Critical Exercises on the Mechanical Problems of Aristotle

Creator: Bernardino Baldi | Date: 1621 | Notes: Original title: In mechanica Aristotelis problemata exercitationes A learned Latin commentary, in the humanist genre of exercitationes, that reworks question by question the Mechanica transmitted under Aristotle's name (a text already suspected in Baldi's circle of spurious authorship). Its governing purpose is to re-found the ancient text on the Archimedean doctrine of the centre of gravity and centrobarics, which Baldi held Aristotle to have lacked, and to reduce the traditional five simple machines (lever, pulley, wheel-and-axle, wedge, and screw) to two near-identical principles, the lever and the balance (vectis et libra). The work argues that mechanics is a facultas in its own right capable of seconding and even surpassing Nature, treats mixed-motion paradoxes (the rhombus and wheel paradoxes), collision and rebound, and the geometry of whirlpools (which Baldi argues are spirals rather than concentric circles, on the basis of his own observations of the Po and Adda), and includes an appendix on finding two mean proportionals. It was printed posthumously at Mainz by the widow of Johann Albin, with a prefatory Vita of the author and dedications within the Cronberg patronage network. 👉 <a href="https://tryleo.ai/collections/exlatinis/baldi-aristotle-mechanics-1621">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 source file</a> This text was transcribed and translated as part of the ExLatinis project—an effort by Leo to make English translations of every published text in Latin in early modern Europe (between 1450 and 1750) available to the public for free online.

Title: Critical Exercises on the Mechanical Problems of Aristotle
Creator: Bernardino Baldi
Date: 1621
Notes: Original title: In mechanica Aristotelis problemata exercitationes A learned Latin commentary, in the humanist genre of exercitationes, that reworks question by question the Mechanica transmitted under Aristotle's name (a text already suspected in Baldi's circle of spurious authorship). Its governing purpose is to re-found the ancient text on the Archimedean doctrine of the centre of gravity and centrobarics, which Baldi held Aristotle to have lacked, and to reduce the traditional five simple machines (lever, pulley, wheel-and-axle, wedge, and screw) to two near-identical principles, the lever and the balance (vectis et libra). The work argues that mechanics is a facultas in its own right capable of seconding and even surpassing Nature, treats mixed-motion paradoxes (the rhombus and wheel paradoxes), collision and rebound, and the geometry of whirlpools (which Baldi argues are spirals rather than concentric circles, on the basis of his own observations of the Po and Adda), and includes an appendix on finding two mean proportionals. It was printed posthumously at Mainz by the widow of Johann Albin, with a prefatory Vita of the author and dedications within the Cronberg patronage network. 👉 <a href="https://tryleo.ai/collections/exlatinis/baldi-aristotle-mechanics-1621">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 source file</a> This text was transcribed and translated as part of the ExLatinis project—an effort by Leo to make English translations of every published text in Latin in early modern Europe (between 1450 and 1750) available to the public for free online.

Document notes

Original title: In mechanica Aristotelis problemata exercitationes A learned Latin commentary, in the humanist genre of exercitationes, that reworks question by question the Mechanica transmitted under Aristotle's name (a text already suspected in Baldi's circle of spurious authorship). Its governing purpose is to re-found the ancient text on the Archimedean doctrine of the centre of gravity and centrobarics, which Baldi held Aristotle to have lacked, and to reduce the traditional five simple machines (lever, pulley, wheel-and-axle, wedge, and screw) to two near-identical principles, the lever and the balance (vectis et libra). The work argues that mechanics is a facultas in its own right capable of seconding and even surpassing Nature, treats mixed-motion paradoxes (the rhombus and wheel paradoxes), collision and rebound, and the geometry of whirlpools (which Baldi argues are spirals rather than concentric circles, on the basis of his own observations of the Po and Adda), and includes an appendix on finding two mean proportionals. It was printed posthumously at Mainz by the widow of Johann Albin, with a prefatory Vita of the author and dedications within the Cronberg patronage network. 👉 Read our introductory primer, full report, and finding guide here 📜 View the original source file This text was transcribed and translated as part of the ExLatinis project—an effort by Leo to make English translations of every published text in Latin in early modern Europe (between 1450 and 1750) available to the public for free online.

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o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] o[mn]es Cuius op[er]a inter p[ro]p[ter] 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Transcription: Translated (English)

The Pope above all Especially Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all Whose works above all

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QV 1

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QV 1

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1 2

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1 2

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Baldus in Mechanica Anstotelis Problemata

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Baldus on Aristotle’s Mechanical Problems

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BERNARDINI BALDI VRBINATIS CVASTALLÆ 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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Bernardino Baldi of Urbino Abbot of Cvastalla On Aristotle’s Mechanical Problems Exercises: with a concise nar- rative of the author’s life and writings. Bibliotheca of the College of the University of Mainz. At Mainz, Printed and 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 Sere- nissimos 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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TO THE MOST NOBLE AND GENEROUS LORD D. ADAM PHILIP BARON OF CRONBERG, KNIGHT, OF THE SACRED CAESAREAN MAJESTY, AND OF THE MOST SERENE PRINCE ARCHDUKE ALBERT, PRIVATE CHAMBERLAIN, ETC. To my most gracious lord. Opportunely, at this very time, when he is preparing his journey into Belgium to the Most Serene Princes, Your Most Noble and Generous Lordship brings forth into the public, through our press, the Commentary of Bernardinus Baldi of Urbino, Abbot of Guastalla, on Aristotle’s Mechanics. He was a man most experienced in every kind of learning, but especially in mathematical disciplines, as many works excellently written by him testify, of which 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 favore 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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LETTER at its proper time to be brought into public light. But since it is not hidden from anyone that it has always been most familiar to the most noble and gracious Lady Your Grace, so that both domestic leisure and especially travels, by which she has encompassed almost the whole of Europe with the highest praise, as well as the perfect use of various languages and knowledge and practice of mathematical disciplines, might be made more delightful, I have no doubt at all that she will also graciously admit Baldus of Urbino, speaking through our press, into her company and protection on this journey, which I pray may be most fortunate through God, with benevolent mind. I most humbly ask and at the same time beseech that she favor this my printing house, which for many years now has boasted of the patronage of the illustrious Cronberg family, and that she kindly look with favor upon the afflicted widow in her fortunes. Thus may God illuminate the Most Noble and Gracious Lady Your Grace with all good things, and may He preserve her for the Most Reverend and Most Illustrious Prince and my most gracious Lord, D. Johann Suicard, Archbishop of Mainz, Prince-Elector, and through Germany Archi-

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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-

Transcription: Translated (English)

DEDICATION. Vice-Chancellor, etc., to his most beloved uncle, may he return safe and flourishing, likewise flourishing and unharmed. At Mainz, from the printing house of the Widow Albiniana, dedicated for ever to the honor of Your Most Noble and Generous Lordship. Year 1621. 26 March. : : 3 PRÆ-

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PRÆFATIO. 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 ex phrasi, & explicatione, quæ Aristotelem sapiunt, tum iudicio subtilitatis & rationum, qui- bus

Transcription: Translated (English)

PREFACE. As I diligently read those questions in which the matters pertaining to the mechanical art are explained, many thoughts came to mind; and first I admired the usefulness, subtlety, and abundance of the things discussed there: then from my heart I grieved that this golden little book was being almost neglected, and was not being kept constantly at hand by those who devote themselves to these most admirable studies: I also have many thanks to render to the Author himself, who has supplied Architects, Mechanics, and almost all craftsmen with so excellent, useful, and well-equipped a store of materials. The commentary is ascribed to the name of Aristotle, although some have hesitated to affirm whether it is the work of that most famous and keen philosopher or not. Yet almost all the better judges agree that it is Aristotle’s; and this is shown both by the style and explanation, which savor of Aristotle, and by the judgment, 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. The questions themselves are most ingeniously resolved. But it seems to me, on closer examination of the matter, highly probable (for I have no supporter of this opinion) that this is a section, and a certain part, of that most noble work which the same author published, On Problems ; and that this one, I know not for what reason—unless perhaps because the treatment is not purely physical—was detached and torn away from the rest of the body of the work. What is certainly pertinent to the matter we know well: that Diogenes Laërtius, among other monuments of Aristotelian genius, also counted the mechanical writings. Considering these things, one is greatly moved 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 indeed they ought to have done so, for it is not unlike the truth that they themselves profited in some measure by it. Yet in truth, since those men were noble-minded and by no means detractors, it is rather to be believed that that little commentary, in their time known to few, lay hidden somewhere in libraries; for even the other writings of Aristotle, after those ancient times, before Alexander of Aphrodisias, were ignored by many.

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PRAEFATIO rata non dubitamus. Habemus siquidem, Sira- bone teste, lib.13. Aristotelis, et Theophrasti bi- bliothecam, post ipsius Theophrasti decessum, ad Neleum quendam Scepsium, Coriscifilium, qui eius fuerat auditor, peruenisse; post hæc libros, blattis olim, et humore corruptos, Apelliconi Te- io venditos, et ab eo Athenas translatos, tum Athenis captis in Syllæ potestatem deuenisse, eos- que 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 cer- tum est, Aristotelem eorum omnium qui de Me- chanicis commentaria edidere, esse longè vetu- stissimum. Pappus enim Heroneiunior, Athe- næus Archimedi æqualis, vterq[ue] enim sub Mar- cello, cui Athenæus suum de bellicis Machinis libellu[m] dedicauit. Archimedes verò circa CXL. Olympiadem floruit, quamobrem post Aristote- lem Olympiadas XL. hoc est, annos ferè CLX. Isthæc autem considerantibus, facile est cognosce- re 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, the library of Aristotle and Theophrastus, after the death of Theophrastus himself, to have come into the hands of one Neleus of Scepsis, son of Coriscus, who had been his pupil; after this, the books, long ago damaged by worms and moisture, were sold to Apellicon of Teos and by him carried to Athens; then, when Athens was captured, they fell into the power of Sulla, and finally, having been received from Sulla, Tyrannion the Grammarian, as best he could, corrected them and published them. From this we gather that it is no wonder that they were unknown to Archimedes, Heron, and others who lived before Sulla. Be that as it may, this much is certain: Aristotle is by far the oldest of all those who published treatises on Mechanics. For Pappus is later than Heron, Athenæus contemporary with Archimedes; both indeed lived under Marcellus, to whom Athenæus dedicated his little book on war machines. Archimedes flourished around the 140th Olympiad; wherefore, Aristotle was forty Olympiads earlier, that is, about 160 years before. Considering these things, it is easy to recognize the nobility and dignity of this science, since the supreme Philosopher not only it 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, sellulariæue, 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)

Of the author. He not only proved it, but also illuminated it with his most acute labors. Aristotle himself taught us that this treatment is, as to its subject, Physics, but as to its demonstrations, Geometry. To him also belong Perspective, Optics, Music, and the other like subordinate faculties, which the Peripatetics indeed call subaltern. Vitruvius makes it, so to speak, a member and a certain portion of Architecture; for he says that the parts of Architecture are three: Building, Gnomonics, and Machinery. It is, however, inferior to Architecture, for the Mechanic is subject to the Architect; yet if you consider the other arts, it is architectural, for nearly all the sedentary and workshop crafts, which the Greeks call banausic, are subordinated to it in due order, and indeed it has very broad limits. But especially it is concerned with that knowledge, and among the rest is almost the principal one, which they called Centrobarica, and which pertains to the gravity of the center and its investigation. In this field, among the ancients, Archimedes first claimed the place for himself, then Hero, and afterwards Pappus; among the moderns, however,

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P RÆFATIO tem Commandinus, qui librum de Centro grauitatis 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 Item Commandinus, who wrote the book On the Center of Gravity of Solids, and after him G. Vbal- dus of Marchion. Montis, who not only composed with the greatest praise of his talent a most complete book of Mechanics, but also elegantly prepared a Paraphrase on Archimedes’ book On Equal-weights, plainly show that this Centrobarica was unknown to Aristotle. For in his mechanical demonstrations he never names the center of gravity, which nevertheless is the chief thing; he considers its nature and force. Now the whole of Mechanics is divided, as Heron testifies in Pappus, book eight, into Rational, that is, Theoretical, and Chirurgical, that is, manual and operative, which we may suitably call Praxis. The Rational part is devoted to speculation and demonstrations from geometrical, arithmetical, and physical principles; the Chirurgical part, however, deals with matter, and spreads itself through various arts: the Art of Money, Carpentry, Sculpture, Painting, Architecture, Machinistry, and Thaumaturgy, and others of that kind. But the parts of Machinistry are Manganaria, by which great 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. the weights are transferred, then the Poliorcetica itself, which builds military machines for the assaulting of cities, as it proclaims even by its very name. But we shall refrain from writing more on this subject, lest we do a thing already done: for whoever desires to know these matters more exactly, let him consult Pappus in the cited book, and Guidus Ubaldus in the Preface which he placed before his Work on Mechanics. But now to return to Aristotle’s little book, of which we have spoken, few are those who before us have devoted pen and labor to it: Leonicenus made it into Latin and adorned it with figures, both very brief and of slight weight, with marginal notes. After him Alexander Picolomineus illuminated it with a most splendid Paraphrase. Recently, as I hear, Simon Sticinus of Holland published certain things, which have by no means come down to us. We at last, being last of all, both in learning and in talent, and also in age, have applied our hand to this work; considering indeed that Aristotle employed other principles, and that the most approved mechanics after him have demonstrated that we shall act in a way pleasing to the students of this discipline, if we treat the 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, boniconsule, & quia fax per manus traditur, tu interim de me accipe, vt alijs tradas. DE

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We would confirm it with mechanical, that is, Archimedean proofs; while wandering through the broadest fields of this art, and introducing other doubts akin to these, we would resolve them. Whatever we have done or accomplished, most excellent Reader, receive it kindly; and since 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 MDCIII. 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 (:(:(: 3

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ON THE LIFE AND WRITINGS OF BERNARDINO BALDI OF URBINO FROM THE LETTERS OF FABRITIUS SCHAR- loncini to the Most Illustrious and Most Reverend Lord Lælius Ruinus, Bishop of Balneo- reggio, formerly Apostolic Nuncio to the King of Poland, etc. Bern. Baldus was born at Urbino of noble parents on the day after the Nones of June in the year 1603. He derived his lineage, as I have often heard from him myself, from the Cantagallina family, which is illustrious among the Perugians; this surname, Baldi having been adopted, as happens in the variety of times, his great-grandfather left behind. From his tenderest years he showed piety toward God; for, as his mother used to recount, he contemplated images of saints and little altars not only with joy but with reverence, even as a child of one year. In his youth he had most praiseworthy teachers, Io. And. Palatio and Io. Antonio Turoneo, the latter more learned, and very dear to Paolo Manutio because of his almost singular proficiency in the Latin and Greek languages. Yet to their diligence he brought such ardor of mind, such force of talent and judgment, that he surpassed not only his peers but the expectation of all. While still a boy he rendered Aratus’ Appearances in Italian verse. His father, moved by this praise and glory of his son, in the year 1573 sent him to Padua to pursue a greater cultivation of his genius. There he immediately came into the intimacy of Emanuel Margunius, to whom moreover he was (:(:(: 3

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V I T A fuit in amoribus. Homeri Iliad. illo Doctore & interpre- te diligentiis 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 Molfetæ 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 went through Homer’s Iliad with that teacher and diligent interpreter, more thoroughly than he had done before. In private study he devoted himself to Anacreon, Pindar, Aeschylus, Euripides, and Sophocles, but above all he spent himself on Theocritus’ Bucolica, to which kind of writing he seemed by nature to be more inclined. He knew by heart hundreds of verses of some Greek poet, and often had them on his lips; in working through the Greek orators he felt some labor, in the poets none. He wrote at Padua a little book on siege engines and their inventors, and when he had fallen in with the friendship of men from beyond the Alps, he considered it a disgrace not to understand those speaking their own language. Therefore, with incredible speed, he learned French and German. Driven from that gymnasium by pestilence, he returned to his homeland, where for a full five years, attached to Federico Commandino, he relearned all parts of mathematics; to that man he lent his hand in drawing figures for the monuments of Euclid, Pappus, and Hero. On Commandino’s death he endured an almost inconsolable grief, and, having taken up the plan of writing his life, he then applied his mind to writing the lives of all mathematicians, which he carried out most successfully for the space of twelve years. But while he was tormented by love of the mathematical disciplines, and after the loss of Commandino his teacher, he found an excellent friend and fellow devotee, Guidobaldo, Marquis of Montis, into whose company he entered: how much he progressed is shown by those commentaries which he wrote in 1582 on Aristotle’s Mechanics. In order later to turn his mind from the severer studies to more agreeable ones, he composed in Italian a poem on navigation. When this was finished, he explained many mathematical paradoxes. Fame having spread abroad regarding Baldus’ talents, Ferdinand Gonzaga, Prince of Molfetta and Lord of Guastalla, began to think of enrolling him in his household, since he was captivated by the same arts in which Baldus 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:

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AUTHORIS. piebat: Thus, by the efforts of Curtij Arditi, he was honorably summoned to the court, but while he lived not a courtly life, wholly devoted to letters in retirement, at the prayers of Vespasiano Gonzaga, Duke of Sabionetta, he was compelled to explain the books of Vitruvius. Hence there then arose a commentary on the meanings of the Vitruvian words, in which it is by no means surprising if he pursued certain minute details that might seem less worthy of a great man: for he was complying with that Prince’s wishes. I know that Adrianus the Roman once said, after returning from Poland, where he had explained Vitruvius for a certain Palatine, that if I could have used Baldus’s commentary in Poland, I would have stripped out the gold I brought with me, because I would have fulfilled the task with no labor. When the necessity arose for his lord Ferrando to go to Spain, he did not think he could make that journey without Baldus, not so much in order to have someone who, by learned speech, might lighten the tedium of the road, as someone to whom he could entrust his secrets, and even from whom he might receive counsel and help. Hardly had they set out when Baldus, falling into a serious illness, was forced to break off the journey: accordingly he turned aside to Milan, where he was both kindly received by St. Charles Borromeo and kept there until he recovered his health. Afterwards he withdrew to Guastalla, where, while the Lord was absent and he enjoyed a freer leisure, he composed six most learned books On the Court by the analytical method. I do not mention the others, because when I have leisure I shall give a catalogue 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 Chaldean. In the year 1593, he composed five books on the new Gnomonics. The following year he translated Onkelos’s Chaldean paraphrase on the Pentateuch and added commentaries; 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 was: in that divination, as he used to say, he spent the spare hours of one month. He wrote excellently on the Firmament and the waters. Cardinal Baronius, who had seen no other work by Baldi, greatly praised the Oeconomia Tropologica in St. Matthew. While he lived in Rome, he scarcely knew what was going on in the Courts: for he diligently devoted himself, together with Io. Baptista Raimondo, to the Arabic language, and by secret industry to the Slavonic language, which he knew perfectly. From Arabic he translated the Hortus Geographicus of an Anonymous writer, whom he judged to have flourished six hundred years earlier. Marcus Velserus, of Augsburg, would indeed have brought this work to light, as he did other books by Baldi, if a somewhat longer enjoyment of this life had been granted him. He also composed an Arabic Dictionary. And since that blessed abundance of genius necessarily had to be continually diffused, in the year 1603 he undertook to describe the whole world; and indeed in such a way that he embraced both what belonged to History and what belonged to Geography: nor did he wish merely to illustrate what the ancients knew, as Ortelius thought, but even all the little towns and villages of which there is some mention in later writers. And indeed he brought the whole work to completion as far as the navel, but did not yet arrange the whole. There were only four or, if I am not mistaken, five volumes arranged in alphabetical order; seven or eight more would have remained to be arranged, as may be inferred from the mass of papers and bundles. In the year 1617, on the fourth Ides of October, after he had been afflicted for 40 days by a violent catarrh, he rendered up his spirit to God, having been duly provided with all the sacraments of the Church. He was of tall stature, with an oblong face and keen eyes, of a swarthy complexion. His limbs were of proper proportion, and his body compact. On all feast days he heard 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. St. Augustine’s books On the City of God three

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AVTHORIS. ter prandium euoluit. Statim à noctis meridie dum ei vires firmiores essent ad lucubrandum surgebat. à prandio Euclidem Arabice editum, vel libellum aliquem germanicum aut gallicum in manus sumebat. Suauitate morum & modestia, etiam si ceteræ 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 cenju 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. Bibliothecam 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 tempora. 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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AUTHOR. He would spend time after lunch in study. Immediately after midnight, while his strength was still firmer for keeping vigil, he would rise. After lunch he would take up Euclid in Arabic edition, or some German or French little book, into his hands. By the sweetness of his character and his modesty, even if all other gifts had been absent, he could have drawn anyone to love him. His speech was restrained, and so was his dress. He never sought any honors, though the most generous ones had been promised to him by Clement VIII; he desired no reward, and was content with his condition. He used to say that one ought to be lenient, especially toward those who had erred in a small matter, since if we were to see those whom we consider the best stripped bare, we should judge none of them undeserving of many blows. He had a library not rich in number, but furnished with select volumes. But to go through everything in detail would be long. It is enough for me to have said a few things about the incomparable learning of Baldi and his great innocence, O rare union, perhaps too many for imitation. CATALOGUE OF THE BOOKS of B. Abb. Baldi. A. Rati Apparitiones, translated from Greek into Italian. On War Machines and their Inventors, book. He translated Heron’s Automata. He wrote the Lives of all the Mathematicians, in three parts, in vol. 2. 1. From Thales to Christ. 2. From Christ to his own time. He composed a Chronological Epitome of those same lives. Commentary on Aristotle’s Mechanics. On Nautics, a 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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S C R I P T A Pious poems, entitled, Anni Corona. On the meaning of Vitruvian words. Various poems and mixed eclogues. One hundred apologies, which he wrote in emulation of Leon Batt. Alberti. A Dialogue on Humanity entitled Goselinus. A comparison of the monastic life with the secular, On the Court, six books. A Dialogue on the happiness of a Prince. On Dignity, Dial. Roman poems. He translated the fable of Musaeus. A Dialogue on the nature of Italian poetry, entitled Tassus. A poem on the universal Flood. Five books of the New Gnomonics. He translated Jeremiah’s Lamentations, and added notes from the Hebrew source. A poem entitled Deiphobe, which he wrote in emulation of Lycophon in Cassandra. Scala coelestis. 1. Pious sermons and poems. He translated Onkelos’s Chaldean paraphrase on the Pentateuch and added abundant commentaries. A Latin paraphrase on Job from the Hebrew source, with Scholies added. On the unequal staves of Vitruvius. On the firmament and the waters. He translated Quintus Calaber’s Paralipomena. Interpretation of the Etruscan Eugubine Tables. Tropological Economy on St. Matthew. An encomium of Urbino. A version from Arabic of the Geographic Gardens. Poems against the Court. Translation of Lucian on the miseries of courtiers. An oration 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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AUTHORIS. A geographical and historical description of the whole world, compiled from seven hundred and more authors. Life of Federico, Duke of Urbino. Life of Guidobaldo, Duke of Urbino. Three books of Epigrams and Odes. A book of other poems. A book of moral sayings. Arabic dictionary. Against Flavio Biondo, on behalf of Procopius. Universal horograph. Other epigrams. Translation of Heron’s book On Ballistas. Exercises on Aristotle’s Mechanics. New description of Ezekiel’s Temple. Book on the antiquities of Guastalla. Laws for writing history. And certain other things. IN

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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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IN MECHANICS OF ARISTOTLE’S PROBLEMS EXERCISES. Description, nature, and end of mechanics. MECHANICS is a certain faculty, which, considering natural matter by means of geometric demonstrations, and by the speculation of the center of gravity, and of those things which are reduced to the lever and the balance, provides for human necessity and convenience, and by its own power either follows Nature herself or surpasses her, and produces various, indeed marvelous works. In this brief definition and description we have comprised almost all those things which have been treated at length by Aristotle, Pappus, Guido Ubaldo, and others on this subject. Object of Mechanics. But the mechanic considers Heavy and Light. Heavy is twofold, by Nature and by Violence. That is called Heavy by Nature which is carried by an inborn tendency into the center of the world. But Heavy by Violence is that which, by weight impressed from without, is driven by the one applying force. Light, on the contrary, is that which is carried by nature away from the center. Moreover, whatever is heavy has a point, which is called the center of gravity; and this is twofold, as weight is twofold, of Nature and of Violence. 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 a threefold magnitude: linear, plane, solid. No one has written about the center of gravity of lines, for that is the simplest matter of contemplation. Archimedes treated the center of gravity of planes excellently in the book On Equal Weights , and in On the Quadrature of the Parabola , and also in the work which he entitled On Things Carried . Archimedes himself had formerly written on the center of gravity of solids, but what he produced was lost through the injury of time; Federico Commandino restored it by his diligence. That there is also a center of lightness in nature is plain. For that is the point according to which light bodies are carried straight upward from the center. But mechanics have made no mention of this, because it either contributes nothing or little to their purpose. Further, Heron defines the center of gravity thus, and Pappus after Heron, in book 8 of the Mathematical Collections : The center of gravity of any body is a certain point placed within it, from which, if the weight be conceived as suspended in the mind, while it is being carried it rests and preserves that position which it had at the beginning, nor does it turn around in the movement itself. Commandino, however, in his book On the Center of Gravity of Solids , puts it in this way: The center of gravity of every solid figure is that point placed within it, around which on all sides parts of equal moments are situated. For if a plane is drawn through such a center, cutting the figure in any way whatever, it divides it into parts of equal weight. We, however, say as briefly as possible: The center of gravity of any magnitude is a point placed within or without the magnitude, through which, if it is divided by a plane, line, or point, it is divided into parts of equal weight. 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 we were including the center of magnitude, of a line, plane, and solid. Therefore, as in the present figure, the center of the line will be A, of the plane B, of the solid however C. But if someone objects that a line and a surface have no gravity, he should know that mathematical bodies also do not have gravity; whereas mechanics takes ropes, rods, and levers for lines; and tablets, and planar things of that kind, refers to the nature of surfaces. We said further, within and 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 equally weighted, A/B/C, EDC. Let GH also be drawn, dividing likewise into equally weighted parts GCH, and GAB, EDH. Let them cut one another at I. Therefore the center I will be outside the boundaries of the figure and its own mass. 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 circumference ACDA. We were saying that there are two centers of gravity, Natu- re, 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, idem 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. Proijciatur 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. rather, by Violence: we affirm only that in this matter it is indeed one and the same thing, and is to be considered only in reason, not however in reality, as if they were two. For let the center of natural gravity be B, of body A, according to which, when released, it will fall of itself by its own nature to 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, motion, in D; yet they are the same in reality, namely one B, but two if violence and nature are considered separately. These centers follow two motions, both straight, namely Natural and Violent. A third is compounded of these, and indeed is not straight, but curved. For if a heavy body A be thrown violently, with violence prevailing, it will be carried straight to B; but as this weakens little by little it is divided by a curved and mixed line into C, inasmuch as it is carried forward, there is violence; insofar as it is carried toward the lower parts, nature. But when it has reached C, violence ceasing, nature however remaining, it is carried straight downward D C D. Moreover, these centers, and these motions, namely natural and violent, are related to one another in different ways. For if a heavy body, by external force applied, is driven toward the center of the world, Nature and Violence will assist one another. But if otherwise, one will resist the other; but in 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 however, to the side, the more they will fight, the more motion is made from the lower parts to the higher. The principal instruments of mechanics. These things having thus been settled, we say that the instruments by which Mechanics use themselves for various operations are indeed different from one another, manifold, and, if you look at their variety, nearly innumerable; and although this is true, Aristotle reduces them all to the lever and the balance, which also G. V. Baldo did in his books on Mechanics. Moreover, the Mechanics who flourished after Aristotle reduced everything to the five so-called Powers. These are, from Hero, Pappus, Guido Vbaldo, who is secure in them, the Lever, Pulley, Axis in Peritrochio, Wedge, Screw. It seems, however, that G. V. Baldo himself adds a sixth, namely the Balance, concerning which he himself first also established a treatise on mechanics. But in truth the Lever and the Balance are almost the same thing, except perhaps that it is called a Balance when the arms are equal. The Lever, however, however they may stand with respect to one another; we thus place before the eyes images of these five Powers. Lever A. Pulley B. Axis in Peritrochio C. Wedge D. Screw E. A 3 Moreover

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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. Vbaldus does as well, who refers the screw to it too, 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 wheel-and-axle. Not at all, however, the wedge and the screw; which we shall indeed show more fully when the wedge is our special subject. Of the Lever and the Balance according to Aristotle. Aristotle at the very beginning of the Mechanical Problems writes thus: “It seems strange that by a small force a great weight is moved,”

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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- tentia 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. Autem, absurdum nullatenus esse, si exre mitabili mirandum quippiam oriatur. In circulo autem qua-

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EXERCISES. 7 indeed, with weight added to weight, since a lever is weight. Therefore there is a double wonder in it: namely, that a small force moves a great weight, and that this is done even with the addition of the lever’s own weight. He seems to have added this second point for the sake of some amplification. For insofar as it concerns the matter itself, if you compare the lever’s own weight with the weights to be moved, you would undoubtedly declare it to be of almost no importance. But this too must be noted: sometimes the lever’s weight brings aid to the mover, which happens when, with the support placed between the moving power and the weight itself, the part of the lever from the support to the power is pressed down. Then, as we said, the lever aids the power by its own weight. On the contrary, it happens when the weight itself is between the support and the power, or the power itself is between the support and the weight. Then indeed the lever is lifted together with the weight. Although these things are true, it does not follow that the lever’s weight accomplishes, or impedes, anything that ought to be cared about in the operation. Moreover, we may define a lever as a certain inflexible length, by which, with a given support and a given power, a given weight is moved. A Balance also, as we said, is a lever: but it is of such a nature that the central support always occupies a middle place between weight and weight. A steelyard, however, is nothing but a lever, if you assign the fixed point as the support; but the movable counterpoise as the moving power. The doctrine of Aristotle concerning the Circle and its nature examined. Aristotle thinks that whatever marvelous thing is accomplished in Mechanics is wholly to be attributed to the admirable nature of the circle. But there is nothing absurd in the fact that something admirable should arise from something admirable. In the circle, however,

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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 constitui, 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 are found worthy of admiration. First, that it is constituted from contraries, namely the mover and the moved. Second, that contraries are found in its circumference, for since it is one 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 although there is but one semidiameter, 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 semidiameter A C; then let point D be taken in it, and likewise point E. Therefore in the same rotation D will be slower than E, and E itself slower than A; and thus that part is always carried more quickly which is taken farther from the moving end. We are restrained from readily assenting to these things not without reason. We say therefore that it seems to us that the circle is not constituted from contraries, as from the remaining and the moved, but simply from the moved. For there is no part of the semidiameter which does not move. But a point which stands still is no part of the semidiameter. And truly why a circle is formed from a moving semidiameter does not happen because one extremity stands still while the other moves, but because the semidiameter perpetually preserves the same length. An ellipse indeed has a center, but from that center to the circumference, only four semidiameters taken in any way are drawn equal. If anyone should therefore give a semidiameter increasing and decreasing in proportion, while one of the extremities remains still, an ellipse would be described. Moreover, the spiral line, which is mixed, is produced with one extremity of the semidiameter remaining still, and the other moving. Thus a law for the circle is pre-

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EXERCITATIONES. 9 præscribit, non quidem quòd hæc extremitas sit, 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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EXERCISES. 9 prescribes, not indeed because this extremity is moved and that one is not, but because by its circulation the semidiameter always preserves the same length, which may even be gathered from the very definition of the circle. To the second marvel, namely, that in a circle the circumference, which is an empty line, is at once concave and convex. Someone might say that this, if it is marvellous at all, belongs not only to the circle but first to any curved line; for both the Ellipsis and Hyperbole, and Parabole, and spirra, then Cyssois, Conchois, and countless other irregular lines are at once concave and convex. But this is also lacking in surfaces as well. To the third, that it is carried by contrary motions, forwards, backwards, upwards and downwards. We say this is easy to resolve. For no one, having examined the matter properly, would affirm that a circle is moved by contrary motions. Let there be a circle ABCD, around center E; let us suppose it rotates, and from A toward B, for example, forwards; but also from B toward C, and from C toward D, then from D toward A. I do not think anyone would say that this circle is carried forwards at the same time, and backwards, nor upwards or downwards; for if someone were to walk along the circumference of that circle, he would certainly always have the center itself on the right hand, or on the left; if on the right, he will go forwards, if on the left, backwards. But neither upwards nor downwards, it is clear. Nothing, however, prevents the same motion from being said to be contrary in different respects, yet that certainly cannot happen, namely that A moves toward B, that is, B forw-

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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 at the same time toward B, that is, backward; for it is contrary to nature. We shall speak there about the fourth miracle of the circle, where first we shall consider those things which the Philosopher brings forward in the midst of discussing the production of the circle. They are of this kind: That the circle is produced in a twofold sense, namely, in one way naturally, and in another way contrary to nature, and therefore the circular line is to be reckoned among mixed motions. A mixed motion, he says, either takes place while proportion is preserved, or not; if proportion is preserved, a straight line is produced; but if it is not preserved, a circular line is produced. Let there be, for example, the rectangle ABCD, whose sides AD and AB are in a given proportion. Let A be moved by a double motion, tending indeed toward B by the one, but by the other, according to the motion of the line AB, let it be carried toward D, while the proportion of the sides is preserved in the meantime. Thus let it be assumed that from the motion from A toward B it has arrived at E, but from the motion by which it is carried proportionally with the line AB, after AB itself has been made into FH, it has arrived at G, 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 is carried by two motions while the proportion of the sides is preserved, will produce a straight line, namely the diameter AGC. And this indeed has no doubt, from the things which Euclid teaches in 1.6, prop. 34. These things being thus demonstrated, the Philosopher seems to use this argu-

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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 fit, 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 A C 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; if never removed, it will make a circle. For if it were sometimes preserved, sometimes not, it would produce partly a straight line and partly not straight. A clever argument indeed, were it not for the flaw it contains. For a mixed motion, which is made when the proportion is never preserved, does not always produce a circle; it can produce an Ellipsis, and indeed any other line of which no part is straight. Piccolomineus saw this difficulty in his Paraphrasis, and tried to solve it, but how well, let others judge. Moreover, it is false to assert that a circle is produced from a mixed motion when the proportion is never preserved. For the mixed motion by which it is produced continually preserves some proportion, if we wish to produce it by a mixed motion, but not the same one. Let there be, then, a straight line AB, to which AC is at right angles. Let A, however, move toward C along the line AC, and at the same time let the line AC move toward B, but in such a way that it is always perpendicular to AB itself. Let it be carried on under this law, namely, that the proportion which the motion of the line AC toward B has to the motion of the point A toward C, the same proportion the motion itself from A toward C shall have to the remainder of the line AB, namely after subtracting that part which the moving line AC has traversed toward B. Now let it be that, when A C by its motion has arrived at D, the point A, likewise carried by its motion through that line, has arrived at E. Therefore, from the mixed motion, not indeed at D nor at E, but at F, there will be [a point], and point F will be on the circumference of a circle, of which the diameter is the line AB itself; and this 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 AC be made in GH, to the motion H along the line

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12 IN MECHAN. ARIST. PROBL. lineam AC, vsque in C, vtse 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 procul dubio 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 produci. 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. Etto, inquit, circulus BCDE & alter in eo minor MNOP circa idem centrum A. Ducanturq; 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, as the proportion of AG to GH and of GH to GB stands, there will be from the mixed motion A to H, namely in the circumference AFHB of the same circle. From which 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 for producing a straight line mixed motion is not necessary, though it can be produced by mixed motion, so neither is it necessary for a circular one; and therefore what the Philosopher asserted is not true, that a circle can never necessarily be produced from mixed motion while the proportion is not preserved. After this Aristotle tries to offer a reason why the parts of a circle are slower, the nearer they are to the center. For he says: if of two things moved by the same power, this one is repelled more and that one less, it is fitting that that thing which is repelled more move more slowly, than that which is less. But a line is drawn away more, whose end is nearer the center, than that which has its end farther from the center. Let there be, he says, the circle BCDE and another smaller one in it MNOP about the same center A. And let the diameters of the larger be drawn, namely CD, EB, and of the smaller MO, NP. Therefore when AB, having been carried around, has reached the point from which it departed, AM itself 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. Therefore let ALF be drawn, and from the point L, perpendicular to AB, Lq, falling in the smal- ler 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. Maiorem 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 BF 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[m] 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, MqO, 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 in the circle, and again from the same point L to the same AB, let the parallel LS be drawn; from S, however, the perpendicular ST to it, and from F the line FX. Therefore qL and ST are equal, namely those by which, according to nature, the points BM are moved; but by the motion of recession toward the center, that is, contrary to nature, M has been moved more than B. For Mq is greater than the line BT itself, which, as is known, Aristotle supposed. But we shall demonstrate this below. If therefore it is made so that, as the motion contrary to nature is to the motion contrary to nature, so the motion according to nature is to the motion according to nature, then the point B, when M shall have been in L, will not be in S, but in F. For then, as the motion FX according to nature is to XB contrary to nature, so is qL according to nature to qM contrary to nature; but BF is greater than ML, therefore, the proportion being preserved, B is moved more swiftly than M about the same center A. This, moreover, is the sum of what Aristotle sets forth. However, we have omitted the parallelogram, which is found in his figure, since it contributes nothing to the demonstration that is given. Now what we promised, namely that BT is less than qM, we demonstrate thus. Since ST, by proposition 13.1,6, 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 proportionals. Wherefore, as TE is to qO, so is Mq 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. Moreover, we do not deny that these things are subtle and ingenious, and this truth can be demonstrated in a far easier and clearer way, namely with those notions rejected, both according to, and contrary

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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 compleat 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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14 IN MECHAN. ARIST. PROBL. concerning the nature of motions, which certainly do not necessarily fall in a simple circle: but they would perhaps do so, if the matter were of a circle described by weights carried around from a fixed center; on which matter G. V. Baldus treats in the Mechanics, in the tract on the balance. For then it can be said that the weight which otherwise would tend straight toward the center of the world is, in circulation, drawn back from the center of the circle; but these things in no way pertain to 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 on A C, D, and with center C, at distance C D, let the circumference D G E I be described. I say that point A moves more quickly than point D, rotated by the same circulation. 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 the line C A, moved around center C, likewise moves C D; therefore if points A and D complete their rotation in the same time, A measures a greater space in the same time than D itself, and therefore is swifter. Thus therefore speed is to speed as circumference is 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 from it by less, which was to be demonstrated. QVÆ.

