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version 1.17, 2002/08/18 16:45:21

version 1.21, 2002/09/09 22:04:06

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<?xml version="1.0"?>

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info>

<author>Commandino, Federico</author>

<title>Liber de centro gravitatis solidorum</title>

<place>Bologna</place>

<cvs_file>comma_centr_01_la_1565</cvs_file>

<cvs_version></cvs_version>

<cvs_version/>

</info> <text> <front> </front> <body> <chap> <pb/>

<s>FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</s>

<s>LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM.</s><figure></figure>

<s>LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM.</s><figure/>

<s>CVM PRIVILEGIO IN ANNOS X.</s>

<s>CVM PRIVILEGIO IN ANNOS X.</s>

<s>BONONIAE,</s>

<s>BONONIAE,</s>

<s>Ex Officina Alexandri Benacii.</s>

<s>Ex Officina Alexandri Benacii.</s>

<s>ALEXANDRO FARNESIO <lb/>CARDINALI AMPLISSIMO. <lb/>ET OPTIMO.</s>

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<s>harum mearum cogitationum &longs;umma fa<lb/>cta, exi&longs;timaui nemini conuenientius de centro graui <lb/>tatis corporum opus dicari oportere, quàm ALE<lb/>XANDRO FARNESIO grauis&longs;imo, ac prudentis&longs;i<lb/>mo Cardinali, quo in uiro &longs;umma fortuna &longs;emper <expan abbr="c&utilde;">cum</expan> <lb/>&longs;umma uirtute certauit. </s>

<s>quid enim maxime in te ad<lb/>mirati debeant homines, ob&longs;curum e&longs;t; u&longs;um'ne re-<pb/>rum, qui pueritiæ tempus extremum principium ha<lb/>bui&longs;ti, & <expan abbr="imperior&utilde;">imperiorum</expan>, & ad Reges, & Imperatores ho<lb/>norificenti&longs;simarum legationum; an excellentiam <lb/>in omni genere literarum, qui vix <expan abbr="adole&longs;c&etilde;tulus">adole&longs;centulus</expan>, quæ <lb/>homines iam confirmata ætate &longs;ummo &longs;tudio, <expan abbr="diu-turnis&qacute;">diu<lb/>turnisque</expan>; laboribus didicerunt, &longs;cientia, & cognitione <lb/>comprehendi&longs;ti: an con&longs;ilium, & &longs;apientiam in re<lb/>gendis, & <expan abbr="gubernãdis">gubernandis</expan> Ciuitatibus, cuius graui&longs;simæ <lb/>&longs;ententiæ in &longs;ancti&longs;simo Reip. </s>

<s>Chr&longs;tianæ con&longs;ilio di<lb/>ctæ, potius diuina oracula, quàm &longs;ententiæ habitæ <lb/>&longs;unt, & habentur. </s>

<s>prætermitto liberalitatem, & mu<lb/>nificentiam tuam, quam in &longs;tudio&longs;i&longs;simo quoque ho <lb/>ne&longs;tando quotidie magis o&longs;tendis, ne videar auribus <lb/>tuis potius, quàm veritati &longs;eruire. </s>

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<s>c&oelig;<lb/>lum igitur digito attingam, &longs;i po&longs;t graui&longs;simas oc-<pb/>cupationes tuas legendo Federici tui libro aliquid <lb/>impertiri temporis non grauaberis: <expan abbr="cum&qacute;">cumque</expan>; in iis, qui <lb/>tibi &longs;emper addicti erunt, numerare. </s>

<s>Federicus Commandinus.</s><pb pagenum="1"/>

<s>FEDERICI COMMANDINI <lb/>VRBINATIS LIBER DE CENTRO <lb/>GRAVITATIS SOLIDORVM.</s>

<s>FEDERICI COMMANDINI <lb/>VRBINATIS LIBER DE CENTRO <lb/>GRAVITATIS SOLIDORVM.</s>

<s>DIFFINITIONES.</s>

<s>Centrvm grauitatis, Pappus <lb/>Alexandrinus in octauo ma<lb/>thematicarum collectionum <lb/>libro ita diffiniuit.</s>

Line 110

<s>&longs;i enim per tale centrum ducatur planum <lb/>figuram quomodocunque &longs;ecans &longs;emper in par<pb/>tes æqueponderantes ip&longs;am diuidet.</s>

<s>Pri&longs;matis, cylindri, & portionis cylindri axem <lb/>appello rectam lineam, quæ oppo&longs;itorum plano<lb/>rum centra grauitatis coniungit.</s>

<s>Pyramidis, coni, & portionis coni axem dico li <lb/>neam, quæ à uertice ad centrum grauitatis ba&longs;is <lb/>perducitur.</s>

<s>Si pyramis, conus, portio coni, uel conoidis &longs;e<lb/>cetur plano ba&longs;i æquidi&longs;tante, pars, quæ e&longs;t ad ba<lb/>&longs;im, fru&longs;tum pyramidis, coni, portionis coni, uel <lb/>conoidis dicetur; quorum plana æquidi&longs;tantia, <lb/>quæ opponuntur &longs;imilia &longs;unt, & inæqualia: axes <lb/>uero &longs;unt axium figurarum partes, quæ in ip&longs;is <lb/>comprehenduntur.</s>

<s>PETITIONES.</s>

<s>PETITIONES.</s>

<s>Solidarum figurarum fimilium centra grauita<lb/>tis &longs;imiliter &longs;unt po&longs;ita.</s>

<s>Solidis figuris &longs;imilibus, & æqualibus inter &longs;e <lb/>aptatis, centra quoque grauitatis ip&longs;arum inter &longs;e <lb/>aptata crunt.</s>

<s>THEOREMA I. PROPOSITIO I.</s>

<s>THEOREMA I. PROPOSITIO I.</s>

<s>Omnis figuræ rectilineæ in circulo de&longs;criptæ, <lb/>quæ æqualibus lateribus, & angulis contine<lb/><gap/><pb pagenum="2"/>tur, centrum grauitatis e&longs;t idem, quod circuli cen<lb/>trum.</s>

<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. </s>

<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. </s>

<s>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. </s>

<s>Et <lb/><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>Et <lb/><figure id="fig1"/><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><arrow.to.target n="marg7"></arrow.to.target><lb/>gulum a b d æquale erit trian <lb/>gulo c b d: & anguli angulis æ<lb/>quales, qui æqualibus lateri<lb/><arrow.to.target n="marg8"></arrow.to.target><lb/>bus &longs;ubtenduntur. </s>

<s><arrow.to.target n="marg7"/><lb/>gulum a b d æquale erit trian <lb/>gulo c b d: & anguli angulis æ<lb/>quales, qui æqualibus lateri<lb/><arrow.to.target n="marg8"/><lb/>bus &longs;ubtenduntur. </s>

<s>ergo angu <lb/>li ad d <expan abbr="utriq;">utrique</expan> recti &longs;unt. </s>

<s>quòd <lb/>cum linea b d &longs;ecet a e bifa<lb/><arrow.to.target n="marg9"></arrow.to.target><lb/>riam, & ad angulos rectos; in <lb/>ip&longs;a b d e&longs;t centrum circuli. </s>

<s>quòd <lb/>cum linea b d &longs;ecet a e bifa<lb/><arrow.to.target n="marg9"/><lb/>riam, & ad angulos rectos; in <lb/>ip&longs;a b d e&longs;t centrum circuli. </s>

<s><lb/>quare in eadem b d linea erit <lb/>centrum grauitatis trianguli, & circuli centrum. </s>

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<s>trianguli igitur a b c centrum gra<lb/>uitatis e&longs;t idem, quod circuli centrum.</s>

<s>Sit quadratum a b c d in cir<lb/>culo de&longs;criptum: & ducantur <lb/>a c, b d, quæ conueniant in e. </s>

<s>er<lb/>go punctum e e&longs;t centrum gra<lb/>uitatis quadrati, ex decima eiu&longs; <lb/>dem libri Archimedis. </s>

<s>er<lb/>go punctum e e&longs;t centrum gra<lb/>uitatis quadrati, ex decima eiu&longs; <lb/>dem libri Archimedis. </s>

<s>Sed cum <lb/>omnes anguli ad a b c d recti <lb/><arrow.to.target n="marg10"></arrow.to.target><lb/>&longs;int; erit a b c femicirculus: <lb/><expan abbr="item&qacute;">itemque</expan>; b c d: & propterea li<lb/>neæ a c, b d diametri circuli: <pb/>quæ quidem in centro conucniunt. </s>

<s>Sed cum <lb/>omnes anguli ad a b c d recti <lb/><arrow.to.target n="marg10"/><lb/>&longs;int; erit a b c femicirculus: <lb/><expan abbr="item&qacute;">itemque</expan>; b c d: & propterea li<lb/>neæ a c, b d diametri circuli: <pb/>quæ quidem in centro conucniunt. </s>

<s>idem igitur e&longs;t centrum <lb/>grauitatis quadrati, & circuli centrum.</s>

<s>Sit pentagonum æquilaterum, & æquiangulum in circu<lb/><figure id="fig2"></figure><lb/>lo de&longs;criptum a b c d e. </s>

<s>Sit pentagonum æquilaterum, & æquiangulum in circu<lb/><figure id="fig2"/><lb/>lo de&longs;criptum a b c d e. </s>

<s>& 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. </s>

<s>Eodem <lb/>modo, quo &longs;upra demon&longs;tra<lb/>bimus angulum b c f æqualem <lb/>e&longs;&longs;e. </s>

<s>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. </s>

<s>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. </s>

<s>Quoniam igi<lb/>tur latera c b, b a, & c d, d e æqualia &longs;unt; & æquales anguli <lb/><arrow.to.target n="marg11"/><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. </s>

<s>ergo & reliquus a c h, reliquo e c h. </s>

<s>e&longs;t au<lb/>tem c h utrique triangulo a c h, e c h communis. </s>

<s>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. </s>

<s>ergo lineæ a e, b d inter &longs;e &longs;e æquidi&longs;tant. </s><lb/>

<s>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. </s>

<s>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"/><lb/>in linea fh, ex ultima eiu&longs;dem libri Archimedis. </s>

<s>Sed trian<lb/>guli b c d centrum grauitatis e&longs;t in linea c f. </s>

Line 210

<s>Sequitur ergo, ut punctum, in quo lineæ c g, e l con<lb/>ueniunt, idem &longs;it centrum circuli, & centrum grauitatis <lb/>pentagoni.</s>

<s><margin.target id="marg11"></margin.target>4. Primi.</s>

<s><margin.target id="marg11"/>4. Primi.</s>

<s><margin.target id="marg12"></margin.target>28. primi.</s>

<s><margin.target id="marg12"/>28. primi.</s>

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

<s><margin.target id="marg13"/>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. </s>

<s>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/></s>

<s>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/><figure id="fig3"></figure><lb/><arrow.to.target n="marg14"></arrow.to.target><lb/>rentiæ circuli. </s>

<s>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/><figure id="fig3"/><lb/><arrow.to.target n="marg14"/><lb/>rentiæ circuli. </s>

<s>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. </s>

Line 230

<s>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/><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. </s>

<s>Sit heptagonum a b c d e f g æquilaterum atque æquian<lb/><figure id="fig4"/><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. </s>

<s>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. </s>

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<s>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/><arrow.to.target n="marg15"></arrow.to.target></s>

<s><margin.target id="marg15"></margin.target><foreign lang="greek"><gap/>wrímw</foreign><gap/></s>

<s><margin.target id="marg15"/><foreign lang="greek"><gap/>wrímw</foreign><gap/></s>

<s>Figuram in circulo plane de&longs;criptam appella<lb/>mus, cuiu&longs;modi e&longs;t ea, quæ in duodecimo elemen <lb/>torum libro, propo&longs;itione &longs;ecunda de&longs;cribitur. <lb/></s>

<s>ex æqualibus enim lateribus, & angulis con&longs;tare <lb/>per&longs;picuum e&longs;t.</s>

<s>THEOREMA II, PROPOSITIO II.</s>

<s>THEOREMA II, PROPOSITIO II.</s>

<s>Omnis figuræ rectilineæ in ellip&longs;i plane de&longs;cri<lb/>ptæ centrum grauitatis e&longs;t idem, quod ellip&longs;is <lb/>centrum.</s>

Line 260

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>ba&longs;is a b ba&longs;i a d; & reliqui anguli reliquis an<lb/>gulis æquales erunt. </s>

<s>eadem quoqueratione o&longs;tendetur b c <lb/><figure id="fig5"></figure><lb/>æqualis c d; & a b ip&longs;i <lb/>b c. </s>

<s>eadem quoqueratione o&longs;tendetur b c <lb/><figure id="fig5"/><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. </s>

<s>quare omnes a b, <lb/>b c, c d, d a &longs;unt æqua<lb/>les. </s>

<s>& 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. </s>

Line 280

<s>Sunt <expan abbr="aut&etilde;">autem</expan> a d, <lb/>b c inter &longs;e &longs;e æquales, & æquidi&longs;tantes. </s>

<s>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. </s>

<s>quare & earum di<lb/><arrow.to.target n="marg17"/><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. </s>

<s><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. </s>

<s>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. </s>

Line 292

<s>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. </s>

<s>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. </s>

<s>quòd cum etram recti <lb/><arrow.to.target n="marg18"/><lb/>&longs;int, qui ad k; æquidi&longs;tabunt lineæ l o, m n axi b d. </s>

