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version 1.1, 2003/09/26 15:46:39

version 1.2, 2003/11/20 19:21:46

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<cvs_file>valer_centr_043_la_1604.xml</cvs_file>

<cvs_version/>

</info> <text> <front> </front> <body> <chap> <pb id="p.0001"/>

</info> <text> <front> </front> <body> <chap> <pb xlink:href="043/01/001.jpg" id="p.0001"/>

<s>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM <lb/>LIBRITRES.</s>

<s>LVCÆ VALERII <lb/><emph type="italics"/>Mathematicæ, & Ciuilis Philo&longs;ophiæ <lb/>in Gymna&longs;io Romano profe&longs;&longs;oris.<emph.end type="italics"/></s><figure/>

<s>LVCÆ VALERII <lb/><emph type="italics"/>Mathematicæ, & Ciuilis Philo&longs;ophiæ <lb/>in Gymna&longs;io Romano profe&longs;&longs;oris.<emph.end type="italics"/></s><figure id="id.043.01.001.1.jpg" xlink:href="043/01/001/1.jpg"/>

<s>ROMÆ, Typis Bartholom ri Bonfadini. </s>

<s>MDC IIII. <lb/>SVPERIORVM PERMISSV.</s><pb/>

<s>MDC IIII. <lb/>SVPERIORVM PERMISSV.</s><pb xlink:href="043/01/002.jpg"/>

<s>Imprimatur </s>

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<s>Apostol. Magi&longs;t.<emph.end type="italics"/></s>

<s>SANCTISSIMO <lb/>DOMINO NOSTRO <lb/>CLEMENTI VIII <lb/>PONT. OPT. MAX.<emph type="italics"/>Lucas Valerius perpetuam felicitatem.<emph.end type="italics"/></s><figure/>

<s>Grata Principi munera, <lb/>P. B. ex Philo&longs;ophiæ late<lb/>bris deprompta, qua&longs;i aurum <lb/>&longs;oli expo&longs;itum illico &longs;plen<lb/>dent, & publicæ vtilitatis <lb/>&longs;pem o&longs;tendunt, magno or<lb/>nata præ&longs;idio in primos liuo<lb/>ris impetus illius approbatione, cuius officium e&longs;t <lb/>alia à rep. </s>

<s>auertere, alia imperare. </s>

<s>Hinc por<lb/>rò factum e&longs;t, vt omnis ferè &longs;criptor exi&longs;ti matio<lb/>nis periculum aditurus, aliquem ex principibus <pb/>viris &longs;ibi deligat, cuius autoritate ip&longs;i dicatum <lb/>opus ab inuidorum mor&longs;ibus &longs;eruetur incolume. <lb/></s>

<s>Hinc por<lb/>rò factum e&longs;t, vt omnis ferè &longs;criptor exi&longs;ti matio<lb/>nis periculum aditurus, aliquem ex principibus <pb xlink:href="043/01/004.jpg"/>viris &longs;ibi deligat, cuius autoritate ip&longs;i dicatum <lb/>opus ab inuidorum mor&longs;ibus &longs;eruetur incolume. <lb/></s>

<s>Hanc ergo con&longs;uetudinem amanti mihi &longs;anè feli<lb/>citer cecidit, vt tu &longs;ola tua propria benignitate <lb/>permotus in tuos me familiares vltro a&longs;criberes. <lb/></s>

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<s>quantum <lb/>enim tuam excel&longs;am &longs;u&longs;picio dignitatem, tantum <lb/>de&longs;picor i&longs;tius doni incredibilem cum illa com<lb/>parati humilitatem: neque id ni&longs;i diuinitus cre<lb/>diderim perpetuam in tuis laudibus famam ha<lb/>biturum. </s>

<s>Quare illud non &longs;olum tibi diuini gre<lb/>gis anti&longs;titi cupio gratum accidere, cuius auto<lb/>ritate protectum in tanta nouarum rerum po&longs;t <lb/>tam graues autores contemptione, minimo meo <lb/>cum rubore in medium prodeat: &longs;ed ip&longs;i diuinita<lb/>ti ex voluntate donum expendenti, penes quam <lb/>e&longs;t æternitas, & cui primum dicata omnia e&longs;&longs;e <lb/>oportet: vt hi, quostuis luminibus dignaris, de <pb/>centro grauitatis &longs;olidorum &longs;terilis ingenij mei <lb/>te&longs;tes libelli à mortis æmula me obliuione defen<lb/>dant. </s>

<s>Quare illud non &longs;olum tibi diuini gre<lb/>gis anti&longs;titi cupio gratum accidere, cuius auto<lb/>ritate protectum in tanta nouarum rerum po&longs;t <lb/>tam graues autores contemptione, minimo meo <lb/>cum rubore in medium prodeat: &longs;ed ip&longs;i diuinita<lb/>ti ex voluntate donum expendenti, penes quam <lb/>e&longs;t æternitas, & cui primum dicata omnia e&longs;&longs;e <lb/>oportet: vt hi, quostuis luminibus dignaris, de <pb xlink:href="043/01/005.jpg"/>centro grauitatis &longs;olidorum &longs;terilis ingenij mei <lb/>te&longs;tes libelli à mortis æmula me obliuione defen<lb/>dant. </s>

<s>Stomacharis hic, arbitror, quòd tantum <lb/>&longs;pectem de nihilo; &longs;ed magis confe&longs;&longs;ionis impu<lb/>dentia. </s>

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<s>Alij tibi co<lb/>lumnas hone&longs;ti&longs;&longs;imis titulis ornatas erigant: &longs;ta <lb/>tuas in foris collocent: magnificas ædes extruant, <lb/>quarum in frontibus grandes marmoreæ tabulæ <lb/>flammantibus auro &longs;yderibus, & peregrinis lapi<lb/>dibus intextæ ea de te viuo referant &longs;axum impu<lb/>dens, quæ verecunda hæc pagina prætermittit. <lb/></s>

<s>Ego incredibilis tuæ benignitatis non tam gra<lb/>uia te&longs;timonia, quæ loco moueri nequeant: &longs;ed <pb/>expeditum hunc nuntium in longi&longs;&longs;ima itinera <lb/>de&longs;tinaui. </s>

<s>Ego incredibilis tuæ benignitatis non tam gra<lb/>uia te&longs;timonia, quæ loco moueri nequeant: &longs;ed <pb xlink:href="043/01/006.jpg"/>expeditum hunc nuntium in longi&longs;&longs;ima itinera <lb/>de&longs;tinaui. </s>

<s>Quem quidem eo minus vereor ne <lb/>non tu, quamobrem Telchines forta&longs;&longs;e aliqui in<lb/>&longs;ectaturi, di&longs;pari &longs;is voluntate protecturus, quòd <lb/>in his tàm reconditis naturæ arcanis geometrica <lb/>demon&longs;tratione patefactis, tanquam in &longs;emine <lb/>multiplicem præ&longs;criptionem, ac normam e&longs;&longs;e in<lb/>telliges ip&longs;e pacis inter tuos greges autor, lupi <lb/>otomani terror, ciuili, & bellicæ architecturæ <lb/>maximè nece&longs;&longs;ariam. </s>

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<s>Quod denique <lb/>&longs;cientiæ ciuilis ip&longs;e periti&longs;&longs;imus omnium optimè <lb/>intelligis, quanti referat ad humanæ &longs;ocietatis for <lb/>mam & candorem, regum, atque optimatum a<lb/>mor in &longs;tudio&longs;os bonarum litterarum. </s>

<s>contrà au<lb/>tem ex de&longs;pectione in hos cadente abijs, quorum <lb/>mores pro legibus haberi &longs;olent, no&longs;ti commu<lb/>nem ingeniorum veternum, mox tyrannidem gi<lb/>gni, magna cu&longs;tode adempta mode&longs;tiæ imperi<lb/>tantium crebra ciuium &longs;apientia, quæ prauis ti<lb/>morem efficit, melioribus pudorem, Quod &longs;i meæ <lb/>expectationi exitus re&longs;pondebit, vt te hoc munu<lb/>&longs;culo vel leuiter lætari &longs;entiam; alia non iniucun-<pb/>da ftatim proferam, qua PETRVS ALDOBRAN<lb/>DINVS tuus nepos, domi fori&longs;que clari&longs;&longs;imus <lb/>Cardinalis, cuius inter familiares itidem, <expan abbr="bene-ficijs&qacute;ue">bene<lb/>ficijsque</expan> deuinctos locum habeo, &longs;uæ erga me hu<lb/>manitatis te&longs;timonia ab inuidiæ &longs;atellite & mi<lb/>ni&longs;tra calumnia tueatur: quando duobus talibus <lb/>viris animi mei captum beneficentia &longs;ua pericli<lb/>tantibus, duplex periculum &longs;ubire &longs;um coactus. <lb/></s>

<s>contrà au<lb/>tem ex de&longs;pectione in hos cadente abijs, quorum <lb/>mores pro legibus haberi &longs;olent, no&longs;ti commu<lb/>nem ingeniorum veternum, mox tyrannidem gi<lb/>gni, magna cu&longs;tode adempta mode&longs;tiæ imperi<lb/>tantium crebra ciuium &longs;apientia, quæ prauis ti<lb/>morem efficit, melioribus pudorem, Quod &longs;i meæ <lb/>expectationi exitus re&longs;pondebit, vt te hoc munu<lb/>&longs;culo vel leuiter lætari &longs;entiam; alia non iniucun-<pb xlink:href="043/01/007.jpg"/>da ftatim proferam, qua PETRVS ALDOBRAN<lb/>DINVS tuus nepos, domi fori&longs;que clari&longs;&longs;imus <lb/>Cardinalis, cuius inter familiares itidem, <expan abbr="bene-ficijs&qacute;ue">bene<lb/>ficijsque</expan> deuinctos locum habeo, &longs;uæ erga me hu<lb/>manitatis te&longs;timonia ab inuidiæ &longs;atellite & mi<lb/>ni&longs;tra calumnia tueatur: quando duobus talibus <lb/>viris animi mei captum beneficentia &longs;ua pericli<lb/>tantibus, duplex periculum &longs;ubire &longs;um coactus. <lb/></s>

<s>Sed iam verbo&longs;æ epi&longs;tolæ, & tuo fa&longs;tidio finem im <lb/>po&longs;iturus peto à te vnum; vt tibi per&longs;uadeas, me <lb/>inter tuos famulos, quos ære proprio, & victu quo<lb/>tidiano liberaliter &longs;u&longs;tentas, eorum, qui pro te <lb/>emori po&longs;&longs;unt, amore, con&longs;tantia, fidelitate nemini <lb/>planè concedere. </s>

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<s>Valet. </s><pb/>

<s>Valet. </s><pb xlink:href="043/01/008.jpg"/>

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<s><foreign lang="greek">*lne/zos ou) kle/yw xa/<gap/>n eu)/fzonos e)<gap/>omo/noi<gap/><lb/>*d<gap/>gm) a)glao\n, <gap/>, <gap/>nomes, kai\ patzi/d<gap/>. </foreign></s>

<s><foreign lang="greek">*os de/ me laqzai<gap/>os dh/z<gap/>, kako/ep<gap/>os a)kou/o<gap/>, <lb/>*lu<gap/>w_n h(=s fqonezh_s a)/zios purkai<gap/>h_s. </foreign></s><pb pagenum="1"/><figure/>

<s>LVC AE <lb/>VALER II <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM</s>

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<s>Propo&longs;itum e&longs;t mihi in hi&longs;ce tribus li<lb/>bris, ò Geometra, cuiu&longs;cumque figuræ <lb/>&longs;olidæ in geometria ratio haberi &longs;olet, <lb/>centrum grauitatis inuenire. </s>

<s>Huius <lb/>autem prouinciæ mihi &longs;u&longs;cipiendæ oc<lb/>ca&longs;io fuit liber ille iam pridem editus <lb/>Federici Commandini Vrbinatis, in <lb/>quo cum ille corporum planis termi<lb/>nis definitorum; necnon cylindri, & coni, & fru&longs;ti conici, <lb/>& &longs;phæræ, & &longs;phæroidis centrum grauitatis o&longs;tendi&longs;&longs;et; <lb/>aliorum autem, quæ &longs;uperficie mixta continentur vno co<lb/>noide parabolico tentato &longs;yllogi&longs;mi iactura operam per<lb/>didi&longs;&longs;et, ego &longs;pe magis, ad quam vir ille exar&longs;erat incita<pb pagenum="2"/>tus, quàm deterritus lap&longs;u, vehementerque dolens geo<lb/>metriæ partem tamdiu de&longs;iderari cognitione digni&longs;&longs;imam; <lb/>cum ante exercitationis cau&longs;a omnium, quæ propo&longs;ui &longs;oli<lb/>dorum, excepto conoide parabolico, centra grauitatis aliis <lb/>viis indaga&longs;&longs;em; po&longs;tea non &longs;olum parabolici, &longs;ed ante me <lb/>tentata nemini, hyperbolici conoidis, & fru&longs;ti vtriu&longs;que, & <lb/>portionis vtriu&longs;que conoidis, & portionis fru&longs;ti, & hemi<lb/>&longs;phærij, & hemi&longs;phæroidis, & cuiu&longs;libet portionis &longs;phæ<lb/>ræ, & &longs;phæroidis vno, & duobus planis parallelis ab&longs;ci&longs;&longs;æ <lb/><expan abbr="c&etilde;tra">centra</expan> grauitatis adinueni, multa autem ex his duplici, quæ<lb/>dam triplici via. </s>

<s>Huius <lb/>autem prouinciæ mihi &longs;u&longs;cipiendæ oc<lb/>ca&longs;io fuit liber ille iam pridem editus <lb/>Federici Commandini Vrbinatis, in <lb/>quo cum ille corporum planis termi<lb/>nis definitorum; necnon cylindri, & coni, & fru&longs;ti conici, <lb/>& &longs;phæræ, & &longs;phæroidis centrum grauitatis o&longs;tendi&longs;&longs;et; <lb/>aliorum autem, quæ &longs;uperficie mixta continentur vno co<lb/>noide parabolico tentato &longs;yllogi&longs;mi iactura operam per<lb/>didi&longs;&longs;et, ego &longs;pe magis, ad quam vir ille exar&longs;erat incita<pb xlink:href="043/01/010.jpg" pagenum="2"/>tus, quàm deterritus lap&longs;u, vehementerque dolens geo<lb/>metriæ partem tamdiu de&longs;iderari cognitione digni&longs;&longs;imam; <lb/>cum ante exercitationis cau&longs;a omnium, quæ propo&longs;ui &longs;oli<lb/>dorum, excepto conoide parabolico, centra grauitatis aliis <lb/>viis indaga&longs;&longs;em; po&longs;tea non &longs;olum parabolici, &longs;ed ante me <lb/>tentata nemini, hyperbolici conoidis, & fru&longs;ti vtriu&longs;que, & <lb/>portionis vtriu&longs;que conoidis, & portionis fru&longs;ti, & hemi<lb/>&longs;phærij, & hemi&longs;phæroidis, & cuiu&longs;libet portionis &longs;phæ<lb/>ræ, & &longs;phæroidis vno, & duobus planis parallelis ab&longs;ci&longs;&longs;æ <lb/><expan abbr="c&etilde;tra">centra</expan> grauitatis adinueni, multa autem ex his duplici, quæ<lb/>dam triplici via. </s>

<s>Taceo nunc alia eiu&longs;dem generis, quæ <lb/>cum vtilia, tum geometriæ &longs;tudio&longs;is non iniucunda, vt arbi<lb/>tror, futura in po&longs;teriores libros di&longs;tribuimus. </s>

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<s>Sed <lb/>ad definitiones accedamus. </s><pb pagenum="3"/>

<s>Sed <lb/>ad definitiones accedamus. </s><pb xlink:href="043/01/011.jpg" pagenum="3"/>

<s>DEFINITIONES.</s>

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<s>Polyedri regularis centrum dicitur punctum, <lb/>in quo omnes rectæ lineæ, quæ ad angulos oppo<lb/>&longs;itos pertinent bifariam diuiduntur. </s><pb pagenum="4"/>

<s>Polyedri regularis centrum dicitur punctum, <lb/>in quo omnes rectæ lineæ, quæ ad angulos oppo<lb/>&longs;itos pertinent bifariam diuiduntur. </s><pb xlink:href="043/01/012.jpg" pagenum="4"/>

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<s>Si qua figura &longs;olida planis parallelis ita &longs;eca<lb/>ri po&longs;&longs;it, vt quæcumque &longs;ectiones centrum ha<lb/>beant, & &longs;int inter &longs;e &longs;imiles; aliqua autem recta <lb/>linea, &longs;iue ad centra ba&longs;ium oppo&longs;itarum prædi<lb/>ctis &longs;ectionibus parallelarum, & &longs;imilium, vt in <lb/>cylindro; &longs;iue ad verticem, & centrum ba&longs;is ter<lb/>minata, vt in cono, hemi&longs;phærio, & conoide, tran<lb/>&longs;eat per centra omnium prædictarum &longs;ectionum; <lb/>ea talis figuræ axis nominetur: ip&longs;a autem figura, <lb/>&longs;olidum circa axim. </s>

<s>Quæ &longs;i vel vnam tantum ha<lb/>beat ba&longs;im, vel duas inæquales, & parallelas: dua<lb/>rum autem quarumlibet prædictarum &longs;ectionum <lb/>vertici, vel minori ba&longs;i propinquior &longs;it minor re-<pb pagenum="5"/>motiori; &longs;olidum circa axem in alteram partem de <lb/>ficiens nominetur: quo nomine &longs;ignificari etiam <lb/>volumus ea &longs;olida, quorum quælibet &longs;ectiones <lb/>ba&longs;i parallelæ quamuis ba&longs;i non &longs;int omnino &longs;imi<lb/>les, tamen ijs figuris deficiunt, quæ &longs;unt &longs;imiles <lb/>ha&longs;i, ac totis ijs, à quibus ip&longs;æ ablatæ intelli<lb/>guntur, ita vt tota figura & ablata habeant com<lb/>mune centrum in vna recta linea ad centrum ba<lb/>&longs;is terminata, quæ & ip&longs;a talis &longs;olidi axis nomi<lb/>netur. </s>

<s>Quæ &longs;i vel vnam tantum ha<lb/>beat ba&longs;im, vel duas inæquales, & parallelas: dua<lb/>rum autem quarumlibet prædictarum &longs;ectionum <lb/>vertici, vel minori ba&longs;i propinquior &longs;it minor re-<pb xlink:href="043/01/013.jpg" pagenum="5"/>motiori; &longs;olidum circa axem in alteram partem de <lb/>ficiens nominetur: quo nomine &longs;ignificari etiam <lb/>volumus ea &longs;olida, quorum quælibet &longs;ectiones <lb/>ba&longs;i parallelæ quamuis ba&longs;i non &longs;int omnino &longs;imi<lb/>les, tamen ijs figuris deficiunt, quæ &longs;unt &longs;imiles <lb/>ha&longs;i, ac totis ijs, à quibus ip&longs;æ ablatæ intelli<lb/>guntur, ita vt tota figura & ablata habeant com<lb/>mune centrum in vna recta linea ad centrum ba<lb/>&longs;is terminata, quæ & ip&longs;a talis &longs;olidi axis nomi<lb/>netur. </s>

<s>Vt in figura, &longs;olidi ABDC deficientis &longs;olido CED <lb/>ba&longs;is e&longs;t circulus AB, terminus ba&longs;i oppo&longs;itus circum<lb/>ferentia circuli CMD. axis communis omnibus EF, <lb/>per cuius quodlibet punctum I plano ba&longs;i AB paralle<lb/>lo &longs;ecante &longs;olidum ABDC, & ablatum CED, & re<lb/>&longs;iduum, e&longs;t totius <lb/>&longs;ectio circulus G <lb/>H, ablati vero cir<lb/>culus KL, & re&longs;i<lb/>dui &longs;ectio reliquum <lb/>circuli GH dem<lb/>pto circulo KL. <lb/>quarum &longs;ectionum <lb/>omnium centrum <lb/>commune e&longs;t I. <lb/></s>

<s>Quod &longs;i &longs;uper duos <lb/><figure id="fig1"/><lb/>circulos GH, KL circa axem communem EI cylin<lb/>dri de&longs;cribantur, (erunt autem eiu&longs;dem altitudinis) erit <lb/>reliquum cylindri GB, dempto cylindro cuius ba&longs;is <lb/>KL, axis EI, con&longs;titutum &longs;uper ba&longs;im G, <emph type="italics"/>K<emph.end type="italics"/>, & circa <lb/>axim EI, quæ &longs;uo loco expectatur cogitatio. </s><pb pagenum="6"/>

<s>Quod &longs;i &longs;uper duos <lb/><figure id="id.043.01.013.1.jpg" xlink:href="043/01/013/1.jpg"/><lb/>circulos GH, KL circa axem communem EI cylin<lb/>dri de&longs;cribantur, (erunt autem eiu&longs;dem altitudinis) erit <lb/>reliquum cylindri GB, dempto cylindro cuius ba&longs;is <lb/>KL, axis EI, con&longs;titutum &longs;uper ba&longs;im G, <emph type="italics"/>K<emph.end type="italics"/>, & circa <lb/>axim EI, quæ &longs;uo loco expectatur cogitatio. </s><pb xlink:href="043/01/014.jpg" pagenum="6"/>

<s>POSTVLATA.</s>

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<s>A quibus longitudinibus duo grauia æquipon<lb/>derant, ab ij&longs;dem alia duo quælibet illis æqualia <lb/>æquiponderare. </s><pb pagenum="7"/>

<s>A quibus longitudinibus duo grauia æquipon<lb/>derant, ab ij&longs;dem alia duo quælibet illis æqualia <lb/>æquiponderare. </s><pb xlink:href="043/01/015.jpg" pagenum="7"/>

<s>PROPOSITIO <lb/>PRIMA.</s>

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<s>Simili<lb/>ter o&longs;tenderemus e&longs;se vt CD ad EF, <lb/>ita DK ad FL; vt igitur BH ad DK, <lb/>ita erit DK ad FL, in proportione, quæ <lb/>e&longs;t AB ad CD, & CD ad EF. </s>

<s>Quod demon&longs;tran<lb/>dum erat. </s><figure/><pb pagenum="8"/>

<s>Quod demon&longs;tran<lb/>dum erat. </s><figure id="id.043.01.015.1.jpg" xlink:href="043/01/015/1.jpg"/><pb xlink:href="043/01/016.jpg" pagenum="8"/>

<s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/></s>

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<s>Et reliquum latus BC <lb/>&longs;ectum e&longs;&longs;e bifariam in puncto G. </s>

<s>Quoniam enim e&longs;t vt BA <lb/>ad AD, ita CA ad AE: hoc e&longs;t, vt triangulum ABC ad <lb/>triangulum ADC, ita triangulum idem ABC ad trian<lb/>gulum AEB; æqualia <lb/>erunt triangula ADC, <lb/>AEB, & ablato trape<lb/>zio DE communi re<lb/>liquum triangulum BD <lb/>F reliquo triangulo C <lb/>EF æquale erit: &longs;ed <lb/>triangulum ADF e&longs;t <lb/>æquale triangulo BDF; <lb/>& triangulum AFE <lb/>triangulo EFC, pro<lb/>pter æquales ba&longs;es, & <lb/><figure id="fig2"/><lb/>communes altitudines; totum igitur triangulum AFB <lb/>toti AFC, triangulo æquale erit: &longs;ed vt triangulum AFB <pb pagenum="9"/>ad triangulum FBG, hoc e&longs;t vt AF ad FG, ita e&longs;t <lb/>triangulum AFC ad triangulum FCG; triangulum er<lb/>go FBG triangulo FCG æquale erit, & ba&longs;is BG ba<lb/>&longs;i GC æqualis. </s>

<s>Quoniam enim e&longs;t vt BA <lb/>ad AD, ita CA ad AE: hoc e&longs;t, vt triangulum ABC ad <lb/>triangulum ADC, ita triangulum idem ABC ad trian<lb/>gulum AEB; æqualia <lb/>erunt triangula ADC, <lb/>AEB, & ablato trape<lb/>zio DE communi re<lb/>liquum triangulum BD <lb/>F reliquo triangulo C <lb/>EF æquale erit: &longs;ed <lb/>triangulum ADF e&longs;t <lb/>æquale triangulo BDF; <lb/>& triangulum AFE <lb/>triangulo EFC, pro<lb/>pter æquales ba&longs;es, & <lb/><figure id="id.043.01.016.1.jpg" xlink:href="043/01/016/1.jpg"/><lb/>communes altitudines; totum igitur triangulum AFB <lb/>toti AFC, triangulo æquale erit: &longs;ed vt triangulum AFB <pb xlink:href="043/01/017.jpg" pagenum="9"/>ad triangulum FBG, hoc e&longs;t vt AF ad FG, ita e&longs;t <lb/>triangulum AFC ad triangulum FCG; triangulum er<lb/>go FBG triangulo FCG æquale erit, & ba&longs;is BG ba<lb/>&longs;i GC æqualis. </s>

<s>Quoniam igitur & AE e&longs;t æqualis <lb/>EC, &longs;imiliter vt ante, o&longs;tenderemus, triangulum BCF, <lb/>triangulo ACF, eademque ratione triangulum ABF, <lb/>triangulo BCF æquale e&longs;&longs;e: igitur vnumquodque trian<lb/>gulorum ABF, ACF, BCF, tertia pars e&longs;t trianguli <lb/>ABC: &longs;ed vt triangulum ABC, ad triangulum BCF, <lb/>ita e&longs;t AG, ad GF; tripla igitur e&longs;t AG ip&longs;ius GF, <lb/>ac proinde AF, ip&longs;ius FG dupla. </s>

Line 244

<s>Quoniam <lb/>igitur D e&longs;t centrum trian <lb/>guli ABC erit triangu<lb/>lum BDC tertia pars <lb/>trianguli ABC. </s>

<s>Eadem <lb/>ratione triangulum BEC <lb/>tertia pars erit trianguli <lb/>ABC; triangulum ergo <lb/>DBC æquale erit trian<lb/>gulo BEC pars toti, quod <lb/>fieri non pote&longs;t, atqui <expan abbr="id&etilde;">idem</expan> <lb/><figure id="fig3"/><lb/>ab&longs;urdum &longs;equitur, &longs;i punctum D cadat in aliquo latere <lb/>triangulorum, quorum vertex E; Manife&longs;tum e&longs;t igitur <lb/>propo&longs;itum. </s><pb pagenum="10"/>

<s>Eadem <lb/>ratione triangulum BEC <lb/>tertia pars erit trianguli <lb/>ABC; triangulum ergo <lb/>DBC æquale erit trian<lb/>gulo BEC pars toti, quod <lb/>fieri non pote&longs;t, atqui <expan abbr="id&etilde;">idem</expan> <lb/><figure id="id.043.01.017.1.jpg" xlink:href="043/01/017/1.jpg"/><lb/>ab&longs;urdum &longs;equitur, &longs;i punctum D cadat in aliquo latere <lb/>triangulorum, quorum vertex E; Manife&longs;tum e&longs;t igitur <lb/>propo&longs;itum. </s><pb xlink:href="043/01/018.jpg" pagenum="10"/>

<s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/></s>

Line 252

<s>Sint triangula &longs;imilia, & &longs;imiliter po&longs;ita ABC, DEF, <lb/>quorum &longs;int centra O, P, in ijs autem triangulis &longs;int pun<lb/>cta &longs;imiliter po&longs;ita K, L, quæ cadant primum in rectis <lb/>BG, EH, quæ ab angulis æqualibus B, E, ba&longs;es bifa<lb/>riam diuidunt. </s>

<s>Dico e&longs;&longs;e OK ad PL, vt e&longs;t latus AB, <lb/>ad latus DE. iunctis enim AK, KC, DL, LF, quo<lb/><figure id="fig4"/><lb/>niam angulus KAC, æqualis e&longs;t angulo LDF, & angu<lb/>lus KCA, angulo LFD, ob &longs;imiliter po&longs;ita puncta K, <lb/>L, triangulum AKC, triangulo LDF &longs;imile erit, & vt <lb/>KA ad AC, ita LD ad DF: &longs;ed vt CA ad AG, ita <lb/>e&longs;t FD ad DH, expræcedenti; vt igitur KA, ad AG <lb/>ita erit LD, ad DH, circa æquales angulos: &longs;imilia igi<lb/>tur &longs;unt triangula AGK, DHL, & angulus AGK, <pb pagenum="11"/>æqualis angulo DHL, & vt KG, ad GA, ita LH, ad <lb/>HD: &longs;ed vt GA, ad AC, ita e&longs;t HD ad DF: & vt <lb/>AC ad AB, ita DF ad DE, ex æquali igitur erit vt <lb/>KG ad AB, ita LH ad DE: &longs;ed vt AB ad BG, ita <lb/>e&longs;t DE ad EH, propter &longs;imilitudinem triangulorum <lb/>ABG, DEH: & vt BG ad GO ita e&longs;t EH ad HP, <lb/>propter triangulorum centra O, P; ex æquali igitur erit <lb/>vt KG ad GO, ita LH ad HP: & permutando vt <lb/>OG ad PH, ide&longs;t vt BG ad EH, ide&longs;t vt AB ad ED, <lb/>ita KG ad LH, & reliqua OK ad reliquam PL. </s>

<s>Dico e&longs;&longs;e OK ad PL, vt e&longs;t latus AB, <lb/>ad latus DE. iunctis enim AK, KC, DL, LF, quo<lb/><figure id="id.043.01.018.1.jpg" xlink:href="043/01/018/1.jpg"/><lb/>niam angulus KAC, æqualis e&longs;t angulo LDF, & angu<lb/>lus KCA, angulo LFD, ob &longs;imiliter po&longs;ita puncta K, <lb/>L, triangulum AKC, triangulo LDF &longs;imile erit, & vt <lb/>KA ad AC, ita LD ad DF: &longs;ed vt CA ad AG, ita <lb/>e&longs;t FD ad DH, expræcedenti; vt igitur KA, ad AG <lb/>ita erit LD, ad DH, circa æquales angulos: &longs;imilia igi<lb/>tur &longs;unt triangula AGK, DHL, & angulus AGK, <pb xlink:href="043/01/019.jpg" pagenum="11"/>æqualis angulo DHL, & vt KG, ad GA, ita LH, ad <lb/>HD: &longs;ed vt GA, ad AC, ita e&longs;t HD ad DF: & vt <lb/>AC ad AB, ita DF ad DE, ex æquali igitur erit vt <lb/>KG ad AB, ita LH ad DE: &longs;ed vt AB ad BG, ita <lb/>e&longs;t DE ad EH, propter &longs;imilitudinem triangulorum <lb/>ABG, DEH: & vt BG ad GO ita e&longs;t EH ad HP, <lb/>propter triangulorum centra O, P; ex æquali igitur erit <lb/>vt KG ad GO, ita LH ad HP: & permutando vt <lb/>OG ad PH, ide&longs;t vt BG ad EH, ide&longs;t vt AB ad ED, <lb/>ita KG ad LH, & reliqua OK ad reliquam PL. </s>

<s>Sed &longs;int puncta &longs;imiliter po&longs;ita M, N, quæ cadant ex<lb/>tra lineas BG, EH, iunctæque OM, PN. </s>

Line 264

<s>Rur&longs;us quo<lb/>niam angulus ABM, æqualis e&longs;t angulo DEN, quorum <lb/>angulus ABG, æqualis e&longs;t angulo DEH: erit reliquus <lb/>angulus OBM, æqualis reliquo angulo PEN: &longs;ed vt MB <lb/>ad BO, ita erat NE ad EP; triangulum igitur OBM <lb/>triangulo PEN, &longs;imile erit, & vt BO ad EP, hoc e&longs;t <lb/>BG ad EH, hoc e&longs;t AB ad DE, ita OM ad PN. <lb/></s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="12"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/020.jpg" pagenum="12"/>

<s><emph type="italics"/>PROPOSITIO IV.<emph.end type="italics"/></s>

Line 274

<s>Dico in triangulo DEF, <lb/>vnum duntaxat punctum puncto G &longs;imiliter po&longs;itum in<lb/>ueniri po&longs;se. </s>

<s>Iunctis enim AG, BG, GC, ponatur <lb/>angulus EDH, æqualis angulo BAG, & angulus DEH, <lb/><figure id="fig5"/><lb/>æqualis angulo ABG, & HF iungatur. </s>

<s>Iunctis enim AG, BG, GC, ponatur <lb/>angulus EDH, æqualis angulo BAG, & angulus DEH, <lb/><figure id="id.043.01.020.1.jpg" xlink:href="043/01/020/1.jpg"/><lb/>æqualis angulo ABG, & HF iungatur. </s>

<s>Manife&longs;tum <lb/>e&longs;t igitur ex præcedentis Theorematis demon&longs;tratione, <lb/>triangula EDH, HDF, FEH, &longs;imilia e&longs;se triangulis <lb/>BAG, GAC, CBG, prout inter &longs;e re&longs;pondent po&longs;i<lb/>tione, quorum &longs;ex triangulorum binis quibu&longs;que binæ ba<lb/>&longs;es homologæ re&longs;pondent: AB ED, AC DF, BC <pb pagenum="13"/>EF. quæ &longs;untin latera homologa duorum triangulorum <lb/>ABC, DEF. </s>

<s>Manife&longs;tum <lb/>e&longs;t igitur ex præcedentis Theorematis demon&longs;tratione, <lb/>triangula EDH, HDF, FEH, &longs;imilia e&longs;se triangulis <lb/>BAG, GAC, CBG, prout inter &longs;e re&longs;pondent po&longs;i<lb/>tione, quorum &longs;ex triangulorum binis quibu&longs;que binæ ba<lb/>&longs;es homologæ re&longs;pondent: AB ED, AC DF, BC <pb xlink:href="043/01/021.jpg" pagenum="13"/>EF. quæ &longs;untin latera homologa duorum triangulorum <lb/>ABC, DEF. </s>

<s>Ex definitione igitur, duo puncta G, H, <lb/>in triangulis ABC, DEF, &longs;imiliter po&longs;ita erunt. </s>

Line 292

<s><emph type="italics"/>PROPOSITIO V.<emph.end type="italics"/></s>

<s>Cuilibet figuræ planæ rectangulum æquale <lb/>pote&longs;t e&longs;&longs;e. </s><pb pagenum="14"/>

<s>Cuilibet figuræ planæ rectangulum æquale <lb/>pote&longs;t e&longs;&longs;e. </s><pb xlink:href="043/01/022.jpg" pagenum="14"/>

<s>Sit quælibet figura plana A. </s>

Line 302

<s>Quoniam igitur e&longs;t vt rectangulum BD, ad <lb/>planam figuram A, ita recta BD, ad aliquam lineam <lb/>rectam &longs;it vt BC, ad A, ita BD, ad DE, & comple<lb/>atur rectan<lb/>gulum EC. <lb/></s>

<s>Quoniam igi <lb/>tur e&longs;t vt BD <lb/>ad DE, ita <lb/>rectangulum <lb/>BC, ad figu<lb/>ram A: &longs;ed <lb/>vt BD, ad <lb/>DE, ita e&longs;t <lb/><figure id="fig6"/><lb/>rectangulum BC, ad rectangulum CE; vt igitur re<lb/>ctangulum BC, ad figuram A, ita e&longs;t rectangulum <lb/>BC, ad rectangulum CE; rectangulum ergo CE, fi<lb/>guræ A, æquale erit. </s>

<s>Quoniam igi <lb/>tur e&longs;t vt BD <lb/>ad DE, ita <lb/>rectangulum <lb/>BC, ad figu<lb/>ram A: &longs;ed <lb/>vt BD, ad <lb/>DE, ita e&longs;t <lb/><figure id="id.043.01.022.1.jpg" xlink:href="043/01/022/1.jpg"/><lb/>rectangulum BC, ad rectangulum CE; vt igitur re<lb/>ctangulum BC, ad figuram A, ita e&longs;t rectangulum <lb/>BC, ad rectangulum CE; rectangulum ergo CE, fi<lb/>guræ A, æquale erit. </s>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s>

Line 312

<s>Semper autem in &longs;imilibus intelli<lb/>ge, eiu&longs;dem generis. </s>

<s>Sit figura plana ABC circa diametrum AD, ad par-<pb pagenum="15"/>tes A deficiens, cuius ba&longs;is BC. </s>

<s>Sit figura plana ABC circa diametrum AD, ad par-<pb xlink:href="043/01/023.jpg" pagenum="15"/>tes A deficiens, cuius ba&longs;is BC. </s>

<s>Dico fieri po&longs;se quod <lb/>proponitur: ducta enim per verticem figuræ A, ba&longs;i BC, <lb/>parallela, atque ideo figuram ip&longs;am contingente, ab&longs;ol<lb/>uatur parallelogrammum BL, &longs;ectaque diametro AD, <lb/>bifariam, & &longs;ingulis eius partibus &longs;emper bifariam, du<lb/>cantur per puncta &longs;ectionum rectæ lineæ ba&longs;i BC, & in<lb/>ter &longs;e parallelæ, atque ita multiplicatæ &longs;int &longs;ectiones, <lb/>vt &longs;ecti parallelogrammi in parallelogramma æqua<lb/>lia, & eiu&longs;dem altitudinis quælibet pars, vt paralle<lb/>logrammum BF, &longs;it minus &longs;uperficie propo&longs;ita, cu<lb/>ius parallelogram<lb/>mi latus EF, &longs;e<lb/>cet figuræ termi<lb/>num BAC, in <lb/>punctis GH, & <lb/>diametrum AD, in <lb/>puncto K. erit igi<lb/>tur GK, æqualis <lb/>KH: per omnia <lb/>igitur puncta &longs;e<lb/>ctionum termini <lb/><figure id="fig7"/><lb/>BAC, quæ à prædictis fiunt lineis parallelis, &longs;i ducan<lb/>tur diametro AD parallelæ, figura quædam ip&longs;i ABC, <lb/>in&longs;cribetur, & altera circum&longs;cribetur ex parallelogram<lb/>mis æqualium altitudinum. </s>

<s>Dico fieri po&longs;se quod <lb/>proponitur: ducta enim per verticem figuræ A, ba&longs;i BC, <lb/>parallela, atque ideo figuram ip&longs;am contingente, ab&longs;ol<lb/>uatur parallelogrammum BL, &longs;ectaque diametro AD, <lb/>bifariam, & &longs;ingulis eius partibus &longs;emper bifariam, du<lb/>cantur per puncta &longs;ectionum rectæ lineæ ba&longs;i BC, & in<lb/>ter &longs;e parallelæ, atque ita multiplicatæ &longs;int &longs;ectiones, <lb/>vt &longs;ecti parallelogrammi in parallelogramma æqua<lb/>lia, & eiu&longs;dem altitudinis quælibet pars, vt paralle<lb/>logrammum BF, &longs;it minus &longs;uperficie propo&longs;ita, cu<lb/>ius parallelogram<lb/>mi latus EF, &longs;e<lb/>cet figuræ termi<lb/>num BAC, in <lb/>punctis GH, & <lb/>diametrum AD, in <lb/>puncto K. erit igi<lb/>tur GK, æqualis <lb/>KH: per omnia <lb/>igitur puncta &longs;e<lb/>ctionum termini <lb/><figure id="id.043.01.023.1.jpg" xlink:href="043/01/023/1.jpg"/><lb/>BAC, quæ à prædictis fiunt lineis parallelis, &longs;i ducan<lb/>tur diametro AD parallelæ, figura quædam ip&longs;i ABC, <lb/>in&longs;cribetur, & altera circum&longs;cribetur ex parallelogram<lb/>mis æqualium altitudinum. </s>

<s>Dico harum figurarum <lb/>in&longs;criptam &longs;uperari à circum&longs;cripta minori &longs;pacio &longs;uper<lb/>ficie propo&longs;ita. </s>

<s>Quoniam enim omnia parallelogramma, <lb/>quibus figura circum&longs;cripta &longs;uperat in&longs;criptam &longs;imul &longs;um<lb/>pta &longs;unt æqualia BF parallelogrammo: &longs;ed parallelo<lb/>grammum BF, e&longs;t minus &longs;uperficie propo&longs;ita: exce&longs;&longs;us <lb/>igitur quo figura circum&longs;cripta in&longs;criptam &longs;uperat, minor <lb/>erit &longs;uperficie propo&longs;ita. </s>

<s>Fieri igitur pote&longs;t, quod propo<lb/>nebatur. </s><pb pagenum="16"/>

<s>Fieri igitur pote&longs;t, quod propo<lb/>nebatur. </s><pb xlink:href="043/01/024.jpg" pagenum="16"/>

<s><emph type="italics"/>PROPOSITIO VII.<emph.end type="italics"/></s>

Line 330

<s>Dico pyramidem ABCD, pyramidi EFGH, æqualem <lb/>e&longs;se. </s>

<s>A punctis enim A, E, manantia latera inferius pro<lb/>ducantur, & prædictis lateribus maiores, inter &longs;e autem <lb/>æquales ab&longs;cindantur AK, AL, AM, EN, EO, EP, <lb/><figure id="fig8"/><lb/>& con&longs;truantur pyramides AKLM, ENOP: pyramides <lb/>igitur hæ æqualibus, & &longs;imilibus triangulis comprehenden <lb/>tur, vt colligitur ex ip&longs;a con&longs;tructione; triangulis igitur inter <lb/>&longs;e æquilateris, & æquiangulis KLM, NOP, inter &longs;e con<lb/>gruentibus non congruat, &longs;i fieri pote&longs;t, pyramis ENOP, <lb/>pyramidi AKLM, &longs;ed cadat vertex E, pyramidis ENOP, <lb/>extra verticem A, pyramidis AKLM, & ex puncto A, <pb pagenum="17"/>ad centrum circuli tran&longs;euntis per tria puncta K, L, M, quod <lb/>&longs;it R, ducatur recta AR, & ER iungatur. </s>

<s>A punctis enim A, E, manantia latera inferius pro<lb/>ducantur, & prædictis lateribus maiores, inter &longs;e autem <lb/>æquales ab&longs;cindantur AK, AL, AM, EN, EO, EP, <lb/><figure id="id.043.01.024.1.jpg" xlink:href="043/01/024/1.jpg"/><lb/>& con&longs;truantur pyramides AKLM, ENOP: pyramides <lb/>igitur hæ æqualibus, & &longs;imilibus triangulis comprehenden <lb/>tur, vt colligitur ex ip&longs;a con&longs;tructione; triangulis igitur inter <lb/>&longs;e æquilateris, & æquiangulis KLM, NOP, inter &longs;e con<lb/>gruentibus non congruat, &longs;i fieri pote&longs;t, pyramis ENOP, <lb/>pyramidi AKLM, &longs;ed cadat vertex E, pyramidis ENOP, <lb/>extra verticem A, pyramidis AKLM, & ex puncto A, <pb xlink:href="043/01/025.jpg" pagenum="17"/>ad centrum circuli tran&longs;euntis per tria puncta K, L, M, quod <lb/>&longs;it R, ducatur recta AR, & ER iungatur. </s>

<s>Quoniam igi<lb/>tur æquales rectæ &longs;unt AK, AL, AM, quæ ex puncto <lb/>A, in &longs;ublimi pertinent ad &longs;ubiectum planum: & punctum <lb/>R, e&longs;t centrum circuli tran&longs;euntis per puncta N, O, P; cadet <lb/>recta AR ad &longs;ubiectum planum perpendicularis. </s>

Line 348

<s><emph type="italics"/>PROPOSITIO VIII.<emph.end type="italics"/></s>

<s>Omnis pyramidis triangulam ba&longs;im habentis <lb/>quatuor axes &longs;ecant &longs;e in vno puncto in ea&longs;dem ra<pb pagenum="18"/>tiones, ita vt &longs;egmenta, quæ ad angulos, eo<lb/>rum, quæ ad oppo&longs;ita triangula, &longs;int tripla; ex quo <lb/>puncto tota pyramis diuiditur in quatuor pyrami <lb/>des æquales. </s>

<s>Omnis pyramidis triangulam ba&longs;im habentis <lb/>quatuor axes &longs;ecant &longs;e in vno puncto in ea&longs;dem ra<pb xlink:href="043/01/026.jpg" pagenum="18"/>tiones, ita vt &longs;egmenta, quæ ad angulos, eo<lb/>rum, quæ ad oppo&longs;ita triangula, &longs;int tripla; ex quo <lb/>puncto tota pyramis diuiditur in quatuor pyrami <lb/>des æquales. </s>

<s>Et in nullo alio puncto quatuor re<lb/>ctæ lineæ ductæ ab angulis ad triangula oppo&longs;ita <lb/>pyramidis &longs;ecant &longs;e&longs;e in ea&longs;dem rationes. </s>

Line 364

<s>Cum igitur &longs;it vt AL ad <lb/>LH, ita CE ad EH, hoc e&longs;t vt triangulum AFL ad <lb/>triangulum FLH, (&longs;i ducatur FH) ita triangulum CFE, <lb/>ad triangulum FEH, erunt inter &longs;e æqualia triangula <lb/>FEH, FLH. </s>

<s>Quare vt triangulum AFH, ad triangu<lb/>lum FLH, hoc e&longs;t vt AH ad HL, ita erit triangulum <lb/>AFH ad triangulum FEH, hoc e&longs;t AF ad FE: &longs;ed re<lb/>cta AH, e&longs;t tripla ip&longs;ius LH; igitur & AF, erit ip&longs;ius FE, <pb pagenum="19"/>tripla: &longs;ed vt AF, ad FE, ita e&longs;t CF, ad FL; tripla igi<lb/>tur erit CF, ip&longs;ius FL. </s>

<s>Quare vt triangulum AFH, ad triangu<lb/>lum FLH, hoc e&longs;t vt AH ad HL, ita erit triangulum <lb/>AFH ad triangulum FEH, hoc e&longs;t AF ad FE: &longs;ed re<lb/>cta AH, e&longs;t tripla ip&longs;ius LH; igitur & AF, erit ip&longs;ius FE, <pb xlink:href="043/01/027.jpg" pagenum="19"/>tripla: &longs;ed vt AF, ad FE, ita e&longs;t CF, ad FL; tripla igi<lb/>tur erit CF, ip&longs;ius FL. </s>

<s>Similiter o&longs;tenderemus rectas <lb/>AE, BM, &longs;ecare &longs;e &longs;e in ea&longs;dem rationes, ita vt &longs;egmen<lb/>ta, quæ ad angulos, &longs;int tripla eorum, quæ &longs;unt ad centra <lb/>E, M, quorum AF, e&longs;t tripla ip&longs;ius FE: in puncto igitur <lb/>F, &longs;ecant &longs;e rectæ lineæ AE, BM. </s>

Line 372

<s>Quatuor igitur <lb/>axes pyramidis ABCD, &longs;ecant&longs;e &longs;e in puncto F, in ea&longs;<lb/>dem rationes, ita vt <lb/>&longs;egmenta ad angulos, <lb/>&longs;int <expan abbr="reliquor&utilde;">reliquorum</expan> tripla. <lb/></s>

<s>Rur&longs;us, quia compo<lb/>nendo, & conuerten<lb/>do, e&longs;t vt FE ad EA, <lb/>ita FL ad LC: hoc <lb/>e&longs;t, vt pyramis BCD <lb/>F, ad pyramidem A <lb/>BCD, ita pyramis <lb/>ABDF, ad pyrami<lb/>dem CBDA, (pro<lb/>pter ba&longs;ium commu<lb/>nitatem, & vertices in <lb/>eadem recta linea) erit <lb/><figure id="fig9"/><lb/>pyramis ABDF, æqualis pyramidi BCDF. </s>

<s>Rur&longs;us, quia compo<lb/>nendo, & conuerten<lb/>do, e&longs;t vt FE ad EA, <lb/>ita FL ad LC: hoc <lb/>e&longs;t, vt pyramis BCD <lb/>F, ad pyramidem A <lb/>BCD, ita pyramis <lb/>ABDF, ad pyrami<lb/>dem CBDA, (pro<lb/>pter ba&longs;ium commu<lb/>nitatem, & vertices in <lb/>eadem recta linea) erit <lb/><figure id="id.043.01.027.1.jpg" xlink:href="043/01/027/1.jpg"/><lb/>pyramis ABDF, æqualis pyramidi BCDF. </s>

<s>Eadem ra<lb/>tione tam pyramis ACDF, quàm pyramis ABCF, æqua <lb/>lis e&longs;t pyramidi BCDF. </s>

Line 382

<s>Si enim fieri pote&longs;t &longs;ecent <lb/>&longs;e tales rectæ in ea&longs;dem rationes in alio puncto S. </s>

<s>Simi<pb pagenum="20"/>liter igitur vt ante o&longs;tenderemus, vnamquamque qua<lb/>tuor pyramidum, quarum communis vertex S, ba&longs;es au<lb/>tem triangula, quæ &longs;unt circa pyramidem ABCD, e&longs;se <lb/>quartam partem pyramidis ABCD. </s>

<s>Simi<pb xlink:href="043/01/028.jpg" pagenum="20"/>liter igitur vt ante o&longs;tenderemus, vnamquamque qua<lb/>tuor pyramidum, quarum communis vertex S, ba&longs;es au<lb/>tem triangula, quæ &longs;unt circa pyramidem ABCD, e&longs;se <lb/>quartam partem pyramidis ABCD. </s>

<s>Siue igitur pun<lb/>ctum S, cadat intra vnam priorum quatuor pyrami<lb/>dum, &longs;iue in earum aliquo latere, &longs;eu triangulo; nece&longs;<lb/>&longs;ario erit pars æquali toti; tam enim tota vna pyramis <lb/>quatuor priorum, quarum communis vertex F, quàm eius <lb/>pars, vna quatuor pyramidum po&longs;teriorum, quarum com<lb/>munis vertex S, erit eiu&longs;dem ABCD, pyramidis pars <lb/>quarta. </s>

Line 400

<s>Du<lb/>cantur enim rectæ DNH, BQH, LN: & po&longs;ita BE, du <lb/>pla ip&longs;ius BH, iungatur DOC, in triangulo DBH, & <lb/>ponatur DP, ip&longs;ius PE, tripla, & connectantur rectæ LP, <lb/>PH. </s>

<s>Quoniam igitur E, e&longs;t centrum trianguli ABC, <pb pagenum="21"/>erit axis DE, pyramidis ABCD, cuius axis &longs;egmentum <lb/>DP e&longs;t triplum ip&longs;ius PE: igitur P centrum erit pyra<lb/>midis ABCD. </s>

<s>Quoniam igitur E, e&longs;t centrum trianguli ABC, <pb xlink:href="043/01/029.jpg" pagenum="21"/>erit axis DE, pyramidis ABCD, cuius axis &longs;egmentum <lb/>DP e&longs;t triplum ip&longs;ius PE: igitur P centrum erit pyra<lb/>midis ABCD. </s>

<s>Et quoniam tres rectæ FK, KH, HF, <lb/>&longs;unt parallelæ tribus BD, DC, CB, pro vt inter &longs;e re&longs;pon<lb/>dent, vt KH, ip&longs;i LG, quoniam vtraque lateri DC, ob <lb/>latera triangulorum &longs;ecta proportionaliter in punctis K, H, <lb/>L, G: & &longs;ic de reliquis; erit pyramis A<emph type="italics"/>K<emph.end type="italics"/>FH, &longs;imilis toti <lb/>pyramidi ABCD. </s>

Line 408

<s>Rur&longs;us, <lb/>quoniam pyramidum <lb/>&longs;imilium latus AD e&longs;t <lb/>duplum lateris AK, ho <lb/>mologi; pyramis AB<lb/>CD, octupla erit py<lb/>ramidis AKFH, ob <lb/>triplicatam laterum ho <lb/>mologorum proportio <lb/>nem. </s>

<s>Similiter <expan abbr="vna-q&utilde;æque">vna<lb/>qunæque</expan> trium reliqua<lb/>rum pyramidum ab&longs;ci&longs; <lb/>&longs;arum erit octaua pars <lb/>pyramidis ABCD; <lb/><figure id="fig10"/><lb/>quatuor igitur pyramides ab&longs;ci&longs;&longs;æ &longs;imul &longs;umptæ dimi<lb/>dium erit pyramidis ABCD: & reliquum igitur &longs;oli<lb/>dum demptis quatuor pyramidibus, dimidium pyramidis <lb/>ABCD. </s>

<s>Similiter <expan abbr="vna-q&utilde;æque">vna<lb/>qunæque</expan> trium reliqua<lb/>rum pyramidum ab&longs;ci&longs; <lb/>&longs;arum erit octaua pars <lb/>pyramidis ABCD; <lb/><figure id="id.043.01.029.1.jpg" xlink:href="043/01/029/1.jpg"/><lb/>quatuor igitur pyramides ab&longs;ci&longs;&longs;æ &longs;imul &longs;umptæ dimi<lb/>dium erit pyramidis ABCD: & reliquum igitur &longs;oli<lb/>dum demptis quatuor pyramidibus, dimidium pyramidis <lb/>ABCD. </s>

<s>Dico reliquum &longs;olidum LKMGFH, e&longs;&longs;e <lb/>octaedrum. </s>

Line 416

<s>bina autem oppo&longs;ita e&longs;&longs;e parallela, & æqualia, <lb/>& &longs;imilia, &longs;ic o&longs;tendimus. </s>

<s>Quoniam enim triangulum <lb/>FGH, e&longs;t in plano trianguli ABC, plano trianguli KLM <lb/>parallelo; erit triangulum FGH, parallelum triangu-<pb pagenum="22"/>lo KLM: &longs;ed triangulum FGH, e&longs;t &longs;imile triangulo <lb/>ABC, & triangulum KLM, &longs;imile eidem triangulo <lb/>ABC; <expan abbr="triangul&utilde;">triangulum</expan> ergo FGH, &longs;imile erit triangulo KLM: <lb/>&longs;ed & æquale propter æqualitatem laterum homologo<lb/>rum. </s>

<s>Quoniam enim triangulum <lb/>FGH, e&longs;t in plano trianguli ABC, plano trianguli KLM <lb/>parallelo; erit triangulum FGH, parallelum triangu-<pb xlink:href="043/01/030.jpg" pagenum="22"/>lo KLM: &longs;ed triangulum FGH, e&longs;t &longs;imile triangulo <lb/>ABC, & triangulum KLM, &longs;imile eidem triangulo <lb/>ABC; <expan abbr="triangul&utilde;">triangulum</expan> ergo FGH, &longs;imile erit triangulo KLM: <lb/>&longs;ed & æquale propter æqualitatem laterum homologo<lb/>rum. </s>

<s>Similiter o&longs;tenderemus reliquum &longs;olidum LKM <lb/>GFH continentia triangula bina oppo&longs;ita æqualia <lb/>inter &longs;e, & &longs;imilia, & parallela; octaedrum e&longs;t igitur <lb/>LKMGFH. </s>

Line 424

<s>Quoniam enim DP, ponitur tripla ip&longs;ius PE, <lb/>& DO, e&longs;t æqualis <lb/>OE (&longs;iquidem planum <lb/>trianguli KLM, plano <lb/><expan abbr="triãguli">trianguli</expan> ABC, paralle <lb/>lum &longs;ecat proportione <lb/><expan abbr="o&etilde;s">oens</expan> rectas lineas, quæ <lb/>ex puncto D, in &longs;ubli<lb/>mi pertinent ad &longs;ubie<lb/>ctum planum trianguli <lb/>ABC) erit OP, ip&longs;i <lb/>PE, æqualis. </s>

<s>Et quo<lb/>niam BH e&longs;t dupla <lb/>ip&longs;ius QH, quarum <lb/>BE e&longs;t dupla ip&longs;ius <lb/><figure id="fig11"/><lb/>EH, &longs;iquidem E e&longs;t centrum trianguli ABC; erit reli<lb/>qua EH reliquæ EQ dupla: & quia e&longs;t vt LD ad DB, <lb/>ita LN ad BH, propter &longs;imilitudinem triangulorum, & <lb/>e&longs;t LD, dimidia ip&longs;ius BD, erit & LN, dimidia ip&longs;ius <lb/>BH: &longs;ed QH e&longs;t dimidia ip&longs;ius BH; æqualis igitur LN <lb/>ip&longs;i QH. </s>

<s>Et quo<lb/>niam BH e&longs;t dupla <lb/>ip&longs;ius QH, quarum <lb/>BE e&longs;t dupla ip&longs;ius <lb/><figure id="id.043.01.030.1.jpg" xlink:href="043/01/030/1.jpg"/><lb/>EH, &longs;iquidem E e&longs;t centrum trianguli ABC; erit reli<lb/>qua EH reliquæ EQ dupla: & quia e&longs;t vt LD ad DB, <lb/>ita LN ad BH, propter &longs;imilitudinem triangulorum, & <lb/>e&longs;t LD, dimidia ip&longs;ius BD, erit & LN, dimidia ip&longs;ius <lb/>BH: &longs;ed QH e&longs;t dimidia ip&longs;ius BH; æqualis igitur LN <lb/>ip&longs;i QH. </s>

<s>Iam igitur quia e&longs;t vt BE ad EH, ita <lb/>LO ad ON: &longs;ed BE, e&longs;t dupla ip&longs;ius EH; dupla igi<lb/>tur LO, erit ip&longs;ius ON: &longs;ed & QH erat dupla ip&longs;ius <lb/>QE; vt igitur LN ad NO, ita erit HQ ad QE: & <pb pagenum="23"/>per conuer&longs;ionem rationis, vt NL ad LO, ita QH, ad <lb/>HE: & permutando, vt LN ad QH, ita LO ad EH: <lb/>&longs;ed LN, o&longs;ten&longs;a e&longs;t æqualis QH; æqualis igitur LO, <lb/>erit ip&longs;i EH; &longs;ed & OP, e&longs;t æqualis ip&longs;i PE, vt o&longs;ten<lb/>dimus: duæ igitur LO, OP, duabus HE, EP æqua<lb/>les erunt altera alteri, & angulos æquales continent LOP, <lb/>PEH, parallelis exi&longs;tentibus LN, BH &longs;ectionibus tri<lb/>anguli DBH, quæ fiunt à duobus planis parallelis; ba<lb/>&longs;is igitur LP, trianguli LOP, æqualis e&longs;t ba&longs;i PH, <lb/>trianguli PEH, & angulus OPL, angulo EPH in pla<lb/>no trianguli DBH, in quo DPE, e&longs;t vna recta linea; <lb/>igitur LPH, erit vna recta linea, quæ cum &longs;it axis octa<lb/>edri LKMGFH, & &longs;ectus &longs;it in puncto P, bifariam, <lb/>erit punctum P, centrum octaedri LKMGEH. &longs;ed & <lb/>centrum pyramidis ABCD. </s>

<s>Iam igitur quia e&longs;t vt BE ad EH, ita <lb/>LO ad ON: &longs;ed BE, e&longs;t dupla ip&longs;ius EH; dupla igi<lb/>tur LO, erit ip&longs;ius ON: &longs;ed & QH erat dupla ip&longs;ius <lb/>QE; vt igitur LN ad NO, ita erit HQ ad QE: & <pb xlink:href="043/01/031.jpg" pagenum="23"/>per conuer&longs;ionem rationis, vt NL ad LO, ita QH, ad <lb/>HE: & permutando, vt LN ad QH, ita LO ad EH: <lb/>&longs;ed LN, o&longs;ten&longs;a e&longs;t æqualis QH; æqualis igitur LO, <lb/>erit ip&longs;i EH; &longs;ed & OP, e&longs;t æqualis ip&longs;i PE, vt o&longs;ten<lb/>dimus: duæ igitur LO, OP, duabus HE, EP æqua<lb/>les erunt altera alteri, & angulos æquales continent LOP, <lb/>PEH, parallelis exi&longs;tentibus LN, BH &longs;ectionibus tri<lb/>anguli DBH, quæ fiunt à duobus planis parallelis; ba<lb/>&longs;is igitur LP, trianguli LOP, æqualis e&longs;t ba&longs;i PH, <lb/>trianguli PEH, & angulus OPL, angulo EPH in pla<lb/>no trianguli DBH, in quo DPE, e&longs;t vna recta linea; <lb/>igitur LPH, erit vna recta linea, quæ cum &longs;it axis octa<lb/>edri LKMGFH, & &longs;ectus &longs;it in puncto P, bifariam, <lb/>erit punctum P, centrum octaedri LKMGEH. &longs;ed & <lb/>centrum pyramidis ABCD. </s>

<s>Manife&longs;tum e&longs;t igitur pro<lb/>po&longs;itum. </s>

Line 434

<s>Omne fru&longs;tum pyramidis triangulam ba&longs;im <lb/>habentis, &longs;iue coni, ad pyramidem, vel conum, cu<lb/>ius ba&longs;is e&longs;t eadem, quæ maior ba&longs;is fru&longs;ti, & ea<lb/>dem altitudo, eam habet proportionem, quam duo <lb/>latera homologa, vel duæ diametri ba&longs;ium ip&longs;ius <lb/>fru&longs;ti, vnà cum tertia minori proportionali ad <lb/>prædicta duo latera, vel diametros; ad maioris ba<lb/>&longs;is latus, vel diametrum. </s>

<s>Ad pri&longs;ma autem, vel <lb/>cylindrum, cuius eadem e&longs;t ba&longs;is, quæ maior ba&longs;is <lb/>fru&longs;ti, & eadem altitudo; vt tres prædictæ deìn<lb/>ceps proportionales &longs;imul, ad triplam lateris, vel <lb/>diametri maioris ba&longs;is. </s><pb pagenum="24"/>

<s>Sit fru&longs;tum ABCFGH, pyramidis, vel coni ABCD, <lb/>cuius ba&longs;is triangulum, vel circulus ABC, axis autem <lb/>DE: & vt e&longs;t AC ad FH, ita &longs;it FH ad N, & fru<lb/>&longs;ti axis EK, nec non idem pyramidis, vel coni AB <lb/>CK, vt &longs;it eadem altitudo. </s>

<s>Dico fru&longs;tum ABCF <lb/>GH, ad pyramidem, vel conum, ABCK, e&longs;se vt <lb/>tres lineas AC, FH, NO, &longs;imul ad ip&longs;ius AC, tri<lb/>plam: ad pri&longs;ma autem, vel cylindrum, cuius ba&longs;is ABC, <lb/>altitudo autem eadem cum fru&longs;to, vttres AC, FH, NO, <lb/>&longs;imul, ad ip&longs;ius AC, triplam. </s>

<s>Nam vt e&longs;t AC ad FH, <lb/>& FH ad NO, ita &longs;it NO ad P: & exce&longs;&longs;us, quo hæ <lb/><figure id="fig12"/><lb/>quatuor lineæ differunt, &longs;int AL, FM, <expan abbr="Oq.">Oque</expan> Ergo <lb/>vt AC ad FH, ita erit AL ad FM, & FM ad <expan abbr="Oq.">Oque</expan> <lb/>Quoniam igitur e&longs;t vt AC ad P, ita pyramis, vel conus <lb/>ABCD, ad &longs;imilem ip&longs;i pyramidem, vel conum DFGH, <lb/>ob triplicatam laterum homologorum proportionem; erit <lb/>diuidendo, vt tres AL, FM, OQ, &longs;imul ad P, ita fru<lb/>&longs;tum ABCFGH, ad pyramidem, vel conum DFGH: <lb/>&longs;ed conuertendo e&longs;t vt P, ad AC, ita pyramis, vel conus <lb/>DFGH, ad pyramidem, vel conum ABCD: ex æquali <lb/>igitur, vt tres AL, FM, OQ, &longs;imul ad AC, ita fru&longs;tum <pb pagenum="25"/>ABCDFGH, ad pyramidem, vel conum ABCD. <lb/></s>

<s>Nam vt e&longs;t AC ad FH, <lb/>& FH ad NO, ita &longs;it NO ad P: & exce&longs;&longs;us, quo hæ <lb/><figure id="id.043.01.032.1.jpg" xlink:href="043/01/032/1.jpg"/><lb/>quatuor lineæ differunt, &longs;int AL, FM, <expan abbr="Oq.">Oque</expan> Ergo <lb/>vt AC ad FH, ita erit AL ad FM, & FM ad <expan abbr="Oq.">Oque</expan> <lb/>Quoniam igitur e&longs;t vt AC ad P, ita pyramis, vel conus <lb/>ABCD, ad &longs;imilem ip&longs;i pyramidem, vel conum DFGH, <lb/>ob triplicatam laterum homologorum proportionem; erit <lb/>diuidendo, vt tres AL, FM, OQ, &longs;imul ad P, ita fru<lb/>&longs;tum ABCFGH, ad pyramidem, vel conum DFGH: <lb/>&longs;ed conuertendo e&longs;t vt P, ad AC, ita pyramis, vel conus <lb/>DFGH, ad pyramidem, vel conum ABCD: ex æquali <lb/>igitur, vt tres AL, FM, OQ, &longs;imul ad AC, ita fru&longs;tum <pb xlink:href="043/01/033.jpg" pagenum="25"/>ABCDFGH, ad pyramidem, vel conum ABCD. <lb/></s>

<s>Rur&longs;us quoniam axis DE, & latera pyramidis, vel coni <lb/>ABCD, &longs;ecantur plano trianguli, vel circuli FGH, ba&longs;i <lb/>ABC, parallelo; erit componendo, vt AD, ad DF, hoc <lb/>e&longs;t, vt AC ad FH, propter &longs;imilitudinem triangulorum, <lb/>hoc e&longs;t vt AC, ad CL, ita ED, ad DK; & per conuer<lb/>&longs;ionem rationis, vt AC, ad AL, ita DE, ad EK: &longs;ed vt <lb/>DE ad EK, ita e&longs;t pyramis, vel conus ABCD, ad py<lb/>ramidem, vel conum ABCK; vt igitur AC, ad AL, <lb/>ita e&longs;t pyramis, vel conus ABCD, ad pyramidem, vel <lb/>conum ABCK; &longs;ed vt tres lineæ AL, FM, OQ &longs;imul <lb/>ad AC, ita erat fru&longs;tum ABCFGH, ad pyramidem, <lb/>vel conum ABCD; ex æquali igitur, erit vt tres lineæ <lb/>AL, FM, OQ, &longs;imul ad AL, ita fru&longs;tum ABCFGH, <lb/>ad pyramidem, vel conum ABCK. Rur&longs;us, quoniam <lb/>tres exce&longs;&longs;us AL, FM, OQ, &longs;unt deinceps proportio<lb/>nales in proportione totidem terminorum AC, FH, NO, <lb/>erunt vt AL, FM, OQ, &longs;imul ad AL, ita AC, FH, <lb/>NO, &longs;imul ad AC: &longs;ed vt AL, FM, OQ, &longs;im ul ad <lb/>AL, ita erat fru&longs;tum ABCFGH, ad pyamidem, vel <lb/>conum ABCK; vt igitur tres lineæ AC, FH, NO, &longs;i<lb/>mul, ad AC, ita erit fru&longs;tum ABCFGH, ad pyrami<lb/>dem, vel conum ABCK. </s>

<s>Sed vt AC, ad &longs;ui triplam, ita <lb/>e&longs;t pyramis, vel conus ABCK ad pri&longs;ma, vel cylindrum, <lb/>cuius e&longs;t eadem ba&longs;is ABC, & eadem altitudo cum py<lb/>ramide, vel cono ABCK; ex æquali igitur, erit vt tres <lb/>lineæ AC, FH, NO, &longs;imul ad ip&longs;ius AC, triplam, ita <lb/>fru&longs;tum ABCFGH, ad pri&longs;ma, vel cylindrum, cu<lb/>ius ba&longs;is ABC, & eadem altitudo pyramidi, vel cono <lb/>ABCK: ide&longs;t eadem, fru&longs;to ABCFGH. </s>

<s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s><pb pagenum="26"/>

<s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s><pb xlink:href="043/01/034.jpg" pagenum="26"/>

<s><emph type="italics"/>PROPOSITIO XI.<emph.end type="italics"/></s>

Line 454

<s>Sit &longs;olidum ABC, circa axim AD, in alteram par<lb/>tem deficiens, cuius vertex A, ba&longs;is autem circulus, vel <lb/>ellyp&longs;is, cuius diameter BC. </s>