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EXERCITATIONES. 15 QVÆSTIONES MECHANICÆ. QVÆSTIO I. Cur maiores libræ exactiores sint minoribus? Prioribus, cèu fundamentis quibusdam iactis, opportunè ad quæstiones proponendas, eas quæ 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 E, eritque minor libra in G, in eadem enim linea sunt CGF. Vtraque igitur ex eodem cen-

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EXERCISES. 15 QUESTIONS OF MECHANICS. QUESTION I. Why are larger balances more accurate than smaller ones? Having laid down certain foundations beforehand, Aristotle suitably turns to proposing questions that are to be answered. Moreover, in the proposed question, at first sight it seems that a cause is sought for a thing that is not; for who has ever affirmed that the scales used by Apothecaries and Butchers, the large ones indeed, are more accurate than those with which Jewellers and Silversmiths weigh the smallest particles of grain and scruple, which are nevertheless very tiny, and, if compared with those, are smallest? Yet the matter is exactly as Aristotle asserts. For it is not because the former are large and the latter small that these are more accurate than those; rather, because the large ones are rough, while the smaller ones are worked with exquisite care and freer from the stubbornness of the material. Therefore, all other things being equal, that larger balances are more accurate, we shall show from Philosophy in this way. Let there be a larger balance AB, with its support C. Let there also be a smaller balance DE, turned around the same support C together with the larger one in imagination. Let any weight be placed on the larger balance at A, and it will incline; for example, toward E, and the smaller balance will be in G, for CGF 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. Cæterum hac eadem de causa, Astronomica instrumenta, puta Astrolabia, Armillæ, & alia 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 of the circle 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. Therefore, as diameter is to diameter, so portion is to portion: but the greater diameter AB is to the diameter DE: therefore the greater portion is AF, than portion DG. And what is greater deceives the eye less; thus we shall recognize the drawing made from the larger AB more accurately than from the smaller DE itself, which was to be shown. Moreover, for the same reason, astronomical instruments, such as astrolabes, armillae, and others of this kind, are judged the larger they are, the more accurate and more certain they are. For let there be a large astrolabe, whose diameter is AB, and a small one CD, about the same center E. Let the straight line EF be drawn from the center, touching the larger circle at F, and cutting the smaller 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 on GD to those marked on BF. Therefore those on FB are greater, and capable of smaller divisions. Hence it appears that whatever instruments are the larger, the more accurate they are also, as Aristotle had proposed in this question concerning the balance. What he adds concerning the frauds of the purple sellers, saying, “Wherefore do those who sell purple contrive to defraud by selling, while to the middle, 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, èquè 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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EXERCITATIONES. 17 not placing them; then by pouring lead into either side of the scale; or by placing the piece of wood that inclined toward the root on the side to which they wish it to be carried, or if it has a knot; for the part in which the root is is heavier, and the knot is in some way a root. Hence one might ask: Whether scales that, when empty of weights, balance, are altogether free from fraud? It might seem to someone that scales which, when empty of weights, balance, are altogether free from fraud; nevertheless this is not so, as we shall examine more carefully, for the matter is of great importance. Let there be a balance AB, divided thus in C, so that AC is of the parts IS, but CB of the same parts IO. Let there be placed on the side A a pan weighing IO, and on the side B a pan weighing 15. Thus, by the inverse proportion, when the balance is suspended in C, it will weigh equally; but if there be added to the pan B a sacoma of 6 ounces, and in pan A be placed purple that has this relation to 6 ounces as IO has to 15, it will again balance equally, but as IO is to 15, so are 4 to 6. Therefore the purple seller, acting fraudulently, placing in pan A 4 ounces of purple, when equilibrium has been achieved will demand the price of 6 ounces, and thus will deceive the buyer, which indeed Aristotle had hinted at, but not demonstrated. These matters, however, will become easier from those things that will be explained in the following questions, where the lever will be discussed. But the fraud is detected if, in weighing, the sacoma is placed alternately now on this pan, now on that. For if the sacoma is placed in pan A, but the purple in B, equilibrium does not occur. 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, sartum 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 ON MECHAN. ARIST. PROBL. QUESTION II. Why, if the support of a balance is above, when a weight is applied to one end it descends, and when that weight is removed it rises again and returns to equilibrium. But if the support is below, why, when depressed, does it not return to equilibrium? The Philosopher states the question in an incomplete form; it ought to be a threefold question, since the support may be placed in three positions: upper, middle, or lower. We will speak to each of them. First part of the question. On a balance having the support above. Aristotle solves the first part of the question thus: Is it because, when the upper part is present, more of the balance lies outside the perpendicular? For the string is a perpendicular line; therefore that which is greater must necessarily move downward, until it reaches the point where the balance is divided in two, namely the perpendicular itself, when the load rests upon the part of the balance drawn upward. Let the balance be straight, that is, placed in equilibrium, B C; and let A D be above; and let the support be at D above. But let the string be projected downward to M, where A D M will be perpendicular. If therefore a weight is placed at B itself, B will be at E, and C at H; wherefore the line that cuts the balance in two will first be D M, that is, the perpendicular itself; but when the weight is applied, it will be D G. Therefore the balance itself will be E H, which is outside

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EXERCITATIONES. 19 extra perpendiculum, est AM, vbi est qP maius est dimidio. Si igitur amoueaturonus 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 diuisamad 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 Derit 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 plumb line, it is AM, where qP is greater than the half. If therefore the load is removed from E, H must necessarily be carried downward, for it is less than E; and if therefore it has had the spar upward, on this account the balance rises. All have drawn this figure very badly, so that it is extremely difficult to grasp the author’s meaning from it. But we set the matter more clearly before the eyes. This then is what Aristotle means, because the part of the beam HDG is greater than the part EDq, it must necessarily, when raised, descend, and again return to equilibrium, divided in two by the perpendicular ADM. We can demonstrate the same thing with a simpler figure. Let there be a balance AB, divided in two at C, with the support however above at D; let the perpendicular DC be produced to E. Therefore, when the balance AB is at rest in equilibrium, the part CH is equal to the part CB. Let a weight be placed at B. The balance will therefore incline, moving around the center D, and 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 at H. Let the placed weight be removed: 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 as far as H, but the same part is taken away from HF, IF will be less than GI. Therefore IF will be overcome by GI, FI will descend, but IF will rise, until once again the balance

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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[n]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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...be divided into equal parts, as before, in C, and equilibrium will be established. 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, considering the following. If a weight is turned around a fixed center, when released it will not remain at rest unless the center of gravity is in the perpendicular which, passing through the center about which it is turned, falls toward the center of the world. But it will stand on that perpendicular in two points, one farthest from the center of the world; the other, however, as near to it as possible. Let there be a body A, whose center of gravity B, resting on the inflexible line B C, with which it freely turns about the center C. Let a perpendicular B C D be drawn through the center of the world. Let the weight A therefore first be according to the center B of the slender body, on the perpendicular itself above the center C, namely in B. Let it be moved and descend to E; after this, however, 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, cutting the perpendicular in two opposite points, B F; I say, the weight when freely released

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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- queponderantes, 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 made to rest at only two points, would by its nature remain supported, B F, at B, first, because when the body A itself, divided by the perpendicular, which is understood in place of the surface A B C D through the center of gravity, is divided into parts of equal weight, it will therefore incline to neither side. It will therefore stand upright, supported by the line itself, the inflexible B C, which rests on the point C. But at E it will not stand, since in that position the center of gravity itself lies outside the perpendicular, and therefore outside the stable support C. But at F it will again stand, hanging from the center C, because there also it is divided by the same perpendicular through the center of gravity into parts of equal weight. Therefore, with respect to B, the point C itself is the support below; with respect to F, however, the support is above. 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, the 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 around the center C, will stand in only two places on the perpendicular line, at the more distant one, as B, and at the one nearest to it, as F. The same thing was excellently demonstrated by G. V bald. in his Mechanics, Treatise on the Balance, prop. 1. But here someone might doubt why, experience teaching, weights which have their support below, as a lance or javelin erected perpendicular to the horizontal plane, although in that case the center of gravity is placed on the perpendicular itself, do not stand, but fall to one side? C 3 Let it be

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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 AB C, 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 the plane AB of the horizon be such that at the point C there is set upright, perpendicular to it, a sarissa DC, whose center of gravity is E, on the same perpendicular. Then it will stand, from what has been premised, and indeed it ought to stand, and would stand were it not for the defect of the material; but it does not stand, because it is very difficult for the center of gravity, naturally indivisible, to be fixed so exactly that it should incline to neither side from the perpendicular. These things, therefore, belong to those speculations which, because of the defect of the material hindering them, can hardly or never be reduced to practice. Hence also that question is solved: why those who try to support an upright sarissa with the tip of a finger do not in fact stand still, but with the motion of the finger follow the motion of the sarissa. Indeed, he who follows with his finger the motion of a wavering sarissa does this: namely, in the very act of motion he continually places the finger beneath the center of gravity of the sarissa, so that, since it never remains outside its support, it never falls. A similar difficulty is likewise solved: namely, why spinning tops, with which boys play, while they are indeed turning, stand upright, but when the rotation ceases, fall. Let there be, then, a top AB, whose center of gravity is C; let DE be the plane of the horizon; let AB C be the line perpendicular to the horizon, passing through the center of gravity C; and let the support be at B. Therefore, since the center of gravity C is on 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, rapido 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 in layers, but because of a defect in the material it will not remain steady. Only, as is usual, let it rotate with rapid motion. I say that the top, while the motion or rotation lasts, stands upright; but as that motion gradually slackens, it tends to fall; and when it has completely ceased, it falls over. For this happens because of the unevenness of the material, or the roughness of the workmanship, or for any other cause: the center of gravity is not in C, but, for example, in F; and let the sides of the top on either side be marked G H. Certainly, when F is outside the perpendicular, the top will fall toward G; but this is prevented by the speed of the motion, by which the center F is carried to the opposite side, where I is; nor does it fall toward H, since by the same speed it is again carried to F. Wherefore, since such a center is continually being shifted around the perpendicular, the top cannot fall to any side; but as the motion weakens, the rotating body gradually begins to lean, until, when it has entirely ceased, it falls to that side toward which the center of gravity inclines away from the perpendicular. Now in rotation the center of gravity, which is not in the middle, describes a small circle, through whose center the perpendicular itself passes. Now returning to the balance, whose support is above, we shall demonstrate, by another principle, namely a mechanical one, why, when depressed, it 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 libra 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 quouis 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, therefore, as above, the balance AB, whose center of gravity is C, and whose support, however, is above at D; the balance itself being joined perpendicularly in C. Now the perpendicular which, passing through the support and the center of gravity, tends to the center of the world is DLE. Therefore, the balance standing in its equality, the center of gravity C will be on that perpendicular itself, indeed below the support D, and as near as possible to the center of the world. Then let a weight be added at B. But the part CB will decline into HF, while in the meantime the part AC is raised into GH. Therefore, the whole balance being moved, it will be moved about the same center around the support 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, if the weight is removed from the scale B, by whose pressure the balance had declined, 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 that (which no one before us has observed) the nature of these balances, which have their support above, is such that they are not moved, or wholly inclined, by any weight whatever added to them. For, while they remain in the same state, let any weight whatever be added to the scale B. Therefore if it should be such as to exceed the resistance, which of 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 this produces at H a center of gravity raised contrary to nature, a certain balance will be moved. But if it is of so small a force that it does not overcome that resistance, the center C remaining at the lowest place, the balance itself will not move, or at least only slightly. Hence we conclude that it is possible for those balances which do not deviate by any, however small, weight to have their support above. To these we add that, other things being equal, the resistance is greater the less the center of gravity is distant from the upper support around 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, so that if, a weight having been applied, the balance has inclined toward B, the point C, as it rises, will describe the portion of the circle C E. Let it be again supported above at the point F, and again incline toward B, and again the point C, as it rises, will describe the portion of the circle C G. But the angle of contact A C E is smaller than the angle A C G; therefore C will be carried upward, that is, toward its natural position, more through C G, from the center F, than through C E, from the center D, which was to be demonstrated. Now this resistance, arising from the same support and the same weight, will be overcome more easily the longer the arm of the balance is. For let there be again a balance A B, whose support is D, and center of gravity C; let there also be another balance, with shorter arms 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 I B G. quod nos ita probamus. Ducatur perpendicularis ipsi DF linea L F contingens circulum FK in F, item ipsi DB, perpendicularis MB, contingens circulum B I in B, & quia angulus contingentiæ maioris circuli minor est angulo contingentiæ minoris, erit K FL maior I B M, Rectiautem sunt DFL, DBM, minor ergo DFK residua ipso DBI residuo. Maior autem DFC ex iam citata propos. quâ DBC, erit igitur residuum C F K, multo minus residuo FBI, sed recti sunt CFH, FBG, ex quibussi detrahantur C F K, FBI, erit residuum K F H, maius residuo I B G, 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 A B, minor E F, quàm commune grauitatis centrum C, fulcimentum ve- ro sursum D. Producatur perpendicularis D C, in G & fiat C G æqualis CB, CH verò æqualis CF. Sunt igitur duo vectes

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the balance will incline toward the side B, than if the same weight were placed in F. For let there be drawn from the point B a perpendicular to the horizon, B G, and likewise from F a perpendicular F H. Then, joining D B, with center D and the same distance D B, let the portion of the circle B I be described; and likewise, joining D F, with the same center D and the distance D F, let the portion of the circle F K be described. But D B is greater than D F, by proposition 21 of Book I of the Elements; therefore the portion of the greater circle is B I rather than F K. And the motion through F K is more oblique, that is, more remote from the perpendicular, than through B I. For the angle K F H is greater than the angle I B G. This we prove thus: draw through the line D F a perpendicular line L F, touching the circle F K at F; likewise, to D B, a perpendicular M B, touching the circle B I at B. And because the angle of contact of the greater circle is less than the angle of contact of the smaller, K F L will be greater than I B M. But D F L and D B M are right angles; therefore the remaining angle D F K is less than the remaining angle D B I. But D F C is greater than D B C, by the proposition already cited, as D B C is greater than D F C; therefore the remaining C F K is much less than the remaining F B I. But C F H and F B G are right angles; from which, if C F K and F B I be subtracted, there remains K F H, greater than the remainder I B G. Therefore more is taken away from the perpendicular by the weight descending through F K than through B I; accordingly, a weight suspended at C will prevail less against the resistance when placed at F than if it be placed at B. Which was to be demonstrated. We can also show the same thing in another way. Let there be separately two balances, the greater A B and the smaller E F, having the common center of gravity C, but the support D above. Let the perpendicular D C be produced to G, and let C G be equal to C B, and C H equal to C F. Therefore there are two levers

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EXERCITATIONES. 27 vectes DG, DH, quorum quidem commune fulcimentum D, pondus verò C, potentia vbi HG. Sunt autem hi vectes eius naturæ, in quibus pòdus est inter fulcimentum & potentiam, itaque vt se habet DC, ad DG, ita potentia in G ad pondus in C, item vt DC ad DH ita potentia in Had idem pondus C, sed minor est propositio DC, ad DG quàm DC ad DH. 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 angulatuum ABC, cuius fulcimentum B, æqualia autem brachia AB, BC, & pondera item vtrinq[ue] appensa æqualia. Demittatur ex puncto B ad mundi centrum perpendicularis BD. Stante igitur libra ABC in æquilibrio, erit eius graui- tatis D 2

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EXERCITATIONES. 27 the levers DG, DH, whose common support is D, the weight indeed C, and the power at HG. Now these levers are of such a nature, in which the weight is between the support and the power; thus, 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 in H to the same weight C, but the proportion DC to DG is smaller than DC to DH. Therefore a smaller power is required in G, that is, in B, than in H, that is, in F. Therefore, the weights being equal, the resistance C is more easily overcome in B than in F: which was to be shown. To the nature of this balance are also reduced those whose beam is not straight, but curved, or bent upward from straight parts into an angle at the support; nor does it matter whether the curvature is any portion of a circle, or an ellipse according to one of its diameters; which we demonstrate thus. Let there be a balance, whose beam is a curved angular ABC, whose support is B, and let the arms AB, BC be equal, and likewise weights hanging on both sides equal. Let the perpendicular BD be drawn from point B to the center of the world. Therefore, the balance ABC standing in equilibrium, there will be its gravity

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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, tursum, 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[ue]; decurrere faciunt, imagun- cula interim erecta & in neutram partem cadente, quod vt figurâ clarius fiat; A B C D E F G H I clinata imaguncula, & conuersa circa punctum B, si de- clinet

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the center of gravity in the perpendicular BD itself, namely in F. Let a weight be placed at C; the balance will then tilt, being in the position FBG. Therefore the center of gravity E, by the line EH, will be in H. Thus the center of gravity rises in H, that is, upward, contrary to its nature; so when the weight is removed from C, the center of gravity, placed outside the perpendicular, will again descend, and the balance ABC will return once more to equilibrium. G. V. Bald. shows the same very clearly in the treatise on the balance, proposition 4. From this depends the explanation of those little figures which are made from crushed paper or some light wooden material, and through both of their hands an iron wire is passed, with equal leaden weights hanging from either side, under the condition that the center of gravity is set below the feet of the figure. For then, by stretching the wire and placing them on it like tightrope-walkers, they are made to run back and forth, while the figure remains upright and falls to neither side, as will be made clearer by the figure; A B C D E F G H I a tilted little figure, turned around the point B, if it tilts

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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 Evtpote 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 toward parts I, the center of gravity will be raised to F. But if it is raised toward parts H, it will be in G. Therefore, since the places FG are farther 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 image will return to the perpendicular by the thread HBE, that is, to the horizon itself. From this also depends the force of Battering Rams and of the demolishing Machines called Tortoises, namely from the proportion 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, and the perpendicular, indeed, which is toward the center of the world, is CDE. Therefore, the machine being at rest in equilibrium, the center of gravity will be on the very perpendicular. If a repelling force is applied somewhere, the center of gravity will therefore be raised through the segment DF of a circle, whose semidiameter is CD, and it will be placed according to the position CF. The Ram, however, will be in GFH. Hence, if the Machine is released, the center E, being heavy, will not remain, but by its own nature will return to D. Now the motion of the Ram is most violent for four reasons: from the force of natural weight, by which it is carried downward; then from the increased speed of the natural motion in descending; then from the force of the impelling power, which assists the natural motion; and then from the speed acquired from the violent motion impelling downward and forward. We also add this, that the blows will be more forceful, the heavier the Machine is, and the greater the distance over 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 adiuantibus. 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 fit libræ, quæ deorsum est pars, quàm quod perpendiculum secet, quapropter non ascendit. eleuata enim pars leuior est. Hæc ille, qui schemate quoquerem aperit, at eo apud interpretes, & Picolomineum Paraphrastem, ita me[n]dosè lineato, vt inde obscuritas lucis loco, legentibus fundatur. 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. is brought about by gravity itself, and by distance, and then the union of forces greatly assists the operation. These things we have wished to say about the lever whose support is above; now we shall speak of that whose support is below. The other part of the question: On the lever whose support is below. If that which rests underneath is below, Aristotle says, it does the opposite of that which has the support above, namely, it does not return to equilibrium. For, he says, more than half of the lever becomes the part that is below, than that which the plumb line cuts through; wherefore it does not rise. For the raised part is lighter. Thus far he, who also opens up the matter by a diagram, but in the way in which, among the interpreters and in Picolomini's paraphrase, it is so corruptly drawn that from it obscurity is produced for readers in place of light. We, as we have also done above, shall make the whole matter, by our figure, brighter than the sun itself, according to Aristotle's own mind. Let there be a straight lever, that is, one set in equilibrium, where NG is. But the plumb line, that is, the perpendicular line that passes to the center of the world, is KLM. Therefore NG is divided in two equal parts. After this, when a weight is placed at N itself, N will indeed be where O is; but G itself where R is; KL however where LP is. Therefore

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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- clinabitque; 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, fietque 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 whereby K O is greater than L R, by the very part P K L. Therefore, when the load is removed, it must remain. For the load presses upon the excess of its half, where F is. The sense, therefore, is that on this account the inclined part of the beam K L O does not return to equilibrium, because the very part K L O that pulls, namely R K L, is greater than the part that is drawn and lifted. This same thing can be demonstrated by a far simpler theme. Let there be a balance A B, whose center is C, but whose support is below at D. Let the perpendicular passing through the center and the support be E F. If a weight is placed at B, it will incline; let it incline, say, to G H. The center C, however, from the fixed support D, will describe a portion of a circle C I, and the balance will cut the perpendicular E F at K. Now I G and I H are equal; but from the side H I, K I is taken away, and added to I G, so the whole K G is therefore greater than the whole K H. Therefore K H does not outweigh K G, but the balance, unless impeded, with center C descending through I to M, carried to the perpendicular itself, will come to rest on the lower part, with the positions exchanged; namely, once the support is made above, it will become equidistant from the horizon according to the position L M N. This demonstration is indeed valid, but not from the proper mechanical principles, namely, from the reason derived from the center of gravity. For with the same conditions remaining, when the center of gravity C comes outside the perpendicular, descending to I, it will never return to C, since it will have risen; but if it were freely turned around the center D, then, descending as said through the circle C I M, the weight B would come to be at L, and A at N, having obtained the position L M N. Why

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32 IN MECHAN. ARIST. PROBL. Cur autem huius libræ, quæ aliâs inutilis est, meminerit Philosophus, ea videtur causa, quòd inde vectis virtutem eliciat, vt suo loco videbimus. Id autem valdemi-rum, 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 sur-sum est, nec deorsum, sed prorsus in medio, nempe in ipso grauitatis centro, vbi brachia & pondera vtrinque appo-sita fuerint æqualia, si ab æquilibrio mouentur, quomo-docunque posita, stare nec ab eo, quem adepta est, situ di-moueri. Quæstionem hanc perperam tractârunt recentio-res quidam, Hieron. Cardanus, Nicolaus Tartalea, & alij nonnulli, qui Iordani Nemoracij assertiones sunt secuti, quorum demonstrationes vel paralogismos potiùs egre-giè confutauit in libr. Mechanicor. Tractatu de libra pro- pos.4. Guid. V bald. ad cuius probatissima scripta Lecto- rem ablegamus. fusissimè enim ibi hac de re & absolutissimè agit. Nos autem quidem paucis ea, quæ ad hanc co-gnitionem pertinent, explicabimus. Esto enim libra A B, cuius brachia æqualia, & centrum grauitatis in C, brachijs verò AC, CB æqualibus, æ- qualia 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 balance, which is otherwise useless, seems to be because from it he derives the power of the lever, as we shall see in its proper place. Now it is truly remarkable that the sharpest man should have said nothing at all about this balance, which has no support either above or below, but exactly in the middle itself, so that the center of gravity lies in the very support. We therefore shall present briefly what may be worthwhile, and what is useful to the matter we are discussing. Of the balance whose support is in the middle. We say, then, that a balance whose support is neither above nor below, but precisely in the middle, namely at the center of gravity itself, when equal arms and equal weights are placed on either side, if it is moved out of equilibrium, remains, however it is positioned, at rest and is not displaced from the position it has obtained. Some more recent writers, Hieronymus Cardanus, Nicolaus Tartalea, and a few others, have handled this question incorrectly, having followed the assertions of Jordanus Nemorarius, whose demonstrations, or rather paralogisms, Guido Ubaldi excellently refuted in his treatise on Mechanics, On the Proposition concerning the Balance, 4. We refer the reader to his most trustworthy writings. For there he treats this matter most fully and most completely. But we shall briefly explain the points that pertain to this understanding. Let there be, then, a balance A B, whose arms are equal, and the center of gravity at C, and let equal weights be placed on either side at the equal arms AC and CB. 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 FCG. 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 cùm per- pendicularis DCE per grauitatis centrum ducatur, cor- pus ipsum ex ponderibus & libra constans ab ea in partes èqueponderantes 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- modòlibet librâ per grauitatis centrum diuisâ, sunt æ- queponderantes 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, where the center of gravity C being applied, through the center itself C let a perpendicular be drawn, which, to the center of the world DCE, and let the balance first be set parallel to the horizon. Then, pressed from the other side, let it move and come to be in the position FCG. I say that when released it will remain so; for since the center of gravity is 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 around a center of motion, but is in the support itself; 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 equally weighted, and therefore, according to the definition of the center of gravity given by Pappus, the body itself, suspended at the center of gravity, while it is carried along, remains at rest and preserves the position which it had at the beginning. And indeed if the parts, however the balance is divided by the center of gravity, are equally weighted, they will neither pull one another nor be pulled, so the balance will stand, and it will preserve the position it has acquired. Yet we do not deny that it is difficult to fabricate balances of this kind out of material, since not all things that are true and made plain by the most evident demonstrations are easily reduced to practice, because of the imperfection of art and material. Moreover, the virtue of these balances is such that with even the least weight applied on one side or the other, they incline; this does not happen with those which have the center above, as we have demonstrated. With regard to these matters, a doubt might arise in someone’s mind as to whether the cords by which the pans are hung could introduce some variation in what has been demonstrated. We say that none results from them: suppose then a balance AB, whose center and support is C, from whose end A hangs the cord AD, and from the other end B the cord BE, E qui-

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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 exigua 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, inquiés, 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 equal-weight scales are hung DE. Let the balance be moved, and let the cords in the scales be in I K, H L. But let the cord I K cut the balance A B at M, and let H L be extended and cut the same at N. Since therefore I C is equal to C H, and K I, L N are parallel, the alternate angles M I C, N H C will be equal; but also the angles at the vertex I G H, B C H are equal, therefore triangle I M C is equal to triangle H N C, and the sides [are equal] to the sides which are subtended by equal angles. Therefore line M C is equal to line N C. Thus if the weights or scales, K L, are conceived to be hung at the points M N, they will weigh equally by reason of the equality of the arms and the weights. which was to be demonstrated. QVAESTIO III. Why do small forces, as he had also said at the beginning, move great weights with a lever; and why, furthermore, do levers take up the load, when it is easier to move a smaller weight, and yet the smaller is without a lever? Aristotle frames the question in such a way as to make it more wonderful by a certain rhetorical flourish. He answers, however, that it is possible for this to happen for the reason that the lever is a balance, of that kind namely which has a support underneath, and therefore under pressure 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; 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 caussa, 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 caussa 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 it presents scarcely explains what it intends. We will propose another to suit its purpose, and one much clearer. Let there be a lever A B, whose support is downward at C, the weight D, the power from the lever, sustaining the weight E. A perpendicular through the support F C G. Thus, since the power at E does not exceed the weight D, nor is exceeded by it, the lever stands with the power equidistant from the horizon, that is, in equilibrium; but the lever at point C is divided into equal-weight parts. Now if the power prevails over the weight, and depresses the lever, let it be in L C H; therefore B will be at L, A at H, D at K, and C F, which divided the lever into equal-weight parts, will be in C I. Now therefore the parts are not equally weighted, since the part of the lever F C I is taken away from the part H C I, and added to the part I C L, which is therefore heavier; hence also the power is aided in raising the weight. He uses here the same demonstration which he had employed in explaining the effect of the balance, whose support is downward. Nor, as we noted above, does there seem to be any other reason why he introduced the consideration of that balance in the earlier question. And indeed what he concludes is true. Nevertheless, that small addition, which is joined to the depressed part of the lever in the very act of depression, is of the least importance for so great a force. Therefore the cause of so great a matter must be sought elsewhere, and this too we shall do hereafter. But Aristotle himself also seems not to have been entirely satisfied with the reason assigned, and therefore adds: since among equal weights that which is farther from the center is moved more quickly; but the two weights, that which moves and that which E 2

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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 weight that is moved is to the moving power, so is length to length; and always, the farther it is from the hypomochlion (that is, the fulcrum), the more easily it will be moved. The cause, however, is that which was mentioned before: namely, that what is farther from the center describes a larger circle. For this reason, by the same power that which is moved, being 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 power has the force and proportion of weights. But he says that powers acquire strength from the length of the arm, and from the velocity that follows from it; for the longer the arms are, the swifter they are at the end, and therefore the motion of the weight to the moving power is related as the length of the arm to the length of the arm. And we call the arms those parts of the lever which extend from the fulcrum to either end of the lever; and therefore, the more the power is distant from the fulcrum, the more easily it will move the weight. This statement is indeed true and most thoroughly established. Nevertheless, we do not affirm that the cause of this wonderful effect is the velocity which follows the length of the arm. For what velocity is there in a thing at rest? The lever and the balance stand still while they remain in equilibrium, and none the less a small power supports a great weight. Someone may reply to this that velocity in the longer arm, if not actually, is at least greater potentially. But I ask, what importance has potentiality in a thing that exists actually? For it is actually sustaining, not merely sustaining in potential. It follows necessarily—this we certainly admit—that the greater velocity of the greater arm accompanies motion; yet it is not the cause why a force, placed where the velocity is greater, moves more. Certainly it is clear that bodies acquire weight from velocity, whether projected or falling, as was also established in question 19, with the Philosopher,

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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, afferramus, 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 in a balance the arms are indeed sustained in act, but they do not move. Moreover, Aristotle seems to have caught a hint of this, which later Archimedes, prince of mechanicians, in proposition 6 of the first On Equal Weights expressly set forth and proved: namely, that in equilibrium weight stands to weight as arm to arm, with the relation inverted. Suppose, then, a lever A B, divided by some support at C in any manner whatever. Let there be hung at A a weight D, and at B a weight E, so that E bears to weight D the same relation as A C itself does to C B. The lever will therefore stand, and will not incline to either side; for the center of gravity will be at C, since the lever is there divided into equally balancing parts. This, after Archimedes, and those illustrious ancient mechanicians, was demonstrated most clearly by G. Vbaldus in the Mechanics , Treatise on the Balance, proposition 6, and also on the Lever, proposition 4. Moreover, so that for the moment we may contribute something of our own, let us ask those excellent men what the cause of this wondrous effect is? They will say, a reversed proportion. I understand, yet I am not satisfied: for I shall ask, Why does that inversion of ratio produce that marvelous effect? What they do not teach, I think we, buried in the sleep of ignorance, have dreamed. Equality of condition to be the cause, no one, as I think, will deny. For the matter is clear in itself. Suppose there is some line A B, and to the extremity A there is applied the force 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 FG, 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 some force which draws the line toward itself, namely toward part A; then in B there is some other force equal to the force in A, which draws the line in like manner toward part B. Therefore, given the equality of these forces, the line AB will be transferred neither toward A nor toward B, but will remain wholly motionless. With these things established, I say that, however a lever may be divided, and weights placed on either side, if the proportion be changed so that they correspond to one another reciprocally, the matter is reduced to equality, and hence a state is produced, that is, equilibrium. Let AB then be a lever divided in any way at C, and let a support be placed under C itself. Let weights also be suspended on either side according to the ratio of the arms AC, CB, corresponding to one another reciprocally, let them be DE. I say that the lever, from equality, will incline to neither side, but will remain in equilibrium. For since the weight D has the same effect 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 has the same effect as the arm AC, let there be added in a straight line to CB the line BG, equal to AC. Therefore, since the parts CA, AF of the whole FG 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. 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 beam FG into equal parts FC, CG at the point of support C. And since equal does not act on equal, the beam will stand and incline to neither side. Again, since toward the part FC, two powers of the arms FA, HC, are hung at point F, two weights H, I, equal to the same DE, likewise at point G, other two weights KL, equal to the same DE, it will again weigh equally, since equal weights are hung on equal arms FCCG HI KL. Why then, with the proportion of the arms and weights preserved by interchange, equilibrium occurs, is clear from what we have demonstrated. But perhaps someone will say: if the arms are weights, or equivalent to weights, the burden on the support will be doubled. D, which is five, will become six; likewise if to arm CB, which is five, a weight E, which is one, is added, it will become six. Therefore the support will bear twelve, which is absurd, from what G. V. Bald. clearly demonstrated in the Mechanical treatise on the Balance, proposition 5. To this we answer that the arms indeed act not by weight, but by power, which is a certain force, not weight. Although this too is true, namely that, given a weighted beam, the support must sustain both the weights hung from it and the weight of the beam itself. This kind of equality, as we have said, being established, 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- temari 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. it 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 the center itself and the fulcrum, and extends to 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 because the oar is a lever, while the fulcrum is the rowlock; for it stands fixed. But the weight is the sea itself, which is driven by the oar, and what moves is the rower himself. Now that which moves always moves a greater weight the farther it is from the fulcrum. For in this way the greater effect is produced from the center; and the rowlock is the center. Moreover, in the middle of the ship the oar is for the most part inside. For there the ship is widest. And the ship is moved because, when it is pushed by the oar, the outermost part of that which is inside is driven forward first, and the ship follows its motion, to which the rowlock is attached. But where the oar divides the most sea, there it is most necessary that it should be pushed on. And the most is divided where the largest part of the oar is away from the rowlock. He did not make the matter easy, in that he explained it by words but not by a figure; but we shall add one. Let the ship be AB, the sea CD, one of the oars, which is at the prow EF, whose rowlock is G; the other, however, in the middle of the ship HI, around the rowlock K. He says, then, that oars are levers, and the rowlocks are fulcrums; 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 rowlock K is, the greater part KH is inside the ship, and the smaller part KL outside. On the other hand, the oars at the prow, namely EF, have the smaller 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, nè- 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, ful- cimenta, quibus remorum extrema in ipsa impulsionenituntur, pondera verò seu pondus pluribus vectibus & po- tentijs impulsum nauis ipsa, quæ scalmis est annexa. Resi- stente igitur mari, cedente autem impulsionibus scalmo, nauis eo transfertur, quo scalmi ab ipsa potentia mouen- te 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 pon- dus 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 au- tem ratio tunc procederet, si stante naui immobili, vt fit vbi à Remoræ occulta vi aut ab alio impedimento reti- netur, 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 na- uim mouere, facilius, melius dixisset. Si enim maximè, quod ait, denotat maximo spatio, & velocius prorsus fal- sum, etenim tardius mouent & minori spatio, quod nos i- ta demonstramus. F Esto

