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| <s>PARAPHRASIS <lb/>Scholijs illu&longs;trata.</s></p> | <s>PARAPHRASIS <lb/>Scholijs illu&longs;trata.</s></p> |
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| <s>PISAVRI <lb/>Apud Hieronymum Concordiam; <lb/>M D LXXXVIII. <lb/><emph type="italics"/>Superiorum Conce&longs;&longs;<gap/>.<emph.end type="italics"/></s></p> | <s>PISAVRI <lb/>Apud Hieronymum Concordiam; <lb/>M D LXXXVIII. <lb/><emph type="italics"/>Superiorum Conce&longs;&longs;u.<emph.end type="italics"/></s></p> |
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| <s>SERENISSIMO <lb/>FRANC.^{CO} MARIAE <lb/>II. VRBINI DVCI.</s></p> | <s>SERENISSIMO <lb/>FRANC.^{CO} MARIAE <lb/>II. VRBINI DVCI.</s></p> |
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| <s>Cæterùm ad meliorem horum notitiam ob&longs;eruandum e&longs;t, <lb/>h&ecedil;c centra aliquando &longs;imul omnia inter &longs;e conuenire, <expan abbr="aliquã">aliquam</expan> <lb/>do nonnulla; aliquando autem minimè. </s><s>&longs;imul verò omnia <lb/>conueniunt. </s><s>vt centrum vniuer&longs;i, centrum magnitudinis ter <lb/>ræ (&longs;ph&ecedil;ræ &longs;cilicet ex aqua, terraquè compo&longs;it&ecedil;, quam nos bre <lb/>uitatis &longs;tudio terram tantùm nuncupabimus) centrum figu­<lb/>r&ecedil; terr&ecedil;; ac centrum grauitatis terr&ecedil;. </s><s>Cùm enim terra &longs;it &longs;phæ­<lb/>rica (vt omnes fatentur.) eius medium erit centrum figur&ecedil;, à <lb/>quo &longs;emidiam etri exeunt. </s><s>idip&longs;um què erit centrum magnitu <lb/>dinis, &longs;iquidem ip&longs;ius figur&ecedil; medium obtinet. </s><s>Pr&ecedil;terea idem <lb/>punctum e&longs;t centrum grauitatis terr&ecedil;. </s><s>& quoniam terra in me <lb/>dio <expan abbr="mūdi">mundi</expan> quie&longs;cit, erit hoc <expan abbr="centrũ">centrum</expan> grauitatis in centro vniuer&longs;i <lb/>collocatum. </s><s>& hoc dun taxat modo centra omnia in <expan abbr="vnũ">vnum</expan> con <lb/>uenire po&longs;&longs;unt. </s><s>quamquam verò &longs;ph&ecedil;ra, qu&ecedil; continet <expan abbr="terrā">terram</expan> & <lb/>aqu&acedil;, compo&longs;ita e&longs;t ex corporibus diuer&longs;&ecedil; &longs;peciei, <expan abbr="differ&etilde;ti&longs;què">differenti&longs;què</expan> <lb/>grauitatis, nimirum ex terra, & aqua; non <expan abbr="tam&etilde;">tamen</expan> efficitur, quin <lb/><expan abbr="mediũ">medium</expan> ip&longs;ius cum centro grauitatis con&longs;piret in vnum. <expan abbr="Nã">Nam</expan> ex <lb/>Ari&longs;to telis &longs;ententia terra circa mundi centrum vn dique <expan abbr="cõ&longs;i">con&longs;i</expan> | <s>Cæterùm ad meliorem horum notitiam ob&longs;eruandum e&longs;t, <lb/>h&ecedil;c centra aliquando &longs;imul omnia inter &longs;e conuenire, <expan abbr="aliquã">aliquam</expan> <lb/>do nonnulla; aliquando autem minimè. </s><s>&longs;imul verò omnia <lb/>conueniunt. </s><s>vt centrum vniuer&longs;i, centrum magnitudinis ter <lb/>ræ (&longs;ph&ecedil;ræ &longs;cilicet ex aqua, terraquè compo&longs;it&ecedil;, quam nos bre <lb/>uitatis &longs;tudio terram tantùm nuncupabimus) centrum figu­<lb/>r&ecedil; terr&ecedil;; ac centrum grauitatis terr&ecedil;. </s><s>Cùm enim terra &longs;it &longs;phæ­<lb/>rica (vt omnes fatentur.) eius medium erit centrum figur&ecedil;, à <lb/>quo &longs;emidiam etri exeunt. </s><s>idip&longs;um què erit centrum magnitu <lb/>dinis, &longs;iquidem ip&longs;ius figur&ecedil; medium obtinet. </s><s>Pr&ecedil;terea idem <lb/>punctum e&longs;t centrum grauitatis terr&ecedil;. </s><s>& quoniam terra in me <lb/>dio <expan abbr="mūdi">mundi</expan> quie&longs;cit, erit hoc <expan abbr="centrũ">centrum</expan> grauitatis in centro vniuer&longs;i <lb/>collocatum. </s><s>& hoc dun taxat modo centra omnia in <expan abbr="vnũ">vnum</expan> con <lb/>uenire po&longs;&longs;unt. </s><s>quamquam verò &longs;ph&ecedil;ra, qu&ecedil; continet <expan abbr="terrā">terram</expan> & <lb/>aqu&acedil;, compo&longs;ita e&longs;t ex corporibus diuer&longs;&ecedil; &longs;peciei, <expan abbr="differ&etilde;ti&longs;què">differenti&longs;què</expan> <lb/>grauitatis, nimirum ex terra, & aqua; non <expan abbr="tam&etilde;">tamen</expan> efficitur, quin <lb/><expan abbr="mediũ">medium</expan> ip&longs;ius cum centro grauitatis con&longs;piret in vnum. <expan abbr="Nã">Nam</expan> ex <lb/>Ari&longs;to telis &longs;ententia terra circa mundi centrum vn dique <expan abbr="cõ&longs;i">con&longs;i</expan> |
| <arrow.to.target n="marg7"></arrow.to.target><lb/>&longs;tit; & Archimedes affirmat, <expan abbr="etiã">etiam</expan> <expan abbr="humidũ">humidum</expan> manens e&longs;&longs;e | <arrow.to.target n="marg7"></arrow.to.target><lb/>&longs;tit; & Archimedes affirmat, <expan abbr="etiã">etiam</expan> <expan abbr="humidũ">humidum</expan> manens e&longs;&longs;e |
| <arrow.to.target n="marg8"></arrow.to.target> <expan abbr="&longs;ph&ecedil;ri-cũ">&longs;ph&ecedil;ri­<lb/>cum</expan>, cuius <expan abbr="c&etilde;trum">centrum</expan> e&longs;t <expan abbr="centrũ">centrum</expan> vniuer&longs;i. </s><s>&longs;i ita que terra, & aqua ma <lb/><expan abbr="n&etilde;t">nent</expan>, <expan abbr="quie&longs;cũtquè">quie&longs;cuntquè</expan> circa <expan abbr="centrũ">centrum</expan> vniuer&longs;i, ergo <expan abbr="centrũ">centrum</expan> <expan abbr="mūdi">mundi</expan> <expan abbr="ip&longs;o-rũ">ip&longs;o­<lb/>rum</expan> &longs;imul <expan abbr="c&etilde;trũ">centrum</expan> grauitatis exi&longs;tit. </s><s>atque adeo quatuorpr&ecedil;dicta <lb/>centra in <expan abbr="vnũ">vnum</expan> &longs;imul conueniunt punctum. </s><s>Quod <expan abbr="aut&etilde;">autem</expan> tria &longs;i. <lb/>mul centra in vnum co<gap/>ant, &longs;atis <expan abbr="con&longs;picuū">con&longs;picuum</expan> e&longs;&longs;e poterit cuiquè | <arrow.to.target n="marg8"></arrow.to.target> <expan abbr="&longs;ph&ecedil;ri-cũ">&longs;ph&ecedil;ri­<lb/>cum</expan>, cuius <expan abbr="c&etilde;trum">centrum</expan> e&longs;t <expan abbr="centrũ">centrum</expan> vniuer&longs;i. </s><s>&longs;i ita que terra, & aqua ma <lb/><expan abbr="n&etilde;t">nent</expan>, <expan abbr="quie&longs;cũtquè">quie&longs;cuntquè</expan> circa <expan abbr="centrũ">centrum</expan> vniuer&longs;i, ergo <expan abbr="centrũ">centrum</expan> <expan abbr="mūdi">mundi</expan> <expan abbr="ip&longs;o-rũ">ip&longs;o­<lb/>rum</expan> &longs;imul <expan abbr="c&etilde;trũ">centrum</expan> grauitatis exi&longs;tit. </s><s>atque adeo quatuorpr&ecedil;dicta <lb/>centra in <expan abbr="vnũ">vnum</expan> &longs;imul conueniunt punctum. </s><s>Quod <expan abbr="aut&etilde;">autem</expan> tria &longs;i. <lb/>mul centra in vnum co<gap/>ant, &longs;atis <expan abbr="con&longs;picuū">con&longs;picuum</expan> e&longs;&longs;e poterit cuiquè |
| <pb pagenum="12"/>&longs;phæram aliquam, putà ligneam, vel al terius (&longs;imilaris <expan abbr="tam&etilde;">tamen</expan>) <lb/>naturæ intuenti; &longs;iquidem eius medium erit centrum magni­<lb/>tudinis, & centrum &longs;iguræ; idemquè punctum crit ip&longs;ius cen­<lb/> | <pb pagenum="12"/>&longs;phæram aliquam, putà ligneam, vel al terius (&longs;imilaris <expan abbr="tam&etilde;">tamen</expan>) <lb/>naturæ intuenti; &longs;iquidem eius medium erit centrum magni­<lb/>tudinis, & centrum figuræ; idemquè punctum crit ip&longs;ius cen­<lb/> |
| <arrow.to.target n="marg9"></arrow.to.target> trum grauitatis; circa quod vndique partes æqueponderant. <lb/>& quoniam hæc &longs;phæra non e&longs;t in centro mundi; propterea <lb/>tria tantùm centra &longs;imul conuenient. </s><s>&longs;i verò &longs;phçra non &longs;imi­<lb/>laris, &longs;ed di&longs;&longs;imilaris fuerit, veluti altera ip&longs;ius meditate plum­<lb/>bea, altera verò medietate lignea exi&longs;tente, tunc eius medium <lb/>erit quippe centrum magnitudinis, & figur&ecedil;, grauitatis verò <lb/>centrum nequaquam. </s><s>Nam partes vndique circa medium æ­<lb/>queponderare non po&longs;&longs;ent; &longs;ed grauitatis centrum ad grauio­<lb/>rem partem, nimirum plumbeam declinabit. </s><s>& hoc modo <lb/>duo tantùm centra inter &longs;e conuenient. </s><s>vt etiam (modo ta­<lb/>men diuer&longs;o) accidit ellip&longs;i; cuius centrum e&longs;t centrum figu­<lb/>r&ecedil;, &longs;iquidem per ip&longs;um tran&longs;eunt diametri; idemquè <expan abbr="punctũ">punctum</expan> <lb/> | <arrow.to.target n="marg9"></arrow.to.target> trum grauitatis; circa quod vndique partes æqueponderant. <lb/>& quoniam hæc &longs;phæra non e&longs;t in centro mundi; propterea <lb/>tria tantùm centra &longs;imul conuenient. </s><s>&longs;i verò &longs;phçra non &longs;imi­<lb/>laris, &longs;ed di&longs;&longs;imilaris fuerit, veluti altera ip&longs;ius meditate plum­<lb/>bea, altera verò medietate lignea exi&longs;tente, tunc eius medium <lb/>erit quippe centrum magnitudinis, & figur&ecedil;, grauitatis verò <lb/>centrum nequaquam. </s><s>Nam partes vndique circa medium æ­<lb/>queponderare non po&longs;&longs;ent; &longs;ed grauitatis centrum ad grauio­<lb/>rem partem, nimirum plumbeam declinabit. </s><s>& hoc modo <lb/>duo tantùm centra inter &longs;e conuenient. </s><s>vt etiam (modo ta­<lb/>men diuer&longs;o) accidit ellip&longs;i; cuius centrum e&longs;t centrum figu­<lb/>r&ecedil;, &longs;iquidem per ip&longs;um tran&longs;eunt diametri; idemquè <expan abbr="punctũ">punctum</expan> <lb/> |
| <arrow.to.target n="marg10"></arrow.to.target> e&longs;t ip&longs;ius centrum grauitatis. </s><s>quod cùm non &longs;it propriè me­<lb/>dium figuræ, non erit quoque centrum magnitudinis. <expan abbr="mediū">medium</expan> <lb/>enim figuræ propriè circulo, ac &longs;phæræ tantùm competit. <lb/>Quare duo centra hoc quoque modo &longs;imul tantùm conue­<lb/>nient. </s><s>In figura paraboles recta linea terminat&ecedil; centrum gra <lb/> | <arrow.to.target n="marg10"></arrow.to.target> e&longs;t ip&longs;ius centrum grauitatis. </s><s>quod cùm non &longs;it propriè me­<lb/>dium figuræ, non erit quoque centrum magnitudinis. <expan abbr="mediū">medium</expan> <lb/>enim figuræ propriè circulo, ac &longs;phæræ tantùm competit. <lb/>Quare duo centra hoc quoque modo &longs;imul tantùm conue­<lb/>nient. </s><s>In figura paraboles recta linea terminat&ecedil; centrum gra <lb/> |
| <arrow.to.target n="marg11"></arrow.to.target> uitatis intra figuram reperitur, quippè quod neque centrum <lb/>figuræ, neque centrum magnitudinis e&longs;&longs;e pote&longs;t. </s><s>etenim in <lb/>hac figura non pote&longs;t dari medium, vnde neque centrum ma <lb/>gnitudinis dabitur, & quoniam in parabole diametri &longs;unt in <lb/>ter&longs;e &ecedil;quidi&longs;tantes, vt ex primo libro conicorum Apollonij <lb/>pergei con&longs;tat; neque etiam centrum figuræ dabitur. </s><s>&longs;ic igi­<lb/>tur centra nullo modo conuenient. </s></p> | <arrow.to.target n="marg11"></arrow.to.target> uitatis intra figuram reperitur, quippè quod neque centrum <lb/>figuræ, neque centrum magnitudinis e&longs;&longs;e pote&longs;t. </s><s>etenim in <lb/>hac figura non pote&longs;t dari medium, vnde neque centrum ma <lb/>gnitudinis dabitur, & quoniam in parabole diametri &longs;unt in <lb/>ter&longs;e &ecedil;quidi&longs;tantes, vt ex primo libro conicorum Apollonij <lb/>pergei con&longs;tat; neque etiam centrum figuræ dabitur. </s><s>&longs;ic igi­<lb/>tur centra nullo modo conuenient. </s></p> |
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| <s>Noui&longs;&longs;e quoque oportet centrum grauitatis communius <lb/>e&longs;&longs;e, in pluribu&longs;què reperiri, quàm centra magnitudinis, & fi­<lb/>guræ: centrum verò figuræ communius e&longs;&longs;e centro magnitu­<lb/>dinis. <expan abbr="Nã">Nam</expan> quodlibet corpus, & qu&ecedil;libet figura nece&longs;&longs;e e&longs;t, vt ha <lb/><expan abbr="beatc&etilde;trũ">beatcentrum</expan> grauitatis in trin&longs;ecùs, vel extrin&longs;ecùs. </s><s>In trin&longs;ecùs vt <lb/><expan abbr="c&etilde;trũ">centrum</expan> grauitatis alicuius corporis regularis, quod e&longs;t in medio <lb/>figuræ, vel alicuius figuræ vt A; cuius centrum grauitatis &longs;it <lb/>in ambitu figuræ, vt in puncto B; extrin &longs;ecùs verò vt figura <lb/>C, cuius centrum grauitatis extrin&longs;ecus &longs;it, vt in D; quod <lb/>e&longs;t in telligendum, &longs;i graue C in centrum mundi ten deret, | <s>Noui&longs;&longs;e quoque oportet centrum grauitatis communius <lb/>e&longs;&longs;e, in pluribu&longs;què reperiri, quàm centra magnitudinis, & fi­<lb/>guræ: centrum verò figuræ communius e&longs;&longs;e centro magnitu­<lb/>dinis. <expan abbr="Nã">Nam</expan> quodlibet corpus, & qu&ecedil;libet figura nece&longs;&longs;e e&longs;t, vt ha <lb/><expan abbr="beatc&etilde;trũ">beatcentrum</expan> grauitatis in trin&longs;ecùs, vel extrin&longs;ecùs. </s><s>In trin&longs;ecùs vt <lb/><expan abbr="c&etilde;trũ">centrum</expan> grauitatis alicuius corporis regularis, quod e&longs;t in medio <lb/>figuræ, vel alicuius figuræ vt A; cuius centrum grauitatis &longs;it <lb/>in ambitu figuræ, vt in puncto B; extrin &longs;ecùs verò vt figura <lb/>C, cuius centrum grauitatis extrin&longs;ecus &longs;it, vt in D; quod <lb/>e&longs;t in telligendum, &longs;i graue C in centrum mundi ten deret, |
| <pb pagenum="13"/>tunc centrum D cum centro mundi <expan abbr="cõ-">con­<lb/></expan> | <pb pagenum="13"/>tunc centrum D cum centro mundi <expan abbr="cõ-">con­<lb/></expan> |
