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version 1.1, 2002/06/18 09:37:13

version 1.2, 2002/06/24 18:03:53

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<!ELEMENT s

<!ATTLIST s

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<info>

<author>Pierre Gassendi</author>

<author>Gassendi, Pierre</author>

<title>De proportione qua gravia decidentia accelerantur</title>

<place>Paris</place>

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<s><emph type="italics"/>Impugnat<emph.end type="italics"/> G<emph type="italics"/>alileus definitionem R. Patri probatam, <lb/>quòd &longs;i velocitates e&longs;&longs;ent, vt emen&longs;a &longs;patia, atque idcircò <lb/>&longs;patium v. c. duplum percurreretur velocitate dupla illius, qua <lb/>dimidium: &longs;equeretur duplum, & dimidium, &longs;eu totum, & <lb/>partem, eodem, aut æquali tempore percurri. Nempe &longs;eu <lb/>motus æquabilis, &longs;eu acceleratus æquabiliter &longs;it, non potest ce<lb/>leritas e&longs;&longs;e dupla per duplum &longs;patij, quin ea ex&longs;i&longs;tente vbique <lb/>dupla, duplæ partes percurrantur quibu&longs;libet temporibus, &longs;ic<lb/>que perueniatur eodem tempore ad dupli, & ad dimidij finem. <lb/>Contendit R. P. committi heic Paralogi&longs;mum: & nullam <lb/>tamen rationem profert, quàm quæ continetur his verbis,<emph.end type="italics"/><lb/>Si graue de&longs;cendens per AB, tempus quodcum<lb/>

<arrow.to.target n="fig1"></arrow.to.target><lb/>que in&longs;umat, putà quadrantem; ac deinde BC <lb/>ip&longs;i AB æquale dimidio quadrante percurrat: <lb/>quis neget in C duplam haberi velocitatem eius, <lb/>quæ fuit in B? & tamen idem graue totam AC, <lb/>& dimidium eius AB non percurreret. <emph type="italics"/>Vbi &longs;anè <lb/>nihil aliud, quàm rem controuer&longs;am &longs;upponit, habetque <lb/>pro principio: videlicet &longs;ecundam partem percurri di<lb/>midio temporis, quo primam. Atque id quidem præter Incom<lb/>modum ex po&longs;itione hac con&longs;equens, quòd cùm oporteat pari <lb/>modo percurri partem tertiam dimidio temporis, quo &longs;ecun<lb/>dam; quartam, quo tertiam, &c. debeat cum effluxu temporis <lb/>&longs;ecundi percurri spatium infinitum: quatenus omnia illa di<lb/>midiorum dimidia, &longs;iue fragmenta temporis non po&longs;&longs;unt<emph.end type="italics"/>

<figure id="fig1"></figure><lb/>que in&longs;umat, putà quadrantem; ac deinde BC <lb/>ip&longs;i AB æquale dimidio quadrante percurrat: <lb/>quis neget in C duplam haberi velocitatem eius, <lb/>quæ fuit in B? & tamen idem graue totam AC, <lb/>& dimidium eius AB non percurreret. <emph type="italics"/>Vbi &longs;anè <lb/>nihil aliud, quàm rem controuer&longs;am &longs;upponit, habetque <lb/>pro principio: videlicet &longs;ecundam partem percurri di<lb/>midio temporis, quo primam. Atque id quidem præter Incom<lb/>modum ex po&longs;itione hac con&longs;equens, quòd cùm oporteat pari <lb/>modo percurri partem tertiam dimidio temporis, quo &longs;ecun<lb/>dam; quartam, quo tertiam, &c. debeat cum effluxu temporis <lb/>&longs;ecundi percurri spatium infinitum: quatenus omnia illa di<lb/>midiorum dimidia, &longs;iue fragmenta temporis non po&longs;&longs;unt<emph.end type="italics"/>

<pb/><emph type="italics"/>æquari vni integro (cuius &longs;emper relinquitur inexhaustum <lb/>aliquid) ni&longs;i priùs omnia, hoc e&longs;t infinita, fuerint numerata.<emph.end type="italics"/><lb/>A. p. 14. in 21. </s>

<s>ART. XIII. XIV. XV. XVI. XVII. <lb/>XVIII. De Po&longs;tulato Galilei circa motum <lb/>&longs;uper æque-altis, non æque-inclinatis planis. </s>

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<s>IIII. Itaque illud heic apponendo, vides primùm

<pb pagenum="6"/>lineas AB, AC, angulum creanteis in A, &longs;ic diui&longs;as e&longs;&longs;e <lb/>heinc inde in parteis æqualeis, ad puncta D, E, F, G, H, <lb/>I, K, L (po&longs;&longs;ent autem in longè plureis continuatæ di<lb/>uidi) vt lineæ ductæ cùm inter ip&longs;a puncta, tùm ex ip&longs;is <lb/>in puncta M, N, O, totum &longs;patium KAL di&longs;pe&longs;cant <lb/>in triangula inter <lb/>

<arrow.to.target n="fig2"></arrow.to.target><lb/>&longs;e &longs;imilia, ac pror<lb/>sù æqualia. Cùm <lb/>po&longs;&longs;imus porto <lb/>habere punctum <lb/>A pro initio tem<lb/>poris, pro initio <lb/>&longs;patij, pro initio <lb/>velocitatis, quæ <lb/>tria heic in motu <lb/>&longs;pectantur, ac vna <lb/>cùm ip&longs;o inci<lb/>piunt; Po&longs;&longs;umus imprimis habere parteis æqualeis al<lb/>terutrius, aut vtriu&longs;que lineæ AB, AC pro partibus, <lb/>&longs;iue momentis æqualibus temporis ab initio fluentis, <lb/>adeòproinde, vt AE, v. g. repræ&longs;entet <expan abbr="prim&utilde;">primum</expan> momen<lb/>tum, EG &longs;ecundum, GI: tertium, IL quartum. <lb/>Po&longs;&longs;umus &longs;ecundò habere æqualia illa triangula pro <lb/>æqualibus &longs;patij partibus, quæ ab initio percurrun<lb/>tur; adeò vt ductâ &longs;eor&longs;im lineâ PQ ca&longs;um refe<lb/>rente per orgyias &longs;exdecim, Triangulum ADE <lb/>repræ&longs;entet primam orgyiam PR, quæ primò mo<lb/>mento percurritur; tria proxima, treis orgyias RS, <lb/>quæ &longs;ecundo; quinque &longs;equentia quinque orgyias <lb/>ST, quæ tertiò; & &longs;eptem &longs;uccedentia &longs;eptem or-

<figure id="fig2"></figure><lb/>&longs;e &longs;imilia, ac pror<lb/>sù æqualia. Cùm <lb/>po&longs;&longs;imus porto <lb/>habere punctum <lb/>A pro initio tem<lb/>poris, pro initio <lb/>&longs;patij, pro initio <lb/>velocitatis, quæ <lb/>tria heic in motu <lb/>&longs;pectantur, ac vna <lb/>cùm ip&longs;o inci<lb/>piunt; Po&longs;&longs;umus imprimis habere parteis æqualeis al<lb/>terutrius, aut vtriu&longs;que lineæ AB, AC pro partibus, <lb/>&longs;iue momentis æqualibus temporis ab initio fluentis, <lb/>adeòproinde, vt AE, v. g. repræ&longs;entet <expan abbr="prim&utilde;">primum</expan> momen<lb/>tum, EG &longs;ecundum, GI: tertium, IL quartum. <lb/>Po&longs;&longs;umus &longs;ecundò habere æqualia illa triangula pro <lb/>æqualibus &longs;patij partibus, quæ ab initio percurrun<lb/>tur; adeò vt ductâ &longs;eor&longs;im lineâ PQ ca&longs;um refe<lb/>rente per orgyias &longs;exdecim, Triangulum ADE <lb/>repræ&longs;entet primam orgyiam PR, quæ primò mo<lb/>mento percurritur; tria proxima, treis orgyias RS, <lb/>quæ &longs;ecundo; quinque &longs;equentia quinque orgyias <lb/>ST, quæ tertiò; & &longs;eptem &longs;uccedentia &longs;eptem or-

<pb pagenum="7"/>gyias TQ, quæ quarto. Con&longs;tat autem exinde &longs;patia <lb/>aggregata ita &longs;e habere, &longs;icut quadrata tempo<lb/>

<arrow.to.target n="fig3"></arrow.to.target><lb/>rum; quandò ADE triangulum (&longs;patiumve <lb/>PR) e&longs;t vnum; quemadmodum quadratum <lb/>ip&longs;ius AE, hoc e&longs;t temporis vnius, e&longs;t vnum; & <lb/>aggregatum AFG (&longs;eu PS) e&longs;t quatuor; quem<lb/>admodum quadratum AG, duorum, e&longs;t qua<lb/>tuor; & aggregatum AHI (&longs;eu PT) e&longs;t nouem; <lb/>quemadmodum quadratum AI trium, e&longs;t no<lb/>uem; & aggregatum AKL (&longs;eu PQ) e&longs;t &longs;ex<lb/>decim; quemadmodum quadratum AL qua<lb/>tuor, e&longs;t &longs;exdecim. Po&longs;&longs;umus tertiò habere li<lb/>neam DE, pro primo gradu velocitatis acqui<lb/>&longs;itæ in fine primi temporis: quatenus, vt pri<lb/>mùm tempus AE non e&longs;t indiuiduum, &longs;ed in <lb/>tot in&longs;tantia, &longs;eu temporula pote&longs;t diuidi, quot <lb/>&longs;unt puncta, particulæve in ip&longs;a AE (aut AD) <lb/>ita neque gradus velocitatis indiuiduus e&longs;t, &longs;eu <lb/>vno in&longs;tanti, acqui&longs;itus totus; &longs;ed ab v&longs;que ini<lb/>tio per totum primum tempus incre&longs;cit, ac re<lb/>præ&longs;entari pote&longs;t per tot lineas, quot po&longs;&longs;unt <lb/>parallelæ duci ip&longs;i DE inter puncta linearum <lb/>AD, & AE; adeò vt quemadmodum illæ lineæ <lb/>continuo incre&longs;cunt à puncto A in lineam DE, &longs;ic <lb/>velocitas à principio motus continuò incre&longs;cat, & re<lb/>præ&longs;entata, qualis e&longs;t in interceptis primi temporis in<lb/>&longs;tantibus, per interceptas lineas, repræ&longs;entetur qualis <lb/>e&longs;t in vltimo in&longs;tanti eiu&longs;dem primi temporis, per <lb/>ip&longs;am DE inter vltima ductam puncta. Et quia ve<lb/>locitas deinceps incre&longs;cere pergens, repræ&longs;entari rur-

<figure id="fig3"></figure><lb/>rum; quandò ADE triangulum (&longs;patiumve <lb/>PR) e&longs;t vnum; quemadmodum quadratum <lb/>ip&longs;ius AE, hoc e&longs;t temporis vnius, e&longs;t vnum; & <lb/>aggregatum AFG (&longs;eu PS) e&longs;t quatuor; quem<lb/>admodum quadratum AG, duorum, e&longs;t qua<lb/>tuor; & aggregatum AHI (&longs;eu PT) e&longs;t nouem; <lb/>quemadmodum quadratum AI trium, e&longs;t no<lb/>uem; & aggregatum AKL (&longs;eu PQ) e&longs;t &longs;ex<lb/>decim; quemadmodum quadratum AL qua<lb/>tuor, e&longs;t &longs;exdecim. Po&longs;&longs;umus tertiò habere li<lb/>neam DE, pro primo gradu velocitatis acqui<lb/>&longs;itæ in fine primi temporis: quatenus, vt pri<lb/>mùm tempus AE non e&longs;t indiuiduum, &longs;ed in <lb/>tot in&longs;tantia, &longs;eu temporula pote&longs;t diuidi, quot <lb/>&longs;unt puncta, particulæve in ip&longs;a AE (aut AD) <lb/>ita neque gradus velocitatis indiuiduus e&longs;t, &longs;eu <lb/>vno in&longs;tanti, acqui&longs;itus totus; &longs;ed ab v&longs;que ini<lb/>tio per totum primum tempus incre&longs;cit, ac re<lb/>præ&longs;entari pote&longs;t per tot lineas, quot po&longs;&longs;unt <lb/>parallelæ duci ip&longs;i DE inter puncta linearum <lb/>AD, & AE; adeò vt quemadmodum illæ lineæ <lb/>continuo incre&longs;cunt à puncto A in lineam DE, &longs;ic <lb/>velocitas à principio motus continuò incre&longs;cat, & re<lb/>præ&longs;entata, qualis e&longs;t in interceptis primi temporis in<lb/>&longs;tantibus, per interceptas lineas, repræ&longs;entetur qualis <lb/>e&longs;t in vltimo in&longs;tanti eiu&longs;dem primi temporis, per <lb/>ip&longs;am DE inter vltima ductam puncta. Et quia ve<lb/>locitas deinceps incre&longs;cere pergens, repræ&longs;entari rur-

<pb pagenum="8"/>&longs;us pote&longs;t per lineas maiores, maiore&longs;que continenter <lb/>ductas inter omnia puncta &longs;uccedentia re&longs;iduarum <lb/>linearum DB, & EC, heinc efficitur, vt linea FG re<lb/>præ&longs;entet velocitatem acqui&longs;itam in fine &longs;ecundi mo<lb/>menti: linea HI acqui&longs;itam in fine tertij, & linea KL <lb/>acqui&longs;itam in fine quarti. Con&longs;tat verò inde, vt ve<lb/>locitates &longs;e habeant &longs;icut tempora; cùm ob triangulos <lb/>anguli communis, & parallelarum ba&longs;ium, notum &longs;it <lb/>e&longs;&longs;e vt DE ad EA, ita FG ad GA, HI ad IA, & KL <lb/>ad LA. Atque hæc quidem, vt clariùs con&longs;tet, qua <lb/>de re inter nos agatur. </s>

<s>V. Iam, Tu initio argumentum Di&longs;&longs;ertationis ita <lb/>partiris, vt duo præ&longs;tanda tibi proponas. Vnum, <lb/><emph type="italics"/>vt Galilei hac inre errores, eorumque fonteis aperias; claré<lb/>que demon&longs;tres ea, quæ ab ip&longs;o de acceleratione Motus in na<lb/>turali de&longs;cen&longs;u grauium Libro &longs;ecundo nouæ &longs;cientiæ, & <lb/>toto Dialogo tertio dicta &longs;unt, non modò &longs;u&longs;picionibus meris, <lb/>vixque probabilibus coniecturis niti, &longs;ed ex principijs etiam <lb/>apertè fal&longs;is, euidentibu&longs;que paralogi&longs;mis omnia concludi; ex <lb/>quo con&longs;equens &longs;it, nouam illam &longs;cientiam euane&longs;cere, quam <lb/>ingenio&longs;o quidem, & plau&longs;ibili, &longs;ed inani tamen, & ca&longs;&longs;o <lb/>apparatu nobis<emph.end type="italics"/> G<emph type="italics"/>alileus exhibuerit.<emph.end type="italics"/> Alterum, <emph type="italics"/>vt reiecta<emph.end type="italics"/><lb/>G<emph type="italics"/>alilei p&longs;eudo-&longs;cientia, veram tu, ac certam in eius locum <lb/>&longs;ubstituas, rationemque, modum, ac men&longs;uram acceleratio<lb/>nis eiu&longs;dem in naturali de&longs;cen&longs;u grauium ex euidentibus, at<lb/>que indubitatis experientiis demonstres.<emph.end type="italics"/> Circa priùs de<lb/>inde caput, duo &longs;unt, in quibus occuparis; nam impe<lb/>tis <emph type="italics"/>primò<emph.end type="italics"/> definitionem Motus æquabiliter accelerati à <lb/>Galileo traditam: & <emph type="italics"/>&longs;ecundò<emph.end type="italics"/> quod idem ait de gradi<lb/>bus velocitatis, qui acquiruntur à mobili, dùm &longs;uper

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<s>VII Ego intereà, optime Vir, neque video De<lb/>&longs;initionem a Galileo in&longs;titutam à te directè impu<lb/>gnari, fal&longs;itati&longs;ve vllius conuinci; neque agno&longs;co quî <lb/>magis vera, congruaque potuerit in&longs;titui. Quippe <lb/>memini&longs;&longs;e, aut potiùs adnota&longs;&longs;e diligenter oportet <lb/>agi heic de motu æquabiliter accelerato, &longs;iue cuius <lb/>celeritas continenter, vniformiterque incre&longs;cat, ne<lb/>que vllum &longs;it momentum con&longs;equentis temporis, in <lb/>quo motus non &longs;it velocior, quàm in quovis antece<lb/>dente, & in quo non eadem ratione ip&longs;a velocitas <lb/>augeatur. Fieri id porrò e&longs;t manife&longs;tum ex ijs, quæ <lb/>deducta iam &longs;unt; &longs;i æqualibus temporibus æqualia <lb/>celeritatis momenta, &longs;eu incrementa acquirantur. <lb/>Nam vt repetamus &longs;uperiorem figuram, nemo dicat <lb/>celeritatem vniformiter incre&longs;centem po&longs;&longs;e meliùs <lb/>repræ&longs;entari, quàm diductione crurum anguli, linea<lb/>rumve AB, AC, quatenus complectuntur &longs;pa<lb/>tium, quod à puncto A magis continenter, vni<lb/>formiterque cre&longs;cere non po&longs;&longs;it: ac aliunde angu<lb/>lus BAC apertior, aut conductior (prout finge<lb/>tur maior, aut minor velocitas) valet v&longs;urpari. <lb/>Vt autem rem magis ob oculos ponam; duco ec<lb/>ce lineam VX, per ip&longs;um apicem A, quæ cum li-

<pb pagenum="11"/>neis AB, AC, angulos con&longs;tituat vtrimque æqua<lb/>

<arrow.to.target n="fig4"></arrow.to.target><lb/>leis, & &longs;eruatâ eorum<lb/>dem angulorum men<lb/>&longs;urâ, ita fluere conci<lb/>piatur, vt totum &longs;pa<lb/>tium BAC peruadat. <lb/>Tunc enim manife<lb/>&longs;tum e&longs;t portiones <lb/>huius lineæ continuò <lb/>veluti re&longs;ectas, inter<lb/>cepta&longs;que à lineis AB, <lb/>AC, cre&longs;cere &longs;emper, <lb/>&longs;eu maiores, maiore&longs;<lb/>que vniformiter e&longs;&longs;e; ac non portiones &longs;emel inter<lb/>ceptas perire, &longs;ed ip&longs;is permanentibus nouas, nouaf<lb/>que heinc inde continenter &longs;uper-acquiri. Cùm <lb/>verò etiam gradus velocitatis con&longs;imiliter cre&longs;cant, <lb/>&longs;iue maiores, maiore&longs;que vniformiter euadant, ac <lb/>&longs;emel acqui&longs;iti non pereant, &longs;ed ip&longs;is &longs;uper&longs;titibus, <lb/>per&longs;euerantibu&longs;que noua, atque noua momenta, &longs;iue <lb/>incrementa velocitatis &longs;uper-addantur; &longs;upere&longs;t, vt <lb/>quemadmodum linearum illarum incrementa fiunt, <lb/>&longs;ic fiant quoque velocitatum. Notum e&longs;t autem. vt <lb/>acceptis partibus æqualibus lineæ AC, verbi cau&longs;sâ, <lb/>incrementa earum portionum, &longs;iue linearum paralle<lb/>larum interceptarum, acquirantur &longs;emper æqualia <lb/>&longs;ub æqualibus illis partibus. Nam, vt &longs;ub AE ac<lb/>qui&longs;ita e&longs;t linea DE, ita &longs;ub EG, acquiritur æqua<lb/>lis alia; cùm ip&longs;a FG &longs;it dupla ip&longs;ius DE; & &longs;ub <lb/>GI iterum alia; cùm ip&longs;a HI &longs;it eiu&longs;dem tripla; &

<figure id="fig4"></figure><lb/>leis, & &longs;eruatâ eorum<lb/>dem angulorum men<lb/>&longs;urâ, ita fluere conci<lb/>piatur, vt totum &longs;pa<lb/>tium BAC peruadat. <lb/>Tunc enim manife<lb/>&longs;tum e&longs;t portiones <lb/>huius lineæ continuò <lb/>veluti re&longs;ectas, inter<lb/>cepta&longs;que à lineis AB, <lb/>AC, cre&longs;cere &longs;emper, <lb/>&longs;eu maiores, maiore&longs;<lb/>que vniformiter e&longs;&longs;e; ac non portiones &longs;emel inter<lb/>ceptas perire, &longs;ed ip&longs;is permanentibus nouas, nouaf<lb/>que heinc inde continenter &longs;uper-acquiri. Cùm <lb/>verò etiam gradus velocitatis con&longs;imiliter cre&longs;cant, <lb/>&longs;iue maiores, maiore&longs;que vniformiter euadant, ac <lb/>&longs;emel acqui&longs;iti non pereant, &longs;ed ip&longs;is &longs;uper&longs;titibus, <lb/>per&longs;euerantibu&longs;que noua, atque noua momenta, &longs;iue <lb/>incrementa velocitatis &longs;uper-addantur; &longs;upere&longs;t, vt <lb/>quemadmodum linearum illarum incrementa fiunt, <lb/>&longs;ic fiant quoque velocitatum. Notum e&longs;t autem. vt <lb/>acceptis partibus æqualibus lineæ AC, verbi cau&longs;sâ, <lb/>incrementa earum portionum, &longs;iue linearum paralle<lb/>larum interceptarum, acquirantur &longs;emper æqualia <lb/>&longs;ub æqualibus illis partibus. Nam, vt &longs;ub AE ac<lb/>qui&longs;ita e&longs;t linea DE, ita &longs;ub EG, acquiritur æqua<lb/>lis alia; cùm ip&longs;a FG &longs;it dupla ip&longs;ius DE; & &longs;ub <lb/>GI iterum alia; cùm ip&longs;a HI &longs;it eiu&longs;dem tripla; &

<pb pagenum="12"/>&longs;ub IL rursùs alia, cùm ip&longs;a KL &longs;it eiu&longs;dem qua<lb/>drupla; atque ita porrò, &longs;eu vlteriùs pergas, &longs;eu <lb/>alia puncta intra ea&longs;dem parteis lineæ AC, alia&longs;<lb/>que parallelas commemoratis interceptas, &longs;ingula&longs;<lb/>que &longs;uis punctis re&longs;pondenteis, accipias. Quare & <lb/>a&longs;&longs;umptis partibus æqualibus temporis per parteis <lb/>æqualeis lineæ AC repræ&longs;entatis, <expan abbr="not&utilde;">notum</expan> e&longs;t momenta, <lb/>&longs;eu incrementa velocitatis per parallelas repræ&longs;entatæ, <lb/>æqualia acquiri &longs;ub huiu&longs;modi partibus; adeò vt <lb/>qualis gradus velocitatis acqui&longs;itus e&longs;t in fine primi <lb/>temporis vnus, talis alius, hoc e&longs;t æqualis, &longs;it ip&longs;i &longs;uper<lb/>acqui&longs;itus in fine &longs;ecundi, ac &longs;int iam duo; & iterum <lb/>æqualis alius in fine tertij, ac &longs;in: iam tres; & rursùs <lb/>alius in fine quarti, ac &longs;int iam quatuor; atque ita de <lb/>cæteris, &longs;iue con&longs;equentibus, &longs;iue inter&longs;umptis. </s>

<s>VIII. Sic itaque mihi videtur Motus æquabili<lb/>ter, hoc e&longs;t continenter, vniformiterque acceleratus <lb/>perquàm appo&longs;itè definiri is, <emph type="italics"/>Qui à quiete recedens <lb/>temporibus æqualibus æqualia celeritatis momenta<emph.end type="italics"/> (aug<lb/>mentave) <emph type="italics"/>acquirat;<emph.end type="italics"/> cùm præ&longs;ertim non videam <lb/>po&longs;&longs;e ip&longs;um alia ratione concipi, aut de&longs;cribi talem. <lb/>Nam quod &longs;pectat quidem ad illam à te laudatam <lb/>definitionem, qua motus æquabiliter acceleratus de&longs;<lb/>cribitur is, <emph type="italics"/>Qui æqualibus spatiis æqualia celeritatis aug<lb/>menta acquirit:<emph.end type="italics"/> dic amabò quanam ratione concipere <lb/>exinde licear acceleratum æquabiliter motum? E&longs;to <lb/>enim &longs;patium percurrendum v. c. linea AB in par<lb/>teis æqualeis diui&longs;a ad puncta C, D, E, F, G, I, K. <lb/>Decidat mobile ex A; & in C fine primæ partis ac<lb/>qui&longs;ierit primum velocitatis gradum; in D autem

<pb pagenum="13"/>&longs;ecundum, quo ad priorem per&longs;euerantem iunctp <lb/>

<arrow.to.target n="fig5"></arrow.to.target><lb/>duo iam &longs;int: in E tertium, quo <lb/>iuncto ad duos &longs;uperiores, per<lb/>&longs;euerantei&longs;que &longs;int tres; in F <lb/>quartum, & ita porrò, quo<lb/>v&longs;que in B acqui&longs;ierit nonum, <lb/>quo iuncto cum octo antece<lb/>dentibus &longs;int nouem. Iam cùm <lb/>quilibet horum graduum la<lb/>titudinem quandam habeat; <lb/>neque enim e&longs;t magis indiui&longs;i<lb/>bilis, aut ex indiui&longs;ibilibus <lb/>con&longs;tans, quàm pars AC, CD, <lb/>DE, quælibet-ve alia: ac idcir<lb/>cò ip&longs;e quoque incre&longs;cat æqua<lb/>biliter, vnoque tenore: repræ&longs;entetur primus gradus <lb/>per triangulum ALC, vt pote à puncto, &longs;eu angulo <lb/>A ad ba&longs;in LC æquabiliter, vnóque tenore cre&longs;cen<lb/>tem. Aptentur deinde ad CD duo triangula æqua<lb/>lia tum inter &longs;e, tum cum ip&longs;o ALC, <expan abbr="quor&utilde;">quorum</expan> CMD <lb/>repræ&longs;entet illum, qui &longs;ecundò acquiritur, LCM au<lb/>tem primò aqui&longs;itum, ac per&longs;euerantem. Nihil e&longs;t <lb/>opus, vt de&longs;ude<gap/> ad o&longs;tendendum non increui&longs;&longs;e <lb/>velocitatem æquabiliter, eodemve tenore ex C in D, <lb/>quo inc&oelig;perat, perrexeratque v&longs;que in D; vt feci&longs;&longs;et <lb/>enim, oporteret de&longs;criptum e&longs;&longs;e non quadrangulum <lb/>LD con&longs;tans ex duobus triangulis; &longs;ed trapezion CN <lb/>con&longs;titutum ex tribus. Eadem autem ratione mani<lb/>fe&longs;tum e&longs;t, &longs;i ad DE aptentur tria triangula, defutura <lb/>duo; &longs;i ad EF quatuor, defutura tria, & ita deinceps,

<figure id="fig5"></figure><lb/>duo iam &longs;int: in E tertium, quo <lb/>iuncto ad duos &longs;uperiores, per<lb/>&longs;euerantei&longs;que &longs;int tres; in F <lb/>quartum, & ita porrò, quo<lb/>v&longs;que in B acqui&longs;ierit nonum, <lb/>quo iuncto cum octo antece<lb/>dentibus &longs;int nouem. Iam cùm <lb/>quilibet horum graduum la<lb/>titudinem quandam habeat; <lb/>neque enim e&longs;t magis indiui&longs;i<lb/>bilis, aut ex indiui&longs;ibilibus <lb/>con&longs;tans, quàm pars AC, CD, <lb/>DE, quælibet-ve alia: ac idcir<lb/>cò ip&longs;e quoque incre&longs;cat æqua<lb/>biliter, vnoque tenore: repræ&longs;entetur primus gradus <lb/>per triangulum ALC, vt pote à puncto, &longs;eu angulo <lb/>A ad ba&longs;in LC æquabiliter, vnóque tenore cre&longs;cen<lb/>tem. Aptentur deinde ad CD duo triangula æqua<lb/>lia tum inter &longs;e, tum cum ip&longs;o ALC, <expan abbr="quor&utilde;">quorum</expan> CMD <lb/>repræ&longs;entet illum, qui &longs;ecundò acquiritur, LCM au<lb/>tem primò aqui&longs;itum, ac per&longs;euerantem. Nihil e&longs;t <lb/>opus, vt de&longs;ude<gap/> ad o&longs;tendendum non increui&longs;&longs;e <lb/>velocitatem æquabiliter, eodemve tenore ex C in D, <lb/>quo inc&oelig;perat, perrexeratque v&longs;que in D; vt feci&longs;&longs;et <lb/>enim, oporteret de&longs;criptum e&longs;&longs;e non quadrangulum <lb/>LD con&longs;tans ex duobus triangulis; &longs;ed trapezion CN <lb/>con&longs;titutum ex tribus. Eadem autem ratione mani<lb/>fe&longs;tum e&longs;t, &longs;i ad DE aptentur tria triangula, defutura <lb/>duo; &longs;i ad EF quatuor, defutura tria, & ita deinceps,

<pb pagenum="14"/>quov&longs;que, &longs;i ad KB aptentur nouem, &longs;int defutura <lb/>octo; vt proinde intelligamus totidem dce&longs;&longs;e ad acce<lb/>lerationis æquabilitatem velocitatis gradus, quot nu<lb/>merare licet triangulos ad læuam è regione cuiu&longs;que <lb/>partis, complendo &longs;ummam triangulorum APB. <lb/>Con&longs;tare ergo videtur Motum æquabiliter accelera<lb/>tum definiti non po&longs;&longs;e illum, <emph type="italics"/>Qui æquabilibus &longs;patiis <lb/>æqualia celeritatis augmenta acquirat;<emph.end type="italics"/> &longs;ed potiùs illum, <lb/><emph type="italics"/>Qui acquirat æqualia æqualibus temporibus:<emph.end type="italics"/> atque idcircò <lb/>definitionem à Galileo traditam e&longs;&longs;e meritò præfe<lb/>rendam. </s>

<s><emph type="center"/><emph type="italics"/>De Paralogi&longs;mo, qui Galileo Definitionem &longs;puriam <lb/>impugnanti obiicitur.<emph.end type="italics"/><emph.end type="center"/></s>

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Line 995

<s>IX. Ac tu id quidem non fers; fed ais, <emph type="italics"/>Mirari to <lb/>&longs;atis non po&longs;&longs;e, quomodo<emph.end type="italics"/> G<emph type="italics"/>alileus vir alioquin perspicacis <lb/>ingenij receptam communi con&longs;en&longs;u motus accelerati defini<lb/>tionem non modò fal&longs;am, atque impoßibilem exi&longs;timauerit; <lb/>&longs;ed patenti quoque, ip&longs;i&longs;que tyronibus obuio paralogi&longs;mo <lb/>eiu&longs;dem fal&longs;itatem palam, atque euidenter demon&longs;tra&longs;&longs;e adeò <lb/>prafidenter a&longs;&longs;eruerit; & quod ampliùs e&longs;t, etiam viru non <lb/>ineruditis per&longs;ua&longs;erit.<emph.end type="italics"/> Tum autem pergis, <emph type="italics"/>Audi igitur; <lb/>mi<emph.end type="italics"/> G<emph type="italics"/>a&longs;&longs;ende, & mecum mirare tanti viri demon&longs;trationem. <lb/>Si acceleratio motus,<emph.end type="italics"/> inquit, <emph type="italics"/>in de&longs;cen&longs;u grauium æquali<lb/>bus &longs;patiis æqualia &longs;umeret velocitatu ineremema, e&longs;&longs;ent <lb/>&longs;ine dubio velocitates inter &longs;e, vt emen&longs;a spatia: At quoite&longs;. <lb/>cúmque velocitates inter &longs;e &longs;unt vt emen&longs;a &longs;patia, debent <lb/>nece&longs;&longs;ariò ea spatia aut eodem, aut æquali tempore percur<lb/>ri. Si igitur velocitas acqui&longs;ita per totam AC eam <lb/>rationem habeat ad velocitatem acqui&longs;itam per AB, quam<emph.end type="italics"/>

<pb pagenum="15"/><emph type="italics"/>spatium AC ad spatium AB, nece&longs;&longs;e e&longs;t, vt spatium<emph.end type="italics"/><lb/>