<s>Igitur &longs;uper hanc ba&longs;im <lb/>circa axim AD, <lb/>intelligatur de&longs;eri <lb/>ptus cylindrus, vel <lb/>cylindri portio <lb/>BL, quæ &longs;olidum <lb/>ABC, compre<lb/>hendet: &longs;ectoque <lb/>cylindro, vel cylin <lb/>dri portione BL, <lb/>planis ba&longs;i paralle <lb/><figure id="fig13"/><lb/>lis in tot cylindros, vel cylindri portiones æqualium al<lb/>ritudinum, vt quilibet eorum &longs;it minor magnitudine <lb/>propo&longs;ita; e&longs;to &longs;olidum ABC, &longs;ectum prædictis planis: <lb/>erunt autem &longs;ectiones circuli, vel ellyp&longs;es fimiles inter <lb/>&longs;e & ba&longs;i BC, &longs;olidi ABC &longs;uper quas &longs;ectiones tam<lb/>quam ba&longs;es cylindris, vel cylindri portionibus æqua<lb/>lium altitudinum intra, atque extra figuram con&longs;titutis, <lb/>quorum bini inter eadem plana parallela inter &longs;e refe-<pb pagenum="27"/>runtur, veluti BF, & GDH, quorum axis communis e&longs;t <lb/>D<emph type="italics"/>K<emph.end type="italics"/>, ba&longs;es autem circuli, vel ellyp&longs;es EF, GH, qua<lb/>rum commune centrum K: &longs;upremus autem, qui ad A, <lb/>ad nullum refertur. </s>

<s>Igitur &longs;uper hanc ba&longs;im <lb/>circa axim AD, <lb/>intelligatur de&longs;eri <lb/>ptus cylindrus, vel <lb/>cylindri portio <lb/>BL, quæ &longs;olidum <lb/>ABC, compre<lb/>hendet: &longs;ectoque <lb/>cylindro, vel cylin <lb/>dri portione BL, <lb/>planis ba&longs;i paralle <lb/><figure id="id.043.01.034.1.jpg" xlink:href="043/01/034/1.jpg"/><lb/>lis in tot cylindros, vel cylindri portiones æqualium al<lb/>ritudinum, vt quilibet eorum &longs;it minor magnitudine <lb/>propo&longs;ita; e&longs;to &longs;olidum ABC, &longs;ectum prædictis planis: <lb/>erunt autem &longs;ectiones circuli, vel ellyp&longs;es fimiles inter <lb/>&longs;e & ba&longs;i BC, &longs;olidi ABC &longs;uper quas &longs;ectiones tam<lb/>quam ba&longs;es cylindris, vel cylindri portionibus æqua<lb/>lium altitudinum intra, atque extra figuram con&longs;titutis, <lb/>quorum bini inter eadem plana parallela inter &longs;e refe-<pb xlink:href="043/01/035.jpg" pagenum="27"/>runtur, veluti BF, & GDH, quorum axis communis e&longs;t <lb/>D<emph type="italics"/>K<emph.end type="italics"/>, ba&longs;es autem circuli, vel ellyp&longs;es EF, GH, qua<lb/>rum commune centrum K: &longs;upremus autem, qui ad A, <lb/>ad nullum refertur. </s>

<s>Quoniam igitur ex con&longs;tructione, <lb/>cylindrus, vel cylindri portio BF, e&longs;t minor magnitudi<lb/>ne propo&longs;ita; exce&longs;sus autem omnes, quibus cylindri, ex <lb/>quibus con&longs;tat figura circum&longs;cripta, excedunt eos, ex qui<lb/>bus con&longs;tat figura in&longs;cripta, pro vt bini inter &longs;e referun<lb/>tur, vna cum &longs;upremo, qui ad nullum refertur, &longs;unt æqua<lb/>les cylindro, vel cylindri portioni BF, figura circum<lb/>&longs;cripta &longs;olido ABC, excedet in&longs;criptam minori exce&longs;<lb/>&longs;u magnitudine propo&longs;ita. </s>

Line 470

<s>Per punctum igitur E, extendatur pla<lb/>num erectum ad lineam ED, & vt e&longs;t DE, ad BC, ita <lb/>fiat ba&longs;is AC, ad quadratum F: & ad punctum E, in <lb/>plano erecto ad lineam ED, quartæ parti quadrati F, <lb/>æquale GE, quadratum de&longs;cribatur, & compleatur <lb/>quadratum GH, quadruplum quadrati EG, &longs;eu qua<lb/>drato F, æquale: & ex puncto K, erecta KL, ip&longs;i EF, <lb/>æquali, & ad &longs;ubiectum planum perpendiculari &longs;uper ba<lb/>&longs;im GH, con&longs;tituatur parallelepipedum GK. </s>

<s>Dico <pb pagenum="28"/>parallelepipedum GK, e&longs;se æquale parallelepipedo AB; <lb/>& rectam DE, axim parallelepipedi GK. </s>

<s>Dico <pb xlink:href="043/01/036.jpg" pagenum="28"/>parallelepipedum GK, e&longs;se æquale parallelepipedo AB; <lb/>& rectam DE, axim parallelepipedi GK. </s>

<s>Iungantur <lb/>enim ba&longs;ium oppo&longs;itarum diametri GH, LK. </s>

<s>Quo<lb/>niam igitur qua<lb/>drata &longs;unt EG, <lb/>GH, communem<lb/>que habent angu<lb/>lum, qui ad G, <lb/>con&longs;i&longs;tent circa di<lb/>ametrum GH; in <lb/>recta igitur GH, <lb/>erit punctum E. <lb/></s>

<s>Et quoniam qua<lb/>dratum GH, e&longs;t <lb/>quadrati EG, qua<lb/>druplum; erit dia<lb/><figure id="fig14"/><lb/>meter GH, diametri EG, dupla; punctum igitur E, <lb/>erit in medio diametri GH. Rur&longs;us, quoniam ob pa<lb/>rallelepipedum GK, recta GL, æqualis e&longs;t, & paral<lb/>lela ip&longs;i KH, erit LH, parallelogrammum: & quia <lb/>vtraque DE, KH, e&longs;t ad &longs;ubiectum planum perpendi<lb/>cularis, parallelæ erunt, & in eodem plano parallelogram<lb/>mi LH; in quo cum LG, &longs;it parallela ip&longs;i KH; erit & <lb/>ED, ip&longs;i LG, parallela: e&longs;t autem, & æqualis vtrilibet <lb/>ip&longs;arum GL, GH, oppo&longs;itarum; punctum igitur D, e&longs;t <lb/>in recta LK, & tam KD, ip&longs;i EH, quàm LD, ip&longs;i <lb/>EG, æqualis erit, & inter &longs;e æquales LD, DK. pun<lb/>ctum igitur D, erit in medio diametri LK; &longs;ed & pun<lb/>ctum E, erat in medio diametri GH; recta igitur ED, <lb/>axis e&longs;t parallelepipedi GK, cuius parallelepipedi cum <lb/>altitudo DE, &longs;it ad BC, altitudinem parallelepipedi AB, <lb/>vt e&longs;t ba&longs;is AC, ad quadratum F, hoc e&longs;t ad ba&longs;im GH, <lb/>parallelepipedi GK; parallelepipedum GK, parallelepipe <lb/>do AB, æquale erit, Factum igitur e&longs;t quod oportebat. </s><pb pagenum="29"/>

<s>Et quoniam qua<lb/>dratum GH, e&longs;t <lb/>quadrati EG, qua<lb/>druplum; erit dia<lb/><figure id="id.043.01.036.1.jpg" xlink:href="043/01/036/1.jpg"/><lb/>meter GH, diametri EG, dupla; punctum igitur E, <lb/>erit in medio diametri GH. Rur&longs;us, quoniam ob pa<lb/>rallelepipedum GK, recta GL, æqualis e&longs;t, & paral<lb/>lela ip&longs;i KH, erit LH, parallelogrammum: & quia <lb/>vtraque DE, KH, e&longs;t ad &longs;ubiectum planum perpendi<lb/>cularis, parallelæ erunt, & in eodem plano parallelogram<lb/>mi LH; in quo cum LG, &longs;it parallela ip&longs;i KH; erit & <lb/>ED, ip&longs;i LG, parallela: e&longs;t autem, & æqualis vtrilibet <lb/>ip&longs;arum GL, GH, oppo&longs;itarum; punctum igitur D, e&longs;t <lb/>in recta LK, & tam KD, ip&longs;i EH, quàm LD, ip&longs;i <lb/>EG, æqualis erit, & inter &longs;e æquales LD, DK. pun<lb/>ctum igitur D, erit in medio diametri LK; &longs;ed & pun<lb/>ctum E, erat in medio diametri GH; recta igitur ED, <lb/>axis e&longs;t parallelepipedi GK, cuius parallelepipedi cum <lb/>altitudo DE, &longs;it ad BC, altitudinem parallelepipedi AB, <lb/>vt e&longs;t ba&longs;is AC, ad quadratum F, hoc e&longs;t ad ba&longs;im GH, <lb/>parallelepipedi GK; parallelepipedum GK, parallelepipe <lb/>do AB, æquale erit, Factum igitur e&longs;t quod oportebat. </s><pb xlink:href="043/01/037.jpg" pagenum="29"/>

<s><emph type="italics"/>PROPOSITIO XIII.<emph.end type="italics"/></s>

Line 490

<s>Quoniam igitur e&longs;t vt <lb/>&longs;olidum BC, ad &longs;olidum A, ita recta linea, &longs;iue latus BD, <lb/>ad aliam rectam lineam; producto latere BD, &longs;it vt BC, <lb/>ad A, ita recta BD, ad rectam DE, & compleatur pa<lb/>rallelepipedum CE. </s>

<s>Quoniam itaque e&longs;t vt BD, ad DE, <lb/>ita parallelogrammum &longs;iue ba&longs;is BG, ad parallelogram<lb/><figure id="fig15"/><lb/>mum, &longs;iue ba&longs;im EG; hoc e&longs;t parallelepipedum BC, ad <lb/>parallelepipedum CE: &longs;ed vt BD, ad DE, ita e&longs;t paral<lb/>lelepipedum BC, ad &longs;olidum A; vt igitur parallelepipe<lb/>dum BC, ad &longs;olidum A, ita erit parallelepipedum BC, <lb/>ad parallelepipedum CE; parallelepipedum igitur CE <lb/>æquale erit &longs;olido A. </s>

<s>Quoniam itaque e&longs;t vt BD, ad DE, <lb/>ita parallelogrammum &longs;iue ba&longs;is BG, ad parallelogram<lb/><figure id="id.043.01.037.1.jpg" xlink:href="043/01/037/1.jpg"/><lb/>mum, &longs;iue ba&longs;im EG; hoc e&longs;t parallelepipedum BC, ad <lb/>parallelepipedum CE: &longs;ed vt BD, ad DE, ita e&longs;t paral<lb/>lelepipedum BC, ad &longs;olidum A; vt igitur parallelepipe<lb/>dum BC, ad &longs;olidum A, ita erit parallelepipedum BC, <lb/>ad parallelepipedum CE; parallelepipedum igitur CE <lb/>æquale erit &longs;olido A. </s>

<s>Quod fieri po&longs;se propo&longs;uimus. </s><pb pagenum="30"/>

<s>Quod fieri po&longs;se propo&longs;uimus. </s><pb xlink:href="043/01/038.jpg" pagenum="30"/>

<s><emph type="italics"/>PROPOSITIO XIV.<emph.end type="italics"/></s>

Line 504

<s>Trianguli enim ABC, &longs;it centrum grauitatis H; iuncta<lb/>que HG, & producta, ponatur GK, æqualis GH, & re<lb/>ctæ à punctis K, H, ad angulos ducantur. </s>

<s>Quoniam igi<lb/>tur AG, e&longs;t æqualis GC, & <lb/>GH, ip&longs;i GK, & angulus <lb/>AGK, æqualis angulo CGH, <lb/>erit ba&longs;is AK, æqualis ba&longs;i <lb/>CH, & angulus GAK, æqua<lb/>lis angulo GCK: &longs;ed totus <lb/>angulus DAK, æqualis e&longs;t to <lb/>ti angulo BCA; reliquus igi<lb/>tur DAK, reliquo BCH, <lb/>æqualis erit, circa quos angu<lb/>los latus BC e&longs;t æquale lateri <lb/>AD, & CH, ip&longs;i AK; angu<lb/>lus igitur CBH, æqualis erit <lb/><figure id="fig16"/><lb/>angulo ADK. </s>

<s>Similiter o&longs;tenderemus angulum CAH, <lb/>angulo ACK, & angulum BAH, angulo DCK, & an<lb/>gulum ABH, angulo CDK, æquales e&longs;se: &longs;ed latera <lb/>triangulorum, cum quibus rectæ ductæ à punctis K, H, ad <lb/>angulos triangulorum &longs;imilium ABC, CDA, &longs;unt ho-<pb pagenum="31"/>mologa; puncta igitur K, H, in prædictis triangulis &longs;unt <lb/>&longs;imiliter po&longs;ita. </s>

<s>Similiter o&longs;tenderemus angulum CAH, <lb/>angulo ACK, & angulum BAH, angulo DCK, & an<lb/>gulum ABH, angulo CDK, æquales e&longs;se: &longs;ed latera <lb/>triangulorum, cum quibus rectæ ductæ à punctis K, H, ad <lb/>angulos triangulorum &longs;imilium ABC, CDA, &longs;unt ho-<pb xlink:href="043/01/039.jpg" pagenum="31"/>mologa; puncta igitur K, H, in prædictis triangulis &longs;unt <lb/>&longs;imiliter po&longs;ita. </s>

<s>Rur&longs;us quoniam angulus ABC, non <lb/>e&longs;t minor recto, acuti erunt reliqui ACB, BAC; igitur <lb/>latus AC, maximum erit: ponitur autem AB maius, <lb/>quàm BC; triangulum igitur ABC, &longs;calenum erit. <lb/></s>

Line 520

<s>Sed &longs;int omnia latera æqualia <expan abbr="parallelogrãmi">parallelogrammi</expan> ABCD, <lb/>Sectisque duobus lateribus AD, BC, bifariam in E, F <lb/>iungantur EF, AE, ED, <lb/>AGC, & per punctum G, <lb/>ducatur ip&longs;i AD, vel BC, <lb/>parallela HGK. </s>

<s>Quoniam <lb/>igitur EC, e&longs;t æqualis <lb/>AF, erit CG æqualis AG, <lb/>& EG, æqualis GF, pro<lb/>pter &longs;imilitudinem triangu <lb/>lorum: nec non EH, ip&longs;i <lb/>AH, & EK, ip&longs;i KD: tres <lb/>igitur diametri AC, AE, <lb/>ED, erunt &longs;ectæ bifariam <lb/><figure id="fig17"/><lb/>in punctis K, G, H: & quoniam ex æquali propter triangu<lb/>la &longs;imilia e&longs;t vt AF, ad FD, ita HG, ad GK, erit HG, <lb/>æqualis ip&longs;i GK: &longs;ed puncta K, H, &longs;unt centra grauitatis <lb/>parallelogrammorum BF, FC; igitur totius parallelo<lb/>grammi ABCD, centrum grauitatis erit G, in medio <pb pagenum="32"/>diametri AG. </s>

<s>Quoniam <lb/>igitur EC, e&longs;t æqualis <lb/>AF, erit CG æqualis AG, <lb/>& EG, æqualis GF, pro<lb/>pter &longs;imilitudinem triangu <lb/>lorum: nec non EH, ip&longs;i <lb/>AH, & EK, ip&longs;i KD: tres <lb/>igitur diametri AC, AE, <lb/>ED, erunt &longs;ectæ bifariam <lb/><figure id="id.043.01.039.1.jpg" xlink:href="043/01/039/1.jpg"/><lb/>in punctis K, G, H: & quoniam ex æquali propter triangu<lb/>la &longs;imilia e&longs;t vt AF, ad FD, ita HG, ad GK, erit HG, <lb/>æqualis ip&longs;i GK: &longs;ed puncta K, H, &longs;unt centra grauitatis <lb/>parallelogrammorum BF, FC; igitur totius parallelo<lb/>grammi ABCD, centrum grauitatis erit G, in medio <pb xlink:href="043/01/040.jpg" pagenum="32"/>diametri AG. </s>

<s>Quod e&longs;t propo&longs;itum. </s>

Line 534

<s>Sit parallelogrammum ABCD, &longs;ectum in duo paral<lb/>lelogramma AE, ED, & <lb/>parallelogrammi AE, &longs;it <lb/>centrum grauitatis H, pa<lb/>rallelogrammi autem ED, <lb/>centrum grauitatis K: & <lb/>parallelogrammi ABCD, <lb/>&longs;it centrum grauitatis G: <lb/>& iungatur KH. </s>

<s>Dico re<lb/>ctam KH, diuidi à puncto <lb/>G, ita vt &longs;it KG, ad G <lb/>H, vt e&longs;t parallelogrammum <lb/>AE, ad parallelogrammum <lb/><figure id="fig18"/><lb/>ED, Iungantur enim diametri AC, AE, ED. </s>

<s>Dico re<lb/>ctam KH, diuidi à puncto <lb/>G, ita vt &longs;it KG, ad G <lb/>H, vt e&longs;t parallelogrammum <lb/>AE, ad parallelogrammum <lb/><figure id="id.043.01.040.1.jpg" xlink:href="043/01/040/1.jpg"/><lb/>ED, Iungantur enim diametri AC, AE, ED. </s>

<s>Igitur <pb pagenum="33"/>per præcedentem &longs;ectæ erunt hæ diametri bifariam in pun<lb/>ctis H, G, K. </s>

<s>Igitur <pb xlink:href="043/01/041.jpg" pagenum="33"/>per præcedentem &longs;ectæ erunt hæ diametri bifariam in pun<lb/>ctis H, G, K. </s>

<s>Quoniam igitur e&longs;t vt EH, ad HA, ita <lb/>EK ad KD, parallela erit KH, ip&longs;i AD; igitur & EC; <lb/>&longs;ed recta KH, &longs;ecat latus AE, trianguli AEC, bifariam <lb/>in puncto H, ergo & latus AC, bifariam &longs;ecabit; igitur <lb/>in puncto G. punctum igitur G, e&longs;t in linea KH. Rur&longs;us, <lb/>quoniam e&longs;t vt GA, ad AC, ita GH, ad EC, propter &longs;i<lb/>militudinem triangulorum; &longs;ed dimidia e&longs;t GA, ip&longs;ius <lb/>AC, igitur & GH, erit dimidia ip&longs;ius EC, hoc e&longs;t ip&longs;ius <lb/>FD. </s>

Line 556

<s>Si autem inæquales, ab&longs;cindatur BD, <lb/>æqualis AC, vt &longs;it AD, ad DB, vt BC, ad CA. </s>

<s>Et quo<lb/>niam &longs;pacio R, rectangulum æquale pote&longs;t e&longs;se; applice<lb/>tur ad lineam BD, rectangulum BDKE, æquale quar<lb/>tæ parti rectanguli æqualis ip&longs;i R, hoc e&longs;t quartæ parti <lb/>ip&longs;ius R; & po&longs;ita DG, æquali, & in directum ip&longs;i DK, <pb pagenum="34"/>ducantur rectæ GBH, GAF, quæ cum KE, produ<lb/>cta conueniant in punctis F, H: & fiant parallelogramma <lb/>FL, AK. </s>

<s>Et quo<lb/>niam &longs;pacio R, rectangulum æquale pote&longs;t e&longs;se; applice<lb/>tur ad lineam BD, rectangulum BDKE, æquale quar<lb/>tæ parti rectanguli æqualis ip&longs;i R, hoc e&longs;t quartæ parti <lb/>ip&longs;ius R; & po&longs;ita DG, æquali, & in directum ip&longs;i DK, <pb xlink:href="043/01/042.jpg" pagenum="34"/>ducantur rectæ GBH, GAF, quæ cum KE, produ<lb/>cta conueniant in punctis F, H: & fiant parallelogramma <lb/>FL, AK. </s>

<s>Quoniam igitur e&longs;t vt N, ad R, ita BC, ad <lb/>CA, hoc e&longs;t AD, ad DB, hoc e&longs;t rectangulum AK, ad <lb/>rectangulum BK; erit permutando vt rectangulum AK, <lb/>ad N, ita rectangulum BK, ad R; &longs;ed rectangulum BK, <lb/>e&longs;t pars quarta ip&longs;ius R, ergo & rectangulum AK, erit <lb/>pars quarta ip&longs;ius N. </s>

<s>Rur&longs;us quia e&longs;t vt GD, ad D<emph type="italics"/>K<emph.end type="italics"/>, <lb/>ita GA, ad AF, & GB, ad BH: &longs;ed GD e&longs;t æqualis <lb/>DK; ergo & GA, ip&longs;i AF, & GB, ip&longs;i BH, æquales <lb/>erunt & centra grauita<lb/>tis A, quidem rectangu<lb/>li MK, B, vero rectan<lb/>guli KL, & rectangulum <lb/>AK, pars quarta ip&longs;ius <lb/>M<emph type="italics"/>K<emph.end type="italics"/>, quemadmodum <lb/>& B<emph type="italics"/>K<emph.end type="italics"/> ip&longs;ius KL; &longs;ed <lb/>N, rectanguli AK, qua<lb/>druplum erat, quemad<lb/>modum & R ip&longs;ius BK; <lb/>igitur rectangulum MK, <lb/>&longs;pacio N, & rectangulum <lb/>KL, &longs;pacio R, æquale <lb/>erit. </s>

<s>Sed vt BC, ad CA, <lb/>ita e&longs;t N, ad R; vt igi<lb/>tur BC, ad CA, ita <lb/><figure id="fig19"/><lb/>rectangulum MK, ad rectangulum KL; &longs;ed A e&longs;t cen<lb/>trum grauitatis rectanguli MK, & B, rectanguli KL; to<lb/>tius ergo rectanguli FL, hoc e&longs;t duorum rectangulorum <lb/>MK, KL, &longs;imul centrum grauitatis erit C. </s>

<s>Sed vt BC, ad CA, <lb/>ita e&longs;t N, ad R; vt igi<lb/>tur BC, ad CA, ita <lb/><figure id="id.043.01.042.1.jpg" xlink:href="043/01/042/1.jpg"/><lb/>rectangulum MK, ad rectangulum KL; &longs;ed A e&longs;t cen<lb/>trum grauitatis rectanguli MK, & B, rectanguli KL; to<lb/>tius ergo rectanguli FL, hoc e&longs;t duorum rectangulorum <lb/>MK, KL, &longs;imul centrum grauitatis erit C. </s>

<s>Sed rectan<lb/>gulo MK, æquale e&longs;t &longs;pacium N; & rectangulo KL, &longs;pa<lb/>cium R. </s>

<s>Igitur &longs;i pro rectangulo MK, &longs;it &longs;u&longs;pen&longs;um N <lb/>&longs;pacium &longs;ecundum centrum grauitatis in puncto A, & pro <lb/>rectangulo KL, &longs;pacium R, &longs;ecundum centrum graui-<pb pagenum="35"/>tatis in puncto B, &longs;pacia N, R, æquiponderabunt à lon<lb/>gitudinibus AC, CB; eritque vtriu&longs;que plani N, R, &longs;i<lb/>mul centrum grauitatis C. </s>

<s>Igitur &longs;i pro rectangulo MK, &longs;it &longs;u&longs;pen&longs;um N <lb/>&longs;pacium &longs;ecundum centrum grauitatis in puncto A, & pro <lb/>rectangulo KL, &longs;pacium R, &longs;ecundum centrum graui-<pb xlink:href="043/01/043.jpg" pagenum="35"/>tatis in puncto B, &longs;pacia N, R, æquiponderabunt à lon<lb/>gitudinibus AC, CB; eritque vtriu&longs;que plani N, R, &longs;i<lb/>mul centrum grauitatis C. </s>

<s>Quod demon&longs;trandum erat. </s>

Line 582

<s>Dico <lb/>reliquæ partis BC, centrum <lb/>grauitatis, quod &longs;it G, e&longs;se in <lb/>linea EH. </s>

<s>Quoniam enim D, <lb/>G, &longs;unt centra grauitatis par<lb/><figure id="fig20"/><lb/>tium AB, BC, cadet totius ABC, centrum grauitatis <pb pagenum="36"/>E, in recta linea, quæ iungit centra D, G; tria igitur pun<lb/>cta D, E, G, &longs;unt in eadem recta linea. </s>

<s>Quoniam enim D, <lb/>G, &longs;unt centra grauitatis par<lb/><figure id="id.043.01.043.1.jpg" xlink:href="043/01/043/1.jpg"/><lb/>tium AB, BC, cadet totius ABC, centrum grauitatis <pb xlink:href="043/01/044.jpg" pagenum="36"/>E, in recta linea, quæ iungit centra D, G; tria igitur pun<lb/>cta D, E, G, &longs;unt in eadem recta linea. </s>

<s>in qua igitur &longs;unt <lb/>puncta D, E, in eadem e&longs;t punctum G; &longs;ed puncta D, E, &longs;unt <lb/>in recta DH; igitur & punctum G, erit in recta DH: &longs;ed <lb/>extra ip&longs;am DE, vt modo o&longs;tendimus, in reliqua igitur <lb/>EH. </s>

Line 602

<s>Quoniam <lb/>igitur partium AC, CD, <lb/>centra grauitatis &longs;unt E, <lb/>F; erit totius AB, in re<lb/>cta EF, centrum graui<lb/>tatis: &longs;ed & in puncto E, <lb/>vnius ergo magnitudinis <lb/>duo centra grauitatis e<lb/>runt. </s>

<s>Quod e&longs;t ab&longs;urdum; <lb/><figure id="fig21"/><lb/>idem igitur E erit centrum grauitatis vtriuslibet partium <lb/>AC, CD. </s>

<s>Quod e&longs;t ab&longs;urdum; <lb/><figure id="id.043.01.044.1.jpg" xlink:href="043/01/044/1.jpg"/><lb/>idem igitur E erit centrum grauitatis vtriuslibet partium <lb/>AC, CD. </s>

<s>Sed vtriuslibet partium AC, CD, &longs;it cen<lb/>trum grauitatis E. </s>

<s>Dico idem E totius AB, e&longs;se cen-<pb pagenum="37"/>trum grauitatis. </s>

<s>Dico idem E totius AB, e&longs;se cen-<pb xlink:href="043/01/045.jpg" pagenum="37"/>trum grauitatis. </s>

<s>Si enim non e&longs;t, erit aliud, e&longs;to G: & <lb/>iunctatur EG, producatur ad partes G, in infinitum v&longs;<lb/>que ìn F. </s>

Line 630

<s>Et quoniam triangulum EDF, <lb/>dimidium e&longs;t cuius vis trium parallelogrammorum AF, <lb/>BD, CE, æqualia inter &longs;e erunt ea parallelogramma <lb/>omnifariam &longs;umpta, quorum centra grauitatis H, K, R; <lb/>intelligantur autem tria parallelogramma AF, BD, CE, <lb/>di&longs;tincta penitus, ita vt inter &longs;e congruant &longs;ecundum tria <lb/>triangula DEF, inter &longs;e congruentia: trium igitur trian <lb/>gulorum DEF, inter &longs;e congruentium & centra grauita<lb/>tis inter &longs;e congruent in puncto M. </s>

<s>Quoniam igitur in<lb/>ter duas parallelas EF, KH, &longs;ecant &longs;e rectæ lineæ FH, <lb/>LR, in puncto G; erit vt FG, ad GH, ita RG, ad GL; <pb pagenum="38"/>dupla igitur RG, e&longs;t ip&longs;ius GL. </s>

<s>Quoniam igitur in<lb/>ter duas parallelas EF, KH, &longs;ecant &longs;e rectæ lineæ FH, <lb/>LR, in puncto G; erit vt FG, ad GH, ita RG, ad GL; <pb xlink:href="043/01/046.jpg" pagenum="38"/>dupla igitur RG, e&longs;t ip&longs;ius GL. </s>

<s>Et quoniam in triangu<lb/>lo AGC, recta GD, &longs;ecat AC, bifariam in puncto D; <lb/>ip&longs;i AC, parallelam KH, bifariam &longs;ecabit in puncto L, <lb/>duorum igitur æqualium parallelogrammorum AF, EG; <lb/>&longs;imul, quorum centra grauitatis &longs;unt K, H, centrum gra<lb/>uitatis erit L. </s>

<s>Sed duo parallelogramma AF, EC, &longs;i<lb/>mul &longs;unt paralle<lb/>logrammi BD, du <lb/>plum; trium igitur <lb/>parallelogrammo<lb/>rum AF, EC, <lb/>BD, &longs;imul: hoc <lb/>e&longs;t <expan abbr="triãguli">trianguli</expan> ABC, <lb/>vnà cum duobus <lb/>trium <expan abbr="triangulor&utilde;">triangulorum</expan> <lb/>inter &longs;e congruen<lb/>tium EDF, cen<lb/>trum grauitatis e<lb/>rit G. </s>

<s>Sed triangu <lb/>li ABC, ponitur <lb/><figure id="fig22"/><lb/>centrum grauitatis N; producta igitur NG, occurret <lb/>centro M, reliquæ partis, ide&longs;t duorum triangulorum DEF; <lb/>quare vt triangulum ABC, ad duo triangula DEF, &longs;i<lb/>mul, ita erit MG, ad GN. </s>

<s>Sed triangu <lb/>li ABC, ponitur <lb/><figure id="id.043.01.046.1.jpg" xlink:href="043/01/046/1.jpg"/><lb/>centrum grauitatis N; producta igitur NG, occurret <lb/>centro M, reliquæ partis, ide&longs;t duorum triangulorum DEF; <lb/>quare vt triangulum ABC, ad duo triangula DEF, &longs;i<lb/>mul, ita erit MG, ad GN. </s>

<s>Sed triangulum ABC, e&longs;t <lb/>duplum duorum triangulorum EDF: igitur & MG, erit <lb/>ip&longs;ius GN, dupla. </s>

Line 646

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

<s>Non igitur centrum <pb pagenum="39"/>grauitatis trianguli ABC, erit aliud à puncto G: pun<lb/>ctum igitur G, erit centrum grauitatis trianguli ABC. <lb/></s>

<s>Non igitur centrum <pb xlink:href="043/01/047.jpg" pagenum="39"/>grauitatis trianguli ABC, erit aliud à puncto G: pun<lb/>ctum igitur G, erit centrum grauitatis trianguli ABC. <lb/></s>

<s>Quod demon&longs;trandum erat. </s>

Line 658

<s>per<lb/>mittitur igitur aduer&longs;ario ponere centrum grauitatis trian<lb/>guli, vbicumque vult intra illius limites. </s>

<s>atqui cum datis <lb/>duobus triangulis i&longs;o&longs;celiis &longs;imilibus, & in altero eorum <lb/>dato puncto, quod non &longs;it in prædicta recta linea, po&longs;sint <lb/>in altero duo puncta prædicto &longs;imiliter po&longs;ita inueniri, quo<lb/>rum vnum duntaxat concedet aduer&longs;arius e&longs;se alterius <lb/>trianguli centrum grauitatis, non autem non &longs;imiliter po<lb/>&longs;itum, ex quo ab&longs;urdum infertur partem anguli æqualem <lb/>e&longs;se toti: quid quod datis duobus triangulis æquilateris, & <lb/>in altero eorum dato puncto, quod non &longs;it centrum trian-<pb pagenum="40"/>guli, &longs;ed aliqua earum, quæ ab angulis ad bipartitorum <lb/>laterum &longs;ectiones cadunt, nece&longs;se e&longs;t in altero triangulo <lb/>tria puncta prædicto puncto e&longs;se &longs;imiliter po&longs;ita? </s>

<s>atqui cum datis <lb/>duobus triangulis i&longs;o&longs;celiis &longs;imilibus, & in altero eorum <lb/>dato puncto, quod non &longs;it in prædicta recta linea, po&longs;sint <lb/>in altero duo puncta prædicto &longs;imiliter po&longs;ita inueniri, quo<lb/>rum vnum duntaxat concedet aduer&longs;arius e&longs;se alterius <lb/>trianguli centrum grauitatis, non autem non &longs;imiliter po<lb/>&longs;itum, ex quo ab&longs;urdum infertur partem anguli æqualem <lb/>e&longs;se toti: quid quod datis duobus triangulis æquilateris, & <lb/>in altero eorum dato puncto, quod non &longs;it centrum trian-<pb xlink:href="043/01/048.jpg" pagenum="40"/>guli, &longs;ed aliqua earum, quæ ab angulis ad bipartitorum <lb/>laterum &longs;ectiones cadunt, nece&longs;se e&longs;t in altero triangulo <lb/>tria puncta prædicto puncto e&longs;se &longs;imiliter po&longs;ita? </s>

<s>quod &longs;i <lb/>etiam extra i&longs;tas lineas cadat vnius trianguli punctum, ne<lb/>ce&longs;se e&longs;t illi &longs;ex puncta in altero triangulo e&longs;se &longs;imiliter po<lb/>&longs;ita: &longs;ed &longs;i quod diximus de i&longs;o&longs;celiis &longs;imilibus, & æquila<lb/>teris triangulis demon&longs;trauerimus, rem velut ante oculos <lb/>expo&longs;uerimus. </s>

Line 666

<s>Datis duobustriangulis i&longs;o&longs;celijs &longs;imilibus, & <lb/>in altero eorum dato puncto extra rectam, quæ à <lb/>vertice ad medium ba&longs;is cadit, duo puncta in re<lb/>liquo triangulo prædicto puncto &longs;imiliter po&longs;ita <lb/>inuenire. </s>

<s>Sint duo triangula i&longs;o&longs;celia, & &longs;imilia ABC, DEF: <lb/>quorum in altero ABC, à vertice A, ad ba&longs;im BC, bi<lb/>partitam in puncto G, cadat recta AG: atque extra hanc <lb/><figure id="fig23"/><lb/>in triangulo ABC, &longs;it quoduis punctum H: & iuncta AH, <lb/>fiat angulus EDK æqualis angulo BAH; & vt BA, ad <pb pagenum="41"/>AH, ita fiat ED, ad DK: & quoniam angulus BAG, <lb/>æqualis e&longs;t angulo EDF: quorum angulus EDK, <lb/>æqualis e&longs;t angulo BAH, erit reliquus angulus <emph type="italics"/>K<emph.end type="italics"/>DF, <lb/>æqualis reliquo angulo HAC; &longs;ed angulus HAC, e&longs;t <lb/>maior angulo BAH; ergo & angulus KDF, maior erit <lb/>angulo BAH; po&longs;ito igitur angulo FDL, æquali an<lb/>gulo BAH, ac proinde minori, quàm &longs;it angulus FD<emph type="italics"/>K<emph.end type="italics"/>, <lb/>fiat vt BA, ad AH, ita FD, ad DL. Dico, in triangu<lb/>lo EDF, duo puncta K, L, &longs;imiliter po&longs;ita e&longs;se ac pun<lb/>ctum H, in triangulo BAC. </s>

<s>Sint duo triangula i&longs;o&longs;celia, & &longs;imilia ABC, DEF: <lb/>quorum in altero ABC, à vertice A, ad ba&longs;im BC, bi<lb/>partitam in puncto G, cadat recta AG: atque extra hanc <lb/><figure id="id.043.01.048.1.jpg" xlink:href="043/01/048/1.jpg"/><lb/>in triangulo ABC, &longs;it quoduis punctum H: & iuncta AH, <lb/>fiat angulus EDK æqualis angulo BAH; & vt BA, ad <pb xlink:href="043/01/049.jpg" pagenum="41"/>AH, ita fiat ED, ad DK: & quoniam angulus BAG, <lb/>æqualis e&longs;t angulo EDF: quorum angulus EDK, <lb/>æqualis e&longs;t angulo BAH, erit reliquus angulus <emph type="italics"/>K<emph.end type="italics"/>DF, <lb/>æqualis reliquo angulo HAC; &longs;ed angulus HAC, e&longs;t <lb/>maior angulo BAH; ergo & angulus KDF, maior erit <lb/>angulo BAH; po&longs;ito igitur angulo FDL, æquali an<lb/>gulo BAH, ac proinde minori, quàm &longs;it angulus FD<emph type="italics"/>K<emph.end type="italics"/>, <lb/>fiat vt BA, ad AH, ita FD, ad DL. Dico, in triangu<lb/>lo EDF, duo puncta K, L, &longs;imiliter po&longs;ita e&longs;se ac pun<lb/>ctum H, in triangulo BAC. </s>

<s>Iungantur enim rectæ AH, <lb/>BH, CH, EK, KF, FL, LE. </s>

Line 674

<s>Rur&longs;us quoniam angulus <lb/>FDL, æqualis e&longs;t angulo BAH, & latus AB, homo<lb/>logum lateri DF, (e&longs;t enim vt BA, ad AC, ita FD, ad <lb/>DE) &longs;ed vt BA, ad AH, ita e&longs;t FD, ad DL, per con<lb/>&longs;tructionem; &longs;imiliter vt ante, o&longs;tenderemus, punctum L, <lb/>in triangulo EDF, &longs;imiliter po&longs;itum e&longs;se puncto H; in<lb/>uenta igitur &longs;unt duo puncta in triangulo DEF, &longs;imili<lb/>ter po&longs;ita ac punctum H, in triangulo BAC. </s>

<s>Quod pro<lb/>po&longs;itum erat. </s><pb pagenum="42"/>

<s>Quod pro<lb/>po&longs;itum erat. </s><pb xlink:href="043/01/050.jpg" pagenum="42"/>

<s><emph type="italics"/>PROPOSITIO XX.<emph.end type="italics"/></s>

Line 688

<s>Secti&longs;que AD, BC, bifa<lb/>riam in punctis F, E, <lb/>iunctaque EF, & &longs;e<lb/>cta in tres partes æ<lb/>quales in punctis K, <lb/>H, fiat vt AD, ad <lb/>BC, ita HG, ad GK. <lb/></s>

<s>Dico G, e&longs;se centrum <lb/>grauitatis trapezij A <lb/>BCD: & vt e&longs;t du<lb/>pla ip&longs;ius AD, vna <lb/>cum BC, ad duplam <lb/>ip&longs;ius BC, vna cum <lb/>AD, ita e&longs;se EG, ad <lb/><figure id="fig24"/><lb/>GF. </s>

<s>Ducta enim per punctum H, ip&longs;is AD, BC, pa-<pb pagenum="43"/>rallela NO, ab&longs;cindantur EL, FM, ip&longs;i GK æquales, & <lb/>iungantur ANE, EOD. </s>

<s>Ducta enim per punctum H, ip&longs;is AD, BC, pa-<pb xlink:href="043/01/051.jpg" pagenum="43"/>rallela NO, ab&longs;cindantur EL, FM, ip&longs;i GK æquales, & <lb/>iungantur ANE, EOD. </s>

<s>Quoniam igitur NO ip&longs;i AD, <lb/>parallela &longs;ecat omnes ip&longs;is AD, EC, interceptas in ea&longs;<lb/>dem rationes, & e&longs;t EH, pars tertia ip&longs;ius EF, erit & EN <lb/>ip&longs;ius EA, & EO, ip&longs;ius ED, pars tertia. </s>

Line 704

<s>Sed vt HG, bis vna cum <lb/>GK, ad GK bis vna cum GH, ita e&longs;t AD, bis vna cum <lb/>BC, ad BC, bis vna cum AB, propterea quod e&longs;t vt <lb/>AD, ad BC, ita HG, ad GK; vt igitur e&longs;t AD, bis vna <lb/>cum BC, ad BC, bis vna cum AD, ita erit EG, ad GF. <lb/></s>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s><pb pagenum="44"/>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s><pb xlink:href="043/01/052.jpg" pagenum="44"/>

<s><emph type="italics"/>PROPOSITIO XXI.<emph.end type="italics"/></s>

Line 716

<s>Quoniam igitur ex decima tertia quar <lb/>ti Elem.

quemadmodum in pentagono, ita in omni præ<lb/>dicto polygono imparium multitudine laterum plane col<lb/>ligitur centrum po<lb/>lygoni e&longs;se in qua<lb/>libet recta, quæ ab <lb/>angulo ad medium <lb/>lateris oppo&longs;iti du<lb/>citur, quoniam ab <lb/>omnibus angulis &longs;ic <lb/>ductæ &longs;ecant &longs;e &longs;e <lb/>in eadem proportio<lb/>ne æqualitatis, ita <lb/>vt eadem &longs;it propor<lb/>tio &longs;egmentorum, <lb/>quæ ad angulos, ad <lb/>ea, quæ ad latera <lb/><figure id="fig25"/><lb/>illis angulis oppo&longs;ita; rectæ AH, CK, &longs;ecabunt &longs;e &longs;e in <lb/>puncto L. </s>

quemadmodum in pentagono, ita in omni præ<lb/>dicto polygono imparium multitudine laterum plane col<lb/>ligitur centrum po<lb/>lygoni e&longs;se in qua<lb/>libet recta, quæ ab <lb/>angulo ad medium <lb/>lateris oppo&longs;iti du<lb/>citur, quoniam ab <lb/>omnibus angulis &longs;ic <lb/>ductæ &longs;ecant &longs;e &longs;e <lb/>in eadem proportio<lb/>ne æqualitatis, ita <lb/>vt eadem &longs;it propor<lb/>tio &longs;egmentorum, <lb/>quæ ad angulos, ad <lb/>ea, quæ ad latera <lb/><figure id="id.043.01.052.1.jpg" xlink:href="043/01/052/1.jpg"/><lb/>illis angulis oppo&longs;ita; rectæ AH, CK, &longs;ecabunt &longs;e &longs;e in <lb/>puncto L. </s>

<s>Rurfus quoniam ex eadem Euclidis angulus <lb/>BAL, æqualis e&longs;t angulo GAL, &longs;ed AB, e&longs;t æqualis <lb/>AG, & AM, communis, erit ba&longs;is BM, æqualis ba&longs;i <pb pagenum="45"/>MG, & angulus ABM, angulo AGM, &longs;ed totus ABC, <lb/>toti AGF, e&longs;t æqualis; reliquus igitur angulus CBG, <lb/>reliquo BGF, æqualis erit: &longs;ed circa hos æquales an<lb/>gulos recta BM, o&longs;ten&longs;a e&longs;t æqualis rectæ MG, & CB, <lb/>e&longs;t æqualis GF; ba&longs;is igitur CM, ba&longs;i GF, & angulus <lb/>CMB, angulo FMG, æqualis erit; &longs;ed totus BMN, <lb/>æqualis e&longs;t toti GMN; quia vterque rectus; reliquus <lb/>igitur CMN, reliquo NMF, æqualis erit, quos circa <lb/>recta CM, e&longs;t æqualis MF, & MN, communis; ba&longs;is <lb/>igitur CN, ba&longs;i NF, & anguli, qui ad N, æquales erunt, <lb/>atque ideo recti: &longs;ed & qui ad M, &longs;unt recti, & BM, e&longs;t <lb/>æqualis GM; parallelæ igitur &longs;unt BG, CF, & trape<lb/>zij CBGF, centrum grauitatis e&longs;t in linea MN: &longs;ed & <lb/>trianguli ABG, centrum grauitatis e&longs;t in linea AM; to<lb/>tius igitur figuræ ABCFG, centrum grauitatis e&longs;t in li<lb/>nea AN; hoc e&longs;t in linea AH. </s>

<s>Rurfus quoniam ex eadem Euclidis angulus <lb/>BAL, æqualis e&longs;t angulo GAL, &longs;ed AB, e&longs;t æqualis <lb/>AG, & AM, communis, erit ba&longs;is BM, æqualis ba&longs;i <pb xlink:href="043/01/053.jpg" pagenum="45"/>MG, & angulus ABM, angulo AGM, &longs;ed totus ABC, <lb/>toti AGF, e&longs;t æqualis; reliquus igitur angulus CBG, <lb/>reliquo BGF, æqualis erit: &longs;ed circa hos æquales an<lb/>gulos recta BM, o&longs;ten&longs;a e&longs;t æqualis rectæ MG, & CB, <lb/>e&longs;t æqualis GF; ba&longs;is igitur CM, ba&longs;i GF, & angulus <lb/>CMB, angulo FMG, æqualis erit; &longs;ed totus BMN, <lb/>æqualis e&longs;t toti GMN; quia vterque rectus; reliquus <lb/>igitur CMN, reliquo NMF, æqualis erit, quos circa <lb/>recta CM, e&longs;t æqualis MF, & MN, communis; ba&longs;is <lb/>igitur CN, ba&longs;i NF, & anguli, qui ad N, æquales erunt, <lb/>atque ideo recti: &longs;ed & qui ad M, &longs;unt recti, & BM, e&longs;t <lb/>æqualis GM; parallelæ igitur &longs;unt BG, CF, & trape<lb/>zij CBGF, centrum grauitatis e&longs;t in linea MN: &longs;ed & <lb/>trianguli ABG, centrum grauitatis e&longs;t in linea AM; to<lb/>tius igitur figuræ ABCFG, centrum grauitatis e&longs;t in li<lb/>nea AN; hoc e&longs;t in linea AH. </s>

<s>Rur&longs;us quoniam omnis <lb/>quadrilateri quatuor anguli &longs;unt æquales quatuor rectis: <lb/>& tres anguli ABM, BMN, MNC, &longs;unt æquales tri<lb/>bus angulis FGM, GMN, MNF, reliquus angulus <lb/>BCF, reliquo CFG, æqualis erit: &longs;ed totus angulus <lb/>BCD, e&longs;t æqualis toti angulo GFE; reliquus ergo <lb/>DCF, reliquo CFE, æqualis erit: &longs;ed linea CN, e&longs;t <lb/>æqualis NF, & anguli, qui ad N, &longs;unt recti; &longs;imiliter <lb/>ergo vt antea, centrum grauitatis trapezij CDEF, erit <lb/>in linea AH: &longs;ed & totius figuræ ABCFG, e&longs;t in li<lb/>nea AH; totius igitur polygoni ABCDEFG, in li<lb/>nea AH, e&longs;t centrum grauitatis, quod idem &longs;imiliter in <lb/>linea CK, e&longs;se oftenderemus; in communi igitur &longs;ectione <lb/>puncto L, e&longs;t centrum grauitatis polygoni ABCDEFG. <lb/></s>

<s>Similiter quotcumque plurium laterum numero impa<lb/>rium e&longs;set polygonum æquilaterum, & æquiangulum, <lb/>&longs;emper deueniendo ab vno triangulo ad quotcumque eius <lb/>trapezia; propo&longs;itum concluderemus. </s><pb pagenum="46"/>

<s>Sed e&longs;to polygonum æquilaterum, & æquiangulum, <lb/>ABCDEF, cuius laterum numerus &longs;it par, & centrum <lb/>e&longs;to G. </s>

Line 736

<s>Quoniam igitur duo<lb/>rum triangulorum CBG, GFE, anguli ad verticem <lb/>BGC, FGE, &longs;unt æquales; & vterlibet angulorum CBG, <lb/>GCB, æqualis e&longs;t vtrilibet ip&longs;orum EFG, GEF; ex <lb/>quarto Elem.

& circa æquales angulos latera proportio<lb/>nalia horum triangu <lb/>lorum &longs;unt æqualia; <lb/>&longs;imilia, & æqualia <lb/>erunt triangula BC <lb/>G, GFE: po&longs;itis <lb/>igitur centris graui<lb/>tatis K, H, duorum <lb/>triangulorum EFG, <lb/>GBC, iunctifque <lb/>KG, GH, erit v<lb/>terlibet angulorum <lb/>BGH, HGC, æ<lb/>qualis vtrilibet an<lb/><figure id="fig26"/><lb/>gulorum CGK, KGE, propter &longs;imilitudinem po&longs;itio<lb/>nis centrorum K, H, in i&longs;o&longs;celijs triangulis CBG, <lb/>GFE: (nam GH, &longs;i produceretur latus BC, bifariam <lb/>&longs;ecaret: &longs;imiliter GK, latus EF) &longs;ed CG, e&longs;t in directum <lb/>po&longs;ita ip&longs;i GF; igitur & GH ip&longs;i GK: & &longs;unt æquales, <lb/>vtpote lateribus triangulorum BCG, GFE, æqualibus <lb/>homologæ; cum igitur eorundem triangulorum centra <lb/>grauitatis &longs;int K, H; centrum grauitatis duorum triangu<lb/>lorum CBG, GFE, &longs;imul, erit punctum G. </s>

& circa æquales angulos latera proportio<lb/>nalia horum triangu <lb/>lorum &longs;unt æqualia; <lb/>&longs;imilia, & æqualia <lb/>erunt triangula BC <lb/>G, GFE: po&longs;itis <lb/>igitur centris graui<lb/>tatis K, H, duorum <lb/>triangulorum EFG, <lb/>GBC, iunctifque <lb/>KG, GH, erit v<lb/>terlibet angulorum <lb/>BGH, HGC, æ<lb/>qualis vtrilibet an<lb/><figure id="id.043.01.054.1.jpg" xlink:href="043/01/054/1.jpg"/><lb/>gulorum CGK, KGE, propter &longs;imilitudinem po&longs;itio<lb/>nis centrorum K, H, in i&longs;o&longs;celijs triangulis CBG, <lb/>GFE: (nam GH, &longs;i produceretur latus BC, bifariam <lb/>&longs;ecaret: &longs;imiliter GK, latus EF) &longs;ed CG, e&longs;t in directum <lb/>po&longs;ita ip&longs;i GF; igitur & GH ip&longs;i GK: & &longs;unt æquales, <lb/>vtpote lateribus triangulorum BCG, GFE, æqualibus <lb/>homologæ; cum igitur eorundem triangulorum centra <lb/>grauitatis &longs;int K, H; centrum grauitatis duorum triangu<lb/>lorum CBG, GFE, &longs;imul, erit punctum G. </s>

<s>Eadem <pb pagenum="47"/>ratione, tam duorum triangulorum ABG, DGE, quàm <lb/>duorum AFG, CDG, &longs;imul, centrum grauitatis erit G; <lb/>totius igitur polygoni ABCDEF; centrum grauitatis <lb/>erit idem G. </s>

<s>Eadem <pb xlink:href="043/01/055.jpg" pagenum="47"/>ratione, tam duorum triangulorum ABG, DGE, quàm <lb/>duorum AFG, CDG, &longs;imul, centrum grauitatis erit G; <lb/>totius igitur polygoni ABCDEF; centrum grauitatis <lb/>erit idem G. </s>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s>

Line 750

<s>Dico centrum grauitatis figuræ <lb/>ABC, e&longs;se in linea BD. &longs;it enim punctum E, generali<lb/>ter extra lineam BD. </s>

<s>Et per puncta E, C, ducantur ip&longs;i <lb/>BD, parallelæ EF, <lb/>CG, & vt e&longs;t CD, <lb/>ad DF, ita ponatur <lb/>figura ABC, ad ali<lb/>quod &longs;pacium M: & <lb/>figuræ ABC, in&longs;cri<lb/>batur figura ex paral<lb/>lelogrammis æqua<lb/>lium altitudinum de<lb/>ficiens à figura ABC, <lb/>minori defectu, quam <lb/>&longs;it &longs;pacium M, quan<lb/>tumcumque illud &longs;it: <lb/>minor igitur propor<lb/><figure id="fig27"/><lb/>tio erit figuræ ABC, ad &longs;pacium M, hoc e&longs;t minor pro<lb/>portio CD, ad DF, quàm figuræ ABC, ad &longs;ui reliquum, <lb/>dempta figura in&longs;cripta. </s>

<s>Et per puncta E, C, ducantur ip&longs;i <lb/>BD, parallelæ EF, <lb/>CG, & vt e&longs;t CD, <lb/>ad DF, ita ponatur <lb/>figura ABC, ad ali<lb/>quod &longs;pacium M: & <lb/>figuræ ABC, in&longs;cri<lb/>batur figura ex paral<lb/>lelogrammis æqua<lb/>lium altitudinum de<lb/>ficiens à figura ABC, <lb/>minori defectu, quam <lb/>&longs;it &longs;pacium M, quan<lb/>tumcumque illud &longs;it: <lb/>minor igitur propor<lb/><figure id="id.043.01.055.1.jpg" xlink:href="043/01/055/1.jpg"/><lb/>tio erit figuræ ABC, ad &longs;pacium M, hoc e&longs;t minor pro<lb/>portio CD, ad DF, quàm figuræ ABC, ad &longs;ui reliquum, <lb/>dempta figura in&longs;cripta. </s>

<s>Quoniam autem diameter BD, <pb pagenum="48"/>bifariam &longs;ecat omnia latera parallelogrammorum in&longs;cri<lb/>ptorum ba&longs;i AC, parallela; erit in diametro BD, eorum <lb/>omnium parallelogrammorum centra grauitatis, atque <lb/>ideo totius figuræ in&longs;criptæ centrum grauitatis, quod &longs;it <lb/>H: & HEK, ducatur. </s>

<s>Quoniam autem diameter BD, <pb xlink:href="043/01/056.jpg" pagenum="48"/>bifariam &longs;ecat omnia latera parallelogrammorum in&longs;cri<lb/>ptorum ba&longs;i AC, parallela; erit in diametro BD, eorum <lb/>omnium parallelogrammorum centra grauitatis, atque <lb/>ideo totius figuræ in&longs;criptæ centrum grauitatis, quod &longs;it <lb/>H: & HEK, ducatur. </s>

<s>Quoniam igitur EF, parallela <lb/>e&longs;t vtrique DH, CK; erit vt CD, ad DF, ita KH, ad <lb/>HE, &longs;ed minor e&longs;t proportio CD, ad DF, quàm figu<lb/>ræ ABC, ad re&longs;i<lb/>duum, dempta figu<lb/>ra in&longs;cripta; ergo & <lb/>KH, ad HE, minor <lb/>erit proportio, quàm <lb/>figuræ ABC, ad præ<lb/>dictum re&longs;iduum: ha<lb/>beat LKH, eandem <lb/>proportione<gap/> ad EH, <lb/>quàm figura ABC, <lb/>ad prædictum re&longs;i<lb/>duum. </s>

<s>Quoniam <lb/>igitur punctum K, <lb/>cadit extra figuram <lb/><figure id="fig28"/><lb/>ABC; multo magis punctum L; non igitur punctum L, <lb/>erit prædicti re&longs;idui centrum grauitatis. </s>

<s>Quoniam <lb/>igitur punctum K, <lb/>cadit extra figuram <lb/><figure id="id.043.01.056.1.jpg" xlink:href="043/01/056/1.jpg"/><lb/>ABC; multo magis punctum L; non igitur punctum L, <lb/>erit prædicti re&longs;idui centrum grauitatis. </s>

<s>Sed punctum <lb/>H, e&longs;t in&longs;criptæ figuræ centrum grauitatis: & vt figura <lb/>in&longs;cripta ad prædictum re&longs;iduum, diuidendo, ita e&longs;t LE, <lb/>ad EH; non igitur E, e&longs;t centrum grauitatis figuræ ABC: <lb/>&longs;ed ponitur E, generaliter punctum extra lineam BD; <lb/>Nullum igitur punctum extra lineam BD, e&longs;t centrum <lb/>grauitatis figuræ ABC; in linea igitur BD, erit figu<lb/>ræ ABC, centrum grauitatis. </s>

<s>Quod demon&longs;trandum <lb/>erat. </s><pb pagenum="49"/>

<s>Quod demon&longs;trandum <lb/>erat. </s><pb xlink:href="043/01/057.jpg" pagenum="49"/>

<s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/></s>

<s>Ex huius theorematis demon&longs;tratione con&longs;tat, <lb/>omnis figuræ planæ, &longs;iue &longs;olidæ, cuius termini <lb/>omnis cauitas &longs;it interior, atque ideo intra ter<lb/>minum centrum grauitatis; & cuius pars aliqua <lb/>e&longs;se po&longs;sit, quæ à tota figura deficiens minori <lb/>defectu quacumque magnitudine propo&longs;ita habe<lb/>at centrum grauitatis in aliqua certa linea recta <lb/>intra terminum figuræ con&longs;tituta, e&longs;&longs;e in ea recta <lb/>linea totius figuræ centrum grauitatis. </s>

<s>Ac proin<lb/>de, cum per vndecimam huius, omni &longs;olido circa <lb/>axim in alteram partem deficienti, & ba&longs;im ha<lb/>benti circulum, vel ellyp&longs;im figura in&longs;cribi po&longs;&longs;it <lb/>ex cylindris, vel cylindri portionibus, à prædicto <lb/>&longs;olido deficiens minori &longs;pacio quacumque ma<lb/>gnitudine propo&longs;ita: talis autem figuræ in&longs;criptæ, <lb/>quemadmodum & circum&longs;criptæ centrum gra<lb/>uitatis &longs;it in axe, vt ex &longs;equentibus patebit, & <lb/>nunc cogitanti facilè patere pote&longs;t; manife&longs;tum <lb/>e&longs;t omnis &longs;olidi circa axim in alteram partem de<lb/>ficientis centrum grauitatis e&longs;&longs;e in axe. </s><pb pagenum="50"/>

<s><emph type="italics"/>PROPOSITIO XXIII.<emph.end type="italics"/></s>

Line 778

<s>Ducantur enim duæ diametri ad rectos inter &longs;e angulos <lb/>AC, BD; in ellyp&longs;i autem &longs;int diametri coniugatæ. <lb/></s>

<s>Quoniam igitur omnes rectæ lineæ, quæ in &longs;emicirculo, <lb/>vel dimidia ellyp&longs;i diametro ducantur parallelæ bifariam <lb/>&longs;ecantur à &longs;emidiametro, & quo à ba&longs;i remotiores, eo &longs;unt <lb/><figure id="fig29"/><lb/>minores; erit centrum grauitatis &longs;emicirculi, &longs;iue dimidiæ <lb/>ellyp&longs;is ABC, in linea BE; &longs;icut & &longs;emicirculi, &longs;iue di<lb/>midiæ ellyp&longs;is ADC, centrum grauitatis in linea DE. <lb/>e&longs;t autem BED, vna recta linea: in diametro igitur BD, <lb/>erit centrum grauitatis circuli, &longs;iue ellyp&longs;is ABCD. <lb/></s>

<s>Quoniam igitur omnes rectæ lineæ, quæ in &longs;emicirculo, <lb/>vel dimidia ellyp&longs;i diametro ducantur parallelæ bifariam <lb/>&longs;ecantur à &longs;emidiametro, & quo à ba&longs;i remotiores, eo &longs;unt <lb/><figure id="id.043.01.058.1.jpg" xlink:href="043/01/058/1.jpg"/><lb/>minores; erit centrum grauitatis &longs;emicirculi, &longs;iue dimidiæ <lb/>ellyp&longs;is ABC, in linea BE; &longs;icut & &longs;emicirculi, &longs;iue di<lb/>midiæ ellyp&longs;is ADC, centrum grauitatis in linea DE. <lb/>e&longs;t autem BED, vna recta linea: in diametro igitur BD, <lb/>erit centrum grauitatis circuli, &longs;iue ellyp&longs;is ABCD. <lb/></s>

<s>Eadem ratione o&longs;tenderemus idem centrum grauitatis e&longs;se <lb/>in altera diametro AC: in communi igitur vtriu&longs;que &longs;e<lb/>ctione puncto E. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="51"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/059.jpg" pagenum="51"/>

<s><emph type="italics"/>PROPOSITIO XXIV.<emph.end type="italics"/></s>

Line 796

<s>E&longs;to enim H, centrum gra <lb/>uitatis pyramidis ABCG, <lb/>& ducta HGK, ponatur <lb/>G<emph type="italics"/>K<emph.end type="italics"/>, æqualis GH, & iun<lb/>gantur EK, KD, BH, <lb/>CH. </s>

<s>Quoniam igitur e&longs;t <lb/>vt HG, ad GK, ita CG, <lb/>ad GE, & proportio e&longs;t <lb/>æqualitatis: & angulus <lb/>HGC, æqualis angulo EG <lb/><emph type="italics"/>K<emph.end type="italics"/>, erit triangulum CGH, <lb/><figure id="fig30"/><lb/>&longs;imile, & æquale triangulo EGK. </s>

<s>Quoniam igitur e&longs;t <lb/>vt HG, ad GK, ita CG, <lb/>ad GE, & proportio e&longs;t <lb/>æqualitatis: & angulus <lb/>HGC, æqualis angulo EG <lb/><emph type="italics"/>K<emph.end type="italics"/>, erit triangulum CGH, <lb/><figure id="id.043.01.059.1.jpg" xlink:href="043/01/059/1.jpg"/><lb/>&longs;imile, & æquale triangulo EGK. </s>

<s>Similiter triangulum <lb/>BGH, trian gulo DGK; & triangulum BGC, triangu<lb/>lo DGE: quare & triangulum BCH, triangulo DEK. <lb/>pyramis igitur BCGH, &longs;imilis, & æqualis e&longs;t pyramidi <lb/>EDGK. </s>

<s>Congruentibus igitur inter &longs;e duobus triangu<pb pagenum="52"/>lis æqualibus, & &longs;imilibus BGC, DGE, & pyramis <lb/>BCGH, pyramidi GDEK congruet, & puncto K, pun<lb/>ctum H: & eadem ratione <lb/>pyramis ABCG, pyra<lb/>midi DEFG. congruente <lb/>igitur pyramide ABCG, <lb/>pyramidi DEFG, & pun<lb/>ctum K, congruet puncto <lb/>H. &longs;ed H, e&longs;t centrum gra<lb/>uitatis pyramidis ABCG: <lb/>igitur K, erit centrum gra <lb/>uitatis pyramidis DEFG: <lb/>&longs;ed e&longs;t GK, æqualis ip<lb/>&longs;i GH; vtriufque igitur <lb/>pyramidis ABCG, DE<lb/>FG, &longs;imul centrum grauitatis erit K; Quod demon&longs;tran<lb/>dum erat. </s><figure/>

<s>Congruentibus igitur inter &longs;e duobus triangu<pb xlink:href="043/01/060.jpg" pagenum="52"/>lis æqualibus, & &longs;imilibus BGC, DGE, & pyramis <lb/>BCGH, pyramidi GDEK congruet, & puncto K, pun<lb/>ctum H: & eadem ratione <lb/>pyramis ABCG, pyra<lb/>midi DEFG. congruente <lb/>igitur pyramide ABCG, <lb/>pyramidi DEFG, & pun<lb/>ctum K, congruet puncto <lb/>H. &longs;ed H, e&longs;t centrum gra<lb/>uitatis pyramidis ABCG: <lb/>igitur K, erit centrum gra <lb/>uitatis pyramidis DEFG: <lb/>&longs;ed e&longs;t GK, æqualis ip<lb/>&longs;i GH; vtriufque igitur <lb/>pyramidis ABCG, DE<lb/>FG, &longs;imul centrum grauitatis erit K; Quod demon&longs;tran<lb/>dum erat. </s><figure id="id.043.01.060.1.jpg" xlink:href="043/01/060/1.jpg"/>

<s><emph type="italics"/>PROPOSITIO XXV.<emph.end type="italics"/></s>

Line 810

<s>Dico K e&longs;se <lb/>centrum grauitatis parallelepipedi ABCDEFGH. <lb/>iungantur enim diametri AG, BH, CE, DF, quæ <lb/>omnes nece&longs;sario tran&longs;ibunt per punctum K, & in eo <lb/>puncto bifariam diuidentur. </s>

<s>Iunctis igitur BD, FH: <lb/>quoniam triangulum EFK, &longs;imile e&longs;t, & æquale trian<lb/>gulo CDK, propter latera circa æquales angulos ad <pb pagenum="53"/>verticem æqualia alterum alteri: eademque ratione, & <lb/>triangulum E<emph type="italics"/>K<emph.end type="italics"/>H, triangulo BCK: & triangulum FKH, <lb/>triangulo BDK; erit pyramis KEFH, &longs;imilis, & æqua<lb/>lis pyramidi KBCD: habent autem tria latera tribus <lb/>lateribus homologis, ide&longs;t æ<lb/>qualibus, in directum, prout <lb/>inter &longs;e re&longs;pondent, con&longs;tituta; <lb/>duarum igitur pyramidum KE <lb/>FH, KBCD, &longs;imul centrum <lb/>grauitatis erit K: non aliter <lb/>duarum pyramidum <emph type="italics"/>K<emph.end type="italics"/>GFH, <lb/>KBDA, &longs;imul centrum gra<lb/>uitatis erit K; totius igitur com <lb/>po&longs;iti ex quatuor pyramidibus; <lb/>ide&longs;t duabus oppo&longs;itis ABC<lb/>DK, EFGHK, centrum gra<lb/>uitatis erit idem K. </s>

<s>Iunctis igitur BD, FH: <lb/>quoniam triangulum EFK, &longs;imile e&longs;t, & æquale trian<lb/>gulo CDK, propter latera circa æquales angulos ad <pb xlink:href="043/01/061.jpg" pagenum="53"/>verticem æqualia alterum alteri: eademque ratione, & <lb/>triangulum E<emph type="italics"/>K<emph.end type="italics"/>H, triangulo BCK: & triangulum FKH, <lb/>triangulo BDK; erit pyramis KEFH, &longs;imilis, & æqua<lb/>lis pyramidi KBCD: habent autem tria latera tribus <lb/>lateribus homologis, ide&longs;t æ<lb/>qualibus, in directum, prout <lb/>inter &longs;e re&longs;pondent, con&longs;tituta; <lb/>duarum igitur pyramidum KE <lb/>FH, KBCD, &longs;imul centrum <lb/>grauitatis erit K: non aliter <lb/>duarum pyramidum <emph type="italics"/>K<emph.end type="italics"/>GFH, <lb/>KBDA, &longs;imul centrum gra<lb/>uitatis erit K; totius igitur com <lb/>po&longs;iti ex quatuor pyramidibus; <lb/>ide&longs;t duabus oppo&longs;itis ABC<lb/>DK, EFGHK, centrum gra<lb/>uitatis erit idem K. </s>

<s>Eadem <lb/>ratione tam duarum pyrami<lb/><figure id="fig31"/><lb/>dum AEHDK, BCGFK, &longs;imul, quàm duarum AB<lb/>FEK, CDHGK, &longs;imul centrum grauitatis erit K. </s>

<s>Eadem <lb/>ratione tam duarum pyrami<lb/><figure id="id.043.01.061.1.jpg" xlink:href="043/01/061/1.jpg"/><lb/>dum AEHDK, BCGFK, &longs;imul, quàm duarum AB<lb/>FEK, CDHGK, &longs;imul centrum grauitatis erit K. </s>

<s>To<lb/>tius igitur parallelepipedi ABCDEFG<emph type="italics"/>K<emph.end type="italics"/>, centrum <lb/>grauitatis erit K. </s>

Line 820

<s><emph type="italics"/>PROPOSITIO XXVI.<emph.end type="italics"/></s>

<s>Si parallelepipedum in duo parallelepipeda <lb/>&longs;ecetur, &longs;egmenta axis à centris grauitatis totius <lb/>parallelepipedi, & partium terminata ex contra<lb/>rio parallelepipedi partibus re&longs;pondent. </s><pb pagenum="54"/>

<s>Si parallelepipedum AB, cuius axis CD, &longs;ectum in <lb/>duo parallelepipeda AE, EN, quare & axis CD, in <lb/>axes CL, LD, parallelepipedorum AE, EN. </s>

Line 830

<s>Iungantur enim diametri ba&longs;ium oppo&longs;itarum, quæ per <lb/>puncta axium D, L, G, tran&longs;ibunt, ADM, KLE, <lb/>NCB; iamque parallelogramma <lb/>erunt AB, AE, EN, DB, DE, <lb/>EC, propter eas, quæ parallelas <lb/>iungunt, & æquales: quorum bi<lb/>na latera oppo&longs;ita &longs;ecta erunt bi<lb/>fariam in punctis C, L, D, per <lb/>definitionem axis: punctum igitur <lb/>F, in medio rectæ CL, oppo&longs;i<lb/>torum laterum bipartitorum &longs;ectio<lb/>nes coniungentis, erit parallelo<lb/>grammi EN, centrum grauitatis. <lb/></s>

<s>Eadem ratione & parallelogram<lb/><figure id="fig32"/><lb/>mi AE, centrum grauitatis erit G, & H, parallelogram <lb/>mi AB. </s>

<s>Eadem ratione & parallelogram<lb/><figure id="id.043.01.062.1.jpg" xlink:href="043/01/062/1.jpg"/><lb/>mi AE, centrum grauitatis erit G, & H, parallelogram <lb/>mi AB. </s>

