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<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd">
<archimedes xmlns:xlink="http://www.w3.org/1999/xlink">
<info>
    <author>Monte, Guidobaldo del</author>
    <title>In Duos Archimedis Aequeponderatium libros paraphrasis</title>
    <date>1588</date>
    <place>Pesaro</place>
    <translator/>
    <lang>la</lang>
    <cvs_file>monte_aeque_077_la_1588.xml</cvs_file>
    <cvs_version/>
    <locator>077.xml</locator>
  </info>
  <text>
    <front>          </front>
    <body>
      <chap id="N10019">
        <pb xlink:href="077/01/001.jpg" id="p.0001"/>
        <p id="N1001D" type="head">
          <s id="N1001F">GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS <lb/>IN DVOS ARCHIMEDIS <lb/>&AElig;QVEPONDERANTIVM <lb/>LIBROS</s>
        </p>
        <p id="N1002B" type="head">
          <s id="N1002D">PARAPHRASIS <lb/>Scholijs illu&longs;trata.</s>
        </p>
        <figure id="id.077.01.001.1.jpg" xlink:href="077/01/001/1.jpg"/>
        <p id="N10034" type="head">
          <s id="N10036">PISAVRI <lb/>Apud Hieronymum Concordiam; <lb/>M D LXXXVIII. <lb/><emph type="italics"/>Superiorum Conce&longs;&longs;u.<emph.end type="italics"/></s>
        </p>
        <pb xlink:href="077/01/002.jpg"/>
        <pb xlink:href="077/01/003.jpg"/>
        <p id="N10046" type="head">
          <s id="N10048">SERENISSIMO <lb/>FRANC.^&lcub;CO} MARIAE <lb/>II. VRBINI DVCI.</s>
        </p>
        <p id="N1004E" type="head">
          <s id="N10050">GVIDVSVBALDVS <lb/>E&apos; MARCHIONIBVS MONTIS S.</s>
        </p>
        <p id="N10054" type="main">
          <s id="N10056">Iam decemnium elap&longs;um e&longs;t, DVX Sere&shy;<lb/>ni&longs;&longs;ime, ex quo de rebus machanicis volu&shy;<lb/>men, veras (ni fallor) mirabilium mechani&shy;<lb/>corum effectuum cau&longs;as manife&longs;tans, in lu&shy;<lb/>cem dedi; vbi non nulla antiquiora, <expan abbr="pr&aelig;ci-puaq;">pr&aelig;ci&shy;<lb/>pua&que;</expan> illu&longs;trium gr&aelig;corum authorum pla&shy;<lb/>cita ad &longs;u&longs;ceptum negotium pertinentia, <lb/>tanquam rect&etail; rationi magis con&longs;entanea amplexatus &longs;um. </s>
          <s id="N1006A"><lb/>quibus &longs;an&egrave;, tanquam &longs;olidi&longs;&longs;imis innixa fundamentis, theo&shy;<lb/>remata multa, ac varia con&longs;truxi. </s>
          <s id="N1006F">quippe qu&aelig;, licet non inua&shy;<lb/>lidis quo&que; demon&longs;trationum pr&aelig;&longs;idijs &agrave; me ip&longs;o munita <lb/>fuerint; pleri&longs;qu&egrave; tamen, qui non admodum forta&longs;&longs;e in huiu&longs;&shy;<lb/>modi rerum cau&longs;is inue&longs;tigandis ver&longs;ati exi&longs;tunt, noua pror&shy;<lb/>&longs;us (vt accepi) ac ferme inaudita, nec &longs;atis (vt opinor) apud eos <lb/>firma, at&que; ideo illis non omnino &longs;atisfeci&longs;&longs;e, vi&longs;a &longs;unt. </s>
          <s id="N1007B">Quo&shy;<lb/>circa cogitanti mihi, qua ratione fieri po&longs;&longs;et, vt opus illud &agrave; <lb/>me editum, qu&agrave;m plurimorum &longs;ibi gratiam in dies magis con<lb/>ciliaret, in mentem venit, non aliunde id mihi oportuni&ugrave;s <expan abbr="c&otilde;tingere">con<lb/>tingere</expan> potui&longs;&longs;e, qu&agrave;m &longs;i pri&longs;cos ip&longs;os, &amp; graui&longs;&longs;imos alioqui <lb/>authores de hac re eleganti&longs;&longs;im&egrave; di&longs;&longs;erentes illis offerrem. </s>
          <s id="N1008B">ra&shy;<lb/>tus, vt &longs;olidi&longs;&longs;im&acirc; eorum doctrin&acirc;, qu&aelig; &agrave; me propo&longs;ita, &amp; ex&shy;<pb xlink:href="077/01/004.jpg"/>plicata fuere theoremata, firmiora redderentur. </s>
          <s id="N10092">&longs;imulqu&egrave; alio<lb/>rum ambiguitati, ne dicam imbecillitam &longs;uccurreretur. </s>
          <s id="N10096">vel &longs;al<lb/>tem ip&longs;i graui&longs;&longs;ima eorum authoritate non nullorum captiua&shy;<lb/>rent intellectum, in ob&longs;equium meli&ugrave;s, recti&ugrave;&longs;qu&egrave; <expan abbr="&longs;entienti&utilde;">&longs;entientium</expan>, <lb/>at&que; intelligentium. </s>
          <s id="N100A2">Nihil enim tam, aut a con&longs;uetudine, aut <lb/>ab opinione remotum e&longs;&longs;e &longs;olet, quod &longs;ola authoritate proba&shy;<lb/>ri non po&longs;&longs;it. </s>
          <s id="N100A8">Ver&ugrave;m ne huiu&longs;modi negotium in recen&longs;endis <lb/>multorum ad propo&longs;itam veritatem confirmandam te&longs;timo&shy;<lb/>nijs lati&ugrave;s, qu&agrave;m par e&longs;&longs;et, protraheretur; mihi con&longs;titui, ex mul<lb/>tis vnicum tant&ugrave;m, eumqu&egrave; reliquorum omnium hac in par <lb/>te facil&egrave; principem deligere: qui, &amp; meam cau&longs;am tueretur: &amp; <lb/>illis, &longs;i fieri po&longs;&longs;et, &longs;atisfaceret: vt&que; grave; cor&agrave;m illis ip&longs;e &longs;e offerens, <lb/>tanquam meo quo&que; nomine mi&longs;&longs;us intelligeretur; quibu&longs;&shy;<lb/>dam meis notis non in&longs;ignitum cert&egrave;, &longs;ed a&longs;&longs;ociatum eundem <lb/>prodire volui. </s>
          <s id="N100BA">E&longs;t autem graui&longs;&longs;imus hic author Syracu&longs;ius ille <lb/>Archimedes de mechanicis elementis con&longs;ulti&longs;&longs;im&egrave; di&longs;&longs;erens. </s>
          <s id="N100BE"><lb/>cuius nimirum dignitati, at&que; authoritati, vt omnes prob&egrave; &agrave; <lb/>me con&longs;ultum intelligerent; decreui, vt &que;madmodum inter <lb/>alios illius ordinis viros primatum obtinet, ita nulli alij, qu&agrave;m <lb/>amplitudini tu&etail; DVX Sereni&longs;&longs;ime, hac no&longs;tra &etail;tate, doctrina, <lb/>rerumqu&egrave; omnium cognitione &longs;ingulari, citra controuer&longs;iam <lb/>Principi &longs;upremo, &longs;uum in primis hoc tempore pr&aelig;&longs;taret ob&longs;e<lb/>quium. </s>
          <s id="N100CD">quod incredibili &longs;an&egrave; animi mei iucunditate conti&shy;<lb/>gi&longs;&longs;e fateor; non &longs;ol&ugrave;m, vt rur&longs;um aliquam &longs;ingularis me&aelig; er&shy;<lb/>ga amplitudinem tuam ob&longs;eruanti&aelig;, ac venerationis, tot, tan&shy;<lb/>ti&longs;qu&egrave; nominibus iam pridem debit&etail; te&longs;tificationem ederem; <lb/>ver&ugrave;m etiam, vt munu&longs;culo illi meo tanto Principi audenti&ugrave;s <lb/>forta&longs;&longs;e antea oblato, ne pror&longs;us pr&etail; &longs;ua tenuitate de&longs;piceretur, <lb/>opem ferret. </s>
          <s id="N100DB">quanquam ne&que; id quidem, pro eximia animi <lb/>tam excel&longs;i magnitudine, &longs;u&longs;picandum fuit. </s>
          <s id="N100DF">Per hunc ergo <expan abbr="t&atilde;">tam</expan> <lb/>celebrem authorem ad te Princeps optime, ac pr&etail;&longs;tanti&longs;&longs;ime <lb/>l&aelig;tabundus accedo. </s>
          <s id="N100E9">Is enim mihi, &que;madmodum &amp; ego ip&longs;i, <lb/>ad te aditum patefeci&longs;&longs;e videtur; &amp; &longs;icut eundem tibi <expan abbr="l&otilde;ge">longe</expan> gra&shy;<lb/>ti&longs;&longs;imum futurum confido; ita me tui amanti&longs;&longs;imum, &amp; ob&longs;er<lb/>uanti&longs;&longs;imum, vt e&acirc;dem, qua con&longs;ueui&longs;ti, benignitate pro&longs;e&shy;<lb/>quaris, oro &longs;uplex, &amp; ob&longs;ecro. </s>
          <s id="N100F3">Aueto dulce pr&aelig;&longs;idium, ac &etail;tatis <lb/>no&longs;tr&aelig; &longs;plendidum decus; &amp; e&longs;to perpetu&ograve; f&etail;lix. </s>
        </p>
        <pb xlink:href="077/01/005.jpg" pagenum="1"/>
        <p id="N100FA" type="head">
          <s id="N100FC">GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS.</s>
        </p>
        <p id="N10102" type="head">
          <s id="N10104">PRAEFATIO:</s>
        </p>
        <p id="N10106" type="main">
          <s id="N10108">Mechanica facultas <expan abbr="n&otilde;">non</expan> &longs;ol&ugrave;m ab imperitis, <lb/>ver&ugrave;m etiam ab eruditis admirabilis &longs;em&shy;<lb/>per habita fuit; eorum enim, qu&etail; in admi&shy;<lb/>rationem homines trahunt, duo e&longs;&longs;e gene&shy;<lb/>ra Ari&longs;toteles in principio <expan abbr="&longs;uar&utilde;">&longs;uarum</expan> <expan abbr="qu&etail;&longs;tion&utilde;">qu&etail;&longs;tionum</expan> <lb/>Meehanicarum a&longs;&longs;eruit; quorum &longs;an&egrave; alte <lb/>rum ad ea pertinet, qu&aelig; natura quidem, <lb/>proximis tamen ip&longs;orum cau&longs;is latentibus in lucem <expan abbr="prode&utilde;t">prodeunt</expan>; <lb/>alterum ver&ograve; &longs;pectat ad ea, qu&etail; pr&etail;ter naturam, &amp; arte fiunt; <lb/>quibus natura &longs;uperari videtur (quamquam &amp; ip&longs;a plurim&ugrave;m <lb/>momenti ad &longs;e ip&longs;am euincendam tune quo&que; afferat) &amp; <lb/>quod natur&etail; uiribus in lucem prodire nequit, id arte fieri con<lb/>tingat, ob idqu&egrave; maiorem adhuc admirationem excitat, qu&ograve;d <lb/>ars natur&etail; &etail;mula, qua&longs;i aduer&longs;us naturam <expan abbr="ipugn&atilde;s">ipugnans</expan>, cam &longs;upe&shy;<lb/>ret, &amp; <expan abbr="tanqu&atilde;">tanquan</expan> vim ip&longs;i in ferre videatur; cuius &longs;an&egrave; operationis <lb/>cau&longs;a quo&que; cognita admirationem parit; c&ugrave;m exigua admo <lb/>dum ad tanti operis productionem appareat. </s>
          <s id="N10142">admirabile e&longs;t &longs;a&shy;<lb/>n&egrave; ip&longs;ius artis magi&longs;terium, c&ugrave;m ade&ograve; potens &longs;it, vt effectus na&shy;<lb/>tur&etail; repugnantes producere tentet. </s>
          <s id="N10148">quipp&egrave; quibus, ni&longs;i ita &longs;en<lb/>&longs;ibus &longs;ubijci&agrave;ntur; vt tangi propemodum, &amp; con&longs;pici po&longs;&longs;int, <lb/>vix fides adhibeatur; idqu&egrave; <expan abbr="n&otilde;">non</expan> &longs;ine admiratione adhuc cogni<lb/>tum, ac per&longs;ua&longs;um nobis e&longs;&longs;e po&longs;&longs;it. </s>
          <s id="N10156">huiu&longs;modi autem mira&shy;<lb/>bilium operum opifex e&longs;t ip&longs;a mechanica di&longs;ciplina, tam na&shy;<lb/>tur&etail; &etail;mula, qu&agrave;m oppugnatrix valida. </s>
          <s id="N1015C">H&etail;c enim grauia pro<lb/>prio ferm&egrave; nutu &longs;ur&longs;um attolli, magnaqu&egrave; pondera ab exigua
<pb xlink:href="077/01/006.jpg" pagenum="2"/>admodum virtute moueri, aliaqu&egrave; id genus huiu&longs;modi &longs;pe&shy;<lb/>ctanda proponit. </s>
          <s id="N10166">vt tum imperitis ex ip&longs;orummet effectuum <lb/>intuitu, tum eruditis in cau&longs;arum varia contemplatione ad&shy;<lb/>mirationem pariat. </s>
          <s id="N1016C">veluti &longs;i ea &longs;pectemus, qu&etail; neruis, vel ali&shy;<lb/>quo mou&eacute;tur in&longs;trumento; vel qu&etail; &longs;piritibus <expan abbr="c&otilde;cinnuntur">concinnuntur</expan>, &amp; <lb/>fiunt; de quibus Heron, &amp; alij pertractarunt; vel deni&que; alijs <lb/>modis. </s>
          <s id="N10178">quamquam nos in ijs, qu&aelig; dicenda &longs;unt, de ea mecha&shy;<lb/>nic&aelig; facultatis parte, qu&aelig; ad <expan abbr="p&otilde;dera">pondera</expan>, <expan abbr="di&longs;t&atilde;tia&longs;&que;">di&longs;tantia&longs;&que;</expan> inter ip&longs;a <expan abbr="exi&longs;t&etilde;tes">exi&shy;<lb/>&longs;tentes</expan> pertinet, <expan abbr="quor&utilde;">quorum</expan> &longs;tatus ad &etail;quilibrium reduci pote&longs;t, ver<lb/>ba faciemus. </s>
          <s id="N10188">qu&aelig; <expan abbr="quid&etilde;">quidem</expan> pars totius mechanic&etail; facultatis prin&shy;<lb/>ceps exi&longs;tit. </s>
          <s id="N1018C">ea enim e&longs;t, in qua artem &longs;uperare naturam aper&shy;<lb/>ti&ugrave;s <expan abbr="c&otilde;&longs;picitur">con&longs;picitur</expan>: quod quidem, qua ratione contingat, hinc pla<lb/>num euadet. </s>
        </p>
        <p id="N10196" type="main">
          <s id="N10198">Ars quippe ex Ari&longs;totele phi&longs;icorum &longs;ecundo, &amp; ex pro&aelig;&shy;<lb/>mio qu&aelig;&longs;tionum mechanicarum triplici modo in &longs;uis opifi&shy;<lb/>cijs &longs;e&longs;e habere videtur. </s>
          <s id="N1019E">Nam vel immitatur naturam; vel ea <lb/>perficit, qu&aelig; natura perficere non pote&longs;t; vel deni&que; ea, qu&aelig; <lb/>pr&etail;ter naturam fiunt, operatur; in quibus tamen omnibus o&shy;<lb/>perandi rationibus, &longs;i diligenter eas con&longs;ideremus, artem &longs;em&shy;<lb/>per immitari naturam per&longs;piciemus. </s>
          <s id="N101A8">Prim&ugrave;m quidem multas <lb/>artes naturam immitari aperte videmus, vt &longs;culpturam, &amp; hu&shy;<lb/>iu&longs;modi alias. </s>
          <s id="N101AE">Quando autem ars ea perficit, qu&aelig; &longs;ola natu&shy;<lb/>ra perficere non pote&longs;t, vt in arte medica euenire &longs;olet; <expan abbr="natur&atilde;">naturam</expan> <lb/>ip&longs;am pariter emulatur, &amp; natur&aelig; a&longs;&longs;ociata, velut in&longs;trumen&shy;<lb/>tum eius, naturalem effectum perficere dicitur: tuncqu&egrave; <expan abbr="eod&etilde;">eodem</expan> <lb/>modo operatur, ac &longs;i natura rem ip&longs;am ab&longs;&que; artis ope perfice <lb/>repo&longs;&longs;et, quod plan&egrave; artis pr&aelig;&longs;tantiam manife&longs;tat: quipp&egrave; <lb/>c&ugrave;m ni&longs;i ars ip&longs;i natur&aelig; <expan abbr="man&utilde;">manum</expan> porrigat, natura ip&longs;a proprios <lb/>effectus perficere ex &longs;e&longs;e minim&egrave; po&longs;&longs;it. </s>
          <s id="N101C6">At ver&ograve; &longs;i ars <expan abbr="natur&atilde;">naturam</expan> <lb/>immitando ip&longs;am &longs;uperauerit; vt ea, qu&aelig; ab arte fiunt, pr&aelig;ter <lb/>naturam eueniant, long&egrave; adhuc pr&aelig;&longs;tanti&ugrave;s artis ingenium <lb/>apparebit. </s>
          <s id="N101D2">&longs;iquidem immitando naturam (paradoxum id for <lb/>t&egrave; videbitur, c&ugrave;m tamen veri&longs;&longs;imum &longs;it) pr&aelig;ter natur&aelig; ordi&shy;<lb/>nem operari dicatur. </s>
          <s id="N101D8">Ars. <expan abbr="n.">enim</expan> mirabili artificio naturam ipsa na<lb/>tura &longs;uperat; ita nimirum res di&longs;ponendo, vt ip&longs;a efficeret na<lb/>tura, &longs;i eiu&longs;modi &longs;ibi producendos &longs;tatueret effectus. </s>
          <s id="N101E2">quod qui <lb/>dem &longs;ubiecto exemplo magis per&longs;picuum fiet. </s>
        </p>
        <pb xlink:href="077/01/007.jpg" pagenum="3"/>
        <p id="N101E9" type="main">
          <s id="N101EB">Sint enim duo pondera <lb/>
<arrow.to.target n="fig1"/><lb/>AB in aliquo vecte, A ma<lb/>ius, B minus; quorum &longs;i&shy;<lb/>mul ita in vecte di&longs;po&longs;ito&shy;<lb/>rum &longs;it centrum grauitatis <lb/>C. &longs;it autem &longs;ub vecte in&shy;<lb/>ter CA fulcimentum in D. <lb/>&amp; quoniam pondera AB penes C grauitatis centrum inclinan<lb/>tur? </s>
          <s id="N10202">tunc C deor&longs;um naturaliter mouebitur; ac per con&longs;e&queacute;s <lb/><expan abbr="p&omacr;dus">pondus</expan> quo&que; B deor&longs;um tendet. </s>
          <s id="N10209">Sed &longs;i B deor&longs;um mouetur, <lb/>A cert&egrave; &longs;ur&longs;um eleuabitur. </s>
          <s id="N1020D">quippe quod, <expan abbr="qu&atilde;uis">quamuis</expan>, vt graue e&longs;t, <lb/>at&que; &longs;olutum ab&longs;&que; connexione ponderis B deor&longs;um tende <lb/>ret; attamen vt adnexum ponderi B, intercedente vecte AB, <lb/>&longs;ur&longs;um mouebitur: &amp; (vt ita dicam) pondus A contra pro&shy;<lb/>priam naturam naturaliter a&longs;cendet. </s>
          <s id="N1021B">Vnd&egrave; <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> e&longs;t, hos <lb/>motus effectus e&longs;&longs;e naturales. </s>
          <s id="N10223">Quid igitur efficit ars ip&longs;a? </s>
          <s id="N10225">nil <lb/>fan&egrave; aliud, qu&agrave;m qu&ograve;d resita di&longs;ponit, &amp; accomodat; vt &longs;imi&shy;<lb/>les effectus inde prodeant at&que; &longs;i naturales omnino exi&longs;tant, <lb/>quare opus erit, ut Ars naturam immitetur, &longs;iquidem effectus <lb/>naturales prouenire debent. </s>
          <s id="N1022F">propterea vectem, fulcimentum&shy;<lb/>qu&egrave; eodem modo di&longs;ponit; &amp; loco ponderis B aliquam con&shy;<lb/><gap/>&longs;tituit potentiam, qu&aelig; pr&etail;mendo parem vim habeat grauita&shy;<lb/>ti ip&longs;ius B; at&que; tunc ip&longs;a potentia mouens, qu&etail; minore&longs;t gra<lb/>uitate ponderis A, ip&longs;um A grauius nihilominus attollet. <lb/>quod quamuis propri&aelig; ip&longs;ius natur&aelig; repugnet, naturaliter <expan abbr="tam&etilde;">ta&shy;<lb/>men</expan> ab ip&longs;a potentia in B exi&longs;tente <expan abbr="&longs;urs&utilde;">&longs;ursum</expan> feretur: res enim ita di<lb/>&longs;po&longs;it&aelig; talem habent naturam, vt A quidem &longs;ur&longs;um, B vero <lb/>deor&longs;um moueri debeant. </s>
          <s id="N10246">qu&etail; &longs;an&egrave; ex no&longs;tro Mechanicorum <lb/>libro, &amp; ex ijs, qu&aelig; in hoc pertractantur; comperti&longs;&longs;im&egrave; red&shy;<lb/>dentur, &amp; quod diximus devecte, de alijs quo&que; in &longs;trumen&shy;<lb/>tis mechanicis intelligendum e&longs;t. </s>
          <s id="N1024E">quorum quidem apparatus <lb/>&longs;unt artis opera, effectus autem ip&longs;ius pen&egrave; natur&aelig;: c&ugrave;m eius <lb/>momenta, inclinationesqu&egrave; &longs;equantur, veluti pr&aelig;cipuas eiu&longs;&shy;<lb/>modi operum effectrices cau&longs;as: quipp&egrave; qu&aelig; &longs;unt omnino ad&shy;<lb/>mirabiles, ac pr&etail;&longs;tanti&longs;&longs;ime; &que;madmodum ex ip&longs;arum <expan abbr="cont&etilde;platione">con<lb/>templatione</expan> patere pote&longs;t. </s>
          <s id="N1025A">cuius rei <expan abbr="argum&etilde;t&utilde;">argumentum</expan> illud indica&longs;&longs;e &longs;at <lb/>e&longs;to, <expan abbr="nimir&utilde;">nimirum</expan> eas &agrave; &longs;ummis uiris, Ari&longs;totele, &amp; Archimede fui&longs;&longs;e
<pb xlink:href="077/01/008.jpg" pagenum="4"/>pertractatas. </s>
          <s id="N1026A">Ari&longs;toteles. <expan abbr="n.">enim</expan> in principio <expan abbr="Qu&etail;&longs;tion&utilde;">Qu&etail;&longs;tionum</expan> <expan abbr="mechanica-r&utilde;">mechanica&shy;<lb/>rum</expan> multa, ea&queacute; pr&etail;cipua ad cau&longs;as rei mechanic&aelig; <expan abbr="digno&longs;c&etilde;das">digno&longs;cendas</expan> <lb/>aperuit; &queacute; &longs;ecutus Archimedes in his libris mechanica prin&shy;<lb/>cipia explicati&ugrave;s patefecit, eaqu&egrave; planiora reddidit. </s>
          <s id="N10282">Nec propte<lb/>rea Ari&longs;toteles diminutus extitit: etenim <expan abbr="eor&utilde;">eorum</expan>, qu&etail; ab ip&longs;o pro<lb/>po&longs;ita, &amp; explicata fuere, problematum cau&longs;as egregi&egrave; patefe&shy;<lb/>cit. </s>
          <s id="N1028E">&longs;ed quoniam Archimedi &longs;copus fuit mechanic&etail; di&longs;ciplin&etail; <lb/>rudimenta explanare; propterea ad magis particularia <expan abbr="enucle&atilde;">enucleam</expan> <lb/>da de&longs;cendere voluit. </s>
          <s id="N10298">Ari&longs;toteles. <expan abbr="n.">enim</expan> (gratia <expan abbr="ex&etilde;pli">exempli</expan>) <expan abbr="qu&etail;r&etilde;s">qu&etail;rens</expan> cur <lb/>vecte magna mouemus pondera? </s>
          <s id="N102A8">cau&longs;am e&longs;&longs;e ait <expan abbr="longitudin&etilde;">longitudinem</expan> <lb/>vectis maiorem ad partem potenti&aelig;: &amp; rect&egrave; quidem; c&ugrave;m ex <lb/>principio ab ip&longs;o con&longs;tituto manife&longs;tum &longs;it, ea, qu&etail; &longs;unt in <lb/>longiori &agrave; centro <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan>, <expan abbr="maior&etilde;">maiorem</expan> quo&que; habere virtut&eacute;. </s>
          <s id="N102BC">Ar&shy;<lb/>chimedes ver&ograve; vlteri&ugrave;s adhuc progredi voluit, hoc admi&longs;&longs;o, <expan abbr="n&etilde;">nem</expan> <lb/>p&egrave; quod e&longs;t in longiori di&longs;tantia maiorem uim habere, qu&agrave;m <lb/>id, quod e&longs;t in breuiori, inquirere etiam voluit, quanta &longs;it vis <lb/>eius, quod e&longs;t in longiori di&longs;tantia ad id, quod e&longs;t in breuiori; <lb/>ita vt inter h&etail;c nota reddatur qualis, &amp; qu&etail; &longs;it eorum propor&shy;<lb/>tio determinata. </s>
          <s id="N102CE">at&que; ideo <expan abbr="fundam&etilde;tum">fundamentum</expan> illud mechanicum <lb/>pr&etail;&longs;tanti&longs;&longs;imum manife&longs;tauit; videlicet ita &longs;e&longs;e habere pon&shy;<lb/>dus ad pondus, vt di&longs;tantia ad in&longs;tantiam, vnde pondera &longs;u&shy;<lb/>&longs;penduntur, &longs;e&longs;e permutatim habet. </s>
          <s id="N102DA">quo ignoto, res mechani&shy;<lb/>c&etail; nullo modo pertractari po&longs;&longs;e videntur. </s>
          <s id="N102DE">quandoquidem <lb/>huic tota mechanica facultas tanquam vnico, pr&etail;cipuo&que; <lb/><expan abbr="fundam&etilde;to">fundamento</expan> innititur. </s>
          <s id="N102E7">Quare Archimedes <expan abbr="Ari&longs;totel&etilde;">Ari&longs;totelem</expan> &longs;equi vide<lb/>tur; quod non &longs;ol&ugrave;m patet exijs, qu&aelig; dicta &longs;unt; ver&ugrave;m etiam <lb/>&longs;i Archimedis po&longs;tulata <expan abbr="c&otilde;&longs;iderauerimus">con&longs;iderauerimus</expan>, quibus <expan abbr="c&otilde;&longs;titu&etilde;dis">con&longs;tituendis</expan>, <lb/>ea, qu&aelig; de principijs mechanicis Ari&longs;toteles patefecit, Archi&shy;<lb/>med&eacute; &longs;upponere <expan abbr="c&otilde;periemus">comperiemus</expan>. vt deinceps &longs;uo loco <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> <lb/>fiet. </s>
          <s id="N10307">In ratione pr&etail;terea, ac modo <expan abbr="c&otilde;&longs;ider&atilde;di">con&longs;iderandi</expan> mechanica, maxi&shy;<lb/>ma ambo affinitate coniuncti in cedere vid&eacute;tur. </s>
          <s id="N1030F">Ari&longs;toteles. <expan abbr="n.">enim</expan> <lb/>
<arrow.to.target n="marg1"/> res mechanicas tum Mathematica, t&uacute; naturalia &longs;apere, ac re&longs;pi<lb/>cere a&longs;&longs;eruit: quod <expan abbr="quid&etilde;">quidem</expan> &amp; Archimedes optim&egrave; nouit: <expan abbr="n&atilde;">nam</expan> qu&etail; <lb/>Mathematic&egrave; &longs;unt con&longs;ideranda, geometric&egrave; demon&longs;trauit, <lb/>vt &longs;unt di&longs;tanti&aelig;, proportiones, &amp; alia huiu&longs;modi: qu&aelig; ver&ograve; <lb/>&longs;unt naturalia, naturaliter <expan abbr="quoq;">quo&que;</expan> <expan abbr="c&otilde;&longs;iderauit">con&longs;iderauit</expan>; vt ea, qu&aelig; ad gra<lb/>uitatis centrum &longs;pectant, &amp; qu&aelig; &longs;ur&longs;um, &amp; qu&etail; deor&longs;um moue
<pb xlink:href="077/01/009.jpg" pagenum="5"/>ri debent; &amp; c&etail;tera huiu&longs;modi. </s>
          <s id="N10337">Ex quibus patet <expan abbr="maxim&utilde;">maximum</expan> e&longs;&longs;e <lb/>inter tantos viros in his pertractandis con&longs;en&longs;um. </s>
          <s id="N1033F">Ambiget <lb/>forta&longs;&longs;e qui&longs;piam, nunquid h&etail;c principia rect&egrave; ab illis fuerint <lb/>pertractata? </s>
          <s id="N10345">&longs;ed &longs;tatim omnis ce&longs;&longs;at dubitandi occa&longs;io, &longs;i tan<lb/>torum virorum pr&etail;&longs;tantia ad memoriam reuocetur; quibus, <lb/>citra controuer&longs;iam in di&longs;ciplinis ab ip&longs;is traditis, omnes eru&shy;<lb/>diti <expan abbr="palm&atilde;">palmam</expan> deferunt. </s>
          <s id="N10351">vt &que;madmodum <expan abbr="ab&longs;q;">ab&longs;&que;</expan> Ari&longs;totele duce, <lb/>at&que; doctore, nemo ad rect&egrave; <expan abbr="philo&longs;oph&atilde;dum">philo&longs;ophandum</expan>, ita ne&que; <expan abbr="eti&atilde;">etiam</expan> <lb/>ad Mathematicam, <expan abbr="pr&etail;cipue&qacute;ue">pr&etail;cipue&que;</expan> Mechanicam di&longs;ciplinam <lb/><expan abbr="ab&longs;q;">ab&longs;&que;</expan> Archimede &longs;e&longs;e <expan abbr="qui&longs;pi&atilde;">qui&longs;piam</expan> di&longs;ponere po&longs;&longs;it: quorum &longs;an&egrave; <lb/>apud peritiores authoritas merit&ograve; ob id &longs;uprema extat; qu&ograve;d <lb/>ab ip&longs;is res eo meliori, <expan abbr="pr&etail;&longs;tantiori&qacute;">pr&etail;&longs;tantiori&que;</expan>; modo pertractat&etail; <expan abbr="fuer&utilde;t">fuerunt</expan>, <lb/>quo ip&longs;arum rerum natura, at&que; doctrin&etail; ratio po&longs;tulabat. &amp; <lb/>qui &longs;cientiarum cupidi &longs;unt, illos &longs;equi, eorum qu&egrave; &longs;cripta &longs;&etail;p&egrave; <lb/>&longs;&etail;pius attent&egrave; perlegere debent. </s>
          <s id="N1037E">Pr&etail;terea philo&longs;ophi&etail;, ac Ma&shy;<lb/>thematic&etail; profe&longs;&longs;ores in hoc conueniunt; qu&ograve;d c&ugrave;m aliqua ad <lb/>philo&longs;ophiam &longs;pectantia tractant; mirum in modum Ari&longs;to&shy;<lb/>telem laudibus extollunt. </s>
          <s id="N10386">qui ver&ograve; Mathematicas pertractare <lb/>&longs;tud&eacute;t, &longs;tatim ad Archimedis laudes pariter &longs;e <expan abbr="c&omacr;fer&utilde;t">conferunt</expan>. tamet&longs;i <lb/>circa ea, qu&etail; n&oacute; &longs;unt Archimedis ver&longs;entur; vt <expan abbr="qu&atilde;">quam</expan> plurimi fece<lb/>re, quod <expan abbr="quid&etilde;">quidem</expan> optimo factum e&longs;t con&longs;ilio. </s>
          <s id="N10396">etenim &longs;i ea, qu&aelig; <lb/>mathematica ope indigent, laudare volunt, ad Archimedem <lb/>confugiendum e&longs;t; vt &longs;i inuentionem, &longs;ubtili&longs;&longs;imum Archi&shy;<lb/>medis inuentum afferant, quo modum adinuenit cogno&longs;cen&shy;<lb/>d&etail; quantitatis argenti, quod erat in corona Regis aurea, vt Vi&shy;<lb/>truuius te&longs;tatur; &amp; alia huiu&longs;modi; &longs;i admirabilia, &longs;tatim affe&shy;<lb/>rant Archimedis &longs;ph&etail;ram in globo vitreo elaboratam, in qua <lb/>omnes c&etail;le&longs;tis &longs;ph&aelig;r&aelig; motus relucebant; ita ut natura poti&ugrave;s <lb/>Archimedem immitata, qu&agrave;m Archimedes naturam illu&longs;i&longs;&longs;e
<arrow.to.target n="marg2"/><lb/>videatur; nauim pr&aelig;terea graui pondere oneratam &egrave; mari in <lb/>littus ab Archimede eductam; aliaqu&egrave; id genus plurima. </s>
          <s id="N103AF">De&shy;<lb/>ni&que; &longs;i res Mathematicas ciuitatibus e&longs;&longs;e vtiles o&longs;tendere vo&shy;<lb/>lunt, ea, qu&aelig; ab Archimede contra Marcellum in defen&longs;io&shy;<lb/>ne patri&aelig; facta fuere, in medium afferant, quo tempore bellica <lb/>opera adeo mirabilia effecit, vt &longs;olus Archimedes contra bel<lb/>lico&longs;i&longs;&longs;imos Romanos pugnare &longs;ufficiens videretur. </s>
          <s id="N103BB">qu&aelig; qui&shy;<lb/>dem omnia Mechanica di&longs;ciplina <expan abbr="c&otilde;fecta">confecta</expan> &longs;unt. </s>
          <s id="N103C3">Quid igitur
<pb xlink:href="077/01/010.jpg" pagenum="6"/>Mechanica admirabilius, &amp; vtilius? </s>
          <s id="N103C9">&egrave; qua tot, tantaqu&egrave; ad <lb/>humani generis vtilitatem conferentia prodeunt? </s>
          <s id="N103CD">eximia cer&shy;<lb/>t&egrave;, &amp; pr&aelig;clara admodum h&aelig;c Archimedis ge&longs;ta fuere; qu&aelig; ta&shy;<lb/>men, &longs;i ad alia quamplurima, qu&aelig; de ip&longs;o dici, ac afferri po&longs;<lb/>&longs;unt, conferantur; exigua &longs;an&egrave; mihi videntur. </s>
          <s id="N103D5">Nam qu&aelig; ha&shy;<lb/>ctenus commemorata &longs;unt, (quamquam forta&longs;&longs;e <expan abbr="n&otilde;">non</expan> omnia) <lb/>multa tamen, huiu&longs;modiqu&egrave; &longs;imilia alij quo&que; effecerunt, <lb/>&amp; adhuc extant forta&longs;&longs;e viri eo ingenij acumine pr&etail;diti, qui <lb/>talia aggredi non vererentur: &longs;ed <expan abbr="n&otilde;nulla">nonnulla</expan> egregia <expan abbr="ext&atilde;t">extant</expan> ip&longs;ius <lb/>Archimedis opera, quorum &longs;imilia, nec antea, nec po&longs;t <expan abbr="ips&utilde;">ipsum</expan> <lb/>facta fuere, ne&que; in futurum facienda fore &agrave; nemine &longs;int ex&shy;<lb/>pectanda. </s>
          <s id="N103F1">omnium enim admirabili&longs;&longs;ima, pr&aelig;&longs;tanti&longs;&longs;ima&shy;<lb/>qu&egrave; &longs;unt eius &longs;cripta, in quibus, &amp; ingenij acumen, inuentio&shy;<lb/>nes &longs;ubtili&longs;&longs;im&aelig;, perfectaqu&egrave; doctrina plan&egrave; con&longs;picitur. </s>
          <s id="N103F7">adeo <lb/>enim his omnibus Archimedis &longs;cripta aliorum &longs;cripta mathe<lb/>maticorum excellunt, &longs;uperantqu&egrave;; vt qu&aelig; aliorum, facil&egrave; <lb/>quidem inter &longs;e&longs;e comparari, cum ijs ver&ograve;, qu&etail; ab Archimede <lb/>nobis relicta fuerunt; nullo modo po&longs;&longs;int. </s>
          <s id="N10401">ut aperti&longs;sim&egrave; <lb/>(alijs interim omi&longs;sis) con&longs;picuum redditur ex ijs, qu&aelig; de <lb/>&longs;ph&etail;ra &amp; cylindro, &amp; ex ijs, qu&etail; de &aelig;&que;ponderantibus &longs;cri&shy;<lb/>pta reliquit: quipp&egrave; qu&etail; ob eorum <expan abbr="pr&etail;&longs;t&atilde;tiam">pr&etail;&longs;tantiam</expan>, ac dignitatem <lb/>merit&ograve; literis aureis e&longs;&longs;ent imprimenda. </s>
          <s id="N1040F">liber enim de &longs;ph&etail;ra, <lb/>&amp; cylindro inter Archimedis &longs;cripta <expan abbr="excell&etilde;s">excellens</expan> ade&ograve; <expan abbr="habit^&lcub;9}">habitus</expan> fuit; <lb/>vt ad eius <expan abbr="&longs;epulcr&utilde;">sepulcrum</expan> appo&longs;ita fuerit &longs;ph&etail;ra, &amp; <expan abbr="cylindr^&lcub;9}">cylindrus</expan>: <expan abbr="quib^&lcub;9}">quibus</expan> a <lb/>Cicerone con&longs;pectis; &longs;tatim illud Archimedis <expan abbr="&longs;epulcr&utilde;">sepulcrum</expan> e&longs;&longs;e in<lb/>tellexit: de cuius inuentione ob uiri <expan abbr="excellenti&atilde;">excellentiam</expan> maxim&egrave; glo&shy;<lb/>riatur: Deind&egrave; qua ratione ip&longs;um &agrave; temerario van&etail; orationis <lb/>proferend&aelig; au&longs;u, (dum &longs;ic loquitur, da mihi vbi &longs;i&longs;tam, ter&shy;<lb/>ramqu&egrave; mouebo) vindicare po&longs;&longs;emus; ni&longs;i h&etail;c, qu&aelig; de &aelig;&que;&shy;<lb/>ponderantibus extant, &longs;cripta reliqui&longs;&longs;et? ex his enim habita <lb/>notitia proportionis ponderum, &amp; di&longs;tantiarum, &longs;it manife&shy;<lb/>&longs;tum non e&longs;&longs;e &agrave; ratione, nequ&egrave; &agrave; natura pror&longs;us alienum, po&longs;&longs;e <lb/>terram moueri, &longs;i daretur con&longs;i&longs;tendi locus. </s>
          <s id="N10433">quod etiam ex <lb/>no&longs;tro volumine Mechanico annis ab hinc aliquot elap&longs;is e&shy;<lb/>dito varijs quoqu&egrave; in&longs;trumentis parere pote&longs;t. <expan abbr="quandoquid&etilde;">quandoquidem</expan> <lb/>multis modis, datum pondus &agrave; data potentia moueri, ibi <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>&longs;ume&longs;t. </s>
          <s id="N10445">vbi demon&longs;trationes &agrave; nobis con&longs;titut&etail; ijs, qu&aelig; apud
<pb xlink:href="077/01/011.jpg" pagenum="7"/>Archimedem pre&longs;enti opere habentur, totam eorum vim fer&shy;<lb/>ri volunt acceptam. </s>
          <s id="N1044D">Etne quidpiam, quod &longs;tudio&longs;is mecha&shy;<lb/>nic&aelig; facultatis prode&longs;&longs;e po&longs;&longs;it, pr&etail;termitteretur, ad horum <lb/>Archimedis librorum interpr&aelig;tationem aliquid operis con&shy;<lb/>tuli&longs;&longs;e placuit; &longs;atisqu&egrave; nobis feci&longs;&longs;e videbimur; &longs;i &longs;altem &longs;tu&shy;<lb/>dio&longs;i nos Archimedis ve&longs;tigia &longs;ecutos fui&longs;&longs;e cognouerint. <lb/>Et quamuis opus hoc fuerit ab Eutocio A&longs;calonita nonnullis <lb/>commentarijs illu&longs;tratum, quia tamen propter Archimedis <lb/><expan abbr="&longs;criptor&utilde;">&longs;criptorum</expan> ob&longs;curitat&eacute; multa adhuc reman&eacute;t ab&longs;tru&longs;a, nec pror<lb/>&longs;us omnibus peruia; pr&etail;&longs;ertim gr&etail;carum literarum experti&shy;<lb/>bus; c&ugrave;m liber hic in latinum ver&longs;us multis in locis ob&longs;curus, <lb/>alijsqu&egrave; pleris&que; quodammodo mancus merit&ograve; &longs;u&longs;picetur; <lb/>ita vt adhuc in tenebris iacere videatur; gr&etail;cusqu&egrave; pr&aelig;terea <lb/>codex impre&longs;&longs;us, &que;m &longs;ecuti &longs;umus, multis in locis aliqua <lb/>correctione egere videatur; idcirco ab huiu&longs;modi munere <lb/>pr&etail;&longs;tando de&longs;i&longs;tere noluimus: quin &longs;imul hos libros in <expan abbr="latin&utilde;">latinum</expan> <lb/>&longs;ermonem verteremus; commentarijsqu&egrave; illu&longs;tratos redde&shy;<lb/>remus. </s>
          <s id="N1046F">C&ugrave;m pr&aelig;&longs;ertim hinc tutus ad mechanicam <expan abbr="di&longs;ciplin&atilde;">di&longs;ciplinam</expan> <lb/>pateat aditus. </s>
          <s id="N10477">Quare vt mens huius pr&etail;clari&longs;&longs;imi Mathema<lb/>tici magis, at&que; magis, qu&agrave;m fieri po&longs;sit, pro virili no&longs;tra <lb/>per&longs;picua reddatur; &amp; huius &longs;cienti&aelig; cupidi in adipi&longs;cendis <lb/>pulcherrimis hi&longs;ce theorematibus min&ugrave;s laborent; &agrave; commu<lb/>ni genere interpr&etail;tandi aliquantulum in pr&aelig;&longs;entia di&longs;cedere <lb/>nobis vi&longs;um e&longs;t oportunum. </s>
          <s id="N10483">Nam qui res mathematicas in&shy;<lb/>terpr&aelig;tati &longs;unt, &longs;uos commentarios &longs;eor&longs;um &agrave; demon&longs;tratio&shy;<lb/>nibus collocauere: nos ver&ograve;, qu&etail; no&longs;tra &longs;unt, verbis ip&longs;ius
<arrow.to.target n="marg3"/><lb/>Archimedis in&longs;eruimus, &amp; hoc tant&ugrave;m in ip&longs;is demon&longs;tra&shy;<lb/>tionibus, non in propo&longs;itionibus, &amp; huiu&longs;modi alijs, hac <lb/>plan&egrave; habita di&longs;tinctione, vt qu&aelig; &longs;unt Archimedis (his, vel <lb/><emph type="italics"/>his literarum notis<emph.end type="italics"/>) cogno&longs;cantur, ip&longs;iusqu&egrave; tant&ugrave;m Ar&shy;<lb/>chimedis e&longs;&longs;e intelligantur. </s>
          <s id="N1049B">Qu&etail; ver&ograve; alterius &longs;unt cha&shy;<lb/>racteris, utqu&etail; huius exi&longs;tent form&aelig;, no&longs;tra e&longs;&longs;e &longs;emper <lb/>&longs;int exi&longs;timanda. </s>
          <s id="N104A1">&amp; quoad fieri potuit, verba omnia, qu&etail; <lb/>nobis declaratione aliqua, nec non correctione indigere vi&longs;a <lb/>&longs;unt (ijs tamen omi&longs;&longs;is, qu&etail; parui, im&ograve; nullius &longs;unt momenti, <lb/>vt e&longs;t literarum immutatio, &amp; huiu&longs;modi alia) dilucid&egrave; expli&shy;<lb/>care, at&que; emendare &longs;tuduimus. </s>
          <s id="N104AB">quibus etiam hanc adhibui
<pb xlink:href="077/01/012.jpg" pagenum="8"/>mus diligentiam, quod quamuis ea, qu&aelig; no&longs;tra, &longs;unt, verbis <lb/>&longs;int Archimedis in&longs;erta; &longs;iquis tamen verba tant&ugrave;m Archi&shy;<lb/>medis legere maluerit, rect&egrave; id a&longs;&longs;equi poterit; &longs;iquidem ne <lb/>verbum quidem Archimedis omi&longs;imus: quinnimo ea ita di&shy;<lb/>&longs;po&longs;uimus, vt &longs;uum pror&longs;us retineant &longs;en&longs;um, po&longs;&longs;intqu&egrave; <expan abbr="c&omacr;tinuat&egrave;">con<lb/>tinuat&egrave;</expan> legi; ac &longs;i nihil inter ip&longs;a in&longs;ertum fuerit. </s>
          <s id="N104BF">quod qui&shy;<lb/>dem &longs;tudio&longs;is non inutile fore iudicauimus; qui ab&longs;&que; no&shy;<lb/>&longs;tris additionibus <expan abbr="Archimed&emacr;">Archimedem</expan> tant&ugrave;m habebunt; <expan abbr="c&utilde;">cum</expan> no&longs;tris <lb/>ver&ograve; additionibus Archimedis demon&longs;trationes continua&shy;<lb/>tas, &amp; explicatas habebunt. </s>
          <s id="N104CD">Huberionis autem doctrin&aelig; gra<lb/>tia permulta adiunximus &longs;cholia, in quibus pa&longs;&longs;im ordinem, <lb/>Authori&longs;qu&egrave; artificium patefecimus; nec non multa lemma<lb/>ta ad Archimedis demon&longs;trationes nece&longs;&longs;aria <expan abbr="dem&otilde;&longs;traui-mus">demon&longs;traui&shy;<lb/>mus</expan>; aliaqu&egrave; nonnulla ad explicationem, &longs;ubiectamqu&egrave; ma<lb/>teriam valde vtilia adiecimus. </s>
          <s id="N104DD">Vt etiam Archimedis dicta <lb/>magis eluce&longs;cant, antequam ad explicationem verborum <lb/>ip&longs;ius accedamus, nonnulla prius declarare oportunum no&shy;<lb/>bis vi&longs;um e&longs;t ad ea, qu&aelig; in his libris Archimedis &longs;upponit <lb/>tanquam cognita. </s>
          <s id="N104E7">Deinde con&longs;iderandus proponitur &longs;copus, <lb/>at&que; intentio Archimedis; diui&longs;io item librorum; huiu&longs;&shy;<lb/>modiqu&egrave; alia, qu&aelig; &longs;ummam afferent facilitatem ad intel<lb/>ligendam: mentem Archimedis. </s>
        </p>
        <p id="N104EF" type="margin">
          <s id="N104F1"><margin.target id="marg1"/><emph type="italics"/>in princip. <lb/>&que;&longs;t. </s>
          <s id="N104F9">Me&shy;<lb/>chan.<emph.end type="italics"/></s>
        </p>
        <p id="N104FF" type="margin">
          <s id="N10501"><margin.target id="marg2"/><emph type="italics"/>Claudianus<emph.end type="italics"/></s>
        </p>
        <p id="N10509" type="margin">
          <s id="N1050B"><margin.target id="marg3"/><emph type="italics"/>declaratio <lb/>huius para <lb/>phra&longs;is.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.012.1.jpg" xlink:href="077/01/012/1.jpg"/>
        <p id="N1051B" type="main">
          <s id="N1051D">C&ugrave;m itaqu&egrave; &longs;upponat, nos exqui&longs;itam habere notitiam <lb/>centri grauitatis; illius definitionem afferre libuit: pro cuius <lb/>tamen faciliori notitia illud quo&que; in primis admonen&shy;<lb/>
<arrow.to.target n="marg4"/> dum duximus; nimirum quatuor reperiri centra. <expan abbr="Centr&utilde;">Centrum</expan> ui&shy;<lb/>delicet vniuer&longs;i, centrum magnitudinis, centrum figur&aelig;, &amp; <lb/>centrum grauitatis, quod quidem grauitatis centrum rect&egrave; <lb/>definitur &agrave; Pappo Alexandrino in octauo libro mathemati&shy;<lb/>carum collectio num hoc pacto. </s>
        </p>
        <p id="N10535" type="margin">
          <s id="N10537"><margin.target id="marg4"/><gap/></s>
        </p>
        <p id="N1053B" type="head">
          <s id="N1053D">DEFINITIO CENTRI GRAVITATIS</s>
        </p>
        <p id="N1053F" type="main">
          <s id="N10541">Centrum grauitatis vniu&longs;cuiu&longs;&que; corporis e&longs;t punctum <lb/>quoddam intra po&longs;itum, &agrave; quo &longs;i graue appen&longs;um mente <lb/>con&ccedil;ipiatur, dum fertur, quie&longs;cit<gap/> &amp; &longs;erua<gap/> eam, quam in <lb/>principio habebat po&longs;itionem, neque in ip&longs;a latione circum-
<pb xlink:href="077/01/013.jpg" pagenum="9"/>uertitur. </s>
        </p>
        <p id="N10553" type="head">
          <s id="N10555">EIVSDEM ALIA DEFINITIO.</s>
        </p>
        <p id="N10557" type="main">
          <s id="N10559">Centrum grauitatis vniu&longs;cuiu&longs;&que; &longs;olid&aelig; figur&aelig; e&longs;t <expan abbr="punct&utilde;">punctum</expan> <lb/>illud intra po&longs;itum, circa quod vndi&que; partes &etail;qualium mo <lb/>mentorum con&longs;i&longs;tunt. </s>
          <s id="N10563">&longs;i. <expan abbr="n.">enim</expan> per tale centrum ducatur <expan abbr="plan&utilde;">planum</expan> fi<lb/>guram quomodo cun&que; &longs;ecans, &longs;emper in partes &aelig;&que;ponde<lb/>rantes ip&longs;am diuidet. </s>
        </p>
        <p id="N10571" type="main">
          <s id="N10573">Hanc po&longs;tremam definitionem, &longs;eu poti&ugrave;s de&longs;criptionem <lb/>tradidit Federicus Commandinus in libro de centro grauita&shy;<lb/>tis &longs;olidorum. </s>
          <s id="N10579">ex quipus &longs;an&egrave; definitionibus eluce&longs;cit natura, <lb/>
<arrow.to.target n="fig2"/><lb/>at&que; facultas <expan abbr="c&etilde;tri">centri</expan> grauitatis. <lb/>vt &longs;i punctum A fuerit <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis corporis BC, tunc <lb/>ex Pappi &longs;ententia, &longs;i BC <expan abbr="&longs;u&longs;p&etilde;">&longs;u&longs;pem</expan> <lb/>datur ex A, magnitudo BC <lb/>eadem, qua reperitur, di&longs;po&shy;<lb/>&longs;itione locata manebit; ne&que; <lb/>partes ullas ip&longs;ius corporis, vt qu&etail; &longs;unt ad <lb/>
<arrow.to.target n="fig3"/><lb/>BC, circumuerti, ne&que; omnino &longs;uum <lb/>mutare &longs;itum depr&aelig;hendetur. </s>
          <s id="N105A5">&longs;i ver&ograve; vt <lb/><expan abbr="C&otilde;mandino">Commandino</expan> placuit, A fuerit centrum <lb/>grauitatis magnitudinis BCD, eadem&shy;<lb/>qu&egrave; per punctum A vtcun&que; <expan abbr="&longs;ec&utilde;d&ugrave;m">&longs;ecund&ugrave;m</expan> <lb/>rectitudinem diuidatur, veluti per EAF. <lb/>tunc pars EBF ip&longs;i ECDF &aelig;&que;ponde&shy;<lb/>rabit, quamuis EBF, &amp; ED &longs;int magni<lb/>tudines in&aelig;quales. </s>
          <s id="N105B8">&longs;&aelig;penumero enim e&shy;<lb/>uenire &longs;olet, vt in diui&longs;ione figur&aelig; per eius centrum graui&shy;<lb/>tatis ip&longs;a aliquando in partes diuidatur &aelig;quales, ali&shy;<lb/>quando in partes in&aelig;quales: vt &longs;uo loco o&longs;tendemus:
<arrow.to.target n="marg5"/><lb/>&longs;emper tamen in partes diuiditur hinc inde &aelig;&que;pon&shy;<lb/>derantes; non tamen &longs;eor&longs;um con&longs;titutas, ab inuicen<lb/>qu&egrave; &longs;eiunctas, &amp; veluti ad &aelig;quilibrium examinatas; vt pu&shy;<lb/>ta &longs;i EBF decem pondo ponderet; ED quo&que; totidem <lb/>pependi&longs;&longs;e oporteat. </s>
          <s id="N105CD">res quippe non &longs;ic &longs;e habet, &longs;ed cas e&longs;&longs;e <lb/>in eo &longs;itu &aelig;&que;ponderantes, in quo reperiuntur; vt neutra
<pb xlink:href="077/01/014.jpg" pagenum="10"/>alteri pr&etail;ponderet. </s>
          <s id="N105D5">ex quibus colligi pote&longs;t, &longs;i graue quidpiam <lb/>in centro mundi collo catum fuerit, oportere centrum graui<lb/>tatis illius in centro mundi con&longs;titutum e&longs;&longs;e: &longs;iquidem vt <lb/>graue illud tunc quie&longs;cat, partes vndi&que; ip&longs;um ambientes &etail;&shy;<lb/>qualium momentorum exi&longs;tere, at&que; manere oporteat. <lb/>Quare dum a&longs;&longs;eritur, graue quod cum&que; naturali propen&shy;<lb/>&longs;ione &longs;edem in mundi centro appetere, nil aliud &longs;ignifica&shy;<lb/>tur, qu&agrave;m qu&ograve;d eiu&longs;modi graue proprium centrum grauitatis <lb/>cum centro vniuer&longs;i coaptare expetit, vt optim&egrave; quie&longs;cere va&shy;<lb/>leat. </s>
          <s id="N105E9">Ex quo &longs;equitur motum deor&longs;um alicuius grauis fieri <lb/>per rectam lineam, qu&aelig; centrum grauitatis ip&longs;ius grauis, cen<lb/>trumqu&egrave; mundi connectit. </s>
          <s id="N105EF">quandoquidem grauia deor&longs;um <lb/>rect&agrave; feruntur. </s>
          <s id="N105F3">Vnde manife&longs;tum e&longs;t, Grauia &longs;ecundum gra<lb/>uitatis centrum deor&longs;um tendere. </s>
          <s id="N105F7">quod nos in no&longs;tro Mecha<lb/>nicorum libro &longs;uppo&longs;uimus. </s>
        </p>
        <p id="N105FB" type="margin">
          <s id="N105FD"><margin.target id="marg5"/><emph type="italics"/>in fine pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.014.1.jpg" xlink:href="077/01/014/1.jpg"/>
        <figure id="id.077.01.014.2.jpg" xlink:href="077/01/014/2.jpg"/>
        <p id="N1060F" type="main">
          <s id="N10611">Ex ijs omnibus, qu&aelig; hactenus de centro grauitatis dicta <lb/>&longs;unt, per&longs;picuum e&longs;t, vnumquod&que; graue in eius centro <lb/>grauitatis propri&egrave; grauitare, veluti nomen ip&longs;um centri gra&shy;<lb/>uitatis idip&longs;um manife&longs;t&egrave; pr&aelig;&longs;eferre videtur. </s>
          <s id="N10619">ita vt tota vis, <lb/>grauita&longs;qu&egrave; ponderis in ip&longs;o grauitatis centro coaceruata, col<lb/>lectaqu&egrave; e&longs;&longs;e, ac tanquam in ip&longs;um vndiqu&egrave; fluere videatur. <lb/>Nam ob <expan abbr="grauitat&etilde;">grauitatem</expan> pondus in <expan abbr="c&etilde;trum">centrum</expan> vniuer&longs;i naturaliter per <lb/>uenire cupit; centrum ver&ograve; graui tatis (exdictis) e&longs;t id, quod <lb/>propri&egrave; in centrum mundi tendit. </s>
          <s id="N1062D">in centro igitur grauitatis <lb/>pondus propri&egrave; grauitat. </s>
          <s id="N10631">Pr&aelig;terea quando aliquod pondus <lb/>ab aliqua potentia in centro grauitatis &longs;u&longs;tinetur; tunc pon&shy;<lb/>dus &longs;tatim manet, totaqu&egrave; ip&longs;ius ponderis grauitas &longs;en&longs;u per&shy;<lb/>cipitur. </s>
          <s id="N10639">quod etiam contingit, &longs;i &longs;u&longs;teneatur pondus in ali&shy;<lb/>quo puncto, &agrave; quo per centrum grauitatis ducta recta linea <lb/>in centrum mundi tendat. </s>
          <s id="N1063F">hoc nam&que; modo idem e&longs;t, ac <lb/>
<arrow.to.target n="marg6"/> &longs;i <expan abbr="p&otilde;dus">pondus</expan> in eius centro grauitatis propri&egrave; &longs;u&longs;tineretur. </s>
          <s id="N1064B">Quod <lb/>quidem non contingit, &longs;i &longs;u&longs;tineatur pondus in alio pun&shy;<lb/>cto. </s>
          <s id="N10651">ne&que; enim pondus manet, quin poti&ugrave;s <expan abbr="antequ&atilde;">antequam</expan> ip&longs;ius <lb/>grauitas percipi po&longs;&longs;it, vertitur vti&que; pondus, donec &longs;imi <lb/>liter &agrave; &longs;u&longs;pen&longs;ionis puncto ad centrum grauitatis ducta re&shy;<lb/>cta linea in vniuer&longs;i centrum recto tramite feratur. <lb/>qu&aelig; quidem ex prima no&longs;trorum Mechanicorum pro-
<pb xlink:href="077/01/015.jpg" pagenum="11"/>po&longs;itione &longs;unt manife&longs;ta, quando autem h&aelig;c linea e&longs;t hori&shy;<lb/>zonti erecta, tunc idem pror&longs;us e&longs;t (vt mox diximus) perinde <lb/>ac &longs;i pondus in centro grauitatis ad vnguem &longs;u&longs;tineretur. <lb/>Quocirca &longs;i p&ograve;nderis grauitas minim&egrave; percipi pote&longs;t, ni&longs;i in <lb/><expan abbr="c&etilde;tro">centro</expan> grauitatis ip&longs;ius, <expan abbr="p&otilde;dus">pondus</expan> cert&egrave; in ip&longs;o propri&egrave; grauitat. </s>
        </p>
        <p id="N10672" type="margin">
          <s id="N10674"><margin.target id="marg6"/><gap/></s>
        </p>
        <p id="N10678" type="main">
          <s id="N1067A">Centrum figur&aelig; apud Mathematicos e&longs;t punctum, &agrave; quo <lb/>&longs;emidiametri exeunt; vel per quod <expan abbr="tr&atilde;&longs;eunt">tran&longs;eunt</expan> diametri, vt circu<lb/>li centrum, &amp; ellip&longs;is, necnon oppo&longs;itarum &longs;ectionum. </s>
        </p>
        <p id="N10684" type="main">
          <s id="N10686">Centrum ver&ograve; magnitudinis e&longs;t id, quod medium figur&aelig; <lb/>obtinet; vel quod &etail;qualiter ab exteriori &longs;uperficie di&longs;tat. </s>
          <s id="N1068A">vt <lb/>&longs;ph&aelig;r&etail; centrum. </s>
        </p>
        <p id="N1068E" type="main">
          <s id="N10690">Centrum deni&que; mundi e&longs;t punctum in medio vniuer&longs;i <lb/>&longs;itum, omniumqu&egrave; rerum infimum. </s>
        </p>
        <p id="N10694" type="main">
          <s id="N10696">C&aelig;ter&ugrave;m ad meliorem horum notitiam ob&longs;eruandum e&longs;t, <lb/>h&etail;c centra aliquando &longs;imul omnia inter &longs;e conuenire, <expan abbr="aliqu&atilde;">aliquam</expan> <lb/>do nonnulla; aliquando autem minim&egrave;. </s>
          <s id="N106A0">&longs;imul ver&ograve; omnia <lb/>conueniunt. </s>
          <s id="N106A4">vt centrum vniuer&longs;i, centrum magnitudinis ter<lb/>r&aelig; (&longs;ph&etail;r&aelig; &longs;cilicet ex aqua, terraqu&egrave; compo&longs;it&etail;, quam nos bre<lb/>uitatis &longs;tudio terram tant&ugrave;m nuncupabimus) centrum figu&shy;<lb/>r&etail; terr&etail;; ac centrum grauitatis terr&etail;. </s>
          <s id="N106AC">C&ugrave;m enim terra &longs;it &longs;ph&aelig;&shy;<lb/>rica (vt omnes fatentur.) eius medium erit centrum figur&etail;, &agrave; <lb/>quo &longs;emidiametri exeunt. </s>
          <s id="N106B2">idip&longs;um qu&egrave; erit centrum magnitu<lb/>dinis, &longs;iquidem ip&longs;ius figur&etail; medium obtinet. </s>
          <s id="N106B6">Pr&etail;terea idem <lb/>punctum e&longs;t centrum grauitatis terr&etail;. </s>
          <s id="N106BA">&amp; quoniam terra in me <lb/>dio <expan abbr="m&umacr;di">mundi</expan> quie&longs;cit, erit hoc <expan abbr="centr&utilde;">centrum</expan> grauitatis in centro vniuer&longs;i <lb/>collocatum. </s>
          <s id="N106C8">&amp; hoc duntaxat modo centra omnia in <expan abbr="vn&utilde;">vnum</expan> con<lb/>uenire po&longs;&longs;unt. </s>
          <s id="N106D0">quamquam ver&ograve; &longs;ph&etail;ra, qu&etail; continet <expan abbr="terr&amacr;">terram</expan> &amp; <lb/>aqu&atail;, compo&longs;ita e&longs;t ex corporibus diuer&longs;&etail; &longs;peciei, <expan abbr="differ&etilde;ti&longs;qu&egrave;">differenti&longs;qu&egrave;</expan> <lb/>grauitatis, nimirum ex terra, &amp; aqua; non <expan abbr="tam&etilde;">tamen</expan> efficitur, quin <lb/><expan abbr="medi&utilde;">medium</expan> ip&longs;ius cum centro grauitatis con&longs;piret in vnum. <expan abbr="N&atilde;">Nam</expan> ex <lb/>Ari&longs;to telis &longs;ententia terra circa mundi centrum vndi&que; <expan abbr="c&otilde;&longs;i">con&longs;i</expan>
<arrow.to.target n="marg7"/><lb/>&longs;tit; &amp; Archimedes affirmat, <expan abbr="eti&atilde;">etiam</expan> <expan abbr="humid&utilde;">humidum</expan> manens e&longs;&longs;e
<arrow.to.target n="marg8"/> <expan abbr="&longs;ph&etail;ri-c&utilde;">&longs;ph&etail;ri&shy;<lb/>cum</expan>, cuius <expan abbr="c&etilde;trum">centrum</expan> e&longs;t <expan abbr="centr&utilde;">centrum</expan> vniuer&longs;i. </s>
          <s id="N10710">&longs;i ita &que; terra, &amp; aqua ma<lb/><expan abbr="n&etilde;t">nent</expan>, <expan abbr="quie&longs;c&utilde;tqu&egrave;">quie&longs;cuntqu&egrave;</expan> circa <expan abbr="centr&utilde;">centrum</expan> vniuer&longs;i, ergo <expan abbr="centr&utilde;">centrum</expan> <expan abbr="m&umacr;di">mundi</expan> <expan abbr="ip&longs;o-r&utilde;">ip&longs;o&shy;<lb/>rum</expan> &longs;imul <expan abbr="c&etilde;tr&utilde;">centrum</expan> grauitatis exi&longs;tit. </s>
          <s id="N10731">at&que; adeo quatuor pr&etail;dicta <lb/>centra in <expan abbr="vn&utilde;">vnum</expan> &longs;imul conueniunt punctum. </s>
          <s id="N10739">Quod <expan abbr="aut&etilde;">autem</expan> tria &longs;i<lb/>mul centra in vnum coeant, &longs;atis <expan abbr="con&longs;picu&umacr;">con&longs;picuum</expan> e&longs;&longs;e poterit cuiqu&egrave;
<pb xlink:href="077/01/016.jpg" pagenum="12"/>&longs;ph&aelig;ram aliquam, put&agrave; ligneam, vel alterius (&longs;imilaris <expan abbr="tam&etilde;">tamen</expan>) <lb/>natur&aelig; intuenti; &longs;iquidem eius medium erit centrum magni&shy;<lb/>tudinis, &amp; centrum figur&aelig;; idemqu&egrave; punctum erit ip&longs;ius cen&shy;<lb/><arrow.to.target n="marg9"/>trum grauitatis; circa quod vndi&que; partes &aelig;&que;ponderant. <lb/>&amp; quoniam h&aelig;c &longs;ph&aelig;ra non e&longs;t in centro mundi; propterea <lb/>tria tant&ugrave;m centra &longs;imul conuenient. </s>
          <s id="N1075D">&longs;i ver&ograve; &longs;ph&etail;ra non &longs;imi&shy;<lb/>laris, &longs;ed di&longs;&longs;imilaris fuerit, veluti altera ip&longs;ius meditate plum&shy;<lb/>bea, altera ver&ograve; medietate lignea exi&longs;tente, tunc eius medium <lb/>erit quippe centrum magnitudinis, &amp; figur&etail;, grauitatis ver&ograve; <lb/>centrum nequaquam. </s>
          <s id="N10767">Nam partes vndi&que; circa medium &aelig;&shy;<lb/>&que;ponderare non po&longs;&longs;ent; &longs;ed grauitatis centrum ad grauio&shy;<lb/>rem partem, nimirum plumbeam declinabit. </s>
          <s id="N1076D">&amp; hoc modo <lb/>duo tant&ugrave;m centra inter &longs;e conuenient. </s>
          <s id="N10771">vt etiam (modo ta&shy;<lb/>men diuer&longs;o) accidit ellip&longs;i; cuius centrum e&longs;t centrum figu&shy;<lb/>r&etail;, &longs;iquidem per ip&longs;um tran&longs;eunt diametri; idemqu&egrave; <expan abbr="punct&utilde;">punctum</expan> <lb/>
<arrow.to.target n="marg10"/> e&longs;t ip&longs;ius centrum grauitatis. </s>
          <s id="N10781">quod c&ugrave;m non &longs;it propri&egrave; me&shy;<lb/>dium figur&aelig;, non erit quo&que; centrum magnitudinis. <expan abbr="medi&umacr;">medium</expan> <lb/>enim figur&aelig; propri&egrave; circulo, ac &longs;ph&aelig;r&aelig; tant&ugrave;m competit. <lb/>Quare duo centra hoc quo&que; modo &longs;imul tant&ugrave;m conue&shy;<lb/>nient. </s>
          <s id="N1078F">In figura paraboles recta linea terminat&etail; centrum gra<lb/>
<arrow.to.target n="marg11"/>uitatis intra figuram reperitur, quipp&egrave; quod ne&que; centrum <lb/>figur&aelig;, ne&que; centrum magnitudinis e&longs;&longs;e pote&longs;t. </s>
          <s id="N10799">etenim in <lb/>hac figura non pote&longs;t dari medium, vnde ne&que; centrum ma<lb/>gnitudinis dabitur, &amp; quoniam in parabole diametri &longs;unt in<lb/>ter&longs;e &etail;quidi&longs;tantes, vt ex primo libro conicorum Apollonij <lb/>Pergei con&longs;tat; ne&que; etiam centrum figur&aelig; dabitur. </s>
          <s id="N107A3">&longs;ic igi&shy;<lb/>tur centra nullo modo conuenient. </s>
        </p>
        <p id="N107A7" type="margin">
          <s id="N107A9"><margin.target id="marg7"/><emph type="italics"/>lib. </s>
          <s id="N107AF">de c&aelig;lo<emph.end type="italics"/></s>
        </p>
        <p id="N107B3" type="margin">
          <s id="N107B5"><margin.target id="marg8"/><emph type="italics"/>lib. </s>
          <s id="N107BB">de iis <lb/>qu&etail; uehun<lb/>tur in aqua<emph.end type="italics"/></s>
        </p>
        <p id="N107C3" type="margin">
          <s id="N107C5"><margin.target id="marg9"/>16 <emph type="italics"/>Federi&shy;<lb/>ci <expan abbr="c&otilde;m">comm</expan>. de <lb/>centro gra<lb/>uitatis &longs;oli <lb/>dorum.<emph.end type="italics"/></s>
        </p>
        <p id="N107DA" type="margin">
          <s id="N107DC"><margin.target id="marg10"/>4. <emph type="italics"/>Fed. </s>
          <s id="N107E3">com<lb/>man. </s>
          <s id="N107E7">de cen<lb/>tro graui&shy;<lb/>tatis &longs;olido <lb/>rum.<emph.end type="italics"/></s>
        </p>
        <p id="N107F1" type="margin">
          <s id="N107F3"><margin.target id="marg11"/><emph type="italics"/>in &longs;ecundo <lb/>libro huius<emph.end type="italics"/></s>
        </p>
        <p id="N107FD" type="main">
          <s id="N107FF">Noui&longs;&longs;e quo&que; oportet centrum grauitatis communius <lb/>e&longs;&longs;e, in pluribu&longs;qu&egrave; reperiri, qu&agrave;m centra magnitudinis, &amp; fi&shy;<lb/>gur&aelig;: centrum ver&ograve; figur&aelig; communius e&longs;&longs;e centro magnitu&shy;<lb/>dinis. <expan abbr="N&atilde;">Nam</expan> quodlibet corpus, &amp; qu&etail;libet figura nece&longs;&longs;e e&longs;t, vt habeat <lb/><expan abbr="c&etilde;tr&utilde;">centrum</expan> grauitatis intrin&longs;ec&ugrave;s, vel extrin&longs;ec&ugrave;s. </s>
          <s id="N10810">intrin&longs;ec&ugrave;s vt <lb/><expan abbr="c&etilde;tr&utilde;">centrum</expan> grauitatis alicuius corporis regularis, quod e&longs;t in medio <lb/>figur&aelig;, vel alicuius figur&aelig; vt A; cuius centrum grauitatis &longs;it <lb/>in ambitu figur&aelig;, vt in puncto B; extrin&longs;ec&ugrave;s ver&ograve; vt figura <lb/>C, cuius centrum grauitatis extrin&longs;ecus &longs;it, vt in D; quod <lb/>e&longs;t intelligendum, &longs;i graue C in centrum mundi tenderet,
<pb xlink:href="077/01/017.jpg" pagenum="13"/>tunc centrum D cum centro mundi <expan abbr="c&otilde;-">con&shy;<lb/></expan>
<arrow.to.target n="fig4"/><lb/>ueniret; figuraqu&egrave; C quie&longs;ceret circa cen<lb/>trum vniuer&longs;i, veluti &longs;e habet circa <expan abbr="c&etilde;trum">centrum</expan> <lb/>D. partes enim figur&aelig; talem po&longs;&longs;unt ha&shy;<lb/>bere &longs;itum, vt inter &longs;e &etail;&que;ponderare po&longs;&shy;<lb/>&longs;int. </s>
          <s id="N10839">vt ex &longs;ubiectis figuris per&longs;picuum e&longs;t. <lb/>&amp; ad huc clari&ugrave;s, &longs;i intelligatur figura, vt <lb/>E circulo tum exteriori, tum interiori ter <lb/>minata, cuius centrum grauitatis extra fi&shy;<lb/>guram erit in F. quod quidem cum cir&shy;<lb/>culorum centro conueniet. </s>
          <s id="N10845">circa quod <lb/>(exi&longs;tente centro F in centro mundi) <lb/>partes vndi&que; &etail;&que;ponderabunt: c&ugrave;m <lb/>omnes &etail;qualiter &agrave; centro grauitatis <expan abbr="di&longs;t&etilde;t">di&longs;tent</expan>. <lb/>pr&aelig;terea in hac figura E centrum graui&shy;<lb/>tatis (quamuis &longs;it extra figuram) cum cen&shy;<lb/>tro figur&aelig;, <expan abbr="c&etilde;troqu&egrave;">centroqu&egrave;</expan> magnitudinis ip&longs;ius <lb/>figur&aelig; conuenire, forta&longs;&longs;e non erit incon&shy;<lb/>ueniens a&longs;&longs;erere. </s>
          <s id="N1085F">At ver&ograve; figur&aelig; AC nul<lb/>lo pacto figur&aelig;, magnitudinisqu&egrave; <expan abbr="centr&utilde;">centrum</expan> <lb/>habebunt. </s>
          <s id="N10869">&amp; quamuis dictum &longs;it <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis corporum regularium e&longs;&longs;e me&shy;<lb/>dium ip&longs;orum, non tamen propterea dicendum e&longs;t, idem e&longs;&longs;e <lb/>centrum magnitudinis, at&que; figur&aelig;, ni&longs;i impropri&egrave;; <expan abbr="medi&utilde;">medium</expan> <lb/>enim his impropri&egrave; attribuitur, &longs;icuti etiam centrum figur&aelig;; <lb/>c&ugrave;m line&aelig; ex ip&longs;o prodeuntes non &longs;int ip&longs;orum corporum <lb/>(quatenus regularia &longs;unt) &longs;emidiametri. </s>
          <s id="N1087F">quare centrum gra&shy;<lb/>uitatis reperiri pote&longs;t ab&longs;&que; alijs centris; at non &egrave; conuer&longs;o. <lb/>Rur&longs;us commune magis e&longs;t <expan abbr="c&etilde;trum">centrum</expan> figur&aelig; centro magnitu&shy;<lb/>dinis; quia pr&aelig;ter circulum, &amp; &longs;ph&aelig;ram, qu&aelig; tam figur&aelig;, <expan abbr="qu&atilde;">quam</expan> <lb/>magnitudinis centrum habent, nonnull&aelig; figur&aelig; &longs;uum ha&shy;<lb/>bent figur&aelig; centrum in ip&longs;is, &amp; extra ip&longs;as; in ip&longs;is, vt ellip&longs;is, <lb/>cuius centrum int&ugrave;s habetur; &longs;emicirculus etiam, dimidia qu&egrave; <lb/>&longs;ph&aelig;ra centrum habent in limbo. </s>
          <s id="N10897">extra figuram ver&ograve; veluti <lb/>hyperbol&aelig; centrum, quod extra figuram exi&longs;tit; vbi nemp&egrave; <lb/>diametri concurrunt. </s>
          <s id="N1089D">Qu&aelig; quidem omnia &longs;unt figur&aelig; cen&shy;<lb/>tra; magnitudinis ver&ograve; minim&egrave;. </s>
          <s id="N108A1">ver&ugrave;m obijciet hoc loco for<pb xlink:href="077/01/018.jpg" pagenum="14"/>ta&longs;&longs;e qui&longs;piam, vel ambas, inquiens, centri grauitatis defini&shy;<lb/>tiones allatas, diminutas e&longs;&longs;e; vel ijs, qu&aelig; mod&ograve; &agrave; nobis de <expan abbr="c&etilde;">cem</expan> <lb/>tro grauitatis dicta &longs;unt, repugnare; c&ugrave;m o&longs;tenderimus cen&shy;<lb/>trum grauitatis aliquando e&longs;&longs;e, vel in ambitu figur&aelig;, vel extra <lb/>figuram; definitiones ver&ograve; allat&etail; &longs;emper &longs;upponunt illud e&longs;&longs;e <lb/>in ip&longs;is intra <expan abbr="po&longs;it&utilde;">po&longs;itum</expan>. <expan abbr="C&otilde;firmaturqu&egrave;">Confirmaturqu&egrave;</expan> difficultas, quandoqui&shy;<lb/>dem, ne&que; huiu&longs;modi centrum extra figuram con&longs;titutum, <lb/>fui&longs;&longs;e Archimedi pror&longs;us ignotum, exi&longs;timare debemus; vt <lb/>colligere licet ex nono po&longs;tulato huius libri; c&ugrave;m inquit. <lb/><emph type="italics"/>Omnis figur&aelig;, cuius perimeter &longs;it ad eandem partem concauus, centrum <lb/>grauitatis intra ip&longs;am e&longs;&longs;e oportet.<emph.end type="italics"/> qua&longs;i non repugnet figur&etail; peri<lb/>metrum non ad eandem partem concauum habenti, extra <lb/>ip&longs;am grauitatis centrum obtinere. </s>
          <s id="N108D0">Cui obiectioni in hunc <lb/>modum occurri poterit, &longs;i dixerimus, qu&ograve;d quamuis exempli <lb/>gratia in figura C dictum &longs;it centrum grauitatis D extra fi<lb/>guram exi&longs;tere, id ip&longs;um etiam intra figuram e&longs;&longs;e affirmati <lb/>poterit. </s>
          <s id="N108DA">&longs;iquidem ambitus figur&etail; C centrum D intra &longs;e <expan abbr="c&otilde;">com</expan> <lb/>tinct; ita vt re&longs;pectu t&ouml;tius &longs;it intra. </s>
          <s id="N108E2">idemqu&egrave; dicendum e&longs;t de <lb/>altera figura A. hoc autem euidenti&longs;&longs;imum e&longs;t in figura E. <lb/>&amp; hic e&longs;t &longs;en&longs;us definitionum centri grauitatis. </s>
          <s id="N108E8">His ita&que; pri<lb/>m&ugrave;m cognitis con&longs;ideranda e&longs;t intentio Archimedis in his li<lb/>bris, qu&etail; quidem vt plurimum &agrave; librorum in&longs;criptionibus e&shy;<lb/>luce&longs;cere &longs;olet. </s>
        </p>
        <figure id="id.077.01.018.1.jpg" xlink:href="077/01/018/1.jpg"/>
        <p id="N108F4" type="head">
          <s id="N108F6">DE SCOPO HORVM LIBRORVM</s>
        </p>
        <p id="N108F8" type="main">
          <s id="N108FA">Si Archimedis propo&longs;itum in his libris ex ip&longs;a operis in&shy;<lb/>&longs;criptione, vt in alijs quo&que; aliorum authorum volumini&shy;<lb/>bus fieri vt plurim&ugrave;m &longs;olet, inue&longs;tigandum erit, partim &longs;an&egrave; <lb/>con&longs;picuum illud e&longs;&longs;e videbitur, partim ver&ograve; ignotum ade&ograve;, <lb/>vt poti&ugrave;s nullius ferm&egrave; rei &longs;e habiturum e&longs;&longs;e &longs;ermonem profi&shy;<lb/>teatur Archimedes. </s>
          <s id="N10906">quid enim (ob&longs;ecro) verbis illis &longs;ignificari <lb/>potuit, &que; primi libri initio ita &longs;e <expan abbr="hab&etilde;t">habent</expan>. <foreign lang="greek">Arximh/dous e)pipe/dwn i)sor&shy;<lb/>ropixw_n, h\ ke/ntra ba/rwn e)pipe/dwn.</foreign> hoc e&longs;t. <emph type="italics"/>Archimedis planorum &aelig;&que;pon<lb/>derantium, vel centra grauitatum planorum.<emph.end type="italics"/> quando quidem vide&shy;<lb/>tur Archimedes rem pror&longs;us <expan abbr="inutil&etilde;">inutilem</expan>, quinnim&ograve; natur&etail; repu&shy;<lb/>gnantem &longs;ibi contemplandam proponere. </s>
          <s id="N10924">d&ugrave;m enim polli-
<pb xlink:href="077/01/019.jpg" pagenum="15"/>cetur &longs;e e&longs;&longs;e pertractaturum de planis &aelig;qu&aelig;ponderantibus, &longs;i<lb/>ue de centris grauitatum planorum; c&ugrave;m ea, qu&aelig; &aelig;&que;ponde <lb/>rare debent, ponderare quo&que; oporteat; &longs;i plana &aelig;&que;ponde<lb/>rare <expan abbr="deb&etilde;t">debent</expan>, grauitate quadam illa pr&aelig;dita e&longs;&longs;e nece&longs;&longs;e e&longs;t. </s>
          <s id="N10934">quod <lb/>vald&egrave; &agrave; planorum natura abhorret, c&ugrave;m grauitas, nonni&longs;i cor<lb/>poribus, ne&que; tamen omnibus competat. </s>
          <s id="N1093A">ip&longs;e tamen, dum <lb/>plana &aelig;&que;ponderantia, vel centra grauitatum planorum &longs;e <lb/>explicaturum pollicetur, apert&egrave; &longs;upponit plana, ac &longs;uperficies <lb/>graues exi&longs;tere, rem &longs;an&egrave; immaginariam pror&longs;us, ip&longs;iusqu&egrave; rei <lb/>natur&aelig; nullatenus re&longs;pondentem. </s>
          <s id="N10944">ita vt Archimedes circa ea, <lb/>qu&aelig; omnino rei natur&aelig; aduer&longs;antur, negotium &longs;ump&longs;i&longs;&longs;e vi&shy;<lb/>deatur. </s>
          <s id="N1094A">Ver&ugrave;m enimuero &longs;i Authoris <expan abbr="m&etilde;tem">mentem</expan> acurati&ugrave;s intuea<lb/>mur, rem plan&egrave; egregiam, natur&aelig;qu&egrave; rei apprim&egrave; con&longs;enta&shy;<lb/>neam ip&longs;um pertractandam &longs;ump&longs;i&longs;&longs;e depr&aelig;hendemus. </s>
          <s id="N10954">Nam <lb/>quamuis plana, quatenus plana &longs;unt, nullam habeant graui&shy;<lb/>tatem, non e&longs;t tamen &agrave; rei natura, ne&que; &agrave; ratione alienum, <lb/>quin po&longs;&longs;imus planorum, &longs;uperficierum qu&egrave; centra grauitatis <lb/>depr&aelig;hendere, ex quibus &longs;i &longs;u&longs;pendantur, planorum partes <lb/>vndiqu&egrave; &etail;qualium momentorum con&longs;i&longs;tentes maneant. <expan abbr="qu&atilde;-doquidem">quan<lb/>doquidem</expan> centrum grauitatis talis e&longs;t natur&aelig;, vt &longs;i mente <expan abbr="c&otilde;-cipiamus">con&shy;<lb/>cipiamus</expan>, rem aliquam in eius centro grauitatis appen&longs;am e&longs;&shy;<lb/>&longs;e, eo pror&longs;us modo, quo reperitur, quie&longs;cat, &amp; maneat. </s>
          <s id="N1096E">vt <lb/>antea declarauimus. </s>
          <s id="N10972">&amp; quamuis re ip&longs;a, act&ugrave;&que; plana <expan abbr="&longs;eors&utilde;">&longs;eorsum</expan> <lb/>&agrave; corporibus reperiri ne&que;ant; in ip&longs;is tamen h&aelig;c ip&longs;orum <lb/>circa centra grauitatis &aelig;&que;ponderatio ad actum facil&egrave; redigi <lb/>poterit. </s>
          <s id="N1097E">Vt &longs;it &longs;olidum AB pri&longs;&shy;<lb/>
<arrow.to.target n="fig5"/><lb/>ma, <expan abbr="cui^&lcub;9}">cuius</expan> latera AE CF DB &longs;int <lb/>horizonti erecta, &longs;uperiorqu&egrave; ba&shy;<lb/>&longs;is ACD, &que;m ad modum &amp; in&shy;<lb/>ferior EFB &longs;it horizonti &aelig;quidi&shy;<lb/>&longs;tans; &longs;it autem plani ACD cen&shy;<lb/>trum grauitatis G, ex quo G &longs;i <lb/>&longs;u&longs;pendatur totum AB patet <lb/>planum ACD horizonti &aelig;qui&shy;<lb/>di&longs;tans permanere, ac propterea <lb/>circa <expan abbr="c&etilde;trum">centrum</expan> grauitatis G &aelig;&que;&shy;<lb/>ponderare. </s>
          <s id="N1099F">quod quidem, quamuis egeat demon&longs;tratione,
<pb xlink:href="077/01/020.jpg" pagenum="16"/>
<arrow.to.target n="marg12"/> in pr&aelig;&longs;entia omittatur; infraqu&egrave; &longs;uo loco o&longs;tendendum. </s>
          <s id="N109A9">&longs;at <lb/>autem nobis nunc &longs;it o&longs;tendi&longs;&longs;e, h&aelig;c ad praxim reduci, ma&shy;<lb/>nibu&longs;qu&egrave; (vt dicitur.) contrectari po&longs;&longs;e. </s>
          <s id="N109AF">Qu&ograve;d &longs;i h&aelig;c ita &longs;e ha<lb/>bent, huiu&longs;modi con&longs;ideratio non erit vana, ne&que; vt inuti&shy;<lb/>lis reijcienda. </s>
          <s id="N109B5">Sed vlteri&ugrave;s adhuc progrediamur, dicamu&longs;&shy;<lb/>qu&egrave;, quoniam planum ACD, quatenus e&longs;t corpori coniun&shy;<lb/>ctum, horizonti &aelig;quidi&longs;tans permanere debet; &longs;i &longs;eor&longs;um &agrave; <lb/>corpore illud intelligamus, vt &longs;i ADC ex eius centro graui&shy;<lb/>tatis G &longs;u&longs;pendatur, tunc quocun&que; modo reperiatur, hoc <lb/>e&longs;t &longs;iue horizonti &etail;quidi&longs;tans, &longs;iu&egrave; <lb/>min&ugrave;s, idip&longs;um perman&longs;urum ni<lb/><arrow.to.target n="fig6"/><lb/>hilominus intelligere po&longs;&longs;umus, <lb/>parte&longs;qu&egrave; vndi&que; &aelig;qualium mo<lb/>mentorum con&longs;i&longs;tentes. </s>
          <s id="N109CE">Ne&que; <lb/>enim Ari&longs;to teles grauibus dunta&shy;<lb/>xat, &longs;ed etiam leuibus momenta <lb/>tribuit, idip&longs;um qu&egrave; (vt Eutocius <lb/>in horum librorum comentarijs <lb/>refert) Ptol&aelig;meo quo&que; placuit, vt habetur in l&iacute;bro (&agrave; nobis <lb/>ramen de&longs;iderato) &que;m de momentis &longs;crip&longs;it. </s>
          <s id="N109DC">Pr&etail;terea alij&shy;<lb/>quo&que; Philo&longs;ophi id ip&longs;um &longs;en&longs;i&longs;&longs;e videntur. </s>
          <s id="N109E0">quod e&longs;t qui&shy;<lb/>dem rationi con&longs;entaneum, &longs;uperuolant enim, qu&aelig; leuia &longs;unt, <lb/>&amp; &longs;i mente concipiatur <expan abbr="ead&etilde;">eadem</expan> figura leuis cuiu&longs;piam e&longs;&longs;e, tunc <lb/>&longs;i detineatur in G, partes vndi&que; &etail;qualium <expan abbr="momentor&utilde;">momentorum</expan> <lb/>con&longs;i&longs;tent, e&longs;&longs;etqu&egrave; G (vt ita dicam) centrum leuitatis. </s>
          <s id="N109F2">Quo&shy;<lb/>niam autem circa centrum grauitatis &etail;&que;ponderationem <lb/>con&longs;ideramus, id circo plana, tanquam no bis apparentia gra&shy;<lb/>uitatem habere, mente concipimus. </s>
          <s id="N109FA">Non e&longs;t igitur &agrave; ratio&shy;<lb/>ne alienum, &aelig;&que;ponderantiam in planis, vt grauibus con&longs;i&shy;<lb/>deratis intelligere, conciperequ&egrave;. </s>
          <s id="N10A00">Nec quicquam nobis offi&shy;<lb/>cit, qu&ograve;d definitiones centri grauitatis pri&ugrave;s allat&aelig; non pla&shy;<lb/>norum, &longs;ed corporum centra explicarunt, ita vt grauitatis <expan abbr="c&etilde;-tr&utilde;">cen&shy;<lb/>trum</expan> ad corpora, <expan abbr="n&otilde;">non</expan> ad plana &longs;it <expan abbr="referend&utilde;">referendum</expan>. Hoc enim ideo fa<lb/><expan abbr="ct&utilde;">ctum</expan> e&longs;t, quia propri&egrave; <expan abbr="centr&utilde;">centrum</expan> grauitatis re&longs;picit corpora; non ta<lb/>men propterea impropri&egrave; re&longs;picit plana, &longs;ed quia prim&ograve; re&longs;pi<lb/>cit corpora; in <expan abbr="quib^&lcub;9}">quibus</expan> actu ine&longs;&longs;e <expan abbr="depr&aelig;h&etilde;ditur">depr&aelig;henditur</expan>. propterea <expan abbr="e&etail;d&etilde;-met">e&etail;dem&shy;<lb/>met</expan> definitiones planis quo&que; in <expan abbr="h&utilde;c">hunc</expan> <expan abbr="mod&utilde;">modum</expan> aptari <expan abbr="poter&utilde;t">poterunt</expan>. </s>
        </p>
        <pb xlink:href="077/01/021.jpg" pagenum="17"/>
        <p id="N10A3B" type="margin">
          <s id="N10A3D"><margin.target id="marg12"/><emph type="italics"/>in fine pri&shy;<lb/>mi libri.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.021.1.jpg" xlink:href="077/01/021/1.jpg"/>
        <figure id="id.077.01.021.2.jpg" xlink:href="077/01/021/2.jpg"/>
        <p id="N10A4F" type="head">
          <s id="N10A51">DEFINITIO CENTRI GRAVITATIS PLANORVM.</s>
        </p>
        <p id="N10A53" type="main">
          <s id="N10A55">Centrum grauitatis vniu&longs;cuiu&longs;&que; plani e&longs;t punctum quod&shy;<lb/>dam intra po&longs;itum, &agrave; quo &longs;i planum appen&longs;um mente con&shy;<lb/>cipiatur, dum fertur, quie&longs;cit; &amp; &longs;eruat eam, quam in princi&shy;<lb/>pio habebat po&longs;itionem, ne&que; in ip&longs;a latione <expan abbr="circ&utilde;uertitur">circumuertitur</expan>. </s>
        </p>
        <p id="N10A61" type="head">
          <s id="N10A63">EIVSDEM ALIA DEFINITIO.</s>
        </p>
        <p id="N10A65" type="main">
          <s id="N10A67">Centrum grauitatis vniu&longs;cuiu&longs;&que; plani e&longs;t punctum il&shy;<lb/>lud intra po&longs;itum, circa quod vndi&que; partes &aelig;qualium mo <lb/>mentorum con&longs;i&longs;tunt. </s>
          <s id="N10A6D">&longs;i enim per tale centrum recta du&shy;<lb/>catur linea figuram quomodocun&que; &longs;ecans, &longs;emper in par<lb/>tes &aelig;&que;ponderantes ip&longs;am diuidet. </s>
        </p>
        <p id="N10A75" type="main">
          <s id="N10A77">Vt Ita&que; in planis quo&que; centrum grauitatis con&longs;ide&shy;<lb/>ratur, ita etiam plana grauitate pr&aelig;dita con&longs;iderare, non e&shy;<lb/>rit ab&longs;urdum. </s>
          <s id="N10A7D">&longs;i enim impo&longs;&longs;ibile e&longs;&longs;et con&longs;iderare plana gra<lb/>uitate pr&aelig;dita, centrum quo&que; grauitatis in ip&longs;is nullo mo&shy;<lb/>do concipi po&longs;&longs;et; at&que; per&longs;picuum e&longs;t, centrum grauitatis in <lb/>ip&longs;is admitti, ac de&longs;ignari po&longs;&longs;e, igitur &amp; plana grauitate in&longs;i<lb/>gnita. </s>
          <s id="N10A87">Et &longs;i mathematicus con&longs;iderat corpora &longs;eclu&longs;a interim <lb/>ip&longs;orum grauitate, &amp; leuitate: &amp; A&longs;tronomus corpora con&longs;i&shy;<lb/>derans c&aelig;le&longs;tia, qu&aelig; ne&que; grauia, ne&que; leuia &longs;unt, non pro&shy;<lb/>pterea <expan abbr="c&otilde;&longs;iderat">con&longs;iderat</expan> ea ex propria <expan abbr="ip&longs;or&utilde;">ip&longs;orum</expan> natura, ne&que; grauia, ne <lb/>&que; leuia e&longs;&longs;e; etenim quamuis grauia, vel leuia e&longs;&longs;ent, nihilo <lb/>minus ne&que; grauia, ne&que; leuia e&longs;&longs;e ea con&longs;ideraret. </s>
          <s id="N10A9B">qu&ograve;d &longs;i <lb/>Mathematicus hoc pacto huiu&longs;modi corpora intelligere po&shy;<lb/>te&longs;t; quid prohibet rur&longs;um <expan abbr="ead&etilde;">eadem</expan>, <expan abbr="qu&atilde;uis">quamuis</expan> vt talia, ne&que; grauia, <lb/>ne&que; leuia &longs;int; vel grauia, vel leuia e&longs;&longs;e concipere? <expan abbr="&quetilde;ad-modum">&que;mad&shy;<lb/>modum</expan> hoc quo&que; <expan abbr="ex&etilde;">exem</expan> <lb/>
<arrow.to.target n="fig7"/><lb/>plo res magis eluce&longs;cet: <lb/>veluti &longs;i intelligamus ex <lb/>AC appen&longs;a e&longs;&longs;e plana <lb/>DE, qu&aelig; &longs;int &aelig;qualia; &longs;u<lb/>&longs;pendaturqu&egrave; AC in me <lb/>dio pror&longs;us in B; cur mente intelligere non po&longs;&longs;umus, <lb/><expan abbr="quantitat&etilde;">quantitatem</expan>, <expan abbr="&longs;paci&utilde;qu&egrave;">&longs;paciumqu&egrave;</expan> D <expan abbr="&aelig;&que;p&otilde;derare">&aelig;&que;ponderare</expan> &longs;pacio E; c&ugrave;m &longs;int &aelig;qua<lb/>lia? <gap/> &longs;i planorum alterum, put&agrave; D, maius e&longs;&longs;et ip&longs;o E; tunc
<pb xlink:href="077/01/022.jpg" pagenum="18"/>&longs;tatim non &longs;ol&ugrave;m &etail;&que;ponderare non po&longs;&longs;e, ver&ugrave;m etiam pla<lb/>num D deor&longs;um tendere concipiemus. </s>
          <s id="N10ADE">&amp; hoc nulla alia de <lb/>cau&longs;a, qu&agrave;m qu&ograve;d c&ugrave;m D maius &longs;it, qu&agrave;m E, &longs;tatim <expan abbr="ips&utilde;">ipsum</expan> <lb/>D, qu&agrave;m E grauius quo&que; e&longs;&longs;e concipimus. </s>
          <s id="N10AE8">Con&longs;iderare <lb/>igitur plana cum grauitate non e&longs;t omnino &agrave; ratione <expan abbr="alien&utilde;">alienum</expan>. <lb/>Quare vtrum &que; titulum, nempe planorum &aelig;&que;ponderan<lb/>tium, vel centra grauitatis <expan abbr="planor&utilde;">planorum</expan>, admittendum duximus. <lb/>Ver&ugrave;m quoniam Archimedes &longs;ecundum librum &longs;implici vo<lb/>cabulo, nimirum (qua&longs;i &longs;imul omnia complectens) <emph type="italics"/>&aelig;&que;pon&shy;<lb/>derantium<emph.end type="italics"/> in &longs;crip&longs;it; idcirco tam primum, qu&agrave;m &longs;ecundum li<lb/>brum (&aelig;&que;ponderantium) in&longs;cribendum exi&longs;timamus. </s>
          <s id="N10B06">eo&shy;<lb/>qu&egrave; libenti&ugrave;s; quoniam ip&longs;emet Eutocius horum quo&que; li&shy;<lb/>brorum explanator ho&longs;ce libros hoc tant&ugrave;m nomine &aelig;&que;&shy;<lb/>ponderantium nuncupauit: alijqu&egrave; omnes, qui hos Archime<lb/>dis libros nominant; hoc titulo de &aelig;&que;ponderantibus nun<lb/>cupant. </s>
          <s id="N10B12">Pr&aelig;terea titulus hic magis operi congruere mihi vide<lb/>tur; quoniam nonnulla Archimedes in principio pertractat, <lb/>qu&aelig; tam &longs;olidis, qu&agrave;m planis communia exi&longs;tunt; quamuis <lb/>c&aelig;tera ad plana &longs;int <expan abbr="tant&utilde;">tantum</expan> <expan abbr="refer&etilde;da">referenda</expan>. in quibus omnibus de re <lb/>admodum vtili, &amp; ad <expan abbr="qu&atilde;plurima">quamplurima</expan> <expan abbr="c&otilde;du&etilde;cti">conduencti</expan> pertractat. <expan abbr="qu&atilde;doqui">quandoqui</expan> <lb/><expan abbr="d&etilde;">dem</expan> ex ijs, qu&aelig; ab Archimede his libris docemur, in <expan abbr="multar&utilde;">multarum</expan> <expan abbr="re-r&utilde;">re&shy;<lb/>rum</expan> <expan abbr="cognition&etilde;">cognitionem</expan> peruenire po&longs;&longs;umus. </s>
          <s id="N10B3F">quod facil&egrave; con&longs;tat inpri<lb/>mis ip&longs;iu&longs;met Archimedis <expan abbr="ex&etilde;plo">exemplo</expan>. <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> hac methodo ip&longs;e <lb/>in libro de quadratura paraboles <expan abbr="c&otilde;par&atilde;do">comparando</expan> plana in libra <expan abbr="c&otilde;">com</expan> <lb/>&longs;tituta, ip&longs;ius paraboles <expan abbr="quadratur&atilde;">quadraturam</expan> miro artificio adinuenit. <lb/>Deinceps ex cognitione <expan abbr="c&etilde;troi&utilde;">centrorum</expan> grauitatis planorum, nos in <lb/>cognitionem centrorum grauitatum &longs;olidorum deducimur. <lb/>Deni&que; adeo proficua e&longs;t h&aelig;c doctrina, quam nobis in his <lb/>libris Archimedes pr&aelig;&longs;tat; vt affirmare non verear, nullum <lb/>e&longs;&longs;e Theorema, nullum qu&egrave; problema ad rem mechanicam <lb/>pertinens, quod in &longs;ui &longs;peculatione peculiare <expan abbr="n&otilde;">non</expan> a&longs;&longs;umat <expan abbr="fun-dam&etilde;tum">fun<lb/>damentum</expan> ex ijs, qu&aelig; Archimedes in his libris edi&longs;&longs;erit. </s>
          <s id="N10B74">&que;m&shy;<lb/>admodum (c&aelig;teris interim omi&longs;&longs;is) patet ex vulgata illa pro&shy;<lb/>po&longs;itione enunciante, ita &longs;e habere pondus ad pondus, vt di<lb/>&longs;tantia ad di&longs;tantiam permutatim &longs;e habet, ex quibus &longs;u&longs;pen<lb/>duntur. </s>
          <s id="N10B7E">qu&aelig; pr&aelig;clari&longs;&longs;im&egrave; ab ip&longs;o in primo libro demon&longs;tra<lb/>tur. </s>
          <s id="N10B82">Et quamuis Iordanus Nemorarius (&que;m &longs;ecutus e&longs;t
<pb xlink:href="077/01/023.jpg" pagenum="19"/>Nicolaus Tartalea, &amp; alij) in libello de ponderibus hanc <expan abbr="e&atilde;-dem">ean&shy;<lb/>dem</expan> propo&longs;itionem quo&que; demon&longs;trare conatus &longs;it; &amp; ad <lb/><expan abbr="c&atilde;">cam</expan> o&longs;tendendam pluribus medijs fuerit v&longs;us; nulli tamen pro<lb/>bationi demon&longs;trationis nomen conuenire pote&longs;t. </s>
          <s id="N10B95">c&ugrave;m vix <lb/>ex probabilibus, &amp; ijs, qu&aelig; nullo modo nece&longs;&longs;itatem <expan abbr="affer&utilde;t">afferunt</expan>, <lb/>&amp; forta&longs;&longs;e ne&que; ex probabilibus &longs;uas componat rationes. <lb/>C&ugrave;m in mathematicis demon&longs;trationes requirantur exqui&longs;i&shy;<lb/>ti&longs;&longs;im&aelig;. </s>
          <s id="N10BA3">ac propterea ne&que; inter Mechanicos videtur mihi <lb/>Iordanus ille e&longs;&longs;e recen&longs;endus. </s>
          <s id="N10BA7">Quapropter ad Archimedem <lb/>confugiendum e&longs;t, &longs;i fundamenta mechanica, veraqu&egrave; huius <lb/>&longs;cienti&aelig; principia perdi&longs;cere cupimus: qui (meo iudicio) ad <lb/>hoc poti&longs;&longs;im&ugrave;m re&longs;pexit; vt elementa mechanica traderet. </s>
          <s id="N10BAF">vt <lb/>etiam Pappus in octauo Mathematicarum collectionum li&shy;<lb/>bro &longs;entit; quod quidem ex diui&longs;ione, ac progre&longs;&longs;u horum li&shy;<lb/>brorum facil&egrave; digno&longs;cetur. </s>
        </p>
        <figure id="id.077.01.023.1.jpg" xlink:href="077/01/023/1.jpg"/>
        <p id="N10BBB" type="head">
          <s id="N10BBD">DE DIVISIONE HORVM LIBRORVM.</s>
        </p>
        <p id="N10BBF" type="main">
          <s id="N10BC1">Diuiditur enim in primis hic tractatus in duos libros diui&shy;<lb/>&longs;us, in po&longs;tulata, &amp; theoremata: theoremata ver&ograve; &longs;ubdiui&shy;<lb/>duntur in duas &longs;ectiones, quarum prima continet priora o&shy;<lb/>cto theoremata; ad alteram ver&ograve; reliqua theoremata <expan abbr="&longs;pect&atilde;t">&longs;pectant</expan>. <lb/>qu&aelig; quidem adhuc in alias duas partes diuidi pote&longs;t; nemp&egrave; <lb/>in theoremata primo libro examinata, &amp; in ea, qu&aelig; &longs;ecun&shy;<lb/>dus liber contemplatur. </s>
          <s id="N10BD3">Hanc autem horum librorum con<lb/>&longs;tituimus diui&longs;ionem, quoniam imprimis Archimedes, (o&shy;<lb/>mi&longs;&longs;is po&longs;tulatis, qu&aelig; primum locum obtinere debent) qu&aelig;&shy;<lb/>dam tractauit communia in prioribus octo theorematibus; <lb/>quorum &longs;copus e&longs;t inuenire fundamentum illud <expan abbr="pr&aelig;cipu&utilde;">pr&aelig;cipuum</expan> <lb/>mechanicum, qu&ograve;d &longs;cilicet ita &longs;e habet grauitas ad grauita&shy;<lb/>tem, vt di&longs;tantia ad di&longs;tantiam permutatim. </s>
          <s id="N10BE5">ad quod <expan abbr="dem&otilde;&longs;trandum">demon<lb/>&longs;trandum</expan> quin&que; pr&aelig;mittit theoremata, qu&aelig; paulatim <lb/>deducunt nos in cognitionem demon&longs;trationis pr&aelig;fati fun<lb/>damenti. </s>
          <s id="N10BED">quo loco illud &longs;ummoper&egrave; notandum e&longs;t, nimi&shy;<lb/>rum fundamentum illud, nec non octo priora theorema&shy;<lb/>ta communia e&longs;&longs;e tam planis, qu&agrave;m &longs;olidis; at&que; promi&longs;&shy;<lb/>cu&egrave; de vtri&longs;&que; <expan abbr="Archimed&etilde;">Archimedem</expan> demon&longs;trare. </s>
          <s id="N10BF9">qu&ograve;d &longs;i quis aliter
<pb xlink:href="077/01/024.jpg" pagenum="20"/>&longs;en&longs;erit, demon&longs;tratione&longs;qu&egrave; tant&ugrave;m de planis <expan abbr="c&otilde;cludere">concludere</expan> exi<lb/>&longs;timauerit, vel de &longs;olidis, non autem <expan abbr="quibu&longs;c&utilde;&que;">quibu&longs;cun&que;</expan>, &longs;ed vel de <lb/>rectilineis, vel de homogeneis tant&ugrave;m, &amp; de ijs, qu&aelig; inter &longs;e <lb/>&longs;unt eiu&longs;dem &longs;peciei, long&egrave; aberrat &agrave; &longs;copo, &amp; mente Archi&shy;<lb/>medis. </s>
          <s id="N10C0F">etenim in his &longs;emper loquitur. </s>
          <s id="N10C11">vel de grauibus &longs;impli<lb/>citer, veluti in primis tribus theorematibus; vel de magnitu<lb/>dinibus, vt in reliquis quin&que; quod quidem nomen tam <lb/>planis, qu&agrave;m &longs;olidis quibu&longs;cun&que; e&longs;t <expan abbr="c&otilde;mune">commune</expan>, vt etiam ij, <lb/>qui par&ugrave;m in Mathematicis ver&longs;ati &longs;unt, &longs;atis norunt. </s>
          <s id="N10C1F">&longs;icu&shy;<lb/>ti etiam Euclides, dum quinti libri propo&longs;itiones pertracta&shy;<lb/>uit, quantitatem continuam &longs;ub nomine magnitudinis <expan abbr="c&otilde;">com</expan> <lb/>prehendit. </s>
          <s id="N10C2B">qu&ograve;d <expan abbr="aut&etilde;">autem</expan> nomen grauis &longs;it <expan abbr="c&otilde;mune">commune</expan>, iam &longs;atis <lb/>per &longs;e con&longs;tat. </s>
          <s id="N10C37">Per&longs;picuum e&longs;t igitur priora h&aelig;c octo Theo<lb/>remata communia e&longs;&longs;e, tam planis, qu&agrave;m &longs;olidis. </s>
          <s id="N10C3B">ac non &longs;o&shy;<lb/>l&ugrave;m &longs;olidis eiu&longs;dem &longs;peciei, &amp; homogeneis, ver&ugrave;m etiam &longs;oli <lb/>dis diuer&longs;&aelig; &longs;peciei, &amp; h&etail;terogeneis, vt &longs;uo loco manife&longs;tum <lb/>fiet. </s>
          <s id="N10C43">Iactoqu&egrave; hoc fundamento, quod Archimedes in <expan abbr="duob^&lcub;9}">duobus</expan> <lb/>propo&longs;itionibus, &longs;exta nemp&egrave;, &amp; &longs;eptima demon&longs;trauit; in o&shy;<lb/>ctaua tanquam corrollarium colligit. </s>
          <s id="N10C49">Deinceps peculiariter <lb/>pertractat de centro grauitatis planorum, nec amplius plana <lb/>nominat magnitudinis nomine, &longs;ed proprijs cuiu&longs;cun&que; <lb/>nominibus; vt parallelogrammi, trianguli, &amp; aliorum huiu&longs;&shy;<lb/>modi. </s>
          <s id="N10C53">&amp; in hac parte de&longs;cendit ad particularia. </s>
          <s id="N10C55">quipp&egrave; c&ugrave;m <lb/>&amp; &longs;i non actu forta&longs;&longs;e, virtute tamen cuiu&longs;libet particularis <lb/>plani centrum grauitatis nos doceat. </s>
          <s id="N10C5B">in primo enim libro <lb/>&longs;at &longs;i bi vi&longs;um e&longs;t o&longs;tendi&longs;&longs;e centra grauitatum <expan abbr="triangulor&utilde;">triangulorum</expan>, <lb/>ac parallelogrammorum, ex quibus c&aelig;terarum figurarum, <lb/>veluti pentagoni, hexagoni, &amp; aliorum &longs;imilium centra gra&shy;<lb/>uitatis inue&longs;tigare non admodum erit difficile. </s>
          <s id="N10C65">&longs;iquidem hu<lb/>iu&longs;modi plana in triangula diuiduntur. </s>
          <s id="N10C69">vt in &longs;ine primi li&shy;<lb/>bri attingemus. </s>
          <s id="N10C6D">In &longs;ecundo autem libro alti&ugrave;s &longs;e extollit, &amp; <lb/>moro &longs;uo circa &longs;ubtili&longs;&longs;ima theoremata ver&longs;atur; nemp&egrave; cir<lb/>ca centrum grauitatis conice &longs;ectionis, qu&aelig; parabole nun&shy;<lb/>cupatur. </s>
          <s id="N10C75">nonnullaqu&egrave; pr&aelig;mittit theoremata, qu&aelig; &longs;unt tan&shy;<lb/>quam pr&aelig;uie di&longs;po&longs;itiones ad inue&longs;tigandam demon&longs;tra&shy;<lb/>tionem centri grauitatis in parabole. </s>
          <s id="N10C7B">Ita&que; per&longs;picuum e&longs;t, <lb/>Archimedem propri&egrave; elementa mechanica tradere. </s>
          <s id="N10C7F">quando-
<pb xlink:href="077/01/025.jpg" pagenum="21"/>quidem duo pertractat, qu&aelig; &longs;unt tanquam elementa huius <lb/>&longs;cienti&aelig;. </s>
          <s id="N10C87">fundamentum nemp&egrave; illud pr&aelig;&longs;tanti&longs;&longs;imum iam <lb/>toties pr&aelig;fatum, deinde centra grauitatis planorum o&longs;tendit. <lb/>&amp; quamuis hi duo Archimedis libelli pauca continere videan<lb/>tur, non tamen pauca docui&longs;&longs;e Archimedem exi&longs;timandum <lb/>e&longs;t. </s>
          <s id="N10C91">multa enim &longs;unt mole exigua, qu&aelig; tamen virtute maxima <lb/>habentur. </s>
          <s id="N10C95">quod plan&egrave; Archimedis &longs;criptis accidit; hi&longs;qu&egrave; pr&etail;<lb/>&longs;ertim, ex quibus patet aditus ad multa, ac pen&egrave; infinita theo&shy;<lb/>remata, problemataqu&egrave; mechanica. </s>
