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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Commandino, Federico</author> <title>Liber de centro gravitatis solidorum</title> <date>1565</date> <place>Bologna</place> <translator></translator> <lang>la</lang> <cvs_file>comma_centr_01_la_1565</cvs_file> <cvs_version></cvs_version> <locator>0000000023.xml</locator> </info> <text> <front> </front> <body> <chap> <pb/><p type="head"> <s>FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</s> <s>LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM.</s></p><figure></figure><p type="head"> <s>CVM PRIVILEGIO IN ANNOS X.</s></p><p type="head"> <s>BONONIAE,</s></p><p type="head"> <s>Ex Officina Alexandri Benacii.</s></p><p type="head"> <s>M D LXV.</s></p><pb/><p type="head"> <s>ALEXANDRO FARNESIO <lb/>CARDINALI AMPLISSIMO. <lb/>ET OPTIMO.</s></p><p type="main"> <s>Cvm multæ res in mathematicis <lb/>di&longs;ciplinis nequaquam &longs;atis ad­<lb/>huc explicatæ &longs;int, tum perdif­<lb/>ficilis, & perob&longs;cura quæ&longs;tio <lb/>e&longs;t de centro grauitatis corpo­<lb/>rum &longs;olidorum; quæ, & ad co­<lb/>gno&longs;cendum pulcherrima e&longs;t, <lb/>& ad multa, quæ à mathematicis proponuntur, præ­<lb/>clare intelligenda maximum affert adiumentum. </s> <s>de <lb/>qua neminem ex mathematicis, neque no&longs;tra, neque <lb/>patrum no&longs;trorum memoria &longs;criptum reliqui&longs;&longs;e &longs;ci­<lb/>mus. </s> <s>& quamuis in earum monumentis literarum <expan abbr="nõ">non</expan> <lb/>nulla reperiantur, ex quibus in hanc &longs;ententiam addu <lb/>ci po&longs;&longs;umus, vt exi&longs;timemus hanc rem ab <expan abbr="ij&longs;d&etilde;">ij&longs;dem</expan> vber­<lb/>rime tractatam e&longs;&longs;e; tamen ne&longs;cio quo fato adhuc <lb/>in eiu&longs;modi librorum ignoratione ver&longs;amur. </s> <s>Archi­<lb/>medes quidem <expan abbr="mathematicorũ">mathematicorum</expan> princeps in libello, <lb/>cuius in&longs;criptio e&longs;t, <foreign lang="greek">ke/ttraba/rwg ipipe/dwg,</foreign> de centro pla­<lb/>norum copio&longs;i&longs;sime, atque acuti&longs;sime con&longs;crip&longs;it: & <lb/>in co explicando <expan abbr="fummã">fummam</expan> ingenii, & &longs;cientiæ <expan abbr="gloriã">gloriam</expan> e&longs;t <lb/><expan abbr="cõ&longs;ecutus">con&longs;ecutus</expan>. </s> <s>Sed de cognitione <expan abbr="c&etilde;tri">centri</expan> grauitatis <expan abbr="corporũ">corporum</expan> <lb/><expan abbr="&longs;olidorũ">&longs;olidorum</expan> nulla in eius libris litera inuenitur. </s> <s>non mul <lb/>tos abhinc annos MARC<gap/>LLVS <gap/>I. PONT. MAX<gap/><pb/>cum adhuc Cardinalis e&longs;&longs;et, mihi, quæ &longs;ua erat hu­<lb/>manitas, libros ciu&longs;dem Archimedis de ijs, quæ ve­<lb/>huntur in aqua, latine redditos dono dedit. </s> <s>hos cum <lb/>ego, ut aliorum &longs;tudia incitarem, <expan abbr="emendãdos">emendandos</expan>, & <expan abbr="cõ-mentariis">con­<lb/>mentariis</expan> illu&longs;trandos &longs;u&longs;cepi&longs;&longs;em, animaduerti dubi <lb/>tari non po&longs;&longs;e, quin Archimedes vel de hac materia <lb/>&longs;crip&longs;i&longs;&longs;et, vel aliorum mathematicorum &longs;cripta per­<lb/>legi&longs;&longs;et. </s> <s>nam in iis tum alia nonnulla, tum maxime <lb/>illam propo&longs;itionem, ut cuidentem, & aliàs proba­<lb/>tam a&longs;&longs;umit, <expan abbr="Centrũ">Centrum</expan> grauitatis in portionibus conoi <lb/>dis rectanguli axem ita diuidere, vt pars, quæ ad verti <lb/>cem terminatur, alterius partis, quæ ad ba&longs;im dupla <lb/>&longs;it. </s> <s>Verum hæc ad cam partem mathematicarum <lb/>di&longs;ciplinarum præcipue refertur, in qua de centro <lb/>grauitatis corporum &longs;olidorum tractatur. </s> <s>non e&longs;t au <lb/>tem con&longs;entaneum Archimedem illum admirabilem <lb/>virum hanc propo&longs;itionem &longs;ibi argumentis con­<lb/>firmandam exi&longs;timaturum non fui&longs;&longs;e, ni&longs;i eam vel <lb/>aliis in locis probaui&longs;&longs;et, vel ab aliis probatam e&longs;&longs;e <lb/>comperi&longs;&longs;et. </s> <s>quamobrem nequid in iis libris intel­<lb/>ligendis de&longs;iderari po&longs;&longs;et, &longs;tatui hanc etiam partem <lb/>vel à veteribus prætermi&longs;&longs;am, vel tractatam quidem, <lb/>fed in tenebris iacentem, non intactam relinquere; <lb/>atque ex a&longs;sidua mathematicorum, præ&longs;ertim Archi­<lb/>medis lectione, quæ mihi in mentem venerunt, ea in <lb/>medium afferre; ut centri grauitatis corporum &longs;oli­<lb/>dorum, &longs;i non perfectam, at certe aliquam noti-<pb/>tiam haberemus. </s> <s>Qucm meum laborem <expan abbr="nõ">non</expan> mathe­<lb/>maticis&longs;olum, verum iis etiam, qui naturæ ob&longs;curi­<lb/>tate delectantur, <expan abbr="nõ">non</expan> iniucundam fore &longs;peraui:multa <lb/>enim <foreign lang="greek">problh/mxta</foreign> cognitione digni&longs;sima, quæ ad <expan abbr="vtrã-que">vtran­<lb/>que</expan> &longs;cientiam attinent, &longs;e&longs;e legentibus obtuli&longs;&longs;ent.</s> <lb/> <s>neque id vlli mirandum videri debet. </s> <s>vt enim in cor­<lb/>poribus no&longs;tris omnia membra, ex quibus certa quæ <lb/>dam officia na&longs;cuntur, diuino quodam ordine inter <lb/>&longs;e implicata, & colligata &longs;unt: in <expan abbr="iis&qacute;">iisque</expan>; admirabilis il­<lb/>la con&longs;piratio, quam <foreign lang="greek">tn/mpnoixn</foreign> græci vocant, eluce&longs;cit, <lb/>ita tres illæ Philo&longs;ophiæ (ut Ari&longs;totelis verbo vtar) <lb/>quæ veritatem &longs;olam propo&longs;itam habent, licet qui­<lb/>bu&longs;dam qua&longs;i finibus &longs;uis regantur: tamen <expan abbr="earũ">earum</expan> vna­<lb/>quæque per &longs;e ip&longs;am quodammodo imperfecta e&longs;t: <lb/>neque altera &longs;ine alterius auxilio plene comprehen­<lb/>di pote&longs;t. </s> <s>complures præterca mathematicorum no­<lb/>di ante hac explicatu difficillimi nullo negotio expe <lb/>diti e&longs;&longs;ent: atque (ut vno verbo complectar) ni&longs;i <lb/>mea vaide amo, tractationem hanc meam &longs;tudio&longs;is <lb/>non mediocrem vtilitatem, & magnam volupta­<lb/>tem allaturam e&longs;&longs;e mihi per&longs;ua&longs;i. </s> <s>cum autem ad hoc <lb/>&longs;cribendum aggre&longs;&longs;us e&longs;sem, allatus e&longs;t ad me liber <lb/>Franci&longs;ci Maurolici Me&longs;&longs;anen&longs;is, in quo vir ille do­<lb/>cti&longs;simus, & in iis di&longs;ciplinis exercitati&longs;simus af­<lb/>firmabat &longs;e de centro grauitatis corporum &longs;olido­<lb/>rum co<gap/>&longs;crip&longs;i&longs;&longs;e. </s> <s>cum hoc intellexi&longs;&longs;em, &longs;u&longs;tinui <lb/>me pauli&longs;per: tacitus'que expectaui, dum opus cla-<pb/>ris&longs;imi uiri, quem &longs;emper honoris cau&longs;&longs;a nomino, <lb/>in lucem proferretur: mihi enim exploratis&longs;imum <lb/>erat: Franci&longs;cum Maurolicum multo doctius, & <lb/>exqui&longs;itius hoc di&longs;ciplinarum genus &longs;criptis &longs;uis tra <lb/>diturum. </s> <s>&longs;ed cum id tardius fieret, hoc e&longs;t, ut ego <lb/>interpretor, diligentius, mihi diutius hac &longs;criptione <lb/>non &longs;uper&longs;edendum e&longs;&longs;e duxi, præ&longs;ertim cum iam li­<lb/>bri Archimedis de iis, quæ uehuntur in aqua, opera <lb/>mea illu&longs;trati typis <expan abbr="excud&etilde;di">excudendi</expan> e&longs;&longs;ent. </s> <s>nec me alia cau&longs; <lb/>&longs;a impuli&longs;&longs;et, ut de centro grauitatis corporum &longs;oli­<lb/>dorum &longs;criberem, ni&longs;i ut hac etiam ratione lux eis <lb/>quàm maxime fieri po&longs;let afferretur. </s> <s><expan abbr="atq;">atque</expan> id eò mihi <lb/>faciendum exi&longs;timaui, quòd in &longs;pem ueniebam fore, <lb/>ut cum ego ex omnibus mathematicis primus, hanc <lb/>materiam explicandam &longs;u&longs;cepi&longs;&longs;em; &longs;i quid errati for <lb/>te à me commi&longs;&longs;um e&longs;&longs;et, boni uiri potius id meæ de <lb/>&longs;tudio&longs;is hominibus bene <expan abbr="mer&etilde;di">merendi</expan> cupiditati, quàm <lb/>arrogantiæ a&longs;criberent. </s> <s>re&longs;tabat ut con&longs;iderarem, cui <lb/>potis&longs;imum ex principibus uiris contemplationem <lb/>hanc, nunc primum memoriæ, ac literis proditam de <lb/>dicarem. </s> <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="cũ">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="imperiorũ">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> <s>quamuis à te in tot <lb/>præclaros viros tanta beneficia collata &longs;unt, & <expan abbr="confe-rũtur">confe­<lb/>runtur</expan>, vt omnibus te&longs;tatum &longs;it, nihil tibi e&longs;&longs;e charius, <lb/>nihil iucundius, quàm eximia tua liberalitate homi­<lb/>nes ad amplexandam virtutem, licet currentes incita­<lb/>re. </s> <s>nihil dico de ceteris virtutibus tuis, quæ tantæ <lb/>&longs;unt, quantæ ne cogitatione quidem comprehendi <lb/>po&longs;&longs;unt. </s> <s>Quamobrem hac præcipue de cau&longs;&longs;a te hu­<lb/>ius meæ lucubrationis patronum e&longs;&longs;e volui, quam ea, <lb/>qua &longs;oles, humanitate accipies. </s> <s>te enim &longs;emper ob <lb/>diuinas virtutes tuas colui, & ob&longs;eruaui: <expan abbr="nihil&qacute;">nihilque</expan>; mi­<lb/>hi fuit optatius; quàm tibi per&longs;pectum e&longs;&longs;e meum <lb/>erga te animum; <expan abbr="&longs;ingularem&qacute;">&longs;ingularemque</expan>; ob&longs;eruantiam. </s> <s>cœ­<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>Vale.</s></p><p type="main"> <s>Federicus Commandinus.</s></p><pb pagenum="1"/><p type="head"> <s>FEDERICI COMMANDINI <lb/>VRBINATIS LIBER DE CENTRO <lb/>GRAVITATIS SOLIDORVM.</s></p><p type="head"> <s>DIFFINITIONES.</s></p><p type="main"> <s><arrow.to.target n="marg1"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1"></margin.target>1</s></p><p type="main"> <s>Centrvm grauitatis, Pappus <lb/>Alexandrinus in octauo ma­<lb/>thematicarum collectionum <lb/>libro ita diffiniuit.</s></p><p type="main"> <s><foreign lang="greek">le/gomen de/ ne/ntron ba/rons i/a/<gap/>ton da/ <lb/>matos <gap/>einai <gap/>mei<gap/>on ti aei/menon e/nto/s, a/f<gap/><lb/>on<gap/> kat'e/po<gap/>nian a\rtn<gap/>e/n to/ ba/ros n/mer<gap/><lb/>fero/menon, nai\ fula/ssei to/n i/x a/r<gap/>sri­<lb/>sin, on\ mn\ peritrep o/menon i/nt<gap/> fora<gap/>.</foreign> hoc e&longs;t,</s></p><p type="main"> <s>Dicimus autem centrum grauitatis uniu&longs;cu­<lb/>iu&longs;que corporis punctum quoddam intra po&longs;i­<lb/>tum, à quo &longs;i graue appen&longs;um mente concipia­<lb/>tur, dum fertur quie&longs;cit; & &longs;eruat eam, quam in <lb/>principio habebat po&longs;itionem: neque in ip&longs;a la­<lb/>tione circumuertitur.</s></p><p type="main"> <s>Po&longs;&longs;umus etiam hoc modo diffinire.</s></p><p type="main"> <s>Centrum grauitatis uniu&longs;cuiu&longs;que &longs;olidæ figu <lb/>ræ e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>undique partes æqualium momentorum con&longs;i­<lb/>&longs;tunt. </s> <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></p><p type="main"> <s><arrow.to.target n="marg2"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg2"></margin.target>2</s></p><p type="main"> <s>Pri&longs;matis, cylindri, & portionis cylindri axem <lb/>appello rectam lineam, quæ oppo&longs;itorum plano­<lb/>rum centra grauitatis coniungit.</s></p><p type="main"> <s><arrow.to.target n="marg3"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg3"></margin.target>3</s></p><p type="main"> <s>Pyramidis, coni, & portionis coni axem dico li <lb/>neam, quæ à uertice ad centrum grauitatis ba&longs;is <lb/>perducitur.</s></p><p type="main"> <s><arrow.to.target n="marg4"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg4"></margin.target>4</s></p><p type="main"> <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></p><p type="head"> <s>PETITIONES.</s></p><p type="main"> <s><arrow.to.target n="marg5"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg5"></margin.target>1</s></p><p type="main"> <s>Solidarum figurarum fimilium centra grauita­<lb/>tis &longs;imiliter &longs;unt po&longs;ita.</s></p><p type="main"> <s><arrow.to.target n="marg6"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg6"></margin.target>2</s></p><p type="main"> <s>Solidis figuris &longs;imilibus, & æqualibus inter &longs;e <lb/>aptatis, centra quoque grauitatis ip&longs;arum inter &longs;e <lb/>aptata crunt.