex unco G dependentis, in altum abripiatur cochlea. Hic vides compositam cum Vecte Cochleam, quæ Vectis vires notabili incremento auget. Est autem vectis hypomochlium in eâ incisæ aut insculptæ crenæ parte, quæ cochleam respicit, quando pondus attollitur, & vectis caput F supra lineam horizontalem elevatur; secùs verò, quando deprimitur vectis infra lineam horizontalem, tunc enim hypomochlium est in D. Sit igitur hypomochlium ad totius longitudinis E F bessem; ac proinde Ratio motûs potentiæ in E, ad motum ponderis in F, est dupla. Ponamus Radij K O extremitatem O ab axe cylindri distare intervallo saltem decuplo intervalli spirarum cochleæ H V: ergo potentia describens peripheriam circuli, cujus diameter est 20, habet motum, qui est, ut minimum, ut 62 ad motum HV, hoc est ad motum extremitatis E: hujus autem motus est duplus motûs ponderis in F: igitur motus potentiæ est ad motum ponderis ut 124 ad 1. Quare ut major sit motus ponderis, poterit hypomochlium minùs abesse ab extremitate E; vix enim tales sunt sarcinæ, quæ ut moveantur, citrà laborem sustentandi, indigeant 60 hominibus. At si loci ratio ferat, suaderem potiùs Vectem secundi generis, ita ut altera vectis extremitas insisteret tigno A B, & pondus inter tignum & cochleam interciperetur: sic enim quò pondus

Transcription: Translated (English)

Of mechanics not only in the four faces are there holes, through which the Radius K O is inserted, but also in the lower part of the base, where it corresponds to the axis of the cylinder shaped into a screw, there should be a pivot, around which it can be wound: this however (as is sufficiently evident) within an iron plate duly plumbed to a stone fixed most firmly in the ground, or certainly made so heavy that it far exceeds all the weight of the loads to be raised, must be so held that from it in no way can it be removed, nor torn away from the prism. But if there is not an enormous interval between floor and ceiling, it will be easier to shape a suitable part of the beam into a screw, and to place it on the base (whose height may provide a convenient turning of Radius K O), and to wind it around two pivots, one subject to the same base, the other fixed in the ceiling. Or at least let a horizontal beam be fixed to the nearest wall, which, projecting outward, may receive the upper pivot, and prevent the screw, by the force of the weight hanging from hook G, from being carried upward. Here you see the Screw combined with the Lever, which greatly increases the power of the Lever. And the lever’s hypomochlion is in that part of the notch cut or carved in it which faces the screw, when the weight is raised, and the head F of the lever is elevated above the horizontal line; otherwise, when the lever is depressed below the horizontal line, then the hypomochlion is at D. Let the hypomochlion therefore be at two-thirds of the whole length E F; and consequently the ratio of the motion of the power at E to the motion of the weight at F is double. Let us suppose the end O of Radius K O to be distant from the axis of the cylinder by an interval at least ten times the interval of the spirals of the screw H V: therefore the power describing the circumference of a circle whose diameter is 20 has a motion which is, at the least, as 62 to the motion HV, that is, to the motion of the extremity E: but this motion is double the motion of the weight at F: therefore the motion of the power is to the motion of the weight as 124 to 1. Wherefore, that the motion of the weight may be greater, the hypomochlion could be farther from the extremity E; for there are scarcely such loads which, in order to be moved, without the labor of sustaining them, require 60 men. But if the nature of the place should allow it, I would rather recommend a Lever of the second kind, so that one end of the lever might rest on beam A B, and the weight be intercepted between the beam and the screw: for thus the more the weight

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Liber octavus. CAPUT III. 777 pondus esset propius cochleæ, ad majorem altitudinem attolle- retur, licèt majore conatu; sed cochleæ vis abundare videtur, & Vectis secundi generis semper auget momenta. Nec dissimili ratione Vectem manu tractabilem ita cochleæ instruere possumus, ut ad ingentia pondera movenda satis sit. Finge siquidem re- vellendas suis è car- dinibus ingentes a- licujus basilicæ val- vas, ut reficiantur; ferreus vectis A B paretur extremita- te A subjiciendus ponderi, & altera extremitas B in ma- tricem cochleæ ex- cavetur: in hanc immittatur cochlea C D, manubrium habens C E: atque ut cochlea faciliùs convertatur, neque pavimentum atterat, pa- ratam habeto ferream laminam H, quæ illi subjiciatur. Nam si ponderi supponatur vectis ex. gr. in F ad totius longitudinis sextantem, manubrij autem longitudo C E sit saltem decupla intervalli spirarum cochleæ, utique peripheria descripta à po- tentiâ in E, ad elevationem extremitatis B, est saltem ut 62 ad 1: elevatio autem ipsius B, ad elevationem ponderis in F, est ut 6 ad 1: igitur motus potentiæ ad motum ponderis est ut 372 ad 1. Quapropter, etsi initio parùm attollantur fores, & subjecto cuneo, ne recidant, atque revolutâ cochleâ deprimen- dus, ac promovendus sit vectis, ut pondus sit in I, puta ad to- tius longitudinis quadrantem aut trientem, adhuc potentiæ momenta erunt ut 248, aut 186 ad 1: quæ sanè exigua non sunt pro simplici hujusmodi machinulâ. Huc pariter spectat præli genus in meâ patriâ vulgare (in lo- cis potissimùm montanis, ubi faciliùs ingentes lapides non pro- cul advehendi suppetunt) quo ex uvæ jam calcatæ reliquiis tortivum mustum exprimitur. Roboris, quoad fieri potest, lon- gissimum truncum unâ cum imo caudice assumunt, & ita ramis omnibus spoliant, ut tamen bifurcum relinquant, quatenus FFfff

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Book Eight. Chapter III. 777 If the weight were brought closer to the screw, it would be raised to a greater height, though with greater effort; but the force of the screw seems to be ample, and a lever of the second kind always increases the moments. Nor can we, by a similar method, fit a hand-worked lever with a screw in such a way that it may suffice to move very great weights. For suppose that the great doors of some basilica are to be torn from their hinges, so that they may be repaired; let an iron lever A B be prepared, with one end A to be placed under the weight, and the other end B hollowed out into a socket for a screw: let the screw C D be inserted into this, having the handle C E; and, so that the screw may be turned more easily and not wear the floor, let an iron plate H be provided to be placed beneath it. For if a lever be placed under the weight, for example at F, at one-sixth of its whole length, and the length of the handle C E be at least ten times the interval of the screw threads, then the circumference described by the force at E, for raising the end B, is at least as 62 to 1: but the raising of B itself, compared with the raising of the weight at F, is as 6 to 1: therefore the movement of the force compared with the movement of the weight is as 372 to 1. Wherefore, although at first the doors are lifted only a little, and, with a wedge placed underneath so that they do not fall back, the screw being turned, the lever must be lowered and advanced, so that the weight may be at I, say at a quarter or a third of the whole length, still the moments of the force will be as 248, or 186 to 1: and these are certainly not small for a simple machine of this kind. To this also belongs a kind of press common in my country (especially in mountainous places, where large stones are more easily at hand to be brought from nearby), by which the pressed must is expressed from the remains of grapes already trodden. They take the longest possible trunk of oak, together with the lower stock, and strip it of all its branches, yet leave it forked, insofar as

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Mechanicorum 778 bifidæ illi extremitati inniti, atque connecti valeat matrix co- chleæ, quæ convertatur circa polum ingenti subjecto lapidi in- sistentem, sed eâ ratione, ut demum etiam lapis attolli queat. Truncum verò illum, qui præli munere fungi debet, præter extremum caudicem crassissimum, dedolant, ut inter bina tigna hinc atque hinc in alvei lateribus ad perpendiculum erecta in- terjectum prælum attolli ac deprimi possit citrâ impedimentum, quod alioqui ipsa rudis asperitas pareret. Porrò tigna illa bina erecta, aut ex adverso rotundis aliquot foraminibus perforant, quibus immitti possit crassiusculus cylindrus, seu ferreus vectis, aut illa incidunt patente crenâ, cui inseri valeat repagulum; eo consilio, ut alterutra præli extremitas pro opportunitate prohi- beatur, ne ascendat, aut descendat. Quare convoluta cochlea attollit matricem, & opposita præli extremitas amotis omnibus subjectis repagulis sensim descendit: ubi autem eò venerit, ut non absit ab altitudine eorum, quæ in torculari calcanda sunt, immittitur superiùs repagulum, ne amplius attolli valeat; Tum revoluta in contrarium cochleâ matricem cum præli extremita- te deorsum trahit: & quoniam reliqua extremitas attolli nequit obtante repagulo, premuntur uvæ, & in Lacum defluit mustum. Ubi demum adeò compressa fuerint vinacea, ut fa- cilius sit lapidem cochleæ adnexum attollere, quàm illa magis comprimere, ex cochleæ conversione attollitur lapis; quem ad mediocrem altitudinem elevatum pendere diutiùs permittunt, ut, lapidis gravitate deorsum conante, à prælo exprimatur, quantulumcumque musti adhuc vinaceis inest. Duplex igitur hîc consideranda est pressio: altera quidem vi potentiæ co- chleam volventis; & hîc cochlea cum vecte secundi generis componitur; est enim prælum vectis, cujus hypomochlium est in eâ extremitate, quæ repagulo prohibetur, ne attollatur; po- tentiæ autem vectem hujusmodi deprimentis vices obit cochlea claviculatim striata; quæ tamen motûs originem non habens sibi insitam, potentiæ munus & nomen relinquit vectiariis il- lam versantibus. Altera pressio fit, cessante convolutione co- chleæ, vi gravitatis lapidis suspensi; & tunc non nisi Ratio vectis intervenit, atque Potentia est ipsa gravitas. Sed quoniam non ubique reperiuntur aut tam ingentes lapi- des, aut tam longæ arbores, communiter universus premendi labor

Transcription: Translated (English)

Mechanics 778 the matrix of the screw may rest upon, and be connected with, that bifid extremity, and may turn about a pivot fixed upon the great stone set beneath it, but in such a way that the stone itself may also at length be raised. That trunk likewise, which is to serve the office of the press, besides the very thick extremity of the stem, they hew down smooth, so that the press, inserted between two beams erected perpendicularly on either side in the sides of the trough, may be raised and lowered without hindrance, which otherwise the roughness itself would cause. Moreover, those two upright beams are either bored opposite each other with a few round holes, into which a somewhat thick cylinder, or iron bar, may be inserted, or they are cut with an open notch, into which a bolt may be fitted; with the purpose that either end of the press, as convenience requires, may be restrained, so that it may not ascend or descend. Wherefore, when the screw is turned, it raises the matrix, and the opposite end of the press, all the supports beneath having been removed, gradually descends: but when it has come to the point where it is not far from the height of the things that are to be pressed in the wine-press, a bolt is inserted above, so that it may not be raised further. Then, the screw being turned back the other way, it draws the matrix and the end of the press downward: and since the other end cannot be raised because of the bolt, the grapes are pressed, and the must flows into the vat. When at last the grape-skins have been so compressed that it is easier to raise the stone attached to the screw than to compress them further, the stone is lifted by the turning of the screw; and, raised to a moderate height, they allow it to hang for a longer time, so that, by the stone’s weight striving downward, it may squeeze out from the press whatever must still remains in the grape-skins. Thus there are here two pressures to be considered: one indeed by the force of the person turning the screw; and here the screw is combined with a lever of the second kind. For the press is a lever, whose fulcrum is at that end which is restrained by the bolt, lest it be raised; but the office of the power depressing such a lever is performed by the screw, grooved in a spiral; which nevertheless, having no motion originating in itself, leaves the function and the name of power to the men turning it. The other pressure takes place, when the screw is no longer turned, by the force of the suspended stone’s gravity; and then only the principle of the lever intervenes, and the power is gravity itself. But since there are not everywhere to be found either stones so huge, or trees so long, the labor of pressing in common is

