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version 1.42, 2002/09/23 21:11:43

version 1.48, 2002/10/09 23:05:34

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<cvs_version/>

</info> <text> <front><section>

<s id="id.000001">BERNARDINI BALDI VRBINATIS <lb/>GVASTALLÆ AB<lb/>BATIS <lb/><emph type="italics"/>IN<emph.end type="italics"/></s>

<s id="id.000002">MECHANICA ARISTOTE<lb/>LIS PROBLEMATA <lb/>EXERCITATIONES:</s>

<s id="id.000003"><emph type="italics"/>ADIECTA SVCCINCTA NAR<lb/>ratione de autoris vita & &longs;criptis.<emph.end type="italics"/></s><figure id="id.007.00.002.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.002.1.jpg"></figure>

<s id="id.000003"><emph type="italics"/>ADIECTA SVCCINCTA NAR<lb/>ratione de autoris vita & &longs;criptis.<emph.end type="italics"/></s><figure id="id.007.01.001.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.001.1.jpg"></figure>

<s id="id.000004"><emph type="italics"/><gap/>NTIAE.<emph.end type="italics"/></s>

<s id="id.000004"><emph type="italics"/>MOGVNTIAE.<emph.end type="italics"/></s>

<s id="id.000005">Typis & Sumptibus Viduæ Ioannis Albini.</s>

</section><section>

<s id="id.000007"><emph type="italics"/>NOBILISSIMO AC GENE<lb/>ROSO DOMINO<emph.end type="italics"/>D. ADAMO PHILIP<lb/>PO BARONI A CRON<lb/>BERG, EQVITI, SACRÆ CÆSA<lb/>REÆ MAIESTATIS, ET SERENISSIMI Principis Archiducis Alberti Camerario intimo & c. <lb/></s>

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<s id="id.000011">Is <lb/>vir in omni &longs;cientiæ genere, at maxime in Ma<lb/>thematicis di&longs;ciplinis fuit ver&longs;ati&longs;&longs;imus, quod <lb/>multa ab eo præclare &longs;cripta te&longs;tantur opera, <lb/>ex quibus paucula edita, reliqua vero &longs;pera<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.004.jpg"/>mus &longs;uo tempore in publicam luce<gap/> produ<lb/>cenda. </s>

<s id="id.000011">Is <lb/>vir in omni &longs;cientiæ genere, at maxime in Ma<lb/>thematicis di&longs;ciplinis fuit ver&longs;ati&longs;&longs;imus, quod <lb/>multa ab eo præclare &longs;cripta te&longs;tantur opera, <lb/>ex quibus paucula edita, reliqua vero &longs;pera<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.003.jpg"/>mus &longs;uo tempore in publicam luce<gap/> produ<lb/>cenda. </s>

<s id="id.000012"> Cum vero nemini &longs;it ob&longs;curum Nobi<lb/>li&longs;&longs;imæ ac Genero&longs;æ Dom. V.^{ræ} id &longs;emper <lb/>extiti&longs;&longs;e familiari&longs;&longs;imum, vt tum dome&longs;ticum <lb/>otium, tum maxime peregrinationes, quibus <lb/>totam pæne Europam &longs;umma cum laude <lb/>circum&longs;crip&longs;it, tum variarum linguarum per<lb/>fecto v&longs;u, tum Mathematicarum di&longs;ciplina<lb/>rum notitia & exercitio redderet <expan abbr="iuc&utilde;diores">iucundiores</expan>, <lb/>nulla me tenet dubitatio quin & Baldum Vr<lb/>binatem no&longs;tris typis loquentem in hoc iti<lb/>nere, quod à Deo felici&longs;&longs;imum Nobili&longs;&longs;imæ <lb/>ac Genero&longs;æ Dom. V.^{ræ} precor, in &longs;uum comi<lb/>tatum ac tutelam beneuolo animo &longs;it admi&longs;<lb/>&longs;ura.</s>

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<s id="id.000014">hanc meam typographiam plurimis iam re<lb/>tro annis de inclytæ familiæ Cronbergicæ tu<lb/>tela gloriantem, &longs;uo fauore pro&longs;equatur, vi<lb/>duæque afflictæ fortunis beneuole ad&longs;piret.</s>

<s id="id.000015"> <lb/>Sic Deus Nobili&longs;&longs;. & Genero&longs;am Dom. V.^{ram} <lb/>illu&longs;tret omnibus bonis, eamque R.^{mo} & Ill.^{mo} <lb/>Principiac Domino meo Clementi&longs;&longs;imo, D. <lb/>Ioanni Suicardo Archiepi&longs;copo Mogunti<lb/>no Principi Electori ac per <emph type="italics"/>G<emph.end type="italics"/>ermaniam Ar-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.005.jpg"/>chicancellario &c. </s>

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<s id="id.000018">Anno 1621.26.Martij. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.006.jpg"/><figure id="id.007.00.006.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.006.1.jpg"></figure>

<s id="id.000018">Anno 1621.26.Martij. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.005.jpg"/><figure id="id.007.01.005.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.005.1.jpg"></figure>

</section>

</front>

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<s id="id.000021">Ari&longs;totelis nomini a&longs;cribitur <lb/>Commentarius, licet nonnulli, &longs;itne Philo&longs;ophi <lb/>illius præclarissimi & acutissimi labor, an non, <lb/>adfirmare &longs;ubdubitauerint. </s>

<s id="id.000022">Ari&longs;totelis tamen <lb/>e&longs;&longs;e omnes ferè meliores con&longs;entiunt: Idque tum <lb/>ex phra&longs;i, & explicatione, quæ Ari&longs;totelem &longs;a<lb/>piunt, tum iudicio &longs;ubtilitatis & rationum, qui-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.007.jpg"/>bus quæ&longs;tiones ip&longs;æ ingenio&longs;issimè diluuntur. </s>

<s id="id.000022">Ari&longs;totelis tamen <lb/>e&longs;&longs;e omnes ferè meliores con&longs;entiunt: Idque tum <lb/>ex phra&longs;i, & explicatione, quæ Ari&longs;totelem &longs;a<lb/>piunt, tum iudicio &longs;ubtilitatis & rationum, qui-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.006.jpg"/>bus quæ&longs;tiones ip&longs;æ ingenio&longs;issimè diluuntur. </s>

<s id="id.000023">Vi<lb/>detur autem mihi, rem accuratius exploranti, &longs;a<lb/>tis veri&longs;imile (nullum enim habeo opinionis hu<lb/>ius a&longs;&longs;ertorem,) &longs;ectionem e&longs;&longs;e hanc, & partem <lb/>quandam eius operis nobilissimi, quod idem au<lb/>ctor De Problematibus edidit, & hanc, ne&longs;cio <lb/>quam ob cau&longs;am; ni&longs;i fortè quod tractatio merè <lb/>Phy&longs;ica non &longs;it, à reliquo corpore di&longs;tractam at<lb/>que reuul&longs;am. </s>

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<s id="id.000025">Quibus con&longs;ideratis magnopere <lb/>&longs;ubit mirari, cur ij qui po&longs;t Ari&longs;totelem floruêre <lb/>atque vixere, Mechanici, Archimedes, Athenæus, <lb/>Heron, Pappus, & cæteri, nullam huius libelli fe<lb/>cerint commemorationem: & &longs;anè debuerunt; <lb/>neque enim à vero est dissimile, ip&longs;os per hunc ali<lb/>quatenus profeci&longs;&longs;e. </s>

<s id="id.000026">Verum enim uero cum inge<lb/>nui illi fuerint homines, & nullatenus obtrecta<lb/>tores, credendum potius est, Comment ariolum i<lb/>&longs;tud, eorum æuo, paucis cognitum, alicubi in Bi<lb/>bliothecis latui&longs;&longs;e: etenim cætera quoque Ari&longs;tote<lb/>lis &longs;cripta, po&longs;t vetu&longs;ta illa tempora, ante Ale<lb/>xandrum Aphrodi&longs;ien&longs;em, à multis fui&longs;&longs;e igno-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.008.jpg"/>rata non dubitamus. </s>

<s id="id.000026">Verum enim uero cum inge<lb/>nui illi fuerint homines, & nullatenus obtrecta<lb/>tores, credendum potius est, Comment ariolum i<lb/>&longs;tud, eorum æuo, paucis cognitum, alicubi in Bi<lb/>bliothecis latui&longs;&longs;e: etenim cætera quoque Ari&longs;tote<lb/>lis &longs;cripta, po&longs;t vetu&longs;ta illa tempora, ante Ale<lb/>xandrum Aphrodi&longs;ien&longs;em, à multis fui&longs;&longs;e igno-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.007.jpg"/>rata non dubitamus. </s>

<s id="id.000027">Habemus &longs;iquidem, Stra<lb/>bone te&longs;te, lib. 13. Ari&longs;totelis, & Theophra&longs;ti bi<lb/>bliothecam, po&longs;t ip&longs;ius Theophra&longs;ti dece&longs;&longs;um, ad <lb/>Neleum quendam Scep&longs;ium, Cori&longs;cifilium, qui <lb/>eius fuerat auditor, perueni&longs;&longs;e; po&longs;t hæc libros, <lb/>blattis olim, & humore corruptos, Apelliconi Te<lb/>io venditos, & ab eo Athenas translatos, tum <lb/>Athenis captis in Syllæ pote&longs;tatem deacni&longs;&longs;e, eo&longs;<lb/>que tandem à Sylla acceptos, Tyrannionem <lb/>Grammaticum, vt potuit meliùs emendatos, <lb/>promulga&longs;&longs;e. </s>

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<s id="id.000032">Olympiadem floruit, quamobrem po&longs;t Ari&longs;tote<lb/>lem Olympiadas XL. hoc est, annos ferè CLX. <lb/></s>

<s id="id.000033">I&longs;thæc autem con&longs;iderantibus, facile e&longs;t cogno&longs;ce<lb/>re facultatis huius nobilitatem, atque dignitatem; <lb/>quippe quod &longs;ummus Philo&longs;ophus non modo eam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.009.jpg"/>probauerit, &longs;ed etiam &longs;uis acutissimis lucubra<lb/>tionibus illu&longs;trauerit. </s>

<s id="id.000033">I&longs;thæc autem con&longs;iderantibus, facile e&longs;t cogno&longs;ce<lb/>re facultatis huius nobilitatem, atque dignitatem; <lb/>quippe quod &longs;ummus Philo&longs;ophus non modo eam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.008.jpg"/>probauerit, &longs;ed etiam &longs;uis acutissimis lucubra<lb/>tionibus illu&longs;trauerit. </s>

<s id="id.000034">Hanc porro tractationem <lb/>&longs;ubiecto quidem Phy&longs;icam e&longs;&longs;e, demon&longs;tratio<lb/>nibus verò Geometricam, ip&longs;emet nos docuit <lb/>Ari&longs;toteles, cuius etiam naturæ &longs;unt Per&longs;pecti<lb/>ua, Specularia, Mu&longs;ica, & cæteræ eiu&longs;dem <lb/>modi facultates, quas quidem &longs;ubalternas Peri<lb/>patetici appellant. </s>

<s id="id.000035">Vitruuius Architecturæ <lb/>membrum, vt ita dicam, & portionem quan<lb/>dam facit, ait enim Architecturæ partes e&longs;&longs;e tres, <lb/>Ædificationem, Gnomonicam, Machinatio<lb/>nem. </s>

<s id="id.000036">Est autem Architecturâ quidem inferior, <lb/>paret enim Architecto Mechanicus; attamen &longs;i <lb/>cæteras artes &longs;pectes, Architectonica; hæc enim <lb/>omnes ferè &longs;edentariæ, &longs;ellulariæue, quas banau<lb/>&longs;as Græci appellant, ordine &longs;ubijciuntur, & &longs;a<lb/>nè latissimos i&longs;thæc habet fines; præcipuè autem <lb/>circa eam ver&longs;atur cognitionem, eamque inter <lb/>cæteras ferè principem, quam dixere Centrobari<lb/>cam, quæ quidem ad Centri grauitatem, eiu&longs;que <lb/>&longs;peculationem pertinet: quà in &longs;pecie inter vete<lb/>res primum &longs;ibi vindicauit locum Archimedes, <lb/>mox Heron, deinde Pappus; inter neotericos au-<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.010.jpg"/><emph type="italics"/>tem Commandinus, qui librum de Centro gra<lb/>uitatis &longs;olidorum &longs;crip&longs;it, & po&longs;t eum G. Vbal<lb/>dus è Marchion. Montis, qui non modò ab<lb/>&longs;olutissimum Mechanicorum librum cum maxi<lb/>ma ingenij &longs;ui laude con&longs;crip&longs;it, &longs;ed & Paraphra<lb/>&longs;in in librum Æqueponderantium Archimedis <lb/>egregiè concinnauit Centrobaricam hanc, igno<lb/>tam fui&longs;&longs;e Ari&longs;toteli, &longs;ætis patet. </s>

<s id="id.000036">Est autem Architecturâ quidem inferior, <lb/>paret enim Architecto Mechanicus; attamen &longs;i <lb/>cæteras artes &longs;pectes, Architectonica; hæc enim <lb/>omnes ferè &longs;edentariæ, &longs;ellulariæue, quas banau<lb/>&longs;as Græci appellant, ordine &longs;ubijciuntur, & &longs;a<lb/>nè latissimos i&longs;thæc habet fines; præcipuè autem <lb/>circa eam ver&longs;atur cognitionem, eamque inter <lb/>cæteras ferè principem, quam dixere Centrobari<lb/>cam, quæ quidem ad Centri grauitatem, eiu&longs;que <lb/>&longs;peculationem pertinet: quà in &longs;pecie inter vete<lb/>res primum &longs;ibi vindicauit locum Archimedes, <lb/>mox Heron, deinde Pappus; inter neotericos au-<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.009.jpg"/><emph type="italics"/>tem Commandinus, qui librum de Centro gra<lb/>uitatis &longs;olidorum &longs;crip&longs;it, & po&longs;t eum G. Vbal<lb/>dus è Marchion. Montis, qui non modò ab<lb/>&longs;olutissimum Mechanicorum librum cum maxi<lb/>ma ingenij &longs;ui laude con&longs;crip&longs;it, &longs;ed & Paraphra<lb/>&longs;in in librum Æqueponderantium Archimedis <lb/>egregiè concinnauit Centrobaricam hanc, igno<lb/>tam fui&longs;&longs;e Ari&longs;toteli, &longs;ætis patet. </s>

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<s id="id.000039">Rationalis, &longs;peculationi & <expan abbr="dem&omacr;&longs;trationibus">demon&longs;trationibus</expan>, ex <lb/>Geometricis, Arithmeticis & Phy&longs;icis rationi<lb/>bus, dat operam; Chirurgica vero materiam <lb/>tractat, & &longs;e&longs;e in varias artes diffundit, Æra<lb/>riam, Lignariam, Sculptoriam, Pictoriam, Æ<lb/>dificatoriam, Machinariam & Thaumaturgi<lb/>cam, cæterasque eiu&longs;modi. </s>

<s id="id.000040">Machinatoriæ au<lb/>tem &longs;unt partes Manganaria, qua ingentia <emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.011.jpg"/><emph type="italics"/>transferuntur pondera, tum ip&longs;a Poliorcetica, <lb/>quæ bellicas Machinas ad vrbium expugnatio<lb/>nes, quod vel ip&longs;o nomine profitetur, ædificat. </s>

<s id="id.000040">Machinatoriæ au<lb/>tem &longs;unt partes Manganaria, qua ingentia <emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.010.jpg"/><emph type="italics"/>transferuntur pondera, tum ip&longs;a Poliorcetica, <lb/>quæ bellicas Machinas ad vrbium expugnatio<lb/>nes, quod vel ip&longs;o nomine profitetur, ædificat. </s>

<s id="id.000041">At<lb/>qui hac dere plura &longs;cribere &longs;uper&longs;edemus, ne a<lb/>ctum agamus: quis quis enim minutè magis hæc <lb/>cogno&longs;cere de&longs;iderat, is Pappum adeat libro cita<lb/>to, & Guidum Vbaldumin Præfatione quam <lb/>&longs;uo Mechanicorum Operi præpo&longs;uit. </s>

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<s id="id.000044">Modo, vt audio, Simon <lb/>Sticinus Hollanden&longs;is quædam edidit, quæ ad <lb/>nos minime peruenêre. </s>

<s id="id.000045">Nos demum, omnium, <lb/>tum &longs;cientia, & ingenio, tum ætate, po&longs;tremi huic <lb/>operi manum admouimus; Con&longs;iderantes enim <lb/>Ari&longs;totelem aliis fecerint Mechanici, demon&longs;tra&longs;&longs;e, <lb/>morem huiu&longs;ce facultatis &longs;tudio&longs;is ge&longs;turos nos <lb/>fore arbitrati &longs;umus, &longs;i ea&longs;dem illas quæ&longs;tiones<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.012.jpg"/><emph type="italics"/>Mechanicis, hoc est, Archimedeis probationi<lb/>bus confirmaremus; dum per latissimos faculta<lb/>tis huius campos vagantes, alias quoque i&longs;tis af<lb/>fines dubitationes introducentes &longs;olueremus. <lb/></s>

<s id="id.000045">Nos demum, omnium, <lb/>tum &longs;cientia, & ingenio, tum ætate, po&longs;tremi huic <lb/>operi manum admouimus; Con&longs;iderantes enim <lb/>Ari&longs;totelem aliis fecerint Mechanici, demon&longs;tra&longs;&longs;e, <lb/>morem huiu&longs;ce facultatis &longs;tudio&longs;is ge&longs;turos nos <lb/>fore arbitrati &longs;umus, &longs;i ea&longs;dem illas quæ&longs;tiones<emph.end type="italics"/><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.011.jpg"/><emph type="italics"/>Mechanicis, hoc est, Archimedeis probationi<lb/>bus confirmaremus; dum per latissimos faculta<lb/>tis huius campos vagantes, alias quoque i&longs;tis af<lb/>fines dubitationes introducentes &longs;olueremus. <lb/></s>

<s id="id.000046"><expan abbr="quicquidaut&emacr;&longs;ecerimus">quicquid autem fecerimus</expan> profecerimu&longs;ue, Lector <lb/>optime, boni con&longs;ule, & quia fax per manus tra<lb/>ditur, tu interim de me accipe, vt alijs tradas.<emph.end type="italics"/></s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.013.jpg"/>

</chap>

<chap>

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<s id="id.000054">Parens hac <lb/>filij laude & gloria motus anno 1573. eum ad maiorem in<lb/>genij cultum cape&longs;&longs;endum Patauium mi&longs;it. </s>

<s id="id.000055">Hîc in Ema<lb/>nuelis Margunij familiaritatem &longs;tatim venit, cui porro <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.014.jpg"/>fuit in amoribus. </s>

<s id="id.000055">Hîc in Ema<lb/>nuelis Margunij familiaritatem &longs;tatim venit, cui porro <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.013.jpg"/>fuit in amoribus. </s>

<s id="id.000056">Homeri Iliad. illo Doctore & interpre<lb/>te diligentius quam feci&longs;&longs;et antea, euoluit. </s>

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<s id="id.000063">quo ab&longs;oluto Paradoxa multa Mathema<lb/>tica explicauit. </s>

<s id="id.000064">Fama de Baldi virtutibus di&longs;&longs;ipata Ferran<lb/>dus Gonzaga Molfetræ Princeps & Gua&longs;tallæ Dominus <lb/>c&oelig;pit de illo in &longs;uam familiam a&longs;ci&longs;cendo cogitare, vt qui <lb/>ij&longs;dem caperetur artibus, quibus excellere Baldus inci-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.015.jpg"/>piebat: Itaque opera Curtij Arditij honorifice fuit in au<lb/>lam euocatus, dum vitam non aulicam viueret totus in <lb/>litteras abditus precibus Ve&longs;pa&longs;iani Gonzagæ Sablonetæ <lb/>Ducis ad explanandos Vitruuij libros adactus fuit. </s>

<s id="id.000064">Fama de Baldi virtutibus di&longs;&longs;ipata Ferran<lb/>dus Gonzaga Molfetræ Princeps & Gua&longs;tallæ Dominus <lb/>c&oelig;pit de illo in &longs;uam familiam a&longs;ci&longs;cendo cogitare, vt qui <lb/>ij&longs;dem caperetur artibus, quibus excellere Baldus inci-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.014.jpg"/>piebat: Itaque opera Curtij Arditij honorifice fuit in au<lb/>lam euocatus, dum vitam non aulicam viueret totus in <lb/>litteras abditus precibus Ve&longs;pa&longs;iani Gonzagæ Sablonetæ <lb/>Ducis ad explanandos Vitruuij libros adactus fuit. </s>

<s id="id.000065">quare <lb/><expan abbr="t&umacr;c">tunc</expan> natus de <expan abbr="Verbor&umacr;">Verborum</expan> Vitruuianorum &longs;ignificatione com<lb/>mentarius; in quo minime mirandum &longs;i minuta quæ dam <lb/>pro&longs;equutus fuit, quæ viro magno minus e&longs;&longs;e digna vi<lb/>deantur:illi enim Principi morem ge&longs;&longs;it. </s>

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<s id="id.000076">Tabulam Etru&longs;cam Eugubinam interptetatus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.016.jpg"/>fuit:in ca autem diuinatione, vt aiebat, &longs;ubci&longs;iuas vnius <lb/>men&longs;is horas con&longs;ump&longs;it. </s>

<s id="id.000076">Tabulam Etru&longs;cam Eugubinam interptetatus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.015.jpg"/>fuit:in ca autem diuinatione, vt aiebat, &longs;ubci&longs;iuas vnius <lb/>men&longs;is horas con&longs;ump&longs;it. </s>

<s id="id.000077">De Firmamento & aquis egre<lb/>gie &longs;crip&longs;it. </s>

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<s id="id.000091">In &longs;tudijs &longs;ic a&longs;&longs;iduus fuit, vt &longs;æpe & legeret <lb/>& comederet. </s>

<s id="id.000092">S.Augu&longs;tini libros de Ciuitate Dei ter in-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.017.jpg"/>ter prandium euoluit. </s>

<s id="id.000092">S.Augu&longs;tini libros de Ciuitate Dei ter in-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.016.jpg"/>ter prandium euoluit. </s>

<s id="id.000093">Statim à noctis meridie dum ei vi<lb/>res firmiores e&longs;&longs;ent ad lucubrandum &longs;urgebat. </s>

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<s id="id.000113">De&longs;criptio Palatij Ducum Vrbinarum quod e&longs;t Vrbini. </s>

<s id="id.000114">Poema cui titulus, Lamus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.018.jpg"/>

<s id="id.000114">Poema cui titulus, Lamus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.017.jpg"/>

<s id="id.000115">Carmina pia, quæ in&longs;cribuntur, Anni Corona. </s>

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<s id="id.000143">Luciani de mi&longs;erijs.Aulicorum ver&longs;io. </s>

<s id="id.000144">Oratio ad Romæ con&longs;eruatores pro antiquitatum eius <lb/>Vrbis cu&longs;todia. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.019.jpg"/>

<s id="id.000145">Vniuer&longs;i orbis geographica & Hi&longs;torica de&longs;criptio con<lb/>texta ex &longs;eptingentis & eo amplius &longs;criptoribus. </s>

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<s id="id.000159">Hi&longs;toriæ &longs;cribendæ leges. </s>

<s id="id.000160">Et alia quædam. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.020.jpg"/><figure id="id.007.00.020.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.020.1.jpg"></figure>

<s id="id.000160">Et alia quædam. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.019.jpg"/><figure id="id.007.01.019.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.019.1.jpg"></figure>

</chap>

<chap>

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<s id="id.000170">Leue contrà, quòd Naturâ à centro fertur. </s>

<s id="id.000171">Cæterùm quicquid graue e&longs;t, &longs;ecundum punctum <lb/>e&longs;t, quod Grauitatis centrum dicitur, & hoc duplex, vt <lb/>duplex e&longs;t grauitas, Naturæ, Violentiæ. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.021.jpg"/>

<s id="id.000172">Grauitatis centrum in triplici magnitudine con&longs;i<lb/>derari pote&longs;t, lineari, planà, &longs;olidâ. </s>

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<s id="id.000180">Centrum grauitatis <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> corporis e&longs;t pun<lb/>ctum quod dam intra po&longs;itum, à quo &longs;i graue, mente ap<lb/>pen&longs;um concipiatur, dum fertur, quie&longs;cit, & &longs;eruat eam <lb/>quam in principio habuit po&longs;itionem; neque in ip&longs;a latio<lb/>ne circumuertitur. </s>

<s id="id.000181">Commandinus verò in lib.

<s id="id.000181">Commandinus verò in lib. de centro <lb/>grauitatis &longs;olidorum hoc pacto: Centrum grauitatis v<lb/>niu&longs;cuiu&longs;que &longs;olidæ figuræ, e&longs;t punctum illud intra po&longs;i<lb/>tum, circa quod vndique partes æqualium momentorum <lb/>ad&longs;i&longs;tunt. </s>

de centro <lb/>grauitatis &longs;olidorum hoc pacto: Centrum grauitatis v<lb/>niu&longs;cuiu&longs;que &longs;olidæ figuræ, e&longs;t punctum illud intra po&longs;i<lb/>tum, circa quod vndique partes æqualium momentorum <lb/>ad&longs;i&longs;tunt. </s>

<s id="id.000182">Si enim per tale centrum ducatur planum, fi<lb/>guram quomodolibet &longs;ecans, in partes æquè ponderantes <lb/>eam diuidit. </s>

<s id="id.000183">Nos verò quàm breui&longs;&longs;imè dicimus: <expan abbr="Centr&umacr;">Centrum</expan> <lb/>grauitatis, <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> magnitudinis punctum e&longs;&longs;e intra <lb/>extraue magnitudinem po&longs;itum, per quod &longs;i plano linea <lb/>punctoue diuidatur, in partes &longs;ecatur æqueponderantes. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.022.jpg"/><figure id="id.007.00.022.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.022.1.jpg"></figure>

<s id="id.000183">Nos verò quàm breui&longs;&longs;imè dicimus: <expan abbr="Centr&umacr;">Centrum</expan> <lb/>grauitatis, <expan abbr="vniu&longs;cuiu&longs;q;">vniu&longs;cuiu&longs;que</expan> magnitudinis punctum e&longs;&longs;e intra <lb/>extraue magnitudinem po&longs;itum, per quod &longs;i plano linea <lb/>punctoue diuidatur, in partes &longs;ecatur æqueponderantes. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.021.jpg"/><figure id="id.007.01.021.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.021.1.jpg"></figure>

<s id="id.000184">Diximus, Magnitudinis vt lineæ, plani &longs;olidique; cen<lb/>trum complecteremur. </s>

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<s id="id.000186">Diximus in&longs;uper, intra extraue. </s>

<s id="id.000187">Aliquando enim <lb/>grauitatis centrum extra molem corporis cuius corporis <lb/>centrum e&longs;t, cadit, vt in &longs;equenti figura. </s><figure id="id.007.00.022.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.022.2.jpg"></figure>

<s id="id.000187">Aliquando enim <lb/>grauitatis centrum extra molem corporis cuius corporis <lb/>centrum e&longs;t, cadit, vt in &longs;equenti figura. </s><figure id="id.007.01.021.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.021.2.jpg"></figure>

<s id="id.000188">E&longs;to corpus aliquod <lb/>&longs;uperficiesue ABCDE, <lb/>ducatur linea CF, <expan abbr="diuid&emacr;s">diuidens</expan> <lb/>figuras in partes hinc inde <lb/>æqueponderantes ABC, <lb/>EDC. </s>

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<s id="id.000190">Attamen licet hoc verum &longs;it, intra e&longs;&longs;e dici <lb/>pote&longs;t, quippe quod imaginario quodam, & vt ita dicam, <lb/>virtuali ambitu ACDA contineatur. </s>

<s id="id.000191">Dicebamus, duplex e&longs;&longs;e grauitatis centrum, Natu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.023.jpg"/>râ, Violentià: affirmamus modò, hæc re quidem vnum e&longs;<lb/>&longs;e, & ratione &longs;olum, non autem re ip&longs;a ac &longs;i duo e&longs;&longs;ent con<lb/>&longs;iderari. </s><figure id="id.007.00.023.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.023.1.jpg"></figure>

<s id="id.000191">Dicebamus, duplex e&longs;&longs;e grauitatis centrum, Natu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.022.jpg"/>râ, Violentià: affirmamus modò, hæc re quidem vnum e&longs;<lb/>&longs;e, & ratione &longs;olum, non autem re ip&longs;a ac &longs;i duo e&longs;&longs;ent con<lb/>&longs;iderari. </s><figure id="id.007.01.022.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.022.1.jpg"></figure>

<s id="id.000192">E&longs;to enim grauitatis na<lb/>turalis centrum B, corporis A, <lb/>&longs;ecundum quod dimi&longs;&longs;um, &longs;ua<lb/>pte naturâ cadet in C, &longs;i verò <lb/>corpus violenter impellatur in <lb/>D, aliud acquiret centrum gra<lb/>uitatis ex violentia &longs;ecundum <lb/>quam fertur, motum, in D, <expan abbr="id&emacr;">idem</expan> <lb/>autem &longs;unt re, nempe vnum B, <lb/>duo autem &longs;i violentia & natura &longs;eor&longs;um con&longs;ideren<lb/>tur. </s>

<s id="id.000193">Hæc centra, duo motus &longs;equuntur, rectus vterque, <lb/>Naturalis videlicet, & Violentus. </s>

<s id="id.000194">Tertius ex his mixtus, & <lb/>is quidem non rectus, &longs;ed curuus. </s><figure id="id.007.00.023.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.023.2.jpg"></figure>

<s id="id.000194">Tertius ex his mixtus, & <lb/>is quidem non rectus, &longs;ed curuus. </s><figure id="id.007.01.022.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.022.2.jpg"></figure>

<s id="id.000195">Proijciatur enim violen<lb/>ter corpus graue A &longs;uperante <lb/>igitur violentia, rectà feretur <lb/>in B; ea autem elangue&longs;cente <lb/>paullatim per curuam & mi<lb/>xtam <expan abbr="lineã">lineam</expan> &longs;ecetur in C, qua<lb/>tenus enim ad anteriora fer<lb/>tur, violentia e&longs;t; quatenus ve<lb/>rò ad inferiores partes, naturæ. </s>

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<s id="id.000197">Cæterùm hæc centra, hiqueue motus, naturalis nem<lb/>pe, & violentus diuer&longs;imode &longs;e habent adinuicem. </s>

<s id="id.000198">Si e<lb/>nim graue corpus externâ vi adhibita, centrum mundi <lb/>ver&longs;us impellatur, adiuuabunt &longs;e inuicem Natura, Vio<lb/>lentia, Si autem contra, altera alteri re&longs;i&longs;tet, in motibus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.024.jpg"/>autem ad latus, eo magis pugnabunt, quo magis ab infe<lb/>rioribus ad &longs;uperiora fiet motus. </s>

<s id="id.000198">Si e<lb/>nim graue corpus externâ vi adhibita, centrum mundi <lb/>ver&longs;us impellatur, adiuuabunt &longs;e inuicem Natura, Vio<lb/>lentia, Si autem contra, altera alteri re&longs;i&longs;tet, in motibus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.023.jpg"/>autem ad latus, eo magis pugnabunt, quo magis ab infe<lb/>rioribus ad &longs;uperiora fiet motus. </s>

</subchap1><subchap1>

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<s id="id.000208">Cuneus D. </s>

<s id="id.000209">Cochlea vero E. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.025.jpg"/><figure id="id.007.00.025.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.025.1.jpg"></figure>

<s id="id.000209">Cochlea vero E. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.024.jpg"/><figure id="id.007.01.024.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.024.1.jpg"></figure>

<s id="id.000210">Porro, Cuneum ad libram reducere conatur Ari<lb/>&longs;toteles, quod facit & G. Vbaldus, qui eò refert & Co<lb/>chleam, quippe quod nihil aliud &longs;it Cochlea, quàm Cu<lb/>neus Cylindro inuolutus. </s>

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<s id="id.000214"><emph type="italics"/>De Vecte & Libra &longs;ecundum Ari<lb/>&longs;totelem.<emph.end type="italics"/></s>

<s id="id.000215">Ari&longs;toteles in ip&longs;o Mechanicorum ingre&longs;&longs;u ita &longs;cri<lb/>bit, Mirum videri ab exigua virtute magnum pondus mo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.026.jpg"/>ueri, addito nimirum ponderi pondere, &longs;iquidem & vectis <lb/>e&longs;t pondus. </s>

<s id="id.000215">Ari&longs;toteles in ip&longs;o Mechanicorum ingre&longs;&longs;u ita &longs;cri<lb/>bit, Mirum videri ab exigua virtute magnum pondus mo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.025.jpg"/>ueri, addito nimirum ponderi pondere, &longs;iquidem & vectis <lb/>e&longs;t pondus. </s>

<s id="id.000216">Duplex ergo illi admiratio, &longs;cilicet quòd exi<lb/>gua potentia moueat ingens pondus, idqueue etiam addito <lb/>vectis ip&longs;ius pondere, fiat. </s>

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<s id="id.000229">Ait autem, ab&longs;urdum nullatenus e&longs;&longs;e, &longs;i ex re <lb/>mirabili mirandum quippiam oriatur. </s>

<s id="id.000230">In circulo autem <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.027.jpg"/>quatuor inueniri qualitates admiratione dignas. </s>

<s id="id.000230">In circulo autem <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.026.jpg"/>quatuor inueniri qualitates admiratione dignas. </s>

<s id="id.000231"><expan abbr="Primã">Primam</expan>, <lb/>quod ex contrarijs con&longs;tituatur, mouente videlicet & <lb/>moto. </s>

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<s id="id.000235">Sit enim circulus AB, cuius centrum C, <lb/>&longs;emidiameter AC, &longs;umatur autem in ea punctum D, i<lb/>temqueue punctum E. </s>

<s id="id.000236">Erit itaque in ip&longs;a circulatione D <lb/>tardius E, ip&longs;um verò E tardius A, & ita citius id feretur <lb/>&longs;emper, quod remotius à mouente termino accipitur. </s><figure id="id.007.00.027.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.027.1.jpg"></figure>

<s id="id.000236">Erit itaque in ip&longs;a circulatione D <lb/>tardius E, ip&longs;um verò E tardius A, & ita citius id feretur <lb/>&longs;emper, quod remotius à mouente termino accipitur. </s><figure id="id.007.01.026.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.026.1.jpg"></figure>

<s id="id.000237">Hæc ex illo, quibus ne vltro a&longs;<lb/>&longs;en&longs;um præbeamus non vnica de cau<lb/>&longs;a cohibemur. </s>

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<s id="id.000244">Præterea & &longs;piralis li<lb/>nea, quæ mixta e&longs;t, altero &longs;emidiametri extremo manen<lb/>te, altero vero moto producitur. </s>

<s id="id.000245">Legem itaque circulo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.028.jpg"/>præ&longs;cribit, non quidem quòd hæc extremitas &longs;ter, illa ve<lb/>rò moueatur, &longs;ed quod &longs;ua circulatione &longs;emper &longs;emidia<lb/>meter eandem &longs;eruet longitudinem, quod vel ex ip&longs;a cir<lb/>culi definitione colligitur. </s>

<s id="id.000245">Legem itaque circulo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.027.jpg"/>præ&longs;cribit, non quidem quòd hæc extremitas &longs;ter, illa ve<lb/>rò moueatur, &longs;ed quod &longs;ua circulatione &longs;emper &longs;emidia<lb/>meter eandem &longs;eruet longitudinem, quod vel ex ip&longs;a cir<lb/>culi definitione colligitur. </s>

<s id="id.000246">Ad &longs;ecundum miraculum, &longs;cilicet, quòd in circulo <lb/>circum ferentia, quæ vacua linea e&longs;t, concaua &longs;imul &longs;it, & <lb/>conuexa. </s>

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<s id="id.000249">Ad tertium, quod contrarijs feratur lationibus, an<lb/>tror&longs;um, retror&longs;um, &longs;ur&longs;um & deor&longs;um. </s>

<s id="id.000250">Dicimus, facilè <lb/>&longs;olui, Nullus enim, re bene per&longs;pectâ, affirmauerit circu<lb/>lum contrarijs lationibus moueri. </s><figure id="id.007.00.028.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.028.1.jpg"></figure>

<s id="id.000250">Dicimus, facilè <lb/>&longs;olui, Nullus enim, re bene per&longs;pectâ, affirmauerit circu<lb/>lum contrarijs lationibus moueri. </s><figure id="id.007.01.027.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.027.1.jpg"></figure>

<s id="id.000251">E&longs;to enim circulus ABCD, <lb/>circa centrum E; ponamus ro<lb/>tari, & A ver&longs;us B, exempli gra<lb/>tiâ, antror&longs;um, mouebitur <expan abbr="aut&emacr;">autem</expan> <lb/>& B ver&longs;us C, & C ver&longs;us D, tum <lb/>D ver&longs;us A. </s>

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<s id="id.000253">Sed nec &longs;ur&longs;um vel deor&longs;um, e&longs;t <lb/>manife&longs;tum. </s>

<s id="id.000254">Nihil autem prohibet eundem motum va<lb/>rio re&longs;pectu contrarium dici po&longs;&longs;e, id tamen profectò fie<lb/>ri nequaquam pote&longs;t, nempe A moueri ver&longs;us B, hoc e&longs;t, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.029.jpg"/>antror&longs;um, & eandem eodem tempore ver&longs;us B, id e&longs;t, re<lb/>tror&longs;um; repugnat enim naturæ. </s>

<s id="id.000254">Nihil autem prohibet eundem motum va<lb/>rio re&longs;pectu contrarium dici po&longs;&longs;e, id tamen profectò fie<lb/>ri nequaquam pote&longs;t, nempe A moueri ver&longs;us B, hoc e&longs;t, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.028.jpg"/>antror&longs;um, & eandem eodem tempore ver&longs;us B, id e&longs;t, re<lb/>tror&longs;um; repugnat enim naturæ. </s>

<s id="id.000255">De quarto circuli miraculo, ibi erit nobis &longs;ermo, vbi <lb/>ca perpenderimus primò, quæ Philo&longs;ophus de Circuli <lb/>productione di&longs;&longs;erens in medium profert. </s>

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<s id="id.000257">Circulum quidem duplici notione produci, Natu<lb/>rali videlicet altera, & altera quæ e&longs;t præter naturam, & <lb/>ideo circularem lineam in ter mixtas computari. </s>

<s id="id.000258">Motus mixtus ait, vel proportione &longs;eruata fit, aut <lb/>non; Si proportione &longs;eruatâ, rectam lineam; ea verò non <lb/>&longs;eruata, circularem lineam produci. </s><figure id="id.007.00.029.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.029.1.jpg"></figure>

<s id="id.000258">Motus mixtus ait, vel proportione &longs;eruata fit, aut <lb/>non; Si proportione &longs;eruatâ, rectam lineam; ea verò non <lb/>&longs;eruata, circularem lineam produci. </s><figure id="id.007.01.028.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.028.1.jpg"></figure>

<s id="id.000259">E&longs;to enim rectangu<lb/>lum ABCD, cuius late<lb/>ra in datâ &longs;int proportio<lb/>ne, AD cum AB. </s>

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24. </s>

<s id="id.000265">His ita demon&longs;tratis hac vti videtur Philo&longs;ophus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.030.jpg"/>argumentatione: Si mixtus motus proportione &longs;emotâ, <lb/>rectam producit, &longs;i nunquam &longs;emota, efficiet circulum; &longs;i <lb/>enim modo &longs;eruaretur, modo non, partim recta partim <lb/>non recta produceretur. </s>

<s id="id.000265">His ita demon&longs;tratis hac vti videtur Philo&longs;ophus <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.029.jpg"/>argumentatione: Si mixtus motus proportione &longs;emotâ, <lb/>rectam producit, &longs;i nunquam &longs;emota, efficiet circulum; &longs;i <lb/>enim modo &longs;eruaretur, modo non, partim recta partim <lb/>non recta produceretur. </s>

<s id="id.000266">Ingenio&longs;a quidem argumenta<lb/>tio, ni vitium contineret. </s>

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<s id="id.000269">Cæterùm fal&longs;um <lb/>e&longs;t, a&longs;&longs;erere circulum ex mixto motu nunquam &longs;eruatâ <lb/>proportione produci. </s>

<s id="id.000270">&longs;eruat enim a&longs;&longs;iduè mixtus motus <lb/>quo producitur (&longs;i cum mixto motu producere velimus) <lb/>aliquam proportionem, &longs;ed non eandem. </s><figure id="id.007.00.030.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.030.1.jpg"></figure>

<s id="id.000270">&longs;eruat enim a&longs;&longs;iduè mixtus motus <lb/>quo producitur (&longs;i cum mixto motu producere velimus) <lb/>aliquam proportionem, &longs;ed non eandem. </s><figure id="id.007.01.029.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.029.1.jpg"></figure>

<s id="id.000271">E&longs;to enim recta AB, cui ad rectos <lb/>angulos AC. </s>

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<s id="id.000276">E&longs;t enim AE <lb/>hoce&longs;t DF media proportionalis inter EF, hoc e&longs;t, AD, <lb/>& DB. </s>

<s id="id.000277">Iterum &longs;i &longs;iat motus AC in GH, ad motum H per <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.031.jpg"/>lineam AC, v&longs;que in C, vt &longs;e habet proportio AG ad <lb/>GH & GH ad GB, erit ex motu mixto A in H, nempe in <lb/>eiu&longs;dem circuli circum ferentia AFHB. ex quibus ha<lb/>bemus, circulum ex mixto motu fieri po&longs;&longs;e proportioni<lb/>bus quidem mediarum &longs;eruatis, &longs;ed nunquam ij&longs;dem. </s>

<s id="id.000277">Iterum &longs;i &longs;iat motus AC in GH, ad motum H per <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.030.jpg"/>lineam AC, v&longs;que in C, vt &longs;e habet proportio AG ad <lb/>GH & GH ad GB, erit ex motu mixto A in H, nempe in <lb/>eiu&longs;dem circuli circum ferentia AFHB. ex quibus ha<lb/>bemus, circulum ex mixto motu fieri po&longs;&longs;e proportioni<lb/>bus quidem mediarum &longs;eruatis, &longs;ed nunquam ij&longs;dem. </s>

<s id="id.000278">Vera hæc procul dubio &longs;unt; nihilominus, veluti ad <lb/>rectam producendam mixtus motus non e&longs;t nece&longs;&longs;arius, <lb/>licet mixto motu produci po&longs;&longs;it, ita ne que ad circularem, <lb/>& ideo verum non e&longs;&longs;e quod a&longs;&longs;erebat Philo&longs;ophus, cir<lb/>culum ex mixto motu proportione nunquam &longs;eruatâ ne<lb/>ce&longs;&longs;ariò produci. </s>

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<s id="id.000280">Ait autem; &longs;i duobus ab eadem potentia latis hoc <lb/>quidem plus repellatur, illud verò minus, æquum e&longs;t tar<lb/>diùs id moueri quod plus repellitur, eo quod minus. </s>

<s id="id.000281">De<lb/>trahi autem plus lineam, cuius extremum propius e&longs;t cen<lb/>tro illa quæ &longs;uum habet terminum à centro remotiorem. </s><figure id="id.007.00.031.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.031.1.jpg"></figure>

<s id="id.000281">De<lb/>trahi autem plus lineam, cuius extremum propius e&longs;t cen<lb/>tro illa quæ &longs;uum habet terminum à centro remotiorem. </s><figure id="id.007.01.030.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.030.1.jpg"></figure>

<s id="id.000282">E&longs;to, inquit, circulus <lb/>BCDE & alter in eo minor <lb/>MNOP circa idem centrum <lb/>A. Ducanturque; Diametri ma<lb/>ioris quidem CD, EB, mino<lb/>ris verò MO, NP. </s>

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<s id="id.000284">Tardiùs autem <lb/>fertur AM, quam AD, pro<lb/>pterea quòd AM à centro <lb/>magis retrahatur quàm ip&longs;a AB. </s>

<s id="id.000285">Ducatur igitur ALF & <lb/>à puncto L, ip&longs;i AB perpendicularis L q, cadens in mino-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.032.jpg"/>ri circulo, & rur&longs;us ab eodem L ip&longs;i AB, parallela duca<lb/>tur LS, Ab S verò eidem perpendicularis ST, & ab F i<lb/>tem FX. </s>

<s id="id.000285">Ducatur igitur ALF & <lb/>à puncto L, ip&longs;i AB perpendicularis L q, cadens in mino-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.031.jpg"/>ri circulo, & rur&longs;us ab eodem L ip&longs;i AB, parallela duca<lb/>tur LS, Ab S verò eidem perpendicularis ST, & ab F i<lb/>tem FX. </s>

<s id="id.000286">Sunt ergo q L, ST, quidem æquales, nempe illæ, <lb/>per quæ, &longs;ecundum naturam, mouentur puncta BM. </s>

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<s id="id.000296">quare, vt TE, ad <lb/>q O, ita M q ad TB, &longs;ed TE maior e&longs;t ip&longs;a q O, quippe <lb/>quòd pars &longs;it q O ip&longs;ius TE, maior ergo & M q ip&longs;a TB, <lb/>quod o&longs;tendendum fuerat. </s>

<s id="id.000297">Cæterùm &longs;ubtilia & ingenio&longs;a i&longs;thæc e&longs;&longs;e non nega<lb/>mus, & longè faciliori & explicatiori modo veritas hæc <lb/>demon&longs;trari pote&longs;t, reiectis nempe illis, &longs;ecundùm, & prae<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.033.jpg"/>ter naturam motibus, qui <expan abbr="quid&emacr;">quidem</expan> in &longs;implici circulo nece&longs;<lb/>&longs;ario non cadunt: caderent autem forta&longs;&longs;e, &longs;i de circulo <lb/>res e&longs;&longs;et à <expan abbr="p&omacr;deribus">ponderibus</expan> circumlatis ex &longs;tabili centro de&longs;eri<lb/>pto, qua de re agit G. Vbaldus in Mechanicis tractatu de <lb/>libra. </s>

<s id="id.000297">Cæterùm &longs;ubtilia & ingenio&longs;a i&longs;thæc e&longs;&longs;e non nega<lb/>mus, & longè faciliori & explicatiori modo veritas hæc <lb/>demon&longs;trari pote&longs;t, reiectis nempe illis, &longs;ecundùm, & prae<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.032.jpg"/>ter naturam motibus, qui <expan abbr="quid&emacr;">quidem</expan> in &longs;implici circulo nece&longs;<lb/>&longs;ario non cadunt: caderent autem forta&longs;&longs;e, &longs;i de circulo <lb/>res e&longs;&longs;et à <expan abbr="p&omacr;deribus">ponderibus</expan> circumlatis ex &longs;tabili centro de&longs;eri<lb/>pto, qua de re agit G. Vbaldus in Mechanicis tractatu de <lb/>libra. </s>

<s id="id.000298">tunc enim dici pote&longs;t, pondus quod aliâs rectà ad <lb/>mundi centrum tenderet, à circuli centro in circulatio<lb/>ne retrahi, &longs;ed hæc ad circuli naturam, quatenus circulus <lb/>e&longs;t, nequaquam &longs;pectant. </s><figure id="id.007.00.033.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.033.1.jpg"></figure>

<s id="id.000298">tunc enim dici pote&longs;t, pondus quod aliâs rectà ad <lb/>mundi centrum tenderet, à circuli centro in circulatio<lb/>ne retrahi, &longs;ed hæc ad circuli naturam, quatenus circulus <lb/>e&longs;t, nequaquam &longs;pectant. </s><figure id="id.007.01.032.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.032.1.jpg"></figure>

<s id="id.000299">E&longs;to igitur circumferentia <lb/>AFBH, cuius centrum C, dia<lb/>meter ACB, &longs;emidiameter AC. <lb/>&longs;umatur in AC punctum quod<lb/>libet, D, & centro C, &longs;patio CD, <lb/>circumferentia de&longs;cribatur <lb/>DGEI. </s>

