Bernardino Baldi's Exercitations on the Mechanical Problems of Aristotle, with an appended brief account of the author's life and writings

Creator: Bernardino Baldi | Date: 1621 | Notes: Original title: In mechanica Aristotelis problemata exercitationes, adiecta succincta narratione de autoris vita & scriptis A Neo-Latin critical commentary on the pseudo-Aristotelian Mechanical Problems, printed posthumously in quarto at Mainz. Composed as an exercitatio, it restates each of some thirty-five Aristotelian quaestiones in turn and re-solves them through original geometric demonstrations grounded in the Archimedean and Guidobaldan theory of centers of gravity. Baldi opens by defining mechanics as a facultas uniting natural matter to geometric demonstration, treating it as a mixed mathematical science (scientia media). He reduces the five simple machines to the lever and balance, refuses the traditional reduction of the wedge to the lever, redefines the screw as a wedge wrapped around a cylinder, and corrects Aristotle's account of vortices as concentric circles in favor of spirals. The work traverses the lever and balance, sailing, motion of round bodies, the sling, fracture of materials, percussion, composite motions, and the motion of bodies in swirling water, concluding with an appendix on finding two mean proportionals. An appended succincta narratio recounts the author's life and writings. 👉 <a href="https://tryleo.ai/collections/exlatinis/the-wedge-wound-round-a-cylinder-bernardino-baldis-line-by-line-audit-of-aristotles-mechanics">Read our introductory primer, full report, and finding guide here</a> 📜 <a href="https://archive.org/details/jbc.bj.uj.edu.pl.NDIGSTDR048711">View the original file on Internet Archive</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: Bernardino Baldi's Exercitations on the Mechanical Problems of Aristotle, with an appended brief account of the author's life and writings
Creator: Bernardino Baldi
Date: 1621
Notes: Original title: In mechanica Aristotelis problemata exercitationes, adiecta succincta narratione de autoris vita & scriptis A Neo-Latin critical commentary on the pseudo-Aristotelian Mechanical Problems, printed posthumously in quarto at Mainz. Composed as an exercitatio, it restates each of some thirty-five Aristotelian quaestiones in turn and re-solves them through original geometric demonstrations grounded in the Archimedean and Guidobaldan theory of centers of gravity. Baldi opens by defining mechanics as a facultas uniting natural matter to geometric demonstration, treating it as a mixed mathematical science (scientia media). He reduces the five simple machines to the lever and balance, refuses the traditional reduction of the wedge to the lever, redefines the screw as a wedge wrapped around a cylinder, and corrects Aristotle's account of vortices as concentric circles in favor of spirals. The work traverses the lever and balance, sailing, motion of round bodies, the sling, fracture of materials, percussion, composite motions, and the motion of bodies in swirling water, concluding with an appendix on finding two mean proportionals. An appended succincta narratio recounts the author's life and writings. 👉 <a href="https://tryleo.ai/collections/exlatinis/the-wedge-wound-round-a-cylinder-bernardino-baldis-line-by-line-audit-of-aristotles-mechanics">Read our introductory primer, full report, and finding guide here</a> 📜 <a href="https://archive.org/details/jbc.bj.uj.edu.pl.NDIGSTDR048711">View the original file on Internet Archive</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: In mechanica Aristotelis problemata exercitationes, adiecta succincta narratione de autoris vita & scriptis A Neo-Latin critical commentary on the pseudo-Aristotelian Mechanical Problems, printed posthumously in quarto at Mainz. Composed as an exercitatio, it restates each of some thirty-five Aristotelian quaestiones in turn and re-solves them through original geometric demonstrations grounded in the Archimedean and Guidobaldan theory of centers of gravity. Baldi opens by defining mechanics as a facultas uniting natural matter to geometric demonstration, treating it as a mixed mathematical science (scientia media). He reduces the five simple machines to the lever and balance, refuses the traditional reduction of the wedge to the lever, redefines the screw as a wedge wrapped around a cylinder, and corrects Aristotle's account of vortices as concentric circles in favor of spirals. The work traverses the lever and balance, sailing, motion of round bodies, the sling, fracture of materials, percussion, composite motions, and the motion of bodies in swirling water, concluding with an appendix on finding two mean proportionals. An appended succincta narratio recounts the author's life and writings. 👉 Read our introductory primer, full report, and finding guide here 📜 View the original file on Internet Archive 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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QV 1

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QV 1

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1862

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1862

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Baldus in Mechanica Anstotelis Problemata

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Baldus in Aristotle's Mechanical Problems

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BERNARDINI BALDI VRBINATIS GVASTALLÆ AB- BATIS IN MECHANICA ARISTOTE- LIS PROBLEMATA EXERCITATIONES: ADIECTA SUCCINCTA NAR- ratione de autoris vita & scriptis. Bibliotheca Collegii Maji- ni Univ[er]siti Bawes. MOGNTIÆ, Typis & Sumptibus Viduæ Ioannis Albini. M. DC. XXI.

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BERNARDINI BALDI OF URBINO ABBOT OF GUASTALLA ON ARISTOTLE'S MECHANICAL PROBLEMS EXERCISES: WITH A BRIEF NAR- rative of the author's life & writings. Library of the Great College of the University of Bawes. MAINZ, Printed & at the expense of the Widow of Ioannes Albinus. 1621.

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593937 11 Mag. 8.2.

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593937 11 Mag. 8.2.

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NOBILISSIMO AC GENEROSO DOMINO D. ADAMO PHILIPPO BARONI A CRONBERG, EQVITI, SACRÆ CAESAREÆ MAIESTATIS, ET SERENISSIMI Principis Archiducis Alberti Camerario intimo &c. Domino meo gratiosissimo. Pportune sub hoc ipsum tempus, quo in Belgium ad Serenissimos Principes iter adornat. Nobilissima & Generosa Dom. V[est]ra, prodit nostris formis in publicum editus Commentarius Bernardini Baldi Vrbinatis Guastallæ Abbatis in Aristotelis Mechanica. Is vir in omni scientiæ genere, at maxime in Mathematicis disciplinis fuit versatissimus, quod multa ab eo præclare scripta testantur opera, ex quibus paucula edita, reliqua vero (peramus ): (2

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NOBLEST AND MOST GENTLE LORD D. ADAM PHILIP BARON OF CRONBERG, KNIGHT, OF HIS SACRED CAESAREAN MAJESTY, AND OF THE MOST SERENE PRINCE ARCHDUKE ALBERT, PRIVATE CHAMBERLAIN, &c. My most gracious lord. Opportunely, at this very time, when he is preparing his journey into Belgium to the Most Serene Princes. Your Nobility and Gentleness has brought into the public through our press a published Commentary of Bernardini Baldi of Urbino, Abbot of Guastalla, on Aristotle’s Mechanics. This man was most skilled in every kind of learning, but especially in the mathematical disciplines, as the many works brilliantly written by him testify, of which only a few have been published, while the rest indeed (we hope ): (2

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EPISTOLA mus suo tempore in publicam lucem producenda. Cum vero nemini sit obscurum Nobilissimæ ac Generosæ Dom. V. ra id semper extitisse familiarissimum, vt tum domesticum otium, tum maxime peregrinationes, quibus totam pæne Europam summa cum laude circumscripsit, tum variarum linguarum perfecto vsu, tum Mathematicarum disciplinarum notitia & exercitio redderet iucudiores, nulla me tenet dubitatio quin & Baldum Vrbinatem nostris typis loquentem in hoc itinere, quod à Deo felicissimum Nobilissimæ ac Generosæ Dom. V. ra precor, in suum comitatum ac tutelam beneuolo animo sit admisura. Id rogo humillime simulque precor, vt hanc meam typographiam plurimis iam retro annis de inclytæ familiæ Cronbergicæ tutela gloriantem, suo fauore prosequatur, viduæque afflictæ fortunis beneuole adspiret. Sic Deus Nobiliss. & Generosam Dom. V. ram illustret omnibus bonis, eamque R. mo & Ill. mo Principi ac Domino meo Clementissimo, D. Ioanni Suicardo Archiepiscopo Moguntino Principi Electori ac per Germaniam Ar- chican-

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EPISTLE to be brought to public light in its own time. Since, however, it is not obscure to anyone that it has always been most familiar to the Most Noble and Generous Lady V. ra, that she might make both domestic leisure and especially travels, by which she has covered almost all of Europe with the highest praise, and likewise the perfect use of various languages, and knowledge and exercise of the mathematical disciplines, the more delightful, I have no doubt that she will also receive Baldus of Urbino speaking in our types on this journey, which I pray to God may be most prosperous for the Most Noble and Generous Lady V. ra, and will in a benevolent spirit admit him into her company and protection. I humbly ask and at the same time beseech that she will favor this my printing press, which for many years past has boasted of the patronage of the illustrious Cronberg family, and will kindly lend aid to the afflicted widow in her fortunes. Thus may God enlighten the Most Noble and Generous Lady V. ra with all blessings, and may He keep her for the Most Reverend and Most Illustrious Prince and my most gracious Lord, D. Johann Schweikard, Archbishop of Mainz, Prince-Elector and throughout Germany Archican-

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DEDICATORIA. chicancellario &c. patruo suo optatissimo saluo florentique redhibeat saluum simili- ter florentem ac incolumem. Moguntiæ è typographeio Viduæ Albinianæ, honori No- bilissimæ ac Generosæ Dom. Vestræ perpe- tuum dicato. Anno 1621.26. Martij. ::(3 PRÆ-

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DEDICATION. the chancellor, etc., returns to his most beloved uncle, safe and flourishing, likewise flourishing and unharmed. At Mainz, from the printing house of the Widow Albinian, for the everlasting honor of Your Most Noble and Generous Lordship, dedicated. In the year 1621, 26 March. ::(3 PRÆ-

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PRAEFATIO. Diligenter legenti mihi quæstiones illas, in quibus ea quæ ad Mechanicam facultatem pertinent, explicantur, multa in mentem veniebant; & primum quidem eorum, quæ ibi disputantur, vtilitatem, subtilitatem, copiam admirabar: Tum ex animo dolebam, aureum hunc libellum propè negligi, & ab iis qui pulcherrimis hisce studiis dant operam, assiduè præ manibus non haberi: Multas autem Auctori ipsi habendas referendasq[ue] esse gratias, qui tam egregiam, vtilem & probè instructam supellectilem Architectis, Mechanicis, & omnibus ferè Artificibus suppeditauerit. Aristotelis nomini ascribitur Commentarius, licet nonnulli, sitne Philosophi illius præclarissimi & acutissimi labor, an non, adfirmare subdubitauerint. Aristotelis tamen esse omnes ferè meliores consentiunt: Idque tum exphrasi, & explicatione, quæ Aristotelem sapiunt, tum iudicio subtilitatis & rationum, qui- bus

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PREFACE. As I was carefully reading those questions, in which those matters are explained which pertain to the mechanical art, many things came to mind; and first I marveled at the usefulness, subtlety, and abundance of what is discussed there: then I was sincerely grieved that this golden little book was almost neglected, and not kept constantly in hand by those who devote themselves to these most excellent studies. I must also give and return many thanks to the Author himself, who has supplied architects, mechanics, and almost all craftsmen with so excellent, useful, and well-equipped an apparatus. A commentary is attributed to the name of Aristotle, although some have hesitated to affirm whether it is the work of that most distinguished and most acute philosopher or not. Yet nearly all the better scholars agree that it is Aristotle’s: and this both from the style and explanation, which savor of Aristotle, and from the judgment of the subtlety and reasoning by which

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PRAEFATIO. bus quæstiones ipsæ ingeniosissimè diluuntur. Vi- detur autem mihi, rem accuratius exploranti, sa- tis verisimile (nullum enim habeo opinionis hu- ius assertorem) sectionem esse hanc, & partem quandam eius operis nobilissimi, quod idem au- ctor De Problematibus edidit, & hanc, nescio quam ob causam; nisi fortè quod tractatio merè Physica non sit, à reliquo corpore distractam at- que reuulsam. Id certè quod ad rem facit, probè nouimus, Diogenem Laërtium inter cætera Ari- stotelici ingenij monumenta Mechanica quoque adnumerasse. Quibus consideratis magnopere subit mirari, cur ij qui post Aristotelem floruêre atq[ue] vixere, Mechanici, Archimedes, Athenæus, Heron, Pappus, & cæteri, nullam huius libelli fe- cerint commemorationem: & sanè debuerunt; neq[ue] enim à vero est dissimile, ipsos per hunc ali- quatenus profecisse. Verum enimuero cum inge- nui illi fuerint homines, & nullatenus obtracta- tores, credendum potius est, Commentariolum i- stud, eorum æuo, paucis cognitum, alicubi in Bi- bliothecis latuisse: etenim cætera quoq[ue] Aristote- lis scripta, post vetusta illa tempora, ante Ale- xandrum Aphrodisensem, à multis fuisse igno- rata

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PREFACE. but the questions themselves are most ingeniously resolved. It seems to me, however, on closer examination of the matter, quite likely (for I have no proponent of this opinion) that this section is a part of that most noble work which the same author published as De Problematibus , and that this one, for some reason or other—unless perhaps because the treatment is not purely physical—was detached and torn away from the rest of the body of the work. At any rate, what is relevant to the matter, we know well enough that Diogenes Laërtius, among the other monuments of Aristotle’s genius, also counted the Mechanica . Considering these things, one is greatly inclined to wonder why those who flourished and lived after Aristotle, the mechanicians, Archimedes, Athenaeus, Heron, Pappus, and the rest, made no mention of this little book: and surely they ought to have done so; for it is not unlike the truth that they themselves may in some measure have profited from it through this work. Yet indeed, since those men were honorable and in no way detractors, it should rather be believed that that little commentary, in their time known to only a few, lay hidden somewhere in libraries; for even the other writings of Aristotle, after those ancient times, before Alexander of Aphrodisias, were unknown to many.

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PRAEFATIO rata non dubitamus. Habemus siquidem, Strabone teste, lib.13. Aristotelis, et Theophrasti bibliothecam, post ipsius Theophrasti decessum, ad Neleum quendam Scepsium, Coriscifilium, qui eius fuerat auditor, peruenisse; post hæc libros, blattis olim, et humore corruptos, Apeliconi Teio venditos, et ab eo Athenas translatos, tum Athenis captis in Syllæ potestatem deuenisse, eosque tandem à Sylla acceptos, Tyrannionem Grammaticum, vt potuit meliùs emendatos, promulgasse. Ex quibus colligimus, mirum non esse, Archimedi, Heroni, et alijs qui ante Syllam vixêre, fuisse incognitos. quicquid sit, illud certum est, Aristotelem eorum omnium qui de Mechanicis commentaria edidere, esse longè vetustissimum. Pappus enim Heroneiunior, Athenæus Archimedi æqualis, vterq[ue] enim sub Marcello, cui Athenæus suum de bellicis Machinis libellu[m] dedicauit. Archimedes verò circa CXL. Olympiadem floruit, quamobrem post Aristotelem Olympiadas XL. hoc est, annos ferè CLX. Isthæc autem considerantibus, facile est cognoscere facultatis huius nobilitatem, atq[ue] dignitatem; quippe quod summus Philosophus non modo eam pro-

Transcription: Translated (English)

PREFACE we do not doubt to be true. For we have, as Strabo testifies, in book 13, that the library of Aristotle and Theophrastus, after the death of Theophrastus himself, came to a certain Neleus of Scepsis, son of Coriscus, who had been his pupil; after this, the books, once corrupted by moths and damp, were sold to Apellicon of Teos and brought by him to Athens, then, when Athens had been captured, they fell into the power of Sulla, and at last, having been received from Sulla, Tyrannion the Grammarian published them, having corrected them as well as he could. From these things we gather that it is no wonder that Archimedes, Hero, and others who lived before Sulla were unknown to him. Be that as it may, this is certain: Aristotle is by far the most ancient of all those who have written commentaries on Mechanics. For Pappus is later than Hero, Athenaeus contemporary with Archimedes; both, indeed, lived under Marcellus, to whom Athenaeus dedicated his little book on war machines. But Archimedes flourished around the 140th Olympiad, wherefore he is forty Olympiads, that is, about 160 years, later than Aristotle. Considering these matters, it is easy to recognize the nobility and dignity of this art; for the supreme Philosopher not only did not pro-

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AUTHORIS. probauerit, sed etiam suis acutissimis lucubrationibus illustrauerit. Hanc porro tractationem subiecto quidem Physicam esse, demonstrationibus verò Geometricam, ipsemet nos docuit Aristoteles, cuius etiam naturæ sunt Perspectiva, Specularia, Musica, & cæteræ eiusdem modifacultates, quas quidem subalternas Peripatetici appellant. Vitruuius Architecturæ membrum, vt ita dicam, & portionem quandam facit, ait enim Architecturæ partes esse tres, Ædificationem, Gnomonicam, Machinationem. Est autem Architecturâ quidem inferior, paret enim Architecto Mechanicus; attamen si cæteras artes spectes, Architectonica; hæc enim omnes ferè sedentariæ, sellulariæque, quas banansas Græci appellant, ordine subijciuntur, & sanè latissimos isthæc habet fines; præcipuè autem circa eam versatur cognitionem, eamque inter cæteras ferè principem, quam dixere Centrobaricam, quæ quidem ad Centri grauitatem, eiusque speculationem pertinet: qua in specie inter veteres primum sibi vindicauit locum Archimedes, mox Heron, deinde Pappus; inter neotericos autem (:(:(::)

Transcription: Translated (English)

AUTHOR'S. not only has he proved it, but also illuminated it with his most acute studies. This treatment, furthermore, Aristotle himself taught us is indeed Physical in subject, but Geometrical in demonstrations; and such also are Perspective, Optics, Music, and the other similar subsidiary sciences, which the Peripatetics indeed call subordinate. Vitruvius makes it, so to speak, a member of Architecture and a certain part of it, for he says that the parts of Architecture are three: Building, Gnomonics, and Mechanics. It is, however, inferior to Architecture, for the Mechanic is subject to the Architect; nevertheless, if you consider the other arts, it is architectonic. For to this all the almost entirely sedentary, chair-bound arts, which the Greeks call banansai, are subordinated in order; and indeed it has the widest bounds. But especially it is occupied with that knowledge, and among the rest is almost the principal one, which they called Centrobaric, which indeed concerns the weight of the center and its speculation. In this field, among the ancients, Archimedes first claimed the place for himself, then Hero, then Pappus; among the more recent however (:(:(::)

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P RÆFATIO tem Commandinus, qui librum de Centro gra- uitatis solidorum scripsit, & post eum G. Vbal- dus è Marchion. Montis, qui non modò ab- solutissimum Mechanicorum librum cum maxi- ma ingenij sui laude conscripsit, sed & Paraphra- sin in librum Æqueponderantium Archimedis egregiè concinnauit Centrobaricam hanc, igno- tam fuisse Aristoteli, satis patet. nunquam enim in Mechanicis demonstrationibus, quod tamen est potissimum, grauitatis centrum nominat, e- iusuenaturam atque vim speculatur. Diuidi- tur autem Mechanice tota, teste Herone apud Pappum libro octauo, in Rationalem, hoc est, Theoricam & Chirurgicam, id est, manu ope- ratricem, quam Praxim aptè dicere valemus. Rationalis, speculationi & demo[n]strationibus, ex Geometricis, Arithmeticis & Physicis rationi- bus, dat operam; Chirurgica vero materiam tractat, & sese in varias artes diffundit, Æra- riam, Lignariam, Sculptoriam, Pictoriam, Æ- dificatoriam, Machinariam & Thaumaturgi- cam, cæterasque eiusmodi. Machinatoriæ au- tem sunt partes Manganaria, qua ingentia trans-

Transcription: Translated (English)

PREFACE Commandinus, who wrote a book on the center of gravity of solids, and after him G. Ubaldus of Marchion. of Montis, who not only composed the most perfect book on Mechanics with the greatest praise of his intellect, but also excellently arranged a Paraphrase on Archimedes’ book On Equal Weights, make it quite clear that this Centrobarica was unknown to Aristotle. For in mechanical demonstrations he never names the center of gravity, which however is the chief matter, but investigates its nature and power. Mechanics as a whole, according to Hero as cited by Pappus in the eighth book, is divided into the Rational, that is, the Theoretical, and the Chirurgical, that is, the manual-working part, which we may suitably call Praxis. The Rational part gives itself to speculation and demonstrations, from geometrical, arithmetical, and physical reasonings; but the Chirurgical part deals with matter and spreads itself through various arts: the art of the treasury, carpentry, sculpture, painting, architecture, machinery, and thaumaturgy, and the rest of this kind. The parts of machinery, however, are the Manganarian, by which immense trans-

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AUTHORIS. transferuntur pondera, tum ipsa Poliorcetica, quæ bellicas Machinas ad vrbium expugnationes, quod vel ipso nomine profitetur, ædificat. Atqui hac de re plura scribere supersedemus, ne actum agamus: quisquis enim minutè magis hæc cognoscere desiderat, is Pappum adeat libro citato, & Guidum Vbaldum in Præfatione quam suo Mechanicorum Operi præposuit. Vt autem ad Aristotelis, de quo egimus, libellum reuertamur, pauci sunt qui ei ante nos stilum & operam commodauerint: Leonicenus Latinum fecit & figuris tum breuissimis, & paruisane ponderis, marginalibus adnotatiunculis, instruxit. Post hunc Alexander Picolomineus luculentissima Paraphrasi illustrauit. Modo, vt audio, Simon Sticinus Hollandensis quædam edidit, quæ ad nos minime peruenêre. Nos demum, omnium, tum scientia, & ingenio, tum ætate, postremi huic operimanum admouimus; Considerantes enim Aristotelem alijs principijs vsum, ac probatissimi post eum fecerint Mechanici, demonstrasse, morem huiusce facultatis studiosis gesturos nos fore arbitratis sumus, si easdem illas quæstiones ):(:(: 2 Me-

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AUTHORIS. weights are transferred, then the Poliorcetica itself, which constructs military machines for the capture of cities, as it declares even by its very name. But on this subject we refrain from writing further, lest we do the same thing again; for whoever more minutely desires to know these matters, let him go to Pappus in the cited book, and Guido Ubaldo in the Preface which he placed before his Mechanical Works. But then let us return to Aristotle’s little book, of which we have spoken. Few are there who before us have applied pen and labor to it: Leonicenus rendered it into Latin and adorned it with figures, both very brief and of slight weight, and with marginal notes. After him Alexander Piccolomini illuminated it with a most splendid Paraphrase. Recently, as I hear, Simon Sticcinus of Holland has published certain things, which have by no means reached us. We finally, of all men, both in knowledge and genius, and also in age, the last, set our hand to this work; for, considering that Aristotle used other principles, and that the most approved Mechanicians after him have shown, we judged that we should be doing a service to the students of this discipline if we should treat those same questions ):(: 2 Me-

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Mechanicis, hoc est, Archimedeis probationibus confirmaremus; dum per latissimos facultatis huius campos vagantes, alias quoque istis affines dubitationes introducentes solueremus. quicquid aut e fecerimus profecerimusue, Lector optime, boni consule, & quia fax per manus traditur, tu interim de me accipe, vt alijs tradas. DE

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We would confirm by mechanical, that is, Archimedean proofs; while wandering through the broadest fields of this discipline, and introducing other doubts akin to these, we would resolve them. Whatever we have done or accomplished, most excellent Reader, take it in good part; and because the torch is handed from hand to hand, you in the meantime receive it from me, that you may pass it on to others. DE

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DE VITA ET SCRIPTIS BERNARDINI BALDI VRBINATIS EX LITERIS FABRITII SCHAR- loncini ad Illustrissimum & Reuerendissimum Dominum Lælium Ruinum Episcopum Bal- neoregiensem ex-Nuntium Apostolicum ad Poloniæ Regem &c. Atus est Bern. Baldus Vrbini nobilibus pa- retibus postridie Non. Iunij anno MdlIII. Genus traxit, quod me sæpè ab eomemini audire, à familia Cantagallina, quæ inter Perusinas illustris: hoc autem cognomen, Baldi accepto, vt in varietate temporum fit, Abauus reliquit, à teneris vnguiculis pietate erga Deum præsetulit; nam vt mater eius narrabat, sanctorum imagi- nes & Altariola non cum lætitia solum, sed cum venera- tione anniculus intuebatur. Præceptoribus in adolescen- tia vsus fuit laudatissimis Io. And. Palatio, & Io. Antonio Turoneo, qui altero doctior, & Paulo Manutio maxime carus ob latinæ & græcæ linguæ peritiam propè singula- rem: ad illorum autem sedulitatem tantum animi ardo- rem attulit, tantam ingenij ac iudicij vim, vt non tantum æqualis sed omnium vicerit expectationem. Puer adhuc Arati apparitiones Italico carmine reddidit. Parenshac filij laude & gloria motus anno 1573. eum ad maiorem in- genij cultum capessendum Patauum misit. Hîc in Ema- nuelis Margunij familiaritatem statim venit, cui porro fuit

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On the life and writings of Bernardino Baldi of Urbino from the letters of Fabrizio Scharloncini to the Most Illustrious and Most Reverend Lord Laelio Ruino, Bishop of Balneoregium, former Apostolic Nuncio to the King of Poland, etc. Bern. Baldi of Urbino was born of noble parents on the day after the Nones of June in the year 1553. He traced his lineage, as I have often heard from him himself, from the Cantagallina family, which is among the illustrious families of Perugia; but this surname, taking the name Baldi, as happens amid the changes of time, his great-grandfather left behind. From his earliest years he showed piety toward God; for, as his mother used to tell, he gazed upon the images of the saints and little altars not only with delight, but with veneration, even as a one-year-old child. In his youth he had as teachers the most highly praised Io. And. Palatius and Io. Antonio Turoneus, the latter more learned, and very dear to Paolo Manuzio because of an almost singular knowledge of the Latin and Greek tongues. To their diligence he brought such ardor of mind, such force of talent and judgment, that he surpassed not only equal but all expectation. While still a boy he rendered Aratus in Italian verse. His father, moved by this praise and glory of his son, sent him in the year 1573 to Padua to pursue a higher cultivation of his talent. There he immediately came into the intimacy of Emanuele Margunio, to whom moreover he was

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V I T A fuit in amoribus. Homeri Iliad. illo Doctore & interpre- te diligentius quam fecisset antea, euoluit. priuato autem studio Anacreonti, Pindaro, Æschyli, Euripidi, Sophocli operam dedit, sed præ cæteris Theocriti Bucolica triuit, ad quod scriptionis genus natura magis ferri videbatur: centenos græci alicuius poëtæ versus memoriter tenebat, sæpeque habebat in ore, in oratoribus græcis versandis laborem se aliquem sentire, in poëtis nullum. Scripsit Pa- tauij libellum de Tormentis Bellicis, & eorum inuentori- bus, & cum in Transalpinorum amicitias incidisset, sibi ducebat dedecori ipsos sua lingua loquentes non intelli- gere. quare incredibili celeritate Gallicam & Germani- cam didicit. Pestilentia ex eo Gymnasio exactus in Pa- triam redijt, vbi quinquennium integrum Federico Co- mandino affixus omnes Matheseos partes perdidicit, cui viro in delineandis figuris ad Euclidis, Pappi, & Heronis monumenta manum commodauit: ex eiusdem obitu do- lorem vix consolabilem sustinuit, susceptoque eius vitam scribendi consilio, subinde ad omnium Mathematicorum vitas conscribendas animum adplicuit, quod & duode- cim annorum spatio præstitit felicissimè. cum vero Ma- thematicarum disciplinarum amore torqueretur, amisso Commandino Præceptore, amicum nactus fuit præstan- tissimum & symmystam Guidum V baldum è Marchioni- bus Montis, in cuius se consuetudinem daret: quantum profecisset, ostendunt ij commentarij quos anno 1582. in Arist. Mechanica scripsit. Vt postea à grauioribus studijs ad amoeniora animum abduceret, de re nautica poëma I- talicè confecit. quo absoluto Paradoxa multa Mathema- tica explicauit. Fama de Baldi virtutibus dissipata Ferran- dus Gonzaga Molfettæ Princeps & Guastallæ Dominus coepit de illo in suam familiam asciscendo cogitare, vt qui ijsdem caperetur artibus, quibus excellere Baldus inci- piebat:

Transcription: Translated (English)

He was involved in love affairs. He read Homer’s Iliad with that teacher and interpreter more diligently than he had done before. He devoted himself in private to Anacreon, Pindar, Aeschylus, Euripides, and Sophocles, but above all he worked through Theocritus’s Bucolica, to which kind of writing his nature seemed more inclined. He could hold in memory, from any Greek poet, hundreds of verses, and often had them on his lips; in working through Greek orators he felt some labor, but in poets none at all. In Padua he wrote a little book on siege engines and their inventors, and when he had fallen in with the Transalpine peoples’ friendship, he considered it a disgrace not to understand those speaking in their own language. For that reason he learned French and German with incredible speed. Driven from that gymnasium by plague, he returned to his native country, where for a full five years he stayed attached to Federico Comandino and learned all parts of mathematics over again; to that man, in drawing figures according to the monuments of Euclid, Pappus, and Hero, he lent his hand. On Comandino’s death he endured a grief scarcely bearable, and having taken up the plan of writing his life, he then applied himself to writing the lives of all mathematicians as well, which he carried out most successfully over a span of twelve years. But when he was tormented by love of mathematical disciplines and, his teacher Comandino having been lost, had found the most excellent friend and fellow initiate Guido Baldo of the Marchesi of Monte, he gave himself into his company. How much he profited is shown by the commentaries he wrote in 1582 on Aristotle’s Mechanics. Later, in order to turn his mind from the more serious studies to more pleasant ones, he composed an Italian poem on navigation. When that was finished, he explained many mathematical paradoxes. Fame having spread concerning Baldo’s virtues, Ferrando Gonzaga, Prince of Molfetta and Lord of Guastalla, began to think about taking him into his household, since he was captivated by the same arts in which Baldo was beginning to excel:

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AUTHORIS. piebat: Itaque opera Curtij Arditi honorifice fuit in aulam euocatus, dum vitam non aulicam viueret totus in litteras abditus precibus Vespasiani Gonzagæ Sablonetæ Ducis ad explanandos Vitruuij libros adactus fuit. quare tûc natus de Verboru[m] Vitruuianorum significatione commentarius; in quo minime mirandum si minuta quædam prosequutus fuit, quæ viro magno minus esse digna videantur: illi enim Principi morem gessit. scio dixisse aliquando Adrianum Romanum è Polonia reuersum, vbi Vitruuium Palatino cuidam explicauerat, si commentarium Baldi in Polonia adhibere potuissem, aurum quod mecum attuli emunxissem, quia satisfecissem muneri labore nullo. Cum Ferrando hero suo obuenisset necessitas Hispanias adeundi, illud iter sine Baldo facere se posse non putabat, non tam, vt haberet, qui erudito eloquio viæ tædium leuaret, quam cui posset arcana committere, atque adeo à quo iuuaretur consilio. Vix viæ se dederant cum Baldus grauem in morbum delapsus itinere cogitur desistere: Mediolanum proinde diuertit, vbi à S. Carolo Borromæo & benignè exceptus, & tamdiu detentus donec valetudinem recuperaret. Guastallam postea se recepit, vbi cum absente Domino liberiori otio frueretur, libros sex de Aula eruditissimos methodo analytica conscripsit. alios non commemoro, quod cum otium erit, omnium syllabum dabo. Anno 1586. ipso nihil postulante eligitur Guastallæ Abbas, à quo tempore Iuri Can. Concilijs, & SS. Patribus totum se dedit. Hebreæ & Chaldææ linguarum discendarum triennium posuit. Anno 1593, nouæ Gnomonices libros quinque composuit. insequenti Chaldæam Onkeli paraphrasin in Pentateuchum vertit & commentarios adiunxit; quo exantlato labore in lob ex Heb. fonte paraphrasin texuit, quam & scholijs illustrauit. Tabulam Etruscam Eugubinam interpretatus fuit:

Transcription: Translated (English)

AUTHOR. He was brought to court with honour by the work of Curtius Arditi; but since he did not live a courtly life, remaining wholly devoted to letters, he was, at the request of Vespasiano Gonzaga, Duke of Sabbioneta, compelled to explain Vitruvius’s books. Hence arose that commentary on the meanings of the Vitruvian words; in which it is by no means surprising if he followed certain minute points that might seem less worthy of a great man: for he was complying with that Prince’s wishes. I know that Adrian of Rome once said, after returning from Poland, where he had explained Vitruvius to a certain Palatino, that if he could have used Baldus’s commentary in Poland, he would have stripped the gold I had brought with me, because I should have fulfilled the task with no labor. When the need arose for his lord Ferrando to go to Spain, he did not think he could make that journey without Baldus, not so much so that he might have someone whose learned eloquence would relieve the tedium of the road, as someone to whom he could entrust secrets, and indeed from whom he might receive counsel. Hardly had they set out when Baldus fell into a serious illness and was forced to stop traveling; he therefore turned aside to Milan, where he was both kindly received by Saint Charles Borromeo and retained until he recovered his health. Later he withdrew to Guastalla, where, while the Lord was absent and he enjoyed freer leisure, he composed, in analytical method, six most learned books on the Court. I do not mention the others, because when there is leisure I shall give a list of them all. In the year 1586, without asking for anything, he was elected Abbot of Guastalla, from which time he devoted himself entirely to Canon Law, Councils, and the Holy Fathers. He spent three years learning Hebrew and Chaldee. In the year 1593 he composed five books on the new Gnomonics. The following year he translated Onkelos’s Chaldee paraphrase on the Pentateuch and added commentaries; and after completing that labor, he composed from the Hebrew source a paraphrase on Job, which he also illustrated with scholia. He interpreted the Etruscan Eugubine Tablet:

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VITA ET SCRIPTA fuit: in ea autem diuinatione, vt aiebat, subcisiuas vnius mensis horas consumpsit. De Firmamento & aquis egre- gie scripsit. Oeconomiam Tropologicam in S. Matthæum Card. Baronius, qui non alia Baldi vidit, vehementer pro- babat. Romæ dum viueret, fere nesciuit quid gereretur in Aulis: Arabicæ enim linguæ cum Io. Baptista Raimon- do diligentissime studuit, & arcana industria Slauonicæ, quam perfecte callebat. Ex Arabico vertit Hortum Geo- graphicum Anonymi, quem ante sexcentos annos flo- ruisse arbitrabatur. Hunc vero extrusisset, vt alios Baldi libros, Marcus Velserus Ilvir Aug. si eo paulo longior huius lucis vsura contigisset. Composuit & Dictionarium Arabicum. atque cum beatissimam illam vbertatem in- genij assidue diffundi necesse esset, anno 1603. orbem vni- uersum describere aggressus fuit; atque ita quidem, vt tam quæ ad Historiam, quam quæ ad Geographiam per- tinerent complecteretur: Neque illustrare solum voluit quæ nouerunt antiqui, quemadmodum visum Ortelio, sed vel oppidula omnia & pagos, de quibus aliqua in po- stremis scriptoribus mentio. & profecto totum opus ad vmbilicum perduxit: non digessit tamen vniuersum. qua- tuor aut ni fallor quinque tantum Tomi fuerunt ordine Alphabetico dispositi: superessent septem aut octo dispo- nendi, quantum ex chartarum & fasciculorum mole con- ijcere licet. Anno 1617. quarto Idus Octob. posteaquam dies 40. vehementi destillatione vexatus fuisset, spiritum Deo reddidit Sacramentis Ecclesiæ omnibus rite muni- tus. Statura procerus fuit, facie oblonga & acribus oculis, colore subfusco. Membrorum ei fuit decens habitudo, & compactum corpus. Diebus festis omnibus sacrum facie- bat, ieiunabat bis in hebdomada, eleemosynisque paupe- res subleuabat. Instudijs sic assiduus fuit, vt sæpe & legeret & comederet. S. Augustinilibros de Ciuitate Dei ter in- ter

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VITA ET SCRIPTA he had spent, as he said, the spare hours of one month in that divination. He wrote excellently on the Firmament and the waters. Cardinal Baronius, who had seen no other works by Baldi, strongly approved the Oeconomia Tropologica in S. Matthew. While living in Rome, he scarcely knew what was going on in the Courts: for he devoted himself most diligently with Giovanni Battista Raimondo to the Arabic language, and by secret industry to Slavonic, which he knew perfectly. From Arabic he translated the Hortus Geographicus of an Anonymous author, whom he believed had flourished six hundred years before. Marcus Velserus of Augsburg would have brought this work forth, as well as Baldi’s other books, if a somewhat longer enjoyment of this life had been granted him. He also composed an Arabic dictionary. And since that most blessed abundance of genius had necessarily to be continually diffused, in the year 1603 he undertook to describe the entire world; and indeed in such a way as to include both what belongs to History and what belongs to Geography. He wished not only to illuminate what the ancients had known, as Ortelius thought, but even all the small towns and villages of which there is some mention in later writers. And indeed he brought the whole work to the navel, but he did not arrange the whole of it. There were only four or, if I am not mistaken, five volumes arranged in alphabetical order; seven or eight more remained to be arranged, as may be conjectured from the mass of papers and bundles. In the year 1617, on the fourth day before the Ides of October, after he had been afflicted for 40 days by a severe catarrh, he returned his spirit to God, having been duly fortified with all the sacraments of the Church. He was of tall stature, with an oblong face and keen eyes, and a somewhat swarthy complexion. His limbs were well proportioned, and his body compact. On all feast days he attended Mass, fasted twice a week, and relieved the poor with alms. He was so assiduous in his studies that he often both read and ate. The books of St. Augustine on the City of God three times in

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AVTHORIS. ter prandium euoluit. Statim à noctis meridie dum ei vi- res firmiores essent ad lucubrandum surgebat. à prandio Euclidem Arabice editum, vel libellum aliquem germa- nicum aut gallicum in manus sumebat. Suauitate morum & modestia, etiam si cetera dotes abfuissent, quemlibet ad amorem sui allicere potuisset. Sermo modicus ei fuit, itemque cultus. Nullos vnquam honores petijt, qui à Clem.8. amplissimi promissi fuerant; nullum emolumentum quæ siuit suo censu contentus. facile parcendum esse dicebat, ijs maxime qui in re leui impregissent, quoniam si quos censemus optimos, nudos conspiceremus, nullum eorum non iudicaremus multis dignum verberibus. Bi- bliothecam habuit non locupletem, sed selectis instructa codicibus. Verum ire per singula longum esset. Satis mihi de incomparabili Baldi doctrina, & summa innocentia, ô rarum connubium, pauca dixisse, quæ forfitan ad imitandum nimis multa. SYLLABVS LIBRORVM omnium B.Abb.Baldi. A Rati apparitiones è gr. in Ital. vertit. De Tormentis Bellicis & eorum Inuentoribus lib. Heronis automata vertit. Vitas omnium Mathematicorum scripsit, & trib. in Tom. 2.1. Ps. à Thalete ad Christum. 2. à Christo ad sua tem- pora. Earumdem vitarum Epitomen Chronologicum confecit. In Aristot. Mechan. Commentar. De Renautica Poëmation. Paradoxorum Mathematicorum liber. Descriptio Palatij Ducum Vrbinarum quod est Vrbini. Poema cui titulus, Lamus. :)(:(:(::) Carmi-

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after dinner he would turn to reading. At once, from the middle of the night, while his strength was more firmly gathered for study, he would rise to keep vigil. After dinner he would take up Euclid published in Arabic, or some little German or French book into his hands. By the sweetness of his character and his modesty, even if his other gifts had been lacking, he could have drawn anyone to love him. His speech was measured, and so was his dress. He never sought any honors, though the most generous had been promised to him by Clem. 8.; nor did he seek any profit, being content with his own means. He used to say that one should readily spare, especially those who had slipped in some slight matter, because if we were to behold those whom we judge the best, stripped bare, we should judge none of them unworthy of many blows. He had a library not rich, but stocked with select codices. But to go through each item would be too long. It is enough for me to have said a few things about the incomparable learning of Baldi, and his highest innocence—O rare union—perhaps too many things for imitation. SYLLABUS OF THE BOOKS of all B. Abb. Baldi. He translated the Apparitions of Aratus from Greek into Italian. On War Machines and their Inventors, book. He translated Heron's automata. He wrote the Lives of all the Mathematicians, and divided them into three parts in one volume. 2. From Thales to Christ. 2. From Christ to his own times. He composed a Chronological Epitome of the same lives. Commentary on Aristotle's Mechanics. De Renautica, a little poem. Book of Mathematical Paradoxes. Description of the Ducal Palace of Urbino, which is at Urbino. A poem entitled Lamus. :)(:(:(::) Carmi-

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S C R I P T A Carmina pia, quæ inscribuntur, Anni Corona. De Verborum Vitruuanorum significatione. Carmina varia & eclogæ mixtæ. Apologi centum, quos scripsit æmulatus Leonem Bapt. Albertum. De Humanitate Dialogus qui inscribitur Goselinus. Comparatio Vitæ Monasticæ cum seculari. De Aula libri sex. De felicitate Principis Dialogus. De Dignitate Dial. Carmina Romana. Musæi fabulam vertit. De Italici carminis natura Dial. qui inscribitur Tassus. De vniuersali Diluui poëmation. Nouæ Gnomonices lib. quinque. Hieremiæ Threnos vertit, & ex Heb. fonte annotat. adiecit. Poemation inscriptum, Deiphobe, quod scripsit æmula- tus Lycophonem in Cassandra. Scala coelestis. 1. Sermones pij & carmina. Onkeli paraphrasin Chaldæam in Pentateuchum vertit & vberes commentarios adiecit. In Iob Paraphrasis latina ex fonte Heb. additis Scholijs. Descamillis imparibus Vitruuij. De firmamento & aquis. Quincti Calabri Paralipomena vertit. Tabulæ Etruscæ Eugubinæ Interpretatio. Oeconomia Tropologica in S. Matthæum. Vrbini encomium. Horti geographicci ex Arab. versio. Aduersus Aulam Carmina. Luciani de miserijs Aulicorum versio. Oratio ad Romæ conseruatores pro antiquitatum eius Vrbis custodia. Vni-

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W R I T I N G S Pious poems, entitled Anni Corona. On the meaning of Vitruvius’s words. Various poems and mixed eclogues. One hundred apologiae, which he wrote in emulation of Leon Batt. Alberto. Dialogue on Humanity, entitled Goselinus. Comparison of Monastic Life with Secular Life. Six books On the Court. Dialogue On the Happiness of the Prince. Dialogue On Dignity. Roman poems. He translated Musaeus’s fable. Dialogue On the Nature of Italian Poetry, entitled Tassus. Poem On the Universal Flood. Five books of New Gnomonics. He translated Jeremiah’s Lamentations, and added notes from the Hebrew source. Poem entitled Deiphobe, which he wrote in emulation of Lycophron in Cassandra. Scala coelestis. 1. Pious sermons and poems. He translated Onkelos’s Chaldean paraphrase on the Pentateuch and added abundant commentaries. Latin paraphrase on Job from the Hebrew source, with scholia added. On Vitruvius’s unequal staircases. On the firmament and the waters. He translated Quintus of Calabria’s Paralipomena. Interpretation of the Etruscan Eugubine Tables. Tropological economy on St. Matthew. Encomium of Urbino. Translation from Arabic of the Geographical Garden. Poems Against the Court. Translation of Lucian on the miseries of courtiers. Speech to the conservators of Rome for the preservation of the antiquities of that city. Uni-

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AUTHORIS. Vniuersi orbis geographica & Historica descriptio contexta ex septingentis & eo amplius scriptoribus. Federici Vrbini Ducis Vita. Guidi Vbaldi Vrbini Ducis Vita. Epigrammaton & Odarum libri tres. Aliorum Carminum liber. Sententiarum moralium liber. Dictionarium Arabicum. Pro Procopio contra Flauium Blondum. Horographium vniuersale. Epigrammata alia. Heronis lib. de Ballistis conuersio. Exercitationes in Aristotelis Mechan. Templi Ezechielis noua descriptio. Antiquitatum Guastallensium liber. Historiæ scribendæ leges. Et alia quædam. IN

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Works. Geographical and historical description of the whole world, compiled from more than seven hundred authors. Life of Federico, Duke of Urbino. Life of Guidi Vbaldi, Duke of Urbino. Three books of Epigrams and Odes. A book of other poems. A book of moral sayings. Arabic dictionary. In defense of Procopius against Flavius Blondus. Universal horograph. Other epigrams. Translation of Heron’s book On Ballistae. Exercises on Aristotle’s Mechanics. New description of the Temple of Ezekiel. Book on the antiquities of Guastalla. Laws for writing history. And certain other works. IN

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I IN MECHANICA ARISTOTE- LIS PROBLEMATA EXERCITATIONES. Mechanices descriptio, natura, finis. MECHANICE, facultas quædam est, quæ naturali materiâ, Geometricisq[ue] demon- strationibus visa, excentrobaricâ, & eoru[m] quæ ad vectem & libram rediguntur, spe- culatione; humanæ consulens necessitati, commoditati que, suapte vi, Naturam i- psam vel secundans, vel superans, varia, ea que mirabilia operatur. Hac diffinitione descriptioneue breuiter ea fe- re omnia complexi sumus, quæ fusissimè ab Aristotele, Pappo, Guido Vbaldo, & alijs hac de re tradita fuêre. Mechanices Obiectum. Considerat autem Mechanicus Graue & Leue. Graue duplex, Naturâ, Violentiâ. Graue Naturâ dicitur, quod insita propensione in centrum mundi fertur. Graue autem Violentiâ, quod im- presso extrinsecus pondere ab impellente pellitur. Leue contrà, quòd Naturâ à centro fertur. Cæterùm quicquid graue est, secundum punctum est, quod Grauitatis centrum dicitur, & hoc duplex, vt duplex est grauitas, Naturæ, Violentiæ. A Gra-

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I IN THE MECHANICAL PROBLEMS OF ARISTOTLE EXERCISES. Description, nature, end of mechanics. MECHANICS is a certain faculty, which, by means of natural matter and geometrical demonstrations, considered from the center of gravity and those things which are reduced to the lever and the balance, providing for human necessity and convenience, by its own power either following nature itself or surpassing it, performs various and marvelous things. By this brief definition and description we have comprised almost all those things which have been more fully treated by Aristotle, Pappus, Guido Ubaldo, and others on this matter. Object of mechanics. The mechanic considers Heavy and Light. Heavy is twofold: by Nature, by Force. Heavy by Nature is said of that which is borne by an innate tendency toward the center of the world. Heavy by Force is that which, when an external weight is applied, is driven by the one impelling it. Light, on the contrary, is that which is borne by Nature away from the center. Moreover, whatever is heavy, has a certain point, which is called the center of gravity, and this is twofold, as gravity is twofold: of Nature, of Force. Of Gra-

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2 IN MECHAN. ARIST. PROBL. Grauitatis centrum in triplici magnitudine considerari potest, lineari, planâ, solidâ. De centro grauitatis linearum nemo scripsit, simplicissimi enim illud est contemplationis. De centro grauitatis linearum planum egregiè tractauit Archimedes in libro Æqueponderantium, & de quadratura Parabole, tum in eo quem de his quæ vehuntur inscripsit. De centro grauitatis solidorum ipsemet olim scripserat Archimedes, sed ea quæ protulit, temporis iniuriâ deperdita, suâ diligentiâ restituit Iedericus Commandinus. Esse autem & Leuitatis centrum in rerum natura, palam est. Punctum enim illud est, secundum quod leuia rectà à centro sursum feruntur. Huius autem non meminêre Mechanici, propterea quod aut nihil, aut parum ad eorum rem faciat. Porro Grauitatis centrum ita definit Heron, & qui ab Herone Pappus 1. 8. Collectionum Mathematicarum. Centrum grauitatis vniuscuiusq; corporis est punctum quoddam intra positum, à quo si graue, mente appensum concipiatur, dum fertur, quiescit, & seruat eam quam in principio habuit positionem; neque in ipsa latione circumuertitur. Commandinus verò in lib. de centro grauitatis solidorum hoc pacto: Centrum grauitatis vniuscuiusque solidæ figuræ, est punctum illud intra positum, circa quod vndique partes æqualium momentorum adsistunt. Si enim per tale centrum ducatur planum, figuram quomodolibet secans, in partes æquè ponderantes eam diuidit. Nos verò quàm breuissimè dicimus: Centru[m] grauitatis, vniuscuiusq; magnitudinis punctum esse intra extraue magnitudinem positum, per quod si plano linea punctoue diuidatur, in partes secatur æqueponderantes. Dixi-

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2 IN MECHAN. ARIST. PROBL. The center of gravity can be considered in three magnitudes: linear, plane, solid. No one has written about the center of gravity of lines, for it is a very simple matter of contemplation. Archimedes treated the center of gravity of planes excellently in the book On Equal Weights, and On the Quadrature of the Parabola, and also in the work which he inscribed On Floating Bodies. As for the center of gravity of solids, Archimedes himself had once written about it, but what he produced was lost through the injury of time, and Iedericus Commandinus restored it by his diligence. Now that there is also a center of lightness in the nature of things is evident. For that is the point according to which light things are carried straight upward from the center. This, however, the Mechanici did not mention, because it either serves not at all, or but little, for their purpose. Moreover, Heron defines the center of gravity thus, and Pappus following Heron, 1. 8. of the Mathematical Collections. The center of gravity of any body is a certain point placed within it, from which, if the weight is conceived as suspended in thought, while it is being carried, it remains at rest, and preserves that position which it had at the beginning; nor does it turn about in the movement itself. But Commandinus in the book On the Center of Gravity of Solids [defines it] in this way: The center of gravity of each solid figure is that point placed within it, around which on every side parts of equal moments stand. For if through such a center a plane be drawn, cutting the figure in whatever way, it divides it into equally weighing parts. But we, in the briefest way, say: The center of gravity is the point of each magnitude, placed within or outside the magnitude, by which, if it be divided by a plane, line, or point, it is cut into equally weighing parts. I said-

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EXERCITATIONES. A B C Diximus, Magnitudinis vt lineæ, plani solidiq[ue] centrum complecteremur. Erit igitur, vt in præsenti figura, lineæ quidem centrum A, plani B, solidi verò C. quod si obijciat quispiam, lineam & superficiem nullam habere grauitatem; is sciat, neq[ue] corpora Mathematica grauitatem habere, Mechanicum verò funes, hastas, vectes pro lineis sumere; tabulas verò, & eiusmodi plana ad superficierum naturam referre. Diximus insuper, intra extraue. Aliquando enim grauitatis centrum extra molem corporis cuius corporis centrum est, cadit, vt in sequenti figura. Esto corpus aliquod superficiesue ABCDE, ducatur linea CF, diuidés figuras in partes hinc inde æqueponderantes A/B/C, EDC. Ducatur & GH. diuidens item in partes æqueponderantes GCH, & GAB, EDH. secent autem seiplas in I. erit igitur centrum I extra figuræ terminos & molem ipsam. Attamen licet hoc verum sit, intra esse dici potest, quippe quod imaginario quodam, & vt ita dicam, virtuali ambitu ACDA contineatur. Dicebamus, duplex esse grauitatis centrum, Natu- ra, Vio- A 2

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EXERCISES. A B C We said that by the center of magnitude we mean the center of the line, plane, and solid. It will therefore be, as in the present figure, that the center of the line is A, of the plane B, and of the solid C. But if anyone objects that a line and a surface have no gravity, let him know that mathematical bodies also have no gravity; the mechanic, however, takes ropes, poles, and levers for lines, and boards and similar planes as pertaining to the nature of surfaces. We said further, within or without. For sometimes the center of gravity falls outside the mass of the body whose center it is, as in the following figure. Let there be some body or surface ABCDE; let the line CF be drawn, dividing the figures into parts on either side that are equally weighted, A/B/C, EDC. Let GH also be drawn, likewise dividing into equally weighted parts GCH, and GAB, EDH. But let them cut one another at I. Thus the center I will be outside the boundaries of the figure and of the mass itself. Yet although this is true, it can be said to be within, since it is contained by a certain imaginary, and, so to speak, virtual, boundary ACDA. We were saying that there is a twofold center of gravity, Natu- ra, Vio- A 2

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4 IN MECHAN. ARIST. PROBL. râ, Violentiâ: affirmamus modò, hæc re quidem vnum esse, & ratione solum, non autem reipsa ac si duo essent considerari. Esto enim grauitatis naturalis centrum B, corporis A, secundum quod dimissum, suapte naturâ cadet in C, si verò corpus violenter impellatur in D, aliud acquiret centrum grauitatis ex violentia secundum quam fertur, motum, in D, idé autem sunt re, nempe vnum B, duo autem si violentia & natura seorsum considerentur. Hæc centra, duo motus sequuntur, rectus vterque, Naturalis videlicet, & Violentus. Tertius ex his mixtus, & is quidem non rectus, sed curuus. Proijciature enim violenter corpus graue A superante igitur violentia, rectà feretur in B; ea autem elanguescente paullatim per curuam & mixtam lineâ secetur in C, quatenus enim ad anteriora fertur, violentia est; quatenus verò ad inferiores partes, naturæ. Vbi verò peruenit in C, violentiâ cessante, naturâ verò manente, rectà deorsum fertur D C D. Cæterùm hæc centra, hi que motus, naturalis nempe, & violentus diuersimode se habent adinuicem. Si enim graue corpus externâ vi adhibita, centrum mundi versus impellatur, adiuvabunt se inuicem Natura, Violentia. Si autem contra, altera alteri resistet, in motibus autem

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4 IN MECHAN. ARIST. PROBL. [...], by violence: we affirm, moreover, that this indeed is one and the same thing, and is to be considered as two only in reason, not however in reality, as if they were two. For let the center of natural gravity be B, of the body A, according to which, if let go, by its own nature it will fall into C; but if the body be violently driven into D, it will acquire another center of gravity from the violence according to which it is carried, the motion being in D; yet in reality they are the same, namely one B, but two if violence and nature are considered separately. These centers, and the two motions that follow, are straight, both of them, namely the Natural and the Violent. The third, made up of these, is indeed not straight, but curved. For if the heavy body A be projected violently, violence prevailing, it will be carried straight into B; but as this grows feebler it will be cut by a curved and mixed line into C, for insofar as it is borne forward, it is violence; but insofar as it is borne toward the lower parts, it is nature. But when it arrives at C, violence ceasing and nature remaining, it is carried straight downward D C D. Moreover, these centers and these motions, namely the natural and the violent, are related to one another in different ways. For if a heavy body, by external force applied, be driven toward the center of the world, Nature and Violence will help one another. But if contrariwise, one will resist the other, but in the motions

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EXERCITATIONES. 5 autem ad latus, eo magis pugnabunt, quo magis ab inferioribus ad superiora fiet motus. Mechanices præcipua instrumenta. His ita constitutis dicimus, instrumenta, quibus ad varias operationes Mechanici vtuntur, esse inter se quidem diuersa, multiplicia, & si varietatem spectes, penè innumerabilia, quod quamuis verum sit, ea omnia Aristoteles ad vectem reducit, & libram: quod etiam G. V baldus in libris Mechanicorum fecit. Cæterum qui post Aristotelem floruere Mechanici, omnia ad quinque, quas appellant, Potentias, redegere. Sunt autem ex Herone, Pappo, Guido V baldo, qui eos securus est, Vectis, Trochlea, Axis in Peritrochio, Cuneus, Cochlea. Videtur autem ipse G. V baldus sextam addere, nempe Libram, de qua & primus ipse Mechanicorum tractatum instituit. Verum enimuero idem ferè sunt Vectis & Libra, nisi fortè quod Libra tunc dicitur, cum brachia sunt æqualia. Vectis vero quomodo cunque ea se habeant; quinque harum Potentiaru[m] imagines ita ob oculos ponimus. Vectis A. Trochlea B, Axis in Peritrochio C. Cuneus D. Cochlea vero E. A 3 Porro

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EXERCISES. 5 but to the side, they will fight all the more, the more the motion is made from the lower parts to the upper. The chief instruments of mechanics. These things being thus established, we say that the instruments by which Mechanics use for various operations are indeed different from one another, many in number, and, if you consider the variety, almost innumerable; and although this is true, Aristotle reduces them all to the lever and the balance; which G. V. Baldus also did in the books on Mechanics. Moreover, those Mechanics who flourished after Aristotle reduced everything to five, as they call them, Powers. These are, from Hero, Pappus, Guido Ubaldo, who is safe among them, the Lever, Pulley, Axis in Peritrochium, Wedge, Screw. But it seems that G. V. Baldus adds a sixth, namely the Balance, concerning which he himself first instituted a treatise on Mechanics. Yet in truth the Lever and the Balance are almost the same thing, except perhaps that the Balance is so called when the arms are equal. The Lever, however, however they may stand; we place before the eyes the images of these five Powers. Lever A. Pulley B. Axis in Peritrochium C. Wedge D. Screw E. A 3 Furthermore

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6 IN MECHAN. ARIST. PROBL. Porro, Cuneum ad libram reducere conatur Aristoteles, quod facit & G. Vbaldus, qui eò refert & Cochleam, quippe quod nihil aliud sit Cochlea, quàm Cuneus Cylindro inuolutus. Nos autem duas tantùm Potentias ad vectem reduci posse arbitramur, Trochleam nempe, & Axem in Peritrochio. Nequaquam autem Cuneum & Cochleam. quod latiùs quidem ostendemus, cùm de Cunco erit nobis sermo peculiaris. De Vecte & Libra secundum Aristotelem. Aristoteles in ipso Mechanicorum ingressu ita scribit, Mirum videri ab exigua virtute magnum pondus mo- ueri,

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6 IN MECHAN. ARIST. PROBL. Furthermore, Aristotle tries to reduce the wedge to the balance, as G. Ubaldus does, who refers the screw to it as well, since a screw is nothing other than a wedge wrapped around a cylinder. But we think that only two powers can be reduced to the lever, namely the pulley and the axis in the peritrochium. Not at all the wedge and the screw, which we shall indeed show more fully when we come to a special discussion of the wedge. On the Lever and Balance according to Aristotle. Aristotle, at the very beginning of the Mechanics, writes thus: “It seems remarkable that a great weight is moved by a small force,”

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EXERCITATIONES. 7 ueri, addito nimirum ponderi pondere, siquidem & vectis est pondus. Duplex ergo illi admiratio, scilicet quòd exigu gua potentia moueat ingens pondus, idque etiam addito vectis ipsius pondere, fiat. Hoc secundum adiecisse vide- tur, amplificationis alicuius gratiâ. Etenim quatenus ad rem pertinet, si mouendis ponderibus vectis ipsius pondus compares, nullius ferè esse momenti proculdu- bio affirmaueris. Sed & illud quoque notandum, aliquan- do vectis pondus mouenti auxilium ferre, quod fit vbi fulcimento inter potentiam mouentem, & pondus ipsum collocato, vectis pars quæ à fulcimento ad potentiam est, premitur. Tunc enim, vt dicebamus, vectis pondere suo potentiam adiuvat. Contra verò accidit, cum pondus i- psum inter fulcimentum est & potentiam vel potentia i- psa inter fulcimentum & pondus. tunc enim vectis vnâ cum pondere attollitur. quæ licet vera sint, non tamen in- desequitur, vectis pondus, quicquam quod curandum sit, in operatione efficere, aut impedire. Porro vectem ita finire possumus, longitudinem es- se quandam inflexibilem, quæ fulcimento dato, datâ po- tentiâ datum pondus mouetur. Ipsa quoque Libra, vt diximus, vectis est: eius autem naturæ, vt semper fulcimentum medium obtineat locum inter pondus & pondus. Statera autem merus est vectis, si sparsum pro fulcimento; appendiculum verò currens pro potentia mouente deputaueris. De Circulo eiusque natura Aristotelis doctri- na examinata. Aristoteles, quicquid mirum in Mechanicis opera- tur, id totum admirabili circuli naturæ esse tribuendum arbitratur. Ait autem, absurdum nullatenus esse, si exre mirabili mirandum quippiam oriatur. In circulo autem qua-

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EXERCISES. 7 indeed, by adding weight to weight, since the lever itself is a weight. There is therefore a double wonder in it: namely, that a very small power moves an enormous weight, and that this is done even with the weight of the lever itself added. It seems that he added this second point for the sake of some amplification. For insofar as it concerns the matter itself, if you compare the weight of the lever with the moving of weights, you would undoubtedly affirm that it is of almost no importance. But this too must be noted: that sometimes the weight of the lever helps the mover, which happens when, a support being placed between the moving power and the weight itself, that part of the lever which lies from the support to the power is pressed down. Then, as we were saying, the lever aids the power by its own weight. The contrary happens when the weight itself is between the support and the power, or when the power itself is between the support and the weight; for then the lever is lifted together with the weight. Although these things are true, it does not follow that the weight of the lever produces or impedes anything worth caring about in the operation. Moreover, we may define a lever as a certain inflexible length, which, a support being given and a power being given, moves a given weight. A balance, too, as we have said, is a lever: but of such a nature that the support always occupies the middle position between weight and weight. A steelyard, however, is a mere lever if you assign the fixed point as the support; and the movable counterweight as the moving power. On the Circle and the examination of Aristotle’s doctrine concerning its nature. Aristotle thinks that whatever is remarkable is done in mechanics is to be attributed wholly to the wonderful nature of the circle. But he says that it is by no means absurd if something admirable arises from a marvelous thing. Now in the circle,

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8 IN MECHAN. ARIST. PROBL. quatuor inueniri qualitates admiratione dignas. Primâ, quod ex contrarijs constituatur, mouente videlicet & moto. Secundam, quòd contraria in eius circumferentia inueniantur, quippe quæ cum vnica linea sit, concaua simul est & conuexa. Tertiam, quod contrarijs feratur motionibus, antrorum nimirum, retrorsum, sursum, atque deorsum. Quartam, quod vnicâ existente semidiametro, nullum in ea punctum sumi possit, æqualis alteri, in latione, velocitatis. Sit enim circulus A B, cuius centrum C, semidiameter A C, sumatur autem in ea punctum D, itemque punctum E. Erit itaque in ipsa circulatione D tardius E, ipsum verò E tardius A, & ita citius id feretur semper, quod remotius à mouente termino accipitur. Hæc ex illo, quibus ne vltro assensum præbeamus non vnica de causa cohibemur. Dicimus igitur, videri nobis, circulum non ex contrarijs co[n]stitui, puta ex manente & moto, sed ex moto simpliciter. Nulla est enim semidiametri pars, quæ non moueatur. Punctum autem, quod stat, semidiametri pars nulla est. Et sanè cur moto semidiametro fiat circulus, non ideo accidit, quod alteru[m] extremum stet, alterum verò moueatur: sed ideo quòd semidiameter perpetuò eandem seruet longitudinem. Ellipsis sanè centrum habet, sed ab eo ad circumferentiam, quatuor tantùm semidiametri quomodolibet sumpti ducuntur æquales. Si quis igitur semidiametrum daret proportione crescentem & decrescentem, stante altero extremorum Ellipsis describeretur. Præterea & spiralis linea, quæ mixta est, altero semidiametri extremo manente, altero vero moto producitur. Legem itaque circulo præ-

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8 IN MECHAN. ARIST. PROBL. four qualities worthy of admiration are found. First, that it is constituted from contraries, namely the moving and the moved. Second, that contraries are found in its circumference, since, though it is a single line, it is at once concave and convex. Third, that it is carried by contrary motions, namely forward and backward, upward and downward. Fourth, that, since there is only one radius, no point in it can be taken that is equal to another in velocity of motion. For let there be a circle A B, whose center is C, and radius A C; let points D and E be taken in it. Thus in the rotation itself D will be slower than E, and E itself slower than A; and so that will always be carried more quickly which is taken farther from the moving terminus. These things from that source are not for us to give assent to without further thought, for we are restrained by more than one reason. We say, therefore, that a circle seems to us not to be constituted from contraries, such as from the stationary and the moved, but simply from the moved. For there is no part of the radius that is not moved. But a point that stands still is no part of the radius. And indeed the reason why a circle comes to be from a moved radius is not that one extremity stands still and the other moves, but because the radius continually preserves the same length. An ellipse indeed has a center, but from it to the circumference only four radii, taken however one may choose, are drawn equal. If someone were to give a radius increasing and decreasing in proportion, with one of the extremities standing still, an ellipse would be described. Moreover, the spiral line, which is mixed, is produced when one extremity of the radius remains fixed and the other moves. Therefore the law for the circle pre-

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EXERCITATIONES. 9 præscribit, non quidem quòd hæc extremitas stet, illa verò moueatur, sed quod sua circulatione semper semidiameter eandem seruet longitudinem, quod vel ex ipsa circuli definitione colligitur. Ad secundum miraculum, scilicet, quòd in circulo circumferentia, quæ vacua linea est, concaua simul sit, & conuexa. Diceret quispiam id, si modò mirabile est non circulari tantum, sed cuilibet curuæ lineæ primo competere, etenim & Ellipsis & Hyperbole, & Parabole, & spirra, tum Cyssois, Conchois, & infinitæ aliæ irregulares concauæ simul sunt & conuexæ. Sed & hæc in superficiebus quoque desiderantur. Ad tertium, quod contrarijs feratur lationibus, antrorsum, retrorsum, sursum & deorsum. Dicimus, facilè solui. Nullus enim, rebene perspectâ, affirmauerit circulum contrarijs lationibus moueri. Esto enim circulus ABCD, circa centrum E; ponamus rotari, & A versus B, exempli gratiâ, antrorsum, mouebitur autem & B versus C, & C versus D, tum D versus A. Non puto quenqua[m] dicturum, circulum hunc antrorsum eodem tempore, & retrorsum ferri nec sursum aut deorsum, si enim quispiam per eius circuli circumferentiam ambularet, is certè centrum ipsum semper ad dexteram haberet, vel ad sinistram, si ad dexteram, antrorsum ibit, si ad sinistram, retrorsum. Sed nec sursum vel deorsum, est manifestum. Nihil autem prohibet eundem motum vario respectu contrarium dici posse, id tamen profectò fierinequaquam potest, nempe A moueri versus B, hoc est, B antror-

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EXERCITATIONS. 9 prescribes, not indeed because this extremity stands still and that one is moved, but because by its circulation the semidiameter always preserves the same length, which may also be gathered from the very definition of a circle. To the second miracle, namely, that in a circle the circumference, which is an empty line, is at the same time concave and convex. One might say that this, if it is indeed marvelous, belongs first not only to a circle, but to any curved line; for even the Ellipsis, and Hyperbole, and Parabole, and spirra, then Cyssois, Conchois, and innumerable other irregular figures are at once concave and convex. But this is also lacking in surfaces. To the third, that it is carried by contrary motions, forward, backward, upward and downward. We say it is easily resolved. For no one, having considered the matter properly, would affirm that a circle is moved by contrary motions. Let there be a circle ABCD, around center E; suppose it turns, and A toward B, for example, will move forward; but B toward C, and C toward D, then D toward A. I do not think anyone would say that this circle is at the same time carried forward and backward, or upward or downward; for if someone were walking along the circumference of that circle, he would certainly always have the center itself on his right hand, or on his left if on the right; if on the right, he will go forward, if on the left, backward. But neither upward nor downward, it is manifest. Yet nothing prevents the same motion from being called contrary in different respects; nevertheless that certainly cannot happen in any way, namely, that A move toward B, that is, B antror-

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10 IN MECHAN. ARIST. PROBL. antrorsum, & eandem eodem tempore versus B, id est, retrorsum; repugnat enim naturæ. De quarto circuli miraculo, ibi erit nobis sermo, vbi ea perpende imus primò, quæ Philosophus de Circuli productione differens in medium profert. Sunt autem eiusmodi: Circulum quidem duplici notione produci, Naturali videlicet altera, & altera quæ est præter naturam, & ideo circularem lineam in ter mixtas computari. Motus mixtus ait, vel proportione seruata fit, aut non; Si proportione seruatâ, rectam lineam; ea verò non seruata, circularem lineam produci. Esto enim rectangulum ABCD, cuius latera in datâ sint proportione, AD cum AB. Moueatur A, duplici motu, Altero quidem tendens in B, altero verò ad motum lineæ AB, feratur versus D, seruata interim laterum proportione. Itaque ponatur ex motu ab A versus B, peruenisse in E, ex motu autem quo proportionaliter fertur cum linea AB, facta ipsa AB, in FH, peruenisse in G, & EG connectatur. Erit igitur Parallelogrammum AEGF, Parallelogrammo ABCD proportionale simile, & circa eandem diametrum AGC. Semper igitur punctum A si duabus lationibus feratur, laterum proportione seruata, lineam producet rectam, diametrum nempe AGC. Ethoc sanè nullam habet dubitationem, ex ijs quæ docet Euclides 1.6. prop.34. His ita demonstratis hac vti videtur Philosophus argu-

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10 IN MECHAN. ARIST. PROBL. forward, and the same at the same time toward B, that is, backward; for it is contrary to nature. Of the fourth wonder of the circle, there we shall speak, when first we weigh what the Philosopher puts forward in the middle concerning the production of the circle. These things are as follows: That the circle is indeed produced in a twofold sense, namely one natural, and the other which is contrary to nature, and therefore the circular line is to be counted among mixed things. He says that a mixed motion is produced either with the proportion preserved, or not; if the proportion is preserved, a straight line is produced; if it is not preserved, a circular line is produced. Let there be the rectangle ABCD, whose sides AD and AB are in a given proportion. Let A be moved by a double motion, one tending toward B, and the other, by the motion of the line AB, let it be carried toward D, while the proportion of the sides is preserved in the meantime. So let it be assumed that, from the motion from A toward B, it has come to E, but from the motion by which it is proportionally carried along with the line AB, the line AB itself having been made, it has come to G in FH, and let EG be joined. There will therefore be the parallelogram AEGF, similar and proportional to the parallelogram ABCD, and around the same diameter AGC. Therefore the point A, if it be carried by two motions while the proportion of the sides is preserved, will always produce a straight line, namely the diameter AGC. And this indeed admits of no doubt, from what Euclid teaches in 1.6. prop. 34. These things being thus demonstrated, the Philosopher seems to use this argument

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EXERCITATIONES. II argumentatione: Si mixtus motus proportione semotâ, rectam producit, si nunquam semota, efficiet circulum; si enim modo seruaretur, modo non, partim recta partim non recta produceretur. Ingeniosa quidem argumentatio, ni vitium contineret. non enim mixtus motus, qui nunquam seruatâ proportione sit, semper circulum producit, sed & Ellipsis potest, & quamlibet aliam lineam, cuius nulla pars sit recta. Hanc difficultatem vidit Pico- lomineus in sua Paraphrasi, & eam soluere conatus est, sed quàm bene, aliorum esto iudicium. Cæterùm falsum est, asserere circulum ex mixto motu nunquam seruatâ proportione produci. seruat enim assiduè mixtus motus quo producitur (si eum mixto motu producere velimus) aliquam proportionem, sed non eandem. Esto enim recta AB, cui ad rectos angulos AC. Moueatur autem A, versus C per lineam AC, & eodem temporelinea AC, versus B, ita tamen, vt semper ipsi AB, sit perpendicularis. feratur autem eâ lege, vt quam proportionem habet motus lineæ AC versus B, ad motum puncti A versus C, eandem habeat ipse motus ab A versus C, ad residuum lineæ AB, demptâ nempe ea parte quam peragrauit linea AC mota versus B. Sit autem, cum AC suo motu peruenerit in D, punctum A, similiter suo motu per eam latum peruenisse in E. erit ergo ex mixto motu, non quidem in D, nec in E, sed in F, eritque punctum F in circumferentia circuli, cuius est diameter ipsa linea AB, quod quidem demonstratur ex conuersa propos. 13. lib. 6. Elem. Est enim AE hoc est DF media proportionalis inter EF, hoc est, AD, & DB. Iterum si fiat motus AC in GH, ad motum H per lineam B 2

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EXERCISES. II by argument: If a mixed motion, with the proportion removed, produces a straight line, and if, when it is never removed, it will produce a circle; for if now it were preserved, now not, it would be produced partly straight and partly not straight. Certainly a clever argument, were it not to contain a fault. For not every mixed motion which never preserves proportion produces a circle, but it can produce an Ellipsis, and any other line, of which no part is straight. Piccolomini saw this difficulty in his Paraphrasis, and tried to solve it, but how well, let the judgment of others be. Moreover it is false to assert that a circle is produced from a mixed motion that never preserves proportion. For the mixed motion by which it is produced always preserves a proportion, if we wish to produce it by a mixed motion, but not the same one. For let AB be a straight line, to which at right angles AC is applied. Let A, however, move toward C by the line AC, and at the same time let the line AC move toward B, yet in such a way that it is always perpendicular to AB itself. Let it be carried by this law, so that whatever proportion the motion of the line AC toward B has to the motion of the point A toward C, the same proportion may the motion itself from A toward C have to the remainder of the line AB, namely after subtracting that part which the moved line AC has traversed toward B. Let it be so, moreover, when AC in its motion has reached D, that point A likewise, carried by its motion through it, has reached E. Therefore from the mixed motion there will be, not indeed at D, nor at E, but at F, and point F will be on the circumference of the circle, of which the diameter is the line AB itself, which indeed is demonstrated from the converse of proposition 13, book 6 of the Elements. For AE that is DF is a mean proportional between EF, that is, AD, and DB. Again, if the motion of AC is made into GH, to the motion H by the line B 2

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12 IN MECHAN. ARIST. PROBL. lineam AC, vsque in C, vt se habet proportio AG ad GH & GH ad GB, erit ex motu mixto A in H, nempe in eiusdem circuli circumferentia AFHB. ex quibus habemus, circulum ex mixto motu fieri posse proportionibus quidem mediarum seruatis, sed nunquam ijsdem. Vera hæc proculdubio sunt; nihilominus, veluti ad rectam producendam mixtus motus non est necessarius, licet mixto motu produci possit, ita neque ad circularem, & ideo verum non esse quod asserebat Philosophus, circulum ex mixto motu proportione nunquam seruatâ necessariò produc[i]o. Conatur post hæc Aristoteles rationem afferre, cur circuli partes, quò propiores centro fuerint, eo sint tardiores. Ait autem; si duobus ab eadem potentia latis hoc quidem plus repellatur, illud verò minus, æquum est tardiùs id moueri quod plus repellitur, eo quod minus. Detrahi autem plus lineam, cuius extremum propius est centro illa quæ suum habet terminum à centro remotiorem. Esse, inquit, circulus BCDE & alter in eo minor MNOP circa idem centrum A. Ducanturq[ue]; Diametri maioris quidem CD, EB, minoris verò MO, NP. Itaque vbi AB circulata eò peruenerit vnde est gressa, ipsa quoque AM eo vnde moueri cæperat, perueniet. Tardiùs autem fertur AM, quam AD, propterea quòd AM à centro magis retrahatur quàm ipsa AB. Ducatur igitur ALF & à puncto L, ipsi AB perpendicularis Lq, cadens in mino- ricir-

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12 IN MECHAN. ARIST. PROBL. line AC, as far as C, so that the proportion of AG to GH is as GH to GB, there will be from the mixed motion A into H, namely, in the circumference AFHB of the same circle. From these things we have that a circle can be produced from mixed motion, the proportions of the intermediates indeed being preserved, but never the same ones. These things are undoubtedly true; nevertheless, just as mixed motion is not necessary for producing a straight line, although it can be produced by mixed motion, so neither is it necessary for a circular line; and therefore it is not true what the Philosopher asserted, that a circle is necessarily produced from mixed motion, with the proportion never preserved. After this Aristotle tries to give a reason why the parts of a circle, the nearer they are to the center, are the slower. For he says: if, of two things moved by the same power, one is repelled more and the other less, it is fair that that which is repelled more should move more slowly than that which is less so. And more is taken away from the line whose end is nearer the center than from that which has its limit farther from the center. Let there be, he says, the circle BCDE, and another smaller one MNOP within it around the same center A. And let the diameters of the greater be drawn, CD and EB, and of the smaller MO and NP. Therefore, when AB, having completed its revolution, has come to the point from which it set out, AM also will come to the point from which it began to move. But AM is carried more slowly than AD, because AM is drawn back from the center more than AB itself. Let ALF therefore be drawn, and from the point L let Lq be drawn perpendicular to AB, falling in the smaller cir-

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EXERCITATIONES. 13 ri circulo, & rursus ab eodem L ipsi AB, parallela duca- tur LS, Ab S verò eidem perpendicularis ST, & ab Fi- tem FX. Suntergo qL, ST, quidem æquales, nempe illæ, per quas, secundum naturam, mouentur puncta BM Mo- tu verò retractionis ad centrum, hoc est, præter naturam, plus motum est M quàm B. Maior enim est Mq, ipsa BT, quod, ceu notum, supposuit Aristoteles. nos autem infà demonstrabimus. Si igitur fiat vt motus præter naturam ad motum præter naturam, ita motus secûdum naturam, ad motum secundum naturam, punctum B; cum M fuerit in L, non erit in S, sed in F. tunc enim, vt est FX motus se- cundùm naturam ad X B, præter naturam, ita est qL se- cundum naturam ad qM præter naturam; sed B F maior est ML, ergo proportione seruatâ, velociùs mouetur B quàm M circa idem centrum A. Hæc autem summa est eorum quæ præfert Aristoteles. Cæterùm nos parallelo- grammum, quod in figura eius habetur prætermisimus, quippe quod nihil ad eam quæ affertur, demonstrationem faciat. Modò quod pollicebamur, nempe minorem esse BT, quàm qM, ita demonstramus. quonia ST. ex prop. 13. 1.6. media proportionalis est inter BT & TE, erit qua- dratum TS æquale parallelográmo seu rectangulo BT, TE, item, quoniam qL media proportionalis est inter Mq, & qO. erit quadratum qL æquale rectangulo Mq, qO, æqualia ergo sunt rectangula BT E, Mq O, itaque reciprocatera habent proportionalia. quare, vt TE, ad qO, ita Mq ad TB, sed TE maior est ipsa qO, quippe quòd pars sit qO ipsius TE, maior ergo & Mq ipsa TB, quod ostendendum fuerat. Cæterùm subtilia & ingeniosa isthæc esse non nega- mus, & longè faciliori & explicatiori modo veritas hæc demonstrari potest, reiectis nempe illis, secundùm, & præ- ter B 3

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EXERCISES. 13 and from the same point L, to the line AB itself, let LS be drawn parallel; from S, moreover, let ST be perpendicular to the same line, and from F, FX. Therefore qL and ST are equal, namely those through which, according to nature, the points B and M are moved; but by the motion of retraction toward the center, that is, contrary to nature, M has moved more than B. For Mq is greater, namely BT itself, as Aristotle has, as is known, assumed. But we shall prove this below. If, then, the motion contrary to nature be as the motion contrary to nature, so let the motion according to nature be to the motion according to nature, point B; when M shall have been in L, it will not be in S, but in F. For then, as FX is the motion according to nature to XB contrary to nature, so qL is the motion according to nature to qM contrary to nature; but BF is greater than ML, therefore, the proportion being preserved, B moves more quickly than M about the same center A. And this is the sum of what Aristotle puts forward. Moreover, we have omitted the parallelogram which appears in his figure, since it contributes nothing to the demonstration that is offered. Now what we promised, namely that BT is less than qM, we demonstrate thus. Since ST, from Proposition 13 of Book 6, is the mean proportional between BT and TE, the square of TS will be equal to the parallelogram or rectangle BT, TE; likewise, since qL is the mean proportional between Mq and qO, the square of qL will be equal to the rectangle Mq, qO. Therefore the rectangles BTE and MqO are equal, and thus they have reciprocal proportions. Whence, as TE is to qO, so Mq is to TB. But TE is greater than qO itself, since qO is a part of TE; therefore Mq is also greater than TB itself, which was to be shown. However, we do not deny that these things are subtle and ingenious, and this truth can be demonstrated in a much easier and clearer way, namely by rejecting those things, according to, and contrary to

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14 IN MECHAN. ARIST. PROBL. ter naturam motibus, qui quidé in simplici circulo necessario non cadunt: caderent autem fortasse, si de circulo res esset à pôderibus circumlatis ex stabili centro descripto; qua de re agit G. V baldus in Mechanicis tractatu de libra. tunc enim dici potest, pondus quod aliâs rectà ad mundi centrum tenderet, à circuli centro in circulatione retrahi, sed hæc ad circuli naturam, quatenus circulus est, nequaquam spectant. Esto igitur circumferentia A F B H, cuius centrum C, diameter A C B, semidiameter A C. sumatur in A C punctum quodlibet, D, & centro C, spatio C D, circumferentia describatur D G E I. Dico punctum A velocius moueri puncto D eâdem circulatione rotato. etenim vt diameter ad diametrum, & semidiameter ad semidiametrum, ita circumferentia ad circumferentiam: igitur vt A C ad C D, ita circumferentia A F H B ad circumferentiam D G E I. At mota linea C A circa centrum C mouetur simul & C D, eodem igitur tempore rotationem complent puncta A D, maius ergo spatium eodem tempore metitur A, ipsa D, quare velocior. Ita igitur se habet velocitas ad velocitatem, vt circumferentia ad circumferentiam, & diameter ad diametrum, quare id quod mouetur in puncto à centro remotiori, velociùs illo mouetur quod ab eo distat minus, quod fuerat demonstrandum. QVÆ.