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Liber octavus. CAPUT III. 779 labor vectiariis incumbit cochleam unam, aut alteram versantibus. Si duæ sint cochleæ ad opposita torcularis latera constitutæ, matricem habent in ipso prælo excavatam, quod suâ conversione deorsum trahunt, ut ex subjectis vinaceis exprimatur mustum: & tunc nihil est, quod Vectis momenta exerceat, sed sola vis Cochleæ habetur. At si unica fuerit cochlea (quemadmodum & in typographorum torculis) præli non est usus; sed transversæ trabi superiori immotæ inseritur per exca- vatas congruentes strias cochlea, quæ in conversione depressa calcat impositum vinaceis planum ex solidis asseribus. Verùm contingere potest ut non sit vectiario spatium expeditum, quando, post modicam & faciliorem compressionem breviore vecte peractam, adhuc longiore Radio utendum esset ad con- torquendam cochleam. Propterea ab Axe in Peritrochio subsidium facile peti potest; si videlicet extra torcularis alveum ligneus cylindrus ad perpendiculum erigatur circa suos polos, alterum subjecto plano, alterum exporrectæ è proximo pariete trabi, infixos versatilis: huic funem adnecte, qui extremo un- co apprehendat annulum Radij, quo cochlea versatur: cylindro enim infixus Radius dum illum volvit, & funem illi circumducit, cochleæ vectem ad se rapit, & vehementiùs premitur subjectum cochleæ planum, quàm si eadem cochlea duplo longiore vecte convolveretur. Unum tamen hìc observandum, videlicet in hujusmodi conversione non eadem esse momenta, quandoquidem funis extentus non eundem semper cum cochleæ vecte angulum constituit; eò autem minora sunt momenta, quò magis hic ab angulo recto recedit, ut ex iis constat, quæ lib.4. cap.7. dicta sunt. Quapropter expedit cylindrum illum versatilem non longiùs à torculari abesse, ut funis minùs acutum angulum cum vecte constituat, quando vectis extremitas incipit suæ peripheriæ arcum describere: atque adeò ita statuendus videtur cylindrus, ut quando funis angulum rectum constituet cum cochleæ vecte, hic jam percurrerit arcum non majorem semirecto angulo, respondentem gradibus 45: sic enim fiet, non nimis acutum esse angulum initio tractionis, & progrediendo augeri, donec fiat rectus: deinde, licèt momenta decrescant angulo in obtusum transeunte, ubi nimis obliquus factus fuerit angulus, poterit in FFff 2

Transcription: Translated (English)

The work falls on the laborers turning one screw or the other. If there are two screws placed on opposite sides of the press, they have a nut hollowed out in the press itself, which by its turning draws downward, so that the must may be pressed out from the grapes beneath: and then there is nothing for the lever to act upon, but only the force of the screw is used. But if there be a single screw (as also in printers’ presses), the press is not used; instead, a screw is inserted into the upper transverse beam, fixed through matching hollow grooves, which, when turned and pressed down, treads upon the platform laid over the grapes, made of solid planks. Yet it may happen that there is not enough room for the laborer’s swing, when, after a slight and easier compression carried out with a shorter lever, one would still need to use a longer radius for turning the screw. For this reason aid may readily be sought from a wheel and axle; namely, if outside the bed of the press a wooden cylinder is set upright on its pivots, one end resting on the floor beneath, the other projecting from the nearby wall-beam, fixed to a turning shaft: attach to this a rope, which with its hooked end grips the ring of the radius by which the screw is turned: for when the radius fixed to the cylinder turns it and draws the rope around it, it pulls the screw’s lever toward itself, and the platform beneath the screw is pressed more powerfully than if the same screw were turned with a lever twice as long. One thing, however, must here be observed: namely, that in such a turning the moments are not the same, since the stretched rope does not always form the same angle with the screw’s lever; and the moments are smaller the more this departs from a right angle, as is clear from what was said in book 4, chapter 7. Therefore it is advantageous that that rotating cylinder not be placed too far from the press, so that the rope forms a less acute angle with the lever when the end of the lever begins to describe an arc of its circumference: and thus the cylinder should seem to be positioned so that when the rope forms a right angle with the screw’s lever, the lever has already traversed an arc no greater than a half-right angle, corresponding to 45 degrees: for in this way it will happen that the angle is not too acute at the beginning of the pull, and as it advances it increases until it becomes right; then, although the moments decrease as the angle passes into an obtuse one, when the angle has become too oblique, it may in

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780 Mechanicorum aliud foramen immitti Radius, laxato priùs fune ex cylindri revolutione. Quapropter ut potentiæ cylindrum volventis momenta ad calculos revoces, maximum momentum est fune ad cochleæ Radium perpendiculari: quamvis autem non semper progrediente motu constituat angulum rectum, tunc tamen perinde computandum est momentum, atque si eandem positionem ad angulum rectum servatura esset potentia per funem Radio applicata; præterita siquidem atque futura applicatio nihil minuit præsentis applicationis virtutem. In eâ verò applicatione perpendiculari, momenti Ratio desumenda est ex circuli à Radio descripti peripheriâ, atque ex intervallo spirarum cochleæ: quæ Ratio componenda est cum Ratione Radij cylindrum volventis ad ejusdem cylindri semidiametrum. Sed ut habeantur momenta aliarum positionum, inquirendus est angulus applicationis funis ad eundem cochleæ vectem. Et primò quidem datur Radij cochleæ infixi longitudo A B, semidiameter cylindri C D, & distantia A D. Inquiratur, in quo puncto accidat positio funis perpendicularis ad Radium cochleæ: hæc utique non fit, nisi fune tangente utramque peripheriam tum circuli à Radio A B descripti, tum cylindri; & erit B C. Quia igitur linea B C utrumque circulum tangit, ductis semidiametris A B & C D, anguli A B C, & D C B sunt recti, ex 18. lib. 3: igitur ex 27. lib. 1. lineæ A B &

Transcription: Translated (English)

780 Mechanicorum another hole is inserted by the Radius, after the rope has first been loosened by the revolution of the cylinder. Therefore, so that you may reduce to calculation the moments of the force turning the cylinder, the greatest moment is when the rope is perpendicular to the Radius of the screw: although, however, it does not always form a right angle in the advancing motion, yet then the moment must be computed as if the force, applied through the rope to the Radius, were about to preserve the same position at a right angle; for the past and future application diminishes nothing of the power of the present application. But in that perpendicular application, the ratio of the moment must be taken from the circumference of the circle described by the Radius, and from the interval of the threads of the screw: which ratio is to be combined with the ratio of the Radius turning the cylinder to the semidiameter of the same cylinder. But in order that the moments of the other positions may be obtained, the angle of application of the rope to the same arm of the screw must be investigated. And first indeed the length of the fixed Radius of the screw A B, the semidiameter of the cylinder C D, and the distance A D are given. Let it be inquired at what point the position of the rope perpendicular to the Radius of the screw occurs: this certainly does not happen unless the rope is touching both circumferences, both of the circle described by the Radius A B, and of the cylinder; and it will be B C. Since therefore the line B C touches both circles, after the semidiameters A B and C D have been drawn, the angles A B C and D C B are right angles, according to 18. lib. 3: therefore, according to 27. lib. 1, the lines A B and

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Liber octavus. CAPUT III. 781 & DC sunt parallelæ, & per 29. lib. 1. anguli alterni B AD, & CDA sunt æquales: sed & anguli ad verticem E sunt æquales: ergo triangula BAE, CDE sunt similia; & per 4. lib. 6. ut BA ad CD, ita AE ad DE; & componendo ut AB plus CD ad CD, ita AD ad DE: innotescit itaque DE. Quare ex quadrato ipsius DE auferatur quadratum lateris DC, & residui Radix erit recta CE. Fiat ergo ut DC ad CE, ita AB ad BE; atque additis BE & CE nota est tota BC. Deinde assumatur positio Radij AF, ita ut arcus FB non sit major gradibus 45. Dato igitur arcu illo, hoc est angulo FAB, noti sunt anguli AFB, ABF trianguli isoscelis ad basim BF, singuli enim habent semissem residui ad duos rectos: atque adeò, si recto CBA addatur notus ABF, innotescit anguli obtusi CBF quantitas: Inventum jam est latus CB, & latus BF subtensa dati arcûs ex Canone Sinuum innotescit in partibus Radij AB; quapropter ex Trigonometria inveniri potest basis CF, & angulus BFC; qui si auferatur ex noto angulo BFA, remanet quæsitus angulus CFA applicationis funis CF ad vectem AF. Habetur itaque ex hujusmodi applicatione ad vectem per angulum acutum CFA Ratio momenti comparati cum momento applicationis ad angulum rectum CBA: est enim ex dictis lib. 4. cap. 7. ut Sinus anguli acuti ad Radium. Fiat igitur ut Radius ad Sinum anguli AFC, ita peripheria descripta à vecte AB ad aliud; & hoc inventum comparandum est cum intervallo spirarum cochleæ, ut habeatur Ratio momenti potentiæ in C constitutæ, & applicatæ ad vectem AF cum directione CF. Componenda deinde est hæc Ratio cum Ratione Radij DH cylindrum volventis, ad ejusdem cylindri semidiametrum DC, & habebitur adæquata Ratio momenti potentiæ in H. Tertiò. Ex puncto C ad A ducatur recta CA: & cum in triangulo ABC rectangulo nota jam sint latera AB & BC circa rectum, invenitur hypothenusa AC, & angulus BAC. Tum ex 33. lib. 3. super rectâ AC descripto circuli segmento capiente angulum obtusum æqualem Supplemento anguli AFC ad duos rectos, in puncto I, ubi hujus segmenti arcus secat peripheriam à Cochleæ vecte descriptam, concurrant duæ rectæ AI & CI. Est igitur triangulum CAI, in quo data sunt FFfff 3

Transcription: Translated (English)

Book Eight. Chapter III. 781 & DC are parallel, and by 29, book 1, the alternate angles BAD & CDA are equal: but also the vertical angles E are equal: therefore the triangles BAE, CDE are similar; and by 4, book 6. as BA is to CD, so is AE to DE; and by composition, as AB plus CD is to CD, so is AD to DE: thus DE becomes known. Therefore from the square of DE let the square of side DC be subtracted, and the root of the remainder will be the straight line CE. Let it therefore be as DC is to CE, so is AB to BE; and by adding BE and CE the whole BC is known. Then let the position of radius AF be assumed, so that the arc FB is not greater than 45 degrees. The arc being given, that is, the angle FAB, the angles AFB, ABF of the isosceles triangle at the base BF are known, for each has half of the remainder to two right angles: and therefore, if to the right angle CBA there be added the known ABF, the quantity of the obtuse angle CBF becomes known: now the side CB has already been found, and the side BF, the chord of the given arc, is made known by the Canon of Sines in parts of the radius AB; wherefore by Trigonometry the base CF, and the angle BFC, can be found; which, if subtracted from the known angle BFA, there remains the sought angle CFA, of the application of the cord CF to the lever AF. Thus from such an application to the lever through the acute angle CFA, the ratio of the moment compared with the moment of application at the right angle CBA is obtained: for, as is stated in book 4, chapter 7, it is as the sine of the acute angle to the radius. Let it therefore be as the radius to the sine of angle AFC, so is the circumference described by the lever AB to another; and this found quantity is to be compared with the spacing of the threads of the screw, so that the ratio of the moment of the power placed at C, and applied to the lever AF with direction CF, may be obtained. Then this ratio must be combined with the ratio of the radius DH turning the cylinder to the semidiameter of the same cylinder DC, and there will be had the equivalent ratio of the moment of the power placed at H. Thirdly. From point C let the straight line CA be drawn to A: and since in the right triangle ABC the sides AB and BC about the right angle are now known, the hypotenuse AC is found, and the angle BAC. Then from 33, book 3, over the straight line AC let there be described a segment of a circle containing an obtuse angle equal to the supplement of angle AFC to two right angles, in the point I, where the arc of this segment cuts the circumference described by the lever of the screw, let two straight lines AI and CI meet. Thus there is the triangle CAI, in which are given FFfff 3