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<s id="id.000302">At mota linea CA circa centrum C mo<lb/>uetur &longs;imul & CD, eodem igitur tempore rotationem <lb/>complent puncta AD, maius ergo &longs;patium eodem tem<lb/>pore metitur A, ip&longs;a D, quare velocior. </s>

<s id="id.000303">Ita igitur &longs;e ha<lb/>bet velocitas ad velocitatem, vt circumferentia ad cir<lb/>cumferentiam, & diameter ad diametrum, quare id quod <lb/>mouetur in puncto à centro remotiori, velocius illo mo<lb/>uetur quod ab eo di&longs;tat minus, quod fuerat <lb/>demon&longs;trandum. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.034.jpg"/>

</subchap1>

</chap>

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<s id="id.000310">Non enim propterea <lb/>quòd illæ magnæ &longs;int, hæ verò exiguæ, hæ &longs;unt illis exa<lb/>ctiores; &longs;ed quoniam magnæ, rudes &longs;unt, minores verò ex<lb/>qui&longs;ita diligentia elaboratæ, & à materiæ pertina cia libe<lb/>riores. </s>

<s id="id.000311">Cæteris ergo paribus, exactiores e&longs;&longs;e maiores, ex <lb/>Philo&longs;ophi mente, ita docebimus. </s><figure id="id.007.00.034.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.034.1.jpg"></figure>

<s id="id.000311">Cæteris ergo paribus, exactiores e&longs;&longs;e maiores, ex <lb/>Philo&longs;ophi mente, ita docebimus. </s><figure id="id.007.01.033.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.033.1.jpg"></figure>

<s id="id.000312">E&longs;to libra maior AB, <lb/>cuius fulcimentum C. <lb/></s>

<s id="id.000313">Minor verò libra DE, <lb/>circa idem <expan abbr="fulcim&emacr;tum">fulcimentum</expan> <lb/>C, vnà cum maiori, ima<lb/>ginatione, conuer&longs;a. </s>

<s id="id.000314">Ap<lb/>ponatur quoduis pon<lb/>dus maiori libræ in A, <lb/>declinetque; exempli gratiâ in F, erit queue minor libra in G, <lb/>in eadem enim linea &longs;unt CGF. <expan abbr="Vnaq;">Vnaque</expan> igitur ex eodem <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.035.jpg"/>centro C portionem circuli de&longs;cribet GD, AF, eritqueue <lb/>ACF &longs;ector circuli, cuius diameter AB, &longs;ed DCG &longs;e<lb/>ctor circuli, cuius diameter DE. </s>

<s id="id.000314">Ap<lb/>ponatur quoduis pon<lb/>dus maiori libræ in A, <lb/>declinetque; exempli gratiâ in F, erit queue minor libra in G, <lb/>in eadem enim linea &longs;unt CGF. <expan abbr="Vnaq;">Vnaque</expan> igitur ex eodem <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.034.jpg"/>centro C portionem circuli de&longs;cribet GD, AF, eritqueue <lb/>ACF &longs;ector circuli, cuius diameter AB, &longs;ed DCG &longs;e<lb/>ctor circuli, cuius diameter DE. </s>

<s id="id.000315">Itaque vt diameter ad <lb/>diametrum, ita portio ad portionem: maior autem dia<lb/>meter AB diametro DE: maior ergo portio AF, portio<lb/>ne DG. quod autem maius e&longs;t, minus obtutum fallit, ex<lb/>qui&longs;itius itaque tractum ex maiori AB quàm ex ip&longs;a mi<lb/>nori DE cogno&longs;cemus, quod fuerat o&longs;tendendum. </s>

<s id="id.000316">Cæterùm hac eadem de cau&longs;&longs;a, A&longs;tronomica in<lb/>&longs;trumenta, puta A&longs;trolabia, Armillæ, & alia eiu&longs;modi, <lb/>quo ampliora eò exqui&longs;itiora, & certiora probantur. </s><figure id="id.007.00.035.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.035.1.jpg"></figure>

<s id="id.000316">Cæterùm hac eadem de cau&longs;&longs;a, A&longs;tronomica in<lb/>&longs;trumenta, puta A&longs;trolabia, Armillæ, & alia eiu&longs;modi, <lb/>quo ampliora eò exqui&longs;itiora, & certiora probantur. </s><figure id="id.007.01.034.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.034.1.jpg"></figure>

<s id="id.000317">E&longs;to enim A<lb/>&longs;trolabium magnum, <lb/>cuius diameter AB, <lb/>paruum autem CD, <lb/>circa idem centrum <lb/>E. </s>

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<s id="id.000319">Hinc <lb/>itaque apparet, <expan abbr="in&longs;trum&emacr;ta">in&longs;trumenta</expan> quælibet quò maiora fuerint, <lb/>eò e&longs;&longs;e & exqui&longs;itiora, quod propo&longs;uerat Ari&longs;toteles, in <lb/>hac quæ&longs;tione de Libra. </s>

<s id="id.000320">Quod autem addit de fraudibus Purpurariorum, <lb/>inquiens; quamobrem machinántur ij qui purpuram ven<lb/>dunt, vt <expan abbr="p&emacr;dendo">pendendo</expan> defraudent, dum ad medium, &longs;partum, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.036.jpg"/>non ponentes; tum plumbum in alterutram libræ partem <lb/>infundentes; aut ligni quod ad radicem vergebat, in eam <lb/>quam deferri volunt partem con&longs;tituentes, aut &longs;i nodum <lb/>habuerit, ligni enim grauior ea e&longs;t pars, in qua e&longs;t radix, <lb/>nodus verò radix quæ dam e&longs;t. </s>

<s id="id.000320">Quod autem addit de fraudibus Purpurariorum, <lb/>inquiens; quamobrem machinántur ij qui purpuram ven<lb/>dunt, vt <expan abbr="p&emacr;dendo">pendendo</expan> defraudent, dum ad medium, &longs;partum, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.035.jpg"/>non ponentes; tum plumbum in alterutram libræ partem <lb/>infundentes; aut ligni quod ad radicem vergebat, in eam <lb/>quam deferri volunt partem con&longs;tituentes, aut &longs;i nodum <lb/>habuerit, ligni enim grauior ea e&longs;t pars, in qua e&longs;t radix, <lb/>nodus verò radix quæ dam e&longs;t. </s>

<s id="id.000321">Hinc quæri po&longs;&longs;et: </s>

<s id="id.000322"><emph type="italics"/>Vtrum libræ quæ ponderibus vacuæ æquilibrant, <lb/>omni pror&longs;us careant fraude?<emph.end type="italics"/></s>

<s id="id.000323">Videri cuipiam po&longs;&longs;et, libras, quæ ponderibus va<lb/>cuæ, æquilibrant, omni pror&longs;us fraude carere, verunta<lb/>men ita non e&longs;t, quod diligentiùs (res enim magni mo<lb/>menti e&longs;t) di&longs;quiremus. </s><figure id="id.007.00.036.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.036.1.jpg"></figure>

<s id="id.000323">Videri cuipiam po&longs;&longs;et, libras, quæ ponderibus va<lb/>cuæ, æquilibrant, omni pror&longs;us fraude carere, verunta<lb/>men ita non e&longs;t, quod diligentiùs (res enim magni mo<lb/>menti e&longs;t) di&longs;quiremus. </s><figure id="id.007.01.035.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.035.1.jpg"></figure>

<s id="id.000324">E&longs;to enim libra AB, ita diui&longs;a <lb/>in C, vt AC &longs;it partium IS, CB ve<lb/>rò earundem &longs;it 10. apponatur parti <lb/>A lanx ponderans 10, parti vero B <lb/>lanx ponderans 15. ex permutata i<lb/>gitur proportione libra &longs;u&longs;pen&longs;a in <lb/>C, aequè ponderabit; &longs;i autem appo<lb/>natur lanci B &longs;acoma vnciarum 6, & in lance A con&longs;titua<lb/>tur purpura, quæ ita &longs;e habeat ad vncias 6, vt 10 ad 15, ite<lb/>rum æqueponderabit, &longs;ed vt 10 ad 15, ita 4 ad 6. Purpura<lb/>rius ergo fraudulentus, ponens in lance A vncias purpuræ <lb/>4, facto æquilibrio petet pretium vnciarum 6, & ita em<lb/>ptorem decipiet, quod &longs;anè innuerat, non autem demon<lb/>&longs;trauerat Ari&longs;toteles. </s>

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<s id="id.000326">Detegitur autem fraus, &longs;i alternatim &longs;acoma in pon<lb/>derando, modo huic, modò illi lanci apponatur. </s>

<s id="id.000327">Si enim <lb/>in lance A con&longs;tituatur &longs;acoma, in B verò purpura non fit <lb/>æquilibrium. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.037.jpg"/>

</subchap1>

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<s id="id.000335">Ari&longs;toteles primam quæ&longs;tionis partem ita &longs;oluit: An <lb/>quia &longs;ur&longs;um parte quidem exi&longs;tente, plus libræ extra per<lb/>pendiculum &longs;it? </s>

<s id="id.000336">Spartum enim perpendiculum e&longs;t: quare <lb/>nece&longs;&longs;e e&longs;t deor&longs;um ferri id quod plus e&longs;t, donec a&longs;cendat <lb/>qua bifariam libram diuidit ad ip&longs;um perpendiculum, <lb/>cum onus incumbat ad libræ partem &longs;ur&longs;us raptam. </s><figure id="id.007.00.037.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.037.1.jpg"></figure>

<s id="id.000336">Spartum enim perpendiculum e&longs;t: quare <lb/>nece&longs;&longs;e e&longs;t deor&longs;um ferri id quod plus e&longs;t, donec a&longs;cendat <lb/>qua bifariam libram diuidit ad ip&longs;um perpendiculum, <lb/>cum onus incumbat ad libræ partem &longs;ur&longs;us raptam. </s><figure id="id.007.01.036.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.036.1.jpg"></figure>

<s id="id.000337">Sit libra recta (hoc e&longs;t, in æquilibrio con&longs;tituta) BC, <lb/>&longs;partum autem AD, <lb/>fulcimentum autem <lb/>D, de&longs;uper: &longs;parto au<lb/>tem deor&longs;um proie<lb/>cto ad M perpendicu<lb/>laris erit vbi ADM. <lb/></s>

<s id="id.000338">Si igitur in ip&longs;o B po<lb/>natur onus, erit B qui<lb/>dem vbi E, C autem <lb/>vbi H, quamobrem <lb/>ea quæ bifariam <expan abbr="librã">libram</expan> <lb/>&longs;ecat, primo quidem erit DM, ip&longs;ius perpendiculi; in<expan abbr="c&umacr;-bente">cum<lb/>bente</expan> <expan abbr="aut&emacr;">autem</expan> onere, erit DG. quare libræ ip&longs;ius EH, quod <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.038.jpg"/>extra perpendiculum, e&longs;t AM, vbi e&longs;t q P maius e&longs;t dimi<lb/>dio. </s>

<s id="id.000338">Si igitur in ip&longs;o B po<lb/>natur onus, erit B qui<lb/>dem vbi E, C autem <lb/>vbi H, quamobrem <lb/>ea quæ bifariam <expan abbr="librã">libram</expan> <lb/>&longs;ecat, primo quidem erit DM, ip&longs;ius perpendiculi; in<expan abbr="c&umacr;-bente">cum<lb/>bente</expan> <expan abbr="aut&emacr;">autem</expan> onere, erit DG. quare libræ ip&longs;ius EH, quod <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.037.jpg"/>extra perpendiculum, e&longs;t AM, vbi e&longs;t q P maius e&longs;t dimi<lb/>dio. </s>

<s id="id.000339">Si igitur amoueatur onus ab E, nece&longs;&longs;e e&longs;t deor&longs;um <lb/>ferri H, minus e&longs;t enim E: &longs;iquidem igitur habuerit &longs;par<lb/>tum &longs;ur&longs;um, propter hoc a&longs;cendit libra. </s>

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<s id="id.000341">Nos autem cla<lb/>rius rem ob oculos ponimus. </s>

<s id="id.000342">Id ergo &longs;ibi vult Ari&longs;toteles, <lb/>propterea quòd pars iugi HDG maior e&longs;t parte ED q, <lb/>eam eleuatam nece&longs;&longs;e e&longs;t de&longs;cendere, & iterum à perpen<lb/>diculari ADM bifariam diui&longs;am ad æquilibrium reuer<lb/>ti, Po&longs;&longs;umus nos idem &longs;impliciori figura demon&longs;trare. </s><figure id="id.007.00.038.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.038.1.jpg"></figure>

<s id="id.000342">Id ergo &longs;ibi vult Ari&longs;toteles, <lb/>propterea quòd pars iugi HDG maior e&longs;t parte ED q, <lb/>eam eleuatam nece&longs;&longs;e e&longs;t de&longs;cendere, & iterum à perpen<lb/>diculari ADM bifariam diui&longs;am ad æquilibrium reuer<lb/>ti, Po&longs;&longs;umus nos idem &longs;impliciori figura demon&longs;trare. </s><figure id="id.007.01.037.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.037.1.jpg"></figure>

<s id="id.000343">E&longs;to libra AB, bi<lb/>fariam, diui&longs;a in G, <lb/><expan abbr="fulciment&umacr;">fulcimentum</expan> verò &longs;ur<lb/>&longs;um vbi D, produca<lb/>tur perpendicularis <lb/>DC in E. </s>

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<s id="id.000346">Punctum vero C eodem motu cir<lb/>ca idem centrum D erit in H. amoueatur pondus appo&longs;i<lb/>tum: Dico libram à &longs;itu FG declinaturam & iterum re<lb/>uer&longs;uram in &longs;itum pri&longs;tinum ACB. quoniam enim parti <lb/>GH, quæ æqualis e&longs;t parti HF, additur pars IH, quæ à <lb/>perpendiculari e&longs;t v&longs;que ad H, ip&longs;i verò HF eadem pars <lb/>detrahitur, erit IF minor GI. </s>

<s id="id.000347">Superabitur ita que IF à <lb/>GI, de&longs;cendetque FI, a&longs;cendet verò IF, donec iterum li<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.039.jpg"/>bra ín partes æquales, vt antea, diuidatur in C, &longs;iat que æ<lb/>quilibrium. </s>

<s id="id.000347">Superabitur ita que IF à <lb/>GI, de&longs;cendetque FI, a&longs;cendet verò IF, donec iterum li<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.038.jpg"/>bra ín partes æquales, vt antea, diuidatur in C, &longs;iat que æ<lb/>quilibrium. </s>

<s id="id.000348">Hæc Philo&longs;ophi demon&longs;tratio e&longs;t vera illa quidem, <lb/>&longs;ed non ex Mechanicis principijs, hoc e&longs;t, ex centri graui<lb/>tatis &longs;peculatione; nos igitur clariùs rem exponemus, his <lb/>quæ &longs;equuntur con&longs;ideratis. </s>

<s id="id.000349">Si pondus circa &longs;tabile centrum conuertatur, dimi&longs;<lb/>&longs;um non &longs;tabit, ni&longs;i &longs;ecundum grauitatis centrum fuerit <lb/>in perpendiculari, quæ per centrum, circa quod conuer<lb/>titur, ad mundi centrum cadit. </s>

<s id="id.000350">Stabit autem in ea per<lb/>pendiculari in duobus punctis, altero à centro mundi <lb/>remoti&longs;&longs;imo; altero verò eidem quantum licuerit pro<lb/>ximo. </s><figure id="id.007.00.039.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.039.1.jpg"></figure>

<s id="id.000350">Stabit autem in ea per<lb/>pendiculari in duobus punctis, altero à centro mundi <lb/>remoti&longs;&longs;imo; altero verò eidem quantum licuerit pro<lb/>ximo. </s><figure id="id.007.01.038.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.038.1.jpg"></figure>

<s id="id.000351">E&longs;to corpus A, cuius graui<lb/>tatis centrum B, nixum lineae in<lb/>flexibili BC, cum qua liberè <lb/>conuertatur circa centrum C. <lb/></s>

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<s id="id.000355">Po&longs;t hæc verò in F, hoc <lb/>e&longs;t iterum in ip&longs;a perpendiculari <lb/>infra centrum C. </s>

<s id="id.000356">De&longs;cribet er<lb/>go circulum ex centro C, nem<lb/>pe BEF &longs;ecantem perpendicu<lb/>larem in duobus punctis oppo<lb/>&longs;itis BF, dico, pondus libe è di-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.040.jpg"/>mi&longs;&longs;um in duobus tantum punctis &longs;uapte naturâ perman<lb/>&longs;urum, BF, in B, primò, quoniam cum corpus ip&longs;um A à <lb/>perpendiculari, quæ &longs;uperficiei loco intelligitur ABCD <lb/>per centrum grauitatis diuidatur, in partes diuiditur æ<lb/>queponderantes, quare in neutram partem inclinabit. <lb/></s>

<s id="id.000356">De&longs;cribet er<lb/>go circulum ex centro C, nem<lb/>pe BEF &longs;ecantem perpendicu<lb/>larem in duobus punctis oppo<lb/>&longs;itis BF, dico, pondus libe è di-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.039.jpg"/>mi&longs;&longs;um in duobus tantum punctis &longs;uapte naturâ perman<lb/>&longs;urum, BF, in B, primò, quoniam cum corpus ip&longs;um A à <lb/>perpendiculari, quæ &longs;uperficiei loco intelligitur ABCD <lb/>per centrum grauitatis diuidatur, in partes diuiditur æ<lb/>queponderantes, quare in neutram partem inclinabit. <lb/></s>

<s id="id.000357">Stabit igitur erectum, lineæ ip&longs;i fultum, inflexibili BC, <lb/>quæ nititur puncto C. </s>

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<s id="id.000365">Ad hæc autem dubitare quis po&longs;&longs;et, cur experientiâ <lb/>docente, pondera quæ infra fulcimentum habent, vt lan<lb/>cea &longs;ari&longs;&longs;aue ad planum horizontis perpendiculariter e<lb/>recta, licet eo ca&longs;u grauitatis centrum in ip&longs;a perpendicu<lb/>lari con&longs;tituatur, non &longs;tet quidem, &longs;ed altrin&longs;ecus ca<lb/>dat? </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.041.jpg"/><figure id="id.007.00.041.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.041.1.jpg"></figure>

<s id="id.000365">Ad hæc autem dubitare quis po&longs;&longs;et, cur experientiâ <lb/>docente, pondera quæ infra fulcimentum habent, vt lan<lb/>cea &longs;ari&longs;&longs;aue ad planum horizontis perpendiculariter e<lb/>recta, licet eo ca&longs;u grauitatis centrum in ip&longs;a perpendicu<lb/>lari con&longs;tituatur, non &longs;tet quidem, &longs;ed altrin&longs;ecus ca<lb/>dat? </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.040.jpg"/><figure id="id.007.01.040.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.040.1.jpg"></figure>

<s id="id.000366">Sit enim horizontis <lb/>planum AB, cui in puncto <lb/>C perpendiculariter ere<lb/>cta &longs;tatuatur &longs;ari&longs;&longs;a DC, <lb/>cuius grauitatis centrum <lb/>E, in ip&longs;a perpendiculari. <lb/></s>

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<s id="id.000370">Id certè agit, qui nutantis &longs;ari&longs;&longs;æ, digito, motum &longs;e<lb/>quitur; vt in ip&longs;o motu digitum a&longs;&longs;iduè centro grauitatis <lb/>&longs;ari&longs;&longs;æ &longs;upponat, vnde &longs;it vt nunquam extra fulcimentum <lb/>permanens, nunquam cadat. </s>

<s id="id.000371">Similis huic alia quo que dubitatio &longs;oluitur: Nempe, <lb/>Cur turbines, quibus pueri ludunt, dum quidem rotan<lb/>tur, &longs;tent erecti, rotatione vero ce&longs;&longs;ante, cadant. </s><figure id="id.007.00.041.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.041.2.jpg"></figure>

<s id="id.000371">Similis huic alia quoque dubitatio &longs;oluitur: Nempe, <lb/>Cur turbines, quibus pueri ludunt, dum quidem rotan<lb/>tur, &longs;tent erecti, rotatione vero ce&longs;&longs;ante, cadant. </s><figure id="id.007.01.040.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.040.2.jpg"></figure>

<s id="id.000372">E&longs;to enim Turbo AB, cu<lb/>ius grauitatis centrum C, planum <lb/>horizontis DE, linea Horizonti <lb/>perpendicularis ABC, tran&longs;iens <lb/>per centrum grauitatis C, &longs;it au<lb/>tem fulcimentum in B. <expan abbr="Itaq;">Itaque</expan> cum <lb/>centrum grauitatis C &longs;it in ip&longs;a <lb/>perpendiculari, &longs;tabit ex demon-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.042.jpg"/>&longs;tratis, at ex vitio materiæ non &longs;tabit. </s>

<s id="id.000372">E&longs;to enim Turbo AB, cu<lb/>ius grauitatis centrum C, planum <lb/>horizontis DE, linea Horizonti <lb/>perpendicularis ABC, tran&longs;iens <lb/>per centrum grauitatis C, &longs;it au<lb/>tem fulcimentum in B. <expan abbr="Itaq;">Itaque</expan> cum <lb/>centrum grauitatis C &longs;it in ip&longs;a <lb/>perpendiculari, &longs;tabit ex demon-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.041.jpg"/>&longs;tratis, at ex vitio materiæ non &longs;tabit. </s>

<s id="id.000373">Modò, vt a&longs;&longs;olet, ra<lb/>pido motu rotetur. </s>

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<s id="id.000378">De&longs;cribit au<lb/>tem in rotatione grauitatis centrum, quod in medio non <lb/>e&longs;t paruum circulum, per cuius centrum ip&longs;a perpendi<lb/>cula<gap/>is pertingit. </s>

<s id="id.000379">Modò redeuntes ad libram, cuius fulcimentum e&longs;t <lb/>&longs;ur&longs;um, alio principio, nempe Mechanico, cur depre&longs;&longs;a <lb/>ad æqualitatem reuertatur, demon&longs;trabimus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.043.jpg"/><figure id="id.007.00.043.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.043.1.jpg"></figure>

<s id="id.000379">Modò redeuntes ad libram, cuius fulcimentum e&longs;t <lb/>&longs;ur&longs;um, alio principio, nempe Mechanico, cur depre&longs;&longs;a <lb/>ad æqualitatem reuertatur, demon&longs;trabimus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.042.jpg"/><figure id="id.007.01.042.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.042.1.jpg"></figure>

<s id="id.000380">Sit igitur, vt &longs;u<lb/>periùs, libra AB, cu<lb/>ius centrum grauita<lb/>tis C, fulcimentum, <lb/>verò &longs;ur&longs;um, in D li<lb/>bræ quidem in C per<lb/>pendiculariter con<lb/>iunctum. </s>

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<s id="id.000385">His ita declaratis, o&longs;tendemus, (quod nullus ante <lb/>nos animaduertit) harum librarum, quæ fulcimentum <lb/>habent &longs;ur&longs;um, eam e&longs;&longs;e naturam, vt non à quouis ponde<lb/>re appo&longs;ito moueantur, vel penitus declinent. </s>

<s id="id.000386">Ij&longs;dem enim &longs;tantibus, addatur quoduis pondus lan<lb/>ci B; Itaque &longs;i tale fuerit quod &longs;uperet re&longs;i&longs;tentiam, quam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.044.jpg"/>illi facit centrum grauitatis contra naturam elatum in H <lb/>mouebitur quædam libra. </s>

<s id="id.000386">Ij&longs;dem enim &longs;tantibus, addatur quoduis pondus lan<lb/>ci B; Itaque &longs;i tale fuerit quod &longs;uperet re&longs;i&longs;tentiam, quam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.043.jpg"/>illi facit centrum grauitatis contra naturam elatum in H <lb/>mouebitur quædam libra. </s>

<s id="id.000387">Sin autem tam parui momenti <lb/>&longs;it, vt eam re&longs;i&longs;tentiam non vincat, &longs;tante circa locum in<lb/>fimum centro C, non mouebitur aut &longs;altem parum, ip&longs;a <lb/>libra. </s>

<s id="id.000388">Hinc colligimus &longs;ieri po&longs;&longs;e, libras illas, quæ non <gap/><lb/>quouis, quantumuis paruo pondere declinant, eas fulci-<gap/><lb/>mentum habere &longs;ur&longs;um. </s>

<s id="id.000389">His addimus, cæteris paribus, re&longs;i&longs;tentiam eò e&longs;&longs;e <lb/>maiorem, quo minus grauitatis centrum di&longs;tat à fulci<lb/>mento &longs;ur&longs;um, circa quod ip&longs;a libra aduertitur. </s><figure id="id.007.00.044.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.044.1.jpg"></figure>

<s id="id.000389">His addimus, cæteris paribus, re&longs;i&longs;tentiam eò e&longs;&longs;e <lb/>maiorem, quo minus grauitatis centrum di&longs;tat à fulci<lb/>mento &longs;ur&longs;um, circa quod ip&longs;a libra aduertitur. </s><figure id="id.007.01.043.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.043.1.jpg"></figure>

<s id="id.000390">E&longs;to libra AB, cuius gra<lb/>uitatis centrum C, & primò <lb/>quidem eius fulcimentum <lb/>&longs;ur&longs;um &longs;it vbi D, itaque &longs;i ap<lb/>po&longs;ito pondere declinauerit <lb/>libra ad partes B, punctum <lb/>C, dum a&longs;cendet de&longs;cribet <lb/>portionem circuli CE. fulciatur iterum &longs;ur&longs;um puncto F, <lb/>& iterum declinet ad partes B, & iterum punctum C, dum <lb/>a&longs;cendet, circuli portionem de&longs;cribet CG. </s>

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<s id="id.000393">E&longs;to enim iterum libra AB, cuius fulcimentum D, <lb/>centrum grauitatis C, &longs;it & alia libra, cuius brachia bre<lb/>uiora EF, idem habens centrum C, & eidem puncto &longs;u<lb/>&longs;pen&longs;a D. </s>

<s id="id.000394">Dico igitur, eodem pondere appo&longs;ito, faciliùs <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.045.jpg"/><figure id="id.007.00.045.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.045.1.jpg"></figure><lb/>declinaturam libram ad <lb/>partes B, quàm &longs;i idem ap<lb/>poneretur in F. </s>

<s id="id.000394">Dico igitur, eodem pondere appo&longs;ito, faciliùs <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.044.jpg"/><figure id="id.007.01.044.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.044.1.jpg"></figure><lb/>declinaturam libram ad <lb/>partes B, quàm &longs;i idem ap<lb/>poneretur in F. </s>

<s id="id.000395">Demit<lb/>tatur enim, à puncto B <lb/>horizonti perpendicula<lb/>ris BG, & ab F item per<lb/>pendicularis FH, Tum <lb/>iuncta DB, centro D, eo<lb/>dem vero &longs;patio DB, circuli portio de&longs;cribatur BI, item <lb/>iuncta DF eodem centro D, &longs;patio DF, portio circuli de<lb/>&longs;cribatu: FK. e&longs;t autem maior DB ip&longs;a DF ex propo&longs;. <lb/></s>

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<s id="id.000403">Producatur perpendicularis DC, in G & fiat <lb/>CG æqualis CB, CH verò æqualis CF. </s>

<s id="id.000404">Sunt igitur duo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.046.jpg"/><figure id="id.007.00.046.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.046.1.jpg"></figure><lb/>vectes DG, DH, quo<lb/>rum quidem commu<lb/>ne fulcimentum D, <lb/>pondus verò C, poten<lb/>tiæ vbi HG. </s>

<s id="id.000404">Sunt igitur duo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.045.jpg"/><figure id="id.007.01.045.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.045.1.jpg"></figure><lb/>vectes DG, DH, quo<lb/>rum quidem commu<lb/>ne fulcimentum D, <lb/>pondus verò C, poten<lb/>tiæ vbi HG. </s>

<s id="id.000405">Sunt au<lb/>tem hi vectes eius na<lb/>turæ, in quibus <expan abbr="p&omacr;dus">pondus</expan> <lb/>e&longs;t inter fulcimentum <lb/>& potentiam, itaque <lb/>vt &longs;e habet DC, ad <lb/>DG, ita potentia in G <lb/>ad pondus in C, item vt DC ad DH ita potentia in H ad <lb/>idem pondus C, &longs;ed minor e&longs;t propo&longs;itio DC, ad DG <lb/>quàm DC ad DH. minor ergo potentia requiritur in G, <lb/>hoc e&longs;t, in B, quàm in H, hoc e&longs;t in F. </s>

<s id="id.000406">Data igitur ponderis <lb/>æqualitate faciliùs &longs;uperabitur re&longs;i&longs;tentia C in B, quàm <lb/>in F: quod o&longs;tendendum fuerat. </s>

<s id="id.000407">Ad huius libræ naturam illæ quoque rediguntur, <lb/>quarum iugum non rectum quidem, &longs;ed curuum, vel ex <lb/>rectis &longs;ur&longs;um in angulum ad fulcimentum detinentibus, <lb/>nec refert vtrum curuitas &longs;it circuli portio quælibet, aut <lb/>ellip&longs;is &longs;ecundum alterum diametrorum; quod ita de<lb/>mon&longs;tramus. </s><figure id="id.007.00.046.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.046.2.jpg"></figure>

<s id="id.000407">Ad huius libræ naturam illæ quoque rediguntur, <lb/>quarum iugum non rectum quidem, &longs;ed curuum, vel ex <lb/>rectis &longs;ur&longs;um in angulum ad fulcimentum detinentibus, <lb/>nec refert vtrum curuitas &longs;it circuli portio quælibet, aut <lb/>ellip&longs;is &longs;ecundum alterum diametrorum; quod ita de<lb/>mon&longs;tramus. </s><figure id="id.007.01.045.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.045.2.jpg"></figure>

<s id="id.000408">E&longs;to libra, cuius iugum <lb/>curuum <expan abbr="angulat&umacr;ue">angulatumue</expan> ABC, <lb/>cuius fulcimentum B, æqua<lb/>lia autem brachia AB, BC, <lb/>& pondera item <expan abbr="vtrinq;">vtrinque</expan> ap<lb/>pen&longs;a æqualia. </s>

<s id="id.000409">Demittatur <lb/>ex puncto B ad mundi cen<lb/>trum perpendicularis BD. <lb/></s>

<s id="id.000410">Stante igitur libra ABC in <lb/>æquilibrio, erit eius graui<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.047.jpg"/>tatis centrum in ip&longs;a perpendiculari BD, puta in E. </s>

<s id="id.000410">Stante igitur libra ABC in <lb/>æquilibrio, erit eius graui<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.046.jpg"/>tatis centrum in ip&longs;a perpendiculari BD, puta in E. </s>

<s id="id.000411">Ap<lb/>ponatur pondus in C, declinabit autem libra, &longs;it autem <lb/>iuxta po&longs;itionem FBG. </s>

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<s id="id.000416">Hinc ratio pendet earum imaguncularum, quas ex <lb/>contu&longs;a papyro ligneaue leui materia compingunt, per<lb/>queue manus earum ambas, ferreum filum trajicientes, v<lb/>trinque plumbea appendunt pondera æqualia, ea <expan abbr="quid&emacr;">quidem</expan> <lb/>lege, vt centrum grauitatis infra pedes imaguncula &longs;ta<lb/>tuatur. </s>

<s id="id.000417">Tunc enim exten&longs;o filo imponentes ceu funam<lb/>bulos per illud, vltrò citroque; decurrere faciunt, imagun<lb/>cula interim erecta & in neutram partem cadente, quod <lb/>vt figurâ clarius fiat; </s><figure id="id.007.00.047.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.047.1.jpg"></figure>

<s id="id.000417">Tunc enim exten&longs;o filo imponentes ceu funam<lb/>bulos per illud, vltrò citroque; decurrere faciunt, imagun<lb/>cula interim erecta & in neutram partem cadente, quod <lb/>vt figurâ clarius fiat; </s><figure id="id.007.01.046.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.046.1.jpg"></figure>

<s id="id.000418">E&longs;to imaguncu<lb/>la AB, per cuius ma<lb/>nus traijciatur filum <lb/>ferreum curuum <expan abbr="c&umacr;">cum</expan> <lb/>æ qualibus ponderi<lb/>bus hinc inde <expan abbr="app&emacr;-&longs;is">appen<lb/>&longs;is</expan> CD. </s>

<s id="id.000419">Nitatur au<lb/>tem pedibus filo HI <lb/>in <emph type="italics"/>B<emph.end type="italics"/>, &longs;itque; totìus ma<lb/>chinæ grauitatis <expan abbr="c&emacr;-trum">cen<lb/>trum</expan> E, &longs;itque <expan abbr="per-p&emacr;dicularis">per<lb/>pendicularis</expan> per gra<lb/>uitatis <expan abbr="centr&umacr;">centrum</expan> tran&longs;i<lb/>ens A<emph type="italics"/>B<emph.end type="italics"/> E. </s>

<s id="id.000420">Itaque in<lb/>clinata imaguncula, & conuer&longs;a circa punctum <emph type="italics"/>B<emph.end type="italics"/>, &longs;i de-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.048.jpg"/>clinet ad partes I, centrum grauitatis eleuabitur in F. </s>

<s id="id.000420">Itaque in<lb/>clinata imaguncula, & conuer&longs;a circa punctum <emph type="italics"/>B<emph.end type="italics"/>, &longs;i de-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.047.jpg"/>clinet ad partes I, centrum grauitatis eleuabitur in F. </s>

<s id="id.000421">Si <lb/>verò ad partes H eleuabitur in G. quare cum FG loca <lb/>&longs;int remotiora à mundi centro, quàm &longs;it E, non &longs;tabit gra<lb/>uitatis centrum in punctis FG, &longs;ed ad infimum locum re<lb/>uertetur, hoc e&longs;t, in ip&longs;a perpendiculari in E, & imagun<lb/>cula ad perpendiculum ip&longs;i H<emph type="italics"/>B<emph.end type="italics"/>E filo, hoc e&longs;t, ip&longs;i hori<lb/>zonti reuertetur. </s>

<s id="id.000422">Hinc etiam Arictum, Te&longs;tudinumqueue demolito<lb/>riatum Machinarum vis pendet, nempe ex ratione libra<lb/>rum, quæ fulcimentum habent &longs;ur&longs;um. </s><figure id="id.007.00.048.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.048.1.jpg"></figure>

<s id="id.000422">Hinc etiam Arictum, Te&longs;tudinumqueue demolito<lb/>riatum Machinarum vis pendet, nempe ex ratione libra<lb/>rum, quæ fulcimentum habent &longs;ur&longs;um. </s><figure id="id.007.01.047.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.047.1.jpg"></figure>

<s id="id.000423">E&longs;to enim Aries A<emph type="italics"/>B<emph.end type="italics"/> <lb/>funi appen&longs;us CD, cu<lb/>ius grauitatis centrum, <lb/>D, perpendicularis verò <lb/>quæ ad mundi centrum <lb/>ip&longs;a CDE. </s>

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<s id="id.000428">Quadruplici autem <lb/>de cau&longs;&longs;a motus Arietis violenti&longs;&longs;imus e&longs;t ex vi naturalis <lb/>ponderis, quo deor&longs;um fertur, tum velocitate naturalis <lb/>motus in de&longs;cendendo auctæ, tum ex vi potentiæ impel<lb/>lentis, & naturalem motum adiuuantis, tum ex velocita<lb/>te ex motu violento deor&longs;um & antror&longs;um impellente <lb/>acqui&longs;itâ. </s>

<s id="id.000429">Id etiam addimus, eo validiores fore ictus, quò <lb/>grauior fuerit Machina, & maius &longs;patium, quo retrotra<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.049.jpg"/>hitur, grauitate ip&longs;a & &longs;patio tum virium vnione opera<gap/><lb/>tionem mirum in modum adiuuantibus. </s>

<s id="id.000429">Id etiam addimus, eo validiores fore ictus, quò <lb/>grauior fuerit Machina, & maius &longs;patium, quo retrotra<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.048.jpg"/>hitur, grauitate ip&longs;a & &longs;patio tum virium vnione opera<gap/><lb/>tionem mirum in modum adiuuantibus. </s>

<s id="id.000430">Hæc nos de Libra &longs;ur&longs;um fulcimentum habente, dí<lb/>cta voluimus, nunc de ea, cuius fulcimentum deor&longs;um, <lb/>e&longs;t, verba faciemus. </s>

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<s id="id.000436">Hæc ille, qui &longs;chemate quo que rem aperit, at eo a<lb/>pud interpretes, & Picolomineum Paraphra&longs;tem, ita <expan abbr="m&emacr;-dosè">men<lb/>dosè</expan> lineato, vt inde ob&longs;curitas lucis loco, legentibus of<lb/>fundatur. </s>

<s id="id.000437">Nos, quod & &longs;uprà quo que fecimus, no&longs;tra fi<lb/>gurâ, &longs;ole ip&longs;o clariorem, ex Ari&longs;to telis ip&longs;ius mente rem <lb/>totam efficiemus. </s><figure id="id.007.00.049.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.049.1.jpg"></figure>

<s id="id.000437">Nos, quod & &longs;uprà quoque fecimus, no&longs;tra fi<lb/>gurâ, &longs;ole ip&longs;o clariorem, ex Ari&longs;to telis ip&longs;ius mente rem <lb/>totam efficiemus. </s><figure id="id.007.01.048.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.048.1.jpg"></figure>

<s id="id.000438">Sit libra recta, (hoc <lb/>e&longs;t, in æquilibrio con<lb/>&longs;tituta) vbi NG. </s>

<s id="id.000439">Per<lb/>pendiculum autem (id <lb/>e&longs;t, perpendicularis <lb/>quæ ad mundi <expan abbr="centr&umacr;">centrum</expan>) <lb/>KLM. </s>

<s id="id.000440">Bifariam igitur <lb/>&longs;ecatur NG. impo&longs;ito <lb/>po&longs;thæc onere in ip&longs;o <lb/>N, erit quidem N, vbi <lb/>O. ip&longs;um autem G vbi <lb/>R. KL autem vbi LP. <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.050.jpg"/>quare maius e&longs;t KO, quam LR, ip&longs;a parte PKL. </s>

<s id="id.000440">Bifariam igitur <lb/>&longs;ecatur NG. impo&longs;ito <lb/>po&longs;thæc onere in ip&longs;o <lb/>N, erit quidem N, vbi <lb/>O. ip&longs;um autem G vbi <lb/>R. KL autem vbi LP. <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.049.jpg"/>quare maius e&longs;t KO, quam LR, ip&longs;a parte PKL. </s>

<s id="id.000441">Amoto <lb/>igitur onere nece&longs;&longs;e e&longs;t manere. </s>

<s id="id.000442">Incumbit enim onus ex<lb/>ce&longs;&longs;us medietatis eius, vbi e&longs;t F. </s>

<s id="id.000443">Sen&longs;us e&longs;t igitur, idcirco <lb/>partem iugi KLO inclinatam, ad æquilibrium non re<lb/>uerti, propterea quòd maior &longs;it ip&longs;a KLO pars quæ tra<lb/>hit, ip&longs;a<gap/> RKL, quæ trahitur & eleuatur. </s><figure id="id.007.00.050.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.050.1.jpg"></figure>

<s id="id.000443">Sen&longs;us e&longs;t igitur, idcirco <lb/>partem iugi KLO inclinatam, ad æquilibrium non re<lb/>uerti, propterea quòd maior &longs;it ip&longs;a KLO pars quæ tra<lb/>hit, ip&longs;a<gap/> RKL, quæ trahitur & eleuatur. </s><figure id="id.007.01.049.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.049.1.jpg"></figure>

<s id="id.000444">Pote&longs;t hoc idem longè <lb/>&longs;impliciori themate demon<lb/>&longs;trari. </s>

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<s id="id.000448">Demon&longs;tratio <expan abbr="quid&emacr;">quidem</expan> e&longs;t hæc, &longs;ed non ex proprijs prin<lb/>cipijs Mechanicis, <expan abbr="n&emacr;pe">nempe</expan> ex ratione <expan abbr="c&emacr;t">cent</expan>ri grauitatis petitâ. <lb/></s>

<s id="id.000449">Ii&longs;dem enim &longs;tantibus, <expan abbr="c&umacr;">cum</expan> centrum grauitatis C fiat extra <lb/>perpendicularem, de&longs;cendens ad I, nunquam reuertetur <lb/>in C, a&longs;cenderet enim; &longs;ed &longs;i liberè circa centrum D con<lb/>uerteretur, de&longs;cendens vt dictum e&longs;t per circulum CIM <lb/>pondus B, fieret in L, A vero in N adepta po&longs;itione <lb/>LMN. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.051.jpg"/>

<s id="id.000450">Cur autem huius libræ, quæ aliàs inutilis e&longs;t, memi<lb/>nerit Philo&longs;ophus, ea videtur cau&longs;&longs;a, quòd inde vectis vir<lb/>tutem eliciat, vt &longs;uo loco videbimus. </s>

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<s id="id.000458">fu&longs;i&longs;&longs;imè enim ibi hac de re & ab&longs;oluti&longs;&longs;i<lb/>mè agit. </s>

<s id="id.000459">Nos autem quidem paucis ea, quæ ad hanc co<lb/>gnitionem pertinent, explicabimus. </s><figure id="id.007.00.051.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.051.1.jpg"></figure>

<s id="id.000459">Nos autem quidem paucis ea, quæ ad hanc co<lb/>gnitionem pertinent, explicabimus. </s><figure id="id.007.01.050.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.050.1.jpg"></figure>

<s id="id.000460">E&longs;to enim libra A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>cuius brachia æqualia, <lb/>& centrum grauitatis <lb/>in C, brachijs verò <lb/>AC, C<emph type="italics"/>B<emph.end type="italics"/> æqualibus, æ<lb/>qualia pondera hinc <lb/>inde <expan abbr="apponãtur">apponantur</expan>. </s>

<s id="id.000461">Tum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.052.jpg"/>fulcimento in medio, hoc e&longs;t, vbi grauitatis centrum C <lb/>applicato per centrum ip&longs;um C ducatur perpendicularis, <lb/>quæ ad mundi centrum, DCE, &longs;itque primum libra æ<lb/>quedi&longs;tans horizonti, con&longs;tituta. </s>

<s id="id.000461">Tum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.051.jpg"/>fulcimento in medio, hoc e&longs;t, vbi grauitatis centrum C <lb/>applicato per centrum ip&longs;um C ducatur perpendicularis, <lb/>quæ ad mundi centrum, DCE, &longs;itque primum libra æ<lb/>quedi&longs;tans horizonti, con&longs;tituta. </s>

<s id="id.000462">Tum ex altera parte <lb/>pre&longs;&longs;a moueatur & fiat iuxta po&longs;itionem FCG. </s>

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<s id="id.000468">Circa hæc po&longs;&longs;et cuipiam oriri Dubium, num chor<lb/>dulæ, quibus lances appenduntur, variationem aliquam <lb/>circa ea quæ demon&longs;trata &longs;unt, inducere valeant. </s>

<s id="id.000469">Dicimus nullam inde fieri: E&longs;to enim libra AB, cu<lb/>ius centrum & fulcimentum C, ab cuius extremitate A <lb/>dependeat, funiculus AD, ab alia verò <emph type="italics"/>B<emph.end type="italics"/>, funiculus <emph type="italics"/>B<emph.end type="italics"/>E, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.053.jpg"/><figure id="id.007.00.053.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.053.1.jpg"></figure><lb/>quibus appen&longs;æ &longs;int æ<lb/>qualis ponderis lances <lb/>DE. </s>

<s id="id.000469">Dicimus nullam inde fieri: E&longs;to enim libra AB, cu<lb/>ius centrum & fulcimentum C, ab cuius extremitate A <lb/>dependeat, funiculus AD, ab alia verò <emph type="italics"/>B<emph.end type="italics"/>, funiculus <emph type="italics"/>B<emph.end type="italics"/>E, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.052.jpg"/><figure id="id.007.01.052.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.052.1.jpg"></figure><lb/>quibus appen&longs;æ &longs;int æ<lb/>qualis ponderis lances <lb/>DE. </s>

<s id="id.000470">Moueatur libra, <lb/>fiatque in ICH, funi<lb/>culi verò in lancibus in <lb/>IK, HL. &longs;ecet autem fu<lb/>niculus IK libram A<emph type="italics"/>B<emph.end type="italics"/>, <lb/>in M, LH verò produ<lb/>catur & eandem &longs;ecer <lb/>in N. quoniam igitur <lb/>IC, æqualis e&longs;t CH, pa<lb/>rallelæ autem KI, LN æquales <expan abbr="er&umacr;t">erunt</expan> alterni anguli MIC, <lb/>NHC, &longs;ed & anguli ad verticem ICH, BCH æquales <lb/>&longs;unt, quare triangulum IMC, æquale triangulo HNC, <lb/>& latera lateribus, quæ æqualibus angulis &longs;ubtenduntur. <lb/></s>

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<s id="id.000476">Ari&longs;toteles ita quæ&longs;tionem proponit, vt eam Rheto<lb/>rico quodam fuco admirabiliorem faciat. </s>

<s id="id.000477">Soluit au<lb/>tem hoc pacto, <expan abbr="inqui&emacr;s">inquiens</expan>, fieri po&longs;&longs;e eam e&longs;&longs;e cau&longs;&longs;am, quod <lb/>vectis &longs;it libra, eius nempe generis quod fulcimentum ha<lb/>bet deor&longs;um, atque id circo in ip&longs;a pre&longs;&longs;ione in partes in<lb/>æquales vectem diuidi. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.054.jpg"/><figure id="id.007.00.054.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.054.1.jpg"></figure>

<s id="id.000477">Soluit au<lb/>tem hoc pacto, <expan abbr="inqui&emacr;s">inquiens</expan>, fieri po&longs;&longs;e eam e&longs;&longs;e cau&longs;&longs;am, quod <lb/>vectis &longs;it libra, eius nempe generis quod fulcimentum ha<lb/>bet deor&longs;um, atque id circo in ip&longs;a pre&longs;&longs;ione in partes in<lb/>æquales vectem diuidi. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.053.jpg"/><figure id="id.007.01.053.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.053.1.jpg"></figure>

<s id="id.000478">Figura quam ex<lb/>hibet, vix ferè quid &longs;i<lb/>bi velit explicat. </s>

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<s id="id.000488">Aliunde igi<lb/>tur tantæ rei cau&longs;&longs;a e&longs;t petenda, quod & nos deinceps fa<lb/>ciemus. </s>

<s id="id.000489">Videtur autem ip&longs;e quoque Ari&longs;toteles non &longs;ibi <lb/>pror&longs;us in a&longs;&longs;ignata ratione &longs;atis feci&longs;&longs;e, & ideo &longs;ubiungit: <lb/>quoniam ab æquali pondere celerius mouetur maior ca<lb/>rum quæ à centro &longs;unt duo verò pondera; quod mouet & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.055.jpg"/>quod mouetur, quod igitur motum pondus ad mouens <lb/>longitudo patitur ad longitudinem, &longs;emper autem <expan abbr="quã-tum">quan<lb/>tum</expan> ab hypomochlio (id e&longs;t, fulcimento) di&longs;tabit magis, <lb/>tanto facilius mouebit. </s>

<s id="id.000489">Videtur autem ip&longs;e quoque Ari&longs;toteles non &longs;ibi <lb/>pror&longs;us in a&longs;&longs;ignata ratione &longs;atis feci&longs;&longs;e, & ideo &longs;ubiungit: <lb/>quoniam ab æquali pondere celerius mouetur maior ca<lb/>rum quæ à centro &longs;unt duo verò pondera; quod mouet & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.054.jpg"/>quod mouetur, quod igitur motum pondus ad mouens <lb/>longitudo patitur ad longitudinem, &longs;emper autem <expan abbr="quã-tum">quan<lb/>tum</expan> ab hypomochlio (id e&longs;t, fulcimento) di&longs;tabit magis, <lb/>tanto facilius mouebit. </s>

<s id="id.000490">Cau&longs;&longs;a autem est, quæ retro com<lb/>memorata e&longs;t, quoniam quæ plus à centro di&longs;tat <expan abbr="maior&emacr;">maiorem</expan> <lb/>de&longs;cribit circulum. </s>

