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DESpecs 2.0 Autumn 2009
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:14:40 +0200
parents 22d6a63640c6
children
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<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd">
<archimedes xmlns:xlink="http://www.w3.org/1999/xlink">
  <info>
    <author>Commandino, Federico</author>
    <title>Liber de centro gravitatis solidorum</title>
    <date>1565</date>
    <place>Bologna</place>
    <translator/>
    <lang>la</lang>
    <cvs_file>comma_centr_023_la_1565.xml</cvs_file>
    <cvs_version/>
    <locator>023.xml</locator>
  </info>
  <text>
    <pb xlink:href="023/01/001.jpg"/>
    <front>
      <section>
        <p type="head">
          <s id="s.000001">FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</s>
          <s id="s.000002">LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM.<!-- KEEP S--></s>
        </p>
        <figure id="id.023.01.001.1.jpg" xlink:href="023/01/001/1.jpg"/>
        <p type="head">
          <s id="s.000003">CVM PRIVILEGIO IN ANNOS X.<!-- KEEP S--></s>
        </p>
        <p type="head">
          <s id="s.000004">BONONIAE,<!-- KEEP S--></s>
        </p>
        <p type="head">
          <s id="s.000005">Ex Officina Alexandri Benacii.<!-- KEEP S--></s>
        </p>
        <p type="head">
          <s id="s.000006">MDLXV.<!-- KEEP S--></s>
        </p>
        <pb xlink:href="023/01/002.jpg"/>
        <pb xlink:href="023/01/003.jpg"/>
      </section>
      <section>
        <p type="head">
          <s id="s.000007">ALEXANDRO FARNESIO <lb/>
CARDINALI AMPLISSIMO. <lb/>
ET OPTIMO.</s>
        </p>
        <p type="main">
          <s id="s.000008">Cvm mult&aelig; res in mathematicis <lb/>
di&longs;ciplinis nequaquam &longs;atis ad&shy;<lb/>
huc explicat&aelig; &longs;int, tum perdif&shy;<lb/>
ficilis, &amp; perob&longs;cura qu&aelig;&longs;tio <lb/>
e&longs;t de centro grauitatis corpo&shy;<lb/>
rum &longs;olidorum; qu&aelig;, &amp; ad co&shy;<lb/>
gno&longs;cendum pulcherrima e&longs;t, <lb/>
&amp; ad multa, qu&aelig; &agrave; mathematicis proponuntur, pr&aelig;&shy;<lb/>
clare intelligenda maximum affert adiumentum. </s>
          <s id="s.000009">de <lb/>
qua neminem ex mathematicis, neque no&longs;tra, neque <lb/>
patrum no&longs;trorum memoria &longs;criptum reliqui&longs;&longs;e &longs;ci&shy;<lb/>
mus. </s>
          <s id="s.000010">&amp; quamuis in earum monumentis literarum <expan abbr="n&otilde;">non</expan> <lb/>
nulla reperiantur, ex quibus in hanc &longs;ententiam addu<lb/>
ci po&longs;&longs;umus, vt exi&longs;timemus hanc rem ab <expan abbr="ij&longs;d&etilde;">ij&longs;dem</expan> vber&shy;<lb/>
rime tractatam e&longs;&longs;e; tamen ne&longs;cio quo fato adhuc <lb/>
in eiu&longs;modi librorum ignoratione ver&longs;amur. </s>
          <s id="s.000011">Archi&shy;<lb/>
medes quidem <expan abbr="mathematicor&utilde;">mathematicorum</expan> princeps in libello, <lb/>
cuius in&longs;criptio e&longs;t, <foreign lang="greek">ke/ntra ba/rwn e&rpar;pipe/dwn</foreign>, de centro pla&shy;<lb/>
norum copio&longs;i&longs;sime, atque acuti&longs;sime con&longs;crip&longs;it: &amp; <lb/>
in eo explicando <expan abbr="&longs;umm&atilde;">&longs;ummam</expan> ingenii, &amp; &longs;cienti&aelig; <expan abbr="glori&atilde;">gloriam</expan> e&longs;t <lb/>
<expan abbr="c&otilde;&longs;ecutus">con&longs;ecutus</expan>. </s>
          <s id="s.000012">Sed de cognitione <expan abbr="c&etilde;tri">centri</expan> grauitatis <expan abbr="corpor&utilde;">corporum</expan> <lb/>
<expan abbr="&longs;olidor&utilde;">&longs;olidorum</expan> nulla in eius libris litera inuenitur. </s>
          <s id="s.000013">non mul<lb/>
tos abhinc annos MARCELLVS II. PONT. MAX.



<pb xlink:href="023/01/004.jpg"/>cum adhuc Cardinalis e&longs;&longs;et, mihi, qu&aelig; &longs;ua erat hu&shy;<lb/>
manitas, libros eiu&longs;dem Archimedis de ijs, qu&aelig; ve&shy;<lb/>
huntur in aqua, latine redditos dono dedit. </s>
          <s id="s.000014">hos cum <lb/>ego, ut aliorum &longs;tudia incitarem, <expan abbr="emend&atilde;dos">emendandos</expan>, &amp; <expan abbr="c&otilde;-mentariis">com&shy;<lb/>
mentariis</expan> illu&longs;trandos &longs;u&longs;cepi&longs;&longs;em, animaduerti dubi <lb/>
tari non po&longs;&longs;e, quin Archimedes vel de hac materia <lb/>
&longs;crip&longs;i&longs;&longs;et, vel aliorum mathematicorum &longs;cripta per&shy;<lb/>
legi&longs;&longs;et. </s>
          <s id="s.000015">nam in iis tum alia nonnulla, tum maxime <lb/>
illam propo&longs;itionem, ut euidentem, &amp; ali&agrave;s proba&shy;<lb/>
tam a&longs;&longs;umit, <expan abbr="Centr&utilde;">Centrum</expan> grauitatis in portionibus conoi<lb/>
dis rectanguli axem ita diuidere, vt pars, qu&aelig; ad verti<lb/>
cem terminatur, alterius partis, qu&aelig; ad ba&longs;im dupla <lb/>
&longs;it. </s>
          <s id="s.000016">Verum h&aelig;c ad eam partem mathematicarum <lb/>
di&longs;ciplinarum pr&aelig;cipue refertur, in qua de centro <lb/>
grauitatis corporum &longs;olidorum tractatur. </s>
          <s id="s.000017">non e&longs;t au<lb/>
tem con&longs;entaneum Archimedem illum admirabilem <lb/>
virum hanc propo&longs;itionem &longs;ibi argumentis con&shy;<lb/>
firmandam exi&longs;timaturum non fui&longs;&longs;e, ni&longs;i eam vel <lb/>
aliis in locis probaui&longs;&longs;et, vel ab aliis probatam e&longs;&longs;e <lb/>
comperi&longs;&longs;et. </s>
          <s id="s.000018">quamobrem nequid in iis libris intel&shy;<lb/>
ligendis de&longs;iderari po&longs;&longs;et, &longs;tatui hanc etiam partem <lb/>
vel &agrave; veteribus pr&aelig;termi&longs;&longs;am, vel tractatam quidem, <lb/>
&longs;ed in tenebris iacentem, non intactam relinquere; <lb/>
atque ex a&longs;sidua mathematicorum, pr&aelig;&longs;ertim Archi&shy;<lb/>
medis lectione, qu&aelig; mihi in mentem venerunt, ea in <lb/>
medium afferre; ut centri grauitatis corporum &longs;oli&shy;<lb/>
dorum, &longs;i non perfectam,  at certe aliquam noti-



<pb xlink:href="023/01/005.jpg"/>tiam haberemus. </s>
          <s id="s.000019">Quem meum laborem <expan abbr="n&otilde;">non</expan> mathe&shy;<lb/>
maticis &longs;olum, verum iis etiam, qui natur&aelig; ob&longs;curi&shy;<lb/>
tate delectantur, <expan abbr="n&otilde;">non</expan> iniucundam fore &longs;peraui: multa <lb/>
enim <foreign lang="greek">problh/mata</foreign> cognitione digni&longs;sima, qu&aelig; ad <expan abbr="vtr&atilde;-que">vtran&shy;<lb/>
que</expan> &longs;cientiam attinent, &longs;e&longs;e legentibus obtuli&longs;&longs;ent.</s>
          <lb/>
          <s id="s.000020">neque id vlli mirandum videri debet. </s>
          <s id="s.000021">vt enim in cor&shy;<lb/>poribus no&longs;tris omnia membra, ex quibus certa qu&aelig;<lb/>
dam officia na&longs;cuntur, diuino quodam ordine inter <lb/>
&longs;e implicata, &amp; colligata &longs;unt: in <expan abbr="iis&qacute;">iisque</expan>; admirabilis il&shy;<lb/>
la con&longs;piratio, quam <foreign lang="greek">su/mpnoian</foreign> gr&aelig;ci vocant, eluce&longs;cit, <lb/>
ita tres ill&aelig; Philo&longs;ophi&aelig; &lpar;ut Ari&longs;totelis verbo vtar&rpar; <lb/>
qu&aelig; veritatem &longs;olam propo&longs;itam habent, licet qui&shy;<lb/>
bu&longs;dam qua&longs;i finibus &longs;uis regantur: tamen <expan abbr="ear&utilde;">earum</expan> vna&shy;<lb/>
qu&aelig;que per &longs;e ip&longs;am quodammodo imperfecta e&longs;t: <lb/>
neque altera &longs;ine alterius auxilio plene comprehen&shy;<lb/>
di pote&longs;t. </s>
          <s id="s.000022">complures pr&aelig;terea mathematicorum no&shy;<lb/>
di ante hac explicatu difficillimi nullo negotio expe<lb/>
diti e&longs;&longs;ent: atque &lpar;ut vno verbo complectar&rpar; ni&longs;i <lb/>
mea valde amo, tractationem hanc meam &longs;tudio&longs;is <lb/>
non mediocrem vtilitatem, &amp; magnam volupta&shy;<lb/>
tem allaturam e&longs;&longs;e mihi per&longs;ua&longs;i. </s>
          <s id="s.000023">cum autem ad hoc <lb/>
&longs;cribendum aggre&longs;&longs;us e&longs;sem, allatus e&longs;t ad me liber <lb/>
Franci&longs;ci Maurolici Me&longs;&longs;anen&longs;is, in quo vir ille do&shy;<lb/>
cti&longs;simus, &amp; in iis di&longs;ciplinis exercitati&longs;simus af&shy;<lb/>
firmabat &longs;e de centro grauitatis corporum &longs;olido&shy;<lb/>
rum con&longs;crip&longs;i&longs;&longs;e. </s>
          <s id="s.000024">cum hoc intellexi&longs;&longs;em, &longs;u&longs;tinui <lb/>
me pauli&longs;per: tacitus que expectaui, dum opus cla-



<pb xlink:href="023/01/006.jpg"/>ris&longs;imi uiri, quem &longs;emper honoris cau&longs;&longs;a nomino, <lb/>
in lucem proferretur: mihi enim exploratis&longs;imum <lb/>
erat: Franci&longs;cum Maurolicum multo doctius, &amp; <lb/>
exqui&longs;itius hoc di&longs;ciplinarum genus &longs;criptis &longs;uis tra<lb/>
diturum. </s>
          <s id="s.000025">&longs;ed cum id tardius fieret, hoc e&longs;t, ut ego <lb/>
interpretor, diligentius, mihi diutius hac &longs;criptione <lb/>
non &longs;uper&longs;edendum e&longs;&longs;e duxi, pr&aelig;&longs;ertim cum iam li&shy;<lb/>
bri Archimedis de iis, qu&aelig; uehuntur in aqua, opera <lb/>
mea illu&longs;trati typis <expan abbr="excud&etilde;di">excudendi</expan> e&longs;&longs;ent. </s>
          <s id="s.000026">nec me alia cau&longs;<lb/>
&longs;a impuli&longs;&longs;et, ut de centro grauitatis corporum &longs;oli&shy;<lb/>
dorum &longs;criberem, ni&longs;i ut hac etiam ratione lux eis <lb/>
qu&agrave;m maxime fieri po&longs;&longs;et afferretur. </s>
          <s id="s.000027"><expan abbr="atq;">atque</expan> id e&ograve; mihi <lb/>
faciendum exi&longs;timaui, qu&ograve;d in &longs;pem ueniebam fore, <lb/>
ut cum ego ex omnibus mathematicis primus, hanc <lb/>
materiam explicandam &longs;u&longs;cepi&longs;&longs;em; &longs;i quid errati for<lb/>
te &agrave; me commi&longs;&longs;um e&longs;&longs;et, boni uiri potius id me&aelig; de <lb/>
&longs;tudio&longs;is hominibus bene <expan abbr="mer&etilde;di">merendi</expan> cupiditati, qu&agrave;m <lb/>
arroganti&aelig; a&longs;criberent. </s>
          <s id="s.000028">re&longs;tabat ut con&longs;iderarem, cui <lb/>
potis&longs;imum ex principibus uiris contemplationem <lb/>
hanc, nunc primum memori&aelig;, ac literis proditam de <lb/>
dicarem. </s>
          <s id="s.000029">harum mearum cogitationum &longs;umma fa&shy;<lb/>
cta, exi&longs;timaui nemini conuenientius de centro graui <lb/>
tatis corporum opus dicari oportere, qu&agrave;m ALE&shy;<lb/>
XANDRO FARNESIO grauis&longs;imo, ac prudentis&longs;i&shy;<lb/>
mo Cardinali, quo in uiro &longs;umma fortuna &longs;emper <expan abbr="c&utilde;">cum</expan> <lb/>
&longs;umma uirtute certauit. </s>
          <s id="s.000030">quid enim maxime in te ad&shy;<lb/>
mirati debeant homines, ob&longs;curum e&longs;t; u&longs;um ne re-



<pb xlink:href="023/01/007.jpg"/>rum, qui pueriti&aelig; tempus extremum principium ha<lb/>
bui&longs;ti, &amp; <expan abbr="imperior&utilde;">imperiorum</expan>, &amp; ad Reges, &amp; Imperatores ho&shy;<lb/>
norificenti&longs;simarum legationum; an excellentiam <lb/>
in omni genere literarum, qui vix <expan abbr="adole&longs;c&etilde;tulus">adole&longs;centulus</expan>, qu&aelig; <lb/>
homines iam confirmata &aelig;tate &longs;ummo &longs;tudio, <expan abbr="diu-turnis&qacute;">diu&shy;<lb/>
turnisque</expan>; laboribus didicerunt, &longs;cientia, &amp; cognitione <lb/>
comprehendi&longs;ti: an con&longs;ilium, &amp; &longs;apientiam in re&shy;<lb/>
gendis, &amp; <expan abbr="gubern&atilde;dis">gubernandis</expan> Ciuitatibus, cuius graui&longs;sim&aelig; <lb/>
&longs;ententi&aelig; in &longs;ancti&longs;simo Reip. <!-- REMOVE S-->Chri&longs;tian&aelig; con&longs;ilio di&shy;<lb/>
ct&aelig;, potius diuina oracula, qu&agrave;m &longs;ententi&aelig; habit&aelig; <lb/>
&longs;unt, &amp; habentur. </s>
          <s id="s.000031">pr&aelig;termitto liberalitatem, &amp; mu&shy;<lb/>
nificentiam tuam, quam in &longs;tudio&longs;i&longs;simo quoque ho<lb/>
ne&longs;tando quotidie magis o&longs;tendis, ne videar auribus <lb/>
tuis potius, qu&agrave;m veritati &longs;eruire. </s>
          <s id="s.000032">quamuis &agrave; te in tot <lb/>
pr&aelig;claros viros tanta beneficia collata &longs;unt, &amp; <expan abbr="confe-r&utilde;tur">confe&shy;<lb/>
runtur</expan>, vt omnibus te&longs;tatum &longs;it, nihil tibi e&longs;&longs;e charius, <lb/>
nihil iucundius, qu&agrave;m eximia tua liberalitate homi&shy;<lb/>
nes ad amplexandam virtutem, licet currentes incita&shy;<lb/>
re. </s>
          <s id="s.000033">nihil dico de ceteris virtutibus tuis, qu&aelig; tant&aelig; <lb/>
&longs;unt, quant&aelig; ne cogitatione quidem comprehendi <lb/>
po&longs;&longs;unt. </s>
          <s id="s.000034">Quamobrem hac pr&aelig;cipue de cau&longs;&longs;a te hu&shy;<lb/>
ius me&aelig; lucubrationis patronum e&longs;&longs;e volui, quam ea, <lb/>
qua &longs;oles, humanitate accipies. </s>
          <s id="s.000035">te enim &longs;emper ob <lb/>
diuinas virtutes tuas colui, &amp; ob&longs;eruaui: <expan abbr="nihil&qacute;">nihilque</expan>; mi&shy;<lb/>
hi fuit optatius; qu&agrave;m tibi per&longs;pectum e&longs;&longs;e meum <lb/>
erga te animum; <expan abbr="&longs;ingularem&qacute;">&longs;ingularemque</expan>; ob&longs;eruantiam. </s>
          <s id="s.000036">c&oelig;&shy;<lb/>
lum igitur digito attingam, &longs;i po&longs;t graui&longs;simas oc-



<pb xlink:href="023/01/008.jpg"/>cupationes tuas legendo Federici tui libro aliquid <lb/>
impertiri temporis non grauaberis: <expan abbr="cum&qacute;">cumque</expan>; in iis, qui <lb/>
tibi &longs;emper addicti erunt, numerare. </s>
          <s id="s.000037">Vale.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000038">Federicus Commandinus.</s>
        </p>
      </section>
      <section>
        <pb pagenum="1" xlink:href="023/01/009.jpg"/>
        <p type="head">
          <s id="s.000039">FEDERICI COMMANDINI <lb/>
VRBINATIS LIBER DE CENTRO <lb/>
GRAVITATIS SOLIDORVM.</s>
        </p>
      </section>
    </front>
    <body>
      <chap>
        <p type="head">
          <s id="s.000040">DIFFINITIONES.</s>
        </p>
        <p type="main">
          <s id="s.000041"><arrow.to.target n="marg1"/></s>
        </p>
        <p type="margin">
          <s id="s.000042"><margin.target id="marg1"/>1</s>
        </p>
        <p type="main">
          <s id="s.000043">Centrvm grauitatis, Pappus <lb/>
Alexandrinus in octauo ma&shy;<lb/>
thematicarum collectionum <lb/>
libro ita diffiniuit.</s>
        </p>
        <p type="main">
          <s id="s.000044"><foreign lang="greek">le/gomen de/ ke/ntron ba/rous e(ka/stou sw/<lb/>
matos e&rpar;=inai shme=ion ti kei/menon e&rpar;nto/s, a/f&apos; <lb/>
o&lpar;/u kat&apos; e/poi/nian a\rtnqe/n to/ ba/ros n&lpar;mere=i<lb/>
fero/menon, kai\ fula/ssei th/n e&rpar;c a&rpar;rxh=s qe/&shy;<lb/>
sin, o&rpar;u mh\ peritrepo/menon e&rpar;n th= fora=</foreign>. hoc e&longs;t,</s>
        </p>
        <p type="main">
          <s id="s.000045">Dicimus autem centrum grauitatis uniu&longs;cu&shy;<lb/>
iu&longs;que corporis punctum quoddam intra po&longs;i&shy;<lb/>
tum, &agrave; quo &longs;i graue appen&longs;um mente concipia&shy;<lb/>
tur, dum fertur quie&longs;cit; &amp; &longs;eruat eam, quam in <lb/>
principio habebat po&longs;itionem: neque in ip&longs;a la&shy;<lb/>
tione circumuertitur.</s>
        </p>
        <p type="main">
          <s id="s.000046">Po&longs;&longs;umus etiam hoc modo diffinire.</s>
        </p>
        <p type="main">
          <s id="s.000047">Centrum grauitatis uniu&longs;cuiu&longs;que &longs;olid&aelig; figu<lb/>
r&aelig; e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>
undique partes &aelig;qualium momentorum con&longs;i&shy;<lb/>
&longs;tunt. </s>
          <s id="s.000048">&longs;i enim per tale centrum ducatur planum <lb/>
figuram quomodocunque &longs;ecans &longs;emper in par&shy;



<pb xlink:href="023/01/010.jpg"/>tes &aelig;queponderantes ip&longs;am diuidet.</s>
        </p>
        <p type="main">
          <s id="s.000049"><arrow.to.target n="marg2"/></s>
        </p>
        <p type="margin">
          <s id="s.000050"><margin.target id="marg2"/>2</s>
        </p>
        <p type="main">
          <s id="s.000051">Pri&longs;matis, cylindri, &amp; portionis cylindri axem <lb/>
appello rectam lineam, qu&aelig; oppo&longs;itorum plano&shy;<lb/>
rum centra grauitatis coniungit.</s>
        </p>
        <p type="main">
          <s id="s.000052"><arrow.to.target n="marg3"/></s>
        </p>
        <p type="margin">
          <s id="s.000053"><margin.target id="marg3"/>3</s>
        </p>
        <p type="main">
          <s id="s.000054">Pyramidis, coni, &amp; portionis coni axem dico li <lb/>
neam, qu&aelig; &agrave; uertice ad centrum grauitatis ba&longs;is <lb/>
perducitur.</s>
        </p>
        <p type="main">
          <s id="s.000055"><arrow.to.target n="marg4"/></s>
        </p>
        <p type="margin">
          <s id="s.000056"><margin.target id="marg4"/>4</s>
        </p>
        <p type="main">
          <s id="s.000057">Si pyramis, conus, portio coni, uel conoidis &longs;e&shy;<lb/>
cetur plano ba&longs;i &aelig;quidi&longs;tante, pars, qu&aelig; e&longs;t ad ba&shy;<lb/>
&longs;im, fru&longs;tum pyramidis, coni, portionis coni, uel <lb/>
conoidis dicetur; quorum plana &aelig;quidi&longs;tantia, <lb/>
qu&aelig; opponuntur &longs;imilia &longs;unt, &amp; in&aelig;qualia: axes <lb/>
uero &longs;unt axium figurarum partes, qu&aelig; in ip&longs;is <lb/>
comprehenduntur.</s>
        </p>
        <p type="head">
          <s id="s.000058">PETITIONES.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000059"><arrow.to.target n="marg5"/></s>
        </p>
        <p type="margin">
          <s id="s.000060"><margin.target id="marg5"/>1</s>
        </p>
        <p type="main">
          <s id="s.000061">Solidarum figurarum &longs;imilium centra grauita&shy;<lb/>
tis &longs;imiliter &longs;unt po&longs;ita.</s>
        </p>
        <p type="main">
          <s id="s.000062"><arrow.to.target n="marg6"/></s>
        </p>
        <p type="margin">
          <s id="s.000063"><margin.target id="marg6"/>2</s>
        </p>
        <p type="main">
          <s id="s.000064">Solidis figuris &longs;imilibus, &amp; &aelig;qualibus inter &longs;e <lb/>
aptatis, centra quoque grauitatis ip&longs;arum inter &longs;e <lb/>
aptata erunt.</s>
        </p>
        <p type="head">
          <s id="s.000065">THEOREMA I. PROPOSITIO I.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000066">Omnis figur&aelig; rectiline&aelig; in circulo de&longs;cript&aelig;, <lb/>
qu&aelig; &aelig;qualibus lateribus, &amp; angulis contine&shy;<lb/>