</s></p><p type="head"> <s>THEOREMA I. PROPOSITIO I.</s></p><p type="main"> <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></p><p type="main"> <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></p><p type="main"> <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>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><lb/>quare in eadem b d linea erit <lb/>centrum grauitatis trianguli, & circuli centrum. </s> <s>Similiter <lb/>diui&longs;a a b bi&longs;ariam in e, & ducta c e, o&longs;tendetur in ip&longs;a <expan abbr="utrũ">utrum</expan> <lb/>que centrum contineri. </s> <s>ergo ea erunt in puncto, in quo li­<lb/>neæ b d, c e conueniunt. </s> <s>trianguli igitur a b c centrum gra<lb/>uitatis e&longs;t idem, quod circuli centrum.</s></p><p type="margin"> <s><margin.target id="marg7"></margin.target><gap/></s></p><p type="margin"> <s><margin.target id="marg8"></margin.target><gap/></s></p><p type="margin"> <s><margin.target id="marg9"></margin.target><gap/></s></p><figure></figure><p type="main"> <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>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>idem igitur e&longs;t centrum <lb/>grauitatis quadrati, & circuli centrum.</s></p><p type="margin"> <s><margin.target id="marg10"></margin.target>3<gap/></s></p><p type="main"> <s>Sit pentagonum æquilaterum, & æquiangulum in circu­<lb/><figure id="fig2"></figure><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>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>Sed trian­<lb/>guli b c d centrum grauitatis e&longs;t in linea c f. </s> <s>ergo in eadem <lb/>linea c h e&longs;t centrum grauitatis trapezij a b d e, & trian­<lb/>guli b c d: hoc e&longs;t pentagoni ip&longs;ius centrum: & centrum <lb/>circuli. </s> <s>Rur&longs;us &longs;i iuncta a d, <expan abbr="bifariam&qacute;">bifariamque</expan>; &longs;ecta in k, duca­<lb/>tur e k l: demon&longs;trabimus in ip&longs;a utrumque centrum in <lb/>e&longs;&longs;e. </s> <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></p><p type="margin"> <s><margin.target id="marg11"></margin.target>4. Primi.</s></p><p type="margin"> <s><margin.target id="marg12"></margin.target>28. primi.</s></p><p type="margin"> <s><margin.target id="marg13"></margin.target>13. Archi­<lb/>medis.</s></p><p type="main"> <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="punctũ">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>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> <s>per&longs;picuum e&longs;tigitur in <lb/>ip&longs;a c f e&longs;&longs;e circuli centrum, & <lb/>centrum grauitatis hexagoni. <lb/></s> <s>Rur&longs;us ducta altera diamctro <lb/>a d, ei&longs;dem rationibus o&longs;tende­<lb/>mus in ip&longs;a utrumque <expan abbr="c&etilde;trum">centrum</expan> <lb/>ine&longs;&longs;e. </s> <s>Centrum ergo grauita­<lb/>tis hexagoni, & centrum circuli idem erit.</s></p><p type="margin"> <s><margin.target id="marg14"></margin.target>13<gap/><lb/>m<gap/><lb/>9.</s></p><p type="main"> <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="pũ">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> <s>Quare & centrum grauitatis heptagoni, & <lb/>centrum circuli in idem punctum conueniunt. </s> <s>Eodem mo <pb/>do in reliquis figuris æquilateris, & æquiangulis, quæ in cir­<lb/>culo de&longs;cribuntur, probabimus <expan abbr="c&etilde;trum">centrum</expan> grauitatis carun<gap/>, <lb/>& centrum circuli idem e&longs;&longs;e. </s> <s>quod quidem demon&longs;trare <lb/>oportebat.</s></p><p type="main"> <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></p><p type="margin"> <s><margin.target id="marg15"></margin.target><foreign lang="greek"><gap/>wrímw</foreign><gap/></s></p><p type="main"> <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></p><p type="head"> <s>THEOREMA II, PROPOSITIO II.</s></p><p type="main"> <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></p><p type="main"> <s>Quo modo figura rectilinea in ellip&longs;i plane de&longs;cribatur, <lb/>docuimus in commentarijs in quintam propo&longs;itionem li­<lb/>bri Archimedis de conoidibus, & &longs;phæroidibus.</s></p><p type="main"> <s>Sit ellip&longs;is a b c d, cuius maior axis a c, minor b d: <expan abbr="iun-gantur&qacute;">iun­<lb/>ganturque</expan>; a b, b c, c d, d a: & bifariam diuidantur in pun­<lb/>ctis e f g h. </s> <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>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>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> <s>ergo c d ip&longs;i b a; <lb/>& a d ip&longs;i b c æquidi­<lb/>&longs;tat. </s> <s>Atuero cum lineæ <lb/>a b, c d inter &longs;e æquidi­<lb/>&longs;tantes bifariam &longs;ecen­<lb/>tur in punctis e g; erit li <lb/>nea l e k g n diameter &longs;e <lb/>ctionis, & linea una, ex <lb/>demon&longs;tratis in uige&longs;i­<lb/>maoctaua &longs;ecundi coni <lb/>corum. </s> <s>Et eadem ratione linea una m f k h o. </s> <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><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> <s>qua <lb/>re & e l in h o, et f m in g n. </s> <s>Atip&longs;a l z in z o; et m <foreign lang="greek">f</foreign> in <foreign lang="greek">f</foreign> n <lb/>cadet. </s> <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>& 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>Itaque dico <lb/>harum figurarum in el­<lb/>lip&longs;i de&longs;criptarum cen­<lb/>trum grauitatis e&longs;&longs;e <expan abbr="pũ-ctum">pun­<lb/>ctum</expan> k, idem quod & el <lb/>lip&longs;is centrum. </s> <s>quadri­<lb/>lateri enim a b c d cen­<lb/>trum e&longs;t k, ex decima e­<lb/>iu&longs;dem libri Archime­<lb/>dis, quippe <expan abbr="cũ">cum</expan> in eo om <lb/>nes diametri <expan abbr="cõueniãt">conueniant</expan>. </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>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="trapeziorũ">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="centrũ">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>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="punctũ">punctum</expan> K, lineis &longs;cilicet a c, <lb/>b d commune, quæ omnia demon&longs;trare oportebat.</s></p><p type="margin"> <s><margin.target id="marg16"></margin.target>8.<gap/></s></p><p type="margin"> <s><margin.target id="marg17"></margin.target>33<gap/></s></p><p type="margin"> <s><margin.target id="marg18"></margin.target>28. primi.</s></p><p type="margin"> <s><margin.target id="marg19"></margin.target>13. Archi <lb/>medis.</s></p><p type="margin"> <s><margin.target id="marg20"></margin.target>Vltima.</s></p><p type="head"> <s>THEOREMA III. PROPOSITIO III.</s></p><p type="main"> <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></p><figure></figure><p type="main"> <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;tratũ">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></p><p type="head"> <s>THEOREMA IIII. PROPOSITIO IIII.</s></p><p type="main"> <s>IN circulo & ellip&longs;i idem e&longs;t figuræ & graui­<lb/>tatis centrum.</s></p><p type="main"> <s>SIT circulus, uel ellip&longs;is, cuius centrum a. </s> <s>Dico a gra­<lb/>uitatis quoque centrum e&longs;&longs;e. </s> <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>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>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>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></p><p type="margin"> <s><margin.target id="marg21"></margin.target>8. quinti</s></p><p type="margin"> <s><margin.target id="marg22"></margin.target>19. quinti <lb/>apud <expan abbr="Cã">Cam</expan> <lb/>panum.</s></p><p type="margin"> <s><margin.target id="marg23"></margin.target>8.<gap/><lb/>m<gap/></s></p><p type="head"> <s>ALITER.</s></p><p type="main"> <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;trũ">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="ĩ">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="punctũ">punctum</expan> g, itaut fit g e æqualis e f. </s> <s>erit g por­<lb/>tionis a b c centrum. </s> <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></p><p type="main"> <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></p><figure></figure><p type="main"> <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></p><p type="head"> <s>THEOREMA V. PROPOSITIO V.</s></p><p type="main"> <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></p><p type="main"> <s>Sit pri&longs;ina, in quo plana oppo&longs;ita &longs;int triangula a b c, <lb/>d e f; axis g h: & &longs;ecetur plano iam dictis planis <expan abbr="æquidi&longs;tã">æquidi&longs;tam</expan> <lb/>te; quod faciat &longs;ectionem k l m; & axi in <expan abbr="pũcto">puncto</expan> n occurrat. </s><lb/> <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>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="parallelogrammũ">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>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. </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> <s>nam cum plana æquidi&longs;tantia &longs;ecen <lb/>tur à plano c q, communes eorum &longs;ectiones c g o, m p, f q <lb/>&longs;ibi ip&longs;is æquidi&longs;tabunt. </s> <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>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>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>Sed & <expan abbr="centrũ">centrum</expan> gra<lb/>uitatis h in g <expan abbr="centrũ">centrum</expan> ca­<lb/>det. </s> <s><expan abbr="trã&longs;ibit">tran&longs;ibit</expan> igitur linea <lb/>f q per h: & planum per <lb/>c o & c f <expan abbr="ductũ">ductum</expan> per <expan abbr="ax&etilde;">axem</expan> <lb/>g h ducetur: <expan abbr="idcircoq;">idcircoque</expan> li <lb/>neam m p <expan abbr="etiã">etiam</expan> per n <expan abbr="trã">tram</expan> <lb/>&longs;ire nece&longs;&longs;e erit. </s> <s>Quo­<lb/>niam ergo fh, c g æqua­<lb/>les &longs;unt, & <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan>: <lb/><expan abbr="itemq;">itemque</expan> h q, g o; rectæ li­<lb/>neæ, quæ ip&longs;as <expan abbr="cõnectũt">connectunt</expan> <lb/>c m f, g n h, o p q æqua­<lb/><gap/><pb pagenum="8"/>æquidi&longs;tant autem c g o, m n p. </s> <s>ergo <expan abbr="parallelogrãma">parallelogramma</expan> &longs;unt <lb/>o n, g m, & linea m n æqualis c g; & n p ip&longs;i g o. </s> <s>aptatis igi­<lb/>tur k l m, a b c <expan abbr="triãgulis">triangulis</expan>, quæ æqualia & &longs;imilia <expan abbr="sũt">sunt</expan>; linea m p <lb/>in c o, & punctum n in g cadet. </s> <s>Quòd <expan abbr="cũ">cum</expan> g &longs;it centrum gra­<lb/>uitatis trianguli a b c, & n trianguli k l m grauitatis cen­<lb/>trum erit id, quod demon&longs;trandum relinquebatur. </s> <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></p><p type="margin"> <s><margin.target id="marg24"></margin.target><gap/><lb/>ci<gap/></s></p><p type="margin"> <s><margin.target id="marg25"></margin.target><gap/></s></p><p type="margin"> <s><margin.target id="marg26"></margin.target>10. unde <lb/>cimi</s></p><p type="margin"> <s><margin.target id="marg27"></margin.target>10. unde­<lb/>cimi</s></p><p type="margin"> <s><margin.target id="marg28"></margin.target>4. &longs;exti</s></p><p type="margin"> <s><margin.target id="marg29"></margin.target>pe<gap/> 5. pe­<lb/>titionem <lb/>Archime <lb/>dis.</s></p><p type="head"> <s>COROLLARIVM.</s></p><p type="main"> <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="ducũtur">ducuntur</expan> æquidi&longs;tare.</s></p><p type="head"> <s>THE OREMA VI. PROPOSITIO VI.</s></p><p type="main"> <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></p><p type="main"> <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>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>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>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>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/> <s>quare ex corolla­<lb/>rio quintæ pro­<lb/>po&longs;itionis primi <lb/>libri Archimedis <lb/>d e centro graui­<lb/>tatis planorum; magnitudinis ex his omnibus compo&longs;itæ <lb/>centrum grauitatis e&longs;t in medio lineæ, quæ magnitudi­<lb/>num mediarum centra coniungit. </s> <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>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></p><p type="margin"> <s><margin.target id="marg30"></margin.target>33.