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

version 1.2, 2002/06/24 18:03:52

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<!ELEMENT s

<!ATTLIST s

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<info>

<author>Federico Commandino</author>

<author>Commandino, Federico</author>

<title>De centro gravitatis solidorum</title>

<place>Bologna</place>

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<s>Sit primo triangulum æquilaterum a b c in circulo de<lb/>&longs;criptum: & diui&longs;a a c bi&longs;ariam in d, ducatur b d. erit in li<lb/>nea b d centrum grauitatis <expan abbr="triãguli">trianguli</expan> a b c, ex tertia decima <lb/>primi libri Archimedis de centro grauitatis planorum. Et <lb/>

<arrow.to.target n="fig1"></arrow.to.target><lb/>quoniam linea a b e&longs;t æqualis <lb/>lineæ b c; & a d ip&longs;i d c; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <lb/>b d utrique communis: trian</s>

<figure id="fig1"></figure><lb/>quoniam linea a b e&longs;t æqualis <lb/>lineæ b c; & a d ip&longs;i d c; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <lb/>b d utrique communis: trian</s>

<s>

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<s>Sit pentagonum æquilaterum, & æquiangulum in circu<lb/>

<arrow.to.target n="fig2"></arrow.to.target><lb/>lo de&longs;criptum a b c d e. & iun<lb/>cta b d, <expan abbr="bifariam&qacute;">bifariamque</expan>; in f diui&longs;a, <lb/>ducatur c f, & producatur ad <lb/>circuli circumferentiam in g; <lb/>quæ lineam a e in h &longs;ecet: de<lb/>indeiungantur a c, c c. Eodem <lb/>modo, quo &longs;upra demon&longs;tra<lb/>bimus angulum b c f æqualem <lb/>e&longs;&longs;e. angulo d c f; & angulos <lb/>ad f utro&longs;que rectos: & idcir<lb/>co lineam c f g per circuli cen <lb/>trum tran&longs;ire. Quoniam igi<lb/>tur latera c b, b a, & c d, d e æqualia &longs;unt; & æquales anguli <lb/>

<figure id="fig2"></figure><lb/>lo de&longs;criptum a b c d e. & iun<lb/>cta b d, <expan abbr="bifariam&qacute;">bifariamque</expan>; in f diui&longs;a, <lb/>ducatur c f, & producatur ad <lb/>circuli circumferentiam in g; <lb/>quæ lineam a e in h &longs;ecet: de<lb/>indeiungantur a c, c c. Eodem <lb/>modo, quo &longs;upra demon&longs;tra<lb/>bimus angulum b c f æqualem <lb/>e&longs;&longs;e. angulo d c f; & angulos <lb/>ad f utro&longs;que rectos: & idcir<lb/>co lineam c f g per circuli cen <lb/>trum tran&longs;ire. Quoniam igi<lb/>tur latera c b, b a, & c d, d e æqualia &longs;unt; & æquales anguli <lb/>

<arrow.to.target n="marg11"></arrow.to.target><lb/>c b a, c d e: erit ba&longs;is c a ba&longs;i: c e, & angulus b c a angulo <lb/>d c e æqualis. ergo & reliquus a c h, reliquo e c h. e&longs;t au<lb/>tem c h utrique triangulo a c h, e c h communis. quare <lb/>ba&longs;is a h æqualis e&longs;t ba&longs;i h c: & anguli, qui ad h recti: <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; <lb/>

<arrow.to.target n="marg12"></arrow.to.target><lb/>recti, qui ad f. ergo lineæ a e, b d inter &longs;e &longs;e æquidi&longs;tant. <lb/>Itaque cum trapezij a b d e latera b d, a e æquidi&longs;tantia à li <lb/>nea fh bifariam diuidantur; centrum grauitatis ip&longs;ius erit <lb/>

<arrow.to.target n="marg13"></arrow.to.target><lb/>in linea fh, ex ultima eiu&longs;dem libri Archimedis. Sed trian<lb/>guli b c d centrum grauitatis e&longs;t in linea c f. ergo in eadem <lb/>linea c h e&longs;t centrum grauitatis trapezij a b d e, & trian<lb/>guli b c d: hoc e&longs;t pentagoni ip&longs;ius centrum: & centrum <lb/>circuli. Rur&longs;us &longs;i iuncta a d, <expan abbr="bifariam&qacute;">bifariamque</expan>; &longs;ecta in k, duca<lb/>tur e k l: demon&longs;trabimus in ip&longs;a utrumque centrum in <lb/>e&longs;&longs;e. Sequitur ergo, ut punctum, in quo lineæ c g, e l con<lb/>ueniunt, idem &longs;it centrum circuli, & centrum grauitatis <lb/>pentagoni.</s>

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<s><margin.target id="marg13"></margin.target>13. Archi<lb/>medis.</s>

<s>Sit hexagonum a b c d e f æquilaterum, & æquiangulum <lb/>in circulo de&longs;ignatum: <expan abbr="iungantur&qacute;">iunganturque</expan>; b d, a e: & bifariam &longs;e

<pb pagenum="3"/>cta b d in g puncto, ducatur c g; & protrahatur ad circuli <lb/>u&longs;que circumferentiam; quæ &longs;ecet a e in h. Similiter conclu <lb/>demus c g per centrum circuli tran&longs;ire: & bifariam &longs;ecate <lb/>lineam a e; <expan abbr="item&qacute;">itemque</expan>; lineas b d, a e inter &longs;e æquidi&longs;tantes e&longs;&longs;e. <lb/>Cumigitur c g per centrum circuli tran&longs;eat; & ad <expan abbr="punct&utilde;">punctum</expan> <lb/>f perueniat nect&longs;&longs;e e&longs;t: quòd c d e f &longs;it dimidium circumfe <lb/>

<arrow.to.target n="fig3"></arrow.to.target><lb/>

<arrow.to.target n="marg14"></arrow.to.target><lb/>rentiæ circuli. Quare in eadem <lb/>diametro c f erunt centra gra <lb/>uitatis triangulorum b c d, <lb/>a f e, & quadrilateri a b d e, ex <lb/>quibus con&longs;tat hexagonum a b <lb/>c d e f. per&longs;picuum e&longs;tigitur in <lb/>ip&longs;a c f e&longs;&longs;e circuli centrum, & <lb/>centrum grauitatis hexagoni. <lb/>Rur&longs;us ducta altera diamctro <lb/>a d, ei&longs;dem rationibus o&longs;tende<lb/>mus in ip&longs;a utrumque <expan abbr="c&etilde;trum">centrum</expan> <lb/>ine&longs;&longs;e. Centrum ergo grauita<lb/>tis hexagoni, & centrum circuli idem erit.</s>

<s>Sit heptagonum a b c d e f g æquilaterum atque æquian<lb/>

<arrow.to.target n="fig4"></arrow.to.target><lb/>gulum in circulo de&longs;criptum: <lb/>& iungantur c e, b f, a g: di<lb/>ui&longs;a autem c e bifariam in <expan abbr="p&utilde;">pum</expan> <lb/>cto h: & iuncta d h produca<lb/>tur in k. non aliter demon<lb/>&longs;trabimus in linea d k e&longs;&longs;e cen <lb/>trum circuli, & centrum gra<lb/>uitatis trianguli c d e, & tra<lb/>peziorum b c e f, a b f g, hoc <lb/>e&longs;t centrum totius heptago<lb/>ni: & rur&longs;us eadem centra in <lb/>alia diametro c l &longs;imiliter du<lb/>cta contineri. Quare & centrum grauitatis heptagoni, & <lb/>centrum circuli in idem punctum conueniunt. Eodem mo

<figure id="fig4"></figure><lb/>gulum in circulo de&longs;criptum: <lb/>& iungantur c e, b f, a g: di<lb/>ui&longs;a autem c e bifariam in <expan abbr="p&utilde;">pum</expan> <lb/>cto h: & iuncta d h produca<lb/>tur in k. non aliter demon<lb/>&longs;trabimus in linea d k e&longs;&longs;e cen <lb/>trum circuli, & centrum gra<lb/>uitatis trianguli c d e, & tra<lb/>peziorum b c e f, a b f g, hoc <lb/>e&longs;t centrum totius heptago<lb/>ni: & rur&longs;us eadem centra in <lb/>alia diametro c l &longs;imiliter du<lb/>cta contineri. Quare & centrum grauitatis heptagoni, & <lb/>centrum circuli in idem punctum conueniunt. Eodem mo

<pb/>do in reliquis figuris æquilateris, & æquiangulis, quæ in cir<lb/>culo de&longs;cribuntur, probabimus <expan abbr="c&etilde;trum">centrum</expan> grauitatis carun<gap/>, <lb/>& centrum circuli idem e&longs;&longs;e. quod quidem demon&longs;trare <lb/>oportebat.</s>

<s>Ex quibus apparet cuiuslibet figuræ rectilincæ <lb/>in circulo plane de&longs;criptæ centrum grauitatis <expan abbr="id&etilde;">idem</expan> <lb/>e&longs;&longs;e, quod & circuli centrum.<lb/>

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<s>Sit ellip&longs;is a b c d, cuius maior axis a c, minor b d: <expan abbr="iun-gantur&qacute;">iun<lb/>ganturque</expan>; a b, b c, c d, d a: & bifariam diuidantur in pun<lb/>ctis e f g h. à centro autem, quod &longs;it k ductæ lineæ k e, k f, <lb/>k g, k h u&longs;que ad &longs;ectionem in puncta l m n o protrahan<lb/>tur: & iungantur l m, m n, n o, o l, ita ut a c &longs;ecet li<lb/>neas l o, m n, in z <foreign lang="greek">f</foreign> punctis; & b d &longs;ecet l m, o n in <foreign lang="greek">x y.</foreign><lb/>erunt l k, k n linea una, <expan abbr="item&qacute;ue">itemque</expan> linea unaip&longs;æ m k, k o: <lb/>& lineæ b a, c d æquidi&longs;tabunt lineæ m o: & b c, a d ip&longs;i <lb/>l n. rur&longs;us l o, m n axi b d æquidi&longs;tabunt: & l m,

<pb pagenum="4"/>o n ip&longs;i a c. Quoniam enim triangulorum a b k, a d k, latus <lb/>b k e&longs;t æquale lateri k d, & a k utrique commune; <expan abbr="anguli&qacute;">angulique</expan>; <lb/>

<arrow.to.target n="marg16"></arrow.to.target><lb/>ad k recti. ba&longs;is a b ba&longs;i a d; & reliqui anguli reliquis an<lb/>gulis æquales erunt. eadem quoqueratione o&longs;tendetur b c <lb/>

<arrow.to.target n="fig5"></arrow.to.target><lb/>æqualis c d; & a b ip&longs;i <lb/>b c. quare omnes a b, <lb/>b c, c d, d a &longs;unt æqua<lb/>les. & quoniam anguli <lb/>ad a æquales &longs;unt angu <lb/>lis ad c; erunt anguli b <lb/>a c, a c d coalterni inter <lb/>&longs;e æquales; <expan abbr="item&qacute;">itemque</expan>; d a c, <lb/>a c b. ergo c d ip&longs;i b a; <lb/>& a d ip&longs;i b c æquidi<lb/>&longs;tat. Atuero cum lineæ <lb/>a b, c d inter &longs;e æquidi<lb/>&longs;tantes bifariam &longs;ecen<lb/>tur in punctis e g; erit li <lb/>nea l e k g n diameter &longs;e <lb/>ctionis, & linea una, ex <lb/>demon&longs;tratis in uige&longs;i<lb/>maoctaua &longs;ecundi coni <lb/>corum. Et eadem ratione linea una m f k h o. Sunt <expan abbr="aut&etilde;">autem</expan> a d, <lb/>b c inter &longs;e &longs;e æquales, & æquidi&longs;tantes. quare & earum di<lb/>

<figure id="fig5"></figure><lb/>æqualis c d; & a b ip&longs;i <lb/>b c. quare omnes a b, <lb/>b c, c d, d a &longs;unt æqua<lb/>les. & quoniam anguli <lb/>ad a æquales &longs;unt angu <lb/>lis ad c; erunt anguli b <lb/>a c, a c d coalterni inter <lb/>&longs;e æquales; <expan abbr="item&qacute;">itemque</expan>; d a c, <lb/>a c b. ergo c d ip&longs;i b a; <lb/>& a d ip&longs;i b c æquidi<lb/>&longs;tat. Atuero cum lineæ <lb/>a b, c d inter &longs;e æquidi<lb/>&longs;tantes bifariam &longs;ecen<lb/>tur in punctis e g; erit li <lb/>nea l e k g n diameter &longs;e <lb/>ctionis, & linea una, ex <lb/>demon&longs;tratis in uige&longs;i<lb/>maoctaua &longs;ecundi coni <lb/>corum. Et eadem ratione linea una m f k h o. Sunt <expan abbr="aut&etilde;">autem</expan> a d, <lb/>b c inter &longs;e &longs;e æquales, & æquidi&longs;tantes. quare & earum di<lb/>

<arrow.to.target n="marg17"></arrow.to.target><lb/>midiæ a h, b f; <expan abbr="item&qacute;">itemque</expan>; h d, f e; & quæ ip&longs;as coniungunt rectæ <lb/>lineæ æquales, & æquidi&longs;tantes erunt. <expan abbr="æquidi&longs;tãt">æquidi&longs;tant</expan> igitur b a, <lb/>c d diametro m o: & pariter a d, b c ip&longs;i l n æquidi&longs;tare o<lb/>&longs;tendemus. Si igitur <expan abbr="man&etilde;te">manente</expan> diametro a c intelligatur a b c <lb/>portio ellip&longs;is ad portionem a d c moueri, cum primum b <lb/>applicuerit ad d, <expan abbr="cõgruet">congruet</expan> tota portio toti portioni, <expan abbr="linea&qacute;">lineaque</expan>; <lb/>b a lineæ a d; & b c ip&longs;i c d congruet: punctum uero e ca<lb/>det in h; f in g: & linea k e in lineam k h: & k f in k g. qua <lb/>re & e l in h o, et f m in g n. Atip&longs;a l z in z o; et m <foreign lang="greek">f</foreign> in <foreign lang="greek">f</foreign> n <lb/>cadet. congruet igitur triangulum l k z triangulo o k z: et

<pb/>triangulum m k <foreign lang="greek">f</foreign> triangulo n k <foreign lang="greek">f.</foreign> ergo anguli l z k, o z k, <lb/>m <foreign lang="greek">f</foreign> k, n <foreign lang="greek">f</foreign> k æquales &longs;unt, ac recti. quòd cum etram recti <lb/>

<arrow.to.target n="marg18"></arrow.to.target><lb/>&longs;int, qui ad k; æquidi&longs;tabunt lineæ l o, m n axi b d. & ita <lb/>denion&longs;trabuntur l m, o n ip&longs;i a c æquidi&longs;tare. Rurfus &longs;i <lb/>iungantur a l, l b, b m, m c, c n, n d, d o, o a: & bitariam di <lb/>uidantur: à centro autem k ad diui&longs;iones ductæ lineæ pro<lb/>trahantur u&longs;que ad &longs;ectionem in puncta p q r s t u x y: & po <lb/>ftremo p y, q x, r u, s t, q r, p s, y t, x u coniungantur. Simili<lb/>

<arrow.to.target n="fig6"></arrow.to.target><lb/>ter o&longs;tendemus lineas <lb/>p y, q x, r u, s t axi b d æ<lb/>quidi&longs;tantes e&longs;&longs;e: & q r, <lb/>p s, y t, x u æquidi&longs;tan<lb/>tesip&longs;i a c. Itaque dico <lb/>harum figurarum in el<lb/>lip&longs;i de&longs;criptarum cen<lb/>trum grauitatis e&longs;&longs;e <expan abbr="p&utilde;-ctum">pun<lb/>ctum</expan> k, idem quod & el <lb/>lip&longs;is centrum. 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="c&utilde;">cum</expan> in eo om <lb/>nes diametri <expan abbr="cõueniãt">conueniant</expan>. <lb/>Sedin figura a l b m c n <lb/>

<figure id="fig6"></figure><lb/>ter o&longs;tendemus lineas <lb/>p y, q x, r u, s t axi b d æ<lb/>quidi&longs;tantes e&longs;&longs;e: & q r, <lb/>p s, y t, x u æquidi&longs;tan<lb/>tesip&longs;i a c. Itaque dico <lb/>harum figurarum in el<lb/>lip&longs;i de&longs;criptarum cen<lb/>trum grauitatis e&longs;&longs;e <expan abbr="p&utilde;-ctum">pun<lb/>ctum</expan> k, idem quod & el <lb/>lip&longs;is centrum. 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="c&utilde;">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. 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. Po&longs;tremo in figura <lb/>a p l q b r m s c t n u d x o y centrum grauitatis trian <lb/>guli p a y, & trapezii p l o y e&longs;t in linea a z: trapeziorum <lb/>uero l q x o, q b d x centrum e&longs;t in linea z k: & <expan abbr="trapezior&utilde;">trapeziorum</expan> <lb/>b r u d, r m n u in k <foreign lang="greek">f:</foreign> & denique trapezii m s t n; & triangu <lb/>li s c t in <foreign lang="greek">f</foreign> c. quare magnitudinis ex his compo&longs;itæ <expan abbr="centr&utilde;">centrum</expan> <lb/>in linea a c con&longs;i&longs;tit. Rur&longs;us trianguli q b r, & trapezii q l <lb/>m r centrum e&longs;t in linea b <foreign lang="greek">x.</foreign> trapeziorum l p s m, p a c s, <lb/>a y t c, y o n t in linea <foreign lang="greek">x f:</foreign> <expan abbr="trapeziiq;">trapeziique</expan> o x u n, & trianguli <lb/>x d u centrum in <foreign lang="greek">y</foreign> d. totius ergo magnitudinis centrum <lb/>e&longs;t in linea b d. ex quo &longs;equitur, centrum grauitatis figuræ <lb/>a p l q b r m s c t n u d x o y e&longs;&longs;e <expan abbr="punct&utilde;">punctum</expan> K, lineis &longs;cilicet a c, <lb/>b d commune, quæ omnia demon&longs;trare oportebat.</s>

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<s><margin.target id="marg20"></margin.target>Vltima.</s>

<s>THEOREMA III. PROPOSITIO III.</s>

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<s>H O C eodem pror&longs;us <lb/>modo demon&longs;trabitur, <lb/>quo in libro de centro gra <lb/>uitatis planorum ab Ar<lb/>chimede <expan abbr="demon&longs;trat&utilde;">demon&longs;tratum</expan> e&longs;t, <lb/>in portione <expan abbr="cõtenta">contenta</expan> recta <lb/>linea, & rectanguli coni &longs;e <lb/>ctione grauitatis <expan abbr="c&etilde;trum">centrum</expan> <lb/>e&longs;&longs;e in diametro portio<lb/>nis. Etita demon&longs;trari po <lb/>

<arrow.to.target n="fig7"></arrow.to.target>

<pb/>te&longs;t in portione, quæ recta linea & obtu&longs;ianguli coni &longs;e<lb/>ctione, &longs;eu hyperbola continetur.</s>

<s>THEOREMA IIII. PROPOSITIO IIII.</s>

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<s>SIT circulus, uel ellip&longs;is, cuius centrum a. Dico a gra<lb/>uitatis quoque centrum e&longs;&longs;e. Si enim fieri pote&longs;t, &longs;it b cen<lb/>trum grauitatis: & iuncta a b extra figuram in c produca <lb/>tur: quam uero proportionem habet linea c a ad a b, ha<lb/>beat circulus a ad alium circulum, in quo d; uel ellip&longs;is ad <lb/>aliam ellip&longs;im: & in circulo, uel ellip&longs;i &longs;igura rectilinea pla<lb/>ne de&longs;cribatur adco, ut tandem relinquantur portiones <lb/>quædam minores circulo, uel ellip&longs;i d; quæ figura &longs;it e f g <lb/>h k l m n. Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con<lb/>

<arrow.to.target n="fig8"></arrow.to.target><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. 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/>

<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. 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

<arrow.to.target n="fig9"></arrow.to.target><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. fiat o b ad b a, <lb/>ut figura rectilinea ad portio<lb/>nes. cum igitur à circulo, uel el<lb/>lip&longs;i, cuius grauitatis centrum <lb/>e&longs;t b, auferatur figura rectilinea <lb/>e f g h k l m n, cuius centrum a; <lb/>reliquæ magnitudinis ex portio <lb/>

<figure id="fig9"></figure><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. fiat o b ad b a, <lb/>ut figura rectilinea ad portio<lb/>nes. cum igitur à circulo, uel el<lb/>lip&longs;i, cuius grauitatis centrum <lb/>e&longs;t b, auferatur figura rectilinea <lb/>e f g h k l m n, cuius centrum a; <lb/>reliquæ magnitudinis ex portio <lb/>

<arrow.to.target n="marg23"></arrow.to.target><lb/>nibus compo&longs;itæ centrum graui <lb/>tatis erit in linea a b producta, <lb/>& in puncto o, extra figuram po <lb/>&longs;ito. quod quidem fieri nullo mo <lb/>do po&longs;&longs;e per&longs;picuum e&longs;t. &longs;equi<lb/>tur ergo, ut circuli & ellip&longs;is cen <lb/>trum grauitatis &longs;it punctum a, <lb/>idem quod figuræ centrum.</s>

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<s>ALITER.</s>

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<s>Sit circulus, uel ellip&longs;is a b c d, <lb/>cuius diameter d b, & centrum e: <expan abbr="ducaturq;">ducaturque</expan> per e rectall <lb/>nea a c, &longs;ecans ip&longs;am d b ad rectos angulos. erunt a d c, <lb/>a b c circuli, uel ellip&longs;is dimidiæ portiones. Itaque quo<lb/>

<arrow.to.target n="fig10"></arrow.to.target><lb/>niam por <lb/><expan abbr="tiõis">tionis</expan> a d c <lb/><expan abbr="c&etilde;tr&utilde;">centrum</expan> gra<lb/>uitatis e&longs;t <lb/>in diame<lb/>tro d e: & <lb/>portionis <lb/>a b c cen<lb/>trum e&longs;t <expan abbr="&itilde;">im</expan> <lb/>ip&longs;a e b: to <lb/>tius circu <lb/>li, uel ellip&longs;is grauitatis centrum erit in diametro d b. <lb/>Sit autem portionis a d c <expan abbr="c&etilde;trum">centrum</expan> grauitatis f: & &longs;umatur

<figure id="fig10"></figure><lb/>niam por <lb/><expan abbr="tiõis">tionis</expan> a d c <lb/><expan abbr="c&etilde;tr&utilde;">centrum</expan> gra<lb/>uitatis e&longs;t <lb/>in diame<lb/>tro d e: & <lb/>portionis <lb/>a b c cen<lb/>trum e&longs;t <expan abbr="&itilde;">im</expan> <lb/>ip&longs;a e b: to <lb/>tius circu <lb/>li, uel ellip&longs;is grauitatis centrum erit in diametro d b. <lb/>Sit autem portionis a d c <expan abbr="c&etilde;trum">centrum</expan> grauitatis f: & &longs;umatur

<pb/>in linea e b <expan abbr="punct&utilde;">punctum</expan> g, itaut fit g e æqualis e f. erit g por<lb/>tionis a b c centrum. nam &longs;i hæ portiones, quæ æquales <lb/>& &longs;imiles &longs;unt, inter &longs;e &longs;e aptentur, ita ut b e cadat in d e, <lb/>& punctum b in d cadet, & g in f: figuris autem æquali<lb/>bus, & &longs;imilibus inter &longs;e aptatis, centra quoque grauitatis <lb/>ip&longs;arum inter &longs;e aptata crunt, ex quinta petitione Archi<lb/>medis in libro de centro grauitatis planorum. Quare cum <lb/>portionis a d c centrum grauitatis &longs;it f: & portionis <lb/>a b c centrum g: magnitudinis; quæ ex utri&longs;que efficitur: <lb/>hoc e&longs;t circuli uel ellip&longs;is grauitatis centrum in medio li<lb/>neæ f g, quod e&longs;t e, con&longs;iftet, ex quarta propo&longs;itione eiu&longs;<lb/>dem libri Archimedis. ergo circuli, uel ellip&longs;is centrum <lb/>grauitatis e&longs;t idem, quod figuræ centrum. atque illud e&longs;t, <lb/>quod demon&longs;trare oportebat.</s>

<s>Ex quibus &longs;equitur portionis circuli, uel ellip<lb/>&longs;is, quæ dimidia maior &longs;it, centrum grauitatis in <lb/>diametro quoque ip&longs;ius con&longs;i&longs;tere.</s>

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<s>Sit pri&longs;ina, in quo plana oppo&longs;ita &longs;int triangula a b c, <lb/>d e f; axis g h: & &longs;ecetur plano iam dictis planis <expan abbr="æquidi&longs;tã">æquidi&longs;tam</expan> <lb/>te; quod faciat &longs;ectionem k l m; & axi in <expan abbr="p&utilde;cto">puncto</expan> n occurrat. <lb/>Dico k l m trian gulum æquale e&longs;&longs;e, & fimile triangulis a b c <lb/>d e f; atque eius grauitatis centrum e&longs;&longs;e punctum n. Quo<lb/>

<arrow.to.target n="fig11"></arrow.to.target><lb/>niam enim plana a b c <lb/>K l m æquidi&longs;tantia <expan abbr="&longs;ecã">&longs;ecam</expan> <lb/>

<figure id="fig11"></figure><lb/>niam enim plana a b c <lb/>K l m æquidi&longs;tantia <expan abbr="&longs;ecã">&longs;ecam</expan> <lb/>

<arrow.to.target n="marg24"></arrow.to.target><lb/>tur a plano a e; rectæ li<lb/>neæ a b, K l, quæ &longs;unt ip <lb/>&longs;orum <expan abbr="cõmunes">communes</expan> &longs;ectio<lb/>nes inter &longs;e &longs;e æquidi<lb/>&longs;tant. Sed æquidi&longs;tant <lb/>a d, b e; cum a e &longs;it para <lb/>lelogrammum, ex pri&longs;<lb/>matis diffinitione. ergo <lb/>& al <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>erit; & propterea linea <lb/>

<arrow.to.target n="marg25"></arrow.to.target><lb/>k l, ip&longs;i a b æqualis. Si<lb/>militer demon&longs;trabitur <lb/>l m æquidi&longs;tans, & æqua <lb/><gap/>

<pb/>Itaque quoniam duæ lineæ K l, l m &longs;e &longs;e tangentes, duab us <lb/>lineis &longs;e &longs;e tangentibus a b, b c æquidi&longs;tant; nec &longs;unt in e o<lb/>dem plano: angulus k l m æqualis e&longs;t angulo a b c: & ita an <lb/>

<arrow.to.target n="marg26"></arrow.to.target><lb/>gulus l m k, angulo b c a, & m k lip&longs;i c a b æqualis prob abi <lb/>tur. triangulum ergo k l m e&longs;t æquale, & &longs;imile triang ulo <lb/>a b c. quare & triangulo d e f. Ducatur linea c g o, & per ip <lb/>&longs;am, & per c f ducatur planum &longs;ecans pri&longs;ma; cuius & paral <lb/>lelogram<gap/> a e communis &longs;ectio &longs;it o p <expan abbr="q.">que</expan> tran&longs;ibit linea <lb/>fq per h, & m p per n. nam cum plana æquidi&longs;tantia &longs;ecen <lb/>tur à plano c q, communes eorum &longs;ectiones c g o, m p, f q <lb/>&longs;ibi ip&longs;is æquidi&longs;tabunt. Sed & æquidi&longs;tant a b, k l, d e. an<lb/>

<arrow.to.target n="marg27"></arrow.to.target><lb/>guli ergo a o c, k p m, d q f inter &longs;e æquales &longs;unt: & &longs;unt <lb/>æquales qui ad puncta a k d con&longs;tituuntur. quare & reliqui <lb/>reliquis æquales; & triangula a c o, K m p, d f q inter &longs;e &longs;imi <lb/>

<arrow.to.target n="marg28"></arrow.to.target><lb/>lia erunt. Vtigitur c a ad a o, ita fd ad d q: & permutando <lb/>ut c a ad f d, ita a o ad d <expan abbr="q.">que</expan> e&longs;t autem c a æqualis f d. ergo & <lb/>a o ip&longs;i d <expan abbr="q.">que</expan> eadem quoque ratione & a o ip&longs;i K p æqualis <lb/>demon&longs;trabitur. Itaque &longs;i triangula, a b c, d e f æqualia & <lb/>

<arrow.to.target n="fig12"></arrow.to.target><lb/>&longs;imilia inter &longs;e <expan abbr="apt&etilde;tur">aptentur</expan>, <lb/>cadet linea f q in lineam <lb/>

<figure id="fig12"></figure><lb/>&longs;imilia inter &longs;e <expan abbr="apt&etilde;tur">aptentur</expan>, <lb/>cadet linea f q in lineam <lb/>

