| version 1.11, 2002/08/18 16:29:50 |
version 1.17, 2002/08/18 16:50:30 |
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| <s>Itaque dico <lb/>harum figurarum in el­<lb/>lip&longs;i de&longs;criptarum cen­<lb/>trum grauitatis e&longs;&longs;e <expan abbr="pũ-ctum">pun­<lb/>ctum</expan> k, idem quod & el <lb/>lip&longs;is centrum. </s> | <s>Itaque dico <lb/>harum figurarum in el­<lb/>lip&longs;i de&longs;criptarum cen­<lb/>trum grauitatis e&longs;&longs;e <expan abbr="pũ-ctum">pun­<lb/>ctum</expan> k, idem quod & el <lb/>lip&longs;is centrum. </s> |
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| <s>quadri­<lb/>lateri enim a b c d cen­<lb/>trum e&longs;t k, ex decima e­<lb/>iu&longs;dem libri Archime­<lb/>dis, quippe <expan abbr="cũ">cum</expan> in eo om <lb/>nes diametri <expan abbr="cõueniãt">conueniant</expan>. <lb/>Sedin figura a l b m c n <lb/><arrow.to.target n="marg19"></arrow.to.target><lb/>do, quoniam trianguli <lb/>a l b centrum grauitatis <lb/><arrow.to.target n="marg20"></arrow.to.target><lb/>e&longs;t in linea l e: <expan abbr="trapezij&qacute;">trapezijque</expan>; a b m o centrum in linea e k: trape <lb/>zij o m c d in k g: & trianguli c n d in ip&longs;a g n: erit magnitu <lb/>dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen <lb/>trum grauitatis in linea l n: & o b candem cau&longs;&longs;am in linea <lb/>o m. </s> | <s>quadri­<lb/>lateri enim a b c d cen­<lb/>trum e&longs;t k, ex decima e­<lb/>iu&longs;dem libri Archime­<lb/>dis, quippe <expan abbr="cũ">cum</expan> in eo om <lb/>nes diametri <expan abbr="cõueniãt">conueniant</expan>. <lb/>Sedin figura a l b m c n <lb/><arrow.to.target n="marg19"></arrow.to.target><lb/>do, quoniam trianguli <lb/>a l b centrum grauitatis <lb/><arrow.to.target n="marg20"></arrow.to.target><lb/>e&longs;t in linea l e: <expan abbr="trapezij&qacute;">trapezijque</expan>; a b m o centrum in linea e k: trape <lb/>zij o m c d in k g: & trianguli c n d in ip&longs;a g n: erit magnitu <lb/>dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen <lb/>trum grauitatis in linea l n: & o b eandem cau&longs;&longs;am in linea <lb/>o m. </s> |
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| <s>e&longs;t enim trianguli a o d centrum in linea o h: trapezij <lb/>a l n d in h k: trapezij l b c n in k f: & trianguli b m c in fm. <lb/>cum ergo figuræ a l b m c n d o centrum grauitatis &longs;it in li­<lb/>nea l n, & in linea o m; erit centrum ip&longs;ius punctum k, in <pb pagenum="5"/>quo &longs;cilicet l n, o m conueniunt. </s> | <s>e&longs;t enim trianguli a o d centrum in linea o h: trapezij <lb/>a l n d in h k: trapezij l b c n in k f: & trianguli b m c in fm. <lb/>cum ergo figuræ a l b m c n d o centrum grauitatis &longs;it in li­<lb/>nea l n, & in linea o m; erit centrum ip&longs;ius punctum k, in <pb pagenum="5"/>quo &longs;cilicet l n, o m conueniunt. </s> |
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| <s>Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con­<lb/><figure id="fig8"></figure><lb/>ftat; at in ellip&longs;i nos demon&longs;tra­<lb/>uinms in commentariis in quin­<lb/>tam propo&longs;itionem Archimedis <lb/>de conoidibus, & &longs;phæroidibus. <lb/>erit igitur a centrum grauitatis <lb/>ip&longs;ius figuræ, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>dimus. </s> | <s>Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con­<lb/><figure id="fig8"></figure><lb/>ftat; at in ellip&longs;i nos demon&longs;tra­<lb/>uinms in commentariis in quin­<lb/>tam propo&longs;itionem Archimedis <lb/>de conoidibus, & &longs;phæroidibus. <lb/>erit igitur a centrum grauitatis <lb/>ip&longs;ius figuræ, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>dimus. </s> |