<gap/></s></p><p type="margin"> <s><margin.target id="marg31"></margin.target>5. b<gap/></s></p><p type="margin"> <s><margin.target id="marg32"></margin.target>1. decimi</s></p><p type="margin"> <s><margin.target id="marg33"></margin.target>5 huius</s></p><p type="head"> <s>THEOREMA VII. PROPOSITIO VII.</s></p><p type="main"> <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></p><pb/><p type="main"> <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>Dico centrum grauita­<lb/>tis cylindri, uel cylindri por­<lb/>tionis e&longs;&longs;e in plano e f. </s> <s>Si <expan abbr="enĩ">enim</expan> <lb/>fieri pote&longs;t, fit centrum g: & <lb/>ducatur g h ip&longs;i a d æquidi­<lb/>&longs;tans, u&longs;que ad e f planum. </s><lb/> <s>Itaque linea a e continenter <lb/>diui&longs;a bifariam, erit tandem <lb/>pars aliqua ip&longs;ius k e, minor <lb/>g h. </s> <s>Diuidantur ergo lineæ <lb/>a e, e d in partes æquales ip&longs;i <lb/>k e: & per diui&longs;iones plana ba <lb/>&longs;ibus æquidi&longs;tantia <expan abbr="ducãtur">ducantur</expan>. </s><lb/> <s>eruntiam &longs;ectiones, figuræ æ­<lb/>quales, & &longs;imiles eis, quæ &longs;unt <lb/>in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: & cy <lb/>lindri portio in portiones æquales, & &longs;imiles ip&longs;i k f. </s> <s>reli­<lb/>qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s></p><p type="main"> <s><gap/></s></p><pb pagenum="10"/><figure></figure><p type="head"> <s>THEOREMA VIII. PROPOSITIO VIII.</s></p><p type="main"> <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></p><p type="main"> <s>Sit primum a f pri&longs;ma æquidi&longs;tantibus planis <expan abbr="contcntũ">contcntum</expan>, <lb/>quod &longs;olidum parallelepipedum appellatur: & oppo&longs;ito­<lb/>rum planorum c f, a h, d a, f g latera bifariam diuidantur in <lb/>punctis k l m n o p q r s t u x: & per diui&longs;iones ducantur <lb/>plana k n, o r, s x. </s> <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="centrũ">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="planorũ">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>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></p><p type="margin"> <s><margin.target id="marg34"></margin.target>6 huius</s></p><p type="main"> <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="planũ">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>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> <s>Dico <lb/>ip&longs;um e&longs;&longs;e in linea k m. </s> <s>Si enim fieri potc&longs;t, &longs;it o centrum; <pb pagenum="11"/>& per o ducatur o p ad k m ip&longs;i h g æquidi&longs;tans. </s> <s>Itaquc li <lb/>nea h m <expan abbr="bifariã">bifariam</expan> u&longs;que cò diuidatur, quoad reliqua &longs;it pars <lb/>quædam q m, minor o p. </s> <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>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>Sed ut m z ad z g, ita k r ad r g: & ut m <foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign> h, ita k s <lb/>ad s h. </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>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 funt inter &longs;e, & &longs;imilia <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>triangulo K m h: habebit triangulum K m h ad <expan abbr="triangulũ">triangulum</expan> <lb/>K <foreign lang="greek">d</foreign> y duplam proportioncm eius, quæ e&longs;t lineæ <foreign lang="greek">k</foreign> h ad K y. </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="unũ">unum</expan>: & ad quatuor triangula k <foreign lang="greek">d</foreign> y, y u, <lb/>u s, s <foreign lang="greek">a</foreign> h habebit eandem, quam &longs;exdecim ad quatuor, hoc <lb/>e&longs;t quam h K ad k y: & &longs;imiliter eandem habere demon&longs;tra <lb/><figure id="fig18"></figure><lb/>bitur trian­<lb/>gulum k m g <lb/>ad quatuor <lb/><expan abbr="triãgula">triangula</expan> K <foreign lang="greek">d</foreign><lb/>x, x <foreign lang="greek">g</foreign> t, t <foreign lang="greek">b</foreign> r, <lb/><arrow.to.target n="marg39"></arrow.to.target><lb/>r z g. </s> <s>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>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="triangulũ">triangulum</expan> g h <foreign lang="greek">k</foreign> ad omnia triangula, ita <expan abbr="to-tũ">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>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>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> <s>Itaque cum à pri&longs;mate a f, cuius <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis e&longs;t o, auferatur magnitudo ex &longs;olidis parallelepi <lb/>pedis y <foreign lang="greek">g,</foreign> u <foreign lang="greek">b,</foreign> s z con&longs;tans: atque ipfius grauitatis centrum <lb/>&longs;it <foreign lang="greek">q:</foreign> reliquæ magnitudinis, quæ ex omnibus pri&longs;matibus <lb/>con&longs;tat, grauitatis centrum erit in linea <foreign lang="greek">q</foreign> o producta: & <lb/><gap/><pb/>medis. </s> <s>ergo punctum <foreign lang="greek">n</foreign> extra p ri&longs;ma a f po&longs;itum, <expan abbr="centrũ">centrum</expan> <lb/>erit magnitudinis <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> e x omnibus pri&longs;matibus g z r, <lb/>r <foreign lang="greek">b</foreign> t, t <foreign lang="greek">g</foreign> x, x <foreign lang="greek">d</foreign> k, k <foreign lang="greek">d</foreign> y, y u, u s, s <foreign lang="greek">a</foreign> h, quod fieri nullo modo po <lb/>te&longs;t. </s> <s>e&longs;t enim ex diffinitione centr um grauitatis &longs;olidæ figu <lb/>ræ intra ip&longs;am po&longs;itum, non extra. </s> <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="pũctum">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> <s>&longs;it autem planorum c f n l, <lb/>a d <foreign lang="greek">p x</foreign> communis &longs;ectio linea <foreign lang="greek">r s t;</foreign> quæ quidem pri&longs;matis <lb/>axis erit, cum tran&longs;eat per centra grauitatis triangulorum <lb/>a b c, g h <foreign lang="greek">k,</foreign> d e f, ex quartadecima eiu&longs;dem. </s> <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></p><p type="margin"> <s><margin.target id="marg35"></margin.target>5.huius</s></p><p type="margin"> <s><margin.target id="marg36"></margin.target>2. &longs;e<gap/><lb/>. </s> <s>q<gap/></s></p><p type="margin"> <s><margin.target id="marg37"></margin.target>2. &longs;e<gap/></s></p><p type="margin"> <s><margin.target id="marg38"></margin.target><gap/>. </s> <s>&longs;exti</s></p><p type="margin"> <s><margin.target id="marg39"></margin.target>uel 127 <lb/>utnti.</s></p><p type="margin"> <s><margin.target id="marg40"></margin.target>3.<gap/></s></p><p type="margin"> <s><margin.target id="marg41"></margin.target>28.<gap/><lb/>cin<gap/></s></p><p type="margin"> <s><margin.target id="marg42"></margin.target>15.<gap/></s></p><p type="margin"> <s><margin.target id="marg43"></margin.target>19.<gap/><lb/>apu<gap/><lb/>pan<gap/></s></p><p type="main"> <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>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>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>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></p><p type="margin"> <s><margin.target id="marg44"></margin.target>5.h<gap/></s></p><p type="main"> <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="tagonũ">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>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> <s>& itidem <expan abbr="iungãtur">iungantur</expan> r t, &longs;u, <lb/>x y. </s> <s>erit in lincar t <expan abbr="c&etilde;trum">centrum</expan> gra<lb/>uitatis pentagoni a b c d e; <lb/>quod &longs;it z: & in linea &longs;u cen­<lb/>trum pentagoni f g h <foreign lang="greek">k</foreign> l:&longs;it au <lb/>tem <foreign lang="greek">x:</foreign> & ducatur z <foreign lang="greek">x,</foreign> quæ di­<lb/>cto plano in <foreign lang="greek">y</foreign> occurrat. </s> <s><expan abbr="Itaq;">Itaque</expan> <lb/>punctum x e&longs;t centrum graui <lb/>tatis trianguli m n q, ac pri&longs;­<lb/>matis a l: & y grauitatis centrum quadrilateri n o p q, ac <lb/>pri&longs;matis b k. </s> <s>quare y centrum erit pentagoni m n o p <expan abbr="q.">que</expan> & <pb pagenum="14"/>&longs;imiliter demon&longs;irabitur totius pri&longs;matis a K grauitatis ef <lb/>&longs;e centrum. </s> <s>Simili ratione & in aliis pri&longs;matibus illud <lb/>idem facile demon&longs;trabitur. </s> <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></p><p type="main"> <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>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>Dico punctum K cylindri quoque, uel cy <lb/>lindri portionis grauitatis centrum e&longs;&longs;e. </s> <s>Si enim fieri po­<lb/>te&longs;t, &longs;it l centrum: <expan abbr="ducaturq;">ducaturque</expan> k l, & extra figuram in m pro­<lb/>ducatur. </s> <s>quam ucro proportionem habet linea m K ad k l <pb/>habcat circulus, uel ellip&longs;is g h ad aliud &longs;pacium, in quo u: <lb/>& in cit culo, uel ellip&longs;i plane de&longs;cribatur rectilinea figura, <lb/>ita ut <expan abbr="tãdem">tandem</expan> <expan abbr="relinquãtur">relinquantur</expan> portiones minores &longs;pacio u, quæ <lb/>&longs;it o p g q r s h t: <expan abbr="de&longs;criptaq;">de&longs;criptaque</expan> &longs;imili figura in oppo&longs;itis pla­<lb/>nis c d, f e, per lineas &longs;ibi ip&longs;is re&longs;pondentes plana <expan abbr="ducãtur">ducantur</expan>. </s><lb/> <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>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> <s>Sed ut circulus <lb/>uel ellip&longs;is g h ad figuram rectilincam in ip&longs;a de&longs;cri­<lb/>ptam, ita e&longs;t cylindrus uel cylindri portio c c ad pri&longs;ma, <lb/>quod rectilineam figuram pro ba&longs;i habet, & altitudinem <lb/>æqualem; id, quod infra demon&longs;trabitur. </s> <s>crgo per conuer <lb/>&longs;ionem rationis, ut circulus, uel ellip&longs;is g h ad portioncs re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad &longs;olidas por­<lb/>tiones, quate cylindrus uel cylindri portio ad &longs;olidas por­<lb/>tiones eandem proportionem habet, quam linea n <foreign lang="greek">k</foreign> ad k <lb/>& diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o­<lb/>lidas portiones eandem proportionem habet, quam n l ad <lb/>l k & quoniam a cylindro uel cylindri portione, cuius gra­<lb/>uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili­<lb/>neam <expan abbr="figurã">figuram</expan>, cuius <expan abbr="centrũ">centrum</expan> grauitatis e&longs;t K: re&longs;iduæ magnitu <lb/>dinis ex &longs;olidis portionibus <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> grauitatis <expan abbr="c&etilde;trũ">centrum</expan> crit <lb/>in linea k l protracta, & in puncto n; quod e&longs;t <expan abbr="ab&longs;urdũ">ab&longs;urdum</expan>. </s> <s>relin <lb/>quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>tionis &longs;it <expan abbr="punctũ">punctum</expan> k. </s> <s>quæ omnia <expan abbr="demon&longs;trãda">demon&longs;tranda</expan> propo&longs;uimus.</s></p><p type="margin"> <s><margin.target id="marg45"></margin.target>4. hu<gap/></s></p><p type="main"> <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></p><p type="main"> <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>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> <s>Dico <expan abbr="conũ">conum</expan> uel coni portione x pyramidi y <expan abbr="æ-qual&etilde;">æ­<lb/>qualem</expan> e&longs;&longs;e. </s> <s>ni&longs;i enim &longs;it æqualis, uel maior, uel minor crit.