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<s id="id.000500">Con&longs;equìtur, (id vtique fatemur) <lb/>nece&longs;&longs;ariò velocitas maior motu brachij maioris; non ta<lb/>men cau&longs;&longs;a e&longs;t cur vis loco vbi velocitas maior &longs;it, appo&longs;i<lb/>ta magis moueat. </s>

<s id="id.000501">Sanè ex velocitate, dum mouentur, <expan abbr="p&omacr;-dus">pon<lb/>dus</expan> acquirere corpora, tum proiecta, tum cadentia cer<lb/>tum e&longs;t, quod etiam in quæ&longs;tione 19. cum Philo&longs;opho <expan abbr="c&omacr;-">con-</expan><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.056.jpg"/>&longs;iderabimus. </s>

<s id="id.000502">Sed hoc ex velocitate & motu &longs;it, quæ &longs;unt <lb/>actu. </s>

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<s id="id.000504">Cæterum videtur Ari&longs;toteles id &longs;ub<lb/>odora&longs;&longs;e, quod po&longs;tea Archimedes, Mechanicorum prin<lb/>ceps, in propo&longs;. </s>

<s id="id.000505">6. primi Æqueponderantium explicitè <lb/>protulit & probauit: nempe in æquilibrio ita e&longs;&longs;e pondus <lb/>ad pondus, vt brachium ad brachium, ratione permutata. </s><figure id="id.007.00.056.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.056.1.jpg"></figure>

<s id="id.000505">6. primi Æqueponderantium explicitè <lb/>protulit & probauit: nempe in æquilibrio ita e&longs;&longs;e pondus <lb/>ad pondus, vt brachium ad brachium, ratione permutata. </s><figure id="id.007.01.055.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.055.1.jpg"></figure>

<s id="id.000506">E&longs;to enim vectis <lb/>AB, quomodolibet <lb/>fulcimento diui&longs;us in <lb/>C. <expan abbr="app&emacr;datur">appendatur</expan> autem <lb/>in A, pondus D, in B <lb/>verò pondus E, ita &longs;e <lb/>habens ad pondus D, vt ip&longs;a AC ad CB. </s>

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<s id="id.000513">Teneo, at nondum acquie&longs;co: pe<lb/>tam enim, Cur ea rationis permutatio mirabilem illum <lb/>effectum pariat. </s>

<s id="id.000514">Hoc quod illi non docent, puto nos, i<lb/>gnorantiæ &longs;omno &longs;epultos, &longs;omnia&longs;&longs;e. </s><figure id="id.007.00.056.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.056.2.jpg"></figure>

<s id="id.000514">Hoc quod illi non docent, puto nos, i<lb/>gnorantiæ &longs;omno &longs;epultos, &longs;omnia&longs;&longs;e. </s><figure id="id.007.01.055.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.055.2.jpg"></figure>

<s id="id.000515">Æqualitatem &longs;tatus <lb/>e&longs;&longs;e cau&longs;&longs;am, nemo, vt <lb/>puto, inficiabitur. </s>

<s id="id.000516">res e&longs;t <lb/>enim per &longs;e clara. </s>

<s id="id.000517">E&longs;to &longs;i<lb/>quidem linea quæpiam AB, applicetur extremitati A po<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.057.jpg"/>tentia quæ dam quæ lineam ad &longs;e trahat ad partes nempe <lb/>A, Tum in B quædam alia potentia ip&longs;i quæ in A potentiae, <lb/>æqualis, quæ lineam trahat &longs;imili modo ad partes B. </s>

<s id="id.000517">E&longs;to &longs;i<lb/>quidem linea quæpiam AB, applicetur extremitati A po<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.056.jpg"/>tentia quæ dam quæ lineam ad &longs;e trahat ad partes nempe <lb/>A, Tum in B quædam alia potentia ip&longs;i quæ in A potentiae, <lb/>æqualis, quæ lineam trahat &longs;imili modo ad partes B. </s>

<s id="id.000518">Datâ <lb/>igitur harum potentiarum æqualitate, linea AB, nec ad <lb/>partes A, nec ad partes B transferetur, &longs;ed pror&longs;us immo<lb/>bilis &longs;tabit. </s>

<s id="id.000519">His ita con&longs;titutis, Dico vecte quomodolibet diui&longs;o, <lb/>ponderibu&longs;que vtrinque appo&longs;itis, permutatâ propor<lb/>tione &longs;ibi inuicem re&longs;pondentibus, rem e&longs;&longs;e redactam ad <lb/>æqualitatem, & inde &longs;tatum fieri, hoc e&longs;t, æquilibrium. </s><figure id="id.007.00.057.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.057.1.jpg"></figure>

<s id="id.000519">His ita con&longs;titutis, Dico vecte quomodolibet diui&longs;o, <lb/>ponderibu&longs;que vtrinque appo&longs;itis, permutatâ propor<lb/>tione &longs;ibi inuicem re&longs;pondentibus, rem e&longs;&longs;e redactam ad <lb/>æqualitatem, & inde &longs;tatum fieri, hoc e&longs;t, æquilibrium. </s><figure id="id.007.01.056.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.056.1.jpg"></figure>

<s id="id.000520">E&longs;to enim vectis AB, quo modo libet diui&longs;us in C, & <lb/>ip&longs;i quidem C fulcimentum &longs;upponatur. </s>

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<s id="id.000524">Igitur cum par<lb/>tes CA, AF totius FC, æquales &longs;int partibus CB, BG, <lb/>totius CG, erit totum FC, toti CG æquale. </s>

<s id="id.000525">Diui&longs;us ita-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.058.jpg"/>que erit vectis FG in partes æquales FC, CG in puncto <lb/>fulcimenti C. </s>

<s id="id.000525">Diui&longs;us ita-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.057.jpg"/>que erit vectis FG in partes æquales FC, CG in puncto <lb/>fulcimenti C. </s>

<s id="id.000526">Et quoniam æquale in æquale non agit, <lb/>&longs;tabit vectis & in neutram partem inclinabit. </s>

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<s id="id.000528">Cur igitur &longs;eruata permutatim brachiorum & <lb/>ponderum proportione fiat æquilibrium, ex his quæ de<lb/>mon&longs;trauimus, clarè patet. </s>

<s id="id.000529">Sed forte dicet qui&longs;piam, &longs;i brachia, pondera &longs;unt, <lb/>vel ponderibus æquipollentia, &longs;u&longs;tinenti duplicabitur <lb/>pondus. </s><figure id="id.007.00.058.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.058.1.jpg"></figure>

<s id="id.000529">Sed forte dicet qui&longs;piam, &longs;i brachia, pondera &longs;unt, <lb/>vel ponderibus æquipollentia, &longs;u&longs;tinenti duplicabitur <lb/>pondus. </s><figure id="id.007.01.057.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.057.1.jpg"></figure>

<s id="id.000530">E&longs;to enim vectis AB, <lb/>ita diui&longs;us in C, vt pars <lb/>maior CB minori AC &longs;it <lb/>in proportione quintu<lb/>pla. </s>

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<s id="id.000535">Et&longs;i & illud verum &longs;it, da<lb/>to vecte pondero&longs;o, fulcimentum rum ponderum appen<lb/>&longs;orum, tum vectis ip&longs;ius pondus &longs;u&longs;tinere. </s>

<s id="id.000536">Iacta huiu&longs;cemodi, quam diximus, æqualitate, &longs;e-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.059.jpg"/>quitur nece&longs;&longs;ariò, centrum grauitatis ip&longs;ius vectis cum <lb/>appen&longs;is ponderibus, ac &longs;i vnum idemqueue e&longs;&longs;et corpus <lb/>cadere in perpendiculari quæ per centrum ip&longs;um & ful<lb/>cimentum tran&longs;iens ad mundi centrum pertingit. </s>

<s id="id.000536">Iacta huiu&longs;cemodi, quam diximus, æqualitate, &longs;e-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.058.jpg"/>quitur nece&longs;&longs;ariò, centrum grauitatis ip&longs;ius vectis cum <lb/>appen&longs;is ponderibus, ac &longs;i vnum idemqueue e&longs;&longs;et corpus <lb/>cadere in perpendiculari quæ per centrum ip&longs;um & ful<lb/>cimentum tran&longs;iens ad mundi centrum pertingit. </s>

</subchap1><subchap1>

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<s id="id.000546">Plurimum autem diuidi vbi plurima <lb/>pars remi à &longs;calmo e&longs;t. </s>

<s id="id.000547">Rem facilem, eo quod verbis potu<lb/>erit, &longs;chemate non declarauit, nos autem apponemus. </s><figure id="id.007.00.059.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.059.1.jpg"></figure>

<s id="id.000547">Rem facilem, eo quod verbis potu<lb/>erit, &longs;chemate non declarauit, nos autem apponemus. </s><figure id="id.007.01.058.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.058.1.jpg"></figure>

<s id="id.000548">E&longs;to enim nauis AB, mare CD, <lb/>remorum alter, qui ad proram EF, cu<lb/>ius &longs;calmus G, alter verò in medio na<lb/>uis, HI, circa &longs;calmum K. </s>

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<s id="id.000550">Itaque quoniam <lb/>nauis lata e&longs;t in medio vbi Scalmus K <lb/>maior pars KH intra nauim e&longs;t, minor <lb/>verò KI, extra. </s>

<s id="id.000551">Contra autem remi ad <lb/>proram, nempe EF pars minor EG <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.060.jpg"/>intra nauim, pars verò maior GF extra nauim e&longs;t. </s>

<s id="id.000551">Contra autem remi ad <lb/>proram, nempe EF pars minor EG <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.059.jpg"/>intra nauim, pars verò maior GF extra nauim e&longs;t. </s>

<s id="id.000552">Pondus <lb/>autem eò faciliùs mouetur, quo maior e&longs;t vectis pars, quæ <lb/>à fulcimento e&longs;t ad mouentem potentiam. </s>

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<s id="id.000561">Addimus, fal&longs;um videri quod a&longs;&longs;erit Ari&longs;toteles, <lb/>nempe illos qui in media naui &longs;unt, remiges, maximè na<lb/>uim mouere; facilius, melius dixi&longs;&longs;et. </s>

<s id="id.000562">Si enim maximè, <lb/>quod ait, denotat, maximo &longs;patio, & velocius pror&longs;us fal<lb/>&longs;um, etenim tardius mouent & minori &longs;patio, quod nos i<lb/>ta demon&longs;tramus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.061.jpg"/><figure id="id.007.00.061.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.061.1.jpg"></figure>

<s id="id.000562">Si enim maximè, <lb/>quod ait, denotat, maximo &longs;patio, & velocius pror&longs;us fal<lb/>&longs;um, etenim tardius mouent & minori &longs;patio, quod nos i<lb/>ta demon&longs;tramus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.060.jpg"/><figure id="id.007.01.060.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.060.1.jpg"></figure>

<s id="id.000563">E&longs;to enim Remus AB <lb/>qui marí fulcitur in B, Scal<lb/>mus remi qui ad <expan abbr="prorã">proram</expan> pup<lb/>pimue C, qui in media naui <lb/>D, maior autem remi pars <lb/>e&longs;t à &longs;calmo Dad A quam i<lb/>p&longs;ius C 2d A, Pellantur remi & &longs;tante ceu centro BA, in <lb/>E. eodem igitur tempore C eritin F, & D in G, &longs;ed maius <lb/>e&longs;t &longs;patium CF &longs;patio DG, Ergo vnica impul&longs;ione, plus <lb/>mouit &longs;calmum, hoc e&longs;t, nauim, potentia ad puppim pro<lb/>ramue remigans, quàm ea quæ operatur in media naui vt <lb/>&longs;entire videbatur (&longs;i modo is e&longs;t eius &longs;en&longs;us) Ari&longs;toteles. <lb/></s>

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<s id="id.000570">fulcimentum enim in contrarium ver&longs;atur, <lb/>mare vetò interius, & illud exterius. </s>

<s id="id.000571">illud autem &longs;equitur <lb/>nauis quæ illi e&longs;t alligata & remus quidem &longs;ecundum la<lb/>titudinem onus propellens & ab eodem repul&longs;us in re-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.062.jpg"/>ctum propellit, Gubernaculum verò, vt obliquum iacet <lb/>hinc inde in obliquum motionem facit. </s>

<s id="id.000571">illud autem &longs;equitur <lb/>nauis quæ illi e&longs;t alligata & remus quidem &longs;ecundum la<lb/>titudinem onus propellens & ab eodem repul&longs;us in re-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.061.jpg"/>ctum propellit, Gubernaculum verò, vt obliquum iacet <lb/>hinc inde in obliquum motionem facit. </s>

<s id="id.000572">in extremo <expan abbr="aut&emacr;">autem</expan>, <lb/>non in medio iacet, quoniam mouenti facillimum e&longs;t mo<lb/>tum mouere: prima enim pars celerrimè fertur, & quo<lb/>niam, quemadmodum in ijs quæ feruntur in fine deficit <lb/>latio, &longs;ic ip&longs;ius continui in finem, imbecillima e&longs;t latio. <lb/></s>

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<s id="id.000578">quamobrem omnia <lb/>ferè quæ de Temone dicenda fuerant, de remo loquens <lb/>proponit. </s>

<s id="id.000579">Ait autem. </s><figure id="id.007.00.062.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.062.1.jpg"></figure>

<s id="id.000579">Ait autem. </s><figure id="id.007.01.061.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.061.1.jpg"></figure>

<s id="id.000580">Sit remus AB, <lb/>&longs;calmus vero C, remi <lb/>in nauigio <expan abbr="principi&umacr;">principium</expan> <lb/>A, palmula autem, <lb/>quæ in mari B. </s>

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<s id="id.000582">erit igitur vbi F, mi<lb/>nor enim BF, ip&longs;a AD, quare ip&longs;o GF ip&longs;a DG. </s>

<s id="id.000583">Hæc <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.063.jpg"/>demon&longs;tratio licet vera videatur, rei ta men, de qua e&longs;t <lb/>&longs;ermo, minimè aptatur. </s>

<s id="id.000583">Hæc <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.062.jpg"/>demon&longs;tratio licet vera videatur, rei ta men, de qua e&longs;t <lb/>&longs;ermo, minimè aptatur. </s>

<s id="id.000584">Si enim aptaretur in ip&longs;ius remi <lb/>motu, cum palmula e&longs;&longs;et in F, &longs;calmus ficret in G, excur<lb/>reret ergo vel &longs;calmus per remum, vel remus per <expan abbr="&longs;calm&umacr;">&longs;calmum</expan>, <lb/>facta nempe eiu&longs;modi translatione de C in G, & &longs;ic intra <lb/>nauim modo e&longs;&longs;et pars remi DC, modò verò GD, quod <lb/>tamen non fieri ipsâ experientia docemur. </s>

<s id="id.000585">Illud quoque <lb/>fal&longs;um e&longs;t, nauim ip&longs;am tantum moueri in aëre, quantum <lb/>e&longs;t &longs;patium AD, hoc e&longs;t, remi extremum quod e&longs;t in naui, <lb/>&longs;iquidem &longs;calmi motu, non autem manubrij remi, nauis <lb/>agatur. </s>

<s id="id.000586">Aliter igitur res &longs;e habet, & forte hoc pacto. </s><figure id="id.007.00.063.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.063.1.jpg"></figure>

<s id="id.000586">Aliter igitur res &longs;e habet, & forte hoc pacto. </s><figure id="id.007.01.062.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.062.1.jpg"></figure>

<s id="id.000587">Sit remus AB, cuíus <lb/>manubrium A, palmula <lb/>B, &longs;calmus C. </s>

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<s id="id.000589">Po&longs;uit autem Ari&longs;toteles &longs;calmum ad medium <lb/>remi, &longs;ed non ad medium collocari &longs;olet, maior enim pars <lb/>in mare propendet puta HB, quo ca&longs;u translationis &longs;pa<lb/>tium fit maius, nempe ab H in I. fit autem motus &longs;calmi ex <lb/>centris qui &longs;unt in &longs;patio ip&longs;o BE, quatenus autem ad te<lb/>monem pertinet, quem remum ait, obliquè puppim ip&longs;am <lb/>propellentem, ita &longs;e res habet. </s>

<s id="id.000590">E&longs;to nauis carina AB, prora A, puppis B, Temonis <lb/>ala BC, gubernaculum BD, cardo verò fulcimentumue <lb/>B; facta itaque impul&longs;ione obliquâ gubernaculi à D in E, <lb/>minor fiet motus in mari à C in F, eritqueue temo vbi EGF, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.064.jpg"/><figure id="id.007.00.064.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.064.1.jpg"></figure><lb/>cardo verò vbi G, translata igitur e<lb/>rit eo motu, puppis ip&longs;a à B in G. facta <lb/>itaque paruâ motione puppis ex B in <lb/>G, prora ip&longs;a quæ longè di&longs;tat à pup<lb/>pi B maiori &longs;patio &longs;uperato translata <lb/>erit in H facta proræ in contrariam <lb/>partem ab ea quæ facta e&longs;t guberna<lb/>culi motione. </s>

<s id="id.000590">E&longs;to nauis carina AB, prora A, puppis B, Temonis <lb/>ala BC, gubernaculum BD, cardo verò fulcimentumue <lb/>B; facta itaque impul&longs;ione obliquâ gubernaculi à D in E, <lb/>minor fiet motus in mari à C in F, eritqueue temo vbi EGF, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.063.jpg"/><figure id="id.007.01.063.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.063.1.jpg"></figure><lb/>cardo verò vbi G, translata igitur e<lb/>rit eo motu, puppis ip&longs;a à B in G. facta <lb/>itaque paruâ motione puppis ex B in <lb/>G, prora ip&longs;a quæ longè di&longs;tat à pup<lb/>pi B maiori &longs;patio &longs;uperato translata <lb/>erit in H facta proræ in contrariam <lb/>partem ab ea quæ facta e&longs;t guberna<lb/>culi motione. </s>

<s id="id.000591">Porrò quod & in præ<lb/>cedente quæ&longs;tione adnotauimus, <expan abbr="l&omacr;-gè">lon<lb/>gè</expan> meliùs procedet demon&longs;tratio &longs;i <lb/><expan abbr="fulciment&umacr;">fulcimentum</expan> mare intelligatur, quàm <lb/>&longs;calmus, neque enim mare ceu pon<lb/>dus, &longs;ed &longs;calmus ip&longs;e Temonisuecardo, ponderum in&longs;tar <lb/>transferuntur. </s>

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<s id="id.000598">Obliqui igitur motus qui validè in puppi &longs;it, cau&longs;&longs;a <lb/>e&longs;t non quidem ex conuer&longs;ione temonis percu&longs;&longs;io maris, <lb/>&longs;ed mare ip&longs;um, cuius fluctus naui currente obliquam te<lb/>monis alam ad eam partem quæ mari obuertitur, impel<lb/>lentes temonem cum puppi ad contrariam partem vali<lb/>di&longs;&longs;imè transferunt. </s>

<s id="id.000599">E&longs;to nauis carina AB, prora B, puppis A, Temo AC, <lb/>gubernaculum AD; Itaque currente naui, Temone in<lb/>terim & guberna culo in eadem carinæ linea exi&longs;tentibus, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.065.jpg"/><figure id="id.007.00.065.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.065.1.jpg"></figure><lb/>Temo quidem mare &longs;ecat, nulla fa<lb/>ctâ in puppi, nauis ad &longs;ini&longs;tram dex<lb/>tramue translatione. </s>

<s id="id.000599">E&longs;to nauis carina AB, prora B, puppis A, Temo AC, <lb/>gubernaculum AD; Itaque currente naui, Temone in<lb/>terim & guberna culo in eadem carinæ linea exi&longs;tentibus, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.064.jpg"/><figure id="id.007.01.064.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.064.1.jpg"></figure><lb/>Temo quidem mare &longs;ecat, nulla fa<lb/>ctâ in puppi, nauis ad &longs;ini&longs;tram dex<lb/>tramue translatione. </s>

<s id="id.000600">Si verò mouea<lb/>tur gubernaculum à D in E, co moto <lb/>mouebitur aliquantulum & puppis <lb/>ad partes E, quod voluit Ari&longs;toteles. <lb/></s>

<s id="id.000600">Si verò mouea<lb/>tur gubernaculum à D in E, eo moto <lb/>mouebitur aliquantulum & puppis <lb/>ad partes E, quod voluit Ari&longs;toteles. <lb/></s>

<s id="id.000601">Sed minimi, vt diximus, ea res ad tan<lb/>tum effectum e&longs;t momenti. </s>

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<s id="id.000605">Vnde <lb/>colligimus naues, quo longiores &longs;unt in mouente ad Te<lb/>monem adhibita maiori facilitate ad dextram &longs;ini&longs;tram<lb/>ue propelli: quod &longs;anè ip&longs;emet con&longs;iderauit Ari&longs;toteles, <lb/>quì idcirco inquit, in extremo, non autem in medio temo<lb/>nem poni eo quod mouenti facilimum &longs;it ab extremo <lb/>motum mouere. </s>

<s id="id.000606">Ex hac no&longs;trâ &longs;peculatione ratio habetur eius ma-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.066.jpg"/>chinationis, quâ in magnis fluminibus, ceu Pado, Abdua <lb/>& &longs;imilibus, Portitores, equos, currus, viatore&longs;que; ip&longs;os, è <lb/>ripa in ripam transferunt. </s>

<s id="id.000606">Ex hac no&longs;trâ &longs;peculatione ratio habetur eius ma-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.065.jpg"/>chinationis, quâ in magnis fluminibus, ceu Pado, Abdua <lb/>& &longs;imilibus, Portitores, equos, currus, viatore&longs;que; ip&longs;os, è <lb/>ripa in ripam transferunt. </s>

<s id="id.000607">Pulcherrima enim res e&longs;t, & <lb/>nobis per&longs;pecti&longs;&longs;ima, qui Gua&longs;tallâ re&longs;identiæ olim no<lb/>&longs;træ oppido ad Padum, Mantuam pergentes &longs;æpi&longs;&longs;imè ad <lb/>Ca&longs;trum Borgi Iu&longs;is ea qua diximus machinatione lati&longs;<lb/>&longs;imum eiu&longs;dem Padi aluum tran&longs;iecimus. </s>

<s id="id.000608">Habet autem <lb/>&longs;e hoc pacto. </s><figure id="id.007.00.066.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.066.1.jpg"></figure>

<s id="id.000608">Habet autem <lb/>&longs;e hoc pacto. </s><figure id="id.007.01.065.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.065.1.jpg"></figure>

<s id="id.000609">E&longs;to fluminis citerior <lb/>ripa AB, vlterior CD. </s>

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<s id="id.000611">Vnde iterum temone in contrariam partem conuer&longs;o, <lb/>aquâ &longs;imiliter temonem propellente, per eandem circuli <lb/>portionem ad ripam citeriorem reuertitur, à qua paullo <lb/>antè di&longs;ce&longs;&longs;erat. </s>

<s id="id.000612">Ex quibus apparet, motus cau&longs;&longs;am non <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.067.jpg"/>e&longs;&longs;e &longs;olam cam, quæ ab ala temonis fit, aquæ <expan abbr="percu&longs;&longs;ion&emacr;">percu&longs;&longs;ionem</expan>, <lb/>vt &longs;en&longs;erat Ari&longs;toteles, &longs;ed currentis a quæ temonis alam <lb/>ferientis impul&longs;ionem: nihil autem referre, vtrum &longs;tante <lb/>naui a qua currat, vel câ currente a qua &longs;tet, vt in mari fit, <lb/>idem enim vtroque modo temo patitur. </s>

<s id="id.000612">Ex quibus apparet, motus cau&longs;&longs;am non <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.066.jpg"/>e&longs;&longs;e &longs;olam cam, quæ ab ala temonis fit, aquæ <expan abbr="percu&longs;&longs;ion&emacr;">percu&longs;&longs;ionem</expan>, <lb/>vt &longs;en&longs;erat Ari&longs;toteles, &longs;ed currentis a quæ temonis alam <lb/>ferientis impul&longs;ionem: nihil autem referre, vtrum &longs;tante <lb/>naui a qua currat, vel câ currente a qua &longs;tet, vt in mari fit, <lb/>idem enim vtroque modo temo patitur. </s>

<s id="id.000613">Vt autem machi<lb/>næ huius & totius negotij &longs;pecies facilius animo concipia<lb/>tur, &longs;chema hoc &longs;tudio&longs;orum oculis &longs;ubijciemus. </s><figure id="id.007.00.067.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.067.1.jpg"></figure>

<s id="id.000613">Vt autem machi<lb/>næ huius & totius negotij &longs;pecies facilius animo concipia<lb/>tur, &longs;chema hoc &longs;tudio&longs;orum oculis &longs;ubijciemus. </s><figure id="id.007.01.066.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.066.1.jpg"></figure>

<s id="id.000614">Lembi nauiculæue ideo appo&longs;itæ &longs;unt, vt oblongum <lb/>funem &longs;u&longs;tineant; id etenim nî fieret, aquæ immer&longs;us a<lb/>quam &longs;cindens machinæ motum impediret, ideo etiam <lb/>apponuntur, ne funis madens celeriter maceretur & pu<lb/>tre&longs;cat. </s>

<s id="id.000615">Huic &longs;peculationi affinis e&longs;t ea, velorum eorum, <lb/>quæ obliquè ventum, excipientia frumentarijs molis <lb/>dant motum, item verticillorum ex papyro, quibus con<lb/>tra ventum currentes per lu&longs;um pueri vtuntur. </s>

<s id="id.000616">vnicum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.068.jpg"/>enim horum omnium principium, & eadem, ratio. </s>

<s id="id.000616">vnicum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.067.jpg"/>enim horum omnium principium, & eadem, ratio. </s>

<s id="id.000617">Diximus enim, Temonem currente naui, lateraliter <lb/>conuer&longs;um obuios fluctus excipientem puppim ip&longs;am ob<lb/>liquè in alteram partem transferre. </s>

<s id="id.000618">Porrò ea vela, de qui<lb/>bus loquimur, ventorum flatibus obliquè oppo&longs;ita ean<lb/>dem ob cau&longs;&longs;am circulariter agitantur, quod vt figurâ eui<lb/>dentius fiat, </s><figure id="id.007.00.068.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.068.1.jpg"></figure>

<s id="id.000618">Porrò ea vela, de qui<lb/>bus loquimur, ventorum flatibus obliquè oppo&longs;ita ean<lb/>dem ob cau&longs;&longs;am circulariter agitantur, quod vt figurâ eui<lb/>dentius fiat, </s><figure id="id.007.01.067.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.067.1.jpg"></figure>

<s id="id.000619">E&longs;to velum AB, brachio <lb/>CE obliquè affixum, ita vt <lb/>angulus ACE maior &longs;it an<lb/>gulo BCE, ventus obliquè <lb/>velum feriens FG. <expan abbr="Itaq;">Itaque</expan> quo<lb/>niam ventus in velum obli<lb/>quum incidit, elabitur velum, <lb/>& circa centrum E vnà cum <lb/>brachio circumuertitur, in <lb/>cuius locum &longs;uccedit velum <lb/>HI, ex qua a&longs;&longs;idua velorum <lb/>&longs;ucce&longs;&longs;ione, brachiorum & a<lb/>xis cui adhærent, rotatio fit <lb/>perpetua. </s>

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<s id="id.000624">Si autem nece&longs;<lb/>&longs;e ei fuerit ad dextram &longs;ini&longs;tramqueue conuerti non pote<lb/>rit, ni&longs;i cauda ip&longs;a iuuetur. </s>

<s id="id.000625">Omnis enim motus progre&longs;&longs;i<lb/>uus quiete indiget, nec <expan abbr="ab&longs;q;">ab&longs;que</expan> &longs;tabili fulcimento progredi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.069.jpg"/><figure id="id.007.00.069.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.069.1.jpg"></figure><lb/>pote&longs;t, quod in libris de ani<lb/>malium ince&longs;&longs;u docet ip&longs;e<lb/>met Philo&longs;ophus. </s>

<s id="id.000625">Omnis enim motus progre&longs;&longs;i<lb/>uus quiete indiget, nec <expan abbr="ab&longs;q;">ab&longs;que</expan> &longs;tabili fulcimento progredi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.068.jpg"/><figure id="id.007.01.068.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.068.1.jpg"></figure><lb/>pote&longs;t, quod in libris de ani<lb/>malium ince&longs;&longs;u docet ip&longs;e<lb/>met Philo&longs;ophus. </s>

<s id="id.000626">Sit igitur, <lb/>pi&longs;cem conuerti velle, & fie<lb/>ri capite in D, deflectet illi<lb/>co caudam in E, caque; aquam <lb/>ceu &longs;tabile quippiam <expan abbr="feri&emacr;s">feriens</expan> <lb/>eiqueue quoddammodo fultus, <lb/>reliquum corpus CA refle<lb/>ctet in D, &longs;i autem conuerti <lb/>velit in F, caudam defle ctet in G, & eadem ratione <gap/> cte<lb/>tur in F. </s>

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<s id="id.000635">quæ &longs;ic figurâ explicamus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.070.jpg"/><figure id="id.007.00.070.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.070.1.jpg"></figure>

<s id="id.000635">quæ &longs;ic figurâ explicamus. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.069.jpg"/><figure id="id.007.01.069.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.069.1.jpg"></figure>

<s id="id.000636">E&longs;to nauis AB, malus CD, <lb/>mali &longs;edes D, locus antennæ <lb/>&longs;ublimior C, depre&longs;&longs;ior E: ita<lb/>que quoniam CD vectis e&longs;t, <lb/>quo mouens remotior fuerit à <lb/>fulcimento D, eo citiùs & vio<lb/>lentiùs pellet, velocius ergo <lb/>nauis mouebitur antenna in <lb/>C, quàm in E, con&longs;tituta. </s>

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<s id="id.000646">Videmus enim, & turres <lb/>quo &longs;ublimiores fuerint, eo magis à ventorum impetuo&longs;is <lb/>flatibus infe&longs;tari, quod &longs;anè ad vectis longitudinem refer<lb/>re, e&longs;&longs;et ridiculum. </s>

<s id="id.000647">Cæterùm quod ad puppis faciliorem <lb/>eleuationem ex mali ip&longs;ius altitudine pertinet, ad vectis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.071.jpg"/>contemplationem reducimus. </s>

<s id="id.000647">Cæterùm quod ad puppis faciliorem <lb/>eleuationem ex mali ip&longs;ius altitudine pertinet, ad vectis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.070.jpg"/>contemplationem reducimus. </s>

<s id="id.000648">e&longs;t enim quæ dam vectium <lb/>&longs;pecies ab alijs non con&longs;iderata, cuius brachia in angu<lb/>lum de&longs;inunt, vt ip&longs;e angulus in operatione &longs;it fulcimen<lb/>tum. </s><figure id="id.007.00.071.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.071.1.jpg"></figure>

<s id="id.000648">e&longs;t enim quæ dam vectium <lb/>&longs;pecies ab alijs non con&longs;iderata, cuius brachia in angu<lb/>lum de&longs;inunt, vt ip&longs;e angulus in operatione &longs;it fulcimen<lb/>tum. </s><figure id="id.007.01.070.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.070.1.jpg"></figure>

<s id="id.000649">E&longs;to enim vectis, de quo agimus, <lb/>ABC, cuius brachia AB, BC. iuncta <lb/>ad angulum B, &longs;itqueue B in operatione <lb/>fulc mentum. </s>

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<s id="id.000659">Ad huius vectis naturam referuntur fabrorum mal<lb/>lei, quibus clauos reuellunt, forcipes item quæ tenaci <lb/>mor&longs;u clauorum capita vmbellasue apprendentes, vio<lb/>lenter è tabulis extrahunt. </s>

<s id="id.000660">In malleo itaque &longs;ubtili, vt in <lb/>figura videre e&longs;t, AB vectis e&longs;t pars quæ à fulcimento ad <lb/>potentiam; ac verò quæ à fulcimento ad pondus, ponderi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.072.jpg"/><figure id="id.007.00.072.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.072.1.jpg"></figure><lb/>&longs;iquidem æquiparatur re&longs;i<lb/>&longs;tentia quae fit in C. </s>

<s id="id.000660">In malleo itaque &longs;ubtili, vt in <lb/>figura videre e&longs;t, AB vectis e&longs;t pars quæ à fulcimento ad <lb/>potentiam; ac verò quæ à fulcimento ad pondus, ponderi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.071.jpg"/><figure id="id.007.01.071.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.071.1.jpg"></figure><lb/>&longs;iquidem æquiparatur re&longs;i<lb/>&longs;tentia quae fit in C. </s>

<s id="id.000661">I dem ob<lb/>&longs;eruamus in forcipe, in quo <lb/>duo quidem brachia AD, <lb/>CB, quatenus ad appren&longs;io<lb/>nem pertinet, fulcimentum, <lb/>habent in ip&longs;o <expan abbr="c&emacr;tro">centro</expan> &longs;eu ver<lb/>tebra, & ideo quo longiores <lb/>fuerint, eo tenaciùs appre<lb/>hendunt & retinent. </s>

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<s id="id.000664">His igitur hoc pacto examinatis, ad nauim & malum <lb/>reuertentes, dicimus, tunc facillimam fieri puppis eleua<lb/>tionem, proræ verò demer&longs;ionem, cum maxima fuerit <lb/>proportio, quam habet altitudo mali, ad eam nauis <expan abbr="part&emacr;">partem</expan> <lb/>quæ à malo ad ip&longs;am puppis extremitatem, pertingit. <lb/></s>

<s id="id.000665">Quamobrem prudentes nauium fabri, vt huic difficultati <lb/>occurrant, malum non in medio quidem nauis, &longs;ed in ter<lb/>tia ferè parte longitudinis quæ à prora e&longs;t, puppim ver&longs;us <lb/>con&longs;tituunt. </s><figure id="id.007.00.072.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.072.2.jpg"></figure>

<s id="id.000665">Quamobrem prudentes nauium fabri, vt huic difficultati <lb/>occurrant, malum non in medio quidem nauis, &longs;ed in ter<lb/>tia ferè parte longitudinis quæ à prora e&longs;t, puppim ver&longs;us <lb/>con&longs;tituunt. </s><figure id="id.007.01.071.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.071.2.jpg"></figure>

<s id="id.000666">E&longs;to enim nauis AB; cuius <lb/>malus CD: prora A: puppis B; <expan abbr="v&emacr;-to">ven<lb/>to</expan> igitur velum impellente, <expan abbr="mal&umacr;">malum</expan> <lb/>ad partem contrariam vergit, pu<lb/>ta in FD. </s>

<s id="id.000667">At <expan abbr="quoniã">quoniam</expan> catche&longs;ium <lb/>funi ad puppim vnitur in B, nauim, <lb/>hoc e&longs;t, ip&longs;am puppim trahat ne<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.073.jpg"/>ce&longs;&longs;e e&longs;t. </s>

<s id="id.000667">At <expan abbr="quoniã">quoniam</expan> catche&longs;ium <lb/>funi ad puppim vnitur in B, nauim, <lb/>hoc e&longs;t, ip&longs;am puppim trahat ne<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.072.jpg"/>ce&longs;&longs;e e&longs;t. </s>

<s id="id.000668">non pote&longs;t autem; quoniam &longs;uburræ grauitas & <lb/>onera, quæ naui impo&longs;ita inter D. & <emph type="italics"/>B.<emph.end type="italics"/> grauitatis centrum <lb/>circa punctum E con&longs;tituunt, quod quidem vi ventorum <lb/>inclinante malo ab E, in G eleuaretur, quo igitur minor <lb/>fuerit proportio CD ad DE & maius pondus ip&longs;um cu<lb/>ius grauitatis centrum in E minus præualebit potentia <lb/>pellens in C ad eleuationem partis nauigij, quæ à mali &longs;e<lb/>de ad puppim intercedit, An igitur malus &longs;it vectis, pes ve<lb/>rò fulcimentum, pondus autem quod vecte mouetur, <expan abbr="ips&umacr;">ipsum</expan> <lb/>nauigium, vt placuit Ari&longs;toteli, & qua item ratione malus <lb/>in nauim vt vectis operetur, exijs quae dicta &longs;unt, facilè pa<lb/>tet. </s>

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<s id="id.000676">Nos autem vt rem lucidiorem <lb/>faciamus, &longs;chema, quod nec ip&longs;e fecit, nec Philo&longs;ophus, <lb/>proponemus. </s>

<s id="id.000677">E&longs;to nauis A <emph type="italics"/>B<emph.end type="italics"/>, cuius prora A, puppis verò D, guber<lb/>naculum C<emph type="italics"/>B<emph.end type="italics"/>, temonis ala <emph type="italics"/>B<emph.end type="italics"/>D, veli &longs;inus EF, velum vero <lb/>ita con&longs;titutum, vt directè ex puppi flantem ventum exci-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.074.jpg"/><figure id="id.007.00.074.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.074.1.jpg"></figure><lb/>piat. </s>

<s id="id.000677">E&longs;to nauis A <emph type="italics"/>B<emph.end type="italics"/>, cuius prora A, puppis verò D, guber<lb/>naculum C<emph type="italics"/>B<emph.end type="italics"/>, temonis ala <emph type="italics"/>B<emph.end type="italics"/>D, veli &longs;inus EF, velum vero <lb/>ita con&longs;titutum, vt directè ex puppi flantem ventum exci-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.073.jpg"/><figure id="id.007.01.073.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.073.1.jpg"></figure><lb/>piat. </s>

<s id="id.000678">Hoc vbi euenerit, naui<lb/>gium, rectâ è puppi mouetur <lb/>in proram; Si autem ventus la<lb/>teraliter &longs;pirat, puta à parte <lb/>G ver&longs;us H & nihilo &longs;ecius na<lb/>uigium, ac &longs;i ventus ex pup<lb/>pi e&longs;&longs;et antror&longs;um propelle<lb/>re volunt, velum quidem obli<lb/>quant partem eius infimam, <lb/>pedem nempe, quæ e&longs;t in F <lb/>contrahentes, Cornu verò <lb/>antennæ vbi E, proram ver&longs;us <lb/>laxantes ventumque; ip&longs;um obliquè excipientes id <expan abbr="effici&umacr;t">efficiunt</expan>, <lb/>vt ventus minus violenter feriat, & minori &longs;ui parte <expan abbr="vel&umacr;">velum</expan> <lb/>impleat, & quoniam ventus velum pellit in partem con<lb/>trariam, nempe in H, ip&longs;i vt vento re&longs;i&longs;tant conuer&longs;o gu<lb/>bernaculo ex C in L, & temone <emph type="italics"/>B<emph.end type="italics"/>D, in <emph type="italics"/>B<emph.end type="italics"/>M compellunt <lb/>proram ad partem à qua ventu<gap/> ip&longs;e &longs;pirat. </s>

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<s id="id.000682">Quod <lb/>autem addit Picolomineus rem ad vectem reduci po&longs;&longs;e, <lb/>non e&longs;t cur &longs;ub &longs;ilentio prætereamus. </s>

<s id="id.000683">Ventus, inquit, pon<lb/>deris gubernaculum mouentis vicem obtinet; centrum <lb/>verò (fulcimentum intelligit) in medio nauis e&longs;t, quod ta-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.075.jpg"/>men ad proram vergit, vt faciliùs ip&longs;i vento re&longs;i&longs;tere po&longs;<lb/>&longs;it. </s>

<s id="id.000683">Ventus, inquit, pon<lb/>deris gubernaculum mouentis vicem obtinet; centrum <lb/>verò (fulcimentum intelligit) in medio nauis e&longs;t, quod ta-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.074.jpg"/>men ad proram vergit, vt faciliùs ip&longs;i vento re&longs;i&longs;tere po&longs;<lb/>&longs;it. </s>

<s id="id.000684">Tunc enim in rectum mouebitur nauis, cum &longs;ibi inui<lb/>cem æ quatæ vires, qua&longs;i libramentum con&longs;tituerint. </s>

<s id="id.000685">Hæc <lb/>ille, cuius &longs;en&longs;um figurâ propo&longs;itâ facilè aperiemus. </s><figure id="id.007.00.075.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.075.1.jpg"></figure>

<s id="id.000685">Hæc <lb/>ille, cuius &longs;en&longs;um figurâ propo&longs;itâ facilè aperiemus. </s><figure id="id.007.01.074.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.074.1.jpg"></figure>

<s id="id.000686">E&longs;to carina AB, cuius prora <lb/>A, puppis, B temo BC, ventus verò <lb/>obliquè feriens H. </s>

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<s id="id.000692"><emph type="italics"/>Quæritur, Cur ex figuris omnibus rotundæ faciliùs <lb/>moueantur?<emph.end type="italics"/></s>

<s id="id.000693">Trifariam, inquit Ari&longs;toteles, circulum rotari contin<lb/>git; Aut &longs;ecundum ab&longs;idem <expan abbr="c&emacr;tro">centro</expan> &longs;imul moto, quem<lb/>admodum plau&longs;tri vertitur rota; aut circa manens cen<lb/>trum, veluti trochleæ puteorum, &longs;tante centro: Aut in pa<lb/>uimento manente centro, &longs;icuti figuli rota conuertitur. <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.076.jpg"/>Cau&longs;&longs;am verò explicans, ait, celerrima eiu&longs;modi corpora <lb/>e&longs;&longs;e, eo quod paruâ &longs;ui parte planum contingunt, vti cir<lb/>culus &longs;ecundum punctum, item quoniam non offen&longs;ant: <lb/>Non offen&longs;andi vero e&longs;&longs;e cau&longs;&longs;am, quod &longs;emotum à terra <lb/>habeant angulum. </s>

<s id="id.000693">Trifariam, inquit Ari&longs;toteles, circulum rotari contin<lb/>git; Aut &longs;ecundum ab&longs;idem <expan abbr="c&emacr;tro">centro</expan> &longs;imul moto, quem<lb/>admodum plau&longs;tri vertitur rota; aut circa manens cen<lb/>trum, veluti trochleæ puteorum, &longs;tante centro: Aut in pa<lb/>uimento manente centro, &longs;icuti figuli rota conuertitur. <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.075.jpg"/>Cau&longs;&longs;am verò explicans, ait, celerrima eiu&longs;modi corpora <lb/>e&longs;&longs;e, eo quod paruâ &longs;ui parte planum contingunt, vti cir<lb/>culus &longs;ecundum punctum, item quoniam non offen&longs;ant: <lb/>Non offen&longs;andi vero e&longs;&longs;e cau&longs;&longs;am, quod &longs;emotum à terra <lb/>habeant angulum. </s>

<s id="id.000694">Item propterea quod corpus, cui fiunt <lb/>obuiam, &longs;ecundum pu&longs;illum tangunt. </s>

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<s id="id.000697">Hæc ferè Philo&longs;ophus, cuius rationes ad eum &longs;olum<lb/>modo circularem motum faciunt, qui fit &longs;ecundum ab&longs;i<lb/>dem, vt in carrorum rotis v&longs;u venit, nec aptantur rotis fi<lb/>gulorum trochlei&longs;queue, cuiu&longs;modi &longs;unt illæ, quæ &longs;upra <lb/>puteos appenduntur. </s>

<s id="id.000698">Nos igitur, ad Ari&longs;totelis mentem, <lb/>primam rotationis &longs;peciem, quæ e&longs;t &longs;ecundum ab&longs;idem, <lb/>examinabimus. </s><figure id="id.007.00.076.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.076.1.jpg"></figure>

<s id="id.000698">Nos igitur, ad Ari&longs;totelis mentem, <lb/>primam rotationis &longs;peciem, quæ e&longs;t &longs;ecundum ab&longs;idem, <lb/>examinabimus. </s><figure id="id.007.01.075.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.075.1.jpg"></figure>

<s id="id.000699">E&longs;to rota &longs;phæ<lb/>raue AB, cuius cen<lb/>trum C; Horizontis <lb/>planum DE; conta<lb/>ctus circuli in plano <lb/>B. <expan abbr="perp&emacr;dicularis">perpendicularis</expan> ho<lb/>rizonti à puncto <expan abbr="c&omacr;-tactus">con<lb/>tactus</expan> B ip&longs;a <emph type="italics"/>B<emph.end type="italics"/>CA, <lb/>tran&longs;iens per <expan abbr="centr&umacr;">centrum</expan> <lb/>C, partes rotæ circa <lb/>perpendicularem AF<emph type="italics"/>B<emph.end type="italics"/>, AG<emph type="italics"/>B<emph.end type="italics"/>, angulus contactus G<emph type="italics"/>B<emph.end type="italics"/>E. <lb/></s>

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<s id="id.000702">Angulum porro, quem à <lb/>terra &longs;emotum dicit, ip&longs;e angulus e&longs;t contingentiae. </s>

<s id="id.000703">eleua<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.077.jpg"/>tur enim ex <emph type="italics"/>B<emph.end type="italics"/> in G. </s>

<s id="id.000703">eleua<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.076.jpg"/>tur enim ex <emph type="italics"/>B<emph.end type="italics"/> in G. </s>

<s id="id.000704">Si autem corpus quodpiam in plano <lb/>fuerit, puta HI in puncto illud tanget ci culus ei occur<lb/>rens, exempli gratiâ in K. </s>

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<s id="id.000714">Angulos autem hîc &longs;ectores ip&longs;os vocat; oportet <lb/>enim circulos tum maiores tum minores circa idem cen<lb/>trum e&longs;&longs;e con&longs;titutos. </s>

<s id="id.000715">Hæc autem non ab&longs;imili ab eo <lb/>quod &longs;uprà po&longs;uimus &longs;chemate explicantur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.078.jpg"/><figure id="id.007.00.078.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.078.1.jpg"></figure>

<s id="id.000715">Hæc autem non ab&longs;imili ab eo <lb/>quod &longs;uprà po&longs;uimus &longs;chemate explicantur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.077.jpg"/><figure id="id.007.01.077.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.077.1.jpg"></figure>

<s id="id.000716">E&longs;to enim circulus <lb/>AB circa centrum, C, <lb/>Horizontis planum DE, <lb/>tangens circulum in B, <lb/>linea verò perpendicu<lb/>laris per centrum BCA. <lb/></s>

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<s id="id.000720">Si enim in figura producatur ad oppo&longs;i<lb/>tam partem &longs;emidiameter HC in K &longs;ecans minorem cir<lb/>culum in L, duos alios &longs;ectores angulosue habebimus, <expan abbr="n&emacr;-pe">nem<lb/>pe</expan> KCB, LCG, ip&longs;is ACHFCI æquales. </s>

<s id="id.000721"><expan abbr="Itaq;">Itaque</expan> quan<lb/>tum adiuuat motum anguli ACH maioris nutus, in de<lb/>&longs;cendendo ad partes B, tantundem retardat anguli item <lb/>maioris KCB, contra nutus (vt ita appellem) in <expan abbr="a&longs;cend&emacr;-do">a&longs;cenden<lb/>do</expan> ad partes A. & &longs;anè quatenus ad rei naturam pertinet <lb/>& ad ip&longs;um æquilibrium, non differunt maiores circuli à <lb/>minoribus, nec &longs;unt maiores minoribus mobiliores, imo <lb/>ex aliqua ratione minores videntur fore ad motum faci<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.079.jpg"/>liores, tum quia data materiæ æqualitate &longs;unt leuiores, <lb/>tum etiam quod maior e&longs;t angulus contactus ad planum <lb/>circumferentiae minoris quàm maioris circuli, vt in &longs;ubie<lb/><figure id="id.007.00.079.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.079.1.jpg"></figure><lb/>cta figura angulus ABC maior <lb/>e&longs;t angulo DBC, in materiali i<lb/>gitur circulo rotaue maiore &longs;ui <lb/>parte tanget planum DB circu<lb/>lus, ip&longs;o AB. quicquid tamen fit, <lb/>mobiliores &longs;unt maiores circuli <lb/>non quidem ex natura circuli, <lb/>quæ tam in maioribus quàm in <lb/>ip&longs;is minoribus e&longs;t par, &longs;ed alijs de cau&longs;&longs;is, quas &longs;uo loco <lb/>examinabimus. </s>