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rather than to natures of motions, which certainly do not in a simple circle necessarily fall: they would perhaps fall, however, if the case were of a circle described by weights moving around from a fixed center; on this matter G. V. Baldus treats in his Mechanics, in the tract on the balance. For then it may be said that a weight, which otherwise would tend directly toward the center of the world, is in revolving drawn back from the center of the circle; but these things in no way concern the nature of the circle, insofar as it is a circle. Let there therefore be the circumference A F B H, whose center is C, diameter A C B, semidiameter A C. Let an arbitrary point be taken in A C, D, and with center C, at distance C D, let the circumference D G E I be described. I say that the point A moves more quickly than the point D, both being rotated by the same revolution. For as diameter is to diameter, and semidiameter to semidiameter, so circumference is to circumference: therefore as A C is to C D, so the circumference A F H B is to the circumference D G E I. But when the line C A is moved around center C, C D is moved at the same time also; therefore the points A and D complete their rotation in the same time, hence A measures a greater space in the same time than D itself, wherefore it is quicker. Thus therefore speed stands to speed, as circumference to circumference, and diameter to diameter, wherefore that which is moved at a point farther from the center is moved more quickly than that which is farther less from it, which was to be demonstrated. QVÆ.

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EXERCITATIONES. 15 QVÆSTIONES MECHANICÆ. QVÆSTIO I. Cur maiores libræ exactiores sint minoribus? Prioribus, ceu fundamentis quibusdam iactis, opportunè ad quæstiones proponendas, eas que diluendas se confert Aristoteles. Porro in proposita quæstione videtur prima fronte causam quæri de re quæ non est: etenim quis affirmauerit vnquam, lances quibus Apothecarij & Macellarij vtuntur, magnas eas quidem, illis exactiores esse quibus Gemmarij, atque Argentarij siliquis, & scrupulis minutissima appendunt, quæ tamen perexiguæ sunt, & si illis comparentur minimæ? Veruntamen, ita prorsus res habet, vt asserit Aristoteles. Non enim propterea quòd illæ magnæ sint, hæ verò exiguae, hæ sunt illis exactiores; sed quoniam magnæ, rudes sunt, minores verò exquisita diligentia elaboratæ, & à materiæ pertinacia libe- riores. Cæteris ergo paribus, exactiores esse maiores, ex Philosophimente, ita docebimus. Esto libra maior AB, cuius fulcimentum C. Minor verò libra DE, circa idem fulcimetum C, vnà cum maiori, imaginatione, conuersa. Apponatur quoduis pondus maiori libræ in A, declinetque; exempli gratiâ in F, eritque minor libra in G, in eadem enim linea sunt CGF. Vtraque igitur ex eodem cen-

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EXERCISES. 15 MECHANICAL QUESTIONS. QUESTION I. Why are larger balances more accurate than smaller ones? Having laid down the foregoing points, as it were certain foundations, Aristotle suitably proceeds to propose and resolve questions. In the present question, however, there seems at first sight to be sought the cause of something that does not exist; for who has ever asserted that the scales used by Apothecaries and Butchers—those large ones, indeed—are more accurate than those with which Jewellers and Silversmiths weigh seeds and scruples, which are nevertheless very small, and, if compared with those, the smallest? Yet the matter is in fact exactly as Aristotle states. For it is not because the former are large and the latter small that the latter are more accurate than the former; rather, because the large ones are crude, while the smaller are fashioned with exquisite care and freer from the stubbornness of the material. Other things being equal, then, we shall show, according to the philosophers’ teaching, that the larger are more accurate. Let there be a larger balance AB, with its support C. And a smaller balance DE, turned in imagination about the same support C, together with the larger one. Let any weight whatever be placed on the larger balance at A, and let it incline; for example, to F, and the smaller balance will be at G, for C, G, and F are on the same line. Therefore both from the same cen-

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16 IN MECHAN. ARIST. PROBL. centro C portionem circuli describet GD, AF, eritque ACF sector circuli, cuius diameter AB, sed DCG sector circuli, cuius diameter DE. Itaque vt diameter ad diametrum, ita portio ad portionem: maior autem diameter AB diametro DE: maior ergo portio AF, portione DG. quod autem maius est, minus obtutum fallit, exquisitius itaque tractum ex maiori AB quàm ex ipsa minori DE cognoscemus, quod fuerat ostendendum. Cæterùm hac eadem de caussa, Astronomica instrumenta, puta Astrolabia, Armillæ, & aliae eiusmodi, quo ampliora eò exquisitiora, & certiora probantur. Esto enim Astrolabium magnum, cuius diameter AB, paruum autem CD, circa idem centrum E. Ducatur à centro recta EF tangens maiorem circulum in F, minore verò secas in G, vt igitur GD ad totum circulum GCD, ita FB. ad totum circulum FAB, vt ergò GD ad FB, ita gradus signati in GD, ad eos qui signantur in BF, maiores ergo sunt qui in FB, & minutarum partium capaciores. Hinc itaque apparet, instruméta quælibet quò maiora fuerint, eò esse & exquisitiora, quod proposuerat Aristoteles, in hac quæstione de Libra. Quod autem addit de fraudibus Purpuratiorum, inquiens; quamobrem machinantur ij qui purpuram vendunt, vt pedendo defraudent, dum ad medium, spartum, non

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16 IN MECHAN. ARIST. PROBL. from center C it will describe the portion GD, AF, and there will be the sector ACF of the circle whose diameter is AB, and the sector DCG of the circle whose diameter is DE. Thus, as diameter is to diameter, so portion is to portion: but the greater diameter AB to the diameter DE: therefore the portion AF is greater than the portion DG. But that which is greater deceives the sight less; therefore a line drawn more exactly from the greater AB than from the lesser itself DE we shall recognize, which was to be shown. Moreover, for this same reason astronomical instruments, such as astrolabes, armillae, and others of this kind, the larger they are, the more exact and trustworthy they are judged to be. For let there be a large astrolabe, whose diameter is AB, and a small one CD, about the same center E. Let a straight line EF be drawn from the center, touching the greater circle at F, and cutting the smaller one at G; therefore as GD is to the whole circle GCD, so is FB to the whole circle FAB; therefore as GD is to FB, so are the degrees marked in GD to those which are marked in BF. Those, therefore, which are in FB are greater, and more capable of minute divisions. Hence it is clear that whatever instruments are the larger, the more exact also they are, as Aristotle proposed in this question concerning the balance. But what he adds about the frauds of dyers, saying, wherefore those who sell purple contrive, in order that they may cheat by hanging, while to the middle, with a spartum, not

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EXERCITATIONES. 17 non ponentes; tum plumbum in alterutram libræ partem infundentes; aut ligni quod ad radicem vergebat, in eam quam deferri volunt partem constituentes, aut si nodum habuerit, ligni enim grauior ea est pars, in qua est radix, nodus verò radix quædam est. Hinc quæri posset: Vtrum libræ quæ ponderibus vacuæ æquilibrant, omni prorsus careant fraude? Videri cuipiam posset, libras, quæ ponderibus vacuæ, æquilibrant, omni prorsus fraude carere, veruntamen ita non est, quod diligentiùs (res enim magni momenti est) disquiremus. Esto enim libra AB, ita diuisa in C, vt AC sit partium IS, CB verò earundem sit IO. apponatur parti A lanx ponderans IO, parti vero B lanx ponderans 15. ex permutata igitur proportione libra suspensa in C, equè ponderabit; si autem apponatur lanci B sacoma vnciarum 6, & in lance A constitutatur purpura, quæ ita se habeat ad vncias 6, vt IO ad 15, iterum æqueponderabit, sed vt IO ad 15, ita 4 ad 6. Purpurarius ergo fraudulentus, ponens in lance A vncias purpuræ 4, facto æquilibrio petet pretium vnciarum 6, & ita emptorem decipiet, quod sanè innuerat, non autem demonstraverat Aristoteles. Hæc autem faciliora fient ex ijs, quæ in sequentibus quæstionibus, vbi de vecte agetur, explicabuntur. Detegitur autem fraus, si alternatim sacoma in ponderando, modo huic, modò illi lanci apponatur. Si enim in lance A constituatur sacoma, in B verò purpura non fit æquilibrium. C QVAE-

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EXERCISES. 17 not being placed; then by pouring lead into either side of the balance; or by placing on that side toward which they wish it to be carried a piece of wood leaning toward the root, or if it has a knot, for the part of the wood in which the root is is heavier, and the knot is, as it were, a kind of root. Hence the question might be asked: Whether balances that, when empty of weights, are in equilibrium, are entirely free from all fraud? It might seem to someone that balances which, when empty of weights, are in equilibrium, are altogether free from fraud; nevertheless this is not so, which we shall inquire into more diligently (for the matter is of great importance). Let there be a balance AB, divided at C in such a way that AC is of the parts IS, but CB of those same parts IO. Let there be added to side A a pan weighing IO, and to side B a pan weighing 15. By the exchanged proportion, therefore, the balance suspended at C will weigh equally; but if to pan B is added a sacoma of 6 ounces, and in pan A is placed purple that bears the same relation to 6 ounces as IO does to 15, again it will weigh equally; but as IO is to 15, so are 4 to 6. Therefore the purveyor of purple, being fraudulent, placing in pan A 4 ounces of purple, having achieved equilibrium, will demand payment for 6 ounces, and thus will deceive the buyer, which, certainly, Aristotle had hinted at, but had not demonstrated. These matters will, however, become easier from those things which will be explained in the following questions, where the lever will be discussed. The fraud is detected, however, if the sacoma in weighing is alternately placed, now on this pan, now on that. For if the sacoma is placed in pan A, but the purple in B, equilibrium is not achieved. C WHAT-

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18 IN MECHAN. ARIST. PROBL. QVÆSTIO II. Cur, si sursum libræ fulcimentum sit, apposito ad alteram partem pondere, descendat libra, & eo amoto, iterum ascendat, & ad æquilibrium reuertatur. Si verò deorsum fulcimentum fuerit, de- pressa ad æquilibrium nonreuertatur? B Imembrem proponit Philosophus quæstionem, quam trimembrem debuit, triplici siquidem loco fulcimentum aptari potest, superiori, medio, inferiori. Nos deo- mnibus verba faciemus. Prima Quæstionis pars. De Libra sursum fulcimentum habente. Aristoteles primam quæstionis partem ita soluit: An quia sursum parte quidem existente, plus libræ extra perpendiculum sit? Spartum enim perpendiculum est: quare necesse est deorsum ferri id quod plus est, donec ascendat qua bifariam libram diuidit ad ipsum perpendiculum, cum onus incumbat ad libræ partem sursus raptam. Sit libra recta (hoc est, in æquilibrio constituta) B C, spartum autem A D, fulcimentum autem D, desuper: sparto autem deorsum proiecto ad M perpendiculairis erit vbi A D M. Si igitur in ipso B ponatur onus, erit B quidem vbi E, C autem vbi H, quamobrem ea quæ bifariam libra secat, primo quidem erit D M, ipsius perpendiculi; incu[m]bente auté onere, erit D G. quare libræ ipsius E H, quod extra

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18 IN MECHAN. ARIST. PROBL. QUESTION II. Why, if the support of the balance be above, when a weight is placed at the other end does the balance descend, and when the weight is removed does it rise again and return to equilibrium? But if the support be below, why, once depressed, does it not return to equilibrium? But the Philosopher proposes the question in a twofold form, whereas it ought to have been a threefold one, since the support can in truth be placed in three positions: upper, middle, and lower. We will speak of all of them. First part of the question. On the balance having its support above. Aristotle solves the first part of the question thus: is it because, the upper part being present, more of the balance lies outside the perpendicular? For the line AD is the perpendicular; therefore what lies farther out must necessarily be carried downward, until it rises to that point which divides the balance in two, since the load lies on the part of the balance drawn upward. Let the balance be straight (that is, placed in equilibrium) BC, and the line AD, and the support D above; but the line being projected downward to M, the perpendicular will be where ADM is. If then a load be placed at B itself, B will be where E is, and C where H; wherefore that which cuts the balance in two, at first indeed will be DM, the perpendicular itself; but when the load is applied, it will be DG. Therefore the balance EH itself, which outside

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EXERCITATIONES. 19 extra perpendiculum, est AM, vbi est qP maius est dimidio. Si igitur amoueatur onus ab E, necesse est deorsum ferri H, minus est enim E: siquidem igitur habuerit spartum sursum, propter hoc ascendit libra. Pessimè omnes schema hoc lineârunt, ita vt difficillimum sit auctoris inde sensum assequi. Nos autem clarius rem ob oculos ponimus. Id ergo sibi vult Aristoteles, propterea quòd pars iugi HDG maior est parte EDq, eam eleuatam necesse est descendere, & iterum à perpendiculari ADM bifariam diuisam ad æquilibrium reuerti. Possumus nos idem simpliciori figura demonstrare. Esto libra AB, bifariam diuisa in C, fulcimentu verò sursum vbi D, producatur perpendicularis DC in E. Stante igitur libra AB, in æquilibrio æqualis est pars CH, ipsi parti CB apponatur pondus in B. Declinabit igitur libra mota circa centrum D, fiat autem in FG, secetque perpendiculararem in I. Punctum vero C eodem motu circa idem centrum D erit in H. amoueatur pondus appositum: Dico libram â situ FG declinaturam & iterum reuersuram in situm pristinum ACB. quoniam enim parti GH, quæ æqualis est parti HF, additur pars IH, quæ à perpendiculari est vsque ad H, ipsi verò HF eadem pars detrahitur, erit IF minor GI. Superabitur itaque IF à GI, descendetque FI, ascendet verò IF, donec iterum libra

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EXERCISES. 19 outside the perpendicular, is AM, where qP is greater than half. If therefore the load is removed from E, it must necessarily be carried downward to H; for E is lower. If, then, it has had a beam upward, for this reason the balance rises. Everyone has drawn this figure very poorly, so that it is most difficult to grasp the author’s meaning from it. We, however, make the matter clearer before the eyes. Aristotle therefore intends this: because the part of the beam HDG is greater than the part EDq, that part, when lifted up, must descend, and again, being divided in two by the perpendicular ADM, return to equilibrium. We can demonstrate the same thing with a simpler figure. Let there be a balance AB, divided in two at C, but let the support be above at D; let the perpendicular DC be produced to E. Therefore, the balance AB, standing in equilibrium, the part CH is equal to the part CB. Let a weight be placed on B. The balance will therefore incline, moving around the center D; let it be in FG, and let it cut the perpendicular in I. But the point C, by the same motion around the same center D, will be in H. Remove the weight that has been added: I say that the balance, from the position FG, will incline and again return to its former position ACB. For since to the part GH, which is equal to the part HF, there is added the part IH, which is from the perpendicular up to H, but from HF the same part is taken away, IF will be less than GI. Therefore IF will be exceeded by GI, and FI will descend, but IF will rise, until the balance again

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20 IN MECHAN. ARIST. PROBL. bra in partes æquales, vt antea, diuidatur in C, fiatque æquilibrium. Hæc Philosophi demonstratio est vera illa quidem, sed non ex Mechanicis principijs, hoc est, ex centri grauitatis speculatione; nos igitur clariùs rem exponemus, his quæ sequuntur consideratis. Si pondus circa stabile centrum conuertatur, dimissum non stabit, nisi secundum grauitatis centrum fuerit in perpendiculari, quæ per centrum, circa quod conuertitur, ad mundi centrum cadit. Stabit autem in ea perpendiculari in duobus punctis, altero à centro mundi remotissimo; altero verò eidem quantum licuerit proximo. Esto corpus A, cuius grauitatis centrum B, nixum lineę inflexibili B C, cum qua liberè conuertatur circa centrum C. Ducatur autem per mundi centrum perpendicularis B C D. Sit igitur primò pondus A secu[m] dum gracilis B centrum, in perpendiculari ipsa supra centrum C, puta in B. Moueatur & descédat in E, Post hæc verò in F, hoc est iterum in ipsa perpendiculari infra centrum C. Describet ergo circulum ex centro C, nempe B E F secantem perpendicularem in duobus punctis oppositis B F, dico, pondus liberè dimissum

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20 IN MECHAN. ARIST. PROBL. the bar in equal parts, as before, be divided at C, and an equilibrium will be made. This demonstration of the Philosopher is indeed true, but not from mechanical principles, that is, from the speculation of the center of gravity; we shall therefore explain the matter more clearly, after considering the following. If a weight be turned about a fixed center, when let go it will not remain at rest unless the center of gravity shall be in the perpendicular which, through the center about which it is turned, falls to the center of the world. But it will remain on that perpendicular in two points, the one farthest from the center of the world; the other indeed as near to the same as may be allowed. Let there be a body A, whose center of gravity is B, resting on the inflexible line B C, with which it freely turns about the center C. Let the perpendicular B C D be drawn through the center of the world. Therefore let the weight A first be according to the slender B center, on the perpendicular itself above the center C, namely at B. Let it be moved and descend to E; afterward to F, that is, again on the same perpendicular below the center C. Therefore it will describe a circle from the center C, namely B E F, intersecting the perpendicular in two opposite points B F; I say, the weight, freely let go

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EXERCITATIONES. 21 missum in duobus tantum punctis suapte naturâ perman- surum, B F, in B, primò, quoniam cum corpus ipsum A à perpendiculari, quæ superficii loco intelligitur A B C D per centrum grauitatis diuidatur, in partes diuiditur æ- que ponderantes, quare in neutram partem inclinabit. Stabit igitur erectum, lineæ ipsi fultum, inflexibili B C, quæ nititur puncto C. In E verò non stabit, quippe quod eo situ centrum ipsum grauitatis sit extra perpendiculara- rem, & ideo extra fulcimentum stabile C. In F verò ite- rum stabit, pendens à centro C, propterea quòd & ibi ab eadem perpendiculari diuidatur per grauitatis centrum in partes æqueponderantes. Est igitur respectu B, ipsum punctum C, fulcimentum deorsum, respectu verò F, ful- cimentum sursum. At quia linea D F C B, à centro mundi, quod est extra circulum, B E F, circulum ipsum per cen- trum C secat, erit pars eius D F quidem breuissima, ipsa verò D B longissima, ex propos. 8. lib. 3. Elem. Pondus igi- tur A conuersum seu liberè motum circa centrum C, in duobus tantum locis perpendicularis lineæ stabit remo- tissimo altero, vt est B, altero verò eidem quamproximo, vt est F. Hoc idem egregiè demonstrauit G. V bald. in suis Mechanicis, Tractatu de Libra prop. 1. Ad hæc autem dubitare quis posset, cur experientiâ docente, pondera quæ infra fulcimentum habent, vt lan- cea sarissaue ad planum horizontis perpendiculariter e- recta, licet eo casu grauitatis centrum in ipsa perpendicu- lari constituatur, non stet quidem, sed altrinsecus ca- dat? C 3 Sit

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EXERCISES. 21 being placed in only two points, by its very nature it would remain supporting, B F, in B, first, because when the body A itself, from the perpendicular, which is understood in place of the surface A B C D, is divided through the center of gravity, it is divided into parts of equal weight, wherefore it will incline to neither side. Therefore it will stand upright, supported by the line itself, the inflexible B C, which rests on the point C. But in E it will not stand, since in that position the center of gravity itself is outside the perpendicular, and therefore outside the stable support C. But in F again it will stand, hanging from the center C, because there too it is divided by the same perpendicular through the center of gravity into parts of equal weight. Therefore, with respect to B, the very point C is the support downward; with respect, however, to F, the support upward. But because the line D F C B, from the center of the world, which is outside the circle B E F, cuts the circle itself through the center C, that part of it D F will indeed be the shortest, but D B the longest, according to proposition 8, book 3 of the Elements. Therefore the weight A, turned or freely moved about the center C, will stand in only two places of the perpendicular line, the farther from the other, as B is, but the nearer to it, as F is. The same was excellently demonstrated by G. V bald. in his Mechanical Works, Treatise on the Balance, proposition 1. To this one might also doubt why, experience teaching, weights which have their support below, as a lance or javelin erected perpendicularly to the horizontal plane, although in that case the center of gravity is placed in the very perpendicular, do not stand, but fall to one side? C 3 Sit

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IN MECHAN. ARIST. PROBL. Sit enim horizontis planum AB, cui in puncto C perpendiculariter erecta statuatur sarissa DC, cuius grauitatis centrum E, in ipsa perpendiculari. Stabit ergo, ex præmissis, & certè stare debuit, staretque, ni vitium obstaret materiæ; non stat autem, quia difficillimum est grauitatis centrum, suapte naturâ indiuisibile, ita ad amussim sistere, vt in neutram partem à perpendiculari declinet. Hæc igitur ex ijs speculationibus est, quæ ad praxim, materiæ vitio impediente, aut vix aut nunquam rediguntur. Hinc autem ea quæstio soluitur, Cur ij qui sarissam erectam digito summo sustinere conantur, non stent quidem, sed digiti motu, sarissæ motum sequantur. Id certè agit, qui nutantis sarissæ, digito, motum sequitur; vt in ipso motu digitum assiduè centro grauitatis sarissæ supponat, vnde sit vt nunquam extra fulcimentum permanens, nunquam cadat. Similis huic alia quoque dubitatio soluitur: Nempe, Curturbines, quibus pueri ludunt, dum quidem rotantur, stent erecti, rotatione vero cessante, cadant. Esto enim Turbo AB, cuius grauitatis centrum C, planum horizontis DE, linea Horizonti perpendicularis ABC, transiens per centrum grauitatis C, sit autem fulcimentum in B. Itaq[ue] cum centrum grauitatis C sit in ipsa perpendiculari, stabit ex demon- stratis,

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IN MECHAN. ARIST. PROBL. Let there be the plane AB of the horizon, upon which, at the point C, there be erected perpendicularly a sarissa DC, whose center of gravity E is in that very perpendicular. Therefore it will stand, from what has been premised, and certainly ought to stand, and would stand, unless some defect of the material hindered it; but in fact it does not stand, because it is most difficult for the center of gravity, being of its nature indivisible, so exactly to be placed that it should incline to neither side from the perpendicular. This therefore is one of those speculations which, because of the defect of the material hindering, are reduced to practice either with great difficulty or never at all. Hence also that question is solved: why those who try to support a standing sarissa with the tip of a finger do not in fact stand, but with the motion of the finger follow the motion of the sarissa. Certainly the one who follows with his finger the motion of a tottering sarissa is doing this: that in the very act of motion he continually places the finger beneath the center of gravity of the sarissa, so that, never remaining outside the support, it never falls. Another similar doubt is also resolved, namely: why tops, with which boys play, while they are spinning stand upright, but when the spinning has ceased, fall. Let there be, then, a top AB, whose center of gravity is C, the plane of the horizon DE, a line ABC perpendicular to the horizon, passing through the center of gravity C; and let the support be at B. So when the center of gravity C is in the very perpendicular, it will stand, from what has been demonstrated,

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EXERCITATIONES. 23 stratis, at ex vitio materiæ non stabit. Modò, vt assolet, ra- pido motu rotetur. Dico, Turbinem, motu seu rotatione durante stare. ea autem paullatim elanguescente in ca- sum vergere; cessante verò penitus cadere. fit enim ex in- æqualitate materiæ, vel operis ruditate, vel aliâ quauis ex caussa, grauitatis centrum non esse in C, sed exempli gratiâ vbi F, notentur autem hinc inde Turbinis latera notis G H. Vtique cum F extra perpendicularem fuerit, cadet Turbo ad partem G; at id ne fiat, efficitur velocita- te motus, quo centrum F transfertur in contrariam par- tem, vbi I. non autem cadit versus H, quoniam eadem ve- locitate iterum transfertur in F, quamobrem cum huius- cemodi centri assidua circa perpendicularem fiat trans- latio, ad nullam partem Turbo cadere potest; elangue- scente verò motu rotans, paullatim incipit inclinari, do- nec eo penitus cessante, ad eam partem cadit, ad quam à perpendiculari grauitatis centrum vergit. Describit au- tem in rotatione grauitatis centrum, quod in medio non est paruum circulum, per cuius centrum ipsa perpendi- cularis pertingit. Modò redeuntes ad libram, cuius fulcimentum est sursum, alio principio, nempe Mechanico, cur depressa ad æqualitatem reuertatur, demonstrabimus, Sit

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EXERCISES. 23 When layered, but from a defect in the material it will not stand. Only let it, as is usual, be rotated with a rapid motion. I say that a top, while the motion or rotation continues, stands; but as that motion gradually dies away it tends toward falling; and when it has ceased entirely, it falls. For from the inequality of the material, or the roughness of the workmanship, or from any other cause, the center of gravity is not at C, but, for example, at F; and let the sides of the top be marked on either side by the signs G and H. Certainly, when F is outside the perpendicular, the top will fall toward the side G; but lest this happen, it is prevented by the velocity of the motion, by which the center F is transferred to the opposite side, where I is. Nor does it fall toward H, since by the same velocity it is again transferred to F. Wherefore, since there is a continual transfer of this kind of the center around the perpendicular, the top cannot fall to any side; but when the motion grows weak, it gradually begins to incline, until, when it has entirely ceased, it falls to that side toward which the center of gravity inclines from the perpendicular. Moreover, in rotation the center of gravity, which is not in the middle, describes a small circle, through whose center the perpendicular itself reaches. Now returning to the balance, whose support is above, we shall demonstrate, by another principle, namely a mechanical one, why the depressed side returns to equality. Let

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IN MECHAN. ARIST. PROBL. Sit igitur, vt superiùs, libra AB, cuius centrum grauitatis C, fulcimentum, verò sursum in D libræ quidem in C perpendiculariter coniunctum. Perpendicularis verò quæ per fulcimentum, & grauitatis cætrum transiens ad mundi centrum tendit DLE, stante igitur librâ in sua æqualitate, erit centrum grauitatis C in ipsa perpendiculari infra quidem fulcimentum D. Loco verò, mundi centro quàm proximo. Pondus posthæc apponatur in B, Declinabit autem pars CB, in HF, eleuatâ interim parte AC, in GH. Mota igitur libra tota, circa fulcimentum D mouebitur circa idem centrum, & grauitatis centrum C, describens portionem circuli CH, fietq[ue] C in H, & quoniam H, hoc est C, extra perpendicularem sit, amoto pondere, ex lance B, cuius pressione libra declinauerat, centrum grauitatis per eandem circuli portionem HC, ad perpendicularem descendet, donec iterum in ea quiescat, quo casu libra AB ad æquilibrium reuertetur: quod fuerat demonstrandum. His ita declaratis, ostendemus, (quod nullus ante nos animaduertit) harum librarum, quæ fulcimentum habent sursum, eam esse naturam, vt non à quousi pondere apposito moueantur, vel penitus declinent. Ijsdem enim stantibus, addatur quoduis pondus lanci B; Itaque si tale fuerit quod superet resistentiam, quam illi

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IN MECHAN. ARIST. PROBL. Let there be, then, as before, the balance AB, whose center of gravity is C, and whose fulcrum, placed above, is at D; the balance itself at C being joined perpendicularly. But the perpendicular which passes through the fulcrum and the center of gravity, tending toward the center of the world, is DLE. Therefore, the balance standing in its equilibrium, the center of gravity C will be on the same perpendicular, below the fulcrum D. And let the place be as near as possible to the center of the world. After this let a weight be added at B. But the part CB will incline into HF, while meanwhile the part AC is raised into GH. The whole balance therefore being moved, it will move around the same center about the fulcrum D, and the center of gravity C, describing the portion of the circle CH, will become C in H; and since H, that is C, is outside the perpendicular, when the weight is removed from the scale B, by whose pressure the balance had inclined, the center of gravity through the same portion of the circle HC will descend to the perpendicular, until it again rests in it, in which case the balance AB will return to equilibrium: which was to be demonstrated. These things having thus been explained, we shall show, (what no one before us has observed) that the nature of these balances, which have the fulcrum above, is such that they are not moved by any weight whatever when applied, nor do they decline entirely. For while they stand in the same position, let any weight whatever be added to the scale B; and so if it be such as to exceed the resistance, which to it

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EXERCITATIONES. 25 illi facit centrum grauitatis contra naturam elatum in H mouebitur quædam libra. Sin autem tam parui momenti sit, vt eam resistentiam non vincat, stante circa locum insimum centro C, non mouebitur aut saltem parum, ipsa libra. Hinc colligimus fieri posse, libras illas, quæ non " quouis, quantumuis paruo pondere declinant, eas fulcimentum habere sursum. His addimus, cæteris paribus, resistentiam eò esse maiorem, quo minus grauitatis centrum distat à fulcimento sursum, circa quod ipsa libra aduertitur. Esto libra A B, cuius grauitatis centrum C, & primò quidem eius fulcimentum sursum sit vbi D, itaque si apposito pondere declinauerit libra ad partes B, punctum C, dum ascendet describet portionem circuli C E. fulciatur iterum sursum puncto F, & iterum declinet ad partes B, & iterum punctum C, dum ascendet, circuli portionem describet C G. Est autem minor angulus contactus A C E, angulo A C G, magis ergo sursum, hoc est, ad naturam sui feretur C, per C G, ex centro F, quàm per C E, ex centro D, quod fuerat demonstrandum. Hæc autem resistentia ex eodem fulcimento & eodem pondere eo faciliùs superabitur, quo longius brachium libræ fuerit. Esto enim iterum libra A B, cuius fulcimentum D, centrum grauitatis C, sit & alia libra, cuius brachia breuiora E F, idem habens centrum C, & eidem puncto suspensa D. Dico igitur, eodem pondere apposito, faciliùs D decli-

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EXERCISES. 25 if it makes the center of gravity, lifted against nature, in H, a certain balance will be moved. But if it be of so small a moment that it does not overcome that resistance, the balance itself, with the center C standing about the lowest place, will not move, or at least but little. Hence we gather that it is possible for those balances which do not deviate by any weight, however small, to have their support above. To this we add that, all other things being equal, the resistance is greater the less the center of gravity is distant from the upper support, about which the balance itself turns. Let there be a balance A B, whose center of gravity is C, and first let its upper support be at D; thus if, a weight being applied, the balance has inclined toward the parts B, the point C, while it rises, will describe the portion of a circle C E. Let it be supported again above at point F, and again incline toward the parts B, and again the point C, while it rises, will describe the portion of a circle C G. But the angle of contact A C E is smaller than the angle A C G; therefore C will be carried more upward, that is, to its own nature, through C G, from the center F, than through C E, from the center D, which was to be demonstrated. Moreover, this resistance, from the same support and the same weight, will be overcome the more easily the longer the arm of the balance has been. For let there be again a balance A B, whose support D, center of gravity C, and let there be also another balance, whose arms are shorter E F, having the same center C, and suspended from the same point D. I say, therefore, with the same weight applied, more easily D decli-

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declinaturam libram ad partes B, quàm si idem ap- poneretur in F. Demit- tatur enim à puncto B horizonti perpendicula- ris B G, & ab Fitem per- pendicularis F H, Tum iuncta DB, centro D, eo- dem vero spatio DB, circuli portio describatur B I, item iuncta DF eodem centro D, spatio DF, portio circuli de- scribatur FK. est autem maior DB ipsa DF ex propos. 21. lib. I. Elem. quare maioris circuli portio est B I quàm FK. Obliquior autem, hoc est, à perpendicularire motior est motus per FK quàm per B I. maior siquidem est angu- lus K F H angulo IB G. quod nos ita probamus. Ducatur perpendicularis ipsi DF linea LF contingens circulum FK in F, item ipsi DB, perpendicularis MB, contingens circulum BI in B, & quia angulus contingentiæ maioris circuli minor est angulo contingentiæ minoris, erit KFL maior IBM, Rectiautem sunt DFL, DBM, minor ergo DFK residua ipso DBI residuo. Maior autem DFC ex iam citata propos. quâ DBC, erit igitur residuum CFK, multo minus residuo FBI, sed recti sunt CFH, FBG, ex quibus si detrahantur CFK, FBI, erit residuum KFH, maius residuo IBG, plus ergo retrahitur à perpendiculari pondus descendens per FK quàm per B I, minus igitur præualebit resistentiæ in C pondus appensum in F, quàm si appendatur in B. quod fuerat demonstrandum. Possumus & idem quoque aliter ostendere. Sint enim seorsum duæ libræ, maior AB, minor EF, quàm commune grauitatis centrum C, fulcimentum ve- ro sursum D. Producatur perpendicularis DC, in G & fiat CG æqualis CB, CH verò æqualis CF. Sunt igitur duo vectes

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the balance will incline toward the parts at B, than if the same weight were placed at F. For let the perpendiculars to the horizon BG be dropped from point B, and FH from point F. Then, joining DB, and with center D and radius DB, let the portion of a circle BI be described; likewise, joining DF, and with the same center D and radius DF, let the portion of a circle FK be described. But DB is greater than DF, by Proposition 21 of Book I of the Elements; therefore the portion BI of the greater circle is greater than FK. And the motion through FK is more oblique, that is, farther from the perpendicular, than the motion through BI. For the angle KFH is greater than the angle IBG. We prove this thus: let the line LF be drawn perpendicular to DF, touching the circle FK at F; likewise, let MB be drawn perpendicular to DB, touching the circle BI at B. And because the angle of contact of the greater circle is less than the angle of contact of the lesser circle, KFL will be greater than IBM. But DFL and DBM are right angles; therefore the angle DFK is less than the angle DBI by the remainder. Now DFC is greater than DBC, by the proposition already cited, since DBC...; therefore the remainder CFK is much less than the remainder FBI. But CFH and FBG are right angles; if CFK and FBI are subtracted from them, the remainder KFH will be greater than the remainder IBG. Therefore less is taken away from the perpendicular by the weight descending through FK than through BI; and thus the weight suspended at F will prevail less against the resistance in C than if it were suspended at B. Which was to be demonstrated. We can also show the same thing in another way. Let there be separately two balances, the larger AB and the smaller EF, with common center of gravity C, but the support above at D. Let the perpendicular DC be prolonged to G, and let CG be made equal to CB, while CH is made equal to CF. Thus there are two levers

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EXERCITATIONES. 27 vectes D G, D H, quorum quidem commune fulcimentum D, pondus verò C, potentiæ vbi H G. Sunt autem hi vectes eius naturæ, in quibus pòdus est inter fulcimentum & potentiam, itaque vt se habet D C, ad D G, ita potentia in G ad pondus in C, item vt D C ad D H ita potentia in H ad idem pondus C, sed minor est propositio D C, ad D G quàm D C ad D H. minor ergo potentia requiritur in G, hoc est, in B, quàm in H, hoc est in F. Data igitur ponderis æqualitate faciliùs superabitur resistentia C in B, quàm in F: quod ostendendum fuerat. Ad huius libræ naturam illæ quoque rediguntur, quarum iugum non rectum quidem, sed curuum, vel ex rectis sursum in angulum ad fulcimentum detinentibus, nec refert vtrum curuitas sit circuli portio quælibet, aut ellipsis secundum alterum diametrorum; quod ita demonstramus. Esto libra, cuius iugum curuum angulatúue ABC, cuius fulcimentum B, æqualia autem brachia A B, B C, & pondera item vtrinq[ue] appensa æqualia. Demittatur ex puncto B ad mundi centrum perpendicularis B D. Stante igitur libra ABC in æquilibrio, erit eius grauitatis D 2

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EXERCISES. 27 levers DG, DH, of which the common fulcrum is D, the weight C, the power where H G. But these levers are of such a nature, in which the weight is between the fulcrum and the power; therefore, as DC is to DG, so is the power at G to the weight at C; likewise, as DC is to DH, so is the power at H to the same weight C; but the proportion DC to DG is smaller than DC to DH. therefore a smaller power is required at G, that is, at B, than at H, that is, at F. Since therefore the weights are equal, the resistance C is more easily overcome at B than at F: which was to be shown. To this nature of the balance are also reduced those whose beam is not straight, but curved, or rising up from straight lines into an angle toward the fulcrum, nor does it matter whether the curvature is any part of a circle, or an ellipse according to one of the diameters; which we demonstrate thus. Let there be a balance, whose beam is curved or angular ABC, whose fulcrum is B, with equal arms AB, BC, and weights likewise hanging equally on both sides. Let the perpendicular BD be let fall from point B to the center of the world. Therefore, the balance ABC standing in equilibrium, the line of its gravity will be D 2

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28 IN MECHAN. ARIST. PROBL. tatis centrum in ipsa perpendiculari BD, puta in F. Ap- ponatur pondus in C, declinabit autem libra, sit autem iuxta positionem FBG. Centrum igitur grauitatis E per portionem EH, erit in H. Ascendit ergo centrum graui- tatis in H, hoc est, iursum, id est, contra eius naturam; a- moto igitur pondere ex C, grauitatis centrum extra per- pendicularem constitutum rursus descendet, & iterum libra ABC ad æquilibrium reuertetur. Hoc idem egre- giè ostendit G. V bald. in tractatu de libra, propos.4. Hinc ratio pendet earum imaguncularum, quas ex contusa papyro ligneaue leui materia compingunt, per- que manus earum ambas, ferreum filum trajicientes, v- trinque plumbea appendunt pondera æqualia, ea quidé lege, vt centrum grauitatis infra pedes imaguncula sta- tuatur. Tunc enim extenso filo imponentes ceu funam- bulos per illud, vltrò citroq; decurrere faciunt, imagun- cula interim erecta & in neutram partem cadente, quod vt figurâ clarius fiat; A H E F G I D clinata imaguncula, & conuersa circa punctum B, si de- clinet

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28 IN MECHAN. ARIST. PROBL. the center in the perpendicular BD itself, say in F. Let a weight be placed at C; the balance will incline, and its position will be like FBG. Therefore the center of gravity E, by the segment EH, will be at H. Thus the center of gravity rises to H, that is, upward, that is, contrary to its nature; therefore, when the weight is removed from C, the center of gravity, set outside the perpendicular, will again descend, and the balance ABC will return once more to equilibrium. G. V. Bald. shows the same thing excellently in the treatise On the Balance, proposition 4. From this depends the explanation of those little figures which they make up from crushed paper or some light wooden material, and by passing an iron wire through both their hands, they hang equal lead weights on each side, on the condition that the center of gravity is positioned below the feet of the little figure. For then, when the wire is stretched, and they place the figure upon it as if on a tightrope, they make it run back and forth along it, while the figure in the meantime remains upright and does not fall to either side, as is made clearer by the figure; A H E F G I D the inclined little figure, having been turned about point B, if it inclines

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EXERCITATIONES. 29 clinet ad partes I, centrum grauitatis eleuabitur in F. Si verò ad partes H eleuabitur in G. quare cum FG loca sint remotiora à mundi centro, quàm sit E, non stabit grauitatis centrum in punctis FG, sed ad infimum locum reuertetur, hoc est, in ipsa perpendiculari in E, & imaguncula ad perpendicularum ipsi HBE filo, hoc est, ipsi horizonti reuertetur. Hinc etiam Arietum, Testudinumque demolitoriarum Machinarum vis pendet, nempe ex ratione librarum, quæ fulcimentum habent sursum. Esto enim Aries AB funi appensus CD, cuius grauitatis centrum D, perpendicularis verò quæ ad mundi centrum ipsa CDE. Stante igitur in æquilibrio machina, centrum grauitatis erit in ipsa perpendiculari. Applicetur alicubi potentia retropellens, eleuabitur igitur centrum grauitatis per circuli portionem DF, cuius semidiameter est CD, fietque iuxta positionem CF. Aries verò in GFH. Dimissa itaque Machina centrum E vtpote graue, non stabit, sed suapte naturâ reuertetur in D. Quadruplici autem de causa motus Arietis violentissimus est ex vi naturalis ponderis, quo deorsum fertur, tum velocitate naturalis motus in descendendo auctæ, tum ex vi potentiæ impellentis, & naturalem motum adiuuantis, tum ex velocitate ex motu violento deorsum & antrorsum impellente acquisitâ. Id etiam addimus, eo validiores fore ictus, quò grauior fuerit Machina, & maius spatium, quo retrotra- hitur, D 3

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EXERCISES. 29 if it inclines to the parts I, the center of gravity will be raised to F. If indeed to the parts H it will be raised to G. Therefore, since FG are places more remote from the center of the world than E is, the center of gravity will not remain in the points FG, but will return to the lowest place, that is, to the very perpendicular in E, and the little figure will return to the perpendicular to itself HBE thread, that is, to the horizon itself. Hence also depends the force of Rams and Tortoises of battering machines, namely from the ratio of the weights which have their support above. For let there be a Ram AB suspended from a rope CD, whose center of gravity is D, but the perpendicular which to the center of the world is itself CDE. Therefore, the machine standing in equilibrium, the center of gravity will be on the very perpendicular. If force is applied driving backward, therefore the center of gravity will be raised through the portion DF of the circle, whose semidiameter is CD, and it will become according to the position CF. The Ram indeed in GFH. When the Machine is thus let go, the center E, being as it were heavy, will not remain, but of its own nature will return to D. But by a fourfold cause the motion of the Ram is most violent, from the force of natural weight, by which it is carried downward, then from the increased speed of its natural motion in descending, then from the force of the impelling power, and assisting the natural motion, then from the speed acquired from the violent motion impelling downward and forward. We add also this, that the heavier the Machine is, the stronger the blows will be, and the greater the space by which it is drawn back, D 3

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30 IN MECHAN. ARIST. PROBL. hitur, grauitate ipsa & spatio tum virium vnione operationem mirum in modum adiuuantibus. Hæc nos de Libra sursum fulcimentum habente, dicta voluimus, nunc de ea, cuius fulcimentum deorsum est, verba faciemus. Altera quæstionis pars: De Libra cuius fulcimentum deorsum est. Si deorsum fuerit, inquit Aristoteles, id quod substat, contrarium facit illi quæ sursum habet, nempe ad æquilibrium non reuertitur. Plus enim, ait, dimidio sit libræ, quæ deorsum est pars, quàm quod perpendiculum secet, quapropter non ascendit. eleuata enim pars leuior est. Hæc ille, qui schemate quoque rem aperit, at eo apud interpretes, & Picolomineum Paraphrastem, ita me[m]dosè lineato, vt inde obscuritas lucis loco, legentibus offundatur. Nos, quod & supra quoque fecimus, nostra figurâ, sole ipso clariorem, ex Aristotelis ipsius mente rem totam efficiemus. Sit libra recta, (hoc est, in æquilibrio constituta) vbi NG. Perpendiculum autem (id est, perpendicularis quæ ad mundicentru[m]) KLM. Bifariam igitur secatur NG. imposito posthæc onere in ipso N, erit quidem N, vbi O. ipsum autem G vbi R. KL autem vbi LP. quare

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30 IN MECHAN. ARIST. PROBL. are carried, by gravity itself and by the distance, with the union of their forces marvelously assisting the operation. What we have wished to say about the balance having its support above, we now shall speak of that whose support is below. The second part of the question: On the balance whose support is below. If it be below, says Aristotle, that which lies beneath acts contrary to that which has it above, namely it does not return to equilibrium. For, he says, the part of the balance which is below is more than half of it, than that which the plumb line cuts, wherefore it does not rise. For the raised part is lighter. Such is his statement, although he also explains the matter by a diagram; but in the interpreters, and in Picolominius the Paraphrast, it is so badly drawn that obscurity is cast upon readers in place of light. We, as we also did above, by our own figure, clearer than the sun itself, shall present the whole matter from Aristotle’s own mind. Let there be a straight balance, that is, one placed in equilibrium, where NG. The plumb line, however, that is, the perpendicular line toward the center of the world, KLM. Therefore NG is divided in two. After a weight is placed here at N, N will be where O is. G itself will be where R is. KL, however, will be where LP. wherefore

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EXERCITATIONES. 31 quare maius est K O, quam L R, ipsa parte P K L. Amoto igitur onere necesse est manere. Incumbit enim onus ex- cessus medietatis eius, vbi est F. Sensus est igitur, idcirco partem iugi K L O inclinatam, ad æquilibrium non re- uerti, propterea quòd maior sit ipsa K L O pars quæ tra- hit, ipsa R K L, quæ trahitur & eleuatur. Potest hoc idem longè simpliciori themate demon- strari. Esto enim libra A B, cuius centrum C, fulcimen- tum vero deorsum D, Per- pendicularis per centrum & fulcimentum transiens E F. Apponatur pondus in B, de- clinabitq[ue] puta ad G H, cen- trum verò C, ex stabili fulci- mento D, circuli portionem describet C I, libra autem secabit E F perpendicularem in K. Æquales autem sunt I G, I H, at ex parte H I desumpta est K I, additaque ipsi I G, maior est ergo tota K G, totâ K H. Non igitur K H habet K G, sed libra, nisi impedita fuerit, cum centro C descendente per I in M, ad ipsam perpendicularem dela- ta, ad inferiorem partem, mutatis vicibus quiescet, facto nempe fulcimento sursum, fietq[ue] horizonti æquedistans iuxta positionem L M N. Demonstratio quidè est hæc, sed non ex proprijs prin- cipijs Mechanicis, nèpe ex ratione cêtri grauitatis petitâ. Iisdem enim stantibus, cu[m] centrum grauitatis C fiat extra perpendicularem, descendens ad I, nunquam reuertetur in C, ascenderit enim; sed si liberè circa centrum D con- uerteretur, descendens vt dictum est per circulum C I M pondus B, fieret in L, A vero in N adepta positione L M N. Cur

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EXERCISES. 31 because K O is greater than L R by the same part P K L. Therefore, when the load is removed, it must remain. For the load rests upon the excess of its half, where F is. The sense is, therefore, that the inclined part of the yoke K L O does not return to equilibrium, because the part K L O which draws, namely R K L, is greater than the part which is drawn and raised. The same thing can be demonstrated from a much simpler theorem. For let there be a balance A B, whose center is C, but whose support is below at D, and a perpendicular passing through the center and the support, E F. Let a weight be placed at B; it will incline, say, toward G H. The center C, however, from the fixed support D, will describe the portion of a circle C I; but the balance will cut the perpendicular E F at K. Now I G and I H are equal, but from the part H I, K I is taken, and added to I G, therefore the whole K G is greater than the whole K H. Therefore K H does not have K G, but the balance, unless it is obstructed, with the center C descending through I to M, carried to the same perpendicular, will rest toward the lower part, with the positions changed; and once the support is made above, and it becomes equally distant from the horizon according to the position L M N. This indeed is a demonstration, but not from the proper principles of Mechanics, namely from the reason of the center of gravity sought. For with the same things standing, when the center of gravity C is made outside the perpendicular, descending to I, it will never return into C, because it will have ascended; but if it were freely turned about the center D, as has been said, the weight B descending through the circle C I M would be in L, and A in N, the position L M N having been attained. Why

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32 IN MECHAN. ARIST. PROBL. Cur autem huius libræ, quæ aliâs in utilis est, meminerit Philosophus, ea videtur caussa, quòd inde vectis virtutem eliciat, vt suo loco videbimus. Id autem valdemirum, hominem acutissimum nihil prorsus de ea libra egisse, quæ fulcimentum nec sursum habet, nec deorsum, sed in ipso exquisitè medio, ita vt centrum grauitatis in ipso- met fulcimento consistat. Nos igitur de hac quod operæ pretium fuerit, & ad rem, qua de agimus, vtile, in medium proferemus. De libra cuius fulcimentum est in medio. Dicimus itaque, libram, cuius fulcimentum nec sursum est, nec deorsum, sed prorsus in medio, nempe in ipso grauitatis centro, vbi brachia & pondera vtrinque apposita fuerint æqualia, si ab æquilibrio mouentur, quomodocunque posita, stare nec ab eo, quem adepta est, situ dimoueri. Quæstionem hanc perperam tractârunt recentiores quidam, Hieron. Cardanus, Nicolaus Tartalea, & alij nonnulli, qui Iordani Nemoracij assertiones sunt secuti, quorum demonstrationes vel paralogismos potiùs egregiè confutauit in libr. Mechanicor. Tractatu de libra propos. 4. Guid. V bald. ad cuius probatissima scripta Lectorem ablegamus. fusissimè enim ibi hac de re & absolutissimè agit. Nos autem quidem paucis ea, quæ ad hanc cognitionem pertinent, explicabimus. Esto enim libra A B, cuius brachia æqualia, & centrum grauitatis in C, brachijs verò A C, C B æqualibus, æqualia pondera hinc inde apponâtur. Tum fulci-

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32 IN MECHAN. ARIST. PROBL. But why the Philosopher should have made mention of this kind of balance, which is otherwise useless, seems to be this: because from it he elicits the power of the lever, as we shall see in its place. Yet it is most surprising that a man of the keenest intellect should have said nothing at all about that sort of balance which has its fulcrum neither above nor below, but exactly in the middle itself, so that the center of gravity lies in the fulcrum itself. We shall therefore set forth, on this point, whatever may be worth the labor, and useful for the matter under discussion. Of the balance whose fulcrum is in the middle. We say, then, that a balance whose fulcrum is neither above nor below, but exactly in the middle, namely in the center of gravity itself, when equal arms and weights are placed on either side, if moved from equilibrium, no matter how it is positioned, remains fixed and is not displaced from the position it has acquired. Some modern writers have treated this question wrongly, Hieronymus Cardanus, Nicolaus Tartalea, and certain others, who followed the assertions of Iordanus Nemoracius, whose demonstrations, or rather paralogisms, Guido Ubaldus excellently refuted in the book Mechanicorum, Treatise on the Balance, proposition 4. We refer the reader to his most authoritative writings. For there he treats this subject very fully and most completely. But we shall explain briefly those things which pertain to this understanding. For let there be a balance A B, whose arms are equal, and whose center of gravity is in C; but with arms A C, C B equal, equal weights are placed on either side. Then the fulcrum-

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EXERCITATIONES. 33 fulcimento in medio, hoc est, vbi grauitatis centrum C applicato per centrum ipsum C ducatur perpendicularis, quæ ad mundi centrum, DCE, sitque primum libra æquedistans horizonti, constituta. Tum ex altera parte pressa moueatur & fiat iuxta positionem F C G. Dico eam dimissam permanere, etenim cum grauitatis centrum sit in ipsa perpendiculari, in neutram partem verget, sed nec vergere potest, quippe quod non circa fulcimentum ceu centrum motus, moueatur grauitatis centrum, sed in ipso sit fulcimento; situm ergo non mutat. Præterea cum per- pendicularis DCE per grauitatis centrum ducatur, cor- pus ipsum ex ponderibus & libra constans ab ea in partes æque ponderantes secatur, & ideo ex centri grauitatis dif- finitione, quam protulit Pappus, corpus ipsum centro grauitatis appensum, dum fertur quiescit, & seruat eam, quam à principio habuit positioné. Et sanè si partes quo- modo libet librâ per grauitatis centrum diuisâ, sunt æ- que ponderantes nec trahent inuicem, nec trahentur, sta- bit ergo libra, & quam adepta fuerat positionem, eam ser- uabit. Id tamen non negamus, difficile esse libras eiusce- modi ex materia fabricare, quippe quod non omnia quæ vera sunt, & euidentissimis demonstrationibus patent, commodè ad praxim, ex artis & materiæ imperfectione, reducuntur. Cæterùm harum librarum ea est virtus, vt vel min- mo pondere altrinsecus apposito, declinet, quod illis quæ centrum sursum habent, non euenire, demonstrauimus. Circa hæc posset cuipiam oriri Dubium, num chor- dulæ, quibus lances appenduntur, variationem aliquam circa ea quæ demonstrata sunt, inducere valeant. Dicimus nullam inde fieri: Esto enim libra AB, cu- ius centrum & fulcimentum C, ab cuius extremitate A dependeat, funiculus AD, ab alia verò B, funiculus BE, E qui-

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EXERCISES. 33 with the support in the middle, that is, when the center of gravity C is applied, let a perpendicular be drawn through the very center C, which to the center of the world, DCE, and let the balance first be placed equidistant from the horizon. Then, pressed from the other side, let it move and be in the position F C G. I say that when released it will remain so, for since the center of gravity lies on that perpendicular itself, it will incline to neither side, nor can it incline, since the center of gravity does not move around the support as its center of motion, but is in the very support; therefore it does not change its position. Moreover, since the perpendicular DCE is drawn through the center of gravity, the body itself, consisting of weights and balance, is divided by it into parts of equal weight, and therefore, from the definition of the center of gravity given by Pappus, the body itself, suspended at its center of gravity, while being carried, remains at rest and preserves the position it had at the beginning. And indeed if the parts, however the balance be divided through the center of gravity, are of equal weight and neither draw one another nor are drawn, the balance will therefore stand, and it will preserve the position it has acquired. We do not deny, however, that it is difficult to fabricate balances of this kind from material, since not everything that is true and made clear by the most evident demonstrations is conveniently reduced to practice, because of the imperfection of art and matter. Furthermore, these balances have this property, that even with the slightest weight applied on either side they incline, which we have shown does not happen in those which have the center above. Regarding these matters, a doubt might arise for someone, namely whether the cords by which the pans are hung can introduce any variation in what has been demonstrated. We say that none comes from this. Let there be a balance AB, whose center and support is C, from whose extremity A hangs the cord AD, and from the other side B, the cord BE, E which...

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IN MECHAN. ARIST. PROBL. quibus appensæ sint æqualis ponderis lances DE. Moucatur libra, fiatque in I CH, funiculi verò in lancibus in IK, HL. secet autem funiculus IK libram AB, in M, LH verò producatur & eandem secet in N. quoniam igitur IC, æqualis est CH, parallelæ autem KI, LN æquales erût alterni anguli MIC, NHC, sed & anguli ad verticem IGH, BCH æquales sunt, quare triangulum IMC, æquale triangulo HNC, & latera lateribus, quæ æqualibus angulis subtenduntur. Æqualis est igitur linea MC lineæ NC. Itaque si pondera lancesue, KL mente concipiantur appensæ in punctis MN, ex brachiorum & ponderum æqualitate æque ponderabunt. quod fuerat demonstrandum. QVÆSTIO III. Cur exiguæ vires (quod etiam à principio dixerat) vectemagna mouent pondera, vectes insuper onus accipientes, cum facilius sit, minorem mouere grauitatem, minor est autem sine vecte? ARistoteles ita quæstionem proponit, vt eam Rhetorico quodam fuco admirabiliorem faciat. Soluit autem hoc pacto, inquies, fieri posse eam esse caussam, quod vectis sit libra, eius nempe generis quod fulcimentum habet deorsum, atque idcirco in ipsa pressione in partes inæquales vectem diuidi. Figu-

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IN MECHAN. ARIST. PROBL. to which scales of equal weight are suspended DE. Let the balance be moved, and let it be so arranged in ICH, the strings on the scales in IK, HL. But let the string IK cut the balance AB in M, and let LH be produced and cut the same in N. Since therefore IC is equal to CH, and the lines KI, LN are parallel, the alternate angles MIC, NHC will be equal; but also the angles at the vertex IGH, BCH are equal, wherefore triangle IMC is equal to triangle HNC, and the sides corresponding to equal angles are equal. Therefore line MC is equal to line NC. Thus if the weights or scales KL are conceived to be suspended at the points MN, by the equality of the arms and weights they will weigh equally. which was to be demonstrated. QUESTION III. Why do small forces, as he had said also at the beginning, move great weights by means of a lever, although levers themselves receive the load, when it is easier to move a smaller weight, and yet one is smaller without a lever? Aristotle proposes the question in such a way as to make it more remarkable by a certain rhetorical coloring. He solves it, you will say, in this way: it may happen that this is the cause, namely that the lever is a balance, of that kind which has its support below, and therefore in the pressure itself the lever is divided into unequal parts. Figu-

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EXERCITATIONES. 35 Figura quam ex- hibet, vix ferè quid si- bi velit explicat. Nos ad eius metem aliam proponemus eamq[ue] longè clariorem. Esto vectis A B, cuius fulcimentum deorsum in C, pon- dus D, potentia ex vecte, pondus sustinens E. Perpendi- cularis per fulcimentum F C G. Itaque quoniam poten- tia in E non superat pondus D, nec ab eo superatur, stat vectis cum potentia Horizonti æquidistans, hoc est, in æ- quilibrio, vectis autem in puncto C diuiditur in partes æ- que ponderantes. Modo præualeat potentia ponderi, & vectem deprimat, fiat autem in L C H, erit igitur B, in L, A in H, D in K, & C F, quæ vectem in partes æque ponde- rantes diuidebat, in C I. Iam igitur non æque ponderant partes, siquidem pars vectis F C I, aufertur parti H C I, & adiungitur parti I C L, quæ ideo sit ponderosior, vnde & potentia ad ponderis eleuationem adiuvatur. Eadem i- gitur vtitur hic demonstratione, quam in explicando ef- fectu libræ, cuius fulcimentum deorsum est, adhibuerat. Nec alia de causa, vt supra notauimus, videtur eius libræ in superiori quæstione, considerationem introduxisse. Et sanè verum est quod concludit, Veruntamen minimi est momenti ad tantam vim parua illa adiectio, quæ parti ve- ctis depressæ in ipsa depressione adiungitur. Aliunde igi- tur tantæ rei causa est petenda, quod & nos deinceps fa- ciemus. Videtur autem ipse quoque Aristoteles non sibi prorsus in assignata ratione satisfecisse, & ideo subiungit: quoniam ab æquali pondere celerius mouetur maior ea- rum quæ à centro sunt: duo verò pondera, quod mouet & quod E 2

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EXERCISES. 35 The figure which he presents scarcely explains what it wants to say. We shall propose another to its purpose, and one much clearer. Let there be a lever AB, whose support is below at C, the weight D, the power from the lever sustaining the weight E. A perpendicular through the support FCG. Since therefore the power at E neither exceeds the weight D, nor is exceeded by it, the lever stands with the power parallel to the horizon, that is, in equilibrium; but the lever is divided at the point C into parts of equal weight. Now if the power prevails over the weight, and depresses the lever, and let this be in LCH, then B will be at L, A at H, D at K, and CF, which was dividing the lever into parts of equal weight, will be CI. Now therefore the parts do not weigh equally, since the part of the lever FCI is taken away from the part HCI, and added to the part ICL, which therefore becomes heavier, whence also the power is assisted toward the raising of the weight. He therefore uses here the same demonstration which he had employed in explaining the effect of the balance, whose support is below. Nor, as we noted above, does it seem for any other reason that he introduced consideration of that balance in the previous question. And indeed what he concludes is true. Nevertheless, that small addition, which is joined in the very act of depression to the depressed part of the lever, is of very little moment in producing so great a force. The cause of so great a thing must therefore be sought elsewhere, and this we shall also do in what follows. But Aristotle himself also seems not to have been entirely satisfied with the reason he assigned, and therefore he adds: since a greater body, among those which are at equal weight, is moved more quickly than that which is nearer the center: but two weights, namely that which moves and that which...

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36 IN MECHAN. ARIST. PROBL. quod mouetur. quod igitur motum pondus ad mouens longitudo patitur ad longitudinem, semper autem quâ- tum ab hypomochsio (id est, fulcimento) distabit magis, tanto facilius mouebit. Causa autem est, quæ retro com- memorata est, quoniam quæ plus à centro distat maiore describit circulum. quare ab eadem potentia plus supera- bitur id quod mouetur, quæ plus à fulcimento distat. Hæc ille, qui asserit duo pondera in vecte considerari, Pondus nempe motum, & mouentem Potentiam (hanc enim po- deris habere vim atq[ue] rationem certum est) Vires autem potentiam acquirere ex brachij longitudine, & ex inde consequenti velocitate, quo enim brachia longiora, eo in extremitate velociora, atque idcirco ita se habere mo- tum pondus ad potentiam mouentem, vt brachij longi- tudo ad brachij longitudinem: brachia autem vocamus, partes illas vectis, quæ à fulcimento ad vtranque vectis extremitatem pertingunt, & ideo quantum à fulcimento potentia distabit magis, eo faciliùs pondus mouebit. Vera vtique & exploratissima hæc assertio est. Ve- runtamen, causam huiusce mirabilis effectus, esse velo- citatem, quæ brachij longitudinem consequitur, non af- firmamus. quæ enim velocitas in restante? Stant autem vectis, & libra dum manent in æquilibrio, & nihilo secius parua potentia ingens sustinet pondus. Dicet ad hæc quispiam, velocitatem in longiori bra- chio si non actu, saltem potentia esse maiorem. At quæso quid in re quæ est actu, momenti habet potentia? actu e- nim sustinet, sustinens. Consequitur, (id vtique fatemur) necessariò velocitas maior motu brachij maioris; non ta- men causa est cur vis loco vbi velocitas maior sit, apposi- ta magis moueat. Sanè ex velocitate, dum mouentur, po- dus acquirere corpora, tum proiecta, tum cadentia cer- tum est, quod etiam in quæstione 19. cum Philosopho co- fide-

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36 IN MECHAN. ARIST. PROBL. that which is moved. Therefore, just as the length of the moved weight bears to the moving power, so the length bears to the length; and, however much it is farther from the hypomochlion (that is, the fulcrum), so much the more easily will it move. And the cause is the one mentioned before, namely, because that which is farther from the center describes a greater circle. Wherefore, from the same power, that which is moved farther from the fulcrum will be overcome more. This is what he says, who asserts that two weights are to be considered in the lever: namely, the moved weight and the moving power (for it is certain that this has the force and ratio of weight). But powers acquire strength from the length of the arm, and from the speed consequent upon it; for the longer the arms are, the swifter they are at the extremity, and therefore the moved weight stands to the moving power in the same way as the length of the arm stands to the length of the arm: and by arms we mean those parts of the lever which extend from the fulcrum to both ends of the lever, and therefore, the more the power is distant from the fulcrum, the more easily it will move the weight. This assertion is indeed true and most certain. Nevertheless, we do not affirm that the cause of this wonderful effect is the speed which follows the length of the arm. For what speed is there in a resting object? Yet the lever and the balance remain standing while in equilibrium, and nevertheless a small power sustains a huge weight. Someone may say in reply that speed in the longer arm, if not actually, is at least greater potentially. But I ask, what importance has potentiality in a thing that exists in actuality? For it actually sustains, while sustaining. It follows then, as we freely admit, that greater speed necessarily accompanies the motion of the greater arm; yet it is not the cause why the force, placed where the speed is greater, moves more. Certainly it is known that bodies, both projected and falling, acquire weight from speed while they are in motion, as also in question 19, with the Philosopher we confi-

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EXERCITATIONES. 37 fiderabimus. Sed hoc ex velocitate & motu fit, quæ sunt actu. At brachia in ipso æquilibrio sustinent actu quidem, sed non mouentur. Cæterùm videtur Aristoteles id sub- odorasse, quod postea Archimedes, Mechanicorum princeps, in propos. 6. primi AEquponderantium explicè protulit & probauit: nempe in æquilibrio ita esse pondus ad pondus, vt brachium ad brachium, ratione permutata. Esto enim vectis A B, quomodolibet fulcimento diuisus in C. appèdatur autem in A, pondus D, in B verò pondus E, ita se habens ad pondus D, vt ipsa A C ad CB. Stabit igitur vectis, & neutram in partem verget, erit enim centrum grauitatis in C, diuiso nempe ibi vecte in partes æqueponderantes. Hoc post Archimedem, & insignes illos veteres Mechanicos præclarissimè demonstrauit G. Vbaldus in Mechanicis, Tractatu de Libra propos. 6. nec non de Vecte propos. 4. Cæterùm vt aliquid interim, quod nostrum sit, afferamus, liceat nobis egregios illos viros interrogare, quænam mirabilis eius effectionis sit caussa? Dicent permutatam proportionem. Teneo, at nondum acquiesco: petam enim, Cur ea rationis permutatio mirabilem illum effectum pariat. Hoc quod illi non docent, puto nos, ignorantiæ somno sepultos, somniasse. Æqualitatem status esse caussam, nemo, vt puto, inficiabitur. res est enim per se clara. Esto si- quidem linea quæpiam AB, applicetur extremitati A potentiæ E 3

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EXERCISES. 37 we shall trust. But this comes from velocity and motion, which are in act. But the arms, in the very state of equilibrium, do indeed sustain, in act, but they do not move. Moreover, Aristotle seems to have half- suspected what Archimedes afterward, the prince of Mechanics, expressly stated and proved in proposition 6 of the first book of the Treatise on Equal Weights: namely, that in equilibrium weight is to weight as arm is to arm, with the proportion reversed. For let there be a lever A B, divided by a sup- port at C in any way whatever. Let there be suspended at A the weight D, and at B the weight E, so related to the weight D as AC itself is to CB. The lever will therefore stand still, and incline to neither side; for the center of gravity will be at C, since the lever has been divided there into equal-weight parts. This, after Archimedes, and those eminent ancient Mechanicians, G. Ubaldus demonstrated most clearly in the Mechanical Works, in the Treatise on the Balance, proposition 6, and also on the Lever, proposition 4. Moreover, in order meanwhile to offer something of our own, let us be allowed to ask those distinguished men what is the cause of so marvelous an effect? They will say: a reversed proportion. I understand; but I am not yet satisfied: for I shall ask why that reversal of ratio produces that marvelous effect. What they do not teach, I think we, buried in the sleep of ignorance, have dreamt. Equality of position is the cause, no one, I think, will deny. The thing is indeed clear in itself. Suppose, then, some line AB; let there be applied to the extremity A a power E 3

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38 IN MECHAN. ARIST. PROBL. tentia quædam quæ lineam ad se trahat ad partes nempe A, Tum in B quædam alia potentia ipsi quæ in A potentiæ, æqualis, quæ lineam trahat simili modo ad partes B. Datâ igitur harum potentiarum æqualitate, linea AB, nec ad partes A, nec ad partes B transferetur, sed prorsus immo- bilis stabit. His ita constitutis, Dico vecte quomodolibet diuiso, ponderibusque vtrinque appositis, permutatâ propor- tione sibi inuicem respondentibus, rem esse redactam ad æqualitatem, & inde statum fieri, hoc est, æquilibrium. Esto enim vectis AB, quomodolibet diuisus in C, & ipsi quidem C fulcimentum supponatur. Appendantur quoque vtrinque pondera ex ratione brachiorum AC, CB, sibi inuicem permutatim respondentia, sint; DE. Dico vectem ex æqualitate, in neutram partem inclina- turu[m], sed permansurum in æquilibrio. quoniam enim Po- dus D idem potest quod brachium CB, addatur in dire- ctum ipsi AC, recta AF æqualis ipsi CB, item quoniam Pondus E id potest quod brachium AC, rectæ CB ad- datur in directum BG, ipsi AC æqualis. Igitur cum par- tes CA, AF totius FC, æquales sint partibus CB, BG, totius CG, erit totum FC, toti CG æquale. Diuisus ita- que

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38 IN MECHAN. ARIST. PROBL. there is a certain power which draws a line to itself toward the parts, namely A; then in B there is another power equal to that power in A, which draws the line in a similar way toward the parts B. Since, therefore, these powers are equal, the line AB will be moved neither toward A nor toward B, but will stand entirely motionless. These things being established, I say that a lever, divided however you please, and weights applied on either side corresponding to one another in a reversed proportion, has been reduced to equality, and thence motion ceases, that is, equilibrium results. Let there be a lever AB divided however you please at C, and let a support be placed beneath C itself. Let weights also be suspended on either side, corresponding to one another by the ratio of the arms AC, CB, let them be DE. I say that the lever, by reason of equality, will incline to neither side, but will remain in equilibrium. For since the weight D can do the same as the arm CB, let there be added in a straight line to AC itself the straight line AF equal to CB; likewise, since the weight E can do the same as the arm AC, let there be added to the straight line CB in a straight line BG, equal to AC. Therefore, since the parts CA, AF of the whole FC are equal to the parts CB, BG of the whole CG, the whole FC will be equal to the whole CG. Thus divided

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EXERCITATIONES. 39 que erit vectis FG in partes æquales FC, CG in puncto fulcimenti C. Et quoniam æquale in æquale non agit, stabit vectis & in neutram partem inclinabit. Rursum quoniam ad partem FC, duæ sunt brachiorum potentiæ FA, HC, appendantur puncto F, duo pondera H, I, ipsis DE æqualia, item puncto G, alia duo pondera ijsdem DE æqualia KL, iterum æque ponderabit, quippe quod æqualibus brachijs FCCG æqualia appensa sint pondera HI KL. Cur igitur seruata permutatim brachiorum & ponderum proportione fiat æquilibrium, ex his quæ de- monstrauiimus, clarè patet. Sed forte dicet quispiam, si brachia, pondera sunt, vel ponderibus æquipollentia, sustinenti duplicabitur pondus. Esto enim vectis AB, ita diuisus in C, vt pars maior CB minori AC sit in proportione quintu- pla. Appendatur autem in A pondus D, quintuplu[m] ponderi E appenso in B. Si igitur brachio AC, quod est vnum, addatur pondus D, quod est quinque, fient sex, item si brachio CB, quod est quinque, addatur pondus E, quod est vnum, fient sex. Fulcimentum igitur sustinebit duodecim, quod est ab- surdum ex ijs quæ clarè demonstrauit G. V bald. in Me- chan. tractatu de Libra propos.5. His respondemus, bra- chia quidem operari non pondere, sed potentiâ, quæ vis quædam est, non autem pondus. Etsi & illud verum sit, da- to vecte ponderoso, fulcimentum tum ponderum appen- sorum, tum vectis ipsius pondus sustinere. Iacta huiuscemodi, quam diximus, æqualitate, se- quitur

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EXERCISES. 39 which will be the lever FG into equal parts FC, CG at the point of support C. And since equal does not act upon equal, the lever will remain still and incline to neither side. Again, since on the part FC there are two powers of the arms, FA, HC, if at point F are hung two weights H, I, equal to the same DE, likewise at point G, two other weights KL, equal to the same DE, it will again weigh equally, since to equal arms FCCG equal weights are hung, HI KL. Why then, by preserving in turn the proportion of the arms and the weights, equilibrium comes about, is clearly shown by the things we have demonstrated. But perhaps someone will say: if the arms are the weights, or are equivalent to the weights, the burden on the support will be doubled. Let there be a lever AB, so divided at C that the larger part CB is to the smaller AC in a fivefold proportion. Let there be hung at A a weight D, five times the weight E hung at B. If then to the arm AC, which is one, there is added the weight D, which is five, there will be six; likewise if to the arm CB, which is five, there is added the weight E, which is one, there will be six. The support therefore will sustain twelve, which is absurd, as G. V. Bald. clearly demonstrated in the Mechanical treatise on the Balance, proposition 5. To this we reply that the arms indeed do not act by weight, but by power, which is a certain force, and not weight. Although it is also true that, when a weighted lever is given, the support must bear both the weights hung on it and the weight of the lever itself. Having established such an equality, as we have said, it follows

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40 IN MECHAN. ARIST. PROBL. quitur necessariò, centrum grauitatis ipsius vectis cum appensis ponderibus, ac si vnum idemque esset corpus cadere in perpendiculari quæ per centrum ipsum & ful- cimentum transiens ad mundi centrum pertingit. QVÆSTIO IV. Quærit hic Aristoteles, cur ij qui in nauis medio sunt remiges ma- ximè nauem moueant? A It, ideo fortasse fieri, quòd remus vectis sit, fulcimen- tum verò scalmus, stat enim. Pondus autem mare i- psum, quod à remo propellitur, mouens verò ipsum remi- gem, semper autem plus mouere ponderis qui mouet, quo magis distat à fulcimento. Ita enim maiorem fieri quæ ex centro; Scalmum verò centrum esse. Cæterùm in medio nauis plurimum remi intus esse. Ibi enim nauem esse latissimam. Moueri autem nauim, quoniam appelle- te mari remo, extremu[m] illius quod intus est anterius pro- mouetur, cuius motum nauis sequitur, cui scalmus alliga- tur. Vbi autem plurimum maris diuidit remus, eo maximè necesse esse propelli. Plurimum autem diuidi vbi plurima pars remi à scalmo est. Rem facilem, eo quod verbis potu- erit, schemate non declarauit, nos autem apponemus. Esto enim nauis AB, mare CD, remorum alter, qui ad proram EF, cu- ius scalmus G, alter verò in medio na- uis, HI, circa scalmum K. Ait igitur, remos esse vectes, scalmos verò fulci- menta, pondus quod remo, ceu vecte, mouetur mare ipsum. Itaque quoniam nauis lata est in medio vbi Scalmus K maior pars KH intra nauim est, minor verò KL, extra. Contra autem remi ad proram, nempe EF pars minor EG intra

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40 IN MECHAN. ARIST. PROBL. there follows necessarily that the center of gravity of the lever itself, together with the weights suspended from it, is as if one and the same body were falling along the perpendicular which passes through that center and the fulcrum, and reaches the center of the world. QUESTION IV. Here Aristotle asks why those rowers who are in the middle of the ship move the ship most strongly? He says that perhaps it is so because the oar is a lever, and the fulcrum is the thole-pin; for it stands still. But the weight is the sea itself, which is driven by the oar, and that which moves is the rower himself; and always the greater the weight that is moved, the more it moves, the farther it is from the fulcrum. For thus the motion becomes greater from the center; and the thole-pin is the center. Moreover, in the middle of the ship the oar is mostly inside. For there the ship is widest. But the ship is moved because, when the sea is struck by the oar, the extremity of that part which is inside is carried forward first, whose motion the ship follows, to which the thole-pin is attached. But where the oar divides the greatest amount of sea, there is the greatest necessity for it to be pushed forward. Now the greatest division occurs where the greater part of the oar is away from the thole-pin. He did not explain the oar, because he could do so in words, by diagram; but we shall add one. Let there be the ship AB, the sea CD, one oar, which is toward the prow EF, whose thole-pin is G, and another in the middle of the ship, HI, around the thole-pin K. He says, then, that the oars are levers, the thole-pins are fulcrums, and the weight moved by the oar, as by a lever, is the sea itself. Therefore, since the ship is broad in the middle, where the thole-pin K is, the greater part KH is inside the ship, the lesser KL outside. On the other hand, with the oars toward the prow, namely EF, the lesser part EG inside

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EXERCITATIONES. 41 intra nauim, pars verò maior GF extra nauim est. Pondus autem eò faciliùs mouetur, quo maior est vectis pars, quæ à fulcimento est ad mouentem potentiam. Acutè sanè Philosophus. Ego autem si per modestiam liceret, dicerem, non quidem esse fulcimentum scalmu[m], sed mare ipsum, pondus vero nauim, ad locum scalmi, ne[m]pe inter mouentem potentiam, & fulcimentum positum, etenim & eo pacto possumus vti vecte, quod obseruat & demonstrat G. V baldus tractatu de vecte propos. 2. Erunt igitur in descripta figura puncta FI, quæ in mar sunt, fulcimenta, quibus remorum extrema in ipsa impulsionenituntur, pondera verò seu pondus pluribus vectibus & potentijs impulsum nauis ipsa, quæ scalmis est annexa. Resistent igitur mari, cedente autem impulsionibus scalmo, nauis eo transfertur, quo scalmi ab ipsa potentia mouente in anteriorem partem pelluntur. quoniam autem vt FG ad FE ita potentia mouens in E ad pondus motum in G. item vt IK ad IH ita potentia mouens in H ad pondus motum in K, maior autem est proportio FG ad FE quàm proportio IK ad IH. Maiori indiget potentia vt pellatur pondus in G quàm pondus in K. Hæc certè vti diximus ita se habent. Philosophi autem ratio tunc procederet, si stante naui immobili, vt fit vbi à Remoræ occulta vi aut ab alio impedimento retinetur, remiges in ipso remigandi actu mare pulsarent, Tunc enim verè scalmus fieret fulcimentum, mare autem pondus, remex verò ipse mouens. Addimus, falsum videri quod asserit Aristoteles, nempe illos qui in media naui sunt, remiges, maximè nauim mouere, facilius, melius dixisset. Si enim maximè, quod ait, denotat maximo spatio, & velocius prorsus falsum, etenim tardius mouent & minori spatio, quod nos ita demonstramus. F Esto

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EXERCISES. 41 within the ship, but the greater part GF is outside the ship. Moreover, the weight is moved the more easily, the greater is the part of the lever which lies from the fulcrum to the moving power. A keen remark indeed of the Philosopher. But if modesty allowed me, I would say that the fulcrum is not indeed the thole-pin, but the sea itself, while the weight is the ship, placed at the place of the thole-pin, namely between the moving power and the fulcrum; and indeed in this way also we may use a lever, as G. V. Baldus observes and demonstrates in his treatise on the lever, proposition 2. Therefore, in the figure described, the points FI, which are in the sea, will be the fulcrums, upon which the ends of the oars rest in the very act of pushing; but the burdens, or weight, driven by several levers and powers, are the ship itself, which is attached to the thole-pins. They therefore resist the sea, but as the thole-pin yields to the thrusts, the ship is transferred in that direction in which the thole-pins are pushed forward by the moving power itself. And since, just as FG is to FE, so is the moving power in E to the moved weight in G; likewise, just as IK is to IH, so is the moving power in H to the moved weight in K; but the proportion of FG to FE is greater than the proportion of IK to IH. A greater power is therefore required if the weight in G is to be driven than the weight in K. These things certainly are as we have said. But the philosopher’s reasoning would then hold if, while the ship stood motionless—as happens when it is held back by the hidden force of the oarsman or by some other impediment—the rowers, in the very act of rowing, were striking the sea. For then indeed the thole-pin would become the fulcrum, the sea the weight, and the rower himself the mover. We add that what Aristotle asserts seems false, namely that those rowers who are in the middle of the ship move the ship most; he would have said more easily, or better. For if “most,” as he says, indicates the greatest distance and also the greatest speed, it is altogether false; for they move more slowly and through a smaller distance, as we demonstrate thus. F Be it

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IN MECHAN. ARIST. PROBL. E F A G D Esto enim Remus AB qui mari fulcitur in B, Scal- mus remi qui ad prorâ pup- pimue C, qui in media naui D, maior autem remi pars est à scalmo D ad A quam i- psius C ad A, Pellantur remi & stante ceu centro B A, in E. eodem igitur tempore C erit in F, & D in G, sed maius est spatium C F spatio D G, Ergo vnica impulsion, plus mouit scalmum, hoc est, nauim, potentia ad puppim pro- ramueremigans, quàm ea quæ operatur in media naui vt sentire videbatur (si modo is est eius sensus) Aristoteles. Necessarium igitur est, quod ait, maximè intelligendum, faciliùs, Veritatem hanc cognoscentes Triremium præ- fecti robustiores quidem remiges ad proram & puppim, inualidiores verò circa mediam triremem collocant. QVÆSTIO V. Dubitatur, Cur paruum existens gubernaculum, & in extremo nauigio tantas habeat vires, vt ab exiguo temone, & ab hominis vnius viribus alioqui modicè vtentis magnæ nauigiorum moueantur moles? AN, inquit, quoniam gubernaculum vectis est, onus autem mare, Gubernator vero mouens est? Non au- tem secundùm latitudinem veluti remus, mare accipit gubernaculum; non enim in ante nauigium mouet, sed i- psum commotum mare accipiens inclinat obliquè. quo- niam enim pondus est mare contrario innixum modo na- uem inclinat. fulcimentum enim in contrarium versatur, mare verò interius, & illud exterius. illud autem sequitur nauis quæ illi est alligata & remus quidem secundum la- titudinem onus propellens & ab eodem repulsus in re- ctum

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IN MECHAN. ARIST. PROBL. E F A G D For let the oar AB be such as is supported by the sea in B, the rowlock of the oar that is at the prow, or the poop, in C, that which is in the middle of the ship in D; moreover, the greater part of the oar is from the rowlock D to A than from C itself to A. If the oars be driven, and B A standing, as it were, as a center, in E, then at the same time C will be in F, and D in G; but the space C F is greater than the space D G. Therefore, by a single impulse, it has moved the rowlock more, that is, the ship, by power toward the poop and the prow, than that which acts in the middle of the ship, as Aristotle seemed to think (if indeed that is his meaning). It is therefore necessary that what he says should be understood in the fullest sense: more easily, those in charge of triremes, recognizing this truth, place the stronger rowers at the prow and poop, but the weaker around the middle of the trireme. QVÆSTIO V. It is doubted why a rudder, being small and at the end of a vessel, should have such great force that, with a small tiller and by the strength of one man otherwise using only moderate effort, the huge masses of ships are moved? AN, he says, because the rudder is a lever, and the load is the sea, while the helmsman is the moving agent? But not in the manner of an oar, taking hold of the sea according to its breadth; for it does not move the ship forward, but, the sea being itself moved, it inclines obliquely. For since the sea is weight, resting in an opposite manner, it inclines the ship. For the support is turned in the opposite direction, the sea inward, and that outward. And the ship, which is attached to it, follows that; and the oar indeed, pushing the load according to its breadth and repelled by the same, in a straight line.