<s>Vt igitur parallelogrammum AE, ad paralle<lb/>logrammum EN, hoc e&longs;t, vt ba&longs;is ME, ad ba&longs;im EB; <lb/>hoc e&longs;t, vt parallelogrammum MO, ad parallelogram<lb/>mum OB: hoc e&longs;t, vt parallelepipedum AE, ad paral<lb/>lelepipedum EN: ita erit FH, ad HG. </s>

<s>Quod de<lb/>mon&longs;trandum erat. </s><pb pagenum="55"/>

<s>Quod de<lb/>mon&longs;trandum erat. </s><pb xlink:href="043/01/063.jpg" pagenum="55"/>

<s><emph type="italics"/>PROPOSIT'IO XXVII.<emph.end type="italics"/></s>

Line 844

<s>Dico &longs;olida A, E, æquiponderare à longitudini<lb/>bus DE, EC; hoc e&longs;t vtriu&longs;que &longs;imul centrum grauita<lb/>tis e&longs;se E. </s>

<s>Nam &longs;i A, B, &longs;int æqualia, manife&longs;tum e&longs;t <lb/>propo&longs;itum: &longs;i au<lb/>tem inæqualia, e&longs;to <lb/>maius A: maior igi <lb/>tur erit DE, quam <lb/>EC. ab&longs;cindatur <lb/>DF, æqualis EC: <lb/>erit igitur DE, æ<lb/>qualis GF: & CD, <lb/>vtrin que producta, <lb/>ponatur DH, æ<lb/>qualis DF: & CG, <lb/>ip&longs;i CF. & circa <lb/>axim, & <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>GH, e&longs;to paralle<lb/>lepipedum KL, æ<lb/>quale duobus &longs;o<lb/><figure id="fig33"/><lb/>lidis A, B, &longs;imul & parallelepipedum KL, &longs;ecetur plano <lb/>per punctum F, oppo&longs;itis planis parallelo, in duo paral<lb/>lelepipeda KN, ML. </s>

<s>Nam &longs;i A, B, &longs;int æqualia, manife&longs;tum e&longs;t <lb/>propo&longs;itum: &longs;i au<lb/>tem inæqualia, e&longs;to <lb/>maius A: maior igi <lb/>tur erit DE, quam <lb/>EC. ab&longs;cindatur <lb/>DF, æqualis EC: <lb/>erit igitur DE, æ<lb/>qualis GF: & CD, <lb/>vtrin que producta, <lb/>ponatur DH, æ<lb/>qualis DF: & CG, <lb/>ip&longs;i CF. & circa <lb/>axim, & <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>GH, e&longs;to paralle<lb/>lepipedum KL, æ<lb/>quale duobus &longs;o<lb/><figure id="id.043.01.063.1.jpg" xlink:href="043/01/063/1.jpg"/><lb/>lidis A, B, &longs;imul & parallelepipedum KL, &longs;ecetur plano <lb/>per punctum F, oppo&longs;itis planis parallelo, in duo paral<lb/>lelepipeda KN, ML. </s>

<s>Quoniam igitur e&longs;t vt GF, ad <lb/>FH, ita parallelepipedum KN, ad parallelepipedum <pb pagenum="56"/>ML, &longs;ed vt GF, ad FH, ita e&longs;t CF, ad FD, hoc e&longs;t DE, ad <lb/>EC, hoc e&longs;t &longs;olidum A, ad &longs;olidum B; erit vt parallelepipe<lb/>dum KN, ad parallelepipedum ML, ita &longs;olidum A, ad &longs;oli<lb/>dum B. componendo igitur, & permutando, vt parallelepi<lb/>pedum KL, ad duo &longs;olida A, B, &longs;imul, ita parallelepi<lb/>pedum ML, ad &longs;olidum B: & reliquum ad reliquum: &longs;ed <lb/>parallelepipedum KL, æquale e&longs;t duobus &longs;olidis A, B, &longs;i<lb/>mul: parallelepipedum igitur KN, &longs;olido A, & paralle<lb/>lepipedum ML, &longs;olido B, æquale erit. </s>

<s>Quoniam igitur e&longs;t vt GF, ad <lb/>FH, ita parallelepipedum KN, ad parallelepipedum <pb xlink:href="043/01/064.jpg" pagenum="56"/>ML, &longs;ed vt GF, ad FH, ita e&longs;t CF, ad FD, hoc e&longs;t DE, ad <lb/>EC, hoc e&longs;t &longs;olidum A, ad &longs;olidum B; erit vt parallelepipe<lb/>dum KN, ad parallelepipedum ML, ita &longs;olidum A, ad &longs;oli<lb/>dum B. componendo igitur, & permutando, vt parallelepi<lb/>pedum KL, ad duo &longs;olida A, B, &longs;imul, ita parallelepi<lb/>pedum ML, ad &longs;olidum B: & reliquum ad reliquum: &longs;ed <lb/>parallelepipedum KL, æquale e&longs;t duobus &longs;olidis A, B, &longs;i<lb/>mul: parallelepipedum igitur KN, &longs;olido A, & paralle<lb/>lepipedum ML, &longs;olido B, æquale erit. </s>

<s>Rur&longs;us, quo<lb/>niam e&longs;t vt GF, ad <lb/>ad FH, ita CF, ad <lb/>FD; hoc e&longs;t DE, <lb/>ad EC: &longs;ed vt GF, <lb/>ad FH, ita e&longs;t <expan abbr="pa-rallelepiped&utilde;">pa<lb/>rallelepipedum</expan> KN, <lb/>ad <expan abbr="parallelepiped&utilde;">parallelepipedum</expan> <lb/>ML; erit vt DE, <lb/>ad EC, ita paralle <lb/>lepipedum KN, ad <lb/>parallelepipedum <lb/>ML; &longs;ed C e&longs;t pa<lb/>rallelepipedi KN, <lb/>& D, parallelepipe <lb/>di ML, centrum <lb/>grauitatis; totius igi <lb/><figure id="fig34"/><lb/>tur parallelepipedi KL, centrum grauitatis erit E. </s>

<s>Rur&longs;us, quo<lb/>niam e&longs;t vt GF, ad <lb/>ad FH, ita CF, ad <lb/>FD; hoc e&longs;t DE, <lb/>ad EC: &longs;ed vt GF, <lb/>ad FH, ita e&longs;t <expan abbr="pa-rallelepiped&utilde;">pa<lb/>rallelepipedum</expan> KN, <lb/>ad <expan abbr="parallelepiped&utilde;">parallelepipedum</expan> <lb/>ML; erit vt DE, <lb/>ad EC, ita paralle <lb/>lepipedum KN, ad <lb/>parallelepipedum <lb/>ML; &longs;ed C e&longs;t pa<lb/>rallelepipedi KN, <lb/>& D, parallelepipe <lb/>di ML, centrum <lb/>grauitatis; totius igi <lb/><figure id="id.043.01.064.1.jpg" xlink:href="043/01/064/1.jpg"/><lb/>tur parallelepipedi KL, centrum grauitatis erit E. </s>

<s>Igi<lb/>tur &longs;olido A, po&longs;ito ad punctum G, &longs;ecundum centrum <lb/>grauitatis A, & &longs;olidum B, ad punctum D, &longs;ecundum <lb/>centrum grauitatis B, quorum A, e&longs;t æquale parallele<lb/>pipedo KN, & B, parallelepipedo ML; ab ij&longs;dem lon<lb/>gitudinibus DE, EC, æquiponderabunt; eritque com<lb/>po&longs;iti ex vtroque &longs;olido A, B, centrum grauitatis E. </s>

<s>Quod <lb/>demon&longs;trandum erat. </s>

<s>Quod &longs;i quis à me quærat, cur non hic vtar quinta illa <pb pagenum="57"/>generali primi Archimedis de planis æquiponderantibus, <lb/>&longs;ed illud idem propo&longs;itum vna demon&longs;tratione in planis, <lb/>altera præ&longs;enti in &longs;olidis demon&longs;trauerim. </s>

<s>Quod &longs;i quis à me quærat, cur non hic vtar quinta illa <pb xlink:href="043/01/065.jpg" pagenum="57"/>generali primi Archimedis de planis æquiponderantibus, <lb/>&longs;ed illud idem propo&longs;itum vna demon&longs;tratione in planis, <lb/>altera præ&longs;enti in &longs;olidis demon&longs;trauerim. </s>

<s>Re&longs;pondeo: <lb/>quia Propo&longs;itio quarta primi Archimedis, ex qua quinta <lb/>nece&longs;&longs;ario pendet, habet, &longs;i quis attendat, aliquas difficul<lb/>tates phy&longs;icas, quæ mathematicis rationibus non facile <lb/>di&longs;&longs;oluantur: quæ cau&longs;a igitur illum adduxit ad &longs;imile quid <lb/><expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan> demon&longs;tratione ad illas duas parabolas ap. <lb/></s>

Line 870

<s>Di <lb/>co quatuor puncta A, B, <lb/>C, E, e&longs;&longs;e in eodem pla<lb/>no. </s>

<s>Iungantur enim re<lb/>ctæ AB, BC, CA: & vt <lb/>e&longs;t A, ad C, ita &longs;it CD, <lb/>ad DA, & BD, iungatur: <lb/><expan abbr="punct&utilde;">punctum</expan> igitur D, erit cen<lb/><figure id="fig35"/><pb pagenum="58"/>trum grauitatis duarum magnitudinum A, C, &longs;imul. <lb/></s>

<s>Iungantur enim re<lb/>ctæ AB, BC, CA: & vt <lb/>e&longs;t A, ad C, ita &longs;it CD, <lb/>ad DA, & BD, iungatur: <lb/><expan abbr="punct&utilde;">punctum</expan> igitur D, erit cen<lb/><figure id="id.043.01.065.1.jpg" xlink:href="043/01/065/1.jpg"/><pb xlink:href="043/01/066.jpg" pagenum="58"/>trum grauitatis duarum magnitudinum A, C, &longs;imul. <lb/></s>

<s>Rur&longs;us quoniam recta BD, coniungit duo centra gra<lb/>uitatis duarum magnitu<lb/>dinum B &longs;cilicet, & AC, <lb/>erit compo&longs;itæ ACB, in <lb/>recta BD, centrum graui <lb/>tatis: e&longs;t autem illud E. <lb/></s>

<s>Quoniam igitur in quo <lb/>plano e&longs;t recta BD, in <lb/>eodem &longs;unt duo puncta <lb/>B, E, in quo autem pla<lb/>no e&longs;t recta BD, in eo<lb/>dem e&longs;t recta AC, & <lb/>puncta A, C; in quo igi<lb/>tur plano &longs;unt puncta A, <lb/>C, in eodem erunt pun<lb/>cta B, E; quatuor igitur puncta A, B, C, E, erunt in eodem <lb/>plano; Quod demon&longs;tr andum erat. </s><figure/>

<s><emph type="italics"/>PROPOSITIO XXIX.<emph.end type="italics"/></s>

<s>Si à cuiuslibet trianguli centro, & tribus an<lb/>gulis quatuor rectæ inter &longs;e parallelæ plano trian <lb/>guli in&longs;i&longs;tant: tres autem magnitudines æquales <lb/>habeant centra grauitatis in ijs tribus, quæ ad <lb/>angulos; trium magnitudinum &longs;imul centrum <lb/>grauitatis erit in ea, quæ ad trianguli centrum <lb/>terminatur. </s>

<s>Sit triangulum ABC, cuius centrum N, à tribus au<lb/>tem angulis A, B, C, & centro N, in&longs;i&longs;tant plano trian-<pb pagenum="59"/>guli ABC, quatuor rectæ inter &longs;e parallelæ AD, BE, <lb/>CF, NM, tres autem magnitudines æquales habeant cen <lb/>tra grauitatis G, H, K, in tribus AD, BE, CF. </s>

<s>Sit triangulum ABC, cuius centrum N, à tribus au<lb/>tem angulis A, B, C, & centro N, in&longs;i&longs;tant plano trian-<pb xlink:href="043/01/067.jpg" pagenum="59"/>guli ABC, quatuor rectæ inter &longs;e parallelæ AD, BE, <lb/>CF, NM, tres autem magnitudines æquales habeant cen <lb/>tra grauitatis G, H, K, in tribus AD, BE, CF. </s>

<s>Di<lb/>co trium magnitudinum &longs;imul, quarum centra grauitatis <lb/>G, H, K, e&longs;&longs;e in linea NM. </s>

Line 890

<s>Et <expan abbr="quoniã">quoniam</expan> planum quadran <lb/>guli PH, &longs;ecat planum trianguli ABC, à communi autem <lb/>&longs;ectione BP, &longs;urgunt <lb/>duæ parallelæ PL, MN; <lb/>quarum PL, e&longs;t in pla<lb/>no quadranguli PH, <lb/>erit etiam MN, in eo<lb/>dem plano quadranguli <lb/>PH: & &longs;ecabit LH. &longs;e<lb/>cet in puncto O: qùare <lb/>vt LO, ad OH, ita erit <lb/>PN, ad NB, propter <lb/>parallelas: &longs;ed PN, e&longs;t <lb/>dimidia ip&longs;ius NB; er<lb/>go & LO, e&longs;t dimidia ip <lb/>&longs;ius OH. </s>

<s>Eadem ratio<lb/>ne, quoniam AP, æqua<lb/><figure id="fig36"/><lb/>lis e&longs;t PC, erit & GL, æqualis LK. </s>

<s>Eadem ratio<lb/>ne, quoniam AP, æqua<lb/><figure id="id.043.01.067.1.jpg" xlink:href="043/01/067/1.jpg"/><lb/>lis e&longs;t PC, erit & GL, æqualis LK. </s>

<s>Duarum igitur <lb/>magnitudinum G, K, &longs;imul centrum grauitatis erit L: &longs;ed <lb/>reliquæ magnitudinis, quæ ad H, e&longs;t centrum grauitatis <lb/>H; & vt compo&longs;itum ex duabus magnitudinibus G, <lb/>K, ad magnitudinem H, ita ex contraria parte e&longs;t HO, <lb/>ad OL; Trium igitur magnitudinum G, H, K, &longs;imul cen<lb/>trum grauitatis erit O, & in linea MN. </s>

<s>Quod demon<lb/>&longs;trandum erat. </s><pb pagenum="60"/>

<s>Quod demon<lb/>&longs;trandum erat. </s><pb xlink:href="043/01/068.jpg" pagenum="60"/>

<s><emph type="italics"/>PROPOSITIO XXX.<emph.end type="italics"/></s>

Line 904

<s>Di<lb/>co G, e&longs;se centrum grauitatis octaedri ABCDEF. <lb/></s>

<s>Ductis enim axibus AC, BD, EF, communis eorum <lb/>&longs;ectio erit centrum G, in quo axes bifariam &longs;ecabuntur: <lb/>omnium autem angulorum, qui ad G, bini qui que ad <lb/>verticem &longs;unt æquales, qui æqualibus altera alteri rectis <lb/>continentur; &longs;imilia igi<lb/>tur, & æqualia erunt trian <lb/>gula, nimirum EBG, <lb/>GDF, & ECG, ip&longs;i <lb/>GFA, & BCG, ip&longs;i <lb/>GDA: igitur & BCE, <lb/>ip&longs;i ADF; pyramis igi<lb/>tur EBCG, &longs;imilis, & <lb/>æqualis e&longs;t pyramidi A <lb/>DFG, quarum latera ho <lb/>mologa &longs;unt indirectum <lb/>inter &longs;e con&longs;tituta; dua<lb/>rum igitur pyramidum <lb/><figure id="fig37"/><lb/>EBCG, ADFG, &longs;imul centrum grauitatis erit G. <lb/></s>

<s>Ductis enim axibus AC, BD, EF, communis eorum <lb/>&longs;ectio erit centrum G, in quo axes bifariam &longs;ecabuntur: <lb/>omnium autem angulorum, qui ad G, bini qui que ad <lb/>verticem &longs;unt æquales, qui æqualibus altera alteri rectis <lb/>continentur; &longs;imilia igi<lb/>tur, & æqualia erunt trian <lb/>gula, nimirum EBG, <lb/>GDF, & ECG, ip&longs;i <lb/>GFA, & BCG, ip&longs;i <lb/>GDA: igitur & BCE, <lb/>ip&longs;i ADF; pyramis igi<lb/>tur EBCG, &longs;imilis, & <lb/>æqualis e&longs;t pyramidi A <lb/>DFG, quarum latera ho <lb/>mologa &longs;unt indirectum <lb/>inter &longs;e con&longs;tituta; dua<lb/>rum igitur pyramidum <lb/><figure id="id.043.01.068.1.jpg" xlink:href="043/01/068/1.jpg"/><lb/>EBCG, ADFG, &longs;imul centrum grauitatis erit G. <lb/></s>

<s>Eadem ratione &longs;ex reliquarum pyramidum binis quibu&longs;<lb/>que oppo&longs;itis &longs;imul &longs;umptis centrum grauitatis erit G. <lb/></s>

<s>Totius igitur octaedri ABCDEF, centrum grauitatis <lb/>erit G. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="61"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/069.jpg" pagenum="61"/>

<s><emph type="italics"/>PROPOSITIO XXXI.<emph.end type="italics"/></s>

Line 924

<s>Octaedrum <lb/>autem FGHKLM, <lb/>quod dimidium erit <lb/>pyramidis ABCD, & <lb/>&longs;int axes pyramidum <lb/>DSN, DS, KO, LP, <lb/>MQ: & ARG, iunga <lb/>tur. </s>

<s>Quoniam igitur <lb/>FH, e&longs;t parallela ip&longs;i <lb/>BC, & &longs;ecta e&longs;t BC, <lb/>bifariam in puncto G, <lb/><expan abbr="trã&longs;ibit">tran&longs;ibit</expan> recta AG, per <lb/>centra <expan abbr="triangulor&utilde;">triangulorum</expan> O, <lb/>& N, ad quæ axes KO, <lb/><figure id="fig38"/><lb/>DN, terminantur; manife&longs;tum hoc e&longs;t ex &longs;uperioribus: <lb/>eritque dupla AO, ip&longs;ius OR, nec non AN, dupla ip&longs;ius <lb/>NG, componendo igitur erit vt AG, ad GN, ita AR, <lb/>ad RO, & permutando, vt AG, ad AR, ita GN, ad <lb/>RO: &longs;ed AG, e&longs;t dupla ip&longs;ius AR, quoniam & AB, ip<lb/>&longs;ius AF; igitur & GN, erit dupla ip&longs;ius RO: &longs;ed & GN, <lb/>e&longs;t dupla ip&longs;ius NR, nam N, e&longs;t centrum trianguli GFH; <lb/>æqualis e&longs;t igitur NR, ip&longs;i RO, atque hinc dupla NO, <pb pagenum="62"/>ip&longs;ius OR; &longs;ed & AO erat dupla ip&longs;ius OR; æqualis <lb/>igit<gap/>r AO erit ip&longs;i ON. quare vt AK, ad KD, ita erit <lb/>AO, ad ON: igitur in triangulo ADN, erit KO, ip&longs;i <lb/>DN, parallela. </s>

<s>Quoniam igitur <lb/>FH, e&longs;t parallela ip&longs;i <lb/>BC, & &longs;ecta e&longs;t BC, <lb/>bifariam in puncto G, <lb/><expan abbr="trã&longs;ibit">tran&longs;ibit</expan> recta AG, per <lb/>centra <expan abbr="triangulor&utilde;">triangulorum</expan> O, <lb/>& N, ad quæ axes KO, <lb/><figure id="id.043.01.069.1.jpg" xlink:href="043/01/069/1.jpg"/><lb/>DN, terminantur; manife&longs;tum hoc e&longs;t ex &longs;uperioribus: <lb/>eritque dupla AO, ip&longs;ius OR, nec non AN, dupla ip&longs;ius <lb/>NG, componendo igitur erit vt AG, ad GN, ita AR, <lb/>ad RO, & permutando, vt AG, ad AR, ita GN, ad <lb/>RO: &longs;ed AG, e&longs;t dupla ip&longs;ius AR, quoniam & AB, ip<lb/>&longs;ius AF; igitur & GN, erit dupla ip&longs;ius RO: &longs;ed & GN, <lb/>e&longs;t dupla ip&longs;ius NR, nam N, e&longs;t centrum trianguli GFH; <lb/>æqualis e&longs;t igitur NR, ip&longs;i RO, atque hinc dupla NO, <pb xlink:href="043/01/070.jpg" pagenum="62"/>ip&longs;ius OR; &longs;ed & AO erat dupla ip&longs;ius OR; æqualis <lb/>igit<gap/>r AO erit ip&longs;i ON. quare vt AK, ad KD, ita erit <lb/>AO, ad ON: igitur in triangulo ADN, erit KO, ip&longs;i <lb/>DN, parallela. </s>

<s>Eadem ratione &longs;i iungerentur rectæ BH, <lb/>CF o&longs;tenderemus & duos reliquos axes LP, MQ, e&longs;<lb/>&longs;e axi DN parallelos: quatuor autem prædicti axes in<lb/>&longs;i&longs;tunt plano trianguli KLM, ita vt DN tran&longs;eat per <lb/>centrum S: reliqui autem KO, LP, MQ, terminentur <lb/>ad angulorum vertices K, L, M, trianguli KLM; igi<lb/>tur &longs;i tres æquales magnitudines habeant centra grauita<lb/>tis in axibus KO, LP, <lb/><expan abbr="Mq;">Mque</expan> compo&longs;iti ex ijs <lb/>tribus magnitudinibus <lb/>in axe DN erit <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis. </s>

<s>Rur&longs;us <lb/>quoniam E ponitur <expan abbr="c&etilde;">cem</expan> <lb/><expan abbr="tr&utilde;">trum</expan> pyramidis ABCD, <lb/>erit idem E centrum <lb/>octaedri FGHKLM, <lb/>idque in axe DN: e&longs;t <lb/>autem idem <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis octaedri, & figu <lb/>ræ: centrum igitur E <lb/>octaedri FCHKLM <lb/>erit in axe DN. </s>

<s>Quod <lb/><figure id="fig39"/><lb/>&longs;i quatuor reliquæ pyramides dempto prædicto octaedro <lb/>&longs;imiliter diuidantur, ac pyramis ABCD diui&longs;a fuit, erunt <lb/>rur&longs;us in &longs;ingulis quatuor prædictarum pyramidum &longs;in<lb/>gula octaedra centrum grauitatis habentia vnumquodque <lb/>in axe &longs;uæ pyramidis: quæ pyramides cum &longs;int inter &longs;e <lb/>æquales, earum dimidia octaedr a ip&longs;is in&longs;cripta inter &longs;e <lb/>erunt æqualia: &longs;unt autem eorum centra grauitatis in axi<lb/>bus ab&longs;ci&longs;sarum pyramidum, DS, KO, LP, MQ <lb/>axis autem DS: e&longs;t in axe DN; per ea igitur, quæ de-<pb pagenum="63"/>mon&longs;trauimus trium octaedrorum, quæ &longs;unt in pyrami<lb/>dibus AFHK, FBGL, GHOM &longs;imul, centrum gra<lb/>uitatis erit in axe D<emph type="italics"/>K<emph.end type="italics"/>: &longs;ed & octaedri in pyramide DK<lb/>LM, & octaedri FGHKLM centra grauitatis &longs;unt <lb/>in axe DN; omnium igitur quinque octaedrorum, quæ <lb/>&longs;unt in tota pyramide ABCD &longs;imul centrum grauitatis <lb/>e&longs;t in axe DN. </s>

<s>Quod <lb/><figure id="id.043.01.070.1.jpg" xlink:href="043/01/070/1.jpg"/><lb/>&longs;i quatuor reliquæ pyramides dempto prædicto octaedro <lb/>&longs;imiliter diuidantur, ac pyramis ABCD diui&longs;a fuit, erunt <lb/>rur&longs;us in &longs;ingulis quatuor prædictarum pyramidum &longs;in<lb/>gula octaedra centrum grauitatis habentia vnumquodque <lb/>in axe &longs;uæ pyramidis: quæ pyramides cum &longs;int inter &longs;e <lb/>æquales, earum dimidia octaedr a ip&longs;is in&longs;cripta inter &longs;e <lb/>erunt æqualia: &longs;unt autem eorum centra grauitatis in axi<lb/>bus ab&longs;ci&longs;sarum pyramidum, DS, KO, LP, MQ <lb/>axis autem DS: e&longs;t in axe DN; per ea igitur, quæ de-<pb xlink:href="043/01/071.jpg" pagenum="63"/>mon&longs;trauimus trium octaedrorum, quæ &longs;unt in pyrami<lb/>dibus AFHK, FBGL, GHOM &longs;imul, centrum gra<lb/>uitatis erit in axe D<emph type="italics"/>K<emph.end type="italics"/>: &longs;ed & octaedri in pyramide DK<lb/>LM, & octaedri FGHKLM centra grauitatis &longs;unt <lb/>in axe DN; omnium igitur quinque octaedrorum, quæ <lb/>&longs;unt in tota pyramide ABCD &longs;imul centrum grauitatis <lb/>e&longs;t in axe DN. </s>

<s>Quod &longs;i rur&longs;us in &longs;ingulis quatuor præ<lb/>dictarum pyramidum modo dicta ratione quina octaedra <lb/>de&longs;cripta intelligantur, &longs;imiliter o&longs;ten&longs;um erit quina octa<lb/>edra in &longs;ingulis quatuor ab&longs;ci&longs;&longs;arum pyramidum, velut <lb/>quatuor magnitudines, centra grauitatis habere in axibus <lb/>quatuor prædictarum pyramidum: &longs;unt autem hæc qua<lb/>tuor compo&longs;ita ex quinis octaedris inter &longs;e æqualia, pro<lb/>pter æqualitatem octaedrorum multitudine æqualium, <lb/>quæ æqualibus &longs;unt pyramidibus ip&longs;orum duplis ord ine <lb/>diui&longs;ionis inter &longs;e re&longs;pondentibus in&longs;cripta; igitur vt ante, <lb/>quater quinorum octaedrorum &longs;imul in axe DN erit <lb/>centrum grauitatis: &longs;ed & octaedri FGHKLM centrum <lb/>grauitatis e&longs;t in axe DN; vnius igitur & viginti octae<lb/>drorum in pyramide ABCD exi&longs;tentium ex hac &longs;ecun<lb/>da diui&longs;ione, tanquàm vnius magnitudinis in axe DN erit <lb/>centrum grauitatis. </s>

Line 938

<s>Sic autem facienti, & <lb/>reliquarum pyramidum demptis præcedentibus octaedris, <lb/>dimidia octaedra &longs;emper auferenti, tandem relinquen<lb/>tur pyramides minores &longs;imul &longs;umptæ quantacumque <lb/>magnitudine propo&longs;ita. </s>

<s>Totius igitur pyramidis ABCD <pb pagenum="64"/>in axe DN, erit centrum grauitatis. </s>

<s>Totius igitur pyramidis ABCD <pb xlink:href="043/01/072.jpg" pagenum="64"/>in axe DN, erit centrum grauitatis. </s>

<s>Eadem ratione in <lb/>quolibet reliquorum trium axium, pyramidis ABCD, ip<lb/>&longs;ius centrum grauitatis e&longs;se o&longs;tenderemus; communis igi<lb/>tur &longs;ectio quatuor axium pyramidis ABCD, quod e&longs;t <lb/>ip&longs;ius centrum E, erit centrum grauitatis pyramidis AB <lb/>CD. </s>

Line 958

<s>Ducta enim AC, &longs;it <lb/>trianguli ABC, centrum grauitatis H, &longs;icut & K, trian<lb/>guli ACD: & iungantur KH, HE, EK: Factaque vt <lb/>EM, ad MF, ita EL ad LH, & EN ad N<emph type="italics"/>K<emph.end type="italics"/>, iun<lb/>gatur LN. </s>

<s>Quoniam igitur EF e&longs;t axis pyramidis <lb/>ABCDE, erit ba&longs;is ABCD centrum grauitatis F. <pb pagenum="65"/>Rur&longs;us quia puncta K, H, &longs;unt centra grauitatis triangu<lb/>lorum ABC, CDA, erunt EH, EK, axes pyramidum <lb/>ABCE, ACDA: quorum EL, e&longs;t tripla ip&longs;ius LH, <lb/>nec non EN, tripla ip&longs;ius EK; pyramidis igitur ABCE, <lb/>centrum grauitatis erit L, &longs;icut & K, pyramidis ACDE. <lb/>Rur&longs;us, quoniam totius quadrilateri ABCD, e&longs;t cen<lb/>trum grauitatis F, cuius magnitudinis partium triangu<lb/>lorum ABC, CDA, centra grauitatis &longs;unt K, H; recta <lb/>KH, à puncto F, &longs;ic <lb/>diuiditur, vt &longs;it HF, ad <lb/>FK, vt triangulum <lb/>ACD, ad triangulum <lb/>ABC, hoc e&longs;t, vt py<lb/>ramis ACDE, ad py <lb/>ramidem ABCE. &longs;ed <lb/>vt HF, ad FK, ita <lb/>e&longs;t LM, ad MN; vt <lb/>igitur e&longs;t pyramis AC <lb/>DE, ad pyramidem <lb/>ABCE, ita erit LM, <lb/>ad MN. </s>

<s>Quoniam igitur EF e&longs;t axis pyramidis <lb/>ABCDE, erit ba&longs;is ABCD centrum grauitatis F. <pb xlink:href="043/01/073.jpg" pagenum="65"/>Rur&longs;us quia puncta K, H, &longs;unt centra grauitatis triangu<lb/>lorum ABC, CDA, erunt EH, EK, axes pyramidum <lb/>ABCE, ACDA: quorum EL, e&longs;t tripla ip&longs;ius LH, <lb/>nec non EN, tripla ip&longs;ius EK; pyramidis igitur ABCE, <lb/>centrum grauitatis erit L, &longs;icut & K, pyramidis ACDE. <lb/>Rur&longs;us, quoniam totius quadrilateri ABCD, e&longs;t cen<lb/>trum grauitatis F, cuius magnitudinis partium triangu<lb/>lorum ABC, CDA, centra grauitatis &longs;unt K, H; recta <lb/>KH, à puncto F, &longs;ic <lb/>diuiditur, vt &longs;it HF, ad <lb/>FK, vt triangulum <lb/>ACD, ad triangulum <lb/>ABC, hoc e&longs;t, vt py<lb/>ramis ACDE, ad py <lb/>ramidem ABCE. &longs;ed <lb/>vt HF, ad FK, ita <lb/>e&longs;t LM, ad MN; vt <lb/>igitur e&longs;t pyramis AC <lb/>DE, ad pyramidem <lb/>ABCE, ita erit LM, <lb/>ad MN. </s>

<s>Sed N, e&longs;t <lb/>centrum grauitatis py<lb/><figure id="fig40"/><lb/>ramidis ACDE, & L pyramidis ABCE; punctum <lb/>igitur M, erit centrum grauitatis pyramidis ABCDE. <lb/></s>

<s>Sed N, e&longs;t <lb/>centrum grauitatis py<lb/><figure id="id.043.01.073.1.jpg" xlink:href="043/01/073/1.jpg"/><lb/>ramidis ACDE, & L pyramidis ABCE; punctum <lb/>igitur M, erit centrum grauitatis pyramidis ABCDE. <lb/></s>

<s>Quod &longs;i pyramis habeat ba&longs;im quinquelateram; po&longs;ito <lb/>rur&longs;us axe totius pyramidis, & ba&longs;i &longs;ecta in triangulum, <lb/>& quadrilaterum, po&longs;itis vtriu&longs;que proprijs centris graui<lb/>tatis, eadem demon&longs;tratione propo&longs;itum concludetur. <lb/></s>

<s>Quemadmodum &longs;i ba&longs;is &longs;it &longs;ex laterum, &longs;ecta ea in quinque <lb/>laterum, & triangulum, & reliquis vt antea po&longs;itis: & &longs;ic &longs;em <lb/>per deinceps. </s>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s><pb pagenum="66"/>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s><pb xlink:href="043/01/074.jpg" pagenum="66"/>

<s><emph type="italics"/>PROPOSITIO XXXIII.<emph.end type="italics"/></s>

Line 980

<s>Quoniam igitur GH, e&longs;t axis pri&longs;matis <lb/>ABCDEF, erit punctum G, centrum grauitatis trian<lb/>guli DEF: &longs;icut & H, trian<lb/>guli ABC; vtraque igitur <lb/>dupla e&longs;t AG, ip&longs;ius GO, <lb/>& CH, ip&longs;ius PH, &longs;ectæ<lb/>que erunt AB, DE, bifa<lb/>riam in punctis P, O: pa<lb/>rallela igitur, & æqualis e&longs;t <lb/>OP, ip&longs;i DA, iamque ip&longs;i <lb/>FC. quæ igitur illas con<lb/>iungunt CP, FO, æqua<lb/>les &longs;unt, & parallelæ, & pa<lb/>rallelogrammum FP. <lb/></s>

<s>Nunc &longs;ecta OP, bifariam in <lb/>puncto N, iungantur GN, <lb/>NF, AF, FH, FB, & fa<lb/>cta FL, tripla ip&longs;ius LH, <lb/><figure id="fig41"/><lb/>à puncto L, per punctum K, ducatur recta LKMR. <lb/></s>

<s>Nunc &longs;ecta OP, bifariam in <lb/>puncto N, iungantur GN, <lb/>NF, AF, FH, FB, & fa<lb/>cta FL, tripla ip&longs;ius LH, <lb/><figure id="id.043.01.074.1.jpg" xlink:href="043/01/074/1.jpg"/><lb/>à puncto L, per punctum K, ducatur recta LKMR. <lb/></s>

<s>Quoniam igitur e&longs;t vt FG, ad GO, ita CH, ad HP, <lb/>& parallelogrammum e&longs;t FCPO; parallelogramma <lb/>etiam erunt CG, GP, angulus igitur FGH, æqualis <lb/>erit angulo NGO, quos circa æquales angulos latera <pb pagenum="67"/>FG, GH, homologa &longs;unt lateribus GO, ON. nam <lb/>dupla e&longs;t FG, ip&longs;ius GO, & GH, ip&longs;ius ON; angulus <lb/>igitur OGN, æqualis erit angulo GFH; parallela igi<lb/>tur GN, ip&longs;i FH, & propter&longs;imilitudinem triangulorum <lb/>dupla erit FH, ip&longs;ius GN. Rur&longs;us, quoniam recta <lb/>OP, &longs;ecat latera oppo&longs;ita parallelogrammi BD, bifa<lb/>riam in punctis O, P, &longs;ecta, & ip&longs;a bifariam in puncto N, <lb/>erit punctum N, parallelogrammi BD, centrum graui<lb/>tatis, atque ideo axis FN, pyramidis ABDEF. qua <lb/>ratione erit quoque axis FH, pyramidis ABCF: &longs;ed <lb/>FL, e&longs;t tripla ip&longs;ius LH; pyramidis igitur ABCF, cen<lb/>trum grauitatis erit L. </s>

<s>Quoniam igitur e&longs;t vt FG, ad GO, ita CH, ad HP, <lb/>& parallelogrammum e&longs;t FCPO; parallelogramma <lb/>etiam erunt CG, GP, angulus igitur FGH, æqualis <lb/>erit angulo NGO, quos circa æquales angulos latera <pb xlink:href="043/01/075.jpg" pagenum="67"/>FG, GH, homologa &longs;unt lateribus GO, ON. nam <lb/>dupla e&longs;t FG, ip&longs;ius GO, & GH, ip&longs;ius ON; angulus <lb/>igitur OGN, æqualis erit angulo GFH; parallela igi<lb/>tur GN, ip&longs;i FH, & propter&longs;imilitudinem triangulorum <lb/>dupla erit FH, ip&longs;ius GN. Rur&longs;us, quoniam recta <lb/>OP, &longs;ecat latera oppo&longs;ita parallelogrammi BD, bifa<lb/>riam in punctis O, P, &longs;ecta, & ip&longs;a bifariam in puncto N, <lb/>erit punctum N, parallelogrammi BD, centrum graui<lb/>tatis, atque ideo axis FN, pyramidis ABDEF. qua <lb/>ratione erit quoque axis FH, pyramidis ABCF: &longs;ed <lb/>FL, e&longs;t tripla ip&longs;ius LH; pyramidis igitur ABCF, cen<lb/>trum grauitatis erit L. </s>

<s>Rur&longs;us quia e&longs;t vt GK, ad KH, <lb/>ita GR, ad LH, propter &longs;imilitudinem triangulorum, <lb/>erit æqualis GR, ip&longs;i LH: &longs;ed e&longs;t FH, quadrupla ip-, <lb/>&longs;ius LH, quadrupla igitur FH, ip&longs;ius GR: &longs;ed FH <lb/>erat dupla ip&longs;ius GN; quadrupla igitur FH, reliquæ <lb/>NR, ac proinde GR, RN, æquales erunt: recta igitur <lb/>FL, tripla erit vtriu&longs;que ip&longs;arum GR, RN, &longs;ed vt FL, <lb/>ad NR, ita e&longs;t FM, ad MN, propter &longs;imilitudinem trian <lb/>gulorum; recta igitur FM, erit ip&longs;ius MN, tripla, &longs;icut <lb/>& LM, ip&longs;ius MR: &longs;ed quia KH, e&longs;t æqualis GK, <lb/>erit & LK, æqualis RK; propter &longs;imilitudinem trian<lb/>gulorum; cum igitur LK, &longs;it tripla ip&longs;ius MR, erit LK, <lb/>ip&longs;ius KM, dupla; vt igitur e&longs;t pyramis ABEDF, ad <lb/>pyramidem ABCF, ita erit LK, ad KM; e&longs;t autem M, <lb/>centrum grauitatis pyramidis ABED, &longs;icut & L, pyrami<lb/>dis ABCF; totius igitur pri&longs;matis ABCDEF, centrum <lb/>grauitatis erit K. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="68"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/076.jpg" pagenum="68"/>

<s><emph type="italics"/>PROPOSITIO XXXIV.<emph.end type="italics"/></s>

Line 996

<s>Dico punctum M, e&longs;se centrum grauitatis pri&longs;<lb/>matis ABCDEFGH. </s>

<s>Iungantur enim rectæ BD, FH, <lb/>vt parallelogrammum &longs;it BH, &longs;ectumque totum pri&longs;ma <lb/>in duo pri&longs;mata, quorum ba<lb/>&longs;es &longs;unt triangula, in quæ &longs;ecta <lb/>&longs;unt quadrilatera AC, EG, <lb/>&longs;int autem axes duorum pri&longs;<lb/>matum triangulas ba&longs;es ha<lb/>bentium NO, <expan abbr="Pq.">Pque</expan> Erunt <lb/>igitur centra grauitatis O, tri<lb/>anguli ABD, & L, quadri<lb/>lateri AC, & Q, trianguli <lb/>BCD, itemque N, trianguli <lb/>EFH, & K, quadrilateri EG, <lb/>& P, trianguli FGH: iun<lb/>ctæ igitur OQ, NP, per pun <lb/><figure id="fig42"/><lb/>cta L, K, tran&longs;ibunt: cumque tres prædicti axes &longs;int <lb/>lateribus pri&longs;matis, atque ideo inter &longs;e quoque paralleli; <lb/>parallelogramma erunt OP, NL, LP. ducta igitur per <lb/>punctum M, ip&longs;i OQ, vel NP, parallela RS, erit vt <lb/>NK, ad KP, ita RM, ad MS: & vt KM, ad ML, ita <lb/>NR, ad RO, & PS, ad SQ: &longs;ed KM, e&longs;t æqualis ML; <lb/>igitur & KR, ip&longs;i RO, & PS, ip&longs;i SQ, æqualis erit: &longs;unt <lb/>autem hæ &longs;egmenta axium NO, <expan abbr="Pq;">Pque</expan> punctum igitur <lb/>R, e&longs;t centrum grauitatis pri&longs;matis ABDEFH: & per <pb pagenum="69"/>punctum S, pri&longs;matis BCDFGH. </s>

<s>Iungantur enim rectæ BD, FH, <lb/>vt parallelogrammum &longs;it BH, &longs;ectumque totum pri&longs;ma <lb/>in duo pri&longs;mata, quorum ba<lb/>&longs;es &longs;unt triangula, in quæ &longs;ecta <lb/>&longs;unt quadrilatera AC, EG, <lb/>&longs;int autem axes duorum pri&longs;<lb/>matum triangulas ba&longs;es ha<lb/>bentium NO, <expan abbr="Pq.">Pque</expan> Erunt <lb/>igitur centra grauitatis O, tri<lb/>anguli ABD, & L, quadri<lb/>lateri AC, & Q, trianguli <lb/>BCD, itemque N, trianguli <lb/>EFH, & K, quadrilateri EG, <lb/>& P, trianguli FGH: iun<lb/>ctæ igitur OQ, NP, per pun <lb/><figure id="id.043.01.076.1.jpg" xlink:href="043/01/076/1.jpg"/><lb/>cta L, K, tran&longs;ibunt: cumque tres prædicti axes &longs;int <lb/>lateribus pri&longs;matis, atque ideo inter &longs;e quoque paralleli; <lb/>parallelogramma erunt OP, NL, LP. ducta igitur per <lb/>punctum M, ip&longs;i OQ, vel NP, parallela RS, erit vt <lb/>NK, ad KP, ita RM, ad MS: & vt KM, ad ML, ita <lb/>NR, ad RO, & PS, ad SQ: &longs;ed KM, e&longs;t æqualis ML; <lb/>igitur & KR, ip&longs;i RO, & PS, ip&longs;i SQ, æqualis erit: &longs;unt <lb/>autem hæ &longs;egmenta axium NO, <expan abbr="Pq;">Pque</expan> punctum igitur <lb/>R, e&longs;t centrum grauitatis pri&longs;matis ABDEFH: & per <pb xlink:href="043/01/077.jpg" pagenum="69"/>punctum S, pri&longs;matis BCDFGH. </s>

<s>Quoniam igitur <lb/>quadrilateri EG, e&longs;t centrum grauitatis K, cuius duorum <lb/>triangulorum centra grauitatis &longs;unt P, N; erit vt triangu<lb/>lum FGH, ad triangulum EFH, hoc e&longs;t vt pri&longs;ma BC<lb/>DFGH, ad pri&longs;ma ABDEFH, ita NK, ad KP, hoc <lb/>e&longs;t RM, ad MS; cum igitur &longs;it R, centrum grauitatis <lb/>pri&longs;matis ABDEFH: &longs;icut & S, pri&longs;matis BCDFGH; <lb/>totius pri&longs;matis ABCDEFGH, centrum grauitatis erit <lb/>M. </s>

Line 1008

<s>Omnis fru&longs;ti pyramidis triangulam ba&longs;im <lb/>ha bentis centrum grauitatis e&longs;t in axe, primum <lb/>ita diui&longs;o, vt &longs;egmentum attingens minorem <lb/>ba&longs;im &longs;it ad reliquum, vt duplum vnius laterum <lb/>maioris ba&longs;is vna cum latere homologo mino<lb/>ris, ad duplum prædicti lateris minoris ba&longs;is, <lb/>vna cum latere homologo maioris. </s>

<s>Deinde <lb/>à puncto &longs;ectionis ab&longs;ci&longs;sa quarta parte &longs;eg<lb/>menti, quod maiorem ba&longs;im attingit, & à pun<lb/>cto, in quo ad minorem ba&longs;im axis termina<lb/>tur &longs;umpta item quarta parte totius axis; in <lb/>eo puncto, in quo &longs;egmentum axis duabus po<lb/>&longs;terioribus &longs;ectionibus finitum &longs;ic diuiditur, vt <pb pagenum="70"/>&longs;egmentum eius maiori ba&longs;i propinquius &longs;it ad to<lb/>tum prædictum interiectum &longs;egmentum, vt tertia <lb/>proportionalis minor ad duo latera homologa ba<lb/>&longs;ium oppo&longs;itarum, ad compo&longs;itam ex his tribus <lb/>deinceps proportionalibus. </s>

<s>Deinde <lb/>à puncto &longs;ectionis ab&longs;ci&longs;sa quarta parte &longs;eg<lb/>menti, quod maiorem ba&longs;im attingit, & à pun<lb/>cto, in quo ad minorem ba&longs;im axis termina<lb/>tur &longs;umpta item quarta parte totius axis; in <lb/>eo puncto, in quo &longs;egmentum axis duabus po<lb/>&longs;terioribus &longs;ectionibus finitum &longs;ic diuiditur, vt <pb xlink:href="043/01/078.jpg" pagenum="70"/>&longs;egmentum eius maiori ba&longs;i propinquius &longs;it ad to<lb/>tum prædictum interiectum &longs;egmentum, vt tertia <lb/>proportionalis minor ad duo latera homologa ba<lb/>&longs;ium oppo&longs;itarum, ad compo&longs;itam ex his tribus <lb/>deinceps proportionalibus. </s>

<s>Sit pyramidis fru&longs;tum, cuius ba&longs;es oppo&longs;itæ, & parallelæ, <lb/>maior triangulum ABC, minor autem triangulum DEF, <lb/>axis autem GH. triangulorum autem ABC, DEF, quæ <lb/>inter &longs;e &longs;imilia e&longs;se nece&longs;se e&longs;t, &longs;int duo latera homologa <lb/>BC, EF: & vt e&longs;t BC, ad EF, ita &longs;it EF, ad X: vt autem e&longs;t <lb/>duplum lateris BC, vna cum latere EF, ad duplum lateris <lb/>EF, vna cum la <lb/>tere BC, ita &longs;it <lb/>HN, ad NG, <lb/>& NO, pars quar <lb/>ta ip&longs;ius NG, & <lb/>HS, pars quar<lb/>ta ip&longs;ius GH; ip <lb/>&longs;ius autem SO, <lb/>&longs;it VO, ad OS, <lb/>vt e&longs;t X, ad com<lb/>po&longs;itam ex tri<lb/>bus BC, EF, X. <lb/></s>

<s>Dico punctum V <lb/>(quod cadet ne<lb/>ce&longs;sario infra <lb/><figure id="fig43"/><lb/>punctum N, quanquam hoc ad demon&longs;trationem nihil re<lb/>fert) e&longs;se centrum grauitatis fru&longs;ti ABCDEF. </s>

<s>Dico punctum V <lb/>(quod cadet ne<lb/>ce&longs;sario infra <lb/><figure id="id.043.01.078.1.jpg" xlink:href="043/01/078/1.jpg"/><lb/>punctum N, quanquam hoc ad demon&longs;trationem nihil re<lb/>fert) e&longs;se centrum grauitatis fru&longs;ti ABCDEF. </s>

<s>Ducta <lb/>enim recta AGL; quoniam GH, e&longs;t axis fru&longs;ti ABCD <lb/>EF, & punctum G, centrum grauitatis trianguli ABC, <lb/>erit punctum L, in medio ba&longs;is BC: &longs;ecto igitur etiam la<lb/>tere EF, bifariam in puncto K, iungantur LK, <emph type="italics"/>K<emph.end type="italics"/>H: & vt <pb pagenum="71"/>vt e&longs;t HN, ad NG, ita fiat KM, ad ML, & GM, iun<lb/>gatur: & vt e&longs;t GO, ad ON, ita fiat GP, ad PM, & iun <lb/>gantur MN, OP, FG, GD, GE. </s>

<s>Ducta <lb/>enim recta AGL; quoniam GH, e&longs;t axis fru&longs;ti ABCD <lb/>EF, & punctum G, centrum grauitatis trianguli ABC, <lb/>erit punctum L, in medio ba&longs;is BC: &longs;ecto igitur etiam la<lb/>tere EF, bifariam in puncto K, iungantur LK, <emph type="italics"/>K<emph.end type="italics"/>H: & vt <pb xlink:href="043/01/079.jpg" pagenum="71"/>vt e&longs;t HN, ad NG, ita fiat KM, ad ML, & GM, iun<lb/>gatur: & vt e&longs;t GO, ad ON, ita fiat GP, ad PM, & iun <lb/>gantur MN, OP, FG, GD, GE. </s>

<s>Quoniam igitur re <lb/>cta KL, &longs;ecat trapezij BCFE, latera parallela bifariam <lb/>in punctis K,L, & e&longs;t vt HN, ad NG, hoc e&longs;t vt duplum <lb/>lateris BC, vna cum latere EF, ad duplum lateris EF, vna <lb/>cum latere BC, ita KM, ad ML; erit punctum M, cen<lb/>trum grauitatis trapezij BCFE, & pyramidis GBCFE, <lb/>axis GM. </s>

Line 1024

<s>Simi<lb/>liter ex puncto O, ad axes duarum pyramidum GABED, <lb/>GACFD, duæ aliæ rectæ lineæ ducerentnr, quas & cen<lb/>tra grauitatis pyramidum habere, & parallelas rectis GQ, <lb/>GR, alteram alteri e&longs;se o&longs;tenderemus, &longs;icut o&longs;tendimus <lb/>OP, habentem centrum grauitatis pyramidis GBCFE, <lb/>ip&longs;i GL, parallelam; &longs;ed tres rectæ GL, GQ, GR, &longs;unt <lb/>in eodem plano trianguli nimirum ABC; tres igitur præ<lb/>dictæ parallelæ, quæ ex puncto O, atque ideo trium præ<lb/>dictarum pyramidum centra grauitatis erunt in eodem pla<lb/>no, per punctum O, & trianguli ABC, parallelo. </s>

<s>Quo<lb/>niam igitur fru&longs;ti ABCDE, centrum grauitatis e&longs;t in axe <lb/>GH; (manife&longs;tum hoc autem ex duobus centris grauitatis <lb/>pyramidis, cuius e&longs;t prædictum fru&longs;tum, & ablatæ, quæ <lb/>centra grauitatis &longs;unt in axe, cuius &longs;egmentum e&longs;t axis <pb pagenum="72"/>GH) erit eiu&longs;dem fru&longs;ti ABCDEF, centrum grauitatis <lb/>O. </s>

<s>Quo<lb/>niam igitur fru&longs;ti ABCDE, centrum grauitatis e&longs;t in axe <lb/>GH; (manife&longs;tum hoc autem ex duobus centris grauitatis <lb/>pyramidis, cuius e&longs;t prædictum fru&longs;tum, & ablatæ, quæ <lb/>centra grauitatis &longs;unt in axe, cuius &longs;egmentum e&longs;t axis <pb xlink:href="043/01/080.jpg" pagenum="72"/>GH) erit eiu&longs;dem fru&longs;ti ABCDEF, centrum grauitatis <lb/>O. </s>

<s>Rur&longs;us quoniam vt tres deinceps proportionales BC, <lb/>EF, X, &longs;imul ad BC, ita e&longs;t fru&longs;tum ABCDEF, ad py<lb/>ramidem; &longs;i de&longs;cribatur ABCH: &longs;ed vt triangulum ABC, <lb/>ad &longs;imile triangulum EDF, hoc e&longs;t vt BC, ad X, ita e&longs;t <lb/>pyramis ABCH, ad pyramidem GDEF; erit ex æqua<lb/>li, vt tres lineæ <lb/>BC, EF, X, &longs;i<lb/>mul ad X, ita fru <lb/>&longs;tum ABCDEF, <lb/>ad pyramidem <lb/>GDEF: & con<lb/>uertendo, vt X, <lb/>ad compo&longs;itam <lb/>ex BC, EF, X, <lb/>hoc e&longs;t vt VO, <lb/>ad OS, ita pyra <lb/>mis GDEF, ad <lb/>fru&longs;tum ABC<lb/>DEF; & diui<lb/>dendo, vt pyra<lb/><figure id="fig44"/><lb/>mis GDEF, ad reliquas tres pyramides fru&longs;ti, ita OV, <lb/>ad VS; &longs;ed S, e&longs;t centrum grauitatis pyramidis GDEF, <lb/>& O, trium reliquarum; fru&longs;ti igitur ABCDEF, cen<lb/>trum grauitatis erit V. </s>

<s>Rur&longs;us quoniam vt tres deinceps proportionales BC, <lb/>EF, X, &longs;imul ad BC, ita e&longs;t fru&longs;tum ABCDEF, ad py<lb/>ramidem; &longs;i de&longs;cribatur ABCH: &longs;ed vt triangulum ABC, <lb/>ad &longs;imile triangulum EDF, hoc e&longs;t vt BC, ad X, ita e&longs;t <lb/>pyramis ABCH, ad pyramidem GDEF; erit ex æqua<lb/>li, vt tres lineæ <lb/>BC, EF, X, &longs;i<lb/>mul ad X, ita fru <lb/>&longs;tum ABCDEF, <lb/>ad pyramidem <lb/>GDEF: & con<lb/>uertendo, vt X, <lb/>ad compo&longs;itam <lb/>ex BC, EF, X, <lb/>hoc e&longs;t vt VO, <lb/>ad OS, ita pyra <lb/>mis GDEF, ad <lb/>fru&longs;tum ABC<lb/>DEF; & diui<lb/>dendo, vt pyra<lb/><figure id="id.043.01.080.1.jpg" xlink:href="043/01/080/1.jpg"/><lb/>mis GDEF, ad reliquas tres pyramides fru&longs;ti, ita OV, <lb/>ad VS; &longs;ed S, e&longs;t centrum grauitatis pyramidis GDEF, <lb/>& O, trium reliquarum; fru&longs;ti igitur ABCDEF, cen<lb/>trum grauitatis erit V. </s>

<s>Quod demon&longs;trandum erat. </s>

<s><emph type="italics"/>PROPOSITIO XXXVI.<emph.end type="italics"/></s>

<s>Omnis fru&longs;ti pyramidis ba&longs;im plu&longs;quam trila<lb/>teram habentis centrum grauitatis e&longs;t punctum <lb/>illud, in quo axis &longs;ic diuiditur, vt axis fru&longs;ti pyra<lb/>midis triangulam ba&longs;im habentis diuiditur ab <lb/>ip&longs;ius centro grauitatis. </s><pb pagenum="73"/>

<s>Sit pyramidis quadrilateram ba&longs;im habentis fru&longs;tum <lb/>ABCDEFGH, cuius axis KL, atque in ip&longs;o centrum <lb/>grauitatis O. </s>

<s>Dico axim KL, &longs;ectum e&longs;se in puncto O, <lb/>vt propo&longs;uimus. </s>

<s>Ductis enim AC, EG, quæ &longs;imilium <lb/>&longs;ectionum angulos æquales &longs;ubtendant B, F, qui late<lb/>ribus homologis continentur, fru&longs;ta erunt pyramidum <lb/>triangulas ba&longs;es habentium AFG, AGH: &longs;it autem fru<lb/>&longs;ti AFG, axis <lb/>TP, & in eo eiu&longs; <lb/>dem fru&longs;ti cen<lb/>trum grauitatis <lb/>M, & fru&longs;ti AG <lb/>H, axis VQ, & <lb/>in eo centrum <lb/>grauitatis N, & <lb/>iungantur TV, <lb/>MN, <expan abbr="Pq.">Pque</expan> Quo <lb/>niam igitur e&longs;t <lb/>pyramidis fru<lb/>&longs;tum, quod pro<lb/>ponitur; omnia <lb/><figure id="fig45"/><lb/>cius producta latera concurrent in vno puncto, qui e&longs;t pyra<lb/>midis vertex: fru&longs;ta igitur, in quæ diui&longs;um e&longs;t fru&longs;tum pro<lb/>po&longs;itum earum &longs;unt pyramidum, quæ verticem habent <lb/>communem cum pyramide, cuius e&longs;t fru&longs;tum propo&longs;itum: <lb/>tres igitur talium fru&longs;torum axes, vt pote &longs;egmenta axium <lb/>trium prædictarum pyramidum in communi illo vertice <lb/>concurrent: quilibet igitur duo trium prædictorum axium <lb/>KL, TP, VQ, erunt in eodem plano: TP, igitur, & <lb/>VQ, &longs;unt in eodem plano. </s>

<s>Ductis enim AC, EG, quæ &longs;imilium <lb/>&longs;ectionum angulos æquales &longs;ubtendant B, F, qui late<lb/>ribus homologis continentur, fru&longs;ta erunt pyramidum <lb/>triangulas ba&longs;es habentium AFG, AGH: &longs;it autem fru<lb/>&longs;ti AFG, axis <lb/>TP, & in eo eiu&longs; <lb/>dem fru&longs;ti cen<lb/>trum grauitatis <lb/>M, & fru&longs;ti AG <lb/>H, axis VQ, & <lb/>in eo centrum <lb/>grauitatis N, & <lb/>iungantur TV, <lb/>MN, <expan abbr="Pq.">Pque</expan> Quo <lb/>niam igitur e&longs;t <lb/>pyramidis fru<lb/>&longs;tum, quod pro<lb/>ponitur; omnia <lb/><figure id="id.043.01.081.1.jpg" xlink:href="043/01/081/1.jpg"/><lb/>cius producta latera concurrent in vno puncto, qui e&longs;t pyra<lb/>midis vertex: fru&longs;ta igitur, in quæ diui&longs;um e&longs;t fru&longs;tum pro<lb/>po&longs;itum earum &longs;unt pyramidum, quæ verticem habent <lb/>communem cum pyramide, cuius e&longs;t fru&longs;tum propo&longs;itum: <lb/>tres igitur talium fru&longs;torum axes, vt pote &longs;egmenta axium <lb/>trium prædictarum pyramidum in communi illo vertice <lb/>concurrent: quilibet igitur duo trium prædictorum axium <lb/>KL, TP, VQ, erunt in eodem plano: TP, igitur, & <lb/>VQ, &longs;unt in eodem plano. </s>

<s>Eadem autem ratione, qua <lb/>vtebamur de pri&longs;mate K, centrum grauitatis K, ba&longs;is <lb/>EH, e&longs;t in linea TV, & L, ba&longs;is BD, centrum grauita<lb/>tis e&longs;t in linea <expan abbr="Pq;">Pque</expan> reliquæ igitur KL, MN, erunt in eo<lb/>dem plano trapezij PTVQ, &longs;eque mutuo &longs;ecabunt: cum <pb pagenum="74"/>igitur M, N, &longs;int centra grauitatis propo&longs;iti pri&longs;matis par <lb/>tium pri&longs;matum AFG, AGH, atque obid O, totius pri&longs;<lb/>matis AFGH, in linea MN, centrum grauitatis; per pun <lb/>ctum O, recta MN, tran&longs;ibit. </s>

<s>Eadem autem ratione, qua <lb/>vtebamur de pri&longs;mate K, centrum grauitatis K, ba&longs;is <lb/>EH, e&longs;t in linea TV, & L, ba&longs;is BD, centrum grauita<lb/>tis e&longs;t in linea <expan abbr="Pq;">Pque</expan> reliquæ igitur KL, MN, erunt in eo<lb/>dem plano trapezij PTVQ, &longs;eque mutuo &longs;ecabunt: cum <pb xlink:href="043/01/082.jpg" pagenum="74"/>igitur M, N, &longs;int centra grauitatis propo&longs;iti pri&longs;matis par <lb/>tium pri&longs;matum AFG, AGH, atque obid O, totius pri&longs;<lb/>matis AFGH, in linea MN, centrum grauitatis; per pun <lb/>ctum O, recta MN, tran&longs;ibit. </s>

<s>Et quoniam planum tra<lb/>pezij PV, &longs;ecatur duobus planis parallelis, erunt TV, PQ, <lb/>fectiones parallelæ. </s>

<s>His demon&longs;tratis, fiat rur&longs;us vt AB, <lb/>bis vna cum EF, ad EF, bis vna cum AB, ita TY, ad <lb/>YP: & &longs;umatur T<foreign lang="greek">w</foreign>, pars quarta ip&longs;ius TP, & YZ, pars <lb/>quarta ip&longs;ius PY, & ad axim KL, ducantur ip&longs;is TV, <lb/>PQ, parallelæ <lb/><foreign lang="greek">w</foreign>S, YR, ZX, <lb/>quæ rectas TP, <lb/>KL, &longs;ecabunt in <lb/><expan abbr="ea&longs;d&etilde;">ea&longs;dem</expan> rationes: <lb/>vt igitur TY, ad <lb/><foreign lang="greek">*u</foreign>P, hoc e&longs;t vt <lb/>AB, bis vna cum <lb/>EF, ad EF bis <lb/>vna cum AB, ita <lb/>erit <emph type="italics"/>K<emph.end type="italics"/>R, ad RL, <lb/>eritque KS, pars <lb/>quarta ip&longs;ius K <lb/>L, qualis & R <lb/><figure id="fig46"/><lb/>X, ip&longs;ius RL. </s>

<s>Et quoniam M, e&longs;t centrum grauitatis fru<lb/>&longs;ti AFG; manife&longs;tum e&longs;t ex tribus prædictis axis TP, &longs;e<lb/>ctionibus <foreign lang="greek">*u, w</foreign>, Z, e&longs;se MZ, ad Z<foreign lang="greek">w</foreign>, hoc e&longs;t OX, ad XS, <lb/>vt e&longs;t 6 ad compo&longs;itam ex tribus deinceps proportionalibus <lb/>AB, EF, 6; Fru&longs;ti igitur ABCDEFGH, centrum gra<lb/>uitatis O, axim KL, ita diuidit, vt propo&longs;uimus. </s>

<s>Quod <lb/>&longs;i fru&longs;tum propo&longs;itum &longs;it pyramidis ba&longs;im habentis quin<lb/>quelateram, & quotcumque plurium deinceps fuerit la<lb/>terum, eadem demon&longs;tratione &longs;emper deinceps, vt in pri&longs;<lb/>mate monuimus, propo&longs;itum concluderemus. </s><pb pagenum="75"/>

<s><emph type="italics"/>PROPOSITIO XXXVII.<emph.end type="italics"/></s>

Line 1058

<s><emph type="italics"/>PROPOSITIO XXXVIII.<emph.end type="italics"/></s>

<s>Data qualibet figura, cuius termini omnis <lb/>cauitas &longs;it interior, &longs;i certum in ea punctum talis <lb/>cius partis centrum grauitatis e&longs;se po&longs;sit, quæ ab <lb/>ca deficiat minori &longs;pacio quantacumque magnitu <lb/>dine propo&longs;ita; illud erit totius figuræ centrum <lb/>grauitatis. </s><pb pagenum="76"/>

<s>E&longs;to figura AB, cuius termini omnis cauitas &longs;it interior <lb/>& certum in ea punctum E, talis partis AB, figuræ qua<lb/>lem diximus centrum grauitatis e&longs;se po&longs;sit. </s>

<s>Dico pun<lb/>ctum E, e&longs;se figuræ AB, centrum grauitatis. </s>

<s>Si enim <lb/>E, non e&longs;t, erit aliud, e&longs;to F: & iuncta EF producatur, <lb/>& &longs;umatur in illa extra figuræ AB, terminum, quodlibet <lb/>punctum G; & vt e&longs;t FE, ad EG, ita &longs;it alia magnitudo <lb/>K, ad figuram AB, & <lb/>ex vi hypothe&longs;is &longs;it pars <lb/>quædam CD, figuræ <lb/>AB, cuius centrum gra<lb/>uitatis E, talis vt abla<lb/>ta relinquat AC, minus <lb/>magnitudine <emph type="italics"/>K.<emph.end type="italics"/></s><s> Mi<lb/>nor igitur proportio erit <lb/>AC, ad AB, quàm K, <lb/>ad AB, hoc e&longs;t quàm <lb/>FE, ad EG; fiat vt <lb/>AC, ad AB, ita EF, <lb/>ad FGH: &longs;ed F, e&longs;t cen <lb/>trum grauitatis totius <lb/>AB, & E, vnius par<lb/>tis CD; reliquæ igitur <lb/><figure id="fig47"/><lb/>partis AC, centrum grauitatis erit H, vltra punctum G: &longs;ed <lb/>G, cadit extra terminum figuræ AC; multo igitur magis H: <lb/>Quod e&longs;t ab&longs;urdum. </s>

<s>Si enim <lb/>E, non e&longs;t, erit aliud, e&longs;to F: & iuncta EF producatur, <lb/>& &longs;umatur in illa extra figuræ AB, terminum, quodlibet <lb/>punctum G; & vt e&longs;t FE, ad EG, ita &longs;it alia magnitudo <lb/>K, ad figuram AB, & <lb/>ex vi hypothe&longs;is &longs;it pars <lb/>quædam CD, figuræ <lb/>AB, cuius centrum gra<lb/>uitatis E, talis vt abla<lb/>ta relinquat AC, minus <lb/>magnitudine <emph type="italics"/>K.<emph.end type="italics"/></s><s> Mi<lb/>nor igitur proportio erit <lb/>AC, ad AB, quàm K, <lb/>ad AB, hoc e&longs;t quàm <lb/>FE, ad EG; fiat vt <lb/>AC, ad AB, ita EF, <lb/>ad FGH: &longs;ed F, e&longs;t cen <lb/>trum grauitatis totius <lb/>AB, & E, vnius par<lb/>tis CD; reliquæ igitur <lb/><figure id="id.043.01.084.1.jpg" xlink:href="043/01/084/1.jpg"/><lb/>partis AC, centrum grauitatis erit H, vltra punctum G: &longs;ed <lb/>G, cadit extra terminum figuræ AC; multo igitur magis H: <lb/>Quod e&longs;t ab&longs;urdum. </s>

<s>Non igitur aliud punctum à puncto <lb/>E; punctum igitur E, figuræ AB, erit centrum grauitatis <lb/>Quod demon&longs;trandum erat. </s><pb pagenum="77"/>

<s>Non igitur aliud punctum à puncto <lb/>E; punctum igitur E, figuræ AB, erit centrum grauitatis <lb/>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/085.jpg" pagenum="77"/>

<s><emph type="italics"/>PROPOSITIO XXXIX.<emph.end type="italics"/></s>

Line 1078

<s>Si enim cono ABC, pyramis <lb/>in&longs;cribatur, cuius ba&longs;is in&longs;cripta circulo AC, æquilatera &longs;it, <lb/>& æquiangula, eius centrum grauitatis erit idem quod & <lb/>figuræ centrum, &longs;ed centrum <lb/>talis figuræ circulo in&longs;criptæ <lb/>idem e&longs;t, quod centrum cir<lb/>culi, vt colligitur ex demon<lb/>&longs;trationibus quarti Elemen<lb/>torum; in&longs;criptæ igitur pyra <lb/>midis erit axis BD, & cen<lb/>trum grauitatis E. talis au<lb/>tem ea pyramis in&longs;cribi po<lb/>te&longs;t, vt à cono deficiat mino<lb/>ri &longs;pacio quantacumque ma <lb/>gnitudine propo&longs;ita; igitur <lb/>ABC, coni centrum graui<lb/>tatis erit E. </s>

<s>Quod demon&longs;trandum erat. </s><figure/>

<s>Quod demon&longs;trandum erat. </s><figure id="id.043.01.085.1.jpg" xlink:href="043/01/085/1.jpg"/>

<s><emph type="italics"/>PROPOSITIO XXXX.<emph.end type="italics"/></s>

<s>Omnis fru&longs;ti conici centrum grauitatis idem <lb/>e&longs;t in axe centro grauitatis fru&longs;ti pyramidis ba&longs;im <lb/>habentis æquilateram, & æquiangul am in &longs;criptæ <lb/>cono, ab &longs;ci&longs;&longs;i eodem plano, quo coni fru&longs;tum. </s><pb pagenum="78"/>

<s>Sit coni fru&longs;tum ABCD, cuius axis EF, fru&longs;to autem <lb/>ABCD, intelligatur in&longs;criptum fru&longs;tum pyramidis in&longs;cri<lb/>ptæ cono AHD, à quo ab&longs;ci&longs;sum e&longs;t fru&longs;tum ABCD, <lb/>ba&longs;im habentis æquilateram, & æquiangulam in&longs;criptam <lb/>circulo AD: quare eius centrum grauitatis, & figuræ erit <lb/>punctum F, vt diximus in præcedenti, axis autem FH, &longs;i<lb/>cut etiam pyramidis ab&longs;ci&longs;sæ vna cum cono BHC, axis <lb/>EH, quare & reliqui fru&longs;ti pyramidis axis erit EF, igi<lb/>tur in EF, &longs;it fru&longs;ti in&longs;cripti fru&longs;to ABCD, centrum gra<lb/>uitatis G. </s>