<s id="id.000721"><expan abbr="Itaq;">Itaque</expan> quan<lb/>tum adiuuat motum anguli ACH maioris nutus, in de<lb/>&longs;cendendo ad partes B, tantundem retardat anguli item <lb/>maioris KCB, contra nutus (vt ita appellem) in <expan abbr="a&longs;cend&emacr;-do">a&longs;cenden<lb/>do</expan> ad partes A. & &longs;anè quatenus ad rei naturam pertinet <lb/>& ad ip&longs;um æquilibrium, non differunt maiores circuli à <lb/>minoribus, nec &longs;unt maiores minoribus mobiliores, imo <lb/>ex aliqua ratione minores videntur fore ad motum faci<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.078.jpg"/>liores, tum quia data materiæ æqualitate &longs;unt leuiores, <lb/>tum etiam quod maior e&longs;t angulus contactus ad planum <lb/>circumferentiae minoris quàm maioris circuli, vt in &longs;ubie<lb/><figure id="id.007.01.078.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.078.1.jpg"></figure><lb/>cta figura angulus ABC maior <lb/>e&longs;t angulo DBC, in materiali i<lb/>gitur circulo rotaue maiore &longs;ui <lb/>parte tanget planum DB circu<lb/>lus, ip&longs;o AB. quicquid tamen fit, <lb/>mobiliores &longs;unt maiores circuli <lb/>non quidem ex natura circuli, <lb/>quæ tam in maioribus quàm in <lb/>ip&longs;is minoribus e&longs;t par, &longs;ed alijs de cau&longs;&longs;is, quas &longs;uo loco <lb/>examinabimus. </s>

<s id="id.000722">Cæterùm vt aliquid de motu qui &longs;ecundum ab&longs;idem <lb/>fit, ex no&longs;tro penu promamus, Dicimus, Circulos, rota&longs;ue, <lb/>quæ hoc pacto mouentur, vel per horizontis planum mo<lb/>ueri, vel per accliue, aut decliue. </s>

<s id="id.000723">Si autem per horizontis <lb/>planum, ideo facilem e&longs;&longs;e motum, quòd nunquam, cæte<lb/>ris paribus, centrum grauitatis ip&longs;ius corporis à centro <lb/>mundi, in ip&longs;a rotatione, fiat remotius. </s><figure id="id.007.00.079.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.079.2.jpg"></figure>

<s id="id.000723">Si autem per horizontis <lb/>planum, ideo facilem e&longs;&longs;e motum, quòd nunquam, cæte<lb/>ris paribus, centrum grauitatis ip&longs;ius corporis à centro <lb/>mundi, in ip&longs;a rotatione, fiat remotius. </s><figure id="id.007.01.078.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.078.2.jpg"></figure>

<s id="id.000724">E&longs;to enim planum, <lb/>horizontis AB, cui circu<lb/>lus in&longs;i&longs;tat AD, circa cen<lb/>trum C, diui&longs;us per <expan abbr="centr&umacr;">centrum</expan> <lb/>ip&longs;um à perpendiculari <lb/>ACD; Ducatur autem per <lb/>centrum C recta linea ho<lb/>rizonti æquidi&longs;tans, ECFG: dum diuidatur circulus vt<lb/>cunque in partes AH, HF, FI, ID, & CI, CH iungan<lb/>tur. </s>

<s id="id.000725">Po&longs;thæc intelligatur circulum &longs;ecundum ab&longs;idem <lb/>moueri ad partes G, erit igitur aliquando punctum H, <lb/>tangens horizontis planum, tangat autem in K, tum F in <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.080.jpg"/>L, I in N. D verò in O. </s>

<s id="id.000725">Po&longs;thæc intelligatur circulum &longs;ecundum ab&longs;idem <lb/>moueri ad partes G, erit igitur aliquando punctum H, <lb/>tangens horizontis planum, tangat autem in K, tum F in <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.079.jpg"/>L, I in N. D verò in O. </s>

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<s id="id.000727">Centrum ergo circuli, quod idem & grauitatis e&longs;t <expan abbr="centr&umacr;">centrum</expan>, <lb/>feretur per rectam CPQRS, &longs;unt enim KP, LQ, NR, <lb/>OS ip&longs;i AC &longs;emidiametro æquales, <expan abbr="n&umacr;quam">nunquam</expan> igitur cen<lb/>trum ip&longs;um C in circuli rotatione ab horizontis plano e<lb/>leuabitur, nec à mundi centro fiet remotius. </s>

<s id="id.000728">Hoc autem longè aliter cæteris figuris contingit, <lb/>quarum motus ideo in æqualis, quòd non &longs;emper in rota<lb/>tione centrum grauitatis eandem &longs;eruet à mundi centro <lb/>di&longs;tantiam. </s><figure id="id.007.00.080.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.080.1.jpg"></figure>

<s id="id.000728">Hoc autem longè aliter cæteris figuris contingit, <lb/>quarum motus ideo in æqualis, quòd non &longs;emper in rota<lb/>tione centrum grauitatis eandem &longs;eruet à mundi centro <lb/>di&longs;tantiam. </s><figure id="id.007.01.079.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.079.1.jpg"></figure>

<s id="id.000729">E&longs;to enim Ellip&longs;is <lb/>ABCD, cuius <expan abbr="c&emacr;trum">centrum</expan> <lb/>E, diameter longior <lb/>BED, breuior AEC, <lb/>Horizontis planum, <lb/>FCG. locus contactus <lb/>C perpendicularis à <lb/>contactu per centrum i<lb/>p&longs;a CEA diuidens El<lb/>lip&longs;im in partes æquales, & æqueponderantes ABC, <lb/>ADC. </s>

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<s id="id.000733">Mota igitur ellip&longs;ià C in K, haud ita difficilis e<lb/>rit motus, quippe quod haud multum EH &longs;uperet EC, at <lb/>difficilior erit translatio in M, difficillima verò in O. Val<lb/>de enim à &longs;itu E, ibi attollitur grauitatis centrum, a&longs;cen<lb/>dens nempe vbi P. </s>

<s id="id.000734">Videmus igitur ex his eandem poten<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.081.jpg"/>tiam in mouendo ellip&longs;im, haud pariter &longs;e habere, vt in <lb/>mouendo circulum. </s>

<s id="id.000734">Videmus igitur ex his eandem poten<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.080.jpg"/>tiam in mouendo ellip&longs;im, haud pariter &longs;e habere, vt in <lb/>mouendo circulum. </s>

<s id="id.000735">ibi enim centrum grauitatis fertur <lb/>per æquidi&longs;tantem horizonti, hic verò modò attollitur, <lb/>modò deprimitur, quod &longs;anè mole&longs;tiam & difficultatem <lb/>facit. </s>

<s id="id.000736">Sed idem alijs figuris contingere, & maximè latera<lb/>tis, ita docebimus. </s><figure id="id.007.00.081.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.081.1.jpg"></figure>

<s id="id.000736">Sed idem alijs figuris contingere, & maximè latera<lb/>tis, ita docebimus. </s><figure id="id.007.01.080.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.080.1.jpg"></figure>

<s id="id.000737">E&longs;to enim triangulum <lb/>æquilaterum ABC, cuius <lb/>grauitatis centrum E hori<lb/>zontis planum BD. </s>

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<s id="id.000745">Ad hanc quæ&longs;tionem illud quoque facit, cur per de<lb/>cliue planum grauiora corpora, & rotunda maximè; ma<lb/>gno impetu dimi&longs;&longs;a, delabantur. </s>

<s id="id.000746">E&longs;to enim rota &longs;phæraue aut Cylindrus CD, cuius <lb/>centrum E, tangens decliue planum AB in D, quæritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.082.jpg"/>cur dimi&longs;&longs;a hæc magno impetu deferantur ad partes B, <lb/>Ducatur per grauitatis centrum E ad horizontem, BK <lb/>perpendicularis FEL &longs;ecans decliue planum in G, cir<lb/>cumferentiam verò in H. opponitur autem EG angulo <lb/>recto EDG, maior ergo EG ip&longs;a ED, hoc e&longs;t, EH, inter <lb/><figure id="id.007.00.082.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.082.1.jpg"></figure><lb/>circumferentiam igitur & pla<lb/>num decliue, &longs;patium interce<lb/>dit HG. </s>

<s id="id.000746">E&longs;to enim rota &longs;phæraue aut Cylindrus CD, cuius <lb/>centrum E, tangens decliue planum AB in D, quæritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.081.jpg"/>cur dimi&longs;&longs;a hæc magno impetu deferantur ad partes B, <lb/>Ducatur per grauitatis centrum E ad horizontem, BK <lb/>perpendicularis FEL &longs;ecans decliue planum in G, cir<lb/>cumferentiam verò in H. opponitur autem EG angulo <lb/>recto EDG, maior ergo EG ip&longs;a ED, hoc e&longs;t, EH, inter <lb/><figure id="id.007.01.081.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.081.1.jpg"></figure><lb/>circumferentiam igitur & pla<lb/>num decliue, &longs;patium interce<lb/>dit HG. </s>

<s id="id.000747">Ducatur item DI ip&longs;i <lb/>FG æquidi&longs;tans. </s>

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<s id="id.000754">E&longs;t igitur punctum M mundi centro propius <lb/>quàm D & G, quare eò non impedita rota ip&longs;a &longs;uo nutu <lb/>feretur, nec &longs;tabit donec in fimum <expan abbr="loc&umacr;">locum</expan> vbi quie&longs;cat nan<lb/>ci&longs;catur. </s>

<s id="id.000756">Po&longs;&longs;umus etiam Rota &longs;phæraue in plano decliui <lb/>collocata, datam potentiam inuenire, quæ extremitati <lb/>diametri ad eam partem qua vergit applicata ip&longs;am rotam <lb/>&longs;phæramue impediatne delabatur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.083.jpg"/><figure id="id.007.00.083.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.083.1.jpg"></figure>

<s id="id.000756">Po&longs;&longs;umus etiam Rota &longs;phæraue in plano decliui <lb/>collocata, datam potentiam inuenire, quæ extremitati <lb/>diametri ad eam partem qua vergit applicata ip&longs;am rotam <lb/>&longs;phæramue impediatne delabatur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.082.jpg"/><figure id="id.007.01.082.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.082.1.jpg"></figure>

<s id="id.000757">E&longs;to planum in clinatum <lb/>AB, cui Rota &longs;phæraue in&longs;i<lb/>&longs;tat tangatque; illud in C. </s>

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<s id="id.000767">His ita con&longs;ideratis, admonet nos locus, vt pulcher<lb/>rimam dubitationem diluamus. </s>

<s id="id.000768">Quæritur, Cur maiores <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.084.jpg"/>rotae impingentes, facilius offendicula &longs;uperent quàm mi<lb/>nores. </s>

<s id="id.000768">Quæritur, Cur maiores <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.083.jpg"/>rotae impingentes, facilius offendicula &longs;uperent quàm mi<lb/>nores. </s>

<s id="id.000769">Neque enim &longs;atisfacere videtur quod ait Ari&longs;tote<lb/>les, ex contactu in puncto eo anguli à plano eleuatione id <lb/>fieri, alijs ergo principijs dubitatio &longs;oluitur. </s><figure id="id.007.00.084.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.084.1.jpg"></figure>

<s id="id.000769">Neque enim &longs;atisfacere videtur quod ait Ari&longs;tote<lb/>les, ex contactu in puncto eo anguli à plano eleuatione id <lb/>fieri, alijs ergo principijs dubitatio &longs;oluitur. </s><figure id="id.007.01.083.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.083.1.jpg"></figure>

<s id="id.000770">E&longs;to rota quidem maior <lb/>AB, circa centrum C minor <lb/>vero DB circa centrum, E, <lb/><expan abbr="tãgentes">tangentes</expan> horizontis planum <lb/>in B. </s>

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<s id="id.000777">Rota verò minor FD, tangens itidem <lb/>planum in D. rotæ autem centrum G, offendiculi verò re<lb/>ctitudo DH. </s>

<s id="id.000778">Ducatur per H ip&longs;i AB horizonti æquidi<lb/>&longs;tans HI &longs;ecans minorem circulum in K, maiorem verò <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.085.jpg"/><figure id="id.007.00.085.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.085.1.jpg"></figure><lb/>in I. </s>

<s id="id.000778">Ducatur per H ip&longs;i AB horizonti æquidi<lb/>&longs;tans HI &longs;ecans minorem circulum in K, maiorem verò <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.084.jpg"/><figure id="id.007.01.084.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.084.1.jpg"></figure><lb/>in I. </s>

<s id="id.000779">Ducantur etiam dia<lb/>metri maioris quidem <lb/>LEM, minoris NGO, <lb/>Tum à puncto K perpen<lb/>dicularis ducatur ad <lb/>GO, ip&longs;a KP, item à pun<lb/>cto I ad EM perpendi<lb/>cularis <expan abbr="Iq.">Ique</expan> Dico EQ ad <lb/>QL, minorem habere <lb/>proportionem quam GP, <lb/>ad PN. </s>

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<s id="id.000790">Minor ergo potentia requi<lb/>ritur in L ad &longs;u&longs;tinendum pondus E ex fulcimento Q hoc <lb/>e&longs;t I, quàm requiratur in N ad &longs;u&longs;tinendum pondus G ex <lb/>fulcimento P, hoc e&longs;t K. </s>

<s id="id.000791">Minor ergo potentia requiritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.086.jpg"/>ad transferendam maiorem retam CD vltra offendicu<lb/>lum IV, hoc e&longs;t, DH, quàm requiratur ad trans ferendam <lb/>minorem vltra offendiculum KT, hoc e&longs;t HD, quod fue<lb/>rat o&longs;tendendum. </s>

<s id="id.000791">Minor ergo potentia requiritur <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.085.jpg"/>ad transferendam maiorem retam CD vltra offendicu<lb/>lum IV, hoc e&longs;t, DH, quàm requiratur ad trans ferendam <lb/>minorem vltra offendiculum KT, hoc e&longs;t HD, quod fue<lb/>rat o&longs;tendendum. </s>

<s id="id.000792">Ad hæc, quæri pote&longs;t, quo pacto plau&longs;trorum rotæ <lb/>in ip&longs;a plau&longs;tri conuer&longs;ione &longs;e habeant, nempe quæ &longs;it li<lb/>nea illa curua, quam in conuer&longs;ione de&longs;cribunt. </s><figure id="id.007.00.086.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.086.1.jpg"></figure>

<s id="id.000792">Ad hæc, quæri pote&longs;t, quo pacto plau&longs;trorum rotæ <lb/>in ip&longs;a plau&longs;tri conuer&longs;ione &longs;e habeant, nempe quæ &longs;it li<lb/>nea illa curua, quam in conuer&longs;ione de&longs;cribunt. </s><figure id="id.007.01.085.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.085.1.jpg"></figure>

<s id="id.000793">E&longs;to rotarum in <lb/>plano orbita, <expan abbr="d&umacr;">dum</expan> plau<lb/>&longs;trum rectâ procedit <lb/>AB, CD, Sunt autem i<lb/>p&longs;æ lineæ, quod o&longs;ten<lb/>demus po&longs;tea, æquedi<lb/>&longs;tantes. </s>

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<s id="id.000800">Et quoniam vel D in ip&longs;a conuer&longs;ione &longs;tat omnino <lb/>nec quicquam progreditur, vt in prima figura, vel non &longs;tat <lb/>vt in &longs;ecunda, quo ca&longs;u portionem parui circuli de&longs;cribit, <lb/>ip&longs;i maiori circulo & exteriori concentricam. </s>

<s id="id.000801">Vnde col<lb/>ligimus, Plau&longs;trorum conuer&longs;iones flexione<gap/>queue &longs;emper <lb/>circa centrum, & concentricorum circulorum portiones <lb/>fieri, <emph type="italics"/>H<emph.end type="italics"/>inc etiam di&longs;cimus, cur veteres, vt ex antiquis co<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.087.jpg"/>gno&longs;cimus ve&longs;tigijs, circos in quibus cur&longs;us quadrigarum <lb/>fiebant ea forma quæ apparet, efformauerint. </s>

<s id="id.000801">Vnde col<lb/>ligimus, Plau&longs;trorum conuer&longs;iones flexione<gap/>queue &longs;emper <lb/>circa centrum, & concentricorum circulorum portiones <lb/>fieri, <emph type="italics"/>H<emph.end type="italics"/>inc etiam di&longs;cimus, cur veteres, vt ex antiquis co<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.086.jpg"/>gno&longs;cimus ve&longs;tigijs, circos in quibus cur&longs;us quadrigarum <lb/>fiebant ea forma quæ apparet, efformauerint. </s>

<s id="id.000802">Hoc etiam <lb/>theorema probamus. </s>

<s id="id.000803">Cylindros, quorum ba&longs;es axi &longs;unt perpendiculares, <lb/>dum in æquato plano conuoluuntur, rectâ incedere & <lb/>per parallelas, quarum di&longs;tantia axis &longs;eu latoris longitudi<lb/>ne præfinitur. </s><figure id="id.007.00.087.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.087.1.jpg"></figure>

<s id="id.000803">Cylindros, quorum ba&longs;es axi &longs;unt perpendiculares, <lb/>dum in æquato plano conuoluuntur, rectâ incedere & <lb/>per parallelas, quarum di&longs;tantia axis &longs;eu latoris longitudi<lb/>ne præfinitur. </s><figure id="id.007.01.086.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.086.1.jpg"></figure>

<s id="id.000804">E&longs;to enim Cylin<lb/>drus ABCD, cuius a<lb/>xis GH, <expan abbr="horiz&omacr;tis">horizontis</expan> pla<lb/>no in&longs;i&longs;tens &longs;ecundum <lb/>latus AB, cui latus op<lb/>po&longs;itum & aequale CD. <lb/></s>

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<s id="id.000814">E&longs;to enim Cylindrus ABCD, cuius ba&longs;es ellip&longs;es <expan abbr="inuic&emacr;">inuicem</expan> <lb/><expan abbr="æquedi&longs;tãtes">æquedi&longs;tantes</expan>, quarum axes longiores AB, CD, Commu<lb/>nis autem &longs;ectio cylindri & plani ad axem & horizontem <lb/>planum perpendicularis EHF. </s>

<s id="id.000815">Diuidatur autem &longs;emicir-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.088.jpg"/>culus EHF in partes æquales quatuor FI, IH, HG, GE. <lb/><figure id="id.007.00.088.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.088.1.jpg"></figure><lb/>Tum per diui&longs;ionum puncta lateri parallelae, rectæ ducan<lb/>tur KGL, M<emph type="italics"/>H<emph.end type="italics"/>N, OIP, quæ quidem <expan abbr="c&umacr;">cum</expan> ba&longs;es AMB, DNC <lb/>parallelæ &longs;int, erunt inuicem æquales, cumqueue circum<lb/>ferentia E<emph type="italics"/>H<emph.end type="italics"/>F æquales, eosqueue rectos angulos <expan abbr="c&omacr;&longs;tituent">con&longs;tituent</expan>. <lb/></s>

<s id="id.000815">Diuidatur autem &longs;emicir-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.087.jpg"/>culus EHF in partes æquales quatuor FI, IH, HG, GE. <lb/><figure id="id.007.01.087.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.087.1.jpg"></figure><lb/>Tum per diui&longs;ionum puncta lateri parallelae, rectæ ducan<lb/>tur KGL, M<emph type="italics"/>H<emph.end type="italics"/>N, OIP, quæ quidem <expan abbr="c&umacr;">cum</expan> ba&longs;es AMB, DNC <lb/>parallelæ &longs;int, erunt inuicem æquales, cumqueue circum<lb/>ferentia E<emph type="italics"/>H<emph.end type="italics"/>F æquales, eosqueue rectos angulos <expan abbr="c&omacr;&longs;tituent">con&longs;tituent</expan>. <lb/></s>

<s id="id.000816">Ducatur po&longs;t hæc &longs;eor&longs;um recta QR, & eidem perpendi<lb/>cularis ST eam &longs;ecans in V. applicetur autem rectæ ST <lb/>æqualis Cylindri lateri BC, ip&longs;a <foreign lang="greek">hz. </foreign></s>

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<s id="id.000818">Tum fiant VX, XY, YZ, Z<foreign lang="greek">a</foreign> æ quales ip&longs;is EG, <lb/>G<emph type="italics"/>H<emph.end type="italics"/>, <emph type="italics"/>H<emph.end type="italics"/>I, IF, & per puncta X, Y, Z, <foreign lang="greek">a</foreign> & paralleli ip&longs;i ST du<lb/>cantur <foreign lang="greek">o a p, n *z c, l g m, k x q</foreign>, tum & his ex altera parte re<lb/>&longs;pondentes parallelæ per puncta <foreign lang="greek">b, g, d, e. </foreign></s>

<s id="id.000819">Sit autem <foreign lang="greek">o a</foreign> æ<lb/>qualis AF, <foreign lang="greek">a</foreign> <11> æqualis FD, item <foreign lang="greek">e</foreign> <10>, æqualis EC, <foreign lang="greek">e s</foreign> æqualis <lb/>EB, &longs;ed & <foreign lang="greek">n *z</foreign> aequalis OI, <foreign lang="greek">*z c</foreign> ip&longs;i P, <foreign lang="greek">l</foreign>y ip&longs;i MH, y <foreign lang="greek">m</foreign> verò ip&longs;i <lb/>HN, <expan abbr="t&umacr;">tum</expan> <foreign lang="greek">k x</foreign> ip&longs;i KG. & <foreign lang="greek">x q</foreign>, ip&longs;i GL & ip&longs;is æquales & aequa<lb/>liter po&longs;itæ ad partes R, aliæ parallelæ <expan abbr="apt&emacr;tur">aptentur</expan> per <foreign lang="greek">b, g, d, c</foreign>, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.089.jpg"/>quibusita di&longs;po&longs;itis per puncta <foreign lang="greek">o, n, l, k, h</foreign>, item per <foreign lang="greek">p, c, m, q, z</foreign>. <lb/></s>

<s id="id.000819">Sit autem <foreign lang="greek">o a</foreign> æ<lb/>qualis AF, <foreign lang="greek">a</foreign> <11> æqualis FD, item <foreign lang="greek">e</foreign> <10>, æqualis EC, <foreign lang="greek">e s</foreign> æqualis <lb/>EB, &longs;ed & <foreign lang="greek">n *z</foreign> aequalis OI, <foreign lang="greek">*z c</foreign> ip&longs;i P, <foreign lang="greek">l</foreign>y ip&longs;i MH, y <foreign lang="greek">m</foreign> verò ip&longs;i <lb/>HN, <expan abbr="t&umacr;">tum</expan> <foreign lang="greek">k x</foreign> ip&longs;i KG. & <foreign lang="greek">x q</foreign>, ip&longs;i GL & ip&longs;is æquales & aequa<lb/>liter po&longs;itæ ad partes R, aliæ parallelæ <expan abbr="apt&emacr;tur">aptentur</expan> per <foreign lang="greek">b, g, d, c</foreign>, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.088.jpg"/>quibusita di&longs;po&longs;itis per puncta <foreign lang="greek">o, n, l, k, h</foreign>, item per <foreign lang="greek">p, c, m, q, z</foreign>. <lb/></s>

<s id="id.000820">ducantur lineæ <foreign lang="greek">oh, pz</foreign>, curuæ quidem & eodem pacto a<lb/>liæ curuæ illis re&longs;pondentes <foreign lang="greek">h <10>, zs</foreign>, Erunt igitur <foreign lang="greek">o, h, <10>, <lb/>p, z, s</foreign>, parallelæ quidem eo quod lineae quæ inter ip&longs;as du<lb/>cuntur, parallelæ &longs;int & æquales, non tamen rectæ illæ, <lb/>&longs;ed curuæ. </s>

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<s id="id.000825">Hæc quo que &longs;peculatio ad motum qui &longs;ecundum <lb/>ab&longs;idem fit, manife&longs;tè pertinet. </s>

<s id="id.000826">Coni, quorum ba&longs;es cir<lb/>culi &longs;unt, &longs;i in plano &longs;ecundum latus rotentur, ba&longs;i circu<lb/>lum de&longs;cribunt, cuius centrum immobile coni ip&longs;ius e&longs;t <lb/>vertex, &longs;emidiameter verò ip&longs;um latus. </s><figure id="id.007.00.089.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.089.1.jpg"></figure>

<s id="id.000826">Coni, quorum ba&longs;es cir<lb/>culi &longs;unt, &longs;i in plano &longs;ecundum latus rotentur, ba&longs;i circu<lb/>lum de&longs;cribunt, cuius centrum immobile coni ip&longs;ius e&longs;t <lb/>vertex, &longs;emidiameter verò ip&longs;um latus. </s><figure id="id.007.01.088.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.088.1.jpg"></figure>

<s id="id.000827">E&longs;to conus ABC cu<lb/>ius vertex C ba&longs;is AB, axis <lb/>DC, ba&longs;is verò centrum, <lb/>D, latus quo planum tan<lb/>git BC, &longs;ecatur itaque Co<lb/>nus per latus BC & axem <lb/>DE à plano horizonti per<lb/>pendiculari, cuius & coni <lb/>communis &longs;ectio e&longs;t ABC <lb/>triangulum, & quoniam coni grauitatis centrum e&longs;t in <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.090.jpg"/>axe ip&longs;o, conus in partes æque <expan abbr="p&omacr;derantes">ponderantes</expan> &longs;ecatur AEBC, <lb/>AFBC, &longs;tat ergo conus &longs;ibimet æquilibris. </s>

<s id="id.000827">E&longs;to conus ABC cu<lb/>ius vertex C ba&longs;is AB, axis <lb/>DC, ba&longs;is verò centrum, <lb/>D, latus quo planum tan<lb/>git BC, &longs;ecatur itaque Co<lb/>nus per latus BC & axem <lb/>DE à plano horizonti per<lb/>pendiculari, cuius & coni <lb/>communis &longs;ectio e&longs;t ABC <lb/>triangulum, & quoniam coni grauitatis centrum e&longs;t in <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.089.jpg"/>axe ip&longs;o, conus in partes æque <expan abbr="p&omacr;derantes">ponderantes</expan> &longs;ecatur AEBC, <lb/>AFBC, &longs;tat ergo conus &longs;ibimet æquilibris. </s>

<s id="id.000828">Si autem à po<lb/>tentia quadam moueatur, puta ab A ver&longs;us F, trahitur &longs;e<lb/>micirculus BEA, à &longs;emicirculo AFB, & ita fit rotatio. </s>

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<s id="id.000835">Modò de tertia motus &longs;pecie nobis erit &longs;ermo; in <lb/>qua quidem &longs;pecie nonnulla perpendemus, quæ omi&longs;it A<lb/>ri&longs;toteles. </s>

<s id="id.000836">Agitur autem hîc de rotundorum corporum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.091.jpg"/>motu, qui fit çirca axem horizonti perpendicularem, axis <lb/>altera extremitate in eodem horizontis plano manente, <lb/>vti videre e&longs;t in ip&longs;is figulorum rotis. </s>

<s id="id.000836">Agitur autem hîc de rotundorum corporum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.090.jpg"/>motu, qui fit çirca axem horizonti perpendicularem, axis <lb/>altera extremitate in eodem horizontis plano manente, <lb/>vti videre e&longs;t in ip&longs;is figulorum rotis. </s>

<s id="id.000837">Hanc motus &longs;peciem in extrema quæ&longs;tionis parte <lb/>cum duabus alijs &longs;peciebus comparans ait, cam quæ in <lb/>obliquo fit motionem (ita enim hanc, de qua agimus, ap<lb/>pellat) ip&longs;am impellere mouentem, hoc e&longs;t, nullum ex &longs;e <lb/>ad motum propen&longs;ionem habere, nutumue, & omnia illi <lb/>e&longs;&longs;e à motore, &longs;ecundum verò eam motionem, quæ &longs;upra <lb/>diametrum e&longs;t, &longs;e ip&longs;um mouere circulum. </s>

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<s id="id.000843">Ad hæc circa axem ita libratur rota, vt <lb/>quantumuis exigua potentia alteri parti applicetur, alte<lb/>ra illico &longs;uperata moueatur. </s>

<s id="id.000844">Licet enim propriè ea <expan abbr="tant&umacr;">tantum</expan> <lb/>corpora æquilibrare dicantur, quæ ob ponderis hinc in de <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.092.jpg"/>æqualitatem horizonti fiunt æquidi&longs;tantes, nihilominus <lb/>& hic aliquam e&longs;&longs;e æquilibrij &longs;imilitudinem patebit. </s><figure id="id.007.00.092.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.092.1.jpg"></figure>

<s id="id.000844">Licet enim propriè ea <expan abbr="tant&umacr;">tantum</expan> <lb/>corpora æquilibrare dicantur, quæ ob ponderis hinc in de <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.091.jpg"/>æqualitatem horizonti fiunt æquidi&longs;tantes, nihilominus <lb/>& hic aliquam e&longs;&longs;e æquilibrij &longs;imilitudinem patebit. </s><figure id="id.007.01.091.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.091.1.jpg"></figure>

<s id="id.000845">E&longs;to enim rota ABCD, <lb/>cuius axis horizonti perpendi<lb/>cularis FEG tran&longs;iens per cen<lb/>trum E, tangens autem planum <lb/>in puncto G. </s>

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<s id="id.000851">Quærimus igitur primò; Cur ea quæ hoc pacto <expan abbr="ro-tãtur">ro<lb/>tantur</expan>, in ip&longs;a rotatione locum non mutent, ni&longs;i extrin&longs;eca <lb/>aliqua id fiat ex cau&longs;&longs;a. </s>

<s id="id.000852">E&longs;to enim rota aut aliud quippiam rotundum ceu <lb/>Turbines &longs;unt, quibus pueri ludunt, quod circa axem ho<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.093.jpg"/><figure id="id.007.00.093.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.093.1.jpg"></figure><lb/>rizonti perpendicularem mo<lb/>ueatur, ABCD, cuius centrum <lb/>E, Diameter AEC. </s>

<s id="id.000852">E&longs;to enim rota aut aliud quippiam rotundum ceu <lb/>Turbines &longs;unt, quibus pueri ludunt, quod circa axem ho<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.092.jpg"/><figure id="id.007.01.092.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.092.1.jpg"></figure><lb/>rizonti perpendicularem mo<lb/>ueatur, ABCD, cuius centrum <lb/>E, Diameter AEC. </s>

<s id="id.000853">Modò circa <lb/>centrum E in finiti imaginentur <lb/>circuli, alij alijs minores v&longs;que <lb/>ad <expan abbr="centr&umacr;">centrum</expan> ip&longs;um, vti &longs;unt FGH; <lb/>ibi enim circuli e&longs;&longs;e de&longs;inunt, <lb/>vbi nullum amplius e&longs;t &longs;patium. <lb/></s>

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<s id="id.000862">Ad hæc alia proponitur, & &longs;oluitur quæ&longs;tio, Cur ro<lb/>tunda corpora huic motionis generi &longs;int aptiora. </s>

<s id="id.000863">Explorati&longs;&longs;imum e&longs;t, corporum, quæ ita mouentur, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.094.jpg"/>partes eo e&longs;&longs;e velociores, quo magis à centro, circa quod <lb/>mouentur, fuerint remotiores. </s>

<s id="id.000863">Explorati&longs;&longs;imum e&longs;t, corporum, quæ ita mouentur, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.093.jpg"/>partes eo e&longs;&longs;e velociores, quo magis à centro, circa quod <lb/>mouentur, fuerint remotiores. </s>

<s id="id.000864">maius enim eodem tem<lb/>pore &longs;patium pertran&longs;eunt. </s>

<s id="id.000865">quo igitur figura ijs partibus, <lb/>quæ longius à centro ab&longs;unt, abundauerit magis, eo faci<lb/>lius, & velocius in circulum rotata mouebitur. </s>

<s id="id.000866">Modò o<lb/>&longs;tendemus, circularem cæteras omnes ea qua diximus <lb/>partium à centro remoti&longs;&longs;imarum copiâ abundare. </s><figure id="id.007.00.094.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.094.1.jpg"></figure>

<s id="id.000866">Modò o<lb/>&longs;tendemus, circularem cæteras omnes ea qua diximus <lb/>partium à centro remoti&longs;&longs;imarum copiâ abundare. </s><figure id="id.007.01.093.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.093.1.jpg"></figure>

<s id="id.000867">E&longs;to triangulum puta æqui<lb/>laterum, ABC circa centrum D. <lb/></s>

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<s id="id.000874">Quæri etiam pote&longs;t, Num axe in clinato, rotæ motus ali<lb/>qualiter impediatur? </s>

<s id="id.000875">Nos negatiuam partem amplecti<lb/>mur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.095.jpg"/><figure id="id.007.00.095.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.095.1.jpg"></figure>

<s id="id.000875">Nos negatiuam partem amplecti<lb/>mur. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.094.jpg"/><figure id="id.007.01.094.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.094.1.jpg"></figure>

<s id="id.000876">E&longs;to enim tota ABCD, cuius cen<lb/>trum E axis inclinitus, circa quem <lb/>conuertitur EGF. </s>

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<s id="id.000890"><emph type="italics"/>Quæritur, Cur ea quæ per maiores cir culos tolluntur, & trahuntur <lb/>faciliùs, & celeriùs moueri contingat, veluti maioribus tro<lb/>chleis, & &longs;cytalis &longs;imiliter?<emph.end type="italics"/></s>

<s id="id.000891">Re&longs;pondet ad hæc Philo&longs;ophus, forteid cuenire, quo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.096.jpg"/>niam quanto maior fucrit illa quæ à centro e&longs;t, in æquali <lb/>tempore maius mouetur &longs;patium. </s>

<s id="id.000891">Re&longs;pondet ad hæc Philo&longs;ophus, forte id euenire, quo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.095.jpg"/>niam quanto maior fuerit illa quæ à centro e&longs;t, in æquali <lb/>tempore maius mouetur &longs;patium. </s>

<s id="id.000892">quamobrem æquali <lb/>exi&longs;tente onere idem faciet. </s>

<s id="id.000893">Ita enim dixerat de <expan abbr="librar&umacr;">librarum</expan> <lb/>natura, & differentijs agens, maiores minoribus exactio<lb/>res e&longs;&longs;e. </s>

<s id="id.000894">Circulos verò jibras, in quibus centrum &longs;partum, <lb/>&longs;emidiametri hinc in de æqualia brachia. </s>

<s id="id.000894">Circulos verò libras, in quibus centrum &longs;partum, <lb/>&longs;emidiametri hinc in de æqualia brachia. </s>

<s id="id.000895">Quod vltimo loco affirmauit, trochleas e&longs;&longs;e in&longs;tar <lb/>librarum, verum e&longs;t. </s>

<s id="id.000896">Quo d autem dixit, faciliùs & cele<lb/>rius mouere maiores libras ijs quæ minores &longs;unt, &longs;i &longs;inipli<lb/>citer intelligatur, fal&longs;um, quippe quod facilitas motus, in <lb/>tractorijs machinis velocitati &longs;it contraria, quod demon<lb/>&longs;trauit Guid. Vbald. in tractatu de Trochlea in 2. Corol<lb/>lario propo&longs;itione vltima. </s>

<s id="id.000896">Quod autem dixit, faciliùs & cele<lb/>rius mouere maiores libras ijs quæ minores &longs;unt, &longs;i &longs;impli<lb/>citer intelligatur, fal&longs;um, quippe quod facilitas motus, in <lb/>tractorijs machinis velocitati &longs;it contraria, quod demon<lb/>&longs;trauit Guid. Vbald. in tractatu de Trochlea in 2. Corol<lb/>lario propo&longs;itione vltima. </s>

<s id="id.000897">Ad id autem quod dixit, quo <expan abbr="maior&emacr;s">maiorens</expan> fuerint tro<lb/>chleæ, eo faciliùs mouere, non e&longs;t, vt dicebamus, &longs;impli<lb/>citer verum, quod facilè o&longs;tendemus. </s><figure id="id.007.00.096.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.096.1.jpg"></figure>

<s id="id.000897">Ad id autem quod dixit, quo <expan abbr="maior&emacr;s">maiores</expan> fuerint tro<lb/>chleæ, eo faciliùs mouere, non e&longs;t, vt dicebamus, &longs;impli<lb/>citer verum, quod facilè o&longs;tendemus. </s><figure id="id.007.01.095.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.095.1.jpg"></figure>

<s id="id.000898">Efto enim trochlea AB circa centrum C, appen&longs;a in <lb/>puncto D, perpendicularis quæ ad mundi centrum DCE, <lb/>pondera æqualia vtrinque appen&longs;a FG. </s>

<s id="id.000899">E&longs;to item alia <lb/>Trochlea, eaque; maior HI, circa centrum K appen&longs;a in L, <lb/>perpen dicularis, quæ ad mundi centrum LKM, æqualia <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.097.jpg"/>pondera vtrinque appen&longs;a N, O. </s>

<s id="id.000899">E&longs;to item alia <lb/>Trochlea, eaque; maior HI, circa centrum K appen&longs;a in L, <lb/>perpendicularis, quæ ad mundi centrum LKM, æqualia <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.096.jpg"/>pondera vtrinque appen&longs;a N, O. </s>

<s id="id.000900">Dico maiorem Hl ip&longs;a <lb/>minori DE facilius pondera non mouere, eo quòd &longs;it ma<lb/>ior, illa verò difficiliùs, propterea quòd &longs;it minor. </s>

<s id="id.000901">Etcnim, <lb/>quoniam vtraque trochlea per centrum grauitatis à per<lb/>pendiculari diui ditur, erunt partes DAE, DBE, æque <expan abbr="p&omacr;-derantes">pon<lb/>derantes</expan>. </s>

<s id="id.000901">Et enim, <lb/>quoniam vtraque trochlea per centrum grauitatis à per<lb/>pendiculari diuiditur, erunt partes DAE, DBE, æque <expan abbr="p&omacr;-derantes">pon<lb/>derantes</expan>. </s>

<s id="id.000902">Eadem ratione ip&longs;æ quoque LHM, L<gap/>M æquè <lb/>ponderabunt. </s>

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<s id="id.000907">Dico igitur, Axium, circa quos tro<lb/>chleæ rotæue conuertuntur ad rotas ip&longs;as, varias habere <lb/>proportiones. </s>

<s id="id.000908">O&longs;tendemus autem <expan abbr="rotã">rotam</expan> illam, trochleam<lb/>ue faciliùs moueri, & mouere pondera, quo rotæ diame<lb/>ter ad axis diametrum maiorem habuerit proportionem, <lb/>& ideo fieri po&longs;&longs;e rotam maiorem ad &longs;uum axem <expan abbr="minor&emacr;">minorem</expan> <lb/>habere proportionem quam rotam minorem ad &longs;uum. </s><figure id="id.007.00.097.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.097.1.jpg"></figure>

<s id="id.000908">O&longs;tendemus autem <expan abbr="rotã">rotam</expan> illam, trochleam<lb/>ue faciliùs moueri, & mouere pondera, quo rotæ diame<lb/>ter ad axis diametrum maiorem habuerit proportionem, <lb/>& ideo fieri po&longs;&longs;e rotam maiorem ad &longs;uum axem <expan abbr="minor&emacr;">minorem</expan> <lb/>habere proportionem quam rotam minorem ad &longs;uum. </s><figure id="id.007.01.096.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.096.1.jpg"></figure>

<s id="id.000909">E&longs;to enim <lb/>trochlea AB cir<lb/>ca centrum C, <lb/>cuius diameter <lb/>DCE &longs;it in ip&longs;a <lb/>quæ ad mundi <lb/>centrum <expan abbr="perp&emacr;-diculari">perpen<lb/>diculari</expan>: &longs;it au<lb/>tem appen&longs;a in D. </s>

<s id="id.000910">Alia &longs;imiliter ei æqualis &longs;it trochlea F <lb/>G circa centrum H, cuius diameter IHK, conueniens <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.098.jpg"/>cum perpen diculari quæ ad mundi centrum. </s>

<s id="id.000910">Alia &longs;imiliter ei æqualis &longs;it trochlea F <lb/>G circa centrum H, cuius diameter IHK, conueniens <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.097.jpg"/>cum perpendiculari quæ ad mundi centrum. </s>

<s id="id.000911">appendatur <lb/>autem in I. </s>

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<s id="id.000913">Hi&longs;i æquales fuerint, proportione non mutatâ idem <lb/>operabuntur. </s>

<s id="id.000914">Modò ponanturinæquales, &longs;itqueue axis ro<lb/>tç AB, cra&longs;&longs;ior axe rotæ FG, &longs;itqueue cra&longs;&longs;ioris quidem &longs;emi<lb/>diameter CL, &longs;ubtilioris autem HM. </s>

<s id="id.000914">Modò ponantur inæquales, &longs;itqueue axis ro<lb/>tæ AB, cra&longs;&longs;ior axe rotæ FG, &longs;itqueue cra&longs;&longs;ioris quidem &longs;emi<lb/>diameter CL, &longs;ubtilioris autem HM. </s>

<s id="id.000915">Dico per tro chleam <lb/>FG facilius attolli pondera æqualia quàm per AB, licet <lb/>altera tro chlearum alteri &longs;it æqualis. </s>

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<s id="id.000922">Sed maior <lb/>e&longs;t proportio CL ad CD quàm HM ad HI. </s>

<s id="id.000923">Maior igitur <lb/>ad &longs;uperandum idem &longs;eu æquale impedimentum poten<lb/>tia requiritur in D, qu<gap/>m in I. </s>

<s id="id.000923">Maior igitur <lb/>ad &longs;uperandum idem &longs;eu æquale impedimentum poten<lb/>tia requiritur in D, quam in I. </s>

<s id="id.000924">Itaque cum vis tota in rota<lb/>rum & axium, diametrorum proportione con&longs;i&longs;tat, fieri <lb/>pote&longs;t, quod dicebamus, minorem trochleam dari, quæ <lb/>maiorem habeat proportionem ad &longs;uum axem, quàm, <lb/>maior ad &longs;uum, quo ca&longs;u minor rota facilius imp edimen<lb/>tum, quod diximus, ip&longs;a maiori rota &longs;eu trochlea &longs;upera<lb/>bit. </s>

<s id="id.000925">Veruntamen quoniam ex materia fiunt tum axes tum <lb/>rotæ, nec rei natura patitur axes &longs;ubtiles, & imbecilles <lb/>magna <expan abbr="p&omacr;dera">pondera</expan> &longs;u&longs;tinere po&longs;&longs;e, idcirco cra&longs;&longs;iores fiunt, quç <lb/>cra&longs;&longs;itudo cum proportione magis à magnarum rotarum <lb/>diametris &longs;uperetur; fit hinc maiores rotas datâ axium pa-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.099.jpg"/>ritate facilius impedimentum &longs;uperare quàm minores, & <lb/>hoc videtur &longs;en&longs;i&longs;&longs;e Philo&longs;ophus in ip&longs;a quæ&longs;tionis huius <lb/>propo&longs;itione, Hinc aurigæ vulgo axungiâ (quæ inde no<lb/>men trahit) axium a&longs;peritates mitigant, vt minor in rotan. <lb/></s>

<s id="id.000925">Veruntamen quoniam ex materia fiunt tum axes tum <lb/>rotæ, nec rei natura patitur axes &longs;ubtiles, & imbecilles <lb/>magna <expan abbr="p&omacr;dera">pondera</expan> &longs;u&longs;tinere po&longs;&longs;e, idcirco cra&longs;&longs;iores fiunt, qu&ecedil; <lb/>cra&longs;&longs;itudo cum proportione magis à magnarum rotarum <lb/>diametris &longs;uperetur; fit hinc maiores rotas datâ axium pa-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.098.jpg"/>ritate facilius impedimentum &longs;uperare quàm minores, & <lb/>hoc videtur &longs;en&longs;i&longs;&longs;e Philo&longs;ophus in ip&longs;a quæ&longs;tionis huius <lb/>propo&longs;itione, Hinc aurigæ vulgo axungiâ (quæ inde no<lb/>men trahit) axium a&longs;peritates mitigant, vt minor in rotan<lb/>do, ex fricatione fiat re&longs;i&longs;tentia. </s>

<s id="id.000926">do, ex fricatione fiat re&longs;i&longs;tentia. </s>

<s id="id.000927">Concludimus igitur, fa<lb/>cillimè trochleam illam pondus trahere, quæ cum maxi<lb/>ma &longs;it, axem habet minimum, cumqueue axungiâ aliaue vn<lb/>ctuo &longs;a materia perfu&longs;um. </s>

<s id="id.000928">De manubrijs, quæ rotarum a<lb/>xibus aptantur, nemo ferè verba fecit; nos igitur de his a<lb/>liquid; &longs;iquidem res ad &longs;peculationem, qua de agimus, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> Mechanicam pertinet. </s>

<s id="id.000928">De manubrijs, quæ rotarum a<lb/>xibus aptantur, nemo ferè verba fecit; nos igitur de his a<lb/>liquid; &longs;iquidem res ad &longs;peculationem, qua de agimus, <expan abbr="n&emacr;-pe">nem<lb/>pe</expan> Mechanicam pertinet. </s>

<s id="id.000929">Manubria vectes &longs;unt, & ad vectium naturam redu<lb/>cuntur, corum &longs;cilicet, in quibus fulcimentum e&longs;tinter <lb/>pondus & potentiam. </s>

<s id="id.000929">Manubria vectes &longs;unt, & ad vectium naturam redu<lb/>cuntur, eorum &longs;cilicet, in quibus fulcimentum e&longs;t inter <lb/>pondus & potentiam. </s>

<s id="id.000930">In his autem attenditur proportio, <lb/>quam habet manubrij longitudo ad ip&longs;um axis &longs;emidia<lb/>metrum, eo enim faciliùs mouent, quo eorum longitudo <lb/>ad axium &longs;emidia metros proportionem, habuerit ma<lb/>iorem. </s>

<s id="id.000931">Duabus autem partibus con&longs;tant, alterâ, quæ ab <lb/>axe ad angulum; quæ verè vectis e&longs;t; alterâ, cui manusi<lb/>p&longs;a admouetur, ex qua res tota manubrium dicitur. </s>

<s id="id.000932">Fiunt <lb/>autem manubria hæc vt plurimum amouibilia, &longs;unt <expan abbr="tam&emacr;">tamen</expan> <lb/>ceu rotarum ip&longs;arum partes, & rotis ip&longs;is commodè affi<lb/>gerentur, ni&longs;i in rotatione à tran&longs;uer&longs;arijs, quibus rotæ &longs;u<lb/>&longs;tinentur, impedimentum fieret. </s><figure id="id.007.00.099.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.099.1.jpg"></figure>

<s id="id.000932">Fiunt <lb/>autem manubria hæc vt plurimum amouibilia, &longs;unt <expan abbr="tam&emacr;">tamen</expan> <lb/>ceu rotarum ip&longs;arum partes, & rotis ip&longs;is commodè affi<lb/>gerentur, ni&longs;i in rotatione à tran&longs;uer&longs;arijs, quibus rotæ &longs;u<lb/>&longs;tinentur, impedimentum fieret. </s><figure id="id.007.01.098.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.098.1.jpg"></figure>

<s id="id.000933">E&longs;to enim rota AB, cu<lb/>ius axis E, terebretur autem <lb/>in F, ibiqueue paxillus affigatur <lb/>FK. </s>

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<s id="id.000935">Sint autem <lb/>rotæ æquales & axes æqua<lb/>les. </s>

<s id="id.000936">Sint etiam æqualia ip&longs;a <lb/>&longs;patia EF, GH, hoc e&longs;t, ma-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.100.jpg"/>nubrij GHI longitudo. </s>

<s id="id.000936">Sint etiam æqualia ip&longs;a <lb/>&longs;patia EF, GH, hoc e&longs;t, ma-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.099.jpg"/>nubrij GHI longitudo. </s>

<s id="id.000937">Dico, câdem facilitate moueri AB <lb/>rotam à potentia in FK, quâ mouetur CB, à potentia po<lb/>&longs;ita in HI, datis ip&longs;i nempe potentijs æqualibus. </s>

<s id="id.000937">Dico, eâdem facilitate moueri AB <lb/>rotam à potentia in FK, quâ mouetur CB, à potentia po<lb/>&longs;ita in HI, datis ip&longs;i nempe potentijs æqualibus. </s>

<s id="id.000938">Produca<lb/>tur enim IH, v&longs;que ad rotæ CD latus in L, & LG ducatur, <lb/>& FE in rota AB iungatur. </s>

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<s id="id.000940">Sunt autem eorum circulorum &longs;emidiametri, qui <lb/>à punctis FL, in ip&longs;a rotatione de&longs;cribuntur. </s>