<pb pagenum="2" xlink:href="023/01/011.jpg"/>tur, centrum grauitatis e&longs;t idem, quod circuli cen<lb/>
trum.</s>
        </p>
        <p type="main">
          <s id="s.000067">Sit primo triangulum &aelig;quilaterum abc in circulo de&shy;<lb/>
&longs;criptum: &amp; diui&longs;a ac bifariam in d, ducatur bd. <!-- KEEP S--></s>
          <s id="s.000068">erit in li&shy;<lb/>
nea bd centrum grauitatis <expan abbr="tri&atilde;guli">trianguli</expan> abc, ex tertia decima <lb/>
primi libri Archimedis de centro grauitatis planorum. </s>
          <s id="s.000069">Et <lb/>
<figure id="id.023.01.011.1.jpg" xlink:href="023/01/011/1.jpg"/><lb/>
quoniam linea ab e&longs;t &aelig;qualis <lb/>
line&aelig; bc; &amp; ad ip&longs;i dc; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <lb/>
bd utrique communis: trian&shy;<arrow.to.target n="marg7"/><lb/>
gulum abd &aelig;quale erit trian<lb/>
gulo cbd: &amp; anguli angulis &aelig;&shy;<lb/>
quales, qui &aelig;qualibus lateri&shy;<lb/>
<arrow.to.target n="marg8"/><lb/>
bus &longs;ubtenduntur. </s>
          <s id="s.000071">ergo angu<lb/>
li ad d <expan abbr="utriq;">utrique</expan> recti &longs;unt. </s>
          <s id="s.000072">qu&ograve;d <lb/>
cum linea bd &longs;ecet ae bifa&shy;<lb/>
<arrow.to.target n="marg9"/><lb/>
riam, &amp; ad angulos rectos; in <lb/>
ip&longs;a bd e&longs;t centrum circuli. </s>
          <s id="s.000073"><lb/>
quare in eadem bd linea erit <lb/>
centrum grauitatis trianguli, &amp; circuli centrum. </s>
          <s id="s.000074">Similiter <lb/>
diui&longs;a ab bifariam in e, &amp; ducta ce, o&longs;tendetur in ip&longs;a <expan abbr="utr&utilde;">utrum</expan> <lb/>
que centrum contineri. </s>
          <s id="s.000075">ergo ea erunt in puncto, in quo li&shy;<lb/>
ne&aelig; bd, ce conueniunt. </s>
          <s id="s.000076">trianguli igitur abc centrum gra<lb/>
uitatis e&longs;t idem, quod circuli centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000077"><margin.target id="marg7"/>8. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000078"><margin.target id="marg8"/>13. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000079"><margin.target id="marg9"/>corol. pri<lb/>
m&aelig; tertii</s>
        </p>
        <figure id="id.023.01.011.2.jpg" xlink:href="023/01/011/2.jpg"/>
        <p type="main">
          <s id="s.000080">Sit quadratum abcd in cir&shy;<lb/>
culo de&longs;criptum: &amp; ducantur <lb/>
ac, bd, qu&aelig; conueniant in e. </s>
          <s id="s.000081">er&shy;<lb/>
go punctum e e&longs;t centrum gra<lb/>
uitatis quadrati, ex decima eiu&longs; <lb/>
dem libri Archimedis. <!-- KEEP S--></s>
          <s id="s.000082">Sed cum <lb/>
omnes anguli ad abcd recti <lb/>
<arrow.to.target n="marg10"/><lb/>
&longs;int; erit abc &longs;emicirculus: <lb/>
<expan abbr="item&qacute;">itemque</expan>; bcd: &amp; propterea li&shy;<lb/>
ne&aelig; ac, bd diametri circuli: 



<pb xlink:href="023/01/012.jpg"/>qu&aelig; quidem in centro conueniunt. </s>
          <s id="s.000083">idem igitur e&longs;t centrum <lb/>
grauitatis quadrati, &amp; circuli centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000084"><margin.target id="marg10"/>31. tertii.</s>
        </p>
        <p type="main">
          <s id="s.000085">Sit pentagonum &aelig;quilaterum, &amp; &aelig;quiangulum in circu&shy;<lb/>
<figure id="id.023.01.012.1.jpg" xlink:href="023/01/012/1.jpg"/><lb/>
lo de&longs;criptum abcd e. </s>
          <s id="s.000086">&amp; iun&shy;<lb/>
cta bd, <expan abbr="bifariam&qacute;">bifariamque</expan>; in f diui&longs;a, <lb/>
ducatur cf, &amp; producatur ad <lb/>
circuli circumferentiam in g; <lb/>
qu&aelig; lineam ae in h &longs;ecet: de&shy;<lb/>
inde iungantur ac, cc. <!-- KEEP S--></s>
          <s id="s.000087">Eodem <lb/>
modo, quo &longs;upra demon&longs;tra&shy;<lb/>
bimus angulum bcf &aelig;qualem <lb/>
e&longs;&longs;e. </s>
          <s id="s.000088">angulo dcf; &amp; angulos <lb/>
ad f utro&longs;que rectos: &amp; idcir&shy;<lb/>co lineam cfg per circuli cen<lb/>
trum tran&longs;ire. </s>
          <s id="s.000089">Quoniam igi&shy;<lb/>
tur latera cb, ba, &amp; cd, de &aelig;qualia &longs;unt; &amp; &aelig;quales anguli <lb/>
<arrow.to.target n="marg11"/><lb/>
cba, cde: erit ba&longs;is ca ba&longs;i: ce, &amp; angulus bca angulo <lb/>
dce &aelig;qualis. </s>
          <s id="s.000090">ergo &amp; reliquus ach, reliquo ech. </s>
          <s id="s.000091">e&longs;t au&shy;<lb/>
tem ch utrique triangulo ach, ech communis. </s>
          <s id="s.000092">quare <lb/>
ba&longs;is ah &aelig;qualis e&longs;t ba&longs;i hc: &amp; anguli, qui ad h recti: <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; <lb/>
<arrow.to.target n="marg12"/><lb/>
recti, qui ad f. </s>
          <s id="s.000093">ergo line&aelig; ae, bd inter &longs;e &longs;e &aelig;quidi&longs;tant. </s>
          <lb/>
          <s id="s.000094">Itaque cum trapezij abde latera bd, ae &aelig;quidi&longs;tantia &agrave; li<lb/>
nea fh bifariam diuidantur; centrum grauitatis ip&longs;ius erit <lb/>
<arrow.to.target n="marg13"/><lb/>
in linea fh, ex ultima eiu&longs;dem libri Archimedis. <!-- KEEP S--></s>
          <s id="s.000095">Sed trian&shy;<lb/>
guli bcd centrum grauitatis e&longs;t in linea cf. </s>
          <s id="s.000096">ergo in eadem <lb/>
linea ch e&longs;t centrum grauitatis trapezij abde, &amp; trian&shy;<lb/>
guli bcd: hoc e&longs;t pentagoni ip&longs;ius centrum: &amp; centrum <lb/>
circuli. </s>
          <s id="s.000097">Rur&longs;us &longs;i iuncta ad, <expan abbr="bifariam&qacute;">bifariamque</expan>; &longs;ecta in k, duca&shy;<lb/>
tur ekl: demon&longs;trabimus in ip&longs;a utrumque centrum in <lb/>
e&longs;&longs;e. </s>
          <s id="s.000098">Sequitur ergo, ut punctum, in quo line&aelig; cg, el con&shy;<lb/>
ueniunt, idem &longs;it centrum circuli, &amp; centrum grauitatis <lb/>
pentagoni.</s>
        </p>
        <p type="margin">
          <s id="s.000099"><margin.target id="marg11"/>4. Primi.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000100"><margin.target id="marg12"/>28. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000101"><margin.target id="marg13"/>13. Archi&shy;<lb/>
medis.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000102">Sit hexagonum abcdef &aelig;quilaterum, &amp; &aelig;quiangulum <lb/>
in circulo de&longs;ignatum: <expan abbr="iungantur&qacute;">iunganturque</expan>; bd, ae: &amp; bifariam &longs;e&shy;



<pb pagenum="3" xlink:href="023/01/013.jpg"/>cta bd in g puncto, ducatur cg; &amp; protrahatur ad circuli <lb/>
u&longs;que circumferentiam; qu&aelig; &longs;ecet ae in h. </s>
          <s id="s.000103">Similiter conclu<lb/>
demus cg per centrum circuli tran&longs;ire: &amp; bifariam &longs;ecate <lb/>
lineam ae; <expan abbr="item&qacute;">itemque</expan>; lineas bd, ae inter &longs;e &aelig;quidi&longs;tantes e&longs;&longs;e. <lb/>
</s>
          <s id="s.000104">Cum igitur cg per centrum circuli tran&longs;eat; &amp; ad <expan abbr="punct&utilde;">punctum</expan> <lb/>
f perueniat nece&longs;&longs;e e&longs;t: qu&ograve;d cdef &longs;it dimidium circumfe<lb/>
<figure id="id.023.01.013.1.jpg" xlink:href="023/01/013/1.jpg"/><lb/>
<arrow.to.target n="marg14"/><lb/>
renti&aelig; circuli. </s>
          <s id="s.000105">Quare in eadem <lb/>
diametro cf erunt centra gra<lb/>
uitatis triangulorum bcd, <lb/>
afe, &amp; quadrilateri abde, ex <lb/>
quibus con&longs;tat hexagonum ab <lb/>
cdef. </s>
          <s id="s.000106">per&longs;picuum e&longs;t igitur in <lb/>
ip&longs;a cf e&longs;&longs;e circuli centrum, &amp; <lb/>
centrum grauitatis hexagoni. <lb/>
</s>
          <s id="s.000107">Rur&longs;us ducta altera diametro <lb/>
ad, ei&longs;dem rationibus o&longs;tende&shy;<lb/>
mus in ip&longs;a utrumque <expan abbr="c&etilde;trum">centrum</expan> <lb/>
ine&longs;&longs;e. </s>
          <s id="s.000108">Centrum ergo grauita&shy;<lb/>
tis hexagoni, &amp; centrum circuli idem erit.</s>
        </p>
        <p type="margin">
          <s id="s.000109"><margin.target id="marg14"/>13 Archi<lb/>
medis.</s>
          <lb/>
          <s id="s.000110">9. <expan abbr="eiusdetilde;">eiusdem</expan> <lb/>
m</s>
        </p>
        <p type="main">
          <s id="s.000111">Sit heptagonum abcdefg &aelig;quilaterum atque &aelig;quian<lb/>
<figure id="id.023.01.013.2.jpg" xlink:href="023/01/013/2.jpg"/><lb/>
gulum in circulo de&longs;criptum: <lb/>
&amp; iungantur ce, bf, ag: di&shy;<lb/>
ui&longs;a autem ce bifariam in <expan abbr="p&utilde;tco">pun<lb/>
cto</expan> h: &amp; iuncta dh produca&shy;<lb/>
tur in k. </s>
          <s id="s.000112">non aliter demon&shy;<lb/>
&longs;trabimus in linea dk e&longs;&longs;e cen<lb/>
trum circuli, &amp; centrum gra&shy;<lb/>
uitatis trianguli cde, &amp; tra&shy;<lb/>
peziorum bcef, abfg, hoc <lb/>
e&longs;t centrum totius heptago&shy;<lb/>
ni: &amp; rur&longs;us eadem centra in <lb/>
alia diametro cl &longs;imiliter du&shy;<lb/>
cta contineri. </s>
          <s id="s.000113">Quare &amp; centrum grauitatis heptagoni, &amp; <lb/>
centrum circuli in idem punctum conueniunt. </s>
          <s id="s.000114">Eodem mo



<pb xlink:href="023/01/014.jpg"/>do in reliquis figuris &aelig;quilateris, &amp; &aelig;quiangulis, qu&aelig; in cir&shy;<lb/>
culo de&longs;cribuntur, probabimus <expan abbr="c&etilde;trum">centrum</expan> grauitatis earum, <lb/>
&amp; centrum circuli idem e&longs;&longs;e. </s>
          <s id="s.000115">quod quidem demon&longs;trare <lb/>
oportebat.</s>
        </p>
        <p type="main">
          <s id="s.000116">Ex quibus apparet cuiuslibet figur&aelig; rectiline&aelig; <lb/>
in circulo plane de&longs;cript&aelig; centrum grauitatis <expan abbr="id&etilde;">idem</expan> <lb/>
e&longs;&longs;e, quod &amp; circuli centrum.<lb/>
<arrow.to.target n="marg15"/></s>
        </p>
        <p type="margin">
          <s id="s.000117"><margin.target id="marg15"/><foreign lang="greek">gnwri/mws</foreign></s>
        </p>
        <p type="main">
          <s id="s.000118">Figuram in circulo plane de&longs;criptam appella&shy;<lb/>
mus, cuiu&longs;modi e&longs;t ea, qu&aelig; in duodecimo elemen<lb/>
torum libro, propo&longs;itione &longs;ecunda de&longs;cribitur. <lb/>
</s>
          <s id="s.000119">ex &aelig;qualibus enim lateribus, &amp; angulis con&longs;tare <lb/>
per&longs;picuum e&longs;t.</s>
        </p>
        <p type="head">
          <s id="s.000120">THEOREMA II, PROPOSITIO II.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000121">Omnis figur&aelig; rectiline&aelig; in ellip&longs;i plane de&longs;cri&shy;<lb/>
pt&aelig; centrum grauitatis e&longs;t idem, quod ellip&longs;is <lb/>
centrum.</s>
        </p>
        <p type="main">
          <s id="s.000122">Quo modo figura rectilinea in ellip&longs;i plane de&longs;cribatur, <lb/>
docuimus in commentarijs in quintam propo&longs;itionem li&shy;<lb/>
bri Archimedis de conoidibus, &amp; &longs;ph&aelig;roidibus.</s>
        </p>
        <p type="main">
          <s id="s.000123">Sit ellip&longs;is abcd, cuius maior axis ac, minor bd: <expan abbr="iun-gantur&qacute;">iun&shy;<lb/>
ganturque</expan>; ab, bc, cd, da: &amp; bifariam diuidantur in pun&shy;<lb/>
ctis efgh. </s>
          <s id="s.000124">&agrave; centro autem, quod &longs;it k duct&aelig; line&aelig; ke, kf, <lb/>
kg, kh u&longs;que ad &longs;ectionem in puncta lmno protrahan&shy;<lb/>
tur: &amp; iungantur lm, mn, no, ol, ita ut ac &longs;ecet li&shy;<lb/>
neas lo, mn, in z<foreign lang="greek">f</foreign> punctis; &amp; bd &longs;ecet lm, on in <foreign lang="greek">xy.</foreign><lb/>
erunt lk, kn linea una, <expan abbr="item&qacute;ue">itemque</expan> linea una ip&longs;&aelig; mk, ko: <lb/>
&amp; line&aelig; ba, cd &aelig;quidi&longs;tabunt line&aelig; mo: &amp; bc, ad ip&longs;i <lb/>
ln. </s>
          <s id="s.000125">rur&longs;us lo, mn axi bd &aelig;quidi&longs;tabunt: &amp; lm, 



<pb pagenum="4" xlink:href="023/01/015.jpg"/>on ip&longs;i ac. <!-- KEEP S--></s>
          <s id="s.000126">Quoniam enim triangulorum abk, adk, latus <lb/>
bk e&longs;t &aelig;quale lateri kd, &amp; ak utrique commune; <expan abbr="anguli&qacute;">angulique</expan>; <lb/>
<arrow.to.target n="marg16"/><lb/>
ad k recti. </s>
          <s id="s.000127">ba&longs;is ab ba&longs;i ad; &amp; reliqui anguli reliquis an&shy;<lb/>
gulis &aelig;quales erunt. </s>
          <s id="s.000128">eadem quoque ratione o&longs;tendetur bc <lb/>
<figure id="id.023.01.015.1.jpg" xlink:href="023/01/015/1.jpg"/><lb/>
&aelig;qualis cd; &amp; ab ip&longs;i <lb/>
bc. <!-- REMOVE S-->quare omnes ab, <lb/>
bc, cd, da &longs;unt &aelig;qua&shy;<lb/>
les. </s>
          <s id="s.000129">&amp; quoniam anguli <lb/>
ad a &aelig;quales &longs;unt angu<lb/>
lis ad c; erunt anguli b <lb/>
ac, acd coalterni inter <lb/>
&longs;e &aelig;quales; <expan abbr="item&qacute;">itemque</expan>; dac, <lb/>
acb. </s>
          <s id="s.000130">ergo cd ip&longs;i ba; <lb/>
&amp; ad ip&longs;i bc &aelig;quidi&shy;<lb/>
&longs;tat. </s>
          <s id="s.000131">At uero cum line&aelig; <lb/>
ab, cd inter &longs;e &aelig;quidi&shy;<lb/>
&longs;tantes bifariam &longs;ecen&shy;<lb/>tur in punctis eg; erit li<lb/>
nea lekgn diameter &longs;e<lb/>
ctionis, &amp; linea una, ex <lb/>
demon&longs;tratis in uige&longs;i&shy;<lb/>
maoctaua &longs;ecundi coni <lb/>
corum. </s>
          <s id="s.000132">Et eadem ratione linea una mfkho. </s>
          <s id="s.000133">Sunt <expan abbr="aut&etilde;">autem</expan> ad, <lb/>
bc inter &longs;e &longs;e &aelig;quales, &amp; &aelig;quidi&longs;tantes. </s>
          <s id="s.000134">quare &amp; earum di&shy;<lb/>
<arrow.to.target n="marg17"/><lb/>
midi&aelig; ah, bf; <expan abbr="item&qacute;">itemque</expan>; hd, fe; &amp; qu&aelig; ip&longs;as coniungunt rect&aelig; <lb/>
line&aelig; &aelig;quales, &amp; &aelig;quidi&longs;tantes erunt. </s>
          <s id="s.000135"><expan abbr="&aelig;quidi&longs;t&atilde;t">&aelig;quidi&longs;tant</expan> igitur ba, <lb/>
cd diametro mo: &amp; pariter ad, bc ip&longs;i ln &aelig;quidi&longs;tare o&shy;<lb/>
&longs;tendemus. </s>
          <s id="s.000136">Si igitur <expan abbr="man&etilde;te">manente</expan> diametro ac intelligatur abc <lb/>
portio ellip&longs;is ad portionem adc moueri, cum primum b <lb/>
applicuerit ad d, <expan abbr="c&otilde;gruet">congruet</expan> tota portio toti portioni, <expan abbr="linea&qacute;">lineaque</expan>; <lb/>
ba line&aelig; ad; &amp; bc ip&longs;i cd congruet: punctum uero e ca&shy;<lb/>
det in h; f in g: &amp; linea ke in lineam kh: &amp; kf in kg. <!-- KEEP S--></s>
          <s id="s.000137">qua <lb/>
re &amp; el in ho, et fm in gn. </s>
          <s id="s.000138">At ip&longs;a lz in zo; et m<foreign lang="greek">f</foreign> in <foreign lang="greek">f</foreign>n <lb/>
cadet. </s>
          <s id="s.000139">congruet igitur triangulum lkz triangulo okz: et 



<pb xlink:href="023/01/016.jpg"/>triangulum mk<foreign lang="greek">f</foreign> triangulo nk<foreign lang="greek">f.</foreign> ergo anguli lzk, ozk, <lb/>
m <foreign lang="greek">f</foreign> k, n<foreign lang="greek">f</foreign>k &aelig;quales &longs;unt, ac recti. </s>
          <s id="s.000140">qu&ograve;d cum etiam recti <lb/>
<arrow.to.target n="marg18"/><lb/>
&longs;int, qui ad k; &aelig;quidi&longs;tabunt line&aelig; lo, mn axi bd. <!-- KEEP S--></s>
          <s id="s.000141">&amp; ita <lb/>
demon&longs;trabuntur lm, on ip&longs;i ac &aelig;quidi&longs;tare. </s>
          <s id="s.000142">Rur&longs;us &longs;i <lb/>iungantur al, lb, bm, mc, cn, nd, do, oa: &amp; bifariam di<lb/>
uidantur: &agrave; centro autem k ad diui&longs;iones duct&aelig; line&aelig; pro&shy;<lb/>
trahantur u&longs;que ad &longs;ectionem in puncta pqrstuxy: &amp; po <lb/>
&longs;tremo py, qx, ru, st, qr, ps, yt, xu coniungantur. </s>
          <s id="s.000143">Simili&shy;<lb/>
<figure id="id.023.01.016.1.jpg" xlink:href="023/01/016/1.jpg"/><lb/>
ter o&longs;tendemus lineas <lb/>
py, qx, ru, st axi bd &aelig;&shy;<lb/>
quidi&longs;tantes e&longs;&longs;e: &amp; qr, <lb/>
ps, yt, xu &aelig;quidi&longs;tan&shy;<lb/>
tes ip&longs;i ac. <!-- KEEP S--></s>
          <s id="s.000144">Itaque dico <lb/>
harum figurarum in el&shy;<lb/>
lip&longs;i de&longs;criptarum cen&shy;<lb/>
trum grauitatis e&longs;&longs;e <expan abbr="p&utilde;-ctum">pun&shy;<lb/>
ctum</expan> k, idem quod &amp; el<lb/>
lip&longs;is centrum. </s>
          <s id="s.000145">quadri&shy;<lb/>
lateri enim abcd cen&shy;<lb/>
trum e&longs;t k, ex decima e&shy;<lb/>
iu&longs;dem libri Archime&shy;<lb/>
dis, quippe <expan abbr="c&utilde;">cum</expan> in eo om<lb/>
nes diametri <expan abbr="c&otilde;ueni&atilde;t">conueniant</expan>. </s>
          <lb/>
          <s id="s.000146">Sed in figura albmcn <lb/>
<arrow.to.target n="marg19"/><lb/>
do, quoniam trianguli <lb/>
alb centrum grauitatis <lb/>
<arrow.to.target n="marg20"/><lb/>
e&longs;t in linea le: <expan abbr="trapezij&qacute;">trapezijque</expan>; abmo centrum in linea ek: trape<lb/>
zij omcd in kg: &amp; trianguli cnd in ip&longs;a gn: erit magnitu<lb/>
dinis ex his omnibus con&longs;tantis, uidelicet totius figur&aelig; cen <lb/>
trum grauitatis in linea ln: &amp; ob eandem cau&longs;&longs;am in linea <lb/>
om. </s>
          <s id="s.000147">e&longs;t enim trianguli aod centrum in linea oh: trapezij <lb/>
alnd in hk: trapezij lbcn in kf: &amp; trianguli bmc in fm. </s>
          <lb/>
          <s id="s.000148">cum ergo figur&aelig; albmcndo centrum grauitatis &longs;it in li&shy;<lb/>
nea ln, &amp; in linea om; erit centrum ip&longs;ius punctum k, in 



<pb pagenum="5" xlink:href="023/01/017.jpg"/>quo &longs;cilicet ln, om conueniunt. </s>
          <s id="s.000149">Po&longs;tremo in figura <lb/>
aplqbrmsctnudxoy centrum grauitatis trian<lb/>
guli pay, &amp; trapezii ploy e&longs;t in linea az: trapeziorum <lb/>
uero lqxo, qbdx centrum e&longs;t in linea zk: &amp; <expan abbr="trapezior&utilde;">trapeziorum</expan> <lb/>
brud, rmnu in k<foreign lang="greek">f:</foreign> &amp; denique trapezii mstn; &amp; triangu<lb/>
li sct in <foreign lang="greek">f</foreign>c. </s>
          <s id="s.000150">quare magnitudinis ex his compo&longs;it&aelig; <expan abbr="centr&utilde;">centrum</expan> <lb/>
in linea ac con&longs;i&longs;tit. </s>
          <s id="s.000151">Rur&longs;us trianguli qbr, &amp; trapezii ql<lb/>
mr centrum e&longs;t in linea b<foreign lang="greek">x.</foreign> trapeziorum lpsm, pacs, <lb/>
aytc, yont in linea <foreign lang="greek">xf:</foreign> <expan abbr="trapeziiq;">trapeziique</expan> oxun, &amp; trianguli <lb/>
xdu centrum in <foreign lang="greek">y</foreign>d. <!-- KEEP S--></s>
          <s id="s.000152">totius ergo magnitudinis centrum <lb/>
e&longs;t in linea bd. <!-- KEEP S--></s>
          <s id="s.000153">ex quo &longs;equitur, centrum grauitatis figur&aelig; <lb/>
aplqbrmsctnudxoy e&longs;&longs;e <expan abbr="punct&utilde;">punctum</expan> K, lineis &longs;cilicet ac, <lb/>
bd commune, qu&aelig; omnia demon&longs;trare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000154"><margin.target id="marg16"/>8. primi</s>
        </p>
        <p type="margin">
          <s id="s.000155"><margin.target id="marg17"/>33.  primi</s>
        </p>
        <p type="margin">
          <s id="s.000156"><margin.target id="marg18"/>28. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000157"><margin.target id="marg19"/>13. Archi<lb/>
medis.</s>
        </p>
        <p type="margin">
          <s id="s.000158"><margin.target id="marg20"/>Vltima.<!-- KEEP S--></s>
        </p>
        <p type="head">
          <s id="s.000159">THEOREMA III. PROPOSITIO III.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000160">Cuiuslibet portio&shy;<lb/>
nis circuli, &amp; ellip&longs;is, <lb/>
qu&aelig; dimidia non &longs;it <lb/>
maior, centrum graui <lb/>
tatis in portionis dia&shy;<lb/>
metro con&longs;i&longs;tit.</s>
        </p>
        <figure id="id.023.01.017.1.jpg" xlink:href="023/01/017/1.jpg"/>
        <p type="main">
          <s id="s.000161">HOC eodem pror&longs;us <lb/>
modo demon&longs;trabitur, <lb/>
quo in libro de centro gra<lb/>
uitatis planorum ab Ar&shy;<lb/>
chimede <expan abbr="demon&longs;trat&utilde;">demon&longs;tratum</expan> e&longs;t, <lb/>
in portione <expan abbr="c&otilde;tenta">contenta</expan> recta <lb/>
linea, &amp; rectanguli coni &longs;e<lb/>
ctione grauitatis <expan abbr="c&etilde;trum">centrum</expan> <lb/>
e&longs;&longs;e in diametro portio&shy;<lb/>
nis. </s>
          <s id="s.000162">Et ita demon&longs;trari po<lb/>