| <arrow.to.target n="fig4"></arrow.to.target><lb/>ueniret; &longs;iguraquè C quie&longs;ceret circa cen <lb/>trum vniuer&longs;i, veluti &longs;e habetcirca <expan abbr="c&etilde;trum">centrum</expan> <lb/>D. partes enim figuræ talem po&longs;&longs;untha­<lb/>bere &longs;itum, vt inter &longs;e &ecedil;queponderare po&longs;­<lb/>&longs;int. </s><s>vt ex &longs;ubiectis figuris per&longs;picuum e&longs;t. <lb/>& ad huc clariùs, &longs;i in telligatur figura, vt <lb/>E circulo tum exteriori, tum interiori ter <lb/>minata, cuius centrum grauitatis extra fi­<lb/>guram eritin F. quod quidem cum cir­<lb/>culorum centro conueniet. </s><s>circa quod <lb/>(exi&longs;tente centro F in centro mundi) <lb/>partes vndique &ecedil;queponderabunt: cùm <lb/>omnes &ecedil;qualiter à centro grauitatis <expan abbr="di&longs;t&etilde;t">di&longs;tent</expan>. <lb/>præterea in hac figura E centrum graui­<lb/>tatis (quamuis &longs;it extra &longs;iguram) cum cen­<lb/>tro figuræ, <expan abbr="c&etilde;troquè">centroquè</expan> magnitudinis ip&longs;ius <lb/>figuræ conuenire, forta&longs;&longs;e non eritincon­<lb/>ueniens a&longs;&longs;erere. </s><s>At verò figuræ AC nul <lb/>lo pacto figuræ, magnitudinisquè <expan abbr="centrũ">centrum</expan> <lb/>habebunt. </s><s>& quamuis dictum &longs;it <expan abbr="centrũ">centrum</expan> <lb/>grauitatis corporum regularium e&longs;&longs;e me­<lb/>dium ip&longs;orum, non tamen propterea dicen dum e&longs;t, idem e&longs;&longs;e <lb/>centrum magnitudinis, atque figuræ, ni&longs;i impropriè; <expan abbr="mediũ">medium</expan> <lb/>enim his impropriè attribuitur, &longs;icuti etiam centrum figuræ; <lb/>cùm lineæ ex ip&longs;o prodeuntes non &longs;int ip&longs;orum corporum <lb/>(quatenus regularia &longs;unt) &longs;emidiametri. </s><s>quare centrum gra­<lb/>uitatis reperiri pote&longs;t ab&longs;que alijs centris; at non è conuer&longs;o. <lb/>Rur&longs;us commune magis e&longs;t <expan abbr="c&etilde;trum">centrum</expan> figuræ centro magnitu­<lb/>dinis; quia præter circulum, & &longs;phæram, quæ tam figuræ, <expan abbr="quã">quam</expan> <lb/>magnitudinis centrum habent, nonnullæ figuræ &longs;uum ha­<lb/>bent figuræ centrum in ip&longs;is, & extra ip&longs;as; in ip&longs;is, vt ellip&longs;is, <lb/>cuius centrum in tùs habetur; &longs;emicirculus etiam, dimidia què <lb/>&longs;phæra centrum habent in limbo. </s><s>extra figuram verò veluti <lb/>hyperbolæ centrum, quod extra figuram exi&longs;tit; vbi nempè <lb/>diametri concurrunt. </s><s>Quæ quidem omnia &longs;unt figuræ cen­<lb/>tra; magnitudinis verò minimè. </s><s>verùm obijciet hoc loco for | <arrow.to.target n="fig4"></arrow.to.target><lb/>ueniret; figuraquè C quie&longs;ceret circa cen<lb/>trum vniuer&longs;i, veluti &longs;e habetcirca <expan abbr="c&etilde;trum">centrum</expan> <lb/>D. partes enim figuræ talem po&longs;&longs;unt ha­<lb/>bere &longs;itum, vt inter &longs;e &ecedil;queponderare po&longs;­<lb/>&longs;int. </s><s>vt ex &longs;ubiectis figuris per&longs;picuum e&longs;t. <lb/>& ad huc clariùs, &longs;i in telligatur figura, vt <lb/>E circulo tum exteriori, tum interiori ter <lb/>minata, cuius centrum grauitatis extra fi­<lb/>guram erit in F. quod quidem cum cir­<lb/>culorum centro conueniet. </s><s>circa quod <lb/>(exi&longs;tente centro F in centro mundi) <lb/>partes vndique &ecedil;queponderabunt: cùm <lb/>omnes &ecedil;qualiter à centro grauitatis <expan abbr="di&longs;t&etilde;t">di&longs;tent</expan>. <lb/>præterea in hac figura E centrum graui­<lb/>tatis (quamuis &longs;it extra figuram) cum cen­<lb/>tro figuræ, <expan abbr="c&etilde;troquè">centroquè</expan> magnitudinis ip&longs;ius <lb/>figuræ conuenire, forta&longs;&longs;e non erit incon­<lb/>ueniens a&longs;&longs;erere. </s><s>At verò figuræ AC nul <lb/>lo pacto figuræ, magnitudinisquè <expan abbr="centrũ">centrum</expan> <lb/>habebunt. </s><s>& quamuis dictum &longs;it <expan abbr="centrũ">centrum</expan> <lb/>grauitatis corporum regularium e&longs;&longs;e me­<lb/>dium ip&longs;orum, non tamen propterea dicen dum e&longs;t, idem e&longs;&longs;e <lb/>centrum magnitudinis, atque figuræ, ni&longs;i impropriè; <expan abbr="mediũ">medium</expan> <lb/>enim his impropriè attribuitur, &longs;icuti etiam centrum figuræ; <lb/>cùm lineæ ex ip&longs;o prodeuntes non &longs;int ip&longs;orum corporum <lb/>(quatenus regularia &longs;unt) &longs;emidiametri. </s><s>quare centrum gra­<lb/>uitatis reperiri pote&longs;t ab&longs;que alijs centris; at non è conuer&longs;o. <lb/>Rur&longs;us commune magis e&longs;t <expan abbr="c&etilde;trum">centrum</expan> figuræ centro magnitu­<lb/>dinis; quia præter circulum, & &longs;phæram, quæ tam figuræ, <expan abbr="quã">quam</expan> <lb/>magnitudinis centrum habent, nonnullæ figuræ &longs;uum ha­<lb/>bent figuræ centrum in ip&longs;is, & extra ip&longs;as; in ip&longs;is, vt ellip&longs;is, <lb/>cuius centrum in tùs habetur; &longs;emicirculus etiam, dimidia què <lb/>&longs;phæra centrum habent in limbo. </s><s>extra figuram verò veluti <lb/>hyperbolæ centrum, quod extra figuram exi&longs;tit; vbi nempè <lb/>diametri concurrunt. </s><s>Quæ quidem omnia &longs;unt figuræ cen­<lb/>tra; magnitudinis verò minimè. </s><s>verùm obijciet hoc loco for |
| <pb pagenum="14"/>ta&longs;&longs;e qui&longs;piam, vel ambas, inquiens, centri grauitatis defini­<lb/>tiones allatas, diminutas e&longs;&longs;e; vel ijs, quæ modò à nobis de <expan abbr="c&etilde;">cem</expan> <lb/>tro grauitatis dicta &longs;unt, repugnare; cùm o&longs;tenderimus cen­<lb/>trum grauitatis aliquando e&longs;&longs;e, vel in ambitu figuræ, vel extra <lb/>figuram; definitiones verò allat&ecedil; &longs;emper &longs;upponunt illud e&longs;&longs;e <lb/>in ip&longs;is intra po&longs;it <expan abbr="ũ">um</expan>. <expan abbr="Cõfirmaturquè">Confirmaturquè</expan> difficultas, quandoqui­<lb/>dem, neque huiu&longs;modi centrum extra figuram con&longs;titutum, <lb/>fui&longs;&longs;e Archimedi pror&longs;usignotum, exi&longs;timare debemus; vt <lb/>colligere licet ex nono po&longs;tulato huius libri; cùm inquit. <lb/><emph type="italics"/>Omnis figuræ, cuius perimeter &longs;it ad eandem partem concauus, centrum <lb/>grauitatis intra ip&longs;am e&longs;&longs;e oportet.<emph.end type="italics"/> qua&longs;i non repugnet figur&ecedil; peri <lb/>metrum non ad eandem partem concauum habenti, extra <lb/>ip&longs;am grauitatis centrum obtinere. </s><s>Cui obiectioni in hunc <lb/>modum occurri poterit, &longs;i dixerimus, quòd quamuis exempli <lb/>gratia in figura C dictum &longs;it centrum grauitatis D extra fi <lb/>guram exi&longs;tere, id ip&longs;um etiam intra figuram e&longs;&longs;e affirmati <lb/>poterit. </s><s>&longs;iquidem ambitus figur&ecedil; C centrum D intra &longs;e <expan abbr="cõ">com</expan> <lb/>tinct; ita vt re&longs;pectu tötius &longs;it intra. </s><s>idemquè dicen dum e&longs;t de <lb/>altera figura A. hoc autem euidenti&longs;&longs;imum e&longs;t in figura E. <lb/>& hic e&longs;t &longs;en&longs;us definitionum centri grauitatis. </s><s>His itaque pri <lb/>mùm cognitis con&longs;ideranda e&longs;t intentio Archimedis in his li <lb/>bris, quç quidem vt plurimum à librorum in&longs;criptionibus e­<lb/>luce&longs;cere &longs;olet. </s></p> | <pb pagenum="14"/>ta&longs;&longs;e qui&longs;piam, vel ambas, inquiens, centri grauitatis defini­<lb/>tiones allatas, diminutas e&longs;&longs;e; vel ijs, quæ modò à nobis de <expan abbr="c&etilde;">cem</expan> <lb/>tro grauitatis dicta &longs;unt, repugnare; cùm o&longs;tenderimus cen­<lb/>trum grauitatis aliquando e&longs;&longs;e, vel in ambitu figuræ, vel extra <lb/>figuram; definitiones verò allat&ecedil; &longs;emper &longs;upponunt illud e&longs;&longs;e <lb/>in ip&longs;is intra po&longs;it <expan abbr="ũ">um</expan>. <expan abbr="Cõfirmaturquè">Confirmaturquè</expan> difficultas, quandoqui­<lb/>dem, neque huiu&longs;modi centrum extra figuram con&longs;titutum, <lb/>fui&longs;&longs;e Archimedi pror&longs;usignotum, exi&longs;timare debemus; vt <lb/>colligere licet ex nono po&longs;tulato huius libri; cùm inquit. <lb/><emph type="italics"/>Omnis figuræ, cuius perimeter &longs;it ad eandem partem concauus, centrum <lb/>grauitatis intra ip&longs;am e&longs;&longs;e oportet.<emph.end type="italics"/> qua&longs;i non repugnet figur&ecedil; peri <lb/>metrum non ad eandem partem concauum habenti, extra <lb/>ip&longs;am grauitatis centrum obtinere. </s><s>Cui obiectioni in hunc <lb/>modum occurri poterit, &longs;i dixerimus, quòd quamuis exempli <lb/>gratia in figura C dictum &longs;it centrum grauitatis D extra fi <lb/>guram exi&longs;tere, id ip&longs;um etiam intra figuram e&longs;&longs;e affirmati <lb/>poterit. </s><s>&longs;iquidem ambitus figur&ecedil; C centrum D intra &longs;e <expan abbr="cõ">com</expan> <lb/>tinct; ita vt re&longs;pectu tötius &longs;it intra. </s><s>idemquè dicen dum e&longs;t de <lb/>altera figura A. hoc autem euidenti&longs;&longs;imum e&longs;t in figura E. <lb/>& hic e&longs;t &longs;en&longs;us definitionum centri grauitatis. </s><s>His itaque pri <lb/>mùm cognitis con&longs;ideranda e&longs;t intentio Archimedis in his li <lb/>bris, quç quidem vt plurimum à librorum in&longs;criptionibus e­<lb/>luce&longs;cere &longs;olet. </s></p> |
| <figure id="fig4"></figure> | <figure id="fig4"></figure> |
| <p type="head"> | <p type="head"> |
| |
| <arrow.to.target n="fig5"></arrow.to.target><lb/>ma, cui^{9} latera AE CF DB &longs;int <lb/>horizonti erecta, &longs;upetiorquè ba­<lb/>&longs;is ACD, quem ad modum & in­<lb/>ferior EFB &longs;it horizonti æquidi­<lb/>&longs;tans; &longs;it autem plani ACD cen­<lb/>trum grauitatis G, exquo G &longs;i <lb/>&longs;u&longs;pendatur totum AB patet <lb/>planum ACD horizonti æqui­<lb/>di&longs;tans permanere, ac plopterea <lb/>circa <expan abbr="c&etilde;trum">centrum</expan> grauitatis G æque­<lb/>ponderare. </s><s>quod quidem, quamuis egeat demon&longs;tratione, | <arrow.to.target n="fig5"></arrow.to.target><lb/>ma, cui^{9} latera AE CF DB &longs;int <lb/>horizonti erecta, &longs;upetiorquè ba­<lb/>&longs;is ACD, quem ad modum & in­<lb/>ferior EFB &longs;it horizonti æquidi­<lb/>&longs;tans; &longs;it autem plani ACD cen­<lb/>trum grauitatis G, exquo G &longs;i <lb/>&longs;u&longs;pendatur totum AB patet <lb/>planum ACD horizonti æqui­<lb/>di&longs;tans permanere, ac plopterea <lb/>circa <expan abbr="c&etilde;trum">centrum</expan> grauitatis G æque­<lb/>ponderare. </s><s>quod quidem, quamuis egeat demon&longs;tratione, |
| <pb pagenum="16"/> | <pb pagenum="16"/> |
| <arrow.to.target n="marg12"></arrow.to.target> in præ&longs;entia omittatur; infraquè &longs;uo loco o&longs;ten den dum. </s><s>&longs;at <lb/>autem nobis nunc &longs;it o&longs;tendi&longs;&longs;e, hæc ad praxim reduci, ma­<lb/>nibu&longs;què (vt dicitur.) contrectari po&longs;&longs;e. </s><s>Quòd &longs;i hæc ita &longs;e ha <lb/>bent, huiu&longs;modi con&longs;ideratio non erit vana, neque vt inuti­<lb/>lis reijcienda. </s><s>Sed vlteriùs adhuc progrediamur, dicamu&longs;­<lb/>què, quoniam planum ACD, quatenuse&longs;t corpori coniun­<lb/>ctum, horizonti æquidi&longs;tans permanere debet; &longs;i &longs;eor&longs;um à <lb/>corpore illud in telligamus, vt &longs;i ADC ex eius centro graui­<lb/>tatis G &longs;u&longs;pendatur, tunc quocunque modo reperiatur, hoc <lb/>e&longs;t &longs;iue horizonti &ecedil;quidi&longs;tans, &longs;iuè <lb/>minùs, idip&longs;um perman&longs;urum ni <lb/> | <arrow.to.target n="marg12"></arrow.to.target> in præ&longs;entia omittatur; infraquè &longs;uo loco o&longs;ten den dum. </s><s>&longs;at <lb/>autem nobis nunc &longs;it o&longs;tendi&longs;&longs;e, hæc ad praxim reduci, ma­<lb/>nibu&longs;què (vt dicitur.) contrectari po&longs;&longs;e. </s><s>Quòd &longs;i hæc ita &longs;e ha <lb/>bent, huiu&longs;modi con&longs;ideratio non erit vana, neque vt inuti­<lb/>lis reijcienda. </s><s>Sed vlteriùs adhuc progrediamur, dicamu&longs;­<lb/>què, quoniam planum ACD, quatenuse&longs;t corpori coniun­<lb/>ctum, horizonti æquidi&longs;tans permanere debet; &longs;i &longs;eor&longs;um à <lb/>corpore illud in telligamus, vt &longs;i ADC ex eius centro graui­<lb/>tatis G &longs;u&longs;pendatur, tunc quocunque modo reperiatur, hoc <lb/>e&longs;t &longs;iue horizonti &ecedil;quidi&longs;tans, &longs;iuè <lb/>minùs, idip&longs;um perman&longs;urum ni <lb/> |
| <arrow.to.target n="fig6"></arrow.to.target><lb/>hilominus in telligere po&longs;&longs;umus, <lb/>parte&longs;què vndique æqualium mo <lb/>men torum con&longs;i&longs;tentes. </s><s>Neque <lb/>enim Ari&longs;to teles grauibus dunta­<lb/>xat, &longs;ed etiam leuibus momenta <lb/>tribuit, idip&longs;um què (vt Eutocius <lb/>in horum librorum comentarijs <lb/>refert) Ptolæmeo quoque placuit, vt habetur in líbro (à nobis <lb/>ramen de &longs;iderato) quem de momen tis &longs;crip&longs;it. </s><s>Pr&ecedil;terea alij­<lb/>quoque Philo&longs;ophi id ip&longs;um &longs;en&longs;i&longs;&longs;evidentur. </s><s>quod e&longs;t qui­<lb/>dem rationi con&longs;en taneum, &longs;uperuolant enim, quæ leuia &longs;unt, <lb/>& &longs;i mente concipiatur <expan abbr="ead&etilde;">eadem</expan> &longs;igura leuis cuiu&longs;piam e&longs;&longs;e, tunc <lb/>&longs;i detineatur in G, partes vndique &ecedil;qualium <expan abbr="momentorũ">momentorum</expan> <lb/>con&longs;i&longs;tent, e&longs;&longs;etquè G (vt ita dicam) centrum leuitatis. </s><s>Quo­<lb/>niam autem circa centrum grauitatis &ecedil;queponderationem <lb/>con&longs;ideramus, id circo plana, tanquam no bis apparentia gra­<lb/>uitatem habere, mente concipimus. </s><s>Non e&longs;t igitur à ratio­<lb/>ne alienum, æqueponderantiam in planis, vt grauibus con&longs;i­<lb/>deratis intelligere, conciperequè. </s><s>Nec quicquam nobis offi­<lb/>cit, quòd definitiones centri grauitatis priùs allatæ non pla­<lb/>norum, &longs;ed corporum centra explicarunt, ita vt grauitatis <expan abbr="c&etilde;-trũ">cen­<lb/>trum</expan> ad corpora, <expan abbr="nõ">non</expan> ad plana &longs;it refe <gap/><expan abbr="ndũ">ndum</expan>. Hoc enim ideo fa <lb/><expan abbr="ctũ">ctum</expan> e&longs;t, quia propriè <expan abbr="centrũ">centrum</expan> grauitatis re&longs;picit corpora; non ta <lb/>men propterea impropriè re&longs;picit plana, &longs;ed quia primò re&longs;pi <lb/>cit corpora; in quib^{9} actu ine&longs;&longs;e <expan abbr="depræh&etilde;ditur">depræhenditur</expan>. propterea <expan abbr="e&ecedil;d&etilde;-met">e&ecedil;den­<lb/>met</expan> definitiones planis quoque in <expan abbr="hũc">hunc</expan> <expan abbr="modũ">modum</expan> aptari <expan abbr="poterũt">poterunt</expan>. </s></p> | <arrow.to.target n="fig6"></arrow.to.target><lb/>hilominus in telligere po&longs;&longs;umus, <lb/>parte&longs;què vndique æqualium mo <lb/>men torum con&longs;i&longs;tentes. </s><s>Neque <lb/>enim Ari&longs;to teles grauibus dunta­<lb/>xat, &longs;ed etiam leuibus momenta <lb/>tribuit, idip&longs;um què (vt Eutocius <lb/>in horum librorum comentarijs <lb/>refert) Ptolæmeo quoque placuit, vt habetur in líbro (à nobis <lb/>ramen de &longs;iderato) quem de momen tis &longs;crip&longs;it. </s><s>Pr&ecedil;terea alij­<lb/>quoque Philo&longs;ophi id ip&longs;um &longs;en&longs;i&longs;&longs;evidentur. </s><s>quod e&longs;t qui­<lb/>dem rationi con&longs;en taneum, &longs;uperuolant enim, quæ leuia &longs;unt, <lb/>& &longs;i mente concipiatur <expan abbr="ead&etilde;">eadem</expan> figura leuis cuiu&longs;piam e&longs;&longs;e, tunc <lb/>&longs;i detineatur in G, partes vndique &ecedil;qualium <expan abbr="momentorũ">momentorum</expan> <lb/>con&longs;i&longs;tent, e&longs;&longs;etquè G (vt ita dicam) centrum leuitatis. </s><s>Quo­<lb/>niam autem circa centrum grauitatis &ecedil;queponderationem <lb/>con&longs;ideramus, id circo plana, tanquam no bis apparentia gra­<lb/>uitatem habere, mente concipimus. </s><s>Non e&longs;t igitur à ratio­<lb/>ne alienum, æqueponderantiam in planis, vt grauibus con&longs;i­<lb/>deratis intelligere, conciperequè. </s><s>Nec quicquam nobis offi­<lb/>cit, quòd definitiones centri grauitatis priùs allatæ non pla­<lb/>norum, &longs;ed corporum centra explicarunt, ita vt grauitatis <expan abbr="c&etilde;-trũ">cen­<lb/>trum</expan> ad corpora, <expan abbr="nõ">non</expan> ad plana &longs;it refe <gap/><expan abbr="ndũ">ndum</expan>. Hoc enim ideo fa <lb/><expan abbr="ctũ">ctum</expan> e&longs;t, quia propriè <expan abbr="centrũ">centrum</expan> grauitatis re&longs;picit corpora; non ta <lb/>men propterea impropriè re&longs;picit plana, &longs;ed quia primò re&longs;pi <lb/>cit corpora; in quib^{9} actu ine&longs;&longs;e <expan abbr="depræh&etilde;ditur">depræhenditur</expan>. propterea <expan abbr="e&ecedil;d&etilde;-met">e&ecedil;den­<lb/>met</expan> definitiones planis quoque in <expan abbr="hũc">hunc</expan> <expan abbr="modũ">modum</expan> aptari <expan abbr="poterũt">poterunt</expan>. </s></p> |