<arrow.to.target n="marg29"></arrow.to.target><lb/>c g o. Sed & <expan abbr="centr&utilde;">centrum</expan> gra <lb/>uitatis h in g <expan abbr="centr&utilde;">centrum</expan> ca<lb/>det. <expan abbr="trã&longs;ibit">tran&longs;ibit</expan> igitur linea <lb/>f q per h: & planum per <lb/>c o & c f <expan abbr="duct&utilde;">ductum</expan> per <expan abbr="ax&etilde;">axem</expan> <lb/>g h ducetur: <expan abbr="idcircoq;">idcircoque</expan> li <lb/>neam m p <expan abbr="etiã">etiam</expan> per n <expan abbr="trã">tram</expan> <lb/>&longs;ire nece&longs;&longs;e erit. Quo<lb/>niam ergo fh, c g æqua<lb/>les &longs;unt, & <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan>: <lb/><expan abbr="itemq;">itemque</expan> h q, g o; rectæ li<lb/>neæ, quæ ip&longs;as <expan abbr="cõnect&utilde;t">connectunt</expan> <lb/>c m f, g n h, o p q æqua<lb/><gap/>

<pb pagenum="8"/>æquidi&longs;tant autem c g o, m n p. ergo <expan abbr="parallelogrãma">parallelogramma</expan> &longs;unt <lb/>o n, g m, & linea m n æqualis c g; & n p ip&longs;i g o. aptatis igi<lb/>tur k l m, a b c <expan abbr="triãgulis">triangulis</expan>, quæ æqualia & &longs;imilia <expan abbr="s&utilde;t">sunt</expan>; linea m p <lb/>in c o, & punctum n in g cadet. Quòd <expan abbr="c&utilde;">cum</expan> g &longs;it centrum gra<lb/>uitatis trianguli a b c, & n trianguli k l m grauitatis cen<lb/>trum erit id, quod demon&longs;trandum relinquebatur. Simili <lb/>ratione idem contingere demon&longs;trabimus in aliis pri&longs;ma<lb/>tibus, &longs;iue quadrilatera, &longs;iue plurilatera habeant plana, <lb/>quæ opponuntur.</s>

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<s><margin.target id="marg29"></margin.target>pe<gap/> 5. pe<lb/>titionem <lb/>Archime <lb/>dis.</s>

<s>COROLLARIVM.</s>

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<pb/>

<arrow.to.target n="marg32"></arrow.to.target><lb/>ergo linea a g continenter in duas partes æquales diui<lb/>&longs;a, relinquetur <expan abbr="tãdem">tandem</expan> pars aliqua n g, quæ minor erit l m. <lb/>Vtraque uero linearum a g, g b diuidatur in partes æqua<lb/>les ip&longs;i n g: & per puncta diui&longs;ionum plana oppo&longs;itis pla<lb/>

<arrow.to.target n="marg33"></arrow.to.target><lb/>nis æquidi&longs;tantia ducantur. erunt &longs;ectiones figuræ æqua<lb/>les, ac &longs;imiles ip&longs;is a c e, b d f: & totum pri&longs;ma diui&longs;um erit <lb/>in pri&longs;mata æqualia, & &longs;imilia: quæ cum inter &longs;e <expan abbr="congruãt">congruant</expan>; <lb/>& grauitatis centra &longs;ibi ip&longs;is congruentia, <expan abbr="re&longs;pondentiaq;">re&longs;pondentiaque</expan> <lb/>

<arrow.to.target n="fig13"></arrow.to.target><lb/>habebunt. Itaq: <lb/>&longs;unt magnitudi<lb/>nes <expan abbr="quædã">quædam</expan> æqua<lb/>les ip&longs;i n h, & nu<lb/>mero pares, qua<lb/>rum centra gra<lb/>uitatis in <expan abbr="ead&etilde;re">eaderre</expan> <lb/>cta linea con&longs;ti<lb/>tuuntur: duæ ue<lb/>ro mediæ æqua<lb/>les &longs;unt: & quæ ex <lb/>utraque parte i<lb/>p&longs;arum &longs;imili-<lb/>ter æquales: & æ<lb/>quales rectæ li<lb/>neæ, quæ inter <lb/>grauitatis centra <lb/>interiiciuntur. <lb/>quare ex corolla<lb/>rio quintæ pro<lb/>po&longs;itionis primi <lb/>libri Archimedis <lb/>d e centro graui<lb/>tatis planorum; magnitudinis ex his omnibus compo&longs;itæ <lb/>centrum grauitatis e&longs;t in medio lineæ, quæ magnitudi<lb/>num mediarum centra coniungit. at qui non ita res ha

<figure id="fig13"></figure><lb/>habebunt. Itaq: <lb/>&longs;unt magnitudi<lb/>nes <expan abbr="quædã">quædam</expan> æqua<lb/>les ip&longs;i n h, & nu<lb/>mero pares, qua<lb/>rum centra gra<lb/>uitatis in <expan abbr="ead&etilde;re">eaderre</expan> <lb/>cta linea con&longs;ti<lb/>tuuntur: duæ ue<lb/>ro mediæ æqua<lb/>les &longs;unt: & quæ ex <lb/>utraque parte i<lb/>p&longs;arum &longs;imili-<lb/>ter æquales: & æ<lb/>quales rectæ li<lb/>neæ, quæ inter <lb/>grauitatis centra <lb/>interiiciuntur. <lb/>quare ex corolla<lb/>rio quintæ pro<lb/>po&longs;itionis primi <lb/>libri Archimedis <lb/>d e centro graui<lb/>tatis planorum; magnitudinis ex his omnibus compo&longs;itæ <lb/>centrum grauitatis e&longs;t in medio lineæ, quæ magnitudi<lb/>num mediarum centra coniungit. at qui non ita res ha

<pb pagenum="9"/>bet, &longs;i quidem l extra medias magnitudines po&longs;itum e&longs;t. <lb/>Con&longs;tat igitur centrum granitatis pri&longs;matis e&longs;&longs;e in plano <lb/>

<arrow.to.target n="fig14"></arrow.to.target><lb/>g h k, quod nos demon&longs;trandum propo&longs;uimus. At&longs;i op<lb/>po&longs;ita plana in pri&longs;mate &longs;int quadrilatera, uel plurilatera, <lb/>eadem erit in omnibus demon&longs;tratio.</s>

<figure id="fig14"></figure><lb/>g h k, quod nos demon&longs;trandum propo&longs;uimus. At&longs;i op<lb/>po&longs;ita plana in pri&longs;mate &longs;int quadrilatera, uel plurilatera, <lb/>eadem erit in omnibus demon&longs;tratio.</s>

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<s><margin.target id="marg33"></margin.target>5 huius</s>

<s>THEOREMA VII. PROPOSITIO VII.</s>

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<s>SIT cylindrus, uel cylindri portio a c: & plano per a<gap/><lb/>xem ducto &longs;ecetur; cuius fectio &longs;it parallelogrammum ab <lb/>c d: & bifariam diui&longs;is a d, b c parallelogrammi lateribus, <lb/>per diui&longs;ionum puncta e f planum ba&longs;i æquidi&longs;tans duca<lb/>tur; quod faciet &longs;ectionem, in cylindro quidem circulum <lb/>æqualem iis, qui &longs;unt in ba&longs;ibus, ut demon&longs;trauit Serenus <lb/>in libro cylindricorum, propo &longs;itione quin ta: in cylindr <lb/>uero portione ellip&longs;im æqualem, & &longs;imilem eis, quæ &longs;unt <lb/>

<arrow.to.target n="fig15"></arrow.to.target><lb/>in oppo&longs;itis planis, quod nos <lb/>demon&longs;trauimus in commen <lb/>tariis in librum Archimedis <lb/>de conoidibus, & &longs;phæroidi<lb/>bus. Dico centrum grauita<lb/>tis cylindri, uel cylindri por<lb/>tionis e&longs;&longs;e in plano e f. Si <expan abbr="en&itilde;">enim</expan> <lb/>fieri pote&longs;t, fit centrum g: & <lb/>ducatur g h ip&longs;i a d æquidi<lb/>&longs;tans, u&longs;que ad e f planum. <lb/>Itaque linea a e continenter <lb/>diui&longs;a bifariam, erit tandem <lb/>pars aliqua ip&longs;ius k e, minor <lb/>g h. Diuidantur ergo lineæ <lb/>a e, e d in partes æquales ip&longs;i <lb/>k e: & per diui&longs;iones plana ba <lb/>&longs;ibus æquidi&longs;tantia <expan abbr="ducãtur">ducantur</expan>. <lb/>eruntiam &longs;ectiones, figuræ æ<lb/>quales, & &longs;imiles eis, quæ &longs;unt <lb/>in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: & cy <lb/>lindri portio in portiones æquales, & &longs;imiles ip&longs;i k f. reli<lb/>qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s>

<figure id="fig15"></figure><lb/>in oppo&longs;itis planis, quod nos <lb/>demon&longs;trauimus in commen <lb/>tariis in librum Archimedis <lb/>de conoidibus, & &longs;phæroidi<lb/>bus. Dico centrum grauita<lb/>tis cylindri, uel cylindri por<lb/>tionis e&longs;&longs;e in plano e f. Si <expan abbr="en&itilde;">enim</expan> <lb/>fieri pote&longs;t, fit centrum g: & <lb/>ducatur g h ip&longs;i a d æquidi<lb/>&longs;tans, u&longs;que ad e f planum. <lb/>Itaque linea a e continenter <lb/>diui&longs;a bifariam, erit tandem <lb/>pars aliqua ip&longs;ius k e, minor <lb/>g h. Diuidantur ergo lineæ <lb/>a e, e d in partes æquales ip&longs;i <lb/>k e: & per diui&longs;iones plana ba <lb/>&longs;ibus æquidi&longs;tantia <expan abbr="ducãtur">ducantur</expan>. <lb/>eruntiam &longs;ectiones, figuræ æ<lb/>quales, & &longs;imiles eis, quæ &longs;unt <lb/>in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: & cy <lb/>lindri portio in portiones æquales, & &longs;imiles ip&longs;i k f. reli<lb/>qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s>

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<s>Sit primum a f pri&longs;ma æquidi&longs;tantibus planis <expan abbr="contcnt&utilde;">contcntum</expan>, <lb/>quod &longs;olidum parallelepipedum appellatur: & oppo&longs;ito<lb/>rum planorum c f, a h, d a, f g latera bifariam diuidantur in <lb/>punctis k l m n o p q r s t u x: & per diui&longs;iones ducantur <lb/>plana k n, o r, s x. communes autem eorum planorum &longs;e<lb/>ctiones &longs;int lineæ y z, <foreign lang="greek">q f, x y:</foreign> quæ in puncto <foreign lang="greek">w</foreign> <expan abbr="conueniãt">conueniant</expan>. <lb/>erit ex decima eiu&longs;dem libri Archimedis parallelogrammi <lb/>c f centrum grauitatis punctum y; parallelogrammi a h

<pb/>centrum z: paraliclogram mi <gap/> d, <foreign lang="greek">q:</foreign> parallclogrammi f g, <foreign lang="greek"><gap/>:</foreign><lb/>

<arrow.to.target n="fig16"></arrow.to.target><lb/>parallclogrammi d h, <foreign lang="greek">x:</foreign> & <lb/>parallclogrammi c g <expan abbr="centr&utilde;">centrum</expan> <lb/><foreign lang="greek">y:</foreign> atque erit <foreign lang="greek">w</foreign> punctum me <lb/>dium uniu&longs;cuiu&longs;que axis, ui <lb/>delicet eius lineæ quæ oppo <lb/>&longs;itorum <expan abbr="planor&utilde;">planorum</expan> centra con <lb/>iungit. Dico <foreign lang="greek">w</foreign> centrum e&longs;&longs;e <lb/>grauitatis ip&longs;ius &longs;olidi. e&longs;t <lb/>

<figure id="fig16"></figure><lb/>parallclogrammi d h, <foreign lang="greek">x:</foreign> & <lb/>parallclogrammi c g <expan abbr="centr&utilde;">centrum</expan> <lb/><foreign lang="greek">y:</foreign> atque erit <foreign lang="greek">w</foreign> punctum me <lb/>dium uniu&longs;cuiu&longs;que axis, ui <lb/>delicet eius lineæ quæ oppo <lb/>&longs;itorum <expan abbr="planor&utilde;">planorum</expan> centra con <lb/>iungit. Dico <foreign lang="greek">w</foreign> centrum e&longs;&longs;e <lb/>grauitatis ip&longs;ius &longs;olidi. e&longs;t <lb/>

<arrow.to.target n="marg34"></arrow.to.target><lb/>enim, ut demon&longs;trauimus, <lb/>&longs;olidi a f centrum grauitatis <lb/>in plano K n; quod oppo&longs;i<lb/>tis planis a d, g f æquidi&longs;tans <lb/>reliquorum planorum late<lb/>ra bifariam diuidit: & &longs;imili <lb/>ratione idem centrum e&longs;t in plano o r, æquidi&longs;tante planis <lb/>a e, b f oppo&longs;itis. ergo in communi ip&longs;orum &longs;ectione: ui<lb/>delicet in linea y z. Sed e&longs;t etiam in plano t u, quod <expan abbr="quid&etilde;">quidem</expan> <lb/>y z &longs;ecatin <foreign lang="greek">w.</foreign> Con&longs;tatigitur centrum grauitatis &longs;olidi e&longs;&longs;e <lb/>punctum <foreign lang="greek">w,</foreign> medium &longs;cilicet axium, hoc e&longs;t linearum, quæ <lb/>planorum oppo&longs;itorum centra coniungunt.</s>

<s><margin.target id="marg34"></margin.target>6 huius</s>

<s>Sit aliud prima a f; & in eo plana, quæ opponuntur, tri<lb/>angula a b c, d e f: <expan abbr="diui&longs;isq;">diui&longs;isque</expan> bifariam parallelogrammorum <lb/>lateribus a d, b e, c f in punctis g h k, per diui&longs;iones <expan abbr="plan&utilde;">planum</expan> <lb/>ducatur, quod oppo&longs;itis planis æquidi&longs;tans faciet <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> <lb/>triangulum g h x æquale, & &longs;imile ip&longs;is a b c, d e f. Rur&longs;us <lb/>diuidatur a b bi&longs;ariam in l: & iuncta c l per ip&longs;am, & per <lb/>c K f planum ducatur pri&longs;ma &longs;ecans, cuius, & <expan abbr="parallelogrã">parallelogram</expan> <lb/>mi a e communis &longs;ectio &longs;it l m n. diuidet punctum m li<lb/>neam g h bifariam; & ita n diuidet lineam d e: quoniam <lb/>

<arrow.to.target n="marg35"></arrow.to.target><lb/>triangula a c l, g <foreign lang="greek">k</foreign> m, d f n æqualia &longs;unt, & &longs;imilia, ut &longs;upra <lb/>demon&longs;trauimus. Iam ex iis, quæ tradita &longs;unt, con&longs;tat cen <lb/>trum greuitatis pri&longs;matis in plano g h <foreign lang="greek">k</foreign> contineri. Dico <lb/>ip&longs;um e&longs;&longs;e in linea k m. Si enim fieri potc&longs;t, &longs;it o centrum;

<pb pagenum="11"/>& per o ducatur o p ad k m ip&longs;i h g æquidi&longs;tans. Itaquc li <lb/>nea h m <expan abbr="bifariã">bifariam</expan> u&longs;que cò diuidatur, quoad reliqua &longs;it pars <lb/>quædam q m, minor o p. deinde h m, m g diuidantur in <lb/>partes æquales ip&longs;i m q: & per diui&longs;iones lineæ ip&longs;i m K <lb/>æquidi&longs;tantes ducantur. puncta uero, in quibus hæ trian<lb/>gulorum latera &longs;ecant, coniungantur ductis lineis r s, t u, <lb/>

<arrow.to.target n="fig17"></arrow.to.target><lb/>x y; quæ ba&longs;i g h æquidi&longs;tabunt. Quoniam enim lineæ g z, <lb/>h <foreign lang="greek">a</foreign> &longs;unt æquales: <expan abbr="itemq;">itemque</expan> æquales g m, m h: ut m g ad g z, <lb/>ita erit m h, ad h <foreign lang="greek">a:</foreign> & diuidendo, ut m z ad z g, ita m <foreign lang="greek">a</foreign> ad <lb/>

<figure id="fig17"></figure><lb/>x y; quæ ba&longs;i g h æquidi&longs;tabunt. Quoniam enim lineæ g z, <lb/>h <foreign lang="greek">a</foreign> &longs;unt æquales: <expan abbr="itemq;">itemque</expan> æquales g m, m h: ut m g ad g z, <lb/>ita erit m h, ad h <foreign lang="greek">a:</foreign> & diuidendo, ut m z ad z g, ita m <foreign lang="greek">a</foreign> ad <lb/>

<arrow.to.target n="marg36"></arrow.to.target><lb/><foreign lang="greek">a</foreign> h. Sed ut m z ad z g, ita k r ad r g: & ut m <foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign> h, ita k s <lb/>ad s h. quare ut k r ad r g, ita <foreign lang="greek">k</foreign> s ad s h. æquidi&longs;tant igitur <lb/>

<arrow.to.target n="marg37"></arrow.to.target><lb/>inter &longs;e &longs;e r s, g h. eadem quoque ratione demon&longs;trabimus

<pb/>t u, x y ip&longs;i g h æquidi&longs;tare. 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="triangul&utilde;">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. 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="un&utilde;">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/>

<arrow.to.target n="fig18"></arrow.to.target><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/>

<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. quare <lb/>totum trian <lb/>gulum K g h <lb/>ad omnia tri <lb/>angula g z r, <lb/>r <foreign lang="greek">b</foreign> t, t <foreign lang="greek">g</foreign> x, x <foreign lang="greek">d</foreign><lb/>K, K <foreign lang="greek">d</foreign> y, y u, <lb/>u s, s <foreign lang="greek">a</foreign> h ita <lb/>erit, ut h k ad <lb/>k y, hoc e&longs;t <lb/>ut h m ad m <lb/><expan abbr="q.">que</expan> Siigitur in <lb/>triangulis a b c, d e f de&longs;cribantur figuræ &longs;imiles ei, quæ de<lb/>&longs;cripta e&longs;t in g h K triangulo: & per lineas &longs;ibi re&longs;ponden<lb/>tes plana ducantur: totum pri&longs;ma a f diui&longs;um erit in tria <lb/>&longs;olida parallelepipeda y <foreign lang="greek">g,</foreign> u <foreign lang="greek">b,</foreign> s z, quorum ba&longs;es &longs;unt æqua <lb/>les & &longs;imiles ip&longs;is parallelogrammis y <foreign lang="greek">g,</foreign> u <foreign lang="greek">b,</foreign> s z: & in octo <lb/>pri&longs;mata g z r, r <foreign lang="greek">b</foreign> t, t <foreign lang="greek">g</foreign> x, x <foreign lang="greek">d</foreign> K, k <foreign lang="greek">d</foreign> y, y u, u s, s <foreign lang="greek">a</foreign> h: quorum <lb/>item ba&longs;es æquales, & &longs;imiles &longs;unt dictis triangulis; altitu<lb/><gap/>

<pb pagenum="12"/>Itaque &longs;olidi parallelepipedi y <foreign lang="greek">g</foreign> centrum grauitatis e&longs;t in <lb/>linea <foreign lang="greek">d <gap/>:</foreign> &longs;olidi <foreign lang="greek"><gap/> b</foreign> centrum e&longs;t in linea <foreign lang="greek"><gap/>:</foreign> & &longs;olidi s z in li <lb/>nea <gap/> m, quæ quidem lineæ axes &longs;unt, cum planorum oppo <lb/>&longs;itorum centra coniungant. ergo magnitudinis ex his &longs;oli <lb/>dis compo&longs;itæ centrum grauitatis e&longs;t in linea <foreign lang="greek">d</foreign> m, quod &longs;it <lb/><foreign lang="greek"><expan abbr="q;">que</expan></foreign> & iuncta <foreign lang="greek">q</foreign> o producatur: à puncto autem h ducatur h <foreign lang="greek">a</foreign><lb/>ip&longs;i m k æquidi&longs;tans, quæ cum <foreign lang="greek">q</foreign> o in <foreign lang="greek">m</foreign> conueniat. triangu <lb/>lum igitur g h k ad omnia triangula g z r, r <foreign lang="greek">b</foreign> t, t <foreign lang="greek">g</foreign> x, x <foreign lang="greek">d</foreign> k, <lb/>k <foreign lang="greek">d</foreign> y, y u, u s, s <foreign lang="greek">a</foreign> h eandem habet proportionem, quam h m <lb/>ad m <expan abbr="q;">que</expan> hoc e&longs;t, quam <foreign lang="greek">m q</foreign> <gap/> <foreign lang="greek">q l:</foreign> nam &longs;i h m, <foreign lang="greek">y q</foreign> produci in <lb/>telligantur, quou&longs;que coeant; erit ob linearum q y, m k æ<gap/><lb/>quidi&longs;tantiam, ut h q ad q m, ita <foreign lang="greek">m l</foreign> ad ad <foreign lang="greek">l q:</foreign> & componen <lb/>do, ut h m ad m q, ita <foreign lang="greek">m q</foreign> ad <foreign lang="greek">q l.</foreign> linea uero <foreign lang="greek">q</foreign> o maior e&longs;t, <lb/>

<arrow.to.target n="marg40"></arrow.to.target><lb/>quàm <foreign lang="greek">q l:</foreign> habebit igitur <foreign lang="greek">m q</foreign> ad <foreign lang="greek">q l</foreign> maiorem proportio<lb/>nem, quàm ad <foreign lang="greek">q</foreign> o. quare triangulum ctiam g h <foreign lang="greek">k</foreign> ad omnia <lb/>iam dicta triangula maiorem <expan abbr="proportion&etilde;">proportionem</expan> habebit, quàm <lb/><foreign lang="greek">m q</foreign> ad <foreign lang="greek">q</foreign> o. &longs;ed ut <expan abbr="triangul&utilde;">triangulum</expan> g h <foreign lang="greek">k</foreign> ad omnia triangula, ita <expan abbr="to-t&utilde;">to<lb/>tum</expan> pri&longs;ma a fad omnia pri&longs;mata g z r, r <foreign lang="greek">b</foreign> t, t <foreign lang="greek">g</foreign> x, x <foreign lang="greek">d k, k d</foreign> y, <lb/>y u, u s, s <foreign lang="greek">a</foreign> h: quoniam enim &longs;olida parallelepipeda æque al <lb/>ta, eandem inter &longs;e proportionem habent, quam ba&longs;es; ut <lb/>ex trige&longs;ima&longs;ecunda undecimi elementorum con&longs;tat. &longs;unt <lb/>

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<arrow.to.target n="marg42"></arrow.to.target><lb/>&longs;es habentium dupla: &longs;equitur, ut etiam huiu&longs;modi pri&longs;<lb/>mata inter &longs;e &longs;int, &longs;icut eorum ba&longs;es. ergo totum pri&longs;ma ad <lb/>omnia pri&longs;mata maiorem proportionem habet, quam <foreign lang="greek">m q</foreign><lb/>

<arrow.to.target n="marg43"></arrow.to.target><lb/>ad <foreign lang="greek">q</foreign> o: & diuidendo &longs;olida parallelepipeda y <foreign lang="greek">g,</foreign> u <foreign lang="greek">b,</foreign> s z ad o<lb/>mnia pri&longs;mata proportionem habent maiorem, quàm <foreign lang="greek">m</foreign> o <lb/>ad o <gap/>. fiat <foreign lang="greek">n</foreign> o ad o <foreign lang="greek">q,</foreign> ut &longs;olida parallelepipeda y <foreign lang="greek">g,</foreign> u <foreign lang="greek">b,</foreign> s z ad <lb/>omnia pri&longs;mata. Itaque cum à pri&longs;mate a f, cuius <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis e&longs;t o, auferatur magnitudo ex &longs;olidis parallelepi <lb/>pedis y <foreign lang="greek">g,</foreign> u <foreign lang="greek">b,</foreign> s z con&longs;tans: atque ipfius grauitatis centrum <lb/>&longs;it <foreign lang="greek">q:</foreign> reliquæ magnitudinis, quæ ex omnibus pri&longs;matibus <lb/>con&longs;tat, grauitatis centrum erit in linea <foreign lang="greek">q</foreign> o producta: & <lb/><gap/>

<pb/>medis. ergo punctum <foreign lang="greek">n</foreign> extra p ri&longs;ma a f po&longs;itum, <expan abbr="centr&utilde;">centrum</expan> <lb/>erit magnitudinis <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> e x omnibus pri&longs;matibus g z r, <lb/>r <foreign lang="greek">b</foreign> t, t <foreign lang="greek">g</foreign> x, x <foreign lang="greek">d</foreign> k, k <foreign lang="greek">d</foreign> y, y u, u s, s <foreign lang="greek">a</foreign> h, quod fieri nullo modo po <lb/>te&longs;t. e&longs;t enim ex diffinitione centr um grauitatis &longs;olidæ figu <lb/>ræ intra ip&longs;am po&longs;itum, non extra. quare relinquitur, ut <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis pri&longs;matis &longs;it in linea K m. Rur&longs;us b c bifa<lb/>riam in diuidatur: & ducta a <foreign lang="greek">x,</foreign> per ip&longs;am, & per lineam <lb/>a g d plan um ducatur; quod pri&longs;ma &longs;ecet: <expan abbr="faciatq;">faciatque</expan> in paral <lb/>lelogrammo b f &longs;ectionem <foreign lang="greek">x p</foreign> diuidet punctum <foreign lang="greek">p</foreign> lineam <lb/>quoque c f bifariam: & erit p lani eius, & trianguli g h K <lb/>communis &longs;ectio g u; quòd <expan abbr="p&utilde;ctum">punctum</expan> u in medio lineæ h K <lb/>

<arrow.to.target n="fig19"></arrow.to.target><lb/>po&longs;itum &longs;i t. Similiter demon&longs;trabimus centrum grauita<lb/>tis pri&longs;m atis in ip&longs;a g u ine&longs;&longs;e. &longs;it autem planorum c f n l, <lb/>a d <foreign lang="greek">p x</foreign> communis &longs;ectio linea <foreign lang="greek">r s t;</foreign> quæ quidem pri&longs;matis <lb/>axis erit, cum tran&longs;eat per centra grauitatis triangulorum <lb/>a b c, g h <foreign lang="greek">k,</foreign> d e f, ex quartadecima eiu&longs;dem. ergo centrum <lb/>grauitatis pri&longs;matis a f e&longs;t punctum <foreign lang="greek">s,</foreign> centrum &longs;cilicet

<figure id="fig19"></figure><lb/>po&longs;itum &longs;i t. Similiter demon&longs;trabimus centrum grauita<lb/>tis pri&longs;m atis in ip&longs;a g u ine&longs;&longs;e. &longs;it autem planorum c f n l, <lb/>a d <foreign lang="greek">p x</foreign> communis &longs;ectio linea <foreign lang="greek">r s t;</foreign> quæ quidem pri&longs;matis <lb/>axis erit, cum tran&longs;eat per centra grauitatis triangulorum <lb/>a b c, g h <foreign lang="greek">k,</foreign> d e f, ex quartadecima eiu&longs;dem. ergo centrum <lb/>grauitatis pri&longs;matis a f e&longs;t punctum <foreign lang="greek">s,</foreign> centrum &longs;cilicet

<pb pagenum="13"/>trianguli g h K, & ip&longs;ius <foreign lang="greek">r t</foreign> axis medium.</s>

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<s>Sit pri&longs;ma a g, cuius oppo&longs;ita plana &longs;int quadrilatera <lb/>a b c d, e f g h: <expan abbr="&longs;ecenturq;">&longs;ecenturque</expan> a c, b f, c g, d h bifariam: & per di<lb/>ui&longs;iones planum ducatur; quod &longs;ectionem faciat quadrila<lb/>terum K l m n. Deinde iuncta a c per lineas a c, a e ducatur <lb/>planum <expan abbr="&longs;ecãs">&longs;ecans</expan> pri&longs;ma, quod ip&longs;um diuidet in duo pri&longs;mata <lb/>triangulares ba&longs;es habentia a b c e f g, a d c e h g. Sint <expan abbr="aut&etilde;">autem</expan> <lb/>

<arrow.to.target n="fig20"></arrow.to.target><lb/>triangulorum a b c, e f g gra<lb/>uitatis centra o p: & triangu<lb/>lorum a d c, e h g centra q r: <lb/><expan abbr="iunganturq;">iunganturque</expan> o p, q r; quæ pla<lb/>no k l m n occurrant in pun<lb/>ctis s t. erit ex iis, quæ demon <lb/>&longs;trauimus, punctum s grauita <lb/>tis centrum trianguli <foreign lang="greek">k</foreign> l m; & <lb/>ip&longs;ius pri&longs;matis a b c e f g: pun <lb/>ctum uero t centrum grauita <lb/>tis trianguli K n m, & pri&longs;ma<lb/>tis a d c, e h g. iunctis igitur <lb/>o q, p r, s t, erit in linea o q <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis quadrilateri <lb/>a b c d, quod &longs;it u: & in linea <lb/>p r <expan abbr="c&etilde;trum">centrum</expan> quadrilateri e f g h <lb/>&longs;it autem x. denique iungatur <lb/>u x, quæ &longs;ecet lineam &longs; t in y. &longs;e<lb/>cabit enim cum &longs;int in eodem <lb/>