<s>Dico punctum G, e&longs;se centrum grauitatis fru<lb/>&longs;ti ABCD. </s>

<s>Ponatur enim <lb/>FL, pars quarta ip&longs;ius FH, <lb/>necnon EK, pars quarta ip<lb/>&longs;ius EH: punctum igitur K, <lb/>e&longs;t centrum grauitatis pyra<lb/>midis, & coni BHC, &longs;icut <lb/>& punctum L, pyramidis, & <lb/>coni AHD. cum igitur fru <lb/>&longs;ti pyramidis fru&longs;to ABCD, <lb/>in&longs;cripti &longs;it centrum grauita<lb/>tis G; erit vt GL, ad LK, <lb/>ita pyramis BHC, ad pyra<lb/>midis fru&longs;tum fru&longs;to ABCD, <lb/>in&longs;criptum: &longs;ed vt pyramis <lb/>BHC, ad pyramidis fru&longs;tum <lb/>fru&longs;to ABCD, in&longs;criptum, <lb/><figure id="fig48"/><lb/>ita e&longs;t diuidendo, conus BHC, ad fru&longs;tum ABCD, pro<lb/>pter eandem triplicatam communium conis, & pyramidi<lb/>bus &longs;imilibus laterum homologorum proportionem; vt igi<lb/>tur GL, ad LK, ita erit conus BHC: ad fru&longs;tum ABCD: <lb/>&longs;ed coni BHC, centrum grauitatis erat K, & coni AHD, <lb/>centrum grauitatis L; fru&longs;ti igitur ABCD, centrum gra<lb/>nitatis erit G. </s>

<s>Ponatur enim <lb/>FL, pars quarta ip&longs;ius FH, <lb/>necnon EK, pars quarta ip<lb/>&longs;ius EH: punctum igitur K, <lb/>e&longs;t centrum grauitatis pyra<lb/>midis, & coni BHC, &longs;icut <lb/>& punctum L, pyramidis, & <lb/>coni AHD. cum igitur fru <lb/>&longs;ti pyramidis fru&longs;to ABCD, <lb/>in&longs;cripti &longs;it centrum grauita<lb/>tis G; erit vt GL, ad LK, <lb/>ita pyramis BHC, ad pyra<lb/>midis fru&longs;tum fru&longs;to ABCD, <lb/>in&longs;criptum: &longs;ed vt pyramis <lb/>BHC, ad pyramidis fru&longs;tum <lb/>fru&longs;to ABCD, in&longs;criptum, <lb/><figure id="id.043.01.086.1.jpg" xlink:href="043/01/086/1.jpg"/><lb/>ita e&longs;t diuidendo, conus BHC, ad fru&longs;tum ABCD, pro<lb/>pter eandem triplicatam communium conis, & pyramidi<lb/>bus &longs;imilibus laterum homologorum proportionem; vt igi<lb/>tur GL, ad LK, ita erit conus BHC: ad fru&longs;tum ABCD: <lb/>&longs;ed coni BHC, centrum grauitatis erat K, & coni AHD, <lb/>centrum grauitatis L; fru&longs;ti igitur ABCD, centrum gra<lb/>nitatis erit G. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="79"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/087.jpg" pagenum="79"/>

<s><emph type="italics"/>PROPOSITIO XLI.<emph.end type="italics"/></s>

Line 1102

<s>Nam &longs;i cylindro AD, in<lb/>&longs;criptum intelligatur pri&longs;ma, <lb/>cuius ba&longs;es oppo&longs;itæ æquilate<lb/>ræ &longs;int, & æquiangulæ; erunt, <lb/>qua ratione &longs;upra diximus, ea<lb/>rum centra figuræ, & grauitatis <lb/>E, F; axis igitur in&longs;cripti pri&longs;<lb/>matis erit EF: & centrum gra<lb/>uitatis G. pote&longs;t autem tale <lb/>pri&longs;ma &longs;ic in&longs;cribi cylindro <lb/>ABCD, vt ab illo deficiat <lb/>minori &longs;pacio quantacumque <lb/>magnitudine propo&longs;ita; cylin<lb/>dri igitur ABCD, centrum <lb/>grauitatis erit G. </s>

<s>Quod demon&longs;trandum erat. </s><figure/>

<s>Quod demon&longs;trandum erat. </s><figure id="id.043.01.087.1.jpg" xlink:href="043/01/087/1.jpg"/>

<s><emph type="italics"/>PROPOSITIO XLII.<emph.end type="italics"/></s>

<s>Sphæræ, & &longs;phæroidis idem e&longs;t centrum gra<lb/>uitatis, & figuræ. </s>

<s>Sit &longs;phæra, vel &longs;phæroides ABCD, cuius centrum E, <pb pagenum="80"/>Dico &longs;phæræ, vel &longs;phæroidis ABCD, centrum grauitatis <lb/>e&longs;se E. </s>

<s>Sit &longs;phæra, vel &longs;phæroides ABCD, cuius centrum E, <pb xlink:href="043/01/088.jpg" pagenum="80"/>Dico &longs;phæræ, vel &longs;phæroidis ABCD, centrum grauitatis <lb/>e&longs;se E. </s>

<s>Sint enim bini axes &longs;phæræ, vel &longs;phæroidis inter <lb/>&longs;e ad rectos angulos; & in &longs;phæroide &longs;it maior diameter <lb/>BD, minor AC, per binos autem hos axes plana tran<lb/>&longs;euntia ad eos axes erecta, &longs;ecent &longs;phæram, vel &longs;phæroidem. <lb/></s>

<s>Qua ratione axes dimidij erunt axes hemi&longs;phærij, vel he<lb/>mi&longs;phæroidis: hemi&longs;phærium autem, & &longs;phæroidis e&longs;t fi<lb/><figure id="fig49"/><lb/>gura circa axim in alteram partem deficiens, qualium om<lb/>nium figurarum centrum grauitatis e&longs;t in axe; igitur hemi<lb/>&longs;phærij, vel hemi&longs;phæroidis ABCD, centrum grauitatis <lb/>e&longs;t in axi BE, &longs;icut & reliqui ADA, in axi ED; totius <lb/>igitur &longs;phæræ, vel &longs;phæroidis ABCD centrum grauitatis <lb/>e&longs;t in axi BD. </s>

<s>Qua ratione axes dimidij erunt axes hemi&longs;phærij, vel he<lb/>mi&longs;phæroidis: hemi&longs;phærium autem, & &longs;phæroidis e&longs;t fi<lb/><figure id="id.043.01.088.1.jpg" xlink:href="043/01/088/1.jpg"/><lb/>gura circa axim in alteram partem deficiens, qualium om<lb/>nium figurarum centrum grauitatis e&longs;t in axe; igitur hemi<lb/>&longs;phærij, vel hemi&longs;phæroidis ABCD, centrum grauitatis <lb/>e&longs;t in axi BE, &longs;icut & reliqui ADA, in axi ED; totius <lb/>igitur &longs;phæræ, vel &longs;phæroidis ABCD centrum grauitatis <lb/>e&longs;t in axi BD. </s>

<s>Eadem ratione & in axi AC; in communi <lb/>igitur &longs;ectione centro E. </s>

<s>Quod demon&longs;trandum erat. </s>

<s>PRIMI LIBRI FINIS.</s><figure/>

<s>PRIMI LIBRI FINIS.</s><figure id="id.043.01.088.2.jpg" xlink:href="043/01/088/2.jpg"/>

<s>LVC AE <lb/>VALERII <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM</s>

Line 1126

<s><emph type="italics"/>PROPOSITIO I.<emph.end type="italics"/></s>

<s>Si duæ magnitudines vnà maio<lb/>res, vel minores prima, & ter <lb/>tia minori exce&longs;&longs;u, vel defe<lb/>ctu <expan abbr="quantacumq;">quantacumque</expan> magnitudi <lb/>ne propo&longs;ita eiu&longs;dem generis <lb/>cum illa, ad quam refertur, <lb/>eandem <expan abbr="proportion&etilde;">proportionem</expan> habue<lb/>rint, maior vel minor prima ad &longs;ecundam, & vnà <lb/>maior, vel minor tertia ad quartam; erit vt prima <lb/>ad &longs;ecundam, ita tertia ad quartam. </s><pb pagenum="2"/>

<s>Sint quatuor magnitudines A prima, B &longs;ecunda, C ter <lb/>tia, & D quarta: quantacumque autem magnitudine propo <lb/>&longs;ita, ex infinitìs quæ proponi po&longs;&longs;unt eiu&longs;dem generis cum <lb/>A, C, vel vna tantum, &longs;i AC &longs;int eiu&longs;dem generis: vel <lb/>vna, & altera; &longs;i vna vnius, altera &longs;it alterius generis; &longs;emper <lb/>aliæ duæ magnitudines vnà maiores, quàm AC, minori <lb/>exce&longs;su magnitudine propo&longs;ita; eandem habeant proportio <lb/>nem, maior quàm A ad B, & maior quàm C ad D. </s>

Line 1136

<s>Quoniam igitur F maior <lb/>e&longs;t, <34>H, maior erit proportio <lb/>ip&longs;ius F quàm H ad D, hoc e&longs;t <lb/>quàm G ad B. </s>

<s>Sed <expan abbr="c&utilde;">cum</expan> G maior <lb/>&longs;it quàm A, maior e&longs;t proportio <lb/><figure id="fig50"/><lb/>G ad B, quàm A ad B, multo igitur erit maior proportio F <lb/>ad D, quàm A ad B. </s>

<s>Sed <expan abbr="c&utilde;">cum</expan> G maior <lb/>&longs;it quàm A, maior e&longs;t proportio <lb/><figure id="id.043.01.089.1.jpg" xlink:href="043/01/089/1.jpg"/><lb/>G ad B, quàm A ad B, multo igitur erit maior proportio F <lb/>ad D, quàm A ad B. </s>

<s>Sed F ponitur maior quàm C, vtcum <lb/>que; nulla igitur magnitudo maior quàm C e&longs;t ad D, vt <lb/>A ad B: &longs;ed E ad D, e&longs;t vt A ad B; non igitur e&longs;t E ma<lb/>ior quàm C; nec maior proportio E ad D, hoc e&longs;t A ad <lb/>B, quàm C ad D. </s>

Line 1144

<s>Sed aliæ duæ magnitudines vnà minores quàm A, C <lb/>minori defectu quantacumque magnitudine propo&longs;ita, <lb/>eandem habeant proportionem, minor quàm A ad B, & <lb/>minor quàm C, ad D. </s>

<s>Dico e&longs;se vt A ad B, ita C ad D. <pb pagenum="3"/>Po&longs;ita enim rur&longs;us E ad D, vt A ad B, & F minori quàm <lb/>C vtcumque, &longs;it G minor quam A, minori defectu magni <lb/>tudine eiu&longs;dem generis cum A, quam quis voluerit, & H <lb/>minor quàm C, & maior quàm F: &longs;it autem vt G ad B, ita <lb/>H ad D. </s>

<s>Dico e&longs;se vt A ad B, ita C ad D. <pb xlink:href="043/01/090.jpg" pagenum="3"/>Po&longs;ita enim rur&longs;us E ad D, vt A ad B, & F minori quàm <lb/>C vtcumque, &longs;it G minor quam A, minori defectu magni <lb/>tudine eiu&longs;dem generis cum A, quam quis voluerit, & H <lb/>minor quàm C, & maior quàm F: &longs;it autem vt G ad B, ita <lb/>H ad D. </s>

<s>Quoniam igitur F minor e&longs;t quàm H, minor erit <lb/>proportio ip&longs;ius F <expan abbr="quã">quam</expan> H ad D, <lb/>hoc e&longs;t <34>G ad B: &longs;ed cum G &longs;it <lb/>minor <34>A, minor e&longs;t propor<lb/>tio G ad B, quàm A ad B; mul <lb/>to ergo minor proportio F ad <lb/>D, quàm A ad B: &longs;ed F poni <lb/>tur minor quàm C vtcumque; <lb/>nulla igitur magnitudo minor <lb/><figure id="fig51"/><lb/>quàm C e&longs;t ad D, vt A ad B: &longs;ed E e&longs;t ad D, vt A ad B: <lb/>non igitur e&longs;t E minor quàm C, nec minor proportio E ad <lb/>D, hoc e&longs;t A ad B, quàm C ad D. eadem autem ratione <lb/>non minor erit proportio C ad D, quàm A ad B; hoc e&longs;t <lb/>non maior A ad B, quàm C ad D; vt igitur A ad B, ita <lb/>e&longs;t C ad D. </s>

<s>Quoniam igitur F minor e&longs;t quàm H, minor erit <lb/>proportio ip&longs;ius F <expan abbr="quã">quam</expan> H ad D, <lb/>hoc e&longs;t <34>G ad B: &longs;ed cum G &longs;it <lb/>minor <34>A, minor e&longs;t propor<lb/>tio G ad B, quàm A ad B; mul <lb/>to ergo minor proportio F ad <lb/>D, quàm A ad B: &longs;ed F poni <lb/>tur minor quàm C vtcumque; <lb/>nulla igitur magnitudo minor <lb/><figure id="id.043.01.090.1.jpg" xlink:href="043/01/090/1.jpg"/><lb/>quàm C e&longs;t ad D, vt A ad B: &longs;ed E e&longs;t ad D, vt A ad B: <lb/>non igitur e&longs;t E minor quàm C, nec minor proportio E ad <lb/>D, hoc e&longs;t A ad B, quàm C ad D. eadem autem ratione <lb/>non minor erit proportio C ad D, quàm A ad B; hoc e&longs;t <lb/>non maior A ad B, quàm C ad D; vt igitur A ad B, ita <lb/>e&longs;t C ad D. </s>

<s>Quod demon&longs;trandum erat. </s>

Line 1156

<s>Si enim fieri pote&longs;t, &longs;it minor <lb/>proportio A ad B quàm C ad D. <lb/>alia igitur aliqua magnitudo G <lb/>maior quàm A, eandem habebit <lb/>proportionem ad B, quam C ad <lb/>D. </s>

<s>Sit autem F maior quam C <lb/>minori exce&longs;su magnitudine, <expan abbr="quã">quam</expan> <lb/>quis voluerit, & E maior quàm <lb/>A, & minor quàm G: vt autem <lb/><figure id="fig52"/><lb/>E ad B, ita F ad D. </s>

<s>Sit autem F maior quam C <lb/>minori exce&longs;su magnitudine, <expan abbr="quã">quam</expan> <lb/>quis voluerit, & E maior quàm <lb/>A, & minor quàm G: vt autem <lb/><figure id="id.043.01.090.2.jpg" xlink:href="043/01/090/2.jpg"/><lb/>E ad B, ita F ad D. </s>

<s>Quoniamigitur F maior e&longs;t quàm <lb/>C, maior erit proportio F ad D, quàm C ad D. </s>

<s>Sed vt <lb/>F ad D, ità e&longs;t E ad B: & vt C ad D, ita G ad B; maior <pb pagenum="4"/>igitur proportio E ad B, quàm G ad B; quamobrem E <lb/>maior erit quàm G minor maiori, quod fieri non pote&longs;t. <lb/></s>

<s>Sed vt <lb/>F ad D, ità e&longs;t E ad B: & vt C ad D, ita G ad B; maior <pb xlink:href="043/01/091.jpg" pagenum="4"/>igitur proportio E ad B, quàm G ad B; quamobrem E <lb/>maior erit quàm G minor maiori, quod fieri non pote&longs;t. <lb/></s>

<s>Non igitur minor e&longs;t proportio A ad B quàm C ad D. <lb/></s>

Line 1170

<s>Sit autê F minor quàm <lb/>C minori defectu magnitudine, <lb/>quam quis voluerit, & E minor <lb/>quàm A, & maior quàm G, vt au<lb/>tem E ad B ita F ad D. </s>

<s>Quoniam <lb/>igitur maior e&longs;t proportio C ad D, <lb/>quàm F ad D: &longs;ed vt C ad D, ita <lb/>e&longs;t G ad B: & vt F ad D, ita E ad <lb/>B: maior erit proportio G ad B <lb/>quàm E ad B; quamobrem erit <lb/>G maior quàm E, minor maiori, <lb/>quod fieri non pote&longs;t; non igitur ma <lb/><figure id="fig53"/><lb/>ior e&longs;t proportio A ad B, quàm C ad D. </s>

<s>Quoniam <lb/>igitur maior e&longs;t proportio C ad D, <lb/>quàm F ad D: &longs;ed vt C ad D, ita <lb/>e&longs;t G ad B: & vt F ad D, ita E ad <lb/>B: maior erit proportio G ad B <lb/>quàm E ad B; quamobrem erit <lb/>G maior quàm E, minor maiori, <lb/>quod fieri non pote&longs;t; non igitur ma <lb/><figure id="id.043.01.091.1.jpg" xlink:href="043/01/091/1.jpg"/><lb/>ior e&longs;t proportio A ad B, quàm C ad D. </s>

<s>Eadem autem ra<lb/>tione non maior erit proportio C ad D, quàm A ad B, hoc <lb/>e&longs;t non minor A ad B, quàm C ad D. </s>

Line 1180

<s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/></s>

<s>Si maior, vel minor prima ad vnà maiorem, vel <lb/>minorem &longs;ecunda, minori <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> exce&longs;&longs;u, vel de<lb/>fectu <expan abbr="quantacumq;">quantacumque</expan> magnitudine propo&longs;ita fue<lb/>rit vt tertia ad quartam; erit vt prima ad &longs;ecun<lb/>dam, ita tertia ad quartam. </s><pb pagenum="5"/>

<s>Sint quatuor magnitudines, A prima, B &longs;ecunda, C ter<lb/>tia, & D quarta: & aliæ duæ magnitudines E <lb/>F vnà maiores quàm A, B minori exce&longs;su <lb/>quantacumque magnitudine propo&longs;ita eiu&longs;<lb/>dem generis cum ip&longs;is A, B. </s>

Line 1194

<s>Idem autem &longs;imiliter o&longs;ten <lb/>deremus po&longs;itis EF minoribus quàm AB, & <lb/>proportionalibus vt <expan abbr="dict&utilde;">dictum</expan> e&longs;t. </s>

<s><expan abbr="Manife&longs;t&utilde;">Manife&longs;tum</expan> e&longs;t igitur <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>. </s><figure/>

<s><expan abbr="Manife&longs;t&utilde;">Manife&longs;tum</expan> e&longs;t igitur <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>. </s><figure id="id.043.01.092.1.jpg" xlink:href="043/01/092/1.jpg"/>

<s><emph type="italics"/>ALITER.<emph.end type="italics"/></s>

<s>Ij&longs;dem po&longs;itis, &longs;i non e&longs;t A ad <lb/>B, vt C ad D; vel igitur ma<lb/>ior vel minor erit proportio A <lb/>ad B quàm C ad D: &longs;it autem <lb/>maior: vt igitur A ad B, ita erit <lb/>eadem A ad <expan abbr="aliã">aliam</expan> maiorem <34>B. <lb/></s>

<s>E&longs;to illa E. &longs;intque aliæ duæ ma <lb/>gnitudines, G maior quàm A <lb/><figure id="fig54"/><lb/>minori exce&longs;su magnitudine eiu&longs;dem generis cum A, <lb/>quam quis voluerit, & F maior quàm B, & minor quàm <lb/>E. &longs;it autem G ad F vt C ad D. </s>

<s>E&longs;to illa E. &longs;intque aliæ duæ ma <lb/>gnitudines, G maior quàm A <lb/><figure id="id.043.01.092.2.jpg" xlink:href="043/01/092/2.jpg"/><lb/>minori exce&longs;su magnitudine eiu&longs;dem generis cum A, <lb/>quam quis voluerit, & F maior quàm B, & minor quàm <lb/>E. &longs;it autem G ad F vt C ad D. </s>

<s>Quoniam igitur & vt <lb/>C ad D, ita e&longs;t A ad E; erit vt G ad F, ita A ad E; & <lb/>permutando vt G ad A, ita F ad E: &longs;ed G e&longs;t maior <pb pagenum="6"/>quàm A: ergo & F maior quàm <lb/>E, minor maiori, quod e&longs;t ab<lb/>&longs;urdum. </s>

<s>Quoniam igitur & vt <lb/>C ad D, ita e&longs;t A ad E; erit vt G ad F, ita A ad E; & <lb/>permutando vt G ad A, ita F ad E: &longs;ed G e&longs;t maior <pb xlink:href="043/01/093.jpg" pagenum="6"/>quàm A: ergo & F maior quàm <lb/>E, minor maiori, quod e&longs;t ab<lb/>&longs;urdum. </s>

<s>Non igitur maior e&longs;t <lb/>proportio A ad B quàm C ad <lb/>D: eadem autem ratione non <lb/>maior erit proportio B ad A <expan abbr="quã">quam</expan> <lb/>D ad C, hoc e&longs;t non minor A <lb/>ad B, quàm C ad D; e&longs;t igitur <lb/>A ad B, vt C ad D. </s><figure/>

<s>Rur&longs;us in &longs;ecunda parte hypothe&longs;is, quæ attinet ad mi<lb/>norem defectum: &longs;i non e&longs;t A ad B vt C ad D; e&longs;to, &longs;i fie<lb/>ri pote&longs;t, minor proportio A ad B quàm C ad D. igitur A <lb/>ad aliam quam B minorem eandem habebit <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam C ad D, e&longs;to illa E: &longs;intque <lb/>aliæ duæ magnitudines, G minor <lb/>quàm A minori defectu magnitudi<lb/>ne eiu&longs;dem generis cum A, quam <lb/>quis voluerit, & F minor quàm B, <lb/>& maior quàm E: &longs;it autem G ad <lb/>F, vt C ad D, hoc e&longs;t vt A ad E. <lb/></s>

<s>Quoniam igitur permutando e&longs;t vt <lb/>G ad A, ita F ad E, & G e&longs;t mi<lb/><figure id="fig55"/><lb/>nor quàm A; erit & F minor quàm E, maior mino<lb/>ri, quod e&longs;t ab&longs;urdum; non igitur minor e&longs;t proportio <lb/>A ad B quàm C ad D: eadem autem ratione non minor <lb/>erit proportio B ad A, quàm D ad C, hoc e&longs;t non maior <lb/>A ad B, quàm C ad D; e&longs;t igitur A ad B vt C ad D. <lb/></s>

<s>Quoniam igitur permutando e&longs;t vt <lb/>G ad A, ita F ad E, & G e&longs;t mi<lb/><figure id="id.043.01.093.2.jpg" xlink:href="043/01/093/2.jpg"/><lb/>nor quàm A; erit & F minor quàm E, maior mino<lb/>ri, quod e&longs;t ab&longs;urdum; non igitur minor e&longs;t proportio <lb/>A ad B quàm C ad D: eadem autem ratione non minor <lb/>erit proportio B ad A, quàm D ad C, hoc e&longs;t non maior <lb/>A ad B, quàm C ad D; e&longs;t igitur A ad B vt C ad D. <lb/></s>

<s>Quod demon&longs;trandum erat. </s>

<s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/></s>

<s>Si maior, vel minor prima ad vnà maiorem, vel <lb/>minorem &longs;ecunda, minori exce&longs;&longs;u, vel defectu <pb pagenum="7"/>quantacumque magnitudine propo&longs;ita, nomina<lb/>tam habuerit proportionem; prima ad &longs;ecundam <lb/>eandem nominatam habebit proportionem. </s>

<s>Si maior, vel minor prima ad vnà maiorem, vel <lb/>minorem &longs;ecunda, minori exce&longs;&longs;u, vel defectu <pb xlink:href="043/01/094.jpg" pagenum="7"/>quantacumque magnitudine propo&longs;ita, nomina<lb/>tam habuerit proportionem; prima ad &longs;ecundam <lb/>eandem nominatam habebit proportionem. </s>

<s>Sint duæ magnitudines A, B duarum autem aliarum <lb/>EF vnà maiorum, vel minorum quàm AB minori ex<lb/>ce&longs;su vel defectu quantacumque magnitudine propo<lb/>&longs;ita, habeat E maior vel minor quàm A ad F vnà <lb/>maiorem, vel minorem quàm B certam ali quam nomina<lb/>tam proportionem, verbi gratia, &longs;e&longs;quialteram. </s>

Line 1222

<s>Quoniam <lb/>enim omnis proportio in aliquibus ma<lb/>gnitudinibus con&longs;i&longs;tit; &longs;it magnitudo C <lb/>ip&longs;ius D &longs;e&longs;quialtera: &longs;ed & E e&longs;t ip&longs;ius <lb/>F &longs;e&longs;quialtera; vtigitur C, tertia ad D <lb/>quartam, ita erit E maior, vel minor quàm <lb/>A prima, ad F vnà maiorem, vel minorem <lb/>&longs;ecunda, minori, vt ponitur, vtriu&longs;que ex<lb/>ce&longs;su, vel defectu magnitudine propo&longs;ita <lb/>eiu&longs;dem generis cum A, B, quæcumque <lb/>illa, & quantacumque &longs;it; erit per præ<lb/>cedentem eadem proportio A ad B, <lb/>quæ C ad D: &longs;ed proportio quam ha<lb/>bet C ad D, e&longs;t &longs;e&longs;quialtera; ergo & A <lb/>ip&longs;ius B erit &longs;e&longs;quialtera. </s>

<s>Similiter quo<lb/>cumque alio nomine notatam proportio<lb/>nem habeat E ad F, eandem habere A <lb/><figure id="fig56"/><lb/>ad B, o&longs;tenderemus, vt duplam, &longs;e&longs;quitertiam, alicuius du <lb/>plicatam, vel triplicatam, & &longs;ic de &longs;ingulis. </s>

<s>Similiter quo<lb/>cumque alio nomine notatam proportio<lb/>nem habeat E ad F, eandem habere A <lb/><figure id="id.043.01.094.1.jpg" xlink:href="043/01/094/1.jpg"/><lb/>ad B, o&longs;tenderemus, vt duplam, &longs;e&longs;quitertiam, alicuius du <lb/>plicatam, vel triplicatam, & &longs;ic de &longs;ingulis. </s>

<s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s>

<s>Hæc autem propo&longs;itio in paucis exemplaribus, quæ do<lb/>no quibu&longs;dam <expan abbr="dederã">dederam</expan>, non extat; po&longs;terius enim eam exco-<pb pagenum="8"/>gitaui, quo &longs;ecunda <expan abbr="anteced&etilde;s">antecedens</expan> hìc in illis tertia facilius &longs;er<lb/>uiret ijs, in quibus certæ proportionis nomen, <expan abbr="terti&utilde;">tertium</expan> & quar <lb/>tum terminum &longs;ubob&longs;curè indicat, vt in &longs;equenti XII iilud, <lb/>proportio dupla. </s>

<s>Hæc autem propo&longs;itio in paucis exemplaribus, quæ do<lb/>no quibu&longs;dam <expan abbr="dederã">dederam</expan>, non extat; po&longs;terius enim eam exco-<pb xlink:href="043/01/095.jpg" pagenum="8"/>gitaui, quo &longs;ecunda <expan abbr="anteced&etilde;s">antecedens</expan> hìc in illis tertia facilius &longs;er<lb/>uiret ijs, in quibus certæ proportionis nomen, <expan abbr="terti&utilde;">tertium</expan> & quar <lb/>tum terminum &longs;ubob&longs;curè indicat, vt in &longs;equenti XII iilud, <lb/>proportio dupla. </s>

<s>Illo autem Lemmate, quod prima propofi<lb/>tio in&longs;cribebatur, nunc ita non egeo, vt primam, & <expan abbr="&longs;ecundã">&longs;ecundam</expan>, <lb/>quæ &longs;ecunda, & tertia erant, & facilius demon&longs;trem, & ea<lb/>rum &longs;en&longs;um paucioribus comprehendam. </s>

Line 1244

<s>Sed DE, LH, &longs;uperant BC exce&longs;sibus <lb/>EG, KH; minor igitur erit DE quàm <lb/>LH, & minor proportio BC ad LH, <lb/>quàm BC ad DE. </s>

<s>Sed vt BC ad LH, <lb/><figure id="fig57"/><lb/>ita e&longs;t A ad BC; minor igitur proportio erit A ad BC, <lb/>quàm BC ad DE. </s>

<s>Sed vt BC ad LH, <lb/><figure id="id.043.01.095.1.jpg" xlink:href="043/01/095/1.jpg"/><lb/>ita e&longs;t A ad BC; minor igitur proportio erit A ad BC, <lb/>quàm BC ad DE. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="9"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/096.jpg" pagenum="9"/>

<s><emph type="italics"/>PROPOSITIO V.<emph.end type="italics"/></s>

Line 1262

<s>Ponatur enim CL, æqua<lb/>lis AB, & EM, æqualis CD. </s>

<s>Quoniam igitur maior e&longs;t <lb/>proportio DL ad LH, quam DL, ad LC; <lb/>erit componendo maior proportio DH ad <lb/>HL, quam DC ad CL. hoc e&longs;t, maior <lb/>proportio DH, ad BG, quam DC, <lb/>ad AB: & conuertendo, minor proportio <lb/>BG ad DH, quam AB, ad CD: hoc e&longs;t <lb/>maior proportio AB, ad CD, quam BG, <lb/>ad DH. Rur&longs;us, quoniam maior e&longs;t pro<lb/>portio CD, ad EF, quam AB, ad CD: <lb/>hoc e&longs;t quam CL, ad EM; erit permutan <lb/>do, maior proportio CD, ad CL, quam <lb/>FE, ad EM: & diuidendo, maior DL, ad <lb/>LC, quam FM, ad ME: & permutando, <lb/><figure id="fig58"/><lb/>maior DL, ad FM, quam CL, ad EM: hoc e&longs;t quam <lb/>AB, ad CD. </s>

<s>Quoniam igitur maior e&longs;t <lb/>proportio DL ad LH, quam DL, ad LC; <lb/>erit componendo maior proportio DH ad <lb/>HL, quam DC ad CL. hoc e&longs;t, maior <lb/>proportio DH, ad BG, quam DC, <lb/>ad AB: & conuertendo, minor proportio <lb/>BG ad DH, quam AB, ad CD: hoc e&longs;t <lb/>maior proportio AB, ad CD, quam BG, <lb/>ad DH. Rur&longs;us, quoniam maior e&longs;t pro<lb/>portio CD, ad EF, quam AB, ad CD: <lb/>hoc e&longs;t quam CL, ad EM; erit permutan <lb/>do, maior proportio CD, ad CL, quam <lb/>FE, ad EM: & diuidendo, maior DL, ad <lb/>LC, quam FM, ad ME: & permutando, <lb/><figure id="id.043.01.096.1.jpg" xlink:href="043/01/096/1.jpg"/><lb/>maior DL, ad FM, quam CL, ad EM: hoc e&longs;t quam <lb/>AB, ad CD. </s>

<s>Sed maior erat proportio AB, ad CD, <lb/>quam BG ad DH; multo igitur maior proportio erit DL, <lb/>ad FM, quam BG, ad DH: hoc e&longs;t quam LH, ad MK: <lb/>& permutando, maior proportio DL, ad LH, quam FM, <lb/>ad MK: & componendo, maior DH, ad HL, quam FK, <pb pagenum="10"/>ad KM: & permutando, maior DH ad F<emph type="italics"/>K<emph.end type="italics"/>, quam LH, ad <lb/>M<emph type="italics"/>K<emph.end type="italics"/>: hoc e&longs;t, quam BG, ad DH: hoc e&longs;t minor propor<lb/>tio BG ad DH, quam DH, ad FK. </s>

<s>Sed maior erat proportio AB, ad CD, <lb/>quam BG ad DH; multo igitur maior proportio erit DL, <lb/>ad FM, quam BG, ad DH: hoc e&longs;t quam LH, ad MK: <lb/>& permutando, maior proportio DL, ad LH, quam FM, <lb/>ad MK: & componendo, maior DH, ad HL, quam FK, <pb xlink:href="043/01/097.jpg" pagenum="10"/>ad KM: & permutando, maior DH ad F<emph type="italics"/>K<emph.end type="italics"/>, quam LH, ad <lb/>M<emph type="italics"/>K<emph.end type="italics"/>: hoc e&longs;t, quam BG, ad DH: hoc e&longs;t minor propor<lb/>tio BG ad DH, quam DH, ad FK. </s>

<s>Quod demon<lb/>&longs;trandum erat. </s>

Line 1282

<s>Igitur G ad C dupli<lb/>cata erit proportio ip&longs;ius G ad <lb/>B, hoc e&longs;t B ad C: &longs;imiliter <lb/>erit H ad B, duplicata propor<lb/>tio ip&longs;ius A ad B. </s>

<s>Vt igitur <lb/>e&longs;t H ad B, ita erit D ad E: & <lb/>vt G ad C, ita E ad F. Rur<lb/>&longs;us, quia minor e&longs;t proportio <lb/><figure id="fig59"/><lb/>A ad B, quam B ad C, &longs;ed vt A ad B, ita e&longs;t H ad A <pb pagenum="11"/>& vt B ad C, ita G ad B; erit ex æquali minor proportio <lb/>H ad B, quam G ad C, &longs;ed vt H ad B, ita erat D, ad <lb/>E: & vt G ad C, ita E ad F; minor igitur proportio erit <lb/>D ad E, quam E ad F. </s>

<s>Vt igitur <lb/>e&longs;t H ad B, ita erit D ad E: & <lb/>vt G ad C, ita E ad F. Rur<lb/>&longs;us, quia minor e&longs;t proportio <lb/><figure id="id.043.01.097.1.jpg" xlink:href="043/01/097/1.jpg"/><lb/>A ad B, quam B ad C, &longs;ed vt A ad B, ita e&longs;t H ad A <pb xlink:href="043/01/098.jpg" pagenum="11"/>& vt B ad C, ita G ad B; erit ex æquali minor proportio <lb/>H ad B, quam G ad C, &longs;ed vt H ad B, ita erat D, ad <lb/>E: & vt G ad C, ita E ad F; minor igitur proportio erit <lb/>D ad E, quam E ad F. </s>

<s>Quod demon&longs;trandum erat. </s>

Line 1300

<s>Sed vt <lb/>AE, ad F, ita erat CG ad H; ex æqua <lb/>li igitur erit vt AE ad BF, ita CG, <lb/>ad DH. </s>

<s>Quod demon&longs;trandum erat. </s><figure/><pb pagenum="12"/>

<s>Quod demon&longs;trandum erat. </s><figure id="id.043.01.098.1.jpg" xlink:href="043/01/098/1.jpg"/><pb xlink:href="043/01/099.jpg" pagenum="12"/>

<s><emph type="italics"/>PROPOSITIO VIII.<emph.end type="italics"/></s>

Line 1310

<s>Dico AD, &longs;imul <lb/>ad BE, &longs;imul mino<lb/>tem e&longs;&longs;e proportionem <lb/>quam BE, &longs;imul ad <lb/>CF, &longs;imul. </s>

<s>E&longs;to enim <lb/>recta quæpiam GH, <lb/>ad aliam rectam &longs;ibi in <lb/>directum po&longs;itam HK, <lb/>vt magnitudo A ad ip <lb/>&longs;ius F duplam (hoc <lb/>enim fieri pote&longs;t) & <lb/><figure id="fig60"/><lb/>&longs;uper ba&longs;im GK; con&longs;tituatur triangulum GLK, atque <lb/>in eo de&longs;cribatur parallelogrammum GHMN: & vt e&longs;t <pb pagenum="13"/>C ad B, ita fiat HM, ad <expan abbr="Mq.">Mque</expan> & vt B ad A, ita QM, ad <lb/>MP, & ip&longs;i GK, parallelæ TPR, VQS, ducantur. <lb/></s>

<s>E&longs;to enim <lb/>recta quæpiam GH, <lb/>ad aliam rectam &longs;ibi in <lb/>directum po&longs;itam HK, <lb/>vt magnitudo A ad ip <lb/>&longs;ius F duplam (hoc <lb/>enim fieri pote&longs;t) & <lb/><figure id="id.043.01.099.1.jpg" xlink:href="043/01/099/1.jpg"/><lb/>&longs;uper ba&longs;im GK; con&longs;tituatur triangulum GLK, atque <lb/>in eo de&longs;cribatur parallelogrammum GHMN: & vt e&longs;t <pb xlink:href="043/01/100.jpg" pagenum="13"/>C ad B, ita fiat HM, ad <expan abbr="Mq.">Mque</expan> & vt B ad A, ita QM, ad <lb/>MP, & ip&longs;i GK, parallelæ TPR, VQS, ducantur. <lb/></s>

<s>Quoniam igitur e&longs;t vt C, ad duplam ip&longs;ius F, ita GH, ad <lb/>HK; erit vt C ad F, ita e&longs;t par llelogrammum GM, ad <lb/>triangulum MHK: &longs;ed vt C, ad B, ita e&longs;t HM, ad <expan abbr="Mq;">Mque</expan> <lb/>hoc e&longs;t parallelogrammum GM, ad parallelogrammum <lb/>MV: & vt F, ad E, ita triangulum MHK, ad triangu<lb/>lum MQS, ob duplicatam proportionem eius, quæ e&longs;t <lb/>HM ad <expan abbr="Mq.">Mque</expan> hoc e&longs;t ip&longs;ius C ad B; vt igitur trapezium <lb/>NK, ad NS trapezium, ita erit, per præcedentem, CF, <lb/>&longs;imul ad BE &longs;imul. </s>

Line 1322

<s>Sed vt trapezium NR, ad trapezium NS, ita e&longs;t conuer<lb/>tendo AD &longs;imul ad BE, &longs;imul: & vt trapezium NS, ad <lb/>trapezium NK, ita BE, &longs;imul ad CF, &longs;imul; minor igi<lb/>tur proportio erit AD, &longs;imul ad BE &longs;imul, quam BE &longs;i<lb/>mul ad CF, &longs;imul. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="14"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/101.jpg" pagenum="14"/>

<s><emph type="italics"/>PROPOSITIO IX.<emph.end type="italics"/></s>

Line 1332

<s>Di<lb/>co cubum ex AB æqualem e&longs;se duobus &longs;olidis rectangu<lb/>lis, quæ fiunt ex AC CB, & quadrato AB. </s>

<s>Quoniam <lb/><figure id="fig61"/><lb/>enim communi altitudine AB, e&longs;t vt rectangulum BAC <lb/>ad quadratum AB, ita &longs;olidum ex AB, & rectangulo <lb/>BAC ad cubum ex AB, eademque ratione vt rectangu<lb/>lum ABC, ad quadratum AB, ita &longs;olidum e&longs;t AB, & <lb/>rectangulo ABC ad cubum ex AB; erunt vt duo rectan<lb/>gula BAC, ABC ad quadratum AB, ita duo &longs;olida <lb/>ex AB, & rectangulis BAC, ABC ad cubum ex AB. <lb/></s>

<s>Quoniam <lb/><figure id="id.043.01.101.1.jpg" xlink:href="043/01/101/1.jpg"/><lb/>enim communi altitudine AB, e&longs;t vt rectangulum BAC <lb/>ad quadratum AB, ita &longs;olidum ex AB, & rectangulo <lb/>BAC ad cubum ex AB, eademque ratione vt rectangu<lb/>lum ABC, ad quadratum AB, ita &longs;olidum e&longs;t AB, & <lb/>rectangulo ABC ad cubum ex AB; erunt vt duo rectan<lb/>gula BAC, ABC ad quadratum AB, ita duo &longs;olida <lb/>ex AB, & rectangulis BAC, ABC ad cubum ex AB. <lb/></s>

<s>Sed duo rectangula BAC, ABC &longs;unt æqualia quadrato <lb/>AC; duo igitur &longs;olida ex AB, & rectangulis BAC, CBA, <lb/>æqualia &longs;unt cubo ex AB. </s>

Line 1340

<s>Si igitur recta linea <lb/>vtcumque &longs;ecta fuerit, &c. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="15"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/102.jpg" pagenum="15"/>

<s><emph type="italics"/>PROPOSITIO X.<emph.end type="italics"/></s>

Line 1348

<s>Sit recta linea AB &longs;ecta vtcumque in puncto C. </s>

<s>Dico <lb/>cubum ex AB æqualem e&longs;se duobus cubis ex AC, CB, <lb/>& duobus &longs;olidis rectangulis, quorum alterum fit ex tripla <lb/><figure id="fig62"/><lb/>ip&longs;ius AC, & quadrato BC; alterum<gap/>autem ex tripla ip<lb/>&longs;ius BC, & quadrato AC. </s>

<s>Dico <lb/>cubum ex AB æqualem e&longs;se duobus cubis ex AC, CB, <lb/>& duobus &longs;olidis rectangulis, quorum alterum fit ex tripla <lb/><figure id="id.043.01.102.1.jpg" xlink:href="043/01/102/1.jpg"/><lb/>ip&longs;ius AC, & quadrato BC; alterum<gap/>autem ex tripla ip<lb/>&longs;ius BC, & quadrato AC. </s>

<s>Quoniam enim quadratum <lb/>ex AB æquale e&longs;t duobus quadratis ex AC, CB, & ei <lb/>quod bis fit ex AC CB: & parallelepipeda elu&longs;dem al<lb/>titudinis inter &longs;e &longs;unt vt ba&longs;es; erit rectangulorum folido<lb/>rum id quod fit ex AC, & quadrato AB æquale cubo ex <lb/>AC, & ei, quod fit ex AC, & rectangulo ACB bis, & <lb/>ei, quod ex AC, & quadrato BC. </s>

Line 1356

<s>Sed cubus ex AB æqualis e&longs;t <lb/>duobus &longs;olidis ex AC CB. & quadrato AB; cubus igi<lb/>tur ex AB æqualis e&longs;t duobus cubis ex AC CB, & &longs;ex <lb/>&longs;olidis, quorum tres fiunt ex AC, & duobus rectangulis <lb/>ex AC CB, & quadrato BC: tria vero ex BC, & duo<lb/>bus rectangulis ex AC CB, & quadrato AC. </s>

<s>Sed quod <lb/>fit ex AC, & rectangulo ACB, e&longs;t quod fit ex BC, & <pb pagenum="16"/>quadrato AC: & quod fit ex BC, & rectangulo ACB, <lb/>e&longs;t quod fit ex AC, & quadrato BC; cubus igitur ex <lb/>AB æqualis e&longs;t duobus cubis ex AC CB, vna cum &longs;ex <lb/>&longs;olidis, quorum tria fiunt ex AC, & BC quadrato, tria <lb/>autem ex BC, & quadrato AC, hoc e&longs;t duobus &longs;olidis, <lb/>quorum alterum fit ex tripla ip&longs;ius AC, & quadrato BC, <lb/>alterum ex tripla ip&longs;ius BC & quadrato AC. </s>

<s>Sed quod <lb/>fit ex AC, & rectangulo ACB, e&longs;t quod fit ex BC, & <pb xlink:href="043/01/103.jpg" pagenum="16"/>quadrato AC: & quod fit ex BC, & rectangulo ACB, <lb/>e&longs;t quod fit ex AC, & quadrato BC; cubus igitur ex <lb/>AB æqualis e&longs;t duobus cubis ex AC CB, vna cum &longs;ex <lb/>&longs;olidis, quorum tria fiunt ex AC, & BC quadrato, tria <lb/>autem ex BC, & quadrato AC, hoc e&longs;t duobus &longs;olidis, <lb/>quorum alterum fit ex tripla ip&longs;ius AC, & quadrato BC, <lb/>alterum ex tripla ip&longs;ius BC & quadrato AC. </s>

<s>Quod de<lb/>mon&longs;trandum erat. </s>

Line 1368

<s>Di<lb/>co cubum ex AB æqualem e&longs;se duobus cubis ex AC, <lb/>CB, vna cum &longs;olido rectangulo ex AC CB, & tripla <lb/>ip&longs;ius AB. </s>

<s>Quoniam enim quod fit ex AC, & rectan<lb/>gulo ACB, e&longs;t id quod fit ex BC, & quadrato AC: & <lb/>quod fit ex BC, & rectangulo ACB, e&longs;t id, quod fit ex <lb/><figure id="fig63"/><lb/>AC & quadrato BC. &longs;ed duo &longs;olida ex AC CB, & re<lb/>ctangulo ACB &longs;unt id, quod fit ex compo&longs;ita vtriu&longs;que <lb/>altitudine AB, et rectangulo ACB; duo igitur prædi<lb/>cta &longs;olida, quæ ex AC CB, & earum quadratis recipro<lb/>ce fiunt æqualia &longs;unt &longs;olido ex AB BC CA, & triplum <lb/>triplo, videlicet duo &longs;olida, quæ fiunt reciproce ex triplis <pb pagenum="17"/>ip&longs;arum AC, CB, & quadratis ex AC CB, æqualia &longs;i<lb/>mul ei, quod ter fit ex AB, BC, CA, hoc e&longs;t ei, quod <lb/>partibus AC CB, & totius AB tripla continetur: additis <lb/>igitur communibus duobus cubis ex AC, CB, erit id, quod <lb/>&longs;it ex AC CB, & tripla ip&longs;ius AB, & duo cubi ex AC <lb/>CB, æqualia duobus &longs;olidis, quæ fiunt reciproce ex triplis <lb/>ip&longs;arum AC, CB, & earundem AC, CB, quadratis, & <lb/>duobus cubis ex AC, CB, hoc e&longs;t cubo ex AC. </s>

<s>Quoniam enim quod fit ex AC, & rectan<lb/>gulo ACB, e&longs;t id quod fit ex BC, & quadrato AC: & <lb/>quod fit ex BC, & rectangulo ACB, e&longs;t id, quod fit ex <lb/><figure id="id.043.01.103.1.jpg" xlink:href="043/01/103/1.jpg"/><lb/>AC & quadrato BC. &longs;ed duo &longs;olida ex AC CB, & re<lb/>ctangulo ACB &longs;unt id, quod fit ex compo&longs;ita vtriu&longs;que <lb/>altitudine AB, et rectangulo ACB; duo igitur prædi<lb/>cta &longs;olida, quæ ex AC CB, & earum quadratis recipro<lb/>ce fiunt æqualia &longs;unt &longs;olido ex AB BC CA, & triplum <lb/>triplo, videlicet duo &longs;olida, quæ fiunt reciproce ex triplis <pb xlink:href="043/01/104.jpg" pagenum="17"/>ip&longs;arum AC, CB, & quadratis ex AC CB, æqualia &longs;i<lb/>mul ei, quod ter fit ex AB, BC, CA, hoc e&longs;t ei, quod <lb/>partibus AC CB, & totius AB tripla continetur: additis <lb/>igitur communibus duobus cubis ex AC, CB, erit id, quod <lb/>&longs;it ex AC CB, & tripla ip&longs;ius AB, & duo cubi ex AC <lb/>CB, æqualia duobus &longs;olidis, quæ fiunt reciproce ex triplis <lb/>ip&longs;arum AC, CB, & earundem AC, CB, quadratis, & <lb/>duobus cubis ex AC, CB, hoc e&longs;t cubo ex AC. </s>

<s>Si igi<lb/>tur recta linea vtcumque &longs;ecta fuerit, &c. </s>

Line 1378

<s>Hemi&longs;phærium duplum e&longs;t coni, cylindri au<lb/>tem &longs;ub&longs;e&longs;quialterum eandem ip&longs;i ba&longs;im, & ean<lb/>dem altitudinem habentium. </s>

<s>E&longs;to hemi&longs;phærium; cuius axis BD, ba&longs;is circulus, cu<lb/>ius diameter AC, &longs;uper quem cylindrus AE, & conus <lb/><figure id="fig64"/><lb/>ABC, quorum communis axis &longs;it BD, ac propterea <lb/>etiam eadem altitudo. </s>

<s>E&longs;to hemi&longs;phærium; cuius axis BD, ba&longs;is circulus, cu<lb/>ius diameter AC, &longs;uper quem cylindrus AE, & conus <lb/><figure id="id.043.01.104.1.jpg" xlink:href="043/01/104/1.jpg"/><lb/>ABC, quorum communis axis &longs;it BD, ac propterea <lb/>etiam eadem altitudo. </s>

<s>Dico hemi&longs;phærium ABC, co<lb/>ni ABC e&longs;se duplum: cylindri autem AE <expan abbr="&longs;ub&longs;e&longs;quialter&utilde;">&longs;ub&longs;e&longs;quialterum</expan>. <lb/></s>

<s>&longs;uper ba&longs;im enim circulum RE, vertice D de&longs;cribatur <pb pagenum="18"/>conus EDR. </s>

<s>&longs;uper ba&longs;im enim circulum RE, vertice D de&longs;cribatur <pb xlink:href="043/01/105.jpg" pagenum="18"/>conus EDR. </s>

<s>Sectoque axe BD primo bifariam, deinde <lb/>&longs;ingulis eius partibus rur&longs;us bifariam, tran&longs;eant per pun<lb/>cta &longs;ectionum plana ba&longs;i hemi&longs;phærij AC æquidi&longs;tantia, <lb/>quæ &longs;ecent hemi&longs;phærium, conum, & cylindrum. </s>

<s>Se<lb/>ctus igitur erit AE cylindrus in cylindros æqualium alti<lb/>tudinum: &longs;uper &longs;ectiones autem coni, atque hemi&longs;phærij <lb/>nempe circulos, quorum centra in axe BD exi&longs;tunt cy<lb/>lindri con&longs;tituti intelligantur binis quibu&longs;que proximis <lb/>æquidi&longs;tantibus planis interiecti, quorum axes omnes <lb/>æquales in BD. </s>

<s>Erit igitur cono EDR in&longs;cripta, & ABC <lb/><figure id="fig65"/><lb/>hemi&longs;phærio circum&longs;cripta figura quædam ex cylindris <lb/>æqualium altitudinum. </s>

<s>Erit igitur cono EDR in&longs;cripta, & ABC <lb/><figure id="id.043.01.105.1.jpg" xlink:href="043/01/105/1.jpg"/><lb/>hemi&longs;phærio circum&longs;cripta figura quædam ex cylindris <lb/>æqualium altitudinum. </s>

<s>Sint autem hæ figuræ ea ratione <lb/>hæc circum&longs;cripta illa in&longs;cripta, vt circum&longs;cripta excedat <lb/>hemi&longs;phærium, minori exce&longs;su, in&longs;cripta vero deficiat à <lb/>cono minori defectu quam &longs;it magnitudo propo&longs;ita, quan<lb/>tacumque illa &longs;it. </s>

<s>His con&longs;titutis, manife&longs;tum e&longs;t, reliquo <lb/>cylindri AE dempto hemi&longs;phærio in&longs;criptam e&longs;se figu<lb/>ram ex re&longs;iduis cylindrorum, in quos cylindrus AE &longs;e<lb/>ctus fuerit, demptis cylindris hemi&longs;phærio circum&longs;criptis, <lb/>deficientem à reliquo cylindri AE dempto hemi&longs;phærio <lb/>minori defectu magnitudine propo&longs;ita, eodem &longs;cilicet, <lb/>quo figura hemi&longs;phærio circum&longs;cripta excedit hemi&longs;phæ<lb/>rium, excepto re&longs;iduo cylindri infimi AS, dempta he<lb/>mi&longs;phærij portione, quam comprehendit. </s>

<s>Sit autem om-<pb pagenum="19"/>nium prædictorum cylindri AE cylindrorum &longs;upremus <lb/>FE, cuius axis BH, & communis &longs;ectio plani per pun<lb/>ctum H tran&longs;euntis ba&longs;i hemi&longs;phærij cum plano per axim <lb/>BD, &longs;it recta FGKHMNL. </s>

<s>Sit autem om-<pb xlink:href="043/01/106.jpg" pagenum="19"/>nium prædictorum cylindri AE cylindrorum &longs;upremus <lb/>FE, cuius axis BH, & communis &longs;ectio plani per pun<lb/>ctum H tran&longs;euntis ba&longs;i hemi&longs;phærij cum plano per axim <lb/>BD, &longs;it recta FGKHMNL. </s>

<s>Quoniam igitur rectan<lb/>gulum DHB bis vna cum duobus quadratis DH, BH, <lb/>æquale e&longs;t BD quadrato: & rectangulum DHB bis <lb/>vna cum quadrato BH, e&longs;t rectangulum ex BD DH tan<lb/>quam vna, & BH; rectangulum ex BD, DH tanquam <lb/>vna & BH, vna cum quadrato DH æquale erit quadra<lb/>to BD, hoc e&longs;t quadrato FH: quorum quadratum KH <lb/>æquale e&longs;t rectangulo ex BD, DH, tanquam vna, & BH; <lb/>reliquum igitur quadrati FH dempto quadrato KH æ<lb/>quale erit reliquo quadrato DH, hoc e&longs;t quadrato GH: <lb/>& quadruplum quadruplo reliquum quadrati FL dempto <lb/>quadrato MK toti GN quadrato, hoc e&longs;t reliquum circu <lb/>li, FL dempto circulo MK, æquale circulo GN. </s>

Line 1402

<s>Similiter o&longs;tenderemus &longs;ingula reliqua cylindrorum eiu&longs;<lb/>dem altitudinis, in quos totus cylindrus AE &longs;ectus fuit, <lb/>demptis cylindris hemi&longs;phærio circum&longs;criptis æqualia e&longs;<lb/>&longs;e &longs;ingulis cylindris cono EDR in&longs;criptis, quæ inter ea<lb/>dem plana interijciuntur. </s>

<s>Tota igitur figura ex prædictis <lb/>cylindrorum re&longs;iduis reliquo cylindri AE, dempto he<lb/>mi&longs;phærio in&longs;cripta æqualis erit figuræ cono EDR in<lb/>&longs;criptæ: deficit autem vtraque harum figurarum hæc à co<lb/>no ADR, illa à re&longs;iduo cylindri AE dempto hemi&longs;phæ<lb/>rio minori exce&longs;su magnitudine vtcumque propo&longs;ita; re<lb/>liquum igitur cylindri AE dempto hemi&longs;phærio æquale <lb/>e&longs;t cono EDR, &longs;ed conus EDR; hoc e&longs;t conus ABC cylin <lb/>dri AE e&longs;t pars tertia; reliquum igitur cylindri AE dem<lb/>pto hemi&longs;phærio, cylindri AE e&longs;t pars tertia, hoc e&longs;t cylin<lb/>drus AE triplus dicti re&longs;idui: <expan abbr="quamobr&etilde;">quamobrem</expan> AE cylindrus &longs;e&longs;<lb/>quialter hemi&longs;phærij ABC: & <expan abbr="cõuertendo">conuertendo</expan>, hemi&longs;phærium <pb pagenum="20"/>cylindri AE &longs;ub&longs;e&longs;quialterum: coni igitur ABC duplum. <lb/></s>

<s>Tota igitur figura ex prædictis <lb/>cylindrorum re&longs;iduis reliquo cylindri AE, dempto he<lb/>mi&longs;phærio in&longs;cripta æqualis erit figuræ cono EDR in<lb/>&longs;criptæ: deficit autem vtraque harum figurarum hæc à co<lb/>no ADR, illa à re&longs;iduo cylindri AE dempto hemi&longs;phæ<lb/>rio minori exce&longs;su magnitudine vtcumque propo&longs;ita; re<lb/>liquum igitur cylindri AE dempto hemi&longs;phærio æquale <lb/>e&longs;t cono EDR, &longs;ed conus EDR; hoc e&longs;t conus ABC cylin <lb/>dri AE e&longs;t pars tertia; reliquum igitur cylindri AE dem<lb/>pto hemi&longs;phærio, cylindri AE e&longs;t pars tertia, hoc e&longs;t cylin<lb/>drus AE triplus dicti re&longs;idui: <expan abbr="quamobr&etilde;">quamobrem</expan> AE cylindrus &longs;e&longs;<lb/>quialter hemi&longs;phærij ABC: & <expan abbr="cõuertendo">conuertendo</expan>, hemi&longs;phærium <pb xlink:href="043/01/107.jpg" pagenum="20"/>cylindri AE &longs;ub&longs;e&longs;quialterum: coni igitur ABC duplum. <lb/></s>

<s>Manife&longs;tum e&longs;t igitur propo&longs;itum. </s>

Line 1414

<s>Sitque <lb/>EF, cylindrus, cu<lb/>ius axis, &longs;iue alti<lb/>tudo eadem BG: <lb/>ba&longs;is autem æqua<lb/>lis circulo maxi<lb/>mo, cuius &longs;emidia<lb/>meter BD. </s>

<s>Dico <lb/>portionem ABC, <lb/>ad cylindrum EF <lb/>eam habere pro<lb/><figure id="fig66"/><lb/>portionem, quam exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;upe<lb/>rat tres BD, DG; & minorem extremam ad ip&longs;as, quæ <lb/>&longs;it M; ad ip&longs;ius BD triplam. </s>

<s>Dico <lb/>portionem ABC, <lb/>ad cylindrum EF <lb/>eam habere pro<lb/><figure id="id.043.01.107.1.jpg" xlink:href="043/01/107/1.jpg"/><lb/>portionem, quam exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;upe<lb/>rat tres BD, DG; & minorem extremam ad ip&longs;as, quæ <lb/>&longs;it M; ad ip&longs;ius BD triplam. </s>

<s>vertice enim D, ba&longs;i cylin<lb/>dri EF, cuius diameter FH de&longs;cribatur conus FDH, cu<lb/>ius intelligatur fru&longs;tum FHKL ab&longs;ci&longs;sum plano, quod ab-<pb pagenum="21"/>&longs;cidit portionem ABC, plano circuli FH parallelum. <lb/></s>

<s>vertice enim D, ba&longs;i cylin<lb/>dri EF, cuius diameter FH de&longs;cribatur conus FDH, cu<lb/>ius intelligatur fru&longs;tum FHKL ab&longs;ci&longs;sum plano, quod ab-<pb xlink:href="043/01/108.jpg" pagenum="21"/>&longs;cidit portionem ABC, plano circuli FH parallelum. <lb/></s>

<s>Quoniam igitur fru&longs;tum FH<emph type="italics"/>K<emph.end type="italics"/>L æquale e&longs;t cylindri EF <lb/>re&longs;iduo, dempta ABC portione, quod ex præcedenti theo <lb/>remate per&longs;picuum e&longs;se debet: erit portio ABC æqualis <lb/>ei, quod relinquitur cylindri EF, &longs;i fru&longs;tum auferatur <lb/>FHKL: &longs;ed hoc reliquum e&longs;t ad cylindrum EF, vt exce&longs;<lb/>&longs;us, quo tripla lineæ FH, &longs;uperat tres deinceps proportio<lb/>nales FH, KL, & minorem extrema, ad triplam lineæ FH: <lb/><gap/>vt FH, ad KL, ita e&longs;t BD ad DG, & DG, ad M; vt igi<lb/>tur exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;uperat tres BD, DG, <lb/>& M, &longs;imul, ad lineæ BD triplam, ita erit portio ABC ad <lb/>cylindrum EF. </s>

Line 1426

<s>Omnis portio &longs;phæræ ab&longs;ci&longs;sa duobus planis <lb/>parallelis alteroper centrum acto ad cylindrum, <lb/>cuius ba&longs;is e&longs;t eadem ba&longs;i portionis, &longs;iue circu<lb/>lo maximo, & eadem altitudo, eam habet pro<lb/>portionem, quam exce&longs;&longs;us, quo maior extrema ad <lb/>&longs;phæræ &longs;emidiametrum, & axim portionis exce<lb/>dit tertiam partem axis portionis; ad maiorem ex<lb/>tremam antedictam. </s>

<s>Sit portio AB <lb/>CD, &longs;phæræ, cu <lb/>ius centrum F, <lb/>ab&longs;ci&longs;&longs;a duobus <lb/>planis parallelis <lb/>altero per <expan abbr="centr&utilde;">centrum</expan> <lb/>F tran&longs;eunte; <lb/>axis autem por<lb/>tionis fit FG: & <lb/><figure id="fig67"/><pb pagenum="22"/>maior ba&longs;is, circulus maximus, cuius diameter AD, minor <lb/>autem, cuius diameter BC: & cylindrus AE, cuius ba&longs;is <lb/>circulus AD, axis FG; & vt FG ad FA, ita &longs;it FA, ad <lb/>MN, à qua ab&longs;cindatur NO, pars tertia ip&longs;ius FG. </s>

<s>Sit portio AB <lb/>CD, &longs;phæræ, cu <lb/>ius centrum F, <lb/>ab&longs;ci&longs;&longs;a duobus <lb/>planis parallelis <lb/>altero per <expan abbr="centr&utilde;">centrum</expan> <lb/>F tran&longs;eunte; <lb/>axis autem por<lb/>tionis fit FG: & <lb/><figure id="id.043.01.108.1.jpg" xlink:href="043/01/108/1.jpg"/><pb xlink:href="043/01/109.jpg" pagenum="22"/>maior ba&longs;is, circulus maximus, cuius diameter AD, minor <lb/>autem, cuius diameter BC: & cylindrus AE, cuius ba&longs;is <lb/>circulus AD, axis FG; & vt FG ad FA, ita &longs;it FA, ad <lb/>MN, à qua ab&longs;cindatur NO, pars tertia ip&longs;ius FG. </s>

<s>Dico <lb/>ABCD <expan abbr="portion&etilde;">portionem</expan> ad cylindrum AE e&longs;&longs;e vt OM ad MN. <lb/></s>

<s>Po&longs;ita enim G <lb/>H, æquali ip&longs;i <lb/>FG, de&longs;criba<lb/>tur circa axim <lb/>FG, cylindrus <lb/>L<emph type="italics"/>K<emph.end type="italics"/>, & conus <lb/>HFK. </s>

<s>Quoniam <lb/>igitur duo cylin <lb/>dri AE, LK, <lb/>&longs;unt eiu&longs;dem al<lb/><figure id="fig68"/><lb/>titudinis, erunt inter &longs;e vt ba&longs;es, AD, KH. hoc e&longs;t cy<lb/>lindrus AE ad cylindrum LK, duplicatam habebit pro<lb/>portionem diametri AD, ad diametrum KH, hoc e&longs;t eius, <lb/>quæ e&longs;t &longs;emidiametri AF ad &longs;emidiametrum GH. hoc e&longs;t <lb/>eam, quæ e&longs;t MN ad GH, &longs;iue FG. </s>

<s>Quoniam <lb/>igitur duo cylin <lb/>dri AE, LK, <lb/>&longs;unt eiu&longs;dem al<lb/><figure id="id.043.01.109.1.jpg" xlink:href="043/01/109/1.jpg"/><lb/>titudinis, erunt inter &longs;e vt ba&longs;es, AD, KH. hoc e&longs;t cy<lb/>lindrus AE ad cylindrum LK, duplicatam habebit pro<lb/>portionem diametri AD, ad diametrum KH, hoc e&longs;t eius, <lb/>quæ e&longs;t &longs;emidiametri AF ad &longs;emidiametrum GH. hoc e&longs;t <lb/>eam, quæ e&longs;t MN ad GH, &longs;iue FG. </s>

<s>Sed vt FG ad tertiam <lb/>&longs;ui partem NO, ita e&longs;t cylindrus KL, ad conum KFH; <lb/>ex æquali igitur, erit vt MN ad NO, ita cylindrus AE <lb/>ad conum <emph type="italics"/>K<emph.end type="italics"/>FH, hoc e&longs;t ad reliquum cylindri AE dem <lb/>pta ABCD portione: & per conuer&longs;ionem rationis, vt <lb/>NM, ad MO, ita cylindrus AE ad portionem ABCD: <lb/>& conuertendo, vt MO ad MN, ita portio ABCD ad <lb/>cylindrum AE. </s>

Line 1440

<s><emph type="italics"/>PROPOSITIO XV.<emph.end type="italics"/></s>

<s>Omnis portio &longs;phæræ ab&longs;ci&longs;&longs;a duobus planis <lb/>parallelis neutro per centrum, nec centrum inter<lb/>cipientibus ad cylindrum, cuius ba&longs;is æqualis e&longs;t <pb pagenum="23"/>circulo maximo, altitudo autem eadem portioni, <lb/>eam <expan abbr="proportion&etilde;">proportionem</expan> habet, quam exce&longs;&longs;us, quo maior <lb/>extrema ad triplas &longs;emidiametri &longs;phæræ, & eius <lb/>quæ inter <expan abbr="centr&utilde;">centrum</expan> &longs;phæræ, & minoris ba&longs;is portio<lb/>nis interijcitur, &longs;uperat tres deinceps <lb/>proportionales, quarum maxima e&longs;t <lb/>quæ inter centra &longs;phæræ, & minoris <lb/>ba&longs;is, media autem, quæ inter cen<lb/>træ &longs;phæræ, & maioris ba&longs;is portio<lb/>nis interijcitur; ad maiorem extre<lb/>mam antedictam. </s>

<s>Omnis portio &longs;phæræ ab&longs;ci&longs;&longs;a duobus planis <lb/>parallelis neutro per centrum, nec centrum inter<lb/>cipientibus ad cylindrum, cuius ba&longs;is æqualis e&longs;t <pb xlink:href="043/01/110.jpg" pagenum="23"/>circulo maximo, altitudo autem eadem portioni, <lb/>eam <expan abbr="proportion&etilde;">proportionem</expan> habet, quam exce&longs;&longs;us, quo maior <lb/>extrema ad triplas &longs;emidiametri &longs;phæræ, & eius <lb/>quæ inter <expan abbr="centr&utilde;">centrum</expan> &longs;phæræ, & minoris ba&longs;is portio<lb/>nis interijcitur, &longs;uperat tres deinceps <lb/>proportionales, quarum maxima e&longs;t <lb/>quæ inter centra &longs;phæræ, & minoris <lb/>ba&longs;is, media autem, quæ inter cen<lb/>træ &longs;phæræ, & maioris ba&longs;is portio<lb/>nis interijcitur; ad maiorem extre<lb/>mam antedictam. </s>

<s>Sit portio ABCD &longs;phæræ, cuius centrum <lb/>E, ab&longs;ci&longs;sa duobus planis parallelis, neutro <lb/>per E tran&longs;eunte, nec E <expan abbr="intercipi&etilde;tibus">intercipientibus</expan>, cuius <lb/>maior ba&longs;is &longs;it circulus, cui diameter AD. <lb/>minor autem cuius diameter BC, axis GH. <lb/>circa quem cylindrus OS, con&longs;i&longs;tat, cuius <lb/>ba&longs;is &longs;it circulus circa SR æqualis circulo <lb/>maximo: &longs;phæræ autem &longs;emidiater &longs;it EHG. <lb/>& vt GE ad EH, ita &longs;it HE ad V: & po<lb/><figure id="fig69"/><lb/>&longs;ita T tripla ip&longs;ius EF, & X itidem tripla ip&longs;ius EG, vt X <pb pagenum="24"/>ad T, ita fiat T ad ZY, cuius Z<foreign lang="greek">w</foreign>, tribus GE, EH, V <lb/>&longs;imul &longs;it æqualis. </s>

<s>Sit portio ABCD &longs;phæræ, cuius centrum <lb/>E, ab&longs;ci&longs;sa duobus planis parallelis, neutro <lb/>per E tran&longs;eunte, nec E <expan abbr="intercipi&etilde;tibus">intercipientibus</expan>, cuius <lb/>maior ba&longs;is &longs;it circulus, cui diameter AD. <lb/>minor autem cuius diameter BC, axis GH. <lb/>circa quem cylindrus OS, con&longs;i&longs;tat, cuius <lb/>ba&longs;is &longs;it circulus circa SR æqualis circulo <lb/>maximo: &longs;phæræ autem &longs;emidiater &longs;it EHG. <lb/>& vt GE ad EH, ita &longs;it HE ad V: & po<lb/><figure id="id.043.01.110.1.jpg" xlink:href="043/01/110/1.jpg"/><lb/>&longs;ita T tripla ip&longs;ius EF, & X itidem tripla ip&longs;ius EG, vt X <pb xlink:href="043/01/111.jpg" pagenum="24"/>ad T, ita fiat T ad ZY, cuius Z<foreign lang="greek">w</foreign>, tribus GE, EH, V <lb/>&longs;imul &longs;it æqualis. </s>

<s>Dico ABCD portio<lb/>nem ad cylindrum SO e&longs;se vt <foreign lang="greek">w*u</foreign> ad <foreign lang="greek">*u</foreign>Z. <lb/></s>

Line 1454

<s>Quoniam igitur &longs;unt ter <lb/>næ deinceps proportionales GE, EF, I, & <lb/>X, T, ZY, e&longs;tque vt FE ad EG ita T ad X; <lb/>erit vt I ad EG, hoc e&longs;t vt cylindrus SO ad <lb/>PN <expan abbr="cylindr&utilde;">cylindrum</expan> ita ZY ad X. </s>