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782 Mechanicorum latera CA & AI unâ cum angulo AIC noto, utpote ex con- structione æquali supplemento ad duos rectos anguli jam noti AFC. Quapropter inveniatur angulus IAC, qui demptus ex jam invento angulo BAC, relinquit angulum BA I; ac propterea notus est arcus BI, qui additus arcui BF dabit totum arcum FI, in quo primùm momenta crescunt ex F in B, dein- de decrescunt ex B in I, ubi angulus obtusus tantumdem exce- dit rectum, quantum à recto deficit acutus AFC; atque adeò in F & I æqualia sunt momenta. Antequam verò praxim hanc exemplo illustrem, ut sub- ductis calculis noverit Machinator, quonam pacto omnia dis- ponenda sint, monendus est lector à me ideò semper idem punctum C assumptum fuisse, quia in re Physicâ nullus subre- pere potest error notabilis. Cæterùm si funem extentum con- sideremus semper quasi lineam tangentem cylindri periphe- riam, satis manifestum est, si linea BC est tangens in puncto C, & angulus BCD est rectus, non posse lineam à puncto F productam ad contactum cadere in punctum C, sed ultrà illud, ita ut demum veniat punctum contactûs in C, quando positio vectis fuerit AB, & iterum punctum contactûs recedat à C, quando positio vectis fiat AI. Verùm quia exiguum est hujus- modi discrimen, propterea unum idemque punctum C as- sumptum est, cum non sequatur physicè ullum incommodum ex hoc Geometricæ accurationis contemptu. Sit igitur ex. gr. spirarum cochleæ intervallum unc. 2; & vectis longitudo AB cubitorum 3, hoc est unc. 36: quare inte- gra peripheria hoc Radio est unc. 226; atque ideò in B, ubi applicatio est ad angulum rectum, Ratio motuum seu momen- torum est ut 226 ad 2, hoc est 113 ad 1. Sit cylindri semidia- meter DC unc. 3, atque DH similiter unc. 36: est igitur Ra- tio motûs seu momenti potentiæ in H, ad motum seu momen- tum in C ut 12 ad 1. Ratio itaque composita ex Rationibus 113 ad 1, & 12 ad 1 est Ratio 1356 ad 1: quæ longè major est, quàm si cochleæ adhiberi potuisset Radius cubitorum 6, non addito Axe in Peritrochio. Ut inveniatur longitudo BC, pri- mùm fiat ut AB plus DC ad DC, hoc est ut unc. 39. ad unc. 3. ita distantia AD data unc. 65, ad ED unc. 5. Igitur in trian- gulo ECD rectangulo, cujus hypothenusa ED unc. 5. la- tus

Transcription: Translated (English)

782 Mechanicorum The sides CA and AI, together with the known angle AIC, are known, since by construction they complete the two right angles to the already known angle AFC. Therefore let the angle IAC be found, which, subtracted from the already found angle BAC, leaves the angle BAI; and therefore the arc BI is known, which, added to the arc BF, will give the whole arc FI, in which the moments first increase from F to B, then decrease from B to I, where the obtuse angle exceeds the right angle by as much as the acute AFC falls short of the right angle; and thus the moments are equal in F and I. But before I illustrate this procedure by example, so that, after the calculations have been set out, the mechanic may know by what method everything is to be arranged, the reader must be warned by me that for this reason the same point C has always been assumed, because in a physical matter no notable error can creep in. Moreover, if we consider the stretched cord always as a tangent line to the circumference of the cylinder, it is sufficiently clear that, if line BC is tangent at point C, and angle BCD is a right angle, the line drawn from point F to the point of contact cannot fall upon point C, but beyond it, so that the point of contact finally comes to C when the position of the lever is AB, and again the point of contact moves away from C when the position of the lever becomes AI. But because this difference is slight, therefore one and the same point C has been assumed, since no physical inconvenience follows from this disregard of geometrical exactness. Let it be, for example, that the interval of the turns of the screw is 2 inches, and the length of the lever AB is 3 cubits, that is, 36 inches: therefore the whole circumference with this radius is 226 inches; and therefore at B, where the application is to a right angle, the ratio of motions or moments is as 226 to 2, that is, 113 to 1. Let the semidiameter of the cylinder DC be 3 inches, and DH likewise 36 inches: therefore the ratio of the motion or moment of the power in H to the motion or moment in C is as 12 to 1. The ratio compounded from the ratios 113 to 1 and 12 to 1 is therefore the ratio 1356 to 1: which is far greater than if a radius of 6 cubits could have been applied to the screw, without adding an axle in the wheel. To find the length BC, first let it be as AB plus DC to DC, that is, as 39 inches to 3 inches, so is the given distance AD, 65 inches, to ED, 5 inches. Therefore in the right triangle ECD, whose hypotenuse ED is 5 inches, the side

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Liber octavus. CAPUT III. 783 tus DC unc. 3. est latus EC unc. 4: atque adeò ut DC 3 ad CE 4, ita AB 36 ad BE 48; cui addita CE 4 dat totam per- pendicularem BC unc. 52. Ponatur arcus FB gr. 45; ergo ejus subtensa 76536 partium, quarum Radius AB unc. 36 est 100000, erit unc. 27 ́: anguli verò AFB, ABF sunt singuli gr. 67. 30. Quare in triangulo FCB datur angulus CBF gr. 157. 30. comprehensus à lateribus CB unc. 52, & BF unc. 27 ́. Invenitur ergo angulus BFC gr. 14. 45, qui ex angulo BFA gr. 67. 30. demptus relinquit angulum AFC gr. 52. 45; cujus Sinus est particularum 79600. Igitur ut Radius 100000 ad 79600, ita momentum Applicationis per angulum rectum, quod erat ut 113, ad 90 proximè, momentum Applicationis per hunc angulum AFC acutum. Compositis itaque Rationibus 90 ad 1, & 12 ad 1, momentum potentiæ in H erit 1080. Demum in triangulo ABC rectangulo ex lateribus AB 36, & BC 52, reperitur hypothenusa AC 63 ́, & angulus BAC gr. 55. 18. Quapropter in triangulo AIC datur latus AC 63 ́, & angulus illi oppositus AIC gr. 127. 15. & præterea latus AI 36: ex quibus invenitur huic oppositus angulus ICA gr. 26. 56. Igitur tertius angulus CAI est gr. 25. 49: qui si auferatur ex angulo BAC gr. 55. 18, reliquus est angulus BAI gr. 29. 29. Igitur totus arcus FI est gr. 74. 29. Ut autem appareat, quid conferat amplitudo arcûs BF, statuatur hic gr. 60. & huic æquales sunt anguli ad basim BF; quæ recta BF est ipsi AB æqualis, hoc est unc. 36. Quare in triangulo FBC datur latus FB unc. 36. & latus BC unc. 52, & angulus ab iis comprehensus gr. 150: invenitur ergo angulus BFC gr. 17. 47; qui ablatus ex BFA gr. 60. relinquit CFA gr. 42, 13: cujus Sinus est partium 67193. Igitur ut Radius 100000 ad 67193, ita 113 ad 76, quod est momentum Applicationis per hunc angulum acutum: atque compositis Rationibus 76 ad 1, & 12 ad 1, momentum potentiæ in H est ut 912. In triangulo verò AIC dantur latera AI unc. 36, & AC unc. 63 ́ & & Supplementum anguli AFC ad duos rectos est angulus AIC gr. 137, 47: ergo invenitur angulus ACI gr. 22. 29. Est igitur angulus IAC gr. 19. 44: qui demptus ex angulo

Transcription: Translated (English)

Book Eight. CHAPTER III. 783 then DC 3 units 3/10, EC is side 4 units: and thus, as DC 3 to CE 4, so AB 36 to BE 48; to which, added CE 4, gives the whole perpendicular BC, 52 units. Let arc FB be 45 degrees; therefore its chord is 76536 parts, of which Radius AB, 36 units, is 100000, and it will be 27 ́ units: and the angles AFB, ABF are each 67 degrees 30 minutes. Hence in triangle FCB the angle CBF is given as 157 degrees 30 minutes, contained by the sides CB, 52 units, and BF, 27 ́ units. Therefore the angle BFC is found to be 14 degrees 45 minutes, which, taken from angle BFA, 67 degrees 30 minutes, leaves angle AFC 52 degrees 45 minutes; the sine of which is 79600 parts. Therefore, as Radius 100000 is to 79600, so the moment of Application through the right angle, which was as 113 to 90 approximately, is the moment of Application through this acute angle AFC. Thus, combining the ratios 90 to 1 and 12 to 1, the moment of the force at H is 1080. Finally, in right triangle ABC, from the sides AB 36 and BC 52, the hypotenuse AC is found to be 63 ́, and the angle BAC to be 55 degrees 18 minutes. Wherefore, in triangle AIC, there are given the side AC 63 ́, and the opposite angle AIC 127 degrees 15 minutes, and moreover the side AI 36: from which is found the opposite angle ICA, 26 degrees 56 minutes. Therefore the third angle CAI is 25 degrees 49 minutes: which, if it be taken away from angle BAC, 55 degrees 18 minutes, there remains angle BAI 29 degrees 29 minutes. Therefore the whole arc FI is 74 degrees 29 minutes. But that it may appear what the breadth of arc BF contributes, let it be set down here as 60 degrees, and the angles at the base BF are equal to this; and this straight line BF is equal to AB itself, that is, 36 units. Wherefore in triangle FBC there are given side FB, 36 units, and side BC, 52 units, and the angle contained by them, 150 degrees: therefore the angle BFC is found to be 17 degrees 47 minutes; which, taken away from BFA, 60 degrees, leaves CFA 42 degrees 13 minutes: the sine of which is 67193 parts. Therefore as Radius 100000 is to 67193, so 113 is to 76, which is the moment of Application through this acute angle: and, combining the ratios 76 to 1 and 12 to 1, the moment of the force at H is as 912. But in triangle AIC there are given the sides AI, 36 units, and AC, 63 ́ units, and the supplement of angle AFC to two right angles is angle AIC, 137 degrees 47 minutes: therefore angle ACI is found to be 22 degrees 29 minutes. Therefore angle IAC is 19 degrees 44 minutes: which, subtracted from angle

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784 Mechanicorum lo B A C gr. 55. 18. superiùs invento, relinquit angulum I A B, hoc est arcum I B gr. 35. 34. Quare totus arcus F I esset gr. 95. 34. Ex quo vides intra eosdem terminos æqualium mo- mentorum, minora esse extrema momenta in F & I, sed per majorem arcum, si incipias motum in majore distantiâ à puncto Applicationis per angulum rectum: propterea satius videtur majora obtinere momenta, & minorem arcum describere: ideò dixi assumendum esse arcum B F non majorem gradibus 45. His similia de Succulâ dicenda sunt, quæ de Axe perpendi- culari diximus, si succulâ potiùs utendum loci & motûs quæ- siti opportunitas suadeat: id quod ita per se clarum est, ut in his diutiùs immorari non sit opus. CAPUT IV. Cochleæ Infinitæ vires explicantur. Validissimam omnium Facultatum Cochleam esse ex supe- rioribus manifestum est: sed illud accidit incommodum, quod nimis brevibus terminis coërcetur; quos nimirum ejus longitudo definit; sivè illa circa suum axem convoluta intrà Matricem immotam moveatur, sivè illa positionem non mu- tans ex convolutione attrahat aut repellat Matricem & pondus ei adnexum. Propterea alius cochleæ usus excogitatus est citrà ullam Matricem, cui inseratur, atque ejusmodi, ut cochleæ conversioni nullus statuatur finis, easdémque semper exerceat vires. Hinc Cochleæ Infinitæ, aut Viti Perpetuæ nomen in- ditum est. Cylindrus circa suum axem, apposito manubrio, versatilis in brevem cochleam deformatur unâ aut alterâ spirâ conten- tus: ita autem ad tympani dentes accommodatur, ut eorum in- tervallum sit spirarum intervallo congruens; hoc est initium spiræ apprehendat unum tympani dentem; dumque ex Co- chleæ convolutione dens primus tantum promovetur, quantum exigit spirarum distantia, unâ conversione absolutâ iterum ini- tium

Transcription: Translated (English)

784 Mechanicorum to B A C gr. 55. 18. having been found above, leaves the angle I A B, that is, the arc I B gr. 35. 34. Therefore the whole arc F I would be gr. 95. 34. From this you see that, within the same limits of equal mo- ments, the extremities in F and I are smaller moments, but through a greater arc, if you begin the motion at a greater distance from the point of Application through a right angle: therefore it seems better to obtain greater moments, and to describe a smaller arc: hence I said that the arc B F ought to be taken no greater than 45 degrees. Similar things must be said of the Handle, which we said of the vertical Axis, if it is rather by means of the handle that the convenience of the place and of the sought motion suggests using it: which is so clear in itself that there is no need to dwell on these matters longer. CAPUT IV. The powers of the Infinite Screw are explained. That the Screw is the strongest of all the Faculties is manifest from what has gone before: but this inconvenience occurs, that it is constrained within too brief limits; which, namely, its length defines; whether it, wound around its own axis, moves within an unmoving Nut, or whether, without changing position, by its winding it draws or repels the Nut and the weight attached to it. Therefore another use of the screw was devised without any Nut into which it might be inserted, and such that no end is set to the turning of the screw, but that it may always exert the same forces. Hence the name of Infinite Screw, or Perpetual Vice, has been given it. A cylinder, revolving around its own axis with a handle attached, is shaped into a short screw, content with one or two spirals: and thus it is adapted to the teeth of a drum, so that the interval of the teeth corresponds to the interval of the spirals; that is, the beginning of the spiral grasps one tooth of the drum; and while from the turning of the screw the first tooth is advanced only as much as the distance of the spirals requires, when one revolution has been completed again the beginning