<s>& ita <lb/>denion&longs;trabuntur l m, o n ip&longs;i a c æquidi&longs;tare. </s>

<s>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. </s>

<s>Simili<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. </s>

<s>Simili<lb/><figure id="fig6"/><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. </s>

<s>Itaque dico <lb/>harum figurarum in el<lb/>lip&longs;i de&longs;criptarum cen<lb/>trum grauitatis e&longs;&longs;e <expan abbr="p&utilde;-ctum">pun<lb/>ctum</expan> k, idem quod & el <lb/>lip&longs;is centrum. </s>

<s>quadri<lb/>lateri enim a b c d cen<lb/>trum e&longs;t k, ex decima e<lb/>iu&longs;dem libri Archime<lb/>dis, quippe <expan abbr="c&utilde;">cum</expan> in eo om <lb/>nes diametri <expan abbr="cõueniãt">conueniant</expan>. </s><lb/>

<s>Sed in figura a l b m c n <lb/><arrow.to.target n="marg19"></arrow.to.target><lb/>do, quoniam trianguli <lb/>a l b centrum grauitatis <lb/><arrow.to.target n="marg20"></arrow.to.target><lb/>e&longs;t in linea l e: <expan abbr="trapezij&qacute;">trapezijque</expan>; a b m o centrum in linea e k: trape <lb/>zij o m c d in k g: & trianguli c n d in ip&longs;a g n: erit magnitu <lb/>dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen <lb/>trum grauitatis in linea l n: & o b eandem cau&longs;&longs;am in linea <lb/>o m. </s>

<s>Sed in figura a l b m c n <lb/><arrow.to.target n="marg19"/><lb/>do, quoniam trianguli <lb/>a l b centrum grauitatis <lb/><arrow.to.target n="marg20"/><lb/>e&longs;t in linea l e: <expan abbr="trapezij&qacute;">trapezijque</expan>; a b m o centrum in linea e k: trape <lb/>zij o m c d in k g: & trianguli c n d in ip&longs;a g n: erit magnitu <lb/>dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen <lb/>trum grauitatis in linea l n: & o b eandem cau&longs;&longs;am in linea <lb/>o m. </s>

<s>e&longs;t enim trianguli a o d centrum in linea o h: trapezij <lb/>a l n d in h k: trapezij l b c n in k f: & trianguli b m c in fm. </s><lb/>

<s>cum ergo figuræ a l b m c n d o centrum grauitatis &longs;it in li<lb/>nea l n, & in linea o m; erit centrum ip&longs;ius punctum k, in <pb pagenum="5"/>quo &longs;cilicet l n, o m conueniunt. </s>

<s>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. </s>

<s>quare magnitudinis ex his compo&longs;itæ <expan abbr="centr&utilde;">centrum</expan> <lb/>in linea a c con&longs;i&longs;tit. </s>

<s>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. </s>

<s>totius ergo magnitudinis centrum <lb/>e&longs;t in linea b d. </s>

<s>totius ergo magnitudinis centrum <lb/>e&longs;t in linea b d. </s>

<s>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>

<s><margin.target id="marg18"></margin.target>28. primi.</s>

<s><margin.target id="marg18"/>28. primi.</s>

<s><margin.target id="marg19"></margin.target>13. Archi <lb/>medis.</s>

<s><margin.target id="marg19"/>13. Archi <lb/>medis.</s>

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

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

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

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

<s>Cuiuslibet portio<lb/>nis circuli, & ellip&longs;is, <lb/>quæ dimidia non &longs;it <lb/>maior, centrum graui <lb/>tatis in portionis dia<lb/>metro con&longs;i&longs;tit.</s><figure></figure>

<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. </s>

<s>Etita demon&longs;trari po <lb/><figure id="fig7"></figure><pb/>te&longs;t in portione, quæ recta linea & obtu&longs;ianguli coni &longs;e<lb/>ctione, &longs;eu hyperbola continetur.</s>

<s>Etita demon&longs;trari po <lb/><figure id="fig7"/><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>

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

<s>IN circulo & ellip&longs;i idem e&longs;t figuræ & graui<lb/>tatis centrum.</s>

Line 348

<s>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. </s>

<s>Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con<lb/><figure id="fig8"></figure><lb/>ftat; at in ellip&longs;i nos demon&longs;tra<lb/>uinms in commentariis in quin<lb/>tam propo&longs;itionem Archimedis <lb/>de conoidibus, & &longs;phæroidibus. </s><lb/>

<s>Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con<lb/><figure id="fig8"/><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. </s><lb/>

<s>erit igitur a centrum grauitatis <lb/>ip&longs;ius figuræ, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>dimus. </s>

<s>Itaque quoniam circulus <lb/>a ad circulum d, uel ellip&longs;is a ad <lb/>ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam linea c a ad a b: <lb/>portiones uero &longs;unt minores cir <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>culo uel ellip&longs;i d: habebit circu<lb/>lus, uel ellip&longs;is ad portiones ma<lb/>iorem proportionem, quàm c a <lb/><arrow.to.target n="marg22"></arrow.to.target><lb/>ad a b: & diuidendo figura recti<lb/>linea e f g h k l m n ad portiones <pb pagenum="6"/><figure id="fig9"></figure><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. </s>

<s>Itaque quoniam circulus <lb/>a ad circulum d, uel ellip&longs;is a ad <lb/>ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam linea c a ad a b: <lb/>portiones uero &longs;unt minores cir <lb/><arrow.to.target n="marg21"/><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"/><lb/>ad a b: & diuidendo figura recti<lb/>linea e f g h k l m n ad portiones <pb pagenum="6"/><figure id="fig9"/><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. </s>

<s>fiat o b ad b a, <lb/>ut figura rectilinea ad portio<lb/>nes. </s>

<s>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. </s>

<s>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"/><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. </s>

<s>quod quidem fieri nullo mo <lb/>do po&longs;&longs;e per&longs;picuum e&longs;t. </s>

<s>&longs;equi<lb/>tur ergo, ut circuli & ellip&longs;is cen <lb/>trum grauitatis &longs;it punctum a, <lb/>idem quod figuræ centrum.</s>

<s><margin.target id="marg21"></margin.target>8. quinti</s>

<s><margin.target id="marg21"/>8. quinti</s>

<s><margin.target id="marg22"></margin.target>19. quinti <lb/>apud <expan abbr="Cã">Cam</expan> <lb/>panum.</s>

<s><margin.target id="marg22"/>19. quinti <lb/>apud <expan abbr="Cã">Cam</expan> <lb/>panum.</s>

<s>ALITER.</s>

<s>ALITER.</s>

<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. </s>

<s>erunt a d c, <lb/>a b c circuli, uel ellip&longs;is dimidiæ portiones. </s>

<s>Itaque quo<lb/><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. </s><lb/>

<s>Itaque quo<lb/><figure id="fig10"/><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. </s><lb/>

<s>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. </s>

Line 382

<s>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. </s>

<s>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. </s>

<s>ergo circuli, uel ellip&longs;is centrum <lb/>grauitatis e&longs;t idem, quod figuræ centrum. </s>

<s>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><figure></figure>

<s>Sit enim maior portio a b c, cu<emph type="italics"/>i<emph.end type="italics"/>us diameter b d, & com<lb/><gap/><pb pagenum="7"/>metrum habens c d. </s>

<s>Sit enim maior portio a b c, cu<emph type="italics"/>i<emph.end type="italics"/>us diameter b d, & com<lb/><gap/><pb pagenum="7"/>metrum habens c d. </s>

<s>Quoniam igitur circuli uel ellip&longs;is <lb/>a e c b grauitatis centrum e&longs;t in diametro b e, & portio<lb/>nis a e c centrum in linea e d: reliquæ portionis, uidelicet <lb/>a b c centrum grauitatis in ip&longs;a b d con&longs;i&longs;tat nece&longs;&longs;e e&longs;t, ex <lb/>octaua propo&longs;itione eiu&longs;dem.</s>

<s>THEOREMA V. PROPOSITIO V.</s>

<s>THEOREMA V. PROPOSITIO V.</s>

<s>SI pri&longs;ma &longs;ecetur plano oppo&longs;itis planis æqui <lb/>di&longs;tante, &longs;ectio erit figura æqualis & &longs;imilis ei, <lb/>quæ e&longs;t oppo&longs;itorum planorum, centrum graui <lb/>tatis in axe habens.</s>

Line 402

<s>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. </s>

<s>Quo<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. </s>

<s>Quo<lb/><figure id="fig11"/><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"/><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. </s>

<s>Sed æquidi&longs;tant <lb/>a d, b e; cum a e &longs;it para <lb/>lelogrammum, ex pri&longs;<lb/>matis diffinitione. </s>

<s>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. </s>

<s>ergo <lb/>& al <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>erit; & propterea linea <lb/><arrow.to.target n="marg25"/><lb/>k l, ip&longs;i a b æqualis. </s>

<s>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"/><lb/>gulus l m k, angulo b c a, & m k lip&longs;i c a b æqualis prob abi <lb/>tur. </s>

<s>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. </s>

<s>triangulum ergo k l m e&longs;t æquale, & &longs;imile triang ulo <lb/>a b c. quare & triangulo d e f. </s>

<s>triangulum ergo k l m e&longs;t æquale, & &longs;imile triang ulo <lb/>a b c. </s>

<s>quare & triangulo d e f. </s>

<s>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. </s>

Line 420

<s>Sed & æquidi&longs;tant a b, k l, d e. </s>

<s>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. </s>

<s>an<lb/><arrow.to.target n="marg27"/><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. </s>

<s>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. </s>

<s>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"/><lb/>lia erunt. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>Itaque &longs;i triangula, a b c, d e f æqualia & <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. </s>

<s>Itaque &longs;i triangula, a b c, d e f æqualia & <lb/><figure id="fig12"/><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"/><lb/>c g o. </s>

<s>Sed & <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis h in g <expan abbr="centr&utilde;">centrum</expan> ca<lb/>det. </s>

Line 444

<s>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>

<s><margin.target id="marg28"></margin.target>4. &longs;exti</s>

<s><margin.target id="marg28"/>4. &longs;exti</s>

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

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

<s>COROLLARIVM.</s>

<s>COROLLARIVM.</s>

<s>Exiam demon&longs;tratis per&longs;picue apparet, cuius <lb/>Iibet pri&longs;matis axem, parallelogrammorum lat eri <lb/>bus, quæ ab oppo&longs;itis planis <expan abbr="duc&utilde;tur">ducuntur</expan> æquidi&longs;tare.</s>

<s>THE OREMA VI. PROPOSITIO VI.</s>

<s>THE OREMA VI. PROPOSITIO VI.</s>

<s>Cuiuslibet pri&longs;matis centrum grauitatis e&longs;t in <lb/>plano, quod oppo&longs;itis planis æquidi&longs;tans, reli<lb/>quorum planorum latera bifariam diuidit.</s>

<s>Sit pri&longs;ma, in quo plana, quæ opponuntur &longs;int trian<lb/>gula a c e, b d f: & parallelogrammorum latera a b, c d, <lb/>e f bifariam <expan abbr="diuidãtur">diuidantur</expan> in punctis g h k: per diui&longs;iones au<lb/><arrow.to.target n="marg30"></arrow.to.target><lb/>tem planum ducatur; cuius &longs;ectio figura g h K. </s>

<s>Sit pri&longs;ma, in quo plana, quæ opponuntur &longs;int trian<lb/>gula a c e, b d f: & parallelogrammorum latera a b, c d, <lb/>e f bifariam <expan abbr="diuidãtur">diuidantur</expan> in punctis g h k: per diui&longs;iones au<lb/><arrow.to.target n="marg30"/><lb/>tem planum ducatur; cuius &longs;ectio figura g h K. </s>

<s>erit linea <lb/>g h æquidi&longs;tans lineis a c, b d & h k ip&longs;is c e, d f. </s>

<s>quare ex <lb/>decimaquinta undecimi elementorum, planum illud pla <lb/>nis a c e, b d f æquidi&longs;tabit, & faciet &longs;ectionem figu<lb/><arrow.to.target n="marg31"></arrow.to.target><lb/>ram ip&longs;is æqualem, & &longs;imilem, ut proxime demon&longs;tra<lb/>uimus. </s>

<s>quare ex <lb/>decimaquinta undecimi elementorum, planum illud pla <lb/>nis a c e, b d f æquidi&longs;tabit, & faciet &longs;ectionem figu<lb/><arrow.to.target n="marg31"/><lb/>ram ip&longs;is æqualem, & &longs;imilem, ut proxime demon&longs;tra<lb/>uimus. </s>

<s>Dico centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>g h k. </s>

<s>Si enim fieri pote&longs;t, &longs;it eius centrum l: & ducatur <lb/>l m u&longs;que ad planum g h k, quæ ip&longs;i a b æquidi&longs;tet. </s><pb/>

<s><arrow.to.target n="marg32"></arrow.to.target>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. </s><lb/>

<s><arrow.to.target n="marg32"/>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. </s><lb/>

<s>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. </s>

<s>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"/><lb/>nis æquidi&longs;tantia ducantur. </s>

<s>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/><figure id="fig13"></figure><lb/>habebunt. </s>