| <pb pagenum="17"/> | <pb pagenum="17"/> |
| <p type="margin"> | <p type="margin"> |
| <s><margin.target id="marg12"></margin.target><emph type="italics"/>in fine pri­<lb/>milibri.<emph.end type="italics"/></s></p> | <s><margin.target id="marg12"></margin.target><emph type="italics"/>in fine pri­<lb/>milibri.<emph.end type="italics"/></s></p> |
| |
| <s>DE DIVISIONE HORVM LIBRORVM.</s></p> | <s>DE DIVISIONE HORVM LIBRORVM.</s></p> |
| <p type="main"> | <p type="main"> |
| <s>Diuiditur enim in primis hic tractatus in duos libros diui­<lb/>&longs;us, in po&longs;tulata, & theoremata: theoremata verò &longs;ubdiui­<lb/>duntur in duas &longs;ectiones, quarum prima continet priora o­<lb/>cto theoremata; ad alteram verò reliqua theoremata <expan abbr="&longs;pectãt">&longs;pectant</expan>. <lb/>quæ quidem adhuc in alias duas partes diuidi pote&longs;t; nempè <lb/>in theoremata primo libro examina ta, & in ea, quæ &longs;ecun­<lb/>dus liber contemplatur. </s><s>Hanc autem horum librorum con <lb/>&longs;tituimus diui&longs;ionem, quoniam imprimis Archimedes, (o­<lb/>mi&longs;&longs;is po&longs;tulatis, quæ primum locum obtinere debent) quæ­<lb/>dam tractauit communia in pricribus octo theorema tibus; <lb/>quorum &longs;copus e&longs;t inuenire fundamentum illud <expan abbr="præcipuũ">præcipuum</expan> <lb/>mechanicum, quòd &longs;cilicet ita &longs;e habet grauitas ad grauita­<lb/>tem, vt di&longs;tan tia ad di&longs;tantiam permutatim. </s><s>ad quod demo <lb/>&longs;trandum quinque præmittit theoremata, quæ paulatim <lb/>deducunt nos in cognitionem demon&longs;tra tionis præfati fun <lb/>damenti. </s><s>quo loco illud &longs;ummoperè notandum e&longs;t, nimi­<lb/>rum &longs;undamentum illud, nec non octo priora theorema­<lb/>ta communia e&longs;&longs;e tam planis, quàm &longs;olidis; atque promi&longs;­<lb/>cuè de vtri&longs;que <expan abbr="Archimed&etilde;">Archimedem</expan> demon&longs;trare. </s><s>quòd &longs;i quis aliter | <s>Diuiditur enim in primis hic tractatus in duos libros diui­<lb/>&longs;us, in po&longs;tulata, & theoremata: theoremata verò &longs;ubdiui­<lb/>duntur in duas &longs;ectiones, quarum prima continet priora o­<lb/>cto theoremata; ad alteram verò reliqua theoremata <expan abbr="&longs;pectãt">&longs;pectant</expan>. <lb/>quæ quidem adhuc in alias duas partes diuidi pote&longs;t; nempè <lb/>in theoremata primo libro examina ta, & in ea, quæ &longs;ecun­<lb/>dus liber contemplatur. </s><s>Hanc autem horum librorum con <lb/>&longs;tituimus diui&longs;ionem, quoniam imprimis Archimedes, (o­<lb/>mi&longs;&longs;is po&longs;tulatis, quæ primum locum obtinere debent) quæ­<lb/>dam tractauit communia in pricribus octo theorema tibus; <lb/>quorum &longs;copus e&longs;t inuenire fundamentum illud <expan abbr="præcipuũ">præcipuum</expan> <lb/>mechanicum, quòd &longs;cilicet ita &longs;e habet grauitas ad grauita­<lb/>tem, vt di&longs;tan tia ad di&longs;tantiam permutatim. </s><s>ad quod demo <lb/>&longs;trandum quinque præmittit theoremata, quæ paulatim <lb/>deducunt nos in cognitionem demon&longs;tra tionis præfati fun <lb/>damenti. </s><s>quo loco illud &longs;ummoperè notandum e&longs;t, nimi­<lb/>rum &longs;undamentum illud, nec non octo priora theorema­<lb/>ta communia e&longs;&longs;e tam planis, quàm &longs;olidis; atque promi&longs;­<lb/>cuè de vtri&longs;que <expan abbr="Archimed&etilde;">Archimedem</expan> demon&longs;trare. </s><s>quòd &longs;i quis aliter |
| <pb pagenum="20"/>&longs;en&longs;erit, demon&longs;tratione&longs;què tan tùm de planis <expan abbr="cõcludere">concludere</expan> exi <lb/>&longs;timauerit, vel de &longs;olidis, non autem <expan abbr="quibu&longs;cũque">quibu&longs;cunque</expan>, &longs;ed vel de <lb/>rectilineis, vel de homogeneis tantùm, & de ijs, quæ inter &longs;e <lb/>&longs;unteiu&longs;dem &longs;peciei, longè aberrat à &longs;copo, & mente Archi­<lb/>medis. </s><s>etenim in his &longs;emper loquitur. </s><s>vel de grauibus &longs;impli <lb/>citer, veluti in primis tribus theorematibus; vel de magnitu <lb/>dinibus, vt in reliquis quinque quod quidem nomen tam <lb/>planis, quàm &longs;olidis quibu&longs;cunque e&longs;t <expan abbr="cõmune">commune</expan>, vt etiam ij, <lb/>qui parùm in Mathematicis ver&longs;ati &longs;unt, &longs;atis norunt. </s><s>ficu­<lb/>ti etiam Euclides, dum quinti libri propo&longs;itiones pertracta­<lb/>uit, quantitatem continuam &longs;ub nomine magnitudinis <expan abbr="cõ">com</expan> <lb/>prehendit. </s><s>quòd <expan abbr="aut&etilde;">autem</expan> nomen grauis &longs;it <expan abbr="cõmune">commune</expan>, iam &longs;atis <lb/>per &longs;e con&longs;tat. </s><s>Per&longs;picuum e&longs;t igitur priora hæc octo Theo <lb/>remata communia e&longs;&longs;e, tam planis, quàm &longs;olidis. </s><s>ac non &longs;o­<lb/>lùm &longs;olidis eiu&longs;dem &longs;peciei, & homogeneis, verùm etiam &longs;oli <lb/>dis diuer&longs;æ &longs;peciei, & hçterogeneis, vt &longs;uo loco manife&longs;tum <lb/>fiet. </s><s>Iactoquè hoc fundamento, quod Archimedes in duob^{9} <lb/>propo&longs;itionibus, &longs;exta nempè, & &longs;eptima demon&longs;trauit; in o­<lb/>ctaua tanquam corrollarium colligit. </s><s>Deinceps peculiariter <lb/>pertractat de centro grauitatis planorum, nec amplius plana <lb/>nominat magnitudinis nomine, &longs;ed proprijs cuiu&longs;cun que <lb/>nominibus; vt parallelogrammi, trianguli, & aliorum huiu&longs;­<lb/>modi. </s><s>& in hac parte de&longs;cendit ad particularia. </s><s>quippè cùm <lb/>& &longs;i non actu forta&longs;&longs;e, virture tamen cuiu&longs;libet particularis <lb/>plani centrum grauitatis nos doceat. </s><s>in primo enim libro <lb/>&longs;at &longs;i bi vi&longs;um e&longs;t o&longs;tendi&longs;&longs;e centra grauitatum trianguloru, <lb/>ac parallelogrammorum, ex quibus cæterarum &longs;igurarum, <lb/>veluti pen tagoni, hexagoni, & aliorum &longs;imilium centra gra­<lb/>uita tis inue&longs;tigare non admodum erit difficile. </s><s>&longs;iquidem hu <lb/>iu&longs;modi plana in triangula diuiduntur. </s><s>vt in &longs;ine primi li­<lb/>bri attingemus. </s><s>In &longs;ecundo autem libro altiùs &longs;e extollit, & <lb/>moro &longs;uo circa &longs;ubtili&longs;&longs;ima theoremata ver&longs;atur; nompè cir <lb/>ca centrum grauitatis conice &longs;ectionis, quæ parabole nun­<lb/>cupatur. </s><s>nonnullaquè præmittit theorema ta, quæ &longs;unt tan­<lb/>quam præuie di&longs;po&longs;itiones ad inue&longs;tigandam demon&longs;tra­<lb/>tionem centri grauitatis in parabole. </s><s>Itaque per&longs;picuum e&longs;t, <lb/>Archimedem propriè elementa mechanica tradere. </s><s>quando- | <pb pagenum="20"/>&longs;en&longs;erit, demon&longs;tratione&longs;què tan tùm de planis <expan abbr="cõcludere">concludere</expan> exi <lb/>&longs;timauerit, vel de &longs;olidis, non autem <expan abbr="quibu&longs;cũque">quibu&longs;cunque</expan>, &longs;ed vel de <lb/>rectilineis, vel de homogeneis tantùm, & de ijs, quæ inter &longs;e <lb/>&longs;unteiu&longs;dem &longs;peciei, longè aberrat à &longs;copo, & mente Archi­<lb/>medis. </s><s>etenim in his &longs;emper loquitur. </s><s>vel de grauibus &longs;impli <lb/>citer, veluti in primis tribus theorematibus; vel de magnitu <lb/>dinibus, vt in reliquis quinque quod quidem nomen tam <lb/>planis, quàm &longs;olidis quibu&longs;cunque e&longs;t <expan abbr="cõmune">commune</expan>, vt etiam ij, <lb/>qui parùm in Mathematicis ver&longs;ati &longs;unt, &longs;atis norunt. </s><s>ficu­<lb/>ti etiam Euclides, dum quinti libri propo&longs;itiones pertracta­<lb/>uit, quantitatem continuam &longs;ub nomine magnitudinis <expan abbr="cõ">com</expan> <lb/>prehendit. </s><s>quòd <expan abbr="aut&etilde;">autem</expan> nomen grauis &longs;it <expan abbr="cõmune">commune</expan>, iam &longs;atis <lb/>per &longs;e con&longs;tat. </s><s>Per&longs;picuum e&longs;t igitur priora hæc octo Theo <lb/>remata communia e&longs;&longs;e, tam planis, quàm &longs;olidis. </s><s>ac non &longs;o­<lb/>lùm &longs;olidis eiu&longs;dem &longs;peciei, & homogeneis, verùm etiam &longs;oli <lb/>dis diuer&longs;æ &longs;peciei, & hçterogeneis, vt &longs;uo loco manife&longs;tum <lb/>fiet. </s><s>Iactoquè hoc fundamento, quod Archimedes in duob^{9} <lb/>propo&longs;itionibus, &longs;exta nempè, & &longs;eptima demon&longs;trauit; in o­<lb/>ctaua tanquam corrollarium colligit. </s><s>Deinceps peculiariter <lb/>pertractat de centro grauitatis planorum, nec amplius plana <lb/>nominat magnitudinis nomine, &longs;ed proprijs cuiu&longs;cun que <lb/>nominibus; vt parallelogrammi, trianguli, & aliorum huiu&longs;­<lb/>modi. </s><s>& in hac parte de&longs;cendit ad particularia. </s><s>quippè cùm <lb/>& &longs;i non actu forta&longs;&longs;e, virture tamen cuiu&longs;libet particularis <lb/>plani centrum grauitatis nos doceat. </s><s>in primo enim libro <lb/>&longs;at &longs;i bi vi&longs;um e&longs;t o&longs;tendi&longs;&longs;e centra grauitatum trianguloru, <lb/>ac parallelogrammorum, ex quibus cæterarum figurarum, <lb/>veluti pen tagoni, hexagoni, & aliorum &longs;imilium centra gra­<lb/>uita tis inue&longs;tigare non admodum erit difficile. </s><s>&longs;iquidem hu <lb/>iu&longs;modi plana in triangula diuiduntur. </s><s>vt in &longs;ine primi li­<lb/>bri attingemus. </s><s>In &longs;ecundo autem libro altiùs &longs;e extollit, & <lb/>moro &longs;uo circa &longs;ubtili&longs;&longs;ima theoremata ver&longs;atur; nompè cir <lb/>ca centrum grauitatis conice &longs;ectionis, quæ parabole nun­<lb/>cupatur. </s><s>nonnullaquè præmittit theorema ta, quæ &longs;unt tan­<lb/>quam præuie di&longs;po&longs;itiones ad inue&longs;tigandam demon&longs;tra­<lb/>tionem centri grauitatis in parabole. </s><s>Itaque per&longs;picuum e&longs;t, <lb/>Archimedem propriè elementa mechanica tradere. </s><s>quando- |
| <pb pagenum="21"/>quidem duo pertractat, quæ &longs;unt tanquam elementa huius <lb/>&longs;cientiæ. </s><s>fundamentum nempè illud præ&longs;tanti&longs;&longs;imum iam <lb/>to ties præfatum, deinde centra grauitatis planorum o&longs;tendit. <lb/>& quamuis hi duo Archimedis libelli pauca continerevidean <lb/>tur, non tamen pauca docui&longs;&longs;e Archimedem exi&longs;timandum <lb/>e&longs;t. </s><s>multa enim &longs;unt mole exigua, quæ tamen virtute maxima <lb/>habentur. </s><s>quod planè Archimedis &longs;criptis accidit; hi&longs;què pr&ecedil; <lb/>&longs;ertim, ex quibus patet aditus ad multa, ac penè in&longs;inita theo­<lb/>remata, problemataquè mechanica. </s><s>nihil enim in hoc gene­<lb/>re demon&longs;trari pote&longs;t, quod his non indigeat &longs;criptis. </s><s>& <lb/>quod admirabilius e&longs;t, nos non &longs;olùm pro fundamento &longs;u­<lb/>&longs;cipere po&longs;&longs;e ad aliquod demon&longs;trandum theoremata in his <lb/>libris demon&longs;trata, verùm etiam ab his demon&longs;trationibus <lb/>perdi&longs;cerere ip&longs;um modum argumentandi, & demon&longs;trandi; <lb/>vt &longs;uis locis o&longs;tendemus. </s><s>ita vt verè concludendum &longs;it, nemi­<lb/>nem pror&longs;us inter mechanicos connumerandum fore, qui <lb/>hæc Archimedis &longs;cripta ignorat. </s><s>ignoratis enim principijs <lb/>nulla e&longs;t &longs;cientia, vt apud omnes &longs;apientes per&longs;picuum e&longs;t. <lb/>Ip&longs;um igitur Archimedem audiamus, eiu&longs;què &longs;cripta diligen <lb/>ti&longs;&longs;imè perpendamus. </s></p> | <pb pagenum="21"/>quidem duo pertractat, quæ &longs;unt tanquam elementa huius <lb/>&longs;cientiæ. </s><s>fundamentum nempè illud præ&longs;tanti&longs;&longs;imum iam <lb/>to ties præfatum, deinde centra grauitatis planorum o&longs;tendit. <lb/>& quamuis hi duo Archimedis libelli pauca continerevidean <lb/>tur, non tamen pauca docui&longs;&longs;e Archimedem exi&longs;timandum <lb/>e&longs;t. </s><s>multa enim &longs;unt mole exigua, quæ tamen virtute maxima <lb/>habentur. </s><s>quod planè Archimedis &longs;criptis accidit; hi&longs;què pr&ecedil; <lb/>&longs;ertim, ex quibus patet aditus ad multa, ac penè in&longs;inita theo­<lb/>remata, problemataquè mechanica. </s><s>nihil enim in hoc gene­<lb/>re demon&longs;trari pote&longs;t, quod his non indigeat &longs;criptis. </s><s>& <lb/>quod admirabilius e&longs;t, nos non &longs;olùm pro fundamento &longs;u­<lb/>&longs;cipere po&longs;&longs;e ad aliquod demon&longs;trandum theoremata in his <lb/>libris demon&longs;trata, verùm etiam ab his demon&longs;trationibus <lb/>perdi&longs;cerere ip&longs;um modum argumentandi, & demon&longs;trandi; <lb/>vt &longs;uis locis o&longs;tendemus. </s><s>ita vt verè concludendum &longs;it, nemi­<lb/>nem pror&longs;us inter mechanicos connumerandum fore, qui <lb/>hæc Archimedis &longs;cripta ignorat. </s><s>ignoratis enim principijs <lb/>nulla e&longs;t &longs;cientia, vt apud omnes &longs;apientes per&longs;picuum e&longs;t. <lb/>Ip&longs;um igitur Archimedem audiamus, eiu&longs;què &longs;cripta diligen <lb/>ti&longs;&longs;imè perpendamus. </s></p> |
| | <pb pagenum="22"/> |
| <pb pagenum="23"/> | <pb pagenum="23"/> |
| <p type="head"> | <p type="head"> |
| <s>GVIDIVBALDI <lb/>EMARCHIONIBVS <lb/>MONTIS. <lb/>IN PRIMVM ARCHIMEDIS <lb/>AEQVEPONDERANTIVM <lb/>LIBRVM <lb/>PARAPHRASIS <lb/>SCHOLIIS ILLVSTRATA.</s></p> | <s>GVIDIVBALDI <lb/>EMARCHIONIBVS <lb/>MONTIS. <lb/>IN PRIMVM ARCHIMEDIS <lb/>AEQVEPONDERANTIVM <lb/>LIBRVM <lb/>PARAPHRASIS <lb/>SCHOLIIS ILLVSTRATA.</s></p> |
| |
| <p type="main"> | <p type="main"> |
| <s>Dvobvs modis grauia in di&longs;tantijs <lb/>collocata in telligi po&longs;&longs;unt. </s><s>quod & <lb/>in cæteris po&longs;tulatis, & in propo&longs;i­<lb/>tionibus intelligendum e&longs;t. </s><s>etenim <lb/>vel grauia <expan abbr="sũt">sunt</expan> appen&longs;a, vt in prima &longs;i­<lb/>gura æqualia grauia AB &longs;unt in CD <lb/>appen&longs;a; ita vt di&longs;tantia EC &longs;it di­<lb/>&longs;tatiæ ED æqualis. </s><s>intelligaturquè <lb/>CD tanquam libra, quæ &longs;u&longs;pendatur <lb/>in E. vel vt in &longs;ecunda figura grauia AB habent ip&longs;orum <lb/>centra grauitatis, quæ &longs;int CD, in ip&longs;a DC linea, in pun- | <s>Dvobvs modis grauia in di&longs;tantijs <lb/>collocata in telligi po&longs;&longs;unt. </s><s>quod & <lb/>in cæteris po&longs;tulatis, & in propo&longs;i­<lb/>tionibus intelligendum e&longs;t. </s><s>etenim <lb/>vel grauia <expan abbr="sũt">sunt</expan> appen&longs;a, vt in prima &longs;i­<lb/>gura æqualia grauia AB &longs;unt in CD <lb/>appen&longs;a; ita vt di&longs;tantia EC &longs;it di­<lb/>&longs;tatiæ ED æqualis. </s><s>intelligaturquè <lb/>CD tanquam libra, quæ &longs;u&longs;pendatur <lb/>in E. vel vt in &longs;ecunda figura grauia AB habent ip&longs;orum <lb/>centra grauitatis, quæ &longs;int CD, in ip&longs;a DC linea, in pun- |