<s>Et quoniam e&longs;t vt <lb/>GE ad EH, ita EH ad V: hoc e&longs;t, vt GK ad <lb/>LH. ita LH ad V: & ponitur X tripla ip&longs;ius <lb/><figure id="fig70"/><lb/>EG, hoc e&longs;t ip&longs;ius GK, vt autem e&longs;t triplaip&longs;ius GK ad <lb/>tres deinceps proportionales GK, LH, V, ita e&longs;t cylin<lb/>drus PN ad fru&longs;tum LKNM; erit vt X ad tres GE, EH, <lb/>V &longs;imul hoc e&longs;t ad lineam <foreign lang="greek">w</foreign>Z, ita cylindrus PN ad fru-<pb pagenum="25"/>&longs;lum KLMN. </s>

<s>Et quoniam e&longs;t vt <lb/>GE ad EH, ita EH ad V: hoc e&longs;t, vt GK ad <lb/>LH. ita LH ad V: & ponitur X tripla ip&longs;ius <lb/><figure id="id.043.01.111.1.jpg" xlink:href="043/01/111/1.jpg"/><lb/>EG, hoc e&longs;t ip&longs;ius GK, vt autem e&longs;t triplaip&longs;ius GK ad <lb/>tres deinceps proportionales GK, LH, V, ita e&longs;t cylin<lb/>drus PN ad fru&longs;tum LKNM; erit vt X ad tres GE, EH, <lb/>V &longs;imul hoc e&longs;t ad lineam <foreign lang="greek">w</foreign>Z, ita cylindrus PN ad fru-<pb xlink:href="043/01/112.jpg" pagenum="25"/>&longs;lum KLMN. </s>

<s>Sed vt ZY ad X, ita erat cylindrus SO <lb/>ad PN cylindrum; ex æquali igitur erit vt ZY ad Z<foreign lang="greek">w</foreign>, <lb/>ita cylindrus SO ad fru&longs;tum KLMN: hoc e&longs;t, ad reli<lb/>quum cylindri SO dempta ABCD portione, & per con<lb/>uer&longs;ionem rationis, vt ZY, ad Y<foreign lang="greek">w</foreign>, ita cylindrus SO ad <lb/><expan abbr="portion&etilde;">portionem</expan> ABCD: & conuertendo vt <foreign lang="greek">w</foreign>Y ad YZ, ita por<lb/>tio ABCD ad SO cylindrum. </s>

Line 1462

<s><emph type="italics"/>PROPOSITIO XVI.<emph.end type="italics"/></s>

<s>Omnis maior &longs;phæræ portio ad cylindrum, cu<lb/>ius ba&longs;is æqualis e&longs;t circulo maximo, altitudo au<lb/>tem eadem portioni eam habet proportionem, <lb/>quam ad axim portionis habet exce&longs;&longs;us, quo &longs;eg<lb/>mentum axis portionis inter &longs;phæræ centrum, & <lb/>ba&longs;im portionis interiectum &longs;uperat tertiam par<lb/>tem minoris extremæ maiori po&longs;ita prædicto axis <lb/>&longs;egmento in proportione &longs;emidiametri &longs;phæræ <lb/>ad prædictum <lb/><expan abbr="&longs;egment&utilde;">&longs;egmentum</expan>, vna <lb/>cum &longs;ub&longs;e&longs;qui <lb/>altera reliqui <lb/>axis &longs;egmenti. </s><figure/>

<s>Sit &longs;phæræ, cu <lb/>ius <expan abbr="centr&utilde;">centrum</expan> G, dia <lb/>meter DGE ma <lb/>ior portio ABC, <lb/>axis autem por<lb/>tionis BGF, com <lb/>munis cylindro <lb/>KH, cuius ba&longs;is æqualis &longs;it circulo maximo; ba&longs;is autem <pb pagenum="26"/>portionis circulus, cuius diameter AC, & vt EG ad GF, <lb/>ita &longs;it GF ad S, & S ad FM, cuius &longs;it pars tertia FN, & <lb/>ponatur ip&longs;ius BG, &longs;ub&longs;e&longs;quialtera GL. </s>

<s>Sit &longs;phæræ, cu <lb/>ius <expan abbr="centr&utilde;">centrum</expan> G, dia <lb/>meter DGE ma <lb/>ior portio ABC, <lb/>axis autem por<lb/>tionis BGF, com <lb/>munis cylindro <lb/>KH, cuius ba&longs;is æqualis &longs;it circulo maximo; ba&longs;is autem <pb xlink:href="043/01/113.jpg" pagenum="26"/>portionis circulus, cuius diameter AC, & vt EG ad GF, <lb/>ita &longs;it GF ad S, & S ad FM, cuius &longs;it pars tertia FN, & <lb/>ponatur ip&longs;ius BG, &longs;ub&longs;e&longs;quialtera GL. </s>

<s>Dico portio<lb/>nem ABC ad cylindrum KH e&longs;se vt LN ad BF. </s>

Line 1472

<s>Et plano per G <lb/>tran&longs;eunte ba&longs;ibus cylindri KH, & ABC portionis pa<lb/>rallelo &longs;ecentur vna cylindrus KH in duos cylindros DH, <lb/>EK: & portio ABC, in portionem ECAD, & DBE <lb/>hemi&longs;phærium. </s>

<s>Quoniam igitur e&longs;t conuertendo, vt PQ <lb/>ad EG, ita EG <lb/>ad GF, & e&longs;t ip<lb/>&longs;ius GF pars ter <lb/>tia QR, erit por<lb/>tio DACE ad <lb/>cylindrum EK, <lb/>vt PR ad <expan abbr="Pq.">Pque</expan> <lb/>Rur&longs;us, quia e&longs;t <lb/>vt EG ad GF: <lb/>hoc e&longs;t vt PQ ad <lb/>EG, ita GF ad <lb/>S, & vt EG ad <lb/>GF, ita e&longs;t S ad <lb/>FM; erit ex æqua <lb/><figure id="fig71"/><lb/>li, vt PQ ad GF, ita GF ad FM. </s>

<s>Sed vt GF ad <gap/>Q, <lb/>ita e&longs;t MF ad FN, tertiam ip&longs;ius MF partem, ex æquali <lb/>igitur erit vt PQ ad QR, ita GF ad FN, & per conuer<lb/>&longs;ionem rationis, & conuertendo, vt PR ad PQ, ita NG ad <lb/>GF. </s>

Line 1480

<s>Sed vt GF ad FB, ita e&longs;t cy<lb/>lindrus EK ad cylindrum KH: ex æquali igitur vt NG <lb/>ad BF, ita portio ECAD, ad cylindrum KH. </s>

<s>Similiter <lb/>o&longs;tenderemus e&longs;se, vt GL ad BF, ita DBE hemi&longs;phæ-<pb pagenum="27"/>rium ad cylindrum KH, cum vt LG ad GB, ita &longs;it he<lb/>mi&longs;phærium DBE ad cylindrum DH. vt igitur prima <lb/>cum quinta ad &longs;ecundam, ita tertia cum &longs;exta ad quartam; <lb/>videlicet, vt tota LN ad BF, ita portio ABC ad cylin<lb/>drum KH. </s>

<s>Similiter <lb/>o&longs;tenderemus e&longs;se, vt GL ad BF, ita DBE hemi&longs;phæ-<pb xlink:href="043/01/114.jpg" pagenum="27"/>rium ad cylindrum KH, cum vt LG ad GB, ita &longs;it he<lb/>mi&longs;phærium DBE ad cylindrum DH. vt igitur prima <lb/>cum quinta ad &longs;ecundam, ita tertia cum &longs;exta ad quartam; <lb/>videlicet, vt tota LN ad BF, ita portio ABC ad cylin<lb/>drum KH. </s>

<s>Quod erat demon&longs;trandum. </s>

Line 1490

<s>Sit portio AB <lb/>CD, &longs;phæræ, cu<lb/>ius centrum G, <lb/>ab&longs;ci&longs;sa duobus <lb/>planis parallelis <lb/>centrum G inter<lb/>cipientibus, quod <lb/>erit in axe portio<lb/>nis, qui &longs;it HK. <lb/></s>

<s>Sectiones autem <lb/><figure id="fig72"/><lb/>factæ à prædictis planis &longs;int circuli, quorum diametri AD, <lb/>BC, qui circuli erunt ba&longs;es oppo&longs;itæ portionis. </s>

<s>Sectiones autem <lb/><figure id="id.043.01.114.1.jpg" xlink:href="043/01/114/1.jpg"/><lb/>factæ à prædictis planis &longs;int circuli, quorum diametri AD, <lb/>BC, qui circuli erunt ba&longs;es oppo&longs;itæ portionis. </s>

<s>Sectaque <lb/>per punctum G, portione ABCD plano ad axim erecto, <pb pagenum="28"/>atque ideo & portionis ba&longs;ibus parallelo; &longs;uper &longs;ectionem, <lb/>quæ erit circulus maximus, cuius diameter LM, duo cylin<lb/>dri de&longs;cripti intelligantur, ad oppo&longs;ita portionis ba&longs;ium pla <lb/>na terminati ex illis autem totus cylindrus compo&longs;itus EF, <lb/>cuius ba&longs;is æqua<lb/>lis circulo maxi<lb/>mo LM. </s>

<s>Sectaque <lb/>per punctum G, portione ABCD plano ad axim erecto, <pb xlink:href="043/01/115.jpg" pagenum="28"/>atque ideo & portionis ba&longs;ibus parallelo; &longs;uper &longs;ectionem, <lb/>quæ erit circulus maximus, cuius diameter LM, duo cylin<lb/>dri de&longs;cripti intelligantur, ad oppo&longs;ita portionis ba&longs;ium pla <lb/>na terminati ex illis autem totus cylindrus compo&longs;itus EF, <lb/>cuius ba&longs;is æqua<lb/>lis circulo maxi<lb/>mo LM. </s>

<s>Deinde <lb/>in &longs;egmento GH <lb/>&longs;umpta OH, ter<lb/>tia parte minoris <lb/>extremæ maiori <lb/>GH in proportio <lb/>ne, quæ e&longs;t LG ad <lb/>GH; & in &longs;egmen <lb/>to GK, &longs;umatur <lb/><figure id="fig73"/><lb/>NK, tertia pars minoris extremæ maiori GK, in propor<lb/>tione, quæ e&longs;t LG ad GK. </s>

<s>Deinde <lb/>in &longs;egmento GH <lb/>&longs;umpta OH, ter<lb/>tia parte minoris <lb/>extremæ maiori <lb/>GH in proportio <lb/>ne, quæ e&longs;t LG ad <lb/>GH; & in &longs;egmen <lb/>to GK, &longs;umatur <lb/><figure id="id.043.01.115.1.jpg" xlink:href="043/01/115/1.jpg"/><lb/>NK, tertia pars minoris extremæ maiori GK, in propor<lb/>tione, quæ e&longs;t LG ad GK. </s>

<s>Dico portionem ABCD <lb/>ad cylindrum EF, e&longs;se vt NO ad KH. </s>

Line 1504

<s><emph type="italics"/>PROPOSITIO XVIII.<emph.end type="italics"/></s>

<s>Omne conoides parabolicum dimidium e&longs;t <lb/>cylindri, coni autem &longs;e&longs;quialterum eandem ip&longs;i <lb/>ba&longs;im, & eandem altitudinem habentium. </s><pb pagenum="29"/>

<s>Sit conoides parabolicum ABC, & cylindrus AE, & <lb/>conus ABC, quorum omnium &longs;it eadem ba&longs;is circulus, <lb/>cuins diameter AC, axis autem BD, ac proinde vna om<lb/>nium altitudo. </s>

Line 1512

<s>Secto eni<gap/>n <lb/>axe BD in tot partes æquales, quarum infima ad ba&longs;im &longs;it <lb/>MD, vt figura ex cylindris æqualium altitudinum conoi<lb/>di ABC circum&longs;cripta, in&longs;criptam &longs;uperet minori &longs;pacio <lb/>quantacu nque magnitudine propo&longs;ita, & &longs;it hoc factum. <lb/></s>

<s>Et quoniam quibus planis parallelis tran&longs;euntibus per præ<lb/><figure id="fig74"/><lb/>dictas &longs;ectiones axis BD &longs;ecatur conoides ABC, ij&longs;dem <lb/>&longs;ecatur triangulum per axim ABC, eruntque &longs;ectiones <lb/>parallelæ: &longs;it triangulo ABC circum&longs;cripta figura ex pa<lb/>rallelogrammis æqualium altitudinum, quæ triangulum & <lb/>ip&longs;a excedat minori &longs;pacio quantacumque magnitudine <lb/>propo&longs;ita. </s>

<s>Et quoniam quibus planis parallelis tran&longs;euntibus per præ<lb/><figure id="id.043.01.116.1.jpg" xlink:href="043/01/116/1.jpg"/><lb/>dictas &longs;ectiones axis BD &longs;ecatur conoides ABC, ij&longs;dem <lb/>&longs;ecatur triangulum per axim ABC, eruntque &longs;ectiones <lb/>parallelæ: &longs;it triangulo ABC circum&longs;cripta figura ex pa<lb/>rallelogrammis æqualium altitudinum, quæ triangulum & <lb/>ip&longs;a excedat minori &longs;pacio quantacumque magnitudine <lb/>propo&longs;ita. </s>

<s>Cylindrorum autem qui &longs;unt circa conoides, & <lb/>parallelogrammorum multitudine æqualium, quæ &longs;unt cir<lb/>ca triangulum ABC, duo proximi ba&longs;i AC cylindri &longs;int <lb/>AF, HL, & totidem parallelogramma illis re&longs;pondentia <lb/>inter eadem plana parallela &longs;int AF, GK. </s>

<s>Quoniam igi-<pb pagenum="30"/>tur in parabola ABC rectis ad diametrum ordinatim ap<lb/>plicatis e&longs;t vt BM ad BD longitudine, ita MH ad AD <lb/>potentia: hoc e&longs;t, ita circulus, cuius diameter HMN, ad <lb/>circulum, cuius diameter ADC, hoc e&longs;t ita cylindrus HL, <lb/>ad cylindrum AF propter æqualitatem altitudinum: &longs;ed <lb/>vt BM ad BD, ita e&longs;t GM ad AD, propter &longs;imilitudinem <lb/>triangulorum, hoc e&longs;t ita <expan abbr="parallelogrãmum">parallelogrammum</expan> GK ad AF, pa<lb/>rallelogrammum; ergo vt parallelogrammum GK ad paral <lb/><expan abbr="lelogrãmum">lelogrammum</expan> AF, ita e&longs;t cylindrus HL ad cylindrum AF. <lb/></s>

<s>Quoniam igi-<pb xlink:href="043/01/117.jpg" pagenum="30"/>tur in parabola ABC rectis ad diametrum ordinatim ap<lb/>plicatis e&longs;t vt BM ad BD longitudine, ita MH ad AD <lb/>potentia: hoc e&longs;t, ita circulus, cuius diameter HMN, ad <lb/>circulum, cuius diameter ADC, hoc e&longs;t ita cylindrus HL, <lb/>ad cylindrum AF propter æqualitatem altitudinum: &longs;ed <lb/>vt BM ad BD, ita e&longs;t GM ad AD, propter &longs;imilitudinem <lb/>triangulorum, hoc e&longs;t ita <expan abbr="parallelogrãmum">parallelogrammum</expan> GK ad AF, pa<lb/>rallelogrammum; ergo vt parallelogrammum GK ad paral <lb/><expan abbr="lelogrãmum">lelogrammum</expan> AF, ita e&longs;t cylindrus HL ad cylindrum AF. <lb/></s>

<s>Similiter o&longs;tenderemus reliqua parallelogramma, quæ &longs;unt <lb/><figure id="fig75"/><lb/>circa <expan abbr="triãgulum">triangulum</expan> ABC e&longs;se cum reliquis cylindris, qui &longs;unt <lb/>circa conoides ABC bina &longs;umpta prout inter &longs;e re&longs;pon<lb/>dent in eadem proportione; &longs;emper igitur componendo, & <lb/>ex æquali erit vt tota figura triangulo ABC circum&longs;cripta <lb/>ad parallelogrammum AF, ita figura conoidi circum&longs;cri<lb/>pta ad AF cylindrum: &longs;ed vt parallelogrammum AF, ad <lb/>parallelogrammum AE, ita e&longs;t cylindrus AF ad cylindrum <lb/>AE, propter æqualitatem omnifariam &longs;umptarum altitu<lb/>dinum; ex æquali igitur erit vt figura triangulo ABC cir<lb/>cum&longs;cripta ad parallelogrammum AE, ita figura conoidi <pb pagenum="31"/>ABC circum&longs;cripta ad AE cylindrum: vtraque autem <lb/>circum&longs;criptarum figurarum excedit &longs;ibi in&longs;criptam mino<lb/>ri &longs;pacio quantacumque magnitudine propo&longs;ita, vt igitur <lb/>triangulum ABC, ad parallelogrammum AE, ita erit co<lb/>noides ABC, ad cylindrum AE. </s>

<s>Similiter o&longs;tenderemus reliqua parallelogramma, quæ &longs;unt <lb/><figure id="id.043.01.117.1.jpg" xlink:href="043/01/117/1.jpg"/><lb/>circa <expan abbr="triãgulum">triangulum</expan> ABC e&longs;se cum reliquis cylindris, qui &longs;unt <lb/>circa conoides ABC bina &longs;umpta prout inter &longs;e re&longs;pon<lb/>dent in eadem proportione; &longs;emper igitur componendo, & <lb/>ex æquali erit vt tota figura triangulo ABC circum&longs;cripta <lb/>ad parallelogrammum AF, ita figura conoidi circum&longs;cri<lb/>pta ad AF cylindrum: &longs;ed vt parallelogrammum AF, ad <lb/>parallelogrammum AE, ita e&longs;t cylindrus AF ad cylindrum <lb/>AE, propter æqualitatem omnifariam &longs;umptarum altitu<lb/>dinum; ex æquali igitur erit vt figura triangulo ABC cir<lb/>cum&longs;cripta ad parallelogrammum AE, ita figura conoidi <pb xlink:href="043/01/118.jpg" pagenum="31"/>ABC circum&longs;cripta ad AE cylindrum: vtraque autem <lb/>circum&longs;criptarum figurarum excedit &longs;ibi in&longs;criptam mino<lb/>ri &longs;pacio quantacumque magnitudine propo&longs;ita, vt igitur <lb/>triangulum ABC, ad parallelogrammum AE, ita erit co<lb/>noides ABC, ad cylindrum AE. </s>

<s>Sed triangulum ABC <lb/>e&longs;t parallelogrammi AE dimidium; igitur conoides ABC <lb/>e&longs;t cylindro AE dimidium: &longs;ed cylindrus AE e&longs;t coni <lb/>ABC, triplum: igitur conoides ABC, erit coni ABC <lb/>&longs;e&longs;quialterum. </s>

Line 1532

<s>Dico pri&longs;matis ABCDEF, cen <lb/>trum grauitatis e&longs;&longs;e K. </s>

<s>Per pun <lb/>ctum enim H ducatur NO ip<lb/>&longs;i AE, vel CD parallela, quæ <lb/>ip&longs;as AC, ED, &longs;ecabit <expan abbr="bifariã">bifariam</expan>: <lb/>iunctisque BN, FO, ducatur per <lb/>punctum <emph type="italics"/>K<emph.end type="italics"/>, ip&longs;i FB, vel NO <lb/><figure id="fig76"/><lb/>parallela LM. </s>

<s>Quoniam igitur e&longs;t vt HK ad KG, ita <lb/>NL ad LB, & OM ad MF, erit NL, ip&longs;ius LB, & OM <pb pagenum="32"/>ip&longs;ius MF dimidia: &longs;ed & rectæ BN, FO, triangulorum <lb/>ba&longs;es AC, ED, bifariam &longs;e<lb/>cant; erunt igitur puncta L, M, <lb/>centra grauitatis triangulorum <lb/>ABC, DEF, oppo&longs;itorum. <lb/></s>

<s>Quoniam igitur e&longs;t vt HK ad KG, ita <lb/>NL ad LB, & OM ad MF, erit NL, ip&longs;ius LB, & OM <pb xlink:href="043/01/119.jpg" pagenum="32"/>ip&longs;ius MF dimidia: &longs;ed & rectæ BN, FO, triangulorum <lb/>ba&longs;es AC, ED, bifariam &longs;e<lb/>cant; erunt igitur puncta L, M, <lb/>centra grauitatis triangulorum <lb/>ABC, DEF, oppo&longs;itorum. <lb/></s>

<s>Pri&longs;matis igitur ABCDEF <lb/>axis erit LM: quare in eius bi<lb/>partiti &longs;ectione pri&longs;matis ABC <lb/>DEF centrum grauitatis: &longs;ectus <lb/>autem e&longs;t axis LM bifariam in <lb/>puncto K; nam ob parallelogram <lb/>ma e&longs;t vt NH ad HO, ita LK <lb/>ad KM; pri&longs;matis igitur ABC <lb/>DEF, centrum grauitatis erit <emph type="italics"/>K.<emph.end type="italics"/><lb/>Quod demon&longs;trandum erat. </s><figure/>

<s><emph type="italics"/>PROPOSITIO XX.<emph.end type="italics"/></s>

Line 1544

<s>Sit pri&longs;ma ABCDEFGH, cuius ba&longs;is trapezium <lb/>ABCD, habens duo latera AD, BC, inter &longs;e paralle<lb/>la, &longs;itque eorum AD maius: parallela igitur erunt inter &longs;e <lb/>duo parallelogramma BG, AH. </s>

<s>Sit parallelogrammi AH <lb/>centrum K, & BG parallelogrammi centrum L, iuncta-<pb pagenum="33"/>que LK, fiat vt dupla ip&longs;ius AD vna cum BC ad du<lb/>plam ip&longs;ius BC vna cum AD, ita LR ad RK. </s>

<s>Sit parallelogrammi AH <lb/>centrum K, & BG parallelogrammi centrum L, iuncta-<pb xlink:href="043/01/120.jpg" pagenum="33"/>que LK, fiat vt dupla ip&longs;ius AD vna cum BC ad du<lb/>plam ip&longs;ius BC vna cum AD, ita LR ad RK. </s>

<s>Dico <lb/>pri&longs;matis AG centrum grauitatis e&longs;se R. </s>

<s>Ducantur enim <lb/>per puncta L, K lateribus pri&longs;matis, atque ideo inter &longs;e <lb/>parallelæ MN, OP, quæ <lb/>ob centra K, L, &longs;ecabunt <lb/>oppo&longs;ita parallelogrammo<lb/>rum latera bifariam, eas <lb/>&longs;ectiones connectant MO, <lb/>NP, ip&longs;ique MN, vel <lb/>OP, parallela ducatur Q <lb/>RS. </s>

<s>Quoniam igitur e&longs;t <lb/>vt LR ad R<emph type="italics"/>K<emph.end type="italics"/>, hoc e&longs;t vt <lb/>dupla ip&longs;ius AD vna cum <lb/>BC ad duplam ip&longs;ius BC <lb/>vna cum AD, ita OQ ad <lb/>QM, & recta MO bifa<lb/><figure id="fig77"/><lb/>riam &longs;ecat AC trapezij latera parallela, punctum Q, AC <lb/>trapezij centrum grauitatis; &longs;imiliter & punctum S erit EG, <lb/>trapezij centrum grauitatis: pri&longs;matis igitur AG axis erit <lb/>QS, & centrum grauitatis R, quod e&longs;t in medio axis. <lb/></s>

<s>Quoniam igitur e&longs;t <lb/>vt LR ad R<emph type="italics"/>K<emph.end type="italics"/>, hoc e&longs;t vt <lb/>dupla ip&longs;ius AD vna cum <lb/>BC ad duplam ip&longs;ius BC <lb/>vna cum AD, ita OQ ad <lb/>QM, & recta MO bifa<lb/><figure id="id.043.01.120.1.jpg" xlink:href="043/01/120/1.jpg"/><lb/>riam &longs;ecat AC trapezij latera parallela, punctum Q, AC <lb/>trapezij centrum grauitatis; &longs;imiliter & punctum S erit EG, <lb/>trapezij centrum grauitatis: pri&longs;matis igitur AG axis erit <lb/>QS, & centrum grauitatis R, quod e&longs;t in medio axis. <lb/></s>

<s>Omnis igitur pri&longs;matis ba&longs;im habentis trapezium, &c. <lb/></s>

Line 1558

<s><emph type="italics"/>PROPOSITIO XXI.<emph.end type="italics"/></s>

<s>Si à quolibet prædicto pri&longs;mate duo pri&longs;mata <lb/>be&longs;es habentia triangulas &longs;int ita ab&longs;ci&longs;&longs;a, vt pa<lb/>rallelepipedum relinquant ba&longs;im habens minus <lb/>parallelogrammorum inter &longs;e parallelorum præ<lb/>dicti pri&longs;matis, maioris autem partes æqualia pa<lb/>rallelogramma ip&longs;um parallelepipedum relin<pb pagenum="34"/>quat, centrum grauitatis vtriu&longs;que ab&longs;ci&longs;si pri&longs;<lb/>matis tamquam vnius magnitudinis rectam line<lb/>lam, quæ prædicti pri&longs;matis parallelorum paral <lb/>lelogrammorum centra iungit, ita diuidit, vt <lb/>pars, quæ minus parallelogrammum attingit &longs;it <lb/>dupla reliquæ. </s>

<s>Si à quolibet prædicto pri&longs;mate duo pri&longs;mata <lb/>be&longs;es habentia triangulas &longs;int ita ab&longs;ci&longs;&longs;a, vt pa<lb/>rallelepipedum relinquant ba&longs;im habens minus <lb/>parallelogrammorum inter &longs;e parallelorum præ<lb/>dicti pri&longs;matis, maioris autem partes æqualia pa<lb/>rallelogramma ip&longs;um parallelepipedum relin<pb xlink:href="043/01/121.jpg" pagenum="34"/>quat, centrum grauitatis vtriu&longs;que ab&longs;ci&longs;si pri&longs;<lb/>matis tamquam vnius magnitudinis rectam line<lb/>lam, quæ prædicti pri&longs;matis parallelorum paral <lb/>lelogrammorum centra iungit, ita diuidit, vt <lb/>pars, quæ minus parallelogrammum attingit &longs;it <lb/>dupla reliquæ. </s>

<s>Sit pri&longs;ma ABCDEFGH, cuius ba&longs;es oppo&longs;itæ tra<lb/>pezia ADHE, BCGF. </s>

Line 1572

<s>Po&longs;ito <lb/>autem centro K <lb/>parallelogrammi <lb/>AC, & L, paral <lb/>lelogrammi EG, <lb/>iunctaque KL, <lb/>ponatur KM, du <lb/>pla ip&longs;ius ML. <lb/></s>

<s>Dico <expan abbr="duor&utilde;">duorum</expan> pri&longs;<lb/>matum BER, <lb/>CVH, &longs;imul cen <lb/>trum grauitatis <lb/><figure id="fig78"/><lb/>e&longs;se M. </s>

<s>Dico <expan abbr="duor&utilde;">duorum</expan> pri&longs;<lb/>matum BER, <lb/>CVH, &longs;imul cen <lb/>trum grauitatis <lb/><figure id="id.043.01.121.1.jpg" xlink:href="043/01/121/1.jpg"/><lb/>e&longs;se M. </s>

<s>Sectis enim AB, CD, bifariam in punctis P, Q, <lb/>&longs;umpti&longs;que parallelogrammorum ES, VG, centris N, O, <lb/>iungantur PN, QO, & po&longs;ita PX dupla ip&longs;ius XN, & QZ <lb/>dupla ip&longs;ius ZO, iungantur rectæ PKQ, XZ, NO. <lb/></s>

<s>Quoniam igitur in quadrilatero PQON, recta XZ, pa<lb/>rallela e&longs;t vtrilibet ip&longs;arum PQ, NO, &longs;ecat ijs parallelis <lb/>interceptas in ea&longs;dem rationes; recta igitur XT per pun-<pb pagenum="35"/>ctum M tran&longs;ibit. </s>

<s>Sed quia PK e&longs;t æqualis KQ, & NL <lb/>ip&longs;i LO, etiam XM æqualis erit ip&longs;i MZ ob parallelas; <lb/>cum igitur pri&longs;matum BER, CVH centra grauitatis &longs;int <lb/>X, Z; erit vtriu&longs;que pri&longs;matis prædicti &longs;imul centrum gra<lb/>uitatis M. </s>

Line 1584

<s><emph type="italics"/>PROPOSITIO XXII.<emph.end type="italics"/></s>

<s>Si &longs;int duæ pyramides æquales, & æque altæ, <lb/>ba&longs;es habentes in eodem plano, quarum vertices <lb/>recta linea connectens cum ea, quæ ba&longs;ium centra <lb/>grauitatis iungit &longs;it in eodem plano; earum cen<lb/>trum grauitatis tamquam vnius magnitudinis re<lb/>ctam lineam, quæ inter vertices, & centra ba&longs;ium <lb/>interiectas bifariam &longs;ecat, itadiuidit, vt pars &longs;u<lb/>perior &longs;it inferioris tripla. </s><figure/>

<s>Sint duæ <lb/>pyramides æ<lb/>quales, & æ<lb/>que altæ, qua<lb/>rum ba&longs;es in <lb/>eodem plano <lb/>AC, DB, ver <lb/>tices autem <lb/>G, H, & ba<lb/>&longs;ium <expan abbr="c&etilde;tra">centra</expan> E, <lb/>F, iunctæque <lb/>EF, GH, quas <lb/>bifariam &longs;ecet recta KL, huius autem pars quarta &longs;it LM. <lb/></s>

<s>Dico vtriu&longs;que pyramidis GAC, HDB, &longs;imul centrum <lb/>grauitatis e&longs;&longs;e M. </s>

<s>Iunctis enim GE, HF, &longs;umantur ea<pb pagenum="36"/>rum quartæ partes EN, FO, & iungatur NO. </s>

<s>Iunctis enim GE, HF, &longs;umantur ea<pb xlink:href="043/01/123.jpg" pagenum="36"/>rum quartæ partes EN, FO, & iungatur NO. </s>

<s>Quoniam <lb/>igitur propter æqualitatem altitudinum, & quia EF, GH, <lb/>&longs;unt in eodem plano, &longs;unt EF, GH, inter &longs;e parallelæ, & <lb/>vt GN ad NE, ita e&longs;t HO ad OF; erit NO ip&longs;i E Fivel <lb/>GH, paralle<lb/>la, quas KL <lb/>bifariam &longs;ecat: <lb/>igitur & ip&longs;am <lb/>NO &longs;ecabit bi <lb/>fariam, iungit <lb/>autem recta <lb/>NO centra <lb/>grauitatis <expan abbr="py-ramid&utilde;">py<lb/>ramidum</expan> æqua<lb/>lium GAC, <lb/>HDB, vtriu&longs;<lb/><figure id="fig79"/><lb/>que ergo pyramidis &longs;imul centrum grauitatis erit in com<lb/>muni &longs;ectione duarum linearum KL, NO, &longs;ed recta NO, <lb/>&longs;ecans &longs;imiliter ip&longs;as GE, KL, HF, ip&longs;am KL, &longs;ecabit <lb/>in puncto M; punctum igitur M, erit prædictarum pyrami<lb/>dum centrum grauitatis. </s>

<s>Quoniam <lb/>igitur propter æqualitatem altitudinum, & quia EF, GH, <lb/>&longs;unt in eodem plano, &longs;unt EF, GH, inter &longs;e parallelæ, & <lb/>vt GN ad NE, ita e&longs;t HO ad OF; erit NO ip&longs;i E Fivel <lb/>GH, paralle<lb/>la, quas KL <lb/>bifariam &longs;ecat: <lb/>igitur & ip&longs;am <lb/>NO &longs;ecabit bi <lb/>fariam, iungit <lb/>autem recta <lb/>NO centra <lb/>grauitatis <expan abbr="py-ramid&utilde;">py<lb/>ramidum</expan> æqua<lb/>lium GAC, <lb/>HDB, vtriu&longs;<lb/><figure id="id.043.01.123.1.jpg" xlink:href="043/01/123/1.jpg"/><lb/>que ergo pyramidis &longs;imul centrum grauitatis erit in com<lb/>muni &longs;ectione duarum linearum KL, NO, &longs;ed recta NO, <lb/>&longs;ecans &longs;imiliter ip&longs;as GE, KL, HF, ip&longs;am KL, &longs;ecabit <lb/>in puncto M; punctum igitur M, erit prædictarum pyrami<lb/>dum centrum grauitatis. </s>

<s>Quod demon&longs;trandum erat. </s>

<s><emph type="italics"/>PROPOSITIO XXIII.<emph.end type="italics"/></s>

<s>Omnis fru&longs;ti pyramidis ba&longs;im habentis paral<lb/>lelogrammum centrum grauitatis maiori ba&longs;i e&longs;t <lb/>propinquius, quam punctum illud, in quo axis &longs;ic <lb/>diuiditur, vt pars minorem ba&longs;im attingens &longs;it ad <lb/>reliquam vt dupla cuiu&longs;uis laterum maioris ba&longs;is <lb/>vna cum latere minoris &longs;ibi re&longs;pondente, ad <expan abbr="duplã">duplam</expan> <lb/>dicti lateris minoris ba&longs;is vna cum maioris &longs;ibi <lb/>re&longs;pondente. </s><pb pagenum="37"/>

<s>Sit pyramidis, cuius ba&longs;is parallelogrammum EFGH, <lb/>fru&longs;tum ABCDEFGH, <expan abbr="eiu&longs;q;">eiu&longs;que</expan> axis KL, quo &longs;ecto in pun <lb/>cto <foreign lang="greek">a</foreign> ita vt K <foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign> L, &longs;it vt laterum homologorum AD <lb/>EH, dupla ip&longs;ius EH vna cum AD ad duplam ip&longs;ius <lb/>AD vna cum EH, & fru&longs;ti ABCDEFGH &longs;it centrum <lb/>grauitatis <foreign lang="greek"><gap/></foreign> nempe in axe KL. </s>

<s>Dico punctum <foreign lang="greek"><gap/></foreign>, cadere <lb/>infra punctum <foreign lang="greek">a. </foreign></s>

<s>A punctis enim A,B,C,D, ducantur <lb/><figure id="fig80"/><lb/>ad maiorem ba&longs;im axi KL, parallelæ AN, BO, CR, DS, <lb/>& parallelepipedum ABCDNORS compleatur, & <lb/>productis ba&longs;is NO lateribus, de&longs;criptæ &longs;int quatuor py<lb/>ramides AEMNZ, BOPFY, CGXRQ, DHVST, <lb/>quarum ba&longs;es erunt parallelogramma circa diametrum <lb/>æqualia, atque &longs;imilia: & quatuor pri&longs;mata triangulas ba<lb/>&longs;es habentia, quorum binorum ex aduer&longs;o inter &longs;e re&longs;pon-<pb pagenum="38"/>dentium parallelogramma in plano EG exi&longs;tentia erunt <lb/>inter &longs;e æqualia, atque &longs;imilia, &longs;cilicet MS ip&longs;i OQ, & <lb/>ZO, ip&longs;is RV: &longs;itque axis KL pars tertia L <foreign lang="greek">b</foreign>, quarta <lb/>autem L <foreign lang="greek">d. </foreign></s>

<s>A punctis enim A,B,C,D, ducantur <lb/><figure id="id.043.01.124.1.jpg" xlink:href="043/01/124/1.jpg"/><lb/>ad maiorem ba&longs;im axi KL, parallelæ AN, BO, CR, DS, <lb/>& parallelepipedum ABCDNORS compleatur, & <lb/>productis ba&longs;is NO lateribus, de&longs;criptæ &longs;int quatuor py<lb/>ramides AEMNZ, BOPFY, CGXRQ, DHVST, <lb/>quarum ba&longs;es erunt parallelogramma circa diametrum <lb/>æqualia, atque &longs;imilia: & quatuor pri&longs;mata triangulas ba<lb/>&longs;es habentia, quorum binorum ex aduer&longs;o inter &longs;e re&longs;pon-<pb xlink:href="043/01/125.jpg" pagenum="38"/>dentium parallelogramma in plano EG exi&longs;tentia erunt <lb/>inter &longs;e æqualia, atque &longs;imilia, &longs;cilicet MS ip&longs;i OQ, & <lb/>ZO, ip&longs;is RV: &longs;itque axis KL pars tertia L <foreign lang="greek">b</foreign>, quarta <lb/>autem L <foreign lang="greek">d. </foreign></s>

<s>Quoniam ìgitur ex &longs;upra demon&longs;tratis pri&longs;<lb/>matis ABCDTMPQ e&longs;t centrum grauitatis <foreign lang="greek">a</foreign>; duo<lb/>rum autem pri&longs;matum oppo&longs;itorum ABYONZ, CDS <lb/>RXV, centrum grauitatis <foreign lang="greek">b</foreign>, erit reliqui ex fru&longs;to AB <lb/><figure id="fig81"/><lb/>CDEFGH demptis quatuor prædictis pyramidibus in <lb/><foreign lang="greek">a b</foreign> centrum grauitatis, quod &longs;it <foreign lang="greek">g. </foreign></s>

<s>Quoniam ìgitur ex &longs;upra demon&longs;tratis pri&longs;<lb/>matis ABCDTMPQ e&longs;t centrum grauitatis <foreign lang="greek">a</foreign>; duo<lb/>rum autem pri&longs;matum oppo&longs;itorum ABYONZ, CDS <lb/>RXV, centrum grauitatis <foreign lang="greek">b</foreign>, erit reliqui ex fru&longs;to AB <lb/><figure id="id.043.01.125.1.jpg" xlink:href="043/01/125/1.jpg"/><lb/>CDEFGH demptis quatuor prædictis pyramidibus in <lb/><foreign lang="greek">a b</foreign> centrum grauitatis, quod &longs;it <foreign lang="greek">g. </foreign></s>

<s>Nam ex primo li<lb/>bro con&longs;tat punctum <foreign lang="greek">a</foreign> cadere &longs;upra punctum <foreign lang="greek">b</foreign>, &longs;i com<lb/>pleatur trapezium ACGE, cuius diameter erit KL. </s>

<s>Sed <lb/>earum quatuor pyramidum e&longs;t centrum grauitatis <foreign lang="greek">d. </foreign></s>

<s>Si <lb/>enim ba&longs;ium, quibus binæ oppo&longs;itæ pyramides in&longs;i&longs;tunt <lb/>centra grauitatis, & bini oppo&longs;iti vertices &longs;ingulis rectis li-<pb pagenum="39"/>neis connectantur, erunt binæ connectentes parallelæ, & <lb/>ab axe <emph type="italics"/>K<emph.end type="italics"/> L bifariam &longs;ecabuntur, vt figuræ de&longs;criptio ina<lb/>nife&longs;tat. </s>

<s>Si <lb/>enim ba&longs;ium, quibus binæ oppo&longs;itæ pyramides in&longs;i&longs;tunt <lb/>centra grauitatis, & bini oppo&longs;iti vertices &longs;ingulis rectis li-<pb xlink:href="043/01/126.jpg" pagenum="39"/>neis connectantur, erunt binæ connectentes parallelæ, & <lb/>ab axe <emph type="italics"/>K<emph.end type="italics"/> L bifariam &longs;ecabuntur, vt figuræ de&longs;criptio ina<lb/>nife&longs;tat. </s>

<s>Totius igitur fru&longs;ti ABCDEFGH, centrum <lb/>grauitatis <foreign lang="greek"><gap/></foreign> in linea <foreign lang="greek">g d</foreign> cadet: &longs;ed punctum <foreign lang="greek">g</foreign> cadit infra <lb/>punctum <foreign lang="greek">a</foreign>, multo ergo inferius, & ba&longs;i EG propinquius <lb/>punctum <foreign lang="greek"><gap/></foreign> quam punctum <foreign lang="greek">a. </foreign></s>

Line 1626

<s><emph type="italics"/>PROPOSITIO XXV.<emph.end type="italics"/></s>

<s>Si &longs;int quotcumque magnitudines, & aliæ illis <lb/>multitudine æquales, binæque &longs;umptæ in eadem <lb/>proportione, quæ commune habeant centrum gra<lb/>uitatis, centra autem grauitatis omnium &longs;int in <lb/>eadem recta linea; primæ & &longs;ecundæ tanquam <pb pagenum="40"/>duæ magnitudines commune habebunt centrum <lb/>grauitatis. </s>

<s>Si &longs;int quotcumque magnitudines, & aliæ illis <lb/>multitudine æquales, binæque &longs;umptæ in eadem <lb/>proportione, quæ commune habeant centrum gra<lb/>uitatis, centra autem grauitatis omnium &longs;int in <lb/>eadem recta linea; primæ & &longs;ecundæ tanquam <pb xlink:href="043/01/127.jpg" pagenum="40"/>duæ magnitudines commune habebunt centrum <lb/>grauitatis. </s>

<s>Sit recta linea AB, & quotcumque magnitudines <lb/>FGH, & totidem KLM, binæ in eadem proportione: <lb/>nimirum vt F ad G ita K ad L: & vt G ad H ita L ad <lb/>M. in recta autem AB, &longs;int communia centra grauitatis, <lb/>C duarum FK, & D duarum GL: & E duarum HM. </s>

Line 1634

<s>Dico & omnium &longs;ecundarum &longs;i<lb/>mul centrum grauitatis e&longs;se O. </s>

<s>Duarum enim FG &longs;i<lb/><figure id="fig82"/><lb/>mul &longs;it centrum grauitatis N. </s>

<s>Duarum enim FG &longs;i<lb/><figure id="id.043.01.127.1.jpg" xlink:href="043/01/127/1.jpg"/><lb/>mul &longs;it centrum grauitatis N. </s>

<s>Vtigitur e&longs;t F ad G, hoc <lb/>e&longs;t, vt K ad L, ita erit DN, ad NC. punctum igitur N <lb/>e&longs;t centrum grauitatis duarum magnitudinum KL &longs;imul. <lb/></s>

Line 1644

<s><emph type="italics"/>PROPOSITIO XXVI.<emph.end type="italics"/></s>

<s>Si &longs;int quotcumque magnitudines, & aliæ ip<lb/>&longs;is multitudine æquales primarum, ex quibus cen <lb/>tra grauitatis in eadem recta linea di&longs;po&longs;ita &longs;int <lb/>alternatim ad centra grauitatis &longs;ecundarum, qua-<pb pagenum="41"/>rum magnitudinum binæ eodem ordine, qui &longs;u<lb/>mitur ab eodem prædictæ lineæ termino vnain <lb/>primis, & alterain &longs;ecundis inter &longs;e &longs;int æquales; <lb/>omnium primarum &longs;imul, ex quibus primæ cen<lb/>trum grauitatis propinquius e&longs;t prædicto lineæ <lb/>termino quàm primæ &longs;ecundarum, propinquius <lb/>erit prædicto lineæ termino quàm omnium &longs;ecun<lb/>darum &longs;imul centrum grauitatis. </s>

<s>Si &longs;int quotcumque magnitudines, & aliæ ip<lb/>&longs;is multitudine æquales primarum, ex quibus cen <lb/>tra grauitatis in eadem recta linea di&longs;po&longs;ita &longs;int <lb/>alternatim ad centra grauitatis &longs;ecundarum, qua-<pb xlink:href="043/01/128.jpg" pagenum="41"/>rum magnitudinum binæ eodem ordine, qui &longs;u<lb/>mitur ab eodem prædictæ lineæ termino vnain <lb/>primis, & alterain &longs;ecundis inter &longs;e &longs;int æquales; <lb/>omnium primarum &longs;imul, ex quibus primæ cen<lb/>trum grauitatis propinquius e&longs;t prædicto lineæ <lb/>termino quàm primæ &longs;ecundarum, propinquius <lb/>erit prædicto lineæ termino quàm omnium &longs;ecun<lb/>darum &longs;imul centrum grauitatis. </s>

<s>Sint quotcumque magnitudines ABC primæ, & toti<lb/>dem &longs;ecundæ DEF, quarum centra grauitatis in recta <lb/>linea TV, primarum quidem G ip&longs;ius A proximum om<lb/><figure id="fig83"/><lb/>nium termino T, à quo &longs;umitur ordo. </s>

<s>Sint quotcumque magnitudines ABC primæ, & toti<lb/>dem &longs;ecundæ DEF, quarum centra grauitatis in recta <lb/>linea TV, primarum quidem G ip&longs;ius A proximum om<lb/><figure id="id.043.01.128.1.jpg" xlink:href="043/01/128/1.jpg"/><lb/>nium termino T, à quo &longs;umitur ordo. </s>

<s>Deinde H ip&longs;ius B, <lb/>& <emph type="italics"/>K<emph.end type="italics"/>, ip&longs;ius C, di&longs;po&longs;ita &longs;int alternatim ad centra &longs;ecun<lb/>darum; videlicet vt centrum grauitatis L, ip&longs;ius D cadat <lb/>inter centra G, H, & M ip&longs;ius E inter centra H, K: & N <lb/>inter puncta <emph type="italics"/>K<emph.end type="italics"/>, V: &longs;int autem æquales binæ AD, BE, <lb/>CF: & omnium ABC &longs;imul centrum grauitatis P, & om<lb/>nium DEF &longs;imul centrum grauitatis O. </s>

Line 1656

<s>Quoniam igitur Q e&longs;t <lb/>centrum grauitatis duarum magnitudinum DE &longs;imal; erit <lb/>vt D ad E, hoc e&longs;t ad B, ita MQ, ad QL: hoc e&longs;t HS, <lb/>ad SL. & componendo, vt ML, ad LQ, ita HL, ad <lb/>LS; & permutando, vt ML ad LH, ita LQ ad LS: <lb/>&longs;ed ML e&longs;t maior quàm LH; ergo & LQ erit maior <lb/>quàm LS. </s>

<s>Eadem ratione quoniam S e&longs;t centrum gra<pb pagenum="42"/>uitatis duarum DB: & R duarum AB: & AD &longs;unt æ<lb/>quales; erit RH maior quàm SH: &longs;ed quia LQ erat ma<lb/>ior quàm LS, e&longs;t & SH maior quàm QH; multo igitur <lb/>maior RH erit quàm QH: atque ideo punctum R pro<lb/>pinquius termino T, quàm punctum <expan abbr="q.">que</expan> Rur&longs;us quo<lb/>niam tota magnitudo AB e&longs;t æqualis toti DE, & C æ<lb/>qualis F; erunt duæ primæ AB, & C, & totidem &longs;ecun<lb/>dæ DE, & F, quarum vnius po&longs;teriorum DE cen<lb/>trum grauitatis Q cadit inter R, K centra grauitatis <lb/>duarum priorum AB, & C, & reliquæ priorum C cen<lb/>trum grauitatis K cadit inter Q, N, duarum po&longs;terio<lb/>rum DE, & F centra grauitatis; erunt vt antea quatuor <lb/>magnitudines binæ proximæ æquales, &longs;cilicet AB, ip&longs;i <lb/><figure id="fig84"/><lb/>DE: & C ip&longs;i F, centra grauitatis habentes di&longs;pofita <lb/>alternatim in eadem recta TV. </s>

<s>Eadem ratione quoniam S e&longs;t centrum gra<pb xlink:href="043/01/129.jpg" pagenum="42"/>uitatis duarum DB: & R duarum AB: & AD &longs;unt æ<lb/>quales; erit RH maior quàm SH: &longs;ed quia LQ erat ma<lb/>ior quàm LS, e&longs;t & SH maior quàm QH; multo igitur <lb/>maior RH erit quàm QH: atque ideo punctum R pro<lb/>pinquius termino T, quàm punctum <expan abbr="q.">que</expan> Rur&longs;us quo<lb/>niam tota magnitudo AB e&longs;t æqualis toti DE, & C æ<lb/>qualis F; erunt duæ primæ AB, & C, & totidem &longs;ecun<lb/>dæ DE, & F, quarum vnius po&longs;teriorum DE cen<lb/>trum grauitatis Q cadit inter R, K centra grauitatis <lb/>duarum priorum AB, & C, & reliquæ priorum C cen<lb/>trum grauitatis K cadit inter Q, N, duarum po&longs;terio<lb/>rum DE, & F centra grauitatis; erunt vt antea quatuor <lb/>magnitudines binæ proximæ æquales, &longs;cilicet AB, ip&longs;i <lb/><figure id="id.043.01.129.1.jpg" xlink:href="043/01/129/1.jpg"/><lb/>DE: & C ip&longs;i F, centra grauitatis habentes di&longs;pofita <lb/>alternatim in eadem recta TV. </s>

<s>Cum igitur primæ prio<lb/>rum AB, centrum grauitatis R &longs;it termino T propin<lb/>quius quàm Q centrum grauitatis primæ po&longs;teriorum, <lb/>quæ e&longs;t tota DE; &longs;imiliter vt ante totius magnitudinis <lb/>ABC centrum grauitatis P erit termino T propinquius <lb/>quàm totius DEF centrum grauitatis O. </s>

<s>Non aliter <lb/>o&longs;tenderemus, quotcumque plures magnitudines, quales <lb/>& quemadmodum diximus ad rectam TV, di&longs;po&longs;itæ <lb/>proponerentur, &longs;emper centrum grauitatis omnium prio<lb/>rum &longs;imul termino T propinquius cadere, quàm omnium <lb/>po&longs;teriorum &longs;imul centrum grauitatis. </s>

<s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s><pb pagenum="43"/>

<s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s><pb xlink:href="043/01/130.jpg" pagenum="43"/>

<s><emph type="italics"/>PROPOSITIO XXVII.<emph.end type="italics"/></s>

Line 1670

<s>Sint quotcumque magnitudines GHI, & totidem <lb/>LMN. </s>

<s>Sitque maior proportio G ad H, quàm L ad M: & <lb/>H ad I, maior quàm M ad N: in recta autem AB &longs;int <lb/>communia centra grauitatis, C duarum magnitudinum <lb/>GL, & D duarum HM, & E duarum IN. omnium <lb/><figure id="fig85"/><lb/>autem primarum GHI &longs;imul &longs;it centrum grauitatis K: at <lb/>&longs;ecundarum omnium LMN centrum grauitatis R. </s>

<s>Sitque maior proportio G ad H, quàm L ad M: & <lb/>H ad I, maior quàm M ad N: in recta autem AB &longs;int <lb/>communia centra grauitatis, C duarum magnitudinum <lb/>GL, & D duarum HM, & E duarum IN. omnium <lb/><figure id="id.043.01.130.1.jpg" xlink:href="043/01/130/1.jpg"/><lb/>autem primarum GHI &longs;imul &longs;it centrum grauitatis K: at <lb/>&longs;ecundarum omnium LMN centrum grauitatis R. </s>

<s>Di<lb/>co centrum K cadere termino A propinquius quàm cen <lb/>trum R. </s>

<s>Maior igitur proportio erit DP <pb pagenum="44"/>ad PC, quàm DQ ad QC: & componendo, maior DC <lb/>ad CP, quàm DC ad CQ: minor igitur CP erit quàm <lb/>CQ: quare DP maior quàm <expan abbr="Dq.">Dque</expan> & communi addita <lb/>ED, erit EP maior quàm <expan abbr="Eq.">Eque</expan> Et quoniam <emph type="italics"/>K<emph.end type="italics"/> e&longs;t cen<lb/>trum grauitatis omnium GHI &longs;imul, & ip&longs;ius GH e&longs;t cen <lb/>trum grauitatis P, & reliquæ magnitudinis I, centrum <lb/>grauitatis E; erit vt GH ad I, ita EK ad KP. eadem <lb/>ratione vt vtraque LM ad N, ita erit ER ad <expan abbr="Rq.">Rque</expan> Rur<lb/><figure id="fig86"/><lb/>&longs;us, quia maior e&longs;t proportio G ad H, quàm L ad M, erit <lb/>componendo, maior proportio GH ad H, quàm LM ad <lb/>M: &longs;ed maior e&longs;t proportio H ad K, quàm M ad N; ex <lb/>æquali igitur, maior erit proportio GH ad I, quàm LM <lb/>ad N, hoc e&longs;t EK ad KP, quàm ER ad <expan abbr="Rq.">Rque</expan> Multo <lb/>ergo maior proportio EK ad KP, quàm ER ad RP: & <lb/>componendo maior proportio EP ad PK quàm EP ad <lb/>PR; minor igitur PK erit quàm PR, at que ideo centrum <lb/>K propinquius termino A quàm centrum R. </s>

<s>Maior igitur proportio erit DP <pb xlink:href="043/01/131.jpg" pagenum="44"/>ad PC, quàm DQ ad QC: & componendo, maior DC <lb/>ad CP, quàm DC ad CQ: minor igitur CP erit quàm <lb/>CQ: quare DP maior quàm <expan abbr="Dq.">Dque</expan> & communi addita <lb/>ED, erit EP maior quàm <expan abbr="Eq.">Eque</expan> Et quoniam <emph type="italics"/>K<emph.end type="italics"/> e&longs;t cen<lb/>trum grauitatis omnium GHI &longs;imul, & ip&longs;ius GH e&longs;t cen <lb/>trum grauitatis P, & reliquæ magnitudinis I, centrum <lb/>grauitatis E; erit vt GH ad I, ita EK ad KP. eadem <lb/>ratione vt vtraque LM ad N, ita erit ER ad <expan abbr="Rq.">Rque</expan> Rur<lb/><figure id="id.043.01.131.1.jpg" xlink:href="043/01/131/1.jpg"/><lb/>&longs;us, quia maior e&longs;t proportio G ad H, quàm L ad M, erit <lb/>componendo, maior proportio GH ad H, quàm LM ad <lb/>M: &longs;ed maior e&longs;t proportio H ad K, quàm M ad N; ex <lb/>æquali igitur, maior erit proportio GH ad I, quàm LM <lb/>ad N, hoc e&longs;t EK ad KP, quàm ER ad <expan abbr="Rq.">Rque</expan> Multo <lb/>ergo maior proportio EK ad KP, quàm ER ad RP: & <lb/>componendo maior proportio EP ad PK quàm EP ad <lb/>PR; minor igitur PK erit quàm PR, at que ideo centrum <lb/>K propinquius termino A quàm centrum R. </s>

<s>Quod de<lb/>mon&longs;trandum erat. </s>

<s><emph type="italics"/>PROPOSITIO XXVIII.<emph.end type="italics"/></s>

<s>Si &longs;int quotcumque magnitudines, & aliæ ip&longs;is <lb/>multitudine æquales, quarum omnium centra <lb/>grauitatis &longs;int in eadem recta linea, & centra pri<lb/>marum ad centra &longs;ecundarum di&longs;po&longs;ita &longs;int alter<lb/>natim: &longs;it autem maior proportio primæ ad &longs;ecun-<pb pagenum="45"/>dam in primis quàm primæ ad &longs;ecundam in &longs;ecun<lb/>dis: & &longs;ecundæ ad tertiam in primis, maior quàm <lb/>&longs;ecundæ ad tertiam in &longs;e cundis, & &longs;ic deinceps v&longs;<lb/>que ad vltimas; erit omnium primarum &longs;imul cen <lb/>trum grauitatis propinquius prædictæ lineæ ter<lb/>mino à quo &longs;umitur ordo omnium &longs;ecundarum <lb/>centrum grauitatis. </s>

<s>Si &longs;int quotcumque magnitudines, & aliæ ip&longs;is <lb/>multitudine æquales, quarum omnium centra <lb/>grauitatis &longs;int in eadem recta linea, & centra pri<lb/>marum ad centra &longs;ecundarum di&longs;po&longs;ita &longs;int alter<lb/>natim: &longs;it autem maior proportio primæ ad &longs;ecun-<pb xlink:href="043/01/132.jpg" pagenum="45"/>dam in primis quàm primæ ad &longs;ecundam in &longs;ecun<lb/>dis: & &longs;ecundæ ad tertiam in primis, maior quàm <lb/>&longs;ecundæ ad tertiam in &longs;e cundis, & &longs;ic deinceps v&longs;<lb/>que ad vltimas; erit omnium primarum &longs;imul cen <lb/>trum grauitatis propinquius prædictæ lineæ ter<lb/>mino à quo &longs;umitur ordo omnium &longs;ecundarum <lb/>centrum grauitatis. </s>

<s>Sit quotcumque magnitudines GHI, & totidem LMN <lb/>primarum autem &longs;int centra grauitatis CDE cum &longs;ecun<lb/>darum centris OPQ in eadem recta AB di&longs;po&longs;ita alter<lb/>natim, vt O cadat inter puncta CD, & P inter puncta <lb/>DE, & E inter puncta <expan abbr="Pq.">Pque</expan> &longs;itque maior proportio G <lb/>ad H, quàm L ad M, & H ad I maior quàm M ad N. <lb/>omnium autem primarum GHI &longs;imul &longs;it centrum gra<lb/>uitatis T; at omnium &longs;ecundarum LMN, &longs;imul, cen<lb/><figure id="fig87"/><lb/>trum grauitatis V. </s>

<s>Dico punctum T e&longs;&longs;e termino A <lb/>propinquius quàm punctum V. </s>

Line 1692

<s>In recta igitur AB om<lb/>nium FKX, &longs;imul centrum grauitatis erit termino A, pro<lb/>pinquius quàm omnium LMN &longs;imul centrum grauitatis. <lb/></s>

<s>Sed & omnium GHI, &longs;imul centrum grauitatis in eadem <lb/>recta AB propinquius e&longs;t termino A quàm omnium <lb/>FKX, &longs;imul centrum grauitatis; multo igitur termino A <lb/>propinquius erit omnium GHI &longs;imul quàm omnium <pb pagenum="46"/>LMN, &longs;imul centrum grauitatis. </s>

<s>Sed & omnium GHI, &longs;imul centrum grauitatis in eadem <lb/>recta AB propinquius e&longs;t termino A quàm omnium <lb/>FKX, &longs;imul centrum grauitatis; multo igitur termino A <lb/>propinquius erit omnium GHI &longs;imul quàm omnium <pb xlink:href="043/01/133.jpg" pagenum="46"/>LMN, &longs;imul centrum grauitatis. </s>

<s>Quod demon&longs;tran<lb/>dum erat. </s>

Line 1702

<s>Sed punctum V in linea AE cadat. <lb/></s>

<s>Veligitur S centrum grauitatis duarum magnitudinum L, <lb/>M, &longs;imul cadit in puncto D, &longs;iue in linea DB, vel in li<lb/>nea AD. &longs;i in puncto D, vel in linea DB; centrum gra<lb/>uitatis R duarum magnitudinum GH erit termino A <lb/>propinquius quàm ip&longs;um S, & recta ER maior quàm ES, <lb/><figure id="fig88"/><lb/>Sed cadat punctum S in linea AD. </s>

<s>Veligitur S centrum grauitatis duarum magnitudinum L, <lb/>M, &longs;imul cadit in puncto D, &longs;iue in linea DB, vel in li<lb/>nea AD. &longs;i in puncto D, vel in linea DB; centrum gra<lb/>uitatis R duarum magnitudinum GH erit termino A <lb/>propinquius quàm ip&longs;um S, & recta ER maior quàm ES, <lb/><figure id="id.043.01.133.1.jpg" xlink:href="043/01/133/1.jpg"/><lb/>Sed cadat punctum S in linea AD. </s>

<s>Quoniam igitur ma<lb/>ior e&longs;t proportio G ad H, quàm L ad M: & vt G ad H, <lb/>ita e&longs;t DR ad RG, & vt L ad M, ita PS ad SO, ma<lb/>ior erit proportio DR ad RC, quàm PS ad SO; mul<lb/>to ergo maior DR ad RC, quàm DS ad SO, & multo <lb/>maior quàm DS ad SC, & componendo maior propor<lb/>tio DC ad CR, quàm DC ad CS; erit igitur CR mi<lb/>nor quàm CS, atque adeo RD maior DS, addita igitur <lb/>ED communi, erit ER maior quàm ES. </s>

<s>Rur&longs;us quia <lb/>componendo, & ex æquali maior e&longs;t proportio totius GH <lb/>ad I quàm totius LM ad N, hoc e&longs;t maior longitudinis <lb/>ET ad TR, quàm QV ad VS, & multo maior quàm <pb pagenum="47"/>EV ad VS, erit componendo, maior proportio ER ad <lb/>RT quàm ES ad SV: & per conuer&longs;ionem rationis mi<lb/>nor proportio FR ad ET; quàm ES ad EV, & permu<lb/>tando minor proportio ER ad ES quàm ET ad EV: &longs;ed <lb/>ER maior erat quàm ES, ergo ET maior erit quàm EV: <lb/>& punctum T propinquius termino A, quàm punctum V. <lb/></s>

<s>Rur&longs;us quia <lb/>componendo, & ex æquali maior e&longs;t proportio totius GH <lb/>ad I quàm totius LM ad N, hoc e&longs;t maior longitudinis <lb/>ET ad TR, quàm QV ad VS, & multo maior quàm <pb xlink:href="043/01/134.jpg" pagenum="47"/>EV ad VS, erit componendo, maior proportio ER ad <lb/>RT quàm ES ad SV: & per conuer&longs;ionem rationis mi<lb/>nor proportio FR ad ET; quàm ES ad EV, & permu<lb/>tando minor proportio ER ad ES quàm ET ad EV: &longs;ed <lb/>ER maior erat quàm ES, ergo ET maior erit quàm EV: <lb/>& punctum T propinquius termino A, quàm punctum V. <lb/></s>

<s>Quod demon&longs;trandum erat. </s>

Line 1718

<s>Et data figu<lb/>ra ip&longs;i ABC figuræ circum&longs;cripta compo&longs;ita ex cylindris, <lb/>vel cylindri portionibus, vel parallelogrammis æqualium <lb/>altitudinum EF, GH, AK. </s>

<s>Dico figuræ ABC alteram <lb/>figuram, qualem diximus po&longs;&longs;e circum&longs;cribi, cuius centrum <pb pagenum="48"/>grauitatis, nempe in linea BD, &longs;it propinquius ba&longs;i AC, <lb/>&longs;iue termino D, quàm prædictæ datæ figuræ circum&longs;criptæ <lb/>centrum grauitatis, Omnium enim cylindrorum, vel cy<lb/>lindri portionum, vel parallelogrammorum, ex quibus con<lb/>&longs;tat prædicta data figura circum&longs;cripta &longs;int axes, vel quæ <lb/>oppo&longs;ita latera coniungunt rectæ BL, LM, MD, qui<lb/>bus &longs;ectis bifariam in punctis N, O, P, ac planis per ea <lb/>&longs;iue rectis tran&longs;euntibus ba&longs;i AC parallelis, &longs;ecantibus<lb/>que dictos cylindros, vel cylindri portiones, vel pa<lb/>rallelogramma, compleatur & figuræ ABC circum&longs;cri<lb/>batur altera figura <lb/>vt prior, quæ ob &longs;e<lb/>ctiones factas com<lb/>ponetur ex duplis <lb/>multitudine cylin<lb/>dris, vel cylindri por<lb/>tionibus, vel paralle<lb/>logrammis <17>qualium <lb/>altitudinum, eorum <lb/>ex quibus con&longs;tat da <lb/>ta figura circum&longs;cri<lb/>pta &longs;in<gap/>autem hi cy<lb/>lindri, aut reliqua, <lb/>quæ diximus QR, <lb/><figure id="fig89"/><lb/>ES, TV, GX, ZI, AY. </s>

<s>Dico figuræ ABC alteram <lb/>figuram, qualem diximus po&longs;&longs;e circum&longs;cribi, cuius centrum <pb xlink:href="043/01/135.jpg" pagenum="48"/>grauitatis, nempe in linea BD, &longs;it propinquius ba&longs;i AC, <lb/>&longs;iue termino D, quàm prædictæ datæ figuræ circum&longs;criptæ <lb/>centrum grauitatis, Omnium enim cylindrorum, vel cy<lb/>lindri portionum, vel parallelogrammorum, ex quibus con<lb/>&longs;tat prædicta data figura circum&longs;cripta &longs;int axes, vel quæ <lb/>oppo&longs;ita latera coniungunt rectæ BL, LM, MD, qui<lb/>bus &longs;ectis bifariam in punctis N, O, P, ac planis per ea <lb/>&longs;iue rectis tran&longs;euntibus ba&longs;i AC parallelis, &longs;ecantibus<lb/>que dictos cylindros, vel cylindri portiones, vel pa<lb/>rallelogramma, compleatur & figuræ ABC circum&longs;cri<lb/>batur altera figura <lb/>vt prior, quæ ob &longs;e<lb/>ctiones factas com<lb/>ponetur ex duplis <lb/>multitudine cylin<lb/>dris, vel cylindri por<lb/>tionibus, vel paralle<lb/>logrammis <17>qualium <lb/>altitudinum, eorum <lb/>ex quibus con&longs;tat da <lb/>ta figura circum&longs;cri<lb/>pta &longs;in<gap/>autem hi cy<lb/>lindri, aut reliqua, <lb/>quæ diximus QR, <lb/><figure id="id.043.01.135.1.jpg" xlink:href="043/01/135/1.jpg"/><lb/>ES, TV, GX, ZI, AY. </s>

<s>Quoniam igitur cylindro<lb/>rum, vel cylindri portionum, vel parallelogrammorum quæ <lb/>&longs;unt circa figuram ABC, minor e&longs;t proportio QR ad ES, <lb/>quàm ES, ad TV, propter &longs;ectiones circulos, vel &longs;imiles <lb/>ellip&longs;es, vel rectas lineas, & <expan abbr="æqualitat&etilde;">æqualitatem</expan> <expan abbr="altitudin&utilde;">altitudinum</expan>, & figuræ <lb/>propo&longs;itæ <expan abbr="naturã">naturam</expan>. </s>

<s>Sed <expan abbr="ead&etilde;">eadem</expan> ratione minor e&longs;t proportio ES <lb/>ad TV, quàm TV, ad GX; multo ergo minor proportio erit <lb/>QR ad ES, quam TV ad GX: & componendo, minor <lb/>proportio QR, ES, &longs;imul ad ES, quàm TV, GX, &longs;imul <lb/>ad GX. &longs;ed vt GX ad GH, ita e&longs;t ES ad EF; ex æqua-<pb pagenum="49"/>li igitur minor erit proportio QR, ES &longs;imul ad EF, <lb/>quàm TV, GX &longs;imul ad GH. & permutando, minor <lb/>proportio QR, ES &longs;imul ad TV, GX &longs;imul quàm EF <lb/>ad GH. & conuertendo, maior proportio GX, TV &longs;i<lb/>mul ad ES, QR &longs;imul, quàm GH ad EF. </s>

<s>Sed <expan abbr="ead&etilde;">eadem</expan> ratione minor e&longs;t proportio ES <lb/>ad TV, quàm TV, ad GX; multo ergo minor proportio erit <lb/>QR ad ES, quam TV ad GX: & componendo, minor <lb/>proportio QR, ES, &longs;imul ad ES, quàm TV, GX, &longs;imul <lb/>ad GX. &longs;ed vt GX ad GH, ita e&longs;t ES ad EF; ex æqua-<pb xlink:href="043/01/136.jpg" pagenum="49"/>li igitur minor erit proportio QR, ES &longs;imul ad EF, <lb/>quàm TV, GX &longs;imul ad GH. & permutando, minor <lb/>proportio QR, ES &longs;imul ad TV, GX &longs;imul quàm EF <lb/>ad GH. & conuertendo, maior proportio GX, TV &longs;i<lb/>mul ad ES, QR &longs;imul, quàm GH ad EF. </s>

<s>Similiter <lb/>o&longs;tenderemus duo ZI, AY, &longs;imul ad TV, GX, &longs;imul, <lb/>maiorem habere proportionem, quàm AK ad rectarum <lb/>GH. </s>

Line 1732

<s>Et duorum ZI, AY &longs;imul centrum grauitatis propin<lb/>quius erit D termino, quàm P ip&longs;ius AK. </s>

<s>Quoniam <lb/>igitur omnia primarum magnitudinum, ex quibus con&longs;tat <lb/>figura &longs;ecundo circum&longs;cripta centra grauitatis in eadem re <lb/>cta linea BD, di&longs;po&longs;ita &longs;unt alternatim ad centra grauita<lb/>tis &longs;ecundarum primis multitudine æqualium, ex quibus <lb/>data figura con&longs;tat ip&longs;i ABC figuræ circum&longs;cripta, &longs;unt <lb/>termino D propinquiora, quàm centra grauitatis &longs;ecunda<lb/>rum, &longs;i bina, prout inter &longs;e re&longs;pondent comparentur: maior <lb/>autem proportio o&longs;ten&longs;a e&longs;t primæ ad &longs;ecundam in primis, <lb/>quàm primæ ad &longs;ecundam in &longs;ecundis: & &longs;ecundæ ad ter<lb/>tiam in primis, quàm &longs;ecundæ ad tertiam in &longs;ecundis, <lb/>&longs;umpto ordine à termino D, erit centrum grauitatis om<lb/>nium primarum &longs;imul, ide&longs;t figuræ ip&longs;i ABC figuræ <lb/>&longs;ecundo circum&longs;criptæ termino D propinquius, quàm <lb/>datæ figuræ eidem ABC figuræ primo circum&longs;criptæ cen<pb pagenum="50"/>trum grauitatis. </s>