<figure id="fig20"></figure><lb/>triangulorum a b c, e f g gra<lb/>uitatis centra o p: & triangu<lb/>lorum a d c, e h g centra q r: <lb/><expan abbr="iunganturq;">iunganturque</expan> o p, q r; quæ pla<lb/>no k l m n occurrant in pun<lb/>ctis s t. erit ex iis, quæ demon <lb/>&longs;trauimus, punctum s grauita <lb/>tis centrum trianguli <foreign lang="greek">k</foreign> l m; & <lb/>ip&longs;ius pri&longs;matis a b c e f g: pun <lb/>ctum uero t centrum grauita <lb/>tis trianguli K n m, & pri&longs;ma<lb/>tis a d c, e h g. iunctis igitur <lb/>o q, p r, s t, erit in linea o q <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis quadrilateri <lb/>a b c d, quod &longs;it u: & in linea <lb/>p r <expan abbr="c&etilde;trum">centrum</expan> quadrilateri e f g h <lb/>&longs;it autem x. denique iungatur <lb/>u x, quæ &longs;ecet lineam &longs; t in y. &longs;e<lb/>cabit enim cum &longs;int in eodem <lb/>

<arrow.to.target n="marg44"></arrow.to.target><lb/>plano: <expan abbr="atq;">atque</expan> erit y grauitatis centrum quadrilateri K l m n. <lb/>Dico idem punctum y centrum quoque gra uitatis e&longs;&longs;e to<lb/>tius pri&longs;matis. Quoniam enim quadrilateri k l m n graui<lb/>tatis centrum e&longs;t y: linea s y ad y t ean dem proportionem <lb/>habebit, quam triangulum <foreign lang="greek">k</foreign> n m ad triangulum k l m, cx 8 <lb/>Archimedis de centro grauitatis planorum. Vt autem <expan abbr="triã">triam</expan> <lb/>gulum <foreign lang="greek">k</foreign> n m ad ip&longs;um <foreign lang="greek">k</foreign> l m, hoc e&longs;t ut triangulum a d c ad <lb/>triangulum a b c, æqualia enim &longs;unt, ita pri&longs;ma a d c e h g

<pb/>ad pri&longs;ma a b c e f g. quare linea s y ad y t eandem propor<lb/>tionem habet, quam pri&longs;ma a d c e h g ad pri&longs;ma a b c e f g. <lb/>Sed pri&longs;matis a b c e f g centrum grauitatis e&longs;t s: & pri&longs;ma<lb/>tis a d c e h g centrum t. magnitudinis igitur ex his compo <lb/>&longs;itæ hoc e&longs;t totius pri&longs;matis a g centrum grauitatis e&longs;t pun <lb/>ctum y; medium &longs;cilicet axis u x, qui oppo&longs;itorum plano<lb/>rum centra coniungit.</s>

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<s>Rur&longs;us &longs;it pri&longs;ma ba&longs;im habens pentagonum a b c d e: <lb/>& quod ei opponitur &longs;it f g h K l: &longs;ec <expan abbr="enturq;">enturque</expan> a f, b g, c h, <lb/>d k, e l bi&longs;ariam: & per diui&longs;iones ducto plano, &longs;ectio &longs;it <expan abbr="p&etilde;">pem</expan> <lb/><expan abbr="tagon&utilde;">tagonum</expan> m n o p <expan abbr="q.">que</expan> deinde iuncta e b per lineas l e, e b aliud <lb/>

<arrow.to.target n="fig21"></arrow.to.target><lb/>planum ducatur, <expan abbr="diuid&etilde;s">diuidens</expan> pri&longs; <lb/>ma a k in duo pri&longs;mata; in pri&longs; <lb/>ma &longs;cilicet al, cuius plana op<lb/>po&longs;ita &longs;int triangula a b e f g l: <lb/>& in prima b k cuius plana op <lb/>po&longs;ita &longs;int quadrilatera b c d e <lb/>g h k l. Sint autem triangulo<lb/>rum a b e, f g l centra grauita <lb/>tis puncta r &longs;: & b c d e, g h k l <lb/>quadrilaterorum centra t u: <lb/><expan abbr="iunganturq;">iunganturque</expan> r s, t u occurren<lb/>tes plano m n o p q in punctis <lb/>x y. & itidem <expan abbr="iungãtur">iungantur</expan> r t, &longs;u, <lb/>x y. erit in lincar t <expan abbr="c&etilde;trum">centrum</expan> gra <lb/>uitatis pentagoni a b c d e; <lb/>quod &longs;it z: & in linea &longs;u cen<lb/>trum pentagoni f g h <foreign lang="greek">k</foreign> l:&longs;it au <lb/>tem <foreign lang="greek">x:</foreign> & ducatur z <foreign lang="greek">x,</foreign> quæ di<lb/>cto plano in <foreign lang="greek">y</foreign> occurrat. <expan abbr="Itaq;">Itaque</expan> <lb/>punctum x e&longs;t centrum graui <lb/>tatis trianguli m n q, ac pri&longs;<lb/>matis a l: & y grauitatis centrum quadrilateri n o p q, ac <lb/>pri&longs;matis b k. quare y centrum erit pentagoni m n o p <expan abbr="q.">que</expan> &

<figure id="fig21"></figure><lb/>planum ducatur, <expan abbr="diuid&etilde;s">diuidens</expan> pri&longs; <lb/>ma a k in duo pri&longs;mata; in pri&longs; <lb/>ma &longs;cilicet al, cuius plana op<lb/>po&longs;ita &longs;int triangula a b e f g l: <lb/>& in prima b k cuius plana op <lb/>po&longs;ita &longs;int quadrilatera b c d e <lb/>g h k l. Sint autem triangulo<lb/>rum a b e, f g l centra grauita <lb/>tis puncta r &longs;: & b c d e, g h k l <lb/>quadrilaterorum centra t u: <lb/><expan abbr="iunganturq;">iunganturque</expan> r s, t u occurren<lb/>tes plano m n o p q in punctis <lb/>x y. & itidem <expan abbr="iungãtur">iungantur</expan> r t, &longs;u, <lb/>x y. erit in lincar t <expan abbr="c&etilde;trum">centrum</expan> gra <lb/>uitatis pentagoni a b c d e; <lb/>quod &longs;it z: & in linea &longs;u cen<lb/>trum pentagoni f g h <foreign lang="greek">k</foreign> l:&longs;it au <lb/>tem <foreign lang="greek">x:</foreign> & ducatur z <foreign lang="greek">x,</foreign> quæ di<lb/>cto plano in <foreign lang="greek">y</foreign> occurrat. <expan abbr="Itaq;">Itaque</expan> <lb/>punctum x e&longs;t centrum graui <lb/>tatis trianguli m n q, ac pri&longs;<lb/>matis a l: & y grauitatis centrum quadrilateri n o p q, ac <lb/>pri&longs;matis b k. quare y centrum erit pentagoni m n o p <expan abbr="q.">que</expan> &

<pb pagenum="14"/>&longs;imiliter demon&longs;irabitur totius pri&longs;matis a K grauitatis ef <lb/>&longs;e centrum. Simili ratione & in aliis pri&longs;matibus illud <lb/>idem facile demon&longs;trabitur. Quo autem pacto in omni <lb/>figura rectilinea centrum grauitatis inueniatur, docuimus <lb/>in commentariis in &longs;extam propo&longs;itionem Archimedis de <lb/>quadratura parabolæ.</s>

<s>Sit cylindrus, uel cylindri portio c e cuius axis a b: &longs;ece<lb/>turq, plano per axem ducto; quod &longs;ectionem faciat paral<lb/>lelogrammum c d e f: & diui&longs;is c f, d e bifariam in punctis <lb/>

<arrow.to.target n="fig22"></arrow.to.target><lb/>g h, per ea ducatur planum ba&longs;i æquidi&longs;tans. erit &longs;ectio g h <lb/>circulus, uel ellip&longs;is, centrum habens in axe; quod &longs;it K at<lb/>

<figure id="fig22"></figure><lb/>g h, per ea ducatur planum ba&longs;i æquidi&longs;tans. erit &longs;ectio g h <lb/>circulus, uel ellip&longs;is, centrum habens in axe; quod &longs;it K at<lb/>

<arrow.to.target n="marg45"></arrow.to.target><lb/>que erunt ex iis, quæ demon&longs;trauimus, centra grauitatis <lb/>planorum oppo&longs;itorum puncta a b: & plani g h ip&longs;um k in <lb/>quo quidem plano e&longs;t centrum grauitatis cylindri, uel cy<lb/>lindri portionis. Dico punctum K cylindri quoque, uel cy <lb/>lindri portionis grauitatis centrum e&longs;&longs;e. Si enim fieri po<lb/>te&longs;t, &longs;it l centrum: <expan abbr="ducaturq;">ducaturque</expan> k l, & extra figuram in m pro<lb/>ducatur. quam ucro proportionem habet linea m K ad k l

<pb/>habcat circulus, uel ellip&longs;is g h ad aliud &longs;pacium, in quo u: <lb/>& in cit culo, uel ellip&longs;i plane de&longs;cribatur rectilinea figura, <lb/>ita ut <expan abbr="tãdem">tandem</expan> <expan abbr="relinquãtur">relinquantur</expan> portiones minores &longs;pacio u, quæ <lb/>&longs;it o p g q r s h t: <expan abbr="de&longs;criptaq;">de&longs;criptaque</expan> &longs;imili figura in oppo&longs;itis pla<lb/>nis c d, f e, per lineas &longs;ibi ip&longs;is re&longs;pondentes plana <expan abbr="ducãtur">ducantur</expan>. <lb/>Itaque cylindrus, uel cylindri portio diuiditur in pri&longs;ma, <lb/>cuius quidem ba&longs;is e&longs;t figura rectilinea iam dicta, centrum <lb/>que grauitatis punctuni K: & in multa &longs;olida, qaæ pro ba&longs;i <lb/>bus habent relictas portiones, quas nos &longs;olidas portiones <lb/>appellabimus. cum igitur portiones &longs;int minores &longs;pacio <lb/>u, circulus, uel ellip&longs;is g h ad portiones maiorem propor<lb/>tionem habebit, quàm linea m k ad K l. fiat n <foreign lang="greek">k</foreign> ad K l, ut <lb/>circulus uel ellip&longs;is g h ad ip&longs;as portiones. 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. 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="centr&utilde;">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;tr&utilde;">centrum</expan> crit <lb/>in linea k l protracta, & in puncto n; quod e&longs;t <expan abbr="ab&longs;urd&utilde;">ab&longs;urdum</expan>. relin <lb/>quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>tionis &longs;it <expan abbr="punct&utilde;">punctum</expan> k. quæ omnia <expan abbr="demon&longs;trãda">demon&longs;tranda</expan> propo&longs;uimus.</s>

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<s>At uero cylindrum, uel cylindri <expan abbr="portion&etilde;">portionem</expan> ce <lb/>ad pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura in &longs;pa<lb/>cio g h de&longs;cripta, & altitudo æqualis; eandem ha

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<s>Intelligatur circulus, uel ellip&longs;is x æqualis figuræ rectili<lb/>neæ in g h &longs;pacio de&longs;criptæ. & ab x con&longs;tituatur conus, uel <lb/>

<arrow.to.target n="fig23"></arrow.to.target><lb/>coni portio, <expan abbr="altitudin&etilde;">altitudinem</expan> habens <expan abbr="eand&etilde;">eandem</expan>, <expan abbr="quã">quam</expan> cylindrus uel cy <lb/>lindri portio c e. Sit deinde rectilinea figura, in qua y eade, <lb/>quæ in &longs;pacio g h de&longs;cripta e&longs;t: & ab hac pyramis æqucalta <lb/>con&longs;tituatur. Dico <expan abbr="con&utilde;">conum</expan> uel coni portione x pyramidi y <expan abbr="æ-qual&etilde;">æ<lb/>qualem</expan> e&longs;&longs;e. ni&longs;i enim &longs;it æqualis, uel maior, uel minor crit.</s>

<figure id="fig23"></figure><lb/>coni portio, <expan abbr="altitudin&etilde;">altitudinem</expan> habens <expan abbr="eand&etilde;">eandem</expan>, <expan abbr="quã">quam</expan> cylindrus uel cy <lb/>lindri portio c e. Sit deinde rectilinea figura, in qua y eade, <lb/>quæ in &longs;pacio g h de&longs;cripta e&longs;t: & ab hac pyramis æqucalta <lb/>con&longs;tituatur. Dico <expan abbr="con&utilde;">conum</expan> uel coni portione x pyramidi y <expan abbr="æ-qual&etilde;">æ<lb/>qualem</expan> e&longs;&longs;e. ni&longs;i enim &longs;it æqualis, uel maior, uel minor crit.</s>

<s>Sit primum maior, et exuperet &longs;olido z. Itaque in circu <lb/>lo, uel ellip&longs;i x de&longs;cribatur figura rectilinea; & in ea pyra<lb/>mis candem, quam conus, uel coni portio altitudinem ha<lb/>bens, ita ut portiones relictæ minores &longs;int &longs;olido z, quem<lb/>admodum docetur in duodecimo libro elementorum pro <lb/>po&longs;itione undecima. erit pyramis x adhuc pyramide y ma <lb/>ior. & quoniam piramides æque altæ inter &longs;e &longs;unt, &longs;icuti ba <lb/>

<arrow.to.target n="marg46"></arrow.to.target><lb/>&longs;es; pyramis x ad piramidem y eandem proportionem ha<lb/>bet, quàm figura rectilinea x ad figuram y. Sed figura recti

<pb/>

<arrow.to.target n="fig24"></arrow.to.target><lb/>linea x cum &longs;it minor circulo, uel cllip&longs;i, e&longs;t etiam minor fi<lb/>gura rectilinca y. ergo pyramis x pyramide y minor erit. <lb/>Sed & maior; quod ficri <expan abbr="nõ">non</expan> pote&longs;t. At &longs;i conus, uel coni por <lb/>tio x ponatur minor pyramide y: &longs;it alter conus æque al<lb/>tus, uel altera coni portio X ip&longs;i pyramidi y æqualis. crit <lb/>eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x, <lb/>quorum exce&longs;&longs;us &longs;it &longs;pacium <foreign lang="greek">w.</foreign> Si igitur in circulo, uel eili<lb/>p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relictæ <lb/>&longs;int <foreign lang="greek">w</foreign> &longs;pacio minores, ciu&longs;modi figura adhuc maior erit cir <lb/>culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. & pyramis in <lb/>ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi<lb/>nor pyramide y. e&longs;t ergo ut X figura rectilinea ad figuram <lb/>rectilineam y, ita pyramis X ad pyramidem y. quare cum <lb/>figura rectilinea X &longs;it maior figura y: erit & pyramis X py<lb/>ramide y maior. &longs;ed erat minor; quod rur&longs;us fieri non po<lb/>te&longs;t. non e&longs;t igitur conus, uel coni portio x neque maior, <lb/>neque minor pyramide y. ergo ip&longs;i nece&longs;&longs;ario e&longs;t æqualis. <lb/>Itaque quoniam ut conus ad conum, uel coni portio ad co

<figure id="fig24"></figure><lb/>linea x cum &longs;it minor circulo, uel cllip&longs;i, e&longs;t etiam minor fi<lb/>gura rectilinca y. ergo pyramis x pyramide y minor erit. <lb/>Sed & maior; quod ficri <expan abbr="nõ">non</expan> pote&longs;t. At &longs;i conus, uel coni por <lb/>tio x ponatur minor pyramide y: &longs;it alter conus æque al<lb/>tus, uel altera coni portio X ip&longs;i pyramidi y æqualis. crit <lb/>eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x, <lb/>quorum exce&longs;&longs;us &longs;it &longs;pacium <foreign lang="greek">w.</foreign> Si igitur in circulo, uel eili<lb/>p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relictæ <lb/>&longs;int <foreign lang="greek">w</foreign> &longs;pacio minores, ciu&longs;modi figura adhuc maior erit cir <lb/>culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. & pyramis in <lb/>ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi<lb/>nor pyramide y. e&longs;t ergo ut X figura rectilinea ad figuram <lb/>rectilineam y, ita pyramis X ad pyramidem y. quare cum <lb/>figura rectilinea X &longs;it maior figura y: erit & pyramis X py<lb/>ramide y maior. &longs;ed erat minor; quod rur&longs;us fieri non po<lb/>te&longs;t. non e&longs;t igitur conus, uel coni portio x neque maior, <lb/>neque minor pyramide y. ergo ip&longs;i nece&longs;&longs;ario e&longs;t æqualis. <lb/>Itaque quoniam ut conus ad conum, uel coni portio ad co

<arrow.to.target n="fig25"></arrow.to.target><lb/>ni portionem, ita e&longs;t cylindrus ad cylindrum, uel cylin<lb/>dri portio ad cylindri portionem: & ut pyramis ad pyra<lb/>midem, ita pri&longs;ma ad pri&longs;ma, cum eadem &longs;it ba&longs;is, & æqua <lb/>lis altitudo; crit cylindrus uel cylindri portio x pri&longs;ma<lb/>ti y æqualis. <expan abbr="e&longs;tq;">e&longs;tque</expan> ut &longs;pacium g h ad &longs;pacium x, ita cylin<lb/>drus, uel cylindri portio c e ad cylindrum, uel cylindri por<lb/>tionem x. Con&longs;tat igitur cylindrum uel cylindri <expan abbr="portion&etilde;">portionem</expan> <lb/>c e, ad pri&longs;ma y, quippe cuius ba&longs;is e&longs;t figura rectilinea in <lb/>

<figure id="fig25"></figure><lb/>ni portionem, ita e&longs;t cylindrus ad cylindrum, uel cylin<lb/>dri portio ad cylindri portionem: & ut pyramis ad pyra<lb/>midem, ita pri&longs;ma ad pri&longs;ma, cum eadem &longs;it ba&longs;is, & æqua <lb/>lis altitudo; crit cylindrus uel cylindri portio x pri&longs;ma<lb/>ti y æqualis. <expan abbr="e&longs;tq;">e&longs;tque</expan> ut &longs;pacium g h ad &longs;pacium x, ita cylin<lb/>drus, uel cylindri portio c e ad cylindrum, uel cylindri por<lb/>tionem x. Con&longs;tat igitur cylindrum uel cylindri <expan abbr="portion&etilde;">portionem</expan> <lb/>c e, ad pri&longs;ma y, quippe cuius ba&longs;is e&longs;t figura rectilinea in <lb/>

<arrow.to.target n="marg47"></arrow.to.target><lb/>&longs;pacio g h de&longs;cripta, eandem proportionem habere, quam <lb/>&longs;pacium g h habet ad &longs;pacium x, hoc e&longs;t ad dictam figuram. <lb/>quod demon&longs;trandum fuerat.</s>

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<s>THEOREMA IX. PROPOSITIO IX.</s>

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<s>SIT pyramis, cuius ba&longs;is triangulum a b c; axis d c: & <lb/>&longs;ecetur plano ba&longs;i æquidi&longs;tante; quod <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> &longs;aciat f g h; <lb/><expan abbr="occurratq;">occurratque</expan> axi in puncto <foreign lang="greek">k.</foreign> Dico f g h triangulum e&longs;&longs;e, ip&longs;i <lb/>a b c &longs;imile; cuius grauitatis centrum e&longs;t K. <expan abbr="Quoniã">Quoniam</expan> enim <lb/>duo plana æquidi&longs;tantia a b c, f g h &longs;ecantur à plano a b d; <lb/>communes corum &longs;ectiones a b, f g æquidi&longs;tantes erunt: & <lb/>cadem ratione æquidi&longs;tantes ip&longs;æ b c, g h: & c a, h f. Quòd <lb/>cum duæ lineæ f g, g h, duabus a b, b c æquidi&longs;tent, nec <lb/>fintin eodem plano; angulus ad g æqualis e&longs;t angulo ad <lb/>b. & &longs;imiliter angulus ad h angulo ad c: <expan abbr="angulusq;">angulusque</expan> ad f ci, <lb/>qui ad a e&longs;t æqualis. triangulum igitur f g h &longs;imile e&longs;t tri<lb/>angulo a b c. Atuero punctum <foreign lang="greek">k</foreign> centrum e&longs;&longs;e grauita<lb/>tis trianguli f g h hoc modo o&longs;tendemus. Ducantur pla<lb/>na per axem, & per lineas d a, d b, d c: erunt communes &longs;e<lb/>ctiones f K, a e æquidi&longs;tantes: <expan abbr="pariterq;">pariterque</expan> <foreign lang="greek">k</foreign> g, e b; & <foreign lang="greek">k</foreign> h, e c: <lb/>quare angulus <foreign lang="greek">k</foreign> f h angulo e a c; & angulus <foreign lang="greek">k</foreign> f g ip&longs;i e a b <lb/>

<arrow.to.target n="fig26"></arrow.to.target><lb/>e&longs;t æqualis. Eadem ratione <lb/>anguli ad g angulis ad b: & <lb/>anguli ad h iis, qui ad c æ<lb/>quales erunt. ergo puncta <lb/>e K in triangulis a b c, f g h <lb/>&longs;imiliter &longs;unt po&longs;ita, per &longs;e<lb/>xtam po&longs;itionem Archime<lb/>dis in libro de centro graui<lb/>tatis planorum. Sed cum e <lb/>&longs;it centrum grauitatis trian <lb/>guli a b c, erit ex undecima <lb/>propo&longs;itione eiu&longs;dem libri, <lb/>& K trianguli f g h grauita <lb/>tis centrum. id quod demon&longs;trare oportebat. Non aliter <lb/>in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra<lb/><gap/></s>

<figure id="fig26"></figure><lb/>e&longs;t æqualis. Eadem ratione <lb/>anguli ad g angulis ad b: & <lb/>anguli ad h iis, qui ad c æ<lb/>quales erunt. ergo puncta <lb/>e K in triangulis a b c, f g h <lb/>&longs;imiliter &longs;unt po&longs;ita, per &longs;e<lb/>xtam po&longs;itionem Archime<lb/>dis in libro de centro graui<lb/>tatis planorum. Sed cum e <lb/>&longs;it centrum grauitatis trian <lb/>guli a b c, erit ex undecima <lb/>propo&longs;itione eiu&longs;dem libri, <lb/>& K trianguli f g h grauita <lb/>tis centrum. id quod demon&longs;trare oportebat. Non aliter <lb/>in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra<lb/><gap/></s>

<s>PROBLEMA I. PROPOSITIO X.</s>

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<s>SIT conus, cuius axis b d: & &longs;ecetur plano per axem <lb/>ducto, 'ut&longs;ectio &longs;it triangulum a b c: <expan abbr="intelligaturq;">intelligaturque</expan> cylin<lb/>drus, qui ba&longs;im eandem, & eundem axem habeat. Hocigi<lb/>tur cylindro continenter bifariam &longs;ecto, relinquetur cylin <lb/>drus minor &longs;olida magnitudine propo&longs;ita. Sit autem is cy <lb/>lindrus, qui ba&longs;im habet circulum circa diametrum a c, & <lb/>axem d e. Itaque diuidatur b d in partes æquales ip&longs;i d e <lb/>in punctis f g h K l m: & per ea ducantur plana conum &longs;e<lb/>cantia; quæ ba&longs;i æquidi&longs;tent. erunt &longs;ectiones circuli, cen<lb/>train axi habentes, ut in primo libro conicorum, propo&longs;i-

<pb pagenum="18"/>tione quarta Apollonius demon&longs;trauit. Si igitur à &longs;ingu<lb/>lis horum circulorum, duo cylindri fiant; unus quidem ad <lb/>ba&longs;is partes; alter ad partes uerticis: in&longs;cripta erit in co<lb/>no &longs;olida quædam figura, & altera circum&longs;cripta ex cylin<lb/>dris æqualem altitudinem habentibus con&longs;tans; quorum <lb/>

<arrow.to.target n="fig27"></arrow.to.target><lb/>unu&longs;qui&longs;que, qui in <lb/>figura in&longs;cripta con<lb/>tinetur æqualis e&longs;t ei, <lb/>qui ab eodem fit cir<lb/>culo in figura <expan abbr="circ&utilde;-&longs;cripta">circun<lb/>&longs;cripta</expan>. Itaque cylin <lb/>drus o p æqualis e&longs;t <lb/>cylindro o n; cylin<lb/>drus r s <expan abbr="cyl&itilde;dro">cylindro</expan> r <expan abbr="q;">que</expan> <lb/>cylindrus u x cylin<lb/>dro u t e&longs;t æqualis; <lb/>& alii aliis &longs;imiliter. <lb/>quare con&longs;tat <expan abbr="circ&utilde;-&longs;criptam">circun<lb/>&longs;criptam</expan> figuram &longs;u<lb/>perare in&longs;criptam cy <lb/>lindro, cuius ba&longs;is e&longs;t <lb/>circulus circa diametrum a c, & axis d e. atque hic e&longs;t mi<lb/>nor &longs;olida magnitudine propo&longs;ita.</s>

<figure id="fig27"></figure><lb/>unu&longs;qui&longs;que, qui in <lb/>figura in&longs;cripta con<lb/>tinetur æqualis e&longs;t ei, <lb/>qui ab eodem fit cir<lb/>culo in figura <expan abbr="circ&utilde;-&longs;cripta">circun<lb/>&longs;cripta</expan>. Itaque cylin <lb/>drus o p æqualis e&longs;t <lb/>cylindro o n; cylin<lb/>drus r s <expan abbr="cyl&itilde;dro">cylindro</expan> r <expan abbr="q;">que</expan> <lb/>cylindrus u x cylin<lb/>dro u t e&longs;t æqualis; <lb/>& alii aliis &longs;imiliter. <lb/>quare con&longs;tat <expan abbr="circ&utilde;-&longs;criptam">circun<lb/>&longs;criptam</expan> figuram &longs;u<lb/>perare in&longs;criptam cy <lb/>lindro, cuius ba&longs;is e&longs;t <lb/>circulus circa diametrum a c, & axis d e. atque hic e&longs;t mi<lb/>nor &longs;olida magnitudine propo&longs;ita.</s>

<s>PROBLEMA III. PROPOSITIO XII.</s>

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<s>SIT pyramis, cuius ba&longs;is triangulum a b c: & axis d e. <lb/>Dico in linea d e ip&longs;ius grauitatis centrum ine&longs;&longs;e. Si enim <lb/>fieri pote&longs;t, &longs;it centrum f: & ab f ducatur ad ba&longs;im pyrami <lb/>dis linea f g, axi æquidi&longs;tans: <expan abbr="iunctaq;">iunctaque</expan> e g ad latera trian<lb/>guli a b c producatur in h. quam uero proportionem ha<lb/>bet linea h e ad e g, habeat pyramis ad aliud &longs;olidum, in <lb/>quo K: <expan abbr="in&longs;cribaturq;">in&longs;cribaturque</expan> in pyramide &longs;olida figura, & altera cir <lb/>cum&longs;cribatur ex pri&longs;matibus æqualem habentibus altitu<lb/>dinem, ita ut circum&longs;cripta in&longs;criptam exuperet magnitu<lb/>dine, quæ &longs;olido k &longs;it minor. Et quoniam in pyramide pla <lb/>num ba&longs;i æquidi&longs;tans ductum &longs;ectionem facit figuram &longs;i<lb/>milem ei, quæ e&longs;t ba&longs;is; <expan abbr="centrumq;">centrumque</expan> grauitatis in axe haben <lb/>tem: erit pri&longs;matis s t grauitatis <expan abbr="centr&utilde;">centrum</expan> in linear <expan abbr="q;">que</expan> pri&longs;<lb/>matis u x centrum in linea q p, pri&longs;inatis y z in linea p o; <lb/>pri&longs;matis <foreign lang="greek">h q</foreign> in linea o n; pri&longs;matis <foreign lang="greek">l <gap/></foreign> in linea n m; pri&longs;<lb/>matis <foreign lang="greek">n p</foreign> in m l; & denique pri&longs;matis <foreign lang="greek">p s</foreign> in l e. quare to-

<pb/>da &longs;igura, & altera circum&longs;cribatur ex cylindris, uel cylin<lb/>dri portionibus, &longs;icuti dictum e&longs;t, ita ut exce&longs;&longs;us, quo &longs;igu<lb/>ra circum&longs;cripta in&longs;criptam &longs;uperat, &longs;it &longs;olido g minor. <lb/>Itaque centrum grauitatis cylindri, uel cylindri portionit <lb/>q r e&longs;t in linea p o; cylindri, uel cylindri portionis s t cen<lb/>trum in linea o n; centrum u x in linea n m; y z in m b; <foreign lang="greek">n q</foreign><lb/>in l k; <foreign lang="greek">l m</foreign> in K h; & denique <foreign lang="greek">v p</foreign> centrum in h d. ergo figu<lb/>