| <pb pagenum="24"/>ctis <expan abbr="n&etilde;pè">nempè</expan> CD <lb/> | <pb pagenum="24"/>ctis <expan abbr="n&etilde;pè">nempè</expan> CD <lb/> |
| <arrow.to.target n="fig8"></arrow.to.target><lb/>con&longs;tituta. </s><s>li­<lb/>braquè &longs;imili­<lb/>ter ex puncto <lb/>E &longs;u&longs;pendatur; <lb/>&longs;itquè di&longs;tátia <lb/>EC di&longs;tantiæ <lb/>ED æqualis. <lb/><expan abbr="erũt">erunt</expan> vtique in <lb/>vtraque figura <lb/>pondera AB <lb/>in di&longs;tantijs &ecedil;­<lb/>qualibus con­<lb/>&longs;tituta. </s><s>ac pro­<lb/>pterea æquepondera bunt, atque manebunt. </s><s>nulla enim ratio <lb/>afferri pote&longs;t, cur ex parte A, vel ex parte B deor&longs;um, vel &longs;ur <lb/>&longs;um fieri debeat motus; cùm omnia &longs;int paria. </s><s>ea verò æque­<lb/>ponderare debere, aliqua ratione manife&longs;tari pote&longs;t ex eo, <lb/>quod o&longs;ten&longs;um e&longs;t à nobis in no&longs;tro mechanicorum libro, <lb/>tractatu de libra: quod quidem ab Ari&longs;to tele quoque in prin <lb/>cipio quæ&longs;tionum mechanicarum elici pote&longs;t: idem &longs;cilicet <lb/>pondus longius a centro grauius e&longs;&longs;e eodem pondere ip&longs;i cen <lb/>tro propinquiori. </s><s>Vnde &longs;i duo e&longs;&longs;ent pondera æqualia alte­<lb/>rum altero propinquius centro, quod remotius e&longs;t, grauius al <lb/>tero appareret. </s><s>&longs;i igitur grauia æqualia à centro æqualiter di­<lb/>&longs;tabunt, æque grauia erunt. </s><s>ac propterea æqueponderabunt. <lb/>quod quidem &longs;upponit Archimedes. </s><s>Punctum autem illud, <lb/>quod Archimedes accipit, vnde &longs;umuntur di&longs;tantiæ, ex qui­<lb/>bus grauia &longs;u&longs;penduntur, veluti punctum E, Ari&longs;toteles cen <lb/>rum appellat. </s><s>& hæc quidem æqueponderatio tam ponderi­<lb/>bus in libra appen&longs;is, quàm in ip&longs;a (vt dictum e&longs;t) con&longs;titutis <lb/>competit: dummodo ea, quibus appenduntur pondera, libe­<lb/>re &longs;emper in centrum mundi tendere po&longs;&longs;int. </s><s>vtroque enim <lb/>modo in punctis CD grauitant, vt diximus etiam in eodem <lb/>uactatu de libra. </s><s>Noui&longs;&longs;e tamen oportet Archimedem in his <lb/>libris potiùs in tellexi&longs;&longs;e pondera e&longs;&longs;e in di&longs;tantijs collocata, vt <lb/>in &longs;ecunda &longs;igura, quàm appen&longs;a; vt ex quarta, & quinta | <arrow.to.target n="fig8"></arrow.to.target><lb/>con&longs;tituta. </s><s>li­<lb/>braquè &longs;imili­<lb/>ter ex puncto <lb/>E &longs;u&longs;pendatur; <lb/>&longs;itquè di&longs;tátia <lb/>EC di&longs;tantiæ <lb/>ED æqualis. <lb/><expan abbr="erũt">erunt</expan> vtique in <lb/>vtraque figura <lb/>pondera AB <lb/>in di&longs;tantijs &ecedil;­<lb/>qualibus con­<lb/>&longs;tituta. </s><s>ac pro­<lb/>pterea æquepondera bunt, atque manebunt. </s><s>nulla enim ratio <lb/>afferri pote&longs;t, cur ex parte A, vel ex parte B deor&longs;um, vel &longs;ur <lb/>&longs;um fieri debeat motus; cùm omnia &longs;int paria. </s><s>ea verò æque­<lb/>ponderare debere, aliqua ratione manife&longs;tari pote&longs;t ex eo, <lb/>quod o&longs;ten&longs;um e&longs;t à nobis in no&longs;tro mechanicorum libro, <lb/>tractatu de libra: quod quidem ab Ari&longs;to tele quoque in prin<lb/>cipio quæ&longs;tionum mechanicarum elici pote&longs;t: idem &longs;cilicet <lb/>pondus longius a centro grauius e&longs;&longs;e eodem pondere ip&longs;i cen<lb/>tro propinquiori. </s><s>Vnde &longs;i duo e&longs;&longs;ent pondera æqualia alte­<lb/>rum altero propinquius centro, quod remotius e&longs;t, grauius al <lb/>tero appareret. </s><s>&longs;i igitur grauia æqualia à centro æqualiter di­<lb/>&longs;tabunt, æque grauia erunt. </s><s>ac propterea æqueponderabunt. <lb/>quod quidem &longs;upponit Archimedes. </s><s>Punctum autem illud, <lb/>quod Archimedes accipit, vnde &longs;umuntur di&longs;tantiæ, ex qui­<lb/>bus grauia &longs;u&longs;penduntur, veluti punctum E, Ari&longs;toteles cen<lb/>rum appellat. </s><s>& hæc quidem æqueponderatio tam ponderi­<lb/>bus in libra appen&longs;is, quàm in ip&longs;a (vt dictum e&longs;t) con&longs;titutis <lb/>competit: dummodo ea, quibus appenduntur pondera, libe­<lb/>re &longs;emper in centrum mundi tendere po&longs;&longs;int. </s><s>vtroque enim <lb/>modo in punctis CD grauitant, vt diximus etiam in eodem <lb/>uactatu de libra. </s><s>Noui&longs;&longs;e tamen oportet Archimedem in his <lb/>libris potiùs in tellexi&longs;&longs;e pondera e&longs;&longs;e in di&longs;tantijs collocata, vt <lb/>in &longs;ecunda figura, quàm appen&longs;a; vt ex quarta, & quinta |
| <pb pagenum="25"/>primi libri propo&longs;itione pater. </s><s>demon&longs;trationes enim cla­<lb/>riores redduntur. </s></p> | <pb pagenum="25"/>primi libri propo&longs;itione pater. </s><s>demon&longs;trationes enim cla­<lb/>riores redduntur. </s></p> |
| <figure id="fig8"></figure> | <figure id="fig8"></figure> |
| <figure></figure> | <figure></figure> |
| |
| <s><margin.target id="marg17"></margin.target>4 <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s></p> | <s><margin.target id="marg17"></margin.target>4 <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s></p> |
| <p type="main"> | <p type="main"> |
| <s><expan abbr="Ducãtur">Ducantur</expan> pr&ecedil;terea à punctis KL ad latera perpendiculares <lb/>KM KN KO KP, LQ LR LS LT. & quoniam anguli <lb/>KMA LQE &longs;unt recti, ac propterea æquales, & KAM LEQ <lb/>&longs;unt æquales, ut o&longs;ten&longs;um e&longs;t; erit reliquus MKA reliquo <lb/>QLE &ecedil;qualis, triangulumquè AKM triangulo ELQ &longs;imile. <lb/>vtigitur AK ad KM; &longs;ic EL ad <expan abbr="Lq.">Lque</expan> & permutando AK | <s><expan abbr="Ducãtur">Ducantur</expan> pr&ecedil;terea à punctis KL ad latera perpendiculares <lb/>KM KN KO KP, LQ LR LS LT. & quoniam anguli <lb/>KMA LQE &longs;unt recti, ac propterea æquales, & KAM LEQ <lb/>&longs;unt æquales, ut o&longs;ten&longs;um e&longs;t; erit reliquus MKA reliquo <lb/>QLE &ecedil;qualis, triangulumquè AKM triangulo ELQ &longs;imile. <lb/>vtigitur AK ad KM; &longs;ic EL ad <expan abbr="Lq.">Lque</expan> & permutando AK |
| <arrow.to.target n="marg18"></arrow.to.target><lb/>ad EL, vt KM ad <expan abbr="Lq.">Lque</expan> pariquè ratione o&longs;tendetur triangu <lb/>lum BKM triangulo FLQ &longs;imile exi&longs;tere; e&longs;&longs;equè BK ad <lb/>FL, vt KM ad <expan abbr="Lq.">Lque</expan> &longs;imiliterquè in alijs triangulis o&longs;ten­<lb/>detur, ita e&longs;&longs;e Bk ad FL, vt KN ad LR; & Ck ad GL e&longs;&longs;e, vt <lb/>kO ad LS; atque kD ad LH, vt kP ad LT. quia verò AK <lb/>EL, Bk FL, Ck GL, Dk HL in eadem &longs;untproportione, vt <lb/>proximè demon&longs;tratum fuit; in eadem quoque proportione <lb/>erit kM ad LQ, & KN ad LR; & KO ad LS, atque kP ad <lb/>LT. ex quibus &longs;equitur centra grauitatis KL, non &longs;olùm ab <lb/>angulis in cadem proportione di&longs;tare; verùm etiam à lateri­<lb/>ribus in eadem quoque proportione di&longs;tare. </s><s>Itaque cognito, <lb/>quomodo intelligar Archimedes centra grauitatis in &longs;imili­<lb/>bus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; nunc con&longs;iderandum e&longs;t præ <lb/>cedens po&longs;tulatum, quatenus nimirum oporteat grauitatis <expan abbr="c&etilde;">cem</expan> <lb/>tra in &longs;imilibus figuris &longs;imiliter e&longs;&longs;e con&longs;tituta. </s><s>Nam inti­<lb/>miùs con&longs;iderando hanc &longs;imilem horum grauitatis <expan abbr="centrorũ">centrorum</expan> <lb/>po&longs;itionem, congruum, & nece&longs;&longs;arium videtur, &longs;imiles &longs;igu­<lb/>ras &longs;ecundùm eandem proportionem e&longs;&longs;e æquepon <expan abbr="derãtes">derantes</expan>; <lb/>eademquè ratione (ob earum &longs;imilitudinem) circa grauita­<lb/>tis centra æqueponderare, veluti &longs;i figuræ: AC EG (quarum <lb/>centra grauitatis &longs;int KL) à rectis lineis PN TR vrcumquè <lb/>diuidantur, quæ percentra KL tran&longs;eant; dummodo in figu <lb/>ris &longs;int &longs;imiliter ductæ; hoc e&longs;t, vellatera, vel angulos in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione di&longs;pe&longs;cant: vt &longs;it AP ad PD, vt ET ad TH. æ­<lb/>queponderabunt vtique partes PABN PNCD, veluti partes <lb/>TEFR TRGH. & hæc non e&longs;t &longs;implex æqueponderatio; ve­<lb/>rùm etiam (vtita dicam) &longs;imilis, & æqualis æqueponderatio. <lb/>cùm &longs;it &longs;ecundùm eandem proportionem, quandoquidem <lb/>e&longs;t PB ip&longs;i TF &longs;imilis, cùm triangula AKB ELF, AKP ELT, <lb/>BKN FLR, &longs;int inter &longs;e &longs;imilia, quæ quidem efficiunt, figuras | <arrow.to.target n="marg18"></arrow.to.target><lb/>ad EL, vt KM ad <expan abbr="Lq.">Lque</expan> pariquè ratione o&longs;tendetur triangu<lb/>lum BKM triangulo FLQ &longs;imile exi&longs;tere; e&longs;&longs;equè BK ad <lb/>FL, vt KM ad <expan abbr="Lq.">Lque</expan> &longs;imiliterquè in alijs triangulis o&longs;ten­<lb/>detur, ita e&longs;&longs;e Bk ad FL, vt KN ad LR; & Ck ad GL e&longs;&longs;e, vt <lb/>kO ad LS; atque kD ad LH, vt kP ad LT. quia verò AK <lb/>EL, Bk FL, Ck GL, Dk HL in eadem &longs;untproportione, vt <lb/>proximè demon&longs;tratum fuit; in eadem quoque proportione <lb/>erit kM ad LQ, & KN ad LR; & KO ad LS, atque kP ad <lb/>LT. ex quibus &longs;equitur centra grauitatis KL, non &longs;olùm ab <lb/>angulis in cadem proportione di&longs;tare; verùm etiam à lateri­<lb/>ribus in eadem quoque proportione di&longs;tare. </s><s>Itaque cognito, <lb/>quomodo intelligar Archimedes centra grauitatis in &longs;imili­<lb/>bus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; nunc con&longs;iderandum e&longs;t præ <lb/>cedens po&longs;tulatum, quatenus nimirum oporteat grauitatis <expan abbr="c&etilde;">cem</expan> <lb/>tra in &longs;imilibus figuris &longs;imiliter e&longs;&longs;e con&longs;tituta. </s><s>Nam inti­<lb/>miùs con&longs;iderando hanc &longs;imilem horum grauitatis <expan abbr="centrorũ">centrorum</expan> <lb/>po&longs;itionem, congruum, & nece&longs;&longs;arium videtur, &longs;imiles figu­<lb/>ras &longs;ecundùm eandem proportionem e&longs;&longs;e æquepon <expan abbr="derãtes">derantes</expan>; <lb/>eademquè ratione (ob earum &longs;imilitudinem) circa grauita­<lb/>tis centra æqueponderare, veluti &longs;i figuræ: AC EG (quarum <lb/>centra grauitatis &longs;int KL) à rectis lineis PN TR vrcumquè <lb/>diuidantur, quæ percentra KL tran&longs;eant; dummodo in figu<lb/>ris &longs;int &longs;imiliter ductæ; hoc e&longs;t, vellatera, vel angulos in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione di&longs;pe&longs;cant: vt &longs;it AP ad PD, vt ET ad TH. æ­<lb/>queponderabunt vtique partes PABN PNCD, veluti partes <lb/>TEFR TRGH. & hæc non e&longs;t &longs;implex æqueponderatio; ve­<lb/>rùm etiam (vtita dicam) &longs;imilis, & æqualis æqueponderatio. <lb/>cùm &longs;it &longs;ecundùm eandem proportionem, quandoquidem <lb/>e&longs;t PB ip&longs;i TF &longs;imilis, cùm triangula AKB ELF, AKP ELT, <lb/>BKN FLR, &longs;int inter &longs;e &longs;imilia, quæ quidem efficiunt, figuras |
| <pb pagenum="32"/>PB TF inter &longs;e &longs;imiles e&longs;&longs;e. </s><s>ob eademquè cau&longs;am e&longs;t PC &longs;i­<lb/>milis TG. quod quidem ex dem on&longs;tratis etiam facilè con­<lb/>&longs;tat. </s><s>cùm anguli &longs;int &ecedil;quales, & latera proportionalia. </s><s>Vtau­<lb/>tem clariùs intelligatur hæc &longs;imilis, & æqualis æquepondera <lb/>rio, adducerelibuit nonnulla ex ijs, quæ po&longs;teriùs tractanda <lb/>&longs;umentur. </s><s>Itaque intelligatur punctum V centrum e&longs;&longs;e gra­<lb/> | <pb pagenum="32"/>PB TF inter &longs;e &longs;imiles e&longs;&longs;e. </s><s>ob eademquè cau&longs;am e&longs;t PC &longs;i­<lb/>milis TG. quod quidem ex dem on&longs;tratis etiam facilè con­<lb/>&longs;tat. </s><s>cùm anguli &longs;int &ecedil;quales, & latera proportionalia. </s><s>Vtau­<lb/>tem clariùs intelligatur hæc &longs;imilis, & æqualis æquepondera <lb/>rio, adducerelibuit nonnulla ex ijs, quæ po&longs;teriùs tractanda <lb/>&longs;umentur. </s><s>Itaque intelligatur punctum V centrum e&longs;&longs;e gra­<lb/> |
| <arrow.to.target n="fig14"></arrow.to.target><lb/>uitatis figuræ PB, X verò centrum grauitatis figure TF. &longs;i <lb/>militer punctum Y centrum e&longs;&longs;e grauitatis figuræ PC, Z <lb/>verò figur&ecedil; TG. Iunganturquè VY XZ. quæ quidem per <lb/>centra grauitatis KL tran&longs;ibunt. </s><s>quòd ex ijs, qu&ecedil; dicenda <lb/>&longs;unt, manife&longs;tum erit, percipuèque ex octaua proportione <lb/>primi huius. </s><s>quod tamen interim &longs;upponatur. </s><s>At verò quo­<lb/>niam PB PC &ecedil;queponderant &longs;ecundùm proportionem, <lb/>quam habet YK ad KV; TF verò & TG &ecedil;queponderant <lb/>&longs;ecundùm proportionem, quam habet ZL ad LX. e&longs;t. <expan abbr="n.">enim</expan> <lb/>ac &longs;i AN e&longs;&longs;et appen&longs;a in V, & PC in Y; ER in X, & <lb/>TG in Z. vt in &longs;equentibus manife&longs;ta erunt. </s><s>Atverò quo­<lb/> | <arrow.to.target n="fig14"></arrow.to.target><lb/>uitatis figuræ PB, X verò centrum grauitatis figure TF. &longs;i <lb/>militer punctum Y centrum e&longs;&longs;e grauitatis figuræ PC, Z <lb/>verò figur&ecedil; TG. Iunganturquè VY XZ. quæ quidem per <lb/>centra grauitatis KL tran&longs;ibunt. </s><s>quòd ex ijs, qu&ecedil; dicenda <lb/>&longs;unt, manife&longs;tum erit, percipuèque ex octaua proportione <lb/>primi huius. </s><s>quod tamen interim &longs;upponatur. </s><s>At verò quo­<lb/>niam PB PC &ecedil;queponderant &longs;ecundùm proportionem, <lb/>quam habet YK ad KV; TF verò & TG &ecedil;queponderant <lb/>&longs;ecundùm proportionem, quam habet ZL ad LX. e&longs;t. <expan abbr="n.">enim</expan> <lb/>ac &longs;i AN e&longs;&longs;et appen&longs;a in V, & PC in Y; ER in X, & <lb/>TG in Z. vt in &longs;equentibus manife&longs;ta erunt. </s><s>Atverò quo­<lb/> |
| <arrow.to.target n="marg19"></arrow.to.target> niam AN &longs;imilis e&longs;t ip&longs;i ER, habebit AN ad ER <expan abbr="duplã">duplam</expan> <lb/>proportionem eius, quam habet latus PN ad TR. pariquè <lb/>ratione quoniam PC &longs;imilis e&longs;t TG, habebit PC ad TG <lb/>duplam proportionem eius, quam habet idem latus PN ad <lb/> | <arrow.to.target n="marg19"></arrow.to.target> niam AN &longs;imilis e&longs;t ip&longs;i ER, habebit AN ad ER <expan abbr="duplã">duplam</expan> <lb/>proportionem eius, quam habet latus PN ad TR. pariquè <lb/>ratione quoniam PC &longs;imilis e&longs;t TG, habebit PC ad TG <lb/>duplam proportionem eius, quam habet idem latus PN ad <lb/> |
| |