<pb xlink:href="023/01/018.jpg"/>te&longs;t in portione, qu&aelig; recta linea &amp; obtu&longs;ianguli coni &longs;e&shy;<lb/>
ctione, &longs;eu hyperbola continetur.</s>
        </p>
        <p type="head">
          <s id="s.000163">THEOREMA IIII. PROPOSITIO IIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000164">IN circulo &amp; ellip&longs;i idem e&longs;t figur&aelig; &amp; graui&shy;<lb/>
tatis centrum.</s>
        </p>
        <p type="main">
          <s id="s.000165">SIT circulus, uel ellip&longs;is, cuius centrum a. </s>
          <s id="s.000166">Dico a gra&shy;<lb/>
uitatis quoque centrum e&longs;&longs;e. </s>
          <s id="s.000167">Si enim fieri pote&longs;t, &longs;it b cen&shy;<lb/>
trum grauitatis: &amp; iuncta ab extra figuram in c produca<lb/>
tur: quam uero proportionem habet linea ca ad ab, ha&shy;<lb/>
beat circulus a ad alium circulum, in quo d; uel ellip&longs;is ad <lb/>
aliam ellip&longs;im: &amp; in circulo, uel ellip&longs;i figura rectilinea pla&shy;<lb/>
ne de&longs;cribatur adco, ut tandem relinquantur portiones <lb/>
qu&aelig;dam minores circulo, uel ellip&longs;i d; qu&aelig; figura &longs;it abcefg<lb/>
hklmn. </s>
          <s id="s.000168">Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>
elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con&shy;<lb/>
<figure id="id.023.01.018.1.jpg" xlink:href="023/01/018/1.jpg"/><lb/>
&longs;tat; at in ellip&longs;i nos demon&longs;tra&shy;<lb/>
uimus in commentariis in quin&shy;<lb/>
tam propo&longs;itionem Archimedis <lb/>
de conoidibus, &amp; &longs;ph&aelig;roidibus. </s>
          <lb/>
          <s id="s.000169">erit igitur a centrum grauitatis <lb/>
ip&longs;ius figur&aelig;, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;ten</expan><lb/>
dimus. </s>
          <s id="s.000170">Itaque quoniam circulus <lb/>
a ad circulum d, uel ellip&longs;is a ad <lb/>
ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>
habet, quam linea ca ad ab: <lb/>
portiones uero &longs;unt minores cir<lb/>
<arrow.to.target n="marg21"/><lb/>
culo uel ellip&longs;i d: habebit circu&shy;<lb/>
lus, uel ellip&longs;is ad portiones ma&shy;<lb/>
iorem proportionem, qu&agrave;m ca <lb/>
<arrow.to.target n="marg22"/><lb/>
ad ab: &amp; diuidendo figura recti&shy;<lb/>
linea abcefghklmn ad portiones 



<pb pagenum="6" xlink:href="023/01/019.jpg"/><figure id="id.023.01.019.1.jpg" xlink:href="023/01/019/1.jpg"/><lb/>
habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam cb ad ba. </s>
          <s id="s.000171">fiat ob ad ba, <lb/>
ut figura rectilinea ad portio&shy;<lb/>
nes. </s>
          <s id="s.000172">cum igitur &agrave; circulo, uel el&shy;<lb/>
lip&longs;i, cuius grauitatis centrum <lb/>
e&longs;t b, auferatur figura rectilinea <lb/>
efghklmn, cuius centrum a; <lb/>
reliqu&aelig; magnitudinis ex portio<lb/>
<arrow.to.target n="marg23"/><lb/>
nibus compo&longs;it&aelig; centrum graui<lb/>
tatis erit in linea ab producta, <lb/>
&amp; in puncto o, extra figuram po<lb/>
&longs;ito. </s>
          <s id="s.000173">quod quidem fieri nullo mo<lb/>
do po&longs;&longs;e per&longs;picuum e&longs;t. </s>
          <s id="s.000174">&longs;equi&shy;<lb/>
tur ergo, ut circuli &amp; ellip&longs;is cen<lb/>
trum grauitatis &longs;it punctum a, <lb/>
idem quod figur&aelig; centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000175"><margin.target id="marg21"/>8. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000176"><margin.target id="marg22"/>19. quinti <lb/>
apud <expan abbr="C&atilde;panum">Cam<lb/>
panum</expan> .</s>
        </p>
        <p type="margin">
          <s id="s.000177"><margin.target id="marg23"/>8. Archi&shy;<lb/>
medis.</s>
        </p>
        <p type="head">
          <s id="s.000178">ALITER.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000179">Sit circulus, uel ellip&longs;is abcd, <lb/>
cuius diameter db, &amp; centrum e: <expan abbr="ducaturq;">ducaturque</expan> per e recta li<lb/>
nea ac, &longs;ecans ip&longs;am db ad rectos angulos. </s>
          <s id="s.000180">erunt adc, <lb/>
abc circuli, uel ellip&longs;is dimidi&aelig; portiones. </s>
          <s id="s.000181">Itaque quo&shy;<lb/>
<figure id="id.023.01.019.2.jpg" xlink:href="023/01/019/2.jpg"/><lb/>
niam por<lb/>
<expan abbr="ti&otilde;is">tionis</expan> adc <lb/>
<expan abbr="c&etilde;tr&utilde;">centrum</expan> gra&shy;<lb/>
uitatis e&longs;t <lb/>
in diame&shy;<lb/>
tro de: &amp; <lb/>
portionis <lb/>
abc cen&shy;<lb/>
trum e&longs;t <expan abbr="&itilde;">im</expan> <lb/>
ip&longs;a eb: to<lb/>
tius circu<lb/>
li, uel ellip&longs;is grauitatis centrum erit in diametro db. </s>
          <lb/>
          <s id="s.000182">Sit autem portionis adc <expan abbr="c&etilde;trum">centrum</expan> grauitatis f: &amp; &longs;umatur 



<pb xlink:href="023/01/020.jpg"/>in linea eb <expan abbr="punct&utilde;">punctum</expan> g, ita ut fit ge &aelig;qualis ef. </s>
          <s id="s.000183">erit g por&shy;<lb/>
tionis abc centrum. </s>
          <s id="s.000184">nam &longs;i h&aelig; portiones, qu&aelig; &aelig;quales <lb/>
&amp; &longs;imiles &longs;unt, inter &longs;e &longs;e aptentur, ita ut be cadat in de, <lb/>
&amp; punctum b in d cadet, &amp; g in f: figuris autem &aelig;quali&shy;<lb/>
bus, &amp; &longs;imilibus inter &longs;e aptatis, centra quoque grauitatis <lb/>
ip&longs;arum inter &longs;e aptata erunt, ex quinta petitione Archi&shy;<lb/>
medis in libro de centro grauitatis planorum. </s>
          <s id="s.000185">Quare cum <lb/>
portionis adc centrum grauitatis &longs;it f: &amp; portionis <lb/>
abc centrum g: magnitudinis; qu&aelig; ex utri&longs;que efficitur: <lb/>
hoc e&longs;t circuli uel ellip&longs;is grauitatis centrum in medio li&shy;<lb/>
ne&aelig; fg, quod e&longs;t e, con&longs;i&longs;tet, ex quarta propo&longs;itione eiu&longs;&shy;<lb/>
dem libri Archimedis. <!-- KEEP S--></s>
          <s id="s.000186">ergo circuli, uel ellip&longs;is centrum <lb/>
grauitatis e&longs;t idem, quod figur&aelig; centrum. </s>
          <s id="s.000187">atque illud e&longs;t, <lb/>
quod demon&longs;trare oportebat.</s>
        </p>
        <p type="main">
          <s id="s.000188">Ex quibus &longs;equitur portionis circuli, uel ellip&shy;<lb/>
&longs;is, qu&aelig; dimidia maior &longs;it, centrum grauitatis in <lb/>
diametro quoque ip&longs;ius con&longs;i&longs;tere.</s>
        </p>
        <figure id="id.023.01.020.1.jpg" xlink:href="023/01/020/1.jpg"/>
        <p type="main">
          <s id="s.000189">Sit enim maior portio abc, cu<emph type="italics"/>i<emph.end type="italics"/>us diameter bd, &amp; com&shy;<lb/>
pleatur circulus, uel ellip&longs;is, ut portio reliqua fit aec, dia



<pb pagenum="7" xlink:href="023/01/021.jpg"/>metrum habens ed. <!-- KEEP S--></s>
          <s id="s.000190">Quoniam igitur circuli uel ellip&longs;is <lb/>
aecb grauitatis centrum e&longs;t in diametro be, &amp; portio&shy;<lb/>
nis aec centrum in linea ed: reliqu&aelig; portionis, uidelicet <lb/>
abc centrum grauitatis in ip&longs;a bd con&longs;i&longs;tat nece&longs;&longs;e e&longs;t, ex <lb/>
octaua propo&longs;itione eiu&longs;dem.</s>
        </p>
        <p type="head">
          <s id="s.000191">THEOREMA V. PROPOSITIO V.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000192">SI pri&longs;ma &longs;ecetur plano oppo&longs;itis planis &aelig;qui <lb/>
di&longs;tante, &longs;ectio erit figura &aelig;qualis &amp; &longs;imilis ei, <lb/>
qu&aelig; e&longs;t oppo&longs;itorum planorum, centrum graui<lb/>
tatis in axe habens.</s>
        </p>
        <p type="main">
          <s id="s.000193">Sit pri&longs;ma, in quo plana oppo&longs;ita &longs;int triangula abc, <lb/>
def; axis gh: &amp; &longs;ecetur plano iam dictis planis <expan abbr="&aelig;quidi&longs;t&atilde;">&aelig;quidi&longs;tan</expan><lb/>
te; quod faciat &longs;ectionem klm; &amp; axi in <expan abbr="p&utilde;cto">puncto</expan> n occurrat. </s>
          <lb/>
          <s id="s.000194">Dico klm triangulum &aelig;quale e&longs;&longs;e, &amp; &longs;imile triangulis abc <lb/>
def; atque eius grauitatis centrum e&longs;&longs;e punctum n. </s>
          <s id="s.000195">Quo&shy;<lb/>
<figure id="id.023.01.021.1.jpg" xlink:href="023/01/021/1.jpg"/><lb/>
niam enim plana abc <lb/>
Klm &aelig;quidi&longs;tantia <expan abbr="&longs;ec&atilde;">&longs;ecan</expan><lb/>
<arrow.to.target n="marg24"/><lb/>
tur a plano ae; rect&aelig; li&shy;<lb/>
ne&aelig; ab, Kl, qu&aelig; &longs;unt ip <lb/>
&longs;orum <expan abbr="c&otilde;munes">communes</expan> &longs;ectio&shy;<lb/>
nes inter &longs;e &longs;e &aelig;quidi&shy;<lb/>
&longs;tant. </s>
          <s id="s.000196">Sed &aelig;quidi&longs;tant <lb/>
ad, be; cum ae &longs;it para<lb/>
lelogrammum, ex pri&longs;&shy;<lb/>
matis diffinitione. </s>
          <s id="s.000197">ergo <lb/>
&amp; al <expan abbr="parallelogramm&utilde;">parallelogrammum</expan> <lb/>
erit; &amp; propterea linea <lb/>
<arrow.to.target n="marg25"/><lb/>
kl, ip&longs;i ab &aelig;qualis. </s>
          <s id="s.000198">Si&shy;<lb/>
militer demon&longs;trabitur <lb/>
lm &aelig;quidi&longs;tans, &amp; &aelig;qua <lb/>
lis bc; &amp; mk ip&longs;i ca.</s>
          <pb xlink:href="023/01/022.jpg"/>
          <s id="s.000199">Itaque quoniam du&aelig; line&aelig; Kl, lm &longs;e &longs;e tangentes, duabus <lb/>
lineis &longs;e &longs;e tangentibus ab, bc &aelig;quidi&longs;tant; nec &longs;unt in e o&shy;<lb/>
dem plano: angulus klm &aelig;qualis e&longs;t angulo abc: &amp; ita an<lb/>
<arrow.to.target n="marg26"/><lb/>
gulus lmk, angulo bca, &amp; mkl ip&longs;i cab &aelig;qualis probabi<lb/>
tur. </s>
          <s id="s.000200">triangulum ergo klm e&longs;t &aelig;quale, &amp; &longs;imile triangulo <lb/>
abc. quare &amp; triangulo def. </s>
          <s id="s.000201">Ducatur linea cgo, &amp; per ip<lb/>
&longs;am, &amp; per cf ducatur planum &longs;ecans pri&longs;ma; cuius &amp; paral<lb/>
lelogrammi ae communis &longs;ectio &longs;it opq.</s>
          <s id="s.000202"> tran&longs;ibit linea <lb/>
fq per h, &amp; mp per n. </s>
          <s id="s.000203">nam cum plana &aelig;quidi&longs;tantia &longs;ecen <lb/>
tur &agrave; plano cq, communes eorum &longs;ectiones cgo, mp, fq <lb/>
&longs;ibi ip&longs;is &aelig;quidi&longs;tabunt. </s>
          <s id="s.000204">Sed &amp; &aelig;quidi&longs;tant ab, kl, de. </s>
          <s id="s.000205">an&shy;<lb/>
<arrow.to.target n="marg27"/><lb/>
guli ergo aoc, kpm, dqf inter &longs;e &aelig;quales &longs;unt: &amp; &longs;unt <lb/>
&aelig;quales qui ad puncta akd con&longs;tituuntur. </s>
          <s id="s.000206">quare &amp; reliqui <lb/>
reliquis &aelig;quales; &amp; triangula aco, Kmp, dfq inter &longs;e &longs;imi <lb/>
<arrow.to.target n="marg28"/><lb/>
lia erunt. </s>
          <s id="s.000207">Vt igitur ca ad ao, ita fd ad dq: &amp; permutando <lb/>
ut ca ad fd, ita ao ad dq.</s>
          <s id="s.000208">e&longs;t autem ca &aelig;qualis fd. <!-- KEEP S--></s>
          <s id="s.000209">ergo &amp; <lb/>
ao ip&longs;i dq.</s>
          <s id="s.000210"> eadem quoque ratione &amp; ao ip&longs;i Kp &aelig;qualis <lb/>
demon&longs;trabitur. </s>
          <s id="s.000211">Itaque &longs;i triangula, abc, def &aelig;qualia &amp; <lb/>
<figure id="id.023.01.022.1.jpg" xlink:href="023/01/022/1.jpg"/><lb/>
&longs;imilia inter &longs;e <expan abbr="apt&etilde;tur">aptentur</expan>, <lb/>
cadet linea fq in lineam <lb/>
<arrow.to.target n="marg29"/><lb/>
cgo. </s>
          <s id="s.000212">Sed &amp; <expan abbr="centr&utilde;">centrum</expan> gra<lb/>
uitatis h in g <expan abbr="centr&utilde;">centrum</expan> ca&shy;<lb/>
det. </s>
          <s id="s.000213"><expan abbr="tr&atilde;&longs;ibit">tran&longs;ibit</expan> igitur linea <lb/>
fq per h: &amp; planum per <lb/>
co &amp; cf <expan abbr="duct&utilde;">ductum</expan> per <expan abbr="ax&etilde;">axem</expan> <lb/>
gh ducetur: <expan abbr="idcircoq;">idcircoque</expan> li <lb/>
neam mp <expan abbr="eti&atilde;">etiam</expan> per n <expan abbr="tr&atilde;">tran</expan><lb/>
&longs;ire nece&longs;&longs;e erit. </s>
          <s id="s.000214">Quo&shy;<lb/>
niam ergo fh, cg &aelig;qua&shy;<lb/>
les &longs;unt, &amp; <expan abbr="&aelig;quidi&longs;t&atilde;tes">&aelig;quidi&longs;tantes</expan>: <lb/>
<expan abbr="itemq;">itemque</expan> hq, go; rect&aelig; li&shy;<lb/>
ne&aelig;, qu&aelig; ip&longs;as <expan abbr="c&otilde;nect&utilde;t">connectunt</expan> <lb/>
cmf, gnh, opq &aelig;qua&shy;<lb/>
les &aelig;quidi&longs;tantes <expan abbr="er&utilde;t">erunt</expan>.</s>
          <pb pagenum="8" xlink:href="023/01/023.jpg"/>
          <s id="s.000215">&aelig;quidi&longs;tant autem cgo, mnp. </s>
          <s id="s.000216">ergo <expan abbr="parallelogr&atilde;ma">parallelogramma</expan> &longs;unt <lb/>
on, gm, &amp; linea mn &aelig;qualis cg; &amp; np ip&longs;i go. </s>
          <s id="s.000217">aptatis igi&shy;<lb/>
tur klm, abc <expan abbr="tri&atilde;gulis">triangulis</expan>, qu&aelig; &aelig;qualia &amp; &longs;imilia <expan abbr="s&utilde;t">sunt</expan>; linea mp <lb/>
in co, &amp; punctum n in g cadet. </s>
          <s id="s.000218">Qu&ograve;d <expan abbr="c&utilde;">cum</expan> g &longs;it centrum gra&shy;<lb/>
uitatis trianguli abc, &amp; n trianguli klm grauitatis cen&shy;<lb/>
trum erit id, quod demon&longs;trandum relinquebatur. </s>
          <s id="s.000219">Simili <lb/>
ratione idem contingere demon&longs;trabimus in aliis pri&longs;ma&shy;<lb/>
tibus, &longs;iue quadrilatera, &longs;iue plurilatera habeant plana, <lb/>
qu&aelig; opponuntur.</s>
        </p>
        <p type="margin">
          <s id="s.000220"><margin.target id="marg24"/>16. unde&shy;<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000221"><margin.target id="marg25"/>34. primi</s>
        </p>
        <p type="margin">
          <s id="s.000222"><margin.target id="marg26"/>10. unde<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000223"><margin.target id="marg27"/>10. unde&shy;<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000224"><margin.target id="marg28"/>4. &longs;exti</s>
        </p>
        <p type="margin">
          <s id="s.000225"><margin.target id="marg29"/>per 5. pe&shy;<lb/>
titionem <lb/>
Archime<lb/>
dis.</s>
        </p>
        <p type="head">
          <s id="s.000226">COROLLARIVM.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000227">Ex iam demon&longs;tratis per&longs;picue apparet, cuius <lb/>
libet pri&longs;matis axem, parallelogrammorum lateri<lb/>
bus, qu&aelig; ab oppo&longs;itis planis <expan abbr="duc&utilde;tur">ducuntur</expan> &aelig;quidi&longs;tare.</s>
        </p>
        <p type="head">
          <s id="s.000228">THEOREMA VI. PROPOSITIO VI.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000229">Cuiuslibet pri&longs;matis centrum grauitatis e&longs;t in <lb/>
plano, quod oppo&longs;itis planis &aelig;quidi&longs;tans, reli&shy;<lb/>
quorum planorum latera bifariam diuidit.</s>
        </p>
        <p type="main">
          <s id="s.000230">Sit pri&longs;ma, in quo plana, qu&aelig; opponuntur &longs;int trian&shy;<lb/>
gula ace, bdf: &amp; parallelogrammorum latera ab, cd, <lb/>
ef bifariam <expan abbr="diuid&atilde;tur">diuidantur</expan> in punctis ghk: per diui&longs;iones au&shy;<lb/>
<arrow.to.target n="marg30"/><lb/>
tem planum ducatur; cuius &longs;ectio figura ghK. <!-- KEEP S--></s>
          <s id="s.000231">erit linea <lb/>
gh &aelig;quidi&longs;tans lineis ac, bd &amp; hk ip&longs;is ce, df. </s>
          <s id="s.000232">quare ex <lb/>
decimaquinta undecimi elementorum, planum illud pla<lb/>
nis ace, bdf &aelig;quidi&longs;tabit, &amp; faciet &longs;ectionem figu&shy;<lb/>
<arrow.to.target n="marg31"/><lb/>
ram ip&longs;is &aelig;qualem, &amp; &longs;imilem, ut proxime demon&longs;tra&shy;<lb/>
uimus. </s>
          <s id="s.000233">Dico centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>
ghk. </s>
          <s id="s.000234">Si enim fieri pote&longs;t, &longs;it eius centrum l: &amp; ducatur <lb/>
lm u&longs;que ad planum ghk, qu&aelig; ip&longs;i ab &aelig;quidi&longs;tet. </s>
          <pb xlink:href="023/01/024.jpg"/>
          <s id="s.000235"><arrow.to.target n="marg32"/>ergo linea ag continenter in duas partes &aelig;quales diui&shy;<lb/>
&longs;a, relinquetur <expan abbr="t&atilde;dem">tandem</expan> pars aliqua ng, qu&aelig; minor erit lm. </s>
          <lb/>
          <s id="s.000236">Vtraque uero linearum ag, gb diuidatur in partes &aelig;qua&shy;<lb/>
les ip&longs;i ng: &amp; per puncta diui&longs;ionum plana oppo&longs;itis pla&shy;<lb/>
<arrow.to.target n="marg33"/><lb/>
nis &aelig;quidi&longs;tantia ducantur. </s>
          <s id="s.000237">erunt &longs;ectiones figur&aelig; &aelig;qua&shy;<lb/>
les, ac &longs;imiles ip&longs;is ace, bdf: &amp; totum pri&longs;ma diui&longs;um erit <lb/>
in pri&longs;mata &aelig;qualia, &amp; &longs;imilia: qu&aelig; cum inter &longs;e <expan abbr="congru&atilde;t">congruant</expan>; <lb/>
&amp; grauitatis centra &longs;ibi ip&longs;is congruentia, <expan abbr="re&longs;pondentiaq;">re&longs;pondentiaque</expan> <lb/>
<figure id="id.023.01.024.1.jpg" xlink:href="023/01/024/1.jpg"/><lb/>
habebunt. </s>
          <s id="s.000238"><expan abbr="Itaq:">Itaque</expan> <lb/>
&longs;unt magnitudi&shy;<lb/>
nes <expan abbr="qu&aelig;d&atilde;">qu&aelig;dam</expan> &aelig;qua&shy;<lb/>
les ip&longs;i nh, &amp; nu&shy;<lb/>
mero pares, qua&shy;<lb/>
rum centra gra&shy;<lb/>
uitatis in <expan abbr="ead&etilde;re">eadem</expan> re<lb/>
cta linea con&longs;ti&shy;<lb/>
tuuntur: du&aelig; ue&shy;<lb/>
ro medi&aelig; &aelig;qua&shy;<lb/>
les &longs;unt: &amp; qu&aelig; ex <lb/>
utraque parte i&shy;<lb/>
p&longs;arum &longs;imili&shy;<lb/>
ter &aelig;quales: &amp; &aelig;&shy;<lb/>
quales rect&aelig; li&shy;<lb/>
ne&aelig;, qu&aelig; inter <lb/>
grauitatis centra <lb/>
interiiciuntur. </s>
          <lb/>
          <s id="s.000239">quare ex corolla&shy;<lb/>
rio quint&aelig; pro&shy;<lb/>
po&longs;itionis primi <lb/>
libri Archimedis <lb/>
de centro graui&shy;<lb/>
tatis planorum; magnitudinis ex his omnibus compo&longs;it&aelig; <lb/>
centrum grauitatis e&longs;t in medio line&aelig;, qu&aelig; magnitudi&shy;<lb/>
num mediarum centra coniungit. </s>
          <s id="s.000240">at qui non ita res ha&shy;