          <s id="N10C9B">nihil enim in hoc gene&shy;<lb/>re demon&longs;trari pote&longs;t, quod his non indigeat &longs;criptis. </s>
          <s id="N10C9F">&amp; <lb/>quod admirabilius e&longs;t, nos non &longs;ol&ugrave;m pro fundamento &longs;u&shy;<lb/>&longs;cipere po&longs;&longs;e ad aliquod demon&longs;trandum theoremata in his <lb/>libris demon&longs;trata, ver&ugrave;m etiam ab his demon&longs;trationibus <lb/>perdi&longs;cerere ip&longs;um modum argumentandi, &amp; demon&longs;trandi; <lb/>vt &longs;uis locis o&longs;tendemus. </s>
          <s id="N10CAB">ita vt ver&egrave; concludendum &longs;it, nemi&shy;<lb/>nem pror&longs;us inter mechanicos connumerandum fore, qui <lb/>h&aelig;c Archimedis &longs;cripta ignorat. </s>
          <s id="N10CB1">ignoratis enim principijs <lb/>nulla e&longs;t &longs;cientia, vt apud omnes &longs;apientes per&longs;picuum e&longs;t. <lb/>Ip&longs;um igitur Archimedem audiamus, eiu&longs;qu&egrave; &longs;cripta diligen<lb/>ti&longs;&longs;im&egrave; perpendamus. </s>
        </p>
        <pb xlink:href="077/01/026.jpg" pagenum="22"/>
        <pb xlink:href="077/01/027.jpg" pagenum="23"/>
        <p id="N10CBF" type="head">
          <s id="N10CC1">GVIDIVBALDI <lb/>EMARCHIONIBVS <lb/>MONTIS. <lb/>IN PRIMVM ARCHIMEDIS <lb/>AEQVEPONDERANTIVM <lb/>LIBRVM <lb/>PARAPHRASIS <lb/>SCHOLIIS ILLVSTRATA.</s>
        </p>
        <p id="N10CD1" type="head">
          <s id="N10CD3">Archimedis tamen huius primi libri <lb/>titulus &longs;ic &longs;e habet.</s>
        </p>
        <p id="N10CD7" type="head">
          <s id="N10CD9"><emph type="italics"/>ARCHIMEDIS PLANORVM AEQVEPONDERANTIVM, <lb/>VEL CENTRA GRAVITATVM PLANORVM.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.027.1.jpg" xlink:href="077/01/027/1.jpg"/>
        <p id="N10CE4" type="head">
          <s id="N10CE6">ARCHIMEDIS POSTVLATA.</s>
        </p>
        <p id="N10CE8" type="head">
          <s id="N10CEA">I.</s>
        </p>
        <p id="N10CEC" type="main">
          <s id="N10CEE">Grauia &aelig;qualia ex &aelig;qualibus di&longs;tantijs &aelig;&que;&shy;<lb/>ponderare. </s>
        </p>
        <p id="N10CF2" type="head">
          <s id="N10CF4">SCHOLIVM.</s>
        </p>
        <p id="N10CF6" type="main">
          <s id="N10CF8">Dvobvs modis grauia in di&longs;tantijs <lb/>collocata intelligi po&longs;&longs;unt. </s>
          <s id="N10CFC">quod &amp; <lb/>in c&aelig;teris po&longs;tulatis, &amp; in propo&longs;i&shy;<lb/>tionibus intelligendum e&longs;t. </s>
          <s id="N10D02">etenim <lb/>vel grauia <expan abbr="s&utilde;t">sunt</expan> appen&longs;a, vt in prima fi&shy;<lb/>gura &aelig;qualia grauia AB &longs;unt in CD <lb/>appen&longs;a; ita vt di&longs;tantia EC &longs;it <expan abbr="di&shy;&longs;t&atilde;ti&aelig;">di&shy;<lb/>&longs;tanti&aelig;</expan> ED &aelig;qualis. </s>
          <s id="N10D10">intelligaturqu&egrave; <lb/>CD tanquam libra, qu&aelig; &longs;u&longs;pendatur <lb/>in E. vel vt in &longs;ecunda figura grauia AB habent ip&longs;orum <lb/>centra grauitatis, qu&aelig; &longs;int CD, in ip&longs;a DC linea, in pun-
<pb xlink:href="077/01/028.jpg" pagenum="24"/>ctis <expan abbr="n&etilde;p&egrave;">nemp&egrave;</expan> CD <lb/>
<arrow.to.target n="fig8"/><lb/>con&longs;tituta. </s>
          <s id="N10D27">li&shy;<lb/>braqu&egrave; &longs;imili&shy;<lb/>ter ex puncto <lb/>E &longs;u&longs;pendatur; <lb/>&longs;itqu&egrave; <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> <lb/>EC di&longs;tanti&aelig; <lb/>ED &aelig;qualis. <lb/><expan abbr="er&utilde;t">erunt</expan> vti&que; in <lb/>vtra&que; figura <lb/>pondera AB <lb/>in di&longs;tantijs &etail;&shy;<lb/>qualibus con&shy;<lb/>&longs;tituta. </s>
          <s id="N10D44">ac pro&shy;<lb/>pterea &aelig;&que;ponderabunt, at&que; manebunt. </s>
          <s id="N10D48">nulla enim ratio <lb/>afferri pote&longs;t, cur ex parte A, vel ex parte B deor&longs;um, vel &longs;ur<lb/>&longs;um fieri debeat motus; c&ugrave;m omnia &longs;int paria. </s>
          <s id="N10D4E">ea ver&ograve; &aelig;&que;&shy;<lb/>ponderare debere, aliqua ratione manife&longs;tari pote&longs;t ex eo, <lb/>quod o&longs;ten&longs;um e&longs;t &agrave; nobis in no&longs;tro mechanicorum libro, <lb/>tractatu de libra: quod quidem ab Ari&longs;to tele quo&que; in prin<lb/>cipio qu&aelig;&longs;tionum mechanicarum elici pote&longs;t: idem &longs;cilicet <lb/>pondus longius a centro grauius e&longs;&longs;e eodem pondere ip&longs;i cen<lb/>tro propinquiori. </s>
          <s id="N10D5C">Vnde &longs;i duo e&longs;&longs;ent pondera &aelig;qualia alte&shy;<lb/>rum altero propinquius centro, quod remotius e&longs;t, grauius al<lb/>tero appareret. </s>
          <s id="N10D62">&longs;i igitur grauia &aelig;qualia &agrave; centro &aelig;qualiter di&shy;<lb/>&longs;tabunt, &aelig;&que; grauia erunt. </s>
          <s id="N10D66">ac propterea &aelig;&que;ponderabunt. <lb/>quod quidem &longs;upponit Archimedes. </s>
          <s id="N10D6A">Punctum autem illud, <lb/>quod Archimedes accipit, vnde &longs;umuntur di&longs;tanti&aelig;, ex qui&shy;<lb/>bus grauia &longs;u&longs;penduntur, veluti punctum E, Ari&longs;toteles cent<lb/>rum appellat. </s>
          <s id="N10D72">&amp; h&aelig;c quidem &aelig;&que;ponderatio tam ponderi&shy;<lb/>bus in libra appen&longs;is, qu&agrave;m in ip&longs;a (vt dictum e&longs;t) con&longs;titutis <lb/>competit: dummodo ea, quibus appenduntur pondera, libe&shy;<lb/>re &longs;emper in centrum mundi tendere po&longs;&longs;int. </s>
          <s id="N10D7A">vtro&que; enim <lb/>modo in punctis CD grauitant, vt diximus etiam in eodem <lb/>tractatu de libra. </s>
          <s id="N10D80">Noui&longs;&longs;e tamen oportet Archimedem in his <lb/>libris poti&ugrave;s intellexi&longs;&longs;e pondera e&longs;&longs;e in di&longs;tantijs collocata, vt <lb/>in &longs;ecunda figura, qu&agrave;m appen&longs;a; vt ex quarta, &amp; quinta
<pb xlink:href="077/01/029.jpg" pagenum="25"/>primi libri propo&longs;itione pater. </s>
          <s id="N10D8A">demon&longs;trationes enim cla&shy;<lb/>riores redduntur. </s>
        </p>
        <figure id="id.077.01.029.1.jpg" xlink:href="077/01/029/1.jpg"/>
        <figure id="id.077.01.029.2.jpg" xlink:href="077/01/029/2.jpg"/>
        <p id="N10D95" type="main">
          <s id="N10D97">Porr&ograve; non ignoran<lb/>dum hoc Archimedis <lb/>po&longs;tulatum verificari <lb/>de ponderibus quocun<lb/>&que; &longs;itu di&longs;po&longs;itis, &longs;iue <lb/>CED fuerit horizonti <lb/><expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan>, &longs;iu&egrave; min&ugrave;s; <lb/>vt in hac prima figura, <lb/>codem modo &longs;emper <lb/>verum e&longs;&longs;e pondera &aelig;&shy;<lb/>qualia CD ex &etail;quali&shy;<lb/>bus di&longs;tantijs EC ED <lb/>&aelig;&que;ponderare, vt in&shy;<lb/>fra (po&longs;t &longs;cilicet <expan abbr="quart&atilde;">quartam</expan> <lb/>huius propo&longs;itionem) <lb/>per&longs;picuum erit. </s>
          <s id="N10DBE">Qua&shy;<lb/>re c&ugrave;m Archimedes <expan abbr="t&atilde;">tam</expan> <lb/>in hoc po&longs;tulato, <expan abbr="qu&atilde;">quam</expan> <lb/>in &longs;e&que;ntibus, &longs;uppo&shy;<lb/>nit pondera in di&longs;tan&shy;<lb/>tijs e&longs;&longs;e collocata, intel&shy;<lb/>ligendum e&longs;t <expan abbr="di&longs;t&atilde;tias">di&longs;tantias</expan> <lb/>ex vtra&que; parte in ea&shy;<lb/>dem recta linea exi&longs;te&shy;<lb/>re. </s>
          <s id="N10DDE">Nam &longs;i (vt in &longs;ecun<lb/>da figura) <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> AB <lb/>fuerit &etail;qualis di&longs;tanti&etail; BC, qu&aelig; non indirectum iaceant, <lb/>&longs;ed angulum con&longs;tituant; tunc pondera AB, quamuis &longs;int <lb/>&etail;qualia, non &etail;&que;ponderabunt. </s>
          <s id="N10DEC">ni&longs;i quando (vt in tertia fi&shy;<lb/>gura) iuncta AC, bifariamqu&egrave; diui&longs;a in D, ductaqu&egrave; BD, <lb/>fuerit h&etail;c horizonti perpendicularis, vt in eodem tractatu <lb/>no&longs;tro expo&longs;uimus. </s>
          <s id="N10DF4">Di&longs;tantias igitur in eadem recta linea <lb/>&longs;emper exi&longs;tere intelligendum e&longs;t. </s>
          <s id="N10DF8">vt ex demon&longs;trationibus <lb/>Archimedis per&longs;picuum e&longs;t. </s>
        </p>
        <pb xlink:href="077/01/030.jpg" pagenum="26"/>
        <p id="N10DFF" type="head">
          <s id="N10E01">II.</s>
        </p>
        <p id="N10E03" type="main">
          <s id="N10E05">Aequalia ver&ograve; grauia ex in&aelig;qualibus <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> <lb/>non &aelig;queponderare, &longs;ed pr&aelig;ponderare ad gra&shy;<lb/>ue ex maiori di&longs;tantia. </s>
        </p>
        <p id="N10E0B" type="head">
          <s id="N10E0D">SCHOLIVM.</s>
        </p>
        <p id="N10E0F" type="main">
          <s id="N10E11">Si enim <expan abbr="di&longs;t&atilde;">di&longs;tam</expan> <lb/>
<arrow.to.target n="fig9"/><lb/>tia EC maior <lb/>fuerit di&longs;tantia <lb/>ED, grauibus <lb/>AB &longs;imiliter &aelig;&shy;<lb/>qualibus <expan abbr="exi&longs;t&etilde;">exi&longs;tem</expan> <lb/>tibus, &amp; in CD po&longs;itis, tunc concedendum videtur graue A <lb/>pr&aelig;ponderare ip&longs;i B, quandoquidem EC longior e&longs;t, qu&agrave;m <lb/>ED. &longs;upponit autem Archimedes hoc po&longs;tulatum re&longs;piciens <lb/>forta&longs;&longs;e ad ea, qu&aelig; Ari&longs;toteles in principio qu&aelig;&longs;tionum me&shy;<lb/>chanicarum o&longs;tendit, vbi colligit Ari&longs;toteles idem pondus ce&shy;<lb/>leri&ugrave;s ferri, qu&ograve; magis &agrave; centro di&longs;tat, vel quod idem e&longs;t, duo <lb/>pondera &aelig;qualia in&aelig;qualiter &agrave; centro di&longs;tantia, quod magis <lb/>di&longs;tat, celeri&ugrave;s ferri. </s>
          <s id="N10E3A">quod autem &aelig;qualium ponderum cele&shy;<lb/>ri&ugrave;s fertur, grauius exi&longs;tit; erit igitur A grauius, qu&agrave;m B. <lb/>quia EC longior e&longs;t, qu&agrave;m ED. Nos quo&que; (vt diximus) <lb/>in libro no&longs;trorum Mechanicorum tractatu de libra, alijs <lb/>quo&que; rationibus o&longs;tendimus, quo pondus e&longs;t in longiori <lb/>di&longs;tantia grauius e&longs;&longs;e. </s>
          <s id="N10E46">ex quibus &longs;equitur propter longiorem <lb/>di&longs;tantiam EC pondus A pr&aelig;ponderare ponderi B. ac pro&shy;<lb/>pterea deor&longs;um ferri. </s>
        </p>
        <figure id="id.077.01.030.1.jpg" xlink:href="077/01/030/1.jpg"/>
        <p id="N10E50" type="head">
          <s id="N10E52">III.</s>
        </p>
        <p id="N10E54" type="main">
          <s id="N10E56">Grauibus ex aliquibus di&longs;tantijs <expan abbr="&aelig;&que;ponder&atilde;tibus">&aelig;&que;ponderan<lb/>tibus</expan>, &longs;i alteri grauium aliquid adijciatur, non &aelig;&shy;<lb/>&que;ponderare; &longs;ed ad graue, cui adiectum fuit, <lb/>deor&longs;um ferri. </s>
        </p>
        <pb xlink:href="077/01/031.jpg" pagenum="27"/>
        <p id="N10E61" type="head">
          <s id="N10E63">SCHOLIVM</s>
        </p>
        <p id="N10E65" type="main">
          <s id="N10E67">Grauia enim <lb/>
<arrow.to.target n="fig10"/><lb/>AB &longs;iu&egrave; &aelig;qua&shy;<lb/>lia, &longs;iue in &etail;qua<lb/>lia &aelig;&que;ponde<lb/>rent ex di&longs;tan&shy;<lb/>tijs AC CB, al&shy;<lb/>teri ver&ograve; gra&shy;<lb/>uium, put&agrave; B, <lb/>adijciatur pon<lb/>dus D. per&longs;picuum e&longs;t pondera BD &longs;imul magis ponderare, <lb/>qu&agrave;m A. &longs;i enim B &etail;&que;ponderat ip&longs;i A; erit pondus B in <lb/>hoc &longs;itu &aelig;&que;graue, vt A: pondera igitur BD in hoc &longs;itu <expan abbr="n&otilde;">non</expan> <lb/>erunt &aelig;&que;grauia, vt pondus A. &longs;ed grauiora exi&longs;tent, qu&agrave;m <lb/>A. quare BD deor&longs;um tendent. </s>
        </p>
        <figure id="id.077.01.031.1.jpg" xlink:href="077/01/031/1.jpg"/>
        <p id="N10E90" type="head">
          <s id="N10E92">IIII.</s>
        </p>
        <p id="N10E94" type="main">
          <s id="N10E96">Similiter autem, &longs;i ab altero grauium auferatur <lb/>aliquid, non &aelig;&que;ponderare; ver&ugrave;m ad graue, &agrave; <lb/>quo nil ablatum e&longs;t, deor&longs;um tendere. </s>
        </p>
        <p id="N10E9C" type="head">
          <s id="N10E9E">SCHOLIVM.</s>
        </p>
        <p id="N10EA0" type="main">
          <s id="N10EA2">Ae&que;ponderent grauia BD &longs;imul, &amp; A &longs;ecund&ugrave;m
<arrow.to.target n="marg13"/> di&shy;<lb/>&longs;tantias CB CA; vt in eadem figura, &amp; ab altero eorum, put&agrave; <lb/>BD, auferatur D, remanebunt grauia BA; eritqu&egrave; A gra&shy;<lb/>uius ip&longs;o B. Nam &longs;i BD &longs;imul &aelig;&que;ponderant ip&longs;i A, B <lb/>tant&ugrave;m eidem A non &aelig;&que;ponderabit, &longs;ed leuius erit. </s>
          <s id="N10EB4">vnde <lb/>&longs;equitur ex parte A motum fieri deor&longs;um. </s>
        </p>
        <pb xlink:href="077/01/032.jpg" pagenum="28"/>
        <p id="N10EBB" type="margin">
          <s id="N10EBD"><margin.target id="marg13"/><emph type="italics"/>eadem figu<lb/>ra.<emph.end type="italics"/></s>
        </p>
        <p id="N10EC7" type="head">
          <s id="N10EC9">V</s>
        </p>
        <p id="N10ECB" type="main">
          <s id="N10ECD">Aequalibus, &longs;imilibu&longs;qu&egrave; figuris planis inter &longs;e <lb/>coaptatis, centra quo&que; grauitatum inter &longs;e coa&shy;<lb/>ptati oportet. </s>
        </p>
        <p id="N10ED3" type="head">
          <s id="N10ED5">SCHOLIVM.</s>
        </p>
        <p id="N10ED7" type="main">
          <s id="N10ED9">Aequales, <expan abbr="&longs;imiles&qacute;">&longs;imiles&que;</expan>; &longs;int <lb/>
<arrow.to.target n="fig11"/><lb/>figur&aelig; ABC DEF, qua&shy;<lb/>rum centra grauitatis &longs;int <lb/>GH; &longs;i ABC &longs;uperpona&shy;<lb/>tur ip&longs;i DEF, &amp; hoc <expan abbr="&longs;ec&utilde;">&longs;ecum</expan> <lb/>d&ugrave;m laterum <expan abbr="&aelig;qualitat&etilde;">&aelig;qualitatem</expan>, <lb/>hoc e&longs;t &longs;i latus AB fuerit <lb/>&aelig;quale lateri DE, tunc <lb/>ponatur AB &longs;uper DE; &longs;imiliter AC &longs;uper DF, &amp; BC &longs;uper <lb/>EF; tunc manife&longs;tum e&longs;t centrum grauitatis G &longs;uper centro <lb/>grauitatis H ad unguem conuenire; ita vt &longs;int vnum tan <expan abbr="t&utilde;">tum</expan> <lb/>punctum. </s>
          <s id="N10F06">Plana enim qu&aelig; &longs;e inuicem contingunt, non ef&shy;<lb/>ficiunt, ni&longs;i vnum tant&ugrave;m planum. </s>
          <s id="N10F0A">Solius autem figur&aelig; ex <lb/>planis ABC DEF inuicen coaptatis, vnum tant&ugrave;m erit cen<lb/>trum grauitatis, vt nos in no&longs;tro mechanicorum libro &longs;up&shy;<lb/>po&longs;uimus; centra igitur grauitatis inter &longs;e&longs;e conuenire nece&longs;&shy;<lb/>&longs;e e&longs;t. </s>
          <s id="N10F14">&longs;i enim centra grauitatis inter &longs;e non conuenirent, v&shy;<lb/>na tant&ugrave;m figura duo po&longs;&longs;et centra grauitatis habere. </s>
          <s id="N10F18">quod <lb/>e&longs;&longs;et omnino <expan abbr="inc&otilde;ueniens">inconueniens</expan>. Dixit autem Archimedes oporte<lb/>re has figuras e&longs;&longs;e &longs;imiles, &amp; &aelig;quales, nam figur&aelig; &aelig;quales, <lb/>&longs;ed non &longs;imiles, item &longs;imiles, &amp; <expan abbr="n&otilde;">non</expan> &aelig;quales e&longs;&longs;e po&longs;&longs;unt. </s>
          <s id="N10F28">qua&shy;<lb/>re, vt inter &longs;e&longs;e coaptari po&longs;&longs;int, &amp; &longs;imiles, &amp; &aelig;quales e&longs;&longs;e ne&shy;<lb/>ce&longs;&longs;e e&longs;t. </s>
        </p>
        <figure id="id.077.01.032.1.jpg" xlink:href="077/01/032/1.jpg"/>
        <p id="N10F32" type="head">
          <s id="N10F34">VI</s>
        </p>
        <p id="N10F36" type="main">
          <s id="N10F38">In&aelig; qualium autem, &longs;ed &longs;imilium centra graui&shy;<lb/>tatum e&longs;&longs;e &longs;imiliter po&longs;ita. </s>
        </p>
        <pb xlink:href="077/01/033.jpg" pagenum="29"/>
        <p id="N10F3F" type="head">
          <s id="N10F41">SCHOLIVM.</s>
        </p>
        <p id="N10F43" type="main">
          <s id="N10F45">In&aelig;quales &longs;int figur&aelig;, &longs;i&shy;<lb/>
<arrow.to.target n="fig12"/><lb/>miles ver&ograve; ABCD EFGH, <lb/>quarum c&eacute;tra grauitatis &longs;int <lb/>KL. &longs;upponit Archimedes <lb/>h&etail;c grauitatis centra KL e&longs;&shy;<lb/>&longs;e in figuris ABCD EFGH <lb/>&longs;imiliter po&longs;ita. c&ugrave;m enim <lb/>&longs;imilium figurarum, &amp; late&shy;<lb/>ra, &amp; &longs;pacia &longs;int &longs;imilia, nece&longs;&longs;e e&longs;t in ip&longs;is &longs;imili quo &que; mo&shy;<lb/>do centra grauitatis e&longs;&longs;e po&longs;ita. </s>
          <s id="N10F62">vt in &longs;e&que;nti clari&ugrave;s apparebit. <lb/>quomodo autem Archimedes intelligat hanc po&longs;itionis &longs;imi&shy;<lb/>litudinem, hoc modo definit. </s>
        </p>
        <figure id="id.077.01.033.1.jpg" xlink:href="077/01/033/1.jpg"/>
        <p id="N10F6C" type="head">
          <s id="N10F6E">VII.</s>
        </p>
        <p id="N10F70" type="main">
          <s id="N10F72">Dicimus quidem puncta in &longs;imilibus figuris e&longs;&shy;<lb/>&longs;e &longs;imiliter po&longs;ita, &agrave; quibus ad &aelig;quales angulos <lb/>duct&aelig; rect&aelig; line&aelig; cum homologis lateribus angu<lb/>los &aelig;quales efficiunt. </s>
        </p>
        <p id="N10F7A" type="head">
          <s id="N10F7C">SCHOLIVM.</s>
        </p>
        <p id="N10F7E" type="main">
          <s id="N10F80">In &longs;imilibus figuris ABCD EFGH &longs;int homologa latera <lb/>AB EF, BCFG, CD GH, AD EH. anguli ver&ograve; &aelig;quales, qui <lb/>ad AE, BF, CG, DH, primum quidem o&longs;tendendum e&longs;t fie<lb/>ri po&longs;&longs;e, ut &agrave; duobus punctis intra figuras con&longs;titutis, duci <lb/>po&longs;&longs;int rect&etail; line&etail; ad angulos &aelig;quales, qu&etail; cum lateribus an&shy;<lb/>gulos &etail;quales efficiant. </s>
          <s id="N10F8C">Qua&longs;i dicat Archimedes, quoniam <lb/>&longs;upponere po&longs;&longs;umus puncta in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter <lb/>po&longs;ita, ideo &longs;upponere quo&que; po&longs;&longs;umus centra grauitatis in <lb/>ip&longs;is e&longs;&longs;e &longs;imiliter po&longs;ita. </s>
          <s id="N10F94">Ita&que; &longs;int figur&aelig; ABCD EFGH &longs;i&shy;<lb/>miles, vt dictum e&longs;t, &longs;umaturqu&egrave; in ABCD vtcum&que; pun&shy;<lb/>ctum K &agrave; quo ducatur KA KB KC KD. deinde fiat an<pb xlink:href="077/01/034.jpg" pagenum="30"/><arrow.to.target n="fig13"/><lb/>gulus FEL angulo BAK &aelig;qualis; &amp; EFL ip&longs;i ABK. Iun<lb/>ganturqu&egrave; GL LH. Dico L e&longs;&longs;e &longs;imiliter po&longs;itum, vt K. <lb/>Quoniam enim anguli BAK ABK &longs;unt angulis FEL EFL <lb/>&aelig;quales, erit reliquus BKA ip&longs;i FLE &aelig;qualis, eritqu&egrave; ob &longs;i&shy;<lb/>
<arrow.to.target n="marg14"/>militudinem triangulorum KA ad AB, vt LE ad EF. e&longs;t <lb/>ver&ograve; AB ad AD, vt EF ad EH propter &longs;imilitudinem fi&shy;<lb/><arrow.to.target n="marg15"/>gurarum, erit igitur ex &aelig;quali AK ad AD, vt LE ad EH, <lb/>&amp; quoniam angulus BAD angulo FEH e&longs;t &aelig;qualis, &amp; BAK <lb/>ip&longs;i FEL &aelig;qualis; erit &amp; reliquus angulus KAD angulo <lb/>
<arrow.to.target n="marg16"/> LEH &aelig;qualis. </s>
          <s id="N10FC1">Quare triangulum KAD triangulo LEH &longs;i<lb/>mile exi&longs;tit, eodemqu&egrave; modo o&longs;tendetur BKG &longs;imile e&longs;&longs;e <lb/>FLG, &amp; KCD ip&longs;i LGH. ex quibus con&longs;tat angulos KBC <lb/>LFG, KCB LGF, &amp; huiu&longs;modi reliquos reliquis &aelig;quales e&longs;&longs;e. <lb/>&amp; ob id puncta KL in figuris ABCD EFGH e&longs;&longs;e &longs;imili&shy;<lb/>ter po&longs;ita. </s>
        </p>
        <p id="N10FCD" type="margin">
          <s id="N10FCF"><margin.target id="marg14"/>4 <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N10FD8" type="margin">
          <s id="N10FDA"><margin.target id="marg15"/>22 <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N10FE3" type="margin">
          <s id="N10FE5"><margin.target id="marg16"/>6 <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.034.1.jpg" xlink:href="077/01/034/1.jpg"/>
        <p id="N10FF2" type="main">
          <s id="N10FF4">Ita&que; demon&longs;trato dari po&longs;&longs;e puncta in figuris &longs;imiliter <lb/>po&longs;ita, potuit &longs;an&egrave; Archimedes antecedens po&longs;tulatum &longs;up&shy;<lb/>ponere, nemp&egrave; in&aelig;qualium, &longs;ed &longs;imilium figurarum centra <lb/>grauitatis e&longs;&longs;e &longs;imiliter po&longs;ita. </s>
          <s id="N10FFC">quod quidem po&longs;tulatum e&longs;t <lb/>rationi valde con&longs;entaneum. </s>
          <s id="N11000">ex dictis enim (&longs;uppo&longs;itis KL <lb/>centris grauitatum) triangulum ABK triangulo EFL &longs;imi&shy;<lb/>
<arrow.to.target n="marg17"/> le exi&longs;tit; veluti BKC ip&longs;i FLG. &amp; reliqua reliquis. </s>
          <s id="N1100A">Quare vt <lb/>AK ad KB, &longs;ic EL ad LF, ac permutando vt AK ad EL, <lb/>ita BK ad FL. &longs;imiliter o&longs;tendetur ita e&longs;&longs;e BK ad FL, vt <lb/>KC ad LG, &amp; KD ad LH. quare centra grauitatis KL
<pb xlink:href="077/01/035.jpg" pagenum="31"/>proportionaliter ab angulis di&longs;tant. </s>
        </p>
        <p id="N11016" type="margin">
          <s id="N11018"><margin.target id="marg17"/>4 <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <p id="N11028" type="main">
          <s id="N1102A"><expan abbr="Duc&atilde;tur">Ducantur</expan> pr&etail;terea &agrave; punctis KL ad latera perpendiculares <lb/>KM KN KO KP, LQ LR LS LT. &amp; quoniam anguli <lb/>KMA LQE &longs;unt recti, ac propterea &aelig;quales, &amp; KAM LEQ <lb/>&longs;unt &aelig;quales, ut o&longs;ten&longs;um e&longs;t; erit reliquus MKA reliquo <lb/>QLE &etail;qualis, triangulumqu&egrave; AKM triangulo ELQ &longs;imile. <lb/>vt igitur AK ad KM; &longs;ic EL ad <expan abbr="Lq.">L&que;</expan> &amp; permutando AK
<arrow.to.target n="marg18"/><lb/>ad EL, vt KM ad <expan abbr="Lq.">L&que;</expan> pariqu&egrave; ratione o&longs;tendetur triangu<lb/>lum BKM triangulo FLQ &longs;imile exi&longs;tere; e&longs;&longs;equ&egrave; BK ad <lb/>FL, vt KM ad <expan abbr="Lq.">L&que;</expan> &longs;imiliterqu&egrave; in alijs triangulis o&longs;ten&shy;<lb/>detur, ita e&longs;&longs;e Bk ad FL, vt KN ad LR; &amp; Ck ad GL e&longs;&longs;e, vt <lb/>kO ad LS; at&que; kD ad LH, vt kP ad LT. quia ver&ograve; AK <lb/>EL, Bk FL, Ck GL, Dk HL in eadem &longs;unt proportione, vt <lb/>proxim&egrave; demon&longs;tratum fuit; in eadem quo&que; proportione <lb/>erit kM ad LQ, &amp; KN ad LR; &amp; KO ad LS, at&que; kP ad <lb/>LT. ex quibus &longs;equitur centra grauitatis KL, non &longs;ol&ugrave;m ab <lb/>angulis in eadem proportione di&longs;tare; ver&ugrave;m etiam &agrave; late&shy;<lb/>ribus in eadem quo&que; proportione di&longs;tare. </s>
          <s id="N1105E">Ita&que; cognito, <lb/>quomodo intelligar Archimedes centra grauitatis in &longs;imili&shy;<lb/>bus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; nunc con&longs;iderandum e&longs;t pr&aelig; <lb/>cedens po&longs;tulatum, quatenus nimirum oporteat grauitatis <expan abbr="c&etilde;">cem</expan> <lb/>tra in &longs;imilibus figuris &longs;imiliter e&longs;&longs;e con&longs;tituta. </s>
          <s id="N1106C">Nam inti&shy;<lb/>mi&ugrave;s con&longs;iderando hanc &longs;imilem horum grauitatis <expan abbr="centror&utilde;">centrorum</expan> <lb/>po&longs;itionem, congruum, &amp; nece&longs;&longs;arium videtur, &longs;imiles figu&shy;<lb/>ras &longs;ecund&ugrave;m eandem proportionem e&longs;&longs;e &aelig;&que;pon<expan abbr="der&atilde;tes">derantes</expan>; <lb/>eademqu&egrave; ratione (ob earum &longs;imilitudinem) circa grauita&shy;<lb/>tis centra &aelig;&que;ponderare, veluti &longs;i figur&aelig;: AC EG (quarum <lb/>centra grauitatis &longs;int KL) &agrave; rectis lineis PN TR vtcumqu&egrave; <lb/>diuidantur, qu&aelig; per centra KL tran&longs;eant; dummodo in figu<lb/>ris &longs;int &longs;imiliter duct&aelig;; hoc e&longs;t, vel latera, vel angulos in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione di&longs;pe&longs;cant: vt &longs;it AP ad PD, vt ET ad TH. &aelig;&shy;<lb/>&que;ponderabunt vti&que; partes PABN PNCD, veluti partes <lb/>TEFR TRGH. &amp; h&aelig;c non e&longs;t &longs;implex &aelig;&que;ponderatio; ve&shy;<lb/>r&ugrave;m etiam (vt ita dicam) &longs;imilis, &amp; &aelig;qualis &aelig;&que;ponderatio. <lb/>c&ugrave;m &longs;it &longs;ecund&ugrave;m eandem proportionem, quandoquidem <lb/>e&longs;t PB ip&longs;i TF &longs;imilis, c&ugrave;m triangula AKB ELF, AKP ELT, <lb/>BKN FLR, &longs;int inter &longs;e &longs;imilia, qu&aelig; quidem efficiunt, figuras
<pb xlink:href="077/01/036.jpg" pagenum="32"/>PB TF inter &longs;e &longs;imiles e&longs;&longs;e. </s>
          <s id="N1109C">ob eademqu&egrave; cau&longs;am e&longs;t PC &longs;i&shy;<lb/>milis TG. quod quidem ex demon&longs;tratis etiam facil&egrave; con&shy;<lb/>&longs;tat. </s>
          <s id="N110A2">c&ugrave;m anguli &longs;int &etail;quales, &amp; latera proportionalia. </s>
          <s id="N110A4">Vt au&shy;<lb/>tem clari&ugrave;s intelligatur h&aelig;c &longs;imilis, &amp; &aelig;qualis &aelig;&que;pondera<lb/>rio, adducere libuit nonnulla ex ijs, qu&aelig; po&longs;teri&ugrave;s tractanda <lb/>&longs;umentur. </s>
          <s id="N110AC">Ita&que; intelligatur punctum V centrum e&longs;&longs;e gra&shy;<lb/>
<arrow.to.target n="fig14"/><lb/>uitatis figur&aelig; PB, X ver&ograve; centrum grauitatis figure TF. &longs;i<lb/>militer punctum Y centrum e&longs;&longs;e grauitatis figur&aelig; PC, Z <lb/>ver&ograve; figur&etail; TG. Iunganturqu&egrave; VY XZ. qu&aelig; quidem per <lb/>centra grauitatis KL tran&longs;ibunt. </s>
          <s id="N110BB">qu&ograve;d ex ijs, qu&etail; dicenda <lb/>&longs;unt, manife&longs;tum erit, percipu&egrave;&que; ex octaua proportione <lb/>primi huius. </s>
          <s id="N110C1">quod tamen interim &longs;upponatur. </s>
          <s id="N110C3">At ver&ograve; quo&shy;<lb/>niam PB PC &etail;&que;ponderant &longs;ecund&ugrave;m proportionem, <lb/>quam habet YK ad KV; TF ver&ograve; &amp; TG &etail;&que;ponderant <lb/>&longs;ecund&ugrave;m proportionem, quam habet ZL ad LX. e&longs;t. <expan abbr="n.">enim</expan> <lb/>ac &longs;i AN e&longs;&longs;et appen&longs;a in V, &amp; PC in Y; ER in X, &amp; <lb/>TG in Z. vt in &longs;e&que;ntibus manife&longs;ta erunt. </s>
          <s id="N110D3">Atver&ograve; quo&shy;<lb/>
<arrow.to.target n="marg19"/>niam AN &longs;imilis e&longs;t ip&longs;i ER, habebit AN ad ER <expan abbr="dupl&atilde;">duplam</expan> <lb/>proportionem eius, quam habet latus PN ad TR. pariqu&egrave; <lb/>ratione quoniam PC &longs;imilis e&longs;t TG, habebit PC ad TG <lb/>duplam proportionem eius, quam habet idem latus PN ad <lb/>
<arrow.to.target n="marg20"/> TR. quare ita &longs;e habet AN ad ER, ut PC ad TG. &amp; per&shy;<lb/>
<arrow.to.target n="marg21"/> mutando AN ad PC, vt ER ad TG. Sed vt AN ad PC, ita e&longs;t <lb/>Y K ad KV, &amp; vt ER ad TG. &longs;ic ZL ad LX. eandem igitur
<pb xlink:href="077/01/037.jpg" pagenum="33"/><expan abbr="proportion&etilde;">proportionem</expan> habebit YK ad KV, quam ZL ad LX. Quare <lb/>AN PC, &amp; ER TG &longs;ecund&ugrave;m eandem proportionem &aelig;&shy;<lb/>&que;ponderabunt. </s>
          <s id="N110FE">quod quidem contingit ex &longs;imilitudine fi&shy;<lb/>gurarum, &amp; ex centris grauitatum KL &longs;imiliter po&longs;itis, qu&etail; <lb/>quidem magnitudines, &longs;i non e&longs;&longs;ent &longs;imiles, diui&longs;&etail; <expan abbr="quid&etilde;">quidem</expan> per <lb/>centrum grauitatis, partes vti&que; &etail;&que;ponderarent; non ta&shy;<lb/>men &longs;emper &longs;ecund&ugrave;m eandem proportionem. </s>
          <s id="N11108">quod tamen <lb/>&longs;emper figuris &longs;imilibus (c&ugrave;m in ip&longs;is grauitatis centra &longs;int &longs;i <lb/>militer po&longs;ita) contingit; dummodo (vt dictum e&longs;t) diui&shy;<lb/>dantur. </s>
          <s id="N11110">Vnde con&longs;tat, quam &longs;it conueniens grauitatis centra <lb/>in figuris hac ratione e&longs;&longs;e con&longs;tituta. </s>
          <s id="N11114">ex quibus omnibus per<lb/>&longs;picuum e&longs;t, centra grauitatis debere in figuris &longs;imilibus e&longs;&longs;e &longs;i <lb/>militer po&longs;ita. </s>
          <s id="N1111A">vt Archimedes in <expan abbr="pr&etail;ced&etilde;ti">pr&etail;cedenti</expan> po&longs;tulato pr&etail;mi&longs;it. </s>
        </p>
        <p id="N1111C" type="margin">
          <s id="N1111E"><margin.target id="marg18"/>4 <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <p id="N1112E" type="margin">
          <s id="N11130"><margin.target id="marg19"/>20 <emph type="italics"/>&longs;exti<emph.end type="italics"/></s>
        </p>
        <p id="N11139" type="margin">
          <s id="N1113B"><margin.target id="marg20"/>11 <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <p id="N11144" type="margin">
          <s id="N11146"><margin.target id="marg21"/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.037.1.jpg" xlink:href="077/01/037/1.jpg"/>
        <p id="N11153" type="head">
          <s id="N11155">VIII.</s>
        </p>
        <p id="N11157" type="main">
          <s id="N11159">Si magnitudines ex &aelig;qualibus di&longs;tantijs &aelig;&que;&shy;<lb/>ponderant, &amp; ip&longs;is &aelig;quales ex ij&longs;dem di&longs;tantijs &aelig;&shy;<lb/>&que;ponderabunt. </s>
        </p>
        <p id="N1115F" type="head">
          <s id="N11161">SCHOLIVM.</s>
        </p>
        <p id="N11163" type="main">
          <s id="N11165">Hoc e&longs;t per&longs;picuum, <expan abbr="n&atilde;">nam</expan> <lb/>
<arrow.to.target n="fig15"/><lb/>&longs;i magnitudines AB ex di&shy;<lb/>&longs;tantijs CA CB &etail;&que;pon&shy;<lb/>derant: &longs;it autem D ip&longs;i A <lb/>&etail;qualis, &amp; E ip&longs;i B. <expan abbr="aufer&atilde;">auferam</expan> <lb/>turqu&egrave; magnitudines AB &agrave; <lb/>linea AB, ip&longs;arumqu&egrave; loco ponatur D in A, &amp; E in B, ma<lb/>gnitudines DE &longs;imiliter <expan abbr="&etail;&que;pond&etail;rab&utilde;t">&etail;&que;pond&etail;rabunt</expan>. qua ratione enim <lb/>magnitudines AB inter &longs;e&longs;e &etail;&que;ponderare dicuntur; eadem <lb/>pror&longs;us, &amp; magnitudines DE ex ij&longs;dem di&longs;tantijs &etail;&que;pon<lb/>derabunt. </s>
          <s id="N1118C">quandoquidem omnia data &longs;unt paria. </s>
          <s id="N1118E">illud ta&shy;<lb/>men non e&longs;t pretereundum, nimirum non oportere DE ip&longs;is <lb/>AB &etail;quales e&longs;&longs;e in magnitudine, &longs;ed in grauitate. </s>
          <s id="N11194">pote&longs;t enim
<pb xlink:href="077/01/038.jpg" pagenum="34"/>magnitudinum in&etail;qualium minor maiore grauior exi&longs;tere, <lb/>ob natur&aelig; diuer&longs;itatem, ac propterea c&ugrave;m inquit Archimedes <lb/><emph type="italics"/>&amp; ip&longs;is aquales<emph.end type="italics"/>, &longs;iue &longs;int magnitudine &aelig;quales, vel in&aelig;quales, in<lb/>telligendum e&longs;t e&longs;&longs;e omnino &aelig;quales in grauitate. </s>
          <s id="N111A5">grauitas. <expan abbr="n.">enim</expan> <lb/>cau&longs;a e&longs;t, vt magnitudines &aelig;&que;ponderare debeant. </s>
        </p>
        <figure id="id.077.01.038.1.jpg" xlink:href="077/01/038/1.jpg"/>
        <p id="N111B1" type="head">
          <s id="N111B3">VIIII,</s>
        </p>
        <p id="N111B5" type="main">
          <s id="N111B7">Omnis figur&aelig;, cuius perimeter &longs;it ad <expan abbr="eand&etilde;">eandem</expan> par<lb/>tem concauus, centrum grauitatis intra figuram <lb/>e&longs;&longs;e oportet. </s>
        </p>
        <p id="N111C1" type="head">
          <s id="N111C3">SCHOLIVM.</s>
        </p>
        <figure id="id.077.01.038.2.jpg" xlink:href="077/01/038/2.jpg"/>
        <p id="N111C8" type="main">
          <s id="N111CA">Quid intelligat Ar&shy;<lb/>chimedes per has figu&shy;<lb/>ras ad eandem partem <lb/>concauas, aperti&ugrave;s &longs;i&shy;<lb/>gnificauit initio libro&shy;<lb/>rum de &longs;ph&etail;ra, &amp; cylin&shy;<lb/>dro. </s>
          <s id="N111D8">vbi prim&ugrave;m vult <lb/>has figuras e&longs;&longs;e termina<lb/>tas; quod non &longs;ol&ugrave;m in<lb/>telligendum e&longs;t decur&shy;<lb/>uilineis, ver&ugrave;m etiam <lb/>de rectilineis, &amp; de mi&shy;<lb/>xtis. </s>
          <s id="N111E6">rectiline&etail; quidem <lb/>erunt trium, quattuor, <lb/>quin&que; &amp; plurium la&shy;<lb/>terum; quamuis latera <lb/>non &longs;int &aelig;qualia, ne&shy;<lb/>&que; anguli &etail;quales, vt
<pb xlink:href="077/01/039.jpg" pagenum="35"/>ABCDE, cuius omnes anguli &longs;unt flexi ad interiorem figur&aelig; <lb/>partem. </s>
          <s id="N111F8">&amp; hoc modo perimeter huius figur&aelig; erit ad eandem <lb/>partem concauus. </s>
          <s id="N111FC">vnde excluduntur figur&aelig;, exempli gratia <lb/>FGHKL; c&ugrave;m angulus K non &longs;it &longs;inuo&longs;us, &amp; concauus ad <lb/>eandem partem, vt reliqui anguli; qui &longs;unt &longs;inuo&longs;i ver&longs;us inte<lb/>riorem partem figur&etail; K vero ad exteriorem. </s>
          <s id="N11206">&longs;imili modo <lb/>intelligendum e&longs;t de curuilineis, vt circuli, ellip&longs;es, vel alterius <lb/>generis figur&aelig;, vt &longs;unt MN, qu&aelig; &longs;uam habent concauitatem <lb/>ad eandem partem: &longs;ed curuline&cedil; OP non &longs;unt ad eandem <lb/>partem concau&etail;. </s>
          <s id="N11214">Mixt&aelig; quo&que; figur&aelig;, ut &longs;unt portiones cir<lb/>culi, hyperbol&etail; ac parabol&etail; rectis linenis terminat&etail;, vel alte<lb/>rius generis figur&etail;, vt &longs;unt QR. h&etail; quidem omnes &longs;unt ad <expan abbr="e&atilde;dem">ean&shy;<lb/>dem</expan> partem concau&etail;. Mixt&aelig; ver&ograve; ST minim&egrave; Regulam au&shy;<lb/>tem quandam vniuer&longs;alem ex verbis Archimedis loco citato <lb/>elicere po&longs;&longs;umus, vt cogno&longs;cere valeamus, an figur&aelig; &longs;int ad <lb/>eandem partem concau&aelig;, vel min&ugrave;s vt &longs;cilicet in oblata figu<lb/>ra vbicum&que; duo &longs;umi po&longs;&longs;int puncta, qu&aelig; &longs;i recta linea <expan abbr="c&otilde;nectantur">con<lb/>nectantur</expan>, tota recta li<lb/><arrow.to.target n="fig16"/><lb/>nea, vel ip&longs;ius pars ali&shy;<lb/>qua extra figuram non <lb/>cadat. </s>
          <s id="N11239">vt in figuris A, <lb/>qu&aelig; &longs;unt ad <expan abbr="eand&etilde;">eandem</expan> par<lb/>tem concau&aelig;, vtcum&shy;<lb/>&que; duo &longs;umantur <expan abbr="p&utilde;-cta">pun&shy;<lb/>cta</expan> BC, qu&aelig; conne&shy;<lb/>ctantur, tota uti&que; re&shy;<lb/>cta linea inter puncta <lb/>BC exi&longs;tens, extra figu<lb/>ram non cadet. </s>
          <s id="N11253">Qu&ograve;d <lb/>&longs;i h&aelig;c linea cum termino, hoc e&longs;t eum latere figur&etail; conueni&shy;<lb/>ret, vt &longs;i figur&aelig; latus fuerit rectum, in quo duo &longs;umantur pun<lb/>cta, nihilominus recta linea inter h&aelig;c puncta extra figuram <lb/>non cadet: quandoquidem figur&aelig; terminus extra figuram mi<lb/>nim&egrave; reperitur at&que; hac ratione quomodocun&que;, &amp; <expan abbr="vbic&utilde;&que;">vbicum<lb/>&que;</expan> in his figuris duo &longs;umantur puncta, idem &longs;emper contin<lb/>get. </s>
          <s id="N11263">Quod tamen figuris D &longs;emper euenite non pote&longs;t in qui<lb/>bus (c&ugrave;m non &longs;int ad eandem partem concau&etail;) duo &longs;umere
<pb xlink:href="077/01/040.jpg" pagenum="36"/>po&longs;&longs;umus puncta EG, inter qu&etail; tota recta linea EG extra <lb/>figuram cadet. </s>
          <s id="N1126D">vel fumere po&longs;&longs;umus puncta FG, ita vt rect&etail; <lb/>line&etail; FG pars EG extra figuram cadat. </s>
          <s id="N11271">figur&etail; igitur, qu&aelig; <lb/>ad eandem partem &longs;unt concau&aelig;, ill&etail; &longs;unt, qu&etail; &longs;inuo&longs;itatem, <lb/>concauitatemqu&egrave; &longs;uam habent &longs;emper interiorem ip&longs;ius fi&shy;<lb/>gur&etail; partem re&longs;picientem. </s>
          <s id="N11279">Harum qu&egrave; rect&egrave; &longs;upponit Archi&shy;<lb/>medes centrum grauitatis &longs;emper e&longs;&longs;e intra ip&longs;am figuram. <lb/>ita vt ne&que; centrum e&longs;&longs;e po&longs;&longs;it in ambitu ip&longs;ius figur&etail; ete&shy;<lb/>nim &longs;i extra figuram, &longs;iue in ambitu ip&longs;ius e&longs;&longs;e po&longs;&longs;et, num&shy;<lb/>quam circa centrum grauitatis partes figur&etail; vndiqu&egrave; <expan abbr="&etail;&que;p&otilde;derarent">&etail;&que;pon<lb/>
<arrow.to.target n="marg22"/>derarent</expan>: ne&que; facta ex grauitatis centro &longs;u&longs;pen&longs;ione figura <lb/>vbicum&que;, &amp; in omni &longs;itu maneret. </s>
          <s id="N1128F">quod ramen ex ratione <lb/>centri grauitatis efficere deberet. </s>
          <s id="N11293">tota nimirum figura ex vna <lb/>e&longs;&longs;et parte, &amp; ex altera nihil e&longs;&longs;et, quod ip&longs;i figur&etail; &etail;&que;ponde<lb/>rare po&longs;&longs;et. </s>
          <s id="N11299">Nece&longs;&longs;e e&longs;t igitur centrum grauitatis cuiu&longs;libet fi&shy;<lb/>gur&etail; ad eandem partem concau&etail; e&longs;&longs;e in &longs;pacio &agrave; figur&etail; ambi<lb/>tu contento. </s>
          <s id="N1129F">vt figur&etail; AB <lb/>
<arrow.to.target n="fig17"/><lb/>centrum grauitatis erit in&shy;<lb/>tra ip&longs;am, put&agrave; in C. quod <lb/>quidem non euenit &longs;emper <lb/>in alijs figuris, qu&etail; &longs;uum <expan abbr="c&otilde;">com</expan> <lb/>cauitatis ambitum interio&shy;<lb/>rem figur&etail; partem <expan abbr="n&otilde;">non</expan> re&longs;pi&shy;<lb/>cientem habent. </s>
          <s id="N112BC">c&ugrave;m varijs <lb/>modis po&longs;&longs;it centrum graui<lb/>tatis in figuris e&longs;&longs;e <expan abbr="collocat&utilde;">collocatum</expan>. <lb/>vt &longs;uperius quo&que; diximus. <lb/>Nam figur&etail; D <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis erit extra ambitum fi<lb/>gur&etail;, vt in E. figura ver&ograve; F <lb/>ita &longs;e habere poterit, vt cen&shy;<lb/>trum grauitatis &longs;it in perime<lb/>tro, vt in G. euenit <expan abbr="aut&etilde;">autem</expan> aliquando vt in figura HK <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis L intra ip&longs;am figuram reperiatur; quamuis conca&shy;<lb/>uitates la torum interiorem partem minim&egrave; <expan abbr="re&longs;pici&atilde;t">re&longs;piciant</expan>. Sed h&etail;c <lb/>po&longs;&longs;unt e&longs;&longs;e, &amp; non e&longs;&longs;e, vt in figura M, cuius centrum extra <lb/>e&longs;&longs;e pote&longs;t in N. quamuis (vt antea diximus) centrum graui-
<pb xlink:href="077/01/041.jpg" pagenum="37"/>tatis intra figuram &longs;emper exi&longs;tere aliquo modo intelligi po&shy;<lb/>te&longs;t. </s>
        </p>
        <p id="N112F2" type="margin">
          <s id="N112F4"><margin.target id="marg22"/><emph type="italics"/>per def. <lb/><expan abbr="c&etilde;t">cent</expan>. grau.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.041.1.jpg" xlink:href="077/01/041/1.jpg"/>
        <figure id="id.077.01.041.2.jpg" xlink:href="077/01/041/2.jpg"/>
        <p id="N11309" type="main">
          <s id="N1130B">Refert Eutocius hoc loco, Geminum rect&egrave; dicere, dum a&longs;&longs;e&shy;<lb/>rit Archimedem dignitates petitiones apellare. </s>
          <s id="N1130F">&aelig;qualia enim <lb/>grauia ex di&longs;tantijs &aelig;qualibus &aelig;&que;ponderare, dignitas eft; &amp; <lb/>qu&aelig; deinceps. <expan abbr="Ver&utilde;">Verum</expan> &longs;i h&aelig;c principia ab Archimede tradita re<lb/>ct&egrave; perpendamus, omnia dignitates e&longs;&longs;e minim&egrave; reperiemus. <lb/>nam &longs;eptimum po&longs;tulatum e&longs;t definitio, non dignitas. </s>
          <s id="N1131D">veluti <lb/>alia forta&longs;&longs;e nonnulla non &longs;unt dignitates, vt &longs;ecundum; quod <lb/>aliquo modo probari pote&longs;t, vt diximus. </s>
          <s id="N11323">&longs;extum quo&que; po&shy;<lb/>ti&ugrave;s e&longs;t &longs;uppo&longs;ito, qu&agrave;m dignitas. </s>
          <s id="N11327">Quoniam autem vt clar&egrave; <lb/>con&longs;picitur Archimedes &longs;ub vno tant&ugrave;m titulo pauca h&aelig;c <lb/>principia complecti voluit; quippe quod in&longs;titutum qu&agrave;m plu<lb/>rimis mathematicis &longs;olemne fuit, qui principia vnico tantum <lb/>nomine nuncuparunt, mod&ograve; vno, mod&ograve; altero; nimirum, <lb/>vel petitionis, vel dignitatis, vt refert Proclus &longs;ecundo libro, &amp; <lb/>tertio &longs;uorum commentariorum in primum elementorum. </s>
          <s id="N11335">Eu<lb/>clidis; qui de Archimede peculiariter mentionem faciens, in&shy;<lb/>quit illum in his libris principia vnico tant&ugrave;m nomine (peti&shy;<lb/>tionis &longs;cilicet) nuncupa&longs;&longs;e. </s>
          <s id="N1133D">H&aelig;c tamen poti&ugrave;s petitionum, <lb/>qu&agrave;m definitionum, vel dignitatum nomine nuncupare vo&shy;<lb/>luit; nam &longs;i dignitates appella&longs;&longs;et; ea principia, qu&aelig; non &longs;unt <lb/>dignitates, inter dignitates mal&egrave; colloca&longs;&longs;et. </s>
          <s id="N11345">nulla quipp&egrave; defi&shy;<lb/>nitio dignitas dici debet; quandoquidem definitio terminos <lb/>declarat, at&que; con&longs;tituit. </s>
          <s id="N1134B">dignitas ver&ograve; notos terminos copu&shy;<lb/>lat. </s>
          <s id="N1134F">Pariqu&egrave; ratione &longs;i definitionis nomine h&aelig;c principia nun<lb/>cupa&longs;&longs;et. </s>
          <s id="N11353">dignitates mal&egrave; &longs;ub hoc nomine complexus fui&longs;&longs;et, <lb/>qu&aelig; nullo modo rem definiunt, &longs;ed c&ugrave;m &longs;int communes no<lb/>tiones, &longs;tatim c&ugrave;m eas intellectus apprehendit, quie&longs;cit. </s>
          <s id="N11359">Qua&shy;<lb/>re omnia &longs;ub petitionum nomine recte collocauit, non e&longs;t. <expan abbr="n.">enim</expan> <lb/>ab&longs;urdum dignitates, definitione&longs;qu&egrave; po&longs;&longs;e apellari petitio&shy;<lb/>nes. </s>
          <s id="N11365">etenim petimus, qu&aelig; &longs;unt concedenda, at&que; dignitates <lb/>&longs;unt concedend&etail;, ergo eas petere quo&que; po&longs;&longs;umus. </s>
          <s id="N11369">Definitio<lb/>nibus ver&ograve; rect&egrave; quo&que; hoc nomen conuenire pote&longs;t. </s>
          <s id="N1136D">Nam <lb/>d&ugrave;m definitio terminos con&longs;tituat, at&que; declaret, cur non pe<lb/>tere po&longs;&longs;umus, terminos &longs;ic &longs;e habere, vel &longs;ice&longs;&longs;e rect&egrave; definitos? <lb/>vt exempli gratia, petit Archimedes puncta in figuris fimiliter
<pb xlink:href="077/01/042.jpg" pagenum="38"/>po&longs;ita, ita &longs;ehabere, vt &longs;unt ab ip&longs;o definita, vel rect&egrave; e&longs;&longs;e defi&shy;<lb/>nita puncta, qu&aelig; &longs;unt in figuris &longs;imilibus po&longs;ita. </s>
          <s id="N1137B">Quapropter <lb/>h&aelig;c principia, quoniam pauca &longs;unt, &longs;ub petitionum nomine <lb/>Archimedes rect&egrave; collocauit. </s>
          <s id="N11381">qu&ograve;d &longs;i multa extiti&longs;&longs;ent, ea for<lb/>ta&longs;&longs;e di&longs;tinxi&longs;&longs;et. </s>
        </p>
        <p id="N11385" type="main">
          <s id="N11387"><emph type="italics"/>His &longs;uppo&longs;itis.<emph.end type="italics"/> <expan abbr="po&longs;tqu&atilde;">po&longs;tquam</expan> Archimedes <expan abbr="pr&itilde;cipia">principia</expan> po&longs;uit, ad theore<lb/>mata &longs;e conuertit, &amp; inquit, <emph type="italics"/>his &longs;uppo&longs;itis<emph.end type="italics"/>, qua&longs;i dicat, ea, qu&aelig; <lb/>po&longs;uimus, &longs;ufficiunt ad o&longs;tendenda theoremata, veluti. </s>
        </p>
        <p id="N113A0" type="head">
          <s id="N113A2">PROPOSITIO. I.</s>
        </p>
        <p id="N113A4" type="main">
          <s id="N113A6">Grauia, qu&aelig; ex &aelig;qualibus di&longs;tantijs &aelig;&que;pon&shy;<lb/>derant, &aelig;qualia &longs;unt. </s>
        </p>
        <p id="N113AA" type="main">
          <s id="N113AC">Sint AD, &amp; B grauia, <lb/>
<arrow.to.target n="fig18"/><lb/>qu&aelig; ex &aelig;qualibus di&longs;tantijs <lb/>CA CB &aelig;&que;ponderent. </s>
          <s id="N113B7">di<lb/>co grauia AD, &amp; B inter&shy;<lb/>&longs;e&longs;e &aelig;qualia e&longs;&longs;e. <emph type="italics"/>&longs;i enim<emph.end type="italics"/> (&longs;i fie<lb/>ri pote&longs;t) <emph type="italics"/>e&longs;&longs;ent in&aelig;qualia<emph.end type="italics"/>; vt &longs;i <lb/>AD e&longs;&longs;et grauius, qu&agrave;m B, <lb/>&longs;it D exce&longs;&longs;us, quo AD grauius e&longs;t, qu&agrave;m B. <emph type="italics"/>ablato<emph.end type="italics"/> ita&que; <lb/><emph type="italics"/>exce&longs;&longs;u<emph.end type="italics"/> D <emph type="italics"/>&agrave; maiori<emph.end type="italics"/> AD, <emph type="italics"/>reliqua<emph.end type="italics"/> grauia, qu&aelig; relinquuntur AB, <lb/>
<arrow.to.target n="marg23"/> erunt inter &longs;e &etail;qualia; qu&etail; ex &etail;qualibus di&longs;tantijs CA CB &aelig;&shy;<lb/>&que;ponderare deberent; tamen <emph type="italics"/>non &aelig;&que;ponderabunt. </s>
          <s id="N113F3">c&ugrave;m<emph.end type="italics"/> enim <lb/>po&longs;itum &longs;it AD B &etail;&que;ponderare, &amp; <emph type="italics"/>ab altero a&que;ponderan-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg24"/> <emph type="italics"/>tium<emph.end type="italics"/> AD <emph type="italics"/>aliquod &longs;it ablatum<emph.end type="italics"/> D; reliqua grauia AB ex &etail;qua<lb/>libus di&longs;tantijs CA CB non &etail;&que;ponderabunt quod fieri <lb/>non pote&longs;t; &longs;iquidem AB inter &longs;e &longs;unt &etail;qualia. <emph type="italics"/>Grauia igitur, <lb/>qu&aelig; ex &aelig;qualibus <expan abbr="dist&atilde;tijs">distantijs</expan> &aelig;&que;ponderant, &aelig;qualia &longs;unt.<emph.end type="italics"/> quod de&shy;<lb/>mon&longs;trare oportebat. </s>
        </p>
        <p id="N11423" type="margin">
          <s id="N11425"><margin.target id="marg23"/>4. <emph type="italics"/>po&longs;tula&shy;<lb/>tum huius<emph.end type="italics"/></s>
        </p>
        <p id="N11430" type="margin">
          <s id="N11432"><margin.target id="marg24"/><emph type="italics"/>contra pri&shy;<lb/>mum post <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.042.1.jpg" xlink:href="077/01/042/1.jpg"/>
        <p id="N11442" type="head">
          <s id="N11444">SCHOLIVM.</s>
        </p>
        <p id="N11446" type="main">
          <s id="N11448">C&ugrave;m &longs;it &longs;copus Archimedis (vt diximus) in primis octo <lb/>theorematibus, fundamentum tradere in hac &longs;cientia pr&aelig;ci-
<pb xlink:href="077/01/043.jpg" pagenum="39"/>puum, nemp&egrave; magnitudinum grauitates inter &longs;e ita &longs;e habe&shy;<lb/>re, vt di&longs;tanti&aelig; permutatim ex quibus &longs;u&longs;penduntur &longs;e <expan abbr="hab&etilde;t">habent</expan>. <lb/>prim&ugrave;m incipit o&longs;tendere, quomodo &longs;e habeant grauia in di<lb/>&longs;tantijs &etail;qualibus po&longs;ita; prim&ugrave;mqu&egrave; in hac prima propo&longs;itio <lb/>ne o&longs;tendit, &longs;i grauia &etail;&que;ponderant ex di&longs;tantijs &etail;qualibus, <lb/>&etail;qualia e&longs;&longs;e. </s>
          <s id="N1145E">in &longs;e&que;nti ver&ograve;, &longs;i grauia &longs;unt in&etail;qualia, ex di&shy;<lb/>&longs;tantijs &etail;qualibus nullo modo &aelig;&que;ponderare o&longs;tendet; &longs;ed <lb/>pr&aelig;ponderare ad maius. </s>
        </p>
        <p id="N11464" type="head">
          <s id="N11466">PROPOSITIO. II.</s>
        </p>
        <p id="N11468" type="main">
          <s id="N1146A">In&aelig;qualia grauia ex &aelig;qualibus di&longs;tantijs non <lb/>&aelig;&que;ponderabunt, &longs;ed pr&aelig;ponderabit ad maius. </s>
        </p>
        <figure id="id.077.01.043.1.jpg" xlink:href="077/01/043/1.jpg"/>
        <p id="N11471" type="main">
          <s id="N11473">Sint gra&shy;<lb/>uia in&etail;qua&shy;<lb/>lia AB C in <lb/>di&longs;tantijs <expan abbr="&etail;&shy;qualib^&lcub;9}">&etail;&shy;<lb/>qualibus</expan> DA <lb/>DC. &longs;itqu&egrave; <lb/>grauius AB, <lb/>qu&agrave;m C. di<lb/>co grauia AB C non &etail;&que;ponderare, &longs;ed maius AB <expan abbr="deors&utilde;">deorsum</expan> <lb/>ferri. </s>
          <s id="N1148B">&longs;it B exce&longs;&longs;us, quo AB &longs;uperat C. <emph type="italics"/>ablato<emph.end type="italics"/> ita&que; &agrave; ma<lb/>iori AB <emph type="italics"/>exce&longs;&longs;u<emph.end type="italics"/> B, reliqua grauia AC &etail;qualia ex di&longs;tantijs <lb/>DA DC <emph type="italics"/>&aelig;&que;ponderabunt. </s>
          <s id="N114A0">c&ugrave;m &aelig;qualia grauia ex distantiis &aelig;quali-<emph.end type="italics"/>
<arrow.to.target n="marg25"/><lb/><emph type="italics"/>bus &aelig;&que;ponderent.<emph.end type="italics"/> &longs;i ita&que; grauia AC &etail;&que;ponderant, <emph type="italics"/>adiecto <lb/>igitur<emph.end type="italics"/> ip&longs;i A <emph type="italics"/>ablato<emph.end type="italics"/> B, <emph type="italics"/>pr&aelig;ponderabit ad maius<emph.end type="italics"/>, hoc e&longs;t ab deor
<arrow.to.target n="marg26"/><lb/>&longs;um tendet. <emph type="italics"/>quoniam &aelig;&que;ponderantium altero<emph.end type="italics"/> nemp&egrave; A <emph type="italics"/>adiectum <lb/>fuit<emph.end type="italics"/> B. Grauius igitur pr&aelig;ponderat leuiori, ambobus in <expan abbr="di&longs;t&atilde;tijs">di&longs;tan<lb/>tijs</expan> &etail;qualibus po&longs;itis. </s>
          <s id="N114DC">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N114DE" type="margin">
          <s id="N114E0"><margin.target id="marg25"/>1 <emph type="italics"/>po&longs;t hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N114EB" type="margin">
          <s id="N114ED"><margin.target id="marg26"/>3 <emph type="italics"/>post hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N114F8" type="head">
          <s id="N114FA">SCHOLIVM.</s>
        </p>
        <p id="N114FC" type="main">
          <s id="N114FE">H&aelig;c duo theoremata in gr&etail;co exemplari impre&longs;&longs;o &longs;equun<lb/>tur <expan abbr="quid&etilde;">quidem</expan> po&longs;tulata, &amp; reliquis theorematibus &longs;unt pr&etail;po&longs;ita.
<pb xlink:href="077/01/044.jpg" pagenum="40"/>quia ver&ograve; inter principia collocari non po&longs;&longs;unt; c&ugrave;m &longs;uas ha&shy;<lb/>beant propo&longs;itiones, &longs;ua&longs;qu&egrave; &longs;eor&longs;um habeant demon&longs;tratio&shy;<lb/>nes, ideo inter propo&longs;itiones ip&longs;a collocare nobis vi&longs;um e&longs;t. <lb/>c&ugrave;m pr&etail;&longs;ertim nonnulla ex &longs;e&que;ntibus theorematibus, po&shy;<lb/>ti&longs;&longs;im&ugrave;m ver&ograve; proximum eiu&longs;dem cum his duobus ordinis, <lb/>&amp; natur&aelig; &longs;int. </s>
          <s id="N11514">Ne&que; enim propterea peruertitur ordo; non <lb/>enim h&etail; propo&longs;itiones in alium transferuntur locum. </s>
          <s id="N11518">&longs;ed <expan abbr="t&atilde;-t&ugrave;m">tan&shy;<lb/>t&ugrave;m</expan> inter alias numeris adnotantur. </s>
          <s id="N11520">exi&longs;timandum enim e&longs;t, <lb/>Archimedem propo&longs;itiones in &longs;erie propo&longs;itionum colloca&longs;&shy;<lb/>&longs;e. </s>
          <s id="N11526">hanc ver&ograve; exiguam mutationem accidi&longs;&longs;e <expan abbr="oblongitudin&etilde;">oblongitudinem</expan> <lb/>temporis; cuius proprium e&longs;t, res poti&ugrave;s de&longs;truere, qu&agrave;m ac&shy;<lb/>comodare. </s>
          <s id="N11530">Hoc autem nobis hanc pr&aelig;bebit commoditatem, <lb/>vt, quando libuerit, has propo&longs;itiones numeris nominare <lb/>po&longs;&longs;imus. </s>
          <s id="N11536">id ip&longs;umqu&egrave; numeri po&longs;tulata di&longs;tinguentes pr&aelig;&shy;<lb/>&longs;tant, quamuis in Gr&etail;co codice po&longs;tulata (Gr&etail;corum more) <lb/>numeris adnotata non &longs;int. </s>
        </p>
        <p id="N1153C" type="head">
          <s id="N1153E">PROPOSITIO. III.</s>
        </p>
        <p id="N11540" type="main">
          <s id="N11542">In&aelig;qualia grauia ex di&longs;tantijs in&aelig;qualibus &aelig;&shy;<lb/>
<arrow.to.target n="marg27"/> &que;ponderabunt, maius quidem ex minori. </s>
        </p>
        <p id="N1154A" type="margin">
          <s id="N1154C"><margin.target id="marg27"/>A</s>
        </p>
        <figure id="id.077.01.044.1.jpg" xlink:href="077/01/044/1.jpg"/>
        <p id="N11553" type="main">
          <s id="N11555"><emph type="italics"/>Sint in &aelig;qualia grauia AD, B<emph.end type="italics"/>; <lb/>
<arrow.to.target n="marg28"/> <emph type="italics"/>&longs;it qu&egrave; maius AD<emph.end type="italics"/>, exce&longs;&longs;us ve <lb/>r&ograve;, quo AD &longs;uperat B, &longs;it <lb/>D. <emph type="italics"/><expan abbr="&aelig;&que;p&otilde;derentqu&egrave;">&aelig;&que;ponderentqu&egrave;</expan><emph.end type="italics"/> AD B <emph type="italics"/>ex <lb/>di&longs;tantiis AC C B. o&longs;tendendum <lb/>e&longs;t, minorem e&longs;&longs;e<emph.end type="italics"/> <expan abbr="diftanti&atilde;">diftantiam</expan> <emph type="italics"/>AC <lb/>ip&longs;a CB. Non &longs;it quidem, &longs;i fie&shy;<lb/>ri potest<emph.end type="italics"/>, AC minor, qu&agrave;m CB; erit nimirum, vel &etail;qualis, <lb/>vel maior. </s>
          <s id="N1158E">Qu&ograve;d &longs;i AC fuerit &etail;qualis ip&longs;i CB, <emph type="italics"/>ablato enim <lb/>exce&longs;&longs;u<emph.end type="italics"/> D, <emph type="italics"/>quo AD &longs;uperat B. c&ugrave;m ab a&que;ponderantium altero ab<emph.end type="italics"/><lb/>
<arrow.to.target n="marg29"/> <emph type="italics"/>latum &longs;it aliquid<emph.end type="italics"/>, grauia AB non &aelig;&que;ponderabunt; &longs;ed <emph type="italics"/>pr&aelig;-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg30"/> <emph type="italics"/>ponderabit ad B. non pr&aelig;ponderabit autem; exi&longs;tente enim AC aqua <lb/>li CB<emph.end type="italics"/>, c&ugrave;m ab in&etail;qualibus grauibus AD B ablatus &longs;it ex&shy;<lb/>ce&longs;&longs;us D, <emph type="italics"/>grauia<emph.end type="italics"/>, qu&aelig; relinquuntur AB, erunt inter &longs;e <emph type="italics"/>&aelig;qualia<emph.end type="italics"/>;
<pb xlink:href="077/01/045.jpg" pagenum="41"/>qu&aelig; <emph type="italics"/>ex di&longs;tantiis &aelig;qualibus<emph.end type="italics"/> AC CB <emph type="italics"/>&aelig;&que;ponderarent.<emph.end type="italics"/> at non &etail;&que; <lb/>ponderant, quod e&longs;t ab&longs;urdum. </s>
          <s id="N115DE">di&longs;tantia igitur AC ip&longs;i CB <lb/>&aelig;qualis e&longs;&longs;e non pote&longs;t. <emph type="italics"/>&longs;i uer&ograve; AC maior fuerit CB<emph.end type="italics"/>; ablato &longs;i&shy;<lb/>militer exce&longs;&longs;u D, nihilominus &etail;qualia grauia AB non &etail;&que; <lb/>ponderabunt, &longs;ed <emph type="italics"/>inclinabitur ad A. &aelig;qualia enim grauia<emph.end type="italics"/> AB <emph type="italics"/>ex <emph.end type="italics"/>
<arrow.to.target n="marg31"/><lb/><emph type="italics"/>distantiis in&aelig;qualibus non &aelig;&que;ponderant, &longs;ed inclinatur ad maiorem <lb/>distantiam<emph.end type="italics"/> AC. ergo totum AD mult&ograve; magis pr&aelig;ponderabit, <lb/>qu&agrave;m B. quod fieri non pote&longs;t. </s>
          <s id="N11609">po&longs;ita enim &longs;unt &aelig;&que;ponde<lb/>rare. </s>
          <s id="N1160D">Quare AC maior e&longs;&longs;e non pote&longs;t, qu&agrave;m CB. &longs;ed o&longs;ten&longs;a <lb/>e&longs;t, ne&que; ip&longs;i CB &aelig;qualis e&longs;&longs;e: <emph type="italics"/>ac propterea minor e&longs;t AC, qu&agrave;m <lb/>CB. Manifestum e&longs;t ita&que; grauia ex distantiis in&aelig;qualibus &aelig;&que;pon&shy;<lb/>derantia, in&aelig;qualia e&longs;&longs;e; maiu&longs;qu&egrave; in minori<emph.end type="italics"/> di&longs;tantia <emph type="italics"/>existere.<emph.end type="italics"/> quod <lb/>oportebat demon&longs;trare. </s>
        </p>
        <p id="N11623" type="margin">
          <s id="N11625"><margin.target id="marg28"/>B</s>
        </p>
        <p id="N11629" type="margin">
          <s id="N1162B"><margin.target id="marg29"/>4 <emph type="italics"/>post hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N11636" type="margin">
          <s id="N11638"><margin.target id="marg30"/>1 <emph type="italics"/>po&longs;t hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N11643" type="margin">
          <s id="N11645"><margin.target id="marg31"/>2 <emph type="italics"/>post hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N11650" type="head">
          <s id="N11652">SCHOLIVM.</s>
        </p>
        <p id="N11654" type="main">
          <s id="N11656">In propo&longs;itione verba illa, <emph type="italics"/>maius quidem ex minori<emph.end type="italics"/>, non <expan abbr="hab&etilde;tur">haben<arrow.to.target n="marg32"/><lb/>tur</expan> integra in codice gr&aelig;co, qui &longs;ic habet, <foreign lang="greek">kai\ to/ a)po\ to_n e)la/ssonos</foreign><lb/>vbi de&longs;iderari videtur <foreign lang="greek">me/izon</foreign>, vt integr&egrave; ita legatur, <foreign lang="greek">kai\ to\ mei/zon <lb/>a)po\ tou_ e)la/ssonos.</foreign></s>
        </p>
        <p id="N11676" type="margin">
          <s id="N11678"><margin.target id="marg32"/>A</s>
        </p>
        <p id="N1167C" type="main">
          <s id="N1167E"><emph type="italics"/>Sitqu&egrave; maius A.<emph.end type="italics"/> Gr&aelig;cus codex, <foreign lang="greek">kai\ e)/sw to\ a</foreign>, vbi &longs;imiliter
<arrow.to.target n="marg33"/> &longs;up&shy;<lb/>plendum e&longs;t, <foreign lang="greek">kai\ e)/sw mei/zon to\ a</foreign> H&aelig;c ver&ograve; ita &longs;unt omnino re&longs;ti<lb/>tuenda, quia in vltima demon&longs;trationis conclu&longs;ione inquit <lb/>Archimedes, <emph type="italics"/>Manife&longs;tum est ita&que; grauia ex di&longs;tantiis in&aelig;qualibus <lb/>&aelig;&que;ponderantia in&aelig;qualia e&longs;&longs;e; maiu&longs;qu&egrave; in minori existere.<emph.end type="italics"/></s>
        </p>
        <p id="N1169E" type="margin">
          <s id="N116A0"><margin.target id="marg33"/>B</s>
        </p>
        <p id="N116A4" type="main">
          <s id="N116A6"><expan abbr="Po&longs;tqu&atilde;">Po&longs;tquam</expan> Archimedes <expan abbr="duab^&lcub;9}">duabus</expan> primis <expan abbr="po&longs;itionib^&lcub;9}">po&longs;itionibus</expan> <expan abbr="o&longs;t&etilde;dit">o&longs;tendit</expan>, <expan abbr="q&utilde;o">quno</expan> <lb/>&longs;e <expan abbr="h&etilde;ant">henant</expan> grauia ex <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> <expan abbr="&etail;qualib^&lcub;9};">&etail;qualibus</expan> in hac tertia <expan abbr="c&otilde;uerti&longs;&longs;e">conuertit&longs;e</expan> ad <lb/><expan abbr="o&longs;t&etilde;d&etilde;d&utilde;">o&longs;tendendum</expan>, <expan abbr="q&utilde;o">quno</expan> &longs;e <expan abbr="h&etilde;nt">hennt</expan> ex <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> <expan abbr="in&etail;qualib^&lcub;9}">in&etail;qualibus</expan>. &amp; <expan abbr="q&mtilde;">quem</expan> in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> <lb/>po&longs;tulato <expan abbr="a&longs;s&utilde;p&longs;it">a&longs;sump&longs;it</expan>, <expan abbr="q&utilde;o">quno</expan> &longs;e <expan abbr="h&etilde;nt">hennt</expan> grauia &etail;qualia in <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> in &etail;&shy;<lb/>qualibus <expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan>; <expan abbr="nimir&utilde;">nimirum</expan> <expan abbr="qd">quod</expan> e&longs;t in <expan abbr="l&otilde;giori">longiori</expan> <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan>, <expan abbr="pr&etail;p&otilde;de-rat">pr&etail;ponde&shy;<lb/>rat</expan> ei, <expan abbr="qd">quod</expan> e&longs;t in breuiori. <expan abbr="n&utilde;c">nunc</expan> <expan abbr="o&longs;t&etilde;dit">o&longs;tendit</expan>, <expan abbr="q&utilde;o">quno</expan> in&etail;qualia grauia &longs;e <lb/><expan abbr="h&etilde;nt">hennt</expan>, ita vt <expan abbr="&etail;&que;p&otilde;der&etilde;t">&etail;&que;ponderent</expan>, in <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> in &etail;qualibus po&longs;ita. <expan abbr="dem&otilde;">demom</expan> <lb/>&longs;tratqu&egrave; graue maius in breuiori <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> <expan abbr="e&etilde;">eem</expan> oportere, <expan abbr="min^&lcub;9}">minus</expan> ve&shy;<lb/>r&ograve; graue in <expan abbr="l&otilde;giori">longiori</expan>. &amp; ecce quomodo Archimedes <expan abbr="paulat&itilde;">paulatim</expan> de <lb/>ducit nos in <expan abbr="cognition&etilde;">cognitionem</expan> principalis <expan abbr="fundam&etilde;ti">fundamenti</expan>, <expan abbr="qd">quod</expan> &longs;cilicet gra<lb/>ue ad graue e&longs;t, vt <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> ad <expan abbr="di&longs;t&atilde;ti&atilde;">di&longs;tantiam</expan> <expan abbr="pmutatim">permutatim</expan>. </s>
          <s id="N11749">Ex hoc. <expan abbr="n.">enim</expan> pri<lb/>m&ugrave;m cogno&longs;cimus grauius in minori, leuius <expan abbr="aut&etilde;">autem</expan> in maiori <lb/>di&longs;tantia e&longs;&longs;e debere, &longs;i &etail;&que;ponderare debent. </s>
        </p>
        <pb xlink:href="077/01/046.jpg" pagenum="42"/>
        <p id="N1175A" type="head">
          <s id="N1175C">PROPOSITIO. IIII.</s>
        </p>
        <p id="N1175E" type="main">
          <s id="N11760">Si due magnitudines &aelig;quales non idem <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis habuerint, magnitudinis ex vtri&longs;&que; <lb/>magnitudinibus compo&longs;it&aelig; centrum grauitatis <lb/>erit medium rect&aelig; line&aelig; grauitatis centra magni<lb/>tudinum coniungentis. </s>
        </p>
        <p id="N1176E" type="main">
          <s id="N11770"><emph type="italics"/>Sit <expan abbr="quid&etilde;">quidem</expan> A <emph.end type="italics"/><lb/>
<arrow.to.target n="fig19"/><lb/><emph type="italics"/><expan abbr="centr&utilde;">centrum</expan> grauita<lb/>tis magnitudi&shy;<lb/>nis A. B uer&ograve; <emph.end type="italics"/><lb/>&longs;it <expan abbr="c&etilde;tr&utilde;">centrum</expan> gra&shy;<lb/>uitatis <emph type="italics"/>magni&shy;<lb/>tudinis B iun&shy;<lb/>staqu&egrave; AB bifariam diuidatur in C. dico magnitudinis ex utri&longs;qu&egrave; ma&shy;<lb/>gnitudinibus compo&longs;it&aelig; centrum<emph.end type="italics"/> grauitatis <emph type="italics"/>e&longs;&longs;e punctum C. &longs;i. <expan abbr="n.">enim</expan> non; &longs;it <lb/>utrarumqu&egrave; magnitudinum AB centrum grauitatis D, &longs;i fieri <expan abbr="p&otilde;t">potest</expan>. Qu&ograve;d <lb/>autem &longs;it in linea AB, pr&aelig;o&longs;ten&longs;um est. </s>
          <s id="N117AF">Quoniam igitur punstum D <expan abbr="c&etilde;">cem</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="marg34"/> <emph type="italics"/><expan abbr="tr&utilde;">trum</expan> e&longs;t grauitatis magnitudinis ex AB <expan abbr="c&otilde;po&longs;it&aelig;">compo&longs;it&aelig;</expan>, <expan abbr="&longs;u&longs;p&etilde;&longs;o">&longs;u&longs;pen&longs;o</expan> <expan abbr="p&utilde;cto">puncto</expan> D<emph.end type="italics"/>, magni<lb/>tudines AB <emph type="italics"/>&aelig;&que;ponderabunt. </s>
          <s id="N117D6">magnitudines igitur AB<emph.end type="italics"/> &etail;quales <emph type="italics"/>&aelig;&que; <lb/>ponderant ex di&longs;tantiis AD DB<emph.end type="italics"/> in &etail;qualibus exi&longs;tentibus; <emph type="italics"/>quod fie<emph.end type="italics"/><lb/>
<arrow.to.target n="marg35"/> <emph type="italics"/>ri non pote&longs;t. </s>
          <s id="N117F1">&aelig;qualia. <expan abbr="n.">enim</expan><emph.end type="italics"/> grauia <emph type="italics"/>ex di&longs;tantiis in a qualibus non <expan abbr="&aelig;&que;p&otilde;de-r&atilde;t">&aelig;&que;ponde&shy;<lb/>rant</expan>.<emph.end type="italics"/> <expan abbr="N&otilde;">non</expan> e&longs;t igitur D <expan abbr="ip&longs;ar&utilde;">ip&longs;arum</expan> <expan abbr="magnitudin&utilde;">magnitudinum</expan> <expan abbr="c&etilde;tr&utilde;">centrum</expan> grauitatis.. <emph type="italics"/>Qua <lb/>re manifestum est punstum C <expan abbr="centr&utilde;">centrum</expan> e&longs;&longs;e grauitatis magnitudinis ex AB <lb/>compo&longs;it&aelig;.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N11823" type="margin">
          <s id="N11825"><margin.target id="marg34"/><emph type="italics"/>def. </s>
          <s id="N1182B">centri <lb/>grauit. <lb/>contra 2. <lb/>post huius<emph.end type="italics"/></s>
        </p>
        <p id="N11835" type="margin">
          <s id="N11837"><margin.target id="marg35"/>2 <emph type="italics"/>post hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.046.1.jpg" xlink:href="077/01/046/1.jpg"/>
        <p id="N11846" type="head">
          <s id="N11848">SCHOLIVM.</s>
        </p>
        <figure id="id.077.01.046.2.jpg" xlink:href="077/01/046/2.jpg"/>
        <p id="N1184D" type="main">
          <s id="N1184F">Po&longs;&longs;unt magnitudines &etail;quales <expan abbr="id&etilde;">idem</expan> <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis habere, vt duo <expan abbr="parallelogr&atilde;ma">parallelogramma</expan> &aelig;&shy;<lb/>qualia ad rectos &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> angulos exi&longs;ten<lb/>tia: <expan abbr="tri&atilde;gul&utilde;">triangulum</expan> quo&que; &amp; <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> in&shy;<lb/>ter&longs;e &aelig;qualia. <expan abbr="&ptilde;terea">propterea</expan> cubos, piramides, cylin<lb/>dros, &amp; huiu&longs;modi alias magnitudines &etail;qua<lb/>les <expan abbr="id&etilde;">idem</expan> grauitatis <expan abbr="c&etilde;tr&utilde;">centrum</expan> <expan abbr="h&etilde;re">herre</expan> intelligere po&longs;&longs;u<lb/>mus. </s>
          <s id="N11887">propterea in propo&longs;itione c&ugrave;m inquit Archimedes <lb/><emph type="italics"/>&longs;i du&aelig; magnitudines &aelig;quales non idem centrum grauitatis <emph.end type="italics"/>
<pb xlink:href="077/01/047.jpg" pagenum="43"/><emph type="italics"/>habuerint.<emph.end type="italics"/> intelligendum e&longs;t his verbis Archimedem &longs;uppo&shy;<lb/>nere magnitudines ita e&longs;&longs;e con&longs;titutas, vt &agrave; centro ad centrum <lb/>duci po&longs;&longs;it recta linea. </s>
          <s id="N1189D">quod idem ob&longs;eruandum e&longs;t in prima <lb/>propo&longs;itione &longs;ecundi libri huius. </s>
        </p>
        <p id="N118A1" type="main">
          <s id="N118A3">S&uacute;moper&egrave; a&uacute;t <expan abbr="animaduert&etilde;da">animaduertenda</expan> &longs;unt <expan abbr="n&otilde;nulla">nonnulla</expan>, quibus vtitur <lb/>Archimedes in hac propo&longs;itione, c&ugrave;m &longs;int communi&longs;&longs;ima, <lb/>&amp; maxim&egrave; vtilia in hac &longs;cientia. </s>
          <s id="N118AD">ac prim&ugrave;m quidem con&longs;ide<lb/>randum occurrit, quid &longs;ibi vult Archimedes per magnitudi<lb/>nem ex vtri&longs;&que; magnitudinibus AB compo&longs;itam. </s>
          <s id="N118B3">Nam ma&shy;<lb/>gnitudines AB &longs;unt inuicem &longs;eparat&etail;, &amp; &longs;unt du&etail;, ip&longs;e autem <lb/>vtram&que; vnam tant&ugrave;m con&longs;iderat. </s>
          <s id="N118B9">quod quidem ita <expan abbr="intelli-gend&utilde;">intelli&shy;<lb/>gendum</expan> e&longs;t. <expan abbr="quoni&atilde;">quoniam</expan> &longs;cilicet recta linea AB eas coniungit; ideo <lb/>Archimedes con&longs;iderat vnam tant&ugrave;m e&longs;&longs;e <expan abbr="magnitudin&etilde;">magnitudinem</expan>; qu&etail; <lb/>con&longs;tat ex ip&longs;is AB, &amp; efficitur vna magnitudo &agrave; linea AB. <lb/>cuius munus e&longs;t non &longs;ol&ugrave;m connectere magnitudines AB, <lb/>ita vtne&que; ad &longs;e ampli&ugrave;s accedere, ne&que; recedere inuicem <lb/>po&longs;&longs;int; &longs;intqu&egrave; ab hac linea qua&longs;i compul&longs;&etail; eundem &longs;emper <lb/>inter&longs;e &longs;eruare &longs;i tum: verum etiam &longs;i &longs;u&longs;pendantur ex C, in&shy;<lb/>telligendum e&longs;t linea AB in rectitudinem iacere, in&longs;uperqu&egrave; <lb/>&longs;u&longs;tinere magnitudines AB. Ne&que; magis vna e&longs;t magnitudo <lb/>quadrilaterum, <expan abbr="p&etilde;tagonum">pentagonum</expan>, cubus, &amp; huiu&longs;modi ali&aelig;, qu&agrave;m <lb/>&longs;it magnitudo, qu&aelig; componitur ex magnitudinibus AB v&shy;<lb/>n&agrave; cum linea AB. qu&ograve;d &longs;i e&longs;t vna tant&ugrave;m magnitudo, ergo <lb/>vnum habet <expan abbr="c&etilde;trum">centrum</expan> grauitatis. </s>
          <s id="N118E9">Archimedes igitur qu&etail;rit cen<lb/>trum grauitatis huiu&longs;ce magnitudinis; demon&longs;tratqu&egrave; cen<lb/>trum e&longs;&longs;e in puncto C. quod e&longs;t medium line&aelig; AB. notan<lb/>dum e&longs;t autem Archimedem non con&longs;iderare grauitatem li&shy;<lb/>ne&etail; AB. vt pot&egrave;, qu&etail; longitudo tant&ugrave;m exi&longs;tat. </s>
          <s id="N118F3">Qu&ograve;d &longs;i quis <lb/>etiam mente concipere vellet lineam AB grauitate <expan abbr="pr&etail;dit&atilde;">pr&etail;ditam</expan> <lb/>e&longs;&longs;e; nihilominus centrum grauitatis line&etail; AB &longs;imiliter e&longs;&longs;et <lb/>in eius medio C. nam longitudo AC longitudini CB e&longs;t <lb/>&aelig;qualis; ac propterea h&etail; quidem longitudines e&longs;&longs;ent inter &longs;e&longs;e <lb/>&etail;&que;ponderantes. </s>
          <s id="N11903">Quare, &longs;iue <expan abbr="c&otilde;&longs;iderata">con&longs;iderata</expan> grauitate line&etail; AB, <lb/>&longs;iue min&ugrave;s, centrum grauitatis magnitudinis ex AB compo<lb/>&longs;it&etail; e&longs;t <expan abbr="medi&utilde;">medium</expan> rect&etail; line&etail;, qu&aelig; centra grauitatis <expan abbr="magnitudin&utilde;">magnitudinum</expan> <lb/>coniungit. </s>
          <s id="N11913">Et hoc modo &longs;i plures etiam e&longs;&longs;ent magnitudines <lb/>&agrave; recta linea coniunct&etail;, eodem modo eas pro vna tant&ugrave;m ma<pb xlink:href="077/01/048.jpg" pagenum="44"/>gnitudine ex <expan abbr="plurib^&lcub;9}">pluribus</expan> magnitudinibus compo&longs;ita accipere po<lb/>terimus, veluti Archimedes in &longs;e&que;ntibus accipiet. </s>
        </p>
        <p id="N1191D" type="main">
          <s id="N1191F">Argumentandi modus in e&longs;t in hac demon&longs;tratione maxi&shy;<lb/>ma con&longs;ideratione dignus, &amp; huius &longs;cienti&aelig; maxim&egrave; pro&shy;<lb/>prius. </s>
          <s id="N11925">c&ugrave;m enim dixi&longs;&longs;et Archimedes po&longs;ito centro grauitatis <lb/>magnitudinis ex AB compo&longs;it&aelig; in puncto D, &longs;tatim infert. <lb/><emph type="italics"/>Quoniam igitur punctum D centrum e&longs;t grauitatis magnitudinis ex <lb/>AB compo&longs;ita, &longs;u&longs;pen&longs;o puncto D, magnitudines AB &aelig;&que;pondera&shy;<lb/>bunt.<emph.end type="italics"/> hoc e&longs;t &longs;i magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur ex <lb/>D, manebit, vt reperitur; nec amplius in alteram partem in cli <lb/>nabit. </s>
          <s id="N11938">quod euenit ob naturam centri grauitatis, quod talis <lb/>e&longs;t natur&aelig; (&longs;icuti initio explicauimus) ut &longs;i graue in eius cen&shy;<lb/>tro grauitatis &longs;u&longs;tineatur, eo modo manet, quo reperitur, <expan abbr="d&utilde;">dum</expan> <lb/>&longs;u&longs;penditur; parte&longs;qu&egrave; undiqu&egrave; &aelig;&que;ponderant. </s>
          <s id="N11944">&amp; ob id &longs;i <lb/>magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur in eius centro gra<lb/>uitatis, manet; parte&longs;qu&egrave; AB &aelig;&que;ponderant. </s>
          <s id="N1194A">ac propterea <lb/>quando in &longs;e&que;ntibus qu&aelig;rit Archimedes, quoniam grauia <lb/>&aelig;&que;ponderare debent, tunc tant&ugrave;m qu&aelig;rit ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis, ut in &longs;exta, &longs;eptimaqu&egrave; propo&longs;itione in quit Archi&shy;<lb/>medes magnitudines &etail;&que;ponderare ex di&longs;tantijs, qu&etail; permu<lb/>tatim proportionem habent, ut ip&longs;arum grauitates, in <expan abbr="dem&otilde;">demom</expan> <lb/>&longs;tratione tamen qu&aelig;rit, vbi nam e&longs;t <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni<lb/>tudinis ex vtrisqu&egrave; compo&longs;it&etail;. </s>
          <s id="N11966">quo inuento, &longs;tatim nece&longs;&longs;ari&ograve; <lb/>&longs;equitur, magnitudines, &longs;i ex ip&longs;o centro &longs;u&longs;pendantur, &aelig;&que; <lb/>ponderare. </s>
        </p>
        <p id="N1196C" type="main">
          <s id="N1196E">Hinc colligere po&longs;&longs;umus alterum argumentandi modum, <lb/>conuer&longs;o nemp&egrave; modo, veluti in eadem figura, &longs;i dicamus <lb/>grauia AB &longs;u&longs;pen&longs;a ex C &aelig;&que;ponderant, &longs;tatim inferre <lb/>po&longs;&longs;umus, punctum C ip&longs;orum &longs;imul grauium, hoc e&longs;t ma<lb/>gnitudinis ex ip&longs;is AB compo&longs;it&etail; centrum e&longs;&longs;e grauitatis. <lb/>Quare ad &longs;e inuicem conuertuntur, hoc punctum e&longs;t horum <lb/>grauium centrum grauitatis; ergo h&etail;c grauia ex hoc puncto <lb/>&aelig;q&ugrave;eponderant; &amp; &egrave; conuer&longs;o, nemp&egrave; h&aelig;c grauia ex hoc pun<lb/>cto &aelig;&que;ponderant, ergo idem punctum e&longs;t ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis. </s>
          <s id="N11986">&longs;ed ad uertendum hanc &longs;equi <expan abbr="conuertibilitat&etilde;">conuertibilitatem</expan>, <expan abbr="qu&atilde;-do">quan&shy;<lb/>do</expan> pr&aelig;fatum punctum e&longs;t in recta linea, qu&aelig; centra grauita&shy;<lb/>tum ponderum coniungit; deinde quando h&etail;c linea non e&longs;t
<pb xlink:href="077/01/049.jpg" pagenum="45"/>horizonti perpendicularis. </s>
          <s id="N11998">&longs;ecus aurem minim&egrave;. </s>
          <s id="N1199A">Nam &longs;i pon<lb/>dera AB &longs;int in libra ADB, qu&etail; &longs;it arcuata, vel angulum <expan abbr="c&omacr;-&longs;tituat">con&shy;<lb/>&longs;tituat</expan>, &longs;iue intelligatur libra recta linea AB, cui affixa &longs;it <lb/>perpendicularis CD. vt in tractatu de libra no&longs;trorum Me&shy;<lb/>chanicorum diximus. </s>
          <s id="N119A8">&longs;u&longs;pendantur autem pondera AB ex <lb/>
<arrow.to.target n="fig20"/><lb/>D, &amp; &aelig;&que;ponderent; <expan abbr="n&otilde;">non</expan> <lb/>&longs;equitur tamen, ergo D <lb/><expan abbr="c&etilde;trum">centrum</expan> e&longs;t grauitatis ma&shy;<lb/>gnitudinis ex AB com&shy;<lb/>po&longs;it&etail;. </s>
          <s id="N119C0">centrum enim gra<lb/>uitatis in linea exi&longs;tit AB <lb/>qu&aelig; centra grauitatis ma<lb/>gnitudinum AB coniun<lb/>git, nempe in C. Ver&ugrave;m coniungat recta linea AB centra <lb/>
<arrow.to.target n="fig21"/><lb/>grauitatis &aelig;qualium ponderum AB, lineaqu&egrave; <lb/>AB, cuius medium &longs;it C, in centrum mundi <expan abbr="t&etilde;-dat">ten&shy;<lb/>dat</expan>, magnitudoqu&egrave; ex ip&longs;is AB compo&longs;ita vbi&shy;<lb/>cun&que; &longs;u&longs;pendatur in linea AB, vt in E; ma<lb/>nebunt vti&que; pondera AB ex E &longs;u&longs;pen&longs;a, vt in <lb/>prima propo&longs;itione de libra no&longs;trorum Mecha&shy;<lb/>nicorum o&longs;tendimus. </s>
          <s id="N119E1">c&ugrave;m C &longs;it ip&longs;orum <expan abbr="centr&umacr;">centrum</expan> <lb/>grauitatis, &amp; EC &longs;it horizonti erecta. </s>
          <s id="N119E9">Et quam&shy;<lb/>uis magnitudo ex ip&longs;is AB compo&longs;ita ex E &longs;u<lb/>&longs;pen&longs;a maneat; non propterea &longs;equitur ergo E <lb/>centrum e&longs;t grauitatis magnitudinis ex ip&longs;is AB <lb/>compo&longs;it&etail;. </s>
          <s id="N119F3">ni&longs;i fort&egrave; accidat &longs;u&longs;pen&longs;io ex puncto <lb/>C. Pr&aelig;terea ver&ograve; aduertendum e&longs;t in hoc ca&longs;u <expan abbr="p&otilde;dera">pon<lb/>dera</expan> AB, dici quidem po&longs;&longs;e, manere, non autem <lb/>&aelig;&que;ponderare. </s>
          <s id="N119FF">omnia nimirum, qu&etail; &aelig;&que;ponderant, ma&shy;<lb/>nent; &longs;ed non &egrave; conuer&longs;o, qu&aelig; manent, &aelig;&que;ponderant. </s>
          <s id="N11A03">Nam <lb/>&longs;i pondus A maius fuerit pondere B; &longs;iue B maius, qu&agrave;m <lb/>A, vbicun&que; fiat &longs;u&longs;pen&longs;io in linea AB, &longs;emper ob <expan abbr="e&atilde;dem">eandem</expan> <lb/>cau&longs;am, quomodocun&que; &longs;int pondera, manebunt; non ta&shy;<lb/>men &aelig;&que;ponderabunt. </s>
          <s id="N11A11">Vt enim pondera &aelig;&que;ponderent, <lb/>requiritur, vt pars parti, virtu&longs;qu&egrave; vnius virtuti alterius hinc <lb/>inde re&longs;i&longs;tere, &amp; &aelig;quipollere po&longs;&longs;it; vt propri&egrave; dici po&longs;&longs;int <expan abbr="p&otilde;">pom</expan> <lb/>dera &aelig;&que;ponderare. </s>
          <s id="N11A1D">&amp; vt hoc euenire po&longs;&longs;it, oportet, vt par&shy;
<pb xlink:href="077/01/050.jpg" pagenum="46"/>tes ex determinatis di&longs;tantijs determinatas quo&que; habeant <lb/>grauitates; &longs;i ex dato puncto &aelig;&que;ponderare debent. </s>
          <s id="N11A25">Qu&ograve;d <lb/>&longs;i in hoc ca&longs;u datum fuerit punctum C, ex quo pondera AB <lb/>ex &aelig;qualibus di&longs;tantijs CA CB &etail;&que;ponderare debeant: o&shy;<lb/>porteret, vt pondera AB (ex demon&longs;tratis) &longs;emper e&longs;&longs;ent &aelig;&shy;<lb/>qualia. <expan abbr="Quoni&atilde;">Quoniam</expan> <expan abbr="aut&etilde;">autem</expan> <expan abbr="quomodoc&utilde;&que;">quomodocun&que;</expan> &longs;int pondera, hoc e&longs;t; &longs;i <lb/>ue pondus A maius, &longs;iue minus fuerit, qu&agrave;m B, manent, &longs;i <lb/>igitur dixerimus, ergo pondus A ponderi B &etail;&que;ponderat; <lb/>e&longs;&longs;et omnino inconueniens. </s>
          <s id="N11A41">c&ugrave;m ex ijsdem di&longs;tantijs <expan abbr="eid&etilde;">eidem</expan> <expan abbr="p&otilde;">pom</expan> <lb/>deri pondus quandoqu&egrave; maius, quandoqu&egrave; minus &etail;&que;pon&shy;<lb/>derare non po&longs;&longs;it; vt in hoc ca&longs;u accidere pote&longs;t. </s>
          <s id="N11A4F">Quocirca <lb/>nec propri&egrave; dici po&longs;&longs;unt pondera, &longs;iue in libra AB, &longs;iue ex <lb/>di&longs;tantijs CA CB con&longs;tituta e&longs;&longs;e. </s>
          <s id="N11A55">Vnd&egrave; ne&que; Archimedis <lb/>propo&longs;itiones in hoc ca&longs;u &longs;unt intelligend&etail; quandoquidem <lb/>in his propri&egrave; qu&aelig;rit ponderum, magnitudinumqu&egrave; &aelig;&que;&shy;<lb/>ponderationes. </s>
          <s id="N11A5D">ne&que; enim in hac quarta demon&longs;tratione in <lb/>hoc ca&longs;u potui&longs;&longs;et Archimedes ab&longs;urdum o&longs;tendere, &longs;i C <expan abbr="n&otilde;">non</expan> <lb/>e&longs;t grauitatis centrum magnitudinis ex AB compo&longs;it&aelig;, &longs;it <lb/>E. facta igitur ex E &longs;u&longs;pen&longs;ione, magnitudines &aelig;quales AB <lb/>ex in &aelig;qualibus di&longs;tantijs EA EB &etail;&que;ponderabunt. </s>
          <s id="N11A6B">quod <lb/>fieri non pote&longs;t. </s>
          <s id="N11A6F">non enim hoc e&longs;t ab&longs;urdum; c&ugrave;m pondera <lb/>ex E &longs;u&longs;pen&longs;a <expan abbr="mane&atilde;t">maneant</expan> idcirco quando linea AB e&longs;t <expan abbr="horiz&otilde;">horizom</expan> <lb/>ti erecta; propri&egrave; ad rem no&longs;tram minim&egrave; pertinet. </s>
          <s id="N11A7D">Ex dictis <lb/>igitur &longs;emper valet con&longs;e&que;ntia, hoc punctum horum pon<lb/>derum centrum e&longs;t grauitatis, ergo &longs;i ex hoc &longs;u&longs;pendantur, <expan abbr="p&otilde;">pom</expan> <lb/>dera &etail;&que;ponderant. </s>
          <s id="N11A89">non autem &egrave; conuer&longs;o. </s>
          <s id="N11A8B">ni&longs;i quando ar&shy;<lb/>gumentatio &longs;umitur &longs;emper ex recta linea, qu&aelig; centra graui<lb/>tatis magnitudinum coniungit, &amp; quando h&etail;c linea non e&longs;t <lb/>
<arrow.to.target n="fig22"/><lb/>horizonti erecta. </s>
          <s id="N11A98">hac enim <lb/>ratione quocun&que; modo <lb/>recta linea &longs;e habeat, &longs;em&shy;<lb/>per &longs;equitur idem. </s>
          <s id="N11AA0">Vt &longs;i li&shy;<lb/>nea AB fuerit, &longs;iue <expan abbr="n&otilde;">non</expan> fue&shy;<lb/>rit horizonti &aelig;quidi&longs;tans, <lb/>ip&longs;ius medium C centrum <lb/>erit grauitatis magnitudi&shy;<lb/>nis ex magnitudinibus AB &aelig;qualibus compo&longs;it&etail;. </s>
          <s id="N11AB0">vnde &longs;equi<pb xlink:href="077/01/051.jpg" pagenum="47"/>tur, &longs;i appendantur pondera AB ex C, &aelig;&que;ponderare. </s>
          <s id="N11AB6">&amp; <lb/>&egrave; conuer&longs;o, &longs;i AB pondera ex C &aelig;&que;ponderant, ergo C <lb/>centrum grauitatis exi&longs;tit. </s>
          <s id="N11ABC">ex quibus &longs;equitur lineam AB, <expan abbr="p&otilde;">pom</expan> <lb/>deraqu&egrave; manere eo modo, quo reperiuntur. </s>
          <s id="N11AC4">vt in no&longs;tro me&shy;<lb/>chanicorum libro in codem tractatu de libra demon&longs;traui&shy;<lb/>mus, &amp; aduer&longs;us illos, qui aliter &longs;entiunt, abund&egrave; &longs;atis
<arrow.to.target n="marg36"/> di&longs;pu&shy;<lb/>tauimus. </s>
        </p>
        <p id="N11AD0" type="margin">
          <s id="N11AD2"><margin.target id="marg36"/><emph type="italics"/>po&longs;t quar&shy;<lb/>tam propo<lb/>&longs;itionem.<emph.end type="italics"/><lb/>*</s>
        </p>
        <figure id="id.077.01.051.1.jpg" xlink:href="077/01/051/1.jpg"/>
        <figure id="id.077.01.051.2.jpg" xlink:href="077/01/051/2.jpg"/>
        <figure id="id.077.01.051.3.jpg" xlink:href="077/01/051/3.jpg"/>
        <p id="N11AEC" type="main">
          <s id="N11AEE">In demon&longs;tratione autem huius quart&aelig; propo&longs;itionis in&shy;<lb/>quit Archimedes. <emph type="italics"/>Qu&ograve;d autem &longs;it in linea AB, pr&aelig;osten&longs;um e&longs;t.<emph.end type="italics"/> qua <lb/>&longs;i dicat Archimedes, &longs;e pri&ugrave;s o&longs;tendi&longs;&longs;e centrum grauitatis ma <lb/>gnitudinis ex AB compo&longs;it&aelig; e&longs;&longs;e in linea AB; quod tamen <lb/>in ijs, qu&aelig; dicta &longs;unt, non videtur expre&longs;&longs;um. </s>
          <s id="N11AFE">virtute tamen &longs;i <lb/>con&longs;ideremus ea, qu&etail; in prima, tertiaqu&egrave; propo&longs;itione dicta <lb/>&longs;unt, facil&egrave; ex his concludi pote&longs;t, centrum grauitatis magni&shy;<lb/>tudinis ex duabus magnitudinibus compo&longs;it&aelig; e&longs;&longs;e in recta li<lb/>nea, qu&aelig; ip&longs;arum centra grauitatis coniungit. </s>
          <s id="N11B08">Quare memi&shy;<lb/>ni&longs;&longs;e oportet eorum, qu&etail; a nobis in expo&longs;itione primi po&longs;tu<lb/>lati huius dicta fuere, nemp&egrave; Archimedem &longs;upponere, di&longs;tan&shy;<lb/>tias e&longs;&longs;e in vna, eademqu&egrave; recta linea con&longs;titutas. </s>
          <s id="N11B10">ideoqu&egrave; in <lb/>prima propo&longs;itio nec inquit, Grauia, qu&etail; ex <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> &etail;quali<lb/>bus <expan abbr="&aelig;&que;p&otilde;der&atilde;t">&aelig;&que;ponderant</expan>, &aelig;qualia e&longs;&longs;e inter &longs;e; Archimedes qu&egrave; <expan abbr="dem&otilde;">demom</expan> <lb/>&longs;trat, qu&ograve;d quando &aelig;&que;ponderant, &longs;unt &aelig;qualia: ex dictis <lb/>&longs;equitur, &longs;i &aelig;&que;ponderant, ergo centrum grauitatis magni&shy;<lb/>tudinis ex ip&longs;is compo&longs;it&etail; erit in eo puncto, vbi &aelig;&que;ponde&shy;<lb/>rant; hoc e&longs;t in medio di&longs;tantiarum, line&etail; &longs;cilicet, qu&etail; <expan abbr="graui&utilde;">grauium</expan> <lb/>centra grauitatis coniungit. </s>
          <s id="N11B30">quod idem e&longs;t, ac &longs;i Archimedes <lb/>dixi&longs;&longs;et. </s>
          <s id="N11B34">Grauia, qu&etail; habent centrum grauitatis in medio li&shy;<lb/>ne&etail;, qu&etail; magnitudinum centra grauitatis coniungit, &etail;qua&shy;<lb/>lia &longs;unt inter &longs;e. </s>
          <s id="N11B3A">cuius quidem h&etail;c quarta propo&longs;itio videtur <lb/>e&longs;&longs;e conuer&longs;a. </s>
          <s id="N11B3E">quamuis Archimedes loco grauium nominet <lb/>magnitudines. </s>
          <s id="N11B42">Pr&etail;terea in tertia propo&longs;itione, quoniam <expan abbr="o&longs;t&etilde;-dit">o&longs;ten&shy;<lb/>dit</expan> Archimedes, in&etail;qualia grauia &etail;&que;ponderare ex <expan abbr="di&longs;t&atilde;tijs">di&longs;tantijs</expan> <lb/>in&etail;qualibus, ita vt grauius &longs;it in minori di&longs;tantia, &longs;equitur er<lb/>go centrum grauitatis e&longs;t in eo puncto, vbi &aelig;&que;ponderant; <lb/>&amp; idem e&longs;t, ac &longs;i dixi&longs;&longs;et, in &aelig;qualium grauium centrum gra&shy;<lb/>uitatis e&longs;t in recta linea, qu&aelig; ip&longs;orum centra grauitatis con&shy;<lb/>iungit; ita vt &longs;it propinquius grauiori, remotius uer&ograve; leuiori.