| <figure></figure> | <figure></figure> |
| <p type="main"> | <p type="main"> |
| <s>Quid intelligat Ar­<lb/>chimedes per has figu­<lb/>ras ad eandem partem <lb/>concauas, apertiùs &longs;i­<lb/>gnificauit initio libro­<lb/>rum de&longs;ph&ecedil;ra, & cylin­<lb/>dro. </s><s>vbi primùm vult <lb/>has figuras e&longs;&longs;e termina <lb/>tas; quod non &longs;olùm in <lb/>telligendum e&longs;t decur­<lb/>uilineis, verùm etiam <lb/>de rectilineis, & de mi­<lb/>xtis. </s><s>rectiline&ecedil; quidem <lb/>erunt trium, quattuor, <lb/>quinque & plurium la­<lb/>terum; quamuis latera <lb/>non &longs;int æqualia, ne­<lb/>que anguli &ecedil;quales, vt | <s>Quid intelligat Ar­<lb/>chimedes per has figu­<lb/>ras ad eandem partem <lb/>concauas, apertiùs &longs;i­<lb/>gnificauit initio libro­<lb/>rum de&longs;ph&ecedil;ra, & cylin­<lb/>dro. </s><s>vbi primùm vult <lb/>has figuras e&longs;&longs;e termina <lb/>tas; quod non &longs;olùm in <lb/>telligendum e&longs;t decur­<lb/>uilineis, verùm etiam <lb/>de rectilineis, & de mi­<lb/>xtis. </s><s>rectiline&ecedil; quidem <lb/>erunt trium, quattuor, <lb/>quinque & plurium la­<lb/>terum; quamuis latera <lb/>non &longs;int æqualia, ne­<lb/>que anguli &ecedil;quales, vt |
| <pb pagenum="35"/>ABCDE, cuiusom nes ang uli&longs;unt flexi ad interiorem figuræ <lb/>partem. </s><s>& hocmodo perimeter huius figuræ erit ad eandom <lb/>partem con cauus. </s><s>vnde excludun tur figuræ, exempli gratia <lb/>FGHKL; cùm angulus K non &longs;it &longs;inuo&longs;us, & con oauus ad <lb/>eandem partem, vt reliquidnguli; qui &longs;unt &longs;in uo&longs;<gap/> ver&longs;us lifte <lb/>riorem pamem &longs;igur&ecedil; K vero bd exterioitem. </s><s>&longs;imili modo <lb/>intelligen dum e&longs;t ded<gap/>lineis, vt dir<gap/>lis ellip&longs;es, vel alteri us <lb/>generis&longs;igræ, vt &longs;unt MN, quæ &longs;uam habent conqau tatem <lb/>adiean dem partem: &longs;ed curuline¸ OP ilnon &longs;unt ad ea n dem <lb/>partem concau&ecedil;. </s><s>Mixtæ quoque figuræ, ut&longs;unt portiones eil <lb/>culi, hyperbab&ecedil; ac para bod&ecedil; rectis linen <gap/>eminat&ecedil;; vel <gap/><lb/>rius gen erisfigur&ecedil;, vt &longs;pnt QR. h&ecedil; quidemom nes&longs;unt ad ea­<lb/>dem partem concauç Mixcæ verò ST minimè Regulgm au­<lb/>tem qua<gap/> vniuer&longs;alemper verbis Archimedislodo qitato <lb/>elicere po&longs;&longs;unus, vtoog nofcere valeam us, an figu<gap/> &longs;int ad <lb/>eandem partem concauæ, vel minùs vt fcilicet inboblata figu <lb/>ra vbicum que duo &longs;umi po&longs;&longs;int puncta, quæ &longs;i rectal<gap/><lb/>nectantur, tota recta li <lb/> | <pb pagenum="35"/>ABCDE, cuiusom nes ang uli&longs;unt flexi ad interiorem figuræ <lb/>partem. </s><s>& hocmodo perimeter huius figuræ erit ad eandom <lb/>partem con cauus. </s><s>vnde excludun tur figuræ, exempli gratia <lb/>FGHKL; cùm angulus K non &longs;it &longs;inuo&longs;us, & con oauus ad <lb/>eandem partem, vt reliquidnguli; qui &longs;unt &longs;in uo&longs;<gap/> ver&longs;us lifte <lb/>riorem pamem figur&ecedil; K vero bd exterioitem. </s><s>&longs;imili modo <lb/>intelligen dum e&longs;t ded<gap/>lineis, vt dir<gap/>lis ellip&longs;es, vel alteri us <lb/>generis&longs;igræ, vt &longs;unt MN, quæ &longs;uam habent conqau tatem <lb/>adiean dem partem: &longs;ed curuline¸ OP ilnon &longs;unt ad ea n dem <lb/>partem concau&ecedil;. </s><s>Mixtæ quoque figuræ, ut&longs;unt portiones eil <lb/>culi, hyperbab&ecedil; ac para bod&ecedil; rectis linen <gap/>eminat&ecedil;; vel <gap/><lb/>rius gen erisfigur&ecedil;, vt &longs;pnt QR. h&ecedil; quidemom nes&longs;unt ad ea­<lb/>dem partem concauç Mixcæ verò ST minimè Regulgm au­<lb/>tem qua<gap/> vniuer&longs;alemper verbis Archimedislodo qitato <lb/>elicere po&longs;&longs;unus, vtoog nofcere valeam us, an figu<gap/> &longs;int ad <lb/>eandem partem concauæ, vel minùs vt fcilicet inboblata figu<lb/>ra vbicum que duo &longs;umi po&longs;&longs;int puncta, quæ &longs;i rectal<gap/><lb/>nectantur, tota recta li <lb/> |
| <arrow.to.target n="fig16"></arrow.to.target><lb/>nea, velip&longs;ius pars ali­<lb/>qua extra figuram non <lb/>cadat. </s><s>vt in figuris A, <lb/>quæ &longs;unt ad <expan abbr="eand&etilde;">eandem</expan> par <lb/>tem concauæ, vtcum­<lb/>que duo &longs;umantur <expan abbr="pũ-cta">pun­<lb/>cta</expan> BC, quæ conne­<lb/>ctantur, tota utique re­<lb/>cta linea inter puncta <lb/>BC exi&longs;tens, extra figu <lb/>ram non cadet. </s><s>Quòd <lb/>&longs;i hæclinea cum termino, hoc e&longs;t eum latere figur&ecedil; conueni­<lb/>ret, vt &longs;i &longs;iguræ latus fueritrectum, in quo duo &longs;umantur pun <lb/>cta, nihilominus recta linea inter hæc puncta extra figuram <lb/>non cadei: quandoquidem figuræ terminus extra figuram mi <lb/>nimè roperitur atque hac ratione quomodocunque, & vbicú <lb/>que in his figuris duo &longs;um a ntur puncta, idem &longs;emper con tin <lb/>get. </s><s>Quod tamen figuris D &longs;emper euenite non pote&longs;t in qui <lb/>bus (cùm non &longs;int ad eandem partem concau&ecedil;) duo &longs;umero | <arrow.to.target n="fig16"></arrow.to.target><lb/>nea, velip&longs;ius pars ali­<lb/>qua extra figuram non <lb/>cadat. </s><s>vt in figuris A, <lb/>quæ &longs;unt ad <expan abbr="eand&etilde;">eandem</expan> par <lb/>tem concauæ, vtcum­<lb/>que duo &longs;umantur <expan abbr="pũ-cta">pun­<lb/>cta</expan> BC, quæ conne­<lb/>ctantur, tota utique re­<lb/>cta linea inter puncta <lb/>BC exi&longs;tens, extra figu<lb/>ram non cadet. </s><s>Quòd <lb/>&longs;i hæclinea cum termino, hoc e&longs;t eum latere figur&ecedil; conueni­<lb/>ret, vt &longs;i figuræ latus fueritrectum, in quo duo &longs;umantur pun <lb/>cta, nihilominus recta linea inter hæc puncta extra figuram <lb/>non cadei: quandoquidem figuræ terminus extra figuram mi <lb/>nimè roperitur atque hac ratione quomodocunque, & vbicú <lb/>que in his figuris duo &longs;um a ntur puncta, idem &longs;emper contin<lb/>get. </s><s>Quod tamen figuris D &longs;emper euenite non pote&longs;t in qui <lb/>bus (cùm non &longs;int ad eandem partem concau&ecedil;) duo &longs;umero |
| <pb pagenum="36"/>po&longs;&longs;umus puncta EG, inter quç tota recta linea EG extra <lb/>&longs;iguram cadet. </s><s>vel fumerepo&longs;&longs;umus puncta FG, ita vt rect&ecedil; <lb/>line&ecedil; FG pars EG extra figuram cadat. </s><s>figur&ecedil; igitur, quæ <lb/>ad ean dem partem &longs;unt concauæ, ill&ecedil; &longs;unt, qu&ecedil; &longs;inuo&longs;itatem, <lb/>concauitatemquè &longs;uam habent &longs;emper interiorem ip&longs;ius fi­<lb/>gur&ecedil; partem re&longs;picientem. </s><s>Harum què rectè &longs;upponit Archi­<lb/>medes centrum grauitatis &longs;emperle&longs;&longs;e intra ip&longs;am figuram. <lb/>ita vt neque centrum e&longs;&longs;e po&longs;&longs;icin ambitu ip&longs;ius figur&ecedil; ete­<lb/>nim &longs;i extra figuram, &longs;iue in ambitu ip&longs;ius e&longs;&longs;e po&longs;&longs;et, num­<lb/>quam circa centrum grauitatis partes figur&ecedil; vndiquè <expan abbr="&ecedil;quepõ">&ecedil;quepom</expan> <lb/> | <pb pagenum="36"/>po&longs;&longs;umus puncta EG, inter quç tota recta linea EG extra <lb/>figuram cadet. </s><s>vel fumerepo&longs;&longs;umus puncta FG, ita vt rect&ecedil; <lb/>line&ecedil; FG pars EG extra figuram cadat. </s><s>figur&ecedil; igitur, quæ <lb/>ad ean dem partem &longs;unt concauæ, ill&ecedil; &longs;unt, qu&ecedil; &longs;inuo&longs;itatem, <lb/>concauitatemquè &longs;uam habent &longs;emper interiorem ip&longs;ius fi­<lb/>gur&ecedil; partem re&longs;picientem. </s><s>Harum què rectè &longs;upponit Archi­<lb/>medes centrum grauitatis &longs;emperle&longs;&longs;e intra ip&longs;am figuram. <lb/>ita vt neque centrum e&longs;&longs;e po&longs;&longs;it in ambitu ip&longs;ius figur&ecedil; ete­<lb/>nim &longs;i extra figuram, &longs;iue in ambitu ip&longs;ius e&longs;&longs;e po&longs;&longs;et, num­<lb/>quam circa centrum grauitatis partes figur&ecedil; vndiquè <expan abbr="&ecedil;quepõ">&ecedil;quepom</expan> <lb/> |
| <arrow.to.target n="marg22"></arrow.to.target> derarent: neque facta ex grauitatis centro &longs;u&longs;pen&longs;ione figura <lb/>vbicumque, & in omni &longs;itu maneret. </s><s>quod ramen ex ratione <lb/>centri grauitatis efficere deberet. </s><s>to ta nimirum figura ex vna <lb/>e&longs;&longs;et parte, & ex altera nihil e&longs;&longs;et, quod ip&longs;i figur&ecedil; &ecedil;queponde <lb/>rare po&longs;&longs;et. </s><s>Nece&longs;&longs;e e&longs;t igitur centrum grauitatis cuiu&longs;libet fi­<lb/>gur&ecedil; ad ean dem partem concau&ecedil; e&longs;&longs;ein &longs;pacio à figur&ecedil; ambi <lb/>tu contento. </s><s>vt figur&ecedil; AB <lb/> | <arrow.to.target n="marg22"></arrow.to.target> derarent: neque facta ex grauitatis centro &longs;u&longs;pen&longs;ione figura <lb/>vbicumque, & in omni &longs;itu maneret. </s><s>quod ramen ex ratione <lb/>centri grauitatis efficere deberet. </s><s>to ta nimirum figura ex vna <lb/>e&longs;&longs;et parte, & ex altera nihil e&longs;&longs;et, quod ip&longs;i figur&ecedil; &ecedil;queponde <lb/>rare po&longs;&longs;et. </s><s>Nece&longs;&longs;e e&longs;t igitur centrum grauitatis cuiu&longs;libet fi­<lb/>gur&ecedil; ad ean dem partem concau&ecedil; e&longs;&longs;ein &longs;pacio à figur&ecedil; ambi <lb/>tu contento. </s><s>vt figur&ecedil; AB <lb/> |
| <arrow.to.target n="fig17"></arrow.to.target><lb/>centrum grauitatis erit in­<lb/>tra ip&longs;am, putà in C. quod <lb/>quidem non euenit &longs;emper <lb/>in alijs figuris, qu&ecedil; &longs;uum <expan abbr="cõ">com</expan> <lb/>cauitatis ambitum interio­<lb/>rem figur&ecedil; partem <expan abbr="nõ">non</expan> re&longs;pi­<lb/>cientem habent. </s><s>cùm varijs <lb/>modis po&longs;&longs;itcentrum graui<lb/>tatis in figuris e&longs;&longs;e <expan abbr="collocatũ">collocatum</expan>. <lb/>vt &longs;uperius quoque diximus. <lb/>Nam &longs;igur&ecedil; D <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis erit extra ambitum fi <lb/>gur&ecedil;, vt in E. figura verò F <lb/>ita &longs;e habere poterit, vt cen­<lb/>trum grauitatis &longs;it in perime <lb/>tro, vt in G. <expan abbr="euenitaut&etilde;">euenitautem</expan> aliquando vt in figura HK <expan abbr="centrũ">centrum</expan> <lb/>grauitatis L intra ip&longs;am figuram reperiatur; quamuis conca­<lb/>uitates la torum interiorem partem minimè <expan abbr="re&longs;piciãt">re&longs;piciant</expan>. Sed h&ecedil;c <lb/>po&longs;&longs;unt e&longs;&longs;e, & non e&longs;&longs;e, vt in figura M, cuius centrum extra <lb/>e&longs;&longs;e pote&longs;t in N. quamuis (vt an tea diximus) centrum graui- | <arrow.to.target n="fig17"></arrow.to.target><lb/>centrum grauitatis erit in­<lb/>tra ip&longs;am, putà in C. quod <lb/>quidem non euenit &longs;emper <lb/>in alijs figuris, qu&ecedil; &longs;uum <expan abbr="cõ">com</expan> <lb/>cauitatis ambitum interio­<lb/>rem figur&ecedil; partem <expan abbr="nõ">non</expan> re&longs;pi­<lb/>cientem habent. </s><s>cùm varijs <lb/>modis po&longs;&longs;itcentrum graui<lb/>tatis in figuris e&longs;&longs;e <expan abbr="collocatũ">collocatum</expan>. <lb/>vt &longs;uperius quoque diximus. <lb/>Nam figur&ecedil; D <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis erit extra ambitum fi <lb/>gur&ecedil;, vt in E. figura verò F <lb/>ita &longs;e habere poterit, vt cen­<lb/>trum grauitatis &longs;it in perime <lb/>tro, vt in G. euenit<expan abbr="aut&etilde;">autem</expan> aliquando vt in figura HK <expan abbr="centrũ">centrum</expan> <lb/>grauitatis L intra ip&longs;am figuram reperiatur; quamuis conca­<lb/>uitates la torum interiorem partem minimè <expan abbr="re&longs;piciãt">re&longs;piciant</expan>. Sed h&ecedil;c <lb/>po&longs;&longs;unt e&longs;&longs;e, & non e&longs;&longs;e, vt in figura M, cuius centrum extra <lb/>e&longs;&longs;e pote&longs;t in N. quamuis (vt an tea diximus) centrum graui- |
| <pb pagenum="37"/>tatis in tra figuram &longs;emper exi&longs;tere aliquo modo intelligi po­<lb/>te&longs;t. </s></p> | <pb pagenum="37"/>tatis in tra figuram &longs;emper exi&longs;tere aliquo modo intelligi po­<lb/>te&longs;t. </s></p> |
| <p type="margin"> | <p type="margin"> |
| <s><margin.target id="marg22"></margin.target><emph type="italics"/>per def. <lb/><expan abbr="c&etilde;t">cent</expan>. grau.<emph.end type="italics"/></s></p> | <s><margin.target id="marg22"></margin.target><emph type="italics"/>per def. <lb/><expan abbr="c&etilde;t">cent</expan>. grau.<emph.end type="italics"/></s></p> |
| |
| <p type="main"> | <p type="main"> |
| <s>Argumen tandi modus in e&longs;t in hac demon&longs;tratione maxi­<lb/>ma con&longs;ideratione dignus, & huius &longs;cientiæ maximè pro­<lb/>prius. </s><s>cùm enim dixi&longs;&longs;et Archimedes po&longs;ito centro grauitatis <lb/>magnitudinis ex AB compo&longs;itæ in puncto D, &longs;tatim infert. <lb/><emph type="italics"/>Quoniam igitur punctum D centrum e&longs;t grauitatis magnitudinis ex <lb/>AB compo&longs;ita, &longs;u&longs;pen&longs;o puncto D, magnitudines AB æquepondera­<lb/>bunt.<emph.end type="italics"/> hoc e&longs;t &longs;i magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur ex <lb/>D, manebit, vt reperitur; nec amplius in alteram partem in cli <lb/>nabit. </s><s>quod euenit ob naturam centri grauitatis, quod talis <lb/>e&longs;t naturæ (&longs;icuti initio explicauimus) ut &longs;i graue in eius cen­<lb/>tro grauitatis &longs;u&longs;tineatur, eo modo manet, quo reperitur, <expan abbr="dũ">dum</expan> <lb/>&longs;u&longs;penditur; parte&longs;què undiquè æqueponderant. </s><s>& ob id &longs;i <lb/>magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur in eius centro gra <lb/>uitatis, manet; parte&longs;què AB æqueponderant. </s><s>ac propterea <lb/>quando in &longs;equentibus quærit Archimedes, quoniam grauia <lb/>æqueponderare debent, tunc tan tùm quærit ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis, utin &longs;exta, &longs;eptimaquè propo&longs;itione in quit Archi­<lb/>medes magnitudines &ecedil;queponderare ex di&longs;tantijs, quç permu <lb/>tatim proportionem habent, utip&longs;arum grauitates, in <expan abbr="demõ">demom</expan> <lb/>&longs;tratione tamen quærit, vbi nam e&longs;t <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni <lb/>tudinis ex vtrisquè compo&longs;it&ecedil;. </s><s>quo inuento, &longs;tarim nece&longs;&longs;ariò <lb/>&longs;equitur, magnitudines, &longs;i ex ip&longs;o centro &longs;u&longs;pendantur, æque <lb/>ponderare. </s></p> | <s>Argumen tandi modus in e&longs;t in hac demon&longs;tratione maxi­<lb/>ma con&longs;ideratione dignus, & huius &longs;cientiæ maximè pro­<lb/>prius. </s><s>cùm enim dixi&longs;&longs;et Archimedes po&longs;ito centro grauitatis <lb/>magnitudinis ex AB compo&longs;itæ in puncto D, &longs;tatim infert. <lb/><emph type="italics"/>Quoniam igitur punctum D centrum e&longs;t grauitatis magnitudinis ex <lb/>AB compo&longs;ita, &longs;u&longs;pen&longs;o puncto D, magnitudines AB æquepondera­<lb/>bunt.<emph.end type="italics"/> hoc e&longs;t &longs;i magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur ex <lb/>D, manebit, vt reperitur; nec amplius in alteram partem in cli <lb/>nabit. </s><s>quod euenit ob naturam centri grauitatis, quod talis <lb/>e&longs;t naturæ (&longs;icuti initio explicauimus) ut &longs;i graue in eius cen­<lb/>tro grauitatis &longs;u&longs;tineatur, eo modo manet, quo reperitur, <expan abbr="dũ">dum</expan> <lb/>&longs;u&longs;penditur; parte&longs;què undiquè æqueponderant. </s><s>& ob id &longs;i <lb/>magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur in eius centro gra <lb/>uitatis, manet; parte&longs;què AB æqueponderant. </s><s>ac propterea <lb/>quando in &longs;equentibus quærit Archimedes, quoniam grauia <lb/>æqueponderare debent, tunc tan tùm quærit ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis, utin &longs;exta, &longs;eptimaquè propo&longs;itione in quit Archi­<lb/>medes magnitudines &ecedil;queponderare ex di&longs;tantijs, quç permu <lb/>tatim proportionem habent, utip&longs;arum grauitates, in <expan abbr="demõ">demom</expan> <lb/>&longs;tratione tamen quærit, vbi nam e&longs;t <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni <lb/>tudinis ex vtrisquè compo&longs;it&ecedil;. </s><s>quo inuento, &longs;tarim nece&longs;&longs;ariò <lb/>&longs;equitur, magnitudines, &longs;i ex ip&longs;o centro &longs;u&longs;pendantur, æque <lb/>ponderare. </s></p> |