<arrow.to.target n="fig28"></arrow.to.target><lb/>ræ in&longs;criptæ centrum e&longs;t in linea p d. Sit autem <foreign lang="greek">r</foreign>: & iun<lb/>cta <foreign lang="greek">r</foreign> e protendatur, ut cum linea, quæ à <expan abbr="p&utilde;cto">puncto</expan> c ducta &longs;ue<lb/>rit axi æquidi&longs;tans, conueniat in <foreign lang="greek">s.</foreign> erit <foreign lang="greek">s r</foreign> ad <foreign lang="greek">r</foreign> e, ut c d <lb/>ad d f: & conus, &longs;eu coni portio ad exce&longs;&longs;um, quo circum<lb/>&longs;cripta figura in&longs;criptam &longs;uperat, habebit maiorem pro<lb/>portionem, quàm <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. ergo ad partem exce&longs;&longs;us, quæ <lb/>intra ip&longs;ius &longs;uperficiem comprehenditur, multo m aiorem <lb/>proportionem habebit. habeat eam, quam <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. erit

<figure id="fig28"></figure><lb/>ræ in&longs;criptæ centrum e&longs;t in linea p d. Sit autem <foreign lang="greek">r</foreign>: & iun<lb/>cta <foreign lang="greek">r</foreign> e protendatur, ut cum linea, quæ à <expan abbr="p&utilde;cto">puncto</expan> c ducta &longs;ue<lb/>rit axi æquidi&longs;tans, conueniat in <foreign lang="greek">s.</foreign> erit <foreign lang="greek">s r</foreign> ad <foreign lang="greek">r</foreign> e, ut c d <lb/>ad d f: & conus, &longs;eu coni portio ad exce&longs;&longs;um, quo circum<lb/>&longs;cripta figura in&longs;criptam &longs;uperat, habebit maiorem pro<lb/>portionem, quàm <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. ergo ad partem exce&longs;&longs;us, quæ <lb/>intra ip&longs;ius &longs;uperficiem comprehenditur, multo m aiorem <lb/>proportionem habebit. habeat eam, quam <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. erit

<pb pagenum="21"/>diuidendo figura &longs;olida in&longs;cripta ad dictam exce&longs;&longs;us par<lb/>tem, ut <foreign lang="greek">t e</foreign> ad c <foreign lang="greek">p.</foreign> & quoniam à cono, &longs;cu coni portione, <lb/>cuius grauitatis centrum e&longs;t e, au&longs;ertur figura in&longs;cripta, <lb/>cuius centrum <foreign lang="greek">r:</foreign> re&longs;iduæ magnitudinis compo&longs;itæ cx par <lb/>te exce&longs;&longs;us, quæ intra coni, uel coni portionis &longs;uper&longs;iciem <lb/>continetur, centrum grauitatis erit in linea e protracta, <lb/>atque in puncto t. quod c&longs;t ab&longs;urdum. <expan abbr="cõ&longs;tat">con&longs;tat</expan> ergo <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis coni, uel coni portionis, e&longs;&longs;e in axc b d: quod de <lb/>mon&longs;trandum propo&longs;uimus.</s>

<s>THEOREMA XI. PROPOSITIO XV.</s>

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<s>Secetur &longs;phæra, uel &longs;phæroides piano per axem ducto; <lb/>quod &longs;ectionem &longs;aciat circulum, uel cllip&longs;im a b c d, cuius <lb/>diameter, & &longs;phæræ, uel &longs;phæroidis axis d b; & centrum e. <lb/>Dico e grauitatis etiam centrum e&longs;&longs;e. &longs;ecetur enim altero <lb/>plano per e, ad planum &longs;ecans recto, cuius &longs;ectio &longs;it circu<lb/>lus cir ca diametrum a ç. erunt a d c, a b c dimidiæ portio<lb/>nes &longs;phæræ, uel &longs;phæroidis. & quoniam portionis a d c gra <lb/>uitatis centrum e&longs;i in linead, & centrum portionis a b c in <lb/>ip&longs;a b c; totius &longs;phæræ, uel &longs;phæroidis grauitatis centrum <lb/>in a<gap/> c d b con&longs;i&longs;iet, Quòd &longs;i portionis a d c centrum <expan abbr="grau&itilde;">grauim</expan> <lb/>tatis ponatur e&longs;&longs;c f & fiatip&longs;i f e æqualis e g. <expan abbr="punct&utilde;">punctum</expan> g por <lb/>

<arrow.to.target n="fig29"></arrow.to.target><lb/>

<arrow.to.target n="marg48"></arrow.to.target><lb/>tionis a b c centrum crit. &longs;olidis cnim figuris &longs;imilibus & <lb/>æq<gap/> alibus inter &longs;e aptatis, & c. ntra g. auitatis ip&longs;arum in<lb/>

<arrow.to.target n="marg49"></arrow.to.target><lb/>ter <gap/>e aptentur nece&longs;le e&longs;t. ex quo fit, ut magnitudinis, quæ <lb/>ex utilique <expan abbr="cõ&longs;lat">con&longs;lat</expan>, hoc e&longs;t ip&longs;ius &longs;phæræ, uel &longs;phæroidis gra <lb/>uitatis centrum &longs;it in medio lincæ f g uid li<gap/> t in c. Sphæ<lb/>ræ igitur, uel &longs;phæroidis grauitatis centrum e&longs;tidem, quod <lb/>centrum figuræ.</s>

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<s><margin.target id="marg49"></margin.target>4 Arcl <lb/>medis.</s>

<s>Ex demon&longs;tratis per&longs;picue apparet, portioni <lb/>&longs;phæræ uel &longs;phæroidis, quæ dimidia maier e&longs;t, <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis in axe con&longs;i&longs;tere.</s>

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<s>Sint &longs;olida parallelepipeda in <expan abbr="ead&etilde;">eadem</expan> ba&longs;i <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> a b c d, <lb/>a b e f: & &longs;it &longs;olidi a b c d altitudo minor: producatur au<lb/>tem planum c d adeo, ut&longs;olidum a b e f &longs;ecet; cuius &longs;ectio <lb/>

<arrow.to.target n="fig30"></arrow.to.target><lb/>

<arrow.to.target n="marg53"></arrow.to.target><lb/>&longs;it gh. <expan abbr="cr&utilde;t">crunt</expan> &longs;oli <lb/>da a b c d, a b g h <lb/>in eadem ba&longs;i, <lb/>& æquali altitu<lb/>dine inter &longs;e æ<lb/>qualia. <expan abbr="Quoniã">Quoniam</expan> <lb/>igitur &longs;olidum <lb/>a b e f &longs;ecatur <lb/>plano ba&longs;ibus <lb/><expan abbr="æquidi&longs;tãte">æquidi&longs;tante</expan>, erit <lb/>

<arrow.to.target n="marg54"></arrow.to.target><lb/>&longs;olidum g h e f <lb/>adip&longs;um a b g h

<pb/>ut altitudo ad altitudinem: & componendo conuertendo <lb/>

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<s><margin.target id="marg55"></margin.target>7. quinti.</s>

<s>Sint &longs;olida parallelcpipeda a b, c d in æqualibus ba&longs;ibus <lb/>con&longs;tituta: <expan abbr="&longs;itq;">&longs;itque</expan> b e altitudo &longs;olidi a b: & &longs;olidi c d altitudo <lb/>d f; quæ quidem maior &longs;it, quàm b e. Dico &longs;olidum a b ad <lb/>&longs;olidum c d eandem habere proportionem, quam b e ad <lb/>d f. ab&longs;cindatur enim à linea d f æqualis ip&longs;i b e, quæ &longs;it g f: <lb/>& per g ducatur planum &longs;ecans &longs;olidum c d; quod ba&longs;ibus <lb/>æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> h. K. crunt &longs;olida a b, c <foreign lang="greek">k</foreign> æque <lb/>

<arrow.to.target n="marg56"></arrow.to.target><lb/>

<arrow.to.target n="fig31"></arrow.to.target><lb/>alta inter <lb/>&longs;e æqualia <lb/><expan abbr="c&utilde;">cum</expan> æqua<lb/>les ba&longs;es <lb/>habeant. <lb/>

<figure id="fig31"></figure><lb/>alta inter <lb/>&longs;e æqualia <lb/><expan abbr="c&utilde;">cum</expan> æqua<lb/>les ba&longs;es <lb/>habeant. <lb/>

<arrow.to.target n="marg57"></arrow.to.target><lb/>Sed <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> <lb/>h d ad &longs;oli <lb/>dum c K <lb/>e&longs;t, ut alti <lb/>tudo d g <lb/>ad g <expan abbr="falti-tudin&etilde;">falti<lb/>tudinem</expan>; &longs;e <lb/>catur enim &longs;olidum c d plano ba&longs;i <lb/>

<arrow.to.target n="fig32"></arrow.to.target><lb/>bus æquidi&longs;tante: & rur&longs;us <expan abbr="cõpo-nende">compo<lb/>nende</expan>, <expan abbr="conuertendoq;">conuertendoque</expan> <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> c k <lb/>

<figure id="fig32"></figure><lb/>bus æquidi&longs;tante: & rur&longs;us <expan abbr="cõpo-nende">compo<lb/>nende</expan>, <expan abbr="conuertendoq;">conuertendoque</expan> <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> c k <lb/>

<arrow.to.target n="marg58"></arrow.to.target><lb/>ad &longs;olidum c d, ut g f ad f d. ergo <lb/>&longs;olidum a b, quod e&longs;t æquale ip&longs;i <lb/>c <foreign lang="greek">k</foreign> ad &longs;olidum c d eam proportio <lb/>nem habet, quam altitudo g f, hoc <lb/>e&longs;t b e ad d f altitudinem.</s>

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<s><margin.target id="marg58"></margin.target>7. quinti.</s>

<s>Sint deinde &longs;olida parallelepipe <lb/>da a b, a c in cadem ba&longs;i; quorum <lb/>axes d e, &longs; e cum ip&longs;a æquales angu

<pb pagenum="24"/>los contineant. Dico &longs;olidum a b ad &longs;olidum a c e idem ha <lb/>bere proportionem, quam axis d e ad axem e f. 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. 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. 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. Quoniam igitur axes cum ba&longs;ibus <lb/>æquales angulos continent, erit d e g angulus æqualis an<lb/>

<arrow.to.target n="fig33"></arrow.to.target><lb/>gulo f e h: & &longs;unt <lb/>anguli ad g h re<lb/>cti, quare & re<lb/>liquus e d g æqua <lb/>lis erit reliquo <lb/>e fh: & triangu<lb/>lum d e g <expan abbr="triãgu-lo">triangu<lb/>lo</expan> f e h &longs;imile. er<lb/>go g d ad d e e&longs;t', <lb/>ut h f ad fe: & per <lb/>mutando g d ad <lb/>h f, ut d e ad c f. <lb/>Sed &longs;olidum a b <lb/>ad &longs;olidum a c <lb/>candem propor<lb/>tionem habet, <lb/>quam d g altitu<lb/>do ad <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>f h. ergo & <expan abbr="ean-d&etilde;">ean<lb/>dem</expan> habebit, <expan abbr="quã">quam</expan> <lb/>axis d e a l e <gap/> <expan abbr="ax&etilde;">axem</expan></s>

<figure id="fig33"></figure><lb/>gulo f e h: & &longs;unt <lb/>anguli ad g h re<lb/>cti, quare & re<lb/>liquus e d g æqua <lb/>lis erit reliquo <lb/>e fh: & triangu<lb/>lum d e g <expan abbr="triãgu-lo">triangu<lb/>lo</expan> f e h &longs;imile. er<lb/>go g d ad d e e&longs;t', <lb/>ut h f ad fe: & per <lb/>mutando g d ad <lb/>h f, ut d e ad c f. <lb/>Sed &longs;olidum a b <lb/>ad &longs;olidum a c <lb/>candem propor<lb/>tionem habet, <lb/>quam d g altitu<lb/>do ad <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>f h. ergo & <expan abbr="ean-d&etilde;">ean<lb/>dem</expan> habebit, <expan abbr="quã">quam</expan> <lb/>axis d e a l e <gap/> <expan abbr="ax&etilde;">axem</expan></s>

<s>Po&longs;tremo &longs;int <lb/>&longs;olidi paral le pi <lb/>peda a b, c d in

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<s>Sint duo pri&longs;mata a e, a f, quorum eadem ba&longs;is quadri<lb/>latera a b c d: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis a e altitudo e g; & pri&longs;matis <lb/>a f altitudo f h. Dico pri&longs;ma a e ad pri&longs;ma a f eam habere <lb/>proportionem, quam e g ad f h. iungatur enim a c: & in <lb/>unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum <lb/>

<arrow.to.target n="fig34"></arrow.to.target><lb/>ba&longs;es &longs;int triangu <lb/>la a b c, a c d. habe <lb/>bunt duo pri&longs;ma<lb/>te in eadem ba&longs;i <lb/>a b c con&longs;tituta, <lb/>proportionem <expan abbr="eã">eam</expan> <lb/>dem, quam ip&longs;o<lb/>rum altitudines e <lb/>g, f h, ex iam de<lb/>mon&longs;tratis. & &longs;i<lb/>militer alia duo, <lb/>quæ &longs;unt in ba&longs;i a <lb/>

<figure id="fig34"></figure><lb/>ba&longs;es &longs;int triangu <lb/>la a b c, a c d. habe <lb/>bunt duo pri&longs;ma<lb/>te in eadem ba&longs;i <lb/>a b c con&longs;tituta, <lb/>proportionem <expan abbr="eã">eam</expan> <lb/>dem, quam ip&longs;o<lb/>rum altitudines e <lb/>g, f h, ex iam de<lb/>mon&longs;tratis. & &longs;i<lb/>militer alia duo, <lb/>quæ &longs;unt in ba&longs;i a <lb/>

<arrow.to.target n="marg62"></arrow.to.target><lb/>c d. quare totum pri&longs;ma a e ad pri&longs;ma a f eandem propor <lb/>tionem habebit, quam altitudo e g ad f h altitudinem. <lb/>Quòd cum pri&longs;mata &longs;int pyramidum tripla, & ip&longs;æ pyrami <lb/>des, quarum eadem e&longs;t ba&longs;is quadrilatera, & altitudo pri&longs;<lb/>matum altitudini æqualis, eam inter &longs;e proportionem ha<lb/>bebunt, quam altitudines.</s>

<s>Si uero pri&longs;mata ba&longs;es æquales habeant, <expan abbr="nõ">non</expan> ea&longs;dem, &longs;int <lb/>duo eiu&longs;modi pri&longs;mata a e, f l: & &longs;it ba&longs;is pri&longs;matis a e qua <lb/>drilaterum a b c d; & pri&longs;matis f l quadrilaterum f g h k. <lb/>Dico pri&longs;ma a e ad pri&longs;ma f l ita e&longs;&longs;e, ut altitudo illius ad <lb/>huius altitudinem. nam &longs;i altitudo &longs;it eadem, <expan abbr="intelligãtur">intelligantur</expan> <lb/>

<arrow.to.target n="marg63"></arrow.to.target><lb/>duæ pyramides a b c d e, f g h k l. quæ <expan abbr="&itilde;tcr&longs;e">intcr&longs;e</expan> æquales <expan abbr="er&utilde;t">erunt</expan>, <lb/>cum æquales ba&longs;es, & altitudinem eandem habeant. quare <lb/>

<arrow.to.target n="marg64"></arrow.to.target><lb/>& pri&longs;mata a e, f l, quæ &longs;unt <expan abbr="har&utilde;">harum</expan> pyramidum tripla, æqua<lb/>lia &longs;int nece&longs;&longs;e e&longs;t. ex quibus per&longs;picue con&longs;tat <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>. <lb/>Si uero altitudo pri&longs;matis f l &longs;it maior, à pri&longs;mate f l ab<lb/>&longs;cindatur pri&longs;ma fm, quod æque altum &longs;it, <expan abbr="atq;">atque</expan> ip&longs;um a e.

<pb/>

<arrow.to.target n="fig35"></arrow.to.target><lb/>erunt eædem ra<lb/>tione pri&longs;mata a <lb/>e, f m inter &longs;e æ<lb/>qualia. quare &longs;i<lb/>militer demon<lb/>&longs;trabitur pri&longs;ma <lb/>f m ad pri&longs;ma f l <lb/>eandem habere <lb/>proportionem, <lb/>quam pri&longs;matis <lb/>f m altitudo ad <lb/>altitudinem ip<lb/>&longs;ius f l. ergo & pri&longs;ma a e ad pri&longs;ma f l eandem propor<lb/>tionem habebit, quam altitudo ad altitudinem. &longs;equitur <lb/>igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;titu&utilde;">con&longs;tituum</expan> <lb/>tur, candem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s>

<figure id="fig35"></figure><lb/>erunt eædem ra<lb/>tione pri&longs;mata a <lb/>e, f m inter &longs;e æ<lb/>qualia. quare &longs;i<lb/>militer demon<lb/>&longs;trabitur pri&longs;ma <lb/>f m ad pri&longs;ma f l <lb/>eandem habere <lb/>proportionem, <lb/>quam pri&longs;matis <lb/>f m altitudo ad <lb/>altitudinem ip<lb/>&longs;ius f l. ergo & pri&longs;ma a e ad pri&longs;ma f l eandem propor<lb/>tionem habebit, quam altitudo ad altitudinem. &longs;equitur <lb/>igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;titu&utilde;">con&longs;tituum</expan> <lb/>tur, candem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s>

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<s>Sint deinde pri&longs;mata a e, a f in eadem ba&longs;i a b c d; <expan abbr="quor&utilde;">quorum</expan> <lb/>axes cum ba&longs;ibus æquales angulos contineant: & &longs;it pri&longs;

<pb pagenum="26"/>matis a e axis g h; & pri&longs;matis a f axis l h. Dico pri&longs;ma <lb/>a e ad pri&longs;ma a f eam proportionem habere, quam g h ad <lb/>h l. ducantur à punctis g l perpendiculares ad ba&longs;is pla<lb/>

<arrow.to.target n="fig36"></arrow.to.target><lb/>num g K, l m: & iungantur k h, <lb/>h m. Itaque quoniam anguli g h <lb/>k, l h m &longs;unt æquales, &longs;imiliter ut <lb/>&longs;upra demon&longs;trabimus, triangu<lb/>la g h K, l h m &longs;imilia e&longs;&longs;e; & ut g <lb/>K ad l m, ita g h ad h l. habet au <lb/>tem pri&longs;ma a e ad pri&longs;ma a f ean <lb/>dem proportionem, quam altitu<lb/>do g K ad altitudinem l m, &longs;icuti <lb/>demon&longs;tratum e&longs;t. ergo & can<lb/>dem habebit, quam g h, ad h l. py <lb/>ramis igitur a b c d g ad pyrami<lb/>dem a b c d l eandem proportio<lb/>nem habebit, quam axis g h ad h l axem.</s>

<figure id="fig36"></figure><lb/>num g K, l m: & iungantur k h, <lb/>h m. Itaque quoniam anguli g h <lb/>k, l h m &longs;unt æquales, &longs;imiliter ut <lb/>&longs;upra demon&longs;trabimus, triangu<lb/>la g h K, l h m &longs;imilia e&longs;&longs;e; & ut g <lb/>K ad l m, ita g h ad h l. habet au <lb/>tem pri&longs;ma a e ad pri&longs;ma a f ean <lb/>dem proportionem, quam altitu<lb/>do g K ad altitudinem l m, &longs;icuti <lb/>demon&longs;tratum e&longs;t. ergo & can<lb/>dem habebit, quam g h, ad h l. py <lb/>ramis igitur a b c d g ad pyrami<lb/>dem a b c d l eandem proportio<lb/>nem habebit, quam axis g h ad h l axem.</s>

<s>Denique &longs;int pri&longs;mata a e, <foreign lang="greek">k</foreign> o in æqualibus ba&longs;ibus a b <lb/>c d, k l m n con&longs;tituta; quorum axes cum ba&longs;ibus æquales <lb/>faciant angulos: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis a e axis f g, & altitudo f h: <lb/>pri&longs;matis autem k o axis p q, & altitudo p r. Dico pri&longs;ma <lb/>a e ad pri&longs;ma <foreign lang="greek">k</foreign> o ita e&longs;&longs;e, ut f g ad p <expan abbr="q.">que</expan> iunctis enim g h,

<pb/>q r, eodem, quo &longs;upra, modo o&longs;tendemns f g ad p q, ut f h <lb/>ad p r. &longs;ed pri&longs;ma a e ad ip&longs;um <foreign lang="greek">k</foreign> o e&longs;t, ut f h ad p r. ergo <lb/>& ut f g axis ad axem p <expan abbr="q.">que</expan> ex quibus &longs;it, ut pyramis a b c d f <lb/>

<arrow.to.target n="fig37"></arrow.to.target><lb/>ad <expan abbr="pyrami-d&etilde;">pyrami<lb/>dem</expan> <foreign lang="greek">k</foreign> l m n p <lb/>eandem ha <lb/>beat pro <lb/>portion&etilde;, <lb/><expan abbr="quã">quam</expan> axis ad <lb/><expan abbr="ax&etilde;">axem</expan>. quod <lb/><expan abbr="demon&longs;trã">demon&longs;tram</expan> <lb/><expan abbr="d&utilde;">dum</expan> &longs;uerat.</s>

<figure id="fig37"></figure><lb/>ad <expan abbr="pyrami-d&etilde;">pyrami<lb/>dem</expan> <foreign lang="greek">k</foreign> l m n p <lb/>eandem ha <lb/>beat pro <lb/>portion&etilde;, <lb/><expan abbr="quã">quam</expan> axis ad <lb/><expan abbr="ax&etilde;">axem</expan>. quod <lb/><expan abbr="demon&longs;trã">demon&longs;tram</expan> <lb/><expan abbr="d&utilde;">dum</expan> &longs;uerat.</s>

<s>Simili ra <lb/>tione in a<lb/>liis pri&longs;ma<lb/>tibus & py <lb/>ramidibus eadem demon&longs;trabuntur.</s>

Line 1406

Line 1404

<s>Sint enim primum e f, m n æquales: & ut ba&longs;is a b c d <lb/>ad ba&longs;im g h <foreign lang="greek">k</foreign> l, ita fiat linea, in qua o ad lineam, in qua p: <lb/>ut autem e f ad m n, ita linea p ad lineam <expan abbr="q.">que</expan> erunt lineæ <lb/>p q inter &longs;e æquales. Itaque pri&longs;ma a e ad pri&longs;ma g m <expan abbr="eã">eam</expan>

<pb pagenum="27"/>proportionem habet, quam ba&longs;is a b c d ad ba&longs;im g h <foreign lang="greek">k</foreign> l: <lb/>&longs;i enim intelligantur duæ pyramides a b c d e, g h k l m, ha<lb/>bebunt hæ inter &longs;e proportionem eandem, quam ip&longs;ar um <lb/>ba&longs;es ex &longs;exta duodecimi elementorum. Sed ut ba&longs;is a b c d <lb/>ad g h K l ba&longs;im, ita linea o ad lineam p; hoc e&longs;t ad lineam q <lb/>ei æqualem. ergo pri&longs;ma a e ad pri&longs;ma g m e&longs;t, ut linea o <lb/>ad lineam <expan abbr="q.">que</expan> proportio autem o ad q copo&longs;ita e&longs;t ex pro<lb/>portione o ad p, & ex proportione p ad <expan abbr="q.">que</expan> quare pri&longs;ma <lb/>a e ad pri&longs;ma g m, & idcirco pyramis a b c d e, ad pyrami<lb/>dem g h K l m proportionem habet ex ei&longs;dem proportio<lb/>nibus compo&longs;itam, uidelicet ex proportione ba&longs;is a b c d <lb/>ad ba&longs;im g h K l, & ex proportione altitudinis e f ad m n al <lb/>titudinem. Quòd &longs;i lineæ e f, m n inæquales ponantur, &longs;it <lb/>e f minor: & ut e f ad m n, ita fiat linea p ad lineam u: de <lb/>

<arrow.to.target n="fig38"></arrow.to.target><lb/>inde ab ip&longs;a m n ab&longs;cindatur r n æqualis e f: & per r duca<lb/>tur planum, quod oppo&longs;itis planis æquidi&longs;tans faciat &longs;e<lb/>ctionem s t. erit pri&longs;ma a e, ad pri&longs;ma g t, ut ba&longs;is a b c d <lb/>ad ba&longs;im g h k l; hoc e&longs;t ut o ad p: ut autem pri&longs;ma g t ad <lb/>

<figure id="fig38"></figure><lb/>inde ab ip&longs;a m n ab&longs;cindatur r n æqualis e f: & per r duca<lb/>tur planum, quod oppo&longs;itis planis æquidi&longs;tans faciat &longs;e<lb/>ctionem s t. erit pri&longs;ma a e, ad pri&longs;ma g t, ut ba&longs;is a b c d <lb/>ad ba&longs;im g h k l; hoc e&longs;t ut o ad p: ut autem pri&longs;ma g t ad <lb/>

<arrow.to.target n="marg65"></arrow.to.target><lb/>pri&longs;ma g m, ita altitudo r n; hoc e&longs;te f ad altitudine m n; <lb/>uidelicet linea p ad lineam u. ergo ex æquali pri&longs;ma a e ad <lb/>pri&longs;ma g m e&longs;t, ut linea o ad ip&longs;am u. Sed proportio o ad <lb/>u <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> e&longs;t ex proportione o ad p, quæ e&longs;t ba&longs;is a b c d <lb/>ad ba&longs;im g h <foreign lang="greek">k</foreign> l; & ex proportione p ad u, quæ e&longs;t altitudi<lb/>nis e f ad altitudinem m n. pri&longs;ma igitur a e ad pri&longs;ma g m

<pb/>compo&longs;itam proportionem habet ex proportione <expan abbr="ba&longs;i&utilde;">ba&longs;ium</expan>, <lb/>& proportione altitudinum. Quare & pyramis, cuius ba<lb/>&longs;is e&longs;t quadrilaterum a b c d, & altitudo e f ad pyramidem, <lb/>

<arrow.to.target n="fig39"></arrow.to.target><lb/>cuius ba&longs;is quadrilaterum g h K l, & altitudo m n, compo&longs;i <lb/>tam habet proportionem ex proportione ba&longs;ium a b c d, <lb/>g h k l, & ex proportione altitudinum e f, m n. quod qui<lb/>dem demon&longs;tra&longs;&longs;e oportebat.</s>

<figure id="fig39"></figure><lb/>cuius ba&longs;is quadrilaterum g h K l, & altitudo m n, compo&longs;i <lb/>tam habet proportionem ex proportione ba&longs;ium a b c d, <lb/>g h k l, & ex proportione altitudinum e f, m n. quod qui<lb/>dem demon&longs;tra&longs;&longs;e oportebat.</s>

<s>Ex iam demon&longs;tratis per&longs;picuum e&longs;t, pri&longs;ma <lb/>ta omnia, & pyramides, in quibus axes cum ba&longs;i<lb/>bus æquales angulos continent, proportionem <lb/>habere compo&longs;itam ex ba&longs;ium proportione, & <lb/>proportione axium. demon&longs;ttatum e&longs;t enim, a<lb/>xes inter &longs;e eandem proportionem habere, quam <lb/>ip&longs;æ altitudines.</s>

Line 1435

Line 1433

<s>Sit pyramis, cuius ba&longs;is triangulum a b c; axis d e; & gra <lb/>uitatis centrum K. Dico lineam d k ip&longs;ius K e triplam e&longs;&longs;e. <lb/>trianguli enim b d c centrum grauitatis &longs;it punctum f; <expan abbr="triã">triam</expan> <lb/>guli a d c <expan abbr="centr&utilde;">centrum</expan> g; & trianguli a d b &longs;it h: & iungantur a f, <lb/>b g, c h. Quoniam igitur <expan abbr="centr&utilde;">centrum</expan> grauitatis pyramidis in axe <lb/>

<arrow.to.target n="marg66"></arrow.to.target><lb/><expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>: <expan abbr="&longs;untq;">&longs;untque</expan> d e, a f, b g, c h <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> pyramidis axes: conue <lb/>nient omnes in <expan abbr="id&etilde;">idem</expan> <expan abbr="punct&utilde;">punctum</expan> k, quod e&longs;t grauitatis centrum. <lb/>Itaque animo concipiamus hanc pyramidem diui&longs;am in <lb/>quatuor pyramides, quarum ba&longs;es &longs;int ip&longs;a pyramidis <lb/>