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| <s>Itaque quoniam circulus <lb/>a ad circulum d, uel ellip&longs;is a ad <lb/>ellip&longs;im d candem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam linea c a ad a b: <lb/>portiones uero &longs;unt minores cir <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>culo uel ellip&longs;i d: habebit circu­<lb/>lus, uel ellip&longs;is ad portiones ma­<lb/>iorem proportionem, quàm c a <lb/><arrow.to.target n="marg22"></arrow.to.target><lb/>ad a b: & diuidendo figura recti­<lb/>linea e f g h k l m n ad portiones <pb pagenum="6"/><figure id="fig9"></figure><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. </s> | <s>Itaque quoniam circulus <lb/>a ad circulum d, uel ellip&longs;is a ad <lb/>ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam linea c a ad a b: <lb/>portiones uero &longs;unt minores cir <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>culo uel ellip&longs;i d: habebit circu­<lb/>lus, uel ellip&longs;is ad portiones ma­<lb/>iorem proportionem, quàm c a <lb/><arrow.to.target n="marg22"></arrow.to.target><lb/>ad a b: & diuidendo figura recti­<lb/>linea e f g h k l m n ad portiones <pb pagenum="6"/><figure id="fig9"></figure><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. </s> |
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| <s>fiat o b ad b a, <lb/>ut figura rectilinea ad portio­<lb/>nes. </s> | <s>fiat o b ad b a, <lb/>ut figura rectilinea ad portio­<lb/>nes. </s> |
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| <s>eadem quoque ratione demon&longs;trabimus <pb/>t u, x y ip&longs;i g h æquidi&longs;tare. </s> | <s>eadem quoque ratione demon&longs;trabimus <pb/>t u, x y ip&longs;i g h æquidi&longs;tare. </s> |
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| <s>Et quoniam triangula, quæ <lb/>fiunt à lineis K y, y u, u s, s h æqualiz funt inter &longs;e, & &longs;imilia <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>triangulo K m h: habebit triangulum K m h ad <expan abbr="triangulũ">triangulum</expan> <lb/>K <foreign lang="greek">d</foreign> y duplam proportioncm eius, quæ e&longs;t lineæ <foreign lang="greek">k</foreign> h ad K y. <lb/>&longs;ed K h po&longs;ita e&longs;t quadrupla ip&longs;ius k y. </s> | <s>Et quoniam triangula, quæ <lb/>fiunt à lineis K y, y u, u s, s h æqualiz &longs;unt inter &longs;e, & &longs;imilia <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>triangulo K m h: habebit triangulum K m h ad <expan abbr="triangulũ">triangulum</expan> <lb/>K <foreign lang="greek">d</foreign> y duplam proportioncm eius, quæ e&longs;t lineæ <foreign lang="greek">k</foreign> h ad K y. <lb/>&longs;ed K h po&longs;ita e&longs;t quadrupla ip&longs;ius k y. </s> |
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| <s>ergo triangulum <lb/>k m h ad triangulum K <foreign lang="greek">d</foreign> y <expan abbr="eãdem">eandem</expan> proportionem habcbit, <lb/>quam &longs;exdecim ad <expan abbr="unũ">unum</expan>: & ad quatuor triangula k <foreign lang="greek">d</foreign> y, y u, <lb/>u s, s <foreign lang="greek">a</foreign> h habebit eandem, quam &longs;exdecim ad quatuor, hoc <lb/>e&longs;t quam h K ad k y: & &longs;imiliter eandem habere demon&longs;tra <lb/><figure id="fig18"></figure><lb/>bitur trian­<lb/>gulum k m g <lb/>ad quatuor <lb/><expan abbr="triãgula">triangula</expan> K <foreign lang="greek">d</foreign><lb/>x, x <foreign lang="greek">g</foreign> t, t <foreign lang="greek">b</foreign> r, <lb/><arrow.to.target n="marg39"></arrow.to.target><lb/>r z g. </s> | <s>ergo triangulum <lb/>k m h ad triangulum K <foreign lang="greek">d</foreign> y <expan abbr="eãdem">eandem</expan> proportionem habcbit, <lb/>quam &longs;exdecim ad <expan abbr="unũ">unum</expan>: & ad quatuor triangula k <foreign lang="greek">d</foreign> y, y u, <lb/>u s, s <foreign lang="greek">a</foreign> h habebit eandem, quam &longs;exdecim ad quatuor, hoc <lb/>e&longs;t quam h K ad k y: & &longs;imiliter eandem habere demon&longs;tra <lb/><figure id="fig18"></figure><lb/>bitur trian­<lb/>gulum k m g <lb/>ad quatuor <lb/><expan abbr="triãgula">triangula</expan> K <foreign lang="greek">d</foreign><lb/>x, x <foreign lang="greek">g</foreign> t, t <foreign lang="greek">b</foreign> r, <lb/><arrow.to.target n="marg39"></arrow.to.target><lb/>r z g. </s> |