| <p type="main"> | <p type="main"> |
| <s>Hinc colligere po&longs;&longs;umus alterum argumentandi modum, <lb/>conuer&longs;o nempè modo, veluti in eadem &longs;igura, &longs;i dicamus <lb/>grauia AB &longs;u&longs;pen&longs;a ex C æqueponderant, &longs;tatim inferre <lb/>po&longs;&longs;umus, punctum C ip&longs;orum &longs;imul grauium, hoc e&longs;t ma <lb/>gnitudinis ex ip&longs;is AB compo&longs;it&ecedil; centrum e&longs;&longs;e grauitatis. <lb/>Quare ad &longs;e inuicem conuertuntur, hoc punctum e&longs;t horum <lb/>grauium cen trum grauitatis; ergo h&ecedil;c grauia ex hoc puncto <lb/>æqùeponderant; & è conuer&longs;o, nempè hæc grauia ex hoc pun <lb/>cto æqueponderant, ergo idem punctum e&longs;t ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis. </s><s>&longs;ed ad uertendum hanc &longs;equi <expan abbr="conuertibilitat&etilde;">conuertibilitatem</expan>, <expan abbr="quã-do">quan­<lb/>do</expan> præfatum punctum e&longs;t in recta linea, quæ centra grauita­<lb/>tum ponderum coniungit; deinde quando h&ecedil;c linea non e&longs;t | <s>Hinc colligere po&longs;&longs;umus alterum argumentandi modum, <lb/>conuer&longs;o nempè modo, veluti in eadem figura, &longs;i dicamus <lb/>grauia AB &longs;u&longs;pen&longs;a ex C æqueponderant, &longs;tatim inferre <lb/>po&longs;&longs;umus, punctum C ip&longs;orum &longs;imul grauium, hoc e&longs;t ma <lb/>gnitudinis ex ip&longs;is AB compo&longs;it&ecedil; centrum e&longs;&longs;e grauitatis. <lb/>Quare ad &longs;e inuicem conuertuntur, hoc punctum e&longs;t horum <lb/>grauium cen trum grauitatis; ergo h&ecedil;c grauia ex hoc puncto <lb/>æqùeponderant; & è conuer&longs;o, nempè hæc grauia ex hoc pun <lb/>cto æqueponderant, ergo idem punctum e&longs;t ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis. </s><s>&longs;ed ad uertendum hanc &longs;equi <expan abbr="conuertibilitat&etilde;">conuertibilitatem</expan>, <expan abbr="quã-do">quan­<lb/>do</expan> præfatum punctum e&longs;t in recta linea, quæ centra grauita­<lb/>tum ponderum coniungit; deinde quando h&ecedil;c linea non e&longs;t |
| <pb pagenum="45"/>horizonti perpendicularis. </s><s>&longs;ecus aurem minimè. </s><s>Nam &longs;i pon <lb/>dera AB &longs;int in libra ADB, qu&ecedil; &longs;itarcuata, vel angulum <expan abbr="cō-&longs;tituat">con­<lb/>&longs;tituat</expan>, &longs;iue intelligatur libra recta linea AB, cui affixa &longs;it <lb/>perpendicularis CD. vt in tractatu de libra no&longs;trorum Me­<lb/>chanicorum diximus. </s><s>&longs;u&longs;pendantur autem pondera AB ex <lb/> | <pb pagenum="45"/>horizonti perpendicularis. </s><s>&longs;ecus aurem minimè. </s><s>Nam &longs;i pon <lb/>dera AB &longs;int in libra ADB, qu&ecedil; &longs;itarcuata, vel angulum <expan abbr="cō-&longs;tituat">con­<lb/>&longs;tituat</expan>, &longs;iue intelligatur libra recta linea AB, cui affixa &longs;it <lb/>perpendicularis CD. vt in tractatu de libra no&longs;trorum Me­<lb/>chanicorum diximus. </s><s>&longs;u&longs;pendantur autem pondera AB ex <lb/> |
| <arrow.to.target n="fig20"></arrow.to.target><lb/>D, & æqueponderent; <expan abbr="nõ">non</expan> <lb/>&longs;equitur tamen, ergo D <lb/><expan abbr="c&etilde;trum">centrum</expan> e&longs;t grauitatis ma­<lb/>gnitudinis ex AB com­<lb/>po&longs;it&ecedil;. </s><s>centrum enim gra <lb/>uita tis in linea exi&longs;tit AB <lb/>quæ centra grauitatis ma <lb/>gnitudinum AB coniun <lb/>git, nempein C. Verùm coniungat recta linea AB centra <lb/> | <arrow.to.target n="fig20"></arrow.to.target><lb/>D, & æqueponderent; <expan abbr="nõ">non</expan> <lb/>&longs;equitur tamen, ergo D <lb/><expan abbr="c&etilde;trum">centrum</expan> e&longs;t grauitatis ma­<lb/>gnitudinis ex AB com­<lb/>po&longs;it&ecedil;. </s><s>centrum enim gra <lb/>uita tis in linea exi&longs;tit AB <lb/>quæ centra grauitatis ma <lb/>gnitudinum AB coniun <lb/>git, nempein C. Verùm coniungat recta linea AB centra <lb/> |
| <arrow.to.target n="fig21"></arrow.to.target><lb/>grauita tis æqualium ponderum AB, lineaquè <lb/>AB, cuius medium &longs;it C, in centrum mundi <expan abbr="t&etilde;-dat">ten­<lb/>dat</expan>, magnitudoquè ex ip&longs;is AB compo&longs;ita vbi­<lb/>cunque &longs;u&longs;pendatur in linea AB, vt in E; ma <lb/>nebuntvtique pondera AB ex E &longs;u&longs;pen&longs;a, vt in <lb/>prima propo&longs;itione de libra no&longs;trorum Mecha­<lb/>nicorum o&longs;ten dimus. </s><s>cùm C &longs;it ip&longs;orum <expan abbr="centrū">centrum</expan> <lb/>grauita tis, & EC &longs;it horizonti erecta. </s><s>Et quam­<lb/>uis magnitudo ex ip&longs;is AB compo&longs;ita ex E &longs;u <lb/>&longs;pen&longs;a maneat; non propterea &longs;equitur ergo E <lb/>centrum e&longs;t grauitatis magnitudinis ex ip&longs;is AB <lb/>compo&longs;it&ecedil;. </s><s>ni&longs;i fortè accidat &longs;u&longs;pen&longs;io ex puncto <lb/>C. Præterea verò aduertendum e&longs;t in hoc ca&longs;u <expan abbr="põ">pom</expan> <lb/>dera AB, dici quidem po&longs;&longs;e, manere, non autem <lb/>æqueponderare. </s><s>omnia nimirum, qu&ecedil; æqueponderant, ma­<lb/>nent; &longs;ed non è conuer&longs;o, quæ manent, æqueponderant. </s><s>Nam <lb/>&longs;i pondus A maius fuerit pondere B; &longs;iue B maius, quàm <lb/>A, vbicunque fiat &longs;u&longs;pen&longs;io in linea AB, &longs;emper ob <expan abbr="eãdem">eandem</expan> <lb/>cau&longs;am, quomodocun que &longs;int pondera, manebunt; non ta­<lb/>men æqueponderabunt. </s><s>Vt enim pondera æqueponderent, <lb/>requiritur, vt pars parti, virtu&longs;què vnius virtuti alterius hinc <lb/>inde re&longs;i&longs;tere, & æquipollere po&longs;&longs;it; vt propriè dici po&longs;&longs;int <expan abbr="põ">pom</expan> <lb/>dera æqueponderare. </s><s>& vt hoc euenire po&longs;&longs;it, oportet, vt par­ | <arrow.to.target n="fig21"></arrow.to.target><lb/>grauita tis æqualium ponderum AB, lineaquè <lb/>AB, cuius medium &longs;it C, in centrum mundi <expan abbr="t&etilde;-dat">ten­<lb/>dat</expan>, magnitudoquè ex ip&longs;is AB compo&longs;ita vbi­<lb/>cunque &longs;u&longs;pendatur in linea AB, vt in E; ma <lb/>nebuntvtique pondera AB ex E &longs;u&longs;pen&longs;a, vt in <lb/>prima propo&longs;itione de libra no&longs;trorum Mecha­<lb/>nicorum o&longs;ten dimus. </s><s>cùm C &longs;it ip&longs;orum <expan abbr="centrū">centrum</expan> <lb/>grauita tis, & EC &longs;it horizonti erecta. </s><s>Et quam­<lb/>uis magnitudo ex ip&longs;is AB compo&longs;ita ex E &longs;u <lb/>&longs;pen&longs;a maneat; non propterea &longs;equitur ergo E <lb/>centrum e&longs;t grauitatis magnitudinis ex ip&longs;is AB <lb/>compo&longs;it&ecedil;. </s><s>ni&longs;i fortè accidat &longs;u&longs;pen&longs;io ex puncto <lb/>C. Præterea verò aduertendum e&longs;t in hoc ca&longs;u <expan abbr="põ">pom</expan> <lb/>dera AB, dici quidem po&longs;&longs;e, manere, non autem <lb/>æqueponderare. </s><s>omnia nimirum, qu&ecedil; æqueponderant, ma­<lb/>nent; &longs;ed non è conuer&longs;o, quæ manent, æqueponderant. </s><s>Nam <lb/>&longs;i pondus A maius fuerit pondere B; &longs;iue B maius, quàm <lb/>A, vbicunque fiat &longs;u&longs;pen&longs;io in linea AB, &longs;emper ob <expan abbr="eãdem">eandem</expan> <lb/>cau&longs;am, quomodocun que &longs;int pondera, manebunt; non ta­<lb/>men æqueponderabunt. </s><s>Vt enim pondera æqueponderent, <lb/>requiritur, vt pars parti, virtu&longs;què vnius virtuti alterius hinc <lb/>inde re&longs;i&longs;tere, & æquipollere po&longs;&longs;it; vt propriè dici po&longs;&longs;int <expan abbr="põ">pom</expan> <lb/>dera æqueponderare. </s><s>& vt hoc euenire po&longs;&longs;it, oportet, vt par­ |
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| <s>In demon&longs;tratione autem huius quartæ propo&longs;itionis in­<lb/>quit Archimedes. <emph type="italics"/>Quòd autem &longs;it in linea AB, præosten&longs;um e&longs;t.<emph.end type="italics"/> qua <lb/>&longs;i dicat Archimedes, &longs;e priùs o&longs;ten di&longs;&longs;e centrum grauitatis ma <lb/>gnitudinis ex AB compo&longs;itæ e&longs;&longs;ein linea AB; quod tamen <lb/>in ijs, quæ dicta &longs;unt, non videtur expre&longs;&longs;um. </s><s>virtute tamen &longs;i <lb/>con&longs;ideremus ea, qu&ecedil; in prima, tertiaquè propo&longs;itione dicta <lb/>&longs;unt, facilè ex his concludi pote&longs;t, centrum grauitatis magni­<lb/>tudinis ex duabus magnitudinibus compo&longs;itæ e&longs;&longs;e in recta li <lb/>nea, quæ ip&longs;arum centra grauitatis coniungit. </s><s>Quare memi­<lb/>ni&longs;&longs;e oportet eorum, qu&ecedil; a nobis in expo&longs;itione primi po&longs;tu <lb/>lati huius dicta fuere, nempè Archimedem &longs;upponere, di&longs;tan­<lb/>tias e&longs;&longs;e in vna, eademquè recta linea con&longs;titutas. </s><s>ideoquè in <lb/>prima propo&longs;itionec inquit, Grauia, qu&ecedil; ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> &ecedil;quali <lb/>bus <expan abbr="æquepõderãt">æqueponderant</expan>, æqualia e&longs;&longs;e inter &longs;e; Archimedes què <expan abbr="demõ">demom</expan> <lb/>&longs;trat, quòd quando æqueponderant, &longs;unt æqualia: ex dictis <lb/>&longs;equitur, &longs;i æqueponderant, ergo centrum grauitatis magni­<lb/>tudinis ex ip&longs;is compo&longs;it&ecedil; erit in eo puncto, vbi æqueponde­<lb/>rant; hoc e&longs;t in medio di&longs;tantiarum, line&ecedil; &longs;cilicet, qu&ecedil; <expan abbr="grauiũ">grauium</expan> <lb/>centra grauitatis coniungit. </s><s>quod idem e&longs;t, ac &longs;i Archimedes <lb/>dixi&longs;&longs;et. </s><s>Grauia, qu&ecedil; habent centrum grauitatis in medio li­<lb/>ne&ecedil;, qu&ecedil; magnitudinum centra grauitatis coniungit, &ecedil;qua­<lb/>lia &longs;unt inter &longs;e. </s><s>cuius quidem h&ecedil;c quarta propo&longs;itio videtur <lb/>e&longs;&longs;e conuer&longs;a. </s><s>quamuis Archimedes loco grauium nominet <lb/>magnitudines. </s><s>Pr&ecedil;terea in tertia propo&longs;itione, quoniam <expan abbr="o&longs;t&etilde;-dit">o&longs;ten­<lb/>dit</expan> Archimedes, in&ecedil;qualia grauia &ecedil;queponderare ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> <lb/>in&ecedil;qualibus, ita vt grauius &longs;it in minori di&longs;tantia, &longs;equitur er <lb/>go centrum grauitatis e&longs;t in eo puncto, vbi æqueponderant; <lb/>& idem e&longs;t, ac &longs;i dixi&longs;&longs;et, in æqualium grauium centrum gra­<lb/>uitatis e&longs;t in recta linea, quæ ip&longs;orum centra grauitatis con­<lb/>iungit; ita vt &longs;it propinquius grauiori, remotius uerò leuiori. | <s>In demon&longs;tratione autem huius quartæ propo&longs;itionis in­<lb/>quit Archimedes. <emph type="italics"/>Quòd autem &longs;it in linea AB, præosten&longs;um e&longs;t.<emph.end type="italics"/> qua <lb/>&longs;i dicat Archimedes, &longs;e priùs o&longs;ten di&longs;&longs;e centrum grauitatis ma <lb/>gnitudinis ex AB compo&longs;itæ e&longs;&longs;ein linea AB; quod tamen <lb/>in ijs, quæ dicta &longs;unt, non videtur expre&longs;&longs;um. </s><s>virtute tamen &longs;i <lb/>con&longs;ideremus ea, qu&ecedil; in prima, tertiaquè propo&longs;itione dicta <lb/>&longs;unt, facilè ex his concludi pote&longs;t, centrum grauitatis magni­<lb/>tudinis ex duabus magnitudinibus compo&longs;itæ e&longs;&longs;e in recta li <lb/>nea, quæ ip&longs;arum centra grauitatis coniungit. </s><s>Quare memi­<lb/>ni&longs;&longs;e oportet eorum, qu&ecedil; a nobis in expo&longs;itione primi po&longs;tu <lb/>lati huius dicta fuere, nempè Archimedem &longs;upponere, di&longs;tan­<lb/>tias e&longs;&longs;e in vna, eademquè recta linea con&longs;titutas. </s><s>ideoquè in <lb/>prima propo&longs;itionec inquit, Grauia, qu&ecedil; ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> &ecedil;quali <lb/>bus <expan abbr="æquepõderãt">æqueponderant</expan>, æqualia e&longs;&longs;e inter &longs;e; Archimedes què <expan abbr="demõ">demom</expan> <lb/>&longs;trat, quòd quando æqueponderant, &longs;unt æqualia: ex dictis <lb/>&longs;equitur, &longs;i æqueponderant, ergo centrum grauitatis magni­<lb/>tudinis ex ip&longs;is compo&longs;it&ecedil; erit in eo puncto, vbi æqueponde­<lb/>rant; hoc e&longs;t in medio di&longs;tantiarum, line&ecedil; &longs;cilicet, qu&ecedil; <expan abbr="grauiũ">grauium</expan> <lb/>centra grauitatis coniungit. </s><s>quod idem e&longs;t, ac &longs;i Archimedes <lb/>dixi&longs;&longs;et. </s><s>Grauia, qu&ecedil; habent centrum grauitatis in medio li­<lb/>ne&ecedil;, qu&ecedil; magnitudinum centra grauitatis coniungit, &ecedil;qua­<lb/>lia &longs;unt inter &longs;e. </s><s>cuius quidem h&ecedil;c quarta propo&longs;itio videtur <lb/>e&longs;&longs;e conuer&longs;a. </s><s>quamuis Archimedes loco grauium nominet <lb/>magnitudines. </s><s>Pr&ecedil;terea in tertia propo&longs;itione, quoniam <expan abbr="o&longs;t&etilde;-dit">o&longs;ten­<lb/>dit</expan> Archimedes, in&ecedil;qualia grauia &ecedil;queponderare ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> <lb/>in&ecedil;qualibus, ita vt grauius &longs;it in minori di&longs;tantia, &longs;equitur er <lb/>go centrum grauitatis e&longs;t in eo puncto, vbi æqueponderant; <lb/>& idem e&longs;t, ac &longs;i dixi&longs;&longs;et, in æqualium grauium centrum gra­<lb/>uitatis e&longs;t in recta linea, quæ ip&longs;orum centra grauitatis con­<lb/>iungit; ita vt &longs;it propinquius grauiori, remotius uerò leuiori. |
