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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Valerio, Luca</author> <title>De centro gravitatis solidorum</title> <date>1604</date> <place>Bologna</place> <translator/> <lang>la</lang> <cvs_file>valer_centr_043_la_1604.xml</cvs_file> <cvs_version/> <locator>043.xml</locator> </info> <text> <front> </front> <body> <chap> <pb/> <pb/> <pb/> <pb/> <pb/><p type="head"> <s>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM <lb/>LIBRITRES.</s></p><p type="head"> <s>LVCÆ VALERII <lb/><emph type="italics"/>Mathematicæ, & Ciuilis Philo&longs;ophiæ <lb/>in Gymna&longs;io Romano profe&longs;&longs;oris.<emph.end type="italics"/></s></p><figure id="id.043.01.001.1.jpg" xlink:href="043/01/001/1.jpg"/><p type="head"> <s>ROMÆ, Typis Bartholom ri Bonfadini. </s> <s>MDC IIII. <lb/>SVPERIORVM PERMISSV.</s></p><pb/><p type="main"> <s>Imprimatur <!-- KEEP S--></s></p><p type="main"> <s>Si placet R. P. <!-- REMOVE S-->Magi&longs;tro S. Palati<gap/> <lb/>B. <!-- REMOVE S--></s> </p><p type="main"> <s>Gyp&longs;ius Vice&longs;ger. <!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Imprimatur<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Fr. <!-- REMOVE S-->Io. <!-- REMOVE S-->Maria Bra&longs;ichellen. <!-- REMOVE S-->Sacri Pal. <lb/></s> <s>Apostol. <!-- REMOVE S-->Magi&longs;t.<emph.end type="italics"/><!-- KEEP S--></s> </p><pb/><figure id="id.043.01.003.1.jpg" xlink:href="043/01/003/1.jpg"/><p type="head"> <s>SANCTISSIMO <lb/>DOMINO NOSTRO <lb/>CLEMENTI VIII <lb/>PONT. OPT. MAX.<!-- REMOVE S--><emph type="italics"/>Lucas Valerius perpetuam felicitatem.<emph.end type="italics"/></s></p><figure id="id.043.01.003.2.jpg" xlink:href="043/01/003/2.jpg"/><p type="main"> <s>Grata Principi munera, <lb/>P. B. ex Philo&longs;ophiæ late­<lb/>bris deprompta, qua&longs;i aurum <lb/>&longs;oli expo&longs;itum illico &longs;plen­<lb/>dent, & publicæ vtilitatis <lb/>&longs;pem o&longs;tendunt, magno or­<lb/>nata præ&longs;idio in primos liuo­<lb/>ris impetus illius approbatione, cuius officium e&longs;t <lb/>alia à rep. </s> <s>auertere, alia imperare. </s> <s>Hinc por­<lb/>rò factum e&longs;t, vt omnis ferè &longs;criptor exi&longs;timatio­<lb/>nis periculum aditurus, aliquem ex principibus <pb/>viris &longs;ibi deligat, cuius autoritate ip&longs;i dicatum <lb/>opus ab inuidorum mor&longs;ibus &longs;eruetur incolume. <lb/></s> <s>Hanc ergo con&longs;uetudinem amanti mihi &longs;anè feli­<lb/>citer cecidit, vt tu &longs;ola tua propria benignitate <lb/>permotus in tuos me familiares vltro a&longs;criberes. <lb/></s> <s>Siue enim ingenij mei debilis partus <expan abbr="magnā">magnam</expan> pa­<lb/>troni de&longs;iderat autoritatem: tu principum orbis <lb/>terrarum princeps &longs;emper digni&longs;&longs;imam principa­<lb/>tu &longs;apientiam præ&longs;titi&longs;ti. </s> <s>Seu tam elatæ dedica­<lb/>tiones &longs;olent alienas à &longs;apientiæ &longs;tudio &longs;pes olere: <lb/>lux tanti patrocinij, <expan abbr="tuorum&qacute;">tuorumque</expan> veterum in me be­<lb/>neficiorum, atram &longs;u&longs;picionem amouebit. </s> <s>Quòd <lb/>verò ad vitam ip&longs;ius operis attinet, quam nulla <lb/>per te velim temporum permutatione terminari: <lb/>vereor vt id &longs;ua luce multis alijs vitali a&longs;piciat <lb/>illa, quæ tua &longs;tudia, & res ge&longs;tas omnium lin­<lb/>guis, & litteris celebrabit æternitas. </s> <s>quantum <lb/>enim tuam excel&longs;am &longs;u&longs;picio dignitatem, tantum <lb/>de&longs;picor i&longs;tius doni incredibilem cum illa com­<lb/>parati humilitatem: neque id ni&longs;i diuinitus cre­<lb/>diderim perpetuam in tuis laudibus famam ha­<lb/>biturum. </s> <s>Quare illud non &longs;olum tibi diuini gre­<lb/>gis anti&longs;titi cupio gratum accidere, cuius auto­<lb/>ritate protectum in tanta nouarum rerum po&longs;t <lb/>tam graues autores contemptione, minimo meo <lb/>cum rubore in medium prodeat: &longs;ed ip&longs;i diuinita­<lb/>ti ex voluntate donum expendenti, penes quam <lb/>e&longs;t æternitas, & cui primum dicata omnia e&longs;&longs;e <lb/>oportet: vt hi, quostuis luminibus dignaris, de <pb/>centro grauitatis &longs;olidorum &longs;terilis ingenij mei <lb/>te&longs;tes libelli à mortis æmula me obliuione defen­<lb/>dant. </s> <s>Stomacharis hic, arbitror, quòd tantum <lb/>&longs;pectem de nihilo; &longs;ed magis confe&longs;&longs;ionis impu­<lb/>dentia. </s> <s>At verò non impetus animi ad gloriam, <lb/>cuius nullum mihi natura &longs;emen impartiuit (&longs;it <lb/>gloriæ loco ignauiæ fugi&longs;&longs;e dedecus) &longs;ed tua er­<lb/>ga me voluntas, meisapta &longs;tudijs liberalitate te­<lb/>&longs;tata hunc ardorem expre&longs;&longs;it. </s> <s>Tanta enim e&longs;t <lb/>venu&longs;tas tuæ virtutis ex mei meriti penuria, vt <lb/>putem &longs;ine me indice illam diminutum &longs;ui &longs;pecta­<lb/>culum po&longs;teris præbituram. </s> <s>Nihil ergo minus <lb/>cogitans quàm quî tua beneficia cumulando per­<lb/>turbatis iudicijs &longs;atisfacerem, &longs;cientia &longs;cilicet, <lb/>& virtute illa, qua maximè &longs;uperbit eneruata, & <lb/>are&longs;cens Mundiætas; nullum opulentiæ meæ, ar­<lb/>tis alienæ &longs;pecimen pro munere gratiæ à te acce­<lb/>pto partem tibi reddidi: &longs;ed ingenij mei partum, <lb/>qualis is cumque e&longs;t; quod & grati animi quæ&longs;i­<lb/>tum monumentum crimine me audaciæ liberet, <lb/>&longs;i quodimpendeat, palam dedicaui. </s> <s>Alij tibi co­<lb/>lumnas hone&longs;ti&longs;&longs;imis titulis ornatas erigant: &longs;ta <lb/>tuas in foris collocent: magnificas ædes extruant, <lb/>quarum in frontibus grandes marmoreæ tabulæ <lb/>flammantibus auro &longs;yderibus, & peregrinis lapi­<lb/>dibus intextæ ea de te viuo referant &longs;axum impu­<lb/>dens, quæ verecunda hæc pagina prætermittit. <lb/></s> <s>Ego incredibilis tuæ benignitatis non tam gra­<lb/>uia te&longs;timonia, quæ loco moueri nequeant: &longs;ed <pb/>expeditum hunc nuntium in longi&longs;&longs;ima itinera <lb/>de&longs;tinaui. </s> <s>Quem quidem eo minus vereor ne <lb/>non tu, quamobrem Telchines forta&longs;&longs;e aliqui in­<lb/>&longs;ectaturi, di&longs;pari &longs;is voluntate protecturus, quòd <lb/>in his tàm reconditis naturæ arcanis geometrica <lb/>demon&longs;tratione patefactis, tanquam in &longs;emine <lb/>multiplicem præ&longs;criptionem, ac normam e&longs;&longs;e in­<lb/>telliges ip&longs;e pacis inter tuos greges autor, lupi <lb/>otomani terror, ciuili, & bellicæ architecturæ <lb/>maximè nece&longs;&longs;ariam. </s> <s>Quòd que, cum ad theologi­<lb/>cam quandam veritatem chri&longs;tiano generi maxi­<lb/>me &longs;alutarem illu&longs;trandam, per Philo&longs;ophi<17> etiam <lb/>campos &longs;apientium hominum corona decoratus, <lb/>nulla tantæ molis, quantam &longs;u&longs;tines negotiorum <lb/>iactura lati&longs;&longs;imè vageris; nempe illam cre&longs;cere, <lb/>atque illu&longs;trari indies magis ex optas, cuius con­<lb/>&longs;uetudine tantopere delectaris. </s> <s>Quod denique <lb/>&longs;cientiæ ciuilis ip&longs;e periti&longs;&longs;imus omnium optimè <lb/>intelligis, quanti referat ad humanæ &longs;ocietatis for <lb/>mam & candorem, regum, atque optimatum a­<lb/>mor in &longs;tudio&longs;os bonarum litterarum. </s> <s>contrà au­<lb/>tem ex de&longs;pectione in hos cadente abijs, quorum <lb/>mores pro legibus haberi &longs;olent, no&longs;ti commu­<lb/>nem ingeniorum veternum, mox tyrannidem gi­<lb/>gni, magna cu&longs;tode adempta mode&longs;tiæ imperi­<lb/>tantium crebra ciuium &longs;apientia, quæ prauis ti­<lb/>morem efficit, melioribus pudorem, Quod &longs;i meæ <lb/>expectationi exitus re&longs;pondebit, vt te hoc munu­<lb/>&longs;culo vel leuiter lætari &longs;entiam; alia non iniucun-<pb/>da ftatim proferam, qua PETRVS ALDOBRAN­<lb/>DINVS tuus nepos, domi fori&longs;que clari&longs;&longs;imus <lb/>Cardinalis, cuius inter familiares itidem, <expan abbr="bene-ficijs&qacute;ue">bene­<lb/>ficijsque</expan> deuinctos locum habeo, &longs;uæ erga me hu­<lb/>manitatis te&longs;timonia ab inuidiæ &longs;atellite & mi­<lb/>ni&longs;tra calumnia tueatur: quando duobus talibus <lb/>viris animi mei captum beneficentia &longs;ua pericli­<lb/>tantibus, duplex periculum &longs;ubire &longs;um coactus. <lb/></s> <s>Sed iam verbo&longs;æ epi&longs;tolæ, & tuo fa&longs;tidio finem im <lb/>po&longs;iturus peto à te vnum; vt tibi per&longs;uadeas, me <lb/>inter tuos famulos, quos ære proprio, & victu quo­<lb/>tidiano liberaliter &longs;u&longs;tentas, eorum, qui pro te <lb/>emori po&longs;&longs;unt, amore, con&longs;tantia, fidelitate nemini <lb/>planè concedere. </s> <s>Sic tua omnia præ&longs;tanti&longs;&longs;ima <lb/>facinora Princeps magnanime, & pietatis colu­<lb/>men, Deus Opt. <!-- REMOVE S-->Max. <!-- REMOVE S-->tibi fortunet, quem ad ma­<lb/>iores in dies res gerendas in longum æuum inco­<lb/>lumen, felicemque con&longs;eruet. </s> <s>Valet. <!-- KEEP S--></s></p><pb/><p type="head"> <s><foreign lang="greek">*l*o*u*k*a *o*u*a*l*e*r*i*o*u <lb/>*e*i*s *t*a *a*u*t*o*u *k*e*n*t*r*a</foreign></s></p><p type="head"> <s><foreign lang="greek">s<gap/>cew=n b<gap/>ze/wn, e)pi/<gap/>mma</foreign>.</s></p><p type="main"> <s><foreign lang="greek">*pai/gnia filo<gap/>fois *loukas_ t<gap/> de ou/m<gap/>loka da/f<gap/>, <lb/>*st<gap/>umo/nos e)gkela/ds <gap/>ei/<gap/>ona p<gap/>lu/<gap/>n. </foreign></s></p><p type="main"> <s><foreign lang="greek">*dw=ron e(/pemya/ pe/<gap/>as d)<gap/>(ze_in ti_s <gap/>u_ <gap/>t) a)/d<gap/><lb/>*b<gap/>qoou/nhs bape/wn ph_ce <gap/>e/meqla fu/<gap/>s. </foreign></s></p><p type="main"> <s><foreign lang="greek">*toi+s pe/zan au)ale/wn <gap/>ndw_n <gap/>i+/aya m<gap/>i/mnas, <lb/>*me/my<gap/> mh\ p/wn tei/rea, mh\ <gap/>u/x<gap/>. </foreign></s></p><p type="main"> <s><foreign lang="greek">*toi_s pnos o)fruo/en plupza/gmonos o)/mma gila/<gap/>as, <lb/>*be/ltion <gap/>gore/hs ke/rdos e(/deiza <gap/>d. </foreign><!-- KEEP S--></s></p><p type="main"> <s><foreign lang="greek">*ei) de/ p tw_n <gap/>o(/<gap/>ws e<gap/>z<gap/>x<gap/>on eu)/<gap/>, <lb/>*p<gap/>i\n qa/naps ma/zyh m): eu)/xom) <gap/>le/tw. </foreign></s></p><p type="main"> <s><foreign lang="greek">*lne/zos ou) kle/yw xa/<gap/>n eu)/fzonos e)<gap/>omo/noi<gap/><lb/>*d<gap/>gm) a)glao\n, <gap/>, <gap/>nomes, kai\ patzi/d<gap/>. </foreign></s></p><p type="main"> <s><foreign lang="greek">*os de/ me laqzai<gap/>os dh/z<gap/>, kako/ep<gap/>os a)kou/o<gap/>, <lb/>*lu<gap/>w_n h(=s fqonezh_s a)/zios purkai<gap/>h_s. </foreign></s></p><pb/><figure id="id.043.01.009.1.jpg" xlink:href="043/01/009/1.jpg"/><p type="head"> <s>LVC AE <lb/>VALER II <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM<!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>LIBER PRIMVS.<emph.end type="italics"/></s></p><p type="main"> <s>Propo&longs;itum e&longs;t mihi in hi&longs;ce tribus li­<lb/>bris, ò Geometra, cuiu&longs;cumque figuræ <lb/>&longs;olidæ in geometria ratio haberi &longs;olet, <lb/>centrum grauitatis inuenire. </s> <s>Huius <lb/>autem prouinciæ mihi &longs;u&longs;cipiendæ oc­<lb/>ca&longs;io fuit liber ille iam pridem editus <lb/>Federici Commandini Vrbinatis, in <lb/>quo cum ille corporum planis termi­<lb/>nis definitorum; necnon cylindri, & coni, & fru&longs;ti conici, <lb/>& &longs;phæræ, & &longs;phæroidis centrum grauitatis o&longs;tendi&longs;&longs;et; <lb/>aliorum autem, quæ &longs;uperficie mixta continentur vno co­<lb/>noide parabolico tentato &longs;yllogi&longs;mi iactura operam per­<lb/>didi&longs;&longs;et, ego &longs;pe magis, ad quam vir ille exar&longs;erat incita­<pb/>tus, quàm deterritus lap&longs;u, vehementerque dolens geo­<lb/>metriæ partem tamdiu de&longs;iderari cognitione digni&longs;&longs;imam; <lb/>cum ante exercitationis cau&longs;a omnium, quæ propo&longs;ui &longs;oli­<lb/>dorum, excepto conoide parabolico, centra grauitatis aliis <lb/>viis indaga&longs;&longs;em; po&longs;tea non &longs;olum parabolici, &longs;ed ante me <lb/>tentata nemini, hyperbolici conoidis, & fru&longs;ti vtriu&longs;que, & <lb/>portionis vtriu&longs;que conoidis, & portionis fru&longs;ti, & hemi­<lb/>&longs;phærij, & hemi&longs;phæroidis, & cuiu&longs;libet portionis &longs;phæ­<lb/>ræ, & &longs;phæroidis vno, & duobus planis parallelis ab&longs;ci&longs;&longs;æ <lb/><expan abbr="c&etilde;tra">centra</expan> grauitatis adinueni, multa autem ex his duplici, quæ­<lb/>dam triplici via. </s> <s>Taceo nunc alia eiu&longs;dem generis, quæ <lb/>cum vtilia, tum geometriæ &longs;tudio&longs;is non iniucunda, vt arbi­<lb/>tror, futura in po&longs;teriores libros di&longs;tribuimus. </s> <s>Quòd autem <lb/>aliquot propo&longs;itiones, alias Archimedis lemmaticas, alias <lb/>Commandini meis rationibus attuli demon&longs;tratas; non tàm <lb/>idcirco id fcci, ne meæ lucubrationes <expan abbr="deperir&etilde;t">deperirent</expan>, quàm quòd <lb/>vel &longs;tylo Euclidis magis con&longs;onæ, vel ad percipiendum eo <lb/>minus laborio&longs;æ, quo ad inueniendum &longs;unt difficiliores, <lb/>vel meo propo&longs;ito aptiores viderentur. </s> <s>Earum propo&longs;itio­<lb/>num, Archimedis duo &longs;unt in primo libro, decimaquarta, <lb/>& &longs;eptima, & &longs;ecunda pars vige&longs;imæ; in &longs;ecundo autem vna. <lb/></s> <s>Omne conoides parabolicum &longs;e&longs;quialterum e&longs;&longs;e coni ean­<lb/>dem ba&longs;im, & eandem altitudinem habentis. </s> <s>Comman­<lb/>dini autem omnes in primo libro nouem; vige&longs;ima tertia, & <lb/>quinta: trige&longs;ima &longs;ecunda, tertia, quarta, &longs;eptima, & nona: <lb/>quadrage&longs;ima prima, & &longs;ecunda. </s> <s>Sed multa hic noua inue­<lb/>nies ita ad præ&longs;ens in&longs;titutum nece&longs;&longs;aria, vt per &longs;e <expan abbr="tam&etilde;">tamen</expan> ip&longs;a <lb/>in geometria locum habere debeant, maxime verò tres pri­<lb/>mæ &longs;ecundi libri propo&longs;itiones, quippe quibus magnam, ac <lb/>perdifficilem geometriæ partem demon&longs;tratione recta, & <lb/>generali ad viam regiam redactam e&longs;se intelliges. </s> <s>Ita Deus <lb/>Opt. <!-- REMOVE S-->Max. <!-- REMOVE S-->cuius auxilio hæc feci, quibus prode&longs;se alicui <lb/>vehementer cupio, reliquis meis conatibus opem ferat. </s> <s>Sed <lb/>ad definitiones accedamus. </s></p><pb/><p type="head"> <s>DEFINITIONES.</s></p><p type="head"> <s>I.<!-- KEEP S--></s></p><p type="main"> <s>Figuræ aliquæ planæ multilateræ centrum ha­<lb/>bere dicuntur punctum illud, in quo omnes rectæ <lb/>lineæ vel angulos oppo&longs;itos iungentes bifariam <lb/>&longs;ecantur, vel ab angulis ductæ ad laterum op­<lb/>po&longs;itorum bipartitas &longs;ectiones in ea&longs;dem ra­<lb/>tiones. </s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>Circa diametrum e&longs;t figura plana, in qua re­<lb/>cta quædam, quæ diameter figuræ dicitur, omnes <lb/>rectas alicui parallelas, à figura terminatas bi­<lb/>fariam diuidit. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>Octaedrum communiter dictum, e&longs;t figura &longs;oli­<lb/>da octo triangulis binis parallelis, æqualibus, & <lb/>&longs;imilibus comprehen&longs;a. </s></p><p type="head"> <s>IIII.<!-- KEEP S--></s></p><p type="main"> <s>Polyedri regularis centrum dicitur punctum, <lb/>in quo omnes rectæ lineæ, quæ ad angulos oppo­<lb/>&longs;itos pertinent bifariam diuiduntur. </s></p><pb/><p type="head"> <s>V.<!-- KEEP S--></s></p><p type="main"> <s>Cuiu&longs;libet figuræ grauis centrum grauitatis <lb/>e&longs;t punctum illud, à quo &longs;u&longs;pen&longs;um graue per&longs;e <lb/>manet partibus quomodocumque circa con&longs;ti­<lb/>tutis. </s></p><p type="head"> <s>VI.<!-- KEEP S--></s></p><p type="main"> <s>Axis pri&longs;matis, & pyramidis & eius fru&longs;ti di­<lb/>citur recta linea, quæ in pyramide à vertice ad <lb/>ba&longs;is centrum figuræ vel grauitatis pertinet: in <lb/>reliquis autem, quæ ba&longs;ium oppo&longs;itarum figuræ <lb/>vel grauitatis centra iungit. </s></p><p type="head"> <s>VII.<!-- KEEP S--></s></p><p type="main"> <s>Si qua figura &longs;olida planis parallelis ita &longs;eca­<lb/>ri po&longs;&longs;it, vt quæcumque &longs;ectiones centrum ha­<lb/>beant, & &longs;int inter &longs;e &longs;imiles; aliqua autem recta <lb/>linea, &longs;iue ad centra ba&longs;ium oppo&longs;itarum prædi­<lb/>ctis &longs;ectionibus parallelarum, & &longs;imilium, vt in <lb/>cylindro; &longs;iue ad verticem, & centrum ba&longs;is ter­<lb/>minata, vt in cono, hemi&longs;phærio, & conoide, tran­<lb/>&longs;eat per centra omnium prædictarum &longs;ectionum; <lb/>ea talis figuræ axis nominetur: ip&longs;a autem figura, <lb/>&longs;olidum circa axim. </s> <s>Quæ &longs;i vel vnam tantum ha­<lb/>beat ba&longs;im, vel duas inæquales, & parallelas: dua­<lb/>rum autem quarumlibet prædictarum &longs;ectionum <lb/>vertici, vel minori ba&longs;i propinquior &longs;it minor re-<pb/>motiori; &longs;olidum circa axem in alteram partem de <lb/>ficiens nominetur: quo nomine &longs;ignificari etiam <lb/>volumus ea &longs;olida, quorum quælibet &longs;ectiones <lb/>ba&longs;i parallelæ quamuis ba&longs;i non &longs;int omnino &longs;imi­<lb/>les, tamen ijs figuris deficiunt, quæ &longs;unt &longs;imiles <lb/>ha&longs;i, ac totis ijs, à quibus ip&longs;æ ablatæ intelli­<lb/>guntur, ita vt tota figura & ablata habeant com­<lb/>mune centrum in vna recta linea ad centrum ba­<lb/>&longs;is terminata, quæ & ip&longs;a talis &longs;olidi axis nomi­<lb/>netur. </s></p><p type="main"> <s>Vt in figura, &longs;olidi ABDC deficientis &longs;olido CED <lb/>ba&longs;is e&longs;t circulus AB, terminus ba&longs;i oppo&longs;itus circum­<lb/>ferentia circuli CMD. axis communis omnibus EF, <lb/>per cuius quodlibet punctum I plano ba&longs;i AB paralle­<lb/>lo &longs;ecante &longs;olidum ABDC, & ablatum CED, & re­<lb/>&longs;iduum, e&longs;t totius <lb/>&longs;ectio circulus G <lb/>H, ablati vero cir­<lb/>culus KL, & re&longs;i­<lb/>dui &longs;ectio reliquum <lb/>circuli GH dem­<lb/>pto circulo KL. <lb/>quarum &longs;ectionum <lb/>omnium centrum <lb/>commune e&longs;t I. <lb/><!-- KEEP S--></s> <s>Quod &longs;i &longs;uper duos <lb/><figure id="id.043.01.013.1.jpg" xlink:href="043/01/013/1.jpg"/><lb/>circulos GH, KL circa axem communem EI cylin­<lb/>dri de&longs;cribantur, (erunt autem eiu&longs;dem altitudinis) erit <lb/>reliquum cylindri GB, dempto cylindro cuius ba&longs;is <lb/>KL, axis EI, con&longs;titutum &longs;uper ba&longs;im G, <emph type="italics"/>K<emph.end type="italics"/>, & circa <lb/>axim EI, quæ &longs;uo loco expectatur cogitatio. </s></p><pb/><p type="head"> <s>POSTVLATA.</s></p><p type="head"> <s>I.<!-- KEEP S--></s></p><p type="main"> <s>Omnis figuræ grauis vnum e&longs;&longs;e centrum gra­<lb/>uitatis. </s></p><p type="head"> <s>II.<!-- KEEP S--></s></p><p type="main"> <s>Omnium figurarum &longs;ibi mutuo congruentium <lb/>centra grauitatis mutuo &longs;ibi congruere. </s></p><p type="head"> <s>III.<!-- KEEP S--></s></p><p type="main"> <s>Omnis figuræ, cuius termini omnis cauitas <lb/>e&longs;t interior, intra terminum e&longs;&longs;e centrum graui­<lb/>tatis. </s></p><p type="head"> <s>IIII.<!-- KEEP S--></s></p><p type="main"> <s>Similium triangulorum &longs;imiliter po&longs;ita e&longs;se <lb/>centra grauitatis. </s> <s>In triangulis autem &longs;imilibus <lb/>&longs;imiliter po&longs;ita puncta e&longs;&longs;e dicuntur, à quibus re­<lb/>ctæ ad angulos æquales ductæ cum lateribus ho­<lb/>mologis angulos æquales faciunt. </s></p><p type="head"> <s>V.<!-- KEEP S--></s></p><p type="main"> <s>Æqualia grauia ab æqualibus longitudinibus <lb/>&longs;ecundum centrum grauitatis &longs;u&longs;pen&longs;a æquipon­<lb/>derare. </s></p><p type="head"> <s>VI.<!-- KEEP S--></s></p><p type="main"> <s>A quibus longitudinibus duo grauia æquipon<lb/>derant, ab ij&longs;dem alia duo quælibet illis æqualia <lb/>æquiponderare. </s></p><pb/><p type="head"> <s>PROPOSITIO <lb/>PRIMA.</s></p><p type="main"> <s>Si &longs;int quotcumque magnitu­<lb/>dines inæquales deinceps <lb/>proportionales; exce&longs;&longs;us, qui <lb/>bus differunt deinceps pro­<lb/>portionales erunt, in propor­<lb/>tione totarum magnitudi­<lb/>num. </s></p><p type="main"> <s>Sint quotcumque inæquales magnitudines deinceps <lb/>proportionales AB, CD, EF, & G, <lb/>differentes exce&longs;&longs;ibus BH, DK, FL, mi­<lb/>nima autem &longs;it G. <!-- KEEP S--></s> <s>Dico BH, DK, FL, <lb/>deinceps proportionales e&longs;se in proportio­<lb/>ne, quæ e&longs;t AB, ad CD, &longs;eu CD, ad <lb/>EF. <!-- KEEP S--></s> <s>Quoniam enim e&longs;t vt AB, ad <lb/>CD, ita CD ad EF; hoc e&longs;t vt AB, ad <lb/>AH, ita CD, ad CK, permutando <lb/>erit, vt AB, ad CD, ita AH, ad CK: <lb/>vt igitur tota AB, ad totam CD, ita <lb/>reliqua BH, ad reliquam DK. </s> <s>Simili­<lb/>ter o&longs;tenderemus e&longs;se vt CD ad EF, <lb/>ita DK ad FL; vt igitur BH ad DK, <lb/>ita erit DK ad FL, in proportione, quæ <lb/>e&longs;t AB ad CD, & CD ad EF. <!-- KEEP S--></s> <s>Quod demon&longs;tran­<lb/>dum erat. </s></p><figure id="id.043.01.015.1.jpg" xlink:href="043/01/015/1.jpg"/><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>In omni triangulo vnum dumtaxat punctum <lb/>e&longs;t, in quo rectæ ab angulis ad latera incidentes <lb/>&longs;ecant &longs;e&longs;e in ea&longs;dem rationes. </s> <s>& &longs;egmenta, quæ <lb/>ad angulos, &longs;unt reliquorum dupla. </s> <s>& prædictæ <lb/>incidentes &longs;ecant trianguli latera bifariam. </s></p><p type="main"> <s>Sit triangulum ABC, cuius duo quælibet latera AB, <lb/>AC, &longs;int bifariam &longs;ecta in punctis D, E, & ductæ rectæ <lb/>lineæ BE, CFD, AFG. </s> <s>Dico CF duplam e&longs;&longs;e ip&longs;ius <lb/>FD, & AF, ip&longs;ius FG, & BF, ip&longs;ius FE. <!-- KEEP S--></s> <s>Et in nullo alio <lb/>puncto à puncto F tres rectas ab angulis ad latera inciden­<lb/>tes &longs;ecare &longs;e &longs;e in ea&longs;dem rationes. </s> <s>Et reliquum latus BC <lb/>&longs;ectum e&longs;&longs;e bifariam in puncto G. <!-- KEEP S--></s> <s>Quoniam enim e&longs;t vt BA <lb/>ad AD, ita CA ad AE: hoc e&longs;t, vt triangulum ABC ad <lb/>triangulum ADC, ita triangulum idem ABC ad trian­<lb/>gulum AEB; æqualia <lb/>erunt triangula ADC, <lb/>AEB, & ablato trape­<lb/>zio DE communi re­<lb/>liquum triangulum BD <lb/>F reliquo triangulo C <lb/>EF æquale erit: &longs;ed <lb/>triangulum ADF e&longs;t <lb/>æquale triangulo BDF; <lb/>& triangulum AFE <lb/>triangulo EFC, pro­<lb/>pter æquales ba&longs;es, & <lb/><figure id="id.043.01.016.1.jpg" xlink:href="043/01/016/1.jpg"/><lb/>communes altitudines; totum igitur triangulum AFB <lb/>toti AFC, triangulo æquale erit: &longs;ed vt triangulum AFB <pb/>ad triangulum FBG, hoc e&longs;t vt AF ad FG, ita e&longs;t <lb/>triangulum AFC ad triangulum FCG; triangulum er­<lb/>go FBG triangulo FCG æquale erit, & ba&longs;is BG ba­<lb/>&longs;i GC æqualis. </s> <s>Quoniam igitur & AE e&longs;t æqualis <lb/>EC, &longs;imiliter vt ante, o&longs;tenderemus, triangulum BCF, <lb/>triangulo ACF, eademque ratione triangulum ABF, <lb/>triangulo BCF æquale e&longs;&longs;e: igitur vnumquodque trian­<lb/>gulorum ABF, ACF, BCF, tertia pars e&longs;t trianguli <lb/>ABC: &longs;ed vt triangulum ABC, ad triangulum BCF, <lb/>ita e&longs;t AG, ad GF; tripla igitur e&longs;t AG ip&longs;ius GF, <lb/>ac proinde AF, ip&longs;ius FG dupla. </s> <s>Eadem ratione <lb/>BE, ip&longs;ius FE, & CF, ip&longs;ius FD, dupla concludetur. </s></p><p type="main"> <s>Sed &longs;int &longs;i fieri pote&longs;t, trianguli ABC duo centra qua­<lb/>lia diximus D, E: & ab ip&longs;is ad &longs;ingulos angulos du­<lb/>cantur binæ rectæ lineæ: <lb/>& eadat D in aliquo trian <lb/>gulo BEC. </s> <s>Quoniam <lb/>igitur D e&longs;t centrum trian <lb/>guli ABC erit triangu­<lb/>lum BDC tertia pars <lb/>trianguli ABC. <!-- KEEP S--></s> <s>Eadem <lb/>ratione triangulum BEC <lb/>tertia pars erit trianguli <lb/>ABC; triangulum ergo <lb/>DBC æquale erit trian­<lb/>gulo BEC pars toti, quod <lb/>fieri non pote&longs;t, atqui <expan abbr="id&etilde;">idem</expan> <lb/><figure id="id.043.01.017.1.jpg" xlink:href="043/01/017/1.jpg"/><lb/>ab&longs;urdum &longs;equitur, &longs;i punctum D cadat in aliquo latere <lb/>triangulorum, quorum vertex E; Manife&longs;tum e&longs;t igitur <lb/>propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>In &longs;imilibus triangulis rectæ lineæ, quæ inter <lb/>centra, & alia in ijs &longs;imiliter po&longs;ita puncta in­<lb/>terijciuntur, proportionales &longs;unt in proportione <lb/>laterum homologorum. </s></p><p type="main"> <s>Sint triangula &longs;imilia, & &longs;imiliter po&longs;ita ABC, DEF, <lb/>quorum &longs;int centra O, P, in ijs autem triangulis &longs;int pun­<lb/>cta &longs;imiliter po&longs;ita K, L, quæ cadant primum in rectis <lb/>BG, EH, quæ ab angulis æqualibus B, E, ba&longs;es bifa­<lb/>riam diuidunt. </s> <s>Dico e&longs;&longs;e OK ad PL, vt e&longs;t latus AB, <lb/>ad latus DE. iunctis enim AK, KC, DL, LF, quo­<lb/><figure id="id.043.01.018.1.jpg" xlink:href="043/01/018/1.jpg"/><lb/>niam angulus KAC, æqualis e&longs;t angulo LDF, & angu­<lb/>lus KCA, angulo LFD, ob &longs;imiliter po&longs;ita puncta K, <lb/>L, triangulum AKC, triangulo LDF &longs;imile erit, & vt <lb/>KA ad AC, ita LD ad DF: &longs;ed vt CA ad AG, ita <lb/>e&longs;t FD ad DH, expræcedenti; vt igitur KA, ad AG <lb/>ita erit LD, ad DH, circa æquales angulos: &longs;imilia igi­<lb/>tur &longs;unt triangula AGK, DHL, & angulus AGK, <pb/>æqualis angulo DHL, & vt KG, ad GA, ita LH, ad <lb/>HD: &longs;ed vt GA, ad AC, ita e&longs;t HD ad DF: & vt <lb/>AC ad AB, ita DF ad DE, ex æquali igitur erit vt <lb/>KG ad AB, ita LH ad DE: &longs;ed vt AB ad BG, ita <lb/>e&longs;t DE ad EH, propter &longs;imilitudinem triangulorum <lb/>ABG, DEH: & vt BG ad GO ita e&longs;t EH ad HP, <lb/>propter triangulorum centra O, P; ex æquali igitur erit <lb/>vt KG ad GO, ita LH ad HP: & permutando vt <lb/>OG ad PH, ide&longs;t vt BG ad EH, ide&longs;t vt AB ad ED, <lb/>ita KG ad LH, & reliqua OK ad reliquam PL. </s></p><p type="main"> <s>Sed &longs;int puncta &longs;imiliter po&longs;ita M, N, quæ cadant ex­<lb/>tra lineas BG, EH, iunctæque OM, PN. <!-- KEEP S--></s> <s>Dico iti­<lb/>dem e&longs;se vt AB ad ED, ita OM ad PN. <!-- KEEP S--></s> <s>Iungantur <lb/>enim rectæ MB, NE, quæ cum quibus lateribus homo­<lb/>logis angulos æquales faciunt, ea &longs;int AB, DE, quod <lb/>propter i&longs;o&longs;celia triangula &longs;it dictum in &longs;imiliter po&longs;itis <lb/>triangulis. </s> <s>igitur etiam angulus BAM, æqualis erit an­<lb/>gulo EDN; &longs;imilia igitur triangula ABM, DEN: & <lb/>vt MB ad BA, ita erit NE ad ED: &longs;ed vt AB ad <lb/>BG, ita e&longs;t DE ad EH, propter &longs;imilitudinem trian­<lb/>gulorum, & vt BG ad BO, ita e&longs;t EH ad EP, ob <lb/>triangulorum &longs;imilium centra O, P: ex æquali igitur <lb/>erit vt MB, ad BO, ita NE ad EP. </s> <s>Rur&longs;us quo­<lb/>niam angulus ABM, æqualis e&longs;t angulo DEN, quorum <lb/>angulus ABG, æqualis e&longs;t angulo DEH: erit reliquus <lb/>angulus OBM, æqualis reliquo angulo PEN: &longs;ed vt MB <lb/>ad BO, ita erat NE ad EP; triangulum igitur OBM <lb/>triangulo PEN, &longs;imile erit, & vt BO ad EP, hoc e&longs;t <lb/>BG ad EH, hoc e&longs;t AB ad DE, ita OM ad PN. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO IV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Datis duobus triangulis &longs;calenis &longs;imilibus, & <lb/>dato puncto in altero eorum, vnum duntaxat pun­<lb/>ctum in reliquo triangulo prædicto puncto &longs;imi­<lb/>liter po&longs;itum pote&longs;t inueniri. </s></p><p type="main"> <s>Sint data duo triangula &longs;calena &longs;imilia ABC, DEF, <lb/>& in triangulio ABC datum punctum G: &longs;int autem <lb/>hæc triangula &longs;imiliter po&longs;ita. </s> <s>Dico in triangulo DEF, <lb/>vnum duntaxat punctum puncto G &longs;imiliter po&longs;itum in­<lb/>ueniri po&longs;se. </s> <s>Iunctis enim AG, BG, GC, ponatur <lb/>angulus EDH, æqualis angulo BAG, & angulus DEH, <lb/><figure id="id.043.01.020.1.jpg" xlink:href="043/01/020/1.jpg"/><lb/>æqualis angulo ABG, & HF iungatur. </s> <s>Manife&longs;tum <lb/>e&longs;t igitur ex præcedentis Theorematis demon&longs;tratione, <lb/>triangula EDH, HDF, FEH, &longs;imilia e&longs;se triangulis <lb/>BAG, GAC, CBG, prout inter &longs;e re&longs;pondent po&longs;i­<lb/>tione, quorum &longs;ex triangulorum binis quibu&longs;que binæ ba­<lb/>&longs;es homologæ re&longs;pondent: AB ED, AC DF, BC <pb/>EF. quæ &longs;untin latera homologa duorum triangulorum <lb/>ABC, DEF. <!-- KEEP S--></s> <s>Ex definitione igitur, duo puncta G, H, <lb/>in triangulis ABC, DEF, &longs;imiliter po&longs;ita erunt. </s> <s>At <lb/>enim &longs;i fieri pote&longs;t &longs;it aliud punctum K, in triangulo <lb/>DEF, &longs;imiliter po&longs;itum puncto G. <!-- KEEP S--></s> <s>Vel igitur punctum <lb/>K in aliquo triangulorum, quorum e&longs;t communis vertex <lb/>H, vel in aliquo eorundem latere cadet. </s> <s>cadat in latere <lb/>FH, & iungatur DK: triangulum ergo DFK, &longs;imile <lb/>erit triangulo ACG. </s> <s>Sed & triangulum EDF, &longs;imile <lb/>e&longs;t triangulo BAC; vtraque igitur horum ad illorum &longs;i­<lb/>bi re&longs;pondens triangulorum duplicatam eorundem late­<lb/>rum homologorum AC, DF, habebunt proportionem: <lb/>vt igitur e&longs;t triangulum EDF, ad triangulum BAC, ita <lb/>erit triangulum DFK, ad triangulum ACG: & per­<lb/>mutando, vt triangulum ACG, ad triangulum ABC, <lb/>ita triangulum DFK, ad triangulum EDF: eadem ra­<lb/>tione, vt triangulum ACG, ad triangulum ABC, ita <lb/>erit triangulum DFH, ad triangulum DEF: vt igitur <lb/>triangulum DFK, ad triangulum EDF; ita erit trian­<lb/>gulum DFH, ad triangulum EDF; triangulum ergo <lb/>DFK, triangulo DFH, æquale erit, pars toti, quod e&longs;t <lb/>ab&longs;urdum: idem autem ab&longs;urdum &longs;equeretur, &longs;i punctum <lb/><emph type="italics"/>K<emph.end type="italics"/>, poneretur in aliquo prædictorum triangulorum, vt in <lb/>triangulo DFH; Non igitur aliud punctum à puncto H, <lb/>in triangulo EDF, &longs;imiliter po&longs;itum erit puncto G. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO V.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Cuilibet figuræ planæ rectangulum æquale <lb/>pote&longs;t e&longs;&longs;e. </s></p><pb/><p type="main"> <s>Sit quælibet figura plana A. <!-- KEEP S--></s> <s>Dico figuræ A, rectan­<lb/>gulum æquale po&longs;se exi&longs;tere. </s> <s>Exponatur enim rectan­<lb/>gulum BC, cuius latus BD, in infinitum producatur <lb/>ver&longs;us E. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt rectangulum BD, ad <lb/>planam figuram A, ita recta BD, ad aliquam lineam <lb/>rectam &longs;it vt BC, ad A, ita BD, ad DE, & comple­<lb/>atur rectan­<lb/>gulum EC. <lb/></s> <s>Quoniam igi <lb/>tur e&longs;t vt BD <lb/>ad DE, ita <lb/>rectangulum <lb/>BC, ad figu­<lb/>ram A: &longs;ed <lb/>vt BD, ad <lb/>DE, ita e&longs;t <lb/><figure id="id.043.01.022.1.jpg" xlink:href="043/01/022/1.jpg"/><lb/>rectangulum BC, ad rectangulum CE; vt igitur re­<lb/>ctangulum BC, ad figuram A, ita e&longs;t rectangulum <lb/>BC, ad rectangulum CE; rectangulum ergo CE, fi­<lb/>guræ A, æquale erit. </s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omni figuræ circa diametrum in alte ram par­<lb/>tem deficienti figura quædam ex parallelogram­<lb/>mis æqualium altitudinum in&longs;cribi pote&longs;t, & al­<lb/>tera circum&longs;cribi, ita vt circum&longs;cripta &longs;uperet in­<lb/>&longs;criptam minori &longs;pacio quantacumque magnitu­<lb/>dine propo&longs;ita. </s> <s>Semper autem in &longs;imilibus intelli­<lb/>ge, eiu&longs;dem generis. </s></p><p type="main"> <s>Sit figura plana ABC circa diametrum AD, ad par-<pb/>tes A deficiens, cuius ba&longs;is BC. <!-- KEEP S--></s> <s>Dico fieri po&longs;se quod <lb/>proponitur: ducta enim per verticem figuræ A, ba&longs;i BC, <lb/>parallela, atque ideo figuram ip&longs;am contingente, ab&longs;ol­<lb/>uatur parallelogrammum BL, &longs;ectaque diametro AD, <lb/>bifariam, & &longs;ingulis eius partibus &longs;emper bifariam, du­<lb/>cantur per puncta &longs;ectionum rectæ lineæ ba&longs;i BC, & in­<lb/>ter &longs;e parallelæ, atque ita multiplicatæ &longs;int &longs;ectiones, <lb/>vt &longs;ecti parallelogrammi in parallelogramma æqua­<lb/>lia, & eiu&longs;dem altitudinis quælibet pars, vt paralle­<lb/>logrammum BF, &longs;it minus &longs;uperficie propo&longs;ita, cu­<lb/>ius parallelogram­<lb/>mi latus EF, &longs;e­<lb/>cet figuræ termi­<lb/>num BAC, in <lb/>punctis GH, & <lb/>diametrum AD, in <lb/>puncto K. erit igi­<lb/>tur GK, æqualis <lb/>KH: per omnia <lb/>igitur puncta &longs;e­<lb/>ctionum termini <lb/><figure id="id.043.01.023.1.jpg" xlink:href="043/01/023/1.jpg"/><lb/>BAC, quæ à prædictis fiunt lineis parallelis, &longs;i ducan­<lb/>tur diametro AD parallelæ, figura quædam ip&longs;i ABC, <lb/>in&longs;cribetur, & altera circum&longs;cribetur ex parallelogram­<lb/>mis æqualium altitudinum. </s> <s>Dico harum figurarum <lb/>in&longs;criptam &longs;uperari à circum&longs;cripta minori &longs;pacio &longs;uper­<lb/>ficie propo&longs;ita. </s> <s>Quoniam enim omnia parallelogramma, <lb/>quibus figura circum&longs;cripta &longs;uperat in&longs;criptam &longs;imul &longs;um­<lb/>pta &longs;unt æqualia BF parallelogrammo: &longs;ed parallelo­<lb/>grammum BF, e&longs;t minus &longs;uperficie propo&longs;ita: exce&longs;&longs;us <lb/>igitur quo figura circum&longs;cripta in&longs;criptam &longs;uperat, minor <lb/>erit &longs;uperficie propo&longs;ita. </s> <s>Fieri igitur pote&longs;t, quod propo­<lb/>nebatur. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO VII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Pyramides &longs;imilibus, & æqualibus triangulis <lb/>comprehen&longs;æ inter &longs;e &longs;unt æquales. </s></p><p type="main"> <s>Sint pyramides ABCD, EFGH, &longs;imilibus, & æqua­<lb/>libus triangulis comprehen&longs;æ, & &longs;i &longs;int &longs;imiliter po&longs;itæ, qua­<lb/>rum vertices A, E, ba&longs;es autem triangula BCD, FGH. <lb/></s> <s>Dico pyramidem ABCD, pyramidi EFGH, æqualem <lb/>e&longs;se. </s> <s>A punctis enim A, E, manantia latera inferius pro­<lb/>ducantur, & prædictis lateribus maiores, inter &longs;e autem <lb/>æquales ab&longs;cindantur AK, AL, AM, EN, EO, EP, <lb/><figure id="id.043.01.024.1.jpg" xlink:href="043/01/024/1.jpg"/><lb/>& con&longs;truantur pyramides AKLM, ENOP: pyramides <lb/>igitur hæ æqualibus, & &longs;imilibus triangulis comprehenden <lb/>tur, vt colligitur ex ip&longs;a con&longs;tructione; triangulis igitur inter <lb/>&longs;e æquilateris, & æquiangulis KLM, NOP, inter &longs;e con­<lb/>gruentibus non congruat, &longs;i fieri pote&longs;t, pyramis ENOP, <lb/>pyramidi AKLM, &longs;ed cadat vertex E, pyramidis ENOP, <lb/>extra verticem A, pyramidis AKLM, & ex puncto A, <pb/>ad centrum circuli tran&longs;euntis per tria puncta K, L, M, quod <lb/>&longs;it R, ducatur recta AR, & ER iungatur. </s> <s>Quoniam igi­<lb/>tur æquales rectæ &longs;unt AK, AL, AM, quæ ex puncto <lb/>A, in &longs;ublimi pertinent ad &longs;ubiectum planum: & punctum <lb/>R, e&longs;t centrum circuli tran&longs;euntis per puncta N, O, P; cadet <lb/>recta AR ad &longs;ubiectum planum perpendicularis. </s> <s>Eadem <lb/>ratione recta ER ducta à vertice E, pyramidis ENOP, <lb/>ad centrum R, circuli tran&longs;euntis per puncta N, O, P, hoc <lb/>e&longs;t, per puncta K, L, M, illis congruentia, cadet ad idem <lb/>planum, ad quod linea AR, perpendicularis; itaque ab <lb/>eodem puncto R, ad idem planum, & ad ea&longs;dem partes duæ <lb/>perpendiculares erunt excitatæ, quod fieri non pote&longs;t: <lb/>punctum igitur E non cadet extra punctum A: quare la­<lb/>tus EN, congruet lateri AK, quorum EF, e&longs;t æqualis <lb/>AK; igitur & EF, ip&longs;i AB, congruet. </s> <s>eadem ratione la­<lb/>tus AG, congruet lateri AC, & latus EH, lateri AD, & <lb/>triangula triangulis, & pyramis EFGH, pyramidi ABC <lb/>D, & ip&longs;i æqualis erit. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hinc facile colligitur omnia &longs;olida, quæ in py <lb/>ramides æqualibus, & &longs;imilibus triangulis com­<lb/>prehen&longs;as multitudine æquales diuidi po&longs;&longs;unt, e&longs; <lb/>&longs;e inter &longs;e æqualia. </s> <s>Quocirca omnia pri&longs;mata, & <lb/>pyramides, & octahedra, omnia denique corpora <lb/>regularia æqualibus, & &longs;imilibus planis compre­<lb/>hen&longs;a inter &longs;e æqualia erunt. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pyramidis triangulam ba&longs;im habentis <lb/>quatuor axes &longs;ecant &longs;e in vno puncto in ea&longs;dem ra­<pb/>tiones, ita vt &longs;egmenta, quæ ad angulos, eo­<lb/>rum, quæ ad oppo&longs;ita triangula, &longs;int tripla; ex quo <lb/>puncto tota pyramis diuiditur in quatuor pyrami <lb/>des æquales. </s> <s>Et in nullo alio puncto quatuor re­<lb/>ctæ lineæ ductæ ab angulis ad triangula oppo&longs;ita <lb/>pyramidis &longs;ecant &longs;e&longs;e in ea&longs;dem rationes. </s> <s>Vocetur <lb/>autem punctum hoc centrum dictæ pyramidis. </s></p><p type="main"> <s>Sit pyramis ABCD, cuius vertex A, ba&longs;is autem <lb/>triangulum BCD, axes AE, BM, CL, DN, vnde qua­<lb/>tuor triangulorum, quæ &longs;unt circa pyramidem ABCD, <lb/>centra erunt grauitatis E, L, M, N. <!-- KEEP S--></s> <s>Dico quatuor li­<lb/>neas AE, BM, CL, DN, &longs;ecare &longs;e &longs;e in vno puncto in <lb/>ea&longs;dem rationes, quas prædixi, & quæ &longs;equuntur. </s> <s>Nam ex <lb/>puncto A, ducatur recta ALH, quæ ob trianguli ABD, <lb/>centrum L, &longs;ecabit latus BD, bifariam in puncto H; iun­<lb/>cta igitur CE, & producta conueniet cum ALH, vt in <lb/>puncto H. eadem ratione iunctæ AM, BE, & productæ <lb/>conuenient in medio lateris CD, conueniant in puncto K, <lb/>necnon AN, DE, in medio ip&longs;ius BC, vt in puncto G. <lb/><!-- KEEP S--></s> <s>Quoniam igitur ob triangulorum centra, e&longs;t vt CE ad EH, <lb/>ita AL ad LH, dupla enim e&longs;t vtraque vtriu&longs;que, &longs;eca­<lb/>bunt &longs;e&longs;e rectæ AE, CL, inter ea&longs;dem parallelas; quare <lb/>vt AF ad FE, ita erit CF ad FL, circum æquales angu <lb/>los ad verticem: triangula igitur AFL, CFE; & reci­<lb/>proca, & æqualia inter &longs;e erunt. </s> <s>Cum igitur &longs;it vt AL ad <lb/>LH, ita CE ad EH, hoc e&longs;t vt triangulum AFL ad <lb/>triangulum FLH, (&longs;i ducatur FH) ita triangulum CFE, <lb/>ad triangulum FEH, erunt inter &longs;e æqualia triangula <lb/>FEH, FLH. </s> <s>Quare vt triangulum AFH, ad triangu­<lb/>lum FLH, hoc e&longs;t vt AH ad HL, ita erit triangulum <lb/>AFH ad triangulum FEH, hoc e&longs;t AF ad FE: &longs;ed re­<lb/>cta AH, e&longs;t tripla ip&longs;ius LH; igitur & AF, erit ip&longs;ius FE, <pb/>tripla: &longs;ed vt AF, ad FE, ita e&longs;t CF, ad FL; tripla igi­<lb/>tur erit CF, ip&longs;ius FL. </s> <s>Similiter o&longs;tenderemus rectas <lb/>AE, BM, &longs;ecare &longs;e &longs;e in ea&longs;dem rationes, ita vt &longs;egmen­<lb/>ta, quæ ad angulos, &longs;int tripla eorum, quæ &longs;unt ad centra <lb/>E, M, quorum AF, e&longs;t tripla ip&longs;ius FE: in puncto igitur <lb/>F, &longs;ecant &longs;e rectæ lineæ AE, BM. </s> <s>Eadem ratione & re <lb/>ctæ AE, DN, &longs;ecent &longs;e in puncto F, nece&longs;se erit: quare <lb/>vt AF ad FE, ita erit DF ad FN. </s> <s>Quatuor igitur <lb/>axes pyramidis ABCD, &longs;ecant&longs;e &longs;e in puncto F, in ea&longs;­<lb/>dem rationes, ita vt <lb/>&longs;egmenta ad angulos, <lb/>&longs;int <expan abbr="reliquorũ">reliquorum</expan> tripla. <lb/></s> <s>Rur&longs;us, quia compo­<lb/>nendo, & conuerten­<lb/>do, e&longs;t vt FE ad EA, <lb/>ita FL ad LC: hoc <lb/>e&longs;t, vt pyramis BCD <lb/>F, ad pyramidem A <lb/>BCD, ita pyramis <lb/>ABDF, ad pyrami­<lb/>dem CBDA, (pro­<lb/>pter ba&longs;ium commu­<lb/>nitatem, & vertices in <lb/>eadem recta linea) erit <lb/><figure id="id.043.01.027.1.jpg" xlink:href="043/01/027/1.jpg"/><lb/>pyramis ABDF, æqualis pyramidi BCDF. <!-- KEEP S--></s> <s>Eadem ra­<lb/>tione tam pyramis ACDF, quàm pyramis ABCF, æqua <lb/>lis e&longs;t pyramidi BCDF. <!-- KEEP S--></s> <s>Quatuor igitur pyramides, qua­<lb/>rum communis vertex punctum F, ba&longs;es autem triangula, <lb/>quæ &longs;unt circa pyramidem ABCD, inter &longs;e æquales <expan abbr="erũt">erunt</expan>, <lb/>& vnaquæque pyramidis ABCD, pars quarta. </s> <s>Dico in <lb/>nullo alio puncto à puncto F, quatuor rectas, quæ ab an­<lb/>gulis ad triangula oppo&longs;ita pyramidis ABCD, ducantur, <lb/>&longs;ecare &longs;e in ea&longs;dem rationes. </s> <s>Si enim fieri pote&longs;t &longs;ecent <lb/>&longs;e tales rectæ in ea&longs;dem rationes in alio puncto S. <!-- KEEP S--></s> <s>Simi­<pb/>liter igitur vt ante o&longs;tenderemus, vnamquamque qua­<lb/>tuor pyramidum, quarum communis vertex S, ba&longs;es au­<lb/>tem triangula, quæ &longs;unt circa pyramidem ABCD, e&longs;se <lb/>quartam partem pyramidis ABCD. <!-- KEEP S--></s> <s>Siue igitur pun­<lb/>ctum S, cadat intra vnam priorum quatuor pyrami­<lb/>dum, &longs;iue in earum aliquo latere, &longs;eu triangulo; nece&longs;­<lb/>&longs;ario erit pars æquali toti; tam enim tota vna pyramis <lb/>quatuor priorum, quarum communis vertex F, quàm eius <lb/>pars, vna quatuor pyramidum po&longs;teriorum, quarum com­<lb/>munis vertex S, erit eiu&longs;dem ABCD, pyramidis pars <lb/>quarta. </s> <s>Ex ab&longs;urdo igitur non in alio puncto à puncto F <lb/>&longs;ecabunt &longs;e in ea&longs;dem rationes quatuor rectæ, quæ ab angu <lb/>lis ad oppo&longs;ita triangula pyramidis ABCD, ducantur. <lb/></s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO IX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pyramis ba&longs;im habens triangulam di­<lb/>uiditur in quatuor pyra mides æquales, & &longs;imiles <lb/>inter &longs;e, & toti, & vnum octaedrum totius pyrami­<lb/>dis dimidium, ip &longs;i que concentricum. </s></p><p type="main"> <s>Sit pyramis ABCD, cuius ba&longs;is triangulum ABC, <lb/>&longs;ectisque omnibus lateribus bifariam, iungantur rectæ FG, <lb/>GH, HF, FK, KL, LM, M<emph type="italics"/>K<emph.end type="italics"/>, KH, HM, GL, LF. <lb/></s> <s>Dico quatuor pyramides DKLM, LFBG, KHFA, <lb/>MHGC, æquales e&longs;se, & &longs;imiles inter &longs;e, & toti pyrami­<lb/>di ABCD: octaedrum autem e&longs;se LFGM<emph type="italics"/>K<emph.end type="italics"/>H, & di­<lb/>midium pyramidis ABCD, ip&longs;ique concentricum. </s> <s>Du­<lb/>cantur enim rectæ DNH, BQH, LN: & po&longs;ita BE, du <lb/>pla ip&longs;ius BH, iungatur DOC, in triangulo DBH, & <lb/>ponatur DP, ip&longs;ius PE, tripla, & connectantur rectæ LP, <lb/>PH. <!-- KEEP S--></s> <s>Quoniam igitur E, e&longs;t centrum trianguli ABC, <pb/>erit axis DE, pyramidis ABCD, cuius axis &longs;egmentum <lb/>DP e&longs;t triplum ip&longs;ius PE: igitur P centrum erit pyra­<lb/>midis ABCD. <!-- KEEP S--></s> <s>Et quoniam tres rectæ FK, KH, HF, <lb/>&longs;unt parallelæ tribus BD, DC, CB, pro vt inter &longs;e re&longs;pon<lb/>dent, vt KH, ip&longs;i LG, quoniam vtraque lateri DC, ob <lb/>latera triangulorum &longs;ecta proportionaliter in punctis K, H, <lb/>L, G: & &longs;ic de reliquis; erit pyramis A<emph type="italics"/>K<emph.end type="italics"/>FH, &longs;imilis toti <lb/>pyramidi ABCD. <!-- KEEP S--></s> <s>Similiter vnaquæque trium aliarum <lb/>pyramidum ab&longs;ci&longs;&longs;arum, videlicet FLBG, GHMC, <lb/>KDLM, &longs;imilis erit pyramidi ABCD, atque ideo in­<lb/>ter &longs;e &longs;imiles. </s> <s>Rur&longs;us, <lb/>quoniam pyramidum <lb/>&longs;imilium latus AD e&longs;t <lb/>duplum lateris AK, ho <lb/>mologi; pyramis AB­<lb/>CD, octupla erit py­<lb/>ramidis AKFH, ob <lb/>triplicatam laterum ho <lb/>mologorum proportio <lb/>nem. </s> <s>Similiter <expan abbr="vna-qũæque">vna­<lb/>qunæque</expan> trium reliqua­<lb/>rum pyramidum ab&longs;ci&longs; <lb/>&longs;arum erit octaua pars <lb/>pyramidis ABCD; <lb/><figure id="id.043.01.029.1.jpg" xlink:href="043/01/029/1.jpg"/><lb/>quatuor igitur pyramides ab&longs;ci&longs;&longs;æ &longs;imul &longs;umptæ dimi­<lb/>dium erit pyramidis ABCD: & reliquum igitur &longs;oli­<lb/>dum demptis quatuor pyramidibus, dimidium pyramidis <lb/>ABCD. <!-- KEEP S--></s> <s>Dico reliquum &longs;olidum LKMGFH, e&longs;&longs;e <lb/>octaedrum. </s> <s>Nam octo triangulis ip&longs;um contineri mani­<lb/>fe&longs;tum e&longs;t. </s> <s>bina autem oppo&longs;ita e&longs;&longs;e parallela, & æqualia, <lb/>& &longs;imilia, &longs;ic o&longs;tendimus. </s> <s>Quoniam enim triangulum <lb/>FGH, e&longs;t in plano trianguli ABC, plano trianguli KLM <lb/>parallelo; erit triangulum FGH, parallelum triangu-<pb/>lo KLM: &longs;ed triangulum FGH, e&longs;t &longs;imile triangulo <lb/>ABC, & triangulum KLM, &longs;imile eidem triangulo <lb/>ABC; <expan abbr="triangulũ">triangulum</expan> ergo FGH, &longs;imile erit triangulo KLM: <lb/>&longs;ed & æquale propter æqualitatem laterum homologo­<lb/>rum. </s> <s>Similiter o&longs;tenderemus reliquum &longs;olidum LKM <lb/>GFH continentia triangula bina oppo&longs;ita æqualia <lb/>inter &longs;e, & &longs;imilia, & parallela; octaedrum e&longs;t igitur <lb/>LKMGFH. <!-- KEEP S--></s> <s>Dico iam punctum P, quod e&longs;t cen­<lb/>trum pyramidis ABCD, e&longs;se centrum octaedri L<emph type="italics"/>K<emph.end type="italics"/><lb/>MGFH. <!-- KEEP S--></s> <s>Quoniam enim DP, ponitur tripla ip&longs;ius PE, <lb/>& DO, e&longs;t æqualis <lb/>OE (&longs;iquidem planum <lb/>trianguli KLM, plano <lb/><expan abbr="triãguli">trianguli</expan> ABC, paralle <lb/>lum &longs;ecat proportione <lb/><expan abbr="o&etilde;s">oens</expan> rectas lineas, quæ <lb/>ex puncto D, in &longs;ubli­<lb/>mi pertinent ad &longs;ubie­<lb/>ctum planum trianguli <lb/>ABC) erit OP, ip&longs;i <lb/>PE, æqualis. </s> <s>Et quo­<lb/>niam BH e&longs;t dupla <lb/>ip&longs;ius QH, quarum <lb/>BE e&longs;t dupla ip&longs;ius <lb/><figure id="id.043.01.030.1.jpg" xlink:href="043/01/030/1.jpg"/><lb/>EH, &longs;iquidem E e&longs;t centrum trianguli ABC; erit reli­<lb/>qua EH reliquæ EQ dupla: & quia e&longs;t vt LD ad DB, <lb/>ita LN ad BH, propter &longs;imilitudinem triangulorum, & <lb/>e&longs;t LD, dimidia ip&longs;ius BD, erit & LN, dimidia ip&longs;ius <lb/>BH: &longs;ed QH e&longs;t dimidia ip&longs;ius BH; æqualis igitur LN <lb/>ip&longs;i QH. </s> <s>Iam igitur quia e&longs;t vt BE ad EH, ita <lb/>LO ad ON: &longs;ed BE, e&longs;t dupla ip&longs;ius EH; dupla igi­<lb/>tur LO, erit ip&longs;ius ON: &longs;ed & QH erat dupla ip&longs;ius <lb/>QE; vt igitur LN ad NO, ita erit HQ ad QE: & <pb/>per conuer&longs;ionem rationis, vt NL ad LO, ita QH, ad <lb/>HE: & permutando, vt LN ad QH, ita LO ad EH: <lb/>&longs;ed LN, o&longs;ten&longs;a e&longs;t æqualis QH; æqualis igitur LO, <lb/>erit ip&longs;i EH; &longs;ed & OP, e&longs;t æqualis ip&longs;i PE, vt o&longs;ten­<lb/>dimus: duæ igitur LO, OP, duabus HE, EP æqua­<lb/>les erunt altera alteri, & angulos æquales continent LOP, <lb/>PEH, parallelis exi&longs;tentibus LN, BH &longs;ectionibus tri­<lb/>anguli DBH, quæ fiunt à duobus planis parallelis; ba­<lb/>&longs;is igitur LP, trianguli LOP, æqualis e&longs;t ba&longs;i PH, <lb/>trianguli PEH, & angulus OPL, angulo EPH in pla­<lb/>no trianguli DBH, in quo DPE, e&longs;t vna recta linea; <lb/>igitur LPH, erit vna recta linea, quæ cum &longs;it axis octa­<lb/>edri LKMGFH, & &longs;ectus &longs;it in puncto P, bifariam, <lb/>erit punctum P, centrum octaedri LKMGEH. &longs;ed & <lb/>centrum pyramidis ABCD. <!-- KEEP S--></s> <s>Manife&longs;tum e&longs;t igitur pro­<lb/>po&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO X.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omne fru&longs;tum pyramidis triangulam ba&longs;im <lb/>habentis, &longs;iue coni, ad pyramidem, vel conum, cu­<lb/>ius ba&longs;is e&longs;t eadem, quæ maior ba&longs;is fru&longs;ti, & ea­<lb/>dem altitudo, eam habet proportionem, quam duo <lb/>latera homologa, vel duæ diametri ba&longs;ium ip&longs;ius <lb/>fru&longs;ti, vnà cum tertia minori proportionali ad <lb/>prædicta duo latera, vel diametros; ad maioris ba­<lb/>&longs;is latus, vel diametrum. </s> <s>Ad pri&longs;ma autem, vel <lb/>cylindrum, cuius eadem e&longs;t ba&longs;is, quæ maior ba&longs;is <lb/>fru&longs;ti, & eadem altitudo; vt tres prædictæ deìn­<lb/>ceps proportionales &longs;imul, ad triplam lateris, vel <lb/>diametri maioris ba&longs;is. </s></p><pb/><p type="main"> <s>Sit fru&longs;tum ABCFGH, pyramidis, vel coni ABCD, <lb/>cuius ba&longs;is triangulum, vel circulus ABC, axis autem <lb/>DE: & vt e&longs;t AC ad FH, ita &longs;it FH ad N, & fru­<lb/>&longs;ti axis EK, nec non idem pyramidis, vel coni AB <lb/>CK, vt &longs;it eadem altitudo. </s> <s>Dico fru&longs;tum ABCF <lb/>GH, ad pyramidem, vel conum, ABCK, e&longs;se vt <lb/>tres lineas AC, FH, NO, &longs;imul ad ip&longs;ius AC, tri­<lb/>plam: ad pri&longs;ma autem, vel cylindrum, cuius ba&longs;is ABC, <lb/>altitudo autem eadem cum fru&longs;to, vttres AC, FH, NO, <lb/>&longs;imul, ad ip&longs;ius AC, triplam. </s> <s>Nam vt e&longs;t AC ad FH, <lb/>& FH ad NO, ita &longs;it NO ad P: & exce&longs;&longs;us, quo hæ <lb/><figure id="id.043.01.032.1.jpg" xlink:href="043/01/032/1.jpg"/><lb/>quatuor lineæ differunt, &longs;int AL, FM, <expan abbr="Oq.">Oque</expan> Ergo <lb/>vt AC ad FH, ita erit AL ad FM, & FM ad <expan abbr="Oq.">Oque</expan> <lb/>Quoniam igitur e&longs;t vt AC ad P, ita pyramis, vel conus <lb/>ABCD, ad &longs;imilem ip&longs;i pyramidem, vel conum DFGH, <lb/>ob triplicatam laterum homologorum proportionem; erit <lb/>diuidendo, vt tres AL, FM, OQ, &longs;imul ad P, ita fru­<lb/>&longs;tum ABCFGH, ad pyramidem, vel conum DFGH: <lb/>&longs;ed conuertendo e&longs;t vt P, ad AC, ita pyramis, vel conus <lb/>DFGH, ad pyramidem, vel conum ABCD: ex æquali <lb/>igitur, vt tres AL, FM, OQ, &longs;imul ad AC, ita fru&longs;tum <pb/>ABCDFGH, ad pyramidem, vel conum ABCD. <lb/><!-- KEEP S--></s> <s>Rur&longs;us quoniam axis DE, & latera pyramidis, vel coni <lb/>ABCD, &longs;ecantur plano trianguli, vel circuli FGH, ba&longs;i <lb/>ABC, parallelo; erit componendo, vt AD, ad DF, hoc <lb/>e&longs;t, vt AC ad FH, propter &longs;imilitudinem triangulorum, <lb/>hoc e&longs;t vt AC, ad CL, ita ED, ad DK; & per conuer­<lb/>&longs;ionem rationis, vt AC, ad AL, ita DE, ad EK: &longs;ed vt <lb/>DE ad EK, ita e&longs;t pyramis, vel conus ABCD, ad py­<lb/>ramidem, vel conum ABCK; vt igitur AC, ad AL, <lb/>ita e&longs;t pyramis, vel conus ABCD, ad pyramidem, vel <lb/>conum ABCK; &longs;ed vt tres lineæ AL, FM, OQ &longs;imul <lb/>ad AC, ita erat fru&longs;tum ABCFGH, ad pyramidem, <lb/>vel conum ABCD; ex æquali igitur, erit vt tres lineæ <lb/>AL, FM, OQ, &longs;imul ad AL, ita fru&longs;tum ABCFGH, <lb/>ad pyramidem, vel conum ABCK. Rur&longs;us, quoniam <lb/>tres exce&longs;&longs;us AL, FM, OQ, &longs;unt deinceps proportio­<lb/>nales in proportione totidem terminorum AC, FH, NO, <lb/>erunt vt AL, FM, OQ, &longs;imul ad AL, ita AC, FH, <lb/>NO, &longs;imul ad AC: &longs;ed vt AL, FM, OQ, &longs;im ul ad <lb/>AL, ita erat fru&longs;tum ABCFGH, ad pyamidem, vel <lb/>conum ABCK; vt igitur tres lineæ AC, FH, NO, &longs;i­<lb/>mul, ad AC, ita erit fru&longs;tum ABCFGH, ad pyrami­<lb/>dem, vel conum ABCK. </s> <s>Sed vt AC, ad &longs;ui triplam, ita <lb/>e&longs;t pyramis, vel conus ABCK ad pri&longs;ma, vel cylindrum, <lb/>cuius e&longs;t eadem ba&longs;is ABC, & eadem altitudo cum py­<lb/>ramide, vel cono ABCK; ex æquali igitur, erit vt tres <lb/>lineæ AC, FH, NO, &longs;imul ad ip&longs;ius AC, triplam, ita <lb/>fru&longs;tum ABCFGH, ad pri&longs;ma, vel cylindrum, cu­<lb/>ius ba&longs;is ABC, & eadem altitudo pyramidi, vel cono <lb/>ABCK: ide&longs;t eadem, fru&longs;to ABCFGH. </s> <s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omni &longs;olido circa axim in alteram partem defi <lb/>cienti, cuius ba&longs;is &longs;it circulus, vel ellyp&longs;is, figura <lb/>quædam ex cylindris, vel cylindri portionibus <lb/>æqualium altitudinum in&longs;cribi poteft, & altera <lb/>circum&longs;cribi, ita vt circum&longs;cripta &longs;uperet in&longs;cri­<lb/>ptam minori exce&longs;&longs;u quacumque magnitudine <lb/>propo&longs;ita. </s></p><p type="main"> <s>Sit &longs;olidum ABC, circa axim AD, in alteram par­<lb/>tem deficiens, cuius vertex A, ba&longs;is autem circulus, vel <lb/>ellyp&longs;is, cuius diameter BC. <!-- KEEP S--></s> <s>Igitur &longs;uper hanc ba&longs;im <lb/>circa axim AD, <lb/>intelligatur de&longs;eri <lb/>ptus cylindrus, vel <lb/>cylindri portio <lb/>BL, quæ &longs;olidum <lb/>ABC, compre­<lb/>hendet: &longs;ectoque <lb/>cylindro, vel cylin <lb/>dri portione BL, <lb/>planis ba&longs;i paralle <lb/><figure id="id.043.01.034.1.jpg" xlink:href="043/01/034/1.jpg"/><lb/>lis in tot cylindros, vel cylindri portiones æqualium al­<lb/>ritudinum, vt quilibet eorum &longs;it minor magnitudine <lb/>propo&longs;ita; e&longs;to &longs;olidum ABC, &longs;ectum prædictis planis: <lb/>erunt autem &longs;ectiones circuli, vel ellyp&longs;es fimiles inter <lb/>&longs;e & ba&longs;i BC, &longs;olidi ABC &longs;uper quas &longs;ectiones tam­<lb/>quam ba&longs;es cylindris, vel cylindri portionibus æqua­<lb/>lium altitudinum intra, atque extra figuram con&longs;titutis, <lb/>quorum bini inter eadem plana parallela inter &longs;e refe-<pb/>runtur, veluti BF, & GDH, quorum axis communis e&longs;t <lb/>D<emph type="italics"/>K<emph.end type="italics"/>, ba&longs;es autem circuli, vel ellyp&longs;es EF, GH, qua­<lb/>rum commune centrum K: &longs;upremus autem, qui ad A, <lb/>ad nullum refertur. </s> <s>Quoniam igitur ex con&longs;tructione, <lb/>cylindrus, vel cylindri portio BF, e&longs;t minor magnitudi­<lb/>ne propo&longs;ita; exce&longs;sus autem omnes, quibus cylindri, ex <lb/>quibus con&longs;tat figura circum&longs;cripta, excedunt eos, ex qui­<lb/>bus con&longs;tat figura in&longs;cripta, pro vt bini inter &longs;e referun­<lb/>tur, vna cum &longs;upremo, qui ad nullum refertur, &longs;unt æqua­<lb/>les cylindro, vel cylindri portioni BF, figura circum­<lb/>&longs;cripta &longs;olido ABC, excedet in&longs;criptam minori exce&longs;­<lb/>&longs;u magnitudine propo&longs;ita. </s> <s>Fieri igitur pote&longs;t quod pro­<lb/>ponebamus. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dato parallelepipedo erecto circa datam re­<lb/>ctam lineam tamquam axim, erectum parallele­<lb/>pipedum æquale con&longs;tituere. </s></p><p type="main"> <s>Sit datum parallelepipedum AB, erectum, cuius ba­<lb/>&longs;is AC, altitudo autem latus BC: & data recta linea <lb/>finita ED. <!-- KEEP S--></s> <s>Oportet circa rectam ED, tamquam axim <lb/>parallelepipedo AB, æquale parallelepipedum erectum <lb/>con&longs;tituere. </s> <s>Per punctum igitur E, extendatur pla­<lb/>num erectum ad lineam ED, & vt e&longs;t DE, ad BC, ita <lb/>fiat ba&longs;is AC, ad quadratum F: & ad punctum E, in <lb/>plano erecto ad lineam ED, quartæ parti quadrati F, <lb/>æquale GE, quadratum de&longs;cribatur, & compleatur <lb/>quadratum GH, quadruplum quadrati EG, &longs;eu qua­<lb/>drato F, æquale: & ex puncto K, erecta KL, ip&longs;i EF, <lb/>æquali, & ad &longs;ubiectum planum perpendiculari &longs;uper ba­<lb/>&longs;im GH, con&longs;tituatur parallelepipedum GK. <!-- KEEP S--></s> <s>Dico <pb/>parallelepipedum GK, e&longs;se æquale parallelepipedo AB; <lb/>& rectam DE, axim parallelepipedi GK. <!-- KEEP S--></s> <s>Iungantur <lb/>enim ba&longs;ium oppo&longs;itarum diametri GH, LK. <!-- KEEP S--></s> <s>Quo­<lb/>niam igitur qua­<lb/>drata &longs;unt EG, <lb/>GH, communem­<lb/>que habent angu­<lb/>lum, qui ad G, <lb/>con&longs;i&longs;tent circa di­<lb/>ametrum GH; in <lb/>recta igitur GH, <lb/>erit punctum E. <lb/><!-- KEEP S--></s> <s>Et quoniam qua­<lb/>dratum GH, e&longs;t <lb/>quadrati EG, qua­<lb/>druplum; erit dia­<lb/><figure id="id.043.01.036.1.jpg" xlink:href="043/01/036/1.jpg"/><lb/>meter GH, diametri EG, dupla; punctum igitur E, <lb/>erit in medio diametri GH. Rur&longs;us, quoniam ob pa­<lb/>rallelepipedum GK, recta GL, æqualis e&longs;t, & paral­<lb/>lela ip&longs;i KH, erit LH, parallelogrammum: & quia <lb/>vtraque DE, KH, e&longs;t ad &longs;ubiectum planum perpendi­<lb/>cularis, parallelæ erunt, & in eodem plano parallelogram­<lb/>mi LH; in quo cum LG, &longs;it parallela ip&longs;i KH; erit & <lb/>ED, ip&longs;i LG, parallela: e&longs;t autem, & æqualis vtrilibet <lb/>ip&longs;arum GL, GH, oppo&longs;itarum; punctum igitur D, e&longs;t <lb/>in recta LK, & tam KD, ip&longs;i EH, quàm LD, ip&longs;i <lb/>EG, æqualis erit, & inter &longs;e æquales LD, DK. pun­<lb/>ctum igitur D, erit in medio diametri LK; &longs;ed & pun­<lb/>ctum E, erat in medio diametri GH; recta igitur ED, <lb/>axis e&longs;t parallelepipedi GK, cuius parallelepipedi cum <lb/>altitudo DE, &longs;it ad BC, altitudinem parallelepipedi AB, <lb/>vt e&longs;t ba&longs;is AC, ad quadratum F, hoc e&longs;t ad ba&longs;im GH, <lb/>parallelepipedi GK; parallelepipedum GK, parallelepipe <lb/>do AB, æquale erit, Factum igitur e&longs;t quod oportebat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Cuilibet figuræ &longs;olidæ <expan abbr="parallelepipedũ">parallelepipedum</expan> æqua­<lb/>le pote&longs;t e&longs;&longs;e. </s></p><p type="main"> <s>Sit quælibet figura &longs;olida A. <!