<s>Quod demon&longs;trandum erat. </s>

Line 1744

<s>Dico cen<lb/>trum grauitatis figuræ ABC <lb/>propinquius e&longs;&longs;e puncto D, <lb/>quàm punctum E. </s>

<s>Si enim <lb/>fieri pote&longs;t, centrum grauita<lb/>tis figuræ ABC, quod &longs;it <lb/>F, non cadat infra punctum <lb/>E, &longs;ed vel &longs;upra, vel con<lb/>gruat puncto E: figuræ ita<lb/>que ABC circum&longs;cribatur <lb/>figura quædam ex cylindris, <lb/>vel cylindri portionibus, vel <lb/>parallelogrammis <17>qualium <lb/>altitudinum, cuius centrum <lb/><figure id="fig90"/><lb/>grauitatis, quod &longs;it G, &longs;it propinquius D puncto, quàm <lb/>punctum E, ac propterea propinquius, quàm punctum F, <lb/>centrum grauitatis figuræ primo circum&longs;criptæ. </s>

<s>Si enim <lb/>fieri pote&longs;t, centrum grauita<lb/>tis figuræ ABC, quod &longs;it <lb/>F, non cadat infra punctum <lb/>E, &longs;ed vel &longs;upra, vel con<lb/>gruat puncto E: figuræ ita<lb/>que ABC circum&longs;cribatur <lb/>figura quædam ex cylindris, <lb/>vel cylindri portionibus, vel <lb/>parallelogrammis <17>qualium <lb/>altitudinum, cuius centrum <lb/><figure id="id.043.01.137.1.jpg" xlink:href="043/01/137/1.jpg"/><lb/>grauitatis, quod &longs;it G, &longs;it propinquius D puncto, quàm <lb/>punctum E, ac propterea propinquius, quàm punctum F, <lb/>centrum grauitatis figuræ primo circum&longs;criptæ. </s>

<s>Rur&longs;us <lb/>multiplicatis cylindris, vel cylindri portionibus, vel paral-<pb pagenum="51"/>lelogrammis circum&longs;cribatur figuræ ABC, altera tertia fi<lb/>gura, quemadmodum diximus in præcedenti, cuius cen<lb/>trum grauitatis H, in linea GD cadat & &longs;it minor pro<lb/>portio re&longs;idui huius tertiæ figuræ circum&longs;criptæ ip&longs;i ABC, <lb/>ad figuram ABC, quàm FG ad GD. </s>

<s>Rur&longs;us <lb/>multiplicatis cylindris, vel cylindri portionibus, vel paral-<pb xlink:href="043/01/138.jpg" pagenum="51"/>lelogrammis circum&longs;cribatur figuræ ABC, altera tertia fi<lb/>gura, quemadmodum diximus in præcedenti, cuius cen<lb/>trum grauitatis H, in linea GD cadat & &longs;it minor pro<lb/>portio re&longs;idui huius tertiæ figuræ circum&longs;criptæ ip&longs;i ABC, <lb/>ad figuram ABC, quàm FG ad GD. </s>

<s>Multo ergo mi<lb/>nor proportio erit dicti re&longs;idui ad figuram ABC quam F <lb/>H ad HD, fiat igitur vt prædictum re&longs;iduum ad figuram <lb/>ABC, ita ex contraria parte FH ad HDK; prædicti igi<lb/>tur re&longs;idui centrum grauitatis erit K, extra ip&longs;ius terminos, <lb/>quod fieri non pote&longs;t: Non igitur F centrum grauitatis fi<lb/>guræ ABC cadit in puncto E, nec &longs;upra; ergo infra pun <lb/>ctum E: & ponitur E centrum grauitatis cuiuslibet figuræ <lb/>ex cylindris, vel cylindri portionibus, vel parallelogrammis <lb/>æqualium altitudinum quo modo diximus ip&longs;i ABC cir<lb/>cum&longs;criptæ. </s>

Line 1754

<s><emph type="italics"/>PROPOSITIO XXXI.<emph.end type="italics"/></s>

<s>Omni prædictæ figuræ figura quædam ex cylin <lb/>dris, vel cylindri portionibus, vel parallelogram<lb/>mis æqualium altitudi <lb/>num circum&longs;cribi po<lb/>te&longs;t, cuius centri graui <lb/>tatis di&longs;tantia à prædi<lb/>ctæ figuræ centro gra<lb/>uitatis &longs;it minor quan<lb/>tacunque longitudine <lb/>propo&longs;ita. </s><figure/>

<s>Sit figura ABC in <expan abbr="alterã">alteram</expan> <lb/>partem <expan abbr="defici&etilde;s">deficiens</expan> &longs;upradicta, <lb/>cuius centrum grauitatis F, propo&longs;ita autem quan<gap/><expan abbr="ac&utilde;que">acunque</expan> <lb/><expan abbr="lõgitudine">longitudine</expan> minor &longs;it FG ip&longs;ius BF. </s>

<s>Dico figuræ ABC figu <pb pagenum="52"/>ram ex cylindris vel cylindri portionibus, vel <expan abbr="parallelogrã-mis">parallelogran<lb/>mis</expan> æqualium <expan abbr="altitudin&utilde;">altitudinum</expan> circum&longs;cribi po&longs;&longs;e, cuius centrum <lb/>grauitatis &longs;it propinquius puncto F, quàm punctum G: figu <lb/>ræ enim ABC figura, qualem diximus circum&longs;cribatur, cu<lb/>ius re&longs;iduum dempta figura ABC, ad figuram ABC mi<lb/>norem habeat proportionem, quàm FG, ad GB, &longs;it autem <lb/>figuræ circum&longs;criptæ centrum grauitatis K, nempe in axe, <lb/>vel di<gap/> metro BD. </s>

<s>Dico figuræ ABC figu <pb xlink:href="043/01/139.jpg" pagenum="52"/>ram ex cylindris vel cylindri portionibus, vel <expan abbr="parallelogrã-mis">parallelogran<lb/>mis</expan> æqualium <expan abbr="altitudin&utilde;">altitudinum</expan> circum&longs;cribi po&longs;&longs;e, cuius centrum <lb/>grauitatis &longs;it propinquius puncto F, quàm punctum G: figu <lb/>ræ enim ABC figura, qualem diximus circum&longs;cribatur, cu<lb/>ius re&longs;iduum dempta figura ABC, ad figuram ABC mi<lb/>norem habeat proportionem, quàm FG, ad GB, &longs;it autem <lb/>figuræ circum&longs;criptæ centrum grauitatis K, nempe in axe, <lb/>vel di<gap/> metro BD. </s>

<s>Dico <lb/>lineam FK minorem e&longs;&longs;e <lb/>quàm FG, atque adeo lon <lb/>gitudine propo&longs;ita. </s>

<s>Quo<lb/>niam enim F e&longs;t centrum <lb/>grauitatis figuræ ABC, <lb/>erit centrum grauitatis <emph type="italics"/>K<emph.end type="italics"/>, <lb/>figuræ circum&longs;criptæ ip&longs;i <lb/>ABC propinquius termi<lb/>no B, quàm punctum F, <lb/>&longs;ed centrum grauitatis fi<lb/>guræ ABC quòd e&longs;t F, & <lb/>figuræ circum&longs;criptæ, quod <lb/>e&longs;t K & eius re&longs;idui dem<lb/><figure id="fig91"/><lb/>pta figura ABC &longs;unt in communi axe, vel diametro BD; <lb/>erit igitur dicti re&longs;idui in linea BK, centrum grauitatis, <lb/>quod &longs;it H. </s>

<s>Quo<lb/>niam enim F e&longs;t centrum <lb/>grauitatis figuræ ABC, <lb/>erit centrum grauitatis <emph type="italics"/>K<emph.end type="italics"/>, <lb/>figuræ circum&longs;criptæ ip&longs;i <lb/>ABC propinquius termi<lb/>no B, quàm punctum F, <lb/>&longs;ed centrum grauitatis fi<lb/>guræ ABC quòd e&longs;t F, & <lb/>figuræ circum&longs;criptæ, quod <lb/>e&longs;t K & eius re&longs;idui dem<lb/><figure id="id.043.01.139.1.jpg" xlink:href="043/01/139/1.jpg"/><lb/>pta figura ABC &longs;unt in communi axe, vel diametro BD; <lb/>erit igitur dicti re&longs;idui in linea BK, centrum grauitatis, <lb/>quod &longs;it H. </s>

<s>Minor autem proportio e&longs;t prædicti re&longs;idui <lb/>ad figuram ABC, hoc e&longs;t ip&longs;ius FK ad KH, quàm FG <lb/>ad GB, & multo minor, quàm FG ad GH; & compo<lb/>nendo minor proportio FH ad HK, quàm FH ad HG; <lb/>ergo KH maior erit, quàm GH; reliqua igitur F <emph type="italics"/>K<emph.end type="italics"/> mi<lb/>nor, quàm FG atque adeo longitudine propo&longs;ita. </s>

<s>Fieri <lb/>ergo pote&longs;t, quod proponebatur. </s><pb pagenum="53"/>

<s>Fieri <lb/>ergo pote&longs;t, quod proponebatur. </s><pb xlink:href="043/01/140.jpg" pagenum="53"/>

<s><emph type="italics"/>PROPOSITIO XXXII.<emph.end type="italics"/></s>

Line 1776

<s>Dico G e&longs;&longs;e <lb/>centrum grauitatis <lb/>figuræ DBE. &longs;i <lb/>enim non e&longs;t, &longs;it a<lb/>liud punctum H, <lb/>quod cadat primo <lb/>&longs;upra punctum G. <lb/></s>

<s>Figuræ igitur AB <lb/>C, figura circum<lb/>&longs;cribatur qualem <lb/>diximus ex cylin<lb/>dris, vel cylindri <lb/>portionibus, vel pa <lb/>rallelogrammis æ<lb/>qualium <expan abbr="altitudin&utilde;">altitudinum</expan> <lb/>cuius centri graui<lb/>tatis <emph type="italics"/>K<emph.end type="italics"/> di&longs;tantia à <lb/><figure id="fig92"/><lb/>centro G, figuræ ABC &longs;it minor quàm recta GH: & figu <lb/>ræ DBE, figura circum&longs;cribatur ex cylindris, vel cylindri <lb/>portionibus vel parallelogrammis æqualium altitudinum, <lb/>multitudine æqualium ijs, ex quibus con&longs;tat ip&longs;i ABC, <pb pagenum="54"/>figura circum&longs;cripta, quæ cum prædictis circa figuram AB <lb/>C erunt bina &longs;umpto ordine à puncto B, in eadem propor<lb/>tione inter eadem plana parallela, vel rectas parallelas <expan abbr="cõ&longs;i-&longs;tentia">con&longs;i<lb/>&longs;tentia</expan>, propter &longs;ectiones, ide&longs;t ba&longs;es, & æquales altitudines: <lb/>binorum autem quorumque homologorum idem erit in li<lb/>nea BF, centrum grauitatis: punctum igitur K, centrum <lb/>grauitatis figuræ ip&longs;i ABC circum&longs;criptæ, idem erit fi<lb/>guræ ip&longs;i DBE, circum&longs;criptæ centrum grauitatis: cadi<gap/><lb/><expan abbr="aut&etilde;">autem</expan> infra centrum <lb/>grauitatis H figu<lb/>ræ DBE, quod e&longs;t <lb/>ab&longs;urdum. </s>

<s>Figuræ igitur AB <lb/>C, figura circum<lb/>&longs;cribatur qualem <lb/>diximus ex cylin<lb/>dris, vel cylindri <lb/>portionibus, vel pa <lb/>rallelogrammis æ<lb/>qualium <expan abbr="altitudin&utilde;">altitudinum</expan> <lb/>cuius centri graui<lb/>tatis <emph type="italics"/>K<emph.end type="italics"/> di&longs;tantia à <lb/><figure id="id.043.01.140.1.jpg" xlink:href="043/01/140/1.jpg"/><lb/>centro G, figuræ ABC &longs;it minor quàm recta GH: & figu <lb/>ræ DBE, figura circum&longs;cribatur ex cylindris, vel cylindri <lb/>portionibus vel parallelogrammis æqualium altitudinum, <lb/>multitudine æqualium ijs, ex quibus con&longs;tat ip&longs;i ABC, <pb xlink:href="043/01/141.jpg" pagenum="54"/>figura circum&longs;cripta, quæ cum prædictis circa figuram AB <lb/>C erunt bina &longs;umpto ordine à puncto B, in eadem propor<lb/>tione inter eadem plana parallela, vel rectas parallelas <expan abbr="cõ&longs;i-&longs;tentia">con&longs;i<lb/>&longs;tentia</expan>, propter &longs;ectiones, ide&longs;t ba&longs;es, & æquales altitudines: <lb/>binorum autem quorumque homologorum idem erit in li<lb/>nea BF, centrum grauitatis: punctum igitur K, centrum <lb/>grauitatis figuræ ip&longs;i ABC circum&longs;criptæ, idem erit fi<lb/>guræ ip&longs;i DBE, circum&longs;criptæ centrum grauitatis: cadi<gap/><lb/><expan abbr="aut&etilde;">autem</expan> infra centrum <lb/>grauitatis H figu<lb/>ræ DBE, quod e&longs;t <lb/>ab&longs;urdum. </s>

<s>Non <lb/>igitur centrum gra<lb/>uitatis figuræ DB <lb/>E, cadit &longs;upra pun <lb/>ctum G. </s>

<s>Sed ca<lb/>dat infra, vt in pun<lb/>cto L. </s>

<s>Rur&longs;us igi <lb/>tur figuræ DBE fi<lb/>gura, qualem dixi<lb/>mus circum&longs;cripta, <lb/>cuius centrum gra<lb/>uitatis M, &longs;it pro<lb/>pinquius centro L, <lb/><figure id="fig93"/><lb/>quàm punctum G, figuræ ABC altera qualem diximus <lb/>figura circum&longs;cribatur, cuius centrum grauitatis &longs;it idem <lb/>punctum M, quod fieri po&longs;&longs;e con&longs;tat ex &longs;uperioribus. </s>

<s>Rur&longs;us igi <lb/>tur figuræ DBE fi<lb/>gura, qualem dixi<lb/>mus circum&longs;cripta, <lb/>cuius centrum gra<lb/>uitatis M, &longs;it pro<lb/>pinquius centro L, <lb/><figure id="id.043.01.141.1.jpg" xlink:href="043/01/141/1.jpg"/><lb/>quàm punctum G, figuræ ABC altera qualem diximus <lb/>figura circum&longs;cribatur, cuius centrum grauitatis &longs;it idem <lb/>punctum M, quod fieri po&longs;&longs;e con&longs;tat ex &longs;uperioribus. </s>

<s>Sed <lb/>G ponitur centrum grauitatis figuræ ABC; ergo centrum <lb/>grauitatis figuræ ip&longs;i ABC, circum&longs;criptæ erit propinquius <lb/>ba&longs;i & puncto F, quàm figuræ ABC centrum grauitatis, <lb/>quod fieri non pote&longs;t. </s>

Line 1790

<s>Sed neque &longs;upra; punctum <lb/>igitur G erit commune duarum figurarum ABC, DBE, <lb/>centrum grauitatis. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="55"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/142.jpg" pagenum="55"/>

<s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/></s>

Line 1798

<s>Tantum aduer<lb/>tendum e&longs;t, vt pro &longs;ectionibus, dicamus compo&longs;ita <lb/>ex binis &longs;ectionibus (quæ &longs;cilicet fiunt ab codem <lb/>plano, vel eadem recta linea) cum de prædicto com <lb/>po&longs;ito &longs;it &longs;ermo: & in demon&longs;tratione, procylin<lb/>dris, vel cylindri portionibus, vel parallelogram<lb/>mis, compo&longs;ita ex binis cylindris, vel cylindri por <lb/>tionibus, vel parallelogrammis(quæ &longs;cilicet &longs;unt <lb/>inter eadem plana parallela, vel lineas parallelas, <lb/>& circa eundem axim, vel diametrum totius vel <lb/>diametri, vel axis partem) &longs;icut & pro figura com<lb/>po&longs;itum ex duabus dictis figuris: pro re&longs;iduo, com <lb/>po&longs;itum ex re&longs;iduis. </s>

<s>Nam cum vtriu&longs;que re&longs;idui <pb pagenum="56"/>figurarum duobus prædictis figuris vnum quid <lb/>componentibus, & circa eundem axim, vel diame<lb/>trum exi&longs;tentibus, qua ratione diximus, circum<lb/>&longs;criptarum, centra grauitatis &longs;int in diametro, vel <lb/>axe; etiam compo&longs;iti ex ijs duobus re&longs;iduis (vt in <lb/>priori libro generaliter demon&longs;trauimus, cen<lb/>trum grauitatis erit in eadem diametro, vel axe: <lb/>vnde vim habent proximæ quatuor anteceden<lb/>tes demon&longs;trationes, exemplum erit in demon<lb/>&longs;tratione trige&longs;imæ quartæ huius. </s>

<s>Nam cum vtriu&longs;que re&longs;idui <pb xlink:href="043/01/143.jpg" pagenum="56"/>figurarum duobus prædictis figuris vnum quid <lb/>componentibus, & circa eundem axim, vel diame<lb/>trum exi&longs;tentibus, qua ratione diximus, circum<lb/>&longs;criptarum, centra grauitatis &longs;int in diametro, vel <lb/>axe; etiam compo&longs;iti ex ijs duobus re&longs;iduis (vt in <lb/>priori libro generaliter demon&longs;trauimus, cen<lb/>trum grauitatis erit in eadem diametro, vel axe: <lb/>vnde vim habent proximæ quatuor anteceden<lb/>tes demon&longs;trationes, exemplum erit in demon<lb/>&longs;tratione trige&longs;imæ quartæ huius. </s>

<s><emph type="italics"/>PROPOSITIO XXXIII.<emph.end type="italics"/></s>

Line 1810

<s>Ab&longs;cindatur enim BK ip&longs;ius BD <lb/>pars quarta: & &longs;uper ba&longs;im eandem hemi&longs;phærij eundem<lb/>que axim BD cylindrus AF con&longs;i&longs;tat, & conus intelli<lb/>gatur EDF, cuius vertex D, ba&longs;is autem circulus circu<lb/>lo AC oppo&longs;itus, cuius diameter EBF. </s>

<s>Sectoque axe <lb/>BD bifariam in puncto H, & &longs;ingulis eius partibus rur<lb/>&longs;us bifariam, quoad BD &longs;ecta &longs;it in partes æquales cu<lb/>iu&longs;cumque libuerit numeri paris, tran&longs;eant per puncta &longs;e<lb/>ctionum plana quædam ba&longs;i AC parallela, & &longs;ecantia, <lb/>hemi&longs;phærium, conum, & cylindrum, quorum omnes &longs;e<lb/>ctiones erunt circuli, terni in codem plano ad aliam atque <pb pagenum="57"/>aliam trium harum figurarum pertinentes. </s>

<s>Sectoque axe <lb/>BD bifariam in puncto H, & &longs;ingulis eius partibus rur<lb/>&longs;us bifariam, quoad BD &longs;ecta &longs;it in partes æquales cu<lb/>iu&longs;cumque libuerit numeri paris, tran&longs;eant per puncta &longs;e<lb/>ctionum plana quædam ba&longs;i AC parallela, & &longs;ecantia, <lb/>hemi&longs;phærium, conum, & cylindrum, quorum omnes &longs;e<lb/>ctiones erunt circuli, terni in codem plano ad aliam atque <pb xlink:href="043/01/144.jpg" pagenum="57"/>aliam trium harum figurarum pertinentes. </s>

<s>Quod &longs;i præ<lb/>terea factæ &longs;ectiones hemi&longs;phærij ABC à cylindri AF <lb/>&longs;ectionibus, circuli à circulis concentricis auferri intelli<lb/>gantur; reliquæ totidem erunt &longs;ectiones reliquæ figuræ &longs;o<lb/>lidæ, dempto ABC hemi&longs;phærio ex toto AF cylin<lb/>dro, circuli deficientes circulis concentricis, hoc e&longs;t prædi<lb/>ctis ABC hemi&longs;phærij &longs;ectionibus prout inter &longs;e re&longs;pon<lb/>dent. </s>

<s>Nunc &longs;uper &longs;ectiones hemi&longs;phærij ABC, & co<lb/>ni EDF cylindris con&longs;titutis circa axes, quæ &longs;unt &longs;eg<lb/>menta æqualia axis BD, intelligantur duæ figuræ ex cy<lb/>lindris æqualium altitudinum, altera in&longs;cripta hemi&longs;phæ<lb/><figure id="fig94"/><lb/>rio ABC, altera cono EDF circum&longs;cripta. </s>

<s>Nunc &longs;uper &longs;ectiones hemi&longs;phærij ABC, & co<lb/>ni EDF cylindris con&longs;titutis circa axes, quæ &longs;unt &longs;eg<lb/>menta æqualia axis BD, intelligantur duæ figuræ ex cy<lb/>lindris æqualium altitudinum, altera in&longs;cripta hemi&longs;phæ<lb/><figure id="id.043.01.144.1.jpg" xlink:href="043/01/144/1.jpg"/><lb/>rio ABC, altera cono EDF circum&longs;cripta. </s>

<s>Si igitur <lb/>à toto AF cylindro auferatur figura, quæ in&longs;cripta e&longs;t <lb/>hemi&longs;phærio ABC, relinquetur figura quædam ex cylin<lb/>dris circa prædictos axes, vt &longs;unt BK, KH, HL, LD, <lb/>deficientibus ijs cylindris, ex quibus con&longs;tat figura in&longs;cri<lb/>pta hemi&longs;phærio ABC, & vno integro &longs;upiemo XF <lb/>cylindro, circum&longs;cripta re&longs;iduo AF cylindri dempto A <lb/>BC hemi&longs;phærio, circum&longs;criptione interna: talis autem <lb/>figuræ circum&longs;criptæ centrum grauitatis, per ea, quæ in <lb/>primo libro, erit in axe BD, quemadmodum & aliarum <lb/>duarum figurarum ex cylindris, quarum altera in&longs;cripta <lb/>e&longs;t hemi&longs;phærio ABC, altera cono EDF circum&longs;cripta. <pb pagenum="58"/>Quoniam igitur quo exce&longs;su hemi&longs;phærium ABC &longs;u<lb/>perat ex cylindris figuram &longs;ibi in&longs;criptam, eodem figura <lb/>circum&longs;cripta reliquo cylindri AF, dempto ABC he<lb/>mi&longs;phærio, &longs;uperat ip&longs;um re&longs;iduum; figura autem in&longs;cripta <lb/>hemi&longs;phærio ABC pote&longs;t e&longs;&longs;e eiu&longs;modi, quæ ab hemi<lb/>&longs;phærio de&longs;iciat minori defectu quantacumque magnitu<lb/>dine propo&longs;ita; poterit figura, quæ prædicto re&longs;iduo cir<lb/>c<gap/> m&longs;cripta e&longs;t e&longs;&longs;e talis, quæ ip&longs;um re&longs;iduum &longs;uperet mi<lb/>no i exce&longs;su quantacumque magnitudine propo&longs;ita. <lb/></s>

<s>Si igitur <lb/>à toto AF cylindro auferatur figura, quæ in&longs;cripta e&longs;t <lb/>hemi&longs;phærio ABC, relinquetur figura quædam ex cylin<lb/>dris circa prædictos axes, vt &longs;unt BK, KH, HL, LD, <lb/>deficientibus ijs cylindris, ex quibus con&longs;tat figura in&longs;cri<lb/>pta hemi&longs;phærio ABC, & vno integro &longs;upiemo XF <lb/>cylindro, circum&longs;cripta re&longs;iduo AF cylindri dempto A <lb/>BC hemi&longs;phærio, circum&longs;criptione interna: talis autem <lb/>figuræ circum&longs;criptæ centrum grauitatis, per ea, quæ in <lb/>primo libro, erit in axe BD, quemadmodum & aliarum <lb/>duarum figurarum ex cylindris, quarum altera in&longs;cripta <lb/>e&longs;t hemi&longs;phærio ABC, altera cono EDF circum&longs;cripta. <pb xlink:href="043/01/145.jpg" pagenum="58"/>Quoniam igitur quo exce&longs;su hemi&longs;phærium ABC &longs;u<lb/>perat ex cylindris figuram &longs;ibi in&longs;criptam, eodem figura <lb/>circum&longs;cripta reliquo cylindri AF, dempto ABC he<lb/>mi&longs;phærio, &longs;uperat ip&longs;um re&longs;iduum; figura autem in&longs;cripta <lb/>hemi&longs;phærio ABC pote&longs;t e&longs;&longs;e eiu&longs;modi, quæ ab hemi<lb/>&longs;phærio de&longs;iciat minori defectu quantacumque magnitu<lb/>dine propo&longs;ita; poterit figura, quæ prædicto re&longs;iduo cir<lb/>c<gap/> m&longs;cripta e&longs;t e&longs;&longs;e talis, quæ ip&longs;um re&longs;iduum &longs;uperet mi<lb/>no i exce&longs;su quantacumque magnitudine propo&longs;ita. <lb/></s>

<s>Ru &longs;us, quia quemadmodum cylindrus AN infimus de<lb/>ficiens cylindro SR, æqualis e&longs;t cylindro TP, ex &longs;upe<lb/><figure id="fig95"/><lb/>rioribus, ita vnu&longs;qui&longs;que aliorum cylindrorum deficien<lb/>tium cylindris, qui &longs;unt in hemi&longs;phærio, ex quibus cylin<lb/>dris deficientibus con&longs;tat dicto re&longs;iduo figura circum&longs;cri<lb/>pta, æqualis e&longs;t cylindrorum circa conum EDF, ei, qui <lb/>cum ip&longs;o e&longs;t inter eadem plena parallela, & circa eundem <lb/>axem; erunt omnes cylindri circa conum EDF, in ea<lb/>dem proportione cum prædictis cylindris deficientibus, <lb/>circa prædictum re&longs;iduum, &longs;i bini &longs;umantur inter eadem <lb/>plana parallela, & circa eundem axem. </s>

<s>Ru &longs;us, quia quemadmodum cylindrus AN infimus de<lb/>ficiens cylindro SR, æqualis e&longs;t cylindro TP, ex &longs;upe<lb/><figure id="id.043.01.145.1.jpg" xlink:href="043/01/145/1.jpg"/><lb/>rioribus, ita vnu&longs;qui&longs;que aliorum cylindrorum deficien<lb/>tium cylindris, qui &longs;unt in hemi&longs;phærio, ex quibus cylin<lb/>dris deficientibus con&longs;tat dicto re&longs;iduo figura circum&longs;cri<lb/>pta, æqualis e&longs;t cylindrorum circa conum EDF, ei, qui <lb/>cum ip&longs;o e&longs;t inter eadem plena parallela, & circa eundem <lb/>axem; erunt omnes cylindri circa conum EDF, in ea<lb/>dem proportione cum prædictis cylindris deficientibus, <lb/>circa prædictum re&longs;iduum, &longs;i bini &longs;umantur inter eadem <lb/>plana parallela, & circa eundem axem. </s>

<s>Quemadmodum <lb/>igitur omnium cylindrorum, qui circa conum EDF mi<lb/>nor e&longs;t proportio primi ad verticem D, ad &longs;ecundum, <lb/>quàm &longs;ecundi ad tertium, & &longs;ecundi ad tertium, quàm ter-<pb pagenum="59"/>tij ad quartum, & &longs;ic &longs;emper deinceps v&longs;que ad vltimum <lb/>XF (duplicatæ enim &longs;unt talium cylindrorum rationes <lb/>earum, quas inter &longs;e habent diametri æqualibus exce&longs;sibus <lb/>differentes circulorum, qui &longs;unt &longs;ectiones coni, & ba&longs;es cy<lb/>lindrorum, ex quibus con&longs;tat figura cono EDF circum<lb/>&longs;cripta, &longs;umpta progre&longs;&longs;ione proportionum eodem ordine <lb/>gradatim à minima diametro v&longs;que ad maximam EF) ita <lb/>erit cylindrorum deficientium, ex quibus con&longs;tat figura <lb/>circum&longs;cripta reliquo cylindri AF, dempto ABC hemi<lb/>&longs;phærio, minimi, cuius axis DL ad &longs;ecundum minor pro<lb/>portio, quàm &longs;ecundi ad tertium, & &longs;ic deinceps, v&longs;que ad <lb/><expan abbr="maxim&utilde;">maximum</expan> XF, communiter ad conum EDF, & prædictum <lb/>re&longs;iduum pertinentem, &longs;icut & eorum ba&longs;es circuli deficien <lb/>tes, quæ &longs;unt dicti re&longs;idui &longs;ectiones. </s>

<s>Quemadmodum <lb/>igitur omnium cylindrorum, qui circa conum EDF mi<lb/>nor e&longs;t proportio primi ad verticem D, ad &longs;ecundum, <lb/>quàm &longs;ecundi ad tertium, & &longs;ecundi ad tertium, quàm ter-<pb xlink:href="043/01/146.jpg" pagenum="59"/>tij ad quartum, & &longs;ic &longs;emper deinceps v&longs;que ad vltimum <lb/>XF (duplicatæ enim &longs;unt talium cylindrorum rationes <lb/>earum, quas inter &longs;e habent diametri æqualibus exce&longs;sibus <lb/>differentes circulorum, qui &longs;unt &longs;ectiones coni, & ba&longs;es cy<lb/>lindrorum, ex quibus con&longs;tat figura cono EDF circum<lb/>&longs;cripta, &longs;umpta progre&longs;&longs;ione proportionum eodem ordine <lb/>gradatim à minima diametro v&longs;que ad maximam EF) ita <lb/>erit cylindrorum deficientium, ex quibus con&longs;tat figura <lb/>circum&longs;cripta reliquo cylindri AF, dempto ABC hemi<lb/>&longs;phærio, minimi, cuius axis DL ad &longs;ecundum minor pro<lb/>portio, quàm &longs;ecundi ad tertium, & &longs;ic deinceps, v&longs;que ad <lb/><expan abbr="maxim&utilde;">maximum</expan> XF, communiter ad conum EDF, & prædictum <lb/>re&longs;iduum pertinentem, &longs;icut & eorum ba&longs;es circuli deficien <lb/>tes, quæ &longs;unt dicti re&longs;idui &longs;ectiones. </s>

<s>Cum igitur tam maxi<lb/>mi cylindri XF communis, quàm binorum quorumque reli <lb/>quorum cylindrorum circa conum EDF, & prædictum re&longs;i <lb/>duum inter eadem plana parallela con&longs;i&longs;tentium, quorum <lb/>axis communis in BD, commune centrum grauitatis in axe <lb/>BD exi&longs;tat, erit ex antecedenti punctum K, quod pono <lb/>centrum grauitatis coni EDF, idem re&longs;idui ex cylindro <lb/>AF, dempto ABC, hemi&longs;phærio centrum grauitatis. <lb/></s>

<s>Quoniam igitur quarum partium e&longs;t octo axis BD talium <lb/>e&longs;t BG quinque, & BK duarum (ponimus enim nunc K <lb/>coni EDF centrum grauitatis) qualium e&longs;t BD octo, ta<lb/>lium erit GK trium: &longs;ed KH e&longs;t æqualis BK; qualium <lb/>igitur partium e&longs;t GK trium, talium erit KH duarum, ta<lb/>li&longs;que vna GH; dupla igitur KH ip&longs;ius GH: &longs;ed ABC <lb/>hemi&longs;phærium duplum e&longs;t prædicti re&longs;idui, cum &longs;it cylin<lb/>dri AF, &longs;ub&longs;e&longs;quialterum; vt igitur e&longs;t <expan abbr="hemi&longs;phæri&utilde;">hemi&longs;phærium</expan> ABC, <lb/>ad prædictum re&longs;iduum, ita ex contraria parte erit <expan abbr="lõgitudo">longitudo</expan> <lb/>KH, adlongitudinem GH: &longs;ed H e&longs;t centrum grauitatis <lb/>totius cylindri AF & K, prædicti re&longs;idui dempto ABC <lb/>hemi&longs;phærio; ergo ABC hemi&longs;phærij centrum grauitatis <lb/>erit G. </s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="60"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/147.jpg" pagenum="60"/>

<s><emph type="italics"/>PROPOSITIO XXXIV.<emph.end type="italics"/></s>

Line 1836

<s>Dico portio<lb/>nis ABC centrum grauitatis e&longs;se O. </s>

<s>Nam circa axim <lb/>BG, &longs;uper ba&longs;im FH &longs;tet cylindrus EF, cuius cen-<pb pagenum="61"/>trum grauitatis erit N, reliqui autem eius dempta <lb/>ABC portione centrum grauitatis M commune fru&longs;to <lb/>KLFH, vt colligitur ex demon&longs;tratione antecedentis. <lb/></s>

<s>Nam circa axim <lb/>BG, &longs;uper ba&longs;im FH &longs;tet cylindrus EF, cuius cen-<pb xlink:href="043/01/148.jpg" pagenum="61"/>trum grauitatis erit N, reliqui autem eius dempta <lb/>ABC portione centrum grauitatis M commune fru&longs;to <lb/>KLFH, vt colligitur ex demon&longs;tratione antecedentis. <lb/></s>

<s>Quoniam igitur e&longs;t vt exce&longs;sus, quo tripla ip&longs;ius BD &longs;u<lb/>perat tres BD, DG, P tanquam vnam, ad ip&longs;ius BD <lb/><figure id="fig96"/><lb/>triplam, hoc e&longs;t vt NM ad MO, ita portio ABC ad <lb/>EF cylindrum, & diuidendo vt MN ad NO, ita por<lb/>tio ABC ad reliquum cylindri EF; & N e&longs;t cylindri <lb/>EF, & M prædicti re&longs;idui centrum grauitatis; erit reli<lb/>quæ portionis ABC centrum grauitatis O. </s>

<s>Quoniam igitur e&longs;t vt exce&longs;sus, quo tripla ip&longs;ius BD &longs;u<lb/>perat tres BD, DG, P tanquam vnam, ad ip&longs;ius BD <lb/><figure id="id.043.01.148.1.jpg" xlink:href="043/01/148/1.jpg"/><lb/>triplam, hoc e&longs;t vt NM ad MO, ita portio ABC ad <lb/>EF cylindrum, & diuidendo vt MN ad NO, ita por<lb/>tio ABC ad reliquum cylindri EF; & N e&longs;t cylindri <lb/>EF, & M prædicti re&longs;idui centrum grauitatis; erit reli<lb/>quæ portionis ABC centrum grauitatis O. </s>

<s>Quod de<lb/>mon&longs;trandum erat. </s>

<s><emph type="italics"/>PROPOSITIO XXXV.<emph.end type="italics"/></s>

<s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla<lb/>nis parallelis, altero per centrum acto, centrum <lb/>grauitatis e&longs;t in axe primum bifariam &longs;ecto: dein<lb/>de &longs;umpta ad minorem ba&longs;im quarta parte axis <lb/>portionis; in eo puncto, in quo dimidius axis mi<lb/>norem ba&longs;im attingens &longs;ic diuiditur, vt pars dua<lb/>bus prædictis &longs;ectionibus intercepta &longs;it ad eam, <pb pagenum="62"/>quæ inter&longs;ecundam, & vltimam &longs;ectionem inter<lb/>ijcitur, vt exce&longs;&longs;us, quo maior extrema ad &longs;phæræ <lb/>&longs;emidiametrum, & axim portionis &longs;uperat ter<lb/>tiam partem axis portionis; ad maiorem extre<lb/>mam antedictam. </s>

<s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla<lb/>nis parallelis, altero per centrum acto, centrum <lb/>grauitatis e&longs;t in axe primum bifariam &longs;ecto: dein<lb/>de &longs;umpta ad minorem ba&longs;im quarta parte axis <lb/>portionis; in eo puncto, in quo dimidius axis mi<lb/>norem ba&longs;im attingens &longs;ic diuiditur, vt pars dua<lb/>bus prædictis &longs;ectionibus intercepta &longs;it ad eam, <pb xlink:href="043/01/149.jpg" pagenum="62"/>quæ inter&longs;ecundam, & vltimam &longs;ectionem inter<lb/>ijcitur, vt exce&longs;&longs;us, quo maior extrema ad &longs;phæræ <lb/>&longs;emidiametrum, & axim portionis &longs;uperat ter<lb/>tiam partem axis portionis; ad maiorem extre<lb/>mam antedictam. </s>

<s>Sit portio ABCD &longs;phæræ, cuius centrum F: axis au<lb/>tem portionis &longs;it EF ab&longs;ci&longs;sæ duobus planis parallelis, <lb/>quorum alterum tran&longs;iens per punctum F faciat &longs;ectio<lb/>num circulum maximum, cuius diameter AD, reliquam <lb/>autem &longs;ectionem minorem circulum, quæ minor ba&longs;is di<lb/>citur, cuius di<lb/>ameter BC: <lb/>& vt e&longs;t EF <lb/>ad AD, ita <lb/>fiat AD ad <lb/>OP, cuius P <lb/>R, &longs;it æqua<lb/>lis tertiæ parti <lb/>axis EF. </s>

<s>Et <lb/>&longs;ecta EF bi<lb/><figure id="fig97"/><lb/>fariam in puncto M, & po&longs;ita EN ip&longs;ius EF quarta <lb/>parte, fiat vt RO ad OP, ita MN ad NL. </s>

<s>Et <lb/>&longs;ecta EF bi<lb/><figure id="id.043.01.149.1.jpg" xlink:href="043/01/149/1.jpg"/><lb/>fariam in puncto M, & po&longs;ita EN ip&longs;ius EF quarta <lb/>parte, fiat vt RO ad OP, ita MN ad NL. </s>

<s>Dico L e&longs;&longs;e <lb/>centrum grauitatis portionis ABCD. </s>

Line 1856

<s>Quoniam igitur e&longs;t vt RO ad OP, hoc e&longs;t vt <lb/>MN ad NL, ita portio ABCD ad reliquum cylindri <lb/>AG, & diuidendo vt NM ad ML, ita portio ABCD ad <lb/>reliquum cylindri AG: & cylindri AG e&longs;t N, prædicti au<lb/>tem re&longs;idui centrum grauitatis M; erit reliquæ portionis <lb/>ABCD centrum grauitatis L. </s>

<s>Quod <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan> erat. </s><pb pagenum="63"/>

<s>Quod <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan> erat. </s><pb xlink:href="043/01/150.jpg" pagenum="63"/>

<s><emph type="italics"/>PROPOSITIO XXXVI.<emph.end type="italics"/></s>

<s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla<lb/>nis parallelis neutro per centrum acto, nec cen<lb/>trum intercipientibus, centrum grauitatis e&longs;t in <lb/>axe primum bifariam &longs;ecto: deinde &longs;ecundum <lb/>centrum grauitatis fru&longs;ti circa eundem axim, <lb/>ab&longs;ci&longs;&longs;i à cono verticem habente centrum &longs;phæ<lb/>ræ; in eo puncto in quo dimidius axis maiorem <lb/>ba&longs;im attingens &longs;ic diuiditur, vt pars duabus præ<lb/>dictis &longs;ectionibus finita &longs;it ad eam, quæ inter &longs;e<lb/>cundam, & vltimam &longs;ectionem interijcitur, vt <lb/>exce&longs;&longs;us, quo maior extrema ad triplas & &longs;emidia <lb/>metri &longs;phæræ, & eius quæ inter centra &longs;phæræ, <lb/>& minorem ba&longs;im portionis interijcitur, &longs;uperat <lb/>tres deinceps proportionales, quarum maxima <lb/>e&longs;t, quæ inter centra &longs;phæræ, & minoris ba&longs;is, <lb/>media autem, quæ inter centra &longs;phæræ, & maio<lb/>ris ba&longs;is portionis interijcitur; ad maiorem extre<lb/>mam antedictam. </s>

<s>Sit portio ABCD, &longs;phæræ, cuius centrum E, ab<lb/>&longs;ci&longs;sa duobus planis parallelis, neutro per E tran&longs;eun<lb/>te, nec E intercipientibus: axis autem portionis &longs;it GH, <lb/>maior ba&longs;is circulus, cuius diameter AD, minor cuius <lb/>diameter BC: producta autem GH v&longs;que in E intel<lb/>ligatur coni KEN rectanguli, cuius axis EG, fru&longs;tum <pb pagenum="64"/>KLMN ab&longs;ci&longs;&longs;um ij&longs;dem planis, quibus por<lb/>tio, & &longs;phæræ &longs;emidiameter &longs;it EHGS: & po<lb/>&longs;ita T tripla ip&longs;ius ES, & V ip&longs;ius EG tri<lb/>pla, e&longs;to vt V ad T ita T ad XZ: & vt GE <lb/>ad EH ita EH ad <foreign lang="greek">w</foreign>, & &longs;it ZY, ip&longs;ius XZ, <lb/>æqualis tribus GE, EH, <foreign lang="greek">w</foreign>, vt &longs;it exce&longs;&longs;us <lb/>XY: & &longs;ecto axe GH bifariam in puncto I, in <lb/>linea GI, &longs;umatur O, centrum grauitatis fru<lb/>&longs;ti KLMN: Et vt <foreign lang="greek">*u</foreign>X ad XZ, ita fiat IO <lb/>ad OIP. </s>

<s>Sit portio ABCD, &longs;phæræ, cuius centrum E, ab<lb/>&longs;ci&longs;sa duobus planis parallelis, neutro per E tran&longs;eun<lb/>te, nec E intercipientibus: axis autem portionis &longs;it GH, <lb/>maior ba&longs;is circulus, cuius diameter AD, minor cuius <lb/>diameter BC: producta autem GH v&longs;que in E intel<lb/>ligatur coni KEN rectanguli, cuius axis EG, fru&longs;tum <pb xlink:href="043/01/151.jpg" pagenum="64"/>KLMN ab&longs;ci&longs;&longs;um ij&longs;dem planis, quibus por<lb/>tio, & &longs;phæræ &longs;emidiameter &longs;it EHGS: & po<lb/>&longs;ita T tripla ip&longs;ius ES, & V ip&longs;ius EG tri<lb/>pla, e&longs;to vt V ad T ita T ad XZ: & vt GE <lb/>ad EH ita EH ad <foreign lang="greek">w</foreign>, & &longs;it ZY, ip&longs;ius XZ, <lb/>æqualis tribus GE, EH, <foreign lang="greek">w</foreign>, vt &longs;it exce&longs;&longs;us <lb/>XY: & &longs;ecto axe GH bifariam in puncto I, in <lb/>linea GI, &longs;umatur O, centrum grauitatis fru<lb/>&longs;ti KLMN: Et vt <foreign lang="greek">*u</foreign>X ad XZ, ita fiat IO <lb/>ad OIP. </s>

<s>Dico portionis ABCD centrum <lb/>grauitatis e&longs;&longs;e P. </s>

<s>Nam circa axim GH pla<lb/>nis ba&longs;ium portionis interceptus &longs;tet cylin<lb/>drus QR, cuius ba&longs;is &longs;it æqualis circulo ma<lb/>ximo. </s>

<s>Quoniam igitur e&longs;t vt YX ad XZ, <lb/>hoc e&longs;t vt IO ad OP, ita portio ABCD <lb/>ad cylindrum QR, & diuidendo vt OI ad <lb/>IP, ita portio ABCD ad reliquum cylindri <lb/>QR: & I e&longs;t cylindri QR, & O prædicti <lb/>re&longs;idui centrum grauitatis; erit reliquæ por<lb/><figure id="fig98"/><lb/>tionis ABCD centrum grauitatis P. </s>

<s>Quoniam igitur e&longs;t vt YX ad XZ, <lb/>hoc e&longs;t vt IO ad OP, ita portio ABCD <lb/>ad cylindrum QR, & diuidendo vt OI ad <lb/>IP, ita portio ABCD ad reliquum cylindri <lb/>QR: & I e&longs;t cylindri QR, & O prædicti <lb/>re&longs;idui centrum grauitatis; erit reliquæ por<lb/><figure id="id.043.01.151.1.jpg" xlink:href="043/01/151/1.jpg"/><lb/>tionis ABCD centrum grauitatis P. </s>

<s>Quod demon<lb/>&longs;trandum erat. </s><pb pagenum="65"/>

<s>Quod demon<lb/>&longs;trandum erat. </s><pb xlink:href="043/01/152.jpg" pagenum="65"/>

<s><emph type="italics"/>LEMMA.<emph.end type="italics"/></s>

Line 1878

<s>Dico OS, minorem e&longs;&longs;e quàm OR.<emph.end type="italics"/></s>

<s>Fiat enim vt PS, ad SO, ita VZ ad ZT; ma<lb/>ìor igitur erit proportio VZ, ad ZT, quàm VR, ad <lb/>RT: & componendo maior proportio VT, ad TZ, <lb/><figure id="fig99"/><lb/>quàm VT, ad TR; minor igitur TZ, quàm TR, ide&longs;t <lb/>maior DZ, quàm DR. </s>

<s>Fiat enim vt PS, ad SO, ita VZ ad ZT; ma<lb/>ìor igitur erit proportio VZ, ad ZT, quàm VR, ad <lb/>RT: & componendo maior proportio VT, ad TZ, <lb/><figure id="id.043.01.152.1.jpg" xlink:href="043/01/152/1.jpg"/><lb/>quàm VT, ad TR; minor igitur TZ, quàm TR, ide&longs;t <lb/>maior DZ, quàm DR. </s>

<s>Rur&longs;us quia componendo e&longs;t <lb/>vt PO ad OS, ita VT ad TZ: &longs;ed vt DO ad OP, ita <lb/>e&longs;t DT ad TV; erit ex æquali, vt DO ad OS, ita DT, <lb/>ad TZ; & per conuer&longs;ionem rationis, vt OD ad DS, <lb/>ita TD ad DZ: & permutando, vt DO ad DT, ita DS <lb/>ad DZ: &longs;ed DO, e&longs;t maior quàm DT, ergo & DS, erit <lb/>maior quàm DZ: &longs;ed DZ maior erat quàm DR; multo <lb/>ergo DS maior quàm DR, vnde minor erit OS quàm <lb/>OR. </s>

Line 1886

<s><emph type="italics"/>PROPOSITIO XXXVII.<emph.end type="italics"/></s>

<s>Si datæ maiori &longs;phæræ portioni cylindrus cir<lb/>cum&longs;cribatur circa eundem axim portionis, cen<lb/>trum grauitatis reliquæ figuræ ex cylindro cir<lb/>cum&longs;cripto ablata portione, propinquius erit ver<lb/>tici portionis, quàm <expan abbr="c&etilde;trum">centrum</expan> grauitatis portionis. </s><pb pagenum="66"/>

<s>Sit &longs;phæræ cuius centrum D maior portio ABC, cu<lb/>ius axis BE, ba&longs;is circulus cuius diameter AC, & por<lb/>tioni ABC, cylindro XH circa axim BE circum&longs;cripto <lb/>vt &longs;upra fecimus: quoniam tam portionis ABC, quàm <lb/>cylindri XH, centrum grauitatis e&longs;t in axe BE; erit reli<lb/>qui ex cylindro XH, in axe BE centrum grauitatis, &longs;int <lb/>in axe BE centra grauitatis Q portionis ABC & S præ<lb/>dicti re&longs;idui. </s>

<s>Dico e&longs;&longs;e punctum S vertici B propinquius <lb/><figure id="fig100"/><lb/>quàm punctum <expan abbr="q.">que</expan> Per centrum enim D tran&longs;iens planum <lb/>ad axim BE erectum &longs;ecet cylindrum XH, & portionem <lb/>ABC in duos cylindros <emph type="italics"/>K<emph.end type="italics"/>H, XL, & hemi&longs;phærium <lb/>KBL, & portionem AKLC, &longs;ectio autem circulus ma<lb/>ximus e&longs;to ille cuius diameter KL: & duo coni rectan<lb/>guli circa axes BD, DE, vertice D communi de&longs;cri<lb/>bantur GDH, MDN, quorum alterius ba&longs;is GH com<lb/>munis erit cylindro XH: alterius autem MDN, minor <lb/>quàm eiu&longs;dem cylindri XH, ba&longs;is GH. </s>

<s>Dico e&longs;&longs;e punctum S vertici B propinquius <lb/><figure id="id.043.01.153.1.jpg" xlink:href="043/01/153/1.jpg"/><lb/>quàm punctum <expan abbr="q.">que</expan> Per centrum enim D tran&longs;iens planum <lb/>ad axim BE erectum &longs;ecet cylindrum XH, & portionem <lb/>ABC in duos cylindros <emph type="italics"/>K<emph.end type="italics"/>H, XL, & hemi&longs;phærium <lb/>KBL, & portionem AKLC, &longs;ectio autem circulus ma<lb/>ximus e&longs;to ille cuius diameter KL: & duo coni rectan<lb/>guli circa axes BD, DE, vertice D communi de&longs;cri<lb/>bantur GDH, MDN, quorum alterius ba&longs;is GH com<lb/>munis erit cylindro XH: alterius autem MDN, minor <lb/>quàm eiu&longs;dem cylindri XH, ba&longs;is GH. </s>

<s>Denique <gap/><pb pagenum="67"/>BE bifariam in puncto R, &longs;ecentur BD, in puncto T, & <lb/>DE, in puncto V, bifariam & &longs;umatur BO, ip&longs;ius BD, <lb/>pars quarta, necnon EP pars quarta ip&longs;ius DE, primum <lb/>itaque quoniam ER e&longs;t maior, quàm ED, erit punctum <lb/>R, in &longs;egmento BD. </s>

<s>Denique <gap/><pb xlink:href="043/01/154.jpg" pagenum="67"/>BE bifariam in puncto R, &longs;ecentur BD, in puncto T, & <lb/>DE, in puncto V, bifariam & &longs;umatur BO, ip&longs;ius BD, <lb/>pars quarta, necnon EP pars quarta ip&longs;ius DE, primum <lb/>itaque quoniam ER e&longs;t maior, quàm ED, erit punctum <lb/>R, in &longs;egmento BD. </s>

<s>Quoniam igitur ex &longs;upra o&longs;ten&longs;is O <lb/>e&longs;t centrum grauitatis commune cono DGH, & reliquo <lb/>cylindri KH dempto ABC hemi&longs;phærio: & eadem ra<lb/>tione punctum P, cum &longs;it centrum grauitatis coni MDN, <lb/>erit idem centrum grauitatis reliqui ex cylindro XL dem<lb/>pta AKLC portione: e&longs;t autem reliquum cylindri KH <lb/>dempto KBL hemi&longs;phærio, æquale cono DGH, qua <lb/>ratione & reliquum cylindri XL, dempta AKLC por<lb/>tione æquale e&longs;t cono MDN; cum igitur S &longs;it centrum <lb/>grauitatis totius reliqui ex toto cylindro XH, dempta <lb/>ABC portione, erit idem S, centrum grauitatis compo<lb/>&longs;iti ex conis GDH, MDL: &longs;unt autem horum conorum <lb/>centra grauitatis O, P; vt igitur conus GDH, ad co<lb/>num MDN, ita erit PS, ad SO: &longs;ed coni GDH ad <lb/>&longs;imilem ip&longs;i conum MDN triplicata e&longs;t proportio axis <lb/>BD, ad axim BE, hoc e&longs;t cylindri KH ad cylindrum <lb/>XL; maior igitur proportio erit PS ad SO, quàm cy<lb/>lindri KH ad cylindrum XL, &longs;ed vt cylindrus KH, ad <lb/>cylindrum XL, ita e&longs;t VR ad RT, ob centra grauiratis <lb/>V, R, T, maior igitur proportio erit PS ad SO, quàm <lb/>VR ad RT: &longs;ed eiu&longs;dem PO e&longs;t vt PD ad DO, ita <lb/>VD ad DT, ob &longs;ectiones axium proportionales; pun<lb/>ctum igitur S propinquius e&longs;t puncto O, quàm punctum <lb/>R, per Lemma. </s>

<s>Quare & Stermino B propinquius quàm <lb/>punctum R: &longs;ed R e&longs;t centrum grauitatis totius cylindri <lb/>XH: & S reliqui ex cylindro XH dempta ABC por<lb/>tione; igitur Q reliquæ portionis ABC, centrum graui<lb/>tatis erit in linea ER, atque ideo à puncto B remotius <lb/>quàm punctnm S. </s>

<s>Quod e&longs;t propo&longs;itum. </s><pb pagenum="68"/>

<s>Quod e&longs;t propo&longs;itum. </s><pb xlink:href="043/01/155.jpg" pagenum="68"/>

<s><emph type="italics"/>COROLLARIV M.<emph.end type="italics"/></s>

<s>Manife&longs;tum e&longs;t autem ex demon&longs;tratione the<gap/><lb/>rematis, omnis re&longs;idui ex cylindro datæ maiori <lb/>&longs;phæræ portioni circum&longs;cripto circa eundem <lb/>axim portionis, cuius ba&longs;is &longs;it æqualis circulo ma <lb/>ximo, centrum grauitatis e&longs;&longs;e in axe ab&longs;ci&longs;&longs;a pri<lb/>mum quarta parte ad verticem portionis termina<lb/>ta &longs;egmenti axis portionis, quod centro &longs;phæræ, <lb/>& vertice portionis, & quarta parte eius quod <lb/>centro &longs;phæræ, & ba&longs;i portionis terminatur; ad <lb/>ba&longs;im terminata in eo puncto, in quo &longs;egmentum <lb/>axis portionis duabus prædictis &longs;ectionibus fini<lb/>tum &longs;ic diuiditur, vt &longs;egmentum propinquius ba&longs;i <lb/>&longs;it ad reliquum, vt cubus &longs;egmenti axis portionis <lb/>centro &longs;phæræ, & vertice portionis terminati ad <lb/>cubum reliqui quod ba&longs;im portionis tangit, &longs;i<lb/>quidem cubi triplicatam inter &longs;e habent laterum <lb/>proportionem, &longs;imul illud manife&longs;tum e&longs;t, hoc <lb/>idem eadem ratione po&longs;&longs;e demon&longs;trari de centro <lb/>grauitatis reliqui ex cylindro dempta &longs;phæræ por <lb/>tione ab&longs;ci&longs;&longs;a duobus planis paralìelis centrum <lb/>&longs;phæræ intercipientibus, ita vt axis portionis à <lb/>centro &longs;phæræ in partes inæquales diuidatur, cu<lb/>ius cylindri circum&longs;cripti &longs;it idem axis, qui & por <lb/>tionis, ba&longs;is autem æqualis circulo maximo. </s>

<s>Si<lb/>militer enim de&longs;criptis duobus conis rectanguli<gap/><pb pagenum="69"/>verticem habentibus communem centrum &longs;phæ<lb/>ræ, ba&longs;es autem minores ba&longs;ibus oppo&longs;itis cylin<lb/>dri circum&longs;cripti: æqualibus circulo maximo, &longs;u<lb/>mentes pro vertice minorem ba&longs;im, pro ba&longs;i, ma<lb/>iorem ba&longs;im portionis immotis reliquis propo&longs;i<lb/>tum demon&longs;traremus. </s>

<s>Si<lb/>militer enim de&longs;criptis duobus conis rectanguli<gap/><pb xlink:href="043/01/156.jpg" pagenum="69"/>verticem habentibus communem centrum &longs;phæ<lb/>ræ, ba&longs;es autem minores ba&longs;ibus oppo&longs;itis cylin<lb/>dri circum&longs;cripti: æqualibus circulo maximo, &longs;u<lb/>mentes pro vertice minorem ba&longs;im, pro ba&longs;i, ma<lb/>iorem ba&longs;im portionis immotis reliquis propo&longs;i<lb/>tum demon&longs;traremus. </s>

<s><emph type="italics"/>PROPOSITIO XXXVIII.<emph.end type="italics"/></s>

<s>Omnis maioris portionis &longs;phæræ centrum gra<lb/>uitatis e&longs;t in axe primum bifariam &longs;ecto: Deinde <lb/>&longs;umpta ad verticem quarta parte &longs;egmenti axis, <lb/>quod centro &longs;phæræ, & portionis vertice finitur: <lb/>itemque ad ba&longs;im quarta parte reliqui &longs;egmenti <lb/>inter centrum &longs;phæræ, & ba&longs;im portionis interie<lb/>cti. </s>

<s>Deinde &longs;egmento axis, inter eas quartas par<lb/>tes interiecto, ita diui&longs;o, vt pats propinquior ba&longs;i <lb/>&longs;it ad reliquam vt cubus &longs;egmenti axis, quod <lb/><expan abbr="c&etilde;tro">centro</expan> &longs;phæræ, & vertice portionis, ad cubum eius <lb/>quod centris &longs;phæræ, & ba&longs;is portionis termina<lb/>tur; in eo puncto, in quo &longs;egmentum axis centro <lb/>&longs;phæræ, & &longs;ectione penultima finitum &longs;ic diuidi<lb/>tur, vt pars prima & penultima &longs;ectione termina<lb/>ta &longs;it ad totam vltima & penultima &longs;ectione termi <lb/>natam, vt exce&longs;&longs;us, quo &longs;egmentum axis portionis <lb/>inter centrum, & ba&longs;im portionis interiectum &longs;u<lb/>perat tertiam partem minoris extremæ maiori po <lb/>&longs;ita dicto axis &longs;egmento in proportione &longs;emidia-<pb pagenum="70"/>metri &longs;phæræ ad prædictum &longs;egmentum, vnà cum <lb/>&longs;ub&longs;e&longs;quialtera reliqui &longs;egmenti, ad axim por<lb/>tionis. </s>

<s>Deinde &longs;egmento axis, inter eas quartas par<lb/>tes interiecto, ita diui&longs;o, vt pats propinquior ba&longs;i <lb/>&longs;it ad reliquam vt cubus &longs;egmenti axis, quod <lb/><expan abbr="c&etilde;tro">centro</expan> &longs;phæræ, & vertice portionis, ad cubum eius <lb/>quod centris &longs;phæræ, & ba&longs;is portionis termina<lb/>tur; in eo puncto, in quo &longs;egmentum axis centro <lb/>&longs;phæræ, & &longs;ectione penultima finitum &longs;ic diuidi<lb/>tur, vt pars prima & penultima &longs;ectione termina<lb/>ta &longs;it ad totam vltima & penultima &longs;ectione termi <lb/>natam, vt exce&longs;&longs;us, quo &longs;egmentum axis portionis <lb/>inter centrum, & ba&longs;im portionis interiectum &longs;u<lb/>perat tertiam partem minoris extremæ maiori po <lb/>&longs;ita dicto axis &longs;egmento in proportione &longs;emidia-<pb xlink:href="043/01/157.jpg" pagenum="70"/>metri &longs;phæræ ad prædictum &longs;egmentum, vnà cum <lb/>&longs;ub&longs;e&longs;quialtera reliqui &longs;egmenti, ad axim por<lb/>tionis. </s>

<s>Sit maior portio ABC &longs;phæræ, cuius centrum D, dia<lb/>meter KH, axis autem portionis &longs;it BE, ba&longs;is circulus, <lb/>cuius diameter AC, & &longs;it axis BE primum bifariam &longs;e<lb/>ctus in puncto G: &longs;umptaque ip&longs;ius BD, quarta parte <lb/>BP, itemque ip&longs;ius DE quarta parte EN, &longs;ecetur inter<lb/>iecta PN, ita in puncto F, vt NF, ad FP, &longs;it vt cubus ex <lb/>BD ad cubum ex DE; punctum igitur F, ex præcedenti <lb/><figure id="fig101"/><lb/>corollario erit centrum grauitatis reliqui ex cylindro LM <lb/>portioni ABC, vt in antecedenti circum&longs;cripto. </s>

<s>Sit maior portio ABC &longs;phæræ, cuius centrum D, dia<lb/>meter KH, axis autem portionis &longs;it BE, ba&longs;is circulus, <lb/>cuius diameter AC, & &longs;it axis BE primum bifariam &longs;e<lb/>ctus in puncto G: &longs;umptaque ip&longs;ius BD, quarta parte <lb/>BP, itemque ip&longs;ius DE quarta parte EN, &longs;ecetur inter<lb/>iecta PN, ita in puncto F, vt NF, ad FP, &longs;it vt cubus ex <lb/>BD ad cubum ex DE; punctum igitur F, ex præcedenti <lb/><figure id="id.043.01.157.1.jpg" xlink:href="043/01/157/1.jpg"/><lb/>corollario erit centrum grauitatis reliqui ex cylindro LM <lb/>portioni ABC, vt in antecedenti circum&longs;cripto. </s>

<s>Quo<lb/>niam igitur & prædicti re&longs;idui, ex antecedenti, & cylindri <lb/>LM, centra grauitatis &longs;unt in axe BE, erit & portionis <lb/>ABC in axe BE centrum grauitatis, quod &longs;it S: manife<lb/>&longs;tum e&longs;t igitur punctum S, cadere &longs;upra centrum D, in li<lb/>nea BD, minori ablata &longs;phæræ portione, cuius ba&longs;is cir-<pb pagenum="71"/>culus AC: centrum autem F propinquius e&longs;&longs;e puncto B, <lb/>quàm centrum S, con&longs;tat ex præcedenti: quare centrum <lb/>G, totius cylindri LM inter puncta F, S cadet. </s>

<s>Quo<lb/>niam igitur & prædicti re&longs;idui, ex antecedenti, & cylindri <lb/>LM, centra grauitatis &longs;unt in axe BE, erit & portionis <lb/>ABC in axe BE centrum grauitatis, quod &longs;it S: manife<lb/>&longs;tum e&longs;t igitur punctum S, cadere &longs;upra centrum D, in li<lb/>nea BD, minori ablata &longs;phæræ portione, cuius ba&longs;is cir-<pb xlink:href="043/01/158.jpg" pagenum="71"/>culus AC: centrum autem F propinquius e&longs;&longs;e puncto B, <lb/>quàm centrum S, con&longs;tat ex præcedenti: quare centrum <lb/>G, totius cylindri LM inter puncta F, S cadet. </s>

<s>Dico <lb/>GF ad FS e&longs;&longs;e vt exce&longs;&longs;us, quo recta DE &longs;uperat tertiam <lb/>partem minoris extremæ maiori po&longs;ita ip&longs;a DE in propor<lb/>tione continua ip&longs;ius DH ad DE vnà cum &longs;ub&longs;e&longs;quial<lb/>tera ip&longs;ius BD, ad axim BE, ita GF ad FS. </s>

Line 1926

<s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla<lb/>nis parallelis centrum intercipientibus, & à cen<lb/>tro æqualiter di&longs;tantibus, centrum grauitatis e&longs;t <lb/>in medio axis, vel idem, quod centrum &longs;phæræ. </s>

<s>Sit portio ABCD, &longs;phæræ, cuius centrum G, ab&longs;ci&longs;sa <lb/>duobus planis parallelis <lb/>centrum G intercipien<lb/>tibus, & æquè ab eo di<lb/>&longs;tantibus: &longs;ectiones <expan abbr="er&utilde;t">erunt</expan> <lb/>circuli minores, quorum <lb/>diametri &longs;int AD, BC <lb/>centra autem F,E, qui<lb/>bus axis portionis termi <lb/>nabitur, eritque ad pla<lb/>na vtriu&longs;que circuli per <lb/><figure id="fig102"/><lb/>pendicularis tran&longs;iens per centrum G: & quia illa plana <pb pagenum="72"/>à centro G, æquè di&longs;tant, erit EG, æqualis GF. </s>

<s>Sit portio ABCD, &longs;phæræ, cuius centrum G, ab&longs;ci&longs;sa <lb/>duobus planis parallelis <lb/>centrum G intercipien<lb/>tibus, & æquè ab eo di<lb/>&longs;tantibus: &longs;ectiones <expan abbr="er&utilde;t">erunt</expan> <lb/>circuli minores, quorum <lb/>diametri &longs;int AD, BC <lb/>centra autem F,E, qui<lb/>bus axis portionis termi <lb/>nabitur, eritque ad pla<lb/>na vtriu&longs;que circuli per <lb/><figure id="id.043.01.158.1.jpg" xlink:href="043/01/158/1.jpg"/><lb/>pendicularis tran&longs;iens per centrum G: & quia illa plana <pb xlink:href="043/01/159.jpg" pagenum="72"/>à centro G, æquè di&longs;tant, erit EG, æqualis GF. </s>

<s>Dico <lb/>portionis ABCD centrum grauitatis e&longs;&longs;e G. </s>

<s>De&longs;cripta <lb/>enim figura, vt &longs;upra fecimus, intelligantur duo coni re<lb/>ctanguli GNO, GPQ, vertice G, communi, axibus <lb/>autem eorum EG, GF: & cylindrus LM, portioni cir<lb/>cum&longs;criptus circa eun<lb/>dem axim EF, cuius ba <lb/>&longs;is æqualis e&longs;t circulo <lb/>maximo: & &longs;umatur EH <lb/>ip&longs;ius EG, pars quar<lb/>ta, itemque FK, pars <lb/>quarta ip&longs;ius FG. </s>

<s>Quo<lb/>niam igitur conorum G <lb/>NO, PGO, axes FG, <lb/>GH, &longs;unt æquales, re<lb/>liquæ KG, GH, æqua <lb/><figure id="fig103"/><lb/>les erunt; centra autem grauitatis conorum &longs;unt K, H; pun<lb/>ctum igitur G e&longs;t centrum grauitatis compo&longs;iti ex duobus <lb/>conis æqualibus GNO, GPQ, hoc e&longs;t reliqui ex cylin<lb/>dro LM, dempta ABCD, portione, ex ante demon&longs;tra<lb/>tis: &longs;ed idem G e&longs;t centrum grauitatis totius cylindri LM; <lb/>reliquæ igitur ABCD, portionis centrum grauitatis erit <lb/>G. </s>

<s>Quo<lb/>niam igitur conorum G <lb/>NO, PGO, axes FG, <lb/>GH, &longs;unt æquales, re<lb/>liquæ KG, GH, æqua <lb/><figure id="id.043.01.159.1.jpg" xlink:href="043/01/159/1.jpg"/><lb/>les erunt; centra autem grauitatis conorum &longs;unt K, H; pun<lb/>ctum igitur G e&longs;t centrum grauitatis compo&longs;iti ex duobus <lb/>conis æqualibus GNO, GPQ, hoc e&longs;t reliqui ex cylin<lb/>dro LM, dempta ABCD, portione, ex ante demon&longs;tra<lb/>tis: &longs;ed idem G e&longs;t centrum grauitatis totius cylindri LM; <lb/>reliquæ igitur ABCD, portionis centrum grauitatis erit <lb/>G. </s>

<s>Quod demon&longs;trandum erat. </s>

<s><emph type="italics"/>PROPOSITIO XL.<emph.end type="italics"/></s>

<s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla<lb/>nis parallelis centrum intercipientibus, & à cen<lb/>tro non æqualiter di&longs;tantibus centrum grauitatis <lb/>e&longs;t in axe primum bifariam &longs;ecto: Deinde &longs;umpta <lb/>ad minorem ba&longs;im portionis quarta parte &longs;egmen <lb/>ti axis, quod minorem ba&longs;im attingit: & ad maio-<pb pagenum="73"/>rem ba&longs;im quarta parte reliqui &longs;egmenti axis eo<lb/>rum, quæ à centro &longs;phæræ fiunt: Deinde recta <lb/>inter has quartas partes interiecta ita diui&longs;a, vt <lb/>pars maiori ba&longs;i propinquior &longs;it ad reliquam vt <lb/>cubus &longs;egmenti axis inter &longs;phæræ centrum, & mi<lb/>norem ba&longs;im, ad cubum eius, quod inter &longs;phæræ <lb/>centrum, & maiorem ba&longs;im portionis interijci<lb/>tur; in eo puncto, in quo &longs;egmentum axis centro <lb/>&longs;phæræ, & penultima &longs;ectione terminatum &longs;ic di<lb/>uiditur, vt pars quæ penultima, & prima &longs;ectione <lb/>terminatur &longs;it ad totam vltima, & penultima &longs;e<lb/>ctione terminatam, vt ad axim portionis e&longs;t exce&longs; <lb/>&longs;us, quo idem axis portionis &longs;uperat <expan abbr="tertiã">tertiam</expan> partem <lb/>compo&longs;itæ ex duabus minoribus extremis, maio<lb/>ribus po&longs;itis duobus axis &longs;egmentis, quæ fiunt à <lb/>centro &longs;phæræ in rationibus &longs;emidiametri &longs;phæ<lb/>ræ ad prædicta &longs;egmenta. </s><figure/>