<arrow.to.target n="fig6"></arrow.to.target><lb/><emph type="italics"/>totum AC eodem, aut æquali tempore decurratur, <lb/>quo spatium AB ab&longs;oluitur. Jmpoßibile est au<lb/>tem, vt corpus graue de&longs;cendens per AC eodem, <lb/>aut æquali tempore percurrat totam AC, quo per<lb/>currit partem eius AB, ni&longs;i motus fiat in instanti. <lb/>Tam impoßibile e&longs;t igitur, vt velocitates in de&longs;cen&longs;u <lb/>grauium inter &longs;e &longs;int, vt emen&longs;a &longs;patia (ac proinde, <lb/>vt etiam æqualibus &longs;patiis cre&longs;cant æqualiter) quàm <lb/>impoßibile e&longs;t motum illum fieri in instanti.<emph.end type="italics"/> Pergis &longs;ub<lb/>inde, <emph type="italics"/>Proh tuam, mi<emph.end type="italics"/> G<emph type="italics"/>a&longs;&longs;ende, Philo&longs;ophorumque omnium, <lb/>ac Mathematicorum fidem! istud<gap/>ne demon&longs;trare e&longs;t? Et <lb/>tamen mirum quantum Galileus de hac, vt putat, &longs;ubtili, <lb/>clara, euidenti, ac Mathematica demon&longs;tratione &longs;ibi applau<lb/><gap/>at, quam integra pagina mirificis laudibus exaggerat. Sed <lb/>illud multò adhûc mirabiùus, quod Lynceus Philo&longs;ophus, ac <lb/>Mathematicus, Lynceorumque princeps in tam aperta <lb/>luce cæcutiat, & vir eius nominis tam facilè deludatur.<emph.end type="italics"/></s>

<figure id="fig6"></figure><lb/><emph type="italics"/>totum AC eodem, aut æquali tempore decurratur, <lb/>quo spatium AB ab&longs;oluitur. Jmpoßibile est au<lb/>tem, vt corpus graue de&longs;cendens per AC eodem, <lb/>aut æquali tempore percurrat totam AC, quo per<lb/>currit partem eius AB, ni&longs;i motus fiat in instanti. <lb/>Tam impoßibile e&longs;t igitur, vt velocitates in de&longs;cen&longs;u <lb/>grauium inter &longs;e &longs;int, vt emen&longs;a &longs;patia (ac proinde, <lb/>vt etiam æqualibus &longs;patiis cre&longs;cant æqualiter) quàm <lb/>impoßibile e&longs;t motum illum fieri in instanti.<emph.end type="italics"/> Pergis &longs;ub<lb/>inde, <emph type="italics"/>Proh tuam, mi<emph.end type="italics"/> G<emph type="italics"/>a&longs;&longs;ende, Philo&longs;ophorumque omnium, <lb/>ac Mathematicorum fidem! istud<gap/>ne demon&longs;trare e&longs;t? Et <lb/>tamen mirum quantum Galileus de hac, vt putat, &longs;ubtili, <lb/>clara, euidenti, ac Mathematica demon&longs;tratione &longs;ibi applau<lb/><gap/>at, quam integra pagina mirificis laudibus exaggerat. Sed <lb/>illud multò adhûc mirabiùus, quod Lynceus Philo&longs;ophus, ac <lb/>Mathematicus, Lynceorumque princeps in tam aperta <lb/>luce cæcutiat, & vir eius nominis tam facilè deludatur.<emph.end type="italics"/></s>

<s>X. Ego verò, ô optime, ac religio&longs;&longs;ime Vir, quo <lb/>me cen&longs;u putem iri habitum, qui non &longs;im ex viris <lb/>non ineruditis, & eandem tamen cum Galileo opi<lb/>nionem per&longs;ua&longs;us &longs;im, ac perinde cæcutiam, perinde <lb/>deludar? Etenim cùm meam quæ&longs;is fidem, fatcor <lb/>ingenüè me non videre quem in eo notas Paralogi&longs;<lb/>mum; ac videri mihi nece&longs;&longs;ariò deduci, fore, vt &longs;i ve<lb/>locitas per totam AC acquiratur dupla illius, quæ <lb/>acquiritur per totam AB, ip&longs;a AC eodem, aut æqua<lb/>li tempore, quo AB percurratur. Rem certe in <lb/>hunc modum concipio. Intelligatur AC diui&longs;a in <lb/>duodecim parteis æqualcis, ac proinde eius dimidium

<pb pagenum="16"/>AB, &longs;eu ip&longs;i æqualrs DE in &longs;ex: &longs;intque primùm <lb/>duo mobilia, quorum vnum di&longs;cedat ex A <lb/>

<arrow.to.target n="fig7"></arrow.to.target><lb/>ver&longs;us C, eodem momento, quo aliud ex D <lb/>ver&longs;us E. Notum e&longs;t, &longs;i vtrumque quidem <lb/>ferretur non accelerato, &longs;ed æquabili motu, <lb/>euenturum e&longs;&longs;e, vt velocitate illius ex&longs;i&longs;tente <lb/>dupla ad velocitatem i&longs;tius, illud perueniret <lb/>in C eodem momento, quo i&longs;tud in E; quo<lb/>niam &longs;patium ab illo &longs;uperatum foret vbi<lb/>que ad &longs;patium ab i&longs;to &longs;uperatum duplum, hoc e&longs;t, <lb/>forent ab illo &longs;uperatæ duæ partes, cùm ab i&longs;to vna; <lb/>ab illo quatuor, cùm ab hoc duæ, &c. quatenus &longs;pa<lb/>tia &longs;e haberent vbique vt velocitates, hoc e&longs;t veloci<lb/>tas per totam AC e&longs;&longs;et vbique dupla velocitatis per <lb/>totam DE. At verò, quoniam heic agitur de motu <lb/>non æquabili, &longs;ed continenter accelerato; ita de&longs;cen<lb/>dant rur&longs;us mobilia eodem tempore, vnum ab A, <lb/>aliud à D, vt &longs;uccrefcentibus continuò velocitatis gra<lb/>dibus, illud perueniendo in C acqui&longs;ierit duodecim, <lb/>hoc perueniendo in E &longs;ex: Quæ&longs;o quid impediat, <lb/>quo minùs illud perueniat in C eodem tempore, quo <lb/>i&longs;tud in E? Nam di&longs;crimen e&longs;t quidem inter mo<lb/>tum acceleratum, & æquabilem, quòd in æquabili <lb/>partes &longs;patiorum æquales percurrantur æqualibus <lb/>temporibus, vt &longs;ingulæ partes lineæ DE &longs;ingulis mi<lb/>nutis, & geminæ partes lineæ AC minutis item &longs;in<lb/>gulis; in accelerato non item: At in eo tamen motus <lb/>conueniunt, quòd vbique velocitas per totam AC <lb/>dupla &longs;it velocitatis per totam DE; & qua ratione <lb/>plures, plure&longs;que ex &longs;ingulis partibus lineæ DE

<figure id="fig7"></figure><lb/>ver&longs;us C, eodem momento, quo aliud ex D <lb/>ver&longs;us E. Notum e&longs;t, &longs;i vtrumque quidem <lb/>ferretur non accelerato, &longs;ed æquabili motu, <lb/>euenturum e&longs;&longs;e, vt velocitate illius ex&longs;i&longs;tente <lb/>dupla ad velocitatem i&longs;tius, illud perueniret <lb/>in C eodem momento, quo i&longs;tud in E; quo<lb/>niam &longs;patium ab illo &longs;uperatum foret vbi<lb/>que ad &longs;patium ab i&longs;to &longs;uperatum duplum, hoc e&longs;t, <lb/>forent ab illo &longs;uperatæ duæ partes, cùm ab i&longs;to vna; <lb/>ab illo quatuor, cùm ab hoc duæ, &c. quatenus &longs;pa<lb/>tia &longs;e haberent vbique vt velocitates, hoc e&longs;t veloci<lb/>tas per totam AC e&longs;&longs;et vbique dupla velocitatis per <lb/>totam DE. At verò, quoniam heic agitur de motu <lb/>non æquabili, &longs;ed continenter accelerato; ita de&longs;cen<lb/>dant rur&longs;us mobilia eodem tempore, vnum ab A, <lb/>aliud à D, vt &longs;uccrefcentibus continuò velocitatis gra<lb/>dibus, illud perueniendo in C acqui&longs;ierit duodecim, <lb/>hoc perueniendo in E &longs;ex: Quæ&longs;o quid impediat, <lb/>quo minùs illud perueniat in C eodem tempore, quo <lb/>i&longs;tud in E? Nam di&longs;crimen e&longs;t quidem inter mo<lb/>tum acceleratum, & æquabilem, quòd in æquabili <lb/>partes &longs;patiorum æquales percurrantur æqualibus <lb/>temporibus, vt &longs;ingulæ partes lineæ DE &longs;ingulis mi<lb/>nutis, & geminæ partes lineæ AC minutis item &longs;in<lb/>gulis; in accelerato non item: At in eo tamen motus <lb/>conueniunt, quòd vbique velocitas per totam AC <lb/>dupla &longs;it velocitatis per totam DE; & qua ratione <lb/>plures, plure&longs;que ex &longs;ingulis partibus lineæ DE

<pb pagenum="17"/>percurruntur æqualibus temporibus, percurruntur <lb/>quoque plures, plure&longs;que ex geminatis lineæ AC. <lb/>Ex hoc autem &longs;it, vt quemadmodum in æquabili mo<lb/>tu, DE percurreb<gap/>tur &longs;ex minutis, & AC &longs;imiliter <lb/>&longs;ex, ob geminas parteis i&longs;tius corre&longs;pondenteis &longs;ingu<lb/>lis illius, ita in accelerato, &longs;i DE percurratur tribus <lb/>minutis, AC percurratur &longs;imiliter tribus; quòd dum <lb/>primo minuto percurritur pars illius vna, percurran<lb/>tur i&longs;tius duæ, ob ge minam velocitatem; & ob ean<lb/>dem cau&longs;&longs;am, dum &longs;ecundo minuto percurruntur <lb/>illius duæ, percurrantur i&longs;tius quatuor, dum tertio <lb/>demùm illius tres, percurrantur i&longs;tius &longs;ex. Nim rùm <lb/>non alia ratione dici po&longs;&longs;ent habere &longs;e velocitates vt <lb/>&longs;patia: neque velocitas per totam AC dupla e&longs;&longs;et ve<lb/>locitatis per totam DE E&longs;to deinde vnicum mobile, <lb/>quod decedens ab A, tendat ver&longs;us C, & &longs;it rursùs <lb/>velocitas per totam AC dupla velocitatis per totam <lb/>AB, patet idem prorsùs dicendum de AC, re&longs;pectu <lb/>AB, quod dictum fuit de eadem re&longs;pectu DE. Nam <lb/>in æquabili quidem motu oporteret mobile percur<lb/>rere &longs;imul, &longs;eu primo minuto primam, & &longs;ecundam <lb/>parteis; &longs;ecundo &longs;ecundam, & quartam; ac ita porrò <lb/>quov&longs;que &longs;exto, percurreret, &longs;eu attingeret &longs;imul <lb/>&longs;extam, atque duo &longs;ecimam. In accelerato verò <lb/>e&longs;t nece&longs;&longs;e, vt percurrat &longs;imul vnam, & duas in pri<lb/>mo; duas, & quatuor in &longs;ecundo; treis, & &longs;ex in tertio; <lb/>atque adeò totam AB, & totam AC tempore eo<lb/>dem. Atque ego quidem rem itaconcipio. </s>

<s>XI Verum tu &longs;i m&longs;tas, <emph type="italics"/>Vt prima illius Paralo<lb/>gi&longs;mi a&longs;&longs;umptio in motu vniformi, ac perpetuò &longs;ibi æquali<emph.end type="italics"/>

<pb pagenum="18"/><emph type="italics"/>vera, & nece&longs;&longs;aria &longs;it; in motu tamen accelerato min<gap/>me <lb/>nece&longs;&longs;aria e&longs;t, & non vno modo tantum, &longs;ed pluribus in<lb/>telligi potest, quo modo velocitates &longs;int inter &longs;e, vt emen&longs;a <lb/>&longs;patia: licet eadem &longs;patia neque eodem, neque æquali tem<lb/>pore percurrantur.<emph.end type="italics"/> Pergis autem, <emph type="italics"/>Vt, &longs;i graue de&longs;cen-<emph.end type="italics"/><lb/><emph type="italics"/>dens per AB tempus quodcumque in&longs;umat, putà qua<lb/>

<arrow.to.target n="fig8"></arrow.to.target><lb/>drantem; ac deinde BC ip&longs;i AB æquale, dimidio <lb/>quadrante percurrat; quis neget in<emph.end type="italics"/> C <emph type="italics"/>duplam ha<lb/>beri velocitatem eius, quæ fuit in B? & tamen <lb/>idem graue totam AC, & dimidium eius AB <lb/>non percurreret.<emph.end type="italics"/> Et hæc e&longs;t quidem tota tua ad <lb/>conuincendum paralogi&longs;mi Galileum proba<lb/>tio, ob quam continenter hæc verba &longs;ubiun<lb/>gis: <emph type="italics"/>A&longs;&longs;umptio igitur Galilei fal&longs;a e&longs;t, & tota eius <lb/>ratiocinatio merus Paralogi&longs;mus id óque nullo modo, vt ip&longs;e <lb/>gloriatur communem, &longs;anioremque aliorum &longs;en&longs;um erroris <lb/>reuincit, qui in naturali grauium de&longs;cen&longs;u volunt æqualibus <lb/>spatijs æqualia velocitatis momenta acquiri.<emph.end type="italics"/> An verò pa<lb/>tietur tua bonitas, &longs;i dicam po&longs;&longs;e cuipiam videri, e&longs;&longs;e <lb/>te potiùs, qui hoc loco incidas in paralogi&longs;mum? Ni<lb/>mirum videris &longs;ic argumentari, vt id, quod contro<lb/>uertitur, a&longs;&longs;umas pro principio, dum nihil aliud, quàm <lb/>&longs;upponis &longs;patium AB, percurri duplo temporis, quo <lb/>&longs;patium BC; & velocitatem in C, e&longs;&longs;e duplam eius, <lb/>quæ fuit in B; quæ ip&longs;a tamen e&longs;t controuer&longs;ia. Et <lb/>cùm &longs;oluenda e&longs;&longs;et ratio, qua conficitur fore, vt AC <lb/>percurratur eodem, aut æquali tempore, quo &longs;patium <lb/>AB, nihil aliud, quam conclu&longs;ionem negas, fore di<lb/>cendo, vt idem graue totam AC, & dimidium eius <lb/>AB eodem tempore non percurreret. Teneri certè

<figure id="fig8"></figure><lb/>drantem; ac deinde BC ip&longs;i AB æquale, dimidio <lb/>quadrante percurrat; quis neget in<emph.end type="italics"/> C <emph type="italics"/>duplam ha<lb/>beri velocitatem eius, quæ fuit in B? & tamen <lb/>idem graue totam AC, & dimidium eius AB <lb/>non percurreret.<emph.end type="italics"/> Et hæc e&longs;t quidem tota tua ad <lb/>conuincendum paralogi&longs;mi Galileum proba<lb/>tio, ob quam continenter hæc verba &longs;ubiun<lb/>gis: <emph type="italics"/>A&longs;&longs;umptio igitur Galilei fal&longs;a e&longs;t, & tota eius <lb/>ratiocinatio merus Paralogi&longs;mus id óque nullo modo, vt ip&longs;e <lb/>gloriatur communem, &longs;anioremque aliorum &longs;en&longs;um erroris <lb/>reuincit, qui in naturali grauium de&longs;cen&longs;u volunt æqualibus <lb/>spatijs æqualia velocitatis momenta acquiri.<emph.end type="italics"/> An verò pa<lb/>tietur tua bonitas, &longs;i dicam po&longs;&longs;e cuipiam videri, e&longs;&longs;e <lb/>te potiùs, qui hoc loco incidas in paralogi&longs;mum? Ni<lb/>mirum videris &longs;ic argumentari, vt id, quod contro<lb/>uertitur, a&longs;&longs;umas pro principio, dum nihil aliud, quàm <lb/>&longs;upponis &longs;patium AB, percurri duplo temporis, quo <lb/>&longs;patium BC; & velocitatem in C, e&longs;&longs;e duplam eius, <lb/>quæ fuit in B; quæ ip&longs;a tamen e&longs;t controuer&longs;ia. Et <lb/>cùm &longs;oluenda e&longs;&longs;et ratio, qua conficitur fore, vt AC <lb/>percurratur eodem, aut æquali tempore, quo &longs;patium <lb/>AB, nihil aliud, quam conclu&longs;ionem negas, fore di<lb/>cendo, vt idem graue totam AC, & dimidium eius <lb/>AB eodem tempore non percurreret. Teneri certè

<pb pagenum="19"/>videbaris ad vberiorem paralogi&longs;mi detectionem, <lb/>&longs;olutionemque, cùm &longs;i i&longs;ta quidem methodus &longs;uffi<lb/>ciat, nihil e&longs;&longs;e videatur facilius, quàm paralogi&longs;mi ar<lb/>guere vniuer&longs;um Euclidem. Et agno&longs;co quidem te <lb/>&longs;upponere tanquam rem nimis euidentem, totum <lb/>&longs;patium AC prolixiore tempore, quàm eius partem <lb/>AB percurri: &longs;ed cùm Galileus non neget e&longs;&longs;e illud <lb/>tempus prolixius, imò tale e&longs;&longs;e reuerâ &longs;upponat; ab <lb/>incommodo tamen arguit, probando prolixius non <lb/>fore, &longs;i velocitas acqui&longs;ita per totam AC dupla defen<lb/>datur illius, quæ acquiritur per totam AB: vnde & <lb/>videtur omnmò obiecta ratio fui&longs;&longs;e &longs;oluenda. Agno&longs;co <lb/>etiam te heinc moueri, quòd non &longs;atis appareat ratio, <lb/>cur &longs;i ex A in B acquiratur vnus velocitatis gradus, <lb/>acquiri alius ex B in C, per&longs;euerante primo, non <lb/>valeat. Sed cau&longs;&longs;a nimirùm intelligitur non modò <lb/>ex dictis in vulgarem definitionem; verùm etiam <lb/>maximè ex incommodo, in quod aliunde incidis, <lb/>dum con&longs;equenter loquens, vis &longs;patium BC percurri <lb/>dimidio temporis, quo AB; vt putà, quod AB vnico <lb/>gradu velocitatis BC, gemino percurratur. </s>

<s>XII. Nam, vt illud paucis deducam, &longs;equitur <lb/>exinde, vt tempore dato, quo decur&longs;a &longs;emel fuerit <lb/>pars AB, tempus aliud ip&longs;i æquale attingi nulla ra<lb/>tione valeat, ni&longs;i &longs;uperato &longs;patio infinito. Intelliga<lb/>tur enim linea AC infinitè producta, diui&longs;aque in <lb/>parteis CD, DF, EF, &c. ip&longs;is AB & BC <lb/>æqualeis. Qua ratione tu vis tempus, quo percurri<lb/>tur AB, e&longs;&longs;e duplum temporis, quo percurritur BC <lb/>velis oportet tempus id, quo percurritur BC e&longs;&longs;e

<pb pagenum="20"/>duplum temporis, quo percurritur CD, & hoc <lb/>

<arrow.to.target n="fig9"></arrow.to.target><lb/>duplum eius, quo DE, & i&longs;tud illius, quo EF, <lb/>& &longs;ic deinceps; neque enim maior vnius, <lb/>quàm alterius e&longs;t ratio; ac in accelerato poti&longs;<lb/>&longs;imù n æquabiliter motu, de quo præ&longs;ertim <lb/>quæ&longs;tio heic e&longs;t. Quare & velis etiam opor<lb/>tet, vt cùm tempus, quo percurritur BC, &longs;it <lb/>dimidium temporis, quo percurritur AB; illud, <lb/>quo percurritur CD, &longs;it quadrans eiu&longs;dem <lb/>primi temporis; illud, quo DE, octans; quo <lb/>EF, pars decima &longs;exta; quo FG, trige&longs;ima &longs;e<lb/>cunda; quo GH &longs;exage&longs;ima quarta, &c. Por<lb/>rò hæc omnia tempora &longs;imul iuncta nunquam <lb/>æquabuntur primo tempori, quo decur&longs;um <lb/>fuerit AB (quandò procedentes hoc modo <lb/>fractiones relinquunt &longs;emper ex integro, to<lb/>tove quidpiam inexhau&longs;tum) ni&longs;i lineam, &longs;eu <lb/>&longs;patium infinitum admi&longs;eris, & parteis æqua<lb/>leis in eo infinitas, quæ infinitis analogis (&longs;eu <lb/>dimidiorum dimidiis in tempore ip&longs;o, aut <lb/>æquali, quo AB percurritur) contineri intel<lb/>lectis, re&longs;pondeant. Adderem heic etiam <lb/>incommodum aliud de &longs;patijs incre&longs;centibus, <lb/>& in fine cuiu&longs;libet æqualis temporis numerandis <lb/>&longs;ecundum rationem non modò duplam, verùm etiam <lb/>triplam, & ampliùs: &longs;ed res erit po&longs;teà vberiùs dicen<lb/>da. Adderem rursùs alia quoque, vt Quòd &longs;equere<lb/>tur lineam proiectorum, & illam &longs;peciatim, quæ de&longs;<lb/>cribitur à lapide &longs;ur&longs;um, & &longs;ecundum mali altitudi<lb/>nem, dum nauis mouetur, proiecto, non e&longs;&longs;e Parabo-

<figure id="fig9"></figure><lb/>duplum eius, quo DE, & i&longs;tud illius, quo EF, <lb/>& &longs;ic deinceps; neque enim maior vnius, <lb/>quàm alterius e&longs;t ratio; ac in accelerato poti&longs;<lb/>&longs;imù n æquabiliter motu, de quo præ&longs;ertim <lb/>quæ&longs;tio heic e&longs;t. Quare & velis etiam opor<lb/>tet, vt cùm tempus, quo percurritur BC, &longs;it <lb/>dimidium temporis, quo percurritur AB; illud, <lb/>quo percurritur CD, &longs;it quadrans eiu&longs;dem <lb/>primi temporis; illud, quo DE, octans; quo <lb/>EF, pars decima &longs;exta; quo FG, trige&longs;ima &longs;e<lb/>cunda; quo GH &longs;exage&longs;ima quarta, &c. Por<lb/>rò hæc omnia tempora &longs;imul iuncta nunquam <lb/>æquabuntur primo tempori, quo decur&longs;um <lb/>fuerit AB (quandò procedentes hoc modo <lb/>fractiones relinquunt &longs;emper ex integro, to<lb/>tove quidpiam inexhau&longs;tum) ni&longs;i lineam, &longs;eu <lb/>&longs;patium infinitum admi&longs;eris, & parteis æqua<lb/>leis in eo infinitas, quæ infinitis analogis (&longs;eu <lb/>dimidiorum dimidiis in tempore ip&longs;o, aut <lb/>æquali, quo AB percurritur) contineri intel<lb/>lectis, re&longs;pondeant. Adderem heic etiam <lb/>incommodum aliud de &longs;patijs incre&longs;centibus, <lb/>& in fine cuiu&longs;libet æqualis temporis numerandis <lb/>&longs;ecundum rationem non modò duplam, verùm etiam <lb/>triplam, & ampliùs: &longs;ed res erit po&longs;teà vberiùs dicen<lb/>da. Adderem rursùs alia quoque, vt Quòd &longs;equere<lb/>tur lineam proiectorum, & illam &longs;peciatim, quæ de&longs;<lb/>cribitur à lapide &longs;ur&longs;um, & &longs;ecundum mali altitudi<lb/>nem, dum nauis mouetur, proiecto, non e&longs;&longs;e Parabo-

<pb pagenum="21"/>licam, neque tantum temporis, ex&longs;cendendo, quan<lb/>tum a&longs;cendendo con&longs;umi; ac proinde lapidem illum <lb/>neque peruenturum ad mali carche&longs;ium, neque reca<lb/>&longs;urum in pedem eiu&longs;dem: verùm i&longs;ta aut colliguntur <lb/>ex ijs, quæ &longs;unt dicta in Epi&longs;tolis, aut in promptu <lb/>&longs;unt, facileque occurrunt. Et de Definitione huc<lb/>v&longs;que. </s>

<s><emph type="center"/><emph type="italics"/>De Po&longs;tulato Galilei circa Motum &longs;uper æquè altis, non <lb/>æquè inclinatis planis.<emph.end type="italics"/><emph.end type="center"/></s>

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<s>XIII. In&longs;ectaris &longs;ecundo loco, tanquam aliam <lb/>erroris cau&longs;&longs;am, Quod Galileus &longs;ibi dari, & gratis <lb/>concedi, inquis, po&longs;tulat, <emph type="italics"/>Gradus velocitatis eiu&longs;dem <lb/>mobilis &longs;uper diuer&longs;as planorum inclinationes acqui&longs;itos tunc <lb/>e&longs;&longs;e æqualeis, cùm eorumdem planorum eleuationes ponuntur<emph.end type="italics"/><lb/><emph type="italics"/>æquales;<emph.end type="italics"/> hoc e&longs;t gradus velo<lb/>

<arrow.to.target n="fig10"></arrow.to.target><lb/>citatis ab eodem globo (exem<lb/>pli gratia) per plana CA, & <lb/>CD de&longs;cendente, in punctis <lb/>A, & D acqui&longs;itos, e&longs;&longs;e inter &longs;e <lb/>æqualeis, quòd æqualem, vel potiùs eandem eleua<lb/>tionem habeant, videlicet BC. <emph type="italics"/>Hoc enim po&longs;tula<lb/>tum,<emph.end type="italics"/> inquis, <emph type="italics"/>cùm neque ex terminis notum &longs;it, neque vlla <lb/>&longs;ufficiente experientia confirmatum; imò cùm rationes etiam <lb/>non de&longs;int, quibus oppo&longs;itum probabilius reddatur (nempe <lb/>gradus velocitatis per longius planum acqui&longs;itos gradi<gap/>us <lb/>per breuius planum acqui&longs;itis e&longs;&longs;e minores) id à Galileo non <lb/>peti, &longs;ed debuerat demon&longs;trari cùm præ&longs;ertim maxima pars <lb/>&longs;ub&longs;equentium theorematum hoc vnico postu ato nitantur. <lb/>Quid enim certi ex incertis concludi pote&longs;t. aut ex principie,<emph.end type="italics"/>

<figure id="fig10"></figure><lb/>citatis ab eodem globo (exem<lb/>pli gratia) per plana CA, & <lb/>CD de&longs;cendente, in punctis <lb/>A, & D acqui&longs;itos, e&longs;&longs;e inter &longs;e <lb/>æqualeis, quòd æqualem, vel potiùs eandem eleua<lb/>tionem habeant, videlicet BC. <emph type="italics"/>Hoc enim po&longs;tula<lb/>tum,<emph.end type="italics"/> inquis, <emph type="italics"/>cùm neque ex terminis notum &longs;it, neque vlla <lb/>&longs;ufficiente experientia confirmatum; imò cùm rationes etiam <lb/>non de&longs;int, quibus oppo&longs;itum probabilius reddatur (nempe <lb/>gradus velocitatis per longius planum acqui&longs;itos gradi<gap/>us <lb/>per breuius planum acqui&longs;itis e&longs;&longs;e minores) id à Galileo non <lb/>peti, &longs;ed debuerat demon&longs;trari cùm præ&longs;ertim maxima pars <lb/>&longs;ub&longs;equentium theorematum hoc vnico postu ato nitantur. <lb/>Quid enim certi ex incertis concludi pote&longs;t. aut ex principie,<emph.end type="italics"/>

<pb pagenum="22"/><emph type="italics"/>vt ip&longs;emet<emph.end type="italics"/> G<emph type="italics"/>alileus agno&longs;cit, veri&longs;imili tantum, ac probabili <lb/>demonstrari?<emph.end type="italics"/> Po&longs;tmodùm autem, vbi hæc præmi&longs;i&longs;ti, <lb/><emph type="italics"/>In &longs;cientiarum, ac demon&longs;trationum principiis euidentiam <lb/>exigimus, &longs;u&longs;piciones, ac veri&longs;imilitudines nulla ratione ad<lb/>mittimus,<emph.end type="italics"/> &longs;ubdis, <emph type="italics"/>Porrò quæ ex his con&longs;equuntur, aut <lb/>inferuntur theoremata, &longs;uis illis principiis certiora, aut eui<lb/>dentiora e&longs;&longs;e non po&longs;&longs;unt, & nominatim &longs;olemne illud, & <lb/>quod totius &longs;cientiæ à Galileo excogitatæ firmamentum est, <lb/>spatia &longs;cilicet æqualibus temporibus emen&longs;a eam inter &longs;e <lb/>rationem ob&longs;eruare, quæ est inter numeros omneis impareis <lb/>continua &longs;erie ab vnitate procedenteis (quamvis aliunde <lb/>fal&longs;um demon&longs;trari non po&longs;&longs;et) neque ex præ&longs;uppo&longs;itis illis <lb/>principiis euidenter, neque aliunde &longs;ufficienter conclude<lb/>retur.<emph.end type="italics"/></s>

<s>XIV. Hoc autem loco non video primùm, quì <lb/>reprchendendus Galileus &longs;it, &longs;i quam propo&longs;itionem <lb/>non demon&longs;tratam, &longs;ed veri&longs;imilem &longs;olùm habuit, <lb/>non vt demon&longs;tratam, &longs;ed vt veri&longs;imilem duntaxat <lb/>exhibuit. Candidè nimirùm videtur egi&longs;&longs;e, neque <lb/>exegi&longs;&longs;e à Lectoribus, vt maiorem, quàm ip&longs;e Po&longs;tu<lb/>lato fidem haberent; &longs;ed illos potius qua&longs;i monui&longs;&longs;e, <lb/>ne ip&longs;um concederent, ni&longs;i deinceps agno&longs;cerent <lb/>con&longs;tabilitum variis ex eo deductis conclu&longs;ionibus, <lb/>quæ cum experientia planè con&longs;entirent. Deinde <lb/>cùm in &longs;cientijs, ac demon&longs;trationibus attinent bus <lb/>ad Mathe&longs;in puram, mera euidentia, non &longs;ola &longs;u&longs;pi<lb/>cio, aut veri&longs;imilitudo admittenda &longs;it: in &longs;cientijs ta<lb/>men Phy&longs;icis, ac mi&longs;ta Mathe&longs;i, quacumque &longs;e&longs;e Phy<lb/>&longs;ica, hoc e&longs;t caligo humanæ mentis in rebus natura<lb/>libus inue&longs;tigandis, ingerit; f&oelig;lices &longs;imus, &longs;i non

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<s>XV. Verùm, ne ad alia excurram, quàm quæ <lb/>ip&longs;emet ex Galileo commemoras, improbas ec<gap/>e ex<lb/>perimentum, quo ille e&longs;t conatus h<gap/>em Po&longs;tulato <lb/>a&longs;&longs;erere, quodque ad&longs;cripta figura &longs;ic refers E <emph type="italics"/>clauo A<emph.end type="italics"/><lb/>

<arrow.to.target n="fig11"></arrow.to.target><lb/><emph type="italics"/>parieti infixo, globus plumbeus, aut alius quilibet tenui filo, <lb/>tribus, aut quatuor digitis à pariete remoto &longs;u&longs;pendatur, &longs;it<lb/>que AB, De&longs;criptaque in pariete recta CD horizonti <lb/>parallela, globus B à perpendiculari eductus v&longs;que ad alti<lb/>tudinem rectæ CD manu altollatur, nempe ad C; indeque <lb/>liberè dimittatur. Tum globus idem in uit<emph.end type="italics"/> G<emph type="italics"/>alileus, non <lb/>&longs;olùm de&longs;cendet ad punctum B, &longs;ed eodem impetu vlteriùs <lb/>v&longs;que ad D, aut proximè ad illud, a&longs;cendet.<emph.end type="italics"/> S<emph type="italics"/>imiliter, &longs;i <lb/>globus idem è puncto E &longs;uspendatur, & item ad altitudinem<emph.end type="italics"/>

<figure id="fig11"></figure><lb/><emph type="italics"/>parieti infixo, globus plumbeus, aut alius quilibet tenui filo, <lb/>tribus, aut quatuor digitis à pariete remoto &longs;u&longs;pendatur, &longs;it<lb/>que AB, De&longs;criptaque in pariete recta CD horizonti <lb/>parallela, globus B à perpendiculari eductus v&longs;que ad alti<lb/>tudinem rectæ CD manu altollatur, nempe ad C; indeque <lb/>liberè dimittatur. Tum globus idem in uit<emph.end type="italics"/> G<emph type="italics"/>alileus, non <lb/>&longs;olùm de&longs;cendet ad punctum B, &longs;ed eodem impetu vlteriùs <lb/>v&longs;que ad D, aut proximè ad illud, a&longs;cendet.<emph.end type="italics"/> S<emph type="italics"/>imiliter, &longs;i <lb/>globus idem è puncto E &longs;uspendatur, & item ad altitudinem<emph.end type="italics"/>