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EXERCISES. 41 inside the ship, but the greater part GF is outside the ship. Moreover, the weight is moved all the more easily, the greater the part of the lever is which lies from the fulcrum to the moving power. Truly, the Philosopher speaks sharply. But I, if modesty allowed, would say that it is not indeed the scalmus that is the fulcrum, but the sea itself, and the weight the ship, at the place of the scalmus, that is, placed between the moving power and the fulcrum; for in that way too we can 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 fulcra, upon which the ends of the oars rest in the very act of pushing, but the weights, or the ship itself burdened by the impulse of many oars and powers, are attached to the scalmuses. Thus, with the sea resisting and the scalmuses yielding to the impulses, the ship is transferred to that place toward which the scalmuses are driven forward by the moving power. But since, as FG is to FE, so is the moving power in E to the weight moved in G; likewise, as IK is to IH, so is the moving power in H to the weight moved in K; and the proportion of FG to FE is greater than the proportion of IK to IH, a greater power is needed for the weight in G to be driven than for the weight in K. These things certainly are as we have said. But the Philosopher’s reasoning would then hold if, with the ship standing motionless, as happens when it is held back by the hidden force of the oarsmen or by some other impediment, the rowers, in the very act of rowing, were striking the sea. For then truly the scalmus 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 should rather have said more easily, better. For if “most,” as he says, denotes with the greatest distance and more quickly, it is altogether false; for they move more slowly and over a smaller distance, as we demonstrate thus. F So 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 G 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 So let Remus be AB, which is supported by the sea at B; the oar’s fulcrum at C, which is at the prow of the ship; D, which is in the middle of the ship; but the greater part of the oar is from the fulcrum D to A than from G itself to A. Let the oars be driven, and with BA standing as if a center, in E. Therefore at the same time C will be in F, and D in G, but the distance CF is greater than the distance DG. Therefore, by one impulse, that power which acts near the stern and prow moved the fulcrum, that is, the ship, more than that which operates in the middle of the ship, as Aristotle seemed to sense, if indeed this is his meaning. It is therefore necessary that what he says be understood in the strongest sense, more easily; knowing this truth, the captains of triremes place the stronger rowers at the prow and stern, but the weaker around the middle of the trireme. QUESTION V. It is asked why the rudder, though small and at the extreme end of the vessel, has so great a force that, by a small tiller and by the strength of a single man otherwise using only moderate effort, the huge masses of ships are moved. Does it mean, he says, that since the rudder is a lever and the load is the sea, while the helmsman is the moving agent? Not, however, by its breadth like an oar does the sea receive the rudder; for it does not move the ship forward, but, taking up the sea already stirred, it inclines obliquely. For since the sea is a weight resting in an opposite manner, it inclines the ship. The support, indeed, turns in the opposite direction; the sea, however, inward, and that outward. But the ship, which is fastened to it, follows; and the oar indeed, by its breadth, thrusting the load and repelled by the same, straightens itself

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EXERCITATIONES. 43 etum propellit, Gubernaculum verò, vt obliquum iacet hinc inde in obliquum motionem facit. in extremo aute[m], non in medio iacet, quoniam mouenti facillimum est motum mouere: prima enim pars celerrimè fertur, & quoniam, 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 vltimo, quia æqualis angulus semper maiorem adspectat, tanta que magis, quanto maiores fuerint illæ, quæ continent. Ex ijs etiam manifestum est, quam ob causam magis in contrarium procedit nauigium, quam remi ipsius palmula, 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 ceu remum, quo nauis non antrorsum, 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 igitur A, vbi D translatum est, non erit B vbi E. æqualis enim B E ipsi A D, æquale igitur translatum erit, sed erat minus erit igitur vbi F, minor enim B F, ipsa A D, quare ipso G F ipsa D G. Hæc F 2 demon-

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EXERCISES. 43 and it propels it; but the Rudder, because it lies obliquely, produces an oblique motion to this side and that. And at the end, not in the middle, it lies, because for the moving agent it is easiest to impart motion: for the first part is carried most swiftly, and because, just as in things that are carried there is in the end a deficiency of carrying, so in the case of the continuous body itself, toward the end, the carrying is very weak. But what is very weak is easy to expel. For these reasons, then, the rudder is placed in the stern; and no less so, because, when a small motion is made there, it becomes much greater at the far end, since an equal angle always looks to the greater, and so much more the greater as those things which contain it are greater. From these things it is also evident why the ship advances more in the opposite direction than 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 in his usual way, since he could have done it in two words, explained it in six hundred. 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 as it were an oar, by which a ship is moved not forward, but obliquely and to the side. Wherefore he sets forth almost all that ought to be said about the tiller, while speaking of the oar. He says thus: 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 transferred, will not be at B when E is there. For the equal B E to A D is equal; therefore the transfer will be equal, but it was smaller; therefore it will be where F is, for B F is smaller than A D, wherefore D G than G F itself. These F 2 demon-

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demonstratio licet vera videatur, rei tamen, de qua est fermo, 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, fiquidem 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 remo vbi E G F, cardo

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The demonstration, although it may seem true, is nevertheless not suited to the matter under discussion. For if it were suited to the motion of the oar itself, when the handgrip is at F, the rowlock would be at G; therefore either the rowlock would run along the oar, or the oar along the rowlock, namely after such a translation from C to G; and thus within the ship now the part of the oar D C would be present, now indeed G D, which, however, we are taught by experience itself does not happen. That too is false, that the ship itself is moved in the air by only as much space as A D, that is, the end of the oar which is in the ship, if indeed the ship is driven by the motion of the rowlock and not by the handle of the oar. Otherwise, then, the matter stands, and perhaps in this way. Let there be an oar A B, whose handle is A, blade B, rowlock C. If A is pushed forward and comes to D, then if the handle and blade were moved equally, the blade itself would come to G; but it is moved less: it will therefore come to E. But the rowlock itself C will have been transferred to F, and the ship will have been 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 over the sea, as for example HB, in which case the space of the translation is greater, namely from H to I. But the motion of the rowlock takes place from the centers which are in the very space B E; and as far as the tiller is concerned, which he says is the oar propelling the stern itself obliquely, the matter stands thus. Let the ship’s keel be A B, prow A, stern B, the wing of the tiller B C, rudder B D, and the hinge or support B; thus when an oblique push of the rudder is made from D to E, there will be a smaller motion in the sea from C to F, and the oar 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 præ- cedente 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 se rem 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 maris, 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 in- terim & gubernaculo in eadem carinæ linea existentibus, F 3 Temo

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EXERCISES. 45 but the pivot is at G; therefore it will be transferred by that motion, the stern itself from B to G. Thus, after a slight movement of the stern from B to G, the prow itself, which is far distant from the stern B, having traversed a greater distance, will be transferred to H, the prow being turned to the opposite side from that which was produced by the motion of the rudder. Moreover, what we noted in the preceding question, the demonstration will proceed much better if by support one understands the sea rather than the oarlock; for it is not the sea, as though it were a weight, but the oarlock itself, or the tiller pivot, which is transferred like weights. However, in this speculation let us be allowed to dissent somewhat from the Philosopher. Certainly, if the shortness of the tiller, projecting from the stern, is considered in relation to the length of the whole ship, and the slight motion that occurs when the tiller or rudder is moved will be of almost no account in the movement made at the prow. We therefore 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 rudder produces no movement, or hardly any at all, of the ship to right or left; but when it is moving, a very great movement is produced, as experience teaches us. Therefore the oblique motion, which is strong at the stern, is not caused by the strike of the sea from the turning of the rudder, but by the sea itself, whose waves, when the ship is moving, drive the oblique blade of the rudder toward that side which faces the sea, and transfer the rudder with the stern most forcefully to the opposite side. Let the ship have keel AB, prow B, stern A, tiller AC, rudder AD; thus, when the ship is moving, while the tiller and rudder 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 nostrâ speculatione ratio habetur eius ma- china-

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IN MECHAN. ARIST. PROBL. Indeed, the rudder cuts the sea without any change being made in the stern, when the ship is moved to the left or the right. But if the tiller is moved from D to E, when it is moved the stern will move somewhat toward E also, as Aristotle wished. Yet this effect, as we said, is of very little importance toward so great an effect. But when the tiller is set obliquely, as AF, while the ship in the meantime, driven by the force of winds or oars, runs forward with the side of the tiller being struck obliquely by the waves, or by the sail in its course, it is transferred to the opposite side, namely to that toward which the rudder itself inclines. Thus the ship is turned, as it were, around a center or centers considered in the keel between stern and bow, to G, and the bow to H. From this it is clearly apparent that two things are necessary for the turning of the ship through the rudder in the stern: namely, the obliquity of the tiller and the ship’s motion, and if one of these is applied without the other, no turning of the ship of any consequence takes place. We also note that the keel in the turning of the ship behaves like a lever, whose part moved toward the stern is the moving power, while the support is about the bow; but the moving power is the sea itself, striking the rudder obliquely in the ship’s course. Whence we infer that the longer ships are, the more easily they are driven to the right or the left by the force applied to the rudder: which Aristotle himself certainly observed, for this reason saying that the rudder should be placed at the stern, not in the middle, because it is easiest for the mover to move what is moved from the end. From this our speculation an account is had of the machine-

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EXERCITATIONES. 47 chinationis, quâ in magnis fluminibus, ceu Pado, Abdua & similibus, Portitores, equos, currus, viatoresq[ue] 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 AB, vlterior CD. Pontones duo tabulis strati, & vna firmiter juncti EF, Temo inter eorum puppes extans GH, locus in ripa stabilis A, funis, quo pontones, & machina tota continetur AI. fluuij decursus versus BD, stantibus itaque pontonibus ad ripam citeriorem AB, Temone in neutra partem pullo, cum aqua decurrens eum resistentem non inueniat, scinditur quidem ab eo, sed non propellit, eo autem conuerso & in GK constituto, alaeius GK ab aqua defluente propulsa machinam secum trahit versus ripam CD, facta motione circa centrum seu stabilem locum A, otiosis interim portitoribus, donec per circuli portionem ML 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 this contrivance, by which, on great rivers such as the Po, the Adda, and the like, ferrymen carry horses, carts, and the travelers themselves from bank to bank. For it is a very beautiful device, and one with which we are most familiar, since we who formerly lived at Guastalla, a town on the Po, when traveling to Mantua, very often crossed the broad channel of that same Po to Castrum Burgi by the mechanism already described. It works as follows. Let the nearer bank of the river be AB, the farther bank CD. Two pontoons decked with planks and firmly joined together EF, the pole projecting between their sterns GH, a fixed place on the bank A, the rope by which the pontoons and the whole machine are held AI. The course of the river is toward BD. Thus, when the pontoons stand at the nearer bank AB, and the pole is not turned to either side, since the flowing water finds no resistance in it, it is indeed struck by the current, but is not driven along; but when it is turned and set in GK, its blade GK, driven by the descending water, draws the machine with it toward the bank CD, the motion being made about the center, or fixed point, A, while the ferrymen remain idle, until by the portion of the circle ML it has come to the farther bank at L. From there, when the pole is again turned in the opposite direction, the water similarly driving the pole, 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, 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 MECHAN. ARIST. PROBL. to be only that which comes from the rowing-blades, by the striking of the water, as Aristotle had perceived, but the thrust of the current water against the blade of the oar: and it makes no difference whether, with the ship standing still, the water runs, or whether, with it running, the water stands still, as is the case at sea; for in either way the oar suffers the same. But 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. The little boats or skiffs are therefore placed there, so that they may support the long rope; for if this were not done, it would be immersed in the water, cutting through the water and hindering the motion of the machine; they are also placed there lest the wet rope, once soaked, should quickly become sodden and rot. Related to this speculation is that of the sails, which, receiving the wind obliquely, give motion to grain mills; likewise of the paper whirlers, with which boys running against the wind use in play. For the only

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EXERCITATIONES. 49 enim horum omnium principium & eadem ratio. Diximus enim, Temonem currente naui, 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 cui- dentius 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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EXERCITATIONES. 49 for the beginning of all these things is the same, and the same reason. For we have said that the tiller, when a ship is under way, being turned sidewise and receiving the oncoming waves, transfers the stern itself obliquely to the other side. Moreover, those sails of which we are speaking, being set obliquely against the blasts of the winds, are for the same reason driven around in a circle, which, that it may be made clearer by a figure, Let AB be a sail, obliquely fixed to the arm CE, so that the angle ACE is greater than the angle BCE, and the wind FG strikes the sail obliquely. Therefore, since the wind falls upon the oblique sail, the sail slips aside, and turns around the center E together with the arm; in its place comes the sail HI, and from this continual succession of sails, the rotation of the arms and of the axis to which they are attached becomes perpetual. But in speaking of the tiller we should not meanwhile be silent about the tails of birds and fishes. For they are, as it were, tillers, placed by Nature itself upon the suitable parts of animals, namely the hindmost ones, although they do not serve only the use of a tiller, as we shall see. Let AB be a fish, whose head is A, and the tail CB. With this, therefore, turned to neither 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 right or left, it will not be able to do so unless the tail itself helps. For every forward motion requires steadiness, 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; aquam ceu stabile quippiam series 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, quę fulcimentum est remotius faciens, id efficiet. Hæc ille, quæ sic figurâ explicamus. Esto

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as may be, as the Philosopher himself teaches in the books On the Motion of Animals. Let it therefore be that a fish wishes to turn and, being turned so that its head is at D, it will deflect its tail toward E; and since the water, as though some stable support, in a way sustains it, the remaining body C A will bend back toward D. But if it wishes to turn toward F, it will deflect its tail toward G, and by the same method it will be deflected toward F. But the tail also serves as a rudder for swimming and flying creatures. For let a fish be straight, that is, moving directly along I K L; let it incline its tail at K M, and thus, from the collision with the water in its very motion, its posterior parts will be driven, as at I N O. Thus let this be enough for us to have considered about the rudder, insofar as it pertains to this question. QUESTION VI. It is asked why, when the mast is higher, ships are carried more swiftly with the same sails and the same wind? The Philosopher solves it, saying: Is it because the mast is indeed the lever, the support of the mast is the seat in which it is placed, and the weight that must be moved is the ship itself; while the moving power is the spirit that stretches the sails? If, then, the farther the support is removed, the more easily the same power moves the same weight, and more quickly, certainly when the antenna is raised higher, it will accomplish this by making the sail farther from the mast-seat, which is the support. Thus he explains it 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, impeditur 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. Responsio 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 Estonauis AB, a bad CD, the seat of the bad D, the place of the mast higher C, lower E: therefore since CD is a lever, the farther the mover shall be from the support D, the more quickly and violently it will strike; therefore the ship will be moved more quickly with the mast placed in C than in E. These things are plausible, but certainly according to truth itself, not true. I ask, if the support, while the lever is moved, is fixed, the center of motion will assuredly be D. Therefore, if a strong wind blows, the mast will be inclined, and it will be done where F G D, which inclination will indeed be made more violently, the wind driving in F than in G, since it is the point farther from the support. When the mast is struck, two things must necessarily follow: either it will break at the very seat D, or the ship itself, turned around the point D, will be raised so that the motion of the mast follows. But the prow will sink, the ship being made in HDI. But if someone should have tied a rope fastened to the top of the mast to the stern itself in B, the inclination of the mast toward the parts F is certainly prevented, and therefore no force at all will be made in D from the reasoning of the lever. Yet nevertheless, the higher the mast shall be, the more easily the stern will be lifted by the blowing wind, inasmuch therefore as the mast is a lever, it performs only this much that we say. But if, on the contrary, it is objected that experience teaches that the higher the mast shall be, the more quickly the ship moves when the breeze blows. The answer is easy, namely, that it is not surprising if the higher part of the mast is struck more strongly by the wind. For we see that towers too, the higher they have been, the more they are assailed by the violent blasts of the winds, which certainly would be ridiculous to refer to the length of a lever. Moreover, as for the easier raising of the stern from the height of the mast itself, to the lever G 2

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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 BG. 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 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 therefore be the lever of which we are speaking, ABC, whose arms AB, BC are joined at the angle B, and let B be the fulcrum in operation. Nor does it matter, so far as use is concerned, whether the angle itself be right, acute, or obtuse. Let it now be right. Let some weight be placed at C, then let some force be applied at A, which drives the very end A of the lever toward D. Therefore AB will be in DB, and the angle preserved, BC in BE. Thus the weight together with the part of the lever BC will be raised to E. In this kind of lever, however, attention is given to the proportion which AB has to BC. For if the force applied at A is such in relation to the weight at C as CB is to BA, equilibrium will result. But if the proportion of the force at A to the weight at C is greater than that which AB has to BC, then, the resistance of the weight having been overcome, motion will occur. The matter is in no other way than if, extended to F, BF were made equal to BG. For then the lever would be reduced to straightness, preserving the proportion, and thus the force at A, with B as fulcrum, would act at F, as it was acting at C. To this nature of lever belong the smiths’ hammers, with which they tear out nails, as well as the tongs which, with a gripping bite, seizing the heads of nails or umbrellas, violently pull them out of boards. In the hammer, then, as can be seen in the figure, AB is the part of the lever from the fulcrum to the force, but that which is from the fulcrum to the weight, to the weight if any...

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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 vnico forceps totus habetur vecte, cuius quidé 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 elevationem, 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 which is at C is balanced. The same we observe in forceps, in which the two arms AD, CB, so far 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 in so far as they serve for extraction, the whole forceps is considered as one lever, of which the part from the power to the fulcrum is AB; and the part from the fulcrum to this is the nail itself which is being pulled out, AC. Very forcibly however do forceps extract, because the proportion of the length of arm BA to that which is from A to C is greatest. These things therefore having been examined in this way, and returning to the ship and the mast, we say that then the raising of the stern becomes most easy, and the sinking of the prow, when the proportion shall be greatest which the height of the mast has to that part of the ship which extends from the mast to the stern itself extremity. 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 reckoned from the prow, toward the stern. Let there be a ship AB; its mast CD; prow A; stern B; therefore if the wind, the sail being driven, bends the mast toward the opposite side, say into FD. But since the topmast is joined to the rope at B, it draws the ship, that is, the stern itself, ne- G 3 cessarily

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54 IN MECHAN. ARIST. PROBL. celle 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 malisede 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. that is so. But it cannot be; for the weight of the bilge and the loads, 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 toward G, be raised. Therefore, the smaller the proportion of CD to DE, and the greater the load itself, whose center of gravity is in E, the less will be the power pushing at C for the elevation of that part of the ship which lies between the mast and the stern. Is the mast then the lever, the foot the fulcrum, and the weight moved by the lever the ship itself, as Aristotle pleased? And in what way the mast operates on the ship as a lever is easily clear from what has been said. QVÆSTIO VII. The question is: Why, when they wish to sail from the stern, and the wind is not blowing from the stern, do they tighten the part of the sail that turns toward the helmsman, but loosen the part that faces the prow, making a foot? Aristotle explains the cause of this remarkable effect. For, he says, Is it because with a strong wind the helm cannot be drawn back, but with a light wind it can, when they tighten it? Thus the wind itself drives on, and in the stern it places the helm by pulling it back and driving the sea; at the same time the sailors themselves struggle against the wind, for they lean themselves toward the opposite side. Thus he. Since the meaning of this is somewhat obscure because of its brevity, Picolomineus explains it with wonderful ease. But we, in order to make the matter clearer, will set forth 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 wing B D, sail-belly E F, and the sail 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 When this has occurred, the ship moves straight from stern to bow. But if the wind blows sideways, say from G toward H, and yet they wish the ship to be driven forward as though the wind were coming from the stern, they certainly slant the sail, narrowing its lower part, namely the foot, which is in F, while loosening the corner of the yardarm where E is, toward the bow, 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 since the wind drives the sail to the opposite side, namely toward H, in order that they may resist the wind they turn the helm from C to L, and by the tiller B D they force it in BM toward the side from which the wind itself blows. Thus there is a contest between the wind and the helm, the former driving the bow to the right, the latter to the left; and so, since neither prevails, the ship necessarily keeps the middle course, which lies between the two, in its sailing. The sailors therefore go to the side of the ship A E B, which faces the wind, in order to provide equilibrium against the wind, lest, if the breeze drives the ship in the opposite direction, it sink it. Besides, what neither Aristotle nor Picolomineus noticed is that a sail set obliquely is driven forward by the wind for the same reason that we mentioned when speaking about the tiller and the sails by which millstones are turned. But what Picolomineus adds, that the matter can be reduced to a lever, should not be passed over in silence. The wind, he says, takes the place of the weight moving the helm; but the center, that is, the support, 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 vicurrente naui repulsus sit in EF tendens versus I, quo casu prora convertitur in D, nempe contra ventu quispirat ex H. sit autem conuersio circa punctum G, quod fulcimenti locum obtinet. Vetus verò ad contrariam partem 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 cætro 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. the head of the keel inclines toward the prow, so that it can more easily resist the wind itself. For then the ship moves straight when its opposing forces are balanced with one another, as though they had established a balance point. This he, whose meaning we shall easily make clear by the figure proposed. Let AB be the keel, of which A is the prow, B the stern, BC the yardarm, but the wind H striking obliquely. Thus the yardarm is turned so that in BC, with the waves rushing against it, the ship is repelled and tends toward EF, toward I, in which case the prow is turned to D, namely against the wind that blows from H. Let the turning be about point G, which serves as the support point. But the old view pushes the prow to the opposite side, resisting the violence of the rudder, which is directed against the prow itself. Therefore AB, or DE, is like a lever, whose support is G, the moving force the sea by which the yardarm EF is driven back, and the weight, the wind pressing at D; therefore the farther the yardarm is from the support G, but the closer D is to the weight, the more the yardarm will overcome the force of the wind. This reasoning of Picolomini, which we have explained, is certainly ingenious; yet in truth, since the support by its nature ought to stand firm, whereas here it has no stability at all, it encounters difficulty. QUESTION VIII. It is asked, Why among all figures do round ones move more easily? In three ways, says Aristotle, can a circle be rotated; either about the axle with the center moved at the same time, as the wheel turns in wagons; or about a stationary center, like the pulleys of wells, the center standing still: or with the center remaining on the ground, as a 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 contingentiæ. eleua- H tur

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EXERCISES. 57 Explaining the cause, he says that bodies of this kind are most swift, because they touch the plane with only a small part of themselves, as a circle does with a point; likewise because they do not meet with resistance. And the cause of their not meeting with resistance is that they have an angle removed from the earth. Likewise, because the body which they encounter they touch only slightly. But in the case of a straight line it is otherwise, since by its straightness it touches much of the plane. To this he adds that, in the direction in which the weight inclines, so does the mover move. These are, in substance, the Philosopher’s words; and his reasons apply only to that circular motion which takes place according to the epicycle, as is the case with carriage wheels, and are not suitable to wheels of pulleys and to the like, such as those which are hung over wells. We therefore, according to Aristotle’s meaning, shall examine the first kind of rotation, which is according to the epicycle. 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 is 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, then, it is evident that a circle touches a plane, or line, at a point. But since we are, as mechanicians, dealing with material circles and wheels or spheres, the Philosopher rightly said not that it touches the plane exactly at a point, but according to its smallest part. Moreover, the angle which he says is removed from the earth is in fact the angle of contact. The eleva-

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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 caussam 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, & suo 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, the circle meeting it would touch it at a point, for example at K. These things, then, happen in the case of a circular figure. In figures with sides, however, it is otherwise; for they touch the plane neither at a point, nor by a small part of themselves, nor do they have an angle set apart from the plane like a circle does; nor, when they strike, do they meet the obstacle at a point. Moreover, he said that the chief cause of ease of motion in rotation, which takes place according to the axle, is this: namely, that to the side toward which the weight inclines, it is driven and moved by the mover. First, then, a 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 the impulse is made from the side of F, the opposite part will sway and incline toward G, and by its sway and motion will carry with it the part A F B, and a progression will result. For if the diameter F C G is drawn, equally distant from the horizon, it will be like a balance, whose weights are on either side A F B, A G B, but whose arms are equal, C F, C G. And the power by which it is drawn or pushed behaves like a weight; if one adds this to one of the parts and the departure from equilibrium is made, motion will follow. Some thought, as the Philosopher reports, that a circular line would be turned by perpetual motion in the same way that things at rest remain at rest because of contrary resistance; for indeed a circle in a plane has no contrary resistance, since, as we said, it is in equilibrium and easy to move in either direction. Yet the reason that the motion of these bodies cannot be perpetual is that what is violent occurs contrary to nature, and therefore is not durable. In addition, the Philosopher adds that larger circles have a certain inclination toward smaller ones; and that the inclination of the larger to that of the smaller is related as angles to angles, and as diameter to diameter. But here he calls the sectors themselves angles; for it is necessary that both larger and smaller circles be constituted around the same center. These matters are explained by a diagram not unlike the one set forth 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 a line perpendicular through the center B C A. Let there also be, about the same cen- ter C, a smaller circle F G, and let C H be drawn, cut- ting the smaller circle at I, but touching the larger at H, and making with the line A C the angle A C H, containing, according to Aristotle’s view, two an- gles, that is, two sectors A C H, F C I. Since therefore the sector or an- gle A C H, in its extent, exceeds the angle or sector F C I, it is easy to see from the motion which the greater has over the smaller, that the greater itself moves the smaller. But the Philosopher seems silently to refer these things to the nature of the lever, of which one extremity is at the center and the other at the circumference, and thus to make the inclination of the greater over the smaller stand in the relation of lever to lever, that is, semidiameter to semidiameter, or sector to sector, which sectors, as we have seen, he calls angles. But what he says about this inclination, though subtle, does not seem to be true. For if in the figure the semidiameter H C be produced to the opposite side in K, cutting the smaller circle at L, we shall have two other sectors or angles, namely K C B, L C G, equal to A C H, F C I themselves. Therefore, insofar as the inclination of the greater angle A C H assists motion, in descending toward the parts B, so much does the inclination of the angle likewise greater K C B retard it, and conversely the inclination (if I may so call it) in ascend- ing toward the parts A. And certainly, so far as it belongs to the nature of the matter, and also to equilibrium itself, larger circles do not differ from smaller ones, nor are they more readily movable than smaller circles; indeed, for some reason smaller ones seem to be easier to move, 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. They are lighter, both because, given equal matter, they are lighter, and also because the angle of contact with the plane of the smaller circumference is greater than that of the larger circle, as in the figure set below the angle ABC is greater than the angle DBC; therefore, in a material circle or wheel, the greater part of the circle will touch the plane DB rather than AB. However that may be, the greater circles are more easily moved, not indeed from the nature of the circle, which is the same in the greater as in the smaller, but for other reasons, which we shall examine in their place. Moreover, to set forth something from our own resources about the motion which is according to the abside, we say: circles, or wheels, which are moved in this way, are moved either across the plane of the horizon, or on an incline, or down a slope. But if across the plane of the horizon, the motion is therefore easy because, ceteris paribus, the center of gravity of the body itself never becomes farther from the center of the world in the rotation itself. Let there be, then, the plane of the horizon AB, upon which the circle AD rests, 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 equally distant from the horizon, ECFG: while the circle is divided however into the parts AH, HF, FI, ID, and CI, CH are joined. After this let the circle be understood to move according to the abside toward the parts G; there will therefore at some time be the point H, touching the plane of the horizon, let it touch however at K, then F at L, I

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EXERCITATIONES. 61 L, I in N. D verò in O. Ducanturque KP, LQ, NR, OS ipsi A C parallelæ horizonti autem perpendiculares. Centrum ergo circuli, quod idem & grauitatis est centru[m], feretur per rectam CPQRS, sunt enim KP, LQ, NR, OS 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 c[æ]trum 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 in N. D indeed in O. And let KP, LQ, NR, OS be drawn parallel to AC itself, but perpendicular to the horizon. Therefore the center of the circle, which is also the center of gravity, will move through the straight line CPQRS, for KP, LQ, NR, OS are equal to the semidiameter AC itself; thus the center C itself will never be raised in the rotation of the circle above the plane of the horizon, nor will it be carried farther from the center of the world. But this happens in a very different way with 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 contact through the center, itself CEA dividing the ellipse into equal and equally weighted parts ABC, ADC. Let in quadrant CD, points HI be taken, then EH, HI joined; and EH will be longer than EC itself, then EI than EH itself, and ED than EI itself. Let the ellipse be rotated according to the axis; thus the point H will be in K, and from the point K let a perpendicular be erected to the horizon, KL, which may be made equal to EH. After this the point I will be in M, and from M a perpendicular, equal to EI. Again let D be in O, and let OP, a perpendicular equal to ED, be drawn. Therefore the ellipse moved from C to K will be not so difficult a motion, since EH does not greatly exceed EC; but the transfer to M will be more difficult, and truly the most difficult to O. For from the position of E there the center of gravity is raised, namely ascending where P. We see therefore 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, fiatq[ue] 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 a line equidistant from the horizon, whereas here it is now raised, now lowered, which certainly causes trouble and difficulty. But that the same thing happens with other figures, and especially with polygonal ones, we shall show thus. Let there be an equilateral triangle ABC, whose center of gravity is E, and the horizontal plane BD. From the vertex A let the perpendicular AF be dropped to the horizon; it will pass through the center E, and will bisect the base BC at F. But the triangles ABF and ACF are equal and of equal weight. Moreover, the angle AFC is right. Let EC be joined; therefore EC will be greater than EF. Let the triangle therefore be rotated about the point C, and let EC be made perpendicular to the horizon, and let it be CH, and through E let EK be drawn parallel to the horizon. When the triangle has been moved, the center of gravity E will have been transferred to H; but KC is equal to EF, and smaller 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 there arises difficulty in the 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 most easily on a horizontal plane, while polygonal bodies and those having unequal semidiameters move with difficulty, is clearly evident from what has been said. To this question also belongs this: why heavier bodies, and especially round ones, when released with great force, run down along an inclined plane. Let there be a wheel, or a sphere, or a cylinder CD, whose center is E, touching the inclined 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, cir- cumferentiam 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, when this is released, are they carried with great force toward the parts B? Let the perpendicular F E L be drawn through the center of gravity E to the horizon B K, 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, E H. Accordingly, there intervenes between the circumference and the inclined plane the space H G. Let D I be drawn likewise parallel to F G. It will therefore not pass through the center E. It will therefore be smaller than the diameter C D; wherefore it will cut the circle into unequal parts, and not through the center of gravity, which is assumed to be the same as the center of the magnitude or figure. Accordingly, when the wheel is let go, it does indeed touch the inclined plane at the point D. But the center of gravity presses downward by the secondary perpendicular line F G; it is not supported, however, at H, since the space H G lies between the plane and the circumference, and H has no place to rest upon. Moreover, the body is thus divided by the line D I, so that the part I F C H D is far greater than D I, and the center falls in that part which is not supported. Thus, by its own inclination, since at D it is outside the support and rapidly rotating along the perpendicular D I toward the lower parts, it descends. But let the perpendicular G L be drawn, parallel to 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 the wheel, not impeded there, will be carried by its own inclination, and will not stop until it has obtained the lowest place where it may come to rest. We can also, for a wheel or sphere placed on an inclined plane, find the given power which, when applied to the end of the diameter on that side toward which it tends, prevents the wheel or sphere from slipping. Let it be supposed

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IN MECHAN. ARIST. PROBL. Esto planum inclinatum AB, cui Rota sphæraue insi- stat tangatq[ue] illud in C. Rota verò ipsa sphæraue DC, cu- ius centrum E, diameter ve- rò DEC ipsi BA ad punctu[m] contactus C, perpendicularis. Ducatur per C ipsi hori- zonti perpendicularis FG circulum secas in G tum per E ipsi CG perpendicularis, ipsi verò BF horizonti æqui- distans HEI ceu vectis, cuius fulcimentum I respondens ipsi C, pondus verò in E, vbi grauitatis est centrum. Ap- plicata igitur potentia in H erit pondus inter fulcimen- tum & potentiam, quare vt IE ad IH ita potentia susti- nens 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 inclinatio- nis 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 mi- nor fuerit inclinationis angulus, eo facilius rotam sphæ- ramue 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 pulcher- rimam dubitationem diluamus. Quæritur, Cur maiores rotæ

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IN MECHAN. ARIST. PROBL. Let there be a plane inclined AB, on which a wheel or sphere rests and touches it at C. Let the wheel or sphere itself be DC, whose center is E, and whose diameter DEC is perpendicular to BA at the point of contact C. Through C draw FG perpendicular to the horizon; it cuts the circle in G. Then through E draw HI 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 power be 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. Something similar is shown by Pappus, 1.8. prop. 9, though on other suppositions and considerations. I say further that, with the same things standing, the angle ECI is equal to the angle of inclination CBF. Let HI be produced, meeting AB itself at K; and it will meet it because 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. Whence it follows that the smaller the angle of inclination, the more easily the wheel or sphere is sustained on the inclined plane. For the smaller the angle ECI is, the less is the side EI, and the smaller the proportion of EI to IH, and therefore the smaller is the sustaining power required at H. Moreover, an ascending and a descending plane differ only in relation. These things being thus considered, the place now admonishes us that we should clear up a most beautiful doubt. The question is, Why 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 perpendicularè 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 rectitudo 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 more easily surmount them than smaller ones. For what Aristotle says does not seem to be satisfactory, namely, that this happens from contact at the point because of the elevation of the angle from the plane; therefore the doubt is resolved by other principles. Let there be a larger wheel A B, with center C, and a smaller one D B, with center E, touching the plane of the horizon at B. The diameter of the larger wheel A B, of the smaller D B, and the obstacle perpendicular to the horizon F G. Through F let there be drawn the line F K parallel to the horizon, cutting the periphery of the smaller wheel at H, but the diameter A B at K; and from the point H let the perpendicular H I be let down to the plane of the horizon: and H I will be equal to the obstacle itself F G, and let B H, B F be joined. Therefore, since B H falls from the extreme point B into the triangle K F B, the angle K H B will be greater than the angle K F B. But K F and B G are parallel, therefore the angles K H B, H B G are equal; likewise K F B, 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 according to the line B H, the larger however according to the line B F; therefore the smaller wheel will be moved with more difficulty, and will overcome the obstacle rather than the larger wheel: which was to be demonstrated. We can show the same thing more mechanically, that is, by reducing the matter to a lever. Let the plane of the horizon be A B, the larger wheel C D touching the plane at D. And the center of the larger wheel be E. But the smaller wheel F D likewise touches the plane at D. And the center of the wheel is G, and the height of the obstacle D H. Through H let there be drawn H I, equidistant 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 k P, 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 k GP, RES ad inuicem sunt æquales. Sed & ESR, GP k recti sunt, quare ER S Gk P 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 EL ad 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 one, LEM, the smaller one, NGO. Then from the point k let a perpendicular be drawn to GO, namely kP; likewise from point I to EM the perpendicular IQ. I say that EQ has to QL a smaller proportion than GP has to PN. Let Gk be connected, and through E let ER be drawn parallel, cutting the greater circle at R, and from R let RS be drawn perpendicular to EM. 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, RES are equal to one another. And also ESR, GPk are right angles, wherefore the angles ERS and GkP are equal, and the triangles GkP, 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. By composition therefore, as NP is to PG, so LS is to SE. Wherefore if the support were at S, the weight at E, the power at L, the same would happen as does happen with the support set at P, the weight at G, and the power, however, placed at N. And this indeed if both wheels 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 the center E than point S itself. Therefore the proportion of EG to LQ is smaller than that of ES to SL. Therefore less power is required in L to support the weight E from the fulcrum Q, that is, I, than is required in N to support the weight G from the fulcrum P, that is, k. Therefore less 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 carrying the larger wheel CD beyond the obstacle IV, that is, DH, than is required for carrying the smaller beyond the obstacle K T, that is, HD, which was to be shown. To this it may be asked in what manner the wheels of wagons behave in the turning of the wagon itself, namely, what is that curved line which they describe in turning. Let there be the path of the wheels on the plane, while the wagon proceeds straight AB, CD. But these lines, as we shall show later, are equidistant. Let the point B therefore be that point at which the wheel moving along AB, when brought there, touches the plane. D is the point of contact of the other wheel and the plane. Therefore, while the wagon is turning, let the point D be the center of the turn. For in the meantime the wheel stands 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 that wheel, remaining fixed, which is at B, passes through a semicircle DEF around the center D; and when the wheel B has come to F, the wagon being turned to the opposite side, the wheel which is at D moves along the line DC, while that at F moves along the straight line FG, and the wagon retreats. And since either D, in the actual turning, stands altogether still and does not advance 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 larger and outer circle, we conclude that the turnings and bendings of wagons are always made about a center and along portions of concentric circles. From this we also learn why the ancients, as may be known from ancient...