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EXERCITATIONES. 43 etum propellit, Gubernaculum verò, vt obliquum iacet hinc inde in obliquum motionem facit. in extre mo auté, non in medio iacet, quoniam mouenti facillimum est mo- tum mouere: prima enim pars celerrimè fertur, & quo- niam, quemadmodum in ijs quæ feruntur in fine deficit latio, sic ipsius continui in finem, imbecillima est latio. Imbecillima autem ad expellendum est facilis. Propter hæc igitur in puppi gubernaculum ponitur, nec minus, quoniam parua ibi motione facta, multo maior fit in vlti- mo, quia æqualis angulus semper maiorem adspectat, ta- toque magis, quanto maiores fuerint illæ, quæ continent. Ex ijs etiam manifestum est, quam ob causam magis in contrarium procedit nauigium, quam remi ipsius palmu- la, eadem enim magnitudo ijsdem mota viribus in aëre plus quàm in aqua progreditur. Hæc Philosophus, qui haudquaquam ex more suo, quod duobus ferè poterat, sexcentis verbis exposuit. Licebat enim id tantum dicere, Gubernaculum (ita vocat id totum quod gubernaculo & temone constat) esse ceuremum, quo nauis non antror- sum, sed obliquè & ad latus mouetur. quamobrem omnia ferè quæ de Temone dicenda fuerant, de remo loquens proponit. Ait autem: Sit remus A B, scalmus vero C, remi in nauigio principiu[m] A, palmula autem quæ in mari B. Si igi- tur A, vbi D transla- tum est, non erit B v- bi E. æqualis enim B E ipsi A D, æquale igitur translatum erit, sed erat minus erit igitur vbi F, mi- nor enim B F, ipsa A D, quare ipso G F ipsa D G. Hæc F 2 demon-

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EXERCITATIONES. 43 The oar propels the vessel, but the rudder, because it lies obliquely, produces an oblique motion this way and that. Moreover, it lies at the extreme end, not in the middle, because it is easiest for what moves to produce motion: for the first part is carried most swiftly, and since, just as in things that are borne along the motion fails at the end, so also in the end of this continuous part the propulsion is weakest. But the weakest thing is easy to expel. For these reasons, therefore, the rudder is placed in the stern; and no less so because, when a small motion is made there, a much greater one results at the end, since an equal angle always looks to a greater distance, and all the more so the greater those things are which contain it. From these things it is also evident why the ship advances more in the opposite direction than does the blade of the oar itself; for the same magnitude, moved by the same forces, advances farther in air than in water. These are the words of the Philosopher, who by no means, according to his custom, explained in six hundred words what he could have done in two. For it would have been enough to say only that the Rudder (thus he calls the whole thing which consists of rudder and tiller) is the means by which the ship is moved, not forward, but obliquely and sideways. Wherefore he sets forth, speaking of the oar, almost everything that should have been said about the tiller. And he says: Let the oar be A B, and the rowlock C, the beginning of the oar in the ship A, but the blade B, which is in the sea. If therefore A, when D has been transposed, will not be B where E is. For E B is equal to A D; therefore an equal thing will have been transferred; but it was less, therefore it will be where F is, for B F is smaller than A D; therefore, by G F, D G. These F 2 demon-

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demonstratio licet vera videatur, rei tamen, de qua est sermo, minimè aptatur. Si enim aptaretur in ipsius remi motu, cum palmula esset in F, scalmus fieret in G, excur- reret ergo vel scalmus per remum, vel remus per scalmu[m], facta nempe eiusmodi translatione de C in G, & sic intra nauim modo esset pars remi D C, modò verò G D, quod tamen non fieri ipsâ experientiâ docemur. Illud quoque falsum est, nauim ipsam tantùm moueri in aëre, quantum est spatium A D, hoc est, remi extremum quod est in naui, siquidem scalmi motu, non autem manubrij remi, nauis agatur. Aliter igitur res se habet, & forte hoc pacto. Sit remus A B, cuius manubrium A, palmula B, scalmus C. Pellatur an- trorsus A, fiatq[ue] in D, tunc si æqualiter mouerentur manubrium & palmula, i- psa palmula fieret in G, at minus mouetur: fiet ergo in E. ipse verò scalmus C translatus erit in F, motaq[ue]; erit nauis à C in F, non autem ab A in D. Posuit autem Aristoteles scalmum ad medium remi, sed non ad medium collocari solet, maior enim pars in mare propendet puta HB, quo casu translationis spa- tium fit maius, nempe ab H in I. fit autem motus scalmi ex centris qui sunt in spatio ipso B E, quatenus autem ad te- monem pertinet, quem remum ait, obliquè puppim ipsam propellentem, ita se res habet. Esto nauis carina A B, prora A, puppis B, Temonis ala B C, gubernaculum B D, cardo verò fulcimentumue B; facta itaque impulsione obliquâ gubernaculi à D in E, minor fiet motus in mari à C in F, eritque temo vbi E G F, cardo

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The demonstration, although it may seem true, is nevertheless not suited to the subject under discussion. For if it were suited to the motion of the oar itself, then, when the blade were at F, the rowlock would be at G; thus either the rowlock would move over the oar, or the oar over the rowlock, after such a transfer from C to G, and thus inside the ship there would at one moment be the part D C of the oar, and at another G D, which, however, we are taught by actual experience does not happen. Nor is it true that the ship itself is moved through the air only by the distance A D, that is, the end of the oar which is in the ship, since the ship is driven by the motion of the rowlock, not by that of the oar’s handle. Therefore the matter is otherwise, and perhaps in this way. Let there be an oar A B, whose handle is A, blade B, rowlock C. Let A be pushed forward and become D; then, if the handle and blade moved equally, the blade itself would come to G, but it is moved less: it will therefore come to E. The rowlock C itself will be carried to F, and the ship will be moved from C to F, but not from A to D. Aristotle, however, placed the rowlock at the middle of the oar, but it is not usually placed at the middle; for the greater part projects into the sea, such as H B, in which case the space of the transfer becomes greater, namely from H to I. Now the motion of the rowlock arises from the centers which are in the space B E itself, but so far as it pertains to the tiller, which, as he says, propels the stern itself obliquely, the matter stands thus. Let the ship’s keel be A B, the prow A, the stern B, the tiller’s wing B C, the rudder B D, and the hinge or support B; then, after an oblique thrust of the rudder from D to E, there will be a smaller motion in the sea from C to F, and the tiller will be E G F, the hinge...

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EXERCITATIONES. 45 cardo verò vbi G, translata igitur e- rit eo motu, puppis ipsa à B in G. facta itaque paruâ motione puppis ex B in G, prora ipsa quæ longè distat à pup- pi B maiori spatio superato translata erit in H facta proræ in contrariam partem ab ea quæ facta est gubernaculi motione. Porrò quod & in præcedente quæstione adnotauimus, lo- gè meliùs procedet demonstratio si fulcimentu[m] mare intelligatur, quàm scalmus, neque enim mare ceu pon- dus, sed scalmus ipse Temonisuecardo, ponderum instar transferuntur. Cæterùm in hac speculatione liceat nobis aliquantulum à Philosopho dissentire. Certè si breuitas Temonis, è puppi eminentis, respectu longitudinis totius nauis consideretur, & parua motio, quæ temone gubernaculo- ue moto sit, nullius ferè momenti erit ad eam quæ in pro. ra fit translationem. aliter ergo serem habere non dubitamus, & quæstionis solutionem aliunde petendam. Naui non currente nullum ferè, aut qui vix curandus sit ex gubernaculi conuersione nauis ad dextram sinistramue motum fieri. at eâ currente maximum, experientiâ doce- mur. Obliqui igitur motus qui validè in puppi sit, caussa est non quidem ex conuersione temonis percussio matis, sed mare ipsum, cuius fluctus naui currente obliquam temonis alam ad eam partem quæ mari obuertitur, impellentes temonem cum puppi ad contrariam partem vali- dissimè transferunt. Esto nauis carina AB, prora B, puppis A, Temo AC, gubernaculum AD; Itaque currente naui, Temone interim & gubernaculo in eadem carinæ linea existentibus, F 3 Temo

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EXERCISES. 45 when the pivot is at G, it will therefore be transferred by that motion to that point, the stern itself from B to G. Thus, by a small movement of the stern from B to G, the prow, which is far distant from the stern B, having traversed a greater space, will be transferred to H, the prow having been made to move in the contrary direction by that motion which has been made by the helm. Moreover, what we noted in the preceding question, the demonstration will proceed much better if the sea is understood as the support, rather than the scalmus; for neither are the sea and the scalmus itself, like the pivot of the rudder-beam, transferred as weights. However, in this speculation it may be allowed us to differ somewhat from the Philosopher. Certainly, if the shortness of the rudder-beam, projecting from the stern, is considered in relation to the length of the whole ship, and the small motion that is made when the rudder-beam or helm is moved, it will be of scarcely any importance to that transfer which is made at the prow. Therefore we do not doubt that the matter is otherwise, and that the solution of the question must be sought elsewhere. When the ship is not moving, the turning of the helm causes hardly any movement of the ship to the right or left, or at most one that scarcely needs to be considered. But when it is sailing, we are taught by experience that it is very great. Therefore the oblique motions which occur strongly at the stern are not caused by the turning of the rudder striking the sea, but by the sea itself, whose waves, when the ship is moving, driving the oblique blade of the rudder toward that side which faces the sea, transfer the rudder with the stern very powerfully to the opposite side. Let the keel of the ship be AB, the prow B, the stern A, the rudder AC, the helm AD; thus, with the ship in motion, while the rudder and helm are meanwhile in the same line of the keel, F 3 Temo

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IN MECHAN. ARIST. PROBL. Temo quidem mare secat, nullâ factâ in puppi, nauis ad sinistram dex- tramue translatione. Si verò moueatur gubernaculum à D in E, eo moto mouebitur aliquantulum & puppis ad partes E, quod voluit Aristoteles. Sedminimi, vt diximus, ea res ad tantum effectum est momenti. Temone autem in obliquum costituto vt A F, naui interim, ventorum aut remorum vi pulsa proram versus currente temonis latus à fluctibus obliquam partem alamue in ipso cursu ferientibus, in contrariam partem transfertur, ad eam nempe, ad quam ipsum gubernaculum vergit. facta igitur nauis ceu circa centrum centraue quæ in carina inter puppim proramue considerantur A, fertur in G, prora verò in H. ex quibus manifestè apparet, duo ad nauis extemone in puppi conuersione motionem esse necessaria; Temonis nempe obliquationem, & nauis cursum, quoru[m] si alterum sine altero adhibeatur, nullam fieri quæ alicuius momenti sit, nauis conuersionem. Illud quoque notamus, carinam in nauis conuersione vectis instar se habere, cuius pars mota ad puppim, & mouens potentia est; fulcimentum verò circa proram, potentia autem mouens mare ipsum, temonem in nauis cursu oblique feriens. Vnde colligimus naues, quo longiores sunt in mouente ad Temonem adhibita maiori facilitate ad dextram sinistramue propelli: quod sanè ipsemet considerauit Aristoteles, qui idcirco inquit, in extremo, non autem in medio temonem poni eo quod mouenti facilimum sit ab extremo motum mouere. Ex hac nostra speculatione ratio habetur eius ma- china-

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IN MECHAN. ARIST. PROBL. Indeed, the rudder cuts through the sea, no change being made in the stern, by the ship’s shifting to the left or to the right. But if the helm is moved from D to E, when it is so moved the stern will be moved somewhat toward E also, as Aristotle intended. But this effect, as we have said, is of very little importance for such a result. But when the tiller is set obliquely, as AF, while the ship in the meantime, driven forward by the force of the winds or oars, the water striking the side of the tiller, or the blade, obliquely in the very course of its motion, it is transferred to the opposite side, namely to that toward which the helm itself inclines. The ship, therefore, being as it were turned about a center, or centers considered in the keel between the stern and the prow, is carried into G, but the prow into H. From these things it is clearly apparent that two motions are necessary for turning a ship about its stern: namely, the obliquity of the tiller and the ship’s forward motion; and if one of these is used without the other, no turning of the ship, of any significance, is produced. We also note that the keel in the turning of a ship acts like a lever, whose moved part toward the stern is the moving power; but the fulcrum is about the prow, while the moving power is the sea itself, striking the helm obliquely in the ship’s course. Hence we gather that ships, the longer they are, are more easily driven to the right or to the left by the moving power applied to the helm: which surely Aristotle himself considered, when for that reason he says that the helm is placed at the stern, not in the middle, because it is easiest for the mover to move the motion from the end. From this our speculation an explanation is obtained of this machine-

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EXERCITATIONES. 47 chinationis, quâ in magnis fluminibus, ceu Pado, Abdua & similibus, Portitores, equos, currus, viatoresq; ipsos, è ripa in ripam transferunt. Pulcherrima enim res est, & nobis perspectissima, qui Guastallâ residentiæ olim nostræ oppido ad Padum, Mantuam pergentes sæpissimè ad Castrum Burgi Iusis ea qua diximus machinatione latissimum eiusdem Padi aluum transiecimus. Habet autem se hoc pacto. Esto fluminis citerior ripa A B, vlterior C D. Pontones duo tabulis strati, & v- nà firmiter juncti E F, Temo inter eorum puppes extans G H, locus in ripa stabilis A, funis, quo pontones, & ma- china tota continetur A I. fluuij decursus versus B D, stantibus itaque pontonibus ad ripam citeriorem A B, Temone in neutrâ partem pul- lo, cum aqua decurrens eum resistentem non inueniat, scinditur quidem ab eo, sed non propellit, eo autem con- uerso & in GK constituto, a- la eius GK ab aqua defluente propulsa machinam secum trahit versus ripam C D, factâ motione circa centrum seu stabilem locum A, otiosis interim portitoribus, donec per circuli portionem M L deuenerit ad vlteriorem ripam in L. Vnde iterum temone in contrariam partem conuerso, aquâ similiter temonem propellente, per eandem circuli portionem ad ripam citeriorem reuertitur, à qua paullo antè discesserat. Ex quibus apparet, motus causam non esse

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EXERCISES. 47 of ferrying devices, by which on great rivers, such as the Po, the Adda and the like, boatmen carry across from bank to bank horses, wagons, and the travelers themselves. For it is a very beautiful contrivance, and one with which we are most familiar, since, when we used to reside at Guastalla, a town on the Po, very often on journeys to Mantua, to the Castle of Borgo Iusis, we crossed the very broad channel of that same Po by the machine described above. It works in this way. Let the nearer bank of the river be A B, the farther C D. Two boats decked with planks and firmly joined together, E F, the oar or pole extending between their sterns G H, a fixed place on the bank A, and the rope by which the boats and the whole machine are held A I. The current of the river runs toward B D; therefore, with the boats standing by the nearer bank A B, if the oar be turned to neither side, since the flowing water finds it offering no resistance, it is indeed split by it, but does not propel it. But when it is turned the other way and set in GK, its blade GK, driven by the water as it flows down, draws the machine along with it toward the bank C D, the motion being made around the center, or fixed point, A, while the boatmen remain idle, until, through the portion of the circle M L, it has reached the farther bank at L. Then, with the oar again turned in the opposite direction, the water likewise driving the oar, it returns by the same portion of the circle to the nearer bank, from which it had a little before departed. From this it is clear that the cause of the motion is not

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48 IN MECHAN. ARIST. PROBL. esse solam eam, quæ ab alatemonis sit, aquæ percussione[m], vt senserat Aristoteles, sed currentis aquæ temonis alam terientis impulsionem: nihil autem referre, vtrum stante naui aqua currat, vel eâ currente aqua stet, vt in mari sit, idem enim vtroque modo temo patitur. Vt autem machinæ huius & totius negotij species facilius animo concipiatur, schema hoc studiosorum oculis subijciemus. Lembi nauiculæue ideo appositæ sunt, vt oblongum funem sustineant; id etenim nî fieret, aquæ immersus a- quam scindens machinæ motum impediret, ideo etiam apponuntur, ne funis madens celeriter maceretur & putrescat. Huic speculationi affinis est ea, velorum eorum, quæ obliquè ventum excipientia frumentarijs molis dant motum, item verticillorum ex papyro, quibus contra ventum currentes per lusum pueri vtuntur. vnicum enim

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48 IN THE MECHANICAL PROBLEMS OF ARISTOTLE the only one, namely that which is from the oar-blades, by the striking of the water, as Aristotle had understood; but the thrust of the oar-blade of the moving water against the water. Nor does it matter whether, the ship standing still, the water runs, or whether, with the water running, the water stands still, as it is at sea; for in either way the oar undergoes the same thing. But so that the appearance of this machine and of the whole matter may be more easily conceived in the mind, we shall set this diagram before the eyes of students. Little boats or skiffs are therefore attached, so that they may support the long rope; for if this were not done, the part immersed in the water, cutting through the water, would hinder the motion of the machine; they are also attached so that the rope, becoming wet, may not be quickly soaked through and rot. Related to this speculation is that concerning sails, which, receiving the wind obliquely, give motion to grain mills; likewise the little whirligigs made of papyrus, with which children use, for play, running against the wind. For a single

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EXERCITATIONES. 49 enim horum omnium principium & eadem ratio. Diximus enim, Temonem currente navi, lateraliter conuersum obuios fluctus excipientem puppim ipsam obliquè in alteram partem transferre. Porrò ea vela, de quibus loquimur, ventorum flatibus obliquè opposita eandem ob causam circulariter agitantur, quod vt figurâ cuidentius fiat, Esto velum AB, brachio CE obliquè affixum ita vt angulus ACE maior sit angulo BCE, ventus obliquè velum feriens FG. Itaq[ue] quoniam ventus in velum obliquum incidit, elabitur velum, & circa centrum E vnà cum brachio circumuertitur, in cuius locum succedit velum HI, ex qua assidua velorum successione, brachiorum & axis cui adhærent, rotatio sit perpetua. Sed enim de Temone agentes non est interim cur de caudis auium pisciumque taceamus. instar enim temonum sunt à Natura ipsa opportunis animalium partibus, postremis videlicet, appositi, quanquam nec solum Temonis vsum præstent, vt videbimus. Esto piscis AB, cuius caput A, cauda verò CB. Hac igitur neutram in partem reflexâ, piscis pinnarum motu rectâ in anteriorem partem progreditur. Siautem necesse ei fuerit ad dextram sinistramque conuerti non poterit, nisi cauda ipsa iuuetur. Omnis enim motus progressius quiete indiget, nec absq[ue] stabili fulcimento progredi potest, G

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EXERCISES. 49 for the principle and reasoning of all these things is the same. For we have said that the rudder, when the ship is moving, being turned sideways and receiving the oncoming waves, transfers the stern itself obliquely to the other side. Moreover, those sails of which we speak, being set obliquely against the blasts of the winds, are for the same reason whirled round in a circular motion; and that this may be made clearer by a figure, let AB be a sail, attached obliquely to the arm CE, so that the angle ACE is greater than the angle BCE, and let the wind striking the sail obliquely be FG. Therefore, since the wind falls upon the sail obliquely, the sail slips away, and together with the arm revolves around the center E, in whose place sail HI succeeds; and from this continual succession of sails, the rotation of the arms and of the axis to which they adhere becomes perpetual. But since we are speaking of the rudder, meanwhile we ought not to be silent about the tails of birds and fishes. For by Nature herself they are set, as it were rudders, in the suitable parts of animals, namely the hindmost parts, although they serve not only the use of a rudder, as we shall see. Let a fish be AB, whose head is A, but whose tail is CB. If this, therefore, is not bent to either side, the fish, by the motion of its fins, advances in a straight line toward the front. But if it should be necessary for it to turn to the right or to the left, it will not be able to do so unless the tail itself helps it. For every progressive motion requires rest, and cannot advance without a firm support,

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potest, quod in libris de animalium incessu docet ipsemet Philosophus. Sit igitur, piscem conuerti velle, & fieri capite in D, deflectet illi co caudam in E, ea q[ui] aquam ceu stabile quippiam serie[m] ei[que] quodammodo fultus, reliquum corpus C A reflectet in D, si autem conuerti velit in F, caudam deflectet in G, & eadem ratione deflectetur in F. Sed & Temonis quoque vsum præstat natatilibus & volatilibus cauda. Sit enim rectus piscis, hoc est, rectâ pergens IKL, caudam obliquet in KM itaque ex aquæ in ipso motu collisione, eius posteriora pellentur vbi INO. Hæc itaque nos de Temone, quatenus ad hanc quæstionem pertinet, considerasse sit satis. QVÆSTIO VI. Dubitatur, Cur quanto Antenna sublimior fuerit, ijsdem velis, & vento eodem celerius ferantur nauigia? Soluit Philosophus, inquiens: An quia malus quidem sit vectis, fulcimentum verò mali sedes, in qua collocatur, pondus autem quod moueri debet, ipsum nauigium: mouens verò is, qui vela tendit spiritus? Si igitur quanto remotior fuerit fulcimentum facilius eadem potentia, & citiùs idem mouet pondus, altiùs certè sublatâ antennâ, velum à mali sede, quæ fulcimentum est remotius faciens, id efficiet. Hæc ille, quæ sic figurâ explicamus. Esto

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it can, as the Philosopher himself teaches in the books On the Motion of Animals . Let it therefore be that, if a fish wishes to turn and to become, with its head in D, it will deflect its tail toward E; since the water, as though something stable, and in a manner supporting it, will hold it up, the remainder of the body C A will be turned back into D. But if it wishes to turn in F, it will deflect its tail toward G, and in the same way it will be turned into F. And the tail also provides the use of a rudder for swimming and flying creatures. For let the fish be straight, that is, moving straight along IKL; if it slants its tail into KM, then, by the collision of the water in its motion, its hind parts will be driven to where INO is. Thus, then, what we have considered about the rudder, insofar as it pertains to this question, may suffice. QUESTION VI. It is doubted why, the higher the yardarm has been raised, with the same sails and the same wind, ships are carried more swiftly. The Philosopher solves it, saying: Is it because the mast is indeed a lever, and the support is the seat of the mast in which it is placed, while the weight that must be moved is the ship itself; but the moving agent is the spirit that stretches the sails? If therefore, the farther removed the support is, the more easily the same power, and more quickly, moves the same weight, then certainly, with the yardarm raised higher, it will achieve this by making the sail farther from the seat of the mast, which is the support. Thus he speaks, which we explain in this figure. Let it be

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EXERCITATIONES. 51 Estonauis AB, malus CD, mali sedes D, locus antennæ sublimior C, depressior E: itaque quoniam CD vectis est, quo mouens remotior fuerit à fulcimento D, eo citiùs & violentiùs pellet, velocius ergo nauis mouebitur antenna in C, quàm in E, constituta. Plausibilia sunt hæc, at certè per veritatem ipsam, non vera. Rogo, Si fulcimentum dum vectis mouetur, cætrum est, centrum vtique motus erit D. spirante igitur validè vento inclinabitur malus, fietq; vbi F G D, quæ quidem inclinatio violentius fiet, vento pellente in F quàm in G, vtpote puncto à fulcimento remotiore. Impulso malo, duo necessariò co[n]sequentur, vel enim ad ipsam sedem D. frangetur vel puppis ipsa circa D punctum conuersa, vt mali sequatur motum eleuabitur. Prora verò submergetur facta naui in HDI. Quod si quispiam funem ad mali summitatem annexam ad ipsam puppim alligauerit in B, impedietur sanè mali inclinatio ad partes F, & ideo nulla vis prorsus fiet in D ex vectis ratione. Attamen nihilo secius, quo sublimior fuerit antenna, eo faciliùs à spirante vento puppis eleuabitur. quatenus igitur malus vectis est, hoc tantum quod dicimus operatur. Quod si contrà obiectum fuerit, experientiam docere, quo sublimior antenna fuerit, eo citiùs nauigium, spiritu flante moueri. Responso facilis, nempe, mirum non esse, si mali pars sublimior validius à vento feriatur. Videmus enim, & turres quo sublimiores fuerint, eo magis à ventorum impetuosis flatibus infestari, quod sanè ad vectis longitudinem referre, esset ridiculum. Cæterùm quod ad puppis faciliorem eleuationem ex mali ipsius altitudine pertinet, ad vectis con- G 2

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EXERCISES. 51 AB is the mast, CD the boom, D the seat of the boom, C the higher point of the mast, E the lower point; thus, since CD is a lever, the farther the moving point is from the fulcrum D, the more quickly and violently it will drive; therefore the ship will move more quickly when the sail is placed in C than when it is placed in E. These things are plausible, but certainly, by the truth itself, they are not true. I ask, if the fulcrum, while the lever is moving, is fixed, the center of motion will certainly be D. Therefore, when a strong wind blows, the mast will incline, and it will happen that where F G D is, that inclination will occur more violently, the wind driving in F than in G, as being a point farther from the fulcrum. If the mast is driven, two things must necessarily follow: either it will break at the very seat D, or the stern itself, turned around the point D, will be raised so that the motion of the mast follows. But the prow will be submerged, and the ship will be made into HDI. If someone were to tie a rope attached to the top of the mast to the stern itself in B, the inclination of the mast toward F would certainly be impeded, and therefore no force at all will be produced in D by reason of the lever. Yet nonetheless, the higher the mast has been, the more easily the stern will be raised by the blowing wind. So far, then, as the mast is a lever, this alone is what it does, as we say. But if, on the contrary, it is objected that experience teaches that the higher the mast has been, the more quickly a ship moves when the wind is blowing. The answer is easy, namely, that it is no wonder if the higher part of the mast is struck more strongly by the wind. For we see that even towers, the higher they are, the more they are assailed by the forceful gusts of winds, which would certainly be ridiculous to refer to the length of a lever. However, as to the easier raising of the stern from the height of the mast itself, to the lever...

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52 IN MECHAN. ARIST. PROBL. contemplationem reducimus. est enim quædam vectium species ab alijs non considerata, cuius brachia in angulum desinunt, vt ipse angulus in operatione sit fulcimentum. Esto enim vectis, de quo agimus, ABC, cuius brachia AB, BC. iuncta ad angulum B, sitque B in operatione fulcimentum. Nec quicquam refert quatenus ad vsum pertinet, vtrum angulus ipse rectus sit, acutus vel obtusus. sit autem modò rectus. Ponaturi- gitur pondus aliquod in C, tum potentia quædam applicetur in A, quæ ipsam vectis extremitatem A propellat in D. erit igitur AB in DB & angulo seruato BC in BE. Pondus igitur cum parte vectis BC eleuabitur in E. In hoc autem vectis genere attenditur proportio quam habet AB ad BC. Si enim potentia quæ applicatur in A ita se habet ad pondus in C vt CB, ipsi BA, fiet æquilibrium. Si maior autem fuerit proportio potentiæ in A, ad pondus in C, ea quam habet AB ad BC, superatâ ponderis resistentiâ fiet motus. Res autem haud aliter se habet, ac si producta in F, fieret BF æqualis BC. Tunc enim vectis ad rectitudinem, seruatâ proportione, redigeretur, & ita potentia in A, fulcimento B operaretur in F, vt operabatur in C. Ad huius vectis naturam referuntur fabrorum mallei, quibus clauos reuellunt, forcipes item quæ tenaci morsu clauorum capita vmbellasue apprendentes, violenter è tabulis extrahunt. In malleo itaque subtili, vt in figura videre est, AB vectis est pars quæ à fulcimento ad potentiam, ac verò quæ à fulcimento ad pondus, ponderi siqui-

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52 IN MECHAN. ARIST. PROBL. We reduce it to contemplation. For there is a certain kind of lever, not considered by others, whose arms end in an angle, so that the angle itself serves as the fulcrum in the operation. Let there be, then, the lever of which we are speaking, ABC, whose arms are AB and BC, joined at the angle B, and let B in operation be the fulcrum. Nor does it make any difference, so far as use is concerned, whether the angle itself be right, acute, or obtuse. Let it, however, be right for the present. Let some weight therefore be placed at C, and then let some power be applied at A, which propels the very end A of the lever into D. Then AB will be to DB, and the angle preserved, as BC is to BE. The weight, therefore, together with the part of the lever BC, will be raised to E. In this kind of lever attention is given to the proportion which AB has to BC. For if the power applied at A is related to the weight at C in such a way that CB is equal to BA, equilibrium will result. But if the proportion of the power at A to the weight at C be greater than that which AB has to BC, when the resistance of the weight is overcome, motion will occur. The matter stands no differently than if, when extended to F, BF were made equal to BC. For then the lever, with the proportion preserved, would be reduced to straightness, and thus the power at A would act on the fulcrum B in F, just as it acted in C. To the nature of this lever are referred the blacksmiths’ hammers, with which they tear out nails, and also the tongs which, with a tenacious bite, seizing the heads of nails or the handles of umbrellas, forcibly draw them from the boards. In the hammer, therefore, as can be seen in the figure, AB is the part of the lever from the fulcrum to the power, but that which is from the fulcrum to the weight, to the weight if in-

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EXERCITATIONES. 53 B D B A C A B D C A B C D E F G liquidem æquiparatur resistentia quę sit in C. Idem obseruamus in forcipe, in quo duo quidem brachia AD, CB, quatenus ad apprensionem pertinet, fulcimentum habent in ipso cētro seu vertebra, & ideo quo longiores fuerint, eo tenaciùs apprehendunt & retinent. quatenus autem ad extractionem facit, pro vncio forceps totus habetur vecte, cuius quidē pars à potentia ad fulcimentum AB. quæ verò à fulciméto ad hoc est clauum ipsum qui reuellitur AC. Violentissimè autem extrahunt forcipes, propterea quod maxima sit proportio longitudinis brachij BA, adeam quæ est ab A ad C. His igitur hoc pacto examinatis, ad nauim & malum reuertentes, dicimus, tunc facillimam fieri puppis eleuationem, proræ verò demersionem, cum maxima fuerit proportio, quam habet altitudo mali, ad eam nauis parté quæ à malo ad ipsam puppis extremitatem pertingit. Quamobrem prudentes nauium fabri, vt huic difficultati occurrant, malum non in medio quidem nauis, sed in tertia ferè parte longitudinis quæ à prora est, puppim versus constituunt. Esto enim nauis AB; cuius malus CD: prora A: puppis B; vēto igitur velum impellente, malu[m] ad partem contrariam vergit, puta in FD. At quoniâ carchesium funi ad puppim vnitur in B, nauim, hoc est, ipsam puppim trahat ne- G 3 cesse

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EXERCISES. 53 B D B A C A B D C A B C D E F G indeed the resistance that is at C is equivalent. The same we observe in the forceps, in which the two arms AD, CB, insofar as grasping is concerned, have their support at the very center or pivot, and therefore the longer they are, the more firmly they seize and hold. But insofar as it serves for extraction, the whole forceps is regarded as a lever, of which one part, from the power to the support, is AB; and the other, from the support to the point, is the nail itself which is pulled out, AC. But forceps pull out most violently, because the proportion of the length of the arm BA to that which is from A to C is greatest. Having therefore examined these matters in this way, returning to the ship and mast, we say that the lifting of the stern is made easiest, and the submerging of the prow, when the proportion is greatest which the height of the mast bears to that part of the ship which extends from the mast to the very end of the stern. For this reason prudent shipbuilders, in order to meet this difficulty, place the mast not indeed in the middle of the ship, but in about the third part of the length measured from the prow, toward the stern. Let the ship be AB; its mast CD; the prow A; the stern B; therefore with the wind driving the sail, the mast inclines to the opposite side, say toward FD. But since the topmast is joined by a rope to the stern at B, it will make the ship, that is, the stern itself, tra- G 3 cessary

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54 IN MECHAN. ARIST. PROBL. cette est. non potest autem; quoniam suburræ grauitas & onera, quæ naui imposita inter D. & B. grauitatis centrum circa punctum E constituunt, quod quidem vi ventorum inclinante malo ab E, in G eleuaretur, quo igitur minor fuerit proportio CD ad DE & maius pondus ipsum cu- ius grauitatis centrum in E minus præualebit potentia pellens in C ad elevationem partis nauigij, quæ à malisse- de ad puppim intercedit. An igitur malus sit vectis, pes ve- rò fulcimentum, pondus autem quod vecte mouetur, ipsu[m] nauigium, vt placuit Aristoteli, & qua item ratione malus in nauim vt vectis operetur, ex ijs quæ dicta sunt, facilè pa- tet. QVÆSTIO VII. Quæritur, Cur quando ex puppi nauigare voluerint, non flante ex puppi vento, veli quidem partem, quæ ad gubernatorem vergit, constringunt; illam verò quæ proram versus est, pedem facientes, relaxant? Mirabilis huius effectionis caussam explicat Aristote- les. inquit enim, An quia retrahere quidem multo existenti vento gubernaculum non potest, pauco autem potest, quem constringunt? propellit igitur quidem ipse ventus, in puppim verò illum constituit gubernaculum retrahens, & mare compellens: simul & nautæ ipsi cum vento contendunt; in contrariam enim se reclinant par- tem. Hæc ille. Cuius sensum breuitate sub obscurum, mirâ facilita- te explicat Picolomineus. Nos autem vt rem lucidiorem faciamus, schema, quod nec ipse fecit, nec Philosophus, proponemus. Estonauis A B, cuius prora A, puppis verò D, guber- naculum C B, temonis ala B D, veli sinus E F, velum vero ita constitutum, vt directè ex puppi flantem ventum exci- piat.

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54 IN MECHAN. ARIST. PROBL. this is so. But it cannot be; for the weight and burden of the cargo, which, placed on the ship between D and B, establish the center of gravity at point E, would, when the mast is inclined by the force of the winds from E to G, be raised. Therefore, the smaller the ratio of CD to DE, and the greater the weight itself whose center of gravity is in E, the less will prevail the force pushing at C to raise that part of the ship which lies from the mast-step to the stern. Is the mast then the lever, the step indeed the fulcrum, and the weight moved by the lever the ship itself, as Aristotle pleased to say? And in what way likewise the mast operates on the ship as a lever is easily made clear from what has been said. QUESTION VII. It is asked: Why, when they have wished to sail from the stern, if the wind is not blowing from the stern, do they tighten the part of the sail which turns toward the helmsman, but loosen the part which faces toward the prow, making it slack? Aristotle explains the cause of this remarkable effect. For he says, Is it because the wind that is present cannot indeed pull the rudder strongly, but can with less force the part which they tighten? Therefore the wind itself propels, and in the stern it places that which pulls the rudder and drives the sea; at the same time the sailors themselves also contend with the wind, for they lean toward the opposite side. So he. The meaning of this, somewhat obscured by brevity, Picolomini explains with wonderful ease. But we, in order to make the matter clearer, shall propose a diagram, which neither he himself made nor the Philosopher. Let there be a ship A B, whose prow is A, stern D, rudder C B, tiller arm B D, sail-bosom E F, and the sail itself so arranged that it directly receives the wind blowing from the stern.

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EXERCITATIONES. 55 piat. Hoc vbi euenerit, nauigium rectâ è puppi mouetur in proram; Si autem ventus lateraliter spirat, puta à parte G versus H & nihilo secius nauigium ac si ventus ex puppi esset antrorum propellere volunt, velum quidem obliquant partem eius infimam, pedem nempe, quæ est in F contrahentes, Cornu verò antennæ vbi E, proram versus laxantes ventumq; ipsum obliquè excipientes id efficiu[n]t, vt ventus minus violenter feriat, & minori sui parte velu[m] impleat, & quoniam ventus velum pellit in partem contrariam, nempe in H, ipsi vt vento resistant conuerso gubernaculo ex C in L, & temone B D, in BM compellunt proram ad partem à qua ventus ipse spirat. Sit igitur inter ventum & temonem pugna, illo proram in dextram, hoc verò eandem in sinistram pellente, itaq; cum neuter præualeat, necessario nauis mediam viam, quæ inter vtramq; est, suo cursu tenet. Nautæ autem ideo in partem nauis A E B, quæ versus ventum est, se conferunt, vt vento æquilibrium faciant, ne scilicet naui in co[n]trariam partem pellente spiritu, eam demergat. Cæterum quod nec Aristoteles nec Picolomineus animaduerterunt, velum obliquè constitutum à vento in anteriora impellitur eandem ob causam, quam retulimus, vbi de temone & velis, quibus farinariæ molæ couertuntur, verba faceremus. Quod autem addit Picolomineus rem ad vectem reduci posse, non est cur sub silentio prætereamus. Ventus, inquit, ponderis gubernaculum mouentis vicem obtinet; centrum verò (fulcimentum intelligit) in medio nauis est, quod ta- men

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EXERCISES. 55 happens. Once this has occurred, the ship is moved straight from the stern to the prow; if, however, the wind blows laterally, say from side G toward H, and nevertheless they wish to drive the ship forward as if the wind were from the stern, they indeed slant the sail by drawing in its lowest part, namely the foot, which is at F, while loosening the horn of the yard where it is at E, toward the prow, and by receiving the wind itself obliquely they bring it about that the wind strikes less violently and fills a smaller part of the sail; and because the wind drives the sail toward the opposite side, namely toward H, they, in order to resist the wind, turn the rudder from C to L, and with the tiller B D, driving it into BM, they force the prow toward the side from which the wind itself blows. Let there therefore be a struggle between the wind and the tiller, the one driving the prow to the right, the other to the left; and thus, since neither prevails, the ship necessarily holds the middle course, which lies between the two, in its voyage. The sailors, however, for that reason gather on the side of the ship A E B, which faces the wind, so as to create equilibrium against the wind, lest, that is, with the breeze driving the ship toward the opposite side, it sink it. Furthermore, what neither Aristotle nor Picolomineus observed is that a sail set obliquely is driven forward by the wind for the same reason that we mentioned when speaking of the tiller and the sails by which the millstones are turned. But what Picolomineus adds, namely that the matter can be reduced to a lever, there is no reason why we should pass over in silence. The wind, he says, takes the place of the weight moving the rudder; but the center (he means the fulcrum) is in the middle of the ship, which nevertheless

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56 IN MECHAN. ARIST. PROBL. men ad proram vergit, vt faciliùs ipsi vento resistere possit. Tunc enim in rectum mouebiturnauis, cum sibi inuicem æquatæ vires, quasi libramentum constituerint. Hæc ille, cuius sensum figurâ propositâ facilè aperiemus. Esto carina AB, cuius prora A, puppis, B temo BC, ventus verò obliquè feriens H. Conuersus itaque temo vt in BC vndarum vi currente naui repulsus sit in EF tendens versus I, quo casu prora convertitur in D, nempe contra ventu qui spirat ex H. sit autem conuersio circa punctum G, quod fulcimenti locum obtinet. Vetus verò ad contrariam parte proram impellit, repugnans Temonis violentiæ contra ipsam proram dirigentis. Est igitur AB, seu DE carina instar vectis, cuius fulcimentum G, vis mouens mare quo temo EF repellitur, pondus vero, ventus premens in D; quo igitur remotior erit temo à fulcimento G, D autem vbi pondus ei vicinius, eo magis temo venti vim superabit. Hæc Picolominei ratio, quam explicauimus, sanè ingeniosa est, verum enimuero, quoniam fulcimentum sui naturâ stare debet, hic verò nullâ habeat stabilitatem, difficultatem patitur. QVÆSTIO VIII. Quæritur, Cur ex figuris omnibus rotundæ faciliùs moueantur? TRifariam, inquit Aristoteles, circulum rotari contingit; Aut secundum absidem cétro simul moto, quemadmodum plaustris vertitur rota; aut circa manens centrum, veluti trochleæ puteorum, stante centro: Aut in pauimento manente centro, sicuti figuli rota convertitur. Caussam

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56 IN MECHAN. ARIST. PROBL. leans toward the prow, so that it can more easily resist the wind itself. For then the ship will move in a straight line, when the forces balanced against one another will, as it were, have established an equilibrium. This he says; his meaning we shall easily make clear by the figure proposed. Let AB be the keel, whose prow A, stern B, the tiller BC, and the wind, however, striking obliquely H. Thus the tiller being turned so that in BC, with the current force of the waves, the ship is repelled and tends toward EF toward I, in which case the prow is turned to D, namely against the wind blowing from H. Let the turning be about the point G, which serves as a support. But the old one drives the prow to the opposite side, resisting the violence of the tiller directing itself against the prow. Therefore AB, or DE, is like a lever, whose support is G, the moving force the sea by which the tiller EF is driven back, but the weight, the wind pressing at D; therefore the farther the tiller will be from the support G, and D, when the weight is nearer to it, the more the tiller will overcome the force of the wind. This reasoning of Picolomini, which we have explained, is indeed ingenious; but truly, since the support by its nature ought to be fixed, here however it has no stability, it meets with difficulty. QUESTION VIII. It is asked, why do round figures move more easily than all others? In three ways, says Aristotle, does it happen that a circle is rotated; either about an arch with the center moved at the same time, as a wheel turns on wagons; or around a center remaining in place, as the pulleys of wells, with the center standing still: or with the center remaining on the floor, as the potter’s wheel turns. The cause

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EXERCITATIONES. 57 Causam verò explicans, ait, celerrima eiusmodi corpora esse, eo quod paruâ sui parte planum contingunt, vti cir- culus secundum punctum, item quoniam non offensant: Non offensandi vero esse causam, quod semotum à terra habeant angulum. Item propterea quod corpus, cui fiunt obuiam, secundum pusillum tangunt. Rectilineo autem aliter euenire, quippe quod rectitudine suâ, multum plani contingat. Ad hæc, quo nutat pondus eo mouentem mouere. Hæc ferè Philosophus, cuius rationes ad eum solum- modo circularem motum faciunt, qui fit secundum absidem, vt in carrorum rotis vsu venit, nec aptantur rotis sigulorum trochleisque, cuiusmodi sunt illæ, quæ supra puteos appenduntur. Nos igitur, ad Aristotelis mentem, primam rotationis speciem, quæ est secundum absidem, examinabimus. Esto rotæ sphæ- raue A B, cuius cen- trum C; Horizontis planum DE; conta- ctus circuli in plano B. perpedicularis ho- rizonti à puncto co- tactus B ipsa B C A, transiens per centru[m] C, partes rotæ circa perpendicularem A F B, A G B, angulus contactus G B E. Primo itaque id constat, circulum in puncto planum, seu lineam contingere. At quoniam, vt Mechanici, de circulis rotisque seu sphæris agimus materialibus, rectè Philoso- phus non in puncto planum præcisè tangere dixit, sed se- cundum partem sui minimam. Angulum porro, quem à terra semotum dicit, ipse angulus est contingentię. eleua- H tur

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EXERCISES. 57 Explaining the cause, he says that such bodies are very swift because they touch the plane with only a small part of themselves, as a circle does at a point, and likewise because they do not strike against it. The cause of their not striking, moreover, is that they have an angle separated from the earth. Likewise because the body which they meet they touch only slightly. But in a rectilinear motion it is otherwise, since by its straightness it makes a great contact with the plane. To these things he adds that, to whichever side the weight inclines, to that side it moves the mover. These are roughly the Philosopher’s views, whose reasons apply only to that circular motion which is made according to the orbit, as is the case with the wheels of carts, and are not adapted to the wheels of wells and to pulleys, such as those which are hung over wells. We therefore, following Aristotle’s intention, shall examine the first kind of rotation, which is according to the orbit. Let there be the spherical wheel A B, whose center is C; the plane of the horizon D E; the circle of contact in the plane B. The perpendicular to the horizon from the point of contact B itself B C A, passing through the center C, the parts of the wheel about the perpendicular A F B, A G B, the angle of contact G B E. First, therefore, it is established that a circle touches a plane, or line, at a point. But since, as Mechanicians, we are dealing with material circles and wheels or spheres, the Philosopher rightly did not say that a plane is touched precisely at a point, but according to its smallest part. And as for the angle, which he says is separated from the earth, that angle itself is the angle of contact. It is raised

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58 IN MECHAN. ARIST. PROBL. tur enim ex B in G. Si autem corpus quodpiam in plano fuerit, puta HI in puncto illud tanget ciculus ei occur- rens, exempli gratiâ in K. Hæc igitur accidunt circulari figuræ. In lateratis autem secus sit, quippe quæ nec in pu- cto seu secundum paruam sui partem, planum tangunt, nec semotum vt circulus à plano habent angulum, nec impingentes offendiculum in puncto tangunt. Cæterùm potissimam facilitatis motus in rotatione quæ fit secun- dum absidem, esse causam dixit, nempe quò nutat pon- dus eò à inouente impelli ac moueri. Primò igitu circu- laris sphæricaue figura in æquilibrio stat; æquales enim sunt partes quæ circa perpendicularem: ceu sunt A F B, A G B. si enim impulsus fiat ex parte F, pars opposita nuta- bit, & propendet in partem G, & tuo nutu motuq[ue] secum trahet partem A F B, fietque progressus. Si enim ducatur F C G diameter, ipsi horizonti æque distans, erit velutili- bra, cuius pondera vtrinque A F B, A G B, brachia verò æqualia C F, C G. Potentia autem quâ trahitur pellitur- ue ad instar ponderis se habet, quo addito partium alteri, factoque recessu ab æquilibrio, sequetur motus. Putauêre quidam, vt refert Philosophus, circularê lineam, ita per- peti motu versatum iri, vt manentia, propter contrarium nixum, manent, neque enim circulus in plano contrarium nixum habet, cum sit, veluti dicebamus, in æquilibrio & facilis in vtramuis partem moueri. Veruntamen perpe- tuum esse non posse horum corporum motum, ea est caus- sa, quod violentum accidat naturæ, & ideo non durabile. Ad hæc, addit Philosophus, Maiores circulos ad minores nutum habere quêdam; & nutum maioris ad minoris nu- tum, se habere vt angulos ad angulos, & diametru[m] ad dia- metrum. Angulos autem hîc sectores ipsos vocat; oportet enim circulos tum maiores tum minores circa idem cen- trum esse constitutos. Hæc autem non absimili ab eo quod supra posuimus schemate explicantur. Esto

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58 IN MECHAN. ARIST. PROBL. for it is from B to G. But if some body were on a plane, as HI perhaps, the circle meeting it would touch it at a point, for example at K. These things therefore happen in the case of a circular figure. In the case of figures with sides, however, it is otherwise; for they neither touch the plane at a point, or by a small part of themselves, nor, like a circle, do they have an angle set apart from the plane, nor, when they strike, do they touch an obstacle at a point. Moreover, he said that the chief cause of ease of motion in rotation, which takes place according to the apsis, is this: namely, that to the extent that the weight inclines, to that extent it is driven and moved by the mover. First, then, the circular or spherical figure stands in equilibrium; for the parts around the perpendicular are equal, as are A F B and A G B. For if an impulse is applied from the side of F, the opposite part will incline and lean toward G, and by its inclination and motion will draw the part A F B with it, and progress will result. For if the diameter F C G is drawn, equally distant from the horizon itself, it will be as a lever, whose weights are on either side A F B and A G B, while the arms are equal, C F and C G. But the power by which it is drawn or pushed is proportioned to weight; and when something is added to one of the parts, and a departure from equilibrium is made, motion will follow. Some, as the Philosopher reports, thought that the circular line, being thus turned by perpetual motion, would remain as things do which remain because of contrary resistance; for the circle in a plane has no contrary resistance, since it is, as we were saying, in equilibrium and easy to move to either side. Yet that the motion of these bodies cannot be perpetual is due to this cause, that violence is contrary to nature, and therefore not durable. To this the Philosopher adds that larger circles have certain inclinations toward smaller ones; and that the inclination of a larger toward a smaller is related as angles to angles, and diameter to diameter. Here he calls the sectors themselves angles; for it is necessary that both the larger and the smaller circles be established around the same center. These matters are explained by a figure not unlike the one set down above. Let it be so.

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EXERCITATIONES. 59 Esto enim circulus A B circa centrum C, Horizontis planum D E, tangens circulum in B, linea verò perpendicu- laris per centrum B C A. Sit autem circa idem cē- trum C, minor circulus F G, ducaturque C H se- cus minorem circulum in I, tangens verò maiorem in H, constituensque cum A C linea angulum A C H, duos an- gulos, ex Aristotelis mente comprehendentem, hoc est, duos sectores A C H, F C I. quoniam igitur sector seu an- gulus A C H, suo spatio superat angulum seu sectorem F C I, facilè ex nutu quem maior supra minorem habet, maior ipse minorem mouet. Videtur autem tacitè Philo- sophus hæc ad vectis naturam referre, cuius altera extre- mitatum in centro sit, altera verò in abside, & ita se habe- re nutum maioris supra minorem, vt vectis ad vectem, hoc est, semidiameter ad semidiametrum, seu sector ad secto- rem, quos quidem sectores, vt vidimus, angulos appellat. Hæc autem quæ de nutu refert, licet subtilia sint, vera es- se non videntur. Si enim in figura producatur ad opposi- tam partem semidiameter H C in K secans minorem cir- culum in L, duos alios sectores angulosue habebimus, nè- pe K C B, L C G, ipsis A C H F C I æquales. Itaq[ue] quan- tum adiuuat motum anguli A C H maioris nutus, in de- scendendo ad partes B, tantundem retardat anguli item maioris K C B, contra nutus (vt ita appellem) in ascendé- do ad partes A. & sanè quatenus ad reinaturam pertinet & ad ipsum æquilibrium, non differunt maiores circuli à minoribus, nec sunt maiores minoribus mobiliores, imo ex aliqua ratione minores videntur fore ad motum faci- liores, H 2

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EXERCISES. 59 Let there be a circle A B around center C, the plane of the horizon D E, touching the circle at B, and the line perpendicu- lar through the center B C A. But let there also be, around the same cen- ter C, a smaller circle F G, and let C H be drawn, cutting the smaller circle at I, but touching the larger at H, and forming with the line A C an angle A C H, which, in Aristotle's view, comprises two an- gles, that is, two sectors A C H, F C I. Since therefore the sector or an- gle A C H, by its own space, exceeds the angle or sector F C I, it is easily moved by the inclination which the larger has over the smaller; the larger itself moves the smaller. But it seems that the Philosopher tacitly refers this to the nature of the lever, one of whose extremities is at the center, while the other is at the rim, and so he makes the inclination of the larger over the smaller be as lever to lever, that is, semidiameter to semidiameter, or sector to sector, which sectors, as we have seen, he calls angles. But these things which he relates concerning inclination, although subtle, do not seem to be true. For if in the figure the semidiameter H C be prolonged to the opposite side into K, cutting the smaller cir- cle at L, we shall have two other sectors or angles, namely K C B, L C G, equal to A C H and F C I themselves. Therefore, as far as the inclination of the greater angle A C H helps motion in descending toward the parts B, so much does the inclination of the greater angle K C B also retard it, on the contrary, by inclination (if I may so call it) in ascending toward the parts A. And indeed, in so far as the matter pertains to resistance and to equilibrium itself, greater circles do not differ from smaller ones, nor are greater circles more easily moved than smaller ones; rather, for some reason, smaller ones seem to be more easily set in motion, H 2

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60 IN MECHAN. ARIST. PROBL. liores, tum quia data materiæ æqualitate sunt leuiores, tum etiam quod maior est angulus contactus ad planum circumferentiæ minoris quàm maioris circuli, vt in subiecta figura angulus ABC maior est angulo DBC, in materiali igitur circulo rotaue maiore sui parte tanger planum D B circulus, ipso A B. quicquid tamen sit, mobiliores sunt maiores circuli non quidem ex natura circuli, quæ tam in maioribus quàm in ipsis minoribus est par, sed alijs de caussis, quas suo loco examinabimus. Cæterùm vt aliquid de motu qui secundum absidem sit, ex nostro penu promamus, Dicimus, Circulos, rotasue, quæ hoc pacto mouentur, vel per horizontis planum moueri, vel per accliue, aut decliue. Siautem per horizontis planum, ideo facilem essemotum, quòd nunquam, cæteris paribus, centrum grauitatis ipsius corporis à centro mundi, in ipsa rotatione, fiat remotius. Esto enim planum horizontis A B, cui circulus insistat A D, circa centrum C, diuisus per centru[m] ipsum à perpendiculari ACD; Ducatur autem per centrum C recta linea horizontiæ quidistans, E C F G: dum diuidatur circulus vt cunque in partes A H, HF, FI, ID, & CI, CH iungantur. Posthæc intelligatur circulum secundum absidem moueri ad partes G, erit igitur aliquando punctum H, tangens horizontis planum, tangat autem in K, tum F in L, I

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60 IN MECHAN. ARIST. PROBL. We say that larger circles are lighter, both because, given equal matter, they are lighter, and also because the angle of contact with the plane of the circumference of the smaller circle is greater than that of the larger, as in the figure below the angle ABC is greater than the angle DBC. Thus, in the material circle or wheel, the larger part of the circle will touch the plane DB, rather than AB. Whatever the case, larger circles are more mobile, not indeed from the nature of the circle, which is the same in the larger as in the smaller, but from other causes, which we shall examine in their proper place. Moreover, in order to bring forward from our own store something about motion that is according to the apsis, we say that circles, or wheels, which are moved in this way, are moved either over a horizontal plane, or over an incline, or over a decline. But if over a horizontal plane, then the motion is easy, because, other things being equal, the center of gravity of the body itself never, in the rotation itself, becomes farther from the center of the world. Let the horizontal plane AB be such that the circle AD rests upon it, about the center C, divided through the center itself by the perpendicular ACD. And let there be drawn through the center C a straight line equidistant from the horizon, ECFG: while the circle is divided in any way into the parts AH, HF, FI, ID, and CI, CH are joined. After this, let the circle be understood to move according to the apsis toward the parts G; there will therefore at some point be the point H, touching the plane of the horizon; let it touch at K, then F at L, I

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EXERCITATIONES. 61 L, I in N. D verò in O. Ducanturque K P, L Q, N R, O S ipsi A C parallelæ horizonti autem perpendiculares. Centrum ergo circuli, quod idem & grauitatis est centru[m], feretur per rectam C P Q R S, sunt enim K P, L Q, N R, O S ipsi A C semidiametro æquales, n[on]quam igitur centrum ipsum C in circuli rotatione ab horizontis plano eleuabitur, nec à mundi centro fiet remotius. Hoc autem longè aliter cæteris figuris contingit, quarum motus ideo inæqualis, quòd non semper in rotatione centrum grauitatis eandem seruet à mundi centro distantiam. Estò enim Ellipsis ABCD, cuius cætrum E, diameter longior BED, breuior AEC, Horizontis planum. FCG. locus contactus C perpendicularis à contactu per centrum ipsa CEA diuidens Ellipsis in partes æquales, & æqueponderantes ABC, ADC. Sumantur in quadrante CD, p[ro]u[n]cta HI, tum EH, HI iungantur, erit autem EH longior ipsa EC, tum EI, ipsa EH & ED, ipsa EI. Rotetur ellipsis secundum absidem, fiet igitur punctum H in K, & à puncto K horizonti perpendicularis erigatur KL, quæ fiat æqualis EH. Post hæc punctum I erit in M, & ab M perpendicularis, æqualis EI. ruisus D fiat in O, & ipsi ED, æqualis perpendicularis OP. Mota igitur ellipsis à C in K, haud ita difficilis erit motus, quippe quod haud multum EH superet EC, at difficilior erit translatio in M, difficillima verò in O. Valde enim à situ E, ibi attollitur grauitatis centrum, ascendens nempe vbi P. Videmus igitur ex his eandem poten- tiam H 3

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EXERCISES. 61 L, I into N. But let K P, L Q, N R, O S be drawn to A C itself parallel to the horizon, but perpendicular. Therefore the center of the circle, which is also the center of gravity, will be carried along the straight line C P Q R S; for K P, L Q, N R, O S are equal to the semidiameter A C itself. Thus the center itself C in the rotation of the circle will never be raised above the plane of the horizon, nor will it be moved farther from the center of the world. But this happens in a very different way with the other figures, whose motion is therefore unequal, because in rotation the center of gravity does not always preserve the same distance from the center of the world. Let there be an ellipse ABCD, whose center is E, the longer diameter BED, the shorter AEC, the plane of the horizon FCG. The place of contact C, perpendicular from the point of contact through the center, itself CEA dividing the ellipse into equal parts, and equally weighted parts ABC, ADC. Let points HI be taken in the quadrant CD, then let EH, HI be joined; and EH will be longer than EC itself, then EI, EH itself and ED, EI itself. Let the ellipse be rotated according to the apsis; therefore point H will be in K, and from point K let a perpendicular KL be raised to the horizon, which shall be made equal to EH. After this point I will be in M, and from M a perpendicular, equal to EI. Again let D be in O, and to ED itself, let a perpendicular OP be equal. Therefore the ellipse having been moved from C to K, the motion will not be so difficult, since EH surpasses EC by not much; but the transfer to M will be more difficult, and to O most difficult. For very much from the position of E, there the center of gravity is raised, namely ascending where P. Thus we see from these things the same power H 3

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62 IN MECHAN. ARIST. PROBL. tiam in mouendo ellipsoid, haud pariter se habere, vt in mouendo circulum. ibi enim centrum grauitatis fertur per æquidistantem horizonti, hic verò modò attollitur, modò deprimitur, quod sanè molestiam & difficultatem facit. Sed idem alijs figuris contingere, & maximè lateratis, ita docebimus. Esto enim triangulum æquilaterum ABC, cuius grauitatis centrum E horizontis planum BD. Demittatur à vertice A perpendicularis horizonti AF transibit autem per centrum E, & bifariam diuidet basim BC in F. Sunt autem trianguli ABF, ACF, æquales & æqueponderantes. angulus verò AFC rectus. Lungatur EC, erit igitur maior EC, ipsa EF. Rotetur itaque triangulum circa punctum C, fiatque EC horizonti perpendicularis, sitque CH, & per E horizonti parallela ducatur EK, moto igitur triangulo, centrum grauitatis E translatum erit in H, sed KC æqualis est EF, minor autem ipsa CH, eleuatur ergo centrum grauitatis ab E in H, nempe supra K, totum spatium KH. ex qua eleuatione fit in motu difficultas. Idem prorsus eadem demonstratione ostenderetur fieri in quadrato & alijs lateratis figuris. Cur igitur in plano horizontis facillimè circularia, difficile aute[m] laterata & quæ inæquales habent semidiametros, moueantur, ex dictis clarè patet. Ad hanc quæstionem illud quoque facit, cur per decliue planum grauiora corpora, & rotunda maximè; magno impetu dimissa, delabantur. Esto enim rota sphæraue aut Cylindrus CD, cuius centrum E, tangens decliue planum AB in D, quæritur cur

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62 IN MECHAN. ARIST. PROBL. thus, in moving an ellipsoid, it does not behave in the same way as in moving a circle. For there the center of gravity is carried along by a line equidistant from the horizon; here, however, it is now raised, now lowered, which indeed causes inconvenience and difficulty. But that the same thing happens with other figures, and especially polygonal ones, we shall show thus. Let there be an equilateral triangle ABC, whose center of gravity E is in the plane of the horizon BD. Let a perpendicular AF be dropped from the vertex A to the horizon; it will pass through the center E, and will bisect the base BC at F. Now the triangles ABF and ACF are equal and of equal weight. But the angle AFC is right. Let EC be extended; therefore EC will be greater than EF itself. Let the triangle then be rotated about point C, and let EC become perpendicular to the horizon, and let it be CH, and through E let EK be drawn parallel to the horizon. When the triangle has therefore been moved, the center of gravity E will have been transferred to H; but KC is equal to EF, and less than CH itself. Therefore the center of gravity is raised from E to H, namely above K, through the whole distance KH. From this elevation difficulty arises in motion. The very same thing would be shown by the same demonstration to happen in a square and in other polygonal figures. Why then circular bodies move on the plane of the horizon most easily, but polygonal bodies and those having unequal semidiameters with difficulty, is clearly evident from what has been said. To this question there is also this consideration: why heavier bodies, and especially round ones, when released with great force, roll down a sloping plane. Let there be a wheel, or sphere, or cylinder CD, whose center is E, touching the sloping plane AB at D; the question is why

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EXERCITATIONES. 63 cur dimissa hæc magno impetu deferantur ad partes B, Ducatur per grauitatis centrum E ad horizontem B K perpendicularis F E L secans decliue planum in G, circumferentiam verò in H. opponitur autem E G angulo recto ED G, maior ergo E G ipsa ED, hoc est, EH, inter circumferentiam igitur & planum decliue, spatium intercedit H G. Ducatur item D I ipsi F G æquidistans. non transibit igitur per centrum E. minor erit igitur diametro CD, quare circulum in partes inæquales secabit, & non per grauitatis centrum, quod idem cum magnitudinis seu figuræ centro supponitur. Dimissa igitur rota, contingit quidem planum decliue in puncto D. At centrum grauitatis premit secundam per lineam perpendicularem F G, non sustentatur autem in H, quippe quod inter planum & circumferentiâ intercedat spatium HG, nec H locum habeat cui innitatur, corpus autem ita per lineam D I est diuisum, vt longè maior sit pars I F C H D ipsa D I, & centrum in ea parte cadat quæ non fulcitur. itaque suopte nutu, cum extra ful cimentum sit D & perpendicularem D I ad inferiores partes rapidè rotans delabitur. Ducatur autem perpendicularis GL, parallela MN, & quoniam B N breuior est B L, erit MN ipsa GL breuior. Est igitur punctum M mundi centro propius quàm D & G, quare eò non impedita rota ipsa suo nutu feretur, nec stabit donec infimum locu[m] vbi quiescat nanciscatur. Possumus etiam Rota sphæraue in plano decluii collocata, datam potentiam inuenire, quæ extremitati diametri ad eam partem quavergit applicata ipsam rotam sphæramue impediat ne delabatur. Esto

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EXERCISES. 63 why, being released, are these, with great impetus, carried to the parts B, Let there be drawn through the center of gravity E to the horizon B K the perpendicular F E L, cutting the inclined plane in G, and the circumference in H. But E G is opposed to the right angle E D G; therefore E G is greater than E D itself, that is, than E H; therefore between the circumference and the inclined plane there intervenes the space H G. Let there be drawn likewise D I parallel to F G. It will therefore not pass through the center E. It will therefore be less than the diameter C D; wherefore it will cut the circle into unequal parts, and not through the center of gravity, which is here supposed to be the same as the center of magnitude or figure. Therefore, when the wheel is released, it does indeed touch the inclined plane at the point D. But the center of gravity presses on the lower side by the perpendicular line F G; it is not, however, supported at H, since there intervenes the space H G between the plane and the circumference, nor does H have a place on which to rest; and the body is thus divided by the line D I, so that the part I F C H D is much greater than D I itself, and the center falls in that part which is not supported. Thus, by its own inclination, since D is outside the support and, rapidly turning, it descends by the perpendicular D I toward the lower parts. Let there be drawn the perpendicular G L, parallel M N; and since B N is shorter than B L, M N itself will be shorter than G L. Therefore the point M is nearer to the center of the world than D and G; wherefore, not being hindered in that direction, the wheel itself will move by its own inclination, nor will it stop until it finds the lowest place where it may rest. We can also, a wheel or sphere being placed on an inclined plane, find the given power which, applied to the end of the diameter on that side toward which it tends, may prevent the wheel or sphere itself from slipping. Let it be

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IN MECHAN. ARIST. PROBL. Esto planum inclinatum AB, cui Rota sphæraue insistat tangatq[ue] illud in C. Rota verò ipsa sphæraue DG, cuius centrum E, diameter verò DEC ipsi BA ad punctu[m] contactus C, perpendicularis. Ducatur per C ipsi horizonti perpendicularis FG circulum secas in G tum per E ipsi CG perpendicularis, ipsi verò BF horizonti æquidistans HEI ceu vectis, cuius fulcimentum I respondens ipsi C, pondus verò in E, vbi grauitatis est centrum. Applicata igitur potentia in H erit pondus inter fulcimentum & potentiam, quare vt IE ad IH ita potentia sustinens in H ad pondus in E, quod demonstrandum fuerat. Quippiam simile ostendit Pappus 1.8. prop. 9. alijs tamen suppositis & consideratis. Dico præterea, ijsdem stantibus angulum EC I æqualem esse angulo inclinationis CBF. Producatur HI concurrens cum ipsa AB in K, concurrer autem propterea, quod CIK rectus sit, ICA minor recto, & quoniam HK parallela est horizonti BF alterni anguli IKC, CBF, æquales erunt. Similes autem sunt EC I, ECK, trianguli, estque EC I angulus æqualis angulo EKC, hoc est, ipsi CBF. vnde sequitur, quo minor fuerit inclinationis angulus, eo facilius rotam sphæramue in plano inclinato sustineri. quo enim minor fuerit angulus EC I, eo minus latus EI & minor proportio EI ad IH, & ideo minor potentia sustinens requiratur in H. Cæterum accliue & decliue planum nihil differunt nisi respectu. His ita consideratis, admonet nos locus, vt pulcherrimam dubitationem diluamus. Quæritur, Cur maiores rotæ

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IN MECHAN. ARIST. PROBL. Let AB be an inclined plane, upon which a wheel or sphere rests and touches it at C. Let the wheel itself, or sphere DG, whose center is E, and whose diameter DEC is perpendicular to BA at the point of contact C, be considered. Through C draw FG perpendicular to the horizon; it cuts the circle at G. Then through E draw HEI, perpendicular to CG, and parallel to the horizon BF, as a lever, whose fulcrum I corresponds to C, and whose weight is at E, where the center of gravity is. Therefore, if the power is applied at H, the weight will be between the fulcrum and the power; wherefore, as IE is to IH, so is the sustaining power at H to the weight at E, which was to be demonstrated. Pappus shows something similar in Book 1, Proposition 9, though with different assumptions and considerations. I say further that, these conditions being the same, the angle ECI is equal to the angle of inclination BCF. Let HI be produced to meet AB itself at K. It will meet it for this reason, that CIK is a right angle, ICA less than a right angle, and since HK is parallel to the horizon BF, the alternate angles IKC and CBF will be equal. But the triangles ECI and ECK are similar, and the angle ECI is equal to the angle EKC, that is, to CBF itself. Hence it follows that the smaller the angle of inclination is, the more easily will a wheel or sphere be sustained on an inclined plane. For the smaller the angle ECI is, the less will be the side EI and the smaller the proportion of EI to IH, and therefore the less sustaining power will be required at H. Moreover, an ascending and a descending plane differ only in relation. These things being thus considered, the place reminds us that we should resolve a very beautiful doubt. The question is: Why are larger wheels

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EXERCITATIONES. 65 rot[us] impingentes, facilius offendicula superent quàm minores. Neque enim satisfacere videtur quod ait Aristoteles, ex contactu in puncto eo anguli à plano eleuatione id fieri, alijs ergo principijs dubitatio soluitur. Esto rota quidem maior A B, circa centrum C minor vero DB circa centrum E, t[ame]gentes horizontis planum in B. Diameter maioris A B, minoris D B, offendiculum horizonti perpendiculare F G. Ducatur per F horizonti parallela F K secans minoris rotæ peripheriam in H, diametrum verò A B in K, & à puncto H ad planu[m] horizontis perpendicularis demittatur H I: erit autem H I æqualis ipsi offendiculo F G, & iungantur B H, B F. Itaq[ue] quoniam B H ab extremo B cadit in triangulum K F B, erit K H B angulus maior angulo K F B. Parallelæ autem sunt K F, B G, pares ergo anguli K H B, H B G, pares item K F B, F B G, Maior ergo H B I, ipso F B C. At minoris rotæ grauitatis centrum mouetur secundum lineam B H, maius verò secundum literam B F, difficilius ergo mouebitur, & superabit offendiculum minor rota, quàm maior: quod fuerat demonstrandum. Possumus idem ostendere magis mechanicè, hoc est, rem ad vectem reducendo. Esto horizontis planum A B, rota maior C D planum tangens in D. rotæ verò maioris centrum E. Rota verò minor F D, tangens itidem planum in D. rotæ autem centrum G, offendiculi verò rectudo D H. Ducatur per H ipsi A B horizonti æquidistans H I secans minorem circulum in K, maiorem verò I in

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EXERCISES. 65 Wheels impinging upon obstacles overcome them more easily than smaller ones. Nor does what Aristotle says seem sufficient, namely, that this happens because of contact at a point, by the elevation of the angle from the plane; therefore the difficulty is resolved by other principles. Let there be a larger wheel A B, around center C, and a smaller one D B, around center E, touching the plane of the horizon at B. The diameter of the larger, A B, and of the smaller, D B, [and] the obstacle FG perpendicular to the horizon. Through F draw F K parallel to the horizon, cutting the circumference of the smaller wheel at H, but the diameter A B at K; and from point H let H I be dropped perpendicular to the plane of the horizon: H I will be equal to the obstacle itself F G. And join B H, B F. Thus, since B H from the extreme point B falls within triangle K F B, the angle K H B will be greater than angle K F B. But K F and B G are parallel; therefore the angles K H B and H B G are equal, and likewise K F B and F B G are equal. Therefore H B I is greater than F B C itself. But the center of gravity of the smaller wheel moves along line B H, whereas that of the larger moves along line B F; therefore the smaller wheel will move more easily and overcome the obstacle than the larger one: which was to be demonstrated. We can show the same thing more mechanically, that is, by reducing the matter to a lever. Let A B be the plane of the horizon, and let the larger wheel C D touch the plane at D. Let E be the center of the larger wheel. But let the smaller wheel F D likewise touch the plane at D. Let G be the center of that wheel, and D H the height of the obstacle. Through H draw H I, parallel to the horizon A B, cutting the smaller circle at K, and the larger at I in

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66 IN MECHAN. ARIST. PROBL. in I. Ducantur etiam diametri maioris quidem LEM, minoris NGO, Tum à puncto k perpendicularis ducatur ad GO, ipsa kP, item à puncto I ad EM perpendicularis IQ. Dico EQ ad QL, minorem habere proportionem quam GP, ad PN. Connectatur Gk, & eiper E parallela ducatur ER, secans maiorem circulum in R, & ab R ipsi EM perpendicularis ducatur RS. quoniam igitur ER parallela est ipsi Gk, erit GER angulus HGk angulo æqualis. Recti autem sunt HGP, GES reliquie ergo kGP, RES ad inuicem sunt æquales. Sed & ESR, GP k recti sunt, quare ER S GkP anguli æquales sunt, & trianguli GP k ESR, per pr. diff. l. 6. similes. Vt ergo Gk hoc est GN ad GP, ita ER hoc est ELad ES. Componendo igitur vt NP ad PG, ita LS ad SE. quamobrem si fulcimentum esset in S, pondus in E, potetia in L, idem fieret ac fiat fulcimento in P, pondere in G, potentia verò in N constituta. & id quidem si eiusdem ponderis vtraque rota supponatur. Rursus quoniam vt Dk ad totum circulum DF, ita DR ad totum DC. Minor est autem proportio DI ad totum circulum DC, ergo minor est DI ipsa DR. Maior ergo MI ipsa MR, maior ergo QI ipsa SR, propius ergo centro E est Q ipso puncto S, minor est igitur proportio EG ad LQ quàm ES ad SL. Minor ergo potentia requiritur in L ad sustinendum pondus E ex fulcimento Q hoc est I, quàm requiratur in N ad sustinendum pondus G ex fulcimento P, hoc est k. Minor ergo potentia requiritur ad

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66 IN MECHAN. ARIST. PROBL. in I. Let the diameters also be drawn, the greater indeed LEM, the lesser NGO. Then from point k let a perpendicular be drawn to GO, namely kP itself; likewise from point I to EM the perpendicular IQ. I say that EQ has a smaller proportion to QL than GP has to PN. Let Gk be joined, and through E let a parallel be drawn ER, cutting the greater circle at R; and from R let RS be drawn perpendicular to EM itself. Since therefore ER is parallel to Gk, the angle GER will be equal to the angle HGk. But HGP and GES are right angles; therefore the remaining angles kGP and RES are equal to one another. And also ESR and GPk are right angles; wherefore the angles ERS and GkP are equal, and the triangles GPk and ESR are similar, by the preceding difference, book 6. Therefore, as Gk, that is GN, is to GP, so ER, that is EL, is to ES. Composing therefore, as NP is to PG, so LS is to SE. Wherefore, if the fulcrum were at S, the weight at E, and the power at L, the same would happen as does happen when the fulcrum is placed at P, the weight at G, and the power indeed at N. And this indeed if each wheel be supposed of the same weight. Again, since as Dk is to the whole circle DF, so is DR to the whole DC. But the proportion of DI to the whole circle DC is smaller; therefore DI itself is smaller than DR. Therefore MI is greater than MR; therefore QI is greater than SR; therefore Q is closer to center E than point S itself. Therefore the proportion of EG to LQ is smaller than that of ES to SL. Therefore a smaller power is required in L to sustain the weight E from the fulcrum Q, that is I, than is required in N to sustain the weight G from the fulcrum P, that is k. Therefore a smaller power is required to

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EXERCITATIONES. 67 ad transferendam maiorem rotam CD vltra offendiculum IV, hoc est, DH, quàm requiratur ad transferendam minorem vltra offendiculum K T, hoc est HD, quod fuerat ostendendum. Ad hæc, quæri potest, quo pacto plaustrorum rotæ in ipsa plaustri conuersione se habeant, nempe quæ sit linea illa curua, quam in conuersione describunt. Esto rotarum in plano orbita, du[m] plaustrum rectâ procedit AB, CD, Sunt autem ipsæ lineæ, quod ostendemus postea, æquedistantes. Sit itaque punctum B illud in quod rota quæ per AB fertur, eò delata planum tangit. D verò alterius rotæ atque plani contactus. Igitur dum plaustris sit conuersio, punctum D conuersionis sit centrum. Stat enim interim rota & circa lineam conuerititur, quæ à puncto contactus D per rotæ centrum ducta horizontis plano est perpendicularis. ea autem stante, rota quæ in B circa centrum D semicirculu[m] pertransit DEF, vbi autem rota B, peruenerit in F, plaustriam in oppositam partem conuerso, rota quæ est in D per lineam DC, quæ verò in F per rectam FG mouetur, plaustrique fit regressus. Et quoniam vel D in ipsa conuersione stat omnino nec quicquam progreditur, vt in prima figura, vel non stat vt in secunda, quo casu portionem parui circuli describit, ipsi maiori circulo & exteriori concentricam. Vnde colligimus, Plaustrorum conuersiones flexionesque semper circa centrum, & concentricorum circulorum portiones fieri. Hinc etiam discimus, cur veteres, vt ex antiquis co- gnosci- I 2