<pb xlink:href="077/01/052.jpg" pagenum="48"/>vnde &longs;equitur centrum grauitatis ip&longs;orum grauium ubicum<lb/>&que; e&longs;&longs;e po&longs;&longs;e in recta linea, qu&etail; ip&longs;orum centra grauitatis <expan abbr="c&otilde;">con<lb/>iungit</expan>. </s>
          <s id="N11B64">Ex quibus concludi pote&longs;t, <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni&shy;<lb/>tudinis ex duabus magnitudinibus compo&longs;it&etail; e&longs;&longs;e in recta li<lb/>nea, qu&aelig; ip&longs;orum centra grauitatis connectit. </s>
        </p>
        <p id="N11B6E" type="main">
          <s id="N11B70">Po&longs;trem&ograve; notandum e&longs;t, Archimedem ea, qu&aelig; in &longs;uperio<lb/>ribus propo&longs;itionibus nuncupauit grauia, in hac quarta pro<lb/>po&longs;itione, veluti etiam in &longs;e&que;ntibus, non ampli&ugrave;s grauia, <lb/>&longs;ed (vti diximus) magnitudines nominare. </s>
          <s id="N11B78">quod quidem his <lb/>de cau&longs;is id ab ip&longs;o factum exi&longs;timo. </s>
          <s id="N11B7C">prim&ugrave;m enim, quia in <lb/>his expre&longs;se qu&aelig;rit centrum grauitatis; quod quidem <expan abbr="c&etilde;trum">centrum</expan>, <lb/>quamuis &longs;it centrum grauitatis, poti&ugrave;s re&longs;picit <expan abbr="magnitudin&etilde;">magnitudinem</expan>, <lb/>qu&agrave;m graue aliquod. </s>
          <s id="N11B8C">Nam c&ugrave;m dicimus centrum grauitatis, <lb/>&longs;tatim innuimus &longs;itum, &longs;itum inqu&agrave;m determinatum figu&shy;<lb/>r&aelig;, in qua e&longs;t; &longs;iquidem centrum grauitatis e&longs;t punctum, &amp; <lb/>(vt ita dicam) punctum grauitatis eius, in quo e&longs;t. </s>
          <s id="N11B94">&amp; ideo, <lb/>quoniam magnitudo formam habet dete mina tam, <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis rect&egrave; pote&longs;t re&longs;picere &longs;itum re&longs;pectu magnitudinis, <lb/>in qua e&longs;t; quod tamen efficere non pote&longs;t re&longs;pectu grauis. <lb/>etenim graue, ut graue e&longs;t, non habet formam determina <expan abbr="t&atilde;">tam</expan>; <lb/>c&ugrave;m eadem grauitas e&longs;&longs;e po&longs;&longs;it in cubo, in piramide, alii&longs;qu&egrave; <lb/>corporibus quibu&longs;cun&que;, mod&ograve; minoribus, mod&ograve; maiori&shy;<lb/>bus, pro ut &longs;unt diuer&longs;arum &longs;pecierum. </s>
          <s id="N11BAC">quare centrum grauita<lb/>tis non pote&longs;t re&longs;picere &longs;itum in grauibus, quatenus grauia <expan abbr="c&otilde;">con<lb/>&longs;iderantur</expan>; &longs;ed quatenus magnitudines exi&longs;tunt. </s>
          <s id="N11BB6">Pr&aelig;terea Ar&shy;<lb/>chimedes loco grauium magnitudines nominat, quia eas di&shy;<lb/>ui&longs;ibiles con&longs;iderat, quod e&longs;t proprium magnitudinis; vt in &longs;e<lb/>xta, &longs;eptima, &amp; octaua propo&longs;itione. </s>
          <s id="N11BBE">&amp; quamuis, dum <expan abbr="diuid&utilde;">diuidum</expan> <lb/>tur magnitudines, grauia quo&que; diui&longs;a proueniant; non ta&shy;<lb/>men propterea grauia diuiduntur, ut grauia. <expan abbr="n&otilde;">non</expan>.n. </s>
          <s id="N11BCC">hoc ip&longs;is <lb/>competit, vt grauibus; &longs;ed vt magnitudinibus, qu&aelig; &longs;unt per <lb/>&longs;e diui&longs;ibiles. </s>
          <s id="N11BD2">Archimedes igitur his de cau&longs;is nomen <expan abbr="graui&utilde;">grauium</expan> <lb/>in magnitudines mutauit. </s>
          <s id="N11BDA">in &longs;uperioribus enim theoremati&shy;<lb/>bus pertractauit, quomodo res &aelig;&que;ponderant ex di&longs;tantijs <lb/>mod&ograve; &aelig;qualibus, mod&ograve; in &aelig;qualibus. </s>
          <s id="N11BE0">&amp; quoniam res <expan abbr="&etail;&que;p&otilde;-derant">&etail;&que;pon<lb/>derant</expan>, pro ut &longs;unt magis grauia, &amp; min&ugrave;s grauia; non ut <expan abbr="s&utilde;t">sunt</expan> <lb/>maiores, vel minores magnitudines, &longs;iquidem talis natur&aelig;
<pb xlink:href="077/01/053.jpg" pagenum="49"/>e&longs;&longs;e pote&longs;t minor magnitudo, qu&etail; maiore magnitudine alte<lb/>rius nature grauior exi&longs;tat; proind&eacute; Archimedes in &longs;uperiori&shy;<lb/>bus rect&egrave; grauia nuncupauit; optim&egrave;qu&egrave; in his magnitudines <lb/>vocat. </s>
          <s id="N11BF8">At ver&ograve; aduertendum e&longs;t, qu&ograve;d quamuis Archimedes <lb/>in his magnitudines nominet, non propterea exi&longs;timandum <lb/>e&longs;t, eum intelligere magnitudines tant&ugrave;m; &longs;ed magnitudines <lb/>grauitate pr&etail;ditas, ita ut in ip&longs;is omnino grauitatem re&longs;piciat. <lb/>Etenim pluribus modis intelligere po&longs;&longs;umus magnitudines, <lb/>vel enim ut &longs;int inter &longs;e eiu&longs;dem &longs;peciei, vel diuer&longs;&aelig;; nec <expan abbr="n&otilde;">non</expan> <lb/>in&longs;uper homogene&aelig;, vel heterogene&aelig;. </s>
          <s id="N11C0A">vt in hac propo&longs;itione <lb/><expan abbr="qu&atilde;do">quando</expan> Archimedes <expan abbr="pponit">proponit</expan> duas magnitudines &etail;quales, <expan abbr="t&utilde;c">tuc</expan> <lb/>intelligere po&longs;&longs;umus eas e&longs;&longs;e eiu&longs;dem &longs;peciei, &amp; homogeneas; <lb/>qu&aelig;, c&ugrave;m &longs;int &aelig;quales, erit &amp; grauitas vnius grauitati alterius <lb/>&aelig;qualis. </s>
          <s id="N11C17">&longs;i ver&ograve; con&longs;ideremus eas e&longs;&longs;e diuer&longs;&aelig; &longs;peciei, &amp; e&shy;<lb/>tiam heterogeneas; tunc quando Archimedes proponit has <lb/>magnitudines &aelig;quales; intelligendum e&longs;t, eas e&longs;&longs;e &aelig;quales in <lb/>grauitate; qu&aelig; quidem efficit, vt demon&longs;tratio, quod propo&shy;<lb/>&longs;itum e&longs;t, concludat. </s>
          <s id="N11C21">vt ex eius demon&longs;tratione patet. </s>
          <s id="N11C23">Et his <lb/>quo&que; modis intelligere po&longs;&longs;umus magnitudines in &longs;e&que;n<lb/>tibus v&longs;&que; ad nonam propo&longs;itionem in quibus &longs;cilicet intel<lb/>ligere po&longs;&longs;umus magnitudines e&longs;&longs;e non &longs;ol&ugrave;m eiu&longs;dem &longs;pe&shy;<lb/>ciei, vel diuer&longs;&aelig;, ver&ugrave;m etiam &amp; homogeneas. </s>
          <s id="N11C2D">&amp; heteroge&shy;<lb/>neas. </s>
          <s id="N11C31">ut po&longs;t &longs;eptimam clari&ugrave;s o&longs;tendemus. </s>
          <s id="N11C33">Ver&ugrave;m de&shy;<lb/>mon&longs;trationes clariores redduntur, &longs;i intelligamus magnitu&shy;<lb/>dines e&longs;&longs;e eiu&longs;dem &longs;peciei, &amp; homogeneas, in quibus graui&shy;<lb/>tas magnitudini re&longs;pondet, vt &longs;i ip&longs;arum altera fuerit alte&shy;<lb/>rius dupla, &amp; grauitas vnius grauitatis alterius dupla exi&longs;tat. <lb/>Qu&ograve;d &longs;i magnitudo fuerit alterius tripla, vel quadrupla, &amp;c. <lb/>erit &amp; grauitas grauitatis tripla, vel quadrupla, &amp; &longs;ic dein&shy;<lb/>ceps. </s>
          <s id="N11C43">deinde &longs;i magnitudo bifariam diui&longs;a fuerit, &amp; ip&longs;ius gra<lb/>uitas in duas &etail;quas partes &longs;it quo&que; diui&longs;a. </s>
          <s id="N11C47">qu&ograve;d &longs;i magnitu&shy;<lb/>do in plures diuidatur partes, &amp; grauitas quo&que; in totidem <lb/>eiu&longs;dem proportionis diui&longs;a proueniat. </s>
        </p>
        <pb xlink:href="077/01/054.jpg" pagenum="50"/>
        <p id="N11C50" type="head">
          <s id="N11C52">PROPOSITIO. V.</s>
        </p>
        <p id="N11C54" type="main">
          <s id="N11C56">Si trium magnitudinum centra grauitatis in re<lb/>cta linea fuerint po&longs;ita, &amp; magnitudines &aelig;qualem <lb/>habuerint grauitatem, acrect&aelig; line&aelig; inter centra <lb/>fuerint &aelig;quales, magnitudinis ex omnibus magni<lb/>tudinibus compo&longs;it&aelig; centrum grauitatis erit <expan abbr="p&utilde;">pum</expan> <lb/>ctum, quod &amp; ip&longs;arum medi&aelig; centrum grauitatis <lb/>exi&longs;tit. </s>
        </p>
        <figure id="id.077.01.054.1.jpg" xlink:href="077/01/054/1.jpg"/>
        <p id="N11C6B" type="main">
          <s id="N11C6D"><emph type="italics"/>Sint tres magnitudines ACB. ip&longs;arum autem centra grauitatis &longs;int <lb/>puncta ACB in resta linea<emph.end type="italics"/> ACB <emph type="italics"/>po&longs;ita. </s>
          <s id="N11C79">&longs;int ver&ograve; magnitudines ACB <lb/>&aelig;quales; rect&aelig;qu&egrave; line&aelig; AC CB<emph.end type="italics"/> inter centra ip&longs;arum <emph type="italics"/>aquales. </s>
          <s id="N11C83">Di<lb/>co magnitudinis ex omnibus<emph.end type="italics"/> ACB <emph type="italics"/>magnitudinibus compo&longs;it&aelig; <expan abbr="centr&utilde;gra">centrungra</expan> <lb/>uitatis e&longs;&longs;e punctum C.<emph.end type="italics"/> quod e&longs;t centrum grauitatis medi&aelig; ma&shy;<lb/>gnitudinis. <emph type="italics"/>Quoniam enim magnitudines AB &aelig;qualem habent graui<emph.end type="italics"/><lb/>
<arrow.to.target n="marg37"/> <emph type="italics"/>tatem<emph.end type="italics"/>; magnitudinis ex vtri&longs;&que; AB compo&longs;it&aelig; <emph type="italics"/>centrum graui<lb/>tatis erit punctum C: c&ugrave;m &longs;int AC CB &aelig;quales.<emph.end type="italics"/> &longs;itqu&egrave; propterea <lb/>punctum C medium rect&aelig; line&etail; AB. <emph type="italics"/>Sed &amp; magnitudinis C <expan abbr="c&emacr;">cem</expan> <lb/>trum grauitatis est<emph.end type="italics"/> idem <emph type="italics"/>punctum C.<emph.end type="italics"/> punctum ergo C <expan abbr="tri&utilde;">trium</expan> ma&shy;<lb/>gnitudinum ABC centrum quo&que; grauitatis erit. <emph type="italics"/>Quare pa<lb/>tet magnitudinis ex omnibus magnitudinibus<emph.end type="italics"/> ACB <emph type="italics"/>compo&longs;it&aelig; centrum <lb/>grauitatis e&longs;&longs;e punctum, quod &amp;<emph.end type="italics"/> magnitudinis <emph type="italics"/>medi&aelig; centrum graui&shy;<lb/>tatis existit.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/055.jpg" pagenum="51"/>
        <p id="N11CE6" type="margin">
          <s id="N11CE8"><margin.target id="marg37"/>4 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N11CF1" type="head">
          <s id="N11CF3">COROLLARIVM. I.</s>
        </p>
        <p id="N11CF5" type="main">
          <s id="N11CF7">Ex hoc autem manife&longs;tum e&longs;t, &longs;i quotcunqu&egrave; <lb/>magnitudinum, &amp; numero imparium, centra
<arrow.to.target n="marg38"/> gra&shy;<lb/>uitatis in recta linea con&longs;tituta fuerint; &amp; magni&shy;<lb/>tudines &aelig;qualem habuerint grauitatem; rect&aelig;qu&egrave; <lb/>line&aelig; inter ip&longs;arum centra fuerint &aelig;quales, ma&shy;<lb/>gnitudinis ex omnibus magnitudinibus compo&longs;i<lb/>t&aelig; centrum grauitatis e&longs;&longs;e punctum, quod &amp; ip&longs;a&shy;<lb/>rum medi&aelig; centrum grauitatis exi&longs;tit. </s>
        </p>
        <p id="N11D0B" type="margin">
          <s id="N11D0D"><margin.target id="marg38"/>*</s>
        </p>
        <p id="N11D11" type="head">
          <s id="N11D13">SCHOLIVM.</s>
        </p>
        <figure id="id.077.01.055.1.jpg" xlink:href="077/01/055/1.jpg"/>
        <p id="N11D18" type="main">
          <s id="N11D1A">Ex demon&longs;tratione colligit Archimedes &longs;i plures fuerint <lb/>magnitudines, <expan abbr="qu&atilde;">quam</expan> tres; dummodo &longs;int numero impares, vt <lb/>ABCDE; quarum centra grauitatis ABCDE reperiantur in li<lb/>nea recta AE. fuerint autem h&etail; magnitudines &aelig;quales in gra<lb/>uitate. </s>
          <s id="N11D28">in&longs;uper rect&etail; line&etail; AB BC CD DE, qu&etail; &longs;unt inter <expan abbr="c&etilde;-tra">cen&shy;<lb/>tra</expan> grauitatis, fuerint &aelig;quales: magnitudinis ex omnibus ma<lb/>gnitudinibus ABCDE compo&longs;it&aelig; centrum grauitatis e&longs;&longs;e <lb/>punctum C. quod e&longs;t centrum grauitatis magnitudinis <lb/>medi&aelig;. </s>
        </p>
        <p id="N11D36" type="main">
          <s id="N11D38">Eodem enim modo, ac prim&ugrave;m quidem ex demon&longs;tratio <lb/>ne patet <expan abbr="punct&utilde;">punctum</expan> C centrum e&longs;&longs;e grauitatis <expan abbr="tri&utilde;">trium</expan> <expan abbr="magnitudin&utilde;">magnitudinum</expan> <lb/>BCD, &amp; quoniam AB BC &longs;unt &aelig;quales ip&longs;is CD DE,
<pb xlink:href="077/01/056.jpg" pagenum="52"/>erit AC ip&longs;i CE &etail;qualis. </s>
          <s id="N11D4E">c&ugrave;m qu&egrave; &longs;it grauitas magnitudinis <lb/>
<arrow.to.target n="marg39"/> A &etail;qualis grauitati ip&longs;ius E, erit itidem punctum C magni<lb/>tudinum AE centrum grauitatis. </s>
          <s id="N11D58">ergo punctum C magni<lb/>tudinis ex omnibus magnitudinibus ABCDE compo&longs;it&aelig; <lb/>centrum grauitatis exi&longs;tit. </s>
        </p>
        <p id="N11D5E" type="margin">
          <s id="N11D60"><margin.target id="marg39"/>4 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N11D69" type="main">
          <s id="N11D6B">Qu&ograve;d &longs;i fuerint ad huc plures magnitudines, impares ver&ograve; <lb/>extiterint; qu&aelig; ita &longs;e habeant, vt expo&longs;itum e&longs;t; &longs;imiliter <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>detur, centrum grauitatis medi&aelig; magnitudinis centrum e&longs;&longs;e <lb/>grauitatis magnitudinis ex omnibus magnitudinibus com&shy;<lb/>po&longs;it&aelig;. </s>
        </p>
        <p id="N11D79" type="main">
          <s id="N11D7B">
<arrow.to.target n="marg40"/> In hoc corollario, verba illa, <emph type="italics"/>&amp; magnitudines &aelig;qualem habue&shy;<lb/>rint grauitatem<emph.end type="italics"/> in greco codice ita habentur. <foreign lang="greek">ei(/ka tate i)/son a)pe/xon&shy;<lb/>ta a)po\ tou= me/sou mege/qeos i)/son ba/ros e)/xwnti</foreign> quorum multa &longs;uperuaca&shy;<lb/>nea nobis vi&longs;a &longs;unt; loco quorum (vt arbitror) rect&egrave; <expan abbr="congru&etilde;t">congruent</expan> <lb/><foreign lang="greek">kai\ ta\ mege/qea i)/son ba/ros e)/xwnti</foreign>, vt vertimus. </s>
          <s id="N11D9B">Nam &longs;i ordinis at&que; <lb/><expan abbr="c&otilde;ditionum">conditionum</expan> propo&longs;it&etail; propo&longs;itionis ratio habenda e&longs;t, opor<lb/>tet vt magnitudines &etail;qualem habeant grauitatem; Nam &amp; <lb/>Archimedes in &longs;e&que;ntibus demon&longs;trationibus ijs vtitur, ut <lb/>&longs;unt &aelig;&que;graues. </s>
          <s id="N11DA8">Adhuc tamen veritatem habebit &longs;i c&aelig;teris <lb/>conditionibus illud quo&que; addere voluerimus, nempe &longs;i <emph type="italics"/>ma<lb/>gnitudines &agrave; media magnitudine &aelig;qualiter di&longs;tantes &aelig;qualem habuerint <lb/>grauitatem<emph.end type="italics"/> eodem modo punctum C centrum erit grauitatis <lb/>
<arrow.to.target n="fig23"/><lb/>magnitudinis ex omnibus ABCDE compo&longs;it&etail;, Nam &longs;i ma&shy;<lb/>gnitudines &agrave; media magnitudine &longs;unt &etail;&que;graues; &etail;qualem <lb/>quo&que; habebunt grauitatem magnitudines AE; veluti ma&shy;<lb/>gnitudines BD, qu&aelig; &aelig;qualiter &agrave; media magnitudine C di&shy;<lb/>&longs;tant. </s>
          <s id="N11DC5">&amp; quam uis non &longs;int omnes &aelig;&que;graues, &longs;ufficit, vt AE <lb/>qu&aelig; &etail;qualiter &agrave; media magnitudine di&longs;tant, &longs;int &etail;&que;graues. <lb/>&longs;imiliter BD &etail;&que;graues. </s>
          <s id="N11DCB">Eadem enim ratione, quoniam <lb/>BD &longs;unt &aelig;&que;graues, &amp; di&longs;tanti&aelig; BC CD &etail;quales; erit C ip&longs;a-
<pb xlink:href="077/01/057.jpg" pagenum="53"/>rum BD centrum grauitatis. </s>
          <s id="N11DD3">pari qu&egrave; ratione C erit centrum <lb/>grauitatis magnitudinum AE &etail;&que;grauium. </s>
          <s id="N11DD7">cum &longs;int AC <lb/>CE &etail;quales, &amp; idem C e&longs;t grauitatis centrum magnitudinis <lb/>C. ergo punctum C magnitudinis ex omnibus magnitudini&shy;<lb/>bus ABCDE compo&longs;it&etail; centrum grauitatis exi&longs;tit. </s>
        </p>
        <p id="N11DDF" type="margin">
          <s id="N11DE1"><margin.target id="marg40"/>*</s>
        </p>
        <figure id="id.077.01.057.1.jpg" xlink:href="077/01/057/1.jpg"/>
        <p id="N11DE9" type="head">
          <s id="N11DEB">COROLLARIVM. II.</s>
        </p>
        <p id="N11DED" type="main">
          <s id="N11DEF">Si ver&ograve; magnitudines fuerint numero pares; <lb/>&amp; ip&longs;arum centra grauitatis in recta linea extite&shy;<lb/>rint, magnitudine&longs;qu&egrave; &aelig;qualem habuerint graui
<arrow.to.target n="marg41"/><lb/>tatem, rect&aelig; qu&egrave; line&aelig; inter centra fuerint &aelig;qua<lb/>les: magnitudinis ex omnibus magnitudinibus <expan abbr="c&otilde;">com</expan> <lb/>po&longs;it&aelig; centrum grauitatis erit medium rect&aelig; li&shy;<lb/>ne&aelig;, qu&aelig; magnitudinum centra grauitatis <expan abbr="coni&utilde;-git">coniun&shy;<lb/>git</expan>. vt in &longs;ubiecta figura. </s>
        </p>
        <p id="N11E0A" type="margin">
          <s id="N11E0C"><margin.target id="marg41"/>*</s>
        </p>
        <figure id="id.077.01.057.2.jpg" xlink:href="077/01/057/2.jpg"/>
        <p id="N11E13" type="head">
          <s id="N11E15">SCHOLIVM.</s>
        </p>
        <p id="N11E17" type="main">
          <s id="N11E19">Colligit pr&aelig;terea Archimedes &longs;i magnitudines ABCDEF <lb/>fuerint numero pares, quarum centra grauitatis ABCDEF in <lb/>recta linea AF &longs;int con&longs;tituta; magnitudine&longs;qu&egrave; &longs;int &aelig;quales <lb/>in grauitate; &longs;intqu&egrave; inter centra line&etail; AB BC CD DE EF <lb/>&aelig; quales. </s>
          <s id="N11E23">diuidatur autem AF bifariam in G. erit punctum <lb/>G centrum grauitatis magnitudinis ex omnibus compo&longs;i&shy;<lb/>t&aelig; quod quidem, figura tant&ugrave;m in&longs;pecta, per&longs;picuum e&longs;t. <lb/>C&ugrave;m enim magnitudines AF &longs;int &aelig;&que;graues, &amp; AG GF
<pb xlink:href="077/01/058.jpg" pagenum="54"/>
<arrow.to.target n="marg42"/> &longs;int &aelig;quales, erit G centrum grauitatis magnitudinis ex AF <lb/>compo&longs;it&aelig;. </s>
          <s id="N11E35">quia ver&ograve; AB e&longs;t ip&longs;i EF &aelig;qualis, reliqua BG <lb/>ip&longs;i GE &aelig;qualis exi&longs;tet. </s>
          <s id="N11E39">&amp; &longs;unt magnitudines BE &etail;&que;gra&shy;<lb/>ues, erit idem G centrum grauitatis <expan abbr="magnitudin&utilde;">magnitudinum</expan> BE. &longs;imili&shy;<lb/>ter c&ugrave;m &longs;it BC &aelig;qualis DE, relin&que;tur CG ip&longs;i GD &etail;qua&shy;<lb/>lis; magnitudinesqu&egrave; CD &longs;unt &etail;&que;graues. </s>
          <s id="N11E45">ergo <expan abbr="p&utilde;ctum">punctum</expan> G <expan abbr="c&etilde;">cem</expan> <lb/>trum e&longs;t quo&que; magnitudinum CD. Vnde &longs;equitur, <expan abbr="punct&utilde;">punctum</expan> <lb/>G magnitudinis ex omnibus magnitudinibus ABCDEF <expan abbr="c&otilde;-po&longs;it&aelig;">con&shy;<lb/>po&longs;it&aelig;</expan> centrum grauitatis exi&longs;tere. </s>
        </p>
        <p id="N11E5D" type="margin">
          <s id="N11E5F"><margin.target id="marg42"/>4 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N11E68" type="main">
          <s id="N11E6A">
<arrow.to.target n="marg43"/> Hoc quo&que; loco verba illa <emph type="italics"/>magnitudine&longs;qu&egrave; &aelig;qualem habuerint <lb/>grauitatem.<emph.end type="italics"/> Gr&aelig;cus codex ita mendos&egrave; legit. <foreign lang="greek">kai\ ta\ me/sa au)/ths i)/son <lb/>ba/ros e)/xwnti</foreign>, qu&aelig; quidem verba hoc modo re&longs;titui po&longs;&longs;unt. <lb/><foreign lang="greek">kai\ ta\ mege/qea i)/son ba/ros e)/xwnti. </foreign></s>
        </p>
        <p id="N11E82" type="margin">
          <s id="N11E84"><margin.target id="marg43"/>*</s>
        </p>
        <p id="N11E88" type="main">
          <s id="N11E8A">In pr&aelig;cedenti propo&longs;itione o&longs;tendit Archimedes, quomo<lb/>do &longs;e habet centrum grauitatis magnitudinis ex duabus ma&shy;<lb/>gnitudinibus &etail;qualibus compo&longs;it&aelig;. </s>
          <s id="N11E90">In hac autem <expan abbr="dem&otilde;&longs;trat">demon&longs;trat</expan>, <lb/>vbi &longs;imiliter grauitatis centrum reperitur inter plures magni&shy;<lb/>tudines &aelig;&que;graues, &amp; inter &longs;e &etail;qualiter di&longs;tantes. </s>
          <s id="N11E9A">ex quibus <lb/>tandem colliget fundamentum &longs;&aelig;pi&ugrave;s dictum. </s>
          <s id="N11E9E">nemp&egrave; &longs;i ma&shy;<lb/>gnitudines &etail;&que;ponderare debent; ita &longs;e habebit magnitudi&shy;<lb/>num grauitas ad grauitatem, ut &longs;e habent di&longs;tanti&aelig; permuta<lb/>tim, ex quibus &longs;u&longs;penduntur. </s>
          <s id="N11EA6">&amp; hoc demon&longs;trat Archimedes <lb/>in duabus &longs;e&que;ntibus propo&longs;itionibus. </s>
          <s id="N11EAA">nam magnitudines, <lb/>vel &longs;unt commen&longs;urabiles inter&longs;e&longs;e, vel incommen&longs;urabiles. <lb/>de commen&longs;urabilibus aget in &longs;e&que;nti: de incommen&longs;urabi<lb/>libus ver&ograve; in &longs;eptima propo&longs;itione. </s>
          <s id="N11EB2">&amp; Archimedes duas <expan abbr="&longs;e&quetilde;-tes">&longs;e&que;n&shy;<lb/>tes</expan> propo&longs;itiones ueluti coniunctas proponit. </s>
          <s id="N11EBA">Nam in &longs;exta <lb/>inquit <emph type="italics"/>Magnitudines commen&longs;urabiles,<emph.end type="italics"/> &amp;c. </s>
          <s id="N11EC4">in &longs;eptima uer&ograve; in&shy;<lb/>quit, <emph type="italics"/>Si autem magnitudines &longs;uerint incommen&longs;urabiles,<emph.end type="italics"/> qua&longs;i vna <expan abbr="t&atilde;">tam</expan> <lb/>t&ugrave;m &longs;it propo&longs;itio in duas partes diui&longs;a. </s>
          <s id="N11ED6">ita ut ne&que; numeris <lb/>e&longs;&longs;ent di&longs;tinguende, &longs;ed pro vna tant&ugrave;m propo&longs;itione <expan abbr="&longs;umm&etilde;-d&aelig;">&longs;ummen<lb/>d&aelig;</expan>, ob&longs;e&que;ntis autem demon&longs;trationis faciliorem <expan abbr="intellig&etilde;-tiam">intelligen<lb/>tiam</expan> h&etail;c pri&ugrave;s pr&aelig;mittimus. </s>
        </p>
        <p id="N11EE7" type="head">
          <s id="N11EE9">LEMMA.</s>
        </p>
        <p id="N11EEB" type="main">
          <s id="N11EED">Si du&etail; fuerint magnitudines in &aelig;quales, quarum maior &longs;it <lb/>alterius dupla, tertia ver&ograve; qu&etail;dam magnitudo minorem me-
<pb xlink:href="077/01/059.jpg" pagenum="55"/>tiatur. </s>
          <s id="N11EF5">maiorem quo&que; in partes numero pares metietur. </s>
        </p>
        <p id="N11EF7" type="main">
          <s id="N11EF9">Sint du&etail; in &etail;quales magni<lb/><arrow.to.target n="fig24"/><lb/>tudines AB, &longs;itqu&egrave; A ip&longs;ius <lb/>B duplex. </s>
          <s id="N11F04">magnitudo <expan abbr="aut&etilde;">autem</expan> <lb/>C <expan abbr="magnitudin&etilde;">magnitudinem</expan> B metia&shy;<lb/>tur. </s>
          <s id="N11F12">Dico C <expan abbr="magnitudin&etilde;">magnitudinem</expan> <lb/>A metiri, men&longs;urationesqu&egrave; numero pares e&longs;&longs;e. </s>
          <s id="N11F1A">Quoniam <lb/>enim C metitur B, eodem numero C metietur medietates <lb/>ip&longs;ius A, qu&aelig; &longs;untip&longs;i B &aelig;quales. </s>
          <s id="N11F20">ergo duplo plures erunt nu<lb/>mero men&longs;urationes ip&longs;ius A, qu&agrave;m ip&longs;ius B. quare men&longs;u&shy;<lb/>rationes ip&longs;ius A &longs;unt numero pares. </s>
          <s id="N11F26">duplum enim &longs;emper <lb/>paritatem &longs;ecum affert. </s>
          <s id="N11F2A">quod demon&longs;trare oportebat. </s>
        </p>
        <figure id="id.077.01.059.1.jpg" xlink:href="077/01/059/1.jpg"/>
        <p id="N11F30" type="main">
          <s id="N11F32">Porr&ograve; maxima in his duabus &longs;e&que;ntibus propo&longs;itionibus <lb/>adhibenda e&longs;t diligentia; quibus tota rerum Mechanicarum <lb/>ratio in nititur. </s>
          <s id="N11F38">Quocirca vt harum propo&longs;itionum demon&shy;<lb/>&longs;trationes perfect&egrave; intelligere po&longs;&longs;imus; pr&aelig;ter eos argumen&shy;<lb/>tandi modos, quorum ante quintam huius propo&longs;itionem <lb/>meminimus; alterum quo&que; modum, quo Archimedes in <lb/>
<arrow.to.target n="fig25"/><lb/>hac &longs;exta propo&longs;itione vtitur, noui&longs;&longs;e oportet. </s>
          <s id="N11F47">vt &longs;cilicet, &longs;i ma<lb/>gnitudo A &aelig;&que;ponderatip&longs;is BC facta &longs;u&longs;pen&longs;ione ex <expan abbr="p&utilde;-cto">pun&shy;<lb/>cto</expan> D; ita &longs;cilicet, vt D &longs;it centrum grauitatis magnitudinis <lb/>ex omnibus ABC magnitudinibus compo&longs;it&aelig;; ip&longs;arum ver&ograve;
<pb xlink:href="077/01/060.jpg" pagenum="56"/>magnitudinum BC, hoc e&longs;t magnitudinis ex BC compo&longs;i&shy;<lb/>t&aelig; centrum grauitatis &longs;it punctum E; auferantur ver&ograve; BC <lb/>&agrave; linea EA, &amp; ip&longs;arum loco ponatur in E magnitudo; <lb/>qu&aelig; &longs;it vtri&longs;&que; &longs;imul BC &etail;qualis, vt in &longs;ecunda figura. </s>
          <s id="N11F5D">Dico <lb/>eodem modo pondera ABC &etail;&que;ponderare in prima figu&shy;<lb/>ra, veluti grauia AE in &longs;ecunda. </s>
        </p>
        <figure id="id.077.01.060.1.jpg" xlink:href="077/01/060/1.jpg"/>
        <p id="N11F67" type="main">
          <s id="N11F69">Primum autem, vthoc recte per <lb/>
<arrow.to.target n="fig26"/><lb/>pendamus, intelligantur pondera <lb/>BC (vt in tertia figura) &longs;eor&longs;um <lb/>&agrave; linea CA, &amp; penes di&longs;tantias EC <lb/>EB con&longs;tituta. </s>
          <s id="N11F78">quorum quidem <expan abbr="p&otilde;-derum">pon&shy;<lb/>derum</expan> &longs;it centrum grauitatis E. &longs;i igitur intelligatur poten <lb/>
<arrow.to.target n="marg44"/> tia in E &longs;u&longs;tinere pondera BC, hoc e&longs;t pondus exip&longs;is BC <lb/>compo&longs;itum: pondera uti&que; manebunt. </s>
          <s id="N11F88">qu&ograve;d &longs;i ambo pe&shy;<lb/>penderint, vt quinquaginta, potentia in E tant&ugrave;m quinqua <lb/>ginta &longs;u&longs;tinebit. </s>
          <s id="N11F8E">quoniam totum &longs;u&longs;tinebit pondus ex ip&longs;is <lb/>compo&longs;itum, auferantur ver&ograve; pondera BC &agrave; &longs;itu BC, intelli <lb/>ganturqu&egrave; pondera e&longs;&longs;e in E con&longs;tituta; hoc e&longs;t vnum &longs;it <lb/>pondus ex ip&longs;is &longs;imul iunctis compo&longs;itum, cuius <expan abbr="c&etilde;trum">centrum</expan> gra&shy;<lb/>uitatis &longs;it in E con&longs;titutum; tunc eadem potentia in E eo&shy;<lb/>dem modo hoc pondus &longs;u&longs;tinebit; propterea quod <expan abbr="eod&etilde;">eodem</expan> mo&shy;<lb/>do quinquaginta tant&ugrave;m &longs;u&longs;tinebit. </s>
          <s id="N11FA4">Quare pondera BC <expan abbr="t&atilde;">tam</expan> <lb/>ex di&longs;tantijs EC EB grauitant, qu&agrave;m &longs;i vtra&que; in E con <lb/>&longs;tituta fuerint; vel quod idem e&longs;t, qu&agrave;m pondus ip&longs;is BC &longs;i&shy;<lb/>mul &aelig;quale in E po&longs;itum. </s>
          <s id="N11FB0">Ex quo patetid, quod initio pr&etail;&shy;<lb/>fati &longs;um us, nempe, vnumquodqu&egrave; graue in eius centro gra&shy;<lb/>uitatis propri&egrave; grauitare. </s>
          <s id="N11FB6">Quocum &que; enim modo <expan abbr="ead&etilde;">eadem</expan> gra<lb/>uia &longs;e&longs;e habent, eodem &longs;emper modo in eius grauitatis <expan abbr="c&etilde;tro">centro</expan> <lb/>grauitant. </s>
        </p>
        <p id="N11FC4" type="margin">
          <s id="N11FC6"><margin.target id="marg44"/><emph type="italics"/>per def. <lb/>cent. </s>
          <s id="N11FCE">grau.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.060.2.jpg" xlink:href="077/01/060/2.jpg"/>
        <p id="N11FD6" type="main">
          <s id="N11FD8">Quibus cognitis, intelligantur nunc grauia BC in linea <lb/>CA po&longs;ita e&longs;&longs;e; ut in &longs;uperiori figura: &amp; ut quod propo&longs;itum <lb/>fuit, o&longs;tendatur; hoc modo argumentari licebit. </s>
          <s id="N11FDE">Quoniam <lb/>enim magnitudines BC &longs;uam habent grauitatem in E, &longs;iqui <lb/>dem pro vna tant&ugrave;m intelliguntur magnitudine ex BC com<lb/>po&longs;ita, cuius punctum E centrum grauitatis exi&longs;tit. </s>
          <s id="N11FE6">in <expan abbr="&longs;ec&utilde;">&longs;ecum</expan> <lb/>da ver&ograve; figura magnitudo E &longs;imiliter &longs;uam habet <expan abbr="grauitat&etilde;">grauitatem</expan> <lb/>in puncto E; quod e&longs;t eius <expan abbr="centr&utilde;">centrum</expan> grauitatis. </s>
          <s id="N11FF8">at&que; magnitu
<pb xlink:href="077/01/061.jpg" pagenum="57"/>do E e&longs;tip&longs;is BC &longs;imul &longs;umptis &etail;qualis. </s>
          <s id="N11FFE">di&longs;tanti&etail; ver&ograve; AD <lb/>DE &longs;unt &aelig;quales, cum &longs;int &etail;edem; erit vti&que; punctum D in <lb/>&longs;ecunda figura centrum grauitatis magnitudinis ex AE com&shy;<lb/>po&longs;it&aelig;, veluti D in prima figura ip&longs;arum ABC centrum gra<lb/>uitatis exi&longs;tit. </s>
          <s id="N12008">ac propterea in vtra&que; figura pondera &aelig;&que;&shy;<lb/>ponderabunt: </s>
        </p>
        <p id="N1200C" type="main">
          <s id="N1200E">C&aelig;terum hoc quo&que; o&longs;tendemus hoc pacto. </s>
        </p>
        <figure id="id.077.01.061.1.jpg" xlink:href="077/01/061/1.jpg"/>
        <p id="N12013" type="main">
          <s id="N12015">Ii&longs;dem nam&que; po&longs;itis; &aelig;&que;ponderarent &longs;cilicet grauia <lb/>ABC facta ex D &longs;u&longs;pen&longs;ione. </s>
          <s id="N12019">&longs;itqu&egrave; punctum E <lb/>centrum grauitatis ponderum CB. qu&aelig; quidem pondera <lb/>CB grauitatis centrum habeant in linea CB. Dico pondus <lb/>A ponderi ip&longs;is CB &longs;imul &longs;umptis &aelig;quali in E con&longs;ti&shy;<lb/>tuto &aelig;&que;ponderare. </s>
          <s id="N12023">Mente concipiamus di&longs;tantias EC <lb/>EB, manente centro E, circa ip&longs;um circumuerti po&longs;&longs;e; <lb/>vt mod&ograve; &longs;int in FEG, mod&ograve; in HEK. &longs;imiliter in&shy;<lb/>telligantur pondera CB, mod&ograve; in FG, mod&ograve; in HK <lb/>exi&longs;tere. </s>
          <s id="N1202D">Quoniam igitur punctum E. centrum e&longs;t <lb/>grauitatis ponderum CB; erit idem E (c&ugrave;m &longs;itum <lb/>nonmutet) centrum grauitatis ponderum in &longs;itu FG, ac <lb/>ponderum in HK exi&longs;tentium. </s>
          <s id="N12035">Quiaver&ograve; vnumquod&shy;<lb/>&que; pondus (ex dictis) propi&egrave; in eius centro grauitatis graui<lb/>tat; pondera &longs;imul CB &longs;iue &longs;int in FG, &longs;iue in HK, proprie <lb/>in puncto E grauitabunt. </s>
          <s id="N1203D">At ver&ograve; quoniam idem
<pb xlink:href="077/01/062.jpg" pagenum="58"/>pondus vnam &amp; eandem &longs;emper habet grauitatem; erit <expan abbr="p&otilde;dus">pondus</expan> <lb/>ex CB compo&longs;itum &aelig;&que;graue, tam in &longs;itu CB, qu&agrave;m in <lb/>FG, &amp; in &longs;itu HK. con&longs;iderando nempe pondera CB (ut <lb/>revera &longs;unt) nilaliud e&longs;&longs;e ni&longs;i vnum tant&ugrave;m pondus ex CB <lb/>compo&longs;itum. </s>
          <s id="N1204F">Ex quibus per&longs;picuum e&longs;t, punctum E eodem <lb/>&longs;emper modo grauitare. </s>
          <s id="N12053">Quare quoniam pondera CB in &longs;i&shy;<lb/>tu CB ip&longs;i A &etail;&que;ponderant, &longs;uamqu&egrave; habent grauitatem <lb/>in puncto E; eadem pondera CB &longs;iue &longs;int in FG, &longs;iue in <lb/>HK, eidem ponderi A &aelig;&que;ponderabunt. </s>
          <s id="N1205B">&longs;iquidem propri&egrave; <lb/>&longs;emper grauitant in E, &amp; eandem &longs;emper habent <expan abbr="grauita-t&etilde;">grauita&shy;<lb/>tem</expan> Intelligatur deni&que; HEK in centrum mundi tendere; e&shy;<lb/>runtvti&que; vtra&que; pondera HK, tanquam in puncto E <expan abbr="c&otilde;">com</expan> <lb/>&longs;tituta, vt ex prima propo&longs;itione no&longs;trorum Mechanicorum <lb/>elici pote&longs;t, quamuis per&longs;e notum &longs;it. </s>
          <s id="N1206F">&longs;iquidem &longs;eor&longs;um pon<lb/>dus H &longs;ecund&ugrave;m eius centrum grauitatis propri&egrave; grauitat &longs;u<lb/>per puncto E; pondus ver&ograve; K e&longs;t, tanquam ex E appen&longs;um; <lb/>vnd&egrave; &amp; in eodem puncto E quo&que; grauitat. </s>
          <s id="N12077">Ita&que; <expan abbr="quoni&atilde;">quoniam</expan> <lb/>ambo propri&egrave; grauitant in E, erunt pondera HK perinde, <lb/>ac&longs;i vnum e&longs;&longs;et pondusip&longs;is HK, hoc e&longs;tip&longs;is CB &aelig;quale, cu<lb/>ius centrum grauitatis &longs;it in E con&longs;titutum. </s>
          <s id="N12083">atver&ograve; pondus <lb/>A ip&longs;is CB in &longs;itu HK exi&longs;tentibus &aelig;&que;ponderat. </s>
          <s id="N12087">ergo <expan abbr="id&etilde;">idem</expan> <lb/>pondus A ip&longs;is CB in E con&longs;titutis, hoc e&longs;t ponderi ip&longs;is CB <lb/>&longs;imul &longs;umptis &etail;quali in E po&longs;ito &aelig;&que;ponderabit. </s>
          <s id="N12091">quod de&shy;<lb/>mon&longs;trare oportebat. </s>
        </p>
        <p id="N12095" type="main">
          <s id="N12097">Quod idem quo&que;, &longs;i plura e&longs;&longs;ent pondera, &longs;imiliter o&shy;<lb/>&longs;tendetur. </s>
        </p>
        <p id="N1209B" type="main">
          <s id="N1209D">Valetita&que; con&longs;e&que;ntia, punctum D centrum e&longs;tgra&shy;<lb/>uitatis magnitudinis ex ponderibus ABC compo&longs;it&etail;; ergoi&shy;<lb/>dem punctum D centrum e&longs;t grauitatis ponderis in A, &amp; <expan abbr="p&otilde;">pom</expan> <lb/>derisip&longs;is BC &longs;imul &etail;qualis in E con&longs;tituti. </s>
          <s id="N120A9">ex quo con&longs;equi&shy;<lb/>tur, qu&ograve;d &longs;i magnitudines ABC ex D &aelig;&que;ponderant, ergo <lb/>ex eodem D magnitudo ip&longs;is BC &longs;imul &aelig;qualis in E po&longs;ita, <lb/>&amp; magnitudo A &aelig;&que;ponderabunt. </s>
          <s id="N120B1">qu&ograve;d &longs;i rect&egrave; perpenda&shy;<lb/>mus, nil aliud &longs;unt pondera in BC, ni&longs;i magnitudo in E con&shy;<lb/>&longs;tituta. </s>
          <s id="N120B7">&longs;iquidem punctum E ip&longs;ius centrum grauitatis <lb/>exi&longs;tit </s>
        </p>
        <p id="N120BB" type="main">
          <s id="N120BD">In no&longs;tro autem Mechanicorum libro in quinta propo&longs;i-
<pb xlink:href="077/01/063.jpg" pagenum="59"/>tione tractatus de libra duas attulimus demon &longs;trationes <expan abbr="o&longs;t&etilde;-tes">o&longs;ten&shy;<lb/>tes</expan> duo pondera vt CB tam in punctis CB ponderare, qu&agrave;m &longs;i <lb/>vtra&que; ex puncto E &longs;u&longs;pendantur. </s>
          <s id="N120CB">At ver&ograve; quo niam demon <lb/>&longs;trationes ibi allat&aelig; ijs indigent, qu&etail; Archimedes in &longs;e&que;n&shy;<lb/>ti &longs;exta propo&longs;itione demon&longs;trauit, idcirco demon&longs;trationes <lb/>ill&aelig; huic loco non &longs;unt oportun&aelig;; vt ex ip&longs;is&longs;umi po&longs;&longs;it tan&shy;<lb/>quam demon&longs;tratum pondera CB, tam in punctis CB pon&shy;<lb/>derare, qu&agrave;m &longs;i vtra&que; ex E &longs;u&longs;pendantur. </s>
          <s id="N120D7">Quare hoc loco h&etail; <lb/>tant&ugrave;m &longs;ufficiant rationes, qu&aelig; dict&aelig; &longs;unt. </s>
          <s id="N120DB">Ex quibus pote&longs;t <lb/>Archime des di&longs;tam con&longs;e&que;ntiam colligere; nemp&egrave; magni&shy;<lb/>tudines ABC ex D &aelig;&que;ponderant, auferantur autem BC, <lb/>&amp; loco ip&longs;arum vtri&longs;&que; &longs;imul &etail;&que;grauis ponatur magnitu&shy;<lb/>do in E; &longs;imiliter h&etail;c magnitudo ip&longs;i A &aelig;&que;ponderabit. </s>
          <s id="N120E5">Po&shy;<lb/>&longs;tea ver&ograve; ex ijs, qu&aelig; Archimedes demon&longs;trauit, fieri pote&longs;t re <lb/>gre&longs;&longs;us; v<gap/>aperti&ugrave;s, manife&longs;ti&ugrave;&longs; qu&egrave; cogno&longs;cere valeamus, pon<lb/>dera BC ita ponderare, ac &longs;i vtra&que; ex puncto E &longs;u&longs;pen&shy;<lb/>dantur. </s>
        </p>
        <figure id="id.077.01.063.1.jpg" xlink:href="077/01/063/1.jpg"/>
        <p id="N120F4" type="main">
          <s id="N120F6">C&etail;terum hoc loco Archimedes non &longs;ol&ugrave;m de duobus, <expan abbr="ver&utilde;">verum</expan> <lb/>etiam de pluribus ponderibus idip&longs;um <expan abbr="intelligend&utilde;">intelligendum</expan> admittit. <lb/>vt &longs;i magnitudines STVXZM &aelig;&que;ponderent facta <expan abbr="&longs;u&longs;p&etilde;&longs;io">&longs;u&longs;pen&longs;io</expan> <lb/>ne ex puncto C. &longs;itqu&egrave; magnitudinum MZ <expan abbr="centr&utilde;">centrum</expan> grauitatis <lb/>D; ip&longs;arum ver&ograve; STVX &longs;it centrum grauitatis E. &longs;i ita&que; ma <lb/>gnitudines STVX, &amp; ZM ex C &aelig;&que;ponderant; auferantur <lb/>STVX, quarum loco ponatur in E magnitudo ip&longs;is STVX &longs;i <lb/>mul &longs;umptis &etail;qualis: auferanturqu&egrave; ZM, at&que; <expan abbr="ip&longs;ar&utilde;">ip&longs;arum</expan> loco po <lb/>natur in D magnitudo ip&longs;is ZM &longs;imul &etail;qualis; tunclicetinfer <lb/>re, ergo h&aelig; magnitudines in ED po&longs;it&aelig; &etail;&que;pondera&shy;<lb/>bunt. </s>
          <s id="N12120">Quod quidem ijsdem pror&longs;us modis o&longs;tendentur. <lb/>pr&aelig;&longs;ertim &longs;i mente concipiamus di&longs;tantias ES EX,
<pb xlink:href="077/01/064.jpg" pagenum="60"/>nec non magnitudines STVX in &longs;uis di&longs;tantijs circa <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis E circumuerti po&longs;&longs;e; veluti di&longs;tantias DZ DM, ma <lb/>gnitudine&longs;qu&egrave; ZM circacentrum D. moueantur autem <lb/>SEX, &amp; ZDM, donec in centrum mundi vergant. </s>
          <s id="N12132">&longs;imiliter <lb/>o&longs;tendetur magnitudines STVX e&longs;&longs;e, ac &longs;i in E e&longs;&longs;ent appen <lb/>&longs;&etail;, &longs;iue con&longs;titut&etail;; magnitudines ver&ograve; ZM ac &longs;i in D po&longs;i&shy;<lb/>t&aelig; fuerint. </s>
          <s id="N1213A">&amp;c. </s>
          <s id="N1213C">Ex quibus &longs;equitur, &longs;i punctum C centrum <lb/>e&longs;t grauitatis magnitudinum STVXZM. ponatur magnitu&shy;<lb/>do ip&longs;is STVX &longs;imul &longs;umptis &etail;qualis in E; magnitudo au<lb/>tem ip&longs;is ZM &longs;imul &aelig;qualis in D; punctum C &longs;imiliter <lb/>ip&longs;arum quo&que; centrum grauitatis exi&longs;tet. </s>
          <s id="N12146">vnde vtro&que; mo <lb/>do &aelig;&que;ponderabunt. </s>
          <s id="N1214A">&amp; ita in alijs, &longs;i plures fuerint magni&shy;<lb/>tudines. </s>
        </p>
        <p id="N1214E" type="head">
          <s id="N12150">PROPOSITIO. VI.</s>
        </p>
        <p id="N12152" type="main">
          <s id="N12154">Magnitudines commen&longs;urabiles ex di&longs;tantijs <lb/>eandem permutatim proportionem habentibus, <lb/>vt grauitates, &aelig;&que;ponderant. </s>
        </p>
        <p id="N1215A" type="main">
          <s id="N1215C"><emph type="italics"/>Commen&longs;urabiles &longs;int magnitudines AB quarum centra<emph.end type="italics"/> grauita&shy;<lb/>tis <emph type="italics"/>AB, &amp; qu&aelig;dam &longs;it di&longs;tantia E D, &amp; vt<emph.end type="italics"/> &longs;e habet grauitas ma&shy;<lb/>gnitudinis <emph type="italics"/>A ad<emph.end type="italics"/> grauitatem magnitudinis <emph type="italics"/>B, ua &longs;it <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> <lb/>DC ad distantiam CE. <expan abbr="ostend&etilde;d&utilde;">ostendendum</expan> e&longs;i<emph.end type="italics"/>, &longs;i centra grauitatis AB fue <lb/>rint in punctis ED con&longs;tituta, hoc e&longs;t A in E, &amp; B in D; <lb/><emph type="italics"/>magnitudinis ex vtri&longs;qu&egrave;<emph.end type="italics"/> magnitudinibus <emph type="italics"/>AB compo&longs;it&aelig; centrum <lb/>grauitatis e&longs;&longs;e punctum C. Quoniam enim ita est<emph.end type="italics"/> magnitudo <emph type="italics"/>A ad<emph.end type="italics"/><lb/>magnitudinem <emph type="italics"/>B, vt DC ad CE. e&longs;t autem<emph.end type="italics"/> magnitudo <emph type="italics"/>A ip&longs;i <lb/>
<arrow.to.target n="marg45"/> B commen&longs;urabilis; erit &amp; CD ip&longs;i CE commen&longs;urabilis; hoc e&longs;t <lb/>recta linea rect&aelig; line&aelig;<emph.end type="italics"/> commen&longs;urabilis exi&longs;tet. <emph type="italics"/>Quare ip&longs;arum EC <lb/>CD communis reperitur men&longs;ura. </s>
          <s id="N121B4">qu&aelig; quidem &longs;it N. deinde ponatur <lb/>ip&longs;i EC &aelig;qualis vtra&que; DG DK; ip&longs;i ver&ograve; DC &aelig;qualis EL. &amp; <lb/>quoniam &aelig;qualis est DG ip&longs;i CE<emph.end type="italics"/>, communi addita CG, <emph type="italics"/>erit DC <lb/>ip&longs;i EG &aelig;qualis<emph.end type="italics"/>; &longs;ed DC e&longs;t ip&longs;i EL &etail;qualis: <emph type="italics"/>erit igitur LE &aelig;qua&shy;<lb/>lis ip&longs;i EG.<emph.end type="italics"/> quare vtra&que; LE EG &etail;qualis e&longs;t ip&longs;i DC. <emph type="italics"/>ac propte<emph.end type="italics"/>
<pb xlink:href="077/01/065.jpg" pagenum="61"/><emph type="italics"/>rea dupla est LG ip&longs;ius DC.<emph.end type="italics"/> quia ver&ograve; vtra&que; DG DK &aelig;qualis <lb/>facta e&longs;t ip&longs;i CE, erit <emph type="italics"/>&amp; ip&longs;a quo&que; GK ip&longs;ius CE<emph.end type="italics"/> dupla. <emph type="italics"/>Quare <lb/>N <expan abbr="vtr&atilde;&que;">vtran&que;</expan> LG Gk metitur, c&ugrave;m &amp; ip&longs;arum medietates<emph.end type="italics"/> DC CE <lb/>
<arrow.to.target n="fig27"/><lb/>metiatur. <emph type="italics"/>Et quoniam<emph.end type="italics"/> magnitudo <emph type="italics"/>A ita e&longs;t ad<emph.end type="italics"/> magnitudinem <lb/><emph type="italics"/>B, vt DC ad CE, ut autem DC ad CE, ita e&longs;t LG ad G<emph.end type="italics"/>K, <emph type="italics"/>utra&que; <lb/>enim vtriu&longs;&que; duplex exi&longs;tit<emph.end type="italics"/> (&longs;iquidem LG dupla e&longs;t ip&longs;ius DC, <lb/>&amp; GK itidem ip&longs;ius CE duplex) <emph type="italics"/>erit<emph.end type="italics"/> magnitudo <emph type="italics"/>A ad<emph.end type="italics"/>
<arrow.to.target n="marg46"/> magni&shy;<lb/>tudinem <emph type="italics"/>B, ut LG ad G<emph.end type="italics"/>k; &amp; conuertendo magnitudo B ad <lb/>magnitudinem A, vt KG ad GL. <emph type="italics"/>Quotuplex autem est LG ip&longs;ius <lb/>N, totuplex &longs;it<emph.end type="italics"/> magnitudo <emph type="italics"/>A ip&longs;ius F, erit vti&que; LG ad N, vt<emph.end type="italics"/><lb/>magnitudo <emph type="italics"/>A ad F, atqui est KG ad LG, vt<emph.end type="italics"/> magnitudo <emph type="italics"/>B ad<emph.end type="italics"/><lb/>magnitudinem <emph type="italics"/>A:<emph.end type="italics"/> LG ver&ograve; ad N e&longs;t, vt magnitudo A ad
<arrow.to.target n="marg47"/> <expan abbr="i-ps&atilde;">i&shy;<lb/>psam</expan> F, <emph type="italics"/>ex &aelig;quali igitur erit KG ad N, vt<emph.end type="italics"/> magnitudo <emph type="italics"/>B ad F quare &aelig;&shy;<lb/>&que;multiplex e&longs;t<emph.end type="italics"/> kG <emph type="italics"/>ip&longs;ius N, veluti<emph.end type="italics"/> magnitudo <emph type="italics"/>B ip&longs;ius F. demon <lb/><expan abbr="&longs;trat&utilde;">&longs;tratum</expan> <expan abbr="a&utilde;t">aunt</expan> e&longs;t<emph.end type="italics"/> <expan abbr="magnitudin&etilde;">magnitudinem</expan> <emph type="italics"/>A ip&longs;ius F multiplicem e&longs;&longs;e<emph.end type="italics"/>, &longs;iquidem e&longs;t <lb/>magnitudo A ad ip&longs;am F, vt LG ad N, qu&aelig; quidem LG mul <lb/>tiplex e&longs;t ip&longs;ius N. <emph type="italics"/>qua propter F ip&longs;arum AB communis existit men <lb/>&longs;ura. </s>
          <s id="N12290">Jta&que; diui&longs;a LG in partes<emph.end type="italics"/> LH, HE, EC, CG, <emph type="italics"/>ip&longs;i N aquales<emph.end type="italics"/>, <lb/>cadent vti&que; diui&longs;iones in punctis EC, quoniam <expan abbr="Nips&atilde;">Nipsam</expan> EC
<arrow.to.target n="marg48"/><lb/>metitur, nec non ip&longs;am quo&que; LE metitur; c&ugrave;m &longs;it LE ip&longs;i <lb/>CD &aelig;qualis. </s>
          <s id="N122A8">eruntqu&egrave; diui&longs;iones LH, HE, EC, CG, numero <lb/>pares; c&ugrave;m N dimidiam ip&longs;ius LG, hoc e&longs;t CD metiatur.
<pb xlink:href="077/01/066.jpg" pagenum="62"/><emph type="italics"/>Aver&ograve;<emph.end type="italics"/> &longs;imiliter diui&longs;a <emph type="italics"/>in partes<emph.end type="italics"/> OP QR <emph type="italics"/>ip&longs;i F &aelig;quales; &longs;ectio&shy;<lb/>nes<emph.end type="italics"/> LH, HE, EC, CG <emph type="italics"/>in LG existentes magnitudini N &aelig;qua&shy;<lb/>les, erunt numero &aelig;quales &longs;ectionibus<emph.end type="italics"/> OPQR <emph type="italics"/>in<emph.end type="italics"/> magnitudine <emph type="italics"/>A <lb/>existentibus ip&longs;i F &aelig;qualibus.<emph.end type="italics"/> Diuidantur &longs;ectiones LH, HE, EC, <lb/>
<arrow.to.target n="fig28"/><lb/>CG bifariam in punctis STVX. <emph type="italics"/>&longs;i it a&que; in vnaqua&que; &longs;estio <lb/>ne ip&longs;ius LG apponatur magnitudo &aelig;qualis ip&longs;i F, qu&aelig; centrum gra&shy;<lb/>uitatis babeat in medio &longs;ectionis<emph.end type="italics"/>; vt &longs;i in LH ponatur magnitudo <lb/>S, in HE magnitudo T, in EC magnitudo V, &amp; in <lb/>CG magnitudo X; ip&longs;arum qu&egrave; vna qu&aelig;&que; STVX &longs;it ip&longs;i <lb/>F &aelig;qualis: habeat ver&ograve; magnitudo S &longs;uum grauitatis <expan abbr="centr&utilde;">centrum</expan>, <lb/>quod &longs;it punctum S, in medio &longs;ectionis LH, nemp&egrave; in <expan abbr="p&utilde;-cto">pun&shy;<lb/>cto</expan> S; &longs;imiliter c&aelig;ter&aelig; magnitudines TVX habeant <expan abbr="c&etilde;rra">cerrra</expan> <lb/>grauitatis; qu&aelig; &longs;int puncta TVX, in medio &longs;ectionum HE, <lb/>EC, CG, in punctis nemp&egrave; TVX, erunt centra grauitatisma <lb/>gnitudinum STVX in recta linea con&longs;tituta, &amp; quoma<gap/>o <lb/>SH dimidia e&longs;t ip&longs;ius LH, veluti HT ip&longs;ius HE, erit ST, <lb/>ip&longs;ius LE dimidia, vnaqu&aelig;&que; ver&ograve; LH HE dimidia <lb/>quo&que; e&longs;t ip&longs;ius LE, &longs;iquidem LH, HE inter &longs;e &longs;unt &etail;qua <lb/>les; erit igitur ST vnicui&que; LH, &amp; HE &aelig;qualis. </s>
          <s id="N12310">eodem qu&egrave; <lb/>pror&longs;us modo o&longs;tendeturi TV &etail;qualem e&longs;&longs;e vnicui&que; HE <lb/>EC. &amp; VX &aelig;qualem EC. &amp; CG. &amp; quoniam omnes
<pb xlink:href="077/01/067.jpg" pagenum="63"/>LH, HE, EC, CG, inter &longs;e &longs;unt &aelig;quales; erunt ST TV VX in<lb/>ter&longs;e &aelig;quales. </s>
          <s id="N1231C">quare line&aelig; inter centra grauitatis magnitudi&shy;<lb/>num STVX exi&longs;tentes &longs;unt inter &longs;e &etail;quales. <emph type="italics"/>omnes ver&ograve; magni<lb/>tudines<emph.end type="italics"/> STVX &longs;imul <emph type="italics"/>&longs;unt &aelig;quales ip&longs;i A<emph.end type="italics"/>, quandoquidem ip&longs;is <lb/>OPQR, &amp; numero, &amp; magnitudine &longs;unt &etail;quales; ergo <emph type="italics"/>magni&shy;<lb/>tudinis ex omnibus<emph.end type="italics"/> magnitudinibus STVX <emph type="italics"/>compo&longs;it&aelig; centrumgra <lb/>uitatis erit punstum E. c&ugrave;m omnes<emph.end type="italics"/> magnitudines STVX <emph type="italics"/>&longs;int nu&shy;<lb/>mero pares.<emph.end type="italics"/> quippe c&ugrave;m &longs;int in &longs;ectionibus LH HE EC CG nu<lb/>mero paribus. </s>
          <s id="N1234A">&amp; <emph type="italics"/>LE ip&longs;i EG &aelig;qualis exi&longs;tat.<emph.end type="italics"/> qu&ograve;d &longs;i LE e&longs;tip&longs;i <lb/>EG &aelig;qualis, demptis &aelig;qualibus LS GX &aelig;qualibus, &longs;iquidem <lb/>&longs;unt dimidi&aelig; &longs;ectionum LH CG &aelig;qualium: erunt SE EX
<arrow.to.target n="marg49"/> in&shy;<lb/>ter&longs;e &aelig;quales, vnde ex pr&aelig;cedenti colligitur, punctum E cen&shy;<lb/>trum e&longs;&longs;e grauitatis magnitudinum STVX. <emph type="italics"/>&longs;imiliter autem <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>detur, qu&ograve;d &longs;i<emph.end type="italics"/> diuidatur GK in partes GD DK ip&longs;i N &aelig;quales; <lb/>cadetvti&que; diui&longs;ionum aliqua in <expan abbr="p&utilde;cto">puncto</expan> D; &longs;iquidem Nip&longs;as <lb/>GD DK metitur; c&ugrave;m vtra&que; &longs;it &aelig;qualisip&longs;i EC. diui&longs;ione&longs;&shy;<lb/>qu&egrave; GD DK numero pares erunt; c&ugrave;m N dimidiam ip&longs;ius
<arrow.to.target n="marg50"/><lb/>GK, ip&longs;am &longs;cilicet EC metiatur. </s>
          <s id="N12379">&longs;i ita&que; diuidatur GD DK <lb/>bifariam in punctis ZM. deinde diuidatur magnitudo B <lb/>in partes ip&longs;i F &aelig;quales; &longs;ectiones GD DH in GK exi&longs;tentes <lb/>ip&longs;i N &aelig;quales, erunt numero &aelig;quales &longs;ectionibus in ma <lb/>gnitudine B exi&longs;tentibus ip&longs;i F &aelig;qualibus. </s>
          <s id="N12383">quare <emph type="italics"/>vnicui&que; <lb/>partium ip&longs;ius GK apponatur magnitudo &aelig;qualis ip&longs;i F; centrum gra&shy;<lb/>uitatis habens in medio &longs;ectionis<emph.end type="italics"/>; vt <expan abbr="pon&atilde;tur">ponantur</expan> magnitudines ZM in <lb/>&longs;ectionibus GD DK, ita vt magnitudinum centra grauita&shy;<lb/>tis, qu&aelig; &longs;int ZM, in medio &longs;ectionum GD DK, in punctis <lb/>nemp&egrave; ZM &longs;int con&longs;tituta, <emph type="italics"/>omnes autem magnitudines<emph.end type="italics"/> ZM &longs;i <lb/>mul <emph type="italics"/>&longs;unt &aelig;quales ip&longs;i B. magnitudinis ex omnibus<emph.end type="italics"/> magnitudinibus <lb/>ZM <emph type="italics"/>compo&longs;it&aelig; centrum grauitatis erit punctum D.<emph.end type="italics"/> c&ugrave;m &longs;it ZD <lb/>&etail;qualis DM. <emph type="italics"/>&longs;ed<emph.end type="italics"/> magnitudines STVX &longs;unt magnitudini A <lb/>&aelig;quales, &amp; ZM ip&longs;i B ergo <emph type="italics"/>magnitudo A e&longs;t<emph.end type="italics"/> tanquam <emph type="italics"/>impo&longs;ita <lb/>ad E, ip&longs;a ver&ograve; B ad D.<emph.end type="italics"/> eodem &longs;cilicet modo &longs;e habebit ma&shy;<lb/>gnitudo A impo&longs;ita ad E, vt &longs;e habent magnitudines STVX; <lb/>ip&longs;a ver&ograve; B &longs;e habebit ad D, vt magnitudines ZM. <emph type="italics"/>&longs;unt au<lb/>tem magnitudines<emph.end type="italics"/> STVXZM <emph type="italics"/>inter &longs;e &aelig;quales<emph.end type="italics"/>, c&ugrave;m vnaqu&aelig; &que; &longs;it <lb/>ip&longs;i F &etail;qualis: &longs;untqu&egrave; omnes, (hoc e&longs;t ip&longs;arum centra graui<lb/>tatis) <emph type="italics"/>inrecta linea po&longs;it&aelig;; quarum centragrauitatis po&longs;ita &longs;unt inter &longs;e<emph.end type="italics"/>
<pb xlink:href="077/01/068.jpg" pagenum="64"/><emph type="italics"/>&aelig;qualiter di&longs;tantia;<emph.end type="italics"/> &longs;iquidem o&longs;ten&longs;um e&longs;t ST TV VX inter&shy;<lb/>&longs;e &aelig;quales e&longs;&longs;e. </s>
          <s id="N123EE">Eodemqu&egrave; modo o&longs;tendetur XZ ZM c&aelig;teris <lb/>&aelig;quales e&longs;&longs;e. <emph type="italics"/>&amp; &longs;unt<emph.end type="italics"/> magnitudines STVXZM <emph type="italics"/>numero pares,<emph.end type="italics"/><lb/>c&ugrave;m &longs;ectiones totius LK, ( in quibus in&longs;unt) ip&longs;i N &aelig;quales <lb/>&longs;int inter &longs;e &etail;quales, &amp; numero pares. </s>
          <s id="N12401">c&ugrave;m o&longs;ten&longs;um &longs;it &longs;ectio <lb/>
<arrow.to.target n="marg51"/> nes in LG, &amp; in Gk exi&longs;tentes numero pares e&longs;&longs;e. <emph type="italics"/>con&longs;tat magni&shy;<lb/>tudinis ex omnibus<emph.end type="italics"/> STVXZM magnitudinibus <emph type="italics"/>compo&longs;it&aelig; centrum<emph.end type="italics"/><lb/>
<arrow.to.target n="marg52"/> <emph type="italics"/>grauitatis e&longs;&longs;e medietatem rest&aelig; line&aelig;, in qua centra grauitatis magnitu<lb/>dinum habentur. </s>
          <s id="N12421">Ita&que; c&ugrave;m LE &longs;it &aelig;qualis C D, EC ver&ograve; ip&longs;i D<emph.end type="italics"/>k, <lb/><emph type="italics"/>tota LC &aelig;qualis erit CK.<emph.end type="italics"/> c&ugrave;m autem &longs;int LHDK &aelig;quales; &longs;i&shy;<lb/>quidem &longs;unt eidem N &aelig;quales, &amp; harum medietates, hoc e&longs;t <lb/>LS ip&longs;i MK &etail;qualis erit. </s>
          <s id="N12431">&amp; ob id SC ip&longs;i CM e&longs;t &aelig;qualis. <lb/>at ver&ograve; linea SM magnitudinum centra grauitatis <expan abbr="coni&utilde;git">coniungit</expan>, <lb/><emph type="italics"/>ergo magnitudinis ex omnibus<emph.end type="italics"/> STVXZM magnitudinibus <emph type="italics"/>compo&longs;i <lb/>t&aelig;centrum grauitatis est punctum C. Quare<emph.end type="italics"/> loco magnitudinum <lb/>STVX, <emph type="italics"/>po&longs;ito<emph.end type="italics"/> centro grauitatis <emph type="italics"/>A ad E, B ver&ograve;<emph.end type="italics"/> loco ip&longs;arum <lb/>ZM po&longs;ito <emph type="italics"/>ad D,<emph.end type="italics"/> erit punctum C grauitatis centrum ma&shy;<lb/>gnitudinis ex vtri&longs;&que; magnitudinibus AB compo&longs;it&aelig;. </s>
          <s id="N12460">ac <lb/>prop terea <emph type="italics"/>ex puncto C &aelig;&que;ponderabunt.<emph.end type="italics"/> ergo magnitudines AB <lb/>ex di&longs;tantijs DC CE, qu&etail; permutatim eandem habent pro. <lb/>portionem, vt grauitates, &etail;&que;ponderant. </s>
          <s id="N1246E">quod demon&longs;trare <lb/>oportebat. </s>
        </p>
        <p id="N12472" type="margin">
          <s id="N12474"><margin.target id="marg45"/><emph type="italics"/>ex<emph.end type="italics"/> 3 <emph type="italics"/>de&shy;<lb/>cimi.<emph.end type="italics"/></s>
        </p>
        <p id="N12484" type="margin">
          <s id="N12486"><margin.target id="marg46"/>11 <emph type="italics"/>quinti. <lb/>cor.<emph.end type="italics"/> 4. <emph type="italics"/>quin<lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p id="N12499" type="margin">
          <s id="N1249B"><margin.target id="marg47"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N124A4" type="margin">
          <s id="N124A6"><margin.target id="marg48"/><emph type="italics"/>iemme.<emph.end type="italics"/></s>
        </p>
        <p id="N124AE" type="margin">
          <s id="N124B0"><margin.target id="marg49"/><emph type="italics"/>ex<emph.end type="italics"/> 2. <emph type="italics"/>cor.<emph.end type="italics"/></s>
        </p>
        <p id="N124BE" type="margin">
          <s id="N124C0"><margin.target id="marg50"/><emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N124C8" type="margin">
          <s id="N124CA"><margin.target id="marg51"/>2.<emph type="italics"/>cor. </s>
          <s id="N124D1">quin<lb/>t&aelig; huius.<emph.end type="italics"/></s>
        </p>
        <p id="N124D7" type="margin">
          <s id="N124D9"><margin.target id="marg52"/>*</s>
        </p>
        <figure id="id.077.01.068.1.jpg" xlink:href="077/01/068/1.jpg"/>
        <figure id="id.077.01.068.2.jpg" xlink:href="077/01/068/2.jpg"/>
        <p id="N124E5" type="head">
          <s id="N124E7">SCHOLIVM.</s>
        </p>
        <p id="N124E9" type="main">
          <s id="N124EB">
<arrow.to.target n="marg53"/> Circa finem Gr&etail;cus codex habet, <foreign lang="greek">ta ke/nt&lt;10&gt;a tw=n me/swn megeqw=n</foreign>, <lb/>qua&longs;i dicat, centrum grauitatis magnitudinis ex omnibus <lb/>magnitudinibus STVXZM compo&longs;it&etail; medietatem e&longs;&longs;e rect&etail; <lb/>line&etail; VX, qu&etail; centra mediarum magnitudinum VX coniun<lb/>git; qu&ograve;d c&ugrave;m &longs;int omnes magnitudines numero pares; <expan abbr="itid&etilde;">itidem</expan> <lb/>e&longs;&longs;et punctum C, &amp; quamuis hoc &longs;it verum, non tamen ad hoc <lb/>re&longs;pexit Archimedes duabus de cau&longs;is. <expan abbr="N&atilde;in">Nanin</expan> &longs;ecudo corollario <lb/>pr&etail;cedentis o&longs;tendit centrum grauitatis omnium magnitu&shy;<lb/>dinum e&longs;&longs;e medietatem rect&etail; line&etail;, qu&etail; grauitatis centra om&shy;<lb/>nia coniungit. </s>
          <s id="N1250F">Deinde concludere volens punctum C <expan abbr="centr&utilde;">centrum</expan> <lb/>e&longs;&longs;e grauitatis omnium magnitudinum, &longs;tatim inquit hoc &longs;e <lb/>qui, quia LC e&longs;t ip&longs;i CK &etail;qualis, qu&etail; &longs;unt medietates totius
<pb xlink:href="077/01/069.jpg" pagenum="65"/>rect&aelig;line&etail; LK. Et non dixit, quia VC &longs;itip&longs;i CX &etail;qualis. <lb/>Quare codicem gr&aelig;cum ita re&longs;tituendum cen&longs;eo. <foreign lang="greek">ta\ke/nt&lt;10&gt;k tw=n <lb/>tou= ba\&lt;10&gt;eos megeqw=n</foreign>, vt vertimus. </s>
        </p>
        <p id="N12525" type="margin">
          <s id="N12527"><margin.target id="marg53"/>*</s>
        </p>
        <p id="N1252B" type="main">
          <s id="N1252D">Ob &longs;e&que;ntis ver&ograve; demon&longs;trationis cognitionem, hoc pro <lb/>blema pri&ugrave;s o&longs;tendemus. </s>
        </p>
        <p id="N12531" type="head">
          <s id="N12533">PROBLEMA.</s>
        </p>
        <p id="N12535" type="main">
          <s id="N12537">Duarum expo&longs;itarum magnitudinum incommen&longs;urabi&shy;<lb/>lium altera vtcum&que; &longs;ecetur; magnitudinem tota &longs;ecta ma&shy;<lb/>gnitudine minorem, &amp; altero &longs;egmentomaiorem, alteri ve&shy;<lb/>r&ograve; expo&longs;it&aelig; magnitudini commen&longs;urabilem inuenire. </s>
        </p>
        <p id="N1253F" type="main">
          <s id="N12541">Sint du&aelig; magnitudi&shy;<lb/>nes incommen&longs;urabiles <lb/>
<arrow.to.target n="fig29"/><lb/>AE BC. &longs;eceturqu&egrave; ip&longs;a&shy;<lb/>rum altera, put&agrave; BC, vt&shy;<lb/>cum&que; in D. oportet <lb/>magnitudinem inuenire <lb/>minorem quidem BC, <lb/>maiorem ver&ograve; BD, qu&aelig; &longs;itip&longs;i AE commen&longs;urabilis. </s>
          <s id="N12556">Au&shy;<lb/>feratur ab AE pars dimidia, rur&longs;us dimidi&aelig; partis ip&longs;ius AE <lb/>dimidia auferatur; &amp; eius, qu&aelig; remanet, adhuc dimidia; idqu&egrave; <lb/>&longs;emper fiat, donec relinquatur magnitudo minor, qu&agrave;m DE. <lb/>quod quidem per&longs;picuum e&longs;t po&longs;&longs;e fieri ex prima decimi Eu&shy;<lb/>clidis propo&longs;itione. </s>
          <s id="N12562">&longs;itita&que; AF, qu&aelig; minor exi&longs;tat, qu&agrave;m <lb/>DC. quippe qu&etail; AF, c&ugrave;m &longs;it abla ta ex AE &longs;emper per dimi <lb/>diam partem, metietur vti&que; AF ip&longs;am AE. Deinde mul&shy;<lb/>tiplicetur AF &longs;uper BD, tum demum multiplicatio vltima, <lb/>vel in puncto D cadet, vel minus. </s>
          <s id="N1256C">&longs;i cadet; &longs;eceturex DE <lb/>magnitudo DG &etail;qualis AF. quod quidem fiet, <expan abbr="quoni&atilde;">quoniam</expan> AF <lb/>minor e&longs;t DC. Quoniam igitur AF metitur BD, &amp; DG; <lb/>metietur AF totam BG. Sed &amp; ip&longs;am AE metitur; etgo <lb/>AF ip&longs;arum BG AE communis exi&longs;tit men&longs;ura, ac propte&shy;<lb/>rea BG ip&longs;i AE commen&longs;urabilis exi&longs;tir; qu&aelig; quidem BG <lb/>minor e&longs;t BC, maior ver&ograve; BD. Si ver&ograve; vltima
<arrow.to.target n="marg54"/> multi&shy;<lb/>plicatio ip&longs;ius AF &longs;uper BD non cadet in D. &longs;ed in H, <lb/>erit vti&que; HD minor AF. nam &longs;i HD ip&longs;i AF e&longs;&longs;et &etail;qualis,
<pb xlink:href="077/01/070.jpg" pagenum="66"/>vltima multiplicatio caderet in D. &longs;i ver&ograve; maior e&longs;&longs;et HD, <lb/>qu&agrave;m AF tunc non e&longs;&longs;et vltima multiplicatio. </s>
          <s id="N1258C">quare c&ugrave;m &longs;it <lb/>DC maior AF; erit &amp; HC ip&longs;a FA maior. </s>
          <s id="N12590">&longs;i ita&que; fiat HK <lb/>&aelig;qualis AF; erit punctum K inter puncta DC. BK igitur <lb/>minor erit, qu&agrave;m BC, &amp; maior BD; eodemqu&egrave; modo o&shy;<lb/>&longs;tendetur AF ip&longs;arum Bk AE communem e&longs;&longs;e men&longs;u&shy;<lb/>ram. </s>
          <s id="N1259A">&amp; obid BK ip&longs;i AF commen&longs;urabilem exi&longs;tere. </s>
          <s id="N1259C">quod <lb/>facere oportebat. </s>
        </p>
        <p id="N125A0" type="margin">
          <s id="N125A2"><margin.target id="marg54"/>1.<emph type="italics"/>def.deci&shy;<lb/>mi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.070.1.jpg" xlink:href="077/01/070/1.jpg"/>
        <p id="N125B1" type="main">
          <s id="N125B3">C&ugrave;m autem verba &longs;e&que;ntis demon&longs;trationis aliquantu&shy;<lb/>lum &longs;int ob&longs;cura, vt vim demon&longs;trationis rect&egrave; petcipiamus, <lb/>hoc quo&que; theorema ex ijs, qu&aelig; ab Archimede hactenus de&shy;<lb/>mon&longs;trata &longs;unt, o&longs;tendemus. </s>
          <s id="N125BB">ad quod demon&longs;trandum com <lb/>muni notione indigemus, quam nos in no&longs;tro Mechanico&shy;<lb/>rum libro po&longs;uimus. </s>
          <s id="N125C1">Nemp&egrave;. </s>
        </p>
        <p id="N125C3" type="main">
          <s id="N125C5">Qu&aelig; eidem &aelig;&que;pondeiant, inter &longs;e &aelig;qu&egrave; &longs;unt grauia. </s>
        </p>
        <p id="N125C7" type="head">
          <s id="N125C9">PROPOSITIO.</s>
        </p>
        <p id="N125CB" type="main">
          <s id="N125CD">Si commen&longs;urabiles magnitudines minorem habuerint <lb/>proportionem, qu&agrave;m di&longs;tanti&etail; permutatim habent; vt &etail;&que;&shy;<lb/>ponderent, maiori opus erit magnitudine, qu&agrave;m &longs;it ea, qu&etail; <lb/>ad alteram magnitudinem minorem proportionem habet. </s>
        </p>
        <figure id="id.077.01.070.2.jpg" xlink:href="077/01/070/2.jpg"/>
        <p id="N125D8" type="main">
          <s id="N125DA">Sint magnitudines AC commen&longs;urabiles, di&longs;tanti&etail; ve&shy;<lb/>r&ograve; &longs;int ED EF. minorem autem habeat pro-
<pb xlink:href="077/01/071.jpg" pagenum="67"/>portionem A ad C, qu&agrave;m ED ad EF. Dico, vt magnitu&shy;<lb/>dines ex di&longs;tantijs ED EF &aelig;&que;ponderent, maiori o&shy;<lb/>pus e&longs;&longs;e magnitudine in F, qu&agrave;m &longs;it magnitudo A; <lb/>ita vt ip&longs;i C in D &aelig;&que;ponderare po&longs;&longs;it. </s>
          <s id="N125E8">fiat ED <lb/>ad EG, vt magnitudo A ad magnitudinem C. <lb/>Deindefiat EK &aelig;qualis EG. exponaturqu&egrave; altera ma&shy;<lb/>gnitudo L ip&longs;i A &etail;qualis. </s>
          <s id="N125F0">Quoniam igitur minorem <lb/>habet proportionem A ad C, qu&agrave;m ED ad EF, &amp; <lb/>vt A ad C, ita ED ad EG; habebit ED ad <lb/>EG minorem proportionem, qu&agrave;m ad EF. ac propterea
<arrow.to.target n="marg55"/><lb/>EF minor e&longs;t, qu&agrave;m EG. quoniam ausem A ad C <lb/>e&longs;t, vt ED ad EG, commen&longs;urabiles magnitudines <lb/>AC ex di&longs;tantijs ED EG &aelig;&que;ponderabunt. </s>
          <s id="N12601">C&ugrave;m
<arrow.to.target n="marg56"/><lb/>ver&ograve; EK &longs;it &aelig;qualis EG, magnitudines AL &aelig;&shy;<lb/>quales ex di&longs;tantis &aelig;qualibus EK EG &longs;imiliter &aelig;&que;&shy;<lb/>ponderabunt. </s>
          <s id="N1260C">At ver&ograve; quoniam C in D &aelig;&que;&shy;<lb/>ponderat ip&longs;i A in G, &longs;imiliter L in K eidem A in <lb/>G &etail;&que;ponderat; &etail;qualem habebit grauitatem C in D, vt
<arrow.to.target n="marg57"/><lb/>L in K. Ita&que; quoniam di&longs;tantia EG &aelig;qualis e&longs;t di&longs;tan<lb/>ti&aelig; Ek, longitudo EK maior erit longitudine EF. ergo <lb/>magnitudines AL &etail;quales ex in&aelig;qualibus di&longs;tantijs EK
<arrow.to.target n="marg58"/><lb/>EF non &etail;&que;ponderabunt. </s>
          <s id="N12620">&longs;ed magnitudo L deor&longs;um ver&shy;<lb/>get. </s>
          <s id="N12624">&longs;i igitur in F collocanda &longs;it magnitudo, qu&aelig; &aelig;&que;pon<lb/>deret ip&longs;i L in K, proculdubi&ograve; h&etail;c magnitudine A ma&shy;<lb/>ior exi&longs;tet. </s>
          <s id="N1262A">In&aelig;qualia enim grauia, nemp&egrave; L, &amp; magnitu
<arrow.to.target n="marg59"/><lb/>do maior, qu&agrave;m A, exin&aelig;qualibus di&longs;tantijs EK EF &aelig;&shy;<lb/>&que;ponderant, dummodo maius, hoc e&longs;t magnitudo maior, <lb/>qu&agrave;m A, &longs;it in di&longs;tantia minori EF. minusver&ograve;, hoc e&longs;t ma&shy;<lb/>gnitudo L, &longs;it in minori EK. Quoniam ita&que; magnitudo <lb/>C in D e&longs;t &etail;&que;grauis, vt L in K, magnitudo, qu&aelig; in F <lb/>ip&longs;i L in K &aelig;&que;ponderat, eadem quo&que; in F ip&longs;i C in D <lb/>&aelig;&que;ponderabit maior ver&ograve; magnitudo, qu&agrave;m &longs;it A, in F ip&longs;i <lb/>L in K &aelig;&que;ponderat, ergo maior magnitudo, qu&agrave;m A in <lb/>F, ip&longs;i C in D &aelig;&que;ponderabit. </s>
          <s id="N12641">quod demon&longs;trare opor&shy;<lb/>tebat. </s>
        </p>
        <p id="N12645" type="margin">
          <s id="N12647"><margin.target id="marg55"/>10. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N12650" type="margin">
          <s id="N12652"><margin.target id="marg56"/>6. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1265B" type="margin">
          <s id="N1265D"><margin.target id="marg57"/><emph type="italics"/><expan abbr="c&otilde;m">comm</expan>. not.<emph.end type="italics"/></s>
        </p>
        <p id="N12668" type="margin">
          <s id="N1266A"><margin.target id="marg58"/>2. <emph type="italics"/>po&longs;t bu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N12675" type="margin">
          <s id="N12677"><margin.target id="marg59"/>3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N12680" type="main">
          <s id="N12682">His cognitis po&longs;&longs;umus ad Archimedis demon&longs;trationem <lb/>accedere. </s>
        </p>
        <pb xlink:href="077/01/072.jpg" pagenum="68"/>
        <p id="N12689" type="head">
          <s id="N1268B">PROPOSITIO. VII.</s>
        </p>
        <p id="N1268D" type="main">
          <s id="N1268F">Si autem magnitudines fuerint incommen&longs;ura <lb/>biles, &longs;imiliter &aelig;&que;ponderabunt ex di&longs;tantijs per <lb/>mutatim eandem, at&que; magnitudines, propor&shy;<lb/>tionem habentibus. </s>
        </p>
        <figure id="id.077.01.072.1.jpg" xlink:href="077/01/072/1.jpg"/>
        <p id="N1269A" type="main">
          <s id="N1269C"><emph type="italics"/>Sint incommen&longs;urabiles magnitudines AB C. Distanti&aelig; ver&ograve; <lb/>DE EF. Habeat autem AB ad C proportionem eandem, quam di <lb/>stantia ED ad ip&longs;am EF. Dico,<emph.end type="italics"/> &longs;i ponatur AB ad F, C ve&shy;<lb/>r&ograve; ad D, <emph type="italics"/>magnitudinis ex vtri&longs;&que; AB C compo&longs;it&aelig; centrum gra<lb/>uitatis e&longs;&longs;e punctum E. &longs;i enim non &aelig;&que;ponderabit<emph.end type="italics"/> (&longs;i fieri pote&longs;t) <lb/><emph type="italics"/>AB po&longs;ita ad F ip&longs;i C po&longs;it&aelig; ad D; velmaior est AB, qu&agrave;m C, ita <lb/>vt<emph.end type="italics"/> AB ad F <emph type="italics"/>&aelig;&que;ponderet ip&longs;i C<emph.end type="italics"/> ad D; <emph type="italics"/>vel non. </s>
          <s id="N126C3">Sit maior<emph.end type="italics"/>; &longs;itqu&egrave; <lb/>exce&longs;&longs;us HL; ita vt KH ad F, &amp; C ad D &etail;&que;ponderent. <lb/>
<arrow.to.target n="marg60"/> <emph type="italics"/>auferaturqu&egrave; ab ip&longs;a AB<emph.end type="italics"/> magnitudo NL, qu&aelig; &longs;it <emph type="italics"/>minor exce&longs;&longs;u<emph.end type="italics"/><lb/>HL, <emph type="italics"/>quo maior est<emph.end type="italics"/> tota <emph type="italics"/>AB, qu&agrave;m C, ita vt &aelig;&que;ponderent<emph.end type="italics"/>; vt <expan abbr="dict&utilde;">dictum</expan> <lb/>e&longs;t. <emph type="italics"/>&amp; &longs;it quidem re&longs;iduum A,<emph.end type="italics"/> hoc e&longs;t KN, <emph type="italics"/>commen&longs;urabile ip&longs;i C.<emph.end type="italics"/><lb/>Et quoniam minor e&longs;t kN qu&agrave;m KM, minorem quo&que;
<pb xlink:href="077/01/073.jpg" pagenum="69"/>habebit proportionem kN ad C, qu&agrave;m kM ad eandem <lb/>C. tota ver&ograve; KM ad C e&longs;t, vt DE ad EF; ergo KN ad <lb/>C minorem habet proportionem; qu&agrave;m DE ad EF. <emph type="italics"/>Quo <lb/>niam igitur magnitudines AC,<emph.end type="italics"/> hoc e&longs;t KN C, <emph type="italics"/>&longs;unt commen&longs;urabi&shy;<lb/>les, &amp; minorem habet proportionem A,<emph.end type="italics"/> hoc e&longs;t kN <emph type="italics"/>ad C, qu&agrave;m DE <lb/>ad EF; non &aelig;&que;ponderabunt A C,<emph.end type="italics"/> hoc e&longs;t KN C, <emph type="italics"/>ex distantiis<emph.end type="italics"/>
<arrow.to.target n="marg61"/><lb/><emph type="italics"/>DE EF, po&longs;ito quidem A,<emph.end type="italics"/> hoc e&longs;t KN <emph type="italics"/>ad F, C ver&ograve; ad D.<emph.end type="italics"/> &amp; <lb/>vt &aelig;&que;ponderent, oporter, vt in F maior &longs;it magnitudo, <lb/>qu&agrave;m KN; ita vt ip&longs;i C in D &aelig;&que;ponderate po&longs;&longs;it. </s>
          <s id="N12736">Ac <lb/>propterea c&ugrave;m &longs;it kH adhuc minor, qu&agrave;m KN, &longs;i igitur <lb/>KH ponatur ad F, &amp; C ad D, nullo modo &aelig;&que;ponde&shy;<lb/>rabunt. </s>
          <s id="N1273E">quod tamen fieri non pote&longs;t. </s>
          <s id="N12740">&longs;upponebatur enim eas <lb/>&aelig;&que;ponderare. </s>
          <s id="N12744">Non igitur magnitudo minor, qu&agrave;m tota <lb/>KM in F magnitudini C in D &aelig;&que;ponderat. <emph type="italics"/>Eadem au&shy;<lb/>tem ratione, ne&que; &longs;i C maior fuerit, qu&agrave;m vt &aelig;&que;ponderet ip&longs;i A<emph.end type="italics"/>B, <lb/>hoc e&longs;t ip&longs;i KM. etenim grauiore <expan abbr="exi&longs;t&etilde;te">exi&longs;tente</expan> C ad D, qu&agrave;m KM <lb/>ad F. prim&ugrave;m auferatur ex C exce&longs;&longs;us, quo C grauior e&longs;t, <lb/>qu&agrave;m KM, ita vt &aelig;&que;ponderet ip&longs;i KM. Deinde rur&longs;us <lb/>auferatur qu&aelig;dam magnitudo minor exce&longs;&longs;u, quo grauior <lb/>e&longs;t C, qu&agrave;m kM, ita vt &aelig;&que;ponderent; re&longs;iduum ver&ograve; &longs;it <lb/>ip&longs;i KM commen&longs;urabile, &amp; c. </s>
          <s id="N12760">&longs;imiliter o&longs;tendetur <expan abbr="null&atilde;">nullam</expan> <lb/>magnitudinem ip&longs;a C minorem po&longs;itam ad D vllo modo <lb/>&aelig;&que;ponderare ip&longs;i KM ad F po&longs;it&aelig;. </s>
          <s id="N1276A">Quare magnitudo <lb/>C ad D, kM ver&ograve; ad F &etail;&que;ponderant. </s>
          <s id="N1276E">Vnde &longs;equitur ma <lb/>gnitudinis ex vtri&longs;&que; magnitudinibus compo&longs;it&aelig; centrum <lb/>grauitatis e&longs;&longs;e punctum E. ac propterea incommen&longs;urabiles <lb/>magnitudines AB C ex di&longs;tantiijs ED EF, qu&aelig; permutatim <lb/>eandem habent proportionem, vt magnitudines, &aelig;&que;pon&shy;<lb/>derare. </s>
          <s id="N1277A">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N1277C" type="margin">
          <s id="N1277E"><margin.target id="marg60"/><emph type="italics"/>ex proxi&shy;<lb/>mo proble&shy;<lb/>mate.<emph.end type="italics"/><lb/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N12791" type="margin">
          <s id="N12793"><margin.target id="marg61"/><emph type="italics"/>ex pr&aelig;ce&shy;<lb/>denti. <lb/>ex prima <lb/>propo&longs;itio&shy;<lb/>ne.<emph.end type="italics"/></s>
        </p>
        <p id="N127A3" type="head">
          <s id="N127A5">SCHOLIVM.</s>
        </p>
        <p id="N127A7" type="main">
          <s id="N127A9">In demon&longs;tratione occurrit ob&longs;eruandum, qu&ograve;d &longs;i exce&longs;&shy;<lb/>&longs;us HL ita diuideret magnitudinem KM, vt re&longs;iduum KH <lb/>fuerit commen&longs;urabile ip&longs;i C; tunc ab&longs;&que; alia con&longs;tructio&shy;<lb/>ne, magnitudines commen&longs;urabiles KH C ex di&longs;tantijs DE <lb/>EF &aelig;&que;ponderarent; quod fieri non pote&longs;t. </s>
          <s id="N127B3">c&ugrave;m minorem
<pb xlink:href="077/01/074.jpg" pagenum="70"/>habeat proportionem KH ad C, qu&agrave;m ED ad EF. <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> <lb/>&longs;upponitur KM ad C ita e&longs;&longs;e, vt ED ad EF. Archimed es ve <lb/>i&ograve;, vt demon&longs;tratio ab&longs;&que; di&longs;tinctione &longs;it vniuer&longs;alis, pr&etail;&shy;<lb/>cipit (exi&longs;tente KH ip&longs;i C commen&longs;urabili, &longs;iue incommen <lb/>&longs;urabili) vt auferatur pars aliqua minor exce&longs;&longs;u HL, ut AL, <lb/>ita tamen, vt reliqua KN &longs;it commen&longs;urabilis ip&longs;i C. quod qui <lb/>dem fieri po&longs;&longs;e o&longs;ten&longs;um e&longs;t in proximo problemate. </s>
          <s id="N127C9">ex tota <lb/>enim magnitudine KM partem ab&longs;cindere po&longs;&longs;umus, vt KN <lb/>minorem quidem tota KM, maiorem ver&ograve; KH, qu&aelig; ip&longs;i <lb/>C commen&longs;urabilis exi&longs;tat. </s>
        </p>
        <p id="N127D1" type="main">
          <s id="N127D3">Cognita Archimedis demon&longs;tratione de incommen&longs;ura&shy;<lb/>bilibus magnitudinibus, idem alio quo&que; modo o&longs;tendere <lb/>po&longs;&longs;umus, applicando nemp&egrave; diui&longs;ibilitatem, &amp; commen&longs;ura <lb/>bilitatem non magnitudinibus, ver&ugrave;m di&longs;tantijs. </s>
          <s id="N127DB">hac autem <lb/>pri&ugrave;s demon&longs;trata propo&longs;itione. </s>
        </p>
        <p id="N127DF" type="head">
          <s id="N127E1">PROPOSITIO.</s>
        </p>
        <p id="N127E3" type="main">
          <s id="N127E5">Si commen&longs;urabiles di&longs;tanti&etail; maiorem habuerint pro&shy;<lb/>portionem, qu&agrave;m magnitudines permutatim habent; vt <lb/>&etail;&que;ponderent, maiori opus erit longitudine, qu&agrave;m &longs;it <lb/>ea, ad quam altera longitudo maiorem habet proportio&shy;<lb/>nem. </s>
        </p>
        <figure id="id.077.01.074.1.jpg" xlink:href="077/01/074/1.jpg"/>
        <p id="N127F2" type="main">
          <s id="N127F4">Sint di&longs;tanti&aelig; DE EH commen&longs;urabiles, magnitudines <lb/>ver&ograve; &longs;int A C. habeatqu&egrave; ED ad EH maiorem proportio&shy;<lb/>nem, qu&agrave;m A ad C. Dico vt AC &etail;&que;ponderent, maiori opus
<pb xlink:href="077/01/075.jpg" pagenum="71"/>e&longs;&longs;e longitudine, qu&agrave;m &longs;it EH. exponatur altera magnitu&shy;<lb/>do G, qu&aelig; ad C eandem habeat proportionem, qu&agrave;m habet <lb/>DE ad EH. erunt vti&que; magnitudines GC inter &longs;e <expan abbr="comm&etilde;">commen</expan> <lb/>&longs;urabiles. </s>
          <s id="N12808">Deinde fiat EK &aelig;qualis EH, exponaturqu&egrave; ma&shy;<lb/>gnitudo L ip&longs;i G &aelig;qualis. </s>
          <s id="N1280C">Quoniam igitur G ad C e&longs;t, <lb/>vt DE ad EH, ob commen&longs;urabilitatem &aelig;&que;pondera bunt
<arrow.to.target n="marg62"/><lb/>G in H, &amp; C in D. &longs;imiliter &aelig;&que;pondera bunt magnitudi&shy;<lb/>nes &aelig;quales GL ex &aelig;qualibus di&longs;tantijs EK EH. C&ugrave;m igitur <lb/>C in D ip&longs;i G in H &aelig;&que;ponderet; L ver&ograve; in K ip&longs;i quo&shy;<lb/>&que; G in H &aelig;&que;ponderet; eandem habebit grauitatem C
<arrow.to.target n="marg63"/><lb/>in D, ut L in K. Quoniam autem maiorem habet propor&shy;<lb/>tionem DE ad EH, qu&agrave;m A ad C, &amp; vt DE ad EH, ita e&longs;t <lb/>G ad C; maiorem habebit proportionem G ad C, qu&agrave;m A <lb/>ad C. ergo maior e&longs;t G, qu&agrave;m A. ac propterea magnitudo A
<arrow.to.target n="marg64"/><lb/>minor e&longs;t magnitudine L. po&longs;ita igitur magnitudine L in K, <lb/>&amp; A in H, non &aelig;&que;pondera bunt; &amp; vt &etail;&que;ponderent, o&shy;<lb/>portet, vt A in longiori &longs;it di&longs;tantia, qu&agrave;m &longs;it EH: In&etail;qualia <lb/>enim grauia LA ex in&etail;qualibus di&longs;tantijs &etail;&que;ponderant,
<arrow.to.target n="marg65"/><lb/>maius quidem L in minori di&longs;tantia EK, minus ver&ograve; graue <lb/>A in maiori, qu&agrave;m &longs;it EK, hoc e&longs;t in maiori, qu&agrave;m &longs;it EH. <lb/>Ita&que; c&ugrave;m &longs;it C in D &aelig;&que;grauis, vt L in k; longitudo, <lb/>qu&aelig; efficit, vt A &aelig;&que;ponderetip&longs;i L in K; eadem pror&longs;us <lb/>efficiet, vt A ip&longs;i C in D &etail;&que;ponderare po&longs;&longs;it. </s>
          <s id="N1283E">A ver&ograve; in <lb/>maiori di&longs;tantia, qu&agrave;m EH, ip&longs;i L in K &etail;&que;ponderat; ergo <lb/>in maiori di&longs;tantia, qu&agrave;m EH, magnitudo A ip&longs;i C in D <lb/>&etail;&que;ponderabit. </s>
          <s id="N12846">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N12848" type="margin">
          <s id="N1284A"><margin.target id="marg62"/>6. <emph type="italics"/>buius.<emph.end type="italics"/></s>
        </p>
        <p id="N12853" type="margin">
          <s id="N12855"><margin.target id="marg63"/><emph type="italics"/><expan abbr="c&otilde;munis">communis</expan> no <lb/>tio &longs;upradi <lb/>cta.<emph.end type="italics"/></s>
        </p>
        <p id="N12864" type="margin">
          <s id="N12866"><margin.target id="marg64"/>10. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1286F" type="margin">
          <s id="N12871"><margin.target id="marg65"/>3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1287A" type="main">
          <s id="N1287C">Hoc demon&longs;trato Archimedis propo&longs;itionem de incom&shy;<lb/>men&longs;urabilibus magnitudinibus aliter o&longs;tendemus hoc <lb/>pacto. </s>
        </p>
        <p id="N12882" type="head">
          <s id="N12884">ALITER.</s>
        </p>
        <p id="N12886" type="main">
          <s id="N12888">Incommen&longs;urabiles magnitudines ex di&longs;tantijs permuta&shy;<lb/>tim eandem, at&que; magnitudines, proportionem habenti&shy;<lb/>bus; &etail;&que;ponderant. </s>
        </p>
        <pb xlink:href="077/01/076.jpg" pagenum="72"/>
        <p id="N12891" type="main">
          <s id="N12893">Sint incom&shy;<lb/>
<arrow.to.target n="fig30"/><lb/><expan abbr="m&etilde;&longs;urabiles">men&longs;urabiles</expan> ma <lb/>gnitudines AC, <lb/>di&longs;tanti&aelig; ver&ograve; <lb/>DE EF. &longs;itqu&egrave; vt <lb/>A ad C, ita DE <lb/>ad EF. Dico A <lb/>in F, C ver&ograve; in <lb/>D &aelig;&que;ponde&shy;<lb/>rare. </s>
          <s id="N128AF">Si autem (&longs;i fieri pote&longs;t) non &aelig;&que;pondera bunt; <expan abbr="di&longs;t&atilde;">di&longs;tam</expan> <lb/>ti&aelig; DE EF aliter &longs;e&longs;e habere debebunt, vt magnitudines AC <lb/>&etail;&que;ponderent. </s>
          <s id="N128B9">Quocirca vel longior e&longs;t EF, qu&agrave;m opus <lb/>&longs;it, vel longior e&longs;t ED. &longs;it EF longior. </s>
          <s id="N128BD">&longs;itqu&egrave; exce&longs;&longs;us GF, ita <lb/>vt po&longs;ita magnitudine A in G ip&longs;i C in D &aelig;&que;ponde&shy;<lb/>
<arrow.to.target n="marg66"/> ret. </s>
          <s id="N128C7">Fiat EH maior EG, minor ver&ograve; EF. &longs;it autem EH <lb/>ip&longs;i ED commen&longs;urabilis. </s>
          <s id="N128CB">Quoniam igitur DE ad EH <lb/>maiorem habet proportionem, qu&agrave;m ad EF; &amp; vt DE ad <lb/>EF, ita e&longs;t A ad C; maiorem habebit proportionem DE <lb/>ad EH, qu&agrave;m A ad C. &longs;untqu&egrave; longitudines ED EH in&shy;<lb/>ter&longs;e commen&longs;urabiles; ergo magnitudo A in H ip&longs;i C in <lb/>
<arrow.to.target n="marg67"/> D non &aelig;&que;ponderabit, &longs;ed vt &etail;&que;ponderet, maiori opus <lb/>e&longs;t longitudine, qu&agrave;m &longs;it EH; ita vt A ip&longs;i C in D &aelig;&que; <lb/>ponderare po&longs;&longs;it. </s>
          <s id="N128DF">at&que; ade&ograve; c&ugrave;m adhuc minor &longs;it EG, qu&agrave;m <lb/>EH; magnitudo A in G magnitudini C in D nullo modo <lb/>&aelig;&que;ponderabit. </s>
          <s id="N128E5">quod fieri non pote&longs;t. </s>
          <s id="N128E7">&longs;upponebatur enim <lb/>A in G, &amp; C in D &etail;&que;ponderare. </s>
          <s id="N128EB">eademqu&egrave; pror&longs;us ra&shy;<lb/>tione, &longs;i ED longior fuerit, qu&agrave;m opus &longs;it, ita vt magnitu&shy;<lb/>dines &aelig;&que;ponderent, o&longs;tendetur <expan abbr="magnitudin&etilde;">magnitudinem</expan> C nullo pa&shy;<lb/>cto &aelig;&que;ponderare po&longs;&longs;e ip&longs;i A in F in minori di&longs;tantia, <lb/>qu&agrave;m DE. Quare magnitudines in commen&longs;urabiles AC ex <lb/>di&longs;tantijs ED EF, qu&aelig; eandem permutatim habent propor&shy;<lb/>tionem, vt magnitudines, &aelig;&que;ponderant. </s>
          <s id="N128FD">quod demon&longs;tra&shy;<lb/>re oportebat. </s>
        </p>
        <p id="N12901" type="margin">
          <s id="N12903"><margin.target id="marg66"/><emph type="italics"/>problema <lb/>ante<emph.end type="italics"/> 7. <emph type="italics"/>bu&shy;<lb/>ius<emph.end type="italics"/> 8. <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <p id="N1291B" type="margin">
          <s id="N1291D"><margin.target id="marg67"/><emph type="italics"/>ex pxima <lb/>ppo&longs;itione<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.076.1.jpg" xlink:href="077/01/076/1.jpg"/>
        <p id="N1292B" type="main">
          <s id="N1292D">In prioribus &longs;ermonibus ante quintam propo&longs;itionem ha&shy;<lb/>bitis, diximus propo&longs;itionum pr&aelig;cedentium demon&longs;tratio&shy;<lb/>nes planiores euadere, &longs;i intelligamus magnitudines eiu&longs;dem <lb/>e&longs;&longs;e &longs;peciei, &amp; homogeneas. </s>
          <s id="N12935">Qu&ograve;d quidem &longs;i Archimedem
<pb xlink:href="077/01/077.jpg" pagenum="73"/>his, vel de rectilineis tant&ugrave;m demon&longs;trationes attuli&longs;&longs;e (vt <expan abbr="n&otilde;-nulli">non&shy;<lb/>nulli</expan> forta&longs;&longs;e fals&ograve; exi&longs;timarunt) intelligeremus; ita vt ex Ar&shy;<lb/>chimedis demon&longs;trationibus non &longs;it adhuc vniuer&longs;aliter de&shy;<lb/>mon&longs;tratum hoc pr&etail;cipuum fundamentum; nemp&egrave; magni&shy;<lb/>tudines ex di&longs;tantijs permutatim <expan abbr="proportion&etilde;">proportionem</expan> habentibus, vt <lb/>ip&longs;arum grauitates, &etail;&que;ponderare; in hoc cert&egrave; rationes ab <lb/>Archimede allatas, ip&longs;arum qu&egrave; demon&longs;trationum vim mini&shy;<lb/>m&egrave; percipiemus. </s>
          <s id="N12951">Quapropter ea, qu&aelig; demon&longs;trauit, omni&shy;<lb/>bus magnitudinibus vniuer&longs;aliter competere ip&longs;um volui&longs;&longs;e <lb/>nullatenus e&longs;t dubitandum. </s>
          <s id="N12957">Ne&que; enim, vt perfect&egrave;, &amp; vni&shy;<lb/>uer&longs;aliter&longs;ciamus, magnitudines &ccedil;&que;ponderare ex di&longs;tantijs <lb/>permutatim proportionem habentibus, vt ip&longs;arum grauita&shy;<lb/>tes, alijs, qu&agrave;m pr&etail;cedentibus propo&longs;itionibus indigemus. <lb/>In hoc enim fundamento demon&longs;trando minim&egrave; diminu&shy;<lb/>tus extitit Archimede. </s>
          <s id="N12963">Nam &longs;i ad propo&longs;itiones ab ip&longs;o alla&shy;<lb/>tas, pr&etail;cipu&egrave;qu&egrave; ad vim demon&longs;trationum re&longs;piciamus, &longs;iu&egrave; <lb/>magnitudines intelligantur eiuldem &longs;peciei, &longs;iue diuer&longs;&etail;, &longs;i&shy;<lb/>ue homogene&etail;, &longs;iue heterogene&etail;, &longs;iue plan&etail;, &longs;iue &longs;olid&etail;, &amp; <lb/>h&etail; quidem, &longs;iue rectiline&etail;, &longs;iue quom odocun&que; mixt&etail;; ni&shy;<lb/>hilominus demon&longs;trationes idem pror&longs;us concludent, ita vt <lb/>Archimedes non de aliquibus magnitudimbus tant&ugrave;m de&shy;<lb/>mon&longs;trationes attulerit; &longs;ed de omnibus pror&longs;us demon&longs;tra&shy;<lb/>uerit. </s>
          <s id="N12975">In his enim Archimedes non ad magnitudines tant&ugrave;m, <lb/>ver&ugrave;m ad magnitudinum grauitates poti&longs;&longs;im&ugrave;m re&longs;pexit. <lb/>quandoquidem loco grauium magnitudines nominat; vt <lb/>po&longs;t quartam huius propo&longs;itionem adnotauimus. </s>
          <s id="N1297D">quod qui&shy;<lb/>dem facil&egrave; ex verbis ip&longs;ius rect&egrave; intellectis apparere pote&longs;t. <expan abbr="N&atilde;">Nam</expan> <lb/>in qu&aelig;rta propo&longs;itione c&ugrave;m inquit, <emph type="italics"/>&longs;i du&aelig; fuerint magnitudines <lb/>&aelig;quales<emph.end type="italics"/>, vt antea diximus, intelligendum e&longs;t eas &etail;quales <lb/>e&longs;&longs;e grauitate. </s>
          <s id="N12991">quod non &longs;ol&ugrave;m ex eius demon&longs;trationeli&shy;<lb/>&que;t, ver&ugrave;m etiam ex modo lo&que;ndi, quo v&longs;us e&longs;t Archime&shy;<lb/>des in alijs propo&longs;itionibus. </s>
          <s id="N12997">In quinta enim propo&longs;itione, <lb/>qu&etail; eiu&longs;dem e&longs;t cum quarta ordinis, &amp; natur&etail;, in quit; <lb/><emph type="italics"/>Sitrium magnitudinum centra grauitatis in recta linea fuerint po&longs;i&shy;<lb/>ta, &amp; magnitudines &aelig;qualem habuerint grauitatem.<emph.end type="italics"/> &longs;imlli&shy;<lb/>ter po&longs;t quintam demon&longs;trationem bis quoqu&egrave; eodem v&shy;<lb/>titur lo&que;ndi modo, nemp&egrave; c&ugrave;m adhuc proponit
<pb xlink:href="077/01/078.jpg" pagenum="74"/>plures magnitudines, inquit, <emph type="italics"/>&amp; magnitudines &aelig;qualem habuerint <lb/>grauitatem.<emph.end type="italics"/> ex quibus con&longs;tat Archimedem ad magnitudinum <lb/>grauitates omnino re&longs;pexi&longs;&longs;e. </s>
          <s id="N129B6">ita vt quando Archimedes in&shy;<lb/>quit, <emph type="italics"/>&amp; magnitudines &aelig;quales<emph.end type="italics"/>, idem e&longs;t, ac &longs;i dixi&longs;&longs;et, <emph type="italics"/>&amp; magnitu&shy;<lb/>dines &aelig;qualem habuerint grauitatem.<emph.end type="italics"/> Pr&aelig;terea in &longs;exta propo&longs;itio <lb/>ne inquit magnitudines &etail;&que;ponderare ex di&longs;tantijs permu&shy;<lb/>t&agrave;tim proportionem habentibus, vt grauitates. </s>
          <s id="N129CC">ita ut cau&longs;a <lb/>huius &aelig;&que;ponderationis &longs;it (vt reuera e&longs;t) magnitudinum <lb/>grauitas. </s>
          <s id="N129D2">&amp; <expan abbr="qu&atilde;quam">quanquam</expan> in hac &longs;eptima propo&longs;itione dicat, ma <lb/>gnitudines &aelig;&que;ponderare ex di&longs;tantijs permutatim propor&shy;<lb/>tionem habentibus, vt magnitudines, &amp; non dixit, vt grauita <lb/>tes; intelligendum tamen e&longs;t, ac &longs;i dixi&longs;&longs;et, eas &etail;&que;pondera&shy;<lb/>re, vt magnitudinum grauitates. </s>
          <s id="N129E0">h&etail;c enim &longs;eptima propo&longs;i&shy;<lb/>tio e&longs;t pars &longs;ext&aelig; propo&longs;itionis, vt iam pr&etail;fati fum^&lcub;9}; vnde &longs;i in <lb/>&longs;exta magnitudines &etail;&que;ponderant ob earum grauitatem, ob <lb/>eandem quo&que; cau&longs;am &amp; in hac &longs;eptima &aelig;&que;ponderare de <lb/>bent. </s>
          <s id="N129EA">Pr&etail;terea in &longs;e&que;nti etiam propo&longs;itione dum proponit <lb/>o&longs;tendere quam proportionem habere debent &longs;ectiones line&etail; <lb/>intercentra grauitatum diui&longs;&etail; magnitudinis <expan abbr="exi&longs;t&etilde;tes">exi&longs;tentes</expan>, inquit, <lb/><emph type="italics"/>quam habet grauitas magnitudinis ablat&aelig; ad grauitatem re&longs;idu&aelig;<emph.end type="italics"/> hoc <lb/>autem deinceps exponens, <expan abbr="n&otilde;">non</expan> inquit oportere &longs;ectiones line&aelig; <lb/>eam habere proportionem, qu&agrave;m grauitas ad grauitatem ha&shy;<lb/>bet; &longs;ed horum loco inquit, qu&agrave;m magnitudo ad magnitudi <lb/>nem. </s>
          <s id="N12A07">ex quibus omnibus clar&egrave; per&longs;picitur, qu&ograve;d quando Ar&shy;<lb/>chimedes magnitudines nominat, omnino magnitudinum <lb/>grauitates vult intelligere. </s>
        </p>
        <p id="N12A0D" type="main">
          <s id="N12A0F">Ad eorum autem <expan abbr="intelligenti&atilde;">intelligentiam</expan>, qu&etail; dicta &longs;unt in &longs;exta, &longs;epti <lb/>maqu&egrave; propo&longs;itione, <expan abbr="ear&utilde;qu&egrave;">earunqu&egrave;</expan> <expan abbr="dem&otilde;&longs;trationibus">demon&longs;trationibus</expan>, <expan abbr="ob&longs;eruand&utilde;">ob&longs;eruandum</expan> <lb/>e&longs;t, qu&ograve;d in &longs;exta propo&longs;itione pro magnitudinibus commen <lb/>&longs;urabilibus intelligere oportet magnitudines grauitate com&shy;<lb/>men&longs;urabiles; ita nempe, vt numeris exprimi po&longs;&longs;int; quam&shy;<lb/>quam non &longs;int mole, &amp; magnitudine commen&longs;urabiles, vt <lb/>in figura &longs;ext&etail; propo&longs;itionis magnitudo A ponderet exempli <lb/>gratia vt XVI. B ver&ograve; vt VIII. <expan abbr="intelligatur&qacute;">intelligatur&que;</expan>; F <expan abbr="magnitudin&utilde;">magnitudinum</expan>
<pb xlink:href="077/01/079.jpg" pagenum="75"/>AB <expan abbr="c&otilde;munis">communis</expan> men&longs;ura in grauitate, ita vt &longs;it &aelig;&que;grauis vni&shy;<lb/>cui&que; parti OPQR, qu&aelig; quidem, &amp; &longs;i non &longs;int magnitu&shy;<lb/>dine inter &longs;e &etail;quales, &longs;ufficit, vt &longs;int &aelig;&que;graues: veluti magni<lb/>
<arrow.to.target n="fig31"/><lb/>tudines quo&que; STVX inter &longs;e, <expan abbr="ip&longs;is&qacute;">ip&longs;is&que;</expan>; OPQR tant&ugrave;m &etail;&que; <lb/>graues; ita ut vnaqu&aelig;&que; ponderet, vt IIII. veluti etiam par <lb/>tes ip&longs;ius B, &amp; vnaqu&aelig;&que; ZM. hi&longs;qu&egrave; ita po&longs;itis <expan abbr="dem&otilde;&longs;tra">demon&longs;tra</expan> <lb/>tio rect&egrave; concludet. </s>
        </p>
        <figure id="id.077.01.079.1.jpg" xlink:href="077/01/079/1.jpg"/>
        <p id="N12A5C" type="main">
          <s id="N12A5E">In hacver&ograve; &longs;eptima Archimedis propo&longs;itione &longs;imiliter
<arrow.to.target n="marg68"/> in&shy;<lb/>telligantur magnitudines kMC incommen&longs;urabiles graui&shy;<lb/>tate, vt in eius figura grauitas ip&longs;ius C ponderet, vt XII. gra<lb/>uitas ver&ograve; ip&longs;ius KM maior &longs;it, qu&agrave;m XX. ita vth&etail; graui&shy;<lb/>tates &longs;int in
commen&longs;urabiles. </s>
          <s id="N12A6C">auferaturqu&egrave; grauitas exce&longs;&longs;us <lb/>HL, qu&aelig; &longs;it vt IIII. ita vt qu&aelig; relinquiturgrauitas, ip&longs;ius <expan abbr="n&etilde;-p&egrave;">nen&shy;<lb/>p&egrave;</expan> KH, qu&etail; quidem maior e&longs;t, qu&agrave;m XVI, in F po&longs;ita, gra<lb/>uitati ip&longs;ius C, qu&aelig; e&longs;t XII, in D po&longs;it&aelig; &aelig;&que;ponderet, <lb/>Auferatur deinde NL minor exce&longs;&longs;u HL; cuius quidem gra<lb/>uitas &longs;it maior, qu&agrave;m II. ita vt grauitas re&longs;idui KN, qu&aelig; <lb/>nimirum &longs;it XVIII, &longs;it commen&longs;urabilis grauitati <lb/>XII. ip&longs;ius C. &amp; <expan abbr="qu&atilde;uis">quamuis</expan> magnitudines KM C, &amp; KN C &longs;int, <lb/>vel <expan abbr="n&otilde;">non</expan> &longs;int inter &longs;e magnitudine <expan abbr="c&otilde;men&longs;urabiles">commen&longs;urabiles</expan>, vel incom&shy;
<pb xlink:href="077/01/080.jpg" pagenum="76"/>men&longs;urabiles; eadem pror&longs;us demon&longs;tratio idem concludet. <lb/>qu&aelig; quidem omnia in &longs;e&que;nti quo&que; propo&longs;itione <expan abbr="con&longs;i-der&atilde;da">con&longs;i&shy;<lb/>deranda</expan> occurrunt. </s>
          <s id="N12A9A">Vnde per&longs;picuum e&longs;t has Archime dis pro <lb/>po&longs;itiones, ac demon&longs;trationes vniuer&longs;ali&longs;&longs;imas e&longs;&longs;e, ar&que; o&shy;<lb/>mnibus, &amp; quibu&longs;cun&que; magnitudinibus conuenientes. </s>
        </p>
        <p id="N12AA0" type="margin">
          <s id="N12AA2"><margin.target id="marg68"/><emph type="italics"/>re&longs;pice <expan abbr="fi-gur&atilde;">fi&shy;<lb/>guram</expan> &longs;epti&shy;<lb/>m&aelig; propo&longs;i <lb/>tionis Ar&shy;<lb/>chimedis.<emph.end type="italics"/></s>
        </p>
        <p id="N12AB6" type="main">
          <s id="N12AB8">Iacto hoc pr&etail;cipuo, ac pr&etail;&longs;tanti&longs;&longs;imo mechanico funda&shy;<lb/>mento; in &longs;e&que;nti propo&longs;itione colligit ex hoc Archimedes, <lb/>quomodo &longs;e habent centra grauitatis magnitudinis diui&longs;&aelig;. </s>
        </p>
        <p id="N12ABE" type="head">
          <s id="N12AC0">PROPOSITIO. VIII.</s>
        </p>
        <p id="N12AC2" type="main">
          <s id="N12AC4">Si ab aliqua magnitudine magnitudo aufera&shy;<lb/>tur; qu&aelig; non habeat idem centrum cum tota; re&shy;<lb/>liqu&aelig; magnitudinis centrum grauitatis e&longs;t in re&shy;<lb/>cta linea, qu&aelig; coniungit centra grauitatum to tius <lb/>magnitudinis, &amp; ablat&aelig;, ad eam partem produ&shy;<lb/>cta, vbi e&longs;t centrum to tius magnitudinis, ita vt a&longs;&shy;<lb/>&longs;umpta aliqua ex producta, qu&aelig; coniungit <expan abbr="c&etilde;tra">centra</expan> <lb/>pr&aelig;dicta eandem habeat proportionem ad eam, <lb/>qu&aelig; e&longs;t inter centra, quam habet grauitas magni&shy;<lb/>tudinis ablat&aelig; ad grauitatem re&longs;idu&aelig;, centrum e&shy;<lb/>rit terminus a&longs;&longs;umpt&aelig;. </s>
        </p>
        <p id="N12ADE" type="main">
          <s id="N12AE0"><emph type="italics"/>Sit alicuius magnitudinis AB centrum grauitatis C. auferatur&shy;<lb/>qu&egrave; ex AB magnitudo AD; cuius centrum grauitatis &longs;it E. coniuncta <lb/>ver&ograve; EC, &amp;<emph.end type="italics"/> ex parte C <emph type="italics"/>producta, a&longs;&longs;umatur CF, qu&aelig; ad CE <expan abbr="e&atilde;">eam</expan> <lb/>dem habeat proportionem, quam habet magnitudo AD ad DG. osten&shy;<lb/>dendum est, magnitudinis DG centrumgrauitatis e&longs;&longs;e punctum F. <expan abbr="N&otilde;">non</expan> <lb/>&longs;it autem; &longs;ed, &longs;i fieri potest, &longs;it punctum H. Quoniam igitur magnitudi&shy;<lb/>nis AD centrum grauitatis est punctum E; magnitudinis ver&ograve; DG <lb/>e&longs;t punctum H; magnitudinis ex vtri&longs;&que; magnitudinibus AD DG,<emph.end type="italics"/><lb/>
<arrow.to.target n="marg69"/> <emph type="italics"/>compo&longs;it&aelig; centrum grauitatis erit in linea EH, ita diui&longs;a, ut pirtes ip&longs;ius <lb/>permutatim eandem <expan abbr="habe&atilde;t">habeant</expan> proportionem, vt magnitudines. </s>
          <s id="N12B11">Quare non<emph.end type="italics"/>
<pb xlink:href="077/01/081.jpg" pagenum="77"/><emph type="italics"/>erit punctum C &longs;ecund&ugrave;m diui&longs;ionem proportione re&longs;pondentem pr&aelig;di&shy;<lb/>et&aelig;.<emph.end type="italics"/> vt &longs;cilicet &longs;it HC ad CE, vt AD ad DG. etenim ut AD <lb/>ad DG; ita <expan abbr="fact&utilde;">factum</expan> fuit FC ad CE. &longs;i igitur &longs;ecetur linea EH &longs;e <lb/>cund&ugrave;m proportionem ip&longs;ius AD ad DG; non terminabit <lb/>
<arrow.to.target n="fig32"/><lb/>diui&longs;io ad punctum C. c&ugrave;m &longs;it impo&longs;&longs;ibile eandem habere <lb/>proportionem FC ad CE, quam. </s>
          <s id="N12B32">HC ad eandem CE. di&shy;<lb/>ui&longs;io igitur ad aliud terminabitur punctum, vt K; ita vt HK
<arrow.to.target n="marg70"/><lb/>ad KE &longs;it, vt AD ad DG. vnde &longs;equitur punctum K cen&shy;<lb/>trum e&longs;&longs;e grauitatis magnitudinis ex AD DG compo&longs;it&aelig;. <lb/><emph type="italics"/>Non e&longs;t igitur punctum C centrum magnitudinis ex AD DG compo <lb/>&longs;it&aelig;; hoc est ip&longs;ius AB. e&longs;t autem; &longs;uppo&longs;itum e&longs;t enim<emph.end type="italics"/> ip&longs;um e&longs;&longs;e. <emph type="italics"/>er&shy;<lb/>go ne&que; punctum H centrum est grauitatis magnitudinis DG.<emph.end type="italics"/> e&longs;t <lb/>igitur punctum F; quod quidem e&longs;t terminus product&etail; line&etail; <lb/>CF; qu&aelig; eandam habet proportionem ad lineam CE inter <lb/>centra exi&longs;tentem; quam habet grauitas magnitudinis AD <lb/>ad grauitatem ip&longs;ius DG. quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N12B56" type="margin">
          <s id="N12B58"><margin.target id="marg69"/><emph type="italics"/>ex pr&aelig;ce&shy;<lb/>dentibus.<emph.end type="italics"/></s>
        </p>
        <p id="N12B62" type="margin">
          <s id="N12B64"><margin.target id="marg70"/><emph type="italics"/>ex pr&aelig;ce&shy;<lb/>dentibus.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.081.1.jpg" xlink:href="077/01/081/1.jpg"/>
        <p id="N12B72" type="head">
          <s id="N12B74">SCHOLIVM.</s>
        </p>
        <p id="N12B76" type="main">
          <s id="N12B78">In hac demon&longs;tratione intelligendum e&longs;t etiam punctum <lb/>H e&longs;&longs;e po&longs;&longs;e extra lineam EF, ita vt EFH non &longs;itirecta linea. <lb/>qu&ograve;d &longs;i H non e&longs;&longs;et in linea EF, idem &longs;equi ab&longs;urdum ade&ograve; <lb/>per&longs;picuum e&longs;t; vt nec demon&longs;tratione egeat. </s>
          <s id="N12B80">Quoniam &longs;i in<lb/>telligatur H extra lineam EF; iuncta EH, &amp; ita diui&longs;a intel&shy;<lb/>ligatur, vt ip&longs;ius partes permutatim grauitatibus magnitudi&shy;<lb/>num AD DG re&longs;pondeant; e&longs;&longs;et vti&que; hoc punctum <expan abbr="inu&etilde;-tum">inuen&shy;<lb/>tum</expan>, quod extra lineam EF reperiretur, centrum grauitatis to
<pb xlink:href="077/01/082.jpg" pagenum="78"/>tius AB quod fieri non pote&longs;t. </s>
          <s id="N12B92">&longs;iquidem e&longs;t punctum C, vt <lb/>&longs;uppo&longs;itum fuit. </s>
          <s id="N12B96">Vnde ne&que; illud punctum H ip&longs;ius DG <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis exi&longs;teret. </s>
        </p>
        <p id="N12B9E" type="main">
          <s id="N12BA0">Hic e&longs;t terminus prim&etail; partis principalis, in qua Archime <lb/>des (vt initio dixim^&lcub;9}) de magnitudinib^&lcub;9}, &amp; degrauibus in <lb/>communi pertractauit; quandoquidem propo&longs;itiones, ac de&shy;<lb/>mon&longs;trationes tam planis, qu&agrave;m &longs;olidis quibu&longs;cun&que; &longs;unt <lb/>accomodat&aelig;; vt manife&longs;tum fecimus. </s>
        </p>
        <p id="N12BAA" type="main">
          <s id="N12BAC">Nunc ita &que; &longs;e conuertit Archimedes ad <expan abbr="inue&longs;tigand&utilde;">inue&longs;tigandum</expan> cen<lb/>tra grauitatis planorum. </s>
          <s id="N12BB4">prim&ugrave;m qu&egrave; perquirit centrum gra&shy;<lb/>uitatis parallelogrammorum; o&longs;tendetqu&egrave; centrum grauitatis <lb/>cuiu&longs;libet parallelogrammi e&longs;&longs;e in recta linea, qu&aelig; coniungit <lb/>oppo&longs;ita latera bifariam diui&longs;a. </s>
          <s id="N12BBC">ob cuius intelligentiam h&aelig;c <lb/>pri&ugrave;s lemmata in vnum collecta noui&longs;&longs;e erit vald&egrave; vtile. </s>
        </p>
        <p id="N12BC0" type="head">
          <s id="N12BC2">LEMMA.</s>
        </p>
        <p id="N12BC4" type="main">
          <s id="N12BC6">Sit parallelogrammum ABCD, cuius oppo&longs;ita latera AB <lb/>CD &longs;int bifariam diui&longs;a in EF. connectaturqu&egrave; EF, qu&aelig; ni <lb/>mirum &aelig;quidi&longs;tans erit ip&longs;is AC BD. Deinde diuidatur v&shy;<lb/>
<arrow.to.target n="fig33"/><lb/>naqu&aelig;&que; AE EB in partes numero pares, &amp; inuicem &etail;qua <lb/>les; vt in AG GE; &amp; EH HB. <expan abbr="duc&atilde;turqu&egrave;">ducanturqu&egrave;</expan> GK HL ip&longs;i <lb/>EF &etail;quidi&longs;tantes. </s>
          <s id="N12BDB">&longs;it ver&ograve; centrum grauitatis ip&longs;ius AK pun<lb/>ctum M. ipfius ver&ograve; GF punctum N, &amp; ip&longs;ius EL pun&shy;<lb/>ctum O deniqu&egrave; ip&longs;ius HD punctum P. Dico prim&ugrave;m <expan abbr="p&utilde;">pum</expan> <lb/>cta MNOP e&longs;&longs;e in linea recta. </s>
          <s id="N12BE7">deinde lineas MN NO OP <lb/>inter centra exi&longs;tentes inter &longs;e &aelig;quales e&longs;&longs;e. </s>
          <s id="N12BEB">Deni&que; centrum <lb/>grauitatis parallelogrammi AD e&longs;&longs;e in linea NO, qu&etail; con <lb/>iungit centra grauitatis &longs;patiorum mediorum; parallelogram <lb/>morum &longs;cilicet GF EL.