-- KEEP S--></s> <s>Dico &longs;olido A, parallele­<lb/>pipedum æquale po&longs;se exi&longs;tere. </s> <s>Exponatur enim paral­<lb/>lelepipedum BC, cuius ba&longs;is BG. </s> <s>Quoniam igitur e&longs;t vt <lb/>&longs;olidum BC, ad &longs;olidum A, ita recta linea, &longs;iue latus BD, <lb/>ad aliam rectam lineam; producto latere BD, &longs;it vt BC, <lb/>ad A, ita recta BD, ad rectam DE, & compleatur pa­<lb/>rallelepipedum CE. </s> <s>Quoniam itaque e&longs;t vt BD, ad DE, <lb/>ita parallelogrammum &longs;iue ba&longs;is BG, ad parallelogram­<lb/><figure id="id.043.01.037.1.jpg" xlink:href="043/01/037/1.jpg"/><lb/>mum, &longs;iue ba&longs;im EG; hoc e&longs;t parallelepipedum BC, ad <lb/>parallelepipedum CE: &longs;ed vt BD, ad DE, ita e&longs;t paral­<lb/>lelepipedum BC, ad &longs;olidum A; vt igitur parallelepipe­<lb/>dum BC, ad &longs;olidum A, ita erit parallelepipedum BC, <lb/>ad parallelepipedum CE; parallelepipedum igitur CE <lb/>æquale erit &longs;olido A. <!-- KEEP S--></s> <s>Quod fieri po&longs;se propo&longs;uimus. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis parallelogtammi centrum grauitatis <lb/>diametrum bifariam diuidit. </s></p><p type="main"> <s>Sit parallelogrammum ABCD, cuius duo latera AB, <lb/>BC, &longs;int primum in æqualia: & <expan abbr="quoniã">quoniam</expan> omne parallelogram­<lb/>mum habet &longs;altem duos angulos oppo&longs;itos non minores <lb/>recto, e&longs;to vterque angulorum B, D, non minor recto, &longs;it­<lb/>que ducta diameter AC, &longs;ectaque in puncto G, bifariam. <lb/></s> <s>Dico G, e&longs;se centrum grauitatis parallelogrammi ABCD. <lb/><!-- KEEP S--></s> <s>Trianguli enim ABC, &longs;it centrum grauitatis H; iuncta­<lb/>que HG, & producta, ponatur GK, æqualis GH, & re­<lb/>ctæ à punctis K, H, ad angulos ducantur. </s> <s>Quoniam igi­<lb/>tur AG, e&longs;t æqualis GC, & <lb/>GH, ip&longs;i GK, & angulus <lb/>AGK, æqualis angulo CGH, <lb/>erit ba&longs;is AK, æqualis ba&longs;i <lb/>CH, & angulus GAK, æqua­<lb/>lis angulo GCK: &longs;ed totus <lb/>angulus DAK, æqualis e&longs;t to <lb/>ti angulo BCA; reliquus igi­<lb/>tur DAK, reliquo BCH, <lb/>æqualis erit, circa quos angu­<lb/>los latus BC e&longs;t æquale lateri <lb/>AD, & CH, ip&longs;i AK; angu­<lb/>lus igitur CBH, æqualis erit <lb/><figure id="id.043.01.038.1.jpg" xlink:href="043/01/038/1.jpg"/><lb/>angulo ADK. </s> <s>Similiter o&longs;tenderemus angulum CAH, <lb/>angulo ACK, & angulum BAH, angulo DCK, & an­<lb/>gulum ABH, angulo CDK, æquales e&longs;se: &longs;ed latera <lb/>triangulorum, cum quibus rectæ ductæ à punctis K, H, ad <lb/>angulos triangulorum &longs;imilium ABC, CDA, &longs;unt ho-<pb/>mologa; puncta igitur K, H, in prædictis triangulis &longs;unt <lb/>&longs;imiliter po&longs;ita. </s> <s>Rur&longs;us quoniam angulus ABC, non <lb/>e&longs;t minor recto, acuti erunt reliqui ACB, BAC; igitur <lb/>latus AC, maximum erit: ponitur autem AB maius, <lb/>quàm BC; triangulum igitur ABC, &longs;calenum erit. <lb/></s> <s>Eadem ratione &longs;calenum e&longs;t triangulum ACD. <!-- KEEP S--></s> <s>Quare <lb/>in triangulo ACD, vnum duntaxat punctum K, &longs;imili­<lb/>ter po&longs;itum erit, ac punctum H, in triangulo ABC. <!-- KEEP S--></s> <s>Cum <lb/>igitur H &longs;it centrum grauitatis trianguli ABC, erit & <lb/>K, centrum grauitatis trianguli ACD. <!-- KEEP S--></s> <s>Sed longitudo <lb/>GK, æqualis e&longs;t longitudini GH; punctum igitur G erit <lb/>centrum grauitatis parallelogrammi ABCD, in quo ni­<lb/>mirum &longs;ecta e&longs;t bifariam diameter AC: quare &longs;i ducatur <lb/>altera diameter BD, in medio etiam diametri BD, erit <lb/>idem centrum grauitatis G. <!-- KEEP S--></s></p><p type="main"> <s>Sed &longs;int omnia latera æqualia <expan abbr="parallelogrãmi">parallelogrammi</expan> ABCD, <lb/>Sectisque duobus lateribus AD, BC, bifariam in E, F <lb/>iungantur EF, AE, ED, <lb/>AGC, & per punctum G, <lb/>ducatur ip&longs;i AD, vel BC, <lb/>parallela HGK. </s> <s>Quoniam <lb/>igitur EC, e&longs;t æqualis <lb/>AF, erit CG æqualis AG, <lb/>& EG, æqualis GF, pro­<lb/>pter &longs;imilitudinem triangu <lb/>lorum: nec non EH, ip&longs;i <lb/>AH, & EK, ip&longs;i KD: tres <lb/>igitur diametri AC, AE, <lb/>ED, erunt &longs;ectæ bifariam <lb/><figure id="id.043.01.039.1.jpg" xlink:href="043/01/039/1.jpg"/><lb/>in punctis K, G, H: & quoniam ex æquali propter triangu­<lb/>la &longs;imilia e&longs;t vt AF, ad FD, ita HG, ad GK, erit HG, <lb/>æqualis ip&longs;i GK: &longs;ed puncta K, H, &longs;unt centra grauitatis <lb/>parallelogrammorum BF, FC; igitur totius parallelo­<lb/>grammi ABCD, centrum grauitatis erit G, in medio <pb/>diametri AG. <!-- KEEP S--></s> <s>Quod e&longs;t propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hinc manife&longs;tum e&longs;t, omnis parallelogrammi <lb/>centrum grauitatis e&longs;&longs;e in medio rectæ, quæ op­<lb/>po&longs;itorum bipartitorum laterum &longs;ectiones iungit. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si quodlibet parallelogrammum in duo paral­<lb/>lelogramma diuidatur, & eorum <expan abbr="c&etilde;tra">centra</expan> grauitatis <lb/>iungantur recta linea; totius diui&longs;i parallelogram­<lb/>mi centrum grauitatis prædictam lineam ita di­<lb/>uidit, vt eius &longs;egmenta è contrario re&longs;pondeant <lb/>prædictis partibus parallelogrammis. </s></p><p type="main"> <s>Sit parallelogrammum ABCD, &longs;ectum in duo paral­<lb/>lelogramma AE, ED, & <lb/>parallelogrammi AE, &longs;it <lb/>centrum grauitatis H, pa­<lb/>rallelogrammi autem ED, <lb/>centrum grauitatis K: & <lb/>parallelogrammi ABCD, <lb/>&longs;it centrum grauitatis G: <lb/>& iungatur KH. <!-- KEEP S--></s> <s>Dico re­<lb/>ctam KH, diuidi à puncto <lb/>G, ita vt &longs;it KG, ad G <lb/>H, vt e&longs;t parallelogrammum <lb/>AE, ad parallelogrammum <lb/><figure id="id.043.01.040.1.jpg" xlink:href="043/01/040/1.jpg"/><lb/>ED, Iungantur enim diametri AC, AE, ED. <!-- KEEP S--></s> <s>Igitur <pb/>per præcedentem &longs;ectæ erunt hæ diametri bifariam in pun­<lb/>ctis H, G, K. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt EH, ad HA, ita <lb/>EK ad KD, parallela erit KH, ip&longs;i AD; igitur & EC; <lb/>&longs;ed recta KH, &longs;ecat latus AE, trianguli AEC, bifariam <lb/>in puncto H, ergo & latus AC, bifariam &longs;ecabit; igitur <lb/>in puncto G. punctum igitur G, e&longs;t in linea KH. Rur&longs;us, <lb/>quoniam e&longs;t vt GA, ad AC, ita GH, ad EC, propter &longs;i­<lb/>militudinem triangulorum; &longs;ed dimidia e&longs;t GA, ip&longs;ius <lb/>AC, igitur & GH, erit dimidia ip&longs;ius EC, hoc e&longs;t ip&longs;ius <lb/>FD. </s> <s>Similiter o&longs;tenderemus dimidiam e&longs;se KH ip&longs;ius <lb/>AD. vt igitur KH, ad AD, ita erit GH, ad FD: & per­<lb/>mutando, vt AD, ad DF, ita KH, ad HG, & diui­<lb/>dendo, vt AF, ad FD, hoc e&longs;t vt parallelogrammum AE, <lb/>ad parallelogrammum ED, ita KG, ad GH. <!-- KEEP S--></s> <s>Quod de­<lb/>mon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Plana grauia æquiponderant à longitudini­<lb/>bus ex contraria parte re&longs;pondentibus. </s></p><p type="main"> <s>Sint plana grauia N, R, quorum centra grauitatis &longs;int <lb/>N, R, & longitudo aliqua AB: & vt e&longs;t N, ad R, ita &longs;it <lb/>BC, ad CA. <!-- KEEP S--></s> <s>Dico &longs;u&longs;pen&longs;is magnitudinibus &longs;ecundum <lb/>centra grauitatis N, in puncto A, & R, in puncto B, vtri­<lb/>u&longs;que magnitudinis N, R, &longs;imul centrum grauitatis e&longs;se <lb/>C. <!-- KEEP S--></s> <s>Nam &longs;i N, R, magnitudines &longs;int æquales, manife&longs;tum <lb/>e&longs;t propo&longs;itum. </s> <s>Si autem inæquales, ab&longs;cindatur BD, <lb/>æqualis AC, vt &longs;it AD, ad DB, vt BC, ad CA. <!-- KEEP S--></s> <s>Et quo­<lb/>niam &longs;pacio R, rectangulum æquale pote&longs;t e&longs;se; applice­<lb/>tur ad lineam BD, rectangulum BDKE, æquale quar­<lb/>tæ parti rectanguli æqualis ip&longs;i R, hoc e&longs;t quartæ parti <lb/>ip&longs;ius R; & po&longs;ita DG, æquali, & in directum ip&longs;i DK, <pb/>ducantur rectæ GBH, GAF, quæ cum KE, produ­<lb/>cta conueniant in punctis F, H: & fiant parallelogramma <lb/>FL, AK. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt N, ad R, ita BC, ad <lb/>CA, hoc e&longs;t AD, ad DB, hoc e&longs;t rectangulum AK, ad <lb/>rectangulum BK; erit permutando vt rectangulum AK, <lb/>ad N, ita rectangulum BK, ad R; &longs;ed rectangulum BK, <lb/>e&longs;t pars quarta ip&longs;ius R, ergo & rectangulum AK, erit <lb/>pars quarta ip&longs;ius N. <!-- KEEP S--></s> <s>Rur&longs;us quia e&longs;t vt GD, ad D<emph type="italics"/>K<emph.end type="italics"/>, <lb/>ita GA, ad AF, & GB, ad BH: &longs;ed GD e&longs;t æqualis <lb/>DK; ergo & GA, ip&longs;i AF, & GB, ip&longs;i BH, æquales <lb/>erunt & centra grauita­<lb/>tis A, quidem rectangu­<lb/>li MK, B, vero rectan­<lb/>guli KL, & rectangulum <lb/>AK, pars quarta ip&longs;ius <lb/>M<emph type="italics"/>K<emph.end type="italics"/>, quemadmodum <lb/>& B<emph type="italics"/>K<emph.end type="italics"/> ip&longs;ius KL; &longs;ed <lb/>N, rectanguli AK, qua­<lb/>druplum erat, quemad­<lb/>modum & R ip&longs;ius BK; <lb/>igitur rectangulum MK, <lb/>&longs;pacio N, & rectangulum <lb/>KL, &longs;pacio R, æquale <lb/>erit. </s> <s>Sed vt BC, ad CA, <lb/>ita e&longs;t N, ad R; vt igi­<lb/>tur BC, ad CA, ita <lb/><figure id="id.043.01.042.1.jpg" xlink:href="043/01/042/1.jpg"/><lb/>rectangulum MK, ad rectangulum KL; &longs;ed A e&longs;t cen­<lb/>trum grauitatis rectanguli MK, & B, rectanguli KL; to­<lb/>tius ergo rectanguli FL, hoc e&longs;t duorum rectangulorum <lb/>MK, KL, &longs;imul centrum grauitatis erit C. <!-- KEEP S--></s> <s>Sed rectan­<lb/>gulo MK, æquale e&longs;t &longs;pacium N; & rectangulo KL, &longs;pa­<lb/>cium R. <!-- KEEP S--></s> <s>Igitur &longs;i pro rectangulo MK, &longs;it &longs;u&longs;pen&longs;um N <lb/>&longs;pacium &longs;ecundum centrum grauitatis in puncto A, & pro <lb/>rectangulo KL, &longs;pacium R, &longs;ecundum centrum graui-<pb/>tatis in puncto B, &longs;pacia N, R, æquiponderabunt à lon­<lb/>gitudinibus AC, CB; eritque vtriu&longs;que plani N, R, &longs;i­<lb/>mul centrum grauitatis C. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hinc manife&longs;tum e&longs;t &longs;i cuiuslibet figuræ pla­<lb/>næ vtcumque &longs;ectæ centra grauitatis partium <lb/>iungantur recta linea, talem lineam à centro gra­<lb/>uitatis totius prædicti plani ita &longs;ecari, vt &longs;egmen­<lb/>ta ex contrario re&longs;pondeant prædictis partibus. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si totum quoduis planum, & pars aliqua non <lb/>habeant idem centrum grauitatis, & eorum cen­<lb/>tra iungantur recta linea; in ea producta ad par­<lb/>tes centri grauitatis totius, erit reliquæ partis cen <lb/>trum grauitatis. </s></p><p type="main"> <s>Sit totum quoduis planum <lb/>ABC, cuius centrum graui­<lb/>tatis E, & pars illius AB, cuius <lb/>aliud centrum D, & iuncta <lb/>DE, producatur ad partes E, <lb/>in infinitum v&longs;que in H. <!-- KEEP S--></s> <s>Dico <lb/>reliquæ partis BC, centrum <lb/>grauitatis, quod &longs;it G, e&longs;se in <lb/>linea EH. <!-- KEEP S--></s> <s>Quoniam enim D, <lb/>G, &longs;unt centra grauitatis par­<lb/><figure id="id.043.01.043.1.jpg" xlink:href="043/01/043/1.jpg"/><lb/>tium AB, BC, cadet totius ABC, centrum grauitatis <pb/>E, in recta linea, quæ iungit centra D, G; tria igitur pun­<lb/>cta D, E, G, &longs;unt in eadem recta linea. </s> <s>in qua igitur &longs;unt <lb/>puncta D, E, in eadem e&longs;t punctum G; &longs;ed puncta D, E, &longs;unt <lb/>in recta DH; igitur & punctum G, erit in recta DH: &longs;ed <lb/>extra ip&longs;am DE, vt modo o&longs;tendimus, in reliqua igitur <lb/>EH. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Sit totum quoduis planum &longs;it vni parti concen <lb/>tricum &longs;ecundum centrum grauitatis, & reliquæ <lb/>erit concentricum. </s> <s>Et &longs;i partes inter &longs;e &longs;int con­<lb/>centricæ, & toti erunt concentricæ. </s></p><p type="main"> <s>Sit totum quoduis planum AB, quod cum vna parte <lb/>AC habeat commune centrum grauitatis E. <!-- KEEP S--></s> <s>Dico & re­<lb/>liquæ partis CD, e&longs;se <lb/>idem centrum grauitatis <lb/>E. <!-- KEEP S--></s> <s>Si enim illud non <lb/>e&longs;t, erit aliud; e&longs;to F, & <lb/>EF iungatur. </s> <s>Quoniam <lb/>igitur partium AC, CD, <lb/>centra grauitatis &longs;unt E, <lb/>F; erit totius AB, in re­<lb/>cta EF, centrum graui­<lb/>tatis: &longs;ed & in puncto E, <lb/>vnius ergo magnitudinis <lb/>duo centra grauitatis e­<lb/>runt. </s> <s>Quod e&longs;t ab&longs;urdum; <lb/><figure id="id.043.01.044.1.jpg" xlink:href="043/01/044/1.jpg"/><lb/>idem igitur E erit centrum grauitatis vtriuslibet partium <lb/>AC, CD. <!-- KEEP S--></s> <s>Sed vtriuslibet partium AC, CD, &longs;it cen­<lb/>trum grauitatis E. <!-- KEEP S--></s> <s>Dico idem E totius AB, e&longs;se cen-<pb/>trum grauitatis. </s> <s>Si enim non e&longs;t, erit aliud, e&longs;to G: & <lb/>iunctatur EG, producatur ad partes G, in infinitum v&longs;­<lb/>que ìn F. <!-- KEEP S--></s> <s>Quoniam igitur E, e&longs;t centrum grauitatis vnius <lb/>partis AC, & G, totius AB; erit reliquæ partis CD, in <lb/>linea GF centrum grauitatis: &longs;ed & in puncto E; eiu&longs;­<lb/>dem igitur magnitudinis AB, duo centra grauitatis erunt. <lb/></s> <s>Quod fieri non pote&longs;t; totius igitur AB, erit centrum gra<lb/>uitatis idem E. <!-- KEEP S--></s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis trianguli rectilinei idem e&longs;t centrum <lb/>grauitatis, & figuræ. </s></p><p type="main"> <s>Sit triangulum rectilineum ABC, cuius centrum G. <lb/><!-- KEEP S--></s> <s>Dico G, e&longs;se centrum grauitatis trianguli ABC. <!-- KEEP S--></s> <s>Si enim <lb/>fieri pote&longs;t, &longs;it aliud punctum N, centrum grauitatis trian <lb/>guli ABC, & per punctum G, ducantur rectæ AF, BD, <lb/>CE, & DHE, ERF, FKD, <emph type="italics"/>K<emph.end type="italics"/>LH, & NG. </s> <s>Quo­<lb/>niam igitur quæ ab angulis A, B, C, ductæ &longs;unt rectæ <lb/>lineæ per G, &longs;ecant bifariam latera AB, BC, CA; erit <lb/>triangulum EDF, &longs;imile triangulo ABC, ob latera pa­<lb/>rallela vt &longs;unt EF, AC. <!-- KEEP S--></s> <s>Et quoniam triangulum EDF, <lb/>dimidium e&longs;t cuius vis trium parallelogrammorum AF, <lb/>BD, CE, æqualia inter &longs;e erunt ea parallelogramma <lb/>omnifariam &longs;umpta, quorum centra grauitatis H, K, R; <lb/>intelligantur autem tria parallelogramma AF, BD, CE, <lb/>di&longs;tincta penitus, ita vt inter &longs;e congruant &longs;ecundum tria <lb/>triangula DEF, inter &longs;e congruentia: trium igitur trian <lb/>gulorum DEF, inter &longs;e congruentium & centra grauita­<lb/>tis inter &longs;e congruent in puncto M. </s> <s>Quoniam igitur in­<lb/>ter duas parallelas EF, KH, &longs;ecant &longs;e rectæ lineæ FH, <lb/>LR, in puncto G; erit vt FG, ad GH, ita RG, ad GL; <pb/>dupla igitur RG, e&longs;t ip&longs;ius GL. <!-- KEEP S--></s> <s>Et quoniam in triangu­<lb/>lo AGC, recta GD, &longs;ecat AC, bifariam in puncto D; <lb/>ip&longs;i AC, parallelam KH, bifariam &longs;ecabit in puncto L, <lb/>duorum igitur æqualium parallelogrammorum AF, EG; <lb/>&longs;imul, quorum centra grauitatis &longs;unt K, H, centrum gra­<lb/>uitatis erit L. <!-- KEEP S--></s> <s>Sed duo parallelogramma AF, EC, &longs;i­<lb/>mul &longs;unt paralle­<lb/>logrammi BD, du <lb/>plum; trium igitur <lb/>parallelogrammo­<lb/>rum AF, EC, <lb/>BD, &longs;imul: hoc <lb/>e&longs;t <expan abbr="triãguli">trianguli</expan> ABC, <lb/>vnà cum duobus <lb/>trium <expan abbr="triangulorũ">triangulorum</expan> <lb/>inter &longs;e congruen­<lb/>tium EDF, cen­<lb/>trum grauitatis e­<lb/>rit G. <!-- KEEP S--></s> <s>Sed triangu <lb/>li ABC, ponitur <lb/><figure id="id.043.01.046.1.jpg" xlink:href="043/01/046/1.jpg"/><lb/>centrum grauitatis N; producta igitur NG, occurret <lb/>centro M, reliquæ partis, ide&longs;t duorum triangulorum DEF; <lb/>quare vt triangulum ABC, ad duo triangula DEF, &longs;i­<lb/>mul, ita erit MG, ad GN. <!-- KEEP S--></s> <s>Sed triangulum ABC, e&longs;t <lb/>duplum duorum triangulorum EDF: igitur & MG, erit <lb/>ip&longs;ius GN, dupla. </s> <s>Rur&longs;us quoniam vtriuslibet duorum <lb/>triangulorum EDF, centrum grauitatis erat M; erit &longs;i­<lb/>militer po&longs;itum M, in triangulo EDF, ac centrum N, in <lb/>triangulo ABC, propter &longs;imilitudinem triangulorum: <lb/>Sed propter hæc &longs;imiliter po&longs;ita centra, quia homologo­<lb/>rum laterum e&longs;t vt AB, ad DF, ita NG, ad GM: & <lb/>AB, e&longs;t dupla ip&longs;ius EB, erit & NG, dupla ip&longs;ius GM. <lb/><!-- KEEP S--></s> <s>Sed GM, erat dupla ip&longs;ius GN: igitur GN, erit &longs;ui ip&longs;ius <lb/>quadrupla. </s> <s>Quod e&longs;t ab&longs;urdum. </s> <s>Non igitur centrum <pb/>grauitatis trianguli ABC, erit aliud à puncto G: pun­<lb/>ctum igitur G, erit centrum grauitatis trianguli ABC. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="main"> <s>Quod autem ex huius theorematis demon&longs;tratione li­<lb/>quet centrum grauitatis trianguli e&longs;se in ea recta linea, <lb/>quæ ab angulo ad bipartiti lateris &longs;ectionem pertinet, <lb/>Archimedes per in&longs;criptionem figuræ ex parallelogram­<lb/>mis demon&longs;trauit, aliter autem per diui&longs;ionem trianguli <lb/>in triangula nequaquam: qua enim ratione hoc ille tentat, <lb/>ea ex nono theoremate eiu&longs;dem prioris libri de æquipon­<lb/>derantibus nece&longs;sario pendet. </s> <s>Cum igitur in illo ante ceden <lb/>ti &longs;it fallacia accipientis latenter &longs;peciem trianguli; &longs;cale­<lb/>num &longs;cilicet pro genere triangulo, neque con&longs;equens erit <lb/>demon&longs;tratum. </s> <s>Quod autem dico manife&longs;tum e&longs;t: Datis <lb/>enim duobus triangulis &longs;imilibus, & in altero eorum dato <lb/>puncto, quod &longs;it trianguli centrum grauitatis, punctum in <lb/>altero triangulo modo &longs;imiliter po&longs;itum &longs;it prædicto pun­<lb/>cto, nititur demon&longs;trare e&longs;se alterius trianguli centrum <lb/>grauitatis: cum autem nondum con&longs;tet centrum graui­<lb/>tatis trianguli e&longs;se in recta, quæ ab angulo latus oppo&longs;i­<lb/>tum bifariam &longs;ecat, &longs;ed ex nono theoremate &longs;it demon&longs;tran <lb/>dum medio decimo, non pote&longs;t illud accipi in nono theo­<lb/>remate, quod ad demon&longs;trationem e&longs;set nece&longs;sarium. </s> <s>per­<lb/>mittitur igitur aduer&longs;ario ponere centrum grauitatis trian­<lb/>guli, vbicumque vult intra illius limites. </s> <s>atqui cum datis <lb/>duobus triangulis i&longs;o&longs;celiis &longs;imilibus, & in altero eorum <lb/>dato puncto, quod non &longs;it in prædicta recta linea, po&longs;sint <lb/>in altero duo puncta prædicto &longs;imiliter po&longs;ita inueniri, quo­<lb/>rum vnum duntaxat concedet aduer&longs;arius e&longs;se alterius <lb/>trianguli centrum grauitatis, non autem non &longs;imiliter po­<lb/>&longs;itum, ex quo ab&longs;urdum infertur partem anguli æqualem <lb/>e&longs;se toti: quid quod datis duobus triangulis æquilateris, & <lb/>in altero eorum dato puncto, quod non &longs;it centrum trian-<pb/>guli, &longs;ed aliqua earum, quæ ab angulis ad bipartitorum <lb/>laterum &longs;ectiones cadunt, nece&longs;se e&longs;t in altero triangulo <lb/>tria puncta prædicto puncto e&longs;se &longs;imiliter po&longs;ita? </s> <s>quod &longs;i <lb/>etiam extra i&longs;tas lineas cadat vnius trianguli punctum, ne­<lb/>ce&longs;se e&longs;t illi &longs;ex puncta in altero triangulo e&longs;se &longs;imiliter po­<lb/>&longs;ita: &longs;ed &longs;i quod diximus de i&longs;o&longs;celiis &longs;imilibus, & æquila­<lb/>teris triangulis demon&longs;trauerimus, rem velut ante oculos <lb/>expo&longs;uerimus. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO.<emph.end type="italics"/></s></p><p type="main"> <s>Datis duobustriangulis i&longs;o&longs;celijs &longs;imilibus, & <lb/>in altero eorum dato puncto extra rectam, quæ à <lb/>vertice ad medium ba&longs;is cadit, duo puncta in re­<lb/>liquo triangulo prædicto puncto &longs;imiliter po&longs;ita <lb/>inuenire. </s></p><p type="main"> <s>Sint duo triangula i&longs;o&longs;celia, & &longs;imilia ABC, DEF: <lb/>quorum in altero ABC, à vertice A, ad ba&longs;im BC, bi­<lb/>partitam in puncto G, cadat recta AG: atque extra hanc <lb/><figure id="id.043.01.048.1.jpg" xlink:href="043/01/048/1.jpg"/><lb/>in triangulo ABC, &longs;it quoduis punctum H: & iuncta AH, <lb/>fiat angulus EDK æqualis angulo BAH; & vt BA, ad <pb/>AH, ita fiat ED, ad DK: & quoniam angulus BAG, <lb/>æqualis e&longs;t angulo EDF: quorum angulus EDK, <lb/>æqualis e&longs;t angulo BAH, erit reliquus angulus <emph type="italics"/>K<emph.end type="italics"/>DF, <lb/>æqualis reliquo angulo HAC; &longs;ed angulus HAC, e&longs;t <lb/>maior angulo BAH; ergo & angulus KDF, maior erit <lb/>angulo BAH; po&longs;ito igitur angulo FDL, æquali an­<lb/>gulo BAH, ac proinde minori, quàm &longs;it angulus FD<emph type="italics"/>K<emph.end type="italics"/>, <lb/>fiat vt BA, ad AH, ita FD, ad DL. Dico, in triangu­<lb/>lo EDF, duo puncta K, L, &longs;imiliter po&longs;ita e&longs;se ac pun­<lb/>ctum H, in triangulo BAC. <!-- KEEP S--></s> <s>Iungantur enim rectæ AH, <lb/>BH, CH, EK, KF, FL, LE. </s> <s>Quoniam igitur an­<lb/>gulus ED<emph type="italics"/>K<emph.end type="italics"/>, e&longs;t æqualis angulo BAH, qui lateribus <lb/>homologis continentur; erit angulus DE<emph type="italics"/>K<emph.end type="italics"/>, æqualis an­<lb/>gulo ABH: &longs;ed totus angulus DEF, æqualis e&longs;t toti an­<lb/>gulo ABC; reliquus igitur angulus KEF, æqualis erit <lb/>reliquo HBC: &longs;ed ex æquali e&longs;t vt CB, ad BH, ita <lb/>FE, ad EK; igitur vt antea erit angulus KFE, æqualis <lb/>angulo HCB, & angulus DFK, æqualis angulo ACH, <lb/>& angulus FDK, æqualis angulo CAH; punctum igi­<lb/>tur K, &longs;imiliter po&longs;itum erit in triangulo EDF, ac pun­<lb/>ctum H, in triangulo ABC. <!-- KEEP S--></s> <s>Rur&longs;us quoniam angulus <lb/>FDL, æqualis e&longs;t angulo BAH, & latus AB, homo­<lb/>logum lateri DF, (e&longs;t enim vt BA, ad AC, ita FD, ad <lb/>DE) &longs;ed vt BA, ad AH, ita e&longs;t FD, ad DL, per con­<lb/>&longs;tructionem; &longs;imiliter vt ante, o&longs;tenderemus, punctum L, <lb/>in triangulo EDF, &longs;imiliter po&longs;itum e&longs;se puncto H; in­<lb/>uenta igitur &longs;unt duo puncta in triangulo DEF, &longs;imili­<lb/>ter po&longs;ita ac punctum H, in triangulo BAC. <!-- KEEP S--></s> <s>Quod pro­<lb/>po&longs;itum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis trapezij habentis duo latera parallela <lb/>centrum grauitatis e&longs;t in illa recta, quæ prædi­<lb/>ctorum bipartitorum laterum &longs;ectiones iungit. <lb/></s> <s>atque in eo puncto, in quo tertia pars eius media <lb/>&longs;ic diuiditur, vt &longs;egmentum propinquius mino­<lb/>ri parallelarum ad reliquum eam proportionem <lb/>habeat, quam maior parallelarum ad minorem. <lb/></s> <s>Talis autem rectæ lineæ &longs;ic diui&longs;æ, &longs;egmentum <lb/>minorem parallelarum attingens e&longs;t ad reliquum, <lb/>vt dupla maioris parallelarum vna cum minori, <lb/>ad duplam minoris vna cum maiori. </s></p><p type="main"> <s>Sit trapezium ABCD, cuius duæ AD, BC, &longs;int pa­<lb/>rallelæ: &longs;itque AD, maior. </s> <s>Secti&longs;que AD, BC, bifa­<lb/>riam in punctis F, E, <lb/>iunctaque EF, & &longs;e­<lb/>cta in tres partes æ­<lb/>quales in punctis K, <lb/>H, fiat vt AD, ad <lb/>BC, ita HG, ad GK. <lb/><!-- KEEP S--></s> <s>Dico G, e&longs;se centrum <lb/>grauitatis trapezij A <lb/>BCD: & vt e&longs;t du­<lb/>pla ip&longs;ius AD, vna <lb/>cum BC, ad duplam <lb/>ip&longs;ius BC, vna cum <lb/>AD, ita e&longs;se EG, ad <lb/><figure id="id.043.01.050.1.jpg" xlink:href="043/01/050/1.jpg"/><lb/>GF. <!-- KEEP S--></s> <s>Ducta enim per punctum H, ip&longs;is AD, BC, pa-<pb/>rallela NO, ab&longs;cindantur EL, FM, ip&longs;i GK æquales, & <lb/>iungantur ANE, EOD. </s> <s>Quoniam igitur NO ip&longs;i AD, <lb/>parallela &longs;ecat omnes ip&longs;is AD, EC, interceptas in ea&longs;­<lb/>dem rationes, & e&longs;t EH, pars tertia ip&longs;ius EF, erit & EN <lb/>ip&longs;ius EA, & EO, ip&longs;ius ED, pars tertia. </s> <s>E&longs;t autem NO, <lb/>parallela ba&longs;ibus BE, EC, duorum triangulorum ABE, <lb/>ECD; in ip&longs;a igitur NO, erunt centra grauitatis duo­<lb/>rum triangulorum ABE, ECD: ergo & compo&longs;iti ex <lb/>vtroque in linea NO, erit centrum grauitatis. </s> <s>Quoniam <lb/>igitur K, centrum grauitatis trianguli AED, e&longs;t in EF, & <lb/>totius trapezij ABCD, centrum grauitatis in eadem linea <lb/>EF; erit & reliquæ partis, duorum &longs;cilicet triangulorum <lb/>ABE, ECD, &longs;imul in linea EF, centrum grauitatis: &longs;ed & <lb/>in linea NO; in puncto igitur H. <!-- KEEP S--></s> <s>Rur&longs;us quoniam triangula <lb/>AED, ABE, ECD, &longs;unt inter ea&longs;dem parallelas, erit <lb/>vt AD, ad BC, ita triangulum AED, ad duo triangu­<lb/>la ABE, ECD, &longs;imul: &longs;ed vt AD, ad BC, ita e&longs;t HG, <lb/>ad GK; vt igitur triangulum AED, ad duo triangula <lb/>ABE, ECD, &longs;imul, ita erit HG, ad GK. &longs;ed K, e&longs;t <lb/>centrum grauitatis trianguli AED: & H, duorum trian <lb/>gulorum ABE, ECD, &longs;imul; totius igitur trapezij AB <lb/>CD, centrum grauitatis erit G. <!-- KEEP S--></s> <s>Rurius quoniam EL, <lb/>e&longs;t æqualis GK, æqualium EH, HK; erit reliqua LH, <lb/>æqualis reliquæ GH; tota igitur EG; erit bis GH, vna <lb/>cum GK: eadem ratione quoniam FM, e&longs;t æqualis GK, <lb/>& MK, æqualis GH, erit FG, bis GK, vna cum GH: <lb/>vt igitur HG, bis vna cum GK, ad GK, bis vna cum <lb/>GH, ita erit EG, ad GF. <!-- KEEP S--></s> <s>Sed vt HG, bis vna cum <lb/>GK, ad GK bis vna cum GH, ita e&longs;t AD, bis vna cum <lb/>BC, ad BC, bis vna cum AB, propterea quod e&longs;t vt <lb/>AD, ad BC, ita HG, ad GK; vt igitur e&longs;t AD, bis vna <lb/>cum BC, ad BC, bis vna cum AD, ita erit EG, ad GF. <lb/><!-- KEEP S--></s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis polygoni æquilateri, & æquianguli <lb/>idem e&longs;t centrum grauitatis, & figuræ. </s></p><p type="main"> <s>Sit polygonum æquilaterum, & æquiangulum ABC <lb/>DEFG, cuius &longs;it primo laterum numerus impar, centrum <lb/>autem &longs;it L. <!-- KEEP S--></s> <s>Dico punctum L, e&longs;se centrum grauitatis <lb/>polygoni ABCDEFG; &longs;ectis enim duobus lateribus <lb/>DE, FG, bifariam in punctis K, H, ducantur ab angulis <lb/>oppo&longs;itis rectæ AH, CK. & rectæ BMG, CNF, CM, <lb/>MF, iungantur. </s> <s>Quoniam igitur ex decima tertia quar <lb/>ti Elem. quemadmodum in pentagono, ita in omni præ­<lb/>dicto polygono imparium multitudine laterum plane col­<lb/>ligitur centrum po­<lb/>lygoni e&longs;se in qua­<lb/>libet recta, quæ ab <lb/>angulo ad medium <lb/>lateris oppo&longs;iti du­<lb/>citur, quoniam ab <lb/>omnibus angulis &longs;ic <lb/>ductæ &longs;ecant &longs;e &longs;e <lb/>in eadem proportio­<lb/>ne æqualitatis, ita <lb/>vt eadem &longs;it propor<lb/>tio &longs;egmentorum, <lb/>quæ ad angulos, ad <lb/>ea, quæ ad latera <lb/><figure id="id.043.01.052.1.jpg" xlink:href="043/01/052/1.jpg"/><lb/>illis angulis oppo&longs;ita; rectæ AH, CK, &longs;ecabunt &longs;e &longs;e in <lb/>puncto L. <!-- KEEP S--></s> <s>Rurfus quoniam ex eadem Euclidis angulus <lb/>BAL, æqualis e&longs;t angulo GAL, &longs;ed AB, e&longs;t æqualis <lb/>AG, & AM, communis, erit ba&longs;is BM, æqualis ba&longs;i <pb/>MG, & angulus ABM, angulo AGM, &longs;ed totus ABC, <lb/>toti AGF, e&longs;t æqualis; reliquus igitur angulus CBG, <lb/>reliquo BGF, æqualis erit: &longs;ed circa hos æquales an­<lb/>gulos recta BM, o&longs;ten&longs;a e&longs;t æqualis rectæ MG, & CB, <lb/>e&longs;t æqualis GF; ba&longs;is igitur CM, ba&longs;i GF, & angulus <lb/>CMB, angulo FMG, æqualis erit; &longs;ed totus BMN, <lb/>æqualis e&longs;t toti GMN; quia vterque rectus; reliquus <lb/>igitur CMN, reliquo NMF, æqualis erit, quos circa <lb/>recta CM, e&longs;t æqualis MF, & MN, communis; ba&longs;is <lb/>igitur CN, ba&longs;i NF, & anguli, qui ad N, æquales erunt, <lb/>atque ideo recti: &longs;ed & qui ad M, &longs;unt recti, & BM, e&longs;t <lb/>æqualis GM; parallelæ igitur &longs;unt BG, CF, & trape­<lb/>zij CBGF, centrum grauitatis e&longs;t in linea MN: &longs;ed & <lb/>trianguli ABG, centrum grauitatis e&longs;t in linea AM; to­<lb/>tius igitur figuræ ABCFG, centrum grauitatis e&longs;t in li­<lb/>nea AN; hoc e&longs;t in linea AH. <!-- KEEP S--></s> <s>Rur&longs;us quoniam omnis <lb/>quadrilateri quatuor anguli &longs;unt æquales quatuor rectis: <lb/>& tres anguli ABM, BMN, MNC, &longs;unt æquales tri­<lb/>bus angulis FGM, GMN, MNF, reliquus angulus <lb/>BCF, reliquo CFG, æqualis erit: &longs;ed totus angulus <lb/>BCD, e&longs;t æqualis toti angulo GFE; reliquus ergo <lb/>DCF, reliquo CFE, æqualis erit: &longs;ed linea CN, e&longs;t <lb/>æqualis NF, & anguli, qui ad N, &longs;unt recti; &longs;imiliter <lb/>ergo vt antea, centrum grauitatis trapezij CDEF, erit <lb/>in linea AH: &longs;ed & totius figuræ ABCFG, e&longs;t in li­<lb/>nea AH; totius igitur polygoni ABCDEFG, in li­<lb/>nea AH, e&longs;t centrum grauitatis, quod idem &longs;imiliter in <lb/>linea CK, e&longs;se oftenderemus; in communi igitur &longs;ectione <lb/>puncto L, e&longs;t centrum grauitatis polygoni ABCDEFG. <lb/></s> <s>Similiter quotcumque plurium laterum numero impa­<lb/>rium e&longs;set polygonum æquilaterum, & æquiangulum, <lb/>&longs;emper deueniendo ab vno triangulo ad quotcumque eius <lb/>trapezia; propo&longs;itum concluderemus. </s></p><pb/><p type="main"> <s>Sed e&longs;to polygonum æquilaterum, & æquiangulum, <lb/>ABCDEF, cuius laterum numerus &longs;it par, & centrum <lb/>e&longs;to G. <!-- KEEP S--></s> <s>Dico idem G, e&longs;se centrum grauitatis polygoni <lb/>ABCDEF. <!-- KEEP S--></s> <s>Iungantur enim angulorum oppo&longs;itorum <lb/>puncta rectis lineis AD, BE, CF. <!-- KEEP S--></s> <s>Ex quarto igitur <lb/>Elem. &longs;ecabunt &longs;e&longs;e hæ rectæ omnes bifariam in vno pun­<lb/>cto, quod talis figuræ centrum definiuimus: &longs;ed G poni­<lb/>tur centrum; in puncto igitur G. <!-- KEEP S--></s> <s>Quoniam igitur duo­<lb/>rum triangulorum CBG, GFE, anguli ad verticem <lb/>BGC, FGE, &longs;unt æquales; & vterlibet angulorum CBG, <lb/>GCB, æqualis e&longs;t vtrilibet ip&longs;orum EFG, GEF; ex <lb/>quarto Elem. & circa æquales angulos latera proportio­<lb/>nalia horum triangu <lb/>lorum &longs;unt æqualia; <lb/>&longs;imilia, & æqualia <lb/>erunt triangula BC <lb/>G, GFE: po&longs;itis <lb/>igitur centris graui­<lb/>tatis K, H, duorum <lb/>triangulorum EFG, <lb/>GBC, iunctifque <lb/>KG, GH, erit v­<lb/>terlibet angulorum <lb/>BGH, HGC, æ­<lb/>qualis vtrilibet an­<lb/><figure id="id.043.01.054.1.jpg" xlink:href="043/01/054/1.jpg"/><lb/>gulorum CGK, KGE, propter &longs;imilitudinem po&longs;itio­<lb/>nis centrorum K, H, in i&longs;o&longs;celijs triangulis CBG, <lb/>GFE: (nam GH, &longs;i produceretur latus BC, bifariam <lb/>&longs;ecaret: &longs;imiliter GK, latus EF) &longs;ed CG, e&longs;t in directum <lb/>po&longs;ita ip&longs;i GF; igitur & GH ip&longs;i GK: & &longs;unt æquales, <lb/>vtpote lateribus triangulorum BCG, GFE, æqualibus <lb/>homologæ; cum igitur eorundem triangulorum centra <lb/>grauitatis &longs;int K, H; centrum grauitatis duorum triangu­<lb/>lorum CBG, GFE, &longs;imul, erit punctum G. <!-- KEEP S--></s> <s>Eadem <pb/>ratione, tam duorum triangulorum ABG, DGE, quàm <lb/>duorum AFG, CDG, &longs;imul, centrum grauitatis erit G; <lb/>totius igitur polygoni ABCDEF; centrum grauitatis <lb/>erit idem G. <!-- KEEP S--></s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis figuræ circa diametrum in alteram par <lb/>tem deficientis, in diametro e&longs;t centrum graui­<lb/>tatis. </s></p><p type="main"> <s>Sit figura ABC, circa diametrum BD, in alteram par <lb/>tem deficiens ver&longs;us B. </s> <s>Dico centrum grauitatis figuræ <lb/>ABC, e&longs;se in linea BD. &longs;it enim punctum E, generali­<lb/>ter extra lineam BD. <!-- KEEP S--></s> <s>Et per puncta E, C, ducantur ip&longs;i <lb/>BD, parallelæ EF, <lb/>CG, & vt e&longs;t CD, <lb/>ad DF, ita ponatur <lb/>figura ABC, ad ali­<lb/>quod &longs;pacium M: & <lb/>figuræ ABC, in&longs;cri­<lb/>batur figura ex paral­<lb/>lelogrammis æqua­<lb/>lium altitudinum de­<lb/>ficiens à figura ABC, <lb/>minori defectu, quam <lb/>&longs;it &longs;pacium M, quan­<lb/>tumcumque illud &longs;it: <lb/>minor igitur propor­<lb/><figure id="id.043.01.055.1.jpg" xlink:href="043/01/055/1.jpg"/><lb/>tio erit figuræ ABC, ad &longs;pacium M, hoc e&longs;t minor pro­<lb/>portio CD, ad DF, quàm figuræ ABC, ad &longs;ui reliquum, <lb/>dempta figura in&longs;cripta. </s> <s>Quoniam autem diameter BD, <pb/>bifariam &longs;ecat omnia latera parallelogrammorum in&longs;cri­<lb/>ptorum ba&longs;i AC, parallela; erit in diametro BD, eorum <lb/>omnium parallelogrammorum centra grauitatis, atque <lb/>ideo totius figuræ in&longs;criptæ centrum grauitatis, quod &longs;it <lb/>H: & HEK, ducatur. </s> <s>Quoniam igitur EF, parallela <lb/>e&longs;t vtrique DH, CK; erit vt CD, ad DF, ita KH, ad <lb/>HE, &longs;ed minor e&longs;t proportio CD, ad DF, quàm figu­<lb/>ræ ABC, ad re&longs;i­<lb/>duum, dempta figu­<lb/>ra in&longs;cripta; ergo & <lb/>KH, ad HE, minor <lb/>erit proportio, quàm <lb/>figuræ ABC, ad præ­<lb/>dictum re&longs;iduum: ha­<lb/>beat LKH, eandem <lb/><expan abbr="proportion&etilde;">proportionem</expan> ad EH, <lb/>quàm figura ABC, <lb/>ad prædictum re&longs;i­<lb/>duum. </s> <s>Quoniam <lb/>igitur punctum K, <lb/>cadit extra figuram <lb/><figure id="id.043.01.056.1.jpg" xlink:href="043/01/056/1.jpg"/><lb/>ABC; multo magis punctum L; non igitur punctum L, <lb/>erit prædicti re&longs;idui centrum grauitatis. </s> <s>Sed punctum <lb/>H, e&longs;t in&longs;criptæ figuræ centrum grauitatis: & vt figura <lb/>in&longs;cripta ad prædictum re&longs;iduum, diuidendo, ita e&longs;t LE, <lb/>ad EH; non igitur E, e&longs;t centrum grauitatis figuræ ABC: <lb/>&longs;ed ponitur E, generaliter punctum extra lineam BD; <lb/>Nullum igitur punctum extra lineam BD, e&longs;t centrum <lb/>grauitatis figuræ ABC; in linea igitur BD, erit figu­<lb/>ræ ABC, centrum grauitatis. </s> <s>Quod demon&longs;trandum <lb/>erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Ex huius theorematis demon&longs;tratione con&longs;tat, <lb/>omnis figuræ planæ, &longs;iue &longs;olidæ, cuius termini <lb/>omnis cauitas &longs;it interior, atque ideo intra ter­<lb/>minum centrum grauitatis; & cuius pars aliqua <lb/>e&longs;se po&longs;sit, quæ à tota figura deficiens minori <lb/>defectu quacumque magnitudine propo&longs;ita habe­<lb/>at centrum grauitatis in aliqua certa linea recta <lb/>intra terminum figuræ con&longs;tituta, e&longs;&longs;e in ea recta <lb/>linea totius figuræ centrum grauitatis. </s> <s>Ac proin­<lb/>de, cum per vndecimam huius, omni &longs;olido circa <lb/>axim in alteram partem deficienti, & ba&longs;im ha­<lb/>benti circulum, vel ellyp&longs;im figura in&longs;cribi po&longs;&longs;it <lb/>ex cylindris, vel cylindri portionibus, à prædicto <lb/>&longs;olido deficiens minori &longs;pacio quacumque ma­<lb/>gnitudine propo&longs;ita: talis autem figuræ in&longs;criptæ, <lb/>quemadmodum & circum&longs;criptæ centrum gra­<lb/>uitatis &longs;it in axe, vt ex &longs;equentibus patebit, & <lb/>nunc cogitanti facilè patere pote&longs;t; manife&longs;tum <lb/>e&longs;t omnis &longs;olidi circa axim in alteram partem de­<lb/>ficientis centrum grauitatis e&longs;&longs;e in axe. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Circuli, & Ellyp&longs;is idem e&longs;t centrum grauita­<lb/>tis, & figuræ. </s></p><p type="main"> <s>Sit circulus, vel ellyp&longs;is ABCD, cuius centrum E. <lb/><!-- KEEP S--></s> <s>Dico centrum grauitatis figuræ ABCD, e&longs;se punctum E. <lb/><!-- KEEP S--></s> <s>Ducantur enim duæ diametri ad rectos inter &longs;e angulos <lb/>AC, BD; in ellyp&longs;i autem &longs;int diametri coniugatæ. <lb/></s> <s>Quoniam igitur omnes rectæ lineæ, quæ in &longs;emicirculo, <lb/>vel dimidia ellyp&longs;i diametro ducantur parallelæ bifariam <lb/>&longs;ecantur à &longs;emidiametro, & quo à ba&longs;i remotiores, eo &longs;unt <lb/><figure id="id.043.01.058.1.jpg" xlink:href="043/01/058/1.jpg"/><lb/>minores; erit centrum grauitatis &longs;emicirculi, &longs;iue dimidiæ <lb/>ellyp&longs;is ABC, in linea BE; &longs;icut & &longs;emicirculi, &longs;iue di­<lb/>midiæ ellyp&longs;is ADC, centrum grauitatis in linea DE. <lb/>e&longs;t autem BED, vna recta linea: in diametro igitur BD, <lb/>erit centrum grauitatis circuli, &longs;iue ellyp&longs;is ABCD. <lb/><!-- KEEP S--></s> <s>Eadem ratione o&longs;tenderemus idem centrum grauitatis e&longs;se <lb/>in altera diametro AC: in communi igitur vtriu&longs;que &longs;e­<lb/>ctione puncto E. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si duarum pyramidum triangul as ba&longs;es haben­<lb/>tium æqualium, & &longs;imilium inter &longs;e, tria latera <lb/>tribus lateribus homologis fuerint in directum <lb/>con&longs;tituta, in vertice communi erit vtriu&longs;que &longs;i­<lb/>mul centrum grauitatis. </s></p><p type="main"> <s>Sint duæ pyramides &longs;imiles, & æquales, quarum ver­<lb/>tex communis G, ba&longs;es autem triangula ABC, DEF. <lb/><!-- KEEP S--></s> <s>Et &longs;int latera homologa pyramidum in directum inter &longs;e <lb/>con&longs;tituta: vt AG, GF: & BG, GD, & CG, GE. <lb/><!-- KEEP S--></s> <s>Dico compo&longs;iti ex duabus pyramidibus ABCG, GDEF, <lb/>ita con&longs;titut is centrum gra<lb/>uitatis e&longs;se in puncto G. <lb/><!-- KEEP S--></s> <s>E&longs;to enim H, centrum gra <lb/>uitatis pyramidis ABCG, <lb/>& ducta HGK, ponatur <lb/>G<emph type="italics"/>K<emph.end type="italics"/>, æqualis GH, & iun­<lb/>gantur EK, KD, BH, <lb/>CH. </s> <s>Quoniam igitur e&longs;t <lb/>vt HG, ad GK, ita CG, <lb/>ad GE, & proportio e&longs;t <lb/>æqualitatis: & angulus <lb/>HGC, æqualis angulo EG <lb/><emph type="italics"/>K<emph.end type="italics"/>, erit triangulum CGH, <lb/><figure id="id.043.01.059.1.jpg" xlink:href="043/01/059/1.jpg"/><lb/>&longs;imile, & æquale triangulo EGK. </s> <s>Similiter triangulum <lb/>BGH, trian gulo DGK; & triangulum BGC, triangu­<lb/>lo DGE: quare & triangulum BCH, triangulo DEK. <lb/>pyramis igitur BCGH, &longs;imilis, & æqualis e&longs;t pyramidi <lb/>EDGK. </s> <s>Congruentibus igitur inter &longs;e duobus triangu­<pb/>lis æqualibus, & &longs;imilibus BGC, DGE, & pyramis <lb/>BCGH, pyramidi GDEK congruet, & puncto K, pun­<lb/>ctum H: & eadem ratione <lb/>pyramis ABCG, pyra­<lb/>midi DEFG. congruente <lb/>igitur pyramide ABCG, <lb/>pyramidi DEFG, & pun­<lb/>ctum K, congruet puncto <lb/>H. &longs;ed H, e&longs;t centrum gra<lb/>uitatis pyramidis ABCG: <lb/>igitur K, erit centrum gra <lb/>uitatis pyramidis DEFG: <lb/>&longs;ed e&longs;t GK, æqualis ip­<lb/>&longs;i GH; vtriufque igitur <lb/>pyramidis ABCG, DE­<lb/>FG, &longs;imul centrum grauitatis erit K; Quod demon&longs;tran­<lb/>dum erat. </s></p><figure id="id.043.01.060.1.jpg" xlink:href="043/01/060/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis parallelepipedi centrum grauitatis e&longs;t in <lb/>medio axis. </s></p><p type="main"> <s>Sit parallelepipedum ABCDEFGH, cuius axis <lb/>LM, isque &longs;ectus bifariam in puncto K. <!-- KEEP S--></s> <s>Dico K e&longs;se <lb/>centrum grauitatis parallelepipedi ABCDEFGH. <lb/>iungantur enim diametri AG, BH, CE, DF, quæ <lb/>omnes nece&longs;sario tran&longs;ibunt per punctum K, & in eo <lb/>puncto bifariam diuidentur. </s> <s>Iunctis igitur BD, FH: <lb/>quoniam triangulum EFK, &longs;imile e&longs;t, & æquale trian­<lb/>gulo CDK, propter latera circa æquales angulos ad <pb/>verticem æqualia alterum alteri: eademque ratione, & <lb/>triangulum E<emph type="italics"/>K<emph.end type="italics"/>H, triangulo BCK: & triangulum FKH, <lb/>triangulo BDK; erit pyramis KEFH, &longs;imilis, & æqua­<lb/>lis pyramidi KBCD: habent autem tria latera tribus <lb/>lateribus homologis, ide&longs;t æ­<lb/>qualibus, in directum, prout <lb/>inter &longs;e re&longs;pondent, con&longs;tituta; <lb/>duarum igitur pyramidum KE <lb/>FH, KBCD, &longs;imul centrum <lb/>grauitatis erit K: non aliter <lb/>duarum pyramidum <emph type="italics"/>K<emph.end type="italics"/>GFH, <lb/>KBDA, &longs;imul centrum gra­<lb/>uitatis erit K; totius igitur com <lb/>po&longs;iti ex quatuor pyramidibus; <lb/>ide&longs;t duabus oppo&longs;itis ABC­<lb/>DK, EFGHK, centrum gra<lb/>uitatis erit idem K. <!-- KEEP S--></s> <s>Eadem <lb/>ratione tam duarum pyrami­<lb/><figure id="id.043.01.061.1.jpg" xlink:href="043/01/061/1.jpg"/><lb/>dum AEHDK, BCGFK, &longs;imul, quàm duarum AB­<lb/>FEK, CDHGK, &longs;imul centrum grauitatis erit K. <!-- KEEP S--></s> <s>To­<lb/>tius igitur parallelepipedi ABCDEFG<emph type="italics"/>K<emph.end type="italics"/>, centrum <lb/>grauitatis erit K. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si parallelepipedum in duo parallelepipeda <lb/>&longs;ecetur, &longs;egmenta axis à centris grauitatis totius <lb/>parallelepipedi, & partium terminata ex contra­<lb/>rio parallelepipedi partibus re&longs;pondent. </s></p><pb/><p type="main"> <s>Si parallelepipedum AB, cuius axis CD, &longs;ectum in <lb/>duo parallelepipeda AE, EN, quare & axis CD, in <lb/>axes CL, LD, parallelepipedorum AE, EN. </s> <s>Et &longs;int <lb/>centra grauitatis; F, parallelepipedi EN, & G, paral­<lb/>lelepipedi AE, & H, parallelepipedi AB, in medio cu­<lb/>iu&longs;que axis ex antecedenti. </s> <s>Dico e&longs;se FH, ad HG, <lb/>vt parallelepipedum AE, ad EN, parallelepipedum. <lb/></s> <s>Iungantur enim diametri ba&longs;ium oppo&longs;itarum, quæ per <lb/>puncta axium D, L, G, tran&longs;ibunt, ADM, KLE, <lb/>NCB; iamque parallelogramma <lb/>erunt AB, AE, EN, DB, DE, <lb/>EC, propter eas, quæ parallelas <lb/>iungunt, & æquales: quorum bi­<lb/>na latera oppo&longs;ita &longs;ecta erunt bi­<lb/>fariam in punctis C, L, D, per <lb/>definitionem axis: punctum igitur <lb/>F, in medio rectæ CL, oppo&longs;i­<lb/>torum laterum bipartitorum &longs;ectio­<lb/>nes coniungentis, erit parallelo­<lb/>grammi EN, centrum grauitatis. <lb/></s> <s>Eadem ratione & parallelogram­<lb/><figure id="id.043.01.062.1.jpg" xlink:href="043/01/062/1.jpg"/><lb/>mi AE, centrum grauitatis erit G, & H, parallelogram <lb/>mi AB. <!-- KEEP S--></s> <s>Vt igitur parallelogrammum AE, ad paralle­<lb/>logrammum EN, hoc e&longs;t, vt ba&longs;is ME, ad ba&longs;im EB; <lb/>hoc e&longs;t, vt parallelogrammum MO, ad parallelogram­<lb/>mum OB: hoc e&longs;t, vt parallelepipedum AE, ad paral­<lb/>lelepipedum EN: ita erit FH, ad HG. <!-- KEEP S--></s> <s>Quod de­<lb/>mon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSIT'IO XXVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Solida grauia æquiponderant à longitudini­<lb/>bus ex contraria parte re&longs;pondentibus. </s></p><p type="main"> <s>Sint &longs;olida grauia A, & B, quorum centra grauitatis <lb/>&longs;int A, B, &longs;ecundum quæ &longs;u&longs;pen&longs;a intelligantur A, in <lb/>puncto C, & B, in puncto D, cuiuslibet rectæ GH, quæ <lb/>&longs;it ita diui&longs;a in puncto E, vt &longs;it DE, ad EC, vt e&longs;t A, <lb/>ad B. </s> <s>Dico &longs;olida A, E, æquiponderare à longitudini­<lb/>bus DE, EC; hoc e&longs;t vtriu&longs;que &longs;imul centrum grauita­<lb/>tis e&longs;se E. <!-- KEEP S--></s> <s>Nam &longs;i A, B, &longs;int æqualia, manife&longs;tum e&longs;t <lb/>propo&longs;itum: &longs;i au­<lb/>tem inæqualia, e&longs;to <lb/>maius A: maior igi <lb/>tur erit DE, quam <lb/>EC. ab&longs;cindatur <lb/>DF, æqualis EC: <lb/>erit igitur DE, æ­<lb/>qualis GF: & CD, <lb/>vtrin que producta, <lb/>ponatur DH, æ­<lb/>qualis DF: & CG, <lb/>ip&longs;i CF. & circa <lb/>axim, & <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>GH, e&longs;to paralle­<lb/>lepipedum KL, æ­<lb/>quale duobus &longs;o­<lb/><figure id="id.043.01.063.1.jpg" xlink:href="043/01/063/1.jpg"/><lb/>lidis A, B, &longs;imul & parallelepipedum KL, &longs;ecetur plano <lb/>per punctum F, oppo&longs;itis planis parallelo, in duo paral­<lb/>lelepipeda KN, ML. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt GF, ad <lb/>FH, ita parallelepipedum KN, ad parallelepipedum <pb/>ML, &longs;ed vt GF, ad FH, ita e&longs;t CF, ad FD, hoc e&longs;t DE, ad <lb/>EC, hoc e&longs;t &longs;olidum A, ad &longs;olidum B; erit vt parallelepipe­<lb/>dum KN, ad parallelepipedum ML, ita &longs;olidum A, ad &longs;oli­<lb/>dum B. componendo igitur, & permutando, vt parallelepi­<lb/>pedum KL, ad duo &longs;olida A, B, &longs;imul, ita parallelepi­<lb/>pedum ML, ad &longs;olidum B: & reliquum ad reliquum: &longs;ed <lb/>parallelepipedum KL, æquale e&longs;t duobus &longs;olidis A, B, &longs;i­<lb/>mul: parallelepipedum igitur KN, &longs;olido A, & paralle­<lb/>lepipedum ML, &longs;olido B, æquale erit. </s> <s>Rur&longs;us, quo­<lb/>niam e&longs;t vt GF, ad <lb/>ad FH, ita CF, ad <lb/>FD; hoc e&longs;t DE, <lb/>ad EC: &longs;ed vt GF, <lb/>ad FH, ita e&longs;t <expan abbr="pa-rallelepipedũ">pa­<lb/>rallelepipedum</expan> KN, <lb/>ad <expan abbr="parallelepipedũ">parallelepipedum</expan> <lb/>ML; erit vt DE, <lb/>ad EC, ita paralle <lb/>lepipedum KN, ad <lb/>parallelepipedum <lb/>ML; &longs;ed C e&longs;t pa­<lb/>rallelepipedi KN, <lb/>& D, parallelepipe <lb/>di ML, centrum <lb/>grauitatis; totius igi <lb/><figure id="id.043.01.064.1.jpg" xlink:href="043/01/064/1.jpg"/><lb/>tur parallelepipedi KL, centrum grauitatis erit E. <!-- KEEP S--></s> <s>Igi­<lb/>tur &longs;olido A, po&longs;ito ad punctum G, &longs;ecundum centrum <lb/>grauitatis A, & &longs;olidum B, ad punctum D, &longs;ecundum <lb/>centrum grauitatis B, quorum A, e&longs;t æquale parallele­<lb/>pipedo KN, & B, parallelepipedo ML; ab ij&longs;dem lon­<lb/>gitudinibus DE, EC, æquiponderabunt; eritque com­<lb/>po&longs;iti ex vtroque &longs;olido A, B, centrum grauitatis E. <!-- KEEP S--></s> <s>Quod <lb/>demon&longs;trandum erat. </s></p><p type="main"> <s>Quod &longs;i quis à me quærat, cur non hic vtar quinta illa <pb/>generali primi Archimedis de planis æquiponderantibus, <lb/>&longs;ed illud idem propo&longs;itum vna demon&longs;tratione in planis, <lb/>altera præ&longs;enti in &longs;olidis demon&longs;trauerim. </s> <s>Re&longs;pondeo: <lb/>quia Propo&longs;itio quarta primi Archimedis, ex qua quinta <lb/>nece&longs;&longs;ario pendet, habet, &longs;i quis attendat, aliquas difficul­<lb/>tates phy&longs;icas, quæ mathematicis rationibus non facile <lb/>di&longs;&longs;oluantur: quæ cau&longs;a igitur illum adduxit ad &longs;imile quid <lb/><expan abbr="demon&longs;trandũ">demon&longs;trandum</expan> demon&longs;tratione ad illas duas parabolas ap. <lb/></s> <s>plicata in &longs;ecundo &longs;uo libro planorum æquiponderantium, <lb/>qua&longs;i qui quartæ, ac quintæ illi generali non &longs;atis acquie­<lb/>&longs;ceret; eadem me compulit ad hoc propo&longs;itum duabus de­<lb/>mon&longs;trationibus generalibus, altera de planis, altera de &longs;o­<lb/>lidis grauibus &longs;ecurius demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Quarumlibet trium magnitudinum eiu&longs;dem <lb/>generis centra grauitatis cum centro magnitudi­<lb/>nis ex ijs compo&longs;itæ &longs;unt in eodem plano. </s></p><p type="main"> <s>Sint quælibet tres ma­<lb/>gnitudines eiu&longs;dem gene <lb/>ris A, B, C: quarum cen­<lb/>tra grauitatis A, B, C. <!-- KEEP S--></s> <s>Ex <lb/>ijs autem compo&longs;itæ &longs;it <lb/>centrum grauitatis E. <!-- KEEP S--></s> <s>Di <lb/>co quatuor puncta A, B, <lb/>C, E, e&longs;&longs;e in eodem pla­<lb/>no. </s> <s>Iungantur enim re­<lb/>ctæ AB, BC, CA: & vt <lb/>e&longs;t A, ad C, ita &longs;it CD, <lb/>ad DA, & BD, iungatur: <lb/><expan abbr="punctũ">punctum</expan> igitur D, erit cen­<lb/><figure id="id.043.01.065.1.jpg" xlink:href="043/01/065/1.jpg"/><pb/>trum grauitatis duarum magnitudinum A, C, &longs;imul. <lb/></s> <s>Rur&longs;us quoniam recta BD, coniungit duo centra gra­<lb/>uitatis duarum magnitu­<lb/>dinum B &longs;cilicet, & AC, <lb/>erit compo&longs;itæ ACB, in <lb/>recta BD, centrum graui <lb/>tatis: e&longs;t autem illud E. <lb/><!-- KEEP S--></s> <s>Quoniam igitur in quo <lb/>plano e&longs;t recta BD, in <lb/>eodem &longs;unt duo puncta <lb/>B, E, in quo autem pla­<lb/>no e&longs;t recta BD, in eo­<lb/>dem e&longs;t recta AC, & <lb/>puncta A, C; in quo igi­<lb/>tur plano &longs;unt puncta A, <lb/>C, in eodem erunt pun­<lb/>cta B, E; quatuor igitur puncta A, B, C, E, erunt in eodem <lb/>plano; Quod demon&longs;tr andum erat. </s></p><figure id="id.043.01.066.1.jpg" xlink:href="043/01/066/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si à cuiuslibet trianguli centro, & tribus an­<lb/>gulis quatuor rectæ inter &longs;e parallelæ plano trian <lb/>guli in&longs;i&longs;tant: tres autem magnitudines æquales <lb/>habeant centra grauitatis in ijs tribus, quæ ad <lb/>angulos; trium magnitudinum &longs;imul centrum <lb/>grauitatis erit in ea, quæ ad trianguli centrum <lb/>terminatur. </s></p><p type="main"> <s>Sit triangulum ABC, cuius centrum N, à tribus au­<lb/>tem angulis A, B, C, & centro N, in&longs;i&longs;tant plano trian-<pb/>guli ABC, quatuor rectæ inter &longs;e parallelæ AD, BE, <lb/>CF, NM, tres autem magnitudines æquales habeant cen <lb/>tra grauitatis G, H, K, in tribus AD, BE, CF. <!-- KEEP S--></s> <s>Di­<lb/>co trium magnitudinum &longs;imul, quarum centra grauitatis <lb/>G, H, K, e&longs;&longs;e in linea NM. <!-- KEEP S--></s> <s>Iungantur enim rectæ GH, <lb/>H<emph type="italics"/>K<emph.end type="italics"/>, GK, BNP; & per punctum P, recta PL, ip&longs;i MN, <lb/>parallela, & iungatur LH. <!-- KEEP S--></s> <s>Quoniam igitur rectæ BP, LH, <lb/>iungunt duas parallelas LP, BH; erunt quatuor rectæ BH, <lb/>LP, BP, LH, in eodem plano. </s> <s>Et <expan abbr="quoniã">quoniam</expan> planum quadran <lb/>guli PH, &longs;ecat planum trianguli ABC, à communi autem <lb/>&longs;ectione BP, &longs;urgunt <lb/>duæ parallelæ PL, MN; <lb/>quarum PL, e&longs;t in pla­<lb/>no quadranguli PH, <lb/>erit etiam MN, in eo­<lb/>dem plano quadranguli <lb/>PH: & &longs;ecabit LH. &longs;e­<lb/>cet in puncto O: qùare <lb/>vt LO, ad OH, ita erit <lb/>PN, ad NB, propter <lb/>parallelas: &longs;ed PN, e&longs;t <lb/>dimidia ip&longs;ius NB; er­<lb/>go & LO, e&longs;t dimidia ip <lb/>&longs;ius OH. <!-- KEEP S--></s> <s>Eadem ratio­<lb/>ne, quoniam AP, æqua­<lb/><figure id="id.043.01.067.1.jpg" xlink:href="043/01/067/1.jpg"/><lb/>lis e&longs;t PC, erit & GL, æqualis LK. <!-- KEEP S--></s> <s>Duarum igitur <lb/>magnitudinum G, K, &longs;imul centrum grauitatis erit L: &longs;ed <lb/>reliquæ magnitudinis, quæ ad H, e&longs;t centrum grauitatis <lb/>H; & vt compo&longs;itum ex duabus magnitudinibus G, <lb/>K, ad magnitudinem H, ita ex contraria parte e&longs;t HO, <lb/>ad OL; Trium igitur magnitudinum G, H, K, &longs;imul cen­<lb/>trum grauitatis erit O, & in linea MN. <!-- KEEP S--></s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis octaedri idem e&longs;t centrum grauitatis, <lb/>& figuræ. </s></p><p type="main"> <s>E&longs;to octaedrum ABCDEF, cuius centrum G. <!-- KEEP S--></s> <s>Di­<lb/>co G, e&longs;se centrum grauitatis octaedri ABCDEF. <lb/><!-- KEEP S--></s> <s>Ductis enim axibus AC, BD, EF, communis eorum <lb/>&longs;ectio erit centrum G, in quo axes bifariam &longs;ecabuntur: <lb/>omnium autem angulorum, qui ad G, bini qui que ad <lb/>verticem &longs;unt æquales, qui æqualibus altera alteri rectis <lb/>continentur; &longs;imilia igi­<lb/>tur, & æqualia erunt trian <lb/>gula, nimirum EBG, <lb/>GDF, & ECG, ip&longs;i <lb/>GFA, & BCG, ip&longs;i <lb/>GDA: igitur & BCE, <lb/>ip&longs;i ADF; pyramis igi­<lb/>tur EBCG, &longs;imilis, & <lb/>æqualis e&longs;t pyramidi A <lb/>DFG, quarum latera ho <lb/>mologa &longs;unt indirectum <lb/>inter &longs;e con&longs;tituta; dua­<lb/>rum igitur pyramidum <lb/><figure id="id.043.01.068.1.jpg" xlink:href="043/01/068/1.jpg"/><lb/>EBCG, ADFG, &longs;imul centrum grauitatis erit G. <lb/><!-- KEEP S--></s> <s>Eadem ratione &longs;ex reliquarum pyramidum binis quibu&longs;­<lb/>que oppo&longs;itis &longs;imul &longs;umptis centrum grauitatis erit G. <lb/><!-- KEEP S--></s> <s>Totius igitur octaedri ABCDEF, centrum grauitatis <lb/>erit G. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pyramidis triangulam ba&longs;im habentis <lb/>idem e&longs;t centrum grauitatis, & figuræ. </s></p><p type="main"> <s>Sit pyramis ABCD, cuius ba&longs;is triangulum ABC, <lb/>centrum autem E. <!-- KEEP S--></s> <s>Dico E, e&longs;&longs;e centrum grauitatis pyra­<lb/>midis ABCD. <!-- KEEP S--></s> <s>Secta enim ABCD, pyramide in quatuor <lb/>pyramides, &longs;imiles, & æquales inter &longs;e, & toti pyramidi <lb/>ABCD, & vnum octaedrum, &longs;int eæ pyramides DKLM, <lb/>MGCH, LBGF, <lb/>AKFH. </s> <s>Octaedrum <lb/>autem FGHKLM, <lb/>quod dimidium erit <lb/>pyramidis ABCD, & <lb/>&longs;int axes pyramidum <lb/>DSN, DS, KO, LP, <lb/>MQ: & ARG, iunga <lb/>tur. </s> <s>Quoniam igitur <lb/>FH, e&longs;t parallela ip&longs;i <lb/>BC, & &longs;ecta e&longs;t BC, <lb/>bifariam in puncto G, <lb/><expan abbr="trã&longs;ibit">tran&longs;ibit</expan> recta AG, per <lb/>centra <expan abbr="triangulorũ">triangulorum</expan> O, <lb/>& N, ad quæ axes KO, <lb/><figure id="id.043.01.069.1.jpg" xlink:href="043/01/069/1.jpg"/><lb/>DN, terminantur; manife&longs;tum hoc e&longs;t ex &longs;uperioribus: <lb/>eritque dupla AO, ip&longs;ius OR, nec non AN, dupla ip&longs;ius <lb/>NG, componendo igitur erit vt AG, ad GN, ita AR, <lb/>ad RO, & permutando, vt AG, ad AR, ita GN, ad <lb/>RO: &longs;ed AG, e&longs;t dupla ip&longs;ius AR, quoniam & AB, ip­<lb/>&longs;ius AF; igitur & GN, erit dupla ip&longs;ius RO: &longs;ed & GN, <lb/>e&longs;t dupla ip&longs;ius NR, nam N, e&longs;t centrum trianguli GFH; <lb/>æqualis e&longs;t igitur NR, ip&longs;i RO, atque hinc dupla NO, <pb/>ip&longs;ius OR; &longs;ed & AO erat dupla ip&longs;ius OR; æqualis <lb/>igitur AO erit ip&longs;i ON. <!--neuer Satz-->quare vt AK, ad KD, ita erit <lb/>AO, ad ON: igitur in triangulo ADN, erit KO, ip&longs;i <lb/>DN, parallela. </s> <s>Eadem ratione &longs;i iungerentur rectæ BH, <lb/>CF o&longs;tenderemus & duos reliquos axes LP, MQ, e&longs;­<lb/>&longs;e axi DN parallelos: quatuor autem prædicti axes in­<lb/>&longs;i&longs;tunt plano trianguli KLM, ita vt DN tran&longs;eat per <lb/>centrum S: reliqui autem KO, LP, MQ, terminentur <lb/>ad angulorum vertices K, L, M, trianguli KLM; igi­<lb/>tur &longs;i tres æquales magnitudines habeant centra grauita­<lb/>tis in axibus KO, LP, <lb/><expan abbr="Mq;">Mque</expan> compo&longs;iti ex ijs <lb/>tribus magnitudinibus <lb/>in axe DN erit <expan abbr="centrũ">centrum</expan> <lb/>grauitatis. </s> <s>Rur&longs;us <lb/>quoniam E ponitur <expan abbr="c&etilde;">cem</expan> <lb/><expan abbr="trũ">trum</expan> pyramidis ABCD, <lb/>erit idem E centrum <lb/>octaedri FGHKLM, <lb/>idque in axe DN: e&longs;t <lb/>autem idem <expan abbr="centrũ">centrum</expan> gra<lb/>uitatis octaedri, & figu <lb/>ræ: centrum igitur E <lb/>octaedri FCHKLM <lb/>erit in axe DN. <!-- KEEP S--></s> <s>Quod <lb/><figure id="id.043.01.070.1.jpg" xlink:href="043/01/070/1.jpg"/><lb/>&longs;i quatuor reliquæ pyramides dempto prædicto octaedro <lb/>&longs;imiliter diuidantur, ac pyramis ABCD diui&longs;a fuit, erunt <lb/>rur&longs;us in &longs;ingulis quatuor prædictarum pyramidum &longs;in­<lb/>gula octaedra centrum grauitatis habentia vnumquodque <lb/>in axe &longs;uæ pyramidis: quæ pyramides cum &longs;int inter &longs;e <lb/>æquales, earum dimidia octaedr a ip&longs;is in&longs;cripta inter &longs;e <lb/>erunt æqualia: &longs;unt autem eorum centra grauitatis in axi­<lb/>bus ab&longs;ci&longs;sarum pyramidum, DS, KO, LP, MQ <lb/>axis autem DS: e&longs;t in axe DN; per ea igitur, quæ de-<pb/>mon&longs;trauimus trium octaedrorum, quæ &longs;unt in pyrami­<lb/>dibus AFHK, FBGL, GHOM &longs;imul, centrum gra­<lb/>uitatis erit in axe D<emph type="italics"/>K<emph.end type="italics"/>: &longs;ed & octaedri in pyramide DK­<lb/>LM, & octaedri FGHKLM centra grauitatis &longs;unt <lb/>in axe DN; omnium igitur quinque octaedrorum, quæ <lb/>&longs;unt in tota pyramide ABCD &longs;imul centrum grauitatis <lb/>e&longs;t in axe DN. <!-- KEEP S--></s> <s>Quod &longs;i rur&longs;us in &longs;ingulis quatuor præ­<lb/>dictarum pyramidum modo dicta ratione quina octaedra <lb/>de&longs;cripta intelligantur, &longs;imiliter o&longs;ten&longs;um erit quina octa­<lb/>edra in &longs;ingulis quatuor ab&longs;ci&longs;&longs;arum pyramidum, velut <lb/>quatuor magnitudines, centra grauitatis habere in axibus <lb/>quatuor prædictarum pyramidum: &longs;unt autem hæc qua­<lb/>tuor compo&longs;ita ex quinis octaedris inter &longs;e æqualia, pro­<lb/>pter æqualitatem octaedrorum multitudine æqualium, <lb/>quæ æqualibus &longs;unt pyramidibus ip&longs;orum duplis ord ine <lb/>diui&longs;ionis inter &longs;e re&longs;pondentibus in&longs;cripta; igitur vt ante, <lb/>quater quinorum octaedrorum &longs;imul in axe DN erit <lb/>centrum grauitatis: &longs;ed & octaedri FGHKLM centrum <lb/>grauitatis e&longs;t in axe DN; vnius igitur & viginti octae­<lb/>drorum in pyramide ABCD exi&longs;tentium ex hac &longs;ecun­<lb/>da diui&longs;ione, tanquàm vnius magnitudinis in axe DN erit <lb/>centrum grauitatis. </s> <s>Ab hoc igitur numero vnius & vi­<lb/>ginti octaedrorum in pyramide ABCD exi&longs;tentium, &longs;i­<lb/>mili diui&longs;ione illius reliquarum quatuor pyramidum primo <lb/>ab&longs;ci&longs;&longs;arum procedentes, & eundem &longs;emper gyrum, quem <lb/>fecimus à quinario repetentes, poterunt e&longs;se in tota AB­<lb/>CD pyramide tot, quemadmodum diximus, de&longs;cripta, <lb/>octaedra, vt eorum numerus &longs;uperet quemcumque propo­<lb/>&longs;itum numerum, & omnium tanquàm vnius magnitudinis <lb/>in axe DN, &longs;it centrum grauitatis. </s> <s>Sic autem facienti, & <lb/>reliquarum pyramidum demptis præcedentibus octaedris, <lb/>dimidia octaedra &longs;emper auferenti, tandem relinquen­<lb/>tur pyramides minores &longs;imul &longs;umptæ quantacumque <lb/>magnitudine propo&longs;ita. </s> <s>Totius igitur pyramidis ABCD <pb/>in axe DN, erit centrum grauitatis. </s> <s>Eadem ratione in <lb/>quolibet reliquorum trium axium, pyramidis ABCD, ip­<lb/>&longs;ius centrum grauitatis e&longs;se o&longs;tenderemus; communis igi­<lb/>tur &longs;ectio quatuor axium pyramidis ABCD, quod e&longs;t <lb/>ip&longs;ius centrum E, erit centrum grauitatis pyramidis AB <lb/>CD. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hinc manife&longs;tum e&longs;t centrum grauitatis pyra­<lb/>midis triangulam ba&longs;im habentis e&longs;&longs;e in eopun­<lb/>cto, in quo axis &longs;ic diuiditur, vt pars quæ ad ver­<lb/>icem &longs;it reliquæ tripla. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Ominis pyramidis ba&longs;im plu&longs;quam trilate­<lb/>ram habentis centrum grauitatis axim ita diui­<lb/>dit, vt pars, quæ e&longs;t ad verticem &longs;it tripla re­<lb/>liquæ. </s></p><p type="main"> <s>Sit pyramis ABCDE, cui vertex E, ba&longs;is autem <lb/>quadrilatera ABCD, & e&longs;to axis EF, &longs;egmentum EM, <lb/>reliqui MF, triplum. </s> <s>Dico punctum M, e&longs;&longs;e centrum <lb/>grauitatis pyramidis ABCDE. <!-- KEEP S--></s> <s>Ducta enim AC, &longs;it <lb/>trianguli ABC, centrum grauitatis H, &longs;icut & K, trian­<lb/>guli ACD: & iungantur KH, HE, EK: Factaque vt <lb/>EM, ad MF, ita EL ad LH, & EN ad N<emph type="italics"/>K<emph.end type="italics"/>, iun­<lb/>gatur LN. </s> <s>Quoniam igitur EF e&longs;t axis pyramidis <lb/>ABCDE, erit ba&longs;is ABCD centrum grauitatis F. <pb/>Rur&longs;us quia puncta K, H, &longs;unt centra grauitatis triangu­<lb/>lorum ABC, CDA, erunt EH, EK, axes pyramidum <lb/>ABCE, ACDA: quorum EL, e&longs;t tripla ip&longs;ius LH, <lb/>nec non EN, tripla ip&longs;ius EK; pyramidis igitur ABCE, <lb/>centrum grauitatis erit L, &longs;icut & K, pyramidis ACDE. <lb/>Rur&longs;us, quoniam totius quadrilateri ABCD, e&longs;t cen­<lb/>trum grauitatis F, cuius magnitudinis partium triangu­<lb/>lorum ABC, CDA, centra grauitatis &longs;unt K, H; recta <lb/>KH, à puncto F, &longs;ic <lb/>diuiditur, vt &longs;it HF, ad <lb/>FK, vt triangulum <lb/>ACD, ad triangulum <lb/>ABC, hoc e&longs;t, vt py­<lb/>ramis ACDE, ad py <lb/>ramidem ABCE. &longs;ed <lb/>vt HF, ad FK, ita <lb/>e&longs;t LM, ad MN; vt <lb/>igitur e&longs;t pyramis AC <lb/>DE, ad pyramidem <lb/>ABCE, ita erit LM, <lb/>ad MN. <!-- KEEP S--></s> <s>Sed N, e&longs;t <lb/>centrum grauitatis py­<lb/><figure id="id.043.01.073.1.jpg" xlink:href="043/01/073/1.jpg"/><lb/>ramidis ACDE, & L pyramidis ABCE; punctum <lb/>igitur M, erit centrum grauitatis pyramidis ABCDE. <lb/><!-- KEEP S--></s> <s>Quod &longs;i pyramis habeat ba&longs;im quinquelateram; po&longs;ito <lb/>rur&longs;us axe totius pyramidis, & ba&longs;i &longs;ecta in triangulum, <lb/>& quadrilaterum, po&longs;itis vtriu&longs;que proprijs centris graui­<lb/>tatis, eadem demon&longs;tratione propo&longs;itum concludetur. <lb/></s> <s>Quemadmodum &longs;i ba&longs;is &longs;it &longs;ex laterum, &longs;ecta ea in quinque <lb/>laterum, & triangulum, & reliquis vt antea po&longs;itis: & &longs;ic &longs;em <lb/>per deinceps. </s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pri&longs;matis triangulam ba&longs;im habentis <lb/>centrum grauitatis e&longs;t in medio axis. </s></p><p type="main"> <s>Sit pri&longs;ma ABCDEF, cuius ba&longs;es oppo&longs;itæ trian­<lb/>gula ABC, DEF, axis autem GH, &longs;ectus &longs;it bifariam <lb/>in puncto K. <!-- KEEP S--></s> <s>Dico punctum K, e&longs;se pri&longs;inatis ABCD <lb/>EF, centrum grauitatis. </s> <s>Ducantur enim rectæ FGO, <lb/>CHP, PO. </s> <s>Quoniam igitur GH, e&longs;t axis pri&longs;matis <lb/>ABCDEF, erit punctum G, centrum grauitatis trian­<lb/>guli DEF: &longs;icut & H, trian­<lb/>guli ABC; vtraque igitur <lb/>dupla e&longs;t AG, ip&longs;ius GO, <lb/>& CH, ip&longs;ius PH, &longs;ectæ­<lb/>que erunt AB, DE, bifa­<lb/>riam in punctis P, O: pa­<lb/>rallela igitur, & æqualis e&longs;t <lb/>OP, ip&longs;i DA, iamque ip&longs;i <lb/>FC. quæ igitur illas con­<lb/>iungunt CP, FO, æqua­<lb/>les &longs;unt, & parallelæ, & pa­<lb/>rallelogrammum FP. <lb/></s> <s>Nunc &longs;ecta OP, bifariam in <lb/>puncto N, iungantur GN, <lb/>NF, AF, FH, FB, & fa­<lb/>cta FL, tripla ip&longs;ius LH, <lb/><figure id="id.043.01.074.1.jpg" xlink:href="043/01/074/1.jpg"/><lb/>à puncto L, per punctum K, ducatur recta LKMR. <lb/></s> <s>Quoniam igitur e&longs;t vt FG, ad GO, ita CH, ad HP, <lb/>& parallelogrammum e&longs;t FCPO; parallelogramma <lb/>etiam erunt CG, GP, angulus igitur FGH, æqualis <lb/>erit angulo NGO, quos circa æquales angulos latera <pb/>FG, GH, homologa &longs;unt lateribus GO, ON. nam <lb/>dupla e&longs;t FG, ip&longs;ius GO, & GH, ip&longs;ius ON; angulus <lb/>igitur OGN, æqualis erit angulo GFH; parallela igi­<lb/>tur GN, ip&longs;i FH, & propter&longs;imilitudinem triangulorum <lb/>dupla erit FH, ip&longs;ius GN. Rur&longs;us, quoniam recta <lb/>OP, &longs;ecat latera oppo&longs;ita parallelogrammi BD, bifa­<lb/>riam in punctis O, P, &longs;ecta, & ip&longs;a bifariam in puncto N, <lb/>erit punctum N, parallelogrammi BD, centrum graui­<lb/>tatis, atque ideo axis FN, pyramidis ABDEF. qua <lb/>ratione erit quoque axis FH, pyramidis ABCF: &longs;ed <lb/>FL, e&longs;t tripla ip&longs;ius LH; pyramidis igitur ABCF, cen­<lb/>trum grauitatis erit L. <!-- KEEP S--></s> <s>Rur&longs;us quia e&longs;t vt GK, ad KH, <lb/>ita GR, ad LH, propter &longs;imilitudinem triangulorum, <lb/>erit æqualis GR, ip&longs;i LH: &longs;ed e&longs;t FH, quadrupla ip-, <lb/>&longs;ius LH, quadrupla igitur FH, ip&longs;ius GR: &longs;ed FH <lb/>erat dupla ip&longs;ius GN; quadrupla igitur FH, reliquæ <lb/>NR, ac proinde GR, RN, æquales erunt: recta igitur <lb/>FL, tripla erit vtriu&longs;que ip&longs;arum GR, RN, &longs;ed vt FL, <lb/>ad NR, ita e&longs;t FM, ad MN, propter &longs;imilitudinem trian <lb/>gulorum; recta igitur FM, erit ip&longs;ius MN, tripla, &longs;icut <lb/>& LM, ip&longs;ius MR: &longs;ed quia KH, e&longs;t æqualis GK, <lb/>erit & LK, æqualis RK; propter &longs;imilitudinem trian­<lb/>gulorum; cum igitur LK, &longs;it tripla ip&longs;ius MR, erit LK, <lb/>ip&longs;ius KM, dupla; vt igitur e&longs;t pyramis ABEDF, ad <lb/>pyramidem ABCF, ita erit LK, ad KM; e&longs;t autem M, <lb/>centrum grauitatis pyramidis ABED, &longs;icut & L, pyrami­<lb/>dis ABCF; totius igitur pri&longs;matis ABCDEF, centrum <lb/>grauitatis erit K. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pri&longs;matis ba&longs;im plu&longs;quam trilateram <lb/>habentis centrum grauitatis e&longs;t in medio axis. </s></p><p type="main"> <s>Sit pri&longs;ma ABCDEFGH, ba&longs;im habens quadrila­<lb/>teram ABCD: axis autem <emph type="italics"/>K<emph.end type="italics"/>L, bifariam &longs;ectus in pun­<lb/>cto M. </s> <s>Dico punctum M, e&longs;se centrum grauitatis pri&longs;­<lb/>matis ABCDEFGH. </s> <s>Iungantur enim rectæ BD, FH, <lb/>vt parallelogrammum &longs;it BH, &longs;ectumque totum pri&longs;ma <lb/>in duo pri&longs;mata, quorum ba­<lb/>&longs;es &longs;unt triangula, in quæ &longs;ecta <lb/>&longs;unt quadrilatera AC, EG, <lb/>&longs;int autem axes duorum pri&longs;­<lb/>matum triangulas ba&longs;es ha­<lb/>bentium NO, <expan abbr="Pq.">Pque</expan> Erunt <lb/>igitur centra grauitatis O, tri­<lb/>anguli ABD, & L, quadri­<lb/>lateri AC, & Q, trianguli <lb/>BCD, itemque N, trianguli <lb/>EFH, & K, quadrilateri EG, <lb/>& P, trianguli FGH: iun­<lb/>ctæ igitur OQ, NP, per pun <lb/><figure id="id.043.01.076.1.jpg" xlink:href="043/01/076/1.jpg"/><lb/>cta L, K, tran&longs;ibunt: cumque tres prædicti axes &longs;int <lb/>lateribus pri&longs;matis, atque ideo inter &longs;e quoque paralleli; <lb/>parallelogramma erunt OP, NL, LP. ducta igitur per <lb/>punctum M, ip&longs;i OQ, vel NP, parallela RS, erit vt <lb/>NK, ad KP, ita RM, ad MS: & vt KM, ad ML, ita <lb/>NR, ad RO, & PS, ad SQ: &longs;ed KM, e&longs;t æqualis ML; <lb/>igitur & KR, ip&longs;i RO, & PS, ip&longs;i SQ, æqualis erit: &longs;unt <lb/>autem hæ &longs;egmenta axium NO, <expan abbr="Pq;">Pque</expan> punctum igitur <lb/>R, e&longs;t centrum grauitatis pri&longs;matis ABDEFH: & per <pb/>punctum S, pri&longs;matis BCDFGH. </s> <s>Quoniam igitur <lb/>quadrilateri EG, e&longs;t centrum grauitatis K, cuius duorum <lb/>triangulorum centra grauitatis &longs;unt P, N; erit vt triangu­<lb/>lum FGH, ad triangulum EFH, hoc e&longs;t vt pri&longs;ma BC­<lb/>DFGH, ad pri&longs;ma ABDEFH, ita NK, ad KP, hoc <lb/>e&longs;t RM, ad MS; cum igitur &longs;it R, centrum grauitatis <lb/>pri&longs;matis ABDEFH: &longs;icut & S, pri&longs;matis BCDFGH; <lb/>totius pri&longs;matis ABCDEFGH, centrum grauitatis erit <lb/>M. </s> <s>Quod &longs;i pri&longs;ma ba&longs;im habeat quinquelateram; ab­<lb/>&longs;ci&longs;so rur&longs;us pri&longs;mate vno triangulam ba&longs;im habente, <lb/>&longs;umpti&longs;que axibus pri&longs;inatum, quorum alterum habebit <lb/>ba&longs;im quadrilateram, eadem demon&longs;tratione propo&longs;itum <lb/>concluderemus, & &longs;ic deinceps in aliis. </s> <s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti pyramidis triangulam ba&longs;im <lb/>ha bentis centrum grauitatis e&longs;t in axe, primum <lb/>ita diui&longs;o, vt &longs;egmentum attingens minorem <lb/>ba&longs;im &longs;it ad reliquum, vt duplum vnius laterum <lb/>maioris ba&longs;is vna cum latere homologo mino­<lb/>ris, ad duplum prædicti lateris minoris ba&longs;is, <lb/>vna cum latere homologo maioris. </s> <s>Deinde <lb/>à puncto &longs;ectionis ab&longs;ci&longs;sa quarta parte &longs;eg­<lb/>menti, quod maiorem ba&longs;im attingit, & à pun­<lb/>cto, in quo ad minorem ba&longs;im axis termina­<lb/>tur &longs;umpta item quarta parte totius axis; in <lb/>eo puncto, in quo &longs;egmentum axis duabus po­<lb/>&longs;terioribus &longs;ectionibus finitum &longs;ic diuiditur, vt <pb/>&longs;egmentum eius maiori ba&longs;i propinquius &longs;it ad to­<lb/>tum prædictum interiectum &longs;egmentum, vt tertia <lb/>proportionalis minor ad duo latera homologa ba­<lb/>&longs;ium oppo&longs;itarum, ad compo&longs;itam ex his tribus <lb/>deinceps proportionalibus. </s></p><p type="main"> <s>Sit pyramidis fru&longs;tum, cuius ba&longs;es oppo&longs;itæ, & parallelæ, <lb/>maior triangulum ABC, minor autem triangulum DEF, <lb/>axis autem GH. triangulorum autem ABC, DEF, quæ <lb/>inter &longs;e &longs;imilia e&longs;se nece&longs;se e&longs;t, &longs;int duo latera homologa <lb/>BC, EF: & vt e&longs;t BC, ad EF, ita &longs;it EF, ad X: vt autem e&longs;t <lb/>duplum lateris BC, vna cum latere EF, ad duplum lateris <lb/>EF, vna cum la <lb/>tere BC, ita &longs;it <lb/>HN, ad NG, <lb/>& NO, pars quar <lb/>ta ip&longs;ius NG, & <lb/>HS, pars quar­<lb/>ta ip&longs;ius GH; ip <lb/>&longs;ius autem SO, <lb/>&longs;it VO, ad OS, <lb/>vt e&longs;t X, ad com­<lb/>po&longs;itam ex tri­<lb/>bus BC, EF, X. <lb/><!-- KEEP S--></s> <s>Dico punctum V <lb/>(quod cadet ne­<lb/>ce&longs;sario infra <lb/><figure id="id.043.01.078.1.jpg" xlink:href="043/01/078/1.jpg"/><lb/>punctum N, quanquam hoc ad demon&longs;trationem nihil re­<lb/>fert) e&longs;se centrum grauitatis fru&longs;ti ABCDEF. <!-- KEEP S--></s> <s>Ducta <lb/>enim recta AGL; quoniam GH, e&longs;t axis fru&longs;ti ABCD <lb/>EF, & punctum G, centrum grauitatis trianguli ABC, <lb/>erit punctum L, in medio ba&longs;is BC: &longs;ecto igitur etiam la­<lb/>tere EF, bifariam in puncto K, iungantur LK, <emph type="italics"/>K<emph.end type="italics"/>H: & vt <pb/>vt e&longs;t HN, ad NG, ita fiat KM, ad ML, & GM, iun­<lb/>gatur: & vt e&longs;t GO, ad ON, ita fiat GP, ad PM, & iun <lb/>gantur MN, OP, FG, GD, GE. <!-- KEEP S--></s> <s>Quoniam igitur re <lb/>cta KL, &longs;ecat trapezij BCFE, latera parallela bifariam <lb/>in punctis K,L, & e&longs;t vt HN, ad NG, hoc e&longs;t vt duplum <lb/>lateris BC, vna cum latere EF, ad duplum lateris EF, vna <lb/>cum latere BC, ita KM, ad ML; erit punctum M, cen­<lb/>trum grauitatis trapezij BCFE, & pyramidis GBCFE, <lb/>axis GM. <!-- KEEP S--></s> <s>Et quoniam vt GO, ad ON, ita e&longs;t GP, ad <lb/>PM, atque ideo GP, tripla ip&longs;ius PM, erit punctum P, <lb/>centrum grauitatis pyramidis GBCFE, atque ideo in <lb/>linea OP. <!-- KEEP S--></s> <s>Rur&longs;us quoniam angulus ACB; æqualis e&longs;t <lb/>angulo DFK: & vt AC, ad CK, ita e&longs;t DF, ad FK: <lb/>e&longs;t autem DF, parallela ip&longs;i AC, & FK, ip&longs;i CL; erit <lb/>reliqua DK, reliquæ AL, parallela; vnum igitur planum <lb/>e&longs;t, ADKL, in quo iacet triangulum GMN; cum igitur <lb/>&longs;it parallela KH, ip&longs;i GL, vtque HN, ad NG, ita <lb/><emph type="italics"/>K<emph.end type="italics"/>M, ad ML; erit MN, ip&longs;i LG, parallela: &longs;ed OP, e&longs;t <lb/>parallela ip&longs;i MN; &longs;ecant enim latera trianguli GMN, <lb/>in ea&longs;dem rationes; igitur OP, erit LG, parallela. </s> <s>Simi­<lb/>liter ex puncto O, ad axes duarum pyramidum GABED, <lb/>GACFD, duæ aliæ rectæ lineæ ducerentur, quas & cen­<lb/>tra grauitatis pyramidum habere, & parallelas rectis GQ, <lb/>GR, alteram alteri e&longs;se o&longs;tenderemus, &longs;icut o&longs;tendimus <lb/>OP, habentem centrum grauitatis pyramidis GBCFE, <lb/>ip&longs;i GL, parallelam; &longs;ed tres rectæ GL, GQ, GR, &longs;unt <lb/>in eodem plano trianguli nimirum ABC; tres igitur præ­<lb/>dictæ parallelæ, quæ ex puncto O, atque ideo trium præ­<lb/>dictarum pyramidum centra grauitatis erunt in eodem pla­<lb/>no, per punctum O, & trianguli ABC, parallelo. </s> <s>Quo­<lb/>niam igitur fru&longs;ti ABCDE, centrum grauitatis e&longs;t in axe <lb/>GH; (manife&longs;tum hoc autem ex duobus centris grauitatis <lb/>pyramidis, cuius e&longs;t prædictum fru&longs;tum, & ablatæ, quæ <lb/>centra grauitatis &longs;unt in axe, cuius &longs;egmentum e&longs;t axis <pb/>GH) erit eiu&longs;dem fru&longs;ti ABCDEF, centrum grauitatis <lb/>O. <!-- KEEP S--></s> <s>Rur&longs;us quoniam vt tres deinceps proportionales BC, <lb/>EF, X, &longs;imul ad BC, ita e&longs;t fru&longs;tum ABCDEF, ad py­<lb/>ramidem; &longs;i de&longs;cribatur ABCH: &longs;ed vt triangulum ABC, <lb/>ad &longs;imile triangulum EDF, hoc e&longs;t vt BC, ad X, ita e&longs;t <lb/>pyramis ABCH, ad pyramidem GDEF; erit ex æqua­<lb/>li, vt tres lineæ <lb/>BC, EF, X, &longs;i­<lb/>mul ad X, ita fru <lb/>&longs;tum ABCDEF, <lb/>ad pyramidem <lb/>GDEF: & con­<lb/>uertendo, vt X, <lb/>ad compo&longs;itam <lb/>ex BC, EF, X, <lb/>hoc e&longs;t vt VO, <lb/>ad OS, ita pyra <lb/>mis GDEF, ad <lb/>fru&longs;tum ABC­<lb/>DEF; & diui­<lb/>dendo, vt pyra­<lb/><figure id="id.043.01.080.1.jpg" xlink:href="043/01/080/1.jpg"/><lb/>mis GDEF, ad reliquas tres pyramides fru&longs;ti, ita OV, <lb/>ad VS; &longs;ed S, e&longs;t centrum grauitatis pyramidis GDEF, <lb/>& O, trium reliquarum; fru&longs;ti igitur ABCDEF, cen­<lb/>trum grauitatis erit V. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti pyramidis ba&longs;im plu&longs;quam trila­<lb/>teram habentis centrum grauitatis e&longs;t punctum <lb/>illud, in quo axis &longs;ic diuiditur, vt axis fru&longs;ti pyra­<lb/>midis triangulam ba&longs;im habentis diuiditur ab <lb/>ip&longs;ius centro grauitatis. </s></p><pb/><p type="main"> <s>Sit pyramidis quadrilateram ba&longs;im habentis fru&longs;tum <lb/>ABCDEFGH, cuius axis KL, atque in ip&longs;o centrum <lb/>grauitatis O. <!-- KEEP S--></s> <s>Dico axim KL, &longs;ectum e&longs;se in puncto O, <lb/>vt propo&longs;uimus. </s> <s>Ductis enim AC, EG, quæ &longs;imilium <lb/>&longs;ectionum angulos æquales &longs;ubtendant B, F, qui late­<lb/>ribus homologis continentur, fru&longs;ta erunt pyramidum <lb/>triangulas ba&longs;es habentium AFG, AGH: &longs;it autem fru­<lb/>&longs;ti AFG, axis <lb/>TP, & in eo eiu&longs; <lb/>dem fru&longs;ti cen­<lb/>trum grauitatis <lb/>M, & fru&longs;ti AG <lb/>H, axis VQ, & <lb/>in eo centrum <lb/>grauitatis N, & <lb/>iungantur TV, <lb/>MN, <expan abbr="Pq.">Pque</expan> Quo <lb/>niam igitur e&longs;t <lb/>pyramidis fru­<lb/>&longs;tum, quod pro­<lb/>ponitur; omnia <lb/><figure id="id.043.01.081.1.jpg" xlink:href="043/01/081/1.jpg"/><lb/>cius producta latera concurrent in vno puncto, qui e&longs;t pyra­<lb/>midis vertex: fru&longs;ta igitur, in quæ diui&longs;um e&longs;t fru&longs;tum pro­<lb/>po&longs;itum earum &longs;unt pyramidum, quæ verticem habent <lb/>communem cum pyramide, cuius e&longs;t fru&longs;tum propo&longs;itum: <lb/>tres igitur talium fru&longs;torum axes, vt pote &longs;egmenta axium <lb/>trium prædictarum pyramidum in communi illo vertice <lb/>concurrent: quilibet igitur duo trium prædictorum axium <lb/>KL, TP, VQ, erunt in eodem plano: TP, igitur, & <lb/>VQ, &longs;unt in eodem plano. </s> <s>Eadem autem ratione, qua <lb/>vtebamur de pri&longs;mate K, centrum grauitatis K, ba&longs;is <lb/>EH, e&longs;t in linea TV, & L, ba&longs;is BD, centrum grauita­<lb/>tis e&longs;t in linea <expan abbr="Pq;">Pque</expan> reliquæ igitur KL, MN, erunt in eo­<lb/>dem plano trapezij PTVQ, &longs;eque mutuo &longs;ecabunt: cum <pb/>igitur M, N, &longs;int centra grauitatis propo&longs;iti pri&longs;matis par <lb/>tium pri&longs;matum AFG, AGH, atque obid O, totius pri&longs;­<lb/>matis AFGH, in linea MN, centrum grauitatis; per pun <lb/>ctum O, recta MN, tran&longs;ibit. </s> <s>Et quoniam planum tra­<lb/>pezij PV, &longs;ecatur duobus planis parallelis, erunt TV, PQ, <lb/>fectiones parallelæ. </s> <s>His demon&longs;tratis, fiat rur&longs;us vt AB, <lb/>bis vna cum EF, ad EF, bis vna cum AB, ita TY, ad <lb/>YP: & &longs;umatur T<foreign lang="greek">w</foreign>, pars quarta ip&longs;ius TP, & YZ, pars <lb/>quarta ip&longs;ius PY, & ad axim KL, ducantur ip&longs;is TV, <lb/>PQ, parallelæ <lb/><foreign lang="greek">w</foreign>S, YR, ZX, <lb/>quæ rectas TP, <lb/>KL, &longs;ecabunt in <lb/><expan abbr="ea&longs;d&etilde;">ea&longs;dem</expan> rationes: <lb/>vt igitur TY, ad <lb/><foreign lang="greek">*u</foreign>P, hoc e&longs;t vt <lb/>AB, bis vna cum <lb/>EF, ad EF bis <lb/>vna cum AB, ita <lb/>erit <emph type="italics"/>K<emph.end type="italics"/>R, ad RL, <lb/>eritque KS, pars <lb/>quarta ip&longs;ius K <lb/>L, qualis & R <lb/><figure id="id.043.01.082.1.jpg" xlink:href="043/01/082/1.jpg"/><lb/>X, ip&longs;ius RL. </s> <s>Et quoniam M, e&longs;t centrum grauitatis fru­<lb/>&longs;ti AFG; manife&longs;tum e&longs;t ex tribus prædictis axis TP, &longs;e­<lb/>ctionibus <foreign lang="greek">*u, w</foreign>, Z, e&longs;se MZ, ad Z<foreign lang="greek">w</foreign>, hoc e&longs;t OX, ad XS, <lb/>vt e&longs;t 6 ad compo&longs;itam ex tribus deinceps proportionalibus <lb/>AB, EF, 6; Fru&longs;ti igitur ABCDEFGH, centrum gra<lb/>uitatis O, axim KL, ita diuidit, vt propo&longs;uimus. </s> <s>Quod <lb/>&longs;i fru&longs;tum propo&longs;itum &longs;it pyramidis ba&longs;im habentis quin­<lb/>quelateram, & quotcumque plurium deinceps fuerit la­<lb/>terum, eadem demon&longs;tratione &longs;emper deinceps, vt in pri&longs;­<lb/>mate monuimus, propo&longs;itum concluderemus. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dodecaedri, & ico&longs;aedri idem e&longs;t centrum gra<lb/>uitatis, & figuræ. </s></p><p type="main"> <s>Nam huiu&longs;modi figuras habere axes, qui omnes &longs;e &longs;e <lb/>bifariam &longs;ecant; (tale autem &longs;ectionis punctum centrum e&longs;t) <lb/>con&longs;tat ex talium corporum in &longs;phæra in&longs;criptione in de­<lb/>cimotertio Euclidis Elemento: nec non omnem pyrami­<lb/>dem, cuius vertex e&longs;t dodecaedri, vel octaedri centrum <lb/>idem cum centro &longs;phæræ, vt con&longs;tat ex ij&longs;dem Euclidis in­<lb/>&longs;criptionibus; ba&longs;is autem triangulum æquilaterum, vel <lb/>pentagonum, vna ex ba&longs;ibus corporum prædictorum, ha­<lb/>bere pyramidem oppo&longs;itam &longs;imilem ip&longs;i, & æqualem, cuius <lb/>latera eius lateribus homologis &longs;unt in directum po&longs;ita, <lb/>ba&longs;is autem triangulum, vel pentagonum, quale diximus; <lb/>Eadem igitur ratione, qua v&longs;i &longs;umus ad demon&longs;trandum <lb/>centrum grauitatis, & parallelepipedi, & octaedri, propo­<lb/>&longs;itum concluderemus. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Data qualibet figura, cuius termini omnis <lb/>cauitas &longs;it interior, &longs;i certum in ea punctum talis <lb/>cius partis centrum grauitatis e&longs;se po&longs;sit, quæ ab <lb/>ca deficiat minori &longs;pacio quantacumque magnitu <lb/>dine propo&longs;ita; illud erit totius figuræ centrum <lb/>grauitatis. </s></p><pb/><p type="main"> <s>E&longs;to figura AB, cuius termini omnis cauitas &longs;it interior <lb/>& certum in ea punctum E, talis partis AB, figuræ qua­<lb/>lem diximus centrum grauitatis e&longs;se po&longs;sit. </s> <s>Dico pun­<lb/>ctum E, e&longs;se figuræ AB, centrum grauitatis. </s> <s>Si enim <lb/>E, non e&longs;t, erit aliud, e&longs;to F: & iuncta EF producatur, <lb/>& &longs;umatur in illa extra figuræ AB, terminum, quodlibet <lb/>punctum G; & vt e&longs;t FE, ad EG, ita &longs;it alia magnitudo <lb/>K, ad figuram AB, & <lb/>ex vi hypothe&longs;is &longs;it pars <lb/>quædam CD, figuræ <lb/>AB, cuius centrum gra<lb/>uitatis E, talis vt abla­<lb/>ta relinquat AC, minus <lb/>magnitudine <emph type="italics"/>K.<emph.end type="italics"/><!-- KEEP S--></s><s> Mi­<lb/>nor igitur proportio erit <lb/>AC, ad AB, quàm K, <lb/>ad AB, hoc e&longs;t quàm <lb/>FE, ad EG; fiat vt <lb/>AC, ad AB, ita EF, <lb/>ad FGH: &longs;ed F, e&longs;t cen <lb/>trum grauitatis totius <lb/>AB, & E, vnius par­<lb/>tis CD; reliquæ igitur <lb/><figure id="id.043.01.084.1.jpg" xlink:href="043/01/084/1.jpg"/><lb/>partis AC, centrum grauitatis erit H, vltra punctum G: &longs;ed <lb/>G, cadit extra terminum figuræ AC; multo igitur magis H: <lb/>Quod e&longs;t ab&longs;urdum. </s> <s>Non igitur aliud punctum à puncto <lb/>E; punctum igitur E, figuræ AB, erit centrum grauitatis <lb/>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis coni centrum grauitatis axim ita diui­<lb/>dit, vt &longs;egmentum ad verticem &longs;it reliqui triplum. </s></p><p type="main"> <s>Sit conus ABC, cuius vertex B, axis autem BD, cu­<lb/>ius BE, &longs;it tripla ip&longs;ius ED. <!-- KEEP S--></s> <s>Dico punctum E, e&longs;se co­<lb/>ni ABC, centrum grauitatis. </s> <s>Si enim cono ABC, pyramis <lb/>in&longs;cribatur, cuius ba&longs;is in&longs;cripta circulo AC, æquilatera &longs;it, <lb/>& æquiangula, eius centrum grauitatis erit idem quod & <lb/>figuræ centrum, &longs;ed centrum <lb/>talis figuræ circulo in&longs;criptæ <lb/>idem e&longs;t, quod centrum cir­<lb/>culi, vt colligitur ex demon­<lb/>&longs;trationibus quarti Elemen­<lb/>torum; in&longs;criptæ igitur pyra <lb/>midis erit axis BD, & cen­<lb/>trum grauitatis E. talis au­<lb/>tem ea pyramis in&longs;cribi po­<lb/>te&longs;t, vt à cono deficiat mino­<lb/>ri &longs;pacio quantacumque ma <lb/>gnitudine propo&longs;ita; igitur <lb/>ABC, coni centrum graui­<lb/>tatis erit E. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.085.1.jpg" xlink:href="043/01/085/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXX.<emph.end type="italics"/></s></p><p type="main"> <s>Omnis fru&longs;ti conici centrum grauitatis idem <lb/>e&longs;t in axe centro grauitatis fru&longs;ti pyramidis ba&longs;im <lb/>habentis æquilateram, & æquiangul am in &longs;criptæ <lb/>cono, ab &longs;ci&longs;&longs;i eodem plano, quo coni fru&longs;tum. </s></p><pb/><p type="main"> <s>Sit coni fru&longs;tum ABCD, cuius axis EF, fru&longs;to autem <lb/>ABCD, intelligatur in&longs;criptum fru&longs;tum pyramidis in&longs;cri­<lb/>ptæ cono AHD, à quo ab&longs;ci&longs;sum e&longs;t fru&longs;tum ABCD, <lb/>ba&longs;im habentis æquilateram, & æquiangulam in&longs;criptam <lb/>circulo AD: quare eius centrum grauitatis, & figuræ erit <lb/>punctum F, vt diximus in præcedenti, axis autem FH, &longs;i­<lb/>cut etiam pyramidis ab&longs;ci&longs;sæ vna cum cono BHC, axis <lb/>EH, quare & reliqui fru&longs;ti pyramidis axis erit EF, igi­<lb/>tur in EF, &longs;it fru&longs;ti in&longs;cripti fru&longs;to ABCD, centrum gra­<lb/>uitatis G. <!-- KEEP S--></s> <s>Dico punctum G, e&longs;se centrum grauitatis fru­<lb/>&longs;ti ABCD. <!-- KEEP S--></s> <s>Ponatur enim <lb/>FL, pars quarta ip&longs;ius FH, <lb/>necnon EK, pars quarta ip­<lb/>&longs;ius EH: punctum igitur K, <lb/>e&longs;t centrum grauitatis pyra­<lb/>midis, & coni BHC, &longs;icut <lb/>& punctum L, pyramidis, & <lb/>coni AHD. cum igitur fru <lb/>&longs;ti pyramidis fru&longs;to ABCD, <lb/>in&longs;cripti &longs;it centrum grauita­<lb/>tis G; erit vt GL, ad LK, <lb/>ita pyramis BHC, ad pyra­<lb/>midis fru&longs;tum fru&longs;to ABCD, <lb/>in&longs;criptum: &longs;ed vt pyramis <lb/>BHC, ad pyramidis fru&longs;tum <lb/>fru&longs;to ABCD, in&longs;criptum, <lb/><figure id="id.043.01.086.1.jpg" xlink:href="043/01/086/1.jpg"/><lb/>ita e&longs;t diuidendo, conus BHC, ad fru&longs;tum ABCD, pro­<lb/>pter eandem triplicatam communium conis, & pyramidi­<lb/>bus &longs;imilibus laterum homologorum proportionem; vt igi­<lb/>tur GL, ad LK, ita erit conus BHC: ad fru&longs;tum ABCD: <lb/>&longs;ed coni BHC, centrum grauitatis erat K, & coni AHD, <lb/>centrum grauitatis L; fru&longs;ti igitur ABCD, centrum gra­<lb/>nitatis erit G. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XLI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis cylindri centrum grauitatis axim bifa­<lb/>riam diuidit. </s></p><p type="main"> <s>Sit cylindrus ABCD, cuius axis EF, & &longs;it &longs;ectus bi­<lb/>fariam in puncto G. <!-- KEEP S--></s> <s>Dico punctum G, e&longs;se centrum <lb/>grauitatis cylindri ABCD. <!-- KEEP S--></s> <s>Nam &longs;i cylindro AD, in­<lb/>&longs;criptum intelligatur pri&longs;ma, <lb/>cuius ba&longs;es oppo&longs;itæ æquilate­<lb/>ræ &longs;int, & æquiangulæ; erunt, <lb/>qua ratione &longs;upra diximus, ea­<lb/>rum centra figuræ, & grauitatis <lb/>E, F; axis igitur in&longs;cripti pri&longs;­<lb/>matis erit EF: & centrum gra<lb/>uitatis G. pote&longs;t autem tale <lb/>pri&longs;ma &longs;ic in&longs;cribi cylindro <lb/>ABCD, vt ab illo deficiat <lb/>minori &longs;pacio quantacumque <lb/>magnitudine propo&longs;ita; cylin­<lb/>dri igitur ABCD, centrum <lb/>grauitatis erit G. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.087.1.jpg" xlink:href="043/01/087/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XLII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Sphæræ, & &longs;phæroidis idem e&longs;t centrum gra­<lb/>uitatis, & figuræ. </s></p><p type="main"> <s>Sit &longs;phæra, vel &longs;phæroides ABCD, cuius centrum E, <pb/>Dico &longs;phæræ, vel &longs;phæroidis ABCD, centrum grauitatis <lb/>e&longs;se E. <!-- KEEP S--></s> <s>Sint enim bini axes &longs;phæræ, vel &longs;phæroidis inter <lb/>&longs;e ad rectos angulos; & in &longs;phæroide &longs;it maior diameter <lb/>BD, minor AC, per binos autem hos axes plana tran­<lb/>&longs;euntia ad eos axes erecta, &longs;ecent &longs;phæram, vel &longs;phæroidem. <lb/></s> <s>Qua ratione axes dimidij erunt axes hemi&longs;phærij, vel he­<lb/>mi&longs;phæroidis: hemi&longs;phærium autem, & &longs;phæroidis e&longs;t fi­<lb/><figure id="id.043.01.088.1.jpg" xlink:href="043/01/088/1.jpg"/><lb/>gura circa axim in alteram partem deficiens, qualium om­<lb/>nium figurarum centrum grauitatis e&longs;t in axe; igitur hemi­<lb/>&longs;phærij, vel hemi&longs;phæroidis ABCD, centrum grauitatis <lb/>e&longs;t in axi BE, &longs;icut & reliqui ADA, in axi ED; totius <lb/>igitur &longs;phæræ, vel &longs;phæroidis ABCD centrum grauitatis <lb/>e&longs;t in axi BD. <!-- KEEP S--></s> <s>Eadem ratione & in axi AC; in communi <lb/>igitur &longs;ectione centro E. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s>PRIMI LIBRI FINIS.<!-- KEEP S--></s></p><figure id="id.043.01.088.2.jpg" xlink:href="043/01/088/2.jpg"/><p type="head"> <pb/><s>LVCAE <lb/>VALERII <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM<!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>LIBER SECVNDVS.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si duæ magnitudines vnà maio<lb/>res, vel minores prima, & ter <lb/>tia minori exce&longs;&longs;u, vel defe­<lb/>ctu <expan abbr="quantacumq;">quantacumque</expan> magnitudi <lb/>ne propo&longs;ita eiu&longs;dem generis <lb/>cum illa, ad quam refertur, <lb/>eandem <expan abbr="proportion&etilde;">proportionem</expan> habue­<lb/>rint, maior vel minor prima ad &longs;ecundam, & vnà <lb/>maior, vel minor tertia ad quartam; erit vt prima <lb/>ad &longs;ecundam, ita tertia ad quartam. </s></p><pb/><p type="main"> <s>Sint quatuor magnitudines A prima, B &longs;ecunda, C ter <lb/>tia, & D quarta: quantacumque autem magnitudine propo <lb/>&longs;ita, ex infinitìs quæ proponi po&longs;&longs;unt eiu&longs;dem generis cum <lb/>A, C, vel vna tantum, &longs;i AC &longs;int eiu&longs;dem generis: vel <lb/>vna, & altera; &longs;i vna vnius, altera &longs;it alterius generis; &longs;emper <lb/>aliæ duæ magnitudines vnà maiores, quàm AC, minori <lb/>exce&longs;su magnitudine propo&longs;ita; eandem habeant proportio <lb/>nem, maior quàm A ad B, & maior quàm C ad D. <!-- KEEP S--></s> <s>Dico <lb/>e&longs;se vt A ad B, ita C ad D. <!-- KEEP S--></s> <s>Po&longs;ita enim E ad D, vt <lb/>A ad B, & F maiori quàm C vtcumque, &longs;int aliæ duæ ma­<lb/>gnitudines, G maior quàm A minori exce&longs;su magnitudine <lb/>eiu&longs;dem generis cum A, quam quis voluerit, & H maior <lb/>quàm C minori exce&longs;su quàm <lb/>quo F &longs;uperat C, ide&longs;t, quæ ma­<lb/>ior &longs;it quàm C, & minor quàm <lb/>F: &longs;it autem vt G ad B, ita H <lb/>ad D. <!-- KEEP S--></s> <s>Quoniam igitur F maior <lb/>e&longs;t, <34>H, maior erit proportio <lb/>ip&longs;ius F quàm H ad D, hoc e&longs;t <lb/>quàm G ad B. </s> <s>Sed <expan abbr="cũ">cum</expan> G maior <lb/>&longs;it quàm A, maior e&longs;t proportio <lb/><figure id="id.043.01.089.1.jpg" xlink:href="043/01/089/1.jpg"/><lb/>G ad B, quàm A ad B, multo igitur erit maior proportio F <lb/>ad D, quàm A ad B. </s> <s>Sed F ponitur maior quàm C, vtcum <lb/>que; nulla igitur magnitudo maior quàm C e&longs;t ad D, vt <lb/>A ad B: &longs;ed E ad D, e&longs;t vt A ad B; non igitur e&longs;t E ma­<lb/>ior quàm C; nec maior proportio E ad D, hoc e&longs;t A ad <lb/>B, quàm C ad D. <!-- KEEP S--></s> <s>Eadem autem ratione nec maior erit <lb/>proportio C ad D quàm A ad B, hoc e&longs;t non minor A <lb/>ad B, quàm C ad D; eadem igitur proportio A ad B, <lb/>quæ C ad D. <!-- KEEP S--></s></p><p type="main"> <s>Sed aliæ duæ magnitudines vnà minores quàm A, C <lb/>minori defectu quantacumque magnitudine propo&longs;ita, <lb/>eandem habeant proportionem, minor quàm A ad B, & <lb/>minor quàm C, ad D. <!-- KEEP S--></s> <s>Dico e&longs;se vt A ad B, ita C ad D. <pb/>Po&longs;ita enim rur&longs;us E ad D, vt A ad B, & F minori quàm <lb/>C vtcumque, &longs;it G minor quam A, minori defectu magni <lb/>tudine eiu&longs;dem generis cum A, quam quis voluerit, & H <lb/>minor quàm C, & maior quàm F: &longs;it autem vt G ad B, ita <lb/>H ad D. <!-- KEEP S--></s> <s>Quoniam igitur F minor e&longs;t quàm H, minor erit <lb/>proportio ip&longs;ius F <expan abbr="quã">quam</expan> H ad D, <lb/>hoc e&longs;t <34>G ad B: &longs;ed cum G &longs;it <lb/>minor <34>A, minor e&longs;t propor­<lb/>tio G ad B, quàm A ad B; mul <lb/>to ergo minor proportio F ad <lb/>D, quàm A ad B: &longs;ed F poni <lb/>tur minor quàm C vtcumque; <lb/>nulla igitur magnitudo minor <lb/><figure id="id.043.01.090.1.jpg" xlink:href="043/01/090/1.jpg"/><lb/>quàm C e&longs;t ad D, vt A ad B: &longs;ed E e&longs;t ad D, vt A ad B: <lb/>non igitur e&longs;t E minor quàm C, nec minor proportio E ad <lb/>D, hoc e&longs;t A ad B, quàm C ad D. eadem autem ratione <lb/>non minor erit proportio C ad D, quàm A ad B; hoc e&longs;t <lb/>non maior A ad B, quàm C ad D; vt igitur A ad B, ita <lb/>e&longs;t C ad D. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>ALITE R.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dico e&longs;se vt A ad B, ita C ad <lb/>D. <!-- KEEP S--></s> <s>Si enim fieri pote&longs;t, &longs;it minor <lb/>proportio A ad B quàm C ad D. <lb/>alia igitur aliqua magnitudo G <lb/>maior quàm A, eandem habebit <lb/>proportionem ad B, quam C ad <lb/>D. <!-- KEEP S--></s> <s>Sit autem F maior quam C <lb/>minori exce&longs;su magnitudine, <expan abbr="quã">quam</expan> <lb/>quis voluerit, & E maior quàm <lb/>A, & minor quàm G: vt autem <lb/><figure id="id.043.01.090.2.jpg" xlink:href="043/01/090/2.jpg"/><lb/>E ad B, ita F ad D. <!-- KEEP S--></s> <s>Quoniamigitur F maior e&longs;t quàm <lb/>C, maior erit proportio F ad D, quàm C ad D. <!-- KEEP S--></s> <s>Sed vt <lb/>F ad D, ità e&longs;t E ad B: & vt C ad D, ita G ad B; maior <pb/>igitur proportio E ad B, quàm G ad B; quamobrem E <lb/>maior erit quàm G minor maiori, quod fieri non pote&longs;t. <lb/></s> <s>Non igitur minor e&longs;t proportio A ad B quàm C ad D. <lb/><!-- KEEP S--></s> <s>Eadem autem ratione non minor erit proportio C ad D, <lb/>quàm A ad B, hoc e&longs;t non maior A ad B, quàm C ad D; <lb/>eadem igitur proportio A ad B, quæ C ad D. <!-- KEEP S--></s></p><p type="main"> <s>In &longs;ecunda autem hypothe&longs;is parte, quæ pertinet ad mi­<lb/>norem <expan abbr="defectũ">defectum</expan>, e&longs;to &longs;i fieri pote&longs;t maior proportio A ad B, <lb/>quàm C ad D. erit igitur, & &longs;it aliqua alia magnitudo G <lb/>minor quàm A ad B, vt C ad D. <!-- KEEP S--></s> <s>Sit autê F minor quàm <lb/>C minori defectu magnitudine, <lb/>quam quis voluerit, & E minor <lb/>quàm A, & maior quàm G, vt au­<lb/>tem E ad B ita F ad D. <!-- KEEP S--></s> <s>Quoniam <lb/>igitur maior e&longs;t proportio C ad D, <lb/>quàm F ad D: &longs;ed vt C ad D, ita <lb/>e&longs;t G ad B: & vt F ad D, ita E ad <lb/>B: maior erit proportio G ad B <lb/>quàm E ad B; quamobrem erit <lb/>G maior quàm E, minor maiori, <lb/>quod fieri non pote&longs;t; non igitur ma <lb/><figure id="id.043.01.091.1.jpg" xlink:href="043/01/091/1.jpg"/><lb/>ior e&longs;t proportio A ad B, quàm C ad D. <!-- KEEP S--></s> <s>Eadem autem ra<lb/>tione non maior erit proportio C ad D, quàm A ad B, hoc <lb/>e&longs;t non minor A ad B, quàm C ad D. <!-- KEEP S--></s> <s>Eadem igitur erit <lb/>proportio A ad B, quæ C ad D. <!-- KEEP S--></s> <s>Quod <expan abbr="demon&longs;trãdum">demon&longs;trandum</expan> erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si maior, vel minor prima ad vnà maiorem, vel <lb/>minorem &longs;ecunda, minori <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> exce&longs;&longs;u, vel de­<lb/>fectu <expan abbr="quantacumq;">quantacumque</expan> magnitudine propo&longs;ita fue­<lb/>rit vt tertia ad quartam; erit vt prima ad &longs;ecun­<lb/>dam, ita tertia ad quartam. </s></p><pb/><p type="main"> <s>Sint quatuor magnitudines, A prima, B &longs;ecunda, C ter­<lb/>tia, & D quarta: & aliæ duæ magnitudines E <lb/>F vnà maiores quàm A, B minori exce&longs;su <lb/>quantacumque magnitudine propo&longs;ita eiu&longs;­<lb/>dem generis cum ip&longs;is A, B. </s> <s>Sit autem E <lb/>maior quàm A, ad F maiorem quàm B, vt <lb/>C ad D. <!-- KEEP S--></s> <s>Dico e&longs;se A ad B, vt C ad <lb/>D. <!-- KEEP S--></s> <s>E&longs;to enim, quod fieri pote&longs;t, alia ma­<lb/>gnitudo G eiu&longs;dem generis cum EF ad <lb/>aliam H, vt C ad D, vel E ad F. <!-- KEEP S--></s> <s>Quoniam <lb/>igitur e&longs;t permutando vt E ad G, ita F ad H, <lb/>& &longs;unt EF vnà maiores quàm AB minori ex­<lb/>ce&longs;su quantacumque magnitudine propo&longs;i­<lb/>ta; erit per antecedentem, vt A ad G, ita B <lb/>ad H: & permutando A ad B, vt G ad H, <lb/>hoc e&longs;t vt C ad D. <!-- KEEP S--></s> <s>Idem autem &longs;imiliter o&longs;ten <lb/>deremus po&longs;itis EF minoribus quàm AB, & <lb/>proportionalibus vt <expan abbr="dictũ">dictum</expan> e&longs;t. </s> <s><expan abbr="Manife&longs;tũ">Manife&longs;tum</expan> e&longs;t igitur <expan abbr="propo&longs;itũ">propo&longs;itum</expan>. </s></p><figure id="id.043.01.092.1.jpg" xlink:href="043/01/092/1.jpg"/><p type="head"> <s><emph type="italics"/>ALITER.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Ij&longs;dem po&longs;itis, &longs;i non e&longs;t A ad <lb/>B, vt C ad D; vel igitur ma­<lb/>ior vel minor erit proportio A <lb/>ad B quàm C ad D: &longs;it autem <lb/>maior: vt igitur A ad B, ita erit <lb/>eadem A ad <expan abbr="aliã">aliam</expan> maiorem <34>B. <lb/><!-- KEEP S--></s> <s>E&longs;to illa E. &longs;intque aliæ duæ ma <lb/>gnitudines, G maior quàm A <lb/><figure id="id.043.01.092.2.jpg" xlink:href="043/01/092/2.jpg"/><lb/>minori exce&longs;su magnitudine eiu&longs;dem generis cum A, <lb/>quam quis voluerit, & F maior quàm B, & minor quàm <lb/>E. &longs;it autem G ad F vt C ad D. <!-- KEEP S--></s> <s>Quoniam igitur & vt <lb/>C ad D, ita e&longs;t A ad E; erit vt G ad F, ita A ad E; & <lb/>permutando vt G ad A, ita F ad E: &longs;ed G e&longs;t maior <pb/>quàm A: ergo & F maior quàm <lb/>E, minor maiori, quod e&longs;t ab­<lb/>&longs;urdum. </s> <s>Non igitur maior e&longs;t <lb/>proportio A ad B quàm C ad <lb/>D: eadem autem ratione non <lb/>maior erit proportio B ad A <expan abbr="quã">quam</expan> <lb/>D ad C, hoc e&longs;t non minor A <lb/>ad B, quàm C ad D; e&longs;t igitur <lb/>A ad B, vt C ad D. <!-- KEEP S--></s></p><figure id="id.043.01.093.1.jpg" xlink:href="043/01/093/1.jpg"/><p type="main"> <s>Rur&longs;us in &longs;ecunda parte hypothe&longs;is, quæ attinet ad mi­<lb/>norem defectum: &longs;i non e&longs;t A ad B vt C ad D; e&longs;to, &longs;i fie­<lb/>ri pote&longs;t, minor proportio A ad B quàm C ad D. igitur A <lb/>ad aliam quam B minorem eandem habebit <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam C ad D, e&longs;to illa E: &longs;intque <lb/>aliæ duæ magnitudines, G minor <lb/>quàm A minori defectu magnitudi­<lb/>ne eiu&longs;dem generis cum A, quam <lb/>quis voluerit, & F minor quàm B, <lb/>& maior quàm E: &longs;it autem G ad <lb/>F, vt C ad D, hoc e&longs;t vt A ad E. <lb/><!-- KEEP S--></s> <s>Quoniam igitur permutando e&longs;t vt <lb/>G ad A, ita F ad E, & G e&longs;t mi­<lb/><figure id="id.043.01.093.2.jpg" xlink:href="043/01/093/2.jpg"/><lb/>nor quàm A; erit & F minor quàm E, maior mino­<lb/>ri, quod e&longs;t ab&longs;urdum; non igitur minor e&longs;t proportio <lb/>A ad B quàm C ad D: eadem autem ratione non minor <lb/>erit proportio B ad A, quàm D ad C, hoc e&longs;t non maior <lb/>A ad B, quàm C ad D; e&longs;t igitur A ad B vt C ad D. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si maior, vel minor prima ad vnà maiorem, vel <lb/>minorem &longs;ecunda, minori exce&longs;&longs;u, vel defectu <pb/>quantacumque magnitudine propo&longs;ita, nomina­<lb/>tam habuerit proportionem; prima ad &longs;ecundam <lb/>eandem nominatam habebit proportionem. </s></p><p type="main"> <s>Sint duæ magnitudines A, B duarum autem aliarum <lb/>EF vnà maiorum, vel minorum quàm AB minori ex­<lb/>ce&longs;su vel defectu quantacumque magnitudine propo­<lb/>&longs;ita, habeat E maior vel minor quàm A ad F vnà <lb/>maiorem, vel minorem quàm B certam ali quam nomina­<lb/>tam proportionem, verbi gratia, &longs;e&longs;quialteram. </s> <s>Dico A <lb/>ad B, eandem nominatam habere proportionem: vt A <lb/>ip&longs;ius B e&longs;se &longs;e&longs;quialteram. </s> <s>Quoniam <lb/>enim omnis proportio in aliquibus ma­<lb/>gnitudinibus con&longs;i&longs;tit; &longs;it magnitudo C <lb/>ip&longs;ius D &longs;e&longs;quialtera: &longs;ed & E e&longs;t ip&longs;ius <lb/>F &longs;e&longs;quialtera; vtigitur C, tertia ad D <lb/>quartam, ita erit E maior, vel minor quàm <lb/>A prima, ad F vnà maiorem, vel minorem <lb/>&longs;ecunda, minori, vt ponitur, vtriu&longs;que ex­<lb/>ce&longs;su, vel defectu magnitudine propo&longs;ita <lb/>eiu&longs;dem generis cum A, B, quæcumque <lb/>illa, & quantacumque &longs;it; erit per præ­<lb/>cedentem eadem proportio A ad B, <lb/>quæ C ad D: &longs;ed proportio quam ha­<lb/>bet C ad D, e&longs;t &longs;e&longs;quialtera; ergo & A <lb/>ip&longs;ius B erit &longs;e&longs;quialtera. </s> <s>Similiter quo­<lb/>cumque alio nomine notatam proportio­<lb/>nem habeat E ad F, eandem habere A <lb/><figure id="id.043.01.094.1.jpg" xlink:href="043/01/094/1.jpg"/><lb/>ad B, o&longs;tenderemus, vt duplam, &longs;e&longs;quitertiam, alicuius du <lb/>plicatam, vel triplicatam, & &longs;ic de &longs;ingulis. </s> <s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s></p><p type="main"> <s>Hæc autem propo&longs;itio in paucis exemplaribus, quæ do­<lb/>no quibu&longs;dam <expan abbr="dederã">dederam</expan>, non extat; po&longs;terius enim eam exco-<pb/>gitaui, quo &longs;ecunda <expan abbr="anteced&etilde;s">antecedens</expan> hìc in illis tertia facilius &longs;er­<lb/>uiret ijs, in quibus certæ proportionis nomen, <expan abbr="tertiũ">tertium</expan> & quar <lb/>tum terminum &longs;ubob&longs;curè indicat, vt in &longs;equenti XII iilud, <lb/>proportio dupla. </s> <s>Illo autem Lemmate, quod prima propofi­<lb/>tio in&longs;cribebatur, nunc ita non egeo, vt primam, & <expan abbr="&longs;ecundã">&longs;ecundam</expan>, <lb/>quæ &longs;ecunda, & tertia erant, & facilius demon&longs;trem, & ea­<lb/>rum &longs;en&longs;um paucioribus comprehendam. </s> <s>priora ergo ita <lb/>non improbo vt hæc ijs anteponam. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO IIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int tres magnitudines &longs;e &longs;e æqualiter exce­<lb/>dentes, minor erit proportio minimæ ad mediam <lb/>quàm mediæ ad maximam. </s></p><p type="main"> <s>Sint tres magnitudines inæquales A, BC, DE, qua­<lb/>rum BC æquè excedat ip&longs;am A, ac DE ip&longs;am BC <lb/>Dico minorem e&longs;se proportionem A, ad <lb/>BC, quàm BC, ad DE. <!-- KEEP S--></s> <s>Nam vt e&longs;t <lb/>A ad BC, ita &longs;it BC ad LH, & au­<lb/>feratur BF æqualis A, & DG, & LK <lb/>æquales BC. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt A, <lb/>hoc e&longs;t FB ad BC, ita BC hoc e&longs;t KL <lb/>ad LH; erit diuidendo vt BF ad FC, <lb/>ita LK ad KH: & componendo, ac per­<lb/>mutando vt BC ad LH, ita FC ad <lb/>KH. &longs;ed BC e&longs;t minor quàm LH; ergo <lb/>& FC hoc e&longs;t EG erit minor quàm KH. <lb/><!-- KEEP S--></s> <s>Sed DE, LH, &longs;uperant BC exce&longs;sibus <lb/>EG, KH; minor igitur erit DE quàm <lb/>LH, & minor proportio BC ad LH, <lb/>quàm BC ad DE. <!-- KEEP S--></s> <s>Sed vt BC ad LH, <lb/><figure id="id.043.01.095.1.jpg" xlink:href="043/01/095/1.jpg"/><lb/>ita e&longs;t A ad BC; minor igitur proportio erit A ad BC, <lb/>quàm BC ad DE. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO V.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;it minor proportio primæ ad &longs;ecundam, <lb/>quàm &longs;ecundæ ad tertiam, ab ip&longs;is autem æquales <lb/>auferantur; erit minor proportio reliquæ primæ <lb/>ad reliquam &longs;ecundæ, quam reliquæ &longs;ecundæ ad <lb/>reliquam tertiæ. </s></p><p type="main"> <s>Sit minor proportio AB, ad CD, quam CD, ad EF. <lb/><!-- KEEP S--></s> <s>Sitque AB, minima. </s> <s>ablatæ autem æquales fint AG, CH, <lb/>EK. <!-- KEEP S--></s> <s>Dico reliquarum minorem e&longs;se proportionem BG, <lb/>ad DH, quam BH, ad FH. <!-- KEEP S--></s> <s>Ponatur enim CL, æqua­<lb/>lis AB, & EM, æqualis CD. <!-- KEEP S--></s> <s>Quoniam igitur maior e&longs;t <lb/>proportio DL ad LH, quam DL, ad LC; <lb/>erit componendo maior proportio DH ad <lb/>HL, quam DC ad CL. hoc e&longs;t, maior <lb/>proportio DH, ad BG, quam DC, <lb/>ad AB: & conuertendo, minor proportio <lb/>BG ad DH, quam AB, ad CD: hoc e&longs;t <lb/>maior proportio AB, ad CD, quam BG, <lb/>ad DH. Rur&longs;us, quoniam maior e&longs;t pro­<lb/>portio CD, ad EF, quam AB, ad CD: <lb/>hoc e&longs;t quam CL, ad EM; erit permutan <lb/>do, maior proportio CD, ad CL, quam <lb/>FE, ad EM: & diuidendo, maior DL, ad <lb/>LC, quam FM, ad ME: & permutando, <lb/><figure id="id.043.01.096.1.jpg" xlink:href="043/01/096/1.jpg"/><lb/>maior DL, ad FM, quam CL, ad EM: hoc e&longs;t quam <lb/>AB, ad CD. <!-- KEEP S--></s> <s>Sed maior erat proportio AB, ad CD, <lb/>quam BG ad DH; multo igitur maior proportio erit DL, <lb/>ad FM, quam BG, ad DH: hoc e&longs;t quam LH, ad MK: <lb/>& permutando, maior proportio DL, ad LH, quam FM, <lb/>ad MK: & componendo, maior DH, ad HL, quam FK, <pb/>ad KM: & permutando, maior DH ad F<emph type="italics"/>K<emph.end type="italics"/>, quam LH, ad <lb/>M<emph type="italics"/>K<emph.end type="italics"/>: hoc e&longs;t, quam BG, ad DH: hoc e&longs;t minor propor­<lb/>tio BG ad DH, quam DH, ad FK. </s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int tres magnitudines inæquales, & aliæ il­<lb/>lis multitudine æquales binæque in duplicata pri <lb/>marum proportione. </s> <s>Sit autem minor proportio <lb/>primæ ad &longs;ecundam, quam &longs;ecundæ ad tertiam in <lb/>primis; erit minor proportio primæ ad &longs;ecundam, <lb/>quam &longs;ecundæ ad tertiam in &longs;ecundis. </s></p><p type="main"> <s>Sint tres magnitudines A, B, C, & aliæ illis multitudine <lb/>æquales D, E, F. quarum ip&longs;ius D ad E proportio &longs;it du­<lb/>plicata eius, quæ e&longs;t A ad B: & E ad F, duplicata eius, <lb/>quæ e&longs;t B ad C. &longs;it autem mi­<lb/>nor proportio A ad B, quam <lb/>B ad C. <!-- KEEP S--></s> <s>Dico minorem e&longs;se <lb/>proportionem D ad E, quam <lb/>E ad F. <!-- KEEP S--></s> <s>Sit enim vt C ad B, <lb/>ita B ad G: & vt B ad A, ita <lb/>A ad H. <!-- KEEP S--></s> <s>Igitur G ad C dupli­<lb/>cata erit proportio ip&longs;ius G ad <lb/>B, hoc e&longs;t B ad C: &longs;imiliter <lb/>erit H ad B, duplicata propor­<lb/>tio ip&longs;ius A ad B. </s> <s>Vt igitur <lb/>e&longs;t H ad B, ita erit D ad E: & <lb/>vt G ad C, ita E ad F. Rur­<lb/>&longs;us, quia minor e&longs;t proportio <lb/><figure id="id.043.01.097.1.jpg" xlink:href="043/01/097/1.jpg"/><lb/>A ad B, quam B ad C, &longs;ed vt A ad B, ita e&longs;t H ad A <pb/>& vt B ad C, ita G ad B; erit ex æquali minor proportio <lb/>H ad B, quam G ad C, &longs;ed vt H ad B, ita erat D, ad <lb/>E: & vt G ad C, ita E ad F; minor igitur proportio erit <lb/>D ad E, quam E ad F. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int octo magnitudines quaternæ propor­<lb/>tionales: tertiæ autem vtriu&longs;que ordinis inter &longs;o <lb/>&longs;int vt primæ; erit vt compo&longs;ita ex primis ad com <lb/>po&longs;itam ex &longs;ecundis, ita compo&longs;ita ex tertiis ad <lb/>compo&longs;itam ex quartis. </s></p><p type="main"> <s>Sint octo magnitudines quaternæ &longs;um­<lb/>ptæ proportionales, vt A ad B, ita C ad <lb/>D. & vt E ad F, ita G ad H. &longs;it autem vt <lb/>A ad E, ita C ad G. <!-- KEEP S--></s> <s>Dico e&longs;se vt AE, ad <lb/>ABF, ita CG, ad DH. <!-- KEEP S--></s> <s>Quoniam enim <lb/>componendo e&longs;t vt AE, ad E, ita, CG, <lb/>ad G; &longs;ed vt E ad F, ita e&longs;t G, ad H; erit <lb/>ex æquali, vt AE, ad F, ita CG, ad H. <lb/><!-- KEEP S--></s> <s>Eadem ratione erit vt AE, ad B, ita CG, <lb/>ad D: & conuertendo, vt B ad AE, ita <lb/>D ad CG. &longs;ed vt AE, ad F, ita erat <lb/>CG ad H; ex æquali igitur erit vt B <lb/>ad F, ita D, ad H: & componendo, vt <lb/>BF ad F, ita DH ad H: & conuerten­<lb/>do, vt F ad BF, ita H, ad DH. <!-- KEEP S--></s> <s>Sed vt <lb/>AE, ad F, ita erat CG ad H; ex æqua <lb/>li igitur erit vt AE ad BF, ita CG, <lb/>ad DH. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.098.1.jpg" xlink:href="043/01/098/1.jpg"/><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO VIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int tres magnitudines &longs;e &longs;e æqualiter exce­<lb/>dentes; & aliæ eiu&longs;dem generis illis multitudine <lb/>æquales, binæque &longs;umptæ in duplicata primarum <lb/>proportione; erit vtriu&longs;que ordinis minor pro­<lb/>portio compo&longs;itæ ex primis ad compo&longs;itam ex &longs;e­<lb/>cundis, quam compo&longs;itæ ex &longs;ecundis ad compo&longs;i­<lb/>tam ex tertijs. </s></p><p type="main"> <s>Sint tres magnitudines A, B, C, quarum C maxima <lb/>æque &longs;uperet B, atque <lb/>B, ip&longs;am A. & totidem <lb/>eiu&longs;dem generis D, E, <lb/>F, &longs;itque F ad E du­<lb/>plicata proportio ip&longs;ius <lb/>C ad B: & E ad D, <lb/>duplicata ip&longs;ius B ad <lb/>A. <!-- KEEP S--></s> <s>Dico AD, &longs;imul <lb/>ad BE, &longs;imul mino­<lb/>tem e&longs;&longs;e proportionem <lb/>quam BE, &longs;imul ad <lb/>CF, &longs;imul. </s> <s>E&longs;to enim <lb/>recta quæpiam GH, <lb/>ad aliam rectam &longs;ibi in <lb/>directum po&longs;itam HK, <lb/>vt magnitudo A ad ip <lb/>&longs;ius F duplam (hoc <lb/>enim fieri pote&longs;t) & <lb/><figure id="id.043.01.099.1.jpg" xlink:href="043/01/099/1.jpg"/><lb/>&longs;uper ba&longs;im GK; con&longs;tituatur triangulum GLK, atque <lb/>in eo de&longs;cribatur parallelogrammum GHMN: & vt e&longs;t <pb/>C ad B, ita fiat HM, ad <expan abbr="Mq.">Mque</expan> & vt B ad A, ita QM, ad <lb/>MP, & ip&longs;i GK, parallelæ TPR, VQS, ducantur. <lb/></s> <s>Quoniam igitur e&longs;t vt C, ad duplam ip&longs;ius F, ita GH, ad <lb/>HK; erit vt C ad F, ita e&longs;t par llelogrammum GM, ad <lb/>triangulum MHK: &longs;ed vt C, ad B, ita e&longs;t HM, ad <expan abbr="Mq;">Mque</expan> <lb/>hoc e&longs;t parallelogrammum GM, ad parallelogrammum <lb/>MV: & vt F, ad E, ita triangulum MHK, ad triangu­<lb/>lum MQS, ob duplicatam proportionem eius, quæ e&longs;t <lb/>HM ad <expan abbr="Mq.">Mque</expan> hoc e&longs;t ip&longs;ius C ad B; vt igitur trapezium <lb/>NK, ad NS trapezium, ita erit, per præcedentem, CF, <lb/>&longs;imul ad BE &longs;imul. </s> <s>Rur&longs;us quoniam e&longs;t conuertendo, vt <lb/>parallelogrammum MV, ad parallelogrammum GM, ita <lb/>B ad C. &longs;ed vt parallelogrammum GM, ad triangulum <lb/>KHM, ita erat C, ad F: & vt triangulum KHM, ad <lb/>triangulum QSM, ita F ad E; erit ex æquali, vt paral­<lb/>lelogrammum MV, ad triangulum SQM, ita B, ad E. <lb/><!-- KEEP S--></s> <s>Similiter ergo vt ante erit vt trapezium NS, ad NR tra­<lb/>pezium, ita EB, &longs;imul ad AD, &longs;imul. </s> <s>Rur&longs;us, quoniam <lb/>æque excedit LV, ip&longs;am LT, atque LG, ip&longs;am LV; <lb/>minor erit proportio LT ad LV, quam LV, ad LG: e&longs;t <lb/>autem trianguli LTR ad triangulum LVS, duplicata <lb/>proportio ip&longs;ius LT, ad LV, & trianguli LVS, ad trian­<lb/>gulum LGK, duplicata ip&longs;ius LV, ad LG, propter &longs;i­<lb/>militudinem triangulorum; minor igitur proportio erit <lb/>trianguli LTR, ad triangulum LVS, quam trianguli <lb/>LVS, ad triangulum LGK; dempto igitur triangulo <lb/>LNM, communi, minor erit proportio trapezij NR, ad <lb/>trapezium NS, quam trapezij NS, ad trapezium NK. <lb/></s> <s>Sed vt trapezium NR, ad trapezium NS, ita e&longs;t conuer­<lb/>tendo AD &longs;imul ad BE, &longs;imul: & vt trapezium NS, ad <lb/>trapezium NK, ita BE, &longs;imul ad CF, &longs;imul; minor igi­<lb/>tur proportio erit AD, &longs;imul ad BE &longs;imul, quam BE &longs;i­<lb/>mul ad CF, &longs;imul. </s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO IX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si recta linea vtcumque &longs;ecta fuerit, cubus qui <lb/>fit à tota æqualis e&longs;t duobus &longs;olidis rectangulis, <lb/>quæ ex partibus, & totius quadrato fiunt. </s></p><p type="main"> <s>Sit recta linea AB &longs;ecta in puncto C vtcumque. </s> <s>Di­<lb/>co cubum ex AB æqualem e&longs;se duobus &longs;olidis rectangu­<lb/>lis, quæ fiunt ex AC CB, & quadrato AB. <!-- KEEP S--></s> <s>Quoniam <lb/><figure id="id.043.01.101.1.jpg" xlink:href="043/01/101/1.jpg"/><lb/>enim communi altitudine AB, e&longs;t vt rectangulum BAC <lb/>ad quadratum AB, ita &longs;olidum ex AB, & rectangulo <lb/>BAC ad cubum ex AB, eademque ratione vt rectangu­<lb/>lum ABC, ad quadratum AB, ita &longs;olidum e&longs;t AB, & <lb/>rectangulo ABC ad cubum ex AB; erunt vt duo rectan­<lb/>gula BAC, ABC ad quadratum AB, ita duo &longs;olida <lb/>ex AB, & rectangulis BAC, ABC ad cubum ex AB. <lb/><!-- KEEP S--></s> <s>Sed duo rectangula BAC, ABC &longs;unt æqualia quadrato <lb/>AC; duo igitur &longs;olida ex AB, & rectangulis BAC, CBA, <lb/>æqualia &longs;unt cubo ex AB. <!-- KEEP S--></s> <s>Sed &longs;olidum ex AB & rectan­<lb/>gulo BAC e&longs;t id quod fit ex AC, & AC & quadrato <lb/>AB; duo igitur &longs;olida ex AC, CB, & quadrato AB &longs;i­<lb/>mul &longs;umpta æqualia &longs;ua cubo ex AB. <!-- KEEP S--></s> <s>Si igitur recta linea <lb/>vtcumque &longs;ecta fuerit, &c. </s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO X.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si recta linea vtcumque &longs;ecta fuerit, cubus qui <lb/>fit à tota æqualis e&longs;t cubis partium, & duobus &longs;o­<lb/>lidis rectangulis, quæ partium triplis, & earun­<lb/>dem quadratis reciproce continentur. </s></p><p type="main"> <s>Sit recta linea AB &longs;ecta vtcumque in puncto C. <!-- KEEP S--></s> <s>Dico <lb/>cubum ex AB æqualem e&longs;se duobus cubis ex AC, CB, <lb/>& duobus &longs;olidis rectangulis, quorum alterum fit ex tripla <lb/><figure id="id.043.01.102.1.jpg" xlink:href="043/01/102/1.jpg"/><lb/>ip&longs;ius AC, & quadrato BC; alterum autem ex tripla ip­<lb/>&longs;ius BC, & quadrato AC. <!-- KEEP S--></s> <s>Quoniam enim quadratum <lb/>ex AB æquale e&longs;t duobus quadratis ex AC, CB, & ei <lb/>quod bis fit ex AC CB: & parallelepipeda elu&longs;dem al­<lb/>titudinis inter &longs;e &longs;unt vt ba&longs;es; erit rectangulorum folido­<lb/>rum id quod fit ex AC, & quadrato AB æquale cubo ex <lb/>AC, & ei, quod fit ex AC, & rectangulo ACB bis, & <lb/>ei, quod ex AC, & quadrato BC. <!-- KEEP S--></s> <s>Eadem ratione erit <lb/>quod fit ex BC, & quadrato AB æquale cubo ex BC, & <lb/>ei, quod fit ex BC, & rectangulo ACB, bis & ei, quod ex <lb/>BC, & quadrato AC. <!-- KEEP S--></s> <s>Sed cubus ex AB æqualis e&longs;t <lb/>duobus &longs;olidis ex AC CB. & quadrato AB; cubus igi­<lb/>tur ex AB æqualis e&longs;t duobus cubis ex AC CB, & &longs;ex <lb/>&longs;olidis, quorum tres fiunt ex AC, & duobus rectangulis <lb/>ex AC CB, & quadrato BC: tria vero ex BC, & duo­<lb/>bus rectangulis ex AC CB, & quadrato AC. <!-- KEEP S--></s> <s>Sed quod <lb/>fit ex AC, & rectangulo ACB, e&longs;t quod fit ex BC, & <pb/>quadrato AC: & quod fit ex BC, & rectangulo ACB, <lb/>e&longs;t quod fit ex AC, & quadrato BC; cubus igitur ex <lb/>AB æqualis e&longs;t duobus cubis ex AC CB, vna cum &longs;ex <lb/>&longs;olidis, quorum tria fiunt ex AC, & BC quadrato, tria <lb/>autem ex BC, & quadrato AC, hoc e&longs;t duobus &longs;olidis, <lb/>quorum alterum fit ex tripla ip&longs;ius AC, & quadrato BC, <lb/>alterum ex tripla ip&longs;ius BC & quadrato AC. <!-- KEEP S--></s> <s>Quod de­<lb/>mon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si recta linea vtcumque &longs;ecta fuerit, cubus qui <lb/>fit à tota æqualis e&longs;t cubis partium vna cum &longs;oli­<lb/>do rectangulo, quod totius tripla, & partibus <lb/>continetur. </s></p><p type="main"> <s>Sit recta linea AB &longs;ecta in puncto C vtcumque. </s> <s>Di­<lb/>co cubum ex AB æqualem e&longs;se duobus cubis ex AC, <lb/>CB, vna cum &longs;olido rectangulo ex AC CB, & tripla <lb/>ip&longs;ius AB. <!-- KEEP S--></s> <s>Quoniam enim quod fit ex AC, & rectan­<lb/>gulo ACB, e&longs;t id quod fit ex BC, & quadrato AC: & <lb/>quod fit ex BC, & rectangulo ACB, e&longs;t id, quod fit ex <lb/><figure id="id.043.01.103.1.jpg" xlink:href="043/01/103/1.jpg"/><lb/>AC & quadrato BC. &longs;ed duo &longs;olida ex AC CB, & re­<lb/>ctangulo ACB &longs;unt id, quod fit ex compo&longs;ita vtriu&longs;que <lb/>altitudine AB, et rectangulo ACB; duo igitur prædi­<lb/>cta &longs;olida, quæ ex AC CB, & earum quadratis recipro­<lb/>ce fiunt æqualia &longs;unt &longs;olido ex AB BC CA, & triplum <lb/>triplo, videlicet duo &longs;olida, quæ fiunt reciproce ex triplis <pb/>ip&longs;arum AC, CB, & quadratis ex AC CB, æqualia &longs;i­<lb/>mul ei, quod ter fit ex AB, BC, CA, hoc e&longs;t ei, quod <lb/>partibus AC CB, & totius AB tripla continetur: additis <lb/>igitur communibus duobus cubis ex AC, CB, erit id, quod <lb/>&longs;it ex AC CB, & tripla ip&longs;ius AB, & duo cubi ex AC <lb/>CB, æqualia duobus &longs;olidis, quæ fiunt reciproce ex triplis <lb/>ip&longs;arum AC, CB, & earundem AC, CB, quadratis, & <lb/>duobus cubis ex AC, CB, hoc e&longs;t cubo ex AC. <!-- KEEP S--></s> <s>Si igi­<lb/>tur recta linea vtcumque &longs;ecta fuerit, &c. </s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hemi&longs;phærium duplum e&longs;t coni, cylindri au­<lb/>tem &longs;ub&longs;e&longs;quialterum eandem ip&longs;i ba&longs;im, & ean­<lb/>dem altitudinem habentium. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærium; cuius axis BD, ba&longs;is circulus, cu­<lb/>ius diameter AC, &longs;uper quem cylindrus AE, & conus <lb/><figure id="id.043.01.104.1.jpg" xlink:href="043/01/104/1.jpg"/><lb/>ABC, quorum communis axis &longs;it BD, ac propterea <lb/>etiam eadem altitudo. </s> <s>Dico hemi&longs;phærium ABC, co­<lb/>ni ABC e&longs;se duplum: cylindri autem AE <expan abbr="&longs;ub&longs;e&longs;quialterũ">&longs;ub&longs;e&longs;quialterum</expan>. <lb/></s> <s>&longs;uper ba&longs;im enim circulum RE, vertice D de&longs;cribatur <pb/>conus EDR. </s> <s>Sectoque axe BD primo bifariam, deinde <lb/>&longs;ingulis eius partibus rur&longs;us bifariam, tran&longs;eant per pun­<lb/>cta &longs;ectionum plana ba&longs;i hemi&longs;phærij AC æquidi&longs;tantia, <lb/>quæ &longs;ecent hemi&longs;phærium, conum, & cylindrum. </s> <s>Se­<lb/>ctus igitur erit AE cylindrus in cylindros æqualium alti­<lb/>tudinum: &longs;uper &longs;ectiones autem coni, atque hemi&longs;phærij <lb/>nempe circulos, quorum centra in axe BD exi&longs;tunt cy­<lb/>lindri con&longs;tituti intelligantur binis quibu&longs;que proximis <lb/>æquidi&longs;tantibus planis interiecti, quorum axes omnes <lb/>æquales in BD. <!-- KEEP S--></s> <s>Erit igitur cono EDR in&longs;cripta, & ABC <lb/><figure id="id.043.01.105.1.jpg" xlink:href="043/01/105/1.jpg"/><lb/>hemi&longs;phærio circum&longs;cripta figura quædam ex cylindris <lb/>æqualium altitudinum. </s> <s>Sint autem hæ figuræ ea ratione <lb/>hæc circum&longs;cripta illa in&longs;cripta, vt circum&longs;cripta excedat <lb/>hemi&longs;phærium, minori exce&longs;su, in&longs;cripta vero deficiat à <lb/>cono minori defectu quam &longs;it magnitudo propo&longs;ita, quan­<lb/>tacumque illa &longs;it. </s> <s>His con&longs;titutis, manife&longs;tum e&longs;t, reliquo <lb/>cylindri AE dempto hemi&longs;phærio in&longs;criptam e&longs;se figu­<lb/>ram ex re&longs;iduis cylindrorum, in quos cylindrus AE &longs;e­<lb/>ctus fuerit, demptis cylindris hemi&longs;phærio circum&longs;criptis, <lb/>deficientem à reliquo cylindri AE dempto hemi&longs;phærio <lb/>minori defectu magnitudine propo&longs;ita, eodem &longs;cilicet, <lb/>quo figura hemi&longs;phærio circum&longs;cripta excedit hemi&longs;phæ­<lb/>rium, excepto re&longs;iduo cylindri infimi AS, dempta he­<lb/>mi&longs;phærij portione, quam comprehendit. </s> <s>Sit autem om-<pb/>nium prædictorum cylindri AE cylindrorum &longs;upremus <lb/>FE, cuius axis BH, & communis &longs;ectio plani per pun­<lb/>ctum H tran&longs;euntis ba&longs;i hemi&longs;phærij cum plano per axim <lb/>BD, &longs;it recta FGKHMNL. </s> <s>Quoniam igitur rectan­<lb/>gulum DHB bis vna cum duobus quadratis DH, BH, <lb/>æquale e&longs;t BD quadrato: & rectangulum DHB bis <lb/>vna cum quadrato BH, e&longs;t rectangulum ex BD DH tan­<lb/>quam vna, & BH; rectangulum ex BD, DH tanquam <lb/>vna & BH, vna cum quadrato DH æquale erit quadra­<lb/>to BD, hoc e&longs;t quadrato FH: quorum quadratum KH <lb/>æquale e&longs;t rectangulo ex BD, DH, tanquam vna, & BH; <lb/>reliquum igitur quadrati FH dempto quadrato KH æ­<lb/>quale erit reliquo quadrato DH, hoc e&longs;t quadrato GH: <lb/>& quadruplum quadruplo reliquum quadrati FL dempto <lb/>quadrato MK toti GN quadrato, hoc e&longs;t reliquum circu <lb/>li, FL dempto circulo MK, æquale circulo GN. <!-- KEEP S--></s> <s>Qua­<lb/>re & GP, cylindrus reliquo cylindri FE dempto QK, <lb/>cylindro æqualis erit, propter æqualitatem altitudinum. <lb/></s> <s>Similiter o&longs;tenderemus &longs;ingula reliqua cylindrorum eiu&longs;­<lb/>dem altitudinis, in quos totus cylindrus AE &longs;ectus fuit, <lb/>demptis cylindris hemi&longs;phærio circum&longs;criptis æqualia e&longs;­<lb/>&longs;e &longs;ingulis cylindris cono EDR in&longs;criptis, quæ inter ea­<lb/>dem plana interijciuntur. </s> <s>Tota igitur figura ex prædictis <lb/>cylindrorum re&longs;iduis reliquo cylindri AE, dempto he­<lb/>mi&longs;phærio in&longs;cripta æqualis erit figuræ cono EDR in­<lb/>&longs;criptæ: deficit autem vtraque harum figurarum hæc à co­<lb/>no ADR, illa à re&longs;iduo cylindri AE dempto hemi&longs;phæ­<lb/>rio minori exce&longs;su magnitudine vtcumque propo&longs;ita; re­<lb/>liquum igitur cylindri AE dempto hemi&longs;phærio æquale <lb/>e&longs;t cono EDR, &longs;ed conus EDR; hoc e&longs;t conus ABC cylin <lb/>dri AE e&longs;t pars tertia; reliquum igitur cylindri AE dem­<lb/>pto hemi&longs;phærio, cylindri AE e&longs;t pars tertia, hoc e&longs;t cylin­<lb/>drus AE triplus dicti re&longs;idui: <expan abbr="quamobr&etilde;">quamobrem</expan> AE cylindrus &longs;e&longs;­<lb/>quialter hemi&longs;phærij ABC: & <expan abbr="cõuertendo">conuertendo</expan>, hemi&longs;phærium <pb/>cylindri AE &longs;ub&longs;e&longs;quialterum: coni igitur ABC duplum. <lb/></s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis minor &longs;phæræ portio, ad cylindrum, <lb/>cuius ba&longs;is æqualis e&longs;t circulo maximo, altitudo <lb/>autem eadem portioni, eam habet proportionem, <lb/>quam exce&longs;&longs;us, quo tripla &longs;emidiametri &longs;phæræ <lb/>excedit tres deinceps proportionales, quarum ma <lb/>xima e&longs;t &longs;phæræ &longs;emidiameter, media vero quæ <lb/>inter centra &longs;phæræ & ba&longs;is portionis interijci­<lb/>tur; ad &longs;emidiametri &longs;phæræ triplam. </s></p><p type="main"> <s>Sit &longs;phæræ, cuius centrum D, &longs;emidiameter BD, mi­<lb/>nor portio ABC, cuius axis BG &longs;egmentum &longs;emidiame­<lb/>tri BD, ba&longs;is autem circulus, cuius diameter AC. <!-- KEEP S--></s> <s>Sitque <lb/>EF, cylindrus, cu­<lb/>ius axis, &longs;iue alti­<lb/>tudo eadem BG: <lb/>ba&longs;is autem æqua­<lb/>lis circulo maxi­<lb/>mo, cuius &longs;emidia­<lb/>meter BD. <!-- KEEP S--></s> <s>Dico <lb/>portionem ABC, <lb/>ad cylindrum EF <lb/>eam habere pro­<lb/><figure id="id.043.01.107.1.jpg" xlink:href="043/01/107/1.jpg"/><lb/>portionem, quam exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;upe­<lb/>rat tres BD, DG; & minorem extremam ad ip&longs;as, quæ <lb/>&longs;it M; ad ip&longs;ius BD triplam. </s> <s>vertice enim D, ba&longs;i cylin­<lb/>dri EF, cuius diameter FH de&longs;cribatur conus FDH, cu­<lb/>ius intelligatur fru&longs;tum FHKL ab&longs;ci&longs;sum plano, quod ab-<pb/>&longs;cidit portionem ABC, plano circuli FH parallelum. <lb/></s> <s>Quoniam igitur fru&longs;tum FH<emph type="italics"/>K<emph.end type="italics"/>L æquale e&longs;t cylindri EF <lb/>re&longs;iduo, dempta ABC portione, quod ex præcedenti theo <lb/>remate per&longs;picuum e&longs;se debet: erit portio ABC æqualis <lb/>ei, quod relinquitur cylindri EF, &longs;i fru&longs;tum auferatur <lb/>FHKL: &longs;ed hoc reliquum e&longs;t ad cylindrum EF, vt exce&longs;­<lb/>&longs;us, quo tripla lineæ FH, &longs;uperat tres deinceps proportio­<lb/>nales FH, KL, & minorem extrema, ad triplam lineæ FH: <lb/><gap/>vt FH, ad KL, ita e&longs;t BD ad DG, & DG, ad M; vt igi­<lb/>tur exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;uperat tres BD, DG, <lb/>& M, &longs;imul, ad lineæ BD triplam, ita erit portio ABC ad <lb/>cylindrum EF. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portio &longs;phæræ ab&longs;ci&longs;sa duobus planis <lb/>parallelis alteroper centrum acto ad cylindrum, <lb/>cuius ba&longs;is e&longs;t eadem ba&longs;i portionis, &longs;iue circu­<lb/>lo maximo, & eadem altitudo, eam habet pro­<lb/>portionem, quam exce&longs;&longs;us, quo maior extrema ad <lb/>&longs;phæræ &longs;emidiametrum, & axim portionis exce­<lb/>dit tertiam partem axis portionis; ad maiorem ex­<lb/>tremam antedictam. </s></p><p type="main"> <s>Sit portio AB <lb/>CD, &longs;phæræ, cu <lb/>ius centrum F, <lb/>ab&longs;ci&longs;&longs;a duobus <lb/>planis parallelis <lb/>altero per <expan abbr="centrũ">centrum</expan> <lb/>F tran&longs;eunte; <lb/>axis autem por­<lb/>tionis fit FG: & <lb/><figure id="id.043.01.108.1.jpg" xlink:href="043/01/108/1.jpg"/><pb/>maior ba&longs;is, circulus maximus, cuius diameter AD, minor <lb/>autem, cuius diameter BC: & cylindrus AE, cuius ba&longs;is <lb/>circulus AD, axis FG; & vt FG ad FA, ita &longs;it FA, ad <lb/>MN, à qua ab&longs;cindatur NO, pars tertia ip&longs;ius FG. <!-- KEEP S--></s> <s>Dico <lb/>ABCD <expan abbr="portion&etilde;">portionem</expan> ad cylindrum AE e&longs;&longs;e vt OM ad MN. <lb/><!-- KEEP S--></s> <s>Po&longs;ita enim G <lb/>H, æquali ip&longs;i <lb/>FG, de&longs;criba­<lb/>tur circa axim <lb/>FG, cylindrus <lb/>L<emph type="italics"/>K<emph.end type="italics"/>, & conus <lb/>HFK. </s> <s>Quoniam <lb/>igitur duo cylin <lb/>dri AE, LK, <lb/>&longs;unt eiu&longs;dem al­<lb/><figure id="id.043.01.109.1.jpg" xlink:href="043/01/109/1.jpg"/><lb/>titudinis, erunt inter &longs;e vt ba&longs;es, AD, KH. hoc e&longs;t cy­<lb/>lindrus AE ad cylindrum LK, duplicatam habebit pro­<lb/>portionem diametri AD, ad diametrum KH, hoc e&longs;t eius, <lb/>quæ e&longs;t &longs;emidiametri AF ad &longs;emidiametrum GH. hoc e&longs;t <lb/>eam, quæ e&longs;t MN ad GH, &longs;iue FG. <!-- KEEP S--></s> <s>Sed vt FG ad tertiam <lb/>&longs;ui partem NO, ita e&longs;t cylindrus KL, ad conum KFH; <lb/>ex æquali igitur, erit vt MN ad NO, ita cylindrus AE <lb/>ad conum <emph type="italics"/>K<emph.end type="italics"/>FH, hoc e&longs;t ad reliquum cylindri AE dem <lb/>pta ABCD portione: & per conuer&longs;ionem rationis, vt <lb/>NM, ad MO, ita cylindrus AE ad portionem ABCD: <lb/>& conuertendo, vt MO ad MN, ita portio ABCD ad <lb/>cylindrum AE. <!-- KEEP S--></s> <s>Quod e&longs;t propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portio &longs;phæræ ab&longs;ci&longs;&longs;a duobus planis <lb/>parallelis neutro per centrum, nec centrum inter­<lb/>cipientibus ad cylindrum, cuius ba&longs;is æqualis e&longs;t <pb/>circulo maximo, altitudo autem eadem portioni, <lb/>eam <expan abbr="proportion&etilde;">proportionem</expan> habet, quam exce&longs;&longs;us, quo maior <lb/>extrema ad triplas &longs;emidiametri &longs;phæræ, & eius <lb/>quæ inter <expan abbr="centrũ">centrum</expan> &longs;phæræ, & minoris ba&longs;is portio­<lb/>nis interijcitur, &longs;uperat tres deinceps <lb/>proportionales, quarum maxima e&longs;t <lb/>quæ inter centra &longs;phæræ, & minoris <lb/>ba&longs;is, media autem, quæ inter cen­<lb/>træ &longs;phæræ, & maioris ba&longs;is portio­<lb/>nis interijcitur; ad maiorem extre­<lb/>mam antedictam. </s></p><p type="main"> <s>Sit portio ABCD &longs;phæræ, cuius centrum <lb/>E, ab&longs;ci&longs;sa duobus planis parallelis, neutro <lb/>per E tran&longs;eunte, nec E <expan abbr="intercipi&etilde;tibus">intercipientibus</expan>, cuius <lb/>maior ba&longs;is &longs;it circulus, cui diameter AD. <lb/>minor autem cuius diameter BC, axis GH. <lb/>circa quem cylindrus OS, con&longs;i&longs;tat, cuius <lb/>ba&longs;is &longs;it circulus circa SR æqualis circulo <lb/>maximo: &longs;phæræ autem &longs;emidiater &longs;it EHG. <lb/>& vt GE ad EH, ita &longs;it HE ad V: & po­<lb/><figure id="id.043.01.110.1.jpg" xlink:href="043/01/110/1.jpg"/><lb/>&longs;ita T tripla ip&longs;ius EF, & X itidem tripla ip&longs;ius EG, vt X <pb/>ad T, ita fiat T ad ZY, cuius Z<foreign lang="greek">w</foreign>, tribus GE, EH, V <lb/>&longs;imul &longs;it æqualis. </s> <s>Dico ABCD portio­<lb/>nem ad cylindrum SO e&longs;se vt <foreign lang="greek">w*u</foreign> ad <foreign lang="greek">*u</foreign>Z. <lb/><!-- KEEP S--></s> <s>Ab&longs;ci&longs;sa enim GK ip&longs;i EG æquali, cylin­<lb/>drus PN circa axim GH, & conus KEN <lb/>con&longs;tituantur vt in præcedenti. </s> <s>planum igi­<lb/>tur ab&longs;cindens portionem facit fru&longs;tum coni <lb/>KEN, quod &longs;it KLMN, cuius minor ba­<lb/>&longs;is circulus, cui diameter LM; maior autem <lb/>cui diameter KN. </s> <s>Et vt e&longs;t GE ad EF, hoc <lb/>e&longs;t GK ad SH, ita &longs;it EF, vel SH, ad I. <lb/>vt igitur in præcedenti, o&longs;tenderemus cylin­<lb/>drum SO ad cylindrum PN e&longs;se vt I ad <lb/>GK &longs;iue ad EG. <!-- KEEP S--></s> <s>Quoniam igitur &longs;unt ter <lb/>næ deinceps proportionales GE, EF, I, & <lb/>X, T, ZY, e&longs;tque vt FE ad EG ita T ad X; <lb/>erit vt I ad EG, hoc e&longs;t vt cylindrus SO ad <lb/>PN <expan abbr="cylindrũ">cylindrum</expan> ita ZY ad X. <!-- KEEP S--></s> <s>Et quoniam e&longs;t vt <lb/>GE ad EH, ita EH ad V: hoc e&longs;t, vt GK ad <lb/>LH. ita LH ad V: & ponitur X tripla ip&longs;ius <lb/><figure id="id.043.01.111.1.jpg" xlink:href="043/01/111/1.jpg"/><lb/>EG, hoc e&longs;t ip&longs;ius GK, vt autem e&longs;t triplaip&longs;ius GK ad <lb/>tres deinceps proportionales GK, LH, V, ita e&longs;t cylin­<lb/>drus PN ad fru&longs;tum LKNM; erit vt X ad tres GE, EH, <lb/>V &longs;imul hoc e&longs;t ad lineam <foreign lang="greek">w</foreign>Z, ita cylindrus PN ad fru-<pb/>&longs;lum KLMN. </s> <s>Sed vt ZY ad X, ita erat cylindrus SO <lb/>ad PN cylindrum; ex æquali igitur erit vt ZY ad Z<foreign lang="greek">w</foreign>, <lb/>ita cylindrus SO ad fru&longs;tum KLMN: hoc e&longs;t, ad reli­<lb/>quum cylindri SO dempta ABCD portione, & per con­<lb/>uer&longs;ionem rationis, vt ZY, ad Y<foreign lang="greek">w</foreign>, ita cylindrus SO ad <lb/><expan abbr="portion&etilde;">portionem</expan> ABCD: & conuertendo vt <foreign lang="greek">w</foreign>Y ad YZ, ita por­<lb/>tio ABCD ad SO cylindrum. </s> <s>Quod <expan abbr="demon&longs;trandũ">demon&longs;trandum</expan> erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis maior &longs;phæræ portio ad cylindrum, cu­<lb/>ius ba&longs;is æqualis e&longs;t circulo maximo, altitudo au­<lb/>tem eadem portioni eam habet proportionem, <lb/>quam ad axim portionis habet exce&longs;&longs;us, quo &longs;eg­<lb/>mentum axis portionis inter &longs;phæræ centrum, & <lb/>ba&longs;im portionis interiectum &longs;uperat tertiam par­<lb/>tem minoris extremæ maiori po&longs;ita prædicto axis <lb/>&longs;egmento in proportione &longs;emidiametri &longs;phæræ <lb/>ad prædictum <lb/><expan abbr="&longs;egmentũ">&longs;egmentum</expan>, vna <lb/>cum &longs;ub&longs;e&longs;qui <lb/>altera reliqui <lb/>axis &longs;egmenti. </s></p><figure id="id.043.01.112.1.jpg" xlink:href="043/01/112/1.jpg"/><p type="main"> <s>Sit &longs;phæræ, cu <lb/>ius <expan abbr="centrũ">centrum</expan> G, dia <lb/>meter DGE ma <lb/>ior portio ABC, <lb/>axis autem por­<lb/>tionis BGF, com <lb/>munis cylindro <lb/>KH, cuius ba&longs;is æqualis &longs;it circulo maximo; ba&longs;is autem <pb/>portionis circulus, cuius diameter AC, & vt EG ad GF, <lb/>ita &longs;it GF ad S, & S ad FM, cuius &longs;it pars tertia FN, & <lb/>ponatur ip&longs;ius BG, &longs;ub&longs;e&longs;quialtera GL. <!-- KEEP S--></s> <s>Dico portio­<lb/>nem ABC ad cylindrum KH e&longs;se vt LN ad BF. <!-- KEEP S--></s> <s>Nam <lb/>vt FG ad GE, &longs;iue ad BG, ita &longs;it EG ad PQ, à qua <lb/>ab&longs;cindatur QR, pars tertia ip&longs;ius FG. <!-- KEEP S--></s> <s>Et plano per G <lb/>tran&longs;eunte ba&longs;ibus cylindri KH, & ABC portionis pa­<lb/>rallelo &longs;ecentur vna cylindrus KH in duos cylindros DH, <lb/>EK: & portio ABC, in portionem ECAD, & DBE <lb/>hemi&longs;phærium. </s> <s>Quoniam igitur e&longs;t conuertendo, vt PQ <lb/>ad EG, ita EG <lb/>ad GF, & e&longs;t ip­<lb/>&longs;ius GF pars ter <lb/>tia QR, erit por­<lb/>tio DACE ad <lb/>cylindrum EK, <lb/>vt PR ad <expan abbr="Pq.">Pque</expan> <lb/>Rur&longs;us, quia e&longs;t <lb/>vt EG ad GF: <lb/>hoc e&longs;t vt PQ ad <lb/>EG, ita GF ad <lb/>S, & vt EG ad <lb/>GF, ita e&longs;t S ad <lb/>FM; erit ex æqua <lb/><figure id="id.043.01.113.1.jpg" xlink:href="043/01/113/1.jpg"/><lb/>li, vt PQ ad GF, ita GF ad FM. </s> <s>Sed vt GF ad RQ, <lb/>ita e&longs;t MF ad FN, tertiam ip&longs;ius MF partem, ex æquali <lb/>igitur erit vt PQ ad QR, ita GF ad FN, & per conuer­<lb/>&longs;ionem rationis, & conuertendo, vt PR ad PQ, ita NG ad <lb/>GF. <!-- KEEP S--></s> <s>Sed vt PR ad PQ, ita erat portio ECAD ad cy­<lb/>lindrum EK; vtigitur NG ad GF, ita erit portio EC <lb/>AD ad cylindrum EK. <!-- KEEP S--></s> <s>Sed vt GF ad FB, ita e&longs;t cy­<lb/>lindrus EK ad cylindrum KH: ex æquali igitur vt NG <lb/>ad BF, ita portio ECAD, ad cylindrum KH. <!-- KEEP S--></s> <s>Similiter <lb/>o&longs;tenderemus e&longs;se, vt GL ad BF, ita DBE hemi&longs;phæ-<pb/>rium ad cylindrum KH, cum vt LG ad GB, ita &longs;it he­<lb/>mi&longs;phærium DBE ad cylindrum DH. vt igitur prima <lb/>cum quinta ad &longs;ecundam, ita tertia cum &longs;exta ad quartam; <lb/>videlicet, vt tota LN ad BF, ita portio ABC ad cylin­<lb/>drum KH. <!-- KEEP S--></s> <s>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portio &longs;phæræ ab&longs;ci&longs;&longs;a duobus planis <lb/>parallelis centrum intercipientibus ad cylin­<lb/>drum, eiu&longs;dem altitudinis, cuius ba&longs;is æqualis e&longs;t <lb/>circulo maximo, eam habet proportionem, quam <lb/>ad axim portionis habet exce&longs;&longs;us, quo axis portio­<lb/>nis &longs;uperat tertiam partem compo&longs;itæ ex duabus <lb/>minoribus extremis, maioribus po&longs;itis duobus <lb/>axis &longs;egmentis, quæ fiunt à centro &longs;phæræ in ra­<lb/>tionibus, &longs;emidiametri &longs;phæræ ad prædicta &longs;eg­<lb/>menta. </s></p><p type="main"> <s>Sit portio AB <lb/>CD, &longs;phæræ, cu­<lb/>ius centrum G, <lb/>ab&longs;ci&longs;sa duobus <lb/>planis parallelis <lb/>centrum G inter­<lb/>cipientibus, quod <lb/>erit in axe portio­<lb/>nis, qui &longs;it HK. <lb/></s> <s>Sectiones autem <lb/><figure id="id.043.01.114.1.jpg" xlink:href="043/01/114/1.jpg"/><lb/>factæ à prædictis planis &longs;int circuli, quorum diametri AD, <lb/>BC, qui circuli erunt ba&longs;es oppo&longs;itæ portionis. </s> <s>Sectaque <lb/>per punctum G, portione ABCD plano ad axim erecto, <pb/>atque ideo & portionis ba&longs;ibus parallelo; &longs;uper &longs;ectionem, <lb/>quæ erit circulus maximus, cuius diameter LM, duo cylin­<lb/>dri de&longs;cripti intelligantur, ad oppo&longs;ita portionis ba&longs;ium pla <lb/>na terminati ex illis autem totus cylindrus compo&longs;itus EF, <lb/>cuius ba&longs;is æqua­<lb/>lis circulo maxi­<lb/>mo LM. <!-- KEEP S--></s> <s>Deinde <lb/>in &longs;egmento GH <lb/>&longs;umpta OH, ter­<lb/>tia parte minoris <lb/>extremæ maiori <lb/>GH in proportio <lb/>ne, quæ e&longs;t LG ad <lb/>GH; & in &longs;egmen <lb/>to GK, &longs;umatur <lb/><figure id="id.043.01.115.1.jpg" xlink:href="043/01/115/1.jpg"/><lb/>NK, tertia pars minoris extremæ maiori GK, in propor­<lb/>tione, quæ e&longs;t LG ad GK. <!-- KEEP S--></s> <s>Dico portionem ABCD <lb/>ad cylindrum EF, e&longs;se vt NO ad KH. <!-- KEEP S--></s> <s>Sumptis enim <lb/>ij&longs;dem, quæ in præcedentis &longs;ump&longs;imus, demon&longs;trationem <lb/>&longs;imiliter o&longs;tenderemus tam portionem LBCM ad cy­<lb/>lindrum EF, e&longs;se vt OG ad <emph type="italics"/>K<emph.end type="italics"/>H, quam portionem LA <lb/>DM ad eundem EF cylindrum, vt NG ad eundem axim <lb/>KH, vt igitur prima cum quinta ad &longs;ecundam, ita tertia <lb/>cum &longs;exta ad quartam: videlicet, vt NO ad KH, ita por <lb/>tio ABCD ad EF cylindrum. </s> <s>Quod demon&longs;trandum <lb/>crat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omne conoides parabolicum dimidium e&longs;t <lb/>cylindri, coni autem &longs;e&longs;quialterum eandem ip&longs;i <lb/>ba&longs;im, & eandem altitudinem habentium. </s></p><pb/><p type="main"> <s>Sit conoides parabolicum ABC, & cylindrus AE, & <lb/>conus ABC, quorum omnium &longs;it eadem ba&longs;is circulus, <lb/>cuins diameter AC, axis autem BD, ac proinde vna om­<lb/>nium altitudo. </s> <s>Dico conoidis ABC e&longs;se cylindri AE <lb/>dimidium, coni autem ABC &longs;e&longs;quialterum. </s> <s>Secto enim <lb/>axe BD in tot partes æquales, quarum infima ad ba&longs;im &longs;it <lb/>MD, vt figura ex cylindris æqualium altitudinum conoi­<lb/>di ABC circum&longs;cripta, in&longs;criptam &longs;uperet minori &longs;pacio <lb/>quantacumque magnitudine propo&longs;ita, & &longs;it hoc factum. <lb/></s> <s>Et quoniam quibus planis parallelis tran&longs;euntibus per præ­<lb/><figure id="id.043.01.116.1.jpg" xlink:href="043/01/116/1.jpg"/><lb/>dictas &longs;ectiones axis BD &longs;ecatur conoides ABC, ij&longs;dem <lb/>&longs;ecatur triangulum per axim ABC, eruntque &longs;ectiones <lb/>parallelæ: &longs;it triangulo ABC circum&longs;cripta figura ex pa­<lb/>rallelogrammis æqualium altitudinum, quæ triangulum & <lb/>ip&longs;a excedat minori &longs;pacio quantacumque magnitudine <lb/>propo&longs;ita. </s> <s>Cylindrorum autem qui &longs;unt circa conoides, & <lb/>parallelogrammorum multitudine æqualium, quæ &longs;unt cir­<lb/>ca triangulum ABC, duo proximi ba&longs;i AC cylindri &longs;int <lb/>AF, HL, & totidem parallelogramma illis re&longs;pondentia <lb/>inter eadem plana parallela &longs;int AF, GK. <!-- KEEP S--></s> <s>Quoniam igi-<pb/>tur in parabola ABC rectis ad diametrum ordinatim ap­<lb/>plicatis e&longs;t vt BM ad BD longitudine, ita MH ad AD <lb/>potentia: hoc e&longs;t, ita circulus, cuius diameter HMN, ad <lb/>circulum, cuius diameter ADC, hoc e&longs;t ita cylindrus HL, <lb/>ad cylindrum AF propter æqualitatem altitudinum: &longs;ed <lb/>vt BM ad BD, ita e&longs;t GM ad AD, propter &longs;imilitudinem <lb/>triangulorum, hoc e&longs;t ita <expan abbr="parallelogrãmum">parallelogrammum</expan> GK ad AF, pa­<lb/>rallelogrammum; ergo vt parallelogrammum GK ad paral <lb/><expan abbr="lelogrãmum">lelogrammum</expan> AF, ita e&longs;t cylindrus HL ad cylindrum AF. <lb/><!-- KEEP S--></s> <s>Similiter o&longs;tenderemus reliqua parallelogramma, quæ &longs;unt <lb/><figure id="id.043.01.117.1.jpg" xlink:href="043/01/117/1.jpg"/><lb/>circa <expan abbr="triãgulum">triangulum</expan> ABC e&longs;se cum reliquis cylindris, qui &longs;unt <lb/>circa conoides ABC bina &longs;umpta prout inter &longs;e re&longs;pon­<lb/>dent in eadem proportione; &longs;emper igitur componendo, & <lb/>ex æquali erit vt tota figura triangulo ABC circum&longs;cripta <lb/>ad parallelogrammum AF, ita figura conoidi circum&longs;cri­<lb/>pta ad AF cylindrum: &longs;ed vt parallelogrammum AF, ad <lb/>parallelogrammum AE, ita e&longs;t cylindrus AF ad cylindrum <lb/>AE, propter æqualitatem omnifariam &longs;umptarum altitu­<lb/>dinum; ex æquali igitur erit vt figura triangulo ABC cir­<lb/>cum&longs;cripta ad parallelogrammum AE, ita figura conoidi <pb/>ABC circum&longs;cripta ad AE cylindrum: vtraque autem <lb/>circum&longs;criptarum figurarum excedit &longs;ibi in&longs;criptam mino­<lb/>ri &longs;pacio quantacumque magnitudine propo&longs;ita, vt igitur <lb/>triangulum ABC, ad parallelogrammum AE, ita erit co­<lb/>noides ABC, ad cylindrum AE. <!-- KEEP S--></s> <s>Sed triangulum ABC <lb/>e&longs;t parallelogrammi AE dimidium; igitur conoides ABC <lb/>e&longs;t cylindro AE dimidium: &longs;ed cylindrus AE e&longs;t coni <lb/>ABC, triplum: igitur conoides ABC, erit coni ABC <lb/>&longs;e&longs;quialterum. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pri&longs;matis triangulam ba&longs;im habentis <lb/>centrum grauitatis rectam lineam, quæ cuiu&longs;libet <lb/>trium laterum bipartiti &longs;ectionem, & oppo&longs;iti pa­<lb/>rallelogrammi centrum iungit, ita diuidit, vt <lb/>pars, quæ attingit latus &longs;it dupla reliquæ. </s></p><p type="main"> <s>Sit pri&longs;ma, quale diximus AB <lb/>CDEF, &longs;ectoque vno ip&longs;ius la­<lb/>tere BF in puncto G, bifariam <lb/>parallelogrammi oppo&longs;iti &longs;it cen <lb/>trum H, & iuncta GH, cuius <lb/>pars GK &longs;it dupla reliquæ <emph type="italics"/>K<emph.end type="italics"/>H. <lb/><!-- KEEP S--></s> <s>Dico pri&longs;matis ABCDEF, cen <lb/>trum grauitatis e&longs;&longs;e K. <!-- KEEP S--></s> <s>Per pun <lb/>ctum enim H ducatur NO ip­<lb/>&longs;i AE, vel CD parallela, quæ <lb/>ip&longs;as AC, ED, &longs;ecabit <expan abbr="bifariã">bifariam</expan>: <lb/>iunctisque BN, FO, ducatur per <lb/>punctum <emph type="italics"/>K<emph.end type="italics"/>, ip&longs;i FB, vel NO <lb/><figure id="id.043.01.118.1.jpg" xlink:href="043/01/118/1.jpg"/><lb/>parallela LM. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt HK ad KG, ita <lb/>NL ad LB, & OM ad MF, erit NL, ip&longs;ius LB, & OM <pb/>ip&longs;ius MF dimidia: &longs;ed & rectæ BN, FO, triangulorum <lb/>ba&longs;es AC, ED, bifariam &longs;e­<lb/>cant; erunt igitur puncta L, M, <lb/>centra grauitatis triangulorum <lb/>ABC, DEF, oppo&longs;itorum. <lb/></s> <s>Pri&longs;matis igitur ABCDEF <lb/>axis erit LM: quare in eius bi­<lb/>partiti &longs;ectione pri&longs;matis ABC <lb/>DEF centrum grauitatis: &longs;ectus <lb/>autem e&longs;t axis LM bifariam in <lb/>puncto K; nam ob parallelogram <lb/>ma e&longs;t vt NH ad HO, ita LK <lb/>ad KM; pri&longs;matis igitur ABC <lb/>DEF, centrum grauitatis erit <emph type="italics"/>K.<emph.end type="italics"/><lb/>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.119.1.jpg" xlink:href="043/01/119/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis pri&longs;matis ba&longs;im habentis trapezium, cu­<lb/>ius duo latera inter &longs;e &longs;int parallela centrum gra­<lb/>uitatis rectam lineam, quæ æque inter &longs;e di&longs;tan­<lb/>tium parallelogrammorum centra iungit, ita di­<lb/>uidit, vt pars, quæ dictorum parallelogrammorum <lb/>minus attingit &longs;it ad reliquam, vt duorum ba&longs;is la <lb/>terum parallelorum dupla maioris vna cum mino<lb/>ri ad duplam minoris vna cum maiori. </s></p><p type="main"> <s>Sit pri&longs;ma ABCDEFGH, cuius ba&longs;is trapezium <lb/>ABCD, habens duo latera AD, BC, inter &longs;e paralle­<lb/>la, &longs;itque eorum AD maius: parallela igitur erunt inter &longs;e <lb/>duo parallelogramma BG, AH. <!-- KEEP S--></s> <s>Sit parallelogrammi AH <lb/>centrum K, & BG parallelogrammi centrum L, iuncta-<pb/>que LK, fiat vt dupla ip&longs;ius AD vna cum BC ad du­<lb/>plam ip&longs;ius BC vna cum AD, ita LR ad RK. </s> <s>Dico <lb/>pri&longs;matis AG centrum grauitatis e&longs;se R. <!-- KEEP S--></s> <s>Ducantur enim <lb/>per puncta L, K lateribus pri&longs;matis, atque ideo inter &longs;e <lb/>parallelæ MN, OP, quæ <lb/>ob centra K, L, &longs;ecabunt <lb/>oppo&longs;ita parallelogrammo­<lb/>rum latera bifariam, eas <lb/>&longs;ectiones connectant MO, <lb/>NP, ip&longs;ique MN, vel <lb/>OP, parallela ducatur Q <lb/>RS. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t <lb/>vt LR ad R<emph type="italics"/>K<emph.end type="italics"/>, hoc e&longs;t vt <lb/>dupla ip&longs;ius AD vna cum <lb/>BC ad duplam ip&longs;ius BC <lb/>vna cum AD, ita OQ ad <lb/>QM, & recta MO bifa­<lb/><figure id="id.043.01.120.1.jpg" xlink:href="043/01/120/1.jpg"/><lb/>riam &longs;ecat AC trapezij latera parallela, punctum Q, AC <lb/>trapezij centrum grauitatis; &longs;imiliter & punctum S erit EG, <lb/>trapezij centrum grauitatis: pri&longs;matis igitur AG axis erit <lb/>QS, & centrum grauitatis R, quod e&longs;t in medio axis. <lb/></s> <s>Omnis igitur pri&longs;matis ba&longs;im habentis trapezium, &c. <lb/></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si à quolibet prædicto pri&longs;mate duo pri&longs;mata <lb/>be&longs;es habentia triangulas &longs;int ita ab&longs;ci&longs;&longs;a, vt pa­<lb/>rallelepipedum relinquant ba&longs;im habens minus <lb/>parallelogrammorum inter &longs;e parallelorum præ­<lb/>dicti pri&longs;matis, maioris autem partes æqualia pa­<lb/>rallelogramma ip&longs;um parallelepipedum relin­<pb/>quat, centrum grauitatis vtriu&longs;que ab&longs;ci&longs;si pri&longs;­<lb/>matis tamquam vnius magnitudinis rectam line­<lb/>lam, quæ prædicti pri&longs;matis parallelorum paral <lb/>lelogrammorum centra iungit, ita diuidit, vt <lb/>pars, quæ minus parallelogrammum attingit &longs;it <lb/>dupla reliquæ. </s></p><p type="main"> <s>Sit pri&longs;ma ABCDEFGH, cuius ba&longs;es oppo&longs;itæ tra­<lb/>pezia ADHE, BCGF. </s> <s>Sint autem AD, EH, paral­<lb/>lelæ, quarum maior EH. <!-- KEEP S--></s> <s>Oppo&longs;ita igitur parallelogram­<lb/>ma AC, EG, inter &longs;e erunt parallela, quorum maius EG. <lb/><!-- KEEP S--></s> <s>At per rectas AB, CD, &longs;ectum &longs;it pri&longs;ma. </s> <s>ABCDEF <lb/>GH, ita vt ab&longs;ci&longs;&longs;a pri&longs;mata ABSFER, CDVHGT, <lb/>relinquant parallelepipedum AT, ip&longs;um autem AT, re­<lb/>linquat duo parallelogramma æqualia ES, TH. <!-- KEEP S--></s> <s>Po&longs;ito <lb/>autem centro K <lb/>parallelogrammi <lb/>AC, & L, paral <lb/>lelogrammi EG, <lb/>iunctaque KL, <lb/>ponatur KM, du <lb/>pla ip&longs;ius ML. <lb/><!-- KEEP S--></s> <s>Dico <expan abbr="duorũ">duorum</expan> pri&longs;­<lb/>matum BER, <lb/>CVH, &longs;imul cen <lb/>trum grauitatis <lb/><figure id="id.043.01.121.1.jpg" xlink:href="043/01/121/1.jpg"/><lb/>e&longs;se M. </s> <s>Sectis enim AB, CD, bifariam in punctis P, Q, <lb/>&longs;umpti&longs;que parallelogrammorum ES, VG, centris N, O, <lb/>iungantur PN, QO, & po&longs;ita PX dupla ip&longs;ius XN, & QZ <lb/>dupla ip&longs;ius ZO, iungantur rectæ PKQ, XZ, NO. <lb/><!-- KEEP S--></s> <s>Quoniam igitur in quadrilatero PQON, recta XZ, pa­<lb/>rallela e&longs;t vtrilibet ip&longs;arum PQ, NO, &longs;ecat ijs parallelis <lb/>interceptas in ea&longs;dem rationes; recta igitur XT per pun-<pb/>ctum M tran&longs;ibit. </s> <s>Sed quia PK e&longs;t æqualis KQ, & NL <lb/>ip&longs;i LO, etiam XM æqualis erit ip&longs;i MZ ob parallelas; <lb/>cum igitur pri&longs;matum BER, CVH centra grauitatis &longs;int <lb/>X, Z; erit vtriu&longs;que pri&longs;matis prædicti &longs;imul centrum gra­<lb/>uitatis M. </s> <s>Quod e&longs;t propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int duæ pyramides æquales, & æque altæ, <lb/>ba&longs;es habentes in eodem plano, quarum vertices <lb/>recta linea connectens cum ea, quæ ba&longs;ium centra <lb/>grauitatis iungit &longs;it in eodem plano; earum cen­<lb/>trum grauitatis tamquam vnius magnitudinis re­<lb/>ctam lineam, quæ inter vertices, & centra ba&longs;ium <lb/>interiectas bifariam &longs;ecat, itadiuidit, vt pars &longs;u­<lb/>perior &longs;it inferioris tripla. </s></p><figure id="id.043.01.122.1.jpg" xlink:href="043/01/122/1.jpg"/><p type="main"> <s>Sint duæ <lb/>pyramides æ­<lb/>quales, & æ­<lb/>que altæ, qua­<lb/>rum ba&longs;es in <lb/>eodem plano <lb/>AC, DB, ver <lb/>tices autem <lb/>G, H, & ba­<lb/>&longs;ium <expan abbr="c&etilde;tra">centra</expan> E, <lb/>F, iunctæque <lb/>EF, GH, quas <lb/>bifariam &longs;ecet recta KL, huius autem pars quarta &longs;it LM. <lb/><!-- KEEP S--></s> <s>Dico vtriu&longs;que pyramidis GAC, HDB, &longs;imul centrum <lb/>grauitatis e&longs;&longs;e M. </s> <s>Iunctis enim GE, HF, &longs;umantur ea­<pb/>rum quartæ partes EN, FO, & iungatur NO. <!-- KEEP S--></s> <s>Quoniam <lb/>igitur propter æqualitatem altitudinum, & quia EF, GH, <lb/>&longs;unt in eodem plano, &longs;unt EF, GH, inter &longs;e parallelæ, & <lb/>vt GN ad NE, ita e&longs;t HO ad OF; erit NO ip&longs;i E Fivel <lb/>GH, paralle­<lb/>la, quas KL <lb/>bifariam &longs;ecat: <lb/>igitur & ip&longs;am <lb/>NO &longs;ecabit bi <lb/>fariam, iungit <lb/>autem recta <lb/>NO centra <lb/>grauitatis <expan abbr="py-ramidũ">py­<lb/>ramidum</expan> æqua­<lb/>lium GAC, <lb/>HDB, vtriu&longs;­<lb/><figure id="id.043.01.123.1.jpg" xlink:href="043/01/123/1.jpg"/><lb/>que ergo pyramidis &longs;imul centrum grauitatis erit in com­<lb/>muni &longs;ectione duarum linearum KL, NO, &longs;ed recta NO, <lb/>&longs;ecans &longs;imiliter ip&longs;as GE, KL, HF, ip&longs;am KL, &longs;ecabit <lb/>in puncto M; punctum igitur M, erit prædictarum pyrami­<lb/>dum centrum grauitatis. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti pyramidis ba&longs;im habentis paral­<lb/>lelogrammum centrum grauitatis maiori ba&longs;i e&longs;t <lb/>propinquius, quam punctum illud, in quo axis &longs;ic <lb/>diuiditur, vt pars minorem ba&longs;im attingens &longs;it ad <lb/>reliquam vt dupla cuiu&longs;uis laterum maioris ba&longs;is <lb/>vna cum latere minoris &longs;ibi re&longs;pondente, ad <expan abbr="duplã">duplam</expan> <lb/>dicti lateris minoris ba&longs;is vna cum maioris &longs;ibi <lb/>re&longs;pondente. </s></p><pb/><p type="main"> <s>Sit pyramidis, cuius ba&longs;is parallelogrammum EFGH, <lb/>fru&longs;tum ABCDEFGH, <expan abbr="eiu&longs;q;">eiu&longs;que</expan> axis KL, quo &longs;ecto in pun <lb/>cto <foreign lang="greek">a</foreign> ita vt K <foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign> L, &longs;it vt laterum homologorum AD <lb/>EH, dupla ip&longs;ius EH vna cum AD ad duplam ip&longs;ius <lb/>AD vna cum EH, & fru&longs;ti ABCDEFGH &longs;it centrum <lb/>grauitatis <foreign lang="greek"><gap/></foreign> nempe in axe KL. <!-- KEEP S--></s> <s>Dico punctum <foreign lang="greek"><gap/></foreign>, cadere <lb/>infra punctum <foreign lang="greek">a. </foreign></s> <s>A punctis enim A,B,C,D, ducantur <lb/><figure id="id.043.01.124.1.jpg" xlink:href="043/01/124/1.jpg"/><lb/>ad maiorem ba&longs;im axi KL, parallelæ AN, BO, CR, DS, <lb/>& parallelepipedum ABCDNORS compleatur, & <lb/>productis ba&longs;is NO lateribus, de&longs;criptæ &longs;int quatuor py­<lb/>ramides AEMNZ, BOPFY, CGXRQ, DHVST, <lb/>quarum ba&longs;es erunt parallelogramma circa diametrum <lb/>æqualia, atque &longs;imilia: & quatuor pri&longs;mata triangulas ba­<lb/>&longs;es habentia, quorum binorum ex aduer&longs;o inter &longs;e re&longs;pon-<pb/>dentium parallelogramma in plano EG exi&longs;tentia erunt <lb/>inter &longs;e æqualia, atque &longs;imilia, &longs;cilicet MS ip&longs;i OQ, & <lb/>ZO, ip&longs;is RV: &longs;itque axis KL pars tertia L <foreign lang="greek">b</foreign>, quarta <lb/>autem L <foreign lang="greek">d. </foreign><!-- KEEP S--></s> <s>Quoniam ìgitur ex &longs;upra demon&longs;tratis pri&longs;­<lb/>matis ABCDTMPQ e&longs;t centrum grauitatis <foreign lang="greek">a</foreign>; duo­<lb/>rum autem pri&longs;matum oppo&longs;itorum ABYONZ, CDS <lb/>RXV, centrum grauitatis <foreign lang="greek">b</foreign>, erit reliqui ex fru&longs;to AB <lb/><figure id="id.043.01.125.1.jpg" xlink:href="043/01/125/1.jpg"/><lb/>CDEFGH demptis quatuor prædictis pyramidibus in <lb/><foreign lang="greek">a b</foreign> centrum grauitatis, quod &longs;it <foreign lang="greek">g. </foreign><!-- KEEP S--></s> <s>Nam ex primo li­<lb/>bro con&longs;tat punctum <foreign lang="greek">a</foreign> cadere &longs;upra punctum <foreign lang="greek">b</foreign>, &longs;i com­<lb/>pleatur trapezium ACGE, cuius diameter erit KL. <!-- KEEP S--></s> <s>Sed <lb/>earum quatuor pyramidum e&longs;t centrum grauitatis <foreign lang="greek">d. </foreign><!-- KEEP S--></s> <s>Si <lb/>enim ba&longs;ium, quibus binæ oppo&longs;itæ pyramides in&longs;i&longs;tunt <lb/>centra grauitatis, & bini oppo&longs;iti vertices &longs;ingulis rectis li-<pb/>neis connectantur, erunt binæ connectentes parallelæ, & <lb/>ab axe <emph type="italics"/>K<emph.end type="italics"/> L bifariam &longs;ecabuntur, vt figuræ de&longs;criptio ina­<lb/>nife&longs;tat. </s> <s>Totius igitur fru&longs;ti ABCDEFGH, centrum <lb/>grauitatis <foreign lang="greek"><gap/></foreign> in linea <foreign lang="greek">g d</foreign> cadet: &longs;ed punctum <foreign lang="greek">g</foreign> cadit infra <lb/>punctum <foreign lang="greek">a</foreign>, multo ergo inferius, & ba&longs;i EG propinquius <lb/>punctum <foreign lang="greek"><gap/></foreign> quam punctum <foreign lang="greek">a. </foreign></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti conici centrum grauitatis pro­<lb/>pinquius e&longs;t maiori ba&longs;i quam punctum illud, in <lb/>quo axis &longs;ic diuiditur, vt pars minorem ba&longs;im <lb/>attingens &longs;it ad reliquam, vt dupla diametri ma­<lb/>ior is ba&longs;is vna cum minoris diametro ad duplam <lb/>diametri minoris ba&longs;is vna cum diametro ma­<lb/>ioris. </s></p><p type="main"> <s>Hoc eadem ratione deducetur ex antecedenti, qua cen­<lb/>trum grauitatis fru&longs;ti conici in extremo primo libro demon <lb/>&longs;trauimus, quandoquidem &longs;imiliter vt ibi fecimus, omnis <lb/>pyramidis centro grauitatis idem probaremus accedere <lb/>quod prædictæ pyramidis in antecedente. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int quotcumque magnitudines, & aliæ illis <lb/>multitudine æquales, binæque &longs;umptæ in eadem <lb/>proportione, quæ commune habeant centrum gra<lb/>uitatis, centra autem grauitatis omnium &longs;int in <lb/>eadem recta linea; primæ & &longs;ecundæ tanquam <pb/>duæ magnitudines commune habebunt centrum <lb/>grauitatis. </s></p><p type="main"> <s>Sit recta linea AB, & quotcumque magnitudines <lb/>FGH, & totidem KLM, binæ in eadem proportione: <lb/>nimirum vt F ad G ita K ad L: & vt G ad H ita L ad <lb/>M. in recta autem AB, &longs;int communia centra grauitatis, <lb/>C duarum FK, & D duarum GL: & E duarum HM. </s> <s>Om­<lb/>nium autem primarum tamquam vnius magnitudinis &longs;it <lb/>centrum grauitatis O. <!-- KEEP S--></s> <s>Dico & omnium &longs;ecundarum &longs;i­<lb/>mul centrum grauitatis e&longs;se O. <!-- KEEP S--></s> <s>Duarum enim FG &longs;i­<lb/><figure id="id.043.01.127.1.jpg" xlink:href="043/01/127/1.jpg"/><lb/>mul &longs;it centrum grauitatis N. <!-- KEEP S--></s> <s>Vtigitur e&longs;t F ad G, hoc <lb/>e&longs;t, vt K ad L, ita erit DN, ad NC. punctum igitur N <lb/>e&longs;t centrum grauitatis duarum magnitudinum KL &longs;imul. <lb/></s> <s>Rur&longs;us, quia componendo, & ex æquali, e&longs;t vt FG &longs;imul <lb/>ad H, ita KL &longs;imul ad M: e&longs;t autem tam duarum FG, <lb/>quam duarum KL &longs;imul centrum grauitatis N, &longs;imiliter <lb/>vt ante o&longs;tenderemus duarum magnitudinum FGH, <lb/>KLM centrum grauitatis e&longs;se O. <!-- KEEP S--></s> <s>Quod e&longs;t propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int quotcumque magnitudines, & aliæ ip­<lb/>&longs;is multitudine æquales primarum, ex quibus cen <lb/>tra grauitatis in eadem recta linea di&longs;po&longs;ita &longs;int <lb/>alternatim ad centra grauitatis &longs;ecundarum, qua-<pb/>rum magnitudinum binæ eodem ordine, qui &longs;u­<lb/>mitur ab eodem prædictæ lineæ termino vnain <lb/>primis, & alterain &longs;ecundis inter &longs;e &longs;int æquales; <lb/>omnium primarum &longs;imul, ex quibus primæ cen­<lb/>trum grauitatis propinquius e&longs;t prædicto lineæ <lb/>termino quàm primæ &longs;ecundarum, propinquius <lb/>erit prædicto lineæ termino quàm omnium &longs;ecun<lb/>darum &longs;imul centrum grauitatis. </s></p><p type="main"> <s>Sint quotcumque magnitudines ABC primæ, & toti­<lb/>dem &longs;ecundæ DEF, quarum centra grauitatis in recta <lb/>linea TV, primarum quidem G ip&longs;ius A proximum om­<lb/><figure id="id.043.01.128.1.jpg" xlink:href="043/01/128/1.jpg"/><lb/>nium termino T, à quo &longs;umitur ordo. </s> <s>Deinde H ip&longs;ius B, <lb/>& <emph type="italics"/>K<emph.end type="italics"/>, ip&longs;ius C, di&longs;po&longs;ita &longs;int alternatim ad centra &longs;ecun­<lb/>darum; videlicet vt centrum grauitatis L, ip&longs;ius D cadat <lb/>inter centra G, H, & M ip&longs;ius E inter centra H, K: & N <lb/>inter puncta <emph type="italics"/>K<emph.end type="italics"/>, V: &longs;int autem æquales binæ AD, BE, <lb/>CF: & omnium ABC &longs;imul centrum grauitatis P, & om­<lb/>nium DEF &longs;imul centrum grauitatis O. <!-- KEEP S--></s> <s>Dico punctum <lb/>P propinquius e&longs;&longs;e termino T, quàm punctum O. <lb/><!-- KEEP S--></s> <s>Duarum enim A, B &longs;it centrum grauitatis R: & S, dua­<lb/>rum DB, & Q, duarum DE. <!-- KEEP S--></s> <s>Quoniam igitur Q e&longs;t <lb/>centrum grauitatis duarum magnitudinum DE &longs;imal; erit <lb/>vt D ad E, hoc e&longs;t ad B, ita MQ, ad QL: hoc e&longs;t HS, <lb/>ad SL. & componendo, vt ML, ad LQ, ita HL, ad <lb/>LS; & permutando, vt ML ad LH, ita LQ ad LS: <lb/>&longs;ed ML e&longs;t maior quàm LH; ergo & LQ erit maior <lb/>quàm LS. </s> <s>Eadem ratione quoniam S e&longs;t centrum gra­<pb/>uitatis duarum DB: & R duarum AB: & AD &longs;unt æ­<lb/>quales; erit RH maior quàm SH: &longs;ed quia LQ erat ma­<lb/>ior quàm LS, e&longs;t & SH maior quàm QH; multo igitur <lb/>maior RH erit quàm QH: atque ideo punctum R pro­<lb/>pinquius termino T, quàm punctum <expan abbr="q.">que</expan> Rur&longs;us quo­<lb/>niam tota magnitudo AB e&longs;t æqualis toti DE, & C æ­<lb/>qualis F; erunt duæ primæ AB, & C, & totidem &longs;ecun­<lb/>dæ DE, & F, quarum vnius po&longs;teriorum DE cen­<lb/>trum grauitatis Q cadit inter R, K centra grauitatis <lb/>duarum priorum AB, & C, & reliquæ priorum C cen­<lb/>trum grauitatis K cadit inter Q, N, duarum po&longs;terio­<lb/>rum DE, & F centra grauitatis; erunt vt antea quatuor <lb/>magnitudines binæ proximæ æquales, &longs;cilicet AB, ip&longs;i <lb/><figure id="id.043.01.129.1.jpg" xlink:href="043/01/129/1.jpg"/><lb/>DE: & C ip&longs;i F, centra grauitatis habentes di&longs;pofita <lb/>alternatim in eadem recta TV. </s> <s>Cum igitur primæ prio­<lb/>rum AB, centrum grauitatis R &longs;it termino T propin­<lb/>quius quàm Q centrum grauitatis primæ po&longs;teriorum, <lb/>quæ e&longs;t tota DE; &longs;imiliter vt ante totius magnitudinis <lb/>ABC centrum grauitatis P erit termino T propinquius <lb/>quàm totius DEF centrum grauitatis O. <!-- KEEP S--></s> <s>Non aliter <lb/>o&longs;tenderemus, quotcumque plures magnitudines, quales <lb/>& quemadmodum diximus ad rectam TV, di&longs;po&longs;itæ <lb/>proponerentur, &longs;emper centrum grauitatis omnium prio­<lb/>rum &longs;imul termino T propinquius cadere, quàm omnium <lb/>po&longs;teriorum &longs;imul centrum grauitatis. </s> <s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int quotcumque magnitudines, & aliæ illis <lb/>multitudine æquales, quæ binæ commune habe­<lb/>ant in eadem recta centrum grauitatis; &longs;umpto au <lb/>tem ordine ab vno eius lineæ termino, maior &longs;it <lb/>proportio primæ ad &longs;ecundam in primis, quàm <lb/>primæ ad &longs;ecundam in &longs;ecundis: & &longs;ecundæ ad <lb/>tertiam in primis maior quàm &longs;ecundæ ad ter­<lb/>tiam in &longs;ecundis, & &longs;ic deinceps v&longs;que ad vltimas; <lb/>erit omnium primarum &longs;imul centrum grauitatis <lb/>propinquius prædicto lineæ termino, à quo &longs;umi­<lb/>tur ordo, quàm omnium &longs;ecundarum. </s></p><p type="main"> <s>Sint quotcumque magnitudines GHI, & totidem <lb/>LMN. </s> <s>Sitque maior proportio G ad H, quàm L ad M: & <lb/>H ad I, maior quàm M ad N: in recta autem AB &longs;int <lb/>communia centra grauitatis, C duarum magnitudinum <lb/>GL, & D duarum HM, & E duarum IN. omnium <lb/><figure id="id.043.01.130.1.jpg" xlink:href="043/01/130/1.jpg"/><lb/>autem primarum GHI &longs;imul &longs;it centrum grauitatis K: at <lb/>&longs;ecundarum omnium LMN centrum grauitatis R. <!-- KEEP S--></s> <s>Di­<lb/>co centrum K cadere termino A propinquius quàm cen <lb/>trum R. <!-- KEEP S--></s> <s>Fiat enim vt G ad H, ita DP ad PC: & vt L <lb/>ad M, ita DQ ad QC. </s> <s>Maior igitur proportio erit DP <pb/>ad PC, quàm DQ ad QC: & componendo, maior DC <lb/>ad CP, quàm DC ad CQ: minor igitur CP erit quàm <lb/>CQ: quare DP maior quàm <expan abbr="Dq.">Dque</expan> & communi addita <lb/>ED, erit EP maior quàm <expan abbr="Eq.">Eque</expan> Et quoniam <emph type="italics"/>K<emph.end type="italics"/> e&longs;t cen­<lb/>trum grauitatis omnium GHI &longs;imul, & ip&longs;ius GH e&longs;t cen <lb/>trum grauitatis P, & reliquæ magnitudinis I, centrum <lb/>grauitatis E; erit vt GH ad I, ita EK ad KP. eadem <lb/>ratione vt vtraque LM ad N, ita erit ER ad <expan abbr="Rq.">Rque</expan> Rur­<lb/><figure id="id.043.01.131.1.jpg" xlink:href="043/01/131/1.jpg"/><lb/>&longs;us, quia maior e&longs;t proportio G ad H, quàm L ad M, erit <lb/>componendo, maior proportio GH ad H, quàm LM ad <lb/>M: &longs;ed maior e&longs;t proportio H ad K, quàm M ad N; ex <lb/>æquali igitur, maior erit proportio GH ad I, quàm LM <lb/>ad N, hoc e&longs;t EK ad KP, quàm ER ad <expan abbr="Rq.">Rque</expan> Multo <lb/>ergo maior proportio EK ad KP, quàm ER ad RP: & <lb/>componendo maior proportio EP ad PK quàm EP ad <lb/>PR; minor igitur PK erit quàm PR, at que ideo centrum <lb/>K propinquius termino A quàm centrum R. <!-- KEEP S--></s> <s>Quod de­<lb/>mon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int quotcumque magnitudines, & aliæ ip&longs;is <lb/>multitudine æquales, quarum omnium centra <lb/>grauitatis &longs;int in eadem recta linea, & centra pri­<lb/>marum ad centra &longs;ecundarum di&longs;po&longs;ita &longs;int alter­<lb/>natim: &longs;it autem maior proportio primæ ad &longs;ecun-<pb/>dam in primis quàm primæ ad &longs;ecundam in &longs;ecun<lb/>dis: & &longs;ecundæ ad tertiam in primis, maior quàm <lb/>&longs;ecundæ ad tertiam in &longs;e cundis, & &longs;ic deinceps v&longs;­<lb/>que ad vltimas; erit omnium primarum &longs;imul cen <lb/>trum grauitatis propinquius prædictæ lineæ ter­<lb/>mino à quo &longs;umitur ordo omnium &longs;ecundarum <lb/>centrum grauitatis. </s></p><p type="main"> <s>Sit quotcumque magnitudines GHI, & totidem LMN <lb/>primarum autem &longs;int centra grauitatis CDE cum &longs;ecun<lb/>darum centris OPQ in eadem recta AB di&longs;po&longs;ita alter­<lb/>natim, vt O cadat inter puncta CD, & P inter puncta <lb/>DE, & E inter puncta <expan abbr="Pq.">Pque</expan> &longs;itque maior proportio G <lb/>ad H, quàm L ad M, & H ad I maior quàm M ad N. <lb/>omnium autem primarum GHI &longs;imul &longs;it centrum gra­<lb/>uitatis T; at omnium &longs;ecundarum LMN, &longs;imul, cen­<lb/><figure id="id.043.01.132.1.jpg" xlink:href="043/01/132/1.jpg"/><lb/>trum grauitatis V. <!-- KEEP S--></s> <s>Dico punctum T e&longs;&longs;e termino A <lb/>propinquius quàm punctum V. <!-- KEEP S--></s> <s>E&longs;to enim F æqualis <lb/>L, & K æqualis M, & X æqualis N, &longs;it autem cen­<lb/>trum grauitatis ip&longs;ius F in puncto C, & ip&longs;ius K in pun­<lb/>cto D, & ip&longs;ius X in puncto E. <!-- KEEP S--></s> <s>In recta igitur AB om­<lb/>nium FKX, &longs;imul centrum grauitatis erit termino A, pro­<lb/>pinquius quàm omnium LMN &longs;imul centrum grauitatis. <lb/></s> <s>Sed & omnium GHI, &longs;imul centrum grauitatis in eadem <lb/>recta AB propinquius e&longs;t termino A quàm omnium <lb/>FKX, &longs;imul centrum grauitatis; multo igitur termino A <lb/>propinquius erit omnium GHI &longs;imul quàm omnium <pb/>LMN, &longs;imul centrum grauitatis. </s> <s>Quod demon&longs;tran­<lb/>dum erat. </s></p><p type="head"> <s><emph type="italics"/>ALITER.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Po&longs;ito enim R centro grauitatis duarum <expan abbr="magnitudinũ">magnitudinum</expan> G, <lb/>H, & S <expan abbr="duarũ">duarum</expan> L,M, vel punctum V cadit in puncto E, vel in <lb/>linea EB, vel in linea AE, &longs;i in puncto E vel in linea EB, <lb/>cum igitur T &longs;it <expan abbr="centrũ">centrum</expan> grauitatis trium <expan abbr="magnitudinũ">magnitudinum</expan> G,H,I <lb/>&longs;imul, & E ip&longs;ius I, erit punctum T propinquius termino <lb/>A quàm punctum V. <!-- KEEP S--></s> <s>Sed punctum V in linea AE cadat. <lb/></s> <s>Veligitur S centrum grauitatis duarum magnitudinum L, <lb/>M, &longs;imul cadit in puncto D, &longs;iue in linea DB, vel in li­<lb/>nea AD. &longs;i in puncto D, vel in linea DB; centrum gra­<lb/>uitatis R duarum magnitudinum GH erit termino A <lb/>propinquius quàm ip&longs;um S, & recta ER maior quàm ES, <lb/><figure id="id.043.01.133.1.jpg" xlink:href="043/01/133/1.jpg"/><lb/>Sed cadat punctum S in linea AD. <!-- KEEP S--></s> <s>Quoniam igitur ma­<lb/>ior e&longs;t proportio G ad H, quàm L ad M: & vt G ad H, <lb/>ita e&longs;t DR ad RG, & vt L ad M, ita PS ad SO, ma­<lb/>ior erit proportio DR ad RC, quàm PS ad SO; mul­<lb/>to ergo maior DR ad RC, quàm DS ad SO, & multo <lb/>maior quàm DS ad SC, & componendo maior propor­<lb/>tio DC ad CR, quàm DC ad CS; erit igitur CR mi­<lb/>nor quàm CS, atque adeo RD maior DS, addita igitur <lb/>ED communi, erit ER maior quàm ES. </s> <s>Rur&longs;us quia <lb/>componendo, & ex æquali maior e&longs;t proportio totius GH <lb/>ad I quàm totius LM ad N, hoc e&longs;t maior longitudinis <lb/>ET ad TR, quàm QV ad VS, & multo maior quàm <pb/>EV ad VS, erit componendo, maior proportio ER ad <lb/>RT quàm ES ad SV: & per conuer&longs;ionem rationis mi­<lb/>nor proportio FR ad ET; quàm ES ad EV, & permu­<lb/>tando minor proportio ER ad ES quàm ET ad EV: &longs;ed <lb/>ER maior erat quàm ES, ergo ET maior erit quàm EV: <lb/>& punctum T propinquius termino A, quàm punctum V. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Datæ figuræ circa diametrum, vel axim in alte <lb/>ram partem deficienti, &longs;uper ba&longs;im rectam lineam <lb/>vel circulum, vel ellip&longs;im; cuius figuræ ba&longs;is, & <lb/>&longs;ectiones omnes parallelæ &longs;egmenta æqualia dia­<lb/>metri vel axis intercipientes ita &longs;e habeant, vt <lb/>quarumlibet trium proximarum minor proportio <lb/>&longs;it minimæ ad mediam, quàm mediæ ad maxi­<lb/>mam; figura quædam ex cylindris, vel cylindri <lb/>portionibus, vel parallelogrammis æqualium al­<lb/>titudinum circum&longs;cribi pote&longs;t, cuius <expan abbr="c&etilde;trum">centrum</expan> gra­<lb/>uitatis &longs;it propinquius ba&longs;i quàm cuiu&longs;libet datæ <lb/>figuræ, qualem diximus quæ prædictæ figuræ cir <lb/>cadiametrum, vel axim circum&longs;cripta &longs;it. </s></p><p type="main"> <s>Sit figura circa diametrum, vel axim in alteram <expan abbr="part&etilde;">partem</expan> de­<lb/>ficiens qualem diximus, cuius bafis circulus, vel ellip&longs;is vel <lb/>recta linea AC, axis autem vel diameter BD. <!-- KEEP S--></s> <s>Et data figu­<lb/>ra ip&longs;i ABC figuræ circum&longs;cripta compo&longs;ita ex cylindris, <lb/>vel cylindri portionibus, vel parallelogrammis æqualium <lb/>altitudinum EF, GH, AK. <!-- KEEP S--></s> <s>Dico figuræ ABC alteram <lb/>figuram, qualem diximus po&longs;&longs;e circum&longs;cribi, cuius centrum <pb/>grauitatis, nempe in linea BD, &longs;it propinquius ba&longs;i AC, <lb/>&longs;iue termino D, quàm prædictæ datæ figuræ circum&longs;criptæ <lb/>centrum grauitatis, Omnium enim cylindrorum, vel cy­<lb/>lindri portionum, vel parallelogrammorum, ex quibus con­<lb/>&longs;tat prædicta data figura circum&longs;cripta &longs;int axes, vel quæ <lb/>oppo&longs;ita latera coniungunt rectæ BL, LM, MD, qui­<lb/>bus &longs;ectis bifariam in punctis N, O, P, ac planis per ea <lb/>&longs;iue rectis tran&longs;euntibus ba&longs;i AC parallelis, &longs;ecantibus­<lb/>que dictos cylindros, vel cylindri portiones, vel pa­<lb/>rallelogramma, compleatur & figuræ ABC circum&longs;cri­<lb/>batur altera figura <lb/>vt prior, quæ ob &longs;e­<lb/>ctiones factas com­<lb/>ponetur ex duplis <lb/>multitudine cylin­<lb/>dris, vel cylindri por­<lb/>tionibus, vel paralle­<lb/>logrammis &ecedil;qualium <lb/>altitudinum, eorum <lb/>ex quibus con&longs;tat da­ <lb/>ta figura circum&longs;cri­<lb/>pta &longs;in<gap/>autem hi cy­<lb/>lindri, aut reliqua, <lb/>quæ diximus QR, <lb/><figure id="id.043.01.135.1.jpg" xlink:href="043/01/135/1.jpg"/><lb/>ES, TV, GX, ZI, AY. </s> <s>Quoniam igitur cylindro­<lb/>rum, vel cylindri portionum, vel parallelogrammorum quæ <lb/>&longs;unt circa figuram ABC, minor e&longs;t proportio QR ad ES, <lb/>quàm ES, ad TV, propter &longs;ectiones circulos, vel &longs;imiles <lb/>ellip&longs;es, vel rectas lineas, & <expan abbr="æqualitat&etilde;">æqualitatem</expan> <expan abbr="altitudinũ">altitudinum</expan>, & figuræ <lb/>propo&longs;itæ <expan abbr="naturã">naturam</expan>. </s> <s>Sed <expan abbr="ead&etilde;">eadem</expan> ratione minor e&longs;t proportio ES <lb/>ad TV, quàm TV, ad GX; multo ergo minor proportio erit <lb/>QR ad ES, quam TV ad GX: & componendo, minor <lb/>proportio QR, ES, &longs;imul ad ES, quàm TV, GX, &longs;imul <lb/>ad GX. &longs;ed vt GX ad GH, ita e&longs;t ES ad EF; ex æqua-<pb/>li igitur minor erit proportio QR, ES &longs;imul ad EF, <lb/>quàm TV, GX &longs;imul ad GH. & permutando, minor <lb/>proportio QR, ES &longs;imul ad TV, GX &longs;imul quàm EF <lb/>ad GH. & conuertendo, maior proportio GX, TV &longs;i­<lb/>mul ad ES, QR &longs;imul, quàm GH ad EF. <!-- KEEP S--></s> <s>Similiter <lb/>o&longs;tenderemus duo ZI, AY, &longs;imul ad TV, GX, &longs;imul, <lb/>maiorem habere proportionem, quàm AK ad rectarum <lb/>GH. <!-- KEEP S--></s> <s>Rur&longs;us quoniam puncta N, O, in medio BL, LM, <lb/>&longs;unt, ip&longs;orum EF, GH, centra grauitatis: duorum autem <lb/>QR, ES &longs;imul centrum grauitatis e&longs;t in linea NL, pro­<lb/>pterea quòd ES maius e&longs;t quàm QR, & æquales BN, <lb/>NL, quas centra grauitatis ip&longs;orum QR, ES bifariam <lb/>diuidunt, cadet ip&longs;orum QR, ES, &longs;imul centrum grauita­<lb/>tis propius termino D, quàm ip&longs;ius EF centrum grauitatis, <lb/>& duobus centris N, O, interijcietur. </s> <s>Eademque ratio­<lb/>ne duorum TV, GX, &longs;imul centrum grauitatis termino <lb/>D erit propinquius quàm ip&longs;ius GH centrum grauitatis, <lb/>& duobus centris O, P, duorum GH, AK interijcietur. <lb/></s> <s>Et duorum ZI, AY &longs;imul centrum grauitatis propin­<lb/>quius erit D termino, quàm P ip&longs;ius AK. <!-- KEEP S--></s> <s>Quoniam <lb/>igitur omnia primarum magnitudinum, ex quibus con&longs;tat <lb/>figura &longs;ecundo circum&longs;cripta centra grauitatis in eadem re <lb/>cta linea BD, di&longs;po&longs;ita &longs;unt alternatim ad centra grauita­<lb/>tis &longs;ecundarum primis multitudine æqualium, ex quibus <lb/>data figura con&longs;tat ip&longs;i ABC figuræ circum&longs;cripta, &longs;unt <lb/>termino D propinquiora, quàm centra grauitatis &longs;ecunda­<lb/>rum, &longs;i bina, prout inter &longs;e re&longs;pondent comparentur: maior <lb/>autem proportio o&longs;ten&longs;a e&longs;t primæ ad &longs;ecundam in primis, <lb/>quàm primæ ad &longs;ecundam in &longs;ecundis: & &longs;ecundæ ad ter­<lb/>tiam in primis, quàm &longs;ecundæ ad tertiam in &longs;ecundis, <lb/>&longs;umpto ordine à termino D, erit centrum grauitatis om­<lb/>nium primarum &longs;imul, ide&longs;t figuræ ip&longs;i ABC figuræ <lb/>&longs;ecundo circum&longs;criptæ termino D propinquius, quàm <lb/>datæ figuræ eidem ABC figuræ primo circum&longs;criptæ cen­<pb/>trum grauitatis. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis prædictæ figuræ centrum grauitatis <lb/>e&longs;t propinquius ba&longs;i, quàm cuiu&longs;libet figuræ ex <lb/>cylindris, vel cylindri portionibus, vel parallelo­<lb/>grammis æqualium altitudinum ip&longs;i circum&longs;cri­<lb/>ptæ. </s></p><p type="main"> <s>Sit prædicta figura ABC, cuius axis vel diameter BD, <lb/>& data intelligatur figura ex quotcumque cylindris, vel cy­<lb/>lindri portionibus, vel parallelogrammis æqualium altitu­<lb/>dinum figuræ ABC circum&longs;cripta, cuius &longs;it centrum gra­<lb/>uitatis E, nempe in axe vel <lb/>diametro BD. <!-- KEEP S--></s> <s>Dico cen­<lb/>trum grauitatis figuræ ABC <lb/>propinquius e&longs;&longs;e puncto D, <lb/>quàm punctum E. <!-- KEEP S--></s> <s>Si enim <lb/>fieri pote&longs;t, centrum grauita­<lb/>tis figuræ ABC, quod &longs;it <lb/>F, non cadat infra punctum <lb/>E, &longs;ed vel &longs;upra, vel con­<lb/>gruat puncto E: figuræ ita­<lb/>que ABC circum&longs;cribatur <lb/>figura quædam ex cylindris, <lb/>vel cylindri portionibus, vel <lb/>parallelogrammis <17>qualium <lb/>altitudinum, cuius centrum <lb/><figure id="id.043.01.137.1.jpg" xlink:href="043/01/137/1.jpg"/><lb/>grauitatis, quod &longs;it G, &longs;it propinquius D puncto, quàm <lb/>punctum E, ac propterea propinquius, quàm punctum F, <lb/>centrum grauitatis figuræ primo circum&longs;criptæ. </s> <s>Rur&longs;us <lb/>multiplicatis cylindris, vel cylindri portionibus, vel paral-<pb/>lelogrammis circum&longs;cribatur figuræ ABC, altera tertia fi­<lb/>gura, quemadmodum diximus in præcedenti, cuius cen­<lb/>trum grauitatis H, in linea GD cadat & &longs;it minor pro­<lb/>portio re&longs;idui huius tertiæ figuræ circum&longs;criptæ ip&longs;i ABC, <lb/>ad figuram ABC, quàm FG ad GD. </s> <s>Multo ergo mi­<lb/>nor proportio erit dicti re&longs;idui ad figuram ABC quam F <lb/>H ad HD, fiat igitur vt prædictum re&longs;iduum ad figuram <lb/>ABC, ita ex contraria parte FH ad HDK; prædicti igi­<lb/>tur re&longs;idui centrum grauitatis erit K, extra ip&longs;ius terminos, <lb/>quod fieri non pote&longs;t: Non igitur F centrum grauitatis fi­<lb/>guræ ABC cadit in puncto E, nec &longs;upra; ergo infra pun <lb/>ctum E: & ponitur E centrum grauitatis cuiuslibet figuræ <lb/>ex cylindris, vel cylindri portionibus, vel parallelogrammis <lb/>æqualium altitudinum quo modo diximus ip&longs;i ABC cir­<lb/>cum&longs;criptæ. </s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omni prædictæ figuræ figura quædam ex cylin <lb/>dris, vel cylindri portionibus, vel parallelogram­<lb/>mis æqualium altitudi <lb/>num circum&longs;cribi po­<lb/>te&longs;t, cuius centri graui <lb/>tatis di&longs;tantia à prædi­<lb/>ctæ figuræ centro gra­<lb/>uitatis &longs;it minor quan­<lb/>tacunque longitudine <lb/>propo&longs;ita. </s></p><figure id="id.043.01.138.1.jpg" xlink:href="043/01/138/1.jpg"/><p type="main"> <s>Sit figura ABC in <expan abbr="alterã">alteram</expan> <lb/>partem <expan abbr="defici&etilde;s">deficiens</expan> &longs;upradicta, <lb/>cuius centrum grauitatis F, propo&longs;ita autem <expan abbr="quantacũque">quantacumque</expan> <lb/><expan abbr="lõgitudine">longitudine</expan> minor &longs;it FG ip&longs;ius BF. <!-- KEEP S--></s> <s>Dico figuræ ABC figu-<pb/>ram ex cylindris vel cylindri portionibus, vel <expan abbr="parallelogrã-mis">parallelogram­<lb/>mis</expan> æqualium <expan abbr="altitudinũ">altitudinum</expan> circum&longs;cribi po&longs;&longs;e, cuius centrum <lb/>grauitatis &longs;it propinquius puncto F, quàm punctum G: figu­<lb/>ræ enim ABC figura, qualem diximus circum&longs;cribatur, cu­<lb/>ius re&longs;iduum dempta figura ABC, ad figuram ABC mi­<lb/>norem habeat proportionem, quàm FG, ad GB, &longs;it autem <lb/>figuræ circum&longs;criptæ centrum grauitatis K, nempe in axe, <lb/>vel diametro BD. <!