<pb pagenum="25"/><emph type="italics"/>eiu&longs;dem rectæ<emph.end type="italics"/> C<emph type="italics"/>D attollatur ad<emph.end type="italics"/> G, <emph type="italics"/>inde liberè dimi&longs;&longs;us, <lb/>pari modo ad eandem rectam CD, aut proximè ad eam <lb/>con&longs;cendet ver&longs;us H. Jmò, &longs;i ex F &longs;u&longs;pen&longs;us attollatur ad <lb/>I, inde feretur, v&longs;que ad K. Per diuer&longs;os igitur illos arcus <lb/>decidens globus, &longs;emper ad æqualem altitudinem con&longs;cendit. <lb/>Ergo è quolibet de&longs;cen&longs;u æqualem acquirit impetum; ni&longs;i <lb/>enim e&longs;&longs;et impetus æqualis, globum ad æqualem altitudinem <lb/>non attolleret. Quid ni igitur idem quoque faciat globus, <lb/>&longs;i per plana CB, GB, IB de&longs;cendat?<emph.end type="italics"/> C<emph type="italics"/>redibile igitur <lb/>etiam e&longs;t globum per illa, aut &longs;imilia plana decidentem, æqua<lb/>lem tali de&longs;cen&longs;u impetum, ac proinde æqualem quoque ve<lb/>locitatis gradum acquirere.<emph.end type="italics"/> Subinde autem, vt o&longs;ten<lb/>das quàm hæc &longs;int incerta, incohæcentia, &c. <emph type="italics"/>Impri<lb/>mis quidem ne&longs;cio,<emph.end type="italics"/> inquis, <emph type="italics"/>an globi ea, qua vult<emph.end type="italics"/> G<emph type="italics"/>alileus <lb/>ratione &longs;u&longs;pen&longs;i, ac librati alitùs in Etruria, quàm in Gal<lb/>lia a&longs;&longs;urgant; at heic neque tam propè ad horizontalem li<lb/>neam, neque per diuer&longs;os arcus ad eam æqualiter accedunt. <lb/>Nempe filo pedum quatuor cum dimidio &longs;u&longs;pen&longs;us globus ad <lb/>lineam horizontalem tribus infra centum pedibus <expan abbr="de&longs;criptã">de&longs;criptam</expan>, <lb/>propiùs quàm duobus digitis nunquam acceßit. At centro <lb/>nouem tantum digitis &longs;upra lineam horizontalem accepto, <lb/>filóque duorum pedum con&longs;tituto, iam globus ad lineam ho<lb/>rizontalem vno digito, quàm anteà propiùs acceßit. Vbi <lb/>verò centrum &longs;eptem infra lineam horizontalem digitis a&longs;<lb/>&longs;umptum est, vix ad quatuor à linea horizontali digitos <lb/>globus a&longs;cendit.<emph.end type="italics"/> Concludis idcircò his verbis, <emph type="italics"/>Qua <lb/>igitur fide<emph.end type="italics"/> G<emph type="italics"/>alileus tam a&longs;&longs;eueranter ait globum ita &longs;u&longs;pen<lb/>&longs;um, ac per quo&longs;cumque arcus librarum, ad æqualem &longs;em<lb/>per altitudinem a&longs;&longs;urgere? aut quomodo ex re adeò euiden<lb/>ter fal&longs;a petere au&longs;us e&longs;t testim<gap/>nium veritatis?<emph.end type="italics"/></s>

<s>XVI. Imprimis porrò non retices ip&longs;e dictum <lb/>e&longs;&longs;e à Galileo demi&longs;&longs;um ex C globum a&longs;cen&longs;urum <lb/><emph type="italics"/>v&longs;que ad D, aut proxime:<emph.end type="italics"/> vt proinde non videatur di<lb/>ctum ab illo a&longs;&longs;eueranter a&longs;&longs;urrecturum globum <lb/>ad eandem altitudinem, aut veritatis te&longs;timonium <lb/>ex re fal&longs;a ab ip&longs;o peti; qua&longs;i intellexerit globum a&longs;<lb/>&longs;equi altitudinem exqui&longs;itè, &longs;eu præcisè eandem. Et <lb/>certè non modò dixit ip&longs;e <emph type="italics"/>qua&longs;i,<emph.end type="italics"/> &longs;eu <emph type="italics"/>ferè,<emph.end type="italics"/> ac <emph type="italics"/>&longs;uperfu<lb/>turum interuallum quoddam perexiguum;<emph.end type="italics"/> &longs;ed etiam cau&longs;<lb/>&longs;am attigit, ob quam ita fiat; referens eam putà ad <lb/>impedimentum partim aëris, partim fili; de quo vtro<lb/>que heic dicerem, ni&longs;i iam dictum &longs;atis copiosè in <lb/>Epi&longs;tolis memoratis foret. Deinde, quód globus ad <lb/>horizontalem lineam propiùs ad H, remotiùs ad <lb/>Ka&longs;cendat, quàm ad ip&longs;um D; videri pote&longs;t cau&longs;&longs;a <lb/>per&longs;picua, neque infringere vim experimenti. Nam <lb/>quod &longs;pectat quidem ad H, res ideò contin git, quòd <lb/>quoties clauus defigitur inter A, & horizontalem <lb/>lineam, breuitas tum fili, tum &longs;pati<gap/> aërei, per quod <lb/>arcus de&longs;cribitur, minùs præ&longs;tet impedimenti: vnde <lb/>& abfui&longs;&longs;et globus adhûc propiùs, &longs;i fui&longs;&longs;et clauus <lb/>infra E defixus, vti & longiùs, &longs;i &longs;upra ip&longs;um Quod <lb/>verò ad K, res minùs e&longs;t mira; quòd quoties clauus <lb/>defigitur infra horizontalem lineam, dimi&longs;&longs;us ex I <lb/>globus non per totum arcum IB decidat, &longs;ed per <lb/>inferiorem &longs;olùm eius partem, in quam perpendicu<lb/>lariter cadit; vnde & minùs adhûc, minu&longs;que re&longs;i<lb/>lii&longs;&longs;et, &longs;i defixi&longs;&longs;es clauum inferiùs, quou&longs;que globus <lb/>non potui&longs;&longs;et ad lineam attolli; veluti & magis, ma<lb/>gi&longs;que, &longs;i &longs;uperiùs, quou&longs;que clauo defixo in linea,

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<s>XIX. Venio igitur ad Po&longs;terius caput, &longs;ecun<lb/>damue partem tuæ Di&longs;&longs;ertationis; in qua &longs;cilicet re<lb/>cepi&longs;ti te veram, ac certam de Motu accelerato &longs;cien<lb/>tiam fal&longs;æ, ac incertæ Galileanæ &longs;ub&longs;tituturum; & in <lb/>qua profe&longs;&longs;us te iterùm omnia, quæ ab illo con&longs;cri<lb/>pta &longs;unt, fal&longs;a, ac inania e&longs;&longs;e demon&longs;traturum; prouo<lb/>cas me primum ad <emph type="italics"/>clara, facilia, indubitata experimenta:<emph.end type="italics"/><lb/>tamet&longs;i ego tenuitatis con&longs;cius per&longs;onam Arbitri, lu<lb/>dici&longs;que, quam mihi humani&longs;&longs;imè iteratò defers, re<lb/>cu&longs;o; iteratò profe&longs;&longs;us nihil aliud à me, quàm rationes <lb/>qua&longs;dam dubitandi ex&longs;pectari po&longs;&longs;e. Et <emph type="italics"/>prima qui<lb/>dem experientia petitur,<emph.end type="italics"/> inquis, <emph type="italics"/>ex impetu, quo globus, <lb/>aut graue aliud corpus quodcumque per aërem sponte natu<lb/>ræ deorsùm cadit, ac percutit. Indubitatum enim e&longs;t,<emph.end type="italics"/><lb/>pergis, <emph type="italics"/>quod ip&longs;emet<emph.end type="italics"/> G<emph type="italics"/>alileus paßim agno&longs;cit, tantam præ<lb/>ci&longs;e percutientis corporis e&longs;&longs;e velocitatem, quantus impetus, <lb/>quantaque ip&longs;a percußio fuerit. Impetus enim omnis, & <lb/>percußio ex velocitate est; imò impetus ip&longs;e velocitas est, <lb/>nulloque hæc abinuicem di&longs;crimine dirimuntur, vt meritò, pro<lb/>inde, qua ratione accre&longs;cit velocitas, eadem impetus, & per<lb/>cußio augeantur.<emph.end type="italics"/> Hactenus nihil e&longs;t, quod non probem. <lb/>Pro&longs;equeris autem: <emph type="italics"/>At facilè experientiâ con&longs;tat corpus <lb/>graue quodcumque ex qualibet altitudine per aërem cadens, <lb/>& percutiens, vt libet, perpetuò ex altitudine dupla duplo<emph.end type="italics"/>

<pb pagenum="33"/><emph type="italics"/>præcisè ampliùs, & ex tripla, quadrupláue di&longs;tantia, triplo, <lb/>quadruplóue fortiùs percutere: velocitas igitur quoque ex <lb/>altitudine dupla, duplò maior e&longs;t, & tripla, aut quadrupla, <lb/>&longs;i tripla, quadrupláue altitudo &longs;uerit: ac proinde velocitas, <lb/>spatiis æqualibus, non autem æqualibus temporibus, æqualia <lb/>momenta acquirit.<emph.end type="italics"/> Quæ&longs;o verò heic patere, religio&longs;i&longs;<lb/>&longs;imè Vir, me meam te&longs;tari h<gap/> betudmem; neque enim <lb/>quod tu a&longs;&longs;umis, <emph type="italics"/>facilè experientiâ con&longs;tare,<emph.end type="italics"/> mihi vllà <lb/>prorsùs experientiâ con&longs;tat; neque tu vllam &longs;pecia<lb/>lem affers, ex qua res, vt tibi, ita mihi con&longs;tet. Ac <lb/>deducis quidem deinceps, qua&longs;i &longs;ecundam experien<lb/>tiam, id, quod in Libra expertus es: &longs;ed interim circa <lb/>hanc primam, cæcutio planè, neque agno&longs;co, qui <lb/>rem facilè exploraris. Et explora&longs;&longs;e tamen quis hæ<lb/>reat, quandò i&longs;thæc &longs;ubi<gap/>cis? <emph type="italics"/>Experientiam hanc Ga<lb/>lileus, nullo (vt credibile e&longs;t) facto ip&longs;ius periculo, tanquam <lb/>fal&longs;am, atque impoßibilem, eodem paralogi&longs;mo re-<emph.end type="italics"/><lb/>

<arrow.to.target n="fig12"></arrow.to.target><lb/><emph type="italics"/>iecit, quo definitionem motus accelerati vulgò re<lb/>ceptam, & ex eadem experientia &longs;ine dubio dedu<lb/>ctam, conatus e&longs;t reuellere. Si ex altitudine dupla, <lb/>inquit, duplò maior percußio e&longs;t, vt puta ex A du<lb/>pla eius, quæ ex B, erit & velocitas dupla. At <lb/>velocitas dupla e&longs;&longs;e non pote&longs;t, ni&longs;i graue, æquali, <lb/>imò eodem tempore, totum spatium AC, & di <lb/>midium eius AB percurrat, quod tamen e&longs;t impoßibile. <lb/>Nec percußio igitur, nec velocitas dupla e&longs;t, ex altitudine <lb/>dupla Do<gap/>eo equidem virum non ignobilem, in re tam <lb/>obuta, & facili adeò turpiter delu&longs;um e&longs;&longs;e; mirorque item <lb/>vehementer tales, támque apertos eius errores, non modò à <lb/>nenune hactenus e&longs;&longs;e reprehen&longs;os, &longs;ed tanquam prima<emph.end type="italics"/>

<figure id="fig12"></figure><lb/><emph type="italics"/>iecit, quo definitionem motus accelerati vulgò re<lb/>ceptam, & ex eadem experientia &longs;ine dubio dedu<lb/>ctam, conatus e&longs;t reuellere. Si ex altitudine dupla, <lb/>inquit, duplò maior percußio e&longs;t, vt puta ex A du<lb/>pla eius, quæ ex B, erit & velocitas dupla. At <lb/>velocitas dupla e&longs;&longs;e non pote&longs;t, ni&longs;i graue, æquali, <lb/>imò eodem tempore, totum spatium AC, & di <lb/>midium eius AB percurrat, quod tamen e&longs;t impoßibile. <lb/>Nec percußio igitur, nec velocitas dupla e&longs;t, ex altitudine <lb/>dupla Do<gap/>eo equidem virum non ignobilem, in re tam <lb/>obuta, & facili adeò turpiter delu&longs;um e&longs;&longs;e; mirorque item <lb/>vehementer tales, támque apertos eius errores, non modò à <lb/>nenune hactenus e&longs;&longs;e reprehen&longs;os, &longs;ed tanquam prima<emph.end type="italics"/>

<pb pagenum="34"/><emph type="italics"/>&longs;cientiæ principia, à viris etiam eruditis e&longs;&longs;e receptos.<emph.end type="italics"/><lb/>Quandò, inquam, hæc &longs;ubiicis, nemo profectò fa<lb/>cilè hæreat, qum ip&longs;e, expertus, illa videris, quæ <lb/>neque Galileus, neque alij viderunt. </s>

<s>XX. Quod meattinet; cùm lapidem video ex vna, <lb/>ex duabus, ex tribus, ex quatuor orgyiis cadentem in <lb/>terram; agno&longs;co quidem e&longs;&longs;e ictum, atque idcircò <lb/><expan abbr="impet&utilde;">impetum</expan>, velocitatem que maiorem ex duabus orgyiis, <lb/>quàm ex vna, ex tribus, quàm ex duabus, ex quatuor <lb/>quàm ex tribus; verùm e&longs;&longs;e duplò præcisè maiorem <lb/>ex duabus, quàm ex vna, triplò ex tribus, quadruplò <lb/>ex quatuor, nulla penitùs ratione agno&longs;co. Neque <lb/>enim po&longs;&longs;um id di&longs;picere ex cauitate in terram facta, <lb/>aut penetratione in ip&longs;am; quoniam neque lapis du<lb/>plò profundiùs cauat, penetratque ex dupla altitudine, <lb/>aut triplò ex tripla; neque cognitus e&longs;t aut gradus <lb/>re&longs;i&longs;tentiæ, quo talis terra obnititur; aut progre&longs;&longs;us, <lb/>quo cre&longs;cit re&longs;i&longs;tentia, dum quò inferiùs tenditur, <lb/>eò partes terræ minùs &longs;eu deor&longs;um, &longs;eu in latera ce<lb/>dere, compelli, ac &longs;ubire po&longs;&longs;unt: vt habita proinde <lb/>ratione huius re&longs;i&longs;tentiæ, colligere valeam id, quod <lb/>ad duplam penetrationem ex altitudine dupla dce&longs;t, <lb/>non aliunde e&longs;&longs;e, quàm ex huiu&longs;modi re&longs;i&longs;tentia. Sic <lb/>cùm video fi&longs;tucam in palum delap&longs;am ex &longs;implici, <lb/>dupla, aut tripla altitudine: quandoquidem neque <lb/>video palum defigi profundiùs in terram, duplo qui<lb/>dem ex dupla, aut triplò ex tripla altitudine; neque <lb/>per&longs;pectum habeo quo gradu, in qualibet profundi<lb/>tatis parte ip&longs;i vrgenti re&longs;i&longs;tatur; aduerto quidem <lb/>maiorem ictum, maioremque impetum, ac veloci-

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<s>XXII. E&longs;t etiam Tertiò heic repetendum, quod <lb/>iam antè dixi de globis ad fila appen&longs;is, & liberè ire, <lb/>redireque permi&longs;&longs;is. Videlicet globus appen&longs;us ex <lb/>vno v. c. pede, dup ò quidem plureis vibrationes per<lb/>agit, quam appen&longs;us ad quatuor, triplo, quàm ap<lb/>pen&longs;us ad nouem, quadruplo, quàm appen&longs;us ad &longs;ex<lb/>decim; &longs;ed interim tamen &longs;ecundus &longs;patium conficit <lb/>dup ò maius, quàm primus, tertius triplò, quartus <lb/>quadruplò eodem tempore; ac velocitas interim ac<lb/>qui&longs;ita, impetu&longs;que ad perpendiculum expre&longs;&longs;us, non <lb/>vt &longs;patium pertran&longs;itum, &longs;ed vt tempus elap&longs;um &longs;e <lb/>

<arrow.to.target n="fig13"></arrow.to.target><lb/>habet. Ego certè rem &longs;ic intelligo. Sit linea per<lb/>pendicularis AB in pariete ducta, diui&longs;aque in &longs;ex<lb/>decim pedes; ac &longs;int appen&longs;i quatuor globi, vnus <lb/>ad primum, alius ad quartum, tertius ad nonum, <lb/>po&longs;tremus ad decimum&longs;extum. Siquidem tamet&longs;i

<figure id="fig13"></figure><lb/>habet. Ego certè rem &longs;ic intelligo. Sit linea per<lb/>pendicularis AB in pariete ducta, diui&longs;aque in &longs;ex<lb/>decim pedes; ac &longs;int appen&longs;i quatuor globi, vnus <lb/>ad primum, alius ad quartum, tertius ad nonum, <lb/>po&longs;tremus ad decimum&longs;extum. Siquidem tamet&longs;i

<pb pagenum="39"/>inter experiundum applicari diuer&longs;is &longs;eor&longs;im lineis <lb/>debeant, ne inter mouendum &longs;e&longs;e interturbent: om<lb/>neis tamen &longs;chemate vno repræ&longs;entari nihil prohibet. <lb/>Ducantur heinc inde duæ lineæ angulum &longs;tatuentes <lb/>in A, qui à perpendiculo bi&longs;ecctur, &longs;intque v. c. IA, <lb/>KA; & centro A; agantur inter illas arcus CD ad <expan abbr="pri-m&utilde;">pri<lb/>mum</expan> pedem, EF ad quartum, GH ad nonum IK ad <lb/>&longs;extum-decimum; qui &longs;imiles proinde erunt, pares <lb/>videlicet portiones &longs;uotum cuiu&longs;que circulorum eo<lb/>dem angulo men&longs;uratæ. Ducantur & &longs;ubten&longs;æ ar<lb/>cuum; & adnotentur qua&longs;i &longs;agittæ, &longs;eu appellati &longs;inus <lb/>ver&longs;i, lineæ nimirùm LM, NO, PQ, RB; cùm <lb/>&longs;int altitudines, quibus globi delabuntur ex linea AI <lb/>(vbi ad illam abducti, ex ea dimittuntur) in ip&longs;um <lb/>perpendiculum; primus putà ex C in M, &longs;ecundus <lb/>ex E in O, tertius ex G in Q, quartus ex I in B. <lb/>Abducantur proinde globi ad memoratam lineam <lb/>AI, vt &longs;uas, exinde dimi&longs;&longs;i, vibrationes peragant, ad <lb/>lineam AK, aut quam-proximè terminandas. Nam & <lb/>quamuis quilibet globus, &longs;eu longiùs, &longs;eu breuiùs di<lb/>mi&longs;&longs;us, & &longs;eu moueri incipiat, &longs;eu de&longs;inat, vibrationes <lb/>omneis æqui-temporaneas &longs;ortiatur, temporibu&longs;ve <lb/>paribus perficiat; proportio tamen &longs;emper e&longs;t, quoties <lb/>&longs;ub æquali, eodemue angulo accipiuntur. Dimittan<lb/>tur & globi &longs;imul, ac peruenire concipiantur ad v&longs;<lb/>que perpendiculum. Quoniam tunc vt filum AO <lb/>quadruplum e&longs;t fili AM, & filum AQ nonuplum, <lb/>filum AB &longs;exdecuplum; ita altitudo NO dupla e&longs;t <lb/>altitudinis LM, & altitudo PQ nonupla, altitudo <lb/>RB &longs;exdecupla; quo pacto interuallum quoque per-

<pb pagenum="40"/>tran&longs;itum EO quadruplum e&longs;t &longs;patij CM, & &longs;pa<lb/>tium GQ nonuplum, &longs;patium IB &longs;exdecuplum: Id<lb/>circò, cùm aliunde ob&longs;eruemus tempus, quo globus <lb/>&longs;ecundus peruenit ad O, e&longs;&longs;e duplum temporis, quo <lb/>primus peruenit ad M, & tempus, quo tertius ad Q, <lb/>triplum; tempus, quo quartus ad B, quadruplum; Id<lb/>circò, inquam, intelligimus, impetum, &longs;eu velocita<lb/>tem, quæ acquiritur ex E, aut N in O, & ex G, aut <lb/>P in <expan abbr="q;">que</expan> & ex I, aut R in B, pari ratione &longs;e habere <lb/>ad velocitatem acqui&longs;itam ex C, aut L, in M, qua <lb/>&longs;e habet impetus, &longs;eu velocitas, quæ acquiritur ex A <lb/>in O, in Q, in B, ad velocitatem acqui&longs;itam ex A in <lb/>M; &longs;eu comparando oppo&longs;itè, vt illam ad illam, &longs;ic <lb/>i&longs;tam ad i&longs;tam. Hoc autem habito, quoniam im<lb/>

<arrow.to.target n="fig14"></arrow.to.target><lb/>petus, &longs;eu velocitas acqui&longs;ita ex E in O non e&longs;t ac<lb/>qui&longs;itæ ex C in M quadrupla, &longs;ed dupla; & acqui&longs;ita

<figure id="fig14"></figure><lb/>petus, &longs;eu velocitas acqui&longs;ita ex E in O non e&longs;t ac<lb/>qui&longs;itæ ex C in M quadrupla, &longs;ed dupla; & acqui&longs;ita

<pb pagenum="41"/>ex G in Q, non nonupla eiu&longs;dem, &longs;ed tripla: & ac<lb/>qui&longs;ita ex I in B, non &longs;exdecupla, &longs;ed quadrupla e&longs;t: <lb/>quatenus quidem experiundo ob&longs;eruare licuit, con<lb/>&longs;titutam pilam &longs;upra planum libellatum, appo&longs;itum<lb/>que ad M, ad O, ad Q, ad B, dum percuteretur, pro<lb/>pellereturque à globis incurrentibus, a&longs;&longs;equi velocita<lb/>tem, excurrereque, non iuxta numeros quadratos, <lb/>quales &longs;unt &longs;patiorum CM, EO, GQ, IB; &longs;ed iuxta <lb/>radices ip&longs;orum, qualia &longs;unt & tempora, vnum, duo, <lb/>tria, quatuor. Quamobrem & fuit iteratò procliue <lb/>intelligere percu&longs;&longs;ionem quoque à re &longs;ecundum per<lb/>pendiculum cadente factam, &longs;equi rationem non <lb/>quadratorum, &longs;eu &longs;patiorum, &longs;ed radicum, &longs;eu tem<lb/>porum; atque ita, quò cadens graue vehementiùs <lb/>feriat duplò, triplò, quadruplò, cadere debere ex al<lb/>titudine non duplò, triplò quadruplò; verùm, quadru<lb/>plò, nonuplò, atque &longs;exdecuplò maiore. </s>

<s><emph type="center"/><emph type="italics"/>De Experimento in Bilance facto ac aliud reuera probante, <lb/>quàm velocitates e&longs;&longs;e &longs;icut spatia.<emph.end type="italics"/><emph.end type="center"/></s>

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<s>XXIII. Verumtamen, hi&longs;ce dimi&longs;&longs;is, acceden<lb/>dum e&longs;t ad &longs;ecundam, peculiaremve experientiam, <lb/>cui totam &longs;cientiam &longs;uper-ex&longs;truis, & de qua in hunc <lb/>modum præfaris. <emph type="italics"/>Atque, vt quam tibi promi&longs;i expe<lb/>rientiam, cum f&oelig;nore etiam exhibeam, adiungam & aliam, <lb/>à nullo mortalium hactenus ob&longs;eruatam, quæ & priorem <lb/>perfectißimè includat, & non rationem &longs;olùm, quâ celeritas <lb/>in naturali de&longs;cen&longs;u grauium augetur, &longs;ed eiu&longs;dem quoque <lb/>celeritatis pene incredibilem modum, ac men&longs;uram, exacti&longs;-<emph.end type="italics"/>

<pb pagenum="42"/><emph type="italics"/>&longs;imè determinet.<emph.end type="italics"/> Pergis declarando ecqua illa &longs;ie <lb/><emph type="italics"/>Aio igitur, ita e&longs;&longs;e à natura con&longs;titutum, vt globus quilibet, <lb/>tuiu&longs;cumque materiæ, ex vnius diametri altitudine cadens, <lb/>duplum &longs;ui ponderis, hoc e&longs;t, præter pondus quod &longs;ine im<lb/>petu in æquilibrio retineret, aliud &longs;ibi æquale attollat; & <lb/>ex altitudine duarum diametrorum, triplum; ex tribus dia<lb/>metris, quadruplum; & ita deinceps: adeo vt ex quauis <lb/>altitudine cadens, &longs;emper (vltra æquilibrium) toties pro<lb/>prium pondus multiplieatum attollat, quot in tota, vnde <lb/>cadit, altitudine diametri continentur.<emph.end type="italics"/> Subiicis, rem exag<lb/>gerando, <emph type="italics"/>Mirum &longs;anè quòd globus, cuius figuram, vt<lb/>pote &longs;implicißimam, capacißimàmque, natura &longs;ingulariter <lb/>amare videtur, men&longs;uram, ac modum, tam velocitatis <lb/>motus in de&longs;cen&longs;u grauium, quàm virtutis eius motricis, <lb/>quæ in eadem velocitate continetur, &longs;uâ nobis diametro exhi<lb/>beat: adeo vt ex decem, aut centum diametrorum altitudine <lb/>decidens, eum acquirat impetum, qui attollendis decem, aut <lb/>centum &longs;imilibus globis, in altera lance impo&longs;itis, &longs;ufficere <lb/>poßit, &longs;i materiæ conditio id patiatur.<emph.end type="italics"/> Addis & quid <lb/>ip&longs;e ob&longs;eruaueris. <emph type="italics"/>Expertus &longs;um ego,<emph.end type="italics"/> inquis, <emph type="italics"/>globum plum<lb/>beum vnius vnciæ, ex altitudine &longs;ex pedum, &longs;iue dia<lb/>metrorum centum quatuordecim cadentem, vncias toti<lb/>dem vltra æquilibrium, hoc est, libras &longs;eptem, & <lb/>vncias tres in altera lance impo&longs;itas, &longs;uo impetu eleua&longs;&longs;e, <lb/>non &longs;ine ingenti eorum qui præ&longs;entes aderant admiratione, <lb/>ac stupore.<emph.end type="italics"/> Tum & hæc habes. <emph type="italics"/>Porrò &longs;i id in paucis <lb/>diametris experiri placuerit, non admodùm magna opus <lb/>erit diligentia: at &longs;i è maiore altitudine idem tentare pla<lb/>cuerit, tum in hac, vt in cæteris Phy&longs;icis experientiis,<emph.end type="italics"/>

<pb pagenum="43"/><emph type="italics"/>accurata, ac &longs;olerti diligentia, atque industria, variis incom<lb/>modis occurrendum erit; vt ne fortè ex conditione materiæ <lb/>effectus impediatur, nó&longs;que ex errore, aut ex ignorantia, id <lb/>impoßibile arbitremur, quod &longs;olius materiæ vitio, ac defectu, <lb/>in certis circum&longs;tantiis minùs ex animi &longs;ententia &longs;uccedit.<emph.end type="italics"/><lb/>

<arrow.to.target n="fig15"></arrow.to.target><lb/>Paginas de<lb/>inceps ali<lb/>quot in&longs;umis, <lb/>vt ea incom<lb/>moda de&longs;cri<lb/>bas, & mo<lb/>dum, quo il<lb/>lis occurra<lb/>tur, tradas; <lb/>depicta &longs;cili<lb/>cet Bilanco, <lb/>quæ ip&longs;i&longs;&longs;i<lb/>ma heic ap<lb/>pingitur, agi<lb/>na putà im<lb/>mobili, & al<lb/>tera lancium <lb/>&longs;u&longs;pen&longs;a in <lb/>aëre, altera <lb/>&longs;upra men<lb/>&longs;am CD qui<lb/>e&longs;cente, cum <lb/>impo&longs;ito pondere, ac &longs;peculatore ad&longs;tante, qui ad <lb/>quamque vel minimam eius elationem attendat (id<lb/>que dum globus manu H dimi&longs;&longs;us incidit in alterius

<figure id="fig15"></figure><lb/>Paginas de<lb/>inceps ali<lb/>quot in&longs;umis, <lb/>vt ea incom<lb/>moda de&longs;cri<lb/>bas, & mo<lb/>dum, quo il<lb/>lis occurra<lb/>tur, tradas; <lb/>depicta &longs;cili<lb/>cet Bilanco, <lb/>quæ ip&longs;i&longs;&longs;i<lb/>ma heic ap<lb/>pingitur, agi<lb/>na putà im<lb/>mobili, & al<lb/>tera lancium <lb/>&longs;u&longs;pen&longs;a in <lb/>aëre, altera <lb/>&longs;upra men<lb/>&longs;am CD qui<lb/>e&longs;cente, cum <lb/>impo&longs;ito pondere, ac &longs;peculatore ad&longs;tante, qui ad <lb/>quamque vel minimam eius elationem attendat (id<lb/>que dum globus manu H dimi&longs;&longs;us incidit in alterius

<pb pagenum="44"/>medium, directione circuli G, cui ob æquilibrium <lb/>re&longs;pondet con&longs;imilis F) ac in&longs;uper vtraque lance ca<lb/>tenulis ferreis à &longs;capo AB per intermedios circulos, <lb/>triangulo&longs;-ve, dependente. Denique autem &longs;ubiun<lb/>gis, <emph type="italics"/>Tam apertam e&longs;&longs;e eius rei demon&longs;trationem, vt nul<lb/>lus,<emph.end type="italics"/> inquis, <emph type="italics"/>intellectus refragari po&longs;&longs;e videatur; dum &longs;emel <lb/>con&longs;tet (quod quilibet &longs;ine tanto apparatu, tantáque diligen<lb/>tia facillimè experiri potest) globum quemcumque, ex vnius <lb/>diametri altitudine, po&longs;&longs;e (vltra æquilibrium) pondus &longs;ibi <lb/>æquale, & ex duabus diametris duplum pondus attollere.<emph.end type="italics"/><lb/>Prætereo autem demon&longs;trationem non alio nixam <lb/>fundamento, quàm ipsâ experientiâ à te &longs;uppo&longs;itâ; <lb/>prætereoque item, quod iterùm &longs;ubdis, <emph type="italics"/>Illud quoque <lb/>pari certitudine constare, quod antè dictum e&longs;t, nihil ad pro<lb/>po&longs;itionis veritatem, atque euidentiam opus e&longs;&longs;e, vt ex al<lb/>tiore di&longs;tantia, tantóque apparatu experientia inquiratur, <lb/>quæ ex aliquot diametrorum altitudine plu&longs;quam abundè; <lb/>ac facillimè habeatur.<emph.end type="italics"/></s>

<s>XXIV. Et tale e&longs;t quidem tuum experimentum. <lb/>Ego autem, humani&longs;&longs;ime Vir, grati&longs;&longs;imo primùm <lb/>animo complector liberali&longs;&longs;imum erga me affectum; <lb/>ac deinde etiam tibi gratulor, quod primus morta<lb/>lium excogitâris quemadmodum negotium vi&longs;um <lb/>difficile reuocari ad trutinam po&longs;&longs;et. Nempe quan<lb/>tumvis res non videatur pro tua &longs;tare &longs;ententia; fuit <lb/>tamen tua &longs;olertia dignum, id in mentem inducere, <lb/>vnde examen improbum, quacumque ex parte huiu&longs;<lb/>modi foret, po&longs;&longs;et ca&longs;tigari. Ac &longs;i res quidem &longs;ic &longs;e <lb/>haberet, vt enarrati abs te videtur, reputari po&longs;&longs;et <lb/>penitus confecta; nullumque e&longs;&longs;et dubium, quin

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<s>XXXI. Subdis con&longs;equenter; <emph type="italics"/>Quod vt certiùs <lb/>fiat, primùm occurrendum est errori, qui facilè obrepere <lb/>potest, &longs;i quæ de cele itatis augmento in &longs;patiis æqualibus <lb/>antè demonstrata &longs;unt, minùs accuratè perpendantur. Cùm <lb/>enim ex &longs;uperioribus iam euidenter con&longs;tet, in naturali gra<lb/>uium de&longs;cen&longs;u, &longs;emper ex dupla di&longs;tantia, celeritatem ha<lb/>beri duplam; & ex tripla distantia, triplam; atque ita <lb/>deinceps, eadem ratione celeritatem augeri: nihil procliuius <lb/>e&longs;&longs;e potest, quàm vt quis exi&longs;timet, accelerationem <gap/><lb/>fieri per &longs;ubdiui&longs;ionem primi cuiu&longs;libet temporis, in paricis <lb/>&longs;emper minores, pro multitudine, & ratione &longs;patiorum æqua<lb/>lium, quæ motu decurruntur; ita videlicet, vt pars &longs;ecunda <lb/>spatij, ab&longs;oluatur dimidia parte temporis, quo prima pars <lb/>decur&longs;a est; & tertia pars &longs;patij, tertia parte eiu&longs;dem primi <lb/>temporis percurratur, & ita de cæteris: maximè cùm in hac <lb/>etiam hypothe&longs;i, &longs;patia & velocitates in eadem e&longs;&longs;e ratione, <lb/>& quod con&longs;equens e&longs;t, ex impeta quoque decidentium corpo<lb/>rum hac ratione inuariato, iidem omnes, quos experientia <lb/>docet effectus haberi, primo a&longs;pectu videantur.<emph.end type="italics"/> Heic pro<lb/>fectò rur&longs;us mirari tuam &longs;agacitatem par e&longs;t, qua<lb/>tenus eam non fugit error qui ex po&longs;itione à te a&longs;<lb/>&longs;erta con&longs;equitur; tamet&longs;i ip&longs;e non con&longs;equi ex iis, <lb/>quæ &longs;ubiicis, contendas. Me quod attinet, is e&longs;t <lb/>ip&longs;emet, quem deduxi aliàs aduer&longs;us Michaelem <lb/>Varronem, qui primus, quod &longs;ciam, eandem po&longs;itio<lb/>nem ante annos plus minùs &longs;exaginta defendit; de<lb/>clarando ex ea id incommodi inter cætera con&longs;equi, <lb/>vt, quemadmodum antè in&longs;inuaui, &longs;patia acqui&longs;ita <lb/>in &longs;ine æqualis cuiu&longs;libet temporis numeranda &longs;int, <lb/>vt difformiter, &longs;ic in plu&longs;quàm tripla ratione. Vt