<arrow.to.target n="marg67"></arrow.to.target><lb/>

<arrow.to.target n="fig40"></arrow.to.target><lb/>triangula; & <emph type="ul"/>axis<emph.end type="ul"/> pun<lb/>ctum k quæ quidem py<lb/>ramides inter &longs;e æquales <lb/>&longs;unt, ut <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan>. <lb/>Ducatur <expan abbr="en&itilde;">enim</expan> per lineas <lb/>d c, d e planum <expan abbr="&longs;ecãs">&longs;ecans</expan>, ut <lb/>&longs;it ip&longs;ius, & ba&longs;is a b c <expan abbr="cõ">com</expan> <lb/>munis &longs;ectio recta linea <lb/>c e l: <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> uero & <expan abbr="triã-guli">trian<lb/>guli</expan> a d b &longs;it linea d h l. <lb/>erit linea al æqualis ip&longs;i <lb/>l b: nam centrum graui<lb/>tatis trianguli con&longs;i&longs;tit <lb/>in linea, quæ ab angulo <lb/>ad dimidiam ba&longs;im per<lb/>ducitur, ex tertia deci<lb/>ma Archimedis. quare <lb/>

<figure id="fig40"></figure><lb/>triangula; & <emph type="ul"/>axis<emph.end type="ul"/> pun<lb/>ctum k quæ quidem py<lb/>ramides inter &longs;e æquales <lb/>&longs;unt, ut <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan>. <lb/>Ducatur <expan abbr="en&itilde;">enim</expan> per lineas <lb/>d c, d e planum <expan abbr="&longs;ecãs">&longs;ecans</expan>, ut <lb/>&longs;it ip&longs;ius, & ba&longs;is a b c <expan abbr="cõ">com</expan> <lb/>munis &longs;ectio recta linea <lb/>c e l: <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> uero & <expan abbr="triã-guli">trian<lb/>guli</expan> a d b &longs;it linea d h l. <lb/>erit linea al æqualis ip&longs;i <lb/>l b: nam centrum graui<lb/>tatis trianguli con&longs;i&longs;tit <lb/>in linea, quæ ab angulo <lb/>ad dimidiam ba&longs;im per<lb/>ducitur, ex tertia deci<lb/>ma Archimedis. quare <lb/>

<arrow.to.target n="marg68"></arrow.to.target><lb/>triangulum a c l æquale <lb/>e&longs;t triangulo b c l: & propterea pyramis, cuius ba&longs;is trian<lb/>gulum a c l, uer tex d, e&longs;t æqualis pyramidi, cuius ba&longs;is b c l <lb/>

<arrow.to.target n="marg69"></arrow.to.target><lb/>triangulum, & idem uertex. pyramides enim, quæ ab <expan abbr="eod&etilde;">eodem</expan>

<pb/>&longs;unt uertice, eandem proportionem habent, quam <expan abbr="ip&longs;ar&utilde;">ip&longs;arum</expan> <lb/>ba&longs;es. eadem ratione pyramis a c l k pyramidi b c l <foreign lang="greek">k:</foreign> & py <lb/>ramis a d l k ip&longs;i b d l <foreign lang="greek">k</foreign> pyramidi æqualis erit. Itaque &longs;i a py <lb/>ramide a c l d auferantur pyramides a c l k, a d l k: & à pyra <lb/>mide b c l d <expan abbr="auferãtur">auferantur</expan> pyramides b c l <foreign lang="greek">k,</foreign> d b l K: quæ relin<lb/>quuntur erunt æqualia. æqualis igitur e&longs;t pyramis a c d <foreign lang="greek">k</foreign><lb/>pyramidi b c d K. Rur&longs;us &longs;i per lineas a d, d e ducatur pla<lb/>num quod pyramidem &longs;ccet: <expan abbr="&longs;itq;">&longs;itque</expan> eius & ba&longs;is communis <lb/>&longs;ectio a e m: &longs;imiliter o&longs;tendetur pyramis a b d K æqualis <lb/>pyramidi a c d k. ducto denique alio plano per lineas c a, <lb/>a f: ut eius, & trianguli c d b communis &longs;ectio &longs;it c fn, py<lb/>ramis a b c k pyramidi a c d <foreign lang="greek">k</foreign> æqualis demon&longs;trabitur. <expan abbr="c&utilde;">cum</expan> <lb/>ergo tres pyramides b c d k, a b d k, a b c k uni, & eidem py <lb/>ramidi a c d k &longs;int æquales, omnes inter &longs;e &longs;e æquales <expan abbr="er&utilde;t">erunt</expan>. <lb/>Sed ut pyramis a b c d ad pyramidem a b c <foreign lang="greek">k,</foreign> ita d e axis ad <lb/>axem <foreign lang="greek">k</foreign> e, ex uige&longs;ima propo&longs;itione huius: &longs;unt enim hæ <lb/>pyramides in eadem ba&longs;i, & axes cum ba&longs;ibus æquales con <lb/>tinent angulos, quòd in eadem recta linea con&longs;tituantur. <lb/>quare diuidendo, ut tres pyramides a c d k, b c d K, a b d K <lb/>ad pyramidem a b c K, ita d k ad K e. con&longs;tat igitur lineam <lb/>d K ip&longs;ius K e triplam e&longs;&longs;e. &longs;ed & a <foreign lang="greek">k</foreign> tripla e&longs;t K f: itemque <lb/>b K ip&longs;ius K g: & c <foreign lang="greek">k</foreign> ip&longs;ius <foreign lang="greek">k</foreign> l tripla. quod eodem modo <lb/>demon&longs;trabimus.</s>

Line 1456

Line 1454

<s>Sit pyramis, cuius ba&longs;is quadrilaterum a b c d; axis e f: <lb/>& diuidatur e fin g, ita ut e g ip&longs;ius g f &longs;it tripla. Dico cen<lb/>trum grauitatis pyramidis e&longs;&longs;e punctum g. ducatur enim <lb/>linea b d diuidens ba&longs;im in duo triangula a b d, b c d: ex <lb/>quibus <expan abbr="intelligãtur">intelligantur</expan> <expan abbr="cõ&longs;titui">con&longs;titui</expan> duæ pyramides a b d e, b c d e: <lb/>&longs;itque pyramidis a b d e axis e h; & pyramidis b c d e axis <lb/>e K: & iungatur h K, quæ per f tran&longs;ibit: e&longs;t enim in ip&longs;a h K <lb/>centrum grauitatis magnitudinis compo&longs;itæ ex triangulis <lb/>a b d, b c d, hoc e&longs;t ip&longs;ius quadrilateri. Itaque centrum gra <lb/>uitatis pyramidis a b d e &longs;it punctum l: & pyramidis b c d e <lb/>

<arrow.to.target n="marg70"></arrow.to.target><lb/>&longs;it m. ducta igitur l m ip&longs;i h m lineæ æquidi&longs;tabit. nam el ad

<pb pagenum="29"/>lbeandem habet proportionem, quam e m ad m k, uideli<lb/>cet triplam. quare lineal m ip&longs;am e f &longs;ecabit in punctog: <lb/>etenim e g ad g f e&longs;t, ut el adlh. præterea quoniam h k, l m <lb/>æquidi&longs;tant, erunt triangula h e f, l e g &longs;imilia: <expan abbr="itemq;">itemque</expan> inter <lb/>&longs;e &longs;imilia fe <foreign lang="greek">k,</foreign> gem: & ut e fad e g, itah fad l g: & ita f K ad <lb/>gm. ergo uth fadl g, ita f <foreign lang="greek">k</foreign> ad g m: & permutando uth f <lb/>ad f K, ital g ad gm. &longs;ed cum h &longs;it centrum trianguli a b d; <lb/>& <foreign lang="greek">k</foreign> <expan abbr="triãguli">trianguli</expan> b c d <expan abbr="punct&utilde;">punctum</expan> uero f totius quadrilateri a b c d <lb/>centrum: erit ex 8. Archimedis de centro grauitatis plano <lb/>rum h fad f <foreign lang="greek">k,</foreign> ut triangulum b c d ad triangulum a b d: ut, <lb/>autem bcd triangulum ad triangulum a b d, ita pyramis <lb/>

<arrow.to.target n="fig41"></arrow.to.target><lb/>b c d e ad pyramidem a b d e. ergo <lb/>linea lg ad gm erit, ut pyramis <lb/>b c d e ad <expan abbr="pyramid&etilde;">pyramidem</expan> a b d e. ex quo <lb/>&longs;equitur, ut totius pyramidis <lb/>a b c d e punctum g &longs;it grauitatis <lb/>centrum. Rur&longs;us &longs;it pyramis ba<lb/>&longs;im habens pentagonum a b c d e: <lb/>& axem f g: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="p&utilde;">pum</expan> <lb/>cto h, ita ut fh ad h g triplam habe <lb/>at proportionem. Dico h grauita<lb/>tis <expan abbr="centr&utilde;">centrum</expan> e&longs;&longs;e pyramidis a b c d e f. <lb/>iungatur enim e b: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>pyramis, cuius uertex f, & ba&longs;is <lb/>triangulum a b e: & alia pyramis <lb/>intelligatur eundem uerticem ha<lb/>bens, & ba&longs;im b c d e <expan abbr="quadrilater&utilde;">quadrilaterum</expan>: <lb/>&longs;it autem pyramidis a b e faxis f <foreign lang="greek">k,</foreign><lb/>& grauitatis centrum l: & pyrami <lb/>dis b c d e faxis f m, & centrum gra <lb/><gap/> <expan abbr="iunganturq;">iunganturque</expan> <foreign lang="greek">k</foreign> m, ln; <lb/>quæ per puncta g h tran&longs;ibunt. <lb/>Rur&longs;us eodemmodo, quo &longs;up ra, <lb/>demon&longs;trabimus lineas K g m, l h n &longs;ibiip&longs;is æquidi&longs;tare:

<figure id="fig41"></figure><lb/>b c d e ad pyramidem a b d e. ergo <lb/>linea lg ad gm erit, ut pyramis <lb/>b c d e ad <expan abbr="pyramid&etilde;">pyramidem</expan> a b d e. ex quo <lb/>&longs;equitur, ut totius pyramidis <lb/>a b c d e punctum g &longs;it grauitatis <lb/>centrum. Rur&longs;us &longs;it pyramis ba<lb/>&longs;im habens pentagonum a b c d e: <lb/>& axem f g: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="p&utilde;">pum</expan> <lb/>cto h, ita ut fh ad h g triplam habe <lb/>at proportionem. Dico h grauita<lb/>tis <expan abbr="centr&utilde;">centrum</expan> e&longs;&longs;e pyramidis a b c d e f. <lb/>iungatur enim e b: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>pyramis, cuius uertex f, & ba&longs;is <lb/>triangulum a b e: & alia pyramis <lb/>intelligatur eundem uerticem ha<lb/>bens, & ba&longs;im b c d e <expan abbr="quadrilater&utilde;">quadrilaterum</expan>: <lb/>&longs;it autem pyramidis a b e faxis f <foreign lang="greek">k,</foreign><lb/>& grauitatis centrum l: & pyrami <lb/>dis b c d e faxis f m, & centrum gra <lb/><gap/> <expan abbr="iunganturq;">iunganturque</expan> <foreign lang="greek">k</foreign> m, ln; <lb/>quæ per puncta g h tran&longs;ibunt. <lb/>Rur&longs;us eodemmodo, quo &longs;up ra, <lb/>demon&longs;trabimus lineas K g m, l h n &longs;ibiip&longs;is æquidi&longs;tare:

<pb/>& denique punctum h pyramidis a b c d e f grauitatis e&longs;&longs;e <lb/>centrum, & ita in alils.</s>

<s><margin.target id="marg70"></margin.target>2. fexti.</s>

<s>Sit conus, uel coni portio axem habens b d: &longs;eceturque <lb/>plano per axem, quod &longs;ectionem faciat triangulum a b c: <lb/>& b d axis diuidatur in c, ita ut be ip&longs;ius ed &longs;it tripla. <lb/>Dico punctum e coni, uel coni portionis, grauitatis <lb/>e&longs;&longs;e centrum. Sienim fieri pote&longs;t, &longs;itcentrum f: & pro<lb/>ducatur e f extra figuram in g. quam uero proportionem <lb/>habet g e ad e f, habeat ba&longs;is coni, uelconi portionis, hoc <lb/>e&longs;t circulus, uel ellip&longs;is circa diametrum ac ad aliud &longs;pa<lb/>cium, in quo h. Itaque in circulo, uel ellip&longs;i plane de&longs;cri<lb/>batur rectilinea figura a x l m c n o p, ita ut quæ <expan abbr="relinqu&utilde;-tur">relinquun<lb/>tur</expan> portiones &longs;int minores &longs;pacio h: & intelligatur pyra<lb/>mis ba&longs;im habens rectilineam figuram a K l m c n o p, & <lb/>axem b d; cuius quidem grauitatis centrum erit punctum <lb/>e, utiam demon&longs;trauimus. Et quoniam portiones &longs;unt <lb/>minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma<lb/>

<arrow.to.target n="fig42"></arrow.to.target><lb/>iorem proportionem habet, quam g e ad e f. &longs;ed ut circu<lb/>lus, uel ellip&longs;is ad figuram rectilineam &longs;ibi in&longs;criptam, ita <lb/>conus, uel coni portio ad pyramidem, quæ figuram rectili<lb/>neam pro ba&longs;i habet; & altitudinem æqualem: etenim &longs;u

<figure id="fig42"></figure><lb/>iorem proportionem habet, quam g e ad e f. &longs;ed ut circu<lb/>lus, uel ellip&longs;is ad figuram rectilineam &longs;ibi in&longs;criptam, ita <lb/>conus, uel coni portio ad pyramidem, quæ figuram rectili<lb/>neam pro ba&longs;i habet; & altitudinem æqualem: etenim &longs;u

<arrow.to.target n="marg71"></arrow.to.target><lb/>pra demon&longs;tratum e&longs;t, ita e&longs;&longs;e cylindrum, uel cylindri por<lb/>tionem ad pri&longs;ma, cuius ba&longs;is rectilinea figura, & æqua<lb/>lis altitudo. ergo per conuer&longs;ionem rationis, ut circulus, <lb/>uel ellip&longs;is ad portiones, ita conus, uel coni portio ad por<lb/>tiones &longs;olidas. quare conus uel coni portio ad portiones <lb/>&longs;olidas maiorem habet proportionem, quam g e ad e f: & <lb/>diuidendo, pyramis ad portiones &longs;olidas maiorem pro<lb/>portionem habet, quam g f ad f e. fiatigitur q f ad f e <lb/>ut pyramis ad dictas portiones. Itaque quoniam a cono <lb/>uel coni portione, cuius grauitatis centrum e&longs;t f, aufer<lb/>tur pyramis, cuius centrum e; reliquæ magnitudinis, <lb/>quæ ex &longs;olidis portionibus con&longs;tat, centrum grauitatis <lb/>erit in linea e f protracta, & in puncto <expan abbr="q.">que</expan> quod fieri <lb/>non pote&longs;t: e&longs;t enim centrum grauitatis intra. Con&longs;tat <lb/>igitur coni, uel coni portionis grauitatis centrum e&longs;&longs;e pun <lb/>ctum e. quæ omnia demon&longs;trare oportebat.</s>

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<s>THEOREMA XIX. PROPOSITIO XXIII.</s>

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<s>Hoc demon&longs;trauit Leonardus Pi&longs;anus in libro, qui de<lb/>praxi geometriæ in&longs;cribitur. Sed quoniam is adhuc im<lb/>pre&longs;&longs;us non e&longs;t, nos ip&longs;ius demon&longs;trationem breuiter <lb/>per&longs;tringemus, rem ip&longs;am &longs;ecuti, non uerba. Sit fru<lb/>&longs;tum pyramidis a b c d e f, cuius maior ba&longs;is triangulum <lb/>a b c, minor d e f: & iunctis ae, cc, cd, per, line<lb/>as a e, e c ducatur planum &longs;ecans fru&longs;tum: itemque per <lb/>lineas e c, c d; & per cd, da alia plana ducantur, quæ <lb/>diuident fru&longs;tum in trcs pyramides a b c e, a d c e, d e f c.

<pb/>Dico eas proportionales e&longs;&longs;e in proportione, quæ e&longs;t la<lb/>teris a b adlatus d e, itaut earum maior &longs;it a b c e, me<lb/>dia a d c e, & minor d e f c. Quoniam enim lineæ d e, <lb/>a b æquidi&longs;tant; & interip&longs;as &longs;unt triangula a b e, a d e; <lb/>

<arrow.to.target n="marg72"></arrow.to.target><lb/>

<arrow.to.target n="fig43"></arrow.to.target><lb/>erit triangulum a b e <lb/>ad triangulum a b e, <lb/>utlinea a b ad lineam <lb/>d e. ut autem triangu <lb/>lum a b e ad triangu<lb/>

<figure id="fig43"></figure><lb/>erit triangulum a b e <lb/>ad triangulum a b e, <lb/>utlinea a b ad lineam <lb/>d e. ut autem triangu <lb/>lum a b e ad triangu<lb/>

<arrow.to.target n="marg73"></arrow.to.target><lb/>lum a b e, ita pyramis <lb/>a b e c ad pyramidem <lb/>a d e c: habent enim <lb/>altitudinem eandem, <lb/>quæ e&longs;tà puncto c ad <lb/>planum, in quo qua<lb/>

<arrow.to.target n="marg74"></arrow.to.target><lb/>drilaterum a b e d. er<lb/>go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c. <lb/>Rur&longs;us quoniam æquidi&longs;tantes &longs;unt a c, d f; erit eadem <lb/>

<arrow.to.target n="marg75"></arrow.to.target><lb/>ratione pyramis a d c e ad pyramidem c d fe, ut ac ad <lb/>d f. Sed ut a c a l d f, ita a b ad d e, quoniam triangula <lb/>a b c, d e f &longs;imilia &longs;unt, ex nona huius. quare ut pyramis <lb/>a b c e ad pyramidem a b c e, ita pyramis a d c e ad ip&longs;an<gap/><lb/>d e f c. fru&longs;tum igitur a b c d e f diuiditur in tres pyramides <lb/>proportionales in ea proportione, quæ e&longs;t lateris a b ad d e <lb/>latus, & earum maior e&longs;t c a b e, media a d c e, & minor <lb/>d e f c. quod demon&longs;trare oportebat.</s>

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<s><margin.target id="marg75"></margin.target>4 &longs;exti.</s>

<s>PROBLEMA V. PROPOSITIO XXIIII.</s>

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<s>SIT fru&longs;tum pyramidis a e, cuius maior ba&longs;is triangu<lb/>lum a b c, minor d e f: & oporteat ip&longs;um plano, quod ba&longs;i <lb/>æquidi&longs;tet, ita &longs;ecare, ut &longs;ectio &longs;it proportionalis inter <expan abbr="triã">triam</expan> <lb/>gula a b c, d e f. Inueniatur inter lineas a b, d e media pro<lb/>portionalis, quæ &longs;it b g: & à puncto g erigatur g h æquidi<lb/>&longs;tans b e, <expan abbr="&longs;ecansq;">&longs;ecansque</expan> a d in h: deinde per h ducatur planum <lb/>ba&longs;ibus æquidi&longs;tans, cuius &longs;ectio &longs;it triangulum h k l. Dico <lb/>triangulum h K l proportionale e&longs;&longs;e inter triangula a b c, <lb/>

<arrow.to.target n="fig44"></arrow.to.target><lb/>d e f, hoc e&longs;t triangulum a b c ad <lb/>triangulum h K l eandem habere <lb/>proportionem, quam <expan abbr="triãgulum">triangulum</expan> <lb/>h K l ad ip&longs;um d e f. <expan abbr="Quoniã">Quoniam</expan> enim <lb/>

<figure id="fig44"></figure><lb/>d e f, hoc e&longs;t triangulum a b c ad <lb/>triangulum h K l eandem habere <lb/>proportionem, quam <expan abbr="triãgulum">triangulum</expan> <lb/>h K l ad ip&longs;um d e f. <expan abbr="Quoniã">Quoniam</expan> enim <lb/>

<arrow.to.target n="marg76"></arrow.to.target><lb/>lineæ a b, h K æquidi&longs;tantium pla <lb/>norum &longs;ectiones inter &longs;e æquidi<lb/>&longs;tant: atque æquidi&longs;tant b k, gh: <lb/>

<arrow.to.target n="marg77"></arrow.to.target><lb/>linea h k ip&longs;i g b e&longs;t æqualis: & pro <lb/>pterea proportionalis inter a b, <lb/>d e. quare ut a b ad h K, ita e&longs;t h <foreign lang="greek">k</foreign><lb/>ad de. fiat ut h k ad d e, ita d e <lb/>ad aliam lineam, in qua &longs;it m. erit <lb/>ex æquali ut a b ad d e, ita h k ad <lb/>

<arrow.to.target n="marg78"></arrow.to.target><lb/>m. Et quoniam triangula a b c, <lb/>h K l, d e f &longs;imilia &longs;unt; <expan abbr="triangul&utilde;">triangulum</expan> <lb/>

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<s>Sit fru&longs;tum coni, uel coni portionis a d: & &longs;ecetur plano <lb/>per axem, cuius &longs;ectio &longs;it a b c d, ita ut maior ip&longs;ius ba&longs;is &longs;it <lb/>circulus, uel ellip&longs;is circa diametrum a b; minor circa c d. <lb/>Rur&longs;us inter lineas a b, c d inueniatur proportionalis b e: <lb/>& ab e ducta e f æquidi&longs;tante b d, quæ lineam c a in f &longs;ecet,

<pb/>per f planum ba&longs;ibus æquidi&longs;tans ducatur, ut &longs;it &longs;ectio cir <lb/>culus, uel ellip&longs;is circa diametrum f g. Dico &longs;ectionem a b <lb/>ad &longs;ectionem f g eandem proportionem habere, quam f g <lb/>ad ip&longs;am c d. 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="quadrat&utilde;">quadratum</expan> <lb/>

<arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. Sed circuli inter &longs;e candem propor<lb/>tionem habent, quam diametrorum quadrata. 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/>

<arrow.to.target n="fig45"></arrow.to.target><lb/>&longs;eptimæ propo&longs;itionis eiu&longs;dem li<lb/>bri. ellip&longs;es enim nunc appello ip<lb/>&longs;a &longs;pacia ellip&longs;ibus contenta. ergo <lb/>circulus, uel ellip&longs;is a b ad <expan abbr="circul&utilde;">circulum</expan>, <lb/>uel ellip&longs;im f g eam proportionem <lb/>habet, quam circulus, uel ellip&longs;is <lb/>f g ad circulum uel ellip&longs;im c d. <lb/>quod quidem faciendum propo<lb/>&longs;uimus.</s>

<figure id="fig45"></figure><lb/>&longs;eptimæ propo&longs;itionis eiu&longs;dem li<lb/>bri. ellip&longs;es enim nunc appello ip<lb/>&longs;a &longs;pacia ellip&longs;ibus contenta. ergo <lb/>circulus, uel ellip&longs;is a b ad <expan abbr="circul&utilde;">circulum</expan>, <lb/>uel ellip&longs;im f g eam proportionem <lb/>habet, quam circulus, uel ellip&longs;is <lb/>f g ad circulum uel ellip&longs;im c d. <lb/>quod quidem faciendum propo<lb/>&longs;uimus.</s>

<s><margin.target id="marg81"></margin.target>2. duode <lb/><gap/>imi</s>

<s>THEOREMA XX. PROPOSITIO XXV.</s>

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<s>SIT <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> pyramidis, uel coni, uel coni portionis a d, <lb/>cuius maior ba&longs;is a b, minor c d. & &longs;ecetur altero plano <lb/>ba&longs;i æquidi&longs;tante, ita ut &longs;ectio e f &longs;it proportionalis inter <lb/>ba&longs;es a b, c d. con&longs;tituatur <expan abbr="aut&etilde;">autem</expan> pyramis, uel conus, uel co<lb/>ni portio a g b, cuius ba&longs;is &longs;it eadem, quæ ba&longs;is maior fru<lb/>

<arrow.to.target n="fig46"></arrow.to.target><lb/>&longs;ti, & altitudo æqualis. Di<lb/>co fru&longs;tum a d ad pyrami<lb/>dem, uel conum, uel coni <lb/>portionem a g b eandem <lb/><expan abbr="proportion&etilde;">proportionem</expan> habere, <expan abbr="quã">quam</expan> <lb/>utræque ba&longs;es, a b, c d unà <lb/>cum e f ad ba&longs;im a b. e&longs;t <lb/>enim fru&longs;tum a d æquale <lb/>pyramidi, uel cono, uel co<lb/>ni portioni, cuius ba&longs;is ex <lb/>tribus ba&longs;ibus a b, e f, c d <lb/>con&longs;tat; & altitudo ip&longs;ius <lb/>altitudini e&longs;t æqualis: quod mox o&longs;tendemus. Sed pyrami <lb/>

<figure id="fig46"></figure><lb/>&longs;ti, & altitudo æqualis. Di<lb/>co fru&longs;tum a d ad pyrami<lb/>dem, uel conum, uel coni <lb/>portionem a g b eandem <lb/><expan abbr="proportion&etilde;">proportionem</expan> habere, <expan abbr="quã">quam</expan> <lb/>utræque ba&longs;es, a b, c d unà <lb/>cum e f ad ba&longs;im a b. e&longs;t <lb/>enim fru&longs;tum a d æquale <lb/>pyramidi, uel cono, uel co<lb/>ni portioni, cuius ba&longs;is ex <lb/>tribus ba&longs;ibus a b, e f, c d <lb/>con&longs;tat; & altitudo ip&longs;ius <lb/>altitudini e&longs;t æqualis: quod mox o&longs;tendemus. Sed pyrami <lb/>

<arrow.to.target n="fig47"></arrow.to.target><lb/>des, coni, uel coni <expan abbr="portiões">portiones</expan>, <lb/>quæ &longs;unt æquali altitudine, <lb/><expan abbr="eãdem">eandem</expan> inter &longs;e, quam ba&longs;es, <lb/>proportionem habent, &longs;icu<lb/>ti demon&longs;tratum e&longs;t, partim <lb/>

<figure id="fig47"></figure><lb/>des, coni, uel coni <expan abbr="portiões">portiones</expan>, <lb/>quæ &longs;unt æquali altitudine, <lb/><expan abbr="eãdem">eandem</expan> inter &longs;e, quam ba&longs;es, <lb/>proportionem habent, &longs;icu<lb/>ti demon&longs;tratum e&longs;t, partim <lb/>

<arrow.to.target n="marg82"></arrow.to.target><lb/>ab Euclide in duodecimo li<lb/>bro elementorum, partim à <lb/>nobis in <expan abbr="cõmentariis">commentariis</expan> in un<lb/>decimam <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> Ar<lb/>chimedis de conoidibus, & <lb/>&longs;phæroidibus. 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. 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. quod demon&longs;trare uolebamus.</s>

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<s><margin.target id="marg82"></margin.target>6. 11.<gap/><lb/>decin<gap/></s>

<s>Fru&longs;tum uero a d æquale e&longs;&longs;e pyramidi, uel co <lb/>no, uel coni portioni, cuius ba&longs;is con&longs;tat ex ba&longs;i<lb/>bus a b, c d, e f, & altitudo fru&longs;ti altitudini e&longs;t æ<lb/>qualis, hoc modo o&longs;tendemus.</s>

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<s>Sit fru&longs;tum pyramidis a b c d e f, cuius maior ba&longs;is trian<lb/>gulum a b c; minor d e f: & &longs;ecetur plano ba&longs;ibus æquidi<lb/>&longs;tante, quod &longs;ectionem faciat triangulum g h <foreign lang="greek">k</foreign> inter trian<lb/>gula a b c, d e f proportionale. Iam ex iis, quæ demon&longs;trata <lb/>&longs;unt in 23. huius, patet fru&longs;tum a b c d e f diuidi in tres pyra <lb/>mides proportionales; & earum maiorem e&longs;&longs;e <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>a b c d <expan abbr="minor&etilde;">minorem</expan> uero d e f b. ergo pyramis à triangulo g h k <lb/>con&longs;tituta, quæ altitudinem habeat fru&longs;ti altitudini æqua<lb/>lem, proportionalis e&longs;t inter pyramides a b c d, d e f b: & <lb/>idcirco fru&longs;tum a b c d e f tribus dictis pyramidibus æqua <lb/>

<arrow.to.target n="fig48"></arrow.to.target><lb/>le erit. Itaque &longs;i intelligatur alia pyra<lb/>mis æque alta, quæ ba&longs;im habeat ex tri <lb/>bus ba&longs;ibus a b c, d e f, g h k con&longs;tan<lb/>tem; per&longs;picuum e&longs;t ip&longs;am ei&longs;dem py<lb/>ramidibus, & propterea ip&longs;i fru&longs;to æ<lb/>qualem e&longs;&longs;e.</s>

<figure id="fig48"></figure><lb/>le erit. Itaque &longs;i intelligatur alia pyra<lb/>mis æque alta, quæ ba&longs;im habeat ex tri <lb/>bus ba&longs;ibus a b c, d e f, g h k con&longs;tan<lb/>tem; per&longs;picuum e&longs;t ip&longs;am ei&longs;dem py<lb/>ramidibus, & propterea ip&longs;i fru&longs;to æ<lb/>qualem e&longs;&longs;e.</s>