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| <s>Sed ut circulus <lb/>uel ellip&longs;is g h ad figuram rectilincam in ip&longs;a de&longs;cri­<lb/>ptam, ita e&longs;t cylindrus uel cylindri portio c c ad pri&longs;ma, <lb/>quod rectilineam figuram pro ba&longs;i habet, & altitudinem <lb/>æqualem; id, quod infra demon&longs;trabitur. </s> | <s>Sed ut circulus <lb/>uel ellip&longs;is g h ad figuram rectilincam in ip&longs;a de&longs;cri­<lb/>ptam, ita e&longs;t cylindrus uel cylindri portio c c ad pri&longs;ma, <lb/>quod rectilineam figuram pro ba&longs;i habet, & altitudinem <lb/>æqualem; id, quod infra demon&longs;trabitur. </s> |
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| <s>crgo per conuer <lb/>&longs;ionem rationis, ut circulus, uel ellip&longs;is g h ad portioncs re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad &longs;olidas por­<lb/>tiones, quate cylindrus uel cylindri portio ad &longs;olidas por­<lb/>tiones eandem proportionem habet, quam linea n <foreign lang="greek">k</foreign> ad k <lb/>& diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o­<lb/>lidas portiones candem proportionem habet, quam n l ad <lb/>l k & quoniam a cylindro uel cylindri portione, cuius gra­<lb/>uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili­<lb/>neam <expan abbr="figurã">figuram</expan>, cuius <expan abbr="centrũ">centrum</expan> grauitatis e&longs;t K: re&longs;iduæ magnitu <lb/>dinis ex &longs;olidis portionibus <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> grauitatis <expan abbr="c&etilde;trũ">centrum</expan> crit <lb/>in linea k l protracta, & in puncto n; quod e&longs;t <expan abbr="ab&longs;urdũ">ab&longs;urdum</expan>. </s> | <s>crgo per conuer <lb/>&longs;ionem rationis, ut circulus, uel ellip&longs;is g h ad portioncs re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad &longs;olidas por­<lb/>tiones, quate cylindrus uel cylindri portio ad &longs;olidas por­<lb/>tiones eandem proportionem habet, quam linea n <foreign lang="greek">k</foreign> ad k <lb/>& diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o­<lb/>lidas portiones eandem proportionem habet, quam n l ad <lb/>l k & quoniam a cylindro uel cylindri portione, cuius gra­<lb/>uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili­<lb/>neam <expan abbr="figurã">figuram</expan>, cuius <expan abbr="centrũ">centrum</expan> grauitatis e&longs;t K: re&longs;iduæ magnitu <lb/>dinis ex &longs;olidis portionibus <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> grauitatis <expan abbr="c&etilde;trũ">centrum</expan> crit <lb/>in linea k l protracta, & in puncto n; quod e&longs;t <expan abbr="ab&longs;urdũ">ab&longs;urdum</expan>. </s> |
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| <s>relin <lb/>quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>tionis &longs;it <expan abbr="punctũ">punctum</expan> k. </s> | <s>relin <lb/>quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>tionis &longs;it <expan abbr="punctũ">punctum</expan> k. </s> |
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| <s>Si enini <lb/>axes in eadem recta linea fuerint con&longs;tituti, bæc dao lo'i­<lb/>da, in unum, atque idem &longs;olidum conuenient. </s> | <s>Si enini <lb/>axes in eadem recta linea fuerint con&longs;tituti, bæc dao lo'i­<lb/>da, in unum, atque idem &longs;olidum conuenient. </s> |
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| <s>quare <gap/>x <lb/>iis, quæ proxime tradita &longs;unt, habebit &longs;olidum a b ad &longs;o­<lb/>lidum a c candem proportionem, quam axis d e ad e f <lb/>axem. </s> | <s>quare <gap/>x <lb/>iis, quæ proxime tradita &longs;unt, habebit &longs;olidum a b ad &longs;o­<lb/>lidum a c eandem proportionem, quam axis d e ad e f <lb/>axem. </s> |
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| <s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, d g, fh: <lb/>& jungantur e g, e h. </s> | <s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, d g, fh: <lb/>& jungantur e g, e h. </s> |
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| <s>ergo & pri&longs;ma a e ad pri&longs;ma f l eandem propor­<lb/>tionem habebit, quam altitudo ad altitudinem. </s> | <s>ergo & pri&longs;ma a e ad pri&longs;ma f l eandem propor­<lb/>tionem habebit, quam altitudo ad altitudinem. </s> |