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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 secen, 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[m] 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 MECH. ARIST. PROBL. we recognize from the traces that they formed the circles in which the courses of the quadrigae were made in that shape which appears. This theorem also we prove. Cylinders whose bases are perpendicular to the axis, while they roll on a level plane, move straight on and through parallels, the distance of which is determined by the length of the axis, or of the side. E D G A C H B Let there be a cylinder ABCD, whose axis GH, resting on the plane of the horizon along the side AB, has the opposite and equal side CD. Let the cylinder rotate until the side CD is in the plane where EF is. Now let the circle CB describe the line BF. And let the circle AD describe the line AE. I say that these are straight lines, and parallel. For if the surfaces of the bases DA, CB are extended so as to cut the plane of the horizon, they will cut it along the lines AE and BF; therefore each is a straight line. But we have also shown that they are parallel to one another. Since the semicircle AD is equal to the semicircle BC, line AE will be equal to line BF; and AB is equal to DC, therefore also to EF. Therefore the opposite sides of the quadrilateral figure ABFE are equal; hence EF is equidistant from AB, and AE from BF, which was to be proved. We shall also prove that if the bases of the cylinder are not perpendicular to the axis, and thus ellipses are described in the very rotation through the plane, they are indeed parallel, but not straight. Let there be a cylinder ABCD, whose bases are mutually equidistant ellipses, whose longer axes are AB, 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 semicir-

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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 cu[m] bases 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, Zα æquales ipsis EG, GH, HI, IF, & per puncta X, Y, Z, α, & paralleli ipsi ST ducantur [sulphur] απ, νz [uncia], λγμ, ηχθ, tum & his ex altera parte respondentes parallelæ per puncta β, γ, δ, ε. Sit autem [sulphur] α, æqualis AF, απ æqualis FD, item [sulphur] [μ], æqualis EC, [sulphur] α æqualis EB, sed & νz [uncia] equalis OI, z [uncia] ipsi P, λγ ipsi MH, yμ verò ipsi HN, [sulphur] αx ipsi KG. & [chi]θ, ipsi GL & ipsis æquales & æqualiter positæ ad partes R, aliæ parallelæ aptetur per β, γ, δ, ε, quibus I 3

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EXERCISES. 69 the circle EHF into four equal parts FI, IH, HG, GE. Then, through the points of division, let lines be drawn parallel to the side, namely KGL, MHN, OIP, which, since they are parallel to the bases AMB, DNG, will be equal to one another, and, since they are equal to the circumference EHF, they will form right angles with it. After this, let the line QR be drawn separately, and the perpendicular ST to it, cutting it at V; and let there be applied to ST a length equal to the side of the cylinder BC, namely, with point E made to coincide with point V, so that VN is equal to EB, and VQ equal to EC. Then let VX, XY, YZ, Zα be made equal to EG, GH, HI, IF, and through the points X, Y, Z, α, and parallel to ST, let απ, νz, λγμ, ηχθ be drawn, and also on the other side the corresponding parallels through the points β, γ, δ, ε. Now let α, be equal to AF, απ equal to FD; likewise μ equal to EC, α equal to EB; but also νz equal to OI, z equal to P, λγ to MH, yμ to HN, αx to KG, and χθ to GL; and let lines equal to these and equally placed toward the parts R, other parallels, 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, μ, 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, μ, θ, ζ], 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, μ, η], ipsa recta [α, η], sed eodem tempore quo semicirculus EHF conficit in rotatione spatiu[m] a V, eodem dimidia ellipsis BMA metitur curuam [α, λ, μ, 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. triangulum, & quoniam coni 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 grauitatis centrum est in axe

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70 IN MECHAN. ARIST. PROBL. When these things have thus been arranged, through the points [a, v, λ, μ, η], and 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]. There will therefore be [a, η, ζ, μ, θ, ζ], parallel indeed in this respect, that the lines drawn between them are parallel and equal, but not straight lines, rather curved ones. Therefore, when the cylinder is moved, the circle EHF will describe a straight line, but the ellipse AMB will describe the curve [α, η], and the ellipse DNC the curve [π, ζ] itself. In this movement of the cylinder there is that remarkable fact, namely, that the ellipses are faster in the actual rotation than the circle EHF itself. For let the straight line [α, η] be drawn, which meets VS at S, and let [α, η] be joined, and a triangle [α, η] S will be formed. But the angle [α, η] is right; therefore [α, η] is greater than [α, η] itself, and the straight [α, η] is equal to [α, v], that is, to the semicircle FHE. But the curved [a, v, λ, μ, η] is much greater than the straight [α, η]; yet in the same time in which the semicircle EHF, in the rotation, traverses the space from V, the half-ellipse BMA measures the curve [α, λ, μ, η]. The ellipse is therefore faster than the circle itself. This consideration likewise clearly pertains to the motion that takes place according to the apsis. Cones whose bases are circles, if they are rotated on the plane along one side, describe with their base a circle, whose immobile center is the vertex of the cone itself, and whose semidiameter is the side itself. A triangle, and because cones. There is the cone ABC, whose vertex is C, base AB, axis DC, but the center of the base D, the side by which the plane touches BC. The cone is therefore cut by the side BC and axis DE by a plane perpendicular to the horizon, and the common section of this and the cone is ABC; the center of gravity 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 if it is cut by the very axis, the cone is divided into equal parts AEBC, AFBC; therefore the cone stands balanced on 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 infinitely many circles parallel to the base, extending even to the vertex, their semicircles in the very motion will both draw and be drawn; but when they cease 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 therefore point C of side BC stands still, while B indeed 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 easily seen in the accompanying figure. A similar demonstration is given by Heron in the little book Automaton, which we Tyrones have still translated into the vernacular from Greek and submitted to the press in Venice. Moreover, if a round cone has an ellipse for its base, that is, a section by a plane not perpendicular to the axis, in the actual rotation, the vertex remaining fixed, the elliptical base describes an ellipse in the plane, whose greater diameter is divided by the point which is the vertex of the cone in such a way that the greater part of the diameter is equal to the largest side; the smaller part, however, to the smallest side. But these things pertain to another inquiry. Having thus considered these matters concerning the motion of round bodies which takes place around an axis, it would remain opportunely to treat of the motion of pulleys, which takes place around a center; but since the Philosopher in the following question speaks about this, we send the reader on to what will there be discussed. Now we shall speak of the third kind of motion; in which kind we shall consider certain things omitted by Aristotle. The discussion here is about the motion of round bodies,

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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. of motion, which is around a axis perpendicular to the horizon, the axis remaining with one end in the same plane of the horizon, as can be seen in the very wheels of potters. Comparing this kind of motion in the last part of the question with the other two kinds, he says that the motion which is on an oblique line (for so he calls this one, of which we are speaking) itself impels the mover, that is, it has from itself no inclination, or impulse, to 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 above near 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 an impulse, and as though continuously moved it is moved by that which moves it. Yet he seems plainly to admit that this third kind of motion is more difficult, because it lacks its own impulse and is moved only by a foreign, as it were, mover. Nevertheless we shall easily show from what follows that this motion does not in any way уступать to the other two in ease. First, because the whole weight of the rotated body, from the center of gravity which is in the very axis, is supported by the plane on which it rests; indeed only a very small part of it touches the plane by the axis itself, whence it comes about that, while the body is rotated, there is scarcely any friction of the parts around the center where it rests. Besides, the center of gravity always stands still, nor is it lifted up even the least in 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 small a force may be applied to one part, the other is immediately overcome and set in motion. For although properly only those bodies are called balanced which because of equal weights on this side and that are equal...

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EXERCITATIONES. 73 æqualitatem horizonti fiunt æquidistantes, nihilominus & hic aliquam 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 equidistant from the horizon, nevertheless it will here appear that there is some likeness to equilibrium. For let there be a wheel ABCD, whose axis, perpendicular to the horizon, FEG, passing through the center E, but touching the plane at the point G. Let the diameter BED be drawn. Thus if through the diameter BED and the axis FEG the body be divided, because the center of gravity is found in the axis, the body itself will be cut into two parts equal both in mass and in weight, namely BAD, BCD. Therefore, no external force being applied, the body will stand in a certain, as we said, state of equilibrium. But with some force, however slight, applied to one of the parts, say at C, the part BCD will prevail, and will either drive or carry along the part BAD, while the other in the meantime yields to its motion. The force, therefore, which at C, finding nothing that hinders it, moves the wheel most swiftly, and this is done the more easily and more rapidly the more the wheel is in motion, and indeed its diameter is larger and the moving force farther from the center; and certainly from the ease of this motion we know that, the mover himself having ceased from the impulse, the wheel retains the impressed motion for the longest time, and does not come completely to rest until after a long rotation. Moreover, since Aristotle, as they say, passed over in silence what pertains to this motion, we shall examine more carefully certain things that contribute to this matter. We ask first, therefore: why things that are rotated in this manner do not change place during the rotation itself, unless this is done for some external cause. For let there be a wheel or some other round thing, such as the tops with which boys play, whose axis around the hori- K zon

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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. may be moved perpendicular to the horizon, ABCD, whose center is E, the diameter AEC. Now around center E let infinite circles be imagined, some smaller than others, all the way to the center itself, such as FGH; for there the circles cease to be, where there is no longer any space. Let power therefore be applied in B, which may turn the wheel toward A. At the same time, then, and all at once, it will move from A toward D, from D toward C, and from C toward B. For as many semicircles as pass beyond the diameter AEC on the side CBA, so many semicircles, which are on the side ADC, will pass over to the parts CBA. But where motion has ceased, there rotation ceases; but where space ceases, motion ceases; but where the cir- cles cease, space ceases; therefore in the center, since there are no circles, space does not cease there, nor motion. For there is no reason why the body itself should be transferred by rotation to another place than that in which it is. Therefore the rotating body stands still, which had to be demonstrated. This demonstration is similar to the one we mentioned above, concerning the rotation of a cone in a plane about its apex, which we said was devised by Hero in his Automata. We add that in this kind of rotation the body becomes lighter in the very motion, and the more so the faster the rotation. The cause is that the lateral motion somehow impedes that motion which arises from natural heaviness toward the center; therefore experience teaches us that the lightest whirling motions are those with which boys play, if they hold the hands with the palm, while the very quickest rota- tion is made. To these another question is proposed and answered, why round bodies are more suited to this kind of motion. It is quite evident that bodies which are moved in this way, par-

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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 the more swift, the farther they are from the center about which they move. For they pass over a greater distance in the same time. Therefore, the more the figure abounds in those parts which lie farther from the center, the more easily and swiftly will it move, being rotated in a circle. We shall now show that the circular figure surpasses all others in that abundance of parts farthest from the center which we have mentioned. Let there be, say, an equilateral triangle ABC around center D. Let perpendiculars be drawn through the center from the opposite angles to the opposite sides, ADG, BDF, CDE; and they will be perpendicular to the sides. Since therefore the sides AD, DB, DC are subtended at right angles, they will be greater than the sides DE, DF, DG. Therefore in this triangle the three longest lines are DA, DB, DC. The three shortest indeed are DE, DG, DF; for which reason, when the triangle is rotated around center D, only three parts of it, ABC, will be the swiftest, and three the slowest, E, G, F. Therefore the triangular figure is less suitable for this motion than the square, in which the parts farthest from the center, and therefore the swiftest, are four. Therefore the more a figure is sided and abounds in angles, the more suitable it will be for this, and for all other circular motions. But a circle, so to speak, has infinite parts farthest from the center, and thus no figure is more convenient or swifter than the circular one in its own rotation. There is also another reason for this: that when a circular figure moves, it does not strike the air with projecting angles around the circle, and from that striking the motion is hindered and made slower. It may also be asked whether, with the axis inclined, the motion of the wheel is in some way hindered. We embrace the negative view. K 2 Let there be

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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 perpedicularem 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, & celeriùs 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, with an inclined axis, about which EGF turns. It is supported by two points of GF. Let both the weight and the center of the figure be at E. But let the perpendicular passing through the lower support be HFI. Therefore, when the wheel is turned, the center of gravity will remain and will not be moved from its position upward or downward. Now the axis FEG is as a lever, on which the weight is at E, the supporting forces at GF; for here, as in a perpendicular axis, the whole weight is not supported 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 in G. And these things are indeed as follows, the center of gravity being placed on the axis itself; for if it were outside, motion is hindered and, the mover ceasing, it quickly comes to rest. Let the center of gravity be in K. Thus, while motion is made about the axis, the center moved in a circle will at some point be in L; and let the diameter AC of the wheel intersect the perpendicular HI at M. Moreover, from the points LK to the same perpendicular, let the lines LN, KO be drawn at right angles. But 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. Therefore the center of gravity K, if left free, will come to rest in K and will be transferred contrary to nature to L. Therefore, when violence ceases and nature prevails, the wheel will quickly come to rest of itself, which was to be shown. QUESTION IX. It is asked why those things that are lifted and drawn by larger circles come to move more easily and more quickly, as with larger pulleys, and similarly with skittles? The Philosopher answers 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 for by how much greater that which is from the center will be, in equal time it will move a greater space. Wherefore, with an equal load, it will do the same. For thus he had said, treating of the nature and differences of balances, that the greater are more exact than the smaller. But circles are balances, in which the center is fixed, the semidiameters on this and that side being equal arms. What he affirmed in the last place, that pulleys are like balances, is true. But what he said, that the greater balances move more easily and more quickly than those which are smaller, if understood simply, is false; for the ease of motion, in drawing machines, is contrary to speed, as Guid. V. Bald. demonstrated in the treatise On the Pulley, in the second corollary of the last proposition. As for what he said, that the greater the pulleys are, the more easily they move, this is not, as we were saying, simply true, as we shall easily show. Let there be a pulley AB about center C, suspended at point D, a perpendicular which, to the center of the world DCE, has equal weights hanging on both sides FG. Let there be also another pulley, a greater one HI, about center K suspended at L, a perpendicular, which, to the center of the world LKM, equal K 3 pon-

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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[ue] 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 minoti quàm in maiori, bra- chia siquidem bifariam diuiduntur, ergo in vtriq[ue]; 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 the weights suspended on either side, N, O. I say that the greater H I more easily does not move the weights than the smaller D E, because it is greater; but the other 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 of equal weight. For the same reason the parts LHM, LIM also will weigh equally. Therefore, if you add however small weights, on either side of each of them equilibrium will be transferred to the other side, nor is a smaller weight required for the smaller pulley to depart from equilibrium than for the larger one. But the dispute can be concluded in a single word: yet in the smaller as in the larger, since the arms are divided in two, therefore in both the same proportion of the arms, and the same ratio of the weights. These things are quite evident. Nevertheless, since the matter itself teaches that what Aristotle writes is true, the cause of this effect must be sought from elsewhere by us, namely from mechanical principles. I therefore say that the axles, around which the pulleys or wheels turn, have various proportions to the wheels themselves. And we shall show that a wheel or pulley is moved more easily, and moves weights, the greater the proportion the diameter of the wheel has to the diameter of the axle; and therefore it can happen that a larger wheel has a smaller proportion to its axle than a smaller wheel to its own. For let there be a pulley AB around center C, whose diameter DCE is in that line which is perpendicular to the center of the world; and let it be suspended at D. Likewise let there be another pulley equal to it, FG around center H, whose diameter IHK, in agreement 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 attached to the center of the world. But let it be in I. Let them also have axes around which they turn. If these are equal, with the proportion unchanged they will perform the same work. But let them now be set as unequal, 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 can be raised more easily than by AB, although one of the pulleys be equal to the other. For since mechanical bodies are not without matter and weight, the suspended loads and the weight of the pulleys themselves will press the axes from above, where are points L and M; and this, the friction of the solid bodies following one upon another, makes the motion itself of the pulleys more difficult and rough. Thus impediment takes the place of weight. Therefore we have two levers, DC and IH, whose fulcra are against C and H themselves. But let the weights be at L and M between the fulcra and the powers. And let the powers be understood as applied at the points D and I. Therefore, by the nature of such a lever, in which the weight lies between the fulcrum and the power, it will be as CL is to CD, so is the power at D to the weight, that is, the resistance of friction which is at L. But the proportion of CL to CD is greater than that of HM to HI. Therefore a greater power is required at D than at I to overcome the same or an equal impediment. And so, since the whole force consists in the proportion of the diameters of wheels and axes, it may happen, as we said, that a smaller pulley is given which has a greater proportion to its axis than a larger one to its own; in which case the smaller wheel will more easily overcome the impediment which we said the larger wheel or pulley itself would overcome. Yet since both the axes and the wheels are made of matter, and the nature of the thing does not allow slender and weak axes to be able to sustain great weights, therefore thicker ones are made; and since this thickness is more overcome by proportion from the diameters of large wheels, it follows that, with the equality of the axes granted, larger wheels are given

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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, nè- 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 amouibilia, 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. to overcome the obstacle more easily than smaller ones, & the Philosopher seems to have understood this in the very proposition of this question. Hence charioteers commonly grease the axles with axle-grease (whence it takes its name), so that the roughness of the axles may be lessened, and less resistance arise from friction in turning. We conclude, therefore, that that pulley draws the weight most easily which, though it be the largest, has the smallest axle, and that axle soaked with grease or some other unctuous substance. As for the handles fitted to the axles of wheels, almost no one has spoken of them; we shall therefore say something about these, since the matter pertains to the speculation we are discussing, 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, moreover, the proportion is to be observed which the length of the handle has to the semidiameter of the axle itself; for the greater that proportion of their length to the semidiameters of the axles, the more easily do they move. They consist of two parts: one, which extends from the axle to the angle, and this is truly a lever; the other, to which the hand itself is applied, and from which the whole thing is called a handle. These handles are for the most part made removable; nevertheless they are, as it were, parts of the wheels themselves, and would be conveniently attached to the wheels themselves, if only, in turning, no impediment were caused by the transverse pieces by which the wheels are supported. For let there be a wheel AB, whose axle is E, and let it be bored in F, and there let a pin FK be fixed. Let there also be another wheel CD, whose axle is G, with a handle placed on the axle GHI. And let the wheels be equal and the axles equal. Let the spaces EF, GH also be equal, 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, costituto, factâ interim pressione motus à centro, ad quod directè fieret pressio, impediretur. Curuitas auté 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 GH. I say that the wheel AB can be moved with the same ease by the power applied in FK as CB is moved by the power applied in HI, namely, given equal powers. For let IH be produced as far as L on the side of the wheel CD, and let LG be drawn, and FE joined in the wheel AB. Therefore FE and LG will be equal to one another. But they are the semidiameters of those circles which are described from the points FL in the very rotation. Thus, therefore, the power applied at L will be related to the diameter or semidiameter of the axis of the wheel CD as the power applied at F is related to the diameter or semidiameter of the axis of the wheel AB; but the spaces are equal and the powers equal, wherefore it makes no difference whether the handle is attached to the side, or whether, separated from the side of the wheel, it is applied to the axis. Now there are two forms of handles; for one consists of straight parts, while the other is entirely curved; but we use straight handles when we grasp them with our hands, whereas curved ones are used so that a place may be assigned to them, and they may be turned by the pressure of the foot, as is usually done in stone mills, by which swords are sharpened. The reason why these handles are made curved seems to be this: namely, lest, the head of the handle being placed above the center on the line that passes through the center, motion from the center, toward which the pressure would be directed, might be impeded by the pressure being applied in the meantime. Curvature, however, has a certain convenience, from which, a slight bending having been made, the head of the axis, while it is pressed, is gently drawn away from the very perpendicular line. Since these things are lacking in handles that are operated by the hand, therefore another form, namely of straight parts, is commonly made. Let it therefore be that which is made of straight parts is AB, the curved one CD, and the line according to which 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, & diutiùs 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 far, then, it is enough for us to have considered these matters concerning handles or levers. Meanwhile, one might ask why, given two wheels of equal size and equal weight, set upon equal axles, the lighter one is moved more easily and comes to rest more quickly; whereas the heavier one is moved with more difficulty and ceases from motion more slowly. The reason seems to be that the heavier, resisting more when it is overcome, receives the impressed force more strongly and retains it longer, which does not happen in the lighter one. QUESTION X. Aristotle wonders why a balance moves more easily when it is without a weight than when it has a weight. Likewise a wheel, and something of that sort, when it is heavier, and also when something larger and heavier is compared with something smaller and lighter? He gives a brief answer. For, he says, is it because what is heavy is moved with difficulty not only in the contrary direction, but also obliquely? For it is difficult to move it in the direction opposite to that toward which the load tends, while in the direction toward which it tends, it is easy. But obliquely it does not tend at all. We, because he did not do this, will make the matter clearer by adding the figure itself. Let there be the balance AB, whose support is C, with weights suspended on both sides of AB, each of which weighs 10. Likewise the balance DE, whose support is F, with weights D, E suspended, each half as light as A, B, namely S. Let a weight G be added to weight B, and a weight H to weight E, each of which likewise weighs S; thus the balances, with the weights attached, will sway, 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 traht 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 at K, and EH indeed at N, but GB is heavier, for it is as IS to EH, which is 10. But BG will descend more difficultly than EH. This is according to Aristotle’s doctrine, because not only does a heavy body move with difficulty in the contrary direction, but also obliquely; for it is difficult to move it in the direction opposite to that toward which the load inclines, but it moves easily toward that direction. In an oblique direction, however, as by the lines BK, EN, the load does not incline. Therefore the weight BG will be moved obliquely with more difficulty than the weight EH itself. Yet in descent both are drawn back, namely from the perpendiculars BI, EM, and indeed the angles of retraction are equal, and the retractions themselves are equal. But the weight GB is heavier. Now that which is heavier descends more forcibly than that which is lighter. Therefore, unless with greater force and impetus, other things being equal, the weight BG will descend more than EH, which is directly contrary to Aristotle’s assertion. The truth, therefore, must be drawn from other principles. We say that this happens because of the inequality of the proportions; for since the one has the sesquialter proportion to 10, but 10 to 5 the double proportion, EH has a greater proportion to the opposite weight D than BG has to weight A; therefore the balance DE more easily draws the lighter weight D than AB does the heavier weight A, which indeed had to be shown. Another cause also, and this one accidental, contributes to producing this effect: namely the friction of the axes 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, this speculation is also concerned with this: why equal and similar bodies placed on equal and similar bases, and supported on the same and similar plane, yet unequal in weight, are not overturned with the same ease, but the heavier among them with greater 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 Primè 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 duomo- uetur & inferiori substrato spatio, & onere superimposi- to,

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84 IN MECHAN. ARIST. PROBL. Let there be, then, a Prism or Cylinder ABCD, whose center of gravity E is in the plane CI, supported by the base CD. And let there be another Cylinder FGHI, whose center of gravity K, supported by the base HI, is indeed equal and similar to the former AD. Let however FGHI be heavier than ABCD. I say that, with equal force impelling both, the Cylinder AD will be overturned more easily than FI. Let EC, KH be drawn, and equal forces applied at the points BG, driving the Cylinders toward the parts AF. But the 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 on the perpendiculars ACFH. Let EN, KO be dropped, perpendicular to CD, HF. And since the angles CNE, HOK are right angles, EC, KH will be greater than EN, KO; wherefore LC, MH are also raised greater than EN, KO, therefore in the very overturning the centers of gravity E into L, K into M. But that which is heavier is moved with more difficulty against its nature; therefore the body FI will be overturned more difficultly than AD, which was to be demonstrated. QVÆSTIO XI. The Philosopher asks why burdens are carried more easily on skytalae than on wagons, although the latter have large wheels and the former very small ones? At first he answers the doubt. Is it, he says, because in skytalae there is no obstruction; but in wagons there is the axle, against which they strike? For they press upon it from above and from the sides. But that which is in skytalae, being moved to these two things, moves both through the lower space beneath and through the load laid upon it,

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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- cer, & 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 to, for in both of these the circle is rolled in those places, and the motion is driven forward. He explained the truth so fittingly in a few words that almost whatever is added besides may seem superfluous. Be that as it may, for greater clarity we shall dwell somewhat longer on this very question. Aristotle here proposes rolling skittles. But that they are joined together with the skittles must be understood, namely, so that the wheels turn together with the skittles. Otherwise there would be a striking of the axles and the wheels, and since Aristotle knew the force and effect of this striking, it is wonderful, if the very passage bears witness to it, that he said nothing about that question in 9, where we discussed this matter most fully. Moreover, as to what the Philosopher writes about rolling skittles, it is to be noted that, indeed, in Pappus, book 8, and among our Mechanics, they are commonly used without wheels, namely in a simple cylindrical and smooth form. Let then Aristotle’s skittle be AB, but Pappus’s, or the vulgar and common one, CD. These are used not only by stonecutters everywhere, but also by sailors and shipwrights in drawing ships out and bringing them to sea, which they call varare in the vernacular, a Spanish word, I think. For that nation calls a smooth piece of wood or stick Vara. One might ask, however, which of these forms is the more useful and convenient? We prefer the rolling ones more on a level and hard surface, for they touch less and strike less; but on a softer and less hard surface we propose the non-rolling ones, since wheels by their nature, when pressed by weight, very easily cut into the ground and are swallowed up. As far as use is concerned. Let the horizon’s L 3 pla-

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86 IN MECHAN. ARIST. PROBL. planum AB, scytale du[m] CD, EF, Pondus verò eis impositum G, tangens ipsas in puctis 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. the plane AB, the spindle CD, EF, and the weight G placed upon them, touching them at the points CE, and the spindle’s plane at the points D, F, let the weight G be driven forward by some power, so that it does not turn toward the parts E. The spindles will therefore rotate, and a certain part of spindle D, being in contact, will rise to I, while C will descend to H, with nothing preventing the movement, since there is no mutual interference of the weight, the spindles, and the plane. Moreover, since the centers of the spindles are equally distant from the plane of the horizon, the weight is moved parallel to the horizon, and therefore its center of gravity is in no way raised in the motion. Furthermore, if the imperfection of the material is removed, it makes no difference to the ease of motion whether the spindles are of greater or lesser diameter, since, other things being equal, larger circles more easily overcome obstacles, as was demonstrated in Question 5. Hence the thicker the spindles, the more useful they are. But as regards wagon motion, to which the Philosopher made the comparison with spindles, we shall show that weights are moved from it with greater difficulty. Let there be the wheel KL of a wagon, whose center is M, and the axle NO around 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 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 draws at P brings the axle against the hub at V. Thus a twofold obstacle arises from friction or collision, namely below, where O is, and at the side, where V is. These impacts 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 semidiameter of the wheel KM to the semidiameter of the axle MO. Why then sledges transfer weights more easily than wagons, we have clearly demonstrated from what has been said, in accordance with Aristotle’s meaning. Moreover, what he himself passed over in silence, we shall state: namely, that very great weights are moved by sledges most effectively if levers are attached to the sledges themselves. And indeed the motion will be extremely slow; yet this slowness is very abundantly compensated by the ease that results from it. Let then AB be the plane of the horizon, CD the sledges, EFGH the holes in the sledges, and IE, KF, LG, MH the levers inserted into the holes. Let the weight placed on the sledges be N. Therefore, if four forces are applied to the extremities of the levers I, K, L, M, and are pushed forward, the rotation of the sledges will occur,

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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 OPQR fiet ex O; centro aute[m] Q circuli quadrans OT. 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 OT, 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, siqui- 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 per quàm appositè, itaq; illi

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88 IN MECHAN. ARIST. PROBL. and the transfer of the weight N to the forward parts B. Let there also be separately the staff PR, whose center is Q, and a lever inserted through the same center O, P, Q, R. Therefore, when the lever is moved through OPQR, there will be from O, with Q as center, the quarter-circle OT. Since therefore O is in T, P will be in S, after the fourth part of the rotation of the staff itself. And because the quadrants PSOT are from the same center, it will be as OQ is to QP, so is the quadrant OT to the quadrant PS. But the proportion OQ to QP is greatest. Therefore the proportion OT to PS is greatest. Thus from the great motion of O to T, the motion of the staff from P to S is small. Therefore the staff progresses more slowly when it is turned by longer levers, yet the force is greatest, since as QP is related, that is, QR to QO, so is the power at O to the weight that presses at P or at V. It is therefore most easily seen how weights are transferred by levers and staffs across a horizontal plane. QVAESTIO XII. It is asked why projectiles are thrown farther by a sling than by hand, especially since to the thrower the weight of the sling is added to the weight of the stone or missile, and the missile is grasped less well when thrown by hand? The Philosopher answers, saying, perhaps it happens thus, because the slinger throws the missile already set in motion from the sling; for inasmuch as he keeps turning the sling in a circle, he casts it forth, whereas from the hand the beginning is from rest. Now all things, when they are in motion, are moved more easily than when they are at rest. He adds further, “And is it also for this cause?” but also no less because, in the use of the sling, the hand is indeed the center, but the sling is what extends outward 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 is shorter in relation to the sling. These are the Philosopher’s words. And indeed, how suitably, thus to them

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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 qua 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 inferentes, 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 wholly agree with him, if I did not know for certain that, in the throw with a sling, the hand itself is not the center of the motion, but rather that part of the arm which is joined to the shoulder; and therefore the motion is made swifter insofar as the line from the shoulder to the top of the sling is longer than that from the shoulder to the hand itself. This too is remarkable, which Aristotle does not observe, namely, that in the very act of whirling the sling, the rotation about the head of the sling is made slow by slingers. Wherefore it must be considered in what way speed comes from slowness. We answer that speed is acquired not from the simple rotation around the slinger’s head, but from that impulse which is in the very discharge of the stone; and this impulse, if it occurs before or after that moment which is seized by the slinger, the casting itself is altogether vain and ineffective. Let there be the sling AB, hand B, arm BC. Thus, as CH is to CB, so is the speed AD to the speed BE; we have seen boys, with reeds split at the end, inserting little stones and turning the reed with the hand, throw the stones very far. Reed FG, stone F, hand G, arm GH. QUESTION XIII. It is asked why, about 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 therefore happen because both the yoke and the sucula are round, but the projecting collopes M are longer.