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Liber octavus. CAPUT IV. 785 tium spiræ apprehendat secundum tympani dentem proximè consequentem, ex tympani convolutione jam constitutum in eodem loco, in quo erat primus dens initio motûs: atque ita deinceps omnes subinde dentes apprehenduntur à cochleâ; semélque revoluto tympano, iterum à primo dente incipit se- cunda illius convolutio. Hinc quia cochleâ hujusmodi, quate- nus ad se pertinet, nullum statuit convolutionibus terminum, etiamsi definitum habet spirarum numerum, immò unicam ha- beat spiram, Infinita dicitur, nam & tympanum orbitam ha- bens in sese redeuntem plurimis sine fine convolutionibus cir- cumagi potest. At si tympani loco rectam apposueris laminam denticulatam, quæ ex Cochleæ hujusmodi conversione alium atque subinde alium dentem apprehendentis adduceretur, aut repelleretur; an illa appellanda esset Cochlea Infinita, quia longiorem atque longiorem sine fine laminam similiter movere posset; iis examinanda relinquatur quæstio, quibus de vocabu- lo disputandi otium est. Tympano autem infixus est Axis, sive ille simplex sit, cui ductarius funis circumvolvatur, sivè striatus fuerit, qui aliud tympanum convertat, prout suo loco, ubi de Axe in Peritro- chio disputatum est. Quapropter vis Cochleæ componitur cum vi tympani, quod ab illâ convertitur: idcirco huic Machinæ Cochleæ Compositæ aliqui nomen fecerunt. Cùm itaque sin- gulis cochleæ conversionibus singuli dentes tympani promo- veantur, toties convertitur cochlea, quot in tympani orbitâ numerantur dentes. Potentiæ igitur motus, quo illa manubrium versans describit circuli peripheriam, ducendus est per den- tium numerum, ut habeatur Ratio motûs Potentiæ, ad motum orbitæ tympani. Cum verò data sit Ratio tympani ad suum Axem, data est Ratio motûs orbitæ tympani ad motum ponde- ris fune ductario attracti. Hæ duæ Rationes componantur, & nota erit Ratio motûs potentiæ ad motum ponderis. Sit co- chleæ manubrium digitorum 7; igitur peripheria circuli à po- tentiâ manubrio applicatâ descripti est ferè digit. 44: tympani semidiameter ad sui Axis semidiametrum sit ut 4 ad 1: Sit au- tem tympani orbita in dentes 24 distincta; ac propterea dum se- mel tympanum cum suo Axe volvitur, motus Potentiæ est digi- torum ferè 44 vicies & quater sumptorum hoc est digit. 1056. GGggs

Transcription: Translated (English)

Book Eight. CHAPTER IV. 785 the thread of the spiral should seize the nearest tooth of the drum, already set in place by the winding of the drum in that same position in which the first tooth was at the beginning of the motion; and thus in succession all the teeth are seized one after another by the screw. And when the drum has been turned once around, the second winding of it begins again from the first tooth. Hence, because a screw of this kind, so far as it concerns itself, establishes no limit to the windings, even though it has a definite number of spirals, indeed even if it has only a single spiral, it is called Infinite; for also a drum having a returning orbit within itself may be turned round through countless windings without end. But if, in place of the drum, you were to apply a straight toothed plate, which by the turning of such a screw would be brought forward, or pushed back, from one tooth after another; whether that ought to be called an Infinite Screw, because it could likewise move a longer and longer plate without end, let the question be left to those to examine who have leisure to dispute about words. An Axis, however, is fixed in the drum, whether it be a simple one, around which the driver-rope may be wound, or a grooved one, which turns another drum, as in its proper place, where the Axis in the Peritrochium was discussed. Wherefore the force of the Screw is combined with the force of the drum, which is turned by it: for this reason some have given the name Compound Screw to this Machine. Since, therefore, at each revolution of the screw, the individual teeth of the drum are advanced, the screw is turned as many times as there are teeth in the orbit of the drum. Therefore the power’s motion, by which it turns the handle and describes the circumference of the circle, must be divided by the number of teeth, so that the Ratio of the motion of the Power to the motion of the orbit of the drum may be obtained. But since the Ratio of the drum to its Axis is given, the Ratio of the motion of the orbit of the drum to the motion of the weight drawn by the driver-rope is given. Let these two Ratios be compounded, and the Ratio of the motion of the Power to the motion of the weight will be known. Let the handle of the screw be 7 digits; therefore the circumference of the circle described by the power applied to the handle is about 44 digits: let the semidiameter of the drum be to the semidiameter of its Axis as 4 to 1: and let the orbit of the drum be divided into 24 teeth; and therefore while the drum is once turned with its Axis, the motion of the Power is about 44 digits taken 24 times, that is, 1056 digits. GGggs

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Mechanicorum 786 Si igitur tympani semidiameter sit digit. 4, & Axis semidiameter dig. 1, illius peripheria est saltem digit. 25, hujus verò peripheria saltem digit. 6 1/4, quantus est ex unâ tympani conversione motus ponderis. Itaque motus Potentiæ ad motum ponderis est ut 1056 ad 6 1/4, hoc est proximè ut 169 ad 1. Hinc si plura fuerint Composita Tympana, eorum Ratio, quæ ex Rationibus diametrorum tympanorum ad suorum Axium diametros componitur, assumenda est, atque attendendum quoties volvatur cochlea, ut primum tympanum cochleæ proximum circumagatur: deinde per numerum dentium primi tympani ducendus est motus potentiæ manubrio cochleæ applicatæ; & ex Ratione tympanorum, atque ex Ratione Cochleæ, componenda est Ratio. Sit cochlea eadem, quæ priùs, eodemque manubrio instructa, adeò ut potentia semel cochleam versans describat circuli peripheriam digitorum ferè 44; & primum tympanum habens peripheriam dig. 25 in dentes 24 distributam, dum semel volvitur, potentia vicies & quater peripheriam dig. 44 describens percurrit digitos 1056. Sit idem primum tympanum ad suum Axem striatum ut 4 ad 1, secundum tympanum ad suum Axem fune ductario involutum sit ut 3 ad 2: Ratio composita horum duorum tympanorum est ut 6 ad 1. Cum verò motus potentiæ manubrio applicatæ ad integrum motum peripheriæ tympani primi sit ut 1056 ad 25 (nam singulæ conversiones manubrij cochleæ ad motum unius dentis sunt ut 44 ad 25/24) componatur hæc Ratio cum Ratione 6 ad 1, & erit motus potentiæ ad motum ponderis axi secundi tympani per funem ductarium applicati, ut 6336 ad 25, hoc est ferè ut 253 1/2 ad 1: atque adeò quo conatu potentia moveret libras decem; hac machinâ movebit libras 2535. Verùm adhuc augeri possunt vires Cochleæ Infinitæ non multiplicatis tympanis dentatis, sed cum illo unico, quod à Cochleâ movetur, componendo Trochleas: si videlicet alteri Trochleæ adnectatur pondus, altera Trochlea alicubi firmetur: tum funis ductarius, qui à Potentiâ arripiendus esset atque trahendus, axi tympani alligetur. Nam si Ratio, quam Trochleæ inferunt, componatur cum Ratione Axis in Peritrochio, atque Ratione Cochleæ, fit Ratio ex tribus Rationibus trium Faculta

Transcription: Translated (English)

Mechanics 786 If, then, the semidiameter of the drum be 4 digits, and the semidiameter of the axis 1 digit, the circumference of the former is at least 25 digits, and that of the latter at least 6 1/4 digits, as much as is the motion of the weight from one revolution of the drum. Therefore the ratio of the motion of the power to the motion of the weight is as 1056 to 6 1/4, that is, approximately as 169 to 1. Hence, if there are several compound drums, their ratio, which is made up from the ratios of the diameters of the drums to the diameters of their axes, must be taken, and it must be observed how many times the screw is turned, so that the first drum nearest the screw is turned about; then the motion of the power applied to the handle of the screw must be multiplied by the number of teeth in the first drum; and from the ratio of the drums, and from the ratio of the screw, the ratio is to be composed. Let the screw be the same as before, and furnished with the same handle, so that the power turning the screw once describes the circumference of a circle of about 44 digits; and let the first drum, having a circumference of 25 digits distributed into 24 teeth, while it is turned once, have the power traversing 1056 digits by describing the circumference of 44 digits twenty-four times. Let the same first drum to its grooved axis be as 4 to 1, and the second drum to its axis, wound with a carrying rope, be as 3 to 2: the compound ratio of these two drums is as 6 to 1. But when the motion of the power applied to the handle is to the complete motion of the circumference of the first drum as 1056 to 25 (for the several revolutions of the screw handle to the motion of one tooth are as 44 to 25/24), let this ratio be combined with the ratio of 6 to 1, and the motion of the power to the motion of the weight applied by the carrying rope to the axis of the second drum will be as 6336 to 25, that is, about as 253 1/2 to 1: and so, with the force by which the power would move ten pounds, this machine will move 2535 pounds. But the powers of the Endless Screw can still be increased, not by multiplying toothed drums, but by combining pulleys with that single one which is moved by the screw: namely, if to one pulley a weight be attached, and the other pulley be fixed somewhere; then the carrying rope, which would have to be seized and pulled by the power, is tied to the axis of the drum. For if the ratio introduced by the pulleys is combined with the ratio of the axis in the wheel and axle, and with the ratio of the screw, there results a ratio from three ratios of three faculties

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Liber octavus. CAPUT IV. 787 Facultatum composita. Sic Ratio Cochleæ sit, ut priùs, 44 ad 35/24, Ratio Tympani ad Axem sit 4 ad 1, Ratio Trochlearum, capite funis ad trochleam ponderis alligato (sint autem Trochleæ bi- norum orbiculorum) sit 5 ad 1: tres hæ Rationes Compositæ constituunt Rationem 845 ad 1. Quare quo conatu moveres libras decem, movebis libras 8450 tam facili & parabili ma- chinâ. Observanda sunt autem tam commoda, quàm incommoda, quæ hujus machinæ, scilicet Cochleæ Infinitæ usum comitan- tur. Neque in postremis illud numerandum est, quod tantula machinula facillimè transferri potest, ad pondera satis magna dimovenda; maximè si in plano raptanda sint suppositis scytalis, & trochleæ adhibeantur, quas non adeò crasso fune connecti oportet, quemadmodum si in sublime attollendum esset pon- dus, & fune ipso retinendum, ne relabatur. Adde non requiri ampliora spatia, ut cochlea hujusmodi infi- nita circumagatur, & vel sedentem hominem solâ, neque mul- tâ, lacertorum manubrium versantium contentione posse mo- tum quæsitum perficere: atque si pondus attollatur, licet poten- tiæ, quandocumque libitum fuerit, cessare à motu, quin pon- dus suspensum recidat, etiamsi neque illi fulcrum subjiciatur, neque cochleæ manubrium retinaculo aliquo firmetur. Verùm in attollendis ingentibus oneribus non expedit hac machinâ uti, nisi tympanum dentatum satis magnum fuerit, ut Axem crassiorem atque validiorem admittat, cui ductarius funis cir- cumduci queat; hic autem funis cum tenuis esse non possit, ne- que exilem Axem exigit. Præterea dissimulandum non est peri- culum, ne cochlea inutilis fiat; si videlicet vel unicus tympano dens excutiatur: ubi enim in conversione ad eam lacunam ventum fuerit, illico cessat tympani conversio, cum nullus ejus dens occurrat cochleæ. Propterea rem prudenter administrare oportet, ut congrua machina eligatur. Porrò non contemnenda utilitas ex Cochleâ hac infinitâ per- cipi potest ad augendas communis Cochleæ vires sivè premen- tis, sivè etiam attrahentis. Eo videlicet loco, ubi aptandus es- set Radius ad Cochleæ conversionem, tympanum dentatum ad- jiciatur, ex cujus centro exeat cylindrus in cochleam deforma- tus, & Matrici insertus: tympani verò dentes congruâ cochleæ GGggg 2

Transcription: Translated (English)