<pb pagenum="58"/>autem iam rem te iudice experiar; ecce a&longs;&longs;umptâ, diui<lb/>&longs;aque linea, quam ip&longs;e v&longs;urpas, AB, & &longs;up<lb/>

<arrow.to.target n="fig16"></arrow.to.target><lb/>po&longs;ito tempore minutorum &longs;ex, quo &longs;uppo<lb/>nis AD primam partem percurri; Con&longs;tat <lb/>omninò, &longs;i &longs;ecunda æqualis pars DE per<lb/>curratur velocitate dupla ad illam, qua per<lb/>curritur AD, non in&longs;umi plus temporis in <lb/>percurrenda parte &longs;ecunda, quàm dimidium <lb/>eius, quod fuerit in&longs;umptum in prima (cùm <lb/>hac ratione tempora velocitatum &longs;ubmultipla <lb/>&longs;int) atque ita, &longs;i prima pars &longs;uperata fuerit mi<lb/>nutis &longs;ex, percurri &longs;ecundam dimidio, &longs;eu mi<lb/>nutis tribus. Eadem autem ratione &longs;i tertia <lb/>æqualis EF percurratur tripla, nece&longs;&longs;e e&longs;t <lb/>percurratur temporis triente, &longs;eu minutis duo<lb/>bus; ac eodem modo quarta FG quadrante, <lb/>&longs;eu &longs;e&longs;quiminuto, & quinta GH quinta par<lb/>te temporis, &longs;eu minuto vno cum &longs;ecundis duodecim; <lb/>ac &longs;exta HB, &longs;extante, &longs;eu minuto vno, & ita deinceps. <lb/>Atque ego quidem hûc v&longs;que nullum video paralo<lb/>gi&longs;mum. Quamobrem re&longs;tat, vt di&longs;quiratur, quot<lb/>nam æqualia &longs;patia, parte&longs;ve &longs;patii æquales tem<lb/>poribus primum con&longs;equentibus, ip&longs;ique æqualibus <lb/>percurrantur. Porrò cùm ad id perno&longs;cendum, ni<lb/>hil oporteat aliud, quàm iungere &longs;imul varia hæc <lb/>fragmenta primi temporis, hoc e&longs;t dimidium, trien<lb/>tem, quadrantem, & porrò parteis quintam, &longs;extam, <lb/>&longs;eptimam, &c. deprehendimus in ip&longs;o fine quarti <lb/>&longs;patij, ex iunctis &longs;imul dimidio, triente & quadrante, <lb/>confectum e&longs;&longs;e &longs;ecundum tempus, &longs;eu iteratò minuta

<figure id="fig16"></figure><lb/>po&longs;ito tempore minutorum &longs;ex, quo &longs;uppo<lb/>nis AD primam partem percurri; Con&longs;tat <lb/>omninò, &longs;i &longs;ecunda æqualis pars DE per<lb/>curratur velocitate dupla ad illam, qua per<lb/>curritur AD, non in&longs;umi plus temporis in <lb/>percurrenda parte &longs;ecunda, quàm dimidium <lb/>eius, quod fuerit in&longs;umptum in prima (cùm <lb/>hac ratione tempora velocitatum &longs;ubmultipla <lb/>&longs;int) atque ita, &longs;i prima pars &longs;uperata fuerit mi<lb/>nutis &longs;ex, percurri &longs;ecundam dimidio, &longs;eu mi<lb/>nutis tribus. Eadem autem ratione &longs;i tertia <lb/>æqualis EF percurratur tripla, nece&longs;&longs;e e&longs;t <lb/>percurratur temporis triente, &longs;eu minutis duo<lb/>bus; ac eodem modo quarta FG quadrante, <lb/>&longs;eu &longs;e&longs;quiminuto, & quinta GH quinta par<lb/>te temporis, &longs;eu minuto vno cum &longs;ecundis duodecim; <lb/>ac &longs;exta HB, &longs;extante, &longs;eu minuto vno, & ita deinceps. <lb/>Atque ego quidem hûc v&longs;que nullum video paralo<lb/>gi&longs;mum. Quamobrem re&longs;tat, vt di&longs;quiratur, quot<lb/>nam æqualia &longs;patia, parte&longs;ve &longs;patii æquales tem<lb/>poribus primum con&longs;equentibus, ip&longs;ique æqualibus <lb/>percurrantur. Porrò cùm ad id perno&longs;cendum, ni<lb/>hil oporteat aliud, quàm iungere &longs;imul varia hæc <lb/>fragmenta primi temporis, hoc e&longs;t dimidium, trien<lb/>tem, quadrantem, & porrò parteis quintam, &longs;extam, <lb/>&longs;eptimam, &c. deprehendimus in ip&longs;o fine quarti <lb/>&longs;patij, ex iunctis &longs;imul dimidio, triente & quadrante, <lb/>confectum e&longs;&longs;e &longs;ecundum tempus, &longs;eu iteratò minuta

<pb pagenum="59"/>&longs;ex, cum &longs;uperante duodecima parte. Ac pari ratio<lb/>ne in fine vndecimi &longs;patij, ex &longs;uper-adiunctis parti<lb/>bus quinta, &longs;exta, &longs;eptima, octaua, nona, decima, <lb/>vndecima, confectum e&longs;&longs;e tertium tempus, &longs;eu ite<lb/>rum minuta &longs;ex, cum &longs;uperante vna &longs;exage&longs;ima par<lb/>te. Et in fine &longs;patij trige&longs;imi primi, ex &longs;uperadiun<lb/>ctis duodecima, decimatertia, &c. confectum quar<lb/>tum, &longs;eu &longs;ex minuta, cum &longs;uperante parte circiter <lb/>quadrage&longs;ima. Et in fine octoge&longs;imi quarti, ex &longs;u<lb/>peradiunctis trige&longs;ima &longs;ecunda, trige&longs;ima tertia, &c. <lb/>confectum quintum, &longs;eu &longs;ex minuta, cum &longs;uperante <lb/>vna parte proximè nonage&longs;ima, atque ita deinceps. <lb/>Vnde licet aduertere, fore vt &longs;patia ea ratione incre&longs;<lb/>cant, quæ a&longs;&longs;umptis quibu&longs;libet æqualibus tempori<lb/>bus deprehendatur excedere, & difformiter quidem, <lb/>&longs;eu inæquabiliter, triplam; quippe procedendo per <lb/>hos numeros, vnum, quatuor, vndecim, triginta <expan abbr="vn&utilde;">vnum</expan>, <lb/>octoginta quatuor, &c. &longs;icque lapide decidente pri<lb/>mo momento per vnam v. c. orgyiam, fore vt &longs;ecun<lb/>do æquali momento decidendo per treis, in tertio <lb/>per &longs;eptem, in quarto per viginti, in quinto per quin<lb/>quaginta duas, deciderit in fine quinti, orgyiis octo<lb/>ginta quatuor, ac breui res &longs;it abitura in immen&longs;um, <lb/>&longs;ecu&longs;que quàm docet ip&longs;a experientia, iuxta quam <lb/>orgyiiæ in fine quinti momenti &longs;uperatæ, colliguntur <lb/>plures e&longs;&longs;e non debere, quàm viginti quinque. </s>

<s>XXXII. Demon&longs;tras ip&longs;e alia ratione (&longs;ed nimi<lb/>rùm aduer&longs;um te) non fieri accelerationem pro &longs;ub<lb/>diui&longs;ione i&longs;ta temporis. <emph type="italics"/>Si igitur,<emph.end type="italics"/> inquis, <emph type="italics"/>accelera io <lb/>motus per eam primi temporis &longs;ubdiui&longs;ionem fieret, de qua<emph.end type="italics"/>

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<s>XXXIV. Etenim illicò &longs;ic habes; <emph type="italics"/>Sed ex his, <lb/>& eadem pror&longs;us ratione aliud demonstratur, quod ingentis, <lb/>atque admirabilis paradoxi loco non immeritò fortaßis habe<lb/>ri poßit, nempe &longs;i spatium, per quod corpus graue quod<lb/>cumque de&longs;cendit, in parteis quotlibet æqualeis diui&longs;um intel<lb/>ligatur, & primæ, ac &longs;upremæ partis etiam de&longs;ignetur<emph.end type="italics"/>

<pb pagenum="64"/><emph type="italics"/>dimidia pars, & tertia, & quarta, ac deinceps cæteræ, inci<lb/>piendo diui&longs;iones i&longs;tas omneis ab infimo eiu&longs;dem primæ par<lb/>tis puncto, donec totidem de&longs;ignatæ &longs;int, quo<gap/> in reliquo &longs;patio <lb/>partes æquales acceptæ fuerint: tum &longs;ingulæ partes buiu&longs;mo<lb/>di æquales tanto præcisè tempore à corpore graui<emph.end type="italics"/><lb/>

<arrow.to.target n="fig17"></arrow.to.target><lb/><emph type="italics"/>de&longs;cendente percurrantur, quanto partes ip&longs;is analo<lb/>gæ, ac respondentes in &longs;uprema parte<emph.end type="italics"/> (&longs;eu infe<lb/>riore eius dimidio) <emph type="italics"/>ab eodem corpore graui de<lb/>cur&longs;æ fuerint.<emph.end type="italics"/> Rem con&longs;equenter ita declaras; <lb/>S<emph type="italics"/>it &longs;patium<emph.end type="italics"/> AB (in &longs;chemate hoc) <emph type="italics"/>per quod <lb/>corpus graue de&longs;cendat, in parteis exempli gratiâ <lb/>&longs;ex æqualeis diui&longs;um in<emph.end type="italics"/> C, D, E, F, & G: <emph type="italics"/>primæ<lb/>que, ac &longs;upremæ partis<emph.end type="italics"/> AC, <emph type="italics"/>ex infimo eius pun<lb/>cto<emph.end type="italics"/> C <emph type="italics"/>de&longs;ignetur primùm media pars<emph.end type="italics"/> CH, <emph type="italics"/>dein<lb/>de tertia<emph.end type="italics"/> CI, <emph type="italics"/>& quarta<emph.end type="italics"/> CK, <emph type="italics"/>itemque quinta, & <lb/>&longs;exta<emph.end type="italics"/> CL, <emph type="italics"/>&<emph.end type="italics"/> CM. <emph type="italics"/>Dico corpus graue de&longs;cen<lb/>dens per<emph.end type="italics"/> AB <emph type="italics"/>tanto præcisè tempore pertran&longs;ire &longs;e<lb/>cundam partem<emph.end type="italics"/> CD, <emph type="italics"/>quanto dimidiam primæ par<lb/>tis<emph.end type="italics"/> HC, <emph type="italics"/>antè pertran&longs;iuit; & &longs;imiliter pari, atque <lb/>æquali tempore partem<emph.end type="italics"/> DE, <emph type="italics"/>quæ ordine tertia est, <lb/>& tertiam prim&ecedil; partis, nempe<emph.end type="italics"/> IC <emph type="italics"/>ab eodem cor<lb/>pore de&longs;cendente tran&longs;curri, & ita de c&ecedil;teris.<emph.end type="italics"/> Tunc <lb/>autem pergis. <emph type="italics"/>Et quidem de &longs;ecunda parte<emph.end type="italics"/> CD, <lb/><emph type="italics"/>eam non longiore tempore decurri, quàm quo prim&ecedil; <lb/>partis posterior dimidia pars tran&longs;ini&longs;&longs;a fuerit, iam <lb/>paulò antè o&longs;ten&longs;um est, nec maiore negotio idem <lb/>de c&ecedil;teris quoque partibus concludetur. Sumpto <lb/>enim<emph.end type="italics"/> CN, <emph type="italics"/>&c.<emph.end type="italics"/></s>

<figure id="fig17"></figure><lb/><emph type="italics"/>de&longs;cendente percurrantur, quanto partes ip&longs;is analo<lb/>gæ, ac respondentes in &longs;uprema parte<emph.end type="italics"/> (&longs;eu infe<lb/>riore eius dimidio) <emph type="italics"/>ab eodem corpore graui de<lb/>cur&longs;æ fuerint.<emph.end type="italics"/> Rem con&longs;equenter ita declaras; <lb/>S<emph type="italics"/>it &longs;patium<emph.end type="italics"/> AB (in &longs;chemate hoc) <emph type="italics"/>per quod <lb/>corpus graue de&longs;cendat, in parteis exempli gratiâ <lb/>&longs;ex æqualeis diui&longs;um in<emph.end type="italics"/> C, D, E, F, & G: <emph type="italics"/>primæ<lb/>que, ac &longs;upremæ partis<emph.end type="italics"/> AC, <emph type="italics"/>ex infimo eius pun<lb/>cto<emph.end type="italics"/> C <emph type="italics"/>de&longs;ignetur primùm media pars<emph.end type="italics"/> CH, <emph type="italics"/>dein<lb/>de tertia<emph.end type="italics"/> CI, <emph type="italics"/>& quarta<emph.end type="italics"/> CK, <emph type="italics"/>itemque quinta, & <lb/>&longs;exta<emph.end type="italics"/> CL, <emph type="italics"/>&<emph.end type="italics"/> CM. <emph type="italics"/>Dico corpus graue de&longs;cen<lb/>dens per<emph.end type="italics"/> AB <emph type="italics"/>tanto præcisè tempore pertran&longs;ire &longs;e<lb/>cundam partem<emph.end type="italics"/> CD, <emph type="italics"/>quanto dimidiam primæ par<lb/>tis<emph.end type="italics"/> HC, <emph type="italics"/>antè pertran&longs;iuit; & &longs;imiliter pari, atque <lb/>æquali tempore partem<emph.end type="italics"/> DE, <emph type="italics"/>quæ ordine tertia est, <lb/>& tertiam prim&ecedil; partis, nempe<emph.end type="italics"/> IC <emph type="italics"/>ab eodem cor<lb/>pore de&longs;cendente tran&longs;curri, & ita de c&ecedil;teris.<emph.end type="italics"/> Tunc <lb/>autem pergis. <emph type="italics"/>Et quidem de &longs;ecunda parte<emph.end type="italics"/> CD, <lb/><emph type="italics"/>eam non longiore tempore decurri, quàm quo prim&ecedil; <lb/>partis posterior dimidia pars tran&longs;ini&longs;&longs;a fuerit, iam <lb/>paulò antè o&longs;ten&longs;um est, nec maiore negotio idem <lb/>de c&ecedil;teris quoque partibus concludetur. Sumpto <lb/>enim<emph.end type="italics"/> CN, <emph type="italics"/>&c.<emph.end type="italics"/></s>

<s>XXXV. Verùm priu&longs;quàm gradus ad <lb/>cæteras fiat, con&longs;i&longs;tendum e&longs;t in hac prima; cùm

<pb pagenum="65"/>non &longs;it nequicquam, quod de ip&longs;a admonui. O&longs;ten<lb/>&longs;um e&longs;&longs;e ais tempus, quo percurritur CD, æquale e&longs;&longs;e <lb/>tempori, quo decur&longs;um fuerit HC: &longs;eu, ne confun<lb/>damus, & rem explicemus, qua&longs;i repetendam ex &longs;u<lb/>periore &longs;chemate, o&longs;ten&longs;um e&longs;&longs;e ais id tempus, quo <lb/>percurritur DE, æquale e&longs;&longs;e tempori, quo de<lb/>

<arrow.to.target n="fig18"></arrow.to.target><lb/>cur&longs;um fuerit SD. At primò, in&longs;inuatum iam <lb/>e&longs;t id e&longs;&longs;e impo&longs;&longs;ibile; cùm quo iure ip&longs;e bi<lb/>&longs;ecui&longs;ti primam partem AD in S, liceat bi<lb/>&longs;ecare primum dimidium AS in P; & rursùs <lb/>primum horum dimidiorum in duo alia, & <lb/>primum i&longs;torum in duo, & ita porrò quoties <lb/>libuerit: Iuxta tuum verò ratiocinium, &longs;e qua<lb/>tur tempus, quo percurritur SD, e&longs;&longs;e æqua<lb/>le tempori, quo decur&longs;um fuerit PS, quate<lb/>nus, vt velocitas per totam DE e&longs;t dupla ve<lb/>locitatis per totam SD, quemadmodum in<lb/>teruallum duplum e&longs;t; ita velocitas per totam <lb/>SD dupla e&longs;t velocitatis per totam PS, &longs;icut <lb/>interuallum e&longs;t itidem duplum; Hoc autem <lb/>po&longs;ito, vlteriùs &longs;equatur tempus per SD, atque <lb/>idcircò per DE ip&longs;i æquale, e&longs;&longs;e non iam minuto<lb/>rum duorum, vt o&longs;tendi&longs;ti, &longs;ed minuti vnius cum <lb/>triente; vt pote coæquatum tempori per PS, quod <lb/>tam e&longs;&longs;e debet triens quatuor minutorum (quibus <lb/><gap/>is AS percurri) quàm tempus per SD &longs;tatuitur à <lb/>te triens minutorum &longs;ex (quibus percurritur AD) <lb/>cùm id tamen factuimpo&longs;&longs;ibile &longs;it, repugnantia putà <lb/>inuoluens, ac tantò magis, quantò ex vlterioribus <lb/>&longs;ubdiui&longs;ionibus probari pote&longs;t eandem DE percurri

<figure id="fig18"></figure><lb/>cur&longs;um fuerit SD. At primò, in&longs;inuatum iam <lb/>e&longs;t id e&longs;&longs;e impo&longs;&longs;ibile; cùm quo iure ip&longs;e bi<lb/>&longs;ecui&longs;ti primam partem AD in S, liceat bi<lb/>&longs;ecare primum dimidium AS in P; & rursùs <lb/>primum horum dimidiorum in duo alia, & <lb/>primum i&longs;torum in duo, & ita porrò quoties <lb/>libuerit: Iuxta tuum verò ratiocinium, &longs;e qua<lb/>tur tempus, quo percurritur SD, e&longs;&longs;e æqua<lb/>le tempori, quo decur&longs;um fuerit PS, quate<lb/>nus, vt velocitas per totam DE e&longs;t dupla ve<lb/>locitatis per totam SD, quemadmodum in<lb/>teruallum duplum e&longs;t; ita velocitas per totam <lb/>SD dupla e&longs;t velocitatis per totam PS, &longs;icut <lb/>interuallum e&longs;t itidem duplum; Hoc autem <lb/>po&longs;ito, vlteriùs &longs;equatur tempus per SD, atque <lb/>idcircò per DE ip&longs;i æquale, e&longs;&longs;e non iam minuto<lb/>rum duorum, vt o&longs;tendi&longs;ti, &longs;ed minuti vnius cum <lb/>triente; vt pote coæquatum tempori per PS, quod <lb/>tam e&longs;&longs;e debet triens quatuor minutorum (quibus <lb/><gap/>is AS percurri) quàm tempus per SD &longs;tatuitur à <lb/>te triens minutorum &longs;ex (quibus percurritur AD) <lb/>cùm id tamen factuimpo&longs;&longs;ibile &longs;it, repugnantia putà <lb/>inuoluens, ac tantò magis, quantò ex vlterioribus <lb/>&longs;ubdiui&longs;ionibus probari pote&longs;t eandem DE percurri

<pb pagenum="66"/>rursùs non vno minuto, ac triente, &longs;ed &longs;ecundis &longs;olùm <lb/>proximè quinquaginta tribus; & rursùs proximè <lb/>octodecim, & rursùs &longs;ex, &c. Deinde, cùm ad id <lb/>o&longs;tendendum v&longs;us fueris ea ratione, quòd nece&longs;&longs;e &longs;it, <lb/><emph type="italics"/>velocitatem in D duplam e&longs;&longs;e velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& veloci<lb/>tatem in E duplam velocitatis in D, &longs;icuti<emph.end type="italics"/> AD <emph type="italics"/>ponitur du<lb/>pla ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>& &longs;imiliter<emph.end type="italics"/> AE <emph type="italics"/>dupla e&longs;t ip&longs;ius<emph.end type="italics"/> AD; &longs;e<lb/>quitur exinde, vt tam tota DE, quàm tota SD per<lb/>curratur non triente, &longs;ed dimidio temporis, quo tota <lb/>AD: cùm vbicumque e&longs;t velocitatis duplum, vbi &longs;it <lb/>dimidium temporis duntaxat, neque tu id infregeris, <lb/>&longs;ed incommodum &longs;olùm attuleris, quod cùm euertat <lb/>con&longs;equutionem de &longs;ubdiui&longs;ione primi temporis, &longs;up<lb/>po&longs;itionem quoque euertit de velocitate dupla in du<lb/>plo &longs;patii, tripla in triplo, &c. vt aliquoties e&longs;t incul<lb/>catum. Nam & quod po&longs;teà a&longs;&longs;umis tempus per <lb/>SD, atque adeò DE e&longs;&longs;e breuius, quàm dimidium <lb/>eius, quo tran&longs;curritur AD, id facis quidem rectè, <lb/>verumtamen iure non tuo; quippe id facis &longs;olùm me<lb/>tu eius incommodi, quod præ&longs;en&longs;iti po&longs;&longs;e vrgeri de <lb/>motus acceleratione vniformiter, continenterque in<lb/>cre&longs;cente: cùm id alioquin & repugnet tuæ &longs;uppo&longs;i<lb/>tioni de velocitate dupla in duplo &longs;patio, tripla in <lb/>triplo, &c. & euertat demon&longs;trationem ip&longs;am, ad <lb/>quam iam recurris, cùm o&longs;ten&longs;um ais tempus, quo <lb/>percurritur &longs;ecunda pars, æquale e&longs;&longs;e tempori, quo <lb/>tran&longs;mittitur dimidium po&longs;terius, &longs;eu inferius primæ: <lb/>neque enim id fuit o&longs;ten&longs;um alio ratiocinio, quam <lb/>quod ip&longs;emet &longs;tatim pernega&longs;ti. </s>

<s>XXXVI. Attamen &longs;upponatur etiam tua huiu&longs;-

Line 1209

Line 1207

<s>XXXVII. Vt enim probes (re&longs;umpto iam recen<lb/>tiore &longs;chemate) tempus per tertiam partem DE æqua<lb/>le e&longs;&longs;e tempori per trientem primæ partis IC, <emph type="italics"/>Sumpto,<emph.end type="italics"/><lb/>inquis, CN <emph type="italics"/>æquali ip&longs;i<emph.end type="italics"/> CE, <emph type="italics"/>erit tota<emph.end type="italics"/> AD <emph type="italics"/>diui&longs;a in <lb/>treis parteis æqualeis<emph.end type="italics"/> AI, IN, <emph type="italics"/>&<emph.end type="italics"/> ND; <emph type="italics"/>eritque veloci<lb/>tas in<emph.end type="italics"/> D <emph type="italics"/>ad velocitatem in<emph.end type="italics"/> I, <emph type="italics"/>vt tota<emph.end type="italics"/> AD <emph type="italics"/>ad ip&longs;am<emph.end type="italics"/> AI, <emph type="italics"/>hoc <lb/>e&longs;t tripla. Cumque ob eandem cau&longs;&longs;am velocitas quoque in<emph.end type="italics"/> E <lb/><emph type="italics"/>tripla etiam &longs;it velocitatis in<emph.end type="italics"/> C, <emph type="italics"/>erit velocitas per totam<emph.end type="italics"/> DE <lb/><emph type="italics"/>tripla velocitatis per totam<emph.end type="italics"/> IC, <emph type="italics"/>&longs;icut tota<emph.end type="italics"/> DE <emph type="italics"/>tripla e&longs;t ip&longs;ius<emph.end type="italics"/><lb/>IC; <emph type="italics"/>ac proinde percurrentur<emph.end type="italics"/> IC, <emph type="italics"/>&<emph.end type="italics"/> DE <emph type="italics"/>æquali tempore.<emph.end type="italics"/>

<pb pagenum="69"/>Et con&longs;equenter, vt pergas probate tempus per <lb/>

<arrow.to.target n="fig19"></arrow.to.target><lb/>quartam partem EF æquale e&longs;&longs;e tempori per <lb/>KC quadrantem primæ, <emph type="italics"/>Similiter,<emph.end type="italics"/> inquis, <emph type="italics"/>di<lb/>ui&longs;a bifariàm<emph.end type="italics"/> CD <emph type="italics"/>in<emph.end type="italics"/> O, <emph type="italics"/>&longs;umptoque quadr inte<emph.end type="italics"/><lb/>DP <emph type="italics"/>æquali ip&longs;i<emph.end type="italics"/> KC, <emph type="italics"/>tota<emph.end type="italics"/> AE <emph type="italics"/>diui&longs;a erit in par<lb/>teis quatuor æqualeis<emph.end type="italics"/> AK KO, OP, PE; <emph type="italics"/>ideó <lb/>que velocitas in<emph.end type="italics"/> E <emph type="italics"/>erit quadrupla velocitatis in<emph.end type="italics"/> K, <emph type="italics"/>vt <lb/>tota<emph.end type="italics"/> AE <emph type="italics"/>quadrupla e&longs;t ip&longs;ius<emph.end type="italics"/> AK. <emph type="italics"/>At velocitas <lb/>quoque in<emph.end type="italics"/> F <emph type="italics"/>ob eandem rationem quadrupla etiam e&longs;t <lb/>velocitatis in<emph.end type="italics"/> C; <emph type="italics"/>velocitas igitur per totam<emph.end type="italics"/> EF <lb/><emph type="italics"/>quadrupla e&longs;t velocitatis per totam<emph.end type="italics"/> KC, <emph type="italics"/>&longs;icut tota<emph.end type="italics"/><lb/>EF <emph type="italics"/>quadrupla e&longs;t ip&longs;ius<emph.end type="italics"/> KC. <emph type="italics"/>Percurrentur igitur<emph.end type="italics"/><lb/>KC, <emph type="italics"/>&<emph.end type="italics"/> EF <emph type="italics"/>æquali tempore.<emph.end type="italics"/> Sequitur, <emph type="italics"/>Ea <lb/>dem autem etiam ratio e&longs;t cæterarum omnium par<lb/>tium, vt facilè quilibet ex i&longs;tis per &longs;e intelliget.<emph.end type="italics"/> Con<lb/>cludis, <emph type="italics"/>Si &longs;patium igitur, per quod corpus quodcum<lb/>que graue de&longs;cendit, ea, qua dictum e&longs;t, ratione diui<lb/>&longs;um intelligatur, &longs;ingulæ partes huiu&longs;modi æquales tanto <lb/>præcisè tempore à corpore graui de&longs;cendente percur<lb/>rentur, quantò partes ip&longs;is analogæ ac re&longs;pondentes <lb/>in &longs;uprema parte<emph.end type="italics"/> (aut inferiore eius dimidio) <emph type="italics"/>de&longs;i <lb/>gnatæ ab eodem corpore graui decur&longs;æ fuerint, vt est <lb/>propo&longs;itum.<emph.end type="italics"/> Prætereo autem, quod &longs;ubinde de<lb/>claras te ad&longs;crip&longs;i&longs;&longs;e fini cuiu&longs;que &longs;ex partium <lb/>numerum integrum, incipiendo ab vnitate, <lb/>ad de&longs;ignandum velocitatis gradus illeic acqui&longs;itos, & <lb/>ex æquo factos cum decur&longs;is partibus; ad&longs;crip&longs;i&longs;&longs;e au<lb/>tem mediis interuallis &longs;ecundæ, & &longs;equentium partium <lb/>fractos numeros, ad de&longs;ignandum tempora, &longs;iue fra<lb/>ctiones temporis primi, quibus vnumquodque &longs;pa-

<figure id="fig19"></figure><lb/>quartam partem EF æquale e&longs;&longs;e tempori per <lb/>KC quadrantem primæ, <emph type="italics"/>Similiter,<emph.end type="italics"/> inquis, <emph type="italics"/>di<lb/>ui&longs;a bifariàm<emph.end type="italics"/> CD <emph type="italics"/>in<emph.end type="italics"/> O, <emph type="italics"/>&longs;umptoque quadr inte<emph.end type="italics"/><lb/>DP <emph type="italics"/>æquali ip&longs;i<emph.end type="italics"/> KC, <emph type="italics"/>tota<emph.end type="italics"/> AE <emph type="italics"/>diui&longs;a erit in par<lb/>teis quatuor æqualeis<emph.end type="italics"/> AK KO, OP, PE; <emph type="italics"/>ideó <lb/>que velocitas in<emph.end type="italics"/> E <emph type="italics"/>erit quadrupla velocitatis in<emph.end type="italics"/> K, <emph type="italics"/>vt <lb/>tota<emph.end type="italics"/> AE <emph type="italics"/>quadrupla e&longs;t ip&longs;ius<emph.end type="italics"/> AK. <emph type="italics"/>At velocitas <lb/>quoque in<emph.end type="italics"/> F <emph type="italics"/>ob eandem rationem quadrupla etiam e&longs;t <lb/>velocitatis in<emph.end type="italics"/> C; <emph type="italics"/>velocitas igitur per totam<emph.end type="italics"/> EF <lb/><emph type="italics"/>quadrupla e&longs;t velocitatis per totam<emph.end type="italics"/> KC, <emph type="italics"/>&longs;icut tota<emph.end type="italics"/><lb/>EF <emph type="italics"/>quadrupla e&longs;t ip&longs;ius<emph.end type="italics"/> KC. <emph type="italics"/>Percurrentur igitur<emph.end type="italics"/><lb/>KC, <emph type="italics"/>&<emph.end type="italics"/> EF <emph type="italics"/>æquali tempore.<emph.end type="italics"/> Sequitur, <emph type="italics"/>Ea <lb/>dem autem etiam ratio e&longs;t cæterarum omnium par<lb/>tium, vt facilè quilibet ex i&longs;tis per &longs;e intelliget.<emph.end type="italics"/> Con<lb/>cludis, <emph type="italics"/>Si &longs;patium igitur, per quod corpus quodcum<lb/>que graue de&longs;cendit, ea, qua dictum e&longs;t, ratione diui<lb/>&longs;um intelligatur, &longs;ingulæ partes huiu&longs;modi æquales tanto <lb/>præcisè tempore à corpore graui de&longs;cendente percur<lb/>rentur, quantò partes ip&longs;is analogæ ac re&longs;pondentes <lb/>in &longs;uprema parte<emph.end type="italics"/> (aut inferiore eius dimidio) <emph type="italics"/>de&longs;i <lb/>gnatæ ab eodem corpore graui decur&longs;æ fuerint, vt est <lb/>propo&longs;itum.<emph.end type="italics"/> Prætereo autem, quod &longs;ubinde de<lb/>claras te ad&longs;crip&longs;i&longs;&longs;e fini cuiu&longs;que &longs;ex partium <lb/>numerum integrum, incipiendo ab vnitate, <lb/>ad de&longs;ignandum velocitatis gradus illeic acqui&longs;itos, & <lb/>ex æquo factos cum decur&longs;is partibus; ad&longs;crip&longs;i&longs;&longs;e au<lb/>tem mediis interuallis &longs;ecundæ, & &longs;equentium partium <lb/>fractos numeros, ad de&longs;ignandum tempora, &longs;iue fra<lb/>ctiones temporis primi, quibus vnumquodque &longs;pa-