<s>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. </s><lb/>

Line 486

<s>at qui non ita res ha<pb pagenum="9"/>bet, &longs;i quidem l extra medias magnitudines po&longs;itum e&longs;t. </s><lb/>

<s>Con&longs;tat igitur centrum granitatis pri&longs;matis e&longs;&longs;e in plano <lb/><figure id="fig14"></figure><lb/>g h k, quod nos demon&longs;trandum propo&longs;uimus. </s>

<s>Con&longs;tat igitur centrum granitatis pri&longs;matis e&longs;&longs;e in plano <lb/><figure id="fig14"/><lb/>g h k, quod nos demon&longs;trandum propo&longs;uimus. </s>

<s>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>

<s><margin.target id="marg32"></margin.target>1. decimi</s>

<s><margin.target id="marg32"/>1. decimi</s>

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

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

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

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

<s>Cuiuslibet cylindri, & cuiuslibet cylindri por <lb/>tionis centrum grauitatis e&longs;t in plano, quod ba&longs;i<lb/>bus æquidi&longs;tans, parallelogrammi per axem late<lb/>ra bifariam &longs;ecat.</s><pb/>

<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/><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. </s>

<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/><figure id="fig15"/><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. </s>

<s>Dico centrum grauita<lb/>tis cylindri, uel cylindri por<lb/>tionis e&longs;&longs;e in plano e f. </s>

Line 516

<s>reli<lb/>qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s>

<s>THEOREMA VIII. PROPOSITIO VIII.</s>

<s>THEOREMA VIII. PROPOSITIO VIII.</s>

<s>Cuiuslibet pri&longs;matis, & cuiuslibet cylindri, uel <lb/>cylindri portionis grauitatis centrum in medio <lb/>ip&longs;ius axis con&longs;i&longs;tit.</s>

Line 526

<s>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>. </s><lb/>

<s>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/><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. </s>

<s>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/><figure id="fig16"/><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. </s>

<s>Dico <foreign lang="greek">w</foreign> centrum e&longs;&longs;e <lb/>grauitatis ip&longs;ius &longs;olidi. </s>

<s>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. </s>

<s>e&longs;t <lb/><arrow.to.target n="marg34"/><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. </s>

<s>ergo in communi ip&longs;orum &longs;ectione: ui<lb/>delicet in linea y z. </s>

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

<s>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. </s>

<s>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. </s>

<s>diuidet punctum m li<lb/>neam g h bifariam; & ita n diuidet lineam d e: quoniam <lb/><arrow.to.target n="marg35"/><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. </s>

<s>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. </s>

Line 554

<s>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. </s>

<s>puncta uero, in quibus hæ trian<lb/>gulorum latera &longs;ecant, coniungantur ductis lineis r s, t u, <lb/><figure id="fig17"></figure><lb/>x y; quæ ba&longs;i g h æquidi&longs;tabunt. </s>

<s>puncta uero, in quibus hæ trian<lb/>gulorum latera &longs;ecant, coniungantur ductis lineis r s, t u, <lb/><figure id="fig17"/><lb/>x y; quæ ba&longs;i g h æquidi&longs;tabunt. </s>

<s>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. </s>

<s>quare ut k r ad r g, ita <foreign lang="greek">k</foreign> s ad s h. </s>

<s>æquidi&longs;tant igitur <lb/><arrow.to.target n="marg37"></arrow.to.target><lb/>inter &longs;e &longs;e r s, g h. </s>

<s>æquidi&longs;tant igitur <lb/><arrow.to.target n="marg37"/><lb/>inter &longs;e &longs;e r s, g h. </s>

<s>eadem quoque ratione demon&longs;trabimus <pb/>t u, x y ip&longs;i g h æquidi&longs;tare. </s>

<s>Et quoniam triangula, quæ <lb/>fiunt à lineis K y, y u, u s, s h æqualiz &longs;unt inter &longs;e, & &longs;imilia <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>triangulo K m h: habebit triangulum K m h ad <expan abbr="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. </s><lb/>

<s>Et quoniam triangula, quæ <lb/>fiunt à lineis K y, y u, u s, s h æqualiz &longs;unt inter &longs;e, & &longs;imilia <lb/><arrow.to.target n="marg38"/><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. </s><lb/>

<s>&longs;ed K h po&longs;ita e&longs;t quadrupla ip&longs;ius k y. </s>

<s>ergo triangulum <lb/>k m h ad triangulum K <foreign lang="greek">d</foreign> y <expan abbr="eãdem">eandem</expan> proportionem habcbit, <lb/>quam &longs;exdecim ad <expan abbr="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/><figure id="fig18"></figure><lb/>bitur trian<lb/>gulum k m g <lb/>ad quatuor <lb/><expan abbr="triãgula">triangula</expan> K <foreign lang="greek">d</foreign><lb/>x, x <foreign lang="greek">g</foreign> t, t <foreign lang="greek">b</foreign> r, <lb/><arrow.to.target n="marg39"></arrow.to.target><lb/>r z g. </s>

<s>ergo triangulum <lb/>k m h ad triangulum K <foreign lang="greek">d</foreign> y <expan abbr="eãdem">eandem</expan> proportionem habcbit, <lb/>quam &longs;exdecim ad <expan abbr="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/><figure id="fig18"/><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"/><lb/>r z g. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>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"/><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. </s>

<s>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. </s>

<s>&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. </s>

<s>&longs;unt <lb/><arrow.to.target n="marg41"></arrow.to.target><lb/>autem &longs;olida parallelepipeda pri&longs;matum triangulares ba<lb/><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. </s>

<s>&longs;unt <lb/><arrow.to.target n="marg41"/><lb/>autem &longs;olida parallelepipeda pri&longs;matum triangulares ba<lb/><arrow.to.target n="marg42"/><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. </s>

<s>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/>. </s>

<s>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"/><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/>. </s>

<s>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. </s>

Line 596

<s>quare relinquitur, ut <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis pri&longs;matis &longs;it in linea K m. </s>

<s>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/><figure id="fig19"></figure><lb/>po&longs;itum &longs;i t. </s>

<s>Similiter demon&longs;trabimus centrum grauita<lb/>tis pri&longs;m atis in ip&longs;a g u ine&longs;&longs;e. </s>

Line 604

<s>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>

<s><margin.target id="marg35"></margin.target>5.huius</s>

<s><margin.target id="marg35"/>5.huius</s>

<s><margin.target id="marg36"></margin.target>2. &longs;e<gap/><lb/>. </s>

<s><margin.target id="marg36"/>2. &longs;e<gap/><lb/>. </s>

<s><margin.target id="marg37"></margin.target>2. &longs;e<gap/></s>

<s><margin.target id="marg37"/>2. &longs;e<gap/></s>

<s>&longs;exti</s>

<s><margin.target id="marg39"></margin.target>uel 127 <lb/>utnti.</s>

<s><margin.target id="marg39"/>uel 127 <lb/>utnti.</s>

<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. </s>

<s>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. </s>

<s>Sint <expan abbr="aut&etilde;">autem</expan> <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. </s>

<s>Sint <expan abbr="aut&etilde;">autem</expan> <lb/><figure id="fig20"/><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. </s>

<s>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. </s>

<s>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. </s>

<s>denique iungatur <lb/>u x, quæ &longs;ecet lineam &longs; t in y. </s>

<s>&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. </s><lb/>

<s>&longs;e<lb/>cabit enim cum &longs;int in eodem <lb/><arrow.to.target n="marg44"/><lb/>plano: <expan abbr="atq;">atque</expan> erit y grauitatis centrum quadrilateri K l m n. </s><lb/>

<s>Dico idem punctum y centrum quoque gra uitatis e&longs;&longs;e to<lb/>tius pri&longs;matis. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s><lb/>

<s>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. </s><lb/>

<s>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. </s>

<s>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>

<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/><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. </s>

<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/><figure id="fig21"/><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. </s>

<s>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. </s>

Line 670

<s>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/><figure id="fig22"></figure><lb/>g h, per ea ducatur planum ba&longs;i æquidi&longs;tans. </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/><figure id="fig22"/><lb/>g h, per ea ducatur planum ba&longs;i æquidi&longs;tans. </s>

<s>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. </s>

<s>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"/><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. </s>

<s>Dico punctum K cylindri quoque, uel cy <lb/>lindri portionis grauitatis centrum e&longs;&longs;e. </s>

Line 682

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>fiat n <foreign lang="greek">k</foreign> ad K l, ut <lb/>circulus uel ellip&longs;is g h ad ip&longs;as portiones. </s>

Line 694

<s>quæ omnia <expan abbr="demon&longs;trãda">demon&longs;tranda</expan> propo&longs;uimus.</s>

<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<pb pagenum="15"/>here proportionem, quam &longs;pacium g h ad <expan abbr="dictã">dictam</expan> <lb/>figuram, hoc modo demon&longs;trabimus.</s>

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

<s>& ab x con&longs;tituatur conus, uel <lb/><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. </s>

<s>& ab x con&longs;tituatur conus, uel <lb/><figure id="fig23"/><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. </s>

<s>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. </s>

Line 714

<s>erit pyramis x adhuc pyramide y ma <lb/>ior. </s>

<s>& 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. </s>

<s>& quoniam piramides æque altæ inter &longs;e &longs;unt, &longs;icuti ba <lb/><arrow.to.target n="marg46"/><lb/>&longs;es; pyramis x ad piramidem y eandem proportionem ha<lb/>bet, quàm figura rectilinea x ad figuram y. </s>

<s>Sed figura recti <pb/><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. </s>

<s>Sed figura recti <pb/><figure id="fig24"/><lb/>linea x cum &longs;it minor circulo, uel cllip&longs;i, e&longs;t etiam minor fi<lb/>gura rectilinca y. </s>

<s>ergo pyramis x pyramide y minor erit. </s><lb/>

Line 738

<s>ergo ip&longs;i nece&longs;&longs;ario e&longs;t æqualis. </s><lb/>

<s>Itaque quoniam ut conus ad conum, uel coni portio ad co <pb pagenum="16"/><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. </s>

<s>Itaque quoniam ut conus ad conum, uel coni portio ad co <pb pagenum="16"/><figure id="fig25"/><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. </s>

<s><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. </s>

<s>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. </s><lb/>

<s>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"/><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. </s><lb/>

<s>quod demon&longs;trandum fuerat.</s>

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

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

<s>Si pyramis &longs;ecetur plano ba&longs;i æquidi&longs;tante; &longs;e<lb/>ctio erit figura &longs;imilis ei, quæ e&longs;t ba&longs;is, centrum <lb/>grauitatis in axe habens.</s><pb/>

Line 760

<s>& &longs;imiliter angulus ad h angulo ad c: <expan abbr="angulusq;">angulusque</expan> ad f ci, <lb/>qui ad a e&longs;t æqualis. </s>

<s>triangulum igitur f g h &longs;imile e&longs;t tri<lb/>angulo a b c. </s>

<s>triangulum igitur f g h &longs;imile e&longs;t tri<lb/>angulo a b c. </s>

<s>Atuero punctum <foreign lang="greek">k</foreign> centrum e&longs;&longs;e grauita<lb/>tis trianguli f g h hoc modo o&longs;tendemus. </s>

<s>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/><figure id="fig26"></figure><lb/>e&longs;t æqualis. </s>

<s>Eadem ratione <lb/>anguli ad g angulis ad b: & <lb/>anguli ad h iis, qui ad c æ<lb/>quales erunt. </s>

Line 776

<s>Non aliter <lb/>in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra<lb/><gap/></s><pb pagenum="17"/>

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

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

<s>DATA qualibet pyramide, fieri pote&longs;t, ut fi<lb/>gura &longs;olida in ip&longs;a in &longs;cribatur, & altera <expan abbr="circ&utilde;&longs;cri-batur">circun&longs;cri<lb/>batur</expan> ex pri&longs;matibus æqualem altitudinem <expan abbr="ha-b&etilde;tibus">ha<lb/>bentibus</expan>, ita ut circum&longs;cripta in&longs;criptam excedat <lb/>magnitudine, quæ minor &longs;it <expan abbr="quac&utilde;que">quacunque</expan> &longs;olida ma <lb/>gnitudine propo&longs;ita.</s><figure></figure>

<s>Sit pyramis, cuius ba&longs;is <lb/><expan abbr="triangul&utilde;">triangulum</expan> a b c; axis d e. </s><lb/>

Line 788

<s>diuidatur d e in <lb/>partes æquales ip&longs;i e f in <lb/>punctis g h <foreign lang="greek">k</foreign> l m n: & per <lb/>diui&longs;iones plana <expan abbr="ducãtur">ducantur</expan>: <lb/>quæ ba&longs;ibus æquidi&longs;tent, <lb/>erunt &longs;ectiones, triangula <lb/>ip&longs;i a b c &longs;imilia, ut proxi<lb/>me o&longs;tendimus. </s>

<s>ab uno <lb/>quoque <expan abbr="aut&etilde;">autem</expan> horum trian <lb/>gulorum duo pri&longs;mata <expan abbr="cõ">com</expan> <lb/>&longs;truantur; unum quidem <lb/>ad partes e; alterum ad <pb/>partes d. </s>