Book Eight. CHAPTER IV. 787 Compound ratios. Thus let the ratio of the screw be, as before, 44 to 35/24; let the ratio of the drum to the axle be 4 to 1; let the ratio of the pulleys, with the end of the rope fastened to the pulley carrying the weight (and let the pulleys be of two sheaves), be 5 to 1: these three compound ratios produce the ratio 845 to 1. Wherefore, with whatever effort you would move ten pounds, you will move 8450 pounds by so easy and convenient a ma- chine. But both the advantages and the disadvantages which accompany the use of this machine, namely of the Infinite Screw, must be observed. Not among the least of these should be counted the fact that so small a machine can very easily be carried to move very great weights; especially if they are to be dragged along a level surface on rollers, and pulleys are used, which need not be connected with so thick a rope as would be necessary if the weight were to be raised aloft and held by the rope itself, so that it should not slide back. Add that no larger space is required, so that a screw of this kind may be turned round, and that even a seated man, by the sole, nor with much exertion, of the arms turning the handle, can accomplish the desired motion: and if the weight is to be raised, it is permissible for the power, whenever it shall please, to cease from motion, yet the suspended weight will not fall, even if neither a support be placed beneath it, nor the handle of the screw be secured with any retaining device. Yet in raising very great burdens it is not expedient to use this machine, unless the toothed drum is large enough to admit a thicker and stronger axle, around which the driving rope may be wound; but here the rope, since it cannot be thin, does not require a slender axle. Moreover, it must not be concealed that there is danger lest the screw become useless; namely, if even a single tooth be knocked out of the drum: for when, in the rotation, it comes to that gap, the turning of the drum immediately ceases, since none of its teeth meets the screw. Therefore the matter must be prudently managed, so that a suitable machine be chosen. Furthermore, no small advantage may be derived from this Infinite Screw for increasing the force of the common screw, whether pressing or also drawing. Namely, in the place where a Radius would have to be fitted for turning the screw, let a toothed drum be added, from the center of which there may come forth a cylinder shaped into a screw, and inserted into the Nut: but the teeth of the drum with a suitable screw GGggg 2

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Mechanicorum 788 infinitæ spirâ excipientur: Manubrio enim versato cochlea in- finita convertitur, & singulis conversionibus singulos tympani dentes, alios subinde atque alios promovens, tympari covolu- tionem efficit, atque cum eo pariter infixa cochlea versatur. Prudenti autem Machinatori non deerit methodus, qua hujus- modi Cochlea infinita applicetur, & simul cum tympano den- tato deprimatur aut attollatur, si opus fuerit. Quapropter Ra- tio peripheriæ tympani ad intervallum spirarum suæ cochleæ, componenda est cum Ratione peripheriæ à manubrio descriptæ ad intervallum spirarum cochleæ infinitæ: ex hoc siquidem in- tervallo pendet motus peripheriæ tympani, cujus dentes ap- prehenduntur; quo enim pressior est cochleæ infinitæ spira, eò tenuiores & frequentiores insunt tympano dentes. Sit ex. gr. spirarum cochleæ prementis intervallum subtriplum semidia- metri tympani, cui illa infixa est: igitur Ratio perimetri tym- pani ad intervallum spirarum est ut 18 84/100 ad 1. At Cochleæ in- finitæ manubrium ad ejusdem spirarum distantiam sit ut 10 ad 1: Motus igitur potentiæ manubrium versantis est ut peripheria descripta 62 83/100 ad motum unius dentis tympani ut 1. Ratio itaque ex his duabus Rationibus Composita est 1183 2/10 ad 1. Ex quo satis innotescit, quanto virium incremento addatur co- chleæ vulgari cochlea hæc infinita tam brevi manubrio in- structa, loco vectis admodum longi, quem spatij angustiæ non caperent. Verùm non ad augendas tantummodo vires, seu, ut veriùs dicam, ad momentorum potentiæ incrementum, adhiberi po- test cochlea infinita, sed ad motum quantumvis exignum: sæpè enim motum extenuare opus est. Sic in automatis horas indicantibus vi laminæ elasticæ longioris in spiram convolutæ, ad rotarum celeritatem aut tarditatem moderandam oportet ipsum elaterem modò intendere, modò remittere: quia verò in vulgaribus horologiis id perficitur convolutione rotæ denta- tæ (cujus axi intimum spiræ elasticæ caput adnectitur, atque ne lamina per vim complicata se in laxiorem spiram restituat, axem ipsum & rotam dentatam revolvendo, obliquis rotæ ejus- dem dentibus, qua parte recti sunt, objicitur virgula elastica) ut minimum dentem unum promovere aut retrahere necesse est.

Transcription: Translated (English)

Mechanics 788 will be received by the infinite screw: for when the handle is turned, the endless screw is turned, and with each revolution, advancing now one and now another of the teeth of the drum, it causes the drum to revolve, and the fixed screw turns along with it. But a prudent machinist will not lack a method by which a screw of this kind may be applied, and at the same time, if needed, may be lowered or raised together with the toothed drum. Wherefore the ratio of the circumference of the drum to the interval between the threads of its screw must be combined with the ratio of the circumference described by the handle to the interval between the threads of the endless screw: for upon this interval indeed depends the motion of the circumference of the drum, whose teeth are engaged; the closer the thread of the endless screw, the thinner and more frequent are the teeth in the drum. Let it be, for example, that the interval between the threads of the pressing screw is three times smaller than the semidiameter of the drum to which it is fixed: therefore the ratio of the perimeter of the drum to the interval between the threads is as 18 84/100 to 1. But let the handle of the endless screw to the distance between its threads be as 10 to 1: therefore the motion of the power turning the handle is as the circumference described, 62 83/100, to the motion of one tooth of the drum, as 1. The composite ratio arising from these two ratios is therefore 1183 2/10 to 1. From this it is sufficiently evident how much increase of force is added by this endless screw, fitted with so short a handle, to an ordinary screw, in place of a very long lever, which the narrowness of the space would not admit. But this endless screw can be employed not only to increase force, or, more properly speaking, to increase moments of power, but also to produce motion however slight: for often motion must be reduced. Thus in automata indicating the hours, by means of a long elastic blade wound into a spiral, in order to regulate the speed or slowness of the wheels it is necessary now to tighten and now to relax the spring itself: but because in ordinary clocks this is accomplished by the turning of a toothed wheel (to whose axle the inner end of the elastic spiral is attached, and lest the strip, once forced into a more lax spiral, should restore itself, by revolving the axle itself and the toothed wheel, a spring is opposed to the oblique teeth of that wheel, on the side where they are straight), it is necessary to advance or retract at least one tooth.

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Liber octavus. CAPUT IV. 789 est. At sæpè contingere potest, ut elasticam laminam jam val- de intentam amplius intendere, quantum fert integra dentis unius conversio, celeriorem motum inferat, quàm temporis ra- tio postularet; propterea scientissimi artifices, rejectâ virgulâ illâ elasticâ, ita rotæ illius dentes conformant, ut cochleolæ infinitæ congruant; hæc enim convoluta valde minutis pro- gressionibus laminam elasticam intendit, aut remittit, & ubi- cunque placuerit, sistitur. Illud quoque non leve commodum (ut paulò superius indi- catum est) in attollendis ponderibus animadversione dignum est, quod sublato pondere atque suspenso, cessare potest po- tentia; & quamvis nec ab illâ, nec ab alio quolibet retinaculo manubrium cochleæ infinitæ retineatur, neque pendenti oneri fulcrum ullum subjiciatur, ipsa per se cochlea tympanum sistit, & suspensum pondus impeditur, ne suâ vi recidat. Id quod in tympanis dentatis, neque in Succulis, neque in Trochleis, ne- que in Vecte obtinetur: quas Facultates si potentia dimiserit, inchoato jam motu, neque illas aliquo retinaculo coërceat, priorem laborem irritum facit gravitas sibi dimissa, ut satis aper- tè constat. Postremò Cocheas infinitas cochleis pariter infinitis coag- mentare si quis voluerit, is profectò momentis potentiæ immen- sam quandam accessionem fecerit. Si enim primi tympani den- tati Axem deformaveris in cochleam, quæ aliud tympanum pariter dentatum moveat, & secundi hujus tympani Axem item in spiralem striam excavaveris, quæ tertium tympanum con- vertat unà cum Axe, cui ductarius funis circumducitur; ecce quot Rationibus componitur Ratio motuum potentiæ & pon- deris. Prima Ratio est Peripheriæ à manubrio descriptæ ad di- stantiam spirarum primæ cochleæ. Secunda Ratio est periphe- riæ primi tympani ad intervallum spirarum secundæ cochleæ. Secunda Ratio est peripheriæ primi tympani ad intervallum spirarum secundæ cochleæ. Tertia Ratio est peripheriæ secun- di tympani ad intervallum spirarum tertæ cochleæ. Quarta demum est Ratio peripheriæ tertij tympani ad ambitum sui Axis. Ponamus singulas peripherias ad suæ cochleæ spirarum intervallum esse ut 30 ad 1, & tertij tympani orbitam ad sui Axis ambitum esse ut 5 ad 1; componendæ sunt tres Rationes GGg g g 3

Transcription: Translated (English)

Liber octavus. CAPUT IV. 789 is. But it can often happen that, when an elastic strip already greatly strained is stretched further, as far as the full turn of a single tooth allows, a quicker motion is produced than the proportion of time would require; therefore the most skilful craftsmen, having rejected that elastic rod, shape the teeth of that wheel so that they correspond with an endless screw; for this, when wound, by very small advances stretches or relaxes the elastic strip, and can be stopped wherever one wishes. That also is no slight advantage, and worthy of notice, in lifting weights, which, as indicated a little above, is this: that when the weight has been raised and suspended, the power can cease; and although neither by that nor by any other retaining device is the handle of the endless screw held, nor is any support placed under the hanging load, the screw itself alone stops the drum, and the suspended weight is prevented from falling back by its own force. This does not obtain in toothed drums, nor in Winches, nor in Pulleys, nor in the Lever: for if the power has let go these faculties, once motion has already begun, and no retaining device restrains them, gravity, once released to itself, makes the previous labor void, as is sufficiently clear. Lastly, if anyone should wish to join endless screws with equally endless screws, he would certainly add a certain immense increase to the moments of power. For if you were to transform the axle of the first toothed drum into a screw, which moves another likewise toothed drum, and likewise were to hollow out the axle of this second drum with a spiral groove, which turns a third drum together with the axle around which the driving rope is wound; behold how many ratios make up the ratio of the motions of power and weight. The first ratio is that of the circumference described by the handle to the distance between the turns of the first screw. The second ratio is that of the circumference of the first drum to the interval between the turns of the second screw. The second ratio is that of the circumference of the first drum to the interval between the turns of the second screw. The third ratio is that of the circumference of the second drum to the interval between the turns of the third screw. The fourth, finally, is the ratio of the circumference of the third drum to the circumference of its axle. Let us suppose each circumference to be to the interval of its screw’s turns as 30 to 1, and the orbit of the third drum to the circumference of its axle to be as 5 to 1; the three ratios must be composed together. GGg g g 3

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790 Mechanicorum trigecuplæ cum unâ quintuplâ, & exurgit Ratio motûs potentiæ manubrio applicatæ, ad motum ponderis ut 135000 ad 1. Quo igitur conatu potentia moveret libras decem, hac trium cochlearum infinitarum complexione movebit millies mille trecentas quinquaginta libras, seu, ut vulgari vocabulo utar, millionem & trecenta quinquaginta millia librarum. Quid autem, si plura tympana cochleas infinitas habentia addantur? utique si primæ cochleæ manubrio agitatæ quatuor consequentia tympana cum suis cochleis addantur, eandem Rationem trigecuplam habentia, & quintum tympanum cum suo Axe Rationem quintuplam habeat, demum potentia momentum obtinebit ut 121.500000: &, si absque machinâ moveret libras decem, hac machinâ ex quinque cochleis cum sibi congruentibus tympanis movere poterit mille ducentos quindecim milliones librarum. Neque sibi quisquam persuadeat opus esse ingentibus tympanis, ut validissimis cochleis respondeant: Experimento enim didicimus valde exiguas cochleas satis esse ad ingentia pondera attollenda, modò axis funi ductario destinatus satis firmus sit ac validus, & ferendo oneri par. Hic autem Axis (quemadmodum & in Ergatâ) si plurimum funem excipere debeat, ne in nimiam longitudinem protendatur, conformari potest in Cylindroides Hyperbolicum: nam ductarius funis illum aliquoties complexus (quantum satis fuerit, ne excurrat) colligi poterit, & in convolutione se ad apicem Hyperbolæ continebit. At, inquis, hujusmodi motus ponderis nimis longa temporis spatia exigit. Ita planè: neque aliter contingere potest, si quidem tam ingens pondus movere volueris: an non præstat tantam molem demum loco cessisse, quam omnino immotam cui cumque conatui reluctari? Sed quid, si opportunissimum se præbeat proximus rivulus perennis? primæ cochleæ apponatur loco manubrij rota cum pinnis, in quas aqua incurrat; illa enim circumacta cochleam & consequentia tympana versabit, ac demum vel dormientibus operis moles ab exiguâ aquâ dimovebitur. Quod si ex pluribus cochleis infinitis compositam machinam tibi construere volueris, ita tamen, ut modò majoribus, modò minoribus ponderibus movendis sit idonea citrà temporis dispendium,