-- KEEP S--></s> <s>Dico <lb/>lineam FK minorem e&longs;&longs;e <lb/>quàm FG, atque adeo lon <lb/>gitudine propo&longs;ita. </s> <s>Quo­<lb/>niam enim F e&longs;t centrum <lb/>grauitatis figuræ ABC, <lb/>erit centrum grauitatis <emph type="italics"/>K<emph.end type="italics"/>, <lb/>figuræ circum&longs;criptæ ip&longs;i <lb/>ABC propinquius termi­<lb/>no B, quàm punctum F, <lb/>&longs;ed centrum grauitatis fi­<lb/>guræ ABC quòd e&longs;t F, & <lb/>figuræ circum&longs;criptæ, quod <lb/>e&longs;t K & eius re&longs;idui dem­<lb/><figure id="id.043.01.139.1.jpg" xlink:href="043/01/139/1.jpg"/><lb/>pta figura ABC &longs;unt in communi axe, vel diametro BD; <lb/>erit igitur dicti re&longs;idui in linea BK, centrum grauitatis, <lb/>quod &longs;it H. <!-- KEEP S--></s> <s>Minor autem proportio e&longs;t prædicti re&longs;idui <lb/>ad figuram ABC, hoc e&longs;t ip&longs;ius FK ad KH, quàm FG <lb/>ad GB, & multo minor, quàm FG ad GH; & compo­<lb/>nendo minor proportio FH ad HK, quàm FH ad HG; <lb/>ergo KH maior erit, quàm GH; reliqua igitur F <emph type="italics"/>K<emph.end type="italics"/> mi­<lb/>nor, quàm FG atque adeo longitudine propo&longs;ita. </s> <s>Fieri <lb/>ergo pote&longs;t, quod proponebatur. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si duarum prædictarum figurarum circa com­<lb/>munem axim, vel diametrum, vel alterius diame­<lb/>trum alterius axim, ba&longs;es, & quotcumque &longs;ectio­<lb/>nes quales diximus, binæ in eodem plano fue­<lb/>rint proportionales; idem punctum in diametro, <lb/>vel axe erit vtriu&longs;que centrum grauitatis. </s></p><p type="main"> <s>Sint duæ prædictæ figuræ ABC, DBE, circa eandem <lb/>diametrum, vel axim BF. figuræ autem ABC &longs;it cen­<lb/>trum grauitatis G, nempe in linea BF. <!-- KEEP S--></s> <s>Dico G e&longs;&longs;e <lb/>centrum grauitatis <lb/>figuræ DBE. &longs;i <lb/>enim non e&longs;t, &longs;it a­<lb/>liud punctum H, <lb/>quod cadat primo <lb/>&longs;upra punctum G. <lb/><!-- KEEP S--></s> <s>Figuræ igitur AB <lb/>C, figura circum­<lb/>&longs;cribatur qualem <lb/>diximus ex cylin­<lb/>dris, vel cylindri <lb/>portionibus, vel pa­<lb/>rallelogrammis æ­<lb/>qualium <expan abbr="altitudinũ">altitudinum</expan> <lb/>cuius centri graui­<lb/>tatis <emph type="italics"/>K<emph.end type="italics"/> di&longs;tantia à <lb/><figure id="id.043.01.140.1.jpg" xlink:href="043/01/140/1.jpg"/><lb/>centro G, figuræ ABC &longs;it minor quàm recta GH: & figu­<lb/>ræ DBE, figura circum&longs;cribatur ex cylindris, vel cylindri <lb/>portionibus vel parallelogrammis æqualium altitudinum, <lb/>multitudine æqualium ijs, ex quibus con&longs;tat ip&longs;i ABC, <pb/>figura circum&longs;cripta, quæ cum prædictis circa figuram AB <lb/>C erunt bina &longs;umpto ordine à puncto B, in eadem propor­<lb/>tione inter eadem plana parallela, vel rectas parallelas <expan abbr="cõ&longs;i-&longs;tentia">con&longs;i­<lb/>&longs;tentia</expan>, propter &longs;ectiones, ide&longs;t ba&longs;es, & æquales altitudines: <lb/>binorum autem quorumque homologorum idem erit in li­<lb/>nea BF, centrum grauitatis: punctum igitur K, centrum <lb/>grauitatis figuræ ip&longs;i ABC circum&longs;criptæ, idem erit fi­<lb/>guræ ip&longs;i DBE, circum&longs;criptæ centrum grauitatis: cadi<gap/><lb/><expan abbr="aut&etilde;">autem</expan> infra centrum <lb/>grauitatis H figu­<lb/>ræ DBE, quod e&longs;t <lb/>ab&longs;urdum.</s> <s>Non <lb/>igitur centrum gra­<lb/>uitatis figuræ DB <lb/>E, cadit &longs;upra pun <lb/>ctum G. <!-- KEEP S--></s> <s>Sed ca­<lb/>dat infra, vt in pun­<lb/>cto L. <!-- KEEP S--></s> <s>Rur&longs;us igi <lb/>tur figuræ DBE fi­<lb/>gura, qualem dixi­<lb/>mus circum&longs;cripta, <lb/>cuius centrum gra­<lb/>uitatis M, &longs;it pro­<lb/>pinquius centro L, <lb/><figure id="id.043.01.141.1.jpg" xlink:href="043/01/141/1.jpg"/><lb/>quàm punctum G, figuræ ABC altera qualem diximus <lb/>figura circum&longs;cribatur, cuius centrum grauitatis &longs;it idem <lb/>punctum M, quod fieri po&longs;&longs;e con&longs;tat ex &longs;uperioribus. </s> <s>Sed <lb/>G ponitur centrum grauitatis figuræ ABC; ergo centrum <lb/>grauitatis figuræ ip&longs;i ABC, circum&longs;criptæ erit propinquius <lb/>ba&longs;i & puncto F, quàm figuræ ABC centrum grauitatis, <lb/>quod fieri non pote&longs;t. </s> <s>Non igitur figuræ DBE centrum gra<lb/>uitatis cadit infra punctum G. <!-- KEEP S--></s> <s>Sed neque &longs;upra; punctum <lb/>igitur G erit commune duarum figurarum ABC, DBE, <lb/>centrum grauitatis. </s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Manife&longs;tum e&longs;t autem omnia proximis qua­<lb/>tuor propo&longs;itionibus <expan abbr="o&longs;tē&longs;a">o&longs;ten&longs;a</expan> de figura circa axim, <lb/>vel diametrum in alteram partem deficienti, ea­<lb/>dem ij&longs;dem rationibus o&longs;ten &longs;a remanere de com­<lb/>po&longs;ito ex duabus figuris circa communem axim <lb/>vel diametrum in alteram partem deficientibus, <lb/>tam per &longs;e con&longs;iderato, quàm ad alteram figuram <lb/>circa eundem axim, vel diametrum cum prædi­<lb/>cto compo&longs;ito, in alteram partem deficiens, ac &longs;i <lb/>e&longs;&longs;ent duæ tantummodo dictæ figuræ, quales in <lb/>præcedenti proxima inter &longs;e comparauimus; ma­<lb/>nente &longs;emper illa conditione, quàm de &longs;ectioni­<lb/>bus in vige&longs;ima huius diximus. </s> <s>Tantum aduer­<lb/>tendum e&longs;t, vt pro &longs;ectionibus, dicamus compo&longs;ita <lb/>ex binis &longs;ectionibus (quæ &longs;cilicet fiunt ab codem <lb/>plano, vel eadem recta linea) cum de prædicto com <lb/>po&longs;ito &longs;it &longs;ermo: & in demon&longs;tratione, procylin­<lb/>dris, vel cylindri portionibus, vel parallelogram­<lb/>mis, compo&longs;ita ex binis cylindris, vel cylindri por <lb/>tionibus, vel parallelogrammis(quæ &longs;cilicet &longs;unt <lb/>inter eadem plana parallela, vel lineas parallelas, <lb/>& circa eundem axim, vel diametrum totius vel <lb/>diametri, vel axis partem) &longs;icut & pro figura com­<lb/>po&longs;itum ex duabus dictis figuris: pro re&longs;iduo, com <lb/>po&longs;itum ex re&longs;iduis. </s> <s>Nam cum vtriu&longs;que re&longs;idui <pb/>figurarum duobus prædictis figuris vnum quid <lb/>componentibus, & circa eundem axim, vel diame<lb/>trum exi&longs;tentibus, qua ratione diximus, circum­<lb/>&longs;criptarum, centra grauitatis &longs;int in diametro, vel <lb/>axe; etiam compo&longs;iti ex ijs duobus re&longs;iduis (vt in <lb/>priori libro generaliter demon&longs;trauimus, cen­<lb/>trum grauitatis erit in eadem diametro, vel axe: <lb/>vnde vim habent proximæ quatuor anteceden­<lb/>tes demon&longs;trationes, exemplum erit in demon­<lb/>&longs;tratione trige&longs;imæ quartæ huius. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hemi&longs;phærij centrum grauitatis e&longs;t punctum <lb/>illud in quo axis &longs;ic diuiditur, vt pars, quæ ad ver­<lb/>ticem &longs;it ad reliquam vt quin que ad tria. </s></p><p type="main"> <s>E&longs;to hemifphærium ABC cuius vertex B, axis BD: <lb/>&longs;it autem BD &longs;ectus in G puncto, ita vt pars BG ad GD <lb/>&longs;it vt quinque ad tria. </s> <s>Dico G e&longs;se centrum grauitatis <lb/>hemi&longs;phærij ABC. <!-- KEEP S--></s> <s>Ab&longs;cindatur enim BK ip&longs;ius BD <lb/>pars quarta: & &longs;uper ba&longs;im eandem hemi&longs;phærij eundem­<lb/>que axim BD cylindrus AF con&longs;i&longs;tat, & conus intelli­<lb/>gatur EDF, cuius vertex D, ba&longs;is autem circulus circu­<lb/>lo AC oppo&longs;itus, cuius diameter EBF. <!-- KEEP S--></s> <s>Sectoque axe <lb/>BD bifariam in puncto H, & &longs;ingulis eius partibus rur­<lb/>&longs;us bifariam, quoad BD &longs;ecta &longs;it in partes æquales cu­<lb/>iu&longs;cumque libuerit numeri paris, tran&longs;eant per puncta &longs;e­<lb/>ctionum plana quædam ba&longs;i AC parallela, & &longs;ecantia, <lb/>hemi&longs;phærium, conum, & cylindrum, quorum omnes &longs;e­<lb/>ctiones erunt circuli, terni in codem plano ad aliam atque <pb/>aliam trium harum figurarum pertinentes. </s> <s>Quod &longs;i præ­<lb/>terea factæ &longs;ectiones hemi&longs;phærij ABC à cylindri AF <lb/>&longs;ectionibus, circuli à circulis concentricis auferri intelli­<lb/>gantur; reliquæ totidem erunt &longs;ectiones reliquæ figuræ &longs;o­<lb/>lidæ, dempto ABC hemi&longs;phærio ex toto AF cylin­<lb/>dro, circuli deficientes circulis concentricis, hoc e&longs;t prædi­<lb/>ctis ABC hemi&longs;phærij &longs;ectionibus prout inter &longs;e re&longs;pon­<lb/>dent. </s> <s>Nunc &longs;uper &longs;ectiones hemi&longs;phærij ABC, & co­<lb/>ni EDF cylindris con&longs;titutis circa axes, quæ &longs;unt &longs;eg­<lb/>menta æqualia axis BD, intelligantur duæ figuræ ex cy­<lb/>lindris æqualium altitudinum, altera in&longs;cripta hemi&longs;phæ­<lb/><figure id="id.043.01.144.1.jpg" xlink:href="043/01/144/1.jpg"/><lb/>rio ABC, altera cono EDF circum&longs;cripta. </s> <s>Si igitur <lb/>à toto AF cylindro auferatur figura, quæ in&longs;cripta e&longs;t <lb/>hemi&longs;phærio ABC, relinquetur figura quædam ex cylin­<lb/>dris circa prædictos axes, vt &longs;unt BK, KH, HL, LD, <lb/>deficientibus ijs cylindris, ex quibus con&longs;tat figura in&longs;cri­<lb/>pta hemi&longs;phærio ABC, & vno integro &longs;upiemo XF <lb/>cylindro, circum&longs;cripta re&longs;iduo AF cylindri dempto A <lb/>BC hemi&longs;phærio, circum&longs;criptione interna: talis autem <lb/>figuræ circum&longs;criptæ centrum grauitatis, per ea, quæ in <lb/>primo libro, erit in axe BD, quemadmodum & aliarum <lb/>duarum figurarum ex cylindris, quarum altera in&longs;cripta <lb/>e&longs;t hemi&longs;phærio ABC, altera cono EDF circum&longs;cripta. <!--neuer Satz--><pb/>Quoniam igitur quo exce&longs;su hemi&longs;phærium ABC &longs;u­<lb/>perat ex cylindris figuram &longs;ibi in&longs;criptam, eodem figura <lb/>circum&longs;cripta reliquo cylindri AF, dempto ABC he­<lb/>mi&longs;phærio, &longs;uperat ip&longs;um re&longs;iduum; figura autem in&longs;cripta <lb/>hemi&longs;phærio ABC pote&longs;t e&longs;&longs;e eiu&longs;modi, quæ ab hemi­<lb/>&longs;phærio deficiat minori defectu quantacumque magnitu­<lb/>dine propo&longs;ita; poterit figura, quæ prædicto re&longs;iduo cir­<lb/>cum&longs;cripta e&longs;t e&longs;&longs;e talis, quæ ip&longs;um re&longs;iduum &longs;uperet mi­<lb/>no i exce&longs;su quantacumque magnitudine propo&longs;ita. <lb/></s> <s>Ru &longs;us, quia quemadmodum cylindrus AN infimus de­<lb/>ficiens cylindro SR, æqualis e&longs;t cylindro TP, ex &longs;upe­<lb/><figure id="id.043.01.145.1.jpg" xlink:href="043/01/145/1.jpg"/><lb/>rioribus, ita vnu&longs;qui&longs;que aliorum cylindrorum deficien­<lb/>tium cylindris, qui &longs;unt in hemi&longs;phærio, ex quibus cylin­<lb/>dris deficientibus con&longs;tat dicto re&longs;iduo figura circum&longs;cri­<lb/>pta, æqualis e&longs;t cylindrorum circa conum EDF, ei, qui <lb/>cum ip&longs;o e&longs;t inter eadem plena parallela, & circa eundem <lb/>axem; erunt omnes cylindri circa conum EDF, in ea­<lb/>dem proportione cum prædictis cylindris deficientibus, <lb/>circa prædictum re&longs;iduum, &longs;i bini &longs;umantur inter eadem <lb/>plana parallela, & circa eundem axem. </s> <s>Quemadmodum <lb/>igitur omnium cylindrorum, qui circa conum EDF mi­<lb/>nor e&longs;t proportio primi ad verticem D, ad &longs;ecundum, <lb/>quàm &longs;ecundi ad tertium, & &longs;ecundi ad tertium, quàm ter-<pb/>tij ad quartum, & &longs;ic &longs;emper deinceps v&longs;que ad vltimum <lb/>XF (duplicatæ enim &longs;unt talium cylindrorum rationes <lb/>earum, quas inter &longs;e habent diametri æqualibus exce&longs;sibus <lb/>differentes circulorum, qui &longs;unt &longs;ectiones coni, & ba&longs;es cy­<lb/>lindrorum, ex quibus con&longs;tat figura cono EDF circum­<lb/>&longs;cripta, &longs;umpta progre&longs;&longs;ione proportionum eodem ordine <lb/>gradatim à minima diametro v&longs;que ad maximam EF) ita <lb/>erit cylindrorum deficientium, ex quibus con&longs;tat figura <lb/>circum&longs;cripta reliquo cylindri AF, dempto ABC hemi­<lb/>&longs;phærio, minimi, cuius axis DL ad &longs;ecundum minor pro­<lb/>portio, quàm &longs;ecundi ad tertium, & &longs;ic deinceps, v&longs;que ad <lb/><expan abbr="maximũ">maximum</expan> XF, communiter ad conum EDF, & prædictum <lb/>re&longs;iduum pertinentem, &longs;icut & eorum ba&longs;es circuli deficien <lb/>tes, quæ &longs;unt dicti re&longs;idui &longs;ectiones. </s> <s>Cum igitur tam maxi­<lb/>mi cylindri XF communis, quàm binorum quorumque reli <lb/>quorum cylindrorum circa conum EDF, & prædictum re&longs;i <lb/>duum inter eadem plana parallela con&longs;i&longs;tentium, quorum <lb/>axis communis in BD, commune centrum grauitatis in axe <lb/>BD exi&longs;tat, erit ex antecedenti punctum K, quod pono <lb/>centrum grauitatis coni EDF, idem re&longs;idui ex cylindro <lb/>AF, dempto ABC, hemi&longs;phærio centrum grauitatis. <lb/></s> <s>Quoniam igitur quarum partium e&longs;t octo axis BD talium <lb/>e&longs;t BG quinque, & BK duarum (ponimus enim nunc K <lb/>coni EDF centrum grauitatis) qualium e&longs;t BD octo, ta­<lb/>lium erit GK trium: &longs;ed KH e&longs;t æqualis BK; qualium <lb/>igitur partium e&longs;t GK trium, talium erit KH duarum, ta­<lb/>li&longs;que vna GH; dupla igitur KH ip&longs;ius GH: &longs;ed ABC <lb/>hemi&longs;phærium duplum e&longs;t prædicti re&longs;idui, cum &longs;it cylin­<lb/>dri AF, &longs;ub&longs;e&longs;quialterum; vt igitur e&longs;t <expan abbr="hemi&longs;phæriũ">hemi&longs;phærium</expan> ABC, <lb/>ad prædictum re&longs;iduum, ita ex contraria parte erit <expan abbr="lõgitudo">longitudo</expan> <lb/>KH, adlongitudinem GH: &longs;ed H e&longs;t centrum grauitatis <lb/>totius cylindri AF & K, prædicti re&longs;idui dempto ABC <lb/>hemi&longs;phærio; ergo ABC hemi&longs;phærij centrum grauitatis <lb/>erit G. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis minoris portionis &longs;phæræ centrum gra<lb/>uitatis e&longs;t in axe primum bifariam &longs;ecto: deinde <lb/>&longs;ecundum centrum grauitatis fru&longs;ti circa eun­<lb/>dem axim, ab&longs;ci&longs;&longs;i à cono verticem habente cen­<lb/>trum &longs;phæræ; in eo puncto, in quo dimidius axis <lb/>portionis ba&longs;im attingens &longs;ic diuiditur, vt pars <lb/>duabus prædictis &longs;ectionibus intercepta &longs;it ad <lb/>eam, quæ inter &longs;ecundam, & tertiam &longs;ectionem <lb/>interijcitur, vt exce&longs;&longs;us, quo tripla &longs;emidiametri <lb/>&longs;phæræ, cuius e&longs;t prædicta portio, &longs;uperattres de­<lb/>inceps proportionales, quarum maxima e&longs;t &longs;phæ­<lb/>ræ &longs;emidiameter, media autem, quæ inter centra <lb/>&longs;phæræ, & ba&longs;is portionis interijcitur; ad &longs;emi­<lb/>diametri &longs;phæræ triplam. </s></p><p type="main"> <s>Sit minor portio ABC, &longs;phæræ, cuius centrum D, <lb/>&longs;emidiameter BD, in qua axis portionis &longs;it BG, ba&longs;is <lb/>autem circulus, cuius diameter AC: & circa axim BD <lb/>de&longs;criptus e&longs;to conus HDF, cuius ba&longs;is circulus FH <lb/>tangens portionem in B puncto &longs;it æqualis circulo ma­<lb/>ximo, & fru&longs;tum coni HDF ab&longs;ci&longs;&longs;um vna cum portio­<lb/>ne ABC &longs;it KHFL, & vt BD ad DG, ita fiat DG <lb/>ad P: &longs;ectoque axe BG bifariam in puncto N, fiat vt <lb/>exce&longs;&longs;us, quo tripla ip&longs;ius BD &longs;uperat tres BD, DG, <lb/>P, tanquam vnam, ita NM, ad MNO. </s> <s>Dico portio­<lb/>nis ABC centrum grauitatis e&longs;se O. <!-- KEEP S--></s> <s>Nam circa axim <lb/>BG, &longs;uper ba&longs;im FH &longs;tet cylindrus EF, cuius cen-<pb/>trum grauitatis erit N, reliqui autem eius dempta <lb/>ABC portione centrum grauitatis M commune fru&longs;to <lb/>KLFH, vt colligitur ex demon&longs;tratione antecedentis. <lb/></s> <s>Quoniam igitur e&longs;t vt exce&longs;sus, quo tripla ip&longs;ius BD &longs;u­<lb/>perat tres BD, DG, P tanquam vnam, ad ip&longs;ius BD <lb/><figure id="id.043.01.148.1.jpg" xlink:href="043/01/148/1.jpg"/><lb/>triplam, hoc e&longs;t vt NM ad MO, ita portio ABC ad <lb/>EF cylindrum, & diuidendo vt MN ad NO, ita por­<lb/>tio ABC ad reliquum cylindri EF; & N e&longs;t cylindri <lb/>EF, & M prædicti re&longs;idui centrum grauitatis; erit reli­<lb/>quæ portionis ABC centrum grauitatis O. <!-- KEEP S--></s> <s>Quod de­<lb/>mon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla­<lb/>nis parallelis, altero per centrum acto, centrum <lb/>grauitatis e&longs;t in axe primum bifariam &longs;ecto: dein­<lb/>de &longs;umpta ad minorem ba&longs;im quarta parte axis <lb/>portionis; in eo puncto, in quo dimidius axis mi­<lb/>norem ba&longs;im attingens &longs;ic diuiditur, vt pars dua­<lb/>bus prædictis &longs;ectionibus intercepta &longs;it ad eam, <pb/>quæ inter&longs;ecundam, & vltimam &longs;ectionem inter­<lb/>ijcitur, vt exce&longs;&longs;us, quo maior extrema ad &longs;phæræ <lb/>&longs;emidiametrum, & axim portionis &longs;uperat ter­<lb/>tiam partem axis portionis; ad maiorem extre­<lb/>mam antedictam. </s></p><p type="main"> <s>Sit portio ABCD &longs;phæræ, cuius centrum F: axis au­<lb/>tem portionis &longs;it EF ab&longs;ci&longs;sæ duobus planis parallelis, <lb/>quorum alterum tran&longs;iens per punctum F faciat &longs;ectio­<lb/>num circulum maximum, cuius diameter AD, reliquam <lb/>autem &longs;ectionem minorem circulum, quæ minor ba&longs;is di­<lb/>citur, cuius di­<lb/>ameter BC: <lb/>& vt e&longs;t EF <lb/>ad AD, ita <lb/>fiat AD ad <lb/>OP, cuius P <lb/>R, &longs;it æqua­<lb/>lis tertiæ parti <lb/>axis EF. <!-- KEEP S--></s> <s>Et <lb/>&longs;ecta EF bi­<lb/><figure id="id.043.01.149.1.jpg" xlink:href="043/01/149/1.jpg"/><lb/>fariam in puncto M, & po&longs;ita EN ip&longs;ius EF quarta <lb/>parte, fiat vt RO ad OP, ita MN ad NL. </s> <s>Dico L e&longs;&longs;e <lb/>centrum grauitatis portionis ABCD. <!-- KEEP S--></s> <s>Nam circa axim <lb/>EF &longs;uper circulum maximum AD de&longs;cribatur cylindrus <lb/>AG, cuius centrum grauitatis erit M: reliqui autem ex <lb/>cylindro AG dempta ABCD portione centrum graui­<lb/>tatis N. <!-- KEEP S--></s> <s>Quoniam igitur e&longs;t vt RO ad OP, hoc e&longs;t vt <lb/>MN ad NL, ita portio ABCD ad reliquum cylindri <lb/>AG, & diuidendo vt NM ad ML, ita portio ABCD ad <lb/>reliquum cylindri AG: & cylindri AG e&longs;t N, prædicti au­<lb/>tem re&longs;idui centrum grauitatis M; erit reliquæ portionis <lb/>ABCD centrum grauitatis L. <!-- KEEP S--></s> <s>Quod <expan abbr="demon&longs;trandũ">demon&longs;trandum</expan> erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla­<lb/>nis parallelis neutro per centrum acto, nec cen­<lb/>trum intercipientibus, centrum grauitatis e&longs;t in <lb/>axe primum bifariam &longs;ecto: deinde &longs;ecundum <lb/>centrum grauitatis fru&longs;ti circa eundem axim, <lb/>ab&longs;ci&longs;&longs;i à cono verticem habente centrum &longs;phæ­<lb/>ræ; in eo puncto in quo dimidius axis maiorem <lb/>ba&longs;im attingens &longs;ic diuiditur, vt pars duabus præ­<lb/>dictis &longs;ectionibus finita &longs;it ad eam, quæ inter &longs;e­<lb/>cundam, & vltimam &longs;ectionem interijcitur, vt <lb/>exce&longs;&longs;us, quo maior extrema ad triplas & &longs;emidia <lb/>metri &longs;phæræ, & eius quæ inter centra &longs;phæræ, <lb/>& minorem ba&longs;im portionis interijcitur, &longs;uperat <lb/>tres deinceps proportionales, quarum maxima <lb/>e&longs;t, quæ inter centra &longs;phæræ, & minoris ba&longs;is, <lb/>media autem, quæ inter centra &longs;phæræ, & maio­<lb/>ris ba&longs;is portionis interijcitur; ad maiorem extre­<lb/>mam antedictam. </s></p><p type="main"> <s>Sit portio ABCD, &longs;phæræ, cuius centrum E, ab­<lb/>&longs;ci&longs;sa duobus planis parallelis, neutro per E tran&longs;eun­<lb/>te, nec E intercipientibus: axis autem portionis &longs;it GH, <lb/>maior ba&longs;is circulus, cuius diameter AD, minor cuius <lb/>diameter BC: producta autem GH v&longs;que in E intel­<lb/>ligatur coni KEN rectanguli, cuius axis EG, fru&longs;tum <pb/>KLMN ab&longs;ci&longs;&longs;um ij&longs;dem planis, quibus por­<lb/>tio, & &longs;phæræ &longs;emidiameter &longs;it EHGS: & po­<lb/>&longs;ita T tripla ip&longs;ius ES, & V ip&longs;ius EG tri­<lb/>pla, e&longs;to vt V ad T ita T ad XZ: & vt GE <lb/>ad EH ita EH ad <foreign lang="greek">w</foreign>, & &longs;it ZY, ip&longs;ius XZ, <lb/>æqualis tribus GE, EH, <foreign lang="greek">w</foreign>, vt &longs;it exce&longs;&longs;us <lb/>XY: & &longs;ecto axe GH bifariam in puncto I, in <lb/>linea GI, &longs;umatur O, centrum grauitatis fru­<lb/>&longs;ti KLMN: Et vt <foreign lang="greek">*u</foreign>X ad XZ, ita fiat IO <lb/>ad OIP. </s> <s>Dico portionis ABCD centrum <lb/>grauitatis e&longs;&longs;e P. <!-- KEEP S--></s> <s>Nam circa axim GH pla­<lb/>nis ba&longs;ium portionis interceptus &longs;tet cylin­<lb/>drus QR, cuius ba&longs;is &longs;it æqualis circulo ma­<lb/>ximo. </s> <s>Quoniam igitur e&longs;t vt YX ad XZ, <lb/>hoc e&longs;t vt IO ad OP, ita portio ABCD <lb/>ad cylindrum QR, & diuidendo vt OI ad <lb/>IP, ita portio ABCD ad reliquum cylindri <lb/>QR: & I e&longs;t cylindri QR, & O prædicti <lb/>re&longs;idui centrum grauitatis; erit reliquæ por­<lb/><figure id="id.043.01.151.1.jpg" xlink:href="043/01/151/1.jpg"/><lb/>tionis ABCD centrum grauitatis P. <!-- KEEP S--></s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>LEMMA.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s><emph type="italics"/>Sit data recta PO, & in ea punctum D, & punctum quod­<lb/>dam R in ip&longs;a DO, ita vt VD ip&longs;ius PD, ad DT ip&longs;ius DO, <lb/>&longs;it vt PD, ad DO: &longs;it autem maior proportio PS ad SO, quàm <lb/>VR, ad RT. </s> <s>Dico OS, minorem e&longs;&longs;e quàm OR.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Fiat enim vt PS, ad SO, ita VZ ad ZT; ma­<lb/>ìor igitur erit proportio VZ, ad ZT, quàm VR, ad <lb/>RT: & componendo maior proportio VT, ad TZ, <lb/><figure id="id.043.01.152.1.jpg" xlink:href="043/01/152/1.jpg"/><lb/>quàm VT, ad TR; minor igitur TZ, quàm TR, ide&longs;t <lb/>maior DZ, quàm DR. </s> <s>Rur&longs;us quia componendo e&longs;t <lb/>vt PO ad OS, ita VT ad TZ: &longs;ed vt DO ad OP, ita <lb/>e&longs;t DT ad TV; erit ex æquali, vt DO ad OS, ita DT, <lb/>ad TZ; & per conuer&longs;ionem rationis, vt OD ad DS, <lb/>ita TD ad DZ: & permutando, vt DO ad DT, ita DS <lb/>ad DZ: &longs;ed DO, e&longs;t maior quàm DT, ergo & DS, erit <lb/>maior quàm DZ: &longs;ed DZ maior erat quàm DR; multo <lb/>ergo DS maior quàm DR, vnde minor erit OS quàm <lb/>OR. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si datæ maiori &longs;phæræ portioni cylindrus cir­<lb/>cum&longs;cribatur circa eundem axim portionis, cen­<lb/>trum grauitatis reliquæ figuræ ex cylindro cir­<lb/>cum&longs;cripto ablata portione, propinquius erit ver­<lb/>tici portionis, quàm <expan abbr="c&etilde;trum">centrum</expan> grauitatis portionis. </s></p><pb/><p type="main"> <s>Sit &longs;phæræ cuius centrum D maior portio ABC, cu­<lb/>ius axis BE, ba&longs;is circulus cuius diameter AC, & por­<lb/>tioni ABC, cylindro XH circa axim BE circum&longs;cripto <lb/>vt &longs;upra fecimus: quoniam tam portionis ABC, quàm <lb/>cylindri XH, centrum grauitatis e&longs;t in axe BE; erit reli­<lb/>qui ex cylindro XH, in axe BE centrum grauitatis, &longs;int <lb/>in axe BE centra grauitatis Q portionis ABC & S præ­<lb/>dicti re&longs;idui. </s> <s>Dico e&longs;&longs;e punctum S vertici B propinquius <lb/><figure id="id.043.01.153.1.jpg" xlink:href="043/01/153/1.jpg"/><lb/>quàm punctum <expan abbr="q.">que</expan> Per centrum enim D tran&longs;iens planum <lb/>ad axim BE erectum &longs;ecet cylindrum XH, & portionem <lb/>ABC in duos cylindros <emph type="italics"/>K<emph.end type="italics"/>H, XL, & hemi&longs;phærium <lb/>KBL, & portionem AKLC, &longs;ectio autem circulus ma­<lb/>ximus e&longs;to ille cuius diameter KL: & duo coni rectan­<lb/>guli circa axes BD, DE, vertice D communi de&longs;cri­<lb/>bantur GDH, MDN, quorum alterius ba&longs;is GH com­<lb/>munis erit cylindro XH: alterius autem MDN, minor <lb/>quàm eiu&longs;dem cylindri XH, ba&longs;is GH. <!-- KEEP S--></s> <s>Denique &longs;ecta <pb/>BE bifariam in puncto R, &longs;ecentur BD, in puncto T, & <lb/>DE, in puncto V, bifariam & &longs;umatur BO, ip&longs;ius BD, <lb/>pars quarta, necnon EP pars quarta ip&longs;ius DE, primum <lb/>itaque quoniam ER e&longs;t maior, quàm ED, erit punctum <lb/>R, in &longs;egmento BD. <!-- KEEP S--></s> <s>Quoniam igitur ex &longs;upra o&longs;ten&longs;is O <lb/>e&longs;t centrum grauitatis commune cono DGH, & reliquo <lb/>cylindri KH dempto ABC hemi&longs;phærio: & eadem ra­<lb/>tione punctum P, cum &longs;it centrum grauitatis coni MDN, <lb/>erit idem centrum grauitatis reliqui ex cylindro XL dem­<lb/>pta AKLC portione: e&longs;t autem reliquum cylindri KH <lb/>dempto KBL hemi&longs;phærio, æquale cono DGH, qua <lb/>ratione & reliquum cylindri XL, dempta AKLC por­<lb/>tione æquale e&longs;t cono MDN; cum igitur S &longs;it centrum <lb/>grauitatis totius reliqui ex toto cylindro XH, dempta <lb/>ABC portione, erit idem S, centrum grauitatis compo­<lb/>&longs;iti ex conis GDH, MDL: &longs;unt autem horum conorum <lb/>centra grauitatis O, P; vt igitur conus GDH, ad co­<lb/>num MDN, ita erit PS, ad SO: &longs;ed coni GDH ad <lb/>&longs;imilem ip&longs;i conum MDN triplicata e&longs;t proportio axis <lb/>BD, ad axim BE, hoc e&longs;t cylindri KH ad cylindrum <lb/>XL; maior igitur proportio erit PS ad SO, quàm cy­<lb/>lindri KH ad cylindrum XL, &longs;ed vt cylindrus KH, ad <lb/>cylindrum XL, ita e&longs;t VR ad RT, ob centra grauiratis <lb/>V, R, T, maior igitur proportio erit PS ad SO, quàm <lb/>VR ad RT: &longs;ed eiu&longs;dem PO e&longs;t vt PD ad DO, ita <lb/>VD ad DT, ob &longs;ectiones axium proportionales; pun­<lb/>ctum igitur S propinquius e&longs;t puncto O, quàm punctum <lb/>R, per Lemma. </s> <s>Quare & Stermino B propinquius quàm <lb/>punctum R: &longs;ed R e&longs;t centrum grauitatis totius cylindri <lb/>XH: & S reliqui ex cylindro XH dempta ABC por­<lb/>tione; igitur Q reliquæ portionis ABC, centrum graui­<lb/>tatis erit in linea ER, atque ideo à puncto B remotius <lb/>quàm punctnm S. <!-- KEEP S--></s> <s>Quod e&longs;t propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>COROLLARIV M.<emph.end type="italics"/></s></p><p type="main"> <s>Manife&longs;tum e&longs;t autem ex demon&longs;tratione thelo­<lb/>rematis, omnis re&longs;idui ex cylindro datæ maiori <lb/>&longs;phæræ portioni circum&longs;cripto circa eundem <lb/>axim portionis, cuius ba&longs;is &longs;it æqualis circulo ma <lb/>ximo, centrum grauitatis e&longs;&longs;e in axe ab&longs;ci&longs;&longs;a pri­<lb/>mum quarta parte ad verticem portionis termina­<lb/>ta &longs;egmenti axis portionis, quod centro &longs;phæræ, <lb/>& vertice portionis, & quarta parte eius quod <lb/>centro &longs;phæræ, & ba&longs;i portionis terminatur; ad <lb/>ba&longs;im terminata in eo puncto, in quo &longs;egmentum <lb/>axis portionis duabus prædictis &longs;ectionibus fini­<lb/>tum &longs;ic diuiditur, vt &longs;egmentum propinquius ba&longs;i <lb/>&longs;it ad reliquum, vt cubus &longs;egmenti axis portionis <lb/>centro &longs;phæræ, & vertice portionis terminati ad <lb/>cubum reliqui quod ba&longs;im portionis tangit, &longs;i­<lb/>quidem cubi triplicatam inter &longs;e habent laterum <lb/>proportionem, &longs;imul illud manife&longs;tum e&longs;t, hoc <lb/>idem eadem ratione po&longs;&longs;e demon&longs;trari de centro <lb/>grauitatis reliqui ex cylindro dempta &longs;phæræ por­<lb/>tione ab&longs;ci&longs;&longs;a duobus planis paralìelis centrum <lb/>&longs;phæræ intercipientibus, ita vt axis portionis à <lb/>centro &longs;phæræ in partes inæquales diuidatur, cu­<lb/>ius cylindri circum&longs;cripti &longs;it idem axis, qui & por <lb/>tionis, ba&longs;is autem æqualis circulo maximo. </s> <s>Si­<lb/>militer enim de&longs;criptis duobus conis rectangulis<pb/>verticem habentibus communem centrum &longs;phæ­<lb/>ræ, ba&longs;es autem minores ba&longs;ibus oppo&longs;itis cylin­<lb/>dri circum&longs;cripti: æqualibus circulo maximo, &longs;u­<lb/>mentes pro vertice minorem ba&longs;im, pro ba&longs;i, ma­<lb/>iorem ba&longs;im portionis immotis reliquis propo&longs;i­<lb/>tum demon&longs;traremus. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis maioris portionis &longs;phæræ centrum gra<lb/>uitatis e&longs;t in axe primum bifariam &longs;ecto: Deinde <lb/>&longs;umpta ad verticem quarta parte &longs;egmenti axis, <lb/>quod centro &longs;phæræ, & portionis vertice finitur: <lb/>itemque ad ba&longs;im quarta parte reliqui &longs;egmenti <lb/>inter centrum &longs;phæræ, & ba&longs;im portionis interie­<lb/>cti. </s> <s>Deinde &longs;egmento axis, inter eas quartas par­<lb/>tes interiecto, ita diui&longs;o, vt pats propinquior ba&longs;i <lb/>&longs;it ad reliquam vt cubus &longs;egmenti axis, quod <lb/><expan abbr="c&etilde;tro">centro</expan> &longs;phæræ, & vertice portionis, ad cubum eius <lb/>quod centris &longs;phæræ, & ba&longs;is portionis termina­<lb/>tur; in eo puncto, in quo &longs;egmentum axis centro <lb/>&longs;phæræ, & &longs;ectione penultima finitum &longs;ic diuidi­<lb/>tur, vt pars prima & penultima &longs;ectione termina­<lb/>ta &longs;it ad totam vltima & penultima &longs;ectione termi <lb/>natam, vt exce&longs;&longs;us, quo &longs;egmentum axis portionis <lb/>inter centrum, & ba&longs;im portionis interiectum &longs;u­<lb/>perat tertiam partem minoris extremæ maiori po <lb/>&longs;ita dicto axis &longs;egmento in proportione &longs;emidia-<pb/>metri &longs;phæræ ad prædictum &longs;egmentum, vnà cum <lb/>&longs;ub&longs;e&longs;quialtera reliqui &longs;egmenti, ad axim por­<lb/>tionis. </s></p><p type="main"> <s>Sit maior portio ABC &longs;phæræ, cuius centrum D, dia­<lb/>meter KH, axis autem portionis &longs;it BE, ba&longs;is circulus, <lb/>cuius diameter AC, & &longs;it axis BE primum bifariam &longs;e­<lb/>ctus in puncto G: &longs;umptaque ip&longs;ius BD, quarta parte <lb/>BP, itemque ip&longs;ius DE quarta parte EN, &longs;ecetur inter­<lb/>iecta PN, ita in puncto F, vt NF, ad FP, &longs;it vt cubus ex <lb/>BD ad cubum ex DE; punctum igitur F, ex præcedenti <lb/><figure id="id.043.01.157.1.jpg" xlink:href="043/01/157/1.jpg"/><lb/>corollario erit centrum grauitatis reliqui ex cylindro LM <lb/>portioni ABC, vt in antecedenti circum&longs;cripto. </s> <s>Quo­<lb/>niam igitur & prædicti re&longs;idui, ex antecedenti, & cylindri <lb/>LM, centra grauitatis &longs;unt in axe BE, erit & portionis <lb/>ABC in axe BE centrum grauitatis, quod &longs;it S: manife­<lb/>&longs;tum e&longs;t igitur punctum S, cadere &longs;upra centrum D, in li­<lb/>nea BD, minori ablata &longs;phæræ portione, cuius ba&longs;is cir-<pb/>culus AC: centrum autem F propinquius e&longs;&longs;e puncto B, <lb/>quàm centrum S, con&longs;tat ex præcedenti: quare centrum <lb/>G, totius cylindri LM inter puncta F, S cadet. </s> <s>Dico <lb/>GF ad FS e&longs;&longs;e vt exce&longs;&longs;us, quo recta DE &longs;uperat tertiam <lb/>partem minoris extremæ maiori po&longs;ita ip&longs;a DE in propor<lb/>tione continua ip&longs;ius DH ad DE vnà cum &longs;ub&longs;e&longs;quial­<lb/>tera ip&longs;ius BD, ad axim BE, ita GF ad FS. <!-- KEEP S--></s> <s>Quoniam <lb/>enim portio ABC ad cylindrum LM e&longs;t vt prædictus ex­<lb/>ce&longs;&longs;us vnà cum &longs;ub&longs;e&longs;quialtera ip&longs;ius BD ad axim BE: <lb/>& vt portio ABC ad LM cylindrum, ita e&longs;t GF ad FS, <lb/>ob centra grauitatis F, G; erit vt prædictus exce&longs;&longs;us vna <lb/>cum &longs;ub&longs;e&longs;quialtera ip&longs;ius BD ad axim BE, ita GF ad <lb/>FS. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla­<lb/>nis parallelis centrum intercipientibus, & à cen­<lb/>tro æqualiter di&longs;tantibus, centrum grauitatis e&longs;t <lb/>in medio axis, vel idem, quod centrum &longs;phæræ. </s></p><p type="main"> <s>Sit portio ABCD, &longs;phæræ, cuius centrum G, ab&longs;ci&longs;sa <lb/>duobus planis parallelis <lb/>centrum G intercipien­<lb/>tibus, & æquè ab eo di­<lb/>&longs;tantibus: &longs;ectiones <expan abbr="erũt">erunt</expan> <lb/>circuli minores, quorum <lb/>diametri &longs;int AD, BC <lb/>centra autem F,E, qui­<lb/>bus axis portionis termi <lb/>nabitur, eritque ad pla­<lb/>na vtriu&longs;que circuli per <lb/><figure id="id.043.01.158.1.jpg" xlink:href="043/01/158/1.jpg"/><lb/>pendicularis tran&longs;iens per centrum G: & quia illa plana <pb/>à centro G, æquè di&longs;tant, erit EG, æqualis GF. <!-- KEEP S--></s> <s>Dico <lb/>portionis ABCD centrum grauitatis e&longs;&longs;e G. <!-- KEEP S--></s> <s>De&longs;cripta <lb/>enim figura, vt &longs;upra fecimus, intelligantur duo coni re­<lb/>ctanguli GNO, GPQ, vertice G, communi, axibus <lb/>autem eorum EG, GF: & cylindrus LM, portioni cir­<lb/>cum&longs;criptus circa eun­<lb/>dem axim EF, cuius ba <lb/>&longs;is æqualis e&longs;t circulo <lb/>maximo: & &longs;umatur EH <lb/>ip&longs;ius EG, pars quar­<lb/>ta, itemque FK, pars <lb/>quarta ip&longs;ius FG. <!-- KEEP S--></s> <s>Quo­<lb/>niam igitur conorum G <lb/>NO, PGO, axes FG, <lb/>GH, &longs;unt æquales, re­<lb/>liquæ KG, GH, æqua <lb/><figure id="id.043.01.159.1.jpg" xlink:href="043/01/159/1.jpg"/><lb/>les erunt; centra autem grauitatis conorum &longs;unt K, H; pun­<lb/>ctum igitur G e&longs;t centrum grauitatis compo&longs;iti ex duobus <lb/>conis æqualibus GNO, GPQ, hoc e&longs;t reliqui ex cylin­<lb/>dro LM, dempta ABCD, portione, ex ante demon&longs;tra­<lb/>tis: &longs;ed idem G e&longs;t centrum grauitatis totius cylindri LM; <lb/>reliquæ igitur ABCD, portionis centrum grauitatis erit <lb/>G. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XL.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla­<lb/>nis parallelis centrum intercipientibus, & à cen­<lb/>tro non æqualiter di&longs;tantibus centrum grauitatis <lb/>e&longs;t in axe primum bifariam &longs;ecto: Deinde &longs;umpta <lb/>ad minorem ba&longs;im portionis quarta parte &longs;egmen <lb/>ti axis, quod minorem ba&longs;im attingit: & ad maio-<pb/>rem ba&longs;im quarta parte reliqui &longs;egmenti axis eo­<lb/>rum, quæ à centro &longs;phæræ fiunt: Deinde recta <lb/>inter has quartas partes interiecta ita diui&longs;a, vt <lb/>pars maiori ba&longs;i propinquior &longs;it ad reliquam vt <lb/>cubus &longs;egmenti axis inter &longs;phæræ centrum, & mi­<lb/>norem ba&longs;im, ad cubum eius, quod inter &longs;phæræ <lb/>centrum, & maiorem ba&longs;im portionis interijci­<lb/>tur; in eo puncto, in quo &longs;egmentum axis centro <lb/>&longs;phæræ, & penultima &longs;ectione terminatum &longs;ic di­<lb/>uiditur, vt pars quæ penultima, & prima &longs;ectione <lb/>terminatur &longs;it ad totam vltima, & penultima &longs;e­<lb/>ctione terminatam, vt ad axim portionis e&longs;t exce&longs; <lb/>&longs;us, quo idem axis portionis &longs;uperat <expan abbr="tertiã">tertiam</expan> partem <lb/>compo&longs;itæ ex duabus minoribus extremis, maio­<lb/>ribus po&longs;itis duobus axis &longs;egmentis, quæ fiunt à <lb/>centro &longs;phæræ in rationibus &longs;emidiametri &longs;phæ­<lb/>ræ ad prædicta &longs;egmenta. </s></p><figure id="id.043.01.160.1.jpg" xlink:href="043/01/160/1.jpg"/><p type="main"> <s>Sit portio ABCD &longs;phæræ, cuius centrum G, abci&longs;&longs;a <lb/>duobus planis parallelis centrum G intercipien<gap/>ibus, & <pb/>ab eo non æqualiter di&longs;tantibus: & axis portionis &longs;it EF, <lb/>qui per centrum G tran&longs;ibit, vtpote parallelorum circu­<lb/>lorum centra iungens: cumque eorum vtrumque &longs;it à cen­<lb/>tro non æqualiter di&longs;tantium perpendicularis, erunt eius <lb/>&longs;egmenta EG, GF, inæqualia. </s> <s>E&longs;to EG, maius: &longs;ectoque <lb/>axe EF bifariam in puncto P, &longs;umptisque ip&longs;arum EG, <lb/>GF, quartis partibus EH, FK, &longs;ecetur interiecta <emph type="italics"/>K<emph.end type="italics"/>H, <lb/>in puncto Q, ita vt KQ, ad QH, &longs;it vt cubus ex EG, <lb/>ad cubum ex GF, & portionis ABCD, &longs;it centrum gra<lb/>uitatis R: quod quidem cum punctis P, Q, e&longs;&longs;e in axe <lb/><figure id="id.043.01.161.1.jpg" xlink:href="043/01/161/1.jpg"/><lb/>EF: & cylindro LM, &longs;uper ba&longs;im æqualem circulo ma­<lb/>ximo circa axim EF, portioni circum&longs;cripto, reliqui eius <lb/>dempta ABCD, portione centrum grauitatis e&longs;se Q, & <lb/>propinquius E puncto, quàm centrum grauitatis R por­<lb/>tionis ABCD, manife&longs;tum e&longs;t ex &longs;upra demon&longs;tratis de <lb/>maioris portionis &longs;phæræ centro grauitatis: portionis autem <lb/>ABCD centrum grauitatis R e&longs;se in &longs;egmento EG &longs;e­<lb/>quitur ex antecedente. </s> <s>Dico PQ ad QR e&longs;se vt ad axim <lb/>EF exce&longs;sus, quo axis EF &longs;uperat tertiam partem com­<lb/>po&longs;itæ <gap/> duabus minoribus extremis altera re&longs;pondente <lb/>maiori extrema EG in proportione continua ip&longs;ius NG <pb/>ad GE, altera maiori extremæ FG in proportione con­<lb/>tinua ip&longs;ius NG ad GF. <!-- KEEP S--></s> <s>Quoniam enim ob centra gra<lb/>uitatis QPR e&longs;t vt QP ad PR, ita portio ABCD ad <lb/>reliquum cylindri LM, erit componendo, & per conuer­<lb/>&longs;ionem rationis, & conuertendo, vt PQ ad QR, ita por­<lb/>tio ABCD ad LM cylindrum: &longs;ed portio ABCD ad <lb/>LM cylindrum e&longs;t vt prædictus exce&longs;&longs;us ad axim EF; <lb/>vtigitur prædictus exce&longs;&longs;us ad axim EF, ita e&longs;t PQ ad <lb/>QR. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XLI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis conoidis parabolici centrum grauita­<lb/>tis e&longs;t punctum illud, in quo axis &longs;ic diuiditur vt <lb/>pars, quæ e&longs;t ad verticem &longs;it dupla reliquæ. </s></p><p type="main"> <s>Sit conoides parabolicum ABC, cuius vertex B, axis <lb/>autem BD &longs;ectus in puncto E ita vt EB &longs;it ip&longs;ius ED <lb/>dupla. </s> <s>Dico E e&longs;se centrum grauitatis conoidis ABC. <lb/><!-- KEEP S--></s> <s>Nam in &longs;ectione per <lb/>axim parabola ABC, <lb/>cuius diameter erit B <lb/>D, de&longs;cribatur rian­<lb/>gulum ABC; &longs;um­<lb/>ptisque ip&longs;ius BD æ­<lb/>qualibus DH, HO, <lb/>per puncta H, O, &longs;e­<lb/>centur vnà parabola <lb/>& triangulum ABC <lb/>duabus rectis FGH <lb/><figure id="id.043.01.162.1.jpg" xlink:href="043/01/162/1.jpg"/><lb/>KL, MNOPQ: & per eas rectas &longs;ecetur conoi­<lb/>des ABC planis ba&longs;i parallelis, factæ autem &longs;e­<lb/>ctiones erunt circuli circa FL, MQ, & in parabola <pb/>ABC tres ad diametrum ordinatim applicatæ AD, <lb/>FH, MO. </s> <s>Quoniam igitur tres rectæ OB, BH, BD <lb/>&longs;e&longs;e qualiter excedunt, quarum minima BO, maxi­<lb/>ma e&longs;t BD, minor erit proportio BO ad BH, quàm <lb/>BH ad BD; hoc e&longs;t NP ad GK, quàm GKad AC. <lb/>&longs;ed vt OB ad BH hoc e&longs;t NO ad GH, vel NP ad <lb/>GK ita e&longs;t quadra­<lb/>tum MO ad quadra­<lb/>tum FH, hoc e&longs;t eo­<lb/>no dis &longs;ectionum cir­<lb/>culus MQ ad circu­<lb/>lum FL: eademque <lb/>ratione vt GK ad <lb/>AC ita circulus FL <lb/>ad circulum AC; mi<lb/>nor igitur proportio <lb/>erit circuli MQ ad <lb/>circulum FL quàm <lb/><figure id="id.043.01.163.1.jpg" xlink:href="043/01/163/1.jpg"/><lb/>circuli FL ad circulum AC. <!-- KEEP S--></s> <s>Similiter autem o&longs;tende­<lb/>remus ternas quaslibet alias ita factas &longs;ectiones trianguli, <lb/>& parabolæ ABC inter &longs;e & ba&longs;i parallelas proportio­<lb/>nales e&longs;se, & minorem proportionem vtrobique minimæ <lb/>ad mediam, quàm mediæ ad maximam. </s> <s>Sed E e&longs;t cen­<lb/>trum grauitatis trianguli ABC, igitur per vige&longs;imamter­<lb/>tiam huius centrum grauitatis conoidis ABC erit idem E. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat, </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XLII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti conoidis parabolici centrum gra<lb/>uitatis axim ita diuidit, vt pars, quæ minorem <lb/>ba&longs;im attingit &longs;it ad reliquam; vt duplum maioris <pb/>ba&longs;is vnà cum minori, ad duplum minoris, vnà <lb/>cum maiori. </s></p><p type="main"> <s>Sit conoidis parabolici ABC, cuius axis BD fru&longs;tum <lb/>AEFC, eius maior ba&longs;is circulus, cuius diameter AC, mi­<lb/>nor, cuius diameter EF: in eadem parabola per axem, axis <lb/><expan abbr="aut&etilde;">autem</expan> DG, in quo fru&longs;ti AEFC &longs;it centrum grauitatis H. <lb/><!-- KEEP S--></s> <s>Dico e&longs;&longs;e vt duplum circuli AC, vnà cum circulo EF, ad <lb/>duplum circuli EF vna cum circulo AC, ita GH, ad HD. <lb/><expan abbr="Iungãtur">Iungantur</expan> enim re­<lb/>ctæ AKB, BLC. <lb/></s> <s>Quoniam igitur <lb/>qua ratione o&longs;ten <lb/>dimus conoides, <lb/>& triangulum A <lb/>BC, commune <lb/>habere in linea <lb/>BD centrum gra<lb/>uitatis, <expan abbr="ead&etilde;">eadem</expan> pror­<lb/>&longs;us remanet de­<lb/>mon&longs;tratum, fru&longs;ti <lb/><figure id="id.043.01.164.1.jpg" xlink:href="043/01/164/1.jpg"/><lb/>AEFC <expan abbr="centrũ">centrum</expan> grauitatis H, idem e&longs;se quod trapezij AK <lb/>FC; erit duarum parallelarum AG, KL vt dupla ip&longs;ius <lb/>AC, vnà cum KL, ad duplam ip&longs;ius KL, vnà cum AC <lb/>ita GH ad HD: &longs;ecat enim DG ip&longs;as AC, KL bifa­<lb/>riam. </s> <s>Sed vt AC ad <emph type="italics"/>K<emph.end type="italics"/>L ita e&longs;t circulus AC ad circu­<lb/>lum EF, ex demon&longs;tratione antecedentis, hoc e&longs;t vt dupla <lb/>ip&longs;ius AC vnà cum KL ad duplam ip&longs;ius KL vnà cum <lb/>AC, ita duplum circuli AC vna cum circulo KL ad du­<lb/>plum circuli KL vnà cum circulo AC; vt igitur e&longs;t du­<lb/>plum circuli AC, vnà cum circulo EF, ad duplum circu­<lb/>li EF, vnà cum circulo AC; ita erit GH ad HD. <lb/></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XLIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis conoidis hyperbolici centrum grauita­<lb/>tis e&longs;t punctum illud, in quo duodecima pars axis <lb/>ordine quarta ab ea, quæ ba&longs;im attingit, &longs;ic diui­<lb/>ditur, vt pars ba&longs;i propinquior &longs;it ad reliquam, vt <lb/>&longs;e&longs;quialtera tran&longs;uer&longs;i lateris hyperboles, quæ <lb/>conoides de&longs;cribit ad axim conoidis. </s></p><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius vertex B, axis <lb/>autem BD, qui etiam erit diameter hyperboles, quæ co­<lb/>noides de&longs;crip&longs;it, ad quam rectæ ordinatim applicantur: <lb/>eiu&longs;dem autem hyperboles tran&longs;uer&longs;um latus &longs;it EB, cu­<lb/>ius &longs;it &longs;e&longs;quialtera BEI, & &longs;umpta DQ quarta parte <lb/>axis BD, & DG, eiu&longs;dem tertia, qua ratione erit FG <lb/>duodecima pars axis BD, & ordine quarta ab ea cuius <lb/>terminus D, fiat vt IB, ad BD, ita QH, ad HG. <lb/><!-- KEEP S--></s> <s>Dico conoidis ABC, centrum grauitatis e&longs;&longs;e H. <!-- KEEP S--></s> <s>Sumpto <lb/>enim in linea AD quolibet puncto M, vt e&longs;t EB ad <lb/>BD longitudine, ita fiat MD, ad DK ip&longs;ius AD po­<lb/>tentia: & ab&longs;cindatur DN, æqualis DM, & DL æqua­<lb/>lis DK; &longs;iue autem &longs;it DK minor, quàm DM, &longs;iue ma­<lb/>ior, &longs;iue eadem illi; omnibus ca&longs;ibus communis erit demon <lb/>&longs;tratio. </s> <s>At per puncta M, N, vertice B, circa diametrum <lb/>BD, de&longs;cribatur parabola MBN, & triangulum KBL. <lb/><!-- KEEP S--></s> <s>Manente igitur BD, & circumductis figuris MBN, <lb/>KBL, de&longs;cribantur conoides parabolicum MBN, & <lb/>conus KBL, quorum communis axis erit BD, ba&longs;es <lb/>autem circuli, quorum diametri KL, MN, in eodem <lb/>plano cum ba&longs;e conoidis ABC. <!-- KEEP S--></s> <s>Rur&longs;us &longs;ecto axe BD <lb/>bifariam, & &longs;ingulis eius partibus &longs;emper bifariam in qua-<pb/>cumque multiplicatione; &longs;int duæ partes æquales proximæ <lb/>ba&longs;i DF, FQ: & per puncta FQ duo plana ba&longs;ium pla­<lb/>no parallela tres prædictas figuras &longs;olidas &longs;ecare intelli­<lb/>gantur: &longs;ecabunt autem & tres figuras per axim, eruntque <lb/>&longs;ectiones rectæ lineæ ad diametrum figurarum ordinatim <lb/>applicatæ propter <lb/>plana &longs;ecantia pa <lb/>rallela: trium au­<lb/>tem &longs;olidorum &longs;e <lb/>ctiones & ba&longs;es <lb/>omnes circuli, ter <lb/>ni in &longs;ingulis pla­<lb/>nis: ac primi qui­<lb/>dem ordinis &longs;int <lb/>ij, quorum diame­<lb/>tri &longs;unt ba&longs;es <expan abbr="triũ">trium</expan> <lb/><expan abbr="figurarũ">figurarum</expan> per axim, <lb/>trianguli &longs;cilicet, <lb/>parabolæ, & hy­<lb/>perboles, quæ præ <lb/>dictas figuras &longs;oli <lb/>das de&longs;cribunt, re <lb/>ctæ lineæ AC, <lb/>MN, KL. <!-- KEEP S--></s> <s>Se­<lb/>cundi verò reten­<lb/>to eodem ordine <lb/><expan abbr="figurarũ">figurarum</expan> tres <foreign lang="greek">az, <lb/>be, gd. </foreign></s> <s>Tertij <lb/>denique ordinis <lb/>SZ, TY, VX. <lb/><figure id="id.043.01.166.1.jpg" xlink:href="043/01/166/1.jpg"/><lb/>Quoniam igitur e&longs;t vt EB, ad BD, ità quadratum MD, <lb/>ad quadratum DK, ide&longs;t conus MBN, &longs;i de&longs;cribatur eo­<lb/>dem vertice B, ad conum KBL. <!-- KEEP S--></s> <s>Et vt IB, ad BE, ità e&longs;t <lb/>conoides MBN, ad conum MBN, in proportione &longs;cili-<pb/>cet &longs;e&longs;quialtera; ex æquali erit vt IB, ad BD, itì conoi­<lb/>des MBN ad conum KBL: Sed vt IB, ad BD, ità <lb/>ponitur QH ad HG; vt igitur conoides MBN, ad co­<lb/>num KBL, ità e&longs;t QH ad HG. <!-- KEEP S--></s> <s>Sed Q e&longs;t centrum <lb/>grauitatis coni KBL, & G conoidis MBN; compo&longs;i­<lb/>ti igitur ex conoi­<lb/>de MBN, & co­<lb/>no KBL <expan abbr="centrũ">centrum</expan> <lb/>grauitatis erit H. <lb/><!-- KEEP S--></s> <s>Rur&longs;us quoniam <lb/>tres rectæ lineæ B <lb/>D, BF, BQ, æ­<lb/>qualibus exce&longs;&longs;i­<lb/>bus inter &longs;e diffe­<lb/>runt, minor erit <lb/>proportio BQ, ad <lb/>BF, quàm BF, <lb/>ad BD, hoc e&longs;t <lb/>rectanguli EBQ, <lb/>ad rectangulum <lb/>EBF, quàm re­<lb/>ctanguli EBF, ad <lb/>rectangulum EB <lb/>D. <!-- KEEP S--></s> <s>Sed quadrati <lb/>BQ, ad quadra­<lb/>tum BF, dupli­<lb/>cata e&longs;t proportio <lb/>lateris BQ ad la­<lb/>tus BF: hoc e&longs;t <lb/>rectanguli EBQ <lb/><figure id="id.043.01.167.1.jpg" xlink:href="043/01/167/1.jpg"/><lb/>ad rectangulum EBF: & quadrati BF, ad quadratum <lb/>BD duplicata eius, quæ e&longs;t rectanguli EBF, ad rectan­<lb/>gulum EBD; compo&longs;itis igitur primis cum &longs;ecundis, mi­<lb/>nor erit proportio rectanguli BQE, ad rectangulum BFE, <pb/>quàm rectanguli BFE, ad rectangulum BDE. <!-- KEEP S--></s> <s>Sed vt <lb/>rectangulum BQE ad rectangulum BFE, ita e&longs;t quadra­<lb/>tum SQ ad quadratum <foreign lang="greek">a</foreign>F: & vt rectangulum BFE <lb/>ad rectangulum BDE, ita quadratum <foreign lang="greek">a</foreign>F, ad quadra­<lb/>tum AD; minor igitur proportio erit quadrati SQ, ad <lb/>quadratum <foreign lang="greek">a</foreign>F, quàm quadrati <foreign lang="greek">a</foreign>F ad quadratum AD. <lb/><!-- KEEP S--></s> <s>Sed vt quadratum SQ ad quadratum <foreign lang="greek">a</foreign>F, ita e&longs;t qua­<lb/>dratum SZ ad quadratum <foreign lang="greek">a</foreign><37>: & vt quadratum <foreign lang="greek">a</foreign>F ad <lb/>quadratum AD ita quadratum <foreign lang="greek">az</foreign> ad quadratum <lb/>AC; minor igitur proportio erit quadrati SZ ad quadra­<lb/>tum <foreign lang="greek">az</foreign>, quàm quadrati <foreign lang="greek">az</foreign>, ad quadratum AC, hoc e&longs;t <lb/>circuli SZ ad circulum <foreign lang="greek">a</foreign><37>, quàm circuli <foreign lang="greek">a</foreign><37>, ad cir­<lb/>culum AC; qui circuli &longs;unt &longs;ectiones conoidis ABC <lb/>po&longs;iti vt in propo&longs;itionibus lemmaticis dicebamus. </s> <s>Rur&longs;us <lb/>quoniam &longs;unt quatuor primæ proportionales; vt rectangu­<lb/>lum DBE ad rectangulum FBE, ita MD quadratum <lb/>ad quadratum <foreign lang="greek">b</foreign>F: & totidem &longs;ecundæ, vt quadratum <lb/>BD, ad quadratum BF, ita quadratum DK, ad quadra­<lb/>tum F<foreign lang="greek">g</foreign>, ob &longs;imilium triangulorum latera proportionalia: <lb/>&longs;ed vt EB, ad BD, hoc e&longs;t rectangulum DBE prima in <lb/>primis ad quadratum BD primam in &longs;ecundis, ita e&longs;t <lb/>quadratum MD tertia in primis ad quadratum DK ter­<lb/>tiam in &longs;ecundis; vt igitur compo&longs;ita ex primis ad com­<lb/>po&longs;itam ex &longs;ecundis, ità erit compo&longs;ita ex tertijs ad com­<lb/>po&longs;itam ex quartis; videlicet vt rectangulum DBE <lb/>vnà cum quadrato BD, hoc e&longs;t rectangulum BDE <lb/>ad rectangulum BFE, hoc e&longs;t vt quadratum AD, ad <lb/>quadratum <foreign lang="greek">a</foreign>F, ità compo&longs;itum ex quadratis MD, DK, <lb/>ad compo&longs;itum ex quadratis <foreign lang="greek">b</foreign>F, F<foreign lang="greek">g</foreign>: & quadrupla vtro­<lb/>rumque, vt quadratum AC, ad quadratum <foreign lang="greek">a</foreign><37>, ità com­<lb/>po&longs;itum ex quadratis MN, KL, ad compo&longs;itum ex qua­<lb/>dratis <foreign lang="greek">be, gd</foreign>; hoc e&longs;t eorum circulorum, qui &longs;unt &longs;ectio­<lb/>nes &longs;olidorum, vt circulus AC, ad circulum <foreign lang="greek">a</foreign><37>, ità com­<lb/>po&longs;itum ex circulis MN, KL, ad compo&longs;itum ex circu­<pb/>lis <foreign lang="greek">be, gd. </foreign></s> <s>Eadem ratione erit vt circulus AC, ad cir­<lb/>culum SZ, ità compo&longs;itum ex circulis MN, KL, ad <lb/>compo&longs;itum ex circulis TY, VX: & conuertendo, & ex <lb/>æquali, vt circulus SZ, ad circulum <foreign lang="greek">a</foreign><37>, ità compo&longs;itum <lb/>ex circulis TY, VX, ad compo&longs;itum ex circulis <foreign lang="greek">be, gd</foreign>: <lb/>& vt circulus <foreign lang="greek">a</foreign><37>, <lb/>ad circulum AC, <lb/>ità <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex <lb/>circulis <foreign lang="greek">be, gd</foreign>, <lb/>ad <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex <lb/>circulis MN, <emph type="italics"/>K<emph.end type="italics"/><lb/>L. <!-- KEEP S--></s> <s>Sunt igitur tria <lb/>compo&longs;ita ex bi­<lb/>nis &longs;ectionibus cir <lb/>culis, & totidem <lb/>alij circuli, quos <lb/>diximus in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione, &longs;i bi­<lb/>na <expan abbr="&longs;umãtur">&longs;umantur</expan> in &longs;in <lb/>gulis planis &longs;ecan <lb/>tibus: eorum au­<lb/>tem minor erat <lb/>proportio circuli <lb/>SZ ad circulum <lb/><foreign lang="greek">a</foreign><37>, quàm circuli <lb/><foreign lang="greek">a</foreign><37>, ad circulum <lb/>AC; minor igitur <lb/>proportio erit <expan abbr="cõ-po&longs;iti">con­<lb/>po&longs;iti</expan> ex circulis <lb/>T<foreign lang="greek">*u</foreign>, VX, ad <expan abbr="cõ-po&longs;itum">con­<lb/>po&longs;itum</expan> ex circu­<lb/><figure id="id.043.01.169.1.jpg" xlink:href="043/01/169/1.jpg"/><lb/>lis <foreign lang="greek">be, gd</foreign>, quàm compo&longs;iti ex circulis <foreign lang="greek">be, gd</foreign>, ad com <lb/>po&longs;itum ex circulis MN, KL. <!-- KEEP S--></s> <s>Hac eadem ratione ad verti­<lb/>cem deinceps progredienti manife&longs;tum erit, omnium com-<pb/>po&longs;itorum ex binis &longs;ectionibus nempe circulis, quorum al­<lb/>ter ad conum KBL pertinet, alter ad conoides MBN, in <lb/>eodem plano &longs;ecante prædictorum inter &longs;e parallelorum <lb/>exi&longs;tentibus, minorem e&longs;&longs;e proportionem incipienti ab eo, <lb/>quod e&longs;t proximum vertici, primi ad &longs;ecundum, quàm &longs;e­<lb/>cundi ad tertium, & &longs;ecundi ad tertium, quàm tertij ad <lb/>quartum, & &longs;ic &longs;emper deinceps v&longs;que ad maximum & vl­<lb/>timum compo&longs;itum ex circulis MN, KL: & eandem di­<lb/>ctas &longs;ectiones compo&longs;itas ex coni, & conoidis parabolici <lb/>&longs;ectionibus inter &longs;e habere proportionem, quàm habent in­<lb/>ter &longs;e circuli &longs;ectiones conoidis ABC, pro vt illis in <lb/>ij&longs;dem planis &longs;ecantibus, & æqualia axis BD &longs;egmenta <lb/>intercipientibus re&longs;pondent: Igitur per trige&longs;imam &longs;ecun­<lb/>dam huius, & &longs;equens eam Corollarium, conoides ABC, <lb/>& compo&longs;itum ex conoide MBN, & cono BKL, com­<lb/>mune habebunt in axe BD centrum grauitatis. </s> <s>Sed H <lb/>erat huius compo&longs;iti centrum grauitatis; Igitur conoidis <lb/>ABC centrum grauitatis erit idem H. <!-- KEEP S--></s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIV M.<emph.end type="italics"/></s></p><p type="main"> <s>Eadem demon&longs;tratione con&longs;tat &longs;i prædicta tria <lb/>&longs;olida ita vt diximus di&longs;po&longs;ita &longs;ecentur plano ba­<lb/>&longs;ibus parallelo; &longs;ru&longs;tum conoidis hyperbolici, & <lb/>compo&longs;itum ex fru&longs;tis coni, & conoidis paraboli­<lb/>ci, commune habere in communi axe centrum <lb/>grauitatis. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XLIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si conus & conoides parabolicum circa eun­<lb/>dem axim &longs;ecentur plano ba&longs;i parallelo; fru&longs;ti co­<lb/>nici ab&longs;ci&longs;&longs;i maiori ba&longs;i propinquius erit quàm <lb/>parabolici centrum grauitatis. </s></p><p type="main"> <s>Sint conus ABC, & conoides parabolicum EBF, <lb/>quorum communis <lb/>axis BD, cuius per <lb/>quoduis punctum M, <lb/>planum &longs;ecans ea cor <lb/>pora plano ba&longs;ium, <lb/>quarum diametri A <lb/>C, EF, parallelo ab­<lb/>&longs;cindat fru&longs;ta AKL <lb/>C, cuius centrum gra<lb/>uitatis N, & EGH <lb/>F, cuius centrum gra <lb/><figure id="id.043.01.171.1.jpg" xlink:href="043/01/171/1.jpg"/><lb/>uitatis O, quorum vtrumque erit in communi axe DM. <lb/><!-- KEEP S--></s> <s>Dico punctum N, propinquius e&longs;se ip&longs;i D quàm punctum <lb/>O. <!-- KEEP S--></s> <s>Quoniam enim e&longs;t parabolicifru&longs;ti EGHF centrum <lb/>grauitatis O; erit vt duplum maioris ba&longs;is, ide&longs;t circuli <lb/>EF vna cum minori circulo GH, ad duplum circuli GH <lb/>vna cum circulo EF, hoc e&longs;t vt duplum quadrati ED vna <lb/>cum quadrato ED ita MO ad OD. </s> <s>Sed vt quadratum <lb/>ED ad quadratum GM in parabola quæ conoides de­<lb/>&longs;cribit, cuius diameter BD, ita e&longs;t DB ad BM, hoc e&longs;t <lb/>AC ad KL; vt igitur e&longs;t dupla ip&longs;ius AC vna cum KL <lb/>ad duplam ip&longs;ius KL vna cum AC ita erit MO ad OD: <lb/>&longs;ed N e&longs;t fru&longs;ti conoici AKLC, centrum grauitatis; pun­<lb/>ctum igitur N, erit maiori ba&longs;i AC propinquius quàm <pb/>punctum O; e&longs;t autem O, fru&longs;ti EGHF centrum graui­<lb/>tatis. </s> <s>Si igitur conus, & conoides parabolicum circa eun­<lb/>dem axim, &c. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XLV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti conoidis hyperbolici centrum <lb/>grauitatis e&longs;t in axe primum &longs;ecto &longs;ecundum cen­<lb/>trum grauitatis cuiu&longs;uis fru&longs;ti conici circa axem <lb/>conoidis communi vertice, ab&longs;ci&longs;&longs;i vnà cum fru­<lb/>&longs;to conoidis: deinde ita vt pars minorem ba&longs;im <lb/>attingens &longs;it ad reliquam, vt dupla axis conoidis <lb/>vna cum reliqua dempto axe fru&longs;ti, ad duplam <lb/>eiu&longs;dem reliquæ vna cum axe conoidis: dein­<lb/>de po&longs;itis quatuor rectis lineis binis propor­<lb/>tionalibus, potentia primis, &longs;ecundis longitu­<lb/>dine, in proportione, quæ e&longs;t inter axem conoi­<lb/>dis, & reliquam dempto axe fru&longs;ti; ita vt ma­<lb/>ior primarum &longs;it media proportionalis inter axem <lb/>conoidis, & tran&longs;uer&longs;um latus hyperboles, quæ fi­<lb/>guram de&longs;cribit, minoris autem potentia &longs;e&longs;qui­<lb/>altera minor &longs;ecundarum; in eo puncto, in quo <lb/>&longs;egmentum axis fru&longs;ti dictis duabus &longs;ectionibus <lb/>terminatum &longs;ic diuiditur, vt pars minori ba&longs;i pro­<lb/>pinquior &longs;it ad reliquam vt cubus, qui fit ab axe <lb/>fru&longs;ti vnà cum &longs;olido rectangulo, quod axe co­<lb/>noidis, & reliqua dempto axe fru&longs;ti, & tripla <lb/>axis conoidis continetur, ad &longs;olidum rectangu­<lb/>lum ex eadem reliqua parte conoidis, & eo, quo <pb/>plus pote&longs;t quadrato maior quàm minor dicta­<lb/>rum &longs;ecundarum. </s></p><p type="main"> <s>Sit conoidis hyperbolici ABC, cuius axis BD; & <lb/>tran&longs;uer&longs;um latus hyperboles, quæ figuram de&longs;cribit EB, <lb/>fru&longs;tum ALMC ab&longs;ci&longs;&longs;um vnà cum axe FD: cuius <lb/><figure id="id.043.01.173.1.jpg" xlink:href="043/01/173/1.jpg"/><lb/>ba&longs;es oppo&longs;itæ, maior circulus circa AC, minor circa LM: <lb/>&longs;ecto autem axe FD primum &longs;ecundum G centrum gra­<lb/>uitatis fru&longs;ti ab&longs;ci&longs;&longs;i vnà cum fru&longs;to ALMC à quouis co <lb/>no, cuius axis BD, & vertex B, deinde in puncto H ita <lb/>vt FH ad HD &longs;it vt dupla ip&longs;ius BD vnà cum BF ad <lb/>duplam ip&longs;ius BF vnà cum BD, quo facto cadet G <lb/>punctum infra punctum H, ponantur vt DB ad BF, <pb/>ita N ad O potentia, & Q ad P longitudine: &longs;it au­<lb/>tem N media proportionalis inter EB, BD, at P ip&longs;ius <lb/>O potentia &longs;e&longs;quialtera: quo autem Q plus pote&longs;t quàm <lb/>P &longs;it quadratum ex R: & vt cubus ex FD vna cum &longs;oli­<lb/>do rectangulo ex BF, FD, & tripla ip&longs;ius BD, ad &longs;oli­<lb/>dum rectangulum ex BF, & quadrato R, ita &longs;it HK ad <lb/>KG. <!