<s>in pyramide igitur in&longs;cripta erit quædam figura, <lb/>ex pri&longs;matibus æqualem altitudinem habentibus <expan abbr="cõ&longs;tans">con&longs;tans</expan>, <lb/>ad partes e: & altera circum&longs;cripta ad partes d. </s>

<s>Sed unum<lb/>quodque eorum pri&longs;matum, quæ in figura in&longs;cripta conti<lb/>nentur, æquale e&longs;t pri&longs;mati, quod ab eodem fit triangulo in <lb/>figura circumícripta: nam pri&longs;ma p q pri&longs;mati p o e&longs;t æ<lb/>quale; pri&longs;ma s t æquale pri&longs;mati s r; pri&longs;ma x y pri&longs;mati <lb/>x u; pri&longs;ma <foreign lang="greek"><gap/> q</foreign> pri&longs;mati <foreign lang="greek"><gap/></foreign> z; pri&longs;ma <foreign lang="greek">m <gap/></foreign> pri&longs;mati <foreign lang="greek">m l;</foreign> pri&longs;<lb/>ma <foreign lang="greek">r s</foreign> pri&longs;mati <foreign lang="greek">r p;</foreign> & pri&longs;ma <foreign lang="greek">f x</foreign> pri&longs;mati <foreign lang="greek">f t</foreign> æquale. </s>

Line 800

<s><expan abbr="Ead&etilde;">Eadem</expan> <lb/>ratione in&longs;cribetur, & circum&longs;cribetur &longs;olida figura in py<lb/>ramide, quæ quadrilateram, uel <expan abbr="plurilaterã">plurilateram</expan> ba&longs;im habeat.</s>

<s>PROBLEMA II. PROPOSITIO XI.</s>

<s>PROBLEMA II. PROPOSITIO XI.</s>

<s>DATO cono, fieri pote&longs;t, ut figura &longs;olida in<lb/>&longs;cribatur, & altera circum&longs;cribatur ex cylindris <lb/>æqualem habentibus altitudinem, ita ut circum<lb/>&longs;cripta &longs;uperet in&longs;criptam, magnitudine, quæ &longs;o<lb/>lida magnitudine propo&longs;ita &longs;it minor.</s>

Line 814

<s>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. </s>

<s>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/><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>. </s>

<s>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/><figure id="fig27"/><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>. </s>

<s>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. </s><lb/>

Line 822

<s>atque hic e&longs;t mi<lb/>nor &longs;olida magnitudine propo&longs;ita.</s>

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

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

<s>DATA coni portione, pote&longs;t &longs;olida quædam <lb/>figura in&longs;cribi, & altera circum&longs;cribi ex cylindri <lb/>portionibus æqualem altitudinem habentibus; <lb/>ita ut circum&longs;cripta in&longs;criptam exuperet, magni <lb/>tudine, quæ minor fit &longs;olida magnitudine pro<lb/>po&longs;ita.</s><pb/>

<s>Figuram ciu&longs;modi, & in&longs;cribemus, & <expan abbr="circ&utilde;&longs;cribemus">circun&longs;cribemus</expan>, ita <lb/>ut in cono dictum e&longs;t.</s><figure></figure>

<s>Figuram ciu&longs;modi, & in&longs;cribemus, & <expan abbr="circ&utilde;&longs;cribemus">circun&longs;cribemus</expan>, ita <lb/>ut in cono dictum e&longs;t.</s><figure/>

<s>PROBLEMA IIII. PROPOSITIO XIII.</s>

<s>PROBLEMA IIII. PROPOSITIO XIII.</s>

<s>DATA &longs;phæræ portione, quæ dimidia &longs;phæ<lb/>ra maior non &longs;it, pote&longs;t &longs;olida quædam portio in<lb/>&longs;cribi & altera circum&longs;cribi ex cylindris æqualem <lb/>altitudinem habentibus, ita ut circum&longs;cripta in<lb/>&longs;criptam excedat magnitudine, quæ &longs;olida ma<lb/>gnitudine propo&longs;ita &longs;it minor.</s>

<s>HOC etiam codem pror&longs;us modo &longs;iet: atque ut ab <lb/>Archimedc traditum e&longs;t in conoidum, & &longs;phæroidum por <lb/>tionibus, propo&longs;itione ulge&longs;imaprima libri de conoidi<lb/>bus, & &longs;phæroidibus.</s><pb pagenum="19"/><figure></figure>

<s>THEOREMA X. PROPOSITIO XIIII.</s>

<s>THEOREMA X. PROPOSITIO XIIII.</s>

<s>Cuiuslibet pyramidis, & cuiuslibet coni, uel <lb/>coni portionis, centrum grauitatis in axe <expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>.</s>

Line 850

<s>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. </s><lb/>

<s>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. </s>

<s>ergo figu<lb/><figure id="fig28"></figure><lb/>ræ in&longs;criptæ centrum e&longs;t in linea p d. </s>

<s>ergo figu<lb/><figure id="fig28"/><lb/>ræ in&longs;criptæ centrum e&longs;t in linea p d. </s>

<s>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. </s>

Line 866

<s><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>

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

<s>Cuiuslibet portionis &longs;phæræ uel &longs;phæroidis, <lb/>quæ dimidia maior non &longs;it:item&qacute;; cuiuslibet por <lb/>tionis conoidis, uel ab&longs;ci&longs;&longs;æ plano ad axem recto, <lb/>uel non recto, centrum grauitatis in axe con<lb/>&longs;i&longs;tit.</s>

<s>Demon&longs;tratio &longs;imilis erit ei, quam &longs;upra in cono, uel co <lb/>ni portione attulimus, ne toties eadem fru&longs;tra iterentur.</s><figure></figure><pb/>

<s>Demon&longs;tratio &longs;imilis erit ei, quam &longs;upra in cono, uel co <lb/>ni portione attulimus, ne toties eadem fru&longs;tra iterentur.</s><figure/><pb/>

<s>THEOREMA XII. PROPOSITIO XVI.</s>

<s>THEOREMA XII. PROPOSITIO XVI.</s>

<s>In &longs;phæra, & &longs;phæroidc idem e&longs;t grauitatis, & <lb/>figuræ centrum.</s>

Line 884

<s>erunt a d c, a b c dimidiæ portio<lb/>nes &longs;phæræ, uel &longs;phæroidis. </s>

<s>& 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. </s>

<s><expan abbr="punct&utilde;">punctum</expan> g por <lb/><figure id="fig29"></figure><lb/><arrow.to.target n="marg48"></arrow.to.target><lb/>tionis a b c centrum crit. </s>

<s>&longs;olidis cnim figuris &longs;imilibus & <lb/>æq<gap/> alibus inter &longs;e aptatis, & c. </s>

<s>&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"/><lb/>ter <gap/>e aptentur nece&longs;le e&longs;t. </s>

<s>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. </s>

<s>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. </s>

<s>Sphæ<lb/>ræ igitur, uel &longs;phæroidis grauitatis centrum e&longs;tidem, quod <lb/>centrum figuræ.</s><pb pagenum="22"/>

<s><margin.target id="marg48"></margin.target>per 2. pe<lb/>tition<gap/>m</s>

<s><margin.target id="marg48"/>per 2. pe<lb/>tition<gap/>m</s>

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

<s><margin.target id="marg49"/>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><figure></figure>

<s>Data cnim <lb/>qualibet maio<lb/>ri <expan abbr="portiõe">portione</expan>, quo <lb/><expan abbr="niã">niam</expan> totius &longs;phæ <lb/>ræ, uel &longs;phæroi <lb/>dis grauitatis <lb/>centrum e&longs;t in <lb/>axe; e&longs;t autem <lb/>& in axe cen<lb/>trum portio<lb/>nis minoris: <lb/>reliquæ portionis uidelicet maioris centrum in axe nece&longs;<lb/>&longs;ario con&longs;i&longs;tet.</s>

<s>THEOREMA XIII. PROPOSITIO XVII.</s><figure></figure>

<s>THEOREMA XIII. PROPOSITIO XVII.</s><figure/>

<s>Cuiuslibet pyramidis <expan abbr="triã">triam</expan> <lb/>gularem ba&longs;im <expan abbr="hab&etilde;tis">habentis</expan> gra<lb/>uitatis centrum e&longs;t in pun<lb/>cto, in quo ip&longs;ius axes con<lb/>ueniunt.</s>

Line 914

<s>crit & a faxis eiu&longs; <lb/>dem pyramidis ex tertia diffini<lb/>tione huius. </s>

<s>Itaque quoniam centrum grauitatis e&longs;t in <lb/>axe d e; e&longs;t autem & in axe a f; &qgrave;uod proxime demon&longs;traui <pb/>mus: erit utique grauitatis centrum pyramidis punctum <lb/>g. </s>

<s>in quo &longs;cilicet ip&longs;i axes conueniunt.</s>

<s>THEOREMA XIIII. PROPOSITIO XVIII.</s>

<s>THEOREMA XIIII. PROPOSITIO XVIII.</s>

<s>SI &longs;olidum parallelepipedum &longs;ecetur plano <lb/>ba&longs;ibus æquidi&longs;tante; erir &longs;olidum ad &longs;olidum, <lb/>&longs;icut altitudo ad altitudinem, uel &longs;icut axis ad <lb/>axem.</s><figure></figure>

<s>Sit &longs;olidum parallelepipe <lb/>dum a b c d e f g h, cuius axis <lb/>k l: <expan abbr="&longs;eceturq;">&longs;eceturque</expan> plano ba&longs;ibus <lb/>æquidi&longs;tante, quod faciat <lb/>&longs;ectionem m n o p; & axi in <lb/><expan abbr="&ptilde;uncto">pnuncto</expan> q occurrat. </s>

<s>Dico <lb/>&longs;olidum g m ad &longs;olidum m c <lb/>eam proportionem habere, <lb/>quam altitudo &longs;olidi g m ha<lb/>bet ad &longs;olidi m c altitudi<lb/>nem; uel quam axis <foreign lang="greek">k</foreign> q ad <lb/>axem q l. </s>

<s>Si enim axis K l ad <lb/>ba&longs;is planum &longs;it perpendicu <lb/>laris, & linea g c, quæ ex quin <lb/>ta huius ip&longs;i k l æquidi&longs;tat, <lb/>perpendicularis erit ad <expan abbr="id&etilde;">idem</expan> <lb/>planum, & &longs;olidi altitudi<lb/><arrow.to.target n="marg50"></arrow.to.target><lb/>nem dimetietur. </s>

<s>Itaque &longs;o<lb/>lidum gm ad &longs;olidum m c <lb/>eam proportionem habet, <lb/>quam parallelogram <expan abbr="n&utilde;">num</expan> g n <lb/>ad parallelo grammum n c, <lb/><arrow.to.target n="marg51"></arrow.to.target><lb/>hoce&longs;t quam linea g o, quæ <pb pagenum="23"/>e&longs;t &longs;olidi g m altitudo ad o e altitudinem &longs;olidi m c, uel <expan abbr="quã">quam</expan> <lb/>axis k q ad q l axem. </s>

<s>Itaque &longs;o<lb/>lidum gm ad &longs;olidum m c <lb/>eam proportionem habet, <lb/>quam parallelogram <expan abbr="n&utilde;">num</expan> g n <lb/>ad parallelo grammum n c, <lb/><arrow.to.target n="marg51"/><lb/>hoce&longs;t quam linea g o, quæ <pb pagenum="23"/>e&longs;t &longs;olidi g m altitudo ad o e altitudinem &longs;olidi m c, uel <expan abbr="quã">quam</expan> <lb/>axis k q ad q l axem. </s>

<s>Si uero axis k l non &longs;it perpendicularis <lb/>ad planum ba&longs;is; ducatur a puncto <foreign lang="greek">k</foreign> ad idem planum per <lb/>pendicularis <foreign lang="greek">k</foreign> r, <expan abbr="occurr&etilde;s">occurrens</expan> plano m n o p in s. </s>

<s>&longs;imiliter <expan abbr="de-mõ&longs;trabimus">de<lb/>mon&longs;trabimus</expan> &longs;olidum g m ad <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> m c ita e&longs;&longs;e, ut axis k q <lb/>ad axem q l. </s>

<s>Sed ut K q ad q l, ita <foreign lang="greek">k</foreign> s altitudo ad altitudi<lb/><arrow.to.target n="marg52"></arrow.to.target><lb/>nem s r; nam lineæ K l, K r à planis æquidi&longs;tantibus in ea&longs;<lb/>dem proportiones &longs;ecantur. </s>

<s>Sed ut K q ad q l, ita <foreign lang="greek">k</foreign> s altitudo ad altitudi<lb/><arrow.to.target n="marg52"/><lb/>nem s r; nam lineæ K l, K r à planis æquidi&longs;tantibus in ea&longs;<lb/>dem proportiones &longs;ecantur. </s>

<s>ergo &longs;olidum g m ad &longs;olidum <lb/>m c <expan abbr="eand&etilde;">eandem</expan> proportionem habet, quam altitudo ad <expan abbr="altitudin&etilde;">altitu<lb/>dinem</expan>, uel quam axis ad axem. </s>

<s>quod <expan abbr="demõ&longs;trare">demon&longs;trare</expan> oportebat.</s>

<s><margin.target id="marg50"></margin.target><gap/> undeci <lb/><gap/>.</s>

<s><margin.target id="marg50"/><gap/> undeci <lb/><gap/>.</s>

<s><margin.target id="marg51"></margin.target><expan abbr="&longs;ext&itilde;">&longs;extim</expan>.</s>