Transcription: Translated (English)

790 Mechanicorum thirtyfold with one fivefold, and the ratio of the power of the motion applied to the handle, to the motion of the weight, arises as 135000 to 1. So then, by whatever effort a power would move ten pounds, by this triple combination of endless screws it will move one thousand three hundred and fifty thousand pounds, or, to use the common term, a million and three hundred and fifty thousand pounds. But what if, more drums having endless screws be added? Certainly if to the first screws moved by the handle, four successive drums with their screws be added, having the same thirtyfold ratio, and if the fifth drum with its axle have a fivefold ratio, then at length the power will obtain a moment of 121.500000: and, if without the machine it would move ten pounds, by this machine, made up of five screws with the drums corresponding to them, it will be able to move one thousand two hundred and fifteen million pounds. Nor should anyone persuade himself that huge drums are needed in order to match very strong screws: for by experiment we have learned that very small screws are sufficient to lift enormous weights, provided only that the axle intended for the guiding rope be sufficiently firm and strong, and equal to bearing the load. This axle, however (as also in the capstan), if it must take in a very great length of rope, so as not to be stretched out to excessive length, can be formed into a hyperbolic cylindroid: for the guiding rope, winding around it several times (as much as shall be enough, lest it run out) can be gathered in, and in its coiling will keep itself toward the apex of the hyperbola. But, you say, motion of this kind requires too long a span of time. Quite so: nor could it happen otherwise, if indeed you wished to move so great a weight: is it not better that so great a mass should at last yield its place, than remain utterly unmoved and resist every effort? But what if a nearby perennial brook should present itself as most convenient? let there be placed on the first screw, in the place of the handle, a wheel with vanes, against which the water shall strike; for that wheel, when turned, will move the screw and the subsequent drums, and at last even while the workmen sleep the mass will be moved from its place by a little water. And if you should wish to construct for yourself a machine made up of several endless screws, yet so that now with greater, now with smaller weights it may be suitable for moving them without loss of time,

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Liber octavus. CAPUT IV. 791 pendium, ubi satis virium habetur in potentiâ; eâ ratione in loculamento dispone singulos axes in cochleam deformatos, ut eorum poli ex loculamento promineant, atque pro re natâ pro- pelli seu retrahi aliquantisper valeat hic aut ille axis, ne ejus stria occurrat subjecti tympani dentibus. Nam si alterius sal- tem poli extremitas in quadratam figuram desinat, quæ inseri possit manubrio, hoc poterit huic aut illi axi aptari, quin supe- riores cochleæ hujus tympani convolutionem impediant. Quod si majora adhuc requirantur potentiæ momenta, proximè su- perior axis suum in locum restituatur, ut cochleæ stria in sub- jecti tympani dentes incurrat. Quapropter ad minora pondera movenda adhibeantur inferiores cochleæ, ad majora superiores. CAPUT V. Cochleæ usus aliqui indicantur. A Deo frequens est & vulgatus apud plerosque artifices co- chleæ usus, ut ex tam variâ ejus cum cæteris complexio- ne unusquisque facilè colligere possit, quid facto sit opus, ubi eâ utendum necessitas aut utilitas suaserit. Ne tamen ab initâ in antecedentibus libris consuetudine in hujus operis calce re- cedam, pauca quædam indicare placuit, quæ in reliquis non admodum dissimilibus facem præferant. PROPOSITIO I. Aërem validè comprimere, aut dilatare. Follibus lusoriis aërem pyulco ingerentes majorem subinde atque majorem difficultatem percipiunt; quo enim magis aër conclusus à naturali raritate recedere cogitur, etiam majo- re nisu resistit, neque solùm magis densari renuit, sed & se la- tiùs explicare molitur. Hinc didicimus & pneumaticos fontes construere, qui Spiritu interno urgente aquam in altum evi- brant,

Transcription: Translated (English)

Book Eight. CHAPTER IV. 791 ...load, where sufficient strength is available in the power; in this way place in the box the separate shafts deformed into a screw, so that their ends project from the box, and so that whichever shaft may be the case can be pushed forward or drawn back for a little while, lest its groove meet the teeth of the underlying drum. For if at least the end of one of the ends should terminate in a square form, so that it may be inserted into a handle, this will be able to be fitted to this shaft or that, without the upper screws hindering the turning of this drum. But if even greater moments of power are required, let the nearest upper shaft be restored to its place, so that the groove of the screw may strike against the teeth of the underlying drum. Therefore, for moving smaller weights, let the lower screws be used; for larger, the upper. CHAPTER V. Some uses of the screw are indicated. The use of the screw is frequent and common with most craftsmen by God, so that from so varied a combination of it with other things each person can easily gather what needs to be done, when necessity or utility shall have suggested that it be used. Yet lest I depart, at the end of this work, from the custom begun in the preceding books, it has pleased me to indicate a few things that in the rest are not altogether dissimilar and may hold up a torch. PROPOSITION I. To compress or expand air strongly. Those who drive air into a vessel with playful bellows gradually experience greater and greater difficulty; for the more the enclosed air is forced to recede from its natural rarity, the more strongly it resists, and not only refuses to be made more dense, but also strives to spread itself out more widely. From this we have learned also to construct pneumatic fountains, which, the internal spirit urging them, drive water upward,

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Mechanicorum 792 brant, & plumbeas glandes fistulis ejaculari, non pulvere nitrato ignem concipiente, sed aëre per vim densato ad antiquas dimensiones recuperandas erumpente. Quoniam verò ingesta jam in conceptaculum non exigua aëris copia difficiliùs comprimitur novâ aëris accessione, quàm ut manus valeat trusillum rectâ impellere; idcirco trusilli hastulam deformatam in helicem, & suæ matrici insertam, adhibere operæ pretium erit: dum enim manubrio agitante contorquetur cochlea, sensim deprimitur embolus, aëremque ingerit. Ne autem morâ longiore opus sit perpetuâ versatione manubrij, ita cochleæ matrix externam vasis faciem contingat, ut illi adnecti, atque ab eo disjungi valeat: initio enim, quando adhuc levis est aëris modicè compressi resistentia, lamella illa suo foramine interiùs claviculatim striato cohærens hastulæ emboli, si à vase disjuncta fuerit, unà cum hastulâ movebitur: deinde verò, quando jam trusillus ægrè impellitur, lamella illa cum vase connectatur, & non nisi versato manubrio adduci atque reduci embolus poterit; id quod satis lentè perficietur. Rem claritatis gratia in fonte pneumatico explicemus. Sit vas A B ex materiâ metallicâ, in cujus superiore parte labrum, ex quo per ramen A immittatur in vas aqua, ita tamen, ut non impleatur; aqua enim in vas modicè inclinatum descendens aërem expellet per tubulum C D. Ubi satis aquæ immissum fuerit, occludatur ramen A diligentissimè cochleolâ congruente, & convoluto epistomio E, tubus D C sit aëri impervius ad vasis latus statuatur modiolus cum embolo congruente H I, & emboli hastula sit connexa cum mobili vasis ansâ H O. Porro

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Mechanics 792 brant, and to shoot leaden bullets through pipes, not by gunpowder kindling the fire, but by air densely compressed by force, bursting forth to recover its former dimensions. But since now the not inconsiderable quantity of air already introduced into the receptacle is compressed with greater difficulty by a fresh accession of air than that the hand can drive the piston straight forward; therefore it will be worth the trouble to use a piston rod fashioned into a spiral, and inserted into its socket: for while the screw is turned by the revolving handle, the piston is gradually depressed, and admits air. But lest, through a longer delay, the work should require the continual turning of the handle, let the socket of the screw touch the outer face of the vessel in such a way that it may be attached to it and detached from it: for at the beginning, when the resistance of the moderately compressed air is still slight, that little plate, engaging within its hole with a small claw-like groove, with the piston rod, if it be separated from the vessel, will move together with the rod: then indeed, when the piston is now driven with difficulty, let that little plate be connected with the vessel, and then only by turning the handle will the piston be able to be drawn in and pushed back; and that will be accomplished rather slowly. Let us explain the matter for the sake of clarity in a pneumatic fountain. Let there be a vessel A B made of metal, in the upper part of which there is a lip, through which, by means of the channel A, water may be admitted into the vessel, but in such a way that it is not filled; for the water, descending into the vessel which is slightly inclined, will drive out the air through the tube C D. When enough water has been introduced, let the channel A be closed most carefully with a matching screw, and with the cap E turned on, let the tube D C be impervious to air; at the side of the vessel let there be placed the barrel with the matching piston H I, and let the piston rod be connected with the movable handle of the vessel H O. Moreover

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Liber octavus. CAPUT V. 793 Porrò hastula H K perforata sit, & continuo ductu usque ad emboli K S fundum pateat aëri ingredienti via H S; sed fo- ramini S adjecta sit valvula, quæ aëri regressum obstruat. Simi- liter modioli fundo in I valvula exteriùs apposita aperiatur in- gesto aëri transitum præbens, sed aëri intrà vas compresso cum nusquam exitus pateat, valvula ipsa modioli foramen I occlu- dit. Hastulæ verò H K exterior facies sit in helicem striata, & lamellæ M N tanquam matrici congruat, quæ in M & N co- chleolis adnecti queat exteriùs vasi, quasi esset ansæ fulcrum. Ubi immissum fuerit quantum satis est aquæ, cochleolis M & N revolutis disjungatur matrix à vase: tum attractâ ansâ H O, unà cum lamellâ M N attrahitur embolus K S, & per apertum ductum H S ingreditur aër modiolum implens. Im- pulso deinde embolo, valvula ad S clauditur, & aër ex medio- lo per patentem valvulam I ingeritur in vas; ex quo nequit exi- re, neque aquam propellere, clauso scilicet epistomio E, & fo- ramine A: quapropter comprimitur, & densatur; ideóque at- tracto denuo embolo K S inclusus vasi aër se latiùs explicare connitens valvulam I valide applicat foramini modioli, sibique exitum obstruit. Toties adducitur atque reducitur embolus, & aër ingeritur, quoad magna premendi difficultas percipiatur; ubi eò ventum fuerit, tunc lamella M N iterum vasi adnecta- tur suis cochleolis; nec jam embolus rectâ adduci potest; sed arreptum in O manubrium versatur, & embolus intrà modio- lum circumactus sensim attollitur, qui deinde revoluto in con- trarium manubrio deprimitur, & multâ vi aër in vase compri- mitur. Laxato demum Epistomio E, compressus in vase aër aquam exprimit per tubum C D, primùm quidem vehemen- tiùs, subinde remissius, prout aëris vis elastica sensim lan- guescit. Hoc idem quod de aëre intra vas comprimendo ad aquam evibrandam comminisci placuit, servatâ analogiâ dicendum est de aëre, tùm conatu manûs rectâ trusillum impellentis, tum ope cochleæ similiter conformatæ, intrà conceptaculum com- primendo, ut ex fistulâ deinde multâ vi emittatur plumbea glans, ubi referatus aëri exitus illum subitò dilatari permiserit. Quin & pneumatica hujusmodi tormenta citrà conceptaculum aëris compressi construere non inutile accidat, si, quemadmo- HHhh