-- KEEP S--></s> <s>Dico fru&longs;ti ALMC centrum grauitatis e&longs;&longs;e K. <lb/><!-- KEEP S--></s> <s>Producta enim quà opus e&longs;t diametro AC ip&longs;i BD æqua­<lb/>les ab&longs;cindantur DS, DV: necnon ip&longs;i N æquales <lb/>DT, DX, vt &longs;it TD ad DS potentia, vt EB, ad <lb/>BD longitudine, & de&longs;cribantur conoides paraboli­<lb/>cum TBX, & conus SBV, quorum vertex commu­<lb/>nis B, axis BD: &longs;ectis autem his tribus &longs;olidis plano <lb/>per axim, &longs;int &longs;ectiones hyperbole ABC, & parabo­<lb/>la TBX, & triangulum SBV, quæ figuras de&longs;cribunt; <lb/>quas planum ba&longs;is fru&longs;ti propo&longs;iti circa LM &longs;ecans vnà <lb/>cum tribus &longs;olidis faciat cum parabola TBX rectam I<foreign lang="greek">g</foreign>, <lb/>& cum triangulo SBV rectam <foreign lang="greek">*u</foreign>Z: conoidis autem TBX, <lb/>& coni SBV &longs;ectiones circulos circa I<foreign lang="greek">g</foreign>, YZ ba&longs;ibus, <lb/>circa SV, TX parallelos; vt &longs;int conoidis TBX fru­<lb/>&longs;tum TI<foreign lang="greek">g</foreign>X, & coni SBV fru&longs;tum SYZV. </s> <s>Rur­<lb/>&longs;us producta I. M, ponatur <37>F, æqualis Q, & ab­<lb/>&longs;cindatur F<foreign lang="greek">d</foreign>, potentia &longs;e&longs;quialtera ip&longs;ius IF, iunctis­<lb/>que IB, B<foreign lang="greek">d</foreign>, B<37>, de&longs;cribantur tres coni <37>B<foreign lang="greek">q</foreign>, <lb/><foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, IB<foreign lang="greek">g</foreign>, quorum omnium ba&longs;es nempe circuli <lb/>erunt in dicto plano &longs;ecante tria &longs;olida per punctum F. <lb/><!-- KEEP S--></s> <s>Quoniam igitur circuli inter &longs;e &longs;unt vt quæ fiunt à diame­<lb/>tris, vel à &longs;emidiametris quadrata, coni autem eiu&longs;dem al­<lb/>titudinis inter &longs;e vt ba&longs;es; erit vt <foreign lang="greek">d</foreign>F ad FI potentia, ita <lb/>conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ad conum IB<foreign lang="greek">g</foreign>; &longs;e&longs;quialter igitur conus <lb/><foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> coni IB<foreign lang="greek">g</foreign>: &longs;ed & conoides parabolicum IB<foreign lang="greek">g</foreign> &longs;e&longs;qui­<lb/>alterum e&longs;t coni IB<foreign lang="greek">g</foreign>; æqualis igitur e&longs;t conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> co­<lb/>noidi IB<foreign lang="greek">g. </foreign></s> <s>Et quoniam in parabola TBX ordinatim <lb/>ad diametrum applicatarum DT e&longs;t ad FI hoc e&longs;t N <pb/>ad O potentia, vt DB ad BF longitudine: &longs;ed TD e&longs;t <lb/>æqualis N; ergo & IF æqualis erit O: cum igitur & <lb/>P ip&longs;ius O, & <foreign lang="greek">d</foreign>F ip&longs;ius FI &longs;it potentia &longs;e&longs;quialtera, erit <lb/>F<foreign lang="greek">d</foreign> æqualis ip&longs;i <foreign lang="greek">*r</foreign>: &longs;ed F<37> e&longs;t æqualis ip&longs;i <expan abbr="q;">que</expan> vt igitur e&longs;t <lb/>Q ad P, hoc e&longs;t DB ad BF, ita erit <37>F ad F<foreign lang="greek">d</foreign>; dupli­<lb/>cata igitur proportio erit quadrati ex F<37> ad quadratum ex <lb/>E<foreign lang="greek">d</foreign> eius, quæ e&longs;t DB ad BF: &longs;ed vt quadratum ex F<37> ad <lb/><figure id="id.043.01.175.1.jpg" xlink:href="043/01/175/1.jpg"/><lb/>quadratum ex F<foreign lang="greek">d</foreign>, ita e&longs;t circulus circa <37><foreign lang="greek">q</foreign> ad circulum <lb/>circa <foreign lang="greek">de</foreign>, hoc e&longs;t conus <37>B<foreign lang="greek">q</foreign> ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>; coni igitur <lb/><37>B<foreign lang="greek">q</foreign> ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, duplicata e&longs;t proportio eius, quæ e&longs;t <lb/>DB ad BF: &longs;ed & conoidis TBX ad conoides IB<foreign lang="greek">g</foreign> du­<lb/>plicata e&longs;t proportio eius, quæ e&longs;t DB ad BF, vt mon­<lb/>&longs;trant alij; eadem igitur proportio e&longs;t coni <37>B<foreign lang="greek">q</foreign> ad co­<lb/>num <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> quæ conoidis TBX ad conoides IB<foreign lang="greek">g</foreign>: &longs;ed <pb/>conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> æqualis e&longs;t conoidi IB<foreign lang="greek">g</foreign>, vtpote in&longs;cripti co­<lb/>ni IB<foreign lang="greek">g</foreign> &longs;e&longs;quialtero, cuius itidem &longs;e&longs;quialter erat conus <lb/><foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>; reliquum igitur coni <37>B<foreign lang="greek">q</foreign> dempto cono <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> æqua­<lb/>le erit conoidis TBX fru&longs;to TI<foreign lang="greek">g</foreign>X. <!-- KEEP S--></s> <s>Rur&longs;us quia e&longs;t vt <lb/>cubus ex BD ad cubum ex BI ita conus SBV ad &longs;ui &longs;i­<lb/>milem conum YBZ, in triplicata &longs;cilicet proportione la­<lb/>terum, &longs;iue axium DB, BF: &longs;ed quia YF e&longs;t æqualis BF, <lb/>propter &longs;imilitudinem triangulorum, e&longs;t vt cubus ex BF ad <lb/>&longs;olidum ex BF & quadrato ex F<foreign lang="greek">d</foreign>, ita quadratum ex FY <lb/>ad quadratum ex F<foreign lang="greek">d</foreign>, hoc e&longs;t circulus circa YZ ad <expan abbr="circulũ">circulum</expan> <lb/>circa <foreign lang="greek">de</foreign>, hoc e&longs;t conus YBZ ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ex æquali <lb/>igitur erit vt cubus ex BD ad &longs;olidum ex BF, & quadra­<lb/>to F<foreign lang="greek">d</foreign>, ita conus SBV ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>: &longs;ed vt &longs;olidum <lb/>ex BF, & quadrato F<foreign lang="greek">d</foreign>, ad &longs;olidum ex BF & quadrato <lb/>F<37>, ita e&longs;t &longs;imiliter vt ante conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ad conum <37>B<foreign lang="greek">q</foreign>; ex <lb/>æquali igitur erit vt cubus ex BD ad &longs;olidum ex BF, & <lb/>quadrato F<37>, ita conus SBV, ad conum <37>B<foreign lang="greek">q</foreign>: &longs;ed con­<lb/>uertendo, & per conuer&longs;ionem rationis, e&longs;t vt &longs;olidum ex <lb/>BF, & quadrato F<37>, ad &longs;olidum ex BF, & quadrato, <lb/>quo plus pote&longs;t F<37> quàm F<foreign lang="greek">d</foreign>, ita conus <37>B<foreign lang="greek">q</foreign> ad &longs;ui reli­<lb/>quum dempto cono <35>B<foreign lang="greek">e</foreign>; ex æquali igitur, vt cubus ex <lb/>BD ad &longs;olidum ex BF & quadrato, quo plus pote&longs;t F<37>, <lb/>quàm F<foreign lang="greek">d</foreign>, hoc e&longs;t, quo plus pote&longs;t Q quàm P quadrato <lb/>ex R, ita erit conus SBV, ad reliquum coni <37>B<foreign lang="greek">q</foreign> dem­<lb/>pto cono <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, hoc e&longs;t ad fru&longs;tum TI<foreign lang="greek">g</foreign>X. <!--neuer Satz-->Rur&longs;us, quo­<lb/>niam duo cubi ex BF, FD, & &longs;olidum ex BF, FD, & <lb/>tripla ip&longs;ius BD, &longs;unt æqualia cubo ex BD; erit id quo <lb/>plus pote&longs;t cubice recta BD quàm BF, cubus ex <lb/>FD, & &longs;olidum ex BF, FD, & tripla ip&longs;ius BD: cum <lb/>igitur &longs;it vt cubus ex BD ad cubum ex BF, ita conus <lb/>SBV ad conum YBZ; erit per conuer&longs;ionem rationis, & <lb/>conuertendo, vt cubus ex FD vna cum &longs;olido ex BF, <lb/>FD, & tripla ip&longs;ius BD ad cubum ex BD, ita fru&longs;tum <lb/>SYZV, ad conum SBV: &longs;ed cubus ex BD, ad &longs;oli-<pb/>dum ex BF & quadrato R, ita erat conus SBV ad fru­<lb/>&longs;tum TI<foreign lang="greek">g</foreign>X: ex æquali igitur, erit vt cubus ex FD vna <lb/>cum &longs;olido ex BF, FD, & tripla ip&longs;ius BD, ad &longs;olidum <lb/>ex BF, & quadrato R, hoc e&longs;t vt H<emph type="italics"/>K<emph.end type="italics"/> ad KG, ita ex <lb/>contraria parte fru&longs;tum SYZV, ad fru&longs;tum TI<foreign lang="greek">g</foreign>X: nam <lb/>fru&longs;ti SYZV e&longs;t centrum grauitatis G: fru&longs;ti autem TI <lb/><figure id="id.043.01.177.1.jpg" xlink:href="043/01/177/1.jpg"/><lb/><foreign lang="greek">g</foreign>X centrum grauitatis H; totius igitur compo&longs;iti ex his <lb/>duobus fru&longs;tis centrum grauitatis erit K: commune autem <lb/>e&longs;t centrum grauitatis compo&longs;iti ex duobus fru&longs;tis SYZV <lb/>& TI<foreign lang="greek">g</foreign>X, fru&longs;to ALMC per antepenultimæ huius co­<lb/>rollarium; fru&longs;ti igitur ALMC, centrum grauitatis erit K. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Ex omnibus demon&longs;trationibus eorum, quæ in <lb/>hoc &longs;ecundo libro propo&longs;uimus, manife&longs;tum e&longs;t <lb/>omnium &longs;upra dictorum corporum centra grauita <lb/>tis inuenire: quæ cum que enim in modum theore­<lb/>matis propo&longs;uimus, eadem tanquam problema­<lb/>ta proponi, & ij&longs;dem demon&longs;trationibus ab&longs;olui <lb/>po&longs;&longs;unt. </s></p><p type="main"> <s>Idem dico de ijs, quæ in primo, & tertio &longs;equenti libro <lb/>demon&longs;trauimus. </s> <s>Porro autem multa lemmata in&longs;tituto <lb/>præcipuo nece&longs;&longs;aria, & alia addita inuentio &longs;atis iucun­<lb/>da centri grauitatis conoidis, & portionis conoidis parabo­<lb/>lici, & hyperbolici, & fru&longs;ti vtriu&longs;que ne &longs;ecundus hic liber <lb/>nimis longus, & confu&longs;us exi&longs;teret, tertium requirebant. <lb/></s> <s>Quem quidem meorum &longs;tudiorum autumnalium fructum <lb/>Anni à partu Virginis MDCIII. cum SS. </s> <s>Clementis <lb/>Pont. <!-- REMOVE S-->Max. <!-- KEEP S--></s> <s>auctoritate, & Petri eius Nepotis Cardinalis <lb/>ampli&longs;&longs;imi Aldobrandini iu&longs;&longs;u bene de me merentium Ma­<lb/>thematicam &longs;cientiam, & Philo&longs;ophiam ciuilem in almo <lb/>Vrbis Gymna&longs;io profiterer, in eorum gratiam compo&longs;ui, <lb/>qui me centra grauitatis portionum &longs;phæroidis imperfe­<lb/>cti operis crimine condemnandum omittere nolebant; cu­<lb/>ius prouinciæ iuuante Deo, & mira Mathematicæ &longs;tudio­<lb/>&longs;is &longs;atisfaciendi voluntate, multas difficultates ita &longs;upe­<lb/>raui, vt vno men&longs;e Octobri plus præ&longs;titerim, quam à me <lb/>requi&longs;i&longs;&longs;ent. </s> <s>&longs;iquidem quæ de &longs;phæræ portionibus in hoc <lb/>libro proprijs eius figuræ rationibus, eadem in &longs;equen­<lb/>ti aliis communibus cuilibet portioni &longs;phæræ, & &longs;phæroi­<lb/>dis tum lati, tum oblongi ab&longs;ci&longs;&longs;æ vno, vel duobus planis <lb/>æque inter &longs;e di&longs;tantibus, & vtcumque in figuram in cideu-<pb/>tibus demon&longs;traui, & temporis breuitatem magna animi in­<lb/>tentione compen&longs;aui, quòd facere non potui&longs;sem ni&longs;i illi, <lb/>quos &longs;upra nominaui meos patronos tranquillum otium <lb/>mihi &longs;ua benignitate peperi&longs;&longs;ent; ego autem quo&longs;dam ad­<lb/>uer&longs;os flatus vehementes in meam vtilitatem verte­<lb/>re didici&longs;sem, cuius rei monumentum flammæ <lb/>vento agitatæ &longs;imulacrum cum illo Ver­<lb/>gilij HOC ACRIOR in fronte <lb/>operis po&longs;ui, vt meus quali&longs;­<lb/>cumque hic labor vel ab <lb/>inuitis in me collati <lb/>bencficij memo­<lb/>riam præ&longs;e­<lb/>ferret. </s></p><p type="head"> <s>SECVNDI LIBRI FINIS.<lb/><figure id="id.043.01.179.1.jpg" xlink:href="043/01/179/1.jpg"/><!-- KEEP S--></s></p><pb/><figure id="id.043.01.180.1.jpg" xlink:href="043/01/180/1.jpg"/><p type="head"> <s>L V C AE <lb/>VALER II <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM <lb/><emph type="italics"/>LIBER TERTIVS.<emph.end type="italics"/></s></p><figure id="id.043.01.180.2.jpg" xlink:href="043/01/180/2.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si recta linea &longs;ecta fuerit bifa­<lb/>riam, & non bifariam; rectan <lb/>gulum partibus in æqualibus <lb/>contentum æquale e&longs;t rectan <lb/>gulo, quod bis fit ex dimidiæ <lb/>&longs;ectæ &longs;egmentis, vna cum <lb/>quadrato non intermedij eo­<lb/>rundem &longs;egmentorum. </s></p><pb/><p type="main"> <s>Sit recta linea AB &longs;ecta in puncto C bi&longs;ariam, & non <lb/>bifariam in puncto D. <!-- KEEP S--></s> <s>Dico rectangulum ADB æqua­<lb/>le e&longs;&longs;e rectangulo BDC bis vnà cum quadrato BD. <lb/><!-- KEEP S--></s> <s>Quoniam enim rectangulum ADB, æquale e&longs;t duobus <lb/>rectangulis, & ex BD, DC, & ex AC, BD, hoc e&longs;t ex <lb/>CB, BD: &longs;ed rectangulum ex CB, BD, e&longs;t rectangu­<lb/>lum ex BD, DC, vnà cum quadrato BD; rectangulum <lb/>igitur ex AD, DB, æquale e&longs;t duobus rectangulis ex <lb/>BD, DC, vnà cum quadiato BD. <!-- KEEP S--></s> <s>Si igitur recta linea <lb/>&longs;ecta fuerit bifariam, & non bifariam, &c. </s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><figure id="id.043.01.181.1.jpg" xlink:href="043/01/181/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si circulum, vel ellip&longs;im duæ rectæ lineæ tan­<lb/>gentes in terminis coniugatarum diametrorum, <lb/>conueniant: & punctum in quo conueniunt, & <lb/>centrum figuræ iungantur recta linea; quæcun­<lb/>que hanc vnà cum prædictæ figuræ termino al­<lb/>terutri diametrorum parallela &longs;ecuerit recta li­<lb/>nea, ita ip&longs;a &longs;ecabitur in duobus punctis, vt re­<lb/>ctangulum bis contentum &longs;egmentis, quorum al­<lb/>terum inter diametrum, & terminum figuræ, al­<lb/>terum inter figuræ terminum & contingentem <lb/>interijcitur, vnà cum huius quadrato, &longs;it æquale <lb/>quadrato reliqui &longs;egmenti inter diametrum, & <pb/>cum quæ tangentium concur&longs;um, & centrum fi­<lb/>guræ iungit interiecta. </s></p><p type="main"> <s>Sit circulus, vel ellip&longs;is ABCD, cuius diametri con­<lb/>iugatæ AC, BED, & figuram tangentes BF, GF, con <lb/>ueniant in puncto F; (parallelæ enim erunt vtraque alteri <lb/>coniugatorum diametrorum:) & recta FE iungatur, & ex <lb/>quolibet puncto G, in recta BE ducatur ip&longs;i AC paral­<lb/>lela GLKH. </s> <s>Dico rectangulum GKH bis vnà cum <lb/>quadrato KH æquale e&longs;&longs;e quadrato GL. <!-- KEEP S--></s> <s>Quoniam <lb/>enim rectangulum BGD æquale e&longs;t rectangulo BGE <lb/><figure id="id.043.01.182.1.jpg" xlink:href="043/01/182/1.jpg"/><lb/>bis vnà cum quadrato BG: & rectangulum BED, e&longs;t <lb/>quadratum BE, erit vt rectangulum BED, ad re­<lb/>ctangulum BGD, ita quadratum BE, ad rectangu­<lb/>lum BGE bis, vnà cum quadrato BG: &longs;ed vt rectangu­<lb/>lum BED, ad rectangulum BGD, ita e&longs;t quadratum EC, <lb/>hoc e&longs;t quadratum GH ad quadratum GK, ex primo <lb/>conicorum, vt igitur e&longs;t quadratum BE ad rectangulum <lb/>BGE bis, vnà cum quadrato BG, ita erit quadratum <lb/>GH ad quadratum GK. <!-- KEEP S--></s> <s>Rur&longs;us quia e&longs;t vt BE ad EG, <lb/>ita BF ad GL, propter &longs;imilitudinem triangulorum; erit <lb/>vt quadratum BE ad quadratum EG, ita quadratum <pb/>BF hoc e&longs;t quadratum GH ad quadratum GL: & per <lb/>conuer&longs;ionem rationis, vt quadratum BE ad rectangu­<lb/>lum BGE bis, vnà cum quadrato BG, ita quadratum <lb/>GH ad rectangulum GLH bis, vnà cum quadrato LH: <lb/>&longs;ed vt quadratum BE ad rectangulum EGB bis, vnà <lb/>cum quadrato BG, ita erat quadratum GH ad quadra­<lb/>tum GK; vt igitur quadratum GH ad quadratum GK, <lb/>ita erit idem quadratum GH ad rectangulum GLH bis, <lb/>vnà cum quadrato LH: quadratum igitur GK æquale <lb/>erit rectangulo GLH bis, vnà cum quadrato LH; demptis <lb/>igitur ab eodem quadrato GH æqualibus quadrato GK, <lb/>& rectangulo GLH bis, vnà cum quadrato LH, erit <lb/>rectangulum GKH, bis vnà cum quadrato KH æquale <lb/>quadrato GL. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.183.1.jpg" xlink:href="043/01/183/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Per data duo puncta in duabus rectis lineis da­<lb/>tum angulum continentibus, in earum plano pa­<lb/>rabola tran&longs;ibit, cuius vertex &longs;it a&longs;&longs;ignatum præ­<lb/>dictorum punctorum, in quo altera linea parabo-<pb/>lam contingat, altera in altero &longs;ecet diametro æ­<lb/>quidi&longs;tans. </s></p><p type="main"> <s>Sint data duo puncta. </s> <s>A, C, in duabus rectis lincis da­<lb/>tum angulum ABC continentibus, &longs;it autem a&longs;&longs;ignatum <lb/>punctum C. <!-- KEEP S--></s> <s>Dico per puncta A, C, parabolam tran&longs;i­<lb/>re, ita vt ip&longs;am linea AC contingat in C puncto, altera <lb/>autem AB &longs;ecet in puncto A, diametro parabolæ æqui­<lb/>di&longs;tans. </s> <s>Completo enim parallelogrammo BD, ad re­<lb/>ctam CD applicetur rectangulum æquale quadrato AD, <lb/>faciens latitudinem E. <!-- KEEP S--></s> <s>Quoniam igitur in plano BD <lb/>parabola inueniri pote&longs;t, cu­<lb/>ius &longs;it vertex C, diameter <lb/>CD, ita vt quædam ex &longs;e­<lb/>ctione ad diametrum CD <lb/>applicata in dato angulo A <lb/>BC, ide&longs;t ADC, qualis <lb/>e&longs;t recta AD, po&longs;&longs;it rectan­<lb/>gulum ex CD, & E, ex <lb/>primo conicorum elemen. <lb/></s> <s>to; &longs;it ea &longs;ectio parabola <lb/><figure id="id.043.01.184.1.jpg" xlink:href="043/01/184/1.jpg"/><lb/>AC; a&longs;&longs;ignatum e&longs;t autem punctum C; per puncta igi­<lb/>tur A, C parabola AC tran&longs;ibit, cuius vertex e&longs;t a&longs;&longs;i­<lb/>gnatum punctum C. <!-- KEEP S--></s> <s>Et quoniam quæ ex vertice recta <lb/>CB e&longs;t applicatæ DA parallela, &longs;ectionem AC in pun­<lb/>cto C continget: e&longs;t autem AB diametro CD æquidi­<lb/>di&longs;tans, ac proinde parabolam &longs;ecabit in puncto A. <!-- KEEP S--></s> <s>Ma­<lb/>nife&longs;tum e&longs;t igitur propo&longs;itum, </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO IV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si recta linea parabolam contingat, omnes re­<lb/>ctælineæ ex &longs;ectione ad contingentem applicatæ <pb/>diametro &longs;ectionis parallelæ inter &longs;e &longs;unt longi­<lb/>tudine, vt inter applicatas & contactum, vel ver­<lb/>ticem interiectæ inter &longs;e potentia. </s> <s>Productis au­<lb/>tem dictis applicatis, erunt inter &longs;ectionem & ba­<lb/>&longs;im interiectæ inter &longs;e longitudine, vt in circulo, <lb/>vel ellip&longs;e ad diametrum ordinatim applicatæ, &longs;e­<lb/>cantesque illam in ea&longs;dem rationes, in quas aliæ <lb/>prædictæ applicatæ &longs;ecant ba&longs;im parabolæ, inter <lb/>&longs;e potentia. </s></p><p type="main"> <s>Sit &longs;ectio parabola ABC, cuius vertex B, diameter <lb/>BD: & recta quadam BE &longs;ectionem contingente in pun­<lb/>cto B, &longs;int quotcumque rectæ lineæ ex &longs;ectione ordinatim <lb/>ad BE contingentem applicatæ diametro BD &longs;ectionis <lb/>parallelæ FG, KH, quibus productis &longs;int ad ba&longs;im &longs;e­<lb/><figure id="id.043.01.185.1.jpg" xlink:href="043/01/185/1.jpg"/><lb/>ctionis applicatæ GN, KO. </s> <s>Et expo&longs;ito primum circu­<lb/>lo, PQRS, cuius diametri ad rectos inter &longs;e angulos &longs;int <lb/>QS, PR; &longs;ecta autem QT in punctis V, X, in ea&longs;­<lb/>dem rationes, in quas &longs;ecta e&longs;t AD in punctis N, O, <lb/>&longs;umpto ordine à punctis D, T, vt &longs;it DO ad ON, <pb/>vt e&longs;t TV ad VX: & vt ON ad NA, ita VX ad <expan abbr="Xq;">Xque</expan> <lb/>applicentur ad &longs;emidiametrum QT rectæ ZV, XY dia­<lb/>metro PR æquidi&longs;tantes. </s> <s>Dico e&longs;&longs;e HK ad FG lon­<lb/>gitudine, vt FB ad BH potentia: & KO ad GN longi­<lb/>tudine, vt ZY ad YX potentia. </s> <s>Iungantur enim KL, <lb/>GM, ba&longs;i AC parallelæ. </s> <s>Quoniam igitur e&longs;t vt MB <lb/>ad BI. longitudine, ita GM ad KL potentia: &longs;ed MB <lb/>e&longs;t æqualis ip&longs;i FG, & BL ip&longs;i KH, & BF ip&longs;i GM, & <lb/>BH ip&longs;i KL in parallelogrammis BG, BK; vt igitur <lb/>FG ad KH longitudine, ita erit BH ad BF potentia: <lb/>&longs;imiliter quotcumque plures e&longs;&longs;ent applicatæ idem o&longs;ten­<lb/>deremus. </s> <s>Rur&longs;us, quoniam e&longs;t vt EA, hoc e&longs;t FN ad FG, <lb/>ita quadratum EB ad BF quadratum, hoc e&longs;t quadra­<lb/>tum AD ad quadratum DN, hoc e&longs;t ita quadratum QT, <lb/>hoc e&longs;t quadratum TY, hoc e&longs;t duo quadrata TX, XY, <lb/>ad quadratum TX; erit per conuer&longs;ionem rationis, vt FN, <lb/>hoc e&longs;t BD ad GN, ita duo quadrata TX, X<foreign lang="greek">*u</foreign> &longs;imul, <lb/>hoc e&longs;t quadratum TY, hoc e&longs;t quadratum TP, ad qua­<lb/>dratum XY. <!-- KEEP S--></s> <s>Similiter o&longs;tenderemus e&longs;&longs;e vt BD ad <lb/>OK, ita quadratum PT ad quadratum VZ. </s> <s>Conuer­<lb/>tendo igitur erit vt OK ad BD, ita quadratum XY ad <lb/>PT quadratum: & ex æquali vt OK ad GN, ita qua­<lb/>dratum VZ ad quadratum XY. <!-- KEEP S--></s> <s>Suntigitur tres rectæ <lb/>lineæ BD, OK, GN, inter &longs;e longitudine, vt in circu­<lb/>lo PQSR totidem PT, ZV, XY inter &longs;e potentia, <lb/>prout inter &longs;e re&longs;pondent. </s> <s>Idem autem &longs;imiliter o&longs;ten­<lb/>deremus de quotcumque aliis in circulo, & &longs;ectione para­<lb/>bola vt prædictæ applicatis multitudine æqualibus. </s> <s>In <lb/>ellip&longs;e autem, ductis diametris quibu&longs;uis coniugatis, & <lb/>totidem quot in circulo ad vnam &longs;emidiametrum rectis li­<lb/>neis ordinatim applicatis &longs;ecundum puncta &longs;ectionum eiu&longs;­<lb/>dem diametri in ea&longs;dem prædictas rationes, eodemque or­<lb/>dine; quoniam ex XXI primi conicorum &longs;tatim apparet re­<lb/>ctarum linearum ita vt diximus in circulo, & ellip&longs;e appli-<pb/>catarum quadrata e&longs;&longs;e inter &longs;e in eadem proportione; erunt<lb/>prædictæ inter &longs;ectionem parabolam, & ba&longs;im interiectæ <lb/>inter &longs;e longitudine, vt in ellip&longs;e ad diametrum &longs;imiliter <lb/>vt diximus applicatæ inter &longs;e potentia. </s> <s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO V.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis figuræ circa axim in alteram partem <lb/>deficientis, cuius &longs;uperficies, excepta ba&longs;e &longs;it to­<lb/>ta interius concaua ba&longs;im habentis circulum, vel <lb/>ellip&longs;im; quælibet tres &longs;ectiones ba&longs;i parallelæ <lb/>æqualia axis &longs;egmenta intercipientes, ita &longs;e ha­<lb/>bent, vt minor &longs;it proportio minimæ ad mediam, <lb/>quam mediæ ad maximam. </s></p><p type="main"> <s>Sit figura ABC circa axem BD in alteram partem de­<lb/>ficiens, qualem diximus: & po&longs;itis in axe BD tribus qui­<lb/>buslibet punctis <lb/>F, E, L, æqualia <lb/>axis &longs;egmenta in­<lb/>tercipientibus, in <lb/>telligatur <expan abbr="&longs;olidũ">&longs;olidum</expan> <lb/>ABC &longs;ectum per <lb/>ea puncta planis <lb/><expan abbr="buibu&longs;dã">buibu&longs;dam</expan> ba&longs;i cir <lb/>culo, vel ellip&longs;i, <lb/>circa AC pa­<lb/>rallelis: quare &longs;e­<lb/>ctiones erunt cir­<lb/><figure id="id.043.01.187.1.jpg" xlink:href="043/01/187/1.jpg"/><lb/>culi, vel ellip&longs;es &longs;imiles ba&longs;i, per definitionem, quarum dia­<lb/>metri eiu&longs;dem rationis in eodem plano per axim &longs;int IK. <pb/>GH, MN. <!-- KEEP S--></s> <s>Dico &longs;olidi ABC &longs;ectionum, minorem e&longs;&longs;e <lb/>proportionem, ip&longs;ius IK ad GH, quàm GH ad MN. <lb/><!-- KEEP S--></s> <s>Iunctis enim MRS, KSN; quoniam tres rectæ IK, <lb/>RS, MN, &longs;e&longs;e æqualiter excedunt in trapezio KM; mi­<lb/>nor erit proportio IK ad RS, quàm RS ad MN: &longs;ed cir <lb/>culi, & &longs;imiles ellip&longs;es duplicatam habent inter &longs;e propor­<lb/>tionem diametrorum eiu&longs;dem rationis; trium igitur præ­<lb/>dictarum &longs;olidi ABC &longs;ectionum minor erit proportio IK <lb/>ad RS quàm RS ad MN: &longs;ed maior e&longs;t proportio circu­<lb/>li, vel ellip&longs;is GH ad circulum, vel ellip&longs;im MN, quàm <lb/>circuli, vel ellip&longs;is RS, ad circulum, vel ellip&longs;im MN; <lb/>multo ergo minor proportio erit circuli, vel ellip&longs;is IK ad <lb/>circulum, vel ellip&longs;im RS, quàm circuli, vel ellip&longs;is GH ad <lb/>circulum, vel ellip&longs;im MN: &longs;ed minor e&longs;t proportio cir­<lb/>culi vel ellip&longs;is I<emph type="italics"/>K<emph.end type="italics"/> ad circulum, vel ellip&longs;im GH, quàm <lb/>eiu&longs;dem circuli, vel ellip&longs;is IK ad circulum, vel ellip&longs;im <lb/>RS; multo ergo minor proportio erit circuli, vel ellip&longs;is <lb/>IK ad circulum, vel ellip&longs;im GH quàm circuli, vel ellip­<lb/>&longs;is GH ad circulum, vel ellip&longs;im MN. <!-- KEEP S--></s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;phæroides &longs;ecetur plano vtcumque præter <lb/>quàm ad axem, circa quem &longs;phæroides de&longs;cribi­<lb/>tur erecto nam tunc circulus fit. </s> <s>&longs;ectio ellip&longs;is erit: <lb/>&longs;imilis autem ip&longs;i alia quæcumque &longs;ectio &longs;phæ­<lb/>roidis eidem parallela: earumque omnes diame­<lb/>tri quæ eiu&longs;dem &longs;unt rationis erunt in eodem pla­<lb/>no per axem. </s></p><p type="main"> <s>Extant hæc demon&longs;trata ab Archimede in &longs;uo de &longs;phæ­<lb/>roidibus, & conoidibus. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO VII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si conoides parabolicum, vel hyperbolicum <lb/>&longs;ecetur plano vtcumque ad axim inclinato, &longs;ectio <lb/>ellip&longs;is erit: &longs;imilis autem ip&longs;i alia quæcumque <lb/>&longs;ectio conoidis eidem parallela: eruntque earum <lb/>omnes diametri, quæ eiu&longs;dem &longs;unt rationis in eo­<lb/>dem plano per axem. </s></p><p type="main"> <s>Manife&longs;ta &longs;unt hæc ex ijs, quæ Federicus Commandinus <lb/>demon&longs;trauit de &longs;ectionibus horum &longs;olidorum, in &longs;uis com­<lb/>mentariis in eundem Archimedis librum de &longs;phæroidibus, <lb/>& conoidibus: quemadmodum & &longs;phæroidis, & conoi­<lb/>dis vtriu&longs;que &longs;ectionem factam à plano ad axim erecto e&longs;­<lb/>&longs;e circulum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Super datam ellip&longs;im, circa datam rectam line­<lb/>am ab eius centro eleuatam tanquam axem, coni, <lb/>& cylindri portionem inuenire. </s> <s>Datoque &longs;phæ­<lb/>roidi, & conoidi, vel conoidis, &longs;phæroidi&longs;ve por­<lb/>tioni circa datum axem &longs;phæroidis, vel cuiuslibet <lb/>dictarum portionum, cylindrus vel cylindri por­<lb/>tio circum&longs;cripta e&longs;&longs;e pote&longs;t: vel comprehendere <lb/>inter eadem plana parallela, ita vt eius ba&longs;is &longs;it &longs;i­<lb/>milis ba&longs;i, vel ba&longs;ibus comprehen&longs;æ portionis, vel <lb/>fru&longs;ti, &longs;i de conoidibus &longs;it &longs;ermo: & diametri, quæ <lb/>eiu&longs;dem &longs;unt rationis &longs;ectæ à centro bifariam &longs;int <lb/>in eadem recta linea. </s></p><pb/><p type="main"> <s>Manife&longs;ta item &longs;unt hæc omnia, ex ijs, quæ in eodem li­<lb/>bro de &longs;phæroidibus, & conoidibus demon&longs;trat Archi­<lb/>medes. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO IX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti pyramidis triangulam ba&longs;im ha­<lb/>bentis ad pri&longs;tina, cuius ba&longs;is e&longs;t maior ba&longs;is fru­<lb/>&longs;ti, & eadem altitudo, cam habet proportionem, <lb/>quàm rectangulum contentum duobus lateribus <lb/>homologis ba&longs;ium oppo&longs;itarum, vnà cum tertia <lb/>parte quadrati differentiæ dictorum laterum, ad <lb/>maioris lateris quadratum. </s> <s>Ad pyramidem autem, <lb/>cuius ba&longs;is e&longs;t maior ba&longs;is fru&longs;ti, & eadem altitu­<lb/>do, vt prædictum rectangulum, vna cum prædicti <lb/>quadrati tertia parte, ad tertiam partem quadrati <lb/>maioris lateris. </s></p><p type="main"> <s>Sit pyramidis triangulam ba&longs;im habentis fru&longs;tum AB <lb/>CD EF: laterum autem homo­<lb/>logorum AB, DE, triangulorum <lb/>&longs;imilium oppo&longs;itorum ABC, D <lb/>EF, &longs;it differentia DG: & eiu&longs;­<lb/>dem altitudinis fru&longs;to &longs;it pri&longs;ma <lb/>DEFCHK: & pyramis intelli­<lb/>gatur ADEF. <!-- KEEP S--></s> <s>Dico fru&longs;tum <lb/>BDF ad pri&longs;ma HKF, e&longs;&longs;e vt <lb/>rectangulum DEG vna cum ter­<lb/>tia parte quadrati DG. </s> <s>Ad qua­<lb/>dratum DE: ad pyramidem au­<lb/>tem ADEF, vt <expan abbr="prædictũ">prædictum</expan> rectan­<lb/><figure id="id.043.01.190.1.jpg" xlink:href="043/01/190/1.jpg"/><lb/>gulum DEG, vnà cum tertia parte quadrati DG, ad ter­<pb/>tiam partem quadrati DE. <!-- KEEP S--></s> <s>Ab&longs;ci&longs;sis enim æqualibus EL <lb/>ip&longs;i BC, & FM ip&longs;i AC, & EG, ip&longs;i AB, con&longs;tituantur <lb/>pri&longs;mata ABCLEG, AGMFCL, ANHDGM, & <lb/>pyramis ADGM, & iungatur ML. <!-- KEEP S--></s> <s>Quoniam igitur ob pa­<lb/>rallelas EF, GM, & DF, GL, &longs;imilia inter &longs;e &longs;unt trian­<lb/>gula DEF, DGM, EGL, duplicatam inter &longs;e habebunt <lb/>laterum ho mologorum DE, DG, GE, proportionem, <lb/>hoc e&longs;t eandem, quæ totidem e&longs;t quadratorum ex ip&longs;is DE, <lb/>DG, GE, prout inter &longs;e re&longs;pondent: vt igitur DG qua­<lb/>dratum ad quadratum DE, ita e&longs;t triangulum DGM <lb/>ad triangulum DEF: eademque ratione vt quadratum <lb/>GE ad DE quadratum, ita trian <lb/>gulum EGL ad triangulum D <lb/>EF: & vt prima cum quinta ad <lb/>&longs;ecundam, ita tertia cum &longs;exta ad <lb/>quartam: videlicet, vt duo qua­<lb/>drata DG, GE, ad quadratum <lb/>DE, ita duo triangula DGM, <lb/>EGL, ad triangulum DEF. & <lb/>conuertendo, & per conuer&longs;ionem <lb/>rationis, vt quadratum DE ad <lb/>rectangulum DGE bis, ita trian­<lb/>gulum DEF, ad parallelogram­<lb/><figure id="id.043.01.191.1.jpg" xlink:href="043/01/191/1.jpg"/><lb/>mum GF: & conuertendo, vt rectangulum DGE bis, ad <lb/>quadratum DE, ita GF parallelogrammum ad triangu­<lb/>lum DEF: & antecedentium dimidia, vt rectangulum <lb/>DGE ad quadratum DE, ita triangulum GML ad <lb/>triangulum DEF; hoc e&longs;t pri&longs;ma, cuius ba&longs;is triangulum <lb/>GLM, altitudo eadem pri&longs;mati H<emph type="italics"/>K<emph.end type="italics"/>F ad pri&longs;ma HKF. <!-- KEEP S--></s></p><p type="main"> <s>Rur&longs;us, quoniam e&longs;t vt quadratum EG ad quadratum <lb/>ED, ita triangulum EGL ad triangulum DEF; erit &longs;i­<lb/>militer vt quadratum EG ad quadratum ED, ita pri&longs;ma <lb/>BGL ad pri&longs;ma HKF: &longs;ed vt rectangulum DGE ad <lb/>quadratum DE, ita pri&longs;ma erat, cuius ba&longs;is triangulum G <pb/>LM altitudo autem eadem pri&longs;mati HKF, hoc e&longs;t pri&longs;ma <lb/>ACGLFM illi æquale per vltimam XI. elem. </s> <s>ad pri&longs;ma <lb/>HKF: vt igitur prima cum quinta, rectangulum DGE <lb/>vna cum quadrato EG, hoc e&longs;t rectangulum DEG, ad <lb/>&longs;ecundam quadratum DE, ita erit tertia cum &longs;exta, duo <lb/>pri&longs;mata BGL, ACGLFM, ad quartam pri&longs;ma HKF. <lb/><!-- KEEP S--></s> <s>Præterea quoniam vt quadratum DG ad quadratum <lb/>DE, ita erat triangulum DGM ad triangulum DEF: &longs;ed <lb/>vt triangulum DGM ad triangulum DEF, ita e&longs;t pri&longs;ma, <lb/>HGM, ad pri&longs;ma HKF: & tertiæ antecedentium par­<lb/>tes, videlicet, vt tertia pars quadrati DG, ad quadra­<lb/>tum DE, ita pyramis ADGM ad pri&longs;ma HKF: &longs;ed <lb/>vt rectangulum DEG ad DE quadratum, ita erant duo <lb/>pri&longs;mata BGL, ACGLFM, ad pri&longs;ma HKF; vt igi­<lb/>tur prima cum quinta, rectangulum DEG vna cum ter­<lb/>tia parte DG quadrati, ad quadratum GD &longs;ecundam, <lb/>ita erit tertia cum &longs;exta, duo pri&longs;mata BGL, ACGLFM <lb/>vna cum pyramide ADGM, hoc e&longs;t integrum fru&longs;tum <lb/>ABCDEF ad pri&longs;ma HKF quartam. </s> <s>Ex hoc patet &longs;e­<lb/>cunda pars propo&longs;iti. </s> <s>Quoniam enim e&longs;t vt rectangulum <lb/>DEG, vna cum tertia parte quadrati DG, ad quadra­<lb/>tum DE, ita fru&longs;tum ABGDEF ad pri&longs;ma HKF: vt <lb/>autem quadratum DE, ad tertiam &longs;ui partem, ita e&longs;t pri&longs;­<lb/>ma HKF ad pyramidem, cuius ba&longs;is triangulum DEF, <lb/>altitudo eadem pri&longs;mati HKF; erit ex æquali vt re­<lb/>ctangulum DEG vna cum tertia parte quadrati DG <lb/>ad tertiam partem quadrati DE, ita fru&longs;tum ABCDEF, <lb/>ad pyramidem &longs;i compleatur ADEF. <!-- KEEP S--></s> <s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hinc manife&longs;tum e&longs;t eadem demon&longs;tratione, <lb/>qua vtimur ad propo&longs;itionem XXXVI. primili­<lb/>bri; fru&longs;tum cuiuslibet pyramidis ba&longs;im habentis <lb/>pluribus quàm tribus lateribus contentam, ad pri&longs; <lb/>ma, &longs;eu pyramidem, cuius ba&longs;is e&longs;t eadem quæ ma­<lb/>ior ba&longs;is fru&longs;ti, & eadem altitudo: & reliquum ip­<lb/>&longs;ius pri&longs;matis dempto fru&longs;to, ad ip&longs;um pri&longs;ma, eas <lb/>habere rationes, quæ à ba&longs;ium fru&longs;ti oppo&longs;itarum <lb/>homologis lateribus eorumque differentia deri­<lb/>uantur eo modo, quo in præcedenti theoremate <lb/>dicebamus. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO X.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omne fru&longs;tum coni, vel portionis conicæ, ad cy <lb/>lindrum, vel cylindri portionem, cuius ba&longs;is e&longs;t ea <lb/>dem, quæ maior ba&longs;is fru&longs;ti, & eadem altitudo, <lb/>eam habet proportionem, quàm rectangulum con <lb/>tentum ba&longs;ium diametris eiu&longs;dem rationis, vnà <lb/>eum tertia parte quadrati differentiæ earumdem <lb/>diametrorum, ad maioris ba&longs;is quadratum. </s> <s>Ad <lb/>conum autem, vel coni portionem, cuius ba&longs;is e&longs;t <lb/>eadem, quæ maior ba&longs;is fru&longs;ti, & eadem altitudo; <lb/>vt prædictum rectangulum, vnà cum prædicti qua <lb/>drati tertia parte, ad tertiam partem quadrati ex <lb/>diametro maioris ba&longs;is. </s> <s>Prædicti autem cylindri, <pb/>vel portionis cylindricæ re&longs;iduum dempto fru&longs;to, <lb/>ad totum cylindrum, vel cylindri portionem; vt <lb/>rectangulum contentum diametro minoris ba&longs;is <lb/>fru&longs;ti, & differentia diametri maioris, vnà cum <lb/>duabus tertiis quadrati differentiæ, ad quadra­<lb/>tum diametri maioris ba&longs;is. </s></p><p type="main"> <s>Sit coni, vel eius portionis fru&longs;tum ABCD, cuius ba&longs;es <lb/>oppo&longs;itæ, circuli vel &longs;imiles ellip&longs;es, quarum diametri mi­<lb/>noris ba&longs;is AB cuius centrum E: maioris autem CD, <lb/>& &longs;uper ba&longs;im circulum, vel ellip&longs;im CD &longs;tet cylindrus, <lb/>vel portio cylindrica CG comprehendens fru&longs;tum AB <lb/>CD, eiu&longs;demque altitudinis cum ip&longs;o, & conus, vel co­<lb/>ni portio ECD. quo autem AC diameter &longs;uperat dia­<lb/>metrum AB, quæ differentia di­<lb/>citur, &longs;it DF. <!-- KEEP S--></s> <s>Dico fru&longs;tum AD <lb/>ad cylindrum, vel portionem cy­<lb/>lindricam CG, e&longs;&longs;e vt rectangu­<lb/>lum DCF vnà cum tertia parte <lb/>quadrati DF, ad quadratum CD. <lb/><!-- KEEP S--></s> <s>Ad conum autem vel coni portio­<lb/>nem ECD, vt rectangulum DCF, <lb/>vna cum tertia parte quadrati DF, <lb/>ad tertiam partem quadrati CD. <lb/><!-- KEEP S--></s> <s>Cylindri autem, vel cylindri por­<lb/>tionis CG re&longs;iduum dempto fru­<lb/><figure id="id.043.01.194.1.jpg" xlink:href="043/01/194/1.jpg"/><lb/>&longs;to AD, ad cylindrum, vel portionem cylindricam CG, <lb/>vt rectangulum CFD vna cum duabus tertiis quadrati <lb/>FD, ad quadratum CD. <!-- KEEP S--></s> <s>Cono enim, vel portioni coni­<lb/>cæ, cuius fru&longs;tum AD, & cylindro, vel portioni cylindri­<lb/>cæ, cuius ba&longs;is e&longs;t circulus, vel ellip&longs;is CD, altitudo au­<lb/>tem eadem completo cono, vel portioni conicæ iam dictæ, <lb/>illi pyramis, huic pri&longs;ma in&longs;cripta intelligantur, quorum <pb/>communis ba&longs;is &longs;it poly gorum in&longs;criptum circulo quidem <lb/>æquilaterum, & æquiangulum; in ellip&longs;e autem, quod pro <lb/>Archimede de&longs;cribit Commandinus, ita vt & à cylindro, <lb/>vel cylindri portione pri&longs;ina, & à cono, vel coni portione <lb/>pyramis deficiat minori &longs;pacio quantacumque magnitudi­<lb/>ne propo&longs;ita: quo modo autem in portione cylindrica, vel <lb/>conica hoc fieri po&longs;&longs;it, eadem quæ de cono atque cylindro <lb/>Euclides in duodecimo docuit manife&longs;tant. </s> <s>Ab&longs;ci&longs;&longs;ione <lb/>igitur facta fru&longs;ti AD, & cylindri, vel portionis cylindricæ <lb/>CG, ab&longs;ci&longs;&longs;a &longs;imul erunt fru&longs;tum pyramidis in&longs;criptum <lb/>fru&longs;to AD, & pri&longs;ma in&longs;criptum cylindro, vel portioni cy­<lb/>lindricæ CG, eiu&longs;dem altitudinis inter &longs;e, & duobus præ­<lb/>dictis &longs;olidis AD, CG, deficien <lb/>tia vnum à fru&longs;to, alterum à cy­<lb/>lindro, vel portione cylindrica <lb/>multo minori &longs;pacio magnitudine <lb/>propo&longs;ita: &longs;ectiones autem pri&longs;ma <lb/>tis, & pyramidis erunt polygona <lb/>circulis, vel ellip&longs;ibus ip&longs;i CD op <lb/>po&longs;itis & &longs;imilibus in&longs;cripta in­<lb/>ter &longs;e &longs;imilia, vt multi o&longs;tendunt. <lb/></s> <s>erunt etiam &longs;imilium polygono­<lb/>rum circulis, vel ellip&longs;ibus &longs;imili­<lb/>bus, quæ &longs;unt ba&longs;es oppo&longs;itæ fru­<lb/><figure id="id.043.01.195.1.jpg" xlink:href="043/01/195/1.jpg"/><lb/>&longs;ti AD, in&longs;criptorum diametri eædem AB, CD. <!-- KEEP S--></s> <s>Quo­<lb/>niam igitur &longs;imilium polygonorum circulis, & &longs;imilibus <lb/>ellip&longs;ibus in&longs;criptorum latera homologa inter &longs;e &longs;unt vt <lb/>diametri dictorum circulorum, vel ellip&longs;ium, eadem erit <lb/>proportio inter duas diametros AB, CD, hoc e&longs;t FC, <lb/>CD, quæ inter duo quælibet latera homologa polyga­<lb/>norum circulis, vel ellip&longs;ibus &longs;imilibus AB, CD in­<lb/>&longs;criptorum. </s> <s>Sed pyramidis fru&longs;tum fru&longs;to CB in&longs;cri­<lb/>ptum ad pri&longs;ma, cuius ba&longs;is e&longs;t maior ba&longs;is fru&longs;ti pyrami­<lb/>dis, & eadem altitudo, &longs;olido CG in&longs;criptum, e&longs;t vt re-<pb/>ctangulum contentum lateribus homologis ba&longs;ium oppo­<lb/>&longs;itarum, vna cum tertia parte quadrati differentiæ, ad ma­<lb/>ioris lateris quadratum; idem igitur fru&longs;tum pyramidis <lb/>ad idem pri&longs;ma, erit vt rectangulum DCF, vna cum <lb/>tertia parte quadrati DF ad quadratum CD: deficit <lb/>autem vtrumque & pyramidis fru&longs;tum fru&longs;to CB in&longs;cri­<lb/>ptum ab ip&longs;o CB fru&longs;to, & pri&longs;ma ip&longs;i CG in&longs;criptum <lb/>ab ìp&longs;o CG, minori &longs;pacio quantacumque propo&longs;ita ma­<lb/>gnitudine; per tertiam igitur huius, erit vt rectangulum <lb/>DCF vna cum tertia parte quadrati DF, ad CD qua­<lb/>dratum, ita fru&longs;tum CB ad cylindrum, vel portionem <lb/>cylindricam CG. </s> <s>Cum igitur conus, vel coni portio E <lb/>CD &longs;it pars tertia cylindri, vel portionis cylindricæ CG, <lb/>erit ex æquali, vt idem rectangulum DCF, vna cum ter­<lb/>tia parte quadrati DF, ad tertiam partem quadrati CD, <lb/>ita fru&longs;tum BC, ad conum vel coni portionem ECD. <!--neuer Satz-->Præ­<lb/>terea, quia quadratum CD æquale e&longs;t duobus quadratis <lb/>ex CF, FD, vna cum rectangulo bis ex CF, FD: quorum <lb/>rectangulo CFD, vna cum quadrato CF æquale e&longs;t rectan­<lb/>gulum DCF; erit quadratum CD æquale rectangulo <lb/>DCF vna cum quadrato DF; demptis igitur rectangu­<lb/>lo DCF, & tertia parte quadrati DF; quod remanet <lb/>CD quadrati erit rectangulum CFD vna cum duabus <lb/>tertiis quadrati DF. quoniam igitur e&longs;t conuertendo vt <lb/>quadratum CD ad rectangulum DCF, vna cum tertia <lb/>parte quadrati DF, ita cylindris, vel portio cylindrica <lb/>CG ad fru&longs;tum CB, erit per conuer&longs;ionem rationis, & <lb/>conuertendo; vt rectangulum CFD vna cum duabus ter­<lb/>tiis DF quadrati, ad quadratum CD, ita reliquum cy­<lb/>lindri, vel portionis cylindricæ CG dempto fru&longs;to CB, <lb/>ad cylindrum, vel portionem cylindricam. </s> <s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;phæra, vel &longs;phæroides &longs;ecetur duobus pla­<lb/>nis parallelis vtcumque, neutro per <expan abbr="c&etilde;trum">centrum</expan> ducto: <lb/>quædam autem ex centro recta linea tran&longs;eat per <lb/>centrum alterutrius &longs;ectionum; per centrum re­<lb/>liquæ tran&longs;ibit. </s></p><p type="main"> <s>Sit &longs;phæra, vel &longs;phæroides &longs;ectum duobus planis pa­<lb/>callelis vtcumque neutro per centrum ducto, quod &longs;it E: <lb/>per &longs;ectionum autem, quæ &longs;unt circuli, vel &longs;imiles el­<lb/>lip&longs;es, alterutrius centrum F tran&longs;iens recta EFB oc­<lb/>currat reliquæ &longs;ectionis plano in puncto G. <!-- KEEP S--></s> <s>Dico reli­<lb/>quæ &longs;ectionis centrum e&longs;&longs;e G. <!-- KEEP S--></s> <s>Planum enim per OB &longs;e­<lb/><figure id="id.043.01.197.1.jpg" xlink:href="043/01/197/1.jpg"/><lb/>cans &longs;phæram, vel &longs;phæroides, faciensque &longs;ectionem circu­<lb/>lum, vel ellip&longs;im ABCD, &longs;ecabit, & &longs;ecet prædictas &longs;e­<lb/>ctiones, circulos inquam, vel &longs;imiles ellip&longs;es parallelas, qua­<lb/>rum alterius centrum ponitur F. <!-- KEEP S--></s> <s>Faciatque &longs;ectiones re­<lb/>ctas parallelas AFC, KGH: &longs;imiliter aliud quodlibet <pb/>planum per BE &longs;ecans &longs;phæram, vel &longs;phæroides faciat &longs;e­<lb/>ctionem circulum, vel ellip&longs;im, & in ea parallelas LFM, <lb/>NGO, communes &longs;ectiones iam factæ &longs;ectionis &longs;phæræ <lb/>vel &longs;phæroidis cum circulis, vel ellip&longs;ibus inter &longs;e paral­<lb/>lelis quarum diametri &longs;unt AC, KH. <!-- KEEP S--></s> <s>Quoniam igitur <lb/>E e&longs;t centrum &longs;phæræ, vel &longs;phæroidis; omnes in eo per <lb/>punctum E, tran&longs;euntes rectæ lineæ bifariam &longs;ecabuntur: <lb/>&longs;ed idem E e&longs;t in &longs;ectione &longs;phæræ, vel &longs;phæroidis, circu­<lb/>lo, vel ellip&longs;e ABCD; omnes igitur in ip&longs;a rectas lineas <lb/>bifariam &longs;ecabit punctum E, & centrum erit circuli, <lb/>vel ellip&longs;is ABCD: quædam igitur ex centro recta EB <lb/>&longs;ecans parallelarum neutrius per centrum ductæ alteram <lb/>AC bifariam in circuli, vel ellip&longs;is ALCM centro F, <lb/>& reliquam in puncto G bifariam &longs;ecabit. </s> <s>Similiter <lb/>o&longs;tenderemus rectam NO &longs;ectam e&longs;se bifariam in pun­<lb/>cto G: atque adeo circuli, vel ellip&longs;is KNHO centrum <lb/>e&longs;&longs;e G. <!-- KEEP S--></s> <s>Recta igitur E, tran&longs;iens per centrum &longs;ectionis <lb/>ALCM, tran&longs;ibit per centrum reliquæ KNHO ip&longs;i <lb/>ALCM parallelæ. </s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hinc manife&longs;tum e&longs;t, &longs;i &longs;phæra, vel &longs;phæroides <lb/>&longs;ecetur plano non per centrum: & recta linea &longs;phæ­<lb/>ræ, vel &longs;phæroidis, & factæ &longs;ectionis centra iun­<lb/>gens ad &longs;uperficiem vtrinque producatur; talis <lb/>axis &longs;egmenta e&longs;&longs;e <gap/> portionum, earumque <lb/>vertices extrema dicti axis, vt in figura theorema­<lb/>tis &longs;unt puncta B, D. <!-- KEEP S--></s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si hemi&longs;phærium, vel hemi&longs;phæroides vtcum­<lb/>que ab &longs;ci&longs;&longs;um: & cylindrus, vel cylindri portio <lb/>illi circum&longs;cripta: & conus, vel coni portio, cu­<lb/>ius ba&longs;is e&longs;t eadem &longs;olido circum&longs;cripto, hemi­<lb/>&longs;phærium, vel hemi&longs;phæroides ad verticem <expan abbr="con-ting&etilde;s">con­<lb/>tingens</expan>, & communis axis; &longs;ecentur vnoplano, ba&longs;i <lb/>hemi&longs;phærij, vel hemi&longs;phæroidis parallelo: &longs;uper <lb/>&longs;ectiones autem prædicti coni, vel portionis coni­<lb/>cæ, & hemi&longs;phærij, vel hemi&longs;phæroidis, circa hu­<lb/>ius ab&longs;ci&longs;sæ portionis axem duo cylindri, vel por­<lb/>tiones cylindricæ con&longs;titerint; reliquum cylindri <lb/>vel portionis cylindricæ prædicto plano ab&longs;ci&longs;sæ, <lb/><expan abbr="d&etilde;pto">dempto</expan> eo cylindro <expan abbr="duorũ">duorum</expan> prædictorum, vel portio­<lb/>ne cylindrica, cuius ba&longs;is e&longs;t &longs;ectio hemi&longs;phærij, <lb/>vel hemi&longs;phæroidis, æquale erit reliquo cylindro, <lb/>vel portioni cylindricæ, cuius ba&longs;is e&longs;t &longs;ectio præ­<lb/>dicti coni, vel portionis conicæ. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cuius <lb/>axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diameter AC. <lb/><!-- KEEP S--></s> <s>Et &longs;olido ABC circum&longs;criptus cylindrus, vel portio cy­<lb/>lindrica, cuius ba&longs;es oppo&longs;itæ erunt circuli, vel &longs;imiles elli­<lb/>p&longs;es, quarum diametri eiu&longs;dem rationis ADC, EF, la­<lb/>tera oppo&longs;ita parallelogrammi per axem AFGC: & &longs;u­<lb/>per ba&longs;im, cuius diameter EF, circa axim BD, de&longs;criptus <lb/>e&longs;to conus, vel coni portio EDF. <!-- KEEP S--></s> <s>Iam tria &longs;olida ABC, <lb/>EDF, AC, &longs;ecentur plano &longs;olidi ABC ba&longs;i parallelo, <lb/>quod &longs;ecabit, & &longs;ecet vnà figuras planas per axim BD <pb/>tribus &longs;olidis communem, po&longs;itas in eodem plano, quæ &longs;unt <lb/>AF parallelogrammum, triangulum EDF, & &longs;emicir­<lb/>culus, vel &longs;emi ellip&longs;is ABC: & &longs;int &longs;ectiones rectæ GO, <lb/>HN, KM: hæ igitnr erunt diametri eiu&longs;dem rationis trium <lb/>&longs;ectionum, &longs;cilicet circulorum, vel ellip&longs;ium &longs;irnilium, qui­<lb/>bus erit commune centrum L, in quo nimirum axis BD <lb/>tres dictas lineas GO, HN, KM, bifariam &longs;ecat. </s> <s>Vt <lb/>igitur de &longs;olido AF diximus, &longs;int circa axem BL, & &longs;uper <lb/>ba&longs;es circulos, vel ellip&longs;es circa HN, KM cylindri, vel <lb/>portiones cylindricæ HP, KQ, qui vnà cum portione <lb/>cylindrica, vel cylindro GF ip&longs;a &longs;ectione facto, erunt inter <lb/>eadem plana paral­<lb/>lela per EF, GO. <lb/><!-- KEEP S--></s> <s>Dico trium cylin­<lb/>drorum, vel cylin­<lb/>dri portionum GF, <lb/>HP, KQ, <expan abbr="reliquũ">reliquum</expan> <lb/>ip&longs;ius GF dempto <lb/>HP, ip&longs;i KQ e&longs;se <lb/><figure id="id.043.01.200.1.jpg" xlink:href="043/01/200/1.jpg"/><lb/>æquale. </s> <s>Quoniam <lb/>enim cylindri, & cy­<lb/>lindri portiones eiu&longs;dem altitudinis inter &longs;e &longs;unt vt ba­<lb/>&longs;es, circuli autem, & &longs;imiles ellip&longs;es; inter &longs;e, vt quæ à <lb/>diametris eiu&longs;dem rationis fiunt quadrata; ex Archime­<lb/>de, hoc e&longs;t vt earum quartæ partes, quæ à &longs;emidiame­<lb/>tris quadrata de&longs;cribuntur; erit vt quadratum LO ad <lb/>quadratum LN, ita cylindrus, vel portio cylindrica <lb/>GF ad cylindrum, vel portionem cylindricam PH: & <lb/>diuidendo, vt rectangulum LNO bis vnà cum quadra­<lb/>to NO, ad quadratum LN, ita reliquum cylindri, vel <lb/>portionis cylindricæ GF, dempto ip&longs;o PH, ad ip&longs;um <lb/>PH: &longs;ed vt quadratum LN ad quadratum LM, ita e&longs;t <lb/>vt &longs;upra, cylindrus, vel portio cylindrica HP ad cylin­<lb/>drum, vel portionem cylindricam KQ, ex æquali igitur, <pb/>erit vt rectangulum LNO bis, vnà cum quadrato NO, <lb/>ad quadratum LM, ita reliquum cylindri, vel portionis <lb/>cylindricæ GF <expan abbr="d&etilde;-pto">den­<lb/>pto</expan> HP, ad cylin­<lb/>drum, vel <expan abbr="portion&etilde;">portionem</expan> <lb/>cylindricam KQ: <lb/>&longs;ed rectangulum L <lb/>NO bis vnà <expan abbr="cũ">cum</expan> qua <lb/>drato NO æquale <lb/>e&longs;t quadrato LM; <lb/>reliquum igitur cy­<lb/><figure id="id.043.01.201.1.jpg" xlink:href="043/01/201/1.jpg"/><lb/>lindri, vel portionis <lb/>cylindricæ GF, <expan abbr="d&etilde;-pto">den­<lb/>pto</expan> HP, æquale erit cylindro, vel portioni cylindricæ <expan abbr="Kq.">Kque</expan> <lb/>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Cylindri, vel portionis cylindricæ hemi&longs;phæ­<lb/>rio, vel hemi&longs;phæroidi circum&longs;criptæ reliquum <lb/>dempto hemi&longs;phærio, vel hemi&longs;phæroide, æqua­<lb/>le e&longs;t cono, vel portioni conicæ eandem ba&longs;im he­<lb/>mi&longs;phærio, vel hemi&longs;phæroidi, & eandem altitu­<lb/>dinem habenti. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærio, vel hemi&longs;phæroidi ABC, cu­<lb/>ius axis BD, ba&longs;is circulus, vel ellip&longs;is circa diametrum <lb/>ADC, circum&longs;criptus cylindrus, vel cylindrica portio <lb/>AE, circa communem &longs;cilicet axim BD. conus autem, <lb/>vel coni portio circa axim BD, ba&longs;im habens commu­<lb/>nem &longs;olido ABC, intelligatur. </s> <s>Dico reliquum &longs;olidi <lb/>AE, dempto hemi&longs;phærio, vel hemi&longs;phæroide ABC æ-<pb/>quale e&longs;se cono, vel portioni conicæ. </s> <s>Nam circa axim <lb/>BD, & &longs;uper ba&longs;im circulum, vel ellip&longs;im, cuius diame­<lb/>ter RE, &longs;imilem & oppo&longs;itam ei, quæ circa AC, de&longs;cri­<lb/>batur conus, vel coni portio RDE. <!-- KEEP S--></s> <s>Deinde axe BD bi­<lb/>fariam &longs;ecto, & &longs;ingulis eius partibus rur&longs;us bifariam, vt <lb/>partes axis BD omnes &longs;int æquales, per puncta &longs;ectio­<lb/>num, quotquot erunt, totidem plana parallela &longs;ecent vnà <lb/>cum &longs;olido AE duas ip&longs;ius partes, &longs;olida ABC, RDE. <lb/><!-- KEEP S--></s> <s>Omnes igitur factæ &longs;ectiones, vel erunt circuli, vel &longs;imiles <lb/>ellip&longs;es ei, quæ e&longs;t circa AC, atque adeo inter &longs;e &longs;imiles: <lb/>talium autem &longs;ectiones communes cum AE parallelo, <lb/><figure id="id.043.01.202.1.jpg" xlink:href="043/01/202/1.jpg"/><lb/>grammo per axim, erunt rectæ lineæ, ternæ in &longs;ingu­<lb/>lis planis &longs;ecantibus, & in eadem recta linea; vt in proxi­<lb/>ma ip&longs;i RE, &longs;unt FL, GN, KM, quæ quidem erunt <lb/>trium circulorum, vel &longs;imilium ellip&longs;ium diametri eiu&longs;dem <lb/>rationis ba&longs;ium trium &longs;olidorum, cylindri &longs;cilicet, vel por­<lb/>tionis cylindricæ FL, fru&longs;ti GL, & portionis KBM, he <lb/>mi&longs;phærij, vel hemi&longs;phæroidis ABC. <!-- KEEP S--></s> <s>Itaque circa axem <lb/>BH cylindri, vel portionis cylindricæ FE, & &longs;uper ba­<lb/>&longs;es circulos, vel ellip&longs;es circa GN, KM, de&longs;cribantur <lb/>cylindri, vel cylindri portiones GP, KQ, qui pat­<lb/>tes erunt totius cylindri, vel portionis cylindricæ FE. <lb/><!-- KEEP S--></s> <s>Idem fiat circa reliquas axis partes BD tamquam axes, <pb/>&longs;uper reliquas &longs;ectiones ternas in &longs;ingulis prædictis planis <lb/>&longs;ecantibus. </s> <s>Hac ratione habebimus iam duas figuras <lb/>compo&longs;itas ex cylindris, vel cylindri portionibus altitudi­<lb/>ne, & multitudine æqualibus, alteram cono, vel portioni <lb/>conicæ RDE in&longs;criptam, alteram hemilphærio, vel he­<lb/>mi&longs;phæroidi ABC circum&longs;criptam: quod ita factum e&longs;­<lb/>&longs;e intelligatur, quemadmodum in primo libro fieri po&longs;se <lb/>demon&longs;trauimus, vt figura cono RDE in&longs;cripta ab eo <lb/>deficiat, hemi&longs;phærio autem, vel hemi&longs;phæroidi ABC <lb/>circum&longs;cripta ip&longs;um excedat minori &longs;pacio magnitudine <lb/>propo&longs;ita quantacumque illa &longs;it. </s> <s>Reliquo itaque cylin­<lb/><figure id="id.043.01.203.1.jpg" xlink:href="043/01/203/1.jpg"/><lb/>dri, vel portionis cylindricæ AE dempto hemi&longs;phærio, vel <lb/>hemi&longs;phæroide ABC figura quædam in&longs;cripta relinque­<lb/>tur ex cylindris, vel portionis cylindricæ re&longs;iduis æqualium <lb/>altitudinum, demptis ijs, ex quibus con&longs;tat figura hemi­<lb/>&longs;phærio, vel hemi&longs;phæroidi ABC circum&longs;cripta, excepto <lb/>infimo cylindro, vel portione cylindrica AS. <!-- KEEP S--></s> <s>Et quo­<lb/>niam (excepto exce&longs;su, quo &longs;olidum AS excedit &longs;ui par­<lb/>tem portionem quandam hemi&longs;phærij, vel hemi&longs;phæroidis <lb/>ABC) quo &longs;pacio figura hemi&longs;phærio, vel hemi&longs;phæroidi <lb/>ABC circum&longs;cripta &longs;uperat ip&longs;um hemi&longs;phærium, vel he <lb/>hemi&longs;phæroides, eodem figura prædicto re&longs;iduo in&longs;cripta de­<lb/><gap/>duo; deficiet ab eodem minori differentia quàm <pb/>&longs;it magnitudo propo&longs;ita,. <!--neuer Satz-->His ita ex po&longs;itis, quoniam ex <lb/>præcedenti, reliquum cylindri, vel portionis cylindricæ <lb/>FE dempto cylindro, vel portione cylindrica KQ, æ­<lb/>quale e&longs;t cylindro, vel portioni cylindricæ GP: eadem­<lb/>que ratione &longs;ingula cylindrorum, vel cylindri portionum <lb/>re&longs;idua, quæ &longs;unt in reliqua figura cylindri, vel portionis <lb/>cylindricæ AE, dempto hemi&longs;phærio, vel hemi&longs;phæroi­<lb/>de ABC, æqualia erunt &longs;ingulis cylindris, vel cylindri <lb/>portionibus, quæ &longs;unt in cono, vel portione conica RDE, <lb/>&longs;i bina &longs;umantur inter eadem plana parallela, vel circa <lb/>eundem axem; tota igitur figura in&longs;cripta prædicto re&longs;iduo, <lb/>toti figuræ in&longs;criptæ cono, vel portioni conicæ RDE æ­<lb/>qualis erit: deficit autem vtraque figura in&longs;cripta à &longs;ibi <lb/>circum&longs;cripta minori &longs;pacio quantacumque magnitudine <lb/>propo&longs;ita; per tertiam igitur huius, reliquum cylindri, vel <lb/>portionis cylindricæ AE, dempto hemi&longs;phærin, vel he­<lb/>mi&longs;phæroide ABC, æquale e&longs;t cono, vel portioni coni­<lb/>cæ RDE, hoc e&longs;t ip&longs;i ABC. <!-- KEEP S--></s> <s>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XIV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si hemi&longs;phærium, vel hemi&longs;phæroides, & cylin <lb/>drus, vel portio cylindrica ip&longs;i circum&longs;cripta, & <lb/>conus, vel coni portio, cuius e&longs;t <expan abbr="id&etilde;">idem</expan> axis portioni, <lb/>ba&longs;is autem qu<17> opponitur communi ba&longs;i duorum <lb/>prædictorum &longs;olidorum, vnà &longs;ecentur duobus <lb/>planis ba&longs;i parallelis; portiones reliquæ figuræ <lb/>ex cylindro, vel cylindri portione hemi&longs;phærio, <lb/>vel hemi&longs;phæroidi circum&longs;cripta dempto hemi­<lb/>&longs;phærio, vel hemi&longs;phæroide, quæ à duobus præ­<lb/>dictis planis &longs;ecantibus fiunt, æquales &longs;unt &longs;in­<pb/>gulæ &longs;ingulis prædicti coni, vel conicæ portionis <lb/>partibus &longs;iue fru&longs;tis inter eadem plana parallela <lb/>re&longs;pondentibus. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cu­<lb/>ius axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diame­<lb/>ter ADC. &longs;olido autem ABC circum&longs;criptus cylindrus, <lb/>vel portio cylindrica AXEC: & conus, vel coni portio <lb/>&longs;it XDE, cuius vertex D, ba&longs;is circulus, vel ellip&longs;is cir­<lb/>ca XBE ba&longs;i &longs;olidi AE, vel ABC, prædictæ oppo&longs;ita, <lb/>&longs;ecto autem &longs;olido AE, atque vnà cum ip&longs;o eius partibus, <lb/>&longs;olidis ABC, XD <lb/>E, duobus planis ba <lb/>&longs;i &longs;olidi AE, vel <lb/>ABC, atque ideo <lb/>inter &longs;e quoque pa­<lb/>rallelis, intelligan­<lb/>tur trium &longs;olidorum <lb/>portiones ternæ in­<lb/><figure id="id.043.01.205.1.jpg" xlink:href="043/01/205/1.jpg"/><lb/>ter eadem plana pa­<lb/>rallela: videlicet in­<lb/>ter duo per XE, <lb/>FN, hemi&longs;phærij, vel hemi&longs;phæroidis minor portio HBL: <lb/>& reliquum cylindri, vel portionis cylindricæ FE dem­<lb/>pta portione HBL: & coni, vel conicæ portionis fru&longs;tum <lb/>XGME. &longs;imiliter inter duo plana per FN, OV &longs;olidi <lb/>ABC portio PHLT, eaque ablata reliquum &longs;olidi ON, <lb/>& fru&longs;tum GQSM. <!-- KEEP S--></s> <s>Denique &longs;olidi ABC portio AP <lb/>TC, eaque ablata, reliquum &longs;olidi AV, & conus, vel <lb/>coni portio QDS. </s> <s>Dico reliquum &longs;olidi FE, dempto <lb/>HBL e&longs;&longs;e æquale fru&longs;to XGME: & reliquum &longs;olidi ON <lb/>dempto PHLT, æquale fru&longs;to GQSM: & reliquum <lb/>&longs;olidi AV dempto &longs;olido APTC æquale &longs;olido QDS. <pb/>Quoniam enim vt &longs;upra o&longs;tendimus, reliquum &longs;olidi AE, <lb/>dempto &longs;olido ABC æquale e&longs;se &longs;olido XDE, &longs;imili­<lb/>ter o&longs;ten&longs;um remanet, tam reliquum &longs;olidi AN, dempto <lb/>&longs;olido AHLC, æquale e&longs;se &longs;olido GDM, quam reli­<lb/>quum &longs;olidi AV dempto &longs;olido APTC æquale &longs;olido <lb/>QDS; erit demptis æqualibus, tam reliquum &longs;olidi FE, <lb/>dempto &longs;olido HBL, æquale &longs;olido XGME; quam <lb/>reliquum &longs;olidi ON, dempto &longs;olido PHLT æquale &longs;o­<lb/>lido GQSM. <!-- KEEP S--></s> <s>At reliquum &longs;olidi AV dempto &longs;oli­<lb/>do APTC &longs;olido QDS æquale erit. </s> <s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hemi&longs;phærium, vel hemi&longs;phæroides &longs;ub&longs;e&longs;qui <lb/>alterum e&longs;t cylindri; vel portionis cylindricæ ip&longs;i <lb/>circum&longs;criptæ. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, <lb/>ip&longs;ique circum&longs;criptus cylindrus, vel portio cylindri­<lb/>ca AE, circa eundem &longs;cilicet axem BD, & &longs;uper can­<lb/>dem ba&longs;im circulum, <lb/>vel ellip&longs;im, circa AC: <lb/>nam hac ratione ba&longs;is <lb/>oppo&longs;ita &longs;olidum ABC <lb/>tanget ad verticem B. <lb/></s> <s>Dico <expan abbr="hemi&longs;phæriũ">hemi&longs;phærium</expan>, vel <lb/>hemi&longs;phæroides ABC <lb/>e&longs;se cylindri, vel portio <lb/>nis cylindricæ AE &longs;ub <lb/><figure id="id.043.01.206.1.jpg" xlink:href="043/01/206/1.jpg"/><lb/>&longs;e&longs;quialterum. </s> <s>Nam <lb/>circa axem BD, &longs;uper prædictam ba&longs;em circa AC, e&longs;to <lb/>de&longs;criptus conus, vel coni portio ABC. <!