<s>Rur&longs;us &longs;it fru&longs;tum pyramidis a g, cu <lb/>ius maior ba&longs;is quadrilaterum a b c d, <lb/>minor e f g h: & &longs;ecetur plano ba&longs;i<lb/>bus æquidi&longs;tante, ita ut fiat &longs;ectio qua<lb/>drilaterum K l m n, quod &longs;it proportio <lb/>nale inter quadrilatera a b c d, e f g h. Dico pyramidem, <lb/>cuius ba&longs;is &longs;it æqualis tribus quadrilateris a b c d, k l m n, <lb/>e f g h, & altitudo æqualis altitudini fru&longs;ti, ip&longs;i fru&longs;to a g <lb/>æqualem e&longs;&longs;e. Ducatur enim planum per lincas f b, h d,

<pb pagenum="33"/>quod diuidat fru&longs;tum in duo fru&longs;ta triangulares ba&longs;es ha<lb/>bentia, uidelicet in fru&longs;tum a b d e f h, & in <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> b c d f g h. <lb/>erit triangulum <foreign lang="greek">k</foreign> l n proportionale inter triangula a b d, <lb/>e f h: & triangulum l m n proportionale inter b c d, f g h. <lb/>&longs;ed pyramis æque alta, cuius ba&longs;is con&longs;tat ex tribus trian<lb/>

<arrow.to.target n="fig49"></arrow.to.target><lb/>gulis a b d, k l n, e f h, demon&longs;trata <lb/>e&longs;t fru&longs;to a b d c f h æqualis. & &longs;i<lb/>militer pyramis, cuius ba&longs;is con<lb/>&longs;tat ex triangulis b c d, l m n, f g h <lb/>æqualis fru&longs;to b c d f g h: compo<lb/>nuntur autem tria quadrilatera a <lb/>b c d, k l m n, e f g h è &longs;ex triangu<lb/>lis iam dictis. pyramis igitur ba<lb/>&longs;im habens æqualem tribus qua<lb/>drilateris, & altitudinem eandem <lb/>ip&longs;i fru&longs;to a g e&longs;t æqualis. Eodem <lb/>modo illud <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan> in aliis <lb/>eiu&longs;modi fru&longs;tis.</s>

<figure id="fig49"></figure><lb/>gulis a b d, k l n, e f h, demon&longs;trata <lb/>e&longs;t fru&longs;to a b d c f h æqualis. & &longs;i<lb/>militer pyramis, cuius ba&longs;is con<lb/>&longs;tat ex triangulis b c d, l m n, f g h <lb/>æqualis fru&longs;to b c d f g h: compo<lb/>nuntur autem tria quadrilatera a <lb/>b c d, k l m n, e f g h è &longs;ex triangu<lb/>lis iam dictis. pyramis igitur ba<lb/>&longs;im habens æqualem tribus qua<lb/>drilateris, & altitudinem eandem <lb/>ip&longs;i fru&longs;to a g e&longs;t æqualis. Eodem <lb/>modo illud <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan> in aliis <lb/>eiu&longs;modi fru&longs;tis.</s>

<s>Sit fru&longs;tum coni, uel coni portionis a d; cuius maior ba<lb/>&longs;is circulus, uel ellip&longs;is circa diametrum a b; minor circa <lb/>c d: & &longs;ecetur plano, quod ba&longs;ibus æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> &longs;e<lb/>ctionem circulum, uel ellip&longs;im circa diametrum e f, ita ut <lb/>inter circulos, uel ellip&longs;es a b, c d &longs;it proportionalis. Dico <lb/>conum, uel coni portionem, cuius ba&longs;is e&longs;t æqualis tribus <lb/>circulis, uel tribus ellip&longs;ibus a b, e f, c d; & altitudo eadem, <lb/>quæ fru&longs;ti a d, ip&longs;i fru&longs;to æqualem e&longs;&longs;e. producatur enim <lb/>fru&longs;ti &longs;uperficies quou&longs;que coeat in unum punctum, quod <lb/>&longs;it g: & coni, uel coni portionis a g b axis &longs;it g h, occurrens <lb/>planis a b, e f, c d in punctis h k l: circa circulum uero de<lb/>&longs;cribatur quadratum m n o p, & circa ellip&longs;im <expan abbr="rectangul&utilde;">rectangulum</expan> <lb/>m n o p, quod ex ip&longs;ius diametris con&longs;tat: <expan abbr="iunctisq;">iunctisque</expan> g m, <lb/>g n, g o, g p, ex eodem uertice intelligatur pyramis ba&longs;im <lb/>habens dictum quadratum, uel rectangulum: & plana in <lb/>quibus &longs;unt circuli, uel ellip&longs;es e f, c d u&longs;que ad eius latera

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<arrow.to.target n="marg83"></arrow.to.target><lb/>æquidi&longs;tantibus, &longs;ectiones &longs;imiles erunt: atque erunt qua<lb/>drata, uel rectangula circa circulos, uel ellip&longs;es de&longs;cripta, <lb/>quemadmodum & in ip&longs;a ba&longs;i. Sed cum circuli inter &longs;e <expan abbr="eã">eam</expan> <lb/>

<arrow.to.target n="marg84"></arrow.to.target><lb/>proportionem habeant, quam diametrorum quadrata: <lb/><expan abbr="itemq;">itemque</expan> ellip&longs;es eam quam rectangula ex ip&longs;arum diametris <lb/>

<arrow.to.target n="marg85"></arrow.to.target><lb/>con&longs;tantia: & &longs;it circulus, uel ellip&longs;is circa diametrum e f <lb/>

<arrow.to.target n="fig50"></arrow.to.target><lb/>proportionalis inter circulos, uel ellip&longs;es a b, c d; erit re<lb/>ctangulum e f etiam inter rectangula a b, c d proportio<lb/>nale: per rectangulum enim nunc breuitatis cau&longs;a <expan abbr="etiã">etiam</expan> ip<lb/>&longs;um quadratum intelligemus. quare ex iis, quæ proxime <lb/>dicta &longs;unt, pyramis ba&longs;im habens æqualem dictis rectangu <lb/>lis, & altitudinem eandem, quam fru&longs;tum a d, ip&longs;i fru&longs;to à <lb/>pyramide ab&longs;ci&longs;&longs;o æqualis probabitur. ut autem rectangu <lb/>lum c d ad <expan abbr="rectangul&utilde;">rectangulum</expan> e f, ita circulus, uel ellip&longs;is c d ad e f <lb/>circulum, uel ellip&longs;im: <expan abbr="componendoq;">componendoque</expan> ut rectangula c d, <lb/>e f, ad e f rectangulum, ita circuli, uel ellip&longs;es e d, e f, ad e f: <lb/>& ut rectangulum e f ad rectangulum a b, ita circulus, uel <lb/>ellip&longs;is e f ad a b circulum, uel ellip&longs;im. ergo ex æquali, & <lb/>componendo, ut <expan abbr="rectãgula">rectangula</expan> c d, e f, a b ad ip&longs;um a b, ita cir

<figure id="fig50"></figure><lb/>proportionalis inter circulos, uel ellip&longs;es a b, c d; erit re<lb/>ctangulum e f etiam inter rectangula a b, c d proportio<lb/>nale: per rectangulum enim nunc breuitatis cau&longs;a <expan abbr="etiã">etiam</expan> ip<lb/>&longs;um quadratum intelligemus. quare ex iis, quæ proxime <lb/>dicta &longs;unt, pyramis ba&longs;im habens æqualem dictis rectangu <lb/>lis, & altitudinem eandem, quam fru&longs;tum a d, ip&longs;i fru&longs;to à <lb/>pyramide ab&longs;ci&longs;&longs;o æqualis probabitur. ut autem rectangu <lb/>lum c d ad <expan abbr="rectangul&utilde;">rectangulum</expan> e f, ita circulus, uel ellip&longs;is c d ad e f <lb/>circulum, uel ellip&longs;im: <expan abbr="componendoq;">componendoque</expan> ut rectangula c d, <lb/>e f, ad e f rectangulum, ita circuli, uel ellip&longs;es e d, e f, ad e f: <lb/>& ut rectangulum e f ad rectangulum a b, ita circulus, uel <lb/>ellip&longs;is e f ad a b circulum, uel ellip&longs;im. ergo ex æquali, & <lb/>componendo, ut <expan abbr="rectãgula">rectangula</expan> c d, e f, a b ad ip&longs;um a b, ita cir

<pb pagenum="34"/>culi, uel ellip&longs;es c d, e f a b ad circulum, uel ellip&longs;im a b. In<lb/>telligatur pyramis q ba&longs;im habens æqualem tribus rectan <lb/>gulis a b, e f, c d; & altitudinem <expan abbr="eãdem">eandem</expan>, quam fru&longs;tum a d. <lb/>intelligatur ctiam conus, uel coni portio q, eadem altitudi<lb/>ne, cuius ba&longs;is &longs;it tribus circulis, uel tribus ellip&longs;ibus a b, <lb/>e f, c d æqualis. po&longs;tremo intelligatur pyramis a l b, cuius. <lb/>ba&longs;is &longs;it rectangulum m n o p, & altitudo eadem, quæ fru<lb/>&longs;ti: itemq, intelligatur conus, uel coni portio a l b, cuius <lb/>ba&longs;is circulus, uel ellip&longs;is circa diametrum a b, & eadem al <lb/>

<arrow.to.target n="marg86"></arrow.to.target><lb/>titudo. utigitur rectangula a b, e f, c d ad rectangulum a b, <lb/>ita pyramis q ad pyramidem a l b; & ut circuli, uel ellip<lb/>&longs;es a b, e f, c d ad a b circulum, uel ellip&longs;im, ita conus, uel co <lb/>ni portio q ad conum, uel coni portionem a l b. conus <lb/>igitur, uel coni portio q ad conum, uel coni portionem <lb/>a l b e&longs;t, ut pyramis q ad pyramidem a l b. &longs;ed pyramis <lb/>a l b ad pyramidem a g b e&longs;t, ut altitudo ad altitudinem, ex <lb/>20. huius: & ita e&longs;t conus, uel coni portio al b ad conum, <lb/>uel coni portionem a g b ex 14. duodccimi elementorum, <lb/>& ex iis, quæ nos demon&longs;trauimus in commentariis in un<lb/>decimam de conoidibus, & &longs;phæroidibus, propo&longs;itione <lb/>quarta. pyramis autem a g b ad pyramidem c g d propor<lb/>tionem habet compo&longs;itam ex proportione ba&longs;ium & pro <lb/>portione altitudinum, ex uige&longs;ima prima huius: & &longs;imili<lb/>ter conus, uel coni portio a g b ad conum, uel coni portio<lb/>nem c g d proportionem habet <expan abbr="compo&longs;itã">compo&longs;itam</expan> ex ei&longs;dem pro<lb/>portíonibus, per ea, quæ in dictis commentariis demon<lb/>&longs;trauimus, propo&longs;itione quinta, & &longs;exta: altitudo enim in<gap/><lb/>utri&longs;que eadem e&longs;t, & ba&longs;es inter &longs;e &longs;e eandem habent pro<lb/>portionem. ergo ut pyramis a g b ad pyramidem c g d, ita <lb/>e&longs;t conus, uel coni portio a g b ad a g d conum, uel coni <lb/>portionem: & per <expan abbr="conuer&longs;ion&etilde;">conuer&longs;ionem</expan> rationis, ut pyramis a g b <lb/>ad <expan abbr="&longs;ru&longs;t&utilde;">&longs;ru&longs;tum</expan> à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio <lb/>a g b ad fru&longs;tum a d. ex æquali igitur, ut pyramis q ad fru<lb/>&longs;tum à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio q ad

<pb/>fru&longs;tum a d. Sed pyramis q æqualis e&longs;t fru&longs;to à pyramide <lb/>ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. ergo & conus, uel coni por<lb/>tio q, cuius ba&longs;is ex tribus circulis, uel ellip&longs;ibus a b, e f, c d <lb/>con&longs;tat, & altitudo eadem, quæ fru&longs;ti: ip&longs;i fru&longs;to a d e&longs;t æ<lb/>qualis. atque illud e&longs;t, quod demon&longs;trare oportebat.</s>

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<s><margin.target id="marg86"></margin.target>6. II. duo <lb/>decimi</s>

<s>THEOREMA XXI. PROPOSITIO XXVI.</s>

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<s>Sit fru&longs;tum a e a pyramide, quæ triangularem ba&longs;im ha<lb/>beat ab&longs;ci&longs;&longs;um: cuius maior ba&longs;is triangulum a b c, minor <lb/>d e f; & axis g h. ducto autem plano per axem & per <expan abbr="lineã">lineam</expan> <lb/>d a, quod &longs;ectionem faciat d a <foreign lang="greek">k</foreign> l quadrilaterum; puncta <lb/>K l lineas b c, e f bifariam &longs;ecabunt. nam cum g h &longs;it axis <lb/>fru&longs;ti: erit h centrum grauitatis trianguli a b c: & g <lb/>

<arrow.to.target n="fig51"></arrow.to.target><lb/>

<arrow.to.target n="marg87"></arrow.to.target><lb/>centrum trianguli d e f: cen<lb/>trum uero cuiuslibet triangu <lb/>li e&longs;t in recta linea, quæ ab an<lb/>gulo ip&longs;ius ad <expan abbr="dimidiã">dimidiam</expan> ba&longs;im <lb/>ducitur ex decimatertia primi <lb/>libri Archimedis de <expan abbr="c&etilde;tro">centro</expan> gra <lb/>

<arrow.to.target n="marg88"></arrow.to.target><lb/>uitatis planorum. quare <expan abbr="cen-tr&utilde;">cen<lb/>trum</expan> grauitatis trapezii b c f e <lb/>e&longs;t in linea K l, quod &longs;it m: & à <lb/>puncto m ad axem ducta m n <lb/>ip&longs;i a k, uel d l æquidi&longs;tante; <lb/>erit axis g h diui&longs;us in portio<lb/>nes g n, n h, quas diximus: ean <lb/>dem enim proportionem ha<lb/>bet g n ad n h, <expan abbr="quã">quam</expan> l m ad m k. <lb/>At l m ad m K habet eam, <expan abbr="quã">quam</expan> <lb/>duplum lateris maioris ba&longs;is <lb/>b c una cum latere minoris e f <lb/>ad duplum lateris e f unà cum <lb/>latere b c, ex ultima eiu&longs;dem <lb/>libri Archimedis. Itaque à li<lb/>nea n g ab&longs;cindatur, quarta <lb/>pars, quæ fit n p: & ab axe h g ab&longs;cindatur itidem <lb/>quarta pars h o: & quam proportionem habet fru&longs;tum ad <lb/>pyramidem, cuius maior ba&longs;is e&longs;t triangulum a b c, & alti<lb/>tudo ip&longs;i æqualis; habeat o p ad p <expan abbr="q.">que</expan> Dico centrum graui<lb/>tatis fru&longs;ti e&longs;&longs;e in linea p o, & in puncto <expan abbr="q.">que</expan> namque ip&longs;um <lb/>e&longs;&longs;e in linea g h manife&longs;te con&longs;tat. protractis enim fru&longs;ti pla

<pb/>nis, quou&longs;que in unum punctum r conueniant; erit pyra<lb/>midis a b c r, & pyramidis d e f r grauitatis centrum in li<lb/>nca r h. ergo & reliquæ magnitudinis, uidelicet fru&longs;ti cen<lb/>trum in eadem linea nece&longs;lario comperietur. Iungantur <lb/>d b, d c, d h, d m: & per lineas d b, d c ducto altero plano <lb/>intelligatur fru&longs;tum in duas pyramides diui&longs;um: in pyra<lb/>midem quidem, cuius ba&longs;is e&longs;t triangulum a b c, uertex d: <lb/>& in eam, cuius idem uertex, & ba&longs;is trapezium b c f e. erit <lb/>igitur pyramidis a b c d axis d h, & pyramidis b c f e d axis <lb/>d m: atque erunt tres axes gh, d h, d m in eodem plano <lb/>d a K l. ducatur præterea per o linea &longs;t ip&longs;i a K <expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan>, <lb/>quæ lineam d h in u &longs;ecet: per p uero ducatur x y æquidi<lb/>

<arrow.to.target n="fig52"></arrow.to.target><lb/>&longs;tans eidem, &longs;ecansque d m in <lb/>z: & iungatur z u, quæ &longs;ecet <lb/>g h in <foreign lang="greek">*f.</foreign> tran&longs;ibit ea per q: & <lb/>erunt <foreign lang="greek">*f</foreign> q unum, atque idem <lb/>punctum; ut inferius appare<lb/>bit. Quoniam igitur linea u o <lb/>

<figure id="fig52"></figure><lb/>&longs;tans eidem, &longs;ecansque d m in <lb/>z: & iungatur z u, quæ &longs;ecet <lb/>g h in <foreign lang="greek">*f.</foreign> tran&longs;ibit ea per q: & <lb/>erunt <foreign lang="greek">*f</foreign> q unum, atque idem <lb/>punctum; ut inferius appare<lb/>bit. Quoniam igitur linea u o <lb/>

<arrow.to.target n="marg89"></arrow.to.target><lb/>æquidi&longs;tat ip&longs;i d g, erit d u ad <lb/>u h, ut g o ad o h. Sed g o tri<lb/>pla e&longs;t o h. quare & d u ip&longs;ius <lb/>u h e&longs;t tripla: & ideo pyrami<lb/>dis a b c d centrum grauitatis <lb/>erit punctum u. Rur&longs;us quo<lb/>niam z y ip&longs;i d l æquidi&longs;tat, d z <lb/>ad z m e&longs;t, ut l y ad y m: e&longs;tque <lb/>l y ad y m, ut g p ad p n. 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. atque ob candem cau&longs;<lb/>&longs;am punctuniz e&longs;t <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis pyramidis b c f e d. 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. Sed fru&longs;ti centrum e&longs;t etiam in a<lb/>xe g h. ergo in puncto <foreign lang="greek">*f,</foreign> in quo lineæ z u, g h conu<gap/>niunt. <lb/>

<arrow.to.target n="marg90"></arrow.to.target><lb/>Itaque u <gap/> ad <foreign lang="greek">*f</foreign> z eam proportionem habet, quam pyramis <lb/>b c f e d ad pyramidem a b c d. & componendo u z ad z <foreign lang="greek">*f</foreign><lb/>eam habet, quam fru&longs;tum ad pyramidem a b c d. Vtuero <lb/>u z ad z <gap/>, ita o p ad p <foreign lang="greek">*f</foreign> ob &longs;imilitudinem triangulorum, <lb/>u o <gap/>, z p <foreign lang="greek">*f.</foreign> quare o p ad p <foreign lang="greek">*f</foreign> e&longs;t ut fru&longs;tum ad pyramidem <lb/>a b c d. &longs;ed ita erat o p ad p <expan abbr="q.">que</expan> æquales igitur &longs;unt p <gap/>, p q: <gap/><lb/>

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<s><margin.target id="marg91"></margin.target>7. quinti.</s>

<s>Sit fru&longs;tum a g à pyramide, quæ quadrangularem ba&longs;im <lb/>habeat ab&longs;ci&longs;&longs;um, cuius maior ba&longs;is a b c d, minor e f g h, <lb/>& axis <foreign lang="greek">k</foreign> l. diuidatur autem <expan abbr="prim&utilde;">primum</expan> k l, ita ut quam propor<lb/>tionem habet duplum lateris a b unà cum latere e f ad du <lb/>plum lateris e f unà cum a b; habeat k m ad m l. deinde à <lb/><expan abbr="p&utilde;cto">puncto</expan> m ad k &longs;umatur quarta pars ip&longs;ius m <foreign lang="greek">k,</foreign> quæ &longs;it m n. <lb/>& rur&longs;us ab l &longs;umatur quarta pars totius axis l k, quæ &longs;it <lb/>l o. po&longs;tremo fiat o n ad n p, ut fru&longs;tum a g ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>cuius ba&longs;is &longs;it eadem, quæ fru&longs;ti, & altitudo æqualis. Dico <lb/>punctum p fru&longs;ti a g grauitatis centrum e&longs;&longs;e. ducantur <lb/>enim a c, e g: & intelligantur duo fru&longs;ta triangulares ba<lb/>&longs;es habentia, quorum alterum l f ex ba&longs;ibus a b c, e f g <expan abbr="cõ-&longs;tet">con<lb/>&longs;tet</expan>; alterum l h ex ba&longs;ibus a c d, e g h. <expan abbr="Sitq;">Sitque</expan> fru&longs;ti l f axis <lb/>q r; in quo grauitatis centrum s: fru&longs;ti uero l h axis t u, & <lb/>x grauitatis centrum: deinde iungantur u r, t q, x s. tran&longs;i<lb/>bit u r per l: quoniam l e&longs;t centrum grauitatis quadran<lb/>guli a b c d: & puncta r u grauitatis centra triangulorum <lb/>a b c, a c d; in quæ quadrangulum ip&longs;um diuiditur. eadem <lb/>quoque ratione t q per punctum k tran&longs;ibit. At uero pro <lb/>portiones, ex quibus fru&longs;torum grauitatis centra inquiri<lb/>mus, eædem &longs;unt in toto fru&longs;to a g, & in fru&longs;tis l f, l h. Sunt <lb/>enim per octauam huius quadrilatera a b c d, e f g h &longs;imilia:

<pb/><expan abbr="itemq;">itemque</expan> &longs;imilia triangula a b c, e f g: & a c d, e g h. <expan abbr="idcir-coq;">idcir<lb/>coque</expan> latera &longs;ibi ip&longs;is re&longs;pondentia eandem inter &longs;e&longs;e pro<lb/>portionem &longs;eruant. Vt igitur duplum lateris a b unà <lb/>cum latere e f ad duplum lateris e f unà cum a b, ita e&longs;t <lb/>

<arrow.to.target n="fig53"></arrow.to.target><lb/>duplum a d late<lb/>ris una cum late<lb/>re e h ad duplum <lb/>e h unà cum a d: <lb/>& ita in aliis. <lb/>Rur&longs;us fru&longs;tum <lb/>a g ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>cuius eadem e&longs;t <lb/>ba&longs;is, & æqualis <lb/>altitudo eandem <lb/><expan abbr="proportion&etilde;">proportionem</expan> ha <lb/>bet, quam <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> <lb/>l f ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>quæ e&longs;t <expan abbr="ead&etilde;">eadem</expan> ba<lb/>&longs;i, & æquali alti<lb/>tudine: & &longs;imili<lb/>ter quam l h fru<lb/>&longs;tum ad pyrami<lb/>dem, quæ ex <expan abbr="ea-d&etilde;">ea<lb/>dem</expan> ba&longs;i, & æquali <lb/>altitudine con<lb/>&longs;tat. nam &longs;i inter <lb/>ip&longs;as ba&longs;es me<lb/>diæ proportio<lb/>nales con&longs;tituan <lb/>tur, tres ba&longs;es &longs;imul &longs;umptæ ad maiorem ba&longs;im in om<lb/>nibus codem modo &longs;e habebunt. Vnde fit, ut axes K l, <lb/>q r, t u à punctis p s x in eandem proportionem &longs;ecen<lb/>

<figure id="fig53"></figure><lb/>duplum a d late<lb/>ris una cum late<lb/>re e h ad duplum <lb/>e h unà cum a d: <lb/>& ita in aliis. <lb/>Rur&longs;us fru&longs;tum <lb/>a g ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>cuius eadem e&longs;t <lb/>ba&longs;is, & æqualis <lb/>altitudo eandem <lb/><expan abbr="proportion&etilde;">proportionem</expan> ha <lb/>bet, quam <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> <lb/>l f ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>quæ e&longs;t <expan abbr="ead&etilde;">eadem</expan> ba<lb/>&longs;i, & æquali alti<lb/>tudine: & &longs;imili<lb/>ter quam l h fru<lb/>&longs;tum ad pyrami<lb/>dem, quæ ex <expan abbr="ea-d&etilde;">ea<lb/>dem</expan> ba&longs;i, & æquali <lb/>altitudine con<lb/>&longs;tat. nam &longs;i inter <lb/>ip&longs;as ba&longs;es me<lb/>diæ proportio<lb/>nales con&longs;tituan <lb/>tur, tres ba&longs;es &longs;imul &longs;umptæ ad maiorem ba&longs;im in om<lb/>nibus codem modo &longs;e habebunt. Vnde fit, ut axes K l, <lb/>q r, t u à punctis p s x in eandem proportionem &longs;ecen<lb/>

<arrow.to.target n="marg92"></arrow.to.target><lb/>tur. 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. 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. 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. Vt autem triangulum a b c ad triangu<lb/>lum a c d, ita pyramis a b c y ad pyramidem a c d y. & ut <lb/>triangulum e f g ad triangulum e g h, ita pyramis e f g y <lb/>ad pyramidem e g h y; ergo ut pyramis a b c y ad <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>

<arrow.to.target n="marg93"></arrow.to.target><lb/>a c d y, ita pyramis e f g y ad pyramidem e g h y. reliquum <lb/>igitur <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> l f ad reliquum <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> l h e&longs;t ut pyramis a b c y <lb/>ad pyramidem a c d y, hoc e&longs;t ut u l ad l r, & ut x p ad p s. <lb/>Quòd cum fru&longs;ti l f centrum grauitatis &longs;its: & fru&longs;ti l h &longs;it <lb/>

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<s><margin.target id="marg94"></margin.target>8. Archi<lb/>medis.</s>

<s>Sit fru&longs;tum a d à cono, uel coni portione ab&longs;ci&longs;&longs;um, eu<lb/>ius maior ba&longs;is circulus, uel ellip&longs;is circa diametrum a b; <lb/>minor circa diametrum c d: & axis e f. diuidatur <expan abbr="aut&etilde;">autem</expan> e f <lb/>in g, ita ut e g ad g f eandem proportionem habeat, quam <lb/>duplum diametri a b unà cum diametro e d ad duplum c d <lb/>unà cum a b. <expan abbr="Sitq;">Sitque</expan> g h quarta pars lineæ g e: & &longs;it &longs; K item <lb/>quarta pars totius f e axis. Rur&longs;us quam proportionem <lb/>habet fru&longs;tum a d ad conum, uel coni portionem, in <expan abbr="cad&etilde;">cadem</expan> <lb/>ba&longs;i, & æquali altitudine, habeat linea K h ad h l. Dico pun<lb/>ctum l fru&longs;ti a d grauitatis centrum e&longs;&longs;e. Si enini fieri po<lb/>te&longs;t, &longs;it m centrum: <expan abbr="producaturq;">producaturque</expan> l m extra fru&longs;tum in n: <lb/>& ut n l ad l m, ita fiat circulus, uel ellip&longs;is circa <expan abbr="diametr&utilde;">diametrum</expan> <lb/>a b ad aliud &longs;pacium, in quo &longs;it o. Itaque in circulo, uel <lb/>ellip&longs;i circa diametrum a b rectilinea figura plane de&longs;cri<lb/>batur, ita ut quæ relinquuntur portiones &longs;int o &longs;pacio mi<lb/>nores: & intelligatur pyramis a p b, ba&longs;im habens rectili<lb/>neam figuram in circulo, uel ellip&longs;i a b de&longs;criptam: à qua

<pb/>fru&longs;tum pyramidis &longs;it ab&longs;ci&longs;&longs;um. erit ex iis quæ proxime <lb/>tradidimus, fru&longs;ti pyramidis a d ceutrum grauitatis l. Quo <lb/>niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir <lb/>

<arrow.to.target n="fig54"></arrow.to.target><lb/>culus, uel ellip&longs;is a b ad <lb/>portiones dictas <expan abbr="maior&etilde;">maiorem</expan> <lb/>proportionem, quàm n l <lb/>ad l m. &longs;ed ut circulus, uel <lb/>ellip&longs;is a b ad portiones, <lb/>ita a p b conus, uel coni <lb/>portio ad &longs;olidas portio<lb/>nes, id quod &longs;upra demon <lb/>&longs;tratum e&longs;t: & ut circulus <lb/>

<figure id="fig54"></figure><lb/>culus, uel ellip&longs;is a b ad <lb/>portiones dictas <expan abbr="maior&etilde;">maiorem</expan> <lb/>proportionem, quàm n l <lb/>ad l m. &longs;ed ut circulus, uel <lb/>ellip&longs;is a b ad portiones, <lb/>ita a p b conus, uel coni <lb/>portio ad &longs;olidas portio<lb/>nes, id quod &longs;upra demon <lb/>&longs;tratum e&longs;t: & ut circulus <lb/>

<arrow.to.target n="marg95"></arrow.to.target><lb/>uel ellip&longs;is c d ad portio<lb/>nes, quæ ip &longs;i in&longs;unt, ita co <lb/>nus, uel coni portio c p d <lb/>ad &longs;olidas ip&longs;ius portio<lb/>nes. Quòd cum figuræ in <lb/>circulis, uel ellip&longs;ibus a b <lb/>c d de&longs;criptæ &longs;imiles &longs;int, <lb/>erit proportio circuli, uel <lb/>ellip&longs;is a b ad &longs;uas portio <lb/>nes, <expan abbr="ead&etilde;">eadem</expan>, quæ circuli uel <lb/>ellip&longs;is c d ad &longs;uas. ergo <lb/>conus, uel coni portio a p <lb/>b ad portiones &longs;olidas <expan abbr="eã-dem">ean<lb/>dem</expan> habet <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam conus, uel coni por <lb/>tio c p d ad &longs;olidas ip&longs;ius <lb/>

<arrow.to.target n="marg96"></arrow.to.target><lb/>portiones. reliquum igi<lb/>tur coni, uel coni portionis <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan>, &longs;cilicet a d ad reliquas <lb/>portiones &longs;olidas in ip&longs;o contentas eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam conus, uel coni portio a p b ad &longs;olidas portio <lb/>nes: hoc e&longs;t eandem, quam circulus, uel ellip&longs;is a b ad por <lb/>tiones planas. quare fru&longs;tum coni, uel coni portionis a d

<pb pagenum="38"/>ad portiones &longs;olidas maiorem habet <expan abbr="proportion&etilde;">proportionem</expan>, quàm <lb/>n l ad l m: & diuidendo fru&longs;tum pyramidis ad dictas por<lb/>tiones maiorem proportionem habet, quàm n m ad m l. <lb/>fiat igitur ut fru&longs;tum pyramidis ad portiones, ita q m ad <lb/>m l. Itaque quoniam à fru&longs;to coni, uel coni portionis a d, <lb/>cuius grauitatis centrum e&longs;tm, au&longs;ertur fru&longs;tum pyrami<lb/>dis habens centrum l; erit reliquæ magnitudinis, quæ ex <lb/>portionibus &longs;olidis con&longs;tat; grauitatis <expan abbr="c&etilde;trum">centrum</expan> in linea l m <lb/>producta, atque in puncto q, extra figuram po&longs;ito: quod <lb/>fieri nullo modo pote&longs;t. relinquitur ergo, ut punctum l &longs;it <lb/>fru&longs;ti a d grauitatis centrum. quz omnia demon&longs;tranda <lb/>proponebantur.</s>

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<s><margin.target id="marg96"></margin.target>19. quínti</s>

<s>THEOREMA XXII. PROPOSITIO XXVII.</s>

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<s>Sit primo a b c d pyramis <expan abbr="&itilde;">im</expan> &longs;phæra de&longs;cripta, cuius &longs;phæ <lb/>ræ centrum &longs;it e. Dico e pyramidis a b c d grauitatis e&longs;&longs;e <lb/>centrum. Si enim iuncta d c producatur ad ba&longs;im a b c in <lb/>f; exiis, quæ demon&longs;trauit Campanus in quartodecimo li <lb/>bro elementorum, propo&longs;itione decima quinta, & decima <lb/>feptima, erit f centrum circuli circa triangulum a b c de<lb/>fcripti: atque erit e f &longs;exta pars ip&longs;ius &longs;phæræ axis. quare <lb/>ex prima huius con&longs;tat trianguli a b c grauitatis centrum <lb/>e&longs;&longs;e punctum f: & idcirco lineam d f e&longs;&longs;e pyramidis axem.