| <pb pagenum="48"/>vnde &longs;equitur centrum grauitatis ip&longs;orum grauium ubicum <lb/>que e&longs;&longs;e po&longs;&longs;e in recta linea, qu&ecedil; ipiorum centra grauitatis <expan abbr="cõ">com</expan> <lb/>iungit. </s><s>Ex quibus concludi potelt, <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni­<lb/>tudinis ex duabus magnitudinibus compo&longs;it&ecedil; e&longs;&longs;e in recta li <lb/>nea, quæ ip&longs;orum centra grauitatis connectit. </s></p> | <pb pagenum="48"/>vnde &longs;equitur centrum grauitatis ip&longs;orum grauium ubicum <lb/>que e&longs;&longs;e po&longs;&longs;e in recta linea, qu&ecedil; ipiorum centra grauitatis <expan abbr="cõ">com</expan> <lb/>iungit. </s><s>Ex quibus concludi potelt, <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni­<lb/>tudinis ex duabus magnitudinibus compo&longs;it&ecedil; e&longs;&longs;e in recta li <lb/>nea, quæ ip&longs;orum centra grauitatis connectit. </s></p> |
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| <s>Po&longs;tremò notandum e&longs;t, Archimedem ea, quæ in &longs;uperio <lb/>ribus propo&longs;itionibus nuncupauit grauia, in hac quarta pro <lb/>po&longs;itione, veluti etiam in &longs;equentibus, non ampliùs grauia, <lb/>&longs;ed (vti diximus) magnitudines nominare. </s><s>quod quidem his <lb/>de cau&longs;is id ab ip&longs;o factum exi&longs;timo. </s><s>primùm enim, quia in <lb/>his expre&longs;se quærit centrum grauitatis; quod quidem <expan abbr="c&etilde;trum">centrum</expan>, <lb/>quamuis &longs;it centrum grauitatis, potiùs re&longs;picit <expan abbr="magnitudin&etilde;">magnitudinem</expan>, <lb/>quàm graue aliquod. </s><s>Nam cùm dicim us centrum grauitatis, <lb/>&longs;tatim innuim us &longs;i tum, &longs;itum inquàm determinatum &longs;igu­<lb/>ræ, in qua e&longs;t; &longs;iquidem centrum grauitatis e&longs;t punctum, & <lb/>(vtita dicam) punctum grauitatis eius, in quo e&longs;t. </s><s>& ideo, <lb/>quoniam magnitudo formam habet dete mina tam, <expan abbr="centrũ">centrum</expan> <lb/>grauitatis rectè pote&longs;t re&longs;picere &longs;itum re&longs;pectu magnitudinis, <lb/>in qua e&longs;t; quod tamen efficere non pote&longs;t re&longs;pectu grauis. <lb/>etenim graue, ut graue e&longs;t, non habet formam determina <expan abbr="tã">tam</expan>; <lb/>cùm eadem grauitas e&longs;&longs;e po&longs;&longs;itin cubo, in piramide, alii&longs;què <lb/>corporibus quibu&longs;cunque, modò minoribus, modò maiori­<lb/>bus, prout &longs;unt diuer&longs;arum &longs;pecierum. </s><s>quare centrum grauita <lb/>tis non pote&longs;t re&longs;picere &longs;itum in grauibus, quatenus grauia <expan abbr="cõ">com</expan> <lb/>&longs;iderantur; &longs;ed quatenus magnitudines exi&longs;tunt. </s><s>Præterea Ar­<lb/>chimedes loco grauium magnitudines nominat, quia eas di­<lb/>ui&longs;ibiles con&longs;iderat, quod e&longs;t proprium magnitudinis; vt in &longs;e <lb/>xta, &longs;eptima, & octaua propo&longs;itione. </s><s>& quamuis, dum <expan abbr="diuidũ">diuidum</expan> <lb/>tur magnitudines, grauia quoque diui&longs;a proueniant; non ta­<lb/>men propterea grauia diuiduntur, ut grauia. <expan abbr="nõ">non</expan>.n. </s><s>hoc ip&longs;is <lb/>competit, vt grauibus; &longs;ed vt magnitudinibus, quæ &longs;unt por <lb/>&longs;e diui&longs;ibiles. </s><s>Archimedes igitur his de cau&longs;is nomen <expan abbr="grauiũ">grauium</expan> <lb/>in magnitudines mutauit. </s><s>in &longs;uperioribus enim theoremati­<lb/>bus pertractauit, quomodo res æqueponderant ex di&longs;tantijs <lb/>modò æqualibus, modò in æqualibus. </s><s>& quoniam res <expan abbr="&ecedil;quepõ">&ecedil;quepom</expan> <lb/>derant, prout &longs;unt magis grauia, & minùs grauia; non ut <expan abbr="sũt">sunt</expan> <lb/>maiores, vel minores magnitudines, &longs;iquidem talis naturæ | <s>Po&longs;tremò notandum e&longs;t, Archimedem ea, quæ in &longs;uperio <lb/>ribus propo&longs;itionibus nuncupauit grauia, in hac quarta pro <lb/>po&longs;itione, veluti etiam in &longs;equentibus, non ampliùs grauia, <lb/>&longs;ed (vti diximus) magnitudines nominare. </s><s>quod quidem his <lb/>de cau&longs;is id ab ip&longs;o factum exi&longs;timo. </s><s>primùm enim, quia in <lb/>his expre&longs;se quærit centrum grauitatis; quod quidem <expan abbr="c&etilde;trum">centrum</expan>, <lb/>quamuis &longs;it centrum grauitatis, potiùs re&longs;picit <expan abbr="magnitudin&etilde;">magnitudinem</expan>, <lb/>quàm graue aliquod. </s><s>Nam cùm dicim us centrum grauitatis, <lb/>&longs;tatim innuim us &longs;i tum, &longs;itum inquàm determinatum figu­<lb/>ræ, in qua e&longs;t; &longs;iquidem centrum grauitatis e&longs;t punctum, & <lb/>(vtita dicam) punctum grauitatis eius, in quo e&longs;t. </s><s>& ideo, <lb/>quoniam magnitudo formam habet dete mina tam, <expan abbr="centrũ">centrum</expan> <lb/>grauitatis rectè pote&longs;t re&longs;picere &longs;itum re&longs;pectu magnitudinis, <lb/>in qua e&longs;t; quod tamen efficere non pote&longs;t re&longs;pectu grauis. <lb/>etenim graue, ut graue e&longs;t, non habet formam determina <expan abbr="tã">tam</expan>; <lb/>cùm eadem grauitas e&longs;&longs;e po&longs;&longs;it in cubo, in piramide, alii&longs;què <lb/>corporibus quibu&longs;cunque, modò minoribus, modò maiori­<lb/>bus, prout &longs;unt diuer&longs;arum &longs;pecierum. </s><s>quare centrum grauita <lb/>tis non pote&longs;t re&longs;picere &longs;itum in grauibus, quatenus grauia <expan abbr="cõ">com</expan> <lb/>&longs;iderantur; &longs;ed quatenus magnitudines exi&longs;tunt. </s><s>Præterea Ar­<lb/>chimedes loco grauium magnitudines nominat, quia eas di­<lb/>ui&longs;ibiles con&longs;iderat, quod e&longs;t proprium magnitudinis; vt in &longs;e <lb/>xta, &longs;eptima, & octaua propo&longs;itione. </s><s>& quamuis, dum <expan abbr="diuidũ">diuidum</expan> <lb/>tur magnitudines, grauia quoque diui&longs;a proueniant; non ta­<lb/>men propterea grauia diuiduntur, ut grauia. <expan abbr="nõ">non</expan>.n. </s><s>hoc ip&longs;is <lb/>competit, vt grauibus; &longs;ed vt magnitudinibus, quæ &longs;unt por <lb/>&longs;e diui&longs;ibiles. </s><s>Archimedes igitur his de cau&longs;is nomen <expan abbr="grauiũ">grauium</expan> <lb/>in magnitudines mutauit. </s><s>in &longs;uperioribus enim theoremati­<lb/>bus pertractauit, quomodo res æqueponderant ex di&longs;tantijs <lb/>modò æqualibus, modò in æqualibus. </s><s>& quoniam res <expan abbr="&ecedil;quepõ">&ecedil;quepom</expan> <lb/>derant, prout &longs;unt magis grauia, & minùs grauia; non ut <expan abbr="sũt">sunt</expan> <lb/>maiores, vel minores magnitudines, &longs;iquidem talis naturæ |
| <pb pagenum="49"/>e&longs;&longs;e pote&longs;t minor magnitudo, qu&ecedil; maiore magnitudine alte <lb/>rius nature grauior exi&longs;tat; proindé Archimedesin &longs;uperiori­<lb/>busrectè grauia nuncupauit; optimèquè in his magnitudines <lb/>vocat. </s><s>Atverò aduertendum e&longs;t, quòd quamuis Archimedes <lb/>in his magnitudines nominet, non propterea exi&longs;tim andum <lb/>e&longs;t, eum intelligere magnitudines tantùm; &longs;ed magnitudines <lb/>grauitate prçditas, ita utin ip&longs;is omnino grauitatem re&longs;piciat. <lb/>Etenim pluribus modis in telligere po&longs;&longs;umus magnitudines, <lb/>vel enim ut &longs;int inter &longs;e eiu&longs;dem &longs;peciei, vel diuer&longs;æ; nec <expan abbr="nõ">non</expan> <lb/>in&longs;uper homogeneæ, vel heterogeneæ. </s><s>vt in hac propo&longs;itione <lb/><expan abbr="quãdo">quando</expan> Archimedes pponit duas magnitudines &ecedil;quales, tuc <lb/>intelligere po&longs;&longs;umus eas e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas; <lb/>quæ, cùm &longs;int æquales, erit & grauitas vnius grauita ti alterius <lb/>æqualis. </s><s>&longs;i verò con&longs;ideremus eas e&longs;&longs;e diuer&longs;æ &longs;peciei, & e­<lb/>tiam heterogeneas; tunc quando Archimedes proponit has <lb/>magnitudines æ quales; intelligendum e&longs;t, eas e&longs;&longs;e æ quales in <lb/>grauita te; quæ quidem efficit, vt demon&longs;tratio, quod propo­<lb/>&longs;itum e&longs;t, concludat. </s><s>vtex eius demon&longs;tratione patet. </s><s>Et his <lb/>quoque modis intelligere po&longs;&longs;umus magnitudines in &longs;equen <lb/>tibus v&longs;que ad nonam propo&longs;itionem in quibus &longs;cilicet intel <lb/>ligere po&longs;&longs;umus magnitudines e&longs;&longs;e non &longs;olùm eiu&longs;dem &longs;pe­<lb/>ciei, vel diuer&longs;æ, verùm etiam & homogeneas. </s><s>& heteroge­<lb/>neas. </s><s>ut po&longs;t &longs;eptimam clariùs o&longs;tendemus. </s><s>Verùm de­<lb/>mon&longs;trationes clariores red duntur, &longs;i intelligamus magnitu­<lb/>dines e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas, in quibus graui­<lb/>tas magnitudini re&longs;pondet, vt &longs;i ip&longs;arum altera fuerit alte­<lb/>rius dupla, & grauitas vnius grauitatis alterius dupla exi&longs;tat. <lb/>Quòd &longs;i magnitudo fuerit alterius tripla, vel quadrupla, &c. <lb/>erit & grauitas grauitatis tripla, vel quadrupla, & &longs;ic dein­<lb/>ceps. </s><s>deinde &longs;i magnitudo bifariam diui&longs;a fuerit, & ip&longs;ius gra <lb/>uitas in duas &ecedil;quas partes &longs;it quoque diui&longs;a. </s><s>quòd &longs;i magnitu­<lb/>do in plures diuidatur partes, & grauitas quoque in totidem <lb/>eiu&longs;dem proportionis diui&longs;a proueniat. </s></p> | <pb pagenum="49"/>e&longs;&longs;e pote&longs;t minor magnitudo, qu&ecedil; maiore magnitudine alte <lb/>rius nature grauior exi&longs;tat; proindé Archimedesin &longs;uperiori­<lb/>busrectè grauia nuncupauit; optimèquè in his magnitudines <lb/>vocat. </s><s>Atverò aduertendum e&longs;t, quòd quamuis Archimedes <lb/>in his magnitudines nominet, non propterea exi&longs;tim andum <lb/>e&longs;t, eum intelligere magnitudines tantùm; &longs;ed magnitudines <lb/>grauitate prçditas, ita utin ip&longs;is omnino grauitatem re&longs;piciat. <lb/>Etenim pluribus modis in telligere po&longs;&longs;umus magnitudines, <lb/>vel enim ut &longs;int inter &longs;e eiu&longs;dem &longs;peciei, vel diuer&longs;æ; nec <expan abbr="nõ">non</expan> <lb/>in&longs;uper homogeneæ, vel heterogeneæ. </s><s>vt in hac propo&longs;itione <lb/><expan abbr="quãdo">quando</expan> Archimedes pponit duas magnitudines &ecedil;quales, tuc <lb/>intelligere po&longs;&longs;umus eas e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas; <lb/>quæ, cùm &longs;int æquales, erit & grauitas vnius grauita ti alterius <lb/>æqualis. </s><s>&longs;i verò con&longs;ideremus eas e&longs;&longs;e diuer&longs;æ &longs;peciei, & e­<lb/>tiam heterogeneas; tunc quando Archimedes proponit has <lb/>magnitudines æ quales; intelligendum e&longs;t, eas e&longs;&longs;e æ quales in <lb/>grauita te; quæ quidem efficit, vt demon&longs;tratio, quod propo­<lb/>&longs;itum e&longs;t, concludat. </s><s>vtex eius demon&longs;tratione patet. </s><s>Et his <lb/>quoque modis intelligere po&longs;&longs;umus magnitudines in &longs;equen <lb/>tibus v&longs;que ad nonam propo&longs;itionem in quibus &longs;cilicet intel <lb/>ligere po&longs;&longs;umus magnitudines e&longs;&longs;e non &longs;olùm eiu&longs;dem &longs;pe­<lb/>ciei, vel diuer&longs;æ, verùm etiam & homogeneas. </s><s>& heteroge­<lb/>neas. </s><s>ut po&longs;t &longs;eptimam clariùs o&longs;tendemus. </s><s>Verùm de­<lb/>mon&longs;trationes clariores red duntur, &longs;i intelligamus magnitu­<lb/>dines e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas, in quibus graui­<lb/>tas magnitudini re&longs;pondet, vt &longs;i ip&longs;arum altera fuerit alte­<lb/>rius dupla, & grauitas vnius grauitatis alterius dupla exi&longs;tat. <lb/>Quòd &longs;i magnitudo fuerit alterius tripla, vel quadrupla, &c. <lb/>erit & grauitas grauitatis tripla, vel quadrupla, & &longs;ic dein­<lb/>ceps. </s><s>deinde &longs;i magnitudo bifariam diui&longs;a fuerit, & ip&longs;ius gra <lb/>uitas in duas &ecedil;quas partes &longs;it quoque diui&longs;a. </s><s>quòd &longs;i magnitu­<lb/>do in plures diuidatur partes, & grauitas quoque in totidem <lb/>eiu&longs;dem proportionis diui&longs;a proueniat. </s></p> |
| <pb pagenum="50"/> | <pb pagenum="50"/> |
| <p type="head"> | <p type="head"> |
| |
| <arrow.to.target n="marg183"></arrow.to.target><lb/>dupla. </s><s>hoc e&longs;t duæ AD ad BC, vt duæ PS ad PR. Itaque in <lb/>eadem &longs;unt proportione duç BC cum AD ad duas AD, vt <lb/>du&ecedil; PR <expan abbr="cũ">cum</expan> PS ad duas PS. &longs;icut verò du&ecedil; AD ad BC, ita du&ecedil; <lb/>PS ad PR. antecedentes igitur ad &longs;uas &longs;imul con&longs;equentes in | <arrow.to.target n="marg183"></arrow.to.target><lb/>dupla. </s><s>hoc e&longs;t duæ AD ad BC, vt duæ PS ad PR. Itaque in <lb/>eadem &longs;unt proportione duç BC cum AD ad duas AD, vt <lb/>du&ecedil; PR <expan abbr="cũ">cum</expan> PS ad duas PS. &longs;icut verò du&ecedil; AD ad BC, ita du&ecedil; <lb/>PS ad PR. antecedentes igitur ad &longs;uas &longs;imul con&longs;equentes in |
| <arrow.to.target n="marg184"></arrow.to.target><lb/>eadem erunt proportione. <emph type="italics"/>Quare &longs;icut duæ BC cum AD ad duas <lb/>AD cum BC, ita duæ RP cum PS ad duas P S cum PR, <lb/>verùm duæ quidem RP cum PS e&longs;t vtraque &longs;imul SR RP.<emph.end type="italics"/> bis <lb/>enim a&longs;&longs;umitur PR, &longs;emel verò PS. Cum autem lineæ DH ES <lb/>à lineis diuidantur &ecedil;quidi&longs;tantibus ED OT HM, erit DK ad | <arrow.to.target n="marg184"></arrow.to.target><lb/>eadem erunt proportione. <emph type="italics"/>Quare &longs;icut duæ BC cum AD ad duas <lb/>AD cum BC, ita duæ RP cum PS ad duas P S cum PR, <lb/>verùm duæ quidem RP cum PS e&longs;t vtraque &longs;imul SR RP.<emph.end type="italics"/> bis <lb/>enim a&longs;&longs;umitur PR, &longs;emel verò PS. Cum autem lineæ DH ES <lb/>à lineis diuidantur &ecedil;quidi&longs;tantibus ED OT HM, erit DK ad |
| <arrow.to.target n="marg185"></arrow.to.target><lb/>KH, vt ER ad CS; kD verò e&longs;t æqualis KH, erit ER ip&longs;i <lb/>RS &ecedil;qualis. </s><s>erit igitur ER cum RP, <emph type="italics"/>hoc est PE<emph.end type="italics"/> ip&longs;is SR RP <lb/>&ecedil;qualis. <emph type="italics"/>duæ verò PS cum PR e&longs;t vtraque PS SR.<emph.end type="italics"/> bis enim a&longs;­<lb/>&longs;umitur PS, &longs;emel què PR. & quoniam FS e&longs;t &ecedil;qualis ip&longs;i SR. <lb/>quod quidem eodem modo o&longs;tendetur, cùm &longs;it FS ad SR, vt <lb/>BH ad Hk. </s><s>erit FS cum SP, <emph type="italics"/>hoc est PF<emph.end type="italics"/> ip&longs;is PS SR æqualis. <lb/>Quare ita &longs;ehabet PE ad PF, vt duæ BC cum AD ad duas <lb/>AD cum BC. Centrum igitur grauitatis P trapezij ABCD <lb/>in linea e&longs;t EF, quæ <expan abbr="cõiungit">coniungit</expan> parallelas AD BC bifariam di | <arrow.to.target n="marg185"></arrow.to.target><lb/>KH, vt ER ad CS; kD verò e&longs;t æqualis KH, erit ER ip&longs;i <lb/>RS &ecedil;qualis. </s><s>erit igitur ER cum RP, <emph type="italics"/>hoc est PE<emph.end type="italics"/> ip&longs;is SR RP <lb/>&ecedil;qualis. <emph type="italics"/>duæ verò PS cum PR e&longs;t vtraque PS SR.<emph.end type="italics"/> bis enim a&longs;­<lb/>&longs;umitur PS, &longs;emel què PR. & quoniam FS e&longs;t &ecedil;qualis ip&longs;i SR. <lb/>quod quidem eodem modo o&longs;tendetur, cùm &longs;it FS ad SR, vt <lb/>BH ad Hk. </s><s>erit FS cum SP, <emph type="italics"/>hoc est PF<emph.end type="italics"/> ip&longs;is PS SR æqualis. <lb/>Quare ita &longs;ehabet PE ad PF, vt duæ BC cum AD ad duas <lb/>AD cum BC. Centrum igitur grauitatis P trapezij ABCD <lb/>in linea e&longs;t EF, quæ <expan abbr="cõiungit">coniungit</expan> parallelas AD BC bifariam di |