<pb xlink:href="077/01/083.jpg" pagenum="79"/>Ducantur &agrave; punctis MN ip&longs;i AGE &etail;quidi&longs;tantes QMR <lb/>SNT. erunt vti&que; AQRG, &amp; GSTE parallelogramma. <lb/>Quoniam igitur parallelogramma AK GF in &aelig;qualibus <lb/>&longs;untba&longs;ibus AG GE, &amp; in ij&longs;dem parallelis; erunt AK GF
<arrow.to.target n="marg71"/><lb/>inter &longs;e &etail;qualia. </s>
          <s id="N12C02">&amp; quoniam AC GK EF &longs;unt <expan abbr="&etail;quidi&longs;t&atilde;tes">&etail;quidi&longs;tantes</expan>; <lb/>erit angulus CAG ip&longs;i KGE &etail;qualis, &amp; KGA ip&longs;i FEG
<arrow.to.target n="marg72"/><lb/>&aelig;qualis; &amp; horum oppo&longs;iti inter &longs;e &longs;unt &etail;quales; ergo
<arrow.to.target n="marg73"/> paralle&shy;<lb/>logrammum GF ip&longs;i AK &etail;quale, &amp; &longs;imile exi&longs;tit. </s>
          <s id="N12C15">Ita&que; <lb/>&longs;i GF collocetur&longs;uper AK, rect&egrave; congruet: eruntqu&egrave; paral&shy;<lb/>lelogramma inuicen coaptata. </s>
          <s id="N12C1B">line&etail;qu&egrave; GE AG, GK AC, &amp; <lb/>reliqu&aelig; coaptat&aelig; erunt. </s>
          <s id="N12C1F">quare eorum centra grauitatis
<arrow.to.target n="marg74"/> inui&shy;<lb/>cem coaptata erunt. </s>
          <s id="N12C27">hoc e&longs;t N erit in puncto M. Quoniam <lb/>autem &agrave; punctis MN (quod nunc intelligitur vnum tantum <lb/>e&longs;&longs;e punctum) duct&aelig; fuerunt ST QR ip&longs;i AGE &aelig;quidi&shy;<lb/>&longs;tantes, linea ST coaptabitur cum QR, quippe c&ugrave;m amb&aelig; <lb/>h&aelig; line&aelig; ab vno puncto prodeuntes ip&longs;i AG &etail;quidi&longs;tantes <lb/>e&longs;&longs;e debeant. </s>
          <s id="N12C33">punctum igitur S in Q, &amp; T in R coaptabi&shy;<lb/>tur. </s>
          <s id="N12C37">eritqu&egrave; QM ip&longs;i SN &etail;qualis, &amp; MR ip&longs;i NT. ac pro <lb/>pterea linea GS parallelogrammi GT erit coaptata in <expan abbr="Aq;">A&que;</expan> <lb/>&amp; ET coaptata erit in GR parallelogrammi AR. Vnde e&shy;<lb/>rit AQ &etail;qualis GS, c&ugrave;m &longs;int coaptat&aelig;; &amp; GR ip&longs;i ET &etail;&shy;<lb/>qualis; c&ugrave;m &longs;int quo&que; coaptat&etail;. </s>
          <s id="N12C45">Quocirca quoniam
<arrow.to.target n="marg75"/> pa&shy;<lb/>rallelogramma AR GT &longs;unt inuicem coaptata, paral&shy;<lb/>lelogrammorumqu&egrave; oppo&longs;ita latera &longs;unt inter &longs;e &etail;qualia, <expan abbr="er&utilde;t">erunt</expan> <lb/>AQ GS GR ET inter &longs;e &etail;qualia. </s>
          <s id="N12C55">Nunc autem <expan abbr="intellig&atilde;tur">intelligantur</expan> <lb/>parallelogramma AK GF non ampli&ugrave;s coaptata. </s>
          <s id="N12C5D">&amp; <expan abbr="quoni&atilde;">quoniam</expan> <lb/>line&etail; QMR, &amp; SNT &longs;untip&longs;i AGE parallel&etail;; &amp; AQ GR, <lb/>GS ET, inter &longs;e &longs;unt&aelig;quales, &amp; &etail;quidi&longs;tantes; puncta RS in <lb/>vnum coincident punctum. </s>
          <s id="N12C69">eritqu&egrave; QST linea recta. </s>
          <s id="N12C6B">ex qui <lb/>bus patet, rectam <expan abbr="line&atilde;">lineam</expan>, qu&aelig; coniungit centra grauitatis MN <lb/>ip&longs;i AGE &aelig;quidi&longs;tantem exi&longs;tere. </s>
          <s id="N12C75">eodemqu&egrave; modo o&longs;tende&shy;<lb/>tur rectas lineas, qu&aelig; coniungunt grauitatis centra NO, cen&shy;<lb/>traqu&egrave; OP, ip&longs;i AB <expan abbr="&aelig;quidi&longs;t&atilde;tes">&aelig;quidi&longs;tantes</expan> e&longs;&longs;e. </s>
          <s id="N12C7F">Vnde &longs;equitur lineam <lb/>MNOP rectam e&longs;&longs;e. </s>
          <s id="N12C83">Quare prim&ugrave;m con&longs;tat grauitatis <expan abbr="c&etilde;tra">centra</expan> <lb/>in recta linea exi&longs;tere. </s>
        </p>
        <p id="N12C8B" type="margin">
          <s id="N12C8D"><margin.target id="marg71"/>36. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N12C96" type="margin">
          <s id="N12C98"><margin.target id="marg72"/>29. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N12CA1" type="margin">
          <s id="N12CA3"><margin.target id="marg73"/>34. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N12CAC" type="margin">
          <s id="N12CAE"><margin.target id="marg74"/>5. <emph type="italics"/>post, hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N12CB9" type="margin">
          <s id="N12CBB"><margin.target id="marg75"/>34. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.083.1.jpg" xlink:href="077/01/083/1.jpg"/>
        <p id="N12CC8" type="main">
          <s id="N12CCA">Quoniam autem o&longs;ten&longs;um e&longs;t QM &aelig;qualem e&longs;&longs;e ip&longs;i SN, <lb/>&amp; MR ip&longs;i NT, eodem quo&que; modo o&longs;tendetur OT &etail;qua-
<pb xlink:href="077/01/084.jpg" pagenum="80"/>lem e&longs;&longs;e ip&longs;i SN. Quoniam igitur OT NS &longs;unt &etail;quales, iti&shy;<lb/>demqu&egrave; TN SM &aelig;quales, erit ON ip&longs;i NM &aelig;qualis. </s>
          <s id="N12CD4">ea&shy;<lb/>demqu&egrave; ratione o&longs;tendetur OP &etail;qualem e&longs;&longs;e ip&longs;i ON. vn&shy;<lb/>de colligitur lineas MN NO OP inter centra exi&longs;tentes in&shy;<lb/>rer&longs;e &etail;quales e&longs;&longs;e. </s>
        </p>
        <p id="N12CDC" type="main">
          <s id="N12CDE">Po&longs;trem&ograve; quoniam parallelogramma AK GF EL HD <lb/>&longs;unt inuicem &aelig;qualia, &amp; numero paria, centraqu&egrave; grauitatis <lb/>&longs;unt in recta linea po&longs;ita. </s>
          <s id="N12CE4">line&etail;qu&egrave; MN NO OP inter cen&shy;<lb/>tra &longs;unt &etail;quales, magnitudinis ex omnibus AK GF EL HD <lb/>
<arrow.to.target n="marg76"/> magnitudinibus compo&longs;it&aelig; centrum grauitatis e&longs;t in linea <lb/>MP bifariam diui&longs;a. </s>
          <s id="N12CF0">Et quoniam MN e&longs;t &aelig;qualis ip&longs;i OP, <lb/>punctum, quod bifariam diuidit MP cadet in linea NO. <lb/>centrum ergo grauitatis omnium magnitudinum AK GF <lb/>EL HD, hoc e&longs;t parallelogrammi AD e&longs;t in linea NO, qu&etail; <lb/>coniungit centra &longs;patiorum mediorum GF EL. qu&etail; <expan abbr="quid&etilde;">quidem</expan> <lb/>omnia o&longs;tendere oportebat. </s>
        </p>
        <p id="N12D00" type="margin">
          <s id="N12D02"><margin.target id="marg76"/>2.<emph type="italics"/>cor. </s>
          <s id="N12D09">quin<lb/>t&aelig; huius.<emph.end type="italics"/></s>
        </p>
        <p id="N12D0F" type="main">
          <s id="N12D11">Quoniam autem centrum grauitatis <expan abbr="parallelogr&atilde;mi">parallelogrammi</expan> AD <lb/>e&longs;t in linea NO, &amp; in linea MP bifariam diui&longs;a; non repu&shy;<lb/>gnare videtur, quin inferri po&longs;&longs;it, hoc centrum e&longs;&longs;e in puncto <lb/>T, in linea EF exi&longs;tente. </s>
          <s id="N12D1D">Qu&ograve;d tamen fal&longs;um e&longs;t. </s>
          <s id="N12D1F">nam po&longs; <lb/>&longs;et quidem concludi centru e&longs;&longs;e in medio line&etail; NO (<expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> <lb/>e&longs;t in medio line&etail; MP, vt <expan abbr="dict&utilde;">dictum</expan> e&longs;t) &longs;ed <expan abbr="n&otilde;">non</expan> in <expan abbr="p&utilde;cto">puncto</expan> T; ex <expan abbr="dem&otilde;">demom</expan> <lb/>&longs;tratione enim o&longs;tenditur NS &aelig;qualem e&longs;&longs;e ip&longs;i TO. at ver&ograve; <lb/>NT &etail;qualem e&longs;&longs;e ip&longs;i TO, nullo modo demon&longs;trari pote&longs;t; <lb/>ni&longs;i &longs;upponeremus centra grauitatis MNOP in parallelogra <lb/>mis ita &longs;e habere, vt MQ MR, &amp; MR RN, &amp; RN NT &amp; <lb/>NT TO, &amp;c. </s>
          <s id="N12D43">inter &longs;e &etail;quales e&longs;&longs;ent. </s>
          <s id="N12D45">quod nullo modo &longs;up&shy;<lb/>poni pote&longs;t nam hoc modo centra grauitatis parallelogram&shy;<lb/>morum AK GF &amp;c. </s>
          <s id="N12D4B">e&longs;&longs;ent in lineis, qu&etail; bifariam &longs;ecant op <lb/>po&longs;ita latera. </s>
          <s id="N12D4F">e&longs;&longs;ent quipp&egrave; in lineis &agrave; punctis MN OP du&shy;<lb/>ctisip&longs;is AC GK EF &amp;c. </s>
          <s id="N12D53">&aelig;quidiftantibus, qu&aelig; oppo&longs;ita la <lb/>tera AG CK, GE KF, EH FL, &amp;c. </s>
          <s id="N12D57">bifariam &longs;ecarent. </s>
          <s id="N12D59">quod <lb/>e&longs;t id, quod Archimedes demon&longs;trare in <expan abbr="&longs;e&quetilde;ti">&longs;e&que;nti</expan> nititur. </s>
          <s id="N12D61">quod <lb/>quidem in cau&longs;a e&longs;t, vt demon&longs;tratione ad impo&longs;&longs;ibile id de&shy;<lb/>ducat. </s>
          <s id="N12D67">&longs;uppo&longs;uimus autem (vt pare&longs;t) parallelogramma cen-
<pb xlink:href="077/01/085.jpg" pagenum="81"/>tra grauitatis habere; ac centra grauitatis MNOP intra pa&shy;<lb/>rallelogramma exi&longs;tere, quoniam parallelogramma &longs;unt
<arrow.to.target n="marg77"/> fi&shy;<lb/>gur&aelig; ad ea&longs;dem partes concau&aelig;. </s>
          <s id="N12D75">quod quidem eodem modo <lb/>ab Archimede in &longs;e&que;nti &longs;upponitur. </s>
        </p>
        <p id="N12D79" type="margin">
          <s id="N12D7B"><margin.target id="marg77"/>9. <emph type="italics"/>po&longs;t hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N12D86" type="head">
          <s id="N12D88">PROPOSITIO. IX.</s>
        </p>
        <p id="N12D8A" type="main">
          <s id="N12D8C">Omnis parallelogrammi centrum grauitatis <lb/>e&longs;t in recta linea, qu&aelig; oppo&longs;ita latera parallelo&shy;<lb/>grammi bifariam diui&longs;a coniungit. </s>
        </p>
        <figure id="id.077.01.085.1.jpg" xlink:href="077/01/085/1.jpg"/>
        <p id="N12D95" type="main">
          <s id="N12D97"><emph type="italics"/>Sit parallelogrammum ABCD, linea ver&ograve; EF bifariam diuidat la <lb/>tera AB CD. Dico parallelogrammi ABCD centrum grauitatis e&longs;&longs;e<emph.end type="italics"/>
<arrow.to.target n="marg78"/><lb/><emph type="italics"/>in linea EF. Non &longs;it quidem, &longs;ed, &longs;i fieri pote&longs;t, &longs;it H. &amp;<emph.end type="italics"/> ab ip&longs;o <expan abbr="v&longs;q;">v&longs;&que;</expan> <lb/>ad lineam EF <emph type="italics"/>ducatur H<gap/> &aelig;quidistansip&longs;i AB. Diui&longs;a ver&ograve; EB <lb/>&longs;emper bifariam<emph.end type="italics"/> in G. rur&longs;u&longs;qu&egrave; EG brfariam in K; id&egrave;qu&egrave; <lb/>&longs;emper fiat, tandem <emph type="italics"/>qu&aelig;dam relin&que;tur linea,<emph.end type="italics"/> put&agrave; EK, <emph type="italics"/>minor <lb/>ip&longs;a HI. Diuidaturqu&egrave; vtra&que; AE EB in partes<emph.end type="italics"/> AN NM ML
<arrow.to.target n="marg79"/><lb/>LE GO OB <emph type="italics"/>ip&longs;i EK &aelig;quales.<emph.end type="italics"/> quod quidem fieri pote&longs;t, quia <lb/>diui&longs;a e&longs;t EB in partes &longs;emper &etail;quales. <emph type="italics"/>&amp; ex<emph.end type="italics"/> his <emph type="italics"/>diui&longs;ionum pun<lb/>ctis ducantur<emph.end type="italics"/> NP MQ LR kS GT OV <emph type="italics"/>ip&longs;i EF &aelig;quidistantes. <lb/>diui&longs;um enim erit totum parallelogrammum in parallelogramma &aelig;qualia <lb/>&amp; &longs;imiliaip&longs;i<emph.end type="italics"/> k<emph type="italics"/>F.<emph.end type="italics"/> c&ugrave;m enim &longs;int parallelogrammorum ba&longs;es <lb/>EL LM MN NA KG GO OB ip&longs;i KE &aelig;quales,
<arrow.to.target n="marg80"/> parallelo&shy;<lb/>grammaqu&egrave; in ij&longs;dem &longs;int parallelis AB CD con&longs;tituta; <lb/>erunt parallelogramma &aelig;qualia. </s>
          <s id="N12DFF">&longs;imilia ver&ograve;, quoniam <lb/>&longs;unt &etail;quiangula. <emph type="italics"/>Parallelogrammis igitur &aelig;qualibus, at&que;<emph.end type="italics"/>
<pb xlink:href="077/01/086.jpg" pagenum="82"/><emph type="italics"/>&longs;imilibus ip&longs;i KF inuicem coaptatis, &amp; centra grauitatis inter &longs;e conue&shy;<lb/>nient.<emph.end type="italics"/> quia ver&ograve; in EB facta e&longs;t diui&longs;io &longs;emper in duas partes <lb/>&etail;quales erunt parallelogramma in ED numero paria. </s>
          <s id="N12E16">ac per <lb/>con&longs;e&que;ns &amp; qu&etail; &longs;unt in EC numero paria. </s>
          <s id="N12E1A">vnde &amp; qu&etail; <expan abbr="s&utilde;">sunt</expan> <lb/>in toto AD numero paria <expan abbr="er&utilde;t">erunt</expan>. <emph type="italics"/>Jta&que; qu&aelig;dam erunt magnitudi&shy;<lb/>nes &aelig;quidi&longs;tantium laterum &aelig;quales ip&longs;i KF numero pares,<emph.end type="italics"/> hoc e&longs;t o&shy;<lb/>
<arrow.to.target n="marg81"/> mnes, qu&aelig; &longs;unt in AD, <emph type="italics"/>centraqu&egrave; grauitatis ip&longs;arum in recta linea<emph.end type="italics"/><lb/>
<arrow.to.target n="marg82"/> <emph type="italics"/>&longs;unt con&longs;tituta, &amp; line&aelig; inter centra &longs;unt a quales magnitudinis ex ip&longs;is <lb/>omnibus compo&longs;it&aelig; centrum grauitatis erit in recta linea, qu&aelig; coniungit <lb/>centra grauitatis mediorum &longs;patiorum,<emph.end type="italics"/> parallelogrammorum &longs;cili&shy;<lb/>cet LF KF. <emph type="italics"/>Non est autem; punctum enim H,<emph.end type="italics"/> quod &longs;upponitur <lb/>e&longs;&longs;e centrum grauitatis omnium magnitudinum, hoc e&longs;t pa <lb/>rallelogrammi AD, <emph type="italics"/>extra media parallelogramma<emph.end type="italics"/> LF KF <emph type="italics"/>exi&longs;tit.<emph.end type="italics"/><lb/>etenim c&ugrave;m &longs;it EK minor HI, linea KS ip&longs;i EF <expan abbr="&etail;quidi&longs;t&atilde;s">&etail;quidi&longs;tans</expan> <lb/>lineam HI ip&longs;i EK &aelig;quidi&longs;tantem &longs;ecabit, quipp&egrave; qu&aelig; re&shy;<lb/>lin&que;t punctum H extra figuram KF, ac per con&longs;e&que;ns ex&shy;<lb/>tra media parallelogramma LF KF. quare punctum H non <lb/>e&longs;t centrum grauitatis parallelogrammi AD, vt &longs;upponeba&shy;<lb/>tur. <emph type="italics"/>ergo con&longs;tat, centrum grauitatis parallelogrammi ABCD e&longs;&longs;e in re <lb/>cta linea EF.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N12E74" type="margin">
          <s id="N12E76"><margin.target id="marg78"/>*</s>
        </p>
        <p id="N12E7A" type="margin">
          <s id="N12E7C"><margin.target id="marg79"/><emph type="italics"/>ex prima <lb/>pr&etail;cedenti<emph.end type="italics"/></s>
        </p>
        <p id="N12E86" type="margin">
          <s id="N12E88"><margin.target id="marg80"/>36. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N12E91" type="margin">
          <s id="N12E93"><margin.target id="marg81"/>*</s>
        </p>
        <p id="N12E97" type="margin">
          <s id="N12E99"><margin.target id="marg82"/><emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N12EA1" type="head">
          <s id="N12EA3">SCHOLIVM.</s>
        </p>
        <p id="N12EA5" type="main">
          <s id="N12EA7">
<arrow.to.target n="marg83"/> Gr&aelig;cus codex po&longs;t verba, <emph type="italics"/>centraqu&egrave; grauitatis ip&longs;arum in recta <lb/>linea &longs;unt constituta,<emph.end type="italics"/> habet, <foreign lang="greek">kai\ ta\ me\sa i)/sa, kai\ w_a\nta ta\ ef) eka/teza <lb/>tw=n me)swn auta/ te i)/sa e)nti/</foreign>, qu&aelig; quidem omnino &longs;uperflua nobis <lb/>ui<gap/>a &longs;unt, &amp; <expan abbr="tanqu&atilde;">tanquam</expan> ab aliquo addita. </s>
          <s id="N12EC3">Nam &longs;i Archimedes di&shy;<lb/>xit omnia parallelogramma e&longs;&longs;e inter &longs;e, &amp; &etail;qualia, &amp; &longs;imilia; <lb/>non opus e&longs;t addere, media LF ES e&longs;&longs;e inter &longs;e &etail;qualia, &amp; <lb/>qu&etail; ab his &longs;unrad vtram&que; partem, vt MR KT, NQ GV, <lb/>AP OD, e&longs;&longs;e inter &longs;e &aelig;qualia; cum omnia (vt dictum e&longs;t) &longs;int <lb/>&etail;qualia. </s>
          <s id="N12ECF">quare verba h&etail;c (meo quidem iudicio) delenda &longs;unt. <lb/>demon&longs;trationes enim mathematic&etail; nullum admittunt &longs;u&shy;<lb/>perfluum. </s>
          <s id="N12ED5">&amp; Archim edes non tant&ugrave;m &longs;uperfluus, quin poti&ugrave;s <lb/>ob cius breuitatem diminutus fer&egrave; videatur. </s>
        </p>
        <pb xlink:href="077/01/087.jpg" pagenum="83"/>
        <p id="N12EDC" type="margin">
          <s id="N12EDE"><margin.target id="marg83"/>*</s>
        </p>
        <p id="N12EE2" type="main">
          <s id="N12EE4">Ex hac nona propo&longs;itione duo corolloria elicere po&longs;&longs;um^&lcub;9}; <lb/>qu&aelig; quidem tanquam valde nota fortaf&longs;e videtur omi&longs;i&longs;&longs;e Ar <lb/>chimedes. </s>
          <s id="N12EEA">quamuis <expan abbr="prim&utilde;">primum</expan> in &longs;e&que;nti <expan abbr="dem&otilde;&longs;tratione">demon&longs;tratione</expan> in&longs;eruit. </s>
        </p>
        <p id="N12EF4" type="head">
          <s id="N12EF6">COROLLARIVM. I.</s>
        </p>
        <p id="N12EF8" type="main">
          <s id="N12EFA">Ex hoc per&longs;picuum e&longs;t cuiu&longs;libet parallelogrammi <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis e&longs;&longs;e punctum, in quo coincidunt rect&aelig; line&aelig;, qu&aelig; <lb/>oppo&longs;ita latera bifariam &longs;ecant. </s>
        </p>
        <p id="N12F04" type="main">
          <s id="N12F06">Nam (vt Archimedes etiam &longs;e <lb/>
<arrow.to.target n="fig34"/><lb/>&que;nti demon&longs;tratione inquit) <lb/>&longs;i parallelogrammi ABCD line&etail; <lb/>EF GH bifariam diuident late&shy;<lb/>ra oppo&longs;ita AB DC, &amp; AD BC. <lb/>patet in EF centrum e&longs;&longs;e graui&shy;<lb/>tatis parallelogrammi AC. &longs;imi <lb/>liter con&longs;tat idem centrum e&longs;&longs;e <lb/>in linea GH, qu&aelig; oppo&longs;ita latera AD BC bifariam &longs;ecat. </s>
          <s id="N12F1D">e&shy;<lb/>ritigitur in K, vbi EF GH &longs;einuicem &longs;ecant. </s>
        </p>
        <figure id="id.077.01.087.1.jpg" xlink:href="077/01/087/1.jpg"/>
        <p id="N12F25" type="head">
          <s id="N12F27">COROLLARIVM. II.</s>
        </p>
        <p id="N12F29" type="main">
          <s id="N12F2B">Ex hoc patet etiam, cuiu&longs;libet parallelogrammi <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis e&longs;&longs;e in medio rect&aelig; line&etail;, qu&aelig; bifariam oppo&longs;ita latera <lb/>di&longs;pe&longs;cit. </s>
        </p>
        <p id="N12F35" type="main">
          <s id="N12F37">C&ugrave;m enim o&longs;ten&longs;um &longs;it centrum grauitatis parallelogram <lb/>mi AC e&longs;&longs;e punctum K. &amp; ob parallelogrammum EH e&longs;t <lb/>EK &aelig;qualis BH. propter parallelogrammum ver&ograve; KC
<arrow.to.target n="marg84"/><lb/>linea KF e&longs;t &aelig;qualis HC. &longs;untqu&egrave; BH HC &aelig;qua&shy;<lb/>les. </s>
          <s id="N12F44">erit EK ip&longs;i KF &aelig;qualis. </s>
          <s id="N12F46">punctum ergo K e&longs;t in medio <lb/>rect&aelig; line&etail; EF, qu&aelig; oppo&longs;ita latera AB DC bifariam diui&shy;<lb/>dit. <expan abbr="Eod&etilde;&qacute;">Eoden&que;</expan>; pror&longs;us modo <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan>, K <expan abbr="medi&utilde;">medium</expan> e&longs;&longs;e rect&etail; line&etail; <lb/>GH, qu&aelig; bifariam &longs;ecat oppo&longs;ita latera AD BC. </s>
        </p>
        <p id="N12F5A" type="margin">
          <s id="N12F5C"><margin.target id="marg84"/>34. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N12F65" type="main">
          <s id="N12F67">In &longs;e&que;nti Archimedes adhuc per&longs;i&longs;tit in inuentione cen&shy;<lb/>tri grauitatis parallelogrammorum, alia tamen methodo. <lb/>nam hoc perip&longs;orum parallelogrammorum diametros duo&shy;<lb/>bus modis a&longs;&longs;equitur. </s>
        </p>
        <pb xlink:href="077/01/088.jpg" pagenum="84"/>
        <p id="N12F72" type="head">
          <s id="N12F74">PROPOSITIO. X.</s>
        </p>
        <p id="N12F76" type="main">
          <s id="N12F78">Omnis parallelogrammi centrum grauitatis <lb/>e&longs;t punctum, in quo diametri coincidunt. </s>
        </p>
        <p id="N12F7C" type="main">
          <s id="N12F7E"><emph type="italics"/>Sit parallelogrammum <lb/>ABCD. &amp; in ip&longs;o &longs;it li&shy;<lb/>nea EF<emph.end type="italics"/> bifariam <emph type="italics"/><expan abbr="&longs;ec&atilde;s">&longs;ecans</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="fig35"/><lb/><emph type="italics"/>latera AB CD. itidem&shy;<lb/>qu&egrave; &longs;it KL <expan abbr="&longs;ec&atilde;s">&longs;ecans</expan> AC BD<emph.end type="italics"/><lb/>bifariam. </s>
          <s id="N12FA3">conueniant&shy;<lb/>qu&egrave; EF kL in H. <emph type="italics"/>est <lb/>vti&que; parallelogrammi<emph.end type="italics"/><lb/>
<arrow.to.target n="marg85"/> <emph type="italics"/>ABCD centrum grauita <lb/>tis in linea EF. hoc enim <lb/>o&longs;ten&longs;um e&longs;t. </s>
          <s id="N12FBB">eadem ver&ograve; de cau&longs;a<emph.end type="italics"/> centrum grauitatis ip&longs;ius AD <emph type="italics"/>est <lb/>etiam in linea<emph.end type="italics"/> K<emph type="italics"/>L. quare punctum H<emph.end type="italics"/> parallelogrammi AD <emph type="italics"/>cen&shy;<lb/>trum grauitatis existit. </s>
          <s id="N12FD3">Ver&ugrave;m in puncio H diametri parallelogram&shy;<lb/>mi concurrunt.<emph.end type="italics"/> ductis enim lineis AH HB CH HD; quoniam <lb/>line&aelig; AE EB EF FD inter &longs;e &longs;unt &etail;quales. </s>
          <s id="N12FDC">&longs;imiliter quo&que; <lb/>AK KC BL LD inter &longs;e &etail;quales; erit EH ip&longs;i HF &etail;qua <lb/>lis, c&ugrave;m &longs;int ip&longs;is BL LD &etail;quales. </s>
          <s id="N12FE2">du&aelig; igitur AE EH dua <lb/>
<arrow.to.target n="marg86"/> bus DF FH &longs;unt &aelig;quales, &amp; angulus AEH angulo DFH <lb/>
<arrow.to.target n="marg87"/> &etail;qualis; erit triangulum AEH triangulo DFH &etail;quale. </s>
          <s id="N12FF0">ac <lb/>propterea angulus EHA angulo FHD &aelig;qualis. </s>
          <s id="N12FF4">c&ugrave;m igitur <lb/>&longs;it EHF recta linea, eruntangnli EHA FHD adverticem, <lb/>&amp; obid AHD recta exi&longs;tit linea. </s>
          <s id="N12FFA">ac per con&longs;e&que;ns diame&shy;<lb/>ter parallelogrammi AD. pariqu&egrave; ratione o&longs;tendetur BHC <lb/>rectam e&longs;&longs;e lineam. </s>
          <s id="N13000">ex quibus patet in puncto H <expan abbr="vtr&atilde;&que;">vtran&que;</expan> dia <lb/>metrum conuenire. </s>
          <s id="N13008">centrum igitur grauitatis parallelogram&shy;<lb/>mi AD e&longs;t <expan abbr="p&utilde;ctum">punctum</expan>, in quo diametri concurrunt. <emph type="italics"/>Quare demon <lb/>stratume&longs;t, quod propo&longs;itum fuit.<emph.end type="italics"/></s>
        </p>
        <pb xlink:href="077/01/089.jpg" pagenum="85"/>
        <p id="N1301A" type="margin">
          <s id="N1301C"><margin.target id="marg85"/>9 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13025" type="margin">
          <s id="N13027"><margin.target id="marg86"/>29, <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N13030" type="margin">
          <s id="N13032"><margin.target id="marg87"/>4. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.089.1.jpg" xlink:href="077/01/089/1.jpg"/>
        <p id="N1303F" type="main">
          <s id="N13041">ALITER. </s>
        </p>
        <p id="N13043" type="main">
          <s id="N13045"><emph type="italics"/>Hoc autem aliter quo&shy;<lb/>&que; o&longs;tendetur. </s>
          <s id="N1304B">&longs;it paralle<emph.end type="italics"/><lb/>
<arrow.to.target n="fig36"/><lb/><emph type="italics"/>logrammum ABCD. <lb/>ip&longs;ius ver&ograve; diameter &longs;it<emph.end type="italics"/>
<arrow.to.target n="marg88"/><lb/><emph type="italics"/>B D. triangula<emph.end type="italics"/> vti&que; <lb/>ABD BDC <emph type="italics"/>erunt in&shy;<lb/>ter&longs;e &aelig;qualia, &amp; &longs;imilia. <lb/>quare triangulis inuicem <lb/>coaptatis; centra quo&que; <lb/>grauitatis ip&longs;orum inuicem coaptabuntur. </s>
          <s id="N13074">Sit autem trianguli ABD cen<emph.end type="italics"/>
<arrow.to.target n="marg89"/><lb/><emph type="italics"/>trum grauitatis punctum E; lineaqu&egrave; BD bifariam &longs;ecetur in H. con <lb/>nectaturqu&egrave; EH, &amp; producatur. </s>
          <s id="N13082">&longs;umaturqu&egrave; FH &aelig;qualisip&longs;i HE. <lb/>Ita&que; coaptato triangulo ABD cumtriangulo B DC, po&longs;itoqu&egrave; latere <lb/>AB in DC,<emph.end type="italics"/> hoc e&longs;t A in C, &amp; B in D. <emph type="italics"/>AD autem<emph.end type="italics"/> po&longs;ito <emph type="italics"/>in <lb/>BC;<emph.end type="italics"/> A &longs;cilicet in C, &amp; D in B. vnde &amp; BD cum ip&longs;amet <lb/>DB coaptatur, B &longs;cilicet in D, &amp; D in B. quia ver&ograve; pun&shy;<lb/>ctum H &longs;ibi ip&longs;i coaptatur, c&ugrave;m fitmedium line&etail; BD. &amp; an <lb/>guli EHD FHB ad verticem &longs;unt &aelig;quales; lineaqu&egrave; EH e&longs;t <lb/>ip&longs;i HF &etail;qualis; <emph type="italics"/>congruet etiam recta HE cum recta FH, &amp; <expan abbr="p&utilde;-ctum">pun&shy;<lb/>ctum</expan> E cum F conueniet, &longs;ed<emph.end type="italics"/> quoniam punctum E centrum <lb/>e&longs;t grauitatis trianguli ABD idem punctum E <emph type="italics"/>cum centro e&shy;<lb/>tiam grauitatis trianguli B DC<emph.end type="italics"/> conueniet. </s>
          <s id="N130B7">ergo punctum F <expan abbr="c&etilde;-trum">cen&shy;<lb/>trum</expan> e&longs;t grauitatis trianguli BDC. Nunc ver&ograve; intelligantur <lb/>triangula non ampli&ugrave;s coaptata. <emph type="italics"/>Quoniam igitur centrum graui&shy;<lb/>tatis trianguli ABD e&longs;t punctum E, ip&longs;ius ver&ograve; DBC est punctum F,<emph.end type="italics"/><lb/>triangulaqu&egrave; ABD DBC &longs;unt &etail;qualia, <emph type="italics"/>patet magnitudinis ex v&shy;<lb/>tri&longs;&que; triangulis compo&longs;it<gap/> centrum grauitatis e&longs;&longs;e medium rect&aelig; line&aelig;<emph.end type="italics"/>
<arrow.to.target n="marg90"/><lb/><emph type="italics"/>EF; quod e&longs;t punctum H,<emph.end type="italics"/> vt factum furt. </s>
          <s id="N130DE">Quoniam autem dia&shy;<lb/>metri cuiu&longs;libet parallelogrammi &longs;e&longs;e bifariam di&longs;pe&longs;cunt, e&shy;<lb/>rit punctum H, vbi diametri parallelogrammi ABCD con&shy;<lb/>currunt. </s>
          <s id="N130E6">ergo punctum H, in quo diametri coincidunt; ip&longs;ius <lb/>ABCD centrum grauitatis exi&longs;tit. </s>
          <s id="N130EA">quod demon&longs;trare opor&shy;<lb/>rebat. </s>
        </p>
        <pb xlink:href="077/01/090.jpg" pagenum="86"/>
        <p id="N130F1" type="margin">
          <s id="N130F3"><margin.target id="marg88"/><emph type="italics"/>ex<emph.end type="italics"/> 34.<emph type="italics"/>pri <lb/>mi.<emph.end type="italics"/></s>
        </p>
        <p id="N13103" type="margin">
          <s id="N13105"><margin.target id="marg89"/>5. <emph type="italics"/>post hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N13110" type="margin">
          <s id="N13112"><margin.target id="marg90"/>4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.090.1.jpg" xlink:href="077/01/090/1.jpg"/>
        <p id="N1311F" type="head">
          <s id="N13121">SCHOLIVM.</s>
        </p>
        <p id="N13123" type="main">
          <s id="N13125">Cognito centro grauitatis cuiu&longs;libet parallelogrammi, <lb/>vult Archimedes o&longs;tendere centrum grauitatis triangulorum. <lb/>&amp; quoniam in hac po&longs;trema demon&longs;tratione a&longs;&longs;ump&longs;it cen&shy;<lb/>trum grauitatis trianguli ABD e&longs;&longs;e punctum E, videtur or <lb/>dinem peruerti&longs;&longs;e, &amp; per ignotiora doctrinam tradidi&longs;&longs;e; c&ugrave;m <lb/>non &longs;it adhuc o&longs;ten&longs;um, in quo &longs;itu dictum centrum in <expan abbr="tri&atilde;-gulis">trian&shy;<lb/>gulis</expan> reperiatur. </s>
          <s id="N13137">quod tamen &longs;i rect&egrave; perpendamus, non ita &longs;e <lb/>habet. </s>
          <s id="N1313B">Nam vis demon&longs;trationis e&longs;t in hoc con&longs;tituta, vt <lb/>&longs;upponatur triangulum habere centrum grauitatis, idqu&egrave; tan <lb/>
<arrow.to.target n="marg91"/> <gap/>&ugrave;m e&longs;&longs;e intra ipsum triangulum, quod quidem &longs;upponi po&shy;<lb/>te&longs;t. </s>
          <s id="N13149">c&ugrave;m triangulum &longs;it figura ad ea&longs;dem partes concaua. </s>
          <s id="N1314B">ne&shy;<lb/>&que; enim refert, &longs;iu&egrave; centrum &longs;it in E, &longs;iu&egrave; in alio &longs;itu, dum&shy;<lb/>modo intra triangulum exi&longs;tat. </s>
          <s id="N13151">demon&longs;tratio enim <expan abbr="eod&etilde;">eodem</expan> mo&shy;<lb/>do &longs;emper concludet punctum H centrum e&longs;&longs;e grauitatis pa <lb/>rallelogrammi AC, quod idem ob&longs;eruandum e&longs;t in <expan abbr="n&otilde;nullis">nonnullis</expan> <lb/>alijs demon&longs;trationibus. </s>
          <s id="N13161">vt in &longs;ecunda demon&longs;tratione deci&shy;<lb/>m&aelig; terti&aelig;, hui^&lcub;9} &amp; in prima &longs;ecundilibri. </s>
          <s id="N13165">Antequam <expan abbr="aut&etilde;">autem</expan> Ar&shy;<lb/>chimedes centrum grauitatis triangulorum o&longs;tendat, nonnul<lb/>las pr&etail;mittit propo&longs;itiones. </s>
        </p>
        <p id="N1316F" type="margin">
          <s id="N13171"><margin.target id="marg91"/>9. <emph type="italics"/>post hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N1317C" type="head">
          <s id="N1317E">PROPOSITIO. XI.</s>
        </p>
        <p id="N13180" type="main">
          <s id="N13182">Si duo triangula inter &longs;e &longs;imilia fuerint, &amp; in i&shy;<lb/>p&longs;is &longs;int puncta ad triangula &longs;imiliter po&longs;ita &amp; alre <lb/>rum punctum trianguli, in quo e&longs;t, centrum fue&shy;<lb/>rit grauitatis, &amp; alterum punctum trianguli, in <lb/>quo e&longs;t, centrum grauitatis exi&longs;tet. </s>
        </p>
        <pb xlink:href="077/01/091.jpg" pagenum="87"/>
        <p id="N1318F" type="main">
          <s id="N13191">Dicimus quidem puncta in &longs;imilibus figuris e&longs;&longs;e <lb/>&longs;imiliter po&longs;ita, &egrave; quibus ad &aelig;quales angulos du&shy;<lb/>ct&aelig; rect&aelig; line&aelig;, &aelig;quales efficiunt angulos ad ho&shy;<lb/>mologalatera. </s>
          <s id="N13199">Vt dictum fuit in &longs;eptimo po&longs;tulato. </s>
        </p>
        <figure id="id.077.01.091.1.jpg" xlink:href="077/01/091/1.jpg"/>
        <p id="N1319E" type="main">
          <s id="N131A0"><emph type="italics"/>Sint duo triangula ABC DEF<emph.end type="italics"/> &longs;imilia. <emph type="italics"/>&longs;it qu&egrave; AC ad DE, vt <lb/>AB ad DE, &amp; BC ad EF. &amp; in pr&aelig;fatis triangulis ABC DEF <lb/>&longs;int puncta HN &longs;imiliter po&longs;ita &longs;itqu&egrave; punctum H centrum grauitatis <lb/>trianguli ABC. Dico &amp; punctum N centrum e&longs;&longs;e grauitatis trianguli <lb/>DEF. non &longs;it quidem, &longs;ed, &longs;i fieripote&longs;t, &longs;it punctum G centrum grauita <lb/>tis trianguli DEF. <expan abbr="connect&atilde;turqu&egrave;">connectanturqu&egrave;</expan> HA HB HC, DN EN FN, <lb/>DG EG FG. Quoniamigitur &longs;imile e&longs;t triangulum ABC triangulo <lb/>DEF, &amp;<emph.end type="italics"/> ip&longs;orum <emph type="italics"/>centra grauitatum &longs;unt puncta HG. &longs;imi&shy;<lb/>lium autem figurarum centra grauitatum &longs;unt &longs;imiliter po&longs;ita; ita vt<emph.end type="italics"/>
<arrow.to.target n="marg92"/><lb/>ab ip&longs;is ad &etail;quales angulos duct&aelig; rect&aelig; line&etail; <emph type="italics"/>&aelig;quales faciant <lb/>angulos ad homologa latera, vnum&que;mqu&egrave; vnicuiqu&egrave;; erit angulus <lb/>GDE ip&longs;i HAB aqualis. </s>
          <s id="N131D3">at ver&ograve; anguius HAB aqualis est angulo <lb/>EDN, c&ugrave;m &longs;int puncta HN &longs;imiliter po&longs;ita: angulus igitur EDG <lb/>angulo EDN &aelig;qualis existit. </s>
          <s id="N131D9">maior minori qu&ograve;d fierinon potest. </s>
          <s id="N131DB">Non <lb/>igitur punctum G centrum e&longs;t grauitatis trianguli DEF. Quare e&longs;t <lb/>punctum N. quod demonstrare oportebat.<emph.end type="italics"/></s>
        </p>
        <pb xlink:href="077/01/092.jpg" pagenum="88"/>
        <p id="N131E6" type="margin">
          <s id="N131E8"><margin.target id="marg92"/>6.&amp; 7 <emph type="italics"/>po&longs;t <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N131F3" type="head">
          <s id="N131F5">SCHOLIVM.</s>
        </p>
        <p id="N131F7" type="main">
          <s id="N131F9">In hac propo&longs;itione &longs;upponit Archimedes dari po&longs;&longs;e pun&shy;<lb/>cta in triangulis &longs;imilib^&lcub;9} &longs;imiliter po&longs;ita, qd <expan abbr="quid&etilde;">quidem</expan> &longs;ieri po&longs;&longs;e <lb/>o&longs;tendimus in &longs;cholijs &longs;eptimi po&longs;tulati. </s>
          <s id="N13203">Pr&aelig;terea idem vide&shy;<lb/>tur Archimedes in triangulis demon&longs;trare, quod in &longs;exto po&shy;<lb/>&longs;tulato vniuer&longs;aliter in figuris &longs;uppo&longs;uit. </s>
          <s id="N13209">Nam &longs;i centra gra&shy;<lb/>uitatis &longs;upponuntur in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; <lb/>&amp; in &longs;imilibus triangulis quo&que; erunt &longs;imiliter po&longs;ita. </s>
          <s id="N1320F">In&shy;<lb/>ter h&etail;c tamen maxima e&longs;t differentia, nam in po&longs;tulato inquit, <lb/>centra grauitatum in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; cu<lb/>ius quidem conuer&longs;um, nemp&egrave; puncta in &longs;imilibus figuris &longs;i&shy;<lb/>militer po&longs;ita e&longs;&longs;e ip&longs;arum centra grauitatis, e&longs;t falium. </s>
          <s id="N13219">quod <lb/>e&longs;t quidem manife&longs;tum ab&longs;&que; alio exemplo. </s>
          <s id="N1321D">ac propterea <lb/>Archimedes hoc in loco inquit, &longs;i duo erunt pun&longs;ta in &longs;imi&shy;<lb/>libus triangulis &longs;imiliter po&longs;ita, &amp; alterum ip&longs;orum fuerit <expan abbr="c&etilde;-trum">cen&shy;<lb/>trum</expan> grauitatis. </s>
          <s id="N13229">&amp; alterum quo&que; <expan abbr="c&etilde;trum">centrum</expan> grauitatis exi&longs;tet. <lb/>Vnde propo&longs;itio h&etail;c poti&ugrave;s e&longs;t conuer&longs;a po&longs;tulati, qu&agrave;m <lb/>eadem. </s>
        </p>
        <p id="N13233" type="main">
          <s id="N13235">Ob demon&longs;trationem autem noui&longs;&longs;e oportet, qu&ograve;d &longs;i pun<lb/>ctum G fuerit in linea DN, tuncanguli EDG EDN e&longs;&longs;ent in<lb/>ter&longs;e &etail;quales, ac propterea demon&longs;tratio nihil ab&longs;urdi conclu<lb/>deret. </s>
          <s id="N1323D">In hoc autem ca&longs;u o&longs;tendendum e&longs;&longs;et, angulum EFG <lb/>ip&longs;i EFN &etail;qualem e&longs;&longs;e, vel FEG ip&longs;i FEN. qu&aelig; quidem eo&shy;<lb/>dem pror&longs;us modo o&longs;tendentur. </s>
          <s id="N13243">comparando nemp&egrave; angu&shy;<lb/>los EFG EFN angulo BCH; angulos ver&ograve; FEG FEN ip&longs;i <lb/>CBH. Qu&ograve;d &longs;i G fuerit in alio &longs;itu, vt in triangulo EDN, <lb/>tuncanguli FDG FDN o&longs;tendentur &etail;quales. </s>
          <s id="N1324B">&amp; ita in alijs <lb/>ca&longs;ibus, vbicun&que; &longs;cilicet fuerit punctum G, &longs;emper ali&shy;<lb/>quod inuenietur huiu&longs;modi ab&longs;urdum. </s>
          <s id="N13251">qu&aelig; quidem omni&shy;<lb/>n&ograve; fieri non po&longs;&longs;unt. </s>
        </p>
        <pb xlink:href="077/01/093.jpg" pagenum="89"/>
        <p id="N13258" type="head">
          <s id="N1325A">PROPOSITIO. XII.</s>
        </p>
        <p id="N1325C" type="main">
          <s id="N1325E">Si duo triangula &longs;imilia fuerint, alterius ver&ograve; <lb/>trianguli centrum grauitatis in rectalinea fuerit, <lb/>qu&aelig; &longs;it ab aliquo angulo ad dimidiam ba&longs;im du&shy;<lb/>cta; &amp; alrerius trianguli centrum grauitatis erit in <lb/>linea &longs;imiliter ducta. </s>
        </p>
        <figure id="id.077.01.093.1.jpg" xlink:href="077/01/093/1.jpg"/>
        <p id="N1326B" type="main">
          <s id="N1326D"><emph type="italics"/>Sint duo triangula ABC DEF<emph.end type="italics"/> &longs;imilia <emph type="italics"/>&longs;itqu&egrave; AC ad DF, vt <lb/>AB ad DE, &amp; BC ad FE. Diui&longs;aqu&egrave; AC bifariam in G, iunga <lb/>tur BG. centrum ver&ograve; grauitatis trianguli ABC &longs;it punctum H in li <lb/>nea BG. Dico centrum grauitatis trianguli EDF e&longs;&longs;e in recta linea &longs;i <lb/>militer ducta. </s>
          <s id="N1327F">&longs;ecetur DF bifariam in puncto M. &amp; iungatur EM. <lb/>&amp; vt BG ad BH, ita fiat ME ad EN. connectanturqu&egrave; AH <lb/>HC, DN NF. Quoniam enim<emph.end type="italics"/> e&longs;t BA ad ED, vt AC ad DF, &amp; <lb/><emph type="italics"/>AG dimidia e&longs;t ip&longs;ius AC; ip&longs;ius ver&ograve; DF dimidiaest DM; erit BA <lb/>ad ED, vt AG ad DM.<emph.end type="italics"/> Quoniam autem ob <expan abbr="tri&atilde;gulorum">triangulorum</expan>
<arrow.to.target n="marg93"/><lb/>ABC DEF &longs;imilitudinem angulus BAC angulo EDF e&longs;t &etail;&shy;<lb/>qualis. </s>
          <s id="N1329C">&amp; vt AB ad DE, ita AG ad DM; <expan abbr="permutando&qacute;">permutando&que;</expan>; AB ad
<arrow.to.target n="marg94"/><lb/>AG, vt DE ad DM; erit <expan abbr="triangul&utilde;">triangulum</expan> ABG <expan abbr="tri&atilde;gulo">triangulo</expan> DEM &longs;imile. <lb/><expan abbr="&longs;imili&utilde;">&longs;imilium</expan> <expan abbr="&atilde;t">ant</expan> <expan abbr="tri&atilde;gulor&utilde;">triangulorum</expan> <expan abbr="&atilde;guli">anguli</expan> <expan abbr="s&utilde;t">sunt</expan> &etail;quales, <emph type="italics"/>et circa &aelig;quales <expan abbr="&atilde;gulos">angulos</expan> late<emph.end type="italics"/>
<pb xlink:href="077/01/094.jpg" pagenum="90"/><emph type="italics"/>ra sut proportionalia. </s>
          <s id="N132D4">erit <lb/>igitur angul^&lcub;9} AGB angulo<emph.end type="italics"/><lb/>
<arrow.to.target n="fig37"/><lb/><emph type="italics"/>DME aqualis, et<emph.end type="italics"/> ABG ip <lb/>&longs;i DEM &aelig;qualis quare <lb/><emph type="italics"/>vt AG ad DM, ita e&longs;t BG<emph.end type="italics"/><lb/>
<arrow.to.target n="marg95"/> <emph type="italics"/>ad EM,<emph.end type="italics"/> &amp; vt AB ad DE, <lb/>ita BG ad EM; &amp; pmu&shy;<lb/>tado AB ad BG, vt DE <lb/>ad EM. <emph type="italics"/>e&longs;t autem BG ad<emph.end type="italics"/><lb/>
<arrow.to.target n="marg96"/> <emph type="italics"/>BH, vt ME ad EN, erit igitur ex &aelig;quali<emph.end type="italics"/> AB ad BH, vt DE ad EN. <lb/>
<arrow.to.target n="marg97"/> rur&longs;u&longs;qu&egrave; permutando <emph type="italics"/>AB ad DE, vt BH ad EN.<emph.end type="italics"/> <expan abbr="quoni&atilde;">quoniam</expan> <lb/>
<arrow.to.target n="marg98"/> autem anguli ABH DEN, quos ip&longs;&aelig; line&etail; continent, &longs;unt <lb/>&aelig;quales, erit triangulun. </s>
          <s id="N13329">ABH triangulo DEN &longs;imile. </s>
          <s id="N1332B">qua <lb/>re anguli &longs;unt inter &longs;e &aelig;quales, <emph type="italics"/>&amp; circa a quales angulos latera &longs;unt <lb/>proportionalia &longs;i autem hoc, angulus BAH angulo EDN est &aelig;qualis. <lb/>Vnde &amp; reliquus angulus HAC angulo NDF &aelig;quolis exi&longs;tit.<emph.end type="italics"/> <gap/>qui&shy;<lb/>dem totius BAC ip&longs;i EDF e&longs;t &aelig;qualis. <emph type="italics"/>Eademqu&egrave; ratione an-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg99"/> <emph type="italics"/>gulus BCH ip&longs;i EFN est &aelig;qualis. </s>
          <s id="N1334B">&amp; angulas HCG angulo NFM <lb/>&aelig;qualis, o&longs;ten&longs;um est autem angulum ABH ip&longs;i DEM aqualem e&longs;&longs;e.<emph.end type="italics"/><lb/>ob &longs;imilitudinem autem riangulorum ABC DEF totus an <lb/>
<arrow.to.target n="marg100"/> gulus ABC e&longs;tip&longs;i DEF &etail; ualis: <emph type="italics"/>ergo &amp; reliquus angulus HBC <lb/>ip&longs;i NEF &aelig;qualis exi&longs;tit. </s>
          <s id="N1335E">Porr&ograve; ex his omnibus patet puncta HN ad <lb/>homologa latera e&longs;&longs;e &longs;imiliter po&longs;ita, &amp;<emph.end type="italics"/> cum ip&longs;is <emph type="italics"/>angulas &aelig;quales effi&shy;<lb/>cere. </s>
          <s id="N1336A">C&ugrave;m igitur puncta HN &longs;int &longs;imiliter po&longs;ita; &amp; punctum H cen&shy;<lb/>trum e&longs;t grauitatis trianguli ABC, &amp; puncium N trianguli DEF <expan abbr="c&etilde;-trum">cen&shy;<lb/>trum</expan> grauitatis existet.<emph.end type="italics"/> exi&longs;tente igitur centro grauitatis H in li <lb/>nea BG ab angulo ad dimidiam ba&longs;im ducta. </s>
          <s id="N13379">&amp; alterum gra<lb/>uitatis centrum N in linea EM &longs;imiliter ducta reperitur. <lb/>quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N1337F" type="margin">
          <s id="N13381"><margin.target id="marg93"/>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1338A" type="margin">
          <s id="N1338C"><margin.target id="marg94"/>6.<emph type="italics"/>&longs;eati.<emph.end type="italics"/></s>
        </p>
        <p id="N13395" type="margin">
          <s id="N13397"><margin.target id="marg95"/>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N133A0" type="margin">
          <s id="N133A2"><margin.target id="marg96"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N133AB" type="margin">
          <s id="N133AD"><margin.target id="marg97"/>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N133B6" type="margin">
          <s id="N133B8"><margin.target id="marg98"/>6. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N133C1" type="margin">
          <s id="N133C3"><margin.target id="marg99"/>7. <emph type="italics"/>post hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N133CE" type="margin">
          <s id="N133D0"><margin.target id="marg100"/>11.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.094.1.jpg" xlink:href="077/01/094/1.jpg"/>
        <p id="N133DD" type="head">
          <s id="N133DF">SCHOLIVM.</s>
        </p>
        <p id="N133E1" type="main">
          <s id="N133E3">In &longs;e&que;nti Archimedes o&longs;tendet, in qua linea reperitur <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis cuiu&longs;libet trianguli. </s>
          <s id="N133EB">quod quidem duobus a&longs;&shy;<lb/>&longs;equitur medijs. </s>
          <s id="N133EF">Diligenter autem omnia &longs;unt con&longs;ideranda; <lb/>quoniam in hoc con&longs;i&longs;tit tota per&longs;crutatio centri grauitatis <lb/>triangulorum. </s>
          <s id="N133F5">Quapropter vt prior demon&longs;tratio appareat <lb/>per&longs;picua, h&etail;c antea demon&longs;trabimus. </s>
        </p>
        <pb xlink:href="077/01/095.jpg" pagenum="91"/>
        <p id="N133FC" type="main">
          <s id="N133FE">LEMMA. I. </s>
        </p>
        <p id="N13400" type="main">
          <s id="N13402">&AElig;quidi&longs;tantes line&aelig; lineas in eadem proportione di&longs;pe&shy;<lb/>&longs;cunt. </s>
        </p>
        <p id="N13406" type="main">
          <s id="N13408">Sintline&etail; AB CD, quas &longs;ecent &aelig;qui&shy;<lb/>
<arrow.to.target n="fig38"/><lb/>di&longs;tantes line&aelig; AC EF BD. Dico ita e&longs;&shy;<lb/>&longs;e BE ad EA, vt DF ad FC. prim&ugrave;m <lb/>quidem AB CD vel &longs;unt &etail;quidi&longs;tantes,
<arrow.to.target n="marg101"/><lb/>vel min&ugrave;s. </s>
          <s id="N1341A">&longs;i &longs;unt &aelig;quidi&longs;tantes, iam habe <lb/>tur intentum. </s>
          <s id="N1341E">Nam BE erit &aelig;qualis DF, <lb/>&amp; EA ip&longs;i FC. vnde &longs;equitur ita e&longs;&longs;e BE <lb/>
<arrow.to.target n="fig39"/><lb/>ad EA, vt DF ad FC. </s>
        </p>
        <p id="N13429" type="margin">
          <s id="N1342B"><margin.target id="marg101"/>34. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.095.1.jpg" xlink:href="077/01/095/1.jpg"/>
        <figure id="id.077.01.095.2.jpg" xlink:href="077/01/095/2.jpg"/>
        <p id="N1343C" type="main">
          <s id="N1343E">Si ver&ograve; AB CD non fuerint &aelig;quidi&shy;<lb/>&longs;tantes, concurrant in G, vt in &longs;ecunda fi&shy;<lb/>
<arrow.to.target n="fig40"/><lb/>gura, &amp; quoniam BD EF &longs;unt
<arrow.to.target n="marg102"/> &aelig;quidi&shy;<lb/>&longs;tantes, erit GB ad BE, vt GD ad DF.
<arrow.to.target n="marg103"/><lb/>&amp; <expan abbr="c&otilde;ponendo">componendo</expan> GE ad EB, vt GF ad FD.
<arrow.to.target n="marg104"/><lb/>conuertendoqu&egrave; BE ad EG, vt DF ad <lb/>FG, rur&longs;us quoniam EF AC &longs;unt &aelig;quidi <lb/>&longs;tantes; erit GE ad EA, vt GF ad FC, e&shy;<lb/>ritigitur ex &aelig;quali BE ad EA, vt DF ad FC. </s>
        </p>
        <p id="N13463" type="margin">
          <s id="N13465"><margin.target id="marg102"/>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N1346E" type="margin">
          <s id="N13470"><margin.target id="marg103"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13479" type="margin">
          <s id="N1347B"><margin.target id="marg104"/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.095.3.jpg" xlink:href="077/01/095/3.jpg"/>
        <p id="N1348F" type="main">
          <s id="N13491">Secent ver&ograve; &longs;e&longs;e line&aelig; AB CD, vt in tertia figura, ob
<arrow.to.target n="marg105"/> &longs;imi&shy;<lb/>litudinem triangulorum BGD EGF, it a erit BG ad GE, vt
<arrow.to.target n="marg106"/><lb/>DG ad GF. &amp; componendo BE ad EG, vt DF ad FG. e&longs;t
<arrow.to.target n="marg107"/><lb/>ver&ograve; GE ad EA, vt GF ad FC. ergo ex &aelig;quali BE ad EA <lb/>erit, vt DF ad FC. quod demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/096.jpg" pagenum="92"/>
        <p id="N134A8" type="margin">
          <s id="N134AA"><margin.target id="marg105"/><emph type="italics"/>ex<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N134B8" type="margin">
          <s id="N134BA"><margin.target id="marg106"/>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N134C3" type="margin">
          <s id="N134C5"><margin.target id="marg107"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N134CE" type="main">
          <s id="N134D0">LEMMA. II. </s>
        </p>
        <p id="N134D2" type="main">
          <s id="N134D4">Sit A ad B, vt C ad D; rur&longs;us A ad E &longs;it, vt C ad F. <lb/>Dico prim&ugrave;m A ad BE &longs;imul ita e&longs;&longs;e, vt C ad DF. </s>
        </p>
        <figure id="id.077.01.096.1.jpg" xlink:href="077/01/096/1.jpg"/>
        <p id="N134DB" type="main">
          <s id="N134DD">
<arrow.to.target n="marg108"/> Quoniam enim A e&longs;t ad B, vt C ad D, erit conuertendo <lb/>
<arrow.to.target n="marg109"/> B ad A, vt D ad C. e&longs;t autem A ad E, vt C ad F; ergo ex &etail;&shy;<lb/>
<arrow.to.target n="marg110"/> quali B erit ad E, vt D ad F. quare componendo BE ad <lb/>
<arrow.to.target n="marg111"/> E, vt DF ad F. quoniam autem A e&longs;t ad E, vt C ad F; e <lb/>
<arrow.to.target n="marg112"/> rit conuertendo E ad A, vt F ad C. rur&longs;us igitur ex &etail;quali <lb/>erit BE ad A, vt DF ad C. ac deni&que; conuertendo A e&shy;<lb/>rit ad BE, vt C ad DF. </s>
        </p>
        <p id="N134FF" type="margin">
          <s id="N13501"><margin.target id="marg108"/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N13511" type="margin">
          <s id="N13513"><margin.target id="marg109"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1351C" type="margin">
          <s id="N1351E"><margin.target id="marg110"/>18. <emph type="italics"/>qninti.<emph.end type="italics"/></s>
        </p>
        <p id="N13527" type="margin">
          <s id="N13529"><margin.target id="marg111"/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N13539" type="margin">
          <s id="N1353B"><margin.target id="marg112"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13544" type="main">
          <s id="N13546">Si ver&ograve; fuerint quattuor magnitudines; vt adhue A (in ea&shy;<lb/>dem figura) ad G &longs;it, vt C ad H. &longs;imili&shy;<lb/>
<arrow.to.target n="fig41"/><lb/>ter o&longs;tendetur A ad omnes BEG &longs;imul <lb/>&longs;umptas ita e&longs;&longs;e, vt C ad omnes &longs;imul <lb/>DFH. &longs;umendo vt in &longs;ecunda figura BE <lb/>pro vna tan &ugrave;m magnitudine, &amp; DF pro <lb/>alia; erunt&que; ex vtra&que; parte tres <expan abbr="t&atilde;t&ugrave;m">tant&ugrave;m</expan> <lb/>magnitudines; eritqu&egrave; A ad BE &longs;imul, <lb/>vt C ad DF &longs;imul, vt o&longs;ten&longs;um e&longs;t, dein<lb/>de A ad G e&longs;t, vt C ad H, erit igitur <lb/>A ad BEG &longs;imul, vt C ad DFH. </s>
        </p>
        <pb xlink:href="077/01/097.jpg" pagenum="93"/>
        <figure id="id.077.01.097.1.jpg" xlink:href="077/01/097/1.jpg"/>
        <p id="N1356C" type="main">
          <s id="N1356E">Pariqu&egrave; ratione &longs;i quin&que; fuerint magnitudines, eodem <lb/>modo tres medi&aelig; <expan abbr="i&utilde;gatur">iungatur</expan> &longs;imul, ita vttres &longs;int <expan abbr="d&utilde;taxat">duntaxat</expan> magni<lb/>tudines. </s>
          <s id="N1357C">&amp; &longs;ic in infinitum. </s>
          <s id="N1357E">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N13580" type="head">
          <s id="N13582">COROLLARIVM.</s>
        </p>
        <p id="N13584" type="main">
          <s id="N13586">Ex hoc elici pote&longs;t. </s>
          <s id="N13588">qu&ograve;d &longs;i fuerint quotcun&que; magnitudi <lb/>nes proportionales; &amp; ali&etail; ip&longs;is numero &aelig;quales, &amp; in eadem <lb/>proportione, vt &longs;cilicet &longs;it (vt in prima figura) A ad B, vt C <lb/>ad D, B ver&ograve; ad E, vt D ad F. deinde vt E ad G, &longs;ic F <lb/>ad H, &amp; ita deinceps, &longs;i plures fuerint magnitudines, &longs;i&shy;<lb/>militer erit A ad omnes BEG &longs;imul &longs;umptas, vt C ad om&shy;<lb/>nes &longs;imul DFH. </s>
        </p>
        <p id="N13596" type="main">
          <s id="N13598">Prim&ugrave;m quidem A e&longs;t ad B, vt C ad D. &amp; quoniam ma <lb/>gnitudines &longs;unt proportionales, ex &etail;quali erit A ad E, vt C
<arrow.to.target n="marg113"/><lb/>ad F. &longs;imiliter A ad G, vt C ad H. Ex quibus &longs;equitur <lb/>A ad BE &longs;imul ita e&longs;&longs;e, vt C ad DF. A ver&ograve; ad omnes <lb/>BEG &longs;imul, vt C ad omnes &longs;imul DFH. &amp; ita &longs;i plures fue<lb/>rint magnitudines. </s>
        </p>
        <p id="N135A7" type="margin">
          <s id="N135A9"><margin.target id="marg113"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N135B2" type="main">
          <s id="N135B4">LEMMA. III. </s>
        </p>
        <p id="N135B6" type="main">
          <s id="N135B8">Sit triangulum ABC, cuiuslatus BC in quotcun&que; di&shy;<lb/>uidatur partes &aelig;quales BE ED DF FC. &amp; a punctis EDF <lb/>ip&longs;i AB equidi&longs;tanres ducantur EG DH FK. rur&longs;us &agrave; pun<lb/>ctis GHK ip&longs;i BC &etail;quidi&longs;tantes ducantur GL HM KN. <lb/>Dico triangulum ABC ad omnia triangula ALG GMH <lb/>HNK KFC &longs;imul&longs;umpta eandem habere proportionem, <lb/>quam habet CA ad AG. </s>
        </p>
        <pb xlink:href="077/01/098.jpg" pagenum="94"/>
        <p id="N135C9" type="main">
          <s id="N135CB">
<arrow.to.target n="marg114"/> <expan abbr="Quoni&atilde;">Quoniam</expan> enim FK &etail;quidi&longs;tans e&longs;tip&longs;i DH; erit CF ad FD, <lb/>vt CK ad KH. <expan abbr="&longs;unt&qacute;">&longs;unt&que;</expan> CF FD &aelig;quales; ergo &amp; CK KH in&shy;<lb/>ter&longs;e &longs;unt &aelig;quales. </s>
          <s id="N135DD">&longs;imiliter propter lineas &aelig;quidi&longs;tantes FK <lb/>
<arrow.to.target n="marg115"/> DH EG, ita e&longs;t KH ad HG, vt FD ad DE; e&longs;t autem FD <lb/>&aelig;qualis DE; erit igitur KH ip&longs;i HG &aelig;qualis. </s>
          <s id="N135E7">Pariqu&egrave; ra&shy;<lb/>
<arrow.to.target n="fig42"/><lb/>tione o&longs;tendetur ob &etail;quidi&longs;tantes lineas DH EG BA, <expan abbr="line&atilde;">lineam</expan> <lb/>HG ip&longs;i GA &aelig;qualem e&longs;&longs;e. </s>
          <s id="N135F6">Ex quibus patet CK KH HG <lb/>GA inter &longs;e &aelig;quales e&longs;&longs;e. </s>
          <s id="N135FA">Quoniam autem trianguloru ABC <lb/>kFC angulus ad C e&longs;t vtri&que; communis; &amp; ABC ip&longs;i kFC, <lb/>
<arrow.to.target n="marg116"/> &amp; BAC ip&longs;i FKC &aelig;qualis, cum &longs;it Fk ip&longs;i AB &aelig;quidi&longs;tans; <lb/>erit triangulum ABC ip&longs;i KFC &longs;imile. </s>
          <s id="N13606">&amp; quonian NK FC, <lb/>&amp; HN KF &longs;unt &etail;quidi&longs;tantes, erunt anguli KCFCkF angu<lb/>lis HkN KHN &etail;quales; ac propterea reliquus CFK reliquo <lb/>KNH &etail;qualis: latus ver&ograve; CK lateri KH e&longs;t &etail;quale; erit igi&shy;<lb/>
<arrow.to.target n="marg117"/> tur triangulum KFC triangulo HNK &longs;imile, &amp; &etail;quale. </s>
          <s id="N13614">&longs;imi <lb/>literqu&egrave; <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan> omnia triangula ALG GMH HNK KFC <lb/>inter&longs;e&longs;e &longs;imilia, &amp; &aelig;qualia e&longs;&longs;e. </s>
          <s id="N1361E">&amp; obid ip&longs;i ABC &longs;imilia e&longs;&longs;e. <lb/>Fiat igit vt AC ad AG, ita AG ad alia O. &longs;imiliterv AC ad GH, <lb/>ita GH ad P. rur&longs;usvt AC ad Hk, ita HK ad <expan abbr="q.">&que;</expan> deniqu&egrave; <lb/>vt AC ad Ck, ita CK ad R. &amp; quoniam AG GH HK KC <lb/>
<arrow.to.target n="marg118"/> &longs;unt &aelig;quales, eadem AC ad vnamquam&que; ip&longs;arum ean&shy;<lb/>dem habebit proportionem, ergo eandem quo&que; habebit <lb/>propo&longs;itionem AG ad O, vt GH ad P, &amp; HK ad Q, &amp;
<pb xlink:href="077/01/099.jpg" pagenum="95"/>KC ad R. ac propterea line&aelig; OPQR inter &longs;e &longs;unt &aelig;quales. <lb/>Atver&ograve; quoniam ita e&longs;t AC ad AG, vt AG ad O, &amp; vt <lb/>AC ad GH, ita GH, hoc e&longs;t AG ip&longs;i &etail;qualis, ad P. rur&longs;us <lb/>vt AC ad HK, ita HK, hoc e&longs;t AG ad <expan abbr="q.">&que;</expan> ac tandem vt <lb/>AC ad KC, ita KC, hoc e&longs;t AG ip&longs;i &etail;qualis, ad R. erit AC
<arrow.to.target n="marg119"/><lb/>ad omnes con&longs;e&que;ntes &longs;imul &longs;umptas AG GH HK KC, <lb/>hoc e&longs;t erit AC ad eandem AC, vt AG ad omnes &longs;imul <lb/>OPQR. vnde &longs;equitur omnes &longs;imul OPQR ip&longs;i AG &etail;qua <lb/>les e&longs;&longs;e. </s>
          <s id="N1364F">Ita&que; quoniam &longs;imilia triangula in dupla &longs;unt
<arrow.to.target n="marg120"/> pro&shy;<lb/>portione laterum homologorum, erit triangulum ABC ad <lb/>ALG, vt AC ad O. eodemqu&egrave; modo erit triangulum ABC <lb/>ad GMH, vt AC ad P. rur&longs;us ABC ad HNK, vt AC ad <lb/>Q, &amp; vt idem ABC ad KFC, ita AC ad R. triangulum <lb/>igitur ABC ad omnes con&longs;e&que;ntes, videlicet ad omnia <expan abbr="tri&atilde;">triam</expan>
<arrow.to.target n="marg121"/><lb/>gula &longs;imul &longs;umpta ALG GMH HNK KFC, eritvt AC ad <lb/>omnes &longs;imul OPQR. hoc e&longs;t ad AG. o&longs;ten&longs;um e&longs;t igitur, <lb/>quod propo&longs;itum fuit. </s>
        </p>
        <p id="N1366C" type="margin">
          <s id="N1366E"><margin.target id="marg114"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13677" type="margin">
          <s id="N13679"><margin.target id="marg115"/>1. <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N13682" type="margin">
          <s id="N13684"><margin.target id="marg116"/>29. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N1368D" type="margin">
          <s id="N1368F"><margin.target id="marg117"/>76. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N13698" type="margin">
          <s id="N1369A"><margin.target id="marg118"/><emph type="italics"/>ex<emph.end type="italics"/> 17 <emph type="italics"/><expan abbr="qu&itilde;i">quini</expan>.<emph.end type="italics"/></s>
        </p>
        <p id="N136AB" type="margin">
          <s id="N136AD"><margin.target id="marg119"/><emph type="italics"/>ex <expan abbr="pr&aelig;ced&etilde;">pr&aelig;cedem</expan> <lb/>ti lemmate<emph.end type="italics"/></s>
        </p>
        <p id="N136BB" type="margin">
          <s id="N136BD"><margin.target id="marg120"/>19.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N136C6" type="margin">
          <s id="N136C8"><margin.target id="marg121"/><emph type="italics"/>ex <expan abbr="pr&aelig;ced&etilde;">pr&aelig;cedem</expan> <lb/>ti lemmate<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.099.1.jpg" xlink:href="077/01/099/1.jpg"/>
        <p id="N136DA" type="head">
          <s id="N136DC">PROPOSITIO. XIII.</s>
        </p>
        <p id="N136DE" type="main">
          <s id="N136E0">Omnis trianguli centrum grauitatis e&longs;t in recta <lb/>linea ab angulo ad dimidiam ba&longs;im ducta. </s>
        </p>
        <p id="N136E4" type="main">
          <s id="N136E6"><emph type="italics"/>Sit triangulum ABC. &amp; in ip&longs;o &longs;it AD<emph.end type="italics"/> ab angulo A <emph type="italics"/>ad dimi&shy;<lb/>diamba&longs;im BC ducta. </s>
          <s id="N136F2">o&longs;tendendum est, centrum grauitatis trianguli <lb/>ABC e&longs;&longs;e in linea AD. Non &longs;it quidem, &longs;ed &longs;i fieri potest &longs;it punctum <lb/>H. &amp; ab ip&longs;o ducatur HI &aelig;quidi&longs;tansip&longs;i BC,<emph.end type="italics"/> qu&aelig; ip&longs;am AD &longs;ecet
<arrow.to.target n="marg122"/><lb/>in I. <emph type="italics"/>Deinde diui&longs;a DC bifariam, idqu&egrave; &longs;emper fiat, dones relinqua&shy;<lb/>tur linea<emph.end type="italics"/> D<foreign lang="greek">w</foreign> <emph type="italics"/>minor ip&longs;a HI. Diuidaturqu&egrave; ip&longs;arum vtra&que; BD DC <lb/>in partes &aelig;quales<emph.end type="italics"/> D<foreign lang="greek">w</foreign>; parte&longs;qu&egrave; in DC exr&longs;tentes &longs;int D<foreign lang="greek">w wb <lb/>b</foreign>Z ZC; quibus re&longs;pondeant &aelig;quales partes D<foreign lang="greek">aazz</foreign>O OB. <emph type="italics"/>&amp; <lb/>a &longs;ectionum punctis ducantur<emph.end type="italics"/> OE <foreign lang="greek">z</foreign>G <foreign lang="greek">a</foreign>L <foreign lang="greek">w</foreign>M <foreign lang="greek">b</foreign>K ZF <emph type="italics"/>&aelig;quidictan<lb/>tes ip&longs;i AD. &amp; connectantur EF G<emph.end type="italics"/>k <emph type="italics"/>LM qu&aelig; nimirum ip&longs;i BC <lb/>&aelig;quidistantes erunt.<emph.end type="italics"/> c&ugrave;m enim &longs;int BD DC inter&longs;e equales, iti&shy;<lb/>dem OB ZC &aelig;quales; erit DO ip&longs;i DZ &etail;qualis. </s>
          <s id="N1374C">quare DO <lb/>ad OB e&longs;t, vt DZ ad ZC. Quoniam autem EO FZ &longs;unt
<pb xlink:href="077/01/100.jpg" pagenum="96"/>ipsi AD &aelig;quidi&longs;tantes, erit AE ad EB, vt DO ad OB; &amp; vt <lb/>
<arrow.to.target n="marg123"/> DZ ad ZC, &longs;ic AF ad FC. at&que; DO ad OB e&longs;t, vt DZ ad <lb/>ZC. erit igitur AE ad EB, vt AF ad FC. quare EF ip&longs;i BC <lb/>
<arrow.to.target n="marg124"/> e&longs;t &aelig;quidi&longs;tans, eodemqu&egrave; modo o&longs;tendetur, ita e&longs;&longs;e AG ad <lb/>
<arrow.to.target n="fig43"/><lb/>GB, vt AK ad KC, &amp; AL ad LB, vt AM ad MC. ex quib^&lcub;9} <lb/>&longs;equitur LM GK EF non &longs;ol&ugrave;m ip&longs;i BC, ver&ugrave;m etiam inter&shy;<lb/>&longs;e&longs;e parallelas e&longs;&longs;e. </s>
          <s id="N1376D">&longs;ecct EF lineas G<foreign lang="greek">z</foreign> K<foreign lang="greek">b</foreign> in X<foreign lang="greek">e</foreign>. ip&longs;am ver&ograve; <lb/>AD in T. lineaqu&egrave; GK &longs;ecet L<foreign lang="greek">a</foreign> M<foreign lang="greek">w</foreign> in N<foreign lang="greek">d</foreign>, &amp; AD in Y. <lb/>linea deniqu&egrave; LM ip&longs;am AD in S di&longs;pe&longs;cat. </s>
          <s id="N1378B">Quoniam au<lb/>tem D<foreign lang="greek">w</foreign> e&longs;t ip&longs;i HI &aelig;quidi&longs;tans, e&longs;tqu&egrave; D<foreign lang="greek">w</foreign> minor <expan abbr="qu&atilde;">quam</expan> HI, li <lb/>nea <foreign lang="greek">w</foreign>M ip&longs;i AL &etail;quidi&longs;tans ip&longs;am HI &longs;ecabir. </s>
          <s id="N137A1">ac propterea <lb/>punctum H centrum grauitatis trianguli ABC extra paral&shy;<lb/>
<arrow.to.target n="marg125"/> lelogrammum DM reperitur. </s>
          <s id="N137AB">At ver&ograve; quoniam LD DM <lb/>&longs;unt para lelogramma, erunt LS <foreign lang="greek">a</foreign>D inter &longs;e &aelig;quales, &longs;imili&shy;<lb/>ter SM D<foreign lang="greek">w</foreign> &etail;quales. </s>
          <s id="N137B9">&longs;untver&ograve; <foreign lang="greek">a</foreign>D D<foreign lang="greek">w</foreign> &etail;quales: ergo &amp; LS <lb/>SM inter &longs;e &longs;unt &etail;quales. </s>
          <s id="N137C5">eademqu&egrave; rarione NY Y<foreign lang="greek">d</foreign> inter &longs;e&shy;<lb/>&longs;e, &amp; ip&longs;is LS SM &etail;quales exi&longs;tent. </s>
          <s id="N137CD">quarelinea SY bifariam <lb/>diuiditlatera oppo&longs;ita parallelogrammi MN. pariqu&egrave; ratio&shy;<lb/>ne o&longs;tendetur lineam YT bifariam diuidere oppo&longs;ita latera <lb/>parallelogrammi KX; lineamqu&egrave; TD latera oppo&longs;ita paral-
<pb xlink:href="077/01/101.jpg" pagenum="97"/>lelogrammi FO bifariam quo&que; diuidere. <emph type="italics"/>Ita&que; parallelogr&agrave; <lb/>mi MN centrum grauitatis est in linea <foreign lang="greek">*u</foreign>S. parallilogrammi ver<gap/><lb/>KX grouitatis centrum est in linea T<foreign lang="greek">*u</foreign>. parallelogrammi autem FO in <lb/>linea TD; magnitudinis igitur ex<emph.end type="italics"/> his <emph type="italics"/>omnibus<emph.end type="italics"/> parallelogrammi <lb/>MN KX FO <emph type="italics"/>compo&longs;it&aelig; centrum grauitatis e&longs;t in recta linea S D. &longs;iv <lb/>ita&que; punctum R.<emph.end type="italics"/> quod quidem erit centrum grauitatis figura <lb/>LNGXEOZF <foreign lang="greek">e</foreign>K<foreign lang="greek">d</foreign>M. <emph type="italics"/><expan abbr="l&utilde;gatur&qacute;">lungatur&que;</expan>; RH, &amp; producatur,<emph.end type="italics"/> qu&aelig; ipsa <foreign lang="greek">w</foreign>M <lb/>&longs;ecet in P. <emph type="italics"/>ip&longs;iqu&egrave; AD<emph.end type="italics"/> a puncto C <emph type="italics"/>&aelig;qui di&longs;tans ducatur CV,<emph.end type="italics"/> qu<gap/><lb/>ip&longs;i RH occurrat in V. <emph type="italics"/><expan abbr="triangul&utilde;">triangulum</expan> ita&que; ADC ad omnia triangu<lb/>la ex AM MK<emph.end type="italics"/> k<emph type="italics"/>F FC de&longs;cripta &longs;imiliaip&longs;i ADC,<emph.end type="italics"/> hoc e&longs;t ad tria <lb/>gula ASM M <foreign lang="greek">d</foreign>K K<foreign lang="greek">e</foreign>F FZC &longs;imul &longs;umpta <emph type="italics"/>eandem habet propor <lb/>tionem, quam habet CA ad AM. &longs;iquidem &longs;unt AM MK<emph.end type="italics"/> k<emph type="italics"/>F FC<emph.end type="italics"/>
<arrow.to.target n="marg126"/><lb/><emph type="italics"/>&aelig;quales quia ver&ograve; &amp; triangulum ADB ad omnia ex AL LG GE <lb/>EB de&longs;cripta triangula &longs;imilia<emph.end type="italics"/> ALS LGN GEX EFO <emph type="italics"/>eandem ha <lb/>bet proportionem, quam &lsquo;BA ad AL<emph.end type="italics"/>: &amp; antecedentes &longs;imul ad
<arrow.to.target n="marg127"/><lb/>omnes con&longs;e&que;ntes, hoc e&longs;t totum triangulum ABC ad on <lb/>nia triangula &longs;imul &longs;umpta, qu&aelig; &longs;unt in AB, &amp; in AC con&longs;ti&shy;<lb/>tuta, eandem habebit proportionem, quam habet AC AB &longs;i<lb/>mul ad AM AL &longs;imul, quia ver&ograve; ob <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> <expan abbr="triangulor&utilde;">triangulorum</expan> <lb/>ABC ALM CA ad AM e&longs;t, vt BA ad AL; erit CA ad AM, vt <lb/>CA BA &longs;imul ad AM AL &longs;imul. <emph type="italics"/>triangulum igitur ABC ad omnia<emph.end type="italics"/>
<arrow.to.target n="marg128"/><lb/><emph type="italics"/>pr&aelig;dicta triangula eandem habet proportionem quam habet CA ad AM. <lb/>At&que; CA ad AM maiorem habet proportionem qu&agrave;m VR ad RH; e&shy;<lb/>tenim proportio ip&longs;ius CA ad AM e&longs;t eadem, qu&aelig; est totius VR <expan abbr="adips&atilde;">adipsam</expan> <lb/>R. p. <expan abbr="qu&atilde;doquid&etilde;">quandoquidem</expan> triangula<emph.end type="italics"/> ACD MC<foreign lang="greek">w</foreign> <emph type="italics"/>&longs;unt &longs;imilia.<emph.end type="italics"/> <expan abbr="&longs;int&qacute;">&longs;int&que;</expan>; AD &amp;
<arrow.to.target n="marg129"/><lb/>M<foreign lang="greek">w</foreign> &etail;quidi&longs;tantes, &longs;itqu&egrave; propterea CA ad AM, vt CD ad <lb/>D<foreign lang="greek">w</foreign>. &amp; quoniam VR DC &agrave;lineis DR <foreign lang="greek">w</foreign>p CV &aelig;quidi&longs;tantib^&lcub;9}
<arrow.to.target n="marg130"/><lb/>diuiduntur; erit C<foreign lang="greek">w</foreign> ad <foreign lang="greek">w</foreign>D, vt VP ad PR. &amp; <expan abbr="c&otilde;ponendo">componendo</expan> CD
<arrow.to.target n="marg131"/><lb/>ad D<foreign lang="greek">w</foreign>, vt VR ad RP. quare vt CA ad AM, ita VR ad RP.
<arrow.to.target n="marg132"/><lb/>quia ver&ograve; VR ad RP maiorem habet proportionem, qu&agrave;m
<arrow.to.target n="marg133"/><lb/>ad RH. maiorem quo&que; habebit proportionem CA ad <lb/>AM, qu&agrave;m VR ad RH. e&longs;t autem CA ad AM, vt <expan abbr="triangul&utilde;">triangulum</expan> <lb/>ABC ad omnia triangula in lineis AC AB. (vt dictum e&longs;t) <lb/>con&longs;tituta; ergo <emph type="italics"/>&amp; triangulum ABC adpr&aelig;dicta<emph.end type="italics"/> triangula <emph type="italics"/>maio <lb/>rem habet proportionem, qu&agrave;m VR ad RH. Quare &amp; diuidendo pa-<emph.end type="italics"/>
<arrow.to.target n="marg134"/><lb/><emph type="italics"/><expan abbr="rallelogr&amacr;ma">rallelogramma</expan> MN<emph.end type="italics"/> k<emph type="italics"/>X FO<emph.end type="italics"/> hoc e&longs;t figura LNGXEOZF <foreign lang="greek">e</foreign>K <foreign lang="greek">d</foreign>M) <emph type="italics"/>ad <lb/>circumrelicta triangula<emph.end type="italics"/> in lineis AC AB con&longs;tituta <emph type="italics"/>maiorem ha-<emph.end type="italics"/>
<pb xlink:href="077/01/102.jpg" pagenum="98"/><emph type="italics"/>bent proportionem, quam NH ad HR.<emph.end type="italics"/> linea igitur, qu&aelig; eandem <lb/>habeat proportionem ad HR, quam parallelogramma MN <lb/>kX FO ad circumrelicta triangula, maior erit, qu&agrave;m VH <lb/><emph type="italics"/>Fiat itaqu&egrave; in eademproportione QH ad HR, ut parallelogramma ad <lb/>triangula;<emph.end type="italics"/> erit vti&que; QH maior, quam VH. <emph type="italics"/>Quoniam igitur e&longs;t <lb/>magnitudo ABC, cuius centrum grauitatis est H, &amp; ab ea magnitudo<emph.end type="italics"/><lb/>
<arrow.to.target n="fig44"/><lb/><emph type="italics"/>auferatur compo&longs;ita ex MN<emph.end type="italics"/> k<emph type="italics"/>X FO parallelogrammis; &amp; magnitudi <lb/>nis ablat&aelig; centrum grauitatis e&longs;t punctum R; magnitudinis reliqu&aelig; ex <lb/>circumrelictis triangulis compo&longs;it&aelig; centrum grauitatis erit in recta li-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg135"/> <emph type="italics"/>nea RH<emph.end type="italics"/> ex parte H <emph type="italics"/>producta, a&longs;&longs;umptaqu&egrave; aliqua<emph.end type="italics"/> vt, QH, <emph type="italics"/>qu&aelig; ad <lb/>HR eam habeat proportionem, quam habet magnnudo<emph.end type="italics"/> ex parallelo&shy;<lb/>grammis MN KX FO con&longs;tans <emph type="italics"/>ad reliquum,<emph.end type="italics"/> hoc e&longs;t ad reli&shy;<lb/>qua triangula, <emph type="italics"/>ergo punctum Q centrum est grauitatis magnitudinis <lb/>ex ip&longs;is circumrelictis<emph.end type="italics"/> triangulis <emph type="italics"/>compo&longs;it&aelig;. </s>
          <s id="N1397F">quoa fieri non pote&longs;i aucta <lb/>enim recta linea <foreign lang="greek">qk</foreign> per Q ip&longs;i AD &aelig;quidistante in<emph.end type="italics"/> eodem <emph type="italics"/>plano<emph.end type="italics"/> <expan abbr="tri&atilde;">triam</expan> <lb/>guli ABC, <emph type="italics"/>in ip&longs;a e&longs;&longs;ent omnia centra<emph.end type="italics"/> grauitatis trian&shy;<lb/>gulorum, <emph type="italics"/>hoc est in vtram&que; partem<emph.end type="italics"/> Q<foreign lang="greek">q</foreign> Q<foreign lang="greek">k</foreign>, centraqu&egrave; <lb/>grauitatis trianguli ALM, ac centrum magnitudinis ex vtri&longs;&shy;<lb/>qu&egrave; triangulis LGN MK <foreign lang="greek">d</foreign> <expan abbr="c&omacr;po&longs;it&etail;">compo&longs;it&etail;</expan> in parte Q<foreign lang="greek">q</foreign> e&longs;&longs;e <expan abbr="deber&etilde;t">deberent</expan>.