-- KEEP S--></s> <s>Quoniam igitur <pb/>cylindri, vel portionis cylindricæ AE reliquum dempto <lb/>hemi&longs;phærio, vel hemi&longs;phæroide ABC æquale e&longs;t cono, <lb/>vel portioni conicæ ABC: & cylindrus, vel portio cylin­<lb/>drica AE tripla e&longs;t co­<lb/>ni, vel portionis conicæ <lb/>ABC; triplus itidem <lb/>erit cylindrus, vel cylin <lb/>drica portio AE dicti <lb/>re&longs;idui dempto hemi­<lb/>&longs;phærio, vel hemi&longs;phæ­<lb/>roide ABC; ac propte­<lb/>rea hemi&longs;phærij, vel he­<lb/><figure id="id.043.01.207.1.jpg" xlink:href="043/01/207/1.jpg"/><lb/>mi&longs;phæroidis ABC <lb/>&longs;e&longs;quialter, hoc e&longs;t hemi&longs;phærium, vel hemi&longs;phæroides <lb/>ABC cylindri, vel portionis cylindricæ AE &longs;ub&longs;e&longs;quial­<lb/>terum. </s> <s>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis minor portio &longs;phæræ, vel &longs;phæroidis ad <lb/>cylindrum, vel cylindri portionem, cuius ba&longs;is <lb/>æqualis e&longs;t circulo maximo, vel æqualis, & &longs;imi­<lb/>lis ellip&longs;i per centrum ba&longs;i portionis parallelæ, <lb/>& eadem altitudo portioni; eam habet proportio­<lb/>nem, quam rectangulum contentum &longs;phæræ, vel <lb/>&longs;phæroidis dimidij axis axi portionis congruen­<lb/>tis ijs, quæ à centro ba&longs;is portionis fiunt <expan abbr="&longs;egmētis">&longs;egmentis</expan>, <lb/>vnà cum duobus tertiis quadrati axis portionis; ad <lb/>&longs;phæræ, vel &longs;phæroidis dimidij axis quadratum. </s></p><p type="main"> <s>Sit minor portio ABC, &longs;phæræ, vel &longs;phæroidis, cuius <lb/>centrum D, axis autem axi portionis congruens BEDR: <pb/>& cylindrus, vel portio cylindrica FG ab&longs;ci&longs;sa vnà cum <lb/>portione ABC ex cylindro, vel portione cylindrica NO <lb/>circum&longs;cripta hemi&longs;phærio, vel hemi&longs;phæroidi NBO, <lb/>cuius ba&longs;is circa diametrum NO, &longs;it ba&longs;i portionis ABC <lb/>parallela: qua ratione ba&longs;is prædicti &longs;olidi FG, erit vel cir <lb/>culus, vel ellip&longs;is æqualis circulo maximo, vel &longs;imilis, & <lb/>æqualis ellip&longs;i circa NO, portionis ABC ba&longs;i paralle­<lb/>læ. </s> <s>Dico portionem ABC ad cylindrum, vel portio­<lb/>nem cylindricam FG, e&longs;se vt rectangulum BED, vnà <lb/>cum duabus tertiis qua­<lb/>drati EB ad quadratum <lb/>BD. <!-- KEEP S--></s> <s>E&longs;to enim conus, <lb/>vel coni portio HDG, <lb/>cuius fru&longs;tum HKLG <lb/>prædicto plano ab&longs;ci&longs;&longs;um: <lb/>& omnino &longs;int <expan abbr="circulorũ">circulorum</expan>, <lb/>vel ellip&longs;ium &longs;imilium dia <lb/>metri eiu&longs;dem rationis <expan abbr="cũ">cum</expan> <lb/>NO, vt ad XII huius, in <lb/><expan abbr="ead&etilde;">eadem</expan> recta linea tres FM, <lb/>AC, KL, &longs;ectæ omnes bi <lb/>fariam in <expan abbr="cõmuni">communi</expan> <expan abbr="c&etilde;tro">centro</expan> E, <lb/><figure id="id.043.01.208.1.jpg" xlink:href="043/01/208/1.jpg"/><lb/>& HBG, in eodem plano per axem. </s> <s>Quoniam igitur ex &longs;u­<lb/>perioribus, reliquum &longs;olidi FG, dempto ABC, æquale e&longs;t <lb/>fru&longs;to HKLG; erit eiu&longs;dem &longs;olidi FG reliquum ABC <lb/>æquale reliquo &longs;olidi FG, dempto HKLG: &longs;ed hoc reli­<lb/>quum dempto HKLG, &longs;upra o&longs;tendimus e&longs;se ad &longs;olidum <lb/>FG, vt rectangulum ex KL, & differentia HG, vnà <lb/>cum duabus tertiis quadrati differentiæ, ad quadratum <lb/>GH: & vt HG ad KL, ita e&longs;t BD ad DE, propter &longs;imi­<lb/>litudinem triangulorum; vt igitur e&longs;t rectangulum BED, <lb/>vnà cum duabus tertiis quadrati BE, ad quadratum BD, <lb/>ita erit portio ABC, ad cylindrum, vel portionem cylin­<lb/>dricam FG. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portio &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;&longs;a <lb/>duobus planis parallelis, alteroper centrum du­<lb/>cto, ad cy lindrum, vel cylindri portionem, cuius <lb/>ba&longs;is e&longs;t eadem, quæ maior ba&longs;is portionis, & <expan abbr="ead&etilde;">eadem</expan> <lb/>altitudo; eam habet proportionem, quam rectan­<lb/>gulum contentum ijs, quæ à centro minoris ba&longs;is <lb/>fiunt axis &longs;phæræ, vel &longs;phæroidis &longs;egmentis, vnà <lb/>cum duabus tertiis quadrati axis portionis; ad <lb/>&longs;phæræ, vel &longs;phæroidis dimidij axis quadratum. </s></p><p type="main"> <s>Sit portio NACO &longs;phæræ, vel &longs;phærodij, cuius cen­<lb/>trum D, axis autem axi portionis congruens BEDR, <lb/>ab&longs;ci&longs;sa duobus planis parallelis altero per centrum D, &longs;e­<lb/>ctionem faciente circulum <lb/>maximum, vel ellip&longs;im, <lb/>cuius diameter NO, & &longs;u­<lb/>per dictam &longs;ectionem, cir­<lb/>ca axem ED, &longs;tet cylin­<lb/>drus, vel portio cylindrica <lb/>NM, ab&longs;ci&longs;sa ij&longs;dem pla­<lb/>nis, quibus portio NAC <lb/>O, à cylindro, vel portio­<lb/>ne cylindrica NG, &longs;it cir­<lb/>cum&longs;cripta hemi&longs;phærio, <lb/>vel hemi&longs;phæroidi NBO: <lb/>qua ratione erit cylindri, <lb/><figure id="id.043.01.209.1.jpg" xlink:href="043/01/209/1.jpg"/><lb/>vel portionis cylindricæ NM ba&longs;is eadem, quæ maior <lb/>ba&longs;is portionis NACO, circulus &longs;cilicet, vel ellip&longs;is cir­<lb/>ca NO, & eadem altitudo portioni. </s> <s>Dico portionem <pb/>NACO, ad cylindrum, vel portionem cylindricam NM, <lb/>e&longs;se vt rectangulum BER, vnà cum duabus tertiis ED <lb/>quadrati, ad quadratum BD. <!-- KEEP S--></s> <s>Ij&longs;dem enim quæ in præce­<lb/>denti con&longs;tructis, & notatis, &longs;it præterea cylindrus, vel por­<lb/>tio cylindrica PL, circa axim ED circum&longs;cripta cono, <lb/>vel portioni conicæ KDL, Quoniam igitur reliquum <lb/>cylindri, vel portionis cylindricæ NM, dempta portione <lb/>NACO æquale e&longs;t cono, vel portioni conicæ <emph type="italics"/>K<emph.end type="italics"/>DL, <lb/>erit reliqua portio NACO æqualis reliquo eiu&longs;dem NM, <lb/>dempto cono, vel portione conica KDL. </s> <s>Et quoniam cir <lb/>culi, & &longs;imiles ellip&longs;es inter &longs;e &longs;unt vt quadrata diametro­<lb/>rum, vel <expan abbr="&longs;emidiametrorũ">&longs;emidiametrorum</expan> eiu&longs;dem rationis: cylindri autem, <lb/>& portiones cylindricæ <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> altitudinis inter &longs;e vt ba&longs;es; <lb/>erit vt quadratum EM, hoc e&longs;t quadratum BG, ad qua­<lb/>dratum EL, hoc e&longs;t vt quadratum BD ad quadratum <lb/>DE, propter &longs;imilitudinem triangulorum, ita &longs;olidum NM <lb/>ad &longs;olidum PL: & per conuer&longs;ionem rationis, vt quadra­<lb/>tum BD ad rectangulum BED bis, vnà cum quadrato <lb/>BE, ita &longs;olidum MN, ad &longs;ui reliquum dempto &longs;olido <lb/>PL: & conuertendo, vt rectangulum BED bis, vnà cum <lb/>quadrato BE, hoc e&longs;t rectangulum BER, ad quadratum <lb/>BD, ita reliquum &longs;olidi NM dempto &longs;olido PL ad &longs;o­<lb/>lidum NM. Rur&longs;us, quoniam e&longs;t vt quadratum EL ad <lb/>quadratum EM, &longs;iue BG, hoc e&longs;t vt quadratum ED ad <lb/>quadratum BD, ita &longs;olidum PL ad &longs;olidum NM, ob <lb/>&longs;imilem rationem &longs;upradictæ: & duæ tertiæ partes &longs;olidi <lb/>PL e&longs;t &longs;olidum KDL; erit ex æquali, vt duæ tertiæ qua­<lb/>drati ED ad quadratum BD, ita reliquum &longs;olidi PL <lb/>dempto &longs;olido KDL, ad &longs;olidum NM: &longs;ed vt rectangu­<lb/>lum BER ad quadratum BD, ita erat &longs;olidi NM reli­<lb/>quum dempto &longs;olido PL, ad &longs;olidum NM; vt igitur pri­<lb/>ma cum quinta ad &longs;ecundam, ita erit tertia cum &longs;exta ad <lb/>quartam; videlicet, vt rectangulum BED, vnà cum dua­<lb/>bus tertiis ED quadrati ad quadratum BD, ita reliquum <pb/>cylindri, vel portionis cylindricæ NM, dempto cono, vel <lb/>portione conica KDL, hoc e&longs;t portio NACO ip&longs;i æqua­<lb/>lis, ad cylindrum, vel portionem cylindricam NM. <lb/><!-- KEEP S--></s> <s>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portio &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;&longs;a <lb/>duobus planis parallelis, neutro per centrum du­<lb/>cto, nec centrum intercipientibus, ad cylindrum, <lb/>vel cylindri portionem, cuius ba&longs;is æqualis e&longs;t <lb/>circulo maximo, vel ellip&longs;i per centrum ba&longs;ibus <lb/>portionis parallelæ &longs;imilis, & æqualis, eam ha­<lb/>bet proportionem, quam duo rectangula; & quod <lb/>&longs;phæræ, vel &longs;phæroidis axis axi portionis <expan abbr="congru&etilde;">congruem</expan> <lb/>tis ijs, quæ à centro minoris ba&longs;is portionis fiunt <lb/><expan abbr="&longs;egm&etilde;tis">&longs;egmentis</expan>, & quod ea, quæ maioris ba&longs;is portionis, <lb/>& &longs;phæræ, vel &longs;phæroidis centra iungit, & axe por <lb/>tionis continetur, vnà cum duabus tertijs quadra­<lb/>ti axis portionis; ad &longs;phæræ, vel &longs;phæroidis dimi­<lb/>dij axis quadratum. </s></p><p type="main"> <s>Sit portio AQTC &longs;phæræ, vel &longs;phæroidis, cuius cen­<lb/>trum D, axis autem axi portionis congruens BSEDR, <lb/>ab&longs;ci&longs;&longs;um duobus planis parallelis, neutro per centrum <lb/>D acto, nec ip&longs;um intercipientibus: & circa portionis <lb/>axim SE &longs;tet cylindrus, vel portio cylindrica FX ab­<lb/>&longs;ci&longs;sa vnà cum portione AQTC ex toto cylindro, vel <lb/>portione cylindrica NG, hemi&longs;phærio, vel hemi&longs;phæroi­<lb/>di NBO circum&longs;cripta, cuius ba&longs;is circulus maximus <pb/>vel ellip&longs;is circa NO ba&longs;ibus AQTC portionis parallelæ <lb/>qua ratione cylindrus, vel portionis cylindricæ FX eiu&longs;­<lb/>dem altitudinis portioni AQTC, ba&longs;is erit circulus <lb/>æqualis circulo maximo, vel ellip&longs;is &longs;imilis, & æqualis ei, <lb/>cuius diameter NDO, ba&longs;ibus AQTC portionis paral­<lb/>lelæ. </s> <s>Dico portionem AQTC ad cylindrum, vel por­<lb/>tionem cylindricam FX, e&longs;&longs;e vt duo rectangula BSR, <lb/>DES, vnà cum duabus tertiis quadrati ES, ad quadra­<lb/>tum BD. <!-- KEEP S--></s> <s>Ij&longs;dem enim con&longs;tructis, & notatis, quæ in an­<lb/>tecedenti, excepto cylindro, vel portione cylindrica, quæ <lb/>circa axim ED &longs;teterat: <lb/>planum præterea minoris <lb/>ba&longs;is QT portionis AQ <lb/>TC extendatur: & &longs;e­<lb/>cans tria &longs;olida, & figuras <lb/>planas per axim po&longs;itas in <lb/>eodem plano, faciat ternas <lb/>&longs;ectiones, circulos, vel elli­<lb/>p&longs;es &longs;imiles ei, quæ e&longs;t cir­<lb/>ca NO: & earum diame­<lb/>tros IX, PV, QT, in <lb/>eadem recta linea commu­<lb/>ni &longs;ectione exten&longs;i plani, & <lb/><figure id="id.043.01.212.1.jpg" xlink:href="043/01/212/1.jpg"/><lb/>eius, quod per axem: quæ quidem diametri &longs;ectæ erunt om­<lb/>nes bifariam in centro S communi trium prædictarum pla­<lb/>narum <expan abbr="&longs;ectionũ">&longs;ectionum</expan>. </s> <s>Denique coni, vel portionis conicæ HDG <lb/>fru&longs;to PKIV ab&longs;ci&longs;&longs;o vnà cum portione AQTC, &longs;it <lb/>circa axim SE circum&longs;criptus cylindrus vel portio cylin­<lb/>drica ZV. </s> <s>Quoniam igitur per XIIII huius, reliquum <lb/>&longs;olidi FX, dempta portione AQTC, æquale e&longs;t fru&longs;to <lb/>PKLV; erit reliqua portio AQTC, reliquo eiu&longs;dem <lb/>&longs;olidi FX, dempto fru&longs;to PKLV æqualis. </s> <s>Et quoniam <lb/>e&longs;t vt PV ad KL, ita SD, DE, propter &longs;imilitudinem <lb/>triangulorum: & vt rectangulum ex KL, & differentia <pb/>ip&longs;ius PV, vnà cum duabus tertiis quadrati eiu&longs;dem dif­<lb/>ferentiæ, ad quadratum PV, ita e&longs;t reliquum &longs;olidi ZV <lb/>dempto fru&longs;to PKLV ad &longs;olidum ZV; erit vt rectangu­<lb/>lum DES, vnà cum duabus tertiis quadrati ES, ad DS <lb/>quadratum, ita &longs;olidi ZV reliquum dempto fru&longs;to PK <lb/>LV ad &longs;olidum ZV: &longs;ed vt quadratum DS ad quadra­<lb/>tum DB, hoc e&longs;t vt quadratum SV ad quadratum BG, <lb/>ide&longs;t ad quadratum SX, ita e&longs;t &longs;olidum ZV, ad &longs;olidum <lb/>FX; ex æquali igitur, vt rectangulum DES, vnà cum <lb/>duabus tertiis ES quadrati, ad quadratum BD, ita e&longs;t <lb/>reliquum &longs;olidi ZV, dem <lb/>pto &longs;olido PKLV ad &longs;o <lb/>lidum FX: &longs;ed vt rectan­<lb/>gulum BSR ad quadra­<lb/>tum BD, ita e&longs;t, eadem <lb/>ratione, qua in præcedenti <lb/>theoremate vtebamur, re­<lb/>liquum &longs;olidi FX dem­<lb/>pto &longs;olido ZV, ad &longs;oli­<lb/>dum FX; vt igitur prima <lb/>cum quinta ad &longs;ecundam, <lb/>ita tertia cum &longs;exta ad <lb/>quartam; videlicet, vt duo <lb/><figure id="id.043.01.213.1.jpg" xlink:href="043/01/213/1.jpg"/><lb/>rectangula BSR, DES, vnà cum duabus tertiis quadra­<lb/>ti ES ad quadratum BD, ita erit totum reliquum cylin­<lb/>dri, vel portionis cylindricæ FX dempto fru&longs;to PKLV: <lb/>hoc e&longs;t &longs;phæræ, vel &longs;phæroidis portio AQTC ad cylin­<lb/>drum, vel portionem cylindricam FX. </s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis maior portio &longs;phæræ, vel &longs;phæroidis, <lb/>ad cylindrum, vel portionem cylindricam, cuius <lb/>ba&longs;is æqualis e&longs;t circulo maximo, vel æqualis, & <lb/>&longs;imilis ellip&longs;i per centrum ba&longs;i portionis paralle­<lb/>læ, altitudo autem eadem portioni, eam habet <lb/>proportionem, quam &longs;olidum rectangulum con­<lb/>tentum axe portionis, & reliquo axis &longs;phæræ, vel <lb/>&longs;phæroidis &longs;egmento, & eo, quod ba&longs;is portionis, <lb/>& &longs;phæræ, vel &longs;phæroidis centraiungit, vnà cum <lb/>binis tertiis partibus duorum cuborum: & eius <lb/>qui à &longs;phæræ, vel &longs;phæroidis axis dimidio; & <lb/>cius qui ab eo, quod &longs;phæræ, vel &longs;phæroidis, & <lb/>ba&longs;is portionis centra iungit &longs;it &longs;egmento; ad &longs;o­<lb/>lidum rectangulum, quod axe portionis, & duo­<lb/>bus &longs;phæræ, vel &longs;phæroidis axis fit dimidijs. </s></p><p type="main"> <s>Sit maior portio AB <lb/>C, &longs;phæræ, vel &longs;phæroi­<lb/>dis ABCF, cuius cen­<lb/>trum D: ba&longs;is <expan abbr="aut&etilde;">autem</expan> por­<lb/>tionis, circulus, vel elli­<lb/>p&longs;is, cuius diameter A <lb/>C: Et &longs;ecta portione <lb/>ABC per centrum D <lb/>plano ba&longs;i AC paral­<lb/>lelo, qua ratione &longs;ectio <lb/>erit circulus maximus, <lb/>vel ellip&longs;is &longs;imilis ba&longs;i <lb/><figure id="id.043.01.214.1.jpg" xlink:href="043/01/214/1.jpg"/><pb/>portionis: e&longs;to ea cuius diameter KL, iungensque recta <lb/>DE &longs;phæræ, vel &longs;phæroidis, & ba&longs;is portionis centra DE, <lb/>atque producta incidat in &longs;phæræ, vel &longs;phæroidis &longs;uperfi­<lb/>ciem ad partes E in puncto F, & ad partes oppo&longs;itas in <lb/>puncto B: &longs;phæræ igitur, vel &longs;phæroidis axis axi portionis <lb/>BE congruens crit BDEF, nam vertex portionis erit B: <lb/>& hemi&longs;phærio, vel hemi&longs;phæroidi KBL &longs;it circum&longs;cri­<lb/>ptas cylindrus, vel cylindrica portio KH, cuius &longs;cilicet <lb/>axis BD, & circa axim DE, alter cylindrus, vel portio <lb/>cylindrica GL portioni KACL circum&longs;cripta: quorum <lb/>circum&longs;criptorum &longs;olido­<lb/>rum vtriulque communis <lb/>ba&longs;is erit circulus, vel <lb/>ellip&longs;is circa KL. <!-- KEEP S--></s> <s>Ita­<lb/>que ex his compo&longs;itus to­<lb/>tus cylindrus, vel cylin­<lb/>dri portio GH erit por­<lb/>tioni ABC circum&longs;cri­<lb/>pta, habens axim BE, at­<lb/>que ideo eandem altitu­<lb/>dinem ABC portioni, <lb/>ba&longs;im autem, cuius dia­<lb/>meter &longs;it GM &longs;imilem <lb/><figure id="id.043.01.215.1.jpg" xlink:href="043/01/215/1.jpg"/><lb/>& æqualem ei, quæ e&longs;t circa KL. <!-- KEEP S--></s> <s>Dico portionem ABC <lb/>ad cylindrum, vel portionem cylindricam GH, e&longs;se vt &longs;o­<lb/>lidum rectangulum contentum ip&longs;is BE, EF, ED, vnà <lb/>cum binis tertiis duorum cuborum, duabus &longs;cilicet cubi <lb/>BD, & totidem cubi ED, ad &longs;olidum rectangulum con­<lb/>tentum ip&longs;is EB, BD, DF. <!-- KEEP S--></s> <s>Quoniam enim parall ele­<lb/>pipeda eiu&longs;dem altitudinis inter &longs;e &longs;unt vt ba&longs;es, erit vt re­<lb/>ctangulum BEF vnà cum duabus tertiis ED quadrati ad <lb/>rectangulum BDF, ide&longs;t ad quadratum BD, &longs;iue DF, <lb/>ita &longs;olidum ex BE, EF, ED, communi altitudine DE, <lb/>vnà cum duabus tertiis cubi ED, ad &longs;olidum ex DE, <pb/>BD, DF: &longs;ed vt rectangulum BEF, vnà cum duabus <lb/>DE quadrati, ad quadratum DF, ita o&longs;tendimus e&longs;&longs;e <lb/>portionem AKLC ad &longs;olidum GL; vt igitur e&longs;t &longs;olidum <lb/>ex BE, EF, ED, vnà cum duabus tertiis cubi ED, com <lb/>muni altitudine DE, ad &longs;olidum ex ED, BD, DF, ita <lb/>erit portio AKLC ad &longs;olidum GL: &longs;ed vt &longs;olidum ex <lb/>ED, DB, DF, hoc e&longs;t id, cuius altitudo ED, ba&longs;is BD <lb/>quadratum, ad &longs;olidum ex EB, BD, DF, hoc e&longs;t ad id, <lb/>cuius altitudo BE, ba&longs;is quadratum BD, ita e&longs;t altitudo, <lb/>vel latus ED, ad altitudinem vel latum BE: hoc e&longs;t &longs;oli­<lb/>dum GL ad &longs;olidum GH; quippe quorum dictæ lineæ <lb/>ED, BE &longs;unt axes; ex æquali igitur, vt &longs;olidum ex BE, <lb/>EF, ED, vnà cum duabus tertiis cubi DE, ad &longs;olidum <lb/>ex EB, BD, DE, cuius altitudo EB, ba&longs;is quadratum <lb/>BD, ita erit portio AKLC ad &longs;olidum GH. Rur&longs;us, <lb/>quoniam &longs;olidum HK e&longs;t hemi&longs;phærij, vel hemi&longs;phæroi­<lb/>dis KBL &longs;e&longs;quialterum; erit vt duæ tertiæ partes cubi BD <lb/>ad cubum BD, ita hemi&longs;phærium, vel hemi&longs;phæroides <lb/>KBL ad &longs;olidum KH: &longs;ed vt cubus BD ad &longs;olidum ex <lb/>BD, DF, & altitudine BE, hoc e&longs;t vt altitudo BD ad <lb/>altitudinem BE, ita e&longs;t &longs;olidum KH ad &longs;olidum GH, quo­<lb/>rum dictæ altitudines BD, BE &longs;unt axes, ex æquali igitur <lb/>erit vt duæ tertiæ partes cubi BD ad &longs;olidum ex EB, BD, <lb/>DF, ita hemi&longs;phærium, vel hemi&longs;phæroides KBL, ad &longs;oli­<lb/>dum GH: &longs;ed vt <expan abbr="&longs;olidũ">&longs;olidum</expan> ex BE, EF, ED, vna cum duabus <lb/>tertiis cubi ED ad &longs;olidum ex EB, BD, DF, erat por­<lb/>tio AKLC ad cylindrum GH; vt igitur prima cum quin <lb/>ta ad &longs;ecundam, ita tertia cum &longs;exta ad quartam, videlicet, <lb/>vt duæ tertiæ cubi BD, vna cum duabus tertiis cubi BE, <lb/>& &longs;olido ex BE, EF, ED ad &longs;olidum ex EB, BD, DF, <lb/>ita erit &longs;phæræ, vel &longs;phæroidis maior portio ABC ad &longs;oli­<lb/>dum, cylindrum &longs;cilicet, vel portionem cylindricam GH. <lb/><!-- KEEP S--></s> <s>Quod erat demon&longs;trandum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portio &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;sa <lb/>duobus planis parallelis centrum intercipienti­<lb/>bus, ad cylindrum, vel cylindri portionem, cuius <lb/>ba&longs;is æqualis e&longs;t circulo maximo, vel &longs;imilis, & <lb/>æqualis ellip&longs;i per centrum ba&longs;ibus portionis pa­<lb/>rallelæ, & eadem altitudo portioni, eam habet <lb/>proportionem, quam duo &longs;olida rectangula ex ter­<lb/>norum &longs;phæræ, vel &longs;phæroidis axis &longs;egmentorum <lb/>eundem terminum habentium alterutrius ba­<lb/>&longs;ium portionis centrum, binis &longs;phæræ, vel &longs;phæ­<lb/>roidis axem complentibus, & &longs;ingulis axis por­<lb/>tionis itidem à centro &longs;phæræ, vel &longs;phæroidis fa­<lb/>ctis, vnà cum binis tertijs partibus duorum cubo­<lb/>rum ex &longs;egmentis axis portionis à centro &longs;phæræ, <lb/>vel &longs;phæroidis factis; ad &longs;olidum rectangulum, <lb/>quod duobus &longs;phæræ, vel &longs;phæroidis axis dimi­<lb/>diis, & axe portionis continetur. </s></p><p type="main"> <s>Sit portio ABCD &longs;phæræ, vel &longs;phæroidis, cuius cen­<lb/>trum E, axis portionis KEH: ip&longs;i autem portioni cir­<lb/>cum&longs;criptus cylindrus, vel cylindrica portio NO, vt in <lb/>antecedenti, cuius communis &longs;ectio cum &longs;phæra, vel &longs;phæ­<lb/>roide AFDG, &longs;it circulus maximus, vel ellip&longs;is circa dia­<lb/>metrum LEM; quamobrem ba&longs;is &longs;olidi NO, eiu&longs;dem <lb/>altitudinis portioni ABCD circulus erit æqualis circu­<lb/>lo maximo, vel ellip&longs;is æqualis, & &longs;imilis ellip&longs;i circa LM <lb/>ba&longs;ibus portionis parallelæ. </s> <s>Dico portionem ABCD <pb/>ad cylindrum, vel cylindri portionem NO, e&longs;se vt duo <lb/>&longs;olida ad rectangula, alterum ex FH, HG, EH: alterum <lb/>ex GK, KF, EK, vnà cum binis tertiis duorum cubo­<lb/>rum ex EK, EH, ad &longs;olidum rectangulum ex GE, <lb/>EF KH, axe enim KH producto vt incidat in &longs;uper­<lb/>ficiem in punctis F, G, &longs;it &longs;phæræ, vel &longs;phæroidis, ex <lb/>demon&longs;tratis, axis FK, EHG. </s> <s>Intelliganturque vt in <lb/>antecedenti duo cylindri, vel cylindri portiones NM, <lb/>LO, totius prædicti &longs;olidi NO: itemque duæ portiones <lb/>&longs;phæræ, vel &longs;phæroidis ALMD, LBCM, quorum qua­<lb/>tuor &longs;olidorum commu <lb/>nis ba&longs;is e&longs;t circulus, vel <lb/>ellip&longs;is circa LEM. <lb/></s> <s>Quoniam igitur vt in <lb/>antecedenti o&longs;tendere­<lb/>mus portionem ALM <lb/>D ad &longs;olidum NM e&longs; <lb/>&longs;e vt &longs;olidum ex FH, <lb/>HG, EH, vnà cum <lb/>duabus tertiis cubi EH <lb/>ad &longs;olidum ex FE, EG, <lb/>EH, communi altitu­<lb/>dine EH: &longs;ed vt &longs;oli­<lb/>dum ex FE, EG, EH, <lb/><figure id="id.043.01.218.1.jpg" xlink:href="043/01/218/1.jpg"/><lb/>altitudine EH, ad &longs;olidum ex FE, EG, KH altitudi­<lb/>ne KH, ita e&longs;t altitudo EH ad altitudinem KH, hoc <lb/>e&longs;t &longs;olidum NM ad &longs;olidum NO, quippe quorum &longs;unt <lb/>axes EH, KH; ex æquali igitur erit vt &longs;olidum ex FH, <lb/>HG, EH, vnà cum duabus tertiis cubi EH, ad &longs;oli­<lb/>dum ex FE, EG, KH, ita portio ALMD, ad &longs;oli­<lb/>dum NO. <!-- KEEP S--></s> <s>Eadem ratione o&longs;tenderemus e&longs;&longs;e, vt &longs;olidum <lb/>ex GK, KF, EK, vnà cum duabus tertiis cubi EK, ad <lb/>&longs;olidum ex FE, EG, KH, ita portionem LBCM, ad <lb/>&longs;olidum NO; vt igitur prima cum quinta ad &longs;ecundam, <pb/>ita tertia cum &longs;exta ad quartam; videlicet, vt duo &longs;oli­<lb/>da, & quod &longs;it ex FH, <lb/>HG, EH, & quod <lb/>ex GK, KF, EK, vnà <lb/>cum duabus tertiis & <lb/>cubi ex EH, & cu­<lb/>bi ex EK, ad &longs;olidum <lb/>ex FE, EG, KH, ita <lb/>erit tota &longs;phæræ, vel <lb/>&longs;phæroidis portio AB <lb/>CD, ad cylindrum, vel <lb/>portionem cylindricam <lb/>NO. <!-- KEEP S--></s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><figure id="id.043.01.219.1.jpg" xlink:href="043/01/219/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis trianguli comprehen&longs;i &longs;ectione para­<lb/>bola, ex duabus rectis lineis, quarum altera &longs;e­<lb/>ctionem tangat, altera in eam incidat diametro <lb/>&longs;ectionis ex contactu æquidi&longs;tans, centrum graui­<lb/>tatis e&longs;t punctum illud, in quo recta linea ex con­<lb/>tactu diuidens incidentem ita vt pars, quæ &longs;ectio­<lb/>nem attingit &longs;it &longs;e&longs;quialtera reliquæ, &longs;ic diui­<lb/>ditur, vt pars quæ e&longs;t ad contactum &longs;it tripla <lb/>reliquæ. </s></p><p type="main"> <s>Sit triangulum ABC comprehen&longs;um &longs;ectione parabo­<lb/>la ADB, & duabus rectis lineis, quarum altera AC tan­<lb/>gat &longs;ectionem in puncto A, reliqua autem BC, in eam <lb/>incidens in puncto B, &longs;ectionis diametro ex puncto A, <lb/>æquidi&longs;tans intelligatur: & per centrum grauitatis trian-<pb/>guli ABC quod &longs;it F, &longs;it ducta recta AFE. </s> <s>Dico AF <lb/>e&longs;&longs;e ip&longs;ius FE triplam: at BE ip&longs;ius EC &longs;e&longs;quialteram. <lb/></s> <s>Completo enim triangulo rectilineo ABC, &longs;ectis que re­<lb/>ctis lineis bifariam AB in puncto H, & AC in puncto K <lb/>ducatur HDK, quæ parallela erit ba&longs;i BC: parabolæ igi­<lb/>tur &longs;egmenti BDA dia meter erit DH; in qua parabolæ <lb/>ADB, cuius vertex D &longs;it centrum grauitatis M: trian­<lb/>guli autem rectilinei ABC centrum grauitatis N, & iun <lb/>gatur MN: producta igitur MN occurret trianguli ABC <lb/>mixti centro grauitatis F. &longs;int igitur centra M, N, F, in <lb/>eadem recta linea: <lb/>& ducta recta AN <lb/>G &longs;ecet ba&longs;im BC <lb/>bifariam in G pun <lb/>cto, nece&longs;&longs;e e&longs;t e­<lb/>nim: & ex puncto <lb/>F ad rectam AG, <lb/>ducatur recta FO <lb/>ip&longs;is BC, KH pa <lb/>rallela, & BD, DA <lb/>iungantur. </s> <s><expan abbr="Quoniã">Quoniam</expan> <lb/>igitur AG &longs;ecat <lb/>BC, KH paral­<lb/>lelas in rectolineo <lb/>triangulo ABC, <lb/><figure id="id.043.01.220.1.jpg" xlink:href="043/01/220/1.jpg"/><lb/>in ea&longs;dem rationes; &longs;ecta erit HK bifariam à linea AG: <lb/>cumque HD diameter parabolæ ADC, cuius vertex D, <lb/>&longs;it parallela diametro parabolæ, cuius vertex A, atque <lb/>ideo etiam BC incidenti parallela, erit DH pars ip&longs;ius <lb/>KH: quoniam igitur in triangulo mixto ABC recta KD <lb/>applicata parallela e&longs;t ip&longs;i BC, quæ itidem e&longs;t parallela <lb/>diametro parabolæ, cuius vertex A; erit vt AC ad AK <lb/>potentia, ita BC ad DK longitudine, quod &longs;upra demon­<lb/>&longs;trauimus: &longs;ed AC quadrupla e&longs;t potentia ip&longs;ius AK; <pb/>quadrupla igitur BC ip&longs;ius DK: cum igitur BC &longs;it <lb/>dupla ip&longs;ius KH, erit DK dimidia eiu&longs;dem KH, & &longs;ecta <lb/>bifariam KH in puncto D: &longs;ed recta AG &longs;ecabat eandem <lb/>KH bi fariam; per punctum igitur D tran&longs;ibit AG. <!-- KEEP S--></s> <s>Quo­<lb/>niam igitur parabola ADC, cuius vertex D, &longs;e&longs;quiter­<lb/>tia e&longs;t per Archimedem trianguli ADB, cuius duplum <lb/>e&longs;t triangulum ABG, &longs;icut & huius triangulum ABC; <lb/>triangulum ABC quadruplum erit trianguli ADB: qua­<lb/>lium igitur partium æqualium e&longs;t triangulum ABC duo­<lb/>decim, talium erit triangulum ADB trium, & parabola <lb/>ADB, cuius ver­<lb/>tex D quatuor: du <lb/>plum igitur erit tri­<lb/>angulum ABC <lb/>mixtum parabolæ <lb/>ADB, cuius ver­<lb/>tex D, & cen­<lb/>trum grauitatis M: <lb/>&longs;ed trianguli ABC <lb/>rectilinei e&longs;t cen­<lb/>trum grauitatis N, <lb/>& F <expan abbr="triãguli">trianguli</expan> ABC <lb/>mixti; dupla igitur <lb/>erit MN ip&longs;ius N <lb/>F, & MD ip&longs;ius <lb/><figure id="id.043.01.221.1.jpg" xlink:href="043/01/221/1.jpg"/><lb/>OF, & DN ip&longs;ius NO, propter &longs;imilitudinem triangulo­<lb/>rum: &longs;ed & tota AN dupla e&longs;t totius NG, ob centrum <lb/>grauitatis N rectilinei trianguli ABC; reliqua igitur AD <lb/>dupla e&longs;t reliquæ GO. cum igitur AG &longs;it dupla ip&longs;ius <lb/>AD, quadrupla erit AG ip&longs;iu&longs;que GO. quare & quadru <lb/>pla AE ip&longs;ius FE ob parallelas: tripla igitur AF ip&longs;ius FE. <lb/><!-- KEEP S--></s> <s>Rur&longs;us quoniam ex Archimede &longs;e&longs;quialtera e&longs;t DM ip&longs;ius <lb/>MH, erit tota DH ad DM vt quinque ad tria, hoc e&longs;t <lb/>vt decem ad &longs;ex: &longs;ed MD erat dupla ip&longs;ius OF; tota igi-<pb/>tur DH ad OF erit vt decem ad tria: &longs;ed GC dupla <lb/>e&longs;t ip&longs;ius DH; igitur GC ad FO vt viginti ad tria: &longs;ed <lb/>quia tripla exi&longs;tente AO ip&longs;ius OG, e&longs;t tota AG ip&longs;ius <lb/>AO &longs;e&longs;quitertia, erit quoque GE, ip&longs;ius OF &longs;e&longs;quiter­<lb/>tia, propter &longs;imilitudinem triangulorum AGE, AOF, <lb/>hoc e&longs;t qualium partium æqualium OF trium, talium GE <lb/>quatuor; qualium e&longs;t GC hoc e&longs;t BG viginti, talium <lb/>erit EG quatuor, & EC &longs;exdecim: dempta igitur EG <lb/>ex GC, & addita ip&longs;i BG, qualium e&longs;t EC &longs;exdecim: <lb/>talium erit BE vigintiquatuor: &longs;ed vt vigintiquatuor ad <lb/>&longs;exdecim, ita &longs;unt tria ad duo, quæ proportio e&longs;t &longs;e&longs;qui­<lb/>altera, &longs;e&longs;quialtera igitur erit BE ip&longs;ius EC, o&longs;ten&longs;a e&longs;t <lb/>autem AF ip&longs;i FE tripla. </s> <s>Manife&longs;tum e&longs;t igitur pro­<lb/>po&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si duo triangula mixta prædicti generis verti­<lb/>cem communem habeant, qui e&longs;t contactus, & <lb/>ba&longs;es æquales in eadem recta linea, vel continuas, <lb/>vel &longs;egmento interiecto, tota extra &longs;iguram ver&longs;a <lb/>cauitate; centrum grauitatis compo&longs;iti ex vtro­<lb/>que e&longs;t pun ctum illud, in quo recta linea à vertice <lb/>ad bipartitæ rectæ prædictis &longs;ectionibus interce­<lb/>ptæ, in qua &longs;unt ba&longs;es dictorum triangulorum &longs;e­<lb/>ctionis punctum pertinens &longs;ic diuiditur; vt pars, <lb/>quæ e&longs;t ad verticem &longs;it tripla reliquæ. </s></p><p type="main"> <s>Sint duo prædicti generis triangula ABC, ADE ha­<lb/>bentia verticem A communem, qui e&longs;t contactus recta. <lb/></s> <s>rum cum parabolis, tangente AB parabolam AC, & <pb/>AD parabolam AE: ba&longs;es autem æquales BC, DE pa­<lb/>rallelas parabolarum diametres per A, & in vna recta li­<lb/>nea CE &longs;egmento BD interiecto: vtriu&longs;que autem &longs;e­<lb/>ctionis AC, AE concauitas &longs;pectet extra figuram ACE: <lb/>&longs;ecta autem CE bifariam in F, iunctaque AF, ponatur <lb/>AG tripla ip&longs;ius GF. <!-- KEEP S--></s> <s>Dico compo&longs;iti ex triangulis A <lb/>BC, ADE centrum grauitatis e&longs;&longs;e G. <!-- KEEP S--></s> <s>Po&longs;ita enimvtra­<lb/>que &longs;e&longs;quialtera, CH ip&longs;ius HB, & EK ip&longs;ius KD, <lb/>iunctisque AH, AK, ducatur per punctum G ip&longs;i CE <lb/>parallela &longs;ecans AH, AK in punctis L, M. </s> <s>Quoniam <lb/>igitur LM ip&longs;i CE parallela &longs;ecat eas quæ ex puncto A <lb/>ad rectam CD du­<lb/>cuntur rectas lineas <lb/>in ea&longs;dem rationes, & <lb/>e&longs;t AG tripla ip&longs;ius <lb/>GF; tripla erit vtra­<lb/>que AL ip&longs;ius LH, <lb/>& AM ip&longs;ius MK: <lb/>&longs;e&longs;quialtera autem e&longs;t <lb/>CH ip&longs;ius HB, & <lb/>EK ip&longs;ius KD; erit <lb/>igitur L centrum gra<lb/>uitatis trianguli AB <lb/>C, & M trianguli A <lb/>DE per præceden­<lb/><figure id="id.043.01.223.1.jpg" xlink:href="043/01/223/1.jpg"/><lb/>tem. </s> <s>Rur&longs;us quoniam ab&longs;oluantur triangula rectilineæ <lb/>ACB, AEK, & æqualia erunt propter æquales ba&longs;es, <lb/>po&longs;ita inter ea&longs;dem parallelas, & vtrumque &longs;e&longs;quialterum <lb/>eius trianguli mixti, quod comprehendit, ex demon&longs;tra­<lb/>tione antecedentis; æqualia igitur erunt triangula mixta <lb/>ABC, ADE, &longs;iquidem &longs;unt æqualium &longs;ub&longs;e&longs;quialtera. <lb/></s> <s>Et quoniam componendo, & permutando e&longs;t vt CB ad <lb/>DE ita BH ad DK, æqualis erit BH ip&longs;i DK: &longs;ed &longs;i ab <lb/>æqualibus po&longs;itis CF, FE ip&longs;as CB, DE æquales au-<pb/>feras, reliquæ BF, FD æquales erunt; tota igitur FH to­<lb/>ti FK æqualis e&longs;t: in triangulo autem AHK recta AF <lb/>&longs;ecat LM, HK parallelas in ea&longs;dem rationes; erit igitur <lb/>LG æqualis ip&longs;i GM; cum igitur æqualium triangulo­<lb/>rum ABC, ADE centra grauitatis &longs;int L, M; erit com <lb/>po&longs;iti ex vtroque centrum grauitatis G. <!-- KEEP S--></s> <s>Idem o&longs;tendere­<lb/>mus, quod proponitur, & &longs;i ba&longs;es prædictorum triangulo­<lb/>rum &longs;int continuæ. </s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si duæ parabolæ in eodem plano circa æqua­<lb/>les diamet ros in directum inter &longs;e con&longs;titutas, ita <lb/>vt vertices &longs;int extrema ex diametris compo&longs;itæ, <lb/>communem habuerint aliquam ordinatim ad dia <lb/>metrum applicatarum, & vertices cum puncto con <lb/>uenientiæ iungantur rectis lineis: centrum gra­<lb/>uitatis v triu&longs;que portionis ijs rectis lineis ab &longs;ci&longs; <lb/>&longs;æ, rectam lineam, quæ terminum communem <lb/>diamctrorum, & concur&longs;um parabolarum iungit <lb/>bifariam diuidit. </s></p><p type="main"> <s>Circa æquales <lb/>diametros AD, <lb/>DC indirectum <lb/>inter &longs;e con&longs;titutas, <lb/>verticibus A, C, <lb/>duæ parabolæ in <lb/>eodem plano <expan abbr="com-mun&etilde;">com­<lb/>munem</expan> habeant ali­<lb/>quam BD ordi­<lb/><figure id="id.043.01.224.1.jpg" xlink:href="043/01/224/1.jpg"/><pb/>natim ad vtramque diametrorum applicatarum, iunctis­<lb/>que AB, BC, &longs;it &longs;ecta BD bifariam in puncto G. <lb/><!-- KEEP S--></s> <s>Dico G e&longs;se centrum grauita tis duarum portionum AEB, <lb/>BFE &longs;imul. </s> <s>Si enim hoc non e&longs;t, &longs;it aliud punctum L. & <lb/>compleantur parallelogramma ANBD, DBRC, hoc <lb/>e&longs;t totum AR parallelogrammum: & &longs;ecta BG bifariam <lb/>in puncto H, ponatur DK ip&longs;ius BD pars tertia, vt pun­<lb/>ctum K &longs;it trianguli ABC centrum grauitatis. </s> <s>Po&longs;ita au­<lb/>tem &longs;e&longs;quialtera BP ip&longs;ius PN, & BQ ip&longs;ius QR, iun­<lb/>ctisque AP, CQ, duoatur per punctum H ip&longs;i AC, vel <lb/>NR parallela, cum ip&longs;is AP, CQ conueniens in punctis <lb/>ST: & iuncta LG, <lb/>&longs;i punctum L non <lb/>&longs;it in linea BD, <lb/>e&longs;to LM quintu­<lb/>pla ip&longs;ius MG. <lb/></s> <s>Quoniam igitur ob <lb/>parallelas AC, P <lb/>Q, ST in trape­<lb/>zio APQC, e&longs;t <lb/>vt DH ad HB, ita <lb/>AS ad SP, & CT <lb/><figure id="id.043.01.225.1.jpg" xlink:href="043/01/225/1.jpg"/><lb/>ad TQ, erit AS ip&longs;ius SP, & CT ip&longs;ius TQ tripla: <lb/>&longs;ed e&longs;t BP &longs;e&longs;quialtera ip&longs;ius PN, & BQ ip&longs;ius QR; <lb/>mixti igitur trianguli ANB centrum grauitatis erit S, & <lb/>trianguli mixti CRB centrum grauitatis T. cum igitur <lb/>BP, BQ proportionales æqualibus NB, BR inter &longs;e <lb/>&longs;int æquales, & &longs;ecta AC bifariam in puncto D; etiam <lb/>ijs parallela ST &longs;ecta erit bifariam in puncto H: iungit <lb/>autem ST centra grauitatis mixtorum triangulorum AN <lb/>B, BRC; compo&longs;iti igitur ex vtroque centrum grauita­<lb/>tis erit H. <!-- KEEP S--></s> <s>Rur&longs;us quoniam ex quadratura parabolæ, &longs;e­<lb/>miparabola ABD &longs;e&longs;quitertia e&longs;t trianguli BDA, erit <lb/>triangulum BDA &longs;e&longs;quialterum mixti trianguli ANB: <pb/>eadem ratione triangulum BDC, trianguli CRB mi xti <lb/>erit &longs;e&longs;quialterum: totum igitur triangulum ABC &longs;e&longs;qui­<lb/>alterum e&longs;t compo&longs;iti ex triangulis mixtis ANB, CRB. <lb/></s> <s>Et quoniam quarta pars e&longs;t GH ip&longs;ius BD, & DK ter­<lb/>tia, DG verò dimidia; qualium duodecim partium æqua­<lb/>lium e&longs;t BD, talium erit DK quatuor, & GH trium, & <lb/>DG &longs;ex, & reliqua KG duarum; &longs;e&longs;quialtera igitur e&longs;t <lb/>GH ip&longs;ius GK: quare vt triangulum ABC ad compo­<lb/>&longs;itum ex prædictis triangulis mixtis, ita ex contraria parte <lb/>e&longs;t HG ad G<emph type="italics"/>K<emph.end type="italics"/>: cum igitur dicti compo&longs;iti &longs;it centrum <lb/>grauitatis H, trianguli autem ABC centrum grauitatis <lb/>K; erit dicti compo&longs;iti, & trianguli ABC &longs;imul centrum <lb/>grauitatis G. Rur&longs;us, quoniam triangulum ABC &longs;e&longs;­<lb/>quialterum e&longs;t compo&longs;iti ex triangulis mixtis &longs;upra dictis, <lb/>& compo&longs;itum ex duabus &longs;emiparabolis ABD, CBD <lb/>&longs;e&longs;quitertium trianguli ABC; crit compo&longs;itum ex trian­<lb/>gulis mixtis vnà cum triangulo ABC, quintuplum com­<lb/>po&longs;iti ex portionibus AEB, BFC; hoc e&longs;t vt ex contra­<lb/>ria parte LM ad MG: cum igitur G &longs;it centrum graui­<lb/>tatis compo&longs;iti ex triangulis mixtis, & triangulo ABC, & <lb/>compo&longs;iti ex portionibus AEB, BFC centrum grauita­<lb/>tis L; erit vtriu&longs;que dicti compo&longs;iti, hoc e&longs;t totius AR <lb/>parallelogrammi centrum grauitatis L: &longs;ed & punctum G <lb/>ex primo libro e&longs;t centrum grauitatis parallelogrammi <lb/>AR; eiu&longs;dem igitur parallelogrammi AR erunt duo cen­<lb/>tra grauitatis G, L. <!-- KEEP S--></s> <s>Quod fieri non pote&longs;t: duarum igitur <lb/>portionum AEB, BFC &longs;imul centrum grauitatis erit G. <lb/><!-- KEEP S--></s> <s>Quod e&longs;t propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis figuræ circa axim in alteram partem de <lb/>ficientis, cuius ba&longs;is e&longs;t circulus, vel ellip&longs;is, &longs;iue-<pb/>ba&longs;es &longs;unt circuli, vel ellip&longs;es, reliqua autem &longs;u­<lb/>perficies tota interius concaua, centrum grauitatis <lb/>e&longs;t in dimidio axis &longs;egmento, quod ba&longs;im, vel ma­<lb/>iorem ba&longs;im attingit. </s></p><p type="main"> <s>Sit figura circa axim in alteram partem deficiens ABC, <lb/>cuius axis BD, ba&longs;is, vel maior ba&longs;is circulus, vel ellip&longs;is <lb/>circa diametrum AC, reliqua autem &longs;uperficies tota inte­<lb/>rius concaua: &longs;ecto autem axe BD bifariam in puncto G, <lb/>&longs;it &longs;olidi ABC centrum grauitatis F nempe in axe BD. <lb/><!-- KEEP S--></s> <s>Dico punctum F e&longs;&longs;e in &longs;egmento ED. <!-- KEEP S--></s> <s>Secto enim &longs;oli­<lb/>do ABC, & figu <lb/>ra per axem pla <lb/>no per <expan abbr="punctũ">punctum</expan> E <lb/>ba&longs;i, vel ba&longs;ibus <lb/>parallelo, fiat &longs;e­<lb/>ctio circulus, vel <lb/>ellip&longs;is &longs;imilis <lb/>ba&longs;i, per diffini­<lb/>tionem, & &longs;ectio­<lb/>nis diameter K <lb/>N: deinde figu­<lb/>ra quædam ex <lb/><figure id="id.043.01.227.1.jpg" xlink:href="043/01/227/1.jpg"/><lb/>duobus cylindris, vel cylindri portionibus KL, AM cir­<lb/>ca axes BE, ED, eiu&longs;dem altitudinis circum&longs;cribatur <lb/>&longs;olido ABC: &longs;ecanturque bifariam BE in puncto G, & <lb/>ED in puncto H. totius autem figuræ circum&longs;criptæ &longs;it <lb/>centrum grauitatis O, nempe in axe BD. <!-- KEEP S--></s> <s>Quoniam igi­<lb/>tur propter bipartitorum axium &longs;ectiones G, H, e&longs;t &longs;olidi <lb/>KL centrum grauitatis G: &longs;olidi autem AM centrum <lb/>grauitatis H, erit in linea GH totius &longs;olidi AL centrum <lb/>grauitatis O, & vt &longs;olidum AM ad &longs;olidum KL, ita GO <lb/>ad OH: &longs;ed maior e&longs;t proportio &longs;olidi AM ad &longs;olidum KL <pb/>quàm GE, ad EH; maior igitur proportio e&longs;t GO ad <lb/>OH, quàm GE ad EH: & componendo, maior pro­<lb/>portio GH ad HO, quàm eiu&longs;dem GH ad HE; mi­<lb/>nor igitur OH erit quàm EH, & punctum O propin­<lb/>quius puncto D quàm punctum E; verum quoniam ex <lb/>ijs, quæ in præcedenti libro demon&longs;trauimus, propo&longs;itæ <lb/>figuræ &longs;olidæ ABC centrum grauitatis e&longs;t puncto D <lb/>propinquius, quàm cuiuslibet figuræ ex cylindris, vel cy <lb/>lindri portionibus æqualium altitudinum ip&longs;i circum&longs;cri­<lb/>ptæ, erit punctum F propinquius puncto D quàm pun­<lb/>ctum O; multo igitur puncto D erit propinquius pun­<lb/>ctum F quàm punctum E; ergo infra punctum E, & in <lb/>linea ED cadet &longs;olidi ABC centrum grauitatis F. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti coni, vel portionis conicæ cen­<lb/>trum grauitatis e&longs;t punctum illud, in quo eius <lb/>axis &longs;ic diuiditur, vt pars quæ minorem ba&longs;im at­<lb/>tingit a&longs;&longs;umens quartam partem axis ablati coni, <lb/>vel portionis conicæ, &longs;it ad eam, quæ inter po&longs;tre­<lb/>mam &longs;ectionem, & quartæ partis ab&longs;ci&longs;&longs;<17> ad ba&longs;im <lb/>axis totius coni terminum interijcitur, vt cubus, <lb/>qui fit ab axe totius, ad cubum qui fit ab axe abla­<lb/>ti coni. </s></p><p type="main"> <s>Sit coni, vel portionis conicæ ABC fru&longs;tum BDEC, <lb/>cuius axis FG: conus autem, vel coni portio ablata AD <lb/>E: &longs;int centra grauitatis H &longs;olidi ABC, & K &longs;olidi <lb/>ADE, & L fru&longs;ti DC: quæ centra præterquam quod <pb/>&longs;unt omnia in axe AG, centrum L cadet infra <lb/>centrum H, ex ijs, quæ in primo libro demon&longs;traui­<lb/>mus. </s> <s>Dico e&longs;&longs;e KL ad LH vt cubum ex AG ad cu­<lb/>bum ex AF. <!-- KEEP S--></s> <s>Quoniam enim <lb/>ob centra grauitatis <emph type="italics"/>K<emph.end type="italics"/>, H, L, <lb/>e&longs;t vt fru&longs;tum DC ad &longs;olidum <lb/>ADE, ita ex contraria parte <lb/>KH ad HL; erit componen­<lb/>do, vt &longs;olidum ABC ad &longs;oli­<lb/>dum ADE, ita KL ad LH: <lb/>&longs;ed vt <expan abbr="&longs;olidũ">&longs;olidum</expan> ABC ad &longs;olidum <lb/>ADE, ita e&longs;t cubus ex AG <lb/>ad cubum ex AF: triplieata <lb/>enim e&longs;t vtraque proportio eiu&longs;­<lb/>dem, quæ e&longs;t ip&longs;ius AG ad ip­<lb/>&longs;am AF, propter &longs;imilitudi­<lb/>nem &longs;olidorum; vt igitur e&longs;t cu <lb/>bus ex AG ad cubum ex AF, <lb/>ita erit KL ad LH. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.229.1.jpg" xlink:href="043/01/229/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Re&longs;idui &longs;olidi ex cylindro, vel portione cylin­<lb/>drica hemi&longs;phærio, vel hemi&longs;phæroidi circum­<lb/>&longs;cripta, dempto hemi&longs;phærio, vel hemi&longs;phæroide, <lb/>centrum grauitatis e&longs;t punctum illud, in quo axis <lb/>&longs;ic diuiditur, vt pars ba&longs;im attingens hemi&longs;phæ­<lb/>rij, vel hemi&longs;phæroidis &longs;it tripla reliquæ. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærio, vel hem&longs;phæroidi ABC, cuius axis <lb/>BD, circum&longs;criptus cylindrus, vel portio cylindrica AF: <lb/>& ponatur D<emph type="italics"/>K<emph.end type="italics"/> ip&longs;ius <emph type="italics"/>K<emph.end type="italics"/>B tripla. </s> <s>Dico reliqui ex &longs;oli-<pb/>do AF dempto ABC, centrum grauitatis e&longs;&longs;e <emph type="italics"/>K.<emph.end type="italics"/><!-- KEEP S--></s><s> Nam <lb/>&longs;uper ba&longs;im circulum, vel ellip&longs;im, cuius diameter EF &longs;i­<lb/>milem, & oppo&longs;itam &longs;olidi ABC, vel AF ba&longs;i, cuius dia­<lb/>meter AC, &longs;tet cylindrus, vel portio cylindrica EDF: vt <lb/>&longs;itaxis BD communis quatuor &longs;olidis ABC, EDF, <lb/>AF, & reliquæ figuræ dempto &longs;olido ABC compre­<lb/>hen&longs;æ &longs;uperficie cylindrica, & circulo, vel ellip&longs;e circa EF, <lb/>& dimidia &longs;uper&longs;icie &longs;phærica interiori, cuius figuræ &longs;oli­<lb/>dæ ponimus centrum grauitatis <emph type="italics"/>K.<emph.end type="italics"/><!-- KEEP S--></s><s> Secto igitur axe <lb/>BD bifariam, & &longs;ingulis eius partibus rur&longs;us bifariam, <lb/>ducti&longs;que per puncta &longs;ectionum planis quibu&longs;dam planis <lb/><figure id="id.043.01.230.1.jpg" xlink:href="043/01/230/1.jpg"/><lb/>prædictarum ba&longs;ium oppo&longs;itarum parallelis, &longs;ecta &longs;int qua­<lb/>tuor prædicta &longs;olida, quorum, excepto propo&longs;ito re&longs;iduo, <lb/>&longs;ectiones omnes erunt circuli, vel ellip&longs;es inter &longs;e &longs;imi­<lb/>les, & in &longs;olido AF etiam æquales, quarum omnium <lb/>diametri eiu&longs;dem rationis erunt in eodem plano, in quo <lb/>&longs;it parallelogrammum per axim AEFC: &longs;olidi autem dicti <lb/>re&longs;idui &longs;ectiones, re&longs;idua &longs;ectionum &longs;olidi ABC. <!-- KEEP S--></s> <s>At circa <lb/><expan abbr="cõmunes">communes</expan> axes inter &longs;e æquales &longs;egmenta axis BD, & inter <lb/><expan abbr="ead&etilde;">eadem</expan> plana parallela, &longs;uper ba&longs;es &longs;ectiones duorum &longs;olido­<lb/>rum ABC, EDF, cylindri, vel portiones cylindricæ con­<lb/>&longs;i&longs;tant altitudine, & multitudine æquales; ita vt duarum fi­<lb/>gurarum ex ijs compofitarum altera fit cirdum&longs;cripta &longs;oli­<pb/>do EDF, altera &longs;olido ABC in&longs;cripta. </s> <s>hac igitur abla­<lb/>ta ex &longs;olido AF, figura relinquetur ex re&longs;iduis cylindro­<lb/>rum, vel cylindri portionum altitudine, & multitudine <lb/>æqualibus ijs cylindris, vel cylindri portionibus, ex quibus <lb/>con&longs;tat alterutra figurarum &longs;olidis ABC, DEF circum­<lb/>&longs;criptarum: eruntque ex &longs;uperius demon&longs;tratis dicta re&longs;i­<lb/>dua, & cylindri vel cylindri portiones, quæ circa &longs;olidum <lb/>EDF, inter &longs;e æqualia proutinter &longs;e re&longs;pondent inter ea­<lb/>dem plana parallela, vt e&longs;t exempli gratia reliquum &longs;oli­<lb/>di AN dempto &longs;olido SR, æquale &longs;olido TP: & &longs;ic de­<lb/>inceps: &longs;ummus autem XF cylindrus, vel portio cylindrica <lb/><figure id="id.043.01.231.1.jpg" xlink:href="043/01/231/1.jpg"/><lb/>e&longs;t communis: Atqui bina hæc iam dicta &longs;olida centrum <lb/>grauitatis habent commune communis bipartiti axis &longs;ectio <lb/>nem in eadem recta linea BD, in qua e&longs;t etiam &longs;olidi XF <lb/>communis centrum grauitatis. </s> <s>duarum igitur dictarum figu <lb/>rarum &longs;olido EDF, & prædicto re&longs;iduo circum&longs;criptarum <lb/>idem aliquod punctum in axe BD erit commune centrum <lb/>grauitatis: &longs;ieri autem pote&longs;ts quod in &longs;ecundo libro demon <lb/>&longs;trauimus, vt duæ dictæ figuræ &longs;uperent vnaquæ que &longs;ibi in­<lb/>&longs;criptam minori &longs;pacio quantacumque magnitudine pro­<lb/>po&longs;ita. </s> <s>ex demon&longs;tratis igitur in primo libro; duo &longs;olida cir­<lb/>ca axem BD in alteram partem deficientia commune ha­<lb/>bebunt in axe BD centrum grauitatis: &longs;ed &longs;olidi, ide&longs;t co-<pb/>ni, vel portionis conicæ EDF e&longs;t centrum grauitatis K: <lb/>reliqui igitur ex cylindro, vel portione cylindrica AF dem <lb/>pto hemi&longs;phærio, vel hemi&longs;phæroide ABC centrum graui <lb/>tatis erit idem K. <!-- KEEP S--></s> <s>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si hemi&longs;phærium, vel hemi&longs;phæroides vna cum <lb/>cylindro, vel cylindri portione ip&longs;i circum&longs;cripta <lb/>&longs;ecetur plano ba&longs;i parallelo; reliqui ex cylindro, <lb/>vel portione cylindrica ab&longs;ci&longs;&longs;a ad partes verti­<lb/>cis, dempta illa quæ ab&longs;ci&longs;&longs;a e&longs;t &longs;imul minori, <lb/>& &longs;phæræ, vel &longs;phæroidis portione, centrum gra­<lb/>uitatis e&longs;t punctum illud, in quo eius axis &longs;ic diui­<lb/>ditur, vt quæ inter hanc po&longs;tremam &longs;ectionem, & <lb/>centrum ba&longs;is vnà ab&longs;ci&longs;&longs;æ portionis interijci­<lb/>tur, a&longs;&longs;umens quartam partem &longs;egmenti, quod di­<lb/>ctæ ba&longs;is, & &longs;phæræ, vel &longs;phæroidis centra iungit, <lb/>&longs;it ad &longs;ui &longs;egmentum, quod inter po&longs;tremam &longs;e­<lb/>ctionem, & quartæ partis axis hemi&longs;phærij, vel <lb/>hemi&longs;phæroidis ad verticem ab&longs;ci&longs;&longs;æ terminum <lb/>interijcitur, vt cubus axis hemi&longs;phærij, vel hemi­<lb/>&longs;phæroidis, ad cubum eius, quæ ba&longs;is portionis & <lb/>hemi&longs;phærij, vel hemi&longs;phæroidis centra iungit. <lb/></s> <s>Reliqui autem ex cylindro, vel portione cylindri­<lb/>ca vnà ab&longs;ci&longs;&longs;a <expan abbr="cũ">cum</expan> reliqua hemi&longs;phærij, vel hemi­<lb/>&longs;phæroidis portione, quæ e&longs;t ad ba&longs;im, dempta hac <lb/>portione centrum, grauitatis e&longs;t punctum illud, <lb/>quod quartam partem ab&longs;cindit axis portionis ad <pb/>cius minorem ba&longs;im terminatam. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærio, vel hemi&longs;phæroidi ABC, cuius axis <lb/>BD, ba&longs;is circulus vel ellip&longs;is, cuius diameter AC cir­<lb/>cum&longs;criptus cylindrus, vel cylindri portio AF, cuius in­<lb/>telligatur reliquum dempto ABC. quæ &longs;olida &longs;ecans pla <lb/>num per AC, BD, faciat &longs;ectiones &longs;emicirculum, vel &longs;e­<lb/>miellip&longs;im ABC, & parallelogrammum per axem AE <lb/>FC; & per quodlibet punctum L axis BD, planum ba&longs;ibus <lb/>AC, EF &longs;olidi AF <expan abbr="parallelũ">parallelum</expan>, &longs;ecans prædicta &longs;olida ABC, <lb/>AF, faciat &longs;ectiones circulos, vel ellip&longs;es &longs;imiles, & in &longs;olido <lb/>AF etiam æquales ijs, quæ circa AC, EF: earum autem dia­<lb/>metros, &longs;ectiones cum <expan abbr="parallelogrãmo">parallelogrammo</expan> AEFC, ip&longs;am GO: <lb/>& cum &longs;emicirculo, vel &longs;emiellip&longs;e ABC, ip&longs;am HN. </s> <s>Ita­<lb/>que habebimus figuram quandam &longs;olidam GHBNO re&longs;i­<lb/>duum cylindri, vel portionis cylindricæ GF dempta mino­<lb/>ri &longs;phæræ, vel &longs;phæroidis portione HBN, cuius axis erit BL. <lb/></s> <s>Sumpta igitur BQ quarta parte axis BD, & LP quarta par <lb/>te ip&longs;ius DL fiat vt cu <lb/>bus ex BD ad cubum ex <lb/>DL, ita PR ad <expan abbr="Rq.">Rque</expan> <lb/>Dico re&longs;idui GHBNO <lb/>centrum grauitatis e&longs;&longs;e <lb/>R. <!-- KEEP S--></s> <s>Reliqui autem ex <lb/>cylindro, vel portione <lb/>cylindrica AO dempta <lb/>portione AHNC, cen­<lb/>trum grauitatis e&longs;&longs;e P. <lb/><figure id="id.043.01.233.1.jpg" xlink:href="043/01/233/1.jpg"/><lb/>Nam &longs;uper ba&longs;im circulum, vel ellip&longs;im EF, &longs;tet conus, vel <lb/>portio conica EDF: &longs;itque prædicto plano per L ab&longs;ci&longs;­<lb/>&longs;us conus, vel coni portio KDM, cuius axis DL, quæ pro­<lb/>pter planum &longs;ecans ba&longs;i EF parallelum, &longs;imilis erit toti <lb/>cono, vel portioni conicæ EDF. <!-- KEEP S--></s> <s>Quoniam igitur BQ <lb/>e&longs;t axis BD pars quarta, & LP pars quarta ip&longs;ius DL; <pb/>erunt centra grauitatis &longs;olidorum, Q ip&longs;ius EDF, & Pip­<lb/>&longs;ius DKM. </s> <s>Et quoniam &longs;olidum DEF ad &longs;olidum D <lb/>KM e&longs;t vt cubus ex BD ad cubum ex DL, hoc e&longs;t vt <lb/>&longs;olidum EDF ad &longs;olidum KLM, & vt PR ad <expan abbr="Rq;">Rque</expan> <lb/>erit diuidendo, vt fru&longs;tum EKMF ad ablatum KDM, <lb/>ita ex contraria parte PQ ad QR: cum igitur &longs;int <lb/>centra grauitatis P &longs;olidi DKM, & Q &longs;olidi DET; <lb/>erit reliqui fru&longs;ti EKMF centrum grauitatis R: &longs;ed <lb/>qua ratione in præcedenti con&longs;tat, reliqui ex &longs;olido AF, <lb/>dempto &longs;olido ABC centrum grauitatis e&longs;&longs;e Q, eadem <lb/>concluditur idem e&longs;&longs;e centrum grauitatis reliqui ex &longs;olido <lb/>GF, dempta portione HBN, quod & fru&longs;ti EKMF, <lb/>nempe punctum R: Et quoniam P e&longs;t centrum grauita­<lb/>tis coni, vel portionis conicæ KDM, crit idem P centrum <lb/>grauitatis ieliqui ex cylindro, vel portione cylindrica <lb/>AO dempta portione AHNC. </s> <s>Manife&longs;tnm e&longs;t igitur <lb/>propo&longs;ituro. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Ij&longs;dem po&longs;itis &longs;olidis, vt in antecedenti, &longs;ectis­<lb/>que per duo quælibet puncta axis duplici plano <lb/>ba&longs;i parallelo, reliqui ex cylindro, vel portione <lb/>cylindrica dictis duobus planis intercepta dem­<lb/>pta &longs;phæræ, vel &longs;phæ roidis portione ip&longs;i inter ea­<lb/>dem plana re&longs;pondente, centrum grauitatis e&longs;t <lb/>punctum illud, in quo eius axis &longs;ic diuiditur, vt <lb/>quæ inter hanc po&longs;tremam &longs;ectionem, & centrum <lb/>maioris ba&longs;is vnà ab&longs;ci&longs;sæ portionis interijcitur, <lb/>a&longs;&longs;umens quartam partem &longs;egmenti, quod prædi­<lb/>ctæ ba&longs;is, & &longs;phæræ vel &longs;phæroidis centra iungit, <pb/>&longs;it ad &longs;ui &longs;egmentum, quod inter po&longs;tremam &longs;ectio <lb/>nem, & quartæ partis eius, quæ &longs;phæræ, vel hemi­<lb/>&longs;phærij, & minoris ba&longs;is portionis centra iungit <lb/>ad minorem ba&longs;im ab&longs;ci&longs;sæ terminum interijci­<lb/>tur, vt cubus eius, quæ minoris ba&longs;is, & &longs;phæræ, <lb/>vel &longs;phæroidis, ad <expan abbr="cubũ">cubum</expan> eius, qu<17> &longs;phæræ, vel &longs;phæ <lb/>roidis, & maioris ba&longs;is portionis centra iungit. </s></p><p type="main"> <s>Ij&longs;dem po&longs;itis &longs;olidis, vtque in antecedenti ponebantur <lb/>ABC, AF; per duo quælibet puncta RQ axis BD &longs;e­<lb/>centur po&longs;ita &longs;olida duobus planis ba&longs;i, quæ circa AC, cir <lb/>culo &longs;cilicet, vel ellip&longs;i parallelis: quibus planis intercepta <lb/>hemi&longs;phærij, vel hemi&longs;phæroidis portio &longs;it MOPN, vnà <lb/>cum cylindro, vel portione cylindrica GL parte ip&longs;ius AF, <lb/><expan abbr="quorũ">quorum</expan> &longs;olidorum <expan abbr="cõmu">commu</expan> <lb/>nis axis vnà ab&longs;ci&longs;&longs;us <lb/>ab axe BD &longs;olidi AB <lb/>C, &longs;it RQ: & &longs;umptis <lb/>quartis partibus RI ip­<lb/>&longs;ius DR, & QZ ip&longs;ius <lb/>DQ, fiat vt cubus ex <lb/>DQ ad cubum ex D <lb/>R, ita IY ad YZ. <lb/></s> <s>Dico reliqui ex cylin­<lb/><figure id="id.043.01.235.1.jpg" xlink:href="043/01/235/1.jpg"/><lb/>dro, vel portione cylindrica GL dempta portione MOP <lb/>N, centrum grauitatis e&longs;&longs;e Y. <!-- KEEP S--></s> <s>Facta enim con&longs;tructione <lb/>coni, vel portionis conicæ EDF, vt in &longs;uperioribus, erunt <lb/>&longs;imilium conorum, vel coni portionum SDT, VDX, ea­<lb/>dem ordine axes DQ, DR: propter igitur factas diui&longs;io­<lb/>nes, erunt <expan abbr="c&etilde;tra">centra</expan> grauitatis Z &longs;olidi SDT & I &longs;olidi VDX, <lb/>& demon&longs;tratio &longs;imilis antecedenti. </s> <s>dicti igitur re&longs;idui <lb/>GMOPMH centrum grauitatis Y. <!-- KEEP S--></s> <s>Quod e&longs;t propo­<lb/>&longs;itum. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;phæra, vel &longs;phæroides vnà cum cylindro, <lb/>vel portione cylindrica ip&longs;i circum&longs;cripta &longs;ecetur <lb/>plano, haud per centrum, ba&longs;ibus &longs;olidi circum­<lb/>&longs;cripti parallelo; reliqui ex cylindro, vel portio­<lb/>ne cylindrica ad maioris portionis &longs;phæræ, vel <lb/>&longs;phæroidis partes ab&longs;ci&longs;&longs;a, dempta &longs;phæræ, vel <lb/>&longs;phæroidis maiori portione, centrum grauita­<lb/>tis e&longs;t punctum illud, in quo dicti reliqui &longs;olidi <lb/>axis &longs;egmentum inter duas quartas partes extre­<lb/>mas &longs;egmentorum eiu&longs;dem axis, quæ à centro <lb/>&longs;phæræ, vel &longs;phæroidis fiunt interiectum, &longs;ic diui­<lb/>ditur, vt pars propinquior ba&longs;i &longs;it ad reliquam, vt <lb/>prædictorum, quæ à centro fiunt axis &longs;egmento­<lb/>rum maioris cubus ad cubum minoris. </s></p><figure id="id.043.01.236.1.jpg" xlink:href="043/01/236/1.jpg"/><p type="main"> <s>Sit &longs;phæræ, vel &longs;phæ­<lb/>roidi ABCD cuius cen­<lb/>trum E, circum&longs;criptus <lb/>cylindrus, vel portio cy­<lb/>lindrica FGHK, cum <lb/>quibus planum per axim <lb/>communem BED, fa­<lb/>ciat &longs;ectiones, parallelo­<lb/>grammum per axim FG <lb/>HK, & circulum, vel el­<lb/>lip&longs;im ABCD: quas fi­<lb/>guras vnà cum dictis &longs;o­<lb/>lidis &longs;ecans planum ba&longs;ibus &longs;olidi circum&longs;cripti paralle-<pb/>lum per quoduis punctum S dimidij axis ED, faciens­<lb/>que &longs;ectiones circulos, vel ellip&longs;es &longs;imiles &longs;cilicet ba­<lb/>&longs;ibus oppo&longs;itis &longs;olidi FH, & &longs;ectionum diametros LM, <lb/>TV, ab&longs;cindat &longs;olidi ABCD maiorem portionem <lb/>LBM, & &longs;olidi FH cylindrum, vel portionem cy­<lb/>lindricam TH, cuius axis BES: duorum autem &longs;egmen­<lb/>corum BE, ES &longs;umptis duabus quartis partibus extre­<lb/>mis BQ PS, fiat vt cubus ex BE ad cubum ex ES, ita <lb/>PR ad RQ. <!--neuer Satz-->Dico reliquæ figuræ ex cylindro, vel por­<lb/>tione cylindrica TH, portioni LBM circum&longs;cripta, dem­<lb/>pta portione LBM, centrum grauitatis e&longs;&longs;e R. <!-- KEEP S--></s> <s>Se­<lb/>ctis enim parallelogrammo TH, & &longs;olidis LBM, TH, <lb/>plano per centrum E, ba&longs;ibus &longs;olidi TH parallelo, &longs;it &longs;e­<lb/>ctio, (vna enim communis erit vtrique &longs;olido) circulus, <lb/>vel ellip&longs;is, cuius diameter AEC in parallelogrammo T <lb/>H diametris TV, GH <lb/>oppo&longs;itarum ba&longs;ium pa­<lb/>rallela. </s> <s>Tum &longs;uper ba­<lb/>&longs;es oppo&longs;itas circulos, vel <lb/>ellip&longs;es circa GH, FK <lb/>&longs;tent coni, vel portiones <lb/>conicæ GEH, FEK: <lb/>& planum per TV ba&longs;i <lb/>circa FK parallelum ab­<lb/>&longs;cindat à &longs;olido FEK <lb/>conum, vel coni portio­<lb/>nem NEO &longs;imilem vti­<lb/>que ip&longs;i FEK, hoc e&longs;t <lb/><figure id="id.043.01.237.1.jpg" xlink:href="043/01/237/1.jpg"/><lb/>ip&longs;i GEH, propter &longs;imiles ba&longs;es, & &longs;imilia triangula per <lb/>axim in eodem parallelogrammo FH. <!-- KEEP S--></s> <s>Solidi itaque <lb/>NEO, ex ijs, quæ in primo libro demon&longs;trauimus, cen­<lb/>trum grauitatis erit P; quemadmodum & Q &longs;olidi <lb/>NEO. </s> <s>Quoniam igitur tàm &longs;olidi GEH ad &longs;oli­<lb/>dum NEO propter &longs;imilitudinem, quàm cubi ex BE <pb/>ad cubum ex ES, triplicata e&longs;t proportio axis, vel la­<lb/><gap/>eris BE, ad axem, vel latus ES; erit vt cubus ex BE <lb/>ad cubum ex ES, ita &longs;olidum GEH ad &longs;olidum NEO, <lb/>hoc e&longs;t in eadem proportione, quæ e&longs;t ex contraria parte ip­<lb/>&longs;ius PR ad RQ. <!