<pb pagenum="9" xlink:href="023/01/025.jpg"/>bet, &longs;i quidem l extra medias magnitudines po&longs;itum e&longs;t. </s>
          <lb/>
          <s id="s.000241">Con&longs;tat igitur centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>
<figure id="id.023.01.025.1.jpg" xlink:href="023/01/025/1.jpg"/><lb/>
ghk, quod nos demon&longs;trandum propo&longs;uimus. </s>
          <s id="s.000242">At &longs;i op&shy;<lb/>
po&longs;ita plana in pri&longs;mate &longs;int quadrilatera, uel plurilatera, <lb/>
eadem erit in omnibus demon&longs;tratio.</s>
        </p>
        <p type="margin">
          <s id="s.000243"><margin.target id="marg30"/>33. primi</s>
        </p>
        <p type="margin">
          <s id="s.000244"><margin.target id="marg31"/>5. huius</s>
        </p>
        <p type="margin">
          <s id="s.000245"><margin.target id="marg32"/>1. decimi</s>
        </p>
        <p type="margin">
          <s id="s.000246"><margin.target id="marg33"/>5 huius</s>
        </p>
        <p type="head">
          <s id="s.000247">THEOREMA VII. PROPOSITIO VII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000248">Cuiuslibet cylindri, &amp; cuiuslibet cylindri por<lb/>
tionis centrum grauitatis e&longs;t in plano, quod ba&longs;i&shy;<lb/>
bus &aelig;quidi&longs;tans, parallelogrammi per axem late&shy;<lb/>
ra bifariam &longs;ecat.</s>
        </p>
        <pb xlink:href="023/01/026.jpg"/>
        <p type="main">
          <s id="s.000249">SIT cylindrus, uel cylindri portio ac: &amp; plano per a&shy;<lb/>
xem ducto &longs;ecetur; cuius &longs;ectio &longs;it parallelogrammum ab<lb/>
cd: &amp; bifariam diui&longs;is ad, bc parallelogrammi lateribus, <lb/>
per diui&longs;ionum puncta ef planum ba&longs;i &aelig;quidi&longs;tans duca&shy;<lb/>
tur; quod faciet &longs;ectionem, in cylindro quidem circulum <lb/>
&aelig;qualem iis, qui &longs;unt in ba&longs;ibus, ut demon&longs;trauit Serenus <lb/>
in libro cylindricorum, propo&longs;itione quinta: in cylindri <lb/>
uero portione ellip&longs;im &aelig;qualem, &amp; &longs;imilem eis, qu&aelig; &longs;unt <lb/>
<figure id="id.023.01.026.1.jpg" xlink:href="023/01/026/1.jpg"/><lb/>
in oppo&longs;itis planis, quod nos <lb/>demon&longs;trauimus in commen<lb/>
tariis in librum Archimedis <lb/>
de conoidibus, &amp; &longs;ph&aelig;roidi&shy;<lb/>
bus. </s>
          <s id="s.000250">Dico centrum grauita&shy;<lb/>
tis cylindri, uel cylindri por&shy;<lb/>
tionis e&longs;&longs;e in plano ef. </s>
          <s id="s.000251">Si <expan abbr="en&itilde;">enim</expan> <lb/>
fieri pote&longs;t, fit centrum g: &amp; <lb/>
ducatur gh ip&longs;i ad &aelig;quidi&shy;<lb/>
&longs;tans, u&longs;que ad ef planum. </s>
          <lb/>
          <s id="s.000252">Itaque linea ae continenter <lb/>
diui&longs;a bifariam, erit tandem <lb/>
pars aliqua ip&longs;ius ke, minor <lb/>
gh. </s>
          <s id="s.000253">Diuidantur ergo line&aelig; <lb/>
ae, ed in partes &aelig;quales ip&longs;i <lb/>
ke: &amp; per diui&longs;iones plana ba<lb/>
&longs;ibus &aelig;quidi&longs;tantia <expan abbr="duc&atilde;tur">ducantur</expan>. </s>
          <lb/>
          <s id="s.000254">erunt iam &longs;ectiones, figur&aelig; &aelig;&shy;<lb/>
quales, &amp; &longs;imiles eis, qu&aelig; &longs;unt <lb/>
in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: &amp; cy<lb/>
lindri portio in portiones &aelig;quales, &amp; &longs;imiles ip&longs;i kf. </s>
          <s id="s.000255">reli&shy;<lb/>
qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s>
        </p>
        <pb pagenum="10" xlink:href="023/01/027.jpg"/>
        <figure id="id.023.01.027.1.jpg" xlink:href="023/01/027/1.jpg"/>
        <p type="head">
          <s id="s.000256">THEOREMA VIII. PROPOSITIO VIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000257">Cuiuslibet pri&longs;matis, &amp; cuiuslibet cylindri, uel <lb/>
cylindri portionis grauitatis centrum in medio <lb/>
ip&longs;ius axis con&longs;i&longs;tit.</s>
        </p>
        <p type="main">
          <s id="s.000258">Sit primum af pri&longs;ma &aelig;quidi&longs;tantibus planis <expan abbr="content&utilde;">contentum</expan>, <lb/>
quod &longs;olidum parallelepipedum appellatur: &amp; oppo&longs;ito&shy;<lb/>
rum planorum cf, ah, da, fg latera bifariam diuidantur in <lb/>
punctis klmnopqrstux: &amp; per diui&longs;iones ducantur <lb/>
plana kn, or, sx. </s>
          <s id="s.000259">communes autem eorum planorum &longs;e&shy;<lb/>
ctiones &longs;int line&aelig; yz, <foreign lang="greek">qf, xy:</foreign> qu&aelig; in puncto <foreign lang="greek">w</foreign> <expan abbr="conueni&atilde;t">conueniant</expan>. </s>
          <lb/>
          <s id="s.000260">erit ex decima eiu&longs;dem libri Archimedis parallelogrammi <lb/>
cf centrum grauitatis punctum y; parallelogrammi ah 



<pb xlink:href="023/01/028.jpg"/>centrum z: parallelogrammi ad, <foreign lang="greek">q:</foreign> parallelogrammi fg, <foreign lang="greek">f:</foreign><lb/>
<figure id="id.023.01.028.1.jpg" xlink:href="023/01/028/1.jpg"/><lb/>
parallelogrammi dh, <foreign lang="greek">x:</foreign> &amp; <lb/>
parallelogrammi cg <expan abbr="centr&utilde;">centrum</expan> <lb/>
<foreign lang="greek">y:</foreign> atque erit <foreign lang="greek">w</foreign> punctum me <lb/>
dium uniu&longs;cuiu&longs;que axis, ui <lb/>
delicet eius line&aelig; qu&aelig; oppo <lb/>
&longs;itorum <expan abbr="planor&utilde;">planorum</expan> centra con <lb/>
iungit. </s>
          <s id="s.000261">Dico <foreign lang="greek">w</foreign> centrum e&longs;&longs;e <lb/>
grauitatis ip&longs;ius &longs;olidi. </s>
          <s id="s.000262">e&longs;t <lb/>
<arrow.to.target n="marg34"/><lb/>
enim, ut demon&longs;trauimus, <lb/>
&longs;olidi af centrum grauitatis <lb/>
in plano Kn; quod oppo&longs;i&shy;<lb/>
tis planis ad, gf &aelig;quidi&longs;tans <lb/>
reliquorum planorum late&shy;<lb/>
ra bifariam diuidit: &amp; &longs;imili <lb/>
ratione idem centrum e&longs;t in plano or, &aelig;quidi&longs;tante planis <lb/>
ae, bf oppo&longs;itis. </s>
          <s id="s.000263">ergo in communi ip&longs;orum &longs;ectione: ui&shy;<lb/>
delicet in linea yz. </s>
          <s id="s.000264">Sed e&longs;t etiam in plano tu, quod <expan abbr="quid&etilde;">quidem</expan> <lb/>
yz &longs;ecatin <foreign lang="greek">w.</foreign> Con&longs;tat igitur centrum grauitatis &longs;olidi e&longs;&longs;e <lb/>
punctum <foreign lang="greek">w,</foreign> medium &longs;cilicet axium, hoc e&longs;t linearum, qu&aelig; <lb/>
planorum oppo&longs;itorum centra coniungunt.</s>
        </p>
        <p type="margin">
          <s id="s.000265"><margin.target id="marg34"/>6 huius</s>
        </p>
        <p type="main">
          <s id="s.000266">Sit aliud prima af; &amp; in eo plana, qu&aelig; opponuntur, tri&shy;<lb/>
angula abc, def: <expan abbr="diui&longs;isq;">diui&longs;isque</expan> bifariam parallelogrammorum <lb/>
lateribus ad, be, cf in punctis ghk, per diui&longs;iones <expan abbr="plan&utilde;">planum</expan> <lb/>
ducatur, quod oppo&longs;itis planis &aelig;quidi&longs;tans faciet <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> <lb/>
triangulum ghx &aelig;quale, &amp; &longs;imile ip&longs;is abc, def. </s>
          <s id="s.000267">Rur&longs;us <lb/>
diuidatur ab bifariam in l: &amp; iuncta cl per ip&longs;am, &amp; per <lb/>
cKf planum ducatur pri&longs;ma &longs;ecans, cuius, &amp; <expan abbr="parallelogr&atilde;">parallelogram</expan><lb/>
mi ae communis &longs;ectio &longs;it lmn. </s>
          <s id="s.000268">diuidet punctum m li&shy;<lb/>
neam gh bifariam; &amp; ita n diuidet lineam de: quoniam <lb/>
<arrow.to.target n="marg35"/><lb/>
triangula acl, gkm, dfn &aelig;qualia &longs;unt, &amp; &longs;imilia, ut &longs;upra <lb/>
demon&longs;trauimus. </s>
          <s id="s.000269">Iam ex iis, qu&aelig; tradita &longs;unt, con&longs;tat cen<lb/>
trum grauitatis pri&longs;matis in plano ghk contineri. </s>
          <s id="s.000270">Dico <lb/>
ip&longs;um e&longs;&longs;e in linea km. </s>
          <s id="s.000271">Si enim fieri pote&longs;t, &longs;it o centrum; 



<pb pagenum="11" xlink:href="023/01/029.jpg"/>&amp; per o ducatur op ad km ip&longs;i hg &aelig;quidi&longs;tans. </s>
          <s id="s.000272">Itaque li<lb/>
nea hm <expan abbr="bifari&atilde;">bifariam</expan> u&longs;que e&ograve; diuidatur, quoad reliqua &longs;it pars <lb/>
qu&aelig;dam qm, minor op. </s>
          <s id="s.000273">deinde hm, mg diuidantur in <lb/>
partes &aelig;quales ip&longs;i mq: &amp; per diui&longs;iones line&aelig; ip&longs;i mK <lb/>
&aelig;quidi&longs;tantes ducantur. </s>
          <s id="s.000274">puncta uero, in quibus h&aelig; trian&shy;<lb/>
gulorum latera &longs;ecant, coniungantur ductis lineis rs, tu, <lb/>
<figure id="id.023.01.029.1.jpg" xlink:href="023/01/029/1.jpg"/><lb/>
xy; qu&aelig; ba&longs;i gh &aelig;quidi&longs;tabunt. </s>
          <s id="s.000275">Quoniam enim line&aelig; gz, <lb/>
h<foreign lang="greek">a</foreign> &longs;unt &aelig;quales: <expan abbr="itemq;">itemque</expan> &aelig;quales gm, mh: ut mg ad gz, <lb/>
ita erit mh, ad h<foreign lang="greek">a:</foreign> &amp; diuidendo, ut mz ad zg, ita m<foreign lang="greek">a</foreign> ad <lb/>
<arrow.to.target n="marg36"/><lb/>
<foreign lang="greek">a</foreign>h. </s>
          <s id="s.000276">Sed ut mz ad zg, ita kr ad rg: &amp; ut m<foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign>h, ita ks <lb/>
ad sh. </s>
          <s id="s.000277">quare ut kr ad rg, ita ks ad sh. </s>
          <s id="s.000278">&aelig;quidi&longs;tant igitur <lb/>
<arrow.to.target n="marg37"/><lb/>
inter &longs;e &longs;e rs, gh. </s>
          <s id="s.000279">eadem quoque ratione demon&longs;trabimus 



<pb xlink:href="023/01/030.jpg"/>tu, xy ip&longs;i gh &aelig;quidi&longs;tare. </s>
          <s id="s.000280">Et quoniam triangula, qu&aelig; <lb/>
fiunt &agrave; lineis Ky, yu, us, sh &aelig;qualia &longs;unt inter &longs;e, &amp; &longs;imilia <lb/>
<arrow.to.target n="marg38"/><lb/>
triangulo Kmh: habebit triangulum Kmh ad <expan abbr="triangul&utilde;">triangulum</expan> <lb/>
K<foreign lang="greek">d</foreign>y duplam proportionem eius, qu&aelig; e&longs;t line&aelig; kh ad Ky. </s>
          <lb/>
          <s id="s.000281">&longs;ed Kh po&longs;ita e&longs;t quadrupla ip&longs;ius ky. </s>
          <s id="s.000282">ergo triangulum <lb/>
kmh ad triangulum K<foreign lang="greek">d</foreign>y <expan abbr="e&atilde;dem">eandem</expan> proportionem habebit, <lb/>
quam &longs;exdecim ad <expan abbr="un&utilde;">unum</expan>: &amp; ad quatuor triangula k<foreign lang="greek">d</foreign>y, yu, <lb/>
us, s<foreign lang="greek">a</foreign>h habebit eandem, quam &longs;exdecim ad quatuor, hoc <lb/>e&longs;t quam hK ad ky: &amp; &longs;imiliter eandem habere demon&longs;tra<lb/>
<figure id="id.023.01.030.1.jpg" xlink:href="023/01/030/1.jpg"/><lb/>
bitur trian&shy;<lb/>
gulum kmg <lb/>
ad quatuor <lb/>
<expan abbr="tri&atilde;gula">triangula</expan> K<foreign lang="greek">d</foreign><lb/>
x, x<foreign lang="greek">g</foreign>t, t<foreign lang="greek">b</foreign>r, <lb/>
<arrow.to.target n="marg39"/><lb/>
rzg. <!-- KEEP S--></s>
          <s id="s.000283">quare <lb/>totum trian<lb/>
gulum Kgh <lb/>
ad omnia tri <lb/>
angula gzr, <lb/>
r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign><lb/>
K, K<foreign lang="greek">d</foreign>y, yu, <lb/>
us, s<foreign lang="greek">a</foreign>h ita <lb/>
erit, ut hk ad <lb/>
ky, hoc e&longs;t <lb/>
ut hm ad m<lb/>
q. </s>
          <s id="s.000284">Si igitur in <lb/>
triangulis abc, def de&longs;cribantur figur&aelig; &longs;imiles ei, qu&aelig; de&shy;<lb/>
&longs;cripta e&longs;t in ghK triangulo: &amp; per lineas &longs;ibi re&longs;ponden&shy;<lb/>
tes plana ducantur: totum pri&longs;ma af diui&longs;um erit in tria <lb/>
&longs;olida parallelepipeda y<foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz, quorum ba&longs;es &longs;unt &aelig;qua <lb/>
les &amp; &longs;imiles ip&longs;is parallelogrammis y <foreign lang="greek">g,</foreign>u<foreign lang="greek">b,</foreign> sz: &amp; in octo <lb/>
pri&longs;mata gzr, r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign><lb/>
K, k<foreign lang="greek">d</foreign>y, yu, us, s<foreign lang="greek">a</foreign>h: quorum <lb/>
item ba&longs;es &aelig;quales, &amp; &longs;imiles &longs;unt dictis triangulis; altitu&shy;<lb/>
do autem in omnibus, totius pri&longs;matis altitudini &aelig;qualis.



<pb pagenum="12" xlink:href="023/01/031.jpg"/>Itaque &longs;olidi parallelepipedi y<foreign lang="greek">g</foreign> centrum grauitatis e&longs;t in <lb/>
linea <foreign lang="greek">de:</foreign> &longs;olidi u<foreign lang="greek">b</foreign> centrum e&longs;t in linea <foreign lang="greek">eh:</foreign> &amp; &longs;olidi sz in li<lb/>
nea <foreign lang="greek">h</foreign>m, qu&aelig; quidem line&aelig; axes &longs;unt, cum planorum oppo<lb/>
&longs;itorum centra coniungant. </s>
          <s id="s.000285">ergo magnitudinis ex his &longs;oli <lb/>
dis compo&longs;it&aelig; centrum grauitatis e&longs;t in linea <foreign lang="greek">d</foreign>m, quod &longs;it <lb/>
<foreign lang="greek">q</foreign>; &amp; iuncta <foreign lang="greek">q</foreign>o producatur: &agrave; puncto autem h ducatur h<foreign lang="greek">a</foreign><lb/>
ip&longs;i mk &aelig;quidi&longs;tans, qu&aelig; cum <foreign lang="greek">q</foreign>o in <foreign lang="greek">m</foreign> conueniat. </s>
          <s id="s.000286">triangu<lb/>
lum igitur ghk ad omnia triangula gzr,  <foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign>k, <lb/>
k<foreign lang="greek">d</foreign>y, yu, us, s<foreign lang="greek">a</foreign>h eandem habet proportionem, quam hm <lb/>
ad mq ; hoc e&longs;t, quam <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql:</foreign> nam &longs;i hm, <foreign lang="greek">mq</foreign> produci in<lb/>
telligantur, quou&longs;que coeant; erit ob linearum qy, mk &aelig;&shy;<lb/>
quidi&longs;tantiam, ut hq ad qm, ita <foreign lang="greek">ml</foreign> ad ad <foreign lang="greek">lq:</foreign> &amp; componen <lb/>
do, ut hm ad mq, ita <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql.</foreign></s>
          <s id="s.000287"> linea uero <foreign lang="greek">q</foreign>o maior e&longs;t, <lb/>
<arrow.to.target n="marg40"/><lb/>
qu&agrave;m <foreign lang="greek">ql:</foreign> habebit igitur <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql</foreign> maiorem proportio&shy;<lb/>
nem, qu&agrave;m ad <foreign lang="greek">q</foreign>o. </s>
          <s id="s.000288">quare triangulum etiam ghk ad omnia <lb/>
iam dicta triangula maiorem <expan abbr="proportion&etilde;">proportionem</expan> habebit, qu&agrave;m <lb/>
<foreign lang="greek">mq</foreign> ad <foreign lang="greek">q</foreign>o. </s>
          <s id="s.000289">&longs;ed ut <expan abbr="triangul&utilde;">triangulum</expan> ghk ad omnia triangula, ita <expan abbr="to-t&utilde;">to&shy;<lb/>
tum</expan> pri&longs;ma afad omnia pri&longs;mata gzr, r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">dk, kd</foreign> y, <lb/>
yu, us, s<foreign lang="greek">a</foreign>h: quoniam enim &longs;olida parallelepipeda &aelig;que al<lb/>
ta, eandem inter &longs;e proportionem habent, quam ba&longs;es; ut <lb/>
ex trige&longs;ima&longs;ecunda undecimi elementorum con&longs;tat. </s>
          <s id="s.000290">&longs;unt <lb/>
<arrow.to.target n="marg41"/><lb/>
autem &longs;olida parallelepipeda pri&longs;matum triangulares ba&shy;<lb/>
<arrow.to.target n="marg42"/><lb/>
&longs;es habentium dupla: &longs;equitur, ut etiam huiu&longs;modi pri&longs;&shy;<lb/>
mata inter &longs;e &longs;int, &longs;icut eorum ba&longs;es. </s>
          <s id="s.000291">ergo totum pri&longs;ma ad <lb/>
omnia pri&longs;mata maiorem proportionem habet, quam <foreign lang="greek">mq</foreign><lb/>
<arrow.to.target n="marg43"/><lb/>
ad <foreign lang="greek">q</foreign>o: &amp; diuidendo &longs;olida parallelepipeda y<foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz ad o&shy;<lb/>
mnia pri&longs;mata proportionem habent maiorem, qu&agrave;m <foreign lang="greek">m</foreign>o <lb/>
ad o<foreign lang="greek">q</foreign>. </s>
          <s id="s.000292">fiat <foreign lang="greek">n</foreign>o ad o<foreign lang="greek">q,</foreign> ut &longs;olida parallelepipeda y <foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz ad <lb/>
omnia pri&longs;mata. </s>
          <s id="s.000293">Itaque cum &agrave; pri&longs;mate af, cuius <expan abbr="c&etilde;trum">centrum</expan> <lb/>
grauitatis e&longs;t o, auferatur magnitudo ex &longs;olidis parallelepi<lb/>
pedis y <foreign lang="greek">g,</foreign>u<foreign lang="greek">b,</foreign>sz con&longs;tans: atque ip&longs;ius grauitatis centrum <lb/>
&longs;it <foreign lang="greek">q:</foreign> reliqu&aelig; magnitudinis, qu&aelig; ex omnibus pri&longs;matibus <lb/>
con&longs;tat, grauitatis centrum erit in linea <foreign lang="greek">q</foreign> o producta: &amp; <lb/>
in puncto <foreign lang="greek">v</foreign>, ex octava propo&longs;itione eiusdem libri Archi&shy;



<pb xlink:href="023/01/032.jpg"/>medis. </s>
          <s id="s.000294">ergo punctum <foreign lang="greek">n</foreign> extra pri&longs;ma af po&longs;itum, <expan abbr="centr&utilde;">centrum</expan> <lb/>
erit magnitudinis <expan abbr="c&otilde;po&longs;it&aelig;">compo&longs;it&aelig;</expan> ex omnibus pri&longs;matibus gzr, <lb/>
r <foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign>k, k<foreign lang="greek">d</foreign> y, yu, us, s<foreign lang="greek">a</foreign>h, quod fieri nullo modo po<lb/>
te&longs;t. </s>
          <s id="s.000295">e&longs;t enim ex diffinitione centrum grauitatis &longs;olid&aelig; figu<lb/>
r&aelig; intra ip&longs;am po&longs;itum, non extra. </s>
          <s id="s.000296">quare relinquitur, ut <expan abbr="c&etilde;trum">cen<lb/>
trum</expan> grauitatis pri&longs;matis &longs;it in linea Km. </s>
          <s id="s.000297">Rur&longs;us bc bifa&shy;<lb/>
riam in diuidatur: &amp; ducta a<foreign lang="greek">x,</foreign> per ip&longs;am, &amp; per lineam <lb/>agd planum ducatur; quod pri&longs;ma &longs;ecet: <expan abbr="faciatq;">faciatque</expan> in paral<lb/>
lelogrammo bf &longs;ectionem <foreign lang="greek">x p</foreign> diuidet punctum <foreign lang="greek">p</foreign> lineam <lb/>
quoque cf bifariam: &amp; erit plani eius, &amp; trianguli ghK <lb/>
communis &longs;ectio gu; qu&ograve;d <expan abbr="p&utilde;ctum">punctum</expan> u in medio line&aelig; hK <lb/>
<figure id="id.023.01.032.1.jpg" xlink:href="023/01/032/1.jpg"/><lb/>
po&longs;itum &longs;it. </s>
          <s id="s.000298">Similiter demon&longs;trabimus centrum grauita&shy;<lb/>
tis pri&longs;matis in ip&longs;a gu ine&longs;&longs;e. </s>
          <s id="s.000299">&longs;it autem planorum cfnl, <lb/>
ad<foreign lang="greek">px</foreign> communis &longs;ectio linea <foreign lang="greek">rst;</foreign> qu&aelig; quidem pri&longs;matis <lb/>
axis erit, cum tran&longs;eat per centra grauitatis triangulorum <lb/>
abc, ghk def, ex quartadecima eiu&longs;dem. </s>
          <s id="s.000300">ergo centrum <lb/>
grauitatis pri&longs;matis af e&longs;t punctum <foreign lang="greek">s,</foreign> centrum &longs;cilicet 



<pb pagenum="13" xlink:href="023/01/033.jpg"/>trianguli ghK, &amp; ip&longs;ius <foreign lang="greek">rt</foreign> axis medium.</s>
        </p>
        <p type="margin">
          <s id="s.000301"><margin.target id="marg35"/>5.huius</s>
        </p>
        <p type="margin">
          <s id="s.000302"><margin.target id="marg36"/>2. &longs;exti.<lb/>
 12 quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000303"><margin.target id="marg37"/>2. &longs;exti.</s>
        </p>
        <p type="margin">
          <s id="s.000304"><margin.target id="marg38"/>
19. &longs;exti</s>
        </p>
        <p type="margin">
          <s id="s.000305"><margin.target id="marg39"/>2. uel 12. <lb/>
quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000306"><margin.target id="marg40"/>8. quinti.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000307"><margin.target id="marg41"/>28. unde<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000308"><margin.target id="marg42"/>15. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000309"><margin.target id="marg43"/>19. quinti<lb/>
apud <expan abbr="C&atilde;panum">Cam<lb/>
panum</expan></s>
        </p>
        <p type="main">
          <s id="s.000310">Sit pri&longs;ma ag, cuius oppo&longs;ita plana &longs;int quadrilatera <lb/>
abcd, efgh: <expan abbr="&longs;ecenturq;">&longs;ecenturque</expan> ac, bf, cg, dh bifariam: &amp; per di&shy;<lb/>
ui&longs;iones planum ducatur; quod &longs;ectionem faciat quadrila&shy;<lb/>
terum Klmn. </s>
          <s id="s.000311">Deinde iuncta ac per lineas ac, ae ducatur <lb/>
planum <expan abbr="&longs;ec&atilde;s">&longs;ecans</expan> pri&longs;ma, quod ip&longs;um diuidet in duo pri&longs;mata <lb/>
triangulares ba&longs;es habentia abcefg, adcehg. </s>
          <s id="s.000312">Sint <expan abbr="aut&etilde;">autem</expan> <lb/>
<figure id="id.023.01.033.1.jpg" xlink:href="023/01/033/1.jpg"/><lb/>
triangulorum abc, efg gra&shy;<lb/>
uitatis centra op: &amp; triangu&shy;<lb/>
lorum adc, ehg centra qr: <lb/>
<expan abbr="iunganturq;">iunganturque</expan> op, qr; qu&aelig; pla&shy;<lb/>
no klmn occurrant in pun&shy;<lb/>
ctis st. </s>
          <s id="s.000313">erit ex iis, qu&aelig; demon<lb/>
&longs;trauimus, punctum s grauita<lb/>
tis centrum trianguli klm; &amp; <lb/>
ip&longs;ius pri&longs;matis abcefg: pun<lb/>
ctum uero t centrum grauita <lb/>
tis trianguli Knm, &amp; pri&longs;ma&shy;<lb/>
tis adc, ehg. <!-- KEEP S--></s>
          <s id="s.000314">iunctis igitur <lb/>
oq, pr, st, erit in linea oq <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis quadrilateri <lb/>
abcd, quod &longs;it u: &amp; in linea <lb/>
pr <expan abbr="c&etilde;trum">centrum</expan> quadrilateri efgh <lb/>
&longs;it autem x. </s>
          <s id="s.000315">denique iungatur <lb/>
u x, qu&aelig; &longs;ecet lineam &longs; t in y. </s>
          <s id="s.000316">&longs;e<lb/>
cabit enim cum &longs;int in eodem <lb/>
<arrow.to.target n="marg44"/><lb/>
plano: <expan abbr="atq;">atque</expan> erit y grauitatis centrum quadrilateri Klmn. </s>
          <lb/>
          <s id="s.000317">Dico idem punctum y centrum quoque gra uitatis e&longs;&longs;e to&shy;<lb/>
tius pri&longs;matis. </s>
          <s id="s.000318">Quoniam enim quadrilateri klmn graui&shy;<lb/>
tatis centrum e&longs;t y: linea sy ad yt ean dem proportionem <lb/>
habebit, quam triangulum knm ad triangulum klm, ex 8 <lb/>
Archimedis de centro grauitatis planorum. </s>
          <s id="s.000319">Vt autem <expan abbr="tri&atilde;">trian</expan><lb/>
gulum knm ad ip&longs;um klm, hoc e&longs;t ut triangulum adc ad <lb/>
triangulum abc, &aelig;qualia enim &longs;unt, ita pri&longs;ma adcehg 