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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, & collopem 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 that are from the center. But the greater wheels move more quickly and more under the same force than the smaller ones do, because by the same force more is transferred to that end which is farther from the center. And in thinner winches, given equal size of the pegs, that part which is farther from the wood is greater. Let there be a larger winch beam, AB, around center C, and a smaller one around the same center, DE. The peg AF, the weight raised by the beam G. Aristotle thus says that the winches, the beams AB, DE, are as centers, from which extends the peg AB, from the larger one indeed with its whole part BF, but from the smaller one EF. Therefore, he says, the longer the projecting peg has been, the greater it is, and therefore the faster, toward part F, through the larger circle FH, will be the motion of the peg and the lifting of the weight; yet the peg EF is greater than BF itself, so the weight will therefore be moved more easily through the winch DE, by peg EF, from the same force, than through the winch AB and peg BF. Aristotle seems to have thought these things, who, with “thick Minerva,” as they say, pursued a beautiful and subtle matter. We say, then, first, that the instrument which the Latins called a sucula, that is, serosula, I suppose from the squeaking sound that occurs in turning, but the Greeks called it an Onager , that is, an Ass, because it bears and carries weights like an ass. The ancient mechanicians called this same machine an Axis in Peritrochium, whose image, taken from Pappus in the 8th Book of the Mathematical Collections, we presented at the very beginning of this work of ours, among the five powers. Among the ancients, Heron examined its force most diligently, and himself—

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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 fractio quam ex FG. Hæc ille ex suis prin- M 2

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EXERCISES. 91 Pappus himself, and among the more recent writers Guilibaldus in that treatise on Power which he inserted among his Mechanica. The sum of the matter is this: that this machine is to be reduced to a lever. Nor is what Aristotle writes true, namely that the yoke or windlass has no center; 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 more easily move the weight at D than the same power at F would move the weight at A, that is, G. Of this nature also are Ergatæ, which our people, using a corrupted Greek term, call Arganos. For they are in truth windlasses, differing from them only in position; for the ergatæ do not lie parallel to the horizontal plane, as windlasses do, and as the axis in the peritrochium, but are made perpendicular to the same. Moreover, we have shown above that ease does not arise from speed. QVAESTIO XIV. A question is proposed: why a piece of wood of the same size is more easily broken at the knee-joint if someone, taking hold of the ends with hands equally stretched apart, breaks it, than if he were to do so near the knee. And if, with the foot applied to the ground and the hand placed above, he breaks it with the hands drawn apart on either side, why it is easier when near. The Philosopher answers in a few words: because there the knee is the center, but here the very point itself? For the farther anything is from the center, the more easily it is moved; but whatever is broken must needs be moved. Let AB be the piece of wood to be broken, C the place of the knee or foot, DE the position of the hands widely spread apart, FG of the less widely spread apart; and so, since DE is farther from the center C than FG, the points DE will be moved more swiftly than FG themselves, therefore the breaking will be accomplished more easily from DE than from FG. Thus he from his 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-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 facto 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. With careful inquiry, if it can be done, let us examine the cause of this effect. Let there therefore be, in the second figure, an oblong piece of wood AB, whose middle is C; let the line CD be drawn perpendicular to AB itself. Let the knee be applied at point C, and the hands, however, be separated at AB; when pressure has thus been made on both sides, the wood will not break unless a separation of the parts joined in CD is made, and let one be at E, the other at F. Therefore the wood will be broken, and, C remaining fixed as the center, the parts, the angle GCH having been formed, will be in GC, HC. But now let the wood be restored to its integrity, and, the knee again applied at point C, let the hands be drawn apart in I, K, which are located nearer to C itself than AB. I say that from this the breaking is more difficult than from AB. For we consider in the intact piece of wood AB the two levers ACD, BCD, whose angles meet at the common support C. Now these are angled levers, and of the kind whose nature we have 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 resistance must be overcome in order that the wood be broken. I say that this will yield more easily if it is done from the points A, B, which are farther away, than from IK, which are closer to point C itself. For as AC is to CD, so is the resistance which is in D to the force in A; likewise, as IC is to CD, so is the resistance in D to the force in I. But the proportion of IC to CD is smaller than that of AC to CD. Therefore the force which is in A will more easily overcome the resistance which is in D than that which is in I,

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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 the part CB; for the reason is the same. Why then longer and more slender pieces of wood are easily broken is clearly apparent from these remarks: namely, because the proportion of length to thickness is greatest, and the space of this thickness takes the place of that part in the lever which extends from the fulcrum to the weight, that is, to the resistance itself. But on this same subject we shall reflect further in explaining question 16. QUESTION XV. It is proposed to investigate why pebbles that are written upon (the Latins call them glareae , or small stones, which Cicero calls umbilici in book 2 of De Oratore ) are of a round shape, when sometimes they are from large stone or shells? The Philosopher says that perhaps this happens because those things which recede more from the middle, in motions, are carried more quickly; the middle being the center, and the interval the distance from the center; and because in always greater motion from equal motion one describes a larger circle; and that which in equal time passes over a greater space is carried more quickly; and those things which are carried more quickly over an equal space strike with greater force; and those which strike are struck more, and therefore those which are farther from the center must be rounded off, which, when pebbles or small stones undergo this, they necessarily become round. So far he, and indeed quite plausibly. But truly the matter seems to be otherwise: for if this were caused by rotation from greater distance from the center, the larger pebbles or stones would certainly be more round, but we see that not only the largest, but also the smallest, are the ones that are more lacking in angles and approach roundness. Besides, that they do not move around a center is obvious; rather, as their shapes vary, so too do their motions, since they are moved by being agitated. This is certainly most evident, 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 caussam, 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 deuitât, vt pote de- biliores, & magis offensionibus obnoxios. quod nec Vi- truiuium 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 prominences there are in bodies, are weaker. For they are exposed to blows, nor do they have the power of resisting. Therefore, in the friction which occurs in their perpetual motion, the prominences are worn down, and the surface itself is gradually made smooth. Let ABCD be an angular stone. When, therefore, it is moved by perpetual motion and constant turning, and is carried along, the prominences and angles, as being weak and feeble, are worn away, and from this a certain irregular figure results, approaching indeed the original form of the stone, yet smooth and lacking every angle, such as E is, the angular prominences of ABCD having been removed. For this same reason, sculptors, before they give the final polish to marbles, first use a toothed hammer; then at last, easily removing the higher particles with a scraper, they make the surface itself smooth and even. Hence also our architects, in forming the bastions of fortresses, avoid sharp angles, as being weaker and more exposed to blows. This was not unnoticed even by Vi- truvius, who therefore writes thus in book 1, chapter 5: “Towers should therefore be made round or polygonal, for square ones machines more quickly destroy; and with angles, the rams, by striking, break them, whereas in round forms, by driving wedges toward the center, they cannot injure them.” Thus he. But why our people reject other useful round forms, must be asked of those who are engaged in that art. Moreover, what also pertains to this same inquiry is that we see ancient statues very 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, & fit tolluntur, 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 fit, 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 all these things being true, no one, as I judge, would say absolutely that what Aristotle intended comes about from swifter rotation and 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 lifted they bend more; although a short piece, if it were two cubits long, is thin, whereas one of a hundred feet is thick? Aristotle resolves this 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, consist in the very length of the wood? For its earlier 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 be curved. Necessarily, therefore, the ends of the lever are raised. If then the lever is flexible, it must bend more as it is lifted, which happens in long pieces of wood; but in short ones the end itself is made almost at the resting hypomochlion. We set this forth before the eyes with the figure below. Let a long and flexible piece of wood AB be raised by the hand at A; thus it will bend at B and incline toward C. For the hand which supports at A takes the place of the support: but the length AB performs 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 by no means bend, because its end D is less removed from the support which is at A. This, therefore, is the sense 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, ea q[ui] 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 reject; but we add something to it. We say, therefore, that matter, insofar as it pertains to this consideration, is of a twofold difference. For either it is incapable of rarefaction and contraction, as we see in steel, nitre, metal, mar- ble; or it is capable indeed, and this in a twofold way: either by nature it is born to a certain straightness, as the shoots of trees and rods, or not so, as tin, lead, and other such things. Suppose first a glass body, slender, tall, round AB, held by the hand in A, so that with the weight of the body itself prevailing toward the parts B, because at point C, which is near the middle, on the upper part there is no rarefaction, nor on the lower part contraction, nor in the meantime is there any penetration of bodies, a fracture occurs from the upper part, and the part CB, torn away and separated from the remaining part AC, falls in D; and this separation succeeds rareafaction. Moreover, in saying that these materials are not flexible, but brittle, we do not therefore deny, even with the senses teaching us, that some bending takes place in them. But if the material were wooden, which is 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 on the upper side rarefaction occurs, but on the lower side contraction, and the part GF will incline to H; and this inclination will proceed as far as the rarefaction and contraction appropriate to the nature of that material which is bent have reached the greatest intensity; then if a greater force should come on, it will break completely: if otherwise the 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, agitatione. 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 Cremotissima enim ea pars 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 N

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EXERCITATIONS. 97 resistance, where rarefaction and condensation, after an upward inclination, will carry the inclined and nodding part; then, tending toward the opposite side, it will be reflected, as can be seen in the rod IN. For, declining in KL, repelled by the condensation of the matter lying below K, by the force acquired from the descent, after reflection it rises into KM, until little by little it returns to its former straightness; and this motion is indeed called vibration, agitation. But if the rod were of lead, not made by nature for straightness, as OP, overcoming its own weight, it will decline toward QS, and in QR it will be rarefied, namely the upper part being condensed, the lower in Q, and it will not be reflected, since its nature readily admits condensation and rarefaction, and was not made for straightness. Moreover, the elevation of these oblong bodies can happen in three ways, namely either by one of the ends, or if by both, if suspended on both sides, or somewhere between the ends. We have already dealt with the former way. Now let it be suspended in the middle, as AB, in C. In that case, therefore, since the support is in C, there will be bending on both sides in D and E, and indeed if the material admits bending; if not, there will be fracture in C. But if the suspension is from the ends, as in AB, then as it were two levers will be formed, whose supports are at the ends AB. The weights, however, are common in the middle, where C is the most remote part from the ends AB. Therefore, if the material yields to its own weight and is inflexible, it will break, and there will be a separation of the parts in C, and from this two bodies AD, 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 trahiu[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 perpendicularum 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 about C, and with condensation in the upper part, will, because of the prevailing weight, be bent; and a piece of wood or anything else of this kind, like ADB, will no longer return to straightness, its weight naturally tending downward. Moreover, why oblong and more slender bodies are broken more easily than those that are in the contrary condition, we have plainly demonstrated from mechanical principles in question 14. Now, in order that from this contemplation, which otherwise seems useless, we may derive some benefit, and that architects may become more prudent from what we shall consider, we shall aptly adapt these very matters, with which we are dealing, to building practice. Let us therefore transfer our thought to that structure of beams which, for supporting straight members, is made of a crosspiece and an upright, with two braces, which our craftsmen, from the Latins by a distorted word, call a biscauterium. For we shall inquire whence it has so much strength for support, and what properties follow upon this structure. For although craftsmen, by mere practice, produce what is useful, we, for the sake of better minds, shall carefully examine the causes of the thing itself and set them forth; nor shall we speak only of this matter, but also of chambers, vaults, and their defects and virtues insofar as they pertain to mechanics. First we ask: why do beams erected perpendicularly sustain superimposed weights most strongly? And indeed everyone knows this, but not the causes. Let there be the plane of the horizon, and let it be the very solid and impenetrable AB; let a beam CD, erected perpendicular to it, be supported at the base where C, the center of gravity, is, and let there be a superimposed weight FG, whose center of gravity is H; and let H and E be on the same perpendicular, HEC, extending toward the center of the world. Therefore, because both the center of the weight and the center of the beam are situated on the perpendicular, while 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 frangeatur lignum AB, propterea quòd maxima sit proportio AC ad CE; resistentia verò fiat in E, addatur vniaturq; N 2 ligno

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EXERCISES. 99 The mass of the heavier weight will rest on C; it will not descend, however, into I, because the plane AB is supposed to be impenetrable. Therefore, in order that the weight H may descend into C, one of two things is necessary: namely, either that the beam underneath be crushed, or that its parts penetrate one another, and that more bodies be in the same place, namely KC; the latter is utterly contrary to nature, while the former is almost impossible. For let the beam be divided into three equal parts by the lines KL; thus the lowest part KC supports the middle KL, this in turn the upper LD, and this again the weight, that is, the superimposed body in H. Thus the parts sustain one another. But the whole consists of those parts. Therefore the whole weight is sustained 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 then easy, when the proportion of length to thickness is greatest. Here the opposite occurs; for MD, the part of the lever which extends from the fulcrum to the power, has the least proportion to the straight line DC, which extends from the fulcrum to the place of breaking, where C is, which, that it may be more clearly evident, Let AB be a beam apart, whose middle point is C. Let a weight D then be placed at point C. The wood AB will therefore easily be broken, because the proportion of AC to CE is greatest; but the resistance will occur 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 trahem 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 that of AC to CE. Let IM also be added. Therefore it will be broken much more difficultly at K, because the proportion of AC to CK is much smaller than that of the same to CE and CG. These things therefore having been considered and demonstrated, we conclude that it is impossible for an upright beam to yield to a weight and be broken. But someone may say: if these things are true, the more slender the support is, the more strongly it will bear, and the less it will be broken; which is quite false. We reply that this does not happen from the nature of proportions, but from the weakness of the material itself. So also in the case set forth we consider not the material, insofar as it pertains to force, but the proportions of the parts. Therefore both are required for the strength of a support: the due 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 violence, in the first case through a perpendicular line tending to the center of the world, in the other laterally and in various ways, the arrangement of supports is different. The sum of this matter is that they must always be placed against the impulse. Let the plane of the horizon be AB, with perpendiculars to it CADB; thus if a weight presses naturally from C, the support shall be placed at AE. If however from F, GE; if from H, it shall be placed next to BE. If however according to I, let KE be opposed to the weight. We have spoken thus about upright supports; now we shall treat of transverse and inclined ones, and first of transverse ones, insofar as the framing of roofs is concerned. Let AB be a transverse beam, 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, aliquid 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. Thus, with the center of gravity tending in the same way, if the weight which presses at E does not outweigh the smaller part of the material itself 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 because the proportion of AF to FE is very great, a break occurs at E, and after the separation of the parts there will be made on either side two beams AH, Bl, whose centers of gravity are KL. Therefore there will be two levers AE, BE, whose supports are MN; wherefore if the proportion of EM to MH should prevail so much that the weight which is at E exceeds the weight of the wall O superimposed, and likewise of the wall P, the beams will indeed fall, and a collapse of the walls will occur on this side and on that. But if the proportion which we mentioned has not prevailed, the beams will remain suspended on both sides, as AHBI. Architects meet this difficulty excellently, however, in this way: Let there be a cross-beam AB, weak because of its slenderness or for some other cause, supported on both sides by walls CD. The beam’s center of gravity is G. So then, having fastened to the beam the timbers EF, they add braces CE, DF, supported on both sides by the wall, bringing their heads to the fastened timbers EF; but the fastening is made very strong also if, between E and F, one whole timber is laid beneath the beam EF at the heads of the braces. Reason N 3

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102 IN MECHAN. 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; 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 MECHAN. ARIST. PROBL. the force of its resistance is evident; for when the center of gravity presses at G, the supports CE, DF come to its aid on either side, which, since they cannot become shorter in themselves, lest penetration be given to the body, resist and very strongly counteract the weight itself laid upon them. But in this work two levers seem to be considered, GH, GB, whose supports are EF, and the pressing force at G on both sides. The weights, however, are the parts of the walls imposed on the heads of the beam at A and B. Since therefore the proportion of GE to EH is small, and the pressing force at G is small, but the weight at A is greatest, it is not possible for the beam to be broken or for the walls on either side to be shaken apart in AB. Three parts of the whole beam may also be considered, AE, EF, FB, with four supports, A, E, F, B. Thus, by dividing the weight and multiplying the supports, it is impossible for the beam to be bent and fail. And in the same way mechanics are accustomed to strengthen the joists of roofs and weak cross-beams, namely by adding a brace to the beam and fastenings. Let there be a cross-beam AB supported on either side by the walls, I, K, and a brace CD. Let there be fastenings on either side AD, BD, inserted both into the cross-beam at AB and into the brace at D, so that they can in no way slip out from there. Then let an iron band EF connect the middle of the cross-beam AB, from its lower side, to the brace itself. But the foot of the brace at C ought to stand somewhat apart from the cross-beam, so that, as it inclines downward under the weight, the brace may not press the cross-beam itself a little. 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 the weight of the cross-beam, which by its own nature presses in the middle where C is, the iron band attached to the upright beam supports the upright of the truss, while the ends AB of the cross-beam are fitted into it. The whole strength, therefore, of such a work consists in this, that the ends of the truss be well inserted into the cross-beam and the upright beam. For if the feet of the trusses are fixed in AB, it will not descend from the parts or ends D; and if those stand, the upright will stand too, so that, being hung from it, the cross-beam tied to it by the iron band will by no means hang down. Thus the whole frame will stand, and, being most strongly joined together by its own force, will bear the weight of the entire roof. But since it commonly happens that trusses, weakened by too great a length, yield sometimes to their own burden and sometimes to an external one, bending downward, architects remedy this weakness by placing braces underneath on both sides, as supports. Let the weak trusses be AB, AC on either side, with the middle upright beam, which we call the Monachus, AD. The middle parts of the trusses E, F, at the points EF, being therefore farthest from the ends, are greatly strained. Therefore, with small upright supports EH, FI placed beneath on both sides, they set against their heads E, F two small braces, resisting themselves at the foot of the upright at D. Once thus arranged, neither E nor F can descend toward H, I. For let any point G be taken between EH, and BG, DG be connected; BG and DG will be shorter than BE and ED, by proposition 21 of the first elements. Then the point E will be brought into G, when BE and ED are made into BG and DG, which, because B and D do not yield, and because the corresponding parts have been made shorter than themselves

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IN MECHAN. ARIST. PROBL 104 bus BE, ED, prorsus est impossibile. stabunt igitur in eo- 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, potentia 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 reuulsio: 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 bus BE, ED, is altogether impossible. Therefore the bars AB, AC will stand in their straightness, and will not spread apart, as was sought to be done. Here, however, those deserve to be condemned who do not insert the feet of the cauteries into the ends of the transverse beam, but instead, by means of some transverse piece, connect the middle cauteries on either side, in the manner of the element A, which fastening they call a capra. Let the cauteries then be AB, AC on either side, whose middle parts are joined by a transverse piece. DE. I say therefore that this fastening is greatly to be disapproved. For AB, AC are levers, whose common support is A, and the powers diverging on either side at B, C, with the weights DE between the support and the powers. Since therefore, as DH is to AB, so is the power at B to the weight at D, a small power in D will draw and overcome the weight; and from this the easiest possible tearing away of the transverse piece from the very braces on either side will occur. And since the center is indeed A, there is a slight divergence in D, E, but a very great one in BC, since the parts are farthest from the center A itself. Therefore the free parts kick near the feet of the cauteries and the upper walls themselves, and by their kicking drive them on, to no small harm of the whole work. Thus much by us concerning beam-work; now we shall turn our pen to the nature of vaults and chambers, for usefulness, indeed necessity itself, urges this. Few before us have treated these matters, and certainly, if they are not well known or are neglected, architects and craftsmen very often fall into enormous and inexplicable difficulties. We say therefore, first, that baked bricks, and stones not cut into wedges and arranged in a straight line, do not stand. Let the walls on either side be AC, BD. Let the line CD be drawn parallel to the horizon, along which bricks or stones not cut into wedges are set in order, EF. We say, the supports 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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EXERCISES. 105 ment, that is, preventing the bricks themselves from falling. For let AC be produced to G, and BD to H, and with CG, DH themselves equal make CI, DK, and let the straight line IK be joined; therefore the space GD will be similar to, and indeed equal to, the space CK, which, since this is so, there is nothing to prevent the whole mass of the side GD from being transferred into the space CK and from collapsing. But if the wedges themselves, or the bricks arranged in a wedge-like manner, are so placed as to tend toward one center, even though they are laid along a straight line, they will not slide down, but will stand; which we shall thus show. Let the wedges or bricks arranged in a wedge-like manner ABCD be directed toward the center, or common point E; let CAE, DBE be drawn, and let the walls on either side resisting the weight CL, DM be such. Let a perpendicular be let fall, cutting AB at G toward the center of the world FGE. Then let GK be made equal to GF, and through K let a line parallel to AGB itself be drawn, HKI enclosing the space AHIB. Since therefore, as EC is to EA, so CD is to AB, by Proposition 4, Book 6, CD will be greater than AB itself, and for the same reason AB greater than HI itself, and thus the space ABDC greater than the space AHIB. Therefore line CD cannot be made into AB, nor AB into HI, nor can the whole space CABD be transferred into the space AHIB without a given cause (which is repugnant to nature itself).

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106 IN MECHAN. ARIST. PROBL. gnat) corporum penetratione. Stabunt ergo cunei, quod fuerat demonstrandum. Verumenimuero, 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 apterur 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 the penetration of bodies. Therefore the wedges will stand, which was to be demonstrated. Nevertheless, this structure is weak, and the more weak the wider the void has been, and the smaller the height of the wedges. For it suffers the same thing as architraves in the kind of Araeostylos, which, as Vitruvius writes, book 3, ch. 2, break because of the great distance of the intervals. It also has this defect, that the wedges, thus arranged, resting upon their own weight, thrust violently from both sides. Yet it can be useful for the lintels of doors and windows that are in the middle of walls, and are opened in a moderate opening. But if the vault be curved to a smaller portion of the circle, that structure indeed will be better than the one we have spoken of; yet not altogether without fault. Since therefore, as EM is to EA, so MGN is to AIB, the line MGN will be greater than the line AIB; for which reason it is impossible for AIB to be fitted, and to descend into its place. Therefore it will stand, the supports on either side not yielding. But we shall show clearly how this form strongly thrusts the places on which it rests. 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, BLH, those indeed being 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 outside the supports, namely the whole AKLB, which therefore, by its own weight tending downward and opening under the vaults on both sides, becomes weak and easily causes damage. Of a similar nature is that species which is made either from the middle or from a lesser ellipse, according to the greater diameter, as a segment. This is nevertheless more useful, especially around the vaults, because the parts are more upright and more supported than that circular one of which we have spoken. Toward the middle, however, it may seem weaker, since the ellipse there bends less than the circle. But that form with which the Barbarians, who invaded Italy as the empire declined, were astonishingly delighted, and by which they disfigured the good and most correct ancient method of building, consists of two portions of a circle; for which reason Albertus, book 3, calls these arches composite. They are drawn in this way: the chord being divided into three equal parts, the compass point is placed at one of the division points and a part of the circle is described; then, with the compass point set 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, from the chord divided into three parts, the arch does not become round, but ends in an acute angle from two portions of a circle. Let there therefore be walls AC, BD, in which on both sides the vaults KA, BI rest. Let the chord AB, parallel to the horizon, be drawn, and divided into three equal parts by the points E, F; then, with the centers EF, portions of circles are described, on this side AG, HK, and on the other BG, O 2 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 constructed. This kind is indeed useful, though not beautiful, because, since it does not press violently, it repels on both sides, and at the top it is suited to bearing great loads. Let CH be produced to N, and DB to O, and thus let the center of gravity of AG be in L, and that of BG in M. Since therefore these centers, because of the elevated arrangement of the parts, are very close to the lines AN, BO, the supports are most strongly upheld, and they press downward rather than laterally. If, however, it has any fault, it is that the centers of gravity, having moment, exert force toward the inner part at PQ; and unless the parts are compressed by a great superimposed weight, the parts around HG, being driven upward, fall away with some straightness acquired for themselves, namely, a separation of the joined parts having been made around L, M. Having thus explained this matter, 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 there be an empty space ABCD, enclosed by walls on either side. Through the tops of the walls let a straight line AD be drawn, parallel to the horizon; this being bisected in E, with the same center E and with radius EA let the semicircle AFD be described, namely 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 pondere 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 sit 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 dubitandu[m] O 3 esset,

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EXERCISES. 109 then, from the same center, with the distance EG let the corresponding curved part of this vault be described GH. After this, with the lines BH, CD extended to OP, let the whole vault be divided into three equal parts AGKM, MNLK, and NDIL, and let the points KME, LNE be joined; and let the centers of gravity of those parts be QRS. Now R is on the very perpendicular HE. Since therefore the parts AGKM, DILN, whose centers of gravity are QS, are in the very lines of the supports OH PD, these parts will stand of their own accord, the supports sustaining them. But the middle part KMNL, tending downward by the line HE itself toward the center of gravity, if it should lean a little or if the parts on either side, AGKM and DILN, should yield, since it is the farthest from the supports, will be carried downward with great force by its own weight. Which of these parts in such semicircular vaults are the more stable, and which are liable to fall, is clearly evident from what we have said. Moreover, the reason why the vault stands while the leaning parts remain is that the outer parts GK, KL, LI are greater than the lower and opposite parts AM, MN, NG, as we have shown above. But if there is any defect in this type, it is indeed that the upper part KMNL, tending downward, by its great force repels the parts that are on either side, from which comes the separation of the solid parts, and thence collapse. In order to meet this difficulty, more skilled architects have devised many remedies. First, indeed, 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 parastatae, as in the figure TX, VY. Moreover, iron nails driven from one arch to another and fastened on both sides most strongly connect the opposite parts, which, as our architects say, restrain the “kicking” feet of the vault, and prevent the solid work from being disjoined. In this case it would be doubtful,

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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 & inuiectissimo 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 trabe 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. whether the key ought to be in the best place, that is, through the center? And surely it seems so, since the force acting upon the abutment is greater there. 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 either side rest upon the abutments, for this reason, that the first impulse from the middle part which hangs over is made there. Yet rarely do good architects usually fit them in that place, because keys of this sort detract from even the most beautiful buildings. Hence it comes that Lucianus that Benuerardus of Laurana, a Dalmatian, is never praised enough, who placed them nowhere visible in that admirable hall of Urbino, which he built for Federico Feltrio, the most fortunate and equally most invincible Duke. Thirdly, finally, they remedy this weakness in the manner to be seen in the following figure, in which the empty space ADBC, the walls on either side AF, BH, and the vault FGH. Thus while they are constructing the walls, they insert upright beams, of oak or other very strong material, into them, such as IFK LHM, of such height that they exceed the summit of the future vault. For when the vault has been completed, though not yet dismantled, they connect by keys in points I, L, the transverse beam, projecting a little from the top ridge of the vault, to the upright beams, then at the 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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EXERCITATIONES. 111 the cross-beam, and with iron uprights, they fasten them with nails. When these have thus been fitted together, the vault having been strongly pressed in G, and the supports F, H being driven outward, the space AB does not become greater. For when the supports are driven back, the walls too must necessarily be pushed outward, and with the walls the inserted beams IK, LM. But they cannot splay apart unless they drag along the points PQ, which cannot happen, because at the points N, O they are strongly held apart. Therefore, since the space AB is not widened, no loosening of the vault itself occurs, which in any case was sought from the beginning as the intended end. But someone may say, “Will not the transverse beam hang in the very drawing apart of the uprights, pressed at the points N, O?” Either very little, or nothing. For since PQ is near the points FH, which, together with the uprights, are separated from the wall, there is great resistance on both sides in them. Since, therefore, matters stand thus, the Architects observed that, because of the enormous weight, vaults labor in that third part which is at the top; as much solidity is added to the third parts on either side, so much is usually taken away from the uppermost 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 CE, DG, thick indeed too, while the top part EFG is more slender than the others. Therefore, with less weight pressing in F, the pressure on the supports becomes less, or if any pressure does arise, it is not ineffectually sustained by the solidity of the parts ACE, BDG. Cete-

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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; fornicis partibus no 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 quasolutæ 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 quadranti- bus 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 QIRM ad infe-

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112 IN MECHAN. ARIST. PROBL. Moreover, this passage reminds us that we should bring forward something about the collapses of vaults: for when the causes of diseases are known, skilled physicians are more easily accustomed to apply remedies. Let there be, then, a semicircular vault ABC, whose center is E, and the perpendicular through the center DBE, the diameter of the semicircle ABC, resting at both ends on A and C. So if there is no repulsion of the supports, the vault will stand; but if there is, it will collapse. Let them therefore be driven toward the outer parts, as in the second figure, H toward F, and C toward G; from this thrust, since a greater space is produced than that which was filled by the entire vault, now that the parts of the vault are drawn apart on both sides it is not filled. Therefore the greater space thus made is divided into three parts, of which the two on either side are filled by the parts of the vault, while the third, which is in the middle, is filled by air introduced so that no vacuum may be left, as can be seen in the figure, in which the separated parts of the vault are HIKF, PMNG, and the middle air filling the space IKMN. Let the individual quadrants FK, GN be divided into three parts, of which the two on either side are FQ, GR, and from the centers, which have been formed by the separated quadrants, at ST let straight lines SQV, TRX be drawn. Since therefore the third parts on both sides, VIKQ, MNRX, pressed down by their own weight, have no support by which they may be upheld, they will indeed fall. But let the straight lines QI, RM be drawn, making with QV, RX equal angles VQI, MRX. Thus, by the centers QR, the parts QIRM to the...

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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 made, where QZ, RZ. But if QI, RM, the perpendiculars drawn from the points QR to the perpendicular DE, are greater, they will meet somewhere on the perpendicular itself, and one will support the other; if they are equal, they will touch and yet a collapse will still occur; if they are smaller, they will neither touch one another, and, with nothing preventing it, they will fall downward. But let them touch at point Z. By what manner, then, arched structures resting on supports giving way in the middle of the opening are loosened and cause a collapse, is clear from what has been stated above. From what has been demonstrated, as it were as a corollary, we have that arches, the thicker they are, for a given separation of the supports, are less liable to collapse than thinner ones; that is, the thicker arches require a greater opening in order 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, a part ABCD of a thicker arch, and EFCD of a thinner arch, about the same center R. Let RM be drawn, cutting CD at G, EF at H, AB at M. Therefore, with center G, the overturning of the portions MD, HD will occur. Let GA and GE be drawn, and AD produced to N; to AN, as a perpendicular, draw GN. Since therefore GE falls within triangle AGN, it will be, by Proposition 21 of Book 1 of the Elements, that GA is greater than GE. Therefore, as the larger portion MD of the arch falls, the straight line GA, with center G, will describe the portion AI; meanwhile, the smaller line GE will describe EL. But the falling angle A meets the opposite portion at the perpendicular IK, at point I, the angle of the opposite portion O; but to E, falling through EL, point P will not meet, nor will the falling line through Pq, because neither of them reaches the perpendicular Ik. Therefore the thinner arch-

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114 IN MECHAN. ARIST. PROBL. cis partes è suis locis auulsæ 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 quam 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. Quarit Aristoteles, Cur paruo existente cuneo magnascindantur 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, will by the same opening bring about a collapse, which does not happen in 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 both sides are ABCD and EFGH, the upper ends of the larger walls AD and EH, the middle parts of the walls KL, and of the vaults the upper part DIE and the middle part kML. I say that the upper walls are more easily forced away around DE than in the middle around KL. For the walls BA and GH are, as it were, certain levers, to whose extremities the force is applied from points farthest from the supports BG, that is, the repulsion of the vault DIE itself pressing upon DE; but the part from the support to the force AB is longer, namely Bk. Therefore, given equal force, that which is at D will act more than that which is 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 on the lower wall kBC, makes it firmer by its own heaviness and less subject to blows. For that which is heavy is harder to move than that which is light, as we showed 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?

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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- natisunt. 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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EXERCITATIONES. 115 has greatly fatigued the minds of the most distinguished men. Of these, Aristotle among the ancients, and Guid. V bald. among the more recent, attempted, as to the nature of the wedge, to reduce the wedge itself to a lever, lest it should be thought that nothing in Mechanics could be reduced to the lever. But we, contending for the truth, if we have not of our own accord passed over to the opinion of these men, shall be judged worthy of much pardon by no unfair judge. G. V-bald. explains Aristotle’s meaning clearly and at length in the Mechan. where he treats especially of the Wedge. Let something, therefore, ABCD, be to be cut apart: the wedge EFG, whose part HFI inserted into the cut HI, having therefore a forceful blow delivered on EG, it will happen that when EG has been in NO, H will be where N is, A where P is, likewise I where O is, but D where Q is; and the cut NSO will be made, namely equal to the whole wedge EFG. Aristotle therefore wishes 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; 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. V bald. enters another path. For either EHF are indeed levers, having a common support at F, but the moving powers in EG. The weights on both sides are between the supports and the powers, where HI is, and it is the same as if EF and GF, considered apart from the wedge at point F, would mutually, being supported and pulled apart, drive the weights H into NP and I into O, Q. But in truth, since the angle of the wedge does not change, and the vertex itself offers absolutely no use of the center, nor are its sides, pulled apart on both sides, drawn off to contrary parts, P 2

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116 IN MECHAN. ARIST. PROBL. cuntur, vectes in cuneo 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. Vbald. ijs quæ de diuaricatis vectibus in medium adduxerat non acquiescens alias quærit causas, 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. it seems to me foreign to the truth to consider levers in this wedge in that way. But that Aristotle’s solution is false is clearly evident. For how can F from the support H drive the part OS of the wood, and likewise F from the support I drive the opposite part NS, if the extremities of the levers at F, contending with one another, do not in any way act upon one another, when there is an impediment? And indeed the falsity of this opinion is evident from the fact that we see the parts of the matter, when split, in the very act of splitting, separated by the distraction caused by the point of the wedge, not touching at all. But that such operations are performed by contact is unknown to no one. That solution, therefore, in my judgment, seems altogether unworthy of so great a Philosopher. Moreover G. Vbald., not being satisfied with the things he had brought forward concerning divergent levers, seeks other causes why a wedge with a smaller angle splits more strongly. And he tries to demonstrate this from a certain lemma, and the figure is more or less as follows. Let there be the wedge ABC, and likewise another DEF. He therefore demonstrated from the assumption that the sharper the angle BIM may be, the more easily weights are moved, and therefore that by the lever AB the weight I is more easily moved than by the lever DE the weight Q. Ingenious indeed. But this has great difficulty for me. For if AB is related to BI as DE is to EQ (for DE and EQ themselves are assumed equal), then the same equal force, or an equal force, will move the weights I and Q equally. which is altogether contrary to the very demonstration itself. Nor indeed do I