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EXERCISES. 67 for moving the larger wheel CD beyond the obstacle IV, that is, DH, than is required for moving the smaller one beyond the obstacle K T, that is, HD, which was to be shown. In addition, one may ask in what manner the wheels of carts behave in the very turning of the cart, namely, what is that curved line which they describe in turning. Let the track of the wheels in the plane be, while the cart moves straight along AB, CD. These lines are, as we shall show later, equidistant. Let therefore the point B be that point at which the wheel carried along by AB, when brought there, touches the plane. D, however, is the point of contact of the other wheel and the plane. Therefore, while the cart is turning, let point D be the center of the turning. For in the meantime the wheel stands still and turns about the line which, drawn from the point of contact D through the center of the wheel, is perpendicular to the plane of the horizon. But the wheel standing thus, the wheel which is in B passes through the semicircle DEF about the center D; and when the wheel B has arrived at F, the cart being turned to the opposite side, the wheel which is in D moves along the line DC, and that which is in F along the straight line FG, and the cart begins to move backward. And since either D in the very turning stands altogether still and advances nothing at all, as in the first figure, or does not stand still, as in the second figure, in which case it describes a portion of a small circle concentric with the greater and outer circle, we conclude that carts’ turnings and bends are always made about the center and along portions of concentric circles. Hence also we learn why the ancients, as may be known from the old I 2

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68 IN MECHAN. ARIST. PROBL. gnoscimus vestigijs, circos in quibus cursus quadrigarum fiebant ea forma quæ apparet, efformauerint. Hoc etiam theorema probamus. Cylindros, quorum bases axi sunt perpendiculares, dum in æquato plano conuoluuntur, rectâ incedere & per parallelas, quarum distantia axis seu latoris longitudine præfinitur. E D G A C H B Esto enim Cylin- drus ABCD, cuius a- xis GH, horizôtis pla- no insistens secundum latus AB, cui latus op- positum & æquale CD. Moueatur Cylindrus rotans, donec latus CD, in plano sit vbi EF. Describat autem circuli CB linea[m] BF. Circulo verò AD lineam AE. Dico eas rectas esse, & parallelas. Si enim superficies basium DA, CB, extendan- tur ita vt horizontis planum secant, illud secabunt iuxta lineas AE BF, recta ergo est vtraque. Sed & parallelas esse ad inuicem ita ostendimus. quoniam semicirculus AD, æqualis est semicirculo BC, erit linea AE, æqualis linea BF, sed & AB, æqualis est ipsi DC, quare & ipsi EF. Oppo- sita igitur quadrilateri figura ABFE latera æqualia sunt, quare EF æquedistat ipsi AB, tum AE ipsi BF, quod fue- rat demonstrandum. Probabimus etiam si cylindri bases axi perpendicu- lares non fuerint, & ideo ellipses in ipsa rotatione per pla- num, parallelas quidem describere, sed non rectas. Esto enim Cylindrus ABCD, cuius bases ellipses inuice[m] æquedistates, quarum axes longiores AB, CD, Commu- nis autem sectio cylindri & plani ad axem & horizontem planum perpendicularis EHF. Diuidatur autem semicir- culus

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68 IN MECHAN. ARIST. PROBL. we recognize by the tracks that the circles in which the chariot races were run had been formed in this shape. We prove this theorem as well. Cylinders, whose bases are perpendicular to the axis, while they are rolled on a level plane, advance in a straight line and along parallels, the distance between which is determined by the length of the axis or side. E D G A C H B Let there be a cylinder ABCD, whose axis is GH, standing on the plane of the horizon along the side AB, with the opposite and equal side CD. Let the cylinder be moved by rotation until the side CD is in the plane where EF is. Now let the line BF be described by the circle CB. And by the circle AD, the line AE. I say that these are straight, and parallel. For if the surfaces of the bases DA, CB are extended so that they cut the plane of the horizon, they will cut it along the lines AE and BF; therefore each is straight. But that they are parallel to one another we have shown as follows: since the semicircle AD is equal to the semicircle BC, the line AE will be equal to the line BF; and AB is also equal to DC, therefore also to EF. Thus the opposite sides of the quadrilateral ABFE are equal; therefore EF is equidistant from AB, and AE from BF, which was to be demonstrated. We shall also prove that if the bases of the cylinder were not perpendicular to the axis, and therefore ellipses are described in the very rotation through the plane, they do indeed describe parallels, but not straight lines. Let there be a cylinder ABCD, whose bases are mutually equidistant ellipses, whose longer axes are AB and CD. But the common section of the cylinder and the plane perpendicular to the axis and the plane of the horizon is EHF. Let the semicircle be divided

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EXERCITATIONES. 69 culus EHF in partes æquales quatuor FI, IH, HG, GE. Tum per diuisionum puncta lateri parallele, rectæ ducantur KGL, MHN, OIP, quæ quidem cùbases AMB, DNG parallelæ sint, erunt inuicem æquales, cumque circumferentia EHF æquales, eosque rectos angulos costituent. Ducatur post hæc seorsum recta QR, & eidem perpendicularis ST eam secans in V. applicetur autem rectæ ST æqualis Cylindrilateri BC, ipsa [uncia] ita tamen vt punctum E congruat puncto V, sitque Vn æqualis EB, Vq[ue] verò æqualis EC. Tum fiant VX, XY, YZ, Za æquales ipsis EG, GH, HI, IF, & per puncta X, Y, Z, α, & paralleli ipsi ST ducantur oαπ, νZ [uncia], λγμ, ηχθ, tum & his ex altera parte respondentes paralleleæ per puncta β, γ, δ, ε. Sit autem oα, æqualis AF, ααæqualis FD, item εγ, æqualis EC, εσæqualis EB, sed & νZ [uncia] æqualis OI, z [uncia] ipsi P, λγ ipsi MH, yμ verò ipsi HN, tūx ipsi KG. & x θ, ipsi GL & ipsis æquales & æqualiter positæ ad partes R, aliæ paralleleæ aptétur per β, γ, δ, ε, quibus I 3

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EXERCISES. 69 Divide the arc EHF into four equal parts FI, IH, HG, GE. Then through the points of division, draw straight lines parallel to the side, KGL, MHN, OIP, which indeed, if the bases AMB, DNG are parallel, will be equal to one another, and with the circumference EHF will form equal right angles. After this let the straight line QR be drawn separately, and let ST, perpendicular to it, cut it at V. To the straight line ST apply a length equal to the cylindrical side BC, in such a way, however, that the point E coincide with the point V, and let VN be equal to EB, and VQ equal to EC. Then let VX, XY, YZ, Za be made equal to EG, GH, HI, IF respectively, and through the points X, Y, Z, α, and parallel to ST, let oαπ, νZ, λγμ, ηχθ be drawn; then also, on the other side, let the corresponding parallels be drawn through the points β, γ, δ, ε. And let oα be equal to AF, αα equal to FD; likewise εγ equal to EC, εσ equal to EB; but also νZ equal to OI, z equal to P, λγ equal to MH, yμ however to HN, then x to KG, and xθ to GL; and let other parallels equal to these, and similarly placed at the parts R, be fitted through β, γ, δ, ε, by which I 3

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70 IN MECHAN. ARIST. PROBL. quibus ita dispositis per puncta [a, v, λ, η, η], item per [π, ζ, μ, θ, ζ] ducantur linea [α, η, ζ], curuæ quidem & eodem pacto aliæ curuæ illis respondentes [η, ζ], Erunt igitur [a, η, ζ, η, ζ, η], parallelæ quidem eo quod lineæ quæ inter ipsas ducuntur, parallelæ sint & æquales, non tamen rectæ illæ, sed curuæ. Moto igitur Cylindro circulus EHF rectam describetæ, ellipsis verò AMB, curuam [α, η], ellipsis autem DNC, ipsam curuam [π, ζ]. In hoc aute[m] Cylin dri motu illud mirabile, velociores nempe, in ipsa rotatione esse ellipses ipso circulo EHF. Ducatur enim recta [α, η] quæ occurrat ipsi VS in S, & [α, η] iungatur, fietque triangulum [α, η] S. est autem angulus [α, η] rectus, maior ergo [α, η] ipsa [α, η], sed recta [α, η] æqualis est ipsi [α, v], hoc est, semicirculo FHE. multo maior est autem curua, [a, v, λ, η, η], ipsa recta [α, η], sed eodem tempore quo semicirculus EHF conficit in rotatione spatiu[m] αV, eodem dimidia ellipsis BMA metitur curuam [α, λ, η]. velocior igitur est ellipsis ipso circulo. Hæc quoque speculatio ad motum qui secundum absidem fit, manifestè pertinet. Coni, quorum bases circuli sunt, si in plano secundum latus rotentur, basi circulum describunt, cuius centrum immobile coni ipsius est vertex, semidiameter verò ipsum latus. Este conus ABC cuius vertex C basis AB, axis DC, basis verò centrum D, latus quo planum tangit BC, secatur itaque Conus per latus BC & axem DE à plano horizonti perpendiculari, cuius & coni communis sectio est ABC triangulum, & quoniam coni grauitatis centrum est in axe

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70 IN MECHAN. ARIST. PROBL. when these things are thus arranged, through the points [a, v, λ, η, η], likewise through [π, ζ, μ, θ, ζ], let the line [α, η, ζ] be drawn; and indeed curved lines, and in the same way other curved lines corresponding to them [η, ζ], will be drawn. Therefore [a, η, ζ, η, ζ, η] will be parallel, because the lines drawn between them are parallel and equal, though not straight lines, but curved ones. When the Cylinder is moved, then the circle EHF will describe a straight line, but the ellipse AMB the curve [α, η], and the ellipse DNC the curve [π, ζ] itself. In this motion of the Cylinder there is that remarkable thing, namely that the ellipses, in the rotation itself, are faster than the circle EHF. For let the straight line [α, η] be drawn, which meets VS itself at S, and let [α, η] be joined; and a triangle [α, η]S will be formed. Now the angle [α, η] is right, therefore [α, η] itself is greater than [α, η], but the straight [α, η] is equal to [α, v], that is, to the semicircle FHE. But the curved line [a, v, λ, η, η] is much greater than the straight [α, η], yet in the same time in which the semicircle EHF completes in rotation the space αV, by the same time the half-ellipse BMA measures the curved line [α, λ, η]. Therefore the ellipse is faster than the circle itself. This speculation also clearly pertains to motion that takes place according to the apse. Cones, whose bases are circles, if they are rotated on a plane by their side, describe a circle by the base, whose immobile center is the vertex of the cone itself, while the semidiameter is the side itself. Let there be the cone ABC, whose vertex is C, base AB, axis DC, and the center of the base D, and the side by which the plane touches it BC; therefore the cone is cut by the side BC and the axis DE by a plane perpendicular to the horizon, whose common section with the cone is the triangle ABC, and since the center of gravity of the cone is on the axis

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EXERCITATIONES. 71 axe ipso, conus in partes æquepôderantes secatur AEBC, AFBC, stat ergo conus sibimet æquilibris. Si autem à po- tentia quadam moueatur, puta ab A versus F, trahitur se- micirculus BEA, à semicirculo AFB, & ita fit rotatio. Ita- que si imaginemur, infinitos vsque ad verticem parallelos basi circulos, eorum semicirculi in ipso motu & trahent & trahentur; at cum ad verticem circuli desinant, nec ibi se- micirculi sint qui trahant & trahantur, motus rotationis prorsus cessat & vertex ipse immobilis fit rotationis cen- trum. Quoniam igitur lateris BC, punctum C stat, B verò circa ipsum mouetur, in ipso motu circulus describitur BH1K, cuius semidiameter BC, & eodem pacto alij cir- culi in cono, qui basi HEBF sunt æquedistantes, circulos in plano circa idem centrum describent, vt facile videre est in obiecto schemate. Huic similem demonstrationem affert Heron in libello Automatum, quem nos Tyrones adhuc vernacule è Græco translatum, Venetijs prælo subiecimus. Porrò si conus rotundus pro basi ellipsis habeat, sectionem videlicet per planum axi non perpendiculare, in ipsa rotatione, stante vertice, ellipsis basis, ellipsis de- scribit in plano, cuius maior diameter à puncto quod co- ni vertex est, ita diuiditur, vt diametri pars maior æqualis sit lateri maximo; minor verò æqualis lateri minimo. Sed hæc ad aliam pertinent speculationem. His itaque de motu rotundorum, qui circa absidem fit, consideratis, reliquum esset de motu trochlearum, qui circa centrum sit, opportunè agere, sed cùm in sequenti quæstione de hoc sermonem faciat Philosophus, ad ea quæ ibi disputabuntur, lectorem ablegamus. Modò de tertia motus specie nobis erit sermo; in qua quidem specie nonnulla perpendemus, quæ omisit A- ristoteles. Agitur autem hîc de rotundorum corporum motu,

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EXERCISES. 71 on the axis itself, the cone is cut into equal parts AEBC, AFBC; thus the cone stands in equilibrium with itself. But if it is moved by some power, say from A toward F, the semicircle BEA is drawn by the semicircle AFB, and thus rotation takes place. Therefore, if we imagine infinite circular lines parallel to the base, extending all the way to the vertex, their semicircles in the very motion both draw and are drawn; but when they come to an end at the vertex of the circles, and there are no semicircles there to draw and be drawn, the motion of rotation entirely ceases, and the vertex itself becomes the immovable center of rotation. Since, then, point C of the side BC remains fixed, while B moves around it, in the very motion the circle BH1K is described, whose semidiameter is BC; and in the same way the other circles in the cone, which are equally distant from the base HEBF, will describe circles in the plane around the same center, as is easy to see in the figure before you. A similar demonstration is given by Heron in the little book Automata, which we, still as beginners, translated from Greek into the vernacular and submitted to the press in Venice. Moreover, if a round cone has an ellipse for its base, that is, if it is a section by a plane not perpendicular to the axis, then in the actual rotation, with the vertex standing still, the elliptical base describes an ellipse in the plane, whose greater diameter is divided by the point that is the vertex of the cone in such a way that the larger part of the diameter is equal to the larger side; and the smaller, equal to the smaller side. But this belongs to another inquiry. With these things, then, considered concerning the motion of round bodies that takes place around the axis, the remaining matter would be to treat suitably of the motion of pulleys, which takes place around a center; but since the Philosopher speaks of this in the following question, we refer the reader to what will be discussed there. Now there will be discourse for us about the third kind of motion, in which kind we shall consider certain things that Aristotle omitted. Here, however, the motion of round bodies is being treated,

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72 IN MECHAN. ARIST. PROBL. motu, qui sit circa axem horizonti perpendicularem, axis altera extremitate in eodem horizontis plano manente, vti videre est in ipsis figulorum rotis. Hanc motus speciem in extrema quæstionis parte cum duabus alijs speciebus comparans ait, eam quæ in obliquo sit motionem (ita enim hanc, de qua agimus, appellat) ipsam impellere mouentem, hoc est, nullum ex se ad motum propensionem habere, nutumue, & omnia illi esse à motore, secundum verò eam motionem, quæ supra diametrum est, se ipsum mouere circulum. Dixerat enim, ea referens quæ superiùs circa principium de circulo verba faciens, examinauerat, circulum ex duabus fieri lationibus, altera præter, altera verò secundum naturam, & ideo hanc semper nutum habere, & ceu continuo motam ab eo moueri qui mouet. Videtur autem clarè profiteri, ideo difficiliorem esse huius tertia speciei motum, eo quòd nutu careat proprio & tantum ab alieno, vt ita dicam, motore, moueatur. Veruntamen motum hunc facilitate alijs illis duo- bus nequaquam cedere, facilè ex sequentibus ostendemus. Primo, quia pondus totum rotati corporis, ex grauitatis centro quod in ipso axe est à plano cui nititur, sustinetur: minima quidem sui parte axe ipso tangente planu[m] vnde fit, nullam ferè dum rotatur corpus, circa centrum vbi nititur, frictionem partium fieri. Præterea grauitatis centrum semper stat, nec minimum quidem in ipsa rotatione attollitur, quod sanè cum naturæ sit repugnans, difficultatem facit. Ad hæc circa axem ita libratur rota, vt quantumuis exigua potentia alteri parti applicetur, altera illico superata moueatur. Licet enim propriè ea tantu[m] corpora æquilibrae dicantur, quæ ob ponderis hinc inde æqua-

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72 IN MECHAN. ARIST. PROBL. motion, which is around an axis perpendicular to the horizon, the axis remaining with one extremity in the same plane of the horizon, as may be seen in the wheels of potters themselves. Comparing this kind of motion in the last part of the question with the other two kinds, he says that that which is in an oblique motion (for thus he calls this one, of which we are speaking) does not itself impel the mover, that is, it has in itself no inclination, or impulse, toward motion, and everything is from the mover; but according to that motion which is above the diameter, the circle moves itself. For he had said, referring to what he had examined earlier around the beginning when speaking of the circle, that the circle is made from two motions, one contrary, the other according to nature, and therefore it always has impulse, and, as if continually moved, is moved by that which moves it. And clearly he seems to admit that this third kind of motion is therefore more difficult, because it lacks its own impulse and is moved only by another, so to speak, mover. Nevertheless, we shall easily show from the following that this motion is in no way inferior in ease to those other two. First, because the whole weight of the rotating body, from the center of gravity which is in the axis itself, is supported by the plane on which it rests; only a very small part of it touching the plane by the axis itself, whence it comes about that, while the body is rotating, there is scarcely any friction of the parts around the center where it rests. Moreover, the center of gravity always remains still, nor is it raised even the least during the rotation itself, which certainly, since it is contrary to nature, causes difficulty. In addition, the wheel is balanced around the axis in such a way that however slight a force be applied to one part, the other is at once overcome and moved. For although properly only those bodies are called equilibrious, which on account of the equal weight on this side and on that

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EXERCITATIONES. 73 æqualitatem horizonti fiunt æquidistantes, nihilominus & hicaliquam esse æquilibrij similitudinem patebit. Esto enim rota ABCD, cuius axis horizonti perpendicularis FEG transiens per centrum E, tangens autem planum in puncto G. Ducatur diameter BED, Itaque si per diametrum BED, & axem FEG corpus diuidatur, eo quòd centru[m] grauitatis in axe inueniatur, corpus ipsum in duas partes tu[m] mole tum pòdere æquales secabitur, nempe BAD, BCD. Nulla igitur adhibita vi extranea stabit corpus in quodâ, vt diximus, æquilibrio. At alteri partium potentiâ quauis licet exigua appositâ, puta in C, præualebit pars BCD, & partem BAD vel impellet vel rapiet, alterâ interim eius motui obsequente. Potentia igitur quæ in C, nullam rem quæ impediat inueniens, velocissimè rotam mouet, quod eo faciliùs velociusque fit, quo magis rota est in motu, eius verò diameter maior & potentia mouens à centro remotior, & sanè motus facilitate inde cognoscimus, quòd ipso impulsore ab impulsu cessante, diutissimè rota impressum motum seruet, nec nisi post longam rotationem omnino quiescat. Cæterùm quia sicco, vt aiunt, pede Aristoteles quæ adhunc motum pertinet pertransijt, nos quædam quæ ad hanc rem faciunt, diligentius expendemus. Quærimus igitur primò; Cur ea quæ hoc pacto rotatûr, in ipsa rotatione locum non mutent, nisi extrinseca aliqua id fiat ex caussa. Esto enim rota aut aliud quippiam rotundum ceu Turbines sunt, quibus pueri ludunt, quod circa axem ho- K rizonti

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EXERCISES. 73 being parallel to the horizon, nevertheless it will be seen that there is in some measure a likeness to equilibrium. For let there be a wheel ABCD, whose axis FEG, perpendicular to the horizon, passes through the center E, but touches the plane at the point G. Let the diameter BED be drawn. Thus if the body be divided by the diameter BED and the axis FEG, because the center of gravity is found on the axis, the body itself will be cut into two parts equal both in magnitude and in weight, namely BAD and BCD. Therefore, with no external force applied, the body will stand in a certain, as we said, state of equilibrium. But if any force whatever, even a slight one, be applied to one of the parts, say at C, the part BCD will prevail, and will either push or carry along the part BAD, the other part meanwhile yielding to its motion. The force therefore which is at C, finding nothing to hinder it, moves the wheel most swiftly, and this is done the more easily and quickly the more the wheel is in motion, and likewise the larger its diameter and the more remote from the center the moving force is; and indeed we know the ease of the motion from this, that even after the impulse itself has ceased, the wheel retains the impressed motion for the longest time, and only after a long rotation comes completely to rest. Moreover, because Aristotle, as they say, passed over what pertains to this motion with dry foot, we shall more carefully consider certain things that bear on this matter. We ask first, therefore: Why things that revolve in this way, during the very rotation, do not change place, unless this be done for some extrinsic cause. For let there be a wheel or something round such as the tops with which boys play, which about a horizontal axis K

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74 IN MECHAN. ARIST. PROBL. rizonti perpendicularem mo- ueatur, ABCD, cuius centrum E, Diameter AEC. Modò circa centrum E infiniti imaginentur circuli, alij alijs minores vsque ad centru[m] ipsum, vti sunt FGH; ibi enim circuli esse desinunt, vbi nullum amplius est spatium. Applicetur itaque potentia in B, quæ rotam v[er]geat versus A. eodem igitur tempore & insimul A versus D, D versus C, & C versus B mouebitur. quantum enim semicirculorum à parte CBA transit vltra diametrum AEC, tantundem semicirculorum, qui sunt ad partem ADC, transibit ad partes CBA. At vbi desierit motus, ibi desinit rotatio; vbi autem desinit spatium, desinit motus, sed vbi desinunt cir- culi, desinit spatium, quare in centro cum non sint circuli, nec spatium ibi desinit motus. nulla enim adest ratio, cur ipsum corpus alio à loco in quo est, ex rotatione transfe- ratur. Stat ergo rotans, quod fuerat demonstrandum. Est autem hæc demonstratio ei similis, quam suprà retuli- mus de coni in plano circa verticem rotatione, quam ab Herone in Automatis excogitatem diximus. Addimus in hoc rotationis genere corpus in ipso motu fieri leuius, idque eo magis, quo rotatio velocior. Causa est, quod lateralis motus eum motum aliqualiter impedit, qui ex naturali grauitate fit ad centrum, idcirco experientiâ docemur, leuissimos esse turbines, quibus pu- eri ludunt, si manus teneantur palmâ, dum citissima rota- tionemouentur. Ad hæc alia proponitur, & soluitur quæstio, Cur ro- tunda corpora huic motionis generi sint aptiora. Exploratissimum est, corporum, quæ ita mouentur, par-

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74 IN MECHAN. ARIST. PROBL. let there be moved perpendicularly to the horizon, ABCD, whose center is E, diameter AEC. Now let infinite circles be imagined around the center E, some smaller than others, down to the center itself, as are FGH; for there the circles cease to be, where there is no longer any space. Let power therefore be applied at B, which turns the wheel toward A. At the same time, therefore, and simultaneously, A will move toward D, D toward C, and C toward B. For as many semicircles as pass beyond the diameter AEC on the side CBA, just so many semicircles, which are on the side ADC, will pass to the side CBA. But where motion ceases, there rotation ceases; but where space ceases, motion ceases; but where the circles cease, space ceases; therefore in the center, since there are no circles, neither does space cease there, nor motion. For there is no reason present why the body itself should be transferred by rotation to another place from the one in which it is. The rotating body therefore stands still, which was to be demonstrated. And this demonstration is similar to that which we mentioned above concerning the rotation of a cone on a plane around its vertex, which we said was devised by Hero in the Automata. We add that in this kind of rotation the body becomes lighter in the very motion itself, and the faster the rotation, the more so. The cause is that lateral motion somehow impedes that motion which arises from natural heaviness toward the center; therefore experience teaches us that the lightest whirlwinds are those with which boys play, if their hands are held palm up while the quickest rotation is being produced. To this another question is proposed and answered, namely, why round bodies are more suited to this kind of motion. It is quite evident that of bodies which are moved in this way,

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EXERCITATIONES. 75 partes eo esse velociores, quo magis à centro, circa quod mouentur, fuerint remotiores. maius enim eodem tempore spatium pertranseunt. quo igitur figura ijs partibus, quæ longius à centro absunt, abundauerit magis, eo facilius, & velocius in circulum rotata mouebitur. Modò ostendemus, circularem cæteras omnes ea qua diximus partium à centro remotissimarum copiâ abundare. Esto triangulum puta æquilaterum ABC circa centrum D. Ducantur Catheti per centrum ab oppositis angulis ad opposita latera ADG, BDF, CDE, erunt autem lateribus perpendiculares. quoniâ igitur latera AD, DB, DC, rectis angulis subtenduntur, maiora erût lateribus DE, DF, DG. tres igitur lineæ in hoc triangulo sunt longissimæ DA, DB, DC. tres verò breuissimæ DE, DG, DF, quamobrem rotato super centrum D triangulo, tres tantum partes eius ABC velocissimæ erunt, tres verò tardissimæ E, G, F. Minus igitur apta est motui huic triangularis figura, quam quadrata, in qua partes à centro remotissimè, & ideo velocissimè sunt quatuor. Itaq[ue] quo magis laterata figura angulis abundabit, eo magis erit ad hunc, & cæteros omnes circulares motus aptior. At circulus infinitas, vt ita dicam, partes à centro remotissimas habet, itaque nulla figura est circulari, in ipsa rotatione, commodior atque velocior. Alia quoque de caussa id sit, quod dum circularis figura mouetur, nullis eminentibus angulis aërem verberet circu[m]stâtem, ex qua verberatione motus impeditus sit tardior. Quæri etiam potest, Num axe inclinato, rotæ motus aliqualiter impediatur? Nos negatiuam partem amplectimur. K 2 Esto

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EXERCISES. 75 the parts are so much the swifter, the farther they are from the center around which they move. For in the same time they pass through a greater space. Therefore, the more that a figure abounds in those parts which lie farther from the center, the more easily and swiftly will it be moved when rotated in a circle. We shall now show that the circular figure surpasses all others in that abundance of the parts farthest from the center which we have mentioned. Let there be, say, an equilateral triangle ABC about the center D. Let perpendiculars be drawn through the center from the opposite angles to the opposite sides, ADG, BDF, CDE; these will be perpendicular to the sides. Since therefore the sides AD, DB, DC subtend right angles, they will be greater than the sides DE, DF, DG. Therefore the three longest lines in this triangle are DA, DB, DC. The three shortest, however, are DE, DG, DF. Wherefore, when the triangle is rotated about the center D, only three parts of it, ABC, will be the swiftest, while three, E, G, F, will be the slowest. Therefore the triangular figure is less suited to this motion than the square, in which the parts farthest from the center, and therefore the swiftest, are four. Thus, the more a figure is furnished with sides and angles, the more suitable it will be for this, and all other circular motions. But the circle has, so to speak, infinite parts farthest from the center; therefore no figure is more convenient or swifter than the circular in rotation itself. There is also another reason for this: when a circular figure moves, it beats the air with no projecting angles against the surrounding air, and by that beating the motion is hindered and made slower. It may also be asked whether, with the axis inclined, the motion of a wheel is in some degree impeded. We hold the negative.

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IN MECHAN. ARIST. PROBL. Esto enim rota ABCD, cuius centrum E axis inclinatus, circa quem conuertitur EGF. Duobus aute[m] punctis fulcitur GF. Sit autem tum grauius tum figuræ centrum E, Perpendicularis vero per inferius fulcimentum transiens HFI. Conuersa igitur rota, grauitatis centrum stabit nec à suo situ sursum deorsumue mouebitur. Est autem axis FEG, ceu vectis in quo pondus in E, potentiæ sustinentes GF; non enim hic vt in axe perpendiculari pondus totum ab inferiori fulcimento sustinetur. quo igitur minor erit proportio FE ad FG, eo minori indigebit potentiâ is qui pondus sustinet in G. Et hæc sanè ita se habent, grauitatis centro in axe ipso constituto, si enim extra fuerit motus impeditur & motore cessante citò quiescit. Esto enim grauitatis centrum in K. Dum igitur circa axem fit motus, centrum circulatum aliquando erit in L; Secet autem rotæ diameter AC perpendicularem HI in M, Porrò à punctis LK ad ipsam perpèdicularem ducantur ad rectos angulos lineæ LN, KO. Maior est autem MK ipsa ML, maior ergo MO, ipsa MN. magis igitur à mundi centro distat punctum N puncto O. Centrum ergo grauitatis K si liberè dimittatur, requiescet in K & contra naturam transferetur in L. Cessante igitur violentiâ & præualente naturâ citò rota suâ sponte quiescet, quod fuerat ostendendum. QVÆSTIO IX. Quæritur, Cur ea quæ per maiores circulos tolluntur, & trahuntur faciliùs, & celerius moueri contingat, veluti maioribus trochleis, & scy talis similiter? Respondet ad hæc Philosophus, forte id euenire, quo- niam

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IN MECHAN. ARIST. PROBL. Let there be a wheel ABCD, whose center E has an inclined axis, around which EGF revolves. It is supported at two points by GF. Let both the center of gravity and the center of the figure be at E; and let the perpendicular through the lower support pass by HFI. When the wheel is turned, therefore, the center of gravity will stand still and will not move up or down from its position. Now the axis FEG is, as it were, a lever on which the weight is at E, and the supports are at GF; for here, unlike an upright axis, the whole weight is not borne by the lower support. Therefore, the smaller the proportion of FE to FG, the less force will be needed by the one who supports the weight at G. And these things are indeed as stated, with the center of gravity placed in the axis itself; for if it is outside, motion is hindered, and when the mover ceases it quickly comes to rest. Let the center of gravity be at K. While the motion is thus taking place around the axis, the moved center will at one time be at L. Let the diameter AC of the wheel cut the perpendicular HI at M. Moreover, from the points LK to the same perpendicular, let the lines LN, KO be drawn at right angles. Now MK is greater than ML, therefore MO is greater than MN. Therefore point N is farther from the center of the world than point O. If, then, the center of gravity K is let go freely, it will come to rest at K and be transferred contrary to nature to L. So when the violence ceases and nature prevails, the wheel will quickly come to rest of itself, which was to be shown. QUESTION IX. The question is why things that are lifted and drawn by larger circles happen to move more easily and more quickly, as with larger pulleys and also with windlasses? The Philosopher replies to these things that this perhaps happens because

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EXERCITATIONES. 77 niam quanto maior fuerit illa quæ à centro est, in æquali tempore maius mouetur spatium. quamobrem æquali existente onere idem faciet. Ita enim dixerat de libraru[m] natura, & differentijs agens, maiores minoribus exactiores esse. Circulos verò libras, in quibus centrum spartum, semidiametri hinc inde æqualia brachia. Quod vltimo loco affirmauit, trochleas esse instar librarum, verum est. Quod autem dixit, faciliùs & celerius mouere maiores libras ijs quæ minores sunt, si simpliciter intelligatur, falsum, quippe quod facilitas motus, in tractorijs machinis velocitati sit contraria, quod demonstrauit Guid. V bald. in tractatu de Trochlea in 2. Corollario propositione vltima. Ad id autem quod dixit, quo maiores fuerint trochleæ, eo faciliùs mouere, non est, vt dicebamus, simpliciter verum, quod facilè ostendemus. Esto enim trochlea AB circa centrum C, appensa in punto D, perpendicularis quæ ad mundi centrum DCE, pondera æqualia vtrinque appensa FG. Esto item alia Trochlea, ea q; maior HI, circa centrum K appensa in L, perpendicularis, quæ ad mundi centrum LKM, æqualia K 3 pon-

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EXERCISES. 77 because, the greater that part may be which is from the center, in an equal time a greater space is moved. Wherefore, with the load being equal, it will do the same. For he had said, while treating of the nature of balances and their differences, that the greater are more exact than the smaller. But circles are balances, in which the center is fixed, and the semidiameters on either side are equal arms. What he affirmed lastly, that pulleys are like balances, is true. But what he said, that greater balances move more easily and more quickly than smaller ones, if understood simply, is false; for the ease of motion in pulling machines is contrary to speed, as Guid. V. Bald. demonstrated in the treatise on the pulley, in the second corollary of the final proposition. As to what he said, that the larger the pulleys are, the more easily they move, it is not, as we were saying, simply true, which we shall easily show. Let there be a pulley AB about center C, suspended at point D, with the perpendicular DCE to the center of the world, equal weights FG hanging on either side. Let there be also another pulley, namely the greater HI, suspended about center K at L, the perpendicular LKM, equal weights K 3 hang-

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78 IN MECHAN. ARIST. PROBL pondera vtrinque appensa N,O. Dico maiorem H I ipsa minori DE facilius pondera non mouere, eo quòd sit ma- ior, illa verò difficiliùs, propterea quòd sit minor. Etenim, quoniam vtraque trochlea per centrum grauitatis à per- pendiculari diuiditur, erunt partes DAE, DBE, æquepó- derantes. Eadem ratione ipsæ quoque LHM, LIM æquè ponderabunt. Itaque si quantumuis pusilla pondera ad- das, vtriq; earum ad alteram partem tolletur æquilibriu[m], nec minus requiritur pondus vt recedat ab æquilibrio Trochlea minor, quàm maior. Vnico autem verbo con- cludi potest disputatio, t[ame]n in minori quàm in maiori, bra- chia siquidem bifariam diuiduntur, ergo in vtriq; eadem brachiorum proportio, & eadem ponderum ratio. Exploratissima sunt hæc. Veruntamen cùm res ipsa doceat, verum esse quod scribit Aristoteles, huius effe- ctus causa aliunde à nobis, nempe à mechanicis princi- pijs, est mutuanda. Dico igitur, Axium, circa quos tro- chleæ rotæue conuertuntur ad rotas ipsas, varias habere proportiones. Ostendemus autem rotâ illam, trochleam- ue faciliùs moueri, & mouere pondera, quo rotæ diam- ter ad axis diametrum maiorem habuerit proportionem, & ideo fieri posse rotam maiorem ad suum axem minore habere proportionem quam rotam minorem ad suum. Esto enim trochlea ABcir- ca centrum C, cuius diameter DCE sit in ipsa quæ ad mundi centrum perpe- diculari: sit au- tem appensa in D. Alia similiter ei æqualis sit trochlea F G circa centrum H, cuius diameter IHK, conueniens cum

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78 IN MECHAN. ARIST. PROBL weights suspended on both sides N,O. I say that the greater H I moves the weights more easily than the smaller DE, because it is greater; but that the smaller one moves them more difficultly, because it is smaller. For since each pulley is divided by the perpendicular through the center of gravity, the parts DAE, DBE will be equally weighted. By the same reasoning the parts LHM, LIM also will weigh equally. Therefore, even if you add however small weights, for each of them equilibrium will be shifted to the other side; and no less weight is required for the smaller pulley to depart from equilibrium than for the greater. But in a single word the argument can be concluded: in the smaller as in the greater, the arms are indeed divided in two parts; therefore in each there is the same proportion of arms, and the same ratio of weights. These things are most certain. Yet since the matter itself shows that what Aristotle writes is true, the cause of this effect must be borrowed from elsewhere by us, namely from mechanical principles. I say, therefore, that the axles around which pulleys or wheels turn have various proportions to the wheels themselves. We shall show, however, that a wheel or pulley is moved and moves weights more easily in proportion as the wheel’s diameter has a greater ratio to the axle’s diameter; and therefore it can happen that a greater wheel has a smaller proportion to its axle than a smaller wheel has to its own axle. Let there be a pulley AB around center C, whose diameter DCE is in the line perpendicular to that which goes to the center of the world: let a weight be suspended at D. Similarly let there be another pulley, equal to it, FG around center H, whose diameter IHK, fitting with

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EXERCITATIONES. 79 cum perpendiculari quæ ad mundi centrum. appendatur autem in I. Habeant autem & axes, circa quos conuertantur. Hi si æquales fuerint, proportione non mutatâ idem operabuntur. Modò ponantur in æquales, sitque axis ro- te AB, crassior axe rotæ FG, sitque crassioris quidem semi- diameter CL, subtilioris autem HM. Dico per trochleam FG facilius attolli pondera æqualia quàm per AB, licet altera trochlearum alteri sit æqualis. Quoniam enim me- chanica corpora sine materia & pondere non sunt, onera appesa & trochlearum ipsarum grauitas ex superiori par- te prement axes, vbi puncta L, M, quæ res, secutâ inuicem corporum solidorum fricatione, motum ipsum trochlea- rum difficiliorem & asperiorem facit. Succedit igitur im- pedimentum loco ponderis. Duos igitur habemus vectes DC, IH, quorum fulcimenta contra ipsa C, H. Pondera verò inter fulcimenta & potentias in L, M. Intelligantur autem potentiæ applicatæ punctis DI. Igitur ex natura e- iusmodi vectis, in quo pondus inter fulcimentum est & potentiam erit vt CL, ad CD, ita potentia in D ad pódus, hoc est, resistentiam fricationis, quæ sit in L. Sed maior est proportio CL ad CD quàm HM ad HI. Maior igitur ad superandum idem seu æquale impedimentum poten- tia requiritur in D, quàm in I. Itaque cum vis tota in rota- rum & axium, diametrorum proportione consistat, fieri potest, quod dicebamus, minorem trochleam dari, quæ maiorem habeat proportionem ad suum axem, quàm maior ad suum, quo casu minor rota facilius impedimen- tum, quod diximus, ipsa maiori rota seu trochlea supera- bit. Veruntamen quoniam ex materia fiunt tum axes tum rotæ, nec rei natura patitur axes subtiles, & imbecilles magna pódera sustinere posse, idcirco crassiores fiunt, quæ crassitudo cum proportione magis à magnarum rotarum diametris superetur; fit hinc maiores rotas datâ axium pa- ritate

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EXERCISES. 79 with the perpendicular which is suspended toward the center of the world. But let it also be in I. And let them have axes, around which they may turn. If these are equal, without change in proportion they will do the same work. Now let them be in equal condition, and let AB be the axis of the wheel, FG the thicker axis of the wheel; and let CL be the semidiameter of the thicker one, and HM of the more slender one. I say that by means of pulley FG equal weights are lifted more easily than by AB, although one of the pulleys is equal to the other. For since mechanical bodies are not without matter and weight, the loads hung on them and the weight of the pulleys themselves will press the axes from above, where the points L, M are, which thing, following the mutual friction of solid bodies, makes the motion itself of the pulleys more difficult and rough. Thus impediment takes the place of weight. We therefore have two levers, DC, IH, whose supports are against C, H themselves. But the weights are between the supports and the powers in L, M. Let the powers, however, be applied at the points DI. Therefore, by the nature of such a lever, in which the weight is between the support and the power, it will be as CL is to CD, so is the power at D to the weight, that is, to the resistance of friction which is in L. But the proportion of CL to CD is greater than that of HM to HI. Therefore a greater power is required at D to overcome the same or equal impediment than at I. And so, since the whole force consists in the proportion of the diameters of the wheels and axles, it can happen, as we said, that there may be a smaller pulley which has a greater proportion to its own axle than a larger one has to its own; in which case the smaller wheel will more easily overcome the impediment mentioned by us than the larger wheel or pulley itself. Yet since both the axles and the wheels are made from matter, and the nature of the thing does not allow slender and weak axles to be able to sustain great weights, therefore they are made thicker; and since this thickness is more greatly surpassed in proportion by the diameters of large wheels, it follows from this that larger wheels, given equal axles

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80 IN MECHAN. ARIST. PROBL. ritate facilius impedimentum superare quàm minores, & hoc videtur sensisse Philosophus in ipsa quæstionis huius propositione, Hinc aurigæ vulgo axungiâ (quæ inde no- men trahit) axium asperitates mitigant, vt minor in rotan- do, ex fricatione fiat resistentia. Concludimus igitur, fa- cillimè trochleam illam pondus trahere, quæ cum maxi- ma sit, axem habet minimum, eumque axungiâ aliaue vn- ctuosa materia perfusum. De manubrijs, quæ rotarum a- xibus aptantur, nemo ferè verba fecit; nos igitur de his a- liquid; siquidem res ad speculationem, qua de agimus, ne- pe Mechanicam pertinet. Manubria vectes sunt, & ad vectium naturam redu- cuntur, eorum scilicet, in quibus fulcimentum est inter pondus & potentiam. In his autem attenditur proportio, quam habet manubrij longitudo ad ipsum axis semidia- metrum, eo enim faciliùs mouent, quo eorum longitudo ad axium semidiametros proportionem habuerit ma- iorem. Duabus autem partibus constant, alterâ, quæ ab axe ad angulum; quæ verè vectis est; alterâ, cui manus i- psa admouetur, ex qua res tota manubrium dicitur. Fiunt autem manubria hæc vt plurimum amouabilia, sunt tamé ceu rotarum ipsarum partes, & rotis ipsis commodè affi- gerentur, nisi in rotatione à transuersarijs, quibus rotæ su- stinentur, impedimentum fieret. Esto enim rota AB, cu- ius axis E, terebretur autem in F, ibique paxillus affigatur FK. Sit & alia rota CD, cu- ius axis G, manubrium axi appositum GHI. Sint autem rotæ æquales & axes æqua- les. Sint etiam æqualia ipsa spatia EF, GH, hoc est, ma- nubrij

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80 IN MECHAN. ARIST. PROBL. it more easily overcomes the impediment than smaller ones, & this seems to have been what the Philosopher understood in the very statement of this question. Hence coachmen commonly grease the axles (from which that name is derived), so that the roughnesses of the axles may be smoothed, and thereby less resistance arise from friction in turning. We conclude therefore that that pulley will draw a weight most easily which, though the largest, has the smallest axle, and that axle smeared with grease or some other unctuous substance. As for the handles which are fitted to the axles of wheels, almost no one has spoken of them; we shall therefore say something about them, since the matter belongs to the speculation with which we are dealing, namely Mechanics. Handles are levers, and are reduced to the nature of levers, namely those in which the fulcrum is between the weight and the power. In these one observes the proportion which the length of the handle has to the radius of the axle itself; for they move the more easily, the greater proportion their length bears to the radii of the axles. They consist of two parts, one which extends from the axle to the angle; this is truly the lever; the other, to which the hand itself is applied, from which the whole thing is called a handle. These handles are for the most part made removable; yet they are, as it were, parts of the wheels themselves, and could be conveniently attached to the wheels themselves, unless in rotation there were some hindrance from the crosspieces by which the wheels are supported. For let there be a wheel AB, whose axle is E; and let it be bored at F, and there let a pin FK be fixed. Let there also be another wheel CD, whose axle is G, with a handle GHI applied to the axle. Let the wheels be equal and the axles equal. Let the spaces EF, GH be equal as well, that is, the handle

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EXERCITATIONES. 81 nubrij GH longitudo. Dico, eâdem facilitate moueri AB rotam à potentia in FK, quâ mouetur CB, à potentia posita in HI, datis ipsi nempe potentijs æqualibus. Producatur enim IH, vsque ad rotæ CD latus in L, & LG ducatur, & FE in rota AB iungatur. Erunt igitur FE LG inter se æquales. Sunt autem eorum circulorum semidiametri, qui à punctis FL, in ipsa rotatione describuntur. Ita igitur se habebit potentia applicata in L ad diametrum semidiametrumue axis rotæ CD, vt se habet potentia applicata in F, ad diametrum semidiametrumue axis E rotæ AB, sed spatia sunt æqualia & potentiæ æquales, quare nihil refert, vtrum manubrium lateri affigatur, vel axi à latero rotæ separatum applicetur. Duplex autem est manubriorum forma; altera enim rectis partibus constat, altera verò curua est tota, sed rectis vtimur vt manibus apprendamus, curuis verò vt locum illis apponamus, & pedis pressione ceu in molis lapideis, quibus gladij acuuntur, fieri assolet, conuertantur. Cur autem manubria hæc curua fiant, ea videtur ratio, ne videlicet manubrij capite supra centrum in linea quæ per centrum transit, cõstituto, factâ interim pressione motus à centro, ad quod directè fieret pressio, impediretur. Curuitas aute[m] facilitatem quandam habet, ex qua factâ modicâ flexione axis caput, dum premitur ab ipsa perpendiculari linea leniter abducitur. quæ cum cessent in manubrijs quæ manu aguntur, ideo alia forma, nempe ex rectis partibus passim fiunt. Esto igitur illud quod ex rectis partibus AB, curuum verò CD, linea verò, secundum quam pede fit pressio L CDE.

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EXERCISES. 81 the length of nubrij GH. I say that the wheel AB is moved by a force placed in FK with the same ease as CB is moved by a force placed in HI, namely, when equal forces are given to it. For let IH be produced to the side of the wheel CD at L, and let LG be drawn, and FE joined on the wheel AB. Therefore FE and LG will be equal to one another. Now these are the semidiameters of the circles which are described from the points FL in the actual rotation. Thus, then, the applied force at L will be related to the diameter or semidiameter of the axis of the wheel CD as the applied force at F is related to the diameter or semidiameter of the axis E of the wheel AB; but the spaces are equal and the forces equal, therefore it makes no difference whether the handle is attached to the side, or applied to the axis separated from the side of the wheel. Now there are two forms of handles; for one consists of straight parts, while the other is entirely curved. But we use straight ones so that we may grasp them with the hands, and curved ones so that we may set them in place there, and by the pressure of the foot, as is usually done in stone mills, by which swords are sharpened, they may be turned. And the reason why these handles are made curved seems to be this: namely, lest, with the head of the handle set above the center in the line passing through the center, the motion from the center, toward which the pressure would be directed straight on, should be impeded when pressure is made in the meantime. But curvature has a certain ease, by which, after a slight bending has been made, the head of the axis, while it is pressed, is gently drawn away from the very perpendicular line. Since these things are absent in handles that are worked by hand, therefore another form, namely of straight parts, is commonly made. Let that therefore be the one which consists of straight parts AB, but the curved one CD, and the line according to which the pressure is made by the foot L CDE.

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82) IN MECHAN. ARIST. PROBL. CDE. Hæc itaque de manubrijs seu vectibus nos considerasse sit satis. Quæri interim posset, Cur duabus datis rotis æqualis magnitudinis in æqualis ponderis, circa æquales axes constitutis leuior faciliùs moueatur & citiùs quiescat; grauior verò difficilius moueatur & tardiùs cesset à motu, ea videtur ratio, quod grauior resistens magis cum superatur impressam vim suscipit, & diutius retinet, quod cessat in leuiore. QVÆSTIO X. Dubitat Aristoteles, Cur faciliùs, quando sine pondere est, moueatur libra, quàm cum pondus habet. Simili modo rota, & eiusmodi quidpiam, quod grauius quidem est, item quod maius & grauius minori, & leuiori? BReuiter autem soluit. ait enim, An quia non solum in contrarium quod graue est, sed in obliquam etiam difficulter mouetur? In contrarium enim ei ad quod vergit onus mouere difficile est, quo autem vergit, est facile. In obliquum autem haud quaquam vergit. Nos quod ipse non fecit figurâ ipsa appositâ rem clariorem faciemus. Esto libra AB, cuius fulcimentum C, pondera vtrinque appensa AB, quorum vtrumque ponderet 10. Item libra DE, cuius fulcimentum F pondere vero appensa D, E, ipsis A, B, dimidio leuiora, nèpe S. Addatur ponderi B pondus G, & ponderi E pondus H, quorum similiter vtrumq[ue] ponderet S, nutabunt igitur libræ ponderibus appositis, & BG

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82) IN MECHAN. ARIST. PROBL. CDE. Thus much, then, may suffice concerning handles or levers. Meanwhile it might be asked why, when two wheels of equal size and equal weight are given, and set upon equal axles, the lighter one is moved more easily and comes to rest more quickly; whereas the heavier one is moved with greater difficulty and ceases from motion more slowly. The reason seems to be that the heavier, offering resistance, receives the impressed force more strongly when it is overcome, and retains it longer, which is not the case in the lighter one. QUESTION X. Aristotle doubts why a balance moves more easily when it has no weight than when it has weight. Likewise a wheel, and anything of this sort, why that which is heavier, and also that which is larger and heavier than what is smaller and lighter? Briefly, however, he solves it. For he says: Is it because what is heavy is moved with difficulty not only in the contrary direction, but also obliquely? For in the direction contrary to that toward which the load tends, motion is difficult; in the direction toward which it tends, it is easy. But obliquely it does not tend in any way whatever. Since he himself did not do this, by adding the figure itself we shall make the matter clearer. Let AB be a balance, with its support at C, and weights hung at both ends AB, each of which weighs 10. Likewise let DE be a balance, with its support at F, and weights hung at D and E, each half as light as A, B, namely S. Let there be added to the weight at B the weight G, and to the weight at E the weight H, each of which likewise weighs S; therefore the balances will rock with the weights added, and BG

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EXERCITATIONES. 83 BG secetur in K, EH verò in N, grauius est autem GB, est enim IS, ipso EH, quod est 10. Difficiliùs autem descendet BG, quàm EH. hoc autem ex doctrina Aristotelis, quia non solum in contrarium quod graue est, sed in obliquum etiam difficulter mouetur, in contrarium enim ei ad quod vergit onus mouere difficile est, quò autem vergit facilè in obliquum autem puta per lineas BK, EN non vergit onus. Difficiliùs ergo in obliquum mouebitur pondus BG ipso pondere EH. vtrumque autem in descensu retrahitur nempe à perpendicularibus BI, EM & retractionis quidem angulis sunt æquales & æquales ipsæ retractiones. Sed grauius est pondus GB. quod autem grauius est, violentius descédit eo quod est leuius. maiori igitur nisi atque impetu cum cætera paria sint, descendet pondus BG, ipso EH, quod è diametro Aristotelis assertioni est contrarium. ex alijs igitur principijs veritas ipsa est eruenda. Dicimus autem id ex proportionum fieri inæqualitate; quia enim is ad 10. proportionem habet sesqualteram, 10. verò ad 5. duplam, maiorem proportionem habet EH ad oppositum pondus D, quàm BG ad pondus A, facilius ergo trahet libra DE leuior pondus D, quàm ipsa AB, grauior pondus A, quod vtique fuerat ostendendum. Alia quoque caussa & hæc accidentalis ad hunc effectum pariendum concurrit, axium nempe ad fulcimenta, in quibus rotantur, fricatio. quo enim maius est pondus cæteris paribus, quod nos in præcedente quæstione demonstrauimus, eò maior sit ipsa collisio. Porrò huius quoq[ue] speculationis est, Cur æqualia & similia corpora in æqualibus similibusque basibus constituta eodem similique plano fulta, ponderibus tamen inæqualia, non eâdem facilitate euertantur, sed horum grauiora difficilius. L 2 Sit

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EXERCISES. 83 If BG is cut in K, and EH indeed in N, but GB is heavier, for it is as IS to EH itself, which is 10. But BG will descend more difficultly than EH. This, however, is from Aristotle’s doctrine, because what is heavy is moved with difficulty not only in the contrary direction, but also obliquely; for in the contrary direction it is difficult to move what weight inclines toward, but where it inclines it moves easily. But obliquely, as through the lines BK, EN, the weight does not incline. Therefore the weight BG will be moved obliquely with more difficulty than the weight EH itself. But both in the descent are drawn back, namely by the perpendiculars BI, EM, and indeed the angles of the retractions are equal and the retractions themselves are equal. But the weight GB is heavier. Now what is heavier descends more violently than what is lighter. Therefore, unless with greater force and impetus, all else being equal, the weight BG will descend, rather than EH itself, which is directly contrary to Aristotle’s assertion. The truth must therefore be drawn from other principles. We say, however, that this occurs from the inequality of the proportions; for since IS has a sesquialteral proportion to 10, but 10 to 5 a double one, EH has a greater proportion to the opposite weight D than BG has to the weight A; therefore the balance DE will more easily draw the lighter weight D than AB will the heavier weight A, which indeed had to be shown. Another cause too, and this one accidental, contributes to producing this effect: namely, the friction of the axles against the supports in which they rotate. For the greater the weight, all other things being equal, as we demonstrated in the preceding question, the greater is the collision itself. Moreover, the subject of this speculation also is: Why do equal and similar bodies placed on equal and similar bases, and supported on the same and similar plane, yet unequal in weight, not overturn with the same ease, but the heavier of these do so with more difficulty? L 2 Sit

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84 IN MECHAN. ARIST. PROBL. Sit enim Prisma seu Cylindrus ABCD, cuius grauitatis centrum E in plano CI, basi fultus CD. Sit & alter Cylindrus FGHI, cuius grauitatis centrum K fultus basi HI æqualis quidem & similis ipsi AD. Sit autem grauior FGHI, ipso ABCD. Dico, pari potentiâ vtrumque impellente, facilius euersum iri Cy- lindrum AD, ipso FI. Ducantur EC, KH, & æquales po- tentia applicentur punctis BG, pellentes Cylindros ad partes AF. Euersio autem non fiet donec facta corporis conuersione circa puncta CH, grauitatis centra E, K træs- feruntur in L, M, in ipsis scilicet perpedicularibus ACFH. Demittantur EN, KO, perpendiculares ipsis CD, HF. Et quoniam CNE, HOK anguli recti sunt, erunt EC KH i- psis EN, KO, maiores, quare & LC, MH ipsis EN KO, ma- iores attolluntur ergo in ipsa euersione, grauitatum cen- tra E in L, K in M. At quod grauius est, difficilius contra sui naturam mouetur, ideo difficilius euertetur corpus FI, ipso AD, quod fuerat demonstrandum. QVÆSTIO XI. Dubitat Philosophus, Cur super scy talas facilius portentur onera quàm super currus, cum tamen ij magnas habeant rotas, illæ verò pusillas? O Timè respondet dubitationi. An, inquiens, quoniam in scy talis nulla est offensatio; in curribus verò axis est, ad quem offensant. Desuper enim illum premunt, & à lateribus. quod autem est in scy talis ad isthæc duo mo- uetur & inferiori substrato spatio, & onere superimposi- to,

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84 IN MECHAN. ARIST. PROBL. Let there be a Prism, or Cylinder ABCD, whose center of gravity E is in the plane CI, supported on the base CD. And let there be another Cylinder FGHI, whose center of gravity K is supported by the base HI, equal indeed and similar to AD. Let FGHI, however, be heavier than ABCD. I say that, with an equal force impelling each, the Cylinder AD will be overturned more easily than FI. Draw EC, KH, and let equal forces be applied at the points BG, driving the Cylinders toward the parts AF. But overturning will not occur until, the body having been turned about the points CH, the centers of gravity E, K are transferred to L, M, namely onto the perpendiculars ACFH. Let EN, KO be drawn perpendicular to CD, HF. And since the angles CNE, HOK are right angles, EC, KH will be greater than EN, KO; therefore LC, MH are raised greater than EN, KO. Thus in the very overturning, the centers of gravity E in L, K in M are raised up. But what is heavier is moved with more difficulty against its nature; therefore the body FI will be overturned with more difficulty than AD, which was to be demonstrated. QUESTION XI. The philosopher asks: Why are loads carried more easily on handbarrows than on carts, although the latter have large wheels and the former very small ones? Timæus answers the difficulty. Is it, he says, because in handbarrows there is no rubbing or obstruction; but in carts there is the axle, against which they rub? For they press upon it from above and from the sides. But that which is in handbarrows is moved by these two things, both by the underlaying space below and by the load placed above,

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EXERCITATIONES. 85 to, invtrisque enim ijs reuoluitur locis circulus, & motus impellitur. Tam appositè paucis verbis veritatem expli- cauit, vt ferè quicquid insuper addatur, superuacaneum videri possit. quicquid tamen sit, ad maiorem claritatem aliquantulum in hac ipsa quæstione immorabimur. Rotatas scy talas proponit hîc Aristoteles. Coniun- ctas autem esse rotas ipsis scy talis est intelligendum, nem- pe, vt simul rotæ cum scy talis conuertantur. Secus enim axium & Rotarum fieret offensatio, cuius offensionis vim & effectum cum nouerit Aristoteles, vel hoc ipso lo- co teste, mirum est, nihil de ea egisse quæstione 9, vbi nos hac de refusissimè tractauimus. Cæterùm quod de rotatis scy talis scribit Philoso- phus, notandum, à Pappo quidem lib.8. & à nostris Me- chanicis passim absque rotis Cylindrica simplici videli- cet, & tereti formâ ad vsum adhiberi. Esto igitur Ari- stotelis' quidem scy tala AB, Pappi verò seu vul- garis, & communis CD. His non modò lapicidæ passim, sed & nautæ na- uiumque fabri subdu- cendis & mari inducen- dis nauibus vtuntur, quod varare dicunt vernaculè, Hi- spanico, vt arbitror, vocabulo. ea enim natio teres lignum baculumue appellat Varam. Quæri autem posset, vtra harum formarum sit vti- lior atque commodior? Nos rotatas laudamus magis in plano duroque solo, minus enim tangunt & minus offen- sant; in molliori autem & minus duro proponimus non rotatas, siquidem rotæ sui naturâ pondere pressæ solum facillimè scindunt & absorbentur. Quatenus autem ad vsum pertinet Esto horizontis L 3 pla-

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EXERCISES. 85 for in both of them the circle is revolved at the places, and the motion is driven onward. He explained the truth so aptly in a few words that almost whatever is added beyond this may seem superfluous. However that may be, for greater clarity we shall linger a little on this very question. Aristotle here proposes rotated scytalae. But that they are connected with the scytalae themselves is to be understood, namely, so that the wheels turn together with the scytalae. Otherwise there would be a collision of the axes and wheels, and since Aristotle knew the force and effect of that collision, as this very passage testifies, it is surprising that he said nothing about it in question 9, where we treated this matter at great length. Moreover, what the Philosopher writes about rotated scytalae should be noted: according to Pappus, in book 8, and according to our Mechanicians, they are commonly used without wheels, namely in a simple cylindrical and round form. Let there therefore be Aristotle's scytala AB, but Pappus's, or the common and ordinary one, CD. Not only stonecutters use these everywhere, but also sailors and shipwrights use them for hauling ships out and putting them into the sea, which in the vernacular they call varare, a Spanish term, as I suppose. For that nation calls a round piece of wood or a staff Vara. One might ask, which of these forms is more useful and more convenient? We praise the wheeled ones more on a hard and level surface, for they touch less and obstruct less; but on softer ground and one that is less firm we prefer the unwheeled ones, since wheels by their nature, when pressed by weight, very easily cut into the ground and become absorbed in it. As far as use is concerned, let this be the hori

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86 IN MECHAN. ARIST. PROBL. planum AB, scytale du[m] CD, EF, Pondus verò eis impositum G, tangens ipsas in p[ro]uctis CE, scytalæ autem planum in punctis D, F, Pellatur à potentia quapiam pôdus Gad anteriora, nè pe ad partes E. rotabuntur igitur scytalæ & pars quædam scytalæ D, in quasit contactus ascendet in I, C verò descendet in H, nulla remotum impediente, quippe quòd nulla ponderis scytalarum, & plani ad inuicem fiat offensatio. Præterea cum scytalarum centra ab horizontis plano æqualiter distent, pondus quidem horizonti æquidistanter mouetur, & ideo eius centrum grauitatis nequaquam, in motu quisit, eleuatur. Cæterùm materiæ imperfectione remota nihil refert ad facilitatem, vtrum maioris minorisue diametri sint scytalæ, vt ea posita eo quod maiores circuli faciliùs offendicula superent, quod demonstratum est in quæstione S. eo vtiliores sunt scytalæ, quo crassiores. Quatenus autem ad plaustrinaturam spectat, cuius ad scytalas Philosophus fecit comparationem, vt ostendamus difficilius ex eo moueri pondera. Esto plaustri rota KL, cuius centrum M, axis verò NO circa quem rota ipsa conuertitur KL. Funis quo rota ex axis centro M trahitur MP, pondus vero QR. Quoniam igitur pondus axem premit in N, axis autem rotæ modiolum in O, & eodem tem-

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86 IN MECHAN. ARIST. PROBL. planum AB, the spindle CD, EF, and the weight G placed upon them, touching them at the projections CE; but the plane of the spindle at the points D, F. If the weight G is pulled forward by some force, so that it tends toward the parts E, the spindles will therefore rotate, and a certain part of the spindle D, being in quasi-contact, will ascend to I, while C will descend to H, with nothing hindering the movement, since there is no mutual interference of the weights of the spindles and of the plane. Moreover, since the centers of the spindles are equally distant from the plane of the horizon, the weight indeed moves parallel to the horizon, and therefore its center of gravity is by no means raised in the motion. Furthermore, if the imperfection of the material is removed, it makes no difference to the ease of movement whether the spindles are of greater or smaller diameter, since, other things being equal, larger circles more easily overcome obstacles, as has been demonstrated in Question 5. Thus the thicker the spindles are, the more useful they are. As regards the construction of carts, to which the Philosopher made the comparison with spindles, let us show that weights are moved more difficultly from it. Let there be the cart wheel KL, whose center is M, and the axle NO about which the wheel KL itself turns. Let the rope by which the wheel is drawn from the center M of the axle be MP, and let the weight be QR. Since therefore the weight presses the axle at N, but the axle of the wheel at O, and at the same ti-

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EXERCITATIONES. 87. tempore potentia quæ trahit in P, axem admouet modio- lo in parte V. duplex itaque fit ex fricatione seu offensa- tione impedimentum, infra nempe, vbi O, & ad latus vbi V. quæ quidem offensiones currus motum reddunt diffi- ciliorem, quæ quidem difficultas eo maior erit, quo ma- ior fuerit pondus axem premens, & minor proportio se- midiametri rotæ KM, ad axis semidiametrum MO. Cur igitur scy talis facilius pondera transferantur quam plau- stris, apertè ex dictis ad Aristotelis mentem demonstra- uimus. Cæterùm quod ipse reticuit, nos dicemus, nempe validissimè enormia pondera per scy talas moueri, si scy- talis ipsis vectes adiungantur. Et sanè motus erit tardissi- mus, veruntamen tarditas ipsa facilitate, quæ inde fit, v- berrimè compensatur. Esto igitur horizontis planum AB, scy talæ CD, fo- ramina in scy talis EFGH, vectes foraminibus inserti IE, KF, LG, MH. Pondus vero scy talis impositum N. Appli- catis igitur quatuor potentijs extremitatibus vectium I, K, L, M, ijsque in anteriora propulsis, fiet scy talarum rota- tio,

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EXERCISES. 87. at the same time the force which acts in P brings the axle against the block on the side V. Thus a double hindrance arises from friction, or rather from rubbing: namely below, where O is, and at the side, where V is. These rubbings indeed make the motion of the cart more difficult; and this difficulty will be greater, the greater the weight pressing on the axle, and the smaller the proportion of the wheel’s semidiameter KM to the axle’s semidiameter MO. Why then burdens are more easily transferred by sledges than by wagons, we have clearly demonstrated from what has been said, according to Aristotle’s meaning. Besides, what he omitted, we shall say: namely, that enormous weights are moved by sledges most powerfully if levers are attached to the sledges themselves. And certainly the motion will be very slow; nevertheless, that slowness itself is most amply compensated by the ease that results from it. Let then AB be the plane of the horizon, CD the sledge, EFGH the holes in the sledge, and the levers inserted into the holes IE, KF, LG, MH. Let the weight placed on the sledge be N. Therefore, if four forces are applied to the extremities of the levers I, K, L, M, and these are pushed forward, the sledge will rotate,

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88 IN MECHAN. ARIST. PROBL. tio, & ponderis N translatio ad anteriores partes B. Esto item seorsum scytala PR, cuius centrum Q, vectis eidem per centrum insertus O, P, Q, R. facto igitur vectis motu O P Q R fiet ex O; centro aute[m] Q circuli quadrans O T. existente igitur O in T erit P in S. facta quartæ partis ipsius scytalæ rotatione. Et quoniam ex eodem centro sunt qua- drantes PSOT. erit vt OQ ad QP. ita quadrans O T, ad quadrantem PS. Maxima autem est proportio OQ, ad QP. Maxima igitur proportio OT ad PS. Ex magno igitur motu O ad T, paruus sit scytalæ motus à P in S. tardius i- gitur progreditur scytala, quæ longioribus vectibus rota- tur, vis tamen maxima, quippe quod vt se habet QP, hoc est, QR ad QO, ita potentia in O ad pondus quod premit in P vel in V. Facillimè itaque pondera vectibus & scyta- lis per horizontis planum transferri, existis patet. QVAESTIO XII. Quæritur, Cur Missilia longius funda mittantur quam manu, præsertim cum proijcienti fundæ pondus addatur lapidis seu missi- lis ponderi: & minus missili, manu proiecto, com- prehendatur? Soluit Philosophus, inquiens, fortè ita fieri, quòd fun- ditor missile proijciat iam ex funda commotum, si qui- dem fundam circulo subinde rotans, iaculatur, ex manu autem à quiete est initium. Omnia autem cum in motu sunt, quàm cum quiescunt, facilius mouentur. Addit præ- terea, An & ob eam caussam est, sed nec minus etiam, quia in funde vsu manus quidem sit centrum, funda verò quod à centro exit: quantò igitur productius fuerit quod à cen- tro est, tanto citiùs mouetur; iactus autem, qui manu sit, fundæ respectu breuiore est. Hæc Philosophus. Et sanè perquàm appositè, itaq; illi

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88 IN MECHAN. ARIST. PROBL. also the translation of weight N to the front parts B. Let there be likewise separately the skytale PR, whose center is Q, and a lever inserted through the same center O, P, Q, R. Therefore, once the lever has been moved, O P Q R will become from O; but with Q as center, a quadrant of the circle O T. Therefore, with O existing at T, P will be at S, when a quarter-turn of the skytale itself has been made. And since the quadrants PSOT are from the same center, it follows that as OQ is to QP, so is the quadrant O T to the quadrant PS. But the greatest ratio is that of OQ to QP. Therefore the greatest ratio is that of OT to PS. Thus from the large motion of O to T, the motion of the skytale from P to S is small. Therefore the skytale progresses more slowly, which is turned by longer levers, yet the force is greatest, because as QP is to QR, so that is, QR to QO, so is the power at O to the weight which presses at P or at V. Thus it is most easily understood that weights are transferred by levers and skytales across the plane of the horizon. QUESTION XII. It is asked why missiles are thrown farther by sling than by hand, especially since to the thrower with the sling the weight of the stone or missile is added to the weight of the missile itself, and a smaller missile is grasped when thrown by hand? The Philosopher solves it, saying that perhaps it happens thus, because the slinger throws the missile already set in motion by the sling, since he hurls it while continually rotating the sling in a circle, but with the hand the beginning is from rest. But everything, when in motion, is moved more easily than when at rest. He adds further, “And is it for that cause also? But no less also because in the use of the sling the hand is indeed the center, but the sling is that which extends from the center: therefore the more extended that which is from the center, the more quickly it is moved; but a throw made by the hand, in relation to the sling, is shorter.” These are the Philosopher’s words. And truly, most aptly, therefore, those

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EXERCITATIONES. 89 illi prorsus assentirer, nisi pro comperto haberem, in lactu qui fundâ sit, non esse manum ipsam motus centrum, sed potius partem illam brachij, quæ humero iungitur, & id- eo motum eo fieri velociorem, quo longior est linea quæ ab humero ad summitatem fundæ est, ea quæ ab humero ad manum ipsam. Illud quoque mirabile est, quod non obseruat Aristoteles, nempe à funditoribus in ipso ciacu- landi actu, tardam fieri circa caput fundæ rotationem. Quamobrem considerandum est, quo pacto fiat à tardi- tate velocitas, Respondemus, velocitatem acquiri non ex simplici, quæ circa funditoris caput sit, rotatione, sed ex eo impetu qui sit in ipsa lapidis emissione, qui quidem im- petus si ante vel post illud tempus fiat, quod à funditore captatur, cassa prorsus & inualida sit ipsa iaculatio. Esto funda AB, manus B, brachium BC. Vt igitur se habet CH, ad CB, ita veloci- tas AD ad velocitatem BE; Vidimus nos pueros, arundi- ni ad caput scissæ, paruos la- pides inserentes, arundinem- que manu rotantes longissi- mè lapides ipsos proijcere; A- rundo FG, lapis F, manus G, brachium GH. QVÆSTIO XIII. Quæritur, Cur circa idem iugum, maiores collopes (vectes sunt, quos alij scy talas appellant, vt Pappus & Heron) faciliùs quàm mi- nores mouentur: & item suculæ, quæ graciliores sunt eadem vi quam crassiores? Ideo hoc fieri posse docet Philosophus, quòd tam iugu[m] quam sucula cætrum sit, prominentes autem collopum M longi-

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EXERCISES. 89 I would agree with him altogether, if I did not know for certain that, in the act of casting with a sling, the hand itself is not the center of motion, but rather that part of the arm which is joined to the shoulder; and therefore the motion becomes swifter in proportion as the line from the shoulder to the top of the sling is longer than that from the shoulder to the hand itself. That too is a remarkable thing, which Aristotle does not observe, namely, that sling-throwers, in the very act of hurling, make the rotation about the head of the sling slow. Wherefore it must be considered in what way speed is produced from slowness. We answer that the speed is acquired not from the simple rotation which is about the slinger’s head, but from that impulse which is in the very release of the stone; and indeed, if that impulse takes place before or after that moment which is seized by the thrower, the cast itself is altogether vain and ineffective. Let there be the sling AB, the hand B, the arm BC. Therefore, as CH is to CB, so is the velocity AD to the velocity BE; we have seen boys inserting small stones into a reed split at the top, and, rotating the reed by hand, hurling those stones very far; reed FG, stone F, hand G, arm GH. QUESTION XIII. It is asked why, around the same yoke, larger collopes (these are levers, which others call scytalas, as Pappus and Heron do) are moved more easily than smaller ones; and likewise why suculae, which are more slender, are moved with the same force as thicker ones? The Philosopher teaches that this can happen because both the yoke and the sucula are of a certain kind, but the protruding parts of the collopes M are long-

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90 IN MECHAN. ARIST. PROBL. longitudines ex lineæ quæ sunt à centro. Celeriùs autem moueri & plus ab eadem vi quæ maiorum sunt circuloru[m] quàm quæ minorum. quippe quod ab eadem vi plus trâs- feratur illud extremum quod longius à centro distat. In gracilioribus verò suculis datâ collopum paritate plus es- se id quod à ligno distat. Esto iugum sucu- laue maior, AB circa centrum C, minor verò circa idem centrû DE, Collops aute[m] AF, pon- dus quod per iugum at- tollitur G. A it igitur A- ristoteles, suculas, iu- gaue AB, DE ceu cen- tra esse, à quibus extat colops AB, ex maiori quidem, totâ sui parte BF, ex minori autem EF. quo igitur, ait, longior fuerit collops extans, eo maior, & ideo velocior ad parte[m] F per maiorem circulum FH, fiet collopis motus & pon- deris eleuatio, at maior est collops EF ipso BF, facilius er- go mouebitur pondus per suculam DE, ex collope EF, ab eadem vi, quam per suculam AB, & collorem BF. Hæc sensisse videtur Aristoteles, qui crassa, vt aiunt, Minerua rem pulchram & subtilem est prosequutus. Di- cimus igitur primò, instrumentum illud quod Latini su- culam, id est, serosulam, à stridore arbitror qui in conuer- sione fit, appellauere, Græci verò , id est, Asinum, quip- pe quod ceu Asinus pondera sustineat portetque. Hanc eandem Machinam veteres Mechanici vocauere Axem in Peritrochio, cuius nos imaginem, è Pappo in 8. Col- lect. Mathematicarum desumptam in ipso huius nostri o- peris initio, inter quinque Potentias proposuimus. Huius vim inter antiquos diligentissime examinauêre Heron, & ipse-

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90 IN MECHAN. ARIST. PROBL. Lengths from the line which are from the center. But they are moved more quickly and by the same force more is transferred in the case of the larger circles than of the smaller; for by the same force that extremity which is farther from the center is carried over more. And in thinner windlasses, given equality of the arms, that part is greater which is farther from the timber. Let the larger beam of the windlass be AB about center C, and the smaller about the same center DE; and let the arm be AF, the weight lifted by the beam G. Aristotle says, then, that the windlasses, beams AB, DE, are as centers, from which the arm extends AB, from the larger indeed through its whole part BF, but from the smaller through EF. Therefore, he says, the longer the projecting arm is, the greater it is, and therefore the swifter, so that toward part F through the larger circle FH the motion of the arm and the lifting of the weight will occur; but the arm EF is larger than BF itself, therefore the weight will be moved more easily through the windlass DE, from arm EF, by the same force, than through the windlass AB and arm BF. Aristotle seems to have thought this, and, as they say, with coarse Minerva he followed a beautiful and subtle matter. We say first, then, that that instrument which the Latins called a sucula, that is, a little winch, I suppose from the creaking that occurs in turning, the Greeks called , that is, an Ass, because, like an ass, it bears and carries weights. The ancient mechanicians called this same machine an Axis in a Peritrochium, the image of which, taken from Pappus in the 8th Collection of Mathematical Matters, we proposed at the very beginning of this work of ours among the five Powers. Its force among the ancients was examined most diligently by Hero, and he-

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EXERCITATIONES. 91 ipsemet Pappus, interiuniores verò Guilibaldus eo Tra- ctatu quem hac de Potentia Mechanicis suis inseruit. Summa est, hanc Machinam ad vectem reduci. Nec ve- rum est quod scribit Aristoteles, iugum suculamue cen- tra esse, hæc enim centrum habent, quod in figura supe- rius posita notatur signo C. igitur vt se habet FC, ad CA, ita pondus G ad potentiam in F; est autem maior propor- tio FC ad GD, quàm FC, ad CA. faciliùs ergo mouebit potentia quæ in F, pondus in D, quàm eadem potentia F, pondus in A, hoc est, G. Huius naturæ sunt quoque Erga- tæ, quas machinas nostri, Græco luxato vocabulo Arganos appellant. Suculæ enim reuera sunt, positione tantu[m] ab eis differentes, non enim plano horizontis ergatæ æ- quidistant, ceu suculæ & Axis in Peritrochio, sed eidem fiunt perpendiculares. Cæterum facilitatem à velocitate non oriri superius demonstrauimus. QVAESTIO XIV. Proponitur dubitatio, Cur eiusdem magnitudinis lignum facilius genu frangatur si quispiam æque diductis manibus extrema com- prehendens fregerit, quàm si iuxta genu. Et si terræ applicans pede superposito manu hinc inde diducta confregerit quàm propè. Soluitur à Philosopho paucis verbis, An quia ibi genu centrum est, hic verò ipse per? quanto autem remotius à centro fuerit, facilius mouetur quodcunque: Moueri autem quod frangitur necesse est. Esto lignum quod frangi debet AB, genu vel pedis locus C, manuum latè diductatum situs DE, minus didu- ctarum FG; Itaque quoniam DE magis à centro C distant quàm FG, velocius mouebuntur puncta DE ipsis FG, er- go inde facilius fiet tractio quam ex FG. Hæc ille ex suis prin- M 2

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EXERCISES. 91 ipsemet Pappus, and later Guilibaldus in that Treatise which he inserted on this Power among his Mechanical works. The sum of it is, that this Machine is reduced to a lever. Nor is what Aristotle writes true, that the yoke or the winch are centers; for these have a center, which in the figure set above is marked with the sign C. Therefore, as FC is to CA, so is the weight G to the power at F; but the proportion of FC to GD is greater than that of FC to CA. Therefore the power which is at F will move the weight in D more easily than the same power at F will move the weight in A, that is, G. Of this nature also are Ergatæ, which our mechanics, using a corrupted Greek word, call Argans. For they are in truth winches, differing from them only in position; for the ergatæ do not lie parallel to the horizon, as do the winches, and the axis in the peritrochium, but are perpendicular to it. Moreover, we have shown above that ease does not arise from speed. QVAESTIO XIV. The question is proposed: Why does wood of the same size break more easily at the knee-joint if someone, taking hold of the ends with hands equally spread apart, breaks it, than if he breaks it near the knee. And likewise, if by applying it to the ground and placing a foot on it, he breaks it with the hands extended on either side, than if he does so near it. It is answered briefly by the Philosopher: because there the knee is the center, but here it is the very point? For the farther anything is from the center, the more easily it is moved; but what is broken must needs be moved. Let AB be the wood that is to be broken, C the place of the knee or foot, the position of the hands widely spread DE, and the less spread FG. Therefore, since DE is farther from the center C than FG, the points DE will be moved more quickly than the points FG themselves; therefore the pulling from DE will be done more easily than from FG. Thus he, from his own prin- M 2

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principijs. Nos diligentiis, si fieri poterit, effectus huius causam perscrutemur. Esto igitur in secunda figura lignum oblongum AB, cuius medium C, linea ducatur CD perpendicularis ipsi AB. Admoueatur genu pucto C, manus verò diuari-centur in AB, facta igitur vtrinque impressione, lignum non fra[n]getur, nisi partium in CD coniunctarum separatio fiat, sitque altera in E, altera verò in F, fractum ergo erit lignu[m], & centro C immobili permanente, partes tacto angulo GCH erunt in GC, HC: Modò lignum suæ integritati restituetur, & denuò admoto genu puncto C, manus diducantur in I, K, quæ locavinciniora sint ipsi C, quam AB, Dico hinc difficilius fractionem fieri quam ex AB. Consideramus enim in integro ligno AB, duos vectes ACD, BCD, quorum anguli concurrunt in commune fulcimentum C, Sunt autem vectes angulati, & eius naturæ, quam examinauimus in quæstiones. Est igitur resistentia, qua ligni partes vniuntur in D, loco ponderis: superanda hæc est, vt ligni fiat fractio. Dico id facilius cessurum, si fiat ex punctis A, B, remotioribus quam ex IK, ipsi puncto C propioribus: etenim vt AC, ad CD, ita resistentia quæ sit in D ad potentiam in A, item vt se habet IC ad CD, ita resistentia in D ad potentiam in I, sed minore est proportio IC ad CD, quam AC ad CD. ergo facilius potentia quæ est in A, resistentiam superabit, quæ est in D, quam ea quæ est in I, quod

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principles. Let us, with diligence, if it can be done, investigate the cause of this effect. Let there therefore be in the second figure a long piece of wood AB, whose middle is C, and let a line CD be drawn perpendicular to AB itself. Let the knee be applied to the point C, and the hands, moreover, be separated along AB; therefore, when pressure has been made on both sides, the wood will not be broken unless separation be made of the parts joined in CD, and let one be in E, the other in F. The wood will therefore be broken, and the center C remaining fixed, the parts, the angle touched GCH, will be in GC, HC. But now let the wood be restored to its integrity, and when the knee is again applied to point C, let the hands be drawn apart into I, K, which are located closer to C itself than AB. I say from this that the breaking is more difficult than from AB. For we consider in the unbroken piece of wood AB, two levers ACD, BCD, whose angles converge at the common support C. Now these are angled levers, and of the nature which we examined in the questions. There is therefore a resistance, by which the parts of the wood are united at D, in the place of a weight; this must be overcome, if the wood is to be broken. I say that this will happen more easily if it be done from the points A, B, which are farther away, than from IK, which are nearer to the point C itself: for as AC is to CD, so is the resistance which is in D to the power in A; likewise, as IC is to CD, so is the resistance in D to the power in I, but the proportion of IC to CD is smaller than that of AC to CD. Therefore the power which is in A will more easily overcome the resistance which is in D, than that which is in I, which