<pb/>

<arrow.to.target n="fig55"></arrow.to.target><lb/>At cum e f &longs;it &longs;exta pars axis <lb/>&longs;phæræ, crit d e tripla e f. ergo <lb/>punctum e e&longs;t grauitatis cen<lb/>trum ip&longs;ius pyramidis: quod <lb/>in uige&longs;ima &longs;ecunda huius de<lb/>mon&longs;tratum &longs;uit. Sed e e&longs;t cen <lb/>trum &longs;phæræ. Sequitur igitur, <lb/>ut centrum grauitatis pyrami<lb/>dis in &longs;phæra de&longs;criptæ idem <lb/>&longs;it, quod ip&longs;ius &longs;phæræ cen<lb/>trum.</s>

<figure id="fig55"></figure><lb/>At cum e f &longs;it &longs;exta pars axis <lb/>&longs;phæræ, crit d e tripla e f. ergo <lb/>punctum e e&longs;t grauitatis cen<lb/>trum ip&longs;ius pyramidis: quod <lb/>in uige&longs;ima &longs;ecunda huius de<lb/>mon&longs;tratum &longs;uit. Sed e e&longs;t cen <lb/>trum &longs;phæræ. Sequitur igitur, <lb/>ut centrum grauitatis pyrami<lb/>dis in &longs;phæra de&longs;criptæ idem <lb/>&longs;it, quod ip&longs;ius &longs;phæræ cen<lb/>trum.</s>

<s>Sit cubus in &longs;phæra de&longs;criptus a b, & oppo&longs;itorum pla<lb/>norum lateribus bifariam diui&longs;is, per puncta diui&longs;ionum <lb/>plana ducantur, ut communis ip&longs;orum &longs;ectio &longs;it rectali<lb/>nea c d. Itaque &longs;i ducatur a b, &longs;olidi &longs;cilicet diameter, lineæ <lb/>a b, c d ex trige&longs;iman onaun decimi&longs;e&longs;e bifariam &longs;ecabunt. <lb/>

<arrow.to.target n="fig56"></arrow.to.target><lb/>&longs;ecent autem in puncto e. erit, <lb/>e <expan abbr="centr&utilde;">centrum</expan> grauitatis &longs;olidi a b, <lb/>id quod demon&longs;tratum e&longs;t in <lb/>octaua huius. Sed quoniam ab <lb/>e&longs;t &longs;phæræ diametro æqualis, <lb/>ut in decima quinta propo&longs;i<lb/>tione tertii decimilibri <expan abbr="elem&etilde;">elemen</expan> <lb/>torum o&longs;tenditur: punctum e <lb/>&longs;phæræ quoque centrum erit. <lb/>Cubi igitur in &longs;phæra de&longs;cri<lb/>pti grauitatis centrum idem <lb/>e&longs;t, quod centrum ip&longs;ius &longs;phæræ.</s>

<figure id="fig56"></figure><lb/>&longs;ecent autem in puncto e. erit, <lb/>e <expan abbr="centr&utilde;">centrum</expan> grauitatis &longs;olidi a b, <lb/>id quod demon&longs;tratum e&longs;t in <lb/>octaua huius. Sed quoniam ab <lb/>e&longs;t &longs;phæræ diametro æqualis, <lb/>ut in decima quinta propo&longs;i<lb/>tione tertii decimilibri <expan abbr="elem&etilde;">elemen</expan> <lb/>torum o&longs;tenditur: punctum e <lb/>&longs;phæræ quoque centrum erit. <lb/>Cubi igitur in &longs;phæra de&longs;cri<lb/>pti grauitatis centrum idem <lb/>e&longs;t, quod centrum ip&longs;ius &longs;phæræ.</s>

<s>Sit octahedrum a b c d e f, in &longs;phæra de&longs;criptum, cuius <lb/>&longs;phæræ centrum &longs;itg. Dico punctum g ip&longs;ius octahedri <lb/>grauitatis centrum e&longs;&longs;e. Con&longs;tat enim ex iis, quæ demon<lb/>&longs;trata &longs;unt à Campano in quinto decimo libro elemento<lb/>rum, propo&longs;itione &longs;extadecima eiu&longs;modi &longs;olidum diuidi <lb/>in duas pyramides æquales, & &longs;imiles; uidelicet in pyrami

<pb pagenum="39"/>dem, cuius ba&longs;is e&longs;t quadratum a b c d, & altitudo e g: & <lb/>in pyramidem, cuius <expan abbr="ead&etilde;">eadem</expan> ba&longs;is, <expan abbr="altitudoq;">altitudoque</expan> f g; ut &longs;int e g, <lb/>g f &longs;emidiametri &longs;phæræ, & linea una. <expan abbr="C&utilde;igitur">Cunigitur</expan> g &longs;it &longs;phæ<lb/>ræ centrum, erit etiam centrum circuli, qui circa <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>a b c d de&longs;cribitur: & propterea eiu&longs;dem quadrati grauita <lb/>tis centrum: quod in prima propo&longs;itione huius demon<lb/>&longs;tratum e&longs;t. quare pyramidis a b c d e axis erit e g: & pyra <lb/>midis a b c d f axis f g. Itaque &longs;ith centrum grauitatis py<lb/>ramidis a b c d e, & pyramidis a b c d f centrum &longs;it <emph type="italics"/>K:<emph.end type="italics"/> per<lb/>&longs;picuum e&longs;t ex uige&longs;ima &longs;ecunda propo&longs;itione huius, <expan abbr="lineã">lineam</expan> <lb/>

<arrow.to.target n="fig57"></arrow.to.target><lb/>c h triplam e&longs;&longs;e h g: <expan abbr="cõ">com</expan> <lb/><expan abbr="ponendoq;">ponendoque</expan> e g ip&longs;ius g <lb/>h quadruplam. & <expan abbr="ead&etilde;">eadem</expan> <lb/>ratione f g <expan abbr="quadruplã">quadruplam</expan> <lb/>ip&longs;ius g <foreign lang="greek">k.</foreign> quod cum e <lb/>g, g f &longs;int æquales, & h <lb/>g, g <emph type="italics"/>K<emph.end type="italics"/> nece&longs;&longs;ario æqua<lb/>les erunt. ergo ex quar <lb/>ta propo&longs;itione primi <lb/>libri Archimedis de <expan abbr="c&etilde;-tro">cen<lb/>tro</expan> grauitatis <expan abbr="planor&utilde;">planorum</expan>, <lb/>totius octahedri, quod <lb/>ex dictis pyramidibus <lb/>con&longs;tat, centrum graui <lb/>tatis erit punctum g idem, quod ip&longs;ius &longs;phæræ centrum.</s>

<figure id="fig57"></figure><lb/>c h triplam e&longs;&longs;e h g: <expan abbr="cõ">com</expan> <lb/><expan abbr="ponendoq;">ponendoque</expan> e g ip&longs;ius g <lb/>h quadruplam. & <expan abbr="ead&etilde;">eadem</expan> <lb/>ratione f g <expan abbr="quadruplã">quadruplam</expan> <lb/>ip&longs;ius g <foreign lang="greek">k.</foreign> quod cum e <lb/>g, g f &longs;int æquales, & h <lb/>g, g <emph type="italics"/>K<emph.end type="italics"/> nece&longs;&longs;ario æqua<lb/>les erunt. ergo ex quar <lb/>ta propo&longs;itione primi <lb/>libri Archimedis de <expan abbr="c&etilde;-tro">cen<lb/>tro</expan> grauitatis <expan abbr="planor&utilde;">planorum</expan>, <lb/>totius octahedri, quod <lb/>ex dictis pyramidibus <lb/>con&longs;tat, centrum graui <lb/>tatis erit punctum g idem, quod ip&longs;ius &longs;phæræ centrum.</s>

<s>Sit ico&longs;ahedrum a d de&longs;criptum in &longs;phæra, cuius <expan abbr="centr&utilde;">centrum</expan> <lb/>&longs;it g. Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. Si <lb/>enim ab angulo a per g ducatur recta linea u&longs;que ad &longs;phæ <lb/>ræ &longs;uperficiem; con&longs;tat ex &longs;exta decima propo&longs;itione libri <lb/>tertii decimi elementorum, cadere eam in angulum ip&longs;i a <lb/>oppo&longs;itum. cadat in d: <expan abbr="&longs;itq;">&longs;itque</expan> una aliqua ba&longs;is ico&longs;ahedri tri<lb/>angulum a b c: & iunctæ b g, c g producantur, & cadant in <lb/>angulos e f, ip&longs;is b c oppo&longs;itos. Itaque per triangula <lb/>a b c<gap/> d e f ducantur plana &longs;phæram &longs;ecantia<gap/> erunt <gap/> &longs;e-

<pb/>ctiones circuli ex prima propo&longs;itione &longs;phæricorum Theo <lb/>do&longs;ii: unus quidem circa triangulum a b c de&longs;criptus: al<lb/>ter uero circa d e f: & quoniam triangula a b c, d e f æqua<lb/>lia &longs;unt, & &longs;imilia; erunt ex prima, & &longs;ecunda propo&longs;itione <lb/>duodecimi libri clementorum, circuli quoque inter &longs;e &longs;e <lb/>æquales. po&longs;tremo a centro g ad circulum a b c perpendi <lb/>cularis ducatur g h; & alia perpendicularis ducatur ad cir <lb/>culum d e f, quæ &longs;it g k; & iungantur a h, d <foreign lang="greek">k.</foreign> per&longs;picuum <lb/>e&longs;t ex corollario primæ &longs;phæricorum Theodo&longs;ii, punctum <lb/>h centrum e&longs;&longs;e circuli a b c, & <foreign lang="greek">k</foreign> centrum circuli d e f. Quo <lb/>niam igitur triangulorum g a h, g d K latus a g e&longs;t æquale la <lb/>teri g d; &longs;unt enim à centro &longs;phæræ ad &longs;uperficiem: atque <lb/>e&longs;t a h æquale d k: & ex &longs;exta propo&longs;itione libri primi &longs;phæ <lb/>ricorum Theodo&longs;ii g h ip&longs;i g K: triangulum g a h æquale <lb/>erit, & &longs;imile g d <foreign lang="greek">k</foreign> triangulo: & angulus a g h æqualis an<lb/>

<arrow.to.target n="marg97"></arrow.to.target><lb/>gulo d g <emph type="italics"/>K.<emph.end type="italics"/> &longs;ed anguli a g h, h g d &longs;unt æquales duobus re<lb/>ctis. crgo & ip&longs;i h g d, d g <foreign lang="greek">k</foreign> duobus rectis æquales erunt. <lb/>

<arrow.to.target n="marg98"></arrow.to.target><lb/>& idcirco h g, g <emph type="italics"/>K<emph.end type="italics"/> una, atque eadem erit linea. cum autem <lb/>

<arrow.to.target n="fig58"></arrow.to.target><lb/>h &longs;it <expan abbr="centr&utilde;">centrum</expan> circuli, & tri<lb/>anguli a b c grauitatis cen <lb/><expan abbr="tr&utilde;">trum</expan> probabitur exiis, quæ <lb/>in prima propo&longs;itione hu <lb/>ius tradita &longs;unt. quare g h <lb/>erit pyramidis a b c g axis. <lb/>& ob eandem cau&longs;&longs;am g k <lb/>axis pyramidis d e f g. lta<lb/>que centrum grauitatls py <lb/>ramidis a b c g &longs;it <expan abbr="p&utilde;ctum">punctum</expan> <lb/>l, & pyramidis d e f g &longs;it m. <lb/>Simillter ut &longs;upra demon<lb/>&longs;trabimus m g, g linter &longs;e æquales e&longs;&longs;e, & punctum g graui <lb/>tatis centrum magnitudinis, quæ ex utri&longs;que pyramidibus <lb/>con&longs;tat. eodem modo demon&longs;trabitur, quarumcunque <lb/>duarum pyramidum, quæ opponuntur, grauitatis <expan abbr="centr&utilde;">centrum</expan>

<figure id="fig58"></figure><lb/>h &longs;it <expan abbr="centr&utilde;">centrum</expan> circuli, & tri<lb/>anguli a b c grauitatis cen <lb/><expan abbr="tr&utilde;">trum</expan> probabitur exiis, quæ <lb/>in prima propo&longs;itione hu <lb/>ius tradita &longs;unt. quare g h <lb/>erit pyramidis a b c g axis. <lb/>& ob eandem cau&longs;&longs;am g k <lb/>axis pyramidis d e f g. lta<lb/>que centrum grauitatls py <lb/>ramidis a b c g &longs;it <expan abbr="p&utilde;ctum">punctum</expan> <lb/>l, & pyramidis d e f g &longs;it m. <lb/>Simillter ut &longs;upra demon<lb/>&longs;trabimus m g, g linter &longs;e æquales e&longs;&longs;e, & punctum g graui <lb/>tatis centrum magnitudinis, quæ ex utri&longs;que pyramidibus <lb/>con&longs;tat. eodem modo demon&longs;trabitur, quarumcunque <lb/>duarum pyramidum, quæ opponuntur, grauitatis <expan abbr="centr&utilde;">centrum</expan>

<pb pagenum="40"/>e&longs;&longs;e punctum g. Sequitur ergo utico&longs;ahedri centrum gra <lb/>uitatis &longs;it idem, quod ip&longs;ius &longs;phæræ centrum.</s>

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<s><margin.target id="marg98"></margin.target>14. primi</s>

<s>Sit dodecahedrum a f in &longs;phæra de&longs;ignatum, &longs;itque &longs;phæ <lb/>ræ centrum m. Dico m centrum e&longs;&longs;e grauitatis ip&longs;ius do<lb/>decahedri. Sit enim pentagonum a b c d e una ex duode<lb/>cim ba&longs;ibus &longs;olidi a f: & iuncta a m producatur ad &longs;phæræ <lb/>&longs;uperficiem. cadet in angulum ip&longs;i a oppo&longs;itum; quod col<lb/>ligitur ex decima &longs;eptima propo&longs;iticne tertiidecimi libri <lb/>clementorum. cadat in f. at &longs;i ab aliis angulis b c d e per <expan abbr="c&etilde;">cem</expan> <lb/>trum itidem lineæ ducantur ad &longs;uperficiem &longs;phæræ in pun <lb/>cta g h <foreign lang="greek">k</foreign> l; cadent hæ in alios angulos ba&longs;is, quæ ip&longs;i a b c d <lb/>ba&longs;i opponitur. tran&longs;eant ergo per pentagona a b c d e, <lb/>f g h K l plana &longs;phæram &longs;ecantia, quæ facient &longs;ectiones cir<lb/>culos æquales inter &longs;e &longs;e: po&longs;tea ducantur ex centro &longs;phæræ <lb/>

<arrow.to.target n="fig59"></arrow.to.target><lb/>m perpendiculares ad pla<lb/>na dictorum <expan abbr="circulor&utilde;">circulorum</expan>; ad <lb/>circulum quidem a b c d e <lb/>perpendicularis m n: & ad <lb/>circulum f g h K l ip&longs;a m o, <lb/>

<figure id="fig59"></figure><lb/>m perpendiculares ad pla<lb/>na dictorum <expan abbr="circulor&utilde;">circulorum</expan>; ad <lb/>circulum quidem a b c d e <lb/>perpendicularis m n: & ad <lb/>circulum f g h K l ip&longs;a m o, <lb/>

<arrow.to.target n="marg99"></arrow.to.target><lb/>erunt puncta n o <expan abbr="circulor&utilde;">circulorum</expan> <lb/>centra: & lineæ m n, m o in <lb/>ter &longs;e æquales: quòd circu<lb/>

<arrow.to.target n="marg100"></arrow.to.target><lb/>li æquales &longs;int. Eodem mo <lb/>do, quo &longs;upra, demon&longs;trabi <lb/>mus lineas m n, m o in <expan abbr="unã">unam</expan> <lb/>atque eandem lineam con<lb/>uenire. ergo cum puncta n o &longs;int centra circulorum, con<lb/>&longs;tat ex prima huius & <expan abbr="pentagonor&utilde;">pentagonorum</expan> grauitatis e&longs;&longs;e centra: <lb/><expan abbr="idcircoq;">idcircoque</expan> m n, m o pyramidum a b c d e m, f g h <emph type="italics"/>K<emph.end type="italics"/> l m axes. <lb/>ponatur a b c d e m pyramidis grauitatis centrum p: & py <lb/>ramidis f g h <foreign lang="greek">k</foreign> l m ip&longs;um q centrum. erunt p m, m q æqua<lb/>les, & punctum m grauitatis centrum magnitudinis, quæ <lb/>ex ip&longs;is pyramidibus con&longs;tat. <expan abbr="eod&etilde;">eodem</expan> modo probabitur qua<lb/>rumlibet pyramidum, quæ è regione opponuntur, <expan abbr="centr&utilde;">centrum</expan>

<pb/>grauitatis e&longs;&longs;e punctum m. patetigitur totius dodecahe<lb/>dri, centrum grauitatis <expan abbr="id&etilde;">idem</expan> e&longs;&longs;e, quod & &longs;phæræ ip&longs;um com <lb/>prehendentis centrum. quæ quidem omnia demon&longs;tra&longs;&longs;e <lb/>oportebat.</s>

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<s><margin.target id="marg100"></margin.target>6. primi <lb/>phærico <lb/>rum.</s>

<s>PROBLEMA VI. PROPOSITIO XXVIII.</s>

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<arrow.to.target n="marg101"></arrow.to.target><lb/>proportionem habet, quàm portio conoidis ad &longs;olidum h; <lb/>hoc e&longs;t maiorem, quàm b c ad g: & b e ad g non minorem <lb/>habet proportionem, quàm ad k e, propterea quod k e non <lb/>ponitur minor ip&longs;a g: habcbit figura circum&longs;cripta ad por <lb/>tiones reliquas maiorem proportionem quàm b e ad e k: <lb/>

<arrow.to.target n="marg102"></arrow.to.target><lb/>& diuidendo portio conoidis ad reliquas portiones habe<lb/>bit maiorem, quàm b <foreign lang="greek">k</foreign> ad K e. quare &longs;i fiat ut portio co

<pb pagenum="41"/>noidis ad portiones reliquas, ita alia linea, quæ &longs;it l <foreign lang="greek">k</foreign> ad <lb/><foreign lang="greek">k</foreign> e: erit l k maior, quam b k: & ideo punctum l extra por<lb/>

<arrow.to.target n="fig60"></arrow.to.target><lb/>tionem cadet. <expan abbr="Quoniã">Quoniam</expan> <lb/>igitur à figura circum<lb/>&longs;cripta, cuius grauitatis <lb/>centrum e&longs;t k, aufertur <lb/>portio conoidis, cuius <lb/>centrum e. <expan abbr="habetq;">habetque</expan> l K <lb/>ad K e eam proportio<lb/>nem, quam portio co<lb/>noidis ad reliquas por<lb/>tiones; erit punctum l <lb/>extra portionem <expan abbr="cad&etilde;s">cadens</expan>, <lb/>centrum magnitudinis <lb/>ex reliquis portionibus compo&longs;itæ. illud autem fieri nullo <lb/>modo pote&longs;t. quare con&longs;tat lineam k e ip&longs;a g linea propo&longs;i <lb/>ta minorem e&longs;&longs;e.</s>

<figure id="fig60"></figure><lb/>tionem cadet. <expan abbr="Quoniã">Quoniam</expan> <lb/>igitur à figura circum<lb/>&longs;cripta, cuius grauitatis <lb/>centrum e&longs;t k, aufertur <lb/>portio conoidis, cuius <lb/>centrum e. <expan abbr="habetq;">habetque</expan> l K <lb/>ad K e eam proportio<lb/>nem, quam portio co<lb/>noidis ad reliquas por<lb/>tiones; erit punctum l <lb/>extra portionem <expan abbr="cad&etilde;s">cadens</expan>, <lb/>centrum magnitudinis <lb/>ex reliquis portionibus compo&longs;itæ. illud autem fieri nullo <lb/>modo pote&longs;t. quare con&longs;tat lineam k e ip&longs;a g linea propo&longs;i <lb/>ta minorem e&longs;&longs;e.</s>

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<s><margin.target id="marg102"></margin.target>29. quínti <lb/>ex tradi<lb/>tione <expan abbr="Cã-l">Can<lb/>l</expan> ani.</s>

<s>Rur&longs;us in&longs;cribatur portioni figura, uidelicet cylindr us <lb/>

<arrow.to.target n="fig61"></arrow.to.target><lb/>m n, ut &longs;it ip&longs;ius altitudo <lb/>æqualis dimidio axis b d: <lb/>& quam proportionem <lb/>habet b e ad g, habeat m n <lb/>cylindrus ad &longs;olidum o. <lb/>in&longs;cribatur deinde eidem <lb/>alia figura, ita ut portio<lb/>nes reliquæ &longs;int &longs;olido o <lb/>minores: & centrum gra <lb/>uitatis figuræ &longs;it p. Dico <lb/>lineam p e ip&longs;a g <expan abbr="minor&etilde;">minorem</expan> <lb/>e&longs;&longs;e. &longs;i enim n<gap/>n &longs;it mi<lb/>nor, codem, quo &longs;upra modo demon&longs;trabimus figuram in <lb/>&longs;criptam ad reliquas portiones maiorem proportionem <lb/>habere, quàm b e ad e p. & &longs;i fiat alia linea l e ad e p, ut e&longs;t <lb/>figura in&longs;cripta ad reliquas portiones, <expan abbr="p&utilde;ctum">punctum</expan> l extra por

<figure id="fig61"></figure><lb/>m n, ut &longs;it ip&longs;ius altitudo <lb/>æqualis dimidio axis b d: <lb/>& quam proportionem <lb/>habet b e ad g, habeat m n <lb/>cylindrus ad &longs;olidum o. <lb/>in&longs;cribatur deinde eidem <lb/>alia figura, ita ut portio<lb/>nes reliquæ &longs;int &longs;olido o <lb/>minores: & centrum gra <lb/>uitatis figuræ &longs;it p. Dico <lb/>lineam p e ip&longs;a g <expan abbr="minor&etilde;">minorem</expan> <lb/>e&longs;&longs;e. &longs;i enim n<gap/>n &longs;it mi<lb/>nor, codem, quo &longs;upra modo demon&longs;trabimus figuram in <lb/>&longs;criptam ad reliquas portiones maiorem proportionem <lb/>habere, quàm b e ad e p. & &longs;i fiat alia linea l e ad e p, ut e&longs;t <lb/>figura in&longs;cripta ad reliquas portiones, <expan abbr="p&utilde;ctum">punctum</expan> l extra por

<pb/>tionem cadet: Itaque cum à portione conoidis, cuius gra<lb/>uitatis centrum e auferatur in&longs;cripta figura, centrum ha<lb/>bens p: & &longs;itl e ad e p, ut figura in&longs;cripta ad portiones reli <lb/>quas: erit magnitudinis, quæ ex reliquis portionibus con <lb/>&longs;tat, centrum grauitatis punctum l, extra portionem ca<lb/>dens. quod ficrinequit. ergo linca p e minor e&longs;tip&longs;a g li<lb/>nea propo&longs;ita.</s>

<s>Ex quibus per&longs;picuum e&longs;t centrum grauitatis <lb/>figuræ in&longs;criptæ, & circum&longs;criptæ eo magis acce <lb/>dere ad portionis centrum, quo pluribus cylin<lb/>dris, uel cylindri portionibus con&longs;tet: <expan abbr="fiat&qacute;">fiatque</expan>; figu <lb/>rain&longs;cripta maior, & circum&longs;cripta minor. & <lb/>quanquam continenter ad portionis <expan abbr="centr&utilde;">centrum</expan> pro<lb/>pius admoueatur: nunquam tamen ad ip&longs;um per <lb/>ueniet. &longs;equeretur enim figuram in&longs;criptam, <expan abbr="nõ">non</expan> <lb/>&longs;olum portioni, &longs;ed etiam circum&longs;criptæ figuræ <lb/>æqualem e&longs;&longs;e. quod e&longs;t ab&longs;urdum.</s>

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<s>SIT portio conoidis rectanguli uel ab&longs;ci&longs;&longs;a plano ad <lb/>axem recto, uel non recto: & &longs;ecta ip&longs;a altero plano per ax<gap/><lb/>&longs;it &longs;uperficici &longs;ectio a b crectanguli coni &longs;ectio, uel parabo <lb/>le; plani ab&longs;cindentis portionem &longs;ectio &longs;it recta linea a c: <lb/>axis portionis, & &longs;ectionis diameter b d. Sumatur autem <lb/>in linea b d punctum e, ita ut b e &longs;itip&longs;ius e d dupla. Dico

<arrow.to.target n="fig62"></arrow.to.target><lb/>e portionis a b <lb/>c grauitatis e&longs;&longs;e <lb/>centrum. Diui<lb/>datur enim b d <lb/>bifariam in m: <lb/>& rur&longs;us d m, m <lb/>b bifariam diui<lb/>dantur in pun<lb/>ctis n, o: <expan abbr="in&longs;cri-baturq;">in&longs;cri<lb/>baturque</expan> portio<lb/>ni figura &longs;olida, <lb/>& altera circum <lb/>&longs;cribatur ex cy<lb/>lindris æqualem <lb/>altitudinem ha<lb/>bentibus, ut&longs;u<lb/>perius <expan abbr="dict&utilde;">dictum</expan> e&longs;t'. <lb/>Sit autem pri<lb/>mum figura in<lb/>&longs;cripta <expan abbr="cyl&itilde;drus">cylindrus</expan> <lb/>f g: & <expan abbr="circ&utilde;&longs;cri-pta">circun&longs;cri<lb/>pta</expan> ex cylindris <lb/>a h, K<gap/> con&longs;tet. <lb/>