<pb pagenum="70"/>tiorum primum con&longs;equentium percurritur. </s>

<s>XXXVIII. Videris itaque imprimis haud abs re <lb/>dixi&longs;&longs;e <emph type="italics"/>admirabile Paradoxum:<emph.end type="italics"/> cùm habui&longs;ti videlicet <lb/>po&longs;terius primæ partis dimidium, vt &longs;calam Prototy<lb/>picam, cuius gradibus coæquetur, exqui&longs;itéque men<lb/>&longs;uretur cæterarum omnium con&longs;equentium duratio; <lb/>tamet&longs;i illud e&longs;&longs;e po&longs;&longs;it parte digiti mille&longs;ima minus, <lb/>& i&longs;tarum aggeries, quanta e&longs;t tota &longs;emidiameter <lb/>mundi. Non &longs;anè, quod negem dari lineas, alia&longs;que <lb/>magnitudines, in quibus arcana prorsùs admiranda, <lb/>& quæ nemo vnquam &longs;u&longs;picatus fui&longs;&longs;et, detegantur à <lb/>Geometris: &longs;ed quod videri iure po&longs;&longs;it penitus incre<lb/>dibile alliga&longs;&longs;e potius naturam ip&longs;a qua&longs;i fat&adot; cætera<lb/>rum partiumip&longs;i po&longs;teriori, quàm prioridimidio; imò <lb/>& non tam toti ip&longs;i dimidio, quàm infimo illius pun<lb/>cto, à quo triens, quadrans, cæteræ fractiones debeant <lb/>&longs;upputari. Ecquod-nam e&longs;&longs;e enim pote&longs;t illius pri<lb/>uilegium; aut quid-nam admi&longs;&longs;um à priore dimidio <lb/>e&longs;t, vt eo excidi&longs;&longs;e, & pro nihilo reputari iure cen&longs;ea<lb/>tur? Quid habere <expan abbr="cõmune">commune</expan> pote&longs;t cente&longs;ima, mille&longs;ima <lb/>decie&longs;que, & centies mille&longs;ima pars <expan abbr="c&utilde;">cum</expan> po&longs;teriore hoc <lb/>dimidio; non habere autem cum illo priore, à quo to<lb/>tus motus dependet; non cùm cæteris partibus perinde <lb/>&longs;uccedentibus; non &longs;altem cum vicinis, quibu&longs;cum <lb/>proximè cohæret? Deinde, ne hac in re hæream, ea<lb/>dem prorsùs incommoda ex triente pro tertia parte, <lb/>ex quadrante pro quarta, atque ita de cæteris, quæ ex <lb/>ip&longs;o dimidio pro &longs;ecunda con&longs;equuntur. Nam, vt <lb/>rem circa ip&longs;um trientem, partemque tertiam &longs;olùm <lb/>atringam, Sicuti primùm totam AC in treis diui&longs;i&longs;ti

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<s>XXXIX. Supere&longs;t po&longs;tremum, &longs;iue tertium mem<lb/>brum, de Ratione continuò dupla, qua pertran&longs;iri <lb/>&longs;patia temporibus continuò æqualibus infers. Primùm <lb/>autem, vbi adnota&longs;ti non po&longs;&longs;e quidem ex deductio<lb/>ne à te mox facta, <emph type="italics"/>ab&longs;olutè colligi quantum præcisè tem<lb/>poris graue ex aßignata altitudine de&longs;cendens in toto de&longs;<lb/>cen&longs;u in&longs;umat, ni&longs;i di&longs;tinctè etiam cogno&longs;eatur tempus de&longs;-<emph.end type="italics"/>

<pb pagenum="73"/><emph type="italics"/>cen&longs;us non tantum per totam primam partem<emph.end type="italics"/> AC (in nu<lb/>pero &longs;chemate) <emph type="italics"/>&longs;ed etiam &longs;eor&longs;um per<emph.end type="italics"/> AH, <emph type="italics"/>& per<emph.end type="italics"/> HC; <lb/><emph type="italics"/>quod multò difficilius e&longs;&longs;e arbitreris, quàm<emph.end type="italics"/> G<emph type="italics"/>alileo videatur; <lb/>at cognitis, aut præ&longs;uppo&longs;itis temporibus illis, facilè deinceps <lb/>totum tempus totius de&longs;cen&longs;us per quamcumque de&longs;ignatam <lb/>altitudinem determinari:<emph.end type="italics"/> id quod <emph type="italics"/>postmodum te osten&longs;u<lb/>rum recipis.<emph.end type="italics"/> Tum &longs;upponens me ex&longs;pectare <lb/>

<arrow.to.target n="fig20"></arrow.to.target><lb/>diutiùs, quid &longs;is dicturus de <emph type="italics"/>Ratione, qua &longs;e ha<lb/>bent &longs;patia æquali tempore emen&longs;a,<emph.end type="italics"/> &longs;ic infis, <emph type="italics"/>Aio <lb/>verò æqualibus temporibus &longs;patia decurri maiora <lb/>&longs;emper, ac maiora in Ratione dupla. Diui&longs;o enim <lb/>spatio<emph.end type="italics"/> AB, <emph type="italics"/>per quod &longs;upponitur fieri de&longs;cen&longs;us, in <lb/>parteis quotcumque æqualeis in<emph.end type="italics"/> C, D, E, F, <emph type="italics"/>&c. iam <lb/>osten&longs;um est partem &longs;ecundam<emph.end type="italics"/> CD, <emph type="italics"/>& primæ par<lb/>tis dimidiam partem inferiorem<emph.end type="italics"/> NC <emph type="italics"/>æquali tempore <lb/>percurri; & ob eam quidem cau&longs;&longs;am, quòd, vt pars<emph.end type="italics"/><lb/>CD <emph type="italics"/>dupla e&longs;t partis<emph.end type="italics"/> NC, <emph type="italics"/>ita velocitas quoque per <lb/>totam<emph.end type="italics"/> CD <emph type="italics"/>dupla &longs;it velocitatis per totam<emph.end type="italics"/> NC. <emph type="italics"/>At <lb/>&longs;imili ratione etiam efficitur, velocitatem per totam<emph.end type="italics"/><lb/>DF <emph type="italics"/>duplam e&longs;&longs;e velocitatis eius, quæ habetur per <lb/>totam<emph.end type="italics"/> CD; <emph type="italics"/>&longs;icut tota<emph.end type="italics"/> DF <emph type="italics"/>dupla e&longs;t ip&longs;ius<emph.end type="italics"/> CD. <lb/>Æ<emph type="italics"/>quali igitur tempore<emph.end type="italics"/> CD, <emph type="italics"/>&<emph.end type="italics"/> DF <emph type="italics"/>decurruntur; <lb/>eademque omninò ratio e&longs;t ip&longs;arum<emph.end type="italics"/> DF, <emph type="italics"/>&<emph.end type="italics"/> FK, <lb/><emph type="italics"/>cæterarumque omnium &longs;e pariter in ratione dupla &longs;u <lb/>perantium; vt &longs;atis manife&longs;tum e&longs;t.<emph.end type="italics"/> S<emph type="italics"/>patia igitur <lb/>æqualibus temporibus emen&longs;a, & velocitates ii&longs;dem <lb/>temporibus æqualibus acqui&longs;itæ &longs;emper augentur in <lb/>continua ratione dupla.<emph.end type="italics"/></s>

<figure id="fig20"></figure><lb/>diutiùs, quid &longs;is dicturus de <emph type="italics"/>Ratione, qua &longs;e ha<lb/>bent &longs;patia æquali tempore emen&longs;a,<emph.end type="italics"/> &longs;ic infis, <emph type="italics"/>Aio <lb/>verò æqualibus temporibus &longs;patia decurri maiora <lb/>&longs;emper, ac maiora in Ratione dupla. Diui&longs;o enim <lb/>spatio<emph.end type="italics"/> AB, <emph type="italics"/>per quod &longs;upponitur fieri de&longs;cen&longs;us, in <lb/>parteis quotcumque æqualeis in<emph.end type="italics"/> C, D, E, F, <emph type="italics"/>&c. iam <lb/>osten&longs;um est partem &longs;ecundam<emph.end type="italics"/> CD, <emph type="italics"/>& primæ par<lb/>tis dimidiam partem inferiorem<emph.end type="italics"/> NC <emph type="italics"/>æquali tempore <lb/>percurri; & ob eam quidem cau&longs;&longs;am, quòd, vt pars<emph.end type="italics"/><lb/>CD <emph type="italics"/>dupla e&longs;t partis<emph.end type="italics"/> NC, <emph type="italics"/>ita velocitas quoque per <lb/>totam<emph.end type="italics"/> CD <emph type="italics"/>dupla &longs;it velocitatis per totam<emph.end type="italics"/> NC. <emph type="italics"/>At <lb/>&longs;imili ratione etiam efficitur, velocitatem per totam<emph.end type="italics"/><lb/>DF <emph type="italics"/>duplam e&longs;&longs;e velocitatis eius, quæ habetur per <lb/>totam<emph.end type="italics"/> CD; <emph type="italics"/>&longs;icut tota<emph.end type="italics"/> DF <emph type="italics"/>dupla e&longs;t ip&longs;ius<emph.end type="italics"/> CD. <lb/>Æ<emph type="italics"/>quali igitur tempore<emph.end type="italics"/> CD, <emph type="italics"/>&<emph.end type="italics"/> DF <emph type="italics"/>decurruntur; <lb/>eademque omninò ratio e&longs;t ip&longs;arum<emph.end type="italics"/> DF, <emph type="italics"/>&<emph.end type="italics"/> FK, <lb/><emph type="italics"/>cæterarumque omnium &longs;e pariter in ratione dupla &longs;u <lb/>perantium; vt &longs;atis manife&longs;tum e&longs;t.<emph.end type="italics"/> S<emph type="italics"/>patia igitur <lb/>æqualibus temporibus emen&longs;a, & velocitates ii&longs;dem <lb/>temporibus æqualibus acqui&longs;itæ &longs;emper augentur in <lb/>continua ratione dupla.<emph.end type="italics"/></s>

<s>XL. Cæterùm, cùm i&longs;te habeatur qua&longs;i <lb/>prouentus quidam eximius totius tuæ Di&longs;&longs;ertationis;

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<s>XLVI. Ad primum porrò quod attinet; videtur <lb/>&longs;anè non &longs;atis idonea refutatio experimenti, quod <lb/>ille te&longs;tatur &longs;e &longs;æpiùs, ac diligenter peregi&longs;&longs;e, ni&longs;i te&longs;te<lb/>ris temetip&longs;um idem tenta&longs;&longs;e, peregi&longs;&longs;eque, ac fui&longs;&longs;e <lb/>rem &longs;ecùs à te ob&longs;eruatam, fal&longs;itati&longs;que adeò con-

<pb pagenum="84"/>uictam. Quod enim e&longs;&longs;e verendum ais, ne halluci<lb/>natus fuerit, hoc &longs;atis profectò non e&longs;t; & quòd argu<lb/>mento e&longs;t tibi error vehemens, quem admi&longs;i&longs;&longs;e illum <lb/>ais, circa progre&longs;&longs;ionem iuxta &longs;eriem numerorum im<lb/>parium; declaratum iam antè e&longs;t, vt ea quoque ip&longs;a in <lb/>re non fuerit erroris conuictus; imò & &longs;uffragium <lb/>etiam tulerit cùm ex aliis experimentis, tum etiam ex <lb/>tuo, hoc e&longs;t in Bilance peracto. Ad &longs;ecundum quod <lb/>&longs;pectat, determinauit ille, quo præcisè tempore &longs;ecun<lb/>da &longs;patij pars, ac dimidium primæ, & quævis alia per<lb/>curreretur, ex a&longs;&longs;ignato tempore, quo pars prima de<lb/>curritur. O&longs;tendit nimirum ex &longs;uis principijs, <emph type="italics"/>Si à <lb/>lationis principio duo quælibet spatia &longs;umantur, tempora <lb/>ip&longs;orum fore inter &longs;e, vt alterum eorum ad spatium medium <lb/>proportionale inter ip&longs;a.<emph.end type="italics"/> Adeò vt, &longs;i inter AB pri<lb/>mam partem, & AC aggregatum primæ cum <lb/>

<arrow.to.target n="fig21"></arrow.to.target><lb/>&longs;ecunda inuenias mediam proportionalem AD, <lb/>tempus ca&longs;us per AB, ad tempus ca&longs;us per AC, <lb/>futurum &longs;it vt AB, ad AD. Nimirùm id <lb/>con&longs;equitur ex eo, quòd &longs;patia &longs;int inter &longs;e <lb/>in duplicata temporum ratione; &longs;eu vt quadra<lb/>ta temporum; quódque &longs;it per&longs;picuum ratio<lb/>nem &longs;patij AC ad &longs;patium AB e&longs;&longs;e duplam <lb/>rationis AC, ad AD, &longs;eu eandem, quam ha<lb/>bent quadrata AC, & AD. Ex quo fiet, vt cùm AB <lb/>&longs;upponas e&longs;&longs;e &longs;ex minutorum, AC compobetur mi<lb/>nutorum octo, & 29. &longs;ecundorum proximè; ac proin<lb/>de tempus per BC &longs;it minutorum duorum, & viginti <lb/>nouem proximè &longs;ecundorum. Eadem autem ratione <lb/>diui&longs;a bifariam prima parte in E, & accepta AF media

<figure id="fig21"></figure><lb/>&longs;ecunda inuenias mediam proportionalem AD, <lb/>tempus ca&longs;us per AB, ad tempus ca&longs;us per AC, <lb/>futurum &longs;it vt AB, ad AD. Nimirùm id <lb/>con&longs;equitur ex eo, quòd &longs;patia &longs;int inter &longs;e <lb/>in duplicata temporum ratione; &longs;eu vt quadra<lb/>ta temporum; quódque &longs;it per&longs;picuum ratio<lb/>nem &longs;patij AC ad &longs;patium AB e&longs;&longs;e duplam <lb/>rationis AC, ad AD, &longs;eu eandem, quam ha<lb/>bent quadrata AC, & AD. Ex quo fiet, vt cùm AB <lb/>&longs;upponas e&longs;&longs;e &longs;ex minutorum, AC compobetur mi<lb/>nutorum octo, & 29. &longs;ecundorum proximè; ac proin<lb/>de tempus per BC &longs;it minutorum duorum, & viginti <lb/>nouem proximè &longs;ecundorum. Eadem autem ratione <lb/>diui&longs;a bifariam prima parte in E, & accepta AF media

<pb pagenum="85"/>proportionali inter AB, & AE, reperietur tempus <lb/>per primum dimidium AE minutorum 4, & &longs;e cun<lb/>dorum 14 <gap/> ac relinquetur proinde tempus per po&longs;te<lb/>rius dimidium FB minuti 1, at &longs;ecundorum 45 Ad <lb/>Tertium nihil e&longs;t, quod dicam, quandò nihil determi<lb/>nas, &longs;ed prouocas <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ad experimentum, quod fieri ab <lb/>alio exoptes. Ac videbatur quidem id <expan abbr="experiment&utilde;">experimentum</expan> abs <lb/>te præ&longs;ertim ex&longs;pectandum, cùm profitereris <emph type="italics"/>nouam <lb/>&longs;cientiam,<emph.end type="italics"/> &longs;eu <emph type="italics"/>demon&longs;trationem, qua ratio, men&longs;ura, modus, <lb/>ac potentia accelerationis motus in naturali de&longs;cen&longs;u grauium <lb/>determinaretur:<emph.end type="italics"/> idque aduer&longs;us eam, quam <emph type="italics"/>excogitatam à <lb/>Galileo p&longs;eudo-&longs;cientiam<emph.end type="italics"/> appellitares; &longs;ed nolo tamen <lb/>hac in re e&longs;&longs;e importunus; addoque &longs;olùm, vbi id ex<lb/>perimentum peractum fuerit, &longs;ucce&longs;&longs;eritque, certum <lb/>me propemodum e&longs;&longs;e, ex ijs, quæ hactenus peregi ip&longs;i <lb/>valdè affinibus, elicitum exinde iri, quod opinionem <lb/>fulciar, non tuam, &longs;ed ex Galileo hactenus expre&longs;&longs;am. </s>

<s>XLVII. Iam ad finem properans, <emph type="italics"/>omittere<emph.end type="italics"/> te dicis, <lb/><emph type="italics"/>quæ etiam de Motu accelerato examinari po&longs;&longs;ent, vt, qu&ecedil;, qua<lb/>li&longs;que &longs;it cau&longs;&longs;a accelerationis in naturali de&longs;cen&longs;u grauium: <lb/>cur corpora, &longs;altem, quæ eiu&longs;dem figuræ, & homogenea &longs;int, <lb/>cuiu&longs;cumque, & quantumlibet in&ecedil;qualis ponderis illa fuerint, <lb/>deorsùm nihilominùs &ecedil;quali celeritate de&longs;cendant: aliàque <lb/>eiu&longs;mo li, qu&ecedil; tibi quidem in promptu &longs;int, & alio loco, ac <lb/>tempore opportuniùs forta&longs;&longs;e proferenda in publicum: &longs;ed in<lb/>terim h&ecedil;c pr&ecedil;libanda puta&longs;&longs;e, qu&ecedil; non mo ò ad reuincendos<emph.end type="italics"/><lb/>G<emph type="italics"/>alilei errores opportuna, &longs;ed ad veram quoque, ac germa<lb/>nam accelerati motus naturam aperiendam nece&longs;&longs;aria vide<lb/>rentur.<emph.end type="italics"/> Quo rursùs loco, nihil e&longs;t, quod addam, neque <lb/>cur importunè rogem, quamobrem non cen&longs;ueris

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<s>XLVIII. Po&longs;tremò anacephalæo&longs;i concludens; <lb/><emph type="italics"/>Ex his enim,<emph.end type="italics"/> inquis, <emph type="italics"/>ni&longs;i vehementer fallor, manifestè iam <lb/>vides, & euidenter agno&longs;cis, quàm non rectè<emph.end type="italics"/> G<emph type="italics"/>alileus mo<lb/>tum æquabiliter acceleratum eum e&longs;&longs;e definierit, qui æqualibus<emph.end type="italics"/>

<pb pagenum="87"/><emph type="italics"/>temporibus, æqualia celeritatis augmenta acquirat: cùm Sole <lb/>clarius iam tibi &longs;it, eumdem motum, æqualibus temporibus, non <lb/>&ecedil;qualia, &longs;ed maiora &longs;emper, ac maiora recipere celeritatis aug<lb/>menta, in continua ratione dupla. Vides item, & pari eui<lb/>dentiâ per&longs;picis, non minus erra&longs;&longs;e<emph.end type="italics"/> G<emph type="italics"/>alileum, cùm &longs;patia <lb/>&ecedil;qualibus temporibus emen&longs;a, eam inter &longs;e rationem ob<lb/>&longs;eruare voluit, quæ inter numeros omnis imparis ab vnitate <lb/>procedenteis reperitur; cùm eadem quoque spatia, clarè ac <lb/>manife<gap/> è ignoueris, &ecedil;qualibus temporibus, maiora &longs;emper <lb/>ac maiora percurri, in eadem continua ratione dupla. Vides <lb/>porrò, ac penè palpas, quàm vana, atque inanis &longs;it noua illa, <lb/>& tantopere ab ip&longs;o<gap/>et auctore laudata, de Motu accelerato <lb/>p&longs;eudo &longs;cientia: cùm non ni&longs;i fal&longs;is, atque erroneis principiis <lb/>innitatur; & quam non immeritò ante annos duos mihi di&longs;<lb/>plicuerit, quòd tu quoque ii&longs;dem illis principiis nounullam fi<lb/>dem adhiberes.<emph.end type="italics"/> C<emph type="italics"/>ertiora nunc habes, & quibus intrepidé <lb/>a&longs;&longs;en&longs;um pr&ecedil;beas, re&longs;titutam &longs;cilicet motus accelerati defini<lb/>tionem & ab iniu&longs;ta Galilei oppugnatione vindicatam. Ha<lb/>bes & veram, germanàmque in naturali de&longs;cen&longs;u grauium <lb/>accelerationis rationem, tam in temporibus, quàm in &longs;patijs <lb/>&ecedil;qualibus con&longs;ideratam. Habes denique eam quoque ratio<lb/>nem, qu&ecedil; inter spatia &ecedil;qualibus temporibus emen&longs;a reperi<lb/>tur, indubitatis experientiis, certis, euidentibú&longs;que rationibus <lb/>demon&longs;tratam. Qu&ecedil; &longs;i, vt &longs;pero, tibi accepta, probataque fue<lb/>rint, non exiguum huius oper&ecedil; pretium me con&longs;ecutum e&longs;&longs;e <lb/>arbitrabor.<emph.end type="italics"/> Ad hæc verò omnia, Optime virorum, <lb/>nihil regerere in animo e&longs;t: cùm illa &longs;atis, &longs;uperque <lb/>&longs;int, quæ circa &longs;ingula edi&longs;&longs;erui. E&longs;t &longs;olùm, quòd <lb/>gratias agam vberes, pro in&longs;igni illo affectu, quem <lb/>ante duos annos in me te&longs;tari dignatus es, quemque

<pb pagenum="88"/>expre&longs;&longs;i&longs;ti nunc etiam, pretium collocans operæ, <lb/>quam meam &longs;pera&longs;ti comprobationem. Quòd &longs;i <lb/>videaris &longs;pe excidi&longs;&longs;e, dum reprobantem potiùs, quàm <lb/>approbantem habes me: at non excidi&longs;ti profectò, <lb/>cùm & &longs;pera&longs;tia ffarite tui reuerent <lb/>amanti&longs;&longs;imum virum; & me eum volui&longs;ti, vt interpre<lb/>tor, qui aliunde ius amicitiæ &longs;eruans illibati&longs;&longs;imum, <lb/>tibi, in veritatis gratiam, non erube&longs;cerem repugnare. <lb/>Et quàm, putas, &longs;æpe expetij po&longs;&longs;e tibi &longs;ub&longs;cribere, vt <lb/>foret non affectum magis, quàm opinionum con&longs;pi<lb/>ratio; verùm ip&longs;emet iudex eris, vbi meas nugas per<lb/>volveris, an-non &longs;altem di&longs;&longs;en&longs;erim cum aliqua &longs;pecie <lb/>probabilitatis. Sic certe habe, fore me &longs;emper com<lb/>parati&longs;&longs;imum a&longs;&longs;entiendo, &longs;i quandò maior mihi ex <lb/>te &longs;imilitudo veri affulgeat, qui &longs;um interim, &longs;i quis <lb/>alius, comparati&longs;&longs;imus ob&longs;equendo. Vale, Pari&longs;ijs, <lb/>Eidib. Mart. M. DC. XLV. <lb/>

<arrow.to.target n="fig22"></arrow.to.target></s>

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<s><emph type="italics"/>Vt autem probes,<emph.end type="italics"/><lb/>

<arrow.to.target n="fig23"></arrow.to.target><lb/><emph type="italics"/>quàm rectè iuxta Galilei <lb/>mentem, acceleratio illa <lb/>motus habeatur, iu<lb/>bes concipere duas li<lb/>neas<emph.end type="italics"/> AB, <emph type="italics"/>&<emph.end type="italics"/> AC, <lb/><emph type="italics"/>angulum con&longs;tituentes in<emph.end type="italics"/><lb/>A, <emph type="italics"/>& tertiam<emph.end type="italics"/> VX <emph type="italics"/>per <lb/>anguli apicem<emph.end type="italics"/> A <emph type="italics"/><expan abbr="ince-dent&etilde;">ince<lb/>dentem</expan>, & cùm prioribus <lb/>duabus angulos vtrim<lb/>que æqualeis con&longs;tituentem: hanc, &longs;eruatâ &longs;emper eadem<emph.end type="italics"/>

<figure id="fig23"></figure><lb/><emph type="italics"/>quàm rectè iuxta Galilei <lb/>mentem, acceleratio illa <lb/>motus habeatur, iu<lb/>bes concipere duas li<lb/>neas<emph.end type="italics"/> AB, <emph type="italics"/>&<emph.end type="italics"/> AC, <lb/><emph type="italics"/>angulum con&longs;tituentes in<emph.end type="italics"/><lb/>A, <emph type="italics"/>& tertiam<emph.end type="italics"/> VX <emph type="italics"/>per <lb/>anguli apicem<emph.end type="italics"/> A <emph type="italics"/><expan abbr="ince-dent&etilde;">ince<lb/>dentem</expan>, & cùm prioribus <lb/>duabus angulos vtrim<lb/>que æqualeis con&longs;tituentem: hanc, &longs;eruatâ &longs;emper eadem<emph.end type="italics"/>

<pb pagenum="99"/><emph type="italics"/>angulorum æqualitate, ita fluentem, ac de&longs;cenden<lb/>tem concipi postulas, vt etiam intelligamus partem inter <lb/>lineas<emph.end type="italics"/> AB, AC, <emph type="italics"/>interceptam continuò ea ratione augeri, <lb/>vt notatis in<emph.end type="italics"/> AB, <emph type="italics"/>&<emph.end type="italics"/> AC, <emph type="italics"/>partibus æqualibus<emph.end type="italics"/> AE, EG, <lb/>GI, IL, <emph type="italics"/>&longs;emper interceptarum parallelarum incrementa <lb/>haberi æqualia aduertamus. Nempe vt<emph.end type="italics"/> AG <emph type="italics"/>dupla est <lb/>ip&longs;ius<emph.end type="italics"/> AE, <emph type="italics"/>&longs;ic<emph.end type="italics"/> GF <emph type="italics"/>dupla est ip&longs;ius<emph.end type="italics"/> ED: <emph type="italics"/>& eadem ratione<emph.end type="italics"/><lb/>IH <emph type="italics"/>eiu&longs;dem<emph.end type="italics"/> ED <emph type="italics"/>e&longs;t tripla, &<emph.end type="italics"/> LK <emph type="italics"/>quadrupla, atque <lb/>ita deinceps. Ex quibus ita concludis:<emph.end type="italics"/> Quare a&longs;&longs;umptis <lb/>partibus æqualibus temporis per parteis æqualeis lineæ <lb/>AC repræ&longs;entatis, notum e&longs;t momenta, &longs;eu incremen<lb/>ta velocitatis per parallelas repræ&longs;entatæ æqualia ac<lb/>quiri &longs;ub huiu&longs;modi partibus. <emph type="italics"/>Hæc &longs;anè vera &longs;unt; <lb/>&longs;ed recordare verißimè quoque à te dictum numero<emph.end type="italics"/> IV. <emph type="italics"/>pun<lb/>ctum<emph.end type="italics"/> A <emph type="italics"/>po&longs;&longs;e non tantum pro initio temporis haberi, &longs;ed etiam <lb/>pro initio spatij, & (vt item addis) pro initio velocitatis.<emph.end type="italics"/></s>

<s>Recordor; &longs;ed adnoto &longs;imul habui&longs;&longs;e me punctum <lb/>A, pro initio temporis quidem æquabiliter &longs;luentis, <lb/>prout comparatur ad lineam AC (aut AB) in parteis <lb/>æqualeis diui&longs;am; pro initio verò &longs;patij in longum de<lb/>currendi, prout comparatur ad aream AKL in trian<lb/>gulos æqualeis di&longs;tinctam; ac pro initio velocitatis <lb/>continenter acquirendæ, prout comparatur cum linea <lb/>parteis æqualeis continuò ad&longs;ci&longs;cente, quov&longs;que c&oelig;<lb/>pta à puncto A, euadat KL. </s>

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<s><emph type="italics"/>Sed ex his porrò etiam vides, quàm non rectè vulgatam <lb/>accelerati motus definitionem inde numero<emph.end type="italics"/> VIII. <emph type="italics"/>reprehen<lb/>das, quòd ea ratione concepta &longs;it, qua vniformis acceleratio <lb/>non exprimatur. Modò enim vtdisti puncto<emph.end type="italics"/> A <emph type="italics"/>pro initio <lb/>&longs;patij con&longs;tituto, cuius partes æquales æqualibus &longs;egmentis<emph.end type="italics"/><lb/>EG, GI, IL <emph type="italics"/>de&longs;ignentur, non tantum recti vniformem <lb/>quampiam, &longs;ed eandem planè velocitatis accelerationem ha<lb/>beri: vt iam ampliùs inquirere tibi non liceat, quomodo ex <lb/>vulgata motus accelerati definitione, qua<emph.end type="italics"/> is dicitur, qui <lb/>æqualibus &longs;patijs æqualia velocitatis augmenta acqui<lb/>rat, <emph type="italics"/>motum concipere liceat æquabiliter acceleratum. Iam <lb/>enim habes concipi pror&longs;us eodem modo, quo tu illum con-<emph.end type="italics"/><lb/>

<arrow.to.target n="fig24"></arrow.to.target><lb/><emph type="italics"/>cipis, atque exprimis ex data<emph.end type="italics"/> G<emph type="italics"/>a<lb/>lilei definitione. Nam quòd eodem <lb/>numero<emph.end type="italics"/> VIII. <emph type="italics"/>de&longs;cripto nouo trian<lb/>gulo<emph.end type="italics"/> APB <emph type="italics"/>accelerationis augmen<lb/>tum minoribus triangulis inter pa<lb/>rallelas<emph.end type="italics"/> CL, DM, EO, <emph type="italics"/>&c. <lb/>con&longs;titutis metiendum putas, non <lb/>rectè id facis. Vt enim velocitas <lb/>acqui&longs;ita per &longs;patium<emph.end type="italics"/> AC <emph type="italics"/>de&longs;igna<lb/>tur per lineam<emph.end type="italics"/> CL, <emph type="italics"/>velocitas ac<lb/>qui&longs;ita in<emph.end type="italics"/> D, <emph type="italics"/>exprimitur linea<emph.end type="italics"/><lb/>DN: <emph type="italics"/>& velocitas acqui&longs;ita in<emph.end type="italics"/> E, <lb/><emph type="italics"/>repræ&longs;entatur linea<emph.end type="italics"/> EQ, <emph type="italics"/>& ita <lb/>de cæteris. Cur enim celeritatis <lb/>augmenta hoc loco triangulis, in &longs;uperiore autem figura<emph.end type="italics"/>

<figure id="fig24"></figure><lb/><emph type="italics"/>cipis, atque exprimis ex data<emph.end type="italics"/> G<emph type="italics"/>a<lb/>lilei definitione. Nam quòd eodem <lb/>numero<emph.end type="italics"/> VIII. <emph type="italics"/>de&longs;cripto nouo trian<lb/>gulo<emph.end type="italics"/> APB <emph type="italics"/>accelerationis augmen<lb/>tum minoribus triangulis inter pa<lb/>rallelas<emph.end type="italics"/> CL, DM, EO, <emph type="italics"/>&c. <lb/>con&longs;titutis metiendum putas, non <lb/>rectè id facis. Vt enim velocitas <lb/>acqui&longs;ita per &longs;patium<emph.end type="italics"/> AC <emph type="italics"/>de&longs;igna<lb/>tur per lineam<emph.end type="italics"/> CL, <emph type="italics"/>velocitas ac<lb/>qui&longs;ita in<emph.end type="italics"/> D, <emph type="italics"/>exprimitur linea<emph.end type="italics"/><lb/>DN: <emph type="italics"/>& velocitas acqui&longs;ita in<emph.end type="italics"/> E, <lb/><emph type="italics"/>repræ&longs;entatur linea<emph.end type="italics"/> EQ, <emph type="italics"/>& ita <lb/>de cæteris. Cur enim celeritatis <lb/>augmenta hoc loco triangulis, in &longs;uperiore autem figura<emph.end type="italics"/>

<pb pagenum="109"/><emph type="italics"/>lineis metienda edicis? Constat autem lineas<emph.end type="italics"/> CL, ND, <lb/>EQ, <emph type="italics"/>&c. vniformi augmento accre&longs;cere, & e&longs;&longs;e, vt<emph.end type="italics"/> AC <emph type="italics"/>ad<emph.end type="italics"/><lb/>CL, <emph type="italics"/>ita<emph.end type="italics"/> AD <emph type="italics"/>ad<emph.end type="italics"/> DN, <emph type="italics"/>&<emph.end type="italics"/> AE <emph type="italics"/>ad<emph.end type="italics"/> EQ, <emph type="italics"/>&c. Non <lb/>rectè igitur cen&longs;es augmentum velocitatis vniforme e&longs;&longs;e non <lb/>po&longs;&longs;e, &longs;i spatijs æqualibus cre&longs;cat æqualiter, & tota illa noui <lb/>istius trianguli difformis &longs;tructura sponte corruit.<emph.end type="italics"/></s>

<s>Heic idem dicendum, quod & paulò antè, quate<lb/>nus parteis lineæ AB contendis habendas pro parti<lb/>bus &longs;patij, quæ comparentur cum parallelis habitis <lb/>pro gradibus velocitatis, nulla habita temporis ratio<lb/>ne. Quod autem quæris, <emph type="italics"/>Cur hoc loco celeritatis aug<lb/>menta triangulis in &longs;uperiore autem figura lineis metienda <lb/>edixerim?<emph.end type="italics"/> Cau&longs;&longs;am ex eo potes intelligere, quòd cùm <lb/>tu duos gradus NM, & MD, v. c. æqualeis facias, <lb/>qua&longs;i acqui&longs;itos ex L, & C, &longs;ecundum duos triangulos <lb/>LNM, & CMD (tamet&longs;i inæquales &longs;int, & MD ex <lb/>LC manente factus, duobus æquiualeat) ideò quem <lb/>defectum exprimere non licuit lineâ ND, exprimere <lb/>placuerit trapezio LD. Nimirùm, cùm velocitas <lb/>ND creetur partim ex LC, promota in MD &longs;ecun<lb/>dum quadrangulum, partim ex additamentis ip&longs;i fa<lb/>ctis &longs;ecundum triangulum LNM; tu ex velocitate hac <lb/>detrahis integrum triangulum LMC: &longs;icque ex tri<lb/>bus &longs;uper&longs;unt tantum partes velocitatis duæ, quas vt <lb/>&longs;imul iunct is repræ&longs;entarem, triangulum à te factum <lb/><expan abbr="vacu&utilde;">vacuum</expan> &longs;upplcui, tran&longs;lato LNM in MCL: cópo&longs;itoque <lb/>inde quadrangulo, locum ip&longs;um trianguli tran&longs;lati re<lb/>liqui inanem. Fandem autem ob cau&longs;&longs;am relicti &longs;unt <lb/>duo trianguli manes ad ordinem tertium tres ad quar<lb/>tum, &c. Exindeque e&longs;t, cur <emph type="italics"/>difformis<emph.end type="italics"/> quidem, &longs;ed