<pb xlink:href="023/01/034.jpg"/>ad pri&longs;ma abcefg. <!-- KEEP S--></s>
          <s id="s.000320">quare linea sy ad yt eandem propor&shy;<lb/>
tionem habet, quam pri&longs;ma adcehg ad pri&longs;ma abcefg. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000321">Sed pri&longs;matis abcefg centrum grauitatis e&longs;t s: &amp; pri&longs;ma&shy;<lb/>
tis adcehg centrum t. </s>
          <s id="s.000322">magnitudinis igitur ex his compo<lb/>
&longs;it&aelig; hoc e&longs;t totius pri&longs;matis ag centrum grauitatis e&longs;t pun<lb/>
ctum y; medium &longs;cilicet axis ux, qui oppo&longs;itorum plano&shy;<lb/>
rum centra coniungit.</s>
        </p>
        <p type="margin">
          <s id="s.000323"><margin.target id="marg44"/>5. huius/&gt;</s>
        </p>
        <p type="main">
          <s id="s.000324">Rur&longs;us &longs;it pri&longs;ma ba&longs;im habens pentagonum abcde: <lb/>
&amp; quod ei opponitur &longs;it fghKl: &longs;ec<expan abbr="enturq;">enturque</expan> af, bg, ch, <lb/>
dk, el bifariam: &amp; per diui&longs;iones ducto plano, &longs;ectio &longs;it <expan abbr="p&etilde;">pen</expan><lb/>
<expan abbr="tagon&utilde;">tagonum</expan> mnopq. deinde iuncta eb per lineas le, eb aliud <lb/>
<figure id="id.023.01.034.1.jpg" xlink:href="023/01/034/1.jpg"/><lb/>
planum ducatur, <expan abbr="diuid&etilde;s">diuidens</expan> pri&longs;<lb/>
ma ak in duo pri&longs;mata; in pri&longs;<lb/>
ma &longs;cilicet al, cuius plana op&shy;<lb/>
po&longs;ita &longs;int triangula abe fgl: <lb/>
&amp; in prima bk cuius plana op<lb/>
po&longs;ita &longs;int quadrilatera bcde <lb/>
ghkl. <!-- KEEP S--></s>
          <s id="s.000325">Sint autem triangulo&shy;<lb/>rum abe, fgl centra grauita<lb/>
tis puncta r &longs;: &amp; bcde, ghkl <lb/>
quadrilaterorum centra tu: <lb/>
<expan abbr="iunganturq;">iunganturque</expan> rs, tu occurren&shy;<lb/>
tes plano mnopq in punctis <lb/>
xy. </s>
          <s id="s.000326">&amp; itidem <expan abbr="iung&atilde;tur">iungantur</expan> rt, &longs;u, <lb/>
xy. </s>
          <s id="s.000327">erit in linea rt <expan abbr="c&etilde;trum">centrum</expan> gra<lb/>
uitatis pentagoni abcde; <lb/>
quod &longs;it z: &amp; in linea &longs;u cen&shy;<lb/>
trum pentagoni fghkl :&longs;it au <lb/>
tem <foreign lang="greek">x:</foreign> &amp; ducatur z<foreign lang="greek">x,</foreign> qu&aelig; di&shy;<lb/>
cto plano in <foreign lang="greek">y</foreign> occurrat. </s>
          <s id="s.000328"><expan abbr="Itaq;">Itaque</expan> <lb/>
punctum x e&longs;t centrum graui <lb/>
tatis trianguli mnq, ac pri&longs;&shy;<lb/>
matis al: &amp; y grauitatis centrum quadrilateri nopq, ac <lb/>
pri&longs;matis bk. </s>
          <s id="s.000329">quare y centrum erit pentagoni mnopq. </s>
          <s id="s.000330"> &amp; 



<pb pagenum="14" xlink:href="023/01/035.jpg"/>&longs;imiliter demon&longs;trabitur totius pri&longs;matis aK grauitatis ef <lb/>
&longs;e centrum. </s>
          <s id="s.000331">Simili ratione &amp; in aliis pri&longs;matibus illud <lb/>
idem facile demon&longs;trabitur. </s>
          <s id="s.000332">Quo autem pacto in omni <lb/>
figura rectilinea centrum grauitatis inueniatur, docuimus <lb/>
in commentariis in &longs;extam propo&longs;itionem Archimedis de <lb/>
quadratura parabol&aelig;.</s>
        </p>
        <p type="main">
          <s id="s.000333">Sit cylindrus, uel cylindri portio ce cuius axis ab: &longs;ece&shy;<lb/>
<expan abbr="turq,">turque</expan> plano per axem ducto; quod &longs;ectionem faciat paral&shy;<lb/>
lelogrammum cdef: &amp; diui&longs;is cf, de bifariam in punctis <lb/>
<figure id="id.023.01.035.1.jpg" xlink:href="023/01/035/1.jpg"/><lb/>
gh, per ea ducatur planum ba&longs;i &aelig;quidi&longs;tans. </s>
          <s id="s.000334">erit &longs;ectio gh <lb/>
circulus, uel ellip&longs;is, centrum habens in axe; quod &longs;it K at&shy;<lb/>
<arrow.to.target n="marg45"/><lb/>
que erunt ex iis, qu&aelig; demon&longs;trauimus, centra grauitatis <lb/>
planorum oppo&longs;itorum puncta ab: &amp; plani gh ip&longs;um k in <lb/>
quo quidem plano e&longs;t centrum grauitatis cylindri, uel cy&shy;<lb/>
lindri portionis. </s>
          <s id="s.000335">Dico punctum K cylindri quoque, uel cy<lb/>
lindri portionis grauitatis centrum e&longs;&longs;e. </s>
          <s id="s.000336">Si enim fieri po&shy;<lb/>
te&longs;t, &longs;it l centrum: <expan abbr="ducaturq;">ducaturque</expan> kl, &amp; extra figuram in m pro&shy;<lb/>
ducatur. </s>
          <s id="s.000337">quam ucro proportionem habet linea mK ad kl 



<pb xlink:href="023/01/036.jpg"/>habeat circulus, uel ellip&longs;is gh ad aliud &longs;pacium, in quo u: <lb/>
&amp; in cit culo, uel ellip&longs;i plane de&longs;cribatur rectilinea figura, <lb/>
ita ut <expan abbr="t&atilde;dem">tandem</expan> <expan abbr="relinqu&atilde;tur">relinquantur</expan> portiones minores &longs;pacio u, qu&aelig; <lb/>
&longs;it opgqrsht: <expan abbr="de&longs;criptaq;">de&longs;criptaque</expan> &longs;imili figura in oppo&longs;itis pla&shy;<lb/>
nis cd, fe, per lineas &longs;ibi ip&longs;is re&longs;pondentes plana <expan abbr="duc&atilde;tur">ducantur</expan>. </s>
          <lb/>
          <s id="s.000338">Itaque cylindrus, uel cylindri portio diuiditur in pri&longs;ma, <lb/>
cuius quidem ba&longs;is e&longs;t figura rectilinea iam dicta, centrum <lb/>que grauitatis punctum K: &amp; in multa &longs;olida, qu&aelig; pro ba&longs;i<lb/>
bus habent relictas portiones, quas nos &longs;olidas portiones <lb/>
appellabimus. </s>
          <s id="s.000339">cum igitur portiones &longs;int minores &longs;pacio <lb/>
u, circulus, uel ellip&longs;is gh ad portiones maiorem propor&shy;<lb/>
tionem habebit, qu&agrave;m linea mk ad Kl. <!-- KEEP S--></s>
          <s id="s.000340">fiat nk ad Kl, ut <lb/>
circulus uel ellip&longs;is gh ad ip&longs;as portiones. </s>
          <s id="s.000341">Sed ut circulus <lb/>
uel ellip&longs;is gh ad figuram rectilineam in ip&longs;a de&longs;cri&shy;<lb/>
ptam, ita e&longs;t cylindrus uel cylindri portio ce ad pri&longs;ma, <lb/>
quod rectilineam figuram pro ba&longs;i habet, &amp; altitudinem <lb/>
&aelig;qualem; id, quod infra demon&longs;trabitur. </s>
          <s id="s.000342">crgo per conuer<lb/>
&longs;ionem rationis, ut circulus, uel ellip&longs;is gh ad portiones re<lb/>
lictas, ita cylindrus, uel cylindri portio ce ad &longs;olidas por&shy;<lb/>
tiones, quate cylindrus uel cylindri portio ad &longs;olidas por&shy;<lb/>
tiones eandem proportionem habet, quam linea nk ad k <lb/>
&amp; diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o&shy;<lb/>
lidas portiones eandem proportionem habet, quam nl ad <lb/>
lk &amp; quoniam a cylindro uel cylindri portione, cuius gra&shy;<lb/>
uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili&shy;<lb/>
neam <expan abbr="figur&atilde;">figuram</expan>, cuius <expan abbr="centr&utilde;">centrum</expan> grauitatis e&longs;t K: re&longs;idu&aelig; magnitu<lb/>
dinis ex &longs;olidis portionibus <expan abbr="c&otilde;po&longs;it&aelig;">compo&longs;it&aelig;</expan> grauitatis <expan abbr="c&etilde;tr&utilde;">centrum</expan> erit <lb/>
in linea kl protracta, &amp; in puncto n; quod e&longs;t <expan abbr="ab&longs;urd&utilde;">ab&longs;urdum</expan>. </s>
          <s id="s.000343">relin<lb/>
quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por<lb/>
tionis &longs;it <expan abbr="punct&utilde;">punctum</expan> k. </s>
          <s id="s.000344">qu&aelig; omnia <expan abbr="demon&longs;tr&atilde;da">demon&longs;tranda</expan> propo&longs;uimus.</s>
        </p>
        <p type="margin">
          <s id="s.000345"><margin.target id="marg45"/>4. huius</s>
        </p>
        <p type="main">
          <s id="s.000346">At uero cylindrum, uel cylindri <expan abbr="portion&etilde;">portionem</expan> ce <lb/>
ad pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura in &longs;pa&shy;<lb/>
cio gh de&longs;cripta, &amp; altitudo &aelig;qualis; eandem ha&shy;



<pb pagenum="15" xlink:href="023/01/037.jpg"/>bere proportionem, quam &longs;pacium gh ad <expan abbr="dict&atilde;">dictam</expan> <lb/>
figuram, hoc modo demon&longs;trabimus.</s>
        </p>
        <p type="main">
          <s id="s.000347">Intelligatur circulus, uel ellip&longs;is x &aelig;qualis figur&aelig; rectili&shy;<lb/>
ne&aelig; in gh &longs;pacio de&longs;cript&aelig;. </s>
          <s id="s.000348">&amp; ab x con&longs;tituatur conus, uel <lb/>
<figure id="id.023.01.037.1.jpg" xlink:href="023/01/037/1.jpg"/><lb/>
coni portio, <expan abbr="altitudin&etilde;">altitudinem</expan> habens <expan abbr="eand&etilde;">eandem</expan>, <expan abbr="qu&atilde;">quam</expan> cylindrus uel cy<lb/>
lindri portio ce. </s>
          <s id="s.000349">Sit deinde rectilinea figura, in qua y <expan abbr="ead&etilde;">eadem</expan>, <lb/>
qu&aelig; in &longs;pacio gh de&longs;cripta e&longs;t: &amp; ab hac pyramis &aelig;quealta <lb/>
con&longs;tituatur. </s>
          <s id="s.000350">Dico <expan abbr="con&utilde;">conum</expan> uel coni portione x pyramidi y <expan abbr="&aelig;-qual&etilde;">&aelig;&shy;<lb/>
qualem</expan> e&longs;&longs;e. </s>
          <s id="s.000351">ni&longs;i enim &longs;it &aelig;qualis, uel maior, uel minor erit.</s>
        </p>
        <p type="main">
          <s id="s.000352">Sit primum maior, et exuperet &longs;olido z. </s>
          <s id="s.000353">Itaque in circu<lb/>
lo, uel ellip&longs;i x de&longs;cribatur figura rectilinea; &amp; in ea pyra&shy;<lb/>
mis eandem, quam conus, uel coni portio altitudinem ha&shy;<lb/>
bens, ita ut portiones relict&aelig; minores &longs;int &longs;olido a, quem&shy;<lb/>
admodum docetur in duodecimo libro elementorum pro<lb/>
po&longs;itione undecima. </s>
          <s id="s.000354">erit pyramis x adhuc pyramide y ma<lb/>
ior. </s>
          <s id="s.000355">&amp; quoniam piramides &aelig;que alt&aelig; inter &longs;e &longs;unt, &longs;icuti ba<lb/>
<arrow.to.target n="marg46"/><lb/>
&longs;es; pyramis x ad piramidem y eandem proportionem ha&shy;<lb/>
bet, qu&agrave;m figura rectilinea x ad figuram y. </s>
          <s id="s.000356">Sed figura recti 



<pb xlink:href="023/01/038.jpg"/><figure id="id.023.01.038.1.jpg" xlink:href="023/01/038/1.jpg"/><lb/>
linea x cum &longs;it minor circulo, uel ellip&longs;i, e&longs;t etiam minor fi&shy;<lb/>
gura rectilinea y. </s>
          <s id="s.000357">ergo pyramis x pyramide y minor erit. </s>
          <lb/>
          <s id="s.000358">Sed &amp; maior; quod fieri <expan abbr="n&otilde;">non</expan> pote&longs;t. </s>
          <s id="s.000359">At &longs;i conus, uel coni por<lb/>
tio x ponatur minor pyramide y: &longs;it alter conus &aelig;que al&shy;<lb/>
tus, uel altera coni portio X ip&longs;i pyramidi y &aelig;qualis. </s>
          <s id="s.000360">erit <lb/>
eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x, <lb/>
quorum exce&longs;&longs;us &longs;it &longs;pacium <foreign lang="greek">w.</foreign> Si igitur in circulo, uel eili&shy;<lb/>
p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relict&aelig; <lb/>
&longs;int <foreign lang="greek">w</foreign> &longs;pacio minores, ciu&longs;modi figura adhuc maior erit cir <lb/>
culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. </s>
          <s id="s.000361">&amp; pyramis in <lb/>
ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi&shy;<lb/>
nor pyramide y. </s>
          <s id="s.000362">e&longs;t ergo ut X figura rectilinea ad figuram <lb/>
rectilineam y, ita pyramis X ad pyramidem y. </s>
          <s id="s.000363">quare cum <lb/>
figura rectilinea X &longs;it maior figura y: erit &amp; pyramis X py&shy;<lb/>
ramide y maior. </s>
          <s id="s.000364">&longs;ed erat minor; quod rur&longs;us fieri non po&shy;<lb/>
te&longs;t. </s>
          <s id="s.000365">non e&longs;t igitur conus, uel coni portio x neque maior, <lb/>
neque minor pyramide y. </s>
          <s id="s.000366">ergo ip&longs;i nece&longs;&longs;ario e&longs;t &aelig;qualis. </s>
          <lb/>
          <s id="s.000367">Itaque quoniam ut conus ad conum, uel coni portio ad co



<pb pagenum="16" xlink:href="023/01/039.jpg"/><figure id="id.023.01.039.1.jpg" xlink:href="023/01/039/1.jpg"/><lb/>
ni portionem, ita e&longs;t cylindrus ad cylindrum, uel cylin&shy;<lb/>
dri portio ad cylindri portionem: &amp; ut pyramis ad pyra&shy;<lb/>
midem, ita pri&longs;ma ad pri&longs;ma, cum eadem &longs;it ba&longs;is, &amp; &aelig;qua <lb/>
lis altitudo; erit cylindrus uel cylindri portio x pri&longs;ma&shy;<lb/>
ti y &aelig;qualis. </s>
          <s id="s.000368"><expan abbr="e&longs;tq;">e&longs;tque</expan> ut &longs;pacium gh ad &longs;pacium x, ita cylin&shy;<lb/>
drus, uel cylindri portio ce ad cylindrum, uel cylindri por&shy;<lb/>
tionem x. </s>
          <s id="s.000369">Con&longs;tat igitur cylindrum uel cylindri <expan abbr="portion&etilde;">portionem</expan> <lb/>
c e, ad pri&longs;ma y, quippe cuius ba&longs;is e&longs;t figura rectilinea in <lb/>
<arrow.to.target n="marg47"/><lb/>
&longs;pacio gh de&longs;cripta, eandem proportionem habere, quam <lb/>
&longs;pacium gh habet ad &longs;pacium x, hoc e&longs;t ad dictam figuram. </s>
          <lb/>
          <s id="s.000370">quod demon&longs;trandum fuerat.</s>
        </p>
        <p type="margin">
          <s id="s.000371"><margin.target id="marg46"/>6. duode<lb/>
cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000372"><margin.target id="marg47"/>7. quinti</s>
        </p>
        <p type="head">
          <s id="s.000373">THEOREMA IX. PROPOSITIO IX.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000374">Si pyramis &longs;ecetur plano ba&longs;i &aelig;quidi&longs;tante; &longs;e&shy;<lb/>
ctio erit figura &longs;imilis ei, qu&aelig; e&longs;t ba&longs;is, centrum <lb/>
grauitatis in axe habens.</s>
        </p>
        <pb xlink:href="023/01/040.jpg"/>
        <p type="main">
          <s id="s.000375">SIT pyramis, cuius ba&longs;is triangulum abc; axis dc: &amp; <lb/>
&longs;ecetur plano ba&longs;i &aelig;quidi&longs;tante; quod <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> faciat fgh; <lb/>
<expan abbr="occurratq;">occurratque</expan> axi in puncto k. Dico fgh triangulum e&longs;&longs;e, ip&longs;i <lb/>
abc &longs;imile; cuius grauitatis centrum e&longs;t K. <expan abbr="Quoni&atilde;">Quoniam</expan> enim <lb/>
duo plana &aelig;quidi&longs;tantia abc, fgh &longs;ecantur &agrave; plano abd; <lb/>
communes eorum &longs;ectiones ab, fg &aelig;quidi&longs;tantes erunt: &amp; <lb/>
eadem ratione &aelig;quidi&longs;tantes ip&longs;&aelig; bc, gh: &amp; ca, hf. </s>
          <s id="s.000376">Qu&ograve;d <lb/>
cum du&aelig; line&aelig; fg, gh, duabus ab, bc &aelig;quidi&longs;tent, nec <lb/>
&longs;int in eodem plano; angulus ad g &aelig;qualis e&longs;t angulo ad <lb/>
b. </s>
          <s id="s.000377">&amp; &longs;imiliter angulus ad h angulo ad c: <expan abbr="angulusq;">angulusque</expan> ad fci, <lb/>
qui ad a e&longs;t &aelig;qualis. </s>
          <s id="s.000378">triangulum igitur fgh &longs;imile e&longs;t tri&shy;<lb/>
angulo abc. <!-- KEEP S--></s>
          <s id="s.000379">Atuero punctum k centrum e&longs;&longs;e grauita&shy;<lb/>
tis trianguli fgh hoc modo o&longs;tendemus. </s>
          <s id="s.000380">Ducantur pla&shy;<lb/>
na per axem, &amp; per lineas da, db, dc: erunt communes &longs;e&shy;<lb/>
ctiones fK, ae &aelig;quidi&longs;tantes: <expan abbr="pariterq;">pariterque</expan> kg, eb; &amp; kh, ec: <lb/>
quare angulus kfh angulo eac; &amp; angulus kfg ip&longs;i eab <lb/>
<figure id="id.023.01.040.1.jpg" xlink:href="023/01/040/1.jpg"/><lb/>
e&longs;t &aelig;qualis. </s>
          <s id="s.000381">Eadem ratione <lb/>
anguli ad g angulis ad b: &amp; <lb/>
anguli ad h iis, qui ad c &aelig;&shy;<lb/>
quales erunt. </s>
          <s id="s.000382">ergo puncta <lb/>
eK in triangulis abc, fgh <lb/>
&longs;imiliter &longs;unt po&longs;ita, per &longs;e&shy;<lb/>
xtam po&longs;itionem Archime&shy;<lb/>
dis in libro de centro graui&shy;<lb/>
tatis planorum. </s>
          <s id="s.000383">Sed cum e <lb/>
&longs;it centrum grauitatis trian<lb/>
guli abc, erit ex undecima <lb/>
propo&longs;itione eiu&longs;dem libri, <lb/>
&amp; K trianguli fgh grauita<lb/>
tis centrum. </s>
          <s id="s.000384">id quod demon&longs;trare oportebat. </s>
          <s id="s.000385">Non aliter <lb/>
in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra&shy;<lb/>
bitur.</s>
        </p>
        <pb pagenum="17" xlink:href="023/01/041.jpg"/>
        <p type="head">
          <s id="s.000386">PROBLEMA I. PROPOSITIO X.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000387">DATA qualibet pyramide, fieri pote&longs;t, ut fi&shy;<lb/>
gura &longs;olida in ip&longs;a in &longs;cribatur, &amp; altera <expan abbr="circ&utilde;&longs;cri-batur">circum&longs;cri&shy;<lb/>
batur</expan> ex pri&longs;matibus &aelig;qualem altitudinem <expan abbr="ha-b&etilde;tibus">ha&shy;<lb/>
bentibus</expan>, ita ut circum&longs;cripta in&longs;criptam excedat <lb/>
magnitudine, qu&aelig; minor &longs;it <expan abbr="quac&utilde;que">quacunque</expan> &longs;olida ma<lb/>
gnitudine propo&longs;ita.</s>
        </p>
        <figure id="id.023.01.041.1.jpg" xlink:href="023/01/041/1.jpg"/>
        <p type="main">
          <s id="s.000388">Sit pyramis, cuius ba&longs;is <lb/>
<expan abbr="triangul&utilde;">triangulum</expan> abc; axis de. </s>
          <lb/>
          <s id="s.000389"><expan abbr="Sitq;">Sitque</expan> pri&longs;ma, quod <expan abbr="eand&etilde;">eandem</expan> <lb/>
ba&longs;im habeat, &amp; axem eun<lb/>
dem. </s>
          <s id="s.000390">Itaque hoc pri&longs;ma&shy;<lb/>
te continenter &longs;ecto bifa&shy;<lb/>
riam, plano ba&longs;i <expan abbr="&aelig;quidi&longs;t&atilde;">&aelig;quidi&longs;tan</expan><lb/>
te,  relinquetur <expan abbr="t&atilde;dem">tandem</expan> pri&longs;<lb/>
ma quoddam minus pro&shy;<lb/>
po&longs;ita magnitudine: quod <lb/>
quidem ba&longs;im eandem ha<lb/>
beat, quam pyramis, &amp; a&shy;<lb/>
xem ef. </s>
          <s id="s.000391">diuidatur de in <lb/>
partes &aelig;quales ip&longs;i ef in <lb/>
punctis ghklmn: &amp; per <lb/>
diui&longs;iones plana <expan abbr="duc&atilde;tur">ducantur</expan>: <lb/>
qu&aelig; ba&longs;ibus &aelig;quidi&longs;tent, <lb/>
erunt &longs;ectiones, triangula <lb/>
ip&longs;i abc &longs;imilia, ut proxi&shy;<lb/>
me o&longs;tendimus. </s>
          <s id="s.000392">ab uno <lb/>
quoque <expan abbr="aut&etilde;">autem</expan> horum trian<lb/>
gulorum duo pri&longs;mata <expan abbr="c&otilde;">con</expan><lb/>
&longs;truantur; unum quidem <lb/>
ad partes e; alterum ad 