</s></p><p type="main"> <s>Sit primum maior, et exuperet &longs;olido z. </s> <s>Itaque in circu <lb/>lo, uel ellip&longs;i x de&longs;cribatur figura rectilinea; & in ea pyra­<lb/>mis candem, quam conus, uel coni portio altitudinem ha­<lb/>bens, ita ut portiones relictæ minores &longs;int &longs;olido z, quem­<lb/>admodum docetur in duodecimo libro elementorum pro <lb/>po&longs;itione undecima. </s> <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>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>ergo pyramis x pyramide y minor erit. </s><lb/> <s>Sed & maior; quod ficri <expan abbr="nõ">non</expan> pote&longs;t. </s> <s>At &longs;i conus, uel coni por <lb/>tio x ponatur minor pyramide y: &longs;it alter conus æque al­<lb/>tus, uel altera coni portio X ip&longs;i pyramidi y æqualis. </s> <s>crit <lb/>eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x, <lb/>quorum exce&longs;&longs;us &longs;it &longs;pacium <foreign lang="greek">w.</foreign> Si igitur in circulo, uel eili­<lb/>p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relictæ <lb/>&longs;int <foreign lang="greek">w</foreign> &longs;pacio minores, ciu&longs;modi figura adhuc maior erit cir <lb/>culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. </s> <s>& pyramis in <lb/>ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi­<lb/>nor pyramide y. </s> <s>e&longs;t ergo ut X figura rectilinea ad figuram <lb/>rectilineam y, ita pyramis X ad pyramidem y. </s> <s>quare cum <lb/>figura rectilinea X &longs;it maior figura y: erit & pyramis X py­<lb/>ramide y maior. </s> <s>&longs;ed erat minor; quod rur&longs;us fieri non po­<lb/>te&longs;t. </s> <s>non e&longs;t igitur conus, uel coni portio x neque maior, <lb/>neque minor pyramide y. </s> <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><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>quod demon&longs;trandum fuerat.</s></p><p type="margin"> <s><margin.target id="marg46"></margin.target>6. duo<gap/><lb/>m.<gap/></s></p><p type="margin"> <s><margin.target id="marg47"></margin.target>7. q<gap/></s></p><p type="head"> <s>THEOREMA IX. PROPOSITIO IX.</s></p><p type="main"> <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></p><pb/><p type="main"> <s>SIT pyramis, cuius ba&longs;is triangulum a b c; axis d c: & <lb/>&longs;ecetur plano ba&longs;i æquidi&longs;tante; quod <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> &longs;aciat f g h; <lb/><expan abbr="occurratq;">occurratque</expan> axi in puncto <foreign lang="greek">k.</foreign> Dico f g h triangulum e&longs;&longs;e, ip&longs;i <lb/>a b c &longs;imile; cuius grauitatis centrum e&longs;t K. <expan abbr="Quoniã">Quoniam</expan> enim <lb/>duo plana æquidi&longs;tantia a b c, f g h &longs;ecantur à plano a b d; <lb/>communes corum &longs;ectiones a b, f g æquidi&longs;tantes erunt: & <lb/>cadem ratione æquidi&longs;tantes ip&longs;æ b c, g h: & c a, h f. </s> <s>Quòd <lb/>cum duæ lineæ f g, g h, duabus a b, b c æquidi&longs;tent, nec <lb/>fintin eodem plano; angulus ad g æqualis e&longs;t angulo ad <lb/>b. </s> <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>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> <s>ergo puncta <lb/>e K in triangulis a b c, f g h <lb/>&longs;imiliter &longs;unt po&longs;ita, per &longs;e­<lb/>xtam po&longs;itionem Archime­<lb/>dis in libro de centro graui­<lb/>tatis planorum. </s> <s>Sed cum e <lb/>&longs;it centrum grauitatis trian <lb/>guli a b c, erit ex undecima <lb/>propo&longs;itione eiu&longs;dem libri, <lb/>& K trianguli f g h grauita <lb/>tis centrum. </s> <s>id quod demon&longs;trare oportebat. </s> <s>Non aliter <lb/>in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra­<lb/><gap/></s></p><pb pagenum="17"/><p type="head"> <s>PROBLEMA I. PROPOSITIO X.</s></p><p type="main"> <s>DATA qualibet pyramide, fieri pote&longs;t, ut fi­<lb/>gura &longs;olida in ip&longs;a in &longs;cribatur, & altera <expan abbr="circũ&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="quacũque">quacunque</expan> &longs;olida ma <lb/>gnitudine propo&longs;ita.</s></p><figure></figure><p type="main"> <s>Sit pyramis, cuius ba&longs;is <lb/><expan abbr="triangulũ">triangulum</expan> a b c; axis d e. </s><lb/> <s><expan abbr="Sitq;">Sitque</expan> pri&longs;ma, quod <expan abbr="eand&etilde;">eandem</expan> <lb/>ba&longs;im habeat, & axem eun <lb/>dem. </s> <s>Itaque hoc pri&longs;ma­<lb/>te continenter &longs;ecto bifa­<lb/>riam, plano ba&longs;i <expan abbr="æquidi&longs;tã">æquidi&longs;tam</expan> <lb/>te, <gap/>relinquetur <expan abbr="tãdem">tandem</expan> pri&longs; <lb/>ma quoddam minus pro­<lb/>po&longs;ita magnitudine: quod <lb/>quidem ba&longs;im eandem ha <lb/>beat, quam pyramis, & a­<lb/>xem e f. </s> <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> <s>re­<lb/>linquitur ergo, ut circum&longs;cripta figura exuperet <expan abbr="in&longs;criptã">in&longs;criptam</expan> <lb/>pri&longs;mate, quod ba&longs;im habet a b c triangulum, & axem e f. </s><lb/> <s>Illud uero minus e&longs;t &longs;olida magnitudine propo&longs;ita. </s> <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></p><p type="head"> <s>PROBLEMA II. PROPOSITIO XI.</s></p><p type="main"> <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></p><p type="main"> <s>SIT conus, cuius axis b d: & &longs;ecetur plano per axem <lb/>ducto, 'ut&longs;ectio &longs;it triangulum a b c: <expan abbr="intelligaturq;">intelligaturque</expan> cylin­<lb/>drus, qui ba&longs;im eandem, & eundem axem habeat. </s> <s>Hocigi­<lb/>tur cylindro continenter bifariam &longs;ecto, relinquetur cylin <lb/>drus minor &longs;olida magnitudine propo&longs;ita. </s> <s>Sit autem is cy <lb/>lindrus, qui ba&longs;im habet circulum circa diametrum a c, & <lb/>axem d e. </s> <s>Itaque diuidatur b d in partes æquales ip&longs;i d e <lb/>in punctis f g h K l m: & per ea ducantur plana conum &longs;e­<lb/>cantia; quæ ba&longs;i æquidi&longs;tent. </s> <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="circũ-&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="cylĩdro">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/> <s>quare con&longs;tat <expan abbr="circũ-&longs;criptam">circun­<lb/>&longs;criptam</expan> figuram &longs;u­<lb/>perare in&longs;criptam cy <lb/>lindro, cuius ba&longs;is e&longs;t <lb/>circulus circa diametrum a c, & axis d e. </s> <s>atque hic e&longs;t mi­<lb/>nor &longs;olida magnitudine propo&longs;ita.</s></p><p type="head"> <s>PROBLEMA III. PROPOSITIO XII.</s></p><p type="main"> <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></p><pb/><p type="main"> <s>Figuram ciu&longs;modi, & in&longs;cribemus, & <expan abbr="circũ&longs;cribemus">circun&longs;cribemus</expan>, ita <lb/>ut in cono dictum e&longs;t.</s></p><figure></figure><p type="head"> <s>PROBLEMA IIII. PROPOSITIO XIII.</s></p><p type="main"> <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></p><p type="main"> <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></p><pb pagenum="19"/><figure></figure><p type="head"> <s>THEOREMA X. PROPOSITIO XIIII.</s></p><p type="main"> <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></p><p type="main"> <s>SIT pyramis, cuius ba&longs;is triangulum a b c: & axis d e. </s><lb/> <s>Dico in linea d e ip&longs;ius grauitatis centrum ine&longs;&longs;e. </s> <s>Si enim <lb/>fieri pote&longs;t, &longs;it centrum f: & ab f ducatur ad ba&longs;im pyrami <lb/>dis linea f g, axi æquidi&longs;tans: <expan abbr="iunctaq;">iunctaque</expan> e g ad latera trian­<lb/>guli a b c producatur in h. </s> <s>quam uero proportionem ha­<lb/>bet linea h e ad e g, habeat pyramis ad aliud &longs;olidum, in <lb/>quo K: <expan abbr="in&longs;cribaturq;">in&longs;cribaturque</expan> in pyramide &longs;olida figura, & altera cir <lb/>cum&longs;cribatur ex pri&longs;matibus æqualem habentibus altitu­<lb/>dinem, ita ut circum&longs;cripta in&longs;criptam exuperet magnitu­<lb/>dine, quæ &longs;olido k &longs;it minor. </s> <s>Et quoniam in pyramide pla <lb/>num ba&longs;i æquidi&longs;tans ductum &longs;ectionem facit figuram &longs;i­<lb/>milem ei, quæ e&longs;t ba&longs;is; <expan abbr="centrumq;">centrumque</expan> grauitatis in axe haben <lb/>tem: erit pri&longs;matis s t grauitatis <expan abbr="centrũ">centrum</expan> in linear <expan abbr="q;">que</expan> pri&longs;­<lb/>matis u x centrum in linea q p, pri&longs;inatis y z in linea p o; <lb/>pri&longs;matis <foreign lang="greek">h q</foreign> in linea o n; pri&longs;matis <foreign lang="greek">l <gap/></foreign> in linea n m; pri&longs;­<lb/>matis <foreign lang="greek">n p</foreign> in m l; & denique pri&longs;matis <foreign lang="greek">p s</foreign> in l e. </s> <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>Sit autem <foreign lang="greek">r</foreign>: & iun­<lb/>cta <foreign lang="greek">r</foreign> e protendatur, ut cum linea, quæ à <expan abbr="pũcto">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> <s>ergo ad partem exce&longs;&longs;us, quæ <lb/>intra ip&longs;ius &longs;uperficiem comprehenditur, multo m aiorem <lb/>proportionem habebit. </s> <s>habeat eam, quam <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. </s> <s>erit <pb pagenum="21"/>diuidendo figura &longs;olida in&longs;cripta ad dictam exce&longs;&longs;us par­<lb/>tem, ut <foreign lang="greek">t e</foreign> ad c <foreign lang="greek">p.</foreign> & quoniam à cono, &longs;cu coni portione, <lb/>cuius grauitatis centrum e&longs;t e, au&longs;ertur figura in&longs;cripta, <lb/>cuius centrum <foreign lang="greek">r:</foreign> re&longs;iduæ magnitudinis compo&longs;itæ cx par <lb/>te exce&longs;&longs;us, quæ intra coni, uel coni portionis &longs;uper&longs;iciem <lb/>continetur, centrum grauitatis erit in linea e protracta, <lb/>atque in puncto t. </s> <s>quod c&longs;t ab&longs;urdum. </s> <s><expan abbr="cõ&longs;tat">con&longs;tat</expan> ergo <expan abbr="centrũ">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></p><p type="head"> <s>THEOREMA XI. PROPOSITIO XV.</s></p><p type="main"> <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></p><p type="main"> <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></p><figure></figure><pb/><p type="head"> <s>THEOREMA XII. PROPOSITIO XVI.</s></p><p type="main"> <s>In &longs;phæra, & &longs;phæroidc idem e&longs;t grauitatis, & <lb/>figuræ centrum.</s></p><p type="main"> <s>Secetur &longs;phæra, uel &longs;phæroides piano per axem ducto; <lb/>quod &longs;ectionem &longs;aciat circulum, uel cllip&longs;im a b c d, cuius <lb/>diameter, & &longs;phæræ, uel &longs;phæroidis axis d b; & centrum e. </s><lb/> <s>Dico e grauitatis etiam centrum e&longs;&longs;e. </s> <s>&longs;ecetur enim altero <lb/>plano per e, ad planum &longs;ecans recto, cuius &longs;ectio &longs;it circu­<lb/>lus cir ca diametrum a ç. </s> <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="grauĩ">grauim</expan> <lb/>tatis ponatur e&longs;&longs;c f & fiatip&longs;i f e æqualis e g. </s> <s><expan abbr="punctũ">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>ntra g. </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></p><pb pagenum="22"/><p type="margin"> <s><margin.target id="marg48"></margin.target>per 2. pe­<lb/>tition<gap/>m</s></p><p type="margin"> <s><margin.target id="marg49"></margin.target>4 Arcl ­<lb/>medis.</s></p><p type="main"> <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></p><figure></figure><p type="main"> <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></p><p type="head"> <s>THEOREMA XIII. PROPOSITIO XVII.</s></p><figure></figure><p type="main"> <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></p><p type="main"> <s>Sit pyramis, cuius ba&longs;is trian <lb/>gulum a b c, axis d e: <expan abbr="&longs;itq;">&longs;itque</expan> trian <lb/>guli b d c grauitatis centrum f: <lb/>& iungatur a &longs;. </s> <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></p><p type="head"> <s>THEOREMA XIIII. PROPOSITIO XVIII.</s></p><p type="main"> <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></p><figure></figure><p type="main"> <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="nũ">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>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;olidũ">&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>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></p><p type="margin"> <s><margin.target id="marg50"></margin.target><gap/> undeci <lb/><gap/>.</s></p><p type="margin"> <s><margin.target id="marg51"></margin.target><expan abbr="&longs;extĩ">&longs;extim</expan>.