<pb xlink:href="077/01/103.jpg" pagenum="99"/>centra ver&ograve; grauitatis magnitudinis ex GEX K<foreign lang="greek">e</foreign>F compo&shy;<lb/>&longs;it&etail;, ac magnitudinis ex. </s>
          <s id="N139CA">EBO FZC compo&longs;&longs;t&aelig;, e&longs;&longs;ent in par <lb/>te Q<foreign lang="greek">k</foreign>, ita vt punctum Q magnitudinis ex omnibus trian&shy;<lb/>gulis compo&longs;it&aelig; centrum e&longs;&longs;et grauitatis. </s>
          <s id="N139D4">qu&aelig; <expan abbr="quid&etilde;&longs;unt">quiden&longs;unt</expan> om&shy;<lb/>nino ab&longs;urda. </s>
          <s id="N139DC">Qu&ograve;d &longs;i ducta linea per Q, non fuerit etiam <lb/>ip&longs;i AD &etail;quidi&longs;tans, eadem &longs;e&que;ntur in conuenientia. <emph type="italics"/>Ma <lb/>ni&longs;estum e&longs;t igitur; quod propo&longs;itum fuerat.<emph.end type="italics"/></s>
        </p>
        <p id="N139E7" type="margin">
          <s id="N139E9"><margin.target id="marg122"/><emph type="italics"/>ex<emph.end type="italics"/> t. <emph type="italics"/>deci&shy;<lb/>mi.<emph.end type="italics"/></s>
        </p>
        <p id="N139F9" type="margin">
          <s id="N139FB"><margin.target id="marg123"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13A04" type="margin">
          <s id="N13A06"><margin.target id="marg124"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13A0F" type="margin">
          <s id="N13A11"><margin.target id="marg125"/>34. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N13A1A" type="margin">
          <s id="N13A1C"><margin.target id="marg126"/>3. <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N13A25" type="margin">
          <s id="N13A27"><margin.target id="marg127"/><emph type="italics"/>ex<emph.end type="italics"/>12.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N13A37" type="margin">
          <s id="N13A39"><margin.target id="marg128"/><emph type="italics"/>ex<emph.end type="italics"/>12.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N13A49" type="margin">
          <s id="N13A4B"><margin.target id="marg129"/><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s>
        </p>
        <p id="N13A59" type="margin">
          <s id="N13A5B"><margin.target id="marg130"/>1. <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N13A64" type="margin">
          <s id="N13A66"><margin.target id="marg131"/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13A6F" type="margin">
          <s id="N13A71"><margin.target id="marg132"/>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13A7A" type="margin">
          <s id="N13A7C"><margin.target id="marg133"/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13A85" type="margin">
          <s id="N13A87"><margin.target id="marg134"/>20. <emph type="italics"/>quinti <lb/>add.<emph.end type="italics"/></s>
        </p>
        <p id="N13A92" type="margin">
          <s id="N13A94"><margin.target id="marg135"/>8.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.103.1.jpg" xlink:href="077/01/103/1.jpg"/>
        <figure id="id.077.01.103.2.jpg" xlink:href="077/01/103/2.jpg"/>
        <p id="N13AA5" type="head">
          <s id="N13AA7">SCHOLIVM.</s>
        </p>
        <p id="N13AA9" type="main">
          <s id="N13AAB">Id ip&longs;um vult ad huc Archimedes aliter o&longs;tendere. </s>
          <s id="N13AAD">ob <expan abbr="&longs;e&quetilde;">&longs;e&que;m</expan> <lb/>tem ver&ograve; demon&longs;trationem hoc pri&ugrave;s cogno&longs;cere oportet. </s>
        </p>
        <p id="N13AB5" type="head">
          <s id="N13AB7">LEMMA.</s>
        </p>
        <p id="N13AB9" type="main">
          <s id="N13ABB">Si intra triangulum vni lateri &etail;quidi&longs;tans ducatur, ab op&shy;<lb/>po&longs;ito autem angulo intra triangulum quoqu&egrave; recta ducatur <lb/>linea, &aelig;quidi&longs;tantes lineas in eadem proportione di&longs;pe&longs;cet. </s>
        </p>
        <p id="N13AC1" type="main">
          <s id="N13AC3">Hoc in &longs;ecundo no&longs;trorum plani&longs;ph&etail;riorum libro in ea <lb/>parte o&longs;tendimus, vbi quomodo conficienda &longs;it ellip&longs;is, in&longs;tru<lb/>mento &agrave; nobis inuento demon&longs;trauimus. </s>
          <s id="N13AC9">hoc nemp&egrave; modo, <lb/>
<arrow.to.target n="fig45"/><lb/>Sit triangulum ABC, ip&longs;iqu&egrave; BC in&shy;<lb/>tra triangulum ducatur vtcumqu&egrave; &aelig;&shy;<lb/>quidi&longs;tans DE. &agrave; punctoqu&egrave; A intra <lb/>triangulum &longs;imiliter quocum&que; du&shy;<lb/>catur AF; qu&aelig; lineam BC &longs;ecet in F; <lb/>lineam ver&ograve; DE in G. Dico ita o&longs;&longs;e <lb/>CF ad FB, vt EG ad GD. <expan abbr="Quoni&atilde;">Quoniam</expan> <lb/>enim GE FC &longs;unt &aelig;quidi&longs;tantes, erit <lb/>triangulum AFC triangulo AGE &aelig;quiangulum, vt igitur
<arrow.to.target n="marg136"/><lb/>AF ad AG, ita CF ad EG. ob eandemqu&egrave; cau&iacute;am ita e&longs;t FA <lb/>ad AG, vt FB ad GD. quare vt CF ad EG, ita e&longs;t FB ad GD.
<arrow.to.target n="marg137"/><lb/>ac permutando, vt CF ad FB, ita EG ad GD. quod demon
<arrow.to.target n="marg138"/><lb/>&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/104.jpg" pagenum="100"/>
        <p id="N13AFA" type="margin">
          <s id="N13AFC"><margin.target id="marg136"/><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s>
        </p>
        <p id="N13B0A" type="margin">
          <s id="N13B0C"><margin.target id="marg137"/>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13B15" type="margin">
          <s id="N13B17"><margin.target id="marg138"/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.104.1.jpg" xlink:href="077/01/104/1.jpg"/>
        <p id="N13B24" type="head">
          <s id="N13B26"><emph type="italics"/>IDEM ALITER.<emph.end type="italics"/></s>
        </p>
        <p id="N13B2C" type="main">
          <s id="N13B2E"><emph type="italics"/>Sit triangulum ABC, ducaturqu&egrave; AD<emph.end type="italics"/> ab angulo A <emph type="italics"/>ad <expan abbr="dimidi&atilde;">dimidiam</expan><emph.end type="italics"/><lb/>ba&longs;im <emph type="italics"/>BC. Dico in linea AD centrum e&longs;&longs;e grauitatis trianguli ABC. <lb/>N on &longs;it autem, &longs;ed &longs;i fieri pote&longs;t; &longs;it H. iunganturqu&egrave; AH HB HC, &amp; <lb/>ED<emph.end type="italics"/> DF <emph type="italics"/>FE ad dimidias BA<emph.end type="italics"/> BC <emph type="italics"/>AC<emph.end type="italics"/> ducantur, &longs;ecetqu&egrave; EF ip&shy;<lb/>&longs;am AD in M. &amp; <emph type="italics"/>ip&longs;i AH &aelig;quidistantes ducantur EK FL. &amp;<emph.end type="italics"/> <lb/><arrow.to.target n="fig46"/><lb/><emph type="italics"/>iungantur KL LD Dk DH<emph.end type="italics"/>; &longs;ecetqu&egrave; DH ip&longs;am KL in N. <lb/>iungaturqu&egrave; <emph type="italics"/>MN. Quoniam igitur triangulum ABC &longs;imile est <expan abbr="tri&atilde;">triam</expan> <lb/>gulo DFC, c&ugrave;m &longs;it BA ip&longs;i FD &aelig;quidistans<emph.end type="italics"/>; &longs;iquidem &longs;unt late&shy;<lb/>
<arrow.to.target n="marg139"/> ra CA CB bifariam diui&longs;a, ideoqu&egrave; &longs;it CF ad FA, vt CD <lb/>ad DB. <emph type="italics"/>trianguliqu&egrave; ABC centrum grauitatis est punctum H; &amp;<emph.end type="italics"/> <lb/>
<arrow.to.target n="marg140"/> <emph type="italics"/>trianguli FDC centrum grauitatis erit punctum L. puncta enim HB <lb/>intra vtrumqu&egrave; triangulum &longs;unt &longs;imiliter po&longs;ita. </s>
          <s id="N13B8E">etenim ad homologa <lb/>latera angulos efficiunt &aelig;quales. </s>
          <s id="N13B92">hoc enim per&longs;picuum. </s>
          <s id="N13B94">est<emph.end type="italics"/> c&ugrave;m enim <lb/>&longs;int triangulorum ABC DFC homologa latera AC FC, <lb/>
<arrow.to.target n="marg141"/> AB FD, BC DC, &longs;intqu&egrave; AH FL &aelig;quidi&longs;tantes; erit an&shy;<lb/>gulus LFC angulo HAC &etail;qualis. </s>
          <s id="N13BA3">&longs;ed angulus CFD e&longs;t ip&longs;i
<pb xlink:href="077/01/105.jpg" pagenum="101"/>CAB &aelig;qualis; reliquus igitur angulus LFD reliquo HAB <lb/>&aelig;qualis exi&longs;tit. </s>
          <s id="N13BAB">&amp; quoniam ita e&longs;t CF ad FA, vt CL ad LH,
<arrow.to.target n="marg142"/><lb/>c&ugrave;m &longs;int FL AH &etail;quidi&longs;tantes. </s>
          <s id="N13BB2">CF ver&ograve; dimidia e&longs;t ip&longs;ius <lb/>CA, erit &amp; CL ip&longs;ius quo&que; CH dimidia. </s>
          <s id="N13BB6">at CD ip&longs;ius <lb/>CB dimidia exi&longs;tit; erit igitur DL ip&longs;i BH &etail;quidi&longs;tans. </s>
          <s id="N13BBA">ac
<arrow.to.target n="marg143"/><lb/>propterea angulus LDC e&longs;t ip&longs;i HBC &etail;qualis, &amp; LDF ip&longs;i
<arrow.to.target n="marg144"/><lb/>HBA &etail;qualis. </s>
          <s id="N13BC6">c&ugrave;m &longs;ittotus CDF toti CBA &etail;qualis; anguli <lb/>ver&ograve; ACH &amp; HCB tam &longs;unt trianguli ABC, qu&agrave;m FDC. <lb/><emph type="italics"/>Obeandem autem rationem trianguli EBD centrum grauitatis est <expan abbr="p&utilde;-">pun-</expan><emph.end type="italics"/>
<arrow.to.target n="marg145"/><lb/><emph type="italics"/>ctum K.<emph.end type="italics"/> &longs;imiliter enim o&longs;tendetur punctum K in triangu&shy;<lb/>lo EBD e&longs;&longs;e &longs;imiliter po&longs;itum, vt H in triangulo ABC. <lb/><emph type="italics"/>Quare magnitudinis ex vtri&longs;qu&egrave; triangulis EBD FDC compo&longs;it&aelig; <lb/>centrum grauitatis e&longs;t in medietate line&aelig;<emph.end type="italics"/> k<emph type="italics"/>L. cum triangula EBD<emph.end type="italics"/>
<arrow.to.target n="marg146"/><lb/><emph type="italics"/>FDC &longs;int &aelig;qualia.<emph.end type="italics"/> &longs;unt enim in &etail;qualibus ba&longs;ibus BD DC,
<arrow.to.target n="marg147"/><lb/>&amp; in ij&longs;dem parallelis EF BC, &longs;iquidem e&longs;t AE ad EB, vt
<arrow.to.target n="marg148"/><lb/>AF ad FC. quipp&egrave; c&ugrave;m latera AB AC &longs;int bifariam diui&shy;<lb/>&longs;a. <emph type="italics"/>medium ver&ograve;ip&longs;ius<emph.end type="italics"/> k<emph type="italics"/>L e&longs;t punctum N; c&ugrave;m &longs;it<emph.end type="italics"/> KE ip&longs;i AH <lb/>&etail;quidi&longs;tans, &amp; ob id &longs;it <emph type="italics"/>BE ad EA, vt B<emph.end type="italics"/>k <emph type="italics"/>ad<emph.end type="italics"/> k<emph type="italics"/>H.<emph.end type="italics"/> &amp; vt BE
<arrow.to.target n="marg149"/><lb/>ad EA, ita CF ad FA; <emph type="italics"/>vt autem CF ad FA, &longs;ic CL ad LH.<emph.end type="italics"/><lb/>quare vt BK ad KH, ita CL ad LH. <emph type="italics"/>Si autem hoc. </s>
          <s id="N13C34">&aelig;quidi-<emph.end type="italics"/>
<arrow.to.target n="marg150"/><lb/><emph type="italics"/>&longs;tans est BC ip&longs;i<emph.end type="italics"/> k<emph type="italics"/>L, &amp; iuncta est DH, erit igitur BD ad DC, vt<emph.end type="italics"/>
<arrow.to.target n="marg151"/><lb/><emph type="italics"/>KN ad NL.<emph.end type="italics"/> D ver&ograve; medium e&longs;t ip&longs;ius BC. ergo &amp; N
<arrow.to.target n="marg152"/> me&shy;<lb/>dium e&longs;t ip&longs;ius KL. <emph type="italics"/>Quare magnitudinis ex vtri&longs;qu&egrave; <expan abbr="dictor&utilde;">dictorum</expan> trian <lb/>gulorum<emph.end type="italics"/> EBD &amp; FDC <emph type="italics"/>compo&longs;it&aelig; centrum<emph.end type="italics"/> grauitatis <emph type="italics"/>est punctum<emph.end type="italics"/>
<arrow.to.target n="marg153"/><lb/><emph type="italics"/>N. parallelogrammi ver&ograve; AEDF centrum grauitatis e&longs;t punctum M,<emph.end type="italics"/><lb/>vbi &longs;imiliter diametri concurrunt, <emph type="italics"/>ac propterea magnitudinis ex<emph.end type="italics"/>
<arrow.to.target n="marg154"/><lb/><emph type="italics"/>omnibus<emph.end type="italics"/> triangulis EBD FDC vna <expan abbr="c&utilde;">cum</expan> parallelogramo AEDF <lb/><emph type="italics"/>compo&longs;it&aelig; centrum grauitatis e&longs;t in linea MN. Ver&ugrave;m<emph.end type="italics"/> <expan abbr="triangulor&utilde;">triangulorum</expan> <lb/>EBD FDC, &longs;imulqu&egrave; parallelogrammi AEDF, hoc e&longs;t totius <lb/><emph type="italics"/>trianguli ABC grauitatis centrum est punctum H; linea igitur MN pro<emph.end type="italics"/>
<arrow.to.target n="marg155"/><lb/><emph type="italics"/>ducta tran&longs;ibit per punctum H. quod e&longs;&longs;e non pote&longs;t.<emph.end type="italics"/> etenim c&ugrave;m &longs;it <lb/>KN ip&longs;i BD &aelig;quidi&longs;tans; erit BK ad KH, vt DN ad <lb/>NH: vt autem BK ad KH, ita e&longs;t BE ad EA, &amp; vt BE ad <lb/>EA, ita e&longs;t DM ad MA, c&ugrave;m &longs;it EM ip&longs;i BD &aelig;quidi&longs;tans. <lb/>erit igitur DM ad MA, vt DN ad NH. quare MN ip&longs;i AH <lb/>e&longs;t &etail;quidi&longs;tans; ideoqu&egrave; MN numquam c&ugrave;m AH conueni&shy;<lb/>re pote&longs;t. <emph type="italics"/>Non est igitur<emph.end type="italics"/> punctum <emph type="italics"/>H centrum grauitatis trianguli<emph.end type="italics"/>
<pb xlink:href="077/01/106.jpg" pagenum="102"/><emph type="italics"/>ABC. quare non e&longs;t extra lineam AD. in ip&longs;i igitur exi&longs;tit.<emph.end type="italics"/> Quod <lb/>demonitrare oportebat. </s>
        </p>
        <p id="N13CD1" type="margin">
          <s id="N13CD3"><margin.target id="marg139"/>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13CDC" type="margin">
          <s id="N13CDE"><margin.target id="marg140"/>11.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13CE7" type="margin">
          <s id="N13CE9"><margin.target id="marg141"/>29. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N13CF2" type="margin">
          <s id="N13CF4"><margin.target id="marg142"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13CFD" type="margin">
          <s id="N13CFF"><margin.target id="marg143"/>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13D08" type="margin">
          <s id="N13D0A"><margin.target id="marg144"/>29. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N13D13" type="margin">
          <s id="N13D15"><margin.target id="marg145"/>11. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13D1E" type="margin">
          <s id="N13D20"><margin.target id="marg146"/>4.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13D29" type="margin">
          <s id="N13D2B"><margin.target id="marg147"/>38. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N13D34" type="margin">
          <s id="N13D36"><margin.target id="marg148"/>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13D3F" type="margin">
          <s id="N13D41"><margin.target id="marg149"/>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13D4A" type="margin">
          <s id="N13D4C"><margin.target id="marg150"/>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N13D55" type="margin">
          <s id="N13D57"><margin.target id="marg151"/>2.<emph type="italics"/>&longs;exti. <lb/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N13D62" type="margin">
          <s id="N13D64"><margin.target id="marg152"/>*</s>
        </p>
        <p id="N13D68" type="margin">
          <s id="N13D6A"><margin.target id="marg153"/>11.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13D73" type="margin">
          <s id="N13D75"><margin.target id="marg154"/>*</s>
        </p>
        <p id="N13D79" type="margin">
          <s id="N13D7B"><margin.target id="marg155"/>A</s>
        </p>
        <figure id="id.077.01.106.1.jpg" xlink:href="077/01/106/1.jpg"/>
        <p id="N13D83" type="head">
          <s id="N13D85">SCHOLIVM.</s>
        </p>
        <p id="N13D87" type="main">
          <s id="N13D89">
<arrow.to.target n="marg156"/> Inquit Archimedes <emph type="italics"/>linea igitur MN producta tran&longs;ibit per pun&shy;<lb/>ctum H. quod e&longs;&longs;e non pote&longs;t,<emph.end type="italics"/> nemp&egrave;, vt non ip&longs;amet linea MN, <lb/>&longs;ed eius pars, &longs;iuead M, &longs;iue ad N producta cum H conue <lb/>nireoporteat. </s>
          <s id="N13D9B">c&ugrave;m tamen ip&longs;amet linea MN per punctum <lb/>H tran&longs;ire debeat. </s>
          <s id="N13D9F">ita vt punctum H &longs;it inter puncta MN; <lb/>hoc e&longs;t in linea MN, &amp; non in eius parte producta. </s>
          <s id="N13DA3">Nam &longs;i <lb/>punctum H centrum e&longs;t grauitatis totius trianguli ABC. <lb/>punctum ver&ograve; N centrum grauitatis magnitudinis ex <expan abbr="tri&atilde;gu">triangu</expan> <lb/>lis EBD FDC compo&longs;it&etail;; at&que; punctum M centrum gra&shy;<lb/>uitatis parallelogrammi AEDF; oportet vt punctum H ita li&shy;<lb/>neam diuidat MN; vt eius partes magnitudinibus permuta&shy;<lb/>tim re&longs;pondeant. </s>
          <s id="N13DB5">vt nimirum pars ad M ad partem ad N &longs;it, <lb/>vt magnitudo ex triangulis EBD FDC con&longs;tans ad parallelo <lb/>grammum AEDF. vt ex &longs;exta, &amp; octaua huius propo&longs;itione <lb/>per&longs;picuum e&longs;t. </s>
          <s id="N13DBD">Quare punctum H in linea MN e&longs;&longs;e debe&shy;<lb/>ret; vt ip&longs;emet Atchimedes paul&ograve; &longs;uperi&ugrave;s affirmauit; c&ugrave;m in&shy;<lb/>
<arrow.to.target n="marg157"/> quit. <emph type="italics"/>ac propterea magnitudinis ex omnibus compo&longs;it&aelig; contrum grauita&shy;<lb/>tis e&longs;t in linea MN.<emph.end type="italics"/> &amp; non dixit in eius parte producta. </s>
          <s id="N13DCF">Quodiv <lb/>ca vel del<gap/>dum e&longs;t verbum illud <emph type="italics"/>producta,<emph.end type="italics"/> tanquam ab aliquo <lb/>additum, vel ideo tamen hoc dixi&longs;&longs;e voluit Archimedes, vt o&shy;<lb/>&longs;tenderet lineam MN nullo modo (etiam &longs;i produceretur) <expan abbr="c&utilde;">cum</expan> <lb/>H conuenire po&longs;&longs;e. </s>
        </p>
        <p id="N13DE5" type="margin">
          <s id="N13DE7"><margin.target id="marg156"/>A</s>
        </p>
        <p id="N13DEB" type="margin">
          <s id="N13DED"><margin.target id="marg157"/>*</s>
        </p>
        <p id="N13DF1" type="head">
          <s id="N13DF3">PROPOSITIO. XIIII.</s>
        </p>
        <p id="N13DF5" type="main">
          <s id="N13DF7">Omnis trianguli centrum grauitatis e&longs;t <expan abbr="punct&utilde;">punctum</expan> <lb/>in quo rect&aelig; line&aelig; ab angulis trianguli ad dimidia <lb/>later a duct&aelig; concurrunt. </s>
        </p>
        <pb xlink:href="077/01/107.jpg" pagenum="103"/>
        <p id="N13E04" type="main">
          <s id="N13E06"><emph type="italics"/>Sit triangulum ABC, &amp;<emph.end type="italics"/> ab angulo A <emph type="italics"/>ducatur AD ad dimi&shy;<lb/>diam BC. BE ver&ograve;<emph.end type="italics"/> ab angulo B <emph type="italics"/>ad dimidiam AC.<emph.end type="italics"/> qu&etail; quidem <lb/>line&etail; AD BE &longs;einuicem &longs;ecent in <expan abbr="p&umacr;">pum</expan> <lb/>
<arrow.to.target n="fig47"/><lb/>cto H. <emph type="italics"/>Quoniam igitur centrum grauita&shy;<lb/>tis trianguli ABC est in vtra&que; linea <lb/>AD BE; hoc enim demonstratum e&longs;t<emph.end type="italics"/> in <lb/>pr&etail;cedenti. </s>
          <s id="N13E34">erit vti&que; centrum graui&shy;<lb/>tatis, vbiline&ccedil; AD BE &longs;e <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="&longs;ec&atilde;t">&longs;ecant</expan>. <lb/>&longs;ecant ver&ograve; &longs;e&longs;e in H. <emph type="italics"/>ergo punctum <lb/>H centrum e&longs;t grauitatis<emph.end type="italics"/> trianguli ABC. <lb/>quod demon&longs;trare oportebat. </s>
        </p>
        <figure id="id.077.01.107.1.jpg" xlink:href="077/01/107/1.jpg"/>
        <p id="N13E50" type="head">
          <s id="N13E52">SCHOLIVM.</s>
        </p>
        <p id="N13E54" type="main">
          <s id="N13E56">Similiter &longs;i ducta fuerit CH, &amp; producta, bifariam &longs;ecaret <lb/>AB. In hac enim linea e&longs;&longs;et centrum grauitatis trianguli; <expan abbr="c&etilde;">cem</expan> <lb/>trum ver&ograve; e&longs;t in linea ab angulo ad dimidiam ba&longs;im ducta: <lb/>ergo h&aelig;c linea ab angulo C ad dimidiam AB ducta e&longs;&longs;et. <lb/>Pr&aelig;terea &longs;i linea &agrave; puncto C ad dimidiam AB ducta <expan abbr="n&otilde;">non</expan> tran <lb/>&longs;iret per H; e&longs;&longs;et vti&que; in hac linea centrum grauitatis; &longs;ed
<arrow.to.target n="marg158"/> <expan abbr="c&etilde;-trum">cen&shy;<lb/>trum</expan> quo&que; grauitatis e&longs;t in linea AD, &amp; in linea BE, ut in <lb/>H; vnius igitur figur&etail; plura darentur centra grauitatis. </s>
          <s id="N13E76">quod <lb/>fieri non pote&longs;t. </s>
          <s id="N13E7A">quod quidem, c&ugrave;m &longs;it in con ueniens, nos in <lb/>no&longs;tro Mechanicorum libro dari non po&longs;&longs;e &longs;uppo&longs;uimus. <lb/>Quare linea CH indirectum ducta, bifariam &longs;ecaret AB. <lb/>quod quidem paul&ograve; infra aliter quo&que; o&longs;tendemus, <expan abbr="n&otilde;nul">nonnul</expan> <lb/>lis prius demon&longs;tratis; qu&aelig; Archimedes ob &longs;e&que;ntem <expan abbr="dem&otilde;-&longs;trationem">demon&shy;<lb/>&longs;trationem</expan>, tanquam demon&longs;trata &longs;upponit. </s>
          <s id="N13E8E">Vult enim Ar&shy;<lb/>chimedes, po&longs;tquam inuenit centrum grauitatis cuiu&longs;libet <lb/>trianguli, centrum quo&que; grauitatis qu&aelig;rere trapetij duo la&shy;<lb/>tera &etail;quidi&longs;tantia habentis. </s>
          <s id="N13E96">quod e&longs;t quidem pars trianguli, <lb/>&amp; tanquam fru&longs;tum a triangulo ab&longs;ci&longs;&longs;um. </s>
          <s id="N13E9A">&longs;upponitqu&egrave; den<lb/>trum grauitatis cuiu&longs;libet trianguli e&longs;&longs;e in recta linea ba&longs;i du<lb/>cta &etail;quidi&longs;tante, qu&aelig; latera ita diuidat, vt partes ad uerticem <lb/>&longs;int reliquarum partium dupl&aelig;. </s>
          <s id="N13EA2">quod quidem ortum ducit <lb/>ex cognitione alterius theorematis o&longs;tendentis centrum gra-
<pb xlink:href="077/01/108.jpg" pagenum="104"/>uitatis cuiu&longs;libet trianguli e&longs;&longs;e in recta linea ab angulo ad di&shy;<lb/>midiam ba&longs;im ducta (vt Archimedes demon&longs;trauit) &amp; in&longs;u&shy;<lb/>per in eo puncto, quod dictam lineam diuidatita, vt pars ad <lb/>angulum reliqu&etail; ad ba&longs;im &longs;it dupla. </s>
          <s id="N13EB0">Quare hoc prius ita <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>demus. </s>
        </p>
        <p id="N13EB8" type="margin">
          <s id="N13EBA"><margin.target id="marg158"/>13.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13EC3" type="head">
          <s id="N13EC5">PROPOSITIO.</s>
        </p>
        <p id="N13EC7" type="main">
          <s id="N13EC9">Omnis trianguli centrum grauitatis e&longs;t punctum in recta <lb/>linea ab angulo ad dimidiam ba&longs;im ducta exi&longs;tens, quod li&shy;<lb/>neam diuidat, ita vt poitio ad angulum reliqu&aelig; ad ba&longs;im, &longs;it <lb/>dupla. </s>
        </p>
        <p id="N13ED1" type="main">
          <s id="N13ED3">Sit triangulum ABC, in quo ab an <lb/>
<arrow.to.target n="fig48"/><lb/>gulo A ad dimidiam ba&longs;im BC re&shy;<lb/>cta ducatur linea AD. Ducaturqu&egrave; <lb/>ab angulo B ad dimidiom ba&longs;im <lb/>AC linea BE, qu&aelig;&longs;ecet AD in F. Et <lb/>quoniam centrum grauitatis <expan abbr="tri&atilde;gu-">triangu&shy;<lb/></expan>
<arrow.to.target n="marg159"/> li ABC e&longs;t punctum F; <expan abbr="o&longs;tendend&utilde;">o&longs;tendendum</expan> <lb/>e&longs;t lineam FA ip&longs;ius FD duplam e&longs;&shy;<lb/>&longs;e. </s>
          <s id="N13EF5">iungatur FC. quoniam enim AE <lb/>e&longs;t equalis ip&longs;i EC, erit triangulum <lb/>
<arrow.to.target n="marg160"/> ABE triangulo EBC &aelig;quale, c&ugrave;m <lb/>&longs;int &longs;ub eadem altitudine. </s>
          <s id="N13F01">Ob eandemqu&egrave; cau&longs;am <expan abbr="triangul&utilde;">triangulum</expan> <lb/>AFE triangulo EFC exi&longs;tit &aelig;quale. </s>
          <s id="N13F09">&longs;i igitur &agrave; triangulo ABE <lb/>auferatur triangulum AFE, &amp; &agrave; triangulo EBC triangulum <lb/>auferatur EFC; relin&que;tur triangulum ABF triangulo BFC <lb/>&aelig;quale. </s>
          <s id="N13F11">Rur&longs;us quoniam BD e&longs;t &aelig;qualis ip&longs;i DC; erit trian&shy;<lb/>
<arrow.to.target n="marg161"/> gulum BFD triangulo DFC &aelig;quale, &longs;iquidem candem ha&shy;<lb/>bentaltitudinem. </s>
          <s id="N13F1B">duplum igitur e&longs;t triangulum BFC <expan abbr="tri&atilde;gu-li">triangu&shy;<lb/>li</expan> BFD. Quare &amp; triangulum ABF trianguli BFD duplum <lb/>
<arrow.to.target n="marg162"/> exi&longs;tit. </s>
          <s id="N13F29">quia ver&ograve; triangula ABF FBD in eadem &longs;unt altitudi <lb/>ne, idcirco &longs;e&longs;e habebunt, vt ba&longs;es AF FD. at&que; triangulum <lb/>ABF. duplum e&longs;t ip&longs;ius FBD; ergo portio AF ip&longs;ius FD dupla <lb/>exi&longs;tit. </s>
          <s id="N13F31">quod demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/109.jpg" pagenum="105"/>
        <p id="N13F36" type="margin">
          <s id="N13F38"><margin.target id="marg159"/>14.<emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13F41" type="margin">
          <s id="N13F43"><margin.target id="marg160"/>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13F4C" type="margin">
          <s id="N13F4E"><margin.target id="marg161"/>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13F57" type="margin">
          <s id="N13F59"><margin.target id="marg162"/>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.109.1.jpg" xlink:href="077/01/109/1.jpg"/>
        <p id="N13F66" type="main">
          <s id="N13F68">ALITER. </s>
        </p>
        <p id="N13F6A" type="main">
          <s id="N13F6C">Sit rur&longs;us triangulum ABC, &amp; AD BE ab angulis ad di <lb/>midias ba&longs;es duct&aelig; &longs;int erit vti&que; punctum, F (vbi &longs;e in ui
<arrow.to.target n="marg163"/><lb/>cen fecant) centrum grauitatis triangulb ABC. Drco AF a&shy;<lb/>p&longs;ius FD duplam e&longs;&longs;e. </s>
          <s id="N13F77">Iungatur DE. Quoniam enim BC <lb/>
<arrow.to.target n="fig49"/><lb/>AC in punctis DE bifariam &longs;ecantur; erit <lb/>CD ad DB, vt CE ad EA. linea igitur <lb/>DE ip&longs;i AB e&longs;t &aelig;quidi&longs;tans. </s>
          <s id="N13F84">quare
<arrow.to.target n="marg164"/> trian&shy;<lb/>gulum ABC &longs;imile e&longs;t triangulo EDC.
<arrow.to.target n="marg165"/><lb/>ac propterea ita e&longs;t BC ad CD, vt AB <lb/>ad DE. e&longs;t autem. </s>
          <s id="N13F93">BC dupla ip&longs;ius CD <lb/>(&longs;iquidem punctum D bifariam diuidit <lb/>BC) erit igitur AB dupla ip&longs;ius DE. At <lb/>vero quoniam AB DE &longs;unt parallel&aelig;, erit triangulum AFB <lb/>triangulo EFD &longs;imile. </s>
          <s id="N13F9D">&amp; vt AB ad ED, ita AF ad FD, e&longs;t
<arrow.to.target n="marg166"/><lb/>autem AB ip&longs;ius ED dupla, ergo AF ip&longs;ius FD dupla <lb/>exi&longs;tit. </s>
          <s id="N13FA6">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N13FA8" type="margin">
          <s id="N13FAA"><margin.target id="marg163"/>14. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N13FB3" type="margin">
          <s id="N13FB5"><margin.target id="marg164"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13FBE" type="margin">
          <s id="N13FC0"><margin.target id="marg165"/>4. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N13FC9" type="margin">
          <s id="N13FCB"><margin.target id="marg166"/>4.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.109.2.jpg" xlink:href="077/01/109/2.jpg"/>
        <p id="N13FD8" type="main">
          <s id="N13FDA">Exijs, qu&aelig; demon&longs;trata &longs;unt, o&longs;tendemus, quod paul&ograve; an<lb/>te propoiuimus, nemp&egrave; c&ugrave;m line&aelig; AD BE bifariam &longs;ecent <lb/>BC CA. Dico lineam CF productam bifariam quo&que; &longs;e&shy;<lb/>care ip&longs;am AB. </s>
        </p>
        <p id="N13FE2" type="main">
          <s id="N13FE4">Producatur enim (ijsdem po&longs;itis) CFGH; qu&aelig; lineam <lb/>
<arrow.to.target n="fig50"/><lb/>AB &longs;ecet in G. &amp; &agrave; puncto B <lb/>ip&longs;i AD &aelig;quidi&longs;tans ducatur <lb/>BH. qu&aelig; ip&longs;i CG occuriat in <lb/>H. Quoniam igitur FD, e&longs;t i&shy;<lb/>p&longs;i BH &etail;quidi&longs;tans, erit CD <lb/>ad DB, vt CF ad FH. CD
<arrow.to.target n="marg167"/> ve&shy;<lb/>r&ograve; e&longs;t &aelig;qualis BD; ergo CF ip&longs;i <lb/>FH &aelig;qualis exi&longs;tit. </s>
          <s id="N13FFF">ac propterea <lb/>CH dupla e&longs;t ip&longs;ius (F. At ve&shy;<lb/>r&ograve; quoniam ob &longs;imilitudinem <lb/><expan abbr="triangulor&utilde;">triangulorum</expan> CBH CDF, ita e&longs;t <lb/>HC ad CF, vt BH ad DF; erit &amp; BH ip&longs;ius FD duplex.
<pb xlink:href="077/01/110.jpg" pagenum="106"/>ver&ugrave;m &amp; AF (ex proxim&egrave; demon&longs;tratis) ip&longs;ius FD duplex <lb/>exi&longs;tit. </s>
          <s id="N14012">erunt igitur BH FA inter &longs;e &etail;quales. </s>
          <s id="N14014">Quoniam autem <lb/>BH e&longs;t &etail;quidi&longs;tans ip&longs;i AF, &aelig;quiangula erunt triagula GBH <lb/>
<arrow.to.target n="marg168"/> GAF. quare vt BH ad AF, ita BG ad GA, quia ver&ograve; BH e&longs;t <lb/>ip&longs;i AF &aelig;qualis; erit &amp; BG ip&longs;i GA &aelig;qualis. </s>
          <s id="N14020">ergo recta li&shy;<lb/>nea EFG bifariam diuidit AB. quod demon&longs;trare oporte&shy;<lb/>bat. </s>
        </p>
        <p id="N14026" type="margin">
          <s id="N14028"><margin.target id="marg167"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N14031" type="margin">
          <s id="N14033"><margin.target id="marg168"/><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.110.1.jpg" xlink:href="077/01/110/1.jpg"/>
        <p id="N14045" type="main">
          <s id="N14047">Reliquum e&longs;t, vt ob &longs;e&que;ntem demon&longs;trationem alteram <lb/>propo&longs;itionem o&longs;tendamus. </s>
        </p>
        <p id="N1404B" type="head">
          <s id="N1404D">PROPOSITIO.</s>
        </p>
        <p id="N1404F" type="main">
          <s id="N14051">Centrum grauitatis cuiu&longs;libet trianguli e&longs;t in recta linea <lb/>ba&longs;i ducta &aelig;quidi&longs;tante, qu&aelig; latus ita diuidat, vt pars ad an&shy;<lb/>gulum reliqu&aelig; ad ba&longs;im &longs;it dupla. </s>
        </p>
        <p id="N14057" type="main">
          <s id="N14059">In trianagulo enim ABC ducta <lb/>&longs;it DE ba&longs;i BC &aelig;quidi&longs;tans, qu&aelig; <lb/>
<arrow.to.target n="fig51"/><lb/>latus AB diuidat in D, ita vt DA <lb/>ip&longs;ius DB &longs;it duplex. </s>
          <s id="N14066">Dico in linea <lb/>DE centrum e&longs;&longs;e grauitatis triangu<lb/>li ABC. Ducatur ab angulo A ad <lb/>dimidiam BC linea AF, qu&aelig; di&shy;<lb/>
<arrow.to.target n="marg169"/> uidat DE in G. erit AD ad DB, <lb/>vt AG ad GF, ac propterea erit <lb/>AG ip&longs;ius GF dupla. </s>
          <s id="N14078">punctum er<lb/>go G centrum e&longs;t grauitatis trian&shy;<lb/>guli ABC. Quare con&longs;tat <expan abbr="centr&utilde;">centrum</expan> <lb/>e&longs;&longs;e in linea DE. quod demon&longs;tra&shy;<lb/>re oportebat </s>
        </p>
        <pb xlink:href="077/01/111.jpg" pagenum="107"/>
        <p id="N14089" type="margin">
          <s id="N1408B"><margin.target id="marg169"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.111.1.jpg" xlink:href="077/01/111/1.jpg"/>
        <p id="N14098" type="head">
          <s id="N1409A">COROLLARIVM.</s>
        </p>
        <p id="N1409C" type="main">
          <s id="N1409E">Ex hoc elici pote&longs;t centrum grauitatis cuiu&longs;libet trianguli <lb/>e&longs;&longs;e in medio duct&aelig; line&aelig; ba&longs;i &aelig;quidi&longs;tantis, qu&etail; latus diui&shy;<lb/>datita, vt portio ad verticem &longs;it reliqu&etail; ad ba&longs;im dupla. </s>
        </p>
        <p id="N140A4" type="main">
          <s id="N140A6">E&longs;t enim DG ad GE, vt BF ad FC. &longs;unt ver&ograve; BF FC
<arrow.to.target n="marg170"/> &aelig;&shy;<lb/>quales; ergo &amp; DG GE inter &longs;e &longs;unt &aelig;quales. </s>
          <s id="N140AE">quare grauita&shy;<lb/>tis centrum G e&longs;t medium line&etail; DE. </s>
        </p>
        <p id="N140B2" type="margin">
          <s id="N140B4"><margin.target id="marg170"/><emph type="italics"/>lemm.<emph.end type="italics"/><lb/>2. <emph type="italics"/>der <lb/>&longs;tratic<emph.end type="italics"/><lb/>13.<emph type="italics"/>hi<emph.end type="italics"/></s>
        </p>
        <p id="N140CC" type="head">
          <s id="N140CE">PROPOSITIO. XV.</s>
        </p>
        <p id="N140D0" type="main">
          <s id="N140D2">Omnis trapezij duo latera inuicem habentis &aelig;&shy;<lb/>quidi&longs;tantia centrum grauitatis e&longs;t in recta linea, <lb/>qu&aelig; latera &aelig;quidi&longs;tantia bifariam &longs;ecta <expan abbr="c&otilde;iungit">coniungit</expan>; <lb/>ita diui&longs;a, vt ip&longs;ius portio terminum habens mino <lb/>rem parallelam bifariam diui&longs;am ad <expan abbr="reliqu&atilde;">reliquam</expan> por&shy;<lb/>tionem eandem habeat proportionem, quam ha <lb/>bet vtra&que; &longs;imul, qu&aelig; &longs;it &aelig;qualis dupl&aelig; maioris <lb/>parallelarum cum minore ad <expan abbr="dupl&atilde;">duplam</expan> minoris cum <lb/>maiore. </s>
        </p>
        <p id="N140F0" type="main">
          <s id="N140F2"><emph type="italics"/>Sit trapezium ABCD habens latera AD BC parallela. </s>
          <s id="N140F6">linea <lb/>ver&ograve; EF bifariam diuidat AD BC. Qu&ograve;d igitur in linea EF &longs;it cen<lb/>trum grauitatis trapezii, per&longs;picuum est. </s>
          <s id="N140FC">productis enim CDG FEG <lb/>BAG, li&que;t in idem punctum,<emph.end type="italics"/> put&agrave; G <emph type="italics"/>concurrere.<emph.end type="italics"/> propterea qu&ograve;d <lb/>c&ugrave;m &longs;it AD &aelig;quidi&longs;tans ip&longs;i BC, nece&longs;&longs;e e&longs;t proportionem
<arrow.to.target n="marg171"/><lb/>ip&longs;ius BA ad AG, ip&longs;iusqu&egrave; FE ad EG, &amp; CD ad DG, qu&aelig; <expan abbr="ni-mir&utilde;">ni&shy;<lb/>mirum</expan> in omnibus <expan abbr="ead&etilde;">eadem</expan> e&longs;t, in <expan abbr="vn&utilde;">vnum</expan> &amp; <expan abbr="id&etilde;">idem</expan> <expan abbr="p&utilde;ct&utilde;">punctum</expan> terminare. <emph type="italics"/><expan abbr="erit&qacute;">erit&que;</expan>; <lb/>trianguli GBC centrum grauitatis in linea GF. &longs;imiliter&que; trianguli<emph.end type="italics"/>
<arrow.to.target n="marg172"/>
<pb xlink:href="077/01/112.jpg" pagenum="108"/>
<arrow.to.target n="marg173"/> <emph type="italics"/>AG D centrum grauitatis in linea EG. ergo reliqui trapezii ABC <lb/>centrum grauitatis erit in linea EF. iungatur ita&que; BD, qu&aelig; int <lb/>&aelig;qua in punctis<emph.end type="italics"/> K<emph type="italics"/>H diuidatur. </s>
          <s id="N1414A">ac per ea <expan abbr="duc&atilde;tur">ducantur</expan> LHM N<emph.end type="italics"/>k<emph type="italics"/>T<gap/><lb/>BC &aelig;quidi&longs;tantes<emph.end type="italics"/>; qu&aelig; lineam EF in punctis RS di&longs;pe&longs;cant <lb/><emph type="italics"/>lungantur&que; DF BE,<emph.end type="italics"/> &longs;ecetqu&egrave; DF lineam LM in X. ip <lb/>ver&ograve; EB &longs;ecet NT in O. Iungaturqu&egrave; <emph type="italics"/>OX<emph.end type="italics"/>, qu&aelig; lineam EF <lb/>
<arrow.to.target n="fig52"/><lb/>
<arrow.to.target n="marg174"/> P &longs;ecet. <emph type="italics"/>erit ita&que; trianguli DBC centrum grauitatis in linea H <lb/>c&ugrave;m &longs;it HB tertia pars ip&longs;ius B D<emph.end type="italics"/>; &longs;itqu&egrave; propterea DH ip&longs;i <lb/>HB dupla. <emph type="italics"/>&amp; per punctum H ducta &longs;it ba&longs;i<emph.end type="italics"/> BC <emph type="italics"/>&aelig;quidi&longs;tans M<emph.end type="italics"/><lb/>
<arrow.to.target n="marg175"/> <emph type="italics"/>e&longs;t autem centrum quo&que; grauitatis trianguli DBC in linea DF<emph.end type="italics"/>; q <lb/>e&longs;t ab angulo D ad dimidiam BC ducta. <emph type="italics"/>Quare dicti triang <lb/>centrum grauitatis est punctum X. Eademqu&egrave; ratione<emph.end type="italics"/> c&ugrave;m &longs;it D <lb/>tertia pars ip&longs;ius DB, ac proptcrea &longs;it BK ip&longs;ius KD dup <lb/>&longs;itqu&egrave; KN &aelig;quidi&longs;tans ip&longs;i AD; erit centrum grauitatis tri <lb/>guli ABD in linea KN; idem ver&ograve; centrum reperitur quo <lb/>in linea BE, c&ugrave;m &longs;it ab angulo B ad dimidiam AD duc <lb/>ergo <emph type="italics"/>punctum O<emph.end type="italics"/>, vbi &longs;e inuicem &longs;ecant, <emph type="italics"/>centrum e&longs;t grauitatist <lb/>guli ABD. magnitudinis igitur ex vtri&longs;&que; triangulis ABD BI <lb/>compo&longs;it&aelig;, qu&aelig; e&longs;t trapezium<emph.end type="italics"/> ABCD, <emph type="italics"/>centrum grauitatis est in rect<emph.end type="italics"/>
<pb xlink:href="077/01/113.jpg" pagenum="109"/><emph type="italics"/>nea OX. dicti autem trapezii centrum gauitatis est etiam in li&shy;<lb/>nea EF, quare trapezii ABCD centrum grauitatis est punctum <lb/>P. At ver&ograve; triangulum BCD ad ABD proportionem habet eam, quam
<arrow.to.target n="marg176"/><lb/>OP ad P<emph.end type="italics"/>X. c&ugrave;m &longs;int puncta OX triangulorum centla graui<lb/>tatis, ac punctum P vtrorum&que; commune centrum. <emph type="italics"/>Sed vt <lb/>triangulum BDC adtriangulum ABD, ita e&longs;t<emph.end type="italics"/> quo&que; ba&longs;is <emph type="italics"/>BC<emph.end type="italics"/>
<arrow.to.target n="marg177"/><lb/><emph type="italics"/>ad<emph.end type="italics"/> ba&longs;im <emph type="italics"/>AD.<emph.end type="italics"/> c&ugrave;m triangula eandem habeant altitudinem, <lb/>&longs;iquidem &longs;unt in ijsdem parallelis AD BC. quare vt BC ad <lb/>AD, ita OP ad PX. <emph type="italics"/>Sed<emph.end type="italics"/> quoniam anguli RPO SPX ad
<arrow.to.target n="marg178"/> ver&shy;<lb/>ticem &longs;unt &etail;quales, &amp; angulus PRO ip&longs;i PSX, veluti angulus
<arrow.to.target n="marg179"/><lb/>ROP angulo PXS e&longs;t &etail;qualis, erit triangulum OPR triangu<lb/>lo XPS &longs;imile; quare <emph type="italics"/>vt OP ad PX, &longs;ic PR ad PS.<emph.end type="italics"/> e&longs;t autem
<arrow.to.target n="marg180"/><lb/>BC ad AD, vt OP ad PX<emph type="italics"/>; vt igitur BC ad AD, ita RP ad PS.<emph.end type="italics"/>
<arrow.to.target n="marg181"/><lb/>&amp; antecedentium dupla, du&aelig; &longs;cilicet BC ad AD, vt du&aelig; PR <lb/>ad PS. &amp; componendo du&aelig; BC cum AD ad AD; vt du&aelig;
<arrow.to.target n="marg182"/><lb/>PR cum PS ad PS. &amp; ad con&longs;e&que;ntium dupla, vt &longs;cilicet <lb/>du&aelig; BC cum AD ad duas AD, ita du&aelig; PR cum PS ad duas <lb/>PS. dictum e&longs;t autem BC ad AD ita e&longs;&longs;e, vt PR ad PS. quare <lb/>conuerrendo AD ad BC erit, vt PS ad PR. &amp; antecedentium
<arrow.to.target n="marg183"/><lb/>dupla. </s>
          <s id="N14232">hoc e&longs;t du&aelig; AD ad BC, vt du&aelig; PS ad PR. Ita&que; in <lb/>eadem &longs;unt proportione du&ccedil; BC cum AD ad duas AD, vt <lb/>du&etail; PR <expan abbr="c&utilde;">cum</expan> PS ad duas PS. &longs;icut ver&ograve; du&etail; AD ad BC, ita du&etail; <lb/>PS ad PR. antecedentes igitur ad &longs;uas &longs;imul con&longs;e&que;ntes in
<arrow.to.target n="marg184"/><lb/>eadem erunt proportione. <emph type="italics"/>Quare &longs;icut du&aelig; BC cum AD ad duas <lb/>AD cum BC, ita du&aelig; RP cum PS ad duas P S cum PR, <lb/>ver&ugrave;m du&aelig; quidem RP cum PS e&longs;t vtra&que; &longs;imul SR RP.<emph.end type="italics"/> bis <lb/>enim a&longs;&longs;umitur PR, &longs;emel ver&ograve; PS. Cum autem line&aelig; DH ES <lb/>&agrave; lineis diuidantur &etail;quidi&longs;tantibus ED OT HM, erit DK ad
<arrow.to.target n="marg185"/><lb/>KH, vt ER ad CS; kD ver&ograve; e&longs;t &aelig;qualis KH, erit ER ip&longs;i <lb/>RS &etail;qualis. </s>
          <s id="N14258">erit igitur ER cum RP, <emph type="italics"/>hoc est PE<emph.end type="italics"/> ip&longs;is SR RP <lb/>&etail;qualis. <emph type="italics"/>du&aelig; ver&ograve; PS cum PR e&longs;t vtra&que; PS SR.<emph.end type="italics"/> bis enim a&longs;&shy;<lb/>&longs;umitur PS, &longs;emel qu&egrave; PR. &amp; quoniam FS e&longs;t &etail;qualis ip&longs;i SR. <lb/>quod quidem eodem modo o&longs;tendetur, c&ugrave;m &longs;it FS ad SR, vt <lb/>BH ad Hk. </s>
          <s id="N1426E">erit FS cum SP, <emph type="italics"/>hoc est PF<emph.end type="italics"/> ip&longs;is PS SR &aelig;qualis. <lb/>Quare ita &longs;ehabet PE ad PF, vt du&aelig; BC cum AD ad duas <lb/>AD cum BC. Centrum igitur grauitatis P trapezij ABCD <lb/>in linea e&longs;t EF, qu&aelig; <expan abbr="c&otilde;iungit">coniungit</expan> parallelas AD BC bifariam di
<pb xlink:href="077/01/114.jpg" pagenum="110"/>ui&longs;as; ita vt pars PE, qu&aelig; e&longs;t ad minorem parallelam AD <lb/>reliquampartem PF eam habet proportionem, quam du <lb/>ip&longs;ius BC, qu&aelig; e&longs;t maior &aelig;quidi&longs;tantium, vna cum min <lb/>AD, ad duplam minoris AD cum maiore BC, <emph type="italics"/>ergo demons<gap/><lb/>ta &longs;unt, qu&aelig; propo&longs;ita fuerant.<emph.end type="italics"/></s>
        </p>
        <p id="N14292" type="margin">
          <s id="N14294"><margin.target id="marg171"/><emph type="italics"/>ex<emph.end type="italics"/> 2.<emph type="italics"/>&longs;<gap/><emph.end type="italics"/></s>
        </p>
        <p id="N142A3" type="margin">
          <s id="N142A5"><margin.target id="marg172"/>13.<emph type="italics"/>hu<gap/><emph.end type="italics"/></s>
        </p>
        <p id="N142AF" type="margin">
          <s id="N142B1"><margin.target id="marg173"/>8. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N142BA" type="margin">
          <s id="N142BC"><margin.target id="marg174"/><emph type="italics"/>ex proxi&shy;<lb/>me demon <lb/>&longs;tratis.<emph.end type="italics"/></s>
        </p>
        <p id="N142C8" type="margin">
          <s id="N142CA"><margin.target id="marg175"/>* <lb/>13. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N142D5" type="margin">
          <s id="N142D7"><margin.target id="marg176"/>6. <emph type="italics"/>hu<emph.end type="italics"/></s>
        </p>
        <p id="N142E0" type="margin">
          <s id="N142E2"><margin.target id="marg177"/>1. <emph type="italics"/>&longs;e.<emph.end type="italics"/></s>
        </p>
        <p id="N142EB" type="margin">
          <s id="N142ED"><margin.target id="marg178"/>15. <emph type="italics"/>p<emph.end type="italics"/></s>
        </p>
        <p id="N142F6" type="margin">
          <s id="N142F8"><margin.target id="marg179"/>29. <emph type="italics"/>p<emph.end type="italics"/></s>
        </p>
        <p id="N14301" type="margin">
          <s id="N14303"><margin.target id="marg180"/><emph type="italics"/>ex<emph.end type="italics"/> 4.</s>
        </p>
        <p id="N1430C" type="margin">
          <s id="N1430E"><margin.target id="marg181"/>11. <emph type="italics"/><expan abbr="q.">&que;</expan><emph.end type="italics"/></s>
        </p>
        <p id="N14319" type="margin">
          <s id="N1431B"><margin.target id="marg182"/>18. <gap/></s>
        </p>
        <p id="N14320" type="margin">
          <s id="N14322"><margin.target id="marg183"/><emph type="italics"/>corol <lb/>quint<emph.end type="italics"/></s>
        </p>
        <p id="N1432C" type="margin">
          <s id="N1432E"><margin.target id="marg184"/><emph type="italics"/>cor.<emph.end type="italics"/>2 <lb/><emph type="italics"/>ma a<gap/><lb/>huius<emph.end type="italics"/></s>
        </p>
        <p id="N14340" type="margin">
          <s id="N14342"><margin.target id="marg185"/>1. <emph type="italics"/>l. <lb/>in<emph.end type="italics"/> 13</s>
        </p>
        <figure id="id.077.01.114.1.jpg" xlink:href="077/01/114/1.jpg"/>
        <p id="N14352" type="head">
          <s id="N14354">SCHOLIVM.</s>
        </p>
        <p id="N14356" type="main">
          <s id="N14358">
<arrow.to.target n="marg186"/> Gr&aelig;cus codex po&longs;t ea verba, <emph type="italics"/>c&ugrave;m &longs;it HB tertia pars ip&longs;ius<emph.end type="italics"/> Z <lb/>habet <foreign lang="greek">kai dia tou_ q sam<gap/>iou w_aza)ll<gap/>los ta) ba\sei o)ux ta_s a( mq</foreign>, qu<gap/><lb/>quidem verba illa <foreign lang="greek">ou)k ta\s</foreign> perperam leguntur; quorum l<gap/><lb/>ponerem <foreign lang="greek">a<gap/>omi\na e)si\</foreign>, ita vt &longs;int hoc modo re&longs;tituenda, <foreign lang="greek">ka<gap/> dia <lb/><gap/> same_iou w_aza/ll<gap/>lws ta_ ba\sei a<gap/>ome\na isi\ a( mq. </foreign></s>
        </p>
        <p id="N1438A" type="margin">
          <s id="N1438C"><margin.target id="marg186"/>*</s>
        </p>
        <p id="N14390" type="main">
          <s id="N14392">H&aelig;c &longs;unt, qu&aelig; de centro grauitatis figurarum rectiline <lb/>Archimedes &longs;cripta reliquit. </s>
          <s id="N14396">Ex quibus maxima cert&egrave; vtil <lb/>habetur; ne&que; ampli&ugrave;s de rectilineis figuris Archimedes p <lb/>tractare voluit. </s>
          <s id="N1439C">ex dictis enim alia omnia dependent. </s>
          <s id="N1439E">Nan <lb/>tra grauitatis rectilinearum figurarum, qu&aelig; &aelig;quales angu<lb/>latera&que; &aelig;qualia habent, ex his in uenire poterimus. </s>
          <s id="N143A4">qu&aelig; <lb/>dem figur&etail; in circulo in&longs;cribi po&longs;&longs;unt. </s>
          <s id="N143A8">Quod &longs;an&egrave; Federi <lb/>Comandinus in eius libro de centro grauitatis &longs;olidorum <lb/>prioribus propo&longs;itionibus pr&aelig;&longs;titit. </s>
          <s id="N143AE">aliaqu&egrave; nonnulla, vt<gap/><lb/>tragrauitatis rectilinearum figurarum in ellip&longs;i, deind&egrave; ip<gap/><lb/>circuli, &amp; ellip&longs;is centra grauitatis in uenit. </s>
          <s id="N143B6">omne&longs;qu&egrave; dem<lb/>&longs;trationes in ijs, qu&aelig; in hoc libro iam demon&longs;trata &longs;unt, <lb/>dauit. </s>
          <s id="N143BC">pr&aelig;terea ex his etiam idem Commandinus in com <lb/>tarijs libri Archimedis de quadratura paraboles, (quo ad p <lb/>xim) grauitatis centrum cuiu&longs;libet figur&etail; rectiline&aelig; ad in<lb/>nit. </s>
          <s id="N143C4">Quod quidem nos quo&que;, vt initio polliciti fuimus, <lb/>nullis mutatis idem o&longs;tendemus. </s>
          <s id="N143C8">hoc prius &longs;uppo&longs;ito. </s>
        </p>
        <p id="N143CA" type="main">
          <s id="N143CC">Triangula in eadem ba&longs;i con&longs;tituta eam inter &longs;e propo<gap/><lb/>nem habent, quam eorum altitudines. </s>
        </p>
        <p id="N143D1" type="main">
          <s id="N143D3">Hoc autem demon&longs;tratum e&longs;t ab excellenti&longs;simis viris, <lb/>ri&longs;qu&egrave; Euclidis interpretibus, Federico <expan abbr="C&otilde;mandino">Commandino</expan>, &amp; Cl <lb/>&longs;tophoro Clauio; qui hanc propo&longs;itionem po&longs;t primam <lb/>ti libri Euclidis demon&longs;trarunt. </s>
        </p>
        <pb xlink:href="077/01/115.jpg" pagenum="111"/>
        <p id="N143E2" type="head">
          <s id="N143E4">PROBLEMA.</s>
        </p>
        <p id="N143E6" type="main">
          <s id="N143E8">Cuiu&longs;libet rectiline&etail; figur&etail; centrum grauitatis inuenire. </s>
        </p>
        <p id="N143EA" type="main">
          <s id="N143EC">Triangulorum centrum grauitatis iam ab Archimede de&shy;<lb/>mon&longs;tratum e&longs;t. </s>
        </p>
        <p id="N143F0" type="main">
          <s id="N143F2">Sit ita&que; prim&ugrave;m quadri <lb/>
<arrow.to.target n="fig53"/><lb/>laterum ABCD, cuius opor&shy;<lb/>teat centrum grauitatis inue <lb/>nire. </s>
          <s id="N143FF">Ducatur AC, qu&aelig; qua <lb/>drilaterum in duo triangula <lb/>ABC ACD diuidet. </s>
          <s id="N14405">&agrave; <expan abbr="p&utilde;cti&longs;-qu&egrave;">puncti&longs;&shy;<lb/>qu&egrave;</expan> BD ad AC perpendicu<lb/>lares ducantur BE DF. In&shy;<lb/>ueniantur deinde ex dictis <expan abbr="c&etilde;">cem</expan> <lb/>tra grauitatis triangulorum <lb/>ABC ACD. &longs;intqu&egrave; puncta <lb/>GH. iungaturqu&egrave; GH, qu&aelig; diuidatur in K, ita vt GK <lb/>ad KH &longs;it, vt DF ad BE. Dico punctum K centrum <lb/>e&longs;&longs;e grauitatis quadrilateri ABCD. Quoniam enim triangu&shy;<lb/>la ABC ACD in eadem &longs;unt ba&longs;i AC, erunt inter &longs;e&longs;e, vt al&shy;<lb/>titudines. </s>
          <s id="N14423">quare triangulum ACD ita &longs;e habet ad <expan abbr="triangul&utilde;">triangulum</expan> <lb/>ABC, vt DF ad BE. hoc e&longs;t GK ad KH. <expan abbr="punct&utilde;">punctum</expan> ergo K <expan abbr="c&etilde;">cem</expan> <lb/>trum e&longs;t grauitatis magnitudinisex vtril qu&egrave; triangulis ABC
<arrow.to.target n="marg187"/><lb/>ACD compo&longs;it&aelig;; hoc e&longs;t quadrilateri ABCD. </s>
        </p>
        <p id="N1443A" type="margin">
          <s id="N1443C"><margin.target id="marg187"/><emph type="italics"/>ex 6.h<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.115.1.jpg" xlink:href="077/01/115/1.jpg"/>
        <p id="N14448" type="main">
          <s id="N1444A">Sit autem pentagonum <lb/>
<arrow.to.target n="fig54"/><lb/>ABCDE. <expan abbr="iung&atilde;turqu&egrave;">iunganturqu&egrave;</expan> AC <lb/>AD. inueniaturqu&egrave; <expan abbr="tri&atilde;gu">triangu</expan> <lb/>li ABC centrum grauitatis <lb/>H. quadrilateri ver&ograve; ACDE <lb/>ex proxim&egrave; <expan abbr="dem&otilde;">demom</expan> &longs;tratis <expan abbr="c&etilde;-trum">cen&shy;<lb/>trum</expan> grauitatis inueniatur <lb/>Iam vti&que; con&longs;tat (du&shy;<lb/>cta HK) centrum grauita <lb/>tis totius ABCDE in linea
<pb xlink:href="077/01/116.jpg" pagenum="112"/>HK exi&longs;tere. </s>
          <s id="N14477">Rurilus trianguli ADE centrum inueniatur F <lb/>quadrilateri ver&ograve; ADCB punctum G. iungaturqu&egrave; GF. e<gap/><lb/>eodem modo centrum grauitatis totius ABCDE in linea F<gap/><lb/>&longs;ed e&longs;t quo&que; in linea HK, ergo vbr&longs;e inuicem &longs;ecant, vt <lb/>L, centrum erit grauitatis pentagoni ABCDE. </s>
        </p>
        <figure id="id.077.01.116.1.jpg" xlink:href="077/01/116/1.jpg"/>
        <p id="N14487" type="main">
          <s id="N14489">In hexagonis &longs;imiliter. <lb/>
<arrow.to.target n="fig55"/><lb/>vt ABCDEF iungantur <lb/>AC AE, deinceps inuenia <lb/>tur trianguli ABC <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis G, pentagoni <lb/>ver&ograve; ACDEF ex dictis cen<lb/>trum &longs;it H. ductaqu&egrave; GH <lb/>centrum grauitatis totius <lb/>ABCDEF erit in linea GH <lb/>&longs;imiliter centrum grauita&shy;<lb/>tis trianguli AFE &longs;it K, <expan abbr="p&etilde;">pem</expan> <lb/>tagoni ver&ograve; AEDCB &longs;it L, iunctaqu&egrave; KL, erit centrum gr <lb/>uitatis totius hexagoni in linea KL. ver&ugrave;m e&longs;t etiam in lin <lb/>GH. ergo errt in M. in quo GH <emph type="italics"/>K<emph.end type="italics"/>L &longs;e inuicem &longs;ecant. </s>
        </p>
        <figure id="id.077.01.116.2.jpg" xlink:href="077/01/116/2.jpg"/>
        <p id="N144BC" type="main">
          <s id="N144BE">Nequ&egrave; aliter in heptago <lb/>
<arrow.to.target n="fig56"/><lb/>no ABCDEFG, in quo du<lb/>cantur BG CE. trianguli <lb/>ver&ograve; ABG centrum graui&shy;<lb/>tatis &longs;it H. hexagoni <expan abbr="aut&etilde;">autem</expan> <lb/>GBCDEF, &longs;it K. deinde <lb/>trianguli CDE <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis &longs;it L, hexagoni ve&shy;<lb/>r&ograve; CEFGAB &longs;it M. iun&shy;<lb/>cti&longs;qu&egrave; HK ML, eadem ra <lb/>tione centrum grauitatis <lb/>
<arrow.to.target n="marg188"/> totius heptagoni erit in vtraqu&egrave; linea Hk LM. ergo erit in </s>
        </p>
        <p id="N144E7" type="margin">
          <s id="N144E9"><margin.target id="marg188"/>*</s>
        </p>
        <figure id="id.077.01.116.3.jpg" xlink:href="077/01/116/3.jpg"/>
        <p id="N144F1" type="main">
          <s id="N144F3">Eodemqu&egrave; pror&longs;us modo in octagono, &amp; in alijs demc<gap/><lb/>figuris centrum graui tatis inuenietur. </s>
          <s id="N144F8">qu&aelig; quidem facere <lb/>portebat. </s>
        </p>
        <pb xlink:href="077/01/117.jpg" pagenum="113"/>
        <p id="N144FF" type="main">
          <s id="N14501">Curautem hoc modo centra grauitatum in pr&aelig;fatis figu&shy;<lb/>ris po&longs;itione tant&ugrave;m, &amp; non determinat&egrave; ea indeterminata, <lb/>linea, &amp; in tali &longs;itu exi&longs;tere inuenerimus, vt in parallelogram <lb/>mis &amp; in triangulis factum fuitab Archimede; explicabitur in <lb/>&longs;ecundo libro po&longs;t tertiam proportionem; vbi o&longs;tendemus, <lb/>in quibus figuris determinat&egrave; inueniri pote&longs;t centrum graui&shy;<lb/>tatis. </s>
        </p>
        <p id="N1450F" type="main">
          <s id="N14511">Antequam autem finem primolibro imponamus, <expan abbr="reliqu&utilde;">reliquum</expan> <lb/>e&longs;t; vt ea qu&aelig; in pr&aelig;fatione &longs;uppo&longs;uimus, o&longs;tendamus. </s>
          <s id="N14519">pri&shy;<lb/>m&ugrave;m qu&egrave; quando &longs;ecund&ugrave;m rectam lineam aliqua diuiditur <lb/>figura per centrum grauitatis, aliquando diuidi in partes &longs;em<lb/>per &etail;quales, &amp; aliquando in partes in&aelig;quales. </s>
        </p>
        <p id="N14521" type="head">
          <s id="N14523">PROPOSITIO.</s>
        </p>
        <p id="N14525" type="main">
          <s id="N14527">Figura dari pote&longs;t, qu&etail; per centrum grauitatis recta li&shy;<lb/>nea diui&longs;a, &longs;emper in partes diuidatur &aelig;quales. </s>
        </p>
        <p id="N1452B" type="main">
          <s id="N1452D">Sit <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>
<arrow.to.target n="fig57"/><lb/>ABCD, cuius <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis E. Ducaturqu&egrave; per <lb/>E <expan abbr="vtcun&qacute;">vtcun&que;</expan>; linea GEF, qu&etail; <lb/>vel diameter e&longs;t, vel min^&lcub;9}. <lb/>&longs;i e&longs;t diameter, iam <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> <lb/><expan abbr="parallelogr&atilde;mum">parallelogrammum</expan> in duo <lb/>&etail;qua e&longs;&longs;e diui&longs;um. </s>
          <s id="N14555">Si ver&ograve; non e&longs;t diameter, <expan abbr="duc&atilde;tur">ducantur</expan> diametri
<arrow.to.target n="marg189"/><lb/>AC BD, qu&aelig; per E tran&longs;ibunt. </s>
          <s id="N14560">Quoniam igitur AF e&longs;t &aelig;qui&shy;<lb/>diftans ip&longs;i CG, eritangulus EAF ip&longs;i ECG, &amp; EFA ip&longs;i EGC
<arrow.to.target n="marg190"/><lb/>&aelig;qualis, e&longs;t autem AEF ip&longs;i GEC ad verticem &aelig;qualis, <expan abbr="latus&qacute;">latus&que;</expan>;
<arrow.to.target n="marg191"/><lb/>AE ip&longs;i EC &aelig;quale; erit triangulum AEF triangulo CEG &etail;qua <lb/>le. </s>
          <s id="N14574">eodemqu&egrave; modo o&longs;tendetur triangulum FEB triangulo <lb/>EGD. &amp; triangulum AED ip&longs;i BEC &aelig;quale. </s>
          <s id="N14578">Ex quibus patet. <lb/>figuram ex tribus triangulis compo&longs;itam, hoc e&longs;t figuram <lb/>FGDA ip&longs;i FGCB &aelig;qualem e&longs;&longs;e. </s>
          <s id="N1457E">diuiditurergo <expan abbr="parallelogr&atilde;-mum">parallelogran&shy;<lb/>mum</expan> &agrave; linea per centrum grauitatis ducta in partes &longs;em per&ccedil;&shy;<lb/>quales. </s>
          <s id="N14588">quod demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/118.jpg" pagenum="114"/>
        <p id="N1458D" type="margin">
          <s id="N1458F"><margin.target id="marg189"/>34.<emph type="italics"/>primi<emph.end type="italics"/></s>
        </p>
        <p id="N14598" type="margin">
          <s id="N1459A"><margin.target id="marg190"/>29. <emph type="italics"/>primi<emph.end type="italics"/></s>
        </p>
        <p id="N145A3" type="margin">
          <s id="N145A5"><margin.target id="marg191"/>15. <emph type="italics"/>primi<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.118.1.jpg" xlink:href="077/01/118/1.jpg"/>
        <p id="N145B2" type="main">
          <s id="N145B4">Hoc idem multis alijs figuris accidet, vt pentagonis, he <lb/>gonis&aelig;quiangulis, &amp; &aelig;quilateris, &amp; alijs. </s>
        </p>
        <p id="N145B8" type="head">
          <s id="N145BA">PROPOSITIO.</s>
        </p>
        <p id="N145BC" type="main">
          <s id="N145BE">Figura dari pote&longs;t, qu&aelig; per centrum grauitatis recta li <lb/>diui&longs;a, non &longs;emper in partes diuidatur &etail;quales. </s>
        </p>
        <p id="N145C2" type="main">
          <s id="N145C4">Habeat triangulum ABC <lb/>
<arrow.to.target n="fig58"/><lb/>latera AB AC &aelig;qualia. </s>
          <s id="N145CD">trian <lb/>guliver&ograve; centrum grauitatis &longs;it <lb/>D. &agrave; quo ip&longs;i BC &etail;quidi&longs;tans <lb/>Ducatur FDG. Dico partem <lb/>AFG <expan abbr="minor&etilde;">minorem</expan> e&longs;&longs;e parte BFGC. <lb/>ducatur ADE, qu&aelig; bifariam <lb/>
<arrow.to.target n="marg192"/> BC diuidet. </s>
          <s id="N145E3">&amp; &agrave; puncto G <lb/>ip&longs;i AE &etail;quidi&longs;tans ducatur <lb/>HGK. compleantur&que; figur&etail; <lb/>EH KF. Quoniam enim FG <lb/>
<arrow.to.target n="marg193"/> &etail;quidi&longs;tans e&longs;t ip&longs;i BC, erit FD ad DG, vt BE ad E<gap/><lb/>&amp; e&longs;t BE ip&longs;i EC &aelig;qualis. </s>
          <s id="N145F4">erit igitur FD ip&longs;i DG &etail;qua <lb/>vt etiam paul&ograve; ante 15. huius o&longs;tendimus. </s>
          <s id="N145F8">quare FG ip <lb/>DG dupla. </s>
          <s id="N145FC">e&longs;t. </s>
          <s id="N145FE">ac propterea <expan abbr="parallelogr&atilde;mum">parallelogrammum</expan> FK dupi <lb/>e&longs;t parallelogrammi DK. quia ver&ograve; AD ip&longs;ius DE du <lb/>exi&longs;tit, erit quoqu&egrave; parallelogrammum DH ip&longs;ius DK <lb/>plum. </s>
          <s id="N1460A">Quare DH ip&longs;i FK e&longs;t &aelig;quale. </s>
          <s id="N1460C">At ver&ograve; quoni <lb/>
<arrow.to.target n="marg194"/> FG dupla e&longs;t ip&longs;ius DG. erit triangulum AFG parallelog <lb/>mo DH &aelig;quale. </s>
          <s id="N14616">triangulum igitur AFG parallelog<gap/><lb/>FK e&longs;t &aelig;quale. </s>
          <s id="N1461B">Quare pars AFG parte BFGC minor <gap/><lb/>&longs;tit. </s>
          <s id="N14620">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N14622" type="margin">
          <s id="N14624"><margin.target id="marg192"/><emph type="italics"/>ex<emph.end type="italics"/> 13. <emph type="italics"/>hui&apos;<emph.end type="italics"/></s>
        </p>
        <p id="N14632" type="margin">
          <s id="N14634"><margin.target id="marg193"/><emph type="italics"/>lemma an&shy;<lb/>te <expan abbr="&longs;ecund&atilde;">&longs;ecundam</expan> <lb/><expan abbr="demon&longs;tra-tion&etilde;">demon&longs;tra&shy;<lb/>tionem</expan><emph.end type="italics"/> 13 <emph type="italics"/>bu <lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N14650" type="margin">
          <s id="N14652"><margin.target id="marg194"/><emph type="italics"/>ex<emph.end type="italics"/> 41.<emph type="italics"/>pri. <lb/>mi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.118.2.jpg" xlink:href="077/01/118/2.jpg"/>
        <p id="N14666" type="main">
          <s id="N14668">Hinc per&longs;picuum e&longs;t, eandem figuram per centrum gra<lb/>tatis diui&longs;am, aliquando in partes in &aelig;quales, aliquando in <lb/>tes &aelig;quales diuidi po&longs;&longs;e. </s>
          <s id="N1466E">in partes in&etail;quales iam o&longs;ten&longs;um <lb/>hocaccidere <expan abbr="perline&atilde;">perlineam</expan> FG. in partes ver&ograve; &aelig;quales patet pe <lb/>neam ADE, qu&aelig; triangulum ABC in duo &etail;qua diuidi<gap/>. t<gap/><lb/>
<arrow.to.target n="marg195"/> gulum enim ABE triangulo: AEC e&longs;t &etail;quale, c&ugrave;m &longs;int<gap/><lb/>eadem altitudine, ba&longs;e&longs;qu&egrave; BE EC inter &longs;e &longs;int &aelig;quales. </s>
        </p>
        <pb xlink:href="077/01/119.jpg" pagenum="115"/>
        <p id="N14687" type="margin">
          <s id="N14689"><margin.target id="marg195"/>1. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N14692" type="main">
          <s id="N14694">Adhuc (veluti initio quo&que; diximus) &longs;i fuerit prisma, vt <lb/>AB, cuius altera ba&longs;is &longs;it AC. tale ver&ograve; &longs;it prisma, vt pl mum <lb/>AC planis CH CK &amp;c. </s>
          <s id="N1469A">&longs;it erectum. </s>
          <s id="N1469C">&longs;it autem ip&longs;ius ba&longs;is <lb/>AC centrum grauitatis E. Dico &longs;i prima &longs;u&longs;pendatur ex pu&shy;<lb/>
<arrow.to.target n="fig59"/><lb/>cto E, ba&longs;im AC horizonti &aelig;quidi&longs;tantem permanere. </s>
          <s id="N146A7">vt co <lb/>gno&longs;camusea, qu&aelig; his libris pertractantur, ad praxim po&longs;&longs;e <lb/>reduci. </s>
          <s id="N146AD">&amp; ne aliquid ab&longs;&que; demon&longs;tratione confirmatum re <lb/>linquamus. </s>
          <s id="N146B1">hoc quo&que; o&longs;tendemus. </s>
          <s id="N146B3">hoc pacto. </s>
        </p>
        <figure id="id.077.01.119.1.jpg" xlink:href="077/01/119/1.jpg"/>
        <p id="N146B9" type="main">
          <s id="N146BB">Prim&ugrave;m quidem exijs, qu&aelig; demon&longs;trata &longs;unt, rectiline&aelig; <lb/>figur&aelig; AC centrum granitatis inueniatur E. eodemqu&egrave; mo <lb/>do figur&aelig; BD centrum grauitatis &longs;it F. Iungaturqu&egrave; EF, <lb/>qu&aelig; bifariam diuidatur in G. Iam patet punctum G cen&shy;<lb/>trum e&longs;&longs;e grauitatis pri&longs;matis AB, ex octaua propo&longs;itione Fe&shy;<lb/>derici <expan abbr="C&otilde;mandini">Commandini</expan> de centro grauitatis &longs;olidorum, &amp; ex corol<lb/>lario quint&aelig; propo&longs;itionis eiu&longs;dem libri, lineam EF late&shy;<lb/>ribus AD CB &etail;quidi&longs;tantem e&longs;&longs;e. </s>
          <s id="N146CF">quoniam <expan abbr="aut&etilde;">autem</expan> plana CH <lb/>CK ad rectos &longs;untangulos plano AC, erit CB eorum commu
<arrow.to.target n="marg196"/><lb/>nis&longs;ectio eidem plano AC perpendicularis. </s>
          <s id="N146DC">acpropterea EF <lb/>ip&longs;i CB &aelig;quidi&longs;tans plano AC perpendicularis exi&longs;tit.
<pb xlink:href="077/01/120.jpg" pagenum="116"/>Ita&que; intelligatur &longs;olidum AB ex E &longs;u&longs;pen&longs;um; tunc ex <lb/>ma propo&longs;itione de libra no&longs;trorum mechanicorum pon <lb/>AB ex E &longs;u&longs;pen&longs;um <expan abbr="numqu&atilde;">numquam</expan> manebit, ni&longs;i recta EG fu <lb/>horizonti perpendicularis. </s>
          <s id="N146EE">Quando autem EF erit horizc <lb/>ti perpendicularis, erit planum AC horizonti &aelig;quidi&longs;tan <lb/>
<arrow.to.target n="marg197"/> tunc. <expan abbr="n.">enim</expan> EF tum horizonti, tum plano AC perpendicul<gap/><lb/>exi&longs;tet. </s>
          <s id="N146FF">Inuento igitur centro grauitatis E ip&longs;ius ba&longs;is A <lb/>&longs;i AB &longs;u&longs;pendatur ex E, linea EGF in centrum mundi to <lb/>det; planumqu&egrave; AC horizonti erit &aelig;quidi&longs;tans. </s>
          <s id="N14705">quod de<gap/><lb/>&longs;trare oportebat. </s>
        </p>
        <p id="N1470A" type="margin">
          <s id="N1470C"><margin.target id="marg196"/>19. <emph type="italics"/>v <lb/>mi.<emph.end type="italics"/></s>
        </p>
        <p id="N14717" type="margin">
          <s id="N14719"><margin.target id="marg197"/>14.<emph type="italics"/>vndeci <lb/>mi.<emph.end type="italics"/></s>
        </p>
        <p id="N14724" type="main">
          <s id="N14726">PRIMI LIBRI FINIS. </s>
        </p>
        <pb xlink:href="077/01/121.jpg" pagenum="117"/>
        <p id="N1472B" type="head">
          <s id="N1472D">GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS.</s>
        </p>
        <p id="N14733" type="head">
          <s id="N14735">In Secundum Archimedis &aelig;&que;ponderan&shy;<lb/>tium Librum.</s>
        </p>
        <p id="N14739" type="head">
          <s id="N1473B">PR&AElig;FATIO.</s>
        </p>
        <p id="N1473D" type="main">
          <s id="N1473F">Secundus Archimedisliber, vtinitio primi <lb/>libri pr&aelig;fati &longs;umus, &longs;ubtili&longs;&longs;ima theo&shy;<lb/>remata &longs;peculatur. </s>
          <s id="N14745">Vultenim Archimedes <lb/>inue&longs;tigare centrum grauitatis plani coni&shy;<lb/>c&aelig;&longs;ectionis, qu&aelig; parabole pa&longs;&longs;im vocatur. <lb/>quamuis Archimedes alio nomine, ac po&shy;<lb/>ti&ugrave;s de&longs;criptione quadam <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> <expan abbr="h&atilde;c">hanc</expan> <expan abbr="n&utilde;-cuparit">nun&shy;<lb/>cuparit</expan>: veluti portio recta linea <expan abbr="rect&atilde;guli&qacute;">rectanguli&que;</expan>; coni&longs;ectione <expan abbr="c&otilde;">com</expan> <expan abbr="t&etilde;">tem</expan> <lb/>ta. </s>
          <s id="N1476B">Refert enim Eutocius A&longs;calonita in principio &longs;ui <expan abbr="comm&etilde;-tarij">commen&shy;<lb/>tarij</expan> in libros conicorum Apollonij Perg&etail;i, ex &longs;ententia Ge&shy;<lb/>mini (cui Pappus etiam ex Ari&longs;t&etail;i &longs;ententia a&longs;&longs;entire videtur) <lb/>qu&ograve;d qui ante Apollonium fuerunt, perfectam, &amp; ab&longs;olutam <lb/>conorum <expan abbr="cognition&etilde;">cognitionem</expan> <lb/>
<arrow.to.target n="fig60"/><lb/>non habuerunt; inter <lb/>quos re&longs;po&longs;uit Archime <lb/>de. <expan abbr="N&atilde;">Nam</expan> inquit <expan abbr="con&utilde;">conum</expan> de&longs;i <lb/>nientes, ip&longs;um per <expan abbr="rect&atilde;">rectam</expan> <lb/>guli <expan abbr="tri&atilde;guli">trianguli</expan> circumuo&shy;<lb/>lutionem manente vno <lb/>eorum, qu&aelig; circa <expan abbr="rect&utilde;">rectum</expan> <lb/><expan abbr="angul&utilde;">angulum</expan> &longs;unt, latere <expan abbr="c&otilde;&longs;i-derarunt">con&longs;i&shy;<lb/>derarunt</expan>. vt habetur in <lb/>definitionibus Euclidis <lb/>vndecimi libri elem <expan abbr="en-tor&utilde;">en&shy;<lb/>torum</expan>. vt Conus ABC fit <lb/>ex <expan abbr="circ&utilde;uoluto">circumuoluto</expan> triangulo rectangulo ADC. conus ver&ograve; EBC <lb/>ex triangulo EDC, &amp; conus FBC ex rectangulo triangulo
<pb xlink:href="077/01/122.jpg" pagenum="118"/>FDC. &amp; &longs;i AD fuerit i&shy;<lb/>
<arrow.to.target n="fig61"/><lb/>p&longs;i DC &aelig;qualis, conus <lb/>ABC vocabit rectan&shy;<lb/>gulus. </s>
          <s id="N147D0">nam vtcumqu&egrave; <lb/>ducto plano per axem, <lb/>
<arrow.to.target n="marg198"/> quod triangulum faciat <lb/>ABC; erit angulus BAC <lb/>ad coniverticem rectus: <lb/>&longs;iquidem DAC recti di <lb/>midius exi&longs;tit, veluti <lb/>DAB. pari ratione &longs;i ED <lb/>fuerit ip&longs;a DC minor; <lb/>erit conus EBC obtu&longs;i <lb/>angulus:nam ducto per axem plano, habebit triangulum <lb/>EBC angulum ad verticem coni BEC obtu&longs;um; c&ugrave;m &longs;it <lb/>
<arrow.to.target n="marg199"/> BEC maior BAC. exi&longs;tenteautem FD ip&longs;a DC maiori, co <lb/>nus FBC acutiangulus nuncupabitur; quoniam <expan abbr="triangul&utilde;">triangulum</expan> <lb/>per axem FBC angulum ad verticem coni F acutum po&longs;&longs;ide <lb/>bit; &longs;iquidem minor e&longs;t BFC, quam BAC. Refert deinde, <lb/>qu&ograve;d vnum&que;mqu&egrave; <lb/>horum conorum <expan abbr="eo-d&etilde;">eo&shy;<lb/>dem</expan> modo pi&longs;ci &longs;ecue&shy;<lb/>
<arrow.to.target n="fig62"/><lb/>runt; vt &longs;it rectangu&shy;<lb/>lus conus ABC; trian <lb/>gulum ver&ograve; per axem <lb/>&longs;it ABC. in latere au&shy;<lb/>tem AC quoduis &longs;u&shy;<lb/>matur punctum D; <lb/>ducaturqu&egrave; DE ad <lb/>AC perpendicularis; <lb/>&amp; per DE ducatur pla <lb/>num plano ABC ere <lb/>ctum, quod quidem conum &longs;ecet, &longs;ectio autem &longs;it FDG. qu&etail; <lb/>&longs;an&egrave; e&longs;t &longs;e ctio, qu&aelig; abip&longs;is vocatur rectanguli coni &longs;ectio, <lb/>quipp&egrave; qu&aelig; &longs;i intelligatur terminata recta linea FG, nuncupa <lb/>tur portio recta linea, rectanguli&que; coni &longs;ectione contenta. </s>
        </p>
        <pb xlink:href="077/01/123.jpg" pagenum="119"/>
        <p id="N1482A" type="margin">
          <s id="N1482C"><margin.target id="marg198"/>3. <emph type="italics"/>primi co <lb/>mcorum A <lb/>pol.<emph.end type="italics"/></s>
        </p>
        <p id="N14839" type="margin">
          <s id="N1483B"><margin.target id="marg199"/>21. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.123.1.jpg" xlink:href="077/01/123/1.jpg"/>
        <figure id="id.077.01.123.2.jpg" xlink:href="077/01/123/2.jpg"/>
        <figure id="id.077.01.123.3.jpg" xlink:href="077/01/123/3.jpg"/>
        <p id="N14850" type="main">
          <s id="N14852">Si ver&ograve; conus <lb/>
<arrow.to.target n="fig63"/><lb/>ABC fuerit obtu <lb/>&longs;iangulus, &longs;itqu&egrave; <lb/>triangulum per <lb/>axem ABC, <expan abbr="eo-d&etilde;">eo&shy;<lb/>dem</expan> modo&agrave; quo&shy;<lb/>uis puncto D, du<lb/>cta DE ad re&shy;<lb/>ctos angulos ip&longs;i <lb/>AC, acper DE <lb/>ducto plano ad <lb/>planum ABC erecto, quod conum &longs;ecet, vt FDG; erit FDG <lb/>obtu&longs;ianguli coni &longs;ectio, qu&aelig; vn&agrave; cum recta FG vocatur por&shy;<lb/>tio recta linea, obtu&longs;ianguliqu&egrave; coni &longs;ectione contenta. </s>
        </p>
        <figure id="id.077.01.123.4.jpg" xlink:href="077/01/123/4.jpg"/>
        <p id="N1487B" type="main">
          <s id="N1487D">Similiter <expan abbr="exi&longs;t&etilde;te">exi&longs;tente</expan> co&shy;<lb/>
<arrow.to.target n="fig64"/><lb/>no acutiangulo ABC, <lb/>cuius triangulum per a&shy;<lb/>xem &longs;it ABC. &amp; &agrave; <expan abbr="p&utilde;cto">puncto</expan> <lb/>D ducta &longs;it DE perpen&shy;<lb/>dicularis ip&longs;i AC, du&shy;<lb/>ctoqu&egrave; plano per DE ad <lb/>planum ABC erecto, e&shy;<lb/>rit DFEG acutianguli <lb/>coni &longs;ectio. </s>
        </p>
        <figure id="id.077.01.123.5.jpg" xlink:href="077/01/123/5.jpg"/>
        <p id="N148A2" type="main">
          <s id="N148A4">Apollonius au-&shy;<lb/>tem Perg&etail;us, qui ab&shy;<lb/>&longs;oluti&longs;&longs;ima commenta&shy;<lb/>ria de conicis &longs;crip&longs;it, <lb/>huiu&longs;modi conos omnesvocauit rectos; ad differentiam coni <lb/>&longs;caleni. </s>
          <s id="N148B0">coni enim rectiaxes habent ba&longs;ibus erectos. </s>
          <s id="N148B2">&longs;caleni ve <lb/>r&ograve; nequaquam. </s>
          <s id="N148B6">&amp; in &longs;calenis latera triangulorum per axem <lb/>non &longs;unt &longs;emper &aelig;qualia. </s>
          <s id="N148BA">quod &longs;emper conis rectis contingit. </s>
        </p>
        <p id="N148BC" type="main">
          <s id="N148BE">Preterea &longs;ectionem rectanguli coni parabolen nominauit; <lb/>obtu&longs;ianguli ver&ograve; coni &longs;ectionem hyperbolen; &longs;ectionem au<lb/>tem acutianguli coni ellip&longs;im nuncupauit. </s>
          <s id="N148C4">&amp; in vnoquo&que; <lb/>cono t&agrave;m recto, qu&agrave;m &longs;caleno has tres ine&longs;&longs;e &longs;ectiones <expan abbr="dem&otilde;">demom</expan>
<pb xlink:href="077/01/124.jpg" pagenum="120"/>&longs;trauit. </s>
          <s id="N148D0">Ex quibus colligit Geminus (&que;m Eutocius, alijqu&egrave; <lb/>complures &longs;ecuti &longs;unt) eos, qui ante Apollonium extitere, <lb/>conostant&ugrave;m rectos cognoui&longs;&longs;e. </s>
          <s id="N148D6">&amp; in vnoquo&que; cono <expan abbr="vn&atilde;">vnam</expan> <lb/>tant&ugrave;m &longs;ectionem animaduerti&longs;&longs;e. </s>
          <s id="N148DE">quod quidem &longs;i de ijs, qui <lb/>ante Archimedem fuere intelligatur; ad mitti forta&longs;&longs;e poterit; <lb/>ac pr&aelig;&longs;ertim de Euclide. </s>
          <s id="N148E4">vt patet ex definitione coni abeo <lb/>tradita. </s>
          <s id="N148E8">At ver&ograve; de Archimede, qui po&longs;t Euclidem, ante ver&ograve; <lb/>Apollonium fuit, non ita facil&egrave; concedendum videtur. <expan abbr="N&atilde;">Nam</expan> ex <lb/>ijs, qu&aelig; &longs;cripta reliquit. </s>
          <s id="N148F2">eum non &longs;ol&ugrave;m notitiam ha-&shy;<lb/>bui&longs;&longs;e de conis rectis; ver&ugrave;m <expan abbr="eti&atilde;">etiam</expan> de &longs;calenis facil&egrave; ex i-&shy;<lb/>p&longs;ius &longs;criptis conijci pote&longs;t. </s>
          <s id="N148FC">In primo enim librode &longs;ph&aelig;&shy;<lb/>ra, &amp; cylindro multis in locis, vt in &longs;eptima, octaua, no <lb/>na, decimaquarta, decimaquinta propo&longs;itione; alijsqu&egrave; in <lb/>locis conos nominat &etail;quicrures, quod quidem &longs;ecundum i&shy;<lb/>p&longs;um &longs;unt, qui in eius &longs;uperficie &aelig;quales habent rectas lineas <lb/>&agrave; vertice coni ad ba&longs;im ductas. </s>
          <s id="N14908">item in epi&longs;tola quo&que; libri <lb/>de conoidibus &amp; &longs;ph&etail;roidibus, quam Archimedes De&longs;itheo <lb/>&longs;cribit. </s>
          <s id="N1490E">c&ugrave;m de obtu&longs;iangulo conoideverba facit, conum vo&shy;<lb/>cat&aelig;quicrurem. </s>
          <s id="N14912">Qu&ograve;d &longs;i Archimedes hos conos vocauit &aelig;&shy;<lb/>quicrures, cui dubium, ip&longs;um eosad differentiam eorum, qui <lb/>non &longs;unt &aelig;quicrures ita nuncupa&longs;&longs;e? </s>
          <s id="N14918">qui ver&ograve; non &longs;unt &aelig;&shy;<lb/>quicrures ex ip&longs;omet Apollonio &longs;unt &longs;caleni; nam &aelig;quicrures <lb/>hoc modo coni axes habent ba&longs;ibus erectos. </s>
          <s id="N1491E">qui igitur non <lb/>erunt &aelig;quicrures, eorum axes &longs;uis ba&longs;ibus nunqu&agrave;m erunt e&shy;<lb/>recti. </s>
          <s id="N14924">Pr&aelig;terea idem quo&que; confirmari pote&longs;t ex demon&shy;<lb/>&longs;tratione vige&longs;im&aelig;quint&aelig; propo&longs;itionis eiu<gap/>dem libri, in qua <lb/>c&ugrave;m nominet Archimehes conum rectum proculdubi&ograve; ad <lb/>differentiam eorum, qui non &longs;untrecti ita eum nuncupauit. <lb/>nam &longs;i Aichimedes (ex illorum &longs;ententia) conos tan &ugrave;m re&shy;<lb/>ctos cognoui&longs;&longs;et; quor&longs;um his in locis conum rectum, vel &aelig;&shy;<lb/>quicrurem nomina&longs;&longs;et? </s>
          <s id="N14934">&longs;at &longs;ibi fui&longs;&longs;et conum tantum dixi&longs;&longs;e. <lb/>Ne&que; ver&ograve; dicendum e&longs;t Archimedem per cono recto intel<lb/>lexi&longs;&longs;e conum rectangulum eo modo, &que;m &longs;upra expo&longs;ui&shy;<lb/>mus. </s>
          <s id="N1493C">nam in ea propo&longs;itione, dum con&longs;tituit hunc conum, <lb/>non con&longs;urgit conus rectangulus, &longs;ed obtu&longs;iangulus quapro <lb/>pter conum rectum nominatad differentiam coni &longs;caleni. </s>
          <s id="N14942">C&etail; <lb/>ter&ugrave;m ut manife&longs;t&egrave; o&longs;tendamus Archimedem conos cogno-
<pb xlink:href="077/01/125.jpg" pagenum="121"/>ui&longs;&longs;e &longs;calenos, con&longs;ideranda e&longs;t octaua propo&longs;itio libri de co&shy;<lb/>noidibus, &amp; &longs;ph &aelig;roidibus, in qua proponit Archimedes co&shy;<lb/>num con&longs;tituere, &amp; inuenire, in quo &longs;it&longs;ectio ellip&longs;is data, ver <lb/>tex autem coni in linea exi&longs;tat a centro ellip&longs;is ad<gap/>ectos angu<lb/>los ellip&longs;is plano erecta. </s>
          <s id="N14954">Exqua con&longs;tructione plan&egrave; apparet, <lb/>Archimedem (vt ex eius demon&longs;tratione con&longs;tat) hoc in lo&shy;<lb/>co &que;rere, &amp; inuenire conum proculdubio &longs;calenum. </s>
          <s id="N1495A">vt <expan abbr="eti&atilde;">etiam</expan> <lb/>ex nona eiu&longs;dem libri propo&longs;itione per&longs;picuum e&longs;&longs;e pote&longs;t; in <lb/>qua vt plurim&ugrave;m conus inuenitur &longs;calenus. </s>
          <s id="N14964">Ex quibus mani&shy;<lb/>fe&longs;ti&longs;&longs;im&egrave; patet Archimedem non &longs;ol&ugrave;m de conis rectis, <expan abbr="ver&utilde;">verum</expan> <lb/>etiam de conis &longs;calenis notitiam habui&longs;&longs;e. </s>
          <s id="N1496E">Porr&ograve; ea verba, qu&etail; <lb/>refert Eutocius ex &longs;ententia Heraclij, qui Archimedis vitam <lb/>literis mandauit; idip&longs;um &longs;atis manife&longs;tant. </s>
          <s id="N14974">Heraclius enim <lb/>inquit Archimedem quidem <expan abbr="prim&utilde;">primum</expan> conica theoremata fui&longs;&longs;e <lb/>aggre&longs;&longs;um; Apollonium ver&ograve;, c&ugrave;m ea inueni&longs;&longs;etab Archime <lb/>de nondum edita; tanquam eius propria edidi&longs;&longs;e. </s>
          <s id="N14980">quod qui&shy;<lb/>dem etiam exip&longs;iusmet Archimedis &longs;criptis <expan abbr="c&otilde;firmari">confirmari</expan> pote&longs;t. <lb/>in libro nam&que; de conoidibus, &amp; &longs;ph&aelig;roidibus ante <expan abbr="quart&atilde;">quartam</expan> <lb/>propo&longs;itionem vbi Archimedes theorema proponit alibi de&shy;<lb/>mon&longs;tratum, inquit, <emph type="italics"/>Hoc autem o&longs;ten &longs;um e&longs;t in conicis elementis.<emph.end type="italics"/> in <lb/>principio etiam libri de quadratura paraboles, c&ugrave;m nonnulla <lb/>propo&longs;ui&longs;&longs;et; po&longs;t tertiam propo&longs;itionem &longs;cilicet, inquit <emph type="italics"/>De&shy;<lb/>mon&longs;trata autem &longs;unt h&aelig;c in elementis conicis.<emph.end type="italics"/> nonneigitur con&longs;tat <lb/>Archimedem <expan abbr="elem&etilde;ta">elementa</expan> conica &longs;crip&longs;i&longs;&longs;e? </s>
          <s id="N149AA">Obijciet ver&ograve; aliquis, <lb/>non propterea con&longs;tare, h&etail;c elementa eonica, quorum me&shy;<lb/>minit Archimedes, ip&longs;iusmet e&longs;&longs;e Archimedis; c&ugrave;m non affir <lb/>met, h&aelig;cfui&longs;&longs;e ab ip&longs;o demon&longs;trata. </s>
          <s id="N149B2">ver&ugrave;m illud in primis ma <lb/>nife&longs;tum e&longs;t, tempore Archimedis conica elementa extiti&longs;&longs;e. <lb/>vt nonnulli Euclidem quatuor conicorum libros edidi&longs;&longs;e <expan abbr="af-firm&atilde;t">af&shy;<lb/>firmant</expan>; &longs;icut Pappus in &longs;eptimo <expan abbr="Mathematicar&utilde;">Mathematicarum</expan> <expan abbr="collectionu&utilde;">collectionuum</expan> <lb/>libro a&longs;&longs;erit. </s>
          <s id="N149C8">Sed ex modo lo&que;ndi Archimedis plan&egrave; <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> <lb/>h&aelig;c fui&longs;&longs;e ab ip&longs;o con&longs;cripta. </s>
          <s id="N149D0">Nam quando Archimedes ali&shy;<lb/>qua &longs;upponitab alijs demon&longs;trata, <expan abbr="t&utilde;c">tunc</expan> addere con&longs;ueuit, illa <lb/>ab alijs demon&longs;trata e&longs;&longs;e; vt in vndecima propo&longs;itionedeco&shy;<lb/>noidibus, &amp; &longs;ph&aelig;roidibus; c&ugrave;m inquit. <emph type="italics"/>omnis coni ad conum pro&shy;<lb/>portionem compo&longs;itam e&longs;&longs;e ex proportione ba&longs;ium, &amp; proportione altitu&shy;<lb/>dinum,<emph.end type="italics"/> quod quidem, quia ab alijs demon&longs;tratum fuerat, &longs;ta&shy;
<pb xlink:href="077/01/126.jpg" pagenum="122"/>tim inquit, <emph type="italics"/>demon&longs;tratum e&longs;t ab iis, qui ante nos fuerunt.<emph.end type="italics"/> &longs;imiliter <lb/>in libro de &longs;ph&etail;ra, &amp; cylindro ante propo&longs;itionem decimam <lb/>&longs;eptimam, c&ugrave;m nonnulla &longs;uppo&longs;uerit ab alijs demon &longs;trata in <lb/>quit. <emph type="italics"/>H&aelig;c autem omnia &agrave; &longs;uperioribus &longs;unt demon&longs;trata.<emph.end type="italics"/> In &longs;ecunda <lb/>ver&ograve; parte <expan abbr="qu&itilde;t&etail;">quint&etail;</expan> propo&longs;itionis hui^&lcub;9} &longs;ecudi libri cu inquit, <emph type="italics"/>De <lb/>mon&longs;tratum e&longs;t enim aliis in locis portiones &longs;e&longs;quitertias e&longs;&longs;e <expan abbr="triangulor&utilde;">triangulorum</expan>.<emph.end type="italics"/><lb/>quod quia ip&longs;emet a&longs;&longs;ecutus e&longs;t in libro de quadratura para&shy;<lb/>boles, idcirc&ograve; non addit ab ip&longs;omethoc o&longs;ten&longs;um fui&longs;&longs;e. </s>
          <s id="N14A11">A&shy;<lb/>liaqu&egrave; huiu&longs;modi loca breuitatis &longs;tudio omitto o&longs;tendentia <lb/>ea, qu&aelig; Archimedes &longs;upponit tanquam demon&longs;trata, <expan abbr="qu&atilde;do">quando</expan> <lb/>non additab alijs o&longs;ten&longs;a e&longs;&longs;e, &agrave; &longs;e ip&longs;o demon&longs;trata fui&longs;&longs;e, vt <lb/>in demon&longs;tratione decim&aelig; quart&etail; propo&longs;itionis primi libri, <lb/>nec non ex octaua huius &longs;ecundi libri demon&longs;tratione; alij&longs;&shy;<lb/>qu&egrave; locis per&longs;picuum e&longs;&longs;e pote&longs;t. </s>
          <s id="N14A23">Quare t&ugrave;m ex pr&aelig;fntis Archi <lb/>medis locis, t&ugrave;m Heraclij te&longs;tim onio manife&longs;t&egrave; elicipote&longs;t, <lb/>Archimedem elementa conica &longs;crip &longs;i&longs;&longs;e. </s>
          <s id="N14A29">Ne&que; ver&ograve; quicqua <lb/>nos turbare debet, qu&ograve;d Apollo nius coni &longs;ectionibus nomina <lb/>impo&longs;uerit; &longs;i tamen ip&longs;e prim us fuit; c&ugrave;m eas proprijs nomi&shy;<lb/>nibus, vt pot&egrave; parabolen, hyperbolen, &amp; ellip&longs;im nuncupet; <lb/>&amp; in quolibet cono omnes agnouerit &longs;ectiones. </s>
          <s id="N14A33">Nam quam&shy;<lb/>uis v&longs;&que; ad Archimedis tempus hi termini nondum extite&shy;<lb/>rint; &amp; in &longs;ingulis conis pri&longs;ci illi vnicam <expan abbr="t&atilde;t&ugrave;m">tant&ugrave;m</expan> cognouerint <lb/>&longs;ectionem; Archimedes tamen vlteri&ugrave;s progre&longs;&longs;us e&longs;t. </s>
          <s id="N14A3F">etenim <lb/>h&aelig;c quo&que; <expan abbr="&longs;ection&utilde;">&longs;ectionum</expan> nomina ip&longs;i forta&longs;se min&ugrave;s ignota fue&shy;<lb/>runt: quandoquidem in demon&longs;tratione non&aelig; propo&longs;itio&shy;<lb/>nis de conoidibus, &amp; &longs;ph&etail;roidibus ellip&longs;im nominat. </s>
          <s id="N14A4B">Pr&etail;te&shy;<lb/>rea non &longs;ol&ugrave;m cognouit Archimedes conos &longs;ecari po&longs;&longs;e pla&shy;<lb/>nis lateribus coni erectis, ver&ugrave;m etiam alijs modis: quod qui&shy;<lb/>dem exemplo ellip&longs;is manife&longs;tari optim&egrave; pote&longs;t. </s>
          <s id="N14A53">Nam in o&shy;<lb/>ctaua propo&longs;itione eiu&longs;dem libri ellip&longs;es latus coni ad angu&shy;<lb/>los rectos minim&egrave; &longs;ecant. </s>
          <s id="N14A59">veluti quo&que; in nona propo&longs;itione <lb/><expan abbr="id&etilde;">idem</expan> &longs;&etail;p&egrave; <expan abbr="c&otilde;tingit">contingit</expan>. At ver&ograve; in <expan abbr="eod&etilde;">eodem</expan> adhuc libro ante <expan abbr="prim&atilde;">primam</expan> pro <lb/>po&longs;itionem inquit Archimedes. <emph type="italics"/>Si conus plano &longs;ecetur cum omnibus <lb/>eius lateribus coeunti, &longs;ectio vel erit circulus, vel acutianguli coni &longs;e&shy;<lb/>ctio.<emph.end type="italics"/> Vnde per&longs;picuum e&longs;t non in vno duntaxat cono acutian <lb/>gulo, ver&ugrave;m in omnibus conis&longs;ectionem ellip&longs;is cognoui&longs;&longs;e. <lb/>Pr&aelig;terea ex hoclo&que;ndi modo li&que;t ip&longs;um &longs;ectionem quo
<pb xlink:href="077/01/127.jpg" pagenum="123"/>&que; noui&longs;&longs;e &longs;ubcontrariam; qu&aelig; c&ugrave;m &longs;it ba&longs;i &longs;ubcontrai&longs;&egrave; po <lb/>&longs;ita, <expan abbr="o&itilde;a">oina</expan> latera coni &longs;ecat; &amp; <expan abbr="t&ntilde;">tnm</expan> <expan abbr="n&omacr;">non</expan> e&longs;t ellip&longs;is, &longs;ed circulus.