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<s><emph type="italics"/>Po&longs;tquàm autem<emph.end type="italics"/> G<emph type="italics"/>alilei definitionem confirmare, & <lb/>communem aliorum de&longs;truere conatus es, aggrederis numero <lb/>x. eam rationem, qua idem Galileus communiorem motus <lb/>vniformiter accelerati definitionem ab&longs;urditatis arguit, à <lb/>paralogi&longs;mo excu&longs;are: &longs;ed fru&longs;tra huius cau&longs;&longs;æ pairocinium <lb/>&longs;u&longs;cipis, cùm nec eam obtinere, nec &longs;atis plau&longs;ibiliter eam de<lb/>fendere poßis. Hæc porrò e&longs;t<emph.end type="italics"/> G<emph type="italics"/>alilei ratio.<emph.end type="italics"/> Si accelera<lb/>tio motus in de&longs;cen&longs;u grauium æqualibus &longs;patiis, <lb/>æqualia &longs;umeret velocitatis incrementa, e&longs;&longs;ent velo-

<pb pagenum="111"/>citates inter &longs;e, vt emen&longs;a &longs;patia: at quoties velocitates <lb/>inter &longs;e &longs;unt, vt emen&longs;a &longs;patia, debent nece&longs;&longs;ariò ea <lb/>&longs;patia aut eodem, aut æquali tempore percurri. Si igi<lb/>tur velocitas acqui&longs;ita per totam AC, eam rationem <lb/>habeat ad velocitatem acqui&longs;itam per AB, <lb/>

<arrow.to.target n="fig25"></arrow.to.target><lb/>quam &longs;patium AC, ad &longs;patium AB, nece&longs;&longs;e e&longs;t, <lb/>vt totum &longs;patium AC eodem, aut æquali tem<lb/>pore decurratur, quo &longs;patium AB ab&longs;oluitur. <lb/>Impo&longs;&longs;ibile e&longs;t autem, vt corpus de&longs;cendens per <lb/>AC, eodem, aut æquali tempore percurrat to<lb/>tam AC, quo percurrit partem eius AB, ni&longs;i mo<lb/>tus fiat in in&longs;tanti. Tam impo&longs;&longs;ibile e&longs;t igitur, <lb/>vt velocitates in de&longs;cen&longs;u grauium inter &longs;e &longs;int, vt <lb/>emen&longs;a &longs;patia, quàm impo&longs;&longs;ibile e&longs;t motum illum fie<lb/>ri in in&longs;tanti. <emph type="italics"/>Hanc ego rationem Paralogi&longs;mum dico, tu <lb/>contendis e&longs;&longs;e veram Demon&longs;trationem. Vitium ego tan<lb/>quam intelligenti breuiùs fortè indicaui: at præoccupato cer<lb/>tè aliunde animo, non &longs;ufficienter illud detexi. Exactiùs igi<lb/>tur (vt po&longs;tulas) &longs;ingulas huius Ratiocinationis propo&longs;itiones <lb/>hoc loco perpendemus. Prima h&ecedil;c e&longs;t,<emph.end type="italics"/> Si acceleratio motus <lb/>in de&longs;cen&longs;u grauium æqualibus &longs;patiis æqualia &longs;umeret <lb/>velocitatis incrementa, e&longs;&longs;ent velocitates inter &longs;e, vt <lb/>emen&longs;a &longs;patia. <emph type="italics"/>Nunc age, quis huius propo&longs;itionis &longs;en&longs;us <lb/>e&longs;&longs;e videtur? Duplicem enim patitur, & quidem valdè di<lb/>uer&longs;um, quorum alter verus, alter fal&longs;us &longs;it; & ni&longs;i po&longs;teriore <lb/>hoc &longs;en&longs;u illam po&longs;t<emph.end type="italics"/> G<emph type="italics"/>alileum v&longs;urpes, concludis omninò ni<lb/>hil. Prior &longs;en&longs;us i&longs;te e&longs;t, Si acceleratio motus in de&longs;cen&longs;u <lb/>grauium æqualibus spatiis æqualia &longs;umat velocitatis augmen<lb/>ta; nece&longs;&longs;e e&longs;t, vt hæc eadem augmenta quibu&longs;libet spatij <lb/>partibus acqui&longs;ita eandem inter &longs;erationem ob&longs;eruent, quàm<emph.end type="italics"/>

<figure id="fig25"></figure><lb/>quam &longs;patium AC, ad &longs;patium AB, nece&longs;&longs;e e&longs;t, <lb/>vt totum &longs;patium AC eodem, aut æquali tem<lb/>pore decurratur, quo &longs;patium AB ab&longs;oluitur. <lb/>Impo&longs;&longs;ibile e&longs;t autem, vt corpus de&longs;cendens per <lb/>AC, eodem, aut æquali tempore percurrat to<lb/>tam AC, quo percurrit partem eius AB, ni&longs;i mo<lb/>tus fiat in in&longs;tanti. Tam impo&longs;&longs;ibile e&longs;t igitur, <lb/>vt velocitates in de&longs;cen&longs;u grauium inter &longs;e &longs;int, vt <lb/>emen&longs;a &longs;patia, quàm impo&longs;&longs;ibile e&longs;t motum illum fie<lb/>ri in in&longs;tanti. <emph type="italics"/>Hanc ego rationem Paralogi&longs;mum dico, tu <lb/>contendis e&longs;&longs;e veram Demon&longs;trationem. Vitium ego tan<lb/>quam intelligenti breuiùs fortè indicaui: at præoccupato cer<lb/>tè aliunde animo, non &longs;ufficienter illud detexi. Exactiùs igi<lb/>tur (vt po&longs;tulas) &longs;ingulas huius Ratiocinationis propo&longs;itiones <lb/>hoc loco perpendemus. Prima h&ecedil;c e&longs;t,<emph.end type="italics"/> Si acceleratio motus <lb/>in de&longs;cen&longs;u grauium æqualibus &longs;patiis æqualia &longs;umeret <lb/>velocitatis incrementa, e&longs;&longs;ent velocitates inter &longs;e, vt <lb/>emen&longs;a &longs;patia. <emph type="italics"/>Nunc age, quis huius propo&longs;itionis &longs;en&longs;us <lb/>e&longs;&longs;e videtur? Duplicem enim patitur, & quidem valdè di<lb/>uer&longs;um, quorum alter verus, alter fal&longs;us &longs;it; & ni&longs;i po&longs;teriore <lb/>hoc &longs;en&longs;u illam po&longs;t<emph.end type="italics"/> G<emph type="italics"/>alileum v&longs;urpes, concludis omninò ni<lb/>hil. Prior &longs;en&longs;us i&longs;te e&longs;t, Si acceleratio motus in de&longs;cen&longs;u <lb/>grauium æqualibus spatiis æqualia &longs;umat velocitatis augmen<lb/>ta; nece&longs;&longs;e e&longs;t, vt hæc eadem augmenta quibu&longs;libet spatij <lb/>partibus acqui&longs;ita eandem inter &longs;erationem ob&longs;eruent, quàm<emph.end type="italics"/>

<pb pagenum="112"/><emph type="italics"/>emen&longs;a &longs;patia, & hic &longs;en&longs;us verus ac nece&longs;&longs;arius e&longs;t. Si <lb/>enim intriangulo æqualia spatia de&longs;ignentur<emph.end type="italics"/> AD, DE, EF, <lb/><emph type="italics"/>&c. & in<emph.end type="italics"/> D <emph type="italics"/>acqui&longs;itus &longs;upponatur vnus<emph.end type="italics"/><lb/>

<arrow.to.target n="fig26"></arrow.to.target><lb/><emph type="italics"/>gradus, & in<emph.end type="italics"/> E <emph type="italics"/>duo, & tres in<emph.end type="italics"/> F, <emph type="italics"/>manife&longs;tum <lb/>e&longs;t duos gradus, ad quos acceleratio perueni&longs;&longs;e <lb/>ponitur in<emph.end type="italics"/> E, <emph type="italics"/>e&longs;&longs;e ad vnum gradum acqui&longs;i<lb/>tum in<emph.end type="italics"/> D, <emph type="italics"/>vt &longs;patium<emph.end type="italics"/> AE <emph type="italics"/>ad spatium<emph.end type="italics"/> AD; <lb/><emph type="italics"/>& &longs;imiliter gradus treis, qui in<emph.end type="italics"/> F <emph type="italics"/>&longs;upponun<lb/>tur terminare celeritatis augmentum, e&longs;&longs;e ad <lb/>gradum vnum ip&longs;ius<emph.end type="italics"/> D, <emph type="italics"/>vel ad duos ip&longs;ius<emph.end type="italics"/> E, <lb/><emph type="italics"/>vt<emph.end type="italics"/> AF, <emph type="italics"/>ad<emph.end type="italics"/> AD, <emph type="italics"/>vel<emph.end type="italics"/> AE. <emph type="italics"/>Et hoc quidem <lb/>&longs;en&longs;u, &longs;i primam illam<emph.end type="italics"/> G<emph type="italics"/>alilei v&longs;urpares, vera omninò e&longs;&longs;et, <lb/>ac nece&longs;&longs;aria, &longs;ed A&longs;&longs;umptio, quæ &longs;ub&longs;umitur, fal&longs;a e&longs;&longs;et at<lb/>que impoßibilis; nempe hæc.<emph.end type="italics"/> At quoties velocitates inter <lb/>&longs;e &longs;unt vt emen&longs;a &longs;patia (<emph type="italics"/>in &longs;en&longs;u proximè aßignato<emph.end type="italics"/>) de<lb/>bent nece&longs;&longs;ariò ea &longs;patia aut eodem, aut æquali tem<lb/>pore percurri: <emph type="italics"/>Sicque iam hac ex parte<emph.end type="italics"/> G<emph type="italics"/>alilei licet ratio<lb/>cinatio corruit. Sed alius item e&longs;&longs;e pote&longs;t primæ illius Propo<lb/>&longs;itionis &longs;en&longs;us, vt &longs;eilicet quoties acceleratio velocitatis in de&longs;<lb/>cen&longs;u grauium æqualibus &longs;patiis æqualia incrementa acquirit, <lb/>integræ velocitates &longs;ecundum &longs;e totas, & qua&longs;libet &longs;ui parteis <lb/>analogas aeceptæ, & con&longs;iderata, & non tantum acqui&longs;itæ <lb/>partibus &longs;patii æqualibus incrementa, eam ab initio ad finem <lb/>inter &longs;e rationem ob&longs;eruent, quam emen&longs;a &longs;patia. Qui &longs;en<lb/>&longs;us à priore longè diuer&longs;us est, & à te non intenditur modò; <lb/>&longs;ed di&longs;tinctè quoque eodem numero x. exprimitur. Ais enim.<emph.end type="italics"/><lb/>Rem certè in hunc modum concipio. Intelligatur <lb/>AC, diui&longs;a in duodecim parteis æqualeis, ac proinde <lb/>eius dimidium AB, &longs;eu ip&longs;i æqualeis DE in <gap/>x; &longs;int<lb/>que primùm duo mobilia, quorum vnum de&longs;cendat

<figure id="fig26"></figure><lb/><emph type="italics"/>gradus, & in<emph.end type="italics"/> E <emph type="italics"/>duo, & tres in<emph.end type="italics"/> F, <emph type="italics"/>manife&longs;tum <lb/>e&longs;t duos gradus, ad quos acceleratio perueni&longs;&longs;e <lb/>ponitur in<emph.end type="italics"/> E, <emph type="italics"/>e&longs;&longs;e ad vnum gradum acqui&longs;i<lb/>tum in<emph.end type="italics"/> D, <emph type="italics"/>vt &longs;patium<emph.end type="italics"/> AE <emph type="italics"/>ad spatium<emph.end type="italics"/> AD; <lb/><emph type="italics"/>& &longs;imiliter gradus treis, qui in<emph.end type="italics"/> F <emph type="italics"/>&longs;upponun<lb/>tur terminare celeritatis augmentum, e&longs;&longs;e ad <lb/>gradum vnum ip&longs;ius<emph.end type="italics"/> D, <emph type="italics"/>vel ad duos ip&longs;ius<emph.end type="italics"/> E, <lb/><emph type="italics"/>vt<emph.end type="italics"/> AF, <emph type="italics"/>ad<emph.end type="italics"/> AD, <emph type="italics"/>vel<emph.end type="italics"/> AE. <emph type="italics"/>Et hoc quidem <lb/>&longs;en&longs;u, &longs;i primam illam<emph.end type="italics"/> G<emph type="italics"/>alilei v&longs;urpares, vera omninò e&longs;&longs;et, <lb/>ac nece&longs;&longs;aria, &longs;ed A&longs;&longs;umptio, quæ &longs;ub&longs;umitur, fal&longs;a e&longs;&longs;et at<lb/>que impoßibilis; nempe hæc.<emph.end type="italics"/> At quoties velocitates inter <lb/>&longs;e &longs;unt vt emen&longs;a &longs;patia (<emph type="italics"/>in &longs;en&longs;u proximè aßignato<emph.end type="italics"/>) de<lb/>bent nece&longs;&longs;ariò ea &longs;patia aut eodem, aut æquali tem<lb/>pore percurri: <emph type="italics"/>Sicque iam hac ex parte<emph.end type="italics"/> G<emph type="italics"/>alilei licet ratio<lb/>cinatio corruit. Sed alius item e&longs;&longs;e pote&longs;t primæ illius Propo<lb/>&longs;itionis &longs;en&longs;us, vt &longs;eilicet quoties acceleratio velocitatis in de&longs;<lb/>cen&longs;u grauium æqualibus &longs;patiis æqualia incrementa acquirit, <lb/>integræ velocitates &longs;ecundum &longs;e totas, & qua&longs;libet &longs;ui parteis <lb/>analogas aeceptæ, & con&longs;iderata, & non tantum acqui&longs;itæ <lb/>partibus &longs;patii æqualibus incrementa, eam ab initio ad finem <lb/>inter &longs;e rationem ob&longs;eruent, quam emen&longs;a &longs;patia. Qui &longs;en<lb/>&longs;us à priore longè diuer&longs;us est, & à te non intenditur modò; <lb/>&longs;ed di&longs;tinctè quoque eodem numero x. exprimitur. Ais enim.<emph.end type="italics"/><lb/>Rem certè in hunc modum concipio. Intelligatur <lb/>AC, diui&longs;a in duodecim parteis æqualeis, ac proinde <lb/>eius dimidium AB, &longs;eu ip&longs;i æqualeis DE in <gap/>x; &longs;int<lb/>que primùm duo mobilia, quorum vnum de&longs;cendat

<pb pagenum="113"/>ex A, ver&longs;us C, eodem momento, quo aliud ex D, <lb/>ver&longs;us E; notum e&longs;t, &longs;i vtrumque quidem fer<lb/>

<arrow.to.target n="fig27"></arrow.to.target><lb/>retur non accelerato, &longs;ed æquabili motu, <lb/>euenturum e&longs;&longs;e, vt velocitate illius ex&longs;i&longs;tente <lb/>dupla ad velocitatem i&longs;tius, illud perueniret <lb/>in C, eodem momento, quo i&longs;tud in E; <lb/>quoniam &longs;patium ab illo &longs;uperatum foret <lb/>vbique ad &longs;patium ab i&longs;to &longs;uperatum, du<lb/>plum; hoc e&longs;t, forent ab illo &longs;uperatæ duæ <lb/>partes, cùm ab i&longs;to vna: ab illo quatuor, cùm ab hoc <lb/>duæ, &c. quatenus &longs;patia &longs;e haberent vbique, vt velo<lb/>citates; hoc e&longs;t, velocitas per totam AC, e&longs;&longs;et vbique <lb/>dupla velocitatis per totam DE. At verò, quoniam <lb/>heic agitur de motu non æquabili, &longs;ed continenter <lb/>accelerato; ita di&longs;cedant rursùs mobilia eodem tem<lb/>pore, vnum ab A, aliud à D, vt &longs;uccre&longs;centibus conti<lb/>nuò velocitatis gradibus, illud perueniendo in C, ac<lb/>qui&longs;ierit duodecim, hoc perueniendo in E, &longs;ex: <emph type="italics"/>Tum <lb/>interrogas,<emph.end type="italics"/> Quid impediat, quò minùs illud perueniat <lb/>in C, eodem tempore, quo i&longs;tud in E? <emph type="italics"/>Ego verò re&longs;<lb/>pondeo, nihil certè impedire, &longs;i tales e&longs;&longs;ent, quales de&longs;cribis <lb/>velocitate: taleis autem &longs;ine dubio de&longs;cribis, qualeis in prima <lb/>illa<emph.end type="italics"/> G<emph type="italics"/>alilei propo&longs;itione &longs;ignificari putas. At hoc &longs;en&longs;u hypo<lb/>thetica illa Galilei propo&longs;itio fal&longs;a e&longs;t, euidenterque impoßi<lb/>bilis: cùm nulla prorsùs ratione con&longs;equens inferi poßit ex <lb/>antecedente. Hoc enim e&longs;t Antecedens,<emph.end type="italics"/> Acceleratio ve<lb/>locitatis in de&longs;cen&longs;u grauium per totam AC, ita con<lb/>tinua &longs;ucce&longs;&longs;ione cre&longs;cit, vt primùm in B acqui&longs;itus <lb/>&longs;upponatur vnus aliquis velocitatis gradus; & vlteriùs <lb/>procedente motu, & continuò incre&longs;cente celeritate,

<figure id="fig27"></figure><lb/>retur non accelerato, &longs;ed æquabili motu, <lb/>euenturum e&longs;&longs;e, vt velocitate illius ex&longs;i&longs;tente <lb/>dupla ad velocitatem i&longs;tius, illud perueniret <lb/>in C, eodem momento, quo i&longs;tud in E; <lb/>quoniam &longs;patium ab illo &longs;uperatum foret <lb/>vbique ad &longs;patium ab i&longs;to &longs;uperatum, du<lb/>plum; hoc e&longs;t, forent ab illo &longs;uperatæ duæ <lb/>partes, cùm ab i&longs;to vna: ab illo quatuor, cùm ab hoc <lb/>duæ, &c. quatenus &longs;patia &longs;e haberent vbique, vt velo<lb/>citates; hoc e&longs;t, velocitas per totam AC, e&longs;&longs;et vbique <lb/>dupla velocitatis per totam DE. At verò, quoniam <lb/>heic agitur de motu non æquabili, &longs;ed continenter <lb/>accelerato; ita di&longs;cedant rursùs mobilia eodem tem<lb/>pore, vnum ab A, aliud à D, vt &longs;uccre&longs;centibus conti<lb/>nuò velocitatis gradibus, illud perueniendo in C, ac<lb/>qui&longs;ierit duodecim, hoc perueniendo in E, &longs;ex: <emph type="italics"/>Tum <lb/>interrogas,<emph.end type="italics"/> Quid impediat, quò minùs illud perueniat <lb/>in C, eodem tempore, quo i&longs;tud in E? <emph type="italics"/>Ego verò re&longs;<lb/>pondeo, nihil certè impedire, &longs;i tales e&longs;&longs;ent, quales de&longs;cribis <lb/>velocitate: taleis autem &longs;ine dubio de&longs;cribis, qualeis in prima <lb/>illa<emph.end type="italics"/> G<emph type="italics"/>alilei propo&longs;itione &longs;ignificari putas. At hoc &longs;en&longs;u hypo<lb/>thetica illa Galilei propo&longs;itio fal&longs;a e&longs;t, euidenterque impoßi<lb/>bilis: cùm nulla prorsùs ratione con&longs;equens inferi poßit ex <lb/>antecedente. Hoc enim e&longs;t Antecedens,<emph.end type="italics"/> Acceleratio ve<lb/>locitatis in de&longs;cen&longs;u grauium per totam AC, ita con<lb/>tinua &longs;ucce&longs;&longs;ione cre&longs;cit, vt primùm in B acqui&longs;itus <lb/>&longs;upponatur vnus aliquis velocitatis gradus; & vlteriùs <lb/>procedente motu, & continuò incre&longs;cente celeritate,

<pb pagenum="114"/>duo iam in C velocitatis gradus habeantur. <emph type="italics"/>Istad <lb/>certè e&longs;t antecedens, & nihil aliud aiunt ij, qui à<emph.end type="italics"/> G<emph type="italics"/>alileo ab<lb/>&longs;urditatis arguuntur. Iam ergo vide, vtrum ex hoc antece<lb/>dente, rectè tuum illud, & Galilei Con&longs;equens inferatur:<emph.end type="italics"/> Ergo <lb/>velocitas de&longs;cen&longs;us per totam AC ab initio ad finem, <lb/>& &longs;ecundum qua&longs;libet eius parteis con&longs;iderata, perpe<lb/>tuò dupla e&longs;t eius velocitatis, qua idem graue per AB <lb/>de&longs;cendit. <emph type="italics"/>Siue enim<emph.end type="italics"/> AB <emph type="italics"/>coniunctam toti<emph.end type="italics"/> AC, <emph type="italics"/>con&longs;ide<lb/>res, &longs;iue vt &longs;eparatam, qualis e&longs;t<emph.end type="italics"/> DE, <emph type="italics"/>&longs;emper velocitas de&longs;<lb/>cen&longs;us per<emph.end type="italics"/> AC, <emph type="italics"/>quandiù percurritur prior eius pars<emph.end type="italics"/> AB, <emph type="italics"/>nec <lb/>&longs;ui-ip&longs;ius, nec velocitatis per<emph.end type="italics"/> DE, <emph type="italics"/>dupla e&longs;t, vt falsò a&longs;&longs;umis, <lb/>&longs;ed planè eadem, aut æqualis omninò est. Nempe volumus, <lb/>& nece&longs;&longs;ariò exigimus (quod ip&longs;a quoque rei natura po&longs;tu<lb/>lat) vt motus, qui per totam<emph.end type="italics"/> AC, <emph type="italics"/>& per partem<emph.end type="italics"/> AB, <emph type="italics"/>&longs;iue <lb/>per æqualem<emph.end type="italics"/> DE, <emph type="italics"/>eadem planè velocitate incipiat, & eadem <lb/>velocitate progrediatur per totam<emph.end type="italics"/> AB, <emph type="italics"/>& per ip&longs;am<emph.end type="italics"/> DE: <lb/><emph type="italics"/>ex<emph.end type="italics"/> B <emph type="italics"/>verò ita velocitas augeatur, vt tandem in<emph.end type="italics"/> C <emph type="italics"/>dupla in<lb/>ueniatur eius, qua fuit in<emph.end type="italics"/> B, <emph type="italics"/>vel in<emph.end type="italics"/> E. H<emph type="italics"/>æc enim nostra, <lb/>& communis aliorum &longs;uppo&longs;itio e&longs;t, & primæ propo&longs;itionis à<emph.end type="italics"/><lb/>G<emph type="italics"/>alileo a&longs;&longs;umptæ antecedens; &longs;i tamen aduer&longs;um nos, & non <lb/>potiùs aduer&longs;us Chimeras, & Tragalaphos depugnet. At <lb/>ex eo antecedente tuumillud, &<emph.end type="italics"/> G<emph type="italics"/>alilei con&longs;equens nece&longs;&longs;a<lb/>ria illatione non priùs inferetur, quàm aliud quodlibet ex <lb/>vero fal&longs;um eruatur. Prima igitur<emph.end type="italics"/> G<emph type="italics"/>alilei Propo&longs;itio, eo <lb/>&longs;en&longs;u, quo ab ip&longs;o v&longs;urpatur, & à te intelligitur, fal&longs;a e&longs;t, <lb/>atque impoßibilis; ideóque tota eius ratiocinatio, non demon<lb/>&longs;tratio, &longs;ed merus Paralogi&longs;mus e&longs;t.<emph.end type="italics"/></s>

<s>An videri potes operosè quidem, &longs;ed nequicquam <lb/>tamen explicare conatum, vt Paralogi&longs;mum o&longs;tendas, <lb/>quem quanta moderatione potueram non fui&longs;&longs;e à te

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<s><emph type="italics"/>Acceleratio velocitatis in de&longs;cen&longs;u grauium per<emph.end type="italics"/><lb/>

<arrow.to.target n="fig28"></arrow.to.target><lb/><emph type="italics"/>totam AC, ita continua &longs;ucceßione cre&longs;cit, vt pri<lb/>mùm in B acqui&longs;itus &longs;upponatur vnus aliquis ve<lb/>locitatis gradus, & vlteriùs procedente motu, & <lb/>continuò incre&longs;cente celeritate duo iam in C velo<lb/>citatis gradus habeantur.<emph.end type="italics"/></s>

<figure id="fig28"></figure><lb/><emph type="italics"/>totam AC, ita continua &longs;ucceßione cre&longs;cit, vt pri<lb/>mùm in B acqui&longs;itus &longs;upponatur vnus aliquis ve<lb/>locitatis gradus, & vlteriùs procedente motu, & <lb/>continuò incre&longs;cente celeritate duo iam in C velo<lb/>citatis gradus habeantur.<emph.end type="italics"/></s>

<s>Ac tum, po&longs;tquàm dixi&longs;ti illos, quià Ga<lb/>lile o arguuntur, nihil aliud dicere, videre <lb/>me iubes, vtrum ex tali Antecedente, rectè inferatur <lb/>tale Con&longs;equens. </s>

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<s>Hui tamen! &longs;iccine exactius, magi&longs;que <lb/>

<arrow.to.target n="fig29"></arrow.to.target><lb/>&longs;ufficienter mihi Paralogi&longs;mum iam de<lb/>tegis, & proptereáne ægrè fers te mihi <lb/>(<expan abbr="|c&utilde;">|cum</expan> intelligentem putares) indica&longs;&longs;e bre<lb/>uiùs ratiocinationis Galileanæ vitium? <lb/><emph type="italics"/>Semper,<emph.end type="italics"/> inquis, <emph type="italics"/>velocitas de&longs;cen&longs;us per<emph.end type="italics"/> AC, <lb/><emph type="italics"/>quandò percurritur prima eius pars<emph.end type="italics"/> AB, <emph type="italics"/>nec <lb/>&longs;ui ip&longs;ius, nec velocitatis per<emph.end type="italics"/> DE <emph type="italics"/>dupla e&longs;t.<emph.end type="italics"/><lb/>Enimverò, non quæritur, vtrum reipsâ dupla &longs;it, &longs;ed <lb/>an duplam e&longs;&longs;e tuo ex principio con&longs;equatur. Nam <lb/>noui quidem ego fal&longs;um e&longs;&longs;e Con&longs;equens; &longs;ed verè ta<lb/>men con&longs;equi ex Antecedente, admi&longs;&longs;o o&longs;tendo. Dicis <lb/><emph type="italics"/>me id falsò a&longs;&longs;umere;<emph.end type="italics"/> ego verò non falsò a&longs;&longs;umo, qui <lb/>ne a&longs;&longs;umo quidem, &longs;ed &longs;olum con&longs;equi demon&longs;tro, vt<lb/>cumque po&longs;tquàm id demon&longs;traui, &longs;ub&longs;umere deinde <lb/>po&longs;&longs;im, vt o&longs;tendam te tibi repugnare, quòd <expan abbr="c&utilde;">cum</expan> fatea<lb/>ris AC, & AB inæqualibus percurri temporibus, prin<lb/>cipium tamen id defendas, ex quo fateri &longs;imul cogaris <lb/>eodem, aut æquali tempore percurri. Itaque cùm <lb/>heic non agatur de veritate Con&longs;equentis, &longs;ed de ne<lb/>ce&longs;&longs;itate, qua con&longs;equitur, ac tu deberes o&longs;tendere <lb/>nece&longs;&longs;ariò non con&longs;equi, & declarare in quo peceá<lb/>rim, inferendo fore velocitatem per totam AC du<lb/>plam velocitatis per totam AB, nihil aliud habes, <lb/><expan abbr="quã">quam</expan>, <emph type="italics"/>non e&longs;&longs;e duplam.<emph.end type="italics"/> Quod perinde e&longs;t, ac &longs;i te ponen<lb/>te illud Antecedens, <emph type="italics"/>Plato e&longs;t lapis,<emph.end type="italics"/> ego inferam i&longs;tud

<figure id="fig29"></figure><lb/>&longs;ufficienter mihi Paralogi&longs;mum iam de<lb/>tegis, & proptereáne ægrè fers te mihi <lb/>(<expan abbr="|c&utilde;">|cum</expan> intelligentem putares) indica&longs;&longs;e bre<lb/>uiùs ratiocinationis Galileanæ vitium? <lb/><emph type="italics"/>Semper,<emph.end type="italics"/> inquis, <emph type="italics"/>velocitas de&longs;cen&longs;us per<emph.end type="italics"/> AC, <lb/><emph type="italics"/>quandò percurritur prima eius pars<emph.end type="italics"/> AB, <emph type="italics"/>nec <lb/>&longs;ui ip&longs;ius, nec velocitatis per<emph.end type="italics"/> DE <emph type="italics"/>dupla e&longs;t.<emph.end type="italics"/><lb/>Enimverò, non quæritur, vtrum reipsâ dupla &longs;it, &longs;ed <lb/>an duplam e&longs;&longs;e tuo ex principio con&longs;equatur. Nam <lb/>noui quidem ego fal&longs;um e&longs;&longs;e Con&longs;equens; &longs;ed verè ta<lb/>men con&longs;equi ex Antecedente, admi&longs;&longs;o o&longs;tendo. Dicis <lb/><emph type="italics"/>me id falsò a&longs;&longs;umere;<emph.end type="italics"/> ego verò non falsò a&longs;&longs;umo, qui <lb/>ne a&longs;&longs;umo quidem, &longs;ed &longs;olum con&longs;equi demon&longs;tro, vt<lb/>cumque po&longs;tquàm id demon&longs;traui, &longs;ub&longs;umere deinde <lb/>po&longs;&longs;im, vt o&longs;tendam te tibi repugnare, quòd <expan abbr="c&utilde;">cum</expan> fatea<lb/>ris AC, & AB inæqualibus percurri temporibus, prin<lb/>cipium tamen id defendas, ex quo fateri &longs;imul cogaris <lb/>eodem, aut æquali tempore percurri. Itaque cùm <lb/>heic non agatur de veritate Con&longs;equentis, &longs;ed de ne<lb/>ce&longs;&longs;itate, qua con&longs;equitur, ac tu deberes o&longs;tendere <lb/>nece&longs;&longs;ariò non con&longs;equi, & declarare in quo peceá<lb/>rim, inferendo fore velocitatem per totam AC du<lb/>plam velocitatis per totam AB, nihil aliud habes, <lb/><expan abbr="quã">quam</expan>, <emph type="italics"/>non e&longs;&longs;e duplam.<emph.end type="italics"/> Quod perinde e&longs;t, ac &longs;i te ponen<lb/>te illud Antecedens, <emph type="italics"/>Plato e&longs;t lapis,<emph.end type="italics"/> ego inferam i&longs;tud

<pb pagenum="122"/>Con&longs;equens, igitur <emph type="italics"/>Plato &longs;en&longs;u caret:<emph.end type="italics"/> & te negante <lb/>con&longs;equutionem, illam ex eo probem, <emph type="italics"/>quòd lapis &longs;en&longs;u <lb/>careat:<emph.end type="italics"/> ac tum dicas Con&longs;equens <emph type="italics"/>non rectè inferri;<emph.end type="italics"/> & <lb/>compul&longs;us ad id probandum, nihil tamen aliud <lb/>quàm hoc dicas, <emph type="italics"/>Plato enim &longs;en&longs;u non caret, vt falsò, <lb/>a&longs;&longs;umis, &longs;ed planè &longs;en&longs;u præditus e&longs;t.<emph.end type="italics"/> Videlicet quæ&longs;tio <lb/>non erit de veritate Con&longs;equentis, nam ego quoque <lb/>Platonem &longs;en&longs;u carere fal&longs;um reputabo: &longs;ed de nece&longs;<lb/>&longs;itate con&longs;equutionis, quam tu infirmare deberes, ne <lb/>euerteret tuum Antecedens; neque ego falsò a&longs;&longs;u<lb/>mam, carere Platonem &longs;en&longs;u, qui ne a&longs;&longs;umam quidem: <lb/>&longs;ed nece&longs;&longs;ariò &longs;olummodò, & vt conclu&longs;ionem dedu<lb/>cam; tamet&longs;i <expan abbr="deduct&utilde;">deductum</expan> a&longs;&longs;umere po&longs;&longs;im, vt o&longs;tendam te <lb/>contradicere tibi ip&longs;i, tanquam coactum a&longs;&longs;erere Pla<lb/>tonem &longs;en&longs;u de&longs;titui, quem a&longs;&longs;eras præditum &longs;en&longs;u. </s>

