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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <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></translator> <lang>la</lang> <cvs_file>valer_centr_043_la_1604.xml</cvs_file> <cvs_version></cvs_version> <locator>043.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="043/01/001.jpg" id="p.0001"></pb><p type="head"> <s>DE CENTRO <lb></lb>GRAVITATIS <lb></lb>SOLIDORVM <lb></lb>LIBRITRES.</s></p><p type="head"> <s>LVCÆ VALERII <lb></lb><emph type="italics"></emph>Mathematicæ, & Ciuilis Philoſophiæ <lb></lb>in Gymnaſio Romano profeſſoris.<emph.end type="italics"></emph.end></s></p><figure id="id.043.01.001.1.jpg" xlink:href="043/01/001/1.jpg"></figure><p type="head"> <s>ROMÆ, Typis Bartholom ri Bonfadini. </s> <s>MDC IIII. <lb></lb>SVPERIORVM PERMISSV.</s></p><pb xlink:href="043/01/002.jpg"></pb><p type="main"> <s>Imprimatur </s></p><p type="main"> <s>Si placet R. P. Magiſtro S. Palati<gap></gap> <lb></lb>B. </s> </p><p type="main"> <s>Gypſius Viceſger. </s></p><p type="main"> <s><emph type="italics"></emph>Imprimatur<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Fr. Io. Maria Braſichellen. Sacri Pal. <lb></lb></s> <s>Apostol. Magiſt.<emph.end type="italics"></emph.end></s> </p><pb xlink:href="043/01/003.jpg"></pb><figure id="id.043.01.003.1.jpg" xlink:href="043/01/003/1.jpg"></figure><p type="head"> <s>SANCTISSIMO <lb></lb>DOMINO NOSTRO <lb></lb>CLEMENTI VIII <lb></lb>PONT. OPT. MAX.<emph type="italics"></emph>Lucas Valerius perpetuam felicitatem.<emph.end type="italics"></emph.end></s></p><figure id="id.043.01.003.2.jpg" xlink:href="043/01/003/2.jpg"></figure><p type="main"> <s>Grata Principi munera, <lb></lb>P. B. ex Philoſophiæ late<lb></lb>bris deprompta, quaſi aurum <lb></lb>ſoli expoſitum illico ſplen<lb></lb>dent, & publicæ vtilitatis <lb></lb>ſpem oſtendunt, magno or<lb></lb>nata præſidio in primos liuo<lb></lb>ris impetus illius approbatione, cuius officium eſt <lb></lb>alia à rep. </s> <s>auertere, alia imperare. </s> <s>Hinc por<lb></lb>rò factum eſt, vt omnis ferè ſcriptor exiſti matio<lb></lb>nis periculum aditurus, aliquem ex principibus <pb xlink:href="043/01/004.jpg"></pb>viris ſibi deligat, cuius autoritate ipſi dicatum <lb></lb>opus ab inuidorum morſibus ſeruetur incolume. <lb></lb></s> <s>Hanc ergo conſuetudinem amanti mihi ſanè feli<lb></lb>citer cecidit, vt tu ſola tua propria benignitate <lb></lb>permotus in tuos me familiares vltro aſcriberes. <lb></lb></s> <s>Siue enim ingenij mei debilis partus <expan abbr="magnā">magnam</expan> pa<lb></lb>troni deſiderat autoritatem: tu principum orbis <lb></lb>terrarum princeps ſemper digniſſimam principa<lb></lb>tu ſapientiam præſtitiſti. </s> <s>Seu tam elatæ dedica<lb></lb>tiones ſolent alienas à ſapientiæ ſtudio ſpes olere: <lb></lb>lux tanti patrocinij, <expan abbr="tuorumq́">tuorumque</expan> veterum in me be<lb></lb>neficiorum, atram ſuſpicionem amouebit. </s> <s>Quòd <lb></lb>verò ad vitam ipſius operis attinet, quam nulla <lb></lb>per te velim temporum permutatione terminari: <lb></lb>vereor vt id ſua luce multis alijs vitali aſpiciat <lb></lb>illa, quæ tua ſtudia, & res geſtas omnium lin<lb></lb>guis, & litteris celebrabit æternitas. </s> <s>quantum <lb></lb>enim tuam excelſam ſuſpicio dignitatem, tantum <lb></lb>deſpicor iſtius doni incredibilem cum illa com<lb></lb>parati humilitatem: neque id niſi diuinitus cre<lb></lb>diderim perpetuam in tuis laudibus famam ha<lb></lb>biturum. </s> <s>Quare illud non ſolum tibi diuini gre<lb></lb>gis antiſtiti cupio gratum accidere, cuius auto<lb></lb>ritate protectum in tanta nouarum rerum poſt <lb></lb>tam graues autores contemptione, minimo meo <lb></lb>cum rubore in medium prodeat: ſed ipſi diuinita<lb></lb>ti ex voluntate donum expendenti, penes quam <lb></lb>eſt æternitas, & cui primum dicata omnia eſſe <lb></lb>oportet: vt hi, quostuis luminibus dignaris, de <pb xlink:href="043/01/005.jpg"></pb>centro grauitatis ſolidorum ſterilis ingenij mei <lb></lb>teſtes libelli à mortis æmula me obliuione defen<lb></lb>dant. </s> <s>Stomacharis hic, arbitror, quòd tantum <lb></lb>ſpectem de nihilo; ſed magis confeſſionis impu<lb></lb>dentia. </s> <s>At verò non impetus animi ad gloriam, <lb></lb>cuius nullum mihi natura ſemen impartiuit (ſit <lb></lb>gloriæ loco ignauiæ fugiſſe dedecus) ſed tua er<lb></lb>ga me voluntas, meisapta ſtudijs liberalitate te<lb></lb>ſtata hunc ardorem expreſſit. </s> <s>Tanta enim eſt <lb></lb>venuſtas tuæ virtutis ex mei meriti penuria, vt <lb></lb>putem ſine me indice illam diminutum ſui ſpecta<lb></lb>culum poſteris præbituram. </s> <s>Nihil ergo minus <lb></lb>cogitans quàm quî tua beneficia cumulando per<lb></lb>turbatis iudicijs ſatisfacerem, ſcientia ſcilicet, <lb></lb>& virtute illa, qua maximè ſuperbit eneruata, & <lb></lb>areſcens Mundiætas; nullum opulentiæ meæ, ar<lb></lb>tis alienæ ſpecimen pro munere gratiæ à te acce<lb></lb>pto partem tibi reddidi: ſed ingenij mei partum, <lb></lb>qualis is cumque eſt; quod & grati animi quæſi<lb></lb>tum monumentum crimine me audaciæ liberet, <lb></lb>ſi quodimpendeat, palam dedicaui. </s> <s>Alij tibi co<lb></lb>lumnas honeſtiſſimis titulis ornatas erigant: ſta <lb></lb>tuas in foris collocent: magnificas ædes extruant, <lb></lb>quarum in frontibus grandes marmoreæ tabulæ <lb></lb>flammantibus auro ſyderibus, & peregrinis lapi<lb></lb>dibus intextæ ea de te viuo referant ſaxum impu<lb></lb>dens, quæ verecunda hæc pagina prætermittit. <lb></lb></s> <s>Ego incredibilis tuæ benignitatis non tam gra<lb></lb>uia teſtimonia, quæ loco moueri nequeant: ſed <pb xlink:href="043/01/006.jpg"></pb>expeditum hunc nuntium in longiſſima itinera <lb></lb>deſtinaui. </s> <s>Quem quidem eo minus vereor ne <lb></lb>non tu, quamobrem Telchines fortaſſe aliqui in<lb></lb>ſectaturi, diſpari ſis voluntate protecturus, quòd <lb></lb>in his tàm reconditis naturæ arcanis geometrica <lb></lb>demonſtratione patefactis, tanquam in ſemine <lb></lb>multiplicem præſcriptionem, ac normam eſſe in<lb></lb>telliges ipſe pacis inter tuos greges autor, lupi <lb></lb>otomani terror, ciuili, & bellicæ architecturæ <lb></lb>maximè neceſſariam. </s> <s>Quòd que, cum ad theologi<lb></lb>cam quandam veritatem chriſtiano generi maxi<lb></lb>me ſalutarem illuſtrandam, per Philoſophi<17> etiam <lb></lb>campos ſapientium hominum corona decoratus, <lb></lb>nulla tantæ molis, quantam ſuſtines negotiorum <lb></lb>iactura latiſſimè vageris; nempe illam creſcere, <lb></lb>atque illuſtrari indies magis ex optas, cuius con<lb></lb>ſuetudine tantopere delectaris. </s> <s>Quod denique <lb></lb>ſcientiæ ciuilis ipſe peritiſſimus omnium optimè <lb></lb>intelligis, quanti referat ad humanæ ſocietatis for <lb></lb>mam & candorem, regum, atque optimatum a<lb></lb>mor in ſtudioſos bonarum litterarum. </s> <s>contrà au<lb></lb>tem ex deſpectione in hos cadente abijs, quorum <lb></lb>mores pro legibus haberi ſolent, noſti commu<lb></lb>nem ingeniorum veternum, mox tyrannidem gi<lb></lb>gni, magna cuſtode adempta modeſtiæ imperi<lb></lb>tantium crebra ciuium ſapientia, quæ prauis ti<lb></lb>morem efficit, melioribus pudorem, Quod ſi meæ <lb></lb>expectationi exitus reſpondebit, vt te hoc munu<lb></lb>ſculo vel leuiter lætari ſentiam; alia non iniucun-<pb xlink:href="043/01/007.jpg"></pb>da ftatim proferam, qua PETRVS ALDOBRAN<lb></lb>DINVS tuus nepos, domi foriſque clariſſimus <lb></lb>Cardinalis, cuius inter familiares itidem, <expan abbr="bene-ficijsq́ue">bene<lb></lb>ficijsque</expan> deuinctos locum habeo, ſuæ erga me hu<lb></lb>manitatis teſtimonia ab inuidiæ ſatellite & mi<lb></lb>niſtra calumnia tueatur: quando duobus talibus <lb></lb>viris animi mei captum beneficentia ſua pericli<lb></lb>tantibus, duplex periculum ſubire ſum coactus. <lb></lb></s> <s>Sed iam verboſæ epiſtolæ, & tuo faſtidio finem im <lb></lb>poſiturus peto à te vnum; vt tibi perſuadeas, me <lb></lb>inter tuos famulos, quos ære proprio, & victu quo<lb></lb>tidiano liberaliter ſuſtentas, eorum, qui pro te <lb></lb>emori poſſunt, amore, conſtantia, fidelitate nemini <lb></lb>planè concedere. </s> <s>Sic tua omnia præſtantiſſima <lb></lb>facinora Princeps magnanime, & pietatis colu<lb></lb>men, Deus Opt. Max. tibi fortunet, quem ad ma<lb></lb>iores in dies res gerendas in longum æuum inco<lb></lb>lumen, felicemque conſeruet. </s> <s>Valet. </s></p><pb xlink:href="043/01/008.jpg"></pb><p type="head"> <s><foreign lang="grc">ΛΟΥΚΑ ΟΥΑΛΕΡΙΟΥ <lb></lb>ΕΙΣ ΤΑ ΑΥΤΟΥ ΚΕΝΤΡΑ</foreign></s></p><p type="head"> <s><foreign lang="grc">σ<gap></gap>ξεῶν β<gap></gap>ζέων, ἐπί<gap></gap>μμα</foreign>.</s></p><p type="main"> <s><foreign lang="grc">Παίγνια φιλο<gap></gap>φοις Λουκας̄ τ<gap></gap> δε ούμ<gap></gap>λοκα δάφ<gap></gap>, <lb></lb>Στ<gap></gap>υμόνος ἐγκελάδς <gap></gap>εί<gap></gap>ονα π<gap></gap>λύ<gap></gap>ν. </foreign></s></p><p type="main"> <s><foreign lang="grc">Δῶρον ἕπεμψά πέ<gap></gap>ας δ̓<gap></gap>̔ζε̄ιν τῑς <gap></gap>ῡ <gap></gap>τ̓ ἄδ<gap></gap><lb></lb>Β<gap></gap>θοούνης βαπέων πη̄ξε <gap></gap>έμεθλα φύ<gap></gap>ς. </foreign></s></p><p type="main"> <s><foreign lang="grc">Τοϊς πέζαν αὐαλέων <gap></gap>νδω̄ν <gap></gap>ΐαψα μ<gap></gap>ίμνας, <lb></lb>Μέμψ<gap></gap> μὴ π́ων τείρεα, μὴ <gap></gap>ύχ<gap></gap>. </foreign></s></p><p type="main"> <s><foreign lang="grc">Τοῑς πνος ὀφρυόεν πλυπζάγμονος ὄμμα γιλά<gap></gap>ας, <lb></lb>Βέλτιον <gap></gap>γορέης κέρδος ἕδειζα <gap></gap>δ. </foreign></s></p><p type="main"> <s><foreign lang="grc">Εἰ δέ π τω̄ν <gap></gap>ὅ<gap></gap>ως ε<gap></gap>ζ<gap></gap>χ<gap></gap>ον εὔ<gap></gap>, <lb></lb>Π<gap></gap>ὶν θάναπς μάζψη μ̓· εὔχομ̓ <gap></gap>λέτω. </foreign></s></p><p type="main"> <s><foreign lang="grc">Λνέζος οὐ κλέψω χά<gap></gap>ν εὔφζονος ἐ<gap></gap>ομόνοι<gap></gap><lb></lb>Δ<gap></gap>γμ̓ ἀγλαὸν, <gap></gap>, <gap></gap>νομες, καὶ πατζίδ<gap></gap>. </foreign></s></p><p type="main"> <s><foreign lang="grc">Ος δέ με λαθζαι<gap></gap>ος δήζ<gap></gap>, κακόεπ<gap></gap>ος ἀκούο<gap></gap>, <lb></lb>Λυ<gap></gap>ω̄ν ἧς φθονεζη̄ς ἄζιος πυρκαι<gap></gap>η̄ς. </foreign></s></p><pb xlink:href="043/01/009.jpg" pagenum="1"></pb><figure id="id.043.01.009.1.jpg" xlink:href="043/01/009/1.jpg"></figure><p type="head"> <s>LVC AE <lb></lb>VALER II <lb></lb>DE CENTRO <lb></lb>GRAVITATIS <lb></lb>SOLIDORVM</s></p><p type="head"> <s><emph type="italics"></emph>LIBER PRIMVS.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Propoſitum eſt mihi in hiſce tribus li<lb></lb>bris, ò Geometra, cuiuſcumque figuræ <lb></lb>ſolidæ in geometria ratio haberi ſolet, <lb></lb>centrum grauitatis inuenire. </s> <s>Huius <lb></lb>autem prouinciæ mihi ſuſcipiendæ oc<lb></lb>caſio fuit liber ille iam pridem editus <lb></lb>Federici Commandini Vrbinatis, in <lb></lb>quo cum ille corporum planis termi<lb></lb>nis definitorum; necnon cylindri, & coni, & fruſti conici, <lb></lb>& ſphæræ, & ſphæroidis centrum grauitatis oſtendiſſet; <lb></lb>aliorum autem, quæ ſuperficie mixta continentur vno co<lb></lb>noide parabolico tentato ſyllogiſmi iactura operam per<lb></lb>didiſſet, ego ſpe magis, ad quam vir ille exarſerat incita<pb xlink:href="043/01/010.jpg" pagenum="2"></pb>tus, quàm deterritus lapſu, vehementerque dolens geo<lb></lb>metriæ partem tamdiu deſiderari cognitione digniſſimam; <lb></lb>cum ante exercitationis cauſa omnium, quæ propoſui ſoli<lb></lb>dorum, excepto conoide parabolico, centra grauitatis aliis <lb></lb>viis indagaſſem; poſtea non ſolum parabolici, ſed ante me <lb></lb>tentata nemini, hyperbolici conoidis, & fruſti vtriuſque, & <lb></lb>portionis vtriuſque conoidis, & portionis fruſti, & hemi<lb></lb>ſphærij, & hemiſphæroidis, & cuiuſlibet portionis ſphæ<lb></lb>ræ, & ſphæroidis vno, & duobus planis parallelis abſciſſæ <lb></lb><expan abbr="cẽtra">centra</expan> grauitatis adinueni, multa autem ex his duplici, quæ<lb></lb>dam triplici via. </s> <s>Taceo nunc alia eiuſdem generis, quæ <lb></lb>cum vtilia, tum geometriæ ſtudioſis non iniucunda, vt arbi<lb></lb>tror, futura in poſteriores libros diſtribuimus. </s> <s>Quòd autem <lb></lb>aliquot propoſitiones, alias Archimedis lemmaticas, alias <lb></lb>Commandini meis rationibus attuli demonſtratas; non tàm <lb></lb>idcirco id fcci, ne meæ lucubrationes <expan abbr="deperirẽt">deperirent</expan>, quàm quòd <lb></lb>vel ſtylo Euclidis magis conſonæ, vel ad percipiendum eo <lb></lb>minus laborioſæ, quo ad inueniendum ſunt difficiliores, <lb></lb>vel meo propoſito aptiores viderentur. </s> <s>Earum propoſitio<lb></lb>num, Archimedis duo ſunt in primo libro, decimaquarta, <lb></lb>& ſeptima, & ſecunda pars vigeſimæ; in ſecundo autem vna. <lb></lb></s> <s>Omne conoides parabolicum ſeſquialterum eſſe coni ean<lb></lb>dem baſim, & eandem altitudinem habentis. </s> <s>Comman<lb></lb>dini autem omnes in primo libro nouem; vigeſima tertia, & <lb></lb>quinta: trigeſima ſecunda, tertia, quarta, ſeptima, & nona: <lb></lb>quadrageſima prima, & ſecunda. </s> <s>Sed multa hic noua inue<lb></lb>nies ita ad præſens inſtitutum neceſſaria, vt per ſe <expan abbr="tamẽ">tamen</expan> ipſa <lb></lb>in geometria locum habere debeant, maxime verò tres pri<lb></lb>mæ ſecundi libri propoſitiones, quippe quibus magnam, ac <lb></lb>perdifficilem geometriæ partem demonſtratione recta, & <lb></lb>generali ad viam regiam redactam eſse intelliges. </s> <s>Ita Deus <lb></lb>Opt. Max. cuius auxilio hæc feci, quibus prodeſse alicui <lb></lb>vehementer cupio, reliquis meis conatibus opem ferat. </s> <s>Sed <lb></lb>ad definitiones accedamus. </s></p><pb xlink:href="043/01/011.jpg" pagenum="3"></pb><p type="head"> <s>DEFINITIONES.</s></p><p type="head"> <s>I.</s></p><p type="main"> <s>Figuræ aliquæ planæ multilateræ centrum ha<lb></lb>bere dicuntur punctum illud, in quo omnes rectæ <lb></lb>lineæ vel angulos oppoſitos iungentes bifariam <lb></lb>ſecantur, vel ab angulis ductæ ad laterum op<lb></lb>poſitorum bipartitas ſectiones in eaſdem ra<lb></lb>tiones. </s></p><p type="head"> <s>II.</s></p><p type="main"> <s>Circa diametrum eſt figura plana, in qua re<lb></lb>cta quædam, quæ diameter figuræ dicitur, omnes <lb></lb>rectas alicui parallelas, à figura terminatas bi<lb></lb>fariam diuidit. </s></p><p type="head"> <s>III.</s></p><p type="main"> <s>Octaedrum communiter dictum, eſt figura ſoli<lb></lb>da octo triangulis binis parallelis, æqualibus, & <lb></lb>ſimilibus comprehenſa. </s></p><p type="head"> <s>IIII.</s></p><p type="main"> <s>Polyedri regularis centrum dicitur punctum, <lb></lb>in quo omnes rectæ lineæ, quæ ad angulos oppo<lb></lb>ſitos pertinent bifariam diuiduntur. </s></p><pb xlink:href="043/01/012.jpg" pagenum="4"></pb><p type="head"> <s>V.</s></p><p type="main"> <s>Cuiuſlibet figuræ grauis centrum grauitatis <lb></lb>eſt punctum illud, à quo ſuſpenſum graue perſe <lb></lb>manet partibus quomodocumque circa conſti<lb></lb>tutis. </s></p><p type="head"> <s>VI.</s></p><p type="main"> <s>Axis priſmatis, & pyramidis & eius fruſti di<lb></lb>citur recta linea, quæ in pyramide à vertice ad <lb></lb>baſis centrum figuræ vel grauitatis pertinet: in <lb></lb>reliquis autem, quæ baſium oppoſitarum figuræ <lb></lb>vel grauitatis centra iungit. </s></p><p type="head"> <s>VII.</s></p><p type="main"> <s>Si qua figura ſolida planis parallelis ita ſeca<lb></lb>ri poſſit, vt quæcumque ſectiones centrum ha<lb></lb>beant, & ſint inter ſe ſimiles; aliqua autem recta <lb></lb>linea, ſiue ad centra baſium oppoſitarum prædi<lb></lb>ctis ſectionibus parallelarum, & ſimilium, vt in <lb></lb>cylindro; ſiue ad verticem, & centrum baſis ter<lb></lb>minata, vt in cono, hemiſphærio, & conoide, tran<lb></lb>ſeat per centra omnium prædictarum ſectionum; <lb></lb>ea talis figuræ axis nominetur: ipſa autem figura, <lb></lb>ſolidum circa axim. </s> <s>Quæ ſi vel vnam tantum ha<lb></lb>beat baſim, vel duas inæquales, & parallelas: dua<lb></lb>rum autem quarumlibet prædictarum ſectionum <lb></lb>vertici, vel minori baſi propinquior ſit minor re-<pb xlink:href="043/01/013.jpg" pagenum="5"></pb>motiori; ſolidum circa axem in alteram partem de <lb></lb>ficiens nominetur: quo nomine ſignificari etiam <lb></lb>volumus ea ſolida, quorum quælibet ſectiones <lb></lb>baſi parallelæ quamuis baſi non ſint omnino ſimi<lb></lb>les, tamen ijs figuris deficiunt, quæ ſunt ſimiles <lb></lb>haſi, ac totis ijs, à quibus ipſæ ablatæ intelli<lb></lb>guntur, ita vt tota figura & ablata habeant com<lb></lb>mune centrum in vna recta linea ad centrum ba<lb></lb>ſis terminata, quæ & ipſa talis ſolidi axis nomi<lb></lb>netur. </s></p><p type="main"> <s>Vt in figura, ſolidi ABDC deficientis ſolido CED <lb></lb>baſis eſt circulus AB, terminus baſi oppoſitus circum<lb></lb>ferentia circuli CMD. axis communis omnibus EF, <lb></lb>per cuius quodlibet punctum I plano baſi AB paralle<lb></lb>lo ſecante ſolidum ABDC, & ablatum CED, & re<lb></lb>ſiduum, eſt totius <lb></lb>ſectio circulus G <lb></lb>H, ablati vero cir<lb></lb>culus KL, & reſi<lb></lb>dui ſectio reliquum <lb></lb>circuli GH dem<lb></lb>pto circulo KL. <lb></lb>quarum ſectionum <lb></lb>omnium centrum <lb></lb>commune eſt I. <lb></lb></s> <s>Quod ſi ſuper duos <lb></lb><figure id="id.043.01.013.1.jpg" xlink:href="043/01/013/1.jpg"></figure><lb></lb>circulos GH, KL circa axem communem EI cylin<lb></lb>dri deſcribantur, (erunt autem eiuſdem altitudinis) erit <lb></lb>reliquum cylindri GB, dempto cylindro cuius baſis <lb></lb>KL, axis EI, conſtitutum ſuper baſim G, <emph type="italics"></emph>K<emph.end type="italics"></emph.end>, & circa <lb></lb>axim EI, quæ ſuo loco expectatur cogitatio. </s></p><pb xlink:href="043/01/014.jpg" pagenum="6"></pb><p type="head"> <s>POSTVLATA.</s></p><p type="head"> <s>I.</s></p><p type="main"> <s>Omnis figuræ grauis vnum eſſe centrum gra<lb></lb>uitatis. </s></p><p type="head"> <s>II.</s></p><p type="main"> <s>Omnium figurarum ſibi mutuo congruentium <lb></lb>centra grauitatis mutuo ſibi congruere. </s></p><p type="head"> <s>III.</s></p><p type="main"> <s>Omnis figuræ, cuius termini omnis cauitas <lb></lb>eſt interior, intra terminum eſſe centrum graui<lb></lb>tatis. </s></p><p type="head"> <s>IIII.</s></p><p type="main"> <s>Similium triangulorum ſimiliter poſita eſse <lb></lb>centra grauitatis. </s> <s>In triangulis autem ſimilibus <lb></lb>ſimiliter poſita puncta eſſe dicuntur, à quibus re<lb></lb>ctæ ad angulos æquales ductæ cum lateribus ho<lb></lb>mologis angulos æquales faciunt. </s></p><p type="head"> <s>V.</s></p><p type="main"> <s>Æqualia grauia ab æqualibus longitudinibus <lb></lb>ſecundum centrum grauitatis ſuſpenſa æquipon<lb></lb>derare. </s></p><p type="head"> <s>VI.</s></p><p type="main"> <s>A quibus longitudinibus duo grauia æquipon<lb></lb>derant, ab ijſdem alia duo quælibet illis æqualia <lb></lb>æquiponderare. </s></p><pb xlink:href="043/01/015.jpg" pagenum="7"></pb><p type="head"> <s>PROPOSITIO <lb></lb>PRIMA.</s></p><p type="main"> <s>Si ſint quotcumque magnitu<lb></lb>dines inæquales deinceps <lb></lb>proportionales; exceſſus, qui <lb></lb>bus differunt deinceps pro<lb></lb>portionales erunt, in propor<lb></lb>tione totarum magnitudi<lb></lb>num. </s></p><p type="main"> <s>Sint quotcumque inæquales magnitudines deinceps <lb></lb>proportionales AB, CD, EF, & G, <lb></lb>differentes exceſſibus BH, DK, FL, mi<lb></lb>nima autem ſit G. </s> <s>Dico BH, DK, FL, <lb></lb>deinceps proportionales eſse in proportio<lb></lb>ne, quæ eſt AB, ad CD, ſeu CD, ad <lb></lb>EF. </s> <s>Quoniam enim eſt vt AB, ad <lb></lb>CD, ita CD ad EF; hoc eſt vt AB, ad <lb></lb>AH, ita CD, ad CK, permutando <lb></lb>erit, vt AB, ad CD, ita AH, ad CK: <lb></lb>vt igitur tota AB, ad totam CD, ita <lb></lb>reliqua BH, ad reliquam DK. </s> <s>Simili<lb></lb>ter oſtenderemus eſse vt CD ad EF, <lb></lb>ita DK ad FL; vt igitur BH ad DK, <lb></lb>ita erit DK ad FL, in proportione, quæ <lb></lb>eſt AB ad CD, & CD ad EF. </s> <s>Quod demonſtran<lb></lb>dum erat. </s></p><figure id="id.043.01.015.1.jpg" xlink:href="043/01/015/1.jpg"></figure><pb xlink:href="043/01/016.jpg" pagenum="8"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO II.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>In omni triangulo vnum dumtaxat punctum <lb></lb>eſt, in quo rectæ ab angulis ad latera incidentes <lb></lb>ſecant ſeſe in eaſdem rationes. </s> <s>& ſegmenta, quæ <lb></lb>ad angulos, ſunt reliquorum dupla. </s> <s>& prædictæ <lb></lb>incidentes ſecant trianguli latera bifariam. </s></p><p type="main"> <s>Sit triangulum ABC, cuius duo quælibet latera AB, <lb></lb>AC, ſint bifariam ſecta in punctis D, E, & ductæ rectæ <lb></lb>lineæ BE, CFD, AFG. </s> <s>Dico CF duplam eſſe ipſius <lb></lb>FD, & AF, ipſius FG, & BF, ipſius FE. </s> <s>Et in nullo alio <lb></lb>puncto à puncto F tres rectas ab angulis ad latera inciden<lb></lb>tes ſecare ſe ſe in eaſdem rationes. </s> <s>Et reliquum latus BC <lb></lb>ſectum eſſe bifariam in puncto G. </s> <s>Quoniam enim eſt vt BA <lb></lb>ad AD, ita CA ad AE: hoc eſt, vt triangulum ABC ad <lb></lb>triangulum ADC, ita triangulum idem ABC ad trian<lb></lb>gulum AEB; æqualia <lb></lb>erunt triangula ADC, <lb></lb>AEB, & ablato trape<lb></lb>zio DE communi re<lb></lb>liquum triangulum BD <lb></lb>F reliquo triangulo C <lb></lb>EF æquale erit: ſed <lb></lb>triangulum ADF eſt <lb></lb>æquale triangulo BDF; <lb></lb>& triangulum AFE <lb></lb>triangulo EFC, pro<lb></lb>pter æquales baſes, & <lb></lb><figure id="id.043.01.016.1.jpg" xlink:href="043/01/016/1.jpg"></figure><lb></lb>communes altitudines; totum igitur triangulum AFB <lb></lb>toti AFC, triangulo æquale erit: ſed vt triangulum AFB <pb xlink:href="043/01/017.jpg" pagenum="9"></pb>ad triangulum FBG, hoc eſt vt AF ad FG, ita eſt <lb></lb>triangulum AFC ad triangulum FCG; triangulum er<lb></lb>go FBG triangulo FCG æquale erit, & baſis BG ba<lb></lb>ſi GC æqualis. </s> <s>Quoniam igitur & AE eſt æqualis <lb></lb>EC, ſimiliter vt ante, oſtenderemus, triangulum BCF, <lb></lb>triangulo ACF, eademque ratione triangulum ABF, <lb></lb>triangulo BCF æquale eſſe: igitur vnumquodque trian<lb></lb>gulorum ABF, ACF, BCF, tertia pars eſt trianguli <lb></lb>ABC: ſed vt triangulum ABC, ad triangulum BCF, <lb></lb>ita eſt AG, ad GF; tripla igitur eſt AG ipſius GF, <lb></lb>ac proinde AF, ipſius FG dupla. </s> <s>Eadem ratione <lb></lb>BE, ipſius FE, & CF, ipſius FD, dupla concludetur. </s></p><p type="main"> <s>Sed ſint ſi fieri poteſt, trianguli ABC duo centra qua<lb></lb>lia diximus D, E: & ab ipſis ad ſingulos angulos du<lb></lb>cantur binæ rectæ lineæ: <lb></lb>& eadat D in aliquo trian <lb></lb>gulo BEC. </s> <s>Quoniam <lb></lb>igitur D eſt centrum trian <lb></lb>guli ABC erit triangu<lb></lb>lum BDC tertia pars <lb></lb>trianguli ABC. </s> <s>Eadem <lb></lb>ratione triangulum BEC <lb></lb>tertia pars erit trianguli <lb></lb>ABC; triangulum ergo <lb></lb>DBC æquale erit trian<lb></lb>gulo BEC pars toti, quod <lb></lb>fieri non poteſt, atqui <expan abbr="idẽ">idem</expan> <lb></lb><figure id="id.043.01.017.1.jpg" xlink:href="043/01/017/1.jpg"></figure><lb></lb>abſurdum ſequitur, ſi punctum D cadat in aliquo latere <lb></lb>triangulorum, quorum vertex E; Manifeſtum eſt igitur <lb></lb>propoſitum. </s></p><pb xlink:href="043/01/018.jpg" pagenum="10"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO III.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>In ſimilibus triangulis rectæ lineæ, quæ inter <lb></lb>centra, & alia in ijs ſimiliter poſita puncta in<lb></lb>terijciuntur, proportionales ſunt in proportione <lb></lb>laterum homologorum. </s></p><p type="main"> <s>Sint triangula ſimilia, & ſimiliter poſita ABC, DEF, <lb></lb>quorum ſint centra O, P, in ijs autem triangulis ſint pun<lb></lb>cta ſimiliter poſita K, L, quæ cadant primum in rectis <lb></lb>BG, EH, quæ ab angulis æqualibus B, E, baſes bifa<lb></lb>riam diuidunt. </s> <s>Dico eſſe OK ad PL, vt eſt latus AB, <lb></lb>ad latus DE. iunctis enim AK, KC, DL, LF, quo<lb></lb><figure id="id.043.01.018.1.jpg" xlink:href="043/01/018/1.jpg"></figure><lb></lb>niam angulus KAC, æqualis eſt angulo LDF, & angu<lb></lb>lus KCA, angulo LFD, ob ſimiliter poſita puncta K, <lb></lb>L, triangulum AKC, triangulo LDF ſimile erit, & vt <lb></lb>KA ad AC, ita LD ad DF: ſed vt CA ad AG, ita <lb></lb>eſt FD ad DH, expræcedenti; vt igitur KA, ad AG <lb></lb>ita erit LD, ad DH, circa æquales angulos: ſimilia igi<lb></lb>tur ſunt triangula AGK, DHL, & angulus AGK, <pb xlink:href="043/01/019.jpg" pagenum="11"></pb>æqualis angulo DHL, & vt KG, ad GA, ita LH, ad <lb></lb>HD: ſed vt GA, ad AC, ita eſt HD ad DF: & vt <lb></lb>AC ad AB, ita DF ad DE, ex æquali igitur erit vt <lb></lb>KG ad AB, ita LH ad DE: ſed vt AB ad BG, ita <lb></lb>eſt DE ad EH, propter ſimilitudinem triangulorum <lb></lb>ABG, DEH: & vt BG ad GO ita eſt EH ad HP, <lb></lb>propter triangulorum centra O, P; ex æquali igitur erit <lb></lb>vt KG ad GO, ita LH ad HP: & permutando vt <lb></lb>OG ad PH, ideſt vt BG ad EH, ideſt vt AB ad ED, <lb></lb>ita KG ad LH, & reliqua OK ad reliquam PL. </s></p><p type="main"> <s>Sed ſint puncta ſimiliter poſita M, N, quæ cadant ex<lb></lb>tra lineas BG, EH, iunctæque OM, PN. </s> <s>Dico iti<lb></lb>dem eſse vt AB ad ED, ita OM ad PN. </s> <s>Iungantur <lb></lb>enim rectæ MB, NE, quæ cum quibus lateribus homo<lb></lb>logis angulos æquales faciunt, ea ſint AB, DE, quod <lb></lb>propter iſoſcelia triangula ſit dictum in ſimiliter poſitis <lb></lb>triangulis. </s> <s>igitur etiam angulus BAM, æqualis erit an<lb></lb>gulo EDN; ſimilia igitur triangula ABM, DEN: & <lb></lb>vt MB ad BA, ita erit NE ad ED: ſed vt AB ad <lb></lb>BG, ita eſt DE ad EH, propter ſimilitudinem trian<lb></lb>gulorum, & vt BG ad BO, ita eſt EH ad EP, ob <lb></lb>triangulorum ſimilium centra O, P: ex æquali igitur <lb></lb>erit vt MB, ad BO, ita NE ad EP. </s> <s>Rurſus quo<lb></lb>niam angulus ABM, æqualis eſt angulo DEN, quorum <lb></lb>angulus ABG, æqualis eſt angulo DEH: erit reliquus <lb></lb>angulus OBM, æqualis reliquo angulo PEN: ſed vt MB <lb></lb>ad BO, ita erat NE ad EP; triangulum igitur OBM <lb></lb>triangulo PEN, ſimile erit, & vt BO ad EP, hoc eſt <lb></lb>BG ad EH, hoc eſt AB ad DE, ita OM ad PN. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/020.jpg" pagenum="12"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Datis duobus triangulis ſcalenis ſimilibus, & <lb></lb>dato puncto in altero eorum, vnum duntaxat pun<lb></lb>ctum in reliquo triangulo prædicto puncto ſimi<lb></lb>liter poſitum poteſt inueniri. </s></p><p type="main"> <s>Sint data duo triangula ſcalena ſimilia ABC, DEF, <lb></lb>& in triangulio ABC datum punctum G: ſint autem <lb></lb>hæc triangula ſimiliter poſita. </s> <s>Dico in triangulo DEF, <lb></lb>vnum duntaxat punctum puncto G ſimiliter poſitum in<lb></lb>ueniri poſse. </s> <s>Iunctis enim AG, BG, GC, ponatur <lb></lb>angulus EDH, æqualis angulo BAG, & angulus DEH, <lb></lb><figure id="id.043.01.020.1.jpg" xlink:href="043/01/020/1.jpg"></figure><lb></lb>æqualis angulo ABG, & HF iungatur. </s> <s>Manifeſtum <lb></lb>eſt igitur ex præcedentis Theorematis demonſtratione, <lb></lb>triangula EDH, HDF, FEH, ſimilia eſse triangulis <lb></lb>BAG, GAC, CBG, prout inter ſe reſpondent poſi<lb></lb>tione, quorum ſex triangulorum binis quibuſque binæ ba<lb></lb>ſes homologæ reſpondent: AB ED, AC DF, BC <pb xlink:href="043/01/021.jpg" pagenum="13"></pb>EF. quæ ſuntin latera homologa duorum triangulorum <lb></lb>ABC, DEF. </s> <s>Ex definitione igitur, duo puncta G, H, <lb></lb>in triangulis ABC, DEF, ſimiliter poſita erunt. </s> <s>At <lb></lb>enim ſi fieri poteſt ſit aliud punctum K, in triangulo <lb></lb>DEF, ſimiliter poſitum puncto G. </s> <s>Vel igitur punctum <lb></lb>K in aliquo triangulorum, quorum eſt communis vertex <lb></lb>H, vel in aliquo eorundem latere cadet. </s> <s>cadat in latere <lb></lb>FH, & iungatur DK: triangulum ergo DFK, ſimile <lb></lb>erit triangulo ACG. </s> <s>Sed & triangulum EDF, ſimile <lb></lb>eſt triangulo BAC; vtraque igitur horum ad illorum ſi<lb></lb>bi reſpondens triangulorum duplicatam eorundem late<lb></lb>rum homologorum AC, DF, habebunt proportionem: <lb></lb>vt igitur eſt triangulum EDF, ad triangulum BAC, ita <lb></lb>erit triangulum DFK, ad triangulum ACG: & per<lb></lb>mutando, vt triangulum ACG, ad triangulum ABC, <lb></lb>ita triangulum DFK, ad triangulum EDF: eadem ra<lb></lb>tione, vt triangulum ACG, ad triangulum ABC, ita <lb></lb>erit triangulum DFH, ad triangulum DEF: vt igitur <lb></lb>triangulum DFK, ad triangulum EDF; ita erit trian<lb></lb>gulum DFH, ad triangulum EDF; triangulum ergo <lb></lb>DFK, triangulo DFH, æquale erit, pars toti, quod eſt <lb></lb>abſurdum: idem autem abſurdum ſequeretur, ſi punctum <lb></lb><emph type="italics"></emph>K<emph.end type="italics"></emph.end>, poneretur in aliquo prædictorum triangulorum, vt in <lb></lb>triangulo DFH; Non igitur aliud punctum à puncto H, <lb></lb>in triangulo EDF, ſimiliter poſitum erit puncto G. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO V.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Cuilibet figuræ planæ rectangulum æquale <lb></lb>poteſt eſſe. </s></p><pb xlink:href="043/01/022.jpg" pagenum="14"></pb><p type="main"> <s>Sit quælibet figura plana A. </s> <s>Dico figuræ A, rectan<lb></lb>gulum æquale poſse exiſtere. </s> <s>Exponatur enim rectan<lb></lb>gulum BC, cuius latus BD, in infinitum producatur <lb></lb>verſus E. </s> <s>Quoniam igitur eſt vt rectangulum BD, ad <lb></lb>planam figuram A, ita recta BD, ad aliquam lineam <lb></lb>rectam ſit vt BC, ad A, ita BD, ad DE, & comple<lb></lb>atur rectan<lb></lb>gulum EC. <lb></lb></s> <s>Quoniam igi <lb></lb>tur eſt vt BD <lb></lb>ad DE, ita <lb></lb>rectangulum <lb></lb>BC, ad figu<lb></lb>ram A: ſed <lb></lb>vt BD, ad <lb></lb>DE, ita eſt <lb></lb><figure id="id.043.01.022.1.jpg" xlink:href="043/01/022/1.jpg"></figure><lb></lb>rectangulum BC, ad rectangulum CE; vt igitur re<lb></lb>ctangulum BC, ad figuram A, ita eſt rectangulum <lb></lb>BC, ad rectangulum CE; rectangulum ergo CE, fi<lb></lb>guræ A, æquale erit. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omni figuræ circa diametrum in alte ram par<lb></lb>tem deficienti figura quædam ex parallelogram<lb></lb>mis æqualium altitudinum inſcribi poteſt, & al<lb></lb>tera circumſcribi, ita vt circumſcripta ſuperet in<lb></lb>ſcriptam minori ſpacio quantacumque magnitu<lb></lb>dine propoſita. </s> <s>Semper autem in ſimilibus intelli<lb></lb>ge, eiuſdem generis. </s></p><p type="main"> <s>Sit figura plana ABC circa diametrum AD, ad par-<pb xlink:href="043/01/023.jpg" pagenum="15"></pb>tes A deficiens, cuius baſis BC. </s> <s>Dico fieri poſse quod <lb></lb>proponitur: ducta enim per verticem figuræ A, baſi BC, <lb></lb>parallela, atque ideo figuram ipſam contingente, abſol<lb></lb>uatur parallelogrammum BL, ſectaque diametro AD, <lb></lb>bifariam, & ſingulis eius partibus ſemper bifariam, du<lb></lb>cantur per puncta ſectionum rectæ lineæ baſi BC, & in<lb></lb>ter ſe parallelæ, atque ita multiplicatæ ſint ſectiones, <lb></lb>vt ſecti parallelogrammi in parallelogramma æqua<lb></lb>lia, & eiuſdem altitudinis quælibet pars, vt paralle<lb></lb>logrammum BF, ſit minus ſuperficie propoſita, cu<lb></lb>ius parallelogram<lb></lb>mi latus EF, ſe<lb></lb>cet figuræ termi<lb></lb>num BAC, in <lb></lb>punctis GH, & <lb></lb>diametrum AD, in <lb></lb>puncto K. erit igi<lb></lb>tur GK, æqualis <lb></lb>KH: per omnia <lb></lb>igitur puncta ſe<lb></lb>ctionum termini <lb></lb><figure id="id.043.01.023.1.jpg" xlink:href="043/01/023/1.jpg"></figure><lb></lb>BAC, quæ à prædictis fiunt lineis parallelis, ſi ducan<lb></lb>tur diametro AD parallelæ, figura quædam ipſi ABC, <lb></lb>inſcribetur, & altera circumſcribetur ex parallelogram<lb></lb>mis æqualium altitudinum. </s> <s>Dico harum figurarum <lb></lb>inſcriptam ſuperari à circumſcripta minori ſpacio ſuper<lb></lb>ficie propoſita. </s> <s>Quoniam enim omnia parallelogramma, <lb></lb>quibus figura circumſcripta ſuperat inſcriptam ſimul ſum<lb></lb>pta ſunt æqualia BF parallelogrammo: ſed parallelo<lb></lb>grammum BF, eſt minus ſuperficie propoſita: exceſſus <lb></lb>igitur quo figura circumſcripta inſcriptam ſuperat, minor <lb></lb>erit ſuperficie propoſita. </s> <s>Fieri igitur poteſt, quod propo<lb></lb>nebatur. </s></p><pb xlink:href="043/01/024.jpg" pagenum="16"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Pyramides ſimilibus, & æqualibus triangulis <lb></lb>comprehenſæ inter ſe ſunt æquales. </s></p><p type="main"> <s>Sint pyramides ABCD, EFGH, ſimilibus, & æqua<lb></lb>libus triangulis comprehenſæ, & ſi ſint ſimiliter poſitæ, qua<lb></lb>rum vertices A, E, baſes autem triangula BCD, FGH. <lb></lb></s> <s>Dico pyramidem ABCD, pyramidi EFGH, æqualem <lb></lb>eſse. </s> <s>A punctis enim A, E, manantia latera inferius pro<lb></lb>ducantur, & prædictis lateribus maiores, inter ſe autem <lb></lb>æquales abſcindantur AK, AL, AM, EN, EO, EP, <lb></lb><figure id="id.043.01.024.1.jpg" xlink:href="043/01/024/1.jpg"></figure><lb></lb>& conſtruantur pyramides AKLM, ENOP: pyramides <lb></lb>igitur hæ æqualibus, & ſimilibus triangulis comprehenden <lb></lb>tur, vt colligitur ex ipſa conſtructione; triangulis igitur inter <lb></lb>ſe æquilateris, & æquiangulis KLM, NOP, inter ſe con<lb></lb>gruentibus non congruat, ſi fieri poteſt, pyramis ENOP, <lb></lb>pyramidi AKLM, ſed cadat vertex E, pyramidis ENOP, <lb></lb>extra verticem A, pyramidis AKLM, & ex puncto A, <pb xlink:href="043/01/025.jpg" pagenum="17"></pb>ad centrum circuli tranſeuntis per tria puncta K, L, M, quod <lb></lb>ſit R, ducatur recta AR, & ER iungatur. </s> <s>Quoniam igi<lb></lb>tur æquales rectæ ſunt AK, AL, AM, quæ ex puncto <lb></lb>A, in ſublimi pertinent ad ſubiectum planum: & punctum <lb></lb>R, eſt centrum circuli tranſeuntis per puncta N, O, P; cadet <lb></lb>recta AR ad ſubiectum planum perpendicularis. </s> <s>Eadem <lb></lb>ratione recta ER ducta à vertice E, pyramidis ENOP, <lb></lb>ad centrum R, circuli tranſeuntis per puncta N, O, P, hoc <lb></lb>eſt, per puncta K, L, M, illis congruentia, cadet ad idem <lb></lb>planum, ad quod linea AR, perpendicularis; itaque ab <lb></lb>eodem puncto R, ad idem planum, & ad eaſdem partes duæ <lb></lb>perpendiculares erunt excitatæ, quod fieri non poteſt: <lb></lb>punctum igitur E non cadet extra punctum A: quare la<lb></lb>tus EN, congruet lateri AK, quorum EF, eſt æqualis <lb></lb>AK; igitur & EF, ipſi AB, congruet. </s> <s>eadem ratione la<lb></lb>tus AG, congruet lateri AC, & latus EH, lateri AD, & <lb></lb>triangula triangulis, & pyramis EFGH, pyramidi ABC <lb></lb>D, & ipſi æqualis erit. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hinc facile colligitur omnia ſolida, quæ in py <lb></lb>ramides æqualibus, & ſimilibus triangulis com<lb></lb>prehenſas multitudine æquales diuidi poſſunt, eſ <lb></lb>ſe inter ſe æqualia. </s> <s>Quocirca omnia priſmata, & <lb></lb>pyramides, & octahedra, omnia denique corpora <lb></lb>regularia æqualibus, & ſimilibus planis compre<lb></lb>henſa inter ſe æqualia erunt. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis pyramidis triangulam baſim habentis <lb></lb>quatuor axes ſecant ſe in vno puncto in eaſdem ra<pb xlink:href="043/01/026.jpg" pagenum="18"></pb>tiones, ita vt ſegmenta, quæ ad angulos, eo<lb></lb>rum, quæ ad oppoſita triangula, ſint tripla; ex quo <lb></lb>puncto tota pyramis diuiditur in quatuor pyrami <lb></lb>des æquales. </s> <s>Et in nullo alio puncto quatuor re<lb></lb>ctæ lineæ ductæ ab angulis ad triangula oppoſita <lb></lb>pyramidis ſecant ſeſe in eaſdem rationes. </s> <s>Vocetur <lb></lb>autem punctum hoc centrum dictæ pyramidis. </s></p><p type="main"> <s>Sit pyramis ABCD, cuius vertex A, baſis autem <lb></lb>triangulum BCD, axes AE, BM, CL, DN, vnde qua<lb></lb>tuor triangulorum, quæ ſunt circa pyramidem ABCD, <lb></lb>centra erunt grauitatis E, L, M, N. </s> <s>Dico quatuor li<lb></lb>neas AE, BM, CL, DN, ſecare ſe ſe in vno puncto in <lb></lb>eaſdem rationes, quas prædixi, & quæ ſequuntur. </s> <s>Nam ex <lb></lb>puncto A, ducatur recta ALH, quæ ob trianguli ABD, <lb></lb>centrum L, ſecabit latus BD, bifariam in puncto H; iun<lb></lb>cta igitur CE, & producta conueniet cum ALH, vt in <lb></lb>puncto H. eadem ratione iunctæ AM, BE, & productæ <lb></lb>conuenient in medio lateris CD, conueniant in puncto K, <lb></lb>necnon AN, DE, in medio ipſius BC, vt in puncto G. <lb></lb></s> <s>Quoniam igitur ob triangulorum centra, eſt vt CE ad EH, <lb></lb>ita AL ad LH, dupla enim eſt vtraque vtriuſque, ſeca<lb></lb>bunt ſeſe rectæ AE, CL, inter eaſdem parallelas; quare <lb></lb>vt AF ad FE, ita erit CF ad FL, circum æquales angu <lb></lb>los ad verticem: triangula igitur AFL, CFE; & reci<lb></lb>proca, & æqualia inter ſe erunt. </s> <s>Cum igitur ſit vt AL ad <lb></lb>LH, ita CE ad EH, hoc eſt vt triangulum AFL ad <lb></lb>triangulum FLH, (ſi ducatur FH) ita triangulum CFE, <lb></lb>ad triangulum FEH, erunt inter ſe æqualia triangula <lb></lb>FEH, FLH. </s> <s>Quare vt triangulum AFH, ad triangu<lb></lb>lum FLH, hoc eſt vt AH ad HL, ita erit triangulum <lb></lb>AFH ad triangulum FEH, hoc eſt AF ad FE: ſed re<lb></lb>cta AH, eſt tripla ipſius LH; igitur & AF, erit ipſius FE, <pb xlink:href="043/01/027.jpg" pagenum="19"></pb>tripla: ſed vt AF, ad FE, ita eſt CF, ad FL; tripla igi<lb></lb>tur erit CF, ipſius FL. </s> <s>Similiter oſtenderemus rectas <lb></lb>AE, BM, ſecare ſe ſe in eaſdem rationes, ita vt ſegmen<lb></lb>ta, quæ ad angulos, ſint tripla eorum, quæ ſunt ad centra <lb></lb>E, M, quorum AF, eſt tripla ipſius FE: in puncto igitur <lb></lb>F, ſecant ſe rectæ lineæ AE, BM. </s> <s>Eadem ratione & re <lb></lb>ctæ AE, DN, ſecent ſe in puncto F, neceſse erit: quare <lb></lb>vt AF ad FE, ita erit DF ad FN. </s> <s>Quatuor igitur <lb></lb>axes pyramidis ABCD, ſecantſe ſe in puncto F, in eaſ<lb></lb>dem rationes, ita vt <lb></lb>ſegmenta ad angulos, <lb></lb>ſint <expan abbr="reliquorũ">reliquorum</expan> tripla. <lb></lb></s> <s>Rurſus, quia compo<lb></lb>nendo, & conuerten<lb></lb>do, eſt vt FE ad EA, <lb></lb>ita FL ad LC: hoc <lb></lb>eſt, vt pyramis BCD <lb></lb>F, ad pyramidem A <lb></lb>BCD, ita pyramis <lb></lb>ABDF, ad pyrami<lb></lb>dem CBDA, (pro<lb></lb>pter baſium commu<lb></lb>nitatem, & vertices in <lb></lb>eadem recta linea) erit <lb></lb><figure id="id.043.01.027.1.jpg" xlink:href="043/01/027/1.jpg"></figure><lb></lb>pyramis ABDF, æqualis pyramidi BCDF. </s> <s>Eadem ra<lb></lb>tione tam pyramis ACDF, quàm pyramis ABCF, æqua <lb></lb>lis eſt pyramidi BCDF. </s> <s>Quatuor igitur pyramides, qua<lb></lb>rum communis vertex punctum F, baſes autem triangula, <lb></lb>quæ ſunt circa pyramidem ABCD, inter ſe æquales <expan abbr="erũt">erunt</expan>, <lb></lb>& vnaquæque pyramidis ABCD, pars quarta. </s> <s>Dico in <lb></lb>nullo alio puncto à puncto F, quatuor rectas, quæ ab an<lb></lb>gulis ad triangula oppoſita pyramidis ABCD, ducantur, <lb></lb>ſecare ſe in eaſdem rationes. </s> <s>Si enim fieri poteſt ſecent <lb></lb>ſe tales rectæ in eaſdem rationes in alio puncto S. </s> <s>Simi<pb xlink:href="043/01/028.jpg" pagenum="20"></pb>liter igitur vt ante oſtenderemus, vnamquamque qua<lb></lb>tuor pyramidum, quarum communis vertex S, baſes au<lb></lb>tem triangula, quæ ſunt circa pyramidem ABCD, eſse <lb></lb>quartam partem pyramidis ABCD. </s> <s>Siue igitur pun<lb></lb>ctum S, cadat intra vnam priorum quatuor pyrami<lb></lb>dum, ſiue in earum aliquo latere, ſeu triangulo; neceſ<lb></lb>ſario erit pars æquali toti; tam enim tota vna pyramis <lb></lb>quatuor priorum, quarum communis vertex F, quàm eius <lb></lb>pars, vna quatuor pyramidum poſteriorum, quarum com<lb></lb>munis vertex S, erit eiuſdem ABCD, pyramidis pars <lb></lb>quarta. </s> <s>Ex abſurdo igitur non in alio puncto à puncto F <lb></lb>ſecabunt ſe in eaſdem rationes quatuor rectæ, quæ ab angu <lb></lb>lis ad oppoſita triangula pyramidis ABCD, ducantur. <lb></lb></s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis pyramis baſim habens triangulam di<lb></lb>uiditur in quatuor pyra mides æquales, & ſimiles <lb></lb>inter ſe, & toti, & vnum octaedrum totius pyrami<lb></lb>dis dimidium, ip ſi que concentricum. </s></p><p type="main"> <s>Sit pyramis ABCD, cuius baſis triangulum ABC, <lb></lb>ſectisque omnibus lateribus bifariam, iungantur rectæ FG, <lb></lb>GH, HF, FK, KL, LM, M<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, KH, HM, GL, LF. <lb></lb></s> <s>Dico quatuor pyramides DKLM, LFBG, KHFA, <lb></lb>MHGC, æquales eſse, & ſimiles inter ſe, & toti pyrami<lb></lb>di ABCD: octaedrum autem eſse LFGM<emph type="italics"></emph>K<emph.end type="italics"></emph.end>H, & di<lb></lb>midium pyramidis ABCD, ipſique concentricum. </s> <s>Du<lb></lb>cantur enim rectæ DNH, BQH, LN: & poſita BE, du <lb></lb>pla ipſius BH, iungatur DOC, in triangulo DBH, & <lb></lb>ponatur DP, ipſius PE, tripla, & connectantur rectæ LP, <lb></lb>PH. </s> <s>Quoniam igitur E, eſt centrum trianguli ABC, <pb xlink:href="043/01/029.jpg" pagenum="21"></pb>erit axis DE, pyramidis ABCD, cuius axis ſegmentum <lb></lb>DP eſt triplum ipſius PE: igitur P centrum erit pyra<lb></lb>midis ABCD. </s> <s>Et quoniam tres rectæ FK, KH, HF, <lb></lb>ſunt parallelæ tribus BD, DC, CB, pro vt inter ſe reſpon<lb></lb>dent, vt KH, ipſi LG, quoniam vtraque lateri DC, ob <lb></lb>latera triangulorum ſecta proportionaliter in punctis K, H, <lb></lb>L, G: & ſic de reliquis; erit pyramis A<emph type="italics"></emph>K<emph.end type="italics"></emph.end>FH, ſimilis toti <lb></lb>pyramidi ABCD. </s> <s>Similiter vnaquæque trium aliarum <lb></lb>pyramidum abſciſſarum, videlicet FLBG, GHMC, <lb></lb>KDLM, ſimilis erit pyramidi ABCD, atque ideo in<lb></lb>ter ſe ſimiles. </s> <s>Rurſus, <lb></lb>quoniam pyramidum <lb></lb>ſimilium latus AD eſt <lb></lb>duplum lateris AK, ho <lb></lb>mologi; pyramis AB<lb></lb>CD, octupla erit py<lb></lb>ramidis AKFH, ob <lb></lb>triplicatam laterum ho <lb></lb>mologorum proportio <lb></lb>nem. </s> <s>Similiter <expan abbr="vna-qũæque">vna<lb></lb>qunæque</expan> trium reliqua<lb></lb>rum pyramidum abſciſ <lb></lb>ſarum erit octaua pars <lb></lb>pyramidis ABCD; <lb></lb><figure id="id.043.01.029.1.jpg" xlink:href="043/01/029/1.jpg"></figure><lb></lb>quatuor igitur pyramides abſciſſæ ſimul ſumptæ dimi<lb></lb>dium erit pyramidis ABCD: & reliquum igitur ſoli<lb></lb>dum demptis quatuor pyramidibus, dimidium pyramidis <lb></lb>ABCD. </s> <s>Dico reliquum ſolidum LKMGFH, eſſe <lb></lb>octaedrum. </s> <s>Nam octo triangulis ipſum contineri mani<lb></lb>feſtum eſt. </s> <s>bina autem oppoſita eſſe parallela, & æqualia, <lb></lb>& ſimilia, ſic oſtendimus. </s> <s>Quoniam enim triangulum <lb></lb>FGH, eſt in plano trianguli ABC, plano trianguli KLM <lb></lb>parallelo; erit triangulum FGH, parallelum triangu-<pb xlink:href="043/01/030.jpg" pagenum="22"></pb>lo KLM: ſed triangulum FGH, eſt ſimile triangulo <lb></lb>ABC, & triangulum KLM, ſimile eidem triangulo <lb></lb>ABC; <expan abbr="triangulũ">triangulum</expan> ergo FGH, ſimile erit triangulo KLM: <lb></lb>ſed & æquale propter æqualitatem laterum homologo<lb></lb>rum. </s> <s>Similiter oſtenderemus reliquum ſolidum LKM <lb></lb>GFH continentia triangula bina oppoſita æqualia <lb></lb>inter ſe, & ſimilia, & parallela; octaedrum eſt igitur <lb></lb>LKMGFH. </s> <s>Dico iam punctum P, quod eſt cen<lb></lb>trum pyramidis ABCD, eſse centrum octaedri L<emph type="italics"></emph>K<emph.end type="italics"></emph.end><lb></lb>MGFH. </s> <s>Quoniam enim DP, ponitur tripla ipſius PE, <lb></lb>& DO, eſt æqualis <lb></lb>OE (ſiquidem planum <lb></lb>trianguli KLM, plano <lb></lb><expan abbr="triãguli">trianguli</expan> ABC, paralle <lb></lb>lum ſecat proportione <lb></lb><expan abbr="oẽs">oens</expan> rectas lineas, quæ <lb></lb>ex puncto D, in ſubli<lb></lb>mi pertinent ad ſubie<lb></lb>ctum planum trianguli <lb></lb>ABC) erit OP, ipſi <lb></lb>PE, æqualis. </s> <s>Et quo<lb></lb>niam BH eſt dupla <lb></lb>ipſius QH, quarum <lb></lb>BE eſt dupla ipſius <lb></lb><figure id="id.043.01.030.1.jpg" xlink:href="043/01/030/1.jpg"></figure><lb></lb>EH, ſiquidem E eſt centrum trianguli ABC; erit reli<lb></lb>qua EH reliquæ EQ dupla: & quia eſt vt LD ad DB, <lb></lb>ita LN ad BH, propter ſimilitudinem triangulorum, & <lb></lb>eſt LD, dimidia ipſius BD, erit & LN, dimidia ipſius <lb></lb>BH: ſed QH eſt dimidia ipſius BH; æqualis igitur LN <lb></lb>ipſi QH. </s> <s>Iam igitur quia eſt vt BE ad EH, ita <lb></lb>LO ad ON: ſed BE, eſt dupla ipſius EH; dupla igi<lb></lb>tur LO, erit ipſius ON: ſed & QH erat dupla ipſius <lb></lb>QE; vt igitur LN ad NO, ita erit HQ ad QE: & <pb xlink:href="043/01/031.jpg" pagenum="23"></pb>per conuerſionem rationis, vt NL ad LO, ita QH, ad <lb></lb>HE: & permutando, vt LN ad QH, ita LO ad EH: <lb></lb>ſed LN, oſtenſa eſt æqualis QH; æqualis igitur LO, <lb></lb>erit ipſi EH; ſed & OP, eſt æqualis ipſi PE, vt oſten<lb></lb>dimus: duæ igitur LO, OP, duabus HE, EP æqua<lb></lb>les erunt altera alteri, & angulos æquales continent LOP, <lb></lb>PEH, parallelis exiſtentibus LN, BH ſectionibus tri<lb></lb>anguli DBH, quæ fiunt à duobus planis parallelis; ba<lb></lb>ſis igitur LP, trianguli LOP, æqualis eſt baſi PH, <lb></lb>trianguli PEH, & angulus OPL, angulo EPH in pla<lb></lb>no trianguli DBH, in quo DPE, eſt vna recta linea; <lb></lb>igitur LPH, erit vna recta linea, quæ cum ſit axis octa<lb></lb>edri LKMGFH, & ſectus ſit in puncto P, bifariam, <lb></lb>erit punctum P, centrum octaedri LKMGEH. ſed & <lb></lb>centrum pyramidis ABCD. </s> <s>Manifeſtum eſt igitur pro<lb></lb>poſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO X.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omne fruſtum pyramidis triangulam baſim <lb></lb>habentis, ſiue coni, ad pyramidem, vel conum, cu<lb></lb>ius baſis eſt eadem, quæ maior baſis fruſti, & ea<lb></lb>dem altitudo, eam habet proportionem, quam duo <lb></lb>latera homologa, vel duæ diametri baſium ipſius <lb></lb>fruſti, vnà cum tertia minori proportionali ad <lb></lb>prædicta duo latera, vel diametros; ad maioris ba<lb></lb>ſis latus, vel diametrum. </s> <s>Ad priſma autem, vel <lb></lb>cylindrum, cuius eadem eſt baſis, quæ maior baſis <lb></lb>fruſti, & eadem altitudo; vt tres prædictæ deìn<lb></lb>ceps proportionales ſimul, ad triplam lateris, vel <lb></lb>diametri maioris baſis. </s></p><pb xlink:href="043/01/032.jpg" pagenum="24"></pb><p type="main"> <s>Sit fruſtum ABCFGH, pyramidis, vel coni ABCD, <lb></lb>cuius baſis triangulum, vel circulus ABC, axis autem <lb></lb>DE: & vt eſt AC ad FH, ita ſit FH ad N, & fru<lb></lb>ſti axis EK, nec non idem pyramidis, vel coni AB <lb></lb>CK, vt ſit eadem altitudo. </s> <s>Dico fruſtum ABCF <lb></lb>GH, ad pyramidem, vel conum, ABCK, eſse vt <lb></lb>tres lineas AC, FH, NO, ſimul ad ipſius AC, tri<lb></lb>plam: ad priſma autem, vel cylindrum, cuius baſis ABC, <lb></lb>altitudo autem eadem cum fruſto, vttres AC, FH, NO, <lb></lb>ſimul, ad ipſius AC, triplam. </s> <s>Nam vt eſt AC ad FH, <lb></lb>& FH ad NO, ita ſit NO ad P: & exceſſus, quo hæ <lb></lb><figure id="id.043.01.032.1.jpg" xlink:href="043/01/032/1.jpg"></figure><lb></lb>quatuor lineæ differunt, ſint AL, FM, <expan abbr="Oq.">Oque</expan> Ergo <lb></lb>vt AC ad FH, ita erit AL ad FM, & FM ad <expan abbr="Oq.">Oque</expan> <lb></lb>Quoniam igitur eſt vt AC ad P, ita pyramis, vel conus <lb></lb>ABCD, ad ſimilem ipſi pyramidem, vel conum DFGH, <lb></lb>ob triplicatam laterum homologorum proportionem; erit <lb></lb>diuidendo, vt tres AL, FM, OQ, ſimul ad P, ita fru<lb></lb>ſtum ABCFGH, ad pyramidem, vel conum DFGH: <lb></lb>ſed conuertendo eſt vt P, ad AC, ita pyramis, vel conus <lb></lb>DFGH, ad pyramidem, vel conum ABCD: ex æquali <lb></lb>igitur, vt tres AL, FM, OQ, ſimul ad AC, ita fruſtum <pb xlink:href="043/01/033.jpg" pagenum="25"></pb>ABCDFGH, ad pyramidem, vel conum ABCD. <lb></lb></s> <s>Rurſus quoniam axis DE, & latera pyramidis, vel coni <lb></lb>ABCD, ſecantur plano trianguli, vel circuli FGH, baſi <lb></lb>ABC, parallelo; erit componendo, vt AD, ad DF, hoc <lb></lb>eſt, vt AC ad FH, propter ſimilitudinem triangulorum, <lb></lb>hoc eſt vt AC, ad CL, ita ED, ad DK; & per conuer<lb></lb>ſionem rationis, vt AC, ad AL, ita DE, ad EK: ſed vt <lb></lb>DE ad EK, ita eſt pyramis, vel conus ABCD, ad py<lb></lb>ramidem, vel conum ABCK; vt igitur AC, ad AL, <lb></lb>ita eſt pyramis, vel conus ABCD, ad pyramidem, vel <lb></lb>conum ABCK; ſed vt tres lineæ AL, FM, OQ ſimul <lb></lb>ad AC, ita erat fruſtum ABCFGH, ad pyramidem, <lb></lb>vel conum ABCD; ex æquali igitur, erit vt tres lineæ <lb></lb>AL, FM, OQ, ſimul ad AL, ita fruſtum ABCFGH, <lb></lb>ad pyramidem, vel conum ABCK. Rurſus, quoniam <lb></lb>tres exceſſus AL, FM, OQ, ſunt deinceps proportio<lb></lb>nales in proportione totidem terminorum AC, FH, NO, <lb></lb>erunt vt AL, FM, OQ, ſimul ad AL, ita AC, FH, <lb></lb>NO, ſimul ad AC: ſed vt AL, FM, OQ, ſim ul ad <lb></lb>AL, ita erat fruſtum ABCFGH, ad pyamidem, vel <lb></lb>conum ABCK; vt igitur tres lineæ AC, FH, NO, ſi<lb></lb>mul, ad AC, ita erit fruſtum ABCFGH, ad pyrami<lb></lb>dem, vel conum ABCK. </s> <s>Sed vt AC, ad ſui triplam, ita <lb></lb>eſt pyramis, vel conus ABCK ad priſma, vel cylindrum, <lb></lb>cuius eſt eadem baſis ABC, & eadem altitudo cum py<lb></lb>ramide, vel cono ABCK; ex æquali igitur, erit vt tres <lb></lb>lineæ AC, FH, NO, ſimul ad ipſius AC, triplam, ita <lb></lb>fruſtum ABCFGH, ad priſma, vel cylindrum, cu<lb></lb>ius baſis ABC, & eadem altitudo pyramidi, vel cono <lb></lb>ABCK: ideſt eadem, fruſto ABCFGH. </s> <s>Manifeſtum <lb></lb>eſt igitur propoſitum. </s></p><pb xlink:href="043/01/034.jpg" pagenum="26"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omni ſolido circa axim in alteram partem defi <lb></lb>cienti, cuius baſis ſit circulus, vel ellypſis, figura <lb></lb>quædam ex cylindris, vel cylindri portionibus <lb></lb>æqualium altitudinum inſcribi poteft, & altera <lb></lb>circumſcribi, ita vt circumſcripta ſuperet inſcri<lb></lb>ptam minori exceſſu quacumque magnitudine <lb></lb>propoſita. </s></p><p type="main"> <s>Sit ſolidum ABC, circa axim AD, in alteram par<lb></lb>tem deficiens, cuius vertex A, baſis autem circulus, vel <lb></lb>ellypſis, cuius diameter BC. </s> <s>Igitur ſuper hanc baſim <lb></lb>circa axim AD, <lb></lb>intelligatur deſeri <lb></lb>ptus cylindrus, vel <lb></lb>cylindri portio <lb></lb>BL, quæ ſolidum <lb></lb>ABC, compre<lb></lb>hendet: ſectoque <lb></lb>cylindro, vel cylin <lb></lb>dri portione BL, <lb></lb>planis baſi paralle <lb></lb><figure id="id.043.01.034.1.jpg" xlink:href="043/01/034/1.jpg"></figure><lb></lb>lis in tot cylindros, vel cylindri portiones æqualium al<lb></lb>ritudinum, vt quilibet eorum ſit minor magnitudine <lb></lb>propoſita; eſto ſolidum ABC, ſectum prædictis planis: <lb></lb>erunt autem ſectiones circuli, vel ellypſes fimiles inter <lb></lb>ſe & baſi BC, ſolidi ABC ſuper quas ſectiones tam<lb></lb>quam baſes cylindris, vel cylindri portionibus æqua<lb></lb>lium altitudinum intra, atque extra figuram conſtitutis, <lb></lb>quorum bini inter eadem plana parallela inter ſe refe-<pb xlink:href="043/01/035.jpg" pagenum="27"></pb>runtur, veluti BF, & GDH, quorum axis communis eſt <lb></lb>D<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, baſes autem circuli, vel ellypſes EF, GH, qua<lb></lb>rum commune centrum K: ſupremus autem, qui ad A, <lb></lb>ad nullum refertur. </s> <s>Quoniam igitur ex conſtructione, <lb></lb>cylindrus, vel cylindri portio BF, eſt minor magnitudi<lb></lb>ne propoſita; exceſsus autem omnes, quibus cylindri, ex <lb></lb>quibus conſtat figura circumſcripta, excedunt eos, ex qui<lb></lb>bus conſtat figura inſcripta, pro vt bini inter ſe referun<lb></lb>tur, vna cum ſupremo, qui ad nullum refertur, ſunt æqua<lb></lb>les cylindro, vel cylindri portioni BF, figura circum<lb></lb>ſcripta ſolido ABC, excedet inſcriptam minori exceſ<lb></lb>ſu magnitudine propoſita. </s> <s>Fieri igitur poteſt quod pro<lb></lb>ponebamus. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Dato parallelepipedo erecto circa datam re<lb></lb>ctam lineam tamquam axim, erectum parallele<lb></lb>pipedum æquale conſtituere. </s></p><p type="main"> <s>Sit datum parallelepipedum AB, erectum, cuius ba<lb></lb>ſis AC, altitudo autem latus BC: & data recta linea <lb></lb>finita ED. </s> <s>Oportet circa rectam ED, tamquam axim <lb></lb>parallelepipedo AB, æquale parallelepipedum erectum <lb></lb>conſtituere. </s> <s>Per punctum igitur E, extendatur pla<lb></lb>num erectum ad lineam ED, & vt eſt DE, ad BC, ita <lb></lb>fiat baſis AC, ad quadratum F: & ad punctum E, in <lb></lb>plano erecto ad lineam ED, quartæ parti quadrati F, <lb></lb>æquale GE, quadratum deſcribatur, & compleatur <lb></lb>quadratum GH, quadruplum quadrati EG, ſeu qua<lb></lb>drato F, æquale: & ex puncto K, erecta KL, ipſi EF, <lb></lb>æquali, & ad ſubiectum planum perpendiculari ſuper ba<lb></lb>ſim GH, conſtituatur parallelepipedum GK. </s> <s>Dico <pb xlink:href="043/01/036.jpg" pagenum="28"></pb>parallelepipedum GK, eſse æquale parallelepipedo AB; <lb></lb>& rectam DE, axim parallelepipedi GK. </s> <s>Iungantur <lb></lb>enim baſium oppoſitarum diametri GH, LK. </s> <s>Quo<lb></lb>niam igitur qua<lb></lb>drata ſunt EG, <lb></lb>GH, communem<lb></lb>que habent angu<lb></lb>lum, qui ad G, <lb></lb>conſiſtent circa di<lb></lb>ametrum GH; in <lb></lb>recta igitur GH, <lb></lb>erit punctum E. <lb></lb></s> <s>Et quoniam qua<lb></lb>dratum GH, eſt <lb></lb>quadrati EG, qua<lb></lb>druplum; erit dia<lb></lb><figure id="id.043.01.036.1.jpg" xlink:href="043/01/036/1.jpg"></figure><lb></lb>meter GH, diametri EG, dupla; punctum igitur E, <lb></lb>erit in medio diametri GH. Rurſus, quoniam ob pa<lb></lb>rallelepipedum GK, recta GL, æqualis eſt, & paral<lb></lb>lela ipſi KH, erit LH, parallelogrammum: & quia <lb></lb>vtraque DE, KH, eſt ad ſubiectum planum perpendi<lb></lb>cularis, parallelæ erunt, & in eodem plano parallelogram<lb></lb>mi LH; in quo cum LG, ſit parallela ipſi KH; erit & <lb></lb>ED, ipſi LG, parallela: eſt autem, & æqualis vtrilibet <lb></lb>ipſarum GL, GH, oppoſitarum; punctum igitur D, eſt <lb></lb>in recta LK, & tam KD, ipſi EH, quàm LD, ipſi <lb></lb>EG, æqualis erit, & inter ſe æquales LD, DK. pun<lb></lb>ctum igitur D, erit in medio diametri LK; ſed & pun<lb></lb>ctum E, erat in medio diametri GH; recta igitur ED, <lb></lb>axis eſt parallelepipedi GK, cuius parallelepipedi cum <lb></lb>altitudo DE, ſit ad BC, altitudinem parallelepipedi AB, <lb></lb>vt eſt baſis AC, ad quadratum F, hoc eſt ad baſim GH, <lb></lb>parallelepipedi GK; parallelepipedum GK, parallelepipe <lb></lb>do AB, æquale erit, Factum igitur eſt quod oportebat. </s></p><pb xlink:href="043/01/037.jpg" pagenum="29"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Cuilibet figuræ ſolidæ <expan abbr="parallelepipedũ">parallelepipedum</expan> æqua<lb></lb>le poteſt eſſe. </s></p><p type="main"> <s>Sit quælibet figura ſolida A. </s> <s>Dico ſolido A, parallele<lb></lb>pipedum æquale poſse exiſtere. </s> <s>Exponatur enim paral<lb></lb>lelepipedum BC, cuius baſis BG. </s> <s>Quoniam igitur eſt vt <lb></lb>ſolidum BC, ad ſolidum A, ita recta linea, ſiue latus BD, <lb></lb>ad aliam rectam lineam; producto latere BD, ſit vt BC, <lb></lb>ad A, ita recta BD, ad rectam DE, & compleatur pa<lb></lb>rallelepipedum CE. </s> <s>Quoniam itaque eſt vt BD, ad DE, <lb></lb>ita parallelogrammum ſiue baſis BG, ad parallelogram<lb></lb><figure id="id.043.01.037.1.jpg" xlink:href="043/01/037/1.jpg"></figure><lb></lb>mum, ſiue baſim EG; hoc eſt parallelepipedum BC, ad <lb></lb>parallelepipedum CE: ſed vt BD, ad DE, ita eſt paral<lb></lb>lelepipedum BC, ad ſolidum A; vt igitur parallelepipe<lb></lb>dum BC, ad ſolidum A, ita erit parallelepipedum BC, <lb></lb>ad parallelepipedum CE; parallelepipedum igitur CE <lb></lb>æquale erit ſolido A. </s> <s>Quod fieri poſse propoſuimus. </s></p><pb xlink:href="043/01/038.jpg" pagenum="30"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis parallelogtammi centrum grauitatis <lb></lb>diametrum bifariam diuidit. </s></p><p type="main"> <s>Sit parallelogrammum ABCD, cuius duo latera AB, <lb></lb>BC, ſint primum in æqualia: & <expan abbr="quoniã">quoniam</expan> omne parallelogram<lb></lb>mum habet ſaltem duos angulos oppoſitos non minores <lb></lb>recto, eſto vterque angulorum B, D, non minor recto, ſit<lb></lb>que ducta diameter AC, ſectaque in puncto G, bifariam. <lb></lb></s> <s>Dico G, eſse centrum grauitatis parallelogrammi ABCD. <lb></lb></s> <s>Trianguli enim ABC, ſit centrum grauitatis H; iuncta<lb></lb>que HG, & producta, ponatur GK, æqualis GH, & re<lb></lb>ctæ à punctis K, H, ad angulos ducantur. </s> <s>Quoniam igi<lb></lb>tur AG, eſt æqualis GC, & <lb></lb>GH, ipſi GK, & angulus <lb></lb>AGK, æqualis angulo CGH, <lb></lb>erit baſis AK, æqualis baſi <lb></lb>CH, & angulus GAK, æqua<lb></lb>lis angulo GCK: ſed totus <lb></lb>angulus DAK, æqualis eſt to <lb></lb>ti angulo BCA; reliquus igi<lb></lb>tur DAK, reliquo BCH, <lb></lb>æqualis erit, circa quos angu<lb></lb>los latus BC eſt æquale lateri <lb></lb>AD, & CH, ipſi AK; angu<lb></lb>lus igitur CBH, æqualis erit <lb></lb><figure id="id.043.01.038.1.jpg" xlink:href="043/01/038/1.jpg"></figure><lb></lb>angulo ADK. </s> <s>Similiter oſtenderemus angulum CAH, <lb></lb>angulo ACK, & angulum BAH, angulo DCK, & an<lb></lb>gulum ABH, angulo CDK, æquales eſse: ſed latera <lb></lb>triangulorum, cum quibus rectæ ductæ à punctis K, H, ad <lb></lb>angulos triangulorum ſimilium ABC, CDA, ſunt ho-<pb xlink:href="043/01/039.jpg" pagenum="31"></pb>mologa; puncta igitur K, H, in prædictis triangulis ſunt <lb></lb>ſimiliter poſita. </s> <s>Rurſus quoniam angulus ABC, non <lb></lb>eſt minor recto, acuti erunt reliqui ACB, BAC; igitur <lb></lb>latus AC, maximum erit: ponitur autem AB maius, <lb></lb>quàm BC; triangulum igitur ABC, ſcalenum erit. <lb></lb></s> <s>Eadem ratione ſcalenum eſt triangulum ACD. </s> <s>Quare <lb></lb>in triangulo ACD, vnum duntaxat punctum K, ſimili<lb></lb>ter poſitum erit, ac punctum H, in triangulo ABC. </s> <s>Cum <lb></lb>igitur H ſit centrum grauitatis trianguli ABC, erit & <lb></lb>K, centrum grauitatis trianguli ACD. </s> <s>Sed longitudo <lb></lb>GK, æqualis eſt longitudini GH; punctum igitur G erit <lb></lb>centrum grauitatis parallelogrammi ABCD, in quo ni<lb></lb>mirum ſecta eſt bifariam diameter AC: quare ſi ducatur <lb></lb>altera diameter BD, in medio etiam diametri BD, erit <lb></lb>idem centrum grauitatis G. </s></p><p type="main"> <s>Sed ſint omnia latera æqualia <expan abbr="parallelogrãmi">parallelogrammi</expan> ABCD, <lb></lb>Sectisque duobus lateribus AD, BC, bifariam in E, F <lb></lb>iungantur EF, AE, ED, <lb></lb>AGC, & per punctum G, <lb></lb>ducatur ipſi AD, vel BC, <lb></lb>parallela HGK. </s> <s>Quoniam <lb></lb>igitur EC, eſt æqualis <lb></lb>AF, erit CG æqualis AG, <lb></lb>& EG, æqualis GF, pro<lb></lb>pter ſimilitudinem triangu <lb></lb>lorum: nec non EH, ipſi <lb></lb>AH, & EK, ipſi KD: tres <lb></lb>igitur diametri AC, AE, <lb></lb>ED, erunt ſectæ bifariam <lb></lb><figure id="id.043.01.039.1.jpg" xlink:href="043/01/039/1.jpg"></figure><lb></lb>in punctis K, G, H: & quoniam ex æquali propter triangu<lb></lb>la ſimilia eſt vt AF, ad FD, ita HG, ad GK, erit HG, <lb></lb>æqualis ipſi GK: ſed puncta K, H, ſunt centra grauitatis <lb></lb>parallelogrammorum BF, FC; igitur totius parallelo<lb></lb>grammi ABCD, centrum grauitatis erit G, in medio <pb xlink:href="043/01/040.jpg" pagenum="32"></pb>diametri AG. </s> <s>Quod eſt propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hinc manifeſtum eſt, omnis parallelogrammi <lb></lb>centrum grauitatis eſſe in medio rectæ, quæ op<lb></lb>poſitorum bipartitorum laterum ſectiones iungit. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si quodlibet parallelogrammum in duo paral<lb></lb>lelogramma diuidatur, & eorum <expan abbr="cẽtra">centra</expan> grauitatis <lb></lb>iungantur recta linea; totius diuiſi parallelogram<lb></lb>mi centrum grauitatis prædictam lineam ita di<lb></lb>uidit, vt eius ſegmenta è contrario reſpondeant <lb></lb>prædictis partibus parallelogrammis. </s></p><p type="main"> <s>Sit parallelogrammum ABCD, ſectum in duo paral<lb></lb>lelogramma AE, ED, & <lb></lb>parallelogrammi AE, ſit <lb></lb>centrum grauitatis H, pa<lb></lb>rallelogrammi autem ED, <lb></lb>centrum grauitatis K: & <lb></lb>parallelogrammi ABCD, <lb></lb>ſit centrum grauitatis G: <lb></lb>& iungatur KH. </s> <s>Dico re<lb></lb>ctam KH, diuidi à puncto <lb></lb>G, ita vt ſit KG, ad G <lb></lb>H, vt eſt parallelogrammum <lb></lb>AE, ad parallelogrammum <lb></lb><figure id="id.043.01.040.1.jpg" xlink:href="043/01/040/1.jpg"></figure><lb></lb>ED, Iungantur enim diametri AC, AE, ED. </s> <s>Igitur <pb xlink:href="043/01/041.jpg" pagenum="33"></pb>per præcedentem ſectæ erunt hæ diametri bifariam in pun<lb></lb>ctis H, G, K. </s> <s>Quoniam igitur eſt vt EH, ad HA, ita <lb></lb>EK ad KD, parallela erit KH, ipſi AD; igitur & EC; <lb></lb>ſed recta KH, ſecat latus AE, trianguli AEC, bifariam <lb></lb>in puncto H, ergo & latus AC, bifariam ſecabit; igitur <lb></lb>in puncto G. punctum igitur G, eſt in linea KH. Rurſus, <lb></lb>quoniam eſt vt GA, ad AC, ita GH, ad EC, propter ſi<lb></lb>militudinem triangulorum; ſed dimidia eſt GA, ipſius <lb></lb>AC, igitur & GH, erit dimidia ipſius EC, hoc eſt ipſius <lb></lb>FD. </s> <s>Similiter oſtenderemus dimidiam eſse KH ipſius <lb></lb>AD. vt igitur KH, ad AD, ita erit GH, ad FD: & per<lb></lb>mutando, vt AD, ad DF, ita KH, ad HG, & diui<lb></lb>dendo, vt AF, ad FD, hoc eſt vt parallelogrammum AE, <lb></lb>ad parallelogrammum ED, ita KG, ad GH. </s> <s>Quod de<lb></lb>monſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Plana grauia æquiponderant à longitudini<lb></lb>bus ex contraria parte reſpondentibus. </s></p><p type="main"> <s>Sint plana grauia N, R, quorum centra grauitatis ſint <lb></lb>N, R, & longitudo aliqua AB: & vt eſt N, ad R, ita ſit <lb></lb>BC, ad CA. </s> <s>Dico ſuſpenſis magnitudinibus ſecundum <lb></lb>centra grauitatis N, in puncto A, & R, in puncto B, vtri<lb></lb>uſque magnitudinis N, R, ſimul centrum grauitatis eſse <lb></lb>C. </s> <s>Nam ſi N, R, magnitudines ſint æquales, manifeſtum <lb></lb>eſt propoſitum. </s> <s>Si autem inæquales, abſcindatur BD, <lb></lb>æqualis AC, vt ſit AD, ad DB, vt BC, ad CA. </s> <s>Et quo<lb></lb>niam ſpacio R, rectangulum æquale poteſt eſse; applice<lb></lb>tur ad lineam BD, rectangulum BDKE, æquale quar<lb></lb>tæ parti rectanguli æqualis ipſi R, hoc eſt quartæ parti <lb></lb>ipſius R; & poſita DG, æquali, & in directum ipſi DK, <pb xlink:href="043/01/042.jpg" pagenum="34"></pb>ducantur rectæ GBH, GAF, quæ cum KE, produ<lb></lb>cta conueniant in punctis F, H: & fiant parallelogramma <lb></lb>FL, AK. </s> <s>Quoniam igitur eſt vt N, ad R, ita BC, ad <lb></lb>CA, hoc eſt AD, ad DB, hoc eſt rectangulum AK, ad <lb></lb>rectangulum BK; erit permutando vt rectangulum AK, <lb></lb>ad N, ita rectangulum BK, ad R; ſed rectangulum BK, <lb></lb>eſt pars quarta ipſius R, ergo & rectangulum AK, erit <lb></lb>pars quarta ipſius N. </s> <s>Rurſus quia eſt vt GD, ad D<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, <lb></lb>ita GA, ad AF, & GB, ad BH: ſed GD eſt æqualis <lb></lb>DK; ergo & GA, ipſi AF, & GB, ipſi BH, æquales <lb></lb>erunt & centra grauita<lb></lb>tis A, quidem rectangu<lb></lb>li MK, B, vero rectan<lb></lb>guli KL, & rectangulum <lb></lb>AK, pars quarta ipſius <lb></lb>M<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, quemadmodum <lb></lb>& B<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ipſius KL; ſed <lb></lb>N, rectanguli AK, qua<lb></lb>druplum erat, quemad<lb></lb>modum & R ipſius BK; <lb></lb>igitur rectangulum MK, <lb></lb>ſpacio N, & rectangulum <lb></lb>KL, ſpacio R, æquale <lb></lb>erit. </s> <s>Sed vt BC, ad CA, <lb></lb>ita eſt N, ad R; vt igi<lb></lb>tur BC, ad CA, ita <lb></lb><figure id="id.043.01.042.1.jpg" xlink:href="043/01/042/1.jpg"></figure><lb></lb>rectangulum MK, ad rectangulum KL; ſed A eſt cen<lb></lb>trum grauitatis rectanguli MK, & B, rectanguli KL; to<lb></lb>tius ergo rectanguli FL, hoc eſt duorum rectangulorum <lb></lb>MK, KL, ſimul centrum grauitatis erit C. </s> <s>Sed rectan<lb></lb>gulo MK, æquale eſt ſpacium N; & rectangulo KL, ſpa<lb></lb>cium R. </s> <s>Igitur ſi pro rectangulo MK, ſit ſuſpenſum N <lb></lb>ſpacium ſecundum centrum grauitatis in puncto A, & pro <lb></lb>rectangulo KL, ſpacium R, ſecundum centrum graui-<pb xlink:href="043/01/043.jpg" pagenum="35"></pb>tatis in puncto B, ſpacia N, R, æquiponderabunt à lon<lb></lb>gitudinibus AC, CB; eritque vtriuſque plani N, R, ſi<lb></lb>mul centrum grauitatis C. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hinc manifeſtum eſt ſi cuiuslibet figuræ pla<lb></lb>næ vtcumque ſectæ centra grauitatis partium <lb></lb>iungantur recta linea, talem lineam à centro gra<lb></lb>uitatis totius prædicti plani ita ſecari, vt ſegmen<lb></lb>ta ex contrario reſpondeant prædictis partibus. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si totum quoduis planum, & pars aliqua non <lb></lb>habeant idem centrum grauitatis, & eorum cen<lb></lb>tra iungantur recta linea; in ea producta ad par<lb></lb>tes centri grauitatis totius, erit reliquæ partis cen <lb></lb>trum grauitatis. </s></p><p type="main"> <s>Sit totum quoduis planum <lb></lb>ABC, cuius centrum graui<lb></lb>tatis E, & pars illius AB, cuius <lb></lb>aliud centrum D, & iuncta <lb></lb>DE, producatur ad partes E, <lb></lb>in infinitum vſque in H. </s> <s>Dico <lb></lb>reliquæ partis BC, centrum <lb></lb>grauitatis, quod ſit G, eſse in <lb></lb>linea EH. </s> <s>Quoniam enim D, <lb></lb>G, ſunt centra grauitatis par<lb></lb><figure id="id.043.01.043.1.jpg" xlink:href="043/01/043/1.jpg"></figure><lb></lb>tium AB, BC, cadet totius ABC, centrum grauitatis <pb xlink:href="043/01/044.jpg" pagenum="36"></pb>E, in recta linea, quæ iungit centra D, G; tria igitur pun<lb></lb>cta D, E, G, ſunt in eadem recta linea. </s> <s>in qua igitur ſunt <lb></lb>puncta D, E, in eadem eſt punctum G; ſed puncta D, E, ſunt <lb></lb>in recta DH; igitur & punctum G, erit in recta DH: ſed <lb></lb>extra ipſam DE, vt modo oſtendimus, in reliqua igitur <lb></lb>EH. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Sit totum quoduis planum ſit vni parti concen <lb></lb>tricum ſecundum centrum grauitatis, & reliquæ <lb></lb>erit concentricum. </s> <s>Et ſi partes inter ſe ſint con<lb></lb>centricæ, & toti erunt concentricæ. </s></p><p type="main"> <s>Sit totum quoduis planum AB, quod cum vna parte <lb></lb>AC habeat commune centrum grauitatis E. </s> <s>Dico & re<lb></lb>liquæ partis CD, eſse <lb></lb>idem centrum grauitatis <lb></lb>E. </s> <s>Si enim illud non <lb></lb>eſt, erit aliud; eſto F, & <lb></lb>EF iungatur. </s> <s>Quoniam <lb></lb>igitur partium AC, CD, <lb></lb>centra grauitatis ſunt E, <lb></lb>F; erit totius AB, in re<lb></lb>cta EF, centrum graui<lb></lb>tatis: ſed & in puncto E, <lb></lb>vnius ergo magnitudinis <lb></lb>duo centra grauitatis e<lb></lb>runt. </s> <s>Quod eſt abſurdum; <lb></lb><figure id="id.043.01.044.1.jpg" xlink:href="043/01/044/1.jpg"></figure><lb></lb>idem igitur E erit centrum grauitatis vtriuslibet partium <lb></lb>AC, CD. </s> <s>Sed vtriuslibet partium AC, CD, ſit cen<lb></lb>trum grauitatis E. </s> <s>Dico idem E totius AB, eſse cen-<pb xlink:href="043/01/045.jpg" pagenum="37"></pb>trum grauitatis. </s> <s>Si enim non eſt, erit aliud, eſto G: & <lb></lb>iunctatur EG, producatur ad partes G, in infinitum vſ<lb></lb>que ìn F. </s> <s>Quoniam igitur E, eſt centrum grauitatis vnius <lb></lb>partis AC, & G, totius AB; erit reliquæ partis CD, in <lb></lb>linea GF centrum grauitatis: ſed & in puncto E; eiuſ<lb></lb>dem igitur magnitudinis AB, duo centra grauitatis erunt. <lb></lb></s> <s>Quod fieri non poteſt; totius igitur AB, erit centrum gra<lb></lb>uitatis idem E. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis trianguli rectilinei idem eſt centrum <lb></lb>grauitatis, & figuræ. </s></p><p type="main"> <s>Sit triangulum rectilineum ABC, cuius centrum G. <lb></lb></s> <s>Dico G, eſse centrum grauitatis trianguli ABC. </s> <s>Si enim <lb></lb>fieri poteſt, ſit aliud punctum N, centrum grauitatis trian <lb></lb>guli ABC, & per punctum G, ducantur rectæ AF, BD, <lb></lb>CE, & DHE, ERF, FKD, <emph type="italics"></emph>K<emph.end type="italics"></emph.end>LH, & NG. </s> <s>Quo<lb></lb>niam igitur quæ ab angulis A, B, C, ductæ ſunt rectæ <lb></lb>lineæ per G, ſecant bifariam latera AB, BC, CA; erit <lb></lb>triangulum EDF, ſimile triangulo ABC, ob latera pa<lb></lb>rallela vt ſunt EF, AC. </s> <s>Et quoniam triangulum EDF, <lb></lb>dimidium eſt cuius vis trium parallelogrammorum AF, <lb></lb>BD, CE, æqualia inter ſe erunt ea parallelogramma <lb></lb>omnifariam ſumpta, quorum centra grauitatis H, K, R; <lb></lb>intelligantur autem tria parallelogramma AF, BD, CE, <lb></lb>diſtincta penitus, ita vt inter ſe congruant ſecundum tria <lb></lb>triangula DEF, inter ſe congruentia: trium igitur trian <lb></lb>gulorum DEF, inter ſe congruentium & centra grauita<lb></lb>tis inter ſe congruent in puncto M. </s> <s>Quoniam igitur in<lb></lb>ter duas parallelas EF, KH, ſecant ſe rectæ lineæ FH, <lb></lb>LR, in puncto G; erit vt FG, ad GH, ita RG, ad GL; <pb xlink:href="043/01/046.jpg" pagenum="38"></pb>dupla igitur RG, eſt ipſius GL. </s> <s>Et quoniam in triangu<lb></lb>lo AGC, recta GD, ſecat AC, bifariam in puncto D; <lb></lb>ipſi AC, parallelam KH, bifariam ſecabit in puncto L, <lb></lb>duorum igitur æqualium parallelogrammorum AF, EG; <lb></lb>ſimul, quorum centra grauitatis ſunt K, H, centrum gra<lb></lb>uitatis erit L. </s> <s>Sed duo parallelogramma AF, EC, ſi<lb></lb>mul ſunt paralle<lb></lb>logrammi BD, du <lb></lb>plum; trium igitur <lb></lb>parallelogrammo<lb></lb>rum AF, EC, <lb></lb>BD, ſimul: hoc <lb></lb>eſt <expan abbr="triãguli">trianguli</expan> ABC, <lb></lb>vnà cum duobus <lb></lb>trium <expan abbr="triangulorũ">triangulorum</expan> <lb></lb>inter ſe congruen<lb></lb>tium EDF, cen<lb></lb>trum grauitatis e<lb></lb>rit G. </s> <s>Sed triangu <lb></lb>li ABC, ponitur <lb></lb><figure id="id.043.01.046.1.jpg" xlink:href="043/01/046/1.jpg"></figure><lb></lb>centrum grauitatis N; producta igitur NG, occurret <lb></lb>centro M, reliquæ partis, ideſt duorum triangulorum DEF; <lb></lb>quare vt triangulum ABC, ad duo triangula DEF, ſi<lb></lb>mul, ita erit MG, ad GN. </s> <s>Sed triangulum ABC, eſt <lb></lb>duplum duorum triangulorum EDF: igitur & MG, erit <lb></lb>ipſius GN, dupla. </s> <s>Rurſus quoniam vtriuslibet duorum <lb></lb>triangulorum EDF, centrum grauitatis erat M; erit ſi<lb></lb>militer poſitum M, in triangulo EDF, ac centrum N, in <lb></lb>triangulo ABC, propter ſimilitudinem triangulorum: <lb></lb>Sed propter hæc ſimiliter poſita centra, quia homologo<lb></lb>rum laterum eſt vt AB, ad DF, ita NG, ad GM: & <lb></lb>AB, eſt dupla ipſius EB, erit & NG, dupla ipſius GM. <lb></lb></s> <s>Sed GM, erat dupla ipſius GN: igitur GN, erit ſui ipſius <lb></lb>quadrupla. </s> <s>Quod eſt abſurdum. </s> <s>Non igitur centrum <pb xlink:href="043/01/047.jpg" pagenum="39"></pb>grauitatis trianguli ABC, erit aliud à puncto G: pun<lb></lb>ctum igitur G, erit centrum grauitatis trianguli ABC. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="main"> <s>Quod autem ex huius theorematis demonſtratione li<lb></lb>quet centrum grauitatis trianguli eſse in ea recta linea, <lb></lb>quæ ab angulo ad bipartiti lateris ſectionem pertinet, <lb></lb>Archimedes per inſcriptionem figuræ ex parallelogram<lb></lb>mis demonſtrauit, aliter autem per diuiſionem trianguli <lb></lb>in triangula nequaquam: qua enim ratione hoc ille tentat, <lb></lb>ea ex nono theoremate eiuſdem prioris libri de æquipon<lb></lb>derantibus neceſsario pendet. </s> <s>Cum igitur in illo ante ceden <lb></lb>ti ſit fallacia accipientis latenter ſpeciem trianguli; ſcale<lb></lb>num ſcilicet pro genere triangulo, neque conſequens erit <lb></lb>demonſtratum. </s> <s>Quod autem dico manifeſtum eſt: Datis <lb></lb>enim duobus triangulis ſimilibus, & in altero eorum dato <lb></lb>puncto, quod ſit trianguli centrum grauitatis, punctum in <lb></lb>altero triangulo modo ſimiliter poſitum ſit prædicto pun<lb></lb>cto, nititur demonſtrare eſse alterius trianguli centrum <lb></lb>grauitatis: cum autem nondum conſtet centrum graui<lb></lb>tatis trianguli eſse in recta, quæ ab angulo latus oppoſi<lb></lb>tum bifariam ſecat, ſed ex nono theoremate ſit demonſtran <lb></lb>dum medio decimo, non poteſt illud accipi in nono theo<lb></lb>remate, quod ad demonſtrationem eſset neceſsarium. </s> <s>per<lb></lb>mittitur igitur aduerſario ponere centrum grauitatis trian<lb></lb>guli, vbicumque vult intra illius limites. </s> <s>atqui cum datis <lb></lb>duobus triangulis iſoſceliis ſimilibus, & in altero eorum <lb></lb>dato puncto, quod non ſit in prædicta recta linea, poſsint <lb></lb>in altero duo puncta prædicto ſimiliter poſita inueniri, quo<lb></lb>rum vnum duntaxat concedet aduerſarius eſse alterius <lb></lb>trianguli centrum grauitatis, non autem non ſimiliter po<lb></lb>ſitum, ex quo abſurdum infertur partem anguli æqualem <lb></lb>eſse toti: quid quod datis duobus triangulis æquilateris, & <lb></lb>in altero eorum dato puncto, quod non ſit centrum trian-<pb xlink:href="043/01/048.jpg" pagenum="40"></pb>guli, ſed aliqua earum, quæ ab angulis ad bipartitorum <lb></lb>laterum ſectiones cadunt, neceſse eſt in altero triangulo <lb></lb>tria puncta prædicto puncto eſse ſimiliter poſita? </s> <s>quod ſi <lb></lb>etiam extra iſtas lineas cadat vnius trianguli punctum, ne<lb></lb>ceſse eſt illi ſex puncta in altero triangulo eſse ſimiliter po<lb></lb>ſita: ſed ſi quod diximus de iſoſceliis ſimilibus, & æquila<lb></lb>teris triangulis demonſtrauerimus, rem velut ante oculos <lb></lb>expoſuerimus. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Datis duobustriangulis iſoſcelijs ſimilibus, & <lb></lb>in altero eorum dato puncto extra rectam, quæ à <lb></lb>vertice ad medium baſis cadit, duo puncta in re<lb></lb>liquo triangulo prædicto puncto ſimiliter poſita <lb></lb>inuenire. </s></p><p type="main"> <s>Sint duo triangula iſoſcelia, & ſimilia ABC, DEF: <lb></lb>quorum in altero ABC, à vertice A, ad baſim BC, bi<lb></lb>partitam in puncto G, cadat recta AG: atque extra hanc <lb></lb><figure id="id.043.01.048.1.jpg" xlink:href="043/01/048/1.jpg"></figure><lb></lb>in triangulo ABC, ſit quoduis punctum H: & iuncta AH, <lb></lb>fiat angulus EDK æqualis angulo BAH; & vt BA, ad <pb xlink:href="043/01/049.jpg" pagenum="41"></pb>AH, ita fiat ED, ad DK: & quoniam angulus BAG, <lb></lb>æqualis eſt angulo EDF: quorum angulus EDK, <lb></lb>æqualis eſt angulo BAH, erit reliquus angulus <emph type="italics"></emph>K<emph.end type="italics"></emph.end>DF, <lb></lb>æqualis reliquo angulo HAC; ſed angulus HAC, eſt <lb></lb>maior angulo BAH; ergo & angulus KDF, maior erit <lb></lb>angulo BAH; poſito igitur angulo FDL, æquali an<lb></lb>gulo BAH, ac proinde minori, quàm ſit angulus FD<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, <lb></lb>fiat vt BA, ad AH, ita FD, ad DL. Dico, in triangu<lb></lb>lo EDF, duo puncta K, L, ſimiliter poſita eſse ac pun<lb></lb>ctum H, in triangulo BAC. </s> <s>Iungantur enim rectæ AH, <lb></lb>BH, CH, EK, KF, FL, LE. </s> <s>Quoniam igitur an<lb></lb>gulus ED<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, eſt æqualis angulo BAH, qui lateribus <lb></lb>homologis continentur; erit angulus DE<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, æqualis an<lb></lb>gulo ABH: ſed totus angulus DEF, æqualis eſt toti an<lb></lb>gulo ABC; reliquus igitur angulus KEF, æqualis erit <lb></lb>reliquo HBC: ſed ex æquali eſt vt CB, ad BH, ita <lb></lb>FE, ad EK; igitur vt antea erit angulus KFE, æqualis <lb></lb>angulo HCB, & angulus DFK, æqualis angulo ACH, <lb></lb>& angulus FDK, æqualis angulo CAH; punctum igi<lb></lb>tur K, ſimiliter poſitum erit in triangulo EDF, ac pun<lb></lb>ctum H, in triangulo ABC. </s> <s>Rurſus quoniam angulus <lb></lb>FDL, æqualis eſt angulo BAH, & latus AB, homo<lb></lb>logum lateri DF, (eſt enim vt BA, ad AC, ita FD, ad <lb></lb>DE) ſed vt BA, ad AH, ita eſt FD, ad DL, per con<lb></lb>ſtructionem; ſimiliter vt ante, oſtenderemus, punctum L, <lb></lb>in triangulo EDF, ſimiliter poſitum eſse puncto H; in<lb></lb>uenta igitur ſunt duo puncta in triangulo DEF, ſimili<lb></lb>ter poſita ac punctum H, in triangulo BAC. </s> <s>Quod pro<lb></lb>poſitum erat. </s></p><pb xlink:href="043/01/050.jpg" pagenum="42"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis trapezij habentis duo latera parallela <lb></lb>centrum grauitatis eſt in illa recta, quæ prædi<lb></lb>ctorum bipartitorum laterum ſectiones iungit. <lb></lb></s> <s>atque in eo puncto, in quo tertia pars eius media <lb></lb>ſic diuiditur, vt ſegmentum propinquius mino<lb></lb>ri parallelarum ad reliquum eam proportionem <lb></lb>habeat, quam maior parallelarum ad minorem. <lb></lb></s> <s>Talis autem rectæ lineæ ſic diuiſæ, ſegmentum <lb></lb>minorem parallelarum attingens eſt ad reliquum, <lb></lb>vt dupla maioris parallelarum vna cum minori, <lb></lb>ad duplam minoris vna cum maiori. </s></p><p type="main"> <s>Sit trapezium ABCD, cuius duæ AD, BC, ſint pa<lb></lb>rallelæ: ſitque AD, maior. </s> <s>Sectiſque AD, BC, bifa<lb></lb>riam in punctis F, E, <lb></lb>iunctaque EF, & ſe<lb></lb>cta in tres partes æ<lb></lb>quales in punctis K, <lb></lb>H, fiat vt AD, ad <lb></lb>BC, ita HG, ad GK. <lb></lb></s> <s>Dico G, eſse centrum <lb></lb>grauitatis trapezij A <lb></lb>BCD: & vt eſt du<lb></lb>pla ipſius AD, vna <lb></lb>cum BC, ad duplam <lb></lb>ipſius BC, vna cum <lb></lb>AD, ita eſse EG, ad <lb></lb><figure id="id.043.01.050.1.jpg" xlink:href="043/01/050/1.jpg"></figure><lb></lb>GF. </s> <s>Ducta enim per punctum H, ipſis AD, BC, pa-<pb xlink:href="043/01/051.jpg" pagenum="43"></pb>rallela NO, abſcindantur EL, FM, ipſi GK æquales, & <lb></lb>iungantur ANE, EOD. </s> <s>Quoniam igitur NO ipſi AD, <lb></lb>parallela ſecat omnes ipſis AD, EC, interceptas in eaſ<lb></lb>dem rationes, & eſt EH, pars tertia ipſius EF, erit & EN <lb></lb>ipſius EA, & EO, ipſius ED, pars tertia. </s> <s>Eſt autem NO, <lb></lb>parallela baſibus BE, EC, duorum triangulorum ABE, <lb></lb>ECD; in ipſa igitur NO, erunt centra grauitatis duo<lb></lb>rum triangulorum ABE, ECD: ergo & compoſiti ex <lb></lb>vtroque in linea NO, erit centrum grauitatis. </s> <s>Quoniam <lb></lb>igitur K, centrum grauitatis trianguli AED, eſt in EF, & <lb></lb>totius trapezij ABCD, centrum grauitatis in eadem linea <lb></lb>EF; erit & reliquæ partis, duorum ſcilicet triangulorum <lb></lb>ABE, ECD, ſimul in linea EF, centrum grauitatis: ſed & <lb></lb>in linea NO; in puncto igitur H. </s> <s>Rurſus quoniam triangula <lb></lb>AED, ABE, ECD, ſunt inter eaſdem parallelas, erit <lb></lb>vt AD, ad BC, ita triangulum AED, ad duo triangu<lb></lb>la ABE, ECD, ſimul: ſed vt AD, ad BC, ita eſt HG, <lb></lb>ad GK; vt igitur triangulum AED, ad duo triangula <lb></lb>ABE, ECD, ſimul, ita erit HG, ad GK. ſed K, eſt <lb></lb>centrum grauitatis trianguli AED: & H, duorum trian <lb></lb>gulorum ABE, ECD, ſimul; totius igitur trapezij AB <lb></lb>CD, centrum grauitatis erit G. </s> <s>Rurius quoniam EL, <lb></lb>eſt æqualis GK, æqualium EH, HK; erit reliqua LH, <lb></lb>æqualis reliquæ GH; tota igitur EG; erit bis GH, vna <lb></lb>cum GK: eadem ratione quoniam FM, eſt æqualis GK, <lb></lb>& MK, æqualis GH, erit FG, bis GK, vna cum GH: <lb></lb>vt igitur HG, bis vna cum GK, ad GK, bis vna cum <lb></lb>GH, ita erit EG, ad GF. </s> <s>Sed vt HG, bis vna cum <lb></lb>GK, ad GK bis vna cum GH, ita eſt AD, bis vna cum <lb></lb>BC, ad BC, bis vna cum AB, propterea quod eſt vt <lb></lb>AD, ad BC, ita HG, ad GK; vt igitur eſt AD, bis vna <lb></lb>cum BC, ad BC, bis vna cum AD, ita erit EG, ad GF. <lb></lb></s> <s>Manifeſtum eſt igitur propoſitum. </s></p><pb xlink:href="043/01/052.jpg" pagenum="44"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis polygoni æquilateri, & æquianguli <lb></lb>idem eſt centrum grauitatis, & figuræ. </s></p><p type="main"> <s>Sit polygonum æquilaterum, & æquiangulum ABC <lb></lb>DEFG, cuius ſit primo laterum numerus impar, centrum <lb></lb>autem ſit L. </s> <s>Dico punctum L, eſse centrum grauitatis <lb></lb>polygoni ABCDEFG; ſectis enim duobus lateribus <lb></lb>DE, FG, bifariam in punctis K, H, ducantur ab angulis <lb></lb>oppoſitis rectæ AH, CK. & rectæ BMG, CNF, CM, <lb></lb>MF, iungantur. </s> <s>Quoniam igitur ex decima tertia quar <lb></lb>ti Elem. quemadmodum in pentagono, ita in omni præ<lb></lb>dicto polygono imparium multitudine laterum plane col<lb></lb>ligitur centrum po<lb></lb>lygoni eſse in qua<lb></lb>libet recta, quæ ab <lb></lb>angulo ad medium <lb></lb>lateris oppoſiti du<lb></lb>citur, quoniam ab <lb></lb>omnibus angulis ſic <lb></lb>ductæ ſecant ſe ſe <lb></lb>in eadem proportio<lb></lb>ne æqualitatis, ita <lb></lb>vt eadem ſit propor<lb></lb>tio ſegmentorum, <lb></lb>quæ ad angulos, ad <lb></lb>ea, quæ ad latera <lb></lb><figure id="id.043.01.052.1.jpg" xlink:href="043/01/052/1.jpg"></figure><lb></lb>illis angulis oppoſita; rectæ AH, CK, ſecabunt ſe ſe in <lb></lb>puncto L. </s> <s>Rurfus quoniam ex eadem Euclidis angulus <lb></lb>BAL, æqualis eſt angulo GAL, ſed AB, eſt æqualis <lb></lb>AG, & AM, communis, erit baſis BM, æqualis baſi <pb xlink:href="043/01/053.jpg" pagenum="45"></pb>MG, & angulus ABM, angulo AGM, ſed totus ABC, <lb></lb>toti AGF, eſt æqualis; reliquus igitur angulus CBG, <lb></lb>reliquo BGF, æqualis erit: ſed circa hos æquales an<lb></lb>gulos recta BM, oſtenſa eſt æqualis rectæ MG, & CB, <lb></lb>eſt æqualis GF; baſis igitur CM, baſi GF, & angulus <lb></lb>CMB, angulo FMG, æqualis erit; ſed totus BMN, <lb></lb>æqualis eſt toti GMN; quia vterque rectus; reliquus <lb></lb>igitur CMN, reliquo NMF, æqualis erit, quos circa <lb></lb>recta CM, eſt æqualis MF, & MN, communis; baſis <lb></lb>igitur CN, baſi NF, & anguli, qui ad N, æquales erunt, <lb></lb>atque ideo recti: ſed & qui ad M, ſunt recti, & BM, eſt <lb></lb>æqualis GM; parallelæ igitur ſunt BG, CF, & trape<lb></lb>zij CBGF, centrum grauitatis eſt in linea MN: ſed & <lb></lb>trianguli ABG, centrum grauitatis eſt in linea AM; to<lb></lb>tius igitur figuræ ABCFG, centrum grauitatis eſt in li<lb></lb>nea AN; hoc eſt in linea AH. </s> <s>Rurſus quoniam omnis <lb></lb>quadrilateri quatuor anguli ſunt æquales quatuor rectis: <lb></lb>& tres anguli ABM, BMN, MNC, ſunt æquales tri<lb></lb>bus angulis FGM, GMN, MNF, reliquus angulus <lb></lb>BCF, reliquo CFG, æqualis erit: ſed totus angulus <lb></lb>BCD, eſt æqualis toti angulo GFE; reliquus ergo <lb></lb>DCF, reliquo CFE, æqualis erit: ſed linea CN, eſt <lb></lb>æqualis NF, & anguli, qui ad N, ſunt recti; ſimiliter <lb></lb>ergo vt antea, centrum grauitatis trapezij CDEF, erit <lb></lb>in linea AH: ſed & totius figuræ ABCFG, eſt in li<lb></lb>nea AH; totius igitur polygoni ABCDEFG, in li<lb></lb>nea AH, eſt centrum grauitatis, quod idem ſimiliter in <lb></lb>linea CK, eſse oftenderemus; in communi igitur ſectione <lb></lb>puncto L, eſt centrum grauitatis polygoni ABCDEFG. <lb></lb></s> <s>Similiter quotcumque plurium laterum numero impa<lb></lb>rium eſset polygonum æquilaterum, & æquiangulum, <lb></lb>ſemper deueniendo ab vno triangulo ad quotcumque eius <lb></lb>trapezia; propoſitum concluderemus. </s></p><pb xlink:href="043/01/054.jpg" pagenum="46"></pb><p type="main"> <s>Sed eſto polygonum æquilaterum, & æquiangulum, <lb></lb>ABCDEF, cuius laterum numerus ſit par, & centrum <lb></lb>eſto G. </s> <s>Dico idem G, eſse centrum grauitatis polygoni <lb></lb>ABCDEF. </s> <s>Iungantur enim angulorum oppoſitorum <lb></lb>puncta rectis lineis AD, BE, CF. </s> <s>Ex quarto igitur <lb></lb>Elem. ſecabunt ſeſe hæ rectæ omnes bifariam in vno pun<lb></lb>cto, quod talis figuræ centrum definiuimus: ſed G poni<lb></lb>tur centrum; in puncto igitur G. </s> <s>Quoniam igitur duo<lb></lb>rum triangulorum CBG, GFE, anguli ad verticem <lb></lb>BGC, FGE, ſunt æquales; & vterlibet angulorum CBG, <lb></lb>GCB, æqualis eſt vtrilibet ipſorum EFG, GEF; ex <lb></lb>quarto Elem. & circa æquales angulos latera proportio<lb></lb>nalia horum triangu <lb></lb>lorum ſunt æqualia; <lb></lb>ſimilia, & æqualia <lb></lb>erunt triangula BC <lb></lb>G, GFE: poſitis <lb></lb>igitur centris graui<lb></lb>tatis K, H, duorum <lb></lb>triangulorum EFG, <lb></lb>GBC, iunctifque <lb></lb>KG, GH, erit v<lb></lb>terlibet angulorum <lb></lb>BGH, HGC, æ<lb></lb>qualis vtrilibet an<lb></lb><figure id="id.043.01.054.1.jpg" xlink:href="043/01/054/1.jpg"></figure><lb></lb>gulorum CGK, KGE, propter ſimilitudinem poſitio<lb></lb>nis centrorum K, H, in iſoſcelijs triangulis CBG, <lb></lb>GFE: (nam GH, ſi produceretur latus BC, bifariam <lb></lb>ſecaret: ſimiliter GK, latus EF) ſed CG, eſt in directum <lb></lb>poſita ipſi GF; igitur & GH ipſi GK: & ſunt æquales, <lb></lb>vtpote lateribus triangulorum BCG, GFE, æqualibus <lb></lb>homologæ; cum igitur eorundem triangulorum centra <lb></lb>grauitatis ſint K, H; centrum grauitatis duorum triangu<lb></lb>lorum CBG, GFE, ſimul, erit punctum G. </s> <s>Eadem <pb xlink:href="043/01/055.jpg" pagenum="47"></pb>ratione, tam duorum triangulorum ABG, DGE, quàm <lb></lb>duorum AFG, CDG, ſimul, centrum grauitatis erit G; <lb></lb>totius igitur polygoni ABCDEF; centrum grauitatis <lb></lb>erit idem G. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis figuræ circa diametrum in alteram par <lb></lb>tem deficientis, in diametro eſt centrum graui<lb></lb>tatis. </s></p><p type="main"> <s>Sit figura ABC, circa diametrum BD, in alteram par <lb></lb>tem deficiens verſus B. </s> <s>Dico centrum grauitatis figuræ <lb></lb>ABC, eſse in linea BD. ſit enim punctum E, generali<lb></lb>ter extra lineam BD. </s> <s>Et per puncta E, C, ducantur ipſi <lb></lb>BD, parallelæ EF, <lb></lb>CG, & vt eſt CD, <lb></lb>ad DF, ita ponatur <lb></lb>figura ABC, ad ali<lb></lb>quod ſpacium M: & <lb></lb>figuræ ABC, inſcri<lb></lb>batur figura ex paral<lb></lb>lelogrammis æqua<lb></lb>lium altitudinum de<lb></lb>ficiens à figura ABC, <lb></lb>minori defectu, quam <lb></lb>ſit ſpacium M, quan<lb></lb>tumcumque illud ſit: <lb></lb>minor igitur propor<lb></lb><figure id="id.043.01.055.1.jpg" xlink:href="043/01/055/1.jpg"></figure><lb></lb>tio erit figuræ ABC, ad ſpacium M, hoc eſt minor pro<lb></lb>portio CD, ad DF, quàm figuræ ABC, ad ſui reliquum, <lb></lb>dempta figura inſcripta. </s> <s>Quoniam autem diameter BD, <pb xlink:href="043/01/056.jpg" pagenum="48"></pb>bifariam ſecat omnia latera parallelogrammorum inſcri<lb></lb>ptorum baſi AC, parallela; erit in diametro BD, eorum <lb></lb>omnium parallelogrammorum centra grauitatis, atque <lb></lb>ideo totius figuræ inſcriptæ centrum grauitatis, quod ſit <lb></lb>H: & HEK, ducatur. </s> <s>Quoniam igitur EF, parallela <lb></lb>eſt vtrique DH, CK; erit vt CD, ad DF, ita KH, ad <lb></lb>HE, ſed minor eſt proportio CD, ad DF, quàm figu<lb></lb>ræ ABC, ad reſi<lb></lb>duum, dempta figu<lb></lb>ra inſcripta; ergo & <lb></lb>KH, ad HE, minor <lb></lb>erit proportio, quàm <lb></lb>figuræ ABC, ad præ<lb></lb>dictum reſiduum: ha<lb></lb>beat LKH, eandem <lb></lb><expan abbr="proportionẽ">proportionem</expan> ad EH, <lb></lb>quàm figura ABC, <lb></lb>ad prædictum reſi<lb></lb>duum. </s> <s>Quoniam <lb></lb>igitur punctum K, <lb></lb>cadit extra figuram <lb></lb><figure id="id.043.01.056.1.jpg" xlink:href="043/01/056/1.jpg"></figure><lb></lb>ABC; multo magis punctum L; non igitur punctum L, <lb></lb>erit prædicti reſidui centrum grauitatis. </s> <s>Sed punctum <lb></lb>H, eſt inſcriptæ figuræ centrum grauitatis: & vt figura <lb></lb>inſcripta ad prædictum reſiduum, diuidendo, ita eſt LE, <lb></lb>ad EH; non igitur E, eſt centrum grauitatis figuræ ABC: <lb></lb>ſed ponitur E, generaliter punctum extra lineam BD; <lb></lb>Nullum igitur punctum extra lineam BD, eſt centrum <lb></lb>grauitatis figuræ ABC; in linea igitur BD, erit figu<lb></lb>ræ ABC, centrum grauitatis. </s> <s>Quod demonſtrandum <lb></lb>erat. </s></p><pb xlink:href="043/01/057.jpg" pagenum="49"></pb><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Ex huius theorematis demonſtratione conſtat, <lb></lb>omnis figuræ planæ, ſiue ſolidæ, cuius termini <lb></lb>omnis cauitas ſit interior, atque ideo intra ter<lb></lb>minum centrum grauitatis; & cuius pars aliqua <lb></lb>eſse poſsit, quæ à tota figura deficiens minori <lb></lb>defectu quacumque magnitudine propoſita habe<lb></lb>at centrum grauitatis in aliqua certa linea recta <lb></lb>intra terminum figuræ conſtituta, eſſe in ea recta <lb></lb>linea totius figuræ centrum grauitatis. </s> <s>Ac proin<lb></lb>de, cum per vndecimam huius, omni ſolido circa <lb></lb>axim in alteram partem deficienti, & baſim ha<lb></lb>benti circulum, vel ellypſim figura inſcribi poſſit <lb></lb>ex cylindris, vel cylindri portionibus, à prædicto <lb></lb>ſolido deficiens minori ſpacio quacumque ma<lb></lb>gnitudine propoſita: talis autem figuræ inſcriptæ, <lb></lb>quemadmodum & circumſcriptæ centrum gra<lb></lb>uitatis ſit in axe, vt ex ſequentibus patebit, & <lb></lb>nunc cogitanti facilè patere poteſt; manifeſtum <lb></lb>eſt omnis ſolidi circa axim in alteram partem de<lb></lb>ficientis centrum grauitatis eſſe in axe. </s></p><pb xlink:href="043/01/058.jpg" pagenum="50"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Circuli, & Ellypſis idem eſt centrum grauita<lb></lb>tis, & figuræ. </s></p><p type="main"> <s>Sit circulus, vel ellypſis ABCD, cuius centrum E. <lb></lb></s> <s>Dico centrum grauitatis figuræ ABCD, eſse punctum E. <lb></lb></s> <s>Ducantur enim duæ diametri ad rectos inter ſe angulos <lb></lb>AC, BD; in ellypſi autem ſint diametri coniugatæ. <lb></lb></s> <s>Quoniam igitur omnes rectæ lineæ, quæ in ſemicirculo, <lb></lb>vel dimidia ellypſi diametro ducantur parallelæ bifariam <lb></lb>ſecantur à ſemidiametro, & quo à baſi remotiores, eo ſunt <lb></lb><figure id="id.043.01.058.1.jpg" xlink:href="043/01/058/1.jpg"></figure><lb></lb>minores; erit centrum grauitatis ſemicirculi, ſiue dimidiæ <lb></lb>ellypſis ABC, in linea BE; ſicut & ſemicirculi, ſiue di<lb></lb>midiæ ellypſis ADC, centrum grauitatis in linea DE. <lb></lb>eſt autem BED, vna recta linea: in diametro igitur BD, <lb></lb>erit centrum grauitatis circuli, ſiue ellypſis ABCD. <lb></lb></s> <s>Eadem ratione oſtenderemus idem centrum grauitatis eſse <lb></lb>in altera diametro AC: in communi igitur vtriuſque ſe<lb></lb>ctione puncto E. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/059.jpg" pagenum="51"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si duarum pyramidum triangul as baſes haben<lb></lb>tium æqualium, & ſimilium inter ſe, tria latera <lb></lb>tribus lateribus homologis fuerint in directum <lb></lb>conſtituta, in vertice communi erit vtriuſque ſi<lb></lb>mul centrum grauitatis. </s></p><p type="main"> <s>Sint duæ pyramides ſimiles, & æquales, quarum ver<lb></lb>tex communis G, baſes autem triangula ABC, DEF. <lb></lb></s> <s>Et ſint latera homologa pyramidum in directum inter ſe <lb></lb>conſtituta: vt AG, GF: & BG, GD, & CG, GE. <lb></lb></s> <s>Dico compoſiti ex duabus pyramidibus ABCG, GDEF, <lb></lb>ita conſtitut is centrum gra<lb></lb>uitatis eſse in puncto G. <lb></lb></s> <s>Eſto enim H, centrum gra <lb></lb>uitatis pyramidis ABCG, <lb></lb>& ducta HGK, ponatur <lb></lb>G<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, æqualis GH, & iun<lb></lb>gantur EK, KD, BH, <lb></lb>CH. </s> <s>Quoniam igitur eſt <lb></lb>vt HG, ad GK, ita CG, <lb></lb>ad GE, & proportio eſt <lb></lb>æqualitatis: & angulus <lb></lb>HGC, æqualis angulo EG <lb></lb><emph type="italics"></emph>K<emph.end type="italics"></emph.end>, erit triangulum CGH, <lb></lb><figure id="id.043.01.059.1.jpg" xlink:href="043/01/059/1.jpg"></figure><lb></lb>ſimile, & æquale triangulo EGK. </s> <s>Similiter triangulum <lb></lb>BGH, trian gulo DGK; & triangulum BGC, triangu<lb></lb>lo DGE: quare & triangulum BCH, triangulo DEK. <lb></lb>pyramis igitur BCGH, ſimilis, & æqualis eſt pyramidi <lb></lb>EDGK. </s> <s>Congruentibus igitur inter ſe duobus triangu<pb xlink:href="043/01/060.jpg" pagenum="52"></pb>lis æqualibus, & ſimilibus BGC, DGE, & pyramis <lb></lb>BCGH, pyramidi GDEK congruet, & puncto K, pun<lb></lb>ctum H: & eadem ratione <lb></lb>pyramis ABCG, pyra<lb></lb>midi DEFG. congruente <lb></lb>igitur pyramide ABCG, <lb></lb>pyramidi DEFG, & pun<lb></lb>ctum K, congruet puncto <lb></lb>H. ſed H, eſt centrum gra<lb></lb>uitatis pyramidis ABCG: <lb></lb>igitur K, erit centrum gra <lb></lb>uitatis pyramidis DEFG: <lb></lb>ſed eſt GK, æqualis ip<lb></lb>ſi GH; vtriufque igitur <lb></lb>pyramidis ABCG, DE<lb></lb>FG, ſimul centrum grauitatis erit K; Quod demonſtran<lb></lb>dum erat. </s></p><figure id="id.043.01.060.1.jpg" xlink:href="043/01/060/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis parallelepipedi centrum grauitatis eſt in <lb></lb>medio axis. </s></p><p type="main"> <s>Sit parallelepipedum ABCDEFGH, cuius axis <lb></lb>LM, isque ſectus bifariam in puncto K. </s> <s>Dico K eſse <lb></lb>centrum grauitatis parallelepipedi ABCDEFGH. <lb></lb>iungantur enim diametri AG, BH, CE, DF, quæ <lb></lb>omnes neceſsario tranſibunt per punctum K, & in eo <lb></lb>puncto bifariam diuidentur. </s> <s>Iunctis igitur BD, FH: <lb></lb>quoniam triangulum EFK, ſimile eſt, & æquale trian<lb></lb>gulo CDK, propter latera circa æquales angulos ad <pb xlink:href="043/01/061.jpg" pagenum="53"></pb>verticem æqualia alterum alteri: eademque ratione, & <lb></lb>triangulum E<emph type="italics"></emph>K<emph.end type="italics"></emph.end>H, triangulo BCK: & triangulum FKH, <lb></lb>triangulo BDK; erit pyramis KEFH, ſimilis, & æqua<lb></lb>lis pyramidi KBCD: habent autem tria latera tribus <lb></lb>lateribus homologis, ideſt æ<lb></lb>qualibus, in directum, prout <lb></lb>inter ſe reſpondent, conſtituta; <lb></lb>duarum igitur pyramidum KE <lb></lb>FH, KBCD, ſimul centrum <lb></lb>grauitatis erit K: non aliter <lb></lb>duarum pyramidum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>GFH, <lb></lb>KBDA, ſimul centrum gra<lb></lb>uitatis erit K; totius igitur com <lb></lb>poſiti ex quatuor pyramidibus; <lb></lb>ideſt duabus oppoſitis ABC<lb></lb>DK, EFGHK, centrum gra<lb></lb>uitatis erit idem K. </s> <s>Eadem <lb></lb>ratione tam duarum pyrami<lb></lb><figure id="id.043.01.061.1.jpg" xlink:href="043/01/061/1.jpg"></figure><lb></lb>dum AEHDK, BCGFK, ſimul, quàm duarum AB<lb></lb>FEK, CDHGK, ſimul centrum grauitatis erit K. </s> <s>To<lb></lb>tius igitur parallelepipedi ABCDEFG<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, centrum <lb></lb>grauitatis erit K. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si parallelepipedum in duo parallelepipeda <lb></lb>ſecetur, ſegmenta axis à centris grauitatis totius <lb></lb>parallelepipedi, & partium terminata ex contra<lb></lb>rio parallelepipedi partibus reſpondent. </s></p><pb xlink:href="043/01/062.jpg" pagenum="54"></pb><p type="main"> <s>Si parallelepipedum AB, cuius axis CD, ſectum in <lb></lb>duo parallelepipeda AE, EN, quare & axis CD, in <lb></lb>axes CL, LD, parallelepipedorum AE, EN. </s> <s>Et ſint <lb></lb>centra grauitatis; F, parallelepipedi EN, & G, paral<lb></lb>lelepipedi AE, & H, parallelepipedi AB, in medio cu<lb></lb>iuſque axis ex antecedenti. </s> <s>Dico eſse FH, ad HG, <lb></lb>vt parallelepipedum AE, ad EN, parallelepipedum. <lb></lb></s> <s>Iungantur enim diametri baſium oppoſitarum, quæ per <lb></lb>puncta axium D, L, G, tranſibunt, ADM, KLE, <lb></lb>NCB; iamque parallelogramma <lb></lb>erunt AB, AE, EN, DB, DE, <lb></lb>EC, propter eas, quæ parallelas <lb></lb>iungunt, & æquales: quorum bi<lb></lb>na latera oppoſita ſecta erunt bi<lb></lb>fariam in punctis C, L, D, per <lb></lb>definitionem axis: punctum igitur <lb></lb>F, in medio rectæ CL, oppoſi<lb></lb>torum laterum bipartitorum ſectio<lb></lb>nes coniungentis, erit parallelo<lb></lb>grammi EN, centrum grauitatis. <lb></lb></s> <s>Eadem ratione & parallelogram<lb></lb><figure id="id.043.01.062.1.jpg" xlink:href="043/01/062/1.jpg"></figure><lb></lb>mi AE, centrum grauitatis erit G, & H, parallelogram <lb></lb>mi AB. </s> <s>Vt igitur parallelogrammum AE, ad paralle<lb></lb>logrammum EN, hoc eſt, vt baſis ME, ad baſim EB; <lb></lb>hoc eſt, vt parallelogrammum MO, ad parallelogram<lb></lb>mum OB: hoc eſt, vt parallelepipedum AE, ad paral<lb></lb>lelepipedum EN: ita erit FH, ad HG. </s> <s>Quod de<lb></lb>monſtrandum erat. </s></p><pb xlink:href="043/01/063.jpg" pagenum="55"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSIT'IO XXVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Solida grauia æquiponderant à longitudini<lb></lb>bus ex contraria parte reſpondentibus. </s></p><p type="main"> <s>Sint ſolida grauia A, & B, quorum centra grauitatis <lb></lb>ſint A, B, ſecundum quæ ſuſpenſa intelligantur A, in <lb></lb>puncto C, & B, in puncto D, cuiuslibet rectæ GH, quæ <lb></lb>ſit ita diuiſa in puncto E, vt ſit DE, ad EC, vt eſt A, <lb></lb>ad B. </s> <s>Dico ſolida A, E, æquiponderare à longitudini<lb></lb>bus DE, EC; hoc eſt vtriuſque ſimul centrum grauita<lb></lb>tis eſse E. </s> <s>Nam ſi A, B, ſint æqualia, manifeſtum eſt <lb></lb>propoſitum: ſi au<lb></lb>tem inæqualia, eſto <lb></lb>maius A: maior igi <lb></lb>tur erit DE, quam <lb></lb>EC. abſcindatur <lb></lb>DF, æqualis EC: <lb></lb>erit igitur DE, æ<lb></lb>qualis GF: & CD, <lb></lb>vtrin que producta, <lb></lb>ponatur DH, æ<lb></lb>qualis DF: & CG, <lb></lb>ipſi CF. & circa <lb></lb>axim, & <expan abbr="altitudinẽ">altitudinem</expan> <lb></lb>GH, eſto paralle<lb></lb>lepipedum KL, æ<lb></lb>quale duobus ſo<lb></lb><figure id="id.043.01.063.1.jpg" xlink:href="043/01/063/1.jpg"></figure><lb></lb>lidis A, B, ſimul & parallelepipedum KL, ſecetur plano <lb></lb>per punctum F, oppoſitis planis parallelo, in duo paral<lb></lb>lelepipeda KN, ML. </s> <s>Quoniam igitur eſt vt GF, ad <lb></lb>FH, ita parallelepipedum KN, ad parallelepipedum <pb xlink:href="043/01/064.jpg" pagenum="56"></pb>ML, ſed vt GF, ad FH, ita eſt CF, ad FD, hoc eſt DE, ad <lb></lb>EC, hoc eſt ſolidum A, ad ſolidum B; erit vt parallelepipe<lb></lb>dum KN, ad parallelepipedum ML, ita ſolidum A, ad ſoli<lb></lb>dum B. componendo igitur, & permutando, vt parallelepi<lb></lb>pedum KL, ad duo ſolida A, B, ſimul, ita parallelepi<lb></lb>pedum ML, ad ſolidum B: & reliquum ad reliquum: ſed <lb></lb>parallelepipedum KL, æquale eſt duobus ſolidis A, B, ſi<lb></lb>mul: parallelepipedum igitur KN, ſolido A, & paralle<lb></lb>lepipedum ML, ſolido B, æquale erit. </s> <s>Rurſus, quo<lb></lb>niam eſt vt GF, ad <lb></lb>ad FH, ita CF, ad <lb></lb>FD; hoc eſt DE, <lb></lb>ad EC: ſed vt GF, <lb></lb>ad FH, ita eſt <expan abbr="pa-rallelepipedũ">pa<lb></lb>rallelepipedum</expan> KN, <lb></lb>ad <expan abbr="parallelepipedũ">parallelepipedum</expan> <lb></lb>ML; erit vt DE, <lb></lb>ad EC, ita paralle <lb></lb>lepipedum KN, ad <lb></lb>parallelepipedum <lb></lb>ML; ſed C eſt pa<lb></lb>rallelepipedi KN, <lb></lb>& D, parallelepipe <lb></lb>di ML, centrum <lb></lb>grauitatis; totius igi <lb></lb><figure id="id.043.01.064.1.jpg" xlink:href="043/01/064/1.jpg"></figure><lb></lb>tur parallelepipedi KL, centrum grauitatis erit E. </s> <s>Igi<lb></lb>tur ſolido A, poſito ad punctum G, ſecundum centrum <lb></lb>grauitatis A, & ſolidum B, ad punctum D, ſecundum <lb></lb>centrum grauitatis B, quorum A, eſt æquale parallele<lb></lb>pipedo KN, & B, parallelepipedo ML; ab ijſdem lon<lb></lb>gitudinibus DE, EC, æquiponderabunt; eritque com<lb></lb>poſiti ex vtroque ſolido A, B, centrum grauitatis E. </s> <s>Quod <lb></lb>demonſtrandum erat. </s></p><p type="main"> <s>Quod ſi quis à me quærat, cur non hic vtar quinta illa <pb xlink:href="043/01/065.jpg" pagenum="57"></pb>generali primi Archimedis de planis æquiponderantibus, <lb></lb>ſed illud idem propoſitum vna demonſtratione in planis, <lb></lb>altera præſenti in ſolidis demonſtrauerim. </s> <s>Reſpondeo: <lb></lb>quia Propoſitio quarta primi Archimedis, ex qua quinta <lb></lb>neceſſario pendet, habet, ſi quis attendat, aliquas difficul<lb></lb>tates phyſicas, quæ mathematicis rationibus non facile <lb></lb>diſſoluantur: quæ cauſa igitur illum adduxit ad ſimile quid <lb></lb><expan abbr="demonſtrandũ">demonſtrandum</expan> demonſtratione ad illas duas parabolas ap. <lb></lb></s> <s>plicata in ſecundo ſuo libro planorum æquiponderantium, <lb></lb>quaſi qui quartæ, ac quintæ illi generali non ſatis acquie<lb></lb>ſceret; eadem me compulit ad hoc propoſitum duabus de<lb></lb>monſtrationibus generalibus, altera de planis, altera de ſo<lb></lb>lidis grauibus ſecurius demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Quarumlibet trium magnitudinum eiuſdem <lb></lb>generis centra grauitatis cum centro magnitudi<lb></lb>nis ex ijs compoſitæ ſunt in eodem plano. </s></p><p type="main"> <s>Sint quælibet tres ma<lb></lb>gnitudines eiuſdem gene <lb></lb>ris A, B, C: quarum cen<lb></lb>tra grauitatis A, B, C. </s> <s>Ex <lb></lb>ijs autem compoſitæ ſit <lb></lb>centrum grauitatis E. </s> <s>Di <lb></lb>co quatuor puncta A, B, <lb></lb>C, E, eſſe in eodem pla<lb></lb>no. </s> <s>Iungantur enim re<lb></lb>ctæ AB, BC, CA: & vt <lb></lb>eſt A, ad C, ita ſit CD, <lb></lb>ad DA, & BD, iungatur: <lb></lb><expan abbr="punctũ">punctum</expan> igitur D, erit cen<lb></lb><figure id="id.043.01.065.1.jpg" xlink:href="043/01/065/1.jpg"></figure><pb xlink:href="043/01/066.jpg" pagenum="58"></pb>trum grauitatis duarum magnitudinum A, C, ſimul. <lb></lb></s> <s>Rurſus quoniam recta BD, coniungit duo centra gra<lb></lb>uitatis duarum magnitu<lb></lb>dinum B ſcilicet, & AC, <lb></lb>erit compoſitæ ACB, in <lb></lb>recta BD, centrum graui <lb></lb>tatis: eſt autem illud E. <lb></lb></s> <s>Quoniam igitur in quo <lb></lb>plano eſt recta BD, in <lb></lb>eodem ſunt duo puncta <lb></lb>B, E, in quo autem pla<lb></lb>no eſt recta BD, in eo<lb></lb>dem eſt recta AC, & <lb></lb>puncta A, C; in quo igi<lb></lb>tur plano ſunt puncta A, <lb></lb>C, in eodem erunt pun<lb></lb>cta B, E; quatuor igitur puncta A, B, C, E, erunt in eodem <lb></lb>plano; Quod demonſtr andum erat. </s></p><figure id="id.043.01.066.1.jpg" xlink:href="043/01/066/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si à cuiuslibet trianguli centro, & tribus an<lb></lb>gulis quatuor rectæ inter ſe parallelæ plano trian <lb></lb>guli inſiſtant: tres autem magnitudines æquales <lb></lb>habeant centra grauitatis in ijs tribus, quæ ad <lb></lb>angulos; trium magnitudinum ſimul centrum <lb></lb>grauitatis erit in ea, quæ ad trianguli centrum <lb></lb>terminatur. </s></p><p type="main"> <s>Sit triangulum ABC, cuius centrum N, à tribus au<lb></lb>tem angulis A, B, C, & centro N, inſiſtant plano trian-<pb xlink:href="043/01/067.jpg" pagenum="59"></pb>guli ABC, quatuor rectæ inter ſe parallelæ AD, BE, <lb></lb>CF, NM, tres autem magnitudines æquales habeant cen <lb></lb>tra grauitatis G, H, K, in tribus AD, BE, CF. </s> <s>Di<lb></lb>co trium magnitudinum ſimul, quarum centra grauitatis <lb></lb>G, H, K, eſſe in linea NM. </s> <s>Iungantur enim rectæ GH, <lb></lb>H<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, GK, BNP; & per punctum P, recta PL, ipſi MN, <lb></lb>parallela, & iungatur LH. </s> <s>Quoniam igitur rectæ BP, LH, <lb></lb>iungunt duas parallelas LP, BH; erunt quatuor rectæ BH, <lb></lb>LP, BP, LH, in eodem plano. </s> <s>Et <expan abbr="quoniã">quoniam</expan> planum quadran <lb></lb>guli PH, ſecat planum trianguli ABC, à communi autem <lb></lb>ſectione BP, ſurgunt <lb></lb>duæ parallelæ PL, MN; <lb></lb>quarum PL, eſt in pla<lb></lb>no quadranguli PH, <lb></lb>erit etiam MN, in eo<lb></lb>dem plano quadranguli <lb></lb>PH: & ſecabit LH. ſe<lb></lb>cet in puncto O: qùare <lb></lb>vt LO, ad OH, ita erit <lb></lb>PN, ad NB, propter <lb></lb>parallelas: ſed PN, eſt <lb></lb>dimidia ipſius NB; er<lb></lb>go & LO, eſt dimidia ip <lb></lb>ſius OH. </s> <s>Eadem ratio<lb></lb>ne, quoniam AP, æqua<lb></lb><figure id="id.043.01.067.1.jpg" xlink:href="043/01/067/1.jpg"></figure><lb></lb>lis eſt PC, erit & GL, æqualis LK. </s> <s>Duarum igitur <lb></lb>magnitudinum G, K, ſimul centrum grauitatis erit L: ſed <lb></lb>reliquæ magnitudinis, quæ ad H, eſt centrum grauitatis <lb></lb>H; & vt compoſitum ex duabus magnitudinibus G, <lb></lb>K, ad magnitudinem H, ita ex contraria parte eſt HO, <lb></lb>ad OL; Trium igitur magnitudinum G, H, K, ſimul cen<lb></lb>trum grauitatis erit O, & in linea MN. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><pb xlink:href="043/01/068.jpg" pagenum="60"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis octaedri idem eſt centrum grauitatis, <lb></lb>& figuræ. </s></p><p type="main"> <s>Eſto octaedrum ABCDEF, cuius centrum G. </s> <s>Di<lb></lb>co G, eſse centrum grauitatis octaedri ABCDEF. <lb></lb></s> <s>Ductis enim axibus AC, BD, EF, communis eorum <lb></lb>ſectio erit centrum G, in quo axes bifariam ſecabuntur: <lb></lb>omnium autem angulorum, qui ad G, bini qui que ad <lb></lb>verticem ſunt æquales, qui æqualibus altera alteri rectis <lb></lb>continentur; ſimilia igi<lb></lb>tur, & æqualia erunt trian <lb></lb>gula, nimirum EBG, <lb></lb>GDF, & ECG, ipſi <lb></lb>GFA, & BCG, ipſi <lb></lb>GDA: igitur & BCE, <lb></lb>ipſi ADF; pyramis igi<lb></lb>tur EBCG, ſimilis, & <lb></lb>æqualis eſt pyramidi A <lb></lb>DFG, quarum latera ho <lb></lb>mologa ſunt indirectum <lb></lb>inter ſe conſtituta; dua<lb></lb>rum igitur pyramidum <lb></lb><figure id="id.043.01.068.1.jpg" xlink:href="043/01/068/1.jpg"></figure><lb></lb>EBCG, ADFG, ſimul centrum grauitatis erit G. <lb></lb></s> <s>Eadem ratione ſex reliquarum pyramidum binis quibuſ<lb></lb>que oppoſitis ſimul ſumptis centrum grauitatis erit G. <lb></lb></s> <s>Totius igitur octaedri ABCDEF, centrum grauitatis <lb></lb>erit G. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/069.jpg" pagenum="61"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis pyramidis triangulam baſim habentis <lb></lb>idem eſt centrum grauitatis, & figuræ. </s></p><p type="main"> <s>Sit pyramis ABCD, cuius baſis triangulum ABC, <lb></lb>centrum autem E. </s> <s>Dico E, eſſe centrum grauitatis pyra<lb></lb>midis ABCD. </s> <s>Secta enim ABCD, pyramide in quatuor <lb></lb>pyramides, ſimiles, & æquales inter ſe, & toti pyramidi <lb></lb>ABCD, & vnum octaedrum, ſint eæ pyramides DKLM, <lb></lb>MGCH, LBGF, <lb></lb>AKFH. </s> <s>Octaedrum <lb></lb>autem FGHKLM, <lb></lb>quod dimidium erit <lb></lb>pyramidis ABCD, & <lb></lb>ſint axes pyramidum <lb></lb>DSN, DS, KO, LP, <lb></lb>MQ: & ARG, iunga <lb></lb>tur. </s> <s>Quoniam igitur <lb></lb>FH, eſt parallela ipſi <lb></lb>BC, & ſecta eſt BC, <lb></lb>bifariam in puncto G, <lb></lb><expan abbr="trãſibit">tranſibit</expan> recta AG, per <lb></lb>centra <expan abbr="triangulorũ">triangulorum</expan> O, <lb></lb>& N, ad quæ axes KO, <lb></lb><figure id="id.043.01.069.1.jpg" xlink:href="043/01/069/1.jpg"></figure><lb></lb>DN, terminantur; manifeſtum hoc eſt ex ſuperioribus: <lb></lb>eritque dupla AO, ipſius OR, nec non AN, dupla ipſius <lb></lb>NG, componendo igitur erit vt AG, ad GN, ita AR, <lb></lb>ad RO, & permutando, vt AG, ad AR, ita GN, ad <lb></lb>RO: ſed AG, eſt dupla ipſius AR, quoniam & AB, ip<lb></lb>ſius AF; igitur & GN, erit dupla ipſius RO: ſed & GN, <lb></lb>eſt dupla ipſius NR, nam N, eſt centrum trianguli GFH; <lb></lb>æqualis eſt igitur NR, ipſi RO, atque hinc dupla NO, <pb xlink:href="043/01/070.jpg" pagenum="62"></pb>ipſius OR; ſed & AO erat dupla ipſius OR; æqualis <lb></lb>igitur AO erit ipſi ON. quare vt AK, ad KD, ita erit <lb></lb>AO, ad ON: igitur in triangulo ADN, erit KO, ipſi <lb></lb>DN, parallela. </s> <s>Eadem ratione ſi iungerentur rectæ BH, <lb></lb>CF oſtenderemus & duos reliquos axes LP, MQ, eſ<lb></lb>ſe axi DN parallelos: quatuor autem prædicti axes in<lb></lb>ſiſtunt plano trianguli KLM, ita vt DN tranſeat per <lb></lb>centrum S: reliqui autem KO, LP, MQ, terminentur <lb></lb>ad angulorum vertices K, L, M, trianguli KLM; igi<lb></lb>tur ſi tres æquales magnitudines habeant centra grauita<lb></lb>tis in axibus KO, LP, <lb></lb><expan abbr="Mq;">Mque</expan> compoſiti ex ijs <lb></lb>tribus magnitudinibus <lb></lb>in axe DN erit <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis. </s> <s>Rurſus <lb></lb>quoniam E ponitur <expan abbr="cẽ">cem</expan> <lb></lb><expan abbr="trũ">trum</expan> pyramidis ABCD, <lb></lb>erit idem E centrum <lb></lb>octaedri FGHKLM, <lb></lb>idque in axe DN: eſt <lb></lb>autem idem <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis octaedri, & figu <lb></lb>ræ: centrum igitur E <lb></lb>octaedri FCHKLM <lb></lb>erit in axe DN. </s> <s>Quod <lb></lb><figure id="id.043.01.070.1.jpg" xlink:href="043/01/070/1.jpg"></figure><lb></lb>ſi quatuor reliquæ pyramides dempto prædicto octaedro <lb></lb>ſimiliter diuidantur, ac pyramis ABCD diuiſa fuit, erunt <lb></lb>rurſus in ſingulis quatuor prædictarum pyramidum ſin<lb></lb>gula octaedra centrum grauitatis habentia vnumquodque <lb></lb>in axe ſuæ pyramidis: quæ pyramides cum ſint inter ſe <lb></lb>æquales, earum dimidia octaedr a ipſis inſcripta inter ſe <lb></lb>erunt æqualia: ſunt autem eorum centra grauitatis in axi<lb></lb>bus abſciſsarum pyramidum, DS, KO, LP, MQ <lb></lb>axis autem DS: eſt in axe DN; per ea igitur, quæ de-<pb xlink:href="043/01/071.jpg" pagenum="63"></pb>monſtrauimus trium octaedrorum, quæ ſunt in pyrami<lb></lb>dibus AFHK, FBGL, GHOM ſimul, centrum gra<lb></lb>uitatis erit in axe D<emph type="italics"></emph>K<emph.end type="italics"></emph.end>: ſed & octaedri in pyramide DK<lb></lb>LM, & octaedri FGHKLM centra grauitatis ſunt <lb></lb>in axe DN; omnium igitur quinque octaedrorum, quæ <lb></lb>ſunt in tota pyramide ABCD ſimul centrum grauitatis <lb></lb>eſt in axe DN. </s> <s>Quod ſi rurſus in ſingulis quatuor præ<lb></lb>dictarum pyramidum modo dicta ratione quina octaedra <lb></lb>deſcripta intelligantur, ſimiliter oſtenſum erit quina octa<lb></lb>edra in ſingulis quatuor abſciſſarum pyramidum, velut <lb></lb>quatuor magnitudines, centra grauitatis habere in axibus <lb></lb>quatuor prædictarum pyramidum: ſunt autem hæc qua<lb></lb>tuor compoſita ex quinis octaedris inter ſe æqualia, pro<lb></lb>pter æqualitatem octaedrorum multitudine æqualium, <lb></lb>quæ æqualibus ſunt pyramidibus ipſorum duplis ord ine <lb></lb>diuiſionis inter ſe reſpondentibus inſcripta; igitur vt ante, <lb></lb>quater quinorum octaedrorum ſimul in axe DN erit <lb></lb>centrum grauitatis: ſed & octaedri FGHKLM centrum <lb></lb>grauitatis eſt in axe DN; vnius igitur & viginti octae<lb></lb>drorum in pyramide ABCD exiſtentium ex hac ſecun<lb></lb>da diuiſione, tanquàm vnius magnitudinis in axe DN erit <lb></lb>centrum grauitatis. </s> <s>Ab hoc igitur numero vnius & vi<lb></lb>ginti octaedrorum in pyramide ABCD exiſtentium, ſi<lb></lb>mili diuiſione illius reliquarum quatuor pyramidum primo <lb></lb>abſciſſarum procedentes, & eundem ſemper gyrum, quem <lb></lb>fecimus à quinario repetentes, poterunt eſse in tota AB<lb></lb>CD pyramide tot, quemadmodum diximus, deſcripta, <lb></lb>octaedra, vt eorum numerus ſuperet quemcumque propo<lb></lb>ſitum numerum, & omnium tanquàm vnius magnitudinis <lb></lb>in axe DN, ſit centrum grauitatis. </s> <s>Sic autem facienti, & <lb></lb>reliquarum pyramidum demptis præcedentibus octaedris, <lb></lb>dimidia octaedra ſemper auferenti, tandem relinquen<lb></lb>tur pyramides minores ſimul ſumptæ quantacumque <lb></lb>magnitudine propoſita. </s> <s>Totius igitur pyramidis ABCD <pb xlink:href="043/01/072.jpg" pagenum="64"></pb>in axe DN, erit centrum grauitatis. </s> <s>Eadem ratione in <lb></lb>quolibet reliquorum trium axium, pyramidis ABCD, ip<lb></lb>ſius centrum grauitatis eſse oſtenderemus; communis igi<lb></lb>tur ſectio quatuor axium pyramidis ABCD, quod eſt <lb></lb>ipſius centrum E, erit centrum grauitatis pyramidis AB <lb></lb>CD. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hinc manifeſtum eſt centrum grauitatis pyra<lb></lb>midis triangulam baſim habentis eſſe in eopun<lb></lb>cto, in quo axis ſic diuiditur, vt pars quæ ad ver<lb></lb>icem ſit reliquæ tripla. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Ominis pyramidis baſim pluſquam trilate<lb></lb>ram habentis centrum grauitatis axim ita diui<lb></lb>dit, vt pars, quæ eſt ad verticem ſit tripla re<lb></lb>liquæ. </s></p><p type="main"> <s>Sit pyramis ABCDE, cui vertex E, baſis autem <lb></lb>quadrilatera ABCD, & eſto axis EF, ſegmentum EM, <lb></lb>reliqui MF, triplum. </s> <s>Dico punctum M, eſſe centrum <lb></lb>grauitatis pyramidis ABCDE. </s> <s>Ducta enim AC, ſit <lb></lb>trianguli ABC, centrum grauitatis H, ſicut & K, trian<lb></lb>guli ACD: & iungantur KH, HE, EK: Factaque vt <lb></lb>EM, ad MF, ita EL ad LH, & EN ad N<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, iun<lb></lb>gatur LN. </s> <s>Quoniam igitur EF eſt axis pyramidis <lb></lb>ABCDE, erit baſis ABCD centrum grauitatis F. <pb xlink:href="043/01/073.jpg" pagenum="65"></pb>Rurſus quia puncta K, H, ſunt centra grauitatis triangu<lb></lb>lorum ABC, CDA, erunt EH, EK, axes pyramidum <lb></lb>ABCE, ACDA: quorum EL, eſt tripla ipſius LH, <lb></lb>nec non EN, tripla ipſius EK; pyramidis igitur ABCE, <lb></lb>centrum grauitatis erit L, ſicut & K, pyramidis ACDE. <lb></lb>Rurſus, quoniam totius quadrilateri ABCD, eſt cen<lb></lb>trum grauitatis F, cuius magnitudinis partium triangu<lb></lb>lorum ABC, CDA, centra grauitatis ſunt K, H; recta <lb></lb>KH, à puncto F, ſic <lb></lb>diuiditur, vt ſit HF, ad <lb></lb>FK, vt triangulum <lb></lb>ACD, ad triangulum <lb></lb>ABC, hoc eſt, vt py<lb></lb>ramis ACDE, ad py <lb></lb>ramidem ABCE. ſed <lb></lb>vt HF, ad FK, ita <lb></lb>eſt LM, ad MN; vt <lb></lb>igitur eſt pyramis AC <lb></lb>DE, ad pyramidem <lb></lb>ABCE, ita erit LM, <lb></lb>ad MN. </s> <s>Sed N, eſt <lb></lb>centrum grauitatis py<lb></lb><figure id="id.043.01.073.1.jpg" xlink:href="043/01/073/1.jpg"></figure><lb></lb>ramidis ACDE, & L pyramidis ABCE; punctum <lb></lb>igitur M, erit centrum grauitatis pyramidis ABCDE. <lb></lb></s> <s>Quod ſi pyramis habeat baſim quinquelateram; poſito <lb></lb>rurſus axe totius pyramidis, & baſi ſecta in triangulum, <lb></lb>& quadrilaterum, poſitis vtriuſque proprijs centris graui<lb></lb>tatis, eadem demonſtratione propoſitum concludetur. <lb></lb></s> <s>Quemadmodum ſi baſis ſit ſex laterum, ſecta ea in quinque <lb></lb>laterum, & triangulum, & reliquis vt antea poſitis: & ſic ſem <lb></lb>per deinceps. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><pb xlink:href="043/01/074.jpg" pagenum="66"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis priſmatis triangulam baſim habentis <lb></lb>centrum grauitatis eſt in medio axis. </s></p><p type="main"> <s>Sit priſma ABCDEF, cuius baſes oppoſitæ trian<lb></lb>gula ABC, DEF, axis autem GH, ſectus ſit bifariam <lb></lb>in puncto K. </s> <s>Dico punctum K, eſse priſinatis ABCD <lb></lb>EF, centrum grauitatis. </s> <s>Ducantur enim rectæ FGO, <lb></lb>CHP, PO. </s> <s>Quoniam igitur GH, eſt axis priſmatis <lb></lb>ABCDEF, erit punctum G, centrum grauitatis trian<lb></lb>guli DEF: ſicut & H, trian<lb></lb>guli ABC; vtraque igitur <lb></lb>dupla eſt AG, ipſius GO, <lb></lb>& CH, ipſius PH, ſectæ<lb></lb>que erunt AB, DE, bifa<lb></lb>riam in punctis P, O: pa<lb></lb>rallela igitur, & æqualis eſt <lb></lb>OP, ipſi DA, iamque ipſi <lb></lb>FC. quæ igitur illas con<lb></lb>iungunt CP, FO, æqua<lb></lb>les ſunt, & parallelæ, & pa<lb></lb>rallelogrammum FP. <lb></lb></s> <s>Nunc ſecta OP, bifariam in <lb></lb>puncto N, iungantur GN, <lb></lb>NF, AF, FH, FB, & fa<lb></lb>cta FL, tripla ipſius LH, <lb></lb><figure id="id.043.01.074.1.jpg" xlink:href="043/01/074/1.jpg"></figure><lb></lb>à puncto L, per punctum K, ducatur recta LKMR. <lb></lb></s> <s>Quoniam igitur eſt vt FG, ad GO, ita CH, ad HP, <lb></lb>& parallelogrammum eſt FCPO; parallelogramma <lb></lb>etiam erunt CG, GP, angulus igitur FGH, æqualis <lb></lb>erit angulo NGO, quos circa æquales angulos latera <pb xlink:href="043/01/075.jpg" pagenum="67"></pb>FG, GH, homologa ſunt lateribus GO, ON. nam <lb></lb>dupla eſt FG, ipſius GO, & GH, ipſius ON; angulus <lb></lb>igitur OGN, æqualis erit angulo GFH; parallela igi<lb></lb>tur GN, ipſi FH, & propterſimilitudinem triangulorum <lb></lb>dupla erit FH, ipſius GN. Rurſus, quoniam recta <lb></lb>OP, ſecat latera oppoſita parallelogrammi BD, bifa<lb></lb>riam in punctis O, P, ſecta, & ipſa bifariam in puncto N, <lb></lb>erit punctum N, parallelogrammi BD, centrum graui<lb></lb>tatis, atque ideo axis FN, pyramidis ABDEF. qua <lb></lb>ratione erit quoque axis FH, pyramidis ABCF: ſed <lb></lb>FL, eſt tripla ipſius LH; pyramidis igitur ABCF, cen<lb></lb>trum grauitatis erit L. </s> <s>Rurſus quia eſt vt GK, ad KH, <lb></lb>ita GR, ad LH, propter ſimilitudinem triangulorum, <lb></lb>erit æqualis GR, ipſi LH: ſed eſt FH, quadrupla ip-, <lb></lb>ſius LH, quadrupla igitur FH, ipſius GR: ſed FH <lb></lb>erat dupla ipſius GN; quadrupla igitur FH, reliquæ <lb></lb>NR, ac proinde GR, RN, æquales erunt: recta igitur <lb></lb>FL, tripla erit vtriuſque ipſarum GR, RN, ſed vt FL, <lb></lb>ad NR, ita eſt FM, ad MN, propter ſimilitudinem trian <lb></lb>gulorum; recta igitur FM, erit ipſius MN, tripla, ſicut <lb></lb>& LM, ipſius MR: ſed quia KH, eſt æqualis GK, <lb></lb>erit & LK, æqualis RK; propter ſimilitudinem trian<lb></lb>gulorum; cum igitur LK, ſit tripla ipſius MR, erit LK, <lb></lb>ipſius KM, dupla; vt igitur eſt pyramis ABEDF, ad <lb></lb>pyramidem ABCF, ita erit LK, ad KM; eſt autem M, <lb></lb>centrum grauitatis pyramidis ABED, ſicut & L, pyrami<lb></lb>dis ABCF; totius igitur priſmatis ABCDEF, centrum <lb></lb>grauitatis erit K. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/076.jpg" pagenum="68"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis priſmatis baſim pluſquam trilateram <lb></lb>habentis centrum grauitatis eſt in medio axis. </s></p><p type="main"> <s>Sit priſma ABCDEFGH, baſim habens quadrila<lb></lb>teram ABCD: axis autem <emph type="italics"></emph>K<emph.end type="italics"></emph.end>L, bifariam ſectus in pun<lb></lb>cto M. </s> <s>Dico punctum M, eſse centrum grauitatis priſ<lb></lb>matis ABCDEFGH. </s> <s>Iungantur enim rectæ BD, FH, <lb></lb>vt parallelogrammum ſit BH, ſectumque totum priſma <lb></lb>in duo priſmata, quorum ba<lb></lb>ſes ſunt triangula, in quæ ſecta <lb></lb>ſunt quadrilatera AC, EG, <lb></lb>ſint autem axes duorum priſ<lb></lb>matum triangulas baſes ha<lb></lb>bentium NO, <expan abbr="Pq.">Pque</expan> Erunt <lb></lb>igitur centra grauitatis O, tri<lb></lb>anguli ABD, & L, quadri<lb></lb>lateri AC, & Q, trianguli <lb></lb>BCD, itemque N, trianguli <lb></lb>EFH, & K, quadrilateri EG, <lb></lb>& P, trianguli FGH: iun<lb></lb>ctæ igitur OQ, NP, per pun <lb></lb><figure id="id.043.01.076.1.jpg" xlink:href="043/01/076/1.jpg"></figure><lb></lb>cta L, K, tranſibunt: cumque tres prædicti axes ſint <lb></lb>lateribus priſmatis, atque ideo inter ſe quoque paralleli; <lb></lb>parallelogramma erunt OP, NL, LP. ducta igitur per <lb></lb>punctum M, ipſi OQ, vel NP, parallela RS, erit vt <lb></lb>NK, ad KP, ita RM, ad MS: & vt KM, ad ML, ita <lb></lb>NR, ad RO, & PS, ad SQ: ſed KM, eſt æqualis ML; <lb></lb>igitur & KR, ipſi RO, & PS, ipſi SQ, æqualis erit: ſunt <lb></lb>autem hæ ſegmenta axium NO, <expan abbr="Pq;">Pque</expan> punctum igitur <lb></lb>R, eſt centrum grauitatis priſmatis ABDEFH: & per <pb xlink:href="043/01/077.jpg" pagenum="69"></pb>punctum S, priſmatis BCDFGH. </s> <s>Quoniam igitur <lb></lb>quadrilateri EG, eſt centrum grauitatis K, cuius duorum <lb></lb>triangulorum centra grauitatis ſunt P, N; erit vt triangu<lb></lb>lum FGH, ad triangulum EFH, hoc eſt vt priſma BC<lb></lb>DFGH, ad priſma ABDEFH, ita NK, ad KP, hoc <lb></lb>eſt RM, ad MS; cum igitur ſit R, centrum grauitatis <lb></lb>priſmatis ABDEFH: ſicut & S, priſmatis BCDFGH; <lb></lb>totius priſmatis ABCDEFGH, centrum grauitatis erit <lb></lb>M. </s> <s>Quod ſi priſma baſim habeat quinquelateram; ab<lb></lb>ſciſso rurſus priſmate vno triangulam baſim habente, <lb></lb>ſumptiſque axibus priſinatum, quorum alterum habebit <lb></lb>baſim quadrilateram, eadem demonſtratione propoſitum <lb></lb>concluderemus, & ſic deinceps in aliis. </s> <s>Manifeſtum eſt <lb></lb>igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti pyramidis triangulam baſim <lb></lb>ha bentis centrum grauitatis eſt in axe, primum <lb></lb>ita diuiſo, vt ſegmentum attingens minorem <lb></lb>baſim ſit ad reliquum, vt duplum vnius laterum <lb></lb>maioris baſis vna cum latere homologo mino<lb></lb>ris, ad duplum prædicti lateris minoris baſis, <lb></lb>vna cum latere homologo maioris. </s> <s>Deinde <lb></lb>à puncto ſectionis abſciſsa quarta parte ſeg<lb></lb>menti, quod maiorem baſim attingit, & à pun<lb></lb>cto, in quo ad minorem baſim axis termina<lb></lb>tur ſumpta item quarta parte totius axis; in <lb></lb>eo puncto, in quo ſegmentum axis duabus po<lb></lb>ſterioribus ſectionibus finitum ſic diuiditur, vt <pb xlink:href="043/01/078.jpg" pagenum="70"></pb>ſegmentum eius maiori baſi propinquius ſit ad to<lb></lb>tum prædictum interiectum ſegmentum, vt tertia <lb></lb>proportionalis minor ad duo latera homologa ba<lb></lb>ſium oppoſitarum, ad compoſitam ex his tribus <lb></lb>deinceps proportionalibus. </s></p><p type="main"> <s>Sit pyramidis fruſtum, cuius baſes oppoſitæ, & parallelæ, <lb></lb>maior triangulum ABC, minor autem triangulum DEF, <lb></lb>axis autem GH. triangulorum autem ABC, DEF, quæ <lb></lb>inter ſe ſimilia eſse neceſse eſt, ſint duo latera homologa <lb></lb>BC, EF: & vt eſt BC, ad EF, ita ſit EF, ad X: vt autem eſt <lb></lb>duplum lateris BC, vna cum latere EF, ad duplum lateris <lb></lb>EF, vna cum la <lb></lb>tere BC, ita ſit <lb></lb>HN, ad NG, <lb></lb>& NO, pars quar <lb></lb>ta ipſius NG, & <lb></lb>HS, pars quar<lb></lb>ta ipſius GH; ip <lb></lb>ſius autem SO, <lb></lb>ſit VO, ad OS, <lb></lb>vt eſt X, ad com<lb></lb>poſitam ex tri<lb></lb>bus BC, EF, X. <lb></lb></s> <s>Dico punctum V <lb></lb>(quod cadet ne<lb></lb>ceſsario infra <lb></lb><figure id="id.043.01.078.1.jpg" xlink:href="043/01/078/1.jpg"></figure><lb></lb>punctum N, quanquam hoc ad demonſtrationem nihil re<lb></lb>fert) eſse centrum grauitatis fruſti ABCDEF. </s> <s>Ducta <lb></lb>enim recta AGL; quoniam GH, eſt axis fruſti ABCD <lb></lb>EF, & punctum G, centrum grauitatis trianguli ABC, <lb></lb>erit punctum L, in medio baſis BC: ſecto igitur etiam la<lb></lb>tere EF, bifariam in puncto K, iungantur LK, <emph type="italics"></emph>K<emph.end type="italics"></emph.end>H: & vt <pb xlink:href="043/01/079.jpg" pagenum="71"></pb>vt eſt HN, ad NG, ita fiat KM, ad ML, & GM, iun<lb></lb>gatur: & vt eſt GO, ad ON, ita fiat GP, ad PM, & iun <lb></lb>gantur MN, OP, FG, GD, GE. </s> <s>Quoniam igitur re <lb></lb>cta KL, ſecat trapezij BCFE, latera parallela bifariam <lb></lb>in punctis K,L, & eſt vt HN, ad NG, hoc eſt vt duplum <lb></lb>lateris BC, vna cum latere EF, ad duplum lateris EF, vna <lb></lb>cum latere BC, ita KM, ad ML; erit punctum M, cen<lb></lb>trum grauitatis trapezij BCFE, & pyramidis GBCFE, <lb></lb>axis GM. </s> <s>Et quoniam vt GO, ad ON, ita eſt GP, ad <lb></lb>PM, atque ideo GP, tripla ipſius PM, erit punctum P, <lb></lb>centrum grauitatis pyramidis GBCFE, atque ideo in <lb></lb>linea OP. </s> <s>Rurſus quoniam angulus ACB; æqualis eſt <lb></lb>angulo DFK: & vt AC, ad CK, ita eſt DF, ad FK: <lb></lb>eſt autem DF, parallela ipſi AC, & FK, ipſi CL; erit <lb></lb>reliqua DK, reliquæ AL, parallela; vnum igitur planum <lb></lb>eſt, ADKL, in quo iacet triangulum GMN; cum igitur <lb></lb>ſit parallela KH, ipſi GL, vtque HN, ad NG, ita <lb></lb><emph type="italics"></emph>K<emph.end type="italics"></emph.end>M, ad ML; erit MN, ipſi LG, parallela: ſed OP, eſt <lb></lb>parallela ipſi MN; ſecant enim latera trianguli GMN, <lb></lb>in eaſdem rationes; igitur OP, erit LG, parallela. </s> <s>Simi<lb></lb>liter ex puncto O, ad axes duarum pyramidum GABED, <lb></lb>GACFD, duæ aliæ rectæ lineæ ducerentur, quas & cen<lb></lb>tra grauitatis pyramidum habere, & parallelas rectis GQ, <lb></lb>GR, alteram alteri eſse oſtenderemus, ſicut oſtendimus <lb></lb>OP, habentem centrum grauitatis pyramidis GBCFE, <lb></lb>ipſi GL, parallelam; ſed tres rectæ GL, GQ, GR, ſunt <lb></lb>in eodem plano trianguli nimirum ABC; tres igitur præ<lb></lb>dictæ parallelæ, quæ ex puncto O, atque ideo trium præ<lb></lb>dictarum pyramidum centra grauitatis erunt in eodem pla<lb></lb>no, per punctum O, & trianguli ABC, parallelo. </s> <s>Quo<lb></lb>niam igitur fruſti ABCDE, centrum grauitatis eſt in axe <lb></lb>GH; (manifeſtum hoc autem ex duobus centris grauitatis <lb></lb>pyramidis, cuius eſt prædictum fruſtum, & ablatæ, quæ <lb></lb>centra grauitatis ſunt in axe, cuius ſegmentum eſt axis <pb xlink:href="043/01/080.jpg" pagenum="72"></pb>GH) erit eiuſdem fruſti ABCDEF, centrum grauitatis <lb></lb>O. </s> <s>Rurſus quoniam vt tres deinceps proportionales BC, <lb></lb>EF, X, ſimul ad BC, ita eſt fruſtum ABCDEF, ad py<lb></lb>ramidem; ſi deſcribatur ABCH: ſed vt triangulum ABC, <lb></lb>ad ſimile triangulum EDF, hoc eſt vt BC, ad X, ita eſt <lb></lb>pyramis ABCH, ad pyramidem GDEF; erit ex æqua<lb></lb>li, vt tres lineæ <lb></lb>BC, EF, X, ſi<lb></lb>mul ad X, ita fru <lb></lb>ſtum ABCDEF, <lb></lb>ad pyramidem <lb></lb>GDEF: & con<lb></lb>uertendo, vt X, <lb></lb>ad compoſitam <lb></lb>ex BC, EF, X, <lb></lb>hoc eſt vt VO, <lb></lb>ad OS, ita pyra <lb></lb>mis GDEF, ad <lb></lb>fruſtum ABC<lb></lb>DEF; & diui<lb></lb>dendo, vt pyra<lb></lb><figure id="id.043.01.080.1.jpg" xlink:href="043/01/080/1.jpg"></figure><lb></lb>mis GDEF, ad reliquas tres pyramides fruſti, ita OV, <lb></lb>ad VS; ſed S, eſt centrum grauitatis pyramidis GDEF, <lb></lb>& O, trium reliquarum; fruſti igitur ABCDEF, cen<lb></lb>trum grauitatis erit V. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti pyramidis baſim pluſquam trila<lb></lb>teram habentis centrum grauitatis eſt punctum <lb></lb>illud, in quo axis ſic diuiditur, vt axis fruſti pyra<lb></lb>midis triangulam baſim habentis diuiditur ab <lb></lb>ipſius centro grauitatis. </s></p><pb xlink:href="043/01/081.jpg" pagenum="73"></pb><p type="main"> <s>Sit pyramidis quadrilateram baſim habentis fruſtum <lb></lb>ABCDEFGH, cuius axis KL, atque in ipſo centrum <lb></lb>grauitatis O. </s> <s>Dico axim KL, ſectum eſse in puncto O, <lb></lb>vt propoſuimus. </s> <s>Ductis enim AC, EG, quæ ſimilium <lb></lb>ſectionum angulos æquales ſubtendant B, F, qui late<lb></lb>ribus homologis continentur, fruſta erunt pyramidum <lb></lb>triangulas baſes habentium AFG, AGH: ſit autem fru<lb></lb>ſti AFG, axis <lb></lb>TP, & in eo eiuſ <lb></lb>dem fruſti cen<lb></lb>trum grauitatis <lb></lb>M, & fruſti AG <lb></lb>H, axis VQ, & <lb></lb>in eo centrum <lb></lb>grauitatis N, & <lb></lb>iungantur TV, <lb></lb>MN, <expan abbr="Pq.">Pque</expan> Quo <lb></lb>niam igitur eſt <lb></lb>pyramidis fru<lb></lb>ſtum, quod pro<lb></lb>ponitur; omnia <lb></lb><figure id="id.043.01.081.1.jpg" xlink:href="043/01/081/1.jpg"></figure><lb></lb>cius producta latera concurrent in vno puncto, qui eſt pyra<lb></lb>midis vertex: fruſta igitur, in quæ diuiſum eſt fruſtum pro<lb></lb>poſitum earum ſunt pyramidum, quæ verticem habent <lb></lb>communem cum pyramide, cuius eſt fruſtum propoſitum: <lb></lb>tres igitur talium fruſtorum axes, vt pote ſegmenta axium <lb></lb>trium prædictarum pyramidum in communi illo vertice <lb></lb>concurrent: quilibet igitur duo trium prædictorum axium <lb></lb>KL, TP, VQ, erunt in eodem plano: TP, igitur, & <lb></lb>VQ, ſunt in eodem plano. </s> <s>Eadem autem ratione, qua <lb></lb>vtebamur de priſmate K, centrum grauitatis K, baſis <lb></lb>EH, eſt in linea TV, & L, baſis BD, centrum grauita<lb></lb>tis eſt in linea <expan abbr="Pq;">Pque</expan> reliquæ igitur KL, MN, erunt in eo<lb></lb>dem plano trapezij PTVQ, ſeque mutuo ſecabunt: cum <pb xlink:href="043/01/082.jpg" pagenum="74"></pb>igitur M, N, ſint centra grauitatis propoſiti priſmatis par <lb></lb>tium priſmatum AFG, AGH, atque obid O, totius priſ<lb></lb>matis AFGH, in linea MN, centrum grauitatis; per pun <lb></lb>ctum O, recta MN, tranſibit. </s> <s>Et quoniam planum tra<lb></lb>pezij PV, ſecatur duobus planis parallelis, erunt TV, PQ, <lb></lb>fectiones parallelæ. </s> <s>His demonſtratis, fiat rurſus vt AB, <lb></lb>bis vna cum EF, ad EF, bis vna cum AB, ita TY, ad <lb></lb>YP: & ſumatur T<foreign lang="grc">ω</foreign>, pars quarta ipſius TP, & YZ, pars <lb></lb>quarta ipſius PY, & ad axim KL, ducantur ipſis TV, <lb></lb>PQ, parallelæ <lb></lb><foreign lang="grc">ω</foreign>S, YR, ZX, <lb></lb>quæ rectas TP, <lb></lb>KL, ſecabunt in <lb></lb><expan abbr="eaſdẽ">eaſdem</expan> rationes: <lb></lb>vt igitur TY, ad <lb></lb><foreign lang="grc">Υ</foreign>P, hoc eſt vt <lb></lb>AB, bis vna cum <lb></lb>EF, ad EF bis <lb></lb>vna cum AB, ita <lb></lb>erit <emph type="italics"></emph>K<emph.end type="italics"></emph.end>R, ad RL, <lb></lb>eritque KS, pars <lb></lb>quarta ipſius K <lb></lb>L, qualis & R <lb></lb><figure id="id.043.01.082.1.jpg" xlink:href="043/01/082/1.jpg"></figure><lb></lb>X, ipſius RL. </s> <s>Et quoniam M, eſt centrum grauitatis fru<lb></lb>ſti AFG; manifeſtum eſt ex tribus prædictis axis TP, ſe<lb></lb>ctionibus <foreign lang="grc">Υ, ω</foreign>, Z, eſse MZ, ad Z<foreign lang="grc">ω</foreign>, hoc eſt OX, ad XS, <lb></lb>vt eſt 6 ad compoſitam ex tribus deinceps proportionalibus <lb></lb>AB, EF, 6; Fruſti igitur ABCDEFGH, centrum gra<lb></lb>uitatis O, axim KL, ita diuidit, vt propoſuimus. </s> <s>Quod <lb></lb>ſi fruſtum propoſitum ſit pyramidis baſim habentis quin<lb></lb>quelateram, & quotcumque plurium deinceps fuerit la<lb></lb>terum, eadem demonſtratione ſemper deinceps, vt in priſ<lb></lb>mate monuimus, propoſitum concluderemus. </s></p><pb xlink:href="043/01/083.jpg" pagenum="75"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Dodecaedri, & icoſaedri idem eſt centrum gra<lb></lb>uitatis, & figuræ. </s></p><p type="main"> <s>Nam huiuſmodi figuras habere axes, qui omnes ſe ſe <lb></lb>bifariam ſecant; (tale autem ſectionis punctum centrum eſt) <lb></lb>conſtat ex talium corporum in ſphæra inſcriptione in de<lb></lb>cimotertio Euclidis Elemento: nec non omnem pyrami<lb></lb>dem, cuius vertex eſt dodecaedri, vel octaedri centrum <lb></lb>idem cum centro ſphæræ, vt conſtat ex ijſdem Euclidis in<lb></lb>ſcriptionibus; baſis autem triangulum æquilaterum, vel <lb></lb>pentagonum, vna ex baſibus corporum prædictorum, ha<lb></lb>bere pyramidem oppoſitam ſimilem ipſi, & æqualem, cuius <lb></lb>latera eius lateribus homologis ſunt in directum poſita, <lb></lb>baſis autem triangulum, vel pentagonum, quale diximus; <lb></lb>Eadem igitur ratione, qua vſi ſumus ad demonſtrandum <lb></lb>centrum grauitatis, & parallelepipedi, & octaedri, propo<lb></lb>ſitum concluderemus. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Data qualibet figura, cuius termini omnis <lb></lb>cauitas ſit interior, ſi certum in ea punctum talis <lb></lb>cius partis centrum grauitatis eſse poſsit, quæ ab <lb></lb>ca deficiat minori ſpacio quantacumque magnitu <lb></lb>dine propoſita; illud erit totius figuræ centrum <lb></lb>grauitatis. </s></p><pb xlink:href="043/01/084.jpg" pagenum="76"></pb><p type="main"> <s>Eſto figura AB, cuius termini omnis cauitas ſit interior <lb></lb>& certum in ea punctum E, talis partis AB, figuræ qua<lb></lb>lem diximus centrum grauitatis eſse poſsit. </s> <s>Dico pun<lb></lb>ctum E, eſse figuræ AB, centrum grauitatis. </s> <s>Si enim <lb></lb>E, non eſt, erit aliud, eſto F: & iuncta EF producatur, <lb></lb>& ſumatur in illa extra figuræ AB, terminum, quodlibet <lb></lb>punctum G; & vt eſt FE, ad EG, ita ſit alia magnitudo <lb></lb>K, ad figuram AB, & <lb></lb>ex vi hypotheſis ſit pars <lb></lb>quædam CD, figuræ <lb></lb>AB, cuius centrum gra<lb></lb>uitatis E, talis vt abla<lb></lb>ta relinquat AC, minus <lb></lb>magnitudine <emph type="italics"></emph>K.<emph.end type="italics"></emph.end></s><s> Mi<lb></lb>nor igitur proportio erit <lb></lb>AC, ad AB, quàm K, <lb></lb>ad AB, hoc eſt quàm <lb></lb>FE, ad EG; fiat vt <lb></lb>AC, ad AB, ita EF, <lb></lb>ad FGH: ſed F, eſt cen <lb></lb>trum grauitatis totius <lb></lb>AB, & E, vnius par<lb></lb>tis CD; reliquæ igitur <lb></lb><figure id="id.043.01.084.1.jpg" xlink:href="043/01/084/1.jpg"></figure><lb></lb>partis AC, centrum grauitatis erit H, vltra punctum G: ſed <lb></lb>G, cadit extra terminum figuræ AC; multo igitur magis H: <lb></lb>Quod eſt abſurdum. </s> <s>Non igitur aliud punctum à puncto <lb></lb>E; punctum igitur E, figuræ AB, erit centrum grauitatis <lb></lb>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/085.jpg" pagenum="77"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis coni centrum grauitatis axim ita diui<lb></lb>dit, vt ſegmentum ad verticem ſit reliqui triplum. </s></p><p type="main"> <s>Sit conus ABC, cuius vertex B, axis autem BD, cu<lb></lb>ius BE, ſit tripla ipſius ED. </s> <s>Dico punctum E, eſse co<lb></lb>ni ABC, centrum grauitatis. </s> <s>Si enim cono ABC, pyramis <lb></lb>inſcribatur, cuius baſis inſcripta circulo AC, æquilatera ſit, <lb></lb>& æquiangula, eius centrum grauitatis erit idem quod & <lb></lb>figuræ centrum, ſed centrum <lb></lb>talis figuræ circulo inſcriptæ <lb></lb>idem eſt, quod centrum cir<lb></lb>culi, vt colligitur ex demon<lb></lb>ſtrationibus quarti Elemen<lb></lb>torum; inſcriptæ igitur pyra <lb></lb>midis erit axis BD, & cen<lb></lb>trum grauitatis E. talis au<lb></lb>tem ea pyramis inſcribi po<lb></lb>teſt, vt à cono deficiat mino<lb></lb>ri ſpacio quantacumque ma <lb></lb>gnitudine propoſita; igitur <lb></lb>ABC, coni centrum graui<lb></lb>tatis erit E. </s> <s>Quod demonſtrandum erat. </s></p><figure id="id.043.01.085.1.jpg" xlink:href="043/01/085/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti conici centrum grauitatis idem <lb></lb>eſt in axe centro grauitatis fruſti pyramidis baſim <lb></lb>habentis æquilateram, & æquiangul am in ſcriptæ <lb></lb>cono, ab ſciſſi eodem plano, quo coni fruſtum. </s></p><pb xlink:href="043/01/086.jpg" pagenum="78"></pb><p type="main"> <s>Sit coni fruſtum ABCD, cuius axis EF, fruſto autem <lb></lb>ABCD, intelligatur inſcriptum fruſtum pyramidis inſcri<lb></lb>ptæ cono AHD, à quo abſciſsum eſt fruſtum ABCD, <lb></lb>baſim habentis æquilateram, & æquiangulam inſcriptam <lb></lb>circulo AD: quare eius centrum grauitatis, & figuræ erit <lb></lb>punctum F, vt diximus in præcedenti, axis autem FH, ſi<lb></lb>cut etiam pyramidis abſciſsæ vna cum cono BHC, axis <lb></lb>EH, quare & reliqui fruſti pyramidis axis erit EF, igi<lb></lb>tur in EF, ſit fruſti inſcripti fruſto ABCD, centrum gra<lb></lb>uitatis G. </s> <s>Dico punctum G, eſse centrum grauitatis fru<lb></lb>ſti ABCD. </s> <s>Ponatur enim <lb></lb>FL, pars quarta ipſius FH, <lb></lb>necnon EK, pars quarta ip<lb></lb>ſius EH: punctum igitur K, <lb></lb>eſt centrum grauitatis pyra<lb></lb>midis, & coni BHC, ſicut <lb></lb>& punctum L, pyramidis, & <lb></lb>coni AHD. cum igitur fru <lb></lb>ſti pyramidis fruſto ABCD, <lb></lb>inſcripti ſit centrum grauita<lb></lb>tis G; erit vt GL, ad LK, <lb></lb>ita pyramis BHC, ad pyra<lb></lb>midis fruſtum fruſto ABCD, <lb></lb>inſcriptum: ſed vt pyramis <lb></lb>BHC, ad pyramidis fruſtum <lb></lb>fruſto ABCD, inſcriptum, <lb></lb><figure id="id.043.01.086.1.jpg" xlink:href="043/01/086/1.jpg"></figure><lb></lb>ita eſt diuidendo, conus BHC, ad fruſtum ABCD, pro<lb></lb>pter eandem triplicatam communium conis, & pyramidi<lb></lb>bus ſimilibus laterum homologorum proportionem; vt igi<lb></lb>tur GL, ad LK, ita erit conus BHC: ad fruſtum ABCD: <lb></lb>ſed coni BHC, centrum grauitatis erat K, & coni AHD, <lb></lb>centrum grauitatis L; fruſti igitur ABCD, centrum gra<lb></lb>nitatis erit G. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/087.jpg" pagenum="79"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis cylindri centrum grauitatis axim bifa<lb></lb>riam diuidit. </s></p><p type="main"> <s>Sit cylindrus ABCD, cuius axis EF, & ſit ſectus bi<lb></lb>fariam in puncto G. </s> <s>Dico punctum G, eſse centrum <lb></lb>grauitatis cylindri ABCD. </s> <s>Nam ſi cylindro AD, in<lb></lb>ſcriptum intelligatur priſma, <lb></lb>cuius baſes oppoſitæ æquilate<lb></lb>ræ ſint, & æquiangulæ; erunt, <lb></lb>qua ratione ſupra diximus, ea<lb></lb>rum centra figuræ, & grauitatis <lb></lb>E, F; axis igitur inſcripti priſ<lb></lb>matis erit EF: & centrum gra<lb></lb>uitatis G. poteſt autem tale <lb></lb>priſma ſic inſcribi cylindro <lb></lb>ABCD, vt ab illo deficiat <lb></lb>minori ſpacio quantacumque <lb></lb>magnitudine propoſita; cylin<lb></lb>dri igitur ABCD, centrum <lb></lb>grauitatis erit G. </s> <s>Quod demonſtrandum erat. </s></p><figure id="id.043.01.087.1.jpg" xlink:href="043/01/087/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Sphæræ, & ſphæroidis idem eſt centrum gra<lb></lb>uitatis, & figuræ. </s></p><p type="main"> <s>Sit ſphæra, vel ſphæroides ABCD, cuius centrum E, <pb xlink:href="043/01/088.jpg" pagenum="80"></pb>Dico ſphæræ, vel ſphæroidis ABCD, centrum grauitatis <lb></lb>eſse E. </s> <s>Sint enim bini axes ſphæræ, vel ſphæroidis inter <lb></lb>ſe ad rectos angulos; & in ſphæroide ſit maior diameter <lb></lb>BD, minor AC, per binos autem hos axes plana tran<lb></lb>ſeuntia ad eos axes erecta, ſecent ſphæram, vel ſphæroidem. <lb></lb></s> <s>Qua ratione axes dimidij erunt axes hemiſphærij, vel he<lb></lb>miſphæroidis: hemiſphærium autem, & ſphæroidis eſt fi<lb></lb><figure id="id.043.01.088.1.jpg" xlink:href="043/01/088/1.jpg"></figure><lb></lb>gura circa axim in alteram partem deficiens, qualium om<lb></lb>nium figurarum centrum grauitatis eſt in axe; igitur hemi<lb></lb>ſphærij, vel hemiſphæroidis ABCD, centrum grauitatis <lb></lb>eſt in axi BE, ſicut & reliqui ADA, in axi ED; totius <lb></lb>igitur ſphæræ, vel ſphæroidis ABCD centrum grauitatis <lb></lb>eſt in axi BD. </s> <s>Eadem ratione & in axi AC; in communi <lb></lb>igitur ſectione centro E. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s>PRIMI LIBRI FINIS.</s></p><figure id="id.043.01.088.2.jpg" xlink:href="043/01/088/2.jpg"></figure><p type="head"> <pb xlink:href="043/01/089.jpg" pagenum="81"></pb><s>LVCAE <lb></lb>VALERII <lb></lb>DE CENTRO <lb></lb>GRAVITATIS <lb></lb>SOLIDORVM</s></p><p type="head"> <s><emph type="italics"></emph>LIBER SECVNDVS.<emph.end type="italics"></emph.end></s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO I.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si duæ magnitudines vnà maio<lb></lb>res, vel minores prima, & ter <lb></lb>tia minori exceſſu, vel defe<lb></lb>ctu <expan abbr="quantacumq;">quantacumque</expan> magnitudi <lb></lb>ne propoſita eiuſdem generis <lb></lb>cum illa, ad quam refertur, <lb></lb>eandem <expan abbr="proportionẽ">proportionem</expan> habue<lb></lb>rint, maior vel minor prima ad ſecundam, & vnà <lb></lb>maior, vel minor tertia ad quartam; erit vt prima <lb></lb>ad ſecundam, ita tertia ad quartam. </s></p><pb xlink:href="043/01/089.jpg" pagenum="2"></pb><p type="main"> <s>Sint quatuor magnitudines A prima, B ſecunda, C ter <lb></lb>tia, & D quarta: quantacumque autem magnitudine propo <lb></lb>ſita, ex infinitìs quæ proponi poſſunt eiuſdem generis cum <lb></lb>A, C, vel vna tantum, ſi AC ſint eiuſdem generis: vel <lb></lb>vna, & altera; ſi vna vnius, altera ſit alterius generis; ſemper <lb></lb>aliæ duæ magnitudines vnà maiores, quàm AC, minori <lb></lb>exceſsu magnitudine propoſita; eandem habeant proportio <lb></lb>nem, maior quàm A ad B, & maior quàm C ad D. </s> <s>Dico <lb></lb>eſse vt A ad B, ita C ad D. </s> <s>Poſita enim E ad D, vt <lb></lb>A ad B, & F maiori quàm C vtcumque, ſint aliæ duæ ma<lb></lb>gnitudines, G maior quàm A minori exceſsu magnitudine <lb></lb>eiuſdem generis cum A, quam quis voluerit, & H maior <lb></lb>quàm C minori exceſsu quàm <lb></lb>quo F ſuperat C, ideſt, quæ ma<lb></lb>ior ſit quàm C, & minor quàm <lb></lb>F: ſit autem vt G ad B, ita H <lb></lb>ad D. </s> <s>Quoniam igitur F maior <lb></lb>eſt, <34>H, maior erit proportio <lb></lb>ipſius F quàm H ad D, hoc eſt <lb></lb>quàm G ad B. </s> <s>Sed <expan abbr="cũ">cum</expan> G maior <lb></lb>ſit quàm A, maior eſt proportio <lb></lb><figure id="id.043.01.089.1.jpg" xlink:href="043/01/089/1.jpg"></figure><lb></lb>G ad B, quàm A ad B, multo igitur erit maior proportio F <lb></lb>ad D, quàm A ad B. </s> <s>Sed F ponitur maior quàm C, vtcum <lb></lb>que; nulla igitur magnitudo maior quàm C eſt ad D, vt <lb></lb>A ad B: ſed E ad D, eſt vt A ad B; non igitur eſt E ma<lb></lb>ior quàm C; nec maior proportio E ad D, hoc eſt A ad <lb></lb>B, quàm C ad D. </s> <s>Eadem autem ratione nec maior erit <lb></lb>proportio C ad D quàm A ad B, hoc eſt non minor A <lb></lb>ad B, quàm C ad D; eadem igitur proportio A ad B, <lb></lb>quæ C ad D. </s></p><p type="main"> <s>Sed aliæ duæ magnitudines vnà minores quàm A, C <lb></lb>minori defectu quantacumque magnitudine propoſita, <lb></lb>eandem habeant proportionem, minor quàm A ad B, & <lb></lb>minor quàm C, ad D. </s> <s>Dico eſse vt A ad B, ita C ad D. <pb xlink:href="043/01/090.jpg" pagenum="3"></pb>Poſita enim rurſus E ad D, vt A ad B, & F minori quàm <lb></lb>C vtcumque, ſit G minor quam A, minori defectu magni <lb></lb>tudine eiuſdem generis cum A, quam quis voluerit, & H <lb></lb>minor quàm C, & maior quàm F: ſit autem vt G ad B, ita <lb></lb>H ad D. </s> <s>Quoniam igitur F minor eſt quàm H, minor erit <lb></lb>proportio ipſius F <expan abbr="quã">quam</expan> H ad D, <lb></lb>hoc eſt <34>G ad B: ſed cum G ſit <lb></lb>minor <34>A, minor eſt propor<lb></lb>tio G ad B, quàm A ad B; mul <lb></lb>to ergo minor proportio F ad <lb></lb>D, quàm A ad B: ſed F poni <lb></lb>tur minor quàm C vtcumque; <lb></lb>nulla igitur magnitudo minor <lb></lb><figure id="id.043.01.090.1.jpg" xlink:href="043/01/090/1.jpg"></figure><lb></lb>quàm C eſt ad D, vt A ad B: ſed E eſt ad D, vt A ad B: <lb></lb>non igitur eſt E minor quàm C, nec minor proportio E ad <lb></lb>D, hoc eſt A ad B, quàm C ad D. eadem autem ratione <lb></lb>non minor erit proportio C ad D, quàm A ad B; hoc eſt <lb></lb>non maior A ad B, quàm C ad D; vt igitur A ad B, ita <lb></lb>eſt C ad D. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>ALITE R.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Dico eſse vt A ad B, ita C ad <lb></lb>D. </s> <s>Si enim fieri poteſt, ſit minor <lb></lb>proportio A ad B quàm C ad D. <lb></lb>alia igitur aliqua magnitudo G <lb></lb>maior quàm A, eandem habebit <lb></lb>proportionem ad B, quam C ad <lb></lb>D. </s> <s>Sit autem F maior quam C <lb></lb>minori exceſsu magnitudine, <expan abbr="quã">quam</expan> <lb></lb>quis voluerit, & E maior quàm <lb></lb>A, & minor quàm G: vt autem <lb></lb><figure id="id.043.01.090.2.jpg" xlink:href="043/01/090/2.jpg"></figure><lb></lb>E ad B, ita F ad D. </s> <s>Quoniamigitur F maior eſt quàm <lb></lb>C, maior erit proportio F ad D, quàm C ad D. </s> <s>Sed vt <lb></lb>F ad D, ità eſt E ad B: & vt C ad D, ita G ad B; maior <pb xlink:href="043/01/091.jpg" pagenum="4"></pb>igitur proportio E ad B, quàm G ad B; quamobrem E <lb></lb>maior erit quàm G minor maiori, quod fieri non poteſt. <lb></lb></s> <s>Non igitur minor eſt proportio A ad B quàm C ad D. <lb></lb></s> <s>Eadem autem ratione non minor erit proportio C ad D, <lb></lb>quàm A ad B, hoc eſt non maior A ad B, quàm C ad D; <lb></lb>eadem igitur proportio A ad B, quæ C ad D. </s></p><p type="main"> <s>In ſecunda autem hypotheſis parte, quæ pertinet ad mi<lb></lb>norem <expan abbr="defectũ">defectum</expan>, eſto ſi fieri poteſt maior proportio A ad B, <lb></lb>quàm C ad D. erit igitur, & ſit aliqua alia magnitudo G <lb></lb>minor quàm A ad B, vt C ad D. </s> <s>Sit autê F minor quàm <lb></lb>C minori defectu magnitudine, <lb></lb>quam quis voluerit, & E minor <lb></lb>quàm A, & maior quàm G, vt au<lb></lb>tem E ad B ita F ad D. </s> <s>Quoniam <lb></lb>igitur maior eſt proportio C ad D, <lb></lb>quàm F ad D: ſed vt C ad D, ita <lb></lb>eſt G ad B: & vt F ad D, ita E ad <lb></lb>B: maior erit proportio G ad B <lb></lb>quàm E ad B; quamobrem erit <lb></lb>G maior quàm E, minor maiori, <lb></lb>quod fieri non poteſt; non igitur ma <lb></lb><figure id="id.043.01.091.1.jpg" xlink:href="043/01/091/1.jpg"></figure><lb></lb>ior eſt proportio A ad B, quàm C ad D. </s> <s>Eadem autem ra<lb></lb>tione non maior erit proportio C ad D, quàm A ad B, hoc <lb></lb>eſt non minor A ad B, quàm C ad D. </s> <s>Eadem igitur erit <lb></lb>proportio A ad B, quæ C ad D. </s> <s>Quod <expan abbr="demonſtrãdum">demonſtrandum</expan> erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO II.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si maior, vel minor prima ad vnà maiorem, vel <lb></lb>minorem ſecunda, minori <expan abbr="vtriuſq;">vtriuſque</expan> exceſſu, vel de<lb></lb>fectu <expan abbr="quantacumq;">quantacumque</expan> magnitudine propoſita fue<lb></lb>rit vt tertia ad quartam; erit vt prima ad ſecun<lb></lb>dam, ita tertia ad quartam. </s></p><pb xlink:href="043/01/092.jpg" pagenum="5"></pb><p type="main"> <s>Sint quatuor magnitudines, A prima, B ſecunda, C ter<lb></lb>tia, & D quarta: & aliæ duæ magnitudines E <lb></lb>F vnà maiores quàm A, B minori exceſsu <lb></lb>quantacumque magnitudine propoſita eiuſ<lb></lb>dem generis cum ipſis A, B. </s> <s>Sit autem E <lb></lb>maior quàm A, ad F maiorem quàm B, vt <lb></lb>C ad D. </s> <s>Dico eſse A ad B, vt C ad <lb></lb>D. </s> <s>Eſto enim, quod fieri poteſt, alia ma<lb></lb>gnitudo G eiuſdem generis cum EF ad <lb></lb>aliam H, vt C ad D, vel E ad F. </s> <s>Quoniam <lb></lb>igitur eſt permutando vt E ad G, ita F ad H, <lb></lb>& ſunt EF vnà maiores quàm AB minori ex<lb></lb>ceſsu quantacumque magnitudine propoſi<lb></lb>ta; erit per antecedentem, vt A ad G, ita B <lb></lb>ad H: & permutando A ad B, vt G ad H, <lb></lb>hoc eſt vt C ad D. </s> <s>Idem autem ſimiliter oſten <lb></lb>deremus poſitis EF minoribus quàm AB, & <lb></lb>proportionalibus vt <expan abbr="dictũ">dictum</expan> eſt. </s> <s><expan abbr="Manifeſtũ">Manifeſtum</expan> eſt igitur <expan abbr="propoſitũ">propoſitum</expan>. </s></p><figure id="id.043.01.092.1.jpg" xlink:href="043/01/092/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>ALITER.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Ijſdem poſitis, ſi non eſt A ad <lb></lb>B, vt C ad D; vel igitur ma<lb></lb>ior vel minor erit proportio A <lb></lb>ad B quàm C ad D: ſit autem <lb></lb>maior: vt igitur A ad B, ita erit <lb></lb>eadem A ad <expan abbr="aliã">aliam</expan> maiorem <34>B. <lb></lb></s> <s>Eſto illa E. ſintque aliæ duæ ma <lb></lb>gnitudines, G maior quàm A <lb></lb><figure id="id.043.01.092.2.jpg" xlink:href="043/01/092/2.jpg"></figure><lb></lb>minori exceſsu magnitudine eiuſdem generis cum A, <lb></lb>quam quis voluerit, & F maior quàm B, & minor quàm <lb></lb>E. ſit autem G ad F vt C ad D. </s> <s>Quoniam igitur & vt <lb></lb>C ad D, ita eſt A ad E; erit vt G ad F, ita A ad E; & <lb></lb>permutando vt G ad A, ita F ad E: ſed G eſt maior <pb xlink:href="043/01/093.jpg" pagenum="6"></pb>quàm A: ergo & F maior quàm <lb></lb>E, minor maiori, quod eſt ab<lb></lb>ſurdum. </s> <s>Non igitur maior eſt <lb></lb>proportio A ad B quàm C ad <lb></lb>D: eadem autem ratione non <lb></lb>maior erit proportio B ad A <expan abbr="quã">quam</expan> <lb></lb>D ad C, hoc eſt non minor A <lb></lb>ad B, quàm C ad D; eſt igitur <lb></lb>A ad B, vt C ad D. </s></p><figure id="id.043.01.093.1.jpg" xlink:href="043/01/093/1.jpg"></figure><p type="main"> <s>Rurſus in ſecunda parte hypotheſis, quæ attinet ad mi<lb></lb>norem defectum: ſi non eſt A ad B vt C ad D; eſto, ſi fie<lb></lb>ri poteſt, minor proportio A ad B quàm C ad D. igitur A <lb></lb>ad aliam quam B minorem eandem habebit <expan abbr="proportionẽ">proportionem</expan>, <lb></lb>quam C ad D, eſto illa E: ſintque <lb></lb>aliæ duæ magnitudines, G minor <lb></lb>quàm A minori defectu magnitudi<lb></lb>ne eiuſdem generis cum A, quam <lb></lb>quis voluerit, & F minor quàm B, <lb></lb>& maior quàm E: ſit autem G ad <lb></lb>F, vt C ad D, hoc eſt vt A ad E. <lb></lb></s> <s>Quoniam igitur permutando eſt vt <lb></lb>G ad A, ita F ad E, & G eſt mi<lb></lb><figure id="id.043.01.093.2.jpg" xlink:href="043/01/093/2.jpg"></figure><lb></lb>nor quàm A; erit & F minor quàm E, maior mino<lb></lb>ri, quod eſt abſurdum; non igitur minor eſt proportio <lb></lb>A ad B quàm C ad D: eadem autem ratione non minor <lb></lb>erit proportio B ad A, quàm D ad C, hoc eſt non maior <lb></lb>A ad B, quàm C ad D; eſt igitur A ad B vt C ad D. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO III.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si maior, vel minor prima ad vnà maiorem, vel <lb></lb>minorem ſecunda, minori exceſſu, vel defectu <pb xlink:href="043/01/094.jpg" pagenum="7"></pb>quantacumque magnitudine propoſita, nomina<lb></lb>tam habuerit proportionem; prima ad ſecundam <lb></lb>eandem nominatam habebit proportionem. </s></p><p type="main"> <s>Sint duæ magnitudines A, B duarum autem aliarum <lb></lb>EF vnà maiorum, vel minorum quàm AB minori ex<lb></lb>ceſsu vel defectu quantacumque magnitudine propo<lb></lb>ſita, habeat E maior vel minor quàm A ad F vnà <lb></lb>maiorem, vel minorem quàm B certam ali quam nomina<lb></lb>tam proportionem, verbi gratia, ſeſquialteram. </s> <s>Dico A <lb></lb>ad B, eandem nominatam habere proportionem: vt A <lb></lb>ipſius B eſse ſeſquialteram. </s> <s>Quoniam <lb></lb>enim omnis proportio in aliquibus ma<lb></lb>gnitudinibus conſiſtit; ſit magnitudo C <lb></lb>ipſius D ſeſquialtera: ſed & E eſt ipſius <lb></lb>F ſeſquialtera; vtigitur C, tertia ad D <lb></lb>quartam, ita erit E maior, vel minor quàm <lb></lb>A prima, ad F vnà maiorem, vel minorem <lb></lb>ſecunda, minori, vt ponitur, vtriuſque ex<lb></lb>ceſsu, vel defectu magnitudine propoſita <lb></lb>eiuſdem generis cum A, B, quæcumque <lb></lb>illa, & quantacumque ſit; erit per præ<lb></lb>cedentem eadem proportio A ad B, <lb></lb>quæ C ad D: ſed proportio quam ha<lb></lb>bet C ad D, eſt ſeſquialtera; ergo & A <lb></lb>ipſius B erit ſeſquialtera. </s> <s>Similiter quo<lb></lb>cumque alio nomine notatam proportio<lb></lb>nem habeat E ad F, eandem habere A <lb></lb><figure id="id.043.01.094.1.jpg" xlink:href="043/01/094/1.jpg"></figure><lb></lb>ad B, oſtenderemus, vt duplam, ſeſquitertiam, alicuius du <lb></lb>plicatam, vel triplicatam, & ſic de ſingulis. </s> <s>Manifeſtum <lb></lb>eſt igitur propoſitum. </s></p><p type="main"> <s>Hæc autem propoſitio in paucis exemplaribus, quæ do<lb></lb>no quibuſdam <expan abbr="dederã">dederam</expan>, non extat; poſterius enim eam exco-<pb xlink:href="043/01/095.jpg" pagenum="8"></pb>gitaui, quo ſecunda <expan abbr="antecedẽs">antecedens</expan> hìc in illis tertia facilius ſer<lb></lb>uiret ijs, in quibus certæ proportionis nomen, <expan abbr="tertiũ">tertium</expan> & quar <lb></lb>tum terminum ſubobſcurè indicat, vt in ſequenti XII iilud, <lb></lb>proportio dupla. </s> <s>Illo autem Lemmate, quod prima propofi<lb></lb>tio inſcribebatur, nunc ita non egeo, vt primam, & <expan abbr="ſecundã">ſecundam</expan>, <lb></lb>quæ ſecunda, & tertia erant, & facilius demonſtrem, & ea<lb></lb>rum ſenſum paucioribus comprehendam. </s> <s>priora ergo ita <lb></lb>non improbo vt hæc ijs anteponam. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint tres magnitudines ſe ſe æqualiter exce<lb></lb>dentes, minor erit proportio minimæ ad mediam <lb></lb>quàm mediæ ad maximam. </s></p><p type="main"> <s>Sint tres magnitudines inæquales A, BC, DE, qua<lb></lb>rum BC æquè excedat ipſam A, ac DE ipſam BC <lb></lb>Dico minorem eſse proportionem A, ad <lb></lb>BC, quàm BC, ad DE. </s> <s>Nam vt eſt <lb></lb>A ad BC, ita ſit BC ad LH, & au<lb></lb>feratur BF æqualis A, & DG, & LK <lb></lb>æquales BC. </s> <s>Quoniam igitur eſt vt A, <lb></lb>hoc eſt FB ad BC, ita BC hoc eſt KL <lb></lb>ad LH; erit diuidendo vt BF ad FC, <lb></lb>ita LK ad KH: & componendo, ac per<lb></lb>mutando vt BC ad LH, ita FC ad <lb></lb>KH. ſed BC eſt minor quàm LH; ergo <lb></lb>& FC hoc eſt EG erit minor quàm KH. <lb></lb></s> <s>Sed DE, LH, ſuperant BC exceſsibus <lb></lb>EG, KH; minor igitur erit DE quàm <lb></lb>LH, & minor proportio BC ad LH, <lb></lb>quàm BC ad DE. </s> <s>Sed vt BC ad LH, <lb></lb><figure id="id.043.01.095.1.jpg" xlink:href="043/01/095/1.jpg"></figure><lb></lb>ita eſt A ad BC; minor igitur proportio erit A ad BC, <lb></lb>quàm BC ad DE. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/096.jpg" pagenum="9"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO V.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſit minor proportio primæ ad ſecundam, <lb></lb>quàm ſecundæ ad tertiam, ab ipſis autem æquales <lb></lb>auferantur; erit minor proportio reliquæ primæ <lb></lb>ad reliquam ſecundæ, quam reliquæ ſecundæ ad <lb></lb>reliquam tertiæ. </s></p><p type="main"> <s>Sit minor proportio AB, ad CD, quam CD, ad EF. <lb></lb></s> <s>Sitque AB, minima. </s> <s>ablatæ autem æquales fint AG, CH, <lb></lb>EK. </s> <s>Dico reliquarum minorem eſse proportionem BG, <lb></lb>ad DH, quam BH, ad FH. </s> <s>Ponatur enim CL, æqua<lb></lb>lis AB, & EM, æqualis CD. </s> <s>Quoniam igitur maior eſt <lb></lb>proportio DL ad LH, quam DL, ad LC; <lb></lb>erit componendo maior proportio DH ad <lb></lb>HL, quam DC ad CL. hoc eſt, maior <lb></lb>proportio DH, ad BG, quam DC, <lb></lb>ad AB: & conuertendo, minor proportio <lb></lb>BG ad DH, quam AB, ad CD: hoc eſt <lb></lb>maior proportio AB, ad CD, quam BG, <lb></lb>ad DH. Rurſus, quoniam maior eſt pro<lb></lb>portio CD, ad EF, quam AB, ad CD: <lb></lb>hoc eſt quam CL, ad EM; erit permutan <lb></lb>do, maior proportio CD, ad CL, quam <lb></lb>FE, ad EM: & diuidendo, maior DL, ad <lb></lb>LC, quam FM, ad ME: & permutando, <lb></lb><figure id="id.043.01.096.1.jpg" xlink:href="043/01/096/1.jpg"></figure><lb></lb>maior DL, ad FM, quam CL, ad EM: hoc eſt quam <lb></lb>AB, ad CD. </s> <s>Sed maior erat proportio AB, ad CD, <lb></lb>quam BG ad DH; multo igitur maior proportio erit DL, <lb></lb>ad FM, quam BG, ad DH: hoc eſt quam LH, ad MK: <lb></lb>& permutando, maior proportio DL, ad LH, quam FM, <lb></lb>ad MK: & componendo, maior DH, ad HL, quam FK, <pb xlink:href="043/01/097.jpg" pagenum="10"></pb>ad KM: & permutando, maior DH ad F<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, quam LH, ad <lb></lb>M<emph type="italics"></emph>K<emph.end type="italics"></emph.end>: hoc eſt, quam BG, ad DH: hoc eſt minor propor<lb></lb>tio BG ad DH, quam DH, ad FK. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint tres magnitudines inæquales, & aliæ il<lb></lb>lis multitudine æquales binæque in duplicata pri <lb></lb>marum proportione. </s> <s>Sit autem minor proportio <lb></lb>primæ ad ſecundam, quam ſecundæ ad tertiam in <lb></lb>primis; erit minor proportio primæ ad ſecundam, <lb></lb>quam ſecundæ ad tertiam in ſecundis. </s></p><p type="main"> <s>Sint tres magnitudines A, B, C, & aliæ illis multitudine <lb></lb>æquales D, E, F. quarum ipſius D ad E proportio ſit du<lb></lb>plicata eius, quæ eſt A ad B: & E ad F, duplicata eius, <lb></lb>quæ eſt B ad C. ſit autem mi<lb></lb>nor proportio A ad B, quam <lb></lb>B ad C. </s> <s>Dico minorem eſse <lb></lb>proportionem D ad E, quam <lb></lb>E ad F. </s> <s>Sit enim vt C ad B, <lb></lb>ita B ad G: & vt B ad A, ita <lb></lb>A ad H. </s> <s>Igitur G ad C dupli<lb></lb>cata erit proportio ipſius G ad <lb></lb>B, hoc eſt B ad C: ſimiliter <lb></lb>erit H ad B, duplicata propor<lb></lb>tio ipſius A ad B. </s> <s>Vt igitur <lb></lb>eſt H ad B, ita erit D ad E: & <lb></lb>vt G ad C, ita E ad F. Rur<lb></lb>ſus, quia minor eſt proportio <lb></lb><figure id="id.043.01.097.1.jpg" xlink:href="043/01/097/1.jpg"></figure><lb></lb>A ad B, quam B ad C, ſed vt A ad B, ita eſt H ad A <pb xlink:href="043/01/098.jpg" pagenum="11"></pb>& vt B ad C, ita G ad B; erit ex æquali minor proportio <lb></lb>H ad B, quam G ad C, ſed vt H ad B, ita erat D, ad <lb></lb>E: & vt G ad C, ita E ad F; minor igitur proportio erit <lb></lb>D ad E, quam E ad F. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint octo magnitudines quaternæ propor<lb></lb>tionales: tertiæ autem vtriuſque ordinis inter ſo <lb></lb>ſint vt primæ; erit vt compoſita ex primis ad com <lb></lb>poſitam ex ſecundis, ita compoſita ex tertiis ad <lb></lb>compoſitam ex quartis. </s></p><p type="main"> <s>Sint octo magnitudines quaternæ ſum<lb></lb>ptæ proportionales, vt A ad B, ita C ad <lb></lb>D. & vt E ad F, ita G ad H. ſit autem vt <lb></lb>A ad E, ita C ad G. </s> <s>Dico eſse vt AE, ad <lb></lb>ABF, ita CG, ad DH. </s> <s>Quoniam enim <lb></lb>componendo eſt vt AE, ad E, ita, CG, <lb></lb>ad G; ſed vt E ad F, ita eſt G, ad H; erit <lb></lb>ex æquali, vt AE, ad F, ita CG, ad H. <lb></lb></s> <s>Eadem ratione erit vt AE, ad B, ita CG, <lb></lb>ad D: & conuertendo, vt B ad AE, ita <lb></lb>D ad CG. ſed vt AE, ad F, ita erat <lb></lb>CG ad H; ex æquali igitur erit vt B <lb></lb>ad F, ita D, ad H: & componendo, vt <lb></lb>BF ad F, ita DH ad H: & conuerten<lb></lb>do, vt F ad BF, ita H, ad DH. </s> <s>Sed vt <lb></lb>AE, ad F, ita erat CG ad H; ex æqua <lb></lb>li igitur erit vt AE ad BF, ita CG, <lb></lb>ad DH. </s> <s>Quod demonſtrandum erat. </s></p><figure id="id.043.01.098.1.jpg" xlink:href="043/01/098/1.jpg"></figure><pb xlink:href="043/01/099.jpg" pagenum="12"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint tres magnitudines ſe ſe æqualiter exce<lb></lb>dentes; & aliæ eiuſdem generis illis multitudine <lb></lb>æquales, binæque ſumptæ in duplicata primarum <lb></lb>proportione; erit vtriuſque ordinis minor pro<lb></lb>portio compoſitæ ex primis ad compoſitam ex ſe<lb></lb>cundis, quam compoſitæ ex ſecundis ad compoſi<lb></lb>tam ex tertijs. </s></p><p type="main"> <s>Sint tres magnitudines A, B, C, quarum C maxima <lb></lb>æque ſuperet B, atque <lb></lb>B, ipſam A. & totidem <lb></lb>eiuſdem generis D, E, <lb></lb>F, ſitque F ad E du<lb></lb>plicata proportio ipſius <lb></lb>C ad B: & E ad D, <lb></lb>duplicata ipſius B ad <lb></lb>A. </s> <s>Dico AD, ſimul <lb></lb>ad BE, ſimul mino<lb></lb>tem eſſe proportionem <lb></lb>quam BE, ſimul ad <lb></lb>CF, ſimul. </s> <s>Eſto enim <lb></lb>recta quæpiam GH, <lb></lb>ad aliam rectam ſibi in <lb></lb>directum poſitam HK, <lb></lb>vt magnitudo A ad ip <lb></lb>ſius F duplam (hoc <lb></lb>enim fieri poteſt) & <lb></lb><figure id="id.043.01.099.1.jpg" xlink:href="043/01/099/1.jpg"></figure><lb></lb>ſuper baſim GK; conſtituatur triangulum GLK, atque <lb></lb>in eo deſcribatur parallelogrammum GHMN: & vt eſt <pb xlink:href="043/01/100.jpg" pagenum="13"></pb>C ad B, ita fiat HM, ad <expan abbr="Mq.">Mque</expan> & vt B ad A, ita QM, ad <lb></lb>MP, & ipſi GK, parallelæ TPR, VQS, ducantur. <lb></lb></s> <s>Quoniam igitur eſt vt C, ad duplam ipſius F, ita GH, ad <lb></lb>HK; erit vt C ad F, ita eſt par llelogrammum GM, ad <lb></lb>triangulum MHK: ſed vt C, ad B, ita eſt HM, ad <expan abbr="Mq;">Mque</expan> <lb></lb>hoc eſt parallelogrammum GM, ad parallelogrammum <lb></lb>MV: & vt F, ad E, ita triangulum MHK, ad triangu<lb></lb>lum MQS, ob duplicatam proportionem eius, quæ eſt <lb></lb>HM ad <expan abbr="Mq.">Mque</expan> hoc eſt ipſius C ad B; vt igitur trapezium <lb></lb>NK, ad NS trapezium, ita erit, per præcedentem, CF, <lb></lb>ſimul ad BE ſimul. </s> <s>Rurſus quoniam eſt conuertendo, vt <lb></lb>parallelogrammum MV, ad parallelogrammum GM, ita <lb></lb>B ad C. ſed vt parallelogrammum GM, ad triangulum <lb></lb>KHM, ita erat C, ad F: & vt triangulum KHM, ad <lb></lb>triangulum QSM, ita F ad E; erit ex æquali, vt paral<lb></lb>lelogrammum MV, ad triangulum SQM, ita B, ad E. <lb></lb></s> <s>Similiter ergo vt ante erit vt trapezium NS, ad NR tra<lb></lb>pezium, ita EB, ſimul ad AD, ſimul. </s> <s>Rurſus, quoniam <lb></lb>æque excedit LV, ipſam LT, atque LG, ipſam LV; <lb></lb>minor erit proportio LT ad LV, quam LV, ad LG: eſt <lb></lb>autem trianguli LTR ad triangulum LVS, duplicata <lb></lb>proportio ipſius LT, ad LV, & trianguli LVS, ad trian<lb></lb>gulum LGK, duplicata ipſius LV, ad LG, propter ſi<lb></lb>militudinem triangulorum; minor igitur proportio erit <lb></lb>trianguli LTR, ad triangulum LVS, quam trianguli <lb></lb>LVS, ad triangulum LGK; dempto igitur triangulo <lb></lb>LNM, communi, minor erit proportio trapezij NR, ad <lb></lb>trapezium NS, quam trapezij NS, ad trapezium NK. <lb></lb></s> <s>Sed vt trapezium NR, ad trapezium NS, ita eſt conuer<lb></lb>tendo AD ſimul ad BE, ſimul: & vt trapezium NS, ad <lb></lb>trapezium NK, ita BE, ſimul ad CF, ſimul; minor igi<lb></lb>tur proportio erit AD, ſimul ad BE ſimul, quam BE ſi<lb></lb>mul ad CF, ſimul. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/101.jpg" pagenum="14"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si recta linea vtcumque ſecta fuerit, cubus qui <lb></lb>fit à tota æqualis eſt duobus ſolidis rectangulis, <lb></lb>quæ ex partibus, & totius quadrato fiunt. </s></p><p type="main"> <s>Sit recta linea AB ſecta in puncto C vtcumque. </s> <s>Di<lb></lb>co cubum ex AB æqualem eſse duobus ſolidis rectangu<lb></lb>lis, quæ fiunt ex AC CB, & quadrato AB. </s> <s>Quoniam <lb></lb><figure id="id.043.01.101.1.jpg" xlink:href="043/01/101/1.jpg"></figure><lb></lb>enim communi altitudine AB, eſt vt rectangulum BAC <lb></lb>ad quadratum AB, ita ſolidum ex AB, & rectangulo <lb></lb>BAC ad cubum ex AB, eademque ratione vt rectangu<lb></lb>lum ABC, ad quadratum AB, ita ſolidum eſt AB, & <lb></lb>rectangulo ABC ad cubum ex AB; erunt vt duo rectan<lb></lb>gula BAC, ABC ad quadratum AB, ita duo ſolida <lb></lb>ex AB, & rectangulis BAC, ABC ad cubum ex AB. <lb></lb></s> <s>Sed duo rectangula BAC, ABC ſunt æqualia quadrato <lb></lb>AC; duo igitur ſolida ex AB, & rectangulis BAC, CBA, <lb></lb>æqualia ſunt cubo ex AB. </s> <s>Sed ſolidum ex AB & rectan<lb></lb>gulo BAC eſt id quod fit ex AC, & AC & quadrato <lb></lb>AB; duo igitur ſolida ex AC, CB, & quadrato AB ſi<lb></lb>mul ſumpta æqualia ſua cubo ex AB. </s> <s>Si igitur recta linea <lb></lb>vtcumque ſecta fuerit, &c. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/102.jpg" pagenum="15"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO X.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si recta linea vtcumque ſecta fuerit, cubus qui <lb></lb>fit à tota æqualis eſt cubis partium, & duobus ſo<lb></lb>lidis rectangulis, quæ partium triplis, & earun<lb></lb>dem quadratis reciproce continentur. </s></p><p type="main"> <s>Sit recta linea AB ſecta vtcumque in puncto C. </s> <s>Dico <lb></lb>cubum ex AB æqualem eſse duobus cubis ex AC, CB, <lb></lb>& duobus ſolidis rectangulis, quorum alterum fit ex tripla <lb></lb><figure id="id.043.01.102.1.jpg" xlink:href="043/01/102/1.jpg"></figure><lb></lb>ipſius AC, & quadrato BC; alterum autem ex tripla ip<lb></lb>ſius BC, & quadrato AC. </s> <s>Quoniam enim quadratum <lb></lb>ex AB æquale eſt duobus quadratis ex AC, CB, & ei <lb></lb>quod bis fit ex AC CB: & parallelepipeda eluſdem al<lb></lb>titudinis inter ſe ſunt vt baſes; erit rectangulorum folido<lb></lb>rum id quod fit ex AC, & quadrato AB æquale cubo ex <lb></lb>AC, & ei, quod fit ex AC, & rectangulo ACB bis, & <lb></lb>ei, quod ex AC, & quadrato BC. </s> <s>Eadem ratione erit <lb></lb>quod fit ex BC, & quadrato AB æquale cubo ex BC, & <lb></lb>ei, quod fit ex BC, & rectangulo ACB, bis & ei, quod ex <lb></lb>BC, & quadrato AC. </s> <s>Sed cubus ex AB æqualis eſt <lb></lb>duobus ſolidis ex AC CB. & quadrato AB; cubus igi<lb></lb>tur ex AB æqualis eſt duobus cubis ex AC CB, & ſex <lb></lb>ſolidis, quorum tres fiunt ex AC, & duobus rectangulis <lb></lb>ex AC CB, & quadrato BC: tria vero ex BC, & duo<lb></lb>bus rectangulis ex AC CB, & quadrato AC. </s> <s>Sed quod <lb></lb>fit ex AC, & rectangulo ACB, eſt quod fit ex BC, & <pb xlink:href="043/01/103.jpg" pagenum="16"></pb>quadrato AC: & quod fit ex BC, & rectangulo ACB, <lb></lb>eſt quod fit ex AC, & quadrato BC; cubus igitur ex <lb></lb>AB æqualis eſt duobus cubis ex AC CB, vna cum ſex <lb></lb>ſolidis, quorum tria fiunt ex AC, & BC quadrato, tria <lb></lb>autem ex BC, & quadrato AC, hoc eſt duobus ſolidis, <lb></lb>quorum alterum fit ex tripla ipſius AC, & quadrato BC, <lb></lb>alterum ex tripla ipſius BC & quadrato AC. </s> <s>Quod de<lb></lb>monſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si recta linea vtcumque ſecta fuerit, cubus qui <lb></lb>fit à tota æqualis eſt cubis partium vna cum ſoli<lb></lb>do rectangulo, quod totius tripla, & partibus <lb></lb>continetur. </s></p><p type="main"> <s>Sit recta linea AB ſecta in puncto C vtcumque. </s> <s>Di<lb></lb>co cubum ex AB æqualem eſse duobus cubis ex AC, <lb></lb>CB, vna cum ſolido rectangulo ex AC CB, & tripla <lb></lb>ipſius AB. </s> <s>Quoniam enim quod fit ex AC, & rectan<lb></lb>gulo ACB, eſt id quod fit ex BC, & quadrato AC: & <lb></lb>quod fit ex BC, & rectangulo ACB, eſt id, quod fit ex <lb></lb><figure id="id.043.01.103.1.jpg" xlink:href="043/01/103/1.jpg"></figure><lb></lb>AC & quadrato BC. ſed duo ſolida ex AC CB, & re<lb></lb>ctangulo ACB ſunt id, quod fit ex compoſita vtriuſque <lb></lb>altitudine AB, et rectangulo ACB; duo igitur prædi<lb></lb>cta ſolida, quæ ex AC CB, & earum quadratis recipro<lb></lb>ce fiunt æqualia ſunt ſolido ex AB BC CA, & triplum <lb></lb>triplo, videlicet duo ſolida, quæ fiunt reciproce ex triplis <pb xlink:href="043/01/104.jpg" pagenum="17"></pb>ipſarum AC, CB, & quadratis ex AC CB, æqualia ſi<lb></lb>mul ei, quod ter fit ex AB, BC, CA, hoc eſt ei, quod <lb></lb>partibus AC CB, & totius AB tripla continetur: additis <lb></lb>igitur communibus duobus cubis ex AC, CB, erit id, quod <lb></lb>ſit ex AC CB, & tripla ipſius AB, & duo cubi ex AC <lb></lb>CB, æqualia duobus ſolidis, quæ fiunt reciproce ex triplis <lb></lb>ipſarum AC, CB, & earundem AC, CB, quadratis, & <lb></lb>duobus cubis ex AC, CB, hoc eſt cubo ex AC. </s> <s>Si igi<lb></lb>tur recta linea vtcumque ſecta fuerit, &c. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hemiſphærium duplum eſt coni, cylindri au<lb></lb>tem ſubſeſquialterum eandem ipſi baſim, & ean<lb></lb>dem altitudinem habentium. </s></p><p type="main"> <s>Eſto hemiſphærium; cuius axis BD, baſis circulus, cu<lb></lb>ius diameter AC, ſuper quem cylindrus AE, & conus <lb></lb><figure id="id.043.01.104.1.jpg" xlink:href="043/01/104/1.jpg"></figure><lb></lb>ABC, quorum communis axis ſit BD, ac propterea <lb></lb>etiam eadem altitudo. </s> <s>Dico hemiſphærium ABC, co<lb></lb>ni ABC eſse duplum: cylindri autem AE <expan abbr="ſubſeſquialterũ">ſubſeſquialterum</expan>. <lb></lb></s> <s>ſuper baſim enim circulum RE, vertice D deſcribatur <pb xlink:href="043/01/105.jpg" pagenum="18"></pb>conus EDR. </s> <s>Sectoque axe BD primo bifariam, deinde <lb></lb>ſingulis eius partibus rurſus bifariam, tranſeant per pun<lb></lb>cta ſectionum plana baſi hemiſphærij AC æquidiſtantia, <lb></lb>quæ ſecent hemiſphærium, conum, & cylindrum. </s> <s>Se<lb></lb>ctus igitur erit AE cylindrus in cylindros æqualium alti<lb></lb>tudinum: ſuper ſectiones autem coni, atque hemiſphærij <lb></lb>nempe circulos, quorum centra in axe BD exiſtunt cy<lb></lb>lindri conſtituti intelligantur binis quibuſque proximis <lb></lb>æquidiſtantibus planis interiecti, quorum axes omnes <lb></lb>æquales in BD. </s> <s>Erit igitur cono EDR inſcripta, & ABC <lb></lb><figure id="id.043.01.105.1.jpg" xlink:href="043/01/105/1.jpg"></figure><lb></lb>hemiſphærio circumſcripta figura quædam ex cylindris <lb></lb>æqualium altitudinum. </s> <s>Sint autem hæ figuræ ea ratione <lb></lb>hæc circumſcripta illa inſcripta, vt circumſcripta excedat <lb></lb>hemiſphærium, minori exceſsu, inſcripta vero deficiat à <lb></lb>cono minori defectu quam ſit magnitudo propoſita, quan<lb></lb>tacumque illa ſit. </s> <s>His conſtitutis, manifeſtum eſt, reliquo <lb></lb>cylindri AE dempto hemiſphærio inſcriptam eſse figu<lb></lb>ram ex reſiduis cylindrorum, in quos cylindrus AE ſe<lb></lb>ctus fuerit, demptis cylindris hemiſphærio circumſcriptis, <lb></lb>deficientem à reliquo cylindri AE dempto hemiſphærio <lb></lb>minori defectu magnitudine propoſita, eodem ſcilicet, <lb></lb>quo figura hemiſphærio circumſcripta excedit hemiſphæ<lb></lb>rium, excepto reſiduo cylindri infimi AS, dempta he<lb></lb>miſphærij portione, quam comprehendit. </s> <s>Sit autem om-<pb xlink:href="043/01/106.jpg" pagenum="19"></pb>nium prædictorum cylindri AE cylindrorum ſupremus <lb></lb>FE, cuius axis BH, & communis ſectio plani per pun<lb></lb>ctum H tranſeuntis baſi hemiſphærij cum plano per axim <lb></lb>BD, ſit recta FGKHMNL. </s> <s>Quoniam igitur rectan<lb></lb>gulum DHB bis vna cum duobus quadratis DH, BH, <lb></lb>æquale eſt BD quadrato: & rectangulum DHB bis <lb></lb>vna cum quadrato BH, eſt rectangulum ex BD DH tan<lb></lb>quam vna, & BH; rectangulum ex BD, DH tanquam <lb></lb>vna & BH, vna cum quadrato DH æquale erit quadra<lb></lb>to BD, hoc eſt quadrato FH: quorum quadratum KH <lb></lb>æquale eſt rectangulo ex BD, DH, tanquam vna, & BH; <lb></lb>reliquum igitur quadrati FH dempto quadrato KH æ<lb></lb>quale erit reliquo quadrato DH, hoc eſt quadrato GH: <lb></lb>& quadruplum quadruplo reliquum quadrati FL dempto <lb></lb>quadrato MK toti GN quadrato, hoc eſt reliquum circu <lb></lb>li, FL dempto circulo MK, æquale circulo GN. </s> <s>Qua<lb></lb>re & GP, cylindrus reliquo cylindri FE dempto QK, <lb></lb>cylindro æqualis erit, propter æqualitatem altitudinum. <lb></lb></s> <s>Similiter oſtenderemus ſingula reliqua cylindrorum eiuſ<lb></lb>dem altitudinis, in quos totus cylindrus AE ſectus fuit, <lb></lb>demptis cylindris hemiſphærio circumſcriptis æqualia eſ<lb></lb>ſe ſingulis cylindris cono EDR inſcriptis, quæ inter ea<lb></lb>dem plana interijciuntur. </s> <s>Tota igitur figura ex prædictis <lb></lb>cylindrorum reſiduis reliquo cylindri AE, dempto he<lb></lb>miſphærio inſcripta æqualis erit figuræ cono EDR in<lb></lb>ſcriptæ: deficit autem vtraque harum figurarum hæc à co<lb></lb>no ADR, illa à reſiduo cylindri AE dempto hemiſphæ<lb></lb>rio minori exceſsu magnitudine vtcumque propoſita; re<lb></lb>liquum igitur cylindri AE dempto hemiſphærio æquale <lb></lb>eſt cono EDR, ſed conus EDR; hoc eſt conus ABC cylin <lb></lb>dri AE eſt pars tertia; reliquum igitur cylindri AE dem<lb></lb>pto hemiſphærio, cylindri AE eſt pars tertia, hoc eſt cylin<lb></lb>drus AE triplus dicti reſidui: <expan abbr="quamobrẽ">quamobrem</expan> AE cylindrus ſeſ<lb></lb>quialter hemiſphærij ABC: & <expan abbr="cõuertendo">conuertendo</expan>, hemiſphærium <pb xlink:href="043/01/107.jpg" pagenum="20"></pb>cylindri AE ſubſeſquialterum: coni igitur ABC duplum. <lb></lb></s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis minor ſphæræ portio, ad cylindrum, <lb></lb>cuius baſis æqualis eſt circulo maximo, altitudo <lb></lb>autem eadem portioni, eam habet proportionem, <lb></lb>quam exceſſus, quo tripla ſemidiametri ſphæræ <lb></lb>excedit tres deinceps proportionales, quarum ma <lb></lb>xima eſt ſphæræ ſemidiameter, media vero quæ <lb></lb>inter centra ſphæræ & baſis portionis interijci<lb></lb>tur; ad ſemidiametri ſphæræ triplam. </s></p><p type="main"> <s>Sit ſphæræ, cuius centrum D, ſemidiameter BD, mi<lb></lb>nor portio ABC, cuius axis BG ſegmentum ſemidiame<lb></lb>tri BD, baſis autem circulus, cuius diameter AC. </s> <s>Sitque <lb></lb>EF, cylindrus, cu<lb></lb>ius axis, ſiue alti<lb></lb>tudo eadem BG: <lb></lb>baſis autem æqua<lb></lb>lis circulo maxi<lb></lb>mo, cuius ſemidia<lb></lb>meter BD. </s> <s>Dico <lb></lb>portionem ABC, <lb></lb>ad cylindrum EF <lb></lb>eam habere pro<lb></lb><figure id="id.043.01.107.1.jpg" xlink:href="043/01/107/1.jpg"></figure><lb></lb>portionem, quam exceſſus, quo tripla ipſius BD, ſupe<lb></lb>rat tres BD, DG; & minorem extremam ad ipſas, quæ <lb></lb>ſit M; ad ipſius BD triplam. </s> <s>vertice enim D, baſi cylin<lb></lb>dri EF, cuius diameter FH deſcribatur conus FDH, cu<lb></lb>ius intelligatur fruſtum FHKL abſciſsum plano, quod ab-<pb xlink:href="043/01/108.jpg" pagenum="21"></pb>ſcidit portionem ABC, plano circuli FH parallelum. <lb></lb></s> <s>Quoniam igitur fruſtum FH<emph type="italics"></emph>K<emph.end type="italics"></emph.end>L æquale eſt cylindri EF <lb></lb>reſiduo, dempta ABC portione, quod ex præcedenti theo <lb></lb>remate perſpicuum eſse debet: erit portio ABC æqualis <lb></lb>ei, quod relinquitur cylindri EF, ſi fruſtum auferatur <lb></lb>FHKL: ſed hoc reliquum eſt ad cylindrum EF, vt exceſ<lb></lb>ſus, quo tripla lineæ FH, ſuperat tres deinceps proportio<lb></lb>nales FH, KL, & minorem extrema, ad triplam lineæ FH: <lb></lb><gap></gap>vt FH, ad KL, ita eſt BD ad DG, & DG, ad M; vt igi<lb></lb>tur exceſſus, quo tripla ipſius BD, ſuperat tres BD, DG, <lb></lb>& M, ſimul, ad lineæ BD triplam, ita erit portio ABC ad <lb></lb>cylindrum EF. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portio ſphæræ abſciſsa duobus planis <lb></lb>parallelis alteroper centrum acto ad cylindrum, <lb></lb>cuius baſis eſt eadem baſi portionis, ſiue circu<lb></lb>lo maximo, & eadem altitudo, eam habet pro<lb></lb>portionem, quam exceſſus, quo maior extrema ad <lb></lb>ſphæræ ſemidiametrum, & axim portionis exce<lb></lb>dit tertiam partem axis portionis; ad maiorem ex<lb></lb>tremam antedictam. </s></p><p type="main"> <s>Sit portio AB <lb></lb>CD, ſphæræ, cu <lb></lb>ius centrum F, <lb></lb>abſciſſa duobus <lb></lb>planis parallelis <lb></lb>altero per <expan abbr="centrũ">centrum</expan> <lb></lb>F tranſeunte; <lb></lb>axis autem por<lb></lb>tionis fit FG: & <lb></lb><figure id="id.043.01.108.1.jpg" xlink:href="043/01/108/1.jpg"></figure><pb xlink:href="043/01/109.jpg" pagenum="22"></pb>maior baſis, circulus maximus, cuius diameter AD, minor <lb></lb>autem, cuius diameter BC: & cylindrus AE, cuius baſis <lb></lb>circulus AD, axis FG; & vt FG ad FA, ita ſit FA, ad <lb></lb>MN, à qua abſcindatur NO, pars tertia ipſius FG. </s> <s>Dico <lb></lb>ABCD <expan abbr="portionẽ">portionem</expan> ad cylindrum AE eſſe vt OM ad MN. <lb></lb></s> <s>Poſita enim G <lb></lb>H, æquali ipſi <lb></lb>FG, deſcriba<lb></lb>tur circa axim <lb></lb>FG, cylindrus <lb></lb>L<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, & conus <lb></lb>HFK. </s> <s>Quoniam <lb></lb>igitur duo cylin <lb></lb>dri AE, LK, <lb></lb>ſunt eiuſdem al<lb></lb><figure id="id.043.01.109.1.jpg" xlink:href="043/01/109/1.jpg"></figure><lb></lb>titudinis, erunt inter ſe vt baſes, AD, KH. hoc eſt cy<lb></lb>lindrus AE ad cylindrum LK, duplicatam habebit pro<lb></lb>portionem diametri AD, ad diametrum KH, hoc eſt eius, <lb></lb>quæ eſt ſemidiametri AF ad ſemidiametrum GH. hoc eſt <lb></lb>eam, quæ eſt MN ad GH, ſiue FG. </s> <s>Sed vt FG ad tertiam <lb></lb>ſui partem NO, ita eſt cylindrus KL, ad conum KFH; <lb></lb>ex æquali igitur, erit vt MN ad NO, ita cylindrus AE <lb></lb>ad conum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>FH, hoc eſt ad reliquum cylindri AE dem <lb></lb>pta ABCD portione: & per conuerſionem rationis, vt <lb></lb>NM, ad MO, ita cylindrus AE ad portionem ABCD: <lb></lb>& conuertendo, vt MO ad MN, ita portio ABCD ad <lb></lb>cylindrum AE. </s> <s>Quod eſt propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portio ſphæræ abſciſſa duobus planis <lb></lb>parallelis neutro per centrum, nec centrum inter<lb></lb>cipientibus ad cylindrum, cuius baſis æqualis eſt <pb xlink:href="043/01/110.jpg" pagenum="23"></pb>circulo maximo, altitudo autem eadem portioni, <lb></lb>eam <expan abbr="proportionẽ">proportionem</expan> habet, quam exceſſus, quo maior <lb></lb>extrema ad triplas ſemidiametri ſphæræ, & eius <lb></lb>quæ inter <expan abbr="centrũ">centrum</expan> ſphæræ, & minoris baſis portio<lb></lb>nis interijcitur, ſuperat tres deinceps <lb></lb>proportionales, quarum maxima eſt <lb></lb>quæ inter centra ſphæræ, & minoris <lb></lb>baſis, media autem, quæ inter cen<lb></lb>træ ſphæræ, & maioris baſis portio<lb></lb>nis interijcitur; ad maiorem extre<lb></lb>mam antedictam. </s></p><p type="main"> <s>Sit portio ABCD ſphæræ, cuius centrum <lb></lb>E, abſciſsa duobus planis parallelis, neutro <lb></lb>per E tranſeunte, nec E <expan abbr="intercipiẽtibus">intercipientibus</expan>, cuius <lb></lb>maior baſis ſit circulus, cui diameter AD. <lb></lb>minor autem cuius diameter BC, axis GH. <lb></lb>circa quem cylindrus OS, conſiſtat, cuius <lb></lb>baſis ſit circulus circa SR æqualis circulo <lb></lb>maximo: ſphæræ autem ſemidiater ſit EHG. <lb></lb>& vt GE ad EH, ita ſit HE ad V: & po<lb></lb><figure id="id.043.01.110.1.jpg" xlink:href="043/01/110/1.jpg"></figure><lb></lb>ſita T tripla ipſius EF, & X itidem tripla ipſius EG, vt X <pb xlink:href="043/01/111.jpg" pagenum="24"></pb>ad T, ita fiat T ad ZY, cuius Z<foreign lang="grc">ω</foreign>, tribus GE, EH, V <lb></lb>ſimul ſit æqualis. </s> <s>Dico ABCD portio<lb></lb>nem ad cylindrum SO eſse vt <foreign lang="grc">ωΥ</foreign> ad <foreign lang="grc">Υ</foreign>Z. <lb></lb></s> <s>Abſciſsa enim GK ipſi EG æquali, cylin<lb></lb>drus PN circa axim GH, & conus KEN <lb></lb>conſtituantur vt in præcedenti. </s> <s>planum igi<lb></lb>tur abſcindens portionem facit fruſtum coni <lb></lb>KEN, quod ſit KLMN, cuius minor ba<lb></lb>ſis circulus, cui diameter LM; maior autem <lb></lb>cui diameter KN. </s> <s>Et vt eſt GE ad EF, hoc <lb></lb>eſt GK ad SH, ita ſit EF, vel SH, ad I. <lb></lb>vt igitur in præcedenti, oſtenderemus cylin<lb></lb>drum SO ad cylindrum PN eſse vt I ad <lb></lb>GK ſiue ad EG. </s> <s>Quoniam igitur ſunt ter <lb></lb>næ deinceps proportionales GE, EF, I, & <lb></lb>X, T, ZY, eſtque vt FE ad EG ita T ad X; <lb></lb>erit vt I ad EG, hoc eſt vt cylindrus SO ad <lb></lb>PN <expan abbr="cylindrũ">cylindrum</expan> ita ZY ad X. </s> <s>Et quoniam eſt vt <lb></lb>GE ad EH, ita EH ad V: hoc eſt, vt GK ad <lb></lb>LH. ita LH ad V: & ponitur X tripla ipſius <lb></lb><figure id="id.043.01.111.1.jpg" xlink:href="043/01/111/1.jpg"></figure><lb></lb>EG, hoc eſt ipſius GK, vt autem eſt triplaipſius GK ad <lb></lb>tres deinceps proportionales GK, LH, V, ita eſt cylin<lb></lb>drus PN ad fruſtum LKNM; erit vt X ad tres GE, EH, <lb></lb>V ſimul hoc eſt ad lineam <foreign lang="grc">ω</foreign>Z, ita cylindrus PN ad fru-<pb xlink:href="043/01/112.jpg" pagenum="25"></pb>ſlum KLMN. </s> <s>Sed vt ZY ad X, ita erat cylindrus SO <lb></lb>ad PN cylindrum; ex æquali igitur erit vt ZY ad Z<foreign lang="grc">ω</foreign>, <lb></lb>ita cylindrus SO ad fruſtum KLMN: hoc eſt, ad reli<lb></lb>quum cylindri SO dempta ABCD portione, & per con<lb></lb>uerſionem rationis, vt ZY, ad Y<foreign lang="grc">ω</foreign>, ita cylindrus SO ad <lb></lb><expan abbr="portionẽ">portionem</expan> ABCD: & conuertendo vt <foreign lang="grc">ω</foreign>Y ad YZ, ita por<lb></lb>tio ABCD ad SO cylindrum. </s> <s>Quod <expan abbr="demonſtrandũ">demonſtrandum</expan> erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis maior ſphæræ portio ad cylindrum, cu<lb></lb>ius baſis æqualis eſt circulo maximo, altitudo au<lb></lb>tem eadem portioni eam habet proportionem, <lb></lb>quam ad axim portionis habet exceſſus, quo ſeg<lb></lb>mentum axis portionis inter ſphæræ centrum, & <lb></lb>baſim portionis interiectum ſuperat tertiam par<lb></lb>tem minoris extremæ maiori poſita prædicto axis <lb></lb>ſegmento in proportione ſemidiametri ſphæræ <lb></lb>ad prædictum <lb></lb><expan abbr="ſegmentũ">ſegmentum</expan>, vna <lb></lb>cum ſubſeſqui <lb></lb>altera reliqui <lb></lb>axis ſegmenti. </s></p><figure id="id.043.01.112.1.jpg" xlink:href="043/01/112/1.jpg"></figure><p type="main"> <s>Sit ſphæræ, cu <lb></lb>ius <expan abbr="centrũ">centrum</expan> G, dia <lb></lb>meter DGE ma <lb></lb>ior portio ABC, <lb></lb>axis autem por<lb></lb>tionis BGF, com <lb></lb>munis cylindro <lb></lb>KH, cuius baſis æqualis ſit circulo maximo; baſis autem <pb xlink:href="043/01/113.jpg" pagenum="26"></pb>portionis circulus, cuius diameter AC, & vt EG ad GF, <lb></lb>ita ſit GF ad S, & S ad FM, cuius ſit pars tertia FN, & <lb></lb>ponatur ipſius BG, ſubſeſquialtera GL. </s> <s>Dico portio<lb></lb>nem ABC ad cylindrum KH eſse vt LN ad BF. </s> <s>Nam <lb></lb>vt FG ad GE, ſiue ad BG, ita ſit EG ad PQ, à qua <lb></lb>abſcindatur QR, pars tertia ipſius FG. </s> <s>Et plano per G <lb></lb>tranſeunte baſibus cylindri KH, & ABC portionis pa<lb></lb>rallelo ſecentur vna cylindrus KH in duos cylindros DH, <lb></lb>EK: & portio ABC, in portionem ECAD, & DBE <lb></lb>hemiſphærium. </s> <s>Quoniam igitur eſt conuertendo, vt PQ <lb></lb>ad EG, ita EG <lb></lb>ad GF, & eſt ip<lb></lb>ſius GF pars ter <lb></lb>tia QR, erit por<lb></lb>tio DACE ad <lb></lb>cylindrum EK, <lb></lb>vt PR ad <expan abbr="Pq.">Pque</expan> <lb></lb>Rurſus, quia eſt <lb></lb>vt EG ad GF: <lb></lb>hoc eſt vt PQ ad <lb></lb>EG, ita GF ad <lb></lb>S, & vt EG ad <lb></lb>GF, ita eſt S ad <lb></lb>FM; erit ex æqua <lb></lb><figure id="id.043.01.113.1.jpg" xlink:href="043/01/113/1.jpg"></figure><lb></lb>li, vt PQ ad GF, ita GF ad FM. </s> <s>Sed vt GF ad RQ, <lb></lb>ita eſt MF ad FN, tertiam ipſius MF partem, ex æquali <lb></lb>igitur erit vt PQ ad QR, ita GF ad FN, & per conuer<lb></lb>ſionem rationis, & conuertendo, vt PR ad PQ, ita NG ad <lb></lb>GF. </s> <s>Sed vt PR ad PQ, ita erat portio ECAD ad cy<lb></lb>lindrum EK; vtigitur NG ad GF, ita erit portio EC <lb></lb>AD ad cylindrum EK. </s> <s>Sed vt GF ad FB, ita eſt cy<lb></lb>lindrus EK ad cylindrum KH: ex æquali igitur vt NG <lb></lb>ad BF, ita portio ECAD, ad cylindrum KH. </s> <s>Similiter <lb></lb>oſtenderemus eſse, vt GL ad BF, ita DBE hemiſphæ-<pb xlink:href="043/01/114.jpg" pagenum="27"></pb>rium ad cylindrum KH, cum vt LG ad GB, ita ſit he<lb></lb>miſphærium DBE ad cylindrum DH. vt igitur prima <lb></lb>cum quinta ad ſecundam, ita tertia cum ſexta ad quartam; <lb></lb>videlicet, vt tota LN ad BF, ita portio ABC ad cylin<lb></lb>drum KH. </s> <s>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portio ſphæræ abſciſſa duobus planis <lb></lb>parallelis centrum intercipientibus ad cylin<lb></lb>drum, eiuſdem altitudinis, cuius baſis æqualis eſt <lb></lb>circulo maximo, eam habet proportionem, quam <lb></lb>ad axim portionis habet exceſſus, quo axis portio<lb></lb>nis ſuperat tertiam partem compoſitæ ex duabus <lb></lb>minoribus extremis, maioribus poſitis duobus <lb></lb>axis ſegmentis, quæ fiunt à centro ſphæræ in ra<lb></lb>tionibus, ſemidiametri ſphæræ ad prædicta ſeg<lb></lb>menta. </s></p><p type="main"> <s>Sit portio AB <lb></lb>CD, ſphæræ, cu<lb></lb>ius centrum G, <lb></lb>abſciſsa duobus <lb></lb>planis parallelis <lb></lb>centrum G inter<lb></lb>cipientibus, quod <lb></lb>erit in axe portio<lb></lb>nis, qui ſit HK. <lb></lb></s> <s>Sectiones autem <lb></lb><figure id="id.043.01.114.1.jpg" xlink:href="043/01/114/1.jpg"></figure><lb></lb>factæ à prædictis planis ſint circuli, quorum diametri AD, <lb></lb>BC, qui circuli erunt baſes oppoſitæ portionis. </s> <s>Sectaque <lb></lb>per punctum G, portione ABCD plano ad axim erecto, <pb xlink:href="043/01/115.jpg" pagenum="28"></pb>atque ideo & portionis baſibus parallelo; ſuper ſectionem, <lb></lb>quæ erit circulus maximus, cuius diameter LM, duo cylin<lb></lb>dri deſcripti intelligantur, ad oppoſita portionis baſium pla <lb></lb>na terminati ex illis autem totus cylindrus compoſitus EF, <lb></lb>cuius baſis æqua<lb></lb>lis circulo maxi<lb></lb>mo LM. </s> <s>Deinde <lb></lb>in ſegmento GH <lb></lb>ſumpta OH, ter<lb></lb>tia parte minoris <lb></lb>extremæ maiori <lb></lb>GH in proportio <lb></lb>ne, quæ eſt LG ad <lb></lb>GH; & in ſegmen <lb></lb>to GK, ſumatur <lb></lb><figure id="id.043.01.115.1.jpg" xlink:href="043/01/115/1.jpg"></figure><lb></lb>NK, tertia pars minoris extremæ maiori GK, in propor<lb></lb>tione, quæ eſt LG ad GK. </s> <s>Dico portionem ABCD <lb></lb>ad cylindrum EF, eſse vt NO ad KH. </s> <s>Sumptis enim <lb></lb>ijſdem, quæ in præcedentis ſumpſimus, demonſtrationem <lb></lb>ſimiliter oſtenderemus tam portionem LBCM ad cy<lb></lb>lindrum EF, eſse vt OG ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>H, quam portionem LA <lb></lb>DM ad eundem EF cylindrum, vt NG ad eundem axim <lb></lb>KH, vt igitur prima cum quinta ad ſecundam, ita tertia <lb></lb>cum ſexta ad quartam: videlicet, vt NO ad KH, ita por <lb></lb>tio ABCD ad EF cylindrum. </s> <s>Quod demonſtrandum <lb></lb>crat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omne conoides parabolicum dimidium eſt <lb></lb>cylindri, coni autem ſeſquialterum eandem ipſi <lb></lb>baſim, & eandem altitudinem habentium. </s></p><pb xlink:href="043/01/116.jpg" pagenum="29"></pb><p type="main"> <s>Sit conoides parabolicum ABC, & cylindrus AE, & <lb></lb>conus ABC, quorum omnium ſit eadem baſis circulus, <lb></lb>cuins diameter AC, axis autem BD, ac proinde vna om<lb></lb>nium altitudo. </s> <s>Dico conoidis ABC eſse cylindri AE <lb></lb>dimidium, coni autem ABC ſeſquialterum. </s> <s>Secto enim <lb></lb>axe BD in tot partes æquales, quarum infima ad baſim ſit <lb></lb>MD, vt figura ex cylindris æqualium altitudinum conoi<lb></lb>di ABC circumſcripta, inſcriptam ſuperet minori ſpacio <lb></lb>quantacumque magnitudine propoſita, & ſit hoc factum. <lb></lb></s> <s>Et quoniam quibus planis parallelis tranſeuntibus per præ<lb></lb><figure id="id.043.01.116.1.jpg" xlink:href="043/01/116/1.jpg"></figure><lb></lb>dictas ſectiones axis BD ſecatur conoides ABC, ijſdem <lb></lb>ſecatur triangulum per axim ABC, eruntque ſectiones <lb></lb>parallelæ: ſit triangulo ABC circumſcripta figura ex pa<lb></lb>rallelogrammis æqualium altitudinum, quæ triangulum & <lb></lb>ipſa excedat minori ſpacio quantacumque magnitudine <lb></lb>propoſita. </s> <s>Cylindrorum autem qui ſunt circa conoides, & <lb></lb>parallelogrammorum multitudine æqualium, quæ ſunt cir<lb></lb>ca triangulum ABC, duo proximi baſi AC cylindri ſint <lb></lb>AF, HL, & totidem parallelogramma illis reſpondentia <lb></lb>inter eadem plana parallela ſint AF, GK. </s> <s>Quoniam igi-<pb xlink:href="043/01/117.jpg" pagenum="30"></pb>tur in parabola ABC rectis ad diametrum ordinatim ap<lb></lb>plicatis eſt vt BM ad BD longitudine, ita MH ad AD <lb></lb>potentia: hoc eſt, ita circulus, cuius diameter HMN, ad <lb></lb>circulum, cuius diameter ADC, hoc eſt ita cylindrus HL, <lb></lb>ad cylindrum AF propter æqualitatem altitudinum: ſed <lb></lb>vt BM ad BD, ita eſt GM ad AD, propter ſimilitudinem <lb></lb>triangulorum, hoc eſt ita <expan abbr="parallelogrãmum">parallelogrammum</expan> GK ad AF, pa<lb></lb>rallelogrammum; ergo vt parallelogrammum GK ad paral <lb></lb><expan abbr="lelogrãmum">lelogrammum</expan> AF, ita eſt cylindrus HL ad cylindrum AF. <lb></lb></s> <s>Similiter oſtenderemus reliqua parallelogramma, quæ ſunt <lb></lb><figure id="id.043.01.117.1.jpg" xlink:href="043/01/117/1.jpg"></figure><lb></lb>circa <expan abbr="triãgulum">triangulum</expan> ABC eſse cum reliquis cylindris, qui ſunt <lb></lb>circa conoides ABC bina ſumpta prout inter ſe reſpon<lb></lb>dent in eadem proportione; ſemper igitur componendo, & <lb></lb>ex æquali erit vt tota figura triangulo ABC circumſcripta <lb></lb>ad parallelogrammum AF, ita figura conoidi circumſcri<lb></lb>pta ad AF cylindrum: ſed vt parallelogrammum AF, ad <lb></lb>parallelogrammum AE, ita eſt cylindrus AF ad cylindrum <lb></lb>AE, propter æqualitatem omnifariam ſumptarum altitu<lb></lb>dinum; ex æquali igitur erit vt figura triangulo ABC cir<lb></lb>cumſcripta ad parallelogrammum AE, ita figura conoidi <pb xlink:href="043/01/118.jpg" pagenum="31"></pb>ABC circumſcripta ad AE cylindrum: vtraque autem <lb></lb>circumſcriptarum figurarum excedit ſibi inſcriptam mino<lb></lb>ri ſpacio quantacumque magnitudine propoſita, vt igitur <lb></lb>triangulum ABC, ad parallelogrammum AE, ita erit co<lb></lb>noides ABC, ad cylindrum AE. </s> <s>Sed triangulum ABC <lb></lb>eſt parallelogrammi AE dimidium; igitur conoides ABC <lb></lb>eſt cylindro AE dimidium: ſed cylindrus AE eſt coni <lb></lb>ABC, triplum: igitur conoides ABC, erit coni ABC <lb></lb>ſeſquialterum. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis priſmatis triangulam baſim habentis <lb></lb>centrum grauitatis rectam lineam, quæ cuiuſlibet <lb></lb>trium laterum bipartiti ſectionem, & oppoſiti pa<lb></lb>rallelogrammi centrum iungit, ita diuidit, vt <lb></lb>pars, quæ attingit latus ſit dupla reliquæ. </s></p><p type="main"> <s>Sit priſma, quale diximus AB <lb></lb>CDEF, ſectoque vno ipſius la<lb></lb>tere BF in puncto G, bifariam <lb></lb>parallelogrammi oppoſiti ſit cen <lb></lb>trum H, & iuncta GH, cuius <lb></lb>pars GK ſit dupla reliquæ <emph type="italics"></emph>K<emph.end type="italics"></emph.end>H. <lb></lb></s> <s>Dico priſmatis ABCDEF, cen <lb></lb>trum grauitatis eſſe K. </s> <s>Per pun <lb></lb>ctum enim H ducatur NO ip<lb></lb>ſi AE, vel CD parallela, quæ <lb></lb>ipſas AC, ED, ſecabit <expan abbr="bifariã">bifariam</expan>: <lb></lb>iunctisque BN, FO, ducatur per <lb></lb>punctum <emph type="italics"></emph>K<emph.end type="italics"></emph.end>, ipſi FB, vel NO <lb></lb><figure id="id.043.01.118.1.jpg" xlink:href="043/01/118/1.jpg"></figure><lb></lb>parallela LM. </s> <s>Quoniam igitur eſt vt HK ad KG, ita <lb></lb>NL ad LB, & OM ad MF, erit NL, ipſius LB, & OM <pb xlink:href="043/01/119.jpg" pagenum="32"></pb>ipſius MF dimidia: ſed & rectæ BN, FO, triangulorum <lb></lb>baſes AC, ED, bifariam ſe<lb></lb>cant; erunt igitur puncta L, M, <lb></lb>centra grauitatis triangulorum <lb></lb>ABC, DEF, oppoſitorum. <lb></lb></s> <s>Priſmatis igitur ABCDEF <lb></lb>axis erit LM: quare in eius bi<lb></lb>partiti ſectione priſmatis ABC <lb></lb>DEF centrum grauitatis: ſectus <lb></lb>autem eſt axis LM bifariam in <lb></lb>puncto K; nam ob parallelogram <lb></lb>ma eſt vt NH ad HO, ita LK <lb></lb>ad KM; priſmatis igitur ABC <lb></lb>DEF, centrum grauitatis erit <emph type="italics"></emph>K.<emph.end type="italics"></emph.end><lb></lb>Quod demonſtrandum erat. </s></p><figure id="id.043.01.119.1.jpg" xlink:href="043/01/119/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis priſmatis baſim habentis trapezium, cu<lb></lb>ius duo latera inter ſe ſint parallela centrum gra<lb></lb>uitatis rectam lineam, quæ æque inter ſe diſtan<lb></lb>tium parallelogrammorum centra iungit, ita di<lb></lb>uidit, vt pars, quæ dictorum parallelogrammorum <lb></lb>minus attingit ſit ad reliquam, vt duorum baſis la <lb></lb>terum parallelorum dupla maioris vna cum mino<lb></lb>ri ad duplam minoris vna cum maiori. </s></p><p type="main"> <s>Sit priſma ABCDEFGH, cuius baſis trapezium <lb></lb>ABCD, habens duo latera AD, BC, inter ſe paralle<lb></lb>la, ſitque eorum AD maius: parallela igitur erunt inter ſe <lb></lb>duo parallelogramma BG, AH. </s> <s>Sit parallelogrammi AH <lb></lb>centrum K, & BG parallelogrammi centrum L, iuncta-<pb xlink:href="043/01/120.jpg" pagenum="33"></pb>que LK, fiat vt dupla ipſius AD vna cum BC ad du<lb></lb>plam ipſius BC vna cum AD, ita LR ad RK. </s> <s>Dico <lb></lb>priſmatis AG centrum grauitatis eſse R. </s> <s>Ducantur enim <lb></lb>per puncta L, K lateribus priſmatis, atque ideo inter ſe <lb></lb>parallelæ MN, OP, quæ <lb></lb>ob centra K, L, ſecabunt <lb></lb>oppoſita parallelogrammo<lb></lb>rum latera bifariam, eas <lb></lb>ſectiones connectant MO, <lb></lb>NP, ipſique MN, vel <lb></lb>OP, parallela ducatur Q <lb></lb>RS. </s> <s>Quoniam igitur eſt <lb></lb>vt LR ad R<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, hoc eſt vt <lb></lb>dupla ipſius AD vna cum <lb></lb>BC ad duplam ipſius BC <lb></lb>vna cum AD, ita OQ ad <lb></lb>QM, & recta MO bifa<lb></lb><figure id="id.043.01.120.1.jpg" xlink:href="043/01/120/1.jpg"></figure><lb></lb>riam ſecat AC trapezij latera parallela, punctum Q, AC <lb></lb>trapezij centrum grauitatis; ſimiliter & punctum S erit EG, <lb></lb>trapezij centrum grauitatis: priſmatis igitur AG axis erit <lb></lb>QS, & centrum grauitatis R, quod eſt in medio axis. <lb></lb></s> <s>Omnis igitur priſmatis baſim habentis trapezium, &c. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si à quolibet prædicto priſmate duo priſmata <lb></lb>beſes habentia triangulas ſint ita abſciſſa, vt pa<lb></lb>rallelepipedum relinquant baſim habens minus <lb></lb>parallelogrammorum inter ſe parallelorum præ<lb></lb>dicti priſmatis, maioris autem partes æqualia pa<lb></lb>rallelogramma ipſum parallelepipedum relin<pb xlink:href="043/01/121.jpg" pagenum="34"></pb>quat, centrum grauitatis vtriuſque abſciſsi priſ<lb></lb>matis tamquam vnius magnitudinis rectam line<lb></lb>lam, quæ prædicti priſmatis parallelorum paral <lb></lb>lelogrammorum centra iungit, ita diuidit, vt <lb></lb>pars, quæ minus parallelogrammum attingit ſit <lb></lb>dupla reliquæ. </s></p><p type="main"> <s>Sit priſma ABCDEFGH, cuius baſes oppoſitæ tra<lb></lb>pezia ADHE, BCGF. </s> <s>Sint autem AD, EH, paral<lb></lb>lelæ, quarum maior EH. </s> <s>Oppoſita igitur parallelogram<lb></lb>ma AC, EG, inter ſe erunt parallela, quorum maius EG. <lb></lb></s> <s>At per rectas AB, CD, ſectum ſit priſma. </s> <s>ABCDEF <lb></lb>GH, ita vt abſciſſa priſmata ABSFER, CDVHGT, <lb></lb>relinquant parallelepipedum AT, ipſum autem AT, re<lb></lb>linquat duo parallelogramma æqualia ES, TH. </s> <s>Poſito <lb></lb>autem centro K <lb></lb>parallelogrammi <lb></lb>AC, & L, paral <lb></lb>lelogrammi EG, <lb></lb>iunctaque KL, <lb></lb>ponatur KM, du <lb></lb>pla ipſius ML. <lb></lb></s> <s>Dico <expan abbr="duorũ">duorum</expan> priſ<lb></lb>matum BER, <lb></lb>CVH, ſimul cen <lb></lb>trum grauitatis <lb></lb><figure id="id.043.01.121.1.jpg" xlink:href="043/01/121/1.jpg"></figure><lb></lb>eſse M. </s> <s>Sectis enim AB, CD, bifariam in punctis P, Q, <lb></lb>ſumptiſque parallelogrammorum ES, VG, centris N, O, <lb></lb>iungantur PN, QO, & poſita PX dupla ipſius XN, & QZ <lb></lb>dupla ipſius ZO, iungantur rectæ PKQ, XZ, NO. <lb></lb></s> <s>Quoniam igitur in quadrilatero PQON, recta XZ, pa<lb></lb>rallela eſt vtrilibet ipſarum PQ, NO, ſecat ijs parallelis <lb></lb>interceptas in eaſdem rationes; recta igitur XT per pun-<pb xlink:href="043/01/122.jpg" pagenum="35"></pb>ctum M tranſibit. </s> <s>Sed quia PK eſt æqualis KQ, & NL <lb></lb>ipſi LO, etiam XM æqualis erit ipſi MZ ob parallelas; <lb></lb>cum igitur priſmatum BER, CVH centra grauitatis ſint <lb></lb>X, Z; erit vtriuſque priſmatis prædicti ſimul centrum gra<lb></lb>uitatis M. </s> <s>Quod eſt propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint duæ pyramides æquales, & æque altæ, <lb></lb>baſes habentes in eodem plano, quarum vertices <lb></lb>recta linea connectens cum ea, quæ baſium centra <lb></lb>grauitatis iungit ſit in eodem plano; earum cen<lb></lb>trum grauitatis tamquam vnius magnitudinis re<lb></lb>ctam lineam, quæ inter vertices, & centra baſium <lb></lb>interiectas bifariam ſecat, itadiuidit, vt pars ſu<lb></lb>perior ſit inferioris tripla. </s></p><figure id="id.043.01.122.1.jpg" xlink:href="043/01/122/1.jpg"></figure><p type="main"> <s>Sint duæ <lb></lb>pyramides æ<lb></lb>quales, & æ<lb></lb>que altæ, qua<lb></lb>rum baſes in <lb></lb>eodem plano <lb></lb>AC, DB, ver <lb></lb>tices autem <lb></lb>G, H, & ba<lb></lb>ſium <expan abbr="cẽtra">centra</expan> E, <lb></lb>F, iunctæque <lb></lb>EF, GH, quas <lb></lb>bifariam ſecet recta KL, huius autem pars quarta ſit LM. <lb></lb></s> <s>Dico vtriuſque pyramidis GAC, HDB, ſimul centrum <lb></lb>grauitatis eſſe M. </s> <s>Iunctis enim GE, HF, ſumantur ea<pb xlink:href="043/01/123.jpg" pagenum="36"></pb>rum quartæ partes EN, FO, & iungatur NO. </s> <s>Quoniam <lb></lb>igitur propter æqualitatem altitudinum, & quia EF, GH, <lb></lb>ſunt in eodem plano, ſunt EF, GH, inter ſe parallelæ, & <lb></lb>vt GN ad NE, ita eſt HO ad OF; erit NO ipſi E Fivel <lb></lb>GH, paralle<lb></lb>la, quas KL <lb></lb>bifariam ſecat: <lb></lb>igitur & ipſam <lb></lb>NO ſecabit bi <lb></lb>fariam, iungit <lb></lb>autem recta <lb></lb>NO centra <lb></lb>grauitatis <expan abbr="py-ramidũ">py<lb></lb>ramidum</expan> æqua<lb></lb>lium GAC, <lb></lb>HDB, vtriuſ<lb></lb><figure id="id.043.01.123.1.jpg" xlink:href="043/01/123/1.jpg"></figure><lb></lb>que ergo pyramidis ſimul centrum grauitatis erit in com<lb></lb>muni ſectione duarum linearum KL, NO, ſed recta NO, <lb></lb>ſecans ſimiliter ipſas GE, KL, HF, ipſam KL, ſecabit <lb></lb>in puncto M; punctum igitur M, erit prædictarum pyrami<lb></lb>dum centrum grauitatis. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti pyramidis baſim habentis paral<lb></lb>lelogrammum centrum grauitatis maiori baſi eſt <lb></lb>propinquius, quam punctum illud, in quo axis ſic <lb></lb>diuiditur, vt pars minorem baſim attingens ſit ad <lb></lb>reliquam vt dupla cuiuſuis laterum maioris baſis <lb></lb>vna cum latere minoris ſibi reſpondente, ad <expan abbr="duplã">duplam</expan> <lb></lb>dicti lateris minoris baſis vna cum maioris ſibi <lb></lb>reſpondente. </s></p><pb xlink:href="043/01/124.jpg" pagenum="37"></pb><p type="main"> <s>Sit pyramidis, cuius baſis parallelogrammum EFGH, <lb></lb>fruſtum ABCDEFGH, <expan abbr="eiuſq;">eiuſque</expan> axis KL, quo ſecto in pun <lb></lb>cto <foreign lang="grc">α</foreign> ita vt K <foreign lang="grc">α</foreign> ad <foreign lang="grc">α</foreign> L, ſit vt laterum homologorum AD <lb></lb>EH, dupla ipſius EH vna cum AD ad duplam ipſius <lb></lb>AD vna cum EH, & fruſti ABCDEFGH ſit centrum <lb></lb>grauitatis <foreign lang="grc"><gap></gap></foreign> nempe in axe KL. </s> <s>Dico punctum <foreign lang="grc"><gap></gap></foreign>, cadere <lb></lb>infra punctum <foreign lang="grc">α. </foreign></s> <s>A punctis enim A,B,C,D, ducantur <lb></lb><figure id="id.043.01.124.1.jpg" xlink:href="043/01/124/1.jpg"></figure><lb></lb>ad maiorem baſim axi KL, parallelæ AN, BO, CR, DS, <lb></lb>& parallelepipedum ABCDNORS compleatur, & <lb></lb>productis baſis NO lateribus, deſcriptæ ſint quatuor py<lb></lb>ramides AEMNZ, BOPFY, CGXRQ, DHVST, <lb></lb>quarum baſes erunt parallelogramma circa diametrum <lb></lb>æqualia, atque ſimilia: & quatuor priſmata triangulas ba<lb></lb>ſes habentia, quorum binorum ex aduerſo inter ſe reſpon-<pb xlink:href="043/01/125.jpg" pagenum="38"></pb>dentium parallelogramma in plano EG exiſtentia erunt <lb></lb>inter ſe æqualia, atque ſimilia, ſcilicet MS ipſi OQ, & <lb></lb>ZO, ipſis RV: ſitque axis KL pars tertia L <foreign lang="grc">β</foreign>, quarta <lb></lb>autem L <foreign lang="grc">δ. </foreign></s> <s>Quoniam ìgitur ex ſupra demonſtratis priſ<lb></lb>matis ABCDTMPQ eſt centrum grauitatis <foreign lang="grc">α</foreign>; duo<lb></lb>rum autem priſmatum oppoſitorum ABYONZ, CDS <lb></lb>RXV, centrum grauitatis <foreign lang="grc">β</foreign>, erit reliqui ex fruſto AB <lb></lb><figure id="id.043.01.125.1.jpg" xlink:href="043/01/125/1.jpg"></figure><lb></lb>CDEFGH demptis quatuor prædictis pyramidibus in <lb></lb><foreign lang="grc">α β</foreign> centrum grauitatis, quod ſit <foreign lang="grc">γ. </foreign></s> <s>Nam ex primo li<lb></lb>bro conſtat punctum <foreign lang="grc">α</foreign> cadere ſupra punctum <foreign lang="grc">β</foreign>, ſi com<lb></lb>pleatur trapezium ACGE, cuius diameter erit KL. </s> <s>Sed <lb></lb>earum quatuor pyramidum eſt centrum grauitatis <foreign lang="grc">δ. </foreign></s> <s>Si <lb></lb>enim baſium, quibus binæ oppoſitæ pyramides inſiſtunt <lb></lb>centra grauitatis, & bini oppoſiti vertices ſingulis rectis li-<pb xlink:href="043/01/126.jpg" pagenum="39"></pb>neis connectantur, erunt binæ connectentes parallelæ, & <lb></lb>ab axe <emph type="italics"></emph>K<emph.end type="italics"></emph.end> L bifariam ſecabuntur, vt figuræ deſcriptio ina<lb></lb>nifeſtat. </s> <s>Totius igitur fruſti ABCDEFGH, centrum <lb></lb>grauitatis <foreign lang="grc"><gap></gap></foreign> in linea <foreign lang="grc">γ δ</foreign> cadet: ſed punctum <foreign lang="grc">γ</foreign> cadit infra <lb></lb>punctum <foreign lang="grc">α</foreign>, multo ergo inferius, & baſi EG propinquius <lb></lb>punctum <foreign lang="grc"><gap></gap></foreign> quam punctum <foreign lang="grc">α. </foreign></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti conici centrum grauitatis pro<lb></lb>pinquius eſt maiori baſi quam punctum illud, in <lb></lb>quo axis ſic diuiditur, vt pars minorem baſim <lb></lb>attingens ſit ad reliquam, vt dupla diametri ma<lb></lb>ior is baſis vna cum minoris diametro ad duplam <lb></lb>diametri minoris baſis vna cum diametro ma<lb></lb>ioris. </s></p><p type="main"> <s>Hoc eadem ratione deducetur ex antecedenti, qua cen<lb></lb>trum grauitatis fruſti conici in extremo primo libro demon <lb></lb>ſtrauimus, quandoquidem ſimiliter vt ibi fecimus, omnis <lb></lb>pyramidis centro grauitatis idem probaremus accedere <lb></lb>quod prædictæ pyramidis in antecedente. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint quotcumque magnitudines, & aliæ illis <lb></lb>multitudine æquales, binæque ſumptæ in eadem <lb></lb>proportione, quæ commune habeant centrum gra<lb></lb>uitatis, centra autem grauitatis omnium ſint in <lb></lb>eadem recta linea; primæ & ſecundæ tanquam <pb xlink:href="043/01/127.jpg" pagenum="40"></pb>duæ magnitudines commune habebunt centrum <lb></lb>grauitatis. </s></p><p type="main"> <s>Sit recta linea AB, & quotcumque magnitudines <lb></lb>FGH, & totidem KLM, binæ in eadem proportione: <lb></lb>nimirum vt F ad G ita K ad L: & vt G ad H ita L ad <lb></lb>M. in recta autem AB, ſint communia centra grauitatis, <lb></lb>C duarum FK, & D duarum GL: & E duarum HM. </s> <s>Om<lb></lb>nium autem primarum tamquam vnius magnitudinis ſit <lb></lb>centrum grauitatis O. </s> <s>Dico & omnium ſecundarum ſi<lb></lb>mul centrum grauitatis eſse O. </s> <s>Duarum enim FG ſi<lb></lb><figure id="id.043.01.127.1.jpg" xlink:href="043/01/127/1.jpg"></figure><lb></lb>mul ſit centrum grauitatis N. </s> <s>Vtigitur eſt F ad G, hoc <lb></lb>eſt, vt K ad L, ita erit DN, ad NC. punctum igitur N <lb></lb>eſt centrum grauitatis duarum magnitudinum KL ſimul. <lb></lb></s> <s>Rurſus, quia componendo, & ex æquali, eſt vt FG ſimul <lb></lb>ad H, ita KL ſimul ad M: eſt autem tam duarum FG, <lb></lb>quam duarum KL ſimul centrum grauitatis N, ſimiliter <lb></lb>vt ante oſtenderemus duarum magnitudinum FGH, <lb></lb>KLM centrum grauitatis eſse O. </s> <s>Quod eſt propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint quotcumque magnitudines, & aliæ ip<lb></lb>ſis multitudine æquales primarum, ex quibus cen <lb></lb>tra grauitatis in eadem recta linea diſpoſita ſint <lb></lb>alternatim ad centra grauitatis ſecundarum, qua-<pb xlink:href="043/01/128.jpg" pagenum="41"></pb>rum magnitudinum binæ eodem ordine, qui ſu<lb></lb>mitur ab eodem prædictæ lineæ termino vnain <lb></lb>primis, & alterain ſecundis inter ſe ſint æquales; <lb></lb>omnium primarum ſimul, ex quibus primæ cen<lb></lb>trum grauitatis propinquius eſt prædicto lineæ <lb></lb>termino quàm primæ ſecundarum, propinquius <lb></lb>erit prædicto lineæ termino quàm omnium ſecun<lb></lb>darum ſimul centrum grauitatis. </s></p><p type="main"> <s>Sint quotcumque magnitudines ABC primæ, & toti<lb></lb>dem ſecundæ DEF, quarum centra grauitatis in recta <lb></lb>linea TV, primarum quidem G ipſius A proximum om<lb></lb><figure id="id.043.01.128.1.jpg" xlink:href="043/01/128/1.jpg"></figure><lb></lb>nium termino T, à quo ſumitur ordo. </s> <s>Deinde H ipſius B, <lb></lb>& <emph type="italics"></emph>K<emph.end type="italics"></emph.end>, ipſius C, diſpoſita ſint alternatim ad centra ſecun<lb></lb>darum; videlicet vt centrum grauitatis L, ipſius D cadat <lb></lb>inter centra G, H, & M ipſius E inter centra H, K: & N <lb></lb>inter puncta <emph type="italics"></emph>K<emph.end type="italics"></emph.end>, V: ſint autem æquales binæ AD, BE, <lb></lb>CF: & omnium ABC ſimul centrum grauitatis P, & om<lb></lb>nium DEF ſimul centrum grauitatis O. </s> <s>Dico punctum <lb></lb>P propinquius eſſe termino T, quàm punctum O. <lb></lb></s> <s>Duarum enim A, B ſit centrum grauitatis R: & S, dua<lb></lb>rum DB, & Q, duarum DE. </s> <s>Quoniam igitur Q eſt <lb></lb>centrum grauitatis duarum magnitudinum DE ſimal; erit <lb></lb>vt D ad E, hoc eſt ad B, ita MQ, ad QL: hoc eſt HS, <lb></lb>ad SL. & componendo, vt ML, ad LQ, ita HL, ad <lb></lb>LS; & permutando, vt ML ad LH, ita LQ ad LS: <lb></lb>ſed ML eſt maior quàm LH; ergo & LQ erit maior <lb></lb>quàm LS. </s> <s>Eadem ratione quoniam S eſt centrum gra<pb xlink:href="043/01/129.jpg" pagenum="42"></pb>uitatis duarum DB: & R duarum AB: & AD ſunt æ<lb></lb>quales; erit RH maior quàm SH: ſed quia LQ erat ma<lb></lb>ior quàm LS, eſt & SH maior quàm QH; multo igitur <lb></lb>maior RH erit quàm QH: atque ideo punctum R pro<lb></lb>pinquius termino T, quàm punctum <expan abbr="q.">que</expan> Rurſus quo<lb></lb>niam tota magnitudo AB eſt æqualis toti DE, & C æ<lb></lb>qualis F; erunt duæ primæ AB, & C, & totidem ſecun<lb></lb>dæ DE, & F, quarum vnius poſteriorum DE cen<lb></lb>trum grauitatis Q cadit inter R, K centra grauitatis <lb></lb>duarum priorum AB, & C, & reliquæ priorum C cen<lb></lb>trum grauitatis K cadit inter Q, N, duarum poſterio<lb></lb>rum DE, & F centra grauitatis; erunt vt antea quatuor <lb></lb>magnitudines binæ proximæ æquales, ſcilicet AB, ipſi <lb></lb><figure id="id.043.01.129.1.jpg" xlink:href="043/01/129/1.jpg"></figure><lb></lb>DE: & C ipſi F, centra grauitatis habentes diſpofita <lb></lb>alternatim in eadem recta TV. </s> <s>Cum igitur primæ prio<lb></lb>rum AB, centrum grauitatis R ſit termino T propin<lb></lb>quius quàm Q centrum grauitatis primæ poſteriorum, <lb></lb>quæ eſt tota DE; ſimiliter vt ante totius magnitudinis <lb></lb>ABC centrum grauitatis P erit termino T propinquius <lb></lb>quàm totius DEF centrum grauitatis O. </s> <s>Non aliter <lb></lb>oſtenderemus, quotcumque plures magnitudines, quales <lb></lb>& quemadmodum diximus ad rectam TV, diſpoſitæ <lb></lb>proponerentur, ſemper centrum grauitatis omnium prio<lb></lb>rum ſimul termino T propinquius cadere, quàm omnium <lb></lb>poſteriorum ſimul centrum grauitatis. </s> <s>Manifeſtum eſt <lb></lb>igitur propoſitum. </s></p><pb xlink:href="043/01/130.jpg" pagenum="43"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint quotcumque magnitudines, & aliæ illis <lb></lb>multitudine æquales, quæ binæ commune habe<lb></lb>ant in eadem recta centrum grauitatis; ſumpto au <lb></lb>tem ordine ab vno eius lineæ termino, maior ſit <lb></lb>proportio primæ ad ſecundam in primis, quàm <lb></lb>primæ ad ſecundam in ſecundis: & ſecundæ ad <lb></lb>tertiam in primis maior quàm ſecundæ ad ter<lb></lb>tiam in ſecundis, & ſic deinceps vſque ad vltimas; <lb></lb>erit omnium primarum ſimul centrum grauitatis <lb></lb>propinquius prædicto lineæ termino, à quo ſumi<lb></lb>tur ordo, quàm omnium ſecundarum. </s></p><p type="main"> <s>Sint quotcumque magnitudines GHI, & totidem <lb></lb>LMN. </s> <s>Sitque maior proportio G ad H, quàm L ad M: & <lb></lb>H ad I, maior quàm M ad N: in recta autem AB ſint <lb></lb>communia centra grauitatis, C duarum magnitudinum <lb></lb>GL, & D duarum HM, & E duarum IN. omnium <lb></lb><figure id="id.043.01.130.1.jpg" xlink:href="043/01/130/1.jpg"></figure><lb></lb>autem primarum GHI ſimul ſit centrum grauitatis K: at <lb></lb>ſecundarum omnium LMN centrum grauitatis R. </s> <s>Di<lb></lb>co centrum K cadere termino A propinquius quàm cen <lb></lb>trum R. </s> <s>Fiat enim vt G ad H, ita DP ad PC: & vt L <lb></lb>ad M, ita DQ ad QC. </s> <s>Maior igitur proportio erit DP <pb xlink:href="043/01/131.jpg" pagenum="44"></pb>ad PC, quàm DQ ad QC: & componendo, maior DC <lb></lb>ad CP, quàm DC ad CQ: minor igitur CP erit quàm <lb></lb>CQ: quare DP maior quàm <expan abbr="Dq.">Dque</expan> & communi addita <lb></lb>ED, erit EP maior quàm <expan abbr="Eq.">Eque</expan> Et quoniam <emph type="italics"></emph>K<emph.end type="italics"></emph.end> eſt cen<lb></lb>trum grauitatis omnium GHI ſimul, & ipſius GH eſt cen <lb></lb>trum grauitatis P, & reliquæ magnitudinis I, centrum <lb></lb>grauitatis E; erit vt GH ad I, ita EK ad KP. eadem <lb></lb>ratione vt vtraque LM ad N, ita erit ER ad <expan abbr="Rq.">Rque</expan> Rur<lb></lb><figure id="id.043.01.131.1.jpg" xlink:href="043/01/131/1.jpg"></figure><lb></lb>ſus, quia maior eſt proportio G ad H, quàm L ad M, erit <lb></lb>componendo, maior proportio GH ad H, quàm LM ad <lb></lb>M: ſed maior eſt proportio H ad K, quàm M ad N; ex <lb></lb>æquali igitur, maior erit proportio GH ad I, quàm LM <lb></lb>ad N, hoc eſt EK ad KP, quàm ER ad <expan abbr="Rq.">Rque</expan> Multo <lb></lb>ergo maior proportio EK ad KP, quàm ER ad RP: & <lb></lb>componendo maior proportio EP ad PK quàm EP ad <lb></lb>PR; minor igitur PK erit quàm PR, at que ideo centrum <lb></lb>K propinquius termino A quàm centrum R. </s> <s>Quod de<lb></lb>monſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint quotcumque magnitudines, & aliæ ipſis <lb></lb>multitudine æquales, quarum omnium centra <lb></lb>grauitatis ſint in eadem recta linea, & centra pri<lb></lb>marum ad centra ſecundarum diſpoſita ſint alter<lb></lb>natim: ſit autem maior proportio primæ ad ſecun-<pb xlink:href="043/01/132.jpg" pagenum="45"></pb>dam in primis quàm primæ ad ſecundam in ſecun<lb></lb>dis: & ſecundæ ad tertiam in primis, maior quàm <lb></lb>ſecundæ ad tertiam in ſe cundis, & ſic deinceps vſ<lb></lb>que ad vltimas; erit omnium primarum ſimul cen <lb></lb>trum grauitatis propinquius prædictæ lineæ ter<lb></lb>mino à quo ſumitur ordo omnium ſecundarum <lb></lb>centrum grauitatis. </s></p><p type="main"> <s>Sit quotcumque magnitudines GHI, & totidem LMN <lb></lb>primarum autem ſint centra grauitatis CDE cum ſecun<lb></lb>darum centris OPQ in eadem recta AB diſpoſita alter<lb></lb>natim, vt O cadat inter puncta CD, & P inter puncta <lb></lb>DE, & E inter puncta <expan abbr="Pq.">Pque</expan> ſitque maior proportio G <lb></lb>ad H, quàm L ad M, & H ad I maior quàm M ad N. <lb></lb>omnium autem primarum GHI ſimul ſit centrum gra<lb></lb>uitatis T; at omnium ſecundarum LMN, ſimul, cen<lb></lb><figure id="id.043.01.132.1.jpg" xlink:href="043/01/132/1.jpg"></figure><lb></lb>trum grauitatis V. </s> <s>Dico punctum T eſſe termino A <lb></lb>propinquius quàm punctum V. </s> <s>Eſto enim F æqualis <lb></lb>L, & K æqualis M, & X æqualis N, ſit autem cen<lb></lb>trum grauitatis ipſius F in puncto C, & ipſius K in pun<lb></lb>cto D, & ipſius X in puncto E. </s> <s>In recta igitur AB om<lb></lb>nium FKX, ſimul centrum grauitatis erit termino A, pro<lb></lb>pinquius quàm omnium LMN ſimul centrum grauitatis. <lb></lb></s> <s>Sed & omnium GHI, ſimul centrum grauitatis in eadem <lb></lb>recta AB propinquius eſt termino A quàm omnium <lb></lb>FKX, ſimul centrum grauitatis; multo igitur termino A <lb></lb>propinquius erit omnium GHI ſimul quàm omnium <pb xlink:href="043/01/133.jpg" pagenum="46"></pb>LMN, ſimul centrum grauitatis. </s> <s>Quod demonſtran<lb></lb>dum erat. </s></p><p type="head"> <s><emph type="italics"></emph>ALITER.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Poſito enim R centro grauitatis duarum <expan abbr="magnitudinũ">magnitudinum</expan> G, <lb></lb>H, & S <expan abbr="duarũ">duarum</expan> L,M, vel punctum V cadit in puncto E, vel in <lb></lb>linea EB, vel in linea AE, ſi in puncto E vel in linea EB, <lb></lb>cum igitur T ſit <expan abbr="centrũ">centrum</expan> grauitatis trium <expan abbr="magnitudinũ">magnitudinum</expan> G,H,I <lb></lb>ſimul, & E ipſius I, erit punctum T propinquius termino <lb></lb>A quàm punctum V. </s> <s>Sed punctum V in linea AE cadat. <lb></lb></s> <s>Veligitur S centrum grauitatis duarum magnitudinum L, <lb></lb>M, ſimul cadit in puncto D, ſiue in linea DB, vel in li<lb></lb>nea AD. ſi in puncto D, vel in linea DB; centrum gra<lb></lb>uitatis R duarum magnitudinum GH erit termino A <lb></lb>propinquius quàm ipſum S, & recta ER maior quàm ES, <lb></lb><figure id="id.043.01.133.1.jpg" xlink:href="043/01/133/1.jpg"></figure><lb></lb>Sed cadat punctum S in linea AD. </s> <s>Quoniam igitur ma<lb></lb>ior eſt proportio G ad H, quàm L ad M: & vt G ad H, <lb></lb>ita eſt DR ad RG, & vt L ad M, ita PS ad SO, ma<lb></lb>ior erit proportio DR ad RC, quàm PS ad SO; mul<lb></lb>to ergo maior DR ad RC, quàm DS ad SO, & multo <lb></lb>maior quàm DS ad SC, & componendo maior propor<lb></lb>tio DC ad CR, quàm DC ad CS; erit igitur CR mi<lb></lb>nor quàm CS, atque adeo RD maior DS, addita igitur <lb></lb>ED communi, erit ER maior quàm ES. </s> <s>Rurſus quia <lb></lb>componendo, & ex æquali maior eſt proportio totius GH <lb></lb>ad I quàm totius LM ad N, hoc eſt maior longitudinis <lb></lb>ET ad TR, quàm QV ad VS, & multo maior quàm <pb xlink:href="043/01/134.jpg" pagenum="47"></pb>EV ad VS, erit componendo, maior proportio ER ad <lb></lb>RT quàm ES ad SV: & per conuerſionem rationis mi<lb></lb>nor proportio FR ad ET; quàm ES ad EV, & permu<lb></lb>tando minor proportio ER ad ES quàm ET ad EV: ſed <lb></lb>ER maior erat quàm ES, ergo ET maior erit quàm EV: <lb></lb>& punctum T propinquius termino A, quàm punctum V. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Datæ figuræ circa diametrum, vel axim in alte <lb></lb>ram partem deficienti, ſuper baſim rectam lineam <lb></lb>vel circulum, vel ellipſim; cuius figuræ baſis, & <lb></lb>ſectiones omnes parallelæ ſegmenta æqualia dia<lb></lb>metri vel axis intercipientes ita ſe habeant, vt <lb></lb>quarumlibet trium proximarum minor proportio <lb></lb>ſit minimæ ad mediam, quàm mediæ ad maxi<lb></lb>mam; figura quædam ex cylindris, vel cylindri <lb></lb>portionibus, vel parallelogrammis æqualium al<lb></lb>titudinum circumſcribi poteſt, cuius <expan abbr="cẽtrum">centrum</expan> gra<lb></lb>uitatis ſit propinquius baſi quàm cuiuſlibet datæ <lb></lb>figuræ, qualem diximus quæ prædictæ figuræ cir <lb></lb>cadiametrum, vel axim circumſcripta ſit. </s></p><p type="main"> <s>Sit figura circa diametrum, vel axim in alteram <expan abbr="partẽ">partem</expan> de<lb></lb>ficiens qualem diximus, cuius bafis circulus, vel ellipſis vel <lb></lb>recta linea AC, axis autem vel diameter BD. </s> <s>Et data figu<lb></lb>ra ipſi ABC figuræ circumſcripta compoſita ex cylindris, <lb></lb>vel cylindri portionibus, vel parallelogrammis æqualium <lb></lb>altitudinum EF, GH, AK. </s> <s>Dico figuræ ABC alteram <lb></lb>figuram, qualem diximus poſſe circumſcribi, cuius centrum <pb xlink:href="043/01/135.jpg" pagenum="48"></pb>grauitatis, nempe in linea BD, ſit propinquius baſi AC, <lb></lb>ſiue termino D, quàm prædictæ datæ figuræ circumſcriptæ <lb></lb>centrum grauitatis, Omnium enim cylindrorum, vel cy<lb></lb>lindri portionum, vel parallelogrammorum, ex quibus con<lb></lb>ſtat prædicta data figura circumſcripta ſint axes, vel quæ <lb></lb>oppoſita latera coniungunt rectæ BL, LM, MD, qui<lb></lb>bus ſectis bifariam in punctis N, O, P, ac planis per ea <lb></lb>ſiue rectis tranſeuntibus baſi AC parallelis, ſecantibus<lb></lb>que dictos cylindros, vel cylindri portiones, vel pa<lb></lb>rallelogramma, compleatur & figuræ ABC circumſcri<lb></lb>batur altera figura <lb></lb>vt prior, quæ ob ſe<lb></lb>ctiones factas com<lb></lb>ponetur ex duplis <lb></lb>multitudine cylin<lb></lb>dris, vel cylindri por<lb></lb>tionibus, vel paralle<lb></lb>logrammis ęqualium <lb></lb>altitudinum, eorum <lb></lb>ex quibus conſtat da <lb></lb>ta figura circumſcri<lb></lb>pta ſin<gap></gap>autem hi cy<lb></lb>lindri, aut reliqua, <lb></lb>quæ diximus QR, <lb></lb><figure id="id.043.01.135.1.jpg" xlink:href="043/01/135/1.jpg"></figure><lb></lb>ES, TV, GX, ZI, AY. </s> <s>Quoniam igitur cylindro<lb></lb>rum, vel cylindri portionum, vel parallelogrammorum quæ <lb></lb>ſunt circa figuram ABC, minor eſt proportio QR ad ES, <lb></lb>quàm ES, ad TV, propter ſectiones circulos, vel ſimiles <lb></lb>ellipſes, vel rectas lineas, & <expan abbr="æqualitatẽ">æqualitatem</expan> <expan abbr="altitudinũ">altitudinum</expan>, & figuræ <lb></lb>propoſitæ <expan abbr="naturã">naturam</expan>. </s> <s>Sed <expan abbr="eadẽ">eadem</expan> ratione minor eſt proportio ES <lb></lb>ad TV, quàm TV, ad GX; multo ergo minor proportio erit <lb></lb>QR ad ES, quam TV ad GX: & componendo, minor <lb></lb>proportio QR, ES, ſimul ad ES, quàm TV, GX, ſimul <lb></lb>ad GX. ſed vt GX ad GH, ita eſt ES ad EF; ex æqua-<pb xlink:href="043/01/136.jpg" pagenum="49"></pb>li igitur minor erit proportio QR, ES ſimul ad EF, <lb></lb>quàm TV, GX ſimul ad GH. & permutando, minor <lb></lb>proportio QR, ES ſimul ad TV, GX ſimul quàm EF <lb></lb>ad GH. & conuertendo, maior proportio GX, TV ſi<lb></lb>mul ad ES, QR ſimul, quàm GH ad EF. </s> <s>Similiter <lb></lb>oſtenderemus duo ZI, AY, ſimul ad TV, GX, ſimul, <lb></lb>maiorem habere proportionem, quàm AK ad rectarum <lb></lb>GH. </s> <s>Rurſus quoniam puncta N, O, in medio BL, LM, <lb></lb>ſunt, ipſorum EF, GH, centra grauitatis: duorum autem <lb></lb>QR, ES ſimul centrum grauitatis eſt in linea NL, pro<lb></lb>pterea quòd ES maius eſt quàm QR, & æquales BN, <lb></lb>NL, quas centra grauitatis ipſorum QR, ES bifariam <lb></lb>diuidunt, cadet ipſorum QR, ES, ſimul centrum grauita<lb></lb>tis propius termino D, quàm ipſius EF centrum grauitatis, <lb></lb>& duobus centris N, O, interijcietur. </s> <s>Eademque ratio<lb></lb>ne duorum TV, GX, ſimul centrum grauitatis termino <lb></lb>D erit propinquius quàm ipſius GH centrum grauitatis, <lb></lb>& duobus centris O, P, duorum GH, AK interijcietur. <lb></lb></s> <s>Et duorum ZI, AY ſimul centrum grauitatis propin<lb></lb>quius erit D termino, quàm P ipſius AK. </s> <s>Quoniam <lb></lb>igitur omnia primarum magnitudinum, ex quibus conſtat <lb></lb>figura ſecundo circumſcripta centra grauitatis in eadem re <lb></lb>cta linea BD, diſpoſita ſunt alternatim ad centra grauita<lb></lb>tis ſecundarum primis multitudine æqualium, ex quibus <lb></lb>data figura conſtat ipſi ABC figuræ circumſcripta, ſunt <lb></lb>termino D propinquiora, quàm centra grauitatis ſecunda<lb></lb>rum, ſi bina, prout inter ſe reſpondent comparentur: maior <lb></lb>autem proportio oſtenſa eſt primæ ad ſecundam in primis, <lb></lb>quàm primæ ad ſecundam in ſecundis: & ſecundæ ad ter<lb></lb>tiam in primis, quàm ſecundæ ad tertiam in ſecundis, <lb></lb>ſumpto ordine à termino D, erit centrum grauitatis om<lb></lb>nium primarum ſimul, ideſt figuræ ipſi ABC figuræ <lb></lb>ſecundo circumſcriptæ termino D propinquius, quàm <lb></lb>datæ figuræ eidem ABC figuræ primo circumſcriptæ cen<pb xlink:href="043/01/137.jpg" pagenum="50"></pb>trum grauitatis. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis prædictæ figuræ centrum grauitatis <lb></lb>eſt propinquius baſi, quàm cuiuſlibet figuræ ex <lb></lb>cylindris, vel cylindri portionibus, vel parallelo<lb></lb>grammis æqualium altitudinum ipſi circumſcri<lb></lb>ptæ. </s></p><p type="main"> <s>Sit prædicta figura ABC, cuius axis vel diameter BD, <lb></lb>& data intelligatur figura ex quotcumque cylindris, vel cy<lb></lb>lindri portionibus, vel parallelogrammis æqualium altitu<lb></lb>dinum figuræ ABC circumſcripta, cuius ſit centrum gra<lb></lb>uitatis E, nempe in axe vel <lb></lb>diametro BD. </s> <s>Dico cen<lb></lb>trum grauitatis figuræ ABC <lb></lb>propinquius eſſe puncto D, <lb></lb>quàm punctum E. </s> <s>Si enim <lb></lb>fieri poteſt, centrum grauita<lb></lb>tis figuræ ABC, quod ſit <lb></lb>F, non cadat infra punctum <lb></lb>E, ſed vel ſupra, vel con<lb></lb>gruat puncto E: figuræ ita<lb></lb>que ABC circumſcribatur <lb></lb>figura quædam ex cylindris, <lb></lb>vel cylindri portionibus, vel <lb></lb>parallelogrammis <17>qualium <lb></lb>altitudinum, cuius centrum <lb></lb><figure id="id.043.01.137.1.jpg" xlink:href="043/01/137/1.jpg"></figure><lb></lb>grauitatis, quod ſit G, ſit propinquius D puncto, quàm <lb></lb>punctum E, ac propterea propinquius, quàm punctum F, <lb></lb>centrum grauitatis figuræ primo circumſcriptæ. </s> <s>Rurſus <lb></lb>multiplicatis cylindris, vel cylindri portionibus, vel paral-<pb xlink:href="043/01/138.jpg" pagenum="51"></pb>lelogrammis circumſcribatur figuræ ABC, altera tertia fi<lb></lb>gura, quemadmodum diximus in præcedenti, cuius cen<lb></lb>trum grauitatis H, in linea GD cadat & ſit minor pro<lb></lb>portio reſidui huius tertiæ figuræ circumſcriptæ ipſi ABC, <lb></lb>ad figuram ABC, quàm FG ad GD. </s> <s>Multo ergo mi<lb></lb>nor proportio erit dicti reſidui ad figuram ABC quam F <lb></lb>H ad HD, fiat igitur vt prædictum reſiduum ad figuram <lb></lb>ABC, ita ex contraria parte FH ad HDK; prædicti igi<lb></lb>tur reſidui centrum grauitatis erit K, extra ipſius terminos, <lb></lb>quod fieri non poteſt: Non igitur F centrum grauitatis fi<lb></lb>guræ ABC cadit in puncto E, nec ſupra; ergo infra pun <lb></lb>ctum E: & ponitur E centrum grauitatis cuiuslibet figuræ <lb></lb>ex cylindris, vel cylindri portionibus, vel parallelogrammis <lb></lb>æqualium altitudinum quo modo diximus ipſi ABC cir<lb></lb>cumſcriptæ. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omni prædictæ figuræ figura quædam ex cylin <lb></lb>dris, vel cylindri portionibus, vel parallelogram<lb></lb>mis æqualium altitudi <lb></lb>num circumſcribi po<lb></lb>teſt, cuius centri graui <lb></lb>tatis diſtantia à prædi<lb></lb>ctæ figuræ centro gra<lb></lb>uitatis ſit minor quan<lb></lb>tacunque longitudine <lb></lb>propoſita. </s></p><figure id="id.043.01.138.1.jpg" xlink:href="043/01/138/1.jpg"></figure><p type="main"> <s>Sit figura ABC in <expan abbr="alterã">alteram</expan> <lb></lb>partem <expan abbr="deficiẽs">deficiens</expan> ſupradicta, <lb></lb>cuius centrum grauitatis F, propoſita autem <expan abbr="quantacũque">quantacumque</expan> <lb></lb><expan abbr="lõgitudine">longitudine</expan> minor ſit FG ipſius BF. </s> <s>Dico figuræ ABC figu-<pb xlink:href="043/01/139.jpg" pagenum="52"></pb>ram ex cylindris vel cylindri portionibus, vel <expan abbr="parallelogrã-mis">parallelogram<lb></lb>mis</expan> æqualium <expan abbr="altitudinũ">altitudinum</expan> circumſcribi poſſe, cuius centrum <lb></lb>grauitatis ſit propinquius puncto F, quàm punctum G: figu<lb></lb>ræ enim ABC figura, qualem diximus circumſcribatur, cu<lb></lb>ius reſiduum dempta figura ABC, ad figuram ABC mi<lb></lb>norem habeat proportionem, quàm FG, ad GB, ſit autem <lb></lb>figuræ circumſcriptæ centrum grauitatis K, nempe in axe, <lb></lb>vel diametro BD. </s> <s>Dico <lb></lb>lineam FK minorem eſſe <lb></lb>quàm FG, atque adeo lon <lb></lb>gitudine propoſita. </s> <s>Quo<lb></lb>niam enim F eſt centrum <lb></lb>grauitatis figuræ ABC, <lb></lb>erit centrum grauitatis <emph type="italics"></emph>K<emph.end type="italics"></emph.end>, <lb></lb>figuræ circumſcriptæ ipſi <lb></lb>ABC propinquius termi<lb></lb>no B, quàm punctum F, <lb></lb>ſed centrum grauitatis fi<lb></lb>guræ ABC quòd eſt F, & <lb></lb>figuræ circumſcriptæ, quod <lb></lb>eſt K & eius reſidui dem<lb></lb><figure id="id.043.01.139.1.jpg" xlink:href="043/01/139/1.jpg"></figure><lb></lb>pta figura ABC ſunt in communi axe, vel diametro BD; <lb></lb>erit igitur dicti reſidui in linea BK, centrum grauitatis, <lb></lb>quod ſit H. </s> <s>Minor autem proportio eſt prædicti reſidui <lb></lb>ad figuram ABC, hoc eſt ipſius FK ad KH, quàm FG <lb></lb>ad GB, & multo minor, quàm FG ad GH; & compo<lb></lb>nendo minor proportio FH ad HK, quàm FH ad HG; <lb></lb>ergo KH maior erit, quàm GH; reliqua igitur F <emph type="italics"></emph>K<emph.end type="italics"></emph.end> mi<lb></lb>nor, quàm FG atque adeo longitudine propoſita. </s> <s>Fieri <lb></lb>ergo poteſt, quod proponebatur. </s></p><pb xlink:href="043/01/140.jpg" pagenum="53"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si duarum prædictarum figurarum circa com<lb></lb>munem axim, vel diametrum, vel alterius diame<lb></lb>trum alterius axim, baſes, & quotcumque ſectio<lb></lb>nes quales diximus, binæ in eodem plano fue<lb></lb>rint proportionales; idem punctum in diametro, <lb></lb>vel axe erit vtriuſque centrum grauitatis. </s></p><p type="main"> <s>Sint duæ prædictæ figuræ ABC, DBE, circa eandem <lb></lb>diametrum, vel axim BF. figuræ autem ABC ſit cen<lb></lb>trum grauitatis G, nempe in linea BF. </s> <s>Dico G eſſe <lb></lb>centrum grauitatis <lb></lb>figuræ DBE. ſi <lb></lb>enim non eſt, ſit a<lb></lb>liud punctum H, <lb></lb>quod cadat primo <lb></lb>ſupra punctum G. <lb></lb></s> <s>Figuræ igitur AB <lb></lb>C, figura circum<lb></lb>ſcribatur qualem <lb></lb>diximus ex cylin<lb></lb>dris, vel cylindri <lb></lb>portionibus, vel pa<lb></lb>rallelogrammis æ<lb></lb>qualium <expan abbr="altitudinũ">altitudinum</expan> <lb></lb>cuius centri graui<lb></lb>tatis <emph type="italics"></emph>K<emph.end type="italics"></emph.end> diſtantia à <lb></lb><figure id="id.043.01.140.1.jpg" xlink:href="043/01/140/1.jpg"></figure><lb></lb>centro G, figuræ ABC ſit minor quàm recta GH: & figu<lb></lb>ræ DBE, figura circumſcribatur ex cylindris, vel cylindri <lb></lb>portionibus vel parallelogrammis æqualium altitudinum, <lb></lb>multitudine æqualium ijs, ex quibus conſtat ipſi ABC, <pb xlink:href="043/01/141.jpg" pagenum="54"></pb>figura circumſcripta, quæ cum prædictis circa figuram AB <lb></lb>C erunt bina ſumpto ordine à puncto B, in eadem propor<lb></lb>tione inter eadem plana parallela, vel rectas parallelas <expan abbr="cõſi-ſtentia">conſi<lb></lb>ſtentia</expan>, propter ſectiones, ideſt baſes, & æquales altitudines: <lb></lb>binorum autem quorumque homologorum idem erit in li<lb></lb>nea BF, centrum grauitatis: punctum igitur K, centrum <lb></lb>grauitatis figuræ ipſi ABC circumſcriptæ, idem erit fi<lb></lb>guræ ipſi DBE, circumſcriptæ centrum grauitatis: cadi<gap></gap><lb></lb><expan abbr="autẽ">autem</expan> infra centrum <lb></lb>grauitatis H figu<lb></lb>ræ DBE, quod eſt <lb></lb>abſurdum.</s> <s>Non <lb></lb>igitur centrum gra<lb></lb>uitatis figuræ DB <lb></lb>E, cadit ſupra pun <lb></lb>ctum G. </s> <s>Sed ca<lb></lb>dat infra, vt in pun<lb></lb>cto L. </s> <s>Rurſus igi <lb></lb>tur figuræ DBE fi<lb></lb>gura, qualem dixi<lb></lb>mus circumſcripta, <lb></lb>cuius centrum gra<lb></lb>uitatis M, ſit pro<lb></lb>pinquius centro L, <lb></lb><figure id="id.043.01.141.1.jpg" xlink:href="043/01/141/1.jpg"></figure><lb></lb>quàm punctum G, figuræ ABC altera qualem diximus <lb></lb>figura circumſcribatur, cuius centrum grauitatis ſit idem <lb></lb>punctum M, quod fieri poſſe conſtat ex ſuperioribus. </s> <s>Sed <lb></lb>G ponitur centrum grauitatis figuræ ABC; ergo centrum <lb></lb>grauitatis figuræ ipſi ABC, circumſcriptæ erit propinquius <lb></lb>baſi & puncto F, quàm figuræ ABC centrum grauitatis, <lb></lb>quod fieri non poteſt. </s> <s>Non igitur figuræ DBE centrum gra<lb></lb>uitatis cadit infra punctum G. </s> <s>Sed neque ſupra; punctum <lb></lb>igitur G erit commune duarum figurarum ABC, DBE, <lb></lb>centrum grauitatis. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/142.jpg" pagenum="55"></pb><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Manifeſtum eſt autem omnia proximis qua<lb></lb>tuor propoſitionibus <expan abbr="oſtēſa">oſtenſa</expan> de figura circa axim, <lb></lb>vel diametrum in alteram partem deficienti, ea<lb></lb>dem ijſdem rationibus oſten ſa remanere de com<lb></lb>poſito ex duabus figuris circa communem axim <lb></lb>vel diametrum in alteram partem deficientibus, <lb></lb>tam per ſe conſiderato, quàm ad alteram figuram <lb></lb>circa eundem axim, vel diametrum cum prædi<lb></lb>cto compoſito, in alteram partem deficiens, ac ſi <lb></lb>eſſent duæ tantummodo dictæ figuræ, quales in <lb></lb>præcedenti proxima inter ſe comparauimus; ma<lb></lb>nente ſemper illa conditione, quàm de ſectioni<lb></lb>bus in vigeſima huius diximus. </s> <s>Tantum aduer<lb></lb>tendum eſt, vt pro ſectionibus, dicamus compoſita <lb></lb>ex binis ſectionibus (quæ ſcilicet fiunt ab codem <lb></lb>plano, vel eadem recta linea) cum de prædicto com <lb></lb>poſito ſit ſermo: & in demonſtratione, procylin<lb></lb>dris, vel cylindri portionibus, vel parallelogram<lb></lb>mis, compoſita ex binis cylindris, vel cylindri por <lb></lb>tionibus, vel parallelogrammis(quæ ſcilicet ſunt <lb></lb>inter eadem plana parallela, vel lineas parallelas, <lb></lb>& circa eundem axim, vel diametrum totius vel <lb></lb>diametri, vel axis partem) ſicut & pro figura com<lb></lb>poſitum ex duabus dictis figuris: pro reſiduo, com <lb></lb>poſitum ex reſiduis. </s> <s>Nam cum vtriuſque reſidui <pb xlink:href="043/01/143.jpg" pagenum="56"></pb>figurarum duobus prædictis figuris vnum quid <lb></lb>componentibus, & circa eundem axim, vel diame<lb></lb>trum exiſtentibus, qua ratione diximus, circum<lb></lb>ſcriptarum, centra grauitatis ſint in diametro, vel <lb></lb>axe; etiam compoſiti ex ijs duobus reſiduis (vt in <lb></lb>priori libro generaliter demonſtrauimus, cen<lb></lb>trum grauitatis erit in eadem diametro, vel axe: <lb></lb>vnde vim habent proximæ quatuor anteceden<lb></lb>tes demonſtrationes, exemplum erit in demon<lb></lb>ſtratione trigeſimæ quartæ huius. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hemiſphærij centrum grauitatis eſt punctum <lb></lb>illud in quo axis ſic diuiditur, vt pars, quæ ad ver<lb></lb>ticem ſit ad reliquam vt quin que ad tria. </s></p><p type="main"> <s>Eſto hemifphærium ABC cuius vertex B, axis BD: <lb></lb>ſit autem BD ſectus in G puncto, ita vt pars BG ad GD <lb></lb>ſit vt quinque ad tria. </s> <s>Dico G eſse centrum grauitatis <lb></lb>hemiſphærij ABC. </s> <s>Abſcindatur enim BK ipſius BD <lb></lb>pars quarta: & ſuper baſim eandem hemiſphærij eundem<lb></lb>que axim BD cylindrus AF conſiſtat, & conus intelli<lb></lb>gatur EDF, cuius vertex D, baſis autem circulus circu<lb></lb>lo AC oppoſitus, cuius diameter EBF. </s> <s>Sectoque axe <lb></lb>BD bifariam in puncto H, & ſingulis eius partibus rur<lb></lb>ſus bifariam, quoad BD ſecta ſit in partes æquales cu<lb></lb>iuſcumque libuerit numeri paris, tranſeant per puncta ſe<lb></lb>ctionum plana quædam baſi AC parallela, & ſecantia, <lb></lb>hemiſphærium, conum, & cylindrum, quorum omnes ſe<lb></lb>ctiones erunt circuli, terni in codem plano ad aliam atque <pb xlink:href="043/01/144.jpg" pagenum="57"></pb>aliam trium harum figurarum pertinentes. </s> <s>Quod ſi præ<lb></lb>terea factæ ſectiones hemiſphærij ABC à cylindri AF <lb></lb>ſectionibus, circuli à circulis concentricis auferri intelli<lb></lb>gantur; reliquæ totidem erunt ſectiones reliquæ figuræ ſo<lb></lb>lidæ, dempto ABC hemiſphærio ex toto AF cylin<lb></lb>dro, circuli deficientes circulis concentricis, hoc eſt prædi<lb></lb>ctis ABC hemiſphærij ſectionibus prout inter ſe reſpon<lb></lb>dent. </s> <s>Nunc ſuper ſectiones hemiſphærij ABC, & co<lb></lb>ni EDF cylindris conſtitutis circa axes, quæ ſunt ſeg<lb></lb>menta æqualia axis BD, intelligantur duæ figuræ ex cy<lb></lb>lindris æqualium altitudinum, altera inſcripta hemiſphæ<lb></lb><figure id="id.043.01.144.1.jpg" xlink:href="043/01/144/1.jpg"></figure><lb></lb>rio ABC, altera cono EDF circumſcripta. </s> <s>Si igitur <lb></lb>à toto AF cylindro auferatur figura, quæ inſcripta eſt <lb></lb>hemiſphærio ABC, relinquetur figura quædam ex cylin<lb></lb>dris circa prædictos axes, vt ſunt BK, KH, HL, LD, <lb></lb>deficientibus ijs cylindris, ex quibus conſtat figura inſcri<lb></lb>pta hemiſphærio ABC, & vno integro ſupiemo XF <lb></lb>cylindro, circumſcripta reſiduo AF cylindri dempto A <lb></lb>BC hemiſphærio, circumſcriptione interna: talis autem <lb></lb>figuræ circumſcriptæ centrum grauitatis, per ea, quæ in <lb></lb>primo libro, erit in axe BD, quemadmodum & aliarum <lb></lb>duarum figurarum ex cylindris, quarum altera inſcripta <lb></lb>eſt hemiſphærio ABC, altera cono EDF circumſcripta. <pb xlink:href="043/01/145.jpg" pagenum="58"></pb>Quoniam igitur quo exceſsu hemiſphærium ABC ſu<lb></lb>perat ex cylindris figuram ſibi inſcriptam, eodem figura <lb></lb>circumſcripta reliquo cylindri AF, dempto ABC he<lb></lb>miſphærio, ſuperat ipſum reſiduum; figura autem inſcripta <lb></lb>hemiſphærio ABC poteſt eſſe eiuſmodi, quæ ab hemi<lb></lb>ſphærio deficiat minori defectu quantacumque magnitu<lb></lb>dine propoſita; poterit figura, quæ prædicto reſiduo cir<lb></lb>cumſcripta eſt eſſe talis, quæ ipſum reſiduum ſuperet mi<lb></lb>no i exceſsu quantacumque magnitudine propoſita. <lb></lb></s> <s>Ru ſus, quia quemadmodum cylindrus AN infimus de<lb></lb>ficiens cylindro SR, æqualis eſt cylindro TP, ex ſupe<lb></lb><figure id="id.043.01.145.1.jpg" xlink:href="043/01/145/1.jpg"></figure><lb></lb>rioribus, ita vnuſquiſque aliorum cylindrorum deficien<lb></lb>tium cylindris, qui ſunt in hemiſphærio, ex quibus cylin<lb></lb>dris deficientibus conſtat dicto reſiduo figura circumſcri<lb></lb>pta, æqualis eſt cylindrorum circa conum EDF, ei, qui <lb></lb>cum ipſo eſt inter eadem plena parallela, & circa eundem <lb></lb>axem; erunt omnes cylindri circa conum EDF, in ea<lb></lb>dem proportione cum prædictis cylindris deficientibus, <lb></lb>circa prædictum reſiduum, ſi bini ſumantur inter eadem <lb></lb>plana parallela, & circa eundem axem. </s> <s>Quemadmodum <lb></lb>igitur omnium cylindrorum, qui circa conum EDF mi<lb></lb>nor eſt proportio primi ad verticem D, ad ſecundum, <lb></lb>quàm ſecundi ad tertium, & ſecundi ad tertium, quàm ter-<pb xlink:href="043/01/146.jpg" pagenum="59"></pb>tij ad quartum, & ſic ſemper deinceps vſque ad vltimum <lb></lb>XF (duplicatæ enim ſunt talium cylindrorum rationes <lb></lb>earum, quas inter ſe habent diametri æqualibus exceſsibus <lb></lb>differentes circulorum, qui ſunt ſectiones coni, & baſes cy<lb></lb>lindrorum, ex quibus conſtat figura cono EDF circum<lb></lb>ſcripta, ſumpta progreſſione proportionum eodem ordine <lb></lb>gradatim à minima diametro vſque ad maximam EF) ita <lb></lb>erit cylindrorum deficientium, ex quibus conſtat figura <lb></lb>circumſcripta reliquo cylindri AF, dempto ABC hemi<lb></lb>ſphærio, minimi, cuius axis DL ad ſecundum minor pro<lb></lb>portio, quàm ſecundi ad tertium, & ſic deinceps, vſque ad <lb></lb><expan abbr="maximũ">maximum</expan> XF, communiter ad conum EDF, & prædictum <lb></lb>reſiduum pertinentem, ſicut & eorum baſes circuli deficien <lb></lb>tes, quæ ſunt dicti reſidui ſectiones. </s> <s>Cum igitur tam maxi<lb></lb>mi cylindri XF communis, quàm binorum quorumque reli <lb></lb>quorum cylindrorum circa conum EDF, & prædictum reſi <lb></lb>duum inter eadem plana parallela conſiſtentium, quorum <lb></lb>axis communis in BD, commune centrum grauitatis in axe <lb></lb>BD exiſtat, erit ex antecedenti punctum K, quod pono <lb></lb>centrum grauitatis coni EDF, idem reſidui ex cylindro <lb></lb>AF, dempto ABC, hemiſphærio centrum grauitatis. <lb></lb></s> <s>Quoniam igitur quarum partium eſt octo axis BD talium <lb></lb>eſt BG quinque, & BK duarum (ponimus enim nunc K <lb></lb>coni EDF centrum grauitatis) qualium eſt BD octo, ta<lb></lb>lium erit GK trium: ſed KH eſt æqualis BK; qualium <lb></lb>igitur partium eſt GK trium, talium erit KH duarum, ta<lb></lb>liſque vna GH; dupla igitur KH ipſius GH: ſed ABC <lb></lb>hemiſphærium duplum eſt prædicti reſidui, cum ſit cylin<lb></lb>dri AF, ſubſeſquialterum; vt igitur eſt <expan abbr="hemiſphæriũ">hemiſphærium</expan> ABC, <lb></lb>ad prædictum reſiduum, ita ex contraria parte erit <expan abbr="lõgitudo">longitudo</expan> <lb></lb>KH, adlongitudinem GH: ſed H eſt centrum grauitatis <lb></lb>totius cylindri AF & K, prædicti reſidui dempto ABC <lb></lb>hemiſphærio; ergo ABC hemiſphærij centrum grauitatis <lb></lb>erit G. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/147.jpg" pagenum="60"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis minoris portionis ſphæræ centrum gra<lb></lb>uitatis eſt in axe primum bifariam ſecto: deinde <lb></lb>ſecundum centrum grauitatis fruſti circa eun<lb></lb>dem axim, abſciſſi à cono verticem habente cen<lb></lb>trum ſphæræ; in eo puncto, in quo dimidius axis <lb></lb>portionis baſim attingens ſic diuiditur, vt pars <lb></lb>duabus prædictis ſectionibus intercepta ſit ad <lb></lb>eam, quæ inter ſecundam, & tertiam ſectionem <lb></lb>interijcitur, vt exceſſus, quo tripla ſemidiametri <lb></lb>ſphæræ, cuius eſt prædicta portio, ſuperattres de<lb></lb>inceps proportionales, quarum maxima eſt ſphæ<lb></lb>ræ ſemidiameter, media autem, quæ inter centra <lb></lb>ſphæræ, & baſis portionis interijcitur; ad ſemi<lb></lb>diametri ſphæræ triplam. </s></p><p type="main"> <s>Sit minor portio ABC, ſphæræ, cuius centrum D, <lb></lb>ſemidiameter BD, in qua axis portionis ſit BG, baſis <lb></lb>autem circulus, cuius diameter AC: & circa axim BD <lb></lb>deſcriptus eſto conus HDF, cuius baſis circulus FH <lb></lb>tangens portionem in B puncto ſit æqualis circulo ma<lb></lb>ximo, & fruſtum coni HDF abſciſſum vna cum portio<lb></lb>ne ABC ſit KHFL, & vt BD ad DG, ita fiat DG <lb></lb>ad P: ſectoque axe BG bifariam in puncto N, fiat vt <lb></lb>exceſſus, quo tripla ipſius BD ſuperat tres BD, DG, <lb></lb>P, tanquam vnam, ita NM, ad MNO. </s> <s>Dico portio<lb></lb>nis ABC centrum grauitatis eſse O. </s> <s>Nam circa axim <lb></lb>BG, ſuper baſim FH ſtet cylindrus EF, cuius cen-<pb xlink:href="043/01/148.jpg" pagenum="61"></pb>trum grauitatis erit N, reliqui autem eius dempta <lb></lb>ABC portione centrum grauitatis M commune fruſto <lb></lb>KLFH, vt colligitur ex demonſtratione antecedentis. <lb></lb></s> <s>Quoniam igitur eſt vt exceſsus, quo tripla ipſius BD ſu<lb></lb>perat tres BD, DG, P tanquam vnam, ad ipſius BD <lb></lb><figure id="id.043.01.148.1.jpg" xlink:href="043/01/148/1.jpg"></figure><lb></lb>triplam, hoc eſt vt NM ad MO, ita portio ABC ad <lb></lb>EF cylindrum, & diuidendo vt MN ad NO, ita por<lb></lb>tio ABC ad reliquum cylindri EF; & N eſt cylindri <lb></lb>EF, & M prædicti reſidui centrum grauitatis; erit reli<lb></lb>quæ portionis ABC centrum grauitatis O. </s> <s>Quod de<lb></lb>monſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ abſciſſæ duobus pla<lb></lb>nis parallelis, altero per centrum acto, centrum <lb></lb>grauitatis eſt in axe primum bifariam ſecto: dein<lb></lb>de ſumpta ad minorem baſim quarta parte axis <lb></lb>portionis; in eo puncto, in quo dimidius axis mi<lb></lb>norem baſim attingens ſic diuiditur, vt pars dua<lb></lb>bus prædictis ſectionibus intercepta ſit ad eam, <pb xlink:href="043/01/149.jpg" pagenum="62"></pb>quæ interſecundam, & vltimam ſectionem inter<lb></lb>ijcitur, vt exceſſus, quo maior extrema ad ſphæræ <lb></lb>ſemidiametrum, & axim portionis ſuperat ter<lb></lb>tiam partem axis portionis; ad maiorem extre<lb></lb>mam antedictam. </s></p><p type="main"> <s>Sit portio ABCD ſphæræ, cuius centrum F: axis au<lb></lb>tem portionis ſit EF abſciſsæ duobus planis parallelis, <lb></lb>quorum alterum tranſiens per punctum F faciat ſectio<lb></lb>num circulum maximum, cuius diameter AD, reliquam <lb></lb>autem ſectionem minorem circulum, quæ minor baſis di<lb></lb>citur, cuius di<lb></lb>ameter BC: <lb></lb>& vt eſt EF <lb></lb>ad AD, ita <lb></lb>fiat AD ad <lb></lb>OP, cuius P <lb></lb>R, ſit æqua<lb></lb>lis tertiæ parti <lb></lb>axis EF. </s> <s>Et <lb></lb>ſecta EF bi<lb></lb><figure id="id.043.01.149.1.jpg" xlink:href="043/01/149/1.jpg"></figure><lb></lb>fariam in puncto M, & poſita EN ipſius EF quarta <lb></lb>parte, fiat vt RO ad OP, ita MN ad NL. </s> <s>Dico L eſſe <lb></lb>centrum grauitatis portionis ABCD. </s> <s>Nam circa axim <lb></lb>EF ſuper circulum maximum AD deſcribatur cylindrus <lb></lb>AG, cuius centrum grauitatis erit M: reliqui autem ex <lb></lb>cylindro AG dempta ABCD portione centrum graui<lb></lb>tatis N. </s> <s>Quoniam igitur eſt vt RO ad OP, hoc eſt vt <lb></lb>MN ad NL, ita portio ABCD ad reliquum cylindri <lb></lb>AG, & diuidendo vt NM ad ML, ita portio ABCD ad <lb></lb>reliquum cylindri AG: & cylindri AG eſt N, prædicti au<lb></lb>tem reſidui centrum grauitatis M; erit reliquæ portionis <lb></lb>ABCD centrum grauitatis L. </s> <s>Quod <expan abbr="demonſtrandũ">demonſtrandum</expan> erat. </s></p><pb xlink:href="043/01/150.jpg" pagenum="63"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ abſciſſæ duobus pla<lb></lb>nis parallelis neutro per centrum acto, nec cen<lb></lb>trum intercipientibus, centrum grauitatis eſt in <lb></lb>axe primum bifariam ſecto: deinde ſecundum <lb></lb>centrum grauitatis fruſti circa eundem axim, <lb></lb>abſciſſi à cono verticem habente centrum ſphæ<lb></lb>ræ; in eo puncto in quo dimidius axis maiorem <lb></lb>baſim attingens ſic diuiditur, vt pars duabus præ<lb></lb>dictis ſectionibus finita ſit ad eam, quæ inter ſe<lb></lb>cundam, & vltimam ſectionem interijcitur, vt <lb></lb>exceſſus, quo maior extrema ad triplas & ſemidia <lb></lb>metri ſphæræ, & eius quæ inter centra ſphæræ, <lb></lb>& minorem baſim portionis interijcitur, ſuperat <lb></lb>tres deinceps proportionales, quarum maxima <lb></lb>eſt, quæ inter centra ſphæræ, & minoris baſis, <lb></lb>media autem, quæ inter centra ſphæræ, & maio<lb></lb>ris baſis portionis interijcitur; ad maiorem extre<lb></lb>mam antedictam. </s></p><p type="main"> <s>Sit portio ABCD, ſphæræ, cuius centrum E, ab<lb></lb>ſciſsa duobus planis parallelis, neutro per E tranſeun<lb></lb>te, nec E intercipientibus: axis autem portionis ſit GH, <lb></lb>maior baſis circulus, cuius diameter AD, minor cuius <lb></lb>diameter BC: producta autem GH vſque in E intel<lb></lb>ligatur coni KEN rectanguli, cuius axis EG, fruſtum <pb xlink:href="043/01/151.jpg" pagenum="64"></pb>KLMN abſciſſum ijſdem planis, quibus por<lb></lb>tio, & ſphæræ ſemidiameter ſit EHGS: & po<lb></lb>ſita T tripla ipſius ES, & V ipſius EG tri<lb></lb>pla, eſto vt V ad T ita T ad XZ: & vt GE <lb></lb>ad EH ita EH ad <foreign lang="grc">ω</foreign>, & ſit ZY, ipſius XZ, <lb></lb>æqualis tribus GE, EH, <foreign lang="grc">ω</foreign>, vt ſit exceſſus <lb></lb>XY: & ſecto axe GH bifariam in puncto I, in <lb></lb>linea GI, ſumatur O, centrum grauitatis fru<lb></lb>ſti KLMN: Et vt <foreign lang="grc">Υ</foreign>X ad XZ, ita fiat IO <lb></lb>ad OIP. </s> <s>Dico portionis ABCD centrum <lb></lb>grauitatis eſſe P. </s> <s>Nam circa axim GH pla<lb></lb>nis baſium portionis interceptus ſtet cylin<lb></lb>drus QR, cuius baſis ſit æqualis circulo ma<lb></lb>ximo. </s> <s>Quoniam igitur eſt vt YX ad XZ, <lb></lb>hoc eſt vt IO ad OP, ita portio ABCD <lb></lb>ad cylindrum QR, & diuidendo vt OI ad <lb></lb>IP, ita portio ABCD ad reliquum cylindri <lb></lb>QR: & I eſt cylindri QR, & O prædicti <lb></lb>reſidui centrum grauitatis; erit reliquæ por<lb></lb><figure id="id.043.01.151.1.jpg" xlink:href="043/01/151/1.jpg"></figure><lb></lb>tionis ABCD centrum grauitatis P. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><pb xlink:href="043/01/152.jpg" pagenum="65"></pb><p type="head"> <s><emph type="italics"></emph>LEMMA.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>Sit data recta PO, & in ea punctum D, & punctum quod<lb></lb>dam R in ipſa DO, ita vt VD ipſius PD, ad DT ipſius DO, <lb></lb>ſit vt PD, ad DO: ſit autem maior proportio PS ad SO, quàm <lb></lb>VR, ad RT. </s> <s>Dico OS, minorem eſſe quàm OR.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Fiat enim vt PS, ad SO, ita VZ ad ZT; ma<lb></lb>ìor igitur erit proportio VZ, ad ZT, quàm VR, ad <lb></lb>RT: & componendo maior proportio VT, ad TZ, <lb></lb><figure id="id.043.01.152.1.jpg" xlink:href="043/01/152/1.jpg"></figure><lb></lb>quàm VT, ad TR; minor igitur TZ, quàm TR, ideſt <lb></lb>maior DZ, quàm DR. </s> <s>Rurſus quia componendo eſt <lb></lb>vt PO ad OS, ita VT ad TZ: ſed vt DO ad OP, ita <lb></lb>eſt DT ad TV; erit ex æquali, vt DO ad OS, ita DT, <lb></lb>ad TZ; & per conuerſionem rationis, vt OD ad DS, <lb></lb>ita TD ad DZ: & permutando, vt DO ad DT, ita DS <lb></lb>ad DZ: ſed DO, eſt maior quàm DT, ergo & DS, erit <lb></lb>maior quàm DZ: ſed DZ maior erat quàm DR; multo <lb></lb>ergo DS maior quàm DR, vnde minor erit OS quàm <lb></lb>OR. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si datæ maiori ſphæræ portioni cylindrus cir<lb></lb>cumſcribatur circa eundem axim portionis, cen<lb></lb>trum grauitatis reliquæ figuræ ex cylindro cir<lb></lb>cumſcripto ablata portione, propinquius erit ver<lb></lb>tici portionis, quàm <expan abbr="cẽtrum">centrum</expan> grauitatis portionis. </s></p><pb xlink:href="043/01/153.jpg" pagenum="66"></pb><p type="main"> <s>Sit ſphæræ cuius centrum D maior portio ABC, cu<lb></lb>ius axis BE, baſis circulus cuius diameter AC, & por<lb></lb>tioni ABC, cylindro XH circa axim BE circumſcripto <lb></lb>vt ſupra fecimus: quoniam tam portionis ABC, quàm <lb></lb>cylindri XH, centrum grauitatis eſt in axe BE; erit reli<lb></lb>qui ex cylindro XH, in axe BE centrum grauitatis, ſint <lb></lb>in axe BE centra grauitatis Q portionis ABC & S præ<lb></lb>dicti reſidui. </s> <s>Dico eſſe punctum S vertici B propinquius <lb></lb><figure id="id.043.01.153.1.jpg" xlink:href="043/01/153/1.jpg"></figure><lb></lb>quàm punctum <expan abbr="q.">que</expan> Per centrum enim D tranſiens planum <lb></lb>ad axim BE erectum ſecet cylindrum XH, & portionem <lb></lb>ABC in duos cylindros <emph type="italics"></emph>K<emph.end type="italics"></emph.end>H, XL, & hemiſphærium <lb></lb>KBL, & portionem AKLC, ſectio autem circulus ma<lb></lb>ximus eſto ille cuius diameter KL: & duo coni rectan<lb></lb>guli circa axes BD, DE, vertice D communi deſcri<lb></lb>bantur GDH, MDN, quorum alterius baſis GH com<lb></lb>munis erit cylindro XH: alterius autem MDN, minor <lb></lb>quàm eiuſdem cylindri XH, baſis GH. </s> <s>Denique ſecta <pb xlink:href="043/01/154.jpg" pagenum="67"></pb>BE bifariam in puncto R, ſecentur BD, in puncto T, & <lb></lb>DE, in puncto V, bifariam & ſumatur BO, ipſius BD, <lb></lb>pars quarta, necnon EP pars quarta ipſius DE, primum <lb></lb>itaque quoniam ER eſt maior, quàm ED, erit punctum <lb></lb>R, in ſegmento BD. </s> <s>Quoniam igitur ex ſupra oſtenſis O <lb></lb>eſt centrum grauitatis commune cono DGH, & reliquo <lb></lb>cylindri KH dempto ABC hemiſphærio: & eadem ra<lb></lb>tione punctum P, cum ſit centrum grauitatis coni MDN, <lb></lb>erit idem centrum grauitatis reliqui ex cylindro XL dem<lb></lb>pta AKLC portione: eſt autem reliquum cylindri KH <lb></lb>dempto KBL hemiſphærio, æquale cono DGH, qua <lb></lb>ratione & reliquum cylindri XL, dempta AKLC por<lb></lb>tione æquale eſt cono MDN; cum igitur S ſit centrum <lb></lb>grauitatis totius reliqui ex toto cylindro XH, dempta <lb></lb>ABC portione, erit idem S, centrum grauitatis compo<lb></lb>ſiti ex conis GDH, MDL: ſunt autem horum conorum <lb></lb>centra grauitatis O, P; vt igitur conus GDH, ad co<lb></lb>num MDN, ita erit PS, ad SO: ſed coni GDH ad <lb></lb>ſimilem ipſi conum MDN triplicata eſt proportio axis <lb></lb>BD, ad axim BE, hoc eſt cylindri KH ad cylindrum <lb></lb>XL; maior igitur proportio erit PS ad SO, quàm cy<lb></lb>lindri KH ad cylindrum XL, ſed vt cylindrus KH, ad <lb></lb>cylindrum XL, ita eſt VR ad RT, ob centra grauiratis <lb></lb>V, R, T, maior igitur proportio erit PS ad SO, quàm <lb></lb>VR ad RT: ſed eiuſdem PO eſt vt PD ad DO, ita <lb></lb>VD ad DT, ob ſectiones axium proportionales; pun<lb></lb>ctum igitur S propinquius eſt puncto O, quàm punctum <lb></lb>R, per Lemma. </s> <s>Quare & Stermino B propinquius quàm <lb></lb>punctum R: ſed R eſt centrum grauitatis totius cylindri <lb></lb>XH: & S reliqui ex cylindro XH dempta ABC por<lb></lb>tione; igitur Q reliquæ portionis ABC, centrum graui<lb></lb>tatis erit in linea ER, atque ideo à puncto B remotius <lb></lb>quàm punctnm S. </s> <s>Quod eſt propoſitum. </s></p><pb xlink:href="043/01/155.jpg" pagenum="68"></pb><p type="head"> <s><emph type="italics"></emph>COROLLARIV M.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Manifeſtum eſt autem ex demonſtratione thelo<lb></lb>rematis, omnis reſidui ex cylindro datæ maiori <lb></lb>ſphæræ portioni circumſcripto circa eundem <lb></lb>axim portionis, cuius baſis ſit æqualis circulo ma <lb></lb>ximo, centrum grauitatis eſſe in axe abſciſſa pri<lb></lb>mum quarta parte ad verticem portionis termina<lb></lb>ta ſegmenti axis portionis, quod centro ſphæræ, <lb></lb>& vertice portionis, & quarta parte eius quod <lb></lb>centro ſphæræ, & baſi portionis terminatur; ad <lb></lb>baſim terminata in eo puncto, in quo ſegmentum <lb></lb>axis portionis duabus prædictis ſectionibus fini<lb></lb>tum ſic diuiditur, vt ſegmentum propinquius baſi <lb></lb>ſit ad reliquum, vt cubus ſegmenti axis portionis <lb></lb>centro ſphæræ, & vertice portionis terminati ad <lb></lb>cubum reliqui quod baſim portionis tangit, ſi<lb></lb>quidem cubi triplicatam inter ſe habent laterum <lb></lb>proportionem, ſimul illud manifeſtum eſt, hoc <lb></lb>idem eadem ratione poſſe demonſtrari de centro <lb></lb>grauitatis reliqui ex cylindro dempta ſphæræ por<lb></lb>tione abſciſſa duobus planis paralìelis centrum <lb></lb>ſphæræ intercipientibus, ita vt axis portionis à <lb></lb>centro ſphæræ in partes inæquales diuidatur, cu<lb></lb>ius cylindri circumſcripti ſit idem axis, qui & por <lb></lb>tionis, baſis autem æqualis circulo maximo. </s> <s>Si<lb></lb>militer enim deſcriptis duobus conis rectangulis<pb xlink:href="043/01/156.jpg" pagenum="69"></pb>verticem habentibus communem centrum ſphæ<lb></lb>ræ, baſes autem minores baſibus oppoſitis cylin<lb></lb>dri circumſcripti: æqualibus circulo maximo, ſu<lb></lb>mentes pro vertice minorem baſim, pro baſi, ma<lb></lb>iorem baſim portionis immotis reliquis propoſi<lb></lb>tum demonſtraremus. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis maioris portionis ſphæræ centrum gra<lb></lb>uitatis eſt in axe primum bifariam ſecto: Deinde <lb></lb>ſumpta ad verticem quarta parte ſegmenti axis, <lb></lb>quod centro ſphæræ, & portionis vertice finitur: <lb></lb>itemque ad baſim quarta parte reliqui ſegmenti <lb></lb>inter centrum ſphæræ, & baſim portionis interie<lb></lb>cti. </s> <s>Deinde ſegmento axis, inter eas quartas par<lb></lb>tes interiecto, ita diuiſo, vt pats propinquior baſi <lb></lb>ſit ad reliquam vt cubus ſegmenti axis, quod <lb></lb><expan abbr="cẽtro">centro</expan> ſphæræ, & vertice portionis, ad cubum eius <lb></lb>quod centris ſphæræ, & baſis portionis termina<lb></lb>tur; in eo puncto, in quo ſegmentum axis centro <lb></lb>ſphæræ, & ſectione penultima finitum ſic diuidi<lb></lb>tur, vt pars prima & penultima ſectione termina<lb></lb>ta ſit ad totam vltima & penultima ſectione termi <lb></lb>natam, vt exceſſus, quo ſegmentum axis portionis <lb></lb>inter centrum, & baſim portionis interiectum ſu<lb></lb>perat tertiam partem minoris extremæ maiori po <lb></lb>ſita dicto axis ſegmento in proportione ſemidia-<pb xlink:href="043/01/157.jpg" pagenum="70"></pb>metri ſphæræ ad prædictum ſegmentum, vnà cum <lb></lb>ſubſeſquialtera reliqui ſegmenti, ad axim por<lb></lb>tionis. </s></p><p type="main"> <s>Sit maior portio ABC ſphæræ, cuius centrum D, dia<lb></lb>meter KH, axis autem portionis ſit BE, baſis circulus, <lb></lb>cuius diameter AC, & ſit axis BE primum bifariam ſe<lb></lb>ctus in puncto G: ſumptaque ipſius BD, quarta parte <lb></lb>BP, itemque ipſius DE quarta parte EN, ſecetur inter<lb></lb>iecta PN, ita in puncto F, vt NF, ad FP, ſit vt cubus ex <lb></lb>BD ad cubum ex DE; punctum igitur F, ex præcedenti <lb></lb><figure id="id.043.01.157.1.jpg" xlink:href="043/01/157/1.jpg"></figure><lb></lb>corollario erit centrum grauitatis reliqui ex cylindro LM <lb></lb>portioni ABC, vt in antecedenti circumſcripto. </s> <s>Quo<lb></lb>niam igitur & prædicti reſidui, ex antecedenti, & cylindri <lb></lb>LM, centra grauitatis ſunt in axe BE, erit & portionis <lb></lb>ABC in axe BE centrum grauitatis, quod ſit S: manife<lb></lb>ſtum eſt igitur punctum S, cadere ſupra centrum D, in li<lb></lb>nea BD, minori ablata ſphæræ portione, cuius baſis cir-<pb xlink:href="043/01/158.jpg" pagenum="71"></pb>culus AC: centrum autem F propinquius eſſe puncto B, <lb></lb>quàm centrum S, conſtat ex præcedenti: quare centrum <lb></lb>G, totius cylindri LM inter puncta F, S cadet. </s> <s>Dico <lb></lb>GF ad FS eſſe vt exceſſus, quo recta DE ſuperat tertiam <lb></lb>partem minoris extremæ maiori poſita ipſa DE in propor<lb></lb>tione continua ipſius DH ad DE vnà cum ſubſeſquial<lb></lb>tera ipſius BD, ad axim BE, ita GF ad FS. </s> <s>Quoniam <lb></lb>enim portio ABC ad cylindrum LM eſt vt prædictus ex<lb></lb>ceſſus vnà cum ſubſeſquialtera ipſius BD ad axim BE: <lb></lb>& vt portio ABC ad LM cylindrum, ita eſt GF ad FS, <lb></lb>ob centra grauitatis F, G; erit vt prædictus exceſſus vna <lb></lb>cum ſubſeſquialtera ipſius BD ad axim BE, ita GF ad <lb></lb>FS. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ abſciſſæ duobus pla<lb></lb>nis parallelis centrum intercipientibus, & à cen<lb></lb>tro æqualiter diſtantibus, centrum grauitatis eſt <lb></lb>in medio axis, vel idem, quod centrum ſphæræ. </s></p><p type="main"> <s>Sit portio ABCD, ſphæræ, cuius centrum G, abſciſsa <lb></lb>duobus planis parallelis <lb></lb>centrum G intercipien<lb></lb>tibus, & æquè ab eo di<lb></lb>ſtantibus: ſectiones <expan abbr="erũt">erunt</expan> <lb></lb>circuli minores, quorum <lb></lb>diametri ſint AD, BC <lb></lb>centra autem F,E, qui<lb></lb>bus axis portionis termi <lb></lb>nabitur, eritque ad pla<lb></lb>na vtriuſque circuli per <lb></lb><figure id="id.043.01.158.1.jpg" xlink:href="043/01/158/1.jpg"></figure><lb></lb>pendicularis tranſiens per centrum G: & quia illa plana <pb xlink:href="043/01/159.jpg" pagenum="72"></pb>à centro G, æquè diſtant, erit EG, æqualis GF. </s> <s>Dico <lb></lb>portionis ABCD centrum grauitatis eſſe G. </s> <s>Deſcripta <lb></lb>enim figura, vt ſupra fecimus, intelligantur duo coni re<lb></lb>ctanguli GNO, GPQ, vertice G, communi, axibus <lb></lb>autem eorum EG, GF: & cylindrus LM, portioni cir<lb></lb>cumſcriptus circa eun<lb></lb>dem axim EF, cuius ba <lb></lb>ſis æqualis eſt circulo <lb></lb>maximo: & ſumatur EH <lb></lb>ipſius EG, pars quar<lb></lb>ta, itemque FK, pars <lb></lb>quarta ipſius FG. </s> <s>Quo<lb></lb>niam igitur conorum G <lb></lb>NO, PGO, axes FG, <lb></lb>GH, ſunt æquales, re<lb></lb>liquæ KG, GH, æqua <lb></lb><figure id="id.043.01.159.1.jpg" xlink:href="043/01/159/1.jpg"></figure><lb></lb>les erunt; centra autem grauitatis conorum ſunt K, H; pun<lb></lb>ctum igitur G eſt centrum grauitatis compoſiti ex duobus <lb></lb>conis æqualibus GNO, GPQ, hoc eſt reliqui ex cylin<lb></lb>dro LM, dempta ABCD, portione, ex ante demonſtra<lb></lb>tis: ſed idem G eſt centrum grauitatis totius cylindri LM; <lb></lb>reliquæ igitur ABCD, portionis centrum grauitatis erit <lb></lb>G. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XL.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ abſciſſæ duobus pla<lb></lb>nis parallelis centrum intercipientibus, & à cen<lb></lb>tro non æqualiter diſtantibus centrum grauitatis <lb></lb>eſt in axe primum bifariam ſecto: Deinde ſumpta <lb></lb>ad minorem baſim portionis quarta parte ſegmen <lb></lb>ti axis, quod minorem baſim attingit: & ad maio-<pb xlink:href="043/01/160.jpg" pagenum="73"></pb>rem baſim quarta parte reliqui ſegmenti axis eo<lb></lb>rum, quæ à centro ſphæræ fiunt: Deinde recta <lb></lb>inter has quartas partes interiecta ita diuiſa, vt <lb></lb>pars maiori baſi propinquior ſit ad reliquam vt <lb></lb>cubus ſegmenti axis inter ſphæræ centrum, & mi<lb></lb>norem baſim, ad cubum eius, quod inter ſphæræ <lb></lb>centrum, & maiorem baſim portionis interijci<lb></lb>tur; in eo puncto, in quo ſegmentum axis centro <lb></lb>ſphæræ, & penultima ſectione terminatum ſic di<lb></lb>uiditur, vt pars quæ penultima, & prima ſectione <lb></lb>terminatur ſit ad totam vltima, & penultima ſe<lb></lb>ctione terminatam, vt ad axim portionis eſt exceſ <lb></lb>ſus, quo idem axis portionis ſuperat <expan abbr="tertiã">tertiam</expan> partem <lb></lb>compoſitæ ex duabus minoribus extremis, maio<lb></lb>ribus poſitis duobus axis ſegmentis, quæ fiunt à <lb></lb>centro ſphæræ in rationibus ſemidiametri ſphæ<lb></lb>ræ ad prædicta ſegmenta. </s></p><figure id="id.043.01.160.1.jpg" xlink:href="043/01/160/1.jpg"></figure><p type="main"> <s>Sit portio ABCD ſphæræ, cuius centrum G, abciſſa <lb></lb>duobus planis parallelis centrum G intercipien<gap></gap>ibus, & <pb xlink:href="043/01/161.jpg" pagenum="74"></pb>ab eo non æqualiter diſtantibus: & axis portionis ſit EF, <lb></lb>qui per centrum G tranſibit, vtpote parallelorum circu<lb></lb>lorum centra iungens: cumque eorum vtrumque ſit à cen<lb></lb>tro non æqualiter diſtantium perpendicularis, erunt eius <lb></lb>ſegmenta EG, GF, inæqualia. </s> <s>Eſto EG, maius: ſectoque <lb></lb>axe EF bifariam in puncto P, ſumptisque ipſarum EG, <lb></lb>GF, quartis partibus EH, FK, ſecetur interiecta <emph type="italics"></emph>K<emph.end type="italics"></emph.end>H, <lb></lb>in puncto Q, ita vt KQ, ad QH, ſit vt cubus ex EG, <lb></lb>ad cubum ex GF, & portionis ABCD, ſit centrum gra<lb></lb>uitatis R: quod quidem cum punctis P, Q, eſſe in axe <lb></lb><figure id="id.043.01.161.1.jpg" xlink:href="043/01/161/1.jpg"></figure><lb></lb>EF: & cylindro LM, ſuper baſim æqualem circulo ma<lb></lb>ximo circa axim EF, portioni circumſcripto, reliqui eius <lb></lb>dempta ABCD, portione centrum grauitatis eſse Q, & <lb></lb>propinquius E puncto, quàm centrum grauitatis R por<lb></lb>tionis ABCD, manifeſtum eſt ex ſupra demonſtratis de <lb></lb>maioris portionis ſphæræ centro grauitatis: portionis autem <lb></lb>ABCD centrum grauitatis R eſse in ſegmento EG ſe<lb></lb>quitur ex antecedente. </s> <s>Dico PQ ad QR eſse vt ad axim <lb></lb>EF exceſsus, quo axis EF ſuperat tertiam partem com<lb></lb>poſitæ <gap></gap> duabus minoribus extremis altera reſpondente <lb></lb>maiori extrema EG in proportione continua ipſius NG <pb xlink:href="043/01/162.jpg" pagenum="75"></pb>ad GE, altera maiori extremæ FG in proportione con<lb></lb>tinua ipſius NG ad GF. </s> <s>Quoniam enim ob centra gra<lb></lb>uitatis QPR eſt vt QP ad PR, ita portio ABCD ad <lb></lb>reliquum cylindri LM, erit componendo, & per conuer<lb></lb>ſionem rationis, & conuertendo, vt PQ ad QR, ita por<lb></lb>tio ABCD ad LM cylindrum: ſed portio ABCD ad <lb></lb>LM cylindrum eſt vt prædictus exceſſus ad axim EF; <lb></lb>vtigitur prædictus exceſſus ad axim EF, ita eſt PQ ad <lb></lb>QR. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis conoidis parabolici centrum grauita<lb></lb>tis eſt punctum illud, in quo axis ſic diuiditur vt <lb></lb>pars, quæ eſt ad verticem ſit dupla reliquæ. </s></p><p type="main"> <s>Sit conoides parabolicum ABC, cuius vertex B, axis <lb></lb>autem BD ſectus in puncto E ita vt EB ſit ipſius ED <lb></lb>dupla. </s> <s>Dico E eſse centrum grauitatis conoidis ABC. <lb></lb></s> <s>Nam in ſectione per <lb></lb>axim parabola ABC, <lb></lb>cuius diameter erit B <lb></lb>D, deſcribatur rian<lb></lb>gulum ABC; ſum<lb></lb>ptisque ipſius BD æ<lb></lb>qualibus DH, HO, <lb></lb>per puncta H, O, ſe<lb></lb>centur vnà parabola <lb></lb>& triangulum ABC <lb></lb>duabus rectis FGH <lb></lb><figure id="id.043.01.162.1.jpg" xlink:href="043/01/162/1.jpg"></figure><lb></lb>KL, MNOPQ: & per eas rectas ſecetur conoi<lb></lb>des ABC planis baſi parallelis, factæ autem ſe<lb></lb>ctiones erunt circuli circa FL, MQ, & in parabola <pb xlink:href="043/01/163.jpg" pagenum="76"></pb>ABC tres ad diametrum ordinatim applicatæ AD, <lb></lb>FH, MO. </s> <s>Quoniam igitur tres rectæ OB, BH, BD <lb></lb>ſeſe qualiter excedunt, quarum minima BO, maxi<lb></lb>ma eſt BD, minor erit proportio BO ad BH, quàm <lb></lb>BH ad BD; hoc eſt NP ad GK, quàm GKad AC. <lb></lb>ſed vt OB ad BH hoc eſt NO ad GH, vel NP ad <lb></lb>GK ita eſt quadra<lb></lb>tum MO ad quadra<lb></lb>tum FH, hoc eſt eo<lb></lb>no dis ſectionum cir<lb></lb>culus MQ ad circu<lb></lb>lum FL: eademque <lb></lb>ratione vt GK ad <lb></lb>AC ita circulus FL <lb></lb>ad circulum AC; mi<lb></lb>nor igitur proportio <lb></lb>erit circuli MQ ad <lb></lb>circulum FL quàm <lb></lb><figure id="id.043.01.163.1.jpg" xlink:href="043/01/163/1.jpg"></figure><lb></lb>circuli FL ad circulum AC. </s> <s>Similiter autem oſtende<lb></lb>remus ternas quaslibet alias ita factas ſectiones trianguli, <lb></lb>& parabolæ ABC inter ſe & baſi parallelas proportio<lb></lb>nales eſse, & minorem proportionem vtrobique minimæ <lb></lb>ad mediam, quàm mediæ ad maximam. </s> <s>Sed E eſt cen<lb></lb>trum grauitatis trianguli ABC, igitur per vigeſimamter<lb></lb>tiam huius centrum grauitatis conoidis ABC erit idem E. <lb></lb></s> <s>Quod demonſtrandum erat, </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti conoidis parabolici centrum gra<lb></lb>uitatis axim ita diuidit, vt pars, quæ minorem <lb></lb>baſim attingit ſit ad reliquam; vt duplum maioris <pb xlink:href="043/01/164.jpg" pagenum="77"></pb>baſis vnà cum minori, ad duplum minoris, vnà <lb></lb>cum maiori. </s></p><p type="main"> <s>Sit conoidis parabolici ABC, cuius axis BD fruſtum <lb></lb>AEFC, eius maior baſis circulus, cuius diameter AC, mi<lb></lb>nor, cuius diameter EF: in eadem parabola per axem, axis <lb></lb><expan abbr="autẽ">autem</expan> DG, in quo fruſti AEFC ſit centrum grauitatis H. <lb></lb></s> <s>Dico eſſe vt duplum circuli AC, vnà cum circulo EF, ad <lb></lb>duplum circuli EF vna cum circulo AC, ita GH, ad HD. <lb></lb><expan abbr="Iungãtur">Iungantur</expan> enim re<lb></lb>ctæ AKB, BLC. <lb></lb></s> <s>Quoniam igitur <lb></lb>qua ratione oſten <lb></lb>dimus conoides, <lb></lb>& triangulum A <lb></lb>BC, commune <lb></lb>habere in linea <lb></lb>BD centrum gra<lb></lb>uitatis, <expan abbr="eadẽ">eadem</expan> pror<lb></lb>ſus remanet de<lb></lb>monſtratum, fruſti <lb></lb><figure id="id.043.01.164.1.jpg" xlink:href="043/01/164/1.jpg"></figure><lb></lb>AEFC <expan abbr="centrũ">centrum</expan> grauitatis H, idem eſse quod trapezij AK <lb></lb>FC; erit duarum parallelarum AG, KL vt dupla ipſius <lb></lb>AC, vnà cum KL, ad duplam ipſius KL, vnà cum AC <lb></lb>ita GH ad HD: ſecat enim DG ipſas AC, KL bifa<lb></lb>riam. </s> <s>Sed vt AC ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>L ita eſt circulus AC ad circu<lb></lb>lum EF, ex demonſtratione antecedentis, hoc eſt vt dupla <lb></lb>ipſius AC vnà cum KL ad duplam ipſius KL vnà cum <lb></lb>AC, ita duplum circuli AC vna cum circulo KL ad du<lb></lb>plum circuli KL vnà cum circulo AC; vt igitur eſt du<lb></lb>plum circuli AC, vnà cum circulo EF, ad duplum circu<lb></lb>li EF, vnà cum circulo AC; ita erit GH ad HD. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/165.jpg" pagenum="78"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis conoidis hyperbolici centrum grauita<lb></lb>tis eſt punctum illud, in quo duodecima pars axis <lb></lb>ordine quarta ab ea, quæ baſim attingit, ſic diui<lb></lb>ditur, vt pars baſi propinquior ſit ad reliquam, vt <lb></lb>ſeſquialtera tranſuerſi lateris hyperboles, quæ <lb></lb>conoides deſcribit ad axim conoidis. </s></p><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius vertex B, axis <lb></lb>autem BD, qui etiam erit diameter hyperboles, quæ co<lb></lb>noides deſcripſit, ad quam rectæ ordinatim applicantur: <lb></lb>eiuſdem autem hyperboles tranſuerſum latus ſit EB, cu<lb></lb>ius ſit ſeſquialtera BEI, & ſumpta DQ quarta parte <lb></lb>axis BD, & DG, eiuſdem tertia, qua ratione erit FG <lb></lb>duodecima pars axis BD, & ordine quarta ab ea cuius <lb></lb>terminus D, fiat vt IB, ad BD, ita QH, ad HG. <lb></lb></s> <s>Dico conoidis ABC, centrum grauitatis eſſe H. </s> <s>Sumpto <lb></lb>enim in linea AD quolibet puncto M, vt eſt EB ad <lb></lb>BD longitudine, ita fiat MD, ad DK ipſius AD po<lb></lb>tentia: & abſcindatur DN, æqualis DM, & DL æqua<lb></lb>lis DK; ſiue autem ſit DK minor, quàm DM, ſiue ma<lb></lb>ior, ſiue eadem illi; omnibus caſibus communis erit demon <lb></lb>ſtratio. </s> <s>At per puncta M, N, vertice B, circa diametrum <lb></lb>BD, deſcribatur parabola MBN, & triangulum KBL. <lb></lb></s> <s>Manente igitur BD, & circumductis figuris MBN, <lb></lb>KBL, deſcribantur conoides parabolicum MBN, & <lb></lb>conus KBL, quorum communis axis erit BD, baſes <lb></lb>autem circuli, quorum diametri KL, MN, in eodem <lb></lb>plano cum baſe conoidis ABC. </s> <s>Rurſus ſecto axe BD <lb></lb>bifariam, & ſingulis eius partibus ſemper bifariam in qua-<pb xlink:href="043/01/166.jpg" pagenum="79"></pb>cumque multiplicatione; ſint duæ partes æquales proximæ <lb></lb>baſi DF, FQ: & per puncta FQ duo plana baſium pla<lb></lb>no parallela tres prædictas figuras ſolidas ſecare intelli<lb></lb>gantur: ſecabunt autem & tres figuras per axim, eruntque <lb></lb>ſectiones rectæ lineæ ad diametrum figurarum ordinatim <lb></lb>applicatæ propter <lb></lb>plana ſecantia pa <lb></lb>rallela: trium au<lb></lb>tem ſolidorum ſe <lb></lb>ctiones & baſes <lb></lb>omnes circuli, ter <lb></lb>ni in ſingulis pla<lb></lb>nis: ac primi qui<lb></lb>dem ordinis ſint <lb></lb>ij, quorum diame<lb></lb>tri ſunt baſes <expan abbr="triũ">trium</expan> <lb></lb><expan abbr="figurarũ">figurarum</expan> per axim, <lb></lb>trianguli ſcilicet, <lb></lb>parabolæ, & hy<lb></lb>perboles, quæ præ <lb></lb>dictas figuras ſoli <lb></lb>das deſcribunt, re <lb></lb>ctæ lineæ AC, <lb></lb>MN, KL. </s> <s>Se<lb></lb>cundi verò reten<lb></lb>to eodem ordine <lb></lb><expan abbr="figurarũ">figurarum</expan> tres <foreign lang="grc">αζ, <lb></lb>βε, γδ. </foreign></s> <s>Tertij <lb></lb>denique ordinis <lb></lb>SZ, TY, VX. <lb></lb><figure id="id.043.01.166.1.jpg" xlink:href="043/01/166/1.jpg"></figure><lb></lb>Quoniam igitur eſt vt EB, ad BD, ità quadratum MD, <lb></lb>ad quadratum DK, ideſt conus MBN, ſi deſcribatur eo<lb></lb>dem vertice B, ad conum KBL. </s> <s>Et vt IB, ad BE, ità eſt <lb></lb>conoides MBN, ad conum MBN, in proportione ſcili-<pb xlink:href="043/01/167.jpg" pagenum="80"></pb>cet ſeſquialtera; ex æquali erit vt IB, ad BD, itì conoi<lb></lb>des MBN ad conum KBL: Sed vt IB, ad BD, ità <lb></lb>ponitur QH ad HG; vt igitur conoides MBN, ad co<lb></lb>num KBL, ità eſt QH ad HG. </s> <s>Sed Q eſt centrum <lb></lb>grauitatis coni KBL, & G conoidis MBN; compoſi<lb></lb>ti igitur ex conoi<lb></lb>de MBN, & co<lb></lb>no KBL <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis erit H. <lb></lb></s> <s>Rurſus quoniam <lb></lb>tres rectæ lineæ B <lb></lb>D, BF, BQ, æ<lb></lb>qualibus exceſſi<lb></lb>bus inter ſe diffe<lb></lb>runt, minor erit <lb></lb>proportio BQ, ad <lb></lb>BF, quàm BF, <lb></lb>ad BD, hoc eſt <lb></lb>rectanguli EBQ, <lb></lb>ad rectangulum <lb></lb>EBF, quàm re<lb></lb>ctanguli EBF, ad <lb></lb>rectangulum EB <lb></lb>D. </s> <s>Sed quadrati <lb></lb>BQ, ad quadra<lb></lb>tum BF, dupli<lb></lb>cata eſt proportio <lb></lb>lateris BQ ad la<lb></lb>tus BF: hoc eſt <lb></lb>rectanguli EBQ <lb></lb><figure id="id.043.01.167.1.jpg" xlink:href="043/01/167/1.jpg"></figure><lb></lb>ad rectangulum EBF: & quadrati BF, ad quadratum <lb></lb>BD duplicata eius, quæ eſt rectanguli EBF, ad rectan<lb></lb>gulum EBD; compoſitis igitur primis cum ſecundis, mi<lb></lb>nor erit proportio rectanguli BQE, ad rectangulum BFE, <pb xlink:href="043/01/168.jpg" pagenum="81"></pb>quàm rectanguli BFE, ad rectangulum BDE. </s> <s>Sed vt <lb></lb>rectangulum BQE ad rectangulum BFE, ita eſt quadra<lb></lb>tum SQ ad quadratum <foreign lang="grc">α</foreign>F: & vt rectangulum BFE <lb></lb>ad rectangulum BDE, ita quadratum <foreign lang="grc">α</foreign>F, ad quadra<lb></lb>tum AD; minor igitur proportio erit quadrati SQ, ad <lb></lb>quadratum <foreign lang="grc">α</foreign>F, quàm quadrati <foreign lang="grc">α</foreign>F ad quadratum AD. <lb></lb></s> <s>Sed vt quadratum SQ ad quadratum <foreign lang="grc">α</foreign>F, ita eſt qua<lb></lb>dratum SZ ad quadratum <foreign lang="grc">α</foreign><37>: & vt quadratum <foreign lang="grc">α</foreign>F ad <lb></lb>quadratum AD ita quadratum <foreign lang="grc">αζ</foreign> ad quadratum <lb></lb>AC; minor igitur proportio erit quadrati SZ ad quadra<lb></lb>tum <foreign lang="grc">αζ</foreign>, quàm quadrati <foreign lang="grc">αζ</foreign>, ad quadratum AC, hoc eſt <lb></lb>circuli SZ ad circulum <foreign lang="grc">α</foreign><37>, quàm circuli <foreign lang="grc">α</foreign><37>, ad cir<lb></lb>culum AC; qui circuli ſunt ſectiones conoidis ABC <lb></lb>poſiti vt in propoſitionibus lemmaticis dicebamus. </s> <s>Rurſus <lb></lb>quoniam ſunt quatuor primæ proportionales; vt rectangu<lb></lb>lum DBE ad rectangulum FBE, ita MD quadratum <lb></lb>ad quadratum <foreign lang="grc">β</foreign>F: & totidem ſecundæ, vt quadratum <lb></lb>BD, ad quadratum BF, ita quadratum DK, ad quadra<lb></lb>tum F<foreign lang="grc">γ</foreign>, ob ſimilium triangulorum latera proportionalia: <lb></lb>ſed vt EB, ad BD, hoc eſt rectangulum DBE prima in <lb></lb>primis ad quadratum BD primam in ſecundis, ita eſt <lb></lb>quadratum MD tertia in primis ad quadratum DK ter<lb></lb>tiam in ſecundis; vt igitur compoſita ex primis ad com<lb></lb>poſitam ex ſecundis, ità erit compoſita ex tertijs ad com<lb></lb>poſitam ex quartis; videlicet vt rectangulum DBE <lb></lb>vnà cum quadrato BD, hoc eſt rectangulum BDE <lb></lb>ad rectangulum BFE, hoc eſt vt quadratum AD, ad <lb></lb>quadratum <foreign lang="grc">α</foreign>F, ità compoſitum ex quadratis MD, DK, <lb></lb>ad compoſitum ex quadratis <foreign lang="grc">β</foreign>F, F<foreign lang="grc">γ</foreign>: & quadrupla vtro<lb></lb>rumque, vt quadratum AC, ad quadratum <foreign lang="grc">α</foreign><37>, ità com<lb></lb>poſitum ex quadratis MN, KL, ad compoſitum ex qua<lb></lb>dratis <foreign lang="grc">βε, γδ</foreign>; hoc eſt eorum circulorum, qui ſunt ſectio<lb></lb>nes ſolidorum, vt circulus AC, ad circulum <foreign lang="grc">α</foreign><37>, ità com<lb></lb>poſitum ex circulis MN, KL, ad compoſitum ex circu<pb xlink:href="043/01/169.jpg" pagenum="82"></pb>lis <foreign lang="grc">βε, γδ. </foreign></s> <s>Eadem ratione erit vt circulus AC, ad cir<lb></lb>culum SZ, ità compoſitum ex circulis MN, KL, ad <lb></lb>compoſitum ex circulis TY, VX: & conuertendo, & ex <lb></lb>æquali, vt circulus SZ, ad circulum <foreign lang="grc">α</foreign><37>, ità compoſitum <lb></lb>ex circulis TY, VX, ad compoſitum ex circulis <foreign lang="grc">βε, γδ</foreign>: <lb></lb>& vt circulus <foreign lang="grc">α</foreign><37>, <lb></lb>ad circulum AC, <lb></lb>ità <expan abbr="cõpoſitum">compoſitum</expan> ex <lb></lb>circulis <foreign lang="grc">βε, γδ</foreign>, <lb></lb>ad <expan abbr="cõpoſitum">compoſitum</expan> ex <lb></lb>circulis MN, <emph type="italics"></emph>K<emph.end type="italics"></emph.end><lb></lb>L. </s> <s>Sunt igitur tria <lb></lb>compoſita ex bi<lb></lb>nis ſectionibus cir <lb></lb>culis, & totidem <lb></lb>alij circuli, quos <lb></lb>diximus in <expan abbr="eadẽ">eadem</expan> <lb></lb>proportione, ſi bi<lb></lb>na <expan abbr="ſumãtur">ſumantur</expan> in ſin <lb></lb>gulis planis ſecan <lb></lb>tibus: eorum au<lb></lb>tem minor erat <lb></lb>proportio circuli <lb></lb>SZ ad circulum <lb></lb><foreign lang="grc">α</foreign><37>, quàm circuli <lb></lb><foreign lang="grc">α</foreign><37>, ad circulum <lb></lb>AC; minor igitur <lb></lb>proportio erit <expan abbr="cõ-poſiti">con<lb></lb>poſiti</expan> ex circulis <lb></lb>T<foreign lang="grc">Υ</foreign>, VX, ad <expan abbr="cõ-poſitum">con<lb></lb>poſitum</expan> ex circu<lb></lb><figure id="id.043.01.169.1.jpg" xlink:href="043/01/169/1.jpg"></figure><lb></lb>lis <foreign lang="grc">βε, γδ</foreign>, quàm compoſiti ex circulis <foreign lang="grc">βε, γδ</foreign>, ad com <lb></lb>poſitum ex circulis MN, KL. </s> <s>Hac eadem ratione ad verti<lb></lb>cem deinceps progredienti manifeſtum erit, omnium com-<pb xlink:href="043/01/170.jpg" pagenum="83"></pb>poſitorum ex binis ſectionibus nempe circulis, quorum al<lb></lb>ter ad conum KBL pertinet, alter ad conoides MBN, in <lb></lb>eodem plano ſecante prædictorum inter ſe parallelorum <lb></lb>exiſtentibus, minorem eſſe proportionem incipienti ab eo, <lb></lb>quod eſt proximum vertici, primi ad ſecundum, quàm ſe<lb></lb>cundi ad tertium, & ſecundi ad tertium, quàm tertij ad <lb></lb>quartum, & ſic ſemper deinceps vſque ad maximum & vl<lb></lb>timum compoſitum ex circulis MN, KL: & eandem di<lb></lb>ctas ſectiones compoſitas ex coni, & conoidis parabolici <lb></lb>ſectionibus inter ſe habere proportionem, quàm habent in<lb></lb>ter ſe circuli ſectiones conoidis ABC, pro vt illis in <lb></lb>ijſdem planis ſecantibus, & æqualia axis BD ſegmenta <lb></lb>intercipientibus reſpondent: Igitur per trigeſimam ſecun<lb></lb>dam huius, & ſequens eam Corollarium, conoides ABC, <lb></lb>& compoſitum ex conoide MBN, & cono BKL, com<lb></lb>mune habebunt in axe BD centrum grauitatis. </s> <s>Sed H <lb></lb>erat huius compoſiti centrum grauitatis; Igitur conoidis <lb></lb>ABC centrum grauitatis erit idem H. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIV M.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Eadem demonſtratione conſtat ſi prædicta tria <lb></lb>ſolida ita vt diximus diſpoſita ſecentur plano ba<lb></lb>ſibus parallelo; ſruſtum conoidis hyperbolici, & <lb></lb>compoſitum ex fruſtis coni, & conoidis paraboli<lb></lb>ci, commune habere in communi axe centrum <lb></lb>grauitatis. </s></p><pb xlink:href="043/01/171.jpg" pagenum="84"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si conus & conoides parabolicum circa eun<lb></lb>dem axim ſecentur plano baſi parallelo; fruſti co<lb></lb>nici abſciſſi maiori baſi propinquius erit quàm <lb></lb>parabolici centrum grauitatis. </s></p><p type="main"> <s>Sint conus ABC, & conoides parabolicum EBF, <lb></lb>quorum communis <lb></lb>axis BD, cuius per <lb></lb>quoduis punctum M, <lb></lb>planum ſecans ea cor <lb></lb>pora plano baſium, <lb></lb>quarum diametri A <lb></lb>C, EF, parallelo ab<lb></lb>ſcindat fruſta AKL <lb></lb>C, cuius centrum gra<lb></lb>uitatis N, & EGH <lb></lb>F, cuius centrum gra <lb></lb><figure id="id.043.01.171.1.jpg" xlink:href="043/01/171/1.jpg"></figure><lb></lb>uitatis O, quorum vtrumque erit in communi axe DM. <lb></lb></s> <s>Dico punctum N, propinquius eſse ipſi D quàm punctum <lb></lb>O. </s> <s>Quoniam enim eſt parabolicifruſti EGHF centrum <lb></lb>grauitatis O; erit vt duplum maioris baſis, ideſt circuli <lb></lb>EF vna cum minori circulo GH, ad duplum circuli GH <lb></lb>vna cum circulo EF, hoc eſt vt duplum quadrati ED vna <lb></lb>cum quadrato ED ita MO ad OD. </s> <s>Sed vt quadratum <lb></lb>ED ad quadratum GM in parabola quæ conoides de<lb></lb>ſcribit, cuius diameter BD, ita eſt DB ad BM, hoc eſt <lb></lb>AC ad KL; vt igitur eſt dupla ipſius AC vna cum KL <lb></lb>ad duplam ipſius KL vna cum AC ita erit MO ad OD: <lb></lb>ſed N eſt fruſti conoici AKLC, centrum grauitatis; pun<lb></lb>ctum igitur N, erit maiori baſi AC propinquius quàm <pb xlink:href="043/01/172.jpg" pagenum="85"></pb>punctum O; eſt autem O, fruſti EGHF centrum graui<lb></lb>tatis. </s> <s>Si igitur conus, & conoides parabolicum circa eun<lb></lb>dem axim, &c. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XLV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti conoidis hyperbolici centrum <lb></lb>grauitatis eſt in axe primum ſecto ſecundum cen<lb></lb>trum grauitatis cuiuſuis fruſti conici circa axem <lb></lb>conoidis communi vertice, abſciſſi vnà cum fru<lb></lb>ſto conoidis: deinde ita vt pars minorem baſim <lb></lb>attingens ſit ad reliquam, vt dupla axis conoidis <lb></lb>vna cum reliqua dempto axe fruſti, ad duplam <lb></lb>eiuſdem reliquæ vna cum axe conoidis: dein<lb></lb>de poſitis quatuor rectis lineis binis propor<lb></lb>tionalibus, potentia primis, ſecundis longitu<lb></lb>dine, in proportione, quæ eſt inter axem conoi<lb></lb>dis, & reliquam dempto axe fruſti; ita vt ma<lb></lb>ior primarum ſit media proportionalis inter axem <lb></lb>conoidis, & tranſuerſum latus hyperboles, quæ fi<lb></lb>guram deſcribit, minoris autem potentia ſeſqui<lb></lb>altera minor ſecundarum; in eo puncto, in quo <lb></lb>ſegmentum axis fruſti dictis duabus ſectionibus <lb></lb>terminatum ſic diuiditur, vt pars minori baſi pro<lb></lb>pinquior ſit ad reliquam vt cubus, qui fit ab axe <lb></lb>fruſti vnà cum ſolido rectangulo, quod axe co<lb></lb>noidis, & reliqua dempto axe fruſti, & tripla <lb></lb>axis conoidis continetur, ad ſolidum rectangu<lb></lb>lum ex eadem reliqua parte conoidis, & eo, quo <pb xlink:href="043/01/173.jpg" pagenum="86"></pb>plus poteſt quadrato maior quàm minor dicta<lb></lb>rum ſecundarum. </s></p><p type="main"> <s>Sit conoidis hyperbolici ABC, cuius axis BD; & <lb></lb>tranſuerſum latus hyperboles, quæ figuram deſcribit EB, <lb></lb>fruſtum ALMC abſciſſum vnà cum axe FD: cuius <lb></lb><figure id="id.043.01.173.1.jpg" xlink:href="043/01/173/1.jpg"></figure><lb></lb>baſes oppoſitæ, maior circulus circa AC, minor circa LM: <lb></lb>ſecto autem axe FD primum ſecundum G centrum gra<lb></lb>uitatis fruſti abſciſſi vnà cum fruſto ALMC à quouis co <lb></lb>no, cuius axis BD, & vertex B, deinde in puncto H ita <lb></lb>vt FH ad HD ſit vt dupla ipſius BD vnà cum BF ad <lb></lb>duplam ipſius BF vnà cum BD, quo facto cadet G <lb></lb>punctum infra punctum H, ponantur vt DB ad BF, <pb xlink:href="043/01/174.jpg" pagenum="87"></pb>ita N ad O potentia, & Q ad P longitudine: ſit au<lb></lb>tem N media proportionalis inter EB, BD, at P ipſius <lb></lb>O potentia ſeſquialtera: quo autem Q plus poteſt quàm <lb></lb>P ſit quadratum ex R: & vt cubus ex FD vna cum ſoli<lb></lb>do rectangulo ex BF, FD, & tripla ipſius BD, ad ſoli<lb></lb>dum rectangulum ex BF, & quadrato R, ita ſit HK ad <lb></lb>KG. </s> <s>Dico fruſti ALMC centrum grauitatis eſſe K. <lb></lb></s> <s>Producta enim quà opus eſt diametro AC ipſi BD æqua<lb></lb>les abſcindantur DS, DV: necnon ipſi N æquales <lb></lb>DT, DX, vt ſit TD ad DS potentia, vt EB, ad <lb></lb>BD longitudine, & deſcribantur conoides paraboli<lb></lb>cum TBX, & conus SBV, quorum vertex commu<lb></lb>nis B, axis BD: ſectis autem his tribus ſolidis plano <lb></lb>per axim, ſint ſectiones hyperbole ABC, & parabo<lb></lb>la TBX, & triangulum SBV, quæ figuras deſcribunt; <lb></lb>quas planum baſis fruſti propoſiti circa LM ſecans vnà <lb></lb>cum tribus ſolidis faciat cum parabola TBX rectam I<foreign lang="grc">γ</foreign>, <lb></lb>& cum triangulo SBV rectam <foreign lang="grc">Υ</foreign>Z: conoidis autem TBX, <lb></lb>& coni SBV ſectiones circulos circa I<foreign lang="grc">γ</foreign>, YZ baſibus, <lb></lb>circa SV, TX parallelos; vt ſint conoidis TBX fru<lb></lb>ſtum TI<foreign lang="grc">γ</foreign>X, & coni SBV fruſtum SYZV. </s> <s>Rur<lb></lb>ſus producta I. M, ponatur <37>F, æqualis Q, & ab<lb></lb>ſcindatur F<foreign lang="grc">δ</foreign>, potentia ſeſquialtera ipſius IF, iunctis<lb></lb>que IB, B<foreign lang="grc">δ</foreign>, B<37>, deſcribantur tres coni <37>B<foreign lang="grc">θ</foreign>, <lb></lb><foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign>, IB<foreign lang="grc">γ</foreign>, quorum omnium baſes nempe circuli <lb></lb>erunt in dicto plano ſecante tria ſolida per punctum F. <lb></lb></s> <s>Quoniam igitur circuli inter ſe ſunt vt quæ fiunt à diame<lb></lb>tris, vel à ſemidiametris quadrata, coni autem eiuſdem al<lb></lb>titudinis inter ſe vt baſes; erit vt <foreign lang="grc">δ</foreign>F ad FI potentia, ita <lb></lb>conus <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> ad conum IB<foreign lang="grc">γ</foreign>; ſeſquialter igitur conus <lb></lb><foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> coni IB<foreign lang="grc">γ</foreign>: ſed & conoides parabolicum IB<foreign lang="grc">γ</foreign> ſeſqui<lb></lb>alterum eſt coni IB<foreign lang="grc">γ</foreign>; æqualis igitur eſt conus <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> co<lb></lb>noidi IB<foreign lang="grc">γ. </foreign></s> <s>Et quoniam in parabola TBX ordinatim <lb></lb>ad diametrum applicatarum DT eſt ad FI hoc eſt N <pb xlink:href="043/01/175.jpg" pagenum="88"></pb>ad O potentia, vt DB ad BF longitudine: ſed TD eſt <lb></lb>æqualis N; ergo & IF æqualis erit O: cum igitur & <lb></lb>P ipſius O, & <foreign lang="grc">δ</foreign>F ipſius FI ſit potentia ſeſquialtera, erit <lb></lb>F<foreign lang="grc">δ</foreign> æqualis ipſi <foreign lang="grc">Ρ</foreign>: ſed F<37> eſt æqualis ipſi <expan abbr="q;">que</expan> vt igitur eſt <lb></lb>Q ad P, hoc eſt DB ad BF, ita erit <37>F ad F<foreign lang="grc">δ</foreign>; dupli<lb></lb>cata igitur proportio erit quadrati ex F<37> ad quadratum ex <lb></lb>E<foreign lang="grc">δ</foreign> eius, quæ eſt DB ad BF: ſed vt quadratum ex F<37> ad <lb></lb><figure id="id.043.01.175.1.jpg" xlink:href="043/01/175/1.jpg"></figure><lb></lb>quadratum ex F<foreign lang="grc">δ</foreign>, ita eſt circulus circa <37><foreign lang="grc">θ</foreign> ad circulum <lb></lb>circa <foreign lang="grc">δε</foreign>, hoc eſt conus <37>B<foreign lang="grc">θ</foreign> ad conum <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign>; coni igitur <lb></lb><37>B<foreign lang="grc">θ</foreign> ad conum <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign>, duplicata eſt proportio eius, quæ eſt <lb></lb>DB ad BF: ſed & conoidis TBX ad conoides IB<foreign lang="grc">γ</foreign> du<lb></lb>plicata eſt proportio eius, quæ eſt DB ad BF, vt mon<lb></lb>ſtrant alij; eadem igitur proportio eſt coni <37>B<foreign lang="grc">θ</foreign> ad co<lb></lb>num <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> quæ conoidis TBX ad conoides IB<foreign lang="grc">γ</foreign>: ſed <pb xlink:href="043/01/176.jpg" pagenum="89"></pb>conus <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> æqualis eſt conoidi IB<foreign lang="grc">γ</foreign>, vtpote inſcripti co<lb></lb>ni IB<foreign lang="grc">γ</foreign> ſeſquialtero, cuius itidem ſeſquialter erat conus <lb></lb><foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign>; reliquum igitur coni <37>B<foreign lang="grc">θ</foreign> dempto cono <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> æqua<lb></lb>le erit conoidis TBX fruſto TI<foreign lang="grc">γ</foreign>X. </s> <s>Rurſus quia eſt vt <lb></lb>cubus ex BD ad cubum ex BI ita conus SBV ad ſui ſi<lb></lb>milem conum YBZ, in triplicata ſcilicet proportione la<lb></lb>terum, ſiue axium DB, BF: ſed quia YF eſt æqualis BF, <lb></lb>propter ſimilitudinem triangulorum, eſt vt cubus ex BF ad <lb></lb>ſolidum ex BF & quadrato ex F<foreign lang="grc">δ</foreign>, ita quadratum ex FY <lb></lb>ad quadratum ex F<foreign lang="grc">δ</foreign>, hoc eſt circulus circa YZ ad <expan abbr="circulũ">circulum</expan> <lb></lb>circa <foreign lang="grc">δε</foreign>, hoc eſt conus YBZ ad conum <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> ex æquali <lb></lb>igitur erit vt cubus ex BD ad ſolidum ex BF, & quadra<lb></lb>to F<foreign lang="grc">δ</foreign>, ita conus SBV ad conum <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign>: ſed vt ſolidum <lb></lb>ex BF, & quadrato F<foreign lang="grc">δ</foreign>, ad ſolidum ex BF & quadrato <lb></lb>F<37>, ita eſt ſimiliter vt ante conus <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign> ad conum <37>B<foreign lang="grc">θ</foreign>; ex <lb></lb>æquali igitur erit vt cubus ex BD ad ſolidum ex BF, & <lb></lb>quadrato F<37>, ita conus SBV, ad conum <37>B<foreign lang="grc">θ</foreign>: ſed con<lb></lb>uertendo, & per conuerſionem rationis, eſt vt ſolidum ex <lb></lb>BF, & quadrato F<37>, ad ſolidum ex BF, & quadrato, <lb></lb>quo plus poteſt F<37> quàm F<foreign lang="grc">δ</foreign>, ita conus <37>B<foreign lang="grc">θ</foreign> ad ſui reli<lb></lb>quum dempto cono <35>B<foreign lang="grc">ε</foreign>; ex æquali igitur, vt cubus ex <lb></lb>BD ad ſolidum ex BF & quadrato, quo plus poteſt F<37>, <lb></lb>quàm F<foreign lang="grc">δ</foreign>, hoc eſt, quo plus poteſt Q quàm P quadrato <lb></lb>ex R, ita erit conus SBV, ad reliquum coni <37>B<foreign lang="grc">θ</foreign> dem<lb></lb>pto cono <foreign lang="grc">δ</foreign>B<foreign lang="grc">ε</foreign>, hoc eſt ad fruſtum TI<foreign lang="grc">γ</foreign>X. Rurſus, quo<lb></lb>niam duo cubi ex BF, FD, & ſolidum ex BF, FD, & <lb></lb>tripla ipſius BD, ſunt æqualia cubo ex BD; erit id quo <lb></lb>plus poteſt cubice recta BD quàm BF, cubus ex <lb></lb>FD, & ſolidum ex BF, FD, & tripla ipſius BD: cum <lb></lb>igitur ſit vt cubus ex BD ad cubum ex BF, ita conus <lb></lb>SBV ad conum YBZ; erit per conuerſionem rationis, & <lb></lb>conuertendo, vt cubus ex FD vna cum ſolido ex BF, <lb></lb>FD, & tripla ipſius BD ad cubum ex BD, ita fruſtum <lb></lb>SYZV, ad conum SBV: ſed cubus ex BD, ad ſoli-<pb xlink:href="043/01/177.jpg" pagenum="90"></pb>dum ex BF & quadrato R, ita erat conus SBV ad fru<lb></lb>ſtum TI<foreign lang="grc">γ</foreign>X: ex æquali igitur, erit vt cubus ex FD vna <lb></lb>cum ſolido ex BF, FD, & tripla ipſius BD, ad ſolidum <lb></lb>ex BF, & quadrato R, hoc eſt vt H<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ad KG, ita ex <lb></lb>contraria parte fruſtum SYZV, ad fruſtum TI<foreign lang="grc">γ</foreign>X: nam <lb></lb>fruſti SYZV eſt centrum grauitatis G: fruſti autem TI <lb></lb><figure id="id.043.01.177.1.jpg" xlink:href="043/01/177/1.jpg"></figure><lb></lb><foreign lang="grc">γ</foreign>X centrum grauitatis H; totius igitur compoſiti ex his <lb></lb>duobus fruſtis centrum grauitatis erit K: commune autem <lb></lb>eſt centrum grauitatis compoſiti ex duobus fruſtis SYZV <lb></lb>& TI<foreign lang="grc">γ</foreign>X, fruſto ALMC per antepenultimæ huius co<lb></lb>rollarium; fruſti igitur ALMC, centrum grauitatis erit K. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/178.jpg" pagenum="91"></pb><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Ex omnibus demonſtrationibus eorum, quæ in <lb></lb>hoc ſecundo libro propoſuimus, manifeſtum eſt <lb></lb>omnium ſupra dictorum corporum centra grauita <lb></lb>tis inuenire: quæ cum que enim in modum theore<lb></lb>matis propoſuimus, eadem tanquam problema<lb></lb>ta proponi, & ijſdem demonſtrationibus abſolui <lb></lb>poſſunt. </s></p><p type="main"> <s>Idem dico de ijs, quæ in primo, & tertio ſequenti libro <lb></lb>demonſtrauimus. </s> <s>Porro autem multa lemmata inſtituto <lb></lb>præcipuo neceſſaria, & alia addita inuentio ſatis iucun<lb></lb>da centri grauitatis conoidis, & portionis conoidis parabo<lb></lb>lici, & hyperbolici, & fruſti vtriuſque ne ſecundus hic liber <lb></lb>nimis longus, & confuſus exiſteret, tertium requirebant. <lb></lb></s> <s>Quem quidem meorum ſtudiorum autumnalium fructum <lb></lb>Anni à partu Virginis MDCIII. cum SS. </s> <s>Clementis <lb></lb>Pont. Max. </s> <s>auctoritate, & Petri eius Nepotis Cardinalis <lb></lb>ampliſſimi Aldobrandini iuſſu bene de me merentium Ma<lb></lb>thematicam ſcientiam, & Philoſophiam ciuilem in almo <lb></lb>Vrbis Gymnaſio profiterer, in eorum gratiam compoſui, <lb></lb>qui me centra grauitatis portionum ſphæroidis imperfe<lb></lb>cti operis crimine condemnandum omittere nolebant; cu<lb></lb>ius prouinciæ iuuante Deo, & mira Mathematicæ ſtudio<lb></lb>ſis ſatisfaciendi voluntate, multas difficultates ita ſupe<lb></lb>raui, vt vno menſe Octobri plus præſtiterim, quam à me <lb></lb>requiſiſſent. </s> <s>ſiquidem quæ de ſphæræ portionibus in hoc <lb></lb>libro proprijs eius figuræ rationibus, eadem in ſequen<lb></lb>ti aliis communibus cuilibet portioni ſphæræ, & ſphæroi<lb></lb>dis tum lati, tum oblongi abſciſſæ vno, vel duobus planis <lb></lb>æque inter ſe diſtantibus, & vtcumque in figuram in cideu-<pb xlink:href="043/01/179.jpg" pagenum="92"></pb>tibus demonſtraui, & temporis breuitatem magna animi in<lb></lb>tentione compenſaui, quòd facere non potuiſsem niſi illi, <lb></lb>quos ſupra nominaui meos patronos tranquillum otium <lb></lb>mihi ſua benignitate peperiſſent; ego autem quoſdam ad<lb></lb>uerſos flatus vehementes in meam vtilitatem verte<lb></lb>re didiciſsem, cuius rei monumentum flammæ <lb></lb>vento agitatæ ſimulacrum cum illo Ver<lb></lb>gilij HOC ACRIOR in fronte <lb></lb>operis poſui, vt meus qualiſ<lb></lb>cumque hic labor vel ab <lb></lb>inuitis in me collati <lb></lb>bencficij memo<lb></lb>riam præſe<lb></lb>ferret. </s></p><p type="head"> <s>SECVNDI LIBRI FINIS.<lb></lb><figure id="id.043.01.179.1.jpg" xlink:href="043/01/179/1.jpg"></figure></s></p><pb xlink:href="043/01/180.jpg" pagenum="1"></pb><figure id="id.043.01.180.1.jpg" xlink:href="043/01/180/1.jpg"></figure><p type="head"> <s>L V C AE <lb></lb>VALER II <lb></lb>DE CENTRO <lb></lb>GRAVITATIS <lb></lb>SOLIDORVM <lb></lb><emph type="italics"></emph>LIBER TERTIVS.<emph.end type="italics"></emph.end></s></p><figure id="id.043.01.180.2.jpg" xlink:href="043/01/180/2.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO I.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si recta linea ſecta fuerit bifa<lb></lb>riam, & non bifariam; rectan <lb></lb>gulum partibus in æqualibus <lb></lb>contentum æquale eſt rectan <lb></lb>gulo, quod bis fit ex dimidiæ <lb></lb>ſectæ ſegmentis, vna cum <lb></lb>quadrato non intermedij eo<lb></lb>rundem ſegmentorum. </s></p><pb xlink:href="043/01/181.jpg" pagenum="2"></pb><p type="main"> <s>Sit recta linea AB ſecta in puncto C biſariam, & non <lb></lb>bifariam in puncto D. </s> <s>Dico rectangulum ADB æqua<lb></lb>le eſſe rectangulo BDC bis vnà cum quadrato BD. <lb></lb></s> <s>Quoniam enim rectangulum ADB, æquale eſt duobus <lb></lb>rectangulis, & ex BD, DC, & ex AC, BD, hoc eſt ex <lb></lb>CB, BD: ſed rectangulum ex CB, BD, eſt rectangu<lb></lb>lum ex BD, DC, vnà cum quadrato BD; rectangulum <lb></lb>igitur ex AD, DB, æquale eſt duobus rectangulis ex <lb></lb>BD, DC, vnà cum quadiato BD. </s> <s>Si igitur recta linea <lb></lb>ſecta fuerit bifariam, & non bifariam, &c. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><figure id="id.043.01.181.1.jpg" xlink:href="043/01/181/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO II.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si circulum, vel ellipſim duæ rectæ lineæ tan<lb></lb>gentes in terminis coniugatarum diametrorum, <lb></lb>conueniant: & punctum in quo conueniunt, & <lb></lb>centrum figuræ iungantur recta linea; quæcun<lb></lb>que hanc vnà cum prædictæ figuræ termino al<lb></lb>terutri diametrorum parallela ſecuerit recta li<lb></lb>nea, ita ipſa ſecabitur in duobus punctis, vt re<lb></lb>ctangulum bis contentum ſegmentis, quorum al<lb></lb>terum inter diametrum, & terminum figuræ, al<lb></lb>terum inter figuræ terminum & contingentem <lb></lb>interijcitur, vnà cum huius quadrato, ſit æquale <lb></lb>quadrato reliqui ſegmenti inter diametrum, & <pb xlink:href="043/01/182.jpg" pagenum="3"></pb>cum quæ tangentium concurſum, & centrum fi<lb></lb>guræ iungit interiecta. </s></p><p type="main"> <s>Sit circulus, vel ellipſis ABCD, cuius diametri con<lb></lb>iugatæ AC, BED, & figuram tangentes BF, GF, con <lb></lb>ueniant in puncto F; (parallelæ enim erunt vtraque alteri <lb></lb>coniugatorum diametrorum:) & recta FE iungatur, & ex <lb></lb>quolibet puncto G, in recta BE ducatur ipſi AC paral<lb></lb>lela GLKH. </s> <s>Dico rectangulum GKH bis vnà cum <lb></lb>quadrato KH æquale eſſe quadrato GL. </s> <s>Quoniam <lb></lb>enim rectangulum BGD æquale eſt rectangulo BGE <lb></lb><figure id="id.043.01.182.1.jpg" xlink:href="043/01/182/1.jpg"></figure><lb></lb>bis vnà cum quadrato BG: & rectangulum BED, eſt <lb></lb>quadratum BE, erit vt rectangulum BED, ad re<lb></lb>ctangulum BGD, ita quadratum BE, ad rectangu<lb></lb>lum BGE bis, vnà cum quadrato BG: ſed vt rectangu<lb></lb>lum BED, ad rectangulum BGD, ita eſt quadratum EC, <lb></lb>hoc eſt quadratum GH ad quadratum GK, ex primo <lb></lb>conicorum, vt igitur eſt quadratum BE ad rectangulum <lb></lb>BGE bis, vnà cum quadrato BG, ita erit quadratum <lb></lb>GH ad quadratum GK. </s> <s>Rurſus quia eſt vt BE ad EG, <lb></lb>ita BF ad GL, propter ſimilitudinem triangulorum; erit <lb></lb>vt quadratum BE ad quadratum EG, ita quadratum <pb xlink:href="043/01/183.jpg" pagenum="4"></pb>BF hoc eſt quadratum GH ad quadratum GL: & per <lb></lb>conuerſionem rationis, vt quadratum BE ad rectangu<lb></lb>lum BGE bis, vnà cum quadrato BG, ita quadratum <lb></lb>GH ad rectangulum GLH bis, vnà cum quadrato LH: <lb></lb>ſed vt quadratum BE ad rectangulum EGB bis, vnà <lb></lb>cum quadrato BG, ita erat quadratum GH ad quadra<lb></lb>tum GK; vt igitur quadratum GH ad quadratum GK, <lb></lb>ita erit idem quadratum GH ad rectangulum GLH bis, <lb></lb>vnà cum quadrato LH: quadratum igitur GK æquale <lb></lb>erit rectangulo GLH bis, vnà cum quadrato LH; demptis <lb></lb>igitur ab eodem quadrato GH æqualibus quadrato GK, <lb></lb>& rectangulo GLH bis, vnà cum quadrato LH, erit <lb></lb>rectangulum GKH, bis vnà cum quadrato KH æquale <lb></lb>quadrato GL. </s> <s>Quod demonſtrandum erat. </s></p><figure id="id.043.01.183.1.jpg" xlink:href="043/01/183/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO III.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Per data duo puncta in duabus rectis lineis da<lb></lb>tum angulum continentibus, in earum plano pa<lb></lb>rabola tranſibit, cuius vertex ſit aſſignatum præ<lb></lb>dictorum punctorum, in quo altera linea parabo-<pb xlink:href="043/01/184.jpg" pagenum="5"></pb>lam contingat, altera in altero ſecet diametro æ<lb></lb>quidiſtans. </s></p><p type="main"> <s>Sint data duo puncta. </s> <s>A, C, in duabus rectis lincis da<lb></lb>tum angulum ABC continentibus, ſit autem aſſignatum <lb></lb>punctum C. </s> <s>Dico per puncta A, C, parabolam tranſi<lb></lb>re, ita vt ipſam linea AC contingat in C puncto, altera <lb></lb>autem AB ſecet in puncto A, diametro parabolæ æqui<lb></lb>diſtans. </s> <s>Completo enim parallelogrammo BD, ad re<lb></lb>ctam CD applicetur rectangulum æquale quadrato AD, <lb></lb>faciens latitudinem E. </s> <s>Quoniam igitur in plano BD <lb></lb>parabola inueniri poteſt, cu<lb></lb>ius ſit vertex C, diameter <lb></lb>CD, ita vt quædam ex ſe<lb></lb>ctione ad diametrum CD <lb></lb>applicata in dato angulo A <lb></lb>BC, ideſt ADC, qualis <lb></lb>eſt recta AD, poſſit rectan<lb></lb>gulum ex CD, & E, ex <lb></lb>primo conicorum elemen. <lb></lb></s> <s>to; ſit ea ſectio parabola <lb></lb><figure id="id.043.01.184.1.jpg" xlink:href="043/01/184/1.jpg"></figure><lb></lb>AC; aſſignatum eſt autem punctum C; per puncta igi<lb></lb>tur A, C parabola AC tranſibit, cuius vertex eſt aſſi<lb></lb>gnatum punctum C. </s> <s>Et quoniam quæ ex vertice recta <lb></lb>CB eſt applicatæ DA parallela, ſectionem AC in pun<lb></lb>cto C continget: eſt autem AB diametro CD æquidi<lb></lb>diſtans, ac proinde parabolam ſecabit in puncto A. </s> <s>Ma<lb></lb>nifeſtum eſt igitur propoſitum, </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si recta linea parabolam contingat, omnes re<lb></lb>ctælineæ ex ſectione ad contingentem applicatæ <pb xlink:href="043/01/185.jpg" pagenum="6"></pb>diametro ſectionis parallelæ inter ſe ſunt longi<lb></lb>tudine, vt inter applicatas & contactum, vel ver<lb></lb>ticem interiectæ inter ſe potentia. </s> <s>Productis au<lb></lb>tem dictis applicatis, erunt inter ſectionem & ba<lb></lb>ſim interiectæ inter ſe longitudine, vt in circulo, <lb></lb>vel ellipſe ad diametrum ordinatim applicatæ, ſe<lb></lb>cantesque illam in eaſdem rationes, in quas aliæ <lb></lb>prædictæ applicatæ ſecant baſim parabolæ, inter <lb></lb>ſe potentia. </s></p><p type="main"> <s>Sit ſectio parabola ABC, cuius vertex B, diameter <lb></lb>BD: & recta quadam BE ſectionem contingente in pun<lb></lb>cto B, ſint quotcumque rectæ lineæ ex ſectione ordinatim <lb></lb>ad BE contingentem applicatæ diametro BD ſectionis <lb></lb>parallelæ FG, KH, quibus productis ſint ad baſim ſe<lb></lb><figure id="id.043.01.185.1.jpg" xlink:href="043/01/185/1.jpg"></figure><lb></lb>ctionis applicatæ GN, KO. </s> <s>Et expoſito primum circu<lb></lb>lo, PQRS, cuius diametri ad rectos inter ſe angulos ſint <lb></lb>QS, PR; ſecta autem QT in punctis V, X, in eaſ<lb></lb>dem rationes, in quas ſecta eſt AD in punctis N, O, <lb></lb>ſumpto ordine à punctis D, T, vt ſit DO ad ON, <pb xlink:href="043/01/186.jpg" pagenum="7"></pb>vt eſt TV ad VX: & vt ON ad NA, ita VX ad <expan abbr="Xq;">Xque</expan> <lb></lb>applicentur ad ſemidiametrum QT rectæ ZV, XY dia<lb></lb>metro PR æquidiſtantes. </s> <s>Dico eſſe HK ad FG lon<lb></lb>gitudine, vt FB ad BH potentia: & KO ad GN longi<lb></lb>tudine, vt ZY ad YX potentia. </s> <s>Iungantur enim KL, <lb></lb>GM, baſi AC parallelæ. </s> <s>Quoniam igitur eſt vt MB <lb></lb>ad BI. longitudine, ita GM ad KL potentia: ſed MB <lb></lb>eſt æqualis ipſi FG, & BL ipſi KH, & BF ipſi GM, & <lb></lb>BH ipſi KL in parallelogrammis BG, BK; vt igitur <lb></lb>FG ad KH longitudine, ita erit BH ad BF potentia: <lb></lb>ſimiliter quotcumque plures eſſent applicatæ idem oſten<lb></lb>deremus. </s> <s>Rurſus, quoniam eſt vt EA, hoc eſt FN ad FG, <lb></lb>ita quadratum EB ad BF quadratum, hoc eſt quadra<lb></lb>tum AD ad quadratum DN, hoc eſt ita quadratum QT, <lb></lb>hoc eſt quadratum TY, hoc eſt duo quadrata TX, XY, <lb></lb>ad quadratum TX; erit per conuerſionem rationis, vt FN, <lb></lb>hoc eſt BD ad GN, ita duo quadrata TX, X<foreign lang="grc">Υ</foreign> ſimul, <lb></lb>hoc eſt quadratum TY, hoc eſt quadratum TP, ad qua<lb></lb>dratum XY. </s> <s>Similiter oſtenderemus eſſe vt BD ad <lb></lb>OK, ita quadratum PT ad quadratum VZ. </s> <s>Conuer<lb></lb>tendo igitur erit vt OK ad BD, ita quadratum XY ad <lb></lb>PT quadratum: & ex æquali vt OK ad GN, ita qua<lb></lb>dratum VZ ad quadratum XY. </s> <s>Suntigitur tres rectæ <lb></lb>lineæ BD, OK, GN, inter ſe longitudine, vt in circu<lb></lb>lo PQSR totidem PT, ZV, XY inter ſe potentia, <lb></lb>prout inter ſe reſpondent. </s> <s>Idem autem ſimiliter oſten<lb></lb>deremus de quotcumque aliis in circulo, & ſectione para<lb></lb>bola vt prædictæ applicatis multitudine æqualibus. </s> <s>In <lb></lb>ellipſe autem, ductis diametris quibuſuis coniugatis, & <lb></lb>totidem quot in circulo ad vnam ſemidiametrum rectis li<lb></lb>neis ordinatim applicatis ſecundum puncta ſectionum eiuſ<lb></lb>dem diametri in eaſdem prædictas rationes, eodemque or<lb></lb>dine; quoniam ex XXI primi conicorum ſtatim apparet re<lb></lb>ctarum linearum ita vt diximus in circulo, & ellipſe appli-<pb xlink:href="043/01/187.jpg" pagenum="8"></pb>catarum quadrata eſſe inter ſe in eadem proportione; erunt<lb></lb>prædictæ inter ſectionem parabolam, & baſim interiectæ <lb></lb>inter ſe longitudine, vt in ellipſe ad diametrum ſimiliter <lb></lb>vt diximus applicatæ inter ſe potentia. </s> <s>Manifeſtum eſt <lb></lb>igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO V.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis figuræ circa axim in alteram partem <lb></lb>deficientis, cuius ſuperficies, excepta baſe ſit to<lb></lb>ta interius concaua baſim habentis circulum, vel <lb></lb>ellipſim; quælibet tres ſectiones baſi parallelæ <lb></lb>æqualia axis ſegmenta intercipientes, ita ſe ha<lb></lb>bent, vt minor ſit proportio minimæ ad mediam, <lb></lb>quam mediæ ad maximam. </s></p><p type="main"> <s>Sit figura ABC circa axem BD in alteram partem de<lb></lb>ficiens, qualem diximus: & poſitis in axe BD tribus qui<lb></lb>buslibet punctis <lb></lb>F, E, L, æqualia <lb></lb>axis ſegmenta in<lb></lb>tercipientibus, in <lb></lb>telligatur <expan abbr="ſolidũ">ſolidum</expan> <lb></lb>ABC ſectum per <lb></lb>ea puncta planis <lb></lb><expan abbr="buibuſdã">buibuſdam</expan> baſi cir <lb></lb>culo, vel ellipſi, <lb></lb>circa AC pa<lb></lb>rallelis: quare ſe<lb></lb>ctiones erunt cir<lb></lb><figure id="id.043.01.187.1.jpg" xlink:href="043/01/187/1.jpg"></figure><lb></lb>culi, vel ellipſes ſimiles baſi, per definitionem, quarum dia<lb></lb>metri eiuſdem rationis in eodem plano per axim ſint IK. <pb xlink:href="043/01/188.jpg" pagenum="9"></pb>GH, MN. </s> <s>Dico ſolidi ABC ſectionum, minorem eſſe <lb></lb>proportionem, ipſius IK ad GH, quàm GH ad MN. <lb></lb></s> <s>Iunctis enim MRS, KSN; quoniam tres rectæ IK, <lb></lb>RS, MN, ſeſe æqualiter excedunt in trapezio KM; mi<lb></lb>nor erit proportio IK ad RS, quàm RS ad MN: ſed cir <lb></lb>culi, & ſimiles ellipſes duplicatam habent inter ſe propor<lb></lb>tionem diametrorum eiuſdem rationis; trium igitur præ<lb></lb>dictarum ſolidi ABC ſectionum minor erit proportio IK <lb></lb>ad RS quàm RS ad MN: ſed maior eſt proportio circu<lb></lb>li, vel ellipſis GH ad circulum, vel ellipſim MN, quàm <lb></lb>circuli, vel ellipſis RS, ad circulum, vel ellipſim MN; <lb></lb>multo ergo minor proportio erit circuli, vel ellipſis IK ad <lb></lb>circulum, vel ellipſim RS, quàm circuli, vel ellipſis GH ad <lb></lb>circulum, vel ellipſim MN: ſed minor eſt proportio cir<lb></lb>culi vel ellipſis I<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ad circulum, vel ellipſim GH, quàm <lb></lb>eiuſdem circuli, vel ellipſis IK ad circulum, vel ellipſim <lb></lb>RS; multo ergo minor proportio erit circuli, vel ellipſis <lb></lb>IK ad circulum, vel ellipſim GH quàm circuli, vel ellip<lb></lb>ſis GH ad circulum, vel ellipſim MN. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſphæroides ſecetur plano vtcumque præter <lb></lb>quàm ad axem, circa quem ſphæroides deſcribi<lb></lb>tur erecto nam tunc circulus fit. </s> <s>ſectio ellipſis erit: <lb></lb>ſimilis autem ipſi alia quæcumque ſectio ſphæ<lb></lb>roidis eidem parallela: earumque omnes diame<lb></lb>tri quæ eiuſdem ſunt rationis erunt in eodem pla<lb></lb>no per axem. </s></p><p type="main"> <s>Extant hæc demonſtrata ab Archimede in ſuo de ſphæ<lb></lb>roidibus, & conoidibus. </s></p><pb xlink:href="043/01/189.jpg" pagenum="10"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si conoides parabolicum, vel hyperbolicum <lb></lb>ſecetur plano vtcumque ad axim inclinato, ſectio <lb></lb>ellipſis erit: ſimilis autem ipſi alia quæcumque <lb></lb>ſectio conoidis eidem parallela: eruntque earum <lb></lb>omnes diametri, quæ eiuſdem ſunt rationis in eo<lb></lb>dem plano per axem. </s></p><p type="main"> <s>Manifeſta ſunt hæc ex ijs, quæ Federicus Commandinus <lb></lb>demonſtrauit de ſectionibus horum ſolidorum, in ſuis com<lb></lb>mentariis in eundem Archimedis librum de ſphæroidibus, <lb></lb>& conoidibus: quemadmodum & ſphæroidis, & conoi<lb></lb>dis vtriuſque ſectionem factam à plano ad axim erecto eſ<lb></lb>ſe circulum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Super datam ellipſim, circa datam rectam line<lb></lb>am ab eius centro eleuatam tanquam axem, coni, <lb></lb>& cylindri portionem inuenire. </s> <s>Datoque ſphæ<lb></lb>roidi, & conoidi, vel conoidis, ſphæroidiſve por<lb></lb>tioni circa datum axem ſphæroidis, vel cuiuslibet <lb></lb>dictarum portionum, cylindrus vel cylindri por<lb></lb>tio circumſcripta eſſe poteſt: vel comprehendere <lb></lb>inter eadem plana parallela, ita vt eius baſis ſit ſi<lb></lb>milis baſi, vel baſibus comprehenſæ portionis, vel <lb></lb>fruſti, ſi de conoidibus ſit ſermo: & diametri, quæ <lb></lb>eiuſdem ſunt rationis ſectæ à centro bifariam ſint <lb></lb>in eadem recta linea. </s></p><pb xlink:href="043/01/190.jpg" pagenum="11"></pb><p type="main"> <s>Manifeſta item ſunt hæc omnia, ex ijs, quæ in eodem li<lb></lb>bro de ſphæroidibus, & conoidibus demonſtrat Archi<lb></lb>medes. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti pyramidis triangulam baſim ha<lb></lb>bentis ad priſtina, cuius baſis eſt maior baſis fru<lb></lb>ſti, & eadem altitudo, cam habet proportionem, <lb></lb>quàm rectangulum contentum duobus lateribus <lb></lb>homologis baſium oppoſitarum, vnà cum tertia <lb></lb>parte quadrati differentiæ dictorum laterum, ad <lb></lb>maioris lateris quadratum. </s> <s>Ad pyramidem autem, <lb></lb>cuius baſis eſt maior baſis fruſti, & eadem altitu<lb></lb>do, vt prædictum rectangulum, vna cum prædicti <lb></lb>quadrati tertia parte, ad tertiam partem quadrati <lb></lb>maioris lateris. </s></p><p type="main"> <s>Sit pyramidis triangulam baſim habentis fruſtum AB <lb></lb>CD EF: laterum autem homo<lb></lb>logorum AB, DE, triangulorum <lb></lb>ſimilium oppoſitorum ABC, D <lb></lb>EF, ſit differentia DG: & eiuſ<lb></lb>dem altitudinis fruſto ſit priſma <lb></lb>DEFCHK: & pyramis intelli<lb></lb>gatur ADEF. </s> <s>Dico fruſtum <lb></lb>BDF ad priſma HKF, eſſe vt <lb></lb>rectangulum DEG vna cum ter<lb></lb>tia parte quadrati DG. </s> <s>Ad qua<lb></lb>dratum DE: ad pyramidem au<lb></lb>tem ADEF, vt <expan abbr="prædictũ">prædictum</expan> rectan<lb></lb><figure id="id.043.01.190.1.jpg" xlink:href="043/01/190/1.jpg"></figure><lb></lb>gulum DEG, vnà cum tertia parte quadrati DG, ad ter<pb xlink:href="043/01/191.jpg" pagenum="12"></pb>tiam partem quadrati DE. </s> <s>Abſciſsis enim æqualibus EL <lb></lb>ipſi BC, & FM ipſi AC, & EG, ipſi AB, conſtituantur <lb></lb>priſmata ABCLEG, AGMFCL, ANHDGM, & <lb></lb>pyramis ADGM, & iungatur ML. </s> <s>Quoniam igitur ob pa<lb></lb>rallelas EF, GM, & DF, GL, ſimilia inter ſe ſunt trian<lb></lb>gula DEF, DGM, EGL, duplicatam inter ſe habebunt <lb></lb>laterum ho mologorum DE, DG, GE, proportionem, <lb></lb>hoc eſt eandem, quæ totidem eſt quadratorum ex ipſis DE, <lb></lb>DG, GE, prout inter ſe reſpondent: vt igitur DG qua<lb></lb>dratum ad quadratum DE, ita eſt triangulum DGM <lb></lb>ad triangulum DEF: eademque ratione vt quadratum <lb></lb>GE ad DE quadratum, ita trian <lb></lb>gulum EGL ad triangulum D <lb></lb>EF: & vt prima cum quinta ad <lb></lb>ſecundam, ita tertia cum ſexta ad <lb></lb>quartam: videlicet, vt duo qua<lb></lb>drata DG, GE, ad quadratum <lb></lb>DE, ita duo triangula DGM, <lb></lb>EGL, ad triangulum DEF. & <lb></lb>conuertendo, & per conuerſionem <lb></lb>rationis, vt quadratum DE ad <lb></lb>rectangulum DGE bis, ita trian<lb></lb>gulum DEF, ad parallelogram<lb></lb><figure id="id.043.01.191.1.jpg" xlink:href="043/01/191/1.jpg"></figure><lb></lb>mum GF: & conuertendo, vt rectangulum DGE bis, ad <lb></lb>quadratum DE, ita GF parallelogrammum ad triangu<lb></lb>lum DEF: & antecedentium dimidia, vt rectangulum <lb></lb>DGE ad quadratum DE, ita triangulum GML ad <lb></lb>triangulum DEF; hoc eſt priſma, cuius baſis triangulum <lb></lb>GLM, altitudo eadem priſmati H<emph type="italics"></emph>K<emph.end type="italics"></emph.end>F ad priſma HKF. </s></p><p type="main"> <s>Rurſus, quoniam eſt vt quadratum EG ad quadratum <lb></lb>ED, ita triangulum EGL ad triangulum DEF; erit ſi<lb></lb>militer vt quadratum EG ad quadratum ED, ita priſma <lb></lb>BGL ad priſma HKF: ſed vt rectangulum DGE ad <lb></lb>quadratum DE, ita priſma erat, cuius baſis triangulum G <pb xlink:href="043/01/192.jpg" pagenum="13"></pb>LM altitudo autem eadem priſmati HKF, hoc eſt priſma <lb></lb>ACGLFM illi æquale per vltimam XI. elem. </s> <s>ad priſma <lb></lb>HKF: vt igitur prima cum quinta, rectangulum DGE <lb></lb>vna cum quadrato EG, hoc eſt rectangulum DEG, ad <lb></lb>ſecundam quadratum DE, ita erit tertia cum ſexta, duo <lb></lb>priſmata BGL, ACGLFM, ad quartam priſma HKF. <lb></lb></s> <s>Præterea quoniam vt quadratum DG ad quadratum <lb></lb>DE, ita erat triangulum DGM ad triangulum DEF: ſed <lb></lb>vt triangulum DGM ad triangulum DEF, ita eſt priſma, <lb></lb>HGM, ad priſma HKF: & tertiæ antecedentium par<lb></lb>tes, videlicet, vt tertia pars quadrati DG, ad quadra<lb></lb>tum DE, ita pyramis ADGM ad priſma HKF: ſed <lb></lb>vt rectangulum DEG ad DE quadratum, ita erant duo <lb></lb>priſmata BGL, ACGLFM, ad priſma HKF; vt igi<lb></lb>tur prima cum quinta, rectangulum DEG vna cum ter<lb></lb>tia parte DG quadrati, ad quadratum GD ſecundam, <lb></lb>ita erit tertia cum ſexta, duo priſmata BGL, ACGLFM <lb></lb>vna cum pyramide ADGM, hoc eſt integrum fruſtum <lb></lb>ABCDEF ad priſma HKF quartam. </s> <s>Ex hoc patet ſe<lb></lb>cunda pars propoſiti. </s> <s>Quoniam enim eſt vt rectangulum <lb></lb>DEG, vna cum tertia parte quadrati DG, ad quadra<lb></lb>tum DE, ita fruſtum ABGDEF ad priſma HKF: vt <lb></lb>autem quadratum DE, ad tertiam ſui partem, ita eſt priſ<lb></lb>ma HKF ad pyramidem, cuius baſis triangulum DEF, <lb></lb>altitudo eadem priſmati HKF; erit ex æquali vt re<lb></lb>ctangulum DEG vna cum tertia parte quadrati DG <lb></lb>ad tertiam partem quadrati DE, ita fruſtum ABCDEF, <lb></lb>ad pyramidem ſi compleatur ADEF. </s> <s>Manifeſtum eſt <lb></lb>igitur propoſitum. </s></p><pb xlink:href="043/01/193.jpg" pagenum="14"></pb><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hinc manifeſtum eſt eadem demonſtratione, <lb></lb>qua vtimur ad propoſitionem XXXVI. primili<lb></lb>bri; fruſtum cuiuslibet pyramidis baſim habentis <lb></lb>pluribus quàm tribus lateribus contentam, ad priſ <lb></lb>ma, ſeu pyramidem, cuius baſis eſt eadem quæ ma<lb></lb>ior baſis fruſti, & eadem altitudo: & reliquum ip<lb></lb>ſius priſmatis dempto fruſto, ad ipſum priſma, eas <lb></lb>habere rationes, quæ à baſium fruſti oppoſitarum <lb></lb>homologis lateribus eorumque differentia deri<lb></lb>uantur eo modo, quo in præcedenti theoremate <lb></lb>dicebamus. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO X.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omne fruſtum coni, vel portionis conicæ, ad cy <lb></lb>lindrum, vel cylindri portionem, cuius baſis eſt ea <lb></lb>dem, quæ maior baſis fruſti, & eadem altitudo, <lb></lb>eam habet proportionem, quàm rectangulum con <lb></lb>tentum baſium diametris eiuſdem rationis, vnà <lb></lb>eum tertia parte quadrati differentiæ earumdem <lb></lb>diametrorum, ad maioris baſis quadratum. </s> <s>Ad <lb></lb>conum autem, vel coni portionem, cuius baſis eſt <lb></lb>eadem, quæ maior baſis fruſti, & eadem altitudo; <lb></lb>vt prædictum rectangulum, vnà cum prædicti qua <lb></lb>drati tertia parte, ad tertiam partem quadrati ex <lb></lb>diametro maioris baſis. </s> <s>Prædicti autem cylindri, <pb xlink:href="043/01/194.jpg" pagenum="15"></pb>vel portionis cylindricæ reſiduum dempto fruſto, <lb></lb>ad totum cylindrum, vel cylindri portionem; vt <lb></lb>rectangulum contentum diametro minoris baſis <lb></lb>fruſti, & differentia diametri maioris, vnà cum <lb></lb>duabus tertiis quadrati differentiæ, ad quadra<lb></lb>tum diametri maioris baſis. </s></p><p type="main"> <s>Sit coni, vel eius portionis fruſtum ABCD, cuius baſes <lb></lb>oppoſitæ, circuli vel ſimiles ellipſes, quarum diametri mi<lb></lb>noris baſis AB cuius centrum E: maioris autem CD, <lb></lb>& ſuper baſim circulum, vel ellipſim CD ſtet cylindrus, <lb></lb>vel portio cylindrica CG comprehendens fruſtum AB <lb></lb>CD, eiuſdemque altitudinis cum ipſo, & conus, vel co<lb></lb>ni portio ECD. quo autem AC diameter ſuperat dia<lb></lb>metrum AB, quæ differentia di<lb></lb>citur, ſit DF. </s> <s>Dico fruſtum AD <lb></lb>ad cylindrum, vel portionem cy<lb></lb>lindricam CG, eſſe vt rectangu<lb></lb>lum DCF vnà cum tertia parte <lb></lb>quadrati DF, ad quadratum CD. <lb></lb></s> <s>Ad conum autem vel coni portio<lb></lb>nem ECD, vt rectangulum DCF, <lb></lb>vna cum tertia parte quadrati DF, <lb></lb>ad tertiam partem quadrati CD. <lb></lb></s> <s>Cylindri autem, vel cylindri por<lb></lb>tionis CG reſiduum dempto fru<lb></lb><figure id="id.043.01.194.1.jpg" xlink:href="043/01/194/1.jpg"></figure><lb></lb>ſto AD, ad cylindrum, vel portionem cylindricam CG, <lb></lb>vt rectangulum CFD vna cum duabus tertiis quadrati <lb></lb>FD, ad quadratum CD. </s> <s>Cono enim, vel portioni coni<lb></lb>cæ, cuius fruſtum AD, & cylindro, vel portioni cylindri<lb></lb>cæ, cuius baſis eſt circulus, vel ellipſis CD, altitudo au<lb></lb>tem eadem completo cono, vel portioni conicæ iam dictæ, <lb></lb>illi pyramis, huic priſma inſcripta intelligantur, quorum <pb xlink:href="043/01/195.jpg" pagenum="16"></pb>communis baſis ſit poly gorum inſcriptum circulo quidem <lb></lb>æquilaterum, & æquiangulum; in ellipſe autem, quod pro <lb></lb>Archimede deſcribit Commandinus, ita vt & à cylindro, <lb></lb>vel cylindri portione priſina, & à cono, vel coni portione <lb></lb>pyramis deficiat minori ſpacio quantacumque magnitudi<lb></lb>ne propoſita: quo modo autem in portione cylindrica, vel <lb></lb>conica hoc fieri poſſit, eadem quæ de cono atque cylindro <lb></lb>Euclides in duodecimo docuit manifeſtant. </s> <s>Abſciſſione <lb></lb>igitur facta fruſti AD, & cylindri, vel portionis cylindricæ <lb></lb>CG, abſciſſa ſimul erunt fruſtum pyramidis inſcriptum <lb></lb>fruſto AD, & priſma inſcriptum cylindro, vel portioni cy<lb></lb>lindricæ CG, eiuſdem altitudinis inter ſe, & duobus præ<lb></lb>dictis ſolidis AD, CG, deficien <lb></lb>tia vnum à fruſto, alterum à cy<lb></lb>lindro, vel portione cylindrica <lb></lb>multo minori ſpacio magnitudine <lb></lb>propoſita: ſectiones autem priſma <lb></lb>tis, & pyramidis erunt polygona <lb></lb>circulis, vel ellipſibus ipſi CD op <lb></lb>poſitis & ſimilibus inſcripta in<lb></lb>ter ſe ſimilia, vt multi oſtendunt. <lb></lb></s> <s>erunt etiam ſimilium polygono<lb></lb>rum circulis, vel ellipſibus ſimili<lb></lb>bus, quæ ſunt baſes oppoſitæ fru<lb></lb><figure id="id.043.01.195.1.jpg" xlink:href="043/01/195/1.jpg"></figure><lb></lb>ſti AD, inſcriptorum diametri eædem AB, CD. </s> <s>Quo<lb></lb>niam igitur ſimilium polygonorum circulis, & ſimilibus <lb></lb>ellipſibus inſcriptorum latera homologa inter ſe ſunt vt <lb></lb>diametri dictorum circulorum, vel ellipſium, eadem erit <lb></lb>proportio inter duas diametros AB, CD, hoc eſt FC, <lb></lb>CD, quæ inter duo quælibet latera homologa polyga<lb></lb>norum circulis, vel ellipſibus ſimilibus AB, CD in<lb></lb>ſcriptorum. </s> <s>Sed pyramidis fruſtum fruſto CB inſcri<lb></lb>ptum ad priſma, cuius baſis eſt maior baſis fruſti pyrami<lb></lb>dis, & eadem altitudo, ſolido CG inſcriptum, eſt vt re-<pb xlink:href="043/01/196.jpg" pagenum="17"></pb>ctangulum contentum lateribus homologis baſium oppo<lb></lb>ſitarum, vna cum tertia parte quadrati differentiæ, ad ma<lb></lb>ioris lateris quadratum; idem igitur fruſtum pyramidis <lb></lb>ad idem priſma, erit vt rectangulum DCF, vna cum <lb></lb>tertia parte quadrati DF ad quadratum CD: deficit <lb></lb>autem vtrumque & pyramidis fruſtum fruſto CB inſcri<lb></lb>ptum ab ipſo CB fruſto, & priſma ipſi CG inſcriptum <lb></lb>ab ìpſo CG, minori ſpacio quantacumque propoſita ma<lb></lb>gnitudine; per tertiam igitur huius, erit vt rectangulum <lb></lb>DCF vna cum tertia parte quadrati DF, ad CD qua<lb></lb>dratum, ita fruſtum CB ad cylindrum, vel portionem <lb></lb>cylindricam CG. </s> <s>Cum igitur conus, vel coni portio E <lb></lb>CD ſit pars tertia cylindri, vel portionis cylindricæ CG, <lb></lb>erit ex æquali, vt idem rectangulum DCF, vna cum ter<lb></lb>tia parte quadrati DF, ad tertiam partem quadrati CD, <lb></lb>ita fruſtum BC, ad conum vel coni portionem ECD. Præ<lb></lb>terea, quia quadratum CD æquale eſt duobus quadratis <lb></lb>ex CF, FD, vna cum rectangulo bis ex CF, FD: quorum <lb></lb>rectangulo CFD, vna cum quadrato CF æquale eſt rectan<lb></lb>gulum DCF; erit quadratum CD æquale rectangulo <lb></lb>DCF vna cum quadrato DF; demptis igitur rectangu<lb></lb>lo DCF, & tertia parte quadrati DF; quod remanet <lb></lb>CD quadrati erit rectangulum CFD vna cum duabus <lb></lb>tertiis quadrati DF. quoniam igitur eſt conuertendo vt <lb></lb>quadratum CD ad rectangulum DCF, vna cum tertia <lb></lb>parte quadrati DF, ita cylindris, vel portio cylindrica <lb></lb>CG ad fruſtum CB, erit per conuerſionem rationis, & <lb></lb>conuertendo; vt rectangulum CFD vna cum duabus ter<lb></lb>tiis DF quadrati, ad quadratum CD, ita reliquum cy<lb></lb>lindri, vel portionis cylindricæ CG dempto fruſto CB, <lb></lb>ad cylindrum, vel portionem cylindricam. </s> <s>Manifeſtum <lb></lb>eſt igitur propoſitum. </s></p><pb xlink:href="043/01/197.jpg" pagenum="18"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſphæra, vel ſphæroides ſecetur duobus pla<lb></lb>nis parallelis vtcumque, neutro per <expan abbr="cẽtrum">centrum</expan> ducto: <lb></lb>quædam autem ex centro recta linea tranſeat per <lb></lb>centrum alterutrius ſectionum; per centrum re<lb></lb>liquæ tranſibit. </s></p><p type="main"> <s>Sit ſphæra, vel ſphæroides ſectum duobus planis pa<lb></lb>callelis vtcumque neutro per centrum ducto, quod ſit E: <lb></lb>per ſectionum autem, quæ ſunt circuli, vel ſimiles el<lb></lb>lipſes, alterutrius centrum F tranſiens recta EFB oc<lb></lb>currat reliquæ ſectionis plano in puncto G. </s> <s>Dico reli<lb></lb>quæ ſectionis centrum eſſe G. </s> <s>Planum enim per OB ſe<lb></lb><figure id="id.043.01.197.1.jpg" xlink:href="043/01/197/1.jpg"></figure><lb></lb>cans ſphæram, vel ſphæroides, faciensque ſectionem circu<lb></lb>lum, vel ellipſim ABCD, ſecabit, & ſecet prædictas ſe<lb></lb>ctiones, circulos inquam, vel ſimiles ellipſes parallelas, qua<lb></lb>rum alterius centrum ponitur F. </s> <s>Faciatque ſectiones re<lb></lb>ctas parallelas AFC, KGH: ſimiliter aliud quodlibet <pb xlink:href="043/01/198.jpg" pagenum="19"></pb>planum per BE ſecans ſphæram, vel ſphæroides faciat ſe<lb></lb>ctionem circulum, vel ellipſim, & in ea parallelas LFM, <lb></lb>NGO, communes ſectiones iam factæ ſectionis ſphæræ <lb></lb>vel ſphæroidis cum circulis, vel ellipſibus inter ſe paral<lb></lb>lelis quarum diametri ſunt AC, KH. </s> <s>Quoniam igitur <lb></lb>E eſt centrum ſphæræ, vel ſphæroidis; omnes in eo per <lb></lb>punctum E, tranſeuntes rectæ lineæ bifariam ſecabuntur: <lb></lb>ſed idem E eſt in ſectione ſphæræ, vel ſphæroidis, circu<lb></lb>lo, vel ellipſe ABCD; omnes igitur in ipſa rectas lineas <lb></lb>bifariam ſecabit punctum E, & centrum erit circuli, <lb></lb>vel ellipſis ABCD: quædam igitur ex centro recta EB <lb></lb>ſecans parallelarum neutrius per centrum ductæ alteram <lb></lb>AC bifariam in circuli, vel ellipſis ALCM centro F, <lb></lb>& reliquam in puncto G bifariam ſecabit. </s> <s>Similiter <lb></lb>oſtenderemus rectam NO ſectam eſse bifariam in pun<lb></lb>cto G: atque adeo circuli, vel ellipſis KNHO centrum <lb></lb>eſſe G. </s> <s>Recta igitur E, tranſiens per centrum ſectionis <lb></lb>ALCM, tranſibit per centrum reliquæ KNHO ipſi <lb></lb>ALCM parallelæ. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hinc manifeſtum eſt, ſi ſphæra, vel ſphæroides <lb></lb>ſecetur plano non per centrum: & recta linea ſphæ<lb></lb>ræ, vel ſphæroidis, & factæ ſectionis centra iun<lb></lb>gens ad ſuperficiem vtrinque producatur; talis <lb></lb>axis ſegmenta eſſe <gap></gap> portionum, earumque <lb></lb>vertices extrema dicti axis, vt in figura theorema<lb></lb>tis ſunt puncta B, D. </s></p><pb xlink:href="043/01/199.jpg" pagenum="20"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si hemiſphærium, vel hemiſphæroides vtcum<lb></lb>que ab ſciſſum: & cylindrus, vel cylindri portio <lb></lb>illi circumſcripta: & conus, vel coni portio, cu<lb></lb>ius baſis eſt eadem ſolido circumſcripto, hemi<lb></lb>ſphærium, vel hemiſphæroides ad verticem <expan abbr="con-tingẽs">con<lb></lb>tingens</expan>, & communis axis; ſecentur vnoplano, baſi <lb></lb>hemiſphærij, vel hemiſphæroidis parallelo: ſuper <lb></lb>ſectiones autem prædicti coni, vel portionis coni<lb></lb>cæ, & hemiſphærij, vel hemiſphæroidis, circa hu<lb></lb>ius abſciſsæ portionis axem duo cylindri, vel por<lb></lb>tiones cylindricæ conſtiterint; reliquum cylindri <lb></lb>vel portionis cylindricæ prædicto plano abſciſsæ, <lb></lb><expan abbr="dẽpto">dempto</expan> eo cylindro <expan abbr="duorũ">duorum</expan> prædictorum, vel portio<lb></lb>ne cylindrica, cuius baſis eſt ſectio hemiſphærij, <lb></lb>vel hemiſphæroidis, æquale erit reliquo cylindro, <lb></lb>vel portioni cylindricæ, cuius baſis eſt ſectio præ<lb></lb>dicti coni, vel portionis conicæ. </s></p><p type="main"> <s>Eſto hemiſphærium, vel hemiſphæroides ABC, cuius <lb></lb>axis BD, baſis circulus, vel ellipſis, cuius diameter AC. <lb></lb></s> <s>Et ſolido ABC circumſcriptus cylindrus, vel portio cy<lb></lb>lindrica, cuius baſes oppoſitæ erunt circuli, vel ſimiles elli<lb></lb>pſes, quarum diametri eiuſdem rationis ADC, EF, la<lb></lb>tera oppoſita parallelogrammi per axem AFGC: & ſu<lb></lb>per baſim, cuius diameter EF, circa axim BD, deſcriptus <lb></lb>eſto conus, vel coni portio EDF. </s> <s>Iam tria ſolida ABC, <lb></lb>EDF, AC, ſecentur plano ſolidi ABC baſi parallelo, <lb></lb>quod ſecabit, & ſecet vnà figuras planas per axim BD <pb xlink:href="043/01/200.jpg" pagenum="21"></pb>tribus ſolidis communem, poſitas in eodem plano, quæ ſunt <lb></lb>AF parallelogrammum, triangulum EDF, & ſemicir<lb></lb>culus, vel ſemi ellipſis ABC: & ſint ſectiones rectæ GO, <lb></lb>HN, KM: hæ igitnr erunt diametri eiuſdem rationis trium <lb></lb>ſectionum, ſcilicet circulorum, vel ellipſium ſirnilium, qui<lb></lb>bus erit commune centrum L, in quo nimirum axis BD <lb></lb>tres dictas lineas GO, HN, KM, bifariam ſecat. </s> <s>Vt <lb></lb>igitur de ſolido AF diximus, ſint circa axem BL, & ſuper <lb></lb>baſes circulos, vel ellipſes circa HN, KM cylindri, vel <lb></lb>portiones cylindricæ HP, KQ, qui vnà cum portione <lb></lb>cylindrica, vel cylindro GF ipſa ſectione facto, erunt inter <lb></lb>eadem plana paral<lb></lb>lela per EF, GO. <lb></lb></s> <s>Dico trium cylin<lb></lb>drorum, vel cylin<lb></lb>dri portionum GF, <lb></lb>HP, KQ, <expan abbr="reliquũ">reliquum</expan> <lb></lb>ipſius GF dempto <lb></lb>HP, ipſi KQ eſse <lb></lb><figure id="id.043.01.200.1.jpg" xlink:href="043/01/200/1.jpg"></figure><lb></lb>æquale. </s> <s>Quoniam <lb></lb>enim cylindri, & cy<lb></lb>lindri portiones eiuſdem altitudinis inter ſe ſunt vt ba<lb></lb>ſes, circuli autem, & ſimiles ellipſes; inter ſe, vt quæ à <lb></lb>diametris eiuſdem rationis fiunt quadrata; ex Archime<lb></lb>de, hoc eſt vt earum quartæ partes, quæ à ſemidiame<lb></lb>tris quadrata deſcribuntur; erit vt quadratum LO ad <lb></lb>quadratum LN, ita cylindrus, vel portio cylindrica <lb></lb>GF ad cylindrum, vel portionem cylindricam PH: & <lb></lb>diuidendo, vt rectangulum LNO bis vnà cum quadra<lb></lb>to NO, ad quadratum LN, ita reliquum cylindri, vel <lb></lb>portionis cylindricæ GF, dempto ipſo PH, ad ipſum <lb></lb>PH: ſed vt quadratum LN ad quadratum LM, ita eſt <lb></lb>vt ſupra, cylindrus, vel portio cylindrica HP ad cylin<lb></lb>drum, vel portionem cylindricam KQ, ex æquali igitur, <pb xlink:href="043/01/201.jpg" pagenum="22"></pb>erit vt rectangulum LNO bis, vnà cum quadrato NO, <lb></lb>ad quadratum LM, ita reliquum cylindri, vel portionis <lb></lb>cylindricæ GF <expan abbr="dẽ-pto">den<lb></lb>pto</expan> HP, ad cylin<lb></lb>drum, vel <expan abbr="portionẽ">portionem</expan> <lb></lb>cylindricam KQ: <lb></lb>ſed rectangulum L <lb></lb>NO bis vnà <expan abbr="cũ">cum</expan> qua <lb></lb>drato NO æquale <lb></lb>eſt quadrato LM; <lb></lb>reliquum igitur cy<lb></lb><figure id="id.043.01.201.1.jpg" xlink:href="043/01/201/1.jpg"></figure><lb></lb>lindri, vel portionis <lb></lb>cylindricæ GF, <expan abbr="dẽ-pto">den<lb></lb>pto</expan> HP, æquale erit cylindro, vel portioni cylindricæ <expan abbr="Kq.">Kque</expan> <lb></lb>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Cylindri, vel portionis cylindricæ hemiſphæ<lb></lb>rio, vel hemiſphæroidi circumſcriptæ reliquum <lb></lb>dempto hemiſphærio, vel hemiſphæroide, æqua<lb></lb>le eſt cono, vel portioni conicæ eandem baſim he<lb></lb>miſphærio, vel hemiſphæroidi, & eandem altitu<lb></lb>dinem habenti. </s></p><p type="main"> <s>Eſto hemiſphærio, vel hemiſphæroidi ABC, cu<lb></lb>ius axis BD, baſis circulus, vel ellipſis circa diametrum <lb></lb>ADC, circumſcriptus cylindrus, vel cylindrica portio <lb></lb>AE, circa communem ſcilicet axim BD. conus autem, <lb></lb>vel coni portio circa axim BD, baſim habens commu<lb></lb>nem ſolido ABC, intelligatur. </s> <s>Dico reliquum ſolidi <lb></lb>AE, dempto hemiſphærio, vel hemiſphæroide ABC æ-<pb xlink:href="043/01/202.jpg" pagenum="23"></pb>quale eſse cono, vel portioni conicæ. </s> <s>Nam circa axim <lb></lb>BD, & ſuper baſim circulum, vel ellipſim, cuius diame<lb></lb>ter RE, ſimilem & oppoſitam ei, quæ circa AC, deſcri<lb></lb>batur conus, vel coni portio RDE. </s> <s>Deinde axe BD bi<lb></lb>fariam ſecto, & ſingulis eius partibus rurſus bifariam, vt <lb></lb>partes axis BD omnes ſint æquales, per puncta ſectio<lb></lb>num, quotquot erunt, totidem plana parallela ſecent vnà <lb></lb>cum ſolido AE duas ipſius partes, ſolida ABC, RDE. <lb></lb></s> <s>Omnes igitur factæ ſectiones, vel erunt circuli, vel ſimiles <lb></lb>ellipſes ei, quæ eſt circa AC, atque adeo inter ſe ſimiles: <lb></lb>talium autem ſectiones communes cum AE parallelo, <lb></lb><figure id="id.043.01.202.1.jpg" xlink:href="043/01/202/1.jpg"></figure><lb></lb>grammo per axim, erunt rectæ lineæ, ternæ in ſingu<lb></lb>lis planis ſecantibus, & in eadem recta linea; vt in proxi<lb></lb>ma ipſi RE, ſunt FL, GN, KM, quæ quidem erunt <lb></lb>trium circulorum, vel ſimilium ellipſium diametri eiuſdem <lb></lb>rationis baſium trium ſolidorum, cylindri ſcilicet, vel por<lb></lb>tionis cylindricæ FL, fruſti GL, & portionis KBM, he <lb></lb>miſphærij, vel hemiſphæroidis ABC. </s> <s>Itaque circa axem <lb></lb>BH cylindri, vel portionis cylindricæ FE, & ſuper ba<lb></lb>ſes circulos, vel ellipſes circa GN, KM, deſcribantur <lb></lb>cylindri, vel cylindri portiones GP, KQ, qui pat<lb></lb>tes erunt totius cylindri, vel portionis cylindricæ FE. <lb></lb></s> <s>Idem fiat circa reliquas axis partes BD tamquam axes, <pb xlink:href="043/01/203.jpg" pagenum="24"></pb>ſuper reliquas ſectiones ternas in ſingulis prædictis planis <lb></lb>ſecantibus. </s> <s>Hac ratione habebimus iam duas figuras <lb></lb>compoſitas ex cylindris, vel cylindri portionibus altitudi<lb></lb>ne, & multitudine æqualibus, alteram cono, vel portioni <lb></lb>conicæ RDE inſcriptam, alteram hemilphærio, vel he<lb></lb>miſphæroidi ABC circumſcriptam: quod ita factum eſ<lb></lb>ſe intelligatur, quemadmodum in primo libro fieri poſse <lb></lb>demonſtrauimus, vt figura cono RDE inſcripta ab eo <lb></lb>deficiat, hemiſphærio autem, vel hemiſphæroidi ABC <lb></lb>circumſcripta ipſum excedat minori ſpacio magnitudine <lb></lb>propoſita quantacumque illa ſit. </s> <s>Reliquo itaque cylin<lb></lb><figure id="id.043.01.203.1.jpg" xlink:href="043/01/203/1.jpg"></figure><lb></lb>dri, vel portionis cylindricæ AE dempto hemiſphærio, vel <lb></lb>hemiſphæroide ABC figura quædam inſcripta relinque<lb></lb>tur ex cylindris, vel portionis cylindricæ reſiduis æqualium <lb></lb>altitudinum, demptis ijs, ex quibus conſtat figura hemi<lb></lb>ſphærio, vel hemiſphæroidi ABC circumſcripta, excepto <lb></lb>infimo cylindro, vel portione cylindrica AS. </s> <s>Et quo<lb></lb>niam (excepto exceſsu, quo ſolidum AS excedit ſui par<lb></lb>tem portionem quandam hemiſphærij, vel hemiſphæroidis <lb></lb>ABC) quo ſpacio figura hemiſphærio, vel hemiſphæroidi <lb></lb>ABC circumſcripta ſuperat ipſum hemiſphærium, vel he <lb></lb>hemiſphæroides, eodem figura prædicto reſiduo inſcripta de<lb></lb><gap></gap>duo; deficiet ab eodem minori differentia quàm <pb xlink:href="043/01/204.jpg" pagenum="25"></pb>ſit magnitudo propoſita,. His ita ex poſitis, quoniam ex <lb></lb>præcedenti, reliquum cylindri, vel portionis cylindricæ <lb></lb>FE dempto cylindro, vel portione cylindrica KQ, æ<lb></lb>quale eſt cylindro, vel portioni cylindricæ GP: eadem<lb></lb>que ratione ſingula cylindrorum, vel cylindri portionum <lb></lb>reſidua, quæ ſunt in reliqua figura cylindri, vel portionis <lb></lb>cylindricæ AE, dempto hemiſphærio, vel hemiſphæroi<lb></lb>de ABC, æqualia erunt ſingulis cylindris, vel cylindri <lb></lb>portionibus, quæ ſunt in cono, vel portione conica RDE, <lb></lb>ſi bina ſumantur inter eadem plana parallela, vel circa <lb></lb>eundem axem; tota igitur figura inſcripta prædicto reſiduo, <lb></lb>toti figuræ inſcriptæ cono, vel portioni conicæ RDE æ<lb></lb>qualis erit: deficit autem vtraque figura inſcripta à ſibi <lb></lb>circumſcripta minori ſpacio quantacumque magnitudine <lb></lb>propoſita; per tertiam igitur huius, reliquum cylindri, vel <lb></lb>portionis cylindricæ AE, dempto hemiſphærin, vel he<lb></lb>miſphæroide ABC, æquale eſt cono, vel portioni coni<lb></lb>cæ RDE, hoc eſt ipſi ABC. </s> <s>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si hemiſphærium, vel hemiſphæroides, & cylin <lb></lb>drus, vel portio cylindrica ipſi circumſcripta, & <lb></lb>conus, vel coni portio, cuius eſt <expan abbr="idẽ">idem</expan> axis portioni, <lb></lb>baſis autem qu<17> opponitur communi baſi duorum <lb></lb>prædictorum ſolidorum, vnà ſecentur duobus <lb></lb>planis baſi parallelis; portiones reliquæ figuræ <lb></lb>ex cylindro, vel cylindri portione hemiſphærio, <lb></lb>vel hemiſphæroidi circumſcripta dempto hemi<lb></lb>ſphærio, vel hemiſphæroide, quæ à duobus præ<lb></lb>dictis planis ſecantibus fiunt, æquales ſunt ſin<pb xlink:href="043/01/205.jpg" pagenum="26"></pb>gulæ ſingulis prædicti coni, vel conicæ portionis <lb></lb>partibus ſiue fruſtis inter eadem plana parallela <lb></lb>reſpondentibus. </s></p><p type="main"> <s>Eſto hemiſphærium, vel hemiſphæroides ABC, cu<lb></lb>ius axis BD, baſis circulus, vel ellipſis, cuius diame<lb></lb>ter ADC. ſolido autem ABC circumſcriptus cylindrus, <lb></lb>vel portio cylindrica AXEC: & conus, vel coni portio <lb></lb>ſit XDE, cuius vertex D, baſis circulus, vel ellipſis cir<lb></lb>ca XBE baſi ſolidi AE, vel ABC, prædictæ oppoſita, <lb></lb>ſecto autem ſolido AE, atque vnà cum ipſo eius partibus, <lb></lb>ſolidis ABC, XD <lb></lb>E, duobus planis ba <lb></lb>ſi ſolidi AE, vel <lb></lb>ABC, atque ideo <lb></lb>inter ſe quoque pa<lb></lb>rallelis, intelligan<lb></lb>tur trium ſolidorum <lb></lb>portiones ternæ in<lb></lb><figure id="id.043.01.205.1.jpg" xlink:href="043/01/205/1.jpg"></figure><lb></lb>ter eadem plana pa<lb></lb>rallela: videlicet in<lb></lb>ter duo per XE, <lb></lb>FN, hemiſphærij, vel hemiſphæroidis minor portio HBL: <lb></lb>& reliquum cylindri, vel portionis cylindricæ FE dem<lb></lb>pta portione HBL: & coni, vel conicæ portionis fruſtum <lb></lb>XGME. ſimiliter inter duo plana per FN, OV ſolidi <lb></lb>ABC portio PHLT, eaque ablata reliquum ſolidi ON, <lb></lb>& fruſtum GQSM. </s> <s>Denique ſolidi ABC portio AP <lb></lb>TC, eaque ablata, reliquum ſolidi AV, & conus, vel <lb></lb>coni portio QDS. </s> <s>Dico reliquum ſolidi FE, dempto <lb></lb>HBL eſſe æquale fruſto XGME: & reliquum ſolidi ON <lb></lb>dempto PHLT, æquale fruſto GQSM: & reliquum <lb></lb>ſolidi AV dempto ſolido APTC æquale ſolido QDS. <pb xlink:href="043/01/206.jpg" pagenum="27"></pb>Quoniam enim vt ſupra oſtendimus, reliquum ſolidi AE, <lb></lb>dempto ſolido ABC æquale eſse ſolido XDE, ſimili<lb></lb>ter oſtenſum remanet, tam reliquum ſolidi AN, dempto <lb></lb>ſolido AHLC, æquale eſse ſolido GDM, quam reli<lb></lb>quum ſolidi AV dempto ſolido APTC æquale ſolido <lb></lb>QDS; erit demptis æqualibus, tam reliquum ſolidi FE, <lb></lb>dempto ſolido HBL, æquale ſolido XGME; quam <lb></lb>reliquum ſolidi ON, dempto ſolido PHLT æquale ſo<lb></lb>lido GQSM. </s> <s>At reliquum ſolidi AV dempto ſoli<lb></lb>do APTC ſolido QDS æquale erit. </s> <s>Manifeſtum eſt <lb></lb>igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hemiſphærium, vel hemiſphæroides ſubſeſqui <lb></lb>alterum eſt cylindri; vel portionis cylindricæ ipſi <lb></lb>circumſcriptæ. </s></p><p type="main"> <s>Eſto hemiſphærium, vel hemiſphæroides ABC, <lb></lb>ipſique circumſcriptus cylindrus, vel portio cylindri<lb></lb>ca AE, circa eundem ſcilicet axem BD, & ſuper can<lb></lb>dem baſim circulum, <lb></lb>vel ellipſim, circa AC: <lb></lb>nam hac ratione baſis <lb></lb>oppoſita ſolidum ABC <lb></lb>tanget ad verticem B. <lb></lb></s> <s>Dico <expan abbr="hemiſphæriũ">hemiſphærium</expan>, vel <lb></lb>hemiſphæroides ABC <lb></lb>eſse cylindri, vel portio <lb></lb>nis cylindricæ AE ſub <lb></lb><figure id="id.043.01.206.1.jpg" xlink:href="043/01/206/1.jpg"></figure><lb></lb>ſeſquialterum. </s> <s>Nam <lb></lb>circa axem BD, ſuper prædictam baſem circa AC, eſto <lb></lb>deſcriptus conus, vel coni portio ABC. </s> <s>Quoniam igitur <pb xlink:href="043/01/207.jpg" pagenum="28"></pb>cylindri, vel portionis cylindricæ AE reliquum dempto <lb></lb>hemiſphærio, vel hemiſphæroide ABC æquale eſt cono, <lb></lb>vel portioni conicæ ABC: & cylindrus, vel portio cylin<lb></lb>drica AE tripla eſt co<lb></lb>ni, vel portionis conicæ <lb></lb>ABC; triplus itidem <lb></lb>erit cylindrus, vel cylin <lb></lb>drica portio AE dicti <lb></lb>reſidui dempto hemi<lb></lb>ſphærio, vel hemiſphæ<lb></lb>roide ABC; ac propte<lb></lb>rea hemiſphærij, vel he<lb></lb><figure id="id.043.01.207.1.jpg" xlink:href="043/01/207/1.jpg"></figure><lb></lb>miſphæroidis ABC <lb></lb>ſeſquialter, hoc eſt hemiſphærium, vel hemiſphæroides <lb></lb>ABC cylindri, vel portionis cylindricæ AE ſubſeſquial<lb></lb>terum. </s> <s>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis minor portio ſphæræ, vel ſphæroidis ad <lb></lb>cylindrum, vel cylindri portionem, cuius baſis <lb></lb>æqualis eſt circulo maximo, vel æqualis, & ſimi<lb></lb>lis ellipſi per centrum baſi portionis parallelæ, <lb></lb>& eadem altitudo portioni; eam habet proportio<lb></lb>nem, quam rectangulum contentum ſphæræ, vel <lb></lb>ſphæroidis dimidij axis axi portionis congruen<lb></lb>tis ijs, quæ à centro baſis portionis fiunt <expan abbr="ſegmētis">ſegmentis</expan>, <lb></lb>vnà cum duobus tertiis quadrati axis portionis; ad <lb></lb>ſphæræ, vel ſphæroidis dimidij axis quadratum. </s></p><p type="main"> <s>Sit minor portio ABC, ſphæræ, vel ſphæroidis, cuius <lb></lb>centrum D, axis autem axi portionis congruens BEDR: <pb xlink:href="043/01/208.jpg" pagenum="29"></pb>& cylindrus, vel portio cylindrica FG abſciſsa vnà cum <lb></lb>portione ABC ex cylindro, vel portione cylindrica NO <lb></lb>circumſcripta hemiſphærio, vel hemiſphæroidi NBO, <lb></lb>cuius baſis circa diametrum NO, ſit baſi portionis ABC <lb></lb>parallela: qua ratione baſis prædicti ſolidi FG, erit vel cir <lb></lb>culus, vel ellipſis æqualis circulo maximo, vel ſimilis, & <lb></lb>æqualis ellipſi circa NO, portionis ABC baſi paralle<lb></lb>læ. </s> <s>Dico portionem ABC ad cylindrum, vel portio<lb></lb>nem cylindricam FG, eſse vt rectangulum BED, vnà <lb></lb>cum duabus tertiis qua<lb></lb>drati EB ad quadratum <lb></lb>BD. </s> <s>Eſto enim conus, <lb></lb>vel coni portio HDG, <lb></lb>cuius fruſtum HKLG <lb></lb>prædicto plano abſciſſum: <lb></lb>& omnino ſint <expan abbr="circulorũ">circulorum</expan>, <lb></lb>vel ellipſium ſimilium dia <lb></lb>metri eiuſdem rationis <expan abbr="cũ">cum</expan> <lb></lb>NO, vt ad XII huius, in <lb></lb><expan abbr="eadẽ">eadem</expan> recta linea tres FM, <lb></lb>AC, KL, ſectæ omnes bi <lb></lb>fariam in <expan abbr="cõmuni">communi</expan> <expan abbr="cẽtro">centro</expan> E, <lb></lb><figure id="id.043.01.208.1.jpg" xlink:href="043/01/208/1.jpg"></figure><lb></lb>& HBG, in eodem plano per axem. </s> <s>Quoniam igitur ex ſu<lb></lb>perioribus, reliquum ſolidi FG, dempto ABC, æquale eſt <lb></lb>fruſto HKLG; erit eiuſdem ſolidi FG reliquum ABC <lb></lb>æquale reliquo ſolidi FG, dempto HKLG: ſed hoc reli<lb></lb>quum dempto HKLG, ſupra oſtendimus eſse ad ſolidum <lb></lb>FG, vt rectangulum ex KL, & differentia HG, vnà <lb></lb>cum duabus tertiis quadrati differentiæ, ad quadratum <lb></lb>GH: & vt HG ad KL, ita eſt BD ad DE, propter ſimi<lb></lb>litudinem triangulorum; vt igitur eſt rectangulum BED, <lb></lb>vnà cum duabus tertiis quadrati BE, ad quadratum BD, <lb></lb>ita erit portio ABC, ad cylindrum, vel portionem cylin<lb></lb>dricam FG. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/209.jpg" pagenum="30"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portio ſphæræ, vel ſphæroidis abſciſſa <lb></lb>duobus planis parallelis, alteroper centrum du<lb></lb>cto, ad cy lindrum, vel cylindri portionem, cuius <lb></lb>baſis eſt eadem, quæ maior baſis portionis, & <expan abbr="eadẽ">eadem</expan> <lb></lb>altitudo; eam habet proportionem, quam rectan<lb></lb>gulum contentum ijs, quæ à centro minoris baſis <lb></lb>fiunt axis ſphæræ, vel ſphæroidis ſegmentis, vnà <lb></lb>cum duabus tertiis quadrati axis portionis; ad <lb></lb>ſphæræ, vel ſphæroidis dimidij axis quadratum. </s></p><p type="main"> <s>Sit portio NACO ſphæræ, vel ſphærodij, cuius cen<lb></lb>trum D, axis autem axi portionis congruens BEDR, <lb></lb>abſciſsa duobus planis parallelis altero per centrum D, ſe<lb></lb>ctionem faciente circulum <lb></lb>maximum, vel ellipſim, <lb></lb>cuius diameter NO, & ſu<lb></lb>per dictam ſectionem, cir<lb></lb>ca axem ED, ſtet cylin<lb></lb>drus, vel portio cylindrica <lb></lb>NM, abſciſsa ijſdem pla<lb></lb>nis, quibus portio NAC <lb></lb>O, à cylindro, vel portio<lb></lb>ne cylindrica NG, ſit cir<lb></lb>cumſcripta hemiſphærio, <lb></lb>vel hemiſphæroidi NBO: <lb></lb>qua ratione erit cylindri, <lb></lb><figure id="id.043.01.209.1.jpg" xlink:href="043/01/209/1.jpg"></figure><lb></lb>vel portionis cylindricæ NM baſis eadem, quæ maior <lb></lb>baſis portionis NACO, circulus ſcilicet, vel ellipſis cir<lb></lb>ca NO, & eadem altitudo portioni. </s> <s>Dico portionem <pb xlink:href="043/01/210.jpg" pagenum="31"></pb>NACO, ad cylindrum, vel portionem cylindricam NM, <lb></lb>eſse vt rectangulum BER, vnà cum duabus tertiis ED <lb></lb>quadrati, ad quadratum BD. </s> <s>Ijſdem enim quæ in præce<lb></lb>denti conſtructis, & notatis, ſit præterea cylindrus, vel por<lb></lb>tio cylindrica PL, circa axim ED circumſcripta cono, <lb></lb>vel portioni conicæ KDL, Quoniam igitur reliquum <lb></lb>cylindri, vel portionis cylindricæ NM, dempta portione <lb></lb>NACO æquale eſt cono, vel portioni conicæ <emph type="italics"></emph>K<emph.end type="italics"></emph.end>DL, <lb></lb>erit reliqua portio NACO æqualis reliquo eiuſdem NM, <lb></lb>dempto cono, vel portione conica KDL. </s> <s>Et quoniam cir <lb></lb>culi, & ſimiles ellipſes inter ſe ſunt vt quadrata diametro<lb></lb>rum, vel <expan abbr="ſemidiametrorũ">ſemidiametrorum</expan> eiuſdem rationis: cylindri autem, <lb></lb>& portiones cylindricæ <expan abbr="eiuſdẽ">eiuſdem</expan> altitudinis inter ſe vt baſes; <lb></lb>erit vt quadratum EM, hoc eſt quadratum BG, ad qua<lb></lb>dratum EL, hoc eſt vt quadratum BD ad quadratum <lb></lb>DE, propter ſimilitudinem triangulorum, ita ſolidum NM <lb></lb>ad ſolidum PL: & per conuerſionem rationis, vt quadra<lb></lb>tum BD ad rectangulum BED bis, vnà cum quadrato <lb></lb>BE, ita ſolidum MN, ad ſui reliquum dempto ſolido <lb></lb>PL: & conuertendo, vt rectangulum BED bis, vnà cum <lb></lb>quadrato BE, hoc eſt rectangulum BER, ad quadratum <lb></lb>BD, ita reliquum ſolidi NM dempto ſolido PL ad ſo<lb></lb>lidum NM. Rurſus, quoniam eſt vt quadratum EL ad <lb></lb>quadratum EM, ſiue BG, hoc eſt vt quadratum ED ad <lb></lb>quadratum BD, ita ſolidum PL ad ſolidum NM, ob <lb></lb>ſimilem rationem ſupradictæ: & duæ tertiæ partes ſolidi <lb></lb>PL eſt ſolidum KDL; erit ex æquali, vt duæ tertiæ qua<lb></lb>drati ED ad quadratum BD, ita reliquum ſolidi PL <lb></lb>dempto ſolido KDL, ad ſolidum NM: ſed vt rectangu<lb></lb>lum BER ad quadratum BD, ita erat ſolidi NM reli<lb></lb>quum dempto ſolido PL, ad ſolidum NM; vt igitur pri<lb></lb>ma cum quinta ad ſecundam, ita erit tertia cum ſexta ad <lb></lb>quartam; videlicet, vt rectangulum BED, vnà cum dua<lb></lb>bus tertiis ED quadrati ad quadratum BD, ita reliquum <pb xlink:href="043/01/211.jpg" pagenum="32"></pb>cylindri, vel portionis cylindricæ NM, dempto cono, vel <lb></lb>portione conica KDL, hoc eſt portio NACO ipſi æqua<lb></lb>lis, ad cylindrum, vel portionem cylindricam NM. <lb></lb></s> <s>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portio ſphæræ, vel ſphæroidis abſciſſa <lb></lb>duobus planis parallelis, neutro per centrum du<lb></lb>cto, nec centrum intercipientibus, ad cylindrum, <lb></lb>vel cylindri portionem, cuius baſis æqualis eſt <lb></lb>circulo maximo, vel ellipſi per centrum baſibus <lb></lb>portionis parallelæ ſimilis, & æqualis, eam ha<lb></lb>bet proportionem, quam duo rectangula; & quod <lb></lb>ſphæræ, vel ſphæroidis axis axi portionis <expan abbr="congruẽ">congruem</expan> <lb></lb>tis ijs, quæ à centro minoris baſis portionis fiunt <lb></lb><expan abbr="ſegmẽtis">ſegmentis</expan>, & quod ea, quæ maioris baſis portionis, <lb></lb>& ſphæræ, vel ſphæroidis centra iungit, & axe por <lb></lb>tionis continetur, vnà cum duabus tertijs quadra<lb></lb>ti axis portionis; ad ſphæræ, vel ſphæroidis dimi<lb></lb>dij axis quadratum. </s></p><p type="main"> <s>Sit portio AQTC ſphæræ, vel ſphæroidis, cuius cen<lb></lb>trum D, axis autem axi portionis congruens BSEDR, <lb></lb>abſciſſum duobus planis parallelis, neutro per centrum <lb></lb>D acto, nec ipſum intercipientibus: & circa portionis <lb></lb>axim SE ſtet cylindrus, vel portio cylindrica FX ab<lb></lb>ſciſsa vnà cum portione AQTC ex toto cylindro, vel <lb></lb>portione cylindrica NG, hemiſphærio, vel hemiſphæroi<lb></lb>di NBO circumſcripta, cuius baſis circulus maximus <pb xlink:href="043/01/212.jpg" pagenum="33"></pb>vel ellipſis circa NO baſibus AQTC portionis parallelæ <lb></lb>qua ratione cylindrus, vel portionis cylindricæ FX eiuſ<lb></lb>dem altitudinis portioni AQTC, baſis erit circulus <lb></lb>æqualis circulo maximo, vel ellipſis ſimilis, & æqualis ei, <lb></lb>cuius diameter NDO, baſibus AQTC portionis paral<lb></lb>lelæ. </s> <s>Dico portionem AQTC ad cylindrum, vel por<lb></lb>tionem cylindricam FX, eſſe vt duo rectangula BSR, <lb></lb>DES, vnà cum duabus tertiis quadrati ES, ad quadra<lb></lb>tum BD. </s> <s>Ijſdem enim conſtructis, & notatis, quæ in an<lb></lb>tecedenti, excepto cylindro, vel portione cylindrica, quæ <lb></lb>circa axim ED ſteterat: <lb></lb>planum præterea minoris <lb></lb>baſis QT portionis AQ <lb></lb>TC extendatur: & ſe<lb></lb>cans tria ſolida, & figuras <lb></lb>planas per axim poſitas in <lb></lb>eodem plano, faciat ternas <lb></lb>ſectiones, circulos, vel elli<lb></lb>pſes ſimiles ei, quæ eſt cir<lb></lb>ca NO: & earum diame<lb></lb>tros IX, PV, QT, in <lb></lb>eadem recta linea commu<lb></lb>ni ſectione extenſi plani, & <lb></lb><figure id="id.043.01.212.1.jpg" xlink:href="043/01/212/1.jpg"></figure><lb></lb>eius, quod per axem: quæ quidem diametri ſectæ erunt om<lb></lb>nes bifariam in centro S communi trium prædictarum pla<lb></lb>narum <expan abbr="ſectionũ">ſectionum</expan>. </s> <s>Denique coni, vel portionis conicæ HDG <lb></lb>fruſto PKIV abſciſſo vnà cum portione AQTC, ſit <lb></lb>circa axim SE circumſcriptus cylindrus vel portio cylin<lb></lb>drica ZV. </s> <s>Quoniam igitur per XIIII huius, reliquum <lb></lb>ſolidi FX, dempta portione AQTC, æquale eſt fruſto <lb></lb>PKLV; erit reliqua portio AQTC, reliquo eiuſdem <lb></lb>ſolidi FX, dempto fruſto PKLV æqualis. </s> <s>Et quoniam <lb></lb>eſt vt PV ad KL, ita SD, DE, propter ſimilitudinem <lb></lb>triangulorum: & vt rectangulum ex KL, & differentia <pb xlink:href="043/01/213.jpg" pagenum="34"></pb>ipſius PV, vnà cum duabus tertiis quadrati eiuſdem dif<lb></lb>ferentiæ, ad quadratum PV, ita eſt reliquum ſolidi ZV <lb></lb>dempto fruſto PKLV ad ſolidum ZV; erit vt rectangu<lb></lb>lum DES, vnà cum duabus tertiis quadrati ES, ad DS <lb></lb>quadratum, ita ſolidi ZV reliquum dempto fruſto PK <lb></lb>LV ad ſolidum ZV: ſed vt quadratum DS ad quadra<lb></lb>tum DB, hoc eſt vt quadratum SV ad quadratum BG, <lb></lb>ideſt ad quadratum SX, ita eſt ſolidum ZV, ad ſolidum <lb></lb>FX; ex æquali igitur, vt rectangulum DES, vnà cum <lb></lb>duabus tertiis ES quadrati, ad quadratum BD, ita eſt <lb></lb>reliquum ſolidi ZV, dem <lb></lb>pto ſolido PKLV ad ſo <lb></lb>lidum FX: ſed vt rectan<lb></lb>gulum BSR ad quadra<lb></lb>tum BD, ita eſt, eadem <lb></lb>ratione, qua in præcedenti <lb></lb>theoremate vtebamur, re<lb></lb>liquum ſolidi FX dem<lb></lb>pto ſolido ZV, ad ſoli<lb></lb>dum FX; vt igitur prima <lb></lb>cum quinta ad ſecundam, <lb></lb>ita tertia cum ſexta ad <lb></lb>quartam; videlicet, vt duo <lb></lb><figure id="id.043.01.213.1.jpg" xlink:href="043/01/213/1.jpg"></figure><lb></lb>rectangula BSR, DES, vnà cum duabus tertiis quadra<lb></lb>ti ES ad quadratum BD, ita erit totum reliquum cylin<lb></lb>dri, vel portionis cylindricæ FX dempto fruſto PKLV: <lb></lb>hoc eſt ſphæræ, vel ſphæroidis portio AQTC ad cylin<lb></lb>drum, vel portionem cylindricam FX. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><pb xlink:href="043/01/214.jpg" pagenum="35"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis maior portio ſphæræ, vel ſphæroidis, <lb></lb>ad cylindrum, vel portionem cylindricam, cuius <lb></lb>baſis æqualis eſt circulo maximo, vel æqualis, & <lb></lb>ſimilis ellipſi per centrum baſi portionis paralle<lb></lb>læ, altitudo autem eadem portioni, eam habet <lb></lb>proportionem, quam ſolidum rectangulum con<lb></lb>tentum axe portionis, & reliquo axis ſphæræ, vel <lb></lb>ſphæroidis ſegmento, & eo, quod baſis portionis, <lb></lb>& ſphæræ, vel ſphæroidis centraiungit, vnà cum <lb></lb>binis tertiis partibus duorum cuborum: & eius <lb></lb>qui à ſphæræ, vel ſphæroidis axis dimidio; & <lb></lb>cius qui ab eo, quod ſphæræ, vel ſphæroidis, & <lb></lb>baſis portionis centra iungit ſit ſegmento; ad ſo<lb></lb>lidum rectangulum, quod axe portionis, & duo<lb></lb>bus ſphæræ, vel ſphæroidis axis fit dimidijs. </s></p><p type="main"> <s>Sit maior portio AB <lb></lb>C, ſphæræ, vel ſphæroi<lb></lb>dis ABCF, cuius cen<lb></lb>trum D: baſis <expan abbr="autẽ">autem</expan> por<lb></lb>tionis, circulus, vel elli<lb></lb>pſis, cuius diameter A <lb></lb>C: Et ſecta portione <lb></lb>ABC per centrum D <lb></lb>plano baſi AC paral<lb></lb>lelo, qua ratione ſectio <lb></lb>erit circulus maximus, <lb></lb>vel ellipſis ſimilis baſi <lb></lb><figure id="id.043.01.214.1.jpg" xlink:href="043/01/214/1.jpg"></figure><pb xlink:href="043/01/215.jpg" pagenum="36"></pb>portionis: eſto ea cuius diameter KL, iungensque recta <lb></lb>DE ſphæræ, vel ſphæroidis, & baſis portionis centra DE, <lb></lb>atque producta incidat in ſphæræ, vel ſphæroidis ſuperfi<lb></lb>ciem ad partes E in puncto F, & ad partes oppoſitas in <lb></lb>puncto B: ſphæræ igitur, vel ſphæroidis axis axi portionis <lb></lb>BE congruens crit BDEF, nam vertex portionis erit B: <lb></lb>& hemiſphærio, vel hemiſphæroidi KBL ſit circumſcri<lb></lb>ptas cylindrus, vel cylindrica portio KH, cuius ſcilicet <lb></lb>axis BD, & circa axim DE, alter cylindrus, vel portio <lb></lb>cylindrica GL portioni KACL circumſcripta: quorum <lb></lb>circumſcriptorum ſolido<lb></lb>rum vtriulque communis <lb></lb>baſis erit circulus, vel <lb></lb>ellipſis circa KL. </s> <s>Ita<lb></lb>que ex his compoſitus to<lb></lb>tus cylindrus, vel cylin<lb></lb>dri portio GH erit por<lb></lb>tioni ABC circumſcri<lb></lb>pta, habens axim BE, at<lb></lb>que ideo eandem altitu<lb></lb>dinem ABC portioni, <lb></lb>baſim autem, cuius dia<lb></lb>meter ſit GM ſimilem <lb></lb><figure id="id.043.01.215.1.jpg" xlink:href="043/01/215/1.jpg"></figure><lb></lb>& æqualem ei, quæ eſt circa KL. </s> <s>Dico portionem ABC <lb></lb>ad cylindrum, vel portionem cylindricam GH, eſse vt ſo<lb></lb>lidum rectangulum contentum ipſis BE, EF, ED, vnà <lb></lb>cum binis tertiis duorum cuborum, duabus ſcilicet cubi <lb></lb>BD, & totidem cubi ED, ad ſolidum rectangulum con<lb></lb>tentum ipſis EB, BD, DF. </s> <s>Quoniam enim parall ele<lb></lb>pipeda eiuſdem altitudinis inter ſe ſunt vt baſes, erit vt re<lb></lb>ctangulum BEF vnà cum duabus tertiis ED quadrati ad <lb></lb>rectangulum BDF, ideſt ad quadratum BD, ſiue DF, <lb></lb>ita ſolidum ex BE, EF, ED, communi altitudine DE, <lb></lb>vnà cum duabus tertiis cubi ED, ad ſolidum ex DE, <pb xlink:href="043/01/216.jpg" pagenum="37"></pb>BD, DF: ſed vt rectangulum BEF, vnà cum duabus <lb></lb>DE quadrati, ad quadratum DF, ita oſtendimus eſſe <lb></lb>portionem AKLC ad ſolidum GL; vt igitur eſt ſolidum <lb></lb>ex BE, EF, ED, vnà cum duabus tertiis cubi ED, com <lb></lb>muni altitudine DE, ad ſolidum ex ED, BD, DF, ita <lb></lb>erit portio AKLC ad ſolidum GL: ſed vt ſolidum ex <lb></lb>ED, DB, DF, hoc eſt id, cuius altitudo ED, baſis BD <lb></lb>quadratum, ad ſolidum ex EB, BD, DF, hoc eſt ad id, <lb></lb>cuius altitudo BE, baſis quadratum BD, ita eſt altitudo, <lb></lb>vel latus ED, ad altitudinem vel latum BE: hoc eſt ſoli<lb></lb>dum GL ad ſolidum GH; quippe quorum dictæ lineæ <lb></lb>ED, BE ſunt axes; ex æquali igitur, vt ſolidum ex BE, <lb></lb>EF, ED, vnà cum duabus tertiis cubi DE, ad ſolidum <lb></lb>ex EB, BD, DE, cuius altitudo EB, baſis quadratum <lb></lb>BD, ita erit portio AKLC ad ſolidum GH. Rurſus, <lb></lb>quoniam ſolidum HK eſt hemiſphærij, vel hemiſphæroi<lb></lb>dis KBL ſeſquialterum; erit vt duæ tertiæ partes cubi BD <lb></lb>ad cubum BD, ita hemiſphærium, vel hemiſphæroides <lb></lb>KBL ad ſolidum KH: ſed vt cubus BD ad ſolidum ex <lb></lb>BD, DF, & altitudine BE, hoc eſt vt altitudo BD ad <lb></lb>altitudinem BE, ita eſt ſolidum KH ad ſolidum GH, quo<lb></lb>rum dictæ altitudines BD, BE ſunt axes, ex æquali igitur <lb></lb>erit vt duæ tertiæ partes cubi BD ad ſolidum ex EB, BD, <lb></lb>DF, ita hemiſphærium, vel hemiſphæroides KBL, ad ſoli<lb></lb>dum GH: ſed vt <expan abbr="ſolidũ">ſolidum</expan> ex BE, EF, ED, vna cum duabus <lb></lb>tertiis cubi ED ad ſolidum ex EB, BD, DF, erat por<lb></lb>tio AKLC ad cylindrum GH; vt igitur prima cum quin <lb></lb>ta ad ſecundam, ita tertia cum ſexta ad quartam, videlicet, <lb></lb>vt duæ tertiæ cubi BD, vna cum duabus tertiis cubi BE, <lb></lb>& ſolido ex BE, EF, ED ad ſolidum ex EB, BD, DF, <lb></lb>ita erit ſphæræ, vel ſphæroidis maior portio ABC ad ſoli<lb></lb>dum, cylindrum ſcilicet, vel portionem cylindricam GH. <lb></lb></s> <s>Quod erat demonſtrandum. </s></p><pb xlink:href="043/01/217.jpg" pagenum="38"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portio ſphæræ, vel ſphæroidis abſciſsa <lb></lb>duobus planis parallelis centrum intercipienti<lb></lb>bus, ad cylindrum, vel cylindri portionem, cuius <lb></lb>baſis æqualis eſt circulo maximo, vel ſimilis, & <lb></lb>æqualis ellipſi per centrum baſibus portionis pa<lb></lb>rallelæ, & eadem altitudo portioni, eam habet <lb></lb>proportionem, quam duo ſolida rectangula ex ter<lb></lb>norum ſphæræ, vel ſphæroidis axis ſegmentorum <lb></lb>eundem terminum habentium alterutrius ba<lb></lb>ſium portionis centrum, binis ſphæræ, vel ſphæ<lb></lb>roidis axem complentibus, & ſingulis axis por<lb></lb>tionis itidem à centro ſphæræ, vel ſphæroidis fa<lb></lb>ctis, vnà cum binis tertijs partibus duorum cubo<lb></lb>rum ex ſegmentis axis portionis à centro ſphæræ, <lb></lb>vel ſphæroidis factis; ad ſolidum rectangulum, <lb></lb>quod duobus ſphæræ, vel ſphæroidis axis dimi<lb></lb>diis, & axe portionis continetur. </s></p><p type="main"> <s>Sit portio ABCD ſphæræ, vel ſphæroidis, cuius cen<lb></lb>trum E, axis portionis KEH: ipſi autem portioni cir<lb></lb>cumſcriptus cylindrus, vel cylindrica portio NO, vt in <lb></lb>antecedenti, cuius communis ſectio cum ſphæra, vel ſphæ<lb></lb>roide AFDG, ſit circulus maximus, vel ellipſis circa dia<lb></lb>metrum LEM; quamobrem baſis ſolidi NO, eiuſdem <lb></lb>altitudinis portioni ABCD circulus erit æqualis circu<lb></lb>lo maximo, vel ellipſis æqualis, & ſimilis ellipſi circa LM <lb></lb>baſibus portionis parallelæ. </s> <s>Dico portionem ABCD <pb xlink:href="043/01/218.jpg" pagenum="39"></pb>ad cylindrum, vel cylindri portionem NO, eſse vt duo <lb></lb>ſolida ad rectangula, alterum ex FH, HG, EH: alterum <lb></lb>ex GK, KF, EK, vnà cum binis tertiis duorum cubo<lb></lb>rum ex EK, EH, ad ſolidum rectangulum ex GE, <lb></lb>EF KH, axe enim KH producto vt incidat in ſuper<lb></lb>ficiem in punctis F, G, ſit ſphæræ, vel ſphæroidis, ex <lb></lb>demonſtratis, axis FK, EHG. </s> <s>Intelliganturque vt in <lb></lb>antecedenti duo cylindri, vel cylindri portiones NM, <lb></lb>LO, totius prædicti ſolidi NO: itemque duæ portiones <lb></lb>ſphæræ, vel ſphæroidis ALMD, LBCM, quorum qua<lb></lb>tuor ſolidorum commu <lb></lb>nis baſis eſt circulus, vel <lb></lb>ellipſis circa LEM. <lb></lb></s> <s>Quoniam igitur vt in <lb></lb>antecedenti oſtendere<lb></lb>mus portionem ALM <lb></lb>D ad ſolidum NM eſ <lb></lb>ſe vt ſolidum ex FH, <lb></lb>HG, EH, vnà cum <lb></lb>duabus tertiis cubi EH <lb></lb>ad ſolidum ex FE, EG, <lb></lb>EH, communi altitu<lb></lb>dine EH: ſed vt ſoli<lb></lb>dum ex FE, EG, EH, <lb></lb><figure id="id.043.01.218.1.jpg" xlink:href="043/01/218/1.jpg"></figure><lb></lb>altitudine EH, ad ſolidum ex FE, EG, KH altitudi<lb></lb>ne KH, ita eſt altitudo EH ad altitudinem KH, hoc <lb></lb>eſt ſolidum NM ad ſolidum NO, quippe quorum ſunt <lb></lb>axes EH, KH; ex æquali igitur erit vt ſolidum ex FH, <lb></lb>HG, EH, vnà cum duabus tertiis cubi EH, ad ſoli<lb></lb>dum ex FE, EG, KH, ita portio ALMD, ad ſoli<lb></lb>dum NO. </s> <s>Eadem ratione oſtenderemus eſſe, vt ſolidum <lb></lb>ex GK, KF, EK, vnà cum duabus tertiis cubi EK, ad <lb></lb>ſolidum ex FE, EG, KH, ita portionem LBCM, ad <lb></lb>ſolidum NO; vt igitur prima cum quinta ad ſecundam, <pb xlink:href="043/01/219.jpg" pagenum="40"></pb>ita tertia cum ſexta ad quartam; videlicet, vt duo ſoli<lb></lb>da, & quod ſit ex FH, <lb></lb>HG, EH, & quod <lb></lb>ex GK, KF, EK, vnà <lb></lb>cum duabus tertiis & <lb></lb>cubi ex EH, & cu<lb></lb>bi ex EK, ad ſolidum <lb></lb>ex FE, EG, KH, ita <lb></lb>erit tota ſphæræ, vel <lb></lb>ſphæroidis portio AB <lb></lb>CD, ad cylindrum, vel <lb></lb>portionem cylindricam <lb></lb>NO. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><figure id="id.043.01.219.1.jpg" xlink:href="043/01/219/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis trianguli comprehenſi ſectione para<lb></lb>bola, ex duabus rectis lineis, quarum altera ſe<lb></lb>ctionem tangat, altera in eam incidat diametro <lb></lb>ſectionis ex contactu æquidiſtans, centrum graui<lb></lb>tatis eſt punctum illud, in quo recta linea ex con<lb></lb>tactu diuidens incidentem ita vt pars, quæ ſectio<lb></lb>nem attingit ſit ſeſquialtera reliquæ, ſic diui<lb></lb>ditur, vt pars quæ eſt ad contactum ſit tripla <lb></lb>reliquæ. </s></p><p type="main"> <s>Sit triangulum ABC comprehenſum ſectione parabo<lb></lb>la ADB, & duabus rectis lineis, quarum altera AC tan<lb></lb>gat ſectionem in puncto A, reliqua autem BC, in eam <lb></lb>incidens in puncto B, ſectionis diametro ex puncto A, <lb></lb>æquidiſtans intelligatur: & per centrum grauitatis trian-<pb xlink:href="043/01/220.jpg" pagenum="41"></pb>guli ABC quod ſit F, ſit ducta recta AFE. </s> <s>Dico AF <lb></lb>eſſe ipſius FE triplam: at BE ipſius EC ſeſquialteram. <lb></lb></s> <s>Completo enim triangulo rectilineo ABC, ſectis que re<lb></lb>ctis lineis bifariam AB in puncto H, & AC in puncto K <lb></lb>ducatur HDK, quæ parallela erit baſi BC: parabolæ igi<lb></lb>tur ſegmenti BDA dia meter erit DH; in qua parabolæ <lb></lb>ADB, cuius vertex D ſit centrum grauitatis M: trian<lb></lb>guli autem rectilinei ABC centrum grauitatis N, & iun <lb></lb>gatur MN: producta igitur MN occurret trianguli ABC <lb></lb>mixti centro grauitatis F. ſint igitur centra M, N, F, in <lb></lb>eadem recta linea: <lb></lb>& ducta recta AN <lb></lb>G ſecet baſim BC <lb></lb>bifariam in G pun <lb></lb>cto, neceſſe eſt e<lb></lb>nim: & ex puncto <lb></lb>F ad rectam AG, <lb></lb>ducatur recta FO <lb></lb>ipſis BC, KH pa <lb></lb>rallela, & BD, DA <lb></lb>iungantur. </s> <s><expan abbr="Quoniã">Quoniam</expan> <lb></lb>igitur AG ſecat <lb></lb>BC, KH paral<lb></lb>lelas in rectolineo <lb></lb>triangulo ABC, <lb></lb><figure id="id.043.01.220.1.jpg" xlink:href="043/01/220/1.jpg"></figure><lb></lb>in eaſdem rationes; ſecta erit HK bifariam à linea AG: <lb></lb>cumque HD diameter parabolæ ADC, cuius vertex D, <lb></lb>ſit parallela diametro parabolæ, cuius vertex A, atque <lb></lb>ideo etiam BC incidenti parallela, erit DH pars ipſius <lb></lb>KH: quoniam igitur in triangulo mixto ABC recta KD <lb></lb>applicata parallela eſt ipſi BC, quæ itidem eſt parallela <lb></lb>diametro parabolæ, cuius vertex A; erit vt AC ad AK <lb></lb>potentia, ita BC ad DK longitudine, quod ſupra demon<lb></lb>ſtrauimus: ſed AC quadrupla eſt potentia ipſius AK; <pb xlink:href="043/01/221.jpg" pagenum="42"></pb>quadrupla igitur BC ipſius DK: cum igitur BC ſit <lb></lb>dupla ipſius KH, erit DK dimidia eiuſdem KH, & ſecta <lb></lb>bifariam KH in puncto D: ſed recta AG ſecabat eandem <lb></lb>KH bi fariam; per punctum igitur D tranſibit AG. </s> <s>Quo<lb></lb>niam igitur parabola ADC, cuius vertex D, ſeſquiter<lb></lb>tia eſt per Archimedem trianguli ADB, cuius duplum <lb></lb>eſt triangulum ABG, ſicut & huius triangulum ABC; <lb></lb>triangulum ABC quadruplum erit trianguli ADB: qua<lb></lb>lium igitur partium æqualium eſt triangulum ABC duo<lb></lb>decim, talium erit triangulum ADB trium, & parabola <lb></lb>ADB, cuius ver<lb></lb>tex D quatuor: du <lb></lb>plum igitur erit tri<lb></lb>angulum ABC <lb></lb>mixtum parabolæ <lb></lb>ADB, cuius ver<lb></lb>tex D, & cen<lb></lb>trum grauitatis M: <lb></lb>ſed trianguli ABC <lb></lb>rectilinei eſt cen<lb></lb>trum grauitatis N, <lb></lb>& F <expan abbr="triãguli">trianguli</expan> ABC <lb></lb>mixti; dupla igitur <lb></lb>erit MN ipſius N <lb></lb>F, & MD ipſius <lb></lb><figure id="id.043.01.221.1.jpg" xlink:href="043/01/221/1.jpg"></figure><lb></lb>OF, & DN ipſius NO, propter ſimilitudinem triangulo<lb></lb>rum: ſed & tota AN dupla eſt totius NG, ob centrum <lb></lb>grauitatis N rectilinei trianguli ABC; reliqua igitur AD <lb></lb>dupla eſt reliquæ GO. cum igitur AG ſit dupla ipſius <lb></lb>AD, quadrupla erit AG ipſiuſque GO. quare & quadru <lb></lb>pla AE ipſius FE ob parallelas: tripla igitur AF ipſius FE. <lb></lb></s> <s>Rurſus quoniam ex Archimede ſeſquialtera eſt DM ipſius <lb></lb>MH, erit tota DH ad DM vt quinque ad tria, hoc eſt <lb></lb>vt decem ad ſex: ſed MD erat dupla ipſius OF; tota igi-<pb xlink:href="043/01/222.jpg" pagenum="43"></pb>tur DH ad OF erit vt decem ad tria: ſed GC dupla <lb></lb>eſt ipſius DH; igitur GC ad FO vt viginti ad tria: ſed <lb></lb>quia tripla exiſtente AO ipſius OG, eſt tota AG ipſius <lb></lb>AO ſeſquitertia, erit quoque GE, ipſius OF ſeſquiter<lb></lb>tia, propter ſimilitudinem triangulorum AGE, AOF, <lb></lb>hoc eſt qualium partium æqualium OF trium, talium GE <lb></lb>quatuor; qualium eſt GC hoc eſt BG viginti, talium <lb></lb>erit EG quatuor, & EC ſexdecim: dempta igitur EG <lb></lb>ex GC, & addita ipſi BG, qualium eſt EC ſexdecim: <lb></lb>talium erit BE vigintiquatuor: ſed vt vigintiquatuor ad <lb></lb>ſexdecim, ita ſunt tria ad duo, quæ proportio eſt ſeſqui<lb></lb>altera, ſeſquialtera igitur erit BE ipſius EC, oſtenſa eſt <lb></lb>autem AF ipſi FE tripla. </s> <s>Manifeſtum eſt igitur pro<lb></lb>poſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si duo triangula mixta prædicti generis verti<lb></lb>cem communem habeant, qui eſt contactus, & <lb></lb>baſes æquales in eadem recta linea, vel continuas, <lb></lb>vel ſegmento interiecto, tota extra ſiguram verſa <lb></lb>cauitate; centrum grauitatis compoſiti ex vtro<lb></lb>que eſt pun ctum illud, in quo recta linea à vertice <lb></lb>ad bipartitæ rectæ prædictis ſectionibus interce<lb></lb>ptæ, in qua ſunt baſes dictorum triangulorum ſe<lb></lb>ctionis punctum pertinens ſic diuiditur; vt pars, <lb></lb>quæ eſt ad verticem ſit tripla reliquæ. </s></p><p type="main"> <s>Sint duo prædicti generis triangula ABC, ADE ha<lb></lb>bentia verticem A communem, qui eſt contactus recta. <lb></lb></s> <s>rum cum parabolis, tangente AB parabolam AC, & <pb xlink:href="043/01/223.jpg" pagenum="44"></pb>AD parabolam AE: baſes autem æquales BC, DE pa<lb></lb>rallelas parabolarum diametres per A, & in vna recta li<lb></lb>nea CE ſegmento BD interiecto: vtriuſque autem ſe<lb></lb>ctionis AC, AE concauitas ſpectet extra figuram ACE: <lb></lb>ſecta autem CE bifariam in F, iunctaque AF, ponatur <lb></lb>AG tripla ipſius GF. </s> <s>Dico compoſiti ex triangulis A <lb></lb>BC, ADE centrum grauitatis eſſe G. </s> <s>Poſita enimvtra<lb></lb>que ſeſquialtera, CH ipſius HB, & EK ipſius KD, <lb></lb>iunctisque AH, AK, ducatur per punctum G ipſi CE <lb></lb>parallela ſecans AH, AK in punctis L, M. </s> <s>Quoniam <lb></lb>igitur LM ipſi CE parallela ſecat eas quæ ex puncto A <lb></lb>ad rectam CD du<lb></lb>cuntur rectas lineas <lb></lb>in eaſdem rationes, & <lb></lb>eſt AG tripla ipſius <lb></lb>GF; tripla erit vtra<lb></lb>que AL ipſius LH, <lb></lb>& AM ipſius MK: <lb></lb>ſeſquialtera autem eſt <lb></lb>CH ipſius HB, & <lb></lb>EK ipſius KD; erit <lb></lb>igitur L centrum gra<lb></lb>uitatis trianguli AB <lb></lb>C, & M trianguli A <lb></lb>DE per præceden<lb></lb><figure id="id.043.01.223.1.jpg" xlink:href="043/01/223/1.jpg"></figure><lb></lb>tem. </s> <s>Rurſus quoniam abſoluantur triangula rectilineæ <lb></lb>ACB, AEK, & æqualia erunt propter æquales baſes, <lb></lb>poſita inter eaſdem parallelas, & vtrumque ſeſquialterum <lb></lb>eius trianguli mixti, quod comprehendit, ex demonſtra<lb></lb>tione antecedentis; æqualia igitur erunt triangula mixta <lb></lb>ABC, ADE, ſiquidem ſunt æqualium ſubſeſquialtera. <lb></lb></s> <s>Et quoniam componendo, & permutando eſt vt CB ad <lb></lb>DE ita BH ad DK, æqualis erit BH ipſi DK: ſed ſi ab <lb></lb>æqualibus poſitis CF, FE ipſas CB, DE æquales au-<pb xlink:href="043/01/224.jpg" pagenum="45"></pb>feras, reliquæ BF, FD æquales erunt; tota igitur FH to<lb></lb>ti FK æqualis eſt: in triangulo autem AHK recta AF <lb></lb>ſecat LM, HK parallelas in eaſdem rationes; erit igitur <lb></lb>LG æqualis ipſi GM; cum igitur æqualium triangulo<lb></lb>rum ABC, ADE centra grauitatis ſint L, M; erit com <lb></lb>poſiti ex vtroque centrum grauitatis G. </s> <s>Idem oſtendere<lb></lb>mus, quod proponitur, & ſi baſes prædictorum triangulo<lb></lb>rum ſint continuæ. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si duæ parabolæ in eodem plano circa æqua<lb></lb>les diamet ros in directum inter ſe conſtitutas, ita <lb></lb>vt vertices ſint extrema ex diametris compoſitæ, <lb></lb>communem habuerint aliquam ordinatim ad dia <lb></lb>metrum applicatarum, & vertices cum puncto con <lb></lb>uenientiæ iungantur rectis lineis: centrum gra<lb></lb>uitatis v triuſque portionis ijs rectis lineis ab ſciſ <lb></lb>ſæ, rectam lineam, quæ terminum communem <lb></lb>diamctrorum, & concurſum parabolarum iungit <lb></lb>bifariam diuidit. </s></p><p type="main"> <s>Circa æquales <lb></lb>diametros AD, <lb></lb>DC indirectum <lb></lb>inter ſe conſtitutas, <lb></lb>verticibus A, C, <lb></lb>duæ parabolæ in <lb></lb>eodem plano <expan abbr="com-munẽ">com<lb></lb>munem</expan> habeant ali<lb></lb>quam BD ordi<lb></lb><figure id="id.043.01.224.1.jpg" xlink:href="043/01/224/1.jpg"></figure><pb xlink:href="043/01/225.jpg" pagenum="46"></pb>natim ad vtramque diametrorum applicatarum, iunctis<lb></lb>que AB, BC, ſit ſecta BD bifariam in puncto G. <lb></lb></s> <s>Dico G eſse centrum grauita tis duarum portionum AEB, <lb></lb>BFE ſimul. </s> <s>Si enim hoc non eſt, ſit aliud punctum L. & <lb></lb>compleantur parallelogramma ANBD, DBRC, hoc <lb></lb>eſt totum AR parallelogrammum: & ſecta BG bifariam <lb></lb>in puncto H, ponatur DK ipſius BD pars tertia, vt pun<lb></lb>ctum K ſit trianguli ABC centrum grauitatis. </s> <s>Poſita au<lb></lb>tem ſeſquialtera BP ipſius PN, & BQ ipſius QR, iun<lb></lb>ctisque AP, CQ, duoatur per punctum H ipſi AC, vel <lb></lb>NR parallela, cum ipſis AP, CQ conueniens in punctis <lb></lb>ST: & iuncta LG, <lb></lb>ſi punctum L non <lb></lb>ſit in linea BD, <lb></lb>eſto LM quintu<lb></lb>pla ipſius MG. <lb></lb></s> <s>Quoniam igitur ob <lb></lb>parallelas AC, P <lb></lb>Q, ST in trape<lb></lb>zio APQC, eſt <lb></lb>vt DH ad HB, ita <lb></lb>AS ad SP, & CT <lb></lb><figure id="id.043.01.225.1.jpg" xlink:href="043/01/225/1.jpg"></figure><lb></lb>ad TQ, erit AS ipſius SP, & CT ipſius TQ tripla: <lb></lb>ſed eſt BP ſeſquialtera ipſius PN, & BQ ipſius QR; <lb></lb>mixti igitur trianguli ANB centrum grauitatis erit S, & <lb></lb>trianguli mixti CRB centrum grauitatis T. cum igitur <lb></lb>BP, BQ proportionales æqualibus NB, BR inter ſe <lb></lb>ſint æquales, & ſecta AC bifariam in puncto D; etiam <lb></lb>ijs parallela ST ſecta erit bifariam in puncto H: iungit <lb></lb>autem ST centra grauitatis mixtorum triangulorum AN <lb></lb>B, BRC; compoſiti igitur ex vtroque centrum grauita<lb></lb>tis erit H. </s> <s>Rurſus quoniam ex quadratura parabolæ, ſe<lb></lb>miparabola ABD ſeſquitertia eſt trianguli BDA, erit <lb></lb>triangulum BDA ſeſquialterum mixti trianguli ANB: <pb xlink:href="043/01/226.jpg" pagenum="47"></pb>eadem ratione triangulum BDC, trianguli CRB mi xti <lb></lb>erit ſeſquialterum: totum igitur triangulum ABC ſeſqui<lb></lb>alterum eſt compoſiti ex triangulis mixtis ANB, CRB. <lb></lb></s> <s>Et quoniam quarta pars eſt GH ipſius BD, & DK ter<lb></lb>tia, DG verò dimidia; qualium duodecim partium æqua<lb></lb>lium eſt BD, talium erit DK quatuor, & GH trium, & <lb></lb>DG ſex, & reliqua KG duarum; ſeſquialtera igitur eſt <lb></lb>GH ipſius GK: quare vt triangulum ABC ad compo<lb></lb>ſitum ex prædictis triangulis mixtis, ita ex contraria parte <lb></lb>eſt HG ad G<emph type="italics"></emph>K<emph.end type="italics"></emph.end>: cum igitur dicti compoſiti ſit centrum <lb></lb>grauitatis H, trianguli autem ABC centrum grauitatis <lb></lb>K; erit dicti compoſiti, & trianguli ABC ſimul centrum <lb></lb>grauitatis G. Rurſus, quoniam triangulum ABC ſeſ<lb></lb>quialterum eſt compoſiti ex triangulis mixtis ſupra dictis, <lb></lb>& compoſitum ex duabus ſemiparabolis ABD, CBD <lb></lb>ſeſquitertium trianguli ABC; crit compoſitum ex trian<lb></lb>gulis mixtis vnà cum triangulo ABC, quintuplum com<lb></lb>poſiti ex portionibus AEB, BFC; hoc eſt vt ex contra<lb></lb>ria parte LM ad MG: cum igitur G ſit centrum graui<lb></lb>tatis compoſiti ex triangulis mixtis, & triangulo ABC, & <lb></lb>compoſiti ex portionibus AEB, BFC centrum grauita<lb></lb>tis L; erit vtriuſque dicti compoſiti, hoc eſt totius AR <lb></lb>parallelogrammi centrum grauitatis L: ſed & punctum G <lb></lb>ex primo libro eſt centrum grauitatis parallelogrammi <lb></lb>AR; eiuſdem igitur parallelogrammi AR erunt duo cen<lb></lb>tra grauitatis G, L. </s> <s>Quod fieri non poteſt: duarum igitur <lb></lb>portionum AEB, BFC ſimul centrum grauitatis erit G. <lb></lb></s> <s>Quod eſt propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis figuræ circa axim in alteram partem de <lb></lb>ficientis, cuius baſis eſt circulus, vel ellipſis, ſiue-<pb xlink:href="043/01/227.jpg" pagenum="48"></pb>baſes ſunt circuli, vel ellipſes, reliqua autem ſu<lb></lb>perficies tota interius concaua, centrum grauitatis <lb></lb>eſt in dimidio axis ſegmento, quod baſim, vel ma<lb></lb>iorem baſim attingit. </s></p><p type="main"> <s>Sit figura circa axim in alteram partem deficiens ABC, <lb></lb>cuius axis BD, baſis, vel maior baſis circulus, vel ellipſis <lb></lb>circa diametrum AC, reliqua autem ſuperficies tota inte<lb></lb>rius concaua: ſecto autem axe BD bifariam in puncto G, <lb></lb>ſit ſolidi ABC centrum grauitatis F nempe in axe BD. <lb></lb></s> <s>Dico punctum F eſſe in ſegmento ED. </s> <s>Secto enim ſoli<lb></lb>do ABC, & figu <lb></lb>ra per axem pla <lb></lb>no per <expan abbr="punctũ">punctum</expan> E <lb></lb>baſi, vel baſibus <lb></lb>parallelo, fiat ſe<lb></lb>ctio circulus, vel <lb></lb>ellipſis ſimilis <lb></lb>baſi, per diffini<lb></lb>tionem, & ſectio<lb></lb>nis diameter K <lb></lb>N: deinde figu<lb></lb>ra quædam ex <lb></lb><figure id="id.043.01.227.1.jpg" xlink:href="043/01/227/1.jpg"></figure><lb></lb>duobus cylindris, vel cylindri portionibus KL, AM cir<lb></lb>ca axes BE, ED, eiuſdem altitudinis circumſcribatur <lb></lb>ſolido ABC: ſecanturque bifariam BE in puncto G, & <lb></lb>ED in puncto H. totius autem figuræ circumſcriptæ ſit <lb></lb>centrum grauitatis O, nempe in axe BD. </s> <s>Quoniam igi<lb></lb>tur propter bipartitorum axium ſectiones G, H, eſt ſolidi <lb></lb>KL centrum grauitatis G: ſolidi autem AM centrum <lb></lb>grauitatis H, erit in linea GH totius ſolidi AL centrum <lb></lb>grauitatis O, & vt ſolidum AM ad ſolidum KL, ita GO <lb></lb>ad OH: ſed maior eſt proportio ſolidi AM ad ſolidum KL <pb xlink:href="043/01/228.jpg" pagenum="49"></pb>quàm GE, ad EH; maior igitur proportio eſt GO ad <lb></lb>OH, quàm GE ad EH: & componendo, maior pro<lb></lb>portio GH ad HO, quàm eiuſdem GH ad HE; mi<lb></lb>nor igitur OH erit quàm EH, & punctum O propin<lb></lb>quius puncto D quàm punctum E; verum quoniam ex <lb></lb>ijs, quæ in præcedenti libro demonſtrauimus, propoſitæ <lb></lb>figuræ ſolidæ ABC centrum grauitatis eſt puncto D <lb></lb>propinquius, quàm cuiuslibet figuræ ex cylindris, vel cy <lb></lb>lindri portionibus æqualium altitudinum ipſi circumſcri<lb></lb>ptæ, erit punctum F propinquius puncto D quàm pun<lb></lb>ctum O; multo igitur puncto D erit propinquius pun<lb></lb>ctum F quàm punctum E; ergo infra punctum E, & in <lb></lb>linea ED cadet ſolidi ABC centrum grauitatis F. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti coni, vel portionis conicæ cen<lb></lb>trum grauitatis eſt punctum illud, in quo eius <lb></lb>axis ſic diuiditur, vt pars quæ minorem baſim at<lb></lb>tingit aſſumens quartam partem axis ablati coni, <lb></lb>vel portionis conicæ, ſit ad eam, quæ inter poſtre<lb></lb>mam ſectionem, & quartæ partis abſciſſ<17> ad baſim <lb></lb>axis totius coni terminum interijcitur, vt cubus, <lb></lb>qui fit ab axe totius, ad cubum qui fit ab axe abla<lb></lb>ti coni. </s></p><p type="main"> <s>Sit coni, vel portionis conicæ ABC fruſtum BDEC, <lb></lb>cuius axis FG: conus autem, vel coni portio ablata AD <lb></lb>E: ſint centra grauitatis H ſolidi ABC, & K ſolidi <lb></lb>ADE, & L fruſti DC: quæ centra præterquam quod <pb xlink:href="043/01/229.jpg" pagenum="50"></pb>ſunt omnia in axe AG, centrum L cadet infra <lb></lb>centrum H, ex ijs, quæ in primo libro demonſtraui<lb></lb>mus. </s> <s>Dico eſſe KL ad LH vt cubum ex AG ad cu<lb></lb>bum ex AF. </s> <s>Quoniam enim <lb></lb>ob centra grauitatis <emph type="italics"></emph>K<emph.end type="italics"></emph.end>, H, L, <lb></lb>eſt vt fruſtum DC ad ſolidum <lb></lb>ADE, ita ex contraria parte <lb></lb>KH ad HL; erit componen<lb></lb>do, vt ſolidum ABC ad ſoli<lb></lb>dum ADE, ita KL ad LH: <lb></lb>ſed vt <expan abbr="ſolidũ">ſolidum</expan> ABC ad ſolidum <lb></lb>ADE, ita eſt cubus ex AG <lb></lb>ad cubum ex AF: triplieata <lb></lb>enim eſt vtraque proportio eiuſ<lb></lb>dem, quæ eſt ipſius AG ad ip<lb></lb>ſam AF, propter ſimilitudi<lb></lb>nem ſolidorum; vt igitur eſt cu <lb></lb>bus ex AG ad cubum ex AF, <lb></lb>ita erit KL ad LH. </s> <s>Quod demonſtrandum erat. </s></p><figure id="id.043.01.229.1.jpg" xlink:href="043/01/229/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Reſidui ſolidi ex cylindro, vel portione cylin<lb></lb>drica hemiſphærio, vel hemiſphæroidi circum<lb></lb>ſcripta, dempto hemiſphærio, vel hemiſphæroide, <lb></lb>centrum grauitatis eſt punctum illud, in quo axis <lb></lb>ſic diuiditur, vt pars baſim attingens hemiſphæ<lb></lb>rij, vel hemiſphæroidis ſit tripla reliquæ. </s></p><p type="main"> <s>Eſto hemiſphærio, vel hemſphæroidi ABC, cuius axis <lb></lb>BD, circumſcriptus cylindrus, vel portio cylindrica AF: <lb></lb>& ponatur D<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ipſius <emph type="italics"></emph>K<emph.end type="italics"></emph.end>B tripla. </s> <s>Dico reliqui ex ſoli-<pb xlink:href="043/01/230.jpg" pagenum="51"></pb>do AF dempto ABC, centrum grauitatis eſſe <emph type="italics"></emph>K.<emph.end type="italics"></emph.end></s><s> Nam <lb></lb>ſuper baſim circulum, vel ellipſim, cuius diameter EF ſi<lb></lb>milem, & oppoſitam ſolidi ABC, vel AF baſi, cuius dia<lb></lb>meter AC, ſtet cylindrus, vel portio cylindrica EDF: vt <lb></lb>ſitaxis BD communis quatuor ſolidis ABC, EDF, <lb></lb>AF, & reliquæ figuræ dempto ſolido ABC compre<lb></lb>henſæ ſuperficie cylindrica, & circulo, vel ellipſe circa EF, <lb></lb>& dimidia ſuperſicie ſphærica interiori, cuius figuræ ſoli<lb></lb>dæ ponimus centrum grauitatis <emph type="italics"></emph>K.<emph.end type="italics"></emph.end></s><s> Secto igitur axe <lb></lb>BD bifariam, & ſingulis eius partibus rurſus bifariam, <lb></lb>ductiſque per puncta ſectionum planis quibuſdam planis <lb></lb><figure id="id.043.01.230.1.jpg" xlink:href="043/01/230/1.jpg"></figure><lb></lb>prædictarum baſium oppoſitarum parallelis, ſecta ſint qua<lb></lb>tuor prædicta ſolida, quorum, excepto propoſito reſiduo, <lb></lb>ſectiones omnes erunt circuli, vel ellipſes inter ſe ſimi<lb></lb>les, & in ſolido AF etiam æquales, quarum omnium <lb></lb>diametri eiuſdem rationis erunt in eodem plano, in quo <lb></lb>ſit parallelogrammum per axim AEFC: ſolidi autem dicti <lb></lb>reſidui ſectiones, reſidua ſectionum ſolidi ABC. </s> <s>At circa <lb></lb><expan abbr="cõmunes">communes</expan> axes inter ſe æquales ſegmenta axis BD, & inter <lb></lb><expan abbr="eadẽ">eadem</expan> plana parallela, ſuper baſes ſectiones duorum ſolido<lb></lb>rum ABC, EDF, cylindri, vel portiones cylindricæ con<lb></lb>ſiſtant altitudine, & multitudine æquales; ita vt duarum fi<lb></lb>gurarum ex ijs compofitarum altera fit cirdumſcripta ſoli<pb xlink:href="043/01/231.jpg" pagenum="52"></pb>do EDF, altera ſolido ABC inſcripta. </s> <s>hac igitur abla<lb></lb>ta ex ſolido AF, figura relinquetur ex reſiduis cylindro<lb></lb>rum, vel cylindri portionum altitudine, & multitudine <lb></lb>æqualibus ijs cylindris, vel cylindri portionibus, ex quibus <lb></lb>conſtat alterutra figurarum ſolidis ABC, DEF circum<lb></lb>ſcriptarum: eruntque ex ſuperius demonſtratis dicta reſi<lb></lb>dua, & cylindri vel cylindri portiones, quæ circa ſolidum <lb></lb>EDF, inter ſe æqualia proutinter ſe reſpondent inter ea<lb></lb>dem plana parallela, vt eſt exempli gratia reliquum ſoli<lb></lb>di AN dempto ſolido SR, æquale ſolido TP: & ſic de<lb></lb>inceps: ſummus autem XF cylindrus, vel portio cylindrica <lb></lb><figure id="id.043.01.231.1.jpg" xlink:href="043/01/231/1.jpg"></figure><lb></lb>eſt communis: Atqui bina hæc iam dicta ſolida centrum <lb></lb>grauitatis habent commune communis bipartiti axis ſectio <lb></lb>nem in eadem recta linea BD, in qua eſt etiam ſolidi XF <lb></lb>communis centrum grauitatis. </s> <s>duarum igitur dictarum figu <lb></lb>rarum ſolido EDF, & prædicto reſiduo circumſcriptarum <lb></lb>idem aliquod punctum in axe BD erit commune centrum <lb></lb>grauitatis: ſieri autem poteſts quod in ſecundo libro demon <lb></lb>ſtrauimus, vt duæ dictæ figuræ ſuperent vnaquæ que ſibi in<lb></lb>ſcriptam minori ſpacio quantacumque magnitudine pro<lb></lb>poſita. </s> <s>ex demonſtratis igitur in primo libro; duo ſolida cir<lb></lb>ca axem BD in alteram partem deficientia commune ha<lb></lb>bebunt in axe BD centrum grauitatis: ſed ſolidi, ideſt co-<pb xlink:href="043/01/232.jpg" pagenum="53"></pb>ni, vel portionis conicæ EDF eſt centrum grauitatis K: <lb></lb>reliqui igitur ex cylindro, vel portione cylindrica AF dem <lb></lb>pto hemiſphærio, vel hemiſphæroide ABC centrum graui <lb></lb>tatis erit idem K. </s> <s>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si hemiſphærium, vel hemiſphæroides vna cum <lb></lb>cylindro, vel cylindri portione ipſi circumſcripta <lb></lb>ſecetur plano baſi parallelo; reliqui ex cylindro, <lb></lb>vel portione cylindrica abſciſſa ad partes verti<lb></lb>cis, dempta illa quæ abſciſſa eſt ſimul minori, <lb></lb>& ſphæræ, vel ſphæroidis portione, centrum gra<lb></lb>uitatis eſt punctum illud, in quo eius axis ſic diui<lb></lb>ditur, vt quæ inter hanc poſtremam ſectionem, & <lb></lb>centrum baſis vnà abſciſſæ portionis interijci<lb></lb>tur, aſſumens quartam partem ſegmenti, quod di<lb></lb>ctæ baſis, & ſphæræ, vel ſphæroidis centra iungit, <lb></lb>ſit ad ſui ſegmentum, quod inter poſtremam ſe<lb></lb>ctionem, & quartæ partis axis hemiſphærij, vel <lb></lb>hemiſphæroidis ad verticem abſciſſæ terminum <lb></lb>interijcitur, vt cubus axis hemiſphærij, vel hemi<lb></lb>ſphæroidis, ad cubum eius, quæ baſis portionis & <lb></lb>hemiſphærij, vel hemiſphæroidis centra iungit. <lb></lb></s> <s>Reliqui autem ex cylindro, vel portione cylindri<lb></lb>ca vnà abſciſſa <expan abbr="cũ">cum</expan> reliqua hemiſphærij, vel hemi<lb></lb>ſphæroidis portione, quæ eſt ad baſim, dempta hac <lb></lb>portione centrum, grauitatis eſt punctum illud, <lb></lb>quod quartam partem abſcindit axis portionis ad <pb xlink:href="043/01/233.jpg" pagenum="54"></pb>cius minorem baſim terminatam. </s></p><p type="main"> <s>Eſto hemiſphærio, vel hemiſphæroidi ABC, cuius axis <lb></lb>BD, baſis circulus vel ellipſis, cuius diameter AC cir<lb></lb>cumſcriptus cylindrus, vel cylindri portio AF, cuius in<lb></lb>telligatur reliquum dempto ABC. quæ ſolida ſecans pla <lb></lb>num per AC, BD, faciat ſectiones ſemicirculum, vel ſe<lb></lb>miellipſim ABC, & parallelogrammum per axem AE <lb></lb>FC; & per quodlibet punctum L axis BD, planum baſibus <lb></lb>AC, EF ſolidi AF <expan abbr="parallelũ">parallelum</expan>, ſecans prædicta ſolida ABC, <lb></lb>AF, faciat ſectiones circulos, vel ellipſes ſimiles, & in ſolido <lb></lb>AF etiam æquales ijs, quæ circa AC, EF: earum autem dia<lb></lb>metros, ſectiones cum <expan abbr="parallelogrãmo">parallelogrammo</expan> AEFC, ipſam GO: <lb></lb>& cum ſemicirculo, vel ſemiellipſe ABC, ipſam HN. </s> <s>Ita<lb></lb>que habebimus figuram quandam ſolidam GHBNO reſi<lb></lb>duum cylindri, vel portionis cylindricæ GF dempta mino<lb></lb>ri ſphæræ, vel ſphæroidis portione HBN, cuius axis erit BL. <lb></lb></s> <s>Sumpta igitur BQ quarta parte axis BD, & LP quarta par <lb></lb>te ipſius DL fiat vt cu <lb></lb>bus ex BD ad cubum ex <lb></lb>DL, ita PR ad <expan abbr="Rq.">Rque</expan> <lb></lb>Dico reſidui GHBNO <lb></lb>centrum grauitatis eſſe <lb></lb>R. </s> <s>Reliqui autem ex <lb></lb>cylindro, vel portione <lb></lb>cylindrica AO dempta <lb></lb>portione AHNC, cen<lb></lb>trum grauitatis eſſe P. <lb></lb><figure id="id.043.01.233.1.jpg" xlink:href="043/01/233/1.jpg"></figure><lb></lb>Nam ſuper baſim circulum, vel ellipſim EF, ſtet conus, vel <lb></lb>portio conica EDF: ſitque prædicto plano per L abſciſ<lb></lb>ſus conus, vel coni portio KDM, cuius axis DL, quæ pro<lb></lb>pter planum ſecans baſi EF parallelum, ſimilis erit toti <lb></lb>cono, vel portioni conicæ EDF. </s> <s>Quoniam igitur BQ <lb></lb>eſt axis BD pars quarta, & LP pars quarta ipſius DL; <pb xlink:href="043/01/234.jpg" pagenum="55"></pb>erunt centra grauitatis ſolidorum, Q ipſius EDF, & Pip<lb></lb>ſius DKM. </s> <s>Et quoniam ſolidum DEF ad ſolidum D <lb></lb>KM eſt vt cubus ex BD ad cubum ex DL, hoc eſt vt <lb></lb>ſolidum EDF ad ſolidum KLM, & vt PR ad <expan abbr="Rq;">Rque</expan> <lb></lb>erit diuidendo, vt fruſtum EKMF ad ablatum KDM, <lb></lb>ita ex contraria parte PQ ad QR: cum igitur ſint <lb></lb>centra grauitatis P ſolidi DKM, & Q ſolidi DET; <lb></lb>erit reliqui fruſti EKMF centrum grauitatis R: ſed <lb></lb>qua ratione in præcedenti conſtat, reliqui ex ſolido AF, <lb></lb>dempto ſolido ABC centrum grauitatis eſſe Q, eadem <lb></lb>concluditur idem eſſe centrum grauitatis reliqui ex ſolido <lb></lb>GF, dempta portione HBN, quod & fruſti EKMF, <lb></lb>nempe punctum R: Et quoniam P eſt centrum grauita<lb></lb>tis coni, vel portionis conicæ KDM, crit idem P centrum <lb></lb>grauitatis ieliqui ex cylindro, vel portione cylindrica <lb></lb>AO dempta portione AHNC. </s> <s>Manifeſtnm eſt igitur <lb></lb>propoſituro. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Ijſdem poſitis ſolidis, vt in antecedenti, ſectis<lb></lb>que per duo quælibet puncta axis duplici plano <lb></lb>baſi parallelo, reliqui ex cylindro, vel portione <lb></lb>cylindrica dictis duobus planis intercepta dem<lb></lb>pta ſphæræ, vel ſphæ roidis portione ipſi inter ea<lb></lb>dem plana reſpondente, centrum grauitatis eſt <lb></lb>punctum illud, in quo eius axis ſic diuiditur, vt <lb></lb>quæ inter hanc poſtremam ſectionem, & centrum <lb></lb>maioris baſis vnà abſciſsæ portionis interijcitur, <lb></lb>aſſumens quartam partem ſegmenti, quod prædi<lb></lb>ctæ baſis, & ſphæræ vel ſphæroidis centra iungit, <pb xlink:href="043/01/235.jpg" pagenum="56"></pb>ſit ad ſui ſegmentum, quod inter poſtremam ſectio <lb></lb>nem, & quartæ partis eius, quæ ſphæræ, vel hemi<lb></lb>ſphærij, & minoris baſis portionis centra iungit <lb></lb>ad minorem baſim abſciſsæ terminum interijci<lb></lb>tur, vt cubus eius, quæ minoris baſis, & ſphæræ, <lb></lb>vel ſphæroidis, ad <expan abbr="cubũ">cubum</expan> eius, qu<17> ſphæræ, vel ſphæ <lb></lb>roidis, & maioris baſis portionis centra iungit. </s></p><p type="main"> <s>Ijſdem poſitis ſolidis, vtque in antecedenti ponebantur <lb></lb>ABC, AF; per duo quælibet puncta RQ axis BD ſe<lb></lb>centur poſita ſolida duobus planis baſi, quæ circa AC, cir <lb></lb>culo ſcilicet, vel ellipſi parallelis: quibus planis intercepta <lb></lb>hemiſphærij, vel hemiſphæroidis portio ſit MOPN, vnà <lb></lb>cum cylindro, vel portione cylindrica GL parte ipſius AF, <lb></lb><expan abbr="quorũ">quorum</expan> ſolidorum <expan abbr="cõmu">commu</expan> <lb></lb>nis axis vnà abſciſſus <lb></lb>ab axe BD ſolidi AB <lb></lb>C, ſit RQ: & ſumptis <lb></lb>quartis partibus RI ip<lb></lb>ſius DR, & QZ ipſius <lb></lb>DQ, fiat vt cubus ex <lb></lb>DQ ad cubum ex D <lb></lb>R, ita IY ad YZ. <lb></lb></s> <s>Dico reliqui ex cylin<lb></lb><figure id="id.043.01.235.1.jpg" xlink:href="043/01/235/1.jpg"></figure><lb></lb>dro, vel portione cylindrica GL dempta portione MOP <lb></lb>N, centrum grauitatis eſſe Y. </s> <s>Facta enim conſtructione <lb></lb>coni, vel portionis conicæ EDF, vt in ſuperioribus, erunt <lb></lb>ſimilium conorum, vel coni portionum SDT, VDX, ea<lb></lb>dem ordine axes DQ, DR: propter igitur factas diuiſio<lb></lb>nes, erunt <expan abbr="cẽtra">centra</expan> grauitatis Z ſolidi SDT & I ſolidi VDX, <lb></lb>& demonſtratio ſimilis antecedenti. </s> <s>dicti igitur reſidui <lb></lb>GMOPMH centrum grauitatis Y. </s> <s>Quod eſt propo<lb></lb>ſitum. </s></p><pb xlink:href="043/01/236.jpg" pagenum="57"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſphæra, vel ſphæroides vnà cum cylindro, <lb></lb>vel portione cylindrica ipſi circumſcripta ſecetur <lb></lb>plano, haud per centrum, baſibus ſolidi circum<lb></lb>ſcripti parallelo; reliqui ex cylindro, vel portio<lb></lb>ne cylindrica ad maioris portionis ſphæræ, vel <lb></lb>ſphæroidis partes abſciſſa, dempta ſphæræ, vel <lb></lb>ſphæroidis maiori portione, centrum grauita<lb></lb>tis eſt punctum illud, in quo dicti reliqui ſolidi <lb></lb>axis ſegmentum inter duas quartas partes extre<lb></lb>mas ſegmentorum eiuſdem axis, quæ à centro <lb></lb>ſphæræ, vel ſphæroidis fiunt interiectum, ſic diui<lb></lb>ditur, vt pars propinquior baſi ſit ad reliquam, vt <lb></lb>prædictorum, quæ à centro fiunt axis ſegmento<lb></lb>rum maioris cubus ad cubum minoris. </s></p><figure id="id.043.01.236.1.jpg" xlink:href="043/01/236/1.jpg"></figure><p type="main"> <s>Sit ſphæræ, vel ſphæ<lb></lb>roidi ABCD cuius cen<lb></lb>trum E, circumſcriptus <lb></lb>cylindrus, vel portio cy<lb></lb>lindrica FGHK, cum <lb></lb>quibus planum per axim <lb></lb>communem BED, fa<lb></lb>ciat ſectiones, parallelo<lb></lb>grammum per axim FG <lb></lb>HK, & circulum, vel el<lb></lb>lipſim ABCD: quas fi<lb></lb>guras vnà cum dictis ſo<lb></lb>lidis ſecans planum baſibus ſolidi circumſcripti paralle-<pb xlink:href="043/01/237.jpg" pagenum="58"></pb>lum per quoduis punctum S dimidij axis ED, faciens<lb></lb>que ſectiones circulos, vel ellipſes ſimiles ſcilicet ba<lb></lb>ſibus oppoſitis ſolidi FH, & ſectionum diametros LM, <lb></lb>TV, abſcindat ſolidi ABCD maiorem portionem <lb></lb>LBM, & ſolidi FH cylindrum, vel portionem cy<lb></lb>lindricam TH, cuius axis BES: duorum autem ſegmen<lb></lb>corum BE, ES ſumptis duabus quartis partibus extre<lb></lb>mis BQ PS, fiat vt cubus ex BE ad cubum ex ES, ita <lb></lb>PR ad RQ. Dico reliquæ figuræ ex cylindro, vel por<lb></lb>tione cylindrica TH, portioni LBM circumſcripta, dem<lb></lb>pta portione LBM, centrum grauitatis eſſe R. </s> <s>Se<lb></lb>ctis enim parallelogrammo TH, & ſolidis LBM, TH, <lb></lb>plano per centrum E, baſibus ſolidi TH parallelo, ſit ſe<lb></lb>ctio, (vna enim communis erit vtrique ſolido) circulus, <lb></lb>vel ellipſis, cuius diameter AEC in parallelogrammo T <lb></lb>H diametris TV, GH <lb></lb>oppoſitarum baſium pa<lb></lb>rallela. </s> <s>Tum ſuper ba<lb></lb>ſes oppoſitas circulos, vel <lb></lb>ellipſes circa GH, FK <lb></lb>ſtent coni, vel portiones <lb></lb>conicæ GEH, FEK: <lb></lb>& planum per TV baſi <lb></lb>circa FK parallelum ab<lb></lb>ſcindat à ſolido FEK <lb></lb>conum, vel coni portio<lb></lb>nem NEO ſimilem vti<lb></lb>que ipſi FEK, hoc eſt <lb></lb><figure id="id.043.01.237.1.jpg" xlink:href="043/01/237/1.jpg"></figure><lb></lb>ipſi GEH, propter ſimiles baſes, & ſimilia triangula per <lb></lb>axim in eodem parallelogrammo FH. </s> <s>Solidi itaque <lb></lb>NEO, ex ijs, quæ in primo libro demonſtrauimus, cen<lb></lb>trum grauitatis erit P; quemadmodum & Q ſolidi <lb></lb>NEO. </s> <s>Quoniam igitur tàm ſolidi GEH ad ſoli<lb></lb>dum NEO propter ſimilitudinem, quàm cubi ex BE <pb xlink:href="043/01/238.jpg" pagenum="59"></pb>ad cubum ex ES, triplicata eſt proportio axis, vel la<lb></lb><gap></gap>eris BE, ad axem, vel latus ES; erit vt cubus ex BE <lb></lb>ad cubum ex ES, ita ſolidum GEH ad ſolidum NEO, <lb></lb>hoc eſt in eadem proportione, quæ eſt ex contraria parte ip<lb></lb>ſius PR ad RQ. Cum igitur P ſit centrum grauitatis <lb></lb>ſolidi NEO, & Q ſolidi GEH; erit compoſiti ex vtro<lb></lb>que centrum grauitatis R. Rurſus, quoniam reliquum ſo<lb></lb>lidi AH dempto hemiſphærio, vel hemiſphæroide ABC, <lb></lb>æquale eſt ſolido GEH: & reliquum ſolidi TC dempto <lb></lb>ſolido ALMC æquale ſolido NEO; erit vt ſolidum <lb></lb>GEH ad ſolidum NEO, ideſt ex contraria parte, vt PR <lb></lb>ad RQ, ita reliquum ſolidi AH dempto ABC, ad re<lb></lb>liquum ſolidi TC, dempto ALMC: ſed reliqui ex ſoli<lb></lb>do AH dempto ABC eſt centrum grauitatis Q: & reli<lb></lb>qui ex ſolido TC dempto ALMC, centrum grauitatis <lb></lb>P, ex ſuperius demonſtratis; totius igitur reliqui ex cy<lb></lb>lindro, vel portione cylindrica TH dempta ſphæræ, vel <lb></lb>ſphæroidis maiori portione LBM centrum grauitatis eſt <lb></lb>R. </s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſphæra, vel ſphæroides vnà cum cylindro, <lb></lb>vel portione cylindrica ipſi circumſcripta, ſece<lb></lb>tur duobus planis baſi ſolidi circumſcripti pa<lb></lb>rallelis, centrum intercipientibus, & ab eo non <lb></lb>æqualiter diſtantibus; reliqui ex cylindro, vel <lb></lb>portione cylindrica dictis planis intercepta, dem<lb></lb>pta portione ſphæræ, vel ſphæroidis ipſi reſpon<lb></lb>dente, centrum grauitatis eſt punctum illud, in <lb></lb>quo prædicti reliqui ſolidi axis ſegmentum in<pb xlink:href="043/01/239.jpg" pagenum="60"></pb>ter quartas partes extremas eiuſdem axis ſeg<lb></lb>mentorum, quæ à centro ſphæræ, vel ſphæroi<lb></lb>dis fiunt interiectum ſic diuiditur, vt pars ma<lb></lb>iori baſi propinquior ſit ad reliquam, vt prædi<lb></lb>ctorum axis ſegmentorum cubus maioris ad cu<lb></lb>bum minoris. </s></p><p type="main"> <s>Ijſdem poſitis, & conſtructis, quæ in antecedenti, rur<lb></lb>ſus per quodlibet axis BE punctum X, ductum planum <lb></lb>baſibus ſolidi FH parallelum, ſecansque vnà cylindrum, <lb></lb>vel portionem cylindricam FH, & ſphæram, vel ſphæroi<lb></lb>des ABCD: eſto duobus planis per TV, ZY, inter ſe pa<lb></lb>rallelis, & centrum E intercipientibus abciſſa ſphæræ, vel <lb></lb>ſphæroidis portio L <foreign lang="grc">δ ε</foreign> M vnà cum cylindro, vel portione <lb></lb>cylindrica TY: & ſumatur ipſius EX pars quarta XQ, <lb></lb>qualis eſt & PS ipſius E <lb></lb>S: & vt eſt cubus ex EX <lb></lb>ad cubum ex ES, ita fiat <lb></lb>PR ad <expan abbr="Rq.">Rque</expan> Dico reli<lb></lb>qui ex cylindro, vel por<lb></lb>tione cylindrica TY dem <lb></lb>pta ſphæræ, vel ſphæroi<lb></lb>dis portione L <foreign lang="grc">δ ξ</foreign> M, cen<lb></lb>trum grauitatis eſſe R. </s> <s>Eſto <lb></lb>enim conus, vel coni por<lb></lb>tio <foreign lang="grc">θ</foreign> E <foreign lang="grc">λ</foreign> abſciſſa prædi<lb></lb>cto plano per ZY, & com <lb></lb>munibus axibus ES, EX, <lb></lb>ſimili igitur demonſtratio<lb></lb>ne antecedentis manifeſtum eſt quod proponebatur. </s></p><figure id="id.043.01.239.1.jpg" xlink:href="043/01/239/1.jpg"></figure><pb xlink:href="043/01/240.jpg" pagenum="61"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Hemiſphærij, vel hemiſphæroidis centrum <lb></lb>grauitatis eſt punctum illud, in quo axis ſit diui<lb></lb>ditur, vt pars ad verticem ſit ad reliquam vt quin <lb></lb>que ad tria. </s></p><p type="main"> <s>Eſto hemiſphærium, vel hemiſphæroides ABC, cuius <lb></lb>axis BD, baſis circulus, vel ellipſis, cuius diameter AD <lb></lb>C: ſitque ſolidi ABC centrum grauitatis G, nempe <lb></lb>in axe BD. </s> <s>Dico BG ad GD eſſe vt quinque ad tria. <lb></lb></s> <s>Nam circa axim BD ſuper baſim circulum, vel ellipſim cir <lb></lb>ca AC, ſtet circumſcri <lb></lb>ptus ſolido ABC cy<lb></lb>lindrus, vel portio cy<lb></lb>lindrica AE, & ſecta <lb></lb>BD bifariam in F, rur <lb></lb>ſus FB bifariam ſece<lb></lb>tur in puncto H. </s> <s>Quo<lb></lb>niam igitur ſolidum A <lb></lb>BC eſt ſolidi AE, ſub<lb></lb>ſeſquialterum, erit di<lb></lb><figure id="id.043.01.240.1.jpg" xlink:href="043/01/240/1.jpg"></figure><lb></lb>uidendo ſolidum ABC reliqui ex ſolido AE duplum <lb></lb>cum igitur ſint centra grauitatis, G ſolidi ABC, & H <lb></lb>prædicti reliqui, & F totius AE; quo fit vt ex con<lb></lb>traria parte ſit vt ſolidum ABC ad prædictum reſiduum, <lb></lb>ita HF ad FG, erit HF dupla ipſius FG; quadrupla <lb></lb>igitur BF ipſius FG: ſed talium quatuor partium eſt BF, <lb></lb>qualium BD eſt octo, cum ſit BF dimidia ipſius BD; <lb></lb>qualium igitur octo eſt BD, talium erit BG quinque, & <lb></lb>GD trium. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/241.jpg" pagenum="62"></pb><p type="head"> <s><emph type="italics"></emph>ALITER.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Dico hemiſphærij, vel hemiſphæroidis ABC cen<lb></lb>trum grauitatis eſſe G. </s> <s>In plano enim ſemicirculi, vel ſe<lb></lb>miellipſis per axem BD deſcriptæ intelligantur duæ pa<lb></lb>rabolæ, quarum diametri AD, DC, & communiter <lb></lb>ad vtranque ordinatim applicata ſit BD: & connectun<lb></lb>tur rectæ AB, BC: ſumptis autem in BD tribus qui<lb></lb>buslibet punctis, æqualia axis ſegmenta XF, FY interci<lb></lb>pientibus, ſecent per ea puncta tres figuras hemiſphærium, <lb></lb>vel hemiſphæroides ABC, & ſemicirculum, vel ſemielli<lb></lb><figure id="id.043.01.241.1.jpg" xlink:href="043/01/241/1.jpg"></figure><lb></lb>pſim per axem, & figuram planam ARBSC, quæ lineis pa <lb></lb>rabolicis ARB, BSC, & recta AC continetur, pla<lb></lb>na quædam baſi hemiſphærij, vel hemiſphæroidis paralle<lb></lb>la. </s> <s>Erunt igitur ſectiones hemiſphærij, vel hemiſphæroidis <lb></lb>circuli, vel ellipſes ſimiles baſi, <expan abbr="quarũ">quarum</expan> diametri ſint KXH, <lb></lb>LFM, N<foreign lang="grc">Υ</foreign>O: figuræ autem ARBSC ſectiones rectæ <lb></lb>lineæ PXQ, RFS, TYV. </s> <s>Quoniamigitur per IV hu<lb></lb>ius eſt vt KH ad LM potentia, ita KQ ad FS hoc <lb></lb>eſt in earum duplis PQ ad RS longitudine; erit vt PQ <lb></lb>ad RS, ita circulus, vel ellipſis KH ad circulum vel ſi<lb></lb>milem ellipſim LM. </s> <s>Eadem ratione erit vt RS ad <lb></lb>TV, ita circulus, vel ellipſis LM ad circulum, vel <pb xlink:href="043/01/242.jpg" pagenum="63"></pb>ellipſim NO. minor autem proportio eſt PQ ad RS, <lb></lb>quàm RS ad TV circuli igitur, vel ellipſis KH ad <expan abbr="circulũ">circulum</expan>, <lb></lb>vel ellipſim LM, minor erit proportio <34> circuli, vel ellipſis <lb></lb>LM ad circulum, vel ellipſim NO: & duæ figuræ hemi<lb></lb>ſphærium, vel hemiſphæroides ABC, & plana ARBSC, <lb></lb>ſunt circa axim, vel diametrum BD in alteram parte m <lb></lb>deficientes, quales definiuimus; vtriuſque igitur dictæ fi<lb></lb>guræ vnum erit commune centrum grauitatis. </s> <s>Rurſus <lb></lb>poſito puncto F in medio axis BD, & FG ipſius GE <lb></lb>tripla, quoniam ponitur BG ad GD vt quinque ad tria; <lb></lb>qualium partium æqualium ipſi EG eſt FG trium, ta<lb></lb>lium erit BG quindecim, & GD nouem, & talis EG <lb></lb>vna: dempta igitur GE ab ipſa DG, & addita ipſi BG, <lb></lb>qualium partium eſt BE ſexdecim, talium erit ED octo; <lb></lb>dupla igitur BE ipſius ED, & trianguli ABC centrum <lb></lb>grauitatis E. </s> <s>Rurſus quoniam ex quadratura parabolæ, <lb></lb>duarum portionum ARB, BSC triangulum ABC eſt <lb></lb>triplum; hoe eſt vt FG ad GE, ita ex contraria parte <lb></lb>triangulum ABC ad duas portiones ARB, BSC: Sed <lb></lb>trianguli ABC eſt centrum grauitatis E, & duarum por <lb></lb>tionum ARB, BSC ſimul per XXIII huius, centrum <lb></lb>grauitatis F, totius igitur figuræ ARBSC centrum gra<lb></lb>uitatis erit G, commune autem hoc centrum grauitatis <lb></lb>eſt hemiſphærio, vel hemiſphæroidi ABC. </s> <s>Manifeſtum <lb></lb>eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis minoris portionis ſphæræ, vel ſphæroi<lb></lb>dis centrum grauitatis eſt in axe primum bifa<lb></lb>riam ſecto: deinde ſecundum centrum grauitatis <lb></lb>reliqui ſolidi dempta portione ex cylindro, vel <pb xlink:href="043/01/243.jpg" pagenum="64"></pb>portione cylindrica abſciſſo, vel abſciſſa vnà cum <lb></lb>portione, ex cylindro, vel portione cylindrica, <lb></lb>ſphær<17>, vel ſphæroidis circa axim axi portionis <expan abbr="cõ">com</expan> <lb></lb>gruentem <expan abbr="circũſcripta">circunſcripta</expan>; in eo puncto, in quo dimi<lb></lb>dius axis portionis baſim <expan abbr="attingẽs">attingens</expan> ſic diuiditur, vt <lb></lb>pars prima, & ſecunda ſectione terminata, ſit ad <lb></lb>totam ſecunda, & poſtrema ſectione terminatam, <lb></lb>vt rectangulum contentum axe portionis, & reli<lb></lb>quo ſphæræ, vel ſphæroidis dimidij axis ſegmen<lb></lb>to, vnà cum duabus tertijs quadrati axis portio<lb></lb>nis, ad ſphæræ, vel ſphæroidis dimidij axis axi <lb></lb>portionis congruentis quadratum. </s></p><p type="main"> <s>Sit ſphæræ, vel ſphæroidis minor portio ABC, cuius <lb></lb>axis BD: & in eo centrum grauitatis F: ſecto autem axe <lb></lb>BD primum bifariam <lb></lb>in puncto G, & rur <lb></lb>ſus BG in puncto <lb></lb>H centro grauitatis <lb></lb>reliqui dempta por<lb></lb>tione ex cylindro, vel <lb></lb>portione cylindrica <lb></lb>KL circa axim BD, <lb></lb>abſciſſo, vel abſciſ<lb></lb>ſa codem plano cum <lb></lb><figure id="id.043.01.243.1.jpg" xlink:href="043/01/243/1.jpg"></figure><lb></lb>portione ABC, & cylindro, vel portione cylindri<lb></lb>ca, quæ circumſcriberetur ſphæræ, vel ſphæroidi, cu<lb></lb>ius eſt portio ABC, circa axim, cuius dimidium BDE. <lb></lb></s> <s>Dico GH ad HF, (nam cadet centrum F infra biparti<lb></lb>ti axis BD ſectionem G, ex XXIII huius) eſſe vt rectan<lb></lb>gulum BDE vnà cum duabus tertijs BD quadrati ad <lb></lb>quadratum BE. </s> <s>Quoniam enim totius ſolidi KL cen-<pb xlink:href="043/01/244.jpg" pagenum="65"></pb>trum grauitatis eſt G, & F portionis ABC, & H reliqui <lb></lb>ex KL dempta ABC portione; erit vt portio ABC ad <lb></lb>prædictum reſiduum, ita ex contraria parte HG ad GF: <lb></lb>& componendo, vt ſolidum KL ad prædictum reſiduum, <lb></lb>ita HF ad FG: & per conuerſionem rationis, vt ſolidum <lb></lb>KL ad portionem ABC, ita FH ad HG: & conuerten <lb></lb>do, vt portio ABC ad ſolidum KL, ita GH ad HE: <lb></lb>ſed vt portio ABC ad ſolidum KL, ita eſt rectangulum <lb></lb>BDE vnà cum duabus tertiis quadrati BD ad quadra<lb></lb>tum EB; vt igitur rectangulum BDE, vnà cum duabus <lb></lb>tertiis quadrati BD, ad quadratum EB, ita erit GH ad <lb></lb>HF. </s> <s>Quod demonftrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ, vel ſphæroidis abſciſ <lb></lb>ſæ duobus planis parallelis, altero per centrum <lb></lb>acto, centrum grauitatis eſt in axe primum bifa<lb></lb>riam ſecto: deinde ſumpta eius quarta parte ad <lb></lb>minorem baſim; in eo puncto, in quo dimidius <lb></lb>axis maiorem baſim attingens ſic diuiditur, vt <lb></lb>pars axis prima, & ſecunda ſectione terminata, <lb></lb>ſit ad eam, quæ prima, & poſtrema ſectione ter<lb></lb>minatur, vt rectangulum contentum ſphæræ, vel <lb></lb>ſphæroidis axis axi portionis congruentis ijs ſeg<lb></lb>mentis, quæ fiunt à centro minoris baſis portio<lb></lb>nis, vnà cum duabus tertiis quadrati axis portio<lb></lb>nis; adſphæræ, vel ſphæroidis dimidij axis qua<lb></lb>dratum. </s></p><p type="main"> <s>Sit ſphæræ, vel ſphæroidis cuius centrum E portio <pb xlink:href="043/01/245.jpg" pagenum="66"></pb>ABCD abſciſsa duobus planis parallelis altero ducto <lb></lb>per E, & ſectionem faciente circulum maximum, vel <lb></lb>ellipſim per centrum, cuius diameter AED: axis autem <lb></lb>portionis ſit EF, cui congruens ſphæræ, vel ſphæroidis axis <lb></lb>GFER: ſit autem FE bifariam ſectus in puncto H: & <lb></lb>FH bifariam in puncto K, ſitque in EH, ſic enim erit, <lb></lb>portionis ABCD centrum grauitatis L. </s> <s>Dico eſſe HK <lb></lb>ad KL, vt rectangulum GFR, vnà cum duabus tertiis <lb></lb>quadrati EF ad quadratum EG. </s> <s>Sit enim cylindrus, vel <lb></lb>portio cylindrica AM circa axim FE abſciſſa ijſdem pla<lb></lb>nis cum portione AB <lb></lb>CD, ex cylindro, vel <lb></lb>portione cylindrica cir <lb></lb>ca axim GR ſphæ<lb></lb>ræ, vel ſphæroidi AG <lb></lb>DR circumſcripta. <lb></lb></s> <s>Quoniam igitur ſolidi <lb></lb>AM eſt centrum gra<lb></lb>uitatis H: reliqui au<lb></lb>tem dempta ABCD <lb></lb>portione centrum gra<lb></lb>uitatis K: & portionis <lb></lb>ABCD ponitur cen<lb></lb>trum grauitatis L; erit <lb></lb><figure id="id.043.01.245.1.jpg" xlink:href="043/01/245/1.jpg"></figure><lb></lb>vt portio ABCD ad reliquum ſolidi AM, ita ex con<lb></lb>traria parte KH ad HL. componendo igitur vt in antece<lb></lb>denti, & per conuerſionem rationis, & conuertendo, erit <lb></lb>vt portio ABCD ad ſolidum AM; hoc eſt vt rectangu<lb></lb>lum GFR, vnà cum duabus tertiis quadrati EF ad qua<lb></lb>dratum EG, ita HK ad KL. </s> <s>Quod demonſtrandum <lb></lb>erat. </s></p><pb xlink:href="043/01/246.jpg" pagenum="67"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ, vel ſphæroidis ab<lb></lb>ſciſſæ duobusplanis parallelis, neutro per cen<lb></lb>trum acto, nec centrum intercipientibus, centrum <lb></lb>grauitatis eſt in axe, primum bifariam ſecto: de<lb></lb>inde ſecundum centrum grauitatis reliqui dem<lb></lb>pta portione ex cylindro, vel portione cylindrica, <lb></lb>abſciſſo, vel abſciſſa vnà cum portione à cylin<lb></lb>dro, vel portione cylindrica ſphæræ, vel ſphæroi<lb></lb>di circa eius axem axi portionis congruentem cir<lb></lb>cumſcripta; in eo puncto, in quo dimidius axis <lb></lb>portionis maiorem baſim attingens ſic diuiditur, <lb></lb>vt pars prima & ſecunda ſectione terminata ſit ad <lb></lb>eam, quæ prima, & poſtrema ſectione terminatur, <lb></lb>vt duo rectangula, alterum contentum duobus <lb></lb>ſphæræ, vel ſphæroidis axis axi portionis <expan abbr="cõgruen">congruen</expan> <lb></lb>tis ijs ſegmentis, quæ fiunt à centro minoris baſis <lb></lb>portionis: alterum axe portionis, & ſegmento, <lb></lb>quod ſphæræ, vel ſphæroidis, & maioris baſis por<lb></lb>tionis centra iungit, vnà cum duabus tertiis qua<lb></lb>drati axis portionis, ad ſphæræ vel ſphæroidis di<lb></lb>midij axis quadratum. </s></p><p type="main"> <s>Sit ſphæræ, vel ſphæroidis, cuius centrum E portio <lb></lb>ABCD, abſciſſa duobus planis parallelis, neutro per E <lb></lb>tranſeunte, nec E intercipientibus: portionis autem axis <lb></lb>ſit FS: maior baſis circulus, vel ellipſis, cuius diame<pb xlink:href="043/01/247.jpg" pagenum="68"></pb>ter AD: & circa axim EF, ſtet cylindrus, vel portio cylin<lb></lb>drica MN abſciſſa ijſdem planis cum portione ABCD <lb></lb>ex cylindro, vel portione cylindrica, ſphæræ, vel ſphæroidi <lb></lb>BCR circa eius axim CFSR circumſcripta, cuius ſit cen <lb></lb>trum grauitatis H, ac propterea ſecta FS bifariam in pun <lb></lb>cto H. reliqui autem <lb></lb>dempta portione AB <lb></lb>CD ex ſolido MN ſit <lb></lb>centrum grauitatis K, <lb></lb>quod cadet in FH, & <lb></lb>portionis ABCD cen <lb></lb>trum grauitatis in ipſa <lb></lb>HS cadet, quod ſit L. <lb></lb></s> <s>Dico eſſe HK ad KL, <lb></lb>vt duo rectangula GF <lb></lb>R, FSE, vnà cum <lb></lb>duabus tertiis quadra<lb></lb>ti FS, ad quadratum <lb></lb>EG. </s> <s>Quoniam enim <lb></lb><figure id="id.043.01.247.1.jpg" xlink:href="043/01/247/1.jpg"></figure><lb></lb>ſimiliter vt ante oſtenderemus eſſe HK ad KL, vt eſt <lb></lb>portio ABCD ad ſolidum MN: ſed portio ABCD <lb></lb>ad ſolidum MN, eſt vt duo rectaugula GFR, ESF, vnà <lb></lb>cum duabus tertiis quadrati FS, ad quadratum EG; vt <lb></lb>igitur duo prædicta rectangula, vnà cum duabus tertiis <lb></lb>quadrati FS ad quadratum EG, ita erit HK ad KL. <lb></lb></s> <s>Quod erat demonſtrandum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXV.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis maioris portionis ſphæræ, vel ſphæroi<lb></lb>dis centrum grauitatis eſt in axe, primum bifa<lb></lb>riam ſecto: deinde ſecundum centrum grauitatis <lb></lb>reliqui dempta portione ex cylindro, vel portione <pb xlink:href="043/01/248.jpg" pagenum="69"></pb>cylindrica, abſciſſo, vel abſciſſa vnà cum portio<lb></lb>ne, à cylindro, vel portione cylindrica, ſphæræ, vel <lb></lb>ſphæroidi circa eius axim axi portionis <expan abbr="cõgruen-tem">congruen<lb></lb>tem</expan> circumſcripta; in eo puncto, in quo axis portio <lb></lb>nis ſic diuiditur, vt pars prima, & ſecunda ſectione <lb></lb>terminata ſit ad eam, quæ prima & poſtrema ſe<lb></lb>ctione terminatur, vt ſolidum rectangulum ex axe <lb></lb>portionis, & reliquo ſegmento axis ſphæræ, vel <lb></lb>ſphæroidis axi portionis congruentis, & eo, quod <lb></lb>ſphæræ, vel ſphæroidis, & baſis portionis centra <lb></lb>iungit, vnà cum binis tertijs duorum cuborum; & <lb></lb>eius, qui à ſphæræ, vel ſphæroidis axis fit dimi<lb></lb>dio: & eius, qui ab ea, quæ ſphæræ, vel ſphæroidis, <lb></lb>& baſis portionis centra iungit; ad ſolidum rectan <lb></lb>gulum, quod duobus ſphæræ, vel ſphæroidis præ<lb></lb>dicti axis dimidijs, & axe portionis continetur. </s></p><figure id="id.043.01.248.1.jpg" xlink:href="043/01/248/1.jpg"></figure><p type="main"> <s>Sit ſphæræ, vel ſphæ <lb></lb>roidis, cuius centrum <lb></lb>E maior portio ABC, <lb></lb>cuius axis BD, baſis <lb></lb>circulus, vel ellipſis, cu <lb></lb>ius diameter AC: & <lb></lb>circa axem BD ſtet <lb></lb>cylindrus, vel portio <lb></lb>cylindrica KL, abſciſ <lb></lb>ſa eodem plano cum <lb></lb>portione ABC, ex cy<lb></lb>lindro, vel portione cy <lb></lb>lindrica, ſphæræ, vel <lb></lb>ſphæroidi ABCR circa eius axim BDR circumſcripta, <pb xlink:href="043/01/249.jpg" pagenum="70"></pb>& ſecta BD bifariam in puncto H: deinde ſecundum G <lb></lb>in ipſa BH, centrum grauitatis reliqui dempta portione ex <lb></lb>ſolido KL, ſit portionis ABC in ipſa DH centrum gra<lb></lb>uitatis F, per vim XXXVII ſecundi. </s> <s>Dico eſſe HG ad GF, <lb></lb>vt ſolidum rectangulum ex BD, DR, DE vnà cum binis <lb></lb>tertiis duorum <expan abbr="cuborũ">cuborum</expan> <lb></lb>ex BE, ED, ad ſoli<lb></lb>dum rectangulum ex <lb></lb>BD, BE, ER. </s> <s>Simi <lb></lb>liter enim vt ſupra de<lb></lb>monſtrato eſſe vt HG <lb></lb>ad GF, ita portionem <lb></lb>ABC ad <expan abbr="ſolidũ">ſolidum</expan> KL; <lb></lb>quoniamportio ABC <lb></lb>ad ſolidum KL eſt vt <lb></lb>ſolidum ex BD, DR, <lb></lb>DE, vnà cum binis ter <lb></lb>tiis duorum <expan abbr="cuborũ">cuborum</expan> ex <lb></lb>BE, & ED, ad ſoli<lb></lb><figure id="id.043.01.249.1.jpg" xlink:href="043/01/249/1.jpg"></figure><lb></lb>dum ex BD, BE, ER; erit vt modo dicta antecedens <lb></lb>magnitudo ad dictam conſequentem, ita HG, ad GF. <lb></lb></s> <s>Quod demonſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis ſphæræ, vel ſphæroidis ab<lb></lb>ſciſſæ duobus planis parallelis centrum interci<lb></lb>pientibus, & ab eo non æqualiter diſtantibus, cen <lb></lb>trum grauitatis eſt in axe, primum bifariam ſecto: <lb></lb>deinde ſecundum <expan abbr="cẽtrum">centrum</expan> grauitatis reliqui dem<lb></lb>pta portione ex cylindro, vel portione cylindrica, <lb></lb>abſciſſo, vel abſciſſa vnà cum portione, à cylin-<pb xlink:href="043/01/250.jpg" pagenum="71"></pb>dro, vel portione cylindrica, ſphæræ, vel ſphæroi<lb></lb>di circa eius axim axi portionis congruentem cir<lb></lb>cumſcripta; in eopuncto, in quo maius ſegmen<lb></lb>tum axis portionis corum, quæ à centro fiunt ſic <lb></lb>diuiditur, vt pars prima & ſecunda ſectione termi <lb></lb>nata ſit ad eam, quæ prima, & poſtrema ſectione <lb></lb>terminatur, vt duo ſolida rectangula; & quod fit <lb></lb>ex duobus ſphæræ, vel ſphæroidis axis axi portio<lb></lb>nis congruentis ijs ſegmentis, quæ fiunt à centro <lb></lb>maioris baſis portionis, & ea, quæ maioris baſis <lb></lb>& ſphæræ, vel ſphæroidis centra iungit: & quod <lb></lb>ex ſphæræ, vel ſphæroidis eiuſdem axis ſegmentis <lb></lb>à centro minoris baſis factis, & ea, quæ minoris ba <lb></lb>ſis, & ſphæræ, vel ſphæroidis centra iungit, vnà <lb></lb>cum binis tertiis partibus duorum cuborum exijs <lb></lb>ſegmentis axis portionis, quæ à centro ſphæræ, <lb></lb>vel ſphæroidis fiunt; ad ſolidum <expan abbr="rectãgulum">rectangulum</expan> quod <lb></lb>duobus ſphæræ, vel ſphæroidis prædicti axis dimi <lb></lb>dijs, & axe portio<lb></lb>nis continetur. </s></p><figure id="id.043.01.250.1.jpg" xlink:href="043/01/250/1.jpg"></figure><p type="main"> <s>Sit ſphæræ, vel ſphæ <lb></lb>roidis, cuius centrum <lb></lb>E, portio ABCD, ab <lb></lb>ſciſſa duobus planis pa <lb></lb>rallelis centrum E in<lb></lb>tercipientibus, & ab eo <lb></lb>non æqualiter diſtan<lb></lb>tibus: axis autem por<lb></lb>tionis ſit GH: maior <pb xlink:href="043/01/251.jpg" pagenum="72"></pb>baſis circulus, vel cllipſis, cuius diameter AD. minor <expan abbr="autẽ">autem</expan>, <lb></lb>cuius diameter ABC: & circa axim GH, ſtet cylindrus, <lb></lb>vel portio cylindrica NO, abſciſſa ijſdem planis cum por<lb></lb>tione ABCD, ex cylindro, vel portione cylindrica ſphæ<lb></lb>ræ, vel ſphæroidi BCR circa axim FGHR circumſcri<lb></lb>pta, cuius ſit centrum grauitatis K, ſectio ſcilicet bipartiti <lb></lb>axis GH: reliqui autem ex ſolido NO dempta portione, <lb></lb>ſit centrum grauitatis L, nempe in axis GH ſegmento <lb></lb>GK, quod minorem <lb></lb>portionis baſim attln<lb></lb>git: portionis autem <lb></lb>ABCD ſit centrum <lb></lb>grauitatis M: quod qui <lb></lb>dem in reliquo ſeg<lb></lb>mento KH cadet. <lb></lb></s> <s>Dico eſſe KL ad LM, <lb></lb>vt duo ſolida rectan<lb></lb>gula ex FH, HR, EH, <lb></lb>& ex RG, GF, GK, <lb></lb>vnà cum binis tertiis <lb></lb>duorum cuborum ex <lb></lb>EG, EH; ad ſolidum <lb></lb><figure id="id.043.01.251.1.jpg" xlink:href="043/01/251/1.jpg"></figure><lb></lb>rectangulum ex GH, EF, ER. </s> <s>Similiter enim vt ſupra <lb></lb>demonſtrato eſſe vt KL ad LM, ita portionem ABCD <lb></lb>ad ſolidum NO; quoniam portio ABCD ad ſolidum <lb></lb>NO, eſt vt duo ſolida rectangula ex GH, HR, EH, & <lb></lb>ex RG, GF, EG, vnà cum binis tertiis duorum cubo<lb></lb>rum ex EH, EG ad ſolidum ex GH, EF, ER, erit <lb></lb>vt totum iam dictum antecedens ad dictum conſequens, <lb></lb>ita KL ad LM. </s> <s>Quod demonſtrandum erat. </s></p><pb xlink:href="043/01/252.jpg" pagenum="73"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis portionis conoidis parabolici centrum <lb></lb>grauitatis eſt punctum illud, in quo axis ſic diui<lb></lb>ditur, vt pars quæ ad verticem ſit eius, quæ ad ba<lb></lb>ſim dupla. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXVIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis fruſti portionis conoidis parabolici cen <lb></lb>trum grauitatis eſt punctum illud, in quo axis ſic <lb></lb>diuiditur, vt pars minorem baſim attingens ſit ad <lb></lb>reliquam, vt duplum maioris baſis vnà cum mino<lb></lb>ri, ad duplum minoris, vnà cum maiori. </s></p><p type="main"> <s>Harum proportionum vtriuſque non alia demonſtratio <lb></lb>eſt ab ea, quam in ſecundo ſcripſimus de centro grauitatis <lb></lb>conoidis parabolici, & eius fruſti: propterea quod omnis por <lb></lb>tionis conoidis parabolici, ſicut & hyperbolici ſectio baſi <lb></lb>parallela ellipſis eſt ſimilis baſi. </s> <s>Ex corollario xv. </s> <s>de conoi<lb></lb>dibus, & ſphæroidibus Archimedis. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO XXXIX.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis conoidis hyperbolici, vel portionis hy<lb></lb>perbolici conoidis centrum grauitatis, eſt pun<lb></lb>ctum illud, in quo duodecima pars axis ordine <lb></lb>quarta ab ea, quæ baſim attingit, ſic diuiditur, vt <lb></lb>pars propinquior baſi ſit ad reliquam vt ſeſquial<pb xlink:href="043/01/253.jpg" pagenum="74"></pb>tera tranſuerſi lateris, hyperboles per axem, ad <lb></lb>axem conoidis. </s></p><figure id="id.043.01.253.1.jpg" xlink:href="043/01/253/1.jpg"></figure><p type="main"> <s>Sit conoides hyperbolicum, vel portio conoidis hyper<lb></lb>bolici ABC, cuius axis BD, qui in portione non erit ad ba<lb></lb>ſim perpendicularij: baſis autem dicti conoidis, vel portio<lb></lb>nis ſit circulus, vel ellipſis, cuius diameter ADC: & hyper<lb></lb>boles ABC, quæ vel conoides deſcribit, vel eſt ſectio tan<lb></lb>tummodo per axem, cuius tranſuerſum latus ſit BE, & <pb xlink:href="043/01/254.jpg" pagenum="75"></pb>huius ſeſquialtera BEF: & ſumpta axis BD quarta par<lb></lb>te DF, & tertia DG: qua ratione erit FG duodecima <lb></lb>pars axis BD quarta ab ea, cuius terminus D; fiat vt <lb></lb>IB ad BD, ita FH ad HG. </s> <s>Dico conoidis, vel portio<lb></lb>nis ABC centrum grauitatis eſſe H. </s> <s>Nam vt eſt EB <lb></lb>ad BD ita fiat DK ad KA: & ponatur KDY ſeſqui<lb></lb>altera ipſius DK, & ex AK abſcindatur KM ſubſeſ<lb></lb>quialtera ipſius AK: & ipſis DK DM, DA, æquales <lb></lb>eodem ordine abſcindantur DL, DN, DC: & deſcri<lb></lb>bantur triangula, KBL, MBN: & per puncta ABC <lb></lb>vertice communi B, tranſeant duæ ſectiones parabolæ <lb></lb>AOB, & BPC, ita vt contingat recta BK parabolam <lb></lb>AOB, recta autem BL parabolam BPC; ſit autem <lb></lb>AKLC, parabolarum diametris parallela,. Deinde <lb></lb>ſecto axe BD bifariam, & ſingulis eius partibus rurſus bi<lb></lb>fariam in quotlibet partes æquales, ſint ex illis duæ <lb></lb>partes DQ, QF: & per puncta QF planis quibuſdam <lb></lb>baſi parallelis ſecentur vnà ſolidum & hyperbole ABC: <lb></lb>ſintque hyperboles ſectiones, quæ continent ſectiones trian <lb></lb>gulorum ABC mixti, & rectilinei KBL, rectæ RTX <lb></lb>ZVS: <foreign lang="grc">αγεζδβ. </foreign></s> <s>ſolidi autem ABC ſectiones erunt cir<lb></lb>culi, vel ellipſes ſimiles baſi circa diametros RS, <foreign lang="grc">αβ</foreign>. <lb></lb></s> <s>Quoniam igitur eſt vt <foreign lang="grc">Υ</foreign>K ad KD, ita AK ad KM; <lb></lb>vtrobique enim eſt proportio ſeſquialtera: erit permutan<lb></lb>do vt YK ad A<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, hoc eſt vt IB ad BD, vel FH, ad <lb></lb>HG, ita D<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>K<emph.end type="italics"></emph.end>M, hoc eſt triangulum BDK ad <lb></lb>triangulum BKM, hoc eſt ad æquale huic ex demon<lb></lb>ſtratis triangulum A<emph type="italics"></emph>K<emph.end type="italics"></emph.end>B mixtum: hoc eſt in duplis ita, <lb></lb>triangulum BKL ad duo mixta rriangula AKB, BLC <lb></lb>ſimul. </s> <s>ſed duorum triangulorum AKB, BLC ſimul eſt <lb></lb>centrum grauitatis F, vt in hoc tertio libro demonſtra<lb></lb>uimus: trianguli autem BKL, vt in primo, centrum gra<lb></lb>uitatis G; totius igitur trianguli ABC centrum graui<lb></lb>tatis erit H. </s> <s>Rurſus quoniam eſt vt BD ad BQ hoc <pb xlink:href="043/01/255.jpg" pagenum="76"></pb>eſt vt rectangulum EBD ad rectangulum EBQ, ita <lb></lb>DK ad QX: & vt quadratum BK ad quadratum BX, <lb></lb>hoc eſt vt quadratum BD ad quadratum BQ, ita eſt <lb></lb>A<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ad TX; erunt octo magnitudines quaternæ propor<lb></lb><figure id="id.043.01.255.1.jpg" xlink:href="043/01/255/1.jpg"></figure><lb></lb>tionales; ſed & earum primæ, & tertiæ ſunt proportiona<lb></lb>les; nam eſt vt EB ad BD, hoc eſt vt rectangulum EBD <lb></lb>prima in primis ad quadratum BD primam in ſecundis, <lb></lb>ita D<emph type="italics"></emph>K<emph.end type="italics"></emph.end> tertia in primis ad AK tertiam in ſecundis; vt <pb xlink:href="043/01/256.jpg" pagenum="77"></pb>igitur compoſita ex primis vtriuſque ordinis ad compo<lb></lb>ſitam ex ſecundis, ita erit compoſita ex tertiis ad com<lb></lb>poſitam ex quartis; videlicet vt rectangulum BDE, quod <lb></lb>æquale eſt rectangulo EBD vna cum quadrato BD, ad <lb></lb>rectangulum BQE, quod æquale eſt rectangulo EBQ <lb></lb>vnà cum quadrato BQ, ita erit tota AD ad totam TQ. <lb></lb>Sed vt rectangulum BDE ad rectangulum BQE ita eſt <lb></lb>AD quadratum, ad quadratum RQ, hoc eſt ita circu<lb></lb>lus, vel ellipſis circa AC, ad circulum, vel ſimilem illi <lb></lb>ellipſem circa RS; vt igitur AD ad TQ, hoc eſt in ea<lb></lb>rum duplis vt AC ad TV, ita erit circulus, vel ellipſis <lb></lb>circa AC ad circulum, vel ellipſem circa RS. </s> <s>Similiter <lb></lb>oſtenderemus eſſe vt AC ad <foreign lang="grc">γδ</foreign>, ita circulnm, vel elli<lb></lb>pſim circa AC, ad circulum, vel ellipſem, circa <foreign lang="grc">αβ</foreign>: con<lb></lb>uertendo igitur, & ex æquali erunt binæ in eadem propor<lb></lb>tione, vt <foreign lang="grc">γδ</foreign> ad TV, ita circulus, vel ellipſis circa <foreign lang="grc">αβ</foreign><lb></lb>ad circulum, vel ellipſim circa RS: & vt TV ad AC, ita <lb></lb>circulus, vel ellipſis circa RS ad circulum, vel ellipſim <lb></lb>circa AC. Rurſus, quoniam tres rectæ lineæ incipienti <lb></lb>à minima <foreign lang="grc">γε</foreign>, TX, A<emph type="italics"></emph>K<emph.end type="italics"></emph.end> ſunt binæ ſumptæ proportio<lb></lb>nales quadratis ex B<foreign lang="grc">ε</foreign>, BX, B<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, hoc eſt quadratis ex <lb></lb>F<foreign lang="grc">ε</foreign>, QX, DK; duplicata erit proportio <foreign lang="grc">γε</foreign> ad TX ip<lb></lb>ſius F<foreign lang="grc">ε</foreign> ad QX, & TX ad AK duplicata ipſius QX ad <lb></lb>D<emph type="italics"></emph>K<emph.end type="italics"></emph.end>: ſed rectæ F<foreign lang="grc">ε</foreign>, QX, DK, ſeſe æqualiter excedunt, <lb></lb>vtpote proportionales ipſis BF, BQ, BD, propter ſi<lb></lb>militudinem triangulorum; minor igitur proportio erit <lb></lb><foreign lang="grc">γ</foreign>F ad TQ, quàm TQ ad AD: quare his proportiona<lb></lb>lium minor erit proportio circuli, vel ellipſis circa <foreign lang="grc">αβ</foreign> ad <lb></lb>circulum, vel cllipſim circa RS, quàm circuli, vel elli<lb></lb>pſis circa RS, ad circulum, vel ellipſim, circa AC. <lb></lb></s> <s>Similiter quæcumque ſectiones per prædicta axis, vel dia<lb></lb>metri BD puncta ſectionum fierent vt dictum eſt ad ver<lb></lb>ticem retrocedenti oſtenderentur quælibet ternæ inter ſe <lb></lb>proximæ, binæque ſumptæ vtriuſque ordinis proportio-<pb xlink:href="043/01/257.jpg" pagenum="78"></pb>nales eſſe, & minor proportio vtrobique minimæ ad me<lb></lb>diam quàm mediæ ad maximam; per XXXII igitur ſe<lb></lb>cundi, triangulum mixtum, & ſolidum ABC, in huius <lb></lb>axe illius autem diametro BD commune habebunt cen<lb></lb><figure id="id.043.01.257.1.jpg" xlink:href="043/01/257/1.jpg"></figure><lb></lb>trum grauitatis. </s> <s>ſed demonſtrauimus H centrum grauita<lb></lb>tis trianguli ABC; conoidis igitur vel portionis ABC <lb></lb>centrum grauitatis erit idem H. </s> <s>Quod demonſtrandum <lb></lb>erat. </s></p><pb xlink:href="043/01/258.jpg" pagenum="79"></pb><p type="main"> <s>Et hic huius tertij Libri finis eſſet; niſi ſecundo iam im<lb></lb>preſſo, alia quædam via magis naturalis me ad conoidis hy <lb></lb>perbolici centrum grauitatis reduxiſſet. </s> <s>Ea igitur in ſecun<lb></lb>dum librum aliàs inſerenda, nunc in ſequenti appendice <lb></lb>ſeptem propoſitionibus expoſita, per ſectionem prædicti <lb></lb>conoidis in conoides parabolicum eodem vertice, & circa <lb></lb>eundem axim, & reliquam figuram ſolidam, abſque com<lb></lb>poſito ex duabus figuris circumſcriptis, quæ ex cylindris <lb></lb>componuntur, propoſitum concludat. </s></p><p type="head"> <s>APPENDIX.</s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO I.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si ſint octo magnitudines quaternæ <lb></lb>totæ, & ablatæ proportionales, fue<lb></lb>rint autem, & primarum vtriuſque <lb></lb>ordinis ablatæ ad reliquas propor<lb></lb>tionales; erunt vtriuſque ordinis re <lb></lb>liquæ proportionales. </s></p><figure id="id.043.01.258.1.jpg" xlink:href="043/01/258/1.jpg"></figure><p type="main"> <s>Sint octo magnitudines quaternæ <lb></lb>proportionales, ac primi quidem ordi<lb></lb>nis totæ, vt AB ad CD, ita EF ad <lb></lb>GH: ſecundi autem ordinis ablatæ, vt <lb></lb>B ad D, ita F ad H: ſit autem vt B <lb></lb>ad A ita F ad E. </s> <s>Dico & reliquas <lb></lb>eſſe proportionales, videlicet vt A ad <lb></lb>C, ita E ad G. </s> <s>Quoniam enim com <lb></lb>ponendo, & conuertendo eſt vt A ad <lb></lb>AB, ita E ad EF: ſed vt AB ad <pb xlink:href="043/01/259.jpg" pagenum="80"></pb>CD, ita eſt EF ad GH; erit ex æquali vt A ad CD, <lb></lb>ad E ad GH: & conuertendo vt <lb></lb>CD ad A, ita GH ad E: & per<lb></lb>mutando CD ad GH, ita A ad E. <lb></lb></s> <s>Rurſus quoniam eſt vt A ad B ita <lb></lb>E ad F: & vt B ad D, ita F ad H; <lb></lb>erit ex æquali, vt A ad D ita E ad <lb></lb>H: ſed vt CD ad A, ita erat GH <lb></lb>ad E; ex æquali igitur erit vt CD ad <lb></lb>D ita GH ad H: & permutando vt <lb></lb>CD ad GH, ita D ad H, & reli<lb></lb>qua C ad reliquam G: ſed vt CD <lb></lb>ad GH ita erat A ad E; vt igitur <lb></lb>A ad C ita erit E ad G. </s> <s>Quod demonſtrandum erat. </s></p><figure id="id.043.01.259.1.jpg" xlink:href="043/01/259/1.jpg"></figure><p type="head"> <s><emph type="italics"></emph>PROPOSITIO II.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si circa datæ hyperboles communem diame<lb></lb>trum parabola deſcripta illius baſim ita diuidat, <lb></lb>vt quadratum dimidiæ baſis parabole ad reli<lb></lb>quum quadrati dimidiæ baſis hyperboles eam <lb></lb>habeat proportionem, quam tranſuerſum latus <lb></lb>ad diametrum hyperboles; omnes in hyperbole <lb></lb>ad diametrum ordinatim applicatas ita ſecabit, <lb></lb>vt exceſſus, quibus quadrata in hyperbole appli<lb></lb>catàrum ſuperant quadrata in parabola ex ſectio<lb></lb>ne applicatarum, inter ſe ſint vt quadrata diame<lb></lb>tri partium inter applicatas, & verticem inter<lb></lb>iectarum. </s></p><p type="main"> <s>Eſto hyperbole ABC, cuius diameter BD, tranſuer-<pb xlink:href="043/01/260.jpg" pagenum="81"></pb>uerſum latus EB. & poſitis in ipſa, BD duobus pun<lb></lb>ctis quibuslibet GH, ordinatim applicentur MG, NH: <lb></lb>& circa diametrum BD ſit deſcripta parabola KBL tali<lb></lb>ter vt ipſius dimidiæ baſis DK quadratum ad reliquum <lb></lb>quadrati AD, ſit vt EB ad BD, & rectas MH, NG <lb></lb>in infinitum productas ſecet parabola KBL in punctis <lb></lb>OP. </s> <s>Dico puncta OP intra hyperbolem cadere: & reli<lb></lb>quum quadrati MG dempto quadrato GO ad reliquum <lb></lb>quadrati NH dempto quadrato PH, eſſe vt quadratum <lb></lb>BG ad quadratum <lb></lb>BH. </s> <s>Quoniam enim <lb></lb>ponitur vt EB ad B <lb></lb>D, hoc eſt vt rectan<lb></lb>gulum EBD ad qua<lb></lb>dratum BD, ita qua<lb></lb>dratum DK ad reli<lb></lb>quum quadrati AD, <lb></lb>erit componendo, & <lb></lb>conueniendo, vt <expan abbr="rectã">rectam</expan> <lb></lb>gulum BDE ad re<lb></lb>ctangulum EBD, ita <lb></lb>quadratum AD ad <lb></lb>quadratum DK: ſed <lb></lb>vt rectangulum BGE <lb></lb>ad <expan abbr="rectãgulum">rectangulum</expan> BDE, <lb></lb><figure id="id.043.01.260.1.jpg" xlink:href="043/01/260/1.jpg"></figure><lb></lb>ita eſt quadratum MG ad quadratum AD; ex æquali <lb></lb>igitur, vt rectangulum BGE ad rectangulum EBD, ita <lb></lb>eſt quadratum MG ad quadratum DK: ſed vt rectan<lb></lb>gulum EBD ad rectangulum EBG, ita eſt quadratum <lb></lb>DK ad GO quadratum; ex æquali igitur vt rectangu<lb></lb>lu m BGE ad rectangulum EBG, ita erit quadratum <lb></lb>MG ad quadratum GO: ſed rectangulum BGE maius <lb></lb>eſt totum parte rectangulo EBG; quadratum igitur MG <lb></lb>quadrato GO maius erit, & recta MG maior quàm <pb xlink:href="043/01/261.jpg" pagenum="82"></pb>GO: ſecat igitur parabola KBL rectam MG in puncto <lb></lb>O. </s> <s>Similiter oſtenderemus eandem parabolam ſecare <lb></lb>quamcumque aliam in hyperbole ABC ordinatim ad dia <lb></lb>metrum applicatarum. </s> <s>Quoniam igitur ſunt octo magni <lb></lb>tudines quaternæ totæ, & ablatæ proportionales; ac pri<lb></lb>mi quidem ordinis, vt rectangulum BDE ad rectangu<lb></lb>lum BGE, ita quadratum AD ad quadratum MG: ſe<lb></lb>cundi autem ordinis, vt rectangulum EBD ad rectangu<lb></lb>lum EBG ita quadra <lb></lb>tum DK ad quadra<lb></lb>tum OGD: ſed vt <lb></lb>EB ad BD, hoc eſt <lb></lb>vt ablata primæ in pri <lb></lb>mis rectangulum EB <lb></lb>D ad reliquum BD <lb></lb>quadratum, ita poni<lb></lb>tur ablata primæ in ſe <lb></lb>cundis, quadratum D <lb></lb>K ad reliquum exceſ <lb></lb>ſum, quo quadratum <lb></lb>AD ſuperat quadra<lb></lb>tum DK; vt igitur eſt <lb></lb>reliqua primæ ad reli<lb></lb>quam ſecundæ in pri<lb></lb><figure id="id.043.01.261.1.jpg" xlink:href="043/01/261/1.jpg"></figure><lb></lb>mis, ita erit in ſecundis; videlicet vt quadratum BD ad <lb></lb>quadratum BG, ita reliquum quadrati AD dempto qua<lb></lb>drato DK, ad reliquum qua rati MG dempto quadra<lb></lb>to GO. </s> <s>Similiter oſtenderemus reliquum quadrati AD <lb></lb>dempto quadrato DK ad reliquum quadrati NH dem<lb></lb>pto quadrato PH, eſſe vt quadratum BD ad quadra<lb></lb>tum BH; conuertendo igitur, & ex æquali erit vt qua<lb></lb>dratum BG ad quadratum BH, ita reliquum quadra <lb></lb>ti MG dempto quadrato GO, ad reliquum quadrati<pb xlink:href="043/01/262.jpg" pagenum="83"></pb>NH dempto quadrato PH. </s> <s>Quod demonſtrandum <lb></lb>erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO III.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omne conoides hyperbolicum diuiditur in <lb></lb>conoides parabolicum circa eundem axim, & re<lb></lb>liquam figuram quandam, ad quam conoides pa<lb></lb>rabolicum eam habet proportionem, quamſeſqui <lb></lb>altera tranſuerſi lateris hyperboles, quæ conoides <lb></lb>deſcribit, ad axem conoidis. </s></p><figure id="id.043.01.262.1.jpg" xlink:href="043/01/262/1.jpg"></figure><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius axis BD: hy<lb></lb>perboles autem, quæ conoides deſcribit tranſuerſum latus <lb></lb>EB, cuius ſit ſeſquialtera BEF: & abſciſſa DG, ita vt <lb></lb>quadratum ex ipſa ad reliquum quadrati AD ſit vt EB <lb></lb>ad BD, vertice B circa diametrum BD deſcripta ſit <pb xlink:href="043/01/263.jpg" pagenum="84"></pb>parabola GBH, eaque circumducta conoides GBH, <lb></lb>Dico conoides GBH comprehendi à conoide ABC & <lb></lb>eſſe ad illius reliquum, vt FB ad BD. </s> <s>Abſciſſa enim <lb></lb>DK ita potentia ſit ad DG, vt DB ad BE longitudine, <lb></lb>circa axim BD deſcribatur conus KBL: & ſecta BD in <lb></lb>multas partes æquales, ductoſque per ea puncta planis <lb></lb>quibuſdam baſi parallelis, ſecentur tria dicta ſolida, conus <lb></lb>ſcilicet & vtrumque conoides: & ſuper ſectiones circulos <lb></lb>deſcribantur cylindri æqualium altitudinum terni cuca <lb></lb><figure id="id.043.01.263.1.jpg" xlink:href="043/01/263/1.jpg"></figure><lb></lb>communes axes partes æquales, in quas axis BD diuiſus <lb></lb>fuit, & inter eadem plana parallela: & omnino triplex figura <lb></lb>ex cylindris, quos diximus ſit tribus dictis ſolidis circumſcri <lb></lb>pta: ſintque circa duos axes infimos DM, MN terni cylin<lb></lb>dri AO, GP, KQ: & proxime ordine ipſis reſpondentes <lb></lb>cylindri TX, SV, RZ, quorum baſes circa diametros <lb></lb>TI, S<foreign lang="grc">β</foreign>, R<foreign lang="grc">α</foreign>, communes ſectiones plani per punctum M, <lb></lb>cum tribus ſolidorum ſectionibus per axem, triangulo ſcili<lb></lb>cet, parabola, & hyperbole in eodem plano, atque ideo tres <pb xlink:href="043/01/264.jpg" pagenum="85"></pb>diametri TI, S<foreign lang="grc">β</foreign>, R<foreign lang="grc">α</foreign>, erunt in vna recta linea. </s> <s>Quoniam <lb></lb>igitur eſt vt EB ad BD, ita quadratum DG ad <expan abbr="reliquũ">reliquum</expan> <lb></lb>quadrati AD, ſecabit parabola GBH omnes in hyperbo<lb></lb>le ABC ad diametrum ordinatim applicatas, quare conoi <lb></lb>des ABC comprehendet conoides GBH: atque ita para<lb></lb>bola ſecabit, vt exceſſus quibus quadrata in hyperbole ap<lb></lb>plicatarum ſuperant partes quadrata in parabola applicata <lb></lb>rum, inter ſe ſint vt quadrata partium diametri BD inter <lb></lb>applicatas & verticem interiectarum, prout vt inter ſe <expan abbr="reſpõ">reſpom</expan> <lb></lb>dent: vt igitur eſt quadratum BD ad quadratum BM, hoc <lb></lb>eſt vt quadratum DK ad quadratum RM, ita erit <expan abbr="reliquũ">reliquum</expan> <lb></lb>AD quadrati dempto quadrato DG ad reliquum quadrati <lb></lb>TM dempto quadrato SM, & permutando. </s> <s>Sed quia qua<lb></lb>dratum DG ad reliquum quadrati AD, & ad quadratum <lb></lb>DK eandem habet proportionem ex vi conſtructionis, reli <lb></lb>quum quadrati AD, dempto quadrato DG æquale eſt <lb></lb>quadrato DK; reliquum igitur quadrati TM dempto qua <lb></lb>drato SM æquale erit quadrato RM: ſi igitur vtriſque ad<lb></lb>dantur ſingula communia, vnis quadratum DG, alteris <lb></lb>quadratum SM, erit & quadratum AD æquale duobus <lb></lb>quadratis GD, DK, & quadratum TM duobus quadra <lb></lb>tis SM, MR æquale. </s> <s>ſed cum cylindri eiuidem altitudi<lb></lb>nis inter ſe ſint vt baſes, ſunt vt quadrata, quæ ab eorundem <lb></lb>baſium ſemidiametris fiunt; cylindiusigitur AO æqualis <lb></lb>eſt duobus cylindris GP, KQ: & cylindrus TX duobus <lb></lb>cylindris S<foreign lang="grc">Υ</foreign>, RZ æqualis. </s> <s>Eadem ratio eſt de reliquis <lb></lb>deinceps. </s> <s>Tota igitur figura conoidi ABC circumſcripta, <lb></lb>vtrique ſimul, conoidi GBH, & cono KBL circumſcri<lb></lb>ptæ æqualis erit. </s> <s>poſſunt autem eæ figuræ ita eſſe dictis ſoli<lb></lb>dis circumſcriptæ per ea quæ alibi oſtendimus, vt ſuperent <lb></lb>inſcriptas minori ſpacio quantacumque magnitudine pro<lb></lb>poſita; per tertiam igitur ſecundi, conoides ABC vtrique <lb></lb>ſimul, conoidi GBH, & cono KBL æquale erit. </s> <s>dempto <lb></lb>igitur <expan abbr="cõmuni">communi</expan> conoide GBH, reliquum <expan abbr="ſolidũ">ſolidum</expan> AGBHC <pb xlink:href="043/01/265.jpg" pagenum="86"></pb>æquale erit cono KBL. </s> <s>Rurſus quia eſt vt EB ad BD, ita <lb></lb>quadratum GD ad quadratum DK, hoc eſt circulus cir<lb></lb>ca GH ad circulum circa KL, hoc eſt conus GBH ſi <lb></lb>deſcribatur ad conum KBL: ſed vt FB ad BE ita eſt co<lb></lb>noides GBH ad conum GBH; ex æquali igitur erit vt <lb></lb>FB ad BD, ita conoides GBH ad conum KBL, hoc <lb></lb>eſt ad ſolidum AGBHC. </s> <s>Manifeſtum eſt igitur <expan abbr="propoſitũ">propoſitum</expan>. </s></p><p type="head"> <s><emph type="italics"></emph>COROLLARIVM.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Ex huius Theorematis demonſtratione manife <lb></lb>ſtum eſt, ijſdem poſitis cylindros deficientes, ex <lb></lb>quibus conſtat exceſſus, quo figura conoidi hyper <lb></lb>bolico circumſcripta ſuperat circumſcriptam co<lb></lb>noidi parabolico, ita ſe habere, vt quorumlibet <lb></lb>trium inter ſe proximorum minor proportio ſit <lb></lb>minimi ad medium, quam medij ad maximum: <lb></lb>æquales enim ſunt ſinguli ſingulis cylindris, ex <lb></lb>quibus conſtat figura cono BKL circumſcripta, <lb></lb>qui ſunt inter eadem plana parallela. </s> <s>Quod ſi <lb></lb>ita eſt, ſimul illud manifeſtum erit, & ex hoc, & <lb></lb>ex ijs, quæ in ſecundo libro demonſtrauimus; præ<lb></lb>dictum exceſſum ex tot cylindris deficientibus <lb></lb>eiuſdem altitudinis, quos diximus componi poſſe, <lb></lb>vt ipſius centrum grauitatis in axe BD diſtet à <lb></lb>centro grauitatis coni KBL, hoc eſt à puncto in <lb></lb>quo axis BD ſic diuiditur, vt pars, quæ ad ver<lb></lb>ticem ſit reliquæ tripla, ea diſtantia, quæ minor <lb></lb>ſit quantacum que longitudine propoſita. </s></p><pb xlink:href="043/01/266.jpg" pagenum="87"></pb><p type="head"> <s><emph type="italics"></emph>PROPOSITIO IIII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Si conoidi parabolico figura circumſcribatur, <lb></lb>& altera inſcribatur ex cylindris æqualium alti<lb></lb>tudinum, binis circa communes axes ſegmenta <lb></lb>axis conoidis, & inter eadem plana parallela, mi<lb></lb>nimo circumſcriptorum ad nullum relato; omnia <lb></lb>reſidua cylindrorum figuræ circumſcriptæ dem<lb></lb>ptis figuræ inſcriptæ cylindris, & inter ſe, & mi<lb></lb>nimo cylindro æqualia erunt. </s></p><p type="main"> <s>Sit conoidi parabolico ABC, cuius axis BD circum<lb></lb>ſcripta figura ex quotcumque cylindris æqualium altitu<lb></lb>dinum, quorum tres deinceps ſint EL minimus ſupremus, <lb></lb>& GQ, IR, quorum baſes eodem ordine circuli, quorum <lb></lb>ſemidiametri ad parabolæ, quæ figuram deſcribit diame<lb></lb>trum BD ordi<lb></lb>natim applicatæ <lb></lb>ſint EF, GH, IK: <lb></lb>& in duplos cre<lb></lb>ſcentibus cylin<lb></lb>dris circa <expan abbr="priorũ">priorum</expan> <lb></lb>axium duplos a<lb></lb>xes BH, IK, HD, <lb></lb>& <gap></gap>c deinceps <lb></lb>quotcumque plu<lb></lb>res eſsent; ſit co<lb></lb>noidi ABC in<lb></lb><figure id="id.043.01.266.1.jpg" xlink:href="043/01/266/1.jpg"></figure><lb></lb>ſcripta figura ex cylindris æqualium altitudinum inter ſe, & <lb></lb>circumſcriptis. </s> <s>Bini itaque circa communes axes inter ea<lb></lb>dem plana parallela interijcientur, minimo EL ad nullum <pb xlink:href="043/01/267.jpg" pagenum="88"></pb>relato: huic autem proximus, & æqualis cylindrorum in<lb></lb>ſcriptorum ſit NM baſim ipſi communem habens circu<lb></lb>lum circa EFM: & conſequenti circumſcriptorum GQ <lb></lb>ſit. </s> <s>inſcriptorum æqualis PO baſim habens ipſi commu<lb></lb>nem circulum circa GHO: ſint autem circulorum qui <lb></lb>ſunt baſes cylindrorum diametri in parabola per axim: <lb></lb>quæ quoniam ſunt communes ſectiones cum parabola per <lb></lb>axim planorum baſi conoidis, & inter ſe parallelorum, <lb></lb>erunt etiam ipſæ inter ſe, & parabolæ baſi AC parallelæ, <lb></lb>earumque dimidiæ vt EF, GH ad diametrum BD or<lb></lb>dinatim applicatæ. </s> <s>Quoniam igitur in parabola ABC <lb></lb>eſt vt HB ad BF ita quadratum GH ad quadratum <lb></lb>EF, duplum erit <lb></lb>quadratum GH <lb></lb>quadrati EF: qua <lb></lb>re & circulus cir<lb></lb>ca GO circuli <lb></lb>circa EM at que <lb></lb>adeo cylindrus <lb></lb>GQ cylindri E <lb></lb>L duplus, pro<lb></lb>pter <17>qualitatem <lb></lb>altitudinum: ſed <lb></lb>& cylindrus NL <lb></lb><figure id="id.043.01.267.1.jpg" xlink:href="043/01/267/1.jpg"></figure><lb></lb>duplus eſt cylindri EL per conſtructionem; cylindrus igi<lb></lb>tur GQ æqualis eſt cylindro NL: & ablato communi <lb></lb>NM cylindro, reliquus GQ deficiens cylindro NM <lb></lb>cylindro EL æqualis. </s> <s>Rurſus quia eſt vt KB ad BH, <lb></lb>ita quadratum IK ad quadratum GH, hoc eſt ita IR <lb></lb>cylindrus ad cylindrum GQ: ſed vt HB ad BF ita <lb></lb>erat cylindrus GQ ad cylindrum EL; tres igitur cy<lb></lb>lindri IR, GQ, EL, tribus lineis BK, BH, BF, eodem <lb></lb>ordine proportionales erunt: ſed tres eædem lineæ ſeſe <lb></lb>æqualiter excedunt; tres igitur dicti cylindri ſeſe æqua-<pb xlink:href="043/01/268.jpg" pagenum="89"></pb>liter excedent, hoc eſt reliquum cylindri IR dempto cylin<lb></lb>dro PO æquale erit reliquo cylindri GQ dempto cylin<lb></lb>dro NM, & reliquum cylindri GQ dempto cylindro <lb></lb>NM æquale cylindro EL. </s> <s>Similiter ad reliquos cylindros <lb></lb>quotcumque plures eſſent deſcendentes oſtenderemus, om <lb></lb>nes exceſſus, quibus cylindri circumſcripti inſcriptos <lb></lb>ſuperant ſibi quique reſpondentes inter ſe & cylindro <lb></lb>EL æquales eſſe. </s> <s>Manifeſtum eſt igitur propoſitum. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO V.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Dato conoide hyperbolico, & ipſius conoi<lb></lb>de parabolico circa eundem axim, quod ad <lb></lb>reliquum hyperbolici conoidis eam proportio<lb></lb>nem habeat, quam ſeſquialtera tranſuerſi late<lb></lb>ris hyperboles, quæ conoides deſcribit, ad axim <lb></lb>conoidis; fieri poteſt vt conoidi parabolico fi<lb></lb>guræ quædam inſcribatur, & altera circumſcri<lb></lb>bantur vt ſupra factum eſt, & hyperbolico alio cir<lb></lb>cumſcribatur omnes ex cylindris æqualium al<lb></lb>titudinum multitudine æqualibus exiſtentibus <lb></lb>ijs, ex quibus conſtant figuræ conoidibus cir<lb></lb>cumſcriptæ, ita vt exceſſus, quo figura conoidi <lb></lb>parabolico circumſcripta inſcriptam ſuperat, <lb></lb>quem breuitatis cauſa voco exceſſum primum, <lb></lb>ad exceſſum, quo figura conoidi hyperbolico cir<lb></lb>cumſcripta ſuperat circumſcriptam parabolico, <lb></lb>quem voco exceſſum ſecundum, minorem habeat <lb></lb>proportionem quacumque propoſita. </s></p><pb xlink:href="043/01/269.jpg" pagenum="90"></pb><p type="main"> <s>Sit conoides hyperbolicum ABC, & pars eius para<lb></lb>bolicum EBF circa eundem axim BD: & conoides <lb></lb>EBF ad reliquum conoidis ABC eam habeat proportio<lb></lb>nem, quam ſeſquialtera tranſuerſi lateris hyperboles per <lb></lb>axim ABC ad axim BD. </s> <s>Dico fieri poſſe quod proponitur. <lb></lb></s> <s>Habeat enim DL ad LB quamcumque proportionem: & <lb></lb>conoides ABC reliquo ſolido AEBFC dempto conoi <lb></lb>de EBF. ſit conus circa axim BD æqualis GBH: & <lb></lb>deſcribatur conus GLH: & ſecta BD bifariam in pun<lb></lb>cto K, & rurſus BK, KD in multitudine, & longitudi<lb></lb>ne æquales inſcribatur conoidi EBF, & altera cirumſcri<lb></lb><figure id="id.043.01.269.1.jpg" xlink:href="043/01/269/1.jpg"></figure><lb></lb>batur, vt in antecedenti factum eſt, figura ex cylindris æ <lb></lb>qualium altitudinum, ita vt exceſſus, quo circumſcripta <lb></lb>ſuperat inſcriptam fit minor cono GLH; & cylindris cre<lb></lb>ſcentibus in latitudinem abſoluatur figura conoidi ABC <lb></lb>circumſcripta ex cylindris altitudine, & multitudine æqua <lb></lb>libus ijs, qui ſunt circa conoides EBF. </s> <s>Quoniam igitur <lb></lb>primus exceſſus eſt minor cono GLH, multo minor crit <lb></lb>pars eius communis ſolido AEBFG, quàm conus GLH: <lb></lb>ſed ſolidum AEBFC æquale eſt cono GBH; reliquum <lb></lb>igitur ſolidi AEBFC dicto communi ablato, maius erit <lb></lb>coni GBH reliquo BGLH; minor igitur proportio eſt <pb xlink:href="043/01/270.jpg" pagenum="91"></pb>primi exceſſus minoris cono GLH, ad dictum reliquum <lb></lb>ſolidi AEBFC, quàm coni GLH ad reliquum coni <lb></lb>GBH: ſed ſecundus exceſſus maior eſt prædicto reliquo <lb></lb>ſolidi AEBFC, ctenim illud comprehendit; multo igitur <lb></lb>minor proportio erit primi exceſſus ad ſecundum, quàm <lb></lb>coni GLH ad reliquum BGLH, hoc eſt minor propor<lb></lb>tio quàm DL ad LB: ponitur autem proportio DL ad <lb></lb>LB qualiſcumque. </s> <s>Fieri igitur poteſt, quod proponitur. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VI.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis reſidui conoidis hyperbolici dempto <lb></lb>conoide parabolico, vt ſupra diximus, centrum <lb></lb>grauitatis eſt punctum illud, in quo axis ſic diui<lb></lb>ditur, vt pars propinquior vertici ſit tripla re<lb></lb>liquæ. </s></p><figure id="id.043.01.270.1.jpg" xlink:href="043/01/270/1.jpg"></figure><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius axis BD, & <lb></lb>ablatum conoides parabolicum EBF circa eundem axim <lb></lb>BD, ita ſit ad reliquum ſolidum AEBFC, vt ſeſquialte <lb></lb>ra tranſuerſi lateris hyperboles, quæ conoides deſcribit ad <lb></lb>axem BD: & ponatur BG ipſius GD tripla. </s> <s>Dico re<pb xlink:href="043/01/271.jpg" pagenum="92"></pb>liqui ſolidi AEBFC centrum grauitatis eſse G. </s> <s>Secta <lb></lb>enim BD bifariam in puncto H, & poſita GK ipſius GH <lb></lb>minori quantacumque longitudine propoſita, ſumptoque <lb></lb>in GK quolibet puncto L, intelligantur id enim (fieri poſ <lb></lb>ſe manifeſtum eſt ex ſupra demonſtratis) tres figuræ vna in<lb></lb>ſcripta conoidi EBF, & duæ circumſcriptæ altera alteri <lb></lb>conoidum, vt ſupra factum eſt, compoſitæ ex cylindris <lb></lb>æqualium altitudinum ita multiplicatis, vt vtrumque illud <lb></lb>accidat; & vt ſecundi exceſſus centrum grauitatis quod ſit <lb></lb>M (omnium autem trium dictorum exceſſuum in axe <lb></lb>BD erunt centra grauitatis) ſit puncto G propinquius <lb></lb><figure id="id.043.01.271.1.jpg" xlink:href="043/01/271/1.jpg"></figure><lb></lb>quàm punctum L: & vt primus exceſſus ad ſecundum mi<lb></lb>norem habeat proportionem ea, quæ eſt LK, ad KH. </s> <s>Dein <lb></lb>de vt HK ad KL, ita ſit HN ad NM, & vt primus <lb></lb>exceſſus ad ſecundum, ita MO ad OH. </s> <s>Quoniam igitur <lb></lb>cylindri omnes deficientes, & ſummus integer, ex quibus <lb></lb>primus exceſſus conſtat, inter ſe ſunt æquales, habentque <lb></lb>in axe BD centra grauitatis æqualibus interuallis à bipar<lb></lb>titi axis BD ſectione H & inter ſe diſtantia; totius pri<lb></lb>mi exceſſus centrum grauitatis erit H: ſecundi autem ex<lb></lb>ceſſus centrum grauitatis ponitur M; cum igitur ſit vt pri<lb></lb>mus exceſſus ad ſecundum, ita ex contraria parte MO <pb xlink:href="043/01/272.jpg" pagenum="93"></pb>ad OH, erit tertij exceſſus ex duobus prioribus compoſi<lb></lb>ti centrum grauitatis O. </s> <s>Quoniam igitur minor propor<lb></lb>tio eſt primi exceſſus ad ſedundum, hoc eſt MO ad OH, <lb></lb>quàm LK ad KH; erit conuertendo maior proportio HO <lb></lb>ad OM, quàm HK ad KL: ſed vt HK ad KL, ita <lb></lb>ponitur HN ad NM; maior igitur proportio eſt HO ad <lb></lb>OM, quàm HN ad NM; eiuſdem igitur lineæ HM <lb></lb>minor erit MO, quàm MN, & punctum O propinquius <lb></lb>puncto G quam punctum N. </s> <s>Rurſus quia vt HK ad <lb></lb>KL, ita eſt HN ad NM; erit componen do & per con<lb></lb>uerſionem rationis, vt LH ad HK ita MH ad HN: & <lb></lb>permutando, vt HM ad HL, ita HN ad HK: ſed HM <lb></lb>eſt maior quàm HL; ergo & HN erit maior quam H<emph type="italics"></emph>K<emph.end type="italics"></emph.end>, <lb></lb>& punctum N propinquius puncto G quàm punctum K: <lb></lb>ſed punctum O propinquius erat puncto G quàm punctum <lb></lb>N; multo igitur erit punctum O propinquius puncto G <lb></lb>quàm punctum K. ponitur autem diſtantia GK minor <lb></lb>quantacumque longitudine propoſita: & eſt O centrum <lb></lb>grauitatis tertij exceſſus reliquo ſolido AEBFC circum<lb></lb>ſcripti; ex ijs igitur, quæ in primo libro demonſtrauimus, <lb></lb>ſolidi AEBFC centrum grauitatis erit G. </s> <s>Quod demon<lb></lb>ſtrandum erat. </s></p><p type="head"> <s><emph type="italics"></emph>PROPOSITIO VII.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Omnis conoidis hyperbolici centrum grauita<lb></lb>tis eſt punctum illud, in quo duodecima pars axis <lb></lb>quarta ab ea, quæ baſim attingit ſic diuiditur, vt <lb></lb>pars propinquior baſi ſit ad reliquam, vt ſeſquial<lb></lb>tera tranſuerſi lateris hyperboles, quæ conoides <lb></lb>deſcribit; ad axem conoidis. </s></p><p type="main"> <s>Sit conoides hyperbolicum ABC, cuius axis BD: <pb xlink:href="043/01/273.jpg" pagenum="94"></pb>tranſuerſum latus hyperboles, quæ conoides deſcribit ſit <lb></lb>BE, huius autem ſeſquialtera BEF: & ſumpta axis BD <lb></lb>tertia parte DG, & quarta DH, qua ratione erit GH <lb></lb>axis BD pars duodecima, ordine quarta ab ea, cuius termi <lb></lb>nus D; eſto vt FB ad BD, ita HK ad KG. </s> <s>Dico conoi<lb></lb>dis ABC centrum grauitatis eſſe K. </s> <s>Diuidatur enim co<lb></lb><figure id="id.043.01.273.1.jpg" xlink:href="043/01/273/1.jpg"></figure><lb></lb>noides ABC in parabolicum conoides LBM, & reliquum <lb></lb>ſolidum ALBMC, ita vt conoides LBM ad ſelidum <lb></lb>ALBMC ſit vt FB ad BD, hoc eſt vt HK GK. </s> <s>Quo<lb></lb>niam igitur G eſt centrum grauitatis conoidis LBM, & H <lb></lb>ſolidi ALBMC; tot us conoidis ABC centrum graui <lb></lb>tatis crit K. </s> <s>Quod demonſtrandum crat. </s></p><p type="head"> <s>TERTII LIBRI FINIS.</s></p> </chap> </body> <back></back> </text></archimedes>