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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 cunco, nec quidem mente concipi posse, clarè paret. 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, area 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, it seems to follow from this judgment, because, from Pappus, those things which are moved on inclined planes are reduced to the balance. For the reason is very different, since the weights which are moved on inclined planes have fixed supports and determinate proportions both of the arms and of the weights, whereas in the wedge all these things, and indeed the very thing itself, cannot even be conceived in thought, as is clearly apparent. Having therefore considered these difficulties, we know for certain that the force of the wedge must be referred to other principles. Let us therefore proceed in this way. By means of a wedge, indeed, a thing is certainly divided. Moreover, those things which are naturally suited to division 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 these are considered by the mathematician without matter. But concerning division that is made from a point, the mechanician does nothing, since he indeed uses bodies, but to whose nature the point, whose parts are none, is not brought. But bodies are divided not only by lines and surfaces, but also by bodies; which is true, and we shall easily teach that 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 to the nature of a line. Such are those rounded points with which we make holes, and for that reason in the vernacular we call them Pantirolos. Likewise sewing needles, and the rest which no less than a line terminate in a point, and contain within themselves some imaginary line as an axis ending in that point. To the line are also referred elongated, edged points, and slender points such as awls, nails, swords, daggers, and the like, which, when driven in, make a strong separation of parts; to refer them to the nature of a wedge would seem great folly. And then, the more these bodies approach the nature of a line, 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. And this same thing is easy to recognize in things produced not by art, but by nature itself. For who does not observe how powerfully the gnat, a very weak animal, and by the smallness it has, pierces the skins of men and other animals with its pointed proboscis? This surely is for no other reason than that it comes as close as possible to the subtlety of an imaginary line. Wasps too, bees, and scorpions use these stings as if they were linear wedges. Nor does it matter, as we have said, whether they are sided like awls and nails, or round, and whether they have more or fewer sides, so long as they end in a point and a pointed edge. There is moreover another kind of wedge, which savors of the nature of a surface, since its edge ends in a line, which is the boundary of a surface. For this reason all those things are referred here which cut by the edge itself, such as properly so-called wedges, about which this passage treats, knives, swords, axes, hatchets, broad chisels, and others of the same kind, by which bodies are cut with an edge. Some add saws to these, but we do not wholly agree. For they divide in another way, just as files also are accustomed to do; for by wearing away, not by cutting, they divide and subdue the hardness of iron, wood, and marble. These matters therefore being considered, if there were given from some unbreakable material a wedge which approached as closely as possible the nature of a surface, it would with little labor most powerfully split the toughest woods; and therefore those blades are best by which things are divided, which more closely approach the nature of a surface. From all these things, unless we are mistaken, it plainly appears why sharper wedges with a more acute angle split more easily than blunter ones, a reason which is far removed from that by which nearly all the rest have hitherto insisted on referring the wedge to the nature of the lever. Furthermore, Nature, most skillful of all, has recognized both kinds of wedges which we have mentioned, and therefore, since things are accomplished either by striking or by piercing or by cutting, [she has assigned] a threefold 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 πεμνης, 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, quilibet 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 the teeth given by nature: Molars, which are called Maxillaries, with which food is crushed, Canines, by which piercing is made, Front teeth, with which food is cut; for this reason the Greeks call them πεμνης, that is, cutters. Molars KK, Canines L, L, cutting teeth M. The round and linear wedge AB, in which the axis is the line, to whose nature the wedge AB approaches; 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 laterally linear wedge, the nail HI. But all wedges act in a twofold manner: for either they are driven in by a hammer, as in those cases where wood is split and as is usually done with chisels, or by thrust and pressure, as happens with swords, daggers, engravers’ gravers, bodkins, and other such tools. Some also are such that they are not driven in by any blow of a hammer, but have the hammer joined to them, such as axes, spades, adzes, and things like these, which, by their own striking, are inserted into the things to be cut and penetrate powerfully. But here we shall refrain from saying something about the force and efficacy of the blow or percussion, since we shall speak on that matter in the following question. We could have added much here concerning the screw, since the screw is a wedge wrapped around a cylinder, which indeed works very powerfully, by the force of the hammer, but joined to itself by the virtue of the lever, and has been applied to countless uses. However, since G. V. Baldus discusses 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 facilior, 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 intentionally omit this discussion; and for this reason as well, namely, that Aristotle said nothing about the screw, as if he had not known it. But in the act of splitting, which is done by a wedge, we may nevertheless consider the lever in another way: not indeed in the wedge itself, but in the very thing that is being split. For let there be something splittable, ABCD, to one of whose extremities, say BD, a wedge EFG is driven, and let the split be made along the length according to the line EH. Thus, when the wedge enters and separates the parts B, it will be driven out at I, and D at K. Let there therefore be the parts of the split material AIBH, CKDH, as it were two levers, whose supports on either side in the body itself are L and M, the forces widening BD on both sides, but the weight being the resistance of the material, at the place of separation where N is. Let NL be drawn; therefore, the greater ratio BN has to LN, the more easily will the resistance which is at N be overcome. But the support is continually changed in the splitting itself, and with the support the ratio itself. For when the split extends to O, the support is at P. In which case the splitting is easier, because BO has a greater ratio to OP than BN has to NL. This, moreover, is experienced by woodcutters, who at the first blows, while the little axe has not yet been well driven in and no notable split has yet been made, feel difficulty; but as soon as separation has once been made, the separation of the whole material becomes little by little very easy. We also experience the same thing without the use of a wedge, when we split a stick or something of that kind with our hands pulled apart. For at first we feel difficulty, then from the ratio we mentioned the splitting itself becomes quite 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 fit 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 exiguapotentia in- gentiamoueantur 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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EXERCITATIONES. 121 We also use a wedge-shaped lever for splitting and opening: for, when the splitting wedge has been driven in, and that by hand and mallet, then by pressure from the other end, a strong separation of the continuous body is effected by the force of the lever. The splittable material AB, the chisel or wedge-shaped lever CD, whose fulcrum is E, the load indeed at C, the power at D; in which case, the greater the proportion of DE to EC, the easier and more manageable is the splitting itself. QUESTION XVIII. Here Aristotle asks: Why are heavy weights moved by a small force through pulleys? On the pulley Pappus, and the ancients; among the more recent writers, G. V. Baldus has examined everything most excellently in the Mechanics. Yet for the moment, after those most distinguished men, we shall bring forth from our own store something that has the flavor of novelty and subtlety. And indeed to add to discoveries is an easy matter, but to add something to what has already been discovered is not so easy to discover. But let us first recall the philosopher’s own words to the balance. He states the question thus: Why is it that if someone, arranging two pulleys, with two hooks joined to each other in an opposite manner according to the pulley’s circle, has passed a rope around them, one end of which is attached to timbers, while the other rests on the pulleys, and has begun to pull from the beginning of the rope, he draws great weights, though he may be of little strength? A most obscure explanation; and if the matter were not commonly known of itself, and if Vitruvius and the mechanicians had not discussed it, it would certainly be difficult to grasp the meaning from his words. Q Timbers

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122 IN MECHAN. ARIST. PROBL. Tigna sanè vocasse videtur ea ligna, quæ à Vitruvio Rechami dicuntur, in quibus nempe ipsi inferuntur 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 pieces of wood tigna, which Vitruvius calls Rechami, in which indeed the little wheels themselves are inserted. Although Picolomini seems to think something else about these tigna. The Greek reading has for tigna , that is, ligna; likewise where Leoniceno’s translation reads, joined together with one another, the text has , that is, they run together from opposite directions. Certainly I would render the whole passage thus: Why, if someone were to make two pulleys on two pieces of wood running toward each other from opposite sides, and were to place a rope around those pulleys, one end of which is attached to one of the pieces of wood, and the other is joined to the pulleys, or is fastened to them, if someone should pull one end of the rope, does he pull great weights, although the force of the pull is slight? We shall clarify the words by the figure, and the figure itself by the words. Let there be two pieces of wood running toward each other from opposite sides, in which are the pulleys, that is, the circles AB, and the hauling rope DABC, one end of which is tied to the piece of wood of pulley A, where C is. Pulley A is fixed in a stable place, where E is. A weight F is hung on the other piece of wood of the pulley. If therefore the rope DABC is pulled, the weight F is raised and drawn. From this it is clearly evident that the Philosopher proposed a pulley provided with only two wheels, which was certainly enough for the explanation. But he says that it is moved more easily by a lever than by the hand. The pulley, however (that is, the little wheel; for thus it must be understood), is a lever, or works by the power of a lever. And he seems to argue thus. 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 is raised with more than double speed. The sum of what has been said is that 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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EXERCITATIONES. 123 that can be lifted, which is certainly true. Yet we shall consider some things. 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 more difficultly by the lever than by the hand. The same will also happen if we use the lever in the manner observed by Guidus V. Bald. Tract. de Vecte prop. 3, namely with the supporting power 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 spoken more suitably if he had said to the balance; for a lever, however it is divided from the fulcrum, is not always so. A balance, however, as also happens with circles from the center, is always divided into two equal parts. In addition, it seems that he refers the power of pulleys to the multiplicity of the little wheels. For, he says, if a weight is easily drawn by one pulley, it will be much more strongly so by several. Nevertheless, we shall show that this does not happen absolutely from the multiplication of the wheels. Let there be two opposite straight lines, namely the beams AB, CD, equally distant from one another and fixed. To the upper one let three pulleys be suspended from the points E, F, G, namely ML, PQ, TV; to the lower one two from the points IH, namely NO, RS. There will therefore be in all five; let the hauling rope KLMNOP QRSTVX be passed through them, from whose end 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 B A 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 rope be drawn in K. I say that by the multiplication of the pulleys, the weight being pulled is in no way diminished. Let the diameters of the pulleys be LM, NO, PQ, RS, TV, and let the power be applied at S. It will therefore be the same as if it were to support a weight equal to X, for the semidiameters of the pulleys TV are equal. Let the power be transferred to q, and so on successively until it comes to K, where the rope itself begins. Therefore it is the same, the equality of the semidiameters always being 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 so arranged, the multiplication of the pulleys contributes nothing to ease. Another reason must therefore be sought, which Aristotle does not seem to have explained sufficiently. We shall prove, however, that there is no diminution of the weights from the upper pulleys, but that the whole force resides in the lower ones. But in the meantime let us propose something that is relevant to the matter. Let there be a point A, to which straight lines BAC are suspended, divided indeed at A; let B be the head of the line BA, and C the head of CA. Now let them be understood as united at A, and let there be a single line hanging from the point A like a cord, BAC. Let a weight B be attached to the head B. And to the head C, a weight C, equal to one another. Therefore the power at A will sustain the two weights BC. But the weights, because they are equal, will equally balance one another. And if B is called the power sustaining the weight C, or C the power sustaining the weight D, or two powers equal to one another, it makes no difference. For however this may be, equilibrium will result. We thus have from these things enough to support a 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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EXERCISES. 125 supposing a force equal to the weight itself to be required by the hanging load. Let another straight line DEF now be conceived in the mind, whose entire length, if it were extended, would be DE, EF. Let a weight equal to the other weights B or C be attached at E; and let there be two forces D, F sustaining the weight E. Therefore each will sustain half of the weight E; but the force that sustained the weight B, in C was equal to B itself, where the attachment of the weight was in the upper part at A; here, however, where the attachment is in the lower part, both forces sustain half of the attached weight. We therefore see that this mode of attachment does not lessen the weight at all, but that one divides the weight itself, split into two parts, among the sustaining forces. These matters, having considered them in lines by means of mathematical abstraction, let us now weigh the same mechanically. Let there be therefore the point A, as in the following figure, a nail or peg, to which is attached the string BAC, and at the ends of the string weights BC; let there also be an annulus D, through which is passed the string EDF. To the annulus, moreover, is joined the weight G. These things therefore being thus arranged, the same results will be demonstrated as above, namely, that equilibrium must be made so that B and C are equal, then the forces which are in EF sustain the weight G divided between them. Furthermore, the Mechanics wishing to 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- psiùs 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. But to move the cords around the peg and the ring for raising and lowering weights, it was indeed inconvenient for them, the peg and ring making the motion difficult. Wherefore, in order to meet the difficulty, they placed a small wheel around the peg itself in the place of the peg, and likewise fitted a small wheel in the place of the ring. In doing these things they did not change the nature of the matter itself, but, as we said, they provided for themselves the greatest convenience and ease from the pulleys. From these principles the whole theory of pulleys depends, which nevertheless can also be examined from another consideration tending to the same end, as indeed the ancients did, and as did he who best imitated the ancients, Guid. V baldus. We have certainly seen that by the power which is at B a weight equal is sustained in C, but the power which is at E sustains half the weight which is at G. We therefore, adhering to the same principles and adding a balance or lever, divided in two, 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 AC is attached, whose end C is fastened to a certain lever CD, divided in the middle where E is; then to the other end D of that same lever a cord DG is attached, and from the point E a weight F is hung, say of a thousand pounds. Then at the point G, in the middle of the lever HI, let the rope DG be tied, and from the other end of the lever, with a rope HK fastened, let it be secured to a fixed place at K; and from the other end of the lever where I is, to the middle of the lever MN where L is, let the rope IL be attached; then from the end M of the lever, let the rope MO be secured to a fixed place at O, and to the other end N the rope NP, which shall be fastened to the middle of the lever QR at P; and from Q let the rope QS be secured to a fixed place at S, and to the other end R of the lever let the rope RT be attached, to which indeed the sustaining power is applied at T. I therefore say that, in these matters,

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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 tantum

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EXERCISES. 127 things being thus arranged, the power at T is related to the weight F as one to sixteen, that is, in the proportion of the sixteenth part. Here there are four levers in the 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, sustains the weight F, the other half of the weight will be sustained by C, and the remaining part by D. Thus 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 sustain five hundred. Therefore each one will sustain two hundred and fifty; but this whole amount is divided in two between the powers O, that is, M, and P, that is, H. Thus there will be on each side one hundred and twenty-five. But that sum is again divided in two, that is, between the powers S, that is, Q, and T, that is, R, so that each sustains sixty-two and a half. But this number stands to a thousand in the same relation as one to sixteen. Hence we infer that the whole weight is divided among the fixed points, namely A, K, O, S, and the power itself which sustains at T, and that fifteen parts of the whole weight are at the fixed points themselves, while the power T is 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/1 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 superq[ue] 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 to be commended. So if from the point V AB were suspended, in X the power, which in X would sustain a thousand, less sixty-two and a half, would indeed be sustained by the power in T; but if another little wheel were added, and there were five, the power in 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; and if likewise another link were added, the power in T would sustain the sixtieth part of the whole weight, that is, fifteen pounds and one-half 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; and the upper are added not of necessity, but for convenience: for without the upper ones the traction and elevation of the weight itself could not be made by a single continuous rope. Thus far, therefore, let this much suffice us, after others, to have considered concerning the nature and force of the pulley. QUESTION XIX. The Philosopher inquires why, if someone place a large axe upon a log and on top of it add a great weight, it does not divide the wood in any part worth considering; but if he raise the axe and strike it, it splits the wood, although otherwise that which strikes has much less weight than that which lies above and presses down? Aristotle, unless we are mistaken, passes over the matter, so to speak, in a shorter and more general way. Namely, why motion adds weight to weight, and why a moved thing acts more effectively from motion than from unmoved weight. He solves it thus. Is it, he says, because all things come to pass with motion, and the heavy body itself takes on more of heaviness’ motion while it is moving than while it is at rest? Therefore, when the inherent motion of the heavy body is pressed upon, it is not moved; but 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 cuneo iterat, nempe ad vectem eius looperationem referri superius confutauimus. Porrò effectus huius, de quo agitur, disputatio illuc spectat, videlicet ad cadentium atque proiectorum naturam. Ad maiorem autem rei euidentiam hæc addimus. Esto libra AB, cuius centrum C, libra ta æqualibus ponderibus DE, apponatur ponderi E pondus F, item ponderi D pondus G ipsi ponderi F æquale, æquilibrabit itidem, 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 considerantur 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 maior. quod vtique efficacius fieret si pondus G non dimittetur modo remoto prohibente, sed proijceretur. Tunc enim tria concurrerent, grauitas naturalis, grauitas acquisita 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, naturalis nempe grauitatis, & acquisitæ. Cæterùm cur mallei & securis istus sit violentissimus, ideo sit quod non ex vnico neque duplici, sed ex triplici grauitate operetur. Esto enim securis A, cuius manubrium AB, brachium vero securi vtentis BC, erit igitur C R locus

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EXERCISES. 129 Will you strike the one which is being cut? This is indeed excellent; but the rest, which he repeats concerning the wedge, namely that it is to be referred to the operation of the lever, we have already refuted above. Moreover, the discussion of the effect in question looks to this, namely to the nature of falling and thrown bodies. For a clearer understanding of the matter, we add the following. Let there be a balance AB, whose center is C, with equal weights DE; if to the weight E be added the weight F, and likewise to the weight D the weight G, equal to the weight F itself, it will likewise remain in equilibrium. But if the weight G be not simply added, but be let fall from H onto the scale A, then certainly it will not remain in equilibrium, but will depress the balance. For in a weight that has been let fall two weights are considered: namely the natural weight, and that which, by motion imparted to the moving body, has been acquired by the weight. Thus the greater the motion has been, as if it were to fall from I, the greater the gravity will become from the greater motion. This would certainly be made more effective if the weight G were not merely let go, with the obstacle removed, but were thrown. For then three things would concur: natural gravity, acquired gravity from natural motion, and that which is added to the natural by violence. Therefore the weight placed upon the axe and the axe’s own natural gravity operate only by natural gravity, and therefore less effectively. To this belong almost the things we mentioned at the beginning concerning the two centers, namely of natural gravity and acquired gravity. Furthermore, the reason why this hammer and axe are most violent is that they act not from a single gravity, nor a double one, but from a triple gravity. For let there be an axe A, whose handle is AB, and the arm of the axe used by the user BC; therefore C will be the place

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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 is joined to the shoulder, the center of its motion, but he who strikes lifts up the axe, and drawing it back to the shoulders with all his force, from the center C swings the axe, describing the arc ADE, and making the blow at 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 upon it by the striker. Now both natural and violent motion become the stronger, the greater the space through which the thing moved is moved; and this especially when violence supports nature itself. Thus the blow is greater in E than in F, and in F greater than in 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, with forces equal, the more strongly does he strike for the same reason. Now an axe is either a wedged hammer, or a wedge fitted with a hammer-handled shaft. But whether the wedge, when struck by a hammer, or when moved with a handle, as in an axe, works more efficiently, with the edge and weight being equal, is difficult to determine. Certainly the splitting is done more strongly and more certainly by wedge and hammer, for this reason, that the wedge, once driven in, and not removed from there, meanwhile preserves 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. Tar- dior ergo ex A in D, velocior ex D, in F, velocissimus ex F in E; quædam quæ ad hanc rem faciunt, egregiè conside- rat Guid. V bald. in calce Tractatus, De Cunco; ipsum consule. Ad hæc succurrit nobis pulcherrima quæstio. Du- bitari enim potest, vtrum ictus ex ense efficacior sit à par- te quæ est circa aciem, aut circa medium ensem, vel pro- pe manubrium capulumue; etenim hinc inde sunt ra- tiones. Esto quidem ensis AB, cuius capulus A, spiculum ve- rò B, centrum grauitatis C, pars capulo proxima D. Libra- to 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 maioris 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 co- tius fiat grauitatis impressio, fieri autem validissimum in G, licet ibi motus sit tardior inde videtur, quod si conside- retur ensis vt vectis, cuius fulcimentum est A, potentia premens in B, ponderis vero loco resistentia rei quæ per- cutitur in D. Maior est autem proportio BA, ad AD, quam BA ad AC, & ideo violentior fieri pressio ex ictu in D, qua[m] in C. Hisce hoc pacto consideratis, putarem ictum effica- ciorem fieri in F ex medio C, quam ex extremis & oppo- sitis 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, when, once driven in, it is withdrawn to make a new blow. We also consider this: the motion of the axe in a circle, from A to D, is to be observed, that is, not according to nature, for what is heavy is carried upward; from D, however, to F it is mixed: but still closer to the natural is that which is from F to E. Therefore it is slower from A to D, faster from D to F, fastest from F to E; certain things that contribute to this matter are excellently considered by Guid. V. Bald. at the end of the Treatise De Cunco; consult him. To this there comes to our aid a most beautiful question. For it may be doubted whether a blow from a sword is more effective from the part that is near the edge, or near the middle of the sword, or near the hilt or handle; indeed, reasons may be given on both sides. Let there be a sword AB, whose hilt is A and point B, center of gravity C, the part nearest the hilt D. Then, when the sword is swung, 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 stronger in E, since, from the larger semidiameter AB, the portion BE is 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 of the weight is made from the center C itself; but it seems that it is made strongest in G, although there the motion is slower, for this reason, that if the sword is considered as a lever, whose support is A, the pressing force being in B, but in place of 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 made more violent in D than in C. Considering these things in this way, I would think that the blow is made more effective in F, from the middle C, than from the extremes and opposite parts EG. For although in B the speed is greater, there the weight is lacking. For if the sword is again considered as a lever, there will be R 2 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, the two supports sustaining the weight at C, where the center of gravity is. If therefore the distances BC, CA are equal, in B there will be half the weight of C; so much, therefore, as it prevails in velocity the blow in B takes away so much of the weight. But D partakes indeed more of the weight, but has the least velocity; in C, however, the velocity is medium, yet the whole impression of its weight is made from the center of gravity. Some, in this connection, so that the blade itself, which is farther from the hilt, might deal a most violent blow, pour into a channel hollowed from the handle to the point mercury, which by nature is indeed the heaviest and most mobile of substances; by which, in the sword’s descent, carried most swiftly to the point, it transfers there its whole weight, wherefore then, with both velocity and weight combining, the blow is made most violent and far strongest. QVAESTIO XX. It is asked why a balance with which meat is weighed, with a small appendage, weighs down heavy loads, although otherwise the whole balance-beam is half-formed, and on the other side alone there is the balance? The Philosopher solves it, saying that a balance is at once both a lever and a scale, and that the centers, or supports, of the scale 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 proportiones 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 where the suspension is. The weights, however, here and there in the scale and the appended weight, that is, in the place of equilibrium, the appended weight taking its place. But he reduces the demonstration to those things which he himself establishes as mechanical principles; namely, to the circle and the power of the circle. He says, therefore, that although the appended weight be of small weight, it is therefore equal in effect to a greater weight, because it is placed farther from the center, that is, from the very support. Whatever may be the case, however, that the balance is a lever is a matter most certainly established. Let there be therefore a balance AB, whose moving weight F, the support or center C, and the scale-pan suspended by a chain E, the space from the place of support to the appendage CF. But that from the support to the chain, from which the scale-pan is hung, AC. Let another support D also be understood, and let the space AD be greater than AC. Further, let the weight in E be to the weight of the appendage F, as the space CF to the space AC, in which case, the proportion of the weights and the arms being preserved by permutation, equilibrium will result. But if, the weights being thus arranged, it be again suspended at D, equilibrium will not arise, 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, when the suspension is made, the weight E will prevail over the weight F. Thus, if equilibrium is to be restored, it is necessary again to make the proportions of the arms, or spaces, equal to the proportions of the weights. Let therefore the appendage itself be transferred to B, the weight of the scale-pan meanwhile remaining unmoved, and let it be so that FC is to CA as BD is to DA. Then the balance will again stand, brought back to that state which I said— 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 of the proportion exchanged between arms and weights. We use balances with a double fulcrum, one nearer, the other farther from the pan, or from the place where the pan is hung; on the farther side we hang the heavier weights, and we weigh not by ounces and pounds, but by pounds only and half-pounds; and this side of the balance because it is divided less minutely, our people commonly call the Gross, that is, the coarse and rough side. But the other side, when the fulcrum is nearer to the place where the pan is suspended, and is divided by pounds, half-pounds, and ounces, on which indeed we hang smaller weights, because it contains a more exact division, is called the subtle side. Rightly therefore did the Philosopher say that there are more pounds in a balance, although this too may be said for the reason that, as many as are the appendages, the transfers from place to place, so many balances are made from the variation of proportions. And this indeed Aristotle seems to have thought. We could also use the balance in another way, namely with the suspending point fixed, but the fulcrum movable. Let there be a balance AB, whose pan C is hung in A, and the fixed suspending point D, hung in B. Let a weight E be placed on the pan C itself. There will thus be made a single body CEABD, consisting of pan, balance, and weights. Therefore this whole has its center of gravity, and where that is is unknown. But when it has been found, if the whole body is hung, the parts will weigh equally. Let it be hung, for example, in G, and let the center of gravity be in H. Since therefore H is outside the fulcrum 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, vltro citroq, 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 forcipis 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 congiuit. Certè vt sententia non sit contraria propositioni, Græca versio ita videtur concinnanda: Vel magis è manu libitur, mollis enim est digitorum caro, ferrum autem circumplectitur, & hæret magis. quicquid sit, Græcam lectionem contrarium ei quod quæritur,

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EXERCITATIONES. 135 the portion HI of the circle, described by the center of gravity in the actual descent. But if the center of gravity should be at K, since there also it is outside the support G, the part GB will descend, while in the meantime the center of gravity K describes the portion KL of a circle. Therefore if we draw or push the whole balance with weights to and fro, back and forth, the suspension-point remaining unmoved, at some time the support will 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 to the weights becomes the same, with the order reversed. But this method is not in use, because it is troublesome to move the scale or balance with the weights back and forth, a difficulty conveniently avoided by the mobility of the suspension-point. QVAESTIO XXI. It is asked, Why do surgeons extract teeth more easily when a forceps has been provided with weight on the tooth-side, than if they use the bare hand alone? Answer, Philosopher: Is it because from the hand, rather than from the tooth-forceps, the tooth slips away more easily? Or does this happen rather from the iron than from the fingers, since they do not grasp the tooth on every side, which the soft flesh of the fingers causes; for it clings and encloses more. This second reason seems to destroy the first, and to assert something altogether contrary to the opinion proposed in the problem. If you render the Greek word for word, it reads thus: Or is the iron itself more liable to slip from the hand, and does it not itself on every side (namely the tooth) enclose it, but the flesh of the fingers, since it is soft, clings more, and on every side adheres. Certainly, if the sense is not to be contrary to the proposition, the Greek version seems to have to be arranged thus: Or rather it slips from the hand, for the flesh of the fingers is soft, but the iron encloses and clings more. Whatever the case may be, the Greek reading, contrary to that which 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 dentiforcipsis 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 dentisforcipsis 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. ...to affirm it is certain. Picolomini says, “Therefore the soft flesh of the fingers,” he says, “is less suited to extraction, because it cannot grasp the whole tooth; iron does this most conveniently, because of its hardness and firmness.” He has rendered the sense from the meaning, where he could not do so from the words. Aristotle adds finally: “Or is it because the tooth-forceps are two contrary levers having a single fulcrum, namely the connection of the parts of the instrument itself? This then they use for extraction, so that they may move more easily.” The Philosopher presents the figure in this way. Let one end of the tooth-forceps be at A, the other end, which extracts, at B; one lever at ADF, the other lever at BCE; the fulcrum, however, at CGD, the connection at G. The tooth is the weight. Therefore, grasping with both levers, at B and F, they move it. So he says. Yet those who examine the matter itself more subtly seem to find it otherwise than he asserts. Indeed, that the arms of the tooth-forceps are levers, whose common fulcrum is in the very center, where the hinge is, no one will deny. But that the tooth is the weight, I would not say absolutely. For the weight here is properly the hardness itself of the tooth, whose resistance is overcome all the more easily, the greater the proportion is of the arms from the hand to the hinge, to that part which is from the hinge to the tooth. But the constriction of the tooth does nothing at all toward breaking off for extraction; rather, what the tooth-forceps achieves is by the length of its arms, namely, that it powerfully compresses the opposite levers and makes extraction convenient and easy. For the whole tooth-forceps does not here act as a single lever, as happens in the forceps which we call Tenaleae, with which nails are pulled out of boards, concerning which 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 quidem A, cuius diameter BC, longitudo vsque ad extremas radices CD, pars dentiforcipis breuior CG, longior BG. Fit ergo vectis BCD, habens fulcimentum in C. Den- te igitur apprehenso in BC, & manu dentiforcipe ceu vecte ad inferiora compresso 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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EXERCITATIONS. 137 If the extraction of a tooth from the dentiforceps is reduced to the lever, it must be considered more carefully, for the matter is not self-evident. We therefore say that both the tooth itself and the dentiforceps are levers, though in various ways and certainly quite different ones. For the tooth is a lever of that kind, namely, whose fulcrum is at the angle; in which case, among the parts of the Dentiforceps by which the tooth is grasped, the longer part takes the place of the moving power, the shorter part serves as the fulcrum, while the resistance of the tooth represents the weight. Suppose the tooth is A, whose diameter is BC, its length as far as the extreme roots CD, the shorter part of the dentiforceps CG, the longer BG. Thus BCD becomes a lever, having its fulcrum at C. Therefore, when the tooth is grasped in BC, and the hand compresses the dentiforceps downward like a lever at C, the fulcrum, or center, is made. For while point C remains fixed, and the power acting at B draws, the motion of B itself is produced through the portion of the circle BE; and of the root D, motion is produced through DF; and hence the extraction of the tooth is easy. Having considered these things, so that we may reduce the matter 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 fulcrum to the power to that which is from the fulcrum 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, that teeth once loosened are more easily drawn out by hand than by an instrument, he proves in no way. But I think that the words found above belong here, namely fer- rum 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 fier 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, one cannot everywhere grasp the tooth, because of the soft flesh of the fingers, which therefore adheres to it and embraces it more. But whether this is so, let others consider; for us it will be enough to have shown the matter with the finger. QUESTION XXII. Here Aristotle asks why nuts are easily broken without a blow by instruments that serve for that use, and this although 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, meeting at a joint or hinge; and therefore the breaking is done with the greater violence, the smaller the space from the nut that is being broken to the hinge. But the greater the space from the hinge to the ends, which are pressed by the hand of the breaker. He says, then, and very aptly, that the force from the levers takes the place of the blow and produces the same effect. Let the instrument, then, of which we are speaking, CDBF, consist of two levers, one CAF, the other DAB, with the hinge or joint A, the place where the nut is broken K, and the handles BF. Therefore, the longer AB, AF are, and the shorter AC, AD are, the more violent will the breaking be. Now the resistance of the nut will be in the place of the weight A, and the support BF in the place of the power. Thus, unless the proportion of the power to the resistance is greater than the proportion 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. And it will overcome so much the more, the greater the...

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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 eiuscemo di scripta & contem- plata reperiuntur, quibus tum ad duarum mediarum pro- portio- S 2

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EXERCISES. 139 the greater the part of the lever from the power to the fulcrum may be. But what Aristotle adds, that the elevation of the lever becomes greater, that is, the opening of the instrument, the more the nut that is being broken is nearer the fulcrum, that is, to the pivot itself, is easily shown from the converse proposition 21 of book 1, Elements. For if from the extremities of one line toward the same parts there are set two longer lines meeting at an angle, and from the same extremities two other shorter ones, which fall within the triangle formed by the longer lines, they will contain a greater angle. But such is the angle which is made in the instrument, when the parts of the lever from the pivot to the nut are shorter. Therefore the levers are more spread apart, and when more spread apart are more compressed; and when more compressed they break with greater force, as Aristotle had said. Moreover, that also which he writes, that the instrument makes the breaking quicker from a heavier and harder material than from a lighter and less hard one, is true indeed on the side of the material, nor does it pertain to proportion, which in instruments of this kind is indeed almost a matter of form. We do not use these instruments. But they are similar to those instruments by which potters have been accustomed to make and shape clay pellets for the use of the chirobalistae. QUESTION XXIII. At this place the Philosopher proposes a most beautiful contemplation, and one pertaining to mixed motions. Now those who know what is found written and contemplated concerning spiral lines, the helix, cissoids, conchoids, and others of that kind, know that the investigation of mixed motions was both useful and familiar to the ancient Mechanicians, by which both for the proportion of two means S 2

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14o 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] pore moueri 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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14o IN MECHAN. ARIST. PROBL. for the discovery of proportional magnitudes, and then for the squaring of the circle, are usually employed. But what Aristotle here asks is as follows. Why, if two extreme points in a rhombus are carried by two motions, does each of them by no means traverse an equal straight line, but one of them far more? Likewise, why does that which is carried along the side traverse less than the side itself? For it is certain that the diagonal is traversed, but the greater side is not, although the latter is moved by one motion and the former by two? This is difficult to understand at first sight, and indeed remarkable, and therefore requires careful consideration. We shall first, with Aristotle, explain the whole matter; then perhaps we shall produce something not to be regretted from our own storehouse. Let there be therefore a rhombus ABCD, whose sides AB, BD, DC, CA, and whose diagonals AD, the greater, and BC, the lesser, intersect one another at the point or center of the figure K. Now from the very nature of a rhombus the sides are equal and parallel, and the angles opposite the greater diagonal are greater than right angles, while those opposite the lesser are less. These things being considered, let it be understood that point A moves by a special and simple motion along the line AB, from A toward B, and at the same time the whole line AB moves toward the line DC, yet under this condition, that it is always carried parallel to the same DC, and that one of its ends is carried along AC, the other along BD. Let it also be understood that point B moves at the same time by its own motion, and that too a simple one, along the same straight line BA, toward A, and that together with it, as has been said, being moved, it is 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 occur- rentia. Itaque cum ex duobus motibus semper propor- tionalibus, hoc est, laterum proportione seruata, recta producatur, vt demonstratum est à principio, vbi produ- ctio circuli ex Philosophi mente est declarata, vtraq[ue] pun- cta quæ eandem laterum proportionem seruantia moué- tur, rectas lineas producét 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 sim- plices, tum punctorum, tum etiam lineæ, à qua puncta ipsa feruntur, æquali velocitate fieri. Illud igitur mirabile est, cuius etiam ratio quæritur, quo pacto eodem tempore ea- demque 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. Im- ginemur igitur A, proprio motu percurrisse spatium AE, nempe ipsius AB lineæ dimidium. Erit igitur in E, item li- neam totam AB eodem tempore pertransisse dimidia op- positarum 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, e- rit 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 parts. Thus, since from two motions always proportional, that is, with the proportion of the sides preserved, a straight line is produced, as was demonstrated from the beginning, where the generation of the circle according to the mind of the Philosopher was declared, both points, preserving the same proportion of the sides, are moved; they produce straight lines, A indeed AD, but B the line BC itself. Let A therefore be carried, both by mixed and by simple motion through the diameter AD. B also, both by mixed motion and by its own motion, through the diameter BC. It is assumed, moreover, that all simple motions, both of the points and also of the line by which the points themselves are carried, take place with equal speed. That thing therefore is remarkable, and the reason for it is also sought, by what means, at the same time and with the same speed, the side A traverses the whole of AD, which is the greater, but B traverses the whole of BC, which is much smaller? Moreover, it was necessary to examine the matter in a rhombus, and not in a square or a rectangle longer on one side, in which the diagonals are equal, which does not happen in a rhombus. 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 at the same time will have crossed the whole line AB by means of half of the opposite lines, ACBD, and will have been transferred, as in FKG. Since therefore the extremity A of the line AB, moved with equal speed, has been transferred to F, and the point A moved through it to E, the space AE will be equal to the space AF. Therefore, lines FKG and EKH being drawn parallel to the sides AB, AC, the figure AEKF will be 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 have been transferred by mixed motion into K. In the same way, since point B is moved with the same speed toward A, and line AB toward CD, when B has been in E, 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. bus EBGK, similar to the rhombus ABCD, about the diameter BKC itself, and therefore GK is equal to the opposite BE, and BG is equal to EK. Since therefore B has completed the distance BE, there will be, from the mixed motion, at K, after the distance BK has been traversed, and that at the same time in which A had run through the whole distance AK. From the equal speed, therefore, of the simple motions, the points have passed through equal spaces AB, a circumstance which affords occasion for the wonder whose explanation is sought. Moreover, what has been demonstrated concerning half-diameters, we may also conclude concerning the whole by the same reasoning, since the proportion of parts to parts is the same as that 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 strange that point B, when it has arrived at C, the end of the line BA, that is, B itself, should have been carried to D, although the line BA is moved equally through the line BD, and point B through the line BA, while 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 motion, that is, the line AB toward the opposite part CD; so that when each motion tends downward, the motion becomes faster. On the other hand, B is carried by its own motion toward A, that is, upward, but by another's motion, that is, by the line BA toward D, that is, downward; and since these motions oppose one another, the motion itself becomes slower. It is therefore not surprising that A should traverse a greater space in the same time than B. This solution not only seems true, but marvelous and most worthy of the Philosopher himself, to whom indeed we should judge it rash to contradict, unless we were dealing with a field in which not probabilities are sought, but demonstrations, but truths. We shall therefore show, with due respect to Aristotle, that this reasoning of his is futile in the following manner. Let there be a square ABCD, whose diagonals AC and BD intersect at E; let BA move in the same manner 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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EXERCISES. 143 likewise A, toward B, and B toward A, so that point A, both by its own motion and by that of the other, that is, the line will drive it downward by a deferred motion, that is, toward CD. The motion therefore will be faster than the motion of point B, which 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 equal time each point accomplishes an equal distance. Therefore, the cause remaining, the effect ought to follow; but it does not follow; therefore Aristotle’s cause is not a cause. We shall also show the same more clearly with the reversed rhombus in this way: let there be a rhombus A B C D, whose diagonals A C and B D intersect at E. If therefore the line A B be moved toward C D, namely downward, and A also downward toward B, but on the other hand B upward toward A, and downward toward C, B will be slower than A, but the opposite happens, since B D, along which B itself is moved, is longer than A C, along which A is moved. Since these explanations do not satisfy us, we shall investigate, if our weakness allows it, the truer cause of this effect. Thus we must contend with arguments and truth against authority and probability: and this we shall do boldly. We say therefore that in every parallelogram, whether that figure be a square or longer on one side, or a rhombus itself, mixed motions are always produced, with 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 æquicrurium triangulorum; itaq[ue] in istis mixti motus per diametros equeveloces 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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being done through diameters. Moreover, it is most certain that the ratios of the diameters to the sides are various, except in squares, in which the same ratio is always found. It is also certain that in rectangles the diameters can never be equal to the sides, however they may be taken; for the diameters are always drawn to the right angles. But in rhombs and rhomboids the proportions of the diameters to the sides vary. For diameters may be greater than the sides, equal to them, and also shorter than the sides themselves. Therefore, since the diameters and sides stand in various mutual relations, if their proportions are considered, a different and varied comparison will arise between mixed and simple motions. In squares, the mixed motion, which is through the diameters, will always be swifter than the simple motion through the sides. The same is also true in the longer side, in which the mixed motions through the diameters will be swifter, but the simple motions through the sides slower; yet among these, the slower is that through the shorter side. In rhombs, however, the mixed motion made through the diameters is unequal: for that through the longer diameter is swifter, and that through the shorter slower. Thus the simple motions of the points through the sides do not stand in the same relation to that which is made through the diameters. Moreover, since rhomboids have various relations of diameters to sides, a varied proportion may also be given. For sometimes the diameters may be greater than the sides, and at other times one of them smaller. But if a rhomb be divided into two triangles, one of the diameters is equal to the equal sides of the isosceles triangles; and so in these the mixed motions through the diameters will be equally swift as the simple motions through the longer sides, but swifter than those through the shorter sides. These things therefore being considered in this way, and not superficially, it is easy, unless we are mistaken, from their proper causes, to solve this Aristotelian and wonderful Problem. Thus far.