<figure id="fig62"></figure><lb/>e portionis a b <lb/>c grauitatis e&longs;&longs;e <lb/>centrum. Diui<lb/>datur enim b d <lb/>bifariam in m: <lb/>& rur&longs;us d m, m <lb/>b bifariam diui<lb/>dantur in pun<lb/>ctis n, o: <expan abbr="in&longs;cri-baturq;">in&longs;cri<lb/>baturque</expan> portio<lb/>ni figura &longs;olida, <lb/>& altera circum <lb/>&longs;cribatur ex cy<lb/>lindris æqualem <lb/>altitudinem ha<lb/>bentibus, ut&longs;u<lb/>perius <expan abbr="dict&utilde;">dictum</expan> e&longs;t'. <lb/>Sit autem pri<lb/>mum figura in<lb/>&longs;cripta <expan abbr="cyl&itilde;drus">cylindrus</expan> <lb/>f g: & <expan abbr="circ&utilde;&longs;cri-pta">circun&longs;cri<lb/>pta</expan> ex cylindris <lb/>a h, K<gap/> con&longs;tet. <lb/>

<arrow.to.target n="marg103"></arrow.to.target><lb/>punctum n erit <lb/>centrum graui<lb/>tatis figuræ in<lb/>fcriptæ, <expan abbr="medi&utilde;">medium</expan> <lb/>&longs;cilicet ip&longs;ius d <lb/>m axis: <expan abbr="atq;">atque</expan> <expan abbr="id&etilde;">idem</expan> <lb/>erit centrum cy <lb/>lindri ah: & cy<lb/>lindri <foreign lang="greek">k</foreign> l <expan abbr="centr&utilde;">centrum</expan> <lb/>o, axis b m me<lb/>dium. quare &longs;i li

<pb/>

<arrow.to.target n="fig63"></arrow.to.target><lb/>neam o n ita di <lb/>ui&longs;c<gap/>imus in p, <lb/>ut <expan abbr="quã">quam</expan> <expan abbr="propor-tion&etilde;">propor<lb/>tionem</expan> habet cy<lb/>lindrus a h ad <lb/>cylindrum <foreign lang="greek">k</foreign> l, <lb/>habeat linea o p <lb/>

<figure id="fig63"></figure><lb/>neam o n ita di <lb/>ui&longs;c<gap/>imus in p, <lb/>ut <expan abbr="quã">quam</expan> <expan abbr="propor-tion&etilde;">propor<lb/>tionem</expan> habet cy<lb/>lindrus a h ad <lb/>cylindrum <foreign lang="greek">k</foreign> l, <lb/>habeat linea o p <lb/>

<arrow.to.target n="marg104"></arrow.to.target><lb/>ad p n: centrum <lb/>grauitatis toti<lb/>us figuræ <expan abbr="circ&utilde;-&longs;criptæ">circun<lb/>&longs;criptæ</expan> erit pun <lb/>

<arrow.to.target n="marg105"></arrow.to.target><lb/>ctum p. Sed cy<lb/>lindri, qui &longs;unt <lb/>æquali altitudi<lb/>ne, eandem in<lb/>ter &longs;e &longs;e, quam <lb/>ba&longs;es propor— <lb/>tionem habent: <lb/><expan abbr="e&longs;tq;">e&longs;tque</expan> ut linea d b <lb/>ad b m, ita <expan abbr="qua-drat&utilde;">qua<lb/>dratum</expan> lineæ a d <lb/>ad <expan abbr="quadrat&utilde;">quadratum</expan> ip<lb/>&longs;ius K m, ex uige <lb/>&longs;ima primi libri <lb/>

<arrow.to.target n="marg106"></arrow.to.target><lb/><expan abbr="conicor&utilde;">conicorum</expan> & ita <lb/>quadratum a c <lb/>ad <expan abbr="quadrat&utilde;">quadratum</expan> K <lb/>

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<pb pagenum="43"/>b m. crgo circulus a c circuli k g: & idcirco cylindrus <lb/>a h cylindri k. l duplus erit. quare & linea o p dupla <lb/>ip&longs;ius p n. Deinde in&longs;cripta & circum&longs;cripta portioni <lb/>alia figura, ita ut in&longs;cripta con&longs;tituatur ex tribus cylin<lb/>dris q r, s g, t u: circum&longs;cripta uero ex quatuor a x, y z, <lb/>K <foreign lang="greek">g, q l:</foreign> diuidantur b o, o m, m n, n d bifariam in punctis <lb/><foreign lang="greek">m g p r.</foreign> Itaque cylindri <foreign lang="greek">q l</foreign> centrum grauitatis e&longs;t punctum <lb/><foreign lang="greek">m:</foreign> & cylindri <foreign lang="greek">k <gap/></foreign> centrum <foreign lang="greek">g.</foreign> ergo &longs;i linea <foreign lang="greek">m g</foreign> diuidatur in <foreign lang="greek">s,</foreign><lb/>ita ut <foreign lang="greek">m s</foreign> ad <foreign lang="greek">s g</foreign> <expan abbr="proportion&etilde;">proportionem</expan> ea habeat, quam cylindrus K <foreign lang="greek"><gap/></foreign><lb/>ad cylindrum <foreign lang="greek">q l,</foreign> uidelicet quam quadratum <foreign lang="greek">k</foreign> nr ad qua<lb/>

<arrow.to.target n="marg108"></arrow.to.target><lb/>dratum <gap/> o, hoc e&longs;t, quam linea m b ad b o: erit <foreign lang="greek">s</foreign> centrum <lb/>magnitudinis compo&longs;itæ ex cylindris <foreign lang="greek">k g, q l.</foreign> & cum linca <lb/>m b &longs;it dupla b o, erit & <foreign lang="greek">m s</foreign> ip&longs;ius <foreign lang="greek">s g</foreign> dupla. præterea quo<lb/>niam cylindri y z centrum grauitatis e&longs;t <foreign lang="greek">p,</foreign> linea <foreign lang="greek">s p</foreign> ita diui <lb/>&longs;ain <foreign lang="greek">t,</foreign> ut <foreign lang="greek">s t</foreign> ad <foreign lang="greek">t p</foreign> eam habeat proportionem, quam cylin <lb/>drus y z ad duos cylindros K <foreign lang="greek">g, q l:</foreign> crit <foreign lang="greek">t</foreign> centrum magnitu <lb/>dinis, quæ ex dictis tribus cylindris con&longs;tat. cylindrus <expan abbr="au-t&etilde;">au<lb/>tem</expan> y z ad cylindrum <foreign lang="greek">q l</foreign> e&longs;t, utlinea n b ad b o, hoc e&longs;t ut 3 <lb/>ad 1: & ad cylindrum k <foreign lang="greek"><gap/>,</foreign> ut n b ad b m, uidelicet ut 3 ad 2. <lb/>quare y z <expan abbr="cyl&itilde;drus">cylindrus</expan> duobus cylindris k <foreign lang="greek">g, q l</foreign> æqualis erit. & <lb/>propterea linea <foreign lang="greek">s t</foreign> æqualis ip&longs;i <foreign lang="greek">t p.</foreign> denique cylindri a x <lb/>centrum grauitatis e&longs;t punctum <foreign lang="greek">r.</foreign> & cum <foreign lang="greek">t r</foreign> diui&longs;a fuerit <lb/>in <expan abbr="eã">eam</expan> proportionem, quam habet cylindrus ax ad tres cy<lb/>lindros y z, k <foreign lang="greek">g, q l:</foreign> erit in eo puncto centrum grauitatis <lb/>totius figuræ <expan abbr="circ&utilde;&longs;criptæ">circun&longs;criptæ</expan>. Sed cylindrus a x ad ip&longs;um y z <lb/>e&longs;t ut linea d b ad b n: hoc e&longs;t ut 4 ad 3: & duo cylindri k <foreign lang="greek">h <lb/>q l</foreign> cylindro y z &longs;unt æquales. cylindrus igitur a x ad tres <lb/>iam dictos cylindros e&longs;t ut 2 ad 3. Sed <expan abbr="quoniã">quoniam</expan> <foreign lang="greek">m s</foreign> e&longs;t dua<lb/>rum partium, & <foreign lang="greek">s g</foreign> unius, qualium <foreign lang="greek">m p</foreign> e&longs;t &longs;ex; erit <foreign lang="greek">s p</foreign> par<lb/>tium quatuor: <expan abbr="proptereaq;">proptereaque</expan> <foreign lang="greek">t p</foreign> duarum, & <foreign lang="greek">g p,</foreign> hoc e&longs;t <foreign lang="greek">p r</foreign><lb/>trium. quare &longs;equitur ut punctum <foreign lang="greek">p</foreign> totius figuræ circum <lb/>&longs;criptæ &longs;it centrum. Itaque fiat <foreign lang="greek">g u</foreign> ad <foreign lang="greek">u p,</foreign> ut <foreign lang="greek">m s</foreign> ad <foreign lang="greek">s g.</foreign> & <foreign lang="greek">u r</foreign><lb/>bifariam diuidatur in <foreign lang="greek">f.</foreign> Similiter utin circum&longs;cripta figu <lb/>ra o&longs;tendetur centrum magnitudinis compo&longs;itæ ex cylin-

<pb/>

<arrow.to.target n="fig64"></arrow.to.target><lb/>dris s g, tu e&longs;&longs;e <lb/>punctum <foreign lang="greek">u:</foreign> & <lb/>totius figuræ in <lb/>&longs;criptæ, quæ <expan abbr="cõ-&longs;tat">con<lb/>&longs;tat</expan> ex cylindris <lb/>q r, &longs; g, t u e&longs;&longs;e <foreign lang="greek">f</foreign><lb/>centrum. Sunt <lb/>enim hi cylindri <lb/>æquales & &longs;imi<lb/>les cylindris y z, <lb/>K <foreign lang="greek">h, q l,</foreign> figuræ <lb/>circum&longs;criptæ. <lb/><expan abbr="Quoniã">Quoniam</expan> igitur <lb/>ut b e ad e d, ita <lb/>e&longs;t o p ad p n; <lb/><expan abbr="utraq;">utraque</expan> enim u<lb/>triu&longs;que e&longs;t du<lb/>pla: erit compo <lb/>nendo, ut b d ad <lb/>d e, ita o n ad n <lb/>p; & permutan <lb/>do, ut b d ad o <lb/>n, ita d e ad n p. <lb/>Sed b d dupla <lb/>e&longs;t o n. ergo & <lb/>e d ip&longs;ius n p du <lb/>pla erit. quòd &longs;i <lb/>e d bifariam di<lb/>uidatur <expan abbr="&itilde;">im</expan> <foreign lang="greek">x,</foreign> erit <lb/><foreign lang="greek">x</foreign> d, uel e <foreign lang="greek">x</foreign> æ<lb/>qualis n p: & <lb/>&longs;ublata e n, quæ <lb/>e&longs;t <expan abbr="cõmunis">communis</expan> u<lb/>trique e <foreign lang="greek">x,</foreign> p n,

<figure id="fig64"></figure><lb/>dris s g, tu e&longs;&longs;e <lb/>punctum <foreign lang="greek">u:</foreign> & <lb/>totius figuræ in <lb/>&longs;criptæ, quæ <expan abbr="cõ-&longs;tat">con<lb/>&longs;tat</expan> ex cylindris <lb/>q r, &longs; g, t u e&longs;&longs;e <foreign lang="greek">f</foreign><lb/>centrum. Sunt <lb/>enim hi cylindri <lb/>æquales & &longs;imi<lb/>les cylindris y z, <lb/>K <foreign lang="greek">h, q l,</foreign> figuræ <lb/>circum&longs;criptæ. <lb/><expan abbr="Quoniã">Quoniam</expan> igitur <lb/>ut b e ad e d, ita <lb/>e&longs;t o p ad p n; <lb/><expan abbr="utraq;">utraque</expan> enim u<lb/>triu&longs;que e&longs;t du<lb/>pla: erit compo <lb/>nendo, ut b d ad <lb/>d e, ita o n ad n <lb/>p; & permutan <lb/>do, ut b d ad o <lb/>n, ita d e ad n p. <lb/>Sed b d dupla <lb/>e&longs;t o n. ergo & <lb/>e d ip&longs;ius n p du <lb/>pla erit. quòd &longs;i <lb/>e d bifariam di<lb/>uidatur <expan abbr="&itilde;">im</expan> <foreign lang="greek">x,</foreign> erit <lb/><foreign lang="greek">x</foreign> d, uel e <foreign lang="greek">x</foreign> æ<lb/>qualis n p: & <lb/>&longs;ublata e n, quæ <lb/>e&longs;t <expan abbr="cõmunis">communis</expan> u<lb/>trique e <foreign lang="greek">x,</foreign> p n,

<pb pagenum="44"/>relinquetur p e ip&longs;i n <foreign lang="greek">x</foreign> æqualis. cum autem b e &longs;it dupla <lb/>e d, & o p dupla p n, hoc e&longs;tip&longs;ius e <foreign lang="greek">x,</foreign> & reliquum, uideli<lb/>

<arrow.to.target n="marg109"></arrow.to.target><lb/>cet b o unà cum p e ip&longs;ius reliqui <foreign lang="greek">x</foreign> d duplnm erit. e&longs;tque <lb/>b o dupla <foreign lang="greek">r</foreign> d. ergo p e, hoc e&longs;t n <foreign lang="greek">x</foreign> ip&longs;ius <foreign lang="greek">x r</foreign> dupla. &longs;ed d n <lb/>dupla e&longs;t n <foreign lang="greek">r.</foreign> reliquaigitur d <foreign lang="greek">x</foreign> dupla reliquæ <foreign lang="greek">x</foreign> n. &longs;unt au<lb/>tem d <foreign lang="greek">x,</foreign> p n inter &longs;e æquales: <expan abbr="itemq;">itemque</expan> æquales <foreign lang="greek">x</foreign> n, p e. qua<lb/>re con&longs;tat n p ip&longs;ius p e duplam e&longs;&longs;e. & idcirco p e ip&longs;i e n <lb/>æqualem. Rur&longs;us cum &longs;it <foreign lang="greek">m g</foreign> duplao <foreign lang="greek"><gap/>,</foreign> & <foreign lang="greek">m s</foreign> dupla <foreign lang="greek">s g;</foreign> erit <lb/>etiam reliqua <foreign lang="greek">g s</foreign> reliquæ <foreign lang="greek">s</foreign> o dupla. Eadem quoque ratione <lb/><expan abbr="cõcludetur">concludetur</expan> <foreign lang="greek">p u</foreign> dupla <foreign lang="greek">u</foreign> m. ergo ut <foreign lang="greek">g s</foreign> ad <foreign lang="greek">s</foreign> o, ita <foreign lang="greek">p u</foreign> ad <foreign lang="greek">u</foreign> m: <lb/><expan abbr="componendoq;">componendoque</expan>, & permutando, ut <foreign lang="greek">g</foreign> o ad <foreign lang="greek">p</foreign> m, ita o <foreign lang="greek">s</foreign> ad <lb/>m <foreign lang="greek">u:</foreign> & &longs;unt æquales <foreign lang="greek">g</foreign> o, <foreign lang="greek">p</foreign> m. quare & o <foreign lang="greek">s,</foreign> m <foreign lang="greek">u</foreign> æquales. præ <lb/>terea <foreign lang="greek">s p</foreign> dupla e&longs;t <foreign lang="greek">p t,</foreign> & <foreign lang="greek">g p</foreign> ip&longs;ius <foreign lang="greek">p</foreign> m. reliquaigitur <foreign lang="greek">s g</foreign> re <lb/>liquæ m <foreign lang="greek">t</foreign> dupla. atque erat <foreign lang="greek">g s</foreign> dupla <foreign lang="greek">s</foreign> o. ergo m <foreign lang="greek">t, s</foreign> o æ<lb/>quales &longs;unt: & ita æquales m <foreign lang="greek">u,</foreign> n <foreign lang="greek">f.</foreign> at o <foreign lang="greek">s,</foreign> e&longs;t æqualis <lb/>m <foreign lang="greek">u.</foreign> Sequitur igitur, ut omnes o <foreign lang="greek">s,</foreign> m <foreign lang="greek">t,</foreign> m <foreign lang="greek">u,</foreign> n <foreign lang="greek">f</foreign> in<lb/>ter &longs;e &longs;int æquales. Sed ut <foreign lang="greek">r p</foreign> ad <foreign lang="greek">p t,</foreign> hoc e&longs;t ut 3 ad 2, ita n d <lb/>ad d <foreign lang="greek">x:</foreign> <expan abbr="permutãdoq;">permutandoque</expan> ut <foreign lang="greek">r p</foreign> ad n d, ita <foreign lang="greek">p t</foreign> ad d <foreign lang="greek">x.</foreign> & <expan abbr="&longs;&utilde;t">&longs;unt</expan> æqua <lb/>les <foreign lang="greek">r p,</foreign> n d. ergo d <foreign lang="greek">x,</foreign> hoc e&longs;t n p, & <foreign lang="greek">p t</foreign> æquales. Sed etiam æ<lb/>quales n <foreign lang="greek">p, p</foreign> m. reliqua igitur <foreign lang="greek">p</foreign> preliquæ m <foreign lang="greek">t,</foreign> hoc e&longs;t ip&longs;i <lb/>n <foreign lang="greek">f</foreign> æqualis erit. quare dempta p <foreign lang="greek">p</foreign> ex p e, & <foreign lang="greek">f</foreign> n dempta ex <lb/>n e, relinquitur p e æqualis e <foreign lang="greek">f.</foreign> Itaque <foreign lang="greek">p, f</foreign> centra <expan abbr="figurar&utilde;">figurarum</expan> <lb/>&longs;ecundo loco de&longs;criptarum a primis centris p n æquali in<lb/>teruallo recedunt. quòd &longs;i rur&longs;us aliæ figuræ de&longs;cribantur, <lb/>codem modo demon&longs;trabimus earum centra æqualiter ab <lb/>his recedere, & ad portionis conoidis centrum propius ad <lb/>moueri. Ex quibus con&longs;tat lineam <foreign lang="greek">p f</foreign> à centro grauitatis <lb/>portionis diuidi in partes æquales. Si enim fieri pote&longs;t, non <lb/>&longs;it centrum in puncto e, quod e&longs;t lineæ <foreign lang="greek">p f</foreign> medium: &longs;ed in <lb/><foreign lang="greek">y:</foreign> & ip&longs;i <foreign lang="greek">p y</foreign> æqualis fiat <foreign lang="greek">f <gap/>.</foreign> Cumigitur in portione &longs;olida <lb/>quædam figura in&longs;cribi pos&longs;it, ita utlinea, quæ inter cen<lb/>trum grauitatis portionis, & in&longs;criptæ figuræ interiicitur, <lb/>qualibet linea propo&longs;ita &longs;it minor, quod proxime demon<lb/>&longs;trauimus: perueniet tandem <foreign lang="greek">f</foreign> centrum in&longs;criptæ figuræ

<pb/>

<arrow.to.target n="fig65"></arrow.to.target>

<pb pagenum="45"/>ad punctum <foreign lang="greek">w.</foreign> Sed quoniam <foreign lang="greek">p</foreign> circum&longs;criptaitidem alia <lb/>figura æquali interuallo ad portionis centrum accedit, ubi <lb/>primum <foreign lang="greek">f</foreign> applicuerit &longs;e ad <foreign lang="greek">w,</foreign> & <foreign lang="greek">p</foreign> ad <expan abbr="punct&utilde;">punctum</expan> <foreign lang="greek">y,</foreign> hoc e&longs;t ad <lb/>portionis centrum &longs;e applicabit. quod fieri nullo modo <lb/>po&longs;&longs;e per&longs;picuum e&longs;t. non aliter idem ab&longs;urdum &longs;equetur, <lb/>fi ponamus centrum portionis recedere à medio ad par<lb/>tes <foreign lang="greek">w;</foreign> e&longs;&longs;et enim aliquando centrum figuræ in&longs;criptæ idem <lb/>quod portionis <expan abbr="centr&utilde;">centrum</expan>. ergo punctum e centrum erit gra <lb/>uitatis portionis a b c. quod demon&longs;trare oportebat.</s>

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<s><margin.target id="marg109"></margin.target>1<gap/> quinti</s>

<s>Quod autem &longs;upra <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t in portione conoi<lb/>dis recta per figuras, quæ ex cylindris æqualem altitudi<lb/>dinem habentibus con&longs;tant, idem &longs;imiliter demon&longs;trabi<lb/>mus per figuras ex cylindri portionibus con<gap/>antes in ea <lb/>portione, quæ plano non ad axem recto ab&longs;cinditur. ut <lb/>enim tradidimus in commentariis in undecimam propo&longs;i <lb/>tionem libri Archimedis de conoidibus & &longs;phæroidibus. <lb/>portiones cylindri, quæ æquali &longs;unt altitudine eam inter &longs;e <lb/>&longs;e proportionem habent, quam ip&longs;arum ba&longs;es: ba&longs;es <expan abbr="aut&etilde;">autem</expan> <lb/>

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<s>ABSCINDATVR à portione conoidis rectanguli <lb/>a b c alia portio e b f, plano ba&longs;i æquidi&longs;tante: & eadem <lb/>portio &longs;ecetur alio plano per axem; ut &longs;uperficiei &longs;ectio &longs;it <lb/>parabole a b c: <expan abbr="planor&utilde;">planorum</expan> portiones ab&longs;cindentium rectæ <lb/>lincæ a c, e f: axis autem portionis, & &longs;ectionis diameter <lb/>b d; quam linea e fin puncto g &longs;ecet. Dico portionem co<lb/>noidis a b c ad portionem e b f duplam proportionem ha<lb/>bere eius, quæ e&longs;t ba&longs;is a c ad ba&longs;im e f; uel axis d b ad b <gap/><lb/>axem. Intelligantur enim duo coni, &longs;eu coni portiones <lb/>a b c, e b f, <expan abbr="eãdem">eandem</expan> ba&longs;im, quam portiones conoidis, & æqua <lb/>lem habentes altitudinem. & quoniam a b c portio conoi <lb/>dis fe&longs;quialtera e&longs;t coni, &longs;eu portionis coni a b c; & portio <lb/>e b f coni feu portionis coni e b f e&longs;t &longs;e&longs;quialtera, quod de<lb/>

<arrow.to.target n="fig66"></arrow.to.target><lb/>mon&longs;trauit Archimedes in propo&longs;itionibus 23, & 24 libri <lb/>de conoidibus, & &longs;phæroidibus: erit conoidis portio ad <lb/>conoidis portionem, ut conus ad conum, uel ut coni por<lb/>tio ad coni portionem. Sed conns, nel coni portio a b c ad <lb/>conum, uel coni portionem e b f compo&longs;itam proportio<lb/>nem habet ex proportione ba&longs;is a c ad ba&longs;im e f, & ex pro<lb/>portione altitudinis coni, uel coni portionis a b c ad alti<lb/>tudinem ipfius e b f, ut nos demon&longs;trauimus in com men<lb/>tariis in undecimam propo&longs;itionem eiu&longs;dem libri A rchi<lb/>medis: altitudo autem ad altitudinem c&longs;t, ut axis ad axem. <lb/>quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue

<figure id="fig66"></figure><lb/>mon&longs;trauit Archimedes in propo&longs;itionibus 23, & 24 libri <lb/>de conoidibus, & &longs;phæroidibus: erit conoidis portio ad <lb/>conoidis portionem, ut conus ad conum, uel ut coni por<lb/>tio ad coni portionem. Sed conns, nel coni portio a b c ad <lb/>conum, uel coni portionem e b f compo&longs;itam proportio<lb/>nem habet ex proportione ba&longs;is a c ad ba&longs;im e f, & ex pro<lb/>portione altitudinis coni, uel coni portionis a b c ad alti<lb/>tudinem ipfius e b f, ut nos demon&longs;trauimus in com men<lb/>tariis in undecimam propo&longs;itionem eiu&longs;dem libri A rchi<lb/>medis: altitudo autem ad altitudinem c&longs;t, ut axis ad axem. <lb/>quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue

<pb pagenum="46"/>ro ita demon&longs;trabitur. Ducatur à puncto b ad planum ba<lb/>&longs;is a c perpendicularis linea b h, quæ ip&longs;am e fin K &longs;ecet. <lb/>erit b h altitudo coni, uel coni portionis a b c: & b K altitu<lb/>

<arrow.to.target n="marg111"></arrow.to.target><lb/>do efg. Quod cum lineæ a c, e f inter &longs;e æquidi&longs;tent, &longs;unt <lb/>enim planorum æquidi&longs;tantium &longs;ectiones: habebit d b ad <lb/>

<arrow.to.target n="marg112"></arrow.to.target><lb/>b g proportionem eandem, quam h b ad b <foreign lang="greek">k.</foreign> quare por<lb/>tio conoidis a b c ad portionem e f g proportionem habet <lb/>compo&longs;itam ex proportione ba&longs;is a c ad ba&longs;im e f; & ex <lb/>

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<s><margin.target id="marg116"></margin.target>20. p<gap/>mi <lb/><expan abbr="comcor&utilde;">comcorum</expan></s>

<s>THEOREMA XXV. PROPOSITIO XXXI.</s>

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<s>SIT fru&longs;tum à portione rectanguli conoidis ab&longs;ci&longs;&longs;um <lb/>a b c d, cuius maior ba&longs;is circulus, uel ellip&longs;is circa diame<lb/>trum b c, minor circa diametrum a d; & axis e f. de&longs;criba<lb/>tur<gap/>atem portio conoidis, à quo illud ab&longs;ci&longs;&longs;um e&longs;t, & pla<lb/>

<arrow.to.target n="fig67"></arrow.to.target><lb/>no per axem ducto &longs;ecetur; ut &longs;uperficiei &longs;ectio &longs;it parabo<lb/>le b g c, cuius diameter, & axis portionis g f: deinde g f diui <lb/>datur in puncto h, ita ut g h &longs;it dupla h f: & rur&longs;us g e in ean <lb/>dem proportionem diuidatur: <expan abbr="&longs;itq;">&longs;itque</expan> g k ip&longs;ius k e dupla. <expan abbr="Iã">Iam</expan> <lb/>ex iis, quæ proxime demon&longs;trauimus, con&longs;tat centrum gra <lb/>uitatis portionis b g c e&longs;&longs;e h punctum: & portionis a g c <lb/>punctum k. &longs;umpto igitur infra h puncto l, ita ut <foreign lang="greek">k</foreign> h ad h l

<figure id="fig67"></figure><lb/>no per axem ducto &longs;ecetur; ut &longs;uperficiei &longs;ectio &longs;it parabo<lb/>le b g c, cuius diameter, & axis portionis g f: deinde g f diui <lb/>datur in puncto h, ita ut g h &longs;it dupla h f: & rur&longs;us g e in ean <lb/>dem proportionem diuidatur: <expan abbr="&longs;itq;">&longs;itque</expan> g k ip&longs;ius k e dupla. <expan abbr="Iã">Iam</expan> <lb/>ex iis, quæ proxime demon&longs;trauimus, con&longs;tat centrum gra <lb/>uitatis portionis b g c e&longs;&longs;e h punctum: & portionis a g c <lb/>punctum k. &longs;umpto igitur infra h puncto l, ita ut <foreign lang="greek">k</foreign> h ad h l

<pb pagenum="47"/>eam proportionem habeat, quam a b c d fru&longs;tum ad por<lb/>tionem a g d; erit punctum l eius fru&longs;ti grauitatis <expan abbr="c&etilde;trum">centrum</expan>: <lb/><expan abbr="habebitq;">habebitque</expan> componendo K l ad l h proportionem eandem, <lb/>

<arrow.to.target n="marg117"></arrow.to.target><lb/>quam portio conoidis b gc ad a g d portionem. <expan abbr="Itaq;">Itaque</expan> quo <lb/>niam quadratum b f ad quadratum a e, hoc e&longs;t quadratum <lb/>b c ad quadratum a d e&longs;t, ut linea fg ad ge: erunt duæ ter<lb/>tiæ quadrati b c ad duas tertias quadrati a d, uth g ad g k: <lb/>& &longs;i à duabus tertiis quadrati b c demptæ fuerint duæ ter<lb/>tiæ quadrati a d: erit <expan abbr="diuid&etilde;do">diuidendo</expan> id, quod relinquitur ad duas <lb/>tertias quadrati a d, ut h k ad k g. Rur&longs;us duæ tertiæ quadra <lb/>ti a d ad duas tertias quadrati b c &longs;unt, ut k g ad g h: & duæ <lb/>tertiæ quadrati b c ad <expan abbr="tertiã">tertiam</expan> <expan abbr="part&etilde;">partem</expan> ip&longs;ius, ut g h ad h f. 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="quadrat&utilde;">quadratum</expan> <lb/>a d, ut K l ad l h. 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. 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. Cum igitur cen<lb/>trum grauitatis fru&longs;ti a b c d &longs;it l, à quo axis e f in eam, <expan abbr="quã">quam</expan> <lb/>diximus, proportionem diuidatur; con&longs;tat <expan abbr="uer&utilde;">uerum</expan> e&longs;&longs;e illud, <lb/>quod demon&longs;trandum propo&longs;uimus.</s>

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<s>FINIS LIBRI DE CENTRO</s>

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