<pb xlink:href="023/01/042.jpg"/>partes d. <!-- KEEP S--></s>
          <s id="s.000393">in pyramide igitur in&longs;cripta erit qu&aelig;dam figura, <lb/>
ex pri&longs;matibus &aelig;qualem altitudinem habentibus <expan abbr="c&otilde;&longs;tans">con&longs;tans</expan>, <lb/>
ad partes e: &amp; altera circum&longs;cripta ad partes d. <!-- KEEP S--></s>
          <s id="s.000394">Sed unum&shy;<lb/>
quodque eorum pri&longs;matum, qu&aelig; in figura in&longs;cripta conti&shy;<lb/>
nentur, &aelig;quale e&longs;t pri&longs;mati, quod ab eodem fit triangulo in <lb/>
figura circum&longs;cripta: nam pri&longs;ma pq pri&longs;mati po e&longs;t &aelig;&shy;<lb/>
quale; pri&longs;ma st &aelig;quale pri&longs;mati sr; pri&longs;ma xy pri&longs;mati <lb/>
xu; pri&longs;ma <foreign lang="greek">hq</foreign> pri&longs;mati <foreign lang="greek">h</foreign>z; pri&longs;ma <foreign lang="greek">mn</foreign> pri&longs;mati <foreign lang="greek">ml;</foreign> pri&longs;&shy;<lb/>
ma <foreign lang="greek">rs</foreign> pri&longs;mati <foreign lang="greek">rp;</foreign> &amp; pri&longs;ma <foreign lang="greek">fx</foreign> pri&longs;mati <foreign lang="greek">ft</foreign> &aelig;quale. </s>
          <s id="s.000395">re&shy;<lb/>
linquitur ergo, ut circum&longs;cripta figura exuperet <expan abbr="in&longs;cript&atilde;">in&longs;criptam</expan> <lb/>
pri&longs;mate, quod ba&longs;im habet abc triangulum, &amp; axem ef. </s>
          <lb/>
          <s id="s.000396">Illud uero minus e&longs;t &longs;olida magnitudine propo&longs;ita. </s>
          <s id="s.000397"><expan abbr="Ead&etilde;">Eadem</expan> <lb/>
ratione in&longs;cribetur, &amp; circum&longs;cribetur &longs;olida figura in py&shy;<lb/>
ramide, qu&aelig; quadrilateram, uel <expan abbr="plurilater&atilde;">plurilateram</expan> ba&longs;im habeat.</s>
        </p>
        <p type="head">
          <s id="s.000398">PROBLEMA II. PROPOSITIO XI.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000399">DATO cono, fieri pote&longs;t, ut figura &longs;olida in&shy;<lb/>
&longs;cribatur, &amp; altera circum&longs;cribatur ex cylindris <lb/>
&aelig;qualem habentibus altitudinem, ita ut circum&shy;<lb/>
&longs;cripta &longs;uperet in&longs;criptam, magnitudine, qu&aelig; &longs;o&shy;<lb/>
lida magnitudine propo&longs;ita &longs;it minor.</s>
        </p>
        <p type="main">
          <s id="s.000400">SIT conus, cuius axis bd: &amp; &longs;ecetur plano per axem <lb/>ducto, &apos;ut &longs;ectio &longs;it triangulum abc: <expan abbr="intelligaturq;">intelligaturque</expan> cylin&shy;<lb/>
drus, qui ba&longs;im eandem, &amp; eundem axem habeat. </s>
          <s id="s.000401">Hoc igi&shy;<lb/>
tur cylindro continenter bifariam &longs;ecto, relinquetur cylin<lb/>
drus minor &longs;olida magnitudine propo&longs;ita. </s>
          <s id="s.000402">Sit autem is cy<lb/>
lindrus, qui ba&longs;im habet circulum circa diametrum ac, &amp; <lb/>
axem de. </s>
          <s id="s.000403">Itaque diuidatur bd in partes &aelig;quales ip&longs;i de <lb/>
in punctis fghKlm: &amp; per ea ducantur plana conum &longs;e&shy;<lb/>
cantia; qu&aelig; ba&longs;i &aelig;quidi&longs;tent. </s>
          <s id="s.000404">erunt &longs;ectiones circuli, cen&shy;<lb/>
tra in axi habentes, ut in primo libro conicorum, propo&longs;i-



<pb pagenum="18" xlink:href="023/01/043.jpg"/>tione quarta Apollonius demon&longs;trauit. </s>
          <s id="s.000405">Si igitur &agrave; &longs;ingu&shy;<lb/>
lis horum circulorum, duo cylindri fiant; unus quidem ad <lb/>
ba&longs;is partes; alter ad partes uerticis: in&longs;cripta erit in co&shy;<lb/>
no &longs;olida qu&aelig;dam figura, &amp; altera circum&longs;cripta ex cylin&shy;<lb/>
dris &aelig;qualem altitudinem habentibus con&longs;tans; quorum <lb/>
<figure id="id.023.01.043.1.jpg" xlink:href="023/01/043/1.jpg"/><lb/>
unu&longs;qui&longs;que, qui in <lb/>
figura in&longs;cripta con&shy;<lb/>
tinetur &aelig;qualis e&longs;t ei, <lb/>
qui ab eodem fit cir&shy;<lb/>
culo in figura <expan abbr="circ&utilde;-&longs;cripta">circum&shy;<lb/>
&longs;cripta</expan>. </s>
          <s id="s.000406">Itaque cylin<lb/>
drus op &aelig;qualis e&longs;t <lb/>
cylindro on; cylin&shy;<lb/>
drus rs <expan abbr="cyl&itilde;dro">cylindro</expan> rq.</s>
          <lb/>
          <s id="s.000407">
cylindrus ux cylin&shy;<lb/>
dro ut e&longs;t &aelig;qualis; <lb/>
&amp; alii aliis &longs;imiliter. </s>
          <lb/>
          <s id="s.000408">quare con&longs;tat <expan abbr="circ&utilde;-&longs;criptam">circum&shy;<lb/>
&longs;criptam</expan> figuram &longs;u&shy;<lb/>
perare in&longs;criptam cy<lb/>
lindro, cuius ba&longs;is e&longs;t <lb/>
circulus circa diametrum ac, &amp; axis de. </s>
          <s id="s.000409">atque hic e&longs;t mi&shy;<lb/>
nor &longs;olida magnitudine propo&longs;ita.</s>
        </p>
        <p type="head">
          <s id="s.000410">PROBLEMA III. PROPOSITIO XII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000411">DATA coni portione, pote&longs;t &longs;olida qu&aelig;dam <lb/>figura in&longs;cribi, &amp; altera circum&longs;cribi ex cylindri <lb/>
portionibus &aelig;qualem altitudinem habentibus; <lb/>
ita ut circum&longs;cripta in&longs;criptam exuperet, magni <lb/>
tudine, qu&aelig; minor fit &longs;olida magnitudine pro&shy;<lb/>
po&longs;ita.</s>
        </p>
        <pb xlink:href="023/01/044.jpg"/>
        <p type="main">
          <s id="s.000412">Figuram cuiu&longs;modi, &amp; in&longs;cribemus, &amp; <expan abbr="circ&utilde;&longs;cribemus">circum&longs;cribemus</expan>, ita <lb/>
ut in cono dictum e&longs;t.</s>
        </p>
        <figure id="id.023.01.044.1.jpg" xlink:href="023/01/044/1.jpg"/>
        <p type="head">
          <s id="s.000413">PROBLEMA IIII. PROPOSITIO XIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000414">DATA &longs;ph&aelig;r&aelig; portione, qu&aelig; dimidia &longs;ph&aelig;&shy;<lb/>
ra maior non &longs;it, pote&longs;t &longs;olida qu&aelig;dam portio in&shy;<lb/>
&longs;cribi &amp; altera circum&longs;cribi ex cylindris &aelig;qualem <lb/>
altitudinem habentibus, ita ut circum&longs;cripta in&shy;<lb/>
&longs;criptam excedat magnitudine, qu&aelig; &longs;olida ma&shy;<lb/>
gnitudine propo&longs;ita &longs;it minor.</s>
        </p>
        <p type="main">
          <s id="s.000415">HOC etiam eodem pror&longs;us modo &longs;iet: atque ut ab <lb/>Archimede traditum e&longs;t in conoidum, &amp; &longs;ph&aelig;roidum por<lb/>
tionibus, propo&longs;itione uige&longs;imaprima libri de conoidi&shy;<lb/>
bus, &amp; &longs;ph&aelig;roidibus.</s>
        </p>
        <pb pagenum="19" xlink:href="023/01/045.jpg"/>
        <figure id="id.023.01.045.1.jpg" xlink:href="023/01/045/1.jpg"/>
        <p type="head">
          <s id="s.000416">THEOREMA X. PROPOSITIO XIIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000417">Cuiuslibet pyramidis, &amp; cuiuslibet coni, uel <lb/>
coni portionis, centrum grauitatis in axe <expan abbr="c&otilde;&longs;i&longs;tit">con&longs;i&longs;tit</expan>.</s>
        </p>
        <p type="main">
          <s id="s.000418">SIT pyramis, cuius ba&longs;is triangulum abc: &amp; axis de. </s>
          <lb/>
          <s id="s.000419">Dico in linea de ip&longs;ius grauitatis centrum ine&longs;&longs;e. </s>
          <s id="s.000420">Si enim <lb/>
fieri pote&longs;t, &longs;it centrum f: &amp; ab f ducatur ad ba&longs;im pyrami<lb/>
dis linea fg, axi &aelig;quidi&longs;tans: <expan abbr="iunctaq;">iunctaque</expan> eg ad latera trian&shy;<lb/>
guli abc producatur in h. </s>
          <s id="s.000421">quam uero proportionem ha&shy;<lb/>
bet linea he ad eg, habeat pyramis ad aliud &longs;olidum, in <lb/>
quo K: <expan abbr="in&longs;cribaturq;">in&longs;cribaturque</expan> in pyramide &longs;olida figura, &amp; altera cir<lb/>
cum&longs;cribatur ex pri&longs;matibus &aelig;qualem habentibus altitu&shy;<lb/>
dinem, ita ut circum&longs;cripta in&longs;criptam exuperet magnitu&shy;<lb/>
dine, qu&aelig; &longs;olido k &longs;it minor. </s>
          <s id="s.000422">Et quoniam in pyramide pla<lb/>
num ba&longs;i &aelig;quidi&longs;tans ductum &longs;ectionem facit figuram &longs;i&shy;<lb/>
milem ei, qu&aelig; e&longs;t ba&longs;is; <expan abbr="centrumq;">centrumque</expan> grauitatis in axe haben<lb/>
tem: erit pri&longs;matis st grauitatis <expan abbr="centr&utilde;">centrum</expan> in linea rq ; <lb/>
matis ux centrum in linea qp, pri&longs;matis yz in linea po; <lb/>
pri&longs;matis <foreign lang="greek">hq</foreign> in linea on; pri&longs;matis <foreign lang="greek">lm</foreign> in linea  nm; pri&longs;&shy;<lb/>
matis <foreign lang="greek">np</foreign> in ml; &amp; denique pri&longs;matis <foreign lang="greek">rs</foreign> in le. </s>
          <s id="s.000423">quare to&shy;



<pb xlink:href="023/01/046.jpg"/>tius figur&aelig; in&longs;cript&aelig; centrum grauitatis e&longs;t in linea re: <lb/>
<figure id="id.023.01.046.1.jpg" xlink:href="023/01/046/1.jpg"/>quod &longs;it <foreign lang="greek">t</foreign>: <expan abbr="i&utilde;">iun</expan>&shy;<lb/>
ctaque <foreign lang="greek">t</foreign>f, &amp; <lb/>
producta, &agrave; <lb/>
puncto h du&shy;<lb/>
catur linea a&shy;<lb/>
xi pyramidis <lb/>
&aelig;quidi&longs;tans, <lb/>
qu&aelig; <expan abbr="c&utilde;">cum</expan> linea <lb/>
<foreign lang="greek">t</foreign>f conueniat <lb/>
in <foreign lang="greek">f</foreign>.</s>
          <s id="s.000424">habebit <lb/>
<foreign lang="greek">ft</foreign> ad <foreign lang="greek">t</foreign>f ean&shy;<lb/>
dem propor&shy;<lb/>
tionem, <expan abbr="qu&atilde;">quam</expan> <lb/>
he ad eg. <lb/>
</s>
        </p>
        <p>
          <s id="s.000425">Quoniam igi<lb/>
tur exce&longs;&longs;us, <lb/>
quo <expan abbr="circ&utilde;">circum</expan>&longs;cri<lb/>
pta figura in&shy;<lb/>
&longs;criptam &longs;upe<lb/>
rat, minor e&longs;t <lb/>
&longs;olido <foreign lang="greek">x</foreign>; py&shy;<lb/>
ramis ad eun&shy;<lb/>
<expan abbr="d&etilde;">dem</expan> <expan abbr="exce&longs;&longs;&utilde;">exce&longs;&longs;um</expan> ma<lb/>
ior&eacute; propor&shy;<lb/>
tion&eacute;  habet, <lb/>
qu&agrave;m ad K &longs;o<lb/>
lidum: uideli<lb/>
cet maiorem, <lb/>
qu&agrave;m linea h<lb/>
e ad eg; hoc <lb/>
e&longs;t qu&agrave;m <foreign lang="greek">ft</foreign> <lb/>
ad <foreign lang="greek">t</foreign>f: &amp; propterea multo maiorem habet ad partem ex&shy;<lb/>
ce&longs;&longs;us, qu&aelig; intra pyrimidem comprehenditur. </s>
          <s id="s.000426">Itaque ha&shy;



<pb pagenum="20" xlink:href="023/01/047.jpg"/>beat eam, quam <foreign lang="greek">xt</foreign> ad <foreign lang="greek">t</foreign>f erit diuidendo ut <foreign lang="greek">x</foreign>f ad f<foreign lang="greek">t</foreign>, ita fi<lb/>
gura &longs;olida in&longs;cripta ad partem exce&longs;&longs;us, qu&aelig; e&longs;t intra pyra<lb/>
midem. </s>
          <s id="s.000427">Cum ergo &agrave; pyramide, cuius grauitatis <expan abbr="ce&utilde;trum">centrum</expan> e&longs;t <lb/>
punctum f, &longs;olida figura in&longs;cripta auferatur, cuius <expan abbr="centr&utilde;trum">centrum</expan> <lb/>
<foreign lang="greek">t</foreign>: reliqua magnitudinis con&longs;tantis ex parte exce&longs;&longs;us, qu&aelig; <lb/>
e&longs;t intra pyramidem, centrum grauitatis erit in linea <foreign lang="greek">t</foreign>f <lb/>
producta, &amp; in puncto <foreign lang="greek">x</foreign>. </s>
          <s id="s.000428">quod fieri non pote&longs;t. </s>
          <s id="s.000429">Sequitur <lb/>
igitur, ut centrum grauitatis pyramidis in linea de; hoc <lb/>
e&longs;t in eius axe con&longs;i&longs;tat.</s>
        </p>
        <p>
          <s id="s.000430">Sit conus, uel coni portio, cuius axis bd: &amp; &longs;ecetur plano <lb/>
per axem, ut &longs;ectio &longs;it triangulum abc. </s>
          <s id="s.000431">Dico centrum gra<lb/>
uitatis ip&longs;ius e&longs;&longs;e in linea bd. </s>
          <s id="s.000432">Sit enim, &longs;i fieri pote&longs;t, <expan abbr="centr&utilde;">centrum</expan> <lb/>
<figure id="id.023.01.047.1.jpg" xlink:href="023/01/047/1.jpg"/>
e: <expan abbr="perq;">perque</expan> e ducatur ef axi &aelig;quidi&longs;tans: &amp; quam propor&shy;<lb/>
tionem habet cd ad df, habeat conus, uel coni portio ad <lb/>
&longs;olidum g. </s>
          <s id="s.000433">in&longs;cribatur ergo in cono, uel coni portione &longs;oli



<pb xlink:href="023/01/048.jpg"/>da figura, &amp; altera circum&longs;cribatur ex cylindris, uel cylin&shy;<lb/>
dri portionibus, &longs;icuti dictum e&longs;t, ita ut exce&longs;&longs;us, quo figu&shy;<lb/>
ra circum&longs;cripta in&longs;criptam &longs;uperat, &longs;it &longs;olido g minor. </s>
          <lb/>
          <s id="s.000434">Itaque centrum grauitatis cylindri, uel cylindri portionis <lb/>
qr e&longs;t in linea po; cylindri, uel cylindri portionis st cen&shy;<lb/>
trum in linea on; centrum ux in linea nm; yz in mb; <foreign lang="greek">nq</foreign><lb/>
in lk; <foreign lang="greek">lm</foreign> in kh; &amp; denique <foreign lang="greek">vp</foreign> centrum in hd. <!-- KEEP S--></s>
          <s id="s.000435">ergo figu&shy;<lb/>
<figure id="id.023.01.048.1.jpg" xlink:href="023/01/048/1.jpg"/><lb/>
r&aelig; in&longs;cript&aelig; centrum e&longs;t in linea pd. <!-- KEEP S--></s>
          <s id="s.000436">Sit autem <foreign lang="greek">r</foreign>: &amp; iun&shy;<lb/>
cta <foreign lang="greek">r</foreign>e protendatur, ut cum linea, qu&aelig; &agrave; <expan abbr="p&utilde;cto">puncto</expan> c ducta &longs;ue&shy;<lb/>
rit axi &aelig;quidi&longs;tans, conueniat in <foreign lang="greek">s.</foreign> erit <foreign lang="greek">s r</foreign> ad <foreign lang="greek">r</foreign>e, ut cd <lb/>
ad df: &amp; conus, &longs;eu coni portio ad exce&longs;&longs;um, quo circum&shy;<lb/>
&longs;cripta figura in&longs;criptam &longs;uperat, habebit maiorem pro&shy;<lb/>
portionem, qu&agrave;m <foreign lang="greek">tr</foreign> ad <foreign lang="greek">r</foreign>e. </s>
          <s id="s.000437">ergo ad partem exce&longs;&longs;us, qu&aelig; <lb/>
intra ip&longs;ius &longs;uperficiem comprehenditur, multo maiorem <lb/>
proportionem habebit. </s>
          <s id="s.000438">habeat eam, quam <foreign lang="greek">tr</foreign> ad <foreign lang="greek">r</foreign>e. </s>
          <s id="s.000439">erit 



<pb pagenum="21" xlink:href="023/01/049.jpg"/>diuidendo figura &longs;olida in&longs;cripta ad dictam exce&longs;&longs;us par&shy;<lb/>
tem, ut <foreign lang="greek">te</foreign> ad c<foreign lang="greek">p.</foreign> &amp; quoniam &agrave; cono, &longs;eu coni portione, <lb/>
cuius grauitatis centrum e&longs;t e, aufertur figura in&longs;cripta, <lb/>
cuius centrum <foreign lang="greek">r:</foreign> re&longs;idu&aelig; magnitudinis compo&longs;it&aelig; cx par <lb/>
te exce&longs;&longs;us, qu&aelig; intra coni, uel coni portionis &longs;uperficiem <lb/>
continetur, centrum grauitatis erit in linea e protracta, <lb/>
atque in puncto t. </s>
          <s id="s.000440">quod e&longs;t ab&longs;urdum. </s>
          <s id="s.000441"><expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> ergo <expan abbr="centr&utilde;">centrum</expan> <lb/>
grauitatis coni, uel coni portionis, e&longs;&longs;e in axe bd: quod de <lb/>
mon&longs;trandum propo&longs;uimus.</s>
        </p>
        <p type="head">
          <s id="s.000442">THEOREMA XI. PROPOSITIO XV.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000443">Cuiuslibet portionis &longs;ph&aelig;r&aelig; uel &longs;ph&aelig;roidis, <lb/>
qu&aelig; dimidia maior non &longs;it: <expan abbr="item&qacute;;">itemque</expan> cuiuslibet por<lb/>
tionis conoidis, uel ab&longs;ci&longs;&longs;&aelig; plano ad axem recto, <lb/>
uel non recto, centrum grauitatis in axe con&shy;<lb/>
&longs;i&longs;tit.</s>
        </p>
        <p type="main">
          <s id="s.000444">Demon&longs;tratio &longs;imilis erit ei, quam &longs;upra in cono, uel co<lb/>
ni portione attulimus, ne toties eadem fru&longs;tra iterentur.</s>
        </p>
        <figure id="id.023.01.049.1.jpg" xlink:href="023/01/049/1.jpg"/>
        <pb xlink:href="023/01/050.jpg"/>
        <p type="head">
          <s id="s.000445">THEOREMA XII. PROPOSITIO XVI.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000446">In &longs;ph&aelig;ra, &amp; &longs;ph&aelig;roide idem e&longs;t grauitatis, &amp; <lb/>
figur&aelig; centrum.</s>
        </p>
        <p type="main">
          <s id="s.000447">Secetur &longs;ph&aelig;ra, uel &longs;ph&aelig;roides plano per axem ducto; <lb/>
quod &longs;ectionem faciat circulum, uel ellip&longs;im abcd, cuius <lb/>
diameter, &amp; &longs;ph&aelig;r&aelig;, uel &longs;ph&aelig;roidis axis db; &amp; centrum e. </s>
          <lb/>
          <s id="s.000448">Dico e grauitatis etiam centrum e&longs;&longs;e. </s>
          <s id="s.000449">&longs;ecetur enim altero <lb/>
plano per e, ad planum &longs;ecans recto, cuius &longs;ectio &longs;it circu&shy;<lb/>
lus circa diametrum ac. </s>
          <s id="s.000450">erunt adc, abc dimidi&aelig; portio&shy;<lb/>
nes &longs;ph&aelig;r&aelig;, uel &longs;ph&aelig;roidis. </s>
          <s id="s.000451">&amp; quoniam portionis adc gra<lb/>
uitatis centrum e&longs;i in linea d, &amp; centrum portionis abc in <lb/>
ip&longs;a be; totius &longs;ph&aelig;r&aelig;, uel &longs;ph&aelig;roidis grauitatis centrum <lb/>in axe db con&longs;i&longs;tet, Qu&ograve;d &longs;i portionis adc centrum graui <lb/>
tatis ponatur e&longs;&longs;e f &amp; fiat ip&longs;i fe &aelig;qualis eg: <expan abbr="punct&utilde;">punctum</expan> g por<lb/>
<figure id="id.023.01.050.1.jpg" xlink:href="023/01/050/1.jpg"/><lb/>
<arrow.to.target n="marg48"/><lb/>
tionis abc centrum erit. </s>
          <s id="s.000452">&longs;olidis enim figuris &longs;imilibus &amp; <lb/>
&aelig;qualibus inter &longs;e aptatis, &amp; centra grauitatis ip&longs;arum in&shy;<lb/>
<arrow.to.target n="marg49"/><lb/>
ter se aptentur nece&longs;&longs;e e&longs;t. </s>
          <s id="s.000453">ex quo fit, ut magnitudinis, qu&aelig; <lb/>
ex utilique <expan abbr="c&otilde;&longs;lat">con&longs;tat</expan>, hoc e&longs;t ip&longs;ius &longs;ph&aelig;r&aelig;, uel &longs;ph&aelig;roidis gra<lb/>
uitatis centrum &longs;it in medio line&aelig; fg uidelicet in e. </s>
          <s id="s.000454">Sph&aelig;&shy;<lb/>
r&aelig; igitur, uel &longs;ph&aelig;roidis grauitatis centrum e&longs;t idem, quod <lb/>
centrum figur&aelig;.</s>
        </p>
        <pb pagenum="22" xlink:href="023/01/051.jpg"/>
        <p type="margin">
          <s id="s.000455"><margin.target id="marg48"/>per 2. pe&shy;<lb/>
titionem</s>
        </p>
        <p type="margin">
          <s id="s.000456"><margin.target id="marg49"/>4 Archi&shy;<lb/>
medis.</s>
        </p>
        <p type="main">
          <s id="s.000457">Ex demon&longs;tratis per&longs;picue apparet, portioni <lb/>
&longs;ph&aelig;r&aelig; uel &longs;ph&aelig;roidis, qu&aelig; dimidia maior e&longs;t, <expan abbr="c&etilde;">cen</expan><lb/>
trum grauitatis in axe con&longs;i&longs;tere.</s>
        </p>
        <figure id="id.023.01.051.1.jpg" xlink:href="023/01/051/1.jpg"/>
        <p type="main">
          <s id="s.000458">Data enim <lb/>
qualibet maio<lb/>
ri <expan abbr="porti&otilde;e">portione</expan>, quo <lb/>
<expan abbr="ni&atilde;">niam</expan> totius &longs;ph&aelig;<lb/>
r&aelig;, uel &longs;ph&aelig;roi<lb/>
dis grauitatis <lb/>
centrum e&longs;t in <lb/>
axe; e&longs;t autem <lb/>
&amp; in axe cen&shy;<lb/>
trum portio&shy;<lb/>
nis minoris: <lb/>
reliqu&aelig; portionis uidelicet maioris centrum in axe nece&longs;&shy;<lb/>
&longs;ario con&longs;i&longs;tet.</s>
        </p>
        <p type="head">
          <s id="s.000459">THEOREMA XIII. PROPOSITIO XVII.<!-- KEEP S--></s>
        </p>
        <figure id="id.023.01.051.2.jpg" xlink:href="023/01/051/2.jpg"/>
        <p type="main">
          <s id="s.000460">Cuiuslibet pyramidis <expan abbr="tri&atilde;gularem">trian<lb/>
gularem</expan> ba&longs;im <expan abbr="hab&etilde;tis">habentis</expan> gra<lb/>
uitatis centrum e&longs;t in pun&shy;<lb/>
cto, in quo ip&longs;ius axes con&shy;<lb/>
ueniunt.</s>
        </p>
        <p type="main">
          <s id="s.000461">Sit pyramis, cuius ba&longs;is trian<lb/>
gulum abc, axis de: <expan abbr="&longs;itq;">&longs;itque</expan> trian<lb/>
guli bdc grauitatis centrum f: <lb/>
&amp; iungatur a &longs;. </s>
          <s id="s.000462">erit &amp; af axis eiu&longs;<lb/>
dem pyramidis ex tertia diffini&shy;<lb/>
tione huius. </s>
          <s id="s.000463">Itaque quoniam centrum grauitatis e&longs;t in <lb/>
axe de; e&longs;t autem &amp; in axe af; &qgrave;uod proxime demon&longs;traui 