--neuer Satz-->Cum igitur P &longs;it centrum grauitatis <lb/>&longs;olidi NEO, & Q &longs;olidi GEH; erit compo&longs;iti ex vtro­<lb/>que centrum grauitatis R. Rur&longs;us, quoniam reliquum &longs;o­<lb/>lidi AH dempto hemi&longs;phærio, vel hemi&longs;phæroide ABC, <lb/>æquale e&longs;t &longs;olido GEH: & reliquum &longs;olidi TC dempto <lb/>&longs;olido ALMC æquale &longs;olido NEO; erit vt &longs;olidum <lb/>GEH ad &longs;olidum NEO, ide&longs;t ex contraria parte, vt PR <lb/>ad RQ, ita reliquum &longs;olidi AH dempto ABC, ad re­<lb/>liquum &longs;olidi TC, dempto ALMC: &longs;ed reliqui ex &longs;oli­<lb/>do AH dempto ABC e&longs;t centrum grauitatis Q: & reli­<lb/>qui ex &longs;olido TC dempto ALMC, centrum grauitatis <lb/>P, ex &longs;uperius demon&longs;tratis; totius igitur reliqui ex cy­<lb/>lindro, vel portione cylindrica TH dempta &longs;phæræ, vel <lb/>&longs;phæroidis maiori portione LBM centrum grauitatis e&longs;t <lb/>R. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;phæra, vel &longs;phæroides vnà cum cylindro, <lb/>vel portione cylindrica ip&longs;i circum&longs;cripta, &longs;ece­<lb/>tur duobus planis ba&longs;i &longs;olidi circum&longs;cripti pa­<lb/>rallelis, centrum intercipientibus, & ab eo non <lb/>æqualiter di&longs;tantibus; reliqui ex cylindro, vel <lb/>portione cylindrica dictis planis intercepta, dem­<lb/>pta portione &longs;phæræ, vel &longs;phæroidis ip&longs;i re&longs;pon­<lb/>dente, centrum grauitatis e&longs;t punctum illud, in <lb/>quo prædicti reliqui &longs;olidi axis &longs;egmentum in­<pb/>ter quartas partes extremas eiu&longs;dem axis &longs;eg­<lb/>mentorum, quæ à centro &longs;phæræ, vel &longs;phæroi­<lb/>dis fiunt interiectum &longs;ic diuiditur, vt pars ma­<lb/>iori ba&longs;i propinquior &longs;it ad reliquam, vt prædi­<lb/>ctorum axis &longs;egmentorum cubus maioris ad cu­<lb/>bum minoris. </s></p><p type="main"> <s>Ij&longs;dem po&longs;itis, & con&longs;tructis, quæ in antecedenti, rur­<lb/>&longs;us per quodlibet axis BE punctum X, ductum planum <lb/>ba&longs;ibus &longs;olidi FH parallelum, &longs;ecansque vnà cylindrum, <lb/>vel portionem cylindricam FH, & &longs;phæram, vel &longs;phæroi­<lb/>des ABCD: e&longs;to duobus planis per TV, ZY, inter &longs;e pa­<lb/>rallelis, & centrum E intercipientibus abci&longs;&longs;a &longs;phæræ, vel <lb/>&longs;phæroidis portio L <foreign lang="greek">d e</foreign> M vnà cum cylindro, vel portione <lb/>cylindrica TY: & &longs;umatur ip&longs;ius EX pars quarta XQ, <lb/>qualis e&longs;t & PS ip&longs;ius E <lb/>S: & vt e&longs;t cubus ex EX <lb/>ad cubum ex ES, ita fiat <lb/>PR ad <expan abbr="Rq.">Rque</expan> Dico reli­<lb/>qui ex cylindro, vel por­<lb/>tione cylindrica TY dem <lb/>pta &longs;phæræ, vel &longs;phæroi­<lb/>dis portione L <foreign lang="greek">d c</foreign> M, cen­<lb/>trum grauitatis e&longs;&longs;e R. <!-- KEEP S--></s> <s>E&longs;to <lb/>enim conus, vel coni por­<lb/>tio <foreign lang="greek">q</foreign> E <foreign lang="greek">l</foreign> ab&longs;ci&longs;&longs;a prædi­<lb/>cto plano per ZY, & com <lb/>munibus axibus ES, EX, <lb/>&longs;imili igitur demon&longs;tratio­<lb/>ne antecedentis manife&longs;tum e&longs;t quod proponebatur. </s></p><figure id="id.043.01.239.1.jpg" xlink:href="043/01/239/1.jpg"/><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Hemi&longs;phærij, vel hemi&longs;phæroidis centrum <lb/>grauitatis e&longs;t punctum illud, in quo axis &longs;it diui­<lb/>ditur, vt pars ad verticem &longs;it ad reliquam vt quin <lb/>que ad tria. </s></p><p type="main"> <s>E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cuius <lb/>axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diameter AD <lb/>C: &longs;itque &longs;olidi ABC centrum grauitatis G, nempe <lb/>in axe BD. <!-- KEEP S--></s> <s>Dico BG ad GD e&longs;&longs;e vt quinque ad tria. <lb/></s> <s>Nam circa axim BD &longs;uper ba&longs;im circulum, vel ellip&longs;im cir <lb/>ca AC, &longs;tet circum&longs;cri <lb/>ptus &longs;olido ABC cy­<lb/>lindrus, vel portio cy­<lb/>lindrica AE, & &longs;ecta <lb/>BD bifariam in F, rur <lb/>&longs;us FB bifariam &longs;ece­<lb/>tur in puncto H. <!-- KEEP S--></s> <s>Quo­<lb/>niam igitur &longs;olidum A <lb/>BC e&longs;t &longs;olidi AE, &longs;ub­<lb/>&longs;e&longs;quialterum, erit di­<lb/><figure id="id.043.01.240.1.jpg" xlink:href="043/01/240/1.jpg"/><lb/>uidendo &longs;olidum ABC reliqui ex &longs;olido AE duplum <lb/>cum igitur &longs;int centra grauitatis, G &longs;olidi ABC, & H <lb/>prædicti reliqui, & F totius AE; quo fit vt ex con­<lb/>traria parte &longs;it vt &longs;olidum ABC ad prædictum re&longs;iduum, <lb/>ita HF ad FG, erit HF dupla ip&longs;ius FG; quadrupla <lb/>igitur BF ip&longs;ius FG: &longs;ed talium quatuor partium e&longs;t BF, <lb/>qualium BD e&longs;t octo, cum &longs;it BF dimidia ip&longs;ius BD; <lb/>qualium igitur octo e&longs;t BD, talium erit BG quinque, & <lb/>GD trium. </s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>ALITER.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dico hemi&longs;phærij, vel hemi&longs;phæroidis ABC cen­<lb/>trum grauitatis e&longs;&longs;e G. <!-- KEEP S--></s> <s>In plano enim &longs;emicirculi, vel &longs;e­<lb/>miellip&longs;is per axem BD de&longs;criptæ intelligantur duæ pa­<lb/>rabolæ, quarum diametri AD, DC, & communiter <lb/>ad vtranque ordinatim applicata &longs;it BD: & connectun­<lb/>tur rectæ AB, BC: &longs;umptis autem in BD tribus qui­<lb/>buslibet punctis, æqualia axis &longs;egmenta XF, FY interci­<lb/>pientibus, &longs;ecent per ea puncta tres figuras hemi&longs;phærium, <lb/>vel hemi&longs;phæroides ABC, & &longs;emicirculum, vel &longs;emielli­<lb/><figure id="id.043.01.241.1.jpg" xlink:href="043/01/241/1.jpg"/><lb/>p&longs;im per axem, & figuram planam ARBSC, quæ lineis pa <lb/>rabolicis ARB, BSC, & recta AC continetur, pla­<lb/>na quædam ba&longs;i hemi&longs;phærij, vel hemi&longs;phæroidis paralle­<lb/>la. </s> <s>Erunt igitur &longs;ectiones hemi&longs;phærij, vel hemi&longs;phæroidis <lb/>circuli, vel ellip&longs;es &longs;imiles ba&longs;i, <expan abbr="quarũ">quarum</expan> diametri &longs;int KXH, <lb/>LFM, N<foreign lang="greek">*u</foreign>O: figuræ autem ARBSC &longs;ectiones rectæ <lb/>lineæ PXQ, RFS, TYV. </s> <s>Quoniamigitur per IV hu­<lb/>ius e&longs;t vt KH ad LM potentia, ita KQ ad FS hoc <lb/>e&longs;t in earum duplis PQ ad RS longitudine; erit vt PQ <lb/>ad RS, ita circulus, vel ellip&longs;is KH ad circulum vel &longs;i­<lb/>milem ellip&longs;im LM. <!-- KEEP S--></s> <s>Eadem ratione erit vt RS ad <lb/>TV, ita circulus, vel ellip&longs;is LM ad circulum, vel <pb/>ellip&longs;im NO. <!--neuer Satz-->minor autem proportio e&longs;t PQ ad RS, <lb/>quàm RS ad TV circuli igitur, vel ellip&longs;is KH ad <expan abbr="circulũ">circulum</expan>, <lb/>vel ellip&longs;im LM, minor erit proportio <34> circuli, vel ellip&longs;is <lb/>LM ad circulum, vel ellip&longs;im NO: & duæ figuræ hemi­<lb/>&longs;phærium, vel hemi&longs;phæroides ABC, & plana ARBSC, <lb/>&longs;unt circa axim, vel diametrum BD in alteram parte m <lb/>deficientes, quales definiuimus; vtriu&longs;que igitur dictæ fi­<lb/>guræ vnum erit commune centrum grauitatis. </s> <s>Rur&longs;us <lb/>po&longs;ito puncto F in medio axis BD, & FG ip&longs;ius GE <lb/>tripla, quoniam ponitur BG ad GD vt quinque ad tria; <lb/>qualium partium æqualium ip&longs;i EG e&longs;t FG trium, ta­<lb/>lium erit BG quindecim, & GD nouem, & talis EG <lb/>vna: dempta igitur GE ab ip&longs;a DG, & addita ip&longs;i BG, <lb/>qualium partium e&longs;t BE &longs;exdecim, talium erit ED octo; <lb/>dupla igitur BE ip&longs;ius ED, & trianguli ABC centrum <lb/>grauitatis E. <!-- KEEP S--></s> <s>Rur&longs;us quoniam ex quadratura parabolæ, <lb/>duarum portionum ARB, BSC triangulum ABC e&longs;t <lb/>triplum; hoe e&longs;t vt FG ad GE, ita ex contraria parte <lb/>triangulum ABC ad duas portiones ARB, BSC: Sed <lb/>trianguli ABC e&longs;t centrum grauitatis E, & duarum por <lb/>tionum ARB, BSC &longs;imul per XXIII huius, centrum <lb/>grauitatis F, totius igitur figuræ ARBSC centrum gra<lb/>uitatis erit G, commune autem hoc centrum grauitatis <lb/>e&longs;t hemi&longs;phærio, vel hemi&longs;phæroidi ABC. <!-- KEEP S--></s> <s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis minoris portionis &longs;phæræ, vel &longs;phæroi­<lb/>dis centrum grauitatis e&longs;t in axe primum bifa­<lb/>riam &longs;ecto: deinde &longs;ecundum centrum grauitatis <lb/>reliqui &longs;olidi dempta portione ex cylindro, vel <pb/>portione cylindrica ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;&longs;a vnà cum <lb/>portione, ex cylindro, vel portione cylindrica, <lb/>&longs;phær<17>, vel &longs;phæroidis circa axim axi portionis <expan abbr="cõ">com</expan> <lb/>gruentem <expan abbr="circũ&longs;cripta">circun&longs;cripta</expan>; in eo puncto, in quo dimi­<lb/>dius axis portionis ba&longs;im <expan abbr="atting&etilde;s">attingens</expan> &longs;ic diuiditur, vt <lb/>pars prima, & &longs;ecunda &longs;ectione terminata, &longs;it ad <lb/>totam &longs;ecunda, & po&longs;trema &longs;ectione terminatam, <lb/>vt rectangulum contentum axe portionis, & reli­<lb/>quo &longs;phæræ, vel &longs;phæroidis dimidij axis &longs;egmen­<lb/>to, vnà cum duabus tertijs quadrati axis portio­<lb/>nis, ad &longs;phæræ, vel &longs;phæroidis dimidij axis axi <lb/>portionis congruentis quadratum. </s></p><p type="main"> <s>Sit &longs;phæræ, vel &longs;phæroidis minor portio ABC, cuius <lb/>axis BD: & in eo centrum grauitatis F: &longs;ecto autem axe <lb/>BD primum bifariam <lb/>in puncto G, & rur <lb/>&longs;us BG in puncto <lb/>H centro grauitatis <lb/>reliqui dempta por­<lb/>tione ex cylindro, vel <lb/>portione cylindrica <lb/>KL circa axim BD, <lb/>ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;­<lb/>&longs;a codem plano cum <lb/><figure id="id.043.01.243.1.jpg" xlink:href="043/01/243/1.jpg"/><lb/>portione ABC, & cylindro, vel portione cylindri­<lb/>ca, quæ circum&longs;criberetur &longs;phæræ, vel &longs;phæroidi, cu­<lb/>ius e&longs;t portio ABC, circa axim, cuius dimidium BDE. <lb/><!-- KEEP S--></s> <s>Dico GH ad HF, (nam cadet centrum F infra biparti­<lb/>ti axis BD &longs;ectionem G, ex XXIII huius) e&longs;&longs;e vt rectan­<lb/>gulum BDE vnà cum duabus tertijs BD quadrati ad <lb/>quadratum BE. </s> <s>Quoniam enim totius &longs;olidi KL cen-<pb/>trum grauitatis e&longs;t G, & F portionis ABC, & H reliqui <lb/>ex KL dempta ABC portione; erit vt portio ABC ad <lb/>prædictum re&longs;iduum, ita ex contraria parte HG ad GF: <lb/>& componendo, vt &longs;olidum KL ad prædictum re&longs;iduum, <lb/>ita HF ad FG: & per conuer&longs;ionem rationis, vt &longs;olidum <lb/>KL ad portionem ABC, ita FH ad HG: & conuerten <lb/>do, vt portio ABC ad &longs;olidum KL, ita GH ad HE: <lb/>&longs;ed vt portio ABC ad &longs;olidum KL, ita e&longs;t rectangulum <lb/>BDE vnà cum duabus tertiis quadrati BD ad quadra­<lb/>tum EB; vt igitur rectangulum BDE, vnà cum duabus <lb/>tertiis quadrati BD, ad quadratum EB, ita erit GH ad <lb/>HF. </s> <s>Quod demonftrandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs; <lb/>&longs;æ duobus planis parallelis, altero per centrum <lb/>acto, centrum grauitatis e&longs;t in axe primum bifa­<lb/>riam &longs;ecto: deinde &longs;umpta eius quarta parte ad <lb/>minorem ba&longs;im; in eo puncto, in quo dimidius <lb/>axis maiorem ba&longs;im attingens &longs;ic diuiditur, vt <lb/>pars axis prima, & &longs;ecunda &longs;ectione terminata, <lb/>&longs;it ad eam, quæ prima, & po&longs;trema &longs;ectione ter­<lb/>minatur, vt rectangulum contentum &longs;phæræ, vel <lb/>&longs;phæroidis axis axi portionis congruentis ijs &longs;eg­<lb/>mentis, quæ fiunt à centro minoris ba&longs;is portio­<lb/>nis, vnà cum duabus tertiis quadrati axis portio­<lb/>nis; ad&longs;phæræ, vel &longs;phæroidis dimidij axis qua­<lb/>dratum. </s></p><p type="main"> <s>Sit &longs;phæræ, vel &longs;phæroidis cuius centrum E portio <pb/>ABCD ab&longs;ci&longs;sa duobus planis parallelis altero ducto <lb/>per E, & &longs;ectionem faciente circulum maximum, vel <lb/>ellip&longs;im per centrum, cuius diameter AED: axis autem <lb/>portionis &longs;it EF, cui congruens &longs;phæræ, vel &longs;phæroidis axis <lb/>GFER: &longs;it autem FE bifariam &longs;ectus in puncto H: & <lb/>FH bifariam in puncto K, &longs;itque in EH, &longs;ic enim erit, <lb/>portionis ABCD centrum grauitatis L. <!-- KEEP S--></s> <s>Dico e&longs;&longs;e HK <lb/>ad KL, vt rectangulum GFR, vnà cum duabus tertiis <lb/>quadrati EF ad quadratum EG. <!-- KEEP S--></s> <s>Sit enim cylindrus, vel <lb/>portio cylindrica AM circa axim FE ab&longs;ci&longs;&longs;a ij&longs;dem pla­<lb/>nis cum portione AB <lb/>CD, ex cylindro, vel <lb/>portione cylindrica cir <lb/>ca axim GR &longs;phæ­<lb/>ræ, vel &longs;phæroidi AG <lb/>DR circum&longs;cripta. <lb/></s> <s>Quoniam igitur &longs;olidi <lb/>AM e&longs;t centrum gra­<lb/>uitatis H: reliqui au­<lb/>tem dempta ABCD <lb/>portione centrum gra­<lb/>uitatis K: & portionis <lb/>ABCD ponitur cen­<lb/>trum grauitatis L; erit <lb/><figure id="id.043.01.245.1.jpg" xlink:href="043/01/245/1.jpg"/><lb/>vt portio ABCD ad reliquum &longs;olidi AM, ita ex con­<lb/>traria parte KH ad HL. componendo igitur vt in antece­<lb/>denti, & per conuer&longs;ionem rationis, & conuertendo, erit <lb/>vt portio ABCD ad &longs;olidum AM; hoc e&longs;t vt rectangu­<lb/>lum GFR, vnà cum duabus tertiis quadrati EF ad qua­<lb/>dratum EG, ita HK ad KL. <!-- KEEP S--></s> <s>Quod demon&longs;trandum <lb/>erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ, vel &longs;phæroidis ab­<lb/>&longs;ci&longs;&longs;æ duobusplanis parallelis, neutro per cen­<lb/>trum acto, nec centrum intercipientibus, centrum <lb/>grauitatis e&longs;t in axe, primum bifariam &longs;ecto: de­<lb/>inde &longs;ecundum centrum grauitatis reliqui dem­<lb/>pta portione ex cylindro, vel portione cylindrica, <lb/>ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;&longs;a vnà cum portione à cylin­<lb/>dro, vel portione cylindrica &longs;phæræ, vel &longs;phæroi­<lb/>di circa eius axem axi portionis congruentem cir­<lb/>cum&longs;cripta; in eo puncto, in quo dimidius axis <lb/>portionis maiorem ba&longs;im attingens &longs;ic diuiditur, <lb/>vt pars prima & &longs;ecunda &longs;ectione terminata &longs;it ad <lb/>eam, quæ prima, & po&longs;trema &longs;ectione terminatur, <lb/>vt duo rectangula, alterum contentum duobus <lb/>&longs;phæræ, vel &longs;phæroidis axis axi portionis <expan abbr="cõgruen">congruen</expan> <lb/>tis ijs &longs;egmentis, quæ fiunt à centro minoris ba&longs;is <lb/>portionis: alterum axe portionis, & &longs;egmento, <lb/>quod &longs;phæræ, vel &longs;phæroidis, & maioris ba&longs;is por­<lb/>tionis centra iungit, vnà cum duabus tertiis qua­<lb/>drati axis portionis, ad &longs;phæræ vel &longs;phæroidis di­<lb/>midij axis quadratum. </s></p><p type="main"> <s>Sit &longs;phæræ, vel &longs;phæroidis, cuius centrum E portio <lb/>ABCD, ab&longs;ci&longs;&longs;a duobus planis parallelis, neutro per E <lb/>tran&longs;eunte, nec E intercipientibus: portionis autem axis <lb/>&longs;it FS: maior ba&longs;is circulus, vel ellip&longs;is, cuius diame­<pb/>ter AD: & circa axim EF, &longs;tet cylindrus, vel portio cylin­<lb/>drica MN ab&longs;ci&longs;&longs;a ij&longs;dem planis cum portione ABCD <lb/>ex cylindro, vel portione cylindrica, &longs;phæræ, vel &longs;phæroidi <lb/>BCR circa eius axim CFSR circum&longs;cripta, cuius &longs;it cen <lb/>trum grauitatis H, ac propterea &longs;ecta FS bifariam in pun <lb/>cto H. reliqui autem <lb/>dempta portione AB <lb/>CD ex &longs;olido MN &longs;it <lb/>centrum grauitatis K, <lb/>quod cadet in FH, & <lb/>portionis ABCD cen <lb/>trum grauitatis in ip&longs;a <lb/>HS cadet, quod &longs;it L. <lb/><!-- KEEP S--></s> <s>Dico e&longs;&longs;e HK ad KL, <lb/>vt duo rectangula GF <lb/>R, FSE, vnà cum <lb/>duabus tertiis quadra­<lb/>ti FS, ad quadratum <lb/>EG. <!-- KEEP S--></s> <s>Quoniam enim <lb/><figure id="id.043.01.247.1.jpg" xlink:href="043/01/247/1.jpg"/><lb/>&longs;imiliter vt ante o&longs;tenderemus e&longs;&longs;e HK ad KL, vt e&longs;t <lb/>portio ABCD ad &longs;olidum MN: &longs;ed portio ABCD <lb/>ad &longs;olidum MN, e&longs;t vt duo rectaugula GFR, ESF, vnà <lb/>cum duabus tertiis quadrati FS, ad quadratum EG; vt <lb/>igitur duo prædicta rectangula, vnà cum duabus tertiis <lb/>quadrati FS ad quadratum EG, ita erit HK ad KL. <lb/><!-- KEEP S--></s> <s>Quod erat demon&longs;trandum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXV.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis maioris portionis &longs;phæræ, vel &longs;phæroi­<lb/>dis centrum grauitatis e&longs;t in axe, primum bifa­<lb/>riam &longs;ecto: deinde &longs;ecundum centrum grauitatis <lb/>reliqui dempta portione ex cylindro, vel portione <pb/>cylindrica, ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;&longs;a vnà cum portio­<lb/>ne, à cylindro, vel portione cylindrica, &longs;phæræ, vel <lb/>&longs;phæroidi circa eius axim axi portionis <expan abbr="cõgruen-tem">congruen­<lb/>tem</expan> circum&longs;cripta; in eo puncto, in quo axis portio <lb/>nis &longs;ic diuiditur, vt pars prima, & &longs;ecunda &longs;ectione <lb/>terminata &longs;it ad eam, quæ prima & po&longs;trema &longs;e­<lb/>ctione terminatur, vt &longs;olidum rectangulum ex axe <lb/>portionis, & reliquo &longs;egmento axis &longs;phæræ, vel <lb/>&longs;phæroidis axi portionis congruentis, & eo, quod <lb/>&longs;phæræ, vel &longs;phæroidis, & ba&longs;is portionis centra <lb/>iungit, vnà cum binis tertijs duorum cuborum; & <lb/>eius, qui à &longs;phæræ, vel &longs;phæroidis axis fit dimi­<lb/>dio: & eius, qui ab ea, quæ &longs;phæræ, vel &longs;phæroidis, <lb/>& ba&longs;is portionis centra iungit; ad &longs;olidum rectan <lb/>gulum, quod duobus &longs;phæræ, vel &longs;phæroidis præ­<lb/>dicti axis dimidijs, & axe portionis continetur. </s></p><figure id="id.043.01.248.1.jpg" xlink:href="043/01/248/1.jpg"/><p type="main"> <s>Sit &longs;phæræ, vel &longs;phæ <lb/>roidis, cuius centrum <lb/>E maior portio ABC, <lb/>cuius axis BD, ba&longs;is <lb/>circulus, vel ellip&longs;is, cu <lb/>ius diameter AC: & <lb/>circa axem BD &longs;tet <lb/>cylindrus, vel portio <lb/>cylindrica KL, ab&longs;ci&longs; <lb/>&longs;a eodem plano cum <lb/>portione ABC, ex cy­<lb/>lindro, vel portione cy <lb/>lindrica, &longs;phæræ, vel <lb/>&longs;phæroidi ABCR circa eius axim BDR circum&longs;cripta, <pb/>& &longs;ecta BD bifariam in puncto H: deinde &longs;ecundum G <lb/>in ip&longs;a BH, centrum grauitatis reliqui dempta portione ex <lb/>&longs;olido KL, &longs;it portionis ABC in ip&longs;a DH centrum gra<lb/>uitatis F, per vim XXXVII &longs;ecundi. </s> <s>Dico e&longs;&longs;e HG ad GF, <lb/>vt &longs;olidum rectangulum ex BD, DR, DE vnà cum binis <lb/>tertiis duorum <expan abbr="cuborũ">cuborum</expan> <lb/>ex BE, ED, ad &longs;oli­<lb/>dum rectangulum ex <lb/>BD, BE, ER. <!-- KEEP S--></s> <s>Simi <lb/>liter enim vt &longs;upra de­<lb/>mon&longs;trato e&longs;&longs;e vt HG <lb/>ad GF, ita portionem <lb/>ABC ad <expan abbr="&longs;olidũ">&longs;olidum</expan> KL; <lb/>quoniamportio ABC <lb/>ad &longs;olidum KL e&longs;t vt <lb/>&longs;olidum ex BD, DR, <lb/>DE, vnà cum binis ter <lb/>tiis duorum <expan abbr="cuborũ">cuborum</expan> ex <lb/>BE, & ED, ad &longs;oli­<lb/><figure id="id.043.01.249.1.jpg" xlink:href="043/01/249/1.jpg"/><lb/>dum ex BD, BE, ER; erit vt modo dicta antecedens <lb/>magnitudo ad dictam con&longs;equentem, ita HG, ad GF. <lb/><!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis &longs;phæræ, vel &longs;phæroidis ab­<lb/>&longs;ci&longs;&longs;æ duobus planis parallelis centrum interci­<lb/>pientibus, & ab eo non æqualiter di&longs;tantibus, cen <lb/>trum grauitatis e&longs;t in axe, primum bifariam &longs;ecto: <lb/>deinde &longs;ecundum <expan abbr="c&etilde;trum">centrum</expan> grauitatis reliqui dem­<lb/>pta portione ex cylindro, vel portione cylindrica, <lb/>ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;&longs;a vnà cum portione, à cylin-<pb/>dro, vel portione cylindrica, &longs;phæræ, vel &longs;phæroi­<lb/>di circa eius axim axi portionis congruentem cir­<lb/>cum&longs;cripta; in eopuncto, in quo maius &longs;egmen­<lb/>tum axis portionis corum, quæ à centro fiunt &longs;ic <lb/>diuiditur, vt pars prima & &longs;ecunda &longs;ectione termi <lb/>nata &longs;it ad eam, quæ prima, & po&longs;trema &longs;ectione <lb/>terminatur, vt duo &longs;olida rectangula; & quod fit <lb/>ex duobus &longs;phæræ, vel &longs;phæroidis axis axi portio­<lb/>nis congruentis ijs &longs;egmentis, quæ fiunt à centro <lb/>maioris ba&longs;is portionis, & ea, quæ maioris ba&longs;is <lb/>& &longs;phæræ, vel &longs;phæroidis centra iungit: & quod <lb/>ex &longs;phæræ, vel &longs;phæroidis eiu&longs;dem axis &longs;egmentis <lb/>à centro minoris ba&longs;is factis, & ea, quæ minoris ba <lb/>&longs;is, & &longs;phæræ, vel &longs;phæroidis centra iungit, vnà <lb/>cum binis tertiis partibus duorum cuborum exijs <lb/>&longs;egmentis axis portionis, quæ à centro &longs;phæræ, <lb/>vel &longs;phæroidis fiunt; ad &longs;olidum <expan abbr="rectãgulum">rectangulum</expan> quod <lb/>duobus &longs;phæræ, vel &longs;phæroidis prædicti axis dimi <lb/>dijs, & axe portio­<lb/>nis continetur. </s></p><figure id="id.043.01.250.1.jpg" xlink:href="043/01/250/1.jpg"/><p type="main"> <s>Sit &longs;phæræ, vel &longs;phæ <lb/>roidis, cuius centrum <lb/>E, portio ABCD, ab <lb/>&longs;ci&longs;&longs;a duobus planis pa <lb/>rallelis centrum E in­<lb/>tercipientibus, & ab eo <lb/>non æqualiter di&longs;tan­<lb/>tibus: axis autem por­<lb/>tionis &longs;it GH: maior <pb/>ba&longs;is circulus, vel cllip&longs;is, cuius diameter AD. minor <expan abbr="aut&etilde;">autem</expan>, <lb/>cuius diameter ABC: & circa axim GH, &longs;tet cylindrus, <lb/>vel portio cylindrica NO, ab&longs;ci&longs;&longs;a ij&longs;dem planis cum por­<lb/>tione ABCD, ex cylindro, vel portione cylindrica &longs;phæ­<lb/>ræ, vel &longs;phæroidi BCR circa axim FGHR circum&longs;cri­<lb/>pta, cuius &longs;it centrum grauitatis K, &longs;ectio &longs;cilicet bipartiti <lb/>axis GH: reliqui autem ex &longs;olido NO dempta portione, <lb/>&longs;it centrum grauitatis L, nempe in axis GH &longs;egmento <lb/>GK, quod minorem <lb/>portionis ba&longs;im attln­<lb/>git: portionis autem <lb/>ABCD &longs;it centrum <lb/>grauitatis M: quod qui <lb/>dem in reliquo &longs;eg­<lb/>mento KH cadet. <lb/></s> <s>Dico e&longs;&longs;e KL ad LM, <lb/>vt duo &longs;olida rectan­<lb/>gula ex FH, HR, EH, <lb/>& ex RG, GF, GK, <lb/>vnà cum binis tertiis <lb/>duorum cuborum ex <lb/>EG, EH; ad &longs;olidum <lb/><figure id="id.043.01.251.1.jpg" xlink:href="043/01/251/1.jpg"/><lb/>rectangulum ex GH, EF, ER. <!-- KEEP S--></s> <s>Similiter enim vt &longs;upra <lb/>demon&longs;trato e&longs;&longs;e vt KL ad LM, ita portionem ABCD <lb/>ad &longs;olidum NO; quoniam portio ABCD ad &longs;olidum <lb/>NO, e&longs;t vt duo &longs;olida rectangula ex GH, HR, EH, & <lb/>ex RG, GF, EG, vnà cum binis tertiis duorum cubo­<lb/>rum ex EH, EG ad &longs;olidum ex GH, EF, ER, erit <lb/>vt totum iam dictum antecedens ad dictum con&longs;equens, <lb/>ita KL ad LM. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis portionis conoidis parabolici centrum <lb/>grauitatis e&longs;t punctum illud, in quo axis &longs;ic diui­<lb/>ditur, vt pars quæ ad verticem &longs;it eius, quæ ad ba­<lb/>&longs;im dupla. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXVIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis fru&longs;ti portionis conoidis parabolici cen <lb/>trum grauitatis e&longs;t punctum illud, in quo axis &longs;ic <lb/>diuiditur, vt pars minorem ba&longs;im attingens &longs;it ad <lb/>reliquam, vt duplum maioris ba&longs;is vnà cum mino<lb/>ri, ad duplum minoris, vnà cum maiori. </s></p><p type="main"> <s>Harum proportionum vtriu&longs;que non alia demon&longs;tratio <lb/>e&longs;t ab ea, quam in &longs;ecundo &longs;crip&longs;imus de centro grauitatis <lb/>conoidis parabolici, & eius fru&longs;ti: propterea quod omnis por <lb/>tionis conoidis parabolici, &longs;icut & hyperbolici &longs;ectio ba&longs;i <lb/>parallela ellip&longs;is e&longs;t &longs;imilis ba&longs;i. </s> <s>Ex corollario xv. </s> <s>de conoi­<lb/>dibus, & &longs;phæroidibus Archimedis. <!-- KEEP S--></s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO XXXIX.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis conoidis hyperbolici, vel portionis hy­<lb/>perbolici conoidis centrum grauitatis, e&longs;t pun­<lb/>ctum illud, in quo duodecima pars axis ordine <lb/>quarta ab ea, quæ ba&longs;im attingit, &longs;ic diuiditur, vt <lb/>pars propinquior ba&longs;i &longs;it ad reliquam vt &longs;e&longs;quial­<pb/>tera tran&longs;uer&longs;i lateris, hyperboles per axem, ad <lb/>axem conoidis. </s></p><figure id="id.043.01.253.1.jpg" xlink:href="043/01/253/1.jpg"/><p type="main"> <s>Sit conoides hyperbolicum, vel portio conoidis hyper­<lb/>bolici ABC, cuius axis BD, qui in portione non erit ad ba­<lb/>&longs;im perpendicularij: ba&longs;is autem dicti conoidis, vel portio­<lb/>nis &longs;it circulus, vel ellip&longs;is, cuius diameter ADC: & hyper­<lb/>boles ABC, quæ vel conoides de&longs;cribit, vel e&longs;t &longs;ectio tan­<lb/>tummodo per axem, cuius tran&longs;uer&longs;um latus &longs;it BE, & <pb/>huius &longs;e&longs;quialtera BEF: & &longs;umpta axis BD quarta par­<lb/>te DF, & tertia DG: qua ratione erit FG duodecima <lb/>pars axis BD quarta ab ea, cuius terminus D; fiat vt <lb/>IB ad BD, ita FH ad HG. <!-- KEEP S--></s> <s>Dico conoidis, vel portio­<lb/>nis ABC centrum grauitatis e&longs;&longs;e H. <!-- KEEP S--></s> <s>Nam vt e&longs;t EB <lb/>ad BD ita fiat DK ad KA: & ponatur KDY &longs;e&longs;qui­<lb/>altera ip&longs;ius DK, & ex AK ab&longs;cindatur KM &longs;ub&longs;e&longs;­<lb/>quialtera ip&longs;ius AK: & ip&longs;is DK DM, DA, æquales <lb/>eodem ordine ab&longs;cindantur DL, DN, DC: & de&longs;cri­<lb/>bantur triangula, KBL, MBN: & per puncta ABC <lb/>vertice communi B, tran&longs;eant duæ &longs;ectiones parabolæ <lb/>AOB, & BPC, ita vt contingat recta BK parabolam <lb/>AOB, recta autem BL parabolam BPC; &longs;it autem <lb/>AKLC, parabolarum diametris parallela,. Deinde <lb/>&longs;ecto axe BD bifariam, & &longs;ingulis eius partibus rur&longs;us bi­<lb/>fariam in quotlibet partes æquales, &longs;int ex illis duæ <lb/>partes DQ, QF: & per puncta QF planis quibu&longs;dam <lb/>ba&longs;i parallelis &longs;ecentur vnà &longs;olidum & hyperbole ABC: <lb/>&longs;intque hyperboles &longs;ectiones, quæ continent &longs;ectiones trian <lb/>gulorum ABC mixti, & rectilinei KBL, rectæ RTX <lb/>ZVS: <foreign lang="greek">agezdb. </foreign></s> <s>&longs;olidi autem ABC &longs;ectiones erunt cir­<lb/>culi, vel ellip&longs;es &longs;imiles ba&longs;i circa diametros RS, <foreign lang="greek">ab</foreign>. <lb/></s> <s>Quoniam igitur e&longs;t vt <foreign lang="greek">*u</foreign>K ad KD, ita AK ad KM; <lb/>vtrobique enim e&longs;t proportio &longs;e&longs;quialtera: erit permutan­<lb/>do vt YK ad A<emph type="italics"/>K<emph.end type="italics"/>, hoc e&longs;t vt IB ad BD, vel FH, ad <lb/>HG, ita D<emph type="italics"/>K<emph.end type="italics"/> ad <emph type="italics"/>K<emph.end type="italics"/>M, hoc e&longs;t triangulum BDK ad <lb/>triangulum BKM, hoc e&longs;t ad æquale huic ex demon­<lb/>&longs;tratis triangulum A<emph type="italics"/>K<emph.end type="italics"/>B mixtum: hoc e&longs;t in duplis ita, <lb/>triangulum BKL ad duo mixta rriangula AKB, BLC <lb/>&longs;imul. </s> <s>&longs;ed duorum triangulorum AKB, BLC &longs;imul e&longs;t <lb/>centrum grauitatis F, vt in hoc tertio libro demon&longs;tra­<lb/>uimus: trianguli autem BKL, vt in primo, centrum gra­<lb/>uitatis G; totius igitur trianguli ABC centrum graui­<lb/>tatis erit H. <!-- KEEP S--></s> <s>Rur&longs;us quoniam e&longs;t vt BD ad BQ hoc <pb/>e&longs;t vt rectangulum EBD ad rectangulum EBQ, ita <lb/>DK ad QX: & vt quadratum BK ad quadratum BX, <lb/>hoc e&longs;t vt quadratum BD ad quadratum BQ, ita e&longs;t <lb/>A<emph type="italics"/>K<emph.end type="italics"/> ad TX; erunt octo magnitudines quaternæ propor­<lb/><figure id="id.043.01.255.1.jpg" xlink:href="043/01/255/1.jpg"/><lb/>tionales; &longs;ed & earum primæ, & tertiæ &longs;unt proportiona­<lb/>les; nam e&longs;t vt EB ad BD, hoc e&longs;t vt rectangulum EBD <lb/>prima in primis ad quadratum BD primam in &longs;ecundis, <lb/>ita D<emph type="italics"/>K<emph.end type="italics"/> tertia in primis ad AK tertiam in &longs;ecundis; vt <pb/>igitur compo&longs;ita ex primis vtriu&longs;que ordinis ad compo­<lb/>&longs;itam ex &longs;ecundis, ita erit compo&longs;ita ex tertiis ad com­<lb/>po&longs;itam ex quartis; videlicet vt rectangulum BDE, quod <lb/>æquale e&longs;t rectangulo EBD vna cum quadrato BD, ad <lb/>rectangulum BQE, quod æquale e&longs;t rectangulo EBQ <lb/>vnà cum quadrato BQ, ita erit tota AD ad totam TQ. <lb/><!--neuer Satz-->Sed vt rectangulum BDE ad rectangulum BQE ita e&longs;t <lb/>AD quadratum, ad quadratum RQ, hoc e&longs;t ita circu­<lb/>lus, vel ellip&longs;is circa AC, ad circulum, vel &longs;imilem illi <lb/>ellip&longs;em circa RS; vt igitur AD ad TQ, hoc e&longs;t in ea­<lb/>rum duplis vt AC ad TV, ita erit circulus, vel ellip&longs;is <lb/>circa AC ad circulum, vel ellip&longs;em circa RS. <!-- KEEP S--></s> <s>Similiter <lb/>o&longs;tenderemus e&longs;&longs;e vt AC ad <foreign lang="greek">gd</foreign>, ita circulnm, vel elli­<lb/>p&longs;im circa AC, ad circulum, vel ellip&longs;em, circa <foreign lang="greek">ab</foreign>: con­<lb/>uertendo igitur, & ex æquali erunt binæ in eadem propor­<lb/>tione, vt <foreign lang="greek">gd</foreign> ad TV, ita circulus, vel ellip&longs;is circa <foreign lang="greek">ab</foreign><lb/>ad circulum, vel ellip&longs;im circa RS: & vt TV ad AC, ita <lb/>circulus, vel ellip&longs;is circa RS ad circulum, vel ellip&longs;im <lb/>circa AC. Rur&longs;us, quoniam tres rectæ lineæ incipienti <lb/>à minima <foreign lang="greek">ge</foreign>, TX, A<emph type="italics"/>K<emph.end type="italics"/> &longs;unt binæ &longs;umptæ proportio­<lb/>nales quadratis ex B<foreign lang="greek">e</foreign>, BX, B<emph type="italics"/>K<emph.end type="italics"/>, hoc e&longs;t quadratis ex <lb/>F<foreign lang="greek">e</foreign>, QX, DK; duplicata erit proportio <foreign lang="greek">ge</foreign> ad TX ip­<lb/>&longs;ius F<foreign lang="greek">e</foreign> ad QX, & TX ad AK duplicata ip&longs;ius QX ad <lb/>D<emph type="italics"/>K<emph.end type="italics"/>: &longs;ed rectæ F<foreign lang="greek">e</foreign>, QX, DK, &longs;e&longs;e æqualiter excedunt, <lb/>vtpote proportionales ip&longs;is BF, BQ, BD, propter &longs;i­<lb/>militudinem triangulorum; minor igitur proportio erit <lb/><foreign lang="greek">g</foreign>F ad TQ, quàm TQ ad AD: quare his proportiona­<lb/>lium minor erit proportio circuli, vel ellip&longs;is circa <foreign lang="greek">ab</foreign> ad <lb/>circulum, vel cllip&longs;im circa RS, quàm circuli, vel elli­<lb/>p&longs;is circa RS, ad circulum, vel ellip&longs;im, circa AC. <lb/><!-- KEEP S--></s> <s>Similiter quæcumque &longs;ectiones per prædicta axis, vel dia­<lb/>metri BD puncta &longs;ectionum fierent vt dictum e&longs;t ad ver­<lb/>ticem retrocedenti o&longs;tenderentur quælibet ternæ inter &longs;e <lb/>proximæ, binæque &longs;umptæ vtriu&longs;que ordinis proportio-<pb/>nales e&longs;&longs;e, & minor proportio vtrobique minimæ ad me­<lb/>diam quàm mediæ ad maximam; per XXXII igitur &longs;e­<lb/>cundi, triangulum mixtum, & &longs;olidum ABC, in huius <lb/>axe illius autem diametro BD commune habebunt cen­<lb/><figure id="id.043.01.257.1.jpg" xlink:href="043/01/257/1.jpg"/><lb/>trum grauitatis. </s> <s>&longs;ed demon&longs;trauimus H centrum grauita­<lb/>tis trianguli ABC; conoidis igitur vel portionis ABC <lb/>centrum grauitatis erit idem H. <!-- KEEP S--></s> <s>Quod demon&longs;trandum <lb/>erat. </s></p><pb/><p type="main"> <s>Et hic huius tertij Libri finis e&longs;&longs;et; ni&longs;i &longs;ecundo iam im­<lb/>pre&longs;&longs;o, alia quædam via magis naturalis me ad conoidis hy <lb/>perbolici centrum grauitatis reduxi&longs;&longs;et. </s> <s>Ea igitur in &longs;ecun<lb/>dum librum aliàs in&longs;erenda, nunc in &longs;equenti appendice <lb/>&longs;eptem propo&longs;itionibus expo&longs;ita, per &longs;ectionem prædicti <lb/>conoidis in conoides parabolicum eodem vertice, & circa <lb/>eundem axim, & reliquam figuram &longs;olidam, ab&longs;que com­<lb/>po&longs;ito ex duabus figuris circum&longs;criptis, quæ ex cylindris <lb/>componuntur, propo&longs;itum concludat. </s></p><p type="head"> <s>APPENDIX.</s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO I.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si &longs;int octo magnitudines quaternæ <lb/>totæ, & ablatæ proportionales, fue­<lb/>rint autem, & primarum vtriu&longs;que <lb/>ordinis ablatæ ad reliquas propor­<lb/>tionales; erunt vtriu&longs;que ordinis re <lb/>liquæ proportionales. </s></p><figure id="id.043.01.258.1.jpg" xlink:href="043/01/258/1.jpg"/><p type="main"> <s>Sint octo magnitudines quaternæ <lb/>proportionales, ac primi quidem ordi­<lb/>nis totæ, vt AB ad CD, ita EF ad <lb/>GH: &longs;ecundi autem ordinis ablatæ, vt <lb/>B ad D, ita F ad H: &longs;it autem vt B <lb/>ad A ita F ad E. <!-- KEEP S--></s> <s>Dico & reliquas <lb/>e&longs;&longs;e proportionales, videlicet vt A ad <lb/>C, ita E ad G. <!-- KEEP S--></s> <s>Quoniam enim com <lb/>ponendo, & conuertendo e&longs;t vt A ad <lb/>AB, ita E ad EF: &longs;ed vt AB ad <pb/>CD, ita e&longs;t EF ad GH; erit ex æquali vt A ad CD, <lb/>ad E ad GH: & conuertendo vt <lb/>CD ad A, ita GH ad E: & per­<lb/>mutando CD ad GH, ita A ad E. <lb/><!-- KEEP S--></s> <s>Rur&longs;us quoniam e&longs;t vt A ad B ita <lb/>E ad F: & vt B ad D, ita F ad H; <lb/>erit ex æquali, vt A ad D ita E ad <lb/>H: &longs;ed vt CD ad A, ita erat GH <lb/>ad E; ex æquali igitur erit vt CD ad <lb/>D ita GH ad H: & permutando vt <lb/>CD ad GH, ita D ad H, & reli­<lb/>qua C ad reliquam G: &longs;ed vt CD <lb/>ad GH ita erat A ad E; vt igitur <lb/>A ad C ita erit E ad G. <!-- KEEP S--></s> <s>Quod demon&longs;trandum erat. </s></p><figure id="id.043.01.259.1.jpg" xlink:href="043/01/259/1.jpg"/><p type="head"> <s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si circa datæ hyperboles communem diame­<lb/>trum parabola de&longs;cripta illius ba&longs;im ita diuidat, <lb/>vt quadratum dimidiæ ba&longs;is parabole ad reli­<lb/>quum quadrati dimidiæ ba&longs;is hyperboles eam <lb/>habeat proportionem, quam tran&longs;uer&longs;um latus <lb/>ad diametrum hyperboles; omnes in hyperbole <lb/>ad diametrum ordinatim applicatas ita &longs;ecabit, <lb/>vt exce&longs;&longs;us, quibus quadrata in hyperbole appli­<lb/>catàrum &longs;uperant quadrata in parabola ex &longs;ectio­<lb/>ne applicatarum, inter &longs;e &longs;int vt quadrata diame­<lb/>tri partium inter applicatas, & verticem inter­<lb/>iectarum. </s></p><p type="main"> <s>E&longs;to hyperbole ABC, cuius diameter BD, tran&longs;uer-<pb/>uer&longs;um latus EB. & po&longs;itis in ip&longs;a, BD duobus pun­<lb/>ctis quibuslibet GH, ordinatim applicentur MG, NH: <lb/>& circa diametrum BD &longs;it de&longs;cripta parabola KBL tali­<lb/>ter vt ip&longs;ius dimidiæ ba&longs;is DK quadratum ad reliquum <lb/>quadrati AD, &longs;it vt EB ad BD, & rectas MH, NG <lb/>in infinitum productas &longs;ecet parabola KBL in punctis <lb/>OP. <!-- KEEP S--></s> <s>Dico puncta OP intra hyperbolem cadere: & reli­<lb/>quum quadrati MG dempto quadrato GO ad reliquum <lb/>quadrati NH dempto quadrato PH, e&longs;&longs;e vt quadratum <lb/>BG ad quadratum <lb/>BH. </s> <s>Quoniam enim <lb/>ponitur vt EB ad B <lb/>D, hoc e&longs;t vt rectan­<lb/>gulum EBD ad qua­<lb/>dratum BD, ita qua­<lb/>dratum DK ad reli­<lb/>quum quadrati AD, <lb/>erit componendo, & <lb/>conueniendo, vt <expan abbr="rectã">rectam</expan> <lb/>gulum BDE ad re­<lb/>ctangulum EBD, ita <lb/>quadratum AD ad <lb/>quadratum DK: &longs;ed <lb/>vt rectangulum BGE <lb/>ad <expan abbr="rectãgulum">rectangulum</expan> BDE, <lb/><figure id="id.043.01.260.1.jpg" xlink:href="043/01/260/1.jpg"/><lb/>ita e&longs;t quadratum MG ad quadratum AD; ex æquali <lb/>igitur, vt rectangulum BGE ad rectangulum EBD, ita <lb/>e&longs;t quadratum MG ad quadratum DK: &longs;ed vt rectan­<lb/>gulum EBD ad rectangulum EBG, ita e&longs;t quadratum <lb/>DK ad GO quadratum; ex æquali igitur vt rectangu­<lb/>lu m BGE ad rectangulum EBG, ita erit quadratum <lb/>MG ad quadratum GO: &longs;ed rectangulum BGE maius <lb/>e&longs;t totum parte rectangulo EBG; quadratum igitur MG <lb/>quadrato GO maius erit, & recta MG maior quàm <pb/>GO: &longs;ecat igitur parabola KBL rectam MG in puncto <lb/>O. <!-- KEEP S--></s> <s>Similiter o&longs;tenderemus eandem parabolam &longs;ecare <lb/>quamcumque aliam in hyperbole ABC ordinatim ad dia <lb/>metrum applicatarum. </s> <s>Quoniam igitur &longs;unt octo magni <lb/>tudines quaternæ totæ, & ablatæ proportionales; ac pri­<lb/>mi quidem ordinis, vt rectangulum BDE ad rectangu­<lb/>lum BGE, ita quadratum AD ad quadratum MG: &longs;e­<lb/>cundi autem ordinis, vt rectangulum EBD ad rectangu­<lb/>lum EBG ita quadra <lb/>tum DK ad quadra­<lb/>tum OGD: &longs;ed vt <lb/>EB ad BD, hoc e&longs;t <lb/>vt ablata primæ in pri <lb/>mis rectangulum EB <lb/>D ad reliquum BD <lb/>quadratum, ita poni­<lb/>tur ablata primæ in &longs;e <lb/>cundis, quadratum D <lb/>K ad reliquum exce&longs; <lb/>&longs;um, quo quadratum <lb/>AD &longs;uperat quadra­<lb/>tum DK; vt igitur e&longs;t <lb/>reliqua primæ ad reli­<lb/>quam &longs;ecundæ in pri­<lb/><figure id="id.043.01.261.1.jpg" xlink:href="043/01/261/1.jpg"/><lb/>mis, ita erit in &longs;ecundis; videlicet vt quadratum BD ad <lb/>quadratum BG, ita reliquum quadrati AD dempto qua­<lb/>drato DK, ad reliquum qua rati MG dempto quadra­<lb/>to GO. <!-- KEEP S--></s> <s>Similiter o&longs;tenderemus reliquum quadrati AD <lb/>dempto quadrato DK ad reliquum quadrati NH dem­<lb/>pto quadrato PH, e&longs;&longs;e vt quadratum BD ad quadra­<lb/>tum BH; conuertendo igitur, & ex æquali erit vt qua­<lb/>dratum BG ad quadratum BH, ita reliquum quadra <lb/>ti MG dempto quadrato GO, ad reliquum quadrati<pb/>NH dempto quadrato PH. <!-- KEEP S--></s> <s>Quod demon&longs;trandum <lb/>erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omne conoides hyperbolicum diuiditur in <lb/>conoides parabolicum circa eundem axim, & re­<lb/>liquam figuram quandam, ad quam conoides pa­<lb/>rabolicum eam habet proportionem, quam&longs;e&longs;qui <lb/>altera tran&longs;uer&longs;i lateris hyperboles, quæ conoides <lb/>de&longs;cribit, ad axem conoidis. </s></p><figure id="id.043.01.262.1.jpg" xlink:href="043/01/262/1.jpg"/><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius axis BD: hy­<lb/>perboles autem, quæ conoides de&longs;cribit tran&longs;uer&longs;um latus <lb/>EB, cuius &longs;it &longs;e&longs;quialtera BEF: & ab&longs;ci&longs;&longs;a DG, ita vt <lb/>quadratum ex ip&longs;a ad reliquum quadrati AD &longs;it vt EB <lb/>ad BD, vertice B circa diametrum BD de&longs;cripta &longs;it <pb/>parabola GBH, eaque circumducta conoides GBH, <lb/>Dico conoides GBH comprehendi à conoide ABC & <lb/>e&longs;&longs;e ad illius reliquum, vt FB ad BD. <!-- KEEP S--></s> <s>Ab&longs;ci&longs;&longs;a enim <lb/>DK ita potentia &longs;it ad DG, vt DB ad BE longitudine, <lb/>circa axim BD de&longs;cribatur conus KBL: & &longs;ecta BD in <lb/>multas partes æquales, ducto&longs;que per ea puncta planis <lb/>quibu&longs;dam ba&longs;i parallelis, &longs;ecentur tria dicta &longs;olida, conus <lb/>&longs;cilicet & vtrumque conoides: & &longs;uper &longs;ectiones circulos <lb/>de&longs;cribantur cylindri æqualium altitudinum terni cuca <lb/><figure id="id.043.01.263.1.jpg" xlink:href="043/01/263/1.jpg"/><lb/>communes axes partes æquales, in quas axis BD diui&longs;us <lb/>fuit, & inter eadem plana parallela: & omnino triplex figura <lb/>ex cylindris, quos diximus &longs;it tribus dictis &longs;olidis circum&longs;cri <lb/>pta: &longs;intque circa duos axes infimos DM, MN terni cylin­<lb/>dri AO, GP, KQ: & proxime ordine ip&longs;is re&longs;pondentes <lb/>cylindri TX, SV, RZ, quorum ba&longs;es circa diametros <lb/>TI, S<foreign lang="greek">b</foreign>, R<foreign lang="greek">a</foreign>, communes &longs;ectiones plani per punctum M, <lb/>cum tribus &longs;olidorum &longs;ectionibus per axem, triangulo &longs;cili­<lb/>cet, parabola, & hyperbole in eodem plano, atque ideo tres <pb/>diametri TI, S<foreign lang="greek">b</foreign>, R<foreign lang="greek">a</foreign>, erunt in vna recta linea. </s> <s>Quoniam <lb/>igitur e&longs;t vt EB ad BD, ita quadratum DG ad <expan abbr="reliquũ">reliquum</expan> <lb/>quadrati AD, &longs;ecabit parabola GBH omnes in hyperbo­<lb/>le ABC ad diametrum ordinatim applicatas, quare conoi <lb/>des ABC comprehendet conoides GBH: atque ita para­<lb/>bola &longs;ecabit, vt exce&longs;&longs;us quibus quadrata in hyperbole ap­<lb/>plicatarum &longs;uperant partes quadrata in parabola applicata <lb/>rum, inter &longs;e &longs;int vt quadrata partium diametri BD inter <lb/>applicatas & verticem interiectarum, prout vt inter &longs;e <expan abbr="re&longs;põ">re&longs;pom</expan> <lb/>dent: vt igitur e&longs;t quadratum BD ad quadratum BM, hoc <lb/>e&longs;t vt quadratum DK ad quadratum RM, ita erit <expan abbr="reliquũ">reliquum</expan> <lb/>AD quadrati dempto quadrato DG ad reliquum quadrati <lb/>TM dempto quadrato SM, & permutando. </s> <s>Sed quia qua­<lb/>dratum DG ad reliquum quadrati AD, & ad quadratum <lb/>DK eandem habet proportionem ex vi con&longs;tructionis, reli <lb/>quum quadrati AD, dempto quadrato DG æquale e&longs;t <lb/>quadrato DK; reliquum igitur quadrati TM dempto qua <lb/>drato SM æquale erit quadrato RM: &longs;i igitur vtri&longs;que ad­<lb/>dantur &longs;ingula communia, vnis quadratum DG, alteris <lb/>quadratum SM, erit & quadratum AD æquale duobus <lb/>quadratis GD, DK, & quadratum TM duobus quadra <lb/>tis SM, MR æquale. </s> <s>&longs;ed cum cylindri eiuidem altitudi­<lb/>nis inter &longs;e &longs;int vt ba&longs;es, &longs;unt vt quadrata, quæ ab eorundem <lb/>ba&longs;ium &longs;emidiametris fiunt; cylindiusigitur AO æqualis <lb/>e&longs;t duobus cylindris GP, KQ: & cylindrus TX duobus <lb/>cylindris S<foreign lang="greek">*u</foreign>, RZ æqualis. </s> <s>Eadem ratio e&longs;t de reliquis <lb/>deinceps. </s> <s>Tota igitur figura conoidi ABC circum&longs;cripta, <lb/>vtrique &longs;imul, conoidi GBH, & cono KBL circum&longs;cri­<lb/>ptæ æqualis erit. </s> <s>po&longs;&longs;unt autem eæ figuræ ita e&longs;&longs;e dictis &longs;oli­<lb/>dis circum&longs;criptæ per ea quæ alibi o&longs;tendimus, vt &longs;uperent <lb/>in&longs;criptas minori &longs;pacio quantacumque magnitudine pro­<lb/>po&longs;ita; per tertiam igitur &longs;ecundi, conoides ABC vtrique <lb/>&longs;imul, conoidi GBH, & cono KBL æquale erit. </s> <s>dempto <lb/>igitur <expan abbr="cõmuni">communi</expan> conoide GBH, reliquum <expan abbr="&longs;olidũ">&longs;olidum</expan> AGBHC <pb/>æquale erit cono KBL. <!-- KEEP S--></s> <s>Rur&longs;us quia e&longs;t vt EB ad BD, ita <lb/>quadratum GD ad quadratum DK, hoc e&longs;t circulus cir­<lb/>ca GH ad circulum circa KL, hoc e&longs;t conus GBH &longs;i <lb/>de&longs;cribatur ad conum KBL: &longs;ed vt FB ad BE ita e&longs;t co­<lb/>noides GBH ad conum GBH; ex æquali igitur erit vt <lb/>FB ad BD, ita conoides GBH ad conum KBL, hoc <lb/>e&longs;t ad &longs;olidum AGBHC. </s> <s>Manife&longs;tum e&longs;t igitur <expan abbr="propo&longs;itũ">propo&longs;itum</expan>. </s></p><p type="head"> <s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Ex huius Theorematis demon&longs;tratione manife <lb/>&longs;tum e&longs;t, ij&longs;dem po&longs;itis cylindros deficientes, ex <lb/>quibus con&longs;tat exce&longs;&longs;us, quo figura conoidi hyper <lb/>bolico circum&longs;cripta &longs;uperat circum&longs;criptam co­<lb/>noidi parabolico, ita &longs;e habere, vt quorumlibet <lb/>trium inter &longs;e proximorum minor proportio &longs;it <lb/>minimi ad medium, quam medij ad maximum: <lb/>æquales enim &longs;unt &longs;inguli &longs;ingulis cylindris, ex <lb/>quibus con&longs;tat figura cono BKL circum&longs;cripta, <lb/>qui &longs;unt inter eadem plana parallela. </s> <s>Quod &longs;i <lb/>ita e&longs;t, &longs;imul illud manife&longs;tum erit, & ex hoc, & <lb/>ex ijs, quæ in &longs;ecundo libro demon&longs;trauimus; præ­<lb/>dictum exce&longs;&longs;um ex tot cylindris deficientibus <lb/>eiu&longs;dem altitudinis, quos diximus componi po&longs;&longs;e, <lb/>vt ip&longs;ius centrum grauitatis in axe BD di&longs;tet à <lb/>centro grauitatis coni KBL, hoc e&longs;t à puncto in <lb/>quo axis BD &longs;ic diuiditur, vt pars, quæ ad ver­<lb/>ticem &longs;it reliquæ tripla, ea di&longs;tantia, quæ minor <lb/>&longs;it quantacum que longitudine propo&longs;ita. </s></p><pb/><p type="head"> <s><emph type="italics"/>PROPOSITIO IIII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Si conoidi parabolico figura circum&longs;cribatur, <lb/>& altera in&longs;cribatur ex cylindris æqualium alti­<lb/>tudinum, binis circa communes axes &longs;egmenta <lb/>axis conoidis, & inter eadem plana parallela, mi­<lb/>nimo circum&longs;criptorum ad nullum relato; omnia <lb/>re&longs;idua cylindrorum figuræ circum&longs;criptæ dem­<lb/>ptis figuræ in&longs;criptæ cylindris, & inter &longs;e, & mi­<lb/>nimo cylindro æqualia erunt. </s></p><p type="main"> <s>Sit conoidi parabolico ABC, cuius axis BD circum­<lb/>&longs;cripta figura ex quotcumque cylindris æqualium altitu­<lb/>dinum, quorum tres deinceps &longs;int EL minimus &longs;upremus, <lb/>& GQ, IR, quorum ba&longs;es eodem ordine circuli, quorum <lb/>&longs;emidiametri ad parabolæ, quæ figuram de&longs;cribit diame­<lb/>trum BD ordi­<lb/>natim applicatæ <lb/>&longs;int EF, GH, IK: <lb/>& in duplos cre­<lb/>&longs;centibus cylin­<lb/>dris circa <expan abbr="priorũ">priorum</expan> <lb/>axium duplos a­<lb/>xes BH, IK, HD, <lb/>& <gap/>c deinceps <lb/>quotcumque plu­<lb/>res e&longs;sent; &longs;it co­<lb/>noidi ABC in­<lb/><figure id="id.043.01.266.1.jpg" xlink:href="043/01/266/1.jpg"/><lb/>&longs;cripta figura ex cylindris æqualium altitudinum inter &longs;e, & <lb/>circum&longs;criptis. </s> <s>Bini itaque circa communes axes inter ea­<lb/>dem plana parallela interijcientur, minimo EL ad nullum <pb/>relato: huic autem proximus, & æqualis cylindrorum in­<lb/>&longs;criptorum &longs;it NM ba&longs;im ip&longs;i communem habens circu­<lb/>lum circa EFM: & con&longs;equenti circum&longs;criptorum GQ <lb/>&longs;it. </s> <s>in&longs;criptorum æqualis PO ba&longs;im habens ip&longs;i commu­<lb/>nem circulum circa GHO: &longs;int autem circulorum qui <lb/>&longs;unt ba&longs;es cylindrorum diametri in parabola per axim: <lb/>quæ quoniam &longs;unt communes &longs;ectiones cum parabola per <lb/>axim planorum ba&longs;i conoidis, & inter &longs;e parallelorum, <lb/>erunt etiam ip&longs;æ inter &longs;e, & parabolæ ba&longs;i AC parallelæ, <lb/>earumque dimidiæ vt EF, GH ad diametrum BD or­<lb/>dinatim applicatæ. </s> <s>Quoniam igitur in parabola ABC <lb/>e&longs;t vt HB ad BF ita quadratum GH ad quadratum <lb/>EF, duplum erit <lb/>quadratum GH <lb/>quadrati EF: qua <lb/>re & circulus cir­<lb/>ca GO circuli <lb/>circa EM at que <lb/>adeo cylindrus <lb/>GQ cylindri E <lb/>L duplus, pro­<lb/>pter <17>qualitatem <lb/>altitudinum: &longs;ed <lb/>& cylindrus NL <lb/><figure id="id.043.01.267.1.jpg" xlink:href="043/01/267/1.jpg"/><lb/>duplus e&longs;t cylindri EL per con&longs;tructionem; cylindrus igi­<lb/>tur GQ æqualis e&longs;t cylindro NL: & ablato communi <lb/>NM cylindro, reliquus GQ deficiens cylindro NM <lb/>cylindro EL æqualis. </s> <s>Rur&longs;us quia e&longs;t vt KB ad BH, <lb/>ita quadratum IK ad quadratum GH, hoc e&longs;t ita IR <lb/>cylindrus ad cylindrum GQ: &longs;ed vt HB ad BF ita <lb/>erat cylindrus GQ ad cylindrum EL; tres igitur cy­<lb/>lindri IR, GQ, EL, tribus lineis BK, BH, BF, eodem <lb/>ordine proportionales erunt: &longs;ed tres eædem lineæ &longs;e&longs;e <lb/>æqualiter excedunt; tres igitur dicti cylindri &longs;e&longs;e æqua-<pb/>liter excedent, hoc e&longs;t reliquum cylindri IR dempto cylin­<lb/>dro PO æquale erit reliquo cylindri GQ dempto cylin­<lb/>dro NM, & reliquum cylindri GQ dempto cylindro <lb/>NM æquale cylindro EL. </s> <s>Similiter ad reliquos cylindros <lb/>quotcumque plures e&longs;&longs;ent de&longs;cendentes o&longs;tenderemus, om <lb/>nes exce&longs;&longs;us, quibus cylindri circum&longs;cripti in&longs;criptos <lb/>&longs;uperant &longs;ibi quique re&longs;pondentes inter &longs;e & cylindro <lb/>EL æquales e&longs;&longs;e. </s> <s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO V.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Dato conoide hyperbolico, & ip&longs;ius conoi­<lb/>de parabolico circa eundem axim, quod ad <lb/>reliquum hyperbolici conoidis eam proportio­<lb/>nem habeat, quam &longs;e&longs;quialtera tran&longs;uer&longs;i late­<lb/>ris hyperboles, quæ conoides de&longs;cribit, ad axim <lb/>conoidis; fieri pote&longs;t vt conoidi parabolico fi­<lb/>guræ quædam in&longs;cribatur, & altera circum&longs;cri­<lb/>bantur vt &longs;upra factum e&longs;t, & hyperbolico alio cir­<lb/>cum&longs;cribatur omnes ex cylindris æqualium al­<lb/>titudinum multitudine æqualibus exi&longs;tentibus <lb/>ijs, ex quibus con&longs;tant figuræ conoidibus cir­<lb/>cum&longs;criptæ, ita vt exce&longs;&longs;us, quo figura conoidi <lb/>parabolico circum&longs;cripta in&longs;criptam &longs;uperat, <lb/>quem breuitatis cau&longs;a voco exce&longs;&longs;um primum, <lb/>ad exce&longs;&longs;um, quo figura conoidi hyperbolico cir­<lb/>cum&longs;cripta &longs;uperat circum&longs;criptam parabolico, <lb/>quem voco exce&longs;&longs;um &longs;ecundum, minorem habeat <lb/>proportionem quacumque propo&longs;ita. </s></p><pb/><p type="main"> <s>Sit conoides hyperbolicum ABC, & pars eius para­<lb/>bolicum EBF circa eundem axim BD: & conoides <lb/>EBF ad reliquum conoidis ABC eam habeat proportio­<lb/>nem, quam &longs;e&longs;quialtera tran&longs;uer&longs;i lateris hyperboles per <lb/>axim ABC ad axim BD. <!-- KEEP S--></s> <s>Dico fieri po&longs;&longs;e quod proponitur. <lb/></s> <s>Habeat enim DL ad LB quamcumque proportionem: & <lb/>conoides ABC reliquo &longs;olido AEBFC dempto conoi <lb/>de EBF. &longs;it conus circa axim BD æqualis GBH: & <lb/>de&longs;cribatur conus GLH: & &longs;ecta BD bifariam in pun­<lb/>cto K, & rur&longs;us BK, KD in multitudine, & longitudi­<lb/>ne æquales in&longs;cribatur conoidi EBF, & altera cirum&longs;cri­<lb/><figure id="id.043.01.269.1.jpg" xlink:href="043/01/269/1.jpg"/><lb/>batur, vt in antecedenti factum e&longs;t, figura ex cylindris æ <lb/>qualium altitudinum, ita vt exce&longs;&longs;us, quo circum&longs;cripta <lb/>&longs;uperat in&longs;criptam fit minor cono GLH; & cylindris cre­<lb/>&longs;centibus in latitudinem ab&longs;oluatur figura conoidi ABC <lb/>circum&longs;cripta ex cylindris altitudine, & multitudine æqua <lb/>libus ijs, qui &longs;unt circa conoides EBF. <!-- KEEP S--></s> <s>Quoniam igitur <lb/>primus exce&longs;&longs;us e&longs;t minor cono GLH, multo minor crit <lb/>pars eius communis &longs;olido AEBFG, quàm conus GLH: <lb/>&longs;ed &longs;olidum AEBFC æquale e&longs;t cono GBH; reliquum <lb/>igitur &longs;olidi AEBFC dicto communi ablato, maius erit <lb/>coni GBH reliquo BGLH; minor igitur proportio e&longs;t <pb/>primi exce&longs;&longs;us minoris cono GLH, ad dictum reliquum <lb/>&longs;olidi AEBFC, quàm coni GLH ad reliquum coni <lb/>GBH: &longs;ed &longs;ecundus exce&longs;&longs;us maior e&longs;t prædicto reliquo <lb/>&longs;olidi AEBFC, ctenim illud comprehendit; multo igitur <lb/>minor proportio erit primi exce&longs;&longs;us ad &longs;ecundum, quàm <lb/>coni GLH ad reliquum BGLH, hoc e&longs;t minor propor­<lb/>tio quàm DL ad LB: ponitur autem proportio DL ad <lb/>LB quali&longs;cumque. </s> <s>Fieri igitur pote&longs;t, quod proponitur. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VI.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis re&longs;idui conoidis hyperbolici dempto <lb/>conoide parabolico, vt &longs;upra diximus, centrum <lb/>grauitatis e&longs;t punctum illud, in quo axis &longs;ic diui­<lb/>ditur, vt pars propinquior vertici &longs;it tripla re­<lb/>liquæ. </s></p><figure id="id.043.01.270.1.jpg" xlink:href="043/01/270/1.jpg"/><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius axis BD, & <lb/>ablatum conoides parabolicum EBF circa eundem axim <lb/>BD, ita &longs;it ad reliquum &longs;olidum AEBFC, vt &longs;e&longs;quialte <lb/>ra tran&longs;uer&longs;i lateris hyperboles, quæ conoides de&longs;cribit ad <lb/>axem BD: & ponatur BG ip&longs;ius GD tripla. </s> <s>Dico re­<pb/>liqui &longs;olidi AEBFC centrum grauitatis e&longs;se G. <!-- KEEP S--></s> <s>Secta <lb/>enim BD bifariam in puncto H, & po&longs;ita GK ip&longs;ius GH <lb/>minori quantacumque longitudine propo&longs;ita, &longs;umptoque <lb/>in GK quolibet puncto L, intelligantur id enim (fieri po&longs;­ <lb/>&longs;e manife&longs;tum e&longs;t ex &longs;upra demon&longs;tratis) tres figuræ vna in­<lb/>&longs;cripta conoidi EBF, & duæ circum&longs;criptæ altera alteri <lb/>conoidum, vt &longs;upra factum e&longs;t, compo&longs;itæ ex cylindris <lb/>æqualium altitudinum ita multiplicatis, vt vtrumque illud <lb/>accidat; & vt &longs;ecundi exce&longs;&longs;us centrum grauitatis quod &longs;it <lb/>M (omnium autem trium dictorum exce&longs;&longs;uum in axe <lb/>BD erunt centra grauitatis) &longs;it puncto G propinquius <lb/><figure id="id.043.01.271.1.jpg" xlink:href="043/01/271/1.jpg"/><lb/>quàm punctum L: & vt primus exce&longs;&longs;us ad &longs;ecundum mi­<lb/>norem habeat proportionem ea, quæ e&longs;t LK, ad KH. <!-- KEEP S--></s> <s>Dein <lb/>de vt HK ad KL, ita &longs;it HN ad NM, & vt primus <lb/>exce&longs;&longs;us ad &longs;ecundum, ita MO ad OH. <!-- KEEP S--></s> <s>Quoniam igitur <lb/>cylindri omnes deficientes, & &longs;ummus integer, ex quibus <lb/>primus exce&longs;&longs;us con&longs;tat, inter &longs;e &longs;unt æquales, habentque <lb/>in axe BD centra grauitatis æqualibus interuallis à bipar­<lb/>titi axis BD &longs;ectione H & inter &longs;e di&longs;tantia; totius pri­<lb/>mi exce&longs;&longs;us centrum grauitatis erit H: &longs;ecundi autem ex­<lb/>ce&longs;&longs;us centrum grauitatis ponitur M; cum igitur &longs;it vt pri­<lb/>mus exce&longs;&longs;us ad &longs;ecundum, ita ex contraria parte MO <pb/>ad OH, erit tertij exce&longs;&longs;us ex duobus prioribus compo&longs;i­<lb/>ti centrum grauitatis O. <!-- KEEP S--></s> <s>Quoniam igitur minor propor­<lb/>tio e&longs;t primi exce&longs;&longs;us ad &longs;edundum, hoc e&longs;t MO ad OH, <lb/>quàm LK ad KH; erit conuertendo maior proportio HO <lb/>ad OM, quàm HK ad KL: &longs;ed vt HK ad KL, ita <lb/>ponitur HN ad NM; maior igitur proportio e&longs;t HO ad <lb/>OM, quàm HN ad NM; eiu&longs;dem igitur lineæ HM <lb/>minor erit MO, quàm MN, & punctum O propinquius <lb/>puncto G quam punctum N. <!-- KEEP S--></s> <s>Rur&longs;us quia vt HK ad <lb/>KL, ita e&longs;t HN ad NM; erit componen do & per con­<lb/>uer&longs;ionem rationis, vt LH ad HK ita MH ad HN: & <lb/>permutando, vt HM ad HL, ita HN ad HK: &longs;ed HM <lb/>e&longs;t maior quàm HL; ergo & HN erit maior quam H<emph type="italics"/>K<emph.end type="italics"/>, <lb/>& punctum N propinquius puncto G quàm punctum K: <lb/>&longs;ed punctum O propinquius erat puncto G quàm punctum <lb/>N; multo igitur erit punctum O propinquius puncto G <lb/>quàm punctum K. ponitur autem di&longs;tantia GK minor <lb/>quantacumque longitudine propo&longs;ita: & e&longs;t O centrum <lb/>grauitatis tertij exce&longs;&longs;us reliquo &longs;olido AEBFC circum­<lb/>&longs;cripti; ex ijs igitur, quæ in primo libro demon&longs;trauimus, <lb/>&longs;olidi AEBFC centrum grauitatis erit G. <!-- KEEP S--></s> <s>Quod demon­<lb/>&longs;trandum erat. </s></p><p type="head"> <s><emph type="italics"/>PROPOSITIO VII.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> <s>Omnis conoidis hyperbolici centrum grauita­<lb/>tis e&longs;t punctum illud, in quo duodecima pars axis <lb/>quarta ab ea, quæ ba&longs;im attingit &longs;ic diuiditur, vt <lb/>pars propinquior ba&longs;i &longs;it ad reliquam, vt &longs;e&longs;quial­<lb/>tera tran&longs;uer&longs;i lateris hyperboles, quæ conoides <lb/>de&longs;cribit; ad axem conoidis. </s></p><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius axis BD: <pb/>tran&longs;uer&longs;um latus hyperboles, quæ conoides de&longs;cribit &longs;it <lb/>BE, huius autem &longs;e&longs;quialtera BEF: & &longs;umpta axis BD <lb/>tertia parte DG, & quarta DH, qua ratione erit GH <lb/>axis BD pars duodecima, ordine quarta ab ea, cuius termi <lb/>nus D; e&longs;to vt FB ad BD, ita HK ad KG. <!-- KEEP S--></s> <s>Dico conoi­<lb/>dis ABC centrum grauitatis e&longs;&longs;e K. <!-- KEEP S--></s> <s>Diuidatur enim co­<lb/><figure id="id.043.01.273.1.jpg" xlink:href="043/01/273/1.jpg"/><lb/>noides ABC in parabolicum conoides LBM, & reliquum <lb/>&longs;olidum ALBMC, ita vt conoides LBM ad &longs;elidum <lb/>ALBMC &longs;it vt FB ad BD, hoc e&longs;t vt HK GK. <!-- KEEP S--></s> <s>Quo­<lb/>niam igitur G e&longs;t centrum grauitatis conoidis LBM, & H <lb/>&longs;olidi ALBMC; tot us conoidis ABC centrum graui <lb/>tatis crit K. <!-- KEEP S--></s> <s>Quod demon&longs;trandum crat. </s></p><p type="head"> <s>TERTII LIBRI FINIS.<!-- KEEP S--></s></p> <pb/> <pb/> <pb/> <pb/> <pb/> </chap> </body> <back/> </text></archimedes>