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EXERCITATIONES. 145 Esto enim Rhombus ABDC, cuius diameter longior AD maior sit tum lateribus, tum etiam altera diametro BC. secant 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 simili- ter 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. propos.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 equa- lis, 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 motu moueri velocius quam mixto, quippe quod, vt T dictum

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EXERCISES. 145 Let there be a rhombus ABDC, whose longer diagonal AD be greater than both the sides and also the other diagonal BC. Let the diagonals intersect at E. And let FG be drawn parallel to AB and CD, cutting the longer diagonal AD at H, and the shorter one BC at I. And through I, parallel to BD and AC, let KIL be drawn. Since then B, by mixed motion through the diagonal BC, will be at I, and A, by the diagonal AD, by a mixed motion will be at H; and because the mixed motions are made through the diagonals, as has been said, as AD is to BC, so is AE to EB, by proposition 15 of book 5 of the Elements; likewise, as AE is to EB, so by proposition 4 of book 6 is AH to BI. For IH is parallel to AB itself. But AH is longer than BI, since AE is longer than EB itself. Therefore the mixed motion of point A through the diagonal AD as far as H is faster than the motion of B, through the diagonal BC as far as I. Therefore, when line AB is moved, the common points of it and of the diagonals BC, AD will be moved, the proportion of the diagonals always being preserved at the points where they are cut. Since these things are 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 smaller diagonal BC. This was to be demonstrated. But as far as the second part of the problem is concerned, we say that the proposition is not universal. For if a rhombus be given, composed of two equilateral triangles, the shorter diagonal will be equal to the sides; wherefore the point will not be moved more quickly by simple motion through the side than by mixed motion through the smaller diagonal, which Aristotle had proposed as something surprising. But if the side itself is longer than the shorter diagonal, it will also not be surprising that it is moved more quickly by simple motion than by mixed motion, since, as T said

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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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IN MECHAN. ARIST. PROBL. it has been said that these motions, from the proportions of the lines along which they move, receive the law of swiftness and slowness. These things, therefore, we may consider enough concerning this marvelous Aristotelian problem. QUESTION XXIV. Aristotle proposes another marvelous question, which likewise pertains to mixed motions. The doubt is for what cause a greater circle, when placed around the same center, describes a line equal to that of a smaller circle. But when they are revolved separately, how does the magnitude of the one relate to the magnitude of the other, and how do their lines correspond to one another? Moreover, with the same center existing for both, at one time the line around which they revolve is as great as the smaller circle revolves by itself; at another time, as great as the larger. He, in order to prove that the greater circle in its rotation traverses a greater line, and the smaller a smaller one, says that by sense it is known that the angle of the greater circle, that is, of the one which has the greater circumference, is greater, and that of the smaller is smaller. And the circumferences are related as the angles are related, and have the same proportion as those along which both the greater and the smaller circle revolve. For a clearer understanding of these matters, it is necessary to recall to memory what he said about the inclination of greater circles toward smaller circles. For here, as he had done there also, he called the sector itself an angle, and said that the greater angle is the sector of the greater circle, and the smaller angle the sector of the smaller circle itself. He concludes, therefore, by 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 by which the circu-

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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 took the demonstration from sense experience. It would have been sufficient if he had said that the circumferences are related as the diameters, or semidiameters are related, and therefore the lines described in rotation are related to one another as the diameters. Aristotle shows this, somewhat obscurely, in his own figure. Therefore, for those who love clarity, we shall present our own, if we are not mistaken, somewhat clearer one. Let there be a larger circle ABCD, a smaller one FGHI, around the same and common center E. Let the larger be revolved toward D. Let the diameters be, of the larger AEC, BED; of the smaller FEH, GEI; and let CD be the quadrant of the larger circle, and HI the quadrant of the smaller circle. Then, when the larger circle is moved in accordance with the epicycle, when D is at K, CK will be equal to CD itself, and DE from point K will become perpendicular to CK; and thus KO will be such that, since point I is on the line DE, I, after the quadrantal rotation, will be on the line KO at L, while the center E will be on the same KO at O. Therefore, after the quadrant of the larger has been revolved and the space CK of the smaller circle completed, the quadrant HI will complete the space HL, which is equal to the space CK itself. But what is true in the quadrants is also true in the whole circles. Therefore, the smaller circle, moving around center E, by a single rotation has equaled the space of the rotation of the larger circle. It is therefore remarkable that the smaller circle, revolved in the same time and around the same center, should have traversed a line equal to the circumference of the larger circle. Nor does it make the marvel less when ro- T 2 tated

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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[u]m 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 moueri sursum, 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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so that by the smaller circle, the line revolved once around the larger circle may measure a circumference equal to the minor circle. For the quarter of the smaller circle HI is rotated along the straight line HL. Thus the point I will be where M is, the straight line HM being equal to the curved line HI. Then, however, when the motion has been completed, the center E will be where P is, EP being equal to HM; and from P, through M, let the perpendicular PMN be dropped to HL, CK. And since D, I, E are on the same line, when E shall have been at PI it will be at M, and D at N. Wherefore, when the fourth part of the smaller circle has been rotated, meanwhile the quarter of the larger circle has made the space CN, equal to HM, this being less than the quarter-circle HI, which is indeed remarkable. Moreover, the Philosopher carefully seeks the cause of this marvelous effect, and having found it explains it accurately. 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 meanwhile, while the smaller is moving, it interposes some pauses in the course of rotation; whereas the smaller, in relation to the motion of the larger, skips over certain intervals, and thus an equalization of spaces comes about. But when he proceeds to the demonstration he assumes these principles: the same, or equal, power moves one magnitude more slowly and another more quickly. And what is naturally fitted to move is moved more slowly if, together with something not naturally fitted to move, it is moved, than if it were moved separately; but more quickly if it is not moved together with it. Suppose then the body A is indeed light and naturally fitted to move upward, to which B is attached, naturally fitted to move downward. If, therefore, someone tries to move body A upward, it will be moved more difficultly 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 is impossible to be moved more than the one 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 sit 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 & progressuum 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, since it is moved not by its own motion, but by the motion of another. Therefore, when the greater circle moves by its own motion, the smaller is moved not by its own motion, but by the motion of the greater circle, and therefore it is moved no more than that circle is moved; but it is moved through a greater space than it would be moved of itself, because the circumference of the greater circle, by which it is carried at the same time, is greater. Likewise, if the smaller circle revolves by its own motion, it will carry the greater with it, and therefore the greater will not be moved any more in its rotation than the smaller circle itself is moved. The sum of the matter is this: one is carried by the other, and, when carried, is moved through the space of the carrier. For although, when one is moved, the other is moved in the meantime, it makes no difference. For it is as if the one that is carried had no motion at all, or if it has it, as if it did not use it at all; which would not be the case if each moved separately around its own center, for then the greater would cover a greater space, the smaller a smaller one. Hence Aristotle deceives the man who thinks that each circle of itself moves in rotation about the same center; for although it may seem so, in reality it is not. For it is certainly true that, when the smaller is carried by the greater circle, the motion occurs around the center of the greater circle. But if the greater is carried by the smaller, the motion occurs around the center of the smaller circle. This is roughly the mind of the Philosopher, whose solution we do not doubt to be most certain, and founded on true causes. Hence we pass to another, and indeed certain, assertion. We say, namely, that no material wheel, fixed to an axle, while it is rotating, can keep the same place unless a hollow channel is made to receive the axle itself, in the crosspieces by which the wheel is supported and which prevent the forward 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 one KL, then a smaller NO, and still smaller QR, around the same center E. Therefore let it rotate according to the absis of the whole quadrant T 3 spa-

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150 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 progressiuis 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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150 IN MECHAN. ARIST. PROBL. space CD, and it will be D in F; likewise if from the wheel GH, from the quadrant HT, there will be T in I. Similarly from the others, smaller ones, in M, P, S. Therefore the longest space will be CF, but the shortest RS. Therefore, when the wheel is moved around the circle or axis QR, the greater wheel will move through the space RS, and if within QR, around the center E, other infinite circles are imagined, the closer they are to the center, the smaller the progress of the greater wheel will be, until one comes to the center, where, since there is no circle, there will be no progressive motion, but it will rotate around that very center without any change of place. But since no material wheel can be turned around an imaginary line or point, therefore into the iron or other material axis around which and with which the wheel revolves, a hollow half-round should be cut, in which the inserted axis, while it turns, does not recede from the place in which it turns. QUESTION XXV. The question is asked, why the posts of little couches are made in a double proportion, the one indeed six feet, or a little more, the other three. Likewise why levers or ropes are not stretched according to the diameter? The Philosopher resolves the first part of the question thus: perhaps couches were formerly made of such size as to be proportionate to bodies, and therefore to be made according to the posts in double length, namely of four cubits, but in width of two. Nostra-

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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 another proportion, namely sesquialtera, which the Greeks call hemioliam; commonly indeed they make them about four feet wide more or less, but long about six. This is done so that two bodies may lie on them more comfortably. But the little beds of which the Philosopher speaks seem to have been made to support only one. Whatever the case may be, the matter has scarcely any doubt on this point. The second section of the question was this: Why are the ropes not stretched along the diameters? The use of cords or ropes in furnishing beds is not among us. For we use ropes only for bier frames, or sandapilae, on which the bodies of the dead are carried, weaving ropes into them for support. For the rest, we lay beds with boards or planks, upon which we place sacks filled with straw, and upon the sacks mattresses and pads, lest the hardness of the boards offend those lying down. And yet Aristotle seems to have labored much over an easy matter, and also to have treated the question too obscurely and too intricately. 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 it appears, are corrupt and before us. Certainly Picolomineus worked excellently here to bring the truth to light, and he made no small progress. Moreover, why the cords are not stretched along the diameter, the Philosopher gives three reasons. The first is so that the wood of the side rails may be less strained. The second, so that the weight may thereby be more conveniently supported. The third, so that less rope or cord may be used up in the actual weaving. As to the first point, why, with the ropes stretched diametrically, the side rails themselves should be strained or even split, neither he nor the others explain. But I would demonstrate it in this way. Let there be a side rail ABCD, whose length is AB, thickness AC, and in it a hole extending through on both sides EF, with a rope 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, siet 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, siet 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 held fast at E by a knot. Let the beam of the bed, however, be such as is naturally cuttable along its length. Let some force be applied at G, with the rope thus stretched, which violently draws the rope itself toward it. The beam will therefore not be split on that account, because the rope is not extended diametrically. But if the head G is transferred to H, and the rope is strongly pulled, and if there is also strong pressure at F, for there the angle creates an impediment, so that the rope itself, while being pulled, does not attain straightness. Thus, force prevailing, but the wood being cuttable and offering less resistance, the rope, having achieved straightness, will split the beam in HIE according to the proportion of the triangle FIE, which was to be demonstrated. But why ropes stretched from angle to angle bear a load less conveniently is sufficiently clear. For the longer a rope is, the weaker it is; and the pressure that is in the middle, that is, in the part farthest from the ends, fatigues the rope more. Now the longest ropes are those that are stretched diametrically. So far as the third reason is concerned, the Philosopher weaves the ropes in this way. Let there be a little bed with its sides AB CD, whose side AD be six feet, but AB three. Let AD be divided in half at E, and BC at 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 BF be distinguished 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 mtila, 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 & FC likewise into three parts FI, IK, KC; then, the other rope being passed through the hole at A and firmly fastened there, let it be inserted through F, then through I, afterward through GHK CE, and let it be securely tied at E: Thus the rope will have four equal parts AF, IG, HK, EC, to which are added the small portions falling outside, which are FI, GH, KC. After this, the beginning of the other rope is passed through the hole, which is at the corner B. Then through E, thence through L, N, O, M, D, F, and let it be securely tied in F, and, a knot having been made, let it be fastened. There will therefore be four other parts of the other rope, both equal to one another and also to the aforesaid ones, namely BE, NL, OM, ED, to which are likewise added those small portions that fall outside, namely EL, NO, MD. Since therefore the square of BA added to AE is equal to the square of BE, the square of BE is 18, whose side or root is 4 1/3 as nearly as possible. Now there are eight ropes of this length, equal to one another. Their combined length therefore will be 34 2/3 feet, or about so much; to which, if there be added the six feet of rope that fall outside, the total extended length of the rope will be 40 2/3 feet, more or less. But Piccolomini says 34 2/3, for he omitted those six portions which, as we said, fall outside. He likewise gives the length of ropes extended diametrically to the same point as 40 1/2 feet. Here too he omits those portions which fall outside. Therefore, with these added, it is plainly evident that more rope is used in the diametrical than in the lateral extension. Moreover, the reasoning by which the Philosopher tries to prove this is so useless, involved, and obscure, that it needs, as they say, a Delian swimmer altogether. Piccolomini saw the inexplicable difficulty of this passage, and for that reason testified that the interpreters in explaining it had gone astray. Certainly the Greek reading is no clearer than the Latin translation itself. Meanwhile, lest we pursue by excessive diligence, and perhaps in vain, this almost useless speculation, let others resolve this difficulty, or, as it were, the Gordian no- knot

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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 obfirmetur, tum alterius funis caput indatur si libet per K, & inde per S, R, L & in L constringatur. 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 by cutting with a sword. But meanwhile it occurs to wonder why the ancients, passing over the more useful method, should have embraced the less useful one. For they could have woven this in a reticulated manner by lines equidistant from the sides. Let there be, for example, a small bed of the same dimensions, ABCD, in the side AD of which there are five holes, E, F, G, H, I, and as many in the opposite side QP, ONM. Again, two in the shorter side AB, namely RS, and as many in the opposite KL. Let the stretching begin from the hole E, through QP, F, GON, HIM, and let the cord be fastened at M; then let the end of another cord be inserted, if one wishes, through K, and thence through S, R, L, and let it be tied at L. Now all the lengths EQ, FP, GO, NN, IM are fifteen feet; if to these KS, RL are added, each of six feet, they will make twenty-seven feet; and if the parts falling outside, QP, FG, ON, HI, and RS, are added, the whole sum will be thirty-two feet. See therefore how much less cord is used this way than in the method which Aristotle approved and proposed as more useful. Moreover, this work of weaving is very strong, and from it there does not arise any true loosening or tearing of the bedcords, to which that method, preferred by the Philosopher himself, is by no means a little exposed. We conclude, then, either that we have not understood his words and meaning, or that the ancients themselves, whose practice he explains, were ignorant of the nature and convenience of the thing which we propose, which, however, is hardly credible. QVÆ-

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EXERCITATIONES. 155 QVÆSTIO XXVI. Proponitur à Philosopho examinandum, Cur difficilius sit, longa liga ab extremo super humeros ferre, quam secundum me- dium, æquali existente pondere? DVohîc considerat, vibrationem, & pondus. Ait enim primo fieri posse, procera ligna vibratione impedien- te, 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 hume- ro, eique hæreat in C, manu vero sustineatur facta compressione in B. Nutet i- gitur & vibretur, in ipsa vibratione, ad partem A. Sit au- tem centrum grauitatis eius D, Lignum igitur in ipsa vi- bratione 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 descende- rat, repulsum per D, nec enim in sua rectitudine stabit, a- scendet in F, facta iterum materiæ constipatione inter C & F. Mouebitur igitur lignum sua grauitate, motu fre- quentissimo, sursum deorsum, & is interim qui lignum hu- mero fert, procedit antrorum, impedit igitur motus iste, qui sit sursum deorsum lationem, quæ sit ad anteriora; La- torem ipsum quodammodo retrahens. Si autem medio ligno supponatur humerus, eo quod vibratio sit minor. breuiores enim partes sunt, quæ à medio ad extrema mi- nus 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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EXERCISES. 155 QUESTION XXVI. The Philosopher proposes for examination: Why is it more difficult to carry a long beam on the shoulder by the end than by the middle, when the weight is the same? He considers here vibration and weight. For he says first that long timbers may be carried with more difficulty because vibration hinders. Someone, however, might ask—and he himself is silent on this—why this vibration is harmful to the carrier. We shall therefore try to explain it. Let there be an oblong beam, flexible, and, so to speak, capable of vibration, AB; let it be laid on the shoulder and rest upon it at C, but be held by the hand through pressure applied at B. Thus let it oscillate and vibrate toward the part A. Let its center of gravity be D. The beam therefore, in the very act of vibrating, will descend under its own weight, driven down to E; then, with the beam compacted in that part which lies below between C and D, and from there resisting, it will be thrown back through D with nearly the same impulse with which it had descended; for it will not remain in its straight position, but will rise to F, the material being again compressed between C and F. The beam will therefore, by its own weight, move with very frequent motion up and down; and meanwhile the one who carries the beam on his shoulder proceeds forward. This motion, therefore, which is an up-and-down movement, hinders the carrying forward movement; thus it in a way draws back the bearer himself. But if the shoulder is placed under the middle of the beam, the vibration is less, for the parts are shorter; and from the middle to the ends the carrier is less delayed by vibration. And since not vibration alone in this manner of carrying, namely from the end of the beam, creates 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 depresse 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 perhaps this happens because, although nothing is bent and it has not 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 as to why it is more easily lifted from the middle, he says the reason 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 places the shoulder who lifts or bears it. Meanwhile, one of the ends being depressed, the other is lifted up. We, however, using mechanical principles, which he did not do, shall make the matter clearer. Let there be, then, an oblong piece of wood AB, on which the shoulder is placed at B, and the hand, pressing, supports at B. Let the larger part of the wood be AC, the smaller CB, and let the proportion of the larger to the smaller, for example, be sixfold. Therefore, in order that equilibrium may occur 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 there be, then, a weight at A, say of six pounds. Therefore the power at B will be such as to support 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 at the middle of the wood, that is, at D, then in order that equilibrium may be achieved, the power at B will need to be equal to the weight at A, which is six; thus the shoulder will support twelve pounds. Whence it is clear that it is far more difficult to carry the wood from the end C than from the middle D; 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 lever 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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EXERCISES. 157 the support B indeed, 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. Balduus, in his treatise On the Lever, proposition 3. Thus, as he shows, it ought to be so proportioned that the sustaining power is to the weight as the whole lever is to that part of it which is from the power to the fulcrum. So therefore the pressure exerted at C is to the weight at A as the whole lever AB is to that part CB of it which lies between the power and the fulcrum. Therefore the power will be seven times the weight, and thus it will support a weight of forty-two pounds, which was to be demonstrated. From this another question related to this is solved: Why is a spear or sarissa, lying on the ground, when grasped by hand at one of its ends very difficult to lift? Let there be, then, a sarissa or spear lying AB, to whose end A the hand is applied for lifting; let the part grasped by the fingers be AC; the question is why the remaining part CB is very difficult to lift. The doubt is easily resolved from what has been previously demonstrated. For C is the fulcrum, since the place beneath the clenched fist for lifting is supplied by the index finger; but the pressing power in A, by which it overcomes the weight of CB, is the wrist of the hand itself, that is, that part of the hand by which the weight is lifted when the pressure is applied. Therefore AB is a lever, whose fulcrum is C, weight B, power A. And thus, since the proportion of BA to AC is greatest, the power lifting the weight at C ought to be greatest. To this also belongs the question, why, when a spear lies on the ground, if one end is lifted by hand, while the other end is whipped up most swiftly upward, and at the same time the hand is placed beneath the spear so whipped, the spear is raised to the perpendicular with no great difficulty.

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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- cimanus 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. QVÆSTIO 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 lursum 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 rod AB, which is taken by hand at B and raised in C, and made to lie in AC, then, when from C the part A is set in rapid motion, the extremity A itself is transferred to D, so that CD is formed; then, by lowering the hand, the extremity C is transferred to E, and EF is made perpendicular to the horizon; when this has been done, the hand itself which supports it, and its center of gravity G, will be on the same line that goes toward the center of the world; therefore the hand itself, by making the vibration, carries only as much as is precisely the weight of the rod itself. QUESTION XXVII. It is doubted why, if the same weight is very tall, it is carried with more difficulty on the shoulders, even if one carries it in the middle, than if it is shorter? This question is related 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 they sway more easily of their own weight. But if they are shorter, that cause ceasing, the vibration becomes less or none at all, and therefore shorter objects are carried more easily. The bearer is impeded by vibration itself in two ways: first, from the cause we considered in the previous question, namely that the continual up-and-down motion hinders forward motion; and also because the bearer’s shoulder is burdened by a double pressure, which the Philosopher did not notice. If then there is a long timber AB, which by 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. QVAESTIO 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 that it may be supported by the lord in C. Therefore the ends AB will waver, being very far from the center C; but they will fall at the same time, A in D, and B in E, trying to draw the middle C along with them; wherefore he who supports it in C bears not only the weight of the wood from the center of gravity, which is in C, but also the force acquired in the very depression of the ends from that violence itself. This we observe closely: that the bearer is, by the vibration, lowered and raised by intervals. Let a beam therefore be vibrated by contrary motion, where FCG. In that case it will relieve the bearer, since the force acquired from the motion itself draws the middle C upward. Thus, when it is in DCE the bearer supports more in ACD, equal, in FCG less, which certainly had to be demonstrated. This question is familiar to that man, which we explained in place 16. QVAESTIO XXVIII. It is asked: Why do they make pulley-wells in the manner in which they are seen? For they add the weight of lead to the wood, although otherwise the vessel itself, both full and empty, has weight. The Philosopher answers excellently that the work of drawing water is divided into two times: namely, while the vessel itself is lowered empty, and while it is pulled up full. But it happens that empty is easily lowered, while full is difficult to draw up. Nevertheless, it is better that it be lowered more slowly, that is, with greater difficulty, so that it may be more easily drawn up, with lead assisting, and indeed the Philosopher’s solution is most clear. But we shall try to bring still some light to the light itself. Let a celonium (in Latin they call it a tolenon) be ABC, whose upright beam is BD, the cross-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, adiuvat. 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 permutatur. Corporis enim proprij pondere, dum premit, adiuvatur 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 around the point or fulcrum B, with a weight, or lead, at A, and the bucket E hung on the rope CE. I say that, these things being thus arranged, the lowering of the empty bucket is indeed difficult, but the raising of the same bucket is easy. The proportion of the lever, the bucket, and the weight, so that equilibrium may be produced, must be such that, just as AB relates to BC, so the weight of the full bucket E shall relate to the weight A itself; therefore the weight at A will outweigh the empty bucket at E, and there will be no equilibrium. Thus, in order that the empty bucket may be lowered, such force must be applied as is equal to the weight of the water with which the bucket is filled; and this force, when applied, makes the lowering of the empty bucket difficult, as we said. But when the bucket is full there is equilibrium, whence, however small a force is applied, the bucket is drawn up, as though rising of itself by the power of the suspended weight. Therefore, as much weight as hinders it when the empty bucket is being lowered, just so much does the full bucket, when it is being raised, assist. Since this is so, if there is equal difficulty in lowering and ease in raising, what use is there in this contrivance? Certainly the empty bucket is easily lowered by hand through the rope, but when full it is difficult to raise; yet by the use of the Celonium the matter is changed. For by the weight of its own body, while it presses down, the person lowering it is assisted, who, by pulling up through a simple rope, was hindered by the same weight of his own body. Indeed this assistance from the weight of the body produces great convenience in raising. Something similar happens to those drawing water from wells by the use of a pulley. Let there be a pulley overhanging the well ABCD, 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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EXERCITATIONES. 161 the bucket is suspended FCABG, but the bucket itself is G. Therefore CED is the diameter, like a balance, wherefore, so that equilibrium may be established, it is necessary to apply to the end of the rope F a power equal to the weight of the bucket filled with water; and thus, by exerting his own bodily strength and adding weight, the one pulling easily draws out the bucket filled with water, from which a great convenience arises for those drawing it out. It is clear, moreover, that the Celonium and the Pulley assist in different ways those who draw; for in the one case the body’s mass is helped by letting down the empty bucket, but here by the one who draws up the bucket full of water. Furthermore, the part BC of the Celonium, which from the support to the rope ought to be much greater than AB, so that the bucket may be able to be let down into the depth, wherefore the weight at A must be related to the weight of the full bucket as the arm, or part BC, is related to part BA. For then, by inverted proportion, equilibrium is achieved. We add that it is nothing new for architects and mechanicians to apply weights of bodies, both of men and of animals, so that they may move machines more conveniently. For the wheels which for this reason they call ambulatory are moved in no other way; the use of these is employed for mangana, for drawing water out of wells, and also for turning flour mills. Moreover, that the war machine called the Tollenon differs in nothing from the Celonium, either in form or in power, may be seen among the ancient mechanicians, Hero of Byzantium and others; among the moderns, however, Daniel Barbarus on Vitruvius, and Iustus Lipsius in the very elegant book which he published on war machines, discuss this matter. X. WHAT.

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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 vnicoli- gno

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162 IN MECHAN. ARIST. PROBL. QUESTION XXIX. It is doubted why, when two men carry upon a beam, or some such thing, the same weight is not pressed equally, but that man more upon whom the weight is nearer? Aristotle answers, saying that the beam is a lever, but the weight a support; the thing that is moved is the one who is nearer to the weight: the mover, however, the one who is farther away. Therefore the farther the mover is from the weight, that is, from the support, the more violently is pressed the one who is moved on the other, shorter, part of the lever. Let there be a beam AB, with a weight placed in E, nearer to the 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 depressed by the weight that it can by no means be carried upward, but rather, around the point E, as around a center, the support itself or the lever turns. The beam AB, therefore, is the lever: the power moving at A; the part of the lever from the power to the support, AE; the other part of the same, from the support to the thing moved, EB; and since the longer the part EA of the lever is than EB, the more easily the power which is in A acts upon that which is in B, if the matter be 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 is to the part EA; but AE itself is greater than EB, therefore it bears the greater part of the weight, and the one who is in E is pressed more, and so is the one who moves in A. Such is, in substance, the philosopher’s opinion. But Picolominius the paraphrast aptly describes two levers in a single wood

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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 ferens 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 the mover and B the moved object; in the second, B is the mover and A the moved object, with the same common fulcrum E in these levers always. And since in the proposed diagram the shorter part of the lever EB is that which, from the mover to the fulcrum, works less than that part which is from the same fulcrum to the moved object, B acts upon A less than A upon B; and therefore the one moved in B is pressed more. Conversely, because the part EA is greater than the part EB, the one acting in A upon B works more than the reverse. And certainly this consideration is subtle and ingenious, and if it is rightly understood, insofar as it pertains to demonstrating the proportions and the effect itself, it does not depart from the truth. However that may be, the question will be more easily settled by mechanics in this way. For we say that the weight is truly a weight, not a fulcrum, as Aristotle imagined; the beam is the lever; and the two supports which bear the weight are to be taken as a double fulcrum, for the lever rests with the suspended weight upon both. One of them can also be taken as the moving power, the other as the fulcrum, and so in turn. Now, however the matter is taken, the weight is between the fulcrum and the power. Therefore, from what G. V. Bald. has demonstrated when speaking of this kind of lever, as AE is to the whole lever AB, so is the power that supports at B to the weight suspended at E; and as BE is to BA, so is the power that supports at A to the weight which is at E. But the proportion of BE to BA is less than that of AE to AB; therefore the weight at E is overcome more by the power that is at A than by the power that is at B, and therefore the supporter at B bears more weight than the supporter at A, which was to be demonstrated. Hence we conclude that, with the weight suspended in the middle of the lever, the supporters bear equally, because the proportion of the whole lever to its 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, xquidistans 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 xquidistans GB, secans perpendicularem CF in I. Quoniam igitur AG xquidistans 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 C alligatum 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 ON MECHAN. ARIST. PROBL. But it may rightly be doubted whether exactly the same thing happens if one of those who are supporting it is indeed taller in stature, while the other is shorter. Let there be a lever AB, in the middle of which a weight H is freely suspended from C; let the taller of the bearers be AD, and the shorter BE. Let DE be the plane of the horizon; from point C let the perpendicular to the horizon be let fall, and let CF, equally distant from AD and BE. But it will pass through H, the center of gravity of the weight itself. I say, therefore, that, so far as the sustaining of the weight is concerned, it makes no difference whether the bearers are equal in stature or not. For draw GB, equally distant from the horizon, cutting the perpendicular CF at I. Since therefore AG is equally distant from CI, it will be as AC is to CB by Prop. 4 of Book 6, so GI is to IB. Therefore GI and IB are equal to one another. Let the weight H, then, be understood as if, being released from point C, it were freely suspended from point I, that is, from the middle of lever GB; thus the weight will be equally divided between the bearers, although one is taller and the other of dwarf-like stature, which was to be demonstrated. But if the weight is 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 incline toward that part which is raised more from the horizon, and toward that very part of the lever the distance from the weight to the supporter will be shorter. For let there be a lever AB, whose middle is C; let the weight CFG be attached to the lever at C, its center of gravity H; let the bearers be AB, the taller, and BE, the shorter; and let DE be the plane of the horizon. Through center H let the perpendicular IHK to the horizon be drawn, cutting the lever indeed at I, and the plane of the horizon at K.

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EXERCITATIONES. 165 num in K. Post hæc intelligatur pon- dus solutum quidem à puncto 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 let it be in K. After this, let the weight be understood as indeed released from point C, but suspended from the point I. It will therefore remain, by the definition of the center of gravity, and will not be moved from its position. But I say that the 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; but let it fall, if possible, at C. Then CHK will be perpendicular to the horizon, and AD perpendicular to the same. Therefore the angles BCK and BAD will be equal to each other; but the angle CIH is equal to angle BAD, and therefore BCH will also be equal to CIH. If then the side IC of triangle ICH is produced, the exterior angle will be equal to the interior opposite angle, which is absurd. Therefore I does not fall at C. But by the same reasoning it will be shown not to fall between C and B; therefore it falls between C and A, and therefore AI itself 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 force will be required at B than at A, and indeed the part IB corresponds to the sustaining force at A, while IA corresponds to the sustaining force at B; but AI itself is smaller than IB. Therefore a greater force is required at B than at A, which was to be demonstrated. This also will be concluded if the bearers are indeed of equal stature, but walk on a plane inclined upward or downward to the horizon. For if the weight hang freely, the proportion of the parts of the lever will not change; but if it does not hang freely, he will labor more who goes first in the ascent, but less who goes first in the descent. Hence also depends the arrangement of wheelbarrows, which are commonly in use with a double handle and a single wheel; for they are regarded as a lever, whose support is at the contact point 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 in- ter fulcimentum est & potentiam. quo igitur minor fue- rit proportio partis vectis quæ à centro grauitatis ad i- plum fulcimentum, ad totum vectem eo facilius pondus eleuabitur. Cur autem difficilime hæ per accliue horizonti pla- num pellantur, duplici fit de caussa, tum quia grauitatis centrum ad ipsum portantem seu pellentem vergit, & id- eo pars quæ à fulcimento ad centrum grauitatis ponderis fit maior, tum etiam quoniam ipsum graue contra sui na- turam sursus pellitur ferturque. Quærere ad hæc quispiam posset, Cur Baiuli ma- gna ferentes pondera, curui incedant? Dixerit autem ali- quis, ponderis grauitate eos deprimentis id fieri. Nos au- tem duplici item de caussa id fieri putamus, tum ea quam considerauiimus, tum etiam alia, nempe vt grauitatis cen- trum 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 tra- hatur, & pondere ipso opprimatur. Eadem de caussa sit quoque vt ij qui magna ponde- ra sinistro ferunt humero, in dextram partem inclinentur, qui vero dextro, contrario modo se habeant, æquatur e- nim pondus eo pacto, & grauitatis centrum in ipsa per- pendiculari collocatur. QVÆSTIO XXX. Cur assurgentes omnes foemori tibiam ad acutum angulum consti- tuamus & pectori thoraciue similiter foemur, quod nî fiat haudquaquam surgere poterunt? A It Philosophus, forte id fieri, quod æqualitas sit o- mnino quietis caussa, rectum vero angulum quietis angu-

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166 IN MECHAN. ARIST. PROBL. ...; and the power to the end of the 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 very fulcrum, compared with the whole lever, the more easily will the weight be lifted. But why these things are very difficult to drive along an inclined plane is for two reasons: both because the center of gravity inclines toward the very bearer or pusher, and therefore the part from the fulcrum to the center of gravity of the weight becomes greater; and also because the heavy thing itself is pushed and carried upward contrary to its nature. In relation to these things one might ask: Why do carriers, when bearing great weights, walk bent over? Someone might say that this happens because of the weight pressing them down. But we think that this also happens for a double reason: both for that reason which we have considered, and also another, namely that they place the center of gravity of the very weight they are supporting on the perpendicular, lest if the one carrying it were placed outside, with the fulcrum placed outside the center, he should be drawn toward that part toward which he inclines, and be oppressed by the weight itself. For the same reason it happens also that those who carry great weights on the left shoulder lean to the right side, while those who carry them on the right shoulder behave in the opposite way; for in that manner the weight is balanced, and the center of gravity is placed on the very perpendicular. QVÆSTIO XXX. Why do all who rise place the tibia with the femur at an acute angle, and likewise the femur with the chest or thorax, without which they cannot rise at all? The Philosopher says that perhaps this happens because equality is altogether the cause of rest, but a right angle is ... 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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EXERCISES. 167 to be at an angle, and to make a stance, and for no other reason is the standing person perpendicular to the earth itself, and therefore has the head and feet in the same line, whereas the seated person does not. Yet rising from sitting takes place when the head and feet are placed in one line, which certainly happens when the chest and thighs, and the thigh itself, make an acute angle. For let there be, in the case of one standing, AB perpendicular to the horizon IBK, whose head is A and feet B; but let the same person sit, and let there then be with the head his thorax CD, the thigh DE, the legs EF, and let CDE and DEF be right angles. These things being so arranged, the head C and feet F are not in the same line. Therefore a seated person 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 person draw back his feet to H, and with the chest inclined form an acute angle GDE with the thigh; in which case the head at G and the feet at H will be in the same straight line, and that line perpendicular to the horizon, and from the nature of that position an easy rising will be brought about by the seated person himself. This is more or less the Philosopher’s view, though explained by him in other words; and although it is true, it is nevertheless not derived from proper, that is, mechanical, principles. This indeed we shall try to do. We say first, however, that the seated person rests not, as Aristotle thinks, because a right angle is the cause of rest, but because the weight of his thorax and also of his thighs is sustained by the seat itself; while the legs and feet therefore do not labor, because they are partly suspended, partly supported by the ground itself. Therefore since the whole body neither sustains itself nor

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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 instar mollem & 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 fulcimérum constituto, non habet centrum stabilem locum cui in actu surrectionis hæreat, & fulciatur, vnde fit 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 rest and relief of fatigue. But that nature wished to make sitting convenient for human beings is apparent from this, that she made the buttocks, on which the whole upper part, and the heavier part, rests, fleshy, and soft and easy like a kind of cushion for the neck. But let us come to the question. Suppose, then, there is a standing man 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 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 is. Let the straight line FG be produced to L, and let FL be perpendicular to the horizon. The center, then, of gravity K is supported by point G, that is, by point L, at which the posterior 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, rising will not occur; and this not because, as the Philosopher says, equality and rectitude of angles are the cause of rest, but because, since the center of gravity is placed outside the support of the feet, it does not have a stable position to which, in the act of rising, it may cling and be supported; whence it comes about that if the seat be removed from beneath the seated person by taking away what hinders him, the seated person falls completely. Now let the person who is at rest draw in his legs and place his feet at M, and from point M let MN be erected perpendicular to the horizon. Thus the support will be at M, but still he will not be able to rise, since the center of gravity is still outside the line MN, which passes through the support. But let the chest be inclined forward, and let it make an acute angle with the thigh; let it be where GO is, and then the center of gravity will be where P is, that is, on the very perpendicular NM; and therefore from this a convenient rise will be made...