<s>Pergis porrò, <emph type="italics"/><expan abbr="N&etilde;pe">Nempe</expan> volumus, & nece&longs;&longs;ariò exigimus (quod <lb/>ip&longs;a quoque rei natura po&longs;tulat) vt motus, qui per totam<emph.end type="italics"/><lb/>AC, <emph type="italics"/>& per partem<emph.end type="italics"/> AB, <emph type="italics"/>&longs;iue per æqualem<emph.end type="italics"/> DE, <emph type="italics"/>eadem <lb/>planè velocitate incipiat, & eadem velocitate progrediatur, <lb/>per totam<emph.end type="italics"/> AB, <emph type="italics"/>& per ip&longs;am<emph.end type="italics"/> DE, <emph type="italics"/>ex<emph.end type="italics"/> B <emph type="italics"/>verò ita velocitas <lb/>augeatur, vt tandem in<emph.end type="italics"/> C <emph type="italics"/>dupla inueniatur eius, quæ fuit in<emph.end type="italics"/><lb/>B, <emph type="italics"/>vel in<emph.end type="italics"/> E. At, optime Vir, quod vis quidem, & <lb/>exigis, vt <emph type="italics"/>c&oelig;ptus ex A motus,<emph.end type="italics"/> &longs;iue progre&longs;&longs;urus &longs;it v&longs;que <lb/>ad C, &longs;iue de&longs;iturus in B, <emph type="italics"/>eadem planè velocitate incipiat, <lb/>& progrediatur v&longs;que ad B,<emph.end type="italics"/> idip&longs;um e&longs;t, quod po&longs;tulat <lb/>ip&longs;a quoque rei natura (vtcumque po&longs;teà cau&longs;&longs;eris me <lb/>non in&longs;pexi&longs;&longs;e illam penitiùs.) Quod autem vis, & <lb/>exigis, vt velocitas ex B ita augeatur, vt tandem in <lb/>C dupla inueniatur eius, quæ fuit in B, idip&longs;um iam <lb/>e&longs;t, quod o&longs;ten&longs;um e&longs;t tantùm auer&longs;ari Naturam,

<pb pagenum="123"/>quantùm auer&longs;atur motum in&longs;tantancum. Quam<lb/>obrem, non &longs;ufficit tibi, vt velis, atque exigas velo<lb/>citat<gap/>m in C duplam e&longs;&longs;e velocitatis in B, &longs;ed re&longs;tat, <lb/>vt illud, &longs;i po&longs;lit, o&longs;tendas. Quomodo verò id vnquam <lb/>po&longs;&longs;is, ni&longs;i volendo, & exigendo, vt quod quæritur, <lb/>tibi concedatur, atque adeò petendo, vt loquuntur, <lb/>principium? Idip&longs;um e&longs;t, quod te feci&longs;&longs;e, circa relata <lb/>verba, obieci articulo XI. & quod tamen iam repetis <lb/>con&longs;tanter. Quippe eò quoque te iam adegit, quem <lb/>exi&longs;tima&longs;ti te po&longs;&longs;e di&longs;tinguere priorem &longs;en&longs;um, à <lb/>po&longs;teriore valde diuer&longs;um. Nam po&longs;tquàm dixi&longs;ti <lb/><expan abbr="e&utilde;">eum</expan> &longs;en&longs;um <emph type="italics"/>verum e&longs;&longs;e, ac nece&longs;&longs;arium,<emph.end type="italics"/> i&longs;thæc <lb/>

<arrow.to.target n="fig30"></arrow.to.target><lb/>verba tua &longs;equuntur, <emph type="italics"/>Si enim in triangulo <lb/>æqualia spatia de&longs;ignentur<emph.end type="italics"/> AD, DE, EF, <lb/><emph type="italics"/>&c. & in<emph.end type="italics"/> D <emph type="italics"/>acqui&longs;itus &longs;upponatur vnus <lb/>gradus, & in<emph.end type="italics"/> E <emph type="italics"/>duo, & tres in<emph.end type="italics"/> F; <emph type="italics"/>manife<lb/>&longs;tum e&longs;t d<gap/>os gradus, ad quos acceleratio per<lb/>ueni&longs;&longs;e ponitur in<emph.end type="italics"/> E, <emph type="italics"/>e&longs;&longs;e ad vnum gradum <lb/>acqui&longs;itum in<emph.end type="italics"/> D, <emph type="italics"/>vt spatium<emph.end type="italics"/> AE, <emph type="italics"/>ad spa<lb/>tium. AD.<emph.end type="italics"/> Deprehendere enim &longs;tatim <lb/>licet, quemadmodum idip&longs;um &longs;upponas, quod pror<lb/>sùs controuertitur: nempe in E e&longs;&longs;e duos gradus, vbi <lb/>vnus fuerit in D. Ni&longs;i verò hoc e&longs;t; quid nam tandem <lb/>e&longs;t, quod dicunt petere principium? Subinnueram <lb/>ego articulo eodem id mouere te, quòd velocitas ac<lb/>qui&longs;ita in C (re&longs;umendo nempelineam ABC) &longs;it re<lb/>uerâ maior, quàm acqui&longs;ita in B; &longs;ed tu attendere <lb/>nolui&longs;ti ex eo, quòd &longs;it maior, non &longs;equi tamen e&longs;&longs;e <lb/>duplam; ratus &longs;cilicet te penitiùs in&longs;pexi&longs;&longs;e rei natu<lb/>ram, ac eo principio &longs;emper abductus, de quo tota e&longs;t

<figure id="fig30"></figure><lb/>verba tua &longs;equuntur, <emph type="italics"/>Si enim in triangulo <lb/>æqualia spatia de&longs;ignentur<emph.end type="italics"/> AD, DE, EF, <lb/><emph type="italics"/>&c. & in<emph.end type="italics"/> D <emph type="italics"/>acqui&longs;itus &longs;upponatur vnus <lb/>gradus, & in<emph.end type="italics"/> E <emph type="italics"/>duo, & tres in<emph.end type="italics"/> F; <emph type="italics"/>manife<lb/>&longs;tum e&longs;t d<gap/>os gradus, ad quos acceleratio per<lb/>ueni&longs;&longs;e ponitur in<emph.end type="italics"/> E, <emph type="italics"/>e&longs;&longs;e ad vnum gradum <lb/>acqui&longs;itum in<emph.end type="italics"/> D, <emph type="italics"/>vt spatium<emph.end type="italics"/> AE, <emph type="italics"/>ad spa<lb/>tium. AD.<emph.end type="italics"/> Deprehendere enim &longs;tatim <lb/>licet, quemadmodum idip&longs;um &longs;upponas, quod pror<lb/>sùs controuertitur: nempe in E e&longs;&longs;e duos gradus, vbi <lb/>vnus fuerit in D. Ni&longs;i verò hoc e&longs;t; quid nam tandem <lb/>e&longs;t, quod dicunt petere principium? Subinnueram <lb/>ego articulo eodem id mouere te, quòd velocitas ac<lb/>qui&longs;ita in C (re&longs;umendo nempelineam ABC) &longs;it re<lb/>uerâ maior, quàm acqui&longs;ita in B; &longs;ed tu attendere <lb/>nolui&longs;ti ex eo, quòd &longs;it maior, non &longs;equi tamen e&longs;&longs;e <lb/>duplam; ratus &longs;cilicet te penitiùs in&longs;pexi&longs;&longs;e rei natu<lb/>ram, ac eo principio &longs;emper abductus, de quo tota e&longs;t

<pb pagenum="124"/>controuer&longs;ia; itemque opinione illa, quod in trian<lb/>gulo, lineæ ba&longs;i parallelæ repræ&longs;entare gradus veloci<lb/>tates valeant, &longs;i partes cruris alterutrius ip&longs;is re&longs;pon<lb/>dentes repræ&longs;entent &longs;patia; non aduertendo, quî i&longs;ti <lb/>gradus inæquales &longs;int, & à &longs;eip&longs;is differant, dum <lb/>acquiruntur, & dum manent; & quid incommodi ex <lb/>hac repræ&longs;entatione trahatur. Videtur &longs;altem occa&longs;io <lb/>dubitandi fieri debui&longs;&longs;e, po&longs;tquàm admonitus à me, <lb/>fal&longs;um deprehendi&longs;ti id Experimentum, cui &longs;oli in<lb/>nixus, prounciâras velocitatem duplam e&longs;&longs;e ex du<lb/>pla altitudine; ac &longs;altem ob&longs;erua&longs;ti globum cadentem <lb/>ex A in C, hoc e&longs;t ex altitudine duarum <lb/>

<arrow.to.target n="fig31"></arrow.to.target><lb/>&longs;ui diametrorum, non eleuare cum oppo&longs;ita <lb/>lance duplum eius ponderis, quod eleuat ex <lb/>A in B, hoc e&longs;t ex altitudine vnius: &longs;ed res e&longs;t <lb/>po&longs;teà fu&longs;iùs dicenda. Heic &longs;olùm moneo, <lb/>quod &longs;ubdis <emph type="italics"/>tuam, & communem aliorum &longs;uppo&longs;i<lb/>tionem e&longs;&longs;e primæ Propo&longs;itionis<emph.end type="italics"/> (&longs;eu &longs;uperioris A&longs;<lb/>&longs;umptionis) G<emph type="italics"/>alilei Antecedens,<emph.end type="italics"/> e&longs;&longs;e <expan abbr="illã">illam</expan> quidem <lb/>tuam, aliorumque &longs;uppo&longs;itionem, ip&longs;amque fal&longs;am, <lb/>ac impo&longs;&longs;ibilem; &longs;ed à Galileo tamen hypotheticè &longs;o<lb/>lùm v&longs;urpari, & Antecedens fieri, vt quid ex ea incom<lb/>modi nece&longs;&longs;ariò &longs;equatur, demon&longs;tret. Vnde & <lb/>quod ais, <emph type="italics"/>ni&longs;i aduer&longs;us<emph.end type="italics"/> C<emph type="italics"/>himæras, & Tragelaphos depugnet,<emph.end type="italics"/><lb/>vides quonam &longs;en&longs;u accipiendum &longs;it; & quod &longs;upere&longs;t, <lb/>ip&longs;e iam agno&longs;cis, an eius rationem <emph type="italics"/>merum e&longs;&longs;e Paralo<lb/>gi&longs;mum<emph.end type="italics"/> probâris vllo argumento. </s>

<figure id="fig31"></figure><lb/>&longs;ui diametrorum, non eleuare cum oppo&longs;ita <lb/>lance duplum eius ponderis, quod eleuat ex <lb/>A in B, hoc e&longs;t ex altitudine vnius: &longs;ed res e&longs;t <lb/>po&longs;teà fu&longs;iùs dicenda. Heic &longs;olùm moneo, <lb/>quod &longs;ubdis <emph type="italics"/>tuam, & communem aliorum &longs;uppo&longs;i<lb/>tionem e&longs;&longs;e primæ Propo&longs;itionis<emph.end type="italics"/> (&longs;eu &longs;uperioris A&longs;<lb/>&longs;umptionis) G<emph type="italics"/>alilei Antecedens,<emph.end type="italics"/> e&longs;&longs;e <expan abbr="illã">illam</expan> quidem <lb/>tuam, aliorumque &longs;uppo&longs;itionem, ip&longs;amque fal&longs;am, <lb/>ac impo&longs;&longs;ibilem; &longs;ed à Galileo tamen hypotheticè &longs;o<lb/>lùm v&longs;urpari, & Antecedens fieri, vt quid ex ea incom<lb/>modi nece&longs;&longs;ariò &longs;equatur, demon&longs;tret. Vnde & <lb/>quod ais, <emph type="italics"/>ni&longs;i aduer&longs;us<emph.end type="italics"/> C<emph type="italics"/>himæras, & Tragelaphos depugnet,<emph.end type="italics"/><lb/>vides quonam &longs;en&longs;u accipiendum &longs;it; & quod &longs;upere&longs;t, <lb/>ip&longs;e iam agno&longs;cis, an eius rationem <emph type="italics"/>merum e&longs;&longs;e Paralo<lb/>gi&longs;mum<emph.end type="italics"/> probâris vllo argumento. </s>

<s><emph type="italics"/>Quòd &longs;i tamen præoccupatus contrariis decretis animus <lb/>tuus, <expan abbr="nond&utilde;">nondum</expan> clarißimam veritatis huius lucem plenè per&longs;picit, <lb/>ac penitùs agno&longs;cit,<emph.end type="italics"/> C<emph type="italics"/>oncipe in triangulo ABC partes lineæ<emph.end type="italics"/>

<pb pagenum="125"/><emph type="italics"/>AC non iam spatij parteis æqualeis de&longs;ignare, &longs;ed temporis. <lb/>Tunc ex tuis, & Galilei principijs facilè agno&longs;ces velocita<lb/>tem in E, hoc e&longs;t in fine &longs;ecundi temporis acqui&longs;itans, veloci<lb/>tat<gap/>s in D acqui&longs;itæ duplam e&longs;&longs;e, perpetuóque<emph.end type="italics"/><lb/>

<arrow.to.target n="fig32"></arrow.to.target><lb/><emph type="italics"/>velocitates, & tempora in eadem e&longs;&longs;e ratione. <lb/>Hoc autem con&longs;tituto, tuis ego, &<emph.end type="italics"/> G<emph type="italics"/>alilei <lb/>armis ita aduer&longs;um te in&longs;urgo.<emph.end type="italics"/></s>

<figure id="fig32"></figure><lb/><emph type="italics"/>velocitates, & tempora in eadem e&longs;&longs;e ratione. <lb/>Hoc autem con&longs;tituto, tuis ego, &<emph.end type="italics"/> G<emph type="italics"/>alilei <lb/>armis ita aduer&longs;um te in&longs;urgo.<emph.end type="italics"/></s>

<s>Et verò opperior. </s>

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Line 1658

<s>Cùm <emph type="italics"/>dicis eodem tempore,<emph.end type="italics"/> hoc e&longs;t <emph type="italics"/>primo,<emph.end type="italics"/> vnde quæ&longs;o <lb/>a&longs;&longs;umis? Nam in A&longs;&longs;umptione quidem non <lb/>

<arrow.to.target n="fig33"></arrow.to.target><lb/>di&longs;tinxeras v<gap/>rius velocitatis re&longs;pectu idem, <lb/>aut æquale acciperes, & procliue fuit, vt <lb/>acciperes re&longs;pectu duplæ, cui re&longs;ponderent <lb/>duo tempora, &longs;iue, vt expre&longs;&longs;i, aggregatum <lb/>temporum duorum, vnde & &longs;ub&longs;umendum <lb/>fuit. </s>

<figure id="fig33"></figure><lb/>di&longs;tinxeras v<gap/>rius velocitatis re&longs;pectu idem, <lb/>aut æquale acciperes, & procliue fuit, vt <lb/>acciperes re&longs;pectu duplæ, cui re&longs;ponderent <lb/>duo tempora, &longs;iue, vt expre&longs;&longs;i, aggregatum <lb/>temporum duorum, vnde & &longs;ub&longs;umendum <lb/>fuit. </s>

<s><emph type="italics"/>Si primo tempore AD, &longs;patium PM per<lb/>curratur à velocitate &longs;ubdupla, futurum vt toto <lb/>tempore AE<emph.end type="italics"/> (&longs;iue duobus temporibus, aut aggregato <lb/>duorum) <emph type="italics"/>percurratur spatium PN &longs;patij PM duplum.<emph.end type="italics"/></s>

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Line 1690

<s><emph type="italics"/>Nam vt modo ratiocinandi tuo etiam vtar, &longs;umpto quo<lb/>piam alio tempore<emph.end type="italics"/> RO, <emph type="italics"/>tempori<emph.end type="italics"/> PM <emph type="italics"/>æquali, con<gap/>mus <lb/>duo mobilia, quorum vnum per<emph.end type="italics"/> PN <emph type="italics"/>eodem &longs;patio moueri<emph.end type="italics"/>

<pb pagenum="129"/><emph type="italics"/>incipiat velocitate dupla, quo alterum per<emph.end type="italics"/> RO <emph type="italics"/>de&longs;cendit <lb/>velocitate &longs;ubdupla; tum nece&longs;&longs;e fuerit, vt quibu&longs;libet &longs;pa<lb/>tij partibus continuò momenta temporis<emph.end type="italics"/> PM <emph type="italics"/>duplò matora <lb/>percurrantur, quam &longs;int momenta temporis<emph.end type="italics"/> RO, <emph type="italics"/>quæ ab <lb/>aliò mobili ii&longs;dem partibus &longs;ubdupla velocitate decurruntur. <lb/>Quo igitur spatio mobile lentius tempus<emph.end type="italics"/> RO <lb/>

<arrow.to.target n="fig34"></arrow.to.target><lb/><emph type="italics"/>totum percurrerit, eodem alterum mobile duplò <lb/>velocius tempus<emph.end type="italics"/> PN <emph type="italics"/>etiam ab&longs;oluerit. Et quo<lb/>niam te iudice alia ratio non est, &longs;iue tempus<emph.end type="italics"/><lb/>RO, <emph type="italics"/>aut<emph.end type="italics"/> PM <emph type="italics"/>à toto tempore<emph.end type="italics"/> PN <emph type="italics"/>&longs;eiunctum, <lb/>&longs;iue eidem coniunctum &longs;upponatur: nece&longs;&longs;arium <lb/>planè fuerit, vt etiam ab vno, eodemque mobili, <lb/>vno eodemque &longs;patio totum tempus<emph.end type="italics"/> PN, <emph type="italics"/>& <lb/>eius dimidium<emph.end type="italics"/> PM <emph type="italics"/>percurratur, quod certum <lb/>e&longs;t e&longs;&longs;e impoßibile, ni&longs;i motus fieret in puncto.<emph.end type="italics"/></s>

<figure id="fig34"></figure><lb/><emph type="italics"/>totum percurrerit, eodem alterum mobile duplò <lb/>velocius tempus<emph.end type="italics"/> PN <emph type="italics"/>etiam ab&longs;oluerit. Et quo<lb/>niam te iudice alia ratio non est, &longs;iue tempus<emph.end type="italics"/><lb/>RO, <emph type="italics"/>aut<emph.end type="italics"/> PM <emph type="italics"/>à toto tempore<emph.end type="italics"/> PN <emph type="italics"/>&longs;eiunctum, <lb/>&longs;iue eidem coniunctum &longs;upponatur: nece&longs;&longs;arium <lb/>planè fuerit, vt etiam ab vno, eodemque mobili, <lb/>vno eodemque &longs;patio totum tempus<emph.end type="italics"/> PN, <emph type="italics"/>& <lb/>eius dimidium<emph.end type="italics"/> PM <emph type="italics"/>percurratur, quod certum <lb/>e&longs;t e&longs;&longs;e impoßibile, ni&longs;i motus fieret in puncto.<emph.end type="italics"/></s>

<s>Hoc &longs;anè modo fui&longs;&longs;es ratiocinatus <emph type="italics"/>variatis tan<lb/>tum terminis,<emph.end type="italics"/> ac facere mihi quandam re&longs;pondendi ne<lb/>ce&longs;&longs;itatem vi&longs;us fui&longs;&longs;es. Nunc autem, cùm terminos <lb/>controuer&longs;os non varies, ac nihil concludas aduer&longs;um <lb/>me, &longs;ed illud omninò, atque eodem modo, quod e&longs;t <lb/>aduer&longs;us te conclu&longs;um: e&longs;t planè cur mirer &longs;ic captare <lb/>te ex teip&longs;o triumphum. Nam & cum alioquin ita <lb/>habes. </s>

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<s><emph type="italics"/>Sed aliam quoque Propo&longs;itionem, optime<emph.end type="italics"/> G<emph type="italics"/>a&longs;&longs;ende, non <lb/>minùs fal&longs;am, atque impoßibilem numero xi. in fine mihi af<lb/>fingis; dum ais m<gap/>ad vulgatam motus accelerati definitionem <lb/>con&longs;equenter loquentem, velle in de&longs;cen&longs;u per totum spatium <lb/>AC bifariam diui&longs;um in B, partem BC tran&longs;curri<emph.end type="italics"/><lb/>

<arrow.to.target n="fig35"></arrow.to.target><lb/><emph type="italics"/>in dimidio eius temporis, quo percurritur AB; ex <lb/>qua fal&longs;a &longs;uppo&longs;itione, &longs;equenti numero vnum In<lb/>commodum, & ad finem Respon&longs;ionis tuæ alia plura <lb/>longè ab&longs;urdißima deducis, quæ tanquam con&longs;ectaria <lb/>ex meis principiis, ac decretis nece&longs;&longs;ariò illata mihi <lb/>obiectas.<emph.end type="italics"/> V<emph type="italics"/>team igitur Propo&longs;itionem &longs;emel tracte<lb/>mus, eius examen in commodiorem locum re&longs;erua<lb/>bimus.<emph.end type="italics"/></s>

<figure id="fig35"></figure><lb/><emph type="italics"/>in dimidio eius temporis, quo percurritur AB; ex <lb/>qua fal&longs;a &longs;uppo&longs;itione, &longs;equenti numero vnum In<lb/>commodum, & ad finem Respon&longs;ionis tuæ alia plura <lb/>longè ab&longs;urdißima deducis, quæ tanquam con&longs;ectaria <lb/>ex meis principiis, ac decretis nece&longs;&longs;ariò illata mihi <lb/>obiectas.<emph.end type="italics"/> V<emph type="italics"/>team igitur Propo&longs;itionem &longs;emel tracte<lb/>mus, eius examen in commodiorem locum re&longs;erua<lb/>bimus.<emph.end type="italics"/></s>

<s>Cùm tu Caput magni momenti per&longs;tringas adeò <lb/>obiter; non po&longs;&longs;um ego non &longs;altem duo, aut tria quæ<lb/>dam adnotare. Quî enim Imprimis intactum præ<lb/>teream, quod <emph type="italics"/>me affingere tibi<emph.end type="italics"/> ais <emph type="italics"/>Propo&longs;itionem fal&longs;am; <lb/>ac impo&longs;ßibilem, dum aio te ad vulgatam motus definitionem <lb/>con&longs;equenter loquentem, velle in de&longs;cen&longs;u per totum &longs;patium<emph.end type="italics"/>

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<s>Priu&longs;quam hæc de Paralogi&longs;mo, quem Galileo ob<lb/>ieci&longs;ti, dimi<gap/>am, memini&longs;&longs;e iuuatobie ci&longs;&longs;e me tibi arti<lb/>culis XXXVI, & XXXVIII. fui&longs;&longs;e tenon &longs;ecus ratiocina-

<pb pagenum="135"/>tum, quàm Galileum: atque idcircò &longs;i ille quidem Pa<lb/>ralogi&longs;mum admi&longs;erit, incidi&longs;&longs;e te, recidi&longs;&longs;eque <lb/>

<arrow.to.target n="fig36"></arrow.to.target><lb/>in cundem: ac o&longs;tendere vel ex ea &longs;ola ratiocina<lb/>tione tua, quæ relata e&longs;t articulo XXXIII. quemad<lb/>modum ex tuis principiis demon&longs;trare liceat, <emph type="italics"/>&longs;i <lb/>velocitates &longs;icut &longs;patia &longs;int,<emph.end type="italics"/> fore <emph type="italics"/>vt totum, & pars <lb/>eodem, aut æquali tempore percurrantur.<emph.end type="italics"/> A&longs;&longs;umptâ <lb/>ergo, quæ illeic, lineâ, ideò probas <emph type="italics"/>&longs;patium<emph.end type="italics"/> DE, <lb/><emph type="italics"/>eodem tempore tran&longs;curri, quo<emph.end type="italics"/> SD; quia <emph type="italics"/>cùm<emph.end type="italics"/> AD <lb/><emph type="italics"/>dupla &longs;it ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>&<emph.end type="italics"/> AE <emph type="italics"/>ip&longs;ius<emph.end type="italics"/> AD, <emph type="italics"/>nece&longs;&longs;e &longs;it ve<lb/>locitatem in<emph.end type="italics"/> D <emph type="italics"/>duplam e&longs;&longs;e velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& veloci<lb/>tatem in<emph.end type="italics"/> E <emph type="italics"/>velocitatis in<emph.end type="italics"/> D. Cùm & aliunde, <emph type="italics"/>velo<lb/>citas per totam<emph.end type="italics"/> DE <emph type="italics"/>dupla &longs;it velocitatis per totam<emph.end type="italics"/><lb/>SD; quatenus <emph type="italics"/>quodlibet &longs;patium inc&oelig;ptum ab<emph.end type="italics"/> A, <lb/><emph type="italics"/>& terminatum inter<emph.end type="italics"/> D, <emph type="italics"/>&<emph.end type="italics"/> E, <emph type="italics"/>duplum e&longs;t alterius <lb/>&longs;patii, quod &longs;it item inc&oelig;ptum ab<emph.end type="italics"/> A, <emph type="italics"/>& terminatum <lb/>inter<emph.end type="italics"/> S, <emph type="italics"/>&<emph.end type="italics"/> D; Dico aut te inde nihil conclude<lb/>re, aut &longs;ic licere argumentari. </s>

<figure id="fig36"></figure><lb/>in cundem: ac o&longs;tendere vel ex ea &longs;ola ratiocina<lb/>tione tua, quæ relata e&longs;t articulo XXXIII. quemad<lb/>modum ex tuis principiis demon&longs;trare liceat, <emph type="italics"/>&longs;i <lb/>velocitates &longs;icut &longs;patia &longs;int,<emph.end type="italics"/> fore <emph type="italics"/>vt totum, & pars <lb/>eodem, aut æquali tempore percurrantur.<emph.end type="italics"/> A&longs;&longs;umptâ <lb/>ergo, quæ illeic, lineâ, ideò probas <emph type="italics"/>&longs;patium<emph.end type="italics"/> DE, <lb/><emph type="italics"/>eodem tempore tran&longs;curri, quo<emph.end type="italics"/> SD; quia <emph type="italics"/>cùm<emph.end type="italics"/> AD <lb/><emph type="italics"/>dupla &longs;it ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>&<emph.end type="italics"/> AE <emph type="italics"/>ip&longs;ius<emph.end type="italics"/> AD, <emph type="italics"/>nece&longs;&longs;e &longs;it ve<lb/>locitatem in<emph.end type="italics"/> D <emph type="italics"/>duplam e&longs;&longs;e velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& veloci<lb/>tatem in<emph.end type="italics"/> E <emph type="italics"/>velocitatis in<emph.end type="italics"/> D. Cùm & aliunde, <emph type="italics"/>velo<lb/>citas per totam<emph.end type="italics"/> DE <emph type="italics"/>dupla &longs;it velocitatis per totam<emph.end type="italics"/><lb/>SD; quatenus <emph type="italics"/>quodlibet &longs;patium inc&oelig;ptum ab<emph.end type="italics"/> A, <lb/><emph type="italics"/>& terminatum inter<emph.end type="italics"/> D, <emph type="italics"/>&<emph.end type="italics"/> E, <emph type="italics"/>duplum e&longs;t alterius <lb/>&longs;patii, quod &longs;it item inc&oelig;ptum ab<emph.end type="italics"/> A, <emph type="italics"/>& terminatum <lb/>inter<emph.end type="italics"/> S, <emph type="italics"/>&<emph.end type="italics"/> D; Dico aut te inde nihil conclude<lb/>re, aut &longs;ic licere argumentari. </s>

<s><emph type="italics"/>Si<emph.end type="italics"/> DE, <emph type="italics"/>&<emph.end type="italics"/> SD, <emph type="italics"/>eodem tempore percurruntur, quia veloci<lb/>tas à<emph.end type="italics"/> D <emph type="italics"/>in<emph.end type="italics"/> E, <emph type="italics"/>dupla e&longs;t velocitatis ab<emph.end type="italics"/> S <emph type="italics"/>in<emph.end type="italics"/> D. </s>

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<s><emph type="italics"/>Quæ verò de Libro Torricellij posteà adiungis (et&longs;i ea non <lb/>viderim) partim vera, partim fal&longs;a, aut &longs;altem incerta e&longs;&longs;e <lb/>non dubito. Duas certè eius Propo&longs;itiones primas ego quo<lb/>que de globis euidenter demonstro; at quomodo ex prioribus <lb/>illis duabus Propo&longs;itionibus po&longs;teriores inferantur, &longs;atis non <lb/>video, ni&longs;i<emph.end type="italics"/> G<emph type="italics"/>alilei principia &longs;upponantur. Cùm enim globi <lb/>pondere æquales<emph.end type="italics"/> E, <emph type="italics"/>&<emph.end type="italics"/><lb/>

<arrow.to.target n="fig37"></arrow.to.target><lb/>F <emph type="italics"/>planis<emph.end type="italics"/> AC, <emph type="italics"/>&<emph.end type="italics"/> CD <lb/>(<emph type="italics"/>vel<emph.end type="italics"/> CB) <emph type="italics"/>in&longs;i&longs;tentes, <lb/>momenta ad de&longs;cen&longs;um <lb/>retineant in reciproca, <lb/>& permutata ratione planorum, ob eamque cau&longs;&longs;am momen<lb/>ta ip&longs;ius<emph.end type="italics"/> E, <emph type="italics"/>&longs;int ad momenta ip&longs;ius<emph.end type="italics"/> F, <emph type="italics"/>vt<emph.end type="italics"/> CB (<emph type="italics"/>&longs;iue<emph.end type="italics"/> CD) <emph type="italics"/>ad<emph.end type="italics"/><lb/>CA; <emph type="italics"/>non apparet vnde euidenter concludi poßit<emph.end type="italics"/> E, <emph type="italics"/>qui pau<lb/>cioribus momentis deor&longs;um voluitur, & magis à motu perpen<lb/>diculari di&longs;trahitur, eundem nihilominus gradum velocitatis <lb/>acquirere in<emph.end type="italics"/> A, <emph type="italics"/>quem globus<emph.end type="italics"/> F <emph type="italics"/>in<emph.end type="italics"/> B <emph type="italics"/>acqui&longs;ierit. Nam quod <lb/>ais tarditatem motus spatij longitudine compen&longs;ari, conie<lb/>ctando quidem a&longs;&longs;eris; at (quod ad Po&longs;tulati per &longs;e, & ex <lb/>terminis minimè euidentis nece&longs;&longs;arium e&longs;&longs;et) nulla id ratione <lb/>demonstras.<emph.end type="italics"/></s>

<figure id="fig37"></figure><lb/>F <emph type="italics"/>planis<emph.end type="italics"/> AC, <emph type="italics"/>&<emph.end type="italics"/> CD <lb/>(<emph type="italics"/>vel<emph.end type="italics"/> CB) <emph type="italics"/>in&longs;i&longs;tentes, <lb/>momenta ad de&longs;cen&longs;um <lb/>retineant in reciproca, <lb/>& permutata ratione planorum, ob eamque cau&longs;&longs;am momen<lb/>ta ip&longs;ius<emph.end type="italics"/> E, <emph type="italics"/>&longs;int ad momenta ip&longs;ius<emph.end type="italics"/> F, <emph type="italics"/>vt<emph.end type="italics"/> CB (<emph type="italics"/>&longs;iue<emph.end type="italics"/> CD) <emph type="italics"/>ad<emph.end type="italics"/><lb/>CA; <emph type="italics"/>non apparet vnde euidenter concludi poßit<emph.end type="italics"/> E, <emph type="italics"/>qui pau<lb/>cioribus momentis deor&longs;um voluitur, & magis à motu perpen<lb/>diculari di&longs;trahitur, eundem nihilominus gradum velocitatis <lb/>acquirere in<emph.end type="italics"/> A, <emph type="italics"/>quem globus<emph.end type="italics"/> F <emph type="italics"/>in<emph.end type="italics"/> B <emph type="italics"/>acqui&longs;ierit. Nam quod <lb/>ais tarditatem motus spatij longitudine compen&longs;ari, conie<lb/>ctando quidem a&longs;&longs;eris; at (quod ad Po&longs;tulati per &longs;e, & ex <lb/>terminis minimè euidentis nece&longs;&longs;arium e&longs;&longs;et) nulla id ratione <lb/>demonstras.<emph.end type="italics"/></s>