<s><margin.target id="marg51"/><expan abbr="&longs;ext&itilde;">&longs;extim</expan>.</s>

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

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

<s>Solida parallelepipeda in eadem ba&longs;i, uel in <lb/>æqualibus ba&longs;ibus con&longs;tituta eam inter &longs;e propor<lb/>tionem habent, quam altitudines: & &longs;i axes ip&longs;o<lb/>rum cum ba&longs;ibus æquales angulos contineant, <lb/>eam quoque, quam axes proportionem <expan abbr="habeb&utilde;t">habebunt</expan>.</s>

<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/><figure id="fig30"></figure><lb/><arrow.to.target n="marg53"></arrow.to.target><lb/>&longs;it gh. </s>

<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/><figure id="fig30"/><lb/><arrow.to.target n="marg53"/><lb/>&longs;it gh. </s>

<s><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. </s>

<s><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/><arrow.to.target n="marg55"></arrow.to.target><lb/>que &longs;olidum a b g h, hoc e&longs;t &longs;olidum a b c d ip&longs;i æquale, ad<lb/>&longs;olidum a b e f, ut altitudo &longs;olidi a b c d ad &longs;olidi a b e f al<lb/>titudinem.</s>

<s><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"/><lb/>&longs;olidum g h e f <lb/>adip&longs;um a b g h <pb/>ut altitudo ad altitudinem: & componendo conuertendo <lb/><arrow.to.target n="marg55"/><lb/>que &longs;olidum a b g h, hoc e&longs;t &longs;olidum a b c d ip&longs;i æquale, ad<lb/>&longs;olidum a b e f, ut altitudo &longs;olidi a b c d ad &longs;olidi a b e f al<lb/>titudinem.</s>

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

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

Line 968

<s>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. </s>

<s>K. crunt &longs;olida a b, c <foreign lang="greek">k</foreign> æque <lb/><arrow.to.target n="marg56"></arrow.to.target><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. </s><lb/>

<s>K. crunt &longs;olida a b, c <foreign lang="greek">k</foreign> æque <lb/><arrow.to.target n="marg56"/><lb/><figure id="fig31"/><lb/>alta inter <lb/>&longs;e æqualia <lb/><expan abbr="c&utilde;">cum</expan> æqua<lb/>les ba&longs;es <lb/>habeant. </s><lb/>

<s><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/><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. </s>

<s><arrow.to.target n="marg57"/><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/><figure id="fig32"/><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"/><lb/>ad &longs;olidum c d, ut g f ad f d. </s>

<s>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>

<s><margin.target id="marg57"></margin.target>18. huius</s>

<s><margin.target id="marg57"/>18. huius</s>

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

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

<s>Sint deinde &longs;olida parallelepipe <lb/>da a b, a c in eadem ba&longs;i; quorum <lb/>axes d e, &longs; e cum ip&longs;a æquales angu <pb pagenum="24"/>los contineant. </s>

<s>Dico &longs;olidum a b ad &longs;olidum a c e idem ha <lb/>bere proportionem, quam axis d e ad axem e f. </s>

Line 990

<s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, d g, fh: <lb/>& jungantur e g, e h. </s>

<s>Quoniam igitur axes cum ba&longs;ibus <lb/>æquales angulos continent, erit d e g angulus æqualis an<lb/><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. </s>

<s>Quoniam igitur axes cum ba&longs;ibus <lb/>æquales angulos continent, erit d e g angulus æqualis an<lb/><figure id="fig33"/><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. </s>

<s>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. </s><lb/>

Line 1002

<s>Dico &longs;olidum a b <expan abbr="ad&longs;olid&utilde;">ad&longs;olidum</expan> c d ita e&longs;&longs;e, ut axis <lb/>e f ad axem g h: nam &longs;i axes ad planum ba&longs;is recti &longs;int, il<lb/>lud per&longs;picue con&longs;tat: quoniam eadem linea, & axem & &longs;oli <lb/>di altitudinem determinabit. </s>

<s>Si uero &longs;int inclinati, à pun<lb/>ctis e g ad &longs;ubiectum planum perpendiculares ducantur <lb/>e k, g l: & iungantur fk, h l. </s>

<s>Si uero &longs;int inclinati, à pun<lb/>ctis e g ad &longs;ubiectum planum perpendiculares ducantur <lb/>e k, g l: & iungantur fk, h l. </s>

<s>rur&longs;us quoniam axes cum ba <lb/>&longs;ibus æquales faciunt angulos, eodem modo demon&longs;trabi <lb/>tur, triangulum e f K triangulo g h l &longs;imile e&longs;&longs;e: & e <foreign lang="greek">k</foreign> ad g l, <lb/>ut e f ad g h. </s>

<s>Solidum autem a b ad &longs;olidum c d e&longs;t, ut <lb/>e K ad g l. </s>

<s>Solidum autem a b ad &longs;olidum c d e&longs;t, ut <lb/>e K ad g l. </s>

<s>quæ omnia de <lb/>mon&longs;trarc oportebat.</s>

<s>Ex iis quæ demon&longs;trata &longs;unt, facile con&longs;tare <lb/>pote&longs;t, pri&longs;mata omnia & pyramides, quæ trian<lb/>gulares ba&longs;es habent, &longs;iue in ei&longs;dem, &longs;iue in æqua <lb/><arrow.to.target n="marg59"></arrow.to.target><lb/>libus ba&longs;ibus con&longs;tituantur, eandem proportio<lb/>nem habere, quam altitudines: & &longs;i axes cum ba <lb/>&longs;ibus æquales angulos contineant, &longs;imiliter ean<lb/>dem, quam axes, habere proportionem: &longs;unt <lb/><arrow.to.target n="marg60"></arrow.to.target><lb/>enim &longs;olida parallelepipeda pri&longs;matum triangula <lb/><arrow.to.target n="marg61"></arrow.to.target><lb/>res ba&longs;es <expan abbr="habenti&utilde;">habentium</expan> dupla; & pyramidum &longs;extupla.</s>

<s>Ex iis quæ demon&longs;trata &longs;unt, facile con&longs;tare <lb/>pote&longs;t, pri&longs;mata omnia & pyramides, quæ trian<lb/>gulares ba&longs;es habent, &longs;iue in ei&longs;dem, &longs;iue in æqua <lb/><arrow.to.target n="marg59"/><lb/>libus ba&longs;ibus con&longs;tituantur, eandem proportio<lb/>nem habere, quam altitudines: & &longs;i axes cum ba <lb/>&longs;ibus æquales angulos contineant, &longs;imiliter ean<lb/>dem, quam axes, habere proportionem: &longs;unt <lb/><arrow.to.target n="marg60"/><lb/>enim &longs;olida parallelepipeda pri&longs;matum triangula <lb/><arrow.to.target n="marg61"/><lb/>res ba&longs;es <expan abbr="habenti&utilde;">habentium</expan> dupla; & pyramidum &longs;extupla.</s>

<s><margin.target id="marg59"></margin.target>15. quinti</s>

<s><margin.target id="marg59"/>15. quinti</s>

<s><margin.target id="marg61"></margin.target>7. duode<lb/>cimi.</s>

<s><margin.target id="marg61"/>7. duode<lb/>cimi.</s>

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

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

<s>Pri&longs;mata omnia & pyramides, quæ in ei&longs;dem, <lb/>uel æqualibus ba&longs;ibus con&longs;tituuntur, eam inter <lb/>&longs;e proportionem habent, quam altitudines: & &longs;i <lb/>axes cum ba&longs;ibus faciant angulos æquales, eam <lb/>etiam, quam axes habent proportionem.</s><pb pagenum="25"/>

Line 1028

<s>Dico pri&longs;ma a e ad pri&longs;ma a f eam habere <lb/>proportionem, quam e g ad f h. </s>

<s>iungatur enim a c: & in <lb/>unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum <lb/><figure id="fig34"></figure><lb/>ba&longs;es &longs;int triangu <lb/>la a b c, a c d. </s>

<s>iungatur enim a c: & in <lb/>unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum <lb/><figure id="fig34"/><lb/>ba&longs;es &longs;int triangu <lb/>la a b c, a c d. </s>

<s>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. </s>

<s>& &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. </s>

<s>& &longs;i<lb/>militer alia duo, <lb/>quæ &longs;unt in ba&longs;i a <lb/><arrow.to.target n="marg62"/><lb/>c d. </s>

<s>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. </s><lb/>

<s>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. </s><lb/>

<s>Dico pri&longs;ma a e ad pri&longs;ma f l ita e&longs;&longs;e, ut altitudo illius ad <lb/>huius altitudinem. </s>

<s>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. </s>

<s>nam &longs;i altitudo &longs;it eadem, <expan abbr="intelligãtur">intelligantur</expan> <lb/><arrow.to.target n="marg63"/><lb/>duæ pyramides a b c d e, f g h k l. </s>

<s>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. </s>

<s>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. </s>

<s>quare <lb/><arrow.to.target n="marg64"/><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. </s>

<s>ex quibus per&longs;picue con&longs;tat <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>. </s><lb/>

<s>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/><figure id="fig35"></figure><lb/>erunt eædem ra<lb/>tione pri&longs;mata a <lb/>e, f m inter &longs;e æ<lb/>qualia. </s>

<s>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/><figure id="fig35"/><lb/>erunt eædem ra<lb/>tione pri&longs;mata a <lb/>e, f m inter &longs;e æ<lb/>qualia. </s>

<s>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. </s>

<s>ergo & pri&longs;ma a e ad pri&longs;ma f l eandem propor<lb/>tionem habebit, quam altitudo ad altitudinem. </s>

<s>&longs;equitur <lb/>igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;titu&utilde;">con&longs;tituum</expan> <lb/>tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s>

<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. </s>

<s>Dico pri&longs;ma <lb/>a e ad pri&longs;ma a f eam proportionem habere, quam g h ad <lb/>h l. </s>

<s>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/><figure id="fig36"/><lb/>num g K, l m: & iungantur k h, <lb/>h m. </s>

<s>ducantur à punctis g l perpendiculares ad ba&longs;is pla<lb/><figure id="fig36"></figure><lb/>num g K, l m: & iungantur k h, <lb/>h m. </s>

<s>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. </s>

<s>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. </s>

<s>ergo & can<lb/>dem habebit, quam g h, ad h l. </s>

<s>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></figure>

<s>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. </s>

Line 1084

<s>&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. </s>

<s>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/><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>. </s>

<s>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/><figure id="fig37"/><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>. </s>

<s>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>

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

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

<s>Pri&longs;mata omnia, & pyramides inter &longs;e propor<lb/>tionem habent compo&longs;itam ex proportione ba<lb/>&longs;ium, & proportione altitudinum.</s>

Line 1106

<s>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. </s>

<s>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/><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. </s>

<s>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/><figure id="fig38"/><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. </s>

<s>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. </s>

<s>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"/><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. </s>

<s>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. </s>

Line 1116

<s>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. </s>

<s>Quare & pyramis, cuius ba<lb/>&longs;is e&longs;t quadrilaterum a b c d, & altitudo e f ad pyramidem, <lb/><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. </s>

<s>Quare & pyramis, cuius ba<lb/>&longs;is e&longs;t quadrilaterum a b c d, & altitudo e f ad pyramidem, <lb/><figure id="fig39"/><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. </s>

<s>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. </s>

<s>demon&longs;ttatum e&longs;t enim, a<lb/>xes inter &longs;e eandem proportionem habere, quam <lb/>ip&longs;æ altitudines.</s>

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

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

<s>CVIVSLIBEt pyramidis, & cuiuslibet coni, <pb pagenum="28"/>uel coni portionis axis à centro grauitatis ita diui <lb/>ditur, ut pars, quæ terminatur ad uerticem reli<lb/>quæ partis, quæ ad ba&longs;im, &longs;it tripla.</s>

<s>Sit pyramis, cuius ba&longs;is triangulum a b c; axis d e; & gra<lb/>uitatis centrum K. </s>

<s>Sit pyramis, cuius ba&longs;is triangulum a b c; axis d e; & gra<lb/>uitatis centrum K. </s>

<s>Dico lineam d k ip&longs;ius K e triplam e&longs;&longs;e. </s><lb/>

<s>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. </s>

<s>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. </s><lb/>

<s>Quoniam igitur <expan abbr="centr&utilde;">centrum</expan> grauitatis pyramidis in axe <lb/><arrow.to.target n="marg66"/><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. </s><lb/>

<s>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"/><lb/><figure id="fig40"/><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>. </s><lb/>

<s>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/><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>. </s><lb/>

<s>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. 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. </s><lb/>

<s>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. </s>

<s>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. </s>

<s>quare <lb/><arrow.to.target n="marg68"/><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"/><lb/>triangulum, & idem uertex. </s>

<s>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. </s>

Line 1152

<s>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. </s>

<s>æqualis igitur e&longs;t pyramis a c d <foreign lang="greek">k</foreign><lb/>pyramidi b c d K. </s>

<s>æqualis igitur e&longs;t pyramis a c d <foreign lang="greek">k</foreign><lb/>pyramidi b c d K. </s>