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Book Eight. CHAPTER V. 793 Moreover, let the tube H K be perforated, and by a continuous channel let there be, as far as the bottom of the barrel K S, a passage H S open to the entering air; but at the opening S let a valve be attached, which shall prevent the air from returning. Likewise, at the bottom of the receiver, a valve externally applied at I shall open to admit incoming air, but when the air is compressed within the vessel and there is nowhere any exit, the valve itself closes the opening I of the receiver. The outer face of the tube H K should, however, be ridged in a spiral, and the plate M N should fit like a die, which can be attached externally to the vessel at M & N by means of screw-pins, as though it were the support of a handle. When as much water as is sufficient has been introduced, after the screw-pins M & N are turned, let the die be detached from the vessel: then, when the handle H O is pulled, together with the plate M N the piston K S is drawn along, and through the open passage H S air enters, filling the receiver. Then, when the piston is pushed in, the valve at S is closed, and air from the receiver is forced through the open valve I into the vessel; from which it cannot escape, nor drive out the water, the cock E and the opening A being closed; wherefore it is compressed and condensed. And thus, when the piston K S is again drawn out, the air enclosed in the vessel, striving to expand itself more broadly, strongly presses the valve I against the opening of the receiver and blocks its own exit. So often as the piston is drawn out and pushed back, and air is introduced, until a great difficulty in pressing is perceived; when this point has been reached, then the plate M N is again attached to the vessel by its screw-pins; and now the piston can no longer be drawn straight out, but the handle seized at O is turned, and the piston, turned around within the receiver, is gradually raised; afterward, with the handle turned back the opposite way, it is depressed, and with great force the air in the vessel is compressed. Finally, when the cock E is loosened, the compressed air in the vessel expels the water through the tube C D, at first indeed more violently, then more gently, as the elastic force of the air gradually weakens. The same thing, which it seemed proper to devise concerning the compression of air within a vessel in order to project water, must, with the same analogy preserved, be said of compressing air, both by the effort of the hand directly driving the rammer, and by means of a similarly fashioned screw, within the receptacle, so that a leaden bullet may afterward be expelled from the barrel with great force, when an opened exit for the air allows it suddenly to expand. Indeed, it may even be useful to construct pneumatic engines of this kind without a compressed-air receptacle, if, as it were,

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Mechanicorum 794 dum nostrates pueri surculos sambuceos fungosâ medullâ exhauriunt, & utráque tubuli extremitate papyraceis globulis obstructâ, alterum globulum congruo cylindro propellunt, at- que inclusum aërem densant, quoad aëris vim elasticam, & im- pellentis manûs conatum non ferens extremus alter globulus edito scloppo expellatur; ita ferream fistulam longiorem para- veris, cujus alteri extremitati immittatur plumbea glans ob- ducta papyro, aut simili materiâ, ut exquisitè tubi osculum implens demum universam aëris vim excipiat, alteram extre- mitatem aliquot spiris ambiat cava cochlea, quam impleat cy- lindrus ferreus in congruentem cochleam deformatus: Si enim hujusmodi cylindrus vix brevior fuerit, quàm fistula, & apto manubrio convolutus in fistulam sensim immittatur, totum aë- rem, quo fistula replebatur, ad exiguas spatij angustias adiget, ex quibus magnâ vi demum, quâ data porta, erumpens ejacu- labitur plumbeum globulum. Quod si aërem non comprimere, sed distrahere atque dil- tare libitum fuerit, eâdem ratione parandus est modiolus cum embolo, ac hastulâ in helicem striatâ, atque perforatâ, & co- chleæ matrici inserta, nisi quod valvulæ contrariam positionem exigunt; nam modioli valvula I intrà ipsum modiolum statuen- da est, ut adducto embolo aperiatur, & ex vase aër in modio- lum attrahatur: Emboli verò valvula non ad S, sed in H ap- ponenda est, ut reducto embolo, aër in modiolum admissus ex- primatur per tubulum S H, sivè manu urgeatur trusillus, sive cochlea convolvatur. Aërem autem, licèt valdè compressum, magis etiam convolutâ cochleâ densari, aut valde rarum ma- gis adhuc dilatari manifestum est; id quod rectâ manûs impul- sione aut attractione nequaquam fieri posset. PROPOSITIO II. Forcipum vires cochleâ augere. Duplicem exerceri à forcipibus vim constat; altera est con- stringendo id, quod illis apprehenditur, & earum vis major aut minor ex eo æstimatur, quod brachia longè à nodo, aut prope illum, arripiantur: altera vis est in extrahendo ali- quid,

Transcription: Translated (English)

Mechanics 794 while our boys, when they remove the pith from elder sticks, and having stopped both ends of the tube with paper pellets, drive one pellet forward with a fitting cylinder, and compress the enclosed air, until the last pellet, unable to bear the elastic force of the air and the effort of the hand that is impelling it, is driven out with a report; so, if you have prepared a longer iron tube, and into one end of it there is inserted a leaden bullet covered with paper, or some similar material, so that it exactly fills the mouth of the tube and at last receives the whole force of the air, while the other end, for a few turns, is embraced by a hollow screw, which is filled by an iron cylinder fashioned into a matching screw: for if such a cylinder be scarcely shorter than the tube, and, when turned by a suitable handle, be gradually introduced into the tube, it will compress all the air with which the tube was filled into very narrow space, from which, at length, bursting forth with great force when the gate is opened, it will shoot out the leaden ball. But if it should be desired not to compress the air, but to draw it apart and rarefy it, a vessel with a piston is to be prepared in the same manner, together with a rod cut into a helix, and perforated, and inserted into the female screw; except that the valves require the opposite position. For the valve I of the vessel must be placed within the vessel itself, so that, when the piston is drawn back, it may open and air may be drawn from the vessel into the vessel; but the valve of the piston must be placed not at S, but at H, so that, when the piston is withdrawn, the air admitted into the vessel may be forced out through the little tube S H, whether the plunger be pressed by hand or the screw be turned. It is evident, however, that air, even when very much compressed, may be made still denser by a further turn of the screw, or, when very rare, may be made still more dilated; and this could in no way be accomplished by a straight push or pull of the hand. PROPOSITION II. To increase the force of pincers by means of the screw. It is clear that pincers exert a twofold force: one is in gripping the object seized by them, and their force is estimated as greater or less according to whether the arms seize it far from the joint, or near it; the other force is in pulling something out,

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Liber octavus. CAPUT V. 795 quid, ut clavum tabulæ aut parieti infixum; cum enim curva sit forceps, qua parte clavum apprehendit, adnexum in ipso flexu habet hypomochlium, & brachia inclinando, pro eorum longitudine, vis extrahendi exercetur quasi per vectem. At aliquando opus est majore conatu, quàm ut solis forcipibus va- leat potentia infixum clavum extrahere; momentum siquidem potentiæ pendet ex Ratione, quam habet distantia potentiæ ad distantiam clavi ab ipso flexu, qui fungitur munere hypomo- chlij. Quare vis extrahendi major communicari potest ope co- chleæ, ita tamen, ut forceps non exerceat munus vectis. Paretur itaque valida & satis crassa lamina chalybea A B, matricem cochleæ habens in C, & sit cochlea F E, manubrium habens E D. Cochleæ verò extremitas in cylindrum desinat, qui crassioris laminæ H I fora- mini exquisitè polito inseratur, & in eo facillimè convolvi valeat. Cylindri ex- tremitas infra laminam H I ita dilatetur, ut eandem laminam H I sustineat, non tamen convolutionem impediat. Porrò laminæ H I adnexi sint duo annuli ita conformati, ut forcipis brachia exci- piant: nam si brachia in hujusmodi an- nulos immittantur, ut hi proximi sint nodo forcipis maximè di- latatæ, antequam apprehendat clavum extrahendum, postmo- dum constrictâ forcipe & clavum apprehendente, elevata la- mina H I annulos secum rapiet, qui per forcipis brachia diva- ricata excurrentes demum validè illa constringent, nec ulte- riùs excurrere poterunt. His paratis utrique extremitati A B subjiciantur fulcra (sivè sint tigillorum frusta, sivè quæcum- que alia) inter ipsam laminam & planum, ex quo educendus est clavus, interjecta: Nam manubrio D E convoluta cochlea ita matricem A B applicabit fulcris, ut firmissimè cohæreant cum subjecto plano. Iam si pergas cochleam contorquere, hæc secum rapiet laminam H I, & adjectos annulos cum forcipe, & clavo, quem revellit. Quod si fortè placuerit forcipem habere peculiarem huic instrumento aptandam, habeat in brachiorum extremitatibus HHhh 2

Transcription: Translated (English)

Book Eight. CHAPTER V. 795 what, as a nail fixed in a plank or wall; for since the forceps is curved, in the part by which it grasps the nail it has the hypomochlion attached in the very bend, and by inclining the arms, according to their length, the force of extraction is exerted as by a lever. But sometimes a greater effort is required than that by which the power of the forceps alone can extract the fixed nail; for the force of the power depends on the ratio which the distance of the power bears to the distance of the nail from the bend itself, which serves the office of the hypomochlion. Wherefore a greater extracting force can be communicated by means of a screw, provided nevertheless that the forceps does not perform the function of a lever. Let therefore a strong and sufficiently thick steel plate A B be prepared, having the nut of a screw in C, and let there be a screw F E, having the handle E D. But let the end of the screw terminate in a cylinder, which is to be inserted into the hole of the thicker plate H I, exquisitely polished, and may very easily be wound therein. Let the end of the cylinder below the plate H I be so enlarged that it may support the same plate H I, but not hinder the winding. Moreover, let there be joined to the plate H I two rings so shaped that they receive the arms of the forceps; for if the arms are introduced into such rings, so that these are near the knot of the forceps at its widest opening, before it grasps the nail to be extracted, and afterward, when the forceps has been tightened and has taken hold of the nail, the plate H I, being raised, will carry the rings along with it, which, running through the diverging arms of the forceps, will at last strongly constrict them, nor will they be able to run farther. These things being prepared, let props be placed beneath both ends of A B, whether they be pieces of small beams or whatever else, inserted between the plate itself and the plane from which the nail is to be drawn out: for the screw, turned by the handle D E, will apply the nut A B to the props in such a way that they will hold most firmly with the underlying plane. Now if you continue to turn the screw, it will draw along with it the plate H I, and the joined rings with the forceps, and the nail which it tears out. But if perhaps it should please you to have a special forceps adapted to this instrument, let it have at the extremities of the arms HHhh 2

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Mechanicorum 796 uncos aut annulos annulis H & I inferendos aut connectendos, eâ tamen ratione dispositos, ut dum lamina H I vi cochleæ tra- hitur, brachia ipsa ad se invicem accedendo forcipem con- stringant. Unum præterea addendum, quod non levis est momenti, & aliàs quoque observari poterit. Contingere potest, ut omnibus modo dicto paratis, potentia se infirmiorem sentiat, quàm ut valeat circumducto manubrio DE cochleam contorquere. Hoc igitur tibi remedium compara: longiorem vectem validis funi- culis colliga cum manubrio DE, & vecte illo quasi manubrio utens experieris pro Ratione longitudinis aucta momenta; am- plior siquidem peripheria, quæ tunc à potentiâ describitur, ad spirarum cochleæ intervallum habet Majorem Rationem. PROPOSITIO III. Numerum passuum aut rotæ conversionem metiri. Hoc idem problema lib.5. cap.9. prop.2. propositum est, & per rotulas dentatas singulis prioris rotæ conversionibus excipientes impulsionem singulorum dentium, in quos promi- nens paxillus incurrat, perfici posse indicatum est. Nunc aliam methodum indicare placet ex iis, quæ superiore capite sunt dicta de Cochleâ Infinitâ. Primam quidem rotulam, cui motus origo inest ex funiculi tractione, prout ibi dictum est, eandem statue, & illius axis extremitas apposito indice tot pas- sus, aut tot rotæ conversiones indicabit, quot in dentes ipsa prima rotula distributa intelligitur. Hujus rotulæ axis in co- chleam infinitam deformetur, cui sua rotula dentata congruat; & singulis primæ rotulæ conversionibus singuli dentes secundæ promoventur: atque adeò quot dentes secundæ huic rotulæ in- sunt, ut hæc integram conversionem perficiat, tot requiruntur prioris rotulæ conversiones. Similiter secundæ rotulæ axis in cochleam infinitam deformetur, & tertiam rotulam dentatam convertat, cujus axis pariter tertiam cochleam infinitam consti- tuere potest, & quartam rotulam cum suo axe & indice convol- vere. Singulorum axium extremitates in facie loculamenti ad- jecto indice ob oculos ponunt numerum revolutionum proxi- mè