</s></p><p type="margin"> <s><margin.target id="marg52"></margin.target>17.<gap/><lb/>cin<gap/></s></p><p type="head"> <s>THEOREMA XV. PROPOSITIO XIX.</s></p><p type="main"> <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="habebũt">habebunt</expan>.</s></p><p type="main"> <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><expan abbr="crũt">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></p><p type="margin"> <s><margin.target id="marg53"></margin.target>29.1<gap/><lb/>cim<gap/></s></p><p type="margin"> <s><margin.target id="marg54"></margin.target>18.<gap/></s></p><p type="margin"> <s><margin.target id="marg55"></margin.target>7. quinti.</s></p><p type="main"> <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> <s>Dico &longs;olidum a b ad <lb/>&longs;olidum c d eandem habere proportionem, quam b e ad <lb/>d f. </s> <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="cũ">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;olidũ">&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;olidũ">&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>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></p><p type="margin"> <s><margin.target id="marg56"></margin.target>31. unde <lb/>cimi</s></p><p type="margin"> <s><margin.target id="marg57"></margin.target>18. huius</s></p><p type="margin"> <s><margin.target id="marg58"></margin.target>7. quinti.</s></p><p type="main"> <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>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> <s>Si enini <lb/>axes in eadem recta linea fuerint con&longs;tituti, bæc dao lo'i­<lb/>da, in unum, atque idem &longs;olidum conuenient. </s> <s>quare <gap/>x <lb/>iis, quæ proxime tradita &longs;unt, habebit &longs;olidum a b ad &longs;o­<lb/>lidum a c eandem proportionem, quam axis d e ad e f <lb/>axem. </s> <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>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/> <s>Sed &longs;olidum a b <lb/>ad &longs;olidum a c <lb/>candem propor­<lb/>tionem habet, <lb/>quam d g altitu­<lb/>do ad <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>f h. </s> <s>ergo & <expan abbr="ean-d&etilde;">ean­<lb/>dem</expan> habebit, <expan abbr="quã">quam</expan> <lb/>axis d e a l e <gap/> <expan abbr="ax&etilde;">axem</expan></s></p><p type="main"> <s>Po&longs;tremo &longs;int <lb/>&longs;olidi paral le pi <lb/>peda a b, c d in <pb/>æqualibus ba&longs;ibus, quorum axes cum ba&longs;ibus æquales an <lb/>gulos faciant. </s> <s>Dico &longs;olidum a b <expan abbr="ad&longs;olidũ">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>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>ergo & ut axis e f ad axem g h. </s> <s>quæ omnia de <lb/>mon&longs;trarc oportebat.</s></p><p type="main"> <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="habentiũ">habentium</expan> dupla; & pyramidum &longs;extupla.</s></p><p type="margin"> <s><margin.target id="marg59"></margin.target>15. quinti</s></p><p type="margin"> <s><margin.target id="marg60"></margin.target>28. unde­<lb/>cimi.</s></p><p type="margin"> <s><margin.target id="marg61"></margin.target>7. duode­<lb/>cimi.</s></p><p type="head"> <s>THEOREMA XVI. PROPOSITIO XX.</s></p><p type="main"> <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></p><pb pagenum="25"/><p type="main"> <s>Sint duo pri&longs;mata a e, a f, quorum eadem ba&longs;is quadri­<lb/>latera a b c d: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis a e altitudo e g; & pri&longs;matis <lb/>a f altitudo f h. </s> <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>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>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></p><p type="margin"> <s><margin.target id="marg62"></margin.target><gap/>2. qu<gap/></s></p><p type="main"> <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>quæ <expan abbr="ĩtcr&longs;e">intcr&longs;e</expan> æquales <expan abbr="erũt">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="harũ">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;itũ">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>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;tituũ">con&longs;tituum</expan> <lb/>tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s></p><p type="margin"> <s><margin.target id="marg63"></margin.target>6. du<gap/><lb/>cimi<gap/></s></p><p type="margin"> <s><margin.target id="marg64"></margin.target>15. qu<gap/></s></p><figure></figure><p type="main"> <s>Sint deinde pri&longs;mata a e, a f in eadem ba&longs;i a b c d; <expan abbr="quorũ">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>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. </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></p><figure></figure><p type="main"> <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> <s>Dico pri&longs;ma <lb/>a e ad pri&longs;ma <foreign lang="greek">k</foreign> o ita e&longs;&longs;e, ut f g ad p <expan abbr="q.">que</expan> iunctis enim g h, <pb/>q r, eodem, quo &longs;upra, modo o&longs;tendemns f g ad p q, ut f h <lb/>ad p r. </s> <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>quod <lb/><expan abbr="demon&longs;trã">demon&longs;tram</expan> <lb/><expan abbr="dũ">dum</expan> &longs;uerat.</s></p><p type="main"> <s>Simili ra <lb/>tione in a­<lb/>liis pri&longs;ma­<lb/>tibus & py <lb/>ramidibus eadem demon&longs;trabuntur.</s></p><p type="head"> <s>THEOREMA XVII. PROPOSITIO XXI.</s></p><p type="main"> <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></p><p type="main"> <s>Sint duo pri&longs;mata a e, g m: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis a e ba&longs;is qua <lb/>drilaterum a b c d, & altitudo e f: pri&longs;matis uero g m ba­<lb/>&longs;is quadrilaterum g h K l, & altitudo m n. </s> <s>Dico pri&longs;ma a e <lb/>ad pri&longs;ma g m proportionem habere compo&longs;itam ex pro <lb/>portione ba&longs;is a b c d ad ba&longs;im g h k l, & ex proportione <lb/>altitudinis e f, ad altitudinem m n.</s></p><p type="main"> <s>Sint enim primum e f, m n æquales: & ut ba&longs;is a b c d <lb/>ad ba&longs;im g h <foreign lang="greek">k</foreign> l, ita fiat linea, in qua o ad lineam, in qua p: <lb/>ut autem e f ad m n, ita linea p ad lineam <expan abbr="q.">que</expan> erunt lineæ <lb/>p q inter &longs;e æquales. </s> <s>Itaque pri&longs;ma a e ad pri&longs;ma g m <expan abbr="eã">eam</expan> <pb pagenum="27"/>proportionem habet, quam ba&longs;is a b c d ad ba&longs;im g h <foreign lang="greek">k</foreign> l: <lb/>&longs;i enim intelligantur duæ pyramides a b c d e, g h k l m, ha­<lb/>bebunt hæ inter &longs;e proportionem eandem, quam ip&longs;ar um <lb/>ba&longs;es ex &longs;exta duodecimi elementorum. </s> <s>Sed ut ba&longs;is a b c d <lb/>ad g h K l ba&longs;im, ita linea o ad lineam p; hoc e&longs;t ad lineam q <lb/>ei æqualem. </s> <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>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>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> <s>Sed proportio o ad <lb/>u <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> e&longs;t ex proportione o ad p, quæ e&longs;t ba&longs;is a b c d <lb/>ad ba&longs;im g h <foreign lang="greek">k</foreign> l; & ex proportione p ad u, quæ e&longs;t altitudi­<lb/>nis e f ad altitudinem m n. </s> <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;iũ">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>quod qui­<lb/>dem demon&longs;tra&longs;&longs;e oportebat.</s></p><p type="margin"> <s><margin.target id="marg65"></margin.target>20. h<gap/></s></p><p type="main"> <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></p><p type="head"> <s>THEOREMA XVIII. PROPOSITIO XXII.</s></p><p type="main"> <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></p><p type="main"> <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="centrũ">centrum</expan> g; & trianguli a d b &longs;it h: & iungantur a f, <lb/>b g, c h. </s> <s>Quoniam igitur <expan abbr="centrũ">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="punctũ">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"></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="enĩ">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. </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>pyramides enim, quæ ab <expan abbr="eod&etilde;">eodem</expan> <pb/>&longs;unt uertice, eandem proportionem habent, quam <expan abbr="ip&longs;arũ">ip&longs;arum</expan> <lb/>ba&longs;es. </s> <s>eadem ratione pyramis a c l k pyramidi b c l <foreign lang="greek">k:</foreign> & py <lb/>ramis a d l k ip&longs;i b d l <foreign lang="greek">k</foreign> pyramidi æqualis erit. </s> <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>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> <s>ducto denique alio plano per lineas c a, <lb/>a f: ut eius, & trianguli c d b communis &longs;ectio &longs;it c fn, py­<lb/>ramis a b c k pyramidi a c d <foreign lang="greek">k</foreign> æqualis demon&longs;trabitur. </s> <s><expan abbr="cũ">cum</expan> <lb/>ergo tres pyramides b c d k, a b d k, a b c k uni, & eidem py <lb/>ramidi a c d k &longs;int æquales, omnes inter &longs;e &longs;e æquales <expan abbr="erũt">erunt</expan>. </s><lb/> <s>Sed ut pyramis a b c d ad pyramidem a b c <foreign lang="greek">k,</foreign> ita d e axis ad <lb/>axem <foreign lang="greek">k</foreign> e, ex uige&longs;ima propo&longs;itione huius: &longs;unt enim hæ <lb/>pyramides in eadem ba&longs;i, & axes cum ba&longs;ibus æquales con <lb/>tinent angulos, quòd in eadem recta linea con&longs;tituantur. </s><lb/> <s>quare diuidendo, ut tres pyramides a c d k, b c d K, a b d K <lb/>ad pyramidem a b c K, ita d k ad K e. </s> <s>con&longs;tat igitur lineam <lb/>d K ip&longs;ius K e triplam e&longs;&longs;e. </s> <s>&longs;ed & a <foreign lang="greek">k</foreign> tripla e&longs;t K f: itemque <lb/>b K ip&longs;ius K g: & c <foreign lang="greek">k</foreign> ip&longs;ius <foreign lang="greek">k</foreign> l tripla. </s> <s>quod eodem modo <lb/>demon&longs;trabimus.</s></p><p type="margin"> <s><margin.target id="marg66"></margin.target>17<gap/></s></p><p type="margin"> <s><margin.target id="marg67"></margin.target><emph type="italics"/>ucrfex<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg68"></margin.target><gap/></s></p><p type="margin"> <s><margin.target id="marg69"></margin.target>5. d<gap/><lb/>cim<gap/></s></p><p type="main"> <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>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>ducta igitur l m ip&longs;i h m lineæ æquidi&longs;tabit. </s> <s>nam el ad <pb pagenum="29"/>lbeandem habet proportionem, quam e m ad m k, uideli­<lb/>cet triplam. </s> <s>quare lineal m ip&longs;am e f &longs;ecabit in punctog: <lb/>etenim e g ad g f e&longs;t, ut el adlh. </s> <s>præterea quoniam h k, l m <lb/>æquidi&longs;tant, erunt triangula h e f, l e g &longs;imilia: <expan abbr="itemq;">itemque</expan> inter <lb/>&longs;e &longs;imilia fe <foreign lang="greek">k,</foreign> gem: & ut e fad e g, itah fad l g: & ita f K ad <lb/>gm. </s> <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="punctũ">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> <s>ex quo <lb/>&longs;equitur, ut totius pyramidis <lb/>a b c d e punctum g &longs;it grauitatis <lb/>centrum. </s> <s>Rur&longs;us &longs;it pyramis ba­<lb/>&longs;im habens pentagonum a b c d e: <lb/>& axem f g: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="pũ">pum</expan> <lb/>cto h, ita ut fh ad h g triplam habe <lb/>at proportionem. </s> <s>Dico h grauita­<lb/>tis <expan abbr="centrũ">centrum</expan> e&longs;&longs;e pyramidis a b c d e f. </s><lb/> <s>iungatur enim e b: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>pyramis, cuius uertex f, & ba&longs;is <lb/>triangulum a b e: & alia pyramis <lb/>intelligatur eundem uerticem ha­<lb/>bens, & ba&longs;im b c d e <expan abbr="quadrilaterũ">quadrilaterum</expan>: <lb/>&longs;it autem pyramidis a b e faxis f <foreign lang="greek">k,</foreign><lb/>& grauitatis centrum l: & pyrami <lb/>dis b c d e faxis f m, & centrum gra <lb/><gap/> <expan abbr="iunganturq;">iunganturque</expan> <foreign lang="greek">k</foreign> m, ln; <lb/>quæ per puncta g h tran&longs;ibunt. </s><lb/> <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></p><p type="margin"> <s><margin.target id="marg70"></margin.target>2. fexti.