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| <s>&longs;equitur <lb/>igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;tituũ">con&longs;tituum</expan> <lb/>tur, candem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s></p><p type="margin"> | <s>&longs;equitur <lb/>igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;tituũ">con&longs;tituum</expan> <lb/>tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s></p><p type="margin"> |
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| <s><margin.target id="marg63"></margin.target>6. du<gap/><lb/>cimi<gap/></s></p><p type="margin"> | <s><margin.target id="marg63"></margin.target>6. du<gap/><lb/>cimi<gap/></s></p><p type="margin"> |
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| <s>Simili enim ratione, qua &longs;upra, demon&longs;trabi­<lb/>tur quadratum a b ad quadratum f g ita e&longs;&longs;e, ut <expan abbr="quadratũ">quadratum</expan> <lb/><arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. </s> | <s>Simili enim ratione, qua &longs;upra, demon&longs;trabi­<lb/>tur quadratum a b ad quadratum f g ita e&longs;&longs;e, ut <expan abbr="quadratũ">quadratum</expan> <lb/><arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. </s> |
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| <s>Sed circuli inter &longs;e candem propor­<lb/>tionem habent, quam diametrorum quadrata. </s> | <s>Sed circuli inter &longs;e eandem propor­<lb/>tionem habent, quam diametrorum quadrata. </s> |
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| <s>ellip&longs;es au­<lb/>tem circa a b, f g, c d, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="cõ-mentariis">con­<lb/>mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>& &longs;phæroidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio­<lb/><figure id="fig45"></figure><lb/>&longs;eptimæ propo&longs;itionis eiu&longs;dem li­<lb/>bri. </s> | <s>ellip&longs;es au­<lb/>tem circa a b, f g, c d, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="cõ-mentariis">con­<lb/>mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>& &longs;phæroidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio­<lb/><figure id="fig45"></figure><lb/>&longs;eptimæ propo&longs;itionis eiu&longs;dem li­<lb/>bri. </s> |
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| <s>quare pyra­<lb/>mis, uel conus, uel coni por­<lb/>tio, cuius ba&longs;is e&longs;t tribus illis <lb/>ba&longs;ibus æqualis ad a g b eam <lb/>habet proportionem, quam <lb/>ba&longs;es a b, e f, c d ad a b ba&longs;im. </s> | <s>quare pyra­<lb/>mis, uel conus, uel coni por­<lb/>tio, cuius ba&longs;is e&longs;t tribus illis <lb/>ba&longs;ibus æqualis ad a g b eam <lb/>habet proportionem, quam <lb/>ba&longs;es a b, e f, c d ad a b ba&longs;im. </s> |
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| <s>Fru&longs;tum igitur a d ad a g b <pb/>pyramidem, uel conuni, uel coni portionem candem pro­<lb/>portionem habet, quam ba&longs;es a b, c d unà cum e f ad ba­<lb/>&longs;im a b. </s> | <s>Fru&longs;tum igitur a d ad a g b <pb/>pyramidem, uel conuni, uel coni portionem eandem pro­<lb/>portionem habet, quam ba&longs;es a b, c d unà cum e f ad ba­<lb/>&longs;im a b. </s> |
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| <s>quod demon&longs;trare uolebamus.</s></p><p type="margin"> | <s>quod demon&longs;trare uolebamus.</s></p><p type="margin"> |
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| <s>ergo <lb/>d z ad z m e&longs;t, ut g p ad p n. <lb/>Quòd cum g p &longs;it tripla p n; <lb/>erit etiam d z ip&longs;ius z m tri­<lb/>pla. </s> | <s>ergo <lb/>d z ad z m e&longs;t, ut g p ad p n. <lb/>Quòd cum g p &longs;it tripla p n; <lb/>erit etiam d z ip&longs;ius z m tri­<lb/>pla. </s> |
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| <s>atque ob candem cau&longs;­<lb/>&longs;am punctuniz e&longs;t <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis pyramidis b c f e d. </s> | <s>atque ob eandem cau&longs;­<lb/>&longs;am punctuniz e&longs;t <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis pyramidis b c f e d. </s> |