<arrow.to.target n="marg200"/> qua&shy;<lb/>propter &longs;i in omnibus conis ellip&longs;is nouit &longs;ectionem; cur in i&shy;<lb/>p&longs;is, &amp; parabolas, &amp; hyperbolas min&ugrave;s animaduertit? </s>
          <s id="N14A96">c&ugrave;m <lb/>&longs;it manife&longs;tum ex dictis in cono obtu&longs;iangulo &amp; <expan abbr="hyperbol&etilde;">hyperbolem</expan> <lb/>&amp; ellip&longs;im; in rectangulo autem parabolem, ellip&longs;imqu&egrave; co&shy;<lb/>gnoui&longs;&longs;e? </s>
          <s id="N14AA2">h&ograve;c cert&egrave; non e&longs;t a&longs;&longs;erendum. </s>
          <s id="N14AA4">Ex hoc enim per&longs;pi&shy;<lb/>cuum e&longs;t Archimedem cognoui&longs;&longs;e conos &longs;ecari po&longs;&longs;e planis, <lb/>qu&aelig; non &longs;int &longs;emper ad coni latus erecta. </s>
          <s id="N14AAA">dormita&longs;&longs;equ&egrave; Eu&shy;<lb/>tocium Geminum, &amp; alios &longs;ecus hac in parte de Archimede <lb/>&longs;entientes. </s>
          <s id="N14AB0">Ampli&ugrave;s <expan abbr="n&otilde;">non</expan> ne cognouit etiam Archimedes &longs;eca&shy;<lb/>ri po&longs;&longs;e rectangulos conoides, itidemqu&egrave; &amp; <expan abbr="obtu&longs;i&atilde;gulos">obtu&longs;iangulos</expan> pla <lb/>nis, qu&aelig; ne&que; &longs;int per axem ducta, ne&que; axi &aelig;quidi&longs;tantia; <lb/>ne&que; &longs;uper axem erecta. </s>
          <s id="N14AC0">vt in duodecima, decimatertia, &amp; <lb/>decima quarta propo&longs;itione eiu&longs;dem libri patet. </s>
          <s id="N14AC4">quomodo i&shy;<lb/>ta&que; his quo&que; modis &que;mlibet conum &longs;ecari po&longs;&longs;e igno&shy;<lb/>rauit? </s>
          <s id="N14ACA">Non e&longs;t igitur ambigendum Archimedem cognoui&longs;&shy;<lb/>&longs;e conos &longs;ecari po&longs;&longs;e planis ad latus coni differentem inclina&shy;<lb/>tionem habentibus. </s>
          <s id="N14AD0">Ex quibus per&longs;picuum e&longs;t, ip&longs;um in om&shy;<lb/>nibus conis omnes ine&longs;&longs;e &longs;ectiones omnino animaduerti&longs;&longs;e. <lb/>At &longs;i concedamus etiam &longs;ua tempe&longs;tate nondum &longs;ectioni&shy;<lb/>bus ip&longs;is propria fui&longs;&longs;e impo&longs;ita nomina; tam eam parabo&shy;<lb/>lem, qu&aelig; erat rectanguli coni &longs;ectio; qu&agrave;m qu&aelig; erat &longs;ectio <lb/>alterius coni, c&ugrave;m &longs;it eadem &longs;ectio, eodem nomine nuncu&shy;<lb/>pabat; nemp&egrave; rectanguli coni &longs;ectionem. </s>
          <s id="N14ADE">Et hoc, quia <lb/>pri&ugrave;s h&aelig;c &longs;ectio cognita &longs;uit in cono rectangulo (vnde &longs;i&shy;<lb/>bi nomen vindicauit) quam in alio. </s>
          <s id="N14AE4">quod idem dicen&shy;<lb/>dum e&longs;t de alijs &longs;ectionibus. </s>
          <s id="N14AE8">Vt manife&longs;tum e&longs;&longs;e pote&longs;t <lb/>exemplo &longs;ectionis acutianguli coni. </s>
          <s id="N14AEC">Archimedes enim eo&shy;<lb/>dem loco, anteprimam &longs;cilicet propo&longs;itionem de conoidi <lb/>bus, &amp; &longs;ph&etail;roidibus inquit, <emph type="italics"/>Si cylindrus duobus planis &aelig;quidi&shy;<lb/>stantibus &longs;ecetur; qu&aelig; cum omnibus ip&longs;ius lateribus coeant, &longs;ectio&shy;<lb/>nes, uelerunt circuli; uel conorum acutiangulorum &longs;ectiones.<emph.end type="italics"/> vo&shy;<lb/>catigitur Archimedes acutianguli coni &longs;ectionem, tam coni <lb/><expan abbr="&longs;ection&etilde;">&longs;ectionem</expan>, qu&agrave;m <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> cylindri. </s>
          <s id="N14B07">veluti <expan abbr="eti&atilde;">etiam</expan> in decimatertia, <lb/>&amp; decimaquarta propo&longs;itione <expan abbr="eiu&longs;d&emacr;">eiu&longs;dem</expan> libri <expan abbr="acuti&atilde;guli">acutianguli</expan> coni &longs;e&shy;<lb/>ctio ab ip&longs;o ea <expan abbr="n&umacr;cupatur">nuncupatur</expan> &longs;ectio, qu&aelig; <expan abbr="o&imacr;a">oina</expan> latera tam conoidis
<pb xlink:href="077/01/128.jpg" pagenum="124"/>rectanguli, qu&agrave;m obtu&longs;ianguli ab&longs;cindit. </s>
          <s id="N14B25">dum modo non &longs;it <lb/>ad axem erecta. </s>
          <s id="N14B29">nullaqu&egrave; alia de cau&longs;a h&aelig; &longs;ectiones omnes i&shy;<lb/>dem acutianguli coni &longs;ectionis nomen obtiuerunt; ni&longs;i quia <lb/>pri&ugrave;s h&aelig;c &longs;ectio &agrave; cono acutiangulo nomen accepit, quando&shy;<lb/>quidem in ip&longs;o forta&longs;se prim&ugrave;m cognita fuit, qua&agrave;m in alijs. <lb/>Ex dictis ita&que; manife&longs;tum e&longs;t, &longs;ententiam Heraclij veram <lb/>e&longs;&longs;e po&longs;&longs;e, &amp; rationi vald&egrave; con&longs;entaneam; Archimedem &longs;cili <lb/>cet elementa conica &longs;crip&longs;i&longs;&longs;e; Apollonium qu&egrave;, c&ugrave;m ea ab Ar <lb/>chimede nondum edita inueni&longs;&longs;et, &longs;icut propria &longs;ua edidi&longs;&longs;e. <lb/>Omitto interim multa ab Archimede in eius libris &longs;upponi, <lb/>qu&aelig; non ni&longs;i in conicis e&longs;&longs;e dcbebant, qu&aelig; quidem <expan abbr="hab&etilde;tur">habentur</expan> <lb/>&longs;ol&ugrave;m in conicis Apolloni. </s>
          <s id="N14B43">Negandum tamen non e&longs;t, vt <lb/>Eutocius quo&que; affirmat, ip&longs;um Apollonium multa auxi&longs;&longs;e, <lb/>multaqu&egrave; ad conica &longs;pectantia adinueni&longs;&longs;e. </s>
          <s id="N14B49">vt ip&longs;emet Apol&shy;<lb/>lonius in epi&longs;tola ad Eudemum fatetur. </s>
          <s id="N14B4D">c&ugrave;m tamen non &longs;it <lb/>&longs;emperfacil&egrave; inuentis addere. </s>
          <s id="N14B51">Sed de his hactenus. </s>
          <s id="N14B53">&longs;at &longs;it au&shy;<lb/>tem noui&longs;&longs;e, Archimedem, <expan abbr="qu&atilde;do">quando</expan> in hoclibro nominat por <lb/>tionem recta linea, rectanguliqu&egrave; coni &longs;ectione contentam, <lb/>eam &longs;ignificare fectionem, qu&aelig; parabole nuncupatur. </s>
        </p>
        <pb xlink:href="077/01/129.jpg" pagenum="125"/>
        <p id="N14B62" type="margin">
          <s id="N14B64"><margin.target id="marg200"/>5. <emph type="italics"/>primi co <lb/><expan abbr="nicor&utilde;">nicorum</expan> A&shy;<lb/>poll.<emph.end type="italics"/></s>
        </p>
        <p id="N14B74" type="head">
          <s id="N14B76">GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS.</s>
        </p>
        <p id="N14B7C" type="head">
          <s id="N14B7E">IN SECVNDVM ARCHIMEDIS <lb/>&AElig;QVEPONDERANTIVM <lb/>LIBRVM.</s>
        </p>
        <p id="N14B84" type="head">
          <s id="N14B86">PARAPHRASIS <lb/>SCHOLIIS ILLVSTRATA.</s>
        </p>
        <figure id="id.077.01.129.1.jpg" xlink:href="077/01/129/1.jpg"/>
        <p id="N14B8D" type="head">
          <s id="N14B8F">PROPOSITIO. I.</s>
        </p>
        <p id="N14B91" type="main">
          <s id="N14B93">Si duo &longs;pacia recta linea, &amp; re <lb/>ctanguli coni &longs;ectione conten<lb/>ta, qu&aelig; ad datam rectam <expan abbr="line&atilde;">lineam</expan> <lb/>applicare po&longs;&longs;umus, non ha <lb/>beantidem grauitatis <expan abbr="centr&utilde;">centrum</expan>; <lb/>magnitudinis ex vtri&longs;&que; i&shy;<lb/>p&longs;orum compo&longs;it&aelig; centrum <lb/>grauitatis erit in recta linea, qu&aelig; ip&longs;orum centra <lb/>grauitatis coniungit; ita diuidens dictam rectam li <lb/>neam, vt ip&longs;ius portiones permutatim eandem ad <lb/>inuicem proportionem habeant, vt &longs;pacia. </s>
        </p>
        <pb xlink:href="077/01/130.jpg" pagenum="126"/>
        <figure id="id.077.01.130.1.jpg" xlink:href="077/01/130/1.jpg"/>
        <p id="N14BB7" type="main">
          <s id="N14BB9"><emph type="italics"/>Sint duo &longs;pacia AB CD, qualia dicta &longs;unt. </s>
          <s id="N14BBD">ip&longs;orum autem centra <lb/>grauitatis &longs;int puncta EF.<emph.end type="italics"/> iungaturqu&egrave; EF, qu&aelig; diuidatur in <lb/>H; <emph type="italics"/>&amp; quam proportionem habet AB ad CD, <expan abbr="e&atilde;dem">eandem</expan> habeat FH <lb/>ad HE. o&longs;tendendum e&longs;t magnitudmis ex utri&longs;qu&egrave; AB CD &longs;pa&shy;<lb/>ciis compo&longs;it&aelig; centrum grauitaias e&longs;&longs;e punctum H. &longs;it quidemip&longs;i EH <lb/>utra&que; ip&longs;arum FG FK &aelig;qualis; ip&longs;i autem FH, hocest GE<emph.end type="italics"/><lb/>(&longs;untenim EH GF &aelig;quales, &agrave; quibus dempta communi <lb/>GH remanent EG HF &etail;quales) <emph type="italics"/>&longs;it &aelig;qualis EL.<emph.end type="italics"/> &amp; <expan abbr="quoni&atilde;">quoniam</expan> <lb/>FH e&longs;t &aelig;qualis LE, &amp; FK ip&longs;i EH, <emph type="italics"/>erit &amp; LH ip&longs;i KH <lb/>&aelig;qualis.<emph.end type="italics"/> C&ugrave;m autem &longs;it FH ad HE, vt AB ad CD; ip&longs;i <lb/>ver&ograve; FH vtra&que; &longs;it &aelig;qualis LE EG. ip&longs;i autem HE vtra&shy;<lb/>&que; &aelig;qualis GF FK, <emph type="italics"/>erit <expan abbr="eti&atilde;">etiam</expan> ut LG ad G<emph.end type="italics"/>k, <emph type="italics"/>ita AB ad CD.<emph.end type="italics"/><lb/>c&ugrave;m &longs;it LG ad GK, vt FH ad HE; <emph type="italics"/>aupla enim est utra&que;<emph.end type="italics"/><lb/>EG GK <emph type="italics"/>utriu&longs;&que;<emph.end type="italics"/> FH HE. <emph type="italics"/>At uer&ograve; circa punctum<emph.end type="italics"/> E <emph type="italics"/>ip&longs;ius <lb/>AB,<emph.end type="italics"/> quod e&longs;t eius centrum grauitatis, <emph type="italics"/>ex utra&que; parte line&aelig; LG, <lb/>ip&longs;i LG &aelig;quidistantes ducantur<emph.end type="italics"/> MO QN, qu&aelig; &aelig;qualiter ab <lb/>LG di&longs;tent, ductis &longs;cilicet MQ ON &aelig;quidi&longs;tantibus, &longs;int <lb/>LM LQ GO GN inter &longs;e &aelig;quales; <emph type="italics"/>ita ut &longs;pacium MN &longs;it <lb/>&longs;pacio AB &aelig;quale<emph.end type="italics"/>: quod quidem applicatum e&longs;t ad <expan abbr="line&atilde;">lineam</expan> LG. <lb/>
<arrow.to.target n="marg201"/> <emph type="italics"/>erit uti&que; ip&longs;ius MN centrum grauitatis punctum E.<emph.end type="italics"/> c&ugrave;m &longs;it <expan abbr="p&utilde;-ctum">pun&shy;<lb/>ctum</expan> E in medio line&aelig; LG, qu&aelig; bifariam diuidit latera <lb/>oppo&longs;ita MQ ON parallelogrammi MN. <emph type="italics"/>compleatur ita&shy;<lb/>&que; &longs;pacium NX. habebit quidem MN. ad NX proportionem,<emph.end type="italics"/>
<pb xlink:href="077/01/131.jpg" pagenum="127"/><emph type="italics"/>quam<emph.end type="italics"/> habet QN ad NP, hoce&longs;t <emph type="italics"/>LG ad GK. habet autem &amp; <lb/>AB ad CD proportionem ip&longs;ius LG ad G<emph.end type="italics"/>K. <emph type="italics"/>ut igitur AB ad<emph.end type="italics"/>
<arrow.to.target n="marg202"/><lb/><emph type="italics"/>CD, &longs;ic est MN ad NX. &amp; permutando<emph.end type="italics"/> vt AB ad MN, ita <lb/>CD ad NX. <emph type="italics"/>&aelig;quale autem est AB ip&longs;i MN, erit igitur &amp; CD <lb/>ip&longs;i NX &aelig;quale. </s>
          <s id="N14C79">Centrum autem grauitatisip&longs;ius<emph.end type="italics"/> NX <emph type="italics"/>est <expan abbr="punot&utilde;">punotum</expan> <lb/>F.<emph.end type="italics"/> propterea quod e&longs;t in medio line&aelig; GK, qu&aelig;
<arrow.to.target n="marg203"/> parallelo&shy;<lb/>grammi NX oppo&longs;ita latera ON XP bifariam &longs;ecat. <emph type="italics"/>&amp; <lb/>quoniam &aelig;qualis e&longs;t LH ip&longs;i HK, totaqu&egrave; LK appa&longs;ita latera<emph.end type="italics"/> MQ <lb/>XP <emph type="italics"/>bifariam diuidit, totius PM <expan abbr="centr&utilde;">centrum</expan> grauitatis erit punctum Hr <lb/>Ver&ugrave;m ip&longs;um MP &aelig;quale est utri&longs;&que; MN NX,<emph.end type="italics"/> quorum, c&ugrave;m <lb/>&longs;int centra grauitatis EF, &aelig;&que;pondera bunt &longs;pacia MN <lb/>NX ex di&longs;tantijs FH HE. &longs;i igitur loco parallelo gram mo&shy;<lb/>rum MN NX ponatur AB in E, &amp; CD in F, c&ugrave;m &longs;it <lb/>AB ip&longs;i MN, &amp; CD ip&longs;i NX &aelig;quale; &longs;pacia AB CD ex
<arrow.to.target n="marg204"/><lb/>di&longs;tantijs FH HE &aelig;&que;ponderabunt. <emph type="italics"/>ac propterea magnitudi <lb/>nis ex utri&longs;&que; AB CD<emph.end type="italics"/> compo&longs;it&aelig; <emph type="italics"/>centrum grauitatis <expan abbr="e&longs;tpunct&utilde;">e&longs;tpunctum</expan> <lb/>H.<emph.end type="italics"/> quod quidem propo&longs;itum fuit. </s>
        </p>
        <p id="N14CC7" type="margin">
          <s id="N14CC9"><margin.target id="marg201"/>2. <emph type="italics"/>cor.<emph.end type="italics"/> 9. <lb/><emph type="italics"/>primihui<emph.end type="italics"/>^&lcub;9}.</s>
        </p>
        <p id="N14CDA" type="margin">
          <s id="N14CDC"><margin.target id="marg202"/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N14CE5" type="margin">
          <s id="N14CE7"><margin.target id="marg203"/>2.<emph type="italics"/>cor.<emph.end type="italics"/> 9. <lb/><emph type="italics"/>primihui<emph.end type="italics"/>^&lcub;9}.</s>
        </p>
        <p id="N14CF8" type="margin">
          <s id="N14CFA"><margin.target id="marg204"/>8.<emph type="italics"/>po&longs;thui<emph.end type="italics"/>^&lcub;9}</s>
        </p>
        <p id="N14D04" type="head">
          <s id="N14D06">SCHOLIVM.</s>
        </p>
        <p id="N14D08" type="main">
          <s id="N14D0A">C&ugrave;m &longs;it intentio Archimedis non nulla pertractare ad pa&shy;<lb/>rabolen &longs;pectantia; prim&ugrave;m iacit fundamentum, parabolas <lb/>nempe ita &longs;e habere, vt permutatim di&longs;tanti&aelig;, ex quibus <lb/>&longs;untcollocat&aelig;, &longs;e habent. </s>
          <s id="N14D12">&amp; <expan abbr="qu&atilde;uis">quamuis</expan> vniuer&longs;im, atqu&egrave; in om&shy;<lb/>nibus mutuam hanc conuenientiam ex dictis ex primo libro <lb/>depr&aelig;hendere liceat, hoc tamen loco peculiariter voluitad <lb/>huberiorem do ctrinam id ip&longs;um in parabolis demon&longs;trare.
<arrow.to.target n="marg205"/><lb/>&amp; quamuis in primo libro dixerit Archimedes magnitudi&shy;<lb/>nes &aelig;&que;ponderare, quando ita &longs;e habent inter &longs;e, ut di&longs;tan&shy;<lb/>ti&aelig; permutatim &longs;e habent; hocautem loco qu&aelig;rit <expan abbr="centr&utilde;">centrum</expan> gra<lb/>uitatis magnitudinis ex parabolis compo&longs;it&aelig;; non &longs;unt <expan abbr="tam&etilde;">tamen</expan> <lb/>propo&longs;itiones diuer&longs;&aelig;. </s>
          <s id="N14D33">nam &amp; in primo libro dum in demon<lb/>&longs;tratio ne qu&aelig;rit proportionem di&longs;tantiarum, o&longs;tendit, vbi <lb/>nam &longs;it centrum grauitatis magnitudinum. </s>
          <s id="N14D39">quare <expan abbr="qu&atilde;nis">quannis</expan> pro <lb/>po&longs;itiones videantur diuer&longs;&aelig;, non &longs;unt tamen diuer&longs;&aelig;, ete&shy;<lb/>nim vt po&longs;t tertiam primi libri propo&longs;itionem adnotauimus,
<pb xlink:href="077/01/132.jpg" pagenum="128"/>h&aelig;c plan&egrave;&longs;e con&longs;equuntur, vt exempli gratia in figura pun&shy;<lb/>ctum H centrum e&longs;t grauitatis magnitudinis ex vtri&longs;&que; <lb/>AB CD compo&longs;it&aelig;. </s>
          <s id="N14D4B">ergo AB, &amp; CD ex di&longs;tantijs HEHF <lb/>&aelig;&que;ponderant. </s>
          <s id="N14D4F">&amp; &egrave; contra. </s>
          <s id="N14D51">hoc e&longs;t AB CD &aelig;&que;ponde&shy;<lb/>rant ex di&longs;tantijs EH HF. ergo punctum H centrum e&longs;t <lb/>grauitatis magnitudinis ex vtri&longs;&que; AB CD compo&longs;rt&aelig;; <expan abbr="c&utilde;">cum</expan> <lb/>&longs;it EHF recta linea. </s>
          <s id="N14D5D">Solent autem mathematici aliquando <lb/>eandem propo&longs;itionem pluribusmedijs demon&longs;trare; idcirco <lb/>con&longs;iderandum e&longs;t, Archimedem in hac propo&longs;itione alio v&shy;<lb/>ti medio ad o&longs;tendendum punctum H centrum e&longs;ie graui&shy;<lb/>tatis, quo u&longs;us e&longs;t in &longs;exta propo&longs;itione primi libri. </s>
          <s id="N14D67">c&ugrave;m in pri <lb/>mo libro per diui&longs;ionem magnitudinum, diui&longs;io nem qu&egrave; di <lb/>&longs;tantiarum vniuer&longs;aliter domon&longs;tret centrum grauitatis ma&shy;<lb/>gnitudinum. </s>
          <s id="N14D6F">hoc autem loco per parallelogramma MN <lb/>NX parabolis &aelig;qualia, &amp; circa centra grauitatis EF con&longs;ti&shy;<lb/>tuta, in uenit centrum grauitatis magnitudinis ex vtri&longs;&que; pa <lb/>
<arrow.to.target n="marg206"/> rallelogrammis MN NX compo&longs;it&aelig;. </s>
          <s id="N14D7B">quod e&longs;t <expan abbr="quid&etilde;">quidem</expan> pun&shy;<lb/>ctum H. medium nemp&egrave; totius parallelogrammi MP. <lb/>quod idem punctum H centrum e&longs;t grauitatis vtriu&longs;&que; pa <lb/>raboles AB CD in EF collocat&aelig;. </s>
        </p>
        <p id="N14D87" type="margin">
          <s id="N14D89"><margin.target id="marg205"/>6.7.<emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N14D94" type="margin">
          <s id="N14D96"><margin.target id="marg206"/><emph type="italics"/>ex<emph.end type="italics"/> 9.<emph type="italics"/>&amp;<emph.end type="italics"/> 10 <lb/><emph type="italics"/>primihui<emph.end type="italics"/>^&lcub;9}.</s>
        </p>
        <p id="N14DAC" type="main">
          <s id="N14DAE">Ex his ob&longs;eruandum occurrit, hanc e&longs;&longs;e peculiarem metho <lb/>dum, qua po&longs;&longs;umus quorumlibet planorum &aelig;&que;pondera&shy;<lb/>tionem o&longs;tendere; hoc e&longs;t plana ex di&longs;tantijs eandem permu<lb/>tatim proportionem habentibus, vt eadem met plana, &aelig;&que;&shy;<lb/>ponderare; dum modo ip&longs;is &aelig;qualia parallelogramma con&longs;ti <lb/>tuere po&longs;&longs;imus. </s>
          <s id="N14DBA">ac propterea &longs;upponit Archimedes, nos po&longs;&longs;e <lb/>applicare ad rectam lineam &longs;pacium &aelig;quale &longs;pacio recta li&shy;<lb/>nea, rcctanguliqu&egrave; coni &longs;ectione contento. </s>
          <s id="N14DC0">quod <expan abbr="quid&etilde;">quidem</expan> &longs;pa&shy;<lb/>cium &longs;upponit parallelogram mum exi&longs;tere, c&ugrave;m pun&shy;<lb/>ctum E centrum &longs;it grauitatis &longs;pacij MN, e&longs;t F <lb/>&longs;pacij NX. punctum ver&ograve; H totius PM. qu&ograve;d &longs;i MN <lb/>NX &amp; MP non e&longs;&longs;ent parallelogramma, ne&que; puncta EFH <lb/>eorum centra grauitatis exi&longs;terent. </s>
          <s id="N14DD0">vt ex demon&longs;tranone pa&shy;<lb/>tet. </s>
          <s id="N14DD4">&longs;uppo&longs;uit tamen Archimedes nos po&longs;&longs;e applicare ad re&shy;<lb/>ctam lineam parallelogrammum &aelig;quale &longs;pacio recta linea, <lb/>rectanguliqu&egrave; coni&longs;ectione contento; quia duplici medio in
<pb xlink:href="077/01/133.jpg" pagenum="129"/>libro de quadratura paraboles, propo&longs;itione &longs;cilicet decima&longs;e <lb/>ptima, &amp; vige&longs;imaquarta, docuit quamlibet portionem recta <lb/>linea, rectanguliqu&egrave; coni &longs;ectione contentam &longs;e&longs;quitertiam <lb/>e&longs;&longs;e trianguli eandem ip&longs;i ba&longs;im habentis, &amp; <expan abbr="altitudin&etilde;">altitudinem</expan> &etail;qua <lb/>lem. </s>
          <s id="N14DEA">Ex qua propo&longs;itione facil&egrave; con&longs;tat nos parabol&etail; <expan abbr="&longs;paci&umacr;">&longs;pacium</expan> <lb/>ad rectam lineam applicare po&longs;&longs;e, vt propo&longs;itum fuit hoc <lb/>modo. </s>
        </p>
        <p id="N14DF4" type="head">
          <s id="N14DF6">PROBLEMA.</s>
        </p>
        <p id="N14DF8" type="main">
          <s id="N14DFA">Ad datam rectam lineam dat&etail; parabol&etail; &etail;quale parallelo&shy;<lb/>grammum applicare, ita vt data linea oppo&longs;ita <expan abbr="parallelogr&atilde;-mi">parallelogran&shy;<lb/>mi</expan> latera bi&longs;ariam diuidat. </s>
        </p>
        <figure id="id.077.01.133.1.jpg" xlink:href="077/01/133/1.jpg"/>
        <p id="N14E07" type="main">
          <s id="N14E09">Data &longs;it parabole <lb/>ABC, &longs;itqu&egrave; data recta <lb/>linea GK. oportet ad <lb/>GK <expan abbr="parallelogr&atilde;mum">parallelogrammum</expan> <lb/>applicare &aelig;quale por&shy;<lb/>tioni ABC, ita vt GK <lb/>bifariam diuidat oppo <lb/>&longs;ita parallelogram mi <lb/>latera. </s>
          <s id="N14E1F">Con&longs;tituatur &longs;u<lb/>per AC <expan abbr="tri&atilde;gul&utilde;">triangulum</expan> ABC, <lb/>qd ba&longs;im habeat AC, <lb/>eandem&que; portionis <lb/><expan abbr="altitudin&etilde;">altitudinem</expan>; quod <expan abbr="quid&etilde;">quidem</expan> <lb/>fiet, <expan abbr="inu&etilde;ta">inuenta</expan> diametro DB, qu&aelig; parabolen in B &longs;ecet, <expan abbr="iuncti&longs;&qacute;">iuncti&longs;&que;</expan>;
<arrow.to.target n="marg207"/><lb/>AB BC. eritvti&que; parabole ABC trianguli ABC &longs;e&longs;quitertia. <lb/>Ita&que; diuidatur AC in tria &etail;qualia, quarum vna pars &longs;it CH.
<arrow.to.target n="marg208"/><lb/>producaturqu&egrave; AC. fiatqu&egrave; CL ip&longs;i CH &etail;qualis<gap/> erit &longs;an&egrave; AL <lb/>ip&longs;ius AC &longs;e&longs;q uitertia. </s>
          <s id="N14E4E">Et obid (iuncta BL) erit triangulum <lb/>ABL trianguli ABC &longs;e&longs;quitertium. </s>
          <s id="N14E52">&longs;unt quipp&egrave; triangula ABL
<arrow.to.target n="marg209"/><lb/>ABC inter &longs;e, vt ba&longs;es AL AC. ac per con&longs;e&que;ns triangulum <lb/>ABL patabol&etail; ABC exi&longs;tit &etail;quale. </s>
          <s id="N14E5B">Applicetur ita&que; ad linea
<arrow.to.target n="marg210"/><lb/>GK <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> GS &etail;quale <expan abbr="tri&atilde;gulo">triangulo</expan> ABL. erit GS parabo&shy;
<pb xlink:href="077/01/134.jpg" pagenum="132"/>
<arrow.to.target n="fig65"/><lb/>l&aelig; ABC &etail;quale. </s>
          <s id="N14E73">deinceps ducatur NP ip&longs;i GK <lb/>&etail;quidi&longs;tans, qu&etail; bifariam diuidat oppo&longs;ita latera GR <lb/>KS. producanturqu&egrave; RG SK. fiantqu&egrave; GO KX &etail;&shy;<lb/>quales ip&longs;is GN KP. iungaturqu&egrave; OX; erit nimi-&shy;<lb/>rum parallelogram mum OP ip&longs;i GS &etail;quale. </s>
          <s id="N14E7D">qua&shy;<lb/>re parallelogram mum OP parabol&etail; ABC exi&longs;tit &etail;&shy;<lb/>quale. </s>
          <s id="N14E83">Applicatum e&longs;t igitur ad GK parallelogram&shy;<lb/>mum expo&longs;it&etail; parabol&etail; &etail;quale. </s>
          <s id="N14E87">lineaqu&egrave; GK paralle&shy;<lb/>logrammi OP bifariam diuidit oppo&longs;ita latera ON <lb/>XP. quod fieri oportebat. </s>
        </p>
        <p id="N14E8D" type="margin">
          <s id="N14E8F"><margin.target id="marg207"/>44. <emph type="italics"/><expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> <lb/>conicorum <lb/>Apoll.<emph.end type="italics"/></s>
        </p>
        <p id="N14E9F" type="margin">
          <s id="N14EA1"><margin.target id="marg208"/>17. 24. <emph type="italics"/>Ar <lb/>ch. </s>
          <s id="N14EAA">dquad. <lb/>patab.<emph.end type="italics"/></s>
        </p>
        <p id="N14EB0" type="margin">
          <s id="N14EB2"><margin.target id="marg209"/>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N14EBB" type="margin">
          <s id="N14EBD"><margin.target id="marg210"/><emph type="italics"/>ex<emph.end type="italics"/> 44.<emph type="italics"/>pri&shy;<lb/>mi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.134.1.jpg" xlink:href="077/01/134/1.jpg"/>
        <p id="N14ED1" type="main">
          <s id="N14ED3">Si in portione recta linea rectanguliqu&egrave; coni <lb/>&longs;ectione contenta triangulum in&longs;cribatur, <expan abbr="eand&etilde;">eandem</expan> <lb/>ba&longs;im cum portione habens, &amp; altitudinem &aelig;qua <lb/>lem: &amp; rur&longs;us in reliquis portionibus triangula in&shy;<lb/>&longs;cribantur, qu&aelig; ea&longs;dem ba&longs;es cum portionibus <lb/>habeant, &amp; altitudinem &aelig;qualem; &longs;emper qu&egrave; in <lb/>re&longs;iduis portionibus triangula eodem modo <lb/>in&longs;cribantur: figura, qu&aelig; in portione oritur, <lb/>plan&egrave; in&longs;cribi dicatur. </s>
          <s id="N14EE9">Patet quidem lineas
<pb xlink:href="077/01/135.jpg" pagenum="131"/>huius figur&aelig; in&longs;cript&aelig; angulos, qui &longs;unt vertici <lb/>portionis proximi, eo&longs;qu&egrave; deinceps coniungen&shy;<lb/>tes, ba&longs;i portionis &aelig;quidi&longs;tantes e&longs;&longs;e; bifariamqu&egrave; <lb/>&agrave; diametro portionis diuidi; diametrum ver&ograve; in <lb/>proportione diuidere numeris deinceps impari&shy;<lb/>bus. </s>
          <s id="N14EF9">vno deno minato ad verticem portionis. </s>
          <s id="N14EFB">Hoc <lb/>autem ordinate o&longs;ten&longs;um e&longs;t. </s>
        </p>
        <p id="N14EFF" type="head">
          <s id="N14F01">SCHOLIVM.</s>
        </p>
        <p id="N14F03" type="main">
          <s id="N14F05">Scopus Archimedis in hoc &longs;ecundo libio, vt initio primi <lb/>diximus, e&longs;t inuenire centrum grauitatis paraboles. </s>
          <s id="N14F09">&amp; vt de&shy;<lb/>ducatnos in hanc cognitionem, quadam vtitur figura rectili&shy;<lb/>nea in parabole in&longs;cripta, qu&etail; plurim&ugrave;m conducit, &amp; e&longs;t <expan abbr="t&atilde;">tam</expan> <lb/>quam medium ad inueniendum hoc grauitatis centrum. </s>
          <s id="N14F15">his <lb/>igitur verbis docet, quo modo in parabole in &longs;cribenda &longs;it h&etail;c <lb/>figura; in quibus multa quo &que; proponit tanquam &longs;it pro&shy;<lb/>po&longs;itio qu&aelig;dam; in qua multa &longs;int o&longs;tendenda. </s>
          <s id="N14F1D">quorum ta&shy;<lb/>m&etail;n demon&longs;trationem omi&longs;it, ac tanquam ab eo alibi de&shy;<lb/>mon&longs;tratam. </s>
          <s id="N14F23">Horum autem ex Apollonij Perg&etail;i conicis <lb/>demon&longs;trationem elicere quidem potui&longs;&longs;emus. </s>
          <s id="N14F27">at quoniam <lb/>Archimedes ip&longs;e non nulla ad h&aelig; c&longs;pectantia alijs in locis de&shy;<lb/>mon&longs;trauit ideo Archimedem per Archimedem declarare o&shy;<lb/>portunum magis nobis vi&longs;um e&longs;t. </s>
        </p>
        <p id="N14F2F" type="main">
          <s id="N14F31">Sit portio contenta recta linea, rectanguliqu&egrave; coni &longs;ectio&shy;<lb/>ne ABC, cuius diameter BD. Iunganturqu&egrave; AB BC, diuida&shy;<lb/>tur deinde AB bifariam in E, a quo ip&longs;i BD &aelig;quidi&longs;tans
<pb xlink:href="077/01/136.jpg" pagenum="132"/>ducatur EF, eritvti &que; punctum F vertex portionis AFB. <lb/>vt Archimedes demon&longs;trauit in libro de quadratura parabo&shy;<lb/>les propo&longs;itione decimaoctaua. </s>
          <s id="N14F3F">iungantur&que; AF FB. rur <lb/>fus bifariam diuidantur AF FB in punctis GH, &agrave; quibus <lb/>ip&longs;i BD ducantur &aelig;quidi&longs;tantes GI HK <gap/>b eandem cau&shy;<lb/>&longs;am erit punctum I vertex portionis AIF. K ver&ograve; portio&shy;<lb/>nis FKB. connectanturqu&egrave; AI IF FK KB. eademqu&egrave; pror <lb/>fus ratione ad alteram partem in&longs;cribantur triangula CLB <lb/>
<arrow.to.target n="fig66"/><lb/>CML, &amp; LNB. Prim&ugrave;m <expan abbr="quid&etilde;">quidem</expan> triangulum ABC dicitur <lb/>plan&egrave; in&longs;criptum, vt Archimedes ip&longs;e infra in demon&longs;tratio&shy;<lb/>nibus quint&aelig;, &longs;ext&aelig;, &amp; octau&aelig; propo&longs;itionis nominat. </s>
          <s id="N14F5C">Dein<lb/>de figura AFBLC, figuraqu&egrave; AIFKBNLMC dicuntur in <lb/>portione plan&egrave; in&longs;cript&aelig;. </s>
          <s id="N14F62">figuraqu&egrave; AFBLC vna cum AC <lb/><expan abbr="pentagon&utilde;">pentagonum</expan> in portione plan&egrave; <expan abbr="in&longs;cript&utilde;">in&longs;criptum</expan> dici <expan abbr="p&otilde;t">pont</expan>. vt Archime <lb/>des in &longs;ecunda parte demon&longs;trationis quint&aelig; propo&longs;itionis <lb/>huius libri nuncupat. </s>
          <s id="N14F75">ide&ograve;qu&egrave; erit AIFKBNLMC nonago&shy;<lb/>num in portione plan&egrave; in&longs;criptum. </s>
          <s id="N14F79">&amp; ita in alijs. <expan abbr="Connect&atilde;">Connectam</expan>
<pb xlink:href="077/01/137.jpg" pagenum="133"/>tur KN FL IM, qu&aelig; diametrum BD &longs;ecent in punctis <lb/>STV. o&longs;tendendum e&longs;t, lineas KN FL IM ba&longs;i AC &etail;qui <lb/>di&longs;tantes e&longs;&longs;e. </s>
          <s id="N14F87">deinde diametrum BD lineas KN FL IM <lb/>bifariam in punctis STV diuidere po&longs;tremo lineas KN F<gap/><lb/>IM ita diametrum BD di&longs;pe&longs;cere, vt po&longs;ito vno BS, linea ST <lb/>&longs;it tria, TV quin&que;; &amp; VD &longs;eptem. </s>
          <s id="N14F90">Producantur FE KH <lb/>ad RX. quoniam enim FR e&longs;t &aelig;quid<gap/>tans BD, erit AE ad
<arrow.to.target n="marg211"/><lb/>EB, vt AR ad RD; e&longs;t&que; AE ip&longs;i EB &aelig;qualis ergo AR i&shy;<lb/>p&longs;i RD &aelig;qualis exi&longs;tit. </s>
          <s id="N14F9D">eodem qu&egrave; modo o&longs;tendetur FX &aelig;&shy;<lb/>qualem e&longs;&longs;e XT. quandoquidem e&longs;t FX ad XT, vt FH ad <lb/>HB. &longs;imiliterqu&egrave; ad alteram partem, exi&longs;tentibus LO NP i&shy;<lb/>p&longs;i BD &aelig;quidi&longs;tantibus, erit DO ip&longs;i OC &aelig;qualis, &amp; TP <lb/>ip&longs;i PL. quod quidem eodem pror&longs;us modo demon&longs;trabi&shy;<lb/>tur. </s>
          <s id="N14FA9">Quoniam autem AC bifariam &agrave; diametro diuiditur in <lb/>puncto D, erit DR ip&longs;i DO &aelig;qualis, c&ugrave;m vnaqu&aelig;&que; &longs;it <lb/>dimidia ip&longs;arum AD DC &aelig;qualium. </s>
          <s id="N14FAF">e&longs;t igitur RD dimidia <lb/>ip&longs;ius AD, qu&aelig; dimidia e&longs;t ba&longs;is AC. quod idem euenit ip&longs;i <lb/>DO. quare BD &longs;e&longs;quitertia e&longs;t ip&longs;ius FR, &amp; ip&longs;ius LO, ex de&shy;<lb/>cimanona Archimedis de quadratura paraboles. </s>
          <s id="N14FB7">ac propterea <lb/>eandem habet proportionem BD ad FR, quam ad LO. vnde
<arrow.to.target n="marg212"/><lb/>&longs;equitur FR &aelig;qualem e&longs;&longs;e ip&longs;i LO. &amp; obid FL ip&longs;i AC <expan abbr="&aelig;-quidi&longs;tant&etilde;">&aelig;&shy;<lb/>quidi&longs;tantem</expan> e&longs;&longs;e. </s>
          <s id="N14FC6">&amp; FT ip&longs;i RD, &amp; TL ip&longs;i DO &etail;qualem. <lb/>vnde FT ip&longs;i TL &etail;qualis exi&longs;tit. </s>
          <s id="N14FCA">eadem qu&egrave;ratione pror&longs;us in <lb/>portione FBL o&longs;tendetur KN ip&longs;i FL, ac per con&longs;e&que;ns i&shy;<lb/>p&longs;i AC &etail;quidi&longs;tantem e&longs;&longs;e. </s>
          <s id="N14FD0">&amp; KS ip&longs;i SN &aelig;qualem exi&longs;te&shy;<lb/>re. </s>
          <s id="N14FD4">Producatur IG ad Z, qu&aelig; ip&longs;am AB &longs;ecet in 9. linea ve&shy;<lb/>r&ograve; LO &longs;ecet BC in <expan abbr="q;">&que;</expan> ductaqu&egrave; MY ip&longs;i BD &aelig;quidi&longs;tans <lb/>ip&longs;am &longs;ecet BC in <foreign lang="greek">a</foreign>. &amp; quoniam IZ e&longs;t &aelig;quidi&longs;tans FR, e&shy;<lb/>rit AG ad GF, ut A9 ad 9E, &amp; AZ ad ZR. &amp; e&longs;t AG ip&longs;i
<arrow.to.target n="marg213"/><lb/>GF &aelig;qualis, erit igitur A9 ip&longs;i 9E, &amp; AZ ip&longs;i ZR &aelig;quaiis. <lb/>Eodemqu&egrave; modo o&longs;tendetur C<foreign lang="greek">a</foreign> ip&longs;i <foreign lang="greek">a</foreign>Q, &amp; CY ip&longs;i YO &etail;&shy;<lb/>qualem e&longs;&longs;e. </s>
          <s id="N14FF5">quo niam autem in portione AFB a dimidia ba&longs;i <lb/>ducta e&longs;t LF, &agrave; puncto autem 9, hoc e&longs;t &agrave; dimidia dimidi&etail; ba <lb/>&longs;is AB (e&longs;t enim E9 dimidia ip&longs;ius AE, qu&aelig; dimidia e&longs;t ba&longs;is <lb/>AB) ducta e&longs;t 9I diametro &aelig;quidi&longs;tans, erit EF &longs;e&longs;quitertiai&shy;<lb/>p&longs;ius I9 pari&que; ratione o&longs;tendetur QL &longs;e&longs;quitereiam e&longs;&longs;e i&shy;<lb/>p&longs;ius M<foreign lang="greek">a</foreign> quare vt FE ad I9, ita LQ ad M<foreign lang="greek">a</foreign>. ob&longs;imilitudinem
<pb xlink:href="077/01/138.jpg" pagenum="134"/>
<arrow.to.target n="marg214"/> autem triangulorum ABD AER ita e&longs;t BD ad ER, vt DA <lb/>ad AR. eadem&que;iatione ita &longs;ehabet BD ad QO, vt DC <lb/>ad CO. Sed vt DA ad AR, ita e&longs;t DC ad CO, e&longs;t quip <lb/>pe DA ip&longs;ius AR dupla, veluti DC ip&longs;ius CO. quare i&shy;<lb/>
<arrow.to.target n="marg215"/> ta erit BD ad ER, vt BD ad QO. ac propterea ER ip&longs;i <lb/>
<arrow.to.target n="marg216"/> QO &etail;qualis exi&longs;tit. </s>
          <s id="N15023">o&longs;ten&longs;a ver&ograve; e&longs;t RF &etail;qualis OL, reli&shy;<lb/>quaigitur EF reliqu&aelig; QL e&longs;t &aelig;qualis, quia ver&ograve; ita e&longs;t FE <lb/>
<arrow.to.target n="marg217"/> ad I9, vt QL ad M<foreign lang="greek">a</foreign>, erit permutando FE ad QL, vt I9 <lb/>
<arrow.to.target n="fig67"/><lb/>ad M<foreign lang="greek">a</foreign>. &longs;untqu&egrave; FE QL &etail;quales, ergo I9 ip&longs;i M<foreign lang="greek">a</foreign> &etail;qua&shy;<lb/>lis exi&longs;tit. </s>
          <s id="N15042">quoniam autem ob triangu&longs;oium &longs;imilitudinem <lb/>AER A9Z, ita e&longs;t AR ad AZ, vt ER ad 9Z. ob &longs;imili&shy;<lb/>tudinem vero triangulorum QOC <foreign lang="greek">a</foreign>YC ita e&longs;t CO ad CY, <lb/>vt QO ad <foreign lang="greek">a</foreign>Y: &amp; e&longs;t RA ad AZ, vt OC ad CY, c&ugrave;m <lb/>
<arrow.to.target n="marg218"/> vtr&etail;&que; in dupla exi&longs;tant proportione; e<gap/>t ER ad 9Z, vt <lb/>QO ad <foreign lang="greek">a</foreign>Y. &amp; permutando ER ad QO vt 9Z ad <foreign lang="greek">a</foreign>Y. e&longs;t <lb/>vero ER ip&longs;i QO, &aelig;qualis, ergo 9Z ip&longs;i <foreign lang="greek">a</foreign>Y &etail;qualis exi&longs;tit. </s>
          <s id="N1506A">at <lb/>vero o&longs;ten&longs;a e&longs;t I9 &etail;qualis M<foreign lang="greek">a</foreign>; to ta igitur IZ ip&longs;i MY e&longs;t &etail;-
<pb xlink:href="077/01/139.jpg" pagenum="135"/>&aelig;qualis, qu&aelig; c&ugrave;m &longs;intip&longs;i BD &aelig;quidi&longs;tantes, erunt &amp; inter &longs;e&shy;<lb/>&longs;e parallel&aelig;. </s>
          <s id="N15078">quare IM ip&longs;i AC e&longs;t &aelig;quidi&longs;tans. </s>
          <s id="N1507A">Quoniam
<arrow.to.target n="marg219"/> ita&shy;<lb/>&que; AR e&longs;t &aelig;qualis CO, &amp; horum dimidia, hoc e&longs;t RZ ip&longs;i <lb/>OY &aelig;qualis erit. </s>
          <s id="N15084">atqui DR e&longs;t ip&longs;i DO &aelig;qualis; ergo DZ ip&longs;i <lb/>DY exi&longs;tit &aelig;qualis. </s>
          <s id="N15088">ip&longs;i ver&ograve; DZ e&longs;t &aelig;qualis IV, &amp; ip&longs;i DY &aelig;&shy;<lb/>qualis VM. eruntigitur IV VM inter &longs;e equales. </s>
          <s id="N1508C">Iam ita&que;
<arrow.to.target n="marg220"/><lb/>o&longs;ten&longs;um e&longs;t, lineas KN FL IM, qu&etail; coniunguntangulos fi <lb/>gur&aelig; in parabole plan&egrave; in&longs;cript&aelig;, ip&longs;i AC &aelig;quidi&longs;tantes e&longs;&longs;e. <lb/>Diametrum qu&egrave; BD ip&longs;as in punctis STV bifariam di&longs;pe&longs;cere. </s>
        </p>
        <p id="N15097" type="margin">
          <s id="N15099"><margin.target id="marg211"/>2. <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N150A2" type="margin">
          <s id="N150A4"><margin.target id="marg212"/>9. <emph type="italics"/>quinti. <lb/>ex<emph.end type="italics"/> 33.34 <lb/><emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N150B6" type="margin">
          <s id="N150B8"><margin.target id="marg213"/>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N150C1" type="margin">
          <s id="N150C3"><margin.target id="marg214"/><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N150D1" type="margin">
          <s id="N150D3"><margin.target id="marg215"/>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N150DC" type="margin">
          <s id="N150DE"><margin.target id="marg216"/>9. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N150E7" type="margin">
          <s id="N150E9"><margin.target id="marg217"/>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N150F2" type="margin">
          <s id="N150F4"><margin.target id="marg218"/><emph type="italics"/>ex<emph.end type="italics"/> 11.<emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> 16.<emph type="italics"/>qu<gap/>u<emph.end type="italics"/></s>
        </p>
        <p id="N1510C" type="margin">
          <s id="N1510E"><margin.target id="marg219"/>33.<emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N15117" type="margin">
          <s id="N15119"><margin.target id="marg220"/>34.<emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.139.1.jpg" xlink:href="077/01/139/1.jpg"/>
        <figure id="id.077.01.139.2.jpg" xlink:href="077/01/139/2.jpg"/>
        <p id="N1512A" type="main">
          <s id="N1512C">Quoniam ita&que; in portione FBL &agrave; dimidia ba&longs;i ducta e&longs;t <lb/>TB, a dimidia ver&ograve; dimidi&aelig; ba&longs;is ducta e&longs;t XK, erit BT
<arrow.to.target n="marg221"/> &longs;e&longs;&shy;<lb/>quitertia ip&longs;ius KX, hoc e&longs;t ip&longs;ius ST. e&longs;t enim KT parallelo&shy;<lb/>grammum, &amp; ST ip&longs;i KX &aelig;qualis. </s>
          <s id="N15138">Si igitur ponatur BT <lb/>quattuor, erit ST tria, &amp; BS vnum. </s>
          <s id="N1513C">&longs;imiliter quoniam BD
<arrow.to.target n="marg222"/><lb/>&longs;e&longs;quitertia e&longs;t ip&longs;ius FR, hoc e&longs;t ip&longs;ius TD, c&ugrave;m &longs;it TD ip&longs;i <lb/>FR &etail;qualis. </s>
          <s id="N15145">&longs;i ita &que; ponatur BD &longs;exdecim, erit vnaqu&aelig;&que; <lb/>FR TD duodecim. </s>
          <s id="N15149">&amp; TB quattuor, vt po&longs;itum fuit. <expan abbr="Quoni&atilde;">Quoniam</expan> <lb/>autem (vt diximus) e&longs;t BD ad ER, vt DA ad AR, erit BD du&shy;<lb/>pla ip&longs;ius RE. quare &longs;i BD e&longs;t &longs;exdecim, erit RE octo. </s>
          <s id="N15153">&amp; quo&shy;<lb/>niam e&longs;t FR duodecim, erit EF quatuor. </s>
          <s id="N15157">e&longs;t autem FE ip&longs;ius <lb/>I9 &longs;e&longs;quitertia, erit igitur I9 tria. </s>
          <s id="N1515B">&amp; quoniam e&longs;t ER ad 9Z, vt <lb/>RA ad AZ, erit ER dupla ip&longs;ius 9Z. ac propterea erit 9Z quat <lb/>tuor, cum &longs;it ER octo, &amp; e&longs;t 9I tria, tota ergo IZ, hoc e&longs;t DV, <lb/>&longs;eptem exi&longs;tet. </s>
          <s id="N15163">&longs;ed quoniam e&longs;t DT duodecim, cuius pars <lb/>DV e&longs;t &longs;eptem, eritreliqua VT quin&que;. </s>
          <s id="N15167">Po&longs;ito igitur BS v&shy;<lb/>no, erit ST tria, TV quin&que;, &amp; VD &longs;eptem. </s>
          <s id="N1516B">quod erat quo&shy;<lb/>&que; demon&longs;trandum. </s>
          <s id="N1516F">Et h&aelig;c &longs;unt qu&etail; ab Archimede pro&shy;<lb/>po&longs;ita fucrant. </s>
        </p>
        <p id="N15173" type="margin">
          <s id="N15175"><margin.target id="marg221"/>19.<emph type="italics"/>Archi&shy;<lb/>medis de <lb/>quad. </s>
          <s id="N15180">pa&shy;<lb/>rab.<emph.end type="italics"/></s>
        </p>
        <p id="N15186" type="margin">
          <s id="N15188"><margin.target id="marg222"/>34. <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p id="N15191" type="main">
          <s id="N15193">Ex his tamen nonnulla quo&que; colligemus ad ea, qu&aelig; &longs;e&shy;<lb/>quuntur nece&longs;&longs;aria. </s>
          <s id="N15197">ac prim&ugrave;m quidem con&longs;tat BD quadru&shy;<lb/>plam e&longs;&longs;e ip&longs;ius BT, &amp; ip&longs;ius FE. </s>
        </p>
        <pb xlink:href="077/01/140.jpg" pagenum="136"/>
        <p id="N1519E" type="main">
          <s id="N151A0">O&longs;ten&longs;um e&longs;t enim BD &longs;exdecim e&longs;&longs;e, &amp; BT quatuor, &amp; FE <lb/>itidem quatuor exi&longs;tere. </s>
          <s id="N151A4">Ex demon&longs;tratione autem Archime <lb/>dis decim&aelig; non&aelig; ptopo&longs;itionis de quadratura paraboles cla&shy;<lb/>r&egrave; elicitur BD quadruplam e&longs;&longs;e ip&longs;ius BT. </s>
        </p>
        <p id="N151AA" type="main">
          <s id="N151AC">Ex quibus etiam &longs;equitur FE QL inter &longs;e &aelig;quales e&longs;&longs;e. </s>
          <s id="N151AE">am&shy;<lb/>bo enim &longs;unt, vt quatuor. </s>
        </p>
        <figure id="id.077.01.140.1.jpg" xlink:href="077/01/140/1.jpg"/>
        <p id="N151B5" type="main">
          <s id="N151B7">Pr&aelig;terea o&longs;tendendum e&longs;t triangulum AFB <expan abbr="tri&atilde;gulo">triangulo</expan> BLC <lb/>&etail;quale e&longs;&longs;e, portionem qu&egrave; paraboles AFB portiom BLC &etail;qua <lb/>lem. </s>
          <s id="N151C1">Ampli&ugrave;s triangulum AIF triangulo CML, &amp; portio&shy;<lb/>nem AIF portioni CML &aelig;qualem e&longs;&longs;e, &amp; reliqua triangula <lb/>reliquis triangulis, acportiones portionibus &etail;quales e&longs;&longs;e. </s>
        </p>
        <p id="N151C7" type="main">
          <s id="N151C9">Ex vige&longs;ima prima propo&longs;itione Archimedis de quadratu&shy;<lb/>ra paraboles triangulum ABC vniu&longs;cuiu&longs;&que; trianguli AFB <lb/>
<arrow.to.target n="marg223"/> BLC e&longs;t <expan abbr="octupl&utilde;">octuplum</expan>. ergo ad ambo <expan abbr="eand&etilde;">eandem</expan> <expan abbr="h&etilde;t">hent</expan> <expan abbr="proportion&etilde;">proportionem</expan>. qua <lb/>re triangula AFB BLC inter &longs;e &longs;unt &etail;qualia. </s>
          <s id="N151E5">At vero <expan abbr="quoni&atilde;">quoniam</expan>
<pb xlink:href="077/01/141.jpg" pagenum="137"/>portio AFB trianguli AFB e&longs;t &longs;e&longs;quitertia, &que;madmodum
<arrow.to.target n="marg224"/><lb/>portio BLC trianguli BLC, eritportio AFB ad triangulum <lb/>AFB, vt portio CLB ad triangulum CLB, &amp; permutando <lb/>portio AFB ad portionem CLB, vt triangulum AFB ad
<arrow.to.target n="marg225"/><lb/>ip&longs;um CLB <expan abbr="tri&atilde;gula">triangula</expan> ver&ograve; &longs;unt &aelig;qualia; ergo portiones AFB <lb/>CLB inter &longs;e &longs;unt &aelig;quales. </s>
          <s id="N15203">Eademqu&egrave; ratione <expan abbr="triangul&utilde;">triangulum</expan> AFB <lb/>octuplum e&longs;t trianguli AIF, &amp; triangulum CLB octuplum <lb/>ip&longs;ius CML. vnde triangula AIF CML &longs;unt &aelig;qualia. </s>
          <s id="N1520D">et ea&shy;<lb/>rum quo&que; portiones AIF CML &longs;unt &aelig;quales, &longs;iquidem <lb/>&longs;unt triangulorum &longs;e&longs;quiterti&aelig;. </s>
          <s id="N15213">Et hoc modo reliqua trian&shy;<lb/>gula FKB LNB, &amp; portiones FKB LNB <expan abbr="o&longs;tend&etilde;tur">o&longs;tendentur</expan> &aelig;qua&shy;<lb/>les. </s>
          <s id="N1521D">c&ugrave;m &longs;it triangulum FBL dictorum triangulorum octu&shy;<lb/>plum. </s>
          <s id="N15221">quod oportebat quo&que; demon&longs;trate. </s>
        </p>
        <p id="N15223" type="margin">
          <s id="N15225"><margin.target id="marg223"/>9. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1522E" type="margin">
          <s id="N15230"><margin.target id="marg224"/>17.24. A<emph type="italics"/>r <lb/>chimedis <lb/>de quad. <lb/>parab.<emph.end type="italics"/></s>
        </p>
        <p id="N1523F" type="margin">
          <s id="N15241"><margin.target id="marg225"/>16. <emph type="italics"/>quimi<emph.end type="italics"/><lb/>21.<emph type="italics"/>Archi&shy;<lb/>medis de <lb/>quad. </s>
          <s id="N15253">pa&shy;<lb/>rab.<emph.end type="italics"/></s>
        </p>
        <p id="N15259" type="main">
          <s id="N1525B">His demon&longs;tratis &longs;equitur Archimedes qua&longs;i connectens &longs;e <lb/>&que;ntem propo&longs;itionem cumijs, qu&aelig; &longs;uppo&longs;ita &longs;unt, inqui&shy;<lb/>ens, <emph type="italics"/>&longs;i autem &amp; in portione<emph.end type="italics"/> &amp;c. </s>
        </p>
        <p id="N15267" type="head">
          <s id="N15269">PROPOSITIO. II.</s>
        </p>
        <p id="N1526B" type="main">
          <s id="N1526D">Si autem &amp; in portione rectalinea, rectangu&shy;<lb/>li&que; coni &longs;ectione contenta, figura rectilinea pla <lb/>ne in&longs;cribatur, in&longs;cript&aelig; figur&aelig; centrum grauita&shy;<lb/>tis erit in diametro portionis. </s>
        </p>
        <pb xlink:href="077/01/142.jpg" pagenum="138"/>
        <p id="N15278" type="main">
          <s id="N1527A"><emph type="italics"/>Sit portio ABC, qualis dicta e&longs;t, &amp; in ip&longs;a plan&egrave; in&longs;cribatur recti&shy;<lb/>linea figura AEFGBHIKC. portionis ver&ograve; diameter &longs;it BD. <expan abbr="o&longs;t&etilde;-">o&longs;ten-</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="marg226"/> <emph type="italics"/>dendum e&longs;t, rectiline&aelig; figur&aelig; centrum grauitatise&longs;&longs;e in linea BD.<emph.end type="italics"/> <expan abbr="i&utilde;">ium</expan> <lb/>gantur GH FI EK. qu&etail; ip&longs;i AC, &amp; inter &longs;e &etail;quidi&longs;tantes <lb/>erunt. </s>
          <s id="N15299">h&etail; ver&ograve; line&aelig; diametrum BD &longs;ecent in punctis NML <lb/>
<arrow.to.target n="fig68"/><lb/><emph type="italics"/>Quoniam enim<emph.end type="italics"/> line&aelig; GH FI EK bifariam &longs;unt &agrave; diame&shy;<lb/>tro BD diui&longs;&aelig; in punctis NML, trapezium AEKC duas <lb/>
<arrow.to.target n="marg227"/> habebit line as &aelig;quidi&longs;tantes AC EK, quas bifariam diuidit <lb/>DL, quare <emph type="italics"/>trapezii AEKC centrum grauitatis est in LD. at<emph.end type="italics"/> ob <lb/>eandem cau&longs;am <emph type="italics"/>trapezii EFIK centrum est in ML; trapezii ver&ograve; <lb/>FGHI centrum est in MN.<emph.end type="italics"/> line&aelig; enim LM MN bifariam <lb/>
<arrow.to.target n="marg228"/> diuidunt parallela latera EK FI GH, <emph type="italics"/>&longs;ed &amp; trianguli etiam <lb/>GBH centrum grauitatis e&longs;t in BN.<emph.end type="italics"/> quipp&egrave; c&ugrave;m BN ip&longs;am <lb/>GH bifariam diuidat. <emph type="italics"/>per&longs;picuum e&longs;t totius rectiline&aelig; figur&aelig;<emph.end type="italics"/><lb/>AEFGBHIKC <emph type="italics"/>centrum grauitatis e&longs;&longs;e in linea BD.<emph.end type="italics"/> quod de&shy;<lb/>mon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/143.jpg" pagenum="139"/>
        <p id="N152E3" type="margin">
          <s id="N152E5"><margin.target id="marg226"/><emph type="italics"/><expan abbr="exdem&otilde;">exdemom</expan> <lb/>stratis.<emph.end type="italics"/></s>
        </p>
        <p id="N152F2" type="margin">
          <s id="N152F4"><margin.target id="marg227"/>15. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N152FF" type="margin">
          <s id="N15301"><margin.target id="marg228"/>13. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.143.1.jpg" xlink:href="077/01/143/1.jpg"/>
        <p id="N15310" type="head">
          <s id="N15312">SCHOLIVM.</s>
        </p>
        <p id="N15314" type="main">
          <s id="N15316">Ecce q&uacute;o Archimedes incipit inue&longs;tigare centrum graui<lb/>tatis paraboles. </s>
          <s id="N1531A">nam ex hoc, quod o&longs;tendit centrum grauita&shy;<lb/>tis figur&aelig; in portione plan&egrave; in&longs;cript&aelig; e&longs;&longs;e in diametro por&shy;<lb/>tionis, &longs;tatim colliget in quarta propo&longs;itione centrum graui&shy;<lb/>tatis paraboles in diametro quo&que; ip&longs;ius portionis exi&longs;tere. <lb/>interponit autem Archimedes &longs;e&que;ntem propo&longs;itionem. <expan abbr="n&atilde;">nam</expan> <lb/>antequam inueniat centrum grauitatis paraboles, opus habet <lb/>prius o&longs;tendere centra grauitatis duarum, &amp; vt ita dicam om <lb/>nium parabol<gap/>rum diametros in eadem proportione &longs;ecare. <lb/>ad quod demon&longs;trandum, hanc <expan abbr="pa&longs;&longs;ion&etilde;">pa&longs;&longs;ionem</expan> figuris plan&egrave; in&longs;cri&shy;<lb/>ptis pri&ugrave;s accidere <expan abbr="o&longs;t&etilde;dit">o&longs;tendit</expan>. potui&longs;&longs;etqu&egrave; Archimedes pri&ugrave;s quar <lb/>tam propo&longs;itionem o&longs;tendere, quam tertiam; &longs;e&que;ntem ve&shy;<lb/>r&ograve; propo&longs;itionem immediat&egrave; po&longs;uit po&longs;t &longs;ecundam, ordo e&shy;<lb/>nim &longs;ic po&longs;tulat. </s>
          <s id="N15342">etenim amb&aelig; deijs pertractant, qu&aelig; rectili&shy;<lb/>neis figuris plane in&longs;criptis accidunt. </s>
          <s id="N15346">Pr&etail;terea earum demon <lb/>&longs;trationes fer&egrave; circa eadem ver&longs;antur, c&ugrave;m ijsdem rectis lineis <lb/>in portionibus eodem modo ductis vtantur; ob &longs;e&que;ntis ve&shy;<lb/>r&ograve; propo&longs;itionis intelligentiam h&etail;c pri&ugrave;s o&longs;tendemus. </s>
        </p>
        <figure id="id.077.01.143.2.jpg" xlink:href="077/01/143/2.jpg"/>
        <p id="N15351" type="head">
          <s id="N15353">LEMMA I.</s>
        </p>
        <p id="N15355" type="main">
          <s id="N15357">Eandem habeat proportionem AB ad CD, quam habet <lb/>GH ad KL. CD ver&ograve; ad EF <expan abbr="e&atilde;">eam</expan>, <expan abbr="qu&atilde;">quam</expan> habet kL ad MN. &longs;intqu&egrave;
<pb xlink:href="077/01/144.jpg" pagenum="136"/>AB CD EF inter &longs;e &etail;quid&longs;tantes. </s>
          <s id="N15367">&longs;imiliter GH KL MN <lb/>&aelig;quidi&longs;tantes, &longs;intantem duct&aelig; BDF HLN rect&aelig; line&aelig;; &longs;it&shy;<lb/>qu&egrave; BD ad DF, vt HL ad LN. &longs;itqu&egrave; maior AB qu&agrave;m <lb/>CD, &amp; CD, qu&agrave;m EF. vnde erit quoqu&egrave; GH maior KL, <lb/>&amp; KL, quam MN. iuncti&longs;qu&egrave; AC CE, &amp; GK KM. <lb/>Dico &longs;pacium ACDB ad &longs;pacium CEFD eandem habere <lb/>proportionem, quam &longs;pacium GKLH ad &longs;pacium KMNL. </s>
        </p>
        <figure id="id.077.01.144.1.jpg" xlink:href="077/01/144/1.jpg"/>
        <p id="N15378" type="main">
          <s id="N1537A">Producantur AC CE, qu&aelig; cum BF conueniant in OP. <lb/>product&aelig;qu&egrave; GK KM cum HN conueniant in QR. <lb/>concurrentenim, quoniam CD KL &longs;unt minores ip&longs;is AB <lb/>
<arrow.to.target n="marg229"/> GH, &amp; EF MN minores ip&longs;is CD KL. Fiatqu&egrave; vt AB <lb/>ad CD, ita CD ad V. &amp; vt GH ad kL, ita KL ad X. <lb/>deinceps CD ad EF, ita EF ad Y. &amp; vt KL ad MN, <lb/>ita MN ad Z. Quoniam igitur triangulum ABO &longs;imile <lb/>e&longs;t triangulo CDO, c&ugrave;m &longs;it CD &aelig;quidi&longs;tansip&longs;i AB. ha <lb/>
<arrow.to.target n="marg230"/> bebit triangulum ABO ad CDO, proportionem, quam ha <lb/>bet AB ad CD duplicatam. </s>
          <s id="N15396">hoc e&longs;t quam hab et AB ad <lb/>V. Eodemqu&egrave; modo o&longs;tendetur <expan abbr="triangul&utilde;">triangulum</expan> GHQ ad KLQ <lb/>ita e&longs;&longs;e, vt GH ad X<gap/> quia ver&ograve; AB CD V ita &longs;e <expan abbr="hab&etilde;t">habent</expan>, <lb/>
<arrow.to.target n="marg231"/> vt GH kL X, erit ex &aelig;quali AB ad V, &amp; GH ad X. <lb/>triangulum igitur ABO eandem habet proportionem ad
<pb xlink:href="077/01/145.jpg" pagenum="129"/>CDO, quam triangulum GHQ ad <expan abbr="KLq.">KL&que;</expan> quare diuiden&shy;<lb/>do &longs;pacium ACDB ad triangulum CDO e&longs;t, vt &longs;pacium
<arrow.to.target n="marg232"/><lb/>GKLH ad triangulum <expan abbr="kLq.">kL&que;</expan> Rur&longs;us quoniam ob triangu<lb/>lorum &longs;imilitudinem ABO CDO, ita e&longs;t AB ad CD, vt
<arrow.to.target n="marg233"/><lb/>BO ad OD. &longs;imiliter ob &longs;imilitudinem <expan abbr="triangulor&utilde;">triangulorum</expan> GHQ <lb/>KLQ ita e&longs;t GH ad kL, vt HQ ad QL. &amp; e&longs;t AB ad CD, <lb/>vt GH ad KL, erit BO ad OD, vt HQ ad QL. &amp;
<arrow.to.target n="marg234"/> diui&shy;<lb/>dendo BD ad DO, vt HL ad <expan abbr="Lq.">L&que;</expan> deinde <expan abbr="conuert&etilde;do">conuertendo</expan> DO <lb/>ad DB, vt LQ ad LH. &amp; e&longs;t BD ad DF, vt HL ad LN, erit
<arrow.to.target n="marg235"/><lb/>ex &etail;quali DO ad DF, vt LQ ad LN. Quoniam autem &longs;imi <lb/>lium triangulorum CDP EFP latus CD ad latus EF ita &longs;e <lb/>habet, vt DP ad PF. &longs;imiliter exi&longs;tentibus &longs;imilibus triangu<lb/>lis KLR MNR ita e&longs;t KL ad MN, vt LR ad RN, &amp; vt CD <lb/>ad EF, ita e&longs;t KL ad MN, erit DP ad PF, vt LR ad RN.
<arrow.to.target n="marg236"/><lb/>&amp; per conuer&longs;ionem rationis PD ad DF, vt RL ad LN. &amp; <lb/>conuertendo DF ad DP, vt LN ad LR. diximus <expan abbr="aut&etilde;">autem</expan> OD <lb/>ad DF ita e&longs;&longs;e, vt QL ad LN, &amp; e&longs;t DF ad DP, vt LN ad <lb/>LR. ergo ex &etail;quali erit OD ad DP, vt QL ad LR. At ver&ograve;
<arrow.to.target n="marg237"/><lb/>quoniam ita e&longs;t OD ad DP, vt triangulum OCD ad PCD, <lb/>&amp; vt QL ad LR, ita e&longs;t triangulum QKL ad <expan abbr="triangul&utilde;">triangulum</expan> RKL, <lb/>erit OCD ad PCD, vt QKL ad RKL. Quoniam <expan abbr="aut&etilde;">autem</expan> <expan abbr="tri&atilde;">triam</expan> <lb/>gula CDP EFP &longs;unt &longs;imilia, triangulum CDP ad triangulum
<arrow.to.target n="marg238"/><lb/>EFP proportionem habebit, quam CD ad EF duplicatam, <lb/>hoc e&longs;t quam habet CD ad Y, c&ugrave;m &longs;int CD EF Y propor&shy;<lb/>tionales. </s>
          <s id="N1541C">&longs;imiliter ob triangulorum KLR MNR &longs;imilitudi&shy;<lb/>nem triangulum KLR ad MNR, ita erit vt KL ad Z, e&longs;t au&shy;<lb/>tem CD ad Y, vt KL ad Z, erit igitur <expan abbr="tri&atilde;gulum">triangulum</expan> CDP ad <lb/>EFP, vt KLR ad MNR, &amp; diuidendo <expan abbr="&longs;paci&utilde;">&longs;pacium</expan> CEFD ad trian
<arrow.to.target n="marg239"/><lb/>gulum EFP, vt &longs;pacium KMNL ad triangulum MNR. &amp; <expan abbr="c&otilde;">com</expan>
<arrow.to.target n="marg240"/><lb/>uertendo triangulum EFP ad &longs;pacium CEFD, vt <expan abbr="triangul&utilde;">triangulum</expan> <lb/>MNR ad &longs;pacium KMNL. Ita&que; quoniam o&longs;ten&longs;um e&longs;t i&shy;<lb/>ta e&longs;&longs;e &longs;pacium ACDB ad triangulum CDO, vt &longs;pacium <lb/>GKLH ad triangulum <expan abbr="KLq.">KL&que;</expan> &amp; vt <expan abbr="triangul&utilde;">triangulum</expan> CDO ad trian <lb/>gulum CDP, ita triangulum KLQ ad <expan abbr="triangul&utilde;">triangulum</expan> KLR, dein<lb/>de, vt triangulum CDP ad triangulum EFP, ita <expan abbr="tri&atilde;gulum">triangulum</expan> <lb/>KLR ad triangulum MNR; deniqu&egrave; vt triangulum EFP ad <lb/>&longs;pacium CEFD, ita triangulum MNR ad &longs;pacium kMNL,
<pb xlink:href="077/01/146.jpg" pagenum="142"/>
<arrow.to.target n="marg241"/> erit ex &aelig;quali &agrave; primo ad vltimum &longs;pacium ACDB ad <expan abbr="&longs;paci&utilde;">&longs;pacium</expan> <lb/>CEFD, vt &longs;pacium GKLH ad &longs;pacium KMNL. quod <expan abbr="dem&otilde;">demom</expan> <lb/>&longs;trare oportebat. </s>
        </p>
        <p id="N15470" type="margin">
          <s id="N15472"><margin.target id="marg229"/>11. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N1547B" type="margin">
          <s id="N1547D"><margin.target id="marg230"/>9. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N15486" type="margin">
          <s id="N15488"><margin.target id="marg231"/>22 <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15491" type="margin">
          <s id="N15493"><margin.target id="marg232"/>17. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1549C" type="margin">
          <s id="N1549E"><margin.target id="marg233"/><emph type="italics"/>e&longs;t<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s>
        </p>
        <p id="N154AC" type="margin">
          <s id="N154AE"><margin.target id="marg234"/>17.<emph type="italics"/>quinti. <lb/>cor.<emph.end type="italics"/>4. <emph type="italics"/><expan abbr="qu&imacr;">quim</expan> <lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p id="N154C4" type="margin">
          <s id="N154C6"><margin.target id="marg235"/>22. <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <p id="N154CF" type="margin">
          <s id="N154D1"><margin.target id="marg236"/><emph type="italics"/>ex<emph.end type="italics"/> 11. <emph type="italics"/><expan abbr="qu&itilde;">quim</expan> <lb/>ti. <lb/>cor.<emph.end type="italics"/> 19. <lb/><emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N154ED" type="margin">
          <s id="N154EF"><margin.target id="marg237"/>22. <emph type="italics"/>quinti <lb/>ex<emph.end type="italics"/> 1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N15500" type="margin">
          <s id="N15502"><margin.target id="marg238"/>19. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N1550B" type="margin">
          <s id="N1550D"><margin.target id="marg239"/><emph type="italics"/>ex quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15515" type="margin">
          <s id="N15517"><margin.target id="marg240"/><emph type="italics"/>cor.<emph.end type="italics"/>4. <emph type="italics"/><expan abbr="qu&imacr;">quim</expan> <lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p id="N1552A" type="margin">
          <s id="N1552C"><margin.target id="marg241"/>22. <emph type="italics"/>quinti<emph.end type="italics"/></s>
        </p>
        <p id="N15535" type="head">
          <s id="N15537">LEMMA II.</s>
        </p>
        <p id="N15539" type="main">
          <s id="N1553B"><expan abbr="&AElig;quidi&longs;t&atilde;tes">&AElig;quidi&longs;tantes</expan> ver&ograve; line&etail; AB CD ita &longs;e habeant, vt &aelig;quidi&shy;<lb/>&longs;tantes EF GH, &longs;itqu&egrave; maior AB, qu&agrave;m CD, &amp; EF, quam <lb/>GH. &amp; &longs;uper CD GH &longs;int triangula CDP GHR, <expan abbr="&longs;int&qacute;">&longs;int&que;</expan>; BDP <lb/>FHR rect&aelig; line&aelig;, &amp; vt BD ad DP, ita &longs;it FH ad HR. <expan abbr="iunctis&qacute;">iunctis&que;</expan>; <lb/>AC EG. Dico &longs;pacium ACDB ad <expan abbr="triangul&utilde;">triangulum</expan> CDP ita e&longs;&longs;e, vt <lb/>&longs;pacium EG HF ad triangulum GHR. </s>
        </p>
        <figure id="id.077.01.146.1.jpg" xlink:href="077/01/146/1.jpg"/>
        <p id="N15559" type="main">
          <s id="N1555B">Eadem enim pror&longs;us ratione productis AC EG, qu&aelig; cum <lb/>BP FR conueniant in OQ, o&longs;tendetur &longs;pacium AD ad trian <lb/>gulum CDO ita e&longs;&longs;e, vt &longs;pacium EH ad triangulum <expan abbr="GHq.">GH&que;</expan> &amp; <lb/>e&longs;&longs;e OD ad DB, ut QH ad HF. &amp; quoniam e&longs;t BD ad DP, vt <lb/>
<arrow.to.target n="marg242"/> FH ad HR, erit ex &etail;quali OD ad DP, vt QH ad HR. &amp; vt OD <lb/>ad DP, ita e&longs;t triangulum CDO ad triangulum CDP, &amp; vt <lb/>QH ad HR, ita triangulum GHQ ad GHR. c&ugrave;m ita&que; &longs;it <lb/>AD ad CDO, vt EH ad GHQ, &amp; vt CDO ad CDP, ita <lb/>
<arrow.to.target n="marg243"/> GHQ ad GHR. ex &aelig;quali erit &longs;pacium AD ad triangulum <lb/>CDP, vt &longs;pacium EH ad triangulum GHR. quod demon&longs;tra <lb/>re oportebat. </s>
        </p>
        <pb xlink:href="077/01/147.jpg" pagenum="143"/>
        <p id="N15580" type="margin">
          <s id="N15582"><margin.target id="marg242"/>22 <emph type="italics"/>quinti.<emph.end type="italics"/><lb/><gap/>. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N15593" type="margin">
          <s id="N15595"><margin.target id="marg243"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1559E" type="head">
          <s id="N155A0">LEMMA. III.</s>
        </p>
        <figure id="id.077.01.147.1.jpg" xlink:href="077/01/147/1.jpg"/>
        <p id="N155A5" type="main">
          <s id="N155A7">Sit A ad CD, vt E ad FG, diuidan <lb/><expan abbr="tur&qacute;">tur&que;</expan>; CD FG in <expan abbr="ead&etilde;">eadem</expan> proportione in HK, <lb/>ita vt &longs;it CH ad HD, vt FK ad KG. <lb/>Dico A ad DH ita e&longs;&longs;e, vt E ad KG. <lb/>A ver&ograve; ad CH, vt E ad Fk. </s>
        </p>
        <p id="N155B8" type="main">
          <s id="N155BA">Quoniam enim ita e&longs;t CH ad HD, vt FK ad kG; e&shy;<lb/>rit componendo CD ad DH, vt FG ad GK. e&longs;t autem A
<arrow.to.target n="marg244"/><lb/>ad CD, vt E ad FG; CD ver&ograve; e&longs;t ad DH, vt FG ad G<emph type="italics"/>K<emph.end type="italics"/>; er <lb/>go ex &aelig;quali A erit ad DH, vt E ad GK. Deinde
<arrow.to.target n="marg245"/> quo&shy;<lb/>niam e&longs;t GH ad HD, vt FK ad kG; erit conuertendo
<arrow.to.target n="marg246"/><lb/>DH ad HC, vt GK ad KF. rur&longs;us igitur ex &aelig;quali A e&shy;<lb/>rit ad CH, vt E ad FK. quod o&longs;tendere oportebat. </s>
        </p>
        <p id="N155D8" type="margin">
          <s id="N155DA"><margin.target id="marg244"/>18.<emph type="italics"/>qumti.<emph.end type="italics"/></s>
        </p>
        <p id="N155E3" type="margin">
          <s id="N155E5"><margin.target id="marg245"/>22 <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N155EE" type="margin">
          <s id="N155F0"><margin.target id="marg246"/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;-ti">quin&shy;<lb/>ti</expan>.<emph.end type="italics"/></s>
        </p>
        <p id="N15603" type="head">
          <s id="N15605">PROPOSITIO. III.</s>
        </p>
        <p id="N15607" type="main">
          <s id="N15609">Si in <expan abbr="vtraq;">vtra&que;</expan> <expan abbr="duar&utilde;">duarum</expan> <expan abbr="&longs;imili&utilde;">&longs;imilium</expan> <expan abbr="portion&utilde;">portionum</expan> recta linea re <lb/>ctanguliqu&egrave; coni &longs;ectione contentarum rectili&shy;<lb/>ne&aelig; figur&aelig; plan&egrave; in&longs;cribantur; figur&aelig; ver&ograve; in&longs;cri&shy;<lb/>pt&aelig; latera inter &longs;e multitudine &aelig;qualia habeant; <lb/>rectilinearum centra grauitatum portionum dia&shy;<lb/>metros &longs;imiliter &longs;ecabunt. </s>
        </p>
        <pb xlink:href="077/01/148.jpg" pagenum="144"/>
        <figure id="id.077.01.148.1.jpg" xlink:href="077/01/148/1.jpg"/>
        <p id="N1562B" type="main">
          <s id="N1562D"><emph type="italics"/>Sint du&aelig; portiones ABC XOP, in ip&longs;i&longs;qu&egrave; plan&egrave; in &longs;cribantur recti&shy;<lb/>line&aelig; figur&aelig;<emph.end type="italics"/> AEFGBHIKC XSYQOZVTP; <emph type="italics"/>qu&aelig; omnia latera <lb/>inter &longs;e numero &aelig;qualia habeanta, Diametri ver&ograve; portionum &longs;int BD<emph.end type="italics"/>
<pb xlink:href="077/01/149.jpg" pagenum="145"/><emph type="italics"/>OR. <expan abbr="iung&atilde;tur&qacute;">iungantur&que;</expan>; E<emph.end type="italics"/>k <emph type="italics"/>FI GH.<emph.end type="italics"/> qu&aelig; inter &longs;e, &amp; ip&longs;i AC <expan abbr="&ccedil;quidi&longs;t&atilde;tes">&ccedil;quidi&longs;tantes</expan>
<arrow.to.target n="marg247"/><lb/>erunt; bifariam qu&egrave; &agrave; diametro BD in punctis LMN diui&longs;&aelig; e&shy;<lb/>runt. </s>
          <s id="N1565C">Iungantur &longs;imiliter <emph type="italics"/>&amp; ST YV QZ<emph.end type="italics"/>, quas bifariam dia&shy;<lb/>meter OR in punctis 9<foreign lang="greek">ab</foreign> diuidet. </s>
          <s id="N1566A">eruntqu&egrave; duct&aelig; line&aelig; ip&longs;i <lb/>XP, &amp; inter &longs;e &aelig;quidi&longs;tantes. <emph type="italics"/>Quoniam igitur BD diuiditur &agrave; lineis <lb/>&aelig;quidi&longs;tantibus<emph.end type="italics"/> GH FI EK <emph type="italics"/>in proportionibus numeris deinceps impa&shy;<lb/>ribus;<emph.end type="italics"/> po&longs;ito enim vno BN, e&longs;t quidem NM tria, ML quin&que;, <lb/>&amp; LD &longs;eptem. </s>
          <s id="N15680">&longs;ed <emph type="italics"/>&amp; RO &longs;imiliter<emph.end type="italics"/> &agrave; lineis QZ YV ST in pro&shy;<lb/>portionibus diuiditur numeris deinceps imparibus, <expan abbr="ead&etilde;">eadem</expan>. <expan abbr="n.">enim</expan> <lb/>ratione &longs;i ponatur O<foreign lang="greek">b</foreign> vnum, erit <foreign lang="greek">ba</foreign> tria, <foreign lang="greek">a</foreign>9 <expan abbr="quin&qacute;">quin&que;</expan>;, &amp; 9R <lb/>&longs;eptem. <emph type="italics"/>&amp; portiones ip&longs;orum<emph.end type="italics"/> diametrorum BD OR <emph type="italics"/>&longs;unt numero &aelig;<lb/>quales.<emph.end type="italics"/> quot.n &longs;unt BN NM ML LD, tot &longs;unt O<foreign lang="greek">b ba a</foreign> 9 9R. <emph type="italics"/>pa <lb/>tet diametrorum portiones in eadem e&longs;&longs;e proportione<emph.end type="italics"/>, vt &que;m <expan abbr="admod&utilde;">admodum</expan> <lb/>e&longs;t BN ad NM, &amp; NM ad ML, &amp; ML ad LD, ita e&longs;&longs;e O<foreign lang="greek">b</foreign> ad <lb/><foreign lang="greek">ba</foreign>, &amp; <foreign lang="greek">ba</foreign> ad <foreign lang="greek">a</foreign>9, &amp; <foreign lang="greek">a</foreign>9 ad 9R. Atver&ograve; quoniam ita e&longs;t DB ad BL, <lb/>vt RO ad O9; (&longs;unt.n.ut &longs;exdecim ad nouem) &amp; ut DB ad BL,
<arrow.to.target n="marg248"/><lb/>ita e&longs;t quadratum ex AD ad <expan abbr="quadrat&utilde;">quadratum</expan> ex EL; &amp; vt RO ad O9, <lb/>ita e&longs;t <expan abbr="quadrat&utilde;">quadratum</expan> ex XR ad quadratum ex S<emph type="italics"/>9<emph.end type="italics"/>; erit <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>AD ad <expan abbr="quadrat&utilde;">quadratum</expan> ex EL, vt <expan abbr="quadrat&utilde;">quadratum</expan> ex XR ad ex S9 <expan abbr="quadrat&utilde;">quadratum</expan>. <lb/>ergo ut AD ad EL, ita XR ad S9. &amp; horum dupla <expan abbr="n&etilde;p&egrave;">nemp&egrave;</expan> AC ad <lb/>EK, vt XP ad ST: <expan abbr="eadem&qacute;">eadem&que;</expan>; pror&longs;us <expan abbr="r&otilde;ne">ronne</expan>, quoniam ita e&longs;t LB
<arrow.to.target n="marg249"/><lb/>ad BM, vt 9O ad O<foreign lang="greek">a</foreign> (&longs;unt.n.ut nouem ad quatuor) o&longs;tendetur <lb/>EL ad FM ita e&longs;&longs;eut S9 ad Y<foreign lang="greek">a</foreign>, &amp; horum dupla, &longs;cilicet EK ad FI <lb/>ita e&longs;&longs;e, ut ST ad YV. <expan abbr="C&ugrave;m&qacute;">C&ugrave;m&que;</expan>; &longs;it MB ad BN, vt <foreign lang="greek">a</foreign>O ad O<foreign lang="greek">b</foreign>, ut &longs;ci <lb/>licet quatuor ad vnum; &longs;imiliter o&longs;tendetur FM ad GN ita e&longs;&longs;e <lb/>vt Y<foreign lang="greek">a</foreign> ad Q<foreign lang="greek">b</foreign>; FI uer&ograve; ad GH, vt YV ad QZ. vnde colligitur <expan abbr="n&otilde;">non</expan> <lb/>&longs;ol&ugrave;m portiones diametrorum (ut dixim us) in eadem e&longs;&longs;e pro&shy;<lb/>portione, &longs;ed <emph type="italics"/>&amp; parallelas<emph.end type="italics"/> AC EK FI GH, &amp; XP ST YV QZ <emph type="italics"/>in <lb/><expan abbr="ead&emacr;">eadem</expan> e&longs;&longs;e proportione. </s>
          <s id="N15753">&amp; T rapeziorum ip&longs;ius quidem AE<emph.end type="italics"/>k<emph type="italics"/>C, &amp; ip&longs;ius<emph.end type="italics"/>
<arrow.to.target n="marg250"/><lb/><emph type="italics"/>XSTP centra grauitatum e&longs;&longs;e in lineis LD 9R &longs;imiliter po&longs;ita, c&ugrave;m <lb/>eandem habeant proportionem AC EK, quam XP ST.<emph.end type="italics"/> line&aelig;qu&egrave; <lb/>LD 9R bifariam diuidant &longs;uas &aelig;quidi&longs;tantes AC EK. <lb/>&amp; XP ST. etenim &longs;i ponatur trapezij AK centrum graui<lb/>tatis <foreign lang="greek">g</foreign>, ip&longs;ius vcr&ograve; XT centrum grauitatis <foreign lang="greek">d</foreign>, erit L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, <lb/>vt dupla ip&longs;ius AC cum EK ad duplam ip&longs;ius EK
<arrow.to.target n="marg251"/><lb/>cum AC. &amp; 9<foreign lang="greek">d</foreign> ad <foreign lang="greek">d</foreign>R erit, vt dupla ip&longs;ius XP cum <lb/>ST ad duplam ST cum XP. quoniam autem ita e&longs;t AC ad EK,
<pb xlink:href="077/01/150.jpg" pagenum="146"/>
<arrow.to.target n="fig69"/>
<pb xlink:href="077/01/151.jpg" pagenum="147"/>vt XP ad ST, &amp; antecedentium dupla, hoc e&longs;t dupla i&shy;<lb/>p&longs;ius AC ad EK erit, vt dupla ip&longs;ius XP ad ST. <lb/>&amp; componendo dupla ip&longs;ius AC cum EK, vt dupla
<arrow.to.target n="marg252"/> i&shy;<lb/>p&longs;ius XP cum ST ad ST. At ver&ograve; EK ad duplam <lb/>ip&longs;ius EK, ita e&longs;t, vt ST ad duplam ip&longs;ius ST, &longs;ed EK
<arrow.to.target n="marg253"/><lb/>ad AC e&longs;t, vt ST ad XP, erit EK ad vtra&longs;&que; con&longs;e&shy;<lb/>&que;ntes &longs;im ul &longs;umptas, hoc e&longs;t ad duplam ip&longs;ius EK cum <lb/>AC, vt ST ad &longs;uas con&longs;e&que;ntes, nempe ad duplam ip&longs;ius <lb/>ST cum XP. Ita&que; quoniam ita e&longs;t dupla ip&longs;ius AC <lb/><expan abbr="c&utilde;">cum</expan> EK ad Ek, vt dupla ip&longs;ius XP cum ST ad ST, &amp; e&longs;t EK <lb/>ad duplam ip&longs;ius EK cum AC, vt ST ad duplam ip&longs;ius <lb/>ST cum XP. erit ex &etail;quali dupla ip&longs;ius AC cum EK ad du
<arrow.to.target n="marg254"/><lb/>plam ip&longs;ius EK cum AC, vt dupla ip&longs;ius XP cum ST ad <lb/>duplam ip&longs;ius ST cum XP. ac propterea ita e&longs;t L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, <lb/>vt 9<foreign lang="greek">d</foreign> ad <foreign lang="greek">d</foreign>R, &amp; ob id centra <foreign lang="greek">gd</foreign> erunt in lineis LD 9R &longs;i&shy;<lb/>militer po&longs;ita. <emph type="italics"/>Rur&longs;us<emph.end type="italics"/> eodem modo (ne eadem &longs;&aelig;pi&ugrave;s repetan<lb/>tur) <emph type="italics"/>Trapeziorum EFI<emph.end type="italics"/>k <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">*g</foreign><emph type="italics"/>VT centragrauitatum<emph.end type="italics"/>, qu&aelig; &longs;int <foreign lang="greek">ez</foreign>, <emph type="italics"/>&longs;i <lb/>militer<emph.end type="italics"/> hoc e&longs;t in eadem proportione <emph type="italics"/>diuident lineas LM<emph.end type="italics"/> 9<foreign lang="greek">a</foreign>, i&shy;<lb/>ta vt &longs;it L<foreign lang="greek">e</foreign> ad <foreign lang="greek">e</foreign>M, vt 9<foreign lang="greek">z</foreign> ad <foreign lang="greek">za</foreign>. <emph type="italics"/>&amp; in trapezits FH<emph.end type="italics"/> <foreign lang="greek">*g</foreign><emph type="italics"/>Z centra <lb/>grauitatum<emph.end type="italics"/> <foreign lang="greek">*hk</foreign> <emph type="italics"/>&longs;imiliter diuident MN<emph.end type="italics"/> <foreign lang="greek">ab</foreign>, ita ut M<foreign lang="greek">*h</foreign> ad <foreign lang="greek">*h</foreign>N &longs;it, vt <lb/><foreign lang="greek">ak</foreign> ad <foreign lang="greek">kb</foreign> <emph type="italics"/>&longs;ed &amp; triangulorum GBH QOZ centra grauitatum<emph.end type="italics"/> <foreign lang="greek">lm</foreign><lb/><emph type="italics"/>in lineis B N<emph.end type="italics"/> O<foreign lang="greek">b</foreign> <emph type="italics"/>erunt &longs;imiliter po&longs;ita<emph.end type="italics"/>, &longs;iquidem B<foreign lang="greek">l</foreign> ad <foreign lang="greek">l</foreign>N e&longs;t, vt
<arrow.to.target n="marg255"/><lb/>O<foreign lang="greek">m</foreign> ad <foreign lang="greek">mb</foreign>; quipp&egrave; c&ugrave;m in dupla &longs;int proportione. <emph type="italics"/>eandem au&shy;<lb/>tem habent proportionem Trapezia, &amp; triangula:<emph.end type="italics"/> Nam c&ugrave;m <lb/>&longs;it AD ad EL, vt XR ad S9, &amp; ut EL ad FM, ita S9 ad Y; <lb/>e&longs;tqu&egrave; DL ad LM, ut R9 ad 9<foreign lang="greek">a</foreign>, c&ugrave;m &longs;int, vt &longs;eptem ad quin <lb/>&que;; erit &longs;pacium AL ad &longs;pacium EM, vt &longs;pacium X9 ad
<arrow.to.target n="marg256"/> &longs;pa&shy;<lb/>cium S. &longs;imiliterqu&egrave; o&longs;tendetur DK ad LI ita e&longs;&longs;e, vt RT <lb/>ad 9V. quare totum trapezium AK ad EI e&longs;t, vt XT ad SV. <lb/>pariqu&egrave; ratione o&longs;tendeturita e&longs;&longs;e trapezium EI ad FH, vt <lb/>SV ad YZ. quia ver&ograve; ita e&longs;t FM ad GN, vt Y<foreign lang="greek">a</foreign> ad Q<foreign lang="greek">d</foreign>, <lb/>e&longs;t autem MN ad NB, vt <foreign lang="greek">ab</foreign> ad <foreign lang="greek">b</foreign>O, &longs;unt quipp&egrave; ut tria ad <lb/>vnum, erit &longs;pacium FN ad triangulum GBN, vt &longs;pacium
<arrow.to.target n="marg257"/><lb/>Y<foreign lang="greek">b</foreign> ad triangulum Q<foreign lang="greek">b</foreign>O. codemqu&egrave; modo o&longs;tendetur ita <lb/>e&longs;&longs;e &longs;pacium IN ad triangulum BNH, vt &longs;pacium V<foreign lang="greek">b</foreign> ad <lb/>triangulum O<foreign lang="greek">b</foreign>Z. Ex quibus &longs;equitur ita e&longs;&longs;e <expan abbr="trapezi&utilde;">trapezium</expan> FH <lb/>ad triangulum BGH, vt trapezium YZ ad <expan abbr="triangul&utilde;">triangulum</expan> OQZ.
<pb xlink:href="077/01/152.jpg" pagenum="148"/>
<arrow.to.target n="fig70"/>
<pb xlink:href="077/01/153.jpg" pagenum="149"/>&longs;i ita&que; diuidatur <foreign lang="greek">ge</foreign> in <foreign lang="greek">n</foreign>, ita ut &longs;it <foreign lang="greek">en</foreign> ad <foreign lang="greek">ng</foreign>, vt <expan abbr="trapezi&utilde;">trapezium</expan> AK <lb/>ad EI. erit punctum <foreign lang="greek">n</foreign> centrum grauitatis figur&etail; AEFIKC.
<arrow.to.target n="marg258"/><lb/>&longs;imiliqu&egrave; modo diuidatur <foreign lang="greek">dz</foreign> in <foreign lang="greek">&lt;10&gt;</foreign>, ita vt &longs;it <foreign lang="greek">z&lt;10&gt;</foreign> ad <foreign lang="greek">&lt;10&gt;d</foreign>, vt trape <lb/>zium XT ad SV; erit punctum <foreign lang="greek">&lt;10&gt;</foreign> grauitatis centrum figur&aelig; <lb/>XSYVTP. quia ver&ograve; ita e&longs;t AK ad EI, vt XT ad SV, erit <foreign lang="greek">en</foreign><lb/>ad <foreign lang="greek">ng</foreign>, vt <foreign lang="greek">z&lt;10&gt;</foreign> ad <foreign lang="greek">&lt;10&gt;d</foreign>. Diuidatur <expan abbr="a&utilde;t">aunt</expan> deinceps <foreign lang="greek">l*h</foreign> in <foreign lang="greek">s</foreign>, <expan abbr="&longs;it&qacute;">&longs;it&que;</expan>; <foreign lang="greek">ls</foreign> ad <foreign lang="greek">s*h</foreign>, vt <lb/>FH ad triangulum BGH, erit punctum <foreign lang="greek">s</foreign> centrum grauitatis <lb/>figur&aelig; FGBHI. eademqu&egrave; ratione diuidatur <foreign lang="greek">mk</foreign> in <foreign lang="greek">t</foreign>, &longs;itqu&egrave; <lb/><foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>, vt YZ ad triangulum OQZ; erit punctum <foreign lang="greek">t</foreign> cen&shy;<lb/>trum grauitatis figur&aelig; YQOZV. &longs;ed e&longs;t FH ad BGD, vt YZ <lb/>ad OQZ, erit igitur <foreign lang="greek">ls</foreign> ad <foreign lang="greek">sh</foreign>, vt <foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>. Quoniam autem <lb/>ita e&longs;t Ak ad EI, vt XT ad SV, erit componendo AEFIKC
<arrow.to.target n="marg259"/><lb/>ad EI, vt figura XSYVTP ad SV; &amp; e&longs;t EI ad FH, vt SV ad
<arrow.to.target n="marg260"/><lb/>YZ. ergo ex &aelig;quali figura AEFIKC erit ad FH, vt figura <lb/>XSYVTP ad YZ. e&longs;t autem FH ad BGH, vt YZ ad OQZ. e&shy;<lb/>ritigitur figura AEFIKC ad &longs;uas con&longs;e&que;ntes, ad figuram
<arrow.to.target n="marg261"/><lb/>&longs;cilicet FGBHI, vt figura XSYVTP ad &longs;uas con&longs;e&que;ntes, hoc <lb/>e&longs;t ad figuram YQOZV. Diuidatur ita&que; <foreign lang="greek">sn</foreign> in <foreign lang="greek">x</foreign>, ita ut <foreign lang="greek">sx</foreign><lb/>ad <foreign lang="greek">x</foreign> &longs;it, vt figura AEFIKC ad figuram FGBHI, erit punctum
<arrow.to.target n="marg262"/><lb/><foreign lang="greek">x</foreign> <expan abbr="centr&utilde;">centrum</expan> grauitatis totius figur&etail; AEFGBHIKC. &longs;imiliter di&shy;<lb/>uidatur <foreign lang="greek">t&lt;10&gt;</foreign> in <foreign lang="greek">c</foreign>, &longs;it&que; <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c&lt;10&gt;</foreign>, ut figura XSYVTP ad figu&shy;<lb/>ram YQOZV, erit punctum <foreign lang="greek">c</foreign> centrum grauitatis totius fi&shy;<lb/>gur&aelig; XSYQOZVTP. quia ver&ograve; ita e&longs;t figura AEFIKC ad fi <lb/>guram FGBHI, vt figura XSYVTP ad figuram YQOZV. e&shy;<lb/>rit <foreign lang="greek">sx</foreign> ad <foreign lang="greek">xn</foreign>, vt <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c&lt;10&gt;</foreign>. Ita&que; quoniam BD ad DL e&longs;t, vt <foreign lang="greek">sn</foreign><lb/>ad R9, c&ugrave;m &longs;in^&lcub;4} ut&longs;exdecim ad &longs;eptem. </s>
          <s id="N159D8">&amp; e&longs;t L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, vt 9<foreign lang="greek">d</foreign><lb/>ad <foreign lang="greek">d</foreign>R, erit BD ad L<foreign lang="greek">g</foreign>, vt <foreign lang="greek">sn</foreign> ad 9<foreign lang="greek">d</foreign>. &amp; vt BD ad <foreign lang="greek">g</foreign>D, ita OR ad
<arrow.to.target n="marg263"/><lb/><foreign lang="greek">d</foreign>R. rur&longs;us quoniam BD ad LM e&longs;t, vt OR ad 9<foreign lang="greek">a</foreign>, nempe vt &longs;ex <lb/>decim ad quin&que;; &amp; e&longs;t L<foreign lang="greek">e</foreign> ad <foreign lang="greek">e</foreign>M, ut 9<foreign lang="greek">z</foreign> ad <foreign lang="greek">za</foreign>, erit BD ad <foreign lang="greek">e</foreign>L, <lb/>vt OR ad 9<foreign lang="greek">z</foreign>. e&longs;t ver&ograve; BD ad L<foreign lang="greek">g</foreign>, vt OR ad 9<foreign lang="greek">d</foreign>; erit igitur BD ad <lb/>vtram &que; &longs;imul <foreign lang="greek">e</foreign>L L<foreign lang="greek">g</foreign>, hoc e&longs;t ad <foreign lang="greek">eg</foreign>, vt OR ad <foreign lang="greek">zd</foreign>. &longs;ed <expan abbr="quoni&atilde;">quoniam</expan>
<arrow.to.target n="marg264"/><lb/>e&longs;t <foreign lang="greek">gn</foreign> ad <foreign lang="greek">ne</foreign>, vt <foreign lang="greek">d&lt;10&gt;</foreign> ad <foreign lang="greek">&lt;10&gt;z</foreign>, erit BD ad <foreign lang="greek">gn</foreign>, vt OR ad <foreign lang="greek">d&lt;10&gt;</foreign>. e&longs;t <expan abbr="aut&etilde;">autem</expan> BD <lb/>ad D<foreign lang="greek">g</foreign>, vt OR ad R<foreign lang="greek">d</foreign>, vt dictum e&longs;t, ergo BD ad D<foreign lang="greek">n</foreign> e&longs;t, vt OR <lb/>ad R<foreign lang="greek">&lt;10&gt;</foreign>. &longs;imiliterqu&egrave; <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan> BD ad BA ita e&longs;&longs;e, vt OR ad O<foreign lang="greek">t</foreign>. <lb/>C&ugrave;m ita&que; &longs;it BD ad DR, &amp; ad B<foreign lang="greek">s</foreign>, ut OR ad R<foreign lang="greek">&lt;10&gt;</foreign>, &amp; ad O<foreign lang="greek">t</foreign>; e&shy;<lb/>rit BD ad DR B<foreign lang="greek">s</foreign> &longs;imul, vt OR ad R<foreign lang="greek">&lt;10&gt;</foreign> O<foreign lang="greek">t</foreign> &longs;imul, &amp; permutan&shy;<lb/>do tota BD ad totam OR, vt ablata D<foreign lang="greek">n</foreign>B<foreign lang="greek">s</foreign> ad ablatam R<foreign lang="greek">&lt;10&gt;ot</foreign>.