| <pb pagenum="110"/>ui&longs;as; ita vt pars PE, quæ e&longs;t ad minorem parallelam AD <lb/>reliquampartem PF eam habet proportionem, quam du <lb/>ip&longs;ius BC, quæ e&longs;t maior æquidi&longs;tautium, vna cum min <lb/>AD, ad duplam minoris AD cum maiore BC, <emph type="italics"/>ergo demons<gap/><lb/>ta &longs;unt, quæ propo&longs;ita fuerant.<emph.end type="italics"/></s></p> | <pb pagenum="110"/>ui&longs;as; ita vt pars PE, quæ e&longs;t ad minorem parallelam AD <lb/>reliquampartem PF eam habet proportionem, quam du <lb/>ip&longs;ius BC, quæ e&longs;t maior æquidi&longs;tantium, vna cum min <lb/>AD, ad duplam minoris AD cum maiore BC, <emph type="italics"/>ergo demons<gap/><lb/>ta &longs;unt, quæ propo&longs;ita fuerant.<emph.end type="italics"/></s></p> |
| <p type="margin"> | <p type="margin"> |
| <s><margin.target id="marg171"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 2.<emph type="italics"/>&longs;<gap/><emph.end type="italics"/></s></p> | <s><margin.target id="marg171"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 2.<emph type="italics"/>&longs;<gap/><emph.end type="italics"/></s></p> |
| <p type="margin"> | <p type="margin"> |
| |
| <s>LEMMA I.</s></p> | <s>LEMMA I.</s></p> |
| <p type="main"> | <p type="main"> |
| <s>Eandem habeat proportionem AB ad CD, quam habet <lb/>GH ad KL. CD verò ad EF <expan abbr="eã">eam</expan>, <expan abbr="quã">quam</expan> habet kL ad MN. &longs;intquè | <s>Eandem habeat proportionem AB ad CD, quam habet <lb/>GH ad KL. CD verò ad EF <expan abbr="eã">eam</expan>, <expan abbr="quã">quam</expan> habet kL ad MN. &longs;intquè |
| <pb pagenum="136"/>AB CD EF inter &longs;e &ecedil;quid&longs;tantes. </s><s>&longs;imiliter GH KL MN <lb/>æquidi&longs;tantes, &longs;intautem ductæ BDF HLN rectæ lineæ; &longs;it­<lb/>què BD ad DF, vt HL ad LN. &longs;itquè maior AB quàm <lb/>CD, & CD, quàm EF. vnde erit quoquè GH maior KL, <lb/>& KL, quam MN. iuncti&longs;què AC CE, & GK KM. <lb/>Dico &longs;pacium ACDB ad &longs;pacium CEFD eandem habere <lb/>proportionem, quam &longs;pacium GKLH ad &longs;pacium KMNL. </s></p> | <pb pagenum="136"/>AB CD EF inter &longs;e &ecedil;quid&longs;tantes. </s><s>&longs;imiliter GH KL MN <lb/>æquidi&longs;tantes, &longs;intantem ductæ BDF HLN rectæ lineæ; &longs;it­<lb/>què BD ad DF, vt HL ad LN. &longs;itquè maior AB quàm <lb/>CD, & CD, quàm EF. vnde erit quoquè GH maior KL, <lb/>& KL, quam MN. iuncti&longs;què AC CE, & GK KM. <lb/>Dico &longs;pacium ACDB ad &longs;pacium CEFD eandem habere <lb/>proportionem, quam &longs;pacium GKLH ad &longs;pacium KMNL. </s></p> |
| <figure></figure> | <figure></figure> |
| <p type="main"> | <p type="main"> |
| <s>Producantur AC CE, quæ cum BF conueniantin OP. <lb/>productæquè GK KM cum HN conueniant in QR. <lb/>concurrentenim, quoniam CD KL &longs;unt minores ip&longs;is AB <lb/> | <s>Producantur AC CE, quæ cum BF conueniantin OP. <lb/>productæquè GK KM cum HN conueniant in QR. <lb/>concurrentenim, quoniam CD KL &longs;unt minores ip&longs;is AB <lb/> |
| |
| <arrow.to.target n="marg258"></arrow.to.target><lb/>&longs;imiliquè modo diuidatur <foreign lang="greek">dz</foreign> in <foreign lang="greek"><10></foreign>, ita vt &longs;it <foreign lang="greek">z<10></foreign> ad <foreign lang="greek"><10>d</foreign>, vt trape <lb/>zium XT ad SV; erit punctum <foreign lang="greek"><10></foreign> grauitatis centrum figuræ <lb/>XSYVTP. quia verò ita e&longs;t AK ad EI, vt XT ad SV, erit <foreign lang="greek">en</foreign><lb/>ad <foreign lang="greek">ng</foreign>, vt <foreign lang="greek">z<10></foreign> ad <foreign lang="greek"><10>d</foreign>. Diuidatur <expan abbr="aũt">aunt</expan> deinceps <foreign lang="greek">l*h</foreign> in <foreign lang="greek">s</foreign>, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; <foreign lang="greek">ls</foreign> ad <foreign lang="greek">s*h</foreign>, vt <lb/>FH ad triangulum BGH, erit punctum <foreign lang="greek">s</foreign> centrum grauitatis <lb/>figuræ FGBHI. eademquè ratione diuidatur <foreign lang="greek">mk</foreign> in <foreign lang="greek">t</foreign>, &longs;itquè <lb/><foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>, vt YZ ad triangulum OQZ; erit punctum <foreign lang="greek">t</foreign> cen­<lb/>trum grauitatis figuræ YQOZV. &longs;ed e&longs;t FH ad BGD, vt YZ <lb/>ad OQZ, eritigitur <foreign lang="greek">ls</foreign> ad <foreign lang="greek">sh</foreign>, vt <foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>. Quoniam autem <lb/>ita e&longs;t Ak ad EI, vt XT ad SV, erit componendo AEFIKC | <arrow.to.target n="marg258"></arrow.to.target><lb/>&longs;imiliquè modo diuidatur <foreign lang="greek">dz</foreign> in <foreign lang="greek"><10></foreign>, ita vt &longs;it <foreign lang="greek">z<10></foreign> ad <foreign lang="greek"><10>d</foreign>, vt trape <lb/>zium XT ad SV; erit punctum <foreign lang="greek"><10></foreign> grauitatis centrum figuræ <lb/>XSYVTP. quia verò ita e&longs;t AK ad EI, vt XT ad SV, erit <foreign lang="greek">en</foreign><lb/>ad <foreign lang="greek">ng</foreign>, vt <foreign lang="greek">z<10></foreign> ad <foreign lang="greek"><10>d</foreign>. Diuidatur <expan abbr="aũt">aunt</expan> deinceps <foreign lang="greek">l*h</foreign> in <foreign lang="greek">s</foreign>, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; <foreign lang="greek">ls</foreign> ad <foreign lang="greek">s*h</foreign>, vt <lb/>FH ad triangulum BGH, erit punctum <foreign lang="greek">s</foreign> centrum grauitatis <lb/>figuræ FGBHI. eademquè ratione diuidatur <foreign lang="greek">mk</foreign> in <foreign lang="greek">t</foreign>, &longs;itquè <lb/><foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>, vt YZ ad triangulum OQZ; erit punctum <foreign lang="greek">t</foreign> cen­<lb/>trum grauitatis figuræ YQOZV. &longs;ed e&longs;t FH ad BGD, vt YZ <lb/>ad OQZ, eritigitur <foreign lang="greek">ls</foreign> ad <foreign lang="greek">sh</foreign>, vt <foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>. Quoniam autem <lb/>ita e&longs;t Ak ad EI, vt XT ad SV, erit componendo AEFIKC |
| <arrow.to.target n="marg259"></arrow.to.target><lb/>ad EI, vt figura XSYVTP ad SV; & e&longs;t EI ad FH, vt SV ad | <arrow.to.target n="marg259"></arrow.to.target><lb/>ad EI, vt figura XSYVTP ad SV; & e&longs;t EI ad FH, vt SV ad |
| <arrow.to.target n="marg260"></arrow.to.target><lb/>YZ. ergo ex æquali figura AEFIKC erit ad FH, vt figura <lb/>XSYVTP ad YZ. e&longs;t autem FH ad BGH, vt YZ ad OQZ. e­<lb/>ritigitur figura AEFIKC ad &longs;uas con&longs;equentes, ad figuram | <arrow.to.target n="marg260"></arrow.to.target><lb/>YZ. ergo ex æquali figura AEFIKC erit ad FH, vt figura <lb/>XSYVTP ad YZ. e&longs;t autem FH ad BGH, vt YZ ad OQZ. e­<lb/>ritigitur figura AEFIKC ad &longs;uas con&longs;equentes, ad figuram |
| <arrow.to.target n="marg261"></arrow.to.target><lb/>&longs;cilicet FGBHI, vt figura XSYVTP ad &longs;uas con&longs;equentes, hoc <lb/>e&longs;t ad figuram YQOZV. Diuidatur itaque <foreign lang="greek">sn</foreign> in <foreign lang="greek">x</foreign>, ita ut <foreign lang="greek">sx</foreign><lb/>ad <foreign lang="greek">x</foreign> &longs;it, vt &longs;igura AEFIKC ad figuram FGBHI, erit punctum | <arrow.to.target n="marg261"></arrow.to.target><lb/>&longs;cilicet FGBHI, vt figura XSYVTP ad &longs;uas con&longs;equentes, hoc <lb/>e&longs;t ad figuram YQOZV. Diuidatur itaque <foreign lang="greek">sn</foreign> in <foreign lang="greek">x</foreign>, ita ut <foreign lang="greek">sx</foreign><lb/>ad <foreign lang="greek">x</foreign> &longs;it, vt figura AEFIKC ad figuram FGBHI, erit punctum |
| <arrow.to.target n="marg262"></arrow.to.target><lb/><foreign lang="greek">x</foreign> <expan abbr="centrũ">centrum</expan> grauitatis totius figur&ecedil; AEFGBHIKC. &longs;imiliter di­<lb/>uidatur <foreign lang="greek">t<10></foreign> in <foreign lang="greek">c</foreign>, &longs;itque <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>, ut figura XSYVTP ad figu­<lb/>ram YQOZV, erit punctum <foreign lang="greek">c</foreign> centrum grauitatis totius fi­<lb/>guræ XSYQOZVTP. quia verò ita e&longs;t figura AEFIKC ad fi <lb/>guram FGBHI, vt figura XSYVTP ad figuram YQOZV. e­<lb/>rit <foreign lang="greek">sx</foreign> ad <foreign lang="greek">xn</foreign>, vt <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>. Itaque quoniam BD ad DL e&longs;t, vt <foreign lang="greek">sn</foreign><lb/>ad R9, cùm &longs;in^{4} ut&longs;exdecim ad &longs;eptem. </s><s>& e&longs;t L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, vt 9<foreign lang="greek">d</foreign><lb/>ad <foreign lang="greek">d</foreign>R, erit BD ad L<foreign lang="greek">g</foreign>, vt <foreign lang="greek">sn</foreign> ad 9<foreign lang="greek">d</foreign>. & vt BD ad <foreign lang="greek">g</foreign>D, ita OR ad | <arrow.to.target n="marg262"></arrow.to.target><lb/><foreign lang="greek">x</foreign> <expan abbr="centrũ">centrum</expan> grauitatis totius figur&ecedil; AEFGBHIKC. &longs;imiliter di­<lb/>uidatur <foreign lang="greek">t<10></foreign> in <foreign lang="greek">c</foreign>, &longs;itque <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>, ut figura XSYVTP ad figu­<lb/>ram YQOZV, erit punctum <foreign lang="greek">c</foreign> centrum grauitatis totius fi­<lb/>guræ XSYQOZVTP. quia verò ita e&longs;t figura AEFIKC ad fi <lb/>guram FGBHI, vt figura XSYVTP ad figuram YQOZV. e­<lb/>rit <foreign lang="greek">sx</foreign> ad <foreign lang="greek">xn</foreign>, vt <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>. Itaque quoniam BD ad DL e&longs;t, vt <foreign lang="greek">sn</foreign><lb/>ad R9, cùm &longs;in^{4} ut&longs;exdecim ad &longs;eptem. </s><s>& e&longs;t L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, vt 9<foreign lang="greek">d</foreign><lb/>ad <foreign lang="greek">d</foreign>R, erit BD ad L<foreign lang="greek">g</foreign>, vt <foreign lang="greek">sn</foreign> ad 9<foreign lang="greek">d</foreign>. & vt BD ad <foreign lang="greek">g</foreign>D, ita OR ad |
| <arrow.to.target n="marg263"></arrow.to.target><lb/><foreign lang="greek">d</foreign>R. rur&longs;us quoniam BD ad LM e&longs;t, vt OR ad 9<foreign lang="greek">a</foreign>, nempe vt &longs;ex <lb/>decim ad quinque; & e&longs;t L<foreign lang="greek">e</foreign> ad <foreign lang="greek">e</foreign>M, ut 9<foreign lang="greek">z</foreign> ad <foreign lang="greek">za</foreign>, erit BD ad <foreign lang="greek">e</foreign>L, <lb/>vt OR ad 9<foreign lang="greek">z</foreign>. e&longs;t verò BD ad L<foreign lang="greek">g</foreign>, vt OR ad 9<foreign lang="greek">d</foreign>; erit igitur BD ad <lb/>vtram que &longs;imul <foreign lang="greek">e</foreign>L L<foreign lang="greek">g</foreign>, hoc e&longs;t ad <foreign lang="greek">eg</foreign>, vt OR ad <foreign lang="greek">zd</foreign>. &longs;ed <expan abbr="quoniã">quoniam</expan> | <arrow.to.target n="marg263"></arrow.to.target><lb/><foreign lang="greek">d</foreign>R. rur&longs;us quoniam BD ad LM e&longs;t, vt OR ad 9<foreign lang="greek">a</foreign>, nempe vt &longs;ex <lb/>decim ad quinque; & e&longs;t L<foreign lang="greek">e</foreign> ad <foreign lang="greek">e</foreign>M, ut 9<foreign lang="greek">z</foreign> ad <foreign lang="greek">za</foreign>, erit BD ad <foreign lang="greek">e</foreign>L, <lb/>vt OR ad 9<foreign lang="greek">z</foreign>. e&longs;t verò BD ad L<foreign lang="greek">g</foreign>, vt OR ad 9<foreign lang="greek">d</foreign>; erit igitur BD ad <lb/>vtram que &longs;imul <foreign lang="greek">e</foreign>L L<foreign lang="greek">g</foreign>, hoc e&longs;t ad <foreign lang="greek">eg</foreign>, vt OR ad <foreign lang="greek">zd</foreign>. &longs;ed <expan abbr="quoniã">quoniam</expan> |
| <arrow.to.target n="marg264"></arrow.to.target><lb/>e&longs;t <foreign lang="greek">gn</foreign> ad <foreign lang="greek">ne</foreign>, vt <foreign lang="greek">d<10></foreign> ad <foreign lang="greek"><10>z</foreign>, erit BD ad <foreign lang="greek">gn</foreign>, vt OR ad <foreign lang="greek">d<10></foreign>. e&longs;t <expan abbr="aut&etilde;">autem</expan> BD <lb/>ad D<foreign lang="greek">g</foreign>, vt OR ad R<foreign lang="greek">d</foreign>, vt dictum e&longs;t, ergo BD ad D<foreign lang="greek">n</foreign> e&longs;t, vt OR <lb/>ad R<foreign lang="greek"><10></foreign>. &longs;imiliterquè <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan> BD ad BA ita e&longs;&longs;e, vt OR ad O<foreign lang="greek">t</foreign>. <lb/>Cùm itaque &longs;it BD ad DR, & ad B<foreign lang="greek">s</foreign>, ut OR ad R<foreign lang="greek"><10></foreign>, & ad O<foreign lang="greek">t</foreign>; e­<lb/>rit BD ad DR B<foreign lang="greek">s</foreign> &longs;imul, vt OR ad R<foreign lang="greek"><10></foreign> O<foreign lang="greek">t</foreign> &longs;imul, & permutan­<lb/>do tota BD ad totam OR, vt ablata D<foreign lang="greek">n</foreign>B<foreign lang="greek">s</foreign> ad ablatam R<foreign lang="greek"><10>ot</foreign>. | <arrow.to.target n="marg264"></arrow.to.target><lb/>e&longs;t <foreign lang="greek">gn</foreign> ad <foreign lang="greek">ne</foreign>, vt <foreign lang="greek">d<10></foreign> ad <foreign lang="greek"><10>z</foreign>, erit BD ad <foreign lang="greek">gn</foreign>, vt OR ad <foreign lang="greek">d<10></foreign>. e&longs;t <expan abbr="aut&etilde;">autem</expan> BD <lb/>ad D<foreign lang="greek">g</foreign>, vt OR ad R<foreign lang="greek">d</foreign>, vt dictum e&longs;t, ergo BD ad D<foreign lang="greek">n</foreign> e&longs;t, vt OR <lb/>ad R<foreign lang="greek"><10></foreign>. &longs;imiliterquè <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan> BD ad BA ita e&longs;&longs;e, vt OR ad O<foreign lang="greek">t</foreign>. <lb/>Cùm itaque &longs;it BD ad DR, & ad B<foreign lang="greek">s</foreign>, ut OR ad R<foreign lang="greek"><10></foreign>, & ad O<foreign lang="greek">t</foreign>; e­<lb/>rit BD ad DR B<foreign lang="greek">s</foreign> &longs;imul, vt OR ad R<foreign lang="greek"><10></foreign> O<foreign lang="greek">t</foreign> &longs;imul, & permutan­<lb/>do tota BD ad totam OR, vt ablata D<foreign lang="greek">n</foreign>B<foreign lang="greek">s</foreign> ad ablatam R<foreign lang="greek"><10>ot</foreign>. |
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