<pb xlink:href="023/01/052.jpg"/>mus: erit utique grauitatis centrum pyramidis punctum <lb/>
g. <!-- REMOVE S-->in quo &longs;cilicet ip&longs;i axes conueniunt.</s>
        </p>
        <p type="head">
          <s id="s.000464">THEOREMA XIIII. PROPOSITIO XVIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000465">SI &longs;olidum parallelepipedum &longs;ecetur plano <lb/>
ba&longs;ibus &aelig;quidi&longs;tante; erit &longs;olidum ad &longs;olidum, <lb/>
&longs;icut altitudo ad altitudinem, uel &longs;icut axis ad <lb/>
axem.</s>
        </p>
        <figure id="id.023.01.052.1.jpg" xlink:href="023/01/052/1.jpg"/>
        <p type="main">
          <s id="s.000466">Sit &longs;olidum parallelepipe<lb/>
dum abcdefgh, cuius axis <lb/>
kl: <expan abbr="&longs;eceturq;">&longs;eceturque</expan> plano ba&longs;ibus <lb/>
&aelig;quidi&longs;tante, quod faciat <lb/>
&longs;ectionem mnop; &amp; axi in <lb/>
puncto q occurrat. </s>
          <s id="s.000467">Dico <lb/>
&longs;olidum gm ad &longs;olidum mc <lb/>
eam proportionem habere, <lb/>
quam altitudo &longs;olidi gm ha&shy;<lb/>
bet ad &longs;olidi mc altitudi&shy;<lb/>
nem; uel quam axis kq ad <lb/>
axem ql. <!-- KEEP S--></s>
          <s id="s.000468">Si enim axis Kl ad <lb/>
ba&longs;is planum &longs;it perpendicu<lb/>
<figure id="id.023.01.052.2.jpg" xlink:href="023/01/052/2.jpg"/><lb/>
laris, &amp; linea gc, qu&aelig; ex quin<lb/>
ta huius ip&longs;i kl &aelig;quidi&longs;tat, <lb/>
perpendicularis erit ad <expan abbr="id&etilde;">idem</expan> <lb/>
planum, &amp; &longs;olidi altitudi&shy;<lb/>
<arrow.to.target n="marg50"/><lb/>
nem dimetietur. </s>
          <s id="s.000469">Itaque &longs;o&shy;<lb/>
lidum gm ad &longs;olidum mc <lb/>
eam proportionem habet, <lb/>
quam parallelogramm<expan abbr="&utilde;">um</expan> gn <lb/>
ad parallelogrammum nc, <lb/>
<arrow.to.target n="marg51"/><lb/>
hoc e&longs;t quam linea go, qu&aelig; 



<pb pagenum="23" xlink:href="023/01/053.jpg"/>e&longs;t &longs;olidi gm altitudo ad oe altitudinem &longs;olidi mc, uel <expan abbr="qu&atilde;">quam</expan> <lb/>
axis kq ad ql axem. </s>
          <s id="s.000470">Si uero axis kl non &longs;it perpendicularis <lb/>
ad planum ba&longs;is; ducatur a puncto k ad idem planum per<lb/>
pendicularis kr, <expan abbr="occurr&etilde;s">occurrens</expan> plano mnop in s. </s>
          <s id="s.000471">&longs;imiliter <expan abbr="de-m&otilde;&longs;trabimus">de&shy;<lb/>
mon&longs;trabimus</expan> &longs;olidum gm ad <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> mc ita e&longs;&longs;e, ut axis kq <lb/>
ad axem ql. </s>
          <s id="s.000472">Sed ut Kq ad ql, ita ks altitudo ad altitudi&shy;<lb/>
<arrow.to.target n="marg52"/><lb/>
nem sr; nam line&aelig; Kl, Kr &agrave; planis &aelig;quidi&longs;tantibus in ea&longs;&shy;<lb/>
dem proportiones &longs;ecantur. </s>
          <s id="s.000473">ergo &longs;olidum gm ad &longs;olidum <lb/>
mc <expan abbr="eand&etilde;">eandem</expan> proportionem habet, quam altitudo ad <expan abbr="altitudin&etilde;">altitu<lb/>
dinem</expan>, uel quam axis ad axem. </s>
          <s id="s.000474">quod <expan abbr="dem&otilde;&longs;trare">demon&longs;trare</expan> oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000475"><margin.target id="marg50"/>25 undeci<lb/>
mi.</s>
        </p>
        <p type="margin">
          <s id="s.000476"><margin.target id="marg51"/><expan abbr="&longs;ext&itilde;">&longs;extim</expan>.</s>
        </p>
        <p type="margin">
          <s id="s.000477"><margin.target id="marg52"/>17. unde&shy;<lb/>
cimi</s>
        </p>
        <p type="head">
          <s id="s.000478">THEOREMA XV. PROPOSITIO XIX.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000479">Solida parallelepipeda in eadem ba&longs;i, uel in <lb/>
&aelig;qualibus ba&longs;ibus con&longs;tituta eam inter &longs;e propor<lb/>
tionem habent, quam altitudines: &amp; &longs;i axes ip&longs;o&shy;<lb/>
rum cum ba&longs;ibus &aelig;quales angulos contineant, <lb/>
eam quoque, quam axes proportionem <expan abbr="habeb&utilde;t">habebunt</expan>.</s>
        </p>
        <p type="main">
          <s id="s.000480">Sint &longs;olida parallelepipeda in <expan abbr="ead&etilde;">eadem</expan> ba&longs;i <expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan> abcd, <lb/>
abef: &amp; &longs;it &longs;olidi abcd altitudo minor: producatur au&shy;<lb/>
tem planum cd adeo, ut &longs;olidum abef &longs;ecet; cuius &longs;ectio <lb/>
<figure id="id.023.01.053.1.jpg" xlink:href="023/01/053/1.jpg"/><lb/>
<arrow.to.target n="marg53"/><lb/>
&longs;it gh. </s>
          <s id="s.000481"><expan abbr="er&utilde;t">erunt</expan> &longs;oli <lb/>
da abcd, abgh <lb/>
in eadem ba&longs;i, <lb/>
&amp; &aelig;quali altitu<lb/>
dine inter &longs;e &aelig;&shy;<lb/>
qualia. </s>
          <s id="s.000482"><expan abbr="Quoni&atilde;">Quoniam</expan> <lb/>
igitur &longs;olidum <lb/>
abef &longs;ecatur <lb/>
plano ba&longs;ibus <lb/>
<expan abbr="&aelig;quidi&longs;t&atilde;te">&aelig;quidi&longs;tante</expan>, erit <lb/>
<arrow.to.target n="marg54"/><lb/>
&longs;olidum ghef <lb/>
adip&longs;um abgh 



<pb xlink:href="023/01/054.jpg"/>ut altitudo ad altitudinem: &amp; componendo conuertendo <lb/>
<arrow.to.target n="marg55"/><lb/>
que &longs;olidum abgh, hoc e&longs;t &longs;olidum abcd ip&longs;i &aelig;quale, ad <lb/>
&longs;olidum abef, ut altitudo &longs;olidi abcd ad &longs;olidi abef al&shy;<lb/>
titudinem.</s>
        </p>
        <p type="margin">
          <s id="s.000483"><margin.target id="marg53"/>29. unde&shy;<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000484"><margin.target id="marg54"/>18. huius</s>
        </p>
        <p type="margin">
          <s id="s.000485"><margin.target id="marg55"/>7. quinti.</s>
        </p>
        <p type="main">
          <s id="s.000486">Sint &longs;olida parallelopipeda ab, cd in &aelig;qualibus ba&longs;ibus <lb/>
con&longs;tituta: <expan abbr="&longs;itq;">&longs;itque</expan> be altitudo &longs;olidi ab: &amp; &longs;olidi cd altitudo <lb/>
d f; qu&aelig; quidem maior &longs;it, qu&agrave;m be. </s>
          <s id="s.000487">Dico &longs;olidum ab ad <lb/>
&longs;olidum cd eandem habere proportionem, quam be ad <lb/>
d f. </s>
          <s id="s.000488">ab&longs;cindatur enim &agrave; linea df &aelig;qualis ip&longs;i be, qu&aelig; &longs;it gf: <lb/>
&amp; per g ducatur planum &longs;ecans &longs;olidum cd; quod ba&longs;ibus <lb/>
&aelig;quidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> hK. </s>
          <s id="s.000489">erunt &longs;olida ab, ck &aelig;que <lb/>
<arrow.to.target n="marg56"/><lb/>
<figure id="id.023.01.054.1.jpg" xlink:href="023/01/054/1.jpg"/><lb/>
alta inter <lb/>
&longs;e &aelig;qualia <lb/>
<expan abbr="c&utilde;">cum</expan> &aelig;qua&shy;<lb/>
les ba&longs;es <lb/>
habeant. </s>
          <lb/>
          <s id="s.000490"><arrow.to.target n="marg57"/><lb/>
Sed <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> <lb/>
hd ad &longs;oli <lb/>
dum cK <lb/>
e&longs;t, ut alti<lb/>
tudo dg <lb/>
ad gf <expan abbr="alti&shy;tudin&etilde;">alti&shy;<lb/>
tudinem</expan>; &longs;e<lb/>
catur enim &longs;olidum cd plano ba&longs;i<lb/>
<figure id="id.023.01.054.2.jpg" xlink:href="023/01/054/2.jpg"/><lb/>
bus &aelig;quidi&longs;tante: &amp; rur&longs;us <expan abbr="c&otilde;po-nende">compo&shy;<lb/>
nende</expan>, <expan abbr="conuertendoq;">conuertendoque</expan> <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> ck <lb/>
<arrow.to.target n="marg58"/><lb/>
ad &longs;olidum cd, ut gf ad fd. <!-- KEEP S--></s>
          <s id="s.000491">ergo <lb/>
&longs;olidum ab, quod e&longs;t &aelig;quale ip&longs;i <lb/>
ck ad &longs;olidum cd eam proportio <lb/>
nem habet, quam altitudo gf, hoc <lb/>
e&longs;t be ad df altitudinem.</s>
        </p>
        <p type="margin">
          <s id="s.000492"><margin.target id="marg56"/>31. unde<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000493"><margin.target id="marg57"/>18. huius</s>
        </p>
        <p type="margin">
          <s id="s.000494"><margin.target id="marg58"/>7. quinti.</s>
        </p>
        <p type="main">
          <s id="s.000495">Sint deinde &longs;olida parallelepipe<lb/>
da ab, ac in eadem ba&longs;i; quorum <lb/>axes de, &longs; e cum ip&longs;a &aelig;quales angu<pb pagenum="24" xlink:href="023/01/055.jpg"/>los contineant. </s>
          <s id="s.000496">Dico &longs;olidum ab ad &longs;olidum ace idem ha<lb/>
bere proportionem, quam axis de ad axem ef. </s>
          <s id="s.000497">Si enim <lb/>
axes in eadem recta linea fuerint con&longs;tituti, h&aelig;c duo &longs;oli&shy;<lb/>
da, in unum, atque idem &longs;olidum conuenient. </s>
          <s id="s.000498">quare ex <lb/>
iis, qu&aelig; proxime tradita &longs;unt, habebit &longs;olidum ab ad &longs;o&shy;<lb/>
lidum ac eandem proportionem, quam axis de ad ef <lb/>
axem. </s>
          <s id="s.000499">Si uero axes non &longs;int in eadem recta linea, demittan<lb/>
tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, dg, fh: <lb/>
&amp; jungantur eg, eh. </s>
          <s id="s.000500">Quoniam igitur axes cum ba&longs;ibus <lb/>
&aelig;quales angulos continent, erit deg angulus &aelig;qualis an&shy;<lb/>
<figure id="id.023.01.055.1.jpg" xlink:href="023/01/055/1.jpg"/><lb/>
gulo feh: &amp; &longs;unt <lb/>
anguli ad gh re&shy;<lb/>
cti, quare &amp; re&shy;<lb/>
liquus edg &aelig;qua<lb/>
lis erit reliquo <lb/>
efh: &amp; triangu&shy;<lb/>
lum deg <expan abbr="tri&atilde;gu-lo">triangu&shy;<lb/>
lo</expan>  feh &longs;imile. </s>
          <s id="s.000501">er&shy;<lb/>
go gd ad de e&longs;t, <lb/>
ut hf ad e: &amp; per <lb/>
mutando gd ad <lb/>
hf, ut de ad cf. </s>
          <lb/>
          <figure id="id.023.01.055.2.jpg" xlink:href="023/01/055/2.jpg"/>
          <lb/>
          <s id="s.000502">Sed &longs;olidum ab <lb/>
ad &longs;olidum ac <lb/>
eandem propor&shy;<lb/>
tionem habet, <lb/>
quam dg altitu&shy;<lb/>
do ad <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>
fh. </s>
          <s id="s.000503">ergo &amp; <expan abbr="ean-d&etilde;">ean&shy;<lb/>
dem</expan> habebit, <expan abbr="qu&atilde;">quam</expan> <lb/>
axis de ad ef <expan abbr="ax&etilde;">axem</expan></s>
        </p>
        <p type="main">
          <s id="s.000504">Po&longs;tremo &longs;int <lb/>
&longs;olidi parallepi<lb/>
peda ab, cd in 



<pb xlink:href="023/01/056.jpg"/>&aelig;qualibus ba&longs;ibus, quorum axes cum ba&longs;ibus &aelig;quales an<lb/>
gulos faciant. </s>
          <s id="s.000505">Dico &longs;olidum ab ad <expan abbr="&longs;olid&utilde;">&longs;olidum</expan> cd ita e&longs;&longs;e, ut axis <lb/>
ef ad axem gh: nam &longs;i axes ad planum ba&longs;is recti &longs;int, il&shy;<lb/>
lud per&longs;picue con&longs;tat: quoniam eadem linea, &amp; axem &amp; &longs;oli<lb/>
di altitudinem determinabit. </s>
          <s id="s.000506">Si uero &longs;int inclinati, &agrave; pun&shy;<lb/>
ctis eg ad &longs;ubiectum planum perpendiculares ducantur <lb/>
ek, gl: &amp; iungantur fk, hl. <!-- KEEP S--></s>
          <s id="s.000507">rur&longs;us quoniam axes cum ba<lb/>
&longs;ibus &aelig;quales faciunt angulos, eodem modo demon&longs;trabi<lb/>
tur, triangulum efK triangulo ghl &longs;imile e&longs;&longs;e: &amp; ek ad gl, <lb/>
ut ef ad gh. </s>
          <s id="s.000508">Solidum autem ab ad &longs;olidum cd e&longs;t, ut <lb/>
eK ad gl. <!-- KEEP S--></s>
          <s id="s.000509">ergo &amp; ut axis ef ad axem gh. </s>
          <s id="s.000510">qu&aelig; omnia de<lb/>
mon&longs;trare oportebat.</s>
        </p>
        <p type="main">
          <s id="s.000511">Ex iis qu&aelig; demon&longs;trata &longs;unt, facile con&longs;tare <lb/>
pote&longs;t, pri&longs;mata omnia &amp; pyramides, qu&aelig; trian&shy;<lb/>gulares ba&longs;es habent, &longs;iue in ei&longs;dem, &longs;iue in &aelig;qua<lb/>
<arrow.to.target n="marg59"/><lb/>
libus ba&longs;ibus con&longs;tituantur, eandem proportio&shy;<lb/>
nem habere, quam altitudines: &amp; &longs;i axes cum ba<lb/>
&longs;ibus &aelig;quales angulos contineant, &longs;imiliter ean&shy;<lb/>
dem, quam axes, habere proportionem: &longs;unt <lb/>
<arrow.to.target n="marg60"/><lb/>
enim &longs;olida parallelepipeda pri&longs;matum triangula<lb/>
<arrow.to.target n="marg61"/><lb/>
res ba&longs;es <expan abbr="habenti&utilde;">habentium</expan> dupla; &amp; pyramidum &longs;extupla.</s>
        </p>
        <p type="margin">
          <s id="s.000512"><margin.target id="marg59"/>15. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000513"><margin.target id="marg60"/>28. unde&shy;<lb/>
cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000514"><margin.target id="marg61"/>7. duode&shy;<lb/>
cimi.</s>
        </p>
        <p type="head">
          <s id="s.000515">THEOREMA XVI. PROPOSITIO XX.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000516">Pri&longs;mata omnia &amp; pyramides, qu&aelig; in ei&longs;dem, <lb/>
uel &aelig;qualibus ba&longs;ibus con&longs;tituuntur, eam inter <lb/>
&longs;e proportionem habent, quam altitudines: &amp; &longs;i <lb/>
axes cum ba&longs;ibus faciant angulos &aelig;quales, eam <lb/>
etiam, quam axes habent proportionem.</s>
        </p>
        <pb pagenum="25" xlink:href="023/01/057.jpg"/>
        <p type="main">
          <s id="s.000517">Sint duo pri&longs;mata ae, af, quorum eadem ba&longs;is quadri&shy;<lb/>
latera abcd: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae altitudo eg; &amp; pri&longs;matis <lb/>
af altitudo fh. </s>
          <s id="s.000518">Dico pri&longs;ma ae ad pri&longs;ma af eam habere <lb/>
proportionem, quam eg ad fh. </s>
          <s id="s.000519">iungatur enim ac: &amp; in <lb/>
unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum <lb/>
<figure id="id.023.01.057.1.jpg" xlink:href="023/01/057/1.jpg"/><lb/>
ba&longs;es &longs;int triangu<lb/>
la abc, acd. </s>
          <s id="s.000520">habe <lb/>
bunt duo pri&longs;ma&shy;<lb/>
te in eadem ba&longs;i <lb/>
abc con&longs;tituta, <lb/>
proportionem <expan abbr="e&atilde;">eam</expan> <lb/>
dem, quam ip&longs;o&shy;<lb/>
rum altitudines e <lb/>
g, fh, ex iam de&shy;<lb/>
mon&longs;tratis. </s>
          <s id="s.000521">&amp; &longs;i&shy;<lb/>
militer alia duo, <lb/>
qu&aelig; &longs;unt in ba&longs;i a <lb/>
<arrow.to.target n="marg62"/><lb/>
c d. <!-- KEEP S--></s>
          <s id="s.000522">quare totum pri&longs;ma ae ad pri&longs;ma af eandem propor<lb/>
tionem habebit, quam altitudo eg ad fh altitudinem. </s>
          <lb/>
          <s id="s.000523">Qu&ograve;d cum pri&longs;mata &longs;int pyramidum tripla, &amp; ip&longs;&aelig; pyrami<lb/>
des, quarum eadem e&longs;t ba&longs;is quadrilatera, &amp; altitudo pri&longs;&shy;<lb/>
matum altitudini &aelig;qualis, eam inter &longs;e proportionem ha&shy;<lb/>
bebunt, quam altitudines.</s>
        </p>
        <p type="margin">
          <s id="s.000524"><margin.target id="marg62"/>12. quinti</s>
        </p>
        <p type="main">
          <s id="s.000525">Si uero pri&longs;mata ba&longs;es &aelig;quales habeant, <expan abbr="n&otilde;">non</expan> ea&longs;dem, &longs;int <lb/>duo eiu&longs;modi pri&longs;mata ae, fl: &amp; &longs;it ba&longs;is pri&longs;matis ae qua<lb/>
drilaterum abcd; &amp; pri&longs;matis fl quadrilaterum fghk. </s>
          <lb/>
          <s id="s.000526">Dico pri&longs;ma ae ad pri&longs;ma fl ita e&longs;&longs;e, ut altitudo illius ad <lb/>
huius altitudinem. </s>
          <s id="s.000527">nam &longs;i altitudo &longs;it eadem, <expan abbr="intellig&atilde;tur">intelligantur</expan> <lb/>
<arrow.to.target n="marg63"/><lb/>
du&aelig; pyramides abcde, fghkl. <!-- KEEP S--></s>
          <s id="s.000528">qu&aelig; <expan abbr="&itilde;tcr&longs;e">inter&longs;e</expan> &aelig;quales <expan abbr="er&utilde;t">erunt</expan>, <lb/>
cum &aelig;quales ba&longs;es, &amp; altitudinem eandem habeant. </s>
          <s id="s.000529">quare <lb/>
<arrow.to.target n="marg64"/><lb/>
&amp; pri&longs;mata ae, fl, qu&aelig; &longs;unt <expan abbr="har&utilde;">harum</expan> pyramidum tripla, &aelig;qua&shy;<lb/>
lia &longs;int nece&longs;&longs;e e&longs;t. </s>
          <s id="s.000530">ex quibus per&longs;picue con&longs;tat <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>. </s>
          <lb/>
          <s id="s.000531">Si uero altitudo pri&longs;matis fl &longs;it maior, &agrave; pri&longs;mate fl ab&shy;<lb/>
&longs;cindatur pri&longs;ma fm, quod &aelig;que altum &longs;it, <expan abbr="atq;">atque</expan> ip&longs;um ae. 