<s id="id.000941">Ita igitur &longs;e <lb/>habebit potentia applicata in L ad diametrum &longs;emidia<lb/>metrumue axis rotæ CD, vt &longs;e habet potentia applicata <lb/>in F, ad diametrum &longs;emidiame trumue axis E rotæ AB, &longs;ed <lb/>&longs;patia &longs;unt æqualia & potentiæ æquales, quare nihil re<lb/>fert, vtrum manubrium lateria ffigatur, vel axi à latere ro<lb/>tæ &longs;eparatum applicetur. </s><figure id="id.007.00.100.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.100.1.jpg"></figure>

<s id="id.000941">Ita igitur &longs;e <lb/>habebit potentia applicata in L ad diametrum &longs;emidia<lb/>metrumue axis rotæ CD, vt &longs;e habet potentia applicata <lb/>in F, ad diametrum &longs;emidiametrumue axis E rotæ AB, &longs;ed <lb/>&longs;patia &longs;unt æqualia & potentiæ æquales, quare nihil re<lb/>fert, vtrum manubrium lateri affigatur, vel axi à latere ro<lb/>tæ &longs;eparatum applicetur. </s><figure id="id.007.01.099.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.099.1.jpg"></figure>

<s id="id.000942">Duplex autem e&longs;t ma<lb/>nubriorum forma; altera e<lb/>nim rectis partibus con&longs;tat, <lb/>altera verò curua e&longs;t tota, <lb/>&longs;ed rectis vtimur vt mani<lb/>bus apprendamus, curuis <lb/>verò vt locum illis appona<lb/>mus, & pedis pre&longs;&longs;ione ceu <lb/>in molis lapideis, quibus <lb/>gladij acuuntur, fieri a&longs;&longs;olet, conuertantnr. </s>

<s id="id.000943">Cur autem <lb/>manubria hæc curua fiant, ea videtur ratio, ne videlicet <lb/>manubrij capite &longs;upra centrum in linea quæ per centrum <lb/>tran&longs;it, <expan abbr="c&omacr;&longs;tituto">con&longs;tituto</expan>, factâ interim pre&longs;&longs;ione motus à centro, <lb/>ad quod directè fieret pre&longs;&longs;io, impediretur. </s>

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<s id="id.000945">quæ cum ce&longs;&longs;ent in manubrijs quæ manu <lb/>aguntur, ideo alia forma, nempe ex rectis partibus pa&longs;&longs;im <lb/>fiunt. </s>

<s id="id.000946">E&longs;to igiturillud quod ex rectis partibus AB, curuum <lb/>verò CD, linea verò, &longs;ecundum quam pede fit pre&longs;&longs;io <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.101.jpg"/>CDE. </s>

<s id="id.000946">E&longs;to igitur illud quod ex rectis partibus AB, curuum <lb/>verò CD, linea verò, &longs;ecundum quam pede fit pre&longs;&longs;io <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.100.jpg"/>CDE. </s>

<s id="id.000947">Hæc itaque de manubrijs &longs;eu vectibus nos con&longs;i<lb/>dera&longs;&longs;e &longs;it &longs;atis. </s>

<s id="id.000948">Quæriinterim po&longs;&longs;et, Cur duabus datis rotis æqua<lb/>lis magnitudinis in æqualis ponderis, circa æquales axes <lb/>con&longs;titutis leuior faciliùs moueatur & citiùs quie&longs;cat; <lb/>grauior verò di&longs;ficilius moucatur & tardiùs ce&longs;&longs;et à mo<lb/>tu, ea videtur ratio, quod grauior re&longs;i&longs;tens magis cum &longs;u<lb/>peratur impre&longs;&longs;am vim &longs;u&longs;cipit, & diutiùs retinet, quod <lb/>ce&longs;&longs;at in leuiore. </s>

<s id="id.000948">Quæri interim po&longs;&longs;et, Cur duabus datis rotis æqua<lb/>lis magnitudinis in æqualis ponderis, circa æquales axes <lb/>con&longs;titutis leuior faciliùs moueatur & citiùs quie&longs;cat; <lb/>grauior verò di&longs;ficilius moueatur & tardiùs ce&longs;&longs;et à mo<lb/>tu, ea videtur ratio, quod grauior re&longs;i&longs;tens magis cum &longs;u<lb/>peratur impre&longs;&longs;am vim &longs;u&longs;cipit, & diutiùs retinet, quod <lb/>ce&longs;&longs;at in leuiore. </s>

</subchap1><subchap1>

<s id="id.000949">QVÆSTIO X.</s>

<s id="id.000950"><emph type="italics"/>Dubitat Ari&longs;toteles, Cur faciliùs, quando &longs;ine pondere est, mouca<lb/>tur libra, quàm cum pondus habet. </s>

<s id="id.000950"><emph type="italics"/>Dubitat Ari&longs;toteles, Cur faciliùs, quando &longs;ine pondere est, mouea<lb/>tur libra, quàm cum pondus habet. </s>

<s id="id.000951">Simili modo rota, & eiu&longs;modi<lb/>quidpiam, quod grauius quidem est, item quod maius & <lb/>grauius minori, & leutori?<emph.end type="italics"/></s>

<s id="id.000951">Simili modo rota, & eiu&longs;modi <lb/>quidpiam, quod grauius quidem est, item quod maius & <lb/>grauius minori, & leuiori?<emph.end type="italics"/></s>

<s id="id.000952">Breuiter autem &longs;oluit, ait enim, An quia non &longs;olum in <lb/>contrarium quod graue e&longs;t, &longs;ed in obliquam etiam dif<lb/>ficulter mouetur? </s>

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<s id="id.000954">In <lb/>obliquum autem haudqua quam vergit. </s>

<s id="id.000955">Nos quod ip&longs;e <lb/>non fecit figurâ ip&longs;a appo&longs;itâ rem clariorem faciemus. </s><figure id="id.007.00.101.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.101.1.jpg"></figure>

<s id="id.000955">Nos quod ip&longs;e <lb/>non fecit figurâ ip&longs;a appo&longs;itâ rem clariorem faciemus. </s><figure id="id.007.01.100.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.100.1.jpg"></figure>

<s id="id.000956">Efto libra AB, cuius ful<lb/>cimentum C, pondera vtrin<lb/>que appen&longs;a AB, quorum v<lb/>trumque ponderet 10. Item <lb/>libra DE, cuius fulcimentum <lb/>F pondere vero appen&longs;a D, E, <lb/>ip&longs;is A, B, dimidio leuiora, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> S. </s>

<s id="id.000956">E&longs;to libra AB, cuius ful<lb/>cimentum C, pondera vtrin<lb/>que appen&longs;a AB, quorum v<lb/>trumque ponderet 10. Item <lb/>libra DE, cuius fulcimentum <lb/>F pondere vero appen&longs;a D, E, <lb/>ip&longs;is A, B, dimidio leuiora, <expan abbr="n&emacr;-pe">nem<lb/>pe</expan> S. </s>

<s id="id.000957">Addatur ponderi B pon<lb/>dus G, & ponderi E pondus <lb/>H, quorum &longs;imiliter <expan abbr="vtrumq;">vtrumque</expan> <lb/>ponderet S, nutabunt igitur <lb/>libræ ponderibus appo&longs;itis, & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.102.jpg"/>BG &longs;ecetur in K, EH verò in N, grauius e&longs;t autem GB, e&longs;t <lb/>enim IS, ip&longs;o EH, quod e&longs;t 10. Difficiliùs autem de&longs;cen<lb/>det BG, quàm EH. hoc autem ex doctrina Ari&longs;totelis, <lb/>quia non &longs;olum in contrarium quod graue e&longs;t, &longs;ed in obli<lb/>quum etiam difficulter mouetur, in contrarium enim ei <lb/>ad quod vergit onus mouere difficile e&longs;t, quò autem ver<lb/>git facilè in obliquum autem puta per lineas BK, EN non <lb/>vergit onus. </s>

<s id="id.000957">Addatur ponderi B pon<lb/>dus G, & ponderi E pondus <lb/>H, quorum &longs;imiliter <expan abbr="vtrumq;">vtrumque</expan> <lb/>ponderet S, nutabunt igitur <lb/>libræ ponderibus appo&longs;itis, & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.101.jpg"/>BG &longs;ecetur in K, EH verò in N, grauius e&longs;t autem GB, e&longs;t <lb/>enim IS, ip&longs;o EH, quod e&longs;t 10. Difficiliùs autem de&longs;cen<lb/>det BG, quàm EH. hoc autem ex doctrina Ari&longs;totelis, <lb/>quia non &longs;olum in contrarium quod graue e&longs;t, &longs;ed in obli<lb/>quum etiam difficulter mouetur, in contrarium enim ei <lb/>ad quod vergit onus mouere difficile e&longs;t, quò autem ver<lb/>git facilè in obliquum autem puta per lineas BK, EN non <lb/>vergit onus. </s>

<s id="id.000958">Difficiliùs ergo in obliquum mouebitur pon<lb/>dus BG ip&longs;o pondere EH. vtrumque autem in de&longs;cen&longs;u <lb/>retrahitur nempe à perpendicularibus BI, EM & retra<lb/>ctionis quidem anguli &longs;unt æquales & æquales ip&longs;æ retra<lb/>ctioncs. </s>

<s id="id.000959">Sedgrauius e&longs;t pondus GB. quod autem grauius <lb/>e&longs;t, violentius <expan abbr="de&longs;c&emacr;dit">de&longs;cendit</expan> eo quod e&longs;t leuius. </s>

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<s id="id.000961">ex alijs igitur principijs veritas ip&longs;a e&longs;t eruen<lb/>da. </s>

<s id="id.000962">Dicimus autem id ex proportionum fieri inæqualita<lb/>te; quia enim is ad 10. proportionem habet &longs;e&longs;quialteram, <lb/>10. verò ad 5. duplam, maiorem proportionem habet EH <lb/>ad oppo&longs;itum pondus D, quàm BG ad pondus A, facilius <lb/>ergo trahet libra DE lcuior pondus D, quàm ip&longs;a AB, gra<lb/>uior pondus A, quod vtique fuerat o&longs;tendendum. </s>

<s id="id.000962">Dicimus autem id ex proportionum fieri inæqualita<lb/>te; quia enim is ad 10. proportionem habet &longs;e&longs;quialteram, <lb/>10. verò ad 5. duplam, maiorem proportionem habet EH <lb/>ad oppo&longs;itum pondus D, quàm BG ad pondus A, facilius <lb/>ergo trahet libra DE leuior pondus D, quàm ip&longs;a AB, gra<lb/>uior pondus A, quod vtique fuerat o&longs;tendendum. </s>

<s id="id.000963">Alia <lb/>quoque cau&longs;&longs;a & hæc accidentalis ad hunc effectum pa<lb/>riendum concurrit, axium nempe ad fulcimenta, in qui<lb/>bus rotantur, fricatio. </s>

<s id="id.000964">quo enim maius e&longs;t pondus cæteris <lb/>paribus, quod nos in præ cedente quæ&longs;tione demon&longs;tra<lb/>uimus, eò maiìor fit ip&longs;a collifio. </s>

<s id="id.000965">Porrò huius <expan abbr="quoq;">quoque</expan> &longs;peculationis e&longs;t, Cur æqualia & <lb/>&longs;imilia corpora in æqualibus &longs;imilibu&longs;queue bafibus con&longs;ti<lb/>tuta eodem &longs;imiliqueue plano fulta, ponderibus tamen in<lb/>æqualia, non eâdem facilitate euertantur, &longs;ed horum gra<lb/>uiora difficilius. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.103.jpg"/><figure id="id.007.00.103.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.103.1.jpg"></figure>

<s id="id.000965">Porrò huius <expan abbr="quoq;">quoque</expan> &longs;peculationis e&longs;t, Cur æqualia & <lb/>&longs;imilia corpora in æqualibus &longs;imilibu&longs;queue ba&longs;ibus con&longs;ti<lb/>tuta eodem &longs;imiliqueue plano fulta, ponderibus tamen in<lb/>æqualia, non eâdem facilitate euertantur, &longs;ed horum gra<lb/>uiora difficilius. </s><pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.102.jpg"/><figure id="id.007.01.102.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.102.1.jpg"></figure>

<s id="id.000966">Sit enim Pri&longs;ma &longs;eu <lb/>Cylindrus ABCD, cuius <lb/>grauitatis centrum E in <lb/>plano Cl, ba&longs;i fultus CD. <lb/></s>

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<s id="id.000971">Demit tantur EN, KO, perpendiculares ip&longs;is CD, HF. </s>

<s id="id.000972">Et <lb/>quoniam CNE, HOK anguli recti &longs;unt, erunt EC KH i<lb/>p&longs;is EN, KO, maiores, quare & LC, MH ip&longs;is EN KO, ma<lb/>iores atto lluntur ergo in ip&longs;a cuer&longs;ione, grauitatum cen<lb/>tra E in L, K in M. </s>

<s id="id.000972">Et <lb/>quoniam CNE, HOK anguli recti &longs;unt, erunt EC KH i<lb/>p&longs;is EN, KO, maiores, quare & LC, MH ip&longs;is EN KO, ma<lb/>iores attolluntur ergo in ip&longs;a euer&longs;ione, grauitatum cen<lb/>tra E in L, K in M. </s>

<s id="id.000973">At quod grauius e&longs;t, difficilius contra <lb/>&longs;ui naturam mouetur, ideo difficilius euertetur corpus <lb/>FI, ip&longs;o AD, quod fuerat demon&longs;trandum. </s>

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<s id="id.000978">De&longs;uper enim illum premunt, & <lb/>à lateribus. </s>

<s id="id.000979">quod autem e&longs;t in &longs;cytalis ad i&longs;thæc duo mo<lb/>uetur & inferiori &longs;ub&longs;trato &longs;patio, & onere &longs;uperimpo&longs;i-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.104.jpg"/>to, in vtri&longs;que enim ijs <gap/>euoluitur locis circulus, & motus <lb/>impellitur. </s>

<s id="id.000979">quod autem e&longs;t in &longs;cytalis ad i&longs;thæc duo mo<lb/>uetur & inferiori &longs;ub&longs;trato &longs;patio, & onere &longs;uperimpo&longs;i-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.103.jpg"/>to, in vtri&longs;que enim ijs <gap/>euoluitur locis circulus, & motus <lb/>impellitur. </s>

<s id="id.000980">Tam appo&longs;itè paucis verbis veritatem expli<lb/>cauit, vt ferè quicquid in&longs;uper ad datur, &longs;uperuacaneum <lb/>videri po&longs;&longs;it. </s>

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<s id="id.000984">Secus enim <lb/>axium & Rotarum fieret offen&longs;atio, cuius offen&longs;ationis <lb/>vim & effectum cum nouerit Ari&longs;toteles, vel hoc ip&longs;o lo<lb/>co te&longs;te, mirum e&longs;t, nihil de ea egi&longs;&longs;e quæ&longs;tione 9. vbi nos <lb/>hac de re fu&longs;i&longs;&longs;imè tractauimus. </s>

<s id="id.000985">Cæterùm quod de rotatis &longs;cytalis &longs;cribit Philo&longs;o<lb/>phus, notandum, à Pappo quidem lib.

<s id="id.000985">Cæterùm quod de rotatis &longs;cytalis &longs;cribit Philo&longs;o<lb/>phus, notandum, à Pappo quidem lib. 8. & à no&longs;tris Me<lb/>chanicis pa&longs;&longs;im ab&longs;que rotis Cylindrica &longs;implici videli<lb/>cet, & tereti formâ ad v&longs;um adhiberi. </s>

8. & à no&longs;tris Me<lb/>chanicis pa&longs;&longs;im ab&longs;que rotis Cylindrica &longs;implici videli<lb/>cet, & tereti formâ ad v&longs;um adhiberi. </s>

<s id="id.000986">E&longs;to igitur Ari<lb/><figure id="id.007.01.103.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.103.1.jpg"></figure><lb/>&longs;totelis quidem &longs;cytala <lb/>AB, Pappi verò &longs;eu vul<lb/>garis, & communis CD. <lb/></s>

<s id="id.000986">E&longs;to igitur Ari<lb/><figure id="id.007.00.104.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.104.1.jpg"></figure><lb/>&longs;totelis quidem &longs;cytala <lb/>AB, Pappi verò &longs;eu vul<lb/>garis, & communis CD. <lb/></s>

<s id="id.000987">His non modò lapicidæ <lb/>pa&longs;&longs;im, &longs;ed & nautæ na<lb/>uiumqueue fabri &longs;ubdu<lb/>cendis & mari inducen<lb/>dis nauibus vtuntur, quod varare dicunt vernaculè, Hi<lb/>&longs;panico, vt arbitror, vocabulo. </s>

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<s id="id.000991">Quatenus autem ad v&longs;um pertinet. </s>

<s id="id.000992">E&longs;to horizontis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.105.jpg"/><figure id="id.007.00.105.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.105.1.jpg"></figure><lb/>planum AB, &longs;cytalae duae<lb/>CD, EF, Pondus verò <lb/>eis impo&longs;itum G, tan<lb/>gens ip&longs;as in <expan abbr="p&umacr;ctis">punctis</expan> CE, <lb/>&longs;cytalæ autem planum <lb/>in punctis D, F, Pellatur <lb/>à potentia quapiam <expan abbr="p&omacr;-dus">pon<lb/>dus</expan> Gad anteriora, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> ad partes E. rotabuntur igitur &longs;cytalæ & pars quædam <lb/>&longs;cytalæ D, in qua &longs;it contactus a&longs;cendet in I, C verò de<lb/>&longs;cendetin H, nulla re motum impediente, quippe quòd <lb/>nulla ponderis &longs;cytalarum, & plani ad inuicem fiat offen<lb/>&longs;atio. </s>

<s id="id.000992">E&longs;to horizontis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.104.jpg"/><figure id="id.007.01.104.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.104.1.jpg"></figure><lb/>planum AB, &longs;cytalae duae<lb/>CD, EF, Pondus verò <lb/>eis impo&longs;itum G, tan<lb/>gens ip&longs;as in <expan abbr="p&umacr;ctis">punctis</expan> CE, <lb/>&longs;cytalæ autem planum <lb/>in punctis D, F, Pellatur <lb/>à potentia quapiam <expan abbr="p&omacr;-dus">pon<lb/>dus</expan> G ad anteriora, <expan abbr="n&emacr;-pe">nen<lb/>pe</expan> ad partes E. rotabuntur igitur &longs;cytalæ & pars quædam <lb/>&longs;cytalæ D, in qua &longs;it contactus a&longs;cendet in I, C verò de<lb/>&longs;cendet in H, nulla re motum impediente, quippe quòd <lb/>nulla ponderis &longs;cytalarum, & plani ad inuicem fiat offen<lb/>&longs;atio. </s>

<s id="id.000993">Præterea cum &longs;cytalarum centra ab horizontis pla<lb/>no æqualiter di&longs;tent, pondus quidem horizonti æquidi<lb/>&longs;tanter mouetur, & ideo eius centrum grauitatis nequa<lb/>quam, in motu qui &longs;it, eleuatur. </s>

<s id="id.000994">Cæterùm materiæ imperfectione remota nihil re<lb/>fert ad facilitatem, vtrum maioris minorisue diametri <lb/>&longs;int&longs;cytalæ, vt ea po&longs;ita eo quod maiores circuli faciliùs <lb/>offendicula &longs;uperent, quod demon&longs;tratum e&longs;t in quæ&longs;tio<lb/>ne 8. eo vtiliores &longs;unt &longs;cytalæ, quo cra&longs;&longs;iores. </s>

<s id="id.000995">Quatenus <lb/>autem ad plau&longs;tri naturam &longs;pectat, cuius ad &longs;cytalas Phi<lb/>lo&longs;ophus fecit comparationem, vt o&longs;ten da mus difficilius <lb/>ex eo moueri pondera. </s><figure id="id.007.00.105.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.105.2.jpg"></figure>

<s id="id.000995">Quatenus <lb/>autem ad plau&longs;tri naturam &longs;pectat, cuius ad &longs;cytalas Phi<lb/>lo&longs;ophus fecit comparationem, vt o&longs;ten damus difficilius <lb/>ex eo moueri pondera. </s><figure id="id.007.01.104.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.104.2.jpg"></figure>

<s id="id.000996">E&longs;to plau&longs;tri rota <lb/>KL, cuius centrum M, a<lb/>xis verò NO circa quem <lb/>rotaip&longs;a conuertiturKL. <lb/></s>

<s id="id.000997">Funis quo rota ex axis <lb/>centro M trahitur MP, <lb/>pondus vero QR. </s>

<s id="id.000998">Quo<lb/>niam igitur pondus axem <lb/>premit in N, axis autem rotæ modiolum in O, & eodem, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.106.jpg"/>tempore potentia quæ trahitin P, axem admouet modio<lb/>lo in parte V. duplex itaque fit ex fricatione &longs;eu offen&longs;a<lb/>tione impedimentum, in fra nempe, vbi O, & ad latus vbi <lb/>V. quæ quidem offen&longs;iones currus motum reddunt diffi<lb/>ciliorem, quæ quidem difficultas eo maior erit, quo ma<lb/>ior fuerit pondus axem premens, & minor proportio &longs;e<lb/>midiametri rotæ KM, ad axis &longs;emidiametrum MO. </s>

<s id="id.000998">Quo<lb/>niam igitur pondus axem <lb/>premit in N, axis autem rotæ modiolum in O, & eodem, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.105.jpg"/>tempore potentia quæ trahit in P, axem admouet modio<lb/>lo in parte V. duplex itaque fit ex fricatione &longs;eu offen&longs;a<lb/>tione impedimentum, infra nempe, vbi O, & ad latus vbi <lb/>V. quæ quidem offen&longs;iones currus motum reddunt diffi<lb/>ciliorem, quæ quidem difficultas eo maior erit, quo ma<lb/>ior fuerit pondus axem premens, & minor proportio &longs;e<lb/>midiametri rotæ KM, ad axis &longs;emidiametrum MO. </s>

<s id="id.000999">Cur <lb/>igitur &longs;cytalis facilius pondera transferantur quam plau<lb/>&longs;tris, apertè ex dictis ad Ari&longs;to telis mentem demon&longs;tra<lb/>uimus. </s>

<s id="id.001000">Cætetùm quod ip&longs;e reticuit, n<gap/>s dicemus, nempe <lb/>validi&longs;&longs;imè enormia pondera per &longs;cytalas moueri, &longs;i &longs;cy<lb/>talis ip&longs;is vectes adiungantur. </s>

<s id="id.001000">Cæterùm quod ip&longs;e reticuit, nos dicemus, nempe <lb/>validi&longs;&longs;imè enormia pondera per &longs;cytalas moueri, &longs;i &longs;cy<lb/>talis ip&longs;is vectes adiungantur. </s>

<s id="id.001001">Et &longs;anè motus erit tar di&longs;&longs;i<lb/>mus, veruntamen tar ditas ip&longs;a facilitate, quæ in de fit, v<lb/>berrimè compen&longs;atur. </s><figure id="id.007.00.106.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.106.1.jpg"></figure>

<s id="id.001001">Et &longs;anè motus erit tardi&longs;&longs;i<lb/>mus, veruntamen tarditas ip&longs;a facilitate, quæ in de fit, v<lb/>berrimè compen&longs;atur. </s><figure id="id.007.01.105.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.105.1.jpg"></figure>

<s id="id.001002">E&longs;to igitur horizontis planum AB, &longs;cytalæ CD, fo<lb/>ramina in &longs;cytalis EFGH, vectes foraminibus in&longs;erti IE, <lb/>KF, LG, MH. </s>

<s id="id.001003">Pondus vero &longs;cytalis impo&longs;itum N. </s>

<s id="id.001004">Appli<lb/>catis igìtur quatuor potentijs extremitatibus vectium I, <lb/><emph type="italics"/>K<emph.end type="italics"/>, L, M, ij&longs;que in anteriora propul&longs;is, fiet &longs;cytalarum rota-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.107.jpg"/>tio, & ponderis N translatio ad anteriores partes B. </s>

<s id="id.001004">Appli<lb/>catis igìtur quatuor potentijs extremitatibus vectium I, <lb/><emph type="italics"/>K<emph.end type="italics"/>, L, M, ij&longs;que in anteriora propul&longs;is, fiet &longs;cytalarum rota-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.106.jpg"/>tio, & ponderis N translatio ad anteriores partes B. </s>

<s id="id.001005">E&longs;to <lb/>item &longs;eor&longs;um &longs;cytala PR, cuius centrum Q, vectis eidem <lb/>per centrum in&longs;ertus O, P, Q, R. facto igitur vectis motu <lb/>O P Q R fiet ex O; centro <expan abbr="aut&emacr;">autem</expan> Q circuli quadrans O T. <lb/>exi&longs;tente igitur O in T erit P in S. facta quartæ partisip&longs;ius <lb/>&longs;cytalæ rotatione. </s>

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<s id="id.001008">Maxima igitur proportio OT ad PS. </s>

<s id="id.001009">Ex magno igitur <lb/>motu O ad T, paruus &longs;it &longs;cytalæ motus à P in S. tardius i<lb/>gitur progreditur &longs;cytala, quæ longioribus vectibus rota<lb/>tur, vis tamen maxima, quippe quod vt &longs;o habet QP, hoc <lb/>e&longs;t, QR ad QO, ita potentia in O ad pondus quod premit <lb/>in P vel in V. </s>

<s id="id.001009">Ex magno igitur <lb/>motu O ad T, paruus &longs;it &longs;cytalæ motus à P in S. tardius i<lb/>gitur progreditur &longs;cytala, quæ longioribus vectibus rota<lb/>tur, vis tamen maxima, quippe quod vt &longs;e habet QP, hoc <lb/>e&longs;t, QR ad QO, ita potentia in O ad pondus quod premit <lb/>in P vel in V. </s>

<s id="id.001010">Facillimè itaque pondera vectibus & &longs;cyta<lb/>lis per horizontis planum transferri, exi&longs;tis patet. </s>

</subchap1><subchap1>

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<s id="id.001013">Soluit Philo&longs;ophus, inquiens, fortè ita fieri, quòd fun<lb/>ditor mi&longs;&longs;ile proijciatiam ex funda commotum, &longs;iqui<lb/>dem fundam circulo &longs;ubinderotans, iaculatur, ex manu <lb/>autem à quiete e&longs;t initium. </s>

<s id="id.001014">Oinnia autem cum in motu <lb/>&longs;unt, quàm cum quie&longs;cunt, facilius mouentur. </s>

<s id="id.001014">Omnia autem cum in motu <lb/>&longs;unt, quàm cum quie&longs;cunt, facilius mouentur. </s>

<s id="id.001015">Addit præ<lb/>terea, An & ob eam cau&longs;&longs;am e&longs;t, &longs;ed nec minus etiam, quia <lb/>in fundç v&longs;a manus quidem fit centrum, funda verò quod <lb/>à centro exit? </s>

<s id="id.001015">Addit præ<lb/>terea, An & ob eam cau&longs;&longs;am e&longs;t, &longs;ed nec minus etiam, quia <lb/>infunde v&longs;u manus quidem fit centrum, funda verò quod <lb/>à centro exit? </s>

<s id="id.001016">quantò igitur productius fuerit quod à cen<lb/>tro e&longs;t, tanto citiùs mouetur; iactus autem, qui manu fit, <lb/>fundæ re&longs;pectu breuior e&longs;t. </s>

<s id="id.001017">Hæc Philo&longs;ophus. </s>

<s id="id.001018">Et &longs;anè perquàm appo&longs;itè, <expan abbr="itaq;">itaque</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.108.jpg"/>illi pror&longs;us a&longs;&longs;entirer, ni&longs;i pro comperto haberem, in lactu <lb/>qui fundâ fit, non e&longs;&longs;e manum ip&longs;am motus centium, &longs;ed <lb/>potius partem illam brachij, quæ humero iungitur, & id<lb/>co motum eo fieri velociotem, quo longior e&longs;t linea quæ <lb/>ab humero ad &longs;ummitatem fundæ e&longs;t, ea quæ ab humero <lb/>ad manum ip&longs;am. </s>

<s id="id.001018">Et &longs;anè perquàm appo&longs;itè, <expan abbr="itaq;">itaque</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.107.jpg"/>illi pror&longs;us a&longs;&longs;entirer, ni&longs;i pro comperto haberem, in iactu <lb/>qui fundâ fit, non e&longs;&longs;e manum ip&longs;am motus centium, &longs;ed <lb/>potius partem illam brachij, quæ humero iungitur, & id<lb/>eo motum eo fieri velociorem, quo longior e&longs;t linea quæ <lb/>ab humero ad &longs;ummitatem fundæ e&longs;t, ea quæ ab humero <lb/>ad manum ip&longs;am. </s>

<s id="id.001019">Illud quo que mirabilc e&longs;t, quod non <lb/>ob&longs;etuat Ari&longs;toteles, nempe à funditoribus in ip&longs;o eiacu<lb/>landi actu, tardam fieri circa caput fundæ rotationem. <lb/></s>

<s id="id.001019">Illud quoque mirabile e&longs;t, quod non <lb/>ob&longs;eruat Ari&longs;toteles, nempe à funditoribus in ip&longs;o eiacu<lb/>landi actu, tardam fieri circa caput fundæ rotationem. <lb/></s>

<s id="id.001020">Quamobrem con&longs;iderandum e&longs;t, quo pacto fiat à tardi<lb/>tate velocitas. </s>

<s id="id.001021">Re&longs;pondemus, velocitatem acquiri non ex <lb/>&longs;implici, quæ circa funditoris caput &longs;it, rotatione, &longs;ed ex <lb/>co impetu qui fit in ip&longs;a lapidis emi&longs;&longs;ione, qui quidem im<lb/>petus &longs;i ante vel po&longs;t illud tempus fiat, quod à funditore <lb/>captatur, ca&longs;&longs;a pror&longs;us & inualida fit ip&longs;a iaculatio. </s><figure id="id.007.00.108.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.108.1.jpg"></figure>

<s id="id.001021">Re&longs;pondemus, velocitatem acquiri non ex <lb/>&longs;implici, quæ circa funditoris caput &longs;it, rotatione, &longs;ed ex <lb/>eo impetu qui fit in ip&longs;a lapidis emi&longs;&longs;ione, qui quidem im<lb/>petus &longs;i ante vel po&longs;t illud tempus fiat, quod à funditore <lb/>captatur, ca&longs;&longs;a pror&longs;us & inualida fit ip&longs;a iaculatio. </s><figure id="id.007.01.107.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.107.1.jpg"></figure>

<s id="id.001022">E&longs;to funda AB, manus <lb/>B, brachium BC. </s>

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<s id="id.001025"><emph type="italics"/>Quæritur, Cur circa idemiugum, maiores collopes (vectes &longs;unt, <lb/>quos alij &longs;cytalas appellant, vt Pappus & Heron) faciliùs quàm mi<lb/>nores mouentur: & item &longs;uculæ, quæ graciliores &longs;unt eadem <lb/>vi quam cra&longs;&longs;iores?<emph.end type="italics"/></s>

<s id="id.001026">Ideo hoc fieri po&longs;&longs;e docet Philo&longs;ophus, quòd <expan abbr="tamiug&umacr;">tamiugum</expan> <lb/>quam &longs;ucula <expan abbr="c&emacr;trum">centrum</expan> &longs;it, prominenres autem collopum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.109.jpg"/>longitudines eæ lineæ quæ &longs;unt à centro. </s>

<s id="id.001026">Ideo hoc fieri po&longs;&longs;e docet Philo&longs;ophus, quòd <expan abbr="tamiug&umacr;">tam iugum</expan> <lb/>quam &longs;ucula <expan abbr="c&emacr;trum">centrum</expan> &longs;it, prominentes autem collopum <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.108.jpg"/>longitudines eæ lineæ quæ &longs;unt à centro. </s>

<s id="id.001027">Celeriùs autem <lb/>moueri & plus ab eadem vi quæ maiorum &longs;unt <expan abbr="circulor&umacr;">circulorum</expan> <lb/>quàm quæ minorum. </s>

<s id="id.001028">quippe quod ab ea dem vi plus <expan abbr="trã&longs;-feratur">tran&longs;<lb/>feratur</expan> illud extremum quod longius à centro di&longs;tat. </s>

<s id="id.001029">In <lb/>gracilioribus verò &longs;uculis datâ collopum paritate plus e&longs;<lb/>&longs;e id quod à ligno di&longs;tat. </s>

<s id="id.001030">E&longs;to iugum &longs;ucu<lb/>laue maior, AB circa <lb/>centrum C, minor verò <lb/>circa idem <expan abbr="centr&umacr;">centrum</expan> DE. <lb/></s>

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<s id="id.001032">A it igitur A<lb/>ri&longs;toteles, &longs;uculas, iu<lb/>gaue AB, DE ceu cen<lb/>tra e&longs;&longs;e, à quibus extat colops AB, ex maiori quidem, totâ <lb/>&longs;ui parte BF, ex minori autem EF. quo igitur, ait, longior <lb/>fuerit collops extans, eo maior, & deo velocior ad <expan abbr="part&emacr;">partem</expan> <lb/>F per maiorem circulum FH, fiet collopis motus & pon<lb/>deris eleuatio, at maior e&longs;t collops EF ip&longs;o BF, facil. </s>

<s id="id.001033">us er<lb/>go mouebitur pondus per &longs;uculam DE, ex collope EF, ab <lb/>cadem vi, quam per &longs;uculam AB, & collopem BF. </s>

<s id="id.001033">us er<lb/>go mouebitur pondus per &longs;uculam DE, ex collope EF, ab <lb/>eadem vi, quam per &longs;uculam AB, & collopem BF. </s>

<s id="id.001034">Hæc &longs;en&longs;i&longs;&longs;e videtur Ari&longs;to teles, qui cra&longs;&longs;a, vt aiunt, <lb/>Minerua rem pulchram & &longs;ubtilem e&longs;t pro&longs;equutus. </s>

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<s id="id.001037">Huius <lb/>vim inter antiquos diligenti&longs;&longs;ime examinauêre Heron, & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.110.jpg"/>ip&longs;emet Pappus, inter iuniores verò Guilibaldus co Tra<lb/>ctatu quem hac de Potentia Mechanicis &longs;uis in&longs;eruit. <lb/></s>

<s id="id.001037">Huius <lb/>vim inter antiquos diligenti&longs;&longs;ime examinauêre Heron, & <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.109.jpg"/>ip&longs;emet Pappus, inter iuniores verò Guilibaldus eo Tra<lb/>ctatu quem hac de Potentia Mechanicis &longs;uis in&longs;eruit. <lb/></s>

<s id="id.001038">Summa e&longs;t, hanc Machinam ad vectem reduci. </s>

<s id="id.001039">Nec ve<lb/>rum e&longs;t quod &longs;cribit Ari&longs;to teles, iugum &longs;uculamue cen<lb/>tra e&longs;&longs;e, hæc enim centrum habent, quod in figura &longs;upe<lb/>rius po&longs;ita notatur &longs;igno C. igitur vt &longs;e habet FC, ad CA, <lb/>ita pondus G ad potentiam in F; e&longs;t autem maior propor<lb/>tio FC ad CD, quàm FC, ad CA. faciliùs ergo mouebit <lb/>potentia quæ in F, pondus in D, quàm eadem potentia F, <lb/>pondus in A, hoc e&longs;t, G. </s>

<s id="id.001040">Huius naturæ &longs;unt quo que Erga<lb/>tæ, quas machinas no&longs;tr<gap/>, Græco luxato vocabulo Arga<lb/>nos appellant. </s>

<s id="id.001040">Huius naturæ &longs;unt quo que Erga<lb/>tæ, quas machinas no&longs;tri, Græco luxato vocabulo Arga<lb/>nos appellant. </s>

<s id="id.001041">Suculæ enim reuera &longs;unt, po&longs;itione <expan abbr="tant&umacr;">tantum</expan> <lb/>ab eis differentes, non enim plano horizontis ergatæ æ<lb/>quidi&longs;tant, ceu &longs;uculæ & Axis in Peritrochio, &longs;ed eidem <lb/>fiunt perpendiculares. </s>

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<s id="id.001045">Et &longs;i terræ applicans pede <lb/>&longs;uperpo&longs;ito manu hinc inde diducta confregerit <lb/>quàm propè.<emph.end type="italics"/></s>

<s id="id.001046">Soluitur à Philo&longs;opho paucis verbis, An quia ibi genu <lb/>centrum e&longs;t, hic verò ip&longs;e pe? </s>

<s id="id.001046">Soluitur à Philo&longs;opho paucis verbis, An quia ibi genu <lb/>centrum e&longs;t, hic verò ip&longs;e pes? </s>

<s id="id.001047">quanto autem remotius <lb/>à centro fuerit, facilius mouetur quodcunque: Moueri <lb/>autem quod frangitur nece&longs;&longs;e e&longs;t. </s>

<s id="id.001048">E&longs;to lignum quod frangi debet AB, genu vel pedis <lb/>locus C, manuum latè diductarum &longs;itus DE, minus didu<lb/>ctarum FG; ltaque quoniam DE magis à centro C di&longs;tant <lb/>quàm FG, velocius mouebuntur puncta DE ip&longs;is FG, er<lb/>go inde facilius fiet fractio quam ex FG. </s>

<s id="id.001049">Hæc ille ex &longs;uis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.111.jpg"/><figure id="id.007.00.111.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.111.1.jpg"></figure><lb/>principijs. </s>

<s id="id.001049">Hæc ille ex &longs;uis <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.110.jpg"/><figure id="id.007.01.110.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.110.1.jpg"></figure><lb/>principijs. </s>

<s id="id.001050">Nos dili<lb/>gentius, &longs;i fieri poterit, <lb/>effectus huius cau&longs;&longs;am <lb/>per&longs;crutemur. </s>

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<s id="id.001054">E&longs;t igitur re&longs;i&longs;tentia, qua ligni <lb/>partes vniuntur in D, loco ponderis: &longs;uperanda hæc e&longs;t, vt <lb/>ligni fiat fractio. </s>

<s id="id.001055">Dico id facilius ce&longs;&longs;urum, &longs;i fiat ex pun<lb/>ctis A, B, remotioribus quam ex IK, ip&longs;i puncto C propio<lb/>ribus: etenim vt AC, ad CD, itare&longs;i&longs;tentia quæ fit in D ad <lb/>potentiam in A, item vt &longs;e habet IC ad CD, ita re&longs;i&longs;tentia <lb/>in Dad potentiam in I, &longs;ed minor e&longs;t proportio IC ad CD, <lb/>quam AC ad CD. ergo facilius potentia quæ e&longs;t in A, re<lb/>&longs;i&longs;tentiam &longs;uperabit, quæ e&longs;t in D, quam ea quæ e&longs;t in I, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.112.jpg"/>quod fuerat demon&longs;trandum. </s>

<s id="id.001055">Dico id facilius ce&longs;&longs;urum, &longs;i fiat ex pun<lb/>ctis A, B, remotioribus quam ex IK, ip&longs;i puncto C propio<lb/>ribus: etenim vt AC, ad CD, ita re&longs;i&longs;tentia quæ fit in D ad <lb/>potentiam in A, item vt &longs;e habet IC ad CD, ita re&longs;i&longs;tentia <lb/>in Dad potentiam in I, &longs;ed minor e&longs;t proportio IC ad CD, <lb/>quam AC ad CD. ergo facilius potentia quæ e&longs;t in A, re<lb/>&longs;i&longs;tentiam &longs;uperabit, quæ e&longs;t in D, quam ea quæ e&longs;t in I, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.111.jpg"/>quod fuerat demon&longs;trandum. </s>

<s id="id.001056">Idem autem <expan abbr="intelligend&umacr;">intelligendum</expan> <lb/>e&longs;t de parte CB; eadem enim e&longs;t ratio. </s>

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<s id="id.001061">A It Philo&longs;ophus, ideo forta&longs;&longs;e fieri, quòd ca quæ à me<lb/>dio magis recedunt, in motionibus, celerius feran<lb/>tur; me dium e&longs;&longs;e centrum, interuallum vero quæ à cen<lb/>tro, &longs;emper autem maiorem ab æ quali motione maiorem <lb/>de&longs;cribere circulum; quod autem maius in æquali tem<lb/>pore &longs;patium tran&longs;it, celerius ferri; quæ autem celerius ex <lb/>æquali feruntur &longs;patio vehementius impetere, quæ <expan abbr="aut&emacr;">autem</expan> <lb/>impetunt, impeti magis, & ideo quæ magis à centro di<lb/>&longs;tant, nece&longs;&longs;e e&longs;&longs;e confringi, quod cum glareæ &longs;eu croc æ <lb/>patiantur, nece&longs;&longs;ariò rotundas fieri. </s>

<s id="id.001062">Hactenus ille, & &longs;anè <lb/>p<gap/>obabiliter. </s>

<s id="id.001062">Hactenus ille, & &longs;anè <lb/>probabiliter. </s>

<s id="id.001063">Verum enimuerò aliter &longs;eres habere vide<lb/>tur: &longs;iquidem enim à rotatione ex maiori à centro di&longs;tan<lb/>tia id fieret, maiores quidem glareæ crocæue e&longs;&longs;ent ro<lb/>tundiores, at nos non maximas modò, &longs;ed & minimas, <lb/>ea&longs;que magis angulis carere, & ad rotunditatem accede<lb/>re videmus. </s>

<s id="id.001064">Præterea non moueri eas circa centrum pa<lb/>lam e&longs;t, imò vt varia &longs;unt figura, ita varijs quo que motio<lb/>nibus, ex agitatione moueri. </s>

<s id="id.001065">Id &longs;anè explorati&longs;&longs;imum e&longs;t, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.113.jpg"/>angulos omnes, & emin entias quaslibet in corporibus e&longs;<lb/>&longs;e infirmiores, offen&longs;ionibus enim expo&longs;itæ &longs;unt, necre&longs;i<lb/>&longs;ten di habent facultatem. </s>

<s id="id.001065">Id &longs;anè explorati&longs;&longs;imum e&longs;t, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.112.jpg"/>angulos omnes, & eminentias quaslibet in corporibus e&longs;<lb/>&longs;e infirmiores, offen&longs;ionibus enim expo&longs;itæ &longs;unt, nec re&longs;i<lb/>&longs;tendi habent facultatem. </s>

<s id="id.001066">Itaque in attritione quæ fit in <lb/>eorum agitatione perpetua, eminentiæ contunduntur, & <lb/>&longs;uperficies ip&longs;a paullatim leuigatur. </s>

<s id="id.001067">E&longs;to angulatus lapis ABCD. <lb/></s>

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<s id="id.001070">Hinc etiam no&longs;trates Architecti, in arcium propu<lb/>gnaculis efformandis a cutos angulos <expan abbr="deuitãt">deuitant</expan>, vtpote de<lb/>biliores, & magis offen&longs;ionibus obnoxios. </s>

<s id="id.001071">quod nec Vi<lb/>truuium latuit, qui ideo lib.

<s id="id.001071">quod nec Vi<lb/>truuium latuit, qui ideo lib. 1. cap. 5. ita &longs;cribit: <emph type="italics"/>Turres itaque<lb/>rotundæ aut polygoniæ &longs;unt faciendæ, quadratas enim machinæ <lb/>celerius di&longs;&longs;ipant; & angulos, Arietes tundendo frangunt, in ro<lb/>tundationibus autem, vti cuneos ad centrum adigendo lædere non <lb/>po&longs;&longs;unt.<emph.end type="italics"/> Hæc ille. </s>

1. cap.