<s>Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla<lb/>nis parallelis centrum intercipientibus, & à cen<lb/>tro non æqualiter di&longs;tantibus centrum grauitatis <lb/>e&longs;t in axe primum bifariam &longs;ecto: Deinde &longs;umpta <lb/>ad minorem ba&longs;im portionis quarta parte &longs;egmen <lb/>ti axis, quod minorem ba&longs;im attingit: & ad maio-<pb xlink:href="043/01/160.jpg" pagenum="73"/>rem ba&longs;im quarta parte reliqui &longs;egmenti axis eo<lb/>rum, quæ à centro &longs;phæræ fiunt: Deinde recta <lb/>inter has quartas partes interiecta ita diui&longs;a, vt <lb/>pars maiori ba&longs;i propinquior &longs;it ad reliquam vt <lb/>cubus &longs;egmenti axis inter &longs;phæræ centrum, & mi<lb/>norem ba&longs;im, ad cubum eius, quod inter &longs;phæræ <lb/>centrum, & maiorem ba&longs;im portionis interijci<lb/>tur; in eo puncto, in quo &longs;egmentum axis centro <lb/>&longs;phæræ, & penultima &longs;ectione terminatum &longs;ic di<lb/>uiditur, vt pars quæ penultima, & prima &longs;ectione <lb/>terminatur &longs;it ad totam vltima, & penultima &longs;e<lb/>ctione terminatam, vt ad axim portionis e&longs;t exce&longs; <lb/>&longs;us, quo idem axis portionis &longs;uperat <expan abbr="tertiã">tertiam</expan> partem <lb/>compo&longs;itæ ex duabus minoribus extremis, maio<lb/>ribus po&longs;itis duobus axis &longs;egmentis, quæ fiunt à <lb/>centro &longs;phæræ in rationibus &longs;emidiametri &longs;phæ<lb/>ræ ad prædicta &longs;egmenta. </s><figure id="id.043.01.160.1.jpg" xlink:href="043/01/160/1.jpg"/>

<s>Sit portio ABCD &longs;phæræ, cuius centrum G, abci&longs;&longs;a <lb/>duobus planis parallelis centrum G intercipien<gap/>ibus, & <pb pagenum="74"/>ab eo non æqualiter di&longs;tantibus: & axis portionis &longs;it EF, <lb/>qui per centrum G tran&longs;ibit, vtpote parallelorum circu<lb/>lorum centra iungens: cumque eorum vtrumque &longs;it à cen<lb/>tro non æqualiter di&longs;tantium perpendicularis, erunt eius <lb/>&longs;egmenta EG, GF, inæqualia. </s>

<s>Sit portio ABCD &longs;phæræ, cuius centrum G, abci&longs;&longs;a <lb/>duobus planis parallelis centrum G intercipien<gap/>ibus, & <pb xlink:href="043/01/161.jpg" pagenum="74"/>ab eo non æqualiter di&longs;tantibus: & axis portionis &longs;it EF, <lb/>qui per centrum G tran&longs;ibit, vtpote parallelorum circu<lb/>lorum centra iungens: cumque eorum vtrumque &longs;it à cen<lb/>tro non æqualiter di&longs;tantium perpendicularis, erunt eius <lb/>&longs;egmenta EG, GF, inæqualia. </s>

<s>E&longs;to EG, maius: &longs;ectoque <lb/>axe EF bifariam in puncto P, &longs;umptisque ip&longs;arum EG, <lb/>GF, quartis partibus EH, FK, &longs;ecetur interiecta <emph type="italics"/>K<emph.end type="italics"/>H, <lb/>in puncto Q, ita vt KQ, ad QH, &longs;it vt cubus ex EG, <lb/>ad cubum ex GF, & portionis ABCD, &longs;it centrum gra<lb/>uitatis R: quod quidem cum punctis P, Q, e&longs;&longs;e in axe <lb/><figure id="fig104"/><lb/>EF: & cylindro LM, &longs;uper ba&longs;im æqualem circulo ma<lb/>ximo circa axim EF, portioni circum&longs;cripto, reliqui eius <lb/>dempta ABCD, portione centrum grauitatis e&longs;se Q, & <lb/>propinquius E puncto, quàm centrum grauitatis R por<lb/>tionis ABCD, manife&longs;tum e&longs;t ex &longs;upra demon&longs;tratis de <lb/>maioris portionis &longs;phæræ centro grauitatis: portionis autem <lb/>ABCD centrum grauitatis R e&longs;se in &longs;egmento EG &longs;e<lb/>quitur ex antecedente. </s>

<s>E&longs;to EG, maius: &longs;ectoque <lb/>axe EF bifariam in puncto P, &longs;umptisque ip&longs;arum EG, <lb/>GF, quartis partibus EH, FK, &longs;ecetur interiecta <emph type="italics"/>K<emph.end type="italics"/>H, <lb/>in puncto Q, ita vt KQ, ad QH, &longs;it vt cubus ex EG, <lb/>ad cubum ex GF, & portionis ABCD, &longs;it centrum gra<lb/>uitatis R: quod quidem cum punctis P, Q, e&longs;&longs;e in axe <lb/><figure id="id.043.01.161.1.jpg" xlink:href="043/01/161/1.jpg"/><lb/>EF: & cylindro LM, &longs;uper ba&longs;im æqualem circulo ma<lb/>ximo circa axim EF, portioni circum&longs;cripto, reliqui eius <lb/>dempta ABCD, portione centrum grauitatis e&longs;se Q, & <lb/>propinquius E puncto, quàm centrum grauitatis R por<lb/>tionis ABCD, manife&longs;tum e&longs;t ex &longs;upra demon&longs;tratis de <lb/>maioris portionis &longs;phæræ centro grauitatis: portionis autem <lb/>ABCD centrum grauitatis R e&longs;se in &longs;egmento EG &longs;e<lb/>quitur ex antecedente. </s>

<s>Dico PQ ad QR e&longs;se vt ad axim <lb/>EF exce&longs;sus, quo axis EF &longs;uperat tertiam partem com<lb/>po&longs;itæ <gap/> duabus minoribus extremis altera re&longs;pondente <lb/>maiori extrema EG in proportione continua ip&longs;ius NG <pb pagenum="75"/>ad GE, altera maiori extremæ FG in proportione con<lb/>tinua ip&longs;ius NG ad GF. </s>

<s>Dico PQ ad QR e&longs;se vt ad axim <lb/>EF exce&longs;sus, quo axis EF &longs;uperat tertiam partem com<lb/>po&longs;itæ <gap/> duabus minoribus extremis altera re&longs;pondente <lb/>maiori extrema EG in proportione continua ip&longs;ius NG <pb xlink:href="043/01/162.jpg" pagenum="75"/>ad GE, altera maiori extremæ FG in proportione con<lb/>tinua ip&longs;ius NG ad GF. </s>

<s>Quoniam enim ob centra gra<lb/>uitatis QPR e&longs;t vt QP ad PR, ita portio ABCD ad <lb/>reliquum cylindri LM, erit componendo, & per conuer<lb/>&longs;ionem rationis, & conuertendo, vt PQ ad QR, ita por<lb/>tio ABCD ad LM cylindrum: &longs;ed portio ABCD ad <lb/>LM cylindrum e&longs;t vt prædictus exce&longs;&longs;us ad axim EF; <lb/>vtigitur prædictus exce&longs;&longs;us ad axim EF, ita e&longs;t PQ ad <lb/>QR. </s>

Line 1958

<s>Dico E e&longs;se centrum grauitatis conoidis ABC. <lb/></s>

<s>Nam in &longs;ectione per <lb/>axim parabola ABC, <lb/>cuius diameter erit B <lb/>D, de&longs;cribatur rian<lb/>gulum ABC; &longs;um<lb/>ptisque ip&longs;ius BD æ<lb/>qualibus DH, HO, <lb/>per puncta H, O, &longs;e<lb/>centur vnà parabola <lb/>& triangulum ABC <lb/>duabus rectis FGH <lb/><figure id="fig105"/><lb/>KL, MNOPQ: & per eas rectas &longs;ecetur conoi<lb/>des ABC planis ba&longs;i parallelis, factæ autem &longs;e<lb/>ctiones erunt circuli circa FL, MQ, & in parabola <pb pagenum="76"/>ABC tres ad diametrum ordinatim applicatæ AD, <lb/>FH, MO. </s>

<s>Nam in &longs;ectione per <lb/>axim parabola ABC, <lb/>cuius diameter erit B <lb/>D, de&longs;cribatur rian<lb/>gulum ABC; &longs;um<lb/>ptisque ip&longs;ius BD æ<lb/>qualibus DH, HO, <lb/>per puncta H, O, &longs;e<lb/>centur vnà parabola <lb/>& triangulum ABC <lb/>duabus rectis FGH <lb/><figure id="id.043.01.162.1.jpg" xlink:href="043/01/162/1.jpg"/><lb/>KL, MNOPQ: & per eas rectas &longs;ecetur conoi<lb/>des ABC planis ba&longs;i parallelis, factæ autem &longs;e<lb/>ctiones erunt circuli circa FL, MQ, & in parabola <pb xlink:href="043/01/163.jpg" pagenum="76"/>ABC tres ad diametrum ordinatim applicatæ AD, <lb/>FH, MO. </s>

<s>Quoniam igitur tres rectæ OB, BH, BD <lb/>&longs;e&longs;e qualiter excedunt, quarum minima BO, maxi<lb/>ma e&longs;t BD, minor erit proportio BO ad BH, quàm <lb/>BH ad BD; hoc e&longs;t NP ad GK, quàm GKad AC. <lb/>&longs;ed vt OB ad BH hoc e&longs;t NO ad GH, vel NP ad <lb/>GK ita e&longs;t quadra<lb/>tum MO ad quadra<lb/>tum FH, hoc e&longs;t eo<lb/>no dis &longs;ectionum cir<lb/>culus MQ ad circu<lb/>lum FL: eademque <lb/>ratione vt GK ad <lb/>AC ita circulus FL <lb/>ad circulum AC; mi<lb/>nor igitur proportio <lb/>erit circuli MQ ad <lb/>circulum FL quàm <lb/><figure id="fig106"/><lb/>circuli FL ad circulum AC. </s>

<s>Quoniam igitur tres rectæ OB, BH, BD <lb/>&longs;e&longs;e qualiter excedunt, quarum minima BO, maxi<lb/>ma e&longs;t BD, minor erit proportio BO ad BH, quàm <lb/>BH ad BD; hoc e&longs;t NP ad GK, quàm GKad AC. <lb/>&longs;ed vt OB ad BH hoc e&longs;t NO ad GH, vel NP ad <lb/>GK ita e&longs;t quadra<lb/>tum MO ad quadra<lb/>tum FH, hoc e&longs;t eo<lb/>no dis &longs;ectionum cir<lb/>culus MQ ad circu<lb/>lum FL: eademque <lb/>ratione vt GK ad <lb/>AC ita circulus FL <lb/>ad circulum AC; mi<lb/>nor igitur proportio <lb/>erit circuli MQ ad <lb/>circulum FL quàm <lb/><figure id="id.043.01.163.1.jpg" xlink:href="043/01/163/1.jpg"/><lb/>circuli FL ad circulum AC. </s>

<s>Similiter autem o&longs;tende<lb/>remus ternas quaslibet alias ita factas &longs;ectiones trianguli, <lb/>& parabolæ ABC inter &longs;e & ba&longs;i parallelas proportio<lb/>nales e&longs;se, & minorem proportionem vtrobique minimæ <lb/>ad mediam, quàm mediæ ad maximam. </s>

Line 1970

<s><emph type="italics"/>PROPOSITIO XLII.<emph.end type="italics"/></s>

<s>Omnis fru&longs;ti conoidis parabolici centrum gra<lb/>uitatis axim ita diuidit, vt pars, quæ minorem <lb/>ba&longs;im attingit &longs;it ad reliquam; vt duplum maioris <pb pagenum="77"/>ba&longs;is vnà cum minori, ad duplum minoris, vnà <lb/>cum maiori. </s>

<s>Omnis fru&longs;ti conoidis parabolici centrum gra<lb/>uitatis axim ita diuidit, vt pars, quæ minorem <lb/>ba&longs;im attingit &longs;it ad reliquam; vt duplum maioris <pb xlink:href="043/01/164.jpg" pagenum="77"/>ba&longs;is vnà cum minori, ad duplum minoris, vnà <lb/>cum maiori. </s>

<s>Sit conoidis parabolici ABC, cuius axis BD fru&longs;tum <lb/>AEFC, eius maior ba&longs;is circulus, cuius diameter AC, mi<lb/>nor, cuius diameter EF: in eadem parabola per axem, axis <lb/><expan abbr="aut&etilde;">autem</expan> DG, in quo fru&longs;ti AEFC &longs;it centrum grauitatis H. <lb/></s>

<s>Dico e&longs;&longs;e vt duplum circuli AC, vnà cum circulo EF, ad <lb/>duplum circuli EF vna cum circulo AC, ita GH, ad HD. <lb/><expan abbr="Iungãtur">Iungantur</expan> enim re<lb/>ctæ AKB, BLC. <lb/></s>

<s>Quoniam igitur <lb/>qua ratione o&longs;ten <lb/>dimus conoides, <lb/>& triangulum A <lb/>BC, commune <lb/>habere in linea <lb/>BD centrum gra<lb/>uitatis, <expan abbr="ead&etilde;">eadem</expan> pror<lb/>&longs;us remanet de<lb/>mon&longs;tratum, fru&longs;ti <lb/><figure id="fig107"/><lb/>AEFC <expan abbr="centr&utilde;">centrum</expan> grauitatis H, idem e&longs;se quod trapezij AK <lb/>FC; erit duarum parallelarum AG, KL vt dupla ip&longs;ius <lb/>AC, vnà cum KL, ad duplam ip&longs;ius KL, vnà cum AC <lb/>ita GH ad HD: &longs;ecat enim DG ip&longs;as AC, KL bifa<lb/>riam. </s>

<s>Quoniam igitur <lb/>qua ratione o&longs;ten <lb/>dimus conoides, <lb/>& triangulum A <lb/>BC, commune <lb/>habere in linea <lb/>BD centrum gra<lb/>uitatis, <expan abbr="ead&etilde;">eadem</expan> pror<lb/>&longs;us remanet de<lb/>mon&longs;tratum, fru&longs;ti <lb/><figure id="id.043.01.164.1.jpg" xlink:href="043/01/164/1.jpg"/><lb/>AEFC <expan abbr="centr&utilde;">centrum</expan> grauitatis H, idem e&longs;se quod trapezij AK <lb/>FC; erit duarum parallelarum AG, KL vt dupla ip&longs;ius <lb/>AC, vnà cum KL, ad duplam ip&longs;ius KL, vnà cum AC <lb/>ita GH ad HD: &longs;ecat enim DG ip&longs;as AC, KL bifa<lb/>riam. </s>

<s>Sed vt AC ad <emph type="italics"/>K<emph.end type="italics"/>L ita e&longs;t circulus AC ad circu<lb/>lum EF, ex demon&longs;tratione antecedentis, hoc e&longs;t vt dupla <lb/>ip&longs;ius AC vnà cum KL ad duplam ip&longs;ius KL vnà cum <lb/>AC, ita duplum circuli AC vna cum circulo KL ad du<lb/>plum circuli KL vnà cum circulo AC; vt igitur e&longs;t du<lb/>plum circuli AC, vnà cum circulo EF, ad duplum circu<lb/>li EF, vnà cum circulo AC; ita erit GH ad HD. <lb/></s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="78"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/165.jpg" pagenum="78"/>

<s><emph type="italics"/>PROPOSITIO XLIII.<emph.end type="italics"/></s>

Line 1996

<s>Manente igitur BD, & circumductis figuris MBN, <lb/>KBL, de&longs;cribantur conoides parabolicum MBN, & <lb/>conus KBL, quorum communis axis erit BD, ba&longs;es <lb/>autem circuli, quorum diametri KL, MN, in eodem <lb/>plano cum ba&longs;e conoidis ABC. </s>

<s>Rur&longs;us &longs;ecto axe BD <lb/>bifariam, & &longs;ingulis eius partibus &longs;emper bifariam in qua-<pb pagenum="79"/>cumque multiplicatione; &longs;int duæ partes æquales proximæ <lb/>ba&longs;i DF, FQ: & per puncta FQ duo plana ba&longs;ium pla<lb/>no parallela tres prædictas figuras &longs;olidas &longs;ecare intelli<lb/>gantur: &longs;ecabunt autem & tres figuras per axim, eruntque <lb/>&longs;ectiones rectæ lineæ ad diametrum figurarum ordinatim <lb/>applicatæ propter <lb/>plana &longs;ecantia pa <lb/>rallela: trium au<lb/>tem &longs;olidorum &longs;e <lb/>ctiones & ba&longs;es <lb/>omnes circuli, ter <lb/>ni in &longs;ingulis pla<lb/>nis: ac primi qui<lb/>dem ordinis &longs;int <lb/>ij, quorum diame<lb/>tri &longs;unt ba&longs;es <expan abbr="tri&utilde;">trium</expan> <lb/><expan abbr="figurar&utilde;">figurarum</expan> per axim, <lb/>trianguli &longs;cilicet, <lb/>parabolæ, & hy<lb/>perboles, quæ præ <lb/>dictas figuras &longs;oli <lb/>das de&longs;cribunt, re <lb/>ctæ lineæ AC, <lb/>MN, KL. </s>

<s>Rur&longs;us &longs;ecto axe BD <lb/>bifariam, & &longs;ingulis eius partibus &longs;emper bifariam in qua-<pb xlink:href="043/01/166.jpg" pagenum="79"/>cumque multiplicatione; &longs;int duæ partes æquales proximæ <lb/>ba&longs;i DF, FQ: & per puncta FQ duo plana ba&longs;ium pla<lb/>no parallela tres prædictas figuras &longs;olidas &longs;ecare intelli<lb/>gantur: &longs;ecabunt autem & tres figuras per axim, eruntque <lb/>&longs;ectiones rectæ lineæ ad diametrum figurarum ordinatim <lb/>applicatæ propter <lb/>plana &longs;ecantia pa <lb/>rallela: trium au<lb/>tem &longs;olidorum &longs;e <lb/>ctiones & ba&longs;es <lb/>omnes circuli, ter <lb/>ni in &longs;ingulis pla<lb/>nis: ac primi qui<lb/>dem ordinis &longs;int <lb/>ij, quorum diame<lb/>tri &longs;unt ba&longs;es <expan abbr="tri&utilde;">trium</expan> <lb/><expan abbr="figurar&utilde;">figurarum</expan> per axim, <lb/>trianguli &longs;cilicet, <lb/>parabolæ, & hy<lb/>perboles, quæ præ <lb/>dictas figuras &longs;oli <lb/>das de&longs;cribunt, re <lb/>ctæ lineæ AC, <lb/>MN, KL. </s>

<s>Se<lb/>cundi verò reten<lb/>to eodem ordine <lb/><expan abbr="figurar&utilde;">figurarum</expan> tres <foreign lang="greek">az, <lb/>be, gd. </foreign></s>

<s>Tertij <lb/>denique ordinis <lb/>SZ, TY, VX. <lb/><figure id="fig108"/><lb/>Quoniam igitur e&longs;t vt EB, ad BD, ità quadratum MD, <lb/>ad quadratum DK, ide&longs;t conus MBN, &longs;i de&longs;cribatur eo<lb/>dem vertice B, ad conum KBL. </s>

<s>Tertij <lb/>denique ordinis <lb/>SZ, TY, VX. <lb/><figure id="id.043.01.166.1.jpg" xlink:href="043/01/166/1.jpg"/><lb/>Quoniam igitur e&longs;t vt EB, ad BD, ità quadratum MD, <lb/>ad quadratum DK, ide&longs;t conus MBN, &longs;i de&longs;cribatur eo<lb/>dem vertice B, ad conum KBL. </s>

<s>Et vt IB, ad BE, ità e&longs;t <lb/>conoides MBN, ad conum MBN, in proportione &longs;cili-<pb pagenum="80"/>cet &longs;e&longs;quialtera; ex æquali erit vt IB, ad BD, itì conoi<lb/>des MBN ad conum KBL: Sed vt IB, ad BD, ità <lb/>ponitur QH ad HG; vt igitur conoides MBN, ad co<lb/>num KBL, ità e&longs;t QH ad HG. </s>

<s>Et vt IB, ad BE, ità e&longs;t <lb/>conoides MBN, ad conum MBN, in proportione &longs;cili-<pb xlink:href="043/01/167.jpg" pagenum="80"/>cet &longs;e&longs;quialtera; ex æquali erit vt IB, ad BD, itì conoi<lb/>des MBN ad conum KBL: Sed vt IB, ad BD, ità <lb/>ponitur QH ad HG; vt igitur conoides MBN, ad co<lb/>num KBL, ità e&longs;t QH ad HG. </s>

<s>Sed Q e&longs;t centrum <lb/>grauitatis coni KBL, & G conoidis MBN; compo&longs;i<lb/>ti igitur ex conoi<lb/>de MBN, & co<lb/>no KBL <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis erit H. <lb/></s>

<s>Rur&longs;us quoniam <lb/>tres rectæ lineæ B <lb/>D, BF, BQ, æ<lb/>qualibus exce&longs;&longs;i<lb/>bus inter &longs;e diffe<lb/>runt, minor erit <lb/>proportio BQ, ad <lb/>BF, quàm BF, <lb/>ad BD, hoc e&longs;t <lb/>rectanguli EBQ, <lb/>ad rectangulum <lb/>EBF, quàm re<lb/>ctanguli EBF, ad <lb/>rectangulum EB <lb/>D. </s>

<s>Sed quadrati <lb/>BQ, ad quadra<lb/>tum BF, dupli<lb/>cata e&longs;t proportio <lb/>lateris BQ ad l<gap/><lb/>tus BF: hoc e&longs;t <lb/>rectanguli EBQ <lb/><figure id="fig109"/><lb/>ad rectangulum EBF: & quadrati BF, ad quadratum <lb/>BD duplicata eius, quæ e&longs;t rectanguli EBF, ad rectan<lb/>gulum EBD; compo&longs;itis igitur primis cum &longs;ecundis, mi<lb/>nor erit proportio rectanguli BQE, ad rectangulum BFE, <pb pagenum="81"/>quàm rectanguli BFE, ad rectangulum BDE. </s>

<s>Sed quadrati <lb/>BQ, ad quadra<lb/>tum BF, dupli<lb/>cata e&longs;t proportio <lb/>lateris BQ ad l<gap/><lb/>tus BF: hoc e&longs;t <lb/>rectanguli EBQ <lb/><figure id="id.043.01.167.1.jpg" xlink:href="043/01/167/1.jpg"/><lb/>ad rectangulum EBF: & quadrati BF, ad quadratum <lb/>BD duplicata eius, quæ e&longs;t rectanguli EBF, ad rectan<lb/>gulum EBD; compo&longs;itis igitur primis cum &longs;ecundis, mi<lb/>nor erit proportio rectanguli BQE, ad rectangulum BFE, <pb xlink:href="043/01/168.jpg" pagenum="81"/>quàm rectanguli BFE, ad rectangulum BDE. </s>

<s>Sed vt <lb/>rectangulum BQE ad rectangulum BFE, ita e&longs;t quadra<lb/>tum SQ ad quadratum <foreign lang="greek">a</foreign>F: & vt rectangulum BFE <lb/>ad rectangulum BDE, ita quadratum <foreign lang="greek">a</foreign>F, ad quadra<lb/>tum AD; minor igitur proportio erit quadrati SQ, ad <lb/>quadratum <foreign lang="greek">a</foreign>F, quàm quadrati <foreign lang="greek">a</foreign>F ad quadratum AD. <lb/></s>

<s>Sed vt quadratum SQ ad quadratum <foreign lang="greek">a</foreign>F, ita e&longs;t qua<lb/>dratum SZ ad quadratum <foreign lang="greek">a</foreign><37>: & vt quadratum <foreign lang="greek">a</foreign>F ad <lb/>quadratum AD ita quadratum <foreign lang="greek">az</foreign> ad quadratum <lb/>AC; minor igitur proportio erit quadrati SZ ad quadra<lb/>tum <foreign lang="greek">az</foreign>, quàm quadrati <foreign lang="greek">az</foreign>, ad quadratum AC, hoc e&longs;t <lb/>circuli SZ ad circulum <foreign lang="greek">a</foreign><37>, quàm circuli <foreign lang="greek">a</foreign><37>, ad cir<lb/>culum AC; qui circuli &longs;unt &longs;ectiones conoidis ABC <lb/>po&longs;iti vt in propo&longs;itionibus lemmaticis dicebamus. </s>

<s>Rur&longs;us <lb/>quoniam &longs;unt quatuor primæ proportionales; vt rectangu<lb/>lum DBE ad rectangulum FBE, ita MD quadratum <lb/>ad quadratum <foreign lang="greek">b</foreign>F: & totidem &longs;ecundæ, vt quadratum <lb/>BD, ad quadratum BF, ita quadratum DK, ad quadra<lb/>tum F<foreign lang="greek">g</foreign>, ob &longs;imilium triangulorum latera proportionalia: <lb/>&longs;ed vt EB, ad BD, hoc e&longs;t rectangulum DBE prima in <lb/>primis ad quadratum BD primam in &longs;ecundis, ita e&longs;t <lb/>quadratum MD tertia in primis ad quadratum DK ter<lb/>tiam in &longs;ecundis; vt igitur compo&longs;ita ex primis ad com<lb/>po&longs;itam ex &longs;ecundis, ità erit compo&longs;ita ex tertijs ad com<lb/>po&longs;itam ex quartis; videlicet vt rectangulum DBE <lb/>vnà cum quadrato BD, hoc e&longs;t rectangulum BDE <lb/>ad rectangulum BFE, hoc e&longs;t vt quadratum AD, ad <lb/>quadratum <foreign lang="greek">a</foreign>F, ità compo&longs;itum ex quadratis MD, DK, <lb/>ad compo&longs;itum ex quadratis <foreign lang="greek">b</foreign>F, F<foreign lang="greek">g</foreign>: & quadrupla vtro<lb/>rumque, vt quadratum AC, ad quadratum <foreign lang="greek">a</foreign><37>, ità com<lb/>po&longs;itum ex quadratis MN, KL, ad compo&longs;itum ex qua<lb/>dratis <foreign lang="greek">be, gd</foreign>; hoc e&longs;t eorum circulorum, qui &longs;unt &longs;ectio<lb/>nes &longs;olidorum, vt circulus AC, ad circulum <foreign lang="greek">a</foreign><37>, ità com<lb/>po&longs;itum ex circulis MN, KL, ad compo&longs;itum ex circu<pb pagenum="82"/>lis <foreign lang="greek">be, gd. </foreign></s>

<s>Rur&longs;us <lb/>quoniam &longs;unt quatuor primæ proportionales; vt rectangu<lb/>lum DBE ad rectangulum FBE, ita MD quadratum <lb/>ad quadratum <foreign lang="greek">b</foreign>F: & totidem &longs;ecundæ, vt quadratum <lb/>BD, ad quadratum BF, ita quadratum DK, ad quadra<lb/>tum F<foreign lang="greek">g</foreign>, ob &longs;imilium triangulorum latera proportionalia: <lb/>&longs;ed vt EB, ad BD, hoc e&longs;t rectangulum DBE prima in <lb/>primis ad quadratum BD primam in &longs;ecundis, ita e&longs;t <lb/>quadratum MD tertia in primis ad quadratum DK ter<lb/>tiam in &longs;ecundis; vt igitur compo&longs;ita ex primis ad com<lb/>po&longs;itam ex &longs;ecundis, ità erit compo&longs;ita ex tertijs ad com<lb/>po&longs;itam ex quartis; videlicet vt rectangulum DBE <lb/>vnà cum quadrato BD, hoc e&longs;t rectangulum BDE <lb/>ad rectangulum BFE, hoc e&longs;t vt quadratum AD, ad <lb/>quadratum <foreign lang="greek">a</foreign>F, ità compo&longs;itum ex quadratis MD, DK, <lb/>ad compo&longs;itum ex quadratis <foreign lang="greek">b</foreign>F, F<foreign lang="greek">g</foreign>: & quadrupla vtro<lb/>rumque, vt quadratum AC, ad quadratum <foreign lang="greek">a</foreign><37>, ità com<lb/>po&longs;itum ex quadratis MN, KL, ad compo&longs;itum ex qua<lb/>dratis <foreign lang="greek">be, gd</foreign>; hoc e&longs;t eorum circulorum, qui &longs;unt &longs;ectio<lb/>nes &longs;olidorum, vt circulus AC, ad circulum <foreign lang="greek">a</foreign><37>, ità com<lb/>po&longs;itum ex circulis MN, KL, ad compo&longs;itum ex circu<pb xlink:href="043/01/169.jpg" pagenum="82"/>lis <foreign lang="greek">be, gd. </foreign></s>

<s>Eadem ratione erit vt circulus AC, ad cir<lb/>culum SZ, ità compo&longs;itum ex circulis MN, KL, ad <lb/>compo&longs;itum ex circulis TY, VX: & conuertendo, & ex <lb/>æquali, vt circulus SZ, ad circulum <foreign lang="greek">a</foreign><37>, ità compo&longs;itum <lb/>ex circulis TY, VX, ad compo&longs;itum ex circulis <foreign lang="greek">be, gd</foreign>: <lb/>& vt circulus <foreign lang="greek">a</foreign><37>, <lb/>ad circulum AC, <lb/>ità <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex <lb/>circulis <foreign lang="greek">be, gd</foreign>, <lb/>ad <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex <lb/>circulis MN, <emph type="italics"/>K<emph.end type="italics"/><lb/>L. </s>

<s>Sunt igitur tria <lb/>compo&longs;ita ex bi<lb/>nis &longs;ectionibus cir <lb/>culis, & totidem <lb/>alij circuli, quos <lb/>diximus in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione, &longs;i bi<lb/>na <expan abbr="&longs;umãtur">&longs;umantur</expan> in &longs;in <lb/>gulis planis &longs;ecan <lb/>tibus: eorum au<lb/>tem minor erat <lb/>proportio circuli <lb/>SZ ad circulum <lb/><foreign lang="greek">a</foreign><37>, quàm circuli <lb/><foreign lang="greek">a</foreign><37>, ad circulum <lb/>AC; minor igitur <lb/>proportio erit <expan abbr="cõ-po&longs;iti">con<lb/>po&longs;iti</expan> ex circulis <lb/>T<foreign lang="greek">*u</foreign>, VX, ad <expan abbr="cõ-po&longs;itum">con<lb/>po&longs;itum</expan> ex circu<lb/><figure id="fig110"/><lb/>lis <foreign lang="greek">be, gd</foreign>, quàm compo&longs;iti ex circulis <foreign lang="greek">be, gd</foreign>, ad com <lb/>po&longs;itum ex circulis MN, KL. </s>

<s>Sunt igitur tria <lb/>compo&longs;ita ex bi<lb/>nis &longs;ectionibus cir <lb/>culis, & totidem <lb/>alij circuli, quos <lb/>diximus in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione, &longs;i bi<lb/>na <expan abbr="&longs;umãtur">&longs;umantur</expan> in &longs;in <lb/>gulis planis &longs;ecan <lb/>tibus: eorum au<lb/>tem minor erat <lb/>proportio circuli <lb/>SZ ad circulum <lb/><foreign lang="greek">a</foreign><37>, quàm circuli <lb/><foreign lang="greek">a</foreign><37>, ad circulum <lb/>AC; minor igitur <lb/>proportio erit <expan abbr="cõ-po&longs;iti">con<lb/>po&longs;iti</expan> ex circulis <lb/>T<foreign lang="greek">*u</foreign>, VX, ad <expan abbr="cõ-po&longs;itum">con<lb/>po&longs;itum</expan> ex circu<lb/><figure id="id.043.01.169.1.jpg" xlink:href="043/01/169/1.jpg"/><lb/>lis <foreign lang="greek">be, gd</foreign>, quàm compo&longs;iti ex circulis <foreign lang="greek">be, gd</foreign>, ad com <lb/>po&longs;itum ex circulis MN, KL. </s>

<s>Hac eadem ratione ad verti<lb/>cem deinceps progredienti manife&longs;tum erit, omnium com-<pb pagenum="83"/>po&longs;itorum ex binis &longs;ectionibus nempe circulis, quorum al<lb/>ter ad conum KBL pertinet, alter ad conoides MBN, in <lb/>eodem plano &longs;ecante prædictorum inter &longs;e parallelorum <lb/>exi&longs;tentibus, minorem e&longs;&longs;e proportionem incipienti ab eo, <lb/>quod e&longs;t proximum vertici, primi ad &longs;ecundum, quàm &longs;e<lb/>cundi ad tertium, & &longs;ecundi ad tertium, quàm tertij ad <lb/>quartum, & &longs;ic &longs;emper deinceps v&longs;que ad maximum & vl<lb/>timum compo&longs;itum ex circulis MN, KL: & eandem di<lb/>ctas &longs;ectiones compo&longs;itas ex coni, & conoidis parabolici <lb/>&longs;ectionibus inter &longs;e habere proportionem, quàm habent in<lb/>ter &longs;e circuli &longs;ectiones conoidis ABC, pro vt illis in <lb/>ij&longs;dem planis &longs;ecantibus, & æqualia axis BD &longs;egmenta <lb/>intercipientibus re&longs;pondent: Igitur per trige&longs;imam &longs;ecun<lb/>dam huius, & &longs;equens eam Corollarium, conoides ABC, <lb/>& compo&longs;itum ex conoide MBN, & cono BKL, com<lb/>mune habebunt in axe BD centrum grauitatis. </s>

<s>Hac eadem ratione ad verti<lb/>cem deinceps progredienti manife&longs;tum erit, omnium com-<pb xlink:href="043/01/170.jpg" pagenum="83"/>po&longs;itorum ex binis &longs;ectionibus nempe circulis, quorum al<lb/>ter ad conum KBL pertinet, alter ad conoides MBN, in <lb/>eodem plano &longs;ecante prædictorum inter &longs;e parallelorum <lb/>exi&longs;tentibus, minorem e&longs;&longs;e proportionem incipienti ab eo, <lb/>quod e&longs;t proximum vertici, primi ad &longs;ecundum, quàm &longs;e<lb/>cundi ad tertium, & &longs;ecundi ad tertium, quàm tertij ad <lb/>quartum, & &longs;ic &longs;emper deinceps v&longs;que ad maximum & vl<lb/>timum compo&longs;itum ex circulis MN, KL: & eandem di<lb/>ctas &longs;ectiones compo&longs;itas ex coni, & conoidis parabolici <lb/>&longs;ectionibus inter &longs;e habere proportionem, quàm habent in<lb/>ter &longs;e circuli &longs;ectiones conoidis ABC, pro vt illis in <lb/>ij&longs;dem planis &longs;ecantibus, & æqualia axis BD &longs;egmenta <lb/>intercipientibus re&longs;pondent: Igitur per trige&longs;imam &longs;ecun<lb/>dam huius, & &longs;equens eam Corollarium, conoides ABC, <lb/>& compo&longs;itum ex conoide MBN, & cono BKL, com<lb/>mune habebunt in axe BD centrum grauitatis. </s>

<s>Sed H <lb/>erat huius compo&longs;iti centrum grauitatis; Igitur conoidis <lb/>ABC centrum grauitatis erit idem H. </s>

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<s><emph type="italics"/>COROLLARIV M.<emph.end type="italics"/></s>

<s>Eadem demon&longs;tratione con&longs;tat &longs;i prædicta tria <lb/>&longs;olida ita vt diximus di&longs;po&longs;ita &longs;ecentur plano ba<lb/>&longs;ibus parallelo; &longs;ru&longs;tum conoidis hyperbolici, & <lb/>compo&longs;itum ex fru&longs;tis coni, & conoidis paraboli<lb/>ci, commune habere in communi axe centrum <lb/>grauitatis. </s><pb pagenum="84"/>

<s><emph type="italics"/>PROPOSITIO XLIV.<emph.end type="italics"/></s>

<s>Si conus & conoides parabolicum circa eun<lb/>dem axim &longs;ecentur plano ba&longs;i parallelo; fru&longs;ti co<lb/>nici ab&longs;ci&longs;&longs;i maiori ba&longs;i propinquius erit quàm <lb/>parabolici centrum grauitatis. </s>

<s>Sint conus ABC, & conoides parabolicum EBF, <lb/>quorum communis <lb/>axis BD, cuius per <lb/>quoduis punctum M, <lb/>planum &longs;ecans ea cor <lb/>pora plano ba&longs;ium, <lb/>quarum diametri A <lb/>C, EF, parallelo ab<lb/>&longs;cindat fru&longs;ta AKL <lb/>C, cuius centrum gra<lb/>uitatis N, & EGH <lb/>F, cuius centrum gra <lb/><figure id="fig111"/><lb/>uitatis O, quorum vtrumque erit in communi axe DM. <lb/></s>

<s>Sint conus ABC, & conoides parabolicum EBF, <lb/>quorum communis <lb/>axis BD, cuius per <lb/>quoduis punctum M, <lb/>planum &longs;ecans ea cor <lb/>pora plano ba&longs;ium, <lb/>quarum diametri A <lb/>C, EF, parallelo ab<lb/>&longs;cindat fru&longs;ta AKL <lb/>C, cuius centrum gra<lb/>uitatis N, & EGH <lb/>F, cuius centrum gra <lb/><figure id="id.043.01.171.1.jpg" xlink:href="043/01/171/1.jpg"/><lb/>uitatis O, quorum vtrumque erit in communi axe DM. <lb/></s>

<s>Dico punctum N, propinquius e&longs;se ip&longs;i D quàm punctum <lb/>O. </s>

<s>Quoniam enim e&longs;t parabolicifru&longs;ti EGHF centrum <lb/>grauitatis O; erit vt duplum maioris ba&longs;is, ide&longs;t circuli <lb/>EF vna cum minori circulo GH, ad duplum circuli GH <lb/>vna cum circulo EF, hoc e&longs;t vt duplum quadrati ED vna <lb/>cum quadrato ED ita MO ad OD. </s>

<s>Sed vt quadratum <lb/>ED ad quadratum GM in parabola quæ conoides de<lb/>&longs;cribit, cuius diameter BD, ita e&longs;t DB ad BM, hoc e&longs;t <lb/>AC ad KL; vt igitur e&longs;t dupla ip&longs;ius AC vna cum KL <lb/>ad duplam ip&longs;ius KL vna cum AC ita erit MO ad OD: <lb/>&longs;ed N e&longs;t fru&longs;ti conoici AKLC, centrum grauitatis; pun<lb/>ctum igitur N, erit maiori ba&longs;i AC propinquius quàm <pb pagenum="85"/>punctum O; e&longs;t autem O, fru&longs;ti EGHF centrum graui<lb/>tatis. </s>

<s>Sed vt quadratum <lb/>ED ad quadratum GM in parabola quæ conoides de<lb/>&longs;cribit, cuius diameter BD, ita e&longs;t DB ad BM, hoc e&longs;t <lb/>AC ad KL; vt igitur e&longs;t dupla ip&longs;ius AC vna cum KL <lb/>ad duplam ip&longs;ius KL vna cum AC ita erit MO ad OD: <lb/>&longs;ed N e&longs;t fru&longs;ti conoici AKLC, centrum grauitatis; pun<lb/>ctum igitur N, erit maiori ba&longs;i AC propinquius quàm <pb xlink:href="043/01/172.jpg" pagenum="85"/>punctum O; e&longs;t autem O, fru&longs;ti EGHF centrum graui<lb/>tatis. </s>

<s>Si igitur conus, & conoides parabolicum circa eun<lb/>dem axim, &c. </s>

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<s><emph type="italics"/>PROPOSITIO XLV.<emph.end type="italics"/></s>

<s>Omnis fru&longs;ti conoidis hyperbolici centrum <lb/>grauitatis e&longs;t in axe primum &longs;ecto &longs;ecundum cen<lb/>trum grauitatis cuiu&longs;uis fru&longs;ti conici circa axem <lb/>conoidis communi vertice, ab&longs;ci&longs;&longs;i vnà cum fru<lb/>&longs;to conoidis: deinde ita vt pars minorem ba&longs;im <lb/>attingens &longs;it ad reliquam, vt dupla axis conoidis <lb/>vna cum reliqua dempto axe fru&longs;ti, ad duplam <lb/>eiu&longs;dem reliquæ vna cum axe conoidis: dein<lb/>de po&longs;itis quatuor rectis lineis binis propor<lb/>tionalibus, potentia primis, &longs;ecundis longitu<lb/>dine, in proportione, quæ e&longs;t inter axem conoi<lb/>dis, & reliquam dempto axe fru&longs;ti; ita vt ma<lb/>ior primarum &longs;it media proportionalis inter axem <lb/>conoidis, & tran&longs;uer&longs;um latus hyperboles, quæ fi<lb/>guram de&longs;cribit, minoris autem potentia &longs;e&longs;qui<lb/>altera minor &longs;ecundarum; in eo puncto, in quo <lb/>&longs;egmentum axis fru&longs;ti dictis duabus &longs;ectionibus <lb/>terminatum &longs;ic diuiditur, vt pars minori ba&longs;i pro<lb/>pinquior &longs;it ad reliquam vt cubus, qui fit ab axe <lb/>fru&longs;ti vnà cum &longs;olido rectangulo, quod axe co<lb/>noidis, & reliqua dempto axe fru&longs;ti, & tripla <lb/>axis conoidis continetur, ad &longs;olidum rectangu<lb/>lum ex eadem reliqua parte conoidis, & eo, quo <pb pagenum="86"/>plus pote&longs;t quadrato maior quàm minor dicta<lb/>rum &longs;ecundarum. </s>

<s>Omnis fru&longs;ti conoidis hyperbolici centrum <lb/>grauitatis e&longs;t in axe primum &longs;ecto &longs;ecundum cen<lb/>trum grauitatis cuiu&longs;uis fru&longs;ti conici circa axem <lb/>conoidis communi vertice, ab&longs;ci&longs;&longs;i vnà cum fru<lb/>&longs;to conoidis: deinde ita vt pars minorem ba&longs;im <lb/>attingens &longs;it ad reliquam, vt dupla axis conoidis <lb/>vna cum reliqua dempto axe fru&longs;ti, ad duplam <lb/>eiu&longs;dem reliquæ vna cum axe conoidis: dein<lb/>de po&longs;itis quatuor rectis lineis binis propor<lb/>tionalibus, potentia primis, &longs;ecundis longitu<lb/>dine, in proportione, quæ e&longs;t inter axem conoi<lb/>dis, & reliquam dempto axe fru&longs;ti; ita vt ma<lb/>ior primarum &longs;it media proportionalis inter axem <lb/>conoidis, & tran&longs;uer&longs;um latus hyperboles, quæ fi<lb/>guram de&longs;cribit, minoris autem potentia &longs;e&longs;qui<lb/>altera minor &longs;ecundarum; in eo puncto, in quo <lb/>&longs;egmentum axis fru&longs;ti dictis duabus &longs;ectionibus <lb/>terminatum &longs;ic diuiditur, vt pars minori ba&longs;i pro<lb/>pinquior &longs;it ad reliquam vt cubus, qui fit ab axe <lb/>fru&longs;ti vnà cum &longs;olido rectangulo, quod axe co<lb/>noidis, & reliqua dempto axe fru&longs;ti, & tripla <lb/>axis conoidis continetur, ad &longs;olidum rectangu<lb/>lum ex eadem reliqua parte conoidis, & eo, quo <pb xlink:href="043/01/173.jpg" pagenum="86"/>plus pote&longs;t quadrato maior quàm minor dicta<lb/>rum &longs;ecundarum. </s>

<s>Sit conoidis hyperbolici ABC, cuius axis BD; & <lb/>tran&longs;uer&longs;um latus hyperboles, quæ figuram de&longs;cribit EB, <lb/>fru&longs;tum ALMC ab&longs;ci&longs;&longs;um vnà cum axe FD: cuius <lb/><figure id="fig112"/><lb/>ba&longs;es oppo&longs;itæ, maior circulus circa AC, minor circa LM: <lb/>&longs;ecto autem axe FD primum &longs;ecundum G centrum gra<lb/>uitatis fru&longs;ti ab&longs;ci&longs;&longs;i vnà cum fru&longs;to ALMC à quouis co <lb/>no, cuius axis BD, & vertex B, deinde in puncto H ita <lb/>vt FH ad HD &longs;it vt dupla ip&longs;ius BD vnà cum BF ad <lb/>duplam ip&longs;ius BF vnà cum BD, quo facto cadet G <lb/>punctum infra punctum H, ponantur vt DB ad BF, <pb pagenum="87"/>ita N ad O potentia, & Q ad P longitudine: &longs;it au<lb/>tem N media proportionalis inter EB, BD, at P ip&longs;ius <lb/>O potentia &longs;e&longs;quialtera: quo autem Q plus pote&longs;t quàm <lb/>P &longs;it quadratum ex R: & vt cubus ex FD vna cum &longs;oli<lb/>do rectangulo ex BF, FD, & tripla ip&longs;ius BD, ad &longs;oli<lb/>dum rectangulum ex BF, & quadrato R, ita &longs;it HK ad <lb/>KG. </s>

<s>Sit conoidis hyperbolici ABC, cuius axis BD; & <lb/>tran&longs;uer&longs;um latus hyperboles, quæ figuram de&longs;cribit EB, <lb/>fru&longs;tum ALMC ab&longs;ci&longs;&longs;um vnà cum axe FD: cuius <lb/><figure id="id.043.01.173.1.jpg" xlink:href="043/01/173/1.jpg"/><lb/>ba&longs;es oppo&longs;itæ, maior circulus circa AC, minor circa LM: <lb/>&longs;ecto autem axe FD primum &longs;ecundum G centrum gra<lb/>uitatis fru&longs;ti ab&longs;ci&longs;&longs;i vnà cum fru&longs;to ALMC à quouis co <lb/>no, cuius axis BD, & vertex B, deinde in puncto H ita <lb/>vt FH ad HD &longs;it vt dupla ip&longs;ius BD vnà cum BF ad <lb/>duplam ip&longs;ius BF vnà cum BD, quo facto cadet G <lb/>punctum infra punctum H, ponantur vt DB ad BF, <pb xlink:href="043/01/174.jpg" pagenum="87"/>ita N ad O potentia, & Q ad P longitudine: &longs;it au<lb/>tem N media proportionalis inter EB, BD, at P ip&longs;ius <lb/>O potentia &longs;e&longs;quialtera: quo autem Q plus pote&longs;t quàm <lb/>P &longs;it quadratum ex R: & vt cubus ex FD vna cum &longs;oli<lb/>do rectangulo ex BF, FD, & tripla ip&longs;ius BD, ad &longs;oli<lb/>dum rectangulum ex BF, & quadrato R, ita &longs;it HK ad <lb/>KG. </s>

<s>Dico fru&longs;ti ALMC centrum grauitatis e&longs;&longs;e K. <lb/></s>

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<s>Quoniam igitur circuli inter &longs;e &longs;unt vt quæ fiunt à diame<lb/>tris, vel à &longs;emidiametris quadrata, coni autem eiu&longs;dem al<lb/>titudinis inter &longs;e vt ba&longs;es; erit vt <foreign lang="greek">d</foreign>F ad FI potentia, ita <lb/>conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ad conum IB<foreign lang="greek">g</foreign>; &longs;e&longs;quialter igitur conus <lb/><foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> coni IB<foreign lang="greek">g</foreign>: &longs;ed & conoides parabolicum IB<foreign lang="greek">g</foreign> &longs;e&longs;qui<lb/>alterum e&longs;t coni IB<foreign lang="greek">g</foreign>; æqualis igitur e&longs;t conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> co<lb/>noidi IB<foreign lang="greek">g. </foreign></s>

<s>Et quoniam in parabola TBX ordinatim <lb/>ad diametrum applicatarum DT e&longs;t ad FI hoc e&longs;t N <pb pagenum="88"/>ad O potentia, vt DB ad BF longitudine: &longs;ed TD e&longs;t <lb/>æqualis N; ergo & IF æqualis erit O: cum igitur & <lb/>P ip&longs;ius O, & <foreign lang="greek">d</foreign>F ip&longs;ius FI &longs;it potentia &longs;e&longs;quialtera, erit <lb/>F<foreign lang="greek">d</foreign> æqualis ip&longs;i <foreign lang="greek">*r</foreign>: &longs;ed F<37> e&longs;t æqualis ip&longs;i <expan abbr="q;">que</expan> vt igitur e&longs;t <lb/>Q ad P, hoc e&longs;t DB ad BF, ita erit <37>F ad F<foreign lang="greek">d</foreign>; dupli<lb/>cata igitur proportio erit quadrati ex F<37> ad quadratum ex <lb/>E<foreign lang="greek">d</foreign> eius, quæ e&longs;t DB ad BF: &longs;ed vt quadratum ex F<37> ad <lb/><figure id="fig113"/><lb/>quadratum ex F<foreign lang="greek">d</foreign>, ita e&longs;t circulus circa <37><foreign lang="greek">q</foreign> ad circulum <lb/>circa <foreign lang="greek">de</foreign>, hoc e&longs;t conus <37>B<foreign lang="greek">q</foreign> ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>; coni igitur <lb/><37>B<foreign lang="greek">q</foreign> ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, duplicata e&longs;t proportio eius, quæ e&longs;t <lb/>DB ad BF: &longs;ed & conoidis TBX ad conoides IB<foreign lang="greek">g</foreign> du<lb/>plicata e&longs;t proportio eius, quæ e&longs;t DB ad BF, vt mon<lb/>&longs;trant alij; eadem igitur proportio e&longs;t coni <37>B<foreign lang="greek">q</foreign> ad co<lb/>num <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> quæ conoidis TBX ad conoides IB<foreign lang="greek">g</foreign>: &longs;ed <pb pagenum="89"/>conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> æqualis e&longs;t conoidi IB<foreign lang="greek">g</foreign>, vtpote in&longs;cripti co<lb/>ni IB<foreign lang="greek">g</foreign> &longs;e&longs;quialtero, cuius itidem &longs;e&longs;quialter erat conus <lb/><foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>; reliquum igitur coni <37>B<foreign lang="greek">q</foreign> dempto cono <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> æqua<lb/>le erit conoidis TBX fru&longs;to TI<foreign lang="greek">g</foreign>X. </s>

<s>Et quoniam in parabola TBX ordinatim <lb/>ad diametrum applicatarum DT e&longs;t ad FI hoc e&longs;t N <pb xlink:href="043/01/175.jpg" pagenum="88"/>ad O potentia, vt DB ad BF longitudine: &longs;ed TD e&longs;t <lb/>æqualis N; ergo & IF æqualis erit O: cum igitur & <lb/>P ip&longs;ius O, & <foreign lang="greek">d</foreign>F ip&longs;ius FI &longs;it potentia &longs;e&longs;quialtera, erit <lb/>F<foreign lang="greek">d</foreign> æqualis ip&longs;i <foreign lang="greek">*r</foreign>: &longs;ed F<37> e&longs;t æqualis ip&longs;i <expan abbr="q;">que</expan> vt igitur e&longs;t <lb/>Q ad P, hoc e&longs;t DB ad BF, ita erit <37>F ad F<foreign lang="greek">d</foreign>; dupli<lb/>cata igitur proportio erit quadrati ex F<37> ad quadratum ex <lb/>E<foreign lang="greek">d</foreign> eius, quæ e&longs;t DB ad BF: &longs;ed vt quadratum ex F<37> ad <lb/><figure id="id.043.01.175.1.jpg" xlink:href="043/01/175/1.jpg"/><lb/>quadratum ex F<foreign lang="greek">d</foreign>, ita e&longs;t circulus circa <37><foreign lang="greek">q</foreign> ad circulum <lb/>circa <foreign lang="greek">de</foreign>, hoc e&longs;t conus <37>B<foreign lang="greek">q</foreign> ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>; coni igitur <lb/><37>B<foreign lang="greek">q</foreign> ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, duplicata e&longs;t proportio eius, quæ e&longs;t <lb/>DB ad BF: &longs;ed & conoidis TBX ad conoides IB<foreign lang="greek">g</foreign> du<lb/>plicata e&longs;t proportio eius, quæ e&longs;t DB ad BF, vt mon<lb/>&longs;trant alij; eadem igitur proportio e&longs;t coni <37>B<foreign lang="greek">q</foreign> ad co<lb/>num <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> quæ conoidis TBX ad conoides IB<foreign lang="greek">g</foreign>: &longs;ed <pb xlink:href="043/01/176.jpg" pagenum="89"/>conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> æqualis e&longs;t conoidi IB<foreign lang="greek">g</foreign>, vtpote in&longs;cripti co<lb/>ni IB<foreign lang="greek">g</foreign> &longs;e&longs;quialtero, cuius itidem &longs;e&longs;quialter erat conus <lb/><foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>; reliquum igitur coni <37>B<foreign lang="greek">q</foreign> dempto cono <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> æqua<lb/>le erit conoidis TBX fru&longs;to TI<foreign lang="greek">g</foreign>X. </s>

<s>Rur&longs;us quia e&longs;t vt <lb/>cubus ex BD ad cubum ex BI ita conus SBV ad &longs;ui &longs;i<lb/>milem conum YBZ, in triplicata &longs;cilicet proportione la<lb/>terum, &longs;iue a<gap/>ium DB, BF: &longs;ed quia YF e&longs;t æqualis BF, <lb/>propter &longs;imilitudinem triangulorum, e&longs;t vt cubus ex BF ad <lb/>&longs;olidum ex BF & quadrato ex F<foreign lang="greek">d</foreign>, ita quadratum ex FY <lb/>ad quadratum ex F<foreign lang="greek">d</foreign>, hoc e&longs;t circulus circa YZ ad <expan abbr="circul&utilde;">circulum</expan> <lb/>circa <foreign lang="greek">de</foreign>, hoc e&longs;t conus YBZ ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ex æquali <lb/>igitur erit vt cubus ex BD ad &longs;olidum ex BF, & quadra<lb/>to F<foreign lang="greek">d</foreign>, ita conus SBV ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>: &longs;ed vt &longs;olidum <lb/>ex BF, & quadrato F<foreign lang="greek">d</foreign>, ad &longs;olidum ex BF & quadrato <lb/>F<37>, ita e&longs;t &longs;imiliter vt ante conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ad conum <37>B<foreign lang="greek">q</foreign>; ex <lb/>æquali igitur erit vt cubus ex BD ad &longs;olidum ex BF, & <lb/>quadrato F<37>, ita conus SBV, ad conum <37>B<foreign lang="greek">q</foreign>: &longs;ed con<lb/>uertendo, & per conuer&longs;ionem rationis, e&longs;t vt &longs;olidum ex <lb/>BF, & quadrato F<37>, ad &longs;olidum ex BF, & quadrato, <lb/>quo plus pote&longs;t F<37> quàm F<foreign lang="greek">d</foreign>, ita conus <37>B<foreign lang="greek">q</foreign> ad &longs;uireli<lb/>quum dempto cono <35>B<foreign lang="greek">e</foreign>; ex æquali igitur, vt cubus ex <lb/>BD ad &longs;olidum ex BF & quadrato, quo plus pote&longs;t F<37>, <lb/>quàm F<foreign lang="greek">d</foreign>, hoc e&longs;t, quo plus pote&longs;t Q quàm P quadrato <lb/>ex R, ita erit conus SBV, ad reliquum coni <37>B<foreign lang="greek">q</foreign> dem<lb/>pto cono <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, hoc e&longs;t ad fru&longs;tum TI<foreign lang="greek">g</foreign>X. Rur&longs;us, quo<lb/>niam duo cubi ex BF, FD, & &longs;olidum ex BF, FD, & <lb/>tripla ip&longs;ius BD, &longs;unt æqualia cubo ex BD; erit id quo <lb/>plus pote&longs;t cubice recta BD quàm BF, cubus ex <lb/>FD, & &longs;olidum ex BF, FD, & tripla ip&longs;ius BD: cum <lb/>igitur &longs;it vt cubus ex BD ad cubum ex BF, ita conus <lb/>SBV ad conum YBZ; erit per conuer&longs;ionem rationis, & <lb/>conuertendo, vt cubus ex FD vna cum &longs;olido ex BF, <lb/>FD, & tripla ip&longs;ius BD ad cubum ex BD, ita fru&longs;tum <lb/>SYZV, ad conum SBV: &longs;ed cubus ex BD, ad &longs;oli-<pb pagenum="90"/>dum ex BF & quadrato R, ita erat conus SBV ad fru<lb/>&longs;tum TI<foreign lang="greek">g</foreign>X: ex æquali igitur, erit vt cubus ex FD vna <lb/>cum &longs;olido ex BF, FD, & tripla ip&longs;ius BD, ad &longs;olidum <lb/>ex BF, & quadrato R, hoc e&longs;t vt H<emph type="italics"/>K<emph.end type="italics"/> ad KG, ita ex <lb/>contraria parte fru&longs;tum SYZV, ad fru&longs;tum TI<foreign lang="greek">g</foreign>X: nam <lb/>fru&longs;ti SYZV e&longs;t centrum grauitatis G: fru&longs;ti autem TI <lb/><figure id="fig114"/><lb/><foreign lang="greek">g</foreign>X centrum grauitatis H; totius igitur compo&longs;iti ex his <lb/>duobus fru&longs;tis centrum grauitatis erit K: commune autem <lb/>e&longs;t centrum grauitatis compo&longs;iti ex duobus fru&longs;tis SYZV <lb/>& TI<foreign lang="greek">g</foreign>X, fru&longs;to ALMC per antepenultimæ huius co<lb/>rollarium; fru&longs;ti igitur ALMC, centrum grauitatis erit K. <lb/></s>

<s>Rur&longs;us quia e&longs;t vt <lb/>cubus ex BD ad cubum ex BI ita conus SBV ad &longs;ui &longs;i<lb/>milem conum YBZ, in triplicata &longs;cilicet proportione la<lb/>terum, &longs;iue a<gap/>ium DB, BF: &longs;ed quia YF e&longs;t æqualis BF, <lb/>propter &longs;imilitudinem triangulorum, e&longs;t vt cubus ex BF ad <lb/>&longs;olidum ex BF & quadrato ex F<foreign lang="greek">d</foreign>, ita quadratum ex FY <lb/>ad quadratum ex F<foreign lang="greek">d</foreign>, hoc e&longs;t circulus circa YZ ad <expan abbr="circul&utilde;">circulum</expan> <lb/>circa <foreign lang="greek">de</foreign>, hoc e&longs;t conus YBZ ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ex æquali <lb/>igitur erit vt cubus ex BD ad &longs;olidum ex BF, & quadra<lb/>to F<foreign lang="greek">d</foreign>, ita conus SBV ad conum <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>: &longs;ed vt &longs;olidum <lb/>ex BF, & quadrato F<foreign lang="greek">d</foreign>, ad &longs;olidum ex BF & quadrato <lb/>F<37>, ita e&longs;t &longs;imiliter vt ante conus <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign> ad conum <37>B<foreign lang="greek">q</foreign>; ex <lb/>æquali igitur erit vt cubus ex BD ad &longs;olidum ex BF, & <lb/>quadrato F<37>, ita conus SBV, ad conum <37>B<foreign lang="greek">q</foreign>: &longs;ed con<lb/>uertendo, & per conuer&longs;ionem rationis, e&longs;t vt &longs;olidum ex <lb/>BF, & quadrato F<37>, ad &longs;olidum ex BF, & quadrato, <lb/>quo plus pote&longs;t F<37> quàm F<foreign lang="greek">d</foreign>, ita conus <37>B<foreign lang="greek">q</foreign> ad &longs;uireli<lb/>quum dempto cono <35>B<foreign lang="greek">e</foreign>; ex æquali igitur, vt cubus ex <lb/>BD ad &longs;olidum ex BF & quadrato, quo plus pote&longs;t F<37>, <lb/>quàm F<foreign lang="greek">d</foreign>, hoc e&longs;t, quo plus pote&longs;t Q quàm P quadrato <lb/>ex R, ita erit conus SBV, ad reliquum coni <37>B<foreign lang="greek">q</foreign> dem<lb/>pto cono <foreign lang="greek">d</foreign>B<foreign lang="greek">e</foreign>, hoc e&longs;t ad fru&longs;tum TI<foreign lang="greek">g</foreign>X. Rur&longs;us, quo<lb/>niam duo cubi ex BF, FD, & &longs;olidum ex BF, FD, & <lb/>tripla ip&longs;ius BD, &longs;unt æqualia cubo ex BD; erit id quo <lb/>plus pote&longs;t cubice recta BD quàm BF, cubus ex <lb/>FD, & &longs;olidum ex BF, FD, & tripla ip&longs;ius BD: cum <lb/>igitur &longs;it vt cubus ex BD ad cubum ex BF, ita conus <lb/>SBV ad conum YBZ; erit per conuer&longs;ionem rationis, & <lb/>conuertendo, vt cubus ex FD vna cum &longs;olido ex BF, <lb/>FD, & tripla ip&longs;ius BD ad cubum ex BD, ita fru&longs;tum <lb/>SYZV, ad conum SBV: &longs;ed cubus ex BD, ad &longs;oli-<pb xlink:href="043/01/177.jpg" pagenum="90"/>dum ex BF & quadrato R, ita erat conus SBV ad fru<lb/>&longs;tum TI<foreign lang="greek">g</foreign>X: ex æquali igitur, erit vt cubus ex FD vna <lb/>cum &longs;olido ex BF, FD, & tripla ip&longs;ius BD, ad &longs;olidum <lb/>ex BF, & quadrato R, hoc e&longs;t vt H<emph type="italics"/>K<emph.end type="italics"/> ad KG, ita ex <lb/>contraria parte fru&longs;tum SYZV, ad fru&longs;tum TI<foreign lang="greek">g</foreign>X: nam <lb/>fru&longs;ti SYZV e&longs;t centrum grauitatis G: fru&longs;ti autem TI <lb/><figure id="id.043.01.177.1.jpg" xlink:href="043/01/177/1.jpg"/><lb/><foreign lang="greek">g</foreign>X centrum grauitatis H; totius igitur compo&longs;iti ex his <lb/>duobus fru&longs;tis centrum grauitatis erit K: commune autem <lb/>e&longs;t centrum grauitatis compo&longs;iti ex duobus fru&longs;tis SYZV <lb/>& TI<foreign lang="greek">g</foreign>X, fru&longs;to ALMC per antepenultimæ huius co<lb/>rollarium; fru&longs;ti igitur ALMC, centrum grauitatis erit K. <lb/></s>

<s>Quod demon&longs;trandum erat. </s><pb pagenum="91"/>

<s>Quod demon&longs;trandum erat. </s><pb xlink:href="043/01/178.jpg" pagenum="91"/>

<s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/></s>

Line 2082

<s>auctoritate, & Petri eius Nepotis Cardinalis <lb/>ampli&longs;&longs;imi Aldobrandini iu&longs;&longs;u bene de me merentium Ma<lb/>thematicam &longs;cientiam, & Philo&longs;ophiam ciuilem in almo <lb/>Vrbis Gymna&longs;io profiterer, in eorum gratiam compo&longs;ui, <lb/>qui me centra grauitatis portionum &longs;phæroidis imperfe<lb/>cti operis crimine condemnandum omittere nolebant; cu<lb/>ius prouinciæ iuuante Deo, & mira Mathematicæ &longs;tudio<lb/>&longs;is &longs;atisfaciendi voluntate, multas difficultates ita &longs;upe<lb/>raui, vt vno men&longs;e Octobri plus præ&longs;titerim, quam à me <lb/>requi&longs;i&longs;&longs;ent. </s>

<s>&longs;iquidem quæ de &longs;phæræ portionibus in hoc <lb/>libro proprijs eius figuræ rationibus, eadem in &longs;equen<lb/>ti aliis communibus cuilibet portioni &longs;phæræ, & &longs;phæroi<lb/>dis tum lati, tum oblongi ab&longs;ci&longs;&longs;æ vno, vel duobus planis <lb/>æque inter &longs;e di&longs;tantibus, & vtcumque in figuram in cideu-<pb pagenum="92"/>tibus demon&longs;traui, & temporis breuitatem magna animi in<lb/>tentione compen&longs;aui, quòd facere non potui&longs;sem ni&longs;i illi, <lb/>quos &longs;upra nominaui meos patronos tranquillum otium <lb/>mihi &longs;ua benignitate peperi&longs;&longs;ent; ego autem quo&longs;dam ad<lb/>uer&longs;os flatus vehementes in meam vtilitatem verte<lb/>re didici&longs;sem, cuius rei monumentum flammæ <lb/>vento agitatæ &longs;imulacrum cum illo Ver<lb/>gilij HOC ACRIOR in fronte <lb/>operis po&longs;ui, vt meus quali&longs;<lb/>cumque hic labor vel ab <lb/>inuitis in me collati <lb/>bencficij memo<lb/>riam præ&longs;e<lb/>ferret. </s>

<s>&longs;iquidem quæ de &longs;phæræ portionibus in hoc <lb/>libro proprijs eius figuræ rationibus, eadem in &longs;equen<lb/>ti aliis communibus cuilibet portioni &longs;phæræ, & &longs;phæroi<lb/>dis tum lati, tum oblongi ab&longs;ci&longs;&longs;æ vno, vel duobus planis <lb/>æque inter &longs;e di&longs;tantibus, & vtcumque in figuram in cideu-<pb xlink:href="043/01/179.jpg" pagenum="92"/>tibus demon&longs;traui, & temporis breuitatem magna animi in<lb/>tentione compen&longs;aui, quòd facere non potui&longs;sem ni&longs;i illi, <lb/>quos &longs;upra nominaui meos patronos tranquillum otium <lb/>mihi &longs;ua benignitate peperi&longs;&longs;ent; ego autem quo&longs;dam ad<lb/>uer&longs;os flatus vehementes in meam vtilitatem verte<lb/>re didici&longs;sem, cuius rei monumentum flammæ <lb/>vento agitatæ &longs;imulacrum cum illo Ver<lb/>gilij HOC ACRIOR in fronte <lb/>operis po&longs;ui, vt meus quali&longs;<lb/>cumque hic labor vel ab <lb/>inuitis in me collati <lb/>bencficij memo<lb/>riam præ&longs;e<lb/>ferret. </s>

<s>SECVNDI LIBRI FINIS.<lb/><figure id="fig115"/></s><pb pagenum="1"/><figure/>

<s>SECVNDI LIBRI FINIS.<lb/><figure id="id.043.01.179.1.jpg" xlink:href="043/01/179/1.jpg"/></s><pb xlink:href="043/01/180.jpg" pagenum="1"/><figure id="id.043.01.180.1.jpg" xlink:href="043/01/180/1.jpg"/>

<s>L V C AE <lb/>VALER II <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM <lb/><emph type="italics"/>LIBER TERTIVS.<emph.end type="italics"/></s><figure/>

<s>L V C AE <lb/>VALER II <lb/>DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM <lb/><emph type="italics"/>LIBER TERTIVS.<emph.end type="italics"/></s><figure id="id.043.01.180.2.jpg" xlink:href="043/01/180/2.jpg"/>

<s><emph type="italics"/>PROPOSITIO I.<emph.end type="italics"/></s>

<s>Si recta linea &longs;ecta fuerit bifa<lb/>riam, & non bifariam; rectan <lb/>gulum partibus in æqualibus <lb/>contentum æquale e&longs;t rectan <lb/>gulo, quod bis fit ex dimidiæ <lb/>&longs;ectæ &longs;egmentis, vna cum <lb/>quadrato non intermedij eo<lb/>rundem &longs;egmentorum. </s><pb pagenum="2"/>

<s>Sit recta linea AB &longs;ecta in puncto C bi&longs;ariam, & non <lb/>bifariam in puncto D. </s>