</s></p><p type="main"> <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>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="relinquũ-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>&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>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> <s>quare conus uel coni portio ad portiones <lb/>&longs;olidas maiorem habet proportionem, quam g e ad e f: & <lb/>diuidendo, pyramis ad portiones &longs;olidas maiorem pro­<lb/>portionem habet, quam g f ad f e. </s> <s>fiatigitur q f ad f e <lb/>ut pyramis ad dictas portiones. </s> <s>Itaque quoniam a cono <lb/>uel coni portione, cuius grauitatis centrum e&longs;t f, aufer­<lb/>tur pyramis, cuius centrum e; reliquæ magnitudinis, <lb/>quæ ex &longs;olidis portionibus con&longs;tat, centrum grauitatis <lb/>erit in linea e f protracta, & in puncto <expan abbr="q.">que</expan> quod fieri <lb/>non pote&longs;t: e&longs;t enim centrum grauitatis intra. </s> <s>Con&longs;tat <lb/>igitur coni, uel coni portionis grauitatis centrum e&longs;&longs;e pun <lb/>ctum e. </s> <s>quæ omnia demon&longs;trare oportebat.</s></p><p type="margin"> <s><margin.target id="marg71"></margin.target>8 h<gap/></s></p><p type="head"> <s>THEOREMA XIX. PROPOSITIO XXIII.</s></p><p type="main"> <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></p><p type="main"> <s>Hoc demon&longs;trauit Leonardus Pi&longs;anus in libro, qui de­<lb/>praxi geometriæ in&longs;cribitur. </s> <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>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>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>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>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></p><p type="margin"> <s><margin.target id="marg72"></margin.target>1. &longs;exti.</s></p><p type="margin"> <s><margin.target id="marg73"></margin.target>5. duodeci <lb/>mi.</s></p><p type="margin"> <s><margin.target id="marg74"></margin.target>11. quinti.</s></p><p type="margin"> <s><margin.target id="marg75"></margin.target>4 &longs;exti.</s></p><p type="head"> <s>PROBLEMA V. PROPOSITIO XXIIII.</s></p><p type="main"> <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></p><pb pagenum="31"/><p type="main"> <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><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>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>Et quoniam triangula a b c, <lb/>h K l, d e f &longs;imilia &longs;unt; <expan abbr="triangulũ">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="triangulũ">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>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></p><p type="margin"> <s><margin.target id="marg76"></margin.target>16.<gap/><lb/>cim<gap/></s></p><p type="margin"> <s><margin.target id="marg77"></margin.target>34. l<gap/></s></p><p type="margin"> <s><margin.target id="marg78"></margin.target>9<gap/> hu<gap/><lb/>coin<gap/></s></p><p type="margin"> <s><margin.target id="marg79"></margin.target>20. &longs;<gap/></s></p><p type="margin"> <s><margin.target id="marg80"></margin.target>11. q<gap/></s></p><p type="main"> <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>Simili enim ratione, qua &longs;upra, demon&longs;trabi­<lb/>tur quadratum a b ad quadratum f g ita e&longs;&longs;e, ut <expan abbr="quadratũ">quadratum</expan> <lb/><arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. </s> <s>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 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="circulũ">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></p><p type="margin"> <s><margin.target id="marg81"></margin.target>2. duode <lb/><gap/>imi</s></p><p type="head"> <s>THEOREMA XX. PROPOSITIO XXV.</s></p><p type="main"> <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="cũ">cum</expan> <lb/>ca, quæ inter ip&longs;as &longs;it proportionalis, ad ba&longs;im ma <lb/>iorem.</s></p><pb pagenum="32"/><p type="main"> <s>SIT <expan abbr="fru&longs;tũ">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>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>quare pyra­<lb/>mis, uel conus, uel coni por­<lb/>tio, cuius ba&longs;is e&longs;t tribus illis <lb/>ba&longs;ibus æqualis ad a g b eam <lb/>habet proportionem, quam <lb/>ba&longs;es a b, e f, c d ad a b ba&longs;im. </s> <s>Fru&longs;tum igitur a d ad a g b <pb/>pyramidem, uel conuni, uel coni portionem eandem pro­<lb/>portionem habet, quam ba&longs;es a b, c d unà cum e f ad ba­<lb/>&longs;im a b. </s> <s>quod demon&longs;trare uolebamus.</s></p><p type="margin"> <s><margin.target id="marg82"></margin.target>6. 11.<gap/><lb/>decin<gap/></s></p><p type="main"> <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></p><p type="main"> <s>Sit fru&longs;tum pyramidis a b c d e f, cuius maior ba&longs;is trian­<lb/>gulum a b c; minor d e f: & &longs;ecetur plano ba&longs;ibus æquidi­<lb/>&longs;tante, quod &longs;ectionem faciat triangulum g h <foreign lang="greek">k</foreign> inter trian­<lb/>gula a b c, d e f proportionale. </s> <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></p><p type="main"> <s>Rur&longs;us &longs;it fru&longs;tum pyramidis a g, cu <lb/>ius maior ba&longs;is quadrilaterum a b c d, <lb/>minor e f g h: & &longs;ecetur plano ba&longs;i­<lb/>bus æquidi&longs;tante, ita ut fiat &longs;ectio qua­<lb/>drilaterum K l m n, quod &longs;it proportio <lb/>nale inter quadrilatera a b c d, e f g h. </s> <s>Dico pyramidem, <lb/>cuius ba&longs;is &longs;it æqualis tribus quadrilateris a b c d, k l m n, <lb/>e f g h, & altitudo æqualis altitudini fru&longs;ti, ip&longs;i fru&longs;to a g <lb/>æqualem e&longs;&longs;e. </s> <s>Ducatur enim planum per lincas f b, h d, <pb pagenum="33"/>quod diuidat fru&longs;tum in duo fru&longs;ta triangulares ba&longs;es ha­<lb/>bentia, uidelicet in fru&longs;tum a b d e f h, & in <expan abbr="fru&longs;tũ">fru&longs;tum</expan> b c d f g h. </s><lb/> <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;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> <s>pyramis igitur ba­<lb/>&longs;im habens æqualem tribus qua­<lb/>drilateris, & altitudinem eandem <lb/>ip&longs;i fru&longs;to a g e&longs;t æqualis. </s> <s>Eodem <lb/>modo illud <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan> in aliis <lb/>eiu&longs;modi fru&longs;tis.</s></p><p type="main"> <s>Sit fru&longs;tum coni, uel coni portionis a d; cuius maior ba­<lb/>&longs;is circulus, uel ellip&longs;is circa diametrum a b; minor circa <lb/>c d: & &longs;ecetur plano, quod ba&longs;ibus æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> &longs;e­<lb/>ctionem circulum, uel ellip&longs;im circa diametrum e f, ita ut <lb/>inter circulos, uel ellip&longs;es a b, c d &longs;it proportionalis. </s> <s>Dico <lb/>conum, uel coni portionem, cuius ba&longs;is e&longs;t æqualis tribus <lb/>circulis, uel tribus ellip&longs;ibus a b, e f, c d; & altitudo eadem, <lb/>quæ fru&longs;ti a d, ip&longs;i fru&longs;to æqualem e&longs;&longs;e. </s> <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="rectangulũ">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>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>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> <s>ut autem rectangu <lb/>lum c d ad <expan abbr="rectangulũ">rectangulum</expan> e f, ita circulus, uel ellip&longs;is c d ad e f <lb/>circulum, uel ellip&longs;im: <expan abbr="componendoq;">componendoque</expan> ut rectangula c d, <lb/>e f, ad e f rectangulum, ita circuli, uel ellip&longs;es e d, e f, ad e f: <lb/>& ut rectangulum e f ad rectangulum a b, ita circulus, uel <lb/>ellip&longs;is e f ad a b circulum, uel ellip&longs;im. </s> <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>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> <s>conus <lb/>igitur, uel coni portio q ad conum, uel coni portionem <lb/>a l b e&longs;t, ut pyramis q ad pyramidem a l b. </s> <s>&longs;ed pyramis <lb/>a l b ad pyramidem a g b e&longs;t, ut altitudo ad altitudinem, ex <lb/>20. huius: & ita e&longs;t conus, uel coni portio al b ad conum, <lb/>uel coni portionem a g b ex 14. duodccimi elementorum, <lb/>& ex iis, quæ nos demon&longs;trauimus in commentariis in un­<lb/>decimam de conoidibus, & &longs;phæroidibus, propo&longs;itione <lb/>quarta. </s> <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;tũ">&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>Sed pyramis q æqualis e&longs;t fru&longs;to à pyramide <lb/>ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. </s> <s>ergo & conus, uel coni por­<lb/>tio q, cuius ba&longs;is ex tribus circulis, uel ellip&longs;ibus a b, e f, c d <lb/>con&longs;tat, & altitudo eadem, quæ fru&longs;ti: ip&longs;i fru&longs;to a d e&longs;t æ­<lb/>qualis. </s> <s>atque illud e&longs;t, quod demon&longs;trare oportebat.</s></p><p type="margin"> <s><margin.target id="marg83"></margin.target>9 huius</s></p><p type="margin"> <s><margin.target id="marg84"></margin.target>2. duode­<lb/>cnni.</s></p><p type="margin"> <s><margin.target id="marg85"></margin.target>7. de co­<lb/>noidibus <lb/>& &longs;phæ­<lb/>roidibus</s></p><p type="margin"> <s><margin.target id="marg86"></margin.target>6. II. duo <lb/>decimi</s></p><p type="head"> <s>THEOREMA XXI. PROPOSITIO XXVI.</s></p><p type="main"> <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="cũ">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></p><pb pagenum="35"/><p type="main"> <s>Sit fru&longs;tum a e a pyramide, quæ triangularem ba&longs;im ha­<lb/>beat ab&longs;ci&longs;&longs;um: cuius maior ba&longs;is triangulum a b c, minor <lb/>d e f; & axis g h. </s> <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>quare <expan abbr="cen-trũ">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> <s>protractis enim fru&longs;ti pla <pb/>nis, quou&longs;que in unum punctum r conueniant; erit pyra­<lb/>midis a b c r, & pyramidis d e f r grauitatis centrum in li­<lb/>nca r h. </s> <s>ergo & reliquæ magnitudinis, uidelicet fru&longs;ti cen­<lb/>trum in eadem linea nece&longs;lario comperietur. </s> <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>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>Sed g o tri­<lb/>pla e&longs;t o h. </s> <s>quare & d u ip&longs;ius <lb/>u h e&longs;t tripla: & ideo pyrami­<lb/>dis a b c d centrum grauitatis <lb/>erit punctum u. </s> <s>Rur&longs;us quo­<lb/>niam z y ip&longs;i d l æquidi&longs;tat, d z <lb/>ad z m e&longs;t, ut l y ad y m: e&longs;tque <lb/>l y ad y m, ut g p ad p n. </s> <s>ergo <lb/>d z ad z m e&longs;t, ut g p ad p n. </s><lb/> <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="centrũ">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> <s>Sed fru&longs;ti centrum e&longs;t etiam in a­<lb/>xe g h. </s> <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>& 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>ex quibus &longs;equitur lineam. </s><lb/> <s>z u &longs;ecare o p in q: & propterea <expan abbr="pũctum">punctum</expan> q ip&longs;ius fru&longs;ti gra­<lb/>uitatis centrum e&longs;&longs;e.</s></p><p type="margin"> <s><margin.target id="marg87"></margin.target>3. diffi. </s> <s>hu <lb/>ius.</s></p><p type="margin"> <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></p><p type="margin"> <s><margin.target id="marg89"></margin.target>2. &longs;exti.</s></p><p type="margin"> <s><margin.target id="marg90"></margin.target><gap/>. </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></p><p type="margin"> <s><margin.target id="marg91"></margin.target>7. quinti.