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EXERCITATIONES. 93 quod fuerat demonstrandum. Idem autem intelligendu[m] est de parte CB; eadem enim est ratio. Cur igitur longiora & graciliora ligna facilè frangantur, existis clare patet: nempe quia maxima est proportio longitudinis ad crassitudinem, cuius quidem crassitudinis spatium loco partis illius in vecte succedit, quæ pertingit à fulcimento ad pôdus, hoc est, ad ipsam resistentiam. Sed nos hac eadem de re nonnulla in declaranda quæstione 16. perpendemus. QVAESTIO XV. Proponitur inuestigandum, Cur litterales crocæ (glareas dicunt Latini, vel calculos, quos vmbilicos appellat Cicero lib. 2. de Orat.) rotundâ sint figurâ, cum aliquando ex magnis sint la- pidibus testisue? A It Philosophus, ideo fortasse fieri, quòd ea quæ à me- dio magis recedunt, in motionibus, celerius ferantur; medium esse centrum, interuallum vero quæ à cen- tro, semper autem maiorem ab æquali motione maiorem describere circulum; quod autem maius in æquali tem- pore spatium transit, celerius ferri; quæ autem celerius ex æquali feruntur spatio vehementius impetere, quæ aute[m] impetunt, impeti magis, & ideo quæ magis à centro di- stant, necesse esse confingi, quod cum glareæ seu crocæ patiantur, necessariò rotundas fieri. Hactenus ille, & sanè probabiliter. Verum enimuerò aliter se res habere vide- tur: siquidem enim à rotatione ex maiori à centro distantia id fieret, maiores quidem glareæ crocæue essent ro- tundiores, at nos non maximas modò, sed & minimas, easque magis angulis carere, & ad rotunditatem accede- re videmus. Præterea non moueri eas circa centrum pa- lam est, imò vt varia sunt figura, ita varijs quoque motio- nibus, ex agitatione moueri. Id sanè exploratissimum est, M 3 angu-

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EXERCISES. 93 which was to be demonstrated. The same, however, must be understood of part CB; for the ratio is the same. Why then longer and more slender pieces of wood are easily broken is clearly evident from this: namely, because the proportion of length to thickness is greatest, and the space of that thickness serves in the place of the part which extends from the fulcrum to the weight, that is, to the resistance itself. But we shall consider some matters on this same subject in the explanation of Question 16. QUESTION XV. It is proposed to inquire why the literal crocæ (the Latins call them glareas, or calculi, which Cicero calls umbilici in book 2 of the De Oratore) are of a round figure, even though they are sometimes of large stones or shells? The Philosopher says that this perhaps happens because things that recede more from the middle are carried more quickly in motions; the middle being the center, but the interval being what is from the center; and that from an equal motion one always describes a larger circle the greater the distance; and that what passes over a larger space in equal time is carried more quickly; and that what is carried more quickly over an equal space strikes more forcefully; and that what strikes is itself struck more; and therefore those things which are farther from the center must necessarily be rounded off, and because the glareas or crocæ undergo this, they necessarily become round. Thus far he, and indeed plausibly enough. But in truth the matter seems to be otherwise: for if this were due to rotation from greater distance from the center, the larger glareas or crocæ would indeed be more round, whereas we see not only the largest but also the smallest, and these more lacking in angles and approaching roundness. Besides, it is plain that they are not moved around a center; rather, as their shapes vary, so also they are moved by various motions, from agitation. That is certainly most clearly established, M 3 angu-

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94 IN MECHAN. ARIST. PROBL. angulos omnes, & eminentias quaslibet in corporibus esse infirmiores. offensionibus enim expositæ sunt, nec resistendi habent facultatem. Itaque in attritione quæ fit in eorum agitatione perpetua, eminentiæ contunduntur, & superficies ipsa paullatim leuigatur. Esto angulatus lapis ABCD. Dum igitur perpeti motione atq[ue] assiduâ versatione agitatur, ferturque, eminentiæ angulique, vt pote debiles & imbecilli, conte- runtur, & inde figura fit quædam irregularis, ad primam quidem la- pidis formâ accedens, leuistamen & quouis angulo carens, qualis est E remotis ABCD, an- gularibus eminentijs. Hanc eandem ob causam, sculptores antequam mar- moribus vltimum læuorem inducant, dentato malleo pri- mum quidem vtuntur, tum demum eminentiores parti- culas radula facilè amouentes superficiem ipsam læuem & adæquatam reddunt. Hinc etiam nostrates Architecti, in arcium propu- gnaculis efformandis acutos angulos deuitat, vt pote de- biliores, & magis offensionibus obnoxios. quod nec Vi- truuium latuit, qui ideo lib. 1. cap. 5. ita scribit: Turres itaq[ue] rotundæ aut polygoniæ sunt faciendæ, quadratas enim machinæ celerius dissipant; & angulos, Arietes tundendo frangunt, in ro- tundationibus autem, vti cuneos ad centrum adigendo lædere non possunt. Hæc ille. Cur autem nostri rotundas figuras alias vtiles reijciant, ab ijs petendum qui in ea facultate ver- santur. Porrò quod ad hanc eandem speculationem facit, videmus, antiquas statuas, vt sæpius auribus, naso, digitis, manibusue atque pedibus carere, quippe quod imbecillæ sint partes, & facilè quouis occursum mutilentur. Quæ o- mnia

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94 IN MECHAN. ARIST. PROBL. all angles, and whatever projections there are in bodies, are weaker. For they are exposed to blows, nor do they have the capacity to resist. Thus, in the rubbing that occurs in their perpetual agitation, the projections are crushed, and the surface itself is gradually smoothed. Let there be an angular stone ABCD. Therefore, while it is stirred by constant motion and continuous turning, and is carried about, the projections and angles, since they are weak and feeble, are worn away, and from this a certain irregular shape is formed, approaching the original form of the stone indeed, yet smoother and lacking every angle, such as E, the angular projections of ABCD having been removed. For this same reason, sculptors, before they bring the final polish to marbles, first use a toothed mallet, then at last, by easily removing the more prominent particles with a scraper, they make the surface itself smooth and even. Hence also our architects, in shaping the battlements of fortresses, avoid sharp angles, since they are weaker and more liable to blows. This did not escape Vitruvius either, who for that reason writes thus in book 1, chapter 5: “Therefore towers should be made round or polygonal; for square ones machines more quickly destroy, and rams, by striking, break the angles; but in round structures they cannot cause damage, since by driving wedges toward the center they are unable to harm them.” Thus he. But why our people reject other useful round forms is to be asked of those who are engaged in that craft. Moreover, what also pertains to this same inquiry is that we see ancient statues often lacking ears, nose, fingers, hands, or feet, because these are weak parts and are easily mutilated by any collision. All these things

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EXERCITATIONES. 95 mnia cùm vera sint, nemo, vt arbitror, dixerit, absolutè, quod voluit Aristoteles, id ex rotatione velociori & par- tium à centro remotione, fieri. QVAESTIO XVI. Dubitatur, quare, quò longiora sunt ligna, tāto imbecilliora fiant, & fitolluntur, inflectuntur magis: tametsi quod breue est ceu bi- cubitum fuerit, tenue, quod verò subitorum cen- tum crassum? EXsuis principijs soluit Aristoteles. Inquit enim: An quia & vectis & onus & hypomochlium, id est, fulci- mentum in leuando, fit ipsa ligni proceritas? Prior namq; illius pars ceu hypomochlium sit, quod verò in extremo est, pondus: quamobrem quanto extensius fuerit id quod à fulcimento est, inflecti necesse est magis; quo enim plus à fulcimento distat, eo magis incuruari necesse est. Ne- cessariò igitur extrema vectis eleuantur. Si igitur flexilis fuerit vectis, ipsum inflecti magis cum extollitur necesse est, quod longis accidit lignis, in breuibus autem quod vl- timum est, quiescenti hypomochlio de propè fit. Hæc subiectâ figurâ ob oculos ponimus. Esto longum ac fle- xile lignum AB, manu ele- uetur in A, flectetur itaq; in B, & declinabit in C. et- enim manus quæ sustinet in A, fulcimenti loco succedit: longitudo vero AB ponde- ris vices refert, atque vectis, quare quo longius abfuerit à fulcimento, id est, manu extremum B, eo magis flectetur; si autem lignum breuius fuerit, nempe terminatum in D, nequaquam flectetur, eò quòd eius extremum D minus à fulcimento quod est in A sit remotum. Hæc igitur est més Ari-

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EXERCISES. 95 although all these things are true, no one, I think, would say absolutely that what Aristotle meant is brought about by quicker rotation and by the removal of the parts from the center. QUESTION XVI. It is doubted why, the longer pieces of wood are, the weaker they become, and when they are lifted they bend more: although if a piece is short, say two cubits, it is thin, but if it is one hundred cubits long, it is thick? Aristotle solves it from his own principles. For he says: Is it because both the lever and the load and the hypomochlion, that is, the support in lifting, are found in the length of the wood itself? For its first part serves as the hypomochlion, while what is at the end is the weight; wherefore, the more extended that part is which is from the support, the more it must bend; for the farther it is from the support, the more it must curve. Necessarily, therefore, the ends of the lever are raised. If, then, the lever is flexible, it must bend more as it is raised, which happens in long pieces of wood; but in short ones, the last part lies very near the resting hypomochlion. We set this before the eyes by the figure below. Let there be a long and flexible piece of wood AB; let it be raised by the hand at A; thus it will bend at B and incline to C. For the hand which supports at A takes the place of the support; the length AB, however, represents the role of the weight and of the lever, so that the farther the end B is from the support, that is, from the hand, the more it will bend; but if the piece of wood is shorter, namely ending at D, it will not bend at all, because its end D is less distant from the support, which is at A. This, therefore, is the opinion of Ari-

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96 IN MECHAN. ARIST. PROBL. Aristotelis, cuius quidem sententiam non damnamus; quippiam tamen addimus. Dicimus autem materiam, quatenus ad hanc contemplationem spectat, in duplici esse differentia. aut enim rarefactionis & constipationis est incapax, vt in chalybe videmus, nitro, metallo, mar- more, aut capax quidem, & hæc duplex: Vel enim natura nata est ad rectitudinem quandam, vt arborum flagella virgæque, aut non item, ceu stannum, plumbum, & cætera eiusmodi. Esto primò vitreum corpus gracile, procerum, teres AB, manu capiatur in A, itaq[ue] pondere ipsius corporis præualente ad partes B, quia in C puncto, quod circa medium est, ex parte superiori non fit rarefactio, nec in inferiori constipatio, nec interim datur penetratio corporum, fit fractio à superiori parte, & pars CB à reliqua parte AC, auulsa & separata cadit in D, succedit autem ipsa separatio rarefactioni. Porrò quod materias hasce non flexibiles diximus, sed frangibiles, non ideo negamus vel sensu docente, aliquam in ijs fieri flexionem. Si autem lignea fuerit materia, eaq[ue] flexibilis, vt EF, si manu eleuetur in E, præualente pondere in F flectetur vbi G. ibi enim à parte superiori fit rarefactio, ab inferiori verò constipatio, & pars GF declinabit in H, quæ declinatio eò vsque procedet, quo rarefactio & constipatio competens naturæ illius materiæ, quæ flectitur ad summam intensionem deuenerint; tunc sivis maior ingruerit, frangetur omnino: si secus facta ibi resisten-

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96 IN MECHAN. ARIST. PROBL. Aristotle’s opinion, indeed, we do not condemn; however, we add something. We say, therefore, that matter, insofar as it pertains to this consideration, is of a twofold difference: either it is incapable of rarefaction and condensation, as we see in steel, nitre, metal, mar- ble; or it is indeed capable, and this too is twofold: for either by nature it is suited to a certain straightness, as the shoots of trees and rods, or not, as tin, lead, and other things of this kind. Let there first be a slender, tall, round glass body AB; let it be taken by the hand at A, and thus, with the weight of the body itself prevailing toward the parts B, because at point C, which is about the middle, from the upper part there is no rarefaction, nor from the lower part condensation, nor in the meantime is penetration of bodies granted, fracture occurs from the upper part, and the part CB, being torn away and separated from the remaining part AC, falls in D; but the separation itself follows rarefaction. Moreover, when we said that these materials are not flexible, but brittle, we do not on that account deny, even experience teaching, that some bending takes place in them. But if the material were wooden, and flexible, as EF, if it is raised by the hand at E, with the weight prevailing at F it will bend at G. For there from the upper part there is rarefaction, but from the lower condensation, and the part GF will incline toward H; and this inclination will proceed so far that the rarefaction and condensation proper to the nature of that material which is bent have reached their highest intensity; then, if a greater force should come upon it, it will break altogether: if otherwise, there would be there resistan-

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EXERCITATIONES. 97 resistentia, vbi rarefactio sit & constipatio post inclinationem sursum feretur pars inclinata & nutans, tum in contrariam partem tendens reflectetur, vt videre est in virga IN. Declinans enim in KL, repellente ea quæ infra K sit materiæ condensatione, impetu ex descensu acqui- sito facta reflexione ascendit in KM, donec paullatim cir- ca pristinam rectitudinem reuertatur, & hic quidem mo- tus vibratio d: citur, agitatioue. Si autem virga lumbea fuerit, naturâ non factâ ad rectitudinem, puta OP, pro- prio vincente pondere, ad partes declinabit QS, fietq; in QR rarefacta, nempe superiori parte ea constipata infe- riori in Q, nec reflectetur, quippe quòd eius natura con- densationem & rarefactionem commodè patiatur, nec facta sit ad rectitudinem. Porrò tripliciter fieri potest horum oblongorum corporum eleuatio, nempe vel extremorum altero, aut si ambobus, si vtrinque suspendatur, vel alicubi inter extre- ma. De priori modo iam egimus. Modò suspendatur in medio vt AB, in C. eo igitur casu cum fulcimentum sit in C, vtrinq; sit flexio in D, & E, & id quidem si materia flexionem patitur: sin minus, fractio sit in C. Si autem ab ex- tremis fiat suspensio, vt in AB, tunc ceu duo vectes fient, quorum fulcimenta in extremis AB. Pondera au- tem communia in medio vbi est ab extremis AB. Cedente igitur materia suomet pon- deri, siquidem inflexibilis fu- erit, frangetur, & fiet partiu[m] separatio in C, duoque inde corpora AD, BE. Si autem flexionis cepax, vt AB in postre- ma

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EXERCISES. 97 where there is rarefaction and condensation; after an inclination upward, the inclined and oscillating part will be carried, and then, tending toward the opposite side, it will be reflected, as can be seen in the rod IN. For, bending in KL, and repelled by the condensation of the matter lying below K, it rises by the impulse acquired from the descent, after reflection, into KM, until little by little it returns to its former straightness; and this motion is indeed called vibration, or agitation. But if the rod were leaden, not made by nature for straightness, as in OP, it will, by its own prevailing weight, incline toward QS, and in QR it will be rarefied, namely with the upper part condensed and the lower part at Q, and it will not be reflected, because its nature readily admits condensation and rarefaction, and it was not made straight. Moreover, the raising of these oblong bodies can take place in three ways: namely, either by one of the ends, or by both, if it be suspended at both ends, or somewhere between the ends. We have already treated the first way. Let it now be suspended in the middle, as AB, at C. In that case, therefore, since the support is at C, there will be bending on both sides at D and E, and this indeed if the material allows bending; if not, there will be fracture at C. But if the suspension is from the ends, as in AB, then it will become as if two levers, whose supports are at the ends AB. The weights, however, are common in the middle, where it is from the ends AB. Therefore, if the material yields to its own weight, if it is inflexible, it will break, and a separation of the parts will occur at C, and from it will arise two bodies, AD and BE. But if it is capable of bending, as AB in the last...

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98 IN MECHAN. ARIST. PROBL. ma figura, facta ex contrario, nempe in inferiori parte cir- ca C rarefactione, in superiori verò condensatione, pon- dere præualente curuabitur, fietq; lignum quidue aliud huiusmodi, vt ADB, nec amplius pondere suapte naturâ inferiùs vergente ad rectitudinem reuertetur. Cæterùm cur oblonga & graciliora corpora facilius illis, quæ contrario se habent modo, frangantur, ex me- chanicis principijs in quæstione 14. apertè demonstraui- mus. Modò vt ex hac contemplatione, quæ aliàs inutilis videtur, aliquam vtilitatem capiamus, & ex his quæ con- templabimur, Architecti prudentiores fiant, isthæc ipsa, de quibus agimus, ad rem ædificatoriam commodè apta- bimus. Transferamus igitur cogitationem ad eam trabiu[m] compagem, quæ ad recta sustinenda ex transuersario ar- rectarioq; sit, & duobus cauterijs, quam nostri à Latinis detorto vocabulo Biscauterium dicunt. Perscrutabimur enim, vnde illi tanta ad sustinendum vis, & quæ compa- gem hanc consequantur passiones. quamuis enim fabri meræ praxi, quod vtile est efficiant, nos meliorum inge- niorum gratiâ, rei ipsius caussas diligenter examinatas in medium proferemus; nec de hac re tantùm agemus, sed de Cameris quoque, fornicibus eorumque vitijs & virtu- tibus quatenus ad Mechanicum pertinet, sermonem ha- bebimus. Quærimus primo, cur perpendiculariter erecte trabes superimposita pondera validissime sustineant? Et sane hoc omnes norunt, sed non per caussas. Esto horizontis planum, illudque solidissimum, & impenetrabile AB, trabs eidem ad perpendiculum erecta CD fulta basi vbi C grauitatis centrum F. pondus super- impositum FG, cuius grauitatis centrum H: Sint autem H & E in eadem perpendiculari, quæ ad mundi centrum HEC. Itaque eo quod tum ponderis tum trabis centra grauitent in perpendiculari, illa verò fulciatur in C, to- tius

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98 IN MECHAN. ARIST. PROBL. the figure, made from the opposite cause, namely with rarefaction in the lower part around C and condensation in the upper part, will bend under the preponderating weight, and a piece of wood or anything else of this kind, like ADB, will no longer return to straightness when its weight naturally tends downward. Moreover, why oblong and slender bodies are more easily broken than those which are in the opposite condition, we have clearly demonstrated from mechanical principles in Question 14. But now, in order that from this contemplation, which otherwise seems useless, we may derive some utility, and that from the things we shall contemplate architects may become more prudent, we shall suitably adapt these very matters, of which we are treating, to the business of building. Let us therefore turn our thought to that frame of beams which is made for supporting straight members, and has a crosspiece and a post, and two cauterii, which our people, by a corrupted Latin term, call Biscauterium. For we shall investigate whence it has so great a power of support, and what affections follow upon this frame. For although craftsmen achieve what is useful by mere practice, we, for the sake of better minds, shall bring forward the causes of the thing itself, carefully examined; nor shall we deal only with this matter, but we shall also speak about chambers, vaults, and their faults and virtues insofar as they pertain to mechanics. First we ask why beams set upright perpendicularly sustain superimposed weights most strongly? And indeed all know this, but not the causes. Let there be a plane of the horizon, and let it be the very solid and impenetrable AB; let the beam CD be supported by it erect perpendicularly, with its base at C, and let F be the center of gravity. Let the superimposed weight be FG, whose center of gravity is H. Let H and E, moreover, be on the same perpendicular, which is toward the center of the world HEC. Therefore, because both the centers of the weight and of the beam gravitate on the perpendicular, and the latter is supported at C, the whole

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EXERCITATIONES. 99 tius ponderis moles recumbet in C: non descendet autem in I, propterea quod supponatur ipsum planum AB, impenetrabile. Igitur vt pondus H descendat in C, alterum duorum est necessarium, nempe vel trabem subiectam comminui, aut eius partes sese penetrare, & plura corpora esse in eodem loco, puta KC, quorum hoc secundum naturæ penitus repugnat, illud vero primum, penè impossibile. Diuidatur enim trabs in partes æquales tres, lineis KL, ipsa igitur KC infima sustinet mediam KL, hæc verò supremam LD, hæc autem podus, ipsum superpositum in H. Se igitur sustinent partes. Sed illud totum partibus constat. ergo pondus totum à trabe tota, hoc est, à se toto sustinetur. Præterea in præcedenti quæstione monstrauimus tunc facilem esse gracilis & oblongi ligni fractionem, cum maxima est longitudinis ad crassitudinem proportio. Hîc verò contrà accidit, etenim MD pars vectis quæ à fulcimento est ad potentiam minimam habet proportionem ad rectam DC, quæ à fulcimento ad locum fractionis extenditur, vbi C, quod vt euidentius pateat, Esto seorsum trabs AB, cuius medium C. Sit autem pondus D impositum puncto C. facilè igitur frangetur lignum AB, propterea quòd maxima sit proportio AC ad CE; resistentia verò fiat in E, addatur vniatur q; N 2 ligno

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EXERCISES. 99 the mass of a greater weight will rest in C: but it will not descend into I, because the plane AB is assumed to be impenetrable. Therefore, for the weight H to descend into C, one of two things is necessary, namely either that the beam beneath be crushed, or that its parts penetrate one another, and that there be several bodies in the same place, as KC, both of which are wholly contrary to nature; the former indeed is almost impossible. For let the beam be divided into three equal parts by the lines KL; thus the lowest KC sustains the middle KL, this again the upper LD, and this the weight, that placed above in H. Thus the parts sustain one another. But that whole consists of parts; therefore the whole weight is supported by the whole beam, that is, by itself as a whole. Moreover, in the preceding question we showed that the breaking of a slender and long piece of wood is easy when the proportion of length to thickness is greatest. Here however the opposite occurs; for MD, the part of the lever which is from the fulcrum to the power, has the smallest proportion to the straight line DC, which extends from the fulcrum to the place of breaking, where C, which that this may appear more clearly, let there be separately the beam AB, whose middle is C. Let the weight D also be placed at the point C. The wood AB will therefore easily break, because the proportion of AC to CE is greatest; but let the resistance be applied in E, and let it be added and joined to the wood.

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100 IN MECHAN. ARIST. PROBI. ligno AB lignum FH. Crassius igitur est totum AL, ipso AH, & ideo minor proportio AC ad CG quàm AC, ad CE. Addatur adhuc & IM. Longè itaque difficiliùs frangetur in K propterea quòd longè minor sit proportio AC ad CK quàm eiusdem ad CE & CG. His igitur consideratis, & demonstratis concludimus, impossibile esse erectam trabem ponderi cedere, & frangi. Dicet autem quispiam, hæc si vera sunt, quo gracilius fuerit fulcrum, eo validiùs sustinebit, & frangetur minus, quod oppido falsum est. Respondemus, id non ex proportionum naturâ, sed ex materiæ ipsius infirmitate fieri. Ita quoque invecte non materiam, quatenus ad vim pertinet, sed proportiones partium consideramus. Vtrumque igitur requiritur ad fulcri validitatem proportio longitudinis ad crassitudinem debita, & materiæ ipsius robur & fortitudo. Præterea, quoniam pondus, cui fulcrum resistit, vel ex natura premit, vel ex violentia, illud quidem per lineam perpendicularem, quæ ad mundi cætrum, hoc autem lateraliter & diuersimodè, varia sit fulcrorum dispositio, Cuius rei summa hæc est, vt semper contra impetum supponantur. Esto enim horizontis planum AB, eide[m] perpendiculares CADB, itaque si naturaliter pondus prematex C, fulcrum supponetur AE. Siautem ex F ipsum GE, si verò ex H, supponaturiuxta BE. Si verò secundum I ponderi opponatur KE. Hæc nos de arrectarijs fulcrisue; nunc de transuersarijs, & inclinatis agemus, & primum de transuersarijs, quatenus ad tectorum trabeationes spectat. Esto transuersaria trabs AB, muris vtrinq[ue] fulta CD, cuius

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100 IN MECHAN. ARIST. PROBI. wood from wood FH. Therefore the whole AL is thicker than AH itself, and therefore the proportion of AC to CG is smaller than AC to CE. Let IM be added as well. Thus it will be much more difficult to break at K, because the proportion of AC to CK is much smaller than that of the same to CE and CG. These things, then, having been considered and demonstrated, we conclude that it is impossible for an upright beam to yield to a weight and break. But someone will say: if these things are true, then the more slender the support is, the more strongly it will sustain, and the less it will break, which is utterly false. We reply that this comes not from the nature of proportions, but from the weakness of the material itself. Likewise, in the structure we have considered, not the material, insofar as it concerns strength, but the proportions of the parts. Therefore both are required for the strength of a support: the proper proportion of length to thickness, and the material’s own firmness and strength. Moreover, since the weight against which the support resists either presses by nature or by force, the one indeed by a perpendicular line directed to the center of the world, the other laterally and in various ways, the arrangement of supports varies accordingly. The sum of this is that they must always be placed opposite the impulse. Let the plane of the horizon be AB, and CADB the perpendiculars to it. So if a weight naturally presses from C, the support will be placed AE. If, however, from F, the support will be GE; if from H, it will be placed next to BE. But if according to I a support is opposed to the weight, KE. These things we have said about upright supports; now we shall deal with transverse and inclined ones, and first with transverse ones, insofar as they concern roof-beams. Let there be a transverse beam AB, supported on both sides by walls CD, of which

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EXERCITATIONES. 101 cuius grauitatis centrum E, in perpediculari FEG, quæ quidem ad mundi centrum vergit. Itaq[ue] eodem tendente grauitatis centro, si pondus quod premit in E, non præualeat vnioni partiù ipsius materiæ quæ est in E, resistet trabs suomet ponderi, nec frangetur. Si autem vel infirmitate materiæ, aut vitio, vel maxima existente proportione AF ad FE, fractio fiet in E, & secutâ partium separatione duæ fient vtrinque trabes AH, Bl, quorum grauitatis centra KL. Erunt igitur duo vectes AE, BE, quorum fulcimenta MN, quamobrem si proportio EM ad MH ita præualeat, vt pondus quod est in E, superet pondus muri O superimpositi, & item muri P, corruent quidem trabes, & murorum fiet hinc inde dissipatio. Si autem non præualuerit ea, quam diximus, proportio, suspensæ remanebunt vtrinque trabes vt AHBI. Huic difficultati egregiè occurrunt Architecti, aliquando autem hoc modo: Esto transuersaria trabs suâ gracilitate, aliaue de caussa imbecilla AB, muri quibus vtrinq[ue] sustinetur CD, Trabis ipsius grauitatis centrum G. Itaque adpactis trabi lignis EF, capreolos addunt muro vtrinque fultos CE, DF, eorum capita adpactis lignis admouentes EF, sed & tunc validissima fit colligatio, si inter E & F capreolorum capita integrum lignum trabi supponatur EF. Ratio N 3

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EXERCISES. 101 whose center of gravity E, in the perpendicular FEG, which indeed tends toward the center of the world. Therefore, the center of gravity being thus inclined, if the weight which presses at E does not prevail over the union of the part of the material which is in E, the beam will resist its own weight, and will not break. But if, either through weakness of the material, or defect, or if the proportion AF to FE be very great, fracture will occur at E, and, after the separation of the parts, there will be two beams on either side, AH, Bl, whose centers of gravity are KL. There will therefore be two levers AE, BE, whose supports are MN; wherefore if the proportion EM to MH so prevails that the weight which is in E exceeds the weight of the wall O laid upon it, and likewise of the wall P, the beams will certainly fall, and a dissipation of the walls will follow on this side and that. But if that proportion which we have mentioned has not prevailed, the beams will remain suspended on both sides, as AHBI. Architects meet this difficulty excellently, sometimes in this way: Let a cross beam by reason of its slenderness, or for some other cause, be weak, AB; the walls by which it is supported on both sides, CD; the beam’s own center of gravity, G. So, by fitting pieces of timber EF to the beam, they add struts supported on both sides by the wall, CE, DF, bringing their heads up to the fitted timbers EF; but the fastening is also made very strong if, between E and F, a whole timber be placed beneath the beam EF, at the heads of the struts. Reason N 3

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102 IN MêCHAN. ARIST. PROBL. tio autem validitatis patet; premente enim grauitatis cê- tro in G, fulcra hinc inde succurrunt CE, DF, quæ cum se- ipsis fieri non valeant breuiora, ne corpori detur penetra- tio, resistunt & robustissimè ipsi ponderi superimposito contranituntur. Videntur autem in hoc opere duo con- siderari vectes, GH, GB, quorum fulcimenta EF, potentia premens vtrinque G. Pondera autem parietum partes ca- pitibus trabis impositæ in A & B. Quoniam igitur parua est proportio GE ad EH, parua potentia premens in G, maximè autem pondus in A, fieri non potest trabem fran- gi aut muros vtrinque dissipare in AB. Possunt etiam to- tius trabis tres partes considerari AE, EF, FB, quarum ful- cimenta quatuor A, E, F, B, Diuiso igitur pondere & mul- tiplicatis fulcimentis impossibile est trabem conuelli & vitium facere. Sed & tectorum contignationes imbecillaq[ue] trans- uersaria Mechanici corroborare solent, additis nempe arrectaria trabe atque cauterijs. Esto enim trans- uersaria trabs AB parietibus vtrinque fulta I, K, arrectariu[m] CD. Cauterij vtrin- que AD, BD, ita transuersariæ trabi in AB, & arrectario in D inserti, vt ne- quaquam inde ela- bi valeant. Tum ferrea fascia EF mediam transuersariam trabem AB, à parte inferiori ipsi arrectario connectens. Debet autem arrectarij pes vbi C, aliquantulum à trans- uersaria trabe distare, ne deorsum ex pondere vergente paululum arrectario ipsam transuersariam premat. His i- gitur

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102 IN MÊCHAN. ARIST. PROBL. but the force of support is clear; for when the center of gravity presses at G, the supports CE, DF come to its aid here and there, which, since they cannot be made shorter with respect to themselves, lest penetration be given to the body, resist and most strongly counteract the weight placed upon them. But in this work two levers seem to be considered, GH, GB, whose supports EF, with the pressing force on either side at G. The weights, however, are the parts of the walls placed at the ends of the beam in A and B. Since therefore the proportion of GE to EH is small, and the pressing force at G is small, but especially the weight at A, it cannot happen that the beam be broken or the walls on either side be scattered in AB. The three parts of the whole beam AE, EF, FB may also be considered, with their four supports A, E, F, B. Thus, the weight being divided and the supports multiplied, it is impossible for the beam to be bent and damaged. And builders are accustomed to strengthen the roof-framing and weak cross-members by adding struts to the beam and tenons. For let there be a cross-beam AB supported on both sides by the walls at I, K, and a strut CD. Let the tenons AD, BD on either side be inserted into the cross-beam at AB and into the strut at D, so that they cannot slip out from there. Then let an iron band EF connect the middle of the cross-beam AB, from the lower side, to the strut itself. But the foot of the strut, where C is, ought to stand a little apart from the cross-beam, so that, when it tends downward from the weight, the strut may not press the cross-beam itself a little. With these things therefore

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EXERCITATIONES. 103 gitur ita constitutis pondus quidem transuersariæ trabis, quod suapte naturâ premit in medio vbi C, ferrea fascia, arrectariæ trabi affixa distinetur, Arrectariam cauterij su- stinent, hos verò transuersariæ capita AB, quibus indun- tur. Tota igitur eiuscemodi operis vis in eo consistit, vt probè cauterij transuersariæ & arrectariæ trabi inseran- tur. fixis enim cauteriorum pedibus in AB, non descendet à partibus seu capitibus D, ijs verò stantibus stabit & arre- ctarium, quo inde suspenso transuersaria trabs ei ex ferrea fascia alligata nequaquam pendebit. Stabit ergo compa- ges tota & suapte vi robustissimè connexa totius tecti pondus sustinebit. Quoniam autem vsu venire solet, cauterios nimia longitudine debiles, aliquando tum proprio tum extra- neo cedentes ponderi deorsum vergentes pandare, Ar- chitecti capreolis hinc inde suppositis, ceu fulcris, huic medentur infirmitati. Sint enim cauterij debiles hinc inde AB, AC, media trabs arre- ctaria, quam Monachu[m] dicimus AD. Cauterio- rum mediæ partes E, F, in punctis igitur EF, vtpote maximè ab extremis distanti- bus debiles cauterij valde laborant. Itaque suppositis v- trinque arrectariolis EH, FI, eorum capitibus E, F, duos cauteriolo sibi ipsis ad pedem arrectarij in D, resistentes apponunt. quibus ita constitutis nec E, nec F ad partes H, I, descendere valent. Capiatur enim inter EH, quoduis punctum G, & BG, DG, connectantur, erunt autem BG, DG ipsis BE ED breuiores ex 21. primi elem. Tunc igitur punctum E fiet in G cum BE, ED fient in BG, DG, quod non cedentibus B, D, & sibi ipsis breuioribus factis parti- bus

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EXERCISES. 103 ...thus established, the weight indeed of the transverse beam, which by its own nature presses in the middle where C, the iron band, affixed to the upright beam, is joined, sustains the upright beam of the cautery; but the ends AB of the transverse beam, on which they are fitted. Therefore the whole force of such a work consists in this, that the cauteries be well inserted into the transverse and upright beams. For when the feet of the cauteries are fixed in AB, it will not descend from the parts or ends D; and these standing, the upright beam will also stand, and the transverse beam, being suspended from it and bound by the iron band, will by no means hang down. Thus the whole frame will stand together and, by its own strength most firmly joined, will sustain the weight of the whole roof. But since it commonly happens that cauteries, weakened by excessive length, sometimes yielding both to their own weight and to an external load, bend downward, the Architects remedy this weakness by placing little props, as supports, here and there beneath them. Let the weak cauteries be AB, AC, and the middle upright beam, which we call the Monacus, AD. The middle parts of the cauteries E, F, are therefore at the points EF, and, being farthest from the ends, the weak cauteries labour greatly. Therefore, with small upright props EH, FI set underneath on both sides, they place against their heads E, F two little cauteries resisting themselves at the foot of the upright beam in D. When these are thus arranged, neither E nor F can descend to the parts H, I. For if any point G be taken between EH, and BG, DG be joined, BG and DG will be shorter than BE and ED themselves, according to proposition 21 of the first book of the Elements. Then therefore the point E will be in G, when BE, ED become BG, DG, which, with B and D not yielding, and the parts themselves made shorter...

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IN MECHAN. ARIST. PROBL 104 bus BE, ED, prorsus est impossibile. stabunt igitur in co- rum rectitudine cauterij AB, AC, nec pandabunt, quod fieri querebatur. Hîc autem damnandi veniunt ij, qui transuersariæ quidem trabis capitibus cauteriorum pedes non inserût, sed ea vice transuersariolo quodam medios cauterios v- trinque connectunt ad instar elementi A, quam compa- gem, capram, appellant. Sint enim cauterij hinc inde AB, AC, quorum medias partes connectit transuersariolum. DE. Dico igitur colligationem istam magnopere impro- bandam. Sunt enim AB, AC vectes, quorum commune fulcimentum A, potentiæ hinc inde diuaricantes B, C, pondera inter fulcimentum & potentias DE. quoniam i- gitur vt DH ad AB, ita potentia in B, ad pondus in D, par- ua quidem potentia, pondus in D distrahet & superabit: facillimaq[ue] inde fiet transuersarioli à capreolis ipsis vtrin- que reuullio: Et quoniam centrum quidem est A, fact. in D, E, parua diuaricatione, maxima fit in BC, vtpote parti- bus ab ipso centro A quam remotis. Calcitrant igitur li- beri prope cauteriorum pedes, & muros ipsos summos, non sine magno operis totius vitio, sua calcitatione pro- pellunt. Hæc nos de trabeationibus, modò ad fornicum ca- merarumq[ue] naturam stilum transferemus; id enim suadet vtilitas, imo & necessitas ipsa. Pauci enim ante nos hæc tractarunt, & sanè his probè non cognitis aut neglectis, Architecti fabrique ingentes persæpe incurrunt, & inex- plicabiles difficultates. Dicimus igitur primò, coctiles la- teres, & non cuneatos lapides ad rectam lineam disposi- tos, non stare. Sint enim muri vtrinque AC, BD. Ducatur hori- zontiæquidistans CD, iuxta quam lateres lapidesue non cuneati, seriatim collocentur EF. Dicimus amoto arma- mento,

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IN MECHAN. ARIST. PROBL 104 but BE, ED, it is altogether impossible. Therefore they will stand in the straightness of the cauteria AB, AC, nor will they spread apart, which was the thing sought. Here, however, those must be condemned who do not insert the feet of the cauteria into the heads of the transverse beam, but instead connect the middle cauteria on either side by means of some transverse piece, like the element A, which fastening they call a capra. Let there be cauteria here and there AB, AC, whose middle parts are connected by a transverse piece. DE. I say, then, that this fastening is greatly to be disapproved. For AB, AC are levers, whose common support is A, the powers diverging on either side B, C, the weights between the support and the powers DE. Since therefore, as DH is to AB, so is the power at B to the weight at D, a small power will indeed draw apart and overpower the weight at D; and from this there will very easily result the tearing away of the transverse piece from the capreoli themselves on both sides. And since the center is at A, a slight divergence in D, E produces the greatest in BC, because the parts are so far removed from the center A itself. Therefore the free ones kick near the feet of the cauteria and, with their kicking, drive the highest walls themselves, to no small harm of the whole work. Thus far we have spoken about beam constructions; now we shall turn our pen to the nature of vaults and chambers, for utility, indeed necessity itself, demands it. For few before us have treated these matters, and certainly without a proper knowledge of them, or if neglected, architects and builders very often fall into great and inexplicable difficulties. We say first, then, that baked bricks and non-wedged stones placed in a straight line do not stand. Let there be walls on either side AC, BD. Let the line CD, parallel to the horizon, be drawn, along which bricks or non-wedged stones are laid in order EF. We say, when the support is removed,

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EXERCITATIONES. 105 mento, hoc est, pro- hibente ipso lateres ruere. Producantur enim AC in G, BD verò in H, cum ipsis CG, DH, æquales fiant CI, DK, & recta IK iungatur, erit igitur GD spatium ipsi CK spatio simile quidem & æquale, quod cùm ita sit, nihil prohibet quin tota laterum GD moles in spatium CK transferatur, & corruat. Si autem cunei ipsi lateresue, cuneatim dispositi, ita sint vt ad vnum centrum tendant, licet ad rectam lineam collocentur, non delabentur, sed stabunt; quod ita ostendemus. Sint cunei lateresue cuneatim dispositi ABCD, tendentes ad centrum, seu commune punctum E, Ducantur CAE, DBE, sintque muri vtrinque ponderi resistentes CL, DM, Demittatur perpendicularis, quæ ad mundi centrum FGE secans AB, in G. Tum fiat GK æqualis GF & per K ipsi AGB parallela ducatur, HKI claudens spatium AHIB. Quoniam igitur vt EC, ad EA, ita CD ad AB per 4. propos. lib. 6. maior erit CD ipsa AB, & eâdem de causâ maior AB, ipsa HI, & idcirco maius ABDC spatium, spatio AHIB. Non igitur potest linea CD, fieri in AB, neque AB, in HI, neque spatium totum CABD, transferri in spatium AHIB non data (quod naturæ ipsi repugnat) O

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EXERCITATIONES. 105 ... that is, preventing the very bricks from falling. For let AC be extended to G, and BD to H; then, with CG and DH, let CI and DK be made equal, and let the straight line IK be joined. It will therefore follow that the space GD is indeed similar and equal to the space CK; and since this is so, nothing prevents the whole mass of bricks GD from being transferred into the space CK, and from falling. But if the wedges themselves, or the bricks disposed wedgewise, are so arranged that they tend to one center, then, although placed upon a straight line, they will not slip down, but will stand; which we shall thus show. Let there be wedges or bricks arranged wedgewise ABCD, tending to the center, or common point E. Let CAE and DBE be drawn, and let the walls resisting the weight on either side be CL, DM. Let the perpendicular be let fall, which, cutting AB at G, goes to the center of the world FGE. Then let GK be made equal to GF, and through K let a line be drawn parallel to AGB, HKI, enclosing the space AHIB. Since therefore, as EC is to EA, so is CD to AB, by proposition 4 of book 6, CD will be greater than AB, and for the same reason AB greater than HI, and therefore the space ABDC greater than the space AHIB. Therefore the line CD cannot be made into AB, nor AB into HI, nor can the whole space CABD be transferred into the space AHIB without something being given (which is repugnant to nature itself).

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106 IN MECHAN. ARIST. PROBL. gnat) corporum penetratione. Stabunt ergo cunei, quod fuerat demonstrandum. Verumenimucro, debilis hæc structura est, & eo debilior, quo vani latitudo fuerit maior, cuneorum verò altitudo minor. Idem enim patitur quod epistylia in specie Aræostyla, quæ, vt scribit Vitruuius lib.3.c.2. propter interuallorum magnitudinem franguntur. Id quoque habet vitij, quod cunei ita dispositi suo pondere incumbas vtrinque violentissimè pellant. Vtilis tamen esse potest ad portarum & fenestrarum, quæ in medijs muris sunt, & mediocri vano aperiuntur, superliminaria. Si verò ad minorem circuli portionem curuetur Camera, vtilior quidem erit structura ea ipsa, de qua locuti sumus; non tamen omninò sine vitio. Quoniam igitur vt EM ad EA, ita MGN ad AIB, maior erit MGN linea ipsa AIB, quamobrem fieri non potest vt aptetur lineæ AIB, & in eius locum descendat. Stabit igitur, incumbis vtrinque non cedentibus. Validè autem speciem hanc, loca quibus incumbit, propellere, ita ostendemus. Producatur in eadem figura CA in K, & DB in L. Partes igitur quæ muris ad perpendiculum fulciuntur, sunt AKF, BLH, minimæ illæ quidem, maxima verò pars est

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106 IN MECHAN. ARIST. PROBL. by penetration of bodies. Therefore the wedges will stand, which was to be demonstrated. But in truth this construction is weak, and the weaker the greater the width of the openings and the smaller the height of the wedges. For it suffers the same thing as architraves in the Araeostyle type, which, as Vitruvius writes in book 3, chapter 2, break because of the magnitude of the intervals. It also has this defect, that the wedges, so arranged that they bear by their weight, are driven on both sides with extreme violence. Yet it can be useful for lintels of doors and windows that are in the middle of walls and open in a moderate opening. But if the vault is curved to a smaller portion of the circle, that very construction of which we have spoken will indeed be more useful; not, however, altogether without defect. Since therefore, as EM is to EA, so MGN is to AIB, the line MGN itself will be greater than AIB; for which reason it is impossible for it to be fitted to the line AIB and descend into its place. Thus it will stand, with the supports on both sides not yielding. And we shall show in what way this form strongly thrusts at the places on which it bears. Let CA in the same figure be produced to K, and DB to L. Therefore the parts which are supported perpendicularly by the walls are AKF and BLH, indeed the smallest, but the greatest part is

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EXERCITATIONES. 107 est extra fulcimenta, nempe tota AKLB quæ idcircó suo- pte pondere deorsum vergens & in incumbas vtrinq; pel- lens aperitur, & facillimè vitium facit. Eiusdem ferè na- turæ ea species est, quæ vel ex media, vel ex minori ellipsis secundum maiorem diametrum fit segmento. Vtilior ta- men hæc est, præcipuè circa incumbas, propterea quod partes habeat erectiores, & circulari illa de qua egimus, magis fultas. circa medium autem potest videri debilior, quippe quod ellipsis ibi circulo curuetur minus. Ea verò forma, qua mirum in modum delectati sunt Barbari, qui declinante imperio Italiam inuaserunt, & bonam emendatissimamque antiquorum ædificandi ra- tionem deturparunt, ex duobus constat circuli portioni- bus, quamobrem Albertus lib.3. hosce arcus, compositos, appellat. Circinantur autem hoc pacto, diuisa nempe subtensa, in partes tres, easque æquales, ponitur circini pes in altero diuisionum puncto & pars circuli describi- tur, mox in altero puncto circini pede collocato alia cir- culi portio lineatur, quibus arcus ipse integratur. Appel- lant autem tertium acutum, eo quod ex subtensa in tres partes diuisa, arcus non fiat rotundus, sed in acutum an- gulum ex duabus circuli portionibus desinens. Sint igitur muri AC, BD, in quibus v- trinque incumbæ KA, BI. Ducatur itaque sub- tensa horizonti æquidi- stans AB, quæ in tres æ- quales partes diuidatur punctis E,F, tum centris EF, circulorum portio- nes describantur hinc AG, HK, inde verò BG, O 2 IH,

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EXERCISES. 107 is without supports, namely the whole AKLB, which therefore, by its own weight tending downward and resting on both sides on the incumbas, opens out and very easily causes a defect. Of the same kind, for the most part, is that species which is formed as a segment either from the middle or from the lesser ellipse according to the greater diameter. This, however, is more useful, especially around the incumbas, because it has more upright parts, and is more supported than that circular one of which we have spoken. Toward the middle, however, it may seem weaker, since there the ellipse bends less toward the circle. But that form, with which the Barbarians, who invaded Italy as the empire declined, were wonderfully delighted, and by which they disfigured the good and most correct method of building of the ancients, consists of two portions of a circle; wherefore Albertus, book 3, calls these arches composite. They are drawn in this way: the chord being divided into three equal parts, the foot of the compass is placed at one of the points of division and a part of the circle is described; then, with the foot of the compass placed at the other point, another portion of the circle is drawn, by which the arch itself is completed. They call it the acute third, because, when the chord is divided into three parts, the arch is not round, but ends in an acute angle from two portions of a circle. Let therefore AC, BD be walls, on which on both sides rest the incumbas KA, BI. Let the chord AB, equidistant from the horizon, be drawn, and let it be divided into three equal parts by the points E, F; then, with EF as centers, portions of circles are described, here AG, HK, and there BG, IH,

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108 IN MECHAN. ARIST. PROBL. IH, ex quibus arcus totus integratur. Vtilis hæc quidem species est, licet inuenusta, propterea quod haud violenter incumbas vtrinque repellat, & in summo magnis sustinendis oneribus sit apta. Producantur CH in N, DB verò in O, sicque centrum grauitatis AG in L, partis vero BG in M. Quoniam igitur centra hæc ob elatam portionum constitutionem quam proxima lineis AN, BO, fulcimentorum fiunt, maximè sustinêrur, & deorsum potius quam lateraliter incumbas ipsas premunt. Si quid tamen habet vitij, illud est quod grauitatis centra momentum habentia ad interiorem partem versus PQ vim faciant, & nisi partes magno superimposito pondere comprimantur, partes quæ sunt circa HG, sursum pellentes aliquali sibi rectitudine comparata corruunt, facta nempe circa L, M, coniunctarum partium separatione. His hoc pacto explicatis de semicirculari fornice agemus, quæ cæteris omnibus vtilior est, & longè pulcherrima, quamobrem Antiquis Architectis omnibus inprimis admodum familiaris: Esto vanum ABCD, muris vtrinque clausum. Ducatur per sumitates muroru[m] horizonti æquidistans recta AD, hac bifariam secta in E, eodem centro E, spatio verò EA semicirculus describatur AFD, concaua nempe ipsius fornicis

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108 IN MECHAN. ARIST. PROBL. from which the whole arch is composed. This kind is indeed useful, though not beautiful, because it resists pressure from either side without yielding violently, and at the crown it is well suited to bearing great loads. Let CH be extended to N, and DB to O, so that the center of gravity of AG is in L, and that of BG in M. Since therefore these centers, because of the raised arrangement of the parts, lie as close as possible to the lines AN and BO, they act as supports and are most strongly sustained; and when pressed downward they press rather than laterally. If it has any fault, it is this: that the centers of gravity, having leverage, exert force toward the inner part at PQ; and unless the parts are compressed by a heavy superimposed weight, the parts around HG, pushing upward against one another in a certain straightness, collapse, namely, by the separation of the joined parts about L and M. Having explained this in this way, we shall speak of the semicircular vault, which is more useful than all the others and far more beautiful, for which reason it was especially familiar to all the ancient architects: Let the empty space ABCD be enclosed on both sides by walls. Let a straight line AD be drawn through the tops of the walls, parallel to the horizon, and having been bisected at E, with the same center E and with the distance EA, let the semicircle AFD be described, that is, the concave part of the vault itself

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EXERCITATIONES. 109 nicis pars; tum eodem centro, spatio verò EG, circinetur GHI eiusdem fornicis pars conuexa. Post hæc productis lineis BH, CD, in OP, secetur fornix tota in tres æquales partes AGKM, MNLK, NDIL, & KME, LNE iungantur, sint autem partium ipsarum grauitatis centra QRS. Est autem R in ipsa perpendiculari HE. Quoniam igitur partium AGKM, DILN, quæ vtrinq; sunt grauitatis centra QS, in ipsis sunt fulcimentorum lineis OH PD. suâ sponte fulcimentis eas sustinentibus partes ipsæ stabunt. Pars autem media KMNL deorsum vergente per ipsam HE lineam grauitatis centro, si parumper vel incumbæ vel partes vtrinque AGKM, DILN cedant, vtpote quæ à fulcimentis est remotissima, magno impetu suopte pon- dere deorsum feretur. quæ igitur in his semicircularibus fornicibus partes stabiliores sint, quæ verò casibus obnoxiæ, ex his quæ diximus, clarè patet. Cæterùm cur incumbis manentibus fornix stet, ea caussa est, quod partes exteriores GK, KL, LI, maiores sint inferioribus & oppositis AM, MN, NG; quod suprà de- monstrauimus. Si quid autem vitij in hac specie est, illud quidem est, quod summa pars KMNL deorsum vergens magnâ vi partes, quæ vtrinque sunt, repellat, ex quare solidarum partium fit solutio, & inde ruina. Huic difficultati vt occurrerent peritiores Archite- cti, plura excogitârunt remedia. Primum enim parietes hinc inde ita solidos, crassos & firmos faciunt, vt suapte vi resistentes dimoueri loco nequeant, vel parastatas addût vt in figura TX, VY. Præterea & ferrea claui ex incumba in incumbam ducta & vtrinque firmata contrarias partes validissimè connectunt, quæ calcitrantes (ita enim lo- quuntur nostrates Architecti,) fornicis pedes cohibent, & solidum ne soluatur impediunt. qua in specie dubitandû O 3 esset,

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EXERCISES. 109 then, from the same center, with the distance EG, let the curved part GHI of the same vault be described. After this, the lines BH and CD being extended into OP, let the whole vault be divided into three equal parts AGKM, MNLK, and NDIL, and let KME and LNE be joined; and let the centers of gravity of these parts be QRS. Now R lies on the very perpendicular HE. Since therefore the parts AGKM and DILN, whose centers of gravity are QS on both sides, are upon the lines OH and PD of the supports, they will of themselves stand, with the supports holding them up. But the middle part KMNL, tending downward through the line HE to its center of gravity, if it should either lean a little or the parts on either side AGKM and DILN yield, as being farthest from the supports, will be carried downward with great force by its own weight. Which parts, therefore, in these semicircular vaults are the more stable and which more liable to fall, from what we have said is clearly evident. Moreover, the reason why the vault stands when the leaning parts remain is this: that the outer parts GK, KL, LI are greater than the lower and opposite AM, MN, NG, as we demonstrated above. But if there is any defect in this form, it is indeed this: that the upper part KMNL, tending downward, repels the parts which are on either side with great force, from which comes the loosening of the solid parts and, from that, collapse. In order to meet this difficulty, more skilled Architects devised several remedies. First, they make the walls on either side so solid, thick, and strong that, resisting by their own force, they cannot be moved from their place; or they add buttresses, as in figure TX, VY. Moreover, iron clamps, drawn from one leaning part to the next and fastened on both sides, connect the opposing parts very strongly; these, “kicking back” (for so our Architects speak), restrain the feet of the vault and prevent the solid structure from being loosened. In this case one would doubt, O 3, whether,

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110 IN MECHAN. ARIST. PROBL. esset, an optimo loco sita sit clauis, quæ per centrum? Et sanè videtur, quippe quod circa incumbas impetus fiat maior. Ego autem vtilius ibi poni arbitror, vbi puncta q. 5. hoc est, in medio tertiarum illarum partium, quæ vtrinque incumbis insistunt, propterea quod primus impulsus ex media parte quæ impendet, ibi fiat. Rarò tamen boni Architecti eo loco aptare solent, eo quòd eiusmodi claues vel pulcherrimis ædificijs minuant gratiam. Vnde fit vt nunquam satis laudetur Lucianus ille Benuerardus Lauranensis Dalmata, qui nullibi apparentes eas posuit in admirabili illa Vrbini Aula, quam Federico Feltrio, felicissimo æquè & inuictissimo Duci, ædificauit. Tertio denique modo huic infirmitati medentur, vt videre est in sequenti figura, in qua vanum ADBC, muri vtrinque AF, BH, fornix verò FGH. Itaque dum muros exstruunt, arrectarias trabes, robore aliaue materia firmissima, illis inserunt, quales sunt IFK LHM, ea proceritate vt futuri fornicis superent summitatem. Consummato enim fornice, nondum tamen exarmato, transuersariam trabé à summo fornicis dorso parumper eminentem in punctis I, L, arrectarijs trabibus validissimis clauibus connectunt, tum punctis NP, Oq, capreolos trans-

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110 IN MECHAN. ARIST. PROBL. would be, whether the key is placed in the best position, namely through the center? And indeed it seems so, since a greater force is exerted around the supports. But I think it is more useful to place it there where the points q. 5. that is, in the middle of those third parts, which on both sides rest upon the supports, for the reason that the first impulse comes from the middle part that hangs there. Very rarely, however, do good architects customarily place it in that location, because keys of this kind diminish the beauty even of the finest buildings. Hence it comes about that Lucianus Benuerardus Lauranensis, a Dalmatian, can never be praised enough, since he placed them nowhere visible in that admirable Hall of Urbino, which he built for Federico Feltrio, a most fortunate and equally unconquered Duke. In the third and final way they remedy this weakness, as can be seen in the following figure, in which the void ADBC, the walls on either side AF, BH, and the vault FGH. Therefore, while they are building the walls, they insert into them upright beams, of oak or some other very strong material, such as IFK, LHM, of such height that they rise above the summit of the future vault. For when the vault has been completed, but before it has been unshored, they connect a transverse beam, projecting slightly from the top of the back of the vault, at points I, L, with the strongest upright beams by nails; then at points NP, Oq, the braces trans-

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EXERCITATIONES. 111 transuersario, & arrectarijs ferreis, clauis affigunt. Qui- bus ita concinnatis, facta fornicis validâ pressione in G, incumbisque F, H, ad exteriora repulsis, AB spatium non fit maius. Repulsis enim incumbis & muros propelli ne- cesse est, & cum muris ipsas insertas trabes, IK, LM. At va- ricari non possunt, nî secum trahant puncta PQ, quod fie- ri non potest, propterea quod in punctis N, O, validè dis- tineantur. Itaque spatio AB non dilatato nulla fit ipsius fornicis dissolutio, quod vtique à principio ceu propositus finis quærebatur. Sed dicet quispiam, Nonne pende- bit transuersaria trabs in ipsa distractione arrectariorum, pressa in punctis N, O? aut parum dicimus, aut nihil. Cum enim PQ proxima sint punctis FH, quæ cum arrectarijs à muro distinentur, magna in ijs sit vtrobique resistentia. Rebus igitur ita se habentibus cum obseruassent Ar- chitecti, ob enormitatem ponderis fornices in tertia illa parte quæ summa est laborare, quatum ter- tijs vtrinque partibus soliditatis addunt, tan- tundem ex illa parte suprema demere solét, vt videre est in subie- cta figura, in qua par- tes A, B, solidæ & cras- siores, quibus hærent partes, quæ CE, DG crassæ quidem & illæ, tum vero summa EFG, alijs subtilior. Minus igitur grauante ponde- re in F, minor fit ad incumbas pressio, aut si qua fit, à partiu[m] ACE, BDG soliditate haud inualidè sustinetur. Cæte-

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EXERCISES. 111 they attach with crosspieces and iron braces, with nails. When these have been thus fitted together, a strong pressure being applied to the arch at G, and the supports F, H being pushed outward, the span AB does not become greater. For when the supports are pushed, it is necessary that the walls also be forced outward, and with the walls the beams inserted in them, IK, LM. But they cannot bend unless they draw with them the points PQ, which cannot happen, because at the points N, O they are strongly held apart. Therefore, since the span AB is not widened, no dissolution of the arch itself takes place; and this was precisely what was sought from the beginning as the intended end. But someone will say, “Will not the crossbeam hang in the very act of the stretching of the uprights, pressed at the points N, O?” We say, either very little, or nothing. For since PQ are close to the points FH, which with the uprights are separated from the wall, there is great resistance on both sides in them. Since then matters stand thus, and the architects had observed that, because of the enormous weight, the arches labor in that third part which is at the top, as much solid material is added to the upper parts on both sides as is subtracted from that highest part, as may be seen in the figure below, in which the parts A, B are solid and thicker, to which are attached the parts which, CE, DG, are indeed thick, but then the uppermost EFG is thinner than the others. Therefore, since the weight presses less in F, there is less pressure upon the supports, or if there is any, it is not ineffectively sustained by the solidity of the parts ACE, BDG. Cæte-

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112 IN MECHAN. ARIST. PROBL. Cæterùm admonet nos locus, vt aliquid de fornicum dissolutionibus in medium afferamus: caussis enim morborum cognitis, facilius periti medici adhibere solent remedia. Esto enim semicircularis fornix ABC, cuius centrum E, perpendicularis verò quæ per centrum DBE, semicirculi ABC, diameter AEC, incumbæ vtrinq; A, C. Itaque si nulla fiat incumbarum repulsio, stabit fornix; si verò fiat, ruinam faciet. Pellantur itaque ad exteriores partes, vt in secunda figura, H in F, & C in G, ex qua pulsione cum maius fiat spatium quod integro fornice implebatur, iam distractis vtrinq; forniciis partibus no[n] impletur, Diuiditur igitur locus maior factus in tres partes, quarum hinc inde duas replent fornicis partes, tertiam verò quæ media est, replet insertus, ne vacuum detur, aër, vt in figura videre est, in qua solutæ vtrinque fornicis partes HIKF, PMNG, aër autem medius spatium replens IKMN. Diuidantur singuli quadrantes FK, GN, in partes tres, quarum duæ sint hinc inde FQ, GR, & à centris, quæ separatis quadrantibus facta sunt in ST, rectæ ducantur SQV. TRX. Quoniam igitur tertiæ partes vtrinque VIKQ MNRX propria grauitate depressæ, nullum quo sustineantur fulcimentum habent, corruent quidem. Ducantur autem rectæ QI, RM, constituentes cum ipsis QV, RX pares angulos VQI MRX. Itaque centris QR partes QI RM ad infe-

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112 IN MECHAN. ARIST. PROBL. Moreover, the passage reminds us that we should bring forward something about the collapse of arches: for, once the causes of diseases are known, skilled physicians are usually more able to apply remedies. Let there be, then, a semicircular arch ABC, whose center is E; and let the perpendicular through the center DBE of the semicircle ABC, the diameter AEC, rest at both ends on A and C. So if there is no repulsion of the supports, the arch will stand; but if there is, it will fall. Let them therefore be driven toward the outer parts, as in the second figure, H toward F and C toward G; and from this pressure, since the space that was filled by the whole arch becomes greater, it is no longer filled when the parts of the arch are drawn apart on both sides. The greater space thus produced is therefore divided into three parts, of which the two at each side are filled by the parts of the arch, while the third, which is in the middle, is filled by air inserted so that no vacuum may exist, as can be seen in the figure, in which the separated parts of the arch on both sides are HIKF and PMNG, while the air in the middle fills the space IKMN. Let each of the quadrants FK and GN be divided into three parts, of which the two on each side be FQ and GR, and from the centers which were formed by the separated quadrants at ST, let the straight lines SQV and TRX be drawn. Since, therefore, the third parts on each side, VIKQ and MNRX, are pressed down by their own weight and have no support by which they may be upheld, they will indeed fall. Let the straight lines QI and RM then be drawn, making with QV and RX equal angles VQI and MRX. Thus, with QR as centers, the parts QI and RM to the lowe-

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EXERCITATIONES. 113 inferiores partes deuoluentur, fientque QI, RM, vbi QZ, RZ. Si autem QI, RM perpendicularibus quæ à punctis QR ad perpendicularem DE ducuntur, fuerint maiores conuenient alicubi in ipsa perpendiculari, & altera alteram sustinebit; si autem æquales tangent se & nihilominus fiet ruina, si minores nec se inuicem tangent, & nullâ re prohibente deorsum corruent. tangant autem se in puto Z. quo pacto igitur fornices incumbis cedentibus in medio aperti, dissoluâtur & ruinam faciant, existis patet. Ex demonstratis quasi ex consectario habemus fornices quo fuerint crassiores dato pari incumbarum secessu, ruinæ minus esse obnoxios quàm tenuiores, hoc est, maiori aperitione indigere ad ruinam crassiores quam tenuiores, quod licet ex iam dictis resultet, nos tamen clarius ex subjecto schemate demonstrabimus. Esto enim crassioris fornicis pars quide ABCD, tenuioris EFCD circa ide centrum R. Ducatur autem RM, secans CD in G. EF in H AB, in M. Centro igitur G fiet euersio portio- num fornicum MD, HD, Ducantur GA, GE & producta AD in N ipsi AN perpendicularis ducatur GN. quoniam igitur GE cadit in triangulo AGN erit ex 21. propos. lib. 1. elem. GA, maior GE. Corruente igitur maioris fornicis portione MD, recta GA centro G punctum A describet portionem AI, minoris interim ex GE, describente EL, at cadenti angulo A occurrit in perpendiculari IK in puncto I angulus oppositæ portionis, O, ipsi autem E cadenti per EL non occurret punctum P, cadens per Pq eo quod neutrum eorum pertingat ad perpendicularem Ik. Tenuioris ergo forni- P cis

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EXERCISES. 113 the lower parts will roll down, and QI, RM will be formed, where QZ, RZ. But if QI, RM, the perpendiculars which are drawn from the points QR to the perpendicular DE, shall have been greater, they will meet somewhere on the perpendicular itself, and one will support the other; but if they are equal they will touch one another and nevertheless a collapse will occur; if smaller, they will neither touch one another, and with no obstacle they will fall downward. Let them touch, however, at point Z. In what way then vaults supported in the middle by yielding parts in the open center are weakened and cause ruin is plain from the foregoing. From what has been demonstrated, as it were by a corollary, we have that vaults, the thicker they are, with the same separation of supports given, are less liable to ruin than thinner ones; that is, thicker vaults require a greater opening to collapse than thinner ones, which although it follows from what has already been said, we shall nevertheless demonstrate more clearly from the figure below. Let there be, then, the part ABCD of the thicker vault, and EFCD of the thinner, about the same center R. Let RM be drawn, cutting CD at G. EF at H, AB at M. Thus from center G there will be the overturning of the portions of the vault MD, HD. Let GA, GE be drawn, and AD produced to N; to AN draw the perpendicular GN. Since, then, GE falls within triangle AGN, it will be, from proposition 21 of book 1 of the Elements, that GA is greater than GE. Therefore, when the portion MD of the greater vault falls, the straight line GA, from center G, will describe the portion AI; meanwhile from GE, the smaller one describing EL, but the falling angle A meets on the perpendicular IK at point I the angle of the opposite portion, O; but to E, falling by EL, point P will not meet, falling by Pq, because neither of them reaches the perpendicular IK. Therefore the thinner vault P of the vault

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114 IN MECHAN. ARIST. PROBL. cis partes è suis locis auullæ ex eadem aperitione ruinam facient, quod non contingit partibus crassioris. quod sanè fuerat declarandum. Quæritur adhuc, quare grauiores fornices in summis ædificijs non sine vitio fiant? Esto ædificium ABGH, cuius vtrinq[ue] muri ABCD, EFGH, maiorum summitates AD, EH, mediæ murorum partes KL, fornicum summus quidem DIE, medius verò kML. Dico, magis cedere pul- sos muros summos circa DE, quam in medio circa KL. Sunt enim muri BA, GH ceu vectes quidam, quoru[m] extremis partibus à fulcimentis BG remotissimis potentia admouetur, hoc est, ipsius fornicis DIE ad DE incumbans repulsio; lon- gior est autem pars à fulcime[m]e- to ad potentiam AB, ipsa Bk. Data igitur paritate potentia- rum plus operabitur ea quæ in D, illa quæ k. facilius ergo re- pellentur muri in DE quàm in KL. Alia quoque ratio intercedit, siquidem pondus muri superioris ADk, premens inferiorem murum kBC, cum sua grauitate firmiorem, & pulsionibus minus obnoxium reddit. Difficilius enim propellitur id quod graue est qua[m] quod leue, vt nos quæstione 10, demonstrauimus. QVÆSTIO XVII. Quærit Aristoteles, Cur paruo existente cuneo magna scindantur pondera & corporum moles, validaq[ue] fiat impressos IN parua rem magnum negotium. Etenim quæstio hæc claris-

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114 IN MECHAN. ARIST. PROBL. those parts torn from their places by the same opening will cause a collapse, which does not happen in the case of thicker parts. This, indeed, should have been explained. It is still asked why heavier vaults in the upper parts of buildings are made without defect? Let there be the building ABGH, whose walls on either side are ABCD, EFGH, the upper ends of the greater walls AD, EH, the middle parts of the walls KL, and the top of the vault DIE, the middle kML. I say that the upper walls are more yielded to when struck around DE than in the middle around KL. For the walls BA, GH are as it were certain levers, at whose extreme parts, farthest from the supports BG, power is applied; that is, the repulsion of the vault DIE itself bearing upon DE; but the part from the support to the power AB is longer, namely BK itself. Therefore, given equal powers, that which is at D will work more than that at k. Thus the walls will more easily be driven back in DE than in KL. Another reason also intervenes, since the weight of the upper wall ADk, pressing upon the lower wall kBC, makes it firmer by its own weight, and less exposed to blows. For that which is heavy is more difficult to drive forward than that which is light, as we demonstrated in question 10. QUESTION XVII. Aristotle asks, Why, when the wedge is small, are great weights and masses of bodies split, and why is a strong impression made IN a small thing, a great matter. For this question clearly...

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EXERCITATIONES. 115 clarissimorum virorum ingenia magnopere fatigauit. Ex quibus Aristoteles inter veteres, Guid. V bald. inter re- centiores ad vectis naturam (ne quid in Mechanicis ad vectem non reduci putaretur) cuneum ipsum trahere co- nati sunt. Nos autem pro veritate certantes, si in horum sententiam vltrò non transierimus, multa venia digni à non iniquo iudice existimabimur. A- ristotelis mentem clarè & fusè explicat G. V- bald. in Mechan. vbi de Cuneo peculiariter a- git. Esto igitur scindendum quippiam ABCD, Cuneus EFG, cuius pars HFI scissuræ inserta HI, facta igitur vali- da percussione in EG, fiet vt cum EG fuerit in NO, H sit v- bi N, A vbi P, itemque I vbi O, D verò vbi Q & facta erit scissio NSO, toti nempe cuneo EFG, æqualis. Vult igitur Aristoteles, duos in cuneo vectes considerariEF, GF, quo- rum alterius, nempe EF, fulcimentum sit in H, pondus ve- ro in F; alterius autem, hoc est, GF fulcimentum quidem sit in I, pondus verò itidem sit in F. His nequaquam con- sentiens G. V bald. aliam viam ingreditur. Aut enim EHF vectes quidem esse, quorum commune fulcimentum F, potentias verò mouentes in EG. Pondera vtrinque inter fulcimenta & potentias, vbi HI, idemq[ue] esse ac si EF, GF, teorsum à cuneo considerati in puncto F, adinuicem fulti atque distracti pondera pellerent H in NP, I verò in O, Q. Verumenimuerò quoniam cunei angulus non muta- tur, nec vertex ipse centri vllum prorsus præbet vsum, nec eius latera vtrinque distracta ad contrarias partes didu- cuntur, P 2

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EXERCISES. 115 have greatly fatigued the minds of the most distinguished men. Among these, Aristotle among the ancients, and Guid. Vbald. among the more recent writers, have tried to make even the wedge itself serve the nature of the lever (so that nothing in Mechanics might be thought incapable of being reduced to the lever). But we, contending for the truth, if we do not readily go over to the opinion of these men, shall be judged by no unfair judge to deserve much indulgence. G. Vbald. clearly and fully explains Aristotle’s meaning in the Mechanics, where he deals especially with the Wedge. Let something to be split, ABCD, be assumed; let the Wedge EFG be such that its part HFI is inserted into the split HI; therefore, after a strong blow is delivered at EG, it will happen that when EG has become NO, H will be where N is, A where P is, likewise I where O is, and D where Q is; and there will be the split NSO, equal in fact to the whole wedge EFG. Aristotle therefore wants two levers to be considered in the wedge, EF and GF, of which the support of the one, namely EF, is in H, but the weight in F; and of the other, that is GF, the support indeed is in I, but the weight likewise is in F. Not agreeing at all with this, G. Vbald. takes another path. For either EHF are indeed levers, whose common support is F, but the moving forces in EG. The weights on both sides between the supports and the forces, where HI is, are the same as if EF and GF, considered apart from the wedge at the point F, supported and pulling one another, were driving the weights, H into NP, but I into O, Q. But truly, since the angle of the wedge does not change, and the vertex itself affords absolutely no use to the center, nor are its sides, drawn apart on both sides, carried off to opposite parts, P 2

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116 IN MECHAN. ARIST. PROBL. cuntur, vectes in cunco hoc pacto considerare videtur à veritate alienum. Aristotelis autem solutionem falsam esse, clarè patet. quo pacto enim F pellet ex fulcimento Hi- psam ligni partem OS, & idem F ex fulcimento I pellet oppositam partem NS, si inuicem contendentes extremæ vectium partes in F, altera alteri ne quicquam operentur, est impedimento? Et sanè opinionis falsitas inde patet, quòd videamus materiæ partes scissas, in ipso scissionis actu facta distractione à cunei vertice nequaquam tangi. At eiusmodi operationes per contactum fieri nulli est ignotum. Solutio igitur ista meo iudicio, tanto Philoso- pho prorsus videtur indigna. Porrò G. V bald. ijs quæ de diuaricatis vectibus in medium adduxerat non acquiescens alias quærit caussas, cur cuneus minoris anguli validiùs scindat. Idq; ex quodam lemmate demonstrare conatur, figura autem eius ita ferè se habet. Esto cuneus ABC, item alius DEF. Demo- strauit igitur ex assum- pto, quo acutior fuerit angulus BIM, eo faciliùs pondera moueri, & ideo facilius ceu vecte AB moueri pondus I quàm vecte DE pondus Q. In- geniosè quidem. At ma- gnam hæc apud me ha- bent difficultatem. Si e- nimita se habet AB, ad BI, vt DE, ad EQ (ipsæ enim DE, EQ supponuntur æquales) ergo eadem æqualisue poten- tia æqualiter mouebit pondera I & Q. quod ipsi eiusdem demonstrationi prorsus concludit contrarium. Nec meo quidem

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116 IN MECHAN. ARIST. PROBL. seems to consider levers in this way with respect to a wedge, which is far from the truth. But that Aristotle’s solution is false is clearly evident. For how can F from the support H-p cast the part OS of the wood, and likewise F from the support I cast the opposite part NS, if the extreme parts of the levers meeting at F do nothing to one another in the contest, as if they were not an impediment? And indeed the falsity of this opinion is evident from the fact that we see the parts of the material, when split, in the very act of splitting, separated by the wedge’s vertex, and by no means touched. But that such operations are performed by contact is unknown to no one. This solution, therefore, in my judgment, seems utterly unworthy of so great a Philosopher. Moreover, G. V. Bald. not content with the things he had brought forward concerning diverging levers, seeks other causes why a wedge with a smaller angle splits more powerfully. And he tries to demonstrate this from a certain lemma, and its figure is more or less as follows. Let there be the wedge ABC, and likewise another DEF. He has therefore demonstrated from the assumption that the sharper the angle BIM, the more easily weights are moved, and thus the weight I is more easily moved, as by the lever AB, than the weight Q by the lever DE. Ingenious indeed. But I find this to involve a great difficulty. For if AB is to BI as DE is to EQ (for DE and EQ themselves are supposed equal), then the same, or an equal power, will move the weights I and Q equally, which is entirely contrary to his own demonstration. Nor do I myself even

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EXERCITATIONES. 117 quidem iudicio id sequi videtur, propterea quod ex Pappo ea quæ in planis inclinatis mouentur, redigantur ad libram. Ratio enim valde est diuersa, siquidem pondera quæ in planis inclinatis mouentur, certa habent fulcimenta & determinatas tum brachiorum tum ponderum proportiones, quæ omnia in cuneo, nec quidem mente concipi posse, clarè patet. His igitur difficultatibus consideratis, Nos cunei vim, ad alia esse principia referendam pro comperto habemus. Ordimur igitur hoc pacto. Cuneo quidem res diuidi certum est. Cæterùm quæ natura diuidere apta sunt, tria sunt, punctum, linea, superficies. Puncto enim linea, lineâ superficies, superficie autem corpus ipsum diuiditur. quæ omnia à Mathematico absque materia considerantur. De diuisione autem quæ fit ex puncto, nihil agit Mechanicus, qui corporibus quidem vtitur, ad cuius naturam non trahitur punctum, cuius partes sunt nullæ. At non lineis & superficiebus modò corpora diuiduntur, sed etiam corporibus, quod verum est, at ea corpora ad linea- rum & superficierum naturam quodammodo aptari facilè docebimus. Dicimus igitur, duplicem esse Cuneorum speciem, linearem vnam, superficialem alteram. linearem appello, quæ ad lineæ naturam magnopere accedit. Tales sunt orbiculares illæ cuspides, quibus ad perforandum vtimur, & ideo vernaculè Pantirolos vocamus. Acus item sutorij, & cætera quæ non secus ac linea in punctum desinunt, & imaginariam quandam lineam ceu axem in eo puncto desinentem continent. Ad lineam quoque referuntur lateratæ cuspides oblongæ, & subtiles ceu subulæ, claui, enses, pugiones, & his similia, quæ cum adacta validam faciant partium separationem ad cunei naturam nô referre magnæ videretur dementiæ. Et tunc quanto magis corpora hæc ad linearem naturam accedunt, eo ma- gis P 3

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EXERCISES. 117 indeed, by judgment, this seems to follow, because from Pappus those things which move on inclined planes are reduced to the balance. For the reasoning is very different, since the weights which move on inclined planes have fixed supports and determined proportions both of the arms and of the weights, all of which in the wedge, it is clearly seen, cannot even be conceived by the mind. These difficulties, then, having been considered, we have it for certain that the force of the wedge must be referred to other principles. We therefore proceed in this way. It is certain that the wedge divides things. But the things which are naturally fit to divide are three: point, line, surface. For by a point a line is divided, by a line a surface, and by a surface the body itself is divided, all of which are considered by the Mathematician without matter. But as to division which is made from a point, the Mechanic does nothing, who indeed makes use of bodies, but is not led to the nature of a point, whose parts are none. But not only by lines and surfaces are bodies divided, but also by bodies, which is true; yet we shall easily teach that these bodies are in some manner adapted to the nature of lines and surfaces. We therefore say that there are two kinds of wedges, one linear, the other superficial. I call linear that which comes greatly near the nature of a line. Such are those round-pointed instruments with which we use for boring, and therefore in the vernacular we call them Pantirolos. Also tailor’s needles, and the rest which, no otherwise than a line, end in a point, and contain in that point a certain imaginary line as an axis ending there. To the line also are referred elongated, thin, and angular points, like awls, nails, swords, daggers, and the like, which, when driven in, make a strong separation of parts; to refer these to the nature of the wedge would seem sheer madness. And then, the more these bodies approach the linear nature, the more P 3

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118 IN MECHAN. ARIST. PROBL. gis penetrant. Sed & hoc idem in rebus non ab arte, sed ab ipsa natura productis facile est cognoscere. Quis enim non experitur, quàm validè culex, infirmissimum animal, & ea paruitate qua est, hominum & cæterorum animaliu[m], cutes aculeata proboscide penetret? Id vtique non alia de caussa sit, quod ad imaginariæ lineæ subtilitatem quam proximè accedat. Vespæ quoque, Apes, Scorpiones aculeis istis ceu linearibus cuneis vtuntur. Nec refert, vt diximus, vt um laterati sint, ceu subulæ, & claui, vel rotundi & vtrum plura paucioraue latera habeant, dummodo in punctum & aculeatam aciem desinant. Altera porro cuneorum species superficiei naturam sapit, acie siquidem in lineam desinit, quæ superficiei est terminus, quâ. obrem huc ea omnia referuntur, quæ acie ipsâ scindunt, ceu sunt cunei propriè dicti, de quibus hoc loco est sermo, cultra, enses, asciæ, secures, scalpra lata, & cætera eiusmodi, quibus corpora acie scinduntur. Quidam his addunt serras, quibus haud prorsus assentimur. Etenim alia ratione diuidunt, sicut & limæ solent, deterendo enim, nô scindendo ferri, ligni, & marmorum duritiem diuidunt & domant. His igitur consideratis, si daretur ex materia quapiam infrangibili cuneus, qui maximè ad superficiei naturam accederet, vel paruo labore tenacissima ligna validissimè scinderet, & ideo optimè res gladijs illis diuiditur, qui magis ad superficiei naturam accedunt. Ex quibus omnibus, nî fallimur, clarè patet, cur acutiores angulo cunei obtusioribus facilius scindant, quæ quidem ratio longè ab ea distat, ex qua cæteri ferè omnes Cuneum ad vectis naturam referre hactenus contenderunt. Cæterùm vtramque eorum quos diximus, cuneoru[m] speciem solertissima cognouit Natura, & ideo quoniam res vel contusione vel perforatione, vel secatione conficiuntur, triplicem dentium qualitatem dentatis animali- bus