5. ita &longs;cribit: <emph type="italics"/>Turres itaque<lb/>rotundæ aut polygoniæ &longs;unt faciendæ, quadrat as enim machinæ <lb/>celerius di&longs;&longs;ipant; & angulos, Arietes tundendo frangunt, inro<lb/>tundationibus autem, vti cuneos ad centrum adigendo lædere non <lb/>po&longs;&longs;unt.<emph.end type="italics"/> Hæc ille. </s>

<s id="id.001072">Cur autem no&longs;tri rotundas figuras alias <lb/>vtiles reijciant, ab ijs petendum qui in ea facultate ver<lb/>&longs;antur. </s>

<s id="id.001073">Porrò quod ad hanc eandem &longs;peculationem facit, <lb/>videmus, antiquas &longs;tatuas, vt &longs;æpius auribus, na&longs;o, digitis, <lb/>manibu&longs;ue atque pedibus carere, quippe quod imbecillæ <lb/>&longs;int partes, & facilè quouis occur&longs;u mutilentur. </s>

<s id="id.001074">Quæo-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.114.jpg"/>m<gap/>a cùm vera &longs;int, nemo, vt arbitror, dixerit, ab&longs;olutè, <lb/>quod voluit Ari&longs;toteles, id ex rotatione velociori & par<lb/>tium à centro remotione, fieri. </s>

<s id="id.001074">Quæ o-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.113.jpg"/>mnia cùm vera &longs;int, nemo, vt arbitror, dixerit, ab&longs;olutè, <lb/>quod voluit Ari&longs;toteles, id ex rotatione velociori & par<lb/>tium à centro remotione, fieri. </s>

</subchap1><subchap1>

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<s id="id.001075">QVAESTIO XVI.</s>

<s id="id.001076"><emph type="italics"/>Dubitatur, quare, quò longiora &longs;unt ligna, <gap/><expan abbr="tãto">tanto</expan> imbecilliora fiant, <lb/>& &longs;itolluntur, inflectuntur magis: tamet&longs;i quod breue est ceu bi<lb/>cubit umfuerit, tenue, quod verò cubitorum cen<lb/>tum cra&longs;&longs;um?<emph.end type="italics"/></s>

<s id="id.001076"><emph type="italics"/>Dubitatur, quare, quò longiora &longs;unt ligna, <expan abbr="tãto">tanto</expan> imbecilliora fiant, <lb/>& &longs;i tolluntur, inflectuntur magis: tamet&longs;i quod breue est ceu bi<lb/>cubitum fuerit, tenue, quod verò cubitorum cen<lb/>tum cra&longs;&longs;um?<emph.end type="italics"/></s>

<s id="id.001077">Ex &longs;uis principijs &longs;oluit Ari&longs;toteles. </s>

<s id="id.001078">Inquit enim: An <lb/>quia & vectis & ont s & hypomochlium, id e&longs;t, fulci<lb/>mentum in leuando, fit ip&longs;a ligni proceritas? </s>

<s id="id.001078">Inquit enim: An <lb/>quia & vectis & onus & hypomochlium, id e&longs;t, fulci<lb/>mentum in leuando, fit ip&longs;a ligni proceritas? </s>

<s id="id.001079">Prior <expan abbr="namq;">namque</expan> <lb/>illius pars ceu hypomochlium fit, quod verò in extremo <lb/>e&longs;t, pondus: quamobrem quanto exten&longs;ius fuerit id quod <lb/>à fulcimento e&longs;t, in flecti nece&longs;&longs;e e&longs;t magis; quo enim plus <lb/>à fulcimento di&longs;tat, eo magis incuruarinece&longs;&longs;e e&longs;t. </s>

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<s id="id.001081">Si igitur flexilis <lb/>fuerit vectis, ip&longs;um inflectimagis cum extollitur nece&longs;&longs;e <lb/>e&longs;t, quod longis accidit lignis, in breuibus autem quod vl<lb/>timum e&longs;t, quie&longs;centi hypomochlio depropè fit. </s>

<s id="id.001082">Hæc <lb/>&longs;ubiectâ figurâ ob oculos ponimus. </s>

<s id="id.001083">E&longs;to longum ac fle<lb/>xile lignum AB, manu ele<lb/>uetur in A, fle ctetur <expan abbr="itaq;">itaque</expan> <lb/>in B, & declinabit in C. et<lb/>enim manus quæ &longs;u&longs;tin et <lb/>in A, fulcimenti loco &longs;uccedit: longitudo vero AB ponde<lb/>ris vices refert, at que vectis, quare quo longius abfuerit à <lb/>fulcimento, id e&longs;t, manu extremum B, eo magis flectetur; <lb/>&longs;i autem lignum breuius fuerit, nempe terminatum in D, <lb/>nequaquam fle ctetur, eò quòd eius extremum D minus à <lb/>fulcimento quod e&longs;t in A &longs;it remotum. </s>

<s id="id.001084">Hæc igitur e&longs;t <expan abbr="m&emacr;s">mens</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.115.jpg"/>Ari&longs;totelis, cuius quidem &longs;ententiam non damnamus; <lb/>quippiam tamen addimus. </s>

<s id="id.001084">Hæc igitur e&longs;t <expan abbr="m&emacr;s">mens</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.114.jpg"/>Ari&longs;totelis, cuius quidem &longs;ententiam non damnamus; <lb/>quippiam tamen addimus. </s>

<s id="id.001085">Dicimus autem materiam, <lb/>quatenus ad hanc contemplationem &longs;pectat, in duplici <lb/>e&longs;&longs;e differentia. </s>

<s id="id.001086">aut enim rarefactionis & con&longs;tipationis <lb/>e&longs;t incapax, vt in chalybe videmus, nitro, metallo, mar<lb/>more, aut capax quidem, & hæc duplex: Vel enim natura <lb/>nata e&longs;t ad rectitudinem quandam, vt ar borum flagella <lb/>virgæque, aut non item, ceu &longs;tannum, plumbum, & cæte<lb/>ra eiu&longs;modi. </s>

<s id="id.001087">E&longs;to primò vitreum <lb/>corpus gracile, procerum, <lb/>teres AB, manu capiaturin <lb/>A, <expan abbr="itaq.">itaque</expan> pondere ip&longs;ius cor<lb/>poris præualente ad partes <lb/>B, quia in C puncto, quod <lb/>circa medium e&longs;t, ex parte <lb/>&longs;uperiori non fit rarefactio, <lb/>nec in in feriori con&longs;tipatio, <lb/>nec interim datur penetra<lb/>tio corporum, fit fractio à <lb/>&longs;uperiori parte, & pars CB à <lb/>reliqua parte AC, auul&longs;a & <lb/>&longs;eparata cadit in D, &longs;uccedit autem ip&longs;a &longs;eparatio rarefa<lb/>ctioni. </s>

<s id="id.001088">Porrò quod materias ha&longs;ce non flexibiles diximus, <lb/>&longs;ed frangibiles, non ideo negamus vel &longs;en&longs;u docente, ali<lb/>quam inijs fieri flexionem. </s>

<s id="id.001089">Si autem lignea fuerit mate<lb/>ria, caque; flexibilis, vt EF, &longs;i manu eleuetur in E, præualen<lb/>te pondere in F flectetur vbi G. ibi enim à parte &longs;uperiori <lb/>fitrarefactio, ab in feriori verò con&longs;tipatio, & pars GF de<lb/>clinabitin H, quæ declinatio eò v&longs;que procedet, quo ra<lb/>refactio & con&longs;tipatio competens naturæ illius materiæ, <lb/>quæ flectitur ad &longs;ummam inten&longs;ionem deuenerint; tunc <lb/>&longs;ivis maior ingruerit, frangetur omnino: &longs;i &longs;ecus facta ibi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.116.jpg"/>re&longs;i&longs;tentia, vbi rarefactio fit & con&longs;tipatio pe&longs;t inclina<lb/>tionem &longs;ur&longs;um ferctur pars in clinata & nutans, tum in <lb/>contrariam partem tendens reflectetur, vt videre e&longs;t in <lb/>virga IN. </s>

<s id="id.001089">Si autem lignea fuerit mate<lb/>ria, caque; flexibilis, vt EF, &longs;i manu eleuetur in E, præualen<lb/>te pondere in F flectetur vbi G. ibi enim à parte &longs;uperiori <lb/>fit rarefactio, ab in feriori verò con&longs;tipatio, & pars GF de<lb/>clinabit in H, quæ declinatio eò v&longs;que procedet, quo ra<lb/>refactio & con&longs;tipatio competens naturæ illius materiæ, <lb/>quæ flectitur ad &longs;ummam inten&longs;ionem deuenerint; tunc <lb/>&longs;i vis maior ingruerit, frangetur omnino: &longs;i &longs;ecus facta ibi <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.115.jpg"/>re&longs;i&longs;tentia, vbi rarefactio fit & con&longs;tipatio per inclina<lb/>tionem &longs;ur&longs;um feretur pars in clinata & nutans, tum in <lb/>contrariam partem tendens reflectetur, vt videre e&longs;t in <lb/>virga IN. </s>

<s id="id.001090">Declinans enim in KL, repellente ea quæ infra <lb/>K fit materiæ conden&longs;atione, impetu ex de&longs;cen&longs;u a cqui<lb/>&longs;ito facta reflexione a&longs;cendit in KM, donec paullatim cir<lb/>ca pri&longs;tinam rectitudinem reuertatur, & hic quidem mo<lb/>tus vibratio dicitur, agitatioue. </s>

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<s id="id.001094">Modò &longs;u&longs;pendatur in <lb/>medio vt AB, in C. eo igitur ca&longs;u cum fulcimentum &longs;it in <lb/>C, <expan abbr="vtrinq;">vtrinque</expan> fit flexio in D, & E, & id quidem &longs;i materia fle<lb/>xionem patitur: &longs;in minus, fractio fit in C. </s>

<s id="id.001095">Si autem ab ex<lb/><figure id="id.007.00.116.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.116.1.jpg"></figure><lb/>tremis fiat &longs;u&longs;pen&longs;io, vt in <lb/>AB, tunc ceu duo vectes <lb/>fient, quorum fulcimenta in <lb/>extremis AB. </s>

<s id="id.001095">Si autem ab ex<lb/><figure id="id.007.01.115.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.115.1.jpg"></figure><lb/>tremis fiat &longs;u&longs;pen&longs;io, vt in <lb/>AB, tunc ceu duo vectes <lb/>fient, quorum fulcimenta in <lb/>extremis AB. </s>

<s id="id.001096">Pondera au<lb/>tem communia in medio vbi <lb/>Cremoti&longs;&longs;ima enim ea pars e&longs;t ab extremis AB. </s>

<s id="id.001097">Cedente <lb/><figure id="id.007.00.116.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.116.2.jpg"></figure><lb/>igitur materia &longs;uomet pon<lb/>deri, &longs;iquidein in flexibilis fu<lb/>erit, frangetur, & fiet <expan abbr="parti&umacr;">partium</expan> <lb/>&longs;eparatio in C, duoque in de <lb/>corpora AD, BE. </s>

<s id="id.001097">Cedente <lb/><figure id="id.007.01.115.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.115.2.jpg"></figure><lb/>igitur materia &longs;uomet pon<lb/>deri, &longs;iquidem in flexibilis fu<lb/>erit, frangetur, & fiet <expan abbr="parti&umacr;">partium</expan> <lb/>&longs;eparatio in C, duoque in de <lb/>corpora AD, BE. </s>

<s id="id.001098">Si autem fle<lb/>xionis capax, vt AB in po&longs;tre<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.117.jpg"/>ma figura, facta ex contrario, nempe in in feriori parte cir<lb/>ca C rarefactione, in &longs;uperiori verò conden&longs;atione, pon<lb/>dere præualente curuabitur, fietque; lignum quidue aliud <lb/>huiu&longs;modi, vt ADB, nec amplius pondere &longs;uapte naturâ <lb/>inferiùs vergente ad rectitudinem reuertetur. </s>

<s id="id.001098">Si autem fle<lb/>xionis capax, vt AB in po&longs;tre<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.116.jpg"/>ma figura, facta ex contrario, nempe in in feriori parte cir<lb/>ca C rarefactione, in &longs;uperiori verò conden&longs;atione, pon<lb/>dere præualente curuabitur, fietque; lignum quidue aliud <lb/>huiu&longs;modi, vt ADB, nec amplius pondere &longs;uapte naturâ <lb/>inferiùs vergente ad rectitudinem reuertetur. </s>

<s id="id.001099">Cæterùm cur oblonga & graciliora corpora facilius <lb/>illis, quæ contrario &longs;e habent modo, frangantur, ex me<lb/>chanicis principijs in quæ&longs;tione 14. apertè demon&longs;traui<lb/>mus. </s>

<s id="id.001100">Modò vt ex hac contemplatione, quæ aliàs inutilis <lb/>videtur, aliquam vtilitatem capiamus, & ex his quæ con<lb/>templabimur, Architecti prudentiotes fiant, i&longs;thæcip&longs;a, <lb/>de quibus agimus, ad rem ædificatoriam commodè apta<lb/>bimus. </s>

<s id="id.001100">Modò vt ex hac contemplatione, quæ aliàs inutilis <lb/>videtur, aliquam vtilitatem capiamus, & ex his quæ con<lb/>templabimur, Architecti prudentiores fiant, i&longs;thæc ip&longs;a, <lb/>de quibus agimus, ad rem ædificatoriam commodè apta<lb/>bimus. </s>

<s id="id.001101">Transferamus igitur cogitationem ad eam <expan abbr="trabi&umacr;">trabium</expan> <lb/>comp gem, quæ ad tecta &longs;u&longs;tinenda ex tran&longs;uer&longs;ario ar<lb/>rectarioque; &longs;it, & duobus cauterijs, quam no&longs;trià Latinis <lb/>detorto vocabulo Bi&longs;cauterium dicunt. </s>

<s id="id.001101">Transferamus igitur cogitationem ad eam <expan abbr="trabi&umacr;">trabium</expan> <lb/>compagem, quæ ad tecta &longs;u&longs;tinenda ex tran&longs;uer&longs;ario ar<lb/>rectarioque; &longs;it, & duobus cauterijs, quam no&longs;trià Latinis <lb/>detorto vocabulo Bi&longs;cauterium dicunt. </s>

<s id="id.001102">Per&longs;crutabimur <lb/>enim, vnde illi tanta ad &longs;u&longs;tin endum vis, & quæ compa<lb/>gem hanc con&longs;equantur pa&longs;&longs;iones. </s>

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<s id="id.001106">E&longs;to horizontis planum, illudqueue &longs;olidi&longs;&longs;imum, & <lb/>impenetrabile AB, trabs eidem ad perpendiculum erecta <lb/>CD fulta ba&longs;i vbi C grauitatis centrum F. pondus &longs;uper<lb/>impo&longs;itum FG, cuius grauitatis centrum H: Sint autem <lb/>H & E in eadem perpendiculari, quæ ad mun di centrum <lb/>HEC. </s>

<s id="id.001107">Itaque eo quod tum penderis tum trabis centra <lb/>grauitent in perpendiculari, illa verò fulciatur in C, to-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.118.jpg"/><figure id="id.007.00.118.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.118.1.jpg"></figure><lb/>tius ponderis moles recumbet <lb/>in C: non de&longs;cendet autem in I, <lb/>propterea quod &longs;upponatur i<lb/>p&longs;um planum AB, impenetrabi<lb/>le. </s>

<s id="id.001107">Itaque eo quod tum penderis tum trabis centra <lb/>grauitent in perpendiculari, illa verò fulciatur in C, to-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.117.jpg"/><figure id="id.007.01.117.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.117.1.jpg"></figure><lb/>tius ponderis moles recumbet <lb/>in C: non de&longs;cendet autem in I, <lb/>propterea quod &longs;upponatur i<lb/>p&longs;um planum AB, impenetrabi<lb/>le. </s>

<s id="id.001108">Igitur vt pondus H de&longs;cen<lb/>dat in C, alterum duorum e&longs;t <lb/>nece&longs;&longs;arium, nempe vel trabem <lb/>&longs;ubiectam comminui, aut eius <lb/>partes &longs;e&longs;e penetrare, & plura <lb/>corpora e&longs;&longs;e in eodem loco, pu<lb/>ta KC, quorum hoc &longs;ecun dum <lb/>naturæ penitus repugnat, illud <lb/>vero primum, penè impo&longs;&longs;ibile. </s>

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<s id="id.001113">Præterea in præcedenti quæ&longs;tione mon&longs;trauimus <lb/>tunc facilem e&longs;&longs;e gracilis & oblongi ligni fractionem, <expan abbr="c&umacr;">cum</expan> <lb/>maxima e&longs;t longitudinis ad cra&longs;&longs;itudinem proportio. </s>

<s id="id.001114">Hîc <lb/>verò contrà accidit, etenim MD pars vectis quæ à fulci<lb/>mento e&longs;t ad potentiam minimam habet proportionem <lb/>ad rectam DC, quæ à fulcimento ad locum fractionis ex<lb/>tenditur, vbi C, quod vt euidentius pateat, </s>

<s id="id.001115">E&longs;to &longs;eor&longs;umtrabs AB, <lb/>cuius medium C. </s>

<s id="id.001116">Sit autem <lb/>pondus D impo&longs;itum pun<lb/>cto C. facilè igitur frange<lb/>tur lignum AB, propterea <lb/>quòd maxima &longs;it proportio <lb/>AC ad CE; re&longs;i&longs;tentia verò <lb/>fiat in E, addatur vniatu que; <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.119.jpg"/>ligno AB lignum FH. </s>

<s id="id.001116">Sit autem <lb/>pondus D impo&longs;itum pun<lb/>cto C. facilè igitur frange<lb/>tur lignum AB, propterea <lb/>quòd maxima &longs;it proportio <lb/>AC ad CE; re&longs;i&longs;tentia verò <lb/>fiat in E, addatur vniaturque; <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.118.jpg"/>ligno AB lignum FH. </s>

<s id="id.001117">Cra&longs;&longs;ius igitur e&longs;t totum AL, ip&longs;o <lb/>A<gap/>, & ideo minor proportio AC ad CG quàm AC, ad <lb/>CE. </s>

<s id="id.001117">Cra&longs;&longs;ius igitur e&longs;t totum AL, ip&longs;o <lb/>AH, & ideo minor proportio AC ad CG quàm AC, ad <lb/>CE. </s>

<s id="id.001118">Addatur adhuc & IM. </s>

<s id="id.001119">Longè itaque difficilius fran<lb/>getur in K propterea quòd longè minor &longs;it proportio AC <lb/>ad CK quàm ciuidem ad CE & CG. </s>

<s id="id.001119">Longè itaque difficilius fran<lb/>getur in K propterea quòd longè minor &longs;it proportio AC <lb/>ad CK quàm eiu&longs;dem ad CE & CG. </s>

<s id="id.001120">His igitur con&longs;ide<lb/>ratis, & demon&longs;tratis concludimus, impo&longs;libile e&longs;&longs;e ere<lb/>ctam trabem ponderi cedere, & frangi. </s>

<s id="id.001120">His igitur con&longs;ide<lb/>ratis, & demon&longs;tratis concludimus, impo&longs;&longs;ibile e&longs;&longs;e ere<lb/>ctam trabem ponderi cedere, & frangi. </s>

<s id="id.001121">Dicet autem qui&longs;piam, haec&longs;i vera &longs;unt, quo gracilius <lb/>fuerit fulcrum, eo validiùs &longs;u&longs;tinebit, & frangetur minus, <lb/>quod oppido fal&longs;um e&longs;t. </s>

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<s id="id.001123">Ita <lb/>quoque invecte non materiam, quatenus ad vim pertinet, <lb/>&longs;ed proportiones partium con&longs;ideramus. </s>

<s id="id.001124">Vtiumqueigi<lb/>turrequiritur ad fulcti validitatem proportio longitudi<lb/>nis ad cra&longs;&longs;itudinem debita, & materiæ ip&longs;ius robur & <lb/>fortitudo. </s>

<s id="id.001124">Vtrumque igi<lb/>tur requiritur ad fulcri validitatem proportio longitudi<lb/>nis ad cra&longs;&longs;itudinem debita, & materiæ ip&longs;ius robur & <lb/>fortitudo. </s>

<s id="id.001125">Præterea, quoniam pondus, cuifulcrum re&longs;i<lb/>&longs;tit, vel ex natura premit, vel ex violentia, illud quidem <lb/>per lincam perpen dicularem, quæ ad mundi <expan abbr="c&emacr;trum">centrum</expan>, hoc <lb/>autem lateraliter & diuer&longs;imodè, varia fit fulcrorum di&longs;<lb/>po&longs;itio. </s>

<s id="id.001125">Præterea, quoniam pondus, cui fulcrum re&longs;i<lb/>&longs;tit, vel ex natura premit, vel ex violentia, illud quidem <lb/>per lineam perpendicularem, quæ ad mundi <expan abbr="c&emacr;trum">centrum</expan>, hoc <lb/>autem lateraliter & diuer&longs;i modè, varia fit fulcrorum di&longs;<lb/>po&longs;itio. </s>

<s id="id.001126">Cuius rei &longs;umma hæce&longs;t, vt &longs;emper contra impe<lb/>tum &longs;upponantur. </s>

<s id="id.001127">E&longs;to enim horizontis planum <lb/>AB, <expan abbr="eid&emacr;">eidem</expan> perpendiculares CADB, <lb/>ítaque &longs;i naturaliter pondus pre<lb/>matex C, fulcrum &longs;upponetur AE. <lb/></s>

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<s id="id.001130">Hæc nos de arrectarijs fulcrisue; <lb/>nunc de tran&longs;uer&longs;arijs, & inclinatis agemus, & primum <lb/>de tran&longs;uer&longs;arijs, quatenus ad tectorum trabeationes &longs;pe<lb/>ctat. </s>

<s id="id.001131">E&longs;to tran&longs;uer&longs;aria trabs AB, muris <expan abbr="vtrinq;">vtrinque</expan> fulta CD, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.120.jpg"/><figure id="id.007.00.120.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.120.1.jpg"></figure><lb/>cuius grauitatis centrum <lb/>E, in <expan abbr="perp&emacr;diculari">perpendiculari</expan> FEG, <lb/>quæ quidem ad mundi <lb/>centrum vergit. </s>

<s id="id.001131">E&longs;to tran&longs;uer&longs;aria trabs AB, muris <expan abbr="vtrinq;">vtrinque</expan> fulta CD, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.119.jpg"/><figure id="id.007.01.119.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.119.1.jpg"></figure><lb/>cuius grauitatis centrum <lb/>E, in <expan abbr="perp&emacr;diculari">perpendiculari</expan> FEG, <lb/>quæ quidem ad mundi <lb/>centrum vergit. </s>

<s id="id.001132"><expan abbr="Itaq;">Itaque</expan> eo<lb/>dem tendente grauitatis <lb/>contro, &longs;i pondus quod <lb/>premit in E, non præua<lb/>leat vnioni <expan abbr="parti&umacr;">partium</expan> ip&longs;ius <lb/>materiæ quæ e&longs;t in E, re&longs;i&longs;tet trabs &longs;uomet ponderi, nec <lb/>frangetur. </s>

<s id="id.001132"><expan abbr="Itaq;">Itaque</expan> eo<lb/>dem tendente grauitatis <lb/>centro, &longs;i pondus quod <lb/>premit in E, non præua<lb/>leat vnioni <expan abbr="parti&umacr;">partium</expan> ip&longs;ius <lb/>materiæ quæ e&longs;t in E, re&longs;i&longs;tet trabs &longs;uomet ponderi, nec <lb/>frangetur. </s>

<s id="id.001133">Si autem vel in firmitate materiæ, aut vitio, vel <lb/>maxima exiftente proportione AF ad FE, fractio fiet in E, <lb/>& &longs;ecutâ partium &longs;epaiatione duæ fient vtrin que trabes <lb/>AH, Bl, quorum grauitatis centra KL. </s>

<s id="id.001133">Si autem vel in firmitate materiæ, aut vitio, vel <lb/>maxima exiftente proportione AF ad FE, fractio fiet in E, <lb/>& &longs;ecutâ partium &longs;eparatione duæ fient vtrin que trabes <lb/>AH, Bl, quorum grauitatis centra KL. </s>

<s id="id.001134">Erunt igitur duo <lb/>vectes AE, BE, quorum fulcimenta MN, quamobrem &longs;i <lb/>proportio EM ad MH ita præualeat, vt pondus quod e &longs;t <lb/>in E, &longs;uperet pondus muri O &longs;uperimpo&longs;iti, & item muri <lb/>P, corruent quidem trabes, & murorum fiet hinc inde di&longs;<lb/>&longs;ipatio. </s>

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<s id="id.001136">Huic difficultati egregiè occurrunt Architecti, ali<lb/>quando autem hoc modo: </s>

<s id="id.001137">E&longs;to tran&longs;uer&longs;aria <lb/>trabs &longs;uâ gracilitate, alia<lb/>ue de cau&longs;&longs;a imbecilla <lb/>AB, muri quibus <expan abbr="vtrinq;">vtrinque</expan> <lb/>&longs;u&longs;tinetur CD, Trabis i<lb/>p&longs;ius grauitatis centrum <lb/>G. </s>

<s id="id.001138">Itaque adpactis trabi <lb/>lignis EF, caprcolos ad<lb/>dunt muro vtrinque ful<lb/>tos CE, DF, corum capita adpactis lignis admouentes EF, <lb/>&longs;ed & tunc validi&longs;&longs;ima fit colligatio, &longs;i inter E & F capreo<lb/>lorum capita integrum lignum trabi &longs;upponatur EF. </s>

<s id="id.001138">Itaque adpactis trabi <lb/>lignis EF, capreolos ad<lb/>dunt muro vtrinque ful<lb/>tos CE, DF, eorum capita adpactis lignis admouentes EF, <lb/>&longs;ed & tunc validi&longs;&longs;ima fit colligatio, &longs;i inter E & F capreo<lb/>lorum capita integrum lignum trabi &longs;upponatur EF. </s>

<s id="id.001139">Ra<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.121.jpg"/>tio autem validitatis patet; premente enim grauitatis <expan abbr="c&emacr;-tro">cen<lb/>tro</expan> in G, fulcra hinc inde &longs;uccurrunt CE, DF, quæ cum &longs;e<lb/>ip&longs;is fieri non valeant breuiora, ne corpori detur penetra<lb/>tio, re&longs;i&longs;tunt & robu&longs;ti&longs;&longs;imè ip&longs;i ponderi &longs;uperimpo&longs;ito <lb/>contranituntur. </s>

<s id="id.001139">Ra<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.120.jpg"/>tio autem validitatis patet; premente enim grauitatis <expan abbr="c&emacr;-tro">cen<lb/>tro</expan> in G, fulcra hinc inde &longs;uccurrunt CE, DF, quæ cum &longs;e<lb/>ip&longs;is fieri non valeant breuiora, ne corpori detur penetra<lb/>tio, re&longs;i&longs;tunt & robu&longs;ti&longs;&longs;imè ip&longs;i ponderi &longs;uperimpo&longs;ito <lb/>contra nituntur. </s>

<s id="id.001140">Videntur autem in hoc opere duo con<lb/>&longs;iderari vectes, GH, GB, quorum fulcimenta EF, potentia <lb/>premens vtrinque G. </s>

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<s id="id.001144">Sed & tectorum contignationes imbecillaque; tran&longs;<lb/>uer&longs;aria Mechanici corroborare &longs;olent, additis nempe <lb/>arrectaria trabe atque cauterijs. </s>

<s id="id.001145">E&longs;to enim tran&longs;<lb/>uer&longs;aria trabs AB <lb/>parietibus vtrinque <lb/>fulta I, K, <expan abbr="arrectari&umacr;">arrectarium</expan> <lb/>CD. </s>

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<s id="id.001147">Tum ferrea fa&longs;cia EF mediam tran&longs;uer&longs;ariam <lb/>trabem AB, à parte inferiori ip&longs;i arrectario connectens, <lb/>Debet autem arrectarij pes vbi C, aliquantulum à tran&longs;<lb/>uer&longs;aria trabe di&longs;tare, ne deor&longs;um ex pondere vergente <lb/>paululum arrectario ip&longs;am tran&longs;uer&longs;ariam premat. </s>

<s id="id.001148">His i-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.122.jpg"/>gitur ita con&longs;titutis pondus quidem tran&longs;uer&longs;ariæ trabis, <lb/>quod &longs;uapte naturâ premit in medio vbi C, ferrea fa&longs;cia, <lb/>arrectariæ trabi affixa di&longs;tinetur, Arrectariam cauterij &longs;u<lb/>&longs;tinent, hos verò tran&longs;uer&longs;ariæ capita AB, quibus indun<lb/>tur. </s>

<s id="id.001148">His i-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.121.jpg"/>gitur ita con&longs;titutis pondus quidem tran&longs;uer&longs;ariæ trabis, <lb/>quod &longs;uapte naturâ premit in medio vbi C, ferrea fa&longs;cia, <lb/>arrectariæ trabi affixa di&longs;tinetur, Arrectariam cauterij &longs;u<lb/>&longs;tinent, hos verò tran&longs;uer&longs;ariæ capita AB, quibus indun<lb/>tur. </s>

<s id="id.001149">Tota igitur eiu&longs;cemodi operis vis in eo con&longs;i&longs;tit, vt <lb/>probè cauterij tran&longs;uersariæ & arrectariæ trabi in&longs;eran<lb/>tur. </s>

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<s id="id.001152">Quoniam autem v&longs;u venire &longs;olet, cauterios nimia <lb/>longitudine debiles, aliquando tum proprio tum extra<lb/>neo cedentes ponderi deor&longs;um vergentes pandare, Ar<lb/>chitecti capreolis hinc inde &longs;uppo&longs;itis, ceu fulcris, huic <lb/>medentur infirmitati. </s>

<s id="id.001153">Sint enim cauterij <lb/>debiles hinc inde AB, <lb/>AC, media trabs arre<lb/>ctaria, quam <expan abbr="Monach&umacr;">Monachum</expan> <lb/>dicimus AD. </s>

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<s id="id.001157">Capiatur enim inter EH, quoduis <lb/>punctum G, & BG, DG, connectantur, erunt autem BG, <lb/>DG ip&longs;is BE ED breuiores ex 21. primi elem. </s>

<s id="id.001158">Tuncigitur <lb/>punctum E fiet in G cum BE, ED fient in BG, DG, quod <lb/>non cedentibus B, D, & &longs;ibi ip&longs;is breuioribus factis parti-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.123.jpg"/>bus BE, ED, pror&longs;us e&longs;t impo&longs;&longs;ibile. </s>

<s id="id.001158">Tunc igitur <lb/>punctum E fiet in G cum BE, ED fient in BG, DG, quod <lb/>non cedentibus B, D, & &longs;ibi ip&longs;is breuioribus factis parti-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.122.jpg"/>bus BE, ED, pror&longs;us e&longs;t impo&longs;&longs;ibile. </s>

<s id="id.001159">&longs;tabuntigitur in co<lb/>rum rectitudine cauterij AB, AC, nec pandabunt, quod <lb/>fieri querebatur. </s>

<s id="id.001159">&longs;tabunt igitur in eo<lb/>rum rectitudine cauterij AB, AC, nec pandabunt, quod <lb/>fieri querebatur. </s>

<s id="id.001160">Hîc autem damnandi veniunt ij, quitran&longs;uer&longs;ariæ <lb/>quidem trabis capitibus cauteriorum pedes non <expan abbr="in&longs;er&umacr;t">in&longs;erunt</expan>, <lb/>&longs;ed ea vice tran&longs;uer&longs;ariolo quodam medios cauterios v<lb/>trin que connectunt ad in&longs;tar elementi A, quam compa<lb/>gem, capram, appellant. </s>

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<s id="id.001170">Ducatur hori<lb/>zonti æquidi&longs;tans CD, iuxta quam lateres lapide&longs;ue non <lb/>cuneati, &longs;eriatim collocentur EF. </s>

<s id="id.001171">Dicimus amoto arma<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.124.jpg"/><figure id="id.007.00.124.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.124.1.jpg"></figure><lb/>mento, hoc e&longs;t, pro<lb/>hibente ip&longs;o lateres <lb/>ruere. </s>

<s id="id.001171">Dicimus amoto arma<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.123.jpg"/><figure id="id.007.01.123.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.123.1.jpg"></figure><lb/>mento, hoc e&longs;t, pro<lb/>hibente ip&longs;o lateres <lb/>ruere. </s>

<s id="id.001172">Producantur <lb/>enim AC in G, BD <lb/>verò in H, cum ip&longs;is <lb/>CG, DH, æquales <lb/>fiant CI, DK, & recta <lb/>IK iungatur, crit igi<lb/>tur GD &longs;patium ip&longs;i <lb/>CK &longs;patio &longs;imile qui<lb/>dem & æquale, quod <lb/>cùm ita &longs;it, nihil prohibet quin tota laterum GD moles in <lb/>&longs;patium CK transferatur, & corruat. </s>

<s id="id.001172">Producantur <lb/>enim AC in G, BD <lb/>verò in H, cum ip&longs;is <lb/>CG, DH, æquales <lb/>fiant CI, DK, & recta <lb/>IK iungatur, erit igi<lb/>tur GD &longs;patium ip&longs;i <lb/>CK &longs;patio &longs;imile qui<lb/>dem & æquale, quod <lb/>cùm ita &longs;it, nihil prohibet quin tota laterum GD moles in <lb/>&longs;patium CK transferatur, & corruat. </s>

<s id="id.001173">Si autem cunei ip&longs;i latere&longs;ue, cuneatim di&longs;po&longs;iti, ita <lb/>&longs;int vt ad vnum centrum tendant, licet ad rectam lineam <lb/>collocentur, non delabentur, &longs;ed &longs;tabunt; quod ita o&longs;ten<lb/>demus. </s>

<s id="id.001174">Sint cunci latere&longs;ue <lb/>cuneatim di&longs;po&longs;iti ABCD, <lb/>tendentes ad centrum, &longs;eu <lb/>commune punctum E, Du<lb/>cantur CAE, DBE, &longs;intqueue <lb/>muri vtrinque ponderire&longs;i<lb/>&longs;tentes CL, DM, Demitta<lb/>tur perpendicularis, quæ ad <lb/>mundi centrum FGE &longs;ecans AB, in G. </s>

<s id="id.001174">Sint cunei latere&longs;ue <lb/>cuneatim di&longs;po&longs;iti ABCD, <lb/>tendentes ad centrum, &longs;eu <lb/>commune punctum E, Du<lb/>cantur CAE, DBE, &longs;intqueue <lb/>muri vtrinque ponderi re&longs;i<lb/>&longs;tentes CL, DM, Demitta<lb/>tur perpendicularis, quæ ad <lb/>mundi centrum FGE &longs;ecans AB, in G. </s>

<s id="id.001175">Tum fiat GK aequa<lb/>lis GF & per K ip &longs;i AGB parallela ducatur, HKI claudens <lb/>&longs;patium AHIB. </s>

<s id="id.001176">Quoniam igitur vt EC, ad EA, ita CD ad <lb/>AB per 4. propo&longs;. </s>

<s id="id.001176">Quoniam igitur vt EC, ad EA, ita CD ad <lb/>AB per 4. propo&longs;. lib. 6. maior erit CD ip&longs;a AB, & eâdem <lb/>de cau&longs;&longs;a maior AB, ip&longs;a HI, & idcirco maius ABDC &longs;pa<lb/>tium, &longs;patio AHIB. </s>

<s id="id.001177">lib.

6. maior erit CD ip&longs;a AB, & eâdem <lb/>de cau&longs;&longs;a maior AB, ip&longs;a HI, & idcirco maius ABDC &longs;pa<lb/>tium, &longs;patio AHIB. </s>

<s id="id.001178">Non igitur pote&longs;t linea CD, fieri in <lb/>AB, neque AB, in HI, neque &longs;patium totum CABD, tran&longs;<lb/>ferri in &longs;patium AHIB non data (quod naturæ ip&longs;i repu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.125.jpg"/>gnat) corporum penetratione. </s>

<s id="id.001178">Non igitur pote&longs;t linea CD, fieri in <lb/>AB, neque AB, in HI, neque &longs;patium totum CABD, tran&longs;<lb/>ferri in &longs;patium AHIB non data (quod naturæ ip&longs;i repu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.124.jpg"/>gnat) corporum penetratione. </s>

<s id="id.001179">Stabunt ergo cunei, quod <lb/>fuerat demon&longs;trandum. </s>

<s id="id.001180">Verumenimuero, debilis hæc &longs;tructura e&longs;t, & eo de<lb/>bilior, quo vani latitudo fucrit maior, cuneorum verò al<lb/>titudo minor. </s>

<s id="id.001180">Verum enimuero, debilis hæc &longs;tructura e&longs;t, & eo de<lb/>bilior, quo vani latitudo fuerit maior, cuneorum verò al<lb/>titudo minor. </s>

<s id="id.001181">Idem enim patitur quod epi&longs;tylia in &longs;pecie <lb/>Aræos&longs;tyla, quæ, vt&longs;cribit Vitruuius lib.

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<s id="id.001182">Id quoque ha<lb/>bet vitij, quod cunei ita di&longs;po&longs;iti &longs;uo pondere in cumbas <lb/>vtrinque violenti&longs;&longs;imè pellant. </s>

<s id="id.001183">Vtilis tamen e&longs;&longs;c pote&longs;t <lb/>ad portarum & fene&longs;trarum, quæ in medijs muris &longs;unt, & <lb/>mediocri vano aperiuntur, &longs;uperliminaria. </s>

<s id="id.001183">Vtilis tamen e&longs;&longs;e pote&longs;t <lb/>ad portarum & fene&longs;trarum, quæ in medijs muris &longs;unt, & <lb/>mediocri vano aperiuntur, &longs;uperliminaria. </s>

<s id="id.001184">Si verò ad minorem circuli portionem curuetur Ca<lb/>mera, vtilior quidem erit &longs;tructura ea ip&longs;a, de qua locuti <lb/>&longs;umus; non tamen omninò &longs;ine vitio. </s>

<s id="id.001185">E&longs;to fornix ex minori <lb/>circuli portione AB, cuius in<lb/>cumbæ AF, BH muris fultæ <lb/>AC, BD. </s>

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<s id="id.001192">Producatur in eadem figura CA in K, & DB in L. <lb/></s>

<s id="id.001193">Partes igitur quæ muris ad perpendiculum fulciuntur, <lb/>&longs;unt AKF, BLH, minimæ illæ quidem, maxima verò pars <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.126.jpg"/>e&longs;t extra fulcimenta, nempetota AKLB quæ idcircó &longs;uo<lb/>pte pondere deor&longs;um vergens & in incumbas <expan abbr="vtrinq;">vtrinque</expan> pel<lb/>lens aperitur, & facillimè vitium facit. </s>

<s id="id.001193">Partes igitur quæ muris ad perpendiculum fulciuntur, <lb/>&longs;unt AKF, BLH, minimæ illæ quidem, maxima verò pars <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.125.jpg"/>e&longs;t extra fulcimenta, nempe tota AKLB quæ idcircó &longs;uo<lb/>pte pondere deor&longs;um vergens & in incumbas <expan abbr="vtrinq;">vtrinque</expan> pel<lb/>lens aperitur, & facillimè vitium facit. </s>

<s id="id.001194">Eiu&longs;dem ferè na<lb/>turæ ea &longs;pecies e&longs;t, quæ vel ex media, vel ex minori ellip&longs;is <lb/>&longs;ecundum maiorem diametrum fit &longs;egmento. </s>

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<s id="id.001196">circa medium autem pote&longs;t videri debilior, <lb/>quippe quod ellip&longs;isibi circulo curuetur minus. </s>

<s id="id.001197">Ea verò forma, qua mirum in modum delectati &longs;unt <lb/>Barbari, qui declinante imperio Italiam inua&longs;erunt, & <lb/>bonam emendati&longs;&longs;imamqueue antiquorum ædificandi ra<lb/>tionem deturparunt, ex duobus con&longs;tat circuli portioni<lb/>bus, quamobrem Albertus lib.

3. ho&longs;ce arcus, compo&longs;itos, <lb/>appellat. </s>

<s id="id.001198">Circinantur autem hoc pacto, diui&longs;a nempe <lb/>&longs;ubten&longs;a, in partes tres, ea&longs;que æquales, ponitur circini <lb/>pes in altero diui&longs;ionum puncto & pars circuli de&longs;cribi<lb/>tur, mox in altero puncto circini pede collocato alia cir<lb/>culi portio lineatur, quibus arcus ip&longs;e integratur. </s>

<s id="id.001199">Appel<lb/>lant autem tertium acutum, eo quod ex &longs;ubten&longs;a in tres <lb/>partes diui&longs;a, arcus non fiatrotundus, &longs;ed in acutum an<lb/>gulum ex duabus circuli portionibus de&longs;inens. </s>

<s id="id.001200">Sint igitur muri <lb/>AC, BD, in quibus v<lb/>trinque incumbæ KA, <lb/>BI. </s>

<s id="id.001201">Ducatur itaque &longs;ub<lb/>ten&longs;a horizonti æquidi<lb/>&longs;tans AP, quæ in tres æ<lb/>quales partes diuidatur <lb/>punctis E, F, tum centris <lb/>EF, circulorum portio<lb/>nes de&longs;cribantur hinc <lb/>AG, HK, inde verò BG, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.127.jpg"/>IH, ex quibus arcus totus integratur. </s>

<s id="id.001201">Ducatur itaque &longs;ub<lb/>ten&longs;a horizonti æquidi<lb/>&longs;tans AP, quæ in tres æ<lb/>quales partes diuidatur <lb/>punctis E, F, tum centris <lb/>EF, circulorum portio<lb/>nes de&longs;cribantur hinc <lb/>AG, HK, inde verò BG, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.126.jpg"/>IH, ex quibus arcus totus integratur. </s>

<s id="id.001202">Vtilis hæc quidem <lb/>&longs;pecies e&longs;t, licet inuenu&longs;ta, propterea quod haud violen<lb/>ter incumbas vtrinque repellat, & in &longs;ummo magnis &longs;u&longs;ti<lb/>nendis oneribus &longs;it apta. </s>

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<s id="id.001204">Quoniam igitur centra hæc ob elatam portionum <lb/>con&longs;titutionem quam proxima lineis AN, BO, fulcimen<lb/>torum fiunt, maximè <expan abbr="&longs;u&longs;tin&emacr;tur">&longs;u&longs;tinentur</expan>, & deor&longs;um potius quam <lb/>lateraliter in cumbas ip&longs;as premunt. </s>

<s id="id.001205">Si quid tamen <expan abbr="hab&emacr;t">habent</expan> <lb/>vitij, illud e&longs;t quod grauitatis centra momentum haben<lb/>tia ad interiorem partem ver&longs;us PQ vim faciant, & nifi <lb/>partes magno &longs;uperimpo&longs;ito pondere comprimantur, <lb/>partes quæ &longs;unt circa HG, &longs;ur&longs;um pellentes aliquali &longs;ibi <lb/>rectitudine comparata corruunt, facta nempe circa L, M, <lb/>coniunctarum partium &longs;eparatione. </s>

<s id="id.001205">Si quid tamen <expan abbr="hab&emacr;t">habent</expan> <lb/>vitij, illud e&longs;t quod grauitatis centra momentum haben<lb/>tia ad interiorem partem ver&longs;us PQ vim faciant, & ni&longs;i <lb/>partes magno &longs;uperimpo&longs;ito pondere comprimantur, <lb/>partes quæ &longs;unt circa HG, &longs;ur&longs;um pellentes aliquali &longs;ibi <lb/>rectitudine comparata corruunt, facta nempe circa L, M, <lb/>coniunctarum partium &longs;eparatione. </s>

<s id="id.001206">His hoc pacto explicatis de &longs;emicirculari fornice a<lb/>gemus, quæ cæteris omnibus vtilior e&longs;t, & longè pulcher<lb/>rima, quamobrem Antiquis Architectis omnibus inpri<lb/>mis admodum familiaris: </s>

<s id="id.001207">E&longs;to vanum <lb/>ABCD, muris v<lb/>trinque clau&longs;um. <lb/></s>

<s id="id.001208">Ducatur per <expan abbr="s&umacr;-mitates">sun<lb/>mitates</expan> <expan abbr="muror&umacr;">murorum</expan> <lb/>horizonti æqui<lb/>di&longs;tans recta AD, <lb/>hac bifariam &longs;e<lb/>cta in E, eodem <lb/>centro E, &longs;patio <lb/>verò EA &longs;emicir<lb/>culus de&longs;cribatur <lb/>AFD, concaua <lb/>nempeip&longs;ius for-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.128.jpg"/>nicis pars; tum eodem centro, &longs;patio verò EG, circinetur <lb/>GHI eiu&longs;dem fornicis pars conuexa. </s>

<s id="id.001208">Ducatur per <expan abbr="s&umacr;-mitates">sum<lb/>mitates</expan> <expan abbr="muror&umacr;">murorum</expan> <lb/>horizonti æqui<lb/>di&longs;tans recta AD, <lb/>hac bifariam &longs;e<lb/>cta in E, eodem <lb/>centro E, &longs;patio <lb/>verò EA &longs;emicir<lb/>culus de&longs;cribatur <lb/>AFD, concaua <lb/>nempe ip&longs;ius for-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.127.jpg"/>nicis pars; tum eodem centro, &longs;patio verò EG, circinetur <lb/>GHI eiu&longs;dem fornicis pars conuexa. </s>

<s id="id.001209">Po&longs;t hæc productis <lb/>lineis BH, CD, in OP, &longs;ecetur fornix tota in tres æquales <lb/>partes AGKM, MNLK, NDIL, & KME, LNE iungantur, <lb/>&longs;int autem partium ip&longs;arum grauitatis centra QRS. </s>

<s id="id.001210">E&longs;t <lb/>autem R in ip&longs;a perpendiculari HE. </s>

<s id="id.001211">Quoniam igitut <lb/>partium AGKM, DILN, quæ <expan abbr="vtrinq;">vtrinque</expan> &longs;unt grauitatis cen<lb/>tra QS, in ip&longs;is &longs;unt fulcimentorum lineis OH PD, &longs;uâ <lb/>&longs;ponte fulcimentis eas &longs;u&longs;tinentibus partes ip&longs;æ &longs;tabunt. <lb/></s>

<s id="id.001211">Quoniam igitur <lb/>partium AGKM, DILN, quæ <expan abbr="vtrinq;">vtrinque</expan> &longs;unt grauitatis cen<lb/>tra QS, in ip&longs;is &longs;unt fulcimentorum lineis OH PD, &longs;uâ <lb/>&longs;ponte fulcimentis eas &longs;u&longs;tinentibus partes ip&longs;æ &longs;tabunt. <lb/></s>

<s id="id.001212">Pars autem media KMNL deor&longs;um vergente perip&longs;am <lb/>HE lineam grauitatis centro, &longs;i parumper vel incumbæ <lb/>vel partes vtrinque AG<emph type="italics"/>K<emph.end type="italics"/>M, DILN cedant, vtpote quæ à <lb/>fulcimentis e&longs;t remoti&longs;&longs;ima, magno impetu &longs;uopte pon<lb/>dere deor&longs;um feretur. </s>

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<s id="id.001218">Præterea & ferrea claui ex incumba <lb/>in incumbam ducta & vtrinque firmata contrarias partes <lb/>validi&longs;&longs;imè connectunt, quæ calcitrantes (ita enim lo<lb/>quuntur no&longs;trates <emph type="italics"/>A<emph.end type="italics"/>rchitecti,) fornicis pedes cohibent, & <lb/>&longs;olidum ne &longs;oluatur impediunt. </s>

<s id="id.001219">qua in &longs;pecie dubitan <expan abbr="d&umacr;">dum</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.129.jpg"/>e&longs;&longs;et, an optimo loco &longs;it a &longs;it clauis, quæ per centrum? </s>

<s id="id.001219">qua in &longs;pecie dubitan<expan abbr="d&umacr;">dum</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.128.jpg"/>e&longs;&longs;et, an optimo loco &longs;it a &longs;it clauis, quæ per centrum? </s>

<s id="id.001220">Et <lb/>&longs;anè videtur, quippe quod circa incumbas impetus fiat <lb/>maior. </s>

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<s id="id.001224">Tertio denique modo huic infirmitati me dentur, <lb/>vt videre e&longs;t in &longs;equenti figura, in qua vanum ADBC, mu<lb/>ri vtrinque AF, BH, fornix verò FGH. </s>

<s id="id.001225">Itaque dum muros <lb/><figure id="id.007.00.129.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.129.1.jpg"></figure><lb/>ex&longs;truunt, arre<lb/>ctarias trabes, ro<lb/>bore aliaue mate<lb/>ria firmi&longs;&longs;ima, illis <lb/>in&longs;erunt, quales <lb/>&longs;unt IF<emph type="italics"/>K<emph.end type="italics"/> LHM, <lb/>ea proceritate vt <lb/>futuri fornicis &longs;u<lb/>perent &longs;ummita<lb/>tem. </s>

<s id="id.001225">Itaque dum muros <lb/><figure id="id.007.01.128.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.128.1.jpg"></figure><lb/>ex&longs;truunt, arre<lb/>ctarias trabes, ro<lb/>bore aliaue mate<lb/>ria firmi&longs;&longs;ima, illis <lb/>in&longs;erunt, quales <lb/>&longs;unt IF<emph type="italics"/>K<emph.end type="italics"/> LHM, <lb/>ea proceritate vt <lb/>futuri fornicis &longs;u<lb/>perent &longs;ummita<lb/>tem. </s>

<s id="id.001226">Con&longs;umma<lb/>to enim fornice, <lb/>nondum tamen, <lb/>exarmato, tran&longs;<lb/>uer&longs;ariam <expan abbr="trab&emacr;">trabem</expan> à <lb/>&longs;ummo fornicis <lb/>dor&longs;o parumper <lb/>eminentem in punctis I, L, arrectarijs trabibus validi&longs;&longs;i<lb/>mis clauibus connectunt, tum punctis NP, Oq, capreolos <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.130.jpg"/>tran&longs;uer&longs;ario, & arrectarijs ferreis, clauis affigunt. </s>

<s id="id.001226">Con&longs;umma<lb/>to enim fornice, <lb/>nondum tamen, <lb/>exarmato, tran&longs;<lb/>uer&longs;ariam <expan abbr="trab&emacr;">trabem</expan> à <lb/>&longs;ummo fornicis <lb/>dor&longs;o parumper <lb/>eminentem in punctis I, L, arrectarijs trabibus validi&longs;&longs;i<lb/>mis clauibus connectunt, tum punctis NP, Oq, capreolos <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.129.jpg"/>tran&longs;uer&longs;ario, & arrectarijs ferreis, clauis affigunt. </s>

<s id="id.001227">Qui<lb/>bus ita concinnatis, facta fornicis validâ pre&longs;&longs;ione in G, <lb/>in cumbi&longs;queue F, H, ad exteriora repul&longs;is, AB &longs;patium non <lb/>fit maius. </s>

<s id="id.001227">Qui<lb/>bus ita concinnatis, facta fornicis validâ pre&longs;&longs;ione in G, <lb/>in cumbi&longs;que F, H, ad exteriora repul&longs;is, AB &longs;patium non <lb/>fit maius. </s>

<s id="id.001228">Repul&longs;is enim in cumbis & muros propelli ne<lb/>ce&longs;&longs;e e&longs;t, & cum muris ip&longs;as in&longs;ertas trabes, I<emph type="italics"/>K<emph.end type="italics"/>, LM. </s>

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<s id="id.001232">Cum <lb/>enim PQ proxima &longs;int punctis FH, quæ cum arrectarijs à <lb/>muro di&longs;tinentur, magna in ijs fit vtrobique re&longs;i&longs;tentia. </s>

<s id="id.001233">Rebus igitur ita &longs;e habentibus cum ob&longs;erua&longs;&longs;ent Ar<lb/>chitecti, ob enormitatem ponderis fornices in tertia illa <lb/><figure id="id.007.00.130.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.130.1.jpg"></figure><lb/>parte quæ &longs;umma e&longs;t <lb/>laborare, <expan abbr="quãtum">quantum</expan> ter<lb/>tijs vtrinque partibus <lb/>&longs;oliditatis addunt, tan<lb/>tundem ex illa parte <lb/>&longs;uprema demere <expan abbr="&longs;ol&emacr;t">&longs;olent</expan>, <lb/>vt videre e&longs;t in &longs;ubie<lb/>cta figura, in qua par<lb/>tes A, B, &longs;olidæ & cra&longs;<lb/>&longs;iores, quibus hærent <lb/>partes, quæ CE, DG <lb/>cra&longs;&longs;æ quidem & illæ, <lb/>tum vero &longs;umma EFG, <lb/>alijs &longs;ubtilior. </s>