<s>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. </s>

Line 1170

<s>quod eodem modo <lb/>demon&longs;trabimus.</s>

<s><margin.target id="marg67"></margin.target><emph type="italics"/>ucrfex<emph.end type="italics"/></s>

<s><margin.target id="marg67"/><emph type="italics"/>ucrfex<emph.end type="italics"/></s>

<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. </s>

<s>Dico cen<lb/>trum grauitatis pyramidis e&longs;&longs;e punctum g. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>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"/><lb/>&longs;it m. </s>

<s>ducta igitur l m ip&longs;i h m lineæ æquidi&longs;tabit. </s>

Line 1196

<s>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. </s>

<s>&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/><figure id="fig41"></figure><lb/>b c d e ad pyramidem a b d e. </s>

<s>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. </s>

Line 1210

<s>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><margin.target id="marg70"/>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. </s><lb/>

<s>Dico punctum e coni, uel coni portionis, grauitatis <lb/>e&longs;&longs;e centrum. </s>

<s>Sienim fieri pote&longs;t, &longs;itcentrum f: & pro<lb/>ducatur e f extra figuram in g. </s>

<s>Sienim fieri pote&longs;t, &longs;itcentrum f: & pro<lb/>ducatur e f extra figuram in g. </s>

<s>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. </s>

<s>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. </s>

<s>Et quoniam portiones &longs;unt <lb/>minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma<lb/><figure id="fig42"></figure><lb/>iorem proportionem habet, quam g e ad e f. </s>

<s>Et quoniam portiones &longs;unt <lb/>minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma<lb/><figure id="fig42"/><lb/>iorem proportionem habet, quam g e ad e f. </s>

<s>&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<pb pagenum="30"/><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. </s>

<s>&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<pb pagenum="30"/><arrow.to.target n="marg71"/><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. </s>

<s>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. </s>

Line 1238

<s>quæ omnia demon&longs;trare oportebat.</s>

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

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

<s>QVODLIBET fru&longs;tum à pyramide, quæ <lb/>triangularem ba&longs;im habeat, ab&longs;ci&longs;&longs;um, diuiditur <lb/>in tres pyramides proportionales, in ea proportio <lb/>ne, quæ e&longs;t lateris maioris ba&longs;is ad latus minoris <lb/>ip&longs;i re&longs;pondens.</s>

Line 1248

<s>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. </s>

<s>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. </s>

<s>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/><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. </s>

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

<s>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. </s>

<s>ut autem triangu <lb/>lum a b e ad triangu<lb/><arrow.to.target n="marg73"/><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"/><lb/>drilaterum a b e d. </s>

<s>er<lb/>go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c. </s><lb/>

<s>er<lb/>go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c. </s><lb/>

<s>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. </s>

<s>Rur&longs;us quoniam æquidi&longs;tantes &longs;unt a c, d f; erit eadem <lb/><arrow.to.target n="marg75"/><lb/>ratione pyramis a d c e ad pyramidem c d fe, ut ac ad <lb/>d f. </s>

<s>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. </s>

<s>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. </s>

<s>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>

<s>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. </s>

<s>quod demon&longs;trare oportebat.</s>

<s><margin.target id="marg72"></margin.target>1. &longs;exti.</s>

<s><margin.target id="marg73"></margin.target>5. duodeci <lb/>mi.</s>

<s><margin.target id="marg72"/>1. &longs;exti.</s>

<s><margin.target id="marg74"></margin.target>11. quinti.</s>

<s><margin.target id="marg73"/>5. duodeci <lb/>mi.</s>

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

<s><margin.target id="marg74"/>11. quinti.</s>

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

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

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

<s>QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>uel coni portionis, plano ba&longs;i æquidi&longs;tanti ita &longs;e<lb/>care, ut &longs;ectio &longs;it proportionalis inter maiorem, <lb/>& minorem ba&longs;im.</s><pb pagenum="31"/>

<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. </s>

<s>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. </s>

<s>Dico <lb/>triangulum h K l proportionale e&longs;&longs;e inter triangula a b c, <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. </s>

<s>Dico <lb/>triangulum h K l proportionale e&longs;&longs;e inter triangula a b c, <lb/><figure id="fig44"/><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. </s>

<s><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. </s>

<s><expan abbr="Quoniã">Quoniam</expan> enim <lb/><arrow.to.target n="marg76"/><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"/><lb/>linea h k ip&longs;i g b e&longs;t æqualis: & pro <lb/>pterea proportionalis inter a b, <lb/>d e. </s>

<s>quare ut a b ad h K, ita e&longs;t h <foreign lang="greek">k</foreign><lb/>ad de. </s>

<s>fiat ut h k ad d e, ita d e <lb/>ad aliam lineam, in qua &longs;it m. </s>

<s>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. </s>

<s>erit <lb/>ex æquali ut a b ad d e, ita h k ad <lb/><arrow.to.target n="marg78"/><lb/>m. </s>

<s>Et quoniam triangula a b c, <lb/>h K l, d e f &longs;imilia &longs;unt; <expan abbr="triangul&utilde;">triangulum</expan> <lb/><arrow.to.target n="marg79"></arrow.to.target><lb/>a b c ad triangulum h <foreign lang="greek">k</foreign> l e&longs;t, ut li<lb/>nea a b ad lineam d e: <expan abbr="triangul&utilde;">triangulum</expan> <lb/><arrow.to.target n="marg80"></arrow.to.target><lb/>autem h <foreign lang="greek">k</foreign> l adip&longs;um d e f e&longs;t, ut h k ad m. </s>

<s>Et quoniam triangula a b c, <lb/>h K l, d e f &longs;imilia &longs;unt; <expan abbr="triangul&utilde;">triangulum</expan> <lb/><arrow.to.target n="marg79"/><lb/>a b c ad triangulum h <foreign lang="greek">k</foreign> l e&longs;t, ut li<lb/>nea a b ad lineam d e: <expan abbr="triangul&utilde;">triangulum</expan> <lb/><arrow.to.target n="marg80"/><lb/>autem h <foreign lang="greek">k</foreign> l adip&longs;um d e f e&longs;t, ut h k ad m. </s>

<s>ergo triangulum <lb/>a b c ad triangulum h k l eandem proportionem habet, <lb/>quam triangulum h K l ad ip&longs;um d e f. </s>

<s>Eodem modo in a<lb/>liis fru&longs;tis pyramidis idem demon&longs;trabitur.</s>

<s><margin.target id="marg79"></margin.target>20. &longs;<gap/></s>

<s><margin.target id="marg79"/>20. &longs;<gap/></s>

<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. </s><lb/>

<s>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. </s>

<s>Dico &longs;ectionem a b <lb/>ad &longs;ectionem f g eandem proportionem habere, quam f g <lb/>ad ip&longs;am c d. </s>

<s>Dico &longs;ectionem a b <lb/>ad &longs;ectionem f g eandem proportionem habere, quam f g <lb/>ad ip&longs;am c d. </s>

<s>Simili enim ratione, qua &longs;upra, demon&longs;trabi<lb/>tur quadratum a b ad quadratum f g ita e&longs;&longs;e, ut <expan abbr="quadrat&utilde;">quadratum</expan> <lb/><arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. </s>

<s>Sed circuli inter &longs;e eandem propor<lb/>tionem habent, quam diametrorum quadrata. </s>

<s>ellip&longs;es au<lb/>tem circa a b, f g, c d, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="cõ-mentariis">con<lb/>mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>& &longs;phæroidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio<lb/><figure id="fig45"></figure><lb/>&longs;eptimæ propo&longs;itionis eiu&longs;dem li<lb/>bri. </s>

<s>ellip&longs;es au<lb/>tem circa a b, f g, c d, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="cõ-mentariis">con<lb/>mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>& &longs;phæroidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio<lb/><figure id="fig45"/><lb/>&longs;eptimæ propo&longs;itionis eiu&longs;dem li<lb/>bri. </s>

<s>ellip&longs;es enim nunc appello ip<lb/>&longs;a &longs;pacia ellip&longs;ibus contenta. </s>

<s>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. </s><lb/>

<s>quod quidem faciendum propo<lb/>&longs;uimus.</s>

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

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

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

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

<s>QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>uel coni portionis ad pyramidem, uel conum, uel <lb/>coni portionem, cuius ba&longs;is eadem e&longs;t, & æqualis <lb/>altitudo, eandem <expan abbr="proportion&etilde;">proportionem</expan> habet, quam utræ <lb/>que ba&longs;es, maior, & minor &longs;imul &longs;umptæ vnà <expan abbr="c&utilde;">cum</expan> <lb/>ca, quæ inter ip&longs;as &longs;it proportionalis, ad ba&longs;im ma <lb/>iorem.</s><pb pagenum="32"/>

<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. </s>

<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. </s>

<s>& &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. </s>

<s>& &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. </s>

<s>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/><figure id="fig46"></figure><lb/>&longs;ti, & altitudo æqualis. </s>

<s>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. </s>

<s>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. </s>

<s>Sed pyrami <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. </s>

<s>Sed pyrami <lb/><figure id="fig47"/><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"/><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. </s>

<s>quare pyra<lb/>mis, uel conus, uel coni por<lb/>tio, cuius ba&longs;is e&longs;t tribus illis <lb/>ba&longs;ibus æqualis ad a g b eam <lb/>habet proportionem, quam <lb/>ba&longs;es a b, e f, c d ad a b ba&longs;im. </s>

Line 1350

<s>quod demon&longs;trare uolebamus.</s>

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

<s><margin.target id="marg82"/>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>

Line 1358

<s>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. </s>

<s>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/><figure id="fig48"></figure><lb/>le erit. </s>

<s>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>

Line 1370

<s>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. </s><lb/>

<s>&longs;ed pyramis æque alta, cuius ba&longs;is con&longs;tat ex tribus trian<lb/><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. </s>

<s>&longs;ed pyramis æque alta, cuius ba&longs;is con&longs;tat ex tribus trian<lb/><figure id="fig49"/><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. </s>

<s>& &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. </s>

Line 1384

<s>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 <pb/>preducantur. </s>

<s>Quoniam igitur pyramis &longs;ecatur planis ba&longs;i <lb/><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. </s>

<s>Quoniam igitur pyramis &longs;ecatur planis ba&longs;i <lb/><arrow.to.target n="marg83"/><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. </s>

<s>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/><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. </s>

<s>Sed cum circuli inter &longs;e <expan abbr="eã">eam</expan> <lb/><arrow.to.target n="marg84"/><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"/><lb/>con&longs;tantia: & &longs;it circulus, uel ellip&longs;is circa diametrum e f <lb/><figure id="fig50"/><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. </s>

<s>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. </s>

Line 1394

<s>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. </s>

<s>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. </s><lb/>

<s>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. </s><lb/>

<s>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. </s>

<s>po&longs;tremo intelligatur pyramis a l b, cuius. </s><lb/>

<s>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. </s>

<s>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. </s>

Line 1410

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>Sed pyramis q æqualis e&longs;t fru&longs;to à pyramide <lb/>ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. </s>

Line 1420

<s>atque illud e&longs;t, quod demon&longs;trare oportebat.</s>

<s><margin.target id="marg83"></margin.target>9 huius</s>

<s><margin.target id="marg83"/>9 huius</s>

<s><margin.target id="marg84"></margin.target>2. duode<lb/>cnni.</s>

<s><margin.target id="marg84"/>2. duode<lb/>cnni.</s>

<s><margin.target id="marg85"></margin.target>7. de co<lb/>noidibus <lb/>& &longs;phæ<lb/>roidibus</s>

<s><margin.target id="marg85"/>7. de co<lb/>noidibus <lb/>& &longs;phæ<lb/>roidibus</s>

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

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

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

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

<s>CVIVSLIBET fru&longs;ti à pyramide, uel cono, <lb/>uel coni portione ab&longs;cis&longs;i, centrum grauitatis e&longs;t <lb/>in axe, ita ut eo primum in duas portiones diui<lb/>&longs;o, portio &longs;uperior, quæ minorem ba&longs;im attingit <lb/>ad portionem reliquam eam habeat proportio<lb/>nem, quam duplum lateris, uel diametri maioris <lb/>ba&longs;is, vnà cum latere, uel diametro minoris, ip&longs;i <lb/>re&longs;pondente, habet ad duplum lateris, uel diame<lb/>tri minoris ba&longs;is vnà <expan abbr="c&utilde;">cum</expan> latere, uel diametro ma<lb/>ioris: deinde à puncto diui&longs;ionis quarta parte &longs;u <lb/>perioris portionis in ip&longs;a &longs;umpta: & rur&longs;us ab in<lb/>ferioris portionis termino, qui e&longs;t ad ba&longs;im maio<lb/>rem, &longs;umpta quarta parte totius axis: centrum &longs;it <lb/>in linea, quæ his finibus continetur, atque in eo li <lb/>neæ puncto, quo &longs;ic diuiditur, ut tota linea ad par <lb/>tem propinquiorem minori ba&longs;i, <expan abbr="eãdem">eandem</expan> propor<lb/>tionem habeat, quam fru&longs;tum ad <expan abbr="pyramid&etilde;">pyramidem</expan>, uel <lb/>conum, uel coni portionem, cuius ba&longs;is &longs;it ea<lb/>dem, quæ ba&longs;is maior, & altitudo fru&longs;ti altitudini <lb/>æqualis.</s><pb pagenum="35"/>