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118 IN MECHAN. ARIST. PROBL. penetrate. But this same thing is easy to recognize in things produced not by art, but by nature herself. For who does not observe how powerfully the gnat, a most feeble animal, and with whatever minuteness it possesses, penetrates the skin of men and other animals with its pointed proboscis? This is certainly for no other reason than that it approaches as closely as possible to the subtlety of an imaginary line. Wasps too, bees, and scorpions make use of such pointed instruments as linear wedges. Nor does it matter, as we said, whether they are sided, like awls and nails, or round, and whether they have more or fewer sides, provided that they end in a point and a sharpened edge. There is also another kind of wedge, which savors of the nature of a surface, since indeed its edge ends in a line, which is the boundary of a surface, wherefore all those things are referred here which cut by the edge itself, such as the wedges properly so called, of which there is discussion in this place, knives, swords, axes, hatchets, broad chisels, and other things of this kind, by which bodies are cut by an edge. Some add saws to these, with which we do not fully agree. For they divide in another way, as files also are accustomed to do; for by wearing away, not by cutting, they divide and subdue the hardness of iron, wood, and marbles. These things, then, being considered, if there were given from some unbreakable material a wedge which approached as nearly as possible to the nature of a surface, it would with little labor most powerfully split the toughest woods, and therefore those swords divide things best which approach more nearly to the nature of a surface. From all these things, if we are not mistaken, it is clearly evident why wedges with sharper angles cut more easily than those with blunter ones; this reason is far removed from that by which almost all the others have hitherto claimed to refer the wedge to the nature of the lever. Moreover, nature herself most skillfully recognized both kinds of wedges that we have mentioned, and therefore, since things are accomplished either by pounding or by piercing or by cutting, she assigned a triple quality of teeth to toothed animals

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EXERCITATIONES. 119 bus dedit, Molares, qui & Maxillares appellantur, quibus cibus contunditur, Canini, quibus fit perforatio, Anteriores, quibus cibus scinditur, quos ideo πιμνης, id est, secantes appellant Græci. Molares KK, CaniniL, L, Temniciseu secantes M. Cuneus orbicularis linearisque AB, in quo axis linea est, ad cuius naturam accedit AB cuneus superficialis CD, accedens ad superficiei naturam, quam vitro imaginamur EFGD, in aciem cunei desinentem GD, Lateratus linearisque cuneus, clauus HI. Cunei autem omnes dupliciter sunt efficaces, vel enim malleo, vt in ijs fit, quibus ligna scinduntur & scalpris fieri solet, adiguntur, vel impulsu & pressione, vt in gladijs fit, pugionibus, cælatorum scalpris, subulis, & cæteris eiusmodi. Quidam etiam sunt, qui licet mallei ictu non adigantur, malleum coniunctum habent, ceu sunt secures, ligones, Asciæ, & his similia, quæ ex percussione semetipsa scindendis rebus inserunt & validè penetrant. De vi autem & efficacia ictus seu percussionis hic supersedemus aliquid, ea de re, in sequenti quæstione verba facturi. Multa hîc addere potuissemus ad Cochleam spectantia, quippe quòd Cochlea cuneus sit Cylindro inuolutus, qui quidem ad mallei, sed vectis virtute sibi adiunctâ, validissimè operatur, & sexcentis inseruit vsibus. Veruntamen cùm de hac specie egregiè disserat G. V baldus, con-

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EXERCISES. 119 those given by the teeth called Molars, or Maxillaries, by which food is crushed; Canines, by which piercing is done; Front teeth, by which food is cut; for which reason the Greeks call them πιμνης, that is, cutters. Molars KK, Canines L, L, incisors M. Wedge circular and linear AB, in which the axis is a line, to whose nature AB corresponds; the superficial wedge CD, approaching the nature of a surface, which we imagine in glass EFGD, ending in the edge of the wedge GD; the sidewise and linear wedge, a nail HI. But all wedges are effective in two ways: for either they are driven in by a hammer, as happens in those things by which wood is split and as is usually done with chisels, or by thrust and pressure, as happens with swords, daggers, engravers’ chisels, awls, and other things of that kind. There are also some which, although they are not driven in by a hammer blow, have a hammer joined to them, such as axes, hoes, adzes, and similar tools, which by striking insert themselves into things to be split and penetrate strongly. As for the force and efficacy of a blow or striking, we here refrain from saying anything, since we shall speak of that matter in the following question. We could have added much here concerning the screw, since a screw is a wedge wrapped around a cylinder, which indeed works most powerfully with the force of the hammer, but joined to it by the virtue of the lever, and has served in countless uses. However, since G. V. Baldus treats of this species excellently, con-

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IN MECHAN. ARIST. PROBL. consultò hanc disputationem omittimus; idque hac quo- que de caussa, quod nihil de cochlea, ac si eam non nouis- set, locutus sit Aristoteles. Possumus autem in actu scissionis, quæ cuneo sit, a- liâ tamen ratione vectem considerare, nempe non in cu- neo quidem, sed in ipsa re quæ scinditur. Esto enim quip- piam scissile ABCD, cui alteri extremita- tum, puta BD, cuneus adigatur EFG, fiatq; scissio per longitudi- nem secundum linea[m] EH. facta igitur ex cunei ingressu partiu[m] separatione B, expelletur in I, D ve- ro in K. sient igitur materiæ scissæ partes AIBH, CKDH, ceu duo vectes, quorum hinc inde in corpore ipso fulci- menta L, M potentiæ vtrinque dilatantes BD, pondus ve- ro materiæ resistentia, in separationis loco vbi N. Duca- tur NL, quanto itaque BN maiorem habebit proportio- nem ad LN, eo faciliùs resistentia quæ in N, superabitur. Mutatur aute[m] assiduè in ipsa scissione fulcimentum, & cu[m] fulcimento ipsa proportio. Pertingente enim scissione in O, fulcimetum sit in P. quo casu scissura est faciliot, quip- pe quod maiorem habeat proportionem BO ad OP, quâ BN adNL. Hoc autem experiuntur materiarij, qui primis ictibus, securiculâ nondum probè adactâ, & nondum fa- ctâ notabili scissione difficultatem sentiunt, mox facta ia[m] separatione facillima paullatim sit materiæ totius separa- tio, Hoc idem & nos absque cunei vsu experimur, cum ba- culum aut quippiam tale manibus diductis scindimus. à principio enim difficultatem sentimus, deinde ex ea quâ diximus proportione scissio ipsa fit apprime facilis. Vti- mur

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IN MECHAN. ARIST. PROBL. we purposely omit this discussion; and this for the further reason, that Aristotle said nothing about the screw, as if he had not known it. But in the act of splitting, when it is done by a wedge, we may consider the lever in another way, namely not in the wedge itself, but in the thing that is being split. Let there be some splittable thing ABCD, to one of its ends, say BD, let a wedge EFG be driven in, and let the split be made lengthwise along the line EH. When, therefore, through the entrance of the wedge the parts are separated at B, D will be expelled to I, and D to K. Let, then, the parts of the split material be AIBH, CKDH, as it were two levers, whose supports on either side in the body itself are L, M, the forces spreading BD apart on both sides, and the weight the resistance of the material, at the place of separation, where N is. Let NL be drawn; accordingly, the greater the proportion that BN has to LN, the more easily will the resistance which is at N be overcome. But in the splitting itself the support continually changes, and with the support the proportion itself changes. For when the split reaches O, the support will be at P. In which case the splitting is easier, since BO has a greater proportion to OP than BN has to NL. This is what material workers observe; for in the first blows, when the little axe has not yet been properly driven in and no notable split has yet been made, they feel difficulty, but once the separation has been made, the division of the whole material becomes gradually very easy. We too experience the same thing without the use of a wedge, when we split a stick or something of that kind with our hands pulling apart. For at first we feel difficulty, then, because of the proportion we mentioned, the actual splitting becomes remarkably easy. We use

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EXERCITATIONES. 121 mur etiam vecte cuncato ad scindendum & aperiendum: adacto enim scissuræ cuneo, idque manu malleoue, tum ab altera extremitate presso, valida sit ex vectis vi cōtinui corporis separatio. Ma- teria scissilis AB scalpru[m] ceu vectis cuneatus CD, cuius fulcimentum E, pondus verò vbi C, po- tentia vbi D, quo casu quo maior est proportio DE ad EC, eo est ipsa scissio leuior & facilior, QVAESTIO XVIII. Quærit hic Aristoteles, Cur per Trochleas ab exigua potentia in- gentia moueantur pondera? DE Trochlea Pappus, & veteres: inter recentiores e- gregiè admodum, vt omnia examinauit in Mechani- cis G. V baldus. Nos tamen interim post clarissimos illos viros aliquid quod nouitatem & subtilitatem sapiat, de nostro penu promemus. Et sanè inuentis quidem addere res est facilis, at quod inuentis addas inuenire haud adeo facile. Sed nos primum Philosophi ipsius dicta ad trutina[m] reuocemus. Ita autem quæstionem proponit; Cur si quis- piam Trochleas componens duas, insignis duobus, ad se inuicem iunctis contrario ad Trochleas modo circulo fu- nem circumduxerit, cuius alterum quidem caput tigno- rum appendatur alteri, alterum verò Trochleis sit innixu[m] & à funis initio trahere cœperit, magna trahit pondera, li- cet imbecillium fuerit virium? Obscurissima expositio, & nî res esset vulgò per se nota, deque ea Vitruuius & Mechanici non egissent, diffi- cile vtique esset ex eius verbis sensum assequi. Q Tigna

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EXERCISES. 121 also with a wedge-shaped lever for splitting and opening: for when the splitting wedge has been driven in, and that by hand or mallet, then by pressing at the other end, a strong separation of the continuous body is made by the force of the lever. Ma- terial to be split AB with the chisel like a wedge-shaped lever CD, whose fulcrum E, the weight however where C, the power where D; in which case the greater the proportion of DE to EC, the easier and more lightly is the splitting itself effected. QUESTION XVIII. Here Aristotle asks: Why, by means of pulleys, are enormous weights moved by a small force? On the pulley, Pappus, and the ancients; among the more recent writers, G. V. Baldus has examined everything in the Mechanical Problems with remarkable thoroughness. Yet for the present, after those most distinguished men, we shall bring forth something from our own store that has a touch of novelty and subtlety. And indeed, to add to discoveries is an easy thing; but to add something to discoveries, to discover it, is by no means so easy. But first let us recall the philosopher’s own words to the balance. He states the question thus: Why is it that if someone, arranging two pulleys, marked with two, joined to one another in opposite fashion, should pass a rope around them in a circle, with one end of it hung from the beams and the other resting on the pulleys, and should begin to pull from the beginning of the rope, he draws great weights, although he may be of weak strength? This is an obscurissime explanation, and unless the matter were commonly known of itself, and Vitruvius and the mechanicians had not spoken of it, it would indeed be difficult to grasp the meaning from his words. Q Tigna

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122 IN MECHAN. ARIST. PROBL. Tigna sanè vocasse videtur ea ligna, quæ à Vitruvio Rechamidicuntur, in quibus nempe ipsi inseruntur orbiculi. Etsi de tignis eiusmodialiud quippiam sentire videatur Picolomineus. Græca lectio pro tignis habet , id est, ligna; item vbi Leoniceni versio legit, ad se inuicem iunctis, textus habet , hoc est, inuicem ex opposito concurrunt. Certè locum totum ita redderem: Cur si quis duas Trochleas fecerit, in duobus lignis sibi ex opposito concurrentibus, eisque Trochleis circumposuerit funem, cuius alterum caput alteri lignorum sit annexum, alterum verò Trochleis cohæreat, vel apponatur. Si quis alterum funis principium trahat, magna trahat pondera, etsi trahens potentia sit exigua? Nos verbis figuram, & figurâ verba ipsa elucidabimus. Sint duo ligna ex opposito concurrentia, in quibus Trochleæ, hoc est, orbiculi AB, funis ductarius DABC, cuius alterum caput religatum est ligno trochleæ A, vbi est C. Trochlea A loco stabili commendata, vbi E. Pondus alteri ligno Trochleæ appensum F. Tracto itaque fune DABC, eleuatur & trahitur pondus F. Ex quibus clarè patet, Philosophu[m] proposuisse Trochleam duobus tantum orbiculis munitam, quod vtique satis erat ad explicationem. Inquit autem, faciliùs vecte quâ manu pondus moueri. Trochleam vero (id est, orbiculum; ita enim est intelligendum) esse vectem, aut vectis virtute operari. Ita autem videtur argumentari. Si vnicâ Trochleâ plus trahitur quàm manu, multo faci ius & velocius id fiet duobus, quibus plus, vt ipse ait, quàm in duplici velocitate pondus leuabitur. Summa dictorum est, ex multiplicatione orbiculorum pondus ipsum imminui, & minori difficul- tate

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122 IN MECHAN. ARIST. PROBL. He seems certainly to have called those beams tigna, which Vitruvius calls Rechamidicuntur, in which, namely, the pulleys themselves are inserted. Although Piccolomini seems to think something different about tigna of this kind. The Greek reading has for tigna, , that is, ligna; likewise where Leoniceno's version reads, ad se invicem iunctis, the text has , that is, they meet one another from opposite sides. Certainly I would render the whole passage thus: Why, if someone has made two pulleys in two beams meeting one another from opposite sides, and has passed a rope around them, one end of which is attached to one of the beams, but the other adheres to the pulleys, or is applied to them: if someone draws the other end of the rope, why does he draw great weights, although the force drawing be slight? I shall elucidate the words by the figure, and the words themselves by the figure. Let there be two beams meeting one another from opposite sides, in which are the pulleys, that is, the little wheels AB, the hauling rope DABC, one end of which is tied to the beam of pulley A, where C is. The pulley A is fixed in a stable place, where E. The weight F is hung from the other beam of the pulley. Therefore, by pulling the rope DABC, the weight F is raised and drawn. From these things it is clearly evident that the Philosopher proposed a pulley furnished with only two little wheels, which was certainly sufficient for the explanation. But he says that it is more easy to move a weight by a lever than by hand. Now the pulley, that is, the little wheel; for so it must be understood, is a lever, or acts by the power of a lever. And thus he seems to reason. If with a single pulley more is drawn than by hand, this will be done much more easily and more quickly with two, by which, as he says, the weight will be raised with more than double speed. The sum of what has been said is this: by the multiplication of the little wheels, the weight itself is diminished, and with less difficulty

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EXERCITATIONES. 123 tate leuari, quod sanè verum est. Nos tamen nonnulla co[n]siderabimus. quod ait, vecte facilius moueri pondera quam manu, semper non est verum. Si enim vectis pars quæ à fulcimento ad manum breuior fuerit illâ, quæ à fulcimento ad pondus difficilius vecte pondus mouebitur quam manu. Idem quoque accidet, si eo modo vecte vtamur, quem obseruat Guidus V bald. Tract. de Vecte prop. 3. Posita nempe inter fulcimentum & pondus sustinente potentiâ. Præterea quod asseruit Aristoteles, Trochleas ad vectem reduci, verum quidem est, sed aptius dixisset ad libram, etenim vectis vtcunque à fulcimento diuiditur. Libra verò quod & orbiculis ex centro accidit, semper bifariam. Ad hæc videtur ille ad orbiculorum multiplicitatem Trochlearum vim referre. Si enim, ait, vnicâ Trochleâ pondus facile trahitur, id multo validius pluribus fiet. Veruntamen non absolutè ex orbiculorum multiplicatione id fieri ita ostendemus. Sint duæ oppositæ lineæ rectæ, vtpote trabes AB, CD, inuicè æquidistantes & ipsæ stabiles: superiori tres appendantur orbiculi ex puctis E, F, G, nèpe ML, PQ, TV, inferiori auté duobus punctis IH, nempe NO, RS. Erunt igitur invniuersum quinque, indatur pereos funis ductarius KLMNOP QRSTVX, ex cuius extremitate pendeat pondus X, Q 2 Tra

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EXERCISES. 123 to be raised, which is certainly true. However, we shall consider some points. What he says, that weights are moved more easily by a lever than by the hand, is not always true. For if the part of the lever from the fulcrum to the hand is shorter than that from the fulcrum to the weight, the weight will be moved by the lever with more difficulty than by the hand. The same thing will also happen if we use the lever in the manner observed by Guidus V. Bald. Tract. de Vecte prop. 3, namely, when the sustaining power is placed between the fulcrum and the weight. Moreover, what Aristotle asserts, that pulleys are reduced to the lever, is indeed true, but he would have said more aptly to the balance; for a lever, however it is divided from the fulcrum, is not always divided into two equal parts. A balance, however, and also wheels from the center, are always divided in halves. Besides this, he seems to refer the force of pulleys to the multiplicity of the wheels. For if, he says, by a single pulley a weight is drawn easily, this will be done much more powerfully by several. Nevertheless, we shall show that this does not happen simply from the multiplication of the wheels. Let there be two opposite straight lines, namely beams AB, CD, equally distant from one another and themselves fixed: to the upper let three pulleys be hung from the points E, F, G, namely ML, PQ, TV; to the lower, however, two from the points IH, namely NO, RS. Thus there will be in all five. Let the carrying rope KLMNOPQRSTVX be passed through them, from the end of which let the weight X hang. Q 2 Tra

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124 IN MECHAN. ARIST. PROBL. Trahatur funis in K. Dico ex multiplicatione orbiculoru[m], trahenti pondus nequaquam minui. Sint autem orbiculorum diametri, LM, NO, PQ, RS, TV, applicetur potentia in S. Erit igitur ad hoc vt sustineat æqualis ponderi X, orbiculi enim TV semidiametri sunt æquales. Transferatur potetia in q, & ita deinceps donec perueniatur in K, vbi funis ipsius est principium, Idem est igitur seruata semper semidiametrorum æqualitate ac si potentia quæ est in K, applicata intelligatur in T vel in V. vbicunque enim collocetur, ponderi erit æqualis. Nihil igitur rebus ita dispositis, orbiculorum multiplicatio ad facilitatem operatur. Alia itaque ratio quærenda est, quam non satis explicasse videtur Aristoteles. Probabimus autem, nullam ex superioribus orbiculis fieri ponderum imminutionem, sed totam vim in inferioribus consistere. At nos interim quippiam quod ad rem faciat, proponamus. Esto punctum A, cui rectæ appendantur lineæ BAC, diuisæ quidem in A, sit autem lineæ BA caput B, ipsius verò CA caput C. Modò intelligantur vnitæ in A, sitque vnicæ linea à puncto A ceu funiculus dependens BAC; Appendatur capiti B pondus B. Capiti vero C, pódus C, inter se æqualia. Potentia igitur in A, duo sustinebit pondera BC. Pondera verò ex æqualitate æque- ponderabunt. Quod si B potentia dicatur sustinens pondus C, aut C potentia sustinens pondus D, vel duæ potentiæ inter se æquales, nihil refert. Vtcunque enim id sit, fiet æquilibrium. Habemus igitur existis ad sustinendum pondus ex superiori parte appen-

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124 IN MECHAN. ARIST. PROBL. Let the cord be drawn in K. I say that by the multiplication of the pulleys, the weight on the drawing side is by no means diminished. Let the diameters of the pulleys be LM, NO, PQ, RS, TV; let the power be applied at S. It will therefore be required, in order that it may support the equal weight X, for the semidiameters of the pulleys TV to be equal. Let the power be transferred to q, and so on successively until it comes to K, where the beginning of the cord itself is. The same thing therefore remains, the equality of the semidiameters being always preserved, as if the power which is in K were understood to be applied at T or at V. For wherever it is placed, it will be equal to the weight. Thus, with things disposed in this way, the multiplication of the pulleys contributes nothing to ease of operation. Some other reason must therefore be sought, which Aristotle does not seem to have explained sufficiently. But we shall prove that no diminution of the weights is produced by the upper pulleys, but that all the force resides in the lower ones. Yet meanwhile let us propose something relevant to the matter. Let there be a point A, to which straight lines BAC are attached, divided indeed at A; let the line BA have B as its head, and let C be the head of CA. Now let them be conceived as united at A, and let there be one line hanging from point A like a cord, BAC. Let a weight B be attached to the head B, and likewise a weight C to the head C, equal to one another. The power in A will therefore support the two weights BC. But the weights, by their equality, will counterbalance one another. And if B be called the power supporting the weight C, or C the power supporting the weight D, or two powers equal to one another, it makes no difference. For however that may be, equilibrium will result. We thus have from these things what is needed for supporting the weight from the upper part appen-

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EXERCITATIONES. 125 appensum potentiam requiri ipsi ponderi æqualem. Animo posthæc concipiatur alia recta linea DEF, cuius integra longitudi si extenderetur, esset DE, EF. Appendatur in E pondus Exquale alteri ponderum B vel C, sint autem duæ potentiæ pondus E sustinentes D, F. Vtraque igitur dimidium sustinebit ponderis E, sed potentia quæ sustinebat pondus B, in C erat ipsi B æqualis, vbi appensio ponderis erat in superiori parte in A, hîc autem, vbi appensio est in parte inferiori, vtraque potentia dimidium sustinet appensi ponderis. Videmus igitur illam appensionem quidem pondus nullatenus imminuere, hanc verò pondus ipsum, bifariam diuisum, sustinentibus potentijs impartiri. Hæc in lineis, Mathematicâ vsi abstractione, considerauimus, nunc verò eadem mechanicè perpendamus. Sit igitur punctum A, vt in sequenti figura clauus paxillusue, cui appensus funiculus BAC, & funiculi capitibus pondera BC, sit quoque anulus D, per quem traiectus funiculus EDF. Anulo autem coiunctum pondus G. His igitur ita constitutis, eadem demonstrabuntur quæ superius, nempe oportere vt fiat æquilibrium B, C, esse æqualia, tum potentias, quæ sunt in EF pondus G inter eas diuisum sustinere. Porrò volentes Mechanici funi- Q 3

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EXERCITATIONES. 125 it is required that a power equal to the weight itself be applied. After this, let another straight line DEF be conceived in the mind, whose entire length, if extended, would be DE, EF. Let there be attached at E a weight equal to the other weights B or C; and let there be two powers D, F sustaining the weight E. Therefore each will sustain half of the weight E; but the power which sustained the weight B, in C was equal to B itself, whereas where the attachment of the weight was in the upper part at A, here however, where the attachment is in the lower part, both powers sustain half of the attached weight. We see, therefore, that the former attachment indeed diminishes the weight in no way, but the latter distributes the weight itself, divided into two parts, to the sustaining powers. These things, using mathematical abstraction, we have considered in lines; now, however, let us examine the same mechanically. Let there therefore be the point A, as in the following figure, a nail or peg, to which is attached the cord BAC, and at the ends of the cord the weights BC; let there also be the ring D, through which the cord EDF is passed. To the ring, moreover, is joined the weight G. Having thus arranged these things, the same results will be demonstrated as above, namely that it is necessary for equilibrium to be made, B, C, to be equal, and then the powers which are in EF are to sustain the weight G divided between them. Further, mechanical men willing funi- Q 3

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126 IN MECHAN. ARIST. PROBL. funiculos circa paxillum, & anulum ad attollenda & de- primenda pondera mouere incommodè illis vtique suc- cedebat, clauo & anulo motum difficilem facientibus. Quamobrem vt difficultati occurrerent, ad locum claui clauo ipsi orbiculum circumposuerunt, & anuli itidem loco orbiculum aptauerunt. Hæc autem agentes rei i- psius naturam non mutauerunt, sed sibi, vt diximus, ex or- biculis maximam commoditatem atq; facilitatem com- parârunt. Ex his principijs tota Trochlearum ratio pendet, quæ tamen alia quoque consideratione in idem tenden- te examinari potest, quod quidem fecere veteres, & ipse, qui veteres optimè imitatus est, Guid. V baldus. Vidimus vtique nos, à potentia quæ est in B, pondus par sustineri in C, Potentiam autem quæ est in E dimidiu[m] sustinere ponderis quod est in G. Nos igitur ijsdem insi- stentes adiecta libra, vecteue, bifariam diuiso rem ipsam ex subiecto diagrammate lucidiorem faciemus. Esto linea quædam stabilis ceu trabshorizonti æ- quedistans AB, cui in A funiculus annectatur AC, cuius extremum C vecti cuidam alligetur CD, in medio diuiso vbi E, tum alteri vectis eiusdem extremitati D, funiculus nectatur DG, & à puncto E pondus appendatur F. puta li- brarum mille, Tum puncto G in medio vectis HI, funis re- ligetur DG, & ex altero vectis extremo alligato fune HK commendetur loco stabili in K, & ab alio capite vectis vbi I ad medium vectis MN, vbi L, funis annectatur IL, tum ex vectis capite M, funis commendetur MO, loco stabili in O, & alteri capiti N, funis NP, qui alligetur medio ve- cti QR in P, & ex Q, funis QS. Commendetur loco stabili in S, & alteri vectis extremo R funis alligetur RT, cui quidem potentia sustinens applicetur in T. Dico igitur, rebus

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126 IN MECHAN. ARIST. PROBL. funicles around a peg, and a ring for raising and lowering weights, it used to happen that they moved them awkwardly, the peg and ring making the motion difficult. For this reason, in order to meet the difficulty, they placed a small wheel itself at the place of the peg, and likewise fitted a small wheel in the place of the ring. But in doing these things they did not change the nature of the thing itself, but, as we said, they secured for themselves the greatest convenience and ease from the little wheels. From these principles the whole theory of pulleys depends, which however may also be examined from another consideration tending to the same end, as indeed the ancients did, and as did Guid. V baldus, who most excellently imitated the ancients. We have indeed seen that by a power which is at B a weight equal to the half is sustained in C, and the power which is at E sustains half of the weight which is at G. Therefore, relying on the same principles, with the addition of a balance or lever, divided in two, we will make the matter itself clearer from the diagram below. Let there be some fixed line, as it were a beam parallel to the horizon, AB, to which at A a cord is attached AC, whose end C is tied to a certain lever CD, divided in the middle where E is; then to the other end D of that same lever a cord is attached DG, and from the point E a weight is hung F, say of a thousand pounds. Then at point G, in the middle of lever HI, the rope DG is tied, and from the other end of the lever, with the rope HK tied, it is secured to a fixed place at K; and from the other end of the lever, where I is, to the middle of lever MN, where L is, a rope is attached IL, then from the end M of the lever the rope MO is secured to a fixed place at O, and to the other end N the rope NP, which is tied to the middle of lever QR at P, and from Q the rope QS. It is secured to a fixed place at S, and to the other end R of the lever the rope RT is tied, to which the sustaining power is applied at T. I say therefore, the things

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EXERCITATIONES. 127 rebus ita dispositis, potentiam in T ita se habere ad pondus F, vt vnum ad sexdecim, hoc est, in proportione esse subsexdecupla. Sunt autem hic vectes quatuor inferiorum cubiculorum loco, CD, HI, MN, QR, quorum centra E, G, L, P. quoniam enim A hoc est, C, vna cum potentia G, hoc est, D, sustinet pondus F alterum ponderis dimidium sustinebit C, alteru[m] vero D. erunt igitur vtrinque libre quin- gentæ. Tum potentia in K, hoc est, in H, vna cum potentia in L, hoc est, in I sustinebunt quingenta. Quare vtraq[ue] ducenta quinquaginta, sed hoc totum bifariam diuiditur inter potentias, O, id est, M, & P, id est H. erunt igitur vtrinque centum viginti quinque. Ea autem summa iteru[m] bifariam diuiditur, hoc est, inter potentias S, id est, Q & T, id est, R, quare vtraque sustinet sexaginta duo cum dimidio. Sed numerus iste ad Millenarium ita se habet vt vnum ad sexdecim. Hinc colligimus, pondus totum inter loca stabilia diuidi, nempe A, K, O, S, & ipsam potentiam quæ sustinet in T, & locis ipsis stabilibus quindecim partes integri ponderis, potentia verò T sextam decimam tantùm

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EXERCISES. 127 the matters being thus arranged, the power at T is thus related to the weight F, as one to sixteen, that is, it is in a subsexdecuple proportion. There are here, however, four beams in place of the lower chambers, CD, HI, MN, QR, whose centers are E, G, L, P. For since A, that is, C, together with the power G, that is, D, supports the weight F, the other half of the weight will be supported by C, the other indeed by D. Therefore there will be five hundred pounds on each side. Then the power at K, that is, at H, together with the power at L, that is, at I, will support five hundred. Wherefore each will be two hundred and fifty; but this whole amount is divided in two between the powers O, that is, M, and P, that is, H. Therefore there will be on each side one hundred and twenty-five. But this sum is again divided in two, that is, between the powers S, that is, Q, and T, that is, R, wherefore each supports sixty-two and a half. But the number is related to a thousand as one to sixteen. Hence we gather that the whole weight is divided among the fixed places, namely A, K, O, S, and the very power which supports at T, and at the fixed places themselves fifteen parts of the whole weight, but the power T only the sixteenth part.

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128 IN MECHAN. ARIST. PROBL tantùm commendari. Itaque si ex puncto V appendere- tur AB, in X potentia, quæ in X sustineret mille, minus sexaginta duo cum dimidio, quod quidem à potentia in T sustinetur; quod si alius adderetur orbiculus, & fierent quinque, potentia in T sustineret trigesimam secundam partem integri ponderis, hoc est, dimidium librarum se- xaginta duarum cum dimidio, nempe triginta & vnam cum quarta parte, si item textus adderetur, potentia in T sexagesimam partem sustineret integri ponderis, hoc est, libras quindecim & s/8 libræ vnius. Vnde patet clarè pon- deris diminutionem fieri ex orbiculis inferioribus, non autem ex superioribus, superiores autem addi non neces- sitatis quidem, sed commoditatis gratiâ: neque enim abs- que superioribus vnico ductario fune fieri posset attractio & ponderis ipsius eleuatio. Hactenus igitur nobis isthæc de Trochlex natura & vi post alios, considerasse sit satis. QVÆSTIO XIX. Dubitat Philosophus, Cur si quis super lignum magnam imponat securim, de super q[uo] magnum adjiciat pondus, ligni quippiam quod curandum sit, non diuidit; si verò securim extollens percutiat, illud scindit, cum alioquin multo minus habeat ponderis id quod percutit, quam illud quod superiacet & premit? P[ræ]terat Aristoteles, nî fallimur, rem breuius & vniuersalius proponere. Scilicet cur motus ponderi addat pondus & efficacius ex motu quam ex immoto pondere mota res operetur. Soluit autem. An, inquiens, ideo sit, quia omnia cum motu fiunt, & graue ipsum grauitatis ma- gis assumit motum, dum mouetur quam dum quiescit? Incumbens igitur connatam graui motionem non moue- tur, motum verò & secundum hanc mouetur & secun- dum

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128 IN MECHAN. ARIST. PROBL only be recommended. Thus if AB were suspended from point V, the power at X, which at X would sustain a thousand, minus sixty-two and a half, which indeed is sustained by the power at T; but if another wheel were added, and there were five, the power at T would sustain the thirty-second part of the whole weight, that is, half of sixty-two and a half pounds, namely thirty-one and a quarter; likewise if a textum were added, the power at T would sustain the sixtieth part of the whole weight, that is, fifteen pounds and 5/8 of a pound. Whence it is clearly apparent that the diminution of the weight is made by the lower little wheels, but not by the upper ones; and the upper ones are added not out of necessity, but for convenience: for without the upper ones the pulling could not be done with a single cord, nor the lifting of the weight itself. So far, then, let it suffice for us to have considered these matters concerning the nature and force of the trochlea, after others. QUESTION XIX. The philosopher asks why, if someone places a great axe upon wood, and on top of it adds a great weight, it does not split any part of the wood that should be cut; but if, raising the axe, he strikes it, it does split it, although that which strikes has much less weight than that which lies above and presses down upon it? Aristotle, unless we are mistaken, has omitted to propose the matter more briefly and more universally. Namely, why motion adds weight to weight, and why a thing moved acts more effectively by motion than by unmoved weight. He answers thus: whether it be because all things are done with motion, and heavy things themselves assume the nature of heaviness more through motion, while they are being moved, than while they are at rest? Therefore, when the movement inherent in a heavy body is pressing down, it is not moved; but the motion, and according to this, it is moved both according to this and according to

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EXERCITATIONES. 129 dum eam quæ est percutiétis? Hæc præclarè quidem, cætera autem, quæ de cunco iterat, nempe ad vectem eius lo- perationem referri superius confutauimus. Porrò effe- ctus huius, de quo agitur, disputatio illuc spectat, videli- cet ad cadentium atque proiectorum naturam. Ad maio- rem autem rei euidentiam hæc addimus. Esto libra AB, cu- ius centrum C, libra- ta æqualibus ponde- ribus DE, apponatur ponderi E pondus F, item ponderi D pon- dus G ipsi ponderi F æquale, æquilibrabit tidem, Modò non apponatur simpliciter pondus G sex ex H in lancem A dimittatur, tunc sanè non æquilibrabit, sed libram deprimet. Duo enim in pondere dimisso con- siderantur pondera; naturale scilicet, & quod motu ipsi moto, ponderi est acquisitum. Itaque quo motus fuerit maior, puta si cadat ex I, grauitas ex maiori motu fiet ma- ior. quod vtique efficacius fieret si pondus G non dimit- tetur modo remoto prohibente, sed proijceretur. Tunc enim tria concurrerent, grauitas naturalis, grauitas ac- quisita ex naturali motu, & ea quæ naturali adjicitur ex violentia. Pondus igitur securi impositum & securis ipsius naturalis grauitas naturali tantum grauitate operantur, & ideo minus efficaciter. Huc autem ea ferè pertinent quæ nos à principio de duobus centris retulimus, natura- lis nempe grauitatis, & acquisitæ. Cæterùm cur mallei & securis ictus sit violentissi- mus, ideo sit quod non ex vnico neque duplici, sed ex tri- plici grauitate operetur. Esto enim securis A, cuius manu- brium AB, brachium vero securi vtentis BC, erit igitur C R locus

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EXERCISES. 129 Do you strike that which is to be struck? These things, indeed, are very clear; but the others, which are repeated from the trunk, namely, are above refuted by referring them to the lever’s operation. Moreover, the discussion of this effect, of which we are speaking, tends in that direction, namely to the nature of falling and projected bodies. For a clearer understanding of the matter we add the following. Let there be a balance AB, whose center is C, balanced by equal weights DE; if to the weight E there be added the weight F, and likewise to the weight D the weight G, equal to the weight F itself, it will remain in equilibrium. But if the weight G be not simply added, but be let fall from H into the pan A, then assuredly it will not remain in equilibrium, but will depress the balance. For in a falling weight two weights are to be considered: namely, the natural weight, and that which has been acquired by motion in the moving body. Thus, the greater the motion has been—for example, if it falls from I—the greater will the gravity become from the greater motion; which would certainly be made more effective if the weight G were not merely let fall by removing the obstacle that prevents it, but were projected. For then three things would concur: natural gravity, gravity acquired from natural motion, and that which is added to the natural by violence. A weight therefore placed upon an axe, and the natural gravity of the axe itself, operate only by natural gravity, and therefore less effectively. To this belong nearly the things which we mentioned at the beginning concerning the two centers, namely of natural gravity and acquired gravity. Moreover, the reason why the blow of the hammer and axe is most violent is this, that it acts not from one gravity alone, nor from two, but from three gravities. For let there be an axe A, whose handle is AB, but the arm of the axe-user BC; thus C will be the place of

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IN MECHAN. ARIST. PROBL. locus vbi humero brachium iungitur, motus ipsius centrum, attollit autem securim is qui percutit, & retro adscapulas reducens totis viribus ex centro C securim vibrat, portionem circuli describens ADE ictumque faciens in E. Vires igitur acquirit securis, tum ex naturali grauitate, cadens ex D, in E, tum ex proprio pondere, tum etiam ex violentia eidem à percutiente impressa. Fiant autem motus tam naturalis quàm violentus eo validiores, quo maius est spatium, quo res mota mouetur, idque præcipuè cum violentia ipsam secundat naturam. Itaque maior sit ictus in E quàm in F, & in F maior quàm in D. Item violentius feriret percutiens, si manubrium esset longius, puta BG. Tunc enim maior esset circulus GH, & motus tum prolixior, tum velocior. quo igitur longiora habet brachia is qui securi malleoue vtitur, data virium paritate, ex eadem ratione validius percellit. Est autem securis, vel malleus cuneatus, vel cuneus malleatus manubrio insertus. An autem operetur efficacius cuneus malleo percussus, aut cum manubrio motus, vt fit in lecuri, data aciei & ponderis æqualitate, difficile est determinare. Certè validius, & certius fieri scissionem ex cuneo & malleo, ea ratio est, quod cuneus adactus, nec inde remotus eam interim seruat, quam antea fecerat partium separationem, quod

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IN MECHAN. ARIST. PROBL. the place where the arm joins the shoulder, is the center of its motion; but he who strikes lifts the axe and, drawing it back toward the shoulder-blades with all his strength, swings the axe from the center C, describing the arc ADE and making the blow in E. The axe therefore acquires force, both from its natural gravity, falling from D to E, and from its own weight, and also from the violence impressed on it by the striker. Both natural and violent motion are made the stronger the greater the space through which the moved thing is moved, and this especially when violence seconding nature itself. Thus the blow at E is greater than that at F, and that at F greater than that at D. Likewise the striker would strike more violently if the handle were longer, say BG. For then the circle GH would be greater, and the motion both longer and swifter. Therefore the longer the arms has he who uses an axe or hammer, given equal strength, the more powerfully does he strike for the same reason. Now an axe is either a wedge with a hammer, or a wedge fitted with a hammer-shaped handle. But whether the wedge struck by the hammer works more effectively, or when moved with a handle, as in the axe, with edge and weight being equal, is difficult to determine. Certainly the splitting is made stronger and surer with the wedge and hammer, for this reason: when the wedge is driven in, and not removed from there, it preserves in the meantime that separation of the parts which it had previously made, which

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EXERCITATIONES. 131 quod quidem securi non accidit, quæ adacta ad nouam percussionem faciendam extrahitur. Hoc etiam consideramus, securis in circulo motum, ex A in D, esse videndum, id est, non secundum naturam, sursum enim fertur quod est graue, ex D verò in F mixtu[m]: magis autem ad naturalem accedere qui sit ex F in E. Tardior ergo ex A in D, velocior ex D, in F, velocissimus ex F in E; quædam quæ ad hanc rem faciunt, egregiè considerat Guid. V bald. in calce Tractatus, De Cunco; ipsum consule. Ad hæc succurrit nobis pulcherrima quæstio. Dubitari enim potest, vtrum ictus ex ense efficacior sit à parte quæ est circa aciem, aut circa medium ensem, vel prope manubrium capulumue; etenim hinc inde sunt rationes. Esto quidem ensis AB, cuius capulus A, spiculum verò B, centrum grauitatis C, pars capulo proxima D. Librato itaque gladio tres fiunt circulorum portiones BE, CF, DG, quæritur quo loco ictus sit validior, nempe in E, in F, vel in G. Videtur validiorem futurum in E, quippe quod ex maiori semidiametro AB, maioris sit circuli portio BE, & ideo velocior motus ex B in E. Contra efficaciorem futurum apparet in F, propterea quod ibi ex centro C totius fiat grauitatis impressio, fieri autem validissimum in G, licet ibi motus sit tardior inde videtur, quod si consideretur ensis vt vectis, cuius fulcimentum est A, potentia premens in B, ponderis vero loco resistentia rei quæ percutitur in D. Maior est autem proportio BA, ad AD, quam BA ad AC, & ideo violentior fieri pressio ex ictu in D, quâ in C. Hisce hoc pacto consideratis, putarem ictum efficaciorem fieri in F ex medio C, quam ex extremis & oppositis partibus EG. Licet enim in B velocitas sit maior, deest ibi pondus. Si enim ensis iterum vt vectis consideretur, e- R 2 runt

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EXERCISES. 131 which indeed does not happen with the axe, which, once driven in to make a new stroke, is withdrawn. We also consider this: that the motion of the axe in a circle, from A to D, is to be regarded, that is, not according to nature, for what is heavy is carried upward; from D however to F mixed: but more nearly natural is that which is from F to E. Therefore slower from A to D, faster from D to F, fastest from F to E; certain things which contribute to this matter are excellently considered by Guid. V. Bald. at the end of the Treatise, De Cunco; consult him. To these points there comes to our aid a very beautiful question. For it may be doubted whether the blow from a sword is more effective at the part near the edge, or around the middle of the sword, or near the hilt or handle; for there are arguments on both sides. Let there be a sword AB, whose hilt is A and whose point is B, the center of gravity C, and the part nearest the hilt D. When the sword is balanced, therefore, three portions of circles are formed, BE, CF, DG; the question is at what place the blow is strongest, namely in E, in F, or in G. It seems that it will be strongest in E, since, because of the larger semidiameter AB, BE is a portion of a larger circle, and therefore the motion from B to E is faster. On the other hand, it appears more effective in F, because there the impression is made from the center C of the whole weight; but it seems most forceful in G, although there the motion is slower. This appears from the fact that if the sword is considered as a lever, whose fulcrum is A, the pressing power is in B, while in place of a weight the resistance of the thing struck is in D. Now the proportion BA to AD is greater than BA to AC, and therefore the pressure from the blow is more violent in D than in C. Having considered these things in this way, I would think that the blow is made more effective in F from the middle C than from the extreme and opposite parts EG. For although the speed is greater in B, the weight is lacking there. For if the sword is again considered as a lever, there will be

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IN MECHAN. ARIST. PROBL. runt AB, duo fulcimenta sustinentia pondus in C, vbi grauitatis est centrum. Si igitur paria fuerint spatia BC, CA, in B erit dimidium ponderis C, quantum ergo velocitate præualet ictus in B, tantu[m] ponderis amittit. D verò plus quidem de pondere participat, sed velocitatis habet minimum, in C verò velocitas est medio-cris, tota tamen ipsius ex grauitatis centro ponderis fit impressio. Quidam, quod huc pertinet, vt exacie ipsa quæ longius à capulo abest, violentissimum facerent ictum, Argentum viuum, quod sui naturâ grauissimum quidem est & mobilissimum in canali à manubrio ad verticem excauato infundunt, quo in gladij descensu ad verticem velocissimè delato illuc transfert grauitatem totam, quare tum velocitate tum grauitate concurrentibus ictus fit violentissimus & longè validissimus. QVAESTIO XX. Dubitatur, Cur statera quacarnes ponderantur, paruoappendiculo, magnatrutinet onera, cum alioqui tota, dimidiata existat libra, altera vero parte sola sit statera? Soluit Philosophus, inquiens, stateram simul, & vectem esse & libram, ipsius verò libræ centra seu fulcimenta esse

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IN MECHAN. ARIST. PROBL. runt AB, two supports sustaining the weight in C, where the center of gravity is. If therefore the spaces BC, CA are equal, in B there will be half of the weight of C; therefore as much as the blow in B prevails by speed, so much does it lose in weight. D indeed shares more of the weight, but has the least speed; in C, however, the speed is moderate, yet the whole impression of the weight is made from its center of gravity. Certain persons, in order to make the blade that part which is farther from the hilt strike more violently, pour quicksilver, which by its nature is indeed the heaviest and most mobile, into a channel hollowed out in the blade from the handle to the point, by which, in the descent of the sword, the weight is carried most swiftly to the point, and there transfers all its gravity; wherefore, with both speed and weight coming together, the blow is made most violent and far more powerful. QVAESTIO XX. It is asked why the steelyard with which meats are weighed, with a small counterweight, supports heavy loads, when otherwise the whole, being halved, is a pound, while on the other hand only one part is the steelyard? The Philosopher solves it, saying that the steelyard is at once both a lever and a balance, but that the centers or supports of the balance itself are

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EXERCITATIONES. 133 esse ibi vbi sit suspensio. Pondera verò hinc inde in lance & appendiculo, loco scilicet æquipondij, appendiculo succedente. Reducit autem demonstrationem ad ea quæ statuit ipse Mechanica principia; nempe ad circulum & circuli virtutem. Ait igitur, appendiculum licet parui pô- deris sit, ideo maiori ponderi virtute æquari, quod lon- gius à centro, hoc est, ab ipso fulcimento sistatur. quic- quid tamen sit, stateram esse vectem, res est exploratif- sima. Esto igitur statera AB, cuius appendiculum cur- rens F, fulcimentum cen- trumue C, lanx quæ cate- na suspenditur E spatium à loco fulcimenti ad ap- pendiculum CF. quod ve- rò à fulcimento ad cate- nam, ex qua lanx appen- ditur AC. Intelligatur autem & aliud fulcimentum D, sit- que maius spacium AD, quam AC. Porrò ita se habeat pondus in E ad appendiculi F pondus, vt CF spatium, ad spatium AC, quo casu seruata, permutatim, ponderum & brachiorum proportione, fiet equilibrium. Si autem pon- deribus ita constitutis iterum suspendatur in D, non fiet æquilibrium, propterea quod minor sit proportio DF ad DA, ea quæ est FC ad CA. Minor ergo est proportio FD ad DA, quam ponderis E ad pondus F, & idcirco facta suspensione præualebit pondus E ponderi F. Itaque vt ite- rum fiat æquilibrium, necesse est iteru proportiones bra- chiorum seu spatiorum proportionibus ponderum æqua- re. Transferatur igitur (lancis interim immoto pondere) ipsum appendiculum in B, fiatque vt FC ad CA, ita BD ad DA. Stabit autem iterum statera ad eam redacta quam dixi- R 3

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EXERCISES. 133 there to be the suspension point. The weights, however, on this side and that in the balance, and the appendage, that is to say in the place of equilibrium, the appendage being substituted. But he reduces the demonstration to those things which he himself lays down as mechanical principles; namely, to the circle and the power of the circle. He says, therefore, that although the appendage is of little weight, it is for that reason equal in force to a greater weight, because it is placed farther from the center, that is, from the fulcrum itself. Whatever the case may be, that the balance is a lever is a most certain matter. Let there therefore be the balance AB, whose moving appendage is F, the fulcrum or center C, and the scale, which hangs by a chain, E; the space from the place of the fulcrum to the appendage CF, and that from the fulcrum to the chain from which the scale hangs AC. Let there also be understood another fulcrum D, and let the space AD be greater than AC. Moreover, let the weight in E be so related to the weight of the ap- pendage F, as the space CF is to the space AC; in which case, the proportion of the weights and arms being preserved and reciprocated, equilibrium will result. But if, the weights having thus been arranged, it is again suspended at D, there will not be equilibrium, because the proportion of DF to DA is less than that of FC to CA. Therefore the proportion of FD to DA is less than that of the weight E to the weight F, and for that reason, the suspension having been made, the weight E will prevail over the weight F. Thus, in order that equilibrium may again come about, it is necessary again to equate the proportions of the arms, or spaces, with the proportions of the weights. Let the appendage itself therefore be transferred to B, the weight of the scale remaining unmoved in the meantime, and let it be so that, as FC is to CA, so BD is to DA. Then the balance, being reduced to this condition, will again stand as I have sai- R 3

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134 IN MECHAN. ARIST. PROBL. diximus brachiorum & ponderum permutatam proportionem. Nos stateris vtimur ex duplici fulcimento, altero propiori, altero à lance seu loco, vbi lanx appenditur, remotiori, illa grauiora appendimus pondera, & non per vncias & libras, sed per libras tantum & selibra ponderamus; & hoc stateræ latus eo quod minus minutè sit diuisum; vulgo nostrates Grossum, hoc est, rude & crassum appellant. Aliud verò, cum fulcimentum est loco appensionis lancis vicinius, & per libras, selibras & vncias diuiditur, quo quidem minora appendimus pondera, eò quod exquisitioré contineat diuisionem, subtile dicunt. Rectè igitur dicebat Philosophus, in statera plures esse libras, quanquam & ea quoque de caussa dici possit, quod, quot sunt appendiculi, è loco in locum translationes, totidem ex proportionum variatione fiant libræ. Et hoc quidem sensisse videtur Aristoteles. Possemus & alio modo statera vti, nempe stabili appendiculo, mobilem autem fulcimento. Esto enim statera AB, cuius lanx C appensa in A, appendiculum verò stabile D, appensum in B, Apponatur ipsi lanci C, pondus E. Vnicum ergo fiet corpus CEABD constans ex lance, libra & ponderibus. Habet ergo hoc totum grauitatis suæ centrum, quod quidem vbi sit est ignotum. Ex illo autem inuento si corpus totum appendatur, partes æqueponderabunt. Appendatur autem, puta in G, sit aute[m] grauitatis centrum in H. Quoniam igitur H est extra fulcimentum G, declinabit stateræ pars GA, centro G per cir-

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134 IN MECHAN. ARIST. PROBL. we have said that the proportion of the arms and the weights is changed. We use balances with a double support, one nearer to the pan or place where the pan is hung, the other farther away; we hang heavier weights on that side, and we weigh not by ounces and pounds, but only by pounds and half-pounds; and this side of the balance, because it is divided less minutely, our people commonly call the Gross, that is, the rude and coarse side. Another, however, when the support is nearer the place where the pan is hung, and is divided into pounds, half-pounds, and ounces, by which indeed we hang smaller weights, because it contains a more exact division, they call the subtle side. Therefore the Philosopher rightly said that there are more pounds in a balance, although this too may be said for the reason that, as often as there are weights hung and translations from place to place, so many pounds arise from the variation of the proportions. And this indeed Aristotle seems to have thought. We could also use the balance in another way, namely with a fixed weight and a movable support. Let there be, then, a balance AB, whose pan C is hung at A, and the fixed weight D is hung at B. Let a weight E be placed upon the pan C. Thus there will be one body CEABD, made up of the pan, the balance, and the weights. Therefore this whole has its center of gravity, though where it is is unknown. But if, once this is found, the whole body be suspended, its parts will balance equally. Let it be suspended, for example, at G, and let the center of gravity be at H. Since therefore H is outside the support G, the part GA of the balance will incline, with center G by cir-

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EXERCITATIONES. 135 circuli portionem HI, à centro grauitatis in ipsa descensione descriptam. Si autem grauitatis centrum fuerit vbi K, eo quod ibi quoque sit extra fulcimentum G, descendet pars GB, describente interim grauitatis centro K, circuli portionem KL. Itaque si stateram totam eum ponderibus trahamus pellamusq[ue], vltro citroq[ue], immoto appendiculo erit aliquando fulcimentum in ea linea perpendiculari vel loco ipso, vbi est grauitatis centrum, quo casu statera stabit, & tunc ita erit diuisa, vt fiat brachiorum & ponderum eadem ratio, ordine permutato. Hic autem modus ideo non est in vsu, quod molestum sit libram seu stateram cum ponderibus vltro citroque transferre, quæ difficultas commodè appendiculi mobilitate vitatur. QVAESTIO XXI. Quæritur, Cur facilius dentes extrahunt Chirurgi, denti forcipsis onere adiecto, quam si sola manu vtantur? Responde Philosophus, An quia ex manu, magis quam Rex dentiforcipe lubrius elabitur dens? An ferro id potius accidit quam digitis, quoniam vndique dentem non comprehendunt, quod mollis facit digitorum caro; adhæret enim & complectitur magis. Hæc secunda ratio videtur primam destruere, & contrarium prorsus sententiæ, quæ in problemate proponitur, asserere. Si Græca ad verbum reddas ita habent: An magis ipsa manu labile est ferrum, & ipsum vndique (dentem nempe) non complectitur, caro autem digitorum cum mollis sit, adhæret magis, & vndique congruit. Certè vt sententia non sit contraria propositioni, Græca versio ita videtur concinnanda: Vel magis è manu labitur, mollis enim est digitorum caro, ferrum autem circumplectitur, & hæret magis. quicquid sit, Græcam lectionem contrarium ei quod quæri- tur,

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EXERCITATIONS. 135 the part HI of the circle, described by the center of gravity in the very descent. But if the center of gravity should be at K, because there too it is outside the support G, part GB will descend, while meanwhile the center of gravity K describes the portion KL of the circle. Therefore, if we draw the whole balance with weights to and fro, back and forth, with the suspension point unmoved, the support will at some time be on that perpendicular line, or at the very place where the center of gravity is, in which case the balance will stand still, and then it will be divided in such a way that the ratio of the arms and the weights becomes the same, the order being reversed. But this method is not in use for this reason, that it is troublesome to move the scale or balance with weights back and forth, a difficulty conveniently avoided by the mobility of the suspension point. QUESTION XXI. It is asked: Why do surgeons more easily extract teeth when the forceps has weight added to it, than when they use only the hand? Answer, philosopher: Is it because from the hand, more than from the tooth-forceps, the tooth slips away more readily? Or does this happen rather to the iron than to the fingers, since they do not grasp the tooth on all sides, because of the soft flesh of the fingers; for it adheres and embraces more. This second reason seems to destroy the first, and to assert something altogether contrary to the opinion proposed in the problem. If you translate the Greek literally, it reads thus: Or is the iron more slippery from the hand itself, and does it not on all sides embrace the tooth, while the flesh of the fingers, since it is soft, adheres more and fits around on all sides. Certainly, so that the sense may not be contrary to the proposition, the Greek version seems to need to be arranged thus: Or rather it slips from the hand, for the flesh of the fingers is soft, but the iron embraces it round about and clings more. Whatever the case, the Greek reading [states] the opposite of what is being asked,

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136 IN MECHAN. ARIST. PROBL. tur, affirmare certum est. Picolomineus, Ideo, inquit, digitorum caro mollis minus aptè extrahit, quod dentem totum comprehendere non potest, quod ferrum ob suam duritiem & constantiam commodissimè facit. Sensum ex mente reddidit, quod ex verbis non poterat. Subiungit denique Aristoteles, An quia dentiforcipes sint duo contrarij vectes vnicum habentes fulcimentum, ipsam scilicet instrumenti partium connexionem. Hoc igitur ad extractionem vtuntur **, vt facilius moueant. Figuram hoc pacto proponit Philosophus. Esto dentiforcipis alterum quidem extremum vbi A, alterum autem quod extrahit B, vectis vbi ADF, alter vectis, vbi BCE, fulcimentum verò CGD connexio vbi G. Dens autem pondus: vtroque igitur vecte B, & F simul comprehendentes mouent, Hæc ille. At tamen rem ipsam subtilius considerantibus aliter videtur habere, ac ipse asserat. Etsanè dentisforcipis brachia vectes esse, quorum commune fulcimentum est in ipso centro vbi vertebra, nemo negauerit. Dentem autem esse pondus, ego quidem absolute non dixerim. Pondus aute[m] hîc proprie est ipsa dentis durities, cuius resistentia eo facilius superatur, quo maior est proportio brachiorum à manu ad vertebram, ad partem illam quæ à vertebra est ad dentem. At dentis ex constrictione fractioni hil facit prorsus ad extractionem: id tamen operatur brachiorum longitudine dentiforceps, quod valide ex vectium oppositorum videntes constringit & extractioni commodum reddit & facilem. Neque enim totus Dentiforceps hic ceu vectis vnicus operatur, quod fit in forcipibus quas Tenaleas vocamus, quibus è tabulis claui reuelluntur, qua de re nos quæstione 6. verba fecimus. Quo pacto aute[m] dentis

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136 IN MECHAN. ARIST. PROBL. therefore, it is certain to affirm. Picolomineus says, “Therefore the soft flesh of the fingers less suitably extracts, because it cannot grasp the whole tooth, which iron, by reason of its hardness and firmness, does most conveniently.” He has rendered the sense from the mind, which could not be done from the words. Aristotle adds, finally, “Or is it because tooth-forceps are two contrary levers having a single fulcrum, namely the connection of the parts of the instrument itself? Therefore they are used for extraction, so that they may move more easily.” In this way the Philosopher presents the figure. Let one extremity of the tooth-forceps be at A, and the other, which extracts, at B; let the lever be ADF, the other lever BCE, the fulcrum or connection CGD, at G. The tooth is the weight. Thus, grasping B and F at the same time with both levers, they move it. So he. Yet to those considering the matter itself more subtly, it seems to be otherwise than he asserts. And indeed no one would deny that the arms of the tooth-forceps are levers, whose common fulcrum is in the very center, where the hinge is. But I would not absolutely say that the tooth is the weight. For the weight here is properly the hardness of the tooth itself, whose resistance is more easily overcome the greater the proportion of the arms from the hand to the hinge is to that part which lies from the hinge to the tooth. But the constriction of the tooth does nothing at all for breaking it in extraction; rather, what the tooth-forceps accomplishes by the length of its arms is that, by the force of opposite levers, it squeezes and makes extraction convenient and easy. For the whole tooth-forceps does not here operate as a single lever, as happens in those forceps which we call Tenaleae, by which nails are torn out of boards, concerning which matter we spoke in question 6. But in what manner the tooth...

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EXERCITATIONES. 137 dentis ex Dentiforcipe extractio ad vectem reducatur, subtilius est perpendendum, neque enim res est in propatulo. Dicimus igitur, tum dentem ipsum, tum dentiforcipem vectes esse, varia tamen ratione & satis sane diuersa. Dens enim sit vectis eius nempe naturæ quæ fulcimentum habet in angulo, quo casu ipsius Dentiforcipis partiu[m], quibus Dens apprehenditur, ea quæ longior est potentiæ mouentis loco succedit, breuior vero fulcimentum, facit, Dentis vero resistentia ponderis vices refert. Esto enim dens qui- dem A, cuius diameter BC, longitudo vsque ad extremas radices CD, pars dentiforcipis breui- or CG, longior BG. Fit ergo vectis BCD, habens fulcimentum in C. Den- te igitur apprehenso in BC, & manu dentiforcipe ceu vecte ad inferiora compressio C, fit fulcimentum centrum- ue. Stante enim puncto C, trahente autem potentia quæ est in B, fit motus ipsius B, per circuli portionem BE, radicis vero D, fit motus per DF, & inde ipsius dentis extractio facilis. Quibus consideratis vt rem ad proportiones quatenus fieri potest reducamus, dicimus, quo maior fuerit proportio BC, ad CD, hoc est, partis vectis, quæ à fulcimento ad potentiam ad eam quæ à fulcimento est ad pondus, eo facilius fieri dentis auulsionem, quod vtique demonstrandum fuerat. Porro quod in calce quæstionis addit Philosophus, Dentes commotos facilius manu extrahi quam instrumento, nulla ratione probat. Ego autem arbitror, huc pertinere ea verba, quæ superius habentur, videlicet fer- rum S

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EXERCISES. 137 If the extraction of a tooth from the tooth-forceps is to be reduced to the lever, it must be considered more carefully, for the matter is not obvious at first sight. We say therefore that both the tooth itself and the tooth-forceps are levers, though in different and indeed quite different ways. For the tooth is a lever of that kind which has its support at an angle; in which case the longer part of the tooth-forceps, by which the tooth is grasped, takes the place of the moving power, the shorter part, however, of the support, and the resistance of the tooth serves in the place of the weight. For let the tooth be A, whose diameter is BC, and whose length to the extreme roots is CD; let the shorter part of the tooth-forceps be CG, the longer BG. Thus there is formed the lever BCD, having its support at C. The tooth therefore being grasped at BC, and the hand compressing the tooth-forceps as a lever downward at C, C becomes the support and center. For the point C being fixed, but the power acting at B, the motion of B itself takes place through the segment of the circle BE, and of the root D through DF; and from this the extraction of the tooth is easy. These matters having been considered, so that we may reduce the thing to proportions as far as possible, we say that the greater the ratio of BC to CD, that is, of the part of the lever which is from the support to the power to that which is from the support to the weight, the more easily will the tooth be torn out, which indeed had to be demonstrated. Moreover, what the Philosopher adds at the end of the question, namely that teeth once loosened are more easily drawn out by hand than by an instrument, he proves in no way. But I think that those words above belong here, namely: iron S

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138 IN MECHAN. ARIST. PROBL. rum quidem non vndique dentem comprehedere, quod mollis facit digitorum caro, quæ idcirco adhæret & com- plectitur magis. An autem ita sit, alij videant, nobis enim digito rem ostendisse fuerit satis. QVÆSTIO XXII. Hîc quærit Aristoteles, Cur nuces absque ictu facile confringuntur instrumentis quæ ad eum faciunt vsum, & hoc licet multum aufe- ratur virium, cessante motu & violentia, quod accidit dum mal- leo confringuntur. Addit præterea, citius fieri confractionem graui, & duro instrumento ferreo vide- licet quàm ligneo. Soluit, inquiens, id fieri quod instrumentum duobus vectibus constet, coëuntibus in connexione seu verte- bra, & idcirco eo violentius fieri confractionem, quo mi- nus est spatium à nuce, quæ frangitur, ad vertebram. ma- ius verò quod à vertebra ad extremitates, quæ confrin- gentis manu comprimuntur. Ait igitur, & id quam oppo- site, vim ex vectibus ictus loco succedere & idem operari. Esto igitur instrumentum, de quo agimus CDBF, ex duo- bus vectibus constans, quorum alter CAF, alter vero DABver- tebra seu connexio A locus v- bi nux frangitur K, manubria vero BF. quo igitur prolixiores erunt AB, AF, breuiores vero ACAD, violentius fiet co- fractio. Erit autem nucis resistentia loco ponderis A, ful- cimentum BF loco potentiæ. Itaque nî maior sit propor- tio potentiæ ad resistentiam, quam brachij à potentia ad fulcimentum ad eam partem quæ à fulcimento est ad nu- cem, non fier confractio. eo autem magis superabit, quo maior

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138 IN MECHAN. ARIST. PROBL. indeed cannot grasp the tooth all around, because the flesh of the fingers is soft, and for this reason adheres and em- braces it more. But whether this is so, let others see; for us it will be enough to have shown the matter by the finger. QUESTION XXII. Here Aristotle asks why nuts are easily broken without a blow by instruments made for this use, and this though much force is taken away, the motion and violence ceasing, as happens when they are broken with a hammer. He adds moreover that the breaking is done more quickly with a heavy and hard iron instrument than with a wooden one. He solves it, saying that this happens because the instrument consists of two levers, joined together at a connection or joint; and therefore the breaking is effected all the more violently, the less the distance from the nut that is broken to the joint. The distance is greater from the joint to the ends, which are compressed by the hand of the one breaking it. He says therefore, and in a way opposite, that the force from the levers takes the place of a blow and produces the same effect. Let then the instrument we are speaking of, CDBF, be made of two levers, of which the one is CAF, the other DAB, the joint or connection A, the place where the nut is broken K, the handles BF. The longer therefore AB and AF are, and the shorter AC and AD are, the more violently the breaking will be done. The resistance of the nut will be at the place of the weight A, the support BF at the place of the power. Therefore if the proportion of the power to the resistance is not greater than that of the arm from the power to the support, to that part which is from the support to the nut, the breaking will not take place. But it will overcome the more, the greater

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EXERCITATIONES. 139 maior fuerit pars vectis quæ à potentia ad fulcimentum. Quod autem addit Aristoteles, eo maiorem fieri vectium elevationem, hoc est, instrumenti aperitionem, quo magis nux quæ frangitur, fuerit propior fulcimento, hoc est, ipsi vertebræ, facile ostenditur ex conuersa 21. propos. lib. 1. Elem. si enim ab extremitatibus vnius lineæ ad easdem partes constituantur duæ lineæ maiores con- currentes in angulo, & ab ijsdem extremitatibus duæ a- liæ minores, quæ intra triangulum à maioribus constitu- tum cadant, maiorem angulum continebunt. At talis est angulus qui fit in instrumento, cum partes vectis à verte- bra adnucem fuerint breuiores. magis ergo dilatantur vectes, & magis dilatati magis comprimuntur, magis au- tem compressi validius frangunt, quod dixerat Aristo- teles. Cæterum & illud quod scribit, ex grauiori & durio- ri materia instrumentum citius fractionem facere, quam ex leuiori & minus dura, ex parte quidem materiæ verum est, nec pertinet ad proportionem, quæ sane in huiusmodi instrumentis formæ ferè habent rationem. Nos hisce in- strumentis non vtimur. Sunt autem similia instrumentis illis, quibus figuli cretaceas pilas ad chirobalistarum vsum facere & efformare consueuerunt. QVÆSTIO XXIII. P[er]Vlcherrimam proponit hoc loco Philosophus con- templationem, eamque ad mixtos motus pertinêtem. Mixtorum autem motuum speculationem antiquis Me- chanicis fuisse tum vtilem tum etiam familiarem, norunt ij qui norunt quæ de lineis spiralibus Helicisue, cyssoidi- bus, conchoidibus & alijs eiuscemodi scripta & contem- plata reperiuntur, quibus tum ad duarum mediarum pro- portio- S 2

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EXERCISES. 139 when the greater part of the lever is from the power to the fulcrum. But what Aristotle adds, that the elevation of the levers, that is, the opening of the instrument, becomes greater the nearer the nut which is being broken is to the fulcrum, that is, to the vertebra itself, is easily shown from the converse of Proposition 21, book 1 of the Elements. For if from the extremities of one line, toward the same parts, two greater lines are set, meeting in an angle, and from the same extremities two other smaller lines, which fall within the triangle formed by the greater ones, they will contain a greater angle. But such is the angle produced in the instrument, when the parts of the lever from the vertebra to the nut have been shorter. Therefore the levers are more spread apart, and being more spread apart are more compressed; and being more compressed they break more strongly, as Aristotle had said. Moreover, what he writes, that an instrument made of heavier and harder material produces fracture more quickly than one made of lighter and less hard material, is true indeed with respect to the matter itself, and does not pertain to proportion, which in instruments of this kind is for the most part regarded as a matter of form. We do not use these kinds of instruments. But they are similar to those instruments with which potters used to make and shape clay balls for the use of the chirobalistae. QUESTION XXIII. Here the Philosopher proposes a most beautiful consideration, and one pertaining to mixed motions. And those who know what has been written and contemplated by the ancients concerning spiral lines, the helix, conchoids, and other things of this kind, know that the study of mixed motions was then both useful and familiar to the old mechanicians, by which matters were sought both for the proportion of two means- S 2

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14 IN MECHAN. ARIST. PROBL. portionalium inuentionem, tum ad circuli quadrationem vti solent. Quod autem hîc quærit Aristoteles, ita se habet. Cur si duo extrema in Rhombo puncta duabus ferantur lationibus, haudquaquam æqualem vtrumque eorum pertransit rectam, sed multo plus alteram? Item cur quod super latus fertur, minus pertranseat quam ipsum latus. Illud enim diametrum pertransire certum est, hoc vero maius latus, licet hoc vnica, illud autem duabus feratur lationibus? Difficile hoc intellectu prima fronte, & sane admirabile, itaque intentam contemplationem requirit. Nos primo cum Aristotele, rem totam explicabimus, tum aliquid fortasse non poenitendum nostro de promptuario proferemus. Esto itaque Rhombus ABCD, cuius latera AB, BD, DC, CA, diametrorum maior AD, minor BC, secantes se inuicem in puncto seu figuræ centro K. Sunt aute[m] ex ipsius Rhombinatura latera æqualia & parallela, Angulorum vero qui maiori diametro opponuntur, recto maiores, qui vero minori minores. His igitur consideratis, intelligatur punctum A moueri peculiari & simplici motu, per lineam AB, ab A versus B, & eodem te[m] poremoueri totam lineam AB, versus lineam DC, hac tamen lege, vt semper eidem DC feratur parallela, & eius alterum extremorum feratur per AC, alterum vero per BD, Intelligatur etiam punctum B moueri eodem tempore proprio motu, eoque simplici, per eandem rectam BA, versus A, & cum eadem, vt dictum est, mota; ferri ver- sus

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14 ON THE MECHANICS OF ARISTOTLE’S PROBLEMS they are accustomed to use both for the discovery of proportional numbers and for the quadrature of the circle. But what Aristotle here asks is as follows. Why, if the two extreme points in a rhombus are carried by two motions, does not each of them traverse an equal straight line, but one much more than the other? Likewise, why does that which is carried along one side traverse less than the side itself? For it is certain that the one traverses the diagonal, but this one the greater side, although the latter is moved by a single motion, whereas the former by two motions? This is difficult to understand at first sight, and indeed remarkable, and therefore requires close consideration. We shall first explain the whole matter with Aristotle, then perhaps bring forth something not to be regretted from our own storehouse. Let there be, then, the rhombus ABCD, whose sides AB, BD, DC, CA, and the greater diagonal AD, the lesser BC, cut one another at the point K, or the center of the figure. Now from the very nature of the rhombus the sides are equal and parallel, and the angles opposite the greater diagonal are greater than a right angle, while those opposite the lesser are less. These things therefore being considered, let point A be understood to move by its own peculiar and simple motion along the line AB, from A toward B, and at the same time let the whole line AB be moved toward the line DC, under this condition, however, that it is always carried parallel to the same DC, and that one of its extremities is carried through AC and the other through BD. Let point B also be understood to move at the same time by its own motion, and that a simple one, along the same straight line BA, toward A, and together with it, as has been said, moved; let it be carried toward

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EXERCITATIONES. 141 sus CD. Erunt autem semper AB puncta in eadem linea quæ mouetur, sibi inuicem ex contrarijs partibus occurrentia. Itaque cum ex duobus motibus semper proportionalibus, hoc est, laterum proportione seruata, recta producatur, vt demonstratum est à principio, vbi produc[ti]o circuli ex Philosophi mente est declarata, vtraq[ue] puncta quæ eandem laterum proportionem seruantia mouëtur, rectas lineas producêt A quidem AD, B autem ipsam BC. Feratur igitur A, tum mixto tum simplici motu per diametrum AD. B vero quoque tum mixto, tum proprio per diametrum BC, supponitur autem motus omnes simplices, tum punctorum, tum etiam lineę, à qua puncta ipsa feruntur, æquali velocitate fieri. Illud igitur mirabile est, cuius etiam ratio quæritur, quo pacto eodem tempore eademque velocitate latum A quidem totam percurrat AD maiorem, B vero totam BC, eamque longe minorem? Porro necesse fuit rem in Rhombo speculari, non autem in quadrato & altera parte longiori rectangulo, in quibus diametri (quod Rhombo non accidit) sunt æquales. Imaginemur igitur A, proprio motu percurrisse spatium AE, nempe ipsius AB lineæ dimidium. Erit igitur in E, item lineam totam AB eodem tempore pertransisse dimidia oppositarum linearum, ACBD, & esse translatam, vbi FKG. Quoniam igitur æquali celeritate lineæ AB extremitas A, translata est in F & A, punctum per eam motum in E, erit spatium AE, æquale spatio AF. Ductis igitur lineis FKG, EKH lateribus AB, AC æquidistantibus, erit figura AEKF. Rhombus similis quidem Rhombo ABCD, recta igitur FK æqualis erit oppositæ AE. quare A punctum translatum erit ex mixto motu in K. Eodem pacto quonia[m] punctum B. eadem velocitate mouetur versus A, & linea AB versus CD, cum B fuerit in E extremum lineæ motæ BA, nèpe B erit in G. æquales ergo sunt BE, BG & Rhom- bus S 3

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EXERCISES. 141 ... CD. But there will always be points AB in the same line which is moved, meeting one another from opposite sides. Therefore, since from two motions always proportional, that is, with the proportion of the sides preserved, a straight line is produced, as has been shown from the beginning, where the production of the circle, according to the mind of the Philosopher, has been declared, both points which, preserving the same proportion of the sides, are moved, produce straight lines: A indeed AD, but B the very line BC. Let A therefore be carried, both by mixed and by simple motion through the diameter AD. But B also, both by mixed and by its own motion through the diameter BC, it is supposed that all simple motions, both of the points and also of the line by which the points themselves are carried, are made with equal speed. Now that is the wonderful thing, for which also the reason is sought: by what means in the same time and with the same speed does the moved A traverse the whole AD, the greater, while B traverses the whole BC, which is far smaller? Moreover, it was necessary to examine the matter in the Rhombus, and not in a square or a rectangle longer on one side, in which the diagonals (which does not happen in a Rhombus) are equal. Let us imagine therefore that A, by its own motion, has traversed the space AE, namely half of the line AB itself. It will therefore be in E, and also will have crossed the whole line AB in the same time, by half of the opposite lines, ACBD, and will be transferred, where FKG. Since therefore, with equal speed, the extremity of the line AB, A, has been transferred to F, and A, the point moved through it, to E, the space AE will be equal to the space AF. Therefore, if lines FKG, EKH are drawn parallel to the sides AB, AC, the figure will be AEKF. A Rhombus indeed similar to the Rhombus ABCD; therefore the straight line FK will be equal to the opposite AE. wherefore the point A will be transferred from mixed motion to K. In the same way, since the point B is moved toward A with the same speed, and the line AB toward CD, when B has been in E at the end of the moved line BA, namely B will be in G. Therefore BE and BG are equal, and the Rhombus S 3

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IN MECHAN. ARIST. PROBL. bus EBGK, circa diametrum BKC ipsi Rhombo ABCD similis, & ideo GK æqualis oppositæ BE & BG æqualis EK. Cum ergo B confecerit spatium BE, erit ex mixto motu in K, superato nempe spatio BK, idque eodem tempore quo A percurrerat totum spatium AK. Ex æquali igitur simplicium motuum velocitate, in æqualia spatia AB puncta pertransierunt, quæ res miraculo, cuius dilutio quæritur, præbet occasionem. Porro quod de dimidijs diametris demonstratum est, possumus & de totis eadem ratione concludere, quippe quod eadem sit proportio partium ad partes, quæ totius ad totum. Hæc igitur prima est pars propositæ quæstionis. Secunda vero dubitatio ita habet; Nempe mirum videri punctum B, cum peruenerit in C, extremum lineæ BA, videlicet ipsum B, translatum esse in D, licet æqualiter moueantur linea BA, per lineam BD, & punctum B per lineam BA. sitque BC ipsa BD maior. Primam dubitationem hoc pacto soluit Philosophus; A fertur tum proprio, tum alieno motu, hoc est, lineæ AB versus oppositam partem CD, Itaque cum vterque motus deorsum vergat, motus fit velocior. Contra vero B proprio quidem motu fertur versus A, hoc est, sursum, alieno vero, hoc est, lineæ BA versus D, hoc est, deorsum, qui motus cum inuicem aduersentur, motus ipse fit tardior, non igitur est mirum, A eodem tempore maius spatium pertransire quam B. Hæc solutio non modo vera videtur, sed mirabilis & ipsomet Philosopho dignissima, cui quidem temerariu[m] iudicaremus contradicere, nî in genere versaremur, in quo non probabilia quæruntur, sed demonstrata, sed vera. Futilem igitur esse rationem hanc ipsius Aristotelis pace, hoc pacto ostendemus. Esto quadratum ABCD, cuius diametri AC BD secantes sese in E, moueatur eodem pacto BA, versus CD, item

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IN MECHAN. ARIST. PROBL. … EBGK, around the diameter BKC itself, similar to the rhombus ABCD, and therefore GK equal to the opposite BE, and BG equal to EK. Since then B has completed the distance BE, there will be, from the mixed motion, in K, namely after the distance BK has been passed, and that at the same time in which A had traversed the whole distance AK. From the equal speed, therefore, of the simple motions, they have passed through equal spaces, the points AB, which circumstance gives occasion to the miracle whose explanation is sought. Moreover, what has been demonstrated about the half diameters, we may by the same reasoning conclude also of the whole, since the proportion of parts to parts is the same as of the whole to the whole. This then is the first part of the proposed question. The second doubt is as follows: namely, it seems wonderful that the point B, when it has come to C, the end of the line BA, that is, B itself, has been transferred to D, although the line BA is moved equally through the line BD, and the point B through the line BA, and BC is greater than BD itself. The Philosopher resolves the first doubt in this way: A is carried both by its own motion and by another’s, that is, the line AB toward the opposite part CD. Thus, since both motions incline downward, the motion becomes faster. But B, on the other hand, is carried by its own motion toward A, that is, upward, but by another’s, that is, by the line BA toward D, that is, downward; and since these motions are opposed to one another, the motion itself becomes slower. Therefore it is not surprising that A traverses a greater space in the same time than B. This solution not only seems true, but admirable and most worthy of the Philosopher himself, to whom indeed we should judge it rash to contradict, were we not engaged in a field in which not probabilities are sought, but demonstrations, and truth. We shall therefore show, with due respect to Aristotle, that this reasoning of his is futile in this way. Let there be a square ABCD, whose diagonals AC and BD, intersecting one another at E, move in the same way; let BA move toward CD, likewise

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EXERCITATIONES. 143 item A, versus B, & B versus A, ita- que punctum A tum proprio tum alieno, hoc est linea illud deferê- tis motu deorsum trudet, hoc est, versus CD. Motus ergo velocior erit motu puncti B, quod lationi- bus fertur ferè contrarijs, hoc est, ex B versus A sursum, cum linea autem B A versus C deorsum. Ve- locius tamen non mouetur, quip- pe quod æquali tempore æquale spatium vtrumque punctum conficiat. Stante igitur caus- sa sequi debuisset effectus; non sequitur autem, Aristote- lis igitur causa non est causa. Rhombo quoque inuerso idem clarius ostendemus hoc pacto: Sit Rhombus A B C D, cuius diametri A C, B D secan- tes sese in E. Mota igitur linea A B versus C D, nempe deorsum & A quoque deorsum versus B, contra vero B quidem sur- sum versus A, deorsum vero versus C, erit B tardior A, sed contrarium fit, quippe quod longior sit B D, per quam mouetur B ipsa A C, per quam mouetur A. His igitur non satisfacientibus veriorem si per im- becillitatem nostram licuerit, huius effectus causam in- uestigabimus. Rationibus igitur & veritate contra aucto- ritatem & probabilitatem est nobis pugnandum: quod & intrepide faciemus. Dicimus igitur, in quouis parallelogrammo sit illud quadratum aut altera parte longius, vel idem Rhombus Rhomboisue semper mixtos motus proportione seruata fieri

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EXERCITATIONES. 143 likewise A toward B, and B toward A; thus point A, both by its own and by the other’s movement, that is, by the motion of the line, will be driven downward, that is, toward CD. Therefore its motion will be swifter than the motion of point B, since it is carried by motions almost contrary, that is, from B toward A upward, and with the line B A however toward C downward. Yet it is not moved more quickly, since in an equal time each point completes an equal distance. If then the cause stood, the effect ought to follow; but it does not follow, therefore Aristotle’s cause is not a cause. We shall show the same thing more clearly by the inverted rhombus in this way: let there be the rhombus A B C D, whose diagonals A C and B D intersect in E. Therefore, when the line A B is moved toward C D, namely downward, and A also downward toward B, while on the other hand B indeed upward toward A, but downward toward C, B will be slower than A; yet the contrary happens, since B D is longer, along which B itself moves, than A C, along which A moves. Since therefore these things do not satisfy us, we shall, if our weakness permits, investigate the cause of this effect more truly. We must therefore contend with reasons and truth against authority and probability; and this we shall do without fear. We say therefore that in every parallelogram, whether that is a square or longer on one side, or likewise a rhombus or rhomboid, mixed motions are always produced with the proportion preserved.