<s id="id.001233">Rebus igitur ita &longs;e habentibus cum ob&longs;erua&longs;&longs;ent Ar<lb/>chitecti, ob enormitatem ponderis fornices in tertia illa <lb/><figure id="id.007.01.129.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.129.1.jpg"></figure><lb/>parte quæ &longs;umma e&longs;t <lb/>laborare, <expan abbr="quãtum">quantum</expan> ter<lb/>tijs vtrinque partibus <lb/>&longs;oliditatis addunt, tan<lb/>tundem ex illa parte <lb/>&longs;uprema demere <expan abbr="&longs;ol&emacr;t">&longs;olent</expan>, <lb/>vt videre e&longs;t in &longs;ubie<lb/>cta figura, in qua par<lb/>tes A, B, &longs;olidæ & cra&longs;<lb/>&longs;iores, quibus hærent <lb/>partes, quæ CE, DG <lb/>cra&longs;&longs;æ quidem & illæ, <lb/>tum vero &longs;umma EFG, <lb/>alijs &longs;ubtilior. </s>

<s id="id.001234">Minus <lb/>igitur grauante ponde<lb/>re in F, minor fit ad in cumbas pre&longs;&longs;io, aut &longs;i qua fit, à <expan abbr="parti&umacr;">partium</expan> <lb/>ACE, BDG &longs;oliditate haud inualidè &longs;u&longs;tinetur. </s>

<s id="id.001235">Cæterùm admonet nos locus, vt aliquid de forni<lb/>cum di&longs;&longs;olutionibus in medium afferamus: cau&longs;&longs;is enim <lb/>morborum cognitis, facilius periti medici adhibere &longs;o<lb/>lent remedia. </s>

<s id="id.001236">E&longs;to enim &longs;emicircula<lb/>ris fornix ABC, cuius cen<lb/>trum E, perpendicularis ve<lb/>rò quæ per centrum DBE, &longs;e<lb/>micirculi ABC, diameter <lb/>AEC, incumbæ <expan abbr="vtrinq;">vtrinque</expan> A, C. <lb/></s>

<s id="id.001237">Itaque &longs;i nulla fiat incumba<lb/>rum repul&longs;io, &longs;tabit fornix; &longs;i verò fiat, ruinam facict. </s>

<s id="id.001237">Itaque &longs;i nulla fiat incumba<lb/>rum repul&longs;io, &longs;tabit fornix; &longs;i verò fiat, ruinam faciet. </s>

<s id="id.001238">Pellanturitaque ad exteriores partes, vt in &longs;ecunda <lb/><figure id="id.007.00.131.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.131.2.jpg"></figure><lb/>figura, H in F, & C in G, <lb/>ex qua pul&longs;ione cum ma<lb/>ius fiat &longs;patium quod in<lb/>tegro fornice impleba<lb/>tur, iam di&longs;tractis <expan abbr="vtrinq;">vtrinque</expan> <lb/>fornicis partibus <expan abbr="n&omacr;">non</expan> im<lb/>pletur, Diuiditur igitur <lb/>locus maior factus in tres partes, quarum hincinde duas <lb/>replent fornicis partes, tertiam verò quæ media e&longs;t, re<lb/>plet in&longs;ertus, ne vacuum detur, aër, vt in figura videre e&longs;t, <lb/>in qua &longs;olutæ vtrinque fornicis partes HIKF, PMNG, aër <lb/>autem medius &longs;patium replens IKMN. </s>

<s id="id.001238">Pellantur itaque ad exteriores partes, vt in &longs;ecunda <lb/><figure id="id.007.01.130.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.130.2.jpg"></figure><lb/>figura, H in F, & C in G, <lb/>ex qua pul&longs;ione cum ma<lb/>ius fiat &longs;patium quod in<lb/>tegro fornice impleba<lb/>tur, iam di&longs;tractis <expan abbr="vtrinq;">vtrinque</expan> <lb/>fornicis partibus <expan abbr="n&omacr;">non</expan> im<lb/>pletur, Diuiditur igitur <lb/>locus maior factus in tres partes, quarum hinc inde duas <lb/>replent fornicis partes, tertiam verò quæ media e&longs;t, re<lb/>plet in&longs;ertus, ne vacuum detur, aër, vt in figura videre e&longs;t, <lb/>in qua &longs;olutæ vtrinque fornicis partes HIKF, PMNG, aër <lb/>autem medius &longs;patium replens IKMN. </s>

<s id="id.001239">Diuidantur &longs;in<lb/>guli quadrantes FK, GN, in partes tres, quarum duæ &longs;int <lb/>hincinde FQ, GR, & à centris, quæ &longs;eparatis quadranti<lb/>bus facta &longs;untin ST, rectæ ducantur SQV. TRX. </s>

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<s id="id.001241">Ducantur autem re<lb/>ctæ QI, RM, con&longs;tituentes cum ip&longs;is QV, RX pares an<lb/>gulos VQI MRX. </s>

<s id="id.001242">Itaque centris QR partes QIRM ad <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.132.jpg"/>inferiores partes deuoluentur, fientqueue QI, RM, vbi QZ, <lb/>RZ. </s>

<s id="id.001242">Itaque centris QR partes QIRM ad <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.131.jpg"/>inferiores partes deuoluentur, fientqueue QI, RM, vbi QZ, <lb/>RZ. </s>

<s id="id.001243">Siautem QI, RM perpendicularibus quæ à punctis <lb/>QR ad perpendicularem DE ducuntur, fuerint maiores <lb/>conuenient alicubi in ip&longs;a perpendiculari, & altera alte<lb/>ram &longs;u&longs;tinebit; &longs;i autem æquales tangent &longs;e & nihilomi<lb/>nus fiet ruina, &longs;i minores nec &longs;e inuicem tangent, & nullà <lb/>re prohibente deor&longs;um corruent. </s>

<s id="id.001244">tangant tautem &longs;e in <expan abbr="p&umacr;-cto">pun<lb/>cto</expan> Z. quo pacto igitur fornices incumbis cedentibus in <lb/>medio aperti, <expan abbr="di&longs;&longs;oluãtur">di&longs;&longs;oluantur</expan> & ruinam faciant, ex i&longs;tis patet. </s>

<s id="id.001244">tangant autem &longs;e in <expan abbr="p&umacr;-cto">pun<lb/>cto</expan> Z. quo pacto igitur fornices incumbis cedentibus in <lb/>medio aperti, <expan abbr="di&longs;&longs;oluãtur">di&longs;&longs;oluantur</expan> & ruinam faciant, ex i&longs;tis patet. </s>

<s id="id.001245">Ex demon&longs;tratis qua&longs;i ex con&longs;ectario habemus for<lb/>nices quo fuerint cra&longs;&longs;iores dato pari incumbarum &longs;ece&longs;<lb/>&longs;u, ruinæ minus e&longs;&longs;e obnoxios quàm tenuiores, hoc e&longs;t, <lb/>maiori aperitione indigere ad ruinam cra&longs;&longs;iores quam te<lb/>nuiores, quod licet ex iam dictis re&longs;ultet, nos tamen cla<lb/>rius ex &longs;ubiecto &longs;chemate demon&longs;trabimus. </s>

<s id="id.001246">E&longs;to enim cra&longs;&longs;ioris <lb/>fornicis pars <expan abbr="quid&emacr;">quidem</expan> ABCD, <lb/>tenuioris EFCD circa <expan abbr="id&emacr;">idem</expan> <lb/>centrum R. </s>

<s id="id.001247">Ducatur au<lb/>tem RM, &longs;ecans CD in G. <lb/>EF in H AB, in M. </s>

<s id="id.001248">Centro <lb/>igitur G fiet euer&longs;io portio<lb/>num fornicum, MD, HD, <lb/>Ducantur GA, GE & producta AD in N ip&longs;i AN perpen<lb/>dicularis ducatur GN. quoniam igitur GE cadit in trian<lb/>gulo AGN erit ex 21. propo&longs;. </s>

<s id="id.001249">lib.

1. elem. </s>

<s id="id.001250">GA, maior GE. <lb/></s>

<s id="id.001251">Corruente igitur maioris fornicis portione MD, recta <lb/>GA centro G punctum A de&longs;cribet portionem AI, mino<lb/>ris interim ex GE, de&longs;cribente EL, at cadenti angulo A <lb/>occurrit in perpendiculari IK in puncto I angulus oppo<lb/>&longs;itæ portionis, O, ip&longs;i autem E cadenti per EL non occur<lb/>ret punctum P, cadens per Pq eo quod neutrum eorum <lb/>pertingat ad perpendicularem IK. Tenuioris ergo forni<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.133.jpg"/>cis partes è &longs;uis locis auul&longs;æ ex eadem aperitione ruinam <lb/>facient, quod non contingit partibus cra&longs;&longs;ioris. </s>

<s id="id.001251">Corruente igitur maioris fornicis portione MD, recta <lb/>GA centro G punctum A de&longs;cribet portionem AI, mino<lb/>ris interim ex GE, de&longs;cribente EL, at cadenti angulo A <lb/>occurrit in perpendiculari IK in puncto I angulus oppo<lb/>&longs;itæ portionis, O, ip&longs;i autem E cadenti per EL non occur<lb/>ret punctum P, cadens per Pq eo quod neutrum eorum <lb/>pertingat ad perpendicularem IK. Tenuioris ergo forni<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.132.jpg"/>cis partes è &longs;uis locis auul&longs;æ ex eadem aperitione ruinam <lb/>facient, quod non contingit partibus cra&longs;&longs;ioris. </s>

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<s id="id.001253">Quæritur adhuc, quare grauiores fornices in &longs;um<lb/>mis ædificijs non &longs;ine vitio fiant? </s>

<s id="id.001254">E&longs;to ædificium ABGH, cuius <expan abbr="vtrinq;">vtrinque</expan> muri ABCD, <lb/>EFGH, maiorum &longs;ummitates AD, EH, mediæ murorum <lb/>partes KL, fornicum &longs;ummus quidem DIE, medius verò <lb/><figure id="id.007.00.133.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.133.1.jpg"></figure><lb/>KML. Dico, magis cedere pul<lb/>&longs;os muros &longs;ummos circa DE, <lb/>quam in medio circa KL. </s>

<s id="id.001254">E&longs;to ædificium ABGH, cuius <expan abbr="vtrinq;">vtrinque</expan> muri ABCD, <lb/>EFGH, maiorum &longs;ummitates AD, EH, mediæ murorum <lb/>partes KL, fornicum &longs;ummus quidem DIE, medius verò <lb/><figure id="id.007.01.132.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.132.1.jpg"></figure><lb/>KML. Dico, magis cedere pul<lb/>&longs;os muros &longs;ummos circa DE, <lb/>quam in medio circa KL. </s>

<s id="id.001255">Sunt <lb/>enim muri BA, GH ceu vectes <lb/>quidam, <expan abbr="quor&umacr;">quorum</expan> extremis par<lb/>tibus à fulcimentis BG remo<lb/>ti&longs;&longs;imis potentia admouetur, <lb/>hoc e&longs;t, ip&longs;ius fornicis DIE ad <lb/>DE in cumbans repul&longs;io; lon<lb/>gior e&longs;t autem pars à <expan abbr="fulcim&emacr;-to">fulcimen<lb/>to</expan> ad potentiam AB, ip&longs;a BK. <lb/></s>

<s id="id.001256">Data igitur paritate potentia<lb/>rum plus operabitur ea quæ in <lb/>D, illa quæ K. facilius crgo re<lb/>pellentur muri in DE quàm in <lb/>KL. </s>

<s id="id.001256">Data igitur paritate potentia<lb/>rum plus operabitur ea quæ in <lb/>D, illa quæ K. facilius ergo re<lb/>pellentur muri in DE quàm in <lb/>KL. </s>

<s id="id.001257">Alia quo que ratio intercedit, &longs;iquidem pondus muri <lb/>&longs;uperioris ADK, premens inferiorem murum KBC, cum <lb/>&longs;ua grauitate firmiorem, & pul&longs;ionibus minus obnoxium <lb/>reddit. </s>

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<s id="id.001261">In parua re magnum negotium. </s>

<s id="id.001262">Etenim quæ&longs;tio hæc <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.134.jpg"/>clari&longs;&longs;imorum virorum ingenia magnopere fatigauit. </s>

<s id="id.001262">Etenim quæ&longs;tio hæc <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.133.jpg"/>clari&longs;&longs;imorum virorum ingenia magnopere fatigauit. </s>

<s id="id.001263">Ex <lb/>quibus Ari&longs;toteles inter veteres, Guid. Vbald. inter re<lb/>centiores ad vectis naturam (ne quid in Mechanicis ad <lb/>vectem non reduci putaretur) cuneum ip&longs;um trahere co<lb/><figure id="id.007.00.134.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.134.1.jpg"></figure><lb/>nati &longs;unt. </s>

<s id="id.001263">Ex <lb/>quibus Ari&longs;toteles inter veteres, Guid. Vbald. inter re<lb/>centiores ad vectis naturam (ne quid in Mechanicis ad <lb/>vectem non reduci putaretur) cuneum ip&longs;um trahere co<lb/><figure id="id.007.01.133.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.133.1.jpg"></figure><lb/>nati &longs;unt. </s>

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<s id="id.001266">E&longs;to igitur &longs;cindendum quippiam ABCD, Cuneus <lb/>EFG, cuius pars HFI &longs;ci&longs;&longs;uræ in&longs;erta HI, facta igitur vali<lb/>da percu&longs;&longs;ione in EG, fiet vt cum EG fuerit in NO, H &longs;it v<lb/>bi N, A vbi P, itemque I vbi O, D verò vbi Q & facta erit <lb/>&longs;ci&longs;&longs;io NSO, toti nempe cuneo EFG, æqualis. </s>

<s id="id.001267">Vultigitur <lb/>Ari&longs;toteles, duos in cunco vectes con&longs;iderari EF, GF, quo<lb/>rum alterius, nempe EF, fulcimentum &longs;it in H, pondus ve<lb/>ro in F; alterius autem, hoc e&longs;t, GF fulcimentum quidem <lb/>&longs;it in I, pondus verò itidem &longs;it in F. </s>

<s id="id.001267">Vult igitur <lb/>Ari&longs;toteles, duos in cuneo vectes con&longs;iderari EF, GF, quo<lb/>rum alterius, nempe EF, fulcimentum &longs;it in H, pondus ve<lb/>ro in F; alterius autem, hoc e&longs;t, GF fulcimentum quidem <lb/>&longs;it in I, pondus verò itidem &longs;it in F. </s>

<s id="id.001268">His nequaquam con<lb/>&longs;entiens G. Vbald. aliam viam ingreditur. </s>

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<s id="id.001269">Ait enim EHF <lb/>vectes quidem e&longs;&longs;e, quorum commune fulcimentum F, <lb/>potentias verò mouentes in EG. </s>

<s id="id.001270">Pondera vtrinque inter <lb/>fulcimenta & potentias, vbi HI, idemque; e&longs;&longs;e ac &longs;i EF, GF, <lb/><gap/>eor&longs;um à cuneo con&longs;iderati in puncto F, adinuicem fulti <lb/>atque di&longs;tracti pondera pellerent H in NP, I verò in O, <lb/><expan abbr="q.">que</expan> Verumenimuerò quoniam cunei angulus non muta<lb/>tur, nec vertex ip&longs;e centri vllum pror&longs;us præbet v&longs;um, nec <lb/>eius latera vtrinque di&longs;tracta ad contrarias partes didu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.135.jpg"/>cuntur, vectes in cuneo hoc pacto con&longs;iderare videtur à <lb/>veritate alienum. </s>

<s id="id.001270">Pondera vtrinque inter <lb/>fulcimenta & potentias, vbi HI, idemque; e&longs;&longs;e ac &longs;i EF, GF, <lb/>&longs;eor&longs;um à cuneo con&longs;iderati in puncto F, adinuicem fulti <lb/>atque di&longs;tracti pondera pellerent H in NP, I verò in O, <lb/><expan abbr="q.">que</expan> Verum enimuerò quoniam cunei angulus non muta<lb/>tur, nec vertex ip&longs;e centri vllum pror&longs;us præbet v&longs;um, nec <lb/>eius latera vtrinque di&longs;tracta ad contrarias partes didu<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.134.jpg"/>cuntur, vectes in cuneo hoc pacto con&longs;iderare videtur à <lb/>veritate alienum. </s>

<s id="id.001271">Ari&longs;totelis autem &longs;olutionem fal&longs;am e&longs;<lb/>&longs;e, clarè patet. </s>

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<s id="id.001274">At eiu&longs;modi operationes per contactum fieri nulli e&longs;t i<lb/>gnotum. </s>

<s id="id.001275">Solutio igitur i&longs;ta mco iudicio, tanto Philo&longs;o<lb/>pho pror&longs;us videtur indigna. </s>

<s id="id.001275">Solutio igitur i&longs;ta meo iudicio, tanto Philo&longs;o<lb/>pho pror&longs;us videtur indigna. </s>

<s id="id.001276">Porrò G. Vbald. ijs quæ de diuaricatis vectibus in <lb/>medium adduxerat non acquie&longs;cens alias quærit cau&longs;&longs;as, <lb/>cur cuneus minoris anguli validiùs &longs;cindat. </s>

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<s id="id.001277">Idque; ex quo<lb/>dam lemmate demon&longs;trare conatur, figura autem eius ita <lb/>ferè &longs;e habet. </s>

<s id="id.001278">E&longs;to cuneus ABC, <lb/>item alius DEF. <expan abbr="Dem&omacr;-&longs;trauit">Demon<lb/>&longs;trauit</expan> igitur ex a&longs;&longs;um<lb/>pto, quo acutior fuerit <lb/>angulus BIM, eo facilius <lb/>pondera moueri, & ideo <lb/>facilius ceu vecte AB <lb/>moueri pondus I quàm <lb/>vecte DE pondus <expan abbr="q.">que</expan> In<lb/>geniosè quidem. </s>

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<s id="id.001280">Si e<lb/>nim ita &longs;e habet AB, ad BI, vt DE, ad EQ (ip&longs;æ enim DE, <lb/>EQ &longs;upponuntur æquales) ergo eadem æquali&longs;ue poten<lb/>tia æqualiter mouebit pondera I & <expan abbr="q.">que</expan> quod ip&longs;i eiu&longs;dem <lb/>demon&longs;trationi pror&longs;us concludit contrarium. </s>

<s id="id.001281">Nec meo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.136.jpg"/>quidem iudicio id &longs;equi videtur, propterea quod ex Pap<lb/>po ea quæ in planis inclinatis mouentur, redigantur ad li<lb/>bram. </s>

<s id="id.001281">Nec meo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.135.jpg"/>quidem iudicio id &longs;equi videtur, propterea quod ex Pap<lb/>po ea quæ in planis inclinatis mouentur, redigantur ad li<lb/>bram. </s>

<s id="id.001282">Ratio enim valde e&longs;t diuer&longs;a, &longs;iquidem pondera <lb/>quæ in planis inclinatis mouentur, certa habent fulci<lb/>menta & determinatas tum brachiorum tum ponderum <lb/>proportiones, quæ omnia in cuneo, nec quidem mente <lb/>concipi po&longs;&longs;e, clarè patet. </s>

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<s id="id.001292">Tales <lb/>&longs;unt orbiculares illæ cu&longs;pides, quibus ad perforandum v<lb/>timur, & ideo vernaculè Pantirolos vocamus. </s>

<s id="id.001293">Acus item <lb/>&longs;utorij, & cætera quæ nen &longs;ecus ac linea in punctum de&longs;i<lb/>nunt, & imagina<gap/>iam quandam lineam ceu axem in eo <lb/>puncto de&longs;inen<gap/>em continent. </s>

<s id="id.001293">Acus item <lb/>&longs;utorij, & cætera quæ non &longs;ecus ac linea in punctum de&longs;i<lb/>nunt, & imaginariam quandam lineam ceu axem in eo <lb/>puncto de&longs;inentem continent. </s>

<s id="id.001294">Ad lineam quo que refe<lb/>runtur lateratæ cu&longs;pides oblongæ, & &longs;ubtiles ceu&longs;ubulæ, <lb/>claui, en&longs;es, pugiones, & his &longs;imilia, quæ cum adacta vali<lb/>dam faciant partium &longs;eparationem ad cunei naturam <expan abbr="n&omacr;">non</expan> <lb/>referre magnæ videretur dementiæ. </s>

<s id="id.001295">Ettunc quantoma<lb/>gis corpora hæc ad linearem naturam accedunt, eo ma<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.137.jpg"/>gis penetrant. </s>

<s id="id.001295">Et tunc quanto ma<lb/>gis corpora hæc ad linearem naturam accedunt, eo ma<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.136.jpg"/>gis penetrant. </s>

<s id="id.001296">Sed & hocidem in rebus non ab arte, &longs;ed <lb/>ab ip&longs;anatura productis facile e&longs;t cogno&longs;cere. </s>

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<s id="id.001299">Ve&longs;pæ quoque, Apes, Scorpiones a<lb/>culeis i&longs;tis ceu linearibus cuneis vtuntur. </s>

<s id="id.001300">Nec refert, vt <lb/>diximus, vt um laterati &longs;int, ceu &longs;ubulæ, & claui, vel ro<lb/>tundi & vtrum plura paucioraue latera habeant, dummo<lb/>do in punctum & aculeatam aciem de&longs;inant. </s>

<s id="id.001300">Nec refert, vt <lb/>diximus, vtrum laterati &longs;int, ceu &longs;ubulæ, & claui, vel ro<lb/>tundi & vtrum plura paucioraue latera habeant, dummo<lb/>do in punctum & aculeatam aciem de&longs;inant. </s>

<s id="id.001301">Altera por<lb/>ro cuneorum &longs;pecies &longs;uperficiei naturam &longs;apit, acie &longs;iqui<lb/>dem in lineam de&longs;init, quæ &longs;uperficiei e&longs;t terminus, <expan abbr="quã">quam</expan>obrem huc ea omnia referuntur, quæ acie ipsâ &longs;cindunt, <lb/>ceu &longs;unt cunei propriè dicti, de quibus hoc loco e&longs;t &longs;er<lb/>mo, cultra, en&longs;es, a&longs;ciæ, &longs;ecures, &longs;calpra lata, & cætera e<lb/>iu&longs;modi, quibus corpora acie &longs;cinduntur. </s>

<s id="id.001302">obrem huc ea omnia referuntur, quæ acie ipsâ &longs;cindunt, <lb/>ceu &longs;unt cunei propriè dicti, de quibus hoc loco e&longs;t &longs;er<lb/>mo, cultra, en&longs;es, a&longs;ciæ, &longs;ecures, &longs;calpra lata, & cætera e<lb/>in&longs;modi, quibus corpora acie &longs;cinduntur. </s>

<s id="id.001303">Quidam his ad<lb/>dunt &longs;erras, quibus haud pror&longs;us a&longs;&longs;entimur. </s>

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<s id="id.001306">Ex quibus o<lb/>mnibus, nî fallimur, clarè patet, curacutiores angulo cu<lb/>nei obtu&longs;ioribus facilius &longs;cindant, quæ quidem ratio lon<lb/>gè ab ea di&longs;tat, ex qua cæteri ferè omnes Cuneum ad ve<lb/>ctis naturam referre hactenus contenderunt. </s>

<s id="id.001307">Cæterùm vtramque eorum quos diximus, <expan abbr="cuneor&umacr;">cuneorum</expan> <lb/>&longs;peciem &longs;olerti&longs;&longs;ima cognouit Natura, & ideo quoniam <lb/>res vel contu&longs;ione vel perforatione, vel &longs;ecatione con&longs;i<lb/>ciuntur, triplicem dentium qualitatem dentatis animali-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.138.jpg"/><figure id="id.007.00.138.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.138.1.jpg"></figure><lb/>bus dedit, Molares, <lb/>qui & Maxillares ap<lb/>pellantur, quibus <lb/>cibus contunditur, <lb/>Canini, quibus fit <lb/>perforatio, Anterio<lb/>res, quibus cibus <lb/>&longs;cinditur, quos ideo <lb/><foreign lang="greek">temnikou\s</foreign>, id e&longs;t, &longs;ecan<lb/>tes appellant Graeci. </s>

<s id="id.001307">Cæterùm vtramque eorum quos diximus, <expan abbr="cuneor&umacr;">cuneorum</expan> <lb/>&longs;peciem &longs;olerti&longs;&longs;ima cognouit Natura, & ideo quoniam <lb/>res vel contu&longs;ione vel perforatione, vel &longs;ecatione confi<lb/>ciuntur, triplicem dentium qualitatem dentatis animali-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.137.jpg"/><figure id="id.007.01.137.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.137.1.jpg"></figure><lb/>bus dedit, Molares, <lb/>qui & Maxillares ap<lb/>pellantur, quibus <lb/>cibus contunditur, <lb/>Canini, quibus fit <lb/>perforatio, Anterio<lb/>res, quibus cibus <lb/>&longs;cinditur, quos ideo <lb/><foreign lang="greek">temnikou\s</foreign>, id e&longs;t, &longs;ecan<lb/>tes appellant Graeci. </s>

<s id="id.001308">Molares KK, <lb/>CaniniL, L, Temni<lb/>ci &longs;eu &longs;ecantes M. </s>

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<s id="id.001313">Multa hîc addere potui&longs;&longs;emus ad Cochleam &longs;pe<lb/>ctantia, quippe quòd Cochlea cuneus &longs;it Cylindro inuo<lb/>lutus, qui quidem ad mallei, &longs;ed vectis virtute &longs;ibi adiun<lb/>ctâ, validi&longs;&longs;imè operatur, & &longs;excentis in&longs;eruit v&longs;ibus. </s>

<s id="id.001314">Ve<lb/>runtamen cùm de hac &longs;pecie egregiè di&longs;&longs;erat G. Vbaldus, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.139.jpg"/>con&longs;ultò hanc di&longs;putationem omittimus; idque hac quo<lb/>que de cau&longs;&longs;a, quod nihil de cochlea, ac &longs;i eam non noui&longs;<lb/>&longs;et, locutus &longs;it Ari&longs;toteles. </s>

<s id="id.001314">Ve<lb/>runtamen cùm de hac &longs;pecie egregiè di&longs;&longs;erat G. Vbaldus, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.138.jpg"/>con&longs;ultò hanc di&longs;putationem omittimus; idque hac quo<lb/>que de cau&longs;&longs;a, quod nihil de cochlea, ac &longs;i eam non noui&longs;<lb/>&longs;et, locutus &longs;it Ari&longs;toteles. </s>

<s id="id.001315">Po&longs;&longs;umus autem in actu &longs;ci&longs;&longs;ionis, quæ cuneo fit, a<lb/>liâ tamen ratione vectem con&longs;iderare, nempe non in cu<lb/>neo quidem, &longs;ed in ip&longs;a re quæ &longs;cinditur. </s>

<s id="id.001316">E&longs;to enim quip<lb/>piam &longs;ci&longs;&longs;ile ABCD, <lb/>cui alteri extremita<lb/>tum, puta BD, cuneus <lb/>adigatur EFG, <expan abbr="fiatq;">fiatque</expan> <lb/>&longs;ci&longs;&longs;io per longitudi<lb/>nem &longs;ecundum <expan abbr="lineã">lineam</expan> <lb/>EH. facta igitur ex <lb/>cunei ingre&longs;&longs;u <expan abbr="parti&umacr;">partium</expan> &longs;eparatione B, expelletur in I, D ve<lb/>rò in K. fient igitur materiæ &longs;ci&longs;&longs;æ partes AIBH, CKDH, <lb/>ceu duo vectes, quorum hinc inde in corpore ip&longs;o fulci<lb/>menta L, M potentiæ vtrinque dilatantes BD, pondus ve<lb/>rò materiæ re&longs;i&longs;tentia, in &longs;eparationis loco vbi N. </s>

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<s id="id.001319">Pertingente enim &longs;ci&longs;&longs;ione in <lb/>O, <expan abbr="fulcim&emacr;tum">fulcimentum</expan> fit in P. quo ca&longs;u &longs;ci&longs;&longs;ura e&longs;t facilior, quip<lb/>pe quod maiorem habeat proportionem BO ad OP, <expan abbr="quã">quam</expan> <lb/>BN ad NL. </s>

<s id="id.001320">Hoc autem experiuntur materiarij, qui primis <lb/>ictibus, &longs;ecuriculâ nondum probè adactâ, & nondum fa<lb/>ctâ notabili &longs;ci&longs;&longs;ione difficultatem &longs;entiunt, mox <expan abbr="factaiã">facta iam</expan> <lb/>&longs;eparatione faoillima paullatim fit materiæ totius &longs;epara<lb/>tio. </s>

<s id="id.001320">Hoc autem experiuntur materiarij, qui primis <lb/>ictibus, &longs;ecuriculâ nondum probè adactâ, & nondum fa<lb/>ctâ notabili &longs;ci&longs;&longs;ione difficultatem &longs;entiunt, mox <expan abbr="factaiã">facta iam</expan> <lb/>&longs;eparatione facillima paullatim fit materiæ totius &longs;epara<lb/>tio. </s>

<s id="id.001321">Hocidem & nos ab&longs;que cunei v&longs;u experimur, cum ba<lb/>culum aut quippiam tale manibus diductis &longs;cin dimus. </s>

<s id="id.001322">à <lb/>principio enim difficultatem &longs;entimus, deinde ex ea <expan abbr="quã">quam</expan> <lb/>diximus proportionc &longs;ci&longs;&longs;io ip&longs;a fit apprime facilis. </s>

<s id="id.001322">à <lb/>principio enim difficultatem &longs;entimus, deinde ex ea <expan abbr="quã">quam</expan> <lb/>diximus proportione &longs;ci&longs;&longs;io ip&longs;a fit apprime facilis. </s>

<s id="id.001323">Vti-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.140.jpg"/>mur etiam vecte cuneato ad &longs;cindendum & aperiendum: <lb/>adacto enim &longs;ci&longs;&longs;uræ cuneo, idqueue manu malleoue, tum <lb/>ab altera extremitate pre&longs;&longs;o, valida fit ex vectis vi <expan abbr="c&omacr;tinui">continui</expan> <lb/><figure id="id.007.00.140.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.140.1.jpg"></figure><lb/>corporis &longs;eparatio. </s>

<s id="id.001323">Vti-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.139.jpg"/>mur etiam vecte cuneato ad &longs;cindendum & aperiendum: <lb/>adacto enim &longs;ci&longs;&longs;uræ cuneo, idqueue manu malleoue, tum <lb/>ab altera extremitate pre&longs;&longs;o, valida fit ex vectis vi <expan abbr="c&omacr;tinui">continui</expan> <lb/><figure id="id.007.01.139.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.139.1.jpg"></figure><lb/>corporis &longs;eparatio. </s>

<s id="id.001324">Ma<lb/>teria &longs;ci&longs;&longs;ilis AB <expan abbr="&longs;calpr&umacr;">&longs;calprum</expan> <lb/>ceu vectis cuneatus CD, <lb/>cuius fulcimentum, E, <lb/>pondus verò vbi C, po<lb/>tentia vbi D, quo ca&longs;u <lb/>quo maior e&longs;t proportio <lb/>DE ad EC, eo e&longs;t ip&longs;a &longs;ci&longs;&longs;io leuior & facilior. </s>

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<s id="id.001332">Ob&longs;euri&longs;&longs;ima expo&longs;itio, & nî res e&longs;&longs;et vulgò per &longs;e <lb/>nota, dequeue ea Vitruuius & Mechanici non egi&longs;&longs;ent, diffi<lb/>cile vtique e&longs;&longs;et ex eius verbis &longs;en&longs;um a&longs;&longs;equi. </s>

<s id="id.001333">Tigna &longs;anè voca&longs;&longs;e videtur ea ligna, quæ à Vitruuio <lb/>Rechami dicuntur, in quibus nempe ip&longs;i in&longs;eruntur orbi<lb/>culi. </s>

<s id="id.001334">Et&longs;i de tignis eiu&longs;mo di aliud quippiam &longs;entire videa<lb/>tur Picolomineus. </s>

<s id="id.001335">Græea lectio pro tignis habet <foreign lang="greek">cu/la</foreign>, id <lb/>e&longs;t, ligna; item vbi Leoniceni ver&longs;io legit, ad &longs;e inuicem <lb/>iunctis, textus habet <foreign lang="greek">snm<gap/>ai/nousin e(autoi_s e)ranti/ws</foreign>, hoc e&longs;t, in<lb/>uicem ex oppo&longs;ito concurrunt. </s>

<s id="id.001335">Græca lectio pro tignis habet <foreign lang="greek">cu/la</foreign>, id <lb/>e&longs;t, ligna; item vbi Leoniceni ver&longs;io legit, ad &longs;e inuicem <lb/>iunctis, textus habet <foreign lang="greek">sumbai/nousin e(autoi=s e)nanti/ws</foreign>, hoc e&longs;t, in<lb/>uicem ex oppo&longs;ito concurrunt. </s>

<s id="id.001336">Certè locum totum ita <lb/>redderem: Cur &longs;i quis duas Trochleas fecerit, in duobus <lb/>lignis &longs;ibi ex oppo&longs;ito concurrentibus, ei&longs;queue Trochleis <lb/>circumpo&longs;uerit funem, cuius alterum caput alteri ligno<lb/>rum &longs;it annexum, alterum verò Trochleis cohæreat, vel <lb/>apponatur. </s>

<s id="id.001337">Si quis alterum funis principium trahat, ma<lb/>gna trahat pondera, et&longs;i trahens potentia &longs;it exigua? </s>

<s id="id.001338">Nos <lb/>verbis figuram, & figurâ verba ip&longs;a elucidabimus. </s>

<s id="id.001339">Sint duo ligna ex oppo&longs;ito concurrentia, <lb/>in quibus Trochleæ, hoc e&longs;t, orbiculi AB, fu<lb/>nis ductarius DABC, cuius alterum caput re<lb/>ligatum e&longs;t ligno trochleæ A, vbi e&longs;t C. </s>

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<s id="id.001347">Si vnicâ Trochleâ plus trahitur <lb/>quàm manu, multo faci ius & velocius id fiet duobus, <lb/>quibus plus, vt ip&longs;e ait, quàm in duplici velocitate pon<lb/>dus leuabitur. </s>

<s id="id.001348">Summa dictorum e&longs;t, ex multiplicatione <lb/>orbiculorum pondus ip&longs;um imminui, & minori difficul-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.142.jpg"/>tate leuari, quod &longs;anè verum e&longs;t. </s>

<s id="id.001348">Summa dictorum e&longs;t, ex multiplicatione <lb/>orbiculorum pondus ip&longs;um imminui, & minori difficul-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.141.jpg"/>tate leuari, quod &longs;anè verum e&longs;t. </s>

<s id="id.001349">Nos tamen nonnulla <expan abbr="c&omacr;-&longs;iderabimus">con<lb/>&longs;iderabimus</expan>. </s>

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<s id="id.001353">de Vecte <lb/>prop.

<s id="id.001353">de Vecte <lb/>prop. 3. Po&longs;ita nempe inter fulcimentum & pondus &longs;u&longs;ti<lb/>nente potentiâ. </s>

3. Po&longs;ita nempe inter fulcimentum & pondus &longs;u&longs;ti<lb/>nente potentiâ. </s>

<s id="id.001354">Præterea quod a&longs;&longs;eruit Ari&longs;toteles, Tro<lb/>chleas ad vectem reduci, verum quidem e&longs;t, &longs;ed aptius di<lb/>xi&longs;&longs;et ad libram, etenim vectis vtcunque à fulcimento di<lb/>uiditur. </s>

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<s id="id.001357">Si enim, ait, <lb/>vnicâ Trochleâ pondus facile trahitur, id multo validius <lb/>pluribus fiet. </s>

<s id="id.001358">Veruntamen non ab&longs;olutè ex orbiculorum <lb/>multiplicationeid fieriita o&longs;tendemus. </s>

<s id="id.001359">Sint duæ op<lb/>po&longs;itæ lineae rectae, <lb/>vtpote trabes AB, <lb/>CD, <expan abbr="inuic&emacr;">inuicem</expan> æqui<lb/>di&longs;tantes & ip&longs;æ <lb/>&longs;tabiles: &longs;uperiori <lb/>tres appendantur <lb/>orbiculi ex <expan abbr="p&umacr;ctis">punctis</expan> <lb/>E, F, G, <expan abbr="n&emacr;pe">nempe</expan> ML, <lb/>PQ, TV. inferiori <lb/><expan abbr="aut&emacr;">autem</expan> duobus pun<lb/>ctis IH, nempe <lb/>NO, RS. </s>

<s id="id.001360">Erunti<lb/>gitur invniuer&longs;um <lb/>quinque, indatur pereos funis ductarius KLMNOP <lb/>QRSTVX, ex cuius extremitate pendeat pondus X, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.143.jpg"/>Trahatur funis in K. </s>

<s id="id.001360">Erunt i<lb/>gitur in vniuer&longs;um <lb/>quinque, indatur per eos funis ductarius KLMNOP <lb/>QRSTVX, ex cuius extremitate pendeat pondus X, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.142.jpg"/>Trahatur funis in K. </s>

<s id="id.001361">Dico ex multiplicatione <expan abbr="orbiculor&umacr;">orbiculorum</expan>, <lb/>trahentipondus nequaquam minui. </s>

<s id="id.001361">Dico ex multiplicatione <expan abbr="orbiculor&umacr;">orbiculorum</expan>, <lb/>trahenti xspondus nequaquam minui. </s>

<s id="id.001362">Sint autem orbicu<lb/>lorum diametri, LM, NO, PQ, RS, TV, applicetur poten<lb/>tîa in S. </s>

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<s id="id.001367">Probabimus autem, nullam <lb/>ex &longs;uperioribus orbiculis fieri ponderum imminutionem, <lb/>&longs;ed totam vim in inferioribus con&longs;i&longs;tere. </s>

<s id="id.001368">At nos interim <lb/>quippiam quod ad rem faciat, proponamus. </s>

<s id="id.001369">E&longs;to punctum A, cuirectæ ap<lb/>pendantur lineæ BAC, diui&longs;æ qui<lb/>dem in A, &longs;it autem lineæ BA caput <lb/>B, ip&longs;ius verò CA caput C. </s>

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<s id="id.001375">Vtcunque enim id &longs;it, fiet æquilibrium. </s>

<s id="id.001376">Habemus <lb/>igitur ex i&longs;tis ad &longs;u&longs;tinendum pondus ex &longs;uperiori parte <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.144.jpg"/>appen&longs;um potentiam requiri ip&longs;i ponderi æqualem. </s>

<s id="id.001376">Habemus <lb/>igitur ex i&longs;tis ad &longs;u&longs;tinendum pondus ex &longs;uperiori parte <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.143.jpg"/>appen&longs;um potentiam requiri ip&longs;i ponderi æqualem. </s>

<s id="id.001377">Ani<lb/>mo po&longs;thæc concipiatur alia recta linea DEF, cuius inte<lb/>gra longitudo &longs;i exten deretur, e&longs;&longs;et DE, EF. </s>

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<s id="id.001380">Videmus igitur illam appen&longs;ionem <lb/>quidem pondus nullatenus imminuere, hanc verò pon<lb/>dus ip&longs;um, bifariam diui&longs;um, &longs;u&longs;tinentibus potentijs im<lb/>partiri. </s>

<s id="id.001381">Hæ cin lineis, Mathematicâ v&longs;i ab&longs;tractione, con<lb/>&longs;iderauimus, nunc verò eadem mechanicè perpenda<lb/>mus. </s>

<s id="id.001382">Sit igitur <lb/>punctum A, vt <lb/>in &longs;e quenti figu<lb/>ra clauus paxil<lb/>lu&longs;ue, cui appen<lb/>&longs;us funiculus <lb/>BAC, & funicu<lb/>li capitibus pon<lb/>dera BC, &longs;it quo<lb/>que anulus D, <lb/>per quem traìe<lb/>ctus funiculus <lb/>EDF. </s>

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<s id="id.001384">His igiturita con&longs;titutis, eadem demon&longs;tra<lb/>buntur quæ &longs;uperius, nempe oportere vt fiat æquilibrium <lb/>B, C, e&longs;&longs;e æqualia, tum potentias, quæ &longs;unt in EF pondus <lb/>G inter eas diui&longs;um &longs;u&longs;tinere. </s>

<s id="id.001385">Porrò volentes Mechanici <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.145.jpg"/>funiculos circa paxillum, & anulum ad attollenda & de<lb/>primenda pondera mouere incommodè illis vtique &longs;uc<lb/>cedebat, clauo & anulo motum difficilem facientibus. <lb/></s>

<s id="id.001385">Porrò volentes Mechanici <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.144.jpg"/>funiculos circa paxillum, & anulum ad attollenda & de<lb/>primenda pondera mouere incommodè illis vtique &longs;uc<lb/>cedebat, clauo & anulo motum difficilem facientibus. <lb/></s>

<s id="id.001386">Quamobrem vt difficultati occurrerent, ad locum claui <lb/>clauo ip&longs;i orbiculum circumpo&longs;uerunt, & anuli itidem <lb/>loco orbiculum aptauerunt. </s>

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<s id="id.001393">Commendetur loco &longs;tabili <lb/>in S, & alteri vectis extremo R funis alligetur RT, cui <lb/>quidem potentia &longs;u&longs;tinens applicetur in T. </s>

<s id="id.001394">Dico igitur, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.146.jpg"/><figure id="id.007.00.146.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.146.1.jpg"></figure><lb/>rebus ita di&longs;po&longs;itis, <lb/>potentiam in T ita <lb/>&longs;e habere ad pondus <lb/>F, vt vnum ad &longs;ex de<lb/>cim, hoc e&longs;t, in pro<lb/>portione e&longs;&longs;e &longs;ub<lb/>&longs;exdecupla. </s>

<s id="id.001394">Dico igitur, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.145.jpg"/><figure id="id.007.01.145.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.145.1.jpg"></figure><lb/>rebus ita di&longs;po&longs;itis, <lb/>potentiam in T ita <lb/>&longs;e habere ad pondus <lb/>F, vt vnum ad &longs;exde<lb/>cim, hoc e&longs;t, in pro<lb/>portione e&longs;&longs;e &longs;ub<lb/>&longs;exdecupla. </s>

<s id="id.001395">Sunt <lb/>autem, hic vectes <lb/>quatuor in feriorum <lb/>cubiculorum, loco, <lb/>CD, HI, MN, QR, <lb/>qucrum, centra E, <lb/>G, L, P. quoniam e<lb/>nim A hoc e&longs;t, C, v<lb/>nà cum potentia G, <lb/>hoc e&longs;t, D, &longs;u&longs;tinet <lb/>pondus F alterum, <lb/>ponderis dimidium <lb/>&longs;u&longs;tinebit C, <expan abbr="alter&umacr;">alterum</expan> <lb/>vero D. erunt igitur <lb/>vtrinque librae quin<lb/>gentæ. </s>

<s id="id.001395">Sunt <lb/>autem, hic vectes <lb/>quatuor in feriorum <lb/>cubiculorum, loco, <lb/>CD, HI, MN, QR, <lb/>quorum, centra E, <lb/>G, L, P. quoniam e<lb/>nim A hoc e&longs;t, C, v<lb/>nà cum potentia G, <lb/>hoc e&longs;t, D, &longs;u&longs;tinet <lb/>pondus F alterum, <lb/>ponderis dimidium <lb/>&longs;u&longs;tinebit C, <expan abbr="alter&umacr;">alterum</expan> <lb/>vero D. erunt igitur <lb/>vtrinque librae quin<lb/>gentæ. </s>

<s id="id.001396">Tum potentia in K, hoc e&longs;t, in H, vna cum poten<lb/>tia in L, hoc e&longs;t, in I &longs;u&longs;tinebunt quingenta. </s>

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<s id="id.001399">Sed numerus i&longs;te ad Millenarium ita &longs;e habet vt v<lb/>num ad &longs;exdecim. </s>

<s id="id.001400">Hinc colligimus, pondus totum inter <lb/>loca &longs;tabilia diuidi, nempe A, K, O, S, & ip&longs;am potentiam <lb/>quæ &longs;u&longs;tinet in T, & locis ip&longs;is &longs;tabilibus quindecim par<lb/>tes integri ponderis, potentia verò T &longs;extam decimam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.147.jpg"/>tantùm commendari. </s>

<s id="id.001400">Hinc colligimus, pondus totum inter <lb/>loca &longs;tabilia diuidi, nempe A, K, O, S, & ip&longs;am potentiam <lb/>quæ &longs;u&longs;tinet in T, & locis ip&longs;is &longs;tabilibus quindecim par<lb/>tes integri ponderis, potentia verò T &longs;extam decimam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.146.jpg"/>tantùm commendari. </s>

<s id="id.001401">Itaque &longs;i ex puncto V appendere<lb/>tur AB, in X potentia, quæ in X &longs;u&longs;tineret mille, minus <lb/>&longs;exaginta duo cum dimidio, quod quidem à potentia in <lb/>T &longs;u&longs;tinetur; quod &longs;i alius adderetur orbiculus, & fierent <lb/>quinque, potentia in T &longs;u&longs;tiner et trige&longs;imam &longs;ecundam <lb/>partem integri ponderis, hoce&longs;t, dimidium librarum &longs;e<lb/>vaginta duarum cum dimidio, nempe triginta & vnam <lb/>cum quarta parte, &longs;i item textus adderetur, potentia in T <lb/>&longs;exage&longs;imam partem &longs;u&longs;tineret integri ponderis, hoc e&longs;t, <lb/>libras quindecim & <gap/> libræ vnius. </s>

<s id="id.001401">Itaque &longs;i ex puncto V appendere<lb/>tur AB, in X potentia, quæ in X &longs;u&longs;tineret mille, minus <lb/>&longs;exaginta duo cum dimidio, quod quidem à potentia in <lb/>T &longs;u&longs;tinetur; quod &longs;i alius adderetur orbiculus, & fierent <lb/>quinque, potentia in T &longs;u&longs;tineret trige&longs;imam &longs;ecundam <lb/>partem integri ponderis, hoc e&longs;t, dimidium librarum &longs;e<lb/>xaginta duarum cum dimidio, nempe triginta & vnam <lb/>cum quarta parte, &longs;i item textus adderetur, potentia in T <lb/>&longs;exage&longs;imam partem &longs;u&longs;tineret integri ponderis, hoc e&longs;t, <lb/>libras quindecim & <gap/> libræ vnius. </s>

<s id="id.001402">Vnde patet clarè pon<lb/>deris diminutionem fieri ex orbiculis inferioribus, non <lb/>autem ex&longs;uperioribus, &longs;uperiores autem addi non nece&longs;<lb/>&longs;itatis quidem, &longs;ed commoditatis gratiâ: neque enim ab&longs;<lb/>que &longs;uperioribus vnico ductario fune fieri po&longs;&longs;et attractio <lb/>& ponderis ip&longs;ius eleuatio. </s>

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<s id="id.001404">QVÆSTIO XIX.</s>

<s id="id.001405"><emph type="italics"/>Dubitat Philo&longs;ophus, Cur &longs;i quis &longs;uper lignum magnam imponat <lb/>&longs;ecurim, de&longs;uperque magnum adijciat pondus, ligni quippiam quod <lb/>curandum &longs;it, non diuidit; &longs;i verò &longs;ecurim extollens percutiat, illud <lb/>&longs;cindit, cum alioquin multo minus habeat ponder is id quod <lb/>percutit, quam illud quod&longs;uperiacet <lb/>& premit?<emph.end type="italics"/></s>

<s id="id.001405"><emph type="italics"/>Dubitat Philo&longs;ophus, Cur &longs;i quis &longs;uper lignum magnam imponat <lb/>&longs;ecurim, de&longs;uperque magnum adijciat pondus, ligni quippiam quod <lb/>curandum &longs;it, non diuidit; &longs;i verò &longs;ecurim extollens percutiat, illud <lb/>&longs;cindit, cum alioquin multo minus habeat ponderis id quod <lb/>percutit, quam illud quod s&longs;uperiacet <lb/>& premit?<emph.end type="italics"/></s>

<s id="id.001406">Poterat Ari&longs;toteles, nî fallimur, rem breuius & vniuer<lb/>&longs;alius proponere. </s>