Line 1436

<s>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. </s>

<s>nam cum g h &longs;it axis <lb/>fru&longs;ti: erit h centrum grauitatis trianguli a b c: & g <lb/><figure id="fig51"></figure><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. </s>

<s>nam cum g h &longs;it axis <lb/>fru&longs;ti: erit h centrum grauitatis trianguli a b c: & g <lb/><figure id="fig51"/><lb/><arrow.to.target n="marg87"/><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"/><lb/>uitatis planorum. </s>

<s>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. </s><lb/>

<s>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. </s>

<s>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. </s>

Line 1450

<s>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. </s>

<s>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. </s>

<s>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/><figure id="fig52"/><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. </s>

<s>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/><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. </s>

<s>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. </s>

<s>Quoniam igitur linea u o <lb/><arrow.to.target n="marg89"/><lb/>æquidi&longs;tat ip&longs;i d g, erit d u ad <lb/>u h, ut g o ad o h. </s>

<s>Sed g o tri<lb/>pla e&longs;t o h. </s>

Line 1466

<s>Quòd cum g p &longs;it tripla p n; <lb/>erit etiam d z ip&longs;ius z m tri<lb/>pla. </s>

<s>atque ob eandem cau&longs;<lb/>&longs;am punctuniz e&longs;t <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis pyramidis b c f e d. </s>

<s>atque ob eandem cau&longs;<lb/>&longs;am punctuniz e&longs;t <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis pyramidis b c f e d. </s>

<s>iun <lb/>ctaigitur z u, in ea erit <expan abbr="c&etilde;trum">centrum</expan> <pb pagenum="36"/>grauitatis magnitudinis, quæ ex utri&longs;que pyramidibus <expan abbr="cõ">com</expan> <lb/>&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s>

Line 1474

<s>ergo in puncto <foreign lang="greek">*f,</foreign> in quo lineæ z u, g h conu<gap/>niunt. </s><lb/>

<s><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. </s>

<s><arrow.to.target n="marg90"/><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. </s>

<s>& componendo u z ad z <foreign lang="greek">*f</foreign><lb/>eam habet, quam fru&longs;tum ad pyramidem a b c d. </s>

<s>& componendo u z ad z <foreign lang="greek">*f</foreign><lb/>eam habet, quam fru&longs;tum ad pyramidem a b c d. </s>

<s>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. </s>

<s>&longs;ed ita erat o p ad p <expan abbr="q.">que</expan> æquales igitur &longs;unt p <gap/>, p q: <gap/><lb/><arrow.to.target n="marg91"></arrow.to.target><lb/>q <foreign lang="greek">*f</foreign> unum atque idem pun ctum. </s>

<s>&longs;ed ita erat o p ad p <expan abbr="q.">que</expan> æquales igitur &longs;unt p <gap/>, p q: <gap/><lb/><arrow.to.target n="marg91"/><lb/>q <foreign lang="greek">*f</foreign> unum atque idem pun ctum. </s>

<s>ex quibus &longs;equitur lineam. </s><lb/>

<s>z u &longs;ecare o p in q: & propterea <expan abbr="p&utilde;ctum">punctum</expan> q ip&longs;ius fru&longs;ti gra<lb/>uitatis centrum e&longs;&longs;e.</s>

<s><margin.target id="marg87"></margin.target>3. diffi. </s>

<s><margin.target id="marg87"/>3. diffi. </s>

<s><margin.target id="marg88"></margin.target>Vltima <expan abbr="e-iu&longs;d&etilde;">e<lb/>iu&longs;dem</expan> libri <lb/>Archime<lb/>dis.</s>

<s><margin.target id="marg88"/>Vltima <expan abbr="e-iu&longs;d&etilde;">e<lb/>iu&longs;dem</expan> libri <lb/>Archime<lb/>dis.</s>

<s><margin.target id="marg89"></margin.target>2. &longs;exti.</s>

<s><margin.target id="marg89"/>2. &longs;exti.</s>

<s>primi <lb/>libri Ar<lb/>chunedis <lb/>de <expan abbr="c&etilde;tro">centro</expan> <lb/>gr<gap/>u ta<lb/>tis plano <lb/>tu<gap/>n</s>

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

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

<s>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. </s>

<s>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. </s><lb/>

Line 1526

<s><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. </s>

<s>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/><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. </s><lb/>

<s>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/><figure id="fig53"/><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. </s><lb/>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>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"/><lb/>tur. </s>

<s>ergo linea x s per p tran&longs;ibit: & lineæ r u, s x, q t in<lb/>ter &longs;e æquidi&longs;tantes erunt. </s>

Line 1542

<s>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. </s>

<s>& 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. </s>

<s>& 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"/><lb/>a c d y, ita pyramis e f g y ad pyramidem e g h y. </s>

<s>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. </s><lb/>

<s>Quòd cum fru&longs;ti l f centrum grauitatis &longs;its: & fru&longs;ti l h &longs;it <lb/><arrow.to.target n="marg94"></arrow.to.target><lb/>centrum x: con&longs;tat punctum p totius fru&longs;ti a g grauitatis <lb/>e&longs;&longs;e centrum. </s>

<s>Quòd cum fru&longs;ti l f centrum grauitatis &longs;its: & fru&longs;ti l h &longs;it <lb/><arrow.to.target n="marg94"/><lb/>centrum x: con&longs;tat punctum p totius fru&longs;ti a g grauitatis <lb/>e&longs;&longs;e centrum. </s>

<s>Eodem modo fret demon&longs;tratio etiam in <lb/>aliis pyramidibus.</s>

<s>&longs;exti.</s>

<s><margin.target id="marg93"></margin.target>19. quinti</s>

<s><margin.target id="marg93"/>19. quinti</s>

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

<s><margin.target id="marg94"/>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. </s>

Line 1564

<s><expan abbr="Sitq;">Sitque</expan> g h quarta pars lineæ g e: & &longs;it &longs; K item <lb/>quarta pars totius f e axis. </s>

<s>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. </s>

<s>Dico pun<lb/>ctum l fru&longs;ti a d grauitatis centrum e&longs;&longs;e. </s>

Line 1572

<s>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. </s>

<s>erit ex iis quæ proxime <lb/>tradidimus, fru&longs;ti pyramidis a d ceutrum grauitatis l. </s>

<s>erit ex iis quæ proxime <lb/>tradidimus, fru&longs;ti pyramidis a d ceutrum grauitatis l. </s>

<s>Quo <lb/>niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir <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. </s>

<s>Quo <lb/>niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir <lb/><figure id="fig54"/><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. </s>

<s>&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. </s>

<s>&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"/><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. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>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. </s><lb/>

<s>fiat igitur ut fru&longs;tum pyramidis ad portiones, ita q m ad <lb/>m l. </s>

<s>fiat igitur ut fru&longs;tum pyramidis ad portiones, ita q m ad <lb/>m l. </s>

<s>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. </s>

Line 1594

<s>quz omnia demon&longs;tranda <lb/>proponebantur.</s>

<s><margin.target id="marg95"></margin.target>22. huius</s>

<s><margin.target id="marg95"/>22. huius</s>

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

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

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

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

<s>OMNIVM &longs;olidorum in &longs;phæra de&longs;cripto<lb/>rum, quæ æqualibus, & &longs;imilibus ba&longs;ibus conti<lb/>nentur, centrum grauitatis e&longs;t idem, quod &longs;phæ<lb/>ræ centrum.</s>

Line 1610

<s>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. </s>

<s>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/><figure id="fig55"></figure><lb/>At cum e f &longs;it &longs;exta pars axis <lb/>&longs;phæræ, crit d e tripla e f. </s>

<s>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/><figure id="fig55"/><lb/>At cum e f &longs;it &longs;exta pars axis <lb/>&longs;phæræ, crit d e tripla e f. </s>

<s>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. </s>

Line 1618

<s>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. </s>

<s>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. </s><lb/>

<s><figure id="fig56"></figure><lb/>&longs;ecent autem in puncto e. </s>

<s><figure id="fig56"/><lb/>&longs;ecent autem in puncto e. </s>

<s>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. </s>

Line 1638

<s><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. </s>

<s>quare pyramidis a b c d e axis erit e g: & pyra <lb/>midis a b c d f axis f g. </s>

<s>quare pyramidis a b c d e axis erit e g: & pyra <lb/>midis a b c d f axis f g. </s>

<s>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/><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. </s>

<s>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/><figure id="fig57"/><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. </s>

<s>& <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. </s>

<s>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. </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. </s>

<s>Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. </s>

Line 1658

<s>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. </s>

<s>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. </s>

<s>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"/><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. </s>

<s>crgo & ip&longs;i h g d, d g <foreign lang="greek">k</foreign> duobus rectis æquales erunt. </s><lb/>

<s><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. </s>

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

<s>cum autem <lb/><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. </s>

<s>cum autem <lb/><figure id="fig58"/><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. </s>

<s>quare g h <lb/>erit pyramidis a b c g axis. </s><lb/>

<s>& ob eandem cau&longs;&longs;am g k <lb/>axis pyramidis d e f g. </s>

<s>& ob eandem cau&longs;&longs;am g k <lb/>axis pyramidis d e f g. </s>

<s>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. </s><lb/>

<s>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. </s>

<s>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. </s>

<s>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. </s>

<s>Sequitur ergo utico&longs;ahedri centrum gra<lb/>uitatis &longs;it idem, quod ip&longs;ius &longs;phæræ centrum.</s>

<s><margin.target id="marg97"></margin.target>13. primi</s>

<s><margin.target id="marg97"/>13. primi</s>

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

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

<s>Sit dodecahedrum a f in &longs;phæra de&longs;ignatum, &longs;itque &longs;phæ <lb/>ræ centrum m. </s>

Line 1694

<s>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. </s>

<s>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/><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. </s>

<s>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/><figure id="fig59"/><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"/><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"/><lb/>li æquales &longs;int. </s>

<s>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. </s>

Line 1710

<s>quæ quidem omnia demon&longs;tra&longs;&longs;e <lb/>oportebat.</s>

<s><margin.target id="marg99"></margin.target>corol. </s>

<s><margin.target id="marg99"/>corol. </s>

<s>pri<lb/>mæ &longs;phæ<lb/>ricorum <lb/>Theod.</s>

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

<s>pri<lb/>mæ &longs;phæ<lb/>ricorum <lb/>Theod.<margin.target id="marg100"/>6. primi <lb/>phærico <lb/>rum.</s>

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

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

<s>DATA qualibet portione conoidis rectangu <lb/>li, ab&longs;ci&longs;&longs;a plano ad axem recto, uel non recto; fie<lb/>ri pote&longs;t, ut portio &longs;olida in&longs;cribatur, uel circum<lb/>&longs;cribatur ex cylindris, uel cylindri portionibus, <lb/>æqualem habentibus altitudinem, ita ut recta li<lb/>nea, quæ inter centrum grauitatis portionis, & <lb/>figuræ in&longs;criptæ, uel circum&longs;criptæ interiicitur, <lb/>&longs;it minor qualibet recta linea propo&longs;ita.</s>

Line 1726

Line 1724

<s>ni&longs;i enim &longs;it minor, uel æqualis, uel maior erit. </s>

<s>& quo<lb/>niam figura circum&longs;cripta ad reliquas portiones maiorem <lb/><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. </s>

<s>& quo<lb/>niam figura circum&longs;cripta ad reliquas portiones maiorem <lb/><arrow.to.target n="marg101"/><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"/><lb/>& diuidendo portio conoidis ad reliquas portiones habe<lb/>bit maiorem, quàm b <foreign lang="greek">k</foreign> ad K e. </s>

<s>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/><figure id="fig60"></figure><lb/>tionem cadet. </s>

<s><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. </s>

Line 1738

Line 1736

<s>quare con&longs;tat lineam k e ip&longs;a g linea propo&longs;i <lb/>ta minorem e&longs;&longs;e.</s>

<s><margin.target id="marg101"></margin.target>8. quínti.</s>

<s><margin.target id="marg101"/>8. quínti.</s>

<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><margin.target id="marg102"/>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/><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. </s><lb/>

<s>Rur&longs;us in&longs;cribatur portioni figura, uidelicet cylindr us <lb/><figure id="fig61"/><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. </s><lb/>

<s>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. </s>

Line 1764

Line 1762

<s>quod e&longs;t ab&longs;urdum.</s>

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

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

<s>CVIVSLIBET portionis conoidis rectangu<lb/>li axis à <expan abbr="c&etilde;tro">centro</expan> grauitatis ita diuiditur, ut pars quæ <lb/>terminatur ad uerticem, reliquæ partis, quæ ad ba <lb/>&longs;im &longs;it dupla.</s>

<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. </s>

<s>Sumatur autem <lb/>in linea b d punctum e, ita ut b e &longs;itip&longs;ius e d dupla. </s>

<s>Dico <pb pagenum="42"/><figure id="fig62"></figure><lb/>e portionis a b <lb/>c grauitatis e&longs;&longs;e <lb/>centrum. </s>

<s>Dico <pb pagenum="42"/><figure id="fig62"/><lb/>e portionis a b <lb/>c grauitatis e&longs;&longs;e <lb/>centrum. </s>

