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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; 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 &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 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 | <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> |
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| <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/> |
| |
| <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 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/> | <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 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 | <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;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/> | <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;e in &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;e in &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 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- | <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> |
| |
| <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;e in 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;e in 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> |
| <p type="main"> | <p type="main"> |
| <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;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 | <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 Archimedes in &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 ut in 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 Archimedes in &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 ut in 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"> |
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