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| <s>iun <lb/>ctaigitur z u, in ea erit <expan abbr="c&etilde;trum">centrum</expan> <pb pagenum="36"/>grauitatis magnitudinis, quæ ex utri&longs;que pyramidibus <expan abbr="cõ">com</expan> <lb/>&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s> | <s>iun <lb/>ctaigitur z u, in ea erit <expan abbr="c&etilde;trum">centrum</expan> <pb pagenum="36"/>grauitatis magnitudinis, quæ ex utri&longs;que pyramidibus <expan abbr="cõ">com</expan> <lb/>&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s> |
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| <s>ergo linea x s per p tran&longs;ibit: & lineæ r u, s x, q t in­<lb/>ter &longs;e æquidi&longs;tantes erunt. </s> | <s>ergo linea x s per p tran&longs;ibit: & lineæ r u, s x, q t in­<lb/>ter &longs;e æquidi&longs;tantes erunt. </s> |
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| <s>Itaque cum fru&longs;ti a g latera pro­<pb pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y cocant, erunt trià <lb/>gala u y l, x y p, t y k inter &longs;e &longs;imilia: & &longs;imilia etiam triangu <lb/>la l y r, p y s, k y <expan abbr="q.">que</expan> quare ut in 19 huius, demon&longs;trabitur <lb/>x p, ad p s: <expan abbr="itemq;">itemque</expan> t <foreign lang="greek">k</foreign> ad k q candem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam u l ad l r. </s> | <s>Itaque cum fru&longs;ti a g latera pro­<pb pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y cocant, erunt trià <lb/>gala u y l, x y p, t y k inter &longs;e &longs;imilia: & &longs;imilia etiam triangu <lb/>la l y r, p y s, k y <expan abbr="q.">que</expan> quare ut in 19 huius, demon&longs;trabitur <lb/>x p, ad p s: <expan abbr="itemq;">itemque</expan> t <foreign lang="greek">k</foreign> ad k q eandem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam u l ad l r. </s> |
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| <s>Sed ut u l ad l <gap/>, ita e&longs;t triangulum a b c ad <lb/>triangulum a c d: & ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s> | <s>Sed ut u l ad l <gap/>, ita e&longs;t triangulum a b c ad <lb/>triangulum a c d: & ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s> |
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| <s>ergo <lb/>ex æquali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ip&longs;is quadrati a d duabus tertiis, ad <expan abbr="tertiã">tertiam</expan> <lb/>partem quadrati b c, ut k h ad h f: & ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>tertiæ partis, ad quam unà cum ip&longs;a portione, duplam pro <lb/>portionem habeat eius, quæ e&longs;t quadrati b c ad <expan abbr="quadratũ">quadratum</expan> <lb/>a d, ut K l ad l h. </s> | <s>ergo <lb/>ex æquali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ip&longs;is quadrati a d duabus tertiis, ad <expan abbr="tertiã">tertiam</expan> <lb/>partem quadrati b c, ut k h ad h f: & ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>tertiæ partis, ad quam unà cum ip&longs;a portione, duplam pro <lb/>portionem habeat eius, quæ e&longs;t quadrati b c ad <expan abbr="quadratũ">quadratum</expan> <lb/>a d, ut K l ad l h. </s> |
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| <s>habet enim K l ad l h candem proportio­<lb/>nem, quam conoidis portio b g c ad portionem a g d: por­<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, quæ e&longs;t ba&longs;is b c ad ba&longs;im a d: hoc e&longs;t quadrati <lb/><arrow.to.target n="marg118"></arrow.to.target><lb/>b c ad quadratum a d; ut proxime demon&longs;tratum e&longs;t. </s> | <s>habet enim K l ad l h eandem proportio­<lb/>nem, quam conoidis portio b g c ad portionem a g d: por­<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, quæ e&longs;t ba&longs;is b c ad ba&longs;im a d: hoc e&longs;t quadrati <lb/><arrow.to.target n="marg118"></arrow.to.target><lb/>b c ad quadratum a d; ut proxime demon&longs;tratum e&longs;t. </s> |
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| <s>quare <lb/>dempto a d quadrato à duabus tertiis quadrati b c, erit id, <lb/>quod relinquitur unà cum dicta portione tertiæ partis ad <lb/>reliquam eiu&longs;dem portionem, ut e l ad l f. </s> | <s>quare <lb/>dempto a d quadrato à duabus tertiis quadrati b c, erit id, <lb/>quod relinquitur unà cum dicta portione tertiæ partis ad <lb/>reliquam eiu&longs;dem portionem, ut e l ad l f. </s> |
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