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<s id="id.001409">An, inquiens, ideo fit, <lb/>quia omnia cum motu fiunt, & graue ip&longs;um grauitatis ma<lb/>gis a&longs;&longs;umit motum, dum mouetur quam dum quie&longs;cit? <lb/></s>

<s id="id.001410">Incumbens igitur connatam graui motionem non moue<lb/>tur, motum verò & &longs;ecundum hanc mouetur & &longs;ecun-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.148.jpg"/>dum eam quæ e&longs;t <expan abbr="percuti&emacr;tis">percutientis</expan>? </s>

<s id="id.001410">Incumbens igitur connatam graui motionem non moue<lb/>tur, motum verò & &longs;ecundum hanc mouetur & &longs;ecun-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.147.jpg"/>dum eam quæ e&longs;t <expan abbr="percuti&emacr;tis">percutientis</expan>? </s>

<s id="id.001411">Hæc præclarè quidem, cæ<lb/>tera autem, quæ de cuneoiterat, nempe ad vectem eiuslo<lb/>perationem referri &longs;uperius confutauimus. </s>

<s id="id.001411">Hæc præclarè quidem, cæ<lb/>tera autem, quæ de cuneo iterat, nempe ad vectem eius o<lb/>perationem referri &longs;uperius confutauimus. </s>

<s id="id.001412">Porrò effe<lb/>ctus huius, de quo agitur, di&longs;putatio illuc &longs;pectat, videli<lb/>cet ad cadentium atque proiectorum naturam. </s>

<s id="id.001413">Ad maio<lb/>rem autem rei euidentiam hæc addimus. </s>

<s id="id.001414">E&longs;to libra AB, cu<lb/>ius centrum C, libra<lb/>ta æqualibus ponde<lb/>ribus DE, apponatur <lb/>ponderi E pondus F, <lb/>item ponderi D pon<lb/>dus G ip&longs;i ponderi F <lb/>æquale, æquilibrabit <lb/>itidem, Modò non apponatur &longs;impliciter pondus G &longs;ex <lb/>ex H in lancem A dimittatur, tunc &longs;anè non æquilibrabit, <lb/>&longs;ed libram deprimet. </s>

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<s id="id.001421">Cæterùm cur mallei & &longs;ecuris ictus &longs;it violenti&longs;&longs;i<lb/>mus, ideo fit quod non ex vnico neque duplici, &longs;ed ex tri<lb/>plici grauitate operetur. </s>

<s id="id.001422">E&longs;to enim &longs;ecuris A, cuius manu<lb/>brium AB, brachium vero &longs;ecuri vtentis BC, erit igitur C <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.149.jpg"/><figure id="id.007.00.149.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.149.1.jpg"></figure><lb/>locus vbi humero <lb/>brachium iungi<lb/>tur, motus ip&longs;ius <lb/>centrum, attollit <lb/>autem &longs;ecurim is <lb/>qui percutit, & re<lb/>tro ad &longs;capulas re<lb/>ducens totis viri<lb/>bus ex centro C <lb/>&longs;ecurim vibrat, <lb/>portionem circuli <lb/>de&longs;cribens ADE <lb/>ictumqueue faciens <lb/>in E. </s>

<s id="id.001422">E&longs;to enim &longs;ecuris A, cuius manu<lb/>brium AB, brachium vero &longs;ecuri vtentis BC, erit igitur C <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.148.jpg"/><figure id="id.007.01.148.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.148.1.jpg"></figure><lb/>locus vbi humero <lb/>brachium iungi<lb/>tur, motus ip&longs;ius <lb/>centrum, attollit <lb/>autem &longs;ecurim is <lb/>qui percutit, & re<lb/>tro ad &longs;capulas re<lb/>ducens totis viri<lb/>bus ex centro C <lb/>&longs;ecurim vibrat, <lb/>portionem circuli <lb/>de&longs;cribens ADE <lb/>ictumqueue faciens <lb/>in E. </s>

<s id="id.001423">Vires igitur acquirit &longs;ecuris, tum ex naturali grauita<lb/>te, cadens ex D, in E, tum ex proprio pondere, tum etiam <lb/>ex violentia eidem à percutiente impre&longs;&longs;a. </s>

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<s id="id.001430">An autem operetur efficacius cuneus malleo percu&longs;<lb/>&longs;us, aut cum manubrio motus, vt fit in &longs;ecuri, data aciei & <lb/>ponderis æqualitate, difficile e&longs;t determinare. </s>

<s id="id.001431">Certè va<lb/>lidius, & certius fieri &longs;ci&longs;&longs;ionem ex cuneo & malleo, ea ra<lb/>tio e&longs;t, quod cuneus adactus, nec inde remotus eam inte <lb/>rim &longs;eruat, quam antea fecerat partium &longs;eparationem, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.150.jpg"/>quod quidem &longs;ecuri non accidit, quæ adacta ad nouam <lb/>percu&longs;&longs;ionem faciendam extrahitur. </s>

<s id="id.001431">Certè va<lb/>lidius, & certius fieri &longs;ci&longs;&longs;ionem ex cuneo & malleo, ea ra<lb/>tio e&longs;t, quod cuneus adactus, nec inde remotus eam inte <lb/>rim &longs;eruat, quam antea fecerat partium &longs;eparationem, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.149.jpg"/>quod quidem &longs;ecuri non accidit, quæ adacta ad nouam <lb/>percu&longs;&longs;ionem faciendam extrahitur. </s>

<s id="id.001432">Hoc etiam con&longs;ideramus, &longs;ecuris in circulo motum, <lb/>ex A in D, e&longs;&longs;e videndum, id e&longs;t, non &longs;ecundum naturam, <lb/>&longs;ur&longs;um enim fertur quod e&longs;t graue, ex D verq in F <expan abbr="mixt&umacr;">mixtum</expan>: <lb/>magis autem ad naturalem accedere qui fit ex F in E. </s>

<s id="id.001432">Hoc etiam con&longs;ideramus, &longs;ecuris in circulo motum, <lb/>ex A in D, e&longs;&longs;e videndum, id e&longs;t, non &longs;ecundum naturam, <lb/>&longs;ur&longs;um enim fertur quod e&longs;t graue, ex D verò in F <expan abbr="mixt&umacr;">mixtum</expan>: <lb/>magis autem ad naturalem accedere qui fit ex F in E. </s>

<s id="id.001433">Tar<lb/>dior ergo ex A in D, velocior ex D, in F, veloci&longs;&longs;imus ex F <lb/>in E; quæ dam quæ ad hanc rem faciunt, egregiè con&longs;ide<lb/>rat Guid, Vbald. in calce Tractatus, De Cuneo; ip&longs;um <lb/>con&longs;ule. </s>

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<s id="id.001442">Licet enim in B velocitas &longs;it maior, dee&longs;t <lb/>ibi pondus. </s>

<s id="id.001443">Si enim en&longs;is iterum vt vectis con&longs;ideretur, e<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.151.jpg"/>runt AB. duo fulcimenta &longs;u&longs;tinentía pondus in C, vbi gra<lb/>uitatis e&longs;t centrum. </s>

<s id="id.001443">Si enim en&longs;is iterum vt vectis con&longs;ideretur, e<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.150.jpg"/>runt AB. duo fulcimenta &longs;u&longs;tinentía pondus in C, vbi gra<lb/>uitatis e&longs;t centrum. </s>

<s id="id.001444">Si igitur paria fuerint &longs;patia BC, CA, <lb/><figure id="id.007.00.151.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.151.1.jpg"></figure><lb/>in B erit dìmidium <lb/>ponderis C, quantum <lb/>ergo velocitate præ<lb/>ualetictus in B, <expan abbr="tant&umacr;">tantum</expan> <lb/>ponderis amittit. </s>

<s id="id.001444">Si igitur paria fuerint &longs;patia BC, CA, <lb/><figure id="id.007.01.150.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.150.1.jpg"></figure><lb/>in B erit dìmidium <lb/>ponderis C, quantum <lb/>ergo velocitate præ<lb/>ualetictus in B, <expan abbr="tant&umacr;">tantum</expan> <lb/>ponderis amittit. </s>

<s id="id.001445">D <lb/>verò plus quidem de <lb/>pondere participat, <lb/>&longs;ed velocitatis habet <lb/>minimum, in C verò <lb/>velocitas e&longs;t medio<lb/>cris, tota tamen ip&longs;ius <lb/>ex grauitatis centro <lb/>ponderis fit impre&longs;<lb/>&longs;io. </s>

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<s id="id.001448"><emph type="italics"/>Dubitatur, Cur &longs;tatera qua carnes ponderantur, paruo appendicu<lb/>lo, magna trutinet onera, cum alioqui tota, dimidiata exi&longs;tat <lb/>libra, altera vero parte &longs;ola&longs;it <lb/>&longs;tatera?<emph.end type="italics"/></s>

<s id="id.001449">Soluit Philo&longs;ophus, inquiens, &longs;tateram &longs;imul, & vectem <lb/>e&longs;&longs;e & libram, ip&longs;ius verò libræ centra &longs;eu fulcimenta <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.152.jpg"/>e&longs;&longs;e ibi vbi fit &longs;u&longs;pen&longs;io. </s>

<s id="id.001449">Soluit Philo&longs;ophus, inquiens, &longs;tateram &longs;imul, & vectem <lb/>e&longs;&longs;e & libram, ip&longs;ius verò libræ centra &longs;eu fulcimenta <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.151.jpg"/>e&longs;&longs;e ibi vbi fit &longs;u&longs;pen&longs;io. </s>

<s id="id.001450">Pondera verò hinc in de in lance <lb/>& appendiculo, loco &longs;cilicet æquipondij, appendiculo <lb/>&longs;uccedente. </s>

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<s id="id.001452">Ait igitur, appendiculum licet parui <expan abbr="p&omacr;-deris">pon<lb/>deris</expan> &longs;it, ideo maiori ponderi virtute æquari, quod lon<lb/>gius à centro, hoc e&longs;t, ab ip&longs;o fulcimento &longs;i&longs;tatur. </s>

<s id="id.001453">quic<lb/>quid tamen &longs;it, &longs;tateram e&longs;&longs;e vectem, res e&longs;t explorati&longs;<lb/>&longs;ima. </s>

<s id="id.001454">E&longs;to igitur &longs;tatera AB, <lb/>cuius appendiculum cur<lb/>rens F, fulcimentum cen<lb/>trumue C, lanx quæ cate<lb/>na &longs;u&longs;penditur E &longs;patium <lb/>à loco fulcimenti ad ap<lb/>pendiculum CF. quod ve<lb/>rò à fulcimento ad cate<lb/>nam, ex qua lanx appen<lb/>ditur AC. </s>

<s id="id.001455">Intelligatur autem & aliud fulcimentum D, &longs;it<lb/>queue maius &longs;pacium AD, quam AC. </s>

<s id="id.001455">Intelligatur autem & aliud fulcimentum D, &longs;it<lb/>queue maius &longs;patium AD, quam AC. </s>

<s id="id.001456">Porrò ita &longs;e habeat <lb/>pondus in E ad appendiculi F pondus, vt CF &longs;patium, ad <lb/>&longs;patium AC, quo ca&longs;u &longs;eruata, permutatim, ponderum & <lb/>brachiorum proportione, fiet ae quilibrium. </s>

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<s id="id.001459">Ita que vt it e<lb/>rum fiat æquilibrium, nece&longs;&longs;e e&longs;t <expan abbr="iter&umacr;">iterum</expan> proportiones bra<lb/>chiorum &longs;eu &longs;patiorum proportionibus ponderum æqua<lb/>re. </s>

<s id="id.001460">Transferatur igitur (lancis interim immoto pondeie) <lb/>ip&longs;um appendiculum in B, fiatque vt FC ad CA, ita BD ad <lb/>DA. </s>

<s id="id.001460">Transferatur igitur (lancis interim immoto pondere) <lb/>ip&longs;um appendiculum in B, fiatque vt FC ad CA, ita BD ad <lb/>DA. </s>

<s id="id.001461">Stabitautem iterum &longs;tatera ad eam redacta quam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.153.jpg"/>diximus brachiorum & ponderum permutatam propor<lb/>tionem. </s>

<s id="id.001461">Stabit autem iterum &longs;tatera ad eam redacta quam <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.152.jpg"/>diximus brachiorum & ponderum permutatam propor<lb/>tionem. </s>

<s id="id.001462">Nos &longs;tateris vtimur ex duplici fulcimento, altero <lb/>propiori, altero à lance &longs;eu loco, vbi lanx appenditur, re<lb/>motiori, illa grauiora appendimus pondera, & non per <lb/>vncias & libras, &longs;ed per libras tantum & &longs;elibra ponde<lb/>ramus; & hoc &longs;tateræ latus eo quod minus minutè &longs;it di<lb/>ui&longs;um; vulgo no&longs;trates Gro&longs;&longs;um, hoc e&longs;t, rude & cra&longs;&longs;um <lb/>appellant. </s>

<s id="id.001463">Aliud verò, cum fulcimentum e&longs;t loco appen<lb/>&longs;ionis lancis vicinius, & per libras, &longs;elibras & vncias diui<lb/>ditur, quo quidem minora appendimus pondera, cò quod <lb/><expan abbr="exqui&longs;itior&emacr;">exqui&longs;itiorem</expan> contineat diui&longs;ionem, &longs;ubtile dicunt. </s>

<s id="id.001463">Aliud verò, cum fulcimentum e&longs;t loco appen<lb/>&longs;ionis lancis vicinius, & per libras, &longs;elibras & vncias diui<lb/>ditur, quo quidem minora appendimus pondera, eò quod <lb/><expan abbr="exqui&longs;itior&emacr;">exqui&longs;itiorem</expan> contineat diui&longs;ionem, &longs;ubtile dicunt. </s>

<s id="id.001464">Rectè <lb/>igitur dicebat Philo&longs;ophus, in &longs;tatera plures e&longs;&longs;e libras, <lb/>quanquam & ea quoque de cau&longs;&longs;a dici po&longs;&longs;it, quod, quot <lb/>&longs;unt appendiculi, è locoin locum translationes, totidem <lb/>ex proportionum variatione fiant libræ. </s>

<s id="id.001464">Rectè <lb/>igitur dicebat Philo&longs;ophus, in &longs;tatera plures e&longs;&longs;e libras, <lb/>quamquam & ea quoque de cau&longs;&longs;a dici po&longs;&longs;it, quod, quot <lb/>&longs;unt appendiculi, è loco in locum translationes, totidem <lb/>ex proportionum variatione fiant libræ. </s>

<s id="id.001465">Et hoc quidem <lb/>&longs;en&longs;i&longs;&longs;e videtur Ari&longs;toteles. </s>

<s id="id.001466">Po&longs;&longs;emus & alio <lb/>modo &longs;tatera vti, nempe <lb/>&longs;tabili appendiculo, mo<lb/>bili autem fulcimento. <lb/></s>

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<s id="id.001471">Appendatur autem, puta in G, &longs;it <expan abbr="aut&emacr;">autem</expan> <lb/>grauitatis centrum in H. </s>

<s id="id.001472">Quoniam igitur He&longs;t extra ful<lb/>cimentum G, declinabit &longs;tateræ pars GA, centro G per <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.154.jpg"/>circuli portionem Hl, à centro grauitatis in ip&longs;a de&longs;cen<lb/>&longs;ione de&longs;criptam. </s>

<s id="id.001472">Quoniam igitur H e&longs;t extra ful<lb/>cimentum G, declinabit &longs;tateræ pars GA, centro G per <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.153.jpg"/>circuli portionem Hl, à centro grauitatis in ip&longs;a de&longs;cen<lb/>&longs;ione de&longs;criptam. </s>

<s id="id.001473">Siautem grauitatis centrum fuerit vbi <lb/>K, eo quodibi quoque &longs;it extra fulcimentum G, de&longs;cen<lb/>detpars GB, de&longs;cribente interim grauitatis centro K, cir<lb/>culi portionem KL. ltaque &longs;i &longs;tateram totam eum ponde<lb/>ribus trahamus <expan abbr="pellamu&longs;q;">pellamu&longs;que</expan> vltro citroque;, immoto appen<lb/>diculo eritaliquando fulcimentum in ea linea perpendi<lb/>culari velloco ip&longs;o, vbi e&longs;t grauitatis centrum, quo ca&longs;u <lb/>&longs;tatera&longs;tabit, & tuncita erit diui&longs;a, vt fiat brachiorum & <lb/>ponderum eadem ratio, ordine permutato. </s>

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<s id="id.001480">Si Græca ad <lb/>verbum reddas ita habent: An magis ip&longs;a manu labile e&longs;t <lb/>ferrum, & ip&longs;um vndique (dentemnempe) non comple<lb/>ctitur, caro autem digitorum cum mollis &longs;it, adhæret ma<lb/>gis, & vndique congruit. </s>

<s id="id.001481">Certè vt &longs;ententia non &longs;it con<lb/>traria propo&longs;itioni, Græca ver&longs;io ita videtur concinnan<lb/>da: Vel magis è m n<gap/>bitur, mollis enim e&longs;t digitorum <lb/>caro, ferrum autem circumplectitur, & haeret magis. </s>

<s id="id.001481">Certè vt &longs;ententia non &longs;it con<lb/>traria propo&longs;itioni, Græca ver&longs;io ita videtur concinnan<lb/>da: Vel magis è manu elabitur, mollis enim e&longs;t digitorum <lb/>caro, ferrum autem circumplectitur, & haeret magis. </s>

<s id="id.001482">quic<lb/>quid &longs;it, Græcam lectionem contrarium ei quod quæri-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.155.jpg"/>tur, affirmare certum e&longs;t. </s>

<s id="id.001482">quic<lb/>quid &longs;it, Græcam lectionem contrarium ei quod quæri-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.154.jpg"/>tur, affirmare certum e&longs;t. </s>

<s id="id.001483">Picolomineus, Ideo, inquit, di<lb/>gitorum caro mollis minus aptè extrahit, quod dentem <lb/>totum comprehendere non pote&longs;t, quod ferrum ob &longs;uam <lb/>durítiem & con&longs;tantiam commodi&longs;&longs;imè facit. </s>

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<s id="id.001486">Hoc igitur ad ex<lb/>tractionem vtuntur^{**}, vt facilius moueant. </s>

<s id="id.001487">Figuram hoc <lb/>pactto proponit Philo&longs;ophus. </s>

<s id="id.001487">Figuram hoc <lb/>pacto proponit Philo&longs;ophus. </s>

<s id="id.001488">E&longs;to dentiforcipis alterum <lb/>quidem extremum vbi A, alte<lb/>rum autem quod extrahit B, ve<lb/>ctis vbi ADF, alter vectis, vbi <lb/>BCE, fulcimentum verò CGD <lb/>connexio vbi G. </s>

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<s id="id.001495">Neque enim totus Dentiforceps <lb/>hic ceu vectis vnicus operatur, quod fit in forcipibus quas <lb/>Tenaleas vocamus, quibus è tabulis claui reuelluntur, <lb/>qua de re nos quae&longs;tione 6. verba fecimus. </s>

<s id="id.001496">Quo pacto <expan abbr="aut&emacr;">autem</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.156.jpg"/>dentis ex Dentiforcipe extractio ad vectem reducatur, <lb/>&longs;ubtilius e&longs;t perpendendum, neque enim res e&longs;t in propa<lb/>tulo. </s>

<s id="id.001496">Quo pacto <expan abbr="aut&emacr;">autem</expan> <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.155.jpg"/>dentis ex Dentiforcipe extractio ad vectem reducatur, <lb/>&longs;ubtilius e&longs;t perpendendum, neque enim res e&longs;t in propa<lb/>tulo. </s>

<s id="id.001497">Dicimus igitur, tum dentem ip&longs;um, tum dentifor<lb/>cipem vectes e&longs;&longs;e, varia tamen ratione & &longs;atis &longs;ane diuer<lb/>&longs;a. </s>

<s id="id.001498">Dens enim fit vectis eius nempe naturæ quæ fulcimen<lb/>tum habet in angulo, quo ca&longs;u ip&longs;ius Dentiforcipis <expan abbr="parti&umacr;">partium</expan>, <lb/>quibus Dens apprehenditur, ea quæ longior e&longs;t poten<lb/>tiæ mouentis loco &longs;uccedit, breuior vero fulcimentum <lb/>facit, Dentis vero re&longs;i&longs;tentia ponderis vices refert. </s>

<s id="id.001499">E&longs;to enim dens qui<lb/>dem A, cuius diameter <lb/>BC, longitudo v&longs;que ad <lb/>extremas radices CD, <lb/>pars dentiforcipis breui<lb/>or CG, longior BG. </s>

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<s id="id.001504">Porro quod in calce quæ&longs;tionis addit Philo&longs;ophus, <lb/>Dentes commotos facilius manu extrahi quam in&longs;tru<lb/>mento, nulla ratione probat. </s>

<s id="id.001505">Ego autem arbitror, huc <lb/>pertinere ea verba, quæ &longs;uperius habentur, videlicet fer<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.157.jpg"/>rum quidem non vndique dentem <expan abbr="compreh&emacr;dere">comprehendere</expan>, quod <lb/>mollisfacit digitorum caro, quæ id circo adhæret & <gap/>om<lb/>plectitur magis. </s>

<s id="id.001505">Ego autem arbitror, huc <lb/>pertinere ea verba, quæ &longs;uperius habentur, videlicet fer<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.156.jpg"/>rum quidem non vndique dentem <expan abbr="compreh&emacr;dere">comprehendere</expan>, quod <lb/>mollis facit digitorum caro, quæ id circo adhæret & com<lb/>plectitur magis. </s>

<s id="id.001506">An autem ita &longs;it, alij videant, nobis enim <lb/>digito rem o&longs;tendi&longs;&longs;e fuerit &longs;atis. </s>

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<s id="id.001511">ma<lb/>ius verò quod à vertebra ad extremitates, quæ confrin<lb/>gentis manu comprimuntur. </s>

<s id="id.001512">Ait igitur, & id quam oppo<lb/>fite, vim ex vectibus ictus loco &longs;uccedere & idem operari. </s>

<s id="id.001513">E&longs;to igitur in &longs;trumentum, <lb/>de quo agimus CDBF, ex duo<lb/>bus vectibus con&longs;tans, quorum <lb/>alter CAF, alter vero DAB ver<lb/>tebra &longs;eu connexio A locus v<lb/>bi nux frangitur K, manubria <lb/>vero BF. quo igitur prolixiores <lb/>erunt AB, AF, breuiores vero ACAD, violentius fiet <expan abbr="c&omacr;-fractio">con<lb/>fractio</expan>. </s>

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<s id="id.001515">Itaque nî maior &longs;it propor<lb/>tio potentiæ ad re&longs;i&longs;tentiam, quam brachij à potentia ad <lb/>fulcimentum ad eam partem quæ à fulcimento e&longs;t ad nu<lb/>cem, non fiet confractio. </s>

<s id="id.001516">eo autem magis &longs;up erabit, quo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.158.jpg"/>maior fuerit pars vectis quæ à potentia ad fulcimentum. </s>

<s id="id.001516">eo autem magis &longs;uperabit, quo <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.157.jpg"/>maior fuerit pars vectis quæ à potentia ad fulcimentum. </s>

<s id="id.001517">Quod autem addit Ari&longs;toteles, eo maiorem fieri <lb/>vectium eleuationem, hoc e&longs;t, in&longs;trumenti aperitionem, <lb/>quo magis nux quæ frangitur, fuerit propior fulcimento, <lb/>hoc e&longs;t, ip&longs;i vertebræ, facile o&longs;tenditur ex conuer&longs;a 21. <lb/>propo&longs;. </s>

<s id="id.001518">lib.

1. Elem.

<s id="001518">&longs;i enim ab extremitatibus vnius lineæ <lb/>ad ea&longs;dem partes con&longs;tituantur duæ lineæ maiores con<lb/>currentes in angulo, & ab ij&longs;dem extremitatibus duæ a<lb/>liæ minores, quæ intra triangulum à maioribus con&longs;titu<lb/>tum cadant, maiorem angulum continebunt. </s>

&longs;i enim ab extremitatibus vnius line æ <lb/>ad ea&longs;dem partes con&longs;tituantur duæ line æ maiores con<lb/>currentes in angulo, & ab ij&longs;dem extremitatibus duæ a<lb/>liæ minores, quæ intra triangulum à maioribus con&longs;titu<lb/>tum cadant, maiorem angulum continebunt. </s>

<s id="id.001519">At talis e&longs;t <lb/>angulus qui fit in in &longs;trumento, cum partes vectis à verte<lb/>bra ad nucem fuerint breuiores. </s>

<s id="id.001520">magìs ergo dilatantur <lb/>vectes, & magis dilatati magis comprimuntur, magis au<lb/>tem compre&longs;li validius frangunt, quod dixerat Ari&longs;to<lb/>teles. </s>

<s id="id.001520">magìs ergo dilatantur <lb/>vectes, & magis dilatati magis comprimuntur, magis au<lb/>tem compre&longs;&longs;i validius frangunt, quod dixerat Ari&longs;to<lb/>teles. </s>

<s id="id.001521">Cæterum & illud quod &longs;cribit, ex grauiori & durio<lb/>ri materia in&longs;trumentum citius fractionem facere, quam <lb/>ex leuiori & minus dura, ex parte quidem materiæ verum <lb/>e&longs;t, nec pertinet ad proportionem, quæ &longs;ane in <expan abbr="huiu&longs;mod&imacr;">huiu&longs;modim</expan> <lb/>in&longs;trumentis formæ ferè habent rationem. </s>

<s id="id.001521">Cæterum & illud quod &longs;cribit, ex grauiori & durio<lb/>ri materia in&longs;trumentum citius fractionem facere, quam <lb/>ex leuiori & minus dura, ex parte quidem materiæ verum <lb/>e&longs;t, nec pertinet ad proportionem, quæ &longs;ane in <expan abbr="huiu&longs;mod&imacr;">huiu&longs;modi</expan> <lb/>in&longs;trumentis formæ ferè habent rationem. </s>

<s id="id.001522">Nos hi&longs;ce in<lb/>&longs;trumentisnon vtimur. </s>

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<s id="id.001525">Pvlcherrimam proponit hoc loco Philo&longs;ophus con<lb/>templationem, eamque ad mixtos motus <expan abbr="pertin&emacr;tem">pertinentem</expan>. <lb/></s>

<s id="id.001526">Mixtorum autem motuum &longs;peculationem antiquis Me<lb/>chanicis fui&longs;&longs;e tum vtilem tum etiam familiarem, norunt <lb/>ij qui norunt quæ de lineis &longs;piralibus Helici&longs;ue, cy&longs;&longs;oidi<lb/>bus, conchoidibus & alijs eiu&longs;cemo di &longs;cripta & contem<lb/>plata reperiuntur, quibus tum ad duarum mediarum pro<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.159.jpg"/>portionalium inuentionem, tum ad circuli quadratio<lb/>nem vti&longs;olent. </s>

<s id="id.001526">Mixtorum autem motuum &longs;peculationem antiquis Me<lb/>chanicis fui&longs;&longs;e tum vtilem tum etiam familiarem, norunt <lb/>ij qui norunt quæ de lineis &longs;piralibus Helici&longs;ue, cy&longs;&longs;oidi<lb/>bus, conchoidibus & alijs eiu&longs;cemodi &longs;cripta & contem<lb/>plata reperiuntur, quibus tum ad duarum mediarum pro<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.158.jpg"/>portionalium inuentionem, tum ad circuli quadratio<lb/>nem vti &longs;olent. </s>

<s id="id.001527">Quod autem hîc quærit Ari&longs;toteles, ita &longs;e <lb/>habet. </s>

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<s id="id.001531">Difficile hoc intellectu prima fronte, & &longs;ane admi<lb/>rabile, itaque in tentam con templationem requirit. </s>

<s id="id.001532">Nos <lb/>primo cum Ari&longs;totele, rem totam explicabimus, tum ali<lb/>quid forta&longs;&longs;e non p&oelig;nitendum no&longs;tro de promptuario <lb/>proferemus. </s>

<s id="id.001533">E&longs;to itaque Rhombus ABCD, <lb/>cuius latera AB, BD, DC, CA, diame<lb/>trorum maior AD, minor BC, &longs;ecan<lb/>tes &longs;e inuicem in puncto &longs;eu figuræ <lb/>centro K. </s>

<s id="id.001534">Sunt <expan abbr="aut&emacr;">autem</expan> ex ip&longs;ius Rhom<lb/>binatura latera æqualia & parallela, <lb/>Angulorum vero qui maiori diame<lb/>tro opponuntur, recto maiores, qui <lb/>vero minori minores. </s>

<s id="id.001535">His igitur con<lb/>&longs;ideratis, intelligatur punctum A mo<lb/>ueri peculiari & &longs;im plici motu, per li<lb/>neam AB, ab A ver&longs;us B, & eodem <expan abbr="t&emacr;-pore">ten<lb/>pore</expan> moueri totam lineam AB, ver&longs;us lineam DC, hac ta<lb/>men lege, vt &longs;emper eidem DC feratur parallela, & eius <lb/>alterum extremorum feratur per AC, alterum vero per <lb/>BD, Intelligatur etiam punctum B moueri eodem tem<lb/>pore proprio motu, eoque &longs;implici, per eandem rectam <lb/>BA, ver&longs;us A, & cum eadem, vt dictum e&longs;t, mota; ferri ver-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.160.jpg"/>&longs;us CD. </s>

<s id="id.001535">His igitur con<lb/>&longs;intelligatur, id eratis punctum A mo<lb/>ueri peculiari & &longs;implici motu, per li<lb/>neam AB, ab A ver&longs;us B, & eodem <expan abbr="t&emacr;-pore">tem<lb/>pore</expan> moueri totam lineam AB, ver&longs;us lineam DC, hac ta<lb/>men lege, vt &longs;emper eidem DC feratur parallela, & eius <lb/>alterum extremorum feratur per AC, alterum vero per <lb/>BD, Intelligatur etiam punctum B moueri eodem tem<lb/>pore proprio motu, eoque &longs;implici, per eandem rectam <lb/>BA, ver&longs;us A, & cum eadem, vt dictum e&longs;t, mota; ferri ver-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.159.jpg"/>&longs;us CD. </s>

<s id="id.001536">Erunt autem &longs;emper AB puncta in eadem linea <lb/>quæ mouetur, &longs;ibi inuicem ex contrarijs partibus occur<lb/>rentia. </s>

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<s id="id.001546">Rhombus &longs;imilis quidem Rhombo ABCD, recta <lb/>igitur FK æqualis erit oppo&longs;itæ AE. quare A punctum <lb/>translatum erit ex mixto motu in K. </s>

<s id="id.001547">Eodem pacto <expan abbr="quoniã">quoniam</expan> <lb/>punctum B. eadem velocitate mouetur ver&longs;us A, & linea <lb/>AB ver&longs;us CD, cum B fuerit in E extremum line æ motæ <lb/>BA, <expan abbr="n&emacr;pe">nempe</expan> B erit in G. æquales ergo &longs;unt BE, BG & Rhom<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.161.jpg"/>bus EBGK, circa diametrum BKC ip&longs;i Rhombo ABCD <lb/>&longs;imilis, & ideo GK æqualis oppo&longs;itæ BE & BG æqualis <lb/>EK. </s>

<s id="id.001547">Eodem pacto <expan abbr="quoniã">quoniam</expan> <lb/>punctum B. eadem velocitate mouetur ver&longs;us A, & linea <lb/>AB ver&longs;us CD, cum B fuerit in E extremum lineæ motæ <lb/>BA, <expan abbr="n&emacr;pe">nempe</expan> B erit in G. æquales ergo &longs;unt BE, BG & Rhom<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.160.jpg"/>bus EBGK, circa diametrum BKC ip&longs;i Rhombo ABCD <lb/>&longs;imilis, & ideo GK æqualis oppo&longs;itæ BE & BG æqualis <lb/>EK. </s>

<s id="id.001548">Cum ergo B confecerit &longs;patium BE, erit ex mixto <lb/>motu in K, &longs;uperato nempe &longs;patio BK, idque eodem tem<lb/>pore quo A percurrerat totum &longs;patium AK. </s>

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<s id="id.001556">Futilem igitur e&longs;&longs;e rationem hanc ip&longs;ius Ari&longs;totelis <lb/>pace, hoc pacto o&longs;tendemus. </s>

<s id="id.001557">E&longs;to quadratum ABCD, cuius diametri ACBD &longs;e<lb/>cantes &longs;e&longs;e in E, moueatur eodem pacto BA, ver&longs;us CD, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.162.jpg"/><figure id="id.007.00.162.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.162.1.jpg"></figure><lb/>item A, ver&longs;us B, & B ver&longs;us A, ita<lb/>que punctum A tum proprio tum <lb/>alieno, hoc e&longs;t lineæ illud <expan abbr="defer&emacr;-tis">deferen<lb/>tis</expan> motu deor&longs;um trudet, hoc e&longs;t, <lb/>ver&longs;us CD. </s>

<s id="id.001557">E&longs;to quadratum ABCD, cuius diametri ACBD &longs;e<lb/>cantes &longs;e&longs;e in E, moueatur eodem pacto BA, ver&longs;us CD, <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.161.jpg"/><figure id="id.007.01.161.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.161.1.jpg"></figure><lb/>item A, ver&longs;us B, & B ver&longs;us A, ita<lb/>que punctum A tum proprio tum <lb/>alieno, hoc e&longs;t lineæ illud <expan abbr="defer&emacr;-tis">deferen<lb/>tis</expan> motu deor&longs;um trudet, hoc e&longs;t, <lb/>ver&longs;us CD. </s>

<s id="id.001558">Motus ergo velocior <lb/>erit motu puncti B, quod lationi<lb/>bus fertur ferè contrarijs, hoc e&longs;t, <lb/>ex B ver&longs;us A &longs;ur&longs;um, cum linea <lb/>autem BA ver&longs;us C deor&longs;um. </s>

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<s id="id.001560">Stante igitur cau&longs;<lb/>&longs;a &longs;equi debui&longs;&longs;et effectus; non &longs;equitur autem, Ari&longs;tote<lb/>lis igitur cau&longs;&longs;a non e&longs;t cau&longs;&longs;a. </s>

<s id="id.001561">Rhombo quoque inuer&longs;o <lb/>idem clarius o&longs;tendemus hoc pacto: Sit Rhombus ABCD, <lb/><figure id="id.007.00.162.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.162.2.jpg"></figure><lb/>cuius diametri AC, BD &longs;ecan<lb/>tes &longs;e&longs;e in E. </s>

<s id="id.001561">Rhombo quoque inuer&longs;o <lb/>idem clarius o&longs;tendemus hoc pacto: Sit Rhombus ABCD, <lb/><figure id="id.007.01.161.2.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.161.2.jpg"></figure><lb/>cuius diametri AC, BD &longs;ecan<lb/>tes &longs;e&longs;e in E. </s>

<s id="id.001562">Mota igitur linea <lb/>AB ver&longs;us CD, nempe deor&longs;um <lb/>& A quoque deor&longs;um ver&longs;us B, <lb/>contra vero B quidem &longs;ur<lb/>&longs;um ver&longs;us A, deor&longs;um vero <lb/>ver&longs;us C, erit B tardior A, &longs;ed <lb/>contrarium fit, quippe quod <lb/>longior &longs;it BD, per quam mouetur B ip&longs;a AC, per quam <lb/>mouetur A. </s>

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<s id="id.001564">Rationibus igitur & veritate contra aucto<lb/>ritatem & probabilitatem e&longs;t nobis pugnandum: quod & <lb/>intrepide faciemus. </s>

<s id="id.001565">Dicimus igitur, in quouis parallelogrammo &longs;itillud <lb/>qua dratum aut altera parte longius, vel idem Rhombus <lb/>Rhomboi&longs;ue &longs;emper mixtos motus proportione &longs;eruata <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.163.jpg"/>fieri per diametros. </s>

<s id="id.001565">Dicimus igitur, in quouis parallelogrammo &longs;it illud <lb/>quadratum aut altera parte longius, vel idem Rhombus <lb/>Rhomboi&longs;ue &longs;emper mixtos motus proportione &longs;eruata <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.162.jpg"/>fieri per diametros. </s>

<s id="id.001566">Cæterum díametrorum ad latera <lb/>proportiones e&longs;&longs;e varias (quadratis exceptis, in quibus ea<lb/>dem e&longs;t &longs;emper) explorati&longs;&longs;imum. </s>

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<s id="id.001573">Velocior enim qui per longiorem diametrum, tardior <lb/>quiperbreuiorem. </s>

<s id="id.001574">Itaque &longs;implices motus punctorum <lb/>per latera ad eum qui fit per diametrosinon eodem pacto <lb/>&longs;e habent. </s>

<s id="id.001574">Itaque &longs;implices motus punctorum <lb/>per latera ad eum qui fit per diametros non eodem pacto <lb/>&longs;e habent. </s>

<s id="id.001575">Porro cum Rhomboides variæ &longs;int <expan abbr="diametror&umacr;">diametrorum</expan> <lb/>ad latera habitudines, varia quoque dari pote&longs;t propor<lb/>tio. </s>

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<s id="id.001577">Si autem Rhombus in <lb/>duos &longs;oluatur triangulos, alter diametrorum datur æqua<lb/>lis æqualibus lateribus æquicrurium triangulorum; <expan abbr="itaq;">itaque</expan> <lb/>in i&longs;tis mixti motus per diametros aequeveloces erunt &longs;im<lb/>plicibus, qui per latera longiora, velociores autem illis <lb/>qui per latera breuiora. </s>

<s id="id.001578">His igitur hoc pacto non perfun<lb/>ctoriè con&longs;ideratis, facile ex proprijs cau&longs;&longs;is, nî fallimu<gap/>, <lb/>hocce Ari&longs;totelicum & mirabile Problema &longs;oluitur. </s>

<s id="id.001578">His igitur hoc pacto non perfun<lb/>ctoriè con&longs;ideratis, facile ex proprijs cau&longs;&longs;is, nî fallimur, <lb/>hocce Ari&longs;totelicum & mirabile Problema &longs;oluitur. </s>

<s id="id.001579">E&longs;to enim Rhombus ABDC, <lb/>cuius diameter longior AD maior &longs;it <lb/>tum lateribus, tum etiam altera dia<lb/>metro BC. &longs;ecent autem &longs;e inuicem <lb/>diametri in E. </s>

<s id="id.001580">Ducatur queue ip&longs;is AB, <lb/>CD, parallela FG &longs;ecans longiorem <lb/>diametrum AD, in H, breuiorem ve<lb/>ro BC in I. & per I ip&longs;is BD AC paral<lb/>lela ducatur KIL, Cum ergo B mixto <lb/>motu per diametrum BC erit in I & <lb/>A per diametrum AD, mixto &longs;imili<lb/>ter motu erit in H, & quia motus mi<lb/>xti fiunt per diametros, vt dictum e&longs;t, <lb/>vt &longs;e habet AD ad BC, ita AE ad EB, per 15. propol. </s>

<s id="id.001582">item vt AE ad EB, ita per 4. propo&longs;. </s>

<s id="id.001583">6. AH ad BI. e&longs;t enim <lb/>IH ip&longs;i AB parallela. </s>

<s id="id.001582">item vt AE ad EB, ita per 4. propo&longs;. 6. AH ad BI. e&longs;t enim <lb/>IH ip&longs;i AB parallela. </s>

<s id="id.001584">Longior e&longs;t autem AH ip&longs;a BI, quip<lb/>pe quod AE longior &longs;it ip&longs;a EB. motus igitur mixtus pun<lb/>cti A per diametrum AD v&longs;que ad H velocior e&longs;t motu B, <lb/>per diametrum BC v&longs;quead I. </s>

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<s id="id.001588">Si enim Rhombus detur, ex duobus æquilateris <lb/>triangulis con&longs;tans, breuior diameter lateribus erit aequa<lb/>lis, quare non mouebitur citius motu &longs;implici punctum <lb/>per latus ac faciat mixto per minorem diametrum, quod <lb/>vt mirum propo&longs;uerat A ri&longs;toteles. </s>

<s id="id.001589">Si autem latus ip&longs;um <lb/>breuiori diametro &longs;it <expan abbr="l&omacr;gius">longius</expan>, nec mirum quoque erit &longs;im<lb/>plici motu moucri velocius quam mixto, quippe quod, vt <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.165.jpg"/>dictum e&longs;t, motus i&longs;ti à proportionibus linearum, per quas <lb/>mouentur, legem velocitatis atque tarditatis accipiant. <lb/></s>

<s id="id.001589">Si autem latus ip&longs;um <lb/>breuiori diametro &longs;it <expan abbr="l&omacr;gius">longius</expan>, nec mirum quoque erit &longs;im<lb/>plici motu moueri velocius quam mixto, quippe quod, vt <pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.164.jpg"/>dictum e&longs;t, motus i&longs;ti à proportionibus linearum, per quas <lb/>mouentur, legem velocitatis atque tarditatis accipiant. <lb/></s>

<s id="id.001590">Hæc igitur nos circa hoc mirabile Ari&longs;totelicum proble<lb/>ma con&longs;iderare &longs;it &longs;atis. </s>

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<s id="id.001600">Hic enim, quod ibi <lb/>quoque fecerat, &longs;ectorem ip&longs;um angulum appellauit, an<lb/>gulum vero maiorem maioris circuli &longs;ectorem, & mino<lb/>rem angulum minoris ip&longs;ius circuli &longs;ectorem dixit. </s>

<s id="id.001601">Clau<lb/>dit igitur dicens: quoniam circumferentiæ &longs;e habent vt <lb/>anguli, hoc e&longs;t, vt &longs;ectores, maior erit circumferentia ma<lb/>ioris circuli, & ex con&longs;equenti maior linea, per quam cir-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.166.jpg"/>cumuoluitur, ea per quam minor. </s>

<s id="id.001601">Clau<lb/>dit igitur dicens: quoniam circumferentiæ &longs;e habent vt <lb/>anguli, hoc e&longs;t, vt &longs;ectores, maior erit circumferentia ma<lb/>ioris circuli, & ex con&longs;equenti maior linea, per quam cir-<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.165.jpg"/>cumuoluitur, ea per quam minor. </s>

<s id="id.001602">Demon&longs;trationem ve<lb/>ro ex &longs;en&longs;u petijt. </s>

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<s id="id.001604">Ob&longs;curiu&longs;culè, hæc &longs;ua figura o&longs;tendit Ari&longs;to<lb/>teles. </s>

<s id="id.001605">Nos igitur claritatem amantibus, no&longs;tram aliquan<lb/>to, nî fallimur, clariorem, proponemus. </s>

<s id="id.001606">E&longs;to circulus <lb/>maior ABCD, mi<lb/>nor FGHI, circa i<lb/>dem, & commune <lb/><expan abbr="c&emacr;trum">centrum</expan> E. </s>

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<s id="id.001613">Mirabile itaque e&longs;t mi<lb/>norem circulum eodem tempore & circa idem centrum <lb/>circumuolutum, lineam pertran&longs;i&longs;&longs;e æqualem circum fe<lb/>rentiæ maioris circuli. </s>

<s id="id.001614">Nec &longs;ecius admirationem facitro<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-00-pageimg/007.00.167.jpg"/>tato minori circulo, maiorem vna <expan abbr="circumuolut&umacr;">circumuolutum</expan> lineam <lb/>metiri circumferentiæ minoris circuli æqualem. </s>

<s id="id.001614">Nec &longs;ecius admirationem facit ro<pb xlink:href="http://archimedes.fas.harvard.edu/images/007-01-pageimg/007.01.166.jpg"/>tato minori circulo, maiorem vna <expan abbr="circumuolut&umacr;">circumuolutum</expan> lineam <lb/>metiri circumferentiæ minoris circuli æqualem. </s>

<s id="id.001615">Rotetur <lb/>enim minoris circuli quadrans HI per rectam HL. erit i<lb/>gitur punctum I vbi M, æquali exi&longs;tente recta HM, ip&longs;i <lb/>curuæ HI. </s>

<s id="id.001616">Tuncautem facto motu centrum E erit vbi P, <lb/>exi&longs;tente EP, ip&longs;i HM æquali, demittatur autem ex P per <lb/>M, ip&longs;is HL CK perpendicularis PMN. </s>

<s id="id.001617">Et quoniam in <lb/>eadem linea &longs;unt DIE, vbi E fuerit in Pleritin M, & D in <lb/>N. quamobrem rotata quarta minori<gap/> circuli parte, ma<lb/>ioris interim circuli quadrans confecit &longs;patium CN æ<lb/>quale ip&longs;i HM, hoc minus circuli quadranti HI, quod vti<lb/>que e&longs;t admirabile. </s>

<s id="id.001617">Et quoniam in <lb/>eadem linea &longs;unt DIE, vbi E fuerit in PI erit in M, & D in <lb/>N. quamobrem rotata quarta minoris circuli parte, ma<lb/>ioris interim circuli quadrans confecit &longs;patium CN æ<lb/>quale ip&longs;i HM, hoc minus circuli quadranti HI, quod vti<lb/>que e&longs;t admirabile. </s>

<s id="id.001618">Porro cau&longs;&longs;am effectus huius mirifici diligenter quæ<lb/>rit Philo&longs;ophus, & inuenram accurate explicat. </s>

<s id="id.001618">Porro cau&longs;&longs;am effectus huius mirifici diligenter quæ<lb/>rit Philo&longs;ophus, & inueneram accurate explicat. </s>

<s id="id.001619">Occur<lb/>rit autem primo ab&longs;urdæ cuidam opinioni. </s>

<s id="id.001620">Diceret enim <lb/>qui&longs;piam, ideo tardius moueri maiorem circulum, ad mo<lb/>tum minoris, quod interim <expan abbr="d&umacr;">dum</expan> minor moucretur, aliquas <lb/>inter rotan dum moras interponeret, minor vero ad mo<lb/>tum maioris &longs;patia aliqua tran&longs;iliret, & ita &longs;patiorum fieri <lb/>ad æquationem. </s>

<s id="id.001620">Diceret enim <lb/>qui&longs;piam, ideo tardius moueri maiorem circulum, ad mo<lb/>tum minoris, quod interim <expan abbr="d&umacr;">dum</expan> minor moueretur, aliquas <lb/>inter rotandum moras interponeret, minor vero ad mo<lb/>tum maioris &longs;patia aliqua tran&longs;iliret, & ita &longs;patiorum fieri <lb/>ad æquationem. </s>

<s id="id.001621">Porro demon&longs;trationem aggre&longs;&longs;urus haec <lb/>a&longs;&longs;umit principia. </s>

<s id="id.001622">Eandem aequalemue potentiam, <expan abbr="aliquã">aliquam</expan> <lb/>magnitudinem tardius quidem mouere, aliquam vero <lb/>celerius. </s>

<s id="id.001623">quod autem natum e&longs;t aptum moueri, tardius <lb/>moueri, &longs;i &longs;imul cum non apto nato moueri, moueatur, <lb/>quam &longs;i &longs;eparatim moueretur, celerius autem &longs;i non &longs;imul <lb/><figure id="id.007.00.167.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-00-figures/007.00.167.1.jpg"></figure><lb/>cum eo moueatur. </s>

<s id="id.001623">quod autem natum e&longs;t aptum moueri, tardius <lb/>moueri, &longs;i &longs;imul cum non apto nato moueri, moueatur, <lb/>quam &longs;i &longs;eparatim moueretur, celerius autem &longs;i non &longs;imul <lb/><figure id="id.007.01.166.1.jpg" xlink:href="http://archimedes.fas.harvard.edu/images/007-01-figures/007.01.166.1.jpg"></figure><lb/>cum eo moueatur. </s>

<s id="id.001624">E&longs;to enim corpus A leue <lb/>quidem & aptum natum moueri&longs;ur&longs;um, cui <lb/>connectatur B, aptum natum moueri deor<lb/>&longs;um, Si quis igitur mouere conetur corpus A <lb/>&longs;ur&longs;um difficilius mouebit, & tardius <expan abbr="iunct&umacr;">iunctum</expan> <lb/>nempeip&longs;i B, quam &longs;i ab ip&longs;o e&longs;&longs;et <expan abbr="&longs;ei&umacr;ctum">&longs;eiunctum</expan>. <lb/></s>