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144 IN MECHAN. ARIST. PROBL. fieri per diametros. Cæterum diametrorum ad latera proportiones esse varias (quadratis exceptis, in quibus eadem est semper) exploratissimum. Illud quoque certum est, in rectangulis nunquam dari posse diametros lateribus vtcunque captis æquales, semper enim diametri rectis angulis subtruduntur. In Rhombis vero & Rhomboidibus diametrorum ad latera proportiones variant. Dari enim possunt diametri lateribus longiores item æquales, & lateribus quoque ipsis breuiores. Itaque diametrorum & laterum varia adinuicem ratione se habentibus, attentis proportionibus, mixtoru[m] & simplicium motuum diuersa fiet, & varia comparatio. in quadratis motus mixtus, qui per diametros semper velocior erit simplici qui per latera, Idem quoque in altera parte longiori, in quo mixti quidem motus per diametros erunt velociores, simplices vero qui per latera, tardiores quidem, sed ex illis tardior qui per latus breuius. In Rhombis autem mixtus motus qui fit per diametros inæqualis. Velocior enim qui per longiorem diametrum, tardior qui per breuiorem. Itaque simplices motus punctorum per latera ad eum qui fit per diametros in non eodem pacto se habent. Porro cum Rhomboides variæ sint diametroru[m] ad latera habitudines, varia quoque dari potest proportio. aliquando enim diametri dari possunt lateribus maiores quandoque, alter eorum minor. Si autem Rhombus in duos soluatur triangulos, alter diametrorum datur æqualis æqualibus lateribus æquicurium triangulorum; itaq[ue] in istis mixti motus per diametros æque veloces erunt simplicibus, qui per latera longiora, velociores autem illis qui per latera breuiora. His igitur hoc pacto non perfunctoriè consideratis, facile ex proprijs caussis, nî fallimur, hocce Aristotelicum & mirabile Problema soluitur. Esto

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144 IN MECHAN. ARIST. PROBL. comes to pass by the diagonals. Moreover, that the proportions of the diagonals to the sides are various (except in squares, in which the same relation is always maintained) is perfectly well known. This too is certain: in rectangles, however they may be taken, it is never possible for the diagonals to be equal to the sides, for the diagonals are always drawn up to right angles. But in rhombs and rhomboids the proportions of the diagonals to the sides vary. For diagonals may be longer than the sides, or equal to them, and also shorter than the sides themselves. Therefore, since the diagonals and sides stand in various ratios to one another, and the proportions are observed, there will be a diverse and varying comparison of mixed and simple motions. In squares, the mixed motion, which takes place by the diagonals, will always be swifter than the simple motion that takes place by the sides. The same also in the longer part, in which the mixed motions through the diagonals will be swifter, while the simple motions through the sides are slower indeed, but among these slower is that by the shorter side. But in rhombs the mixed motion that takes place by the diagonals is unequal: for the one through the longer diagonal is swifter, the one through the shorter is slower. And so the simple motions of the points through the sides do not stand in the same way in relation to that which takes place by the diagonals. Moreover, since rhomboids have various relations of diagonals to sides, a varied proportion may also be given. For sometimes the diagonals may be larger than the sides, and at other times one of them smaller. But if a rhombus is divided into two triangles, one of the diagonals is equal to the equal sides of the equilateral triangles; and thus in these the mixed motions through the diagonals will be as swift as the simple motions through the longer sides, but swifter than those through the shorter sides. Therefore, these matters being considered in this way, not superficially, it is easy, unless we are mistaken, to solve this Aristotelian and marvellous Problem from its proper causes. Let it be so.

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EXERCITATIONES. 145 Esto enim Rhombus ABDC, cuius diameter longior AD maior sit tum lateribus, tum etiam altera diametro BC. secent autem se inuicem diametri in E. Ducaturque ipsis AB, CD, parallela FG secans longiorem diametrum AD, in H, breuiorem vero BC in I. & per I ipsis BD AC parallela ducatur KIL, Cum ergo B mixto motu per diametrum BC erit in I & A per diametrum AD, mixto similter motu erit in H, & quia motus mixti fiunt per diametros, vt dictum est, vt se habet AD ad BC, ita AE ad EB, per 15. propol.5. elem. item vt AE ad EB, ita per 4. propos.6. AH ad BI. est enim IH ipsi AB parallela. Longior est autem AH ipsa BI, quippe quod AE longior sit ipsa EB. motus igitur mixtus puncti A per diametrum AD vsque ad H velocior est motu B, per diametrum BC vsque ad I. Mota igitur linea AB mouebuntur communia eius & diametrorum BC, AD puncta, quibus secantur semper diametrorum proportione seruata. Quibus ita se habentibus, nil mirum est punctum A motum per AD velociorem esse mixto motu puncti B, quod per minorem diametrum fertur BC. quod fuerat demonstrandum. quatenus vero ad secundam problematis partem pertinet, dicimus Propositionem non esse vniuersalem. Si enim Rhombus detur, ex duobus æquilateris triangulis constans, breuior diameter lateribus erit æqualis, quare non mouebitur citius motu simplici punctum per latus ac faciat mixto per minorem diametrum, quod vt mirum proposuerat Aristoteles. Si autem latus ipsum breuiori diametro sit logius, nec mirum quoque erit simplici motum moueri velocius quam mixto, quippe quod, vt T dictum

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EXERCISES. 145 Let there be the rhombus ABDC, whose longer diameter AD is greater both than the sides and also than the other diameter BC. Let the diameters cut one another at E. And let FG be drawn parallel to AB and CD, cutting the longer diameter AD at H, but the shorter BC at I. And through I let KIL be drawn parallel to BD and AC. Since therefore B, with mixed motion through diameter BC, will be at I, and A through diameter AD, with similarly mixed motion, will be at H, and because the mixed motions are along the diameters, as has been said, as AD is to BC, so AE is to EB, by Proposition 15 of Book 5 of the Elements; likewise as AE is to EB, so by Proposition 4 of Book 6 AH is to BI. For IH is parallel to AB itself. Now AH is greater than BI, because AE is greater than EB. Therefore the mixed motion of point A through diameter AD as far as H is faster than the motion of B through diameter BC as far as I. Therefore, when line AB is moved, the common points of it and of the diameters BC, AD will be moved, with the ratio of the diameters always preserved at the points where they are cut. Since this is so, it is no wonder that point A, moved through AD, is faster than the mixed motion of point B, which is carried through the lesser diameter BC. This was to be demonstrated. So far as the second part of the problem is concerned, we say that the proposition is not universal. For if a rhombus be given consisting of two equilateral triangles, the shorter diameter will be equal to the sides, and therefore a point moving simply along the side will not move faster than by a mixed motion through the lesser diameter, which Aristotle had put forward as a surprising thing. But if the side itself is longer than the shorter diameter, it will likewise not be surprising that simple motion should move faster than mixed, since, as has been said

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146 IN MECHAN. ARIST. PROBL. dictum est, motus isti à proportionibus linearum, per quas mouentur, legem velocitatis atque tarditatis accipiant. Hæc igitur nos circa hoc mirabile Aristotelicum proble- ma considerare sit satis. QVÆSTIO XXIV. Mirabilem aliam quæstionem proponit Aristoteles, quæ itidem ad mixtos motus pertinet. Dubitatio est, quam ob caussam maior circulus æqualem minori circulo circumvoluitur lineam, quando circa idem centrum fue- rint positi. Seorsum autem revoluti quemadmodum alterius ma- gnitudo ad alterius magnitudinem se habet, ita & illorum adin- nicem fiunt lineæ? Præterea vno etiam & eodem vtrisque existen- te centro. Aliquando quidem tanta sit linea, quam conuoluuntur, quantum minor per se conuoluitur circulus, quandoq[ue] vero quan- tum maior. Hæc ille, qui vt probet maiorem circulum in sua ro- tatione maiorem lineam pertransire, minorem vero mi- norem; ait sensu cognosci angulum maioris circuli, id est, eius qui maiorem habet circumferentiam, esse maiorem, eius vero qui minorem, minorem. Ita autem se habere cir- cumferentias vt se habent anguli, & eandem proportione habere per quas tum maior, tum minor circulus circum- uoluuntur. Ad quorum clariorem intelligentiam ea re- uocare oportet in memoriam, quæ dixit de maiorum cir- culorum ad minores circulos nutu. Hic enim, quod ibi quoque fecerat, sectorem ipsum angulum appellauit, an- gulum vero maiorem maioris circuli sectorem, & mino- rem angulum minoris ipsius circuli sectorem dixit. Clau- dit igitur dicens: quoniam circumferentiæ se habent vt anguli, hoc est, vt sectores, maior erit circumferentia ma- ioris circuli, & ex consequenti maior linea, per quam cir- cum-

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146 IN MECHAN. ARIST. PROBL. it has been said that these motions receive the law of speed and slowness from the proportions of the lines through which they move. Let this then suffice for us concerning this wonderful Aristotelian problem. QUESTION XXIV. Aristotle proposes another marvelous question, which likewise pertains to mixed motions. The doubt is for what cause the greater circle, when set around the same center, describes an equal line with the lesser circle. But when they are turned separately, in what way is the magnitude of the one related to the magnitude of the other, and thus do their lines become proportional to one another? Moreover, since the center is one and the same for both, at one time the line described is as much as the lesser circle describes by itself, but at another time as much as the greater. This is what he says. In order to prove that the greater circle, in its rotation, passes through a greater line and the lesser through a lesser, he says that by sense it is known that the angle of the greater circle, that is, of the one which has the larger circumference, is greater, and that of the lesser is less. And that the circumferences are related as the angles are related, and have the same proportion as that by which both the greater and the lesser circle are rolled around. For a clearer understanding of these points, it is necessary to recall what he said about the inclination of the greater circles to the lesser. For here too, as he had done there, he called the sector itself an angle; and he said that the greater angle is the sector of the greater circle, and the lesser angle the sector of the lesser circle itself. He concludes, therefore, saying: since the circumferences are related as the angles, that is, as the sectors, the circumference of the greater circle will be greater, and consequently the line through which it is turned will be greater.

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EXERCITATIONES. 147 cumuoluitur, ea per quam minor. Demonstrationem vero ex sensu petijt. Satautem erat si dixisset, ita se habere circumferentias vt se habent diametri seu semidiametri, & ideo lineas in rotatione descriptas inuicem se habere vt diametros. Obscuriusculè, hæc sua figura ostendit Aristoteles. Nos igitur claritatem amantibus, nostram aliquanto, nî fallimur, clariorem, proponemus. Esto circulus maior ABCD, minor FGHI, circa idem, & commune cætrum E. Circumuoluatur maior ad partes D. Sint auté diametri, maioris quide AEC, BED, minoris verò FEH, GEI, sitque CD, quadrans maioris, HI vero minoris circuli. Moto igitur maiori circulo secu[m] dum absidem, cum D fuerit in K erit CK ipsi CD æqualis, fietq[ue] DE ex puncto K perpendicularis ipsi CK, eritq[ue] vbi KO, & quia punctum I est in linea DE, erit I facta quadratis rotatione in linea KO vbi L, centrum vero E in ipsa KO, vbi O. Reuoluto igitur quadrante maioris, & confecto spatio CK minoris circuli quadrans HI conficiet spatium HL, quod ipsi CK spatio est æquale. quod autem in quadrantibus sit, in totis etiam sit circulis. Motus igitur minor circulus circa centrum E, vnica rotatione æquauit spatium rotationis maioris circuli. Mirabile itaque est minorem circulum eodem tempore & circa idem centrum circumuolutum, lineam pertransisse æqualem circumferentiæ maioris circuli. Nec secius admirationem facit ro- T 2 tato

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EXERCISES. 147 is rolled around, that by which it is smaller. But he sought the demonstration from sense. It would have been enough if he had said that the circumferences are related as the diameters, or semidiameters, and therefore the lines described in rotation are related to one another as the diameters. Aristotle presents this somewhat obscurely in his figure. We therefore, for those who love clarity, shall propose ours, a little clearer, if we are not mistaken. Let there be a larger circle ABCD, a smaller FGHI, about the same and common center E. Let the larger be rotated toward D. Let the diameters of the larger be AEC, BED, and of the smaller FEH, GEI, and let CD be the quadrant of the larger, and HI the quadrant of the smaller circle. Then, when the larger circle is moved according to the circumference, when D has come to K, CK will be equal to CD itself, and DE will become perpendicular from point K to CK, and where KO is, and since point I is on the line DE, I, after the rotation of the quadrants, will be made into the line KO where L is, but the center E itself will be on the line KO, where O is. Therefore, when the quadrant of the larger circle has been rolled through, and the space CK of the smaller circle has been completed, the quadrant HI will complete the space HL, which is equal to the space CK. But what holds for the quadrants also holds for the whole circles. Therefore, when the smaller circle moves around center E, by a single rotation it has made equal the space of the rotation of the larger circle. It is thus remarkable that the smaller circle, rolled about at the same time and around the same center, has traversed a line equal to the circumference of the larger circle. Nor is the remark less surprising— T 2 rolled

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148 IN MECHAN. ARIST. PROBL. tato minori circulo, maiorem vna circumuolutu[m] lineam metiri circumferentiæ minoris circuli æqualem. Rotetur enim minoris circuli quadrans HI per rectam HL. erit igitur punctum I vbi M, æquali existente recta HM, ipsi curuæ HI. Tunc autem facto motu centrum E erit vbi P, existente EP, ipsi HM æquali, demittatur autem ex P per M, ipsis HL CK perpendicularis PMN. Et quoniam in eadem linea sunt DIE, vbi E fuerit in PI erit in M, & D in N. quamobrem rotata quarta minoris circuli parte, maioris interim circuli quadrans confecit spatium CN æquale ipsi HM, hoc minus circuli quadranti HI, quod vti-que est admirabile. Porro causam effectus huius mirifici diligenter quærit Philosophus, & inuentam accurate explicat. Occurrit autem primo absurdæ cuidam opinioni. Diceret enim quispiam, ideo tardius moueri maiorem circulum, ad motum minoris, quod interim d[omi]nus minor moueretur, aliquas inter rotandum moras interponeret, minor vero ad motum maioris spatia aliqua transliret, & ita spatiorum fieri adæquationem. Porro demonstrationem aggressurus hæc assumit principia. Eandem æqualemue potentiam, aliqua[m] magnitudinem tardius quidem mouere, aliquam vero celerius. quod autem natum est aptum moueri, tardius moueri, si simul cum non apto nato moueri, moueatur, quam si separatim moueretur, celerius autem si non simul cum eo moueatur. Esto enim corpus A leue quidem & aptum natum mouerisursum, cui connectatur B, aptum natum moueri deorsum, Si quis igitur mouere conetur corpus A sursum difficilius mouebit, & tardius iunctu[m] nempe ipsi B, quam si ab ipso esset seiu[n]ctum. Præterea quod non suo, sed alieno motu mouetur, impossibile esse plus eo moueri qui mouet,

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148 IN MECHAN. ARIST. PROBL. by the smaller circle, measuring with the line wound once around the larger circle a line equal to the circumference of the smaller circle. For let the quadrant HI of the smaller circle rotate along the straight line HL. Thus the point I will be where M is, the straight line HM being equal to the curve HI. Then, when the motion has been made, the center E will be where P is, EP being equal to HM; and from P through M let PMN be drawn perpendicular to HL and CK. And since DIE are on the same line, where E has been there will be in PI, it will be in M, and D in N. Wherefore, when one quarter of the smaller circle has been rotated, meanwhile the quadrant of the larger circle has completed the space CN equal to HM, this less than the quadrant HI of the circle, which is indeed marvelous. The Philosopher then carefully seeks the cause of this wondrous effect, and explains it accurately once found. But first he encounters a certain absurd opinion. For someone might say that the larger circle moves more slowly, in relation to the motion of the smaller, because in the meantime the lord of the smaller circle, while rotating, would interpose some delays, whereas the smaller, in relation to the motion of the larger, would leap across some intervals, and thus an equalization of distances would come about. Now, when he undertakes the demonstration, he assumes these principles. The same or equal power moves some magnitude more slowly, but another more quickly. And that which is naturally fitted to be moved is moved more slowly if, along with something not naturally fitted to be moved, it is moved than if it were moved separately; but more quickly if it is not moved along with that thing. For let body A be light, and naturally fitted to be moved upward, to which let B be attached, naturally fitted to be moved downward. If someone therefore tries to move body A upward, he will move it with greater difficulty and more slowly when joined to B than if it were separated from it. Moreover, that which is moved not by its own motion but by another’s motion cannot possibly be moved more than the mover who moves it.

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EXERCITATIONES. 149 mouet, siquidem non suo, sed alieno motu mouetur. Mo- to igitur suo motu maiori circulo, minor non suo moue- tur, sed motu maioris circuli, & ideo non plus mouetur quam ille moueatur, mouetur autem maiori spatio quam ex se moueretur, propterea quod maior sit maioris circu- li, à quo simul defertur, circumferentia. Item si minor suo motu circumuoluatur, maiorem feret secum, & ideo non plus in sua rotatione mouebitur maior, quam ipse minor circulus moueatur. Summa rei hæc est, alterum ferri ab al- tero & latum ad ferentis spatium moueri. Licet enim al- tero moto, alter interim moueatur, nihil refert. Est enim ac si is qui fertur, nullam habeat motionem, aut si eam ha- beat, ipsa nequaquam vtatur. quod non fit si vterque se- paratim circa proprium centrum moueatur, tunc enim magnus magnum, paruus vero paruum spatium conficit. Hinc decipiait Aristoteles illum, qui putat vtrumque cir- culum per se super idem centrum in rotatione moueri, li- cet enim videatur, revera non est. Id enim vtique certum est, cum à maiori circulo minor fertur, circa maioris cen- trum motum fieri. Si vero maior à minori feratur circa mi- noris circuli centrum motum fieri. Hæc ferè Philosophi est mens, cuius solutionem esse certissimam, & ex veris caussis non dubitamus. Hinc ad aliam eamque certam assertionem transi- mus. Dicimus enim, nullam materialem rotâ circa axem eidem affixum, dum rotatur, posse eundem locum seruare, nisi cauum fiat, quod axem ipsum recipiat, in transuersa- rijs quibus rota sustinetur & progressiuum axis motum impediat. Esto enim rota ABCD, cuius centrum E, diametri AEC, BED, esto alia minor rota GH, item minor KL, tum minor NO, & adhuc minor QR, circa idem centrum E. Rotetur itaque secundum absidem integri quadrantis T 3 spa-

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EXERCISES. 149 moves, inasmuch as it is moved not by its own motion, but by the motion of another. Therefore, when it is moved by its own motion in a larger circle, the smaller circle is not moved by its own motion, but by the motion of the larger circle; and thus it is no more moved than that other moves, but it is moved through a greater space than it would be moved of itself, because its circumference is greater, by which, together with the larger circle, it is carried around. Likewise, if the smaller circle is rotated by its own motion, it will carry the larger along with it; and thus the larger will not move in its own rotation any more than the smaller circle itself moves. The sum of the matter is this: one is carried by the other, and, being carried, is moved through the space of the carrier. For although, one being moved, the other in the meantime is moved, it makes no difference. For it is as if the one that is carried had no motion at all, or if it has one, did not use it at all; which does not happen if each moves separately around its own center, for then the large one traverses a large space and the small one a small space. Hence Aristotle deceives that man who thinks that both circles, of themselves, move in rotation upon the same center; for although it seems so, in reality it is not. For this is certainly true: when the smaller is carried by the larger circle, the motion takes place around the center of the larger; but if the larger is carried by the smaller, the motion takes place around the center of the smaller circle. This is nearly the meaning of the Philosopher, whose solution we do not doubt to be most certain and drawn from true causes. From here we pass to another, and indeed certain, assertion. We say that no material wheel fixed to an axle can, while it is rotating, preserve the same place, unless a hollow be made to receive the axle itself, in the transverse parts in which the wheel is supported and which impede the progressive motion of the axle. Let there be a wheel ABCD, whose center is E, the diameters AEC, BED; let there be another smaller wheel GH, likewise a smaller KL, then still smaller NO, and still smaller QR, all about the same center E. Therefore let it rotate according to the position of the whole quadrant T 3 spa-

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170 IN MECHAN. ARIST. PROBL. spatium CD, eritque D, in F, item si ex rota GH, ex quadrante HT, erit T in I. Exalijs item minoribus in M, P, S. erit itaq[ue] longissimu[m] spatium CF, breuissimu[m] vero RS, Mota igitur rota circa circulu[m] seu axem, QR, maior rota spatio mouebitur RS, quod si intra QR, circa centrum E alij infiniti imaginentur circuli, quo propiores centro fuerint, eo maioris rotæ progressus erit minor, donec ad centrum deueniatur, vbi cum non sit circulus, nullus fiet progressiuus motus, sed circa ipsum centrum nulla facta loci mutatione rotabitur. At cum nulla materialis rota circa lineam punctumue imaginarium conuerti possit, ideo axi ferreo alteriusue materiæ circa quem & cum quo circumuoluatur rota, cauum semirotundum incidere oportet, in quo insertus axis dum conuertitur à loco in quo conuertitur, non recedat. QVÆSTIO XXV. Quæritur, Cur lectulorum spondas secundum duplam faciant proportionem, hanc quidem sex pedum, vel paulo ampliorem, illam vero trium. Item cur vectes funesue non secundum diametrum extendantur? PRimam quæstionis partem ita diluit Philosophus, fortasse tantæ fieri solitos magnitudinis lectulos vt corporibus sint proportionem habentes, & ideo fieri secundum spondas dupli longitudine nempe cubitorum quatuor, latitudine vero duorum. Nostra-

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170 IN MECHAN. ARIST. PROBL. space CD, and it will be D, in F; likewise if from the wheel GH, from the quadrant HT, it will be T in I. From the others likewise smaller in M, P, S. Therefore the longest space will be CF, the shortest truly RS. Therefore, when the wheel is moved around the circle or axis, QR, the larger wheel will be moved through the space RS, but if within QR, around center E, other infinite circles are imagined, the nearer they are to the center, the smaller will be the progress of the larger wheel, until it comes to the center, where, since there is no circle, there will be no progressive motion, but it will rotate about that center itself, with no change of place. But since no material wheel can be turned about an imaginary line or point, therefore in an iron axle or one of other material, around which and with which the wheel turns, a hollow semicircular groove must be cut, in which the inserted axle, while it is turned, does not depart from the place in which it turns. QUESTION XXV. It is asked why the staves of beds are made according to a double proportion, this one of six feet, or a little more, and that one of three. Also why levers or ropes are not stretched according to the diameter? The first part of the question the Philosopher thus explains, perhaps because beds used to be made of such great size that they might be proportionate to bodies, and therefore made according to staves with a double length, namely of four cubits, but in width of two. Our-

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EXERCITATIONES. 151 Nostrates alia vtuntur proportione, sesquialtera, videlicet, quam Græci Hemioliam dicunt, communiter enim pedes quatuor latos faciunt plus minusue, longos vero circiter sex. quod ideo fit vt in eis duo corpora commodius cubare possint. Lectuli autem, de quibus loquitur Philosophus, ad vnum tantummodo sustinendum facti videntur, quicquid tamen sit, nullam ferè habet res ex hac parte dubitationem. Secunda quæstionis sectio ea erat, Cur non secundu[m] diametros funes extendantur? Restium funiumue in lectulis muniendis vsus non est apud nos. etenim feretra tantum, seu sandapilas, quibus defunctorum corpora offeruntur, funibus ad ea sustinenda inteximus. Cæterum lectos tabulis seu asseribus sternimus, quibus saccos paleis plenos imponimus, saccis vero culcitras, & tormenta, ne tabularum durities cubantes offendat. Atqui in re facili multum laborasse videtur Aristoteles, tum etiam obscure & inuolute nimis quæstionem tractasse. Difficilem enim apud eum habet hæc explicationem, tum ea quam diximus de caussa, tum etiam quod Græca lectio & Latina versio corrupta, vt apparet, præ manibus habeantur. Sane vt veritatem hoc loco vindicaret in lucem, egregie laborauit Picolomineus nec parum profecit. Cæterum currestes non secundum diametrum extrudantur, triplicem affert Philosophus rationem. Prima est vt spondarum ligna, minus distrahantur. Secunda, vt podus inde commodius sustineatur. Tertia, vt in ipsa textura minus restium funiumue absumatur. Ad primam, cur extensis diametraliter funibus spodæ ipsæ distrahantur discindanturue, nec ille nec alij docent. Ego autem demonstrarem hoc pacto. Esto sponda ABCD, cuius longitudo AB, crassitudo AC, in ea foramen vtrinque pertinens EF, restis per foramen

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EXERCISES. 151 Our people use a different proportion, namely sesquialteral, which the Greeks call hemiolia; commonly they make them about four feet broad, more or less, and about six feet long. This is done so that two bodies may more conveniently lie in them. But the little beds of which the Philosopher speaks seem to have been made to support only one person. However that may be, there is scarcely any doubt about the matter from this side. The second section of the question was this: Why are not the cords stretched along the diagonals? The use of ropes or cords in making up beds is not among us. For we use ropes only for biers, or sandapilae, on which the bodies of the dead are carried, weaving cords into them for support. Otherwise, we spread the beds with boards or planks, on which we place sacks filled with straw, and on the sacks mattresses and cushions, lest the hardness of the boards should hurt those lying on them. And yet Aristotle seems to have laboured much over an easy matter, and to have treated the question too obscurely and confusedly. For with him this explanation is difficult, both for the reason we have mentioned, and also because the Greek text and the Latin translation, as appears, are corrupt and at hand. Certainly, in order to vindicate the truth in this place, Picolominius worked excellently and made no small progress. But why are the cords not stretched out along the diagonal? The Philosopher gives three reasons. The first is that the timbers of the sides may be less strained. The second is that the burden may then be more conveniently supported. The third is that less rope or cord is used up in the actual weaving. To the first point, namely why, when the cords are stretched diagonally, the sides themselves are strained and torn apart, neither he nor others explain. I myself, however, would demonstrate it in this way. Let ABCD be a frame, whose length is AB and thickness AC; in it let there be a hole passing through on both sides EF, and let the cord through the hole...

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men inditus GFE, sitque E pars seu caput exterius, quod nodo in E distinetur. Sit autem spondæ lignum iuxta longitudinem vt natura assolet scissile. Vis quædam, fune ita extento applicetur in G, quæ funem ipsum ad se violenter trahat. non discindetur idcirco sponda eo quod non diametraliter funis extendatur. Modo facta capitis G translatione in H, trahatur valide funis, fiet autem pressio valida in F. ibi enim impedimentum facit angulus, ne funis ipsa dum trahitur, rectitudinem assequatur. Itaque vi præualente, ligno vero scissili, minus resistente, funis, assecuta rectitudine, fiet in HIE scissa sponda ad quæritatem trianguli FIE, quod fuerat demonstrandum. Cur autem funes ab angulo in angulum extensæ minus commode pondus sustineant, satis patet. quo enim funis logior, eo debilior, & pressio quæ in medio sit, ea videlicet parte quæ ab extremis est remotissima, magis funem fatigat. Longiores autem funes sunt quæ diametraliter extenduntur. Quatenus ad tertiâ rationem pertinet, hoc pacto funes intexit Philosopho. Esto lectulus cum suis spodis AB CD, cuius sponda AD, sit pedum sex, AB vero triu, Diuidatur AD bifariam in E & BC in F. item AE in tres AG, GH, HE & in totidem ED, nempe EL, LM, MD. Similiter medietas alterius spodæ BF in tres partes distinguatur BN, NO, OF, & FC

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...but let GFE be inserted, and let E be the outer part or head, which is distinguished by the knot in E. Now let the wood of the side piece be, according to its length, split as nature usually allows. Let some force be applied at G with the rope thus stretched, which violently pulls the rope itself toward it. For that reason the side piece will not be split, because the rope is not extended diametrically. But when the head G is transferred to H and the rope is strongly pulled, then there will be strong compression at F. For there the angle creates an impediment, so that the rope itself, while being pulled, does not attain straightness. Thus, the force prevailing and the wood, being split and offering less resistance, the rope, having attained straightness, will cause the side piece to be split in HIE according to the form of triangle FIE, which was to be demonstrated. But why ropes extended from angle to angle bear a weight less conveniently is clear enough. For the longer the rope, the weaker it is, and the compression that is in the middle—that is, in the part farthest from the ends—wears the rope down more. Now the longer ropes are those which are extended diametrically. As for the third reason, the Philosopher weaves ropes in this way. Let there be a little bed with its side pieces AB CD, of which the side AD be six feet, but AB three. Let AD be divided into two equal parts in E, and BC in F. Likewise let AE be divided into three parts AG, GH, HE, and ED into the same number, namely EL, LM, MD. Similarly, let the half of the other side piece BF be divided into three parts BN, NO, OF, and FC

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EXERCITATIONES. 153 & FC similiter in tres FI, IK, KC, tum alterofunis capite inducto per foramen A, ibique probe firmato, indatur per F, inde per I, postea per GHK CE, & in E probe alligetur: Erunt igitur funis quatuor partes æquales AF, IG, HK, EC, quibus adjiciuntur particulæ cadentes extra, quæ sunt FI, GH, KC. Post hæc alterius funis principium per foramen traijcitur, quod est in angulo B. Deinde per E, inde per L, N, O, M, D, F & in F probe vincitur, & nodo facto obfirmatur. Erunt igitur aliæ quatuor alterius funis partes, tum inter se, tum etiam supradictis æquales, nempe BE, NL, OM, ED, quibus illæ pariter adjiciuntur particulæ, quæ cadunt extra, videlicet EL, NO, MD. quonia[m] igitur quadratis ex BA, AE æquale est quadratum BE, erit BE quadratum 18. cuius latus radixue 4 1/3 quam proxime. Sunt autem huius longitudinis funes æquales octo. Eaarum igitur simul sumptarum longitudo erit pedum 34 2/3 vel circiter, quibus si addantur pedes sex funium qui cadunt extra, erit restis totius longitudo expansa pedum 40 2/3 plus minusue. Picolomineus vero ait 34 2/3, omisit enim particulas illas sex, quæ, vt diximus, cadunt extra. Idem rationem funium diametraliter extensarum in idem, ait esse longitudinis pedum 40 1/2. Hic autem eas quoq[ue] particulas prætermittit, quæ extra cadunt. Itaque his additis clare patet, plus restium insumi diametraliter ipsis, quam lateraliter extensis. Cæterum ratio, qua Philosophus hæc probare conatur, adeo est mutila, inuoluta, obscura, vt Delio prorsus, vt aiunt, indigeat natatore. Huius loci inexplicabilem difficultatem, vidit Picolomineus, qui idcirco attestatus est, interpretes in hac exponenda fuisse hallucinatos. Certe Græca lectio versione ipsa Latina non est clarior. Nos interim ne inutilem ferè speculationem nimia diligentia, eaque fortasse frustranea prosequamur, alijs difficultatem hanc dissoluendam aut ceu Gordij no- V dum

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EXERCISES. 153 And similarly the cord FC into three parts FI, IK, KC; then the other end of the cord being passed through the hole at A, and there firmly secured, let it be passed through F, then through I, afterward through GHK CE, and let it be well tied at E. Thus there will be four equal parts of the cord, AF, IG, HK, EC, to which are added the small portions falling outside, namely FI, GH, KC. After this, the beginning of the other cord is carried through the hole which is at the angle B. Then through E, then through L, N, O, M, D, F, and let it be firmly tied at F, and made secure with a knot. Thus there will be four other parts of the other cord, equal both to one another and also to those mentioned above, namely BE, NL, OM, ED, to which likewise are added those portions that fall outside, namely EL, NO, MD. Since therefore, as the square of BA, AE is equal to the square BE, the square BE will be 18, the side or root of which is 4 1/3 as nearly as possible. Now there are eight cords of this length equal to one another. Their total length taken together will therefore be 34 2/3 feet, or about that; if to these are added the six feet of cord which fall outside, the length of the whole rope extended will be 40 2/3 feet, more or less. But Picolomini says 34 2/3, for he omitted those six small portions which, as we said, fall outside. Likewise he says that the length of the cords extended diametrically in the same way is 40 1/2 feet. Here too he passes over those portions that fall outside. Therefore, if these are added, it is clearly evident that more rope is used when extended diametrically than when extended laterally. Moreover, the reasoning by which the Philosopher attempts to prove this is so incomplete, involved, and obscure that it is, as they say, in need of a Delian swimmer altogether. Picolomini saw the inexplicable difficulty of this passage, and on that account testified that the interpreters had gone astray in explaining it. Certainly the Greek reading is not clearer than the Latin translation itself. Meanwhile, lest we pursue by excessive diligence, and perhaps in vain, a speculation that is almost useless, leaving this difficulty to others to be resolved, or as though to be cut as the Gordian kno- V tied

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154 IN MECHAN. ARIST. PROBL. dum gladio scindendo relinquemus. Sed interim subit mirari, cur veteres vtiliori modo prætermisso, inutiliore fuerint amplexati. Poterant enim reticulatim hoc per li- neas lateribus æquidistantes intexere. Esto enim lectulus eiusdem dimensionis ABCD, in cuius latere AD sint foramina quin- que E, F, G, H, I, totidem in latere opposito QP, ONM. Duo vero in la- tere breuiori AB, nempe RS, & totidem in opposito KL incipiatur extensio à fora- mine E, per QP, F, GON, HIM & in M funis obsfirmetur, tum alterius funis caput indatur si libet per K, & inde per S, R, L & in L constingatur. Sunt autem omnes EQ, FP, GO, NN, IM, pedum quindecim, quibus si addantur KS, RL, singuli pedum sex erunt pedum xxvii. quibus adiectis particulis extra cadentibus QP, FG, ON, HI, & RS, erit integra summa pedum xxxii. Vide igitur quantum hinc minus insumatur restium quam eo modo, quem proba- uit, & ceu vtiliorem proposuit Aristoteles. Præterea vali- dissimum est hoc texturæ opus nec ex eo fit vera sponda- rum distractio scissioue, quibus haud parum obnoxia est ea ratio, quam præfert ipse Philosophus. Concludimus i- gitur, aut nos eius verba & sensum non intellexisse, aut veteres ipsos, quorum vsum ipse explicat, rei, quam nos proponimus, naturam & commoditatem (quod ta- men vix credibile est) igno- rare. QVÆ-

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154 IN MECHAN. ARIST. PROBL. which we shall leave to be cut with a sword. But in the meantime one may wonder why the ancients, passing over the more useful method, should have adopted the less useful one. For they could have woven this in a netlike fashion through lines equidistant from the sides. For let there be, then, a bed of the same dimensions, ABCD, in whose side AD there are five holes, E, F, G, H, I, and as many in the opposite side QP, ONM. And in the shorter side AB, namely RS, and as many in the opposite KL, let the extension begin from hole E, through QP, F, GON, HIM, and let the rope be fastened in M; then let the end of another rope be inserted, if you wish, through K, and thence through S, R, L, and let it be tied fast in L. Now all the lengths EQ, FP, GO, NN, IM are fifteen feet; to these if KS, RL be added, each will be six feet, making twenty-seven feet; and with the added portions falling outside, QP, FG, ON, HI, and RS, the total sum will be thirty-two feet. See then how much less rope is used by this method than by that which Aristotle approved and proposed as more useful. Moreover, this woven work is very strong, and by it there is no true stretching of the bedsprings by cutting, to which that method, which the Philosopher himself prefers, is not a little exposed. We therefore conclude either that we have not understood his words and meaning, or that the ancients themselves, whose usage he explains, were ignorant of the nature and convenience of the thing which we propose (which, however, is hardly credible).

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EXERCITATIONES. 155 QVÆSTIO XXVI. Proponitur à Philosopho examinandum, Cur difficilius sit, longa liga ab extremo super humeros ferre, quam secundum medium, æquali existente pondere? DVohîc considerat, vibrationem, & pondus. Ait enim primo fieri posse, procera ligna vibratione impediente, difficilius ferri. Quæreret autem quispiam, (ipse enim id reticet) cur vibratio hæc ferenti sit nocua. Nos itaque id explicare conabimur. Esto igitur lignum oblongum, flexile, & vt ita dicam, vibrabile AB, imponatur humero, eique hæreat in C, manu vero sustineatur facta compressione in B. Nutet igitur & vibretur, in ipsa vibratione, ad partem A. Sit autem centrum grauitatis eius D, Lignum igitur in ipsa vibratione descendet sua pressus grauitate in E, tum facta ligni constipatione in ea parte quæ est inferius inter C & D, & inde resistentia, eodem fere impetu quo descenderat, repulsum per D, nec enim in sua rectitudine stabit, ascendet in F, facta iterum materiæ constipatione inter C & F. Mouebitur igitur lignum sua grauitate, motu frequentissimo, sursum deorsum, & is interim qui lignum humero fert, procedit antrorum, impedit igitur motus iste, qui sit sursum deorsum lationem, quæ sit ad anteriora; Latorem ipsum quodammodo retrahens. Si autem medio ligno supponatur humerus, eo quod vibratio sit minor. breuiores enim partes sunt, quæ à medio ad extrema minus à vibratione remorabitur ferens. Quoniam autem non sola vibratio in hoc lationis modo, nempe ex ligni extremitate difficultatem facit, ait V 2 Phi-

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EXERCITATIONS. 155 QUESTION XXVI. The Philosopher proposes for examination: Why is it more difficult to carry a long pole from one end over the shoulders than from the middle, the weight being equal? He considers two things here: vibration and weight. For he says, first, that long poles may be harder to carry because vibration hinders them. Someone might ask, however, why this vibration is harmful to the bearer; for he himself passes it over in silence. We shall therefore try to explain it. Let there be, then, an oblong, flexible, and, so to speak, vibratory pole AB, laid upon the shoulder and resting against it at C, while it is held by the hand, which, by pressure, sustains it at B. It will therefore sway and vibrate, in the very act of vibration, toward part A. Let its center of gravity be D. The pole, then, in the vibration itself, will descend by its own weight, pressing down in E; then, by the compression of the pole in that part which lies below between C and D, and from the resulting resistance, it is thrown back through D, with nearly the same force with which it had descended; nor, indeed, will it remain in its straight position, but will rise in F, the material again being compressed between C and F. The pole will therefore be moved by its own weight, in a very frequent motion, up and down; and meanwhile the man who carries the pole on his shoulder advances forward. This motion, then, hinders the forward carrying, since it is a motion up and down, while the carrying is toward the front; thus, in a manner, pulling back the bearer himself. But if the shoulder is placed under the middle of the pole, since the vibration is less, for the parts are shorter, the bearer will be less delayed by vibration, from the middle to the ends. And since not vibration alone in this mode of carrying, namely from the end of the pole, causes the difficulty, he says V 2 Phi-

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156 IN MECHAN. ARIST. PROBL. Philosophus, forte id fieri, quoniam licet nihil inflectatur, neque multam habeat longitudinem, difficilius tam e sit ad ferendum ab extremo, eo quod facilius eleuetur ex medio quam ab extremis, & ideo sic ferre sit facilius. Cur autem ex medio facilius eleuetur, caussam esse ait, quod eleuato medio ligno extrema sese inuicem suspendant, & altera pars alteram bene subleuet. Medium enim fieri velut centrum, vbi is supponit humerum qui eleuat aut fert. Extremorum autem interim altero depresso alterum sustolli. Nos interim Mechanicis principijs, quod ipse non fecit, rem clariorem efficiemus. Esto enim oblongum lignum AB, cui humerus supponatur in B, manus vero premendo sustinens in B. sit autem ligni pars maxima AC, minima CB, maioris autem ad minorem proportio exempli gratia sit sexcupla. Ad hoc igitur vt fiat æquilibrium inter potentiam sustinentem in B, & pondus comprimens in A, ita se habere oportet potentiam in B, ad pondus in A, vt se habet pars ligni AC ad partem CD. Esto igitur pondus in A, puta librarum sex. Erit igitur potentia quæ in B ad hoc vt sustineat librarum triginta sex, quas si addas poderi in A, fiet humerus in C sustinens pondus librarum quadraginta duo. Si autem humerus medio ligno, hoc est, in D supponatur, ad hoc vt fiat æquilibrium, necesse erit potentiam in B esse æqualem ponderi in A, quod est sex, quare humerus sustinebit duodecim. Vnde patet, longe difficilius portari lignum ex C extremo, quam ex D medio; quod Mechanice fuerat demonstrandum. Possumus & aliter idem ostendere. Intelligatur enim ijsdem suppositis, vectem quidem esse AB, cuius fulcimentum

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156 IN MECHAN. ARIST. PROBL. The philosopher says that this perhaps happens because, although it is not bent at all, nor has much length, it is more difficult to carry it from the end, since it is more easily lifted from the middle than from the ends, and therefore to carry it thus is easier. But why it is more easily lifted from the middle, he says the cause is that when the middle of the wood is lifted, the ends suspend one another, and one part well supports the other. For the middle becomes as it were a center, where he who lifts or carries it places his shoulder. But of the ends, meanwhile, one being depressed, the other is raised. We, meanwhile, by mechanical principles, which he did not do, shall make the matter clearer. Let there be, then, an oblong piece of wood AB, with the shoulder placed under it at B, and the hand supporting it by pressing in B. Let the larger part of the wood be AC, the smaller CB, and let the proportion of the larger to the smaller be, for example, sixfold. Therefore, in order that there may be equilibrium between the supporting power at B and the compressing weight at A, the power at B must be to the weight at A as the part of the wood AC is to the part CD. Let the weight at A therefore be, say, six pounds. There will therefore be a power at B which, in order to support it, will be equal to thirty-six pounds; and if you add these to the weight at A, the shoulder at C will be supporting a weight of forty-two pounds. But if the shoulder is placed under the middle of the wood, that is, in D, then in order that equilibrium may occur, it will be necessary for the power at B to be equal to the weight at A, which is six; wherefore the shoulder will support twelve pounds. Hence it is clear that it is much more difficult to carry the wood from the end at C than from the middle at D; and this was what had to be demonstrated mechanically. We can also show the same thing in another way. For let it be understood, under the same assumptions, that the beam is indeed AB, whose support

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EXERCITATIONES. 157 cimentum quidem B, pondus A, potentia sustinens in C, nempe inter fulcimentum & pondus. Res igitur ad eum vectis vsum reducitur, de quo G. V baldus tractatu de Vecte, propos. 3. Quare vt ille ostendit, ita se habere oportet potentiam sustinentem ad pondus, vt totus vectis ad partem eius quæ à potentia ad fulcimentum. Ita igitur se habebit pressio, quæ fit in C ad pondus in A, vt totus vectis AB ad partem eius CB, quæ à potentia ad fulcimentum. Erit igitur potentia septupla ponderi, & ideo sustinebit pondus librarum quadraginta duarum. quod fuerat ostendendum. Hinc alia quæstio huic affinis soluitur, Cur hasta sarissaue solo iacens manu ad alteram extremitatum apprensa difficillime extollatur? Esto igitur sarissa ha- staue iacens AB, cuius extremitati A manus ad sustollendum applicetur, sit autem pars quæ digitis capitur AC, quæritur cur pars reliqua CB difficillime sustollatur? Facile dubitatio ex prædemonstratis soluitur. Est enim C fulcimentum, supponitur enim loco, pugno ad sustollendum clauso, digitus index, potentia autem premens in A, vt superet grauitatem CB, est manus ipsius carpus, hoc est illa manus ipsius pars, qua pondus facta suppressione sustollitur. Est igitur AB vectis, cuius fulcimentum C, pondus B, potentia A, Itaq[ue] quoniam maxima est proportio BA ad AC, maximam esse oportet potentiam pondus sustollentem in C. Huc etiam illud pertinet, Cur hasta solo iacente, si alterum extremorum manu sustollatur, alterum vero velocissime sursum vibretur, & eodem tempore manus hastæ sic vibratæ supponatur, haud magna difficultate hastæ ad perpendiculum sit erectio. V 3 Sit

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EXERCITATIONES. 157 the fulcrum indeed B, the weight A, the sustaining power in C, namely between the fulcrum and the weight. The matter therefore is reduced to the use of the lever, concerning which G. V. Baldus treats in his tract De Vecte, proposition 3. Wherefore, as he shows, the sustaining power ought to bear the same relation to the weight as the whole lever does to that part of it which lies from the power to the fulcrum. Thus therefore the pressure which occurs in C to the weight in A will be as the whole lever AB to that part CB of it which lies from the power to the fulcrum. Therefore the power will be seven times the weight, and thus it will sustain a weight of forty-two pounds, which was to be shown. Hence another question akin to this is solved: Why is a spear or sarissa lying on the ground, when grasped by the hand at one end, raised with the greatest difficulty? Let there therefore be a sarissa or spear lying AB, to one end of which A the hand is applied for lifting; and let the part which is grasped by the fingers be AC. The question is why the remaining part CB is raised with such difficulty? The doubt is easily resolved from what has already been demonstrated. For C is the fulcrum, since the index finger is placed beneath it in the position of a fist closed for lifting; the power pressing at A, in order to overcome the weight of CB, is the wrist of the hand itself, that is, that part of the hand itself by which the weight, when pressed down, is lifted. AB is therefore a lever, whose fulcrum is C, whose weight is B, whose power is A. And since the proportion of BA to AC is very great, the power lifting the weight in C must be very great. To this also belongs the question why, when a spear lies on the ground, if one end is lifted by the hand, while the other end is most swiftly whirled upward, and at the same time the hand is suddenly removed from under the spear thus whirled, the spear is raised upright with little difficulty. V 3 Sit

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IN MECHAN. ARIST. PROBL. Sitenim hasta AB, quæ manu ex B capta eleuetur in C, & fiat in AC, tum facta ex C partis A veloci vibratione, ipsa extremitas A transferatur in D, sitque vbi CD, tum velo- ci manus depressione extremitas C transferatur in E, fiatq; EF horizonti perpendicularis; quod vbi factum fuerit, erunt in eadem linea quæ ad centrum mundi, manus ipsa quæ sustinet, & grauitatis ipsius centrum G, quare manus ipsa facta vibratione tantum portat, quantum præcise ipsius est hastæ pondus. QVAESTIO XXVII. Dubitatur, Cur si valde procerum fuerit idem pondus, difficilius super humeros gestatur, etiamsi medium quispiam illud ferat quam si breuius sit? QVæstio hæc superiori est affinis. Ait autem Philosop[hu]s, caussam non esse id, quod in præcedenti quæstione dixerat, sed vibrationem: quo enim longiora sunt ligna, eo magis eorum extrema vibrantur, debiliora enim sunt & à medio remotiora, quare suopte pondere facilius nutant. Si autem breuiora sint ea causa cessante minor fit aut nulla vibratio, quare breuiora feruntur facilius. Dupliciter autem vibratione ipsa, portans offenditur, tum ex causa quam in superiori quæstione consideraui- mus, nempe quod motus sursum deorsum assiduus, pro- gredientis motum impediat, tum etiam quod duplici pressione grauetur ferentis humerus, quod Philosophus non animaduertit. Sitenim oblongum lignum AB, quod humero me- dio

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IN MECHAN. ARIST. PROBL. If a staff AB, which being taken by the hand from B is raised into C, and becomes AC; then, when the part A is set in motion by a rapid vibration from C, the very extremity A is transferred to D, and so on to CD; then by a rapid lowering of the hand the extremity C is transferred to E, and EF is made perpendicular to the horizon; and when this has been done, there will be in the same line which extends to the center of the world both the hand itself that supports it and its center of gravity G; wherefore the hand itself, by the vibration, carries only so much as is precisely the weight of the staff itself. QUESTION XXVII. The doubt is, why, if the same weight be very long, is it carried more difficultly upon the shoulders, even if someone carries it in the middle, than if it is shorter? This question is akin to the preceding one. But the Philosopher says that the cause is not that which he had stated in the previous question, but vibration: for the longer the timbers are, the more their extremities vibrate; for they are weaker and farther from the middle, and therefore, by their own weight, they sway more easily. But if they are shorter, when that cause ceases, the vibration becomes less or none at all, and therefore shorter things are carried more easily. Moreover, in two ways does the vibration itself trouble the carrier: first, from the cause we considered in the previous question, namely that the continual up-and-down motion hinders the motion of the walker; and also because the carrier’s shoulder is burdened with a double pressure, which the Philosopher did not notice. If, then, there is an oblong timber AB, which on the middle shoulder

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EXERCITATIONES. 159 d[omi]no loco sustineatur in C. nutabunt ergo extrema AB, à centro C, valde remota, cadent autem simul A in D, & B in E trahere secum conantes medium C, quare is qui in C sustinet, non modo ligni sustinet pondus ex grauitatis centro quod est in C, sed impetum quoque in ipsa extremorum depressione acquisitum ex ipsa violentia. Illud autem subtiliter consideramus, portantem ex vibratione per interualla deprimi & subleuari fiat enim vibratum lignum ex contrario motu, vbi FCG. alleuiabit igitur eo casu portantem, siquidem impetus ex motu ipso acquisitus, medium C trahat ad superiora. Itaq[ue] cum est in DCE portans plus sustinet in ACD, æquale, in FCG minus, quod vtique demonstrandum fuerat. Est autem quæstio hæc illi familiaris, quam 16. loco explicauimus. QVÆSTIO XXVIII. Quæritur, Cur iuxta puteos celonia faciunt eo quo visuntur modo? Ligno enim plumbi adiungunt pondus, cum alioquin vas ipsum & plenum & vacuum pondus habeat. Respondet optime Philosophus, hauriendi opus duo- bus temporibus diuidi, nempe dum vas ipsum vacuum demittitur, dumque extrahitur plenum: Contingere autem, vacuum facile demitti, plenum autem difficulter extrahi. Expedire nihilominus tardius, hoc est difficilius dimitti vt facilius extrahatur, plumbo nempe coadiuuante, & sane Philosophi solutio est lucidissima. Nos autem luci ipsi lucem aliquam adhuc afferre conabimur. Esto Celonium (Latine Tolenonem appellant) ABC, cuius arrectarium BD, transuersum lignum AC, quod con-

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EXERCISES. 159 Let it be supported at C by the lord. Therefore the extremes AB will swing, being far from center C; but A will fall into D and B into E, while trying to drag the middle C along with them, wherefore the one who supports at C not only bears the weight of the wood from the center of gravity, which is at C, but also the force acquired in the very sinking of the ends from the violence itself. This we consider more subtly: the bearer is depressed and raised by vibration at intervals, for let the wood be vibrated by contrary motion, where FCG. Thus it will lighten the bearer in that case, since the impetus acquired from the motion itself draws the middle C upward. Therefore, when it is in DCE, the bearer bears more; in ACD, equally; in FCG, less, which certainly had to be demonstrated. And this question is familiar to that one, which we explained in the 16th place. QUESTION XXVIII. The question is asked: Why do they make bucket-machines near wells in the way they are seen to be made? For they attach a lead weight to the wood, although otherwise the vessel itself, whether full or empty, has weight. The Philosopher answers very well that the work of drawing water is divided into two times: namely, while the vessel itself, empty, is lowered, and while it is drawn up, full. But it happens that the empty vessel is easily lowered, but the full one is difficult to draw up. Nevertheless, it is advantageous for it to be let down more slowly, that is, more difficultly, so that it may be drawn up more easily, with the lead assisting, and indeed the Philosopher’s solution is most clear. We, however, shall try to bring some light even to the light itself. Let the bucket-machine (which they call, in Latin, a Tolenon) be ABC, whose upright support is BD, the transverse beam AC, which con-

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160 IN MECHAN. ARIST. PROBL. conuertitur, circa puctum seu fulcimentum B, pondus, plumbumue, vbi A, situla E, funi appensa CE. Dico rebus ita constitutis difficilem quidem esse vacuæ situlæ demissionem, facile vero eiusdem extractionem. Vectis diuisi, situlæ, ac ponderis, ad hoc vt fiat æquilibrium, ea debet esse proportio, vt quemadmodum se habet AB ad BC, ita se habeat plenæ situlæ pondus E ad ipsum pondus A, superabit ergo pondus in A situlam vacuam in E nec fiet æquilibrium, itaque vt vacua situla demittatur, tanta vis adhibenda est quantum est ipsius aquæ, qua situla impletur pondus, quæ vis dum apponitur difficilem, vt dicebamus, efficit situlæ vacuæ demissionem. Plena vero situla sit æquilibrium, vnde quantumuis pusilla vi adhibita, situla extrahitur, quasi ex semetipsa ponderis appensi virtute ascendens. Quantum igitur pondus dum vacua demittitur impedit, tantundem plena dum extrahitur, adiuuat. Quæ cum ita sint, si paria sunt difficultas in demittendo, & facilitas in extrahendo, quæ ratio hoc in negotio vtilitatis? Sane situla vacua, manu per funem facile demittitur, plena vero difficile extrahitur, vsu autem Celonij res permutâtur. Corporis enim proprij pondere, dum premit, adiuatur demittens, qui per funem simplicem extrahendo, ab eodem proprij corporis pondere impediebatur. quod quidem ex corporis pondere, auxilium, ingentem parit in extrahendo commoditatem. Quippiam simile accidit, aquas è puteis extrahentibus vsu trochleæ. Sit enim trochlea puteo imminens ABC D, cuius centrum E suspensa quidem in A, funis, cui situla

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160 IN MECHAN. ARIST. PROBL. is turned about, around the point or fulcrum B, the weight, or lead, where A is, and the bucket E hanging from the rope CE. I say that things being thus arranged, the lowering of the empty bucket is indeed difficult, but its lifting is easy. For the lever being divided, the bucket and the weight, in order that equilibrium may be established, the proportion must be such that, just as AB is to BC, so also shall the weight of the full bucket E be to the weight A itself; therefore the weight at A will overcome the empty bucket at E, and equilibrium will not be achieved; thus, in order that the empty bucket may be lowered, a force must be applied equal to the weight of the water with which the bucket is filled, which force, when applied, makes the lowering of the empty bucket difficult, as we said. But the full bucket is in equilibrium, wherefore, however small a force is applied, the bucket is drawn up, as though of itself, rising by the virtue of the attached weight. Therefore whatever weight, while the empty bucket is lowered, hinders it, the same amount, while the full bucket is lifted, assists it. Since these things are so, if there is equal difficulty in lowering and ease in lifting, what is the reason for the usefulness in this matter? Certainly the empty bucket is easily lowered by hand through the rope, while the full one is difficult to lift; but by the use of the Celonium the matter is changed. For by the weight proper to the body, while it presses down, the person lowering is assisted, who, by means of a simple rope, was hindered in lifting by the very weight of his own body. And indeed this assistance from the weight of the body produces a great convenience in lifting. Something similar happens to those drawing water from wells by the use of a pulley. Let there be a pulley overhanging the well ABC D, whose center E, with a rope suspended at A, to which the bucket

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EXERCITATIONES. 161 situla suspenditur FCABG, situla vero G. Est igitur diameter CED, instar libræ, quare vt fiat æquilibrium necesse est capiti funis F, potentiam applicare, quæ sit æqualis pondere situlæ aqua plenæ, itaque extrahens proprijs viribus corporis pondus adjiciens facile situlam aqua plenam extrahit, ex qua re magna extrahentibus fit commoditas. Patet autem diuerso modo extrahentes iuuare Celonium & Trochleam, ibi enim corporis mole adiuuatur demittens vacuam, hic vero qui extrahit plenam aqua situlam. Cæterum Celonij partem BC, qui à fulcimento ad funem longe maiorem esse oportet, ipsa AB, vt situla in profundum possit demitti, quamobrem ita se debet habere pondus in A, ad pondus situlæ plenæ, vt se habet brachium seu pars BC, ad partem BA. Tunc enim ex permutata proportione efficitur æquilibrium. Illud addimus, nouum non esse Architectis Mechanicisque, tum hominum tum animalium vt commodius machinas moueant, adhibere pondera corporum. Nec enim alia ratione mouentur Rotæ illæ, quas ob hanc causam ambulatorias vocant; quarum vsus ad Mangana, ad extrahendas è puteis aquas, & ad farinarias quoque molas agitandas adhibetur. Porro Tollenonem bellicam Machinam à Celonio tum forma tum potestate nihil differre, videre est apud veteres Mechanicos, Heronem Byzantium, & alios, apud neotericos vero hac de re agunt Daniel Barbarus in Vitruuium, & lustus Lipsius in librum quem de bellicis machinis edidit, elegantissimum. X QVÆ-

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EXERCISES. 161 the bucket is suspended at FCABG, and the bucket itself at G. Therefore CED is the diameter, like a balance beam; wherefore, for equilibrium to be made, it is necessary to apply to the head of the rope F a power equal to the weight of the bucket full of water. Thus the person pulling, by adding the weight of his own body to his strength, easily draws out the bucket full of water, by which action great convenience comes to those who pull. But clearly Celonium and the Pulley aid those drawing in different ways: for there the body’s mass assists in lowering the empty bucket, here however the one who draws up the bucket full of water. Moreover, the part BC of the Celonium, which from the support to the rope ought to be much greater, must be to AB itself, so that the bucket may be let down into the depths; wherefore the weight at A ought to relate to the weight of the full bucket as the arm, or part BC, relates to part BA. For then equilibrium is produced by the inverted proportion. We add this: it is not new to Architects and Mechanics to apply the weights of bodies, both of men and of animals, so that machines may be moved more conveniently. For the carts called ambulatory are moved by no other means; their use is employed for Mangana, for drawing water out of wells, and also for turning flour mills. Furthermore, that the Tollenon, a military machine, differs in nothing from the Celonium either in form or in power, may be seen among the ancient Mechanicians, Heron of Byzantium, and others; among the more recent, Daniel Barbarus in Vitruvius, and Justus Lipsius in the book he published on military machines, most elegantly. X W H A-

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162 IN MECHAN. ARIST. PROBL. QVAESTIO XXIX. Dubitatur, Cur quando super ligno, aut huiusmodi quopiam, duo portauerint homines, idem pondus non æqualiter premuntur, sed ille magis cui vicinius fuerit pondus? Soluit Aristoteles, inquiens, lignum esse vectem, pondus vero fulcimentum; res quæ mouetur is qui ponderi est proximior: mouens vero qui remotior. Itaque quo magis remotus est à pondere, hoc est, à fulcimento is qui mouet, eo violentius is premitur qui altera vectis parte eaque breuiori, mouetur. Esto lignum AB, pondus Cappensum in E, vicinius extremo B quam ipsi A, sit auté portatium alter quidem AF, alter vero BG, Imaginemur itaque locum E à pondere ita figi & deprimi, vt sursum quidem ferri nequaquam possit, circa vero punctum E, ceu circa centrum fulcimentum ue ipsum vectem conuerti. Lignum ergo AB vectis: mouens potentia A, pars vectis à potentia ad fulcimentum AE pars eiusdem quæ à fulcimento ad rem motam EB, & quoniam quanto longior est pars vectis EA ipsa EB, eo facilius potentia quæ est in A, operatur in id quod est in B, si res ad proportiones redigatur, erit potentia in A, ad id quod mouetur seu premitur in B, vt pars vectis EB ad partem EA, sed maior est AE ipsa EB, ergo maiorem partem sustinet ponderis, & plus premitur is qui in E, & qui mouet in A. Hæc fere Philosophi est sententia: Picolomineus vero Paraphrastes apposite duos vectes in vnico li- gno

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162 IN MECHAN. ARIST. PROBL. QUESTION XXIX. It is doubted why, when two men have carried something on a beam, or on some such thing, the same weight is not pressed equally, but rather the one is more burdened who is nearer to the weight? Aristotle solves it by saying that the beam is the lever, and the weight the prop; the thing that is moved is the one nearer to the weight, but the mover is the one farther away. Therefore, the farther the mover is from the weight, that is, from the support, the more forcibly is the one pressed who is moved on the other part of the lever, and on the shorter one. Let AB be the beam, with the weight suspended at E, nearer to end B than to A itself; let there be carriers, one AF, the other BG. Let us imagine, therefore, that the place E is so fixed and pressed by the weight that it can by no means be carried upward, but rather that around the point E, as though around the center, the support itself or the lever is turned. Therefore the beam AB is the lever; the moving power is A; the part of the lever from the power to the support is AE; the part of the same from the support to the thing moved is EB; and since the longer the part of the lever EA is than EB, the more easily the power which is in A acts upon that which is in B, if the matter is reduced to proportions, the power in A will be to that which is moved or pressed in B as the part of the lever EB to the part EA. But AE is greater than EB; therefore it sustains the greater part of the weight, and the one who is in E is pressed more, and the one who moves in A. This is more or less the Philosopher’s opinion; but Picolomineus, the Paraphrast, aptly makes two levers in a single beam.

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EXERCITATIONES. 163 gno considerat, alterum AB, alterum BA, in primo A est mouens B, motum in secundo B, mouens A vero motum in quibus vectibus semper idem & commune fulcimentum E. Et quoniam in proposito diagrammate breuior est pars vectis EB, quæque à mouente ad fulcimentum, parte illa quæ ab eodem fulcimento ad rem motam, minus operatur B in A, quam A in B, & ideo qui in B mouetur plus premitur, contra vero quia maior est pars EA ipsa parte EB, magis operatur qui in A in ipsum B, quam econtra. Et sane consideratio hæc subtilis est & ingeniosa, & quæ si recte intelligatur, quatenus ad proportiones & effectum ipsum demonstrandum pertinet, à veritate ipsa non abhorret, Quicquid tamen sit, Mechanice magis hoc pacto quæstio diluetur. Dicimus enim, pondus quidem vere esse pondus, non autem fulcimentum, vt sibi fingebat Aristoteles: lignum vero vectem, duo autem qui pondus sustinent pro duplici fulcimento haberi, vtrisque enim vectis cum appenso pondere innititur. Potest etiam alter eorum pro potentia mouente, alter vero pro fulcimento, & sic vicissim. Est autem, quomodo cunque res accipiatur, pondus inter fulcimentum & potentiam. Quare ex ijs quæ demonstrauit G. V bald. de hoc vectis genereloquens, vt se habet AE pars ad AB vectem totum, ita potentia quæ sustinet in B, ad pondus appensum in E, & vt BE ad BA ita potentia quæ sustinet in A ad pondus quod in E. At minor est proportio BE, ad BA, quam AE ad AB, quare magis superatur pondus in E à potentia quæ in A, quam à potentia quæ in B, & ideo plus ponderis sustinet terens in B, quam ferens in A, quod fuerat demonstrandum. Hinc colligimus, pondere in medio vecte appenso ferentes æqualiter sustinere, propterea quod totius vectis ad partes ipsas proportio sit eadem, vel æqualis. X 2 Pul-

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EXERCISES. 163 he considers one AB, the other BA; in the first A is moving B, in the second B moving A. Yet in these levers there is always the same and common fulcrum, E. And since in the proposed diagram the shorter is part of the lever EB, that which is from the mover to the fulcrum, than that part which is from the same fulcrum to the thing moved, it works less B in A than A in B, and therefore he who is moved in B is pressed more; on the contrary, because the part EA is greater than the part EB, he who acts in A upon B itself works more than the reverse. And indeed this consideration is subtle and ingenious, and if rightly understood, so far as it concerns the proportions and the demonstration of the effect itself, it does not depart from the truth itself. However that may be, the question will be more conveniently resolved in this mechanical way. For we say that the weight is truly a weight, but not a fulcrum, as Aristotle imagined; the wood is the lever, and the two supports of the weight are to be taken as a double fulcrum, for in both cases the lever rests with the attached weight upon them. One of them can also be taken as the moving power, the other as the fulcrum, and so vice versa. Now, however the matter is taken, the weight lies between the fulcrum and the power. Wherefore, from what G. V. Bald. has demonstrated on this kind of lever, as AE is to AB, the whole lever, so is the power which sustains in B to the weight suspended in E; and as BE is to BA, so is the power which sustains in A to the weight which is in E. But the proportion of BE to BA is smaller than that of AE to AB, therefore the weight in E is overcome more by the power which is in A than by the power which is in B; and therefore the one bearing in B sustains more weight than the one bearing in A, which was to be demonstrated. Hence we conclude that, with the weight suspended in the middle of the lever, the supports bear it equally, because the proportion of the whole lever to those parts is the same, or equal. X 2 Pul-

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164 IN MECHAN. ARIST. PROBL. Pulchre autem dubitari potest, an idem prorsus contingat, si alterum eorum qui sustinent, sit statura quidem procerior, alter vero humilior. Sit enim vectis AB, in cuius medio pondus H liberè appensum ex C, alter portantium procerior AD, humilior vero BE. sit autem horizontis planum DE, demittatur à puncto Cad horizontem perpendicularis, ipsis vero AD, BE, æquidistans CF. Transibit autem per ipsius ponderis, grauitatis centrum H. Dico igitur, nil referre quatenus ad pondus sustinendum pertinet, vtrum portantes sint statura pares vel ne. Ducatur enim horizonti æquidistans GB, secans perpendicularem CF in I. Quoniam igitur AG æquidistans est ipsi CI erit vt AC ad CB per 4. sexti elem, ita GI ad IB. Sunt ergo GI, IB inter se æquales. Intelligatur itaque pondus H, solutu[m] à puncto C appensum esse libere ex puncto I, hoc est, ex medio vectis GB, æqualiter ergo diuisum erit pondus inter portantes, licet alter procerior, alter vero statura pumilio, quod fuerat demonstrandum. Si autem pondus ita vecti alligatum sit vt libere non pendeat, vecte ex vna parte eleuato, ex altera vero depresso, grauitatis centrum ad eam partem verget quæ magis ab horizonte attollitur, & ad eam ipsam partem vectis à pondere ad sustinentem sit breuior. Esto enim vectis AB, cuius medium C, pondus vecti in Calligatum CFG, cuius grauitatis centrum H eorum qui portant procerior AB, humilior BE, horizontis planu[m] DE. Demittatur per centrum H horizonti perpendicularis IHK, secans vectem quidem in I, horizontis vero planum

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164 IN MECHAN. ARIST. PROBL. And it can rightly be doubted whether the same thing happens if one of those who support it is indeed taller in stature, but the other shorter. Let there be a lever AB, in the middle of which a weight H is freely suspended from C; one bearer is taller, AD, the shorter BE. Let the plane of the horizon be DE; from point C let a perpendicular be dropped to the horizon, ipsis and let CF be equally distant from AD and BE. It will pass through H, the center of gravity of the weight itself. I say, therefore, that as far as bearing the weight is concerned, it makes no difference whether the bearers are equal in stature or not. For let GB be drawn parallel to the horizon, cutting the perpendicular CF at I. Since AG is parallel to CI, by prop. 4 of book 6 of the Elements it follows that as AC is to CB, so GI is to IB. Therefore GI and IB are equal to one another. Let it then be understood that the weight H, being released from point C, is freely suspended from point I, that is, from the middle of the lever GB; thus the weight is equally divided between the bearers, although one is taller and the other a dwarf in stature, which was to be demonstrated. But if the weight be attached to the lever in such a way that it does not hang freely, the lever being raised on one side and depressed on the other, the center of gravity will tend toward that part which is lifted higher above the horizon, and toward that same part of the lever the distance from the weight to the bearer will be shorter. Let there be a lever AB, whose middle is C, and a weight CFG attached to the lever, whose center of gravity is H; let the bearers be the taller AB and the shorter BE, and the plane of the horizon DE. Through the center H let the perpendicular IHK be dropped to the horizon, cutting the lever at I, and the plane of the horizon...

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EXERCITATIONES. 165 num in K. Post hæc intelligatur pon- dus solutum quidem àpuncto C, ap- pensum vero expuncto I. Stabit igitur ex definitione centri grauitatis nec si- tu suo mouebitur. Dico autem par- tem AI ipsa IB esse breuiorem, hoc est, punctum I cadere inter C & A. Si e- nim non cadat, vel cadet in C, aut in- ter C & B, cadat autem si fieri potest in C. Erit igitur CHK horizonti perpendicularis, sed ei- dem perpendicularis AD. Erunt igitur BCK BAD anguli inter se æquales, sed ipsi BAD angulo æqualis est CIH, quare & BCH ipsi CIH æqualis erit. Producto igitur la- tere IC trianguli ICH erit exterior angulus æqualis inte- riori ex opposito, quod est absurdum. non ergo I cadet in C. Eadem autem ratione monstrabitur non cadere inter CB, cadet ergo inter CA, & ideo minor AI ipsa IB. Itaque vt se habet BI ad BA, ita potentia in A ad pondus in I, sed maiorem proportionem habet BI ad BA, quam IA ad AB. Ergo minor potentia requiretur in B quam in A, & sane pars IB respondet potentiæ sustinenti in A, at IA potentiæ sustinenti in B, minor est autem AI ipsa IB. ergo maior po- tentia requiritur in B, quam in A, quod fuerat demon- strandum. Hoc item concludetur, si portantes statura quidem pares fuerint, sed per planum ambulent horizonti accliue aut decliue. Si enim pondus libere pendeat, vectis partiu[m] proportio non mutabitur; si autem libere non pendeat, is magis laborabit qui in ascensu præibit, minus vero qui in descensu. Hinc quoque Carrucarum ratio pendet, quæ dupli- ci manubrio vnica rota vulgo sunt in vsu, pro vecte enim habentur, cuius fulcimentum ad contactum plani & ro- tæ; X 3

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EXERCISES. 165 Now let the weight afterward be understood as loosened indeed from point C, but hung up from point I. It will therefore stand, by definition of the center of gravity, and will not move from its place. But I say that part AI itself is shorter than IB, that is, that point I falls between C and A. For if it does not fall there, it will either fall at C, or between C and B; let it then fall, if possible, at C. Therefore CHK will be perpendicular to the horizon, but AD is perpendicular to the same. Therefore the angles BCK and BAD will be equal to one another, but CIH is equal to the angle BAD; wherefore BCH also will be equal to CIH. Therefore, when side IC of triangle ICH is produced, the exterior angle will be equal to the interior opposite one, which is absurd. Therefore I will not fall at C. And by the same reasoning it will be shown not to fall between C and B; therefore it will fall between C and A, and so AI is less than IB. Thus, as BI is to BA, so is the power at A to the weight at I; but BI has a greater proportion to BA than IA has to AB. Therefore a smaller power will be required at B than at A; and indeed part IB corresponds to the sustaining power at A, while IA corresponds to the sustaining power at B, but AI itself is less than IB. Therefore a greater power is required at B than at A, which was to be demonstrated. This likewise will be concluded if the bearers are equal in stature, but walk on a plane inclined upward or downward to the horizon. For if the weight hang freely, the proportion of the lever’s parts will not be changed; but if it does not hang freely, he who goes before on the ascent will labor more, but he who goes before on the descent will labor less. Hence also depends the arrangement of wheelbarrows, which are commonly in use with a double handle and a single wheel; for they are held to be a kind of lever, whose support is at the point of contact of the plane and the wheel; X 3

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166 IN MECHAN. ARIST. PROBL. tæ; potentiæ vero ad extremitatem duplicis manubrij. Reducitur enim ad idem genus vectis, in quo pondus inter fulcimentum est & potentiam. quo igitur minor fuerit proportio partis vectis quæ à centro grauitatis ad ipsum fulcimentum, ad totum vectem eo facilius pondus eleuabitur. Cur autem difficilime hæ per accliue horizonti planum pellantur, duplici fit de caussa, tum quia grauitatis centrum ad ipsum portantem seu pellentem vergit, & ideo pars quæ à fulcimento ad centrum grauitatis ponderis fit maior, tum etiam quoniam ipsum graue contra sui naturam sursus pellitur ferturque. Quærere ad hæc quispiam posset, Cur Baiuli magna ferentes pondera, curui incedant? Dixerit autem aliquis, ponderis grauitate eos deprimentis id fieri. Nos autem duplici item de caussa id fieri putamus, tum ea quam considerauiimus, tum etiam alia, nempe vt grauitatis centrum ipsius ponderis quod sustinent, in perpendiculari collocent, ne si extra ponatur is qui fert à centro extra fulcimentum posito, ad eam partem ad quam vergit trahatur, & pondere ipso opprimatur. Eadem de caussa sit quoque vt ij qui magna pondera sinistro ferunt humero, in dextram partem inclinentur, qui vero dextro, contrario modo se habeant, æquatur enim pondus eo pacto, & grauitatis centrum in ipsa perpendiculari collocatur. QVÆSTIO XXX. Cur assurgentes omnes foemori tibiam ad acutum angulum constituamus & pectori thoraciue similiter foemur, quod nî fiat haudquaquam surgere poterunt? A It Philosophus, forte id fieri, quod æqualitas sit omnino quietis caussa, rectum vero angulum quietis angu-

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166 IN MECHAN. ARIST. PROBL. and the power at the end of a double handle. For it is reduced to the same kind of lever, in which the weight is between the fulcrum and the power. Therefore, the smaller the proportion of that part of the lever which is from the center of gravity to the fulcrum, in relation to the whole lever, the more easily the weight will be lifted. But why are these things most difficultly driven uphill on a level plane? This happens for two reasons: both because the center of gravity inclines toward the carrier or pusher himself, and therefore the part from the fulcrum to the center of gravity of the weight becomes greater; and also because the heavy object itself is pushed and carried upward against its nature. In addition to this, someone might ask: why do porters, carrying heavy weights, walk bent over? Someone might say that this is due to the weight pressing them down. But we think it happens for two reasons as well: both for the reason we have considered, and also for another, namely, so that they may place the center of gravity of the very weight they are supporting on the perpendicular, lest, if the one who carries it is placed outside the fulcrum with his center outside it, he be drawn toward that side to which he inclines and be oppressed by the weight itself. For the same reason, those who carry heavy loads on the left shoulder incline to the right side, while those who carry them on the right shoulder do the opposite; for in this way the weight is balanced, and the center of gravity is placed on the perpendicular itself. QUESTION XXX. Why is it that, when rising, we make the leg and shin at an acute angle, and likewise the thigh with the chest or torso; for if this is not done, they will by no means be able to get up? The Philosopher says that perhaps this happens because equality is altogether the cause of rest, but a right angle of rest...