<pb xlink:href="077/01/154.jpg" pagenum="150"/>
<arrow.to.target n="marg265"/> ergo &amp; reliqua <foreign lang="greek">sn</foreign> ad reliquam <foreign lang="greek">t&lt;10&gt;</foreign> e&longs;t, ut tota BD ad <expan abbr="tot&atilde;">totam</expan> OR. <lb/>rur&longs;u&longs;qu&egrave; permutando <foreign lang="greek">sn</foreign> ad BD ut <foreign lang="greek">t&lt;10&gt;</foreign> ad OR, <expan abbr="conuertendo&qacute;">conuertendo&que;</expan>; <lb/>BD ad <foreign lang="greek">sn</foreign> e&longs;t, ut OR ad <foreign lang="greek">t&lt;10&gt;</foreign>, Quia ver&ograve; ita e&longs;t <foreign lang="greek">sx</foreign> ad <foreign lang="greek">xn</foreign>, ut <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c&lt;10&gt;</foreign>; <lb/>
<arrow.to.target n="marg266"/> erit BD ad <foreign lang="greek">sx</foreign>, vt OR ad <foreign lang="greek">tc</foreign> atver&ograve; BD ad b<foreign lang="greek">s</foreign> e&longs;t, vt OR ad O<foreign lang="greek">t</foreign>. <lb/>erit igitur BD ad B<foreign lang="greek">x</foreign>, ut O<foreign lang="greek">g</foreign> ad O<foreign lang="greek">c</foreign>. ac propterea diuidendo D<foreign lang="greek">x</foreign><lb/>ita &longs;e habet ad <foreign lang="greek">x</foreign>B, vt R<foreign lang="greek">c</foreign> ad <foreign lang="greek">c</foreign>O. <emph type="italics"/>Quare manifestum est totius recti&shy;<lb/>line&aelig; figur&aelig; in portione ABC in&longs;cript&aelig; centrum grauitatis<emph.end type="italics"/> <foreign lang="greek">x</foreign> <emph type="italics"/>in eadem <lb/>proportione diuidere BD, veluti centrum grauitatis<emph.end type="italics"/> <foreign lang="greek">c</foreign> <emph type="italics"/>figur&aelig; rectiline&aelig; <lb/>in portione XOP<emph.end type="italics"/> in&longs;cript&aelig; <emph type="italics"/>ip&longs;am OR<emph.end type="italics"/> diametrum. <emph type="italics"/>quod demonstra&shy;<lb/>re oportebat.<emph.end type="italics"/></s>
        </p>
        <p id="N15B46" type="margin">
          <s id="N15B48"><margin.target id="marg247"/><emph type="italics"/>ex iis qu&etail; <lb/>po&longs;t <gap/> pri&shy;<lb/>mi huius <lb/>demon&longs;tra <lb/>ta &longs;unt.<emph.end type="italics"/></s>
        </p>
        <p id="N15B5A" type="margin">
          <s id="N15B5C"><margin.target id="marg248"/>3. A<emph type="italics"/>rchi. <lb/>de quad. <lb/>parab. </s>
          <s id="N15B67">&amp;<emph.end type="italics"/><lb/>20, <emph type="italics"/>primi <lb/>conicorum <lb/>Apoll.<emph.end type="italics"/></s>
        </p>
        <p id="N15B76" type="margin">
          <s id="N15B78"><margin.target id="marg249"/>22. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N15B81" type="margin">
          <s id="N15B83"><margin.target id="marg250"/>15. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N15B8E" type="margin">
          <s id="N15B90"><margin.target id="marg251"/>15. <emph type="italics"/>primi <lb/>buius.<emph.end type="italics"/></s>
        </p>
        <p id="N15B9B" type="margin">
          <s id="N15B9D"><margin.target id="marg252"/>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15BA6" type="margin">
          <s id="N15BA8"><margin.target id="marg253"/>2. <emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> an<lb/>te<emph.end type="italics"/> 13. <emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N15BBE" type="margin">
          <s id="N15BC0"><margin.target id="marg254"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15BC9" type="margin">
          <s id="N15BCB"><margin.target id="marg255"/><emph type="italics"/><expan abbr="&atilde;te">ante</expan><emph.end type="italics"/> 13.<emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N15BDD" type="margin">
          <s id="N15BDF"><margin.target id="marg256"/>1.<emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N15BE8" type="margin">
          <s id="N15BEA"><margin.target id="marg257"/>2.<emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N15BF3" type="margin">
          <s id="N15BF5"><margin.target id="marg258"/><emph type="italics"/>ex<emph.end type="italics"/> 6. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N15C05" type="margin">
          <s id="N15C07"><margin.target id="marg259"/>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15C10" type="margin">
          <s id="N15C12"><margin.target id="marg260"/>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15C1B" type="margin">
          <s id="N15C1D"><margin.target id="marg261"/><emph type="italics"/>cor.<emph.end type="italics"/> 2. <emph type="italics"/>lem <lb/>ma m<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui<emph.end type="italics"/>^&lcub;9}</s>
        </p>
        <p id="N15C35" type="margin">
          <s id="N15C37"><margin.target id="marg262"/><emph type="italics"/>ex<emph.end type="italics"/> 6. <emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N15C47" type="margin">
          <s id="N15C49"><margin.target id="marg263"/>3. <emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N15C52" type="margin">
          <s id="N15C54"><margin.target id="marg264"/>2. <emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> an<lb/>te<emph.end type="italics"/> 13. <emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/><lb/>3. <emph type="italics"/>lcmma.<emph.end type="italics"/><lb/>2. <emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> an<lb/>te<emph.end type="italics"/> 13. <emph type="italics"/>pri&shy;<lb/>mi huius<emph.end type="italics"/><lb/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15C8C" type="margin">
          <s id="N15C8E"><margin.target id="marg265"/>19.<emph type="italics"/>quinti. <lb/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&imacr;ti">quinti</expan>.<emph.end type="italics"/><lb/>3.<emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N15CA9" type="margin">
          <s id="N15CAB"><margin.target id="marg266"/>2. <emph type="italics"/>lemma <lb/>ante<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui<emph.end type="italics"/>^&lcub;9} <lb/>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.154.1.jpg" xlink:href="077/01/154/1.jpg"/>
        <figure id="id.077.01.154.2.jpg" xlink:href="077/01/154/2.jpg"/>
        <p id="N15CCD" type="head">
          <s id="N15CCF">SCHOLIVM.</s>
        </p>
        <p id="N15CD1" type="main">
          <s id="N15CD3">Hinc colligere licet parabolas omnes inter &longs;e &longs;imiles e&longs;&longs;e. </s>
          <s id="N15CD5">Re <lb/>fert enim Eutocius hoc in loco, Apollonium perg&etail;um in &longs;ex <lb/>to Conicorum libro. (qui nondum in lucem prodijt) &longs;imiles <lb/>coni &longs;ectiones dixi&longs;&longs;e eas e&longs;&longs;e, quando in vnaqua&que; &longs;ectione <lb/>line&etail; <expan abbr="duc&utilde;tur">ducuntur</expan> ba&longs;i <expan abbr="&aelig;quidi&longs;t&atilde;tes">&aelig;quidi&longs;tantes</expan> numero pares; hoc e&longs;t tot in v&shy;<lb/>na, quot in alia; vt in &longs;uperioribus figuris duct&aelig; fuerunt, in v&shy;<lb/>na quidem EK FI GH ip&longs;i AC &aelig;quidi&longs;tantes; &amp; in altera ST <lb/>YV QZ ip&longs;i PX &aelig;quidi&longs;tantes; qu&etail; quidem efficiant, vt dia&shy;<lb/>metri in eadem proportione diui&longs;&aelig; proueniant; vt &longs;unt BN <lb/>NM ML LD; &amp; O<foreign lang="greek">b ba a</foreign>9 9R. Deinde <expan abbr="&aelig;quidi&longs;t&atilde;tes">&aelig;quidi&longs;tantes</expan> AC EK <lb/>FI GH in eadem &longs;int proportione ip&longs;arum XP ST YV QZ. <lb/>&amp; quoniam h&aelig; conditiones in omnibus po&longs;&longs;unt accidere pa <lb/>rabolis; vt ex ijs, qu&aelig; demon&longs;trata &longs;unt, manife&longs;tum e&longs;t; id&shy;<lb/>circo parabol&aelig; omnes &longs;unt &longs;imiles. </s>
          <s id="N15D01">Ne&que; ver&ograve; <expan abbr="exi&longs;timand&utilde;">exi&longs;timandum</expan> <lb/>e&longs;t, quoniam parabol&aelig; &longs;unt &longs;imiles, figur as quo&que; plan&egrave; <lb/>in&longs;criptas, vt AEFGBHIKC &amp; XSYQOZVTP &longs;imiles e&longs;&longs;e in<lb/>ter &longs;e, ea pr&aelig;&longs;ertim &longs;imilitudine, qua &longs;unt figur&aelig; rectiline&aelig;; <lb/>vt &longs;cilicet anguli &longs;int &aelig;quales, &amp; circum &etail;quales angulos late&shy;<lb/>ra proportionalia. </s>
          <s id="N15D11">in parabolis <expan abbr="n&otilde;">non</expan> attenditur h&etail;c &longs;imilitudo. <lb/>&longs;atenim e&longs;t, vt pr&aelig;fat&aelig; ad&longs;int conditiones; ex quibus &longs;equi&shy;<lb/>tur (vt o&longs;tendimus) trapezia AK EI FH, triangulum qu&egrave; <lb/>BGH in eadem e&longs;&longs;e proportione trapeziorum XT SV YZ, ac
<pb xlink:href="077/01/155.jpg" pagenum="151"/>trianguli OQZ. ac propterea quando Archimedes in propo<lb/>&longs;itione inquit <emph type="italics"/>&longs;i in vtra&que; &longs;imilium portionum rectalmea, rectangu&shy;<lb/>liqu&egrave; coni &longs;ectione contentarum,<emph.end type="italics"/> non propterda exi&longs;timandum e&longs;t <lb/>reperiri po&longs;&longs;e aliquas parabolas recta linea terminatas no e&longs;&longs;e <lb/>&longs;imiles inter &longs;e; ea nimirumiam explicata &longs;imilitudine. </s>
          <s id="N15D2F">&longs;unte&shy;<lb/>nim Archimedis verba hoc modo intelligenda, nemp&egrave;, &longs;i in <lb/>vtra&que; portionum recta linea rectanguliqu&egrave; coni &longs;ectione <lb/>contentarum, qu&aelig; omnes &longs;unt &longs;imiles, &amp; c. </s>
          <s id="N15D37">veluti &longs;i dicere&shy;<lb/>mus. </s>
          <s id="N15D3B">In &longs;imilibus &longs;emicirculis anguli omnes &longs;untrecti. </s>
          <s id="N15D3D">non <lb/>e&longs;t intelligendum nonnullos &longs;emicirculos inter &longs;e di&longs;&longs;imiles <lb/>exi&longs;tere po&longs;&longs;e. </s>
          <s id="N15D43">&longs;ed hoc modo; in &longs;emicirculis, qui omnes &longs;unt <lb/>&longs;imiles, anguli&longs;unt recti. </s>
          <s id="N15D47">Et hoc modo &longs;emperintelligere o&shy;<lb/>portet, quando in &longs;e&que;ntibus Archimedes parabolas &longs;imiles <lb/>nominat. </s>
          <s id="N15D4D">Nam &amp; Archimedes cognouit omnes parabolas <lb/>inter &longs;e &longs;imiles e&longs;&longs;e; vt ip&longs;e in demon&longs;tratione octau&aelig; propo&longs;i <lb/>tionis huius &longs;upponere videtur. </s>
          <s id="N15D53">Oportebatenim aliquam in <lb/>parabolis demon&longs;trare &longs;imilitudinem, vt demon&longs;trari po&longs;&longs;et <lb/>centrum grauitatis in omnibus parabolis e&longs;&longs;e in certo, ac de&shy;<lb/>terminato &longs;itu ip&longs;ius figur&aelig;. </s>
          <s id="N15D5B">in figuris enim, qu&aelig; aliquam in&shy;<lb/>ter&longs;e non habent &longs;imilitudinem, in ip&longs;is centrum grauitatis <lb/>determinari minim&egrave; po&longs;&longs;e videtur. </s>
          <s id="N15D61">Dicet autem forta&longs;&longs;e ali&shy;<lb/>quis, determinatur tamen centrum grauitatis in omnibus <expan abbr="tri&atilde;">triam</expan> <lb/>gulis, qu&aelig; quidem inter&longs;e non &longs;unt&longs;imilia. </s>
          <s id="N15D6B">Cui re&longs;ponden&shy;<lb/>dum; triangula omnia inter &longs;e &longs;imilia non e&longs;&longs;e &longs;imilitudine <lb/>rectilinearum figurarum, nemp&egrave; vt anguli &longs;int&aelig;quales, &amp; cir&shy;<lb/>cum &aelig;qualesangulos latera proportionalia. </s>
          <s id="N15D73">qu&ograve;d tamen nul&shy;<lb/>lam inter &longs;e&longs;e habeant conuenientiam, omnino negatur. <expan abbr="n&atilde;">nam</expan> <lb/>triangula omnia &longs;imul quodam modo illam habent conue&shy;<lb/>nientiam, &amp; &longs;imilitudinem; qu&aelig; parabolis accidit. </s>
        </p>
        <p id="N15D7F" type="main">
          <s id="N15D81">In triangulis enim ABC DEF duct&etail; &longs;int AG DH ab angu&shy;<lb/>lis ad dimidias ba&longs;es. </s>
          <s id="N15D85">&longs;intqu&egrave; diui&longs;a triangulorum latera in ea <lb/>dem proportione, in punctis kL, OP. &amp; vt AK KL LB, ita &longs;it <lb/>AM MN NC, &amp; DQ QR RF. ducti&longs;qu&egrave; KM LN OQ PR,
<arrow.to.target n="marg267"/><lb/>qu&aelig; lineas AG DH &longs;ecent in punctis ST VX; prim&ugrave;m <expan abbr="quid&etilde;">quidem</expan> <lb/>erunt KM LN OQ PR ba&longs;ibus BC EF &aelig;quidi&longs;tantes; quas <lb/>line&aelig; AG DH in punctis ST VX bifariam diuident, c&ugrave;m &longs;it
<pb xlink:href="077/01/156.jpg" pagenum="152"/>BG ad GC, vt LT ad TN, &amp; KS ad SM. &amp; ut EH ad HF ita <lb/>PX ad XR, &amp; OV ad <expan abbr="Vq.">V&que;</expan> Deinde erunt AG DH &agrave; lineis KM <lb/>LN OQ PR in eadem proportione diui&longs;&aelig;; &longs;iquidem ita e&longs;t <lb/>AS ST TG, ut DV VX XH. c&ugrave;m &longs;int, ut expo&longs;it&aelig; propor&shy;<lb/>tiones AK KL LB, &amp; DO OP PE. Pr&aelig;terea erit &longs;pacium, <lb/>BN ad LM, vt ER ad PQ, &amp; LM ad triangulum AK M, <lb/>
<arrow.to.target n="fig71"/><lb/>vt PQ ad triangulum <expan abbr="DOq.">DO&que;</expan> Nam quoniam triangulu AEC <lb/>&longs;imile e&longs;t triangulo ALN, oblatus LN ip&longs;i BC &aelig;quidi&longs;tans; <lb/>erit ABC ad ALN, ut AB ad AL duplicata. </s>
          <s id="N15DB9">eodemqu&egrave; modo <lb/>erit DEF ad DPR, vt DE ad DP duplicata; eandem aut<gap/>m, <lb/>habet proportionem AB ad AL, quam DE ad DP: quadoqui <lb/>dem latera AB DE in eadem &longs;unt proportione diui&longs;a; erit igi&shy;<lb/>tur triangulum ABC ad ALN, vt triangulum DEF ad DPR. <lb/>&longs;imiliterqu&egrave; o&longs;tendetur ALN ad AkM ita e&longs;&longs;e, ut DPR ad <lb/><expan abbr="DOq.">DO&que;</expan> Quoniam autem ABC e&longs;t ad ALN, ut DEF ad DPR, <lb/>
<arrow.to.target n="marg268"/> diuidendo erit BN ad ALN, ut ER ad DPR. Atver&ograve; <expan abbr="quoni&atilde;">quoniam</expan> <lb/>ALN ad AKM e&longs;t, vt DPR ad <expan abbr="DOq;">DO&que;</expan> erit per conuer&longs;io&shy;<lb/>nem rationis ALN ad LM, vt DPR ad <expan abbr="Pq.">P&que;</expan> qua&shy;<lb/>
<arrow.to.target n="marg269"/> re ex &etail;quali BN e&longs;t ad LM, ut ER ad <expan abbr="Pq.">P&que;</expan> C&ugrave;m au<gap/>em &longs;it <lb/>ALN ad AKM, ut DPR ad <expan abbr="DOq;">DO&que;</expan> erit diuidendo LM ad <lb/>AKM, vt PQ ad <expan abbr="DOq.">DO&que;</expan> Quocirca erit &longs;pacium BN ad <lb/>LM, vt ER ad PQ, &amp; LM ad triangulum AKM, <lb/>vt PQ ad triangulum <expan abbr="DOq.">DO&que;</expan> Ex quibus per&longs;picuum <lb/>e&longs;t omnia triangula aliquam inter &longs;e habere &longs;imilitudinem, <lb/>ex qua po&longs;&longs;ibile fuit determinare in omnibus &longs;itum, vb<gap/>epe-
<pb xlink:href="077/01/157.jpg" pagenum="153"/>ritur centrum graurtatis. </s>
          <s id="N15E0C">Qu&ograve;d &longs;i figur&etail; nullam conuenien&shy;<lb/>tiam, nullamqu&egrave; &longs;imilitudinem inter &longs;e habuerint; ut in qua <lb/>drilateris, pentagonis, &amp; reliquis figuris, qu&aelig; inter &longs;e ne&que; <lb/>latera ne&que; angulos &etail;quales <expan abbr="habe&atilde;t">habeant</expan>; &amp; propterea nullam in&shy;<lb/>ter&longs;e conuenientiam, &amp; &longs;imilitudinem habere po&longs;&longs;unt; im&shy;<lb/>po&longs;&longs;ibile quidem e&longs;&longs;et in ip&longs;is determinare &longs;itum <expan abbr="c&etilde;tri">centri</expan> grauita <lb/>tis; ita vt omnibus quadrilateris, ac omnibus pentagonis quo <lb/>modo cun&que; factis, &amp; ita c&etail;teris figuris de&longs;eruire po&longs;&longs;it. </s>
          <s id="N15E24">Cum <lb/>exempli gratia in pentagonis mod&ograve; in vno, mod&ograve; in alio &longs;i&shy;<lb/>tu centrum reperiatur; prout &longs;unt diuer&longs;&etail; figur&aelig;. </s>
          <s id="N15E2A">Po&longs;&longs;umus <lb/>quidem in vnaqua&que; figura reperire punctum po&longs;itione, <lb/>quod &longs;it quidem centrum grauitatis illius determinat&aelig; figu&shy;<lb/>r&etail;t. </s>
          <s id="N15E32">vt in fine primilibri o&longs;tendimus. </s>
          <s id="N15E34">e&longs;&longs;et tamen impo&longs;&longs;ibile <lb/>in omnibus proprium certum, ac determinatum &longs;itum repe&shy;<lb/>rire; vt &longs;cilicet &longs;it in tali linea, taliqu&egrave; modo diui&longs;a, vtomnib^&lcub;9} <lb/>pentagonis, &amp; hexagonis, c&aelig;teri&longs;qu&egrave; huiu&longs;modi de&longs;eruire <lb/>po&longs;&longs;it. </s>
          <s id="N15E3E">vt determinatur in triangulis, &amp; vt determinari pote&longs;t <lb/>in quadrilateris; qu&aelig; vel &longs;int parallelogramma, vel duo <expan abbr="&longs;alt&etilde;">&longs;altem</expan> <lb/>latera &longs;int &aelig;quidi&longs;tantia. </s>
          <s id="N15E48">c&ugrave;m in his conuenientia, qu&agrave;m <lb/>triangulis accidere o&longs;tendimus, reperiatur; quandoquidem <lb/>&longs;unt <expan abbr="tri&atilde;gulorum">triangulorum</expan> portiones. </s>
          <s id="N15E52">&longs;imiliter in parallelogrammis fa <lb/>cil&egrave; erit o&longs;tendere aliquam inter &longs;e &longs;imilitudinem exi&longs;tere. <expan abbr="p&etilde;-tagona">pen&shy;<lb/>tagona</expan> ver&ograve; hexagona, &amp; c&aelig;ter&aelig; figur&aelig;, qu&aelig; angulos &aelig;qua&shy;<lb/>les, &amp; &aelig;qualia latera habent; iam con&longs;tat &longs;imilia e&longs;&longs;e inter &longs;e. <lb/>pr&aelig;terea circuliomnes &longs;unt &longs;imiles. </s>
          <s id="N15E60">Ellip&longs;es quo&que; inter &longs;e <lb/>aliquam habent &longs;imilitudinem, in quibus de&longs;cribitur figura, <lb/>plan&egrave; in&longs;cripta. </s>
          <s id="N15E66">vt per&longs;picuum e&longs;t in libro Federici Comman<lb/>dini de centro grauitatis &longs;olidorum. </s>
          <s id="N15E6A">ac propterea in his, &amp; in <lb/>alijs, quibus inter &longs;e aliqua &longs;imililudo reperiri pote&longs;t, centrum <lb/>quo&que; grauitatis determinari poterit. </s>
        </p>
        <p id="N15E70" type="margin">
          <s id="N15E72"><margin.target id="marg267"/><emph type="italics"/>ex<emph.end type="italics"/> 2.<emph type="italics"/>&longs;exti <lb/>ex l&egrave;mate <lb/><expan abbr="&itilde;">im</expan> <expan abbr="&longs;ec&utilde;d&atilde;">&longs;ecundam</expan> d <lb/><expan abbr="m&otilde;&longs;tratio-ne">mon&longs;tratio&shy;<lb/>ne</expan><gap/>. pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N15E95" type="margin">
          <s id="N15E97"><margin.target id="marg268"/>17. <emph type="italics"/>quinti. <lb/>coro.<emph.end type="italics"/> 19. <lb/><emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15EA9" type="margin">
          <s id="N15EAB"><margin.target id="marg269"/>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.157.1.jpg" xlink:href="077/01/157/1.jpg"/>
        <p id="N15EB8" type="head">
          <s id="N15EBA">LEMMA.</s>
        </p>
        <p id="N15EBC" type="main">
          <s id="N15EBE">Sint quatuor magnitudines ABCD. &longs;itqu&egrave; A maior B; <lb/>&amp;C maior D. Dico A ad D maiorem habere proportio&shy;<lb/>nem, qu&agrave;m habet B ad C. </s>
        </p>
        <pb xlink:href="077/01/158.jpg" pagenum="154"/>
        <p id="N15EC7" type="main">
          <s id="N15EC9">Hoc &agrave; nobis o&longs;ten&longs;um fuitinitio tractatus devecte in no&shy;<lb/>&longs;tris mechanicishoc pacto. </s>
        </p>
        <figure id="id.077.01.158.1.jpg" xlink:href="077/01/158/1.jpg"/>
        <p id="N15ED0" type="main">
          <s id="N15ED2">
<arrow.to.target n="marg270"/> Quoniam enim A ad C maiorem habet pro<gap/><lb/>portionem, quam B ad C; &amp; A ad D maiorem <lb/>quo&que; habet proportionem, qu&agrave;m habetad C; <lb/>A igitur ad D maiorem habebit, qu&agrave;m B ad C. <lb/>quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N15EE1" type="margin">
          <s id="N15EE3"><margin.target id="marg270"/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N15EEC" type="head">
          <s id="N15EEE">PROPOSITIO. IIII.</s>
        </p>
        <p id="N15EF0" type="main">
          <s id="N15EF2">Omnis portionis recta linea, rectanguliqu&egrave; co <lb/>ni &longs;ectione content&aelig;, centrum grauitatis e&longs;t in dia<lb/>metro portionis. </s>
        </p>
        <figure id="id.077.01.158.2.jpg" xlink:href="077/01/158/2.jpg"/>
        <p id="N15EFB" type="main">
          <s id="N15EFD"><emph type="italics"/>Sit portio, vt dicta e&longs;t, ABC; cuius diameter &longs;it BD. demon&shy;<lb/>&longs;trandum est dict&aelig; portionis centrum grauitatis e&longs;&longs;e in linea BD. &longs;i.n. <lb/>non, &longs;it punctum E. &amp; ab ip&longs;o ducatur ip&longs;i BD aquidistans EF; at&shy;<lb/>&que; in portione in&longs;cribatur triangulum ABC eandem ba&longs;im<emph.end type="italics"/> AC <lb/><emph type="italics"/>habens, &amp; altitudinem<emph.end type="italics"/> portioni <emph type="italics"/>&aelig;qualem. </s>
          <s id="N15F14">&amp; quam proportionem <lb/>habet CF ad FD, eandem habeat triangulum ABC ad &longs;pacium<emph.end type="italics"/>
<pb xlink:href="077/01/159.jpg" pagenum="155"/>k. <emph type="italics"/>in portione autem plan&egrave; in&longs;cribatur figura rectilinea<emph.end type="italics"/> AGBNC, <emph type="italics"/>ita <lb/>vt relict&aelig; portiones<emph.end type="italics"/> AOG GPB BQN NRC &longs;imul <emph type="italics"/>&longs;int minores<emph.end type="italics"/>
<arrow.to.target n="marg271"/><lb/><emph type="italics"/>ip&longs;o K. in&longs;cript&aelig; quidem rectiline&aelig; figur&aelig; centrum grauitatis est in linea <lb/>B D. &longs;it punctum H. connectaturqu&egrave; HE, &amp; producatur; &amp;<emph.end type="italics"/> &agrave; pun<lb/>cto C <emph type="italics"/>ip&longs;i B D ducatur &aelig;quidistans CL.<emph.end type="italics"/> Quoniam autem por <lb/>tiones AOG GPB BQN NRC &longs;imul &longs;unt ip&longs;o K mino&shy;<lb/>res; maiorem habebit proportionem triangulum ABC ad
<arrow.to.target n="marg272"/> di&shy;<lb/>ctas portiones, qu&agrave;m ad K; in&longs;cripta ver&ograve; figura AGBNC ma <lb/>ior e&longs;t triangulo ABC, K ver&ograve; maius e&longs;t reliquis portionibus. <lb/><emph type="italics"/>Mani&longs;e&longs;tum est<emph.end type="italics"/> igitur <emph type="italics"/>figuram rectilineam<emph.end type="italics"/> ACBNC <emph type="italics"/>in portione in-<emph.end type="italics"/>
<arrow.to.target n="marg273"/><lb/><emph type="italics"/>&longs;criptam <expan abbr="maior&etilde;">maiorem</expan> habere proportionem adreliquas portiones<emph.end type="italics"/> AOG GPB <lb/>BQN, NRC, <emph type="italics"/>qu&agrave;m triangulum ABC ad K. &longs;ed vt triangulum <lb/>ABC ad K, ita est CF ad FD; figura igitur in&longs;cripta ad reliquas por&shy;<lb/>tiones maiorem habebit proportionem, quam CF ad FD; hoc e&longs;t LE ad <lb/>EH.<emph.end type="italics"/> C&ugrave;m &longs;int LH CD &agrave; lineis &aelig;quidi&longs;tantibus LC EF
<arrow.to.target n="marg274"/><lb/>HD drui&longs;&aelig;. </s>
          <s id="N15F87">quare c&ugrave;m figura in&longs;cripta ad reliquas portio&shy;<lb/>nes maiotem habeat proportionem, qu&agrave;m LE ad EH; linea, <lb/>qu&aelig; ad EH eandem habeat <expan abbr="proportion&etilde;">proportionem</expan>, qu&agrave;m figura in&longs;cri&shy;<lb/>pta ad reliquas portiones, maior erit, <expan abbr="qu&atilde;">quam</expan> LE. <emph type="italics"/>Habeat igitur ME<emph.end type="italics"/>
<arrow.to.target n="marg275"/><lb/><emph type="italics"/>ad EH <expan abbr="proportion&etilde;">proportionem</expan> eam, <expan abbr="qu&atilde;">quam</expan> figura in&longs;cripta ad portiones. </s>
          <s id="N15FAC">Quoniam igi&shy;<lb/>tur punctum E centrum e&longs;t grauitatis totius portionis, figur&aelig; <expan abbr="aut&etilde;">autem</expan> in ip&longs;a <lb/>in&longs;cript&aelig;<emph.end type="italics"/> centrum grauitatis <emph type="italics"/>est punctum H: constat reliqu&aelig; magni&shy;<lb/>tudinis ex circumrelictis portionibus compo&longs;it&aelig; centrum grauitatis e&longs;&longs;e in <lb/>linea HE producta; ita vt a&longs;&longs;umpta aliqua recta linea<emph.end type="italics"/> ME <emph type="italics"/>eam proportio <lb/>nem habeat ad EH, quam figura in&longs;cripta ad circumrelictas portiones. <lb/>Quare magnitudinis ex circumrelictis portionibus compo&longs;it&aelig; centrum gra<lb/>uitatis e&longs;t punctum M. quod est ab&longs;urdum. </s>
          <s id="N15FCC">Ducta enim linea<emph.end type="italics"/> ST <emph type="italics"/>per <lb/>punctum M ip&longs;i BD &aelig;quidi&longs;tante, in ea omnes circumrelict&aelig; portiones <lb/>centra grauitatis habebunt.<emph.end type="italics"/> hoc e&longs;t magnitudinis ex portioni&shy;<lb/>bus BPG-BQN compo&longs;it&aelig; centrum grauitatis e&longs;&longs;et in parte <lb/>MS. centrum ver&ograve; grauitatis portionum AOG CRN e&longs;&longs;et in <lb/>parte MX; ita ut M omnium dictarum portionum e&longs;&longs;et gra&shy;<lb/>uitatis centrum. </s>
          <s id="N15FE3">qu&aelig; &longs;untquidem inconuenientia. </s>
          <s id="N15FE5">quipp&egrave; <lb/>qu&aelig; etiam eodem modo &longs;e&que;ntur, &longs;i ST ip&longs;i BD <expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan> <lb/>non e&longs;&longs;et. <emph type="italics"/>Patet igitur centrum grauitatis<emph.end type="italics"/> portionis ABC <emph type="italics"/>e&longs;&longs;e in <lb/>linea BD.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/160.jpg" pagenum="156"/>
        <p id="N16000" type="margin">
          <s id="N16002"><margin.target id="marg271"/>2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1600B" type="margin">
          <s id="N1600D"><margin.target id="marg272"/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16016" type="margin">
          <s id="N16018"><margin.target id="marg273"/><emph type="italics"/>lemma.<emph.end type="italics"/></s>
        </p>
        <p id="N16020" type="margin">
          <s id="N16022"><margin.target id="marg274"/><emph type="italics"/>1: <expan abbr="tem-&itilde;">tem-im</expan><emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui<emph.end type="italics"/></s>
        </p>
        <p id="N16034" type="margin">
          <s id="N16036"><margin.target id="marg275"/>8. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16041" type="head">
          <s id="N16043">SCHOLIVM.</s>
        </p>
        <p id="N16045" type="main">
          <s id="N16047">In hac demon&longs;tratione ob&longs;eruandum e&longs;t; qu&ograve;d <expan abbr="qu&atilde;do">quando</expan> Ar&shy;<lb/>chimedes inquit, <emph type="italics"/>in portione autem plan&egrave; in&longs;cribatur figura<emph.end type="italics"/> &amp;c. </s>
          <s id="N16055">in&shy;<lb/>telligendum e&longs;t, in&longs;cribatur prim&ograve; pentagonum AGBNC <lb/>in portione plan&egrave; in&longs;criptum; quod quidem relin&que;t por&shy;<lb/>tiones AOG GPB BQN NRC, qu&aelig; &longs;imul uel erunt minores <lb/>&longs;pacio K, vel min&ugrave;s. </s>
          <s id="N1605F">&longs;i non, rur&longs;us plan&egrave; adhuc in&longs;cribatur <lb/>in portione ABC nonagonum; deinde alia figura; idqu&egrave; &longs;em&shy;<lb/>per fiat, donec circumrelict&aelig; portiones &longs;imul &longs;int &longs;pacio K <lb/>minores. </s>
          <s id="N16067">quod quidem fieri po&longs;&longs;e ex prima decimi Euclidis <lb/>
<arrow.to.target n="marg276"/> patet. </s>
          <s id="N1606F">Aufertur enim &longs;emper maius, <expan abbr="qu&atilde;">quam</expan> dimidium. </s>
          <s id="N16075">C&ugrave;m qu&aelig; <lb/>libet portio paraboles trianguli plane in ip&longs;a in&longs;eripti &longs;it &longs;e&longs;&shy;<lb/>quitertia. </s>
          <s id="N1607B">Vnde triangulum ABC maius e&longs;t, qu&agrave;m <expan abbr="dimidi&utilde;">dimidium</expan> <lb/>portionis ABC. triangulum qu&egrave; AGB maius, qu&agrave;m <expan abbr="dimidi&utilde;">dimidium</expan> <lb/>portionis AGB. &longs;imiliter triangulum BNC portionis BNC &amp; <lb/>ita in alijs. </s>
          <s id="N1608B">Qu&aelig; quidem omnia &longs;untquo&que; manife&longs;ta ex vi <lb/>ge&longs;ima propo&longs;itione, eiu&longs;qu&egrave; demon&longs;tratione de quadratura <lb/>paraboles Archimedis. </s>
        </p>
        <p id="N16091" type="margin">
          <s id="N16093"><margin.target id="marg276"/>17. <emph type="italics"/>Archi. <lb/>de quad. <lb/>parab.<emph.end type="italics"/></s>
        </p>
        <p id="N160A0" type="main">
          <s id="N160A2">Demon&longs;trato centro grauitatis cuiu&longs;libet paraboles in eius <lb/>diametro exi&longs;tere; o&longs;tendet Archimedes, (vt diximus) in pa&shy;<lb/>rabolis grauitatum centra in eadem proportione diametros <lb/>di&longs;pe&longs;cere. </s>
          <s id="N160AA">antequam autem hoc demon&longs;tret, duas pr&etail;mittit <lb/>&longs;e&que;ntes propo&longs;itiones ad demon&longs;trationem nece&longs;&longs;arias. </s>
        </p>
        <p id="N160AE" type="head">
          <s id="N160B0">PROPOSITIO. V.</s>
        </p>
        <p id="N160B2" type="main">
          <s id="N160B4">Si in portione recta linea, rectanguliqu&egrave; coni <lb/>&longs;ectione contenta rectilinea figura plan&egrave; in&longs;criba <lb/>tur, totius portionis <expan abbr="centr&utilde;">centrum</expan> grauitatis <expan abbr="prop&itilde;quius">propinquius</expan> <lb/>e&longs;t vertici portionis, <expan abbr="qu&atilde;">quam</expan> <expan abbr="centr&utilde;">centrum</expan> figur&aelig; in&longs;cript&aelig;. </s>
        </p>
        <pb xlink:href="077/01/161.jpg" pagenum="157"/>
        <p id="N160CF" type="main">
          <s id="N160D1"><emph type="italics"/>Sit portio ABC, qualis dictaest, ip&longs;ius ver&ograve; diameter &longs;it BD. <lb/>prim&ugrave;mqu&egrave; in ip&longs;a plan&egrave; in&longs;eribatur triangulum ABC. &amp; diuidatur<emph.end type="italics"/>
<arrow.to.target n="marg277"/><lb/><emph type="italics"/>BD in E, ita vt dupla &longs;it BE ip&longs;ius ED. erit vtiqu&egrave; trtanguli ABC <lb/>centrum grauitatis punctum E. Diuidatur it&agrave;&que; bi&longs;ariam vtra&que; <lb/>AB BC in punctis FG. &amp; <gap/>punctis FG ip&longs;i BD ducantur &aelig;quidi&shy;<lb/>&longs;tantes FK GL. erit &longs;an&egrave; portionis A<emph.end type="italics"/>k<emph type="italics"/>B centrum grauitatis in linea<emph.end type="italics"/>
<arrow.to.target n="marg278"/><lb/><emph type="italics"/>F<emph.end type="italics"/>k. <emph type="italics"/>portionis ver&ograve; BLC centrum grauit atis erit in linea GL. &longs;int ita&shy;<lb/>&que; puncta HI. connectanturqu&egrave; HI FG.<emph.end type="italics"/> qu&aelig; BD &longs;ecent in QN. <lb/>
<arrow.to.target n="fig72"/><lb/>erit vti&que; punctum Q vertici B propinquius, qu&agrave;m N. quia
<arrow.to.target n="marg279"/><lb/>ver&ograve; e&longs;t BF ad FA, vt BG ad GC, erit FG <expan abbr="&aelig;quidi&longs;t&atilde;sip&longs;i">&aelig;quidi&longs;tansip&longs;i</expan> AC, <lb/>eritqu&egrave; FN ad NG, vt AD ad DC. e&longs;t ver&ograve; AD ip&longs;i DC &aelig;qua&shy;<lb/>lis, ergo FN NG inter &longs;e &longs;unt &aelig;quales. </s>
          <s id="N16118">quoniam autem FN <lb/>e&longs;t ip&longs;i AD &aelig;quidi&longs;tans, erit AF ad FB, vt DN ad NB. e&longs;t au
<arrow.to.target n="marg280"/><lb/>tem AF dimidia ip&longs;ius AB; c&ugrave;m &longs;int AF FB &etail;quales ergo &amp; <lb/>DN dimidia e&longs;t ip&longs;ius DB. at ver&ograve; quoniam DE terria e&longs;t <lb/>pars ip&longs;ius DB, &longs;iquidem e&longs;t BE ip&longs;ius ED dupla, erit pun&shy;<lb/>ctum N propinquius vertici B portionis, qu&agrave;m pun&shy;<lb/>ctum E. <emph type="italics"/>Et quoniam parallelogrammum est HFGI. &amp; &aelig;qualis est <lb/>FN ip&longs;i NG, erit QH ip&longs;i QI &aelig;qualis. </s>
          <s id="N1612E">ac propterea magnitudinis ex <lb/>vtri&longs;&que; A<emph.end type="italics"/>k<emph type="italics"/>B BLC portionibus compo&longs;it&aelig; centrum grauitatis e&longs;t in<emph.end type="italics"/>
<arrow.to.target n="marg281"/><lb/><emph type="italics"/>medio line&aelig; HI, c&ugrave;m portiones<emph.end type="italics"/> AKB BLC <emph type="italics"/>&longs;int &aelig;quales. </s>
          <s id="N16148">erit &longs;cilicet <lb/>punctum <expan abbr="q.">&que;</expan> Quoniam autem trianguli ABC centrum grauitatis e&longs;t <lb/>punctum E, magnitudinis ver&ograve; ex vtri&longs;qu&egrave; A<emph.end type="italics"/>k<emph type="italics"/>B BLC compo&longs;is&aelig;<emph.end type="italics"/>
<pb xlink:href="077/01/162.jpg" pagenum="158"/><emph type="italics"/>e&longs;t punctum <expan abbr="q.">&que;</expan> con&longs;tat totius portionis ABC centrum grauitatis e&longs;&longs;e<emph.end type="italics"/><lb/>
<arrow.to.target n="marg282"/> <emph type="italics"/>in linea QE. hoc est inter puncta QE. Quare totius portionis <expan abbr="c&emacr;trum">centrum</expan> <lb/>grauitatis propinquius e&longs;t vertici portionis, quam<emph.end type="italics"/> centrum grauitatis <lb/><emph type="italics"/>trianguli plan&egrave; in&longs;cripti.<emph.end type="italics"/></s>
        </p>
        <p id="N1617F" type="margin">
          <s id="N16181"><margin.target id="marg277"/><emph type="italics"/>ante pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1618B" type="margin">
          <s id="N1618D"><margin.target id="marg278"/>4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16196" type="margin">
          <s id="N16198"><margin.target id="marg279"/>2. <emph type="italics"/>&longs;exti&shy;<lb/>lemma ta <lb/>aliter<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui^&lcub;9}<emph.end type="italics"/></s>
        </p>
        <p id="N161AC" type="margin">
          <s id="N161AE"><margin.target id="marg280"/>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N161B7" type="margin">
          <s id="N161B9"><margin.target id="marg281"/>4. <emph type="italics"/>primi <lb/>buius. <lb/>ex its qu&aelig; <lb/>ante<emph.end type="italics"/> 2. <emph type="italics"/>hu<lb/>ius demon <lb/>&longs;trata &longs;unt. <lb/>ex<emph.end type="italics"/> 8. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N161DC" type="margin">
          <s id="N161DE"><margin.target id="marg282"/>*</s>
        </p>
        <figure id="id.077.01.162.1.jpg" xlink:href="077/01/162/1.jpg"/>
        <figure id="id.077.01.162.2.jpg" xlink:href="077/01/162/2.jpg"/>
        <p id="N161E9" type="main">
          <s id="N161EB"><emph type="italics"/>Rur&longs;us in portione pent agonum rectilineum AKBLC plan&egrave; in&longs;cri&shy;<lb/>batur. </s>
          <s id="N161F1">&longs;itqu&egrave; totius portionis diameter BD, vtriu&longs;&que; autem portionis<emph.end type="italics"/><lb/>AKB. BLC <emph type="italics"/>diameter &longs;it vtra&que; KF LG. &amp; quoniam in portione <lb/>AKB plan&egrave; in&longs;cripta est figura rectilinea<emph.end type="italics"/> trilatera AKB, <emph type="italics"/>totius por <lb/>tionis<emph.end type="italics"/> AKB <emph type="italics"/>centrum grauitatis est propinquius vertici<emph.end type="italics"/> K, <emph type="italics"/>quam <lb/>centrum rectiline&aelig; figur&aelig;<emph.end type="italics"/> AKB. <emph type="italics"/>&longs;it ita&que; portionis A<emph.end type="italics"/>k<emph type="italics"/>B centrum <lb/>grauitatis punctum H; trianguli ver&ograve; punctum 1. Rur&longs;us autem &longs;it por <lb/>tionis BLC centrum grauitatis punctum M. trianguli ver&ograve;<emph.end type="italics"/> BLC <emph type="italics"/>pun<lb/>ctum N. iunganturqu&egrave; HM JN<emph.end type="italics"/>; qu&aelig; BD &longs;ecent in punctis <lb/>QT. erit vti&que; punctum Q vertici B propinquius, <expan abbr="qu&atilde;">quam</expan> <lb/>T. &amp; quoniam (&longs;i ducta e&longs;&longs;et FG) line&aelig; HM IN FG ab &aelig;<lb/>
<arrow.to.target n="marg283"/> quidi&longs;tantibus lineis KF BD LG in eadem <expan abbr="diuid&utilde;tur">diuiduntur</expan> pro&shy;<lb/>portione. </s>
          <s id="N16241">FG ver&ograve;, vt o&longs;ten&longs;um e&longs;t, bifariam &agrave; linea BD di&shy;<lb/>uideretur; ergo &amp; line&aelig; HM IN bifariam diui&longs;&etail; <expan abbr="proucni&etilde;t">proucnient</expan>. <lb/><emph type="italics"/>&aelig;qualis est igitur HQ ip&longs;i QM; &amp; IT ip&longs;i TN. &longs;ed triangulo <lb/>AKB &aelig;quale est triangulum BLC; portio vero A<emph.end type="italics"/>k<emph type="italics"/>B portioni <lb/>BLC e&longs;t &aelig;qualis. </s>
          <s id="N16257">Demonstratum e&longs;t enim alis in lo&ccedil;is portiones<emph.end type="italics"/>
<pb xlink:href="077/01/163.jpg" pagenum="159"/><emph type="italics"/>&longs;e&longs;quitertias e&longs;&longs;e triangulorum, erit igitur magnitudinis ex vtri&longs;&que; por-<emph.end type="italics"/>
<arrow.to.target n="marg284"/><lb/><emph type="italics"/>tionibus A<emph.end type="italics"/>k<emph type="italics"/>B BLC compo&longs;it&aelig; centrum grauitatis punctum <expan abbr="q.">&que;</expan> magni&shy;<lb/>tudinis ver&ograve; ex vtri&longs;&que; triangulis AKB BLC compo&longs;it&aelig; punctum <lb/>T. Rur&longs;us ita&que; quoniam trianguli ABC centrum grauitatis e&longs;t <expan abbr="punct&umacr;">punctum</expan> <lb/>E, magnitudinis ver&ograve; ex vtri&longs;&que; A<emph.end type="italics"/>k<emph type="italics"/>B BLC portionibus punctum <lb/><expan abbr="q.">&que;</expan> manifestum e&longs;t totius portionis A<emph.end type="italics"/>B<emph type="italics"/>C centrum grauitatis e&longs;&longs;e in linea <lb/>QE ita diui&longs;a<emph.end type="italics"/> in O puncto, <emph type="italics"/>vt quam proportionem habet trian&shy;<lb/>gulum ABC ad vtra&longs;&que; portiones A<emph.end type="italics"/>k<emph type="italics"/>B BLC, eandem habeat por<emph.end type="italics"/>
<arrow.to.target n="marg285"/><lb/><emph type="italics"/>tio ip&longs;ius terminum habens punctum Q,<emph.end type="italics"/> hoc e&longs;t OQ <emph type="italics"/>ad portionem <lb/>minorem<emph.end type="italics"/> OE. <emph type="italics"/>pentagoni autem AKBLC,<emph.end type="italics"/> hoc e&longs;t magnitudinis <lb/>ex triangulo ABC, trianguli&longs;qu&egrave; AKB BLC compo&longs;it&aelig; <lb/><emph type="italics"/>centrum grauitatis e&longs;t in linea ET &longs;ic diui&longs;a<emph.end type="italics"/> in S, <emph type="italics"/>vt quam habet <lb/>proportionem triangulum ABC ad triangula AKB BLC, eande ha&shy;<lb/>beat portio ip&longs;ius ad T terminata,<emph.end type="italics"/> hoc e&longs;t ST <emph type="italics"/>ad reliquam<emph.end type="italics"/> SE. <lb/><emph type="italics"/>Quoniam igitur maiorem habet proportionem triangulum ABC ad <expan abbr="tri&atilde;">triam</expan><emph.end type="italics"/>
<arrow.to.target n="marg286"/><lb/><emph type="italics"/>gula KAB LBC, quam ad portiones<emph.end type="italics"/> AKB BLC; minora enim <lb/>&longs;unt triangula portionibus. </s>
          <s id="N162EB">habebit TS ad SE <expan abbr="mior&etilde;">miorem</expan> pro&shy;<lb/>portio nem, quam QO ad OE ac propterea erit <expan abbr="punct&utilde;">punctum</expan> S <lb/>propinquiusip&longs;i E, qu&agrave;m O. Nam &longs;i punctum S prim&ugrave;m <lb/>e&longs;&longs;et in eodem puncto O, tunc TO ad OE, non quidem <lb/>maiorem, &longs;ed minorem haberet proportionem, qu&agrave;m QO
<arrow.to.target n="marg287"/><lb/>ad OE, c&ugrave;m &longs;it TO minor QO. &longs;imiliter ob eadem cau<lb/>&longs;am &longs;i punctum S e&longs;&longs;et inter OT, minorem haberet
<arrow.to.target n="marg288"/> pro&shy;<lb/>portionem TS ad SE, qu&agrave;m QS ad SE, quare &amp; ad huc <lb/>maiorem haberet proportionem QO ad OE, qu&agrave;m TS <lb/>ad SE. nece&longs;&longs;e e&longs;t igitur punctum S e&longs;&longs;e inter puncta OE. <lb/>Itaqu&egrave; c&ugrave;m punctum O &longs;it <expan abbr="centr&utilde;">centrum</expan> grauitatis portionis ABC, <lb/>punctum ver&ograve; S centrum &longs;it grauitatis rectiline&aelig; figur&aelig; <lb/>AK BLC; <emph type="italics"/>constat portionis ABC centrum grauitatis propinquius <lb/>e&longs;&longs;e vertici B, qu&agrave;m centrum rectiline&aelig; figur&aelig; in&longs;cript&aelig;. </s>
          <s id="N1631D">Et in om&shy;<lb/>nibus rectilineis figuris in portionibus plan&egrave; in&longs;criptis eadem e&longs;t ratio.<emph.end type="italics"/><lb/>quod demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/164.jpg" pagenum="160"/>
        <p id="N16328" type="margin">
          <s id="N1632A"><margin.target id="marg283"/><emph type="italics"/>prima lem <lb/>ma in<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi bui^&lcub;9}.<emph.end type="italics"/></s>
        </p>
        <p id="N1633B" type="margin">
          <s id="N1633D"><margin.target id="marg284"/>4. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16348" type="margin">
          <s id="N1634A"><margin.target id="marg285"/><emph type="italics"/>ex<emph.end type="italics"/> 8. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1635A" type="margin">
          <s id="N1635C"><margin.target id="marg286"/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16365" type="margin">
          <s id="N16367"><margin.target id="marg287"/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16370" type="margin">
          <s id="N16372"><margin.target id="marg288"/>8.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1637B" type="head">
          <s id="N1637D">SCHOLIVM.</s>
        </p>
        <p id="N1637F" type="main">
          <s id="N16381">
<arrow.to.target n="marg289"/> In fine prim&aelig; demon&longs;trationis in vltima conclu&longs;ione <expan abbr="qu&atilde;-do">quan&shy;<lb/>do</expan> inquit Archimedes. <emph type="italics"/>Quare totius portionis centrum propinquius <lb/>e&longs;t vertici portionis, <expan abbr="qu&atilde;">quam</expan> trianguli plan&egrave; in &longs;cripti<emph.end type="italics"/> Gra cus codexita &longs;e <lb/>habet <foreign lang="greek">w)\s t) e)\ihka e)ggute&lt;10&gt;on ta=s tou= tma/matos ko&lt;10&gt;ufa=s to\ ke/nt&lt;10&gt;on tou= o)\lou <lb/>tma/matos, h)/ tou= e)gg&lt;10&gt;afome/nou t&lt;10&gt;igw/nou gnw&lt;10&gt;i/mws</foreign>. verbaqu&egrave; <foreign lang="greek">e)/ih ka</foreign> mal&egrave; in<lb/>terpo&longs;ita &longs;unt, nullumqu&egrave; cum alijs rectum &longs;en&longs;um habent, <lb/>quare horum loco ponerem <foreign lang="greek">e)si/</foreign>, vt &longs;en&longs;us &longs;it, <foreign lang="greek">w)/s te)/ggu/te&lt;10&gt;on e)si ta=s <lb/>tou= tma/matos</foreign>, &amp;c. </s>
        </p>
        <p id="N163B3" type="margin">
          <s id="N163B5"><margin.target id="marg289"/>*</s>
        </p>
        <figure id="id.077.01.164.1.jpg" xlink:href="077/01/164/1.jpg"/>
        <p id="N163BC" type="main">
          <s id="N163BE">Ob&longs;eruandum autem occurrit in demon&longs;trationibus, ab <lb/>Archimede allatis; qu&ograve;d in prima demon&longs;tratione &longs;upponit <lb/>Archimedes, HFGI e&longs;&longs;e parallelogrammum. </s>
          <s id="N163C4">qu&ograve;d vt &longs;it pa&shy;<lb/>rallelogrammum, nece&longs;&longs;e e&longs;t &longs;upponere centra grauitatis HI <lb/>&longs;ecare lineas KF LG in partes inuicem proportionales. </s>
          <s id="N163CA">quod <lb/>tamen &longs;upponi po&longs;&longs;e minim&egrave; videtur. </s>
          <s id="N163CE">Et &longs;i quis ex quinto <lb/>po&longs;tulato obijceret, centragrauitatis in &aelig;qualibus, &longs;imilibu&longs;&shy;<lb/>qu&egrave; figuris e&longs;&longs;e &aelig;qualiter po&longs;ita; admitti quidem pote&longs;t; quo-
<pb xlink:href="077/01/165.jpg" pagenum="161"/>niam figur&aelig;, ipforum qu&egrave; centra inter &longs;e coaptari po&longs;&longs;unt. </s>
          <s id="N163D8">vt <lb/>omnibus figuris rectilineis &etail;qualibus, &amp; &longs;imilib^&lcub;9} accidere po&shy;<lb/>te&longs;t. </s>
          <s id="N163DE">Hoc tam&eacute; contingere po&longs;&longs;e in parabolis, vt AKB BLC, vi <lb/>detur in <expan abbr="c&otilde;ueni&eacute;s">conueni&eacute;s</expan>. <expan abbr="N&atilde;">Nam</expan> quamuis AKB BLC &longs;int &aelig;quales, &amp; &longs;int <lb/><expan abbr="eti&atilde;">etiam</expan> &longs;imiles; non &longs;unt tamen &longs;imiles ea &longs;i militudine, vt &longs;untre <lb/>ctiline&aelig; figur&aelig;; vtantea diximus. </s>
          <s id="N163F1">Quod etiam <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> fit ex <lb/>hoc, quia non &longs;emper coaptari porei&longs;t portio AKB <expan abbr="c&utilde;">cum</expan> portio&shy;<lb/>ne BLC. <expan abbr="n&otilde;">non</expan>. <expan abbr="n.">enim</expan>&longs;emper recta linea BC erit &aelig;qualisip&longs;i BA; <expan abbr="ne&qacute;">ne&que;</expan>; <lb/>&longs;ectionis linea BLC &longs;ectionis line&etail; BKA &etail;qualis exi&longs;tet. <expan abbr="C&utilde;">Cum</expan> <expan abbr="n&otilde;">non</expan> <lb/>&longs;emper AC, &amp; qu&aelig; &longs;untip&longs;i AC &aelig;quidi&longs;tates ad rectos &longs;int an <lb/>gulos diametro BD. &longs;i.n. </s>
          <s id="N16419">&etail;quidi&longs;tantes line&etail; diametro fuerint <lb/>perpendiculares, tunc AB BC inter &longs;e &etail;quales e&longs;&longs;ent; <expan abbr="portio&qacute;">portio&que;</expan>; <lb/>AKB <expan abbr="c&utilde;">cum</expan> portione BLC coaptari po&longs;&longs;et: &longs;ec&ugrave;s autem minim&egrave;. <lb/>Quare centra grauiratis HI lineas KFLG in eadem proportio <lb/>ne &longs;ecare minim&egrave;&longs;upponi po&longs;&longs;e videtur; t&ugrave;m exijs, qu&aelig; dicta <lb/>&longs;unt; t&uacute; quia hoc o&longs;tendet Archimedes in &longs;eptima propo&longs;itio <lb/>ne. </s>
          <s id="N1642F">qu&ograve;d &longs;i adhuc non e&longs;t <expan abbr="dem&otilde;&longs;trat&uacute;">demon&longs;trat&uacute;</expan>, <expan abbr="n&otilde;">non</expan> pote&longs;t <expan abbr="quo&qacute;">quo&que;</expan>; &longs;uppo <lb/>ni; pr&aelig;&longs;ertim c&ugrave;m &longs;it demon&longs;trabile. </s>
          <s id="N1643F">ac propterea <expan abbr="dem&otilde;&longs;tra-tio">demon&longs;tra&shy;<lb/>tio</expan> nullam videturvim haberead <expan abbr="o&longs;tendend&utilde;">o&longs;tendendum</expan>, quod propo&longs;i&shy;<lb/>t&uacute; fuit. </s>
          <s id="N1644D">Huic <expan abbr="tam&etilde;">tamen</expan> occurri po&longs;&longs;evidetur <expan abbr="c&utilde;">cum</expan> Eutocio in exphca <lb/>tione huiusloci dicendo, hoc &longs;upponere Archimed&eacute;, quia por <lb/>tiones AKBBLC &longs;unt&etail;quales, quar&uacute; diametri KFLG &longs;unt &etail;&shy;<lb/>quales, &amp; <expan abbr="&etail;quidi&longs;t&atilde;tes">&etail;quidi&longs;tantes</expan>, qu&aelig; &longs;imiliter diuiduntur &agrave; punctis HI; <lb/>vnde erit kG ad HF, vt LI ad IG. ex quibus colligit HF ip&longs;i IG <lb/><expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> e&longs;&longs;e; ac propterea HG <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> exiltere. </s>
          <s id="N1646C">Qu&aelig; <expan abbr="t&ntilde;">tnm</expan> <lb/>re&longs;pon&longs;io <expan abbr="n&otilde;">non</expan> e&longs;t Eutocio digna. </s>
          <s id="N16478">c&ugrave;m ex dictis <expan abbr="n&otilde;">non</expan> &longs;it omnin&ograve; <lb/>demon&longs;tratiua, vtres mathematic&etail; <expan abbr="requir&utilde;t">requirunt</expan>; quapropter omit <lb/>tenda e&longs;t.hac.n.ratione&longs;upponitur centra HI lineas KFLG in <lb/>eadem proportione &longs;ecare.quod nullo modo &longs;upponi pote&longs;t. <lb/>Quare dici poterit, &amp; forta&longs;le recti&ugrave;s, qu&ograve;d vis demon&longs;tratio&shy;<lb/>nis videtur in hoc e&longs;&longs;e con&longs;tituta, vt &longs;upponatur puncta HI <expan abbr="v-bicun&qacute;">v&shy;<lb/>bicun&que;</expan>; e&longs;&longs;e po&longs;&longs;e in lineis KFLG; ita vt &longs;iue ducta HI fuerit, <lb/>&longs;iue etiam non fuerit ip&longs;i FG &aelig;quidi&longs;tans, demon&longs;tratio <expan abbr="tam&etilde;">tamen</expan> <lb/>&longs;uam &longs;emper habebit vim, <expan abbr="id&etilde;&qacute;">iden&que;</expan>; concludet. </s>
          <s id="N1649E">Nam ex <expan abbr="pr&aelig;ced&etilde;">pr&aelig;cedem</expan>. <lb/>ti patet centra grauitatis portionum AKB BLC e&longs;&longs;e in lineis <lb/>KF LG; hoce&longs;t inter puncta KF, &amp; LG. <expan abbr="&longs;uppon&atilde;turita&qacute;">&longs;upponanturita&que;</expan>; <expan abbr="c&etilde;-tra">cen&shy;<lb/>tra</expan> grauitatis <expan abbr="portion&utilde;">portionum</expan> AKB BLC e&longs;&longs;e puncta HI <expan abbr="vbic&utilde;&qacute;">vbicun&que;</expan>; po&shy;
<pb xlink:href="077/01/166.jpg" pagenum="162"/>&longs;ita, <expan abbr="d&utilde;modo">dummodo</expan> &longs;int in lineis KF LG, veluti Archimedes ip&longs;e in <lb/>demon&longs;tratione &longs;upponit. <expan abbr="Ducatur&qacute;">Ducatur&que;</expan>; HI; qu&aelig; vel ip&longs;i FG &aelig;&shy;<lb/>quidi&longs;tans erit, vel min&ugrave;s: &longs;i e&longs;t &aelig;quidi&longs;tans, <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> <lb/>e&longs;t HFGI, &amp; vera e&longs;t demon&longs;tratio Archimedis. </s>
          <s id="N164D0">&longs;i ver&ograve; <expan abbr="n&otilde;">non</expan> e&longs;t <lb/><expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan>, nihilominus veri&longs;&longs;ima e&longs;t eadem <expan abbr="dem&otilde;&longs;tratio">demon&longs;tratio</expan>. <expan abbr="N&atilde;">Nam</expan> <lb/>&longs;i HI ip&longs;i FG <expan abbr="n&otilde;">non</expan> e&longs;t <expan abbr="&etail;quidi&longs;t&atilde;s">&etail;quidi&longs;tans</expan>, patet in primis <expan abbr="p&utilde;ct&utilde;">punctum</expan> Q propin<lb/>quius e&longs;&longs;e vertici B portionis ABC, <expan abbr="qu&atilde;">quam</expan> <expan abbr="punct&utilde;">punctum</expan> N, ac per con&shy;<lb/>&longs;e&que;ns, <expan abbr="qu&atilde;">quam</expan> punctum E centrum grauitatis trianguli ABC. <lb/>Etquoniam line&aelig; HI FG &agrave; lineis diuiduntur KF BN LG &etail; <lb/>
<arrow.to.target n="fig73"/><lb/>
<arrow.to.target n="marg290"/> quidi&longs;tantibus, erit HQ ad QI, vt FN ad NG. e&longs;t autem FN i&shy;<lb/>pGNG &etail;qualis, ergo HQ ip&longs;i QI &etail;qualis quo&que; erit. </s>
          <s id="N16510">ita&que; <lb/>quoniam portiones AKBBLC &longs;unt &aelig;quales, erit magnitudi&shy;<lb/>nis ex vtri&longs;&que; AKB BLC portionibus compo&longs;it&etail; <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis in medio line&etail; HI. ergo eritpunctum <expan abbr="q.">&que;</expan> quo cognito <lb/>eadem demon&longs;tratio Archimedis o&longs;tendet centrum grauita&shy;<lb/>tis portionis ABC e&longs;&longs;e inter puncta <expan abbr="Eq.">E&que;</expan> Nam ex verbis ip&longs;ius, <lb/>c&ugrave;m ait, <emph type="italics"/>Quoniam autem trianguli ABC centrum grauitatis est <lb/>punctum E magnitudinis ver&ograve; ex vtri&longs;&que; AkB BLC compo&longs;ic&aelig; <lb/>est punctum <expan abbr="q;">&que;</expan> constat totius portionis ABC centrum grauitatis <lb/>e&longs;&longs;e in in linea QE. hoc est inter puncta QE. Quare totius portionis <lb/>centrum grauitatis propinquius e&longs;t vertici portionis, qu&agrave;m trian&shy;<lb/>guli plan&egrave; in&longs;cripti.<emph.end type="italics"/> <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t igitur centrum grauitatis por <lb/>tionis ABC, &longs;iu&egrave; &longs;it HI ip&longs;i FG &aelig;quidi&longs;tans, &longs;iue non &aelig;. <lb/>quidi&longs;tans, propinquius e&longs;&longs;e vertici B portionis, qu&agrave;m <expan abbr="c&etilde;trum">centrum</expan>
<pb xlink:href="077/01/167.jpg" pagenum="163"/>grauitatis trianguli ABC<gap/> Quare cuca <gap/>erba demon&longs;tratio&shy;<lb/>nis, c&ugrave;m inquit Archimedes, <emph type="italics"/>&amp; quoniam parallelogrammum est <lb/>HFGJ, &amp; &aelig;qualisest FN ip&longs;i NG.<emph.end type="italics"/> &amp;c. </s>
          <s id="N1655C">immitando &longs;ecun&shy;<lb/>dam Archimedis demon&longs;trationem huius propo&longs;itionis, vel <lb/>delenda &longs;untverba, <emph type="italics"/>parallelogrammum e&longs;t HFGI, &amp;<emph.end type="italics"/> tamquam <lb/>ab aliquo ad dita; ita vt verba &longs;int hoc modo vniuer&longs;alia, <emph type="italics"/>&amp; <lb/>quoniam &aelig;qualis e&longs;t FN ip&longs;i NG,<emph.end type="italics"/> &amp; qu&aelig; &longs;equuntur. </s>
          <s id="N16572">vel &longs;at for&shy;<lb/>ta&longs;&longs;e Archimedi vi&longs;um e&longs;t. </s>
          <s id="N16576">&longs;e o&longs;tendi&longs;&longs;e hoc contingere exi&shy;<lb/>&longs;tente HI ip&longs;i FG &aelig;quidi&longs;tante. </s>
          <s id="N1657A">qu&ograve;d &longs;i etiam non fuerit HI <lb/>&aelig;quidi&longs;tans FG, idem &longs;equi tanquam notum omi&longs;it. </s>
          <s id="N1657E">c&ugrave;m per <lb/>facilis &longs;it demon&longs;tratio, vt dictum e&longs;t. </s>
          <s id="N16582">Archimede&longs;qu&egrave; res val <lb/>d&egrave; notas &longs;&etail;p&egrave; pr&aelig;termittere&longs;olet. </s>
        </p>
        <p id="N16586" type="margin">
          <s id="N16588"><margin.target id="marg290"/>1.<emph type="italics"/><expan abbr="l&etilde;wa&itilde;">lenwaim</expan><emph.end type="italics"/> 15 <lb/><emph type="italics"/>primu hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.167.1.jpg" xlink:href="077/01/167/1.jpg"/>
        <p id="N165A0" type="main">
          <s id="N165A2">Hocidem etiam con&longs;iderari pote&longs;t in &longs;ecunda demon&longs;tra <lb/>tione quamuis verba hanc difficultatem non habeant. <expan abbr="n&atilde;">nam</expan> ea&shy;<lb/>dem &longs;equltur demon&longs;tratio, &longs;iu&egrave;&longs;it HM line&aelig; IN &etail;quidi&longs;t&aacute;s, <lb/>vel non &aelig;quidi&longs;tans, vt ex verbis Archimedis per&longs;picuum e&longs;t.
<arrow.to.target n="marg291"/><lb/>etenim manife&longs;tum e&longs;t centra grauitatis portionum AKB <lb/>BLC e&longs;&longs;einlineis KF LG. &longs;imiliter centra grauitatis
<arrow.to.target n="marg292"/> trian&shy;<lb/>gulorum AKB BLC in ijsdem e&longs;&longs;e lineis KF LG. vt in <expan abbr="p&utilde;-ctis">pun&shy;<lb/>ctis</expan> IN; qu&aelig; nece&longs;&longs;ari&ograve; diuidunt KF LG in partes propor&shy;<lb/>tionales, vnde FI GN euadunt &aelig;quales. </s>
          <s id="N165C3">&amp; quoniam por&shy;<lb/>tionum centra HM &longs;unt propinquiora verticibus KL, quam <lb/>triangulorum centra IN; ideo nece&longs;&longs;e e&longs;t <expan abbr="p&utilde;cta">puncta</expan> HM in lineis <lb/>KI LN exi&longs;tere. </s>
          <s id="N165CF">quare &longs;int puncta HM vbic&uacute;&que; in lineis KI <lb/>LN con&longs;tituta; <expan abbr="ducta&qacute;">ducta&que;</expan>; HM, qu&aelig; &longs;iu&egrave; &longs;it ip&longs;i IN &etail;quidi&longs;tans, <lb/>&longs;iuenon &aelig;quidi&longs;tans, &longs;em per erit <expan abbr="p&utilde;ctum">punctum</expan> Qpropinquius ver <lb/>tici B, quam T. eodem qu&egrave; modo erit punctum Q <expan abbr="medi&utilde;">medium</expan> li&shy;<lb/>ne&aelig; HM <expan abbr="centr&utilde;">centrum</expan> grauitatis magnitudinis ex portionib^&lcub;9} AKB <lb/>BLC compo&longs;it&aelig;. </s>
          <s id="N165EB">&longs;iquidem portiones &longs;unt &etail;quales. </s>
          <s id="N165ED">qu&etail; <expan abbr="quid&etilde;">quidem</expan> <lb/>omnia ex ip&longs;amet demon&longs;tratione &longs;unt manife&longs;ta. </s>
          <s id="N165F5">&longs;untqu&egrave; <lb/>h&aelig;c <expan abbr="ead&etilde;">eadem</expan> <expan abbr="ob&longs;eru&atilde;da">ob&longs;eruanda</expan> in duabus <expan abbr="&longs;e&quetilde;tibus">&longs;e&que;ntibus</expan> <expan abbr="dem&otilde;&longs;trationib^&lcub;9}">demon&longs;trationib^&lcub;9}</expan>. </s>
        </p>
        <p id="N16609" type="margin">
          <s id="N1660B"><margin.target id="marg291"/>4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16614" type="margin">
          <s id="N16616"><margin.target id="marg292"/><emph type="italics"/>ante<emph.end type="italics"/> 15. <lb/><emph type="italics"/>primi hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N16627" type="head">
          <s id="N16629">PROPOSITIO. VI.</s>
        </p>
        <p id="N1662B" type="main">
          <s id="N1662D">Data portione rectalinea, rectanguliqu&egrave; coni <lb/>&longs;ectione <expan abbr="c&otilde;tenta">contenta</expan>, in portione figurarectilinea pla <lb/>ne in&longs;cribi pote&longs;t; ita vt linea inter centrum graui&shy;
<pb xlink:href="077/01/168.jpg" pagenum="164"/>tatis portionis, &amp; figur&aelig; rectiline&aelig; in&longs;cript&aelig;, mi&shy;<lb/>nor &longs;it propo&longs;ita recta linea. </s>
        </p>
        <figure id="id.077.01.168.1.jpg" xlink:href="077/01/168/1.jpg"/>
        <p id="N16640" type="main">
          <s id="N16642"><emph type="italics"/>Data &longs;it portio ABC, qualis dicta est. </s>
          <s id="N16646">cuius centrum grauitatis &longs;it <lb/>punctum H. &amp; in ip&longs;a plan&egrave; in&longs;cribatur triangulum ABC. &longs;itqu&egrave; pro <lb/>po&longs;ita recta linea F. &amp; quam proportionem habet BH ad F, eandem <lb/>habeat triangulum ABC ad &longs;pacium<emph.end type="italics"/> k. <emph type="italics"/>inportione autem ABC pla<emph.end type="italics"/><lb/>
<arrow.to.target n="marg293"/> <emph type="italics"/>n&egrave; in&longs;cribatur figura rectilinea AGB LC, ita vt circumrelict&aelig; portio <lb/>nes<emph.end type="italics"/> ANG GOB BPL LQC &longs;imul &longs;umpt&etail; <emph type="italics"/>&longs;int minoresip&longs;o<emph.end type="italics"/> k<emph type="italics"/>: <lb/>ip&longs;iu&longs;qu&egrave; figur&aelig; in&longs;cript&aelig; centrum grauitatis &longs;it punctum E. Dico li&shy;<lb/>neam HE minorem e&longs;&longs;e ip&longs;a F. N am&longs;i non, vel &aelig;qualis est, vel <lb/>maior. </s>
          <s id="N16673">Quoniam autem<emph.end type="italics"/> maior e&longs;t figura rectilinea AGBLC, <lb/>qu&agrave;m triangulum ABC, maius ver&ograve; e&longs;t &longs;pacium K portio&shy;<lb/>nibus ANG GOB BPL LQC &longs;imul &longs;umptis, ideo <emph type="italics"/>rectili-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg294"/> <emph type="italics"/>nea figura AGBLC ad circumrelictas portiones maiorem habet pro&shy;<lb/>portionem, qu&agrave;m triangulum ABC ad K. hoc est HB ad F. at ue <lb/>r&ograve; BH nonhabet minorem proportionem ad F, qu&agrave;m habet ad HE. <lb/>c&ugrave;mnon &longs;it minor HE ip&longs;a F.<emph.end type="italics"/> &longs;i enim ponatur HE ip&longs;i F
<pb xlink:href="077/01/169.jpg" pagenum="165"/>&aelig;qualis, eandem habebit proportionem BH ad HE, <expan abbr="qu&atilde;">quam</expan>
<arrow.to.target n="marg295"/><lb/>ad F. qu&aelig; e&longs;t proportio trianguli ABC ad. </s>
          <s id="N166A0">K. vnde figu&shy;<lb/>ra rectilinea AGBLC ad circumrelictas portiones maiorem, <lb/>habebit proportionem, qu&agrave;m BH ad HE. &longs;i ver&ograve; ponatur <lb/>HE maior, qu&agrave;m F, habebit BH ad F, hoc e&longs;t <expan abbr="triangul&utilde;">triangulum</expan>
<arrow.to.target n="marg296"/><lb/>ABC ad K maiorem proportionem, qu&agrave;m BH ad HE. <lb/><emph type="italics"/>multo igitur maiorem habet proportionem figura rectilinea AGBLC ad <lb/>circumrelictas portiones, qu&agrave;m BH ad HE. Quare &longs;i fiat ut rectili&shy;<lb/>linea figura AGBLC ad circumrelictas portiones, &longs;ic alia qu&aelig;dam li&shy;<lb/>nea ad HE. erit maior, qu&agrave;m BH. &longs;itqu&egrave; HM. C&ugrave;m enim portio&shy;<lb/>nis ABC centrum grauitatis &longs;it H. figur&aelig; ver&ograve; rectiline&aelig; AGBLC <lb/>punctum E. producta EH, a&longs;&longs;umptaqu&egrave; aliqua recta linea proportione <lb/>babente ad EH, quam rectilineum AGBLC ad circumtelictas por&shy;<lb/>tiones; maior erit qu&agrave;m HB. habeat igitur<emph.end type="italics"/> (vt dictum e&longs;t) <emph type="italics"/>MH ad <lb/>HE<emph.end type="italics"/> proportionem eam, quam habet figura AGBLC ad reli
<arrow.to.target n="marg297"/><lb/>quas portiones, <emph type="italics"/>ergopunctum M centrum est grauit atis magnitudi&shy;<lb/>nis ex circumrelictis portionibus compo&longs;it&aelig;. </s>
          <s id="N166D8">quod e&longs;&longs;e non pote&longs;t. </s>
          <s id="N166DA">Ducta <lb/>enimrecta linea<emph.end type="italics"/> RS <emph type="italics"/>per M ip&longs;i AC &aelig;quidistante, inip&longs;a &longs;unt centra <lb/>grauitatis vnicuiqu&egrave; portioni re&longs;pondentia<emph.end type="italics"/>; ita &longs;cilicet vt centrum <lb/>magnitudinis ex portionibus ANG GOB compo&longs;it&aelig; &longs;it in <lb/>linea RS. &longs;ed in parte MR. in partever&ograve; MS &longs;it grauitatis <lb/>centrum magnitudinis ex reliquis portionibus BPL LQC <lb/>compo&longs;it&aelig;; ita vt punctum M magnitudinis ex omnibus <lb/>portionibus compo&longs;it&aelig; centrum grauitatisexi&longs;tat. </s>
          <s id="N166F3">qu&aelig; <expan abbr="tam&etilde;">tamen</expan> <lb/>e&longs;&longs;e non po&longs;&longs;unt. </s>
          <s id="N166FB">quod idem accideret, &longs;i etiam RS ip&longs;i AC <lb/>&aelig;quidi&longs;tans non e&longs;&longs;et. <emph type="italics"/>Patetigitur HE minorem e&longs;&longs;e, quam F.<emph.end type="italics"/><lb/>c&ugrave;m ne&que; maior, ne&que; &etail;qualis e&longs;&longs;e po&longs;&longs;it. <emph type="italics"/>quod quidem de&shy;<lb/>mon&longs;trare oportebat.<emph.end type="italics"/></s>
        </p>
        <p id="N1670D" type="margin">
          <s id="N1670F"><margin.target id="marg293"/>A</s>
        </p>
        <p id="N16713" type="margin">
          <s id="N16715"><margin.target id="marg294"/><emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> in<emph.end type="italics"/> 4. <lb/><emph type="italics"/><expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> hui<emph.end type="italics"/>^&lcub;9}</s>
        </p>
        <p id="N1672B" type="margin">
          <s id="N1672D"><margin.target id="marg295"/>7. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16736" type="margin">
          <s id="N16738"><margin.target id="marg296"/>8.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16741" type="margin">
          <s id="N16743"><margin.target id="marg297"/>8.<emph type="italics"/>primi hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N1674E" type="head">
          <s id="N16750">SCHOLIVM.</s>
        </p>
        <p id="N16752" type="main">
          <s id="N16754">In hac quo&que; demon&longs;tratione ob&longs;eruandum e&longs;t, quod
<arrow.to.target n="marg298"/><lb/>po&longs;t quartam huius adnotauimus; nimirum &longs;i pentagonum <lb/>AGBLC in portione plan&egrave;in&longs;criptum relin&que;ret portiones <lb/>ANG GOB BPL LQC, qu&aelig; &longs;imul maiores, vel etiam &aelig;-
<pb xlink:href="077/01/170.jpg" pagenum="166"/>quales e&longs;&longs;ent &longs;pacio K. Rur&longs;us plan&egrave; adhuc in&longs; cribatur in <lb/>portione ABC nonagonum, deinde altera figura, idqu&egrave; &longs;em<lb/>per fiat, donec circumrelict&etail; portiones &longs;imul &longs;int &longs;pacio K <lb/>minores. </s>
          <s id="N16769">quod quidem fieri po&longs;&longs;e ibidem o&longs;tendimus: </s>
        </p>
        <p id="N1676B" type="margin">
          <s id="N1676D"><margin.target id="marg298"/>A</s>
        </p>
        <p id="N16771" type="head">
          <s id="N16773">PROPOSITIO. VII.</s>
        </p>
        <p id="N16775" type="main">
          <s id="N16777">Duabus portionibus &longs;imilibus recta linea, re&shy;<lb/>ctanguliqu&egrave; coni &longs;ectione contentis, centra gra&shy;<lb/>uitatum diametros in eadem proportione di&longs;pe&shy;<lb/>&longs;cunt. </s>
        </p>
        <figure id="id.077.01.170.1.jpg" xlink:href="077/01/170/1.jpg"/>
        <p id="N16782" type="main">
          <s id="N16784"><emph type="italics"/>Sint du&aelig; portiones, quales dict&aelig; &longs;unt ABC EFG. quarum diame&shy;<lb/>tri BD FH. &longs;itqu&egrave; portionis ABC centrum grauitatis punctum K. <lb/>ip&longs;ius ver&ograve; EFG punctum L. Demonstrandum est, puncta<emph.end type="italics"/> k<emph type="italics"/>L in <lb/>eadem proportione diametros diuidere,<emph.end type="italics"/> ita vt BK ad KD &longs;it, vt FL
<pb xlink:href="077/01/171.jpg" pagenum="167"/>ad LH. <emph type="italics"/>&longs;i autemnon.<emph.end type="italics"/> &longs;i fieri pote&longs;t, <emph type="italics"/>&longs;it BK ad<emph.end type="italics"/> k<emph type="italics"/>D, vt FM ad <lb/>MH. &amp; in portione EFG rectilineum plan&egrave; in&longs;cribatur, ita vt linea <lb/>inter centrum<emph.end type="italics"/> grauitatis <emph type="italics"/>portionis, &amp;<emph.end type="italics"/> centrum grauitatis <emph type="italics"/>figur&aelig;<emph.end type="italics"/>
<arrow.to.target n="marg299"/><lb/><emph type="italics"/>in&longs;cript&aelig; minor &longs;it, qu&agrave;m LM. &longs;itqu&egrave; figur&aelig; in&longs;cript&aelig; centrum graui&shy;<lb/>tatis punctum X.<emph.end type="italics"/> eritvtiqu&egrave; punctum L propinquius vertici
<arrow.to.target n="marg300"/><lb/>F, qu&agrave;m punctum X. &amp; quoniam LX minor e&longs;t, qu&agrave;m <lb/>LM, erit quo&que; punctum X vertici F propinquius, qu&agrave;m <lb/>M. <emph type="italics"/>Jn portione autem ABC in&longs;cribatur figura rectilinea &longs;imilis figu<lb/>r&aelig; in portione EFG in&longs;cript&aelig;. </s>
          <s id="N167D7">hoc est &longs;imiliter plan&egrave;,<emph.end type="italics"/> (ita nemp&egrave; vt <lb/>figur&etail; latera multitudine &etail;qualia habeant) <emph type="italics"/>cuius centrum graui&shy;<lb/>tatis<emph.end type="italics"/> &longs;it punctum N. &amp; quoniam figur&aelig; in porrionibus pla&shy;<lb/>n&egrave;in&longs;cript&etail; habentlatera multitudine &aelig;qualia, ip&longs;arum cen&shy;<lb/>tra grauitatis diametros BD FH in eadem proportione di&longs;pe&shy;<lb/>&longs;cent. </s>
          <s id="N167EC">quare erit BN ad ND, vt FX ad XH. po&longs;itum <expan abbr="aut&etilde;">autem</expan>
<arrow.to.target n="marg301"/><lb/>fuitita e&longs;&longs;e FM ad MH, vt BK ad KD. &longs;i ita&que; <expan abbr="punct&utilde;">punctum</expan> <lb/>X propinquius e&longs;t ip&longs;i F, qu&agrave;m M; erit &amp; punctum N i&shy;<lb/>p&longs;i B propinquius, qu&agrave;m K. e&longs;tver&ograve; punctum K <expan abbr="centr&utilde;">centrum</expan> <lb/>grauitatis portionis ABC, punctum ver&ograve; N centrum figur&aelig; <lb/>in&longs;cripte; ergo centrum grauitatis figur&etail; in&longs;cript&aelig; <emph type="italics"/>propinquius <lb/>erit vertici portionis,<emph.end type="italics"/> quam centrum ip&longs;ius portionis. <emph type="italics"/>quod fieri <expan abbr="n&otilde;">non</expan> <lb/>potest. </s>
          <s id="N16818">Manife&longs;tum est igitur eandem habere proportionem BK ad KD. <lb/>quam FL ad LH.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N1681F" type="margin">
          <s id="N16821"><margin.target id="marg299"/>6. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1682A" type="margin">
          <s id="N1682C"><margin.target id="marg300"/>5. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16835" type="margin">
          <s id="N16837"><margin.target id="marg301"/>3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16840" type="head">
          <s id="N16842">SCHOLIVM.</s>
        </p>
        <p id="N16844" type="main">
          <s id="N16846">Pr&etail;&longs;ens demon&longs;tratio ea tant&ugrave;m ratione e&longs;&longs;icax e&longs;&longs;e vide&shy;<lb/>tur, quatenus &longs;upponitur punctum L vertici F propinqui^&lcub;9} <lb/>e&longs;&longs;e, qu&agrave;m M. ex hoc enim &longs;equitur punctum X e&longs;&longs;e ip&longs;i F <lb/>propinquius, qu&agrave;m M. vnde euenitab&longs;urdum, nemp&egrave;, <expan abbr="p&utilde;">pum</expan> <lb/>ctum N e&longs;&longs;evertici B propinquius, qu&agrave;m K. Qu&ograve;d &longs;i &longs;up <lb/>po&longs;itum fuerit Bk ad KD ita e&longs;&longs;e, vt FP ad PH; fuerit <lb/>autem P inter LF; tunc centrum grauitatis figur&etail; in EFG
<pb xlink:href="077/01/172.jpg" pagenum="168"/>plan&egrave; in&longs;cript&aelig; e&longs;&longs;etinter puncta PH; vnde centrum ctiam <lb/>figur&etail; in ABC &longs;imiliter plan&egrave; in&longs;cript&etail; inter KD eueniret; <lb/>e&longs;&longs;etqu&egrave; centrum grauitatis portionis ABC vertici B propin&shy;<lb/>quius, quam centrum figur&aelig; plan&egrave; in&longs;cript&aelig;. </s>
          <s id="N16862">ideoqu&egrave; <expan abbr="null&utilde;">nullum</expan> <lb/>accideret ab&longs;urdum. </s>
          <s id="N1686A">Quare &longs;i &longs;uppo&longs;itum fuerit FP ad PH <lb/>e&longs;&longs;e, vt BK ad KD, tunc (vt eadem demon&longs;tratio rei propo <lb/>&longs;it&aelig; in&longs;eruire po&longs;&longs;et) diuidenda e&longs;&longs;et diameter BD in <expan abbr="q;">&que;</expan> i&shy;<lb/>ta vt BQ ad QD &longs;it, vt FL ad LH. &amp; quoniam maio&shy;<lb/>
<arrow.to.target n="marg302"/> rem habet proportionem FL ad LH, qu&agrave;m FP ad PH; &longs;iqui&shy;<lb/>dem maior e&longs;t FL, qu&agrave;m FP, &amp; PH maior, qu&agrave;m LH. Vtver&ograve; <lb/>FL ad LH, ita e&longs;t BQ ad QD; &amp; vt FP ad PH. ita BK ad KD; <lb/>maiorem quo&que; habebit proportionem BQ ad QD, qu&agrave;m <lb/>
<arrow.to.target n="marg303"/> BK ad KD. &amp; componendo BD ad DQ maiorem, qu&agrave;m ea<lb/>
<arrow.to.target n="marg304"/>dem BD ad Dk. </s>
          <s id="N1688E">Quare maior e&longs;t DK, qu&agrave;m <expan abbr="Dq.">D&que;</expan> &amp; ob id <lb/>punctum K propinquius erit vertici B, qu&agrave;m <expan abbr="q.">&que;</expan> Deinde <lb/>plan&egrave; in&longs;cribenda e&longs;&longs;et figura in portione ABC, ita vt linea <lb/>inter centrum figur&aelig; in&longs;cript&aelig;, &amp; centrum portionis minor <lb/>e&longs;&longs;et, qu&agrave;m <expan abbr="Kq;">K&que;</expan> &amp; reliqua qu&aelig; &longs;equuntur, ita tamen, vt qu&etail; <lb/>facta &longs;unt in EFG, fiant in ABC; &amp; qu&aelig; in ABC, <expan abbr="fi&atilde;t">fiant</expan> in EFG. <lb/>o&longs;tendeturqu&egrave; centrum figur&etail; in&longs;cript&etail; in portione EFG pro <lb/>pinquius e&longs;&longs;e vertici F, qu&agrave;m centrum grauitatis ip&longs;ius portio <lb/>nis EFG. quod quidem fieri non pote&longs;t. </s>
          <s id="N168B0">Ex quibus perlpi&shy;<lb/>cuum fit demon&longs;trationem e&longs;&longs;e vniuer&longs;alem. </s>
          <s id="N168B4">&amp; hanc <expan abbr="dem&otilde;">demom</expan> <lb/>&longs;trationis partem Archimedem omi&longs;i&longs;&longs;e, vt notam. </s>
          <s id="N168BC">Etvt an&shy;<lb/>tea admonuimus, qu&ograve;d centra grauitatis diametros in eadem <lb/>proportione diuidunt, omnibus parabolis competere intelli&shy;<lb/>gendum e&longs;t. </s>
          <s id="N168C4">&longs;iquidem omnes &longs;unt&longs;imiles. </s>
          <s id="N168C6">quo demon&longs;trato, <lb/>in &longs;e&que;nti, quo in loco, &amp; in qua diametri parte reperitur <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis paraboles demon&longs;trat, qu&ograve;d vt res per&longs;picua <lb/>reddatur; h&aelig;c pri&ugrave;s demon&longs;trabimus. </s>
        </p>
        <p id="N168D2" type="margin">
          <s id="N168D4"><margin.target id="marg302"/><emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> in<emph.end type="italics"/> 4. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N168E6" type="margin">
          <s id="N168E8"><margin.target id="marg303"/>28.<emph type="italics"/>quinti. <lb/>addi.<emph.end type="italics"/></s>
        </p>
        <p id="N168F3" type="margin">
          <s id="N168F5"><margin.target id="marg304"/>10.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N168FE" type="head">
          <s id="N16900">LEMMA. I.</s>
        </p>
        <p id="N16902" type="main">
          <s id="N16904">Si magnitudo magnitudinis fuerit quadrupla, minorver&ograve; <lb/>magnitudo alterius magnitudinis &longs;it tripla, maior magnitu&shy;<lb/>do vtrarum qu&egrave; &longs;imul magnitudinum tripla erit. </s>
        </p>
        <pb xlink:href="077/01/173.jpg" pagenum="169"/>
        <p id="N1690D" type="main">
          <s id="N1690F">Quadrupla &longs;it magnitudo A magnitudinis BC. <lb/>
<arrow.to.target n="fig74"/><lb/>&longs;it ver&ograve; BC alterius magnitudinis CD tripla. </s>
          <s id="N16918">Di <lb/>co magnitudinem A vtrarumqu&egrave; &longs;imul BC CD, <lb/>hoc e&longs;t BD triplam e&longs;se. </s>
          <s id="N1691E">Quoniam enim BC tri&shy;<lb/>pla e&longs;t ip&longs;ius CD, erit componendo BC cum CD, <lb/>hoc e&longs;t BD ip&longs;ius CD quadrupla. </s>
          <s id="N16924">&longs;ed magnitudo <lb/>quo&que; A quadrupla e&longs;t ip&longs;ius BC, eandem igitur <lb/>habetproportionem A ad BC, vt BD ad CD. &amp; <lb/>permutando A ad BD, vt BC ad CD. &amp; e&longs;t
<arrow.to.target n="marg305"/> qui&shy;<lb/>dem BC tripla ip&longs;ius CD, ergo A ip&longs;ius BD tri&shy;<lb/>pla exi&longs;tit. </s>
          <s id="N16934">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N16936" type="margin">
          <s id="N16938"><margin.target id="marg305"/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.173.1.jpg" xlink:href="077/01/173/1.jpg"/>
        <p id="N16945" type="head">
          <s id="N16947">LEMMA. II.</s>
        </p>
        <p id="N16949" type="main">
          <s id="N1694B">Si magnitudo magnitudinis fuerit &longs;e&longs;quitertia, erit magni&shy;<lb/>tudo minor ip&longs;ius exce&longs;&longs;us tripla. </s>
        </p>
        <p id="N1694F" type="main">
          <s id="N16951">Sit magnitudo AB magnitudinis C &longs;e&longs;quiter <lb/>
<arrow.to.target n="fig75"/><lb/>tia; exce&longs;&longs;us ver&ograve;, quo AB &longs;uperat C, &longs;it BD. Dico <lb/><expan abbr="magnitudin&etilde;">magnitudinem</expan> C ip&longs;ius BD triplam e&longs;&longs;e. </s>
          <s id="N1695F">quod qui <lb/>dem ex &longs;e patet. </s>
          <s id="N16963">Nam quoniam BD e&longs;t exce&longs;&shy;<lb/>&longs;us, quo AB &longs;uperat C. magnitudo autem AB i&shy;<lb/>p&longs;am C &longs;uperat tertia ip&longs;ius C parte, cum &longs;it AB <lb/>ip&longs;ius C &longs;e&longs;quitertia. </s>
          <s id="N1696B">erit igitur BD tertia pars i&shy;<lb/>&longs;ius C. quare magnitudo C ip&longs;ius BD tripla <lb/>exi&longs;tit. </s>
          <s id="N16971">quod o&longs;tendere oportebat. </s>
        </p>
        <figure id="id.077.01.173.2.jpg" xlink:href="077/01/173/2.jpg"/>
        <p id="N16977" type="head">
          <s id="N16979">LEMMA III.</s>
        </p>
        <p id="N1697B" type="main">
          <s id="N1697D">Sit magnitudo AB ip&longs;ius BC &longs;extupla. </s>
          <s id="N1697F">&longs;it ver&ograve; AD tripla <lb/>ip&longs;ius AC. Dico BD ip&longs;ius BA &longs;e&longs;quialteram e&longs;se. </s>
        </p>
        <pb xlink:href="077/01/174.jpg" pagenum="170"/>
        <p id="N16986" type="main">
          <s id="N16988"><expan abbr="Quoni&atilde;">Quoniam</expan>. <expan abbr="n.">enim</expan> AD multiplex e&longs;t ip&longs;ius AG, erit AC pars ip&longs;i^&lcub;9} <lb/>AD. ac propterea ip&longs;am AD metictur. </s>
          <s id="N16993">rur&longs;us quoniam AB, <lb/>hoc e&longs;t AC vn&agrave; cum CB &longs;extupla e&longs;t ip&longs;ius BC, erit <expan abbr="diuid&etilde;do">diuidendo</expan> <lb/>AC ip&longs;ius CB quintupla. </s>
          <s id="N1699D">vnd&egrave; CB ip&longs;am AC, ac propterea <expan abbr="eti&atilde;">etiam</expan> <lb/>ip&longs;am AB metietur. </s>
          <s id="N169A5">Vta utem AC ad AD, ita fiat <lb/>
<arrow.to.target n="fig76"/><lb/>CB ad aliam <expan abbr="magnitudin&etilde;">magnitudinem</expan> G; eritvti <expan abbr="&qacute;">&que;</expan>; CB ip&longs;ius <lb/>G pars tertia, c&ugrave;m &longs;it AC ip&longs;ius AD pars quo&que; <lb/>tertia. </s>
          <s id="N169BA">Ita&que; quoniam CB ad G e&longs;t, vt AC ad AD, <lb/>
<arrow.to.target n="marg306"/> erit perm utando CB ad CA, vt G ad AD. BC ver&ograve; <lb/>ip&longs;am CA metitur, eiu&longs;qu&egrave; e&longs;t pars quinta; ergo <lb/>Gip&longs;am quo&que; AD metietur, eritqu&egrave; ip&longs;ius pars <lb/>quinta. </s>
          <s id="N169C8">Quoniam autem BC ip&longs;am BA metitur, <lb/>eademqu&egrave; BC ip&longs;am quo&que; G metitur, erit BC <lb/>ip&longs;arum AB G communis men&longs;ura. </s>
          <s id="N169CE">quia ver&ograve; AB <lb/>&longs;extupla e&longs;t ip&longs;ius CB, G ver&ograve; e&longs;t eiu&longs;dem CB tri&shy;<lb/>pla, erit AB ad G, ut &longs;extupla ad triplam. </s>
          <s id="N169D4">hoc e&longs;t <lb/>&longs;e habebunt in dupla proportione. </s>
          <s id="N169D8">quapropter <lb/>AB dupla e&longs;t ip&longs;ius G; ac per con&longs;e&que;ns Gip&longs;am <lb/>AB metitur. </s>
          <s id="N169DE">Quoniam igitur G totam AD metitur, &amp; <lb/>ablatam AB quo&que; metitur; metietur G reliquam BD. G <lb/>igitur ip&longs;arum AB BD communis exi&longs;tit men&longs;ura. </s>
          <s id="N169E4">&amp; <expan abbr="quoni&atilde;">quoniam</expan> <lb/>AB dupla e&longs;t ip&longs;ius G, tota ver&ograve; AD eiu&longs;dem G quintupla <lb/>exi&longs;tit, erit reliqua BD tripla ip&longs;ius G. Ex quibus&longs;equitur <lb/>DB ad BA ita &longs;e habere, vt tripla ad duplam. </s>
          <s id="N169F0">Quare DB <lb/>ip&longs;ius BA &longs;e&longs;quialtera exi&longs;tit. </s>
          <s id="N169F4">quod o&longs;tendere oportebat. </s>
        </p>
        <p id="N169F6" type="margin">
          <s id="N169F8"><margin.target id="marg306"/>16,<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.174.1.jpg" xlink:href="077/01/174/1.jpg"/>
        <p id="N16A05" type="head">
          <s id="N16A07">PROPOSITIO. VIII.</s>
        </p>
        <p id="N16A09" type="main">
          <s id="N16A0B">Omnis portionis recta linea, rectanguliqu&egrave; co <lb/>ni &longs;ectione content&aelig; centrum grauitatis diame&shy;<lb/>trum portionis ita diuidit, vt pars ip&longs;ius ad verti&shy;<lb/>cem portionis reliqu&aelig; ad ba&longs;im &longs;it &longs;e&longs;quialtera. </s>
        </p>
        <pb xlink:href="077/01/175.jpg" pagenum="171"/>
        <p id="N16A16" type="main">
          <s id="N16A18"><emph type="italics"/>Sit portio ABC, qualis dicta est. </s>
          <s id="N16A1C">ip&longs;ius ver&ograve; diameter &longs;it BD. cen&shy;<lb/>trum autem grauitatis &longs;it punctum H. o&longs;tendendum e&longs;t BH ip&longs;ius HD <lb/>&longs;e&longs;quialteram e&longs;&longs;e. </s>
          <s id="N16A22">Plan&egrave; in&longs;cribatur in portione ABC triangulum ABC. <lb/>cuius centrum grauitatis &longs;it punctum E. bi&longs;ariamqu&egrave; diuidatur vtra&shy;<lb/>qu&egrave; AB BC in punctis FG. &amp; ip&longs;i BD &aelig;quidi&longs;tantes ducantur F<emph.end type="italics"/>k <lb/><emph type="italics"/>GL. erunt vti&que;<emph.end type="italics"/> FK GL <emph type="italics"/>diametri portionum A<emph.end type="italics"/>k<emph type="italics"/>B BLC. &longs;it ita&shy;<lb/>&que; portionis A<emph.end type="italics"/>k<emph type="italics"/>B centrum grauitatis M; portionis ver&ograve; BLC pun&shy;<lb/>ctum N. connectantur&que; FG MN<emph.end type="italics"/> k<emph type="italics"/>L<emph.end type="italics"/>, qu&aelig; diametrum BD &longs;e&shy;<lb/>
<arrow.to.target n="fig77"/><lb/>cent in punctis OQS. Quoniam igitur puncta MN in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione diuidunt KF LG, erit KM ad MF, vt LN ad
<arrow.to.target n="marg307"/><lb/>NG. &amp; componendo KF ad FM, vt LG ad GN. &amp;
<arrow.to.target n="marg308"/> per&shy;<lb/>mutando KF ad LG, vt FM ad GN. &longs;untqu&egrave; KF LG <lb/>&aelig;quales; erit FM ip&longs;i GN &etail;qualis; &amp; reliqua Mk reliqu&aelig;
<arrow.to.target n="marg309"/><lb/>LN &aelig;qualis. </s>
          <s id="N16A6D">&amp; quoniam FM GN, &amp; Mk NL &longs;unt
<arrow.to.target n="marg310"/> &etail;qui&shy;<lb/>di&longs;tantes, erunt FG MN KL inter &longs;e &etail;quales, &amp;
<arrow.to.target n="marg311"/> <expan abbr="&aelig;quidi&longs;t&atilde;-tes">&aelig;quidi&longs;tan&shy;<lb/>tes</expan>. &amp; e&longs;t BD &aelig;quidi&longs;tans KF, erit igitur SQ ip&longs;i KM &aelig;&shy;<lb/>qualis. </s>
          <s id="N16A81">quia ver&ograve; KF BD LG &longs;unt &aelig;quidi&longs;tantes, erit MQ ad
<arrow.to.target n="marg312"/><lb/>QN, vt FO ad OG. C&ugrave;m autem &longs;it BF ad FA, vt BG ad GC,
<pb xlink:href="077/01/176.jpg" pagenum="172"/>
<arrow.to.target n="marg313"/> crit FG ip&longs;i AC &etail;quidi&longs;tans. </s>
          <s id="N16A90">&amp; vt AD ad DC, ita FO ad <lb/>OG. &longs;unt autem AD DC &aelig;quales, ergo FO OG, ac per con&shy;<lb/>&longs;e&que;ns MQ QN inter &longs;e &longs;unt &aelig;quales. </s>
          <s id="N16A96">ita&que; quoniam por <lb/>
<arrow.to.target n="marg314"/> tiones AKB BLC &longs;unt &aelig;quales, <emph type="italics"/>magnitudinis ex vtri&longs;&que; portio&shy;<lb/>nibus<emph.end type="italics"/> AKB BLC <emph type="italics"/>compo&longs;it&aelig; centrum grauitatis erit<emph.end type="italics"/> in medio li&shy;<lb/>
<arrow.to.target n="marg315"/> ne&ccedil; MN; hoc e&longs;t erit <emph type="italics"/>punctum <expan abbr="q.">&que;</expan> &amp; quoniam BH ad HD est,<emph.end type="italics"/><lb/>
<arrow.to.target n="marg316"/> <emph type="italics"/>vt KM ad MF<emph.end type="italics"/> (centra enim grauitatum portionum in ea&shy;<lb/>
<arrow.to.target n="fig78"/><lb/>dem proportione diametros &longs;ecare nece&longs;&longs;e e&longs;t) <emph type="italics"/>&amp; componendo<emph.end type="italics"/><lb/>
<arrow.to.target n="marg317"/> BD ad DH, vt KF ad FM. <emph type="italics"/>permutandoqu&egrave; vt BD ad KF,<emph.end type="italics"/><lb/>
<arrow.to.target n="marg318"/> <emph type="italics"/>ita HD ad MF. at ver&ograve; BD quadrupla est ip&longs;ius KF. Hoc enim<emph.end type="italics"/><lb/>
<arrow.to.target n="marg319"/> <emph type="italics"/>in fine demon&longs;tratum est, vbi est &longs;ignum hoc, H. quadrupla igitur est<emph.end type="italics"/><lb/>
<arrow.to.target n="marg320"/> <emph type="italics"/>&amp; DH ip&longs;ius MF. Quare &amp; reliqua BH reliqu&aelig;<emph.end type="italics"/> k<emph type="italics"/>M, hoc est i&shy;<lb/>p&longs;ius SQ, quadrupla existit.<emph.end type="italics"/> exi&longs;tente autem tota BH, qu&aelig; <expan abbr="c&otilde;">com</expan>. <lb/>po&longs;ita e&longs;t ex BS QH, &amp; SQ, quadrupla ip&longs;ius <expan abbr="Sq;">S&que;</expan> dempta <lb/>SQ ab ip&longs;is BS QH SQ, <emph type="italics"/>reliqua igitur ex vtri&longs;&que; BS QH<emph.end type="italics"/><lb/>con&longs;tans <emph type="italics"/>tripla est ip&longs;ius <expan abbr="Sq.">S&que;</expan> &longs;it BS tripla ip&longs;ius SX.<emph.end type="italics"/> &amp; <expan abbr="quoni&atilde;">quoniam</expan> <lb/>tota HQ cum SB ad totam QS e&longs;t, vt ablata BS ad ab&shy;<lb/>
<arrow.to.target n="marg321"/> latam SX; &longs;unt quidem tripl&etail;; erit reliqua HQ ad <expan abbr="reliqu&atilde;">reliquam</expan> <lb/>
<arrow.to.target n="marg322"/> QX in eadem proportione. <emph type="italics"/>ergo &amp; QH ip&longs;ius XQ e&longs;t tripla. <lb/>Et quoniam quadrupla est BD ip&longs;ius BS. hoc enim demon&longs;tratum<emph.end type="italics"/><lb/>
<arrow.to.target n="marg323"/> <emph type="italics"/>e&longs;t. </s>
          <s id="N16B4B">ip&longs;a ver&ograve; BS ip&longs;ius SX e&longs;t tripla<emph.end type="italics"/>; erit BD ip&longs;ius BX tripla.
<pb xlink:href="077/01/177.jpg" pagenum="173"/>ac propterea <emph type="italics"/>erit XB ip&longs;ius BD pars tertia. </s>
          <s id="N16B57">Ver&ugrave;m ED ip&longs;ius <lb/>DB parstertia existit. </s>
          <s id="N16B5B">C&ugrave;m centrum grauitatis trianguli ABC &longs;it <lb/>p<gap/>nctum E.<emph.end type="italics"/> quod ita diuidit BD, vt BE ip&longs;ius ED &longs;itdupla.