<s>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'. </s><lb/>

<s>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. </s><lb/>

<s><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. </s>

<s><arrow.to.target n="marg103"/><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. </s>

<s>quare &longs;i li <pb/><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. </s>

<s>quare &longs;i li <pb/><figure id="fig63"/><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"/><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"/><lb/>ctum p. </s>

<s>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/><arrow.to.target n="marg107"></arrow.to.target><lb/>g: hoc e&longs;t circu<lb/>lus circa diame<lb/>trum a c ad cir<lb/>culum circa dia <lb/>metrum k g. </s>

<s>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"/><lb/><expan abbr="conicor&utilde;">conicorum</expan> & ita <lb/>quadratum a c <lb/>ad <expan abbr="quadrat&utilde;">quadratum</expan> K <lb/><arrow.to.target n="marg107"/><lb/>g: hoc e&longs;t circu<lb/>lus circa diame<lb/>trum a c ad cir<lb/>culum circa dia <lb/>metrum k g. </s>

<s>du <lb/>pla e&longs;t autem li<lb/>nca d b lineæ <pb pagenum="43"/>b m. </s>

Line 1792

Line 1790

<s>quare & linea o p dupla <lb/>ip&longs;ius p n. </s>

<s>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. </s>

<s>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"/><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. </s>

<s>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. </s>

<s>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. </s><lb/>

<s>quare y z <expan abbr="cyl&itilde;drus">cylindrus</expan> duobus cylindris k <foreign lang="greek">g, q l</foreign> æqualis erit. </s>

Line 1808

Line 1806

<s>quare &longs;equitur ut punctum <foreign lang="greek">p</foreign> totius figuræ circum <lb/>&longs;criptæ &longs;it centrum. </s>

<s>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/><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. </s>

<s>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/><figure id="fig64"/><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. </s>

<s>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. </s><lb/>

Line 1820

Line 1818

<s>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. </s>

<s>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. </s>

<s>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"/><lb/>cet b o unà cum p e ip&longs;ius reliqui <foreign lang="greek">x</foreign> d duplnm erit. </s>

<s>e&longs;tque <lb/>b o dupla <foreign lang="greek">r</foreign> d. </s>

<s>e&longs;tque <lb/>b o dupla <foreign lang="greek">r</foreign> d. </s>

<s>ergo p e, hoc e&longs;t n <foreign lang="greek">x</foreign> ip&longs;ius <foreign lang="greek">x r</foreign> dupla. </s>

Line 1850

Line 1848

<s>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. </s>

<s>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. </s>

<s>ergo d <foreign lang="greek">x,</foreign> hoc e&longs;t n p, & <foreign lang="greek">p t</foreign> æquales. </s>

Line 1864

Line 1862

<s>Ex quibus con&longs;tat lineam <foreign lang="greek">p f</foreign> à centro grauitatis <lb/>portionis diuidi in partes æquales. </s>

<s>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/><figure id="fig65"></figure><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. </s>

<s>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/><figure id="fig65"/><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. </s>

<s>quod fieri nullo modo <lb/>po&longs;&longs;e per&longs;picuum e&longs;t. </s>

<s>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>. </s>

<s>ergo punctum e centrum erit gra<lb/>uitatis portionis a b c. </s>

<s>ergo punctum e centrum erit gra<lb/>uitatis portionis a b c. quod demon&longs;trare oportebat.</s>

<s>quod demon&longs;trare oportebat.</s>

<s><margin.target id="marg103"></margin.target>7. huius</s>

<s><margin.target id="marg103"/>7. huius</s>

<s><margin.target id="marg108"></margin.target>20. primi <lb/>conicor</s>

<s><margin.target id="marg108"/>20. primi <lb/>conicor</s>

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

<s><margin.target id="marg109"/>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. </s>

<s>ut <lb/>enim tradidimus in commentariis in undecimam propo&longs;i <lb/>tionem libri Archimedis de conoidibus & &longs;phæroidibus. </s><lb/>

<s>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/><arrow.to.target n="marg110"></arrow.to.target><lb/>quæ &longs;unt ellip&longs;es &longs;imiles eandem proportionem habere, <lb/>quam quadrata diametrorum eiu&longs;dem rationis, ex corol<lb/>lario &longs;eptlmæ propo&longs;itionis libri de conoidibus, & &longs;phæ<lb/>roidibus, manife&longs;te apparet.</s>

<s>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/><arrow.to.target n="marg110"/><lb/>quæ &longs;unt ellip&longs;es &longs;imiles eandem proportionem habere, <lb/>quam quadrata diametrorum eiu&longs;dem rationis, ex corol<lb/>lario &longs;eptlmæ propo&longs;itionis libri de conoidibus, & &longs;phæ<lb/>roidibus, manife&longs;te apparet.</s>

<s><margin.target id="marg110"></margin.target>corol. <gap/><lb/>de conoi<lb/>dibus & <lb/>&longs;phæroi<lb/>dibus.</s>

<s><margin.target id="marg110"/>corol. <gap/><lb/>de conoi<lb/>dibus & <lb/>&longs;phæroi<lb/>dibus.</s>

<s>THEOREMA XXIIII. PROPOSITIO XXX.</s>

<s>THEOREMA XXIIII. PROPOSITIO XXX.</s>

<s>Sr à portione conoidis rectanguli alia portio <lb/>ab&longs;cindatur, plano ba&longs;i æquidi&longs;tante; habebit <lb/>portio tota ad eam, quæ ab&longs;ci&longs;&longs;a e&longs;t, duplam pro <lb/>portion em eius, quæ e&longs;t ba&longs;is maioris portionis <lb/>ad ba&longs;i m minoris, uel quæ axis maioris ad axem <lb/>minoris.</s><pb/>

Line 1906

Line 1904

<s>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. </s>

<s>& 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/><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. </s>

<s>& 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/><figure id="fig66"/><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. </s>

<s>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. </s><lb/>

Line 1914

Line 1912

<s>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. </s><lb/>

<s>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. </s>

<s>erit b h altitudo coni, uel coni portionis a b c: & b K altitu<lb/><arrow.to.target n="marg111"/><lb/>do efg. </s>

<s>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/><arrow.to.target n="marg113"></arrow.to.target><lb/>proportione d b axis ad axem b g. </s>

<s>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"/><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/><arrow.to.target n="marg113"/><lb/>proportione d b axis ad axem b g. </s>

<s>Sed circulus, uel <lb/>ellip&longs;is circa diametrum a c ad circulum, uel ellip&longs;im <lb/><arrow.to.target n="marg114"></arrow.to.target><lb/>circa e f, e&longs;t ut quadratum a c ad quadratum e f; hoc e&longs;t ut <lb/><expan abbr="quadrat&utilde;">quadratum</expan> a d ad <expan abbr="quadrat&utilde;">quadratum</expan> e g. </s>

<s>Sed circulus, uel <lb/>ellip&longs;is circa diametrum a c ad circulum, uel ellip&longs;im <lb/><arrow.to.target n="marg114"/><lb/>circa e f, e&longs;t ut quadratum a c ad quadratum e f; hoc e&longs;t ut <lb/><expan abbr="quadrat&utilde;">quadratum</expan> a d ad <expan abbr="quadrat&utilde;">quadratum</expan> e g. & quadratum a d ad quadra <lb/>tum e g e&longs;t, ut linea d b ad lineam b g. </s>

<s>& quadratum a d ad quadra <lb/>tum e g e&longs;t, ut linea d b ad lineam b g. </s>

<s>circulus igitur, uel el <lb/><arrow.to.target n="marg115"></arrow.to.target><lb/>lip&longs;is circa diametrum a c ad <expan abbr="circul&utilde;">circulum</expan>, uel el<gap/>&longs;im circa e f, <lb/><arrow.to.target n="marg116"></arrow.to.target><lb/>hoc e&longs;t ba&longs;is ad ba&longs;im eandem proportion<gap/>t, <expan abbr="quã">quam</expan> <lb/>d b axis ad axem b g. </s>

<s>circulus igitur, uel el <lb/><arrow.to.target n="marg115"/><lb/>lip&longs;is circa diametrum a c ad <expan abbr="circul&utilde;">circulum</expan>, uel el<gap/>&longs;im circa e f, <lb/><arrow.to.target n="marg116"/><lb/>hoc e&longs;t ba&longs;is ad ba&longs;im eandem proportion<gap/>t, <expan abbr="quã">quam</expan> <lb/>d b axis ad axem b g. </s>

<s>ex quibus &longs;equitur por<gap/>em a b c <lb/>ad portionem e b f habere proportionem duplam eius, <lb/>quæ e&longs;t ba&longs;is a c ad ba&longs;im e f: uel axis d b ad b g axem. </s>

<s>quod <lb/>demon&longs;trandum proponebatur.</s>

<s><margin.target id="marg113"></margin.target>2. duode <lb/>cimi</s>

<s><margin.target id="marg113"/>2. duode <lb/>cimi</s>

<s><margin.target id="marg114"></margin.target>7. de co<lb/>noidibus <lb/>& &longs;phæ<lb/>roidibus</s>

<s><margin.target id="marg114"/>7. de co<lb/>noidibus <lb/>& &longs;phæ<lb/>roidibus</s>

<s>quinti</s>

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

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

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

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

<s>Cuiuslibet fru&longs;ti à portione rectanguli conoi <lb/>dis ab&longs;cis&longs;i, centrum grauitatis e&longs;t in axe, ita ut <lb/>demptis primum à quadrato, quod fit ex diame<lb/>tro maioris ba&longs;is, tertia ip&longs;ius parte, & duabus <lb/>tertiis quadrati, quod fit ex diamerro ba&longs;is mino<lb/>ris: deinde à tertia parte quadrati maioris ba&longs;is <lb/>rur&longs;us dempta portione, ad quam reliquum qua <lb/>drati ba&longs;is maioris unà cum dicta portione <expan abbr="duplã">duplam</expan> <lb/>proportionem habeat eius, quæ e&longs;t quadrati ma<pb/>ioris ba&longs;is ad quadratum minoris: centrum &longs;it in <lb/>eo axis puncto, quo ita diuiditur ut pars, quæ mi<lb/>norem ba&longs;im attingit ad alteram partem eandem <lb/>proportionem habeat, quam dempto quadrato <lb/>minoris ba&longs;is à duabus tertiis quadrati maioris, <lb/>habet id, quod reliquum e&longs;t unà cum portione à <lb/>tertia quadrati maioris parte dempta, ad <expan abbr="reliquã">reliquam</expan> <lb/>eiu&longs;dem tertiæ portionem.</s>

<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. </s>

<s>de&longs;criba<lb/>tur<gap/>atem portio conoidis, à quo illud ab&longs;ci&longs;&longs;um e&longs;t, & pla<lb/><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. </s>

<s>de&longs;criba<lb/>tur<gap/>atem portio conoidis, à quo illud ab&longs;ci&longs;&longs;um e&longs;t, & pla<lb/><figure id="fig67"/><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. </s>

<s><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. </s>

<s>&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. </s>

<s>&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"/><lb/>quam portio conoidis b gc ad a g d portionem. </s>

<s><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. </s>

<s>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. </s>

<s>ergo <lb/>ex æquali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ip&longs;is quadrati a d duabus tertiis, ad <expan abbr="tertiã">tertiam</expan> <lb/>partem quadrati b c, ut k h ad h f: & ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>tertiæ partis, ad quam unà cum ip&longs;a portione, duplam pro <lb/>portionem habeat eius, quæ e&longs;t quadrati b c ad <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>a d, ut K l ad l h. </s>

<s>habet enim K l ad l h eandem proportio<lb/>nem, quam conoidis portio b g c ad portionem a g d: por<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, quæ e&longs;t ba&longs;is b c ad ba&longs;im a d: hoc e&longs;t quadrati <lb/><arrow.to.target n="marg118"></arrow.to.target><lb/>b c ad quadratum a d; ut proxime demon&longs;tratum e&longs;t. </s>

<s>habet enim K l ad l h eandem proportio<lb/>nem, quam conoidis portio b g c ad portionem a g d: por<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, quæ e&longs;t ba&longs;is b c ad ba&longs;im a d: hoc e&longs;t quadrati <lb/><arrow.to.target n="marg118"/><lb/>b c ad quadratum a d; ut proxime demon&longs;tratum e&longs;t. </s>

<s>quare <lb/>dempto a d quadrato à duabus tertiis quadrati b c, erit id, <lb/>quod relinquitur unà cum dicta portione tertiæ partis ad <lb/>reliquam eiu&longs;dem portionem, ut e l ad l f. </s>

<s>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>

<s><margin.target id="marg117"></margin.target>20. 1. con<gap/><lb/>corum.</s>

<s><margin.target id="marg117"/>20. 1. con<gap/><lb/>corum.</s>

<s>FINIS LIBRI DE CENTRO</s>

<s>GRAVITATIS SOLIDORVM.</s>

<s>FINIS LIBRI DE CENTROGRAVITATIS SOLIDORVM.</s>

<s>Impre&longs;&longs;. </s>

<s>Impre&longs;&longs;. Bononiæ cum licentia Superiorum, <lb/><gap/><lb/> </s>

<s>Bononiæ cum licentia Superiorum, <lb/><gap/><lb/> </s> </chap> </body> <back></back> </text></archimedes>

</chap> </body> <back/> </text></archimedes>

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