Line 2100

<s>Si igitur recta linea <lb/>&longs;ecta fuerit bifariam, & non bifariam, &c. </s>

<s>Quod demon<lb/>&longs;trandum erat. </s><figure/>

<s>Quod demon<lb/>&longs;trandum erat. </s><figure id="id.043.01.181.1.jpg" xlink:href="043/01/181/1.jpg"/>

<s><emph type="italics"/>PROPOSITIO II.<emph.end type="italics"/></s>

<s>Si circulum, vel ellip&longs;im duæ rectæ lineæ tan<lb/>gentes in terminis coniugatarum diametrorum, <lb/>conueniant: & punctum i<gap/> quo conueniunt, & <lb/>centrum figuræ <gap/>ungantur recta linea; quæcun<lb/>que hanc vnà cum prædictæ figuræ termino al<lb/>terutri diametrorum parallela &longs;ecuerit recta li<lb/>nea, ita ip&longs;a &longs;ecabitur in duobus punctis, vt re<lb/>ctangulum bis contentum &longs;egmentis, quorum al<lb/>terum inter diametrum, & terminum figuræ, al<lb/>terum inter figuræ terminum & contingentem <lb/>interijcitur, vnà cum huius quadrato, &longs;it æquale <lb/>quadrato reliqui &longs;egmenti inter diametrum, & <pb pagenum="3"/>cum quæ tangentium concur&longs;um, & centrum fi<lb/>guræ iungit interiecta. </s>

<s>Si circulum, vel ellip&longs;im duæ rectæ lineæ tan<lb/>gentes in terminis coniugatarum diametrorum, <lb/>conueniant: & punctum i<gap/> quo conueniunt, & <lb/>centrum figuræ <gap/>ungantur recta linea; quæcun<lb/>que hanc vnà cum prædictæ figuræ termino al<lb/>terutri diametrorum parallela &longs;ecuerit recta li<lb/>nea, ita ip&longs;a &longs;ecabitur in duobus punctis, vt re<lb/>ctangulum bis contentum &longs;egmentis, quorum al<lb/>terum inter diametrum, & terminum figuræ, al<lb/>terum inter figuræ terminum & contingentem <lb/>interijcitur, vnà cum huius quadrato, &longs;it æquale <lb/>quadrato reliqui &longs;egmenti inter diametrum, & <pb xlink:href="043/01/182.jpg" pagenum="3"/>cum quæ tangentium concur&longs;um, & centrum fi<lb/>guræ iungit interiecta. </s>

<s>Sit circulus, vel ellip&longs;is ABCD, cuius diametri con<lb/>iugatæ AC, BED, & figuram tangentes BF, GF, con <lb/>ueniant in puncto F; (parallelæ enim erunt vtraque alteri <lb/>coniugatorum diametrorum:) & recta FE iungatur, & ex <lb/>quolibet puncto G, in recta BE ducatur ip&longs;i AC paral<lb/>lela GLKH. </s>

<s>Dico rectangulum GKH bis vnà cum <lb/>quadrato KH æquale e&longs;&longs;e quadrato GL. </s>

<s>Quoniam <lb/>enim rectangulum BGD æquale e&longs;t rectangulo BGE <lb/><figure id="fig116"/><lb/>bis vnà cum quadrato BG: & rectangulum BED, e&longs;t <lb/>quadratum BE, erit vt rectangulum BED, ad re<lb/>ctangulum BGD, ita quadratum BE, ad rectangu<lb/>lum BGE bis, vnà cum quadrato BG: &longs;ed vt rectangu<lb/>lum BED, ad rectangulum BGD, ita e&longs;t quadratum EC, <lb/>hoc e&longs;t quadratum GH ad quadratum GK, ex primo <lb/>conicorum, vt igitur e&longs;t quadratum BE ad rectangulum <lb/>BGE bis, vnà cum quadrato BG, ita erit quadratum <lb/>GH ad quadratum GK. </s>

<s>Quoniam <lb/>enim rectangulum BGD æquale e&longs;t rectangulo BGE <lb/><figure id="id.043.01.182.1.jpg" xlink:href="043/01/182/1.jpg"/><lb/>bis vnà cum quadrato BG: & rectangulum BED, e&longs;t <lb/>quadratum BE, erit vt rectangulum BED, ad re<lb/>ctangulum BGD, ita quadratum BE, ad rectangu<lb/>lum BGE bis, vnà cum quadrato BG: &longs;ed vt rectangu<lb/>lum BED, ad rectangulum BGD, ita e&longs;t quadratum EC, <lb/>hoc e&longs;t quadratum GH ad quadratum GK, ex primo <lb/>conicorum, vt igitur e&longs;t quadratum BE ad rectangulum <lb/>BGE bis, vnà cum quadrato BG, ita erit quadratum <lb/>GH ad quadratum GK. </s>

<s>Rur&longs;us quia e&longs;t vt BE ad EG, <lb/>ita BF ad GL, propter &longs;imilitudinem triangulorum; erit <lb/>vt quadratum BE ad quadratum EG, ita quadratum <pb pagenum="4"/>BF hoc e&longs;t quadratum GH ad quadratum GL: & per <lb/>conuer&longs;ionem rationis, vt quadratum BE ad rectangu<lb/>lum BGE bis, vnà cum quadrato BG, ita quadratum <lb/>GH ad rectangulum GLH bis, vnà cum quadrato LH: <lb/>&longs;ed vt quadratum BE ad rectangulum EGB bis, vnà <lb/>cum quadrato BG, ita erat quadratum GH ad quadra<lb/>tum GK; vt igitur quadratum GH ad quadratum GK, <lb/>ita erit idem quadratum GH ad rectangulum GLH bis, <lb/>vnà cum quadrato LH: quadratum igitur GK æquale <lb/>erit rectangulo GLH bis, vnà cum quadrato LH; demptis <lb/>igitur ab eodem quadrato GH æqualibus quadrato GK, <lb/>& rectangulo GLH bis, vnà cum quadrato LH, erit <lb/>rectangulum GKH, bis vnà cum quadrato KH æquale <lb/>quadrato GL. </s>

<s>Rur&longs;us quia e&longs;t vt BE ad EG, <lb/>ita BF ad GL, propter &longs;imilitudinem triangulorum; erit <lb/>vt quadratum BE ad quadratum EG, ita quadratum <pb xlink:href="043/01/183.jpg" pagenum="4"/>BF hoc e&longs;t quadratum GH ad quadratum GL: & per <lb/>conuer&longs;ionem rationis, vt quadratum BE ad rectangu<lb/>lum BGE bis, vnà cum quadrato BG, ita quadratum <lb/>GH ad rectangulum GLH bis, vnà cum quadrato LH: <lb/>&longs;ed vt quadratum BE ad rectangulum EGB bis, vnà <lb/>cum quadrato BG, ita erat quadratum GH ad quadra<lb/>tum GK; vt igitur quadratum GH ad quadratum GK, <lb/>ita erit idem quadratum GH ad rectangulum GLH bis, <lb/>vnà cum quadrato LH: quadratum igitur GK æquale <lb/>erit rectangulo GLH bis, vnà cum quadrato LH; demptis <lb/>igitur ab eodem quadrato GH æqualibus quadrato GK, <lb/>& rectangulo GLH bis, vnà cum quadrato LH, erit <lb/>rectangulum GKH, bis vnà cum quadrato KH æquale <lb/>quadrato GL. </s>

<s>Quod demon&longs;trandum erat. </s><figure/>

<s>Quod demon&longs;trandum erat. </s><figure id="id.043.01.183.1.jpg" xlink:href="043/01/183/1.jpg"/>

<s><emph type="italics"/>PROPOSITIO III.<emph.end type="italics"/></s>

<s>Per data duo puncta in duabus rectis lineis da<lb/>tum angulum continentibus, in earum plano pa<lb/>rabola tran&longs;ibit, cuius vertex &longs;it a&longs;&longs;ignatum præ<lb/>dictorum punctorum, in quo altera linea parabo-<pb pagenum="5"/>lam contingat, altera in altero &longs;ecet diametro æ<lb/>quidi&longs;tans. </s>

<s>Per data duo puncta in duabus rectis lineis da<lb/>tum angulum continentibus, in earum plano pa<lb/>rabola tran&longs;ibit, cuius vertex &longs;it a&longs;&longs;ignatum præ<lb/>dictorum punctorum, in quo altera linea parabo-<pb xlink:href="043/01/184.jpg" pagenum="5"/>lam contingat, altera in altero &longs;ecet diametro æ<lb/>quidi&longs;tans. </s>

<s>Sint data duo puncta. </s>

Line 2130

<s>Quoniam igitur in plano BD <lb/>parabola inueniri pote&longs;t, cu<lb/>ius &longs;it vertex C, diameter <lb/>CD, ita vt quædam ex &longs;e<lb/>ctione ad diametrum CD <lb/>applicata in dato angulo A <lb/>BC, ide&longs;t ADC, qualis <lb/>e&longs;t recta AD, po&longs;&longs;it rectan<lb/>gulum ex CD, & E, ex <lb/>primo conicorum elemen. <lb/></s>

<s>to; &longs;it ea &longs;ectio parabola <lb/><figure id="fig117"/><lb/>AC; a&longs;&longs;ignatum e&longs;t autem punctum C; per puncta igi<lb/>tur A, C parabola AC tran&longs;ibit, cuius vertex e&longs;t a&longs;&longs;i<lb/>gnatum punctum C. </s>

<s>to; &longs;it ea &longs;ectio parabola <lb/><figure id="id.043.01.184.1.jpg" xlink:href="043/01/184/1.jpg"/><lb/>AC; a&longs;&longs;ignatum e&longs;t autem punctum C; per puncta igi<lb/>tur A, C parabola AC tran&longs;ibit, cuius vertex e&longs;t a&longs;&longs;i<lb/>gnatum punctum C. </s>

<s>Et quoniam quæ ex vertice recta <lb/>CB e&longs;t applicatæ DA parallela, &longs;ectionem AC in pun<lb/>cto C continget: e&longs;t autem AB diametro CD æquidi<lb/>di&longs;tans, ac proinde parabolam &longs;ecabit in puncto A. </s>

Line 2138

<s><emph type="italics"/>PROPOSITIO IV.<emph.end type="italics"/></s>

<s>Si recta linea parabolam contingat, omnes re<lb/>ctælineæ ex &longs;ectione ad contingentem applicatæ <pb pagenum="6"/>diametro &longs;ectionis parallelæ inter &longs;e &longs;unt longi<lb/>tudine, vt inter applicatas & contactum, vel ver<lb/>ticem interiectæ inter &longs;e potentia. </s>

<s>Si recta linea parabolam contingat, omnes re<lb/>ctælineæ ex &longs;ectione ad contingentem applicatæ <pb xlink:href="043/01/185.jpg" pagenum="6"/>diametro &longs;ectionis parallelæ inter &longs;e &longs;unt longi<lb/>tudine, vt inter applicatas & contactum, vel ver<lb/>ticem interiectæ inter &longs;e potentia. </s>

<s>Productis au<lb/>tem dictis applicatis, erunt inter &longs;ectionem & ba<lb/>&longs;im interiectæ inter &longs;e longitudine, vt in circulo, <lb/>vel ellip&longs;e ad diametrum ordinatim applicatæ, &longs;e<lb/>cantesque illam in ea&longs;dem rationes, in quas aliæ <lb/>prædictæ applicatæ &longs;ecant ba&longs;im parabolæ, inter <lb/>&longs;e potentia. </s>

<s>Sit &longs;ectio parabola ABC, cuius vertex B, diameter <lb/>BD: & recta quadam BE &longs;ectionem contingente in pun<lb/>cto B, &longs;int quotcumque rectæ lineæ ex &longs;ectione ordinatim <lb/>ad BE contingentem applicatæ diametro BD &longs;ectionis <lb/>parallelæ FG, KH, quibus productis &longs;int ad ba&longs;im &longs;e<lb/><figure id="fig118"/><lb/>ctionis applicatæ GN, KO. </s>

<s>Et expo&longs;ito primum circu<lb/>lo, PQRS, cuius diametri ad rectos inter &longs;e angulos &longs;int <lb/>QS, PR; &longs;ecta autem QT in punctis V, X, in ea&longs;<lb/>dem rationes, in quas &longs;ecta e&longs;t AD in punctis N, O, <lb/>&longs;umpto or<gap/>ine à punctis D, T, vt &longs;it DO ad ON, <pb pagenum="7"/>vt e&longs;t TV ad VX: & vt ON ad NA, ita VX ad <expan abbr="Xq;">Xque</expan> <lb/>applicentur ad &longs;emidiametrum QT rectæ ZV, XY dia<lb/>metro PR æquidi&longs;tantes. </s>

<s>Et expo&longs;ito primum circu<lb/>lo, PQRS, cuius diametri ad rectos inter &longs;e angulos &longs;int <lb/>QS, PR; &longs;ecta autem QT in punctis V, X, in ea&longs;<lb/>dem rationes, in quas &longs;ecta e&longs;t AD in punctis N, O, <lb/>&longs;umpto or<gap/>ine à punctis D, T, vt &longs;it DO ad ON, <pb xlink:href="043/01/186.jpg" pagenum="7"/>vt e&longs;t TV ad VX: & vt ON ad NA, ita VX ad <expan abbr="Xq;">Xque</expan> <lb/>applicentur ad &longs;emidiametrum QT rectæ ZV, XY dia<lb/>metro PR æquidi&longs;tantes. </s>

<s>Dico e&longs;&longs;e HK ad FG lon<lb/>git<gap/>dine, vt FB ad BH potentia: & KO ad GN longi<lb/>tudine, vt ZY ad YX potentia. </s>

Line 2162

<s>Idem autem &longs;imiliter o&longs;ten<lb/>deremus de quotcumque aliis in circulo, & &longs;ectione para<lb/>bola vt prædictæ applicatis multitudine æqualibus. </s>

<s>In <lb/>ellip&longs;e autem, ductis diametris quibu&longs;uis coniugatis, & <lb/>totidem quot in circulo ad vnam &longs;emidiametrum rectis li<lb/>neis ordinatim applicatis &longs;ecundum puncta &longs;ectionum eiu&longs;<lb/>dem diametri in ea&longs;dem prædictas rationes, codemque or<lb/>dine; quoniam ex XXI primi conicorum &longs;tatim apparet re<lb/>ctarum linearum ita vt diximus in circulo, & ellip&longs;e appli-<pb pagenum="8"/>catarum quadrata e&longs;&longs;e inter &longs;e in eadem proportione; erun<gap/><lb/>prædictæ inter &longs;ectionem parabolam, & ba&longs;im interiectæ <lb/>inter &longs;e longitudine, vt in ellip&longs;e ad diametrum &longs;imiliter <lb/>vt diximus applicatæ inter &longs;e potentia. </s>

<s>In <lb/>ellip&longs;e autem, ductis diametris quibu&longs;uis coniugatis, & <lb/>totidem quot in circulo ad vnam &longs;emidiametrum rectis li<lb/>neis ordinatim applicatis &longs;ecundum puncta &longs;ectionum eiu&longs;<lb/>dem diametri in ea&longs;dem prædictas rationes, codemque or<lb/>dine; quoniam ex XXI primi conicorum &longs;tatim apparet re<lb/>ctarum linearum ita vt diximus in circulo, & ellip&longs;e appli-<pb xlink:href="043/01/187.jpg" pagenum="8"/>catarum quadrata e&longs;&longs;e inter &longs;e in eadem proportione; erun<gap/><lb/>prædictæ inter &longs;ectionem parabolam, & ba&longs;im interiectæ <lb/>inter &longs;e longitudine, vt in ellip&longs;e ad diametrum &longs;imiliter <lb/>vt diximus applicatæ inter &longs;e potentia. </s>

<s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s>

Line 2170

<s>Omnis figuræ circa axim in alteram partem <lb/>deficientis, cuius &longs;uperficies, excepta ba&longs;e &longs;it to<lb/>ta interius concaua ba&longs;im habentis circulum, vel <lb/>ellip&longs;im; quælibet tres &longs;ectiones ba&longs;i parallelæ <lb/>æqualia axis &longs;egmenta intercipientes, ita &longs;e ha<lb/>bent, vt minor &longs;it proportio minimæ ad mediam, <lb/>quam mediæ ad maximam. </s>

<s>Sit figura ABC circa axem BD in alteram partem de<lb/>ficiens, qualem diximus: & po&longs;itis in axe BD tribus qui<lb/>buslibet punctis <lb/>F, E, L, æqualia <lb/>axis &longs;egmenta in<lb/>tercipientibus, in <lb/>telligatur <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> <lb/>ABC &longs;ectum per <lb/>ea puncta planis <lb/><expan abbr="buibu&longs;dã">buibu&longs;dam</expan> ba&longs;i cir <lb/>culo, vel ellip&longs;i, <lb/>circa AC pa<lb/>rallelis: quare &longs;e<lb/>ctiones erunt cir<lb/><figure id="fig119"/><lb/>culi, vel ellip&longs;es &longs;imiles ba&longs;i, per definitionem, quarum dia<lb/>metri eiu&longs;dem rationis in eodem plano per axim &longs;int IK. <pb pagenum="9"/>GH, MN. </s>

<s>Sit figura ABC circa axem BD in alteram partem de<lb/>ficiens, qualem diximus: & po&longs;itis in axe BD tribus qui<lb/>buslibet punctis <lb/>F, E, L, æqualia <lb/>axis &longs;egmenta in<lb/>tercipientibus, in <lb/>telligatur <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> <lb/>ABC &longs;ectum per <lb/>ea puncta planis <lb/><expan abbr="buibu&longs;dã">buibu&longs;dam</expan> ba&longs;i cir <lb/>culo, vel ellip&longs;i, <lb/>circa AC pa<lb/>rallelis: quare &longs;e<lb/>ctiones erunt cir<lb/><figure id="id.043.01.187.1.jpg" xlink:href="043/01/187/1.jpg"/><lb/>culi, vel ellip&longs;es &longs;imiles ba&longs;i, per definitionem, quarum dia<lb/>metri eiu&longs;dem rationis in eodem plano per axim &longs;int IK. <pb xlink:href="043/01/188.jpg" pagenum="9"/>GH, MN. </s>

<s>Dico &longs;olidi ABC &longs;ectionum, minorem e&longs;&longs;e <lb/>proportionem, ip&longs;ius IK ad GH, quàm GH ad MN. <lb/></s>

Line 2184

<s>&longs;ectio ellip&longs;is erit: <lb/>&longs;imilis autem ip&longs;i alia quæcumque &longs;ectio &longs;phæ<lb/>roidis eidem parallela: earumque omnes diame<lb/>tri quæ eiu&longs;dem &longs;unt rationis erunt in eodem pla<lb/>no per axem. </s>

<s>Extant hæc demon&longs;trata ab Archimede in &longs;uo de &longs;phæ<lb/>roidibus, & conoidibus. </s><pb pagenum="10"/>

<s>Extant hæc demon&longs;trata ab Archimede in &longs;uo de &longs;phæ<lb/>roidibus, & conoidibus. </s><pb xlink:href="043/01/189.jpg" pagenum="10"/>

<s><emph type="italics"/>PROPOSITIO VII.<emph.end type="italics"/></s>

Line 2196

<s>Super datam ellip&longs;im, circa datam rectam line<lb/>am ab eius centro eleuatam tanquam axem, coni, <lb/>& cylindri portionem inuenire. </s>

<s>Datoque &longs;phæ<lb/>roidi, & conoidi, vel conoidis, &longs;phæroidi&longs;ve por<lb/>tioni circa datum axem &longs;phæroidis, vel cuiuslibet <lb/>dictarum portionum, cylindrus vel cylindri por<lb/>tio circum&longs;cripta e&longs;&longs;e pote&longs;t: vel comprehendere <lb/>inter eadem plana parallela, ita vt eius ba&longs;is &longs;it &longs;i<lb/>milis ba&longs;i, vel ba&longs;ibus comprehen&longs;æ portionis, vel <lb/>fru&longs;ti, &longs;i de conoidibus &longs;it &longs;ermo: & diametri, quæ <lb/>eiu&longs;dem &longs;unt rationis &longs;ectæ à centro bifariam &longs;int <lb/>in eadem recta linea. </s><pb pagenum="11"/>

<s>Manife&longs;ta item &longs;unt hæc omnia, ex ijs, quæ in eodem li<lb/>bro de &longs;phæroidibus, & conoidibus demon&longs;trat Archi<lb/>medes. </s>

Line 2210

<s>Dico fru&longs;tum <lb/>BDF ad pri&longs;ma HKF, e&longs;&longs;e vt <lb/>rectangulum DEG vna cum ter<lb/>tia parte quadrati DG. </s>

<s>Ad qua<lb/>dratum DE: ad pyramidem au<lb/>tem ADEF, vt <expan abbr="prædict&utilde;">prædictum</expan> rectan<lb/><figure id="fig120"/><lb/>gulum DEG, vnà cum tertia parte quadrati DG, ad ter<pb pagenum="12"/>tiam partem quadrati DE. </s>

<s>Ad qua<lb/>dratum DE: ad pyramidem au<lb/>tem ADEF, vt <expan abbr="prædict&utilde;">prædictum</expan> rectan<lb/><figure id="id.043.01.190.1.jpg" xlink:href="043/01/190/1.jpg"/><lb/>gulum DEG, vnà cum tertia parte quadrati DG, ad ter<pb xlink:href="043/01/191.jpg" pagenum="12"/>tiam partem quadrati DE. </s>

<s>Ab&longs;ci&longs;sis enim æqualibus EL <lb/>ip&longs;i BC, & FM ip&longs;i AC, & EG, ip&longs;i AB, con&longs;tituantur <lb/>pri&longs;mata ABCLEG, AGMFCL, ANHDGM, & <lb/>pyramis ADGM, & iungatur ML. </s>

<s>Quoniam igitur ob pa<lb/>rallelas EF, GM, & DF, GL, &longs;imilia inter &longs;e &longs;unt trian<lb/>gula DEF, DGM, EGL, duplicatam inter &longs;e habebunt <lb/>laterum ho mologorum DE, DG, GE, proportionem, <lb/>hoc e&longs;t eandem, quæ totidem e&longs;t quadratorum ex ip&longs;is DE, <lb/>DG, GE, prout inter &longs;e re&longs;pondent: vt igitur DG qua<lb/>dratum ad quadratum DE, ita e&longs;t triangulum DGM <lb/>ad triangulum DEF: eademque ratione vt quadratum <lb/>GE ad DE quadratum, ita trian <lb/>gulum EGL ad triangulum D <lb/>EF: & vt prima cum quinta ad <lb/>&longs;ecundam, ita tertia cum &longs;exta ad <lb/>quartam: videlicet, vt duo qua<lb/>drata DG, GE, ad quadratum <lb/>DE, ita duo triangula DGM, <lb/>EGL, ad triangulum DEF. & <lb/>conuertendo, & per conuer&longs;ionem <lb/>rationis, vt quadratum DE ad <lb/>rectangulum DGE bis, ita trian<lb/>gulum DEF, ad parallelogram<lb/><figure id="fig121"/><lb/>mum GF: & conuertendo, vt rectangulum DGE bis, ad <lb/>quadratum DE, ita GF parallelogrammum ad triangu<lb/>lum DEF: & antecedentium dimidia, vt rectangulum <lb/>DGE ad quadratum DE, ita triangulum GML ad <lb/>triangulum DEF; hoc e&longs;t pri&longs;ma, cuius ba&longs;is triangulum <lb/>GLM, altitudo eadem pri&longs;mati H<emph type="italics"/>K<emph.end type="italics"/>F ad pri&longs;ma HKF. </s>

<s>Quoniam igitur ob pa<lb/>rallelas EF, GM, & DF, GL, &longs;imilia inter &longs;e &longs;unt trian<lb/>gula DEF, DGM, EGL, duplicatam inter &longs;e habebunt <lb/>laterum ho mologorum DE, DG, GE, proportionem, <lb/>hoc e&longs;t eandem, quæ totidem e&longs;t quadratorum ex ip&longs;is DE, <lb/>DG, GE, prout inter &longs;e re&longs;pondent: vt igitur DG qua<lb/>dratum ad quadratum DE, ita e&longs;t triangulum DGM <lb/>ad triangulum DEF: eademque ratione vt quadratum <lb/>GE ad DE quadratum, ita trian <lb/>gulum EGL ad triangulum D <lb/>EF: & vt prima cum quinta ad <lb/>&longs;ecundam, ita tertia cum &longs;exta ad <lb/>quartam: videlicet, vt duo qua<lb/>drata DG, GE, ad quadratum <lb/>DE, ita duo triangula DGM, <lb/>EGL, ad triangulum DEF. & <lb/>conuertendo, & per conuer&longs;ionem <lb/>rationis, vt quadratum DE ad <lb/>rectangulum DGE bis, ita trian<lb/>gulum DEF, ad parallelogram<lb/><figure id="id.043.01.191.1.jpg" xlink:href="043/01/191/1.jpg"/><lb/>mum GF: & conuertendo, vt rectangulum DGE bis, ad <lb/>quadratum DE, ita GF parallelogrammum ad triangu<lb/>lum DEF: & antecedentium dimidia, vt rectangulum <lb/>DGE ad quadratum DE, ita triangulum GML ad <lb/>triangulum DEF; hoc e&longs;t pri&longs;ma, cuius ba&longs;is triangulum <lb/>GLM, altitudo eadem pri&longs;mati H<emph type="italics"/>K<emph.end type="italics"/>F ad pri&longs;ma HKF. </s>

<s>Rur&longs;us, quoniam e&longs;t vt quadratum EG ad quadratum <lb/>ED, ita triangulum EGL ad triangulum DEF; erit &longs;i<lb/>militer vt quadratum EG ad quadratum ED, ita pri&longs;ma <lb/>BGL ad pri&longs;ma HKF: &longs;ed vt rectangulum DGE ad <lb/>quadratum DE, ita pri&longs;ma erat, cuius ba&longs;is triangulum G <pb pagenum="13"/>LM altitudo autem eadem pri&longs;mati HKF, hoc e&longs;t pri&longs;ma <lb/>ACGLFM illi æquale per vltimam XI. elem. </s>

<s>Rur&longs;us, quoniam e&longs;t vt quadratum EG ad quadratum <lb/>ED, ita triangulum EGL ad triangulum DEF; erit &longs;i<lb/>militer vt quadratum EG ad quadratum ED, ita pri&longs;ma <lb/>BGL ad pri&longs;ma HKF: &longs;ed vt rectangulum DGE ad <lb/>quadratum DE, ita pri&longs;ma erat, cuius ba&longs;is triangulum G <pb xlink:href="043/01/192.jpg" pagenum="13"/>LM altitudo autem eadem pri&longs;mati HKF, hoc e&longs;t pri&longs;ma <lb/>ACGLFM illi æquale per vltimam XI. elem. </s>

<s>ad pri&longs;ma <lb/>HKF: vt igitur prima cum quinta, rectangulum DGE <lb/>vna cum quadrato EG, hoc e&longs;t rectangulum DEG, ad <lb/>&longs;ecundam quadratum DE, ita erit tertia cum &longs;exta, duo <lb/>pri&longs;mata BGL, ACGLFM, ad quartam pri&longs;ma HKF. <lb/></s>

Line 2226

<s>Quoniam enim e&longs;t vt rectangulum <lb/>DEG, vna cum tertia parte quadrati DG, ad quadra<lb/>tum DE, ita fru&longs;tum ABGDEF ad pri&longs;ma HKF: vt <lb/>autem quadratum DE, ad tertiam &longs;ui partem, ita e&longs;t pri&longs;<lb/>ma HKF ad pyramidem, cuius ba&longs;is triangulum DEF, <lb/>altitudo eadem pri&longs;mati HKF; erit ex æquali vt re<lb/>ctangulum DEG vna cum tertia parte quadrati DG <lb/>ad tertiam partem quadrati DE, ita fru&longs;tum ABCDEF, <lb/>ad pyramidem &longs;i compleatur ADEF. </s>

<s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s><pb pagenum="14"/>

<s>Manife&longs;tum e&longs;t <lb/>igitur propo&longs;itum. </s><pb xlink:href="043/01/193.jpg" pagenum="14"/>

<s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/></s>

Line 2238

<s>Ad <lb/>conum autem, vel coni portionem, cuius ba&longs;is e&longs;t <lb/>eadem, quæ maior ba&longs;is fru&longs;ti, & eadem altitudo; <lb/>vt prædictum rectangulum, vnà cum prædicti qua <lb/>drati tertia parte, ad tertiam partem quadrati ex <lb/>diametro maioris ba&longs;is. </s>

<s>Prædicti autem cylindri, <pb pagenum="15"/>vel portionis cylindricæ re&longs;iduum dempto fru&longs;to, <lb/>ad totum cylindrum, vel cylindri portionem; vt <lb/>rectangulum contentum diametro minoris ba&longs;is <lb/>fru&longs;ti, & differentia diametri maioris, vnà cum <lb/>duabus tertiis quadrati differentiæ, ad quadra<lb/>tum diametri maioris ba&longs;is. </s>

<s>Prædicti autem cylindri, <pb xlink:href="043/01/194.jpg" pagenum="15"/>vel portionis cylindricæ re&longs;iduum dempto fru&longs;to, <lb/>ad totum cylindrum, vel cylindri portionem; vt <lb/>rectangulum contentum diametro minoris ba&longs;is <lb/>fru&longs;ti, & differentia diametri maioris, vnà cum <lb/>duabus tertiis quadrati differentiæ, ad quadra<lb/>tum diametri maioris ba&longs;is. </s>

<s>Sit coni, vel eius portionis fru&longs;tum ABCD, cuius ba&longs;es <lb/>oppo&longs;itæ, circuli vel &longs;imiles ellip&longs;es, quarum diametri mi<lb/>noris ba&longs;is AB cuius centrum E: maioris autem CD, <lb/>& &longs;uper ba&longs;im circulum, vel ellip&longs;im CD &longs;tet cylindrus, <lb/>vel portio cylindrica CG comprehendens fru&longs;tum AB <lb/>CD, eiu&longs;demque altitudinis cum ip&longs;o, & conus, vel co<lb/>ni portio ECD. quo autem AC diameter &longs;uperat dia<lb/>metrum AB, quæ differentia di<lb/>citur, &longs;it DF. </s>

Line 2246

<s>Ad conum autem vel coni portio<lb/>nem ECD, vt rectangulum DCF, <lb/>vna cum tertia parte quadrati DF, <lb/>ad tertiam partem quadrati CD. <lb/></s>

<s>Cylindri autem, vel cylindri por<lb/>tionis CG re&longs;iduum dempto fru<lb/><figure id="fig122"/><lb/>&longs;to AD, ad cylindrum, vel portionem cylindricam CG, <lb/>vt rectangulum CFD vna cum duabus tertiis quadrati <lb/>FD, ad quadratum CD. </s>

<s>Cylindri autem, vel cylindri por<lb/>tionis CG re&longs;iduum dempto fru<lb/><figure id="id.043.01.194.1.jpg" xlink:href="043/01/194/1.jpg"/><lb/>&longs;to AD, ad cylindrum, vel portionem cylindricam CG, <lb/>vt rectangulum CFD vna cum duabus tertiis quadrati <lb/>FD, ad quadratum CD. </s>

<s>Cono enim, vel portioni coni<lb/>cæ, cuius fru&longs;tum AD, & cylindro, vel portioni cylindri<lb/>cæ, cuius ba&longs;is e&longs;t circulus, vel ellip&longs;is CD, altitudo au<lb/>tem eadem completo cono, vel portioni conicæ iam dictæ, <lb/>illi pyramis, huic pri&longs;ma in&longs;cripta intelligantur, quorum <pb pagenum="16"/>communis ba&longs;is &longs;it poly gorum in&longs;criptum circulo quidem <lb/>æquilaterum, & æquiangulum; in ellip&longs;e autem, quod pro <lb/>Archimede de&longs;cribit Commandinus, ita vt & à cylindro, <lb/>vel cylindri portione pri&longs;ina, & à cono, vel coni portione <lb/>pyramis deficiat minori &longs;pacio quantacumque magnitudi<lb/>ne propo&longs;ita: quo modo autem in portione cylindrica, vel <lb/>conica hoc fieri po&longs;&longs;it, eadem quæ de cono atque cylindro <lb/>Euclides in duodecimo docuit manife&longs;tant. </s>

<s>Cono enim, vel portioni coni<lb/>cæ, cuius fru&longs;tum AD, & cylindro, vel portioni cylindri<lb/>cæ, cuius ba&longs;is e&longs;t circulus, vel ellip&longs;is CD, altitudo au<lb/>tem eadem completo cono, vel portioni conicæ iam dictæ, <lb/>illi pyramis, huic pri&longs;ma in&longs;cripta intelligantur, quorum <pb xlink:href="043/01/195.jpg" pagenum="16"/>communis ba&longs;is &longs;it poly gorum in&longs;criptum circulo quidem <lb/>æquilaterum, & æquiangulum; in ellip&longs;e autem, quod pro <lb/>Archimede de&longs;cribit Commandinus, ita vt & à cylindro, <lb/>vel cylindri portione pri&longs;ina, & à cono, vel coni portione <lb/>pyramis deficiat minori &longs;pacio quantacumque magnitudi<lb/>ne propo&longs;ita: quo modo autem in portione cylindrica, vel <lb/>conica hoc fieri po&longs;&longs;it, eadem quæ de cono atque cylindro <lb/>Euclides in duodecimo docuit manife&longs;tant. </s>

<s>Ab&longs;ci&longs;&longs;ione <lb/>igitur facta fru&longs;ti AD, & cylindri, vel portionis cylindricæ <lb/>CG, ab&longs;ci&longs;&longs;a &longs;imul erunt fru&longs;tum pyramidis in&longs;criptum <lb/>fru&longs;to AD, & pri&longs;ma in&longs;criptum cylindro, vel portioni cy<lb/>lindricæ CG, eiu&longs;dem altitudinis inter &longs;e, & duobus præ<lb/>dictis &longs;olidis AD, CG, deficien <lb/>tia vnum à fru&longs;to, alterum à cy<lb/>lindro, vel portione cylindrica <lb/>multo minori &longs;pacio magnitudine <lb/>propo&longs;ita: &longs;ectiones autem pri&longs;ma <lb/>tis, & pyramidis erunt polygona <lb/>circulis, vel ellip&longs;ibus ip&longs;i CD op <lb/>po&longs;itis & &longs;imilibus in&longs;cripta in<lb/>ter &longs;e &longs;imilia, vt multi o&longs;tendunt. <lb/></s>

<s>erunt etiam &longs;imilium polygono<lb/>rum circulis, vel ellip&longs;ibus &longs;imili<lb/>bus, quæ &longs;unt ba&longs;es oppo&longs;itæ fru<lb/><figure id="fig123"/><lb/>&longs;ti AD, in&longs;criptorum diametri eædem AB, CD. </s>

<s>erunt etiam &longs;imilium polygono<lb/>rum circulis, vel ellip&longs;ibus &longs;imili<lb/>bus, quæ &longs;unt ba&longs;es oppo&longs;itæ fru<lb/><figure id="id.043.01.195.1.jpg" xlink:href="043/01/195/1.jpg"/><lb/>&longs;ti AD, in&longs;criptorum diametri eædem AB, CD. </s>

<s>Quo<lb/>niam igitur &longs;imilium polygonorum circulis, & &longs;imilibus <lb/>ellip&longs;ibus in&longs;criptorum latera homologa inter &longs;e &longs;unt vt <lb/>diametri dictorum circulorum, vel ellip&longs;ium, eadem erit <lb/>proportio inter duas diametros AB, CD, hoc e&longs;t FC, <lb/>CD, quæ inter duo quælibet latera homologa polyga<lb/>norum circulis, vel ellip&longs;ibus &longs;imilibus AB, CD in<lb/>&longs;criptorum. </s>

<s>Sed pyramidis fru&longs;tum fru&longs;to CB in&longs;cri<lb/>ptum ad pri&longs;ma, cuius ba&longs;is e&longs;t maior ba&longs;is fru&longs;ti pyrami<lb/>dis, & eadem altitudo, &longs;olido CG in&longs;criptum, e&longs;t vt re-<pb pagenum="17"/>ctangulum contentum lateribus homologis ba&longs;ium oppo<lb/>&longs;itarum, vna cum tertia parte quadrati differentiæ, ad ma<lb/>ioris lateris quadratum; idem igitur fru&longs;tum pyramidis <lb/>ad idem pri&longs;ma, erit vt rectangulum DCF, vna cum <lb/>tertia parte quadrati DF ad quadratum CD: deficit <lb/>autem vtrumque & pyramidis fru&longs;tum fru&longs;to CB in&longs;cri<lb/>ptum ab ip&longs;o CB fru&longs;to, & pri&longs;ma ip&longs;i CG in&longs;criptum <lb/>ab ìp&longs;o CG, minori &longs;pacio quantacumque propo&longs;ita ma<lb/>gnitudine; per tertiam igitur huius, erit vt rectangulum <lb/>DCF vna cum tertia parte quadrati DF, ad CD qua<lb/>dratum, ita fru&longs;tum CB ad cylindrum, vel portionem <lb/>cylindricam CG. </s>

<s>Sed pyramidis fru&longs;tum fru&longs;to CB in&longs;cri<lb/>ptum ad pri&longs;ma, cuius ba&longs;is e&longs;t maior ba&longs;is fru&longs;ti pyrami<lb/>dis, & eadem altitudo, &longs;olido CG in&longs;criptum, e&longs;t vt re-<pb xlink:href="043/01/196.jpg" pagenum="17"/>ctangulum contentum lateribus homologis ba&longs;ium oppo<lb/>&longs;itarum, vna cum tertia parte quadrati differentiæ, ad ma<lb/>ioris lateris quadratum; idem igitur fru&longs;tum pyramidis <lb/>ad idem pri&longs;ma, erit vt rectangulum DCF, vna cum <lb/>tertia parte quadrati DF ad quadratum CD: deficit <lb/>autem vtrumque & pyramidis fru&longs;tum fru&longs;to CB in&longs;cri<lb/>ptum ab ip&longs;o CB fru&longs;to, & pri&longs;ma ip&longs;i CG in&longs;criptum <lb/>ab ìp&longs;o CG, minori &longs;pacio quantacumque propo&longs;ita ma<lb/>gnitudine; per tertiam igitur huius, erit vt rectangulum <lb/>DCF vna cum tertia parte quadrati DF, ad CD qua<lb/>dratum, ita fru&longs;tum CB ad cylindrum, vel portionem <lb/>cylindricam CG. </s>

<s>Cum igitur conus, vel coni portio E <lb/>CD &longs;it pars tertia cylindri, vel portionis cylindricæ CG, <lb/>erit ex æquali, vt idem rectangulum DCF, vna cum ter<lb/>tia parte quadrati DF, ad tertiam partem quadrati CD, <lb/>ita fru&longs;tum BC, ad conum vel coni portionem ECD. P<gap/>æ<lb/>terea, quia quadratum CD æquale e&longs;t duobus quadratis <lb/>ex CF, FD, vna cum rectangulo bis ex CF, FD: quorum <lb/>rectangulo CFD, vna cum quadrato CF æquale e&longs;t rectan <lb/>gulum DCF; erit quadratum CD æquale rectangulo <lb/>DCF vna cum quadrato DF; demptis igitur rectangu<lb/>lo DCF, & tertia parte quadrati DF; quod remanet <lb/>CD quadrati erit rectangulum CFD vna cum duabus <lb/>tertiis quadrati DF. quoniam igitur e&longs;t conuertendo vt <lb/>quadratum CD ad rectangulum DCF, vna cum tertia <lb/>parte quadrati DF, ita cylindris, vel portio cylindrica <lb/>CG ad fru&longs;tum CB, erit per conuer&longs;ionem rationis, & <lb/>conuertendo; vt rectangulum CFD vna cum duabus ter<lb/>tiis DF quadrati, ad quadratum CD, ita reliquum cy<lb/>lindri, vel portionis cylindricæ CG d<gap/>pto fru&longs;to CB, <lb/>ad cylindrum, vel portionem cylindricam. </s>

<s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s><pb pagenum="18"/>

<s>Manife&longs;tum <lb/>e&longs;t igitur propo&longs;itum. </s><pb xlink:href="043/01/197.jpg" pagenum="18"/>

<s><emph type="italics"/>PROPOSITIO XI.<emph.end type="italics"/></s>

Line 2270

<s>Dico reli<lb/>quæ &longs;ectionis centrum e&longs;&longs;e G. </s>

<s>Planum enim per OB &longs;e<lb/><figure id="fig124"/><lb/>cans &longs;phæram, vel &longs;phæroides, faciensque &longs;ectionem circu<lb/>lum, vel ellip&longs;im ABCD, &longs;ecabit, & &longs;ecet prædictas &longs;e<lb/>ctiones, circulos inquam, vel &longs;imiles ellip&longs;es parallelas, qua<lb/>rum alterius centrum ponitur F. </s>

<s>Planum enim per OB &longs;e<lb/><figure id="id.043.01.197.1.jpg" xlink:href="043/01/197/1.jpg"/><lb/>cans &longs;phæram, vel &longs;phæroides, faciensque &longs;ectionem circu<lb/>lum, vel ellip&longs;im ABCD, &longs;ecabit, & &longs;ecet prædictas &longs;e<lb/>ctiones, circulos inquam, vel &longs;imiles ellip&longs;es parallelas, qua<lb/>rum alterius centrum ponitur F. </s>

<s>Faciatque &longs;ectiones re<lb/>ctas parallelas AFC, KGH: &longs;imiliter aliud quodlibet <pb pagenum="19"/>planum per BE &longs;ecans &longs;phæram, vel &longs;phæroides faciat &longs;e<lb/>ctionem circulum, vel ellip&longs;im, & in ea parallelas LFM, <lb/>NGO, communes &longs;ectiones iam factæ &longs;ectionis &longs;phæræ <lb/>vel &longs;phæroidis cum circulis, vel ellip&longs;ibus inter &longs;e paral<lb/>lelis quarum diametri &longs;unt AC, KH. </s>

<s>Faciatque &longs;ectiones re<lb/>ctas parallelas AFC, KGH: &longs;imiliter aliud quodlibet <pb xlink:href="043/01/198.jpg" pagenum="19"/>planum per BE &longs;ecans &longs;phæram, vel &longs;phæroides faciat &longs;e<lb/>ctionem circulum, vel ellip&longs;im, & in ea parallelas LFM, <lb/>NGO, communes &longs;ectiones iam factæ &longs;ectionis &longs;phæræ <lb/>vel &longs;phæroidis cum circulis, vel ellip&longs;ibus inter &longs;e paral<lb/>lelis quarum diametri &longs;unt AC, KH. </s>

<s>Quoniam igitur <lb/>E e&longs;t centrum &longs;phæræ, vel &longs;phæroidis; omnes in eo per <lb/>punctum E, tran&longs;euntes rectæ lineæ bifariam &longs;ecabuntur: <lb/>&longs;ed idem E e&longs;t in &longs;ectione &longs;phæræ, vel &longs;phæroidis, circu<lb/>lo, vel ellip&longs;e ABCD; omnes igitur in ip&longs;a rectas lineas <lb/>bifariam &longs;ecabit punctum E, & centrum erit circuli, <lb/>vel ellip&longs;is ABCD: quædam igitur ex centro recta EB <lb/>&longs;ecans parallelarum neutrius per centrum ductæ alteram <lb/>AC bifariam in circuli, vel ellip&longs;is ALCM centro F, <lb/>& reliquam in puncto G bifariam &longs;ecabit. </s>

Line 2284

<s><emph type="italics"/>COROLLARIVM.<emph.end type="italics"/></s>

<s>Hinc manife&longs;tum e&longs;t, &longs;i &longs;phæra, vel &longs;phæroides <lb/>&longs;ecetur plano non per centrum: & recta linea &longs;phæ <lb/>ræ, vel &longs;phæroidis, & factæ &longs;ectionis centra iun<lb/>gens ad &longs;uperficiem vtrinque producatur; talis <lb/>axis &longs;egmenta e&longs;&longs;e <gap/> portionum, earumque <lb/>vertices extrema dicti axis, vt in figura theorema<lb/>tis &longs;unt puncta B, D. </s><pb pagenum="20"/>

<s><emph type="italics"/>PROPOSITIO XII.<emph.end type="italics"/></s>

Line 2294

<s>Et &longs;olido ABC circum&longs;criptus cylindrus, vel portio cy<lb/>lindrica, cuius ba&longs;es oppo&longs;itæ erunt circuli, vel &longs;imiles elli<lb/>p&longs;es, quarum diametri eiu&longs;dem rationis ADC, EF, la<lb/>tera oppo&longs;ita parallelogrammi per axem AFGC: & &longs;u<lb/>per ba&longs;im, cuius diameter EF, circa axim BD, de&longs;criptus <lb/>e&longs;to conus, vel coni portio EDF. </s>

<s>Iam tria &longs;olida ABC, <lb/>EDF, AC, &longs;ecentur plano &longs;olidi ABC ba&longs;i parallelo, <lb/>quod &longs;ecabit, & &longs;ecet vnà figuras planas per axim BD <pb pagenum="21"/>tribus &longs;olidis communem, po&longs;itas in eodem plano, quæ &longs;unt <lb/>AF parallelogrammum, triangulum EDF, & &longs;emicir<lb/>culus, vel &longs;emi ellip&longs;is ABC: & &longs;int &longs;ectiones rectæ GO, <lb/>HN, KM: hæ igitnr erunt diametri eiu&longs;dem rationis trium <lb/>&longs;ectionum, &longs;cilicet circulorum, vel ellip&longs;ium &longs;irnilium, qui<lb/>bus erit commune centrum L, in quo nimirum axis BD <lb/>tres dictas lineas GO, HN, KM, bifariam &longs;ecat. </s>

<s>Iam tria &longs;olida ABC, <lb/>EDF, AC, &longs;ecentur plano &longs;olidi ABC ba&longs;i parallelo, <lb/>quod &longs;ecabit, & &longs;ecet vnà figuras planas per axim BD <pb xlink:href="043/01/200.jpg" pagenum="21"/>tribus &longs;olidis communem, po&longs;itas in eodem plano, quæ &longs;unt <lb/>AF parallelogrammum, triangulum EDF, & &longs;emicir<lb/>culus, vel &longs;emi ellip&longs;is ABC: & &longs;int &longs;ectiones rectæ GO, <lb/>HN, KM: hæ igitnr erunt diametri eiu&longs;dem rationis trium <lb/>&longs;ectionum, &longs;cilicet circulorum, vel ellip&longs;ium &longs;irnilium, qui<lb/>bus erit commune centrum L, in quo nimirum axis BD <lb/>tres dictas lineas GO, HN, KM, bifariam &longs;ecat. </s>

<s>Vt <lb/>igitur de &longs;olido AF diximus, &longs;int circa axem BL, & &longs;uper <lb/>ba&longs;es circulos, vel ellip&longs;es circa HN, KM cylindri, vel <lb/>portiones cylindricæ HP, KQ, qui vnà cum portione <lb/>cylindrica, vel cylindro GF ip&longs;a &longs;ectione facto, erunt inter <lb/>eadem plana paral<lb/>lela per EF, GO. <lb/></s>

<s>Dico trium cylin<lb/>drorum, vel cylin<lb/>dri portionum GF, <lb/>HP, KQ, <expan abbr="reliqu&utilde;">reliquum</expan> <lb/>ip&longs;ius GF dempto <lb/>HP, ip&longs;i KQ e&longs;se <lb/><figure id="fig125"/><lb/>æquale. </s>

<s>Quoniam <lb/>enim cylindri, & cy<lb/>lindri portiones eiu&longs;dem altitudinis inter &longs;e &longs;unt vt ba<lb/>&longs;es, circuli autem, & &longs;imiles ellip&longs;es; inter &longs;e, vt quæ à <lb/>diametris eiu&longs;dem rationis fiunt quadrata; ex Archime<lb/>de, hoc e&longs;t vt earum quartæ partes, quæ à &longs;emidiame<lb/>tris quadrata de&longs;cribuntur; erit vt quadratum LO ad <lb/>quadratum LN, ita cylindrus, vel portio cylindrica <lb/>GF ad cylindrum, vel portionem cylindricam PH: & <lb/>diuidendo, vt rectangulum LNO bis vnà cum quadra<lb/>to NO, ad quadratum LN, ita reliquum cylindri, vel <lb/>portionis cylindricæ GF, dempto ip&longs;o PH, ad ip&longs;um <lb/>PH: &longs;ed vt quadratum LN ad quadratum LM, ita e&longs;t <lb/>vt &longs;upra, cylindrus, vel portio cylindrica HP ad cylin<lb/>drum, vel portionem cylindricam KQ, ex æquali igitur, <pb pagenum="22"/>erit vt rectangulum LNO bis, vnà cum quadrato NO, <lb/>ad quadratum LM, ita reliquum cylindri, vel portionis <lb/>cylindricæ GF <expan abbr="d&etilde;-pto">den<lb/>pto</expan> HP, ad cylin<lb/>drum, vel <expan abbr="portion&etilde;">portionem</expan> <lb/>cylindricam KQ: <lb/>&longs;ed rectangulum L <lb/>NO bis vnà <expan abbr="c&utilde;">cum</expan> qua <lb/>drato NO æquale <lb/>e&longs;t quadrato LM; <lb/>reliquum igitur cy<lb/><figure id="fig126"/><lb/>lindri, vel portionis <lb/>cylindricæ GF, <expan abbr="d&etilde;-pto">den<lb/>pto</expan> HP, æquale erit cylindro, vel portioni cylindricæ <expan abbr="Kq.">Kque</expan> <lb/>Quod erat demon&longs;trandum. </s>

<s>Quoniam <lb/>enim cylindri, & cy<lb/>lindri portiones eiu&longs;dem altitudinis inter &longs;e &longs;unt vt ba<lb/>&longs;es, circuli autem, & &longs;imiles ellip&longs;es; inter &longs;e, vt quæ à <lb/>diametris eiu&longs;dem rationis fiunt quadrata; ex Archime<lb/>de, hoc e&longs;t vt earum quartæ partes, quæ à &longs;emidiame<lb/>tris quadrata de&longs;cribuntur; erit vt quadratum LO ad <lb/>quadratum LN, ita cylindrus, vel portio cylindrica <lb/>GF ad cylindrum, vel portionem cylindricam PH: & <lb/>diuidendo, vt rectangulum LNO bis vnà cum quadra<lb/>to NO, ad quadratum LN, ita reliquum cylindri, vel <lb/>portionis cylindricæ GF, dempto ip&longs;o PH, ad ip&longs;um <lb/>PH: &longs;ed vt quadratum LN ad quadratum LM, ita e&longs;t <lb/>vt &longs;upra, cylindrus, vel portio cylindrica HP ad cylin<lb/>drum, vel portionem cylindricam KQ, ex æquali igitur, <pb xlink:href="043/01/201.jpg" pagenum="22"/>erit vt rectangulum LNO bis, vnà cum quadrato NO, <lb/>ad quadratum LM, ita reliquum cylindri, vel portionis <lb/>cylindricæ GF <expan abbr="d&etilde;-pto">den<lb/>pto</expan> HP, ad cylin<lb/>drum, vel <expan abbr="portion&etilde;">portionem</expan> <lb/>cylindricam KQ: <lb/>&longs;ed rectangulum L <lb/>NO bis vnà <expan abbr="c&utilde;">cum</expan> qua <lb/>drato NO æquale <lb/>e&longs;t quadrato LM; <lb/>reliquum igitur cy<lb/><figure id="id.043.01.201.1.jpg" xlink:href="043/01/201/1.jpg"/><lb/>lindri, vel portionis <lb/>cylindricæ GF, <expan abbr="d&etilde;-pto">den<lb/>pto</expan> HP, æquale erit cylindro, vel portioni cylindricæ <expan abbr="Kq.">Kque</expan> <lb/>Quod erat demon&longs;trandum. </s>

<s><emph type="italics"/>PROPOSITIO XIII.<emph.end type="italics"/></s>

Line 2308

<s>E&longs;to hemi&longs;phærio, vel hemi&longs;phæroidi ABC, cu<lb/>ius axis BD, ba&longs;is circulus, vel ellip&longs;is circa diametrum <lb/>ADC, circum&longs;criptus cylindrus, vel cylindrica portio <lb/>AE, circa communem &longs;cilicet axim BD. conus autem, <lb/>vel coni portio circa axim BD, ba&longs;im habens commu<lb/>nem &longs;olido ABC, intelligatur. </s>

<s>Dico reliquum &longs;olidi <lb/>AE, dempto hemi&longs;phærio, vel hemi&longs;phæroide ABC æ-<pb pagenum="23"/>quale e&longs;se cono, vel portioni conicæ. </s>

<s>Dico reliquum &longs;olidi <lb/>AE, dempto hemi&longs;phærio, vel hemi&longs;phæroide ABC æ-<pb xlink:href="043/01/202.jpg" pagenum="23"/>quale e&longs;se cono, vel portioni conicæ. </s>

<s>Nam circa axim <lb/>BD, & &longs;uper ba&longs;im circulum, vel ellip&longs;im, cuius diame<lb/>ter RE, &longs;imilem & oppo&longs;itam ei, quæ circa AC, de&longs;cri<lb/>batur conus, vel coni portio RDE. </s>

<s>Deinde axe BD bi<lb/>fariam &longs;ecto, & &longs;ingulis eius partibus rur&longs;us bifariam, vt <lb/>partes axis BD omnes &longs;int æquales, per puncta &longs;ectio<lb/>num, quotquot erunt, totidem plana parallela &longs;ecent vnà <lb/>cum &longs;olido AE duas ip&longs;ius partes, &longs;olida ABC, RDE. <lb/></s>

<s>Omnes igitur factæ &longs;ectiones, vel erunt circuli, vel &longs;imiles <lb/>ellip&longs;es ei, quæ e&longs;t circa AC, atque adeo inter &longs;e &longs;imiles: <lb/>talium autem &longs;ectiones communes cum AE parallelo, <lb/><figure id="fig127"/><lb/>grammo per axim, erunt rectæ lineæ, ternæ in &longs;ingu<lb/>lis planis &longs;ecantibus, & in eadem recta linea; vt in proxi<lb/>ma ip&longs;i RE, &longs;unt FL, GN, KM, quæ quidem erunt <lb/>trium circulorum, vel &longs;imilium ellip&longs;ium diametri eiu&longs;dem <lb/>rationis ba&longs;ium trium &longs;olidorum, cylindri &longs;cilicet, vel por<lb/>tionis cylindricæ FL, fru&longs;ti GL, & portionis KBM, he <lb/>mi&longs;phærij, vel hemi&longs;phæroidis ABC. </s>

<s>Omnes igitur factæ &longs;ectiones, vel erunt circuli, vel &longs;imiles <lb/>ellip&longs;es ei, quæ e&longs;t circa AC, atque adeo inter &longs;e &longs;imiles: <lb/>talium autem &longs;ectiones communes cum AE parallelo, <lb/><figure id="id.043.01.202.1.jpg" xlink:href="043/01/202/1.jpg"/><lb/>grammo per axim, erunt rectæ lineæ, ternæ in &longs;ingu<lb/>lis planis &longs;ecantibus, & in eadem recta linea; vt in proxi<lb/>ma ip&longs;i RE, &longs;unt FL, GN, KM, quæ quidem erunt <lb/>trium circulorum, vel &longs;imilium ellip&longs;ium diametri eiu&longs;dem <lb/>rationis ba&longs;ium trium &longs;olidorum, cylindri &longs;cilicet, vel por<lb/>tionis cylindricæ FL, fru&longs;ti GL, & portionis KBM, he <lb/>mi&longs;phærij, vel hemi&longs;phæroidis ABC. </s>

<s>Itaque circa axem <lb/>BH cylindri, vel portionis cylindricæ FE, & &longs;uper ba<lb/>&longs;es circulos, vel ellip&longs;es circa GN, KM, de&longs;cribantur <lb/>cylindri, vel cylindri portiones GP, KQ, qui pat<lb/>tes erunt totius cylindri, vel portionis cylindricæ FE. <lb/></s>

<s>Idem fiat circa reliquas axis partes BD tamquam axes, <pb pagenum="24"/>&longs;uper reliquas &longs;ectiones ternas in &longs;ingulis prædictis planis <lb/>&longs;ecantibus. </s>

<s>Idem fiat circa reliquas axis partes BD tamquam axes, <pb xlink:href="043/01/203.jpg" pagenum="24"/>&longs;uper reliquas &longs;ectiones ternas in &longs;ingulis prædictis planis <lb/>&longs;ecantibus. </s>

<s>Hac ratione habebimus iam duas figuras <lb/>compo&longs;itas ex cylindris, vel cylindri portionibus altitudi<lb/>ne, & multitudine æqualibus, alteram cono, vel portioni <lb/>conicæ RDE in&longs;criptam, alteram hemilphærio, vel he<lb/>mi&longs;phæroidi ABC circum&longs;criptam: quod ita factum e&longs;<lb/>&longs;e intelligatur, quemadmodum in primo libro fieri po&longs;se <lb/>demon&longs;trauimus, vt figura cono RDE in&longs;cripta ab eo <lb/>deficiat, hemi&longs;phærio autem, vel hemi&longs;phæroidi ABC <lb/>circum&longs;cripta ip&longs;um excedat minori &longs;pacio magnitudine <lb/>propo&longs;ita quantacumque illa &longs;it. </s>

<s>Reliquo itaque cylin<lb/><figure id="fig128"/><lb/>dri, vel portionis cylindricæ AE dempto hemi&longs;phærio, vel <lb/>hemi&longs;phæroide ABC figura quædam in&longs;cripta relinque<lb/>tur ex cylindris, vel portionis cylindricæ re&longs;iduis æqualium <lb/>altitudinum, demptis ijs, ex quibus con&longs;tat figura hemi<lb/>&longs;phærio, vel hemi&longs;phæroidi ABC circum&longs;cripta, excepto <lb/>infimo cylindro, vel portione cylindrica AS. </s>

<s>Reliquo itaque cylin<lb/><figure id="id.043.01.203.1.jpg" xlink:href="043/01/203/1.jpg"/><lb/>dri, vel portionis cylindricæ AE dempto hemi&longs;phærio, vel <lb/>hemi&longs;phæroide ABC figura quædam in&longs;cripta relinque<lb/>tur ex cylindris, vel portionis cylindricæ re&longs;iduis æqualium <lb/>altitudinum, demptis ijs, ex quibus con&longs;tat figura hemi<lb/>&longs;phærio, vel hemi&longs;phæroidi ABC circum&longs;cripta, excepto <lb/>infimo cylindro, vel portione cylindrica AS. </s>

<s>Et quo<lb/>niam (excepto exce&longs;su, quo &longs;olidum AS excedit &longs;ui par<lb/>tem portionem quandam hemi&longs;phærij, vel hemi&longs;phæroidis <lb/>ABC) quo &longs;pacio figura hemi&longs;phærio, vel hemi&longs;phæroidi <lb/>ABC circum&longs;cripta &longs;uperat ip&longs;um hemi&longs;phærium, vel he <lb/><gap/>æroides, eodem figura prædicto re&longs;iduo in&longs;cripta de<lb/><gap/>duo; deficiet ab eodem minori differentia quàm <pb pagenum="25"/>&longs;it magnitudo propo&longs;ita,. His ita ex po&longs;itis, quoniam ex <lb/>præcedenti, reliquum cylindri, vel portionis cylindricæ <lb/>FE dempto cylindro, vel portione cylindrica KQ, æ<lb/>quale e&longs;t cylindro, vel portioni cylindricæ GP: eadem<lb/>que ratione &longs;ingula cylindrorum, vel cylindri portionum <lb/>re&longs;idua, quæ &longs;unt in reliqua figura cylindri, vel portionis <lb/>cylindricæ AE, dempto hemi&longs;phærio, vel hemi&longs;phæroi<lb/>de ABC, æqualia erunt &longs;ingulis cylindris, vel cylindri <lb/>portionibus, quæ &longs;unt in cono, vel portione conica RDE, <lb/>&longs;i bina &longs;umantur inter eadem plana parallela, vel circa <lb/>cundem axem; tota igitur figura in&longs;cripta prædicto re&longs;iduo, <lb/>toti figuræ in&longs;criptæ cono, vel portioni conicæ RDE æ<lb/>qualis erit: deficit autem vtraque figura in&longs;cripta à &longs;ibi <lb/>circum&longs;cripta minori &longs;pacio quantacumque magnitudine <lb/>propo&longs;ita; per tertiam igitur huius, reliquum cylindri, vel <lb/>portionis cylindricæ AE, dempto hemi&longs;phærin, vel he<lb/>mi&longs;phæroide ABC, æquale e&longs;t cono, vel portioni coni<lb/>cæ RDE, hoc e&longs;t ip&longs;i ABC. </s>

<s>Et quo<lb/>niam (excepto exce&longs;su, quo &longs;olidum AS excedit &longs;ui par<lb/>tem portionem quandam hemi&longs;phærij, vel hemi&longs;phæroidis <lb/>ABC) quo &longs;pacio figura hemi&longs;phærio, vel hemi&longs;phæroidi <lb/>ABC circum&longs;cripta &longs;uperat ip&longs;um hemi&longs;phærium, vel he <lb/><gap/>æroides, eodem figura prædicto re&longs;iduo in&longs;cripta de<lb/><gap/>duo; deficiet ab eodem minori differentia quàm <pb xlink:href="043/01/204.jpg" pagenum="25"/>&longs;it magnitudo propo&longs;ita,. His ita ex po&longs;itis, quoniam ex <lb/>præcedenti, reliquum cylindri, vel portionis cylindricæ <lb/>FE dempto cylindro, vel portione cylindrica KQ, æ<lb/>quale e&longs;t cylindro, vel portioni cylindricæ GP: eadem<lb/>que ratione &longs;ingula cylindrorum, vel cylindri portionum <lb/>re&longs;idua, quæ &longs;unt in reliqua figura cylindri, vel portionis <lb/>cylindricæ AE, dempto hemi&longs;phærio, vel hemi&longs;phæroi<lb/>de ABC, æqualia erunt &longs;ingulis cylindris, vel cylindri <lb/>portionibus, quæ &longs;unt in cono, vel portione conica RDE, <lb/>&longs;i bina &longs;umantur inter eadem plana parallela, vel circa <lb/>cundem axem; tota igitur figura in&longs;cripta prædicto re&longs;iduo, <lb/>toti figuræ in&longs;criptæ cono, vel portioni conicæ RDE æ<lb/>qualis erit: deficit autem vtraque figura in&longs;cripta à &longs;ibi <lb/>circum&longs;cripta minori &longs;pacio quantacumque magnitudine <lb/>propo&longs;ita; per tertiam igitur huius, reliquum cylindri, vel <lb/>portionis cylindricæ AE, dempto hemi&longs;phærin, vel he<lb/>mi&longs;phæroide ABC, æquale e&longs;t cono, vel portioni coni<lb/>cæ RDE, hoc e&longs;t ip&longs;i ABC. </s>

<s>Quod erat demon&longs;trandum. </s>

<s><emph type="italics"/>PROPOSITIO XIV.<emph.end type="italics"/></s>

<s>Si hemi&longs;phærium, vel hemi&longs;phæroides, & cylin <lb/>drus, vel portio cylindrica ip&longs;i circum&longs;cripta, & <lb/>conus, vel coni portio, cuius e&longs;t <expan abbr="id&etilde;">idem</expan> axis portioni, <lb/>ba&longs;is autem qu<17> opponitur communi ba&longs;i duorum <lb/>prædictorum &longs;olidorum, vnà &longs;ecentur duobus <lb/>planis ba&longs;i parallelis; portiones reliquæ figuræ <lb/>ex cylindro, vel cylindri portione hemi&longs;phærio, <lb/>vel hemi&longs;phæroidi circum&longs;cripta dempto hemi<lb/>&longs;phærio, vel hemi&longs;phæroide, quæ à duobus præ<lb/>dictis planis &longs;ecantibus fiunt, æquales &longs;unt &longs;in<pb pagenum="26"/>gulæ &longs;ingulis prædicti coni, vel conicæ portionis <lb/>partibus &longs;iue fru&longs;tis inter eadem plana parallela <lb/>re&longs;pondentibus. </s>

<s>Si hemi&longs;phærium, vel hemi&longs;phæroides, & cylin <lb/>drus, vel portio cylindrica ip&longs;i circum&longs;cripta, & <lb/>conus, vel coni portio, cuius e&longs;t <expan abbr="id&etilde;">idem</expan> axis portioni, <lb/>ba&longs;is autem qu<17> opponitur communi ba&longs;i duorum <lb/>prædictorum &longs;olidorum, vnà &longs;ecentur duobus <lb/>planis ba&longs;i parallelis; portiones reliquæ figuræ <lb/>ex cylindro, vel cylindri portione hemi&longs;phærio, <lb/>vel hemi&longs;phæroidi circum&longs;cripta dempto hemi<lb/>&longs;phærio, vel hemi&longs;phæroide, quæ à duobus præ<lb/>dictis planis &longs;ecantibus fiunt, æquales &longs;unt &longs;in<pb xlink:href="043/01/205.jpg" pagenum="26"/>gulæ &longs;ingulis prædicti coni, vel conicæ portionis <lb/>partibus &longs;iue fru&longs;tis inter eadem plana parallela <lb/>re&longs;pondentibus. </s>

<s>E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cu<lb/>ius axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diame<lb/>ter ADC. &longs;olido autem ABC circum&longs;criptus cylindrus, <lb/>vel portio cylindrica AXEC: & conus, vel coni portio <lb/>&longs;it XDE, cuius vertex D, ba&longs;is circulus, vel ellip&longs;is cir<lb/>ca XBE ba&longs;i &longs;olidi AE, vel ABC, prædictæ oppo&longs;ita, <lb/>&longs;ecto autem &longs;olido AE, atque vnà cum ip&longs;o eius partibus, <lb/>&longs;olidis ABC, XD <lb/>E, duobus planis ba <lb/>&longs;i &longs;olidi AE, vel <lb/>ABC, atque ideo <lb/>inter &longs;e quoque pa<lb/>rallelis, intelligan<lb/>tur trium &longs;olidorum <lb/>portiones ternæ in<lb/><figure id="fig129"/><lb/>ter eadem plana pa<lb/>rallela: videlicet in<lb/>ter duo per XE, <lb/>FN, hemi&longs;phærij, vel hemi&longs;phæroidis minor portio HBL: <lb/>& reliquum cylindri, vel portionis cylindricæ FE dem<lb/>pta portione HBL: & coni, vel conicæ portionis fru&longs;tum <lb/>XGME. &longs;imiliter inter duo plana per FN, OV &longs;olidi <lb/>ABC portio PHLT, eaque ablata reliquum &longs;olidi ON, <lb/>& fru&longs;tum GQSM. </s>

<s>E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cu<lb/>ius axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diame<lb/>ter ADC. &longs;olido autem ABC circum&longs;criptus cylindrus, <lb/>vel portio cylindrica AXEC: & conus, vel coni portio <lb/>&longs;it XDE, cuius vertex D, ba&longs;is circulus, vel ellip&longs;is cir<lb/>ca XBE ba&longs;i &longs;olidi AE, vel ABC, prædictæ oppo&longs;ita, <lb/>&longs;ecto autem &longs;olido AE, atque vnà cum ip&longs;o eius partibus, <lb/>&longs;olidis ABC, XD <lb/>E, duobus planis ba <lb/>&longs;i &longs;olidi AE, vel <lb/>ABC, atque ideo <lb/>inter &longs;e quoque pa<lb/>rallelis, intelligan<lb/>tur trium &longs;olidorum <lb/>portiones ternæ in<lb/><figure id="id.043.01.205.1.jpg" xlink:href="043/01/205/1.jpg"/><lb/>ter eadem plana pa<lb/>rallela: videlicet in<lb/>ter duo per XE, <lb/>FN, hemi&longs;phærij, vel hemi&longs;phæroidis minor portio HBL: <lb/>& reliquum cylindri, vel portionis cylindricæ FE dem<lb/>pta portione HBL: & coni, vel conicæ portionis fru&longs;tum <lb/>XGME. &longs;imiliter inter duo plana per FN, OV &longs;olidi <lb/>ABC portio PHLT, eaque ablata reliquum &longs;olidi ON, <lb/>& fru&longs;tum GQSM. </s>