</s></p><p type="main"> <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. </s> <s>diuidatur autem <expan abbr="primũ">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="pũcto">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/> <s>& rur&longs;us ab l &longs;umatur quarta pars totius axis l k, quæ &longs;it <lb/>l o. </s> <s>po&longs;tremo fiat o n ad n p, ut fru&longs;tum a g ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>cuius ba&longs;is &longs;it eadem, quæ fru&longs;ti, & altitudo æqualis. </s> <s>Dico <lb/>punctum p fru&longs;ti a g grauitatis centrum e&longs;&longs;e. </s> <s>ducantur <lb/>enim a c, e g: & intelligantur duo fru&longs;ta triangulares ba­<lb/>&longs;es habentia, quorum alterum l f ex ba&longs;ibus a b c, e f g <expan abbr="cõ-&longs;tet">con­<lb/>&longs;tet</expan>; alterum l h ex ba&longs;ibus a c d, e g h. </s> <s><expan abbr="Sitq;">Sitque</expan> fru&longs;ti l f axis <lb/>q r; in quo grauitatis centrum s: fru&longs;ti uero l h axis t u, & <lb/>x grauitatis centrum: deinde iungantur u r, t q, x s. </s> <s>tran&longs;i­<lb/>bit u r per l: quoniam l e&longs;t centrum grauitatis quadran­<lb/>guli a b c d: & puncta r u grauitatis centra triangulorum <lb/>a b c, a c d; in quæ quadrangulum ip&longs;um diuiditur. </s> <s>eadem <lb/>quoque ratione t q per punctum k tran&longs;ibit. </s> <s>At uero pro <lb/>portiones, ex quibus fru&longs;torum grauitatis centra inquiri­<lb/>mus, eædem &longs;unt in toto fru&longs;to a g, & in fru&longs;tis l f, l h. </s> <s>Sunt <lb/>enim per octauam huius quadrilatera a b c d, e f g h &longs;imilia: <pb/><expan abbr="itemq;">itemque</expan> &longs;imilia triangula a b c, e f g: & a c d, e g h. </s> <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>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;tũ">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>ergo linea x s per p tran&longs;ibit: & lineæ r u, s x, q t in­<lb/>ter &longs;e æquidi&longs;tantes erunt. </s> <s>Itaque cum fru&longs;ti a g latera pro­<pb pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y cocant, erunt trià <lb/>gala u y l, x y p, t y k inter &longs;e &longs;imilia: & &longs;imilia etiam triangu <lb/>la l y r, p y s, k y <expan abbr="q.">que</expan> quare ut in 19 huius, demon&longs;trabitur <lb/>x p, ad p s: <expan abbr="itemq;">itemque</expan> t <foreign lang="greek">k</foreign> ad k q eandem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam u l ad l r. </s> <s>Sed ut u l ad l <gap/>, ita e&longs;t triangulum a b c ad <lb/>triangulum a c d: & ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s> <s>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>reliquum <lb/>igitur <expan abbr="fru&longs;tũ">fru&longs;tum</expan> l f ad reliquum <expan abbr="fru&longs;tũ">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>Eodem modo fret demon&longs;tratio etiam in <lb/>aliis pyramidibus.</s></p><p type="margin"> <s><margin.target id="marg92"></margin.target>a. </s> <s>&longs;exti.</s></p><p type="margin"> <s><margin.target id="marg93"></margin.target>19. quinti</s></p><p type="margin"> <s><margin.target id="marg94"></margin.target>8. Archi­<lb/>medis.</s></p><p type="main"> <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> <s>diuidatur <expan abbr="aut&etilde;">autem</expan> e f <lb/>in g, ita ut e g ad g f eandem proportionem habeat, quam <lb/>duplum diametri a b unà cum diametro e d ad duplum c d <lb/>unà cum a b. </s> <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> <s>Si enini fieri po­<lb/>te&longs;t, &longs;it m centrum: <expan abbr="producaturq;">producaturque</expan> l m extra fru&longs;tum in n: <lb/>& ut n l ad l m, ita fiat circulus, uel ellip&longs;is circa <expan abbr="diametrũ">diametrum</expan> <lb/>a b ad aliud &longs;pacium, in quo &longs;it o. </s> <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>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>&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>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;tũ">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>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> <s>relinquitur ergo, ut punctum l &longs;it <lb/>fru&longs;ti a d grauitatis centrum. </s> <s>quz omnia demon&longs;tranda <lb/>proponebantur.</s></p><p type="margin"> <s><margin.target id="marg95"></margin.target>22. huius</s></p><p type="margin"> <s><margin.target id="marg96"></margin.target>19. quínti</s></p><p type="head"> <s>THEOREMA XXII. PROPOSITIO XXVII.</s></p><p type="main"> <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></p><p type="main"> <s>Solida eiu&longs;modi corpora regularia appellare &longs;olent, de <lb/>quibus agitur in tribus ultimis libris elementorum: &longs;unt <lb/>autem numero quinque, tetrahedrum, uel pyramis, hexa­<lb/>hedrum, uel cubus, octahedrum, dodecahedrum, & ico&longs;a­<lb/>hedrum.</s></p><p type="main"> <s>Sit primo a b c d pyramis <expan abbr="ĩ">im</expan> &longs;phæra de&longs;cripta, cuius &longs;phæ <lb/>ræ centrum &longs;it e. </s> <s>Dico e pyramidis a b c d grauitatis e&longs;&longs;e <lb/>centrum. </s> <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>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> <s>Sed e e&longs;t cen <lb/>trum &longs;phæræ. </s> <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></p><p type="main"> <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>erit, <lb/>e <expan abbr="centrũ">centrum</expan> grauitatis &longs;olidi a b, <lb/>id quod demon&longs;tratum e&longs;t in <lb/>octaua huius. </s> <s>Sed quoniam ab <lb/>e&longs;t &longs;phæræ diametro æqualis, <lb/>ut in decima quinta propo&longs;i­<lb/>tione tertii decimilibri <expan abbr="elem&etilde;">elemen</expan> <lb/>torum o&longs;tenditur: punctum e <lb/>&longs;phæræ quoque centrum erit. </s><lb/> <s>Cubi igitur in &longs;phæra de&longs;cri­<lb/>pti grauitatis centrum idem <lb/>e&longs;t, quod centrum ip&longs;ius &longs;phæræ.</s></p><p type="main"> <s>Sit octahedrum a b c d e f, in &longs;phæra de&longs;criptum, cuius <lb/>&longs;phæræ centrum &longs;itg. </s> <s>Dico punctum g ip&longs;ius octahedri <lb/>grauitatis centrum e&longs;&longs;e. </s> <s>Con&longs;tat enim ex iis, quæ demon­<lb/>&longs;trata &longs;unt à Campano in quinto decimo libro elemento­<lb/>rum, propo&longs;itione &longs;extadecima eiu&longs;modi &longs;olidum diuidi <lb/>in duas pyramides æquales, & &longs;imiles; uidelicet in pyrami­<pb pagenum="39"/>dem, cuius ba&longs;is e&longs;t quadratum a b c d, & altitudo e g: & <lb/>in pyramidem, cuius <expan abbr="ead&etilde;">eadem</expan> ba&longs;is, <expan abbr="altitudoq;">altitudoque</expan> f g; ut &longs;int e g, <lb/>g f &longs;emidiametri &longs;phæræ, & linea una. </s> <s><expan abbr="Cũigitur">Cunigitur</expan> g &longs;it &longs;phæ­<lb/>ræ centrum, erit etiam centrum circuli, qui circa <expan abbr="quadratũ">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>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>& <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="planorũ">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></p><p type="main"> <s>Sit ico&longs;ahedrum a d de&longs;criptum in &longs;phæra, cuius <expan abbr="centrũ">centrum</expan> <lb/>&longs;it g. </s> <s>Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. </s> <s>Si <lb/>enim ab angulo a per g ducatur recta linea u&longs;que ad &longs;phæ <lb/>ræ &longs;uperficiem; con&longs;tat ex &longs;exta decima propo&longs;itione libri <lb/>tertii decimi elementorum, cadere eam in angulum ip&longs;i a <lb/>oppo&longs;itum. </s> <s>cadat in d: <expan abbr="&longs;itq;">&longs;itque</expan> una aliqua ba&longs;is ico&longs;ahedri tri­<lb/>angulum a b c: & iunctæ b g, c g producantur, & cadant in <lb/>angulos e f, ip&longs;is b c oppo&longs;itos. </s> <s>Itaque per triangula <lb/>a b c<gap/> d e f ducantur plana &longs;phæram &longs;ecantia<gap/> erunt <gap/> &longs;e-<pb/>ctiones circuli ex prima propo&longs;itione &longs;phæricorum Theo <lb/>do&longs;ii: unus quidem circa triangulum a b c de&longs;criptus: al­<lb/>ter uero circa d e f: & quoniam triangula a b c, d e f æqua­<lb/>lia &longs;unt, & &longs;imilia; erunt ex prima, & &longs;ecunda propo&longs;itione <lb/>duodecimi libri clementorum, circuli quoque inter &longs;e &longs;e <lb/>æquales. </s> <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>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>cum autem <lb/><figure id="fig58"></figure><lb/>h &longs;it <expan abbr="centrũ">centrum</expan> circuli, & tri­<lb/>anguli a b c grauitatis cen <lb/><expan abbr="trũ">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>lta­<lb/>que centrum grauitatls py <lb/>ramidis a b c g &longs;it <expan abbr="pũctum">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="centrũ">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></p><p type="margin"> <s><margin.target id="marg97"></margin.target>13. primi</s></p><p type="margin"> <s><margin.target id="marg98"></margin.target>14. primi</s></p><p type="main"> <s>Sit dodecahedrum a f in &longs;phæra de&longs;ignatum, &longs;itque &longs;phæ <lb/>ræ centrum m. </s> <s>Dico m centrum e&longs;&longs;e grauitatis ip&longs;ius do­<lb/>decahedri. </s> <s>Sit enim pentagonum a b c d e una ex duode­<lb/>cim ba&longs;ibus &longs;olidi a f: & iuncta a m producatur ad &longs;phæræ <lb/>&longs;uperficiem. </s> <s>cadet in angulum ip&longs;i a oppo&longs;itum; quod col­<lb/>ligitur ex decima &longs;eptima propo&longs;iticne tertiidecimi libri <lb/>clementorum. </s> <s>cadat in f. </s> <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="circulorũ">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="circulorũ">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>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> <s>ergo cum puncta n o &longs;int centra circulorum, con­<lb/>&longs;tat ex prima huius & <expan abbr="pentagonorũ">pentagonorum</expan> grauitatis e&longs;&longs;e centra: <lb/><expan abbr="idcircoq;">idcircoque</expan> m n, m o pyramidum a b c d e m, f g h <emph type="italics"/>K<emph.end type="italics"/> l m axes. </s><lb/> <s>ponatur a b c d e m pyramidis grauitatis centrum p: & py <lb/>ramidis f g h <foreign lang="greek">k</foreign> l m ip&longs;um q centrum. </s> <s>erunt p m, m q æqua­<lb/>les, & punctum m grauitatis centrum magnitudinis, quæ <lb/>ex ip&longs;is pyramidibus con&longs;tat. </s> <s><expan abbr="eod&etilde;">eodem</expan> modo probabitur qua­<lb/>rumlibet pyramidum, quæ è regione opponuntur, <expan abbr="centrũ">centrum</expan> <pb/>grauitatis e&longs;&longs;e punctum m. </s> <s>patetigitur totius dodecahe­<lb/>dri, centrum grauitatis <expan abbr="id&etilde;">idem</expan> e&longs;&longs;e, quod & &longs;phæræ ip&longs;um com <lb/>prehendentis centrum. </s> <s>quæ quidem omnia demon&longs;tra&longs;&longs;e <lb/>oportebat.</s></p><p type="margin"> <s><margin.target id="marg99"></margin.target>corol. </s> <s>pri<lb/>mæ &longs;phæ<lb/>ricorum <lb/>Theod.</s></p><p type="margin"> <s><margin.target id="marg100"></margin.target>6. primi <lb/>phærico <lb/>rum.</s></p><p type="head"> <s>PROBLEMA VI. PROPOSITIO XXVIII.</s></p><p type="main"> <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></p><p type="main"> <s>Sit portio conoidis rectanguli a b c, cuius axis b d, <expan abbr="gra-uitatisq;">gra­<lb/>uitatisque</expan> centrum e: & &longs;it g recta linea propo&longs;ita. </s> <s>quam ue <lb/>ro proportionem habet linea b e ad lineam g, eandem ha­<lb/>beat portio conoidis ad &longs;olidum h: & circum&longs;cribatur por <lb/>tioni figura, &longs;icuti dictum e&longs;t, ita ut portiones reliquæ &longs;int <lb/>&longs;olido h minores: cuius quidem figuræ centrum grauitatis <lb/>&longs;it punctum <foreign lang="greek">k.</foreign> Dico <expan abbr="lineã">lineam</expan> k e minorem e&longs;&longs;e linea g propo­<lb/>&longs;ita. </s> <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>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> <s><expan abbr="habetq;">habetque</expan> l K <lb/>ad K e eam proportio­<lb/>nem, quam portio co­<lb/>noidis ad reliquas por­<lb/>tiones; erit punctum l <lb/>extra portionem <expan abbr="cad&etilde;s">cadens</expan>, <lb/>centrum magnitudinis <lb/>ex reliquis portionibus compo&longs;itæ. </s> <s>illud autem fieri nullo <lb/>modo pote&longs;t. </s> <s>quare con&longs;tat lineam k e ip&longs;a g linea propo&longs;i <lb/>ta minorem e&longs;&longs;e.</s></p><p type="margin"> <s><margin.target id="marg101"></margin.target>8. quínti.</s></p><p type="margin"> <s><margin.target id="marg102"></margin.target>29. quínti <lb/>ex tradi­<lb/>tione <expan abbr="Cã-l">Can­<lb/>l</expan> ani.