<s>Quod de Propo&longs;itionibus Torricellij ais, cogno&longs;ces

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<s>Quomodo quæ&longs;o, optimus, & non pe&longs;&longs;imus po<lb/>tiùs e&longs;&longs;em, &longs;i is quidem, quem tu me hoc articulo de<lb/>pingis, forem? Verùm, quia num tibi quid affinxerim: <lb/>num quædam à te reiecta vt fal&longs;a, ex tuis principiis &longs;e<lb/>qui non o&longs;tenderim; num po&longs;tquam tu quædam ex<lb/>pugna&longs;ti vt fal&longs;a, ego non tam illa, quàm ex illis vt <lb/>tuis, cætera &longs;uperfluè impugnauerim; quia, inquam <lb/>num i&longs;ta &longs;e verè, fal&longs;óve habeant, cogno&longs;cendum e&longs;t <lb/>ex ijs, quæ à te &longs;ubiiciuntur; idcircò nihil e&longs;t, <lb/>

<arrow.to.target n="fig38"></arrow.to.target><lb/>quod ad i&longs;ta regeri generatim debeat. </s>

<figure id="fig38"></figure><lb/>quod ad i&longs;ta regeri generatim debeat. </s>

<s><emph type="italics"/>Atque in primis, eandem iterùm mihi Propo&longs;itio<lb/>nem affingis, quam iam &longs;upra numero<emph.end type="italics"/> 7. <emph type="italics"/>indicaui, <lb/>& in hunc locum examinandam reieci. Vis enim, <lb/>& &longs;iue demon&longs;tratione,s&s&longs;ine vlla vel probabili ratione <lb/>ais, & qua&longs;i pro tuo iure &longs;upponis,<emph.end type="italics"/> graue de&longs;cen<lb/>dens per &longs;patium AB in parteis quotlibet <lb/>æqualeis diui&longs;um, percurrere partem &longs;ecundam <lb/>DE, in dimidio eius temporis, quo percurritur <lb/>prior pars AD, <emph type="italics"/>& quod idem est,<emph.end type="italics"/> partem DE per<lb/>curri velocitate dupla eius velocitatis, qua tran&longs;<lb/>curritur AD. </s>

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<s><emph type="italics"/>Vt porrò ex tuis etiam, &<emph.end type="italics"/> G<emph type="italics"/>alilei principiis rem euincam, <lb/>& eapropemodum omnia, quæ tam acriter nobis obiectas, & <lb/>tanquam ab&longs;urda tantopere in&longs;ectaris, tibi nobi&longs;cum e&longs;&longs;e <lb/>communia demon&longs;trem. Concipe in triangulo AB<emph.end type="italics"/>C, <emph type="italics"/>& <lb/>in eius latere AC parteis æqualeis non iam<emph.end type="italics"/><lb/>

<arrow.to.target n="fig39"></arrow.to.target><lb/><emph type="italics"/>&longs;patij, &longs;ed temporis<emph.end type="italics"/> AD, DE, EF, <emph type="italics"/>&c. <lb/>& in<emph.end type="italics"/> D <emph type="italics"/>acqui&longs;itum quem volueris gra<lb/>dum velocitatis, expre&longs;&longs;um linea<emph.end type="italics"/> DG; <emph type="italics"/>erit <lb/>ex tuis, &<emph.end type="italics"/> G<emph type="italics"/>alilei principiis acqui&longs;itus in<emph.end type="italics"/><lb/>E <emph type="italics"/>gradus &longs;ecundus expre&longs;&longs;us linea<emph.end type="italics"/> EH, <emph type="italics"/>&longs;ic<lb/>que velocitas in<emph.end type="italics"/> E, <emph type="italics"/>dupla erit velocitatis in <lb/>D.<emph.end type="italics"/></s>

<figure id="fig39"></figure><lb/><emph type="italics"/>&longs;patij, &longs;ed temporis<emph.end type="italics"/> AD, DE, EF, <emph type="italics"/>&c. <lb/>& in<emph.end type="italics"/> D <emph type="italics"/>acqui&longs;itum quem volueris gra<lb/>dum velocitatis, expre&longs;&longs;um linea<emph.end type="italics"/> DG; <emph type="italics"/>erit <lb/>ex tuis, &<emph.end type="italics"/> G<emph type="italics"/>alilei principiis acqui&longs;itus in<emph.end type="italics"/><lb/>E <emph type="italics"/>gradus &longs;ecundus expre&longs;&longs;us linea<emph.end type="italics"/> EH, <emph type="italics"/>&longs;ic<lb/>que velocitas in<emph.end type="italics"/> E, <emph type="italics"/>dupla erit velocitatis in <lb/>D.<emph.end type="italics"/></s>

<s>Quamobrem igitur aggre&longs;&longs;us ar<lb/>gumentari ad hominem, & demon<lb/>&longs;trare tanta volens aduer&longs;us Galileum, <lb/>ac me, in ip&longs;o limineaberras? Scilicet neque Galilei,

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<s>Nego, <emph type="italics"/>videre me perinde, atque eadem ratione.<emph.end type="italics"/> Nam <lb/>in mea quidem, & Galilei &longs;ententia &longs;umendo has <lb/>parteis DE, AD, SD pro partibus <lb/>

<arrow.to.target n="fig40"></arrow.to.target><lb/>temporis, res e&longs;t nece&longs;&longs;aria; at in tua, <lb/>&longs;umendo ea&longs;dem pro partibus &longs;patij, <lb/>o&longs;ten&longs;um iam e&longs;t velocitatem per DE <lb/>debere e&longs;&longs;e duplam velocitatis per AD. <lb/>Et o&longs;tendetur paulò pò&longs;t non probari <lb/>duplam velocitatis per SD, ni&longs;i ip&longs;um<lb/>met petendo principium. </s>

<figure id="fig40"></figure><lb/>temporis, res e&longs;t nece&longs;&longs;aria; at in tua, <lb/>&longs;umendo ea&longs;dem pro partibus &longs;patij, <lb/>o&longs;ten&longs;um iam e&longs;t velocitatem per DE <lb/>debere e&longs;&longs;e duplam velocitatis per AD. <lb/>Et o&longs;tendetur paulò pò&longs;t non probari <lb/>duplam velocitatis per SD, ni&longs;i ip&longs;um<lb/>met petendo principium. </s>

<s><emph type="italics"/>Eadem quoque ratione cogéris fateri ve<lb/>locitatem tertio tempore<emph.end type="italics"/> EF <emph type="italics"/>acqui&longs;itam ve<lb/>locitatis primo tempore acqui&longs;itæ triplam &longs;o<lb/>lùm non e&longs;&longs;e, &longs;ed triplo maiorem.<emph.end type="italics"/></s>

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<s>A&longs;&longs;umptâ illeic lineâ AB, diui&longs;a in parteis æqua<lb/>leis AD, DE, &c. ac prima parte bi&longs;ecti in S, <lb/>

<arrow.to.target n="fig41"></arrow.to.target><lb/>expre&longs;&longs;a &longs;ententia hi&longs;ce verbis ex&longs;tat <emph type="italics"/>Tota<emph.end type="italics"/> DE <lb/><emph type="italics"/>eodem præcisè tempore, quo pars<emph.end type="italics"/> SD <emph type="italics"/>tran&longs;curritur.<emph.end type="italics"/><lb/>Tum probatio hæc additur, <emph type="italics"/>Cùm enim<emph.end type="italics"/> AD <emph type="italics"/>du<lb/>pla ponatur ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>& &longs;imiliter<emph.end type="italics"/> AE <emph type="italics"/>dupla &longs;it <lb/>ip&longs;ius<emph.end type="italics"/> AD, <emph type="italics"/>nece&longs;&longs;e e&longs;t, vt velocitas in<emph.end type="italics"/> D, <emph type="italics"/>dupla &longs;it <lb/>velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& velocitas in<emph.end type="italics"/> E <emph type="italics"/>eodem modo du<lb/>pla reperiatur velocitatis in D.<emph.end type="italics"/> Deducis con&longs;e<lb/>quenter velocitatem per totam DE e&longs;&longs;e du<lb/>plam velocitatis per totam SD: &longs;ed quod caput <lb/>e&longs;t, peruideamus. Nece&longs;&longs;e dicis <emph type="italics"/>velocitatem in<emph.end type="italics"/><lb/>D, <emph type="italics"/>e&longs;&longs;e duplam velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& velocitatem in<emph.end type="italics"/> E <lb/><emph type="italics"/>velocitatis in<emph.end type="italics"/> D, eo argumento, <emph type="italics"/>quòd<emph.end type="italics"/> AD <emph type="italics"/>dupla <lb/>&longs;it ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>&<emph.end type="italics"/> AE <emph type="italics"/>ip&longs;ius<emph.end type="italics"/> AD. Quomodo <lb/>ergo nece&longs;&longs;itatem huius con&longs;equutionis pro<lb/>bas? Nam hoc opus, hic labor e&longs;t. Quomodò, <lb/>inquam, ex eo, quòd <emph type="italics"/>spatium<emph.end type="italics"/> AD <emph type="italics"/>duplum &longs;it spatij<emph.end type="italics"/><lb/>AS, <emph type="italics"/>&longs;patium<emph.end type="italics"/> AE <emph type="italics"/>spatij<emph.end type="italics"/> AD, &longs;equi nece&longs;&longs;ariò, <emph type="italics"/>vt <lb/>velocitas in D dupla &longs;it velocitatis in S, & velocitas in<emph.end type="italics"/>

<figure id="fig41"></figure><lb/>expre&longs;&longs;a &longs;ententia hi&longs;ce verbis ex&longs;tat <emph type="italics"/>Tota<emph.end type="italics"/> DE <lb/><emph type="italics"/>eodem præcisè tempore, quo pars<emph.end type="italics"/> SD <emph type="italics"/>tran&longs;curritur.<emph.end type="italics"/><lb/>Tum probatio hæc additur, <emph type="italics"/>Cùm enim<emph.end type="italics"/> AD <emph type="italics"/>du<lb/>pla ponatur ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>& &longs;imiliter<emph.end type="italics"/> AE <emph type="italics"/>dupla &longs;it <lb/>ip&longs;ius<emph.end type="italics"/> AD, <emph type="italics"/>nece&longs;&longs;e e&longs;t, vt velocitas in<emph.end type="italics"/> D, <emph type="italics"/>dupla &longs;it <lb/>velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& velocitas in<emph.end type="italics"/> E <emph type="italics"/>eodem modo du<lb/>pla reperiatur velocitatis in D.<emph.end type="italics"/> Deducis con&longs;e<lb/>quenter velocitatem per totam DE e&longs;&longs;e du<lb/>plam velocitatis per totam SD: &longs;ed quod caput <lb/>e&longs;t, peruideamus. Nece&longs;&longs;e dicis <emph type="italics"/>velocitatem in<emph.end type="italics"/><lb/>D, <emph type="italics"/>e&longs;&longs;e duplam velocitatis in<emph.end type="italics"/> S, <emph type="italics"/>& velocitatem in<emph.end type="italics"/> E <lb/><emph type="italics"/>velocitatis in<emph.end type="italics"/> D, eo argumento, <emph type="italics"/>quòd<emph.end type="italics"/> AD <emph type="italics"/>dupla <lb/>&longs;it ip&longs;ius<emph.end type="italics"/> AS, <emph type="italics"/>&<emph.end type="italics"/> AE <emph type="italics"/>ip&longs;ius<emph.end type="italics"/> AD. Quomodo <lb/>ergo nece&longs;&longs;itatem huius con&longs;equutionis pro<lb/>bas? Nam hoc opus, hic labor e&longs;t. Quomodò, <lb/>inquam, ex eo, quòd <emph type="italics"/>spatium<emph.end type="italics"/> AD <emph type="italics"/>duplum &longs;it spatij<emph.end type="italics"/><lb/>AS, <emph type="italics"/>&longs;patium<emph.end type="italics"/> AE <emph type="italics"/>spatij<emph.end type="italics"/> AD, &longs;equi nece&longs;&longs;ariò, <emph type="italics"/>vt <lb/>velocitas in D dupla &longs;it velocitatis in S, & velocitas in<emph.end type="italics"/>

<pb pagenum="184"/><emph type="italics"/>E, velocitatis in D?<emph.end type="italics"/> Id &longs;anè non probas, &longs;ed &longs;olùm <lb/>&longs;upponis ex eo vim habere, <emph type="italics"/>quòd velocitates &longs;icut spa<lb/>tia &longs;int.<emph.end type="italics"/> Atqui hæc ip&longs;a e&longs;t controuer&longs;ia, <emph type="italics"/>an velocita<lb/>tes inter &longs;e &longs;icut &longs;patia &longs;int:<emph.end type="italics"/> neque idip&longs;um, quod opus <lb/>e&longs;t, probas, &longs;ed omninò principium petis, &longs;eu ip&longs;am <lb/>quæ&longs;tionem, remve controuer&longs;am pro principio a&longs;&longs;u<lb/>mis. An verò dices te proba&longs;&longs;e, <emph type="italics"/>velocitates e&longs;&longs;e vt &longs;pa<lb/>tia?<emph.end type="italics"/> Sed quæ&longs;o, vbi-nam? Id enim nu&longs;quam de<lb/>prehendetur? An vbi volui&longs;ti arguere Galileum Pa<lb/>ralogi&longs;mi? At abundè iam <expan abbr="declarat&utilde;">declaratum</expan> e&longs;t te potiùs v&longs;um <lb/>paralogi&longs;mo, cùm rur&longs;us nihil aliud, quàm principium <lb/>petieris: vt taceam te, &longs;i quid habui&longs;ti probationis, id<lb/>po&longs;teà nega&longs;&longs;e: quando nega&longs;ti <emph type="italics"/>&longs;i prima pars spatij per<lb/>curratur quadrante, percurri &longs;ecundam dimidio quadrantis,<emph.end type="italics"/><lb/>An vbi demùm protuli&longs;ti <emph type="italics"/>tuum de Bilance Experimen<lb/>tum?<emph.end type="italics"/> Certè i&longs;ta vna probatio fuit tua, neque potes pro<lb/>ferre locum, in quo alia ratione probaueris <emph type="italics"/>velocitates <lb/>&longs;e habere vt spatia.<emph.end type="italics"/> Atqui & fal&longs;um deprehen&longs;um Ex<lb/>perimentum tuum e&longs;t; & quæ germana fuit Experien<lb/>tia fal&longs;um conuicit <emph type="italics"/>velocitates &longs;e inter &longs;e habere vt <lb/>spatia.<emph.end type="italics"/> Itaque ex his &longs;equitur, non modo nece&longs;&longs;e non <lb/>e&longs;&longs;e: &longs;ed fal&longs;um etiam, atque adeò prorsùs impo&longs;&longs;ibile, <lb/>vt <emph type="italics"/>velocitas in D, dupla &longs;it velocitatis in S, & velocitas in <lb/>E, velocitatis in D.<emph.end type="italics"/></s>

<s>Cùm ergo non modò probatum non &longs;it, &longs;ed fal<lb/>&longs;um quoque & impo&longs;&longs;ibile ip&longs;a Experientia &longs;uffra<lb/>gante o&longs;ten&longs;um, <emph type="italics"/>velocitatem tota &longs;ecunda parte<emph.end type="italics"/> DE <lb/><emph type="italics"/>acqui&longs;itam pr&ecedil;cisè duplam e&longs;&longs;e totius velocitatis per inferio<lb/>rem primæ partis medietatem<emph.end type="italics"/> SD, <emph type="italics"/>acqui&longs;itæ, &longs;imiliterque, &c.<emph.end type="italics"/><lb/>Quæ&longs;o quî adhûc valeas talem &longs;ententiam defendere,

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<s><emph type="italics"/>Numero &longs;equente xxxviij. quicquid concludis, ex du<lb/>plici fal&longs;a hypothe&longs;i &longs;imiliter concludis. Vna e&longs;t, quód ve<lb/>locitates totæ, atque integræ quibu&longs;libet partibus acqui&longs;itæ <lb/>eam inter &longs;e rationem ob&longs;eruent, quam partes ip&longs;<gap/> quibus<emph.end type="italics"/>

<pb pagenum="188"/><emph type="italics"/>&longs;unt acqui&longs;itæ; atque ita, vt quemadmodum<emph.end type="italics"/> AE <emph type="italics"/>dupla e&longs;t <lb/>ip&longs;ius<emph.end type="italics"/> AD, <emph type="italics"/>ita velocitas acqui&longs;ita per totam<emph.end type="italics"/> AE, <emph type="italics"/>præ-<emph.end type="italics"/><lb/>

<arrow.to.target n="fig42"></arrow.to.target><lb/><emph type="italics"/>cisè dupla &longs;it totius velocitatis acqui&longs;itæ per<emph.end type="italics"/> AD; <lb/><emph type="italics"/>quod tamen ex tuis, &<emph.end type="italics"/> G<emph type="italics"/>alilei principijs fal&longs;um e&longs;&longs;e <lb/>&longs;uperiùs numero<emph.end type="italics"/> 6. <emph type="italics"/>euici.<emph.end type="italics"/></s>

<figure id="fig42"></figure><lb/><emph type="italics"/>cisè dupla &longs;it totius velocitatis acqui&longs;itæ per<emph.end type="italics"/> AD; <lb/><emph type="italics"/>quod tamen ex tuis, &<emph.end type="italics"/> G<emph type="italics"/>alilei principijs fal&longs;um e&longs;&longs;e <lb/>&longs;uperiùs numero<emph.end type="italics"/> 6. <emph type="italics"/>euici.<emph.end type="italics"/></s>

<s>Qaid euiceris, & cuiu&longs;-nam fuerit hypothe<lb/>&longs;is fal&longs;a, memini&longs;&longs;e potes. </s>

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<s>Quàm egregiè elaberis? Ego enim cætera inter, <lb/>quæ non attigi&longs;ti, ex eo, quòd priùs admi&longs;i&longs;&longs;es, pro<lb/>pter progre&longs;&longs;ionem Arithmeticam, duorum graduum <lb/>acqui&longs;itorum in D, vnum acqui&longs;itum ab A, in C, alte<lb/>rum à C, in D, conclu&longs;i te velle velocitatem acqui&longs;i<lb/>tam ab N in C, e&longs;&longs;e æqualem acqui&longs;itæ à C, in D, <lb/>quandoquidem tum velles ip&longs;am CD tanto tempore, <lb/>quanto NC, cuius e&longs;&longs;et dupla, percurri. Et quia vi<lb/>deris non attendi&longs;&longs;e ad con&longs;equutionem, quæ &longs;atis

<pb pagenum="201"/>tamen in&longs;inuabatur in duobus con&longs;ectariis, ecce quo<lb/>modò ea probetur. NC, & CD debent, per <lb/>

<arrow.to.target n="fig43"></arrow.to.target><lb/>te, e&longs;&longs;e æquales; quoniam vtraque, per te, de<lb/>bet e&longs;&longs;e æqualis lineæ AC: quare &, &longs;i æquali <lb/>percurrantur tempore, æquali percurrentur ve<lb/>locitate. Et de æqualitate quidem lineæ CD, <lb/>cum AC, non e&longs;t dubium; probo de NC, AC, & <lb/>NC eadem velocitate &longs;ecundum &longs;e totas per<lb/>curruntur, ex te; Ergo &longs;unt inter &longs;e æquales. Pro <lb/>batur antecedens. Ad velocitatem acqui&longs;itam <lb/>per totam AC, addita quæ acquiritur per CD, <lb/>facit velocitatem in D duplam velocitatis in <lb/>C; ad volocitatem acqui&longs;itam per totam NC, <lb/>addita quæ acquiritur per candem CD, facit <lb/>itidem velocitatem in D, velocitatis in C du<lb/>plam; & vtrumque quidem per te. Igitur per te <lb/>AC, & NC, &longs;ecundum &longs;e totas velocitate ea<lb/>dem percurruntur; &longs;untque idcircò inter &longs;e <lb/>æquales. Tu ergo vim con&longs;equutionis ex inte<lb/>gro antecedente diffimulans, infers &longs;olùm ex <lb/>duplo &longs;patij, & temporis æqualitate, velocitates <lb/>e&longs;&longs;e vt &longs;patia: ac retices quod intere&longs;t, quodque <lb/>&longs;ubnotatum fuit, cùm &longs;ubiunxi, <emph type="italics"/>con&longs;equi, vt <lb/>velocitates per AC, & NC, acqui&longs;itæ &longs;int pror&longs;us <lb/>æquales: ac proinde vt AC, & NC, hoc est totum, & <lb/>pars eodem tempore percurrantur, &c.<emph.end type="italics"/> Sequitur. </s>

<figure id="fig43"></figure><lb/>te, e&longs;&longs;e æquales; quoniam vtraque, per te, de<lb/>bet e&longs;&longs;e æqualis lineæ AC: quare &, &longs;i æquali <lb/>percurrantur tempore, æquali percurrentur ve<lb/>locitate. Et de æqualitate quidem lineæ CD, <lb/>cum AC, non e&longs;t dubium; probo de NC, AC, & <lb/>NC eadem velocitate &longs;ecundum &longs;e totas per<lb/>curruntur, ex te; Ergo &longs;unt inter &longs;e æquales. Pro <lb/>batur antecedens. Ad velocitatem acqui&longs;itam <lb/>per totam AC, addita quæ acquiritur per CD, <lb/>facit velocitatem in D duplam velocitatis in <lb/>C; ad volocitatem acqui&longs;itam per totam NC, <lb/>addita quæ acquiritur per candem CD, facit <lb/>itidem velocitatem in D, velocitatis in C du<lb/>plam; & vtrumque quidem per te. Igitur per te <lb/>AC, & NC, &longs;ecundum &longs;e totas velocitate ea<lb/>dem percurruntur; &longs;untque idcircò inter &longs;e <lb/>æquales. Tu ergo vim con&longs;equutionis ex inte<lb/>gro antecedente diffimulans, infers &longs;olùm ex <lb/>duplo &longs;patij, & temporis æqualitate, velocitates <lb/>e&longs;&longs;e vt &longs;patia: ac retices quod intere&longs;t, quodque <lb/>&longs;ubnotatum fuit, cùm &longs;ubiunxi, <emph type="italics"/>con&longs;equi, vt <lb/>velocitates per AC, & NC, acqui&longs;itæ &longs;int pror&longs;us <lb/>æquales: ac proinde vt AC, & NC, hoc est totum, & <lb/>pars eodem tempore percurrantur, &c.<emph.end type="italics"/> Sequitur. </s>

<s><emph type="italics"/>Numero xli. etiam peccas; cùm, vt impugnes con&longs;titu<lb/>tam à me cæterarum partium toto spatio decurrendo de&longs;igna<lb/>tarum respectum ad analoga primæ partis &longs;egmenta, non &longs;o<lb/>lùm earumdem partium re&longs;pectum ad parteis temporis exigis,<emph.end type="italics"/>

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<s>Ego verò nihil affinxi; &longs;ed idip&longs;um fuit, quod de<lb/>duxeram articulo XL. ea parte, quam non attigi&longs;ti:

<pb pagenum="206"/>o&longs;tendendo videlicet, cùm CN foret duorum minu<lb/>torum, & CD illius dupla, &longs;imiliter minutorum <lb/>

<arrow.to.target n="fig44"></arrow.to.target><lb/>duorum, atque ideò tota ND, minutorum qua<lb/>tuor: forè vt AN ex tuis principiis minutorum <lb/>quatuor, & eius tripla ND, tempore eodem <lb/>percurrerentur. </s>

<figure id="fig44"></figure><lb/>duorum, atque ideò tota ND, minutorum qua<lb/>tuor: forè vt AN ex tuis principiis minutorum <lb/>quatuor, & eius tripla ND, tempore eodem <lb/>percurrerentur. </s>

<s><emph type="italics"/>Sed cùm iis partibus omißis, rectè compares<emph.end type="italics"/> XC <lb/><emph type="italics"/>vltimum trientem &longs;upremæ partis<emph.end type="italics"/> AC <emph type="italics"/>cum tertia <lb/>parte<emph.end type="italics"/> DE: <emph type="italics"/>non rectè con&longs;equenter hanc tertiam par<lb/>tem comparas cum tribus &longs;equentibus<emph.end type="italics"/> EH, <emph type="italics"/>quæ non <lb/>eodem tempore, aut æquali cum tertia parte<emph.end type="italics"/> DE, <emph type="italics"/>&longs;ed <lb/>tempore longiore percurruntur. At rectiùs cum ea<lb/>dem tertia parte<emph.end type="italics"/> DE <emph type="italics"/>compararentur &longs;eptima, octa<lb/>ua, & nona, nempe<emph.end type="italics"/> HL: <emph type="italics"/>&longs;ed talis progreßio per to<lb/>tum &longs;patium decurrendum continua non e&longs;t, vt vides: <lb/>interrupta enim primùm e&longs;t inter<emph.end type="italics"/> XC, <emph type="italics"/>&<emph.end type="italics"/> DE, <emph type="italics"/>& <lb/>inde ab<emph.end type="italics"/> E <emph type="italics"/>ad<emph.end type="italics"/> H: <emph type="italics"/>& &longs;i vlteriùs procedendum e&longs;&longs;et, tum <lb/>à nona parte interrumperetur v&longs;que ad decimam octa<lb/>uam, & ita deinceps: progreßio autem in ratione dupla <lb/>&longs;ola per totum spatium continua e&longs;t.<emph.end type="italics"/></s>

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<s>Cum nihil habeas circa alios, qui heinc ad con<lb/>e&longs;u&longs;ionem v&longs;que inter&longs;eruntur, articulos, & ego in ip<lb/>&longs;is, occa&longs;ione Phy&longs;icæ cau&longs;&longs;æ, de qua te dicturum alio <lb/>loco, temporéque receperas, in&longs;inuâtim lap&longs;um, à quo <lb/>non &longs;atis caueram, in priore Epi&longs;tolatum de Motu im<lb/>pre&longs;&longs;o à motore tran&longs;lato (imò & in Epi&longs;tola ante duos <lb/>annos ad te data) eam ob rem heic locus erit opportu<lb/>nus, vt rem paulò fu&longs;iùs deducam, ac memetip&longs;um cor<lb/>rigam, dicturus &longs;imùl de Phy&longs;ica cau&longs;&longs;a, quam tu pla<lb/>cere tibi dixi&longs;ti in articulum XXXV. Itaque, cùm ego <lb/>demirarer illam accelerati motus progre&longs;&longs;ionem &longs;e<lb/>cundum &longs;eriem numerorum ab vnitate imparium, Ra<lb/>tiocinatus &longs;um imprimis, lapidem v.g. non habere ex <lb/>&longs;e huiu&longs;modi motum; quoniam &longs;i &longs;olus lapis non ex<lb/>&longs;i&longs;tente Mundo foret, aut aliàs in vacuo &longs;ic con&longs;titue<lb/>retur, vt nihil cum Mundo communicaret; non e&longs;&longs;et <lb/>prorsùs, quamobrem eò, quò iam mouetur tenderet; <lb/>vnde & conclu&longs;i cau&longs;&longs;am huiu&longs;modi motus e&longs;&longs;e debe<lb/>re extrin&longs;ecam, ip&longs;amque aut impellentem, aut attra-

<pb pagenum="210"/>hentem, aut vtramque &longs;imùl. Secundò, vi&longs;um mihi <lb/>e&longs;t a&longs;&longs;umi po&longs;&longs;e aërem, qui infernè pre&longs;&longs;us, re&longs;ilien&longs;<lb/>que ad latera, in locum po&longs;terum &longs;uccederet, &longs;uperné<lb/>que in&longs;taret, pro cau&longs;&longs;a impellente; & magneticam Ter<lb/>ræ vim, quæ hamulorum, catenularumque in&longs;tar, in<lb/>fernè pelliceret, pro cau&longs;&longs;a trahente. Tertiò vi&longs;um e&longs;t <lb/>aërem non po&longs;&longs;e inchoare motum, quatenus &longs;i nulla <lb/>e&longs;&longs;et Terra, & &longs;olus aër in vniuer&longs;o, & ip&longs;e quidem in<lb/>finitè fu&longs;us, vnaque lapis intra ip&longs;um, tunc lapis in <lb/>hanc potiùs partem, quàm in aliam non moueretur: <lb/>quare & motum lapidis videri debere ab attractione <lb/>incipere, ac tum po&longs;&longs;e ab aëre &longs;uccedente adiutari. <lb/>Quartò, cùm fui&longs;&longs;em ratiocinatus motum &longs;emel im<lb/>pre&longs;&longs;um futurum perpetuum, & æquabilem, ni&longs;i e&longs;&longs;et <lb/>cau&longs;&longs;a, quæ illum aut retundendo minueret, aut vrgen<lb/>do acceleraret; conclu&longs;i motum lapidis accelerari, quòd <lb/>&longs;tatim à prima attractione, & qua&longs;i po&longs;t primum <expan abbr="ict&utilde;">ictum</expan>, <lb/>&longs;uccederent continuò ictus alij, qui impre&longs;&longs;ione facta, <lb/>manenteque facerent motum celeriorem. Quintò vi<lb/>&longs;um e&longs;t neque attractionem &longs;olam, neque impul&longs;io<lb/>nem &longs;olam, neque vllam aliam &longs;implicem cau&longs;&longs;am &longs;uf<lb/>ficere ad memoratam progre&longs;&longs;ionem: quia cùm hoc <lb/>modo ictus continenter facti progre&longs;&longs;uri e&longs;&longs;ent &longs;ecun<lb/>dum vnitatum &longs;eriem, con&longs;entaneum videbatur, vt & <lb/>velocitatis gradus acquirerentur, & &longs;patia quoque per<lb/>currerentur &longs;ecundum eandem &longs;eriem: &longs;icque in fine <lb/>cuiu&longs;que momenti &longs;patia aggregata non forent vt <lb/>quadrata temporum, vnum, quatuor, nouem, &longs;ex de<lb/>cim, & quadrata cætera; &longs;ed vt vnum tria, &longs;ex, decem, <lb/>& con&longs;equentia aggregata. Sextò vi&longs;um e&longs;t ergo

<pb pagenum="211"/>potiùs vtramque cau&longs;&longs;am &longs;ic coniungendam, vt pri<lb/>mo momento &longs;ola Terra vi attractice ageret, vnumque <lb/>ictum imprimeret: vnde & vnus gradus velocitatis im<lb/>primeretur, quo mobile certum &longs;uperaret &longs;patium: &longs;e<lb/>cundo autem momento tum Terra attrahere pergeret, <lb/>tum aër pellere inciperet; &longs;icque duo, ex duobus velut <lb/>ictibus, noui e&longs;&longs;ent velocitatis gradus, qui cum primo <lb/>manente e&longs;&longs;ent tres, vnde & tria &longs;patia primo æqualia <lb/>percurrerentur; & ita porrò continenter. Po&longs;tremò, <lb/>cùm mihi viderer cau&longs;&longs;am dicere, quamobrem &longs;patia <lb/>percurrerentur &longs;ecundum &longs;eriem numerorum ab vni<lb/>tate imparium, & aggregata &longs;patiorum e&longs;&longs;ent &longs;icut <lb/>quadrata temporum; rem totam &longs;ic expo&longs;ui, repetito <lb/>iam aliquoties triangulo, vt partes æquales alterutrius <lb/>cturum pro momentis, &longs;eu partibus æqualibus temporis <lb/>habens, intercepta <lb/>

<arrow.to.target n="fig45"></arrow.to.target><lb/>illa ip&longs;i triangula <lb/>per interductas li<lb/>neas partim ba&longs;i, <lb/>partim cruribus pa<lb/>rallelas creata, ha<lb/>beri po&longs;&longs;e &longs;imùl <lb/>cen&longs;uerim, & pro <lb/>partibus æquali<lb/>bus &longs;patij decur&longs;i; <lb/>& pro æqualibus <lb/>gradibus veloci<lb/>tatis acqui&longs;itæ. </s>

<figure id="fig45"></figure><lb/>illa ip&longs;i triangula <lb/>per interductas li<lb/>neas partim ba&longs;i, <lb/>partim cruribus pa<lb/>rallelas creata, ha<lb/>beri po&longs;&longs;e &longs;imùl <lb/>cen&longs;uerim, & pro <lb/>partibus æquali<lb/>bus &longs;patij decur&longs;i; <lb/>& pro æqualibus <lb/>gradibus veloci<lb/>tatis acqui&longs;itæ. </s>

<s>Iam lap&longs;us fuit, quatenus proinde velocitates vt <lb/>&longs;patia habere &longs;e admi&longs;i imprudens. Quia enim non

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<s>Ego verò &longs;anctè affirmo, nihil me vnquam remi&longs;<lb/>&longs;urum neque de &longs;umma veneratione, qua te pro&longs;e<lb/>quor, neque de &longs;incero affectu, quo toto ex corde &longs;um <lb/>Tuus. Vale. Pari&longs;ijs, Non. Maij, M. DC. XLV. <lb/>

<arrow.to.target n="fig46"></arrow.to.target></s>

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<s><emph type="center"/>FINIS.<emph.end type="center"/><lb/>

<arrow.to.target n="fig47"></arrow.to.target>

<s><emph type="center"/>VIRO Nobili&longs;&longs;imo PHILIBERTO DE LA MARE <lb/>Senatori Diuionen&longs;i, <emph type="italics"/>P.GASSENDVS S.<emph.end type="italics"/><emph.end type="center"/></s>

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