<arrow.to.target n="marg324"/><lb/>At ver&ograve; quoniam totius line&aelig; BD (qu&aelig; compo&longs;ita e&longs;t ex DE <lb/>EX XB) tertia pars e&longs;t ip&longs;a DE. &amp; tertia quo&que; ip&longs;a BX; <lb/><emph type="italics"/>reliqua igitur XE tertia est pars ip&longs;ius BD. &amp; quoniam totius por&shy;<lb/>tionis centrum grauitatis est punctum H; magnitudinis ver&ograve; ex v&shy;<lb/>tr<gap/>&que; portionibus A<emph.end type="italics"/>k<emph type="italics"/>B BLC compo&longs;it&aelig; centrum grauitatis est pun<lb/>ctum <expan abbr="q;">&que;</expan> trianguli ver&ograve; ABC est punctum E; erit triangulum ABC <lb/>ad circumrelictas portiones<emph.end type="italics"/> AKB BLC, <emph type="italics"/>vt QH ad HE, <expan abbr="tripl&utilde;">triplum</expan><emph.end type="italics"/>
<arrow.to.target n="marg325"/><lb/><emph type="italics"/>autem e&longs;t triangulum ABC portionum. </s>
          <s id="N16B96">C&ugrave;m totaportio<emph.end type="italics"/> ABC <emph type="italics"/>&longs;e&longs;qui&shy;<lb/>tertia &longs;it trianguli ABC<emph.end type="italics"/>, exce&longs;&longs;us ver&ograve;, quo portio ABC
<arrow.to.target n="marg326"/> &longs;upe&shy;<lb/>rat triangulum ABC, &longs;int portiones AKB BLC &longs;imul &longs;um<lb/>pt&aelig;. <emph type="italics"/>tripla igitur est QH ip&longs;ius HE. osten&longs;a ver&ograve; e&longs;t etiam QH <lb/>tripla ip&longs;ius QX.<emph.end type="italics"/> quare erit QX ip&longs;i HE &aelig;qualis. </s>
          <s id="N16BB3">&amp;
<arrow.to.target n="marg327"/> quo&shy;<lb/>niam HQ e&longs;t tripla ip&longs;ius QX, erit HQ cum QX, hoc <lb/>e&longs;t tota BX quadrupla ip&longs;ius QX, hoc e&longs;t ip&longs;ius HE. &longs;i&shy;<lb/>militer quoniam XH quadrupla e&longs;t ip&longs;ius HE; <emph type="italics"/>quintupla i&shy;<lb/>gitur e&longs;t<emph.end type="italics"/> XH cum HE, tota &longs;cilicet <emph type="italics"/>XE ip&longs;ius EH; hoc est <lb/>DE ip&longs;ius EH. inuicem enim &longs;unt &aelig;quales<emph.end type="italics"/> EX ED, vt o&longs;ten&shy;<lb/>&longs;um e&longs;t. </s>
          <s id="N16BD1">C&ugrave;m ita&que; &longs;it DE ip&longs;ius EH quintupla; erit DE <lb/>cum EH &longs;extupla ip&longs;ius EH. <emph type="italics"/>Quare &longs;extupla est<emph.end type="italics"/> tota <emph type="italics"/>DH <lb/>ip&longs;ius HE. &amp; e&longs;t BD ip&longs;ius DE tripla; &longs;equialtera igitur e&longs;t BH<emph.end type="italics"/>
<arrow.to.target n="marg328"/><lb/><emph type="italics"/>ip&longs;ius HD.<emph.end type="italics"/> Quare centrum grauitatis H ita diuidit diame&shy;<lb/>trum BD, vtpars BH ad HD &longs;e&longs;quialtera exi&longs;tit. </s>
          <s id="N16BEF">quod de <lb/>mon&longs;trare oportebat. </s>
        </p>
        <p id="N16BF3" type="margin">
          <s id="N16BF5"><margin.target id="marg307"/>7. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16BFE" type="margin">
          <s id="N16C00"><margin.target id="marg308"/>18.16 <emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N16C0B" type="margin">
          <s id="N16C0D"><margin.target id="marg309"/><emph type="italics"/>po&longs;t <expan abbr="prim&atilde;">primam</expan> <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16C1B" type="margin">
          <s id="N16C1D"><margin.target id="marg310"/>33. <emph type="italics"/>primi<emph.end type="italics"/></s>
        </p>
        <p id="N16C26" type="margin">
          <s id="N16C28"><margin.target id="marg311"/>34, <emph type="italics"/>primi<emph.end type="italics"/></s>
        </p>
        <p id="N16C31" type="margin">
          <s id="N16C33"><margin.target id="marg312"/>1. <emph type="italics"/>lemma <lb/>in<emph.end type="italics"/> 13. <emph type="italics"/>pri <lb/>mi huius<emph.end type="italics"/></s>
        </p>
        <p id="N16C46" type="margin">
          <s id="N16C48"><margin.target id="marg313"/><emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> in ali <lb/>ter<emph.end type="italics"/> 13 <emph type="italics"/>pri <lb/>mi huius<emph.end type="italics"/></s>
        </p>
        <p id="N16C5D" type="margin">
          <s id="N16C5F"><margin.target id="marg314"/><emph type="italics"/>po&longs;t <expan abbr="prim&atilde;">primam</expan> <lb/>huius<emph.end type="italics"/></s>
        </p>
        <p id="N16C6D" type="margin">
          <s id="N16C6F"><margin.target id="marg315"/>4. <emph type="italics"/>primi hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N16C7A" type="margin">
          <s id="N16C7C"><margin.target id="marg316"/>7. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16C85" type="margin">
          <s id="N16C87"><margin.target id="marg317"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16C90" type="margin">
          <s id="N16C92"><margin.target id="marg318"/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16C9B" type="margin">
          <s id="N16C9D"><margin.target id="marg319"/>A</s>
        </p>
        <p id="N16CA1" type="margin">
          <s id="N16CA3"><margin.target id="marg320"/>19 <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16CAC" type="margin">
          <s id="N16CAE"><margin.target id="marg321"/>19.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16CB7" type="margin">
          <s id="N16CB9"><margin.target id="marg322"/>B</s>
        </p>
        <p id="N16CBD" type="margin">
          <s id="N16CBF"><margin.target id="marg323"/>1.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hui^&lcub;9}<emph.end type="italics"/></s>
        </p>
        <p id="N16CCB" type="margin">
          <s id="N16CCD"><margin.target id="marg324"/><emph type="italics"/>ante<emph.end type="italics"/> 1;.<emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16CDD" type="margin">
          <s id="N16CDF"><margin.target id="marg325"/>8.<emph type="italics"/>primi hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N16CEA" type="margin">
          <s id="N16CEC"><margin.target id="marg326"/>2.<emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N16CF7" type="margin">
          <s id="N16CF9"><margin.target id="marg327"/>9.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16D02" type="margin">
          <s id="N16D04"><margin.target id="marg328"/>3.<emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.177.1.jpg" xlink:href="077/01/177/1.jpg"/>
        <figure id="id.077.01.177.2.jpg" xlink:href="077/01/177/2.jpg"/>
        <p id="N16D17" type="head">
          <s id="N16D19">SCHOLIVM.</s>
        </p>
        <p id="N16D1B" type="main">
          <s id="N16D1D">Ea verba in demon&longs;tratione po&longs;ita nemp&egrave; <emph type="italics"/>Hoc enim in fine<emph.end type="italics"/>
<arrow.to.target n="marg329"/><lb/><emph type="italics"/>demon&longs;tratum e&longs;t, vbi est &longs;ignum hoc, H.<emph.end type="italics"/> ita credo e&longs;&longs;e intell igen&shy;<lb/>da, qu&ograve;d &longs;cilicet Archimedes alicubi, &amp; in fine, &longs;iue huius, &longs;i&shy;<lb/>ue alicuius alterius demon&longs;trationis, demon&longs;trauerit linea in
<pb xlink:href="077/01/178.jpg" pagenum="174"/>BD quadruplam e&longs;&longs;e ip&longs;ius KF. &amp; vbi hoc demon&longs;tratum <lb/>erat, ibi quo&que; pro &longs;igno po&longs;ita fuerit littera H. quod qui&shy;<lb/>dem o&longs;ten&longs;um e&longs;t &agrave; nobis paul&ograve; ante &longs;ecundam huius propo&longs;i <lb/>tionem; vbi etiam appo&longs;uim us pro &longs;igno hanc literam H. </s>
        </p>
        <p id="N16D3D" type="margin">
          <s id="N16D3F"><margin.target id="marg329"/>A</s>
        </p>
        <p id="N16D43" type="main">
          <s id="N16D45">
<arrow.to.target n="marg330"/> Rur&longs;um in demon&longs;tratione paul&ograve; infra Archimedes dixit, <lb/><emph type="italics"/>Hoc enim demonstratum e&longs;t<emph.end type="italics"/>, &longs;cilicet BD ip&longs;ius BS quadruplam <lb/>e&longs;&longs;e. </s>
          <s id="N16D54">&longs;upponit autem hoc tanquam demon&longs;tratum po&longs;t pri&shy;<lb/>mam <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> huius, vbi tota BD e&longs;t &longs;exdccim, &amp; BS qua <lb/>tuor, vt eodem in loco o&longs;ten&longs;um fuit&agrave; nobis. </s>
          <s id="N16D5E">Vel ad ea re&shy;<lb/>&longs;pexit Archimedes, qu&aelig; ab ip&longs;o in decimanona propo&longs;itione <lb/>de quadratura paraboles demon&longs;tra ta fuerunt. </s>
          <s id="N16D64">vbi circa <expan abbr="fin&etilde;">finem</expan> <lb/>demon&longs;trationis o&longs;tendit BD quadruplam e&longs;&longs;e ip&longs;ius BS. </s>
        </p>
        <p id="N16D6C" type="margin">
          <s id="N16D6E"><margin.target id="marg330"/>B</s>
        </p>
        <p id="N16D72" type="main">
          <s id="N16D74">Inuento ita&que; centro grauitatis paraboles, vult Archime&shy;<lb/>des in ue&longs;tigare centrum grauitatis fru&longs;ti &agrave; parabole ab&longs;ci&longs;&longs;i. <lb/>&que;madmodum in primo libro po&longs;t inuentionem centri gra<lb/>uitatis trianguli, adinuenit etiam centrum grauitatis trapezij. <lb/>quod e&longs;t tan quam fru&longs;tum &agrave; triangulo ab&longs;ci&longs;sum. </s>
          <s id="N16D7E">quare duo <lb/>adhuc theoremata proponit, in quorum po&longs;tremo, vbi &longs;it <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis fru&longs;ti demon&longs;trat. </s>
          <s id="N16D88">in &longs;e&que;nri ver&ograve; qu&aelig;dam <lb/>demon&longs;trat nece&longs;&longs;aria, vt huiu&longs;modi centrum determinare <lb/>po&longs;&longs;it. </s>
          <s id="N16D8E">Quoniam autem &longs;e&que;ns theorema arduum, difficile&shy;<lb/>qu&egrave; &longs;e&longs;e offert; non nulla pri&ugrave;s quibu&longs;dam lemmatibus o&longs;ten<lb/>demus, ne &longs;i in demon&longs;tratione ea in&longs;ererentur, longa nimis <lb/>euaderet, ac t&aelig;dio&longs;a demon&longs;tratio. </s>
          <s id="N16D96">qu&aelig; quidem &longs;umma indi<lb/>get attentione. </s>
          <s id="N16D9A">quamqu&agrave;m in hoc theoremate explicando ad <lb/>vitandam ob&longs;curitatem copio&longs;um &longs;ermonem adhibendum <lb/>curauimus; ne breuitati &longs;tudentes ob&longs;curiores e&longs;&longs;emus. </s>
        </p>
        <p id="N16DA0" type="head">
          <s id="N16DA2">LEMMA. I.</s>
        </p>
        <p id="N16DA4" type="main">
          <s id="N16DA6">Si qua tuor magnitudines in continua fuerint proportione, <lb/>&amp; earum exce&longs;&longs;us in eadem erunt proportione <expan abbr="magnitudin&utilde;">magnitudinum</expan>. </s>
        </p>
        <pb xlink:href="077/01/179.jpg" pagenum="175"/>
        <p id="N16DB1" type="main">
          <s id="N16DB3">Sint quatuor magnitudines AF BH CL D in continua <lb/>proportione; vt &longs;cilicet &longs;it AF ad BH, vt BH ad CL; &amp; CL <lb/>ad D. exce&longs;&longs;us ver&ograve;, quo AF &longs;uperat BH, &longs;it EF. &amp; exce&longs;&longs;us, quo <lb/>BH &longs;uperat CL, &longs;it GH. exce&longs;&longs;us deni&que;, quo CL &longs;uperat <lb/>D, &longs;it KL. eruntuti&que; AE BH inter &longs;e &etail;quales, itidemqu&egrave; <lb/>
<arrow.to.target n="fig79"/><lb/>BG CL &aelig;quales. </s>
          <s id="N16DC4">Dico EF GH KL in eadem e&longs;&longs;e proportio <lb/>ne, vt &longs;unt magnitudines AF BH CL, &amp; vt BH CL D. Quo&shy;<lb/>niam enim tota AF ad totam BH e&longs;t, vt BH ad CL; hoc e&longs;t
<arrow.to.target n="marg331"/><lb/>vt ablata EA ad ablatam GB. erit reliqua EF ad reliquam GH; <lb/>vt AF ad BH. Pariqu&egrave; ratione o&longs;tendetur GH ad kL ita e&longs;&shy;<lb/>&longs;e, vt BH ad CL. ergo exce&longs;&longs;us EF GH KL in eadem &longs;unt <lb/>proportione, vt magnitudines AF BH CL. qu&aelig; c&ugrave;m &longs;int, vt <lb/>magnitudines BH CL D, &longs;iquidem omnes in continua &longs;unt <lb/>proportione; exce&longs;&longs;us igitur EF GH KL in eadem quo&que; <lb/>&longs;unt proportione, vt magnitudines BH CL D. qu&aelig; quidem <lb/>demon&longs;trare oportebat. </s>
        </p>
        <pb xlink:href="077/01/180.jpg" pagenum="176"/>
        <p id="N16DE0" type="margin">
          <s id="N16DE2"><margin.target id="marg331"/>19.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.180.1.jpg" xlink:href="077/01/180/1.jpg"/>
        <p id="N16DEF" type="head">
          <s id="N16DF1">LEMMA. II.</s>
        </p>
        <p id="N16DF3" type="main">
          <s id="N16DF5">Si tres fuerint magnitudines, &amp; ali&aelig; ip&longs;is numero &aelig;quales, <lb/>&amp; in eadem proportione, in primis magnitudinibus prima; <lb/>&amp; &longs;ecunda ad tertiam erunt, vt in &longs;ecundis magnitudinibus <lb/>prima &amp; &longs;ecunda ad tertiam. </s>
        </p>
        <figure id="id.077.01.180.2.jpg" xlink:href="077/01/180/2.jpg"/>
        <p id="N16E00" type="main">
          <s id="N16E02">Sint tres magnitudines ABC, &amp; ali&aelig; tres DEF in <expan abbr="ead&etilde;">eadem</expan> pro&shy;<lb/>portione. </s>
          <s id="N16E0A">Dico AB &longs;imul ad C ita e&longs;&longs;e, vt DE &longs;imul ad F. <lb/>
<arrow.to.target n="marg332"/> Quoniam enim A ad B e&longs;t, ut D ad E, erit <expan abbr="compon&etilde;do">componendo</expan> AB <lb/>
<arrow.to.target n="marg333"/> ad B, ut DE ad E. &longs;ed vt B ad C, ita e&longs;t E ad F. ergo ex &etail;quali <lb/>AB &longs;imul ad C e&longs;t, vt DE &longs;imul ad F. quod demon&longs;trare opor <lb/>tebat. </s>
        </p>
        <p id="N16E20" type="margin">
          <s id="N16E22"><margin.target id="marg332"/>18,<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16E2B" type="margin">
          <s id="N16E2D"><margin.target id="marg333"/>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16E36" type="head">
          <s id="N16E38">LEMMA. III.</s>
        </p>
        <p id="N16E3A" type="main">
          <s id="N16E3C">Si fuerit AB ad AC, vt DE ad DF. Dico exce&longs;&longs;um BC ad <lb/>
<arrow.to.target n="marg334"/> CA ita e&longs;&longs;e, vt exce&longs;&longs;us EF ad FD. </s>
        </p>
        <p id="N16E44" type="margin">
          <s id="N16E46"><margin.target id="marg334"/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N16E56" type="main">
          <s id="N16E58">Quoniam enim e&longs;t AB ad AC, vt DE ad DF, erit con-
<pb xlink:href="077/01/181.jpg" pagenum="177"/>
<arrow.to.target n="fig80"/><lb/>uertendo CA ad AB, vt FD ad DE. &amp; per conuer <lb/>&longs;ionem rationis AC ad CB, vt DF ad FE. &amp; rur&longs;us
<arrow.to.target n="marg335"/><lb/>conuertendo CB ad CA, vt FE ad FD. quod <expan abbr="dem&otilde;-&longs;trare">demon&shy;<lb/>&longs;trare</expan> oportebat. </s>
        </p>
        <p id="N16E70" type="margin">
          <s id="N16E72"><margin.target id="marg335"/><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan>.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.181.1.jpg" xlink:href="077/01/181/1.jpg"/>
        <p id="N16E87" type="head">
          <s id="N16E89">ALITER.</s>
        </p>
        <p id="N16E8B" type="main">
          <s id="N16E8D">Quoniam enim AB e&longs;t ad AC, vt DE ad DF, erit conuer&shy;<lb/>tendo AC ad AB, vt DF ad DE. diuidendoqu&egrave; CB ad BA, vt <lb/>FE ad ED. e&longs;t autem AB ad AC, vt DE ad DF, erit igitur
<arrow.to.target n="marg336"/><lb/>ex &aelig;quali BC ad CA, vt EF ad FD. quod demon&longs;trare opor
<arrow.to.target n="marg337"/><lb/>tebat. </s>
        </p>
        <p id="N16E9D" type="margin">
          <s id="N16E9F"><margin.target id="marg336"/>17.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16EA8" type="margin">
          <s id="N16EAA"><margin.target id="marg337"/>22,<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16EB3" type="head">
          <s id="N16EB5">LEMMA IIII.</s>
        </p>
        <figure id="id.077.01.181.2.jpg" xlink:href="077/01/181/2.jpg"/>
        <p id="N16EBA" type="main">
          <s id="N16EBC">Si fuerint quotcun&que; magnitudines ABC, &amp; nli&etail; ip&longs;is nu<lb/>mero &aelig;quales DEF, &amp; in <expan abbr="ead&etilde;">eadem</expan> proportione. </s>
          <s id="N16EC4">Dico vtram&que; <lb/>&longs;imul AD ad vtram&que; &longs;imul BE, &amp; vtram&que; &longs;imul BE ad v&shy;<lb/>tram&que; &longs;imul CF eandem habere proportionem, quam ha&shy;<lb/>bet A ad B, &amp; B ad C. </s>
        </p>
        <pb xlink:href="077/01/182.jpg" pagenum="178"/>
        <p id="N16ECF" type="main">
          <s id="N16ED1">
<arrow.to.target n="marg338"/> Quoniam enim e&longs;t A ad B, ut D ad E; erit AD &longs;imul ad <lb/>BE &longs;imul, vt A ad B. &longs;imiliter quoniam B ad C e&longs;t, vt E ad <lb/>F, erit BE &longs;imul ad CF &longs;imul, vt B ad C. in eadem igitur &longs;unt <lb/>proportione AD &longs;imul, &amp; BE &longs;imul, &amp; CF &longs;imul, vt ABC. <lb/>quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N16EDF" type="margin">
          <s id="N16EE1"><margin.target id="marg338"/>12.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N16EEA" type="head">
          <s id="N16EEC">LEMMA. V.</s>
        </p>
        <p id="N16EEE" type="main">
          <s id="N16EF0">Si magnitudo magnitudinis fuerit &longs;e&longs;quialtera ad tres quin<lb/>tas eiu&longs;dem erit duplex &longs;e&longs;quialtera. </s>
        </p>
        <figure id="id.077.01.182.1.jpg" xlink:href="077/01/182/1.jpg"/>
        <p id="N16EF7" type="main">
          <s id="N16EF9">Sit AB ip&longs;ius CD &longs;e&longs;quialtera. </s>
          <s id="N16EFB">&longs;it uer&ograve; CE tres quint&aelig; <lb/>ip&longs;ius CD. Dico AB ad CE ita e&longs;&longs;e, vt quin&que; ad duo. </s>
          <s id="N16EFF">Fiat EF <lb/>&etail;qualis EC, erit CF &longs;ex quint&aelig; ip&longs;ius CD. &amp; quoniam AB i&shy;<lb/>p&longs;ius CD e&longs;t &longs;e&longs;quialtera, &longs;uperabit AB ip&longs;am CD dimidia <lb/>ip&longs;ius CD. erit igitur AB &longs;eptem quint&aelig; cum dimidia i&shy;<lb/>p&longs;ius CD. quare CF minor e&longs;t AB. fiat igitur AG &aelig;qua&shy;<lb/>lis CF. erit vti&que; AG &longs;ex quint&etail; ip&longs;ius CD. &amp; ob id GB <lb/>ip&longs;ius CD quinta e&longs;t pars cum dimidia. </s>
          <s id="N16F0D">&amp; quoniam CE e&longs;t <lb/>eiu&longs;dem CD tres quint&aelig;, erit BG dimidia ip&longs;ius CE. qua&shy;<lb/>re GB ip&longs;am CE bis metietur. </s>
          <s id="N16F13">Et quoniam EF e&longs;t &aelig;qua&shy;<lb/>lis ip&longs;i EC, ip&longs;a BG bis quo&que; metietur ip&longs;am EF. quare
<pb xlink:href="077/01/183.jpg" pagenum="179"/>totam CF, hoc e&longs;t ip&longs;am AG quater metietur. </s>
          <s id="N16F1B">at ver&ograve; GB &longs;ei&shy;<lb/>p&longs;am &longs;emel metitur ip&longs;a igitur GB totam AB quinquies metie&shy;<lb/>tur. </s>
          <s id="N16F21">Ex quibus li&que;t GB ip&longs;arum ABCE communem e&longs;&longs;e <lb/>men&longs;uram. </s>
          <s id="N16F25">Et e&longs;t quidem AB quintupla ip&longs;ius BG; ip&longs;a ver&ograve; <lb/>CE eiu&longs;dem BG dupla. </s>
          <s id="N16F29">erit AB ad CE, vt quintupla ad <expan abbr="dupl&atilde;">duplam</expan>. <lb/>hoc e&longs;t duplex &longs;e&longs;quialtera. </s>
          <s id="N16F31">quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N16F33" type="head">
          <s id="N16F35">PROPOSITIO. VIIII.</s>
        </p>
        <p id="N16F37" type="main">
          <s id="N16F39">Si quatuor line&aelig; in continua fuerint proportio&shy;<lb/>ne, &amp; quam proportionem habet minima ad exce&longs; <lb/>&longs;um, quo maxima minimam &longs;uperat; eandem ha&shy;<lb/>beat qu&aelig;dam a&longs;&longs;umpta linea ad tres quintas exce&longs;&shy;<lb/>&longs;us, quo maxima proportionalium tertiam exce&shy;<lb/>dit: quam ver&ograve; proportionem habet linea &aelig;qualis <lb/>dupl&aelig; maxim&aelig; proportionalium, &amp; quadrupl&aelig; &longs;e <lb/>cund&aelig;, &amp; &longs;extupl&aelig; terti&aelig;, &amp; tripl&aelig; quart&aelig; ad <expan abbr="line&atilde;">lineam</expan> <lb/>&aelig;qualem quintupl&aelig; maxim&aelig;, &amp; decupl&aelig; &longs;ecund&aelig;, <lb/>&amp; decupl&aelig; terti&aelig;, &amp; quintupl&aelig; quart&aelig;, ean-&shy;<lb/>dem habeat qu&aelig;dam a&longs;&longs;umpta linea ad ex ce&longs;&longs;um, <lb/>quo maxima proportionalium tertiam &longs;uperat; <lb/>vtr&aelig;&que; &longs;imul a&longs;&longs;umpt&aelig; line&aelig; erunt du&aelig; quin&shy;<lb/>t&aelig; maxim&aelig;. <lb/></s>
        </p>
        <pb xlink:href="077/01/184.jpg" pagenum="180"/>
        <p id="N16F5D" type="main">
          <s id="N16F5F"><emph type="italics"/>Sint quatuor line&aelig; proportionales AB BC BD BE,<emph.end type="italics"/> ita vt AB <lb/>ad BC &longs;it, vt BC ad BD. &amp; vt BC ad BD, ita &longs;it BD ad BE. <emph type="italics"/>&amp; <lb/>quam proportionem habet BE ad E A, eandem habeat FG adtres quin<lb/>tas ip&longs;ius AD. quam autem proportionem habet linea &aelig;qualis dupl&aelig; i&shy;<lb/>p&longs;ius AB, &amp; quidrupl&aelig; ip&longs;ius BC, &amp; &longs;extupl&aelig; ip&longs;i^&lcub;9} BD, &amp; tripl&aelig; ip&longs;i^&lcub;9} <lb/>BE, ad <expan abbr="line&atilde;">lineam</expan> <expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> <expan abbr="qu&itilde;tupl&aelig;">quintupl&aelig;</expan> ip&longs;i^&lcub;9} AB, ot decupl&aelig; ip&longs;i^&lcub;9} CB, &amp; decupl&aelig; <lb/>ip&longs;i^&lcub;9} B D, &amp; quintupl&aelig; ip&longs;ius BE, eandem habeat GH ad AD. O&longs;teden<lb/>dum est FH duasquintas e&longs;&longs;e ip&longs;ius AB. Quoniam enim proportiona&shy;<lb/>les &longs;unt AB BC BD BE, &amp;<emph.end type="italics"/> ip&longs;arum exce&longs;&longs;us <emph type="italics"/>AC CD DE in<emph.end type="italics"/><lb/>
<arrow.to.target n="fig81"/><lb/>
<arrow.to.target n="marg339"/> <emph type="italics"/>eadem erunt proportione. </s>
          <s id="N16F9B">&amp;<emph.end type="italics"/> quoniam magnitudines AB BC BD <lb/>in continua &longs;unt proportione, &amp; earum exce&longs;&longs;us AC CD DE <lb/>in eadem erunt proportione. </s>
          <s id="N16FA4">quia ver&ograve; tres &longs;unt magnitudi&shy;<lb/>nes proportionales AB BC BD; &amp; ali&etail; ip&longs;is numero &ccedil;quales, &amp; <lb/>
<arrow.to.target n="marg340"/> in eadem proportione AC CD DE, erit in primis magnitu&shy;<lb/>dinibus prima, &amp; &longs;ecunda ad tertiam, vt in &longs;ecundis magni&shy;<lb/>tudinibus prima, &amp; &longs;ecunda ad tertiam; hoc e&longs;t <emph type="italics"/>vtra&que; &longs;imul <lb/>AB BC ad BD eandem habebit proportionem, quam<emph.end type="italics"/> vtra&que; &longs;imul <lb/>
<arrow.to.target n="marg341"/> AC CD, hoc e&longs;t <emph type="italics"/>AD ad DE; &amp;<emph.end type="italics"/> ob eandem rationem cum <lb/>
<arrow.to.target n="marg342"/> tres &longs;int proportionales magnitudines AC CD DE, ali&ccedil;qu&egrave; <lb/>eodem modo proportionales BC BD BE; crit vtra&que; &longs;imul
<pb xlink:href="077/01/185.jpg" pagenum="181"/>AC CD, hoc e&longs;t AD ad DE, vt <emph type="italics"/>vtra&que; &longs;imul BC BD ad EB. <lb/>&amp; omnes adomnes,<emph.end type="italics"/> quoniam &longs;cilicet e&longs;t vtra&que; &longs;imul AB BC <lb/>ad BD, vt horum dupla; erit vtra&que; &longs;imul AB BC ad BD, vt <lb/>dupla vtriu&longs;&que; &longs;imul AB BC ad duplam ip&longs;ius BD. e&longs;t <expan abbr="aut&etilde;">autem</expan> <lb/>vtra&que; &longs;imul AB BC ad BD, vt AD ad DE. erit igitur AD ad <lb/>DE, vt dupla vtriu&longs;&que; &longs;imul AB BC ad duplam ip&longs;ius BD. <lb/>quia ver&ograve;ita etiam e&longs;t AD ad DE, vtvtra&que; &longs;imul CB BD ad <lb/>BE; erit dupla vtriu&longs;&que; &longs;imul AB BC ad duplam ip&longs;ius BD, vt
<arrow.to.target n="marg343"/><lb/>vtra&que; &longs;imul CB BD ad BE. &amp; vtra&que; antecedentia ad
<arrow.to.target n="marg344"/> vtra&shy;<lb/>&que; con&longs;e&que;ntia in eadem erunt proportione: eruntqu&egrave; in <lb/>antecedenti du&etail; AB, tres BC, &amp; &longs;ola BD. in con&longs;e&que;nti ver&ograve; <lb/>erunt du&aelig; BD cum &longs;ola BE. erit igitur dupla ip&longs;ius AB, &amp; tri <lb/>pla ip&longs;ius CB cum &longs;ola BD ad duplam ip&longs;ius BD cum &longs;ola BE, <lb/>vt vtra&que; &longs;imul CB BD ad BE. vtra&que; ver&ograve; &longs;imul CB BD <lb/>ad BE e&longs;t, vt AD ad DE. <emph type="italics"/>eandem ergo proportionem habet AD ad<emph.end type="italics"/>
<arrow.to.target n="marg345"/><lb/><emph type="italics"/>DE, quam linea &aelig;qualis dupl&aelig; ip&longs;ius AB, &amp; tripl&aelig;ip&longs;ius CB, &amp;<emph.end type="italics"/> &longs;oli <lb/><emph type="italics"/>DB adlineam &aelig;qualem dupl&aelig; ip&longs;ius BD &amp;<emph.end type="italics"/> &longs;oli <emph type="italics"/>BE.<emph.end type="italics"/> Quoniam au&shy;<lb/>tem linea compo&longs;ita ex dupla ip&longs;ius AB, &amp; quadrupla ip&longs;ius <lb/>CB, &amp; quadrupla ip&longs;ius BD, &amp; dupla ip&longs;ius BE, maior e&longs;t ea, <lb/>qu&aelig; compo&longs;ita e&longs;t ex dupla ip&longs;ius AB, &amp; tripla ip&longs;ius CB, &amp; <lb/>&longs;ola BD; maiorem habebit proportionem compo&longs;ita ex
<arrow.to.target n="marg346"/> du&shy;<lb/>pla ip&longs;ius AB, &amp; quadrupla ip&longs;ius CB, &amp; quadrupla ip&longs;ius BD, <lb/>&amp; dupla ip&longs;ius BE ad compo&longs;itam ex dupla ip&longs;ius BD cum <lb/>&longs;ola BE, quam compo&longs;ita ex dupla ip&longs;ius AB, &amp; tripla ip&longs;ius <lb/>CB cum &longs;ola BD ad eandem compo&longs;itam ex dupla ip&longs;ius BD <lb/>cum &longs;ola EB. compo&longs;ita ver&ograve; ex dupla ip&longs;ius AB, &amp; tripla <lb/>ip&longs;ius BC cum &longs;ola BD ad duplam ip&longs;ius BD cum &longs;ola BE ita <lb/>o&longs;ten&longs;a e&longs;t &longs;e habere AD ad DE. compo&longs;ita igitur ex dupla i&shy;<lb/>p&longs;ius AB, &amp; quadrupla ip&longs;ius BC, &amp; quadrupla ip&longs;ius BD, &amp; <lb/>dupla ip&longs;ius BE ad compo&longs;itam ex dupla ip&longs;ius BD cum &longs;ola <lb/>BE maiorem habebit proportionem, quam AD ad DE. <emph type="italics"/>Quam <lb/>ita&que; proportionem habet linea &aelig;qualis dupl&aelig; ip&longs;ius AB, &amp; quadrupl&aelig; <lb/>ip&longs;ius BC, &amp; quadrupl&aelig; ip&longs;ius BD, &amp; dupl&aelig; ip&longs;ius BE ad <expan abbr="line&atilde;">lineam</expan> <expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> <lb/>dupl&aelig; ip&longs;ius DB, &amp; ad EB, eandem habebit AD adminorem ip&longs;a DE.<emph.end type="italics"/>
<arrow.to.target n="marg347"/><lb/><emph type="italics"/>habeat igitur ad DO.<emph.end type="italics"/> &amp; <expan abbr="quoni&atilde;ita">quonianita</expan> &longs;e habet AD ad DO, vt <expan abbr="c&otilde;po">compo</expan> <lb/>&longs;ita ex dupla ip&longs;ius AB, &amp; quadrupla ip&longs;ius BC, &amp; quadrupla <lb/>ip&longs;ius BD, &amp; dupla ip&longs;ius BE, hoc e&longs;t <expan abbr="c&otilde;po&longs;ita">compo&longs;ita</expan> ex dupla vtriu&longs;-
<pb xlink:href="077/01/186.jpg" pagenum="182"/>&que; &longs;imul AB BE, &amp; quadrupla vtriu&longs;&que; &longs;imul BC BD. (bis <lb/>enim a&longs;&longs;umitur AB, &amp; bis BE, quater ver&ograve; BC, &amp; quater BD) <lb/>
<arrow.to.target n="marg348"/> ad compo&longs;itam ex dupla ip&longs;ius BD cum &longs;ola BE; erit conuer&shy;<lb/>rendo, ut OD ad DA, ita compo&longs;ita ex dupla ip&longs;ius BD <expan abbr="c&utilde;">cum</expan> &longs;o&shy;<lb/>la BE ad <expan abbr="c&otilde;po&longs;itam">compo&longs;itam</expan> ex dupla utriu&longs;&que; &longs;imul AB BE, &amp; qua&shy;<lb/>
<arrow.to.target n="marg349"/> drupla vtriu&longs;&que; &longs;imul BCBD. <emph type="italics"/>et vtr&aelig;&que; ad primas eandem habe <lb/>bunt proportionem.<emph.end type="italics"/> hoc e&longs;t componendo erit OA ad AD, vt <expan abbr="c&otilde;-po&longs;ita">con&shy;<lb/>po&longs;ita</expan> ex dupla ip&longs;ius BD cum &longs;ola BE, &amp; dupla vtriu&longs;&que; &longs;i&shy;<lb/>mul AB BE, &amp; quadrupla vtriu&longs;&que; &longs;imul BC BD ad compo&shy;<lb/>
<arrow.to.target n="fig82"/><lb/>&longs;itam ex dupla vtriu&longs;&que; &longs;imul AB BE, &amp; quadrupla <expan abbr="vtrius&qacute;">vtrius&que;</expan>; <lb/>&longs;imul BC BD. In hoc autem antecedente bis&longs;umitur AB, qua <lb/>ter BC, &longs;exies ver&ograve; BD, &amp; ter BE. <emph type="italics"/>habebit igitur OA ad AD ean&shy;<lb/>demproportionem, quam linea &aelig;qualis dupl&aelig;ip&longs;ius AB, et quadrupl&aelig;i&shy;<lb/>p&longs;ius CB, et &longs;extupl&aelig; ip&longs;ius BD, ettripl&aelig; ip&longs;ius BE ad lineam compo&longs;i&shy;<lb/>tam ex dupla vtriu&longs;&que; &longs;imul AB EB, et quadrupla vtriu&longs;&que; &longs;imul <lb/>CB BD. babet autem<emph.end type="italics"/> (vt &longs;uppo&longs;itum e&longs;t) GH ad AD eandem <lb/>proportionem, quam linea &aelig;qualis dupl&aelig; ip&longs;ius AB, &amp; qua&shy;<lb/>drupl&aelig; ip&longs;ius BC, &amp; &longs;extupl&aelig; ip&longs;ius BD, &amp; tripl&aelig; ip&longs;ius BE <lb/>ad lineam &aelig;qualem quintupl&aelig; ip&longs;ius AB, &amp; decupl&aelig; ip&longs;ius <lb/>CB, &amp; decupl&aelig; ip&longs;ius BD, &amp; quintupl&aelig; ip&longs;ius BE, hoc e&longs;t ad
<pb xlink:href="077/01/187.jpg" pagenum="183"/><expan abbr="quintupl&atilde;">quintuplam</expan> vtriu&longs;&que;&longs;imul AB BE <expan abbr="c&utilde;">cum</expan> decupla vtriu&longs;qu&egrave; &longs;imul <lb/>CB BD. In <expan abbr="c&otilde;&longs;e&quetilde;ti">con&longs;e&que;nti</expan>.n.quinquies <expan abbr="a&longs;s&utilde;pta">a&longs;sumpta</expan> e&longs;t AB, &amp; quinquies <lb/>BE, decies CB, &amp; decies BD. &amp; conuettendo habebit <emph type="italics"/>AD ad<emph.end type="italics"/>
<arrow.to.target n="marg350"/><lb/><emph type="italics"/>GH eandem proportionem, quam quintupla vtriu&longs;&que; &longs;imul AB BE <lb/><expan abbr="c&utilde;">cum</expan> decupla vtriu&longs;&que; &longs;imul CB BD ad lineam compo&longs;itam ex dupla i&shy;<lb/>p&longs;ius AB, &amp; quadrupla ip&longs;ius CB, &amp; &longs;extuplaip&longs;ius BD, &amp; triplai&shy;<lb/>p&longs;ius EB. Di&longs;similiter autem qu&agrave;m in proportionibus ordinatis, hoc est <lb/>in perturbata proportione<emph.end type="italics"/> quoniam in primis magnitudinibus ita <lb/>&longs;e habet antecedens OA ad con&longs;e&que;ns AD, vt in &longs;ecundis ma <lb/>gnitudinibus antecedens compo&longs;ita nemp&egrave; ex dupla ip&longs;ius <lb/>AB, &amp; quadrupla ip&longs;ius BC, &amp; &longs;extupla ip&longs;ius BD, &amp; tripla <lb/>ip&longs;ius BE, ad con&longs;e&que;ns lineam &longs;cilic et compo&longs;itam ex du&shy;<lb/>pla vtriu&longs;&que; &longs;imul AB BE, &amp; quadrupla vtriu&longs;&que; &longs;imul CB <lb/>BD: ut autem in primis magnitudinibus con&longs;e&que;ns AD ad <lb/>aliud quippiam GH, ita in &longs;ecundis magnitudinibus aliud <lb/>quippiam, nemp&egrave;linea compo&longs;ita ex quintupla vtriu&longs;&que; &longs;i&shy;<lb/>mul AB BE cum decupla vtriu&longs;&que; &longs;imul CB BD ad antece&shy;<lb/>dens, hoc e&longs;t ad compo&longs;itam ex dupla ip&longs;ius AB, &amp; quadru&shy;<lb/>pla ip&longs;ius CB, &amp; &longs;extupla ip&longs;ius BD, &amp; tripla ip&longs;ius BE. quare <lb/><emph type="italics"/>ex &aelig;quali eandemhabet proportionem OA ad GH, quam quintupla v-<emph.end type="italics"/>
<arrow.to.target n="marg351"/><lb/><emph type="italics"/>triu&longs;&que; &longs;imul AB BE cum decupla<emph.end type="italics"/> vtriu&longs;&que; &longs;imul <emph type="italics"/>CB BD ad <lb/><expan abbr="c&otilde;po&longs;it&atilde;">compo&longs;itam</expan> ex dupla <expan abbr="vtrius&qacute;">vtrius&que;</expan>; &longs;imul AB BE, et quadrupla <expan abbr="vtrius&qacute;">vtrius&que;</expan>; &longs;imul <lb/>CB BD. At ver&ograve;<emph.end type="italics"/> quoniam quintupla ip&longs;ius AB ad duplam <lb/>eiu&longs;dem AB e&longs;t, vt quin&que; ad duo; &longs;imiliter quintupla ip&longs;i^&lcub;9} <lb/>BE ad duplam eiu&longs;dem BE e&longs;t, vt quin&que; ad duo, erit quin&shy;<lb/>tupla vtriu&longs;&que; &longs;imul AB BE ad duplam vtriu&longs;&que; &longs;imul AB <lb/>BE, vt quin&que; ad duo. </s>
          <s id="N17133">pariqu&egrave; ratione decupla vtriu&longs;&que; &longs;i&shy;<lb/>mul CB BD ad quadruplam vtriu&longs;&que; &longs;imul CB BD e&longs;t, vt <lb/>decem ad quatuor, hoc e&longs;t vt quin&que; ad duo. </s>
          <s id="N17139">&amp; <expan abbr="anteced&etilde;tia">antecedentia</expan>
<arrow.to.target n="marg352"/><lb/>ad con&longs;e&que;ntia in eadem erunt proportione, hoce&longs;t <emph type="italics"/>compo&longs;i&shy;<lb/>ta ex quintupla vtriu&longs;&que; &longs;imul AB BE cum decupla vtriu&longs;&que; &longs;imul <lb/>CB BD ad compo&longs;itam ex dupla vtriu&longs;&que; &longs;imul AB BE, &amp; quadru<lb/>pla vtriu&longs;&que; &longs;imul CB BD proportionem habet, quam quin&que; ad duo <lb/>Quare OA ad GH proportionem habet, quam quin&que; ad duo. </s>
          <s id="N1714F">Rur&longs;us<emph.end type="italics"/><lb/>factum fuit AD ad DO, vt compo&longs;ita ex dupla vtriu&longs;&que; &longs;i&shy;<lb/>mul AB BE cum quadrupla vtriu&longs;&que; &longs;imul CB BD ad <expan abbr="line&atilde;">lineam</expan> <lb/>BE vn&agrave; cum dupla ip&longs;ius BD. conuertendo etiam <emph type="italics"/>quoniam<emph.end type="italics"/>
<arrow.to.target n="marg353"/>
<pb xlink:href="077/01/188.jpg" pagenum="184"/>in primis magnitudinibus antecedens <emph type="italics"/>OD ad<emph.end type="italics"/> con&longs;e&que;ns <emph type="italics"/>DA <lb/>eandem habet proportionem, quam<emph.end type="italics"/> in &longs;ecundis magnitudinibus an<lb/>tecedens <emph type="italics"/>EB cum dupla ip&longs;ius BD ad<emph.end type="italics"/> con&longs;e&que;ns, <emph type="italics"/>lineam<emph.end type="italics"/> &longs;cilicet <emph type="italics"/>&aelig;&shy;<lb/>qualem line&aelig; compo&longs;it&aelig; ex dupla vtriu&longs;&que; &longs;imul AB BE cum quadru&shy;<lb/>pla vtriu&longs;&que; &longs;imul CB BD; est autem<emph.end type="italics"/> (vt antea o&longs;ten&longs;um e&longs;t) &amp; <lb/>in primis magnitudinibus con&longs;e&que;ns <emph type="italics"/>AD ad<emph.end type="italics"/> aliud <expan abbr="quippi&atilde;">quippiam</expan> <lb/><emph type="italics"/>DE, vt<emph.end type="italics"/> in &longs;ecundis magnitudinibus aliud quippiam, linea <lb/>&longs;cilicet <emph type="italics"/>compo&longs;ita ex dupla ip&longs;ius AB, &amp; tripla ip&longs;ius CB, &amp;<emph.end type="italics"/> &longs;ola <emph type="italics"/>BD <lb/>ad<emph.end type="italics"/> antecedens, nemp&egrave; <emph type="italics"/>lineam <expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> ip&longs;i EB, &amp; dupl&aelig; ip&longs;ius BD.<emph.end type="italics"/><lb/>
<arrow.to.target n="fig83"/><lb/><emph type="italics"/>Non igitur perinde, vt in proportione ordinata; hoc est, perturbata <expan abbr="exi&longs;t&etilde;">exi&longs;tem</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="marg354"/> <emph type="italics"/>te proportione, ex &aelig;qualiest OD ad DE, vt duplaip&longs;ius AB cum tripla <lb/>ip&longs;ius BC &amp;<emph.end type="italics"/> &longs;ola <emph type="italics"/>BD ad <expan abbr="c&otilde;po&longs;itam">compo&longs;itam</expan> ex dupla vtriu&longs;&que; &longs;imul AB BE, <lb/>&amp; quadrupla vtriu&longs;&que; &longs;imul CB BD.<emph.end type="italics"/> &longs;uperat ver&ograve; DE ip&longs;am <lb/>DO exce&longs;&longs;u OE; linea ver&ograve; <expan abbr="c&otilde;po&longs;ita">compo&longs;ita</expan> ex dupla vtriu&longs;&que; &longs;imul <lb/>AB BE, &amp; quadrupla vtriu&longs;&que; &longs;imul CB BD lineam excedit <lb/>compo&longs;itam ex dupla ip&longs;ius AB cum tripla ip&longs;ius BC, ac &longs;ola <lb/>BD, exce&longs;&longs;u line&aelig;, qu&aelig; &longs;it &aelig;qualis &longs;oli CB cum tripla ip&longs;ius <lb/>
<arrow.to.target n="marg355"/> BD, &amp; dupla ip&longs;ius BE. <emph type="italics"/>Quare est EO ad ED, vt CB cum tripla <lb/>ip&longs;ius BD, &amp; dupla ip&longs;ius EB ad duplam vtriu&longs;&que; &longs;imul AB BE, <lb/>&amp; quadruplam vtriu&longs;&que; &longs;imul CB BD. est autem<emph.end type="italics"/> in lineis pro-
<pb xlink:href="077/01/189.jpg" pagenum="185"/>portionalibus initio expo&longs;itis; c&ugrave;m in continua &longs;int propor&shy;<lb/>tione, tertia in ordine BD ad quartam BE, vt prima AB ad <lb/>&longs;ecundam BC, quare diuidendo vt DE ad EB, ita AC ad
<arrow.to.target n="marg356"/><lb/>CB. Rur&longs;us quoniam in lineis proportionalibus ob eandem <lb/>cau&longs;am CB ad BD ita e&longs;t, vt DB ad BE; erit diuidendo, vt <lb/>CD ad DB, ita DE ad EB. ego <emph type="italics"/>vt DE ad EB, ita AC ad<emph.end type="italics"/>
<arrow.to.target n="marg357"/><lb/><emph type="italics"/>CB, &amp; CD ad DB. ac propterea &longs;ecundum<emph.end type="italics"/> multiplicem <emph type="italics"/>compo&longs;itio <lb/>nemtripla ip&longs;ius CD, adtriplam ip&longs;ius DB<emph.end type="italics"/> e&longs;t, vt &longs;ola CD ad &longs;o&shy;<lb/>lam DB. <emph type="italics"/>&amp; dupla ip&longs;ius DE ad duplam ip&longs;ius EB<emph.end type="italics"/> e&longs;t, <lb/>vt DE ad EB. e&longs;t ver&ograve; CD ad DB, vt DE ad <lb/>EB, &amp; AC ad CB; erit igitur AC ad CB, vt tripla ip&longs;ius <lb/>CD ad triplam ip&longs;ius DB; &amp; vt dupla ip&longs;ius DE ad <lb/>duplam ip&longs;ius EB. <emph type="italics"/>Quare &amp;<emph.end type="italics"/> tria antecedentia &longs;imul ad
<arrow.to.target n="marg358"/><lb/>tria &longs;imul con&longs;e&que;ntia, hoc e&longs;t, <emph type="italics"/>compo&longs;ita ex AC, &amp; <lb/>tripla ip&longs;ius CD, &amp; dupla ip&longs;ius DE ad compo&longs;itam ex CB, <lb/>&amp; tripla ip&longs;ius DB, &amp; dupla ip&longs;ius EB<emph.end type="italics"/> ita erit, vt AC <lb/>ad CB, hoc e&longs;t, DE ad EB. <emph type="italics"/>Rur&longs;us ita&que; di&longs;simili modo, <lb/>qu&agrave;m in proportionibus ordinatis, hoc est in perturbata proportione,<emph.end type="italics"/><lb/>quoniam e&longs;t in primis magnitudinibus antecedens OE ad <lb/>con&longs;e&que;ns ED, ita in &longs;ecundis magnitudinibus an <expan abbr="teced&etilde;s">tecedens</expan> <lb/>compo&longs;ita &longs;cilicet ex CB, cum tripla ip&longs;ius BD, &amp; dupla ip&shy;<lb/>&longs;ius EB, ad con&longs;e&que;ns nem p&egrave; compo&longs;itam ex dupla vtriu&longs;&shy;<lb/>&que; &longs;imul AB BE, cum quadrupla vtriu&longs;&que; &longs;imul CB BD: <lb/>in primis ver&ograve; magnitudinibus con&longs;e&que;ns DE ad aliud quip <lb/>piam EB e&longs;t, vt in &longs;ecundis magnitudinibus aliud quippia, <lb/>hoc e&longs;t compo&longs;ita ex AC cum tripla ip&longs;ius CD, &amp; dupla ip&shy;<lb/>&longs;ius DE ad antecedens, lineam &longs;cilicet compo&longs;itam ex CB cum <lb/>tripla ip&longs;ius BD, &amp; dupla ip&longs;ius EB. <emph type="italics"/>ex &aelig;quali eandem<emph.end type="italics"/>
<arrow.to.target n="marg359"/><lb/><emph type="italics"/>habebit proportionem EO ad EB, quam AC cum tri <lb/>pla ip&longs;ius CD, &amp; dupla ip&longs;ius DE ad duplam vtriu&longs; <lb/>&que; &longs;imul AB BE cum qnadrupla vtriu&longs;&que; &longs;imul CB <lb/>BD.<emph.end type="italics"/> &amp; componendo erit OB ad BE, vtlinea AC
<arrow.to.target n="marg360"/><lb/>cum tripla ip&longs;ius CD, &amp; dupla ip&longs;ius DE, &amp; dupla <lb/>vtriu&longs;&que; &longs;imul AB BE, &amp; quadrupla vtriu&longs;&que; &longs;i&shy;<lb/>mul CB BD, ad duplam vtriu&longs;&que; &longs;imul AB BE <lb/>cum quadrupla vtriu&longs;&que; &longs;rmul CB BD. In hoc autem
<pb xlink:href="077/01/190.jpg" pagenum="186"/>antecedente a&longs;&longs;umitur &longs;ola AC, ter CD, bis DE, bis AB, <lb/>bis BE, quater CB, &amp; quater BD. Du&aelig; ver&ograve; AB vn&agrave; <lb/>cum &longs;ola AC, &amp; &longs;ola. </s>
          <s id="N17299">CB, ex quatuor vicibus, quibus ip&shy;<lb/>&longs;a CB &longs;umitur, &longs;unt &aelig;quales tribus AB. tres autem CB, <lb/>qu&aelig; relict&aelig; &longs;unt, vn&agrave; cum tribus CD, &amp; tribus BD <lb/>ex quatuor vicibus, quibus ip&longs;a BD &longs;umitur, &longs;unt &aelig;&shy;<lb/>quales &longs;ex CB. &longs;ola ver&ograve; BD, qu&aelig; relicta fuit, vn&agrave; <lb/>cum duabus DE, &amp; duabus BE, e&longs;t &aelig;qualis tribus <lb/>BD. linea nimirum AC cum tripla ip&longs;ius CD, &amp; <lb/>dupla ip&longs;ius DE, &amp; dupla vtriu&longs;&que; &longs;imul AB BE, <lb/>&amp; quadrupla vtriu&longs;&que; &longs;imul CB BD, &aelig;qualis erit tri&shy;<lb/>pl&aelig; ip&longs;ius AB, cum &longs;extupla ip&longs;ius CB, &amp; tripla ip&shy;<lb/>&longs;ius BD. <emph type="italics"/>Tota igitur OB ad EB eandem habet proportio&shy;<lb/>nem, quam linea &aelig;qualis tripl&aelig; ip&longs;ius AB cum &longs;extupla ip&shy;<lb/>&longs;ius CB &amp; tripla ip&longs;ius BD ad duplam vtriu&longs;&que; &longs;imul <lb/>AB BE cum quadrupla vtriu&longs;&que; &longs;imul CB BD. &amp; <lb/>quoniam<emph.end type="italics"/> initio o&longs;ten&longs;um fuit lineas AC CD DE in eadem <lb/>e&longs;&longs;e proportione, vt &longs;unt quatuor line&aelig; continu&egrave; pro&shy;<lb/>portionales AB BC BD BE; erunt tres AC CD <lb/>DE, &amp; tres AB BC BD, &amp; tres BC BD BE <lb/>
<arrow.to.target n="marg361"/> in eadem proportione. </s>
          <s id="N172C9">conuertendo igitur in eadem quo&shy;<lb/>&que; erunt proportione. </s>
          <s id="N172CD">quare tres <emph type="italics"/>ED DC CA,<emph.end type="italics"/> &amp; <lb/>tres BE BD BC, &amp; tres BD BC BA <emph type="italics"/>in eadem &longs;unt proportione.<emph.end type="italics"/><lb/>
<arrow.to.target n="marg362"/> Quoniam autem BE BD BC ita &longs;e habent, vt BD BC BA; <lb/>vtra&que; &longs;imul BE BD advtram&que; &longs;imul BD BC, &amp; <lb/>vtra&que; &longs;imul BD BC ad vtram&que; &longs;imul BC BA <lb/>ita &longs;e habebunt, vt BE BD BC. h&aelig; ver&ograve; <emph type="italics"/>B<emph.end type="italics"/>E <emph type="italics"/>B<emph.end type="italics"/>D <lb/>BC &longs;unt, vt ED DC CA. ergo <emph type="italics"/>&amp; vtra&que; &longs;imul <lb/>vna&que;&que; ip&longs;arum EB BD, DB BC, CB BA<emph.end type="italics"/>, ita &longs;e <lb/>
<arrow.to.target n="marg363"/> habebunt, vt ED DC CA. quare <emph type="italics"/>erit &amp;<emph.end type="italics"/> antecedens <lb/><emph type="italics"/>ED<emph.end type="italics"/> ad &longs;uas con&longs;e&que;ntes DC CA &longs;imul &longs;umptas, <lb/>hoc e&longs;t <emph type="italics"/>ad DA, vt<emph.end type="italics"/> antecedens <emph type="italics"/>vtra&que; &longs;imul EB BD<emph.end type="italics"/><lb/>ad &longs;uas con&longs;e&que;ntes, nemp&egrave; <emph type="italics"/>ad <expan abbr="vtr&amacr;&que;">vtran&que;</expan> &longs;imul DB BC<emph.end type="italics"/><lb/>
<arrow.to.target n="marg364"/> <emph type="italics"/>cum vtra&que; &longs;imul CB BA. &amp; componendo EA ad AD, <lb/>vt vtra&que; &longs;imul EB BD cum vtra&que; &longs;imul AB BC, <lb/>&amp; vtra&que; &longs;imul CB BD<emph.end type="italics"/> ad vtram&que; &longs;imul BD BC
<pb xlink:href="077/01/191.jpg" pagenum="187"/>cum vtra<gap/>ue &longs;imul CB BA. In hoc autem antecedenti &longs;emel <lb/>&longs;umitur EB, &amp; &longs;emel AB, bis BD, &amp; bis BC. in con&longs;e&que;ntive <lb/>r&ograve; &longs;umitur <gap/>ola BD, &longs;olaqu&egrave; BA, &amp; bis BC. Proportio igitur <lb/>ip&longs;arum EA AD e&longs;t eadem, <emph type="italics"/>qu&aelig; est vtra&que; &longs;imul EB BA cum du<lb/>pla vtriu&longs;&que; &longs;imul DB BC ad vtram&que; &longs;imul BD BA cum dupla <lb/>ip&longs;ius BC. Quare &amp; dupla ad duplam eandem habebit <expan abbr="proportion&etilde;">proportionem</expan> hoc <lb/>est, vt EA ad AD, ita dupla vtriu&longs;&que; &longs;imul EB BA cum quadru&shy;<lb/>pla vtriu&longs;&que; &longs;imul CB BD ad duplam vtriu&longs;&que; &longs;imul AB BD cum<emph.end type="italics"/><lb/>
<arrow.to.target n="fig84"/><lb/><emph type="italics"/>quadrupla ip&longs;ius CB. Quapropter EA adtres quintas ip&longs;ius AD e&longs;t, vt <lb/>compo&longs;ita ex dupla vtriu&longs;&que; &longs;imul AB BE, &amp; qua-<emph.end type="italics"/>
<arrow.to.target n="marg365"/><lb/><emph type="italics"/>drupla utrivs&que; &longs;imul CB BD ad tres quintas line&aelig; com<lb/>po&longs;it&aelig; ex dupla vtriu&longs;&que; &longs;imul AB BD, &amp; quadruplaip&longs;ius CB. Ve&shy;<lb/>r&ugrave;m<emph.end type="italics"/> quia initio a&longs;&longs;umptum fuitita e&longs;&longs;e BE ad EA, vt FG ad <lb/>tres quintas ip&longs;ius AD, erit conuertendo EA ad EB, vt
<arrow.to.target n="marg366"/><lb/>tres quint&aelig; ip&longs;ius AD ad FG; permutandoqu&egrave; <emph type="italics"/>vt EA ad <lb/>tres quintasip&longs;ius AD, &longs;ic e&longs;t EB ad FG, vtigitur EB ad FG, <lb/>&longs;ic dupla vtriu&longs;&que; &longs;imul AB BE cum quadrupla vtriu&longs;&que;<emph.end type="italics"/>
<pb xlink:href="077/01/192.jpg" pagenum="188"/><emph type="italics"/>&longs;imul DB BC ad tres quintas line&aelig; compo&longs;it&aelig; ex dupla vtriu&longs;&que; &longs;i&shy;<lb/>mul AB BD cum quadrupla ip&longs;ius CB. osten &longs;um e&longs;t aut<gap/> OB ad EB <lb/>ita e&longs;&longs;e, vt<emph.end type="italics"/> tripla ip&longs;ius AB cum &longs;extupla ip&longs;ius CB, &amp; tripla i&shy;<lb/>p&longs;ius BD ad duplam vtriu&longs;&que; &longs;imul AB BE cum quadrupla <lb/>vtriu&longs;&que; &longs;imul CB BD. At in hoc antecedente ter a&longs;&longs;umpta <lb/>e&longs;t AB, terqu&egrave; BD, &amp; &longs;exies CB. erit ita&que; in primis magni&shy;<lb/>tudinibus antecedens OB ad con&longs;e&que;ns EB, vt in &longs;ecundis <lb/>magnitudinibus an recedens <emph type="italics"/>tripla<emph.end type="italics"/> &longs;cilicet <emph type="italics"/>vtriu&longs;&que; &longs;imul AB <lb/>BD cum &longs;extupla ip&longs;ius CB ad<emph.end type="italics"/> con&longs;e&que;ns nemp&egrave; <emph type="italics"/>duplam v&shy;<lb/>triu&longs;&que; &longs;imul AB BE, &amp; quadruplam vtriu&longs;&que; &longs;imul CB BD.<emph.end type="italics"/><lb/>
<arrow.to.target n="fig85"/><lb/>in primis ver&ograve; magnitudinibus e&longs;t con&longs;e&que;ns EB ad aliud <lb/>quippiam FG, ut in &longs;ecundis magnitudinibus con&longs;e&que;ns, <lb/>hoc e&longs;t dupla vtriu&longs;&que; &longs;imul AB BE cum quadrupla vtriu&longs;&shy;<lb/>&que; &longs;imul DB BC ad aliud quippiam, nemp&egrave; ad tres quintas <lb/>line&aelig; <expan abbr="c&otilde;po&longs;it&etail;">compo&longs;it&etail;</expan> ex dupla vtri^&lcub;9} <expan abbr="&qacute;">&que;</expan>; &longs;imul AB BD <expan abbr="c&utilde;">cum</expan> quadrupla i&shy;<lb/>
<arrow.to.target n="marg367"/> p&longs;i^&lcub;9} CB. <emph type="italics"/>Ex &aelig;quali igitur e&longs;t, ut OB ad FG, ita linea compo&longs;ita ex tripla<emph.end type="italics"/><lb/>
<arrow.to.target n="marg368"/> <emph type="italics"/><expan abbr="utrius&qacute;">utrius&que;</expan> &longs;imul AB BD, et &longs;extuplaip&longs;i^&lcub;9} CB ad tres <expan abbr="qu&itilde;tas">quintas</expan> line&aelig; <expan abbr="c&otilde;po&longs;i">compo&longs;i</expan> <lb/>t&aelig; ex dupla utri^&lcub;9} <expan abbr="&qacute;">&que;</expan>; &longs;imul AB BD, &amp; quadrupla ip&longs;ius CB. at uer&ograve;<emph.end type="italics"/> tri <lb/>pla ip&longs;ius AB ad <expan abbr="dupl&atilde;">duplam</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> AB e&longs;t, vt tria ad duo. </s>
          <s id="N17400">&longs;imiliter <lb/>tripla ip&longs;ius BD ad duplam eiu&longs;dem BD e&longs;t, vt tria ad duo.
<pb xlink:href="077/01/193.jpg" pagenum="189"/>pariqu&egrave; ratione &longs;extupla ip&longs;ius CB ad quadruplam ciu&longs;dem, <lb/>CB ita &longs;e habet, vt &longs;ex ad quatuor, hoce&longs;t tria ad duo, &amp; om&shy;<lb/>nesad omnes, hoc e&longs;t <emph type="italics"/>compo&longs;ita ex tripla vtriu&longs;&que; &longs;imul AB BD, <lb/>et &longs;extupla ip&longs;ius CB ad compo&longs;itam ex dupla vtriu&longs;&que; &longs;imul AB BD, <lb/>&amp; quadrupla ip&longs;ius CB proportionem habet, quam tria ad duo.<emph.end type="italics"/> vt exem
<arrow.to.target n="marg369"/><lb/>pli gratia quindecim ad decem, <emph type="italics"/>&longs;ed<emph.end type="italics"/> eadem compo&longs;ita ex tri&shy;<lb/>pla vtriu&longs;&que; &longs;imul AB BD, &amp; &longs;extupla ip&longs;ius CB <emph type="italics"/>ad tres quin&shy;<lb/>tas eiu&longs;dem<emph.end type="italics"/> compofit&aelig; ex dupla vtriu&longs;&que; &longs;imul AB BD, &amp; qua
<arrow.to.target n="marg370"/><lb/>drupla ip&longs;ius, CB, qu&aelig; po&longs;ita e&longs;t decem, <emph type="italics"/>proportionem habet, quam <lb/>quin&que; ad duo.<emph.end type="italics"/> hoc e&longs;t ut quindecim ad &longs;ex, tres enim quint&aelig; <lb/>ip&longs;ius decem &longs;unt &longs;ex. </s>
          <s id="N1743A">at ver&ograve; proportio, quam habet linea <expan abbr="c&otilde;">com</expan> <lb/>po&longs;ita ex tripla vtriu&longs;&que; &longs;imul AB BD, &amp; &longs;extupla ip&longs;ius CB <lb/>ad tres quintas line&aelig; compo&longs;it&etail; ex dupla vtriu&longs;&que; &longs;imul AB <lb/>BD cum quadrupla ip&longs;ius CB, e&longs;t &aelig;qualis ei, quam habet OB <lb/>ad FG. ergo erit OB ad FG, vtquin&que; ad duo. <emph type="italics"/><expan abbr="Demonstrat&umacr;">Demonstratum</expan> <lb/>autem e&longs;t, &amp; AO ad GH proportionem habere, quam quin&que; ad duo; <lb/>totaigitur BA ad totam FH proportionem habet, quam quin&que; ad duo.<emph.end type="italics"/>
<arrow.to.target n="marg371"/><lb/><emph type="italics"/>&longs;iautem hoc, e&longs;t quidem FH du&aelig; quint&aelig; ip&longs;ius AB. Quod oportebat <lb/>demon&longs;trare.<emph.end type="italics"/></s>
        </p>
        <p id="N17460" type="margin">
          <s id="N17462"><margin.target id="marg339"/>1.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N17470" type="margin">
          <s id="N17472"><margin.target id="marg340"/>2. <emph type="italics"/>lemma <lb/>buius.<emph.end type="italics"/></s>
        </p>
        <p id="N1747D" type="margin">
          <s id="N1747F"><margin.target id="marg341"/>1.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N1748D" type="margin">
          <s id="N1748F"><margin.target id="marg342"/>2. <emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N1749A" type="margin">
          <s id="N1749C"><margin.target id="marg343"/>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N174A5" type="margin">
          <s id="N174A7"><margin.target id="marg344"/>12. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N174B0" type="margin">
          <s id="N174B2"><margin.target id="marg345"/>11, <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N174BB" type="margin">
          <s id="N174BD"><margin.target id="marg346"/>8.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N174C6" type="margin">
          <s id="N174C8"><margin.target id="marg347"/><emph type="italics"/>ex<emph.end type="italics"/> 8. <emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N174D8" type="margin">
          <s id="N174DA"><margin.target id="marg348"/><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N174EA" type="margin">
          <s id="N174EC"><margin.target id="marg349"/>18, <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N174F5" type="margin">
          <s id="N174F7"><margin.target id="marg350"/><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan>.<emph.end type="italics"/></s>
        </p>
        <p id="N17508" type="margin">
          <s id="N1750A"><margin.target id="marg351"/>23. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17513" type="margin">
          <s id="N17515"><margin.target id="marg352"/>12, <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1751E" type="margin">
          <s id="N17520"><margin.target id="marg353"/><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan><emph.end type="italics"/></s>
        </p>
        <p id="N17530" type="margin">
          <s id="N17532"><margin.target id="marg354"/>23.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1753B" type="margin">
          <s id="N1753D"><margin.target id="marg355"/>3.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N1754B" type="margin">
          <s id="N1754D"><margin.target id="marg356"/>17. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17556" type="margin">
          <s id="N17558"><margin.target id="marg357"/>A</s>
        </p>
        <p id="N1755C" type="margin">
          <s id="N1755E"><margin.target id="marg358"/>12.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17567" type="margin">
          <s id="N17569"><margin.target id="marg359"/>23.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17572" type="margin">
          <s id="N17574"><margin.target id="marg360"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N1757D" type="margin">
          <s id="N1757F"><margin.target id="marg361"/><emph type="italics"/>cor.4.qu&itilde; <lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p id="N17589" type="margin">
          <s id="N1758B"><margin.target id="marg362"/>4.<emph type="italics"/>lema hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N17596" type="margin">
          <s id="N17598"><margin.target id="marg363"/><emph type="italics"/>cor.2.lem. <lb/>in<emph.end type="italics"/> 13. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N175AA" type="margin">
          <s id="N175AC"><margin.target id="marg364"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N175B5" type="margin">
          <s id="N175B7"><margin.target id="marg365"/>B</s>
        </p>
        <p id="N175BB" type="margin">
          <s id="N175BD"><margin.target id="marg366"/><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="qu&itilde;ti">quinti</expan>.<emph.end type="italics"/><lb/>16,<emph type="italics"/>quinti.<emph.end type="italics"/><lb/>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N175DC" type="margin">
          <s id="N175DE"><margin.target id="marg367"/>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N175E7" type="margin">
          <s id="N175E9"><margin.target id="marg368"/>C</s>
        </p>
        <p id="N175ED" type="margin">
          <s id="N175EF"><margin.target id="marg369"/>D</s>
        </p>
        <p id="N175F3" type="margin">
          <s id="N175F5"><margin.target id="marg370"/>5.<emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p id="N17600" type="margin">
          <s id="N17602"><margin.target id="marg371"/>12.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.193.1.jpg" xlink:href="077/01/193/1.jpg"/>
        <figure id="id.077.01.193.2.jpg" xlink:href="077/01/193/2.jpg"/>
        <figure id="id.077.01.193.3.jpg" xlink:href="077/01/193/3.jpg"/>
        <figure id="id.077.01.193.4.jpg" xlink:href="077/01/193/4.jpg"/>
        <figure id="id.077.01.193.5.jpg" xlink:href="077/01/193/5.jpg"/>
        <p id="N1761F" type="head">
          <s id="N17621">SCHOLIVM.</s>
        </p>
        <p id="N17623" type="main">
          <s id="N17625">Gr&aelig;cus codex po&longs;t ea verba, <emph type="italics"/>vt DE ad EB, ita AC ad CB,<emph.end type="italics"/>
<arrow.to.target n="marg372"/><lb/>non habet, <emph type="italics"/>&amp; CD ad DB,<emph.end type="italics"/> qu&aelig; ob ea, qu&aelig; &longs;equuntur, omnin&ograve; <lb/>nece&longs;&longs;aria videntur. </s>
          <s id="N1763A">ideo po&longs;t gr&etail;ca verba, <foreign lang="greek">e)/side\ka<gap/> w(s de w_&lt;10&gt;o\s eb, <lb/>ou)/tws a)/te ag w_&lt;10&gt;o\s, gb</foreign> de&longs;iderarividentur. <foreign lang="greek">ka<gap/> a(= gd w_&lt;10&gt;o/s db. </foreign></s>
        </p>
        <p id="N17649" type="margin">
          <s id="N1764B"><margin.target id="marg372"/><emph type="italics"/>A<emph.end type="italics"/></s>
        </p>
        <p id="N17653" type="main">
          <s id="N17655">Vbiautem &longs;untverba, <emph type="italics"/>vt <expan abbr="c&otilde;po&longs;ita">compo&longs;ita</expan> ex dupla vtriu&longs;&que; &longs;imul,<emph.end type="italics"/> Gr&aelig;
<arrow.to.target n="marg373"/><lb/>cus codex tant&ugrave;m habet, <foreign lang="greek">ou)\tws a) sugkeime/na e)\xte ta=s suuamfote&lt;10&gt;ou</foreign>. <lb/>In quibus de&longs;ideratur illa particula, <emph type="italics"/>dupla,<emph.end type="italics"/> ideo corrigendus e&longs;t <lb/>hoc modo, <foreign lang="greek">ou)/tws a\ sugkeime/na e)\kte ta=s b suuamfote/&lt;10&gt;ou</foreign>, &amp;c. </s>
        </p>
        <pb xlink:href="077/01/194.jpg" pagenum="190"/>
        <p id="N1767B" type="margin">
          <s id="N1767D"><margin.target id="marg373"/>B</s>
        </p>
        <p id="N17681" type="main">
          <s id="N17683">Pr&aelig;terea c&ugrave;m inquit, <emph type="italics"/>ex &aelig;qualiigitur e&longs;t vt OB ad FG,<emph.end type="italics"/> Gr&aelig;&shy;<lb/>
<arrow.to.target n="marg374"/> cus non habet, <emph type="italics"/>ad FG,<emph.end type="italics"/> idcirco po&longs;t ea verba <foreign lang="greek">kai\ di\<gap/>sou a)/&lt;10&gt;a e)sin co(s <lb/>a<gap/> ob</foreign> addenda &longs;unt <foreign lang="greek">w_&lt;10&gt;o\s zk. </foreign></s>
        </p>
        <p id="N176A4" type="margin">
          <s id="N176A6"><margin.target id="marg374"/>C</s>
        </p>
        <p id="N176AA" type="main">
          <s id="N176AC">Similiter quando in quit <emph type="italics"/>ad compo&longs;itam ex dupla vtriu&longs;&que; &longs;imul<emph.end type="italics"/><lb/>
<arrow.to.target n="marg375"/> <emph type="italics"/>AB BD, &amp; quadrupla ip&longs;ius CB,<emph.end type="italics"/> gr&aelig;ca verba &longs;unt <foreign lang="greek">w_&lt;10&gt;o=s me\n ta\n sug&shy;<lb/>keim<gap/>nan e)/kte ta=s b sunamfote\&lt;10&gt;ou ta=s ab bd ta=s *gb</foreign>, in quib^&lcub;9} &longs;imiliter deli&shy;<lb/>deratur, <emph type="italics"/>&amp; quadrupla.<emph.end type="italics"/> quare ita corrigendus videtur. <foreign lang="greek">w_&lt;10&gt;o\s me\n ta/n <lb/>sugkeime\nan e)/ k te tas b sunamfote/&lt;10&gt;ou ta=s ab bd, kai\ d ta)/s *gb</foreign>, </s>
        </p>
        <p id="N176D5" type="margin">
          <s id="N176D7"><margin.target id="marg375"/>D</s>
        </p>
        <p id="N176DB" type="main">
          <s id="N176DD">Po&longs;tremum theorema, &amp; &longs;i non habeat <expan abbr="t&atilde;tam">tantam</expan> <expan abbr="ob&longs;curitat&etilde;">ob&longs;curitatem</expan>, <lb/>veluti pr&etail;cedens, non e&longs;t tamen &longs;ine aliqua ob&longs;curitate, ob cu<lb/>ius intelligentiam hanc pri&ugrave;s propo &longs;itionem o&longs;tendemus. </s>
        </p>
        <p id="N176EB" type="head">
          <s id="N176ED">PROPOSITIO.</s>
        </p>
        <p id="N176EF" type="main">
          <s id="N176F1">Si du&aelig; fuerint rect&aelig; line&etail; in para bolc ad diametrum ordi <lb/>natim applicat&aelig;, erit maior parabole ad <expan abbr="minor&etilde;">minorem</expan>, vt cubus ex <lb/>dimidia line&etail; maioris ad cubum ex dimidia minoris. </s>
        </p>
        <figure id="id.077.01.194.1.jpg" xlink:href="077/01/194/1.jpg"/>
        <p id="N176FE" type="main">
          <s id="N17700">In parabole ABC, cuius diameter BF, du&aelig; &longs;int rect&aelig; line&aelig; <lb/>ad diametrum applicat&aelig; AC DE. Dico parabolen ABC ad <lb/>parabolen DBE eandem habere proportionem, quam cub^&lcub;9} <lb/>ex AF ad cubum ex DG. lungantur AB BC DB BE; &longs;ecet-
<pb xlink:href="077/01/195.jpg" pagenum="191"/>qu&egrave; AB ip&longs;am DG in H. Quoniam enim parabole ABC
<arrow.to.target n="marg376"/><lb/>&longs;e&longs;quitertia e&longs;t trianguli ABC, itidemqu&egrave; parabole DBE <lb/>trianguli DBE &longs;e&longs;quitertia exi&longs;tit, erit parabole ABC ad trian <lb/>gulum ABC, vt parabole DBE ad triangulum DBE. &amp;
<arrow.to.target n="marg377"/> per&shy;<lb/>mutando parabole ABC ad parabolen DBE, vt triangulum <lb/>ABC ad triangulum DBE. Quoniam autem AC ordina&shy;<lb/>tim e&longs;t applicata, vnde AF ip&longs;i FC e&longs;t &aelig;qualis, ac per con&longs;e&shy;<lb/>&que;ns AF e&longs;t ip&longs;ius AC dimidia. </s>
          <s id="N17721">erit triangulum ABF dimi&shy;<lb/>dium trianguli ABC. c&ugrave;m vtraqu&egrave; &longs;ub eadem &longs;int altitudine.
<arrow.to.target n="marg378"/><lb/>eademqu&egrave; ratione triangulum DBG trianguli DBE dimi&shy;<lb/>dium exi&longs;tit. </s>
          <s id="N1772C">quare vt triangulum ABF ad triangulum <lb/>DBG, ita e&longs;t triangulum ABC ad DBE triangulum, ac pro&shy;<lb/>pterea triangulum ABF ad DBG triangulum e&longs;t, vt parabo&shy;<lb/>le ABC ad parabolen DBE. C&ugrave;m autem &longs;it HG &aelig;quidi&longs;tans <lb/>ip&longs;i AF, &longs;iquidem &longs;unt ordinatim applicat&aelig;, ob <expan abbr="triangulor&utilde;">triangulorum</expan>
<arrow.to.target n="marg379"/><lb/>&longs;imilitudinem ABF HBG, ita erit FB ad BG, vt AF ad HG <lb/>vt autem FB ad BG, ita quadratum ex AF ad quadratum ex
<arrow.to.target n="marg380"/><lb/>DG, erit igitur quadratum ex AF ad quadratum ex DG, vt AF <lb/>ad HG. quare line&etail; AF DG HG &longs;unt proportionales. </s>
          <s id="N17748">Pro&shy;<lb/>ducatur FB, ducaturqu&egrave; &agrave; puncto D ip&longs;i AB &aelig;quidi&longs;tans <lb/>DK, erit vtiqu&egrave; triangulorum ABF DKG anguli ABF <lb/>DHG &aelig;quales, &amp; angulus AFB angulo DGK e&longs;t &aelig;qualis, erit <lb/>igitur, &amp; reliquus BAF reliquo KDG &aelig;qualis, ac propterea <lb/>triangulum ABF e&longs;t triangulo DKG &longs;imile. </s>
          <s id="N17754">quare triangu&shy;<lb/>lum ABF ad triangulum DKG eam habet proportionem, <lb/>qu&agrave;m AF ad DG duplicatam, hoc e&longs;t qu&agrave;m AF ad HG, qu&etail; <lb/>e&longs;t ea, qu&agrave;m habet FB ad BG. atqui triangulum ABF ad <lb/>DKG eam quo&que; habet proportionem, quam FB ad GK <lb/>duplicatam. </s>
          <s id="N17760">tres igitur line&etail; FB GK GB &longs;unt proportiona&shy;<lb/>les. </s>
          <s id="N17764">ex quibus &longs;equiturita e&longs;&longs;e FB ad GK, vt AF ad DG; &amp; <lb/>GK ad GB, vt DG ad GH. &longs;ed quoniam triangulum GDK <lb/>ad GDB (c&ugrave;m &longs;int &longs;ub eadem altitudine) ita e&longs;t, vt KG ad
<arrow.to.target n="marg381"/><lb/>BG, &longs;i igitur fiat HG ad L, vt KG ad BG, erit triangulum <lb/>GDK ad triangulum GDB, vt HG ad L. C&ugrave;m autem &longs;it <expan abbr="tri&atilde;">triam</expan> <lb/>gulum ABF ad DKG, vt AF ad HG, e&longs;tqu&egrave; <expan abbr="triangul&utilde;">triangulum</expan> DKG <lb/>ad DBG, vt HG ad L, erit ex &etail;quali triangulum ABF ad <lb/>triangulum DBG, vt AF ad L. ac propterea parabole ABC
<pb xlink:href="077/01/196.jpg" pagenum="192"/>ad parabolen DBE eam habet proportionem, quam linea <lb/>AF ad lineam L. Quoniam autem ita e&longs;t KG ad GB, vt <lb/>HG ad L, &amp; vt eadem KG ad GB, ita e&longs;t DG ad GH. vt <lb/>ver&ograve; DG ad GH, ita e&longs;t AF ad DG; crunt quatuor line&aelig; AF <lb/>DG HG L in continua proportione. </s>
          <s id="N1778B">&amp; quoniam cubi in tri&shy;<lb/>pla &longs;unt proportione laterum, erit cubus ex AF ad cubum ex <lb/>DG, vt AF ad L. cubus ergo ex AF ad cubum ex DG eam <lb/>habet proportionem, quam parabole ABC ad parabolen <lb/>DBE. quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N17795" type="margin">
          <s id="N17797"><margin.target id="marg376"/>17.34. A<emph type="italics"/>r <lb/>ch.de qua. <lb/>par.<emph.end type="italics"/></s>
        </p>
        <p id="N177A4" type="margin">
          <s id="N177A6"><margin.target id="marg377"/>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N177AF" type="margin">
          <s id="N177B1"><margin.target id="marg378"/><emph type="italics"/>ex prima <lb/>&longs;extt.<emph.end type="italics"/></s>
        </p>
        <p id="N177BB" type="margin">
          <s id="N177BD"><margin.target id="marg379"/><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N177CB" type="margin">
          <s id="N177CD"><margin.target id="marg380"/>20. <emph type="italics"/>primi <lb/>conicorum <lb/>Apoll. </s>
          <s id="N177D8">&amp; <lb/>ex<emph.end type="italics"/>3. A<emph type="italics"/>rch. <lb/>de quad. <lb/>parab. <lb/>ex cor.<emph.end type="italics"/> 20. <lb/><emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N177F1" type="margin">
          <s id="N177F3"><margin.target id="marg381"/>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/><lb/>11.<emph type="italics"/>quintl.<emph.end type="italics"/></s>
        </p>
        <p id="N17803" type="main">
          <s id="N17805">Oportet autem banc quoqu&egrave; <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> nobis e&longs;&longs;e cogni <lb/>tam, nem p&egrave; qu&ograve;d &longs;olida parallelepipeda in eadem ba&longs;i con&longs;ti <lb/>tuta eam inter &longs;e proportionem habent, quam ip&longs;arum alti&shy;<lb/>tudines. </s>
        </p>
        <p id="N17811" type="main">
          <s id="N17813">Hoc quidem &agrave; Federico Commandino in eius libro de cen<lb/>tro grauitatis &longs;olidorum propo&longs;itione decimanona demon&shy;<lb/>&longs;tratum fuit. </s>
        </p>
        <p id="N17819" type="head">
          <s id="N1781B">PROPOSITIO. X.</s>
        </p>
        <p id="N1781D" type="main">
          <s id="N1781F">Omnis fru&longs;ti &agrave; rectanguli coni portione ab&longs;ci&longs;&longs;i <lb/>centrum grauitatis e&longs;t in recta linea, qu&aelig; fru&longs;ti dia&shy;<lb/>meter exi&longs;tit, ita po&longs;itum, vt diui&longs;a linea in quin&shy;<lb/>&que; partes &aelig;quales, &longs;it in quinta parte media; ita <lb/>vt ip&longs;ius portio propinquior minoriba&longs;i fru&longs;ti ad <lb/>reliquam portionem eandem habeat proportio&shy;<lb/>nem, quam habet &longs;olidum ba&longs;im habens quadra&shy;<lb/>tumex dimidia maioris ba&longs;is fru&longs;ti, altitudinem au<lb/>tem lineam &aelig;qualem vtri&que; &longs;imul dupl&aelig; mino&shy;<lb/>ris ba&longs;is, &amp; maiori ad &longs;olidum ba&longs;im habens qua&shy;<lb/>dratum ex dimidia minoris ba&longs;is fru&longs;ti, <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>autem lineam &aelig;qualem vtri&que; dupl&aelig; maioris, &amp; <lb/>minori. </s>
        </p>
        <pb xlink:href="077/01/197.jpg" pagenum="193"/>
        <figure id="id.077.01.197.1.jpg" xlink:href="077/01/197/1.jpg"/>
        <p id="N17843" type="main">
          <s id="N17845"><emph type="italics"/>Sit in rectanguli coni portione<emph.end type="italics"/> ABC <emph type="italics"/>du&aelig; rect&aelig; line&aelig; AC DE<emph.end type="italics"/><lb/>&aelig;quidi&longs;tantes. <emph type="italics"/>diameter ver&ograve; portionis ABC &longs;it BF.<emph.end type="italics"/> Intelli&shy;<lb/>gaturqu&egrave; fru&longs;tum ADEC &agrave; portione ABC ab&longs;ci&longs;&longs;um. </s>
          <s id="N1785B">om&shy;<lb/>nes vti&que; line&aelig; ip&longs;is AC DE &aelig;quidi&longs;tantes in fru&longs;to AD <lb/>EC duct&aelig;, erunt &agrave; linea GF bifartam diui&longs;&aelig;, ex quo <emph type="italics"/>pa&shy;<lb/>tet quidem &amp; ip&longs;ius ADEC diametrum e&longs;&longs;e GF, lineasqu&egrave; AC <lb/>DE line&aelig; portionem in B contingenti &aelig;quidistantes e&longs;&longs;e. </s>
          <s id="N17868">Recta<emph.end type="italics"/>
<arrow.to.target n="marg382"/><lb/><emph type="italics"/>ver&ograve; linea FG in quin&que; partes &aelig;quales diui&longs;a, quinta pars me&shy;<lb/>dia &longs;it HK. at&que;<emph.end type="italics"/> diuidatur HK in I, ita vt <emph type="italics"/>HI ad <lb/>IK eandem habeat proportionem, quam habet &longs;olidum ba&longs;im habens <lb/>quadratum ex AF, altitudinem ver&ograve; lineam &aelig;qualem vtri&longs;&que; <lb/>&longs;imul dupl&aelig; ip&longs;ius DG, &amp; ip&longs;i AF, ad &longs;olidum, quod <lb/>ba&longs;im habeat quadratum ex DG, altitudinem autem lineam &aelig;qua-<emph.end type="italics"/>
<pb xlink:href="077/01/198.jpg" pagenum="194"/>
<arrow.to.target n="fig86"/><lb/><emph type="italics"/>lem vtri&longs;&que; dupl&aelig; ip&longs;ius AF, &amp; ip&longs;i DG. ostenden&shy;<lb/>dum est frusti ADEC centrum grauitatis e&longs;&longs;e punctum 1.<emph.end type="italics"/><lb/>
<arrow.to.target n="marg383"/> <emph type="italics"/>&longs;it quidem ip&longs;i FB &aelig;qualis MN, ip&longs;i ver&ograve; GB &aelig;qualis NO. <lb/>&longs;umaturqu&egrave; ip&longs;arum MN NO media proportionalis NX. <lb/>quarta ver&ograve; proportionalis TN.<emph.end type="italics"/> line&aelig; nimirum MN NX <lb/>NO NT in continua erunt proportione. <emph type="italics"/>&amp; vt TM <lb/>ad TN, ita<emph.end type="italics"/> fiat <emph type="italics"/>FH ad quandam lineam &agrave; puncto I, vt <gap/>R, vbi&shy;<lb/>cun&que; perueniat alterum punctum<emph.end type="italics"/> R. <emph type="italics"/>nihil enim refert, &longs;iue inter <lb/>FG, &longs;iue inter GB cadat. </s>
          <s id="N178BF">&amp; quoniam in portione rectanguli coni<emph.end type="italics"/><lb/>ABC <emph type="italics"/>diameter portionis est FB; at ver&ograve; BF, vel prin&shy;<lb/>cipalis est diameter portionis, vel ducta diametro &aelig;quidistans. <lb/>line&aelig; ver&ograve; AF DG ad ip&longs;am ordinatim &longs;unt ap&shy;<lb/>plicat&aelig;, c&ugrave;m &longs;int &aelig;quidistantes contingenti portionem<emph.end type="italics"/>
<pb xlink:href="077/01/199.jpg" pagenum="195"/><emph type="italics"/>in puncto B. &longs;i autem hoc, est vt AF ad DG potentia,<emph.end type="italics"/>
<arrow.to.target n="marg384"/><lb/><emph type="italics"/>&longs;ic FB ad BG longitudine, hoc est MN ad NO. <lb/>vt autem MN ad NO longitudine, itaest MN ad Nx potentia.<emph.end type="italics"/><lb/>quandoquidem tresline&aelig; MN NX NO &longs;unt proportio&shy;<lb/>nales. <emph type="italics"/>vt igitur AF ad DG potentia, ita est MN ad N X<emph.end type="italics"/>
<arrow.to.target n="marg385"/><lb/><emph type="italics"/>potentia. </s>
          <s id="N178F6">quare, &amp; longitudine in eadem &longs;unt proportione<emph.end type="italics"/>; vt &longs;cili <lb/>cet AF ad DG, ita MN ad NX. <emph type="italics"/>&longs;ieist ita&que; cubus ex AF<emph.end type="italics"/>
<arrow.to.target n="marg386"/><lb/><emph type="italics"/>ad cubum ex DG, ita cubus ex MN ad cubum ex NX. Ver&ugrave;m<emph.end type="italics"/>
<arrow.to.target n="marg387"/><lb/><emph type="italics"/>vt cubus ex AF adcubum ex DG, &longs;ic portio ABC ad portio&shy;<lb/>nem DBE.<emph.end type="italics"/> vtigitur cubus ex MN ad cubum ex NX, ita <lb/>portio ABC ad portionem DBE. <emph type="italics"/>&longs;icut autem cubus ex MN <lb/>ad culum ex Nx, ita MN ad NT.<emph.end type="italics"/> c&ugrave;m &longs;int quatuor line&aelig; <lb/>MN NX NO NT in continua proportione. </s>
          <s id="N17925">ac propterea <lb/>eritportio ABC ad portionem DBE, vt MN ad NT. <lb/><emph type="italics"/>Quare &amp; diuidendo frustum ADEC ad portionem DBE e&longs;t, vt<emph.end type="italics"/>
<arrow.to.target n="marg388"/><lb/><emph type="italics"/>MT ad NT.<emph.end type="italics"/> Quia vero, vt factum fuit, it&agrave; e&longs;t MT ad TN, <lb/>vt FH ad IR, e&longs;t ver&ograve; FH ip&longs;ius FG tresquint&aelig;, erit fru&shy;<lb/>&longs;tum ADEC ad portionem DBE, vt FH ad IR <emph type="italics"/>hoc est <lb/>tres quint&aelig; ip&longs;ius GF ad IR. &amp; quoniam &longs;olidum ba&longs;im habens qua&shy;<lb/>dratum ex AF, altitudinem ver&ograve; lineam compo&longs;itam ex dupla ip&longs;ius <lb/>DG, &amp; ip&longs;a AF, ad cubum ex AF proportionem habet,<emph.end type="italics"/> quam &longs;o <lb/>lidi altitudo ad altitudinem cubi, &longs;iquidem &longs;unt in eadem ba <lb/>&longs;i. </s>
          <s id="N1794E">t&agrave;m emm &longs;olidum, qu&agrave;m cubus ba&longs;im habet quadratum <lb/>ex AF. idcirco &longs;olidum ba&longs;im habens quadratum ex AF, <lb/>altitudinem ver&ograve; lineam compo&longs;itam ex dupla ip&longs;ius DG, &amp; <lb/>ip&longs;a AF ad cubum ex AF eam proportio nem habebit, <emph type="italics"/>quam<emph.end type="italics"/><lb/>&longs;olidi altitudo, <emph type="italics"/>dupla,<emph.end type="italics"/> &longs;cilicet <emph type="italics"/>ip&longs;ius DG cumlinea AF<emph.end type="italics"/> ad alci&shy;<lb/>tudinem cubi, hoc e&longs;t <emph type="italics"/>ad FA.<emph.end type="italics"/> Atver&ograve; quoniam o&longs;ten&longs;um e&longs;t <lb/>ita e&longs;&longs;e AF ad DG, vt MN ad NX, eritconuertendo DG <lb/>ad AF, vt NX ad MN, &amp; antecedentium dupla, hoc e&longs;t du<lb/>pla ip&longs;ius DG ad AF, vt dupla ip&longs;ius NX ad MN. &amp; com&shy;<lb/>ponendo dupla ip&longs;ius DG cum AF ad AF, vt dupla ip&longs;ius
<arrow.to.target n="marg389"/><lb/>NX cum MN ad MN. <emph type="italics"/>Quare &amp; vt<emph.end type="italics"/> &longs;olidum ba&longs;im habens <lb/>quadratum ex AF, altitudinem ver&ograve; lineam compo&longs;itam ex <lb/>dupla ip&longs;ius DG cum AF ad cubum ex AF, ita <emph type="italics"/>dupla ip&longs;ius NX <lb/>cum linea NM ad NM. est autem<emph.end type="italics"/> cubus ex AF adcubum <lb/>ex DG, vt cubus ex MN ad cubum ex NX, vt o&longs;ten&longs;um e&longs;t,
<pb xlink:href="077/01/200.jpg" pagenum="196"/>
<arrow.to.target n="fig87"/><lb/><emph type="italics"/>cubusver&ograve; ex MN ad cubum ex NX e&longs;t, vt MN ad N<emph.end type="italics"/>T; <lb/>erit <emph type="italics"/>&amp; vt cubus ex AF ad cubum ex DG, ita MN ad NT. <lb/>&longs;icut autem cubus ex DG ad &longs;olidum ba&longs;im habens quadratum ex DG, <lb/>altitudinem ver&ograve; lineam compo&longs;itam ex dupla ip&longs;ius AF, cum linea <lb/>DG,<emph.end type="italics"/> ita altitudo cubi ad altitudinem &longs;olidi, cum &longs;int in ea&shy;<lb/>dem ba&longs;i, quadrato nemp&egrave; ex DG. erit igitur vt cubus ex <lb/>DG ad &longs;olidum ba&longs;im habens quadratum ex DG, altitudi&shy;<lb/>nem ver&ograve; lineam compo&longs;itam ex dupla ip&longs;ius AF cum linea <lb/>DG, <emph type="italics"/>ita<emph.end type="italics"/> cubi altitudo <emph type="italics"/>DG ad<emph.end type="italics"/> altitudinem &longs;olidi, ad <lb/>lineam &longs;cilicet <emph type="italics"/>compo&longs;itam ex dupla ip&longs;ius AF, &amp; linea <lb/>DG.<emph.end type="italics"/> Quoniam autem ita e&longs;t AF ad DG, vt <lb/>MN ad NX, vt ver&ograve; MN ad NX, ita NO <lb/>ad NT. c&ugrave;m &longs;int MN NX NO NT in continua proportio <lb/>
<arrow.to.target n="marg390"/> ne, crit AF ad DG, vt NO ad NT. &amp; antecedentium dupla,
<pb xlink:href="077/01/201.jpg" pagenum="197"/>hoc e&longs;t, dupla ip&longs;ius AF ad DG, vt dupla ip&longs;ius NO ad <lb/>NT, &amp; componendo, dupla ip&longs;ius AF cum DG ad
<arrow.to.target n="marg391"/><lb/>DG, vt dupla ip&longs;ius NO cum NT ad NT. &amp; conuer&shy;<lb/>tendo DG ad duplam ip&longs;ius AF cum DG, vt NT ad
<arrow.to.target n="marg392"/> du&shy;<lb/>plam ip&longs;ius NO cum NT. <emph type="italics"/>Quare &amp; vt<emph.end type="italics"/> &longs;e habet cubus ex <lb/>DG ad &longs;olidum ba&longs;im habens quadratum ex DG, altitu&shy;<lb/>dinem ver&ograve; compo&longs;itam ex dupla ip&longs;ius AF cum DG, ita <lb/>e&longs;t <emph type="italics"/>TN ad compo&longs;itam ex dupla ip&longs;ius ON, &amp; linea TN.<emph.end type="italics"/> Ita&shy;<lb/>&que; ex ijs, qu&aelig; dicta &longs;unt, ita &longs;e habet &longs;olidum ba&longs;im ha&shy;<lb/>bens quadratum ex AF, altitudinem ver&ograve; lineam com&shy;<lb/>po&longs;itam ex dupla ip&longs;ius DG, &amp; linea AF ad cubum <lb/>ex AF, vt dupla ip&longs;ius NX cum NM ad MN, <lb/>cubus ver&ograve; ex AF ad cubum ex DG e&longs;t, vt MN ad <lb/>NT; ita deinde &longs;e habetcubus ex DG ad &longs;olidum ba&shy;<lb/>&longs;im habens quadratum ex DG, altitudinem ver&ograve; lineam <lb/>compo&longs;itam ex dupla ip&longs;ius AF, &amp; ip&longs;a DG, vt <lb/>NT ad compo&longs;itam ex dupla ip&longs;ius NO, &amp; ip&longs;a NT. <lb/><emph type="italics"/>Sunt igitur quatuor magnitudines &longs;olidum ba&longs;im habens quadratum <lb/>ex AF, altitudinem ver&ograve; lineam compo&longs;itam ex dupla ip&longs;ius <lb/>DG, &amp; linea AF, &amp; cubus ex AF, &amp; cubus ex <lb/>DG, &amp; &longs;olidum ba&longs;im habens quadratum ex DG, altitu<lb/>dinem ver&ograve; lineam compo&longs;itam: ex dupla ip&longs;ius AF, &amp; ip&longs;a <lb/>DG, quatuor magnitudinibus proportionales, duabus &longs;imul &longs;umptis <lb/>tine&aelig; compo&longs;it&aelig; ex dupla ip&longs;ius NX<gap/> &amp; ip&longs;a NM; &amp; alte&shy;<lb/>ri magnitudini MN; aliiqu&egrave; deinceps NT, ac tandem line&aelig; <lb/>compo&longs;it&aelig; ex duplaip&longs;ius NO, &amp; ip&longs;a NT. ex &aelig;quali igitur <lb/>erit, vt &longs;olidum ba&longs;im habens quadratum ex AF, altitudinem<emph.end type="italics"/>
<arrow.to.target n="marg393"/><lb/><emph type="italics"/>autem lineam compo&longs;itam ex dupla ip&longs;ius DG, &amp; ip&longs;a AE, ad <lb/>&longs;olidum ba&longs;im habens quadratum ex DG, altitudinem ver&ograve; lt&shy;<lb/>neam compo&longs;itam ex dupla ip&longs;ius AF, &amp; ip&longs;a DG, ita <lb/>compo&longs;ita ex dupla ip&longs;ius NX, &amp; ip&longs;a MN ad compo&longs;itam <lb/>ex dupla ip&longs;ius NO, &amp; ip&longs;a NT &longs;ed vt pr&aelig;fatum &longs;oii&shy;<lb/>dum<emph.end type="italics"/> ba&longs;im habens quadratum ex AF, altitudinem ver&ograve; <lb/>lineam compo&longs;itam ex dupla ip&longs;ius DG, &amp; ip&longs;a AF <emph type="italics"/>ad <lb/>dictum &longs;olidum<emph.end type="italics"/> ba&longs;im habens quadratum ex DG, altitudi&shy;<lb/>nem ver&ograve; compo&longs;itam ex dupla ip&longs;ius AF &amp; ip&longs;a DG,
<arrow.to.target n="marg394"/><lb/><emph type="italics"/>ita<emph.end type="italics"/> factum fuit <emph type="italics"/>HI ad IK. vt igitur HI ad IK, &longs;u<emph.end type="italics"/>
<pb xlink:href="077/01/202.jpg" pagenum="198"/>
<arrow.to.target n="fig88"/><lb/><emph type="italics"/>compo&longs;ita<emph.end type="italics"/> ex dupla ip&longs;ius NX cum MN <emph type="italics"/>ad compo&longs;itam<emph.end type="italics"/> ex dupla <lb/>
<arrow.to.target n="marg395"/> ip&longs;ius NO cum NT. <emph type="italics"/>quare &amp; componendo<emph.end type="italics"/> HK ad KI, vt <lb/>dupla ip&longs;ius NX cum MN, &amp; dupla ip&longs;ius NO cum NT ad <lb/>compo&longs;itam ex dupla ip&longs;ius NO cum NT, quia ver&ograve; in hoc <lb/>antecedenti &longs;emel &longs;umitur MN, &amp; &longs;emel NT, bis ver&ograve; NX, <lb/>&amp; bis NO, erit HK ad KI, vt vtra&que; &longs;imul MN NT, &amp; du&shy;<lb/>pla vtriu&longs;&que; &longs;imul NX NO ad duplam ip&longs;ius NO, &amp; ip&longs;am <lb/>NT. <emph type="italics"/>&amp; antecedentium quintupla.<emph.end type="italics"/> quintupla ver&ograve; antecedentis <lb/>HK e&longs;t FG, quintupla ver&ograve; alterius antecedentis MN NT, <lb/>&amp; dupl&aelig; vtriu&longs;&que; &longs;imul NX NO e&longs;t quintupla vtriu&longs;&que; &longs;i&shy;<lb/>mul MN NT, &amp; decupla vtriu&longs;&que; &longs;imul NX NO. decu&shy;<lb/>pla enim e&longs;t quintupla dupl&aelig;. <emph type="italics"/>e&longs;t igitur FG ad IK, vt quintupla <lb/>vtriu&longs;&que; &longs;imul MN NT, &amp; decupla vtriu&longs;&que; &longs;imul NX NO ad du<lb/>plam ip&longs;ius ON, &amp; ip&longs;am NT. &amp; vt FG ad FK, qu&aelig;e&longs;t du&aelig; quin<lb/>t&aelig; ip&longs;ius<emph.end type="italics"/> FG, <emph type="italics"/>ita quintupla vtriu&longs;&que; &longs;imul MN NT, &amp; decupla <lb/>vtriu&longs;&que; &longs;imul NX NO ad duplam vtriu&longs;&que; &longs;imul MN NT,<emph.end type="italics"/>
<pb xlink:href="077/01/203.jpg" pagenum="199"/><emph type="italics"/>&amp; quadruplam vtriu&longs;&que; &longs;imul NX NO.<emph.end type="italics"/> c&ugrave;m hoc quidem con <lb/>&longs;e&que;ns &longs;itdu&aelig; quint&aelig; ip&longs;ius antecedentis. </s>
          <s id="N17AB1">etenim dupla v&shy;<lb/>triu&longs;&que; &longs;imul MN NT quintupl&aelig; earumdem &longs;imul MN <lb/>NT du&aelig; quint&aelig; exi&longs;tit. </s>
          <s id="N17AB7">&amp; quadrupla vtriu&longs;&que; &longs;imul NX <lb/>NO e&longs;t du&aelig; quint&aelig; decupl&aelig; earumdem NX NO. quadru&shy;<lb/>pla enim decupl&aelig; e&longs;t du&aelig; quint&aelig;. </s>
          <s id="N17ABD">Quoniam ita&que; ita e&longs;t FG <lb/>ad FK, vt quintupla vtriu&longs;&que; &longs;imul MN NT, &amp; decupla <lb/>vtriu&longs;&que; &longs;imul NX NO ad duplam vtriu&longs;&que; &longs;imul MN <lb/>NT, &amp; quadruplam vtriu&longs;&que; &longs;imul NX NO, &amp; vt FG ad <lb/>KI, ita quintupla vtriu&longs;&que; &longs;imul MN NT, &amp; decupla vtriu&longs; <lb/>&que; &longs;imul NX NO ad duplam ip&longs;ius ON, &amp; ip&longs;am NT: <lb/>erit FG ad &longs;uas con&longs;e&que;ntes &longs;imul &longs;umptas FK KI, hoc
<arrow.to.target n="marg396"/><lb/>e&longs;t FI, vt quintupla vtriu&longs;&que; &longs;imul MN NT, &amp; decupla <lb/>vtriu&longs;&que; &longs;imul NX NO ad duplam vtriu&longs;&que; &longs;imul MN <lb/>NT, &amp; quadruplam vtriu&longs;&que; &longs;imul NX NO, &amp; duplam <lb/>ip&longs;ius ON, &amp; ip&longs;am NT. &longs;ed in hoc con&longs;e&que;nti bis &longs;umi&shy;<lb/>tur MN, quater NX, &longs;exies NO, &amp; ter NT. <emph type="italics"/>erit igitur vt <lb/>FG &aelig;d FI, ita quintupla vtriu&longs;&que; &longs;imul MN NT, &amp; decupla v&shy;<lb/>triu&longs;&que; &longs;imul NX NO ad compo&longs;itam ex dupla ip&longs;ius MN, &amp; qua&shy;<lb/>drupla ip&longs;ius NX, &amp; &longs;extupla ip&longs;ius NO, &amp; tripla ip&longs;ius NT.<emph.end type="italics"/> &amp; <lb/>conuertendo FI ad FG, vt compo&longs;ita ex dupla ip&longs;ius MN,
<arrow.to.target n="marg397"/><lb/>&amp; quadrupla ip&longs;ius NX, &amp; &longs;extupla ip&longs;rus NO, &amp; tripla ip&shy;<lb/>&longs;i&uacute;s NT ad quintuplam vtriu&longs;&que; &longs;imul MN NT, &amp; decu&shy;<lb/>plam vtriu&longs;&que; &longs;imul NX NO. <emph type="italics"/>Quoniam ita&que; quatuor rect&aelig; li <lb/>ne&aelig; MN NX NO NT &longs;unt continu&egrave; proportionales.<emph.end type="italics"/> factaqu&egrave; <lb/>fuit MN &aelig;qualis ip&longs;i FB, &amp; NO ip&longs;i GB; crit reliqua OM <lb/>ip&longs;i FG &aelig;qualis. </s>
          <s id="N17AFB">&amp; vt TM ad TN ita factum fuit FH, <lb/>hoc e&longs;t tres quint&aelig; ip&longs;ius FG, tres &longs;cilicet quint&aelig; ip&longs;ius MO <lb/>ad IR. quare &amp; conuertendo <emph type="italics"/>vt NT ad TM, ita qu&aelig;dam a&longs;&longs;um&shy;<lb/>pta linea NI ad tres quintas ip&longs;ius FG, hoc e&longs;t ip&longs;ius MO. vt autem <lb/>compo&longs;ita ex dupla ip&longs;ius NM, &amp; quadrupla ip&longs;ius NX, &amp; &longs;extupla ip&shy;<lb/>&longs;ius NO &amp; tripla ip&longs;ius NT ad lineam compo&longs;itam ex quintupla vtrius&shy;<lb/>&que; &longs;imul MN NT, &amp; decupla vtriu&longs;&que; &longs;imul XN NO, &longs;ic altera qu&aelig; <lb/>dam a&longs;&longs;umpta linea IF ad FG, hoc est ad MO, erit ex &longs;uperioribus RF<emph.end type="italics"/>
<arrow.to.target n="marg398"/><lb/><emph type="italics"/>du&aelig; quint&aelig; ip&longs;ius MN, hoc est ip&longs;ius FB.<emph.end type="italics"/> ac propterea reliqua RB <lb/>erit tres quint&aelig; ip&longs;ius FB. &amp; obid BR ad. </s>
          <s id="N17B1D">RF e&longs;t, vt tria ad
<arrow.to.target n="marg399"/><lb/>duo. <emph type="italics"/>Quare punctum R centrum est grauitatis portionis ABC. &longs;it<emph.end type="italics"/>
<pb xlink:href="077/01/204.jpg" pagenum="200"/>
<arrow.to.target n="fig89"/><lb/>
<arrow.to.target n="marg400"/> <emph type="italics"/>quidem portionis DBE centrum grauitatis punctum Q frusti AD <lb/>EC centrum grauitatis erit in linea QR<emph.end type="italics"/> producta, <emph type="italics"/>qu&aelig;<emph.end type="italics"/> quiden QR <lb/><emph type="italics"/>adip&longs;ain<emph.end type="italics"/> productam <emph type="italics"/>eandem habeat proportionem quam habet fru&longs;ium<emph.end type="italics"/><lb/>ADEC <emph type="italics"/>ad reliquam portionem<emph.end type="italics"/> DBE. <emph type="italics"/>est autem punctum I. nam.<emph.end type="italics"/><lb/>c&ugrave;m &longs;it tota FB ad totam BR, vt ablata BG ad ablatam <lb/>
<arrow.to.target n="marg401"/> BQ, &longs;unt enim vt quin&que; ad tria, erit &amp; reliqua FG ad reli&shy;<lb/>quam QR, vt FB ad BR. ita&que; <emph type="italics"/>quoniam tres quint&aelig; ip&longs;ius FB <lb/>linea e&longs;i BR; ip&longs;ius ver&ograve; GB tres quint&aelig; linea est <expan abbr="Bq.">B&que;</expan> &amp; reliqu&aelig; <lb/>igitur GF est tres quint&aelig; QR. quoniamigitur est, vt fru&longs;tum AD <lb/>EC adportionem DBE, ita MT ad NT,<emph.end type="italics"/> vt o&longs;ten&longs;um fuit; <emph type="italics"/>&longs;ed vt <lb/>MN ad NT, &longs;ic<emph.end type="italics"/> factum fuit FH ad IR, hoc e&longs;t <emph type="italics"/>tres quint&aelig; ip&longs;ius <lb/>GF; qu&aelig; est QR ad RI. erit igitur vt fru&longs;tum ADEC adportionem <lb/>DBE, ita QR ad RI. &amp; est quidem totius portionis<emph.end type="italics"/> ABC <emph type="italics"/>centrum<emph.end type="italics"/><lb/>
<arrow.to.target n="marg402"/> <emph type="italics"/>grauitatis punctum R; ip&longs;ius ver&ograve; DBE centrum grauitatis punctum <lb/>Q: manife&longs;tum est igitur fru&longs;ti ADEC centrum grauitatis e&longs;&longs;e <expan abbr="pun-ct&utilde;">pun&shy;<lb/>ctum</expan> l.<emph.end type="italics"/> quod <expan abbr="quid&etilde;">quidem</expan> e&longs;t in quinta parte media HK ip&longs;ius FG ab
<pb xlink:href="077/01/205.jpg" pagenum="201"/>eo ita diui&longs;a, vt HI ad IK &longs;it, vt &longs;olidum ba&longs;im habens qua&shy;<lb/>dratum ex AF, altitudinem autem duplam ip&longs;ius DG cum <lb/>AF ad &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem <lb/>ver&ograve; duplam ip&longs;ius AF <expan abbr="c&utilde;">cum</expan> DG. quod demon&longs;trare oportebat. </s>
        </p>
        <p id="N17BB5" type="margin">
          <s id="N17BB7"><margin.target id="marg382"/>1 <emph type="italics"/>Arch de <lb/>quad. </s>
          <s id="N17BC0">pa&shy;<lb/>rab. </s>
          <s id="N17BC4">&amp; <lb/><expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> coni <lb/>corum A&shy;<lb/>poll.<emph.end type="italics"/></s>
        </p>
        <p id="N17BD1" type="margin">
          <s id="N17BD3"><margin.target id="marg383"/>13.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N17BDC" type="margin">
          <s id="N17BDE"><margin.target id="marg384"/>3.<emph type="italics"/>Arch.de <lb/>quad. </s>
          <s id="N17BE7">pa&shy;<lb/>rab. </s>
          <s id="N17BEB">&amp;<emph.end type="italics"/> 20. <lb/><emph type="italics"/>pilmi coni <lb/>corum A&shy;<lb/>poil.<emph.end type="italics"/></s>
        </p>
        <p id="N17BFA" type="margin">
          <s id="N17BFC"><margin.target id="marg385"/>2.<emph type="italics"/>cor.<emph.end type="italics"/> 20. <lb/><emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C0C" type="margin">
          <s id="N17C0E"><margin.target id="marg386"/>22.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C17" type="margin">
          <s id="N17C19"><margin.target id="marg387"/>37. <emph type="italics"/>vndeci <lb/>mi.<emph.end type="italics"/></s>
        </p>
        <p id="N17C24" type="margin">
          <s id="N17C26"><margin.target id="marg388"/>17.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C2F" type="margin">
          <s id="N17C31"><margin.target id="marg389"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C3A" type="margin">
          <s id="N17C3C"><margin.target id="marg390"/>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C45" type="margin">
          <s id="N17C47"><margin.target id="marg391"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C50" type="margin">
          <s id="N17C52"><margin.target id="marg392"/><emph type="italics"/>cor<emph.end type="italics"/> 4.<emph type="italics"/>quin<lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C62" type="margin">
          <s id="N17C64"><margin.target id="marg393"/>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C6D" type="margin">
          <s id="N17C6F"><margin.target id="marg394"/>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C78" type="margin">
          <s id="N17C7A"><margin.target id="marg395"/>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17C83" type="margin">
          <s id="N17C85"><margin.target id="marg396"/><emph type="italics"/>cor.<emph.end type="italics"/>2.<emph type="italics"/>lem&shy;<lb/>in<emph.end type="italics"/> 13. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p id="N17C9D" type="margin">
          <s id="N17C9F"><margin.target id="marg397"/><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/>quin<lb/>ti.<emph.end type="italics"/></s>
        </p>
        <p id="N17CAF" type="margin">
          <s id="N17CB1"><margin.target id="marg398"/><emph type="italics"/>ex pr&aelig;ce&shy;<lb/>denti.<emph.end type="italics"/></s>
        </p>
        <p id="N17CBB" type="margin">
          <s id="N17CBD"><margin.target id="marg399"/>8.<emph type="italics"/>buius.<emph.end type="italics"/></s>
        </p>
        <p id="N17CC6" type="margin">
          <s id="N17CC8"><margin.target id="marg400"/>8.<emph type="italics"/>prim hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p id="N17CD3" type="margin">
          <s id="N17CD5"><margin.target id="marg401"/>19.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17CDE" type="margin">
          <s id="N17CE0"><margin.target id="marg402"/>8 <emph type="italics"/>prim.hu<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <figure id="id.077.01.205.1.jpg" xlink:href="077/01/205/1.jpg"/>
        <figure id="id.077.01.205.2.jpg" xlink:href="077/01/205/2.jpg"/>
        <figure id="id.077.01.205.3.jpg" xlink:href="077/01/205/3.jpg"/>
        <figure id="id.077.01.205.4.jpg" xlink:href="077/01/205/4.jpg"/>
        <p id="N17CFB" type="head">
          <s id="N17CFD">SCHOLIVM.</s>
        </p>
        <p id="N17CFF" type="main">
          <s id="N17D01">In hoc Theoremate prim&ugrave;m ob&longs;eruanda occurrunt verba <lb/>propo&longs;itionis, quibus Archimedes pr&etail;cipit pottionem HK <lb/>in I ita diui&longs;am e&longs;&longs;e oportere, vt HI ad IK eam habeat pro&shy;<lb/>portionem, quam habet &longs;olidum ba&longs;im habens quadratum <lb/>ex dimidia maioris ba&longs;is fru&longs;ti, altitudinem autem lineam &aelig;&shy;<lb/>qualem vtri&que; &longs;imul dupl&aelig; minoris ba&longs;is, &amp; maiori ad &longs;oli&shy;<lb/>dum ba&longs;im habens quadratum ex dimidia minoris ba&longs;is fru&shy;<lb/>&longs;ti, altitudinem autem lineam &aelig;qualem vtri&longs;&que;, dupl&aelig; &longs;cili&shy;<lb/>cet ba&longs;is maioris, &amp; minori. </s>
          <s id="N17D13">hoc e&longs;t &longs;it HI ad IK, vt &longs;olidum <lb/>ba&longs;im habens quadratum ex AF, altitudinem ver&ograve; lineam &aelig;&shy;<lb/>qualem dupl&aelig; ip&longs;ius DE cum AC ad &longs;olidum ba&longs;im habens <lb/>quadratum ex DG, altitudinem ver&ograve; lineam &aelig;qualem <expan abbr="vtriq;">vtri&que;</expan> <lb/>&longs;imul dupl&aelig; ip&longs;ius AC, &amp; ip&longs;i DE. In con&longs;tructione autem <lb/>hunc propo&longs;itionis locum explicans, &amp; in pergre&longs;&longs;u totius <expan abbr="de-m&otilde;&longs;trationis">de&shy;<lb/>mon&longs;trationis</expan>, inquit HI ad IK <expan abbr="e&atilde;">eam</expan> debere proportionem habe&shy;<lb/>re, quam habet &longs;olidum ba&longs;im habens quadratum ex AF, alti <lb/>tudinem ver&ograve; lineam &aelig;qualem <expan abbr="vtriq;">vtri&que;</expan> &longs;imul dupl&aelig; ip&longs;ius DG, <lb/>&amp; ip&longs;i AF ad &longs;olidum ba&longs;im habens quadratum ex DG, al&shy;<lb/>titudinem ver&ograve; lineam &aelig;qualem vtri&que; &longs;imul dupl&aelig; ip&longs;ius <lb/>AF, &amp; DG. Quoniam autem &longs;olida parallelepipeda (vt pr&aelig;&shy;<lb/>fata &longs;olida &longs;unt) in eadem ba&longs;i exi&longs;tentia ita &longs;e habent inter&longs;e, <lb/>vt corum altitudine; &longs;olidum, quod ba&longs;im habet quadratum <lb/>ex AF, altitudinem autem duplam ip&longs;ius DE cum AC, du<lb/>plum erit &longs;olidi ba&longs;im habentis quadratum ex AF, altitudi&shy;<lb/>nem ver&ograve; duplam ip&longs;ius DG cum AF. Nam h&aelig;c &longs;olida ean<lb/>dem habent ba&longs;im, quadratum nemp&egrave; ex AF; ip&longs;orumqu&egrave; <lb/>alterum habet altitudinem duplam. </s>
          <s id="N17D49">quia c&ugrave;m &longs;it DE dupla <lb/>ip&longs;ius DG, erit dupla ip&longs;ius DE dupla ip&longs;ius dupl&aelig; DG;
<pb xlink:href="077/01/206.jpg" pagenum="202"/>&amp; AC dupla e&longs;t ip&longs;ius AF. altitudines igitur horum <expan abbr="&longs;olidor&utilde;">&longs;olidorum</expan> <lb/>in dupla &longs;unt proportione. </s>
          <s id="N17D57">hoc e&longs;t altitudo, linea &longs;cilicet du&shy;<lb/>pla ip&longs;ius DE cum AC altitudinis nemp&egrave; line&aelig; dupl&aelig; ip&longs;ius <lb/>DG cum AF dupla exi&longs;tit. </s>
          <s id="N17D5D">Quare &longs;olidum ba&longs;im habens qua&shy;<lb/>dratum ex AF, altitudinem ver&ograve; duplam ip&longs;ius DE cum AC <lb/>duplum e&longs;t &longs;olidi, quod ba&longs;im habeatidem quadratum ex AF, <lb/>altitudinem ver&ograve; duplam ip&longs;ius DG cum AF. cademqu&egrave; ratio <lb/>neo&longs;tendetur <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> ba&longs;im habens quadratum ex DG, altitu<lb/>dinem ver&ograve; duplam ip&longs;ius AC cum DE duplum e&longs;&longs;e &longs;olidi ba <lb/>&longs;im habentis quadratum ex eadem DG, altitudinem autem du<lb/>plam ip&longs;ius AF cum DG. &longs;olidum igitur ba&longs;im habens qua&shy;<lb/>dratum ex AF, altitudinem autem duplam ip&longs;ius DE cum AC <lb/>ad &longs;olidum quadtatum habens ba&longs;im ex AF, altitudinent ver&ograve; <lb/>duplam ip&longs;ius DG cum AF eam habet proportionem, quam <lb/>habet &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem <lb/>ver&ograve; duplam ip&longs;ius AC cum AE ad &longs;olidum ba&longs;im <expan abbr="hab&etilde;s">habens</expan> qua <lb/>dratum ex DG, altitudinem ver&ograve; duplam ip&longs;ius AF cum DG. <lb/>
<arrow.to.target n="marg403"/> quare permutando <expan abbr="prim&utilde;">primum</expan> &longs;olidum ba&longs;im habens quadratum <lb/>ex AF, altitudinem ver&ograve; duplam ip&longs;ius DE cum AC ad &longs;ecun&shy;<lb/>dum &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem <lb/>autem duplam ip&longs;ius AC cum DE eandem habet proportio&shy;<lb/>nem, quam habet tertium &longs;olidum ba&longs;im habens quadratum <lb/>ex AF, altitudinem autem duplam ip&longs;ius DG cum AF ad quar <lb/>tum &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem ve <lb/>r&ograve; duplam ip&longs;ius AF cum DG. Quapropter Archimedes loco <lb/>primi, &amp; &longs;ecundi &longs;olidi in propo&longs;itione propo&longs;iti rect&egrave; potuit <lb/>in demon&longs;tratione accipere tertium, &amp; quartum &longs;olidum. </s>
          <s id="N17D9D">co <lb/>dem enim modo, &amp; in eadem proportione linea HK in pun&shy;<lb/>cto I diui&longs;a prouenit: quod quidem punctum fru&longs;ti ACED <lb/>centrum grauitatis exi&longs;tit. </s>
        </p>
        <p id="N17DA5" type="margin">
          <s id="N17DA7"><margin.target id="marg403"/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s>
        </p>
        <p id="N17DB0" type="head">
          <s id="N17DB2">Secundi libri Finis.</s>
        </p>
        <pb xlink:href="077/01/207.jpg"/>
        <p id="N17DB6" type="head">
          <s id="N17DB8">Erratorum quorundam re&longs;titutio.</s>
        </p>
        <p id="N17DBA" type="main">
          <s id="N17DBC">Pagina 8, ver&longs;u 18, Archimedes. </s>
          <s id="N17DBE">&lt;33&gt; 10, 7, &longs;ione. </s>
          <s id="N17DC0">&lt;33&gt; 18, 20, conducenti. </s>
          <s id="N17DC2">&lt;33&gt; 21, 14, per <lb/>di&longs;cere ip&longs;um. </s>
          <s id="N17DC6">&lt;33&gt; 39, 25, hoc e&longs;t AB. &lt;33&gt; 43, 19, lineam. </s>
          <s id="N17DC8">&lt;33&gt; 47, 20, c&uacute;m inquit, &lt;33&gt; 63, <lb/>20, GD DK in. </s>
          <s id="N17DCC">&lt;33&gt; 65, 21, DC. Ibidem, 27, ex DC. &lt;33&gt; 67, 29, in maiori. </s>
          <s id="N17DCE">&lt;33&gt; 69, in <lb/>po&longs;til: ex proxima propo&longs;itione. </s>
          <s id="N17DD2">&lt;33&gt; 70, 5, vt NL &lt;33&gt; 73, 1, de his, vel. </s>
          <s id="N17DD4">&lt;33&gt; 84, 8, AEEB <lb/>CF FD. &lt;33&gt; 90, 17, totus. </s>
          <s id="N17DD8">&lt;33&gt; 98, 1, quam VH. Ibidem, 7, aufertur. </s>
          <s id="N17DDA">&lt;33&gt; 11<gap/>, 21, repo&shy;<lb/>&longs;uit. </s>
          <s id="N17DE0">&lt;33&gt; 124, 19, <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan>, &lt;33&gt; 140, 1, <expan abbr="&aelig;quidi&longs;t&atilde;tes">&aelig;quidi&longs;tantes</expan> &lt;33&gt; 143, 11, e&longs;t CH &lt;33&gt; 147, 3, <expan abbr="c&utilde;">cum</expan> EK ad EK, vt. <lb/>Ibide, 25, &longs;ta S 9, ad Y<foreign lang="greek">a</foreign> &lt;33&gt; 149, 19, ad <foreign lang="greek">xn</foreign>. Ibidem, 25, e&longs;t, vt OR. Ibidem, 27, L<foreign lang="greek">*g</foreign>, vt <lb/>OR ad. </s>
          <s id="N17DFE">Ibidem, 31, vt OR ad <foreign lang="greek">zd</foreign> Ibidem, 32, vt <foreign lang="greek">d&lt;10&gt;</foreign> ad <foreign lang="greek">&lt;10&gt;z</foreign> Ibidem, 34, BD ad B<foreign lang="greek">s</foreign>, <lb/>ita. </s>
          <s id="N17E12">Ibidem, 35, &longs;it BD ad D<foreign lang="greek">n</foreign> Ibidem, 36, BD ad D<foreign lang="greek">n</foreign> B<foreign lang="greek">s</foreign>. &lt;33&gt; 150, 5, vt OR ad O<foreign lang="greek">c</foreign> &lt;33&gt; 153, <lb/>13, r&aelig;, vt. </s>
          <s id="N17E26">&lt;33&gt; 157, in po&longs;till ante 15, primi Ibidem, 17, maiorem. </s>
          <s id="N17E28">&lt;33&gt; 161, 24, erit KH. <lb/>&lt;33&gt; 167, 34, efficax. </s>
          <s id="N17E2C">&lt;33&gt; 170, 1, ip&longs;ius AC erit. </s>
          <s id="N17E2E">&lt;33&gt; 181, 36, ex dupla ip&longs;ius AB, &lt;33&gt; 191, <lb/>21, erunt. </s>
          <s id="N17E32">Ibidem, 22, DKG &aelig;quales. </s>
        </p>
        <p id="N17E34" type="head">
          <s id="N17E36">REGISTRVM.</s>
        </p>
        <p id="N17E38" type="main">
          <s id="N17E3A">&lt;12&gt; ABCDEFGHIKLMNOPQRSTVXYZ, <lb/>AA BB. </s>
        </p>
        <p id="N17E3E" type="head">
          <s id="N17E40">Omnes duerniones, pr&aelig;ter, BB, ternionem.</s>
        </p>
        <p id="N17E42" type="head">
          <s id="N17E44">PISAVRI. <lb/>Apud Hieronymum Concordiam, <lb/>M. D. LXXXVII.</s>
        </p>
      </chap>
    </body>
    <back/>
  </text>
</archimedes>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Wed, 30 Jul 2014 15:58:21 +0200
parents	22d6a63640c6
children