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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 upright posture, because they have been made in the same line, the center of gravity P, and the support itself M. That a sharp angle is necessary in an upright posture is clearly evident, but not that it is the cause of the effect, as Aristotle seems to have thought; unless we say that the cause is the cause of the cause, since the sharp angles that are formed place the center and the feet in the same line. Whatever the case may be, we therefore say that upright posture occurs because, when the angles are changed, the center of gravity is placed above the support, and the support beneath the very center of gravity; and this is the proximate cause. So much for Aristotle. Now we shall propose certain other questions, neither useless nor unpleasing. First, therefore, we ask why the feet of human beings and of other animals, which sometimes walk with the body erect, are not short and round, but rather longer and extended toward the lower part? Likewise, why are they extended more toward the toes than toward the heel? Let there be a man or any animal standing AB, whose foot is CD, the part of the foot toward the toes BC, while that toward the heel BD is the vertebra of the thigh E, and the center of gravity of the body itself F. First, therefore, it must be established that man and nearly all other animals are made by Nature so that they move toward the front, and therefore nearly all, as is clearly seen in the elderly, incline toward the front by the very disposition of the body. Thus when one standing is perfectly perpendicular to the horizon, the center of gravity F is located on the very perpendicular which goes to the center of the world AB, and therefore the mass and weight of the body is supported at point B. But if, from vertebra E of the chest AE, there is an inclination toward the front, in GE, and the center of gravity D is diluted into H, and through H a perpendicular is let fall Hl, it will not be ** outside the support of the foot-

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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. support BC. Therefore he will stand who is inclined in this way, and will not fall; but if he leans still more, and comes into KE, the center of gravity being established in M, let the perpendicular ML be drawn through M; therefore, since the line ML falls outside the support of the foot, he who is inclined in that manner will fall and will not be sustained. Why therefore nature has made for animals that walk with upright bodies their feet extended forward, from this it is clearly evident. From this too we have as a consequence why men, if they are pushed, are more prone to fall backward than forward. And also runners, bears, and if there are any other animals of this kind that cannot walk for long with an upright body, namely because the mass of their bodies leans greatly forward, and is not supported so suitably, as happens with human bodies, by the bases of their feet. We may also not inappropriately ask why stilt-walkers do not stand upright unless they are constantly moving. The solution is easy. For the stilts touch the ground only at two points, and cannot make use of the advantage of being extended forward, which occurs in those who walk. Wherefore the center of gravity is outside the support, and therefore stilt-walkers are compelled by continual motion to place the support under the center of gravity, and when 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 can, with nothing impeding it, fall to any side whatever, for the center of motion is indeed a point. That which is supported by a line has a fall to only 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 superficiei grauitatis centrum I. à quo ad horizontis planum perpendicularis demittatur IK. Cadet autem in lineam FG. per propos. 38. vndecimi elem. & anguli IKG IKF recti erunt. 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 púcta 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 perpendiculairis 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 Y 2

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EXERCISES. 171 Let ABCD be the plane of the horizon, upon which let the surface EFGH stand at right angles, along the side FG. And let the center of gravity of that surface be I. From this point let the perpendicular IK be let fall to the plane of the horizon. But it will fall on the line FG, by proposition 38 of the eleventh book of the Elements, and the angles IKG and IKF will be right angles. Therefore, if the surface EFGH is moved about the line FKG as about an axis, the point I will describe the circumference LIM; and if it falls toward the parts CD, the center of gravity will be at M. But if toward the parts AB, it will be at L. Now the points LKM are in the straight line LKM, which is the common section of the plane of the horizon and of the plane passing through IKLM. The same also we say of a body placed on one side. For let there be a cube LO, whose center of gravity is R, and the side on which it is supported, NO. If it is placed in such a way that the inner surface LNOQ is set at right angles to the horizon, and a perpendicular is let fall from the point R, it will fall at S, on the line NSO itself. Therefore, as the body falls, there will be motion about the line NO, while the center of gravity meanwhile describes the circumference TRV. From this it may be observed why provident Nature has given to no living creature a single foot, but either four feet, or at least two; and even those two she has virtually made four, since in every foot of biped animals Y 2

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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[m], 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 a foot at least two points are considered, by which the animal itself is supported. Let the traces of a human foot be A, B, C, D; therefore in each there are two points considered, A, B, C, D, namely those toward the toes and those toward the heel. The same is also observed in the feet of birds, from which we conclude that the support of all bipeds is quadruple. Moreover, since quadrupeds, because the whole mass of their body inclines downward, are prepared by Nature herself with four supports, distinct and conveniently separated from one another. We observe the same also in artificial things. Let there be some vessel ABC, whose single foot, and that round BC, has the center of gravity at D. Since, therefore, in the perimeter of the foot itself infinite points are understood, one may in a manner say that the vessel itself is supported by almost infinite points, although it has only one foot. But some artificial bodies are supported by four feet, as certain tables, some even by three, as tripods, which derive their name from the very number of their 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 on only two, it will not stand. For if IK is drawn, and it rests only on 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. Sivero 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 capi, 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 longeminorem. 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, vt pote 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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EXERCITATIONES. 173 without the extra support IK, it will incline, yielding toward the parts L. But if it rests on the points IL, it will fall toward the parts K. And if on KL itself, it will fall toward the parts I. From which it appears that inanimate bodies, whether with a single foot having the strength of several, or at least with three actually supporting them, need support. Hence it is also clear why the infirm, the crooked, and those with feet seized, need the support of staffs or of staffs and canes; for since these are weak and greatly incline toward the front part, lest the center of gravity be 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 in that position with difficulty is that the center of gravity, situated in the thorax, is supported by the two knees and presses upon them; and these knees, because by nature they are not fitted, like feet, to sustain the mass of the body, labor under the burden, and the more so because, since they are bony, the skin placed between the hardness of the bones is compressed, and thus causes pain and discomfort to those kneeling. But if someone rests on only one knee, he will feel a lesser difficulty. For in that case the kneeling person is supported by a threefold support. For let the kneeling person be ABCDE, whose center of gravity is F. Let the right knee, on which he rests, be D, the left, which is raised, B. Thus the kneeling person is sustained by three supports, as we said, CDE. The weight is therefore divided into three parts, and thus each is less fatigued. Yet point D labors more, since it is the point on which the center of gravity F bears perpendicularly. Likewise, it is also marvelous that sleeping birds are supported by only one foot, and what is more astonishing, 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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the mind can do, which is difficult even for those who are awake. I think the reason 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 warmed by the natural heat; wherefore, leaning toward that side, they lift the foot in order to keep their balance for the time being, and in that position draw it back and hang it up as if it were useless. Another reason is added, namely, that sleeping birds warm the foot itself with their native heat. It is also asked: why do those who lean forward, in order to lift 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, and whose center of gravity is E, and let it be desired to lift something from the ground located at F. Let there be the perpendicular GEH, passing through the center of gravity. Therefore, when it leans forward, the center is moved away from the perpendicular; wherefore, Nature teaching, the leg BC is extended forward, that is, toward the object to be lifted, in order to support that very center. It is also part of this inquiry to investigate why quadrupeds, while walking, move their feet diametrically opposite one another. The Philosopher himself also discussed this in book On the Progression of Animals, chapter 12. But we shall demonstrate, more fully, using mechanical principles what he established on physical principles. Let there be two parallels in a plane, AB and CD, on which are the feet of a quadruped, E, F, B, D, of which EF are the front feet, while BD are the hind feet. Let BDEF be joined; there will be a parallelogram EBDF, 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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EXERCISES. 175 ED, BF, intersecting at G, where the center of gravity is. If therefore the posterior left foot B is moved to K, and at the same time the anterior E were moved to I, while DF meanwhile remained standing as supports, the center G would be outside the supports toward the side BE. It would therefore fall toward the side BE. But if at the same time the right ones were moved in the same manner, the center, placed outside the supports, would fall toward the very side DF. 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 on the diagonals HI, KL, that is, at M, indeed suspended by the feet K, L, H, I themselves. In this way, therefore, it is transferred back and forth, while the supports, transferred at the same time together with the center of gravity, correspond to one another diametrically; which, of course, was to be demonstrated. Indeed, bipeds too, by walking alternately, transfer the center of gravity. For while the right leg is lifted, the center is supported by the left, and vice versa. These things are natural; but in artificial matters one might ask why architects place the walls of fortresses not erected perpendicularly, but inclined inward? They certainly do this so that they may be less inclined to collapse. Let there be, then, a wall slanting toward the interior, ABCD, whose center of gravity E; let the base BC be erected from the point B by the horizontal perpendicular BF, and from the center of gravity E let EM be drawn down to the same, and then let BE be joined. After this, from the point BG let an angle be made with the line of the horizon BK greater than a right angle. Thus a wall thus constructed leans by its own weight toward the interior, and cannot fall, either because the li-

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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 perpendicularem 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. If it is leaning on a rock to which it happens to cling, let it be supported, or let it rest on buttresses, which our people call sperones and, against strong walls, supports. But it will not fall forward either, since, if it is about to collapse, the center of gravity must necessarily be carried with it in the perpendicular BF, and finally into that which lies beyond the perpendicular, BG, namely by a rotation around B as around a center. Let there also be motion taken from the semidiameter BE, and let the portion of a circle described with center B be 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 shorter than BK, that is, than BI. Let LB be cut off from BI, equal to EM. Therefore the point L will lie below the point I, that is, closer than I itself to the center of the world. It will therefore be necessary, in order that the wall may fall, for the center of gravity E, after a rotation about B, to come at some point into I, so that it may finally be transferred to H; but I is farther from the center of the world than E itself and L are. Therefore the heavy body would rise 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 Campanile tower seen at Pisa, and also another at Bologna in the marketplace near the Tower of the Asinelli, which, built long ago by the noble family of the Carisendi, they call Carisenda, and which the great poet Dante also mentions in his Comedy. It leans, however, to one side, and leans in such a way that the perpendicular dropped from the top of the leaning part to the ground falls far away from the base on which it rests, which indeed seems marvelous, namely that walls inclined toward ruin do not cause ruin. For let the tower be ABCD, supported by the base BC, the plane of the horizon BCF, the sides AB and DC, and the center of gravity of the whole mass E. Let it incline toward the side DC from the angle DCF. And being thus situated, let it be understood 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 C. 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 considerauiimus, 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 quidâ con- Z

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EXERCISES. 177 if the body E outside the base BC does not fall, but instead falls at C. Since therefore the mass ABCD is divided by the center of gravity at E, it is cut into equal-weight parts, and the center of gravity does not fall outside the support; therefore neither the part ACD will draw the part ABC, nor will the center placed outside the support seek a place closer to the center of the world. Why therefore Carisenda stands, and that remarkable bell tower which at Pisa, near the summit of the Temple, seems splendidly built of marble, though they threaten ruin, stand forever, and do not fall, from what we have considered, is clearly evident. QUESTION XXXI. Why is a body that has been set in motion more easily moved than one at rest, as carts already moving are driven more quickly than when they are beginning to move? This is the question. This problem is purely physical; nevertheless, since it pertains to local motion, of which Mechanics also treats, this inquiry is inserted among these questions. Aristotle answers, saying that perhaps this happens because it is very difficult to move a weight that is moved in the opposite direction. For something is removed by the force of the mover, though the mover itself, when once moved, is far more powerful and quicker. For it must necessarily be that that moves more slowly which is repelled. These words, although they seem to have been spoken of that force which repels a moved thing in the contrary direction, are nevertheless also suited to that which attempts at the beginning to move an immobile thing. For there is a resistance in the thing which is transferred from rest to motion, as it were a certai- Z

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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 in the very act of motion: for in that case the mover is greatly aided by the motion of the thing itself, since the motion cooperates with the mover, acting upon the very thing moved. And the moved thing somehow increases the power of the mover; for what it would suffer from the mover, the thing that is moved acts of itself. Let the plane AB of the horizon be given, upon which a certain mass CD rests. Now let a certain force be applied at E, which drives the mass 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; but once this resistance is overcome, the mass, having begun to move, is carried to F and is moved. Therefore the force, which at the beginning had overcome the resistance of the unmoved thing, goes on to push the moved thing more easily: for there are, in a certain sense, two movers, namely the mover itself and the motion by which the moved thing is moved. Thus it will 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 swifter in its very speed, the more it is moved within the motion itself. From this is solved that most difficult question among the physicists: namely, why in natural motion the speed continually increases; for there Nature is the mover, and the same is inseparable from the thing moved, so it presses continually, at first indeed more slowly, but afterward, for the cause we have mentioned, ever and ever more swiftly. Motion therefore takes place in motion; and since this motion is always increased by the mover and by motion itself, it grows in its progress to an immense degree. Certainly, no one, as I think, will deny that the cause of the increased speed is this: that the moving power moves the moved thing in the motion itself; for the moved body acquires weighty...

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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 Natura 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, id tum 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 together with motion, makes the motion itself easier, and thereby faster. Simplicius discusses this in book 7 of the Physics, chapter II, in his exposition of Aristotle’s books On Nature. QUESTION XXXII. Here the question is asked: Why do those things which are thrown cease from their motion? This too is a purely physical problem. To it belong the matters treated by the Philosopher in book 8 of the Natural Things and in book 1 of On the Heavens. He proposes three reasons, though hesitantly: whether because the impelling force ceases, or because of a pulling back, or because of the inclination of the thing thrown, when that is stronger than the thrower’s power? Whatever the Philosopher may say, this is certainly most clear: thrown things cease from motion because the impression, by whose impulse and power they are carried, is not a natural motion, but merely accidental and violent; and whatever is accidental and violent, and therefore not natural, is not perpetual. Therefore that accidental impression ceases, and as it gradually ceases the motion of the thrown object grows weak, until it comes to rest altogether. We also note this, what we have seen many fail to observe: namely, that violent motion, when violence predominates, does not differ from natural motion, and therefore is slower at the beginning; afterward it becomes faster in the very motion itself, and finally, as the impressed violence gradually slackens, slower, until the impetus, and with the impetus the motion, disappears, and the thing moved itself attains rest. Hence we are also taught by experience that this is done more violently with thrown objects if it occurs a little farther from the beginning, and that then it is altogether most harmless when it occurs there where the thrown object, from motion fully acquired, has attained the highest 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 vtar, 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, perquam familia- re, quamobrem ex ijsdem prorsus soluitur principijs. QVAE-

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180 IN MECHAN. ARIST. PROBL. them. Hence we see that even boys themselves, taught by Nature, when they try to break nuts or something else by striking them against a wall, step back from the wall by a moderate distance. But if you ask them why they do this, they will answer, so that the blow may be made stronger and more effective. Picolomineus 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 along in no special manner when the impeller is otherwise not continuing? He solves it, saying, namely, that because the first thing, that is, the impeller itself, brings it about that the other, namely the projectile itself, impels that other; and this latter (that is, the projectile) impels the other, that is, the air itself or the medium, which is pushed away by the projectile. Motion ceases when the matter has reached the point where the motion impressed on it by the thrower can no longer move the projected body any further, and likewise the thing itself, namely the air, can no longer be repelled. Or also when the weight of the very projectile declines more by its own inclination than the force of the impeller is advancing onward. In any case the matter is sufficiently clear. For the impressed motion is accidental; but the principle that resists violent motion is natural and inseparable from the mover itself. Therefore, when what is natural prevails, what is accidental gradually slackens, and from this comes the rest of the projectile. This too is a purely physical problem, and one very familiar to the preceding question, which we have just discussed, and for that reason it is solved from exactly the same principles. QVAE-

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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 antorsus 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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EXERCITATIONES. 181 QUESTION XXXIV. Why can neither very small things be thrown far, nor very large things either, but the things thrown must have some proportion to the strength of the thrower? He elegantly dispels the doubt, saying: Is it not because what is thrown and propelled must resist that from which it is propelled? But that which yields nothing in its own magnitude, or resists nothing because of its weakness, does not produce throwing nor propulsion. For what far exceeds the strength of the mover does not yield at all. But what is far weaker offers no resistance and does not receive the impression. He adds another reason as well, namely, that as much air is moved by that which is carried as it moves downward into the depth (that is, into that farther part of the air to which it is carried); for indeed a projectile at the beginning, while it is being carried, drives the air away, but it would not drive it if nothing were moving. It therefore happens, as the Philosopher concludes, that these projectiles are moved less because of contrary causes: for what is very small moves nothing, its weakness hindering it; but what is very large moves nothing for the opposite reason, namely because it is not moved at all on account of its own size. Whence it follows that there must above all be a proportion between the projectile and the thrower in regard to motion. Picolomineus explains these same matters excellently in his Paraphrase. To this question of projectiles we add the following: Why are bodies thrown, when they are irregular in figure, not equally heavy to themselves in their parts, so that in the very motion they are carried forward without violence with the heavier part foremost, and downward unnaturally, and, while they are turning in their course, emit a sound. Let there be a ball ABCD, whose center E is composed of unequal matter, namely BCD light, and ABD heavy. Therefore it 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 quæ reuerberatione fit corporis rotatio dum fertur, & ipse sonitus, quem Græci poizov Rhœzum appellant. Ad hanc quoque speculationem pertinet, Curlapides 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, & inde 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. let there be the center of gravity and the center of mass; let the center of gravity be F. Let the body descend, the restraint having been removed, along the straight line AG. And since heavier things tend more downward, if, from the beginning of the motion, the heavier part is above, then in the very descent the ball will turn, and it will not keep its position until the upper part, namely that which is heavier, has become downward, as may be seen in the ball HIK, whose center is G, the heavier part HIK. But if the same ball is forcibly driven by lateral motion toward N, the heavier part also will turn to that side. For when the center of mass or magnitude has been made 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, in their motion they will turn, and their heavier part will come to the front. Moreover, the sound produced in the motion itself is caused by this, that the irregular body begins to turn from the start, and in the turning itself, while it is carried along, it strikes the air, and is in turn struck back by it; from this striking back there comes the rotation of the body while it is carried along, and the sound itself, which the Greeks call poizov Rhœzum. To this speculation also belongs the fact that pebbles thrown onto the surface of water do not sink at once, but after skimming along the surface of the water several times, rebound from it. Let AB be the surface of the water, the thrown stone C, touching the surface of the water at D, and from there rebounding to E, soon again touching the same surface at F, and rebounding to G, until, the violent motion ceasing, it sinks. Surely the stone C, thrown toward 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- cto 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 it were repelled by the denser medium, namely water, it would penetrate through D into H. But since this resists it, and the impulse still continuing, it is carried into E at nearly equal angles. I say “nearly,” since ADC is greater than EDF itself, because the force is not the same, but less than that which drives it from D into E. Therefore, while the impulse by which it is driven among the cavities continues, the rebounds themselves take place; and when that ceases, the rebounds cease, and the stone by its own weight sinks down. This also relates to the question, Why does a playing ball thrown onto the plane of the horizon rebound at equal, that is, right, angles? Let AB be the plane of the horizon, onto which, from point C, through the perpendicular line CE, the ball DE, whose center of gravity is F, falls or is thrown. But it touches the plane at the 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 on the plane of the horizon, where E is, and at the same point it rebounds; wherefore, since the falling and the acting are divided into equal and equally weighted parts, and likewise the reacting and rebounding are likewise divided into equal and equally weighted parts, thus it rebounds by reacting as it acted in falling, that is, at equal angles; which was to be demonstrated. Now let there be some plane inclined to the horizon, such as GH, and onto it let the same ball fall or be thrown. I say that from the same inclined plane it rebounds at equal angles, but not right angles. Thus-

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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 reactione. Et sane fieri non potest,

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184 IN MECHAN. ARIST. PROBL. Certainly, a falling ball will not touch the plane in E; for there would be GH, where AB touches it. 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. Likewise, through I let IK be drawn parallel to EC, cutting the circumference of the ball at K. Therefore the ball acts and reacts at the point I unequally in unequal ways. For the parts KDLEI and IK are unequal, since IK cuts the circle not through the center; therefore, in rebounding, it is repelled unequally, but according to the inequality of those parts. But let the straight line in the circle LI be equal to IK. Therefore LEI will be equal to IK, and the whole KDLI equal to the whole IKDL. Thus, as the action is through descent along the straight line KI, so the rebound is through ascent from IL. And 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 IK is equal to the segment IEL, and the straight line LI is equal to the straight line IK, and LF equal to FK, and FI common, the triangle LFI is equal to the triangle IFK. Wherefore the angle FIL is also equal to the angle FIK; but G1F and H1F are right angles, therefore the remaining angles LIG and kIH are equal to one another, and less than a right angle, which was to be shown. From this we conclude that the more the plane departs from the horizontality of the horizon, 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 equally distant from the horizon, the ball is projected into it not perpendicularly but according to some angle. These things are indeed shown to happen by the demonstration. Nevertheless, since the projected ball is material, and therefore neither equal, nor equally weighted and resisting with its own gravity, it does not rebound at equal angles according to the rule, but at somewhat smaller ones in the rebound, the force being diminished, as it were, in the reaction itself. And indeed it cannot happen,

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EXERCITATIONES. 18 potest, pilam à plano resiliententem eo peruenire vnde à principio discesserat; Id enim si daretur, æterna quoque pilæ ipsius daretur resilitio, & paullatim vi & impetu remittente per parua interuallamotus 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. 18 can, the ball rebounding from the plane may arrive at the point from which it had originally departed; for if this were granted, the ball’s rebound would also be granted as eternal, and little by little, as force and impulse diminished, it would move by small intervals, until the thing that was being moved should come to rest entirely. QUESTION XXXV. In this final Problem Aristotle asks why the things that are carried in whirling waters are all at last carried to the middle? He solves it by three reasons; of which the first is this: Whatever is carried has a magnitude, whose ends are in two circles, one in the smaller, the other in the larger. And since the larger is faster, the middle magnitude is not carried equally, but is driven away by the larger, indeed, and drawn back by the smaller; whence the motion of the magnitude becomes transverse, and the magnitude itself is driven toward the inner circle, and so in the same way, being thrust from the larger into the smaller, it is carried only to the center, and there it rests. Let there be a vortex AB, whose center is C, the magnitude that is carried is AD, the larger circle AFB, the smaller DHEG. The velocity therefore at A is greater than the velocity which is at D; therefore the upper end A of the magnitude is more swiftly carried off in A than its lower end D is in D. The greater velocity therefore of the larger circle drives A toward F. But the slowness of the smaller circle draws D back toward the parts G. Thus the magnitude is turned by the intervening and drawing-back circle, until the extremi- 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 grauitatem 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 quide[m], attamen haudquaquam 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. the point A would be in the smaller circle where H is, D indeed where I is, and so on in the same way where KL is, until it is gradually carried to the center C, that is, by a passage from the larger into the smaller cir- cle. The second reason is this: because that which is carried has a similar relation to all circles about the center, that is, in whatever circle it is carried around the same center. But all circles move, while the center stands still; therefore, by motion, at length that which is moved must come to the place of rest, that is, to the center itself. Third, because the speed, and therefore the force, of the circles which are formed in vortices is not equal, but the outer is always swifter and more violent than the inner, and because a body moved with equal gravity always stands in different relation to the circles by which it is moved, it is therefore sometimes overcome and sometimes overcomes: it is overcome by the swifter circles, it overcomes the slower ones. Therefore, since, resisting its own gravity, it does not follow the motion of the larger circle at all, it is thrown back to a slower one, that is, an inner one, and so on, until at last it reaches the center itself, where, neither overcoming nor overcome, it rests. These are the reasons, though set forth most obscurely, which, as we have said, Aristotle uses. Truly keen they are indeed, yet by no means to be accepted without further ado. For first it seems false that he asserts vortices to be circles, and to be formed and revolve around the same center. For they are rather spirals, which, beginning from the outer and more remote part, being wound round spirally, at length come down to the innermost part, which is the middle and serves the place of the center. When this truth is known, every difficulty is completely removed. For since the things that are carried are carried by the water, but the water itself is carried spirally, it is necessary that those things 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 impe- tu 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, quia quam ex ripæ ipsius obiectu contineat quiescentem, Cursus vero fluminis liber & rectus, sit inter lineas AC, DE. Itaque dum aqua 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 flumina nauigantes obseruauimus, non eodem permanent loco, sed rapientis aquæ motum secundantes, paullatim in currentem aqua delati A a 2

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EXERCISES. 187 …. These things will be clearer, however, if anyone considers in what way vortices are formed. Let there be a curved and deep bank ABCD of some river. Now the mass of water EFDC, rushing along, because it is carried with great force toward C, follows the nature of the bank itself and is whirled around in a turbinating motion; and when it has gone beyond the place or bank B, following the beginning of the rotation, it twists spirally into itself, and produces a vortex GHFIK, whose center is at K. A spiral or vortex is formed also for another reason, namely, from quiet and moving water. For let there be a bank of a river ABC, making a hollow, because it contains within the obstacle of the bank itself still water, while the course of the river is free and straight between the lines AC, DE. Thus, while the water AC is swiftly carried toward the parts A, it laterally drives the still water of ABC along line CA, and indeed carries along with itself the part it touches, for example from F to G. Therefore, when the water has been carried onward and, flowing from F toward G, the still water resists it laterally in some measure, and repels the current from G to H. Once the spiral motion has begun, the water is whirled around according to the line GHK, until it reaches the center I, where the parts of the whirling water touch one another. Moreover, these vortices or spirals, which we have observed while sailing along the Po, the Adda, and great rivers, do not remain in the same place, but, following the motion of the water that carries them, are gradually carried along 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 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 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 AE repellit, nec E retrahit, siquidem eodem tempore quo A permeauit circulu ADB, eodem & E per- currit circulum EFG. Itaq; 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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188 IN MECHAN. ARIST. PROBL. are carried along, vortices of this kind are also very formidable to sailors, even in the sea, concerning which the Poet in the first book of the Aeneid: --- but there the whirlpool likewise Whirls it, driving it round, and the rapid vortex devours it in the sea. And the same also concerning the vortices which are formed in rivers, in book 7: --- the Tiber, fair with its winding streams, With swift whirlpools, and rich in yellow sand, Breaks forth into the sea. Now such vortices arise in the sea partly from hidden causes, partly also from the force of waters moving against one another and agitating each other. But, these matters having been explained, we shall suitably return to what Aristotle had said. We say, therefore, that his first reason seems of little weight, since the water is not carried round through actually distinct circles, but through its own whole mass all at once. For let there be the vortex AB, whose center is C, and semidiameter CA; let the rotation of the whole water be made from CA toward the parts D, and in the line AC let there be some body AE carried around by the rotation of the water, between the greater circle ADB and the smaller EFG. But ADB is moved more quickly than EFG itself; therefore the upper part of that body, where A is, is carried more quickly than the lower part, where E is. But this neither drives back AE nor draws back E, since at the same time that A has passed through the circle ADB, E also traverses the circle EFG. Thus when A has returned to A and E, the returned point will be in E, with no change having been made in the body E as to position, which Aristotle intended. To

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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, rotatio- nem 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 we reply, however, that it is not for the reason that all circles are equally carried around the center, unless some other foreign force intervene, which, driving them from the outer circles, acts toward the middle. The third reason also seems to fail. For let there be a vortex AB, whose center is C, and let there be some body E, whose nature is somewhat fitted to resist rotation. Since, therefore, its resistance is somewhat overcome by the water dragging 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, transferred to H, the body itself will not be in H, but in G. Thus the body will complete the rotation more slowly than the water itself; nevertheless, unless some other cause is present, it will not on that account be carried to the middle. Moreover, the effect and cause of these vortices may be observed if, in some vessel filled with water, we shall have stirred the water itself in a circular motion with a stick or with the hand; for a vortex will be formed, and if we cast into the moving water something light, by the cause we have mentioned it will be carried into the motion itself, that is, to the center of the vortex or whirl. We set these things forth as true, and perhaps we are mistaken. Certainly to contradict a Philosopher of such authority seems to be great boldness, or rather madness. Yet whatever the case may be, to have labored for the sake of the most beautiful truth will not, I think, be wholly lacking in 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 slight experience in mechanical studies know that the method of finding two mean proportional terms is not only useful, but altogether necessary. For there is no other way by which bodies may be increased or diminished proportionally, while preserving shape and similarity. For this reason it came about that, in the discovery of these means, very eminent men labored greatly, both in the most ancient times and also in later ages. For Plato, Eudoxus (whose method Eutocius rejected), Hero of Alexandria, Philon of Byzantium, Apollonius, most distinguished geometers, Diocles, Pappus, Sporus, Menaechmus, Archytas of Tarentum, Plato’s contemporary; Eratosthenes and Nicomedes devised various methods for finding them, all of whose procedures, instruments, and demonstrations Eutocius himself diligently collected and set forth in those Commentaries which he wrote on Archimedes’ books On the Sphere and Cylinder. We, however, after having examined all these carefully and weighed them most diligently, have found that almost all of them accomplish the task by trial, which is indeed very laborious and very troublesome for those who carry it out. Therefore, since we had found a method, a practical one, by which the worker is most safely and most easily guided to the required means themselves, we judged it shameful to withhold this from students of this very beautiful discipline. And if anyone should say that the use of ballistae, catapults, scorpions, and other machines of this sort has long since ceased among us, and that therefore this problem seems superfluous, we reply that there is no other way by which the balls of bronze artillery may be increased or diminished while preserving the ratio of weight, and that there are countless things which, if they are to be properly carried out, are wholly in need of this investigation. We present something 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 It seemed not inappropriate to annex to our mechanical Exercises these things also. But it is time, with these few remarks premised, to proceed conveniently to the matter itself. From two proportional terms, the first and fourth, to find two means between them in continued proportion. LET the first of the given terms be AB, the fourth BC, between which the second and third are to be found. Let the straight line DE be drawn, to which from a point F, taken at will, let the perpendicular FG be let fall. Then from F toward D let the fourth BC be doubled, and let FH be such; afterwards from H let HI be drawn parallel to FG, and from HF let HK be cut off, equal to half of the fourth BC itself. After this, with the point K, and at a space equal to half of the first given term, let a point L be marked on the line HI, and let FM be made equal to HL, and let KM be joined. These things being thus arranged, let there be prepared separately a sheet or some ruler NO, on whose side let OP be taken, equal to half of the first given term, that is to say, to KL itself. Then let the side of the ruler be applied at point L, but let the extremity O be carried continually along the straight line EK, toward K, meanwhile never

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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 fere lineis 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 the rule is moved away from point L, and this until point P, meeting it, falls on line KM, namely where Q, the farther end, is found at R. Therefore, when point R is marked on line EK, what was sought will be obtained. Therefore AB will be the first, RK the second, QL the third, BC the fourth. This procedure is demonstrated by the same principles by which Nicomedes showed his own method with the Conchoid. He makes an instrument, by which he draws the Conchoid, from which afterward he seeks the two mean proportionals. But we neither construct an instrument nor draw a Conchoid, and with hardly two lines we complete the matter, so that almost no one would not say that this thing which we teach is altogether foreign to the Nicomedian method. But in order that a demonstration of the operation, which we have shown, may be had, we shall here set out 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, let two mean proportionals be taken in continued proportion in this way. Let parallelogram ABCD be completed, and let each of AB and BC be bisected at the points L and E; and joining LD, let it 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, which is equal to AL. Let FG be joined moreover, and let CH be parallel to it; and the angle KCH will be equal to 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 when kD is joined and produced, let it meet BA, produced, at point M. I say that as DC is to Ck, so Ck is to MA, and MA to AD. For since BC is bisected at E, and Ck is added to it, the rectangle BkC by sec. 6 of book 2: 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 a square from EF being added to the common one from Ex., 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 twice 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 also by composition, as ML is to LA, so FK is to KH; but AL is posited equal to HK, since also to CF; therefore also FK by 9. book 5. will be equal to ML, and the square from ML equal to the square from FK. But to the square from ML the rectangle BMA together with the square from AL is equal, and the square from Fk has been shown equal to the rectangle BkC together with the square from BC

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194 IN MECH. ARIST. PROBL. EXERCIT. quadrato ex CF, quorum quidem quadratum ex AL æ- quale est quadrato ex CF, ponitur enim AL, ipsi CF æ- qualis, ergo reliquum BMA rectangulum æquale est reli- quo Bk C. 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, ea ratio est, quod LR vt in prima figura est, sit æqualis ipsi LM secundæ figuræ, in demonstratione Pappi, ex quibus deemptis QR & LA, quæ sunt æqua- les, reliqua QL primæ figuræ æqualis est AM secundæ fi- guræ, hoc est, ipsi tertiæ proportionali: Est igitur, vt in pri- ma figura dicehamus, AB prima, kR secunda, QL tertia, BC quarta. Vides igitur tu quilegis, nos ex Nicomedis demon- stratione (quatenus ad praxin pertinet) superflua resecas- se, & absque Conchoidis instrumento lineaue rem ipsam confecisse, idque non tentantes, vt alij, sed progre- dientes, & quasi manuductos quæsi- tum inuestigasse. FINIS.

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194 IN MECH. ARIST. PROBL. EXERCIT. quadrate from CF, whose square from AL is equal to the square from CF; for AL is set equal to CF itself, therefore the remaining rectangle BMA is equal to the remaining BkC. Thus, as MB is to Bk, so is Ck to MA. But as MD is to Bk, so is DC to Ck. Hence, as DC is to Ck, so Ck is to MA. And as MD is to Bk, so is MA to AD. Therefore, as DC, the first, is to Ck the second, so is Ck the second to MA the third, and MA the third to AD the fourth, which was to be demonstrated. This is Pappus. But what we said in our practice, namely that QL is the third, the reason is that LR, as it is in the first figure, is equal to LM of the second figure in Pappus’s demonstration; and from these, after QR and LA are removed, which are equal, the remaining QL of the first figure is equal to AM of the second figure, that is, to the third proportional. Therefore, as we said in the first figure, AB is the first, kR the second, QL the third, BC the fourth. You see, then, reader, that we have, from Nicomedes’ demonstration, insofar as it pertains to practice, cut away what was superfluous, and without the instrument of the conchoid or a line have accomplished the thing itself; and that not by merely attempting it, as others do, but by proceeding, and, as it were, led by the hand, seeking out the desired result. FINIS.