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Mechanics 796 hooks or rings to be inserted into or connected with the rings H & I, yet so arranged that, while the plate H I is drawn by the force of the screw, the arms themselves, by approaching one another, tighten the forceps. One further point should be added, which is not of slight importance, and may also be observed elsewhere. It may happen that, with everything prepared in the manner just described, the power feels itself too weak to be able to turn the screw with the handle DE rotated around. Provide yourself therefore with this remedy: bind a longer lever with strong cords to the handle DE, and using that lever as if it were a handle, you will find the efforts increased in proportion to the length; for the greater periphery which is then described by the power bears a greater ratio to the distance between the spirals of the screw. PROPOSITION III. To measure the number of paces or the rotation of a wheel. The same problem was proposed in lib. 5, cap. 9, prop. 2, and it was indicated that it could be accomplished by means of toothed wheels, receiving at each revolution of the former wheel the impulse of the individual teeth against which the projecting peg strikes. Now I wish to indicate another method from those things which were said in the preceding chapter about the Infinite Screw. Let the first wheel, in which the origin of motion lies from the traction of the cord, be set up as there described, and let the end of its axle, with an index attached, show as many paces, or as many revolutions of the wheel, as are understood to be distributed into the teeth of that first wheel. Let the axle of this wheel be shaped into an infinite screw, with which its toothed wheel corresponds; and at each revolution of the first wheel the individual teeth of the second are advanced: and thus, as many teeth as are in this second wheel, so many revolutions of the first wheel are required for it to complete one full rotation. Likewise let the axle of the second wheel be shaped into an infinite screw, and let it turn the third toothed wheel, whose axle likewise may form a third infinite screw, and may revolve a fourth wheel with its axle and index. The ends of the individual axles, on the face of the housing, with an index attached, place before the eyes the number of revolutions, nearely

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Liber octavus. CAPUT V. 797 mè antecedentis rotulæ. Quapropter numerus à postremâ ro- tulâ indicatus multiplicandus est per numerum omnium den- tium penultimæ rotulæ, & productus per numerum dentium antepenultimæ ducendus; atque iterum hunc productum per numerum omnium dentium antecedentis rotulæ multiplicare oportet, ut omnium passuum, aut conversionum rotæ currûs, nu- merus innotescat. Quare artificis industria in hoc requiritur, ut rotularum dentibus eos numeros statuat, quorum rationem inire non sit nimis operosum. Illud autem, commodum-ne dixeris? an incommodum? in cochlearum infinitarum complexione contingit necessariò, quod axes sunt in planis invicem rectis, ac proinde indices non in eâdem loculamenti facie constitui possunt: cum enim unus- quisque axis ad planum sui tympani dentati, cui infigitur, sit rectus, ipsum verò tympanum sit in eodem plano, in quo est cochlea infinita, à qua convertitur, manifestum est plana ipsa, in quibus sunt axes, esse invicem recta, atque idcirco non ad eandem loculamenti faciem pertinere eorum indices. PROPOSITIO IV. Lunæ motum & phases in automato indicare. Quæ communiter parantur automata horas indicantia, in- dicem habent horis duodecim perficientem integrum cir- cuitum: quapropter lunæ motum, ejusque ætatem ob oculos ponere cupiens, satis erit, si axem, cui horarum index inseri- tur, in cochleam infinitam deformaveris, quæ convertat tym- panum in dentes 59 distributum; axis enim tympani indicem convertens ætatem lunæ commonstrabit in dexterâ, aut in si- nistrâ facie loculamenti, cui automatum includitur. Cum enim lunaris mensis Synodicus complectatur dies 29 ́, index autem horarum semissem diei perficiat, erunt indicis hujus conversiones 59, dum semel index lunæ suam conversionem absolvit. Si igitur index lunæ sit lamina rotundum habens fo- ramen propè indicis lingulam, per quod appareat pictus in sub- jectâ facie circulus centrum habens extra indicis centrum, adeò ut primâ die lunæ nihil illius circuli appareat, & die decima- HHhhh 3

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Liber octavus. CAPUT V. 797 of the preceding wheel. Therefore the number indicated by the last wheel must be multiplied by the number of all the teeth of the next-to-last wheel, and the product carried through by the number of the teeth of the third-to-last; and again this product must be multiplied by the number of all the teeth of the preceding wheel, so that the number of all the steps, or revolutions of the carriage wheel, may be made known. Wherefore the skill of the artificer is required in this, that he assign to the teeth of the wheels such numbers, the ratio of which is not too troublesome to compute. But that thing, would you call it an advantage? or a disadvantage? in the arrangement of infinite screws it necessarily happens that the axes are in planes mutually perpendicular, and therefore the pointers cannot be placed on the same face of the case: for since each axis is perpendicular to the plane of its toothed wheel, into which it is fixed, but the wheel itself lies in the same plane in which is the infinite screw by which it is turned, it is clear that the planes themselves in which the axes are are mutually perpendicular, and for that reason their pointers do not belong to the same face of the case. PROPOSITION IV. To indicate the motion and phases of the Moon in an automaton. Those automata commonly made for indicating the hours have a pointer completing a full circuit in twelve hours: wherefore, if you wish to set before the eyes the motion of the Moon and its age, it will be enough if you have transformed the axis into which the hour hand is inserted into an infinite screw, which turns a wheel distributed into 59 teeth; for the axis of the wheel turning the pointer will show the age of the Moon on the right or on the left face of the case in which the automaton is enclosed. For since the synodic lunar month comprises 29 days and a half, and the hour hand completes half a day, the revolutions of this pointer will be 59, while once the lunar pointer completes its revolution. If therefore the lunar pointer be a plate having a round opening near the little tongue of the pointer, through which there may appear a painted circle on the surface beneath, having its center outside the center of the pointer, so that on the first day of the Moon nothing of that circle appears, and on the tenth day ... HHhhh 3

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Mechanicorum 798 quintâ foramen integrum exhibeat ejusdem circuli colorem, lunæ Phases à foramine, & ejus ætas à lingulâ indicabuntur. Quod si placuerit in eâdem facie, in qua descriptæ sunt ho- ræ, etiam lunæ phases & motum apparerere, oportebit axi in- dicis horarum aptatam rotulam denticulos habere ad perpendi- culum infixos, qui curriculum, seu Vertebram striatam con- vertant, ita ut vertebræ hujusmodi una conversio planè iso- chrona sit uni conversioni indicis horarum. Curriculi autem axis in cochleam infinitam deformatus convertat tympanum in dentes 59 distinctum, quod collocetur faciei loculamenti pa- rallelum; hujus siquidem conversio in eâdem loculamenti fa- cie, in qua & horæ indicantur, repræsentabit lunæ phases. At si fortasse volueris in eâdem Automati facie ita apparere horas & lunæ ætatem, ut proximè saltem indicetur, quotâ ho- râ accidat Novilunium aut Plenilunium, postquam semel jux- ta Ephemerides conciliaveris indices horarum & lunæ; non sa- tis erit in dentes 59 distinxisse tympanum, cujus singuli dentes horis 12 promoveantur; siquidem mensis lunaris Synodicus complectitur dies 29, horas 12, minuta 44, hoc est ferè tres horæ quadrantes; atque adeò post duos menses index lunæ in- dicaret Novilunium sesquiorâ citiùs, quàm par fuerit, & post annum index anteverteret verum Novilunium novem horis. Quare axi horas indicanti non esset copulandus axis cochleæ infinitæ, cujus tympanum aliam exigeret dentium multitudi- nem; sed peculiaris axis statuendus esset, cujus conversio ita temperaretur, ut horis undecim cum quadrante absolveretur; tympanum verò, ex cujus conversione convolveretur index lu- næ, distribuendum esset in dentes 63; hujus enim unica con- versio responderet conversionibus 63 axis, cujus singulæ con- versiones perficerentur horis 11 1/4: quapropter index lunæ suam conversionem absolveret horis 708 3/4, hoc est diebus 29, horis 12, minutis 45. Esset igitur in singulis lunationibus pau- lò tardior non nisi uno minuto; sed demum absolutis duode- cim lunationibus exiguum esset discrimen. Quod si rotulæ ho- ras indicantis faciem interiorem in partes 16 distinxeris, & denticulos ad perpendiculum erexeris, qui Curriculum con- vertant, ita tamen, ut curriculum unâ conversione excipiat so- lùm quindecim denticulos, utique una curriculi conversio perficietur

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Mechanicorum 798 the fifth hole should display the full color of the same circle, the phases of the moon will be indicated by the hole, and its age by the tongue. But if it should please you, on the same face, on which the hours are marked, also to show the phases and motion of the moon, it will be necessary for the wheel fitted to the shaft of the hour-hand to have teeth set vertically, which turn the carriage, or striated vertebra, so that one revolution of such a vertebra is exactly isochronous with one revolution of the hour-hand. But the axis of the carriage, fashioned into an endless screw, turns a drum divided into 59 teeth, which should be placed parallel to the face of the case; for its revolution on the same face of the case on which the hours are also indicated will represent the phases of the moon. But if perhaps you should wish the hours and the age of the moon to appear on the same face of the Automaton in such a way that it may at least be approximately indicated at what hour New Moon or Full Moon occurs, after you have once, in accordance with the Ephemerides, adjusted the hour-hand and moon-hand; it will not be enough to have divided the drum into 59 teeth, each of which is advanced by 12 hours; for the synodic lunar month comprises 29 days, 12 hours, 44 minutes, that is, almost three quarter-hours; and thus after two months the moon-indicator would show New Moon an hour and a half earlier than ought to be, and after a year the indicator would anticipate the true New Moon by nine hours. Therefore to the shaft indicating the hours there should not be coupled the shaft of the endless screw, whose drum would require a different number of teeth; but a special shaft should be established, whose revolution should be so regulated that it is completed in eleven hours and a quarter; and the drum, from whose revolution the moon-indicator is driven, should be divided into 63 teeth; for one revolution of this drum would correspond to 63 revolutions of the shaft, each of whose revolutions would be completed in 11 1/4 hours: wherefore the moon-indicator would complete its revolution in 708 3/4 hours, that is, in 29 days, 12 hours, 45 minutes. It would therefore be slower in each lunation by no more than one minute; but at last, after twelve lunations are completed, the discrepancy would be slight. If you have divided the inner face of the wheel indicating the hours into 16 parts, and raised vertically small teeth that turn the carriage, nevertheless in such a way that the carriage receives only fifteen teeth in one revolution, then indeed one revolution of the carriage will be completed

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Liber octavus. CAPUT V. 799 perficietur horis 11 1/4, hoc est 15/16 horarum duodecim, seu horarum quadrantibus 45; qui per 63 multiplicati dant horæ quadrantes 2835, quot una lunatio complectitur. PROPOSITIO V. Pancratium ad onera Vecte attollenda opportunum construere. Sæpè contingit Vecte secundi generis attollendum esse ali- quod onus, cui impar sit potentia: idcirco præstò esse potest instrumentum (cui Pancratio nomen fieri posse ostendit vis satis magna) plures in alios usus accommodatum, quod & facillimè quocumque in loco collocari valet, & quocumque transferri. Cochlea infinita cum suo tympano dentato congruente paretur: tympani axis sit excavatus in tres aut quatuor strias convenientes dentibus laminæ rectæ chalybeæ dentatæ satis solidæ, cujusmodi illa est, quam lib. 5. cap. 6. exhibui. Nam si hæc includantur capsulæ paulò longiori, quàm sit lamina illa dentata, & cochleæ axis extra loculamentum promineat, ut ei aptari possit manubrium; ex Cochleæ conversione volvitur tympanum, & unà cum illo ejusdem axis striatus, qui dentes laminæ chalybeæ subiens illam elevat. Et quoniam hujus laminæ caput sinuatum subjicitur vecti, etiam vectis attollitur, & cum eo pondus. Quanta sit cochleæ infinitæ cum suo tympano & axe vis ad elevandam laminam, constat ex dictis: Componenda est autem hæc Ratio cum Ratione Vectis, ut habeatur momentum Potentiæ manubrio applicatæ comparatæ cum onere. FINIS.

Transcription: Translated (English)

The eighth book. CHAPTER V. 799 will be completed in 11 1/4 hours, that is, 15/16 of twelve hours, or 45 quarter-hours; which, multiplied by 63, give 2835 quarter-hours, as many as one lunation contains. PROPOSITION V. To construct a Pancratium suitable for raising loads by a lever. It often happens that with a lever of the second kind some load must be raised for which the power is unequal: therefore there may be at hand an instrument (which a force great enough shows may be called a Pancratium), suited to several other uses, and which can most easily be set up in any place, and transported anywhere. Let an endless screw be provided with its matching toothed drum: let the axis of the drum be hollowed into three or four grooves fitting the teeth of a sufficiently solid straight steel toothed plate, of the sort shown in book 5, chapter 6. For if this is enclosed in a case a little longer than that toothed plate, and the axis of the screw projects beyond the housing, so that a handle may be fitted to it; then by turning the screw the drum is rotated, and together with it the grooved axis, which, passing under the teeth of the steel plate, raises it. And since the hooked head of this plate is placed under the lever, the lever is also raised, and with it the weight. How great the force of the endless screw with its drum and axis is for lifting the plate, is clear from what has been said: but this ratio must be combined with the ratio of the lever, so that the moment of the power applied to the handle compared with the load may be obtained. END.