</s></p><p type="main"> <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>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> <s>Dico <lb/>lineam p e ip&longs;a g <expan abbr="minor&etilde;">minorem</expan> <lb/>e&longs;&longs;e. </s> <s>&longs;i enim n<gap/>n &longs;it mi­<lb/>nor, codem, quo &longs;upra modo demon&longs;trabimus figuram in <lb/>&longs;criptam ad reliquas portiones maiorem proportionem <lb/>habere, quàm b e ad e p. </s> <s>& &longs;i fiat alia linea l e ad e p, ut e&longs;t <lb/>figura in&longs;cripta ad reliquas portiones, <expan abbr="pũctum">punctum</expan> l extra por <pb/>tionem cadet: Itaque cum à portione conoidis, cuius gra­<lb/>uitatis centrum e auferatur in&longs;cripta figura, centrum ha­<lb/>bens p: & &longs;itl e ad e p, ut figura in&longs;cripta ad portiones reli <lb/>quas: erit magnitudinis, quæ ex reliquis portionibus con <lb/>&longs;tat, centrum grauitatis punctum l, extra portionem ca­<lb/>dens. </s> <s>quod ficrinequit. </s> <s>ergo linca p e minor e&longs;tip&longs;a g li­<lb/>nea propo&longs;ita.</s></p><p type="main"> <s>Ex quibus per&longs;picuum e&longs;t centrum grauitatis <lb/>figuræ in&longs;criptæ, & circum&longs;criptæ eo magis acce <lb/>dere ad portionis centrum, quo pluribus cylin­<lb/>dris, uel cylindri portionibus con&longs;tet: <expan abbr="fiat&qacute;">fiatque</expan>; figu <lb/>rain&longs;cripta maior, & circum&longs;cripta minor. </s> <s>& <lb/>quanquam continenter ad portionis <expan abbr="centrũ">centrum</expan> pro­<lb/>pius admoueatur: nunquam tamen ad ip&longs;um per <lb/>ueniet. </s> <s>&longs;equeretur enim figuram in&longs;criptam, <expan abbr="nõ">non</expan> <lb/>&longs;olum portioni, &longs;ed etiam circum&longs;criptæ figuræ <lb/>æqualem e&longs;&longs;e. </s> <s>quod e&longs;t ab&longs;urdum.</s></p><p type="head"> <s>THEOREMA XXIII. PROPOSITIO XXIX.</s></p><p type="main"> <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></p><p type="main"> <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>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="dictũ">dictum</expan> e&longs;t'. </s><lb/> <s>Sit autem pri­<lb/>mum figura in­<lb/>&longs;cripta <expan abbr="cylĩdrus">cylindrus</expan> <lb/>f g: & <expan abbr="circũ&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="mediũ">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="centrũ">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="circũ-&longs;criptæ">circun­<lb/>&longs;criptæ</expan> erit pun <lb/><arrow.to.target n="marg105"></arrow.to.target><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-dratũ">qua­<lb/>dratum</expan> lineæ a d <lb/>ad <expan abbr="quadratũ">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="conicorũ">conicorum</expan> & ita <lb/>quadratum a c <lb/>ad <expan abbr="quadratũ">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>du <lb/>pla e&longs;t autem li­<lb/>nca d b lineæ <pb pagenum="43"/>b m. </s> <s>crgo circulus a c circuli k g: & idcirco cylindrus <lb/>a h cylindri k. </s> <s>l duplus erit. </s> <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>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="cylĩdrus">cylindrus</expan> duobus cylindris k <foreign lang="greek">g, q l</foreign> æqualis erit. </s> <s>& <lb/>propterea linea <foreign lang="greek">s t</foreign> æqualis ip&longs;i <foreign lang="greek">t p.</foreign> denique cylindri a x <lb/>centrum grauitatis e&longs;t punctum <foreign lang="greek">r.</foreign> & cum <foreign lang="greek">t r</foreign> diui&longs;a fuerit <lb/>in <expan abbr="eã">eam</expan> proportionem, quam habet cylindrus ax ad tres cy­<lb/>lindros y z, k <foreign lang="greek">g, q l:</foreign> erit in eo puncto centrum grauitatis <lb/>totius figuræ <expan abbr="circũ&longs;criptæ">circun&longs;criptæ</expan>. </s> <s>Sed cylindrus a x ad ip&longs;um y z <lb/>e&longs;t ut linea d b ad b n: hoc e&longs;t ut 4 ad 3: & duo cylindri k <foreign lang="greek">h <lb/>q l</foreign> cylindro y z &longs;unt æquales. </s> <s>cylindrus igitur a x ad tres <lb/>iam dictos cylindros e&longs;t ut 2 ad 3. Sed <expan abbr="quoniã">quoniam</expan> <foreign lang="greek">m s</foreign> e&longs;t dua­<lb/>rum partium, & <foreign lang="greek">s g</foreign> unius, qualium <foreign lang="greek">m p</foreign> e&longs;t &longs;ex; erit <foreign lang="greek">s p</foreign> par­<lb/>tium quatuor: <expan abbr="proptereaq;">proptereaque</expan> <foreign lang="greek">t p</foreign> duarum, & <foreign lang="greek">g p,</foreign> hoc e&longs;t <foreign lang="greek">p r</foreign><lb/>trium. </s> <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>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/> <s><expan abbr="Quoniã">Quoniam</expan> igitur <lb/>ut b e ad e d, ita <lb/>e&longs;t o p ad p n; <lb/><expan abbr="utraq;">utraque</expan> enim u­<lb/>triu&longs;que e&longs;t du­<lb/>pla: erit compo <lb/>nendo, ut b d ad <lb/>d e, ita o n ad n <lb/>p; & permutan <lb/>do, ut b d ad o <lb/>n, ita d e ad n p. </s><lb/> <s>Sed b d dupla <lb/>e&longs;t o n. </s> <s>ergo & <lb/>e d ip&longs;ius n p du <lb/>pla erit. </s> <s>quòd &longs;i <lb/>e d bifariam di­<lb/>uidatur <expan abbr="ĩ">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>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> <s>&longs;ed d n <lb/>dupla e&longs;t n <foreign lang="greek">r.</foreign> reliquaigitur d <foreign lang="greek">x</foreign> dupla reliquæ <foreign lang="greek">x</foreign> n. </s> <s>&longs;unt au­<lb/>tem d <foreign lang="greek">x,</foreign> p n inter &longs;e æquales: <expan abbr="itemq;">itemque</expan> æquales <foreign lang="greek">x</foreign> n, p e. </s> <s>qua­<lb/>re con&longs;tat n p ip&longs;ius p e duplam e&longs;&longs;e. </s> <s>& idcirco p e ip&longs;i e n <lb/>æqualem. </s> <s>Rur&longs;us cum &longs;it <foreign lang="greek">m g</foreign> duplao <foreign lang="greek"><gap/>,</foreign> & <foreign lang="greek">m s</foreign> dupla <foreign lang="greek">s g;</foreign> erit <lb/>etiam reliqua <foreign lang="greek">g s</foreign> reliquæ <foreign lang="greek">s</foreign> o dupla. </s> <s>Eadem quoque ratione <lb/><expan abbr="cõcludetur">concludetur</expan> <foreign lang="greek">p u</foreign> dupla <foreign lang="greek">u</foreign> m. </s> <s>ergo ut <foreign lang="greek">g s</foreign> ad <foreign lang="greek">s</foreign> o, ita <foreign lang="greek">p u</foreign> ad <foreign lang="greek">u</foreign> m: <lb/><expan abbr="componendoq;">componendoque</expan>, & permutando, ut <foreign lang="greek">g</foreign> o ad <foreign lang="greek">p</foreign> m, ita o <foreign lang="greek">s</foreign> ad <lb/>m <foreign lang="greek">u:</foreign> & &longs;unt æquales <foreign lang="greek">g</foreign> o, <foreign lang="greek">p</foreign> m. </s> <s>quare & o <foreign lang="greek">s,</foreign> m <foreign lang="greek">u</foreign> æquales. </s> <s>præ <lb/>terea <foreign lang="greek">s p</foreign> dupla e&longs;t <foreign lang="greek">p t,</foreign> & <foreign lang="greek">g p</foreign> ip&longs;ius <foreign lang="greek">p</foreign> m. </s> <s>reliquaigitur <foreign lang="greek">s g</foreign> re <lb/>liquæ m <foreign lang="greek">t</foreign> dupla. </s> <s>atque erat <foreign lang="greek">g s</foreign> dupla <foreign lang="greek">s</foreign> o. </s> <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;ũt">&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> <s>Sed etiam æ­<lb/>quales n <foreign lang="greek">p, p</foreign> m. </s> <s>reliqua igitur <foreign lang="greek">p</foreign> preliquæ m <foreign lang="greek">t,</foreign> hoc e&longs;t ip&longs;i <lb/>n <foreign lang="greek">f</foreign> æqualis erit. </s> <s>quare dempta p <foreign lang="greek">p</foreign> ex p e, & <foreign lang="greek">f</foreign> n dempta ex <lb/>n e, relinquitur p e æqualis e <foreign lang="greek">f.</foreign> Itaque <foreign lang="greek">p, f</foreign> centra <expan abbr="figurarũ">figurarum</expan> <lb/>&longs;ecundo loco de&longs;criptarum a primis centris p n æquali in­<lb/>teruallo recedunt. </s> <s>quòd &longs;i rur&longs;us aliæ figuræ de&longs;cribantur, <lb/>codem modo demon&longs;trabimus earum centra æqualiter ab <lb/>his recedere, & ad portionis conoidis centrum propius ad <lb/>moueri. </s> <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="punctũ">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="centrũ">centrum</expan>. </s> <s>ergo punctum e centrum erit gra<lb/>uitatis portionis a b c. </s> <s>quod demon&longs;trare oportebat.</s></p><p type="margin"> <s><margin.target id="marg103"></margin.target>7. huius</s></p><p type="margin"> <s><margin.target id="marg104"></margin.target><gap/>imi <lb/><gap/> Ar­<lb/><gap/>dis</s></p><p type="margin"> <s><margin.target id="marg105"></margin.target><gap/>o. <lb/><gap/>li.</s></p><p type="margin"> <s><margin.target id="marg106"></margin.target><gap/>inti</s></p><p type="margin"> <s><margin.target id="marg107"></margin.target><gap/>de-</s></p><p type="margin"> <s><margin.target id="marg108"></margin.target>20. primi <lb/>conicor</s></p><p type="margin"> <s><margin.target id="marg109"></margin.target>1<gap/> quinti</s></p><p type="main"> <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></p><p type="margin"> <s><margin.target id="marg110"></margin.target>corol. <gap/><lb/>de conoi­<lb/>dibus & <lb/>&longs;phæroi­<lb/>dibus.</s></p><p type="head"> <s>THEOREMA XXIIII. PROPOSITIO XXX.</s></p><p type="main"> <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></p><pb/><p type="main"> <s>ABSCINDATVR à portione conoidis rectanguli <lb/>a b c alia portio e b f, plano ba&longs;i æquidi&longs;tante: & eadem <lb/>portio &longs;ecetur alio plano per axem; ut &longs;uperficiei &longs;ectio &longs;it <lb/>parabole a b c: <expan abbr="planorũ">planorum</expan> portiones ab&longs;cindentium rectæ <lb/>lincæ a c, e f: axis autem portionis, & &longs;ectionis diameter <lb/>b d; quam linea e fin puncto g &longs;ecet. </s> <s>Dico portionem co­<lb/>noidis a b c ad portionem e b f duplam proportionem ha­<lb/>bere eius, quæ e&longs;t ba&longs;is a c ad ba&longs;im e f; uel axis d b ad b <gap/><lb/>axem. </s> <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>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/> <s>quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue <pb pagenum="46"/>ro ita demon&longs;trabitur. </s> <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>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>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="quadratũ">quadratum</expan> a d ad <expan abbr="quadratũ">quadratum</expan> e 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="circulũ">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>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></p><p type="margin"> <s><margin.target id="marg111"></margin.target>16. unde­<lb/>cim<gap/></s></p><p type="margin"> <s><margin.target id="marg112"></margin.target>4 <gap/>.</s></p><p type="margin"> <s><margin.target id="marg113"></margin.target>2. duode <lb/>cimi</s></p><p type="margin"> <s><margin.target id="marg114"></margin.target>7. de co­<lb/>noidibus <lb/>& &longs;phæ­<lb/>roidibus</s></p><p type="margin"> <s><margin.target id="marg115"></margin.target><gap/>. </s> <s>quinti</s></p><p type="margin"> <s><margin.target id="marg116"></margin.target>20. p<gap/>mi <lb/><expan abbr="comcorũ">comcorum</expan></s></p><p type="head"> <s>THEOREMA XXV. PROPOSITIO XXXI.</s></p><p type="main"> <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></p><p type="main"> <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><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><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="quadratũ">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>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="uerũ">uerum</expan> e&longs;&longs;e illud, <lb/>quod demon&longs;trandum propo&longs;uimus.</s></p><p type="margin"> <s><margin.target id="marg117"></margin.target>20. 1. con<gap/><lb/>corum.</s></p><p type="margin"> <s><margin.target id="marg118"></margin.target>30 hui<gap/></s></p><p type="head"> <s>FINIS LIBRI DE CENTRO</s></p><p type="head"> <s>GRAVITATIS SOLIDORVM.</s></p><p type="main"> <s>Impre&longs;&longs;. </s> <s>Bononiæ cum licentia Superiorum, <lb/><gap/><lb/> </s></p> </chap> </body> <back></back> </text></archimedes>