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EXERCITATIONES. 167 angulum esse, & stationem facere, nec alia de caussa stantem ipsi terræ esse perpendicularem, & ideo caput & pedes in eadem linea habere, sedentem vero non item. T[ame]n autem à sessione surrectionem fieri, cum caput & pedes in vna linea collocantur, quod sane sit cum pectus & crura acutum cum ipso foemore angulum faciunt. Esto enim stans AB horizonti IBK perpendicularis, cuius caput A, pedes vero B, sedeat modo sitque eius cum capite Thorax CD, foemur DE, crura EF, sintque CDE, DEF anguli recti, quibus ita constitutis non sunt in eadem linea caput C & pedes F. Surgere itaque non poterit sedens, propterea quod partes omnes corporis non sint horizonti perpendiculares. Ad hoc autem vt surrectio fiat, necesse est vt sedens retrahat quidem pedes in H, & pectore inclinato acutum cum foemore angulum constituat GDE, quo casu fient in eadem recta linea, eaque horizonti perpendiculari caput in G, & pedes in H, ex cuius situs natura commoda fiet ab ipso sedente surrectio. Hæc fere, licet alijs ab eo verbis explicata, ipsius est Philosophi sententia; quæ licet vera sit, non tamen ex proprijs, hoc est, Mechanicis principijs est petita. quod quidem nos facere conabimur. Dicimus autem primo, sedentem non ideo quiescere, vt sentit Aristoteles, quod rectus angulus quietis sit caussa, sed propterea quod eius thoracis tum etiam foemorum pondus ab ipsa sede sustineantur; crura vero & pedes ideo non laborent, quod partim suspensa sint, partim solo ipsi innitantur. Quare cum corpus totum nec se susti-

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EXERCITATIONES. 167 that it is an angle, and that standing is to be explained by this, and by no other cause, namely, that one who stands is perpendicular to the earth, and therefore has head and feet in the same line, whereas one who sits does not. Yet from sitting there is a rising, when head and feet are placed in one line, which surely happens when the chest and the legs, together with the thigh itself, make an acute angle. For let a standing man AB be perpendicular to the horizon IBK, whose head be A and feet B; let him sit, and let his thorax be CD with the head, the thigh DE, and the legs EF, and let CDE and DEF be right angles. When these are so arranged, the head C and feet F are not in the same line. The seated man therefore cannot rise, because all the parts of the body are not perpendicular to the horizon. But for rising to occur, it is necessary that the seated man should draw back the feet indeed to H, and with the chest inclined form an acute angle GDE with the thigh; in which case the head will be at G and the feet at H in the same straight line, and that line perpendicular to the horizon; and from the nature of this position the seated person will conveniently rise by himself. This is, in substance, the philosopher's opinion, though explained by him in different words; and although it is true, it is nevertheless not derived from proper, that is, mechanical principles, which we shall endeavor to do. We say first, then, that the seated man is at rest not for the reason Aristotle thinks, namely, that the right angle is the cause of rest, but rather because the weight of his thorax and also of his thighs is supported by the seat itself; while the legs and feet do not labor because they are partly suspended and partly rest upon the ground. Therefore, when the whole body neither sustains it-

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168 IN MECHAN. ARIST. PROBL. sustineat, nec à pedibus sustineatur, sit quies & lassitudinis alleuatio. Natura autem ideo commodam hominibus sessionem facere voluisse inde apparet, quod clunes, quibus tota superior pars, & grauiornititur, carnosam fecerit, & ceruicalis cuiusdam instarmollem & facilem. Sed nos ad quæstionem. Esto enim stans AB, cuius caput A, Thorax AC, foemora CD, crura DB, pedes vero B, centrum vero grauitatis in ipso Thorace E. Modo sedeat, sitque caput in F, Thorax FG, foemora GH, crura HI, pedes I, grauitatis vero centrum vbi K. Producatur recta FG in L, sitque FL horizonti perpendicularis. Centrum ergo grauitatis K fulcitur puncto G, hoc est, puncto L, in quo posteriores pedes ipsius sedis solo hærent. efficit autem sedens duos rectos angulos FGH, GHI. Rebus igitur ita dispositis seruatis rectis angulis, non fiet surrectio, & id quidem non ideo quod, vt ait Philosophus, æqualitas & rectitudo angulorum quietis sit caussa, sed propterea quod centro grauitatis extra pedum fulcimé-tum constituto, non habet centrum stabilem locum cui in actu surrectionis hæreat, & fulciatur, vnde sit vt si sedenti subtrahatur sedes remoto prohibente, sedens prorsus corruat. Modo retrahat quiesedet crura, & pedes ponat in M, à puncto autem M horizonti perpendicularis erigatur MN. erit ergo fulcimentum in M, sed adhuc surgere non poterit, centro grauitatis adhuc extra lineam MN, quæ per fulcimentum est, constituto. Reclinetur autem pectus ad anteriora, & cum foemore acutum angulum faciat sitque vbi GO, erit igitur grauitatis centrum vbi P, hoc est, in ipsa perpendiculari NM, fiet igitur inde commoda surre-

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168 IN MECHAN. ARIST. PROBL. it is supported, and is not supported by the feet, it is a rest and relief from fatigue. But nature appears to have wished to make sitting convenient for human beings for this reason, that she has made the buttocks, on which the whole upper part, and the heavier part, rests, fleshy, and, as it were, soft and easy like a kind of cervical support. But let us return to the question. Let there be, then, a standing body AB, whose head is A, the thorax AC, the thighs CD, the legs DB, the feet B, and the center of gravity in the thorax itself at E. Now let him sit, and let the head be at F, the thorax FG, the thighs GH, the legs HI, the feet I, and the center of gravity where K. Let the straight line FG be produced to L, and let FL be perpendicular to the horizon. Therefore the center of gravity K is supported by the point G, that is, by the point L, where the rear legs of the seat itself rest on the ground. The seated person, however, makes two right angles, FGH and GHI. Therefore, with things thus arranged and the right angles preserved, no rising will occur, and this indeed not because, as the Philosopher says, equality and straightness of angles are the cause of rest, but because, the center of gravity having been placed outside the support of the feet, it has no fixed place in which to cling and be supported during the act of rising; whence it follows that if the seat is taken away from one who is sitting, the obstacle being removed, the sitter falls entirely. But if the one at rest draws back his legs and places his feet at M, and from the point M a perpendicular MN to the horizon is erected, then the support will be at M, but he still will not be able to rise, since the center of gravity is still placed outside the line MN, which is through the support. But let the chest be leaned forward, and let it make an acute angle with the thigh, and let it be where GO; the center of gravity will then be where P, that is, on the very perpendicular NM; from this there will therefore be a convenient ris-

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EXERCITATIONES. 169 surrectio, propterea quod in eadem linea facta sint, grauitatis centrum P, & fulcimentum ipsum M. Acutum vero angulum in surrectione necessarium esse clare patet, non autem effectus ipsius esse causam, vt videtur sensisse Aristoteles; nisi dicamus, causam esse causæ, siquidem acuti qui fiunt anguli centrum & pedes in eadem linea collocant, quicquid tamen sit, nos ideo surrectionem fieri dicens, quod immutatis angulis centrum grauitatis supra fulcimentum, fulcimento vero sub ipso grauitatis centro collocetur, & hæc est causa proxima. Hæc nos ad Aristotelem. Modo quasdam alias quæstiones, nec inutiles sed & eas non iniucundas quoque proponemus. Primum igitur quærimus, Cur hominum & cæterorum animalium, quæ aliquando erecto corpore incedunt, pedes non quidem breues sint & rotundi, sed longiores potius, & in inferiorem partem porrecti? Item cur magis ad digitos quam ad calcaneum porrigantur? Esto homo animalue quodpiam stans AB, cuius pes CD, pedis pars quæ ad digitos BC. quæ vero ad calcaneum BD foemoris vertebra E, centrum vero grauitatis ipsius corporis F. Primum igitur statuendum est, hominem & cætera fere animalia à Natura facta esse vt ad anteriora moueantur, & ideo omnes fere quod in senioribus manifeste apparet, ad anteriora ex ipsa corporis dispositione vergant. Itaque dum qui stat horizonti prorsus est perpendicularis, grauitatis centrum F in ipsa perpendiculari constituitur quæ ad mundi centrum AB, & ideo corporis moles pondusque fulcitur puncto B. Modo fiat ex vertebra E thoracis AE, inclinatio in anteriora, in GE & grauitatis centrum D diluetur in H, & per H perpendicularis demittatur Hl, non erit ** extra pedis ful- cimen- Y

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EXERCISES. 169 rising, because they have been made in the same line, the center of gravity P, and the support itself M. But that an acute angle is necessary in rising is clearly apparent, yet not that it is the cause of the effect itself, as Aristotle seems to have thought; unless we say that the cause is the cause of a cause, since the acute angles that are formed place the center and feet in the same line. Whatever the case, we say that rising takes place for this reason, because, the angles being changed, the center of gravity is placed above the support, and the support itself beneath the center of gravity; and this is the proximate cause. Thus much on Aristotle. Now we shall propose certain other questions, not useless, and not unpleasant either. First, then, we ask: Why do the feet of men and of the other animals that at times walk upright not indeed tend to be short and round, but rather longer, and extending downward? Likewise, why do they extend more toward the toes than toward the heel? Let there be a man or some animal standing AB, whose foot is CD, the part of the foot toward the toes BC, and that toward the heel BD; the vertebra of the thigh E, and the center of gravity of the body itself F. First, then, it must be established that man and the other animals nearly all were made by Nature so that they move toward the front, and therefore nearly all, as is plainly seen in the elderly, incline toward the front by the very disposition of the body. Therefore, when one who is standing is altogether perpendicular to the horizon, the center of gravity F is placed upon the very perpendicular line which goes to the center of the world AB, and therefore the mass and weight of the body are supported by point B. Now if from the vertebra E of the chest AE there be an inclination toward the front, in GE, and the center of gravity D be displaced into H, and through H a perpendicular be let down Hl, there will not be ** outside the support of the foot- Y

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170 IN MECHAN. ARIST. PROBL. cimentum BC. Stabit ergo qui ita inclinatur, nec corruet: si autem adhuc propendeat magis, fiatque in KE, centro grauitatis constituto in M, ducatur per M perpendicularis ML, quare quoniam linea ML extra pedis fulcimentum cadit, corruet qui eo pacto inclinatur nec sustinebitur. Cur igitur natura animalibus quę erecto corpore ambulant, pedes in anteriora porrectos fecerit, hinc clare patet. Hinc etiam ceu consectarium habemus, cur homines si impellantur, magis ad casum in posteriora quam in anteriora sint proni. Nec non etiam cursimæ, vrsi, & si quæ cætera eiusmodi animalia diutius erecto corpore ambulare nequeant, nempe ideo quodeorum corporum moles valde in anteriora propendeat, nec ita commodo, vt humanis euenit corporibus, pedum ipsorum basibus fulciantur. Quærere item haud importune possumus, Cur grallatores non stent erecti, nisi assidue moueantur? Solutio facilis. grallæ etenim duobus tantum punctis solum tangunt, nec porrecti beneficio, quod ambulantibus accidit, vti possunt. quamobrem grauitatis centrum sit extra fulcimentum, & ideo coguntur grallatores assiduo motu grauitatis centro fulcimentum supponere, quod dum sit, à casu prohibentur. Potest autem id quod fulcitur, tripliciter fulciri, nè pe aut puncto, aut linea, aut superficie. Quod puncto fulcitur, nulla re impediente ad quamuis partem cadere potest, centrum siquidem, motus, punctum est. Quod linea fulcitur ad duas tantum partes, easque oppositas, habet casum. sit illud superficies, corpusue in latus constitutum. Esto

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170 IN MECHAN. ARIST. PROBL. cement BC. Therefore the one who is inclined in this way will stand and will not fall. But if he leans still more, and comes into KE, the center of gravity having been established at M, let the perpendicular ML be drawn through M; wherefore, since the line ML falls outside the support of the foot, the one who is inclined in that way will fall and will not be sustained. Why then nature should have made the feet of animals that walk with an upright body extend forward is clearly evident from this. From this we also have, as a consequence, why men, if they are pushed, are more prone to fall backward than forward. And also why racehorses, bears, and if there are any other such animals, are unable to walk for long with an upright body: namely because the mass of their bodies leans very much forward, and is not supported in so convenient a way, as happens with human bodies, by the bases of their feet. We may also not inappropriately inquire why stilt-walkers do not stand upright unless they are continually moving? The solution is easy. For the stilts touch the ground only at two points, and they cannot make use of the advantage of being extended, which occurs to those who walk. Wherefore, since the center of gravity lies outside the support, they are compelled by continual motion to place the support under the center of gravity, and while this is done, they are prevented from falling. Now that which is supported can be supported in three ways: namely, by a point, or by a line, or by a surface. That which is supported by a point, if nothing hinders it, can fall to any side whatever; for the center of motion is a point. That which is supported by a line has the possibility of falling only to two sides, and those opposite; let it be a surface, or a body placed on its side. Let it be

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EXERCITATIONES. 171 Esto horizontis planum ABCD, cui ad rectos angulos insistat superficies EFGH, secundum latus FG. Sit autem ipsius superficie grauitatis centrum I. à quo ad horizontis planum perpendicularis demittatur IK. Cadet autem in lineam FG. per propos. 38. vndecimi elem. & anguli IKG IKF rectierunt. Itaque superficie EFGH circa lineam FKG ceu circa axem mota punctum I peripheriam describet LIM, & siquidem cadat ad partes CD, grauitatis centrum erit vbi M. Si vero ad partes AB, fiet vbi L. Sunt autem LKM p[ro]p[ter] eta in recta LKM, quæ quidem communis sectio est plani horizontis, & plani per IKLM, transeuntis. Idem quoque de corpore dicimus in latus collocato. Esto enim cubus LO, cuius grauitatis centrum R, latus vero quo fulcitur, NO, Si enimita collocetur, vt interna superficies LNOQ ad rectos angulos horizonti sit constituta, demissa perpendicularis à puncto R, cadet in S, in ipsa linea NSO. Cadente igitur corpore fiet motus circa lineam NO, centro grauitatis interim peripheriam TRV. describente. Hinc animaduertere licet, Cur prouidissima Naturanulli animantium vnicum dederit pedem, sed aut quaternos, aut saltem binos, & binos quidem ipsos virtute quaternos, siquidem in quolibet animantium bipedum pede

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EXERCISES. 171 Let ABCD be the plane of the horizon, and let the surface EFGH stand upon it at right angles, along the side FG. Let the center of gravity of this surface be I. From this point let the perpendicular IK be let fall to the plane of the horizon. It will fall on the line FG, by proposition 38 of the eleventh book of the Elements, and the angles IKG, IKF are right angles. Therefore, if the surface EFGH is moved around the line FKG, as around an axis, the point I will describe the circumference LIM; and if it falls toward the parts CD, the center of gravity will be where M is. But if toward the parts AB, it will be where L is. Now LKM are in a straight line LKM, which indeed is the common section of the plane of the horizon and the plane passing through IKLM. The same thing also we say of a body placed on its side. For let LO be a cube, whose center of gravity is R, and the side on which it is supported, NO. For if it is so placed that the inner surface LNOQ is set at right angles to the horizon, the perpendicular let fall from point R will fall at S, on the very line NSO. Therefore, when the body falls, there will be motion around the line NO, while the center of gravity in the meantime describes the circumference TRV. From this it may be noticed why provident Nature has given to no animals a single foot, but either four, or at least two; and even two in such a way as to have the force of four, since in every foot of biped animals

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IN MECHAN. ARIST. PROBL. pede duo saltem puncta considerantur, quibus ipsum animal fulcitur. Sint enim humani pedis vestigia A, B, C, D, in vtroque igitur duo puncta considerantur, A, B, C, D, illa quidem addigiros, hæc autem ad calcaneum. Idem quoque in auium pedibus obseruatur, ex quibus concludimus, bipedum omnium fulcimentum esse quadruplex. Porro quadrupedia eo quod tota corporis mole ad inferiora vergant, quatuor fulcimenta, eaque distincta, & commode ab inuicem remota eademmet Natura præparauit. Eadem quoque in artificialibus consideramus. Sit enim vas quodpiam ABC, cuius pes vnicus, isque rotundus BC, grauitatis vero centrum D. Quoniam igitur in pedis ipsius peripheria, infinita puncta intelligantur, dici quodammodo potest vas ipsum infinitis fere punctis, licet pes vnicus sit, sustineri. Nonnulla autem corpora artificialia quatuor pedibus sustinentur, vt mensæ quæda, nonnulla etiam tribus, vt tripodes, qui nomen ab ipso pedum numero sortiuntur. Sit enim triangulum EFG, cuius centrum grauitatis H, nitatur autem tribus punctis I, K, L, stabit igitur. Si autem duobus tantum; non stabit. ducta enim IK si punctis tantum IK innitatur, constituto grauitatis centro extra

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IN MECHAN. ARIST. PROBL. In the foot at least two points are considered, on which the animal itself is supported. For let the footprints of the human foot be A, B, C, D; therefore in each there are considered two points, A, B, C, D, those indeed toward the toes, these however toward the heel. The same is also observed in the feet of birds, from which we conclude that the support of all bipeds is fourfold. Moreover, quadrupeds, because with the whole mass of the body they incline downward, Nature itself has prepared four supports, and those distinct, and conveniently separated from one another. The same things we also consider in artificial things. Let there be some vessel ABC, whose foot is single, and round BC, while the center of gravity is D. Since therefore in the periphery of the foot itself infinite points are understood, it can in a manner be said that the vessel itself is supported by almost infinite points, although the foot is single. But some artificial bodies are supported by four feet, as certain tables; some also by three, as tripods, which derive their name from the very number of feet. Let there be a triangle EFG, whose center of gravity is H, and let it rest on three points I, K, L; it will therefore stand. But if only on two, it will not stand. For if IK is drawn, and it rests only on the points IK, with the center of gravity established outside

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EXERCITATIONES. 173 extra fulcimentum IK, verget cedens versus partes, L, Si autem innitatur punctis IL, cadet ad partes K. Si vero ipsis KL, cadet ad partes I. Ex quibus apparet, inanimata corpora aut vnico pede plurium virtutem habente, aut saltem tribus actu, vt sustineantur, indigere. Hinc etiam patet, cursenes, imbecilles, curui, & pedibus capti, baculi baculorumue fulcimento egeant, etenim cum hi debiles sint, & in anteriorem partem magnopere propendeant, ne grauitatis centrum extra fulcimentum fiat, baculo vel baculis indigent, quibus centrum ipsum fulciatur. Cæterum cur duplici genu ingeniculati difficile in eo situ permaneant, ea causa est, quod grauitatis centrum in thorace constitutum, duobus genibus fulciatur, eosque premat. quæ quidem genua eo quod natura apta nata non sint, veluti pedes, ad sustinendam corporis molem laborant, idque eo magis, quod cum ossea sint, cutem inter ossium & plani duritiem constitutam, accidit arctari, & ideo dolorem & molestiam ingeniculatis facere. Si autem vnico tantum genu quispiam nitatur, difficultatem sentiet longe minorem. Triplici enim fulcimento eo casu ingeniculatus fulcitur. Sit enim ingeniculatus ABCDE, cuius grauitatis centrum F. dextrum vero genu, cui nititur D, sinistrum vero, quod eleuatur B. Tribus ergo fulcimentis ingeniculatus vt diximus, sustinetur, CDE. Diuiditur itaque pondus in tres partes, & ideo singulæ minus fatigantur. Magis tamen laborat punctum D, vtpote illud, cui ad perpendiculum F grauitatis centrum innititur. Vti que illud quoque mirabile est, Aues dormientes vnico tantum pede fulciri, & quod magis mirum est, dormientes Y 3

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EXERCISES. 173 without the support IK, it will lean, yielding toward the parts L. But if it rests on the points IL, it will fall toward K. If, however, on KL itself, it will fall toward I. From which it is evident that inanimate bodies, whether with a single foot having the strength of several, or at least with three acting at once, need to be sustained. Hence it is also clear that the old, the weak, the crooked, and those deprived of feet need the support of a staff or staffs; for since these are weak and incline greatly toward the front, lest the center of gravity be placed outside the support, they need a staff or staffs by which the center itself may be supported. Moreover, the reason why those who kneel on both knees remain with difficulty in that position is that the center of gravity, being situated in the thorax, is supported by the two knees and presses upon them. And indeed those knees, since they are not by nature made fit, like feet, to sustain the weight of the body, labor all the more; and this especially because, being bony, they are enclosed between the bone and the toughness of the skin, and so are squeezed, and therefore cause pain and discomfort to those kneeling. If, however, someone rests on only one knee, he will feel a far lesser difficulty. For in that case the person kneeling is supported by a threefold support. Let the kneeling figure be ABCDE, whose center of gravity is F. The right knee, on which it rests, is D; the left, which is raised, is B. Thus the kneeling person, as we said, is supported by three supports, CDE. The weight is therefore divided into three parts, and thus the individual parts are less fatigued. Nevertheless, point D labors more, since it is that on which the center of gravity F rests perpendicularly. Likewise, it is also marvelous that sleeping birds are supported by only one foot, and what is even more marvelous, sleeping Y 3

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mientes posse, quod vel ipsis vigilantibus est difficile. Cur id Natura docente faciant, eam puto esse causam, quod dum dormiunt, caput sinistræ alæ, vt naturali calore iuuentur, supponunt, quapropter ad eam partem declinantes, vt interim æquilibrium faciant, pedem subleuant, & eo casu ceu inutilem retrahunt atque suspendunt: addita item alia caussa, nempe vt pedem ipsum dormientes natiuo calore confoueant. Quæritur etiam, Cur ij qui inclinantur, vt re quampiam a solo sustollant, alterum crurium ad anteriora, nepe versus manum ipsam, quam porrigunt, extendant? Esto enim quispiam ABCD, cuius crura BC, BD, grauitatis centrum E, velit autem quippiam a solo tollere quod sit in F. sit perpendicularis, quæ per grauitatis centrum GEH. Dum igitur ad anteriora inclinatur, centrum amouet a perpendiculari, quamobrem docente Natura, crus BC ad centrum ipsum fulciendum ad anteriora, hoc est, versus rem sustollendam porrigitur. Huius quoque speculationis est inuestigare, Cur quadrupedia dum gradiuntur, pedes diametraliter moueant. Cuius rei verba fecit ipse quoque Philosophus lib. de animalium incessu cap. 12. Nos autem ad maiorem declarationem, quod ipse Physicis principijs fecit, mechanicis demonstrabimus. Sint duæ in plano parallelæ AB, CD, in quibus quadrupedis pedes E, F, B, D, quorum EF, anteriores, BD vero posteroses. iungantur BDEF, eritque EBDF parallelogrammum altera parte longius, cuius diametri ducantur ED,

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to be able to remain still, which is difficult even for them when awake. The reason, I think, why Nature teaches them to do this is that, while they sleep, they place the head under the left wing, so that they may be helped by their natural heat; wherefore, leaning toward that side, they lift the foot, in order meanwhile to make an equilibrium, and in that case draw it back and hold it suspended as though useless: another reason is also added, namely, that sleeping, they warm the foot itself with their native heat. It is also asked why those who lean forward, in order to raise something from the ground, extend one of their legs toward the front, namely toward the hand itself which they stretch out? Let there be, for example, ABCD, whose legs are BC and BD, with the center of gravity E, and let him wish to lift from the ground something which is at F. Let there be a perpendicular line GEH passing through the center of gravity. Therefore, when he inclines forward, he removes the center from the perpendicular; wherefore, Nature teaching, the leg BC is extended forward, that is, toward the thing to be lifted, in order to support that center itself. It is also part of this inquiry to investigate why quadrupeds, while walking, move their feet diametrically opposite one another. The Philosopher himself also spoke of this in book de animalium incessu, chapter 12. But we shall demonstrate, by mechanical principles, what he did by physical principles, for a fuller explanation. Let there be in a plane two parallels AB and CD, on which are the feet of a quadruped, E, F, B, D, of which EF are the front feet, but BD the hind feet. Let BDEF be joined, and there will be a parallelogram longer on one side, whose diagonals are drawn ED,

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EXERCITATIONES. 175 ED, BF, secantes sese in G, vbi & grauitatis centrum. Moto igitur posteriori sinistro pede B in K, si anteriorem E, eodem tempore moueret in I, stantibus interim DF, ceu fulcimentis, centrum G extra fulcimenta fieret ad partes BE. Caderet igitur ad partes BE. Si autem eodem tempore moueret dextros eodem pacto centrum extra fulcimenta positum caderet ad partes ipsas DF. Si autem moto pede B in K, & eodem tempore F in L, & D in H, E, in I, centrum erit in diametris HI, KL, hoc est, vbi M, sultum quidem ab ipsis pedibus K, L, H, I. Hoc igitur pacto transfertur vicissim cum grauitatis centro simul translatis fulcimentis sese diametraliter respondentibus; quod vtique demonstrandum fuerat. Sane & bipedia quoque alternatim gradiendo grauitatis centrum transferunt. Dum enim dextrum crus eleuatur, centrum sinistro fulcitur, & econtra. Naturalia isthæc sunt; in artificialibus autem quæri posset, Cur Architecti, Arcium muros non ad perpendiculum erectos, sed introrsum inclinatos constituant? Vtique hoc faciunt, vt minus sint ad ruinam proni. Esto enim murus ad interiorem partem vergens ABCD, Cuius grauitatis centrum E basis BC erigatur à puncto B horizonti perpendicularis BF, & ad eundem à centro grauitatis E demittatur EM, tum BE iungatur. Post hæc à puncto BG angulum cum linea horizontis BK faciens recto maiorem. Itaque murus hoc pacto constitutus ad interiorem partem suo pondere vergit, cadere autem non potest, vel quod viuæ ru-

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EXERCITATIONES. 175 ED, BF, intersecting at G, where is the center of gravity. Therefore, if with the rear left foot B moved to K, it should at the same time move the front E to I, while in the meantime DF remain standing as supports, the center G would become outside the supports toward BE. Therefore it would fall toward the parts BE. But if at the same time it should move the right feet in the same way, the center placed outside the supports would fall toward the parts DF themselves. But if, with foot B moved to K, and at the same time F to L, and D to H, E to I, the center will be in the diagonals HI, KL, that is, where M, supported indeed by the feet K, L, H, I. In this way therefore it is transferred alternately, the supports themselves being moved together with the center of gravity and responding to one another diametrically; which certainly had to be demonstrated. Indeed, bipeds too transfer the center of gravity by walking alternately. For while the right leg is lifted, the center is supported by the left, and vice versa. These are natural matters; but in artificial things one might ask why architects set the walls of fortresses not upright perpendicular, but inclined inward? They surely do this so that they are less prone to collapse. Let there be, then, a wall ABCD inclining toward the interior, whose center of gravity E, let the base BC be raised from the point B by the perpendicular BF to the horizon, and from the center of gravity E let EM be let down to the same, then let BE be joined. After this from point BG making an angle with the line of the horizon BK greater than a right angle. Thus the wall established in this way leans toward the interior by its own weight, but cannot fall, either because of the living ru-

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176 IN MECHAN. ARIST. PROBL. rupi, cui forte hæret, fulciatur, vel antistatis, quos no- strates sperones & contra fortes appellant, innitatur. Sed nec in anteriora corruet, quandoquidem ruinam factu- rus, necesse est vt grauitatis centrum secum trahat in per- pendiculari BF, & demum in eam quæ vltra perpendicu- larem est BG, facta nempe circa B, ceu circa centrum, co- uersione. Moueatur autem & ex semidiametro BE cen- tro B portio circuli describatur EH, quæ secet BG in H, & BF in I; Et quia EM semidiametro BK perpendicularis per B, centrum non transit, erit EM ipsa BK, hoc est, BI breuior. Abscindatur ex BI, ipsi EM æqualis LB. Erit igi- tur punctum L infra punctum I, hoc est, ipso I, mundi cen- tro propius. Necesse igitur erit ad hoc vt murus corruat, centrum grauitatis E facta circa B, conuersione aliquan- do fieri in I, vt demum transferri possit in H, sed I remo- tius est à mundi centro ipsis E, L, ascendet igitur graue contra sui naturam ex E in I, at hoc est impossibile; quod fuerat demonstrandum. Ex his ijsdem principijs alia soluitur quæstio, Cur scilicet Campanaria turris quæ Pisis visitur, nec non alia Bononiæ in foro prope Asellorum turrim, quam à nobili olim Carisendorum familia exstructam, Carisendam vo- cant, cuius meminit & Dantes Poëta summus in sua Co- mædia. Propendet autem hæc in latus, & ita propendet vt perpendicularis, quæ à summo inclinatæ partis in so- lum demittitur, longe cadat ab ipsa, cui nititur, basi, quod sane mirabile videtur, muros nempe, in ruinam pronos, ruinam non facere. Esto enim turris ABCD, basi fulta BC, horizontis planum BCF latera AB, DC, centrum vero grauitatis to- tius molis E. Propendeat autem ad partes DC ex angulo DCF. Ita autem constituta intelligatur vt perpendicularis ab A, in planum horizontis demissa per grauitatis cen- trum

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176 IN MECHAN. ARIST. PROBL. by the stone on which it chance to rest, let it be supported, or let it lean upon buttresses, which our people call sperones and contra fortes. But it will also not fall forward, since in order for a collapse to occur, the center of gravity must necessarily be carried with it in the perpendicular BF, and then ultimately into that line which is beyond the perpendicular, BG, namely by a rotation around B, as around a center. But let us move from the semidiameter BE a portion of a circle described with center B, EH, which cuts BG at H, and BF at I; and because EM, perpendicular to the semidiameter BK through B, does not pass through the center, EM will be itself shorter than BK, that is, than BI. Let LB be cut off from BI, equal to EM. Therefore the point L will be below the point I, that is, nearer to the center of the world than I itself. Therefore it will be necessary, if the wall is to fall, for the center of gravity E, after a rotation around B, to be brought at some point to I, so that it may finally be transferred to H; but I is farther from the center of the world than E and L themselves, therefore the heavy body will ascend contrary to its nature from E to I, but this is impossible; which was to be demonstrated. From these same principles another question is solved, namely why the Campanaria tower, which is seen at Pisa, and also another one at Bologna in the market-place near the Tower of the Asinelli, which they call the Garisenda, having been built long ago by the noble family of the Garisendi, whom the great poet Dante also mentions in his Comedy. Now this leans to one side, and leans so that the perpendicular, which is let down from the top of the leaning part to the ground, falls far from the base on which it rests, which indeed seems marvelous, namely that walls, inclined toward ruin, do not produce a ruin. For let there be a tower ABCD, supported by the base BC, the plane of the horizon BCF, the sides AB, DC, and the center of gravity of the whole mass E. Let it lean, however, toward the parts DC from the angle DCF. And let it be so placed that the perpendicular dropped from A to the plane of the horizon through the center of gravity

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EXERCITATIONES. 177 trum E extra basim BC, non cadat, cadat autem in G. Quoniam igitur ABCD moles per E grauitatis centrum diuiditur, in partes secatur æqueponderantes, sed & centrum grauitatis extra fulcimentum non cadit, quare nec pars ACD, trahet partem ABC, nec centrum extra fulcimentum positum locum petet centro mundi viciniorem. Cur igitur Carisenda stet, & e-gregia illa turris campanaria quæ Pisis prope summum Templum marmoribus præclare exstructa videtur, licet ruinam minentur, stent æternum, nec cadant, ex his quæ considerauimus, liquido patet. QVAESTIO XXXI. Cur facilius moueatur commotum quam manens, veluti currus commotos citius agitant, quam moueri incipientes? Hoc quæritur. Problema hoc est mere Physicum; verumtamen quo-niam ad localem motum pertinet, de quo ipse quoque Mechanicus agit, Hisce quæstionibus contemplatio hæc interferitur. Soluit autem Aristoteles inquiens, id fortasse ea de caussa fieri, quod difficillimum sit pondus mouere, quod in contrarium mouetur. Demit enim quippiam demotoris potentia resistens, licet mouens ipso moto sit longe potentius atque velocius. necesse enim esse id tardius moueri quod repellitur. Hæc verba licet de ea potentia dicta videantur, quæ rem motam in contrariam partem repellit, nihilominus illi quoque aptantur quæ rem immobilem à principio mouere conatur. est enim resistentia rei quæ à statu ad motum transfertur ceu quida[m] Z con-

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EXERCISES. 177 so that the weight does not fall from E outside the base BC, but falls at G. Since therefore the mass ABCD is divided by E, the center of gravity, into equal-weight parts, and yet the center of gravity does not fall outside the support, therefore neither will the part ACD draw along the part ABC, nor will the center placed outside the support seek a place nearer to the center of the world. Why therefore Carisenda stands, and that noble bell tower which at Pisa near the summit of the Temple is seen to have been splendidly built of marble, although they threaten ruin, stand forever, and do not fall, is plainly clear from what we have considered. QUESTION XXXI. Why is something already moving more easily moved than when at rest, as wagons, once set in motion, are driven more quickly than when beginning to move? This is the question. This problem is purely physical; nevertheless, since it pertains to local motion, of which the Mechanic too treats, this contemplation is inserted among these questions. Aristotle solves it, saying that perhaps this happens because it is very difficult to move a weight that is being moved in the opposite direction. For something is taken away by the power of the mover, resisting, although the mover itself, once in motion, is much more powerful and swifter. For it is necessary that that should move more slowly which is repelled. These words, although they seem to have been said of that power which repels a moved thing to the contrary side, are nevertheless also applied to that which from the beginning tries to move a thing at rest. For there is resistance in the thing which is transferred from a state to motion, as a certain Z con-

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178 IN MECHAN. ARIST. PROBL. contrarius motus. Contra autem accidit illi qui rem mo- tam mouet in ipso motu: eo enim casu mouens ab ipso rei motu magnopere iuuatur, cooperatur enim motus moto- ri, in ipsam rem motam operanti. Auget autem res mota quodammodo mouentis potentiam. quod enim à mouen- te pateretur, ex se ipsa agit res quæ mouetur. Esto horizontis pla- num AB, cui moles quæ- dam insistat, CD. Modo potentia quædam appli- cetur vbi E, quæ molem in anteriora propellat, id est, versus B. Primum igitur, quoniam à quiete ad motum fit transitus, resistit sua quiete corpus graue, potentiæ im- pellenti, superata demum resistentia moles quæ moueri coept, fertur in F & mouetur, quare potentia quæ à prin- cipio resistentiam rei non motæ superauerat, pellendo rem motam pergens facilius pellit: Duo enim sunt quo- dammodo motores, mouens videlicet ipse, & motus quo res mota mouetur. facilius ergo pelletur ex F in G, quam ex D in F, & ex G in B, quam ex F in G, & eo motus fiet in progressu facilior atque in ipsa velocitate velocior, quo magis in ipsa motione mouetur. Hinc soluitur ea quæstio apud Physicos difficillima, Cur nempe in motu naturali velocitas vsque augeatur; etenim ibi Natura mouens est, atque eadem inseparabilis à remota, vrget igitur assidue, à principio quidem tardius, post hæc autem ea quam diximus, de caussa vsque & vsque velocius. Motus ergo fit in motu, qui motus cum semper à motore, & motu ipso augeatur, crescit ex progressu in im- mensum. Certe caussam velocitatis auctæ eam esse, quod potentia mouens rem motam in motu ipso moueat, nemo vt arbitror, inficias ibit, acquirit enim corpus motum po- derosi

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178 IN MECHAN. ARIST. PROBL. contrary motion. But the opposite happens to the one who moves the moved thing while it is in motion: in that case the mover is greatly helped by the motion of the thing itself, for the motion cooperates with the mover, acting upon the thing that is being moved. And the moved thing in a way increases the power of the mover. For what it would suffer from the mover, the thing that is moved does of itself. Let there be the plane AB of the horizon, upon which some body CD stands. Now let some force be applied at E, which drives the body forward, that is, toward B. First, then, since there is a transition from rest to motion, the heavy body resists the impelling force by its own rest; when this resistance is finally overcome, the mass, which has begun to move, is carried to F and is moved. Thus the force, which from the beginning had overcome the resistance of the body not yet in motion, continues, by pushing the moved body, to push more easily: for in a way there are two movers, namely the mover itself and the motion by which the moved body is moved. It will therefore be pushed more easily from F to G than from D to F, and from G to B than from F to G, and in this way the motion will become easier in its progress and faster in its speed, the more it is moved within the very act of moving. Hence that most difficult question among the physicists is solved, namely why in natural motion the speed continually increases; for there Nature is the mover, and being inseparable from the thing moved, it urges continually: at first indeed more slowly, but afterward, for the reason we have said, ever and ever more swiftly. Motion therefore occurs in motion, and since this motion is always increased both by the mover and by the motion itself, it grows in its progress to an immense degree. Certainly, as I think, no one will deny that the cause of the increased speed is this: that the moving power moves the moved thing in the motion itself; for a body in motion acquires power.

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EXERCITATIONES. 179 derositatem quandam accidentalem, quæ cum ex motu perinde augeatur, ipsum motum faciliorem, coque velo- ciorem facit. Disputat hæc & Simplicius lib.7. Physic.c. II. Aristotelis de Natural libros exponens. QVAESTIO XXXII. Quæritur hîc, Cur ea quæ proijciuntur, cessent à latione? Hocitidem problema est mere Physicum. Ad quod ea pertinent quæ à Philosopho tractantur libro Natu- ralium 8. & lib.1. de Coelo. Tres autem affert subdubitan- do rationes, An quia impellens desinit potentia, vel pro- pter retractionem, vel propter rei proiectæ inclinatione, quando ea valentior fuerit quam proijcientis vires? Quicquid dicat Philosophus, id vtique exploratis- simum est. Proiecta ideo à motu cessare, propterea quod impressio, cuius impetu & virtute feruntur, non sit proie- ctus quidem naturalis, sed mere accidentalis & violenta, at nullum accidentale & violentum quodque, non natu- rale est, perpetuum est. Cessat ergo accidentalis illa im- pressio, eaque paullatim cessante proiecti motus elan- guescit, donec quietem prorsus adipiscatur. Illud quoque notamus, quod à multis vidimus non obseruatum, nempe violentum motum violentia præualente non differre à naturali, & ideo tardiorem esse à principio post hæc, in i- pso motu fieri velociorem, remittente demum paullatim impressa violentia, tardiorem, donec impetus, & cum im- petu motus euanescat, & res ipsa mota quietem adipisca- tur. Vnde etiam experientia docemur, istum ex proiectis violentius fieri, si fiat paullo remotior à principio, & tunc demum esse innocentissimum, cum ibi fit, vbi proiectum ex motu plene acquisito, summam adeptum est velocita- tem. Z 2

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EXERCISES. 179 a certain accidental roughness, which, as it increases along with the motion, makes the motion itself easier, and thereby swifter. This is discussed by Simplicius, book 7, on Physics, ch. II., in his exposition of Aristotle’s books On Nature. QUESTION XXXII. Here it is asked: Why do those things that are thrown cease from motion? This problem is likewise purely physical. To it belong those matters treated by the Philosopher in book 8 of the Natural Things, and in book 1 of On the Heaven. He gives, however, three doubtful reasons: whether because the mover’s power ceases, or because of a retraction, or because of the inclination of the thing thrown, when that should be stronger than the thrower’s force? Whatever the Philosopher may say, this is certainly most clearly established: things thrown cease from motion because the impression, by whose impetus and force they are carried, is not a natural motion of the thrown object, but merely accidental and violent; but nothing accidental and violent, since it is not natural, is perpetual. Therefore that accidental impression ceases, and as it gradually ceases, the motion of the thrown body grows languid, until it completely attains rest. We also note what we have seen many fail to observe, namely that violent motion, when violence prevails, does not differ from natural motion, and therefore is slower at the beginning; afterward, in the motion itself, it becomes swifter; finally the impressed violence gradually diminishing, it becomes slower, until the impetus, and with the impetus the motion, vanish, and the thing moved itself attains rest. Hence we are also taught by experience that such motion from projectiles becomes more violent if it be a little more remote from the beginning, and then is most harmless when it happens there where the projectile, having fully acquired its motion, has attained the greatest speed. Z 2

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180 IN MECHAN. ARIST. PROBL. tem. Hinc videmus, vel pueros ipsos, docente Natura cu[m] nuces, vel aliud quippiam, parieti allisum frangere cona[n]tur, à pariete moderato aliquo spatio recedere. Si autem eos interroges, cur id faciant, respondebunt, vt inde ictus valentius fiat atque efficacius. Eleganter ex Simplicij & Alexandri Aphrodisiensis doctrina, quæ lucidissima est, quæstionem hanc in sua Paraphrasi explicat Picolomineus. QVAESTIO XXXIII. Dubitatur, Cur proiecta moueantur, licet impellens à proiectis se- paretur; vel vt verbis Philosophi utar, Cur quippiam non pecu- liarem sibi fertur lationem impulsore alioquin non consequente? Soluit, inquiens, an videlicet, quoniam primum, id est, impellens ipse, id efficit vt alterum, nempe proiectum ipsum impellat, illud vero (hoc est proiectum) alterum impellat, hoc est, aërem ipsum mediumue, quod à proie- cto repelletur. Cessare autem motum, cum res eo deue- nit, vt motus eidem à proijciente impressus, non possit amplius rem proiectam mouere, & itidem rem ipsam, aë- rem videlicet non possit amplius repellere. Vel etiam quando ipsius lari grauitas nutu suo declinat magis quam impellentis in ante sit potentia. Vtique res per se satis cla- ra. etenim motus impressus accidentalis est, quod vero la- tioni violentæ resistit principium, naturale, & ab ipso mo- to inseparabile, vincente igitur quod natura est, paullatim remittitur quod ex accidenti est, & inde proiecti sit quies. Est autem & hoc quoque Problema pure physicum, & superiori, de quo immediate egimus, per quam familia- re, quamobrem ex ijsdem prorsus soluitur principijs. QVAE-

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180 IN MECHAN. ARIST. PROBL. etc. Hence we see that even children themselves, taught by Nature, when they try to break nuts, or anything else, by striking them against a wall, withdraw from the wall by some moderate distance. But if you ask them why they do this, they will reply, so that the blow may become stronger and more effective. Picolomini elegantly explains this question in his Paraphrase, from the doctrine of Simplicius and Alexander of Aphrodisias, which is very clear. QUESTION XXXIII. It is doubted why projectiles move, although the mover is separated from the projectiles; or, to use the Philosopher’s words, why something is carried with a motion peculiar to itself, when the mover does not continue with it. He answers, saying, that because the first thing, that is, the mover itself, brings it about that the other, namely the projectile itself, impels the next thing; but this other thing (that is, the projectile) impels yet another, that is, the air itself or the medium, which will be repelled by the projectile. And the motion ceases when the matter has reached the point where the motion impressed on it by the thrower can no longer move the projected thing any further, and likewise the thing itself, namely the air, can no longer be repelled. Or also when the weight of the very thing descends more by its own inclination than the power of the mover continues forward. In any case the matter is quite clear in itself; for the impressed motion is accidental, but the principle that resists violent motion is natural and inseparable from the thing itself; thus, when what is natural prevails, what is accidental gradually diminishes, and from this comes the rest of the projectile. This problem too is purely physical, and very closely related to the preceding one, which we have just discussed, and therefore it is solved from exactly the same principles. QUAE-

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EXERCITATIONES. 181 QVÆSTIO XXXIV. Cur neque parua multum, neq[ue] magna nimis longe proijci queunt, sed proportionem quandam habere oportet proiecta ipsa ad eius vires qui proijcit? PVlchre dubitationem diluit, inquiens, An quia necesse est quod proijcitur, & impellitur contraniti ei vnde impellitur. Quod autem magnitudine sua nihil cedit, aut imbecillitate nihil contranititur, non efficit proiectione[m] neque impulsionem. quod enim multo impellentis excedit vires, haudquaquam cedit. Quod vero est multo imbecillius, nihil contranititur, & impressionem non suscipit. Aliam quoque adiungit rationem, videlicet, Tantum ferri id quod fertur quantum aeris mouerit ad profundu[m] (hoc est, ad eam partem aeris remotiorem, ad quam fertur) etenim proiectum à principio dum fertur aerem pellit, non pellit autem si nihil mouetur. Accidit igitur vt concludit Philosophus, proiecta isthæc contrarijs ex causis minus moueri. quod enim valde paruum est nihil mouet imbecillitate sua impediente. quod vero valde magnum est, ex contraria caussa nihil mouet, nempe quod ob magnitudinem suam nihil moueatur. Vnde fit proportionem inter proiectum & proijcientem esse inprimis ad motum, necessariam. Hæc eadem præclare in sua Paraphrasi explicat Picolomineus. Huic nos, de proiectis quæstioni, hæc addimus. Cur proiecta corpora non sibimet ipsis secundum partes æque grauia, si fuerint irregularis figuræ in ipso motu, secundum grauiorem partem antrorsus inuiolento, & deorsum innaturali ferantur, & dum in latione conueruntur, sonitum edant. Esto pila ABCD, cuius centrum E concinnata ex dispari materia leui, nempe BCD, & graui ABD. non ergo erit Z 3

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EXERCITATIONS. 181 QUESTION XXXIV. Why can neither very small nor very large bodies be projected very far, but the things thrown themselves must have a certain proportion to the power of the thrower? He neatly dissolves the doubt by saying that it is necessary that what is thrown and driven should resist that from which it is driven. But what yields nothing because of its own size, or resists nothing because of its weakness, neither makes a projection nor receives an impulse. For that which greatly exceeds the strength of the thrower does not yield at all. But that which is much too weak resists nothing, and does not receive the impression. He adds another reason also, namely: that just as much air is moved by the thing carried as it carries down to the depth (that is, to that farther part of the air to which it is carried); for the projectile, at the beginning while it is being carried, drives the air, but it would not drive it if nothing were moved. It therefore happens, as the Philosopher concludes, that these projectiles are moved less from contrary causes. For what is very small moves nothing, its weakness preventing it. But what is very large moves nothing from the opposite cause, namely because, on account of its size, nothing is moved. Whence it follows that a proportion between the projectile and the thrower is chiefly necessary for motion. Picolomini explains these same points excellently in his Paraphrase. To this question about projectiles, we add the following. Why do bodies thrown, though not equally heavy in their parts, if they are of irregular shape, in the very motion move with the heavier part foremost, in an unharmed and downward unnatural way, and while they are turning over in their course, give off a sound. Let there be a ball ABCD, whose center E is arranged from unequal matter, namely BCD light, and ABD heavy. Therefore there will not be Z 3

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182 IN MECHAN. ARIST. PROBL. erit centru[m] grauitatis & centrum molis, sit autem grauitatis centrum F. Descendat corpus prohibente remoto per rectam AG. Et quoniam grauiora deorsum tendunt magis, si à principio motus grauior pars fuerit supra in ipso descensu conuertetur pila, & situm non seruabit donec superior pars ea quæ grauior, deorsum fiat, vt videre est in pila HIK, cuius centrum est G. pars grauior HIK. Si autem eadem pila, laterali motu violenter feratur versus N, ad eam quoque partem conuertetur pars grauior. facto enim molis seu magnitudinis centro vbi L, grauior pars fiet in MNO; quæcunque igitur sunt corpora ita co[n]stituta, vt in illis non sit idem molis & grauitatis centrum in ipsa latione conuertentur, & eorum pars grauior antorsus fiet. Sonitus porro in ipso motu editi ea est caussa, quod irregulare corpus à principio incipit conuerti, & in ipsa conuersione dum fertur aërem verberat, & ab eodem vicissim reuerberatur, ex qua reuerberatione fit corporis rotatio dum fertur, & ipse sonitus, quem Græci poÿson Rhœzum appellant. Ad hanc quoque speculationem pertinet, Cur lapides ad superficiem aquæ proiecti non statim demergantur, sed aliquot vicibus aquæ superficiem radentes, ab eadem resiliant. Esto aquæ superficies AB, lapis proiectus C, tangens aquæ superficiem in D, & indere resiliens in E, mox iterum eandem tangens in F, & resiliens in G, donec violéto motu cessante demergatur. Vtique lapis C, proiectus in D, nisi

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182 IN MECHAN. ARIST. PROBL. there is the center of gravity and the center of the body; let the center of gravity be F. Let the body descend, the hindrance removed, along the straight line AG. And since heavier things tend more downward, if at the beginning of the motion the heavier part is above, then during the descent the ball will turn about, and will not keep the same position until the upper part, namely the heavier one, comes to be below, as may be seen in the ball HIK, whose center is G, the heavier part being HIK. But if the same ball is violently driven by a lateral motion toward N, the heavier part will also turn toward that side. For when the center of the mass, or of the magnitude, is at L, the heavier part will be MNO. Therefore whatever bodies are so constituted that in them the center of mass and the center of gravity are not the same will, in the very act of motion, turn about, and their heavier part will come to the front. The sound, moreover, produced in the motion itself is due to this cause, that the irregular body begins to turn from the beginning, and in that turning, while it is being carried along, it strikes the air, and is in turn struck back by it; from this striking back the body’s rotation while moving is produced, and also the sound itself, which the Greeks call poÿson Rhœzum. To this speculation also belongs the question why stones thrown upon the surface of water do not at once sink, but, skimming the surface of the water several times, rebound from it. Let AB be the surface of the water, C the stone thrown, touching the surface of the water at D, and then rebounding to E, soon touching the same surface again at F, and rebounding to G, until, when the violent motion has ceased, it sinks. Certainly the stone C, thrown to D, unless

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EXERCITATIONES. 183 nisi medio densiori, aqua videlicet, repelleretur, penetraret per D, in H. At eo resistente, & adhuc vigente impetu, fertur in E ad angulos fere pares. Dico autem fere, siquidem maior est ADC ipso EDF, propterea quod vis non sit eadem, sed minor ea quæ ex D pellit in E. Durante igitur impetu quo pellitur antrorum, fiunt ipsæ resilitiones, & eo cessante, resilitiones cessant, & lapis suapte grauitate demergitur. Huc quoque spectat, Cur pila lusoria in horizontis planum proiecta ad pares resiliat, angulos nempe rectos? Esto horizontis planum AB, in quod à puncto C per lineam perpendicularem CE cadat proijciaturue pila DE, cuius grauitatis centrum F. Tangit autem planum in pú- p[er]to E. Perpendicularis ergo EC, circulum DE per centru[m] secat, hoc est, in partes æquales & æqueponderantes, sed dum pila cadit proijciturue, agit in planum horizontis, vbi E, & in eodem puncto repetitur, quare cum cadens & agens diuidatur in partes æquales & æqueponderantes & item repatiens & resiliens diuidatur item in partes æquales & æqueponderantes, ita resilit repatiendo, vti egerat in cadendo, hoc est, ad angulos pares; quod fuerat demonstrandum. Modo sit planu[m] aliquod ita ad horizontem inclinatum, vt GH, & in illud cadat proijciaturue eadem pila. Dico eam ab eodem inclinato plano ad pares angulos resilire, non tamen rectos. Vti-

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EXERCISES. 183 unless, by the denser middle, namely the water, it were repelled, it would penetrate through D to H. But because this resists, and the impetus still continues, it is carried to E at nearly equal angles. I say “nearly,” since ADC is greater than EDF itself, for the reason that the force is not the same, but less than that which drives it from D to E. Therefore, while the impetus by which it is driven through the chambers continues, the rebounds themselves occur, and when that ceases, the rebounds cease, and the stone, by its own weight, sinks. To this too belongs the question, Why does a game ball, thrown onto the plane of the horizon, rebound at equal, that is, right, angles? Let AB be the plane of the horizon, on which from point C through the perpendicular line CE let the ball DE be dropped or thrown, whose center of gravity is F. It touches the plane at point E. Therefore the perpendicular EC cuts the circle DE through the center, that is, into equal and equally weighted parts; but while the ball falls or is thrown, it acts upon the plane of the horizon at E, and at the same point it is repeated, wherefore, since the falling and acting body is divided into equal and equally weighted parts, and likewise the part receiving the action and rebounding is likewise divided into equal and equally weighted parts, thus it rebounds in the same way as it acted in falling, that is, at equal angles; which was to be demonstrated. Now let there be some plane so inclined to the horizon as GH, and upon it let the same ball fall or be thrown. I say that it rebounds from the same inclined plane at equal angles, but not right ones. As-

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184 IN MECHAN. ARIST. PROBL. Vtique pila cadens, planum non tanget in E. esset enim GH, vbi AB, Tangat autem in I, & à centro F ad contin- gentiæ punctum I, recta ducatur FI. Erit igitur FI (prop. 18. lib.3. elem.) ipsi GH plano perpendicularis. Ducatur item per I, ipsi EC, parallela IK, secans pilæ circumferen- tiam in K. Agit ergo & repatitur pila in puncto I non æ- qualiter inæquales. etenim sunt partes KDLEI, & IK, eo quod IK secet circulum non per centrum, repellitur ergo in repatiendo non æqualiter, sed iuxta inæqualitatem ea- rundem partium. Ducatur autem recta in circulo LI æ- qualis ipsi IK. Erit igitur LEI, æqualis IK, & tota KDLI æ- qualis toti IKDL. Vt igitur actio est per descensum iuxta rectam KI, ita est repassio per ascensum ex IL. Dico autem angulos KIH, LIG esse æquales & singulos recto minores. Connectantur FL, FK. Quoniam igitur IK portio æqualis est portioni IEL, & recta LI æqualis rectæ IK, & LF æqua- lis ipsi Fk, & FI communis, triangulum LFI, æquale est triangulo IFk. Quare & angulus FIL æqualis angulo FIk, sed G1F, H1F recti sunt, ergo residui LIG, kIH æquales sunt inter se comparati, & recto minores; quod fuerat o- stendendum. Hinc colligimus, quo magis planum ab æquidistan- tia horizontis recesserit, eo pilam in eo proiectam in par- tes inæqualiores diuidi & ad minores ipsi plano angulos resilire. Nihil autem refert, vtrum planum, in quod pila cadit, ad horizontem sit inclinatum, vel eodem horizonti æquedistante pila non ad perpendiculas, sed iuxta aliqué angulum in illud proijciatur. Hæc sane ita ex demonstra- tione fieri ostenduntur. Veruntamen quoniam proiecta pila materialis est, & ideo nec æqualis, nec æqueponde- rans & sua grauitate resistens, non ad pares ex amussi resi- lit angulos, sed minores aliquantulum in resilitione, re- mittente nimirum vi in ipsa rea ctione. Et sane fieri non potest,

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184 IN MECHAN. ARIST. PROBL. Thus a falling ball will not touch the plane in E; for there would be GH, where AB touches. But let it touch at I, and from the center F to the point I of contact let the straight line FI be drawn. Therefore FI (prop. 18, lib. 3, elem.) will be perpendicular to the plane GH itself. Let there also be drawn through I, parallel to EC, the line IK, intersecting the circumference of the ball at K. Therefore the ball acts and reacts at the point I not equally, but unequally. For the parts KDLEI and IK are unequal, since IK cuts the circle not through the center; therefore in recoiling it is repelled not equally, but according to the inequality of those parts. But let a straight line in the circle LI be drawn equal to IK. Thus LEI will be equal to IK, and the whole KDLI equal to the whole IKDL. As, therefore, the action is by descent along the straight line KI, so the reaction is by ascent from IL. But I say that the angles KIH and LIG are equal, and each less than a right angle. Let FL and FK be joined. Since therefore the segment IK is equal to the segment IEL, and the straight line LI equal to the straight line IK, and LF equal to FK, and FI common, the triangle LFI is equal to the triangle IFK. Wherefore also the angle FIL is equal to the angle FIK; but G1F and H1F are right angles, therefore the remaining angles LIG and kIH are equal to each other, and less than a right angle; which was to be shown. From this we gather that the more the plane has departed from horizontality, the more the ball projected upon it is divided into unequal parts, and rebounds making smaller angles with the plane itself. Nor does it matter whether the plane on which the ball falls is inclined to the horizon, or, being at the same distance from the horizon, the ball is projected into it not perpendicularly but at some angle. These things, indeed, are shown to happen from the demonstration. Yet since the projected ball is material, and therefore neither equal nor equally weighing, and resisting by its own gravity, it does not rebound by equal angles according to rule, but by somewhat smaller ones in the rebound, as the force is diminished in the very reaction. And indeed it cannot happen,

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EXERCITATIONES. 185 potest, pilam à plano resilientem eo peruenire vnde à principio discesserat; Id enim si daretur, æterna quoque pilæ ipsius daretur resilitio, & paullatim vi & impetu remittente per parua interualla motus esset, donec res quæ mouebatur, omnino quiescat. QVÆSTIO XXXV. Quærit hoc ultimo Problemate Aristoteles, Cur ea quæ in vorticosis feruntur aquis, ad medium tandem agantur omnia? TRibus rationibus soluit, quarum prima est: Quicquid fertur, magnitudinem habet, cuius extrema in duo- bus sunt circulis, hoc in minori, illud in maiori. Et quoniam maior velocior est, magnitudo media, non æqualiter fertur, sed à maiori quidem pellitur, à minori vero retrahitur, vnde transuersus fit magnitudinis motus, & ipsa magnitudo ad interiorem propellitur circulum, itaque eodem pacto, è maiori in minorem propulsa in centrum tantum fertur, & ibi quiescit. Esto vortex AB, cuius centrum C, magnitudo quæ fertur AD, maior circulus AFB, minor DHEG. Velocitas igitur in A maior est velocitate quæ in D, magnitudinis ergo extremum A, velocius rapitur in A quam eiusdem extremum inferius D, in D. Velocitas igitur maioris circuli pellit Aversus F. tarditas vero minoris circuli D retrahit ad partes G. conuertitur itaque magnitudo interpellentem & retrahentem circulum, donec extremi- Aa

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EXERCISES. 185 can, the ball rebounding from the plane can come to the place from which it had departed at the beginning; for if this were granted, an eternal rebound of the ball itself would also be granted, and as the force and impetus gradually lessen, the motion would proceed by small intervals, until the thing that was moving comes entirely to rest. QUESTION XXXV. In this last Problem Aristotle asks why those things that are carried in whirling waters are all finally driven to the middle? He resolves it by three reasons, the first of which is: Whatever is carried along has a magnitude, whose extremities are in two circles, this one in the smaller, that one in the larger. And since the larger is faster, the middle magnitude is not carried equally, but is indeed pushed by the larger circle, while drawn back by the smaller; whence the motion of the magnitude becomes transverse, and the magnitude itself is driven toward the inner circle, and thus, being pushed from the larger into the smaller, it is carried only to the center, and there it comes to rest. Let there be a vortex AB, whose center is C, and the magnitude carried along is AD, the larger circle AFB, the smaller DHEG. Therefore the velocity in A is greater than the velocity which is in D; therefore the extremity A of the magnitude is carried away more quickly in A than its lower extremity D in D. Therefore the greater velocity of the larger circle pushes A toward F, while the slowness of the smaller circle draws D back toward G. Thus the magnitude is turned by the circle that impels and draws back, until the extrem- Aa

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186 IN MECHAN. ARIST. PROBL. tremitas A in circulo minori fuerit vbi H, D vero vbi I, & ita deinceps eadem ratione vbi KL, donec paullatim feratur in centrum C, facto nempe à maiori in minorem cir- culum transitu. Secunda ratio ita habet, quia quod fertur, simili se habet modo ad omnes circulos propter centrum, hoc est, in quouis circulo, qui circa idem centrum fertur. Omnes autem circuli mouentur, centrum vero stat, necesse est à motu tandem id quod mouetur ad quietis locum, hoc est, in centrum ipsum peruenire. Tertia, quoniam circulorum, qui in vorticibus fiunt, velocitas, & ideo impetus non est æqualis, sed semper ex- terior est interiore velocior & violentior, Æqualis autem semper in mota magnitudine, grauitas, diuersimode se habet ad circulos, à quibus mouetur, & ideo modo vin- citur, modovincit: vincitur autem à velocioribus circulis, vincit autem tardiores. Itaque quoniam sua grauitate re- sistens, maioris circuli motum prorsus non sequitur, ad tardiorem reijcitur, hoc est, interiorem, & sic deinceps, donec tandem centrum ipsum nanciscatur, in quo nec su- perans, nec superata quiescit. Hæ sunt rationes, licet obscurissime propositæ, qui- bus, vt diximus, vtitur Aristoteles. acutæ sane illæ quidé, attamen haud quaquam vltro admittendæ. Primo enim falsum videtur, quod asserit, vortices circulos esse, & circa idem centrum fieri atque rotari. Spi- ræ enim potius sunt, quæ ab exteriori parte remotioreq; incipientes spiraliter circumuolutæ, ad intimam tandem partem, quæ media est & centri vices gerit, deueniunt. qua veritate cognita, omnis prorsus difficultas tollitur, Cum enim ea quæ feruntur, ab aqua ferantur, aqua vero feratur spiraliter, ea quoque spiraliter ferri, est necessa- rium.

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186 IN MECHAN. ARIST. PROBL. and thus in turn, in the same way, where KL is, until little by little it is carried into the center C, namely by a passage from the greater circle into the smaller. The second reason is this: because that which is carried has a similar relation to all circles by means of the center, that is, to any circle which is carried around the same center. But all the circles move, while the center remains at rest; therefore, by motion, what is moved must at last reach the place of rest, that is, the center itself. The third reason is that the speed, and therefore the impulse, of the circles formed in vortices is not equal, but the outer is always faster and more violent than the inner. But equal gravity, always in the moved body, is related in different ways to the circles by which it is moved, and therefore is sometimes overcome, sometimes overcomes: it is overcome by the faster circles, and overcomes the slower ones. Therefore, since resisting by its own gravity, it does not follow the motion of the greater circle at all, it is thrown back to the slower one, that is, the more inward one, and so on, until at last it attains the center itself, where, neither overcoming nor being overcome, it rests. These are the reasons, though put forward most obscurely, which, as we said, Aristotle uses. Certainly they are acute, yet by no means to be admitted outright. For first it seems false that he asserts vortices to be circles, and to be formed and rotated around the same center. Rather they are spirals, which, beginning from the outer and more remote part, winding around in a spiral, at last reach the inner part, which is middle and serves in place of the center. Once this truth is recognized, every difficulty is altogether removed. For since those things which are carried are carried by water, and the water is carried spirally, it is necessary that they too be carried spirally.

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EXERCITATIONES. 187 rium. Hæc autem clariora erunt si quo pacto vortices fiant, quispiam considerauerit. Esto fluminis cuiuspiam curua eademque profunda ripa ABCD. Aquæ vero moles rapida EFDC, quæ quidem eo quod magno impetu deferatur in C, ripæ ipsius natura sequens turbinatim circumuoluitur, egressa autem extra locum seuripam B rotationis principium secundans, in seipsam spiraliter contorquetur, & vorticem efficit GHFIK, cuius quidem centrum est vbi K. Alia quoque de caussa, ex quiescente nimirum, & mota aqua fiunt spiræ vorticesue. Esto enim fluminis ripa ABC, sinum efficiens, quiaquam ex ripæ ipsius obiectu contineat quiescentem, Cursus vero fluminis liber & rectus, sit inter lineas AC, DE. Itaque dumaqua AC rapide fertur ad partes A, quiescentem ABC iuxta lineam CA lateraliter propellit, & eius quidem partem quam tangit, secum rapit, puta ex F in G. Delata igitur aqua & currente ex F versus G quiescens lateraliter eidem sese aliqualiter opponit, & currentem repellit ex G in H. Cæpto itaq[ue] spirali motu aqua circumuoluitur secundum lineam GHK, donec perueniat ad centrum I, vbi circumuolutæ aquæ partes sese inuicem tangunt. Porro vortices isti spiræue, quod nos per Padum, Abduam, & magna fluminanauigantes obseruauimus, non eodem permanent loco, sed rapientis aquæ motum secundantes, paullatim in currentem aqua delati A a 2

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EXERCISES. 187 …; but these things will be clearer if someone considers in what way vortices are formed. Let ABCD be the curved and likewise deep bank of some river. Now the mass of water EFDC, being carried swiftly with great force toward C, follows the very nature of the bank and is whirled round in a circular motion; but, after leaving the place or bank B, following the beginning of rotation, it twists itself spirally, and forms the vortex GHFIK, whose center is at K. A vortex or spiral is also formed for another reason, namely, from water that is at rest and water that is in motion. Let there be, then, the bank ABC of a river, forming a bay, which by reason of the obstruction of the bank itself contains still water; while the free and straight course of the river lies between the lines AC, DE. Thus, while the water AC is carried rapidly toward the parts A, it laterally drives the still water ABC along the line CA, and carries along with it the part it touches, for example from F to G. Therefore, as the water is carried and flows from F toward G, the still water laterally resists it somewhat and repels the flowing water from G to H. Once the spiral motion has begun, the water is whirled round along the line GHK until it reaches the center I, where the parts of the whirled water touch one another. Moreover, these vortices or spirals, which we observed while navigating on the Po, the Adda, and the great rivers, do not remain in the same place, but, following the motion of the water that carries them, are gradually carried away into the current of the water. A a 2

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188 IN MECHAN. ARIST. PROBL. delatieuanescunt, fiunt etiam eiuscemodi vortices nau- tis quidem valde formidabiles etiam in mari, de quibus Poëta libro Æneidos primo. - - - ast illam ter fluctus ibidem Torquet agens circum, & rapidus vorat æquore vortex. Sed & idem quoque de vorticibus, qui in fluminibus fiunt libro 7. - - - hunc inter fluuo Tiberinus amoen[us] Vorticibus rapidis, & multa flauus arena In mare prorumpit. Fiunt autem in mari partim occultis de caussis, partim etiam ex violentia aquarum sibi inuicem obuiantium a- gitatione. Sed nos hisce explicatis commode ad ea quæ dixerat Aristoteles, reuertemur. Dicimus igitur, primam eius rationem haud magni videri ponderis, siquidem non per circulos actu distinctos aqua circumfertur, sed ipsamet sua mole tota simul. Esto enim vortex AB, cu- ius centrum C, semidiameter CA, fiat autem rotatio totius a- quæ CA ad partes D, in linea autem AC, sit corpus aliquod a- quæ rotatione circumlatu[m] AE, inter circulos maiorem ADB, minorem EFG. velocius autem mouetur ADB, ipso EFG, citius ergo fertur pars superior ipsius corporis vbi A, quam inferior vbi E. At id nec A repellit, nec E retrahit, siquidem eodem tempore quo A permeauit circulu[m] ADB, eodem & E per- currit circulum EFG. Itaq[ue] A reuerso in A & E, punctum reuersum erit in E, nulla facta corporis E quoad situm, mutatione quod voluit Aristoteles. Ad

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... Such vortices also arise, and are indeed very formidable to sailors even at sea, of which the Poet speaks in the first book of the Aeneid: - - - but there the whirling vortex Twirls it three times, driving it round, and swiftly devours it in the tide. And the same is said also of the vortices that are formed in rivers, in book 7: - - - amid these the pleasant Tiberinus With rapid vortices, and with much yellow sand, Bursts forth into the sea. Now they arise in the sea partly from hidden causes, partly also from the violence of the waters striking against one another and their agitation. But, these matters having been explained, we shall return suitably to what Aristotle had said. We say, then, that his first reason seems of little weight, since the water is not carried around by actually distinct circles, but by its own bulk all at once. For let there be a vortex AB, whose center is C, and semidiameter CA; let the rotation of the whole water be toward D from CA, and in the line AC let there be some body of water carried around by the rotation, AE, between the larger circle ADB and the smaller EFG. But ADB moves faster than EFG itself; therefore the upper part of that body, where A is, is carried more quickly than the lower, where E is. Yet this neither pushes back A nor draws back E, since at the same time that A has traversed the circle ADB, E also traverses the circle EFG. Thus, when A has returned to A and E, the point will have returned to E, with no change having been made in the position of the body E, which is what Aristotle intended. Ad

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EXERCITATIONES. 189 Ad secundam vero dicimus, non ideo quod omnes circuli æqualiter circa centrum ferantur, nisi alia quæpiâ extranea vis intercesserit, quæ ea ab exterioribus circulis pellens agat in medium. Tertia quoque ratio la- borare videtur. Esto enim vortex AB, cuius centrum C, sit autem corpus aliquod E, cuius na- tura apta sit rotationi aliqua- tenus resistere. Quoniam i- gitur eius resistentia aliquâ- tulum ab aqua rapiente su- peratur in ipsa rotatione, par- tim aque impetum sequetur, partim suapte natura retardabitur. Quamobrem aqua quæ est in A, translata in H, corpus ipsum non erit in H, sed in G. Tardius igitur corpus quam aqua ipsa, rotationem complebit, non tamen propterea, nisi alia quæpiam adsit caussa, feretur in medium. Cæterum horum vorticum effectum & caussam ob- seruare licet, si vase quopiam aqua pleno aquam ipsam baculo manuue circulariter agitauerimus, fiet enim vor- tex, & si quippiam quod leue sit, in aquam motam proie- cerimus, ea quam diximus de caussa in motum ipsum, hoc est, vorticis spiræue, centrum feretur. Hæc nos, vt vera proponimus, & fortasse decipimur. Certe Philosopho tantæ auctoritatis contradicere, ma- gnæ videtur audaciæ, aut potius insaniæ. Quicquid ta- men sit, pro pulcherrima veritate laborasse, à parte aliqua laudis non fuerit prorsus, vt arbitror, alienum. APPEN- A a 3

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EXERCISES. 189 To the second, however, we say that this is not so, because all the circles are not carried equally around the center, unless some other external force has intervened, which, by driving them from the outer circles, acts toward the middle. The third reason also seems to labor. For let there be a vortex AB, whose center is C, and let there be some body E, whose na- ture is such as to resist rotation to some extent. Since, therefore, its resistance is in some degree overcome by the water carrying it along in the very rotation, it will partly follow the impulse of the water, and partly be delayed by its own nature. Wherefore the water which is in A, having been transferred to H, the body itself will not be in H, but in G. Therefore the body will complete the rotation more slowly than the water itself; yet for that reason, unless some other cause is present, it will not be carried to the middle. Moreover, the effect and cause of these vortices may be observed, if in some vessel full of water we have stirred the water itself circularly with a stick or with the hand; for a vortex will be formed, and if we throw into the moving water something that is light, for the reason we have mentioned it will be carried into the motion itself, that is, to the center of the spiral or vortex. We present these things as true, and perhaps we are deceived. Certainly to contradict a Philosopher of such authority seems to be of great boldness, or rather madness. Whatever the case may be, however, to have labored for the sake of a most beautiful truth will not, as I judge, be altogether without some share of praise. APPEN- A a 3

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190 IN MECHAN. ARIST. PROBL. APPENDIX. Modum inueniendarum duarum mediarum propor- tionalium non tantum vtilem esse, sed prorsus neces- sarium, illi norunt, qui in Mechanicis disciplinis vel paru[m] fuerint versati. Nulla enim alia ratio est, qua corpore[m] ma- gnitudines seruata figura & similitudine augeri propor- tionaliter imminuiue possint. Quamobrem factum est vt in his inueniendis tum vetustissimo tum etiam inferiori æ- uo, clarissimi Viri magnopere laborauerint. Plato etenim, Eudoxus (cuius modum repudiauit Eutocius) Heron A- lexandrinus, Philon Byzantius, Apollonius, clarissimi Geometræ, Diocles, Pappus, Sporus, Menæchmus, Ar- chytas Tarentinus, Platoni æqualis: Eratosthenes, & Ni- comedes ad has inueniendas varias rationes excogitaru[n]t, quorum omnium modos, & instrumenta, demonstrationesq; diligentissime collegit, & in illos Comentarios con- iecit idemmet Eutocius, quos elegantissimos in Archime- dis libros de Sphæra & Cylindro scripsit. Nos autem ijs o- mnibus accurate perspectis, & diligentissime ponderatis, inuenimus eos fere omnes tentando negotium absolue- re, quod sane laboriosum valde est & operantibus permo- lestum. Itaque cum modum praximue inuenissemus, ex qua is qui operatur tutissime & facillime ad quæ sitas ipsas medias manuducitur, hunc pulcherrimæ huius facultatis studiosis inuidere nefarium iudicauimus. Quod si quispiâ dixerit, Ballistarum, Catapultarum, Scorpionum, & cæ- terarum eiuscemodi Machinarum vsum, olim apud nos desijsse, & ideo Problema hoc videri superuacaneum, Re- spondemus, nulla alia ratione æneorum tormentorum pi- las augeri imminuiue seruata ponderis ratione posse, in- numeraque esse, quæ vt rite perficiantur, hæc penitus in- digen speculatione. Nos rem Mechanicis vtilem, Me- chanicis

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190 IN MECHAN. ARIST. PROBL. APPENDIX. Those who have had even a little experience in the mechanical sciences know that the method of finding two mean proportionals is not only useful, but absolutely necessary. For there is no other way by which body sizes, while preserving shape and similarity, can be increased or decreased proportionally. Wherefore it came about that, both in very ancient times and also in later ages, the most eminent men labored greatly over finding them. For Plato, Eudoxus (whose method Eutocius rejected), Heron of Alexandria, Philo of Byzantium, Apollonius, most distinguished geometers, Diocles, Pappus, Sporus, Menaechmus, Archytas of Tarentum, a contemporary of Plato, Eratosthenes, and Nicomedes devised various methods for finding these; Eutocius himself most diligently collected the methods, instruments, and demonstrations of all of them, and set them down in those Commentaries which he wrote on the most elegant books of Archimedes, On the Sphere and Cylinder. But after having examined all these things accurately and weighed them most carefully, we found that they almost all attempted to complete the task by trial, which is certainly very laborious and extremely troublesome for those who work at it. Therefore, since we had found a method in practice, by which the person operating is most safely and easily guided to the required means themselves, we judged it impious to withhold this from students of this most beautiful art. And if anyone should say that the use of ballistae, catapults, scorpions, and other machines of that kind has long since ceased among us, and that therefore this problem seems superfluous, we reply that by no other means can the balls of bronze artillery be increased or diminished while preserving the ratio of weight, and that there are countless things which, in order to be properly completed, are utterly in need of this investigation. We present a matter useful to mechanics, mechanics

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EXERCITATIONES. 191 chanicis nostris Exercitationibus annectere, haud importunum iudicauimus. Sed tempus est, vt his breuiter præfatis, ad rem ipsam explicandâ commode accedamus. Datis duabus proportionalibus prima, & quarta duas inter eas medias in continua proportione inuenire. ESTo prima datarum AB, quarta BC, inter quas secundâ & tertiam oportet inuenire. Ducatur recta DE, cui à puncto F, vtcunque sumpto, perpendicularis demittatur FG, Tum ab F versus D duplicetur quarta BC, sitque FH, deinde ab H ipsi FG parallela demittatur HI, & ab HF abscindatur HK, ipsius BC quartæ medietati æqualis. Posthæc puncto K spatio autem medietati, primæ datarum æquali, in linea HI notetur punctum L, & ipsi HL fiat æqualis FM, & KM iungatur. His ita constitutis paretur seorsum scheda regulaue quæpiam NO, in cuius latere accipiatur OP, æqualis medietati primæ datarum seu ipsi KL. Tum regulæ latus aptetur puncto L, extremum vero O, feratur assidue per rectam EK, versus K, nunquam interim

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EXERCISES. 191 We have judged it not inappropriate to append these to our mechanical Exercises. But it is time, after this brief preface, to proceed conveniently to the matter itself. From two proportional quantities, the first and the fourth, to find two means between them in continued proportion. Let AB be the first of the given lines, BC the fourth, between which the second and third are to be found. Let the straight line DE be drawn, to which, from any point F taken at pleasure, let the perpendicular FG be let fall. Then, from F toward D, let the fourth BC be doubled, and let FH be such. Next, from H let HI be drawn parallel to FG, and from HF let HK be cut off, equal to half of the fourth BC. After this, let point L be marked on line HI at a distance from K equal to half of the first of the given lines, and let FM be made equal to HL, and let KM be joined. When these things have thus been arranged, prepare separately some sheet or rule NO, on whose side let OP be taken equal to half of the first of the given lines, or to KL itself. Then let the side of the rule be applied at point L, and its extremity O be steadily carried along the straight line EK, toward K, never in the meantime

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192 IN MECHAN. ARIST. PROBL. interim regulæ latere ON amoto à puncto L, idque donec punctum P, obuians incidat in lineam KM, puta vbi Q extreum vero O inueniatur in R. notato igitur in linea EK puncto R habebitur, quod quærebatur. Erunt igitur AB prima, RK secunda, QL tertia, BC quarta. Hæc praxis ijsdem principijs demonstratur, quibus suam ex Conchoide ostendit Nicomedes. Conficit ille instrumentum, ex quo describit Conchoide, ex qua postea duas medias venatur. Nos autem nec instrumentum construimus nec Conchoidem describimus, & duabus ferelineis rem absoluimus, vt nemo fere non dixerit, hoc istud quod docemus, à Nicomedea praxi esse prorsus alienum. Sed nos, vt eius, quam ostendimus, operationis demonstratio habeatur; ipsius Nicomedis ex Pappi libro 3. propos. 5. desumptam in medio afferemus, quippe quod isthæc ea quam in suis in Archimedem commentarijs refert Eutocius, sit lucidior. Datis duabus rectis lineis CD, DA; duæ mediæ in continua proportione hoc modo assumuntur. Compleatur ABCD parallelogrammum, & vtraq[ue] ipsarum AB, BC, bifariam secetur in punctis L, E, iunctaque LD producatur; & occurrat productæ CB, in G, ipsi vero BC ad rectos angulos ducatur EF, & CF iungatur, quæ sit æqualis AL. Iungatur præterea FG & ipsi parallela sit CH, eritque angulus KCH, æqualis angulo CGF. Tum à dato puncto F ducatur FHk, quæ faciat kH æqualem ipsi AL vel CF. Hoc enim per lineam Conchoidem fieri posse ostendit Nicomedes, & iuncta kD producatur, occurratque ipsi BA, productæ in puncto M. Dico vt DC ad Ck ita Ck ad MA & MA ad AD. Quoniam enim BC bifariam secta est in E, & ipsi adjicitur Ck. Rectangulum BkC per 6. secundi: vna cum quadrato ex CE, æquale est quadra-

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192 IN MECHAN. ARIST. PROBL. in the meantime let the rule be moved away from the point L, and this until the point P, meeting it, falls on the line KM, namely where Q, with the far end O, is found in R. Therefore, noting on the line EK the point R, that which was sought will be obtained. The first will therefore be AB, the second RK, the third QL, the fourth BC. This procedure is demonstrated by the same principles by which Nicomedes showed his own method by means of the Conchoid. He constructs an instrument, by which he describes the Conchoid, from which afterward he hunts for two mean proportionals. But we construct neither an instrument nor describe a Conchoid, and with almost nothing but straight lines we complete the matter, so that almost no one would not say that what we teach is entirely alien to the Nicomedian method. But so that we may have a demonstration of that operation which we have shown, we shall set forth in the middle Nicomedes’ own proof, taken from Pappus, book 3, proposition 5, since this is clearer than that which Eutocius relates in his commentaries on Archimedes. Given two straight lines CD, DA, two mean proportionals are taken in this way. Let the parallelogram ABCD be completed, and let each of AB, BC be bisected at the points L, E; and let LD, when joined, be produced, and let it meet the produced CB at G; and to BC itself, at right angles, let EF be drawn, and let CF be joined, and let it be equal to AL. Further, let FG be joined and let CH be parallel to it; then the angle KCH will be equal to the angle CGF. Then from the given point F let FHk be drawn, which makes kH equal to AL itself or CF. For Nicomedes shows that this can be done by means of the Conchoid; and let kD be joined and produced, and let it meet BA, produced, at the point M. I say that as DC is to Ck, so Ck is to MA, and MA to AD. For since BC has been bisected at E, and Ck is added to it, the rectangle BkC, by book 6 of the second [book], together with the square on CE, is equal to the squa-

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EXERCITATIONES. 193 quadrato ex Ex. commune apponatur ex EF quadratum, ergo rectangulum BkC vna cum quadrato CF æquale est quadratis ex kE, EF, hoc est, quadrato ex Fk. Et quoniam vt MA ad AB, ita est MD ad DK, vt autem MD ad Dk per 2. sexti, ita BC ad CK erit vt MA ad AB, ita BC ad CK. Atque est ipsius AB dimidia AL, & ipsius BC, dupla CG, est igitur vt MA ad AL, ita GC ad CK. Sed vt GC ad CK, ita FH ad HK propter lineas parallelas GF, CH. quare & componendo vt ML, ad LA, ita FK ad KH, sed AL ponitur æqualis HK, quoniam & ipsi CF, ergo & ML per 9.lib.5. æqualis erit FK, & quadratum ex ML, æquale quadrato ex FK. est autem quadrato ex ML, æquale rectangulum BMA vna cum quadrato ex AL & quadrato ex Fk æquale ostensum est rectangulum BkC vna cum Bb quadrato

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EXERCISES. 193 Since to the square from Ex. the common is added the square from EF, therefore the rectangle BkC together with the square CF is equal to the squares from kE, EF, that is, to the square from Fk. And since as MA is to AB, so is MD to DK, but as MD to Dk by 2. of the sixth, so BC to CK will be as MA to AB, so BC to CK. And AB itself is half AL, and BC itself double CG; therefore as MA is to AL, so GC is to CK. But as GC to CK, so FH to HK, because of the parallel lines GF, CH. Wherefore, by composition also, as ML is to LA, so is FK to KH; but AL is set equal to HK, since it is also equal to CF; therefore, by 9. book 5., ML will also be equal to FK, and the square from ML equal to the square from FK. Now the square from ML is equal to the rectangle BMA together with the square from AL; and the square from Fk has been shown equal to the rectangle BkC together with the square Bb.

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194 IN MECH. ARIST. PROBL. EXERCIT. quadrato ex CF, quorum quidem quadratum ex AL æquale est quadrato ex CF, ponitur enim AL, ipsi CF æqualis, ergo reliquum BMA rectangulum æquale est reliquo BkC. Vt igitur MB ad Bk, ita Ck ad MA. Sed vt MD ad Bk, ita DC ad Ck. quare vt DC ad Ck, ita est Ck ad MA. vt autem MD ad Bk, ita MA, ad AD. Ergo vt DC, prima, ad Ck secundam, ita Ck secunda ad MA tertiam, & MA tertia ad AD quartam, quod fuerat demonstrandum. Hæc Pappus. Quod autem in nostra Praxi diximus, QL esse tertiam, earatio est, quod LR vt in prima figura est, sit æqualis ipsi LM secundæ figuræ, in demonstratione Pappi, ex quibus deemptis QR & LA, quæ sunt æquales, reliqua QL primæ figuræ æqualis est AM secundæ figuræ, hoc est, ipsi tertiæ proportionali: Est igitur, vt in prima figura dicehamus, AB prima, kR secunda, QL tertia, BC quarta. Vides igitur tu quilegis, nos ex Nicomedis demonstratione (quatenus ad praxin pertinet) superflua resecasse, & absque Conchoidis instrumento lineaue rem ipsam confecisse, idque non tentantes, vt alij, sed progredientes, & quasi manuductos quæsitum inuestigasse. FINIS.

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194 IN MECH. ARIST. PROBL. EXERCIT. quadrangle from CF, of which indeed the square from AL is equal to the square from CF, for AL is posited equal to CF; therefore the remaining rectangle BMA is equal to the remaining BkC. Thus, as MB is to Bk, so Ck is to MA. But as MD is to Bk, so DC is to Ck. Wherefore, as DC is to Ck, so Ck is to MA. But as MD is to Bk, so MA is to AD. Therefore, as DC, the first, is to Ck the second, so Ck the second is to MA the third, and MA the third to AD the fourth, which was to be demonstrated. This is Pappus. But what in our practice we said, that QL is the third, the reason is that LR, as in the first figure, is equal to LM itself in the second figure, in Pappus’ demonstration; and after subtracting from these QR and LA, which are equal, the remaining QL of the first figure is equal to AM of the second figure, that is, to the third proportional itself. Therefore, as in the first figure we said, AB is the first, kR the second, QL the third, BC the fourth. You see then, reader, that we have, from Nicomedes’ demonstration (so far as it concerns practice), cut away what was superfluous, and, without the instrument or line of the Conchoid, accomplished the thing itself, and that not by trial, as others do, but by proceeding onward, and, as if led by the hand, investigating the sought thing. END.