<pb xlink:href="023/01/058.jpg"/><figure id="id.023.01.058.1.jpg" xlink:href="023/01/058/1.jpg"/><lb/>
erunt e&aelig;dem ra&shy;<lb/>
tione pri&longs;mata a <lb/>
e, fm inter &longs;e &aelig;&shy;<lb/>
qualia. </s>
          <s id="s.000532">quare &longs;i&shy;<lb/>
militer demon&shy;<lb/>
&longs;trabitur pri&longs;ma <lb/>
fm ad pri&longs;ma fl <lb/>
eandem habere <lb/>
proportionem, <lb/>
quam pri&longs;matis <lb/>
fm altitudo ad <lb/>
altitudinem ip&shy;<lb/>
&longs;ius fl. <!-- KEEP S--></s>
          <s id="s.000533">ergo &amp; pri&longs;ma ae ad pri&longs;ma fl eandem propor&shy;<lb/>
tionem habebit, quam altitudo ad altitudinem. </s>
          <s id="s.000534">&longs;equitur <lb/>
igitur ut &amp; pyramides, qu&aelig; in &aelig;qualibus ba&longs;ibus <expan abbr="con&longs;titu&utilde;">con&longs;tituun</expan><lb/>
tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>
habeant.</s>
        </p>
        <p type="margin">
          <s id="s.000535"><margin.target id="marg63"/>6. duode<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000536"><margin.target id="marg64"/>25. quinti</s>
        </p>
        <figure id="id.023.01.058.2.jpg" xlink:href="023/01/058/2.jpg"/>
        <p type="main">
          <s id="s.000537">Sint deinde pri&longs;mata ae, af in eadem ba&longs;i abcd; <expan abbr="quor&utilde;">quorum</expan> <lb/>
axes cum ba&longs;ibus &aelig;quales angulos contineant: &amp; &longs;it pri&longs;&shy;



<pb pagenum="26" xlink:href="023/01/059.jpg"/>matis ae axis gh; &amp; pri&longs;matis af axis lh. </s>
          <s id="s.000538">Dico pri&longs;ma <lb/>
ae ad pri&longs;ma af eam proportionem habere, quam gh ad <lb/>
h l. <!-- REMOVE S-->ducantur &agrave; punctis gl perpendiculares ad ba&longs;is pla&shy;<lb/>
<figure id="id.023.01.059.1.jpg" xlink:href="023/01/059/1.jpg"/><lb/>
num gK, lm: &amp; iungantur kh, <lb/>
h m. </s>
          <s id="s.000539">Itaque quoniam anguli gh <lb/>
k, lhm &longs;unt &aelig;quales, &longs;imiliter ut <lb/>
&longs;upra demon&longs;trabimus, triangu&shy;<lb/>
la ghK, lhm &longs;imilia e&longs;&longs;e; &amp; ut g <lb/>
K ad lm, ita gh ad hl. </s>
          <s id="s.000540">habet au<lb/>
tem pri&longs;ma ae ad pri&longs;ma af ean <lb/>
dem proportionem, quam altitu<lb/>
do gK ad altitudinem lm, &longs;icuti <lb/>
demon&longs;tratum e&longs;t. </s>
          <s id="s.000541">ergo &amp; ean&shy;<lb/>
dem habebit, quam gh, ad hl. <!-- REMOVE S-->py<lb/>
ramis igitur abcdg ad pyrami&shy;<lb/>
dem abcdl eandem proportio&shy;<lb/>
nem habebit, quam axis gh ad hl axem.</s>
        </p>
        <figure id="id.023.01.059.2.jpg" xlink:href="023/01/059/2.jpg"/>
        <p type="main">
          <s id="s.000542">Denique &longs;int pri&longs;mata ae, ko in &aelig;qualibus ba&longs;ibus ab <lb/>
cd, klmn con&longs;tituta; quorum axes cum ba&longs;ibus &aelig;quales <lb/>
faciant angulos: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae axis fg, &amp; altitudo fh: <lb/>
pri&longs;matis autem ko axis pq, &amp; altitudo pr. </s>
          <s id="s.000543">Dico pri&longs;ma <lb/>
ae ad pri&longs;ma ko ita e&longs;&longs;e, ut fg ad pq. </s>
          <s id="s.000544">iunctis enim gh, 



<pb xlink:href="023/01/060.jpg"/>qr, eodem, quo &longs;upra, modo o&longs;tendemus<!--ostendemns (orig.)--> fg ad pq, ut fh <lb/>
ad pr. </s>
          <s id="s.000545">&longs;ed pri&longs;ma ae ad ip&longs;um ko e&longs;t, ut fh ad pr. </s>
          <s id="s.000546">ergo <lb/>
&amp; ut fg axis ad axem pq.</s>
          <s id="s.000547"> ex quibus &longs;it, ut pyramis abcdf <lb/>
<figure id="id.023.01.060.1.jpg" xlink:href="023/01/060/1.jpg"/><lb/>
ad <expan abbr="pyrami-d&etilde;">pyrami&shy;<lb/>
dem</expan> klmnp <lb/>
eandem ha<lb/>
beat pro &shy;<lb/>
portion&etilde;, <lb/>
<expan abbr="qu&atilde;">quam</expan> axis ad <lb/>
<expan abbr="ax&etilde;">axem</expan>. </s>
          <s id="s.000548">quod <lb/>
<expan abbr="demon&longs;tr&atilde;">demon&longs;tran</expan> <lb/>
<expan abbr="d&utilde;">dum</expan> &longs;uerat.</s>
        </p>
        <p type="main">
          <s id="s.000549">Simili ra<lb/>
tione in a&shy;<lb/>
liis pri&longs;ma&shy;<lb/>
tibus &amp; py<lb/>
ramidibus eadem demon&longs;trabuntur.</s>
        </p>
        <p type="head">
          <s id="s.000550">THEOREMA XVII. PROPOSITIO XXI.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000551">Pri&longs;mata omnia, &amp; pyramides inter &longs;e propor<lb/>
tionem habent compo&longs;itam ex proportione ba&shy;<lb/>
&longs;ium, &amp; proportione altitudinum.</s>
        </p>
        <p type="main">
          <s id="s.000552">Sint duo pri&longs;mata ae, gm: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae ba&longs;is qua<lb/>
drilaterum abcd, &amp; altitudo ef: pri&longs;matis uero gm ba&shy;<lb/>
&longs;is quadrilaterum ghKl, &amp; altitudo mn. </s>
          <s id="s.000553">Dico pri&longs;ma ae <lb/>
ad pri&longs;ma gm proportionem habere compo&longs;itam ex pro<lb/>
portione ba&longs;is abcd ad ba&longs;im ghkl, &amp; ex proportione <lb/>
altitudinis ef, ad altitudinem mn.</s>
        </p>
        <p type="main">
          <s id="s.000554">Sint enim primum ef, mn &aelig;quales: &amp; ut ba&longs;is abcd <lb/>
ad ba&longs;im ghkl, ita fiat linea, in qua o ad lineam, in qua p: <lb/>
ut autem ef ad mn, ita linea p ad lineam q.</s>
          <s id="s.000555"> erunt line&aelig; <lb/>
pq inter &longs;e &aelig;quales. </s>
          <s id="s.000556">Itaque pri&longs;ma ae ad pri&longs;ma gm <expan abbr="e&atilde;">eam</expan> 



<pb pagenum="27" xlink:href="023/01/061.jpg"/>proportionem habet, quam ba&longs;is abcd ad ba&longs;im ghkl: <lb/>
&longs;i enim intelligantur du&aelig; pyramides abcde, ghklm, ha&shy;<lb/>
bebunt h&aelig; inter &longs;e proportionem eandem, quam ip&longs;arum <lb/>
ba&longs;es ex &longs;exta duodecimi elementorum. </s>
          <s id="s.000557">Sed ut ba&longs;is abcd <lb/>
ad ghKl ba&longs;im, ita linea o ad lineam p; hoc e&longs;t ad lineam q <lb/>
ei &aelig;qualem. </s>
          <s id="s.000558">ergo pri&longs;ma ae ad pri&longs;ma gm e&longs;t, ut linea o <lb/>
ad lineam q.</s>
          <s id="s.000559"> proportio autem o ad q copo&longs;ita e&longs;t ex pro&shy;<lb/>
portione o ad p, &amp; ex proportione p ad q.</s>
          <s id="s.000560"> quare pri&longs;ma <lb/>
ae ad pri&longs;ma gm, &amp; idcirco pyramis abcde, ad pyrami&shy;<lb/>
dem ghKlm proportionem habet ex ei&longs;dem proportio&shy;<lb/>
nibus compo&longs;itam, uidelicet ex proportione ba&longs;is abcd <lb/>
ad ba&longs;im ghKl, &amp; ex proportione altitudinis ef ad mn al<lb/>
titudinem. </s>
          <s id="s.000561">Qu&ograve;d &longs;i line&aelig; ef, mn in&aelig;quales ponantur, &longs;it <lb/>
ef minor: &amp; ut ef ad mn, ita fiat linea p ad lineam u: de <lb/>
<figure id="id.023.01.061.1.jpg" xlink:href="023/01/061/1.jpg"/><lb/>
inde ab ip&longs;a mn ab&longs;cindatur rn &aelig;qualis ef: &amp; per r duca&shy;<lb/>
tur planum, quod oppo&longs;itis planis &aelig;quidi&longs;tans faciat &longs;e&shy;<lb/>
ctionem st. </s>
          <s id="s.000562">erit pri&longs;ma ae, ad pri&longs;ma gt, ut ba&longs;is abcd <lb/>
ad ba&longs;im ghkl; hoc e&longs;t ut o ad p: ut autem pri&longs;ma gt ad <lb/>
<arrow.to.target n="marg65"/><lb/>
pri&longs;ma gm, ita altitudo rn; hoc e&longs;t ef ad altitudine mn; <lb/>
uidelicet linea p ad lineam u. </s>
          <s id="s.000563">ergo ex &aelig;quali pri&longs;ma ae ad <lb/>
pri&longs;ma gm e&longs;t, ut linea o ad ip&longs;am u. </s>
          <s id="s.000564">Sed proportio o ad <lb/>
u <expan abbr="c&otilde;po&longs;ita">compo&longs;ita</expan> e&longs;t ex proportione o ad p, qu&aelig; e&longs;t ba&longs;is abcd <lb/>
ad ba&longs;im ghkl; &amp; ex proportione p ad u, qu&aelig; e&longs;t altitudi&shy;<lb/>
nis ef ad altitudinem mn. </s>
          <s id="s.000565">pri&longs;ma igitur ae ad pri&longs;ma gm 



<pb xlink:href="023/01/062.jpg"/>compo&longs;itam proportionem habet ex proportione <expan abbr="ba&longs;i&utilde;">ba&longs;ium</expan>, <lb/>
&amp; proportione altitudinum. </s>
          <s id="s.000566">Quare &amp; pyramis, cuius ba&shy;<lb/>
&longs;is e&longs;t quadrilaterum abcd, &amp; altitudo ef ad pyramidem, <lb/>
<figure id="id.023.01.062.1.jpg" xlink:href="023/01/062/1.jpg"/><lb/>
cuius ba&longs;is quadrilaterum ghKl, &amp; altitudo mn, compo&longs;i<lb/>
tam habet proportionem ex proportione ba&longs;ium abcd, <lb/>
ghkl, &amp; ex proportione altitudinum ef, mn. </s>
          <s id="s.000567">quod qui&shy;<lb/>
dem demon&longs;tra&longs;&longs;e oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000568"><margin.target id="marg65"/>20. huius</s>
        </p>
        <p type="main">
          <s id="s.000569">Ex iam demon&longs;tratis per&longs;picuum e&longs;t, pri&longs;ma <lb/>
ta omnia, &amp; pyramides, in quibus axes cum ba&longs;i&shy;<lb/>
bus &aelig;quales angulos continent, proportionem <lb/>
habere compo&longs;itam ex ba&longs;ium proportione, &amp; <lb/>
proportione axium. </s>
          <s id="s.000570">demon&longs;tratum e&longs;t enim, a&shy;<lb/>
xes inter &longs;e eandem proportionem habere, quam <lb/>
ip&longs;&aelig; altitudines.</s>
        </p>
        <p type="head">
          <s id="s.000571">THEOREMA XVIII. PROPOSITIO XXII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000572">CVIVSLIBEt pyramidis, &amp; cuiuslibet coni, 



<pb pagenum="28" xlink:href="023/01/063.jpg"/>uel coni portionis axis &agrave; centro grauitatis ita diui <lb/>
ditur, ut pars, qu&aelig; terminatur ad uerticem reli&shy;<lb/>
qu&aelig; partis, qu&aelig; ad ba&longs;im, &longs;it tripla.</s>
        </p>
        <p type="main">
          <s id="s.000573">Sit pyramis, cuius ba&longs;is triangulum abc; axis de; &amp; gra<lb/>
uitatis centrum K. <!-- KEEP S--></s>
          <s id="s.000574">Dico lineam dk ip&longs;ius Ke triplam e&longs;&longs;e. </s>
          <lb/>
          <s id="s.000575">trianguli enim bdc centrum grauitatis &longs;it punctum f; <expan abbr="tri&atilde;">trian</expan><lb/>
guli adc <expan abbr="centr&utilde;">centrum</expan> g; &amp; trianguli adb &longs;it h: &amp; iungantur af, <lb/>
b g, ch. </s>
          <s id="s.000576">Quoniam igitur <expan abbr="centr&utilde;">centrum</expan> grauitatis pyramidis in axe <lb/>
<arrow.to.target n="marg66"/><lb/>
<expan abbr="c&otilde;&longs;i&longs;tit">con&longs;i&longs;tit</expan>: <expan abbr="&longs;untq;">&longs;untque</expan> de, af, bg, ch <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> pyramidis axes: conue<lb/>
nient omnes in <expan abbr="id&etilde;">idem</expan> <expan abbr="punct&utilde;">punctum</expan> k, quod e&longs;t grauitatis centrum. </s>
          <lb/>
          <s id="s.000577">Itaque animo concipiamus hanc pyramidem diui&longs;am in <lb/>
quatuor pyramides, quarum ba&longs;es &longs;int ip&longs;a pyramidis <lb/>
<arrow.to.target n="marg67"/><lb/>
<figure id="id.023.01.063.1.jpg" xlink:href="023/01/063/1.jpg"/><lb/>
triangula; &amp; <emph type="ul"/>axis<emph.end type="ul"/> pun&shy;<lb/>
ctum k qu&aelig; quidem py&shy;<lb/>
ramides inter &longs;e &aelig;quales <lb/>
&longs;unt, ut <expan abbr="dem&otilde;&longs;trabitur">demon&longs;trabitur</expan>. </s>
          <lb/>
          <s id="s.000578">Ducatur <expan abbr="en&itilde;">enim</expan> per lineas <lb/>
dc, de planum <expan abbr="&longs;ec&atilde;s">&longs;ecans</expan>, ut <lb/>
&longs;it ip&longs;ius, &amp; ba&longs;is abc <expan abbr="c&otilde;">com</expan><lb/>
munis &longs;ectio recta linea <lb/>
cel: <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> uero &amp; <expan abbr="tri&atilde;-guli">trian&shy;<lb/>
guli</expan> adb &longs;it linea dhl. <!-- REMOVE S-->erit linea al &aelig;qualis ip&longs;i <lb/>
lb: nam centrum graui&shy;<lb/>
tatis trianguli con&longs;i&longs;tit <lb/>
in linea, qu&aelig; ab angulo <lb/>
ad dimidiam ba&longs;im per&shy;<lb/>
ducitur, ex tertia deci&shy;<lb/>
ma Archimedis. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000579">quare <lb/>
<arrow.to.target n="marg68"/><lb/>
triangulum acl &aelig;quale <lb/>
e&longs;t triangulo bcl: &amp; propterea pyramis, cuius ba&longs;is trian&shy;<lb/>
gulum acl, uertex d, e&longs;t &aelig;qualis pyramidi, cuius ba&longs;is bcl <lb/>
<arrow.to.target n="marg69"/><lb/>
triangulum, &amp; idem uertex. </s>
          <s id="s.000580">pyramides enim, qu&aelig; ab <expan abbr="eod&etilde;">eodem</expan> 



<pb xlink:href="023/01/064.jpg"/>&longs;unt uertice, eandem proportionem habent, quam <expan abbr="ip&longs;ar&utilde;">ip&longs;arum</expan> <lb/>
ba&longs;es. </s>
          <s id="s.000581">eadem ratione pyramis aclk pyramidi bclk &amp; py<lb/>
ramis adlk ip&longs;i bdlk pyramidi &aelig;qualis erit. </s>
          <s id="s.000582">Itaque &longs;i a py<lb/>ramide acld auferantur pyramides aclk, adlk: &amp; &agrave; pyra<lb/>
mide bcld <expan abbr="aufer&atilde;tur">auferantur</expan> pyramides bclk dblK: qu&aelig; relin&shy;<lb/>
quuntur erunt &aelig;qualia. </s>
          <s id="s.000583">&aelig;qualis igitur e&longs;t pyramis acdk <lb/>
pyramidi bcdK. <!-- KEEP S--></s>
          <s id="s.000584">Rur&longs;us &longs;i per lineas ad, de ducatur pla&shy;<lb/>
num quod pyramidem &longs;ccet: <expan abbr="&longs;itq;">&longs;itque</expan> eius &amp; ba&longs;is communis <lb/>
&longs;ectio aem: &longs;imiliter o&longs;tendetur pyramis abdK &aelig;qualis <lb/>
pyramidi acdk. </s>
          <s id="s.000585">ducto denique alio plano per lineas ca, <lb/>
af: ut eius, &amp; trianguli cdb communis &longs;ectio &longs;it cfn, py&shy;<lb/>
ramis abck pyramidi acdk &aelig;qualis demon&longs;trabitur. </s>
          <s id="s.000586"><expan abbr="c&utilde;">cum</expan> <lb/>
ergo tres pyramides bcdk, abdk, abck uni, &amp; eidem py<lb/>
ramidi acdk &longs;int &aelig;quales, omnes inter &longs;e &longs;e &aelig;quales <expan abbr="er&utilde;t">erunt</expan>. </s>
          <lb/>
          <s id="s.000587">Sed ut pyramis abcd ad pyramidem abck ita de axis ad <lb/>
axem ke, ex uige&longs;ima propo&longs;itione huius: &longs;unt enim h&aelig; <lb/>
pyramides in eadem ba&longs;i, &amp; axes cum ba&longs;ibus &aelig;quales con<lb/>
tinent angulos, qu&ograve;d in eadem recta linea con&longs;tituantur. </s>
          <lb/>
          <s id="s.000588">quare diuidendo, ut tres pyramides acdk, bcdK, abdK <lb/>
ad pyramidem abcK, ita dk ad Ke. </s>
          <s id="s.000589">con&longs;tat igitur lineam <lb/>
dK ip&longs;ius Ke triplam e&longs;&longs;e. </s>
          <s id="s.000590">&longs;ed &amp; ak tripla e&longs;t Kf: itemque <lb/>
bK ip&longs;ius kg: &amp; ck ip&longs;ius kl tripla. </s>
          <s id="s.000591">quod eodem modo <lb/>
demon&longs;trabimus.</s>
        </p>
        <p type="margin">
          <s id="s.000592"><margin.target id="marg66"/>17 huius</s>
        </p>
        <p type="margin">
          <s id="s.000593"><margin.target id="marg67"/><emph type="italics"/>ucrfex<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000594"><margin.target id="marg68"/>1. sexti.</s>
        </p>
        <p type="margin">
          <s id="s.000595"><margin.target id="marg69"/>5. duode&shy;<lb/>
cimi.</s>
        </p>
        <p type="main">
          <s id="s.000596">Sit pyramis, cuius ba&longs;is quadrilaterum abcd; axis ef: <lb/>
&amp; diuidatur ef in g, ita ut eg ip&longs;ius gf &longs;it tripla. </s>
          <s id="s.000597">Dico cen&shy;<lb/>
trum grauitatis pyramidis e&longs;&longs;e punctum g. <!-- REMOVE S-->ducatur enim <lb/>
linea bd diuidens ba&longs;im in duo triangula abd, bcd: ex <lb/>
quibus <expan abbr="intellig&atilde;tur">intelligantur</expan> <expan abbr="c&otilde;&longs;titui">con&longs;titui</expan> du&aelig; pyramides abde, bcde: <lb/>
&longs;itque pyramidis abde axis eh; &amp; pyramidis bcde axis <lb/>
eK: &amp; iungatur hK, qu&aelig; per f tran&longs;ibit: e&longs;t enim in ip&longs;a hK <lb/>
centrum grauitatis magnitudinis compo&longs;it&aelig; ex triangulis <lb/>
abd, bcd, hoc e&longs;t ip&longs;ius quadrilateri. </s>
          <s id="s.000598">Itaque centrum gra<lb/>
uitatis pyramidis abde &longs;it punctum l: &amp; pyramidis bcde <lb/>
<arrow.to.target n="marg70"/><lb/>
&longs;it m. </s>
          <s id="s.000599">ducta igitur lm ip&longs;i hm line&aelig; &aelig;quidi&longs;tabit. </s>
          <s id="s.000600">nam el ad 



<pb pagenum="29" xlink:href="023/01/065.jpg"/>lh eandem habet proportionem, quam em ad mk, uideli&shy;<lb/>
cet triplam. </s>
          <s id="s.000601">quare linea lm ip&longs;am ef &longs;ecabit in puncto g: <lb/>
etenim eg ad gf e&longs;t, ut el ad lh. </s>
          <s id="s.000602">pr&aelig;terea quoniam hk, lm <lb/>
&aelig;quidi&longs;tant, erunt triangula hef, leg &longs;imilia: <expan abbr="itemq;">itemque</expan> inter <lb/>
&longs;e &longs;imilia fek gem: &amp; ut ef ad eg, ita hf ad lg: &amp; ita fK ad <lb/>
gm. </s>
          <s id="s.000603">ergo ut hf ad lg, ita fk ad gm: &amp; permutando ut hf <lb/>
ad fK, ita lg ad gm. </s>
          <s id="s.000604">&longs;ed cum h &longs;it centrum trianguli abd; <lb/>
&amp; k <expan abbr="tri&atilde;guli">trianguli</expan> bcd <expan abbr="punct&utilde;">punctum</expan> uero f totius quadrilateri abcd <lb/>
centrum: erit ex 8. Archimedis de centro grauitatis plano<lb/>
rum hf ad fk ut triangulum bcd ad triangulum abd: ut, <lb/>
autem bcd triangulum ad triangulum abd, ita pyramis <lb/>
<figure id="id.023.01.065.1.jpg" xlink:href="023/01/065/1.jpg"/><lb/>
bcde ad pyramidem abde. </s>
          <s id="s.000605">ergo <lb/>
linea lg ad gm erit, ut pyramis <lb/>
bcde ad <expan abbr="pyramid&etilde;">pyramidem</expan> abde. </s>
          <s id="s.000606">ex quo <lb/>
&longs;equitur, ut totius pyramidis <lb/>
abcde punctum g &longs;it grauitatis <lb/>
centrum. </s>
          <s id="s.000607">Rur&longs;us &longs;it pyramis ba&shy;<lb/>
&longs;im habens pentagonum abcde: <lb/>
&amp; axem fg: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="p&utilde;">pun</expan><lb/>
cto h, ita ut fh ad hg triplam habe<lb/>
at proportionem. </s>
          <s id="s.000608">Dico h grauita&shy;<lb/>
tis <expan abbr="centr&utilde;">centrum</expan> e&longs;&longs;e pyramidis abcdef. </s>
          <lb/>
          <s id="s.000609">iungatur enim eb: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>
pyramis, cuius uertex f, &amp; ba&longs;is <lb/>
triangulum abe: &amp; alia pyramis <lb/>
intelligatur eundem uerticem ha&shy;<lb/>
bens, &amp; ba&longs;im bcde <expan abbr="quadrilater&utilde;">quadrilaterum</expan>: <lb/>
&longs;it autem pyramidis abef axis fk<lb/>
&amp; grauitatis centrum l: &amp; pyrami<lb/>
dis bcdef axis fm, &amp; centrum gra <lb/>
uitatis n:<expan abbr="iunganturq;">iunganturque</expan> km, ln; <lb/>
qu&aelig; per puncta gh tran&longs;ibunt. </s>
          <lb/>
          <s id="s.000610">Rur&longs;us eodem modo, quo &longs;up ra, <lb/>
demon&longs;trabimus lineas Kgm, lhn &longs;ibi ip&longs;is &aelig;quidi&longs;tare: 



<pb xlink:href="023/01/066.jpg"/>&amp; denique punctum h pyramidis abcdef grauitatis e&longs;&longs;e <lb/>
centrum, &amp; ita in aliis.</s>
        </p>
        <p type="margin">
          <s id="s.000611"><margin.target id="marg70"/>2. fexti.</s>
        </p>
        <p type="main">
          <s id="s.000612">Sit conus, uel coni portio axem habens bd: &longs;eceturque <lb/>
plano per axem, quod &longs;ectionem faciat triangulum abc: <lb/>
&amp; bd axis diuidatur in c, ita ut be ip&longs;ius ed &longs;it tripla. </s>
          <lb/>
          <s id="s.000613">Dico punctum e coni, uel coni portionis, grauitatis <lb/>
e&longs;&longs;e centrum. </s>
          <s id="s.000614">Si enim fieri pote&longs;t, &longs;it centrum f: &amp; pro&shy;<lb/>
ducatur ef extra figuram in g. <!-- KEEP S--></s>
          <s id="s.000615">quam uero proportionem <lb/>
habet ge ad ef, habeat ba&longs;is coni, uelconi portionis, hoc <lb/>
e&longs;t circulus, uel ellip&longs;is circa diametrum ac ad aliud &longs;pa&shy;<lb/>
cium, in quo h. </s>
          <s id="s.000616">Itaque in circulo, uel ellip&longs;i plane de&longs;cri&shy;<lb/>
batur rectilinea figura axlmcnop, ita ut qu&aelig; <expan abbr="relinqu&utilde;-tur">relinquun&shy;<lb/>
tur</expan> portiones &longs;int minores &longs;pacio h: &amp; intelligatur pyra&shy;<lb/>
mis ba&longs;im habens rectilineam figuram aKlmcnop, &amp; <lb/>
axem bd; cuius quidem grauitatis centrum erit punctum <lb/>
e, ut iam demon&longs;trauimus. </s>
          <s id="s.000617">Et quoniam portiones &longs;unt <lb/>
minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma&shy;<lb/>
<figure id="id.023.01.066.1.jpg" xlink:href="023/01/066/1.jpg"/><lb/>
iorem proportionem habet, quam ge ad ef. </s>
          <s id="s.000618">&longs;ed ut circu&shy;<lb/>
lus, uel ellip&longs;is ad figuram rectilineam &longs;ibi in&longs;criptam, ita <lb/>
conus, uel coni portio ad pyramidem, qu&aelig; figuram rectili&shy;<lb/>
neam pro ba&longs;i habet; &amp; altitudinem &aelig;qualem: etenim &longs;u&shy;



<pb pagenum="30" xlink:href="023/01/067.jpg"/><arrow.to.target n="marg71"/><lb/>
pra demon&longs;tratum e&longs;t, ita e&longs;&longs;e cylindrum, uel cylindri por&shy;<lb/>
tionem ad pri&longs;ma, cuius ba&longs;is rectilinea figura, &amp; &aelig;qua&shy;<lb/>
lis altitudo. </s>
          <s id="s.000619">ergo per conuer&longs;ionem rationis, ut circulus, <lb/>
uel ellip&longs;is ad portiones, ita conus, uel coni portio ad por&shy;<lb/>
tiones &longs;olidas. </s>
          <s id="s.000620">quare conus uel coni portio ad portiones <lb/>
&longs;olidas maiorem habet proportionem, quam ge ad ef: &amp; <lb/>
diuidendo, pyramis ad portiones &longs;olidas maiorem pro&shy;<lb/>
portionem habet, quam gf ad fe. </s>
          <s id="s.000621">fiat igitur qf ad fe <lb/>
ut pyramis ad dictas portiones. </s>
          <s id="s.000622">Itaque quoniam a cono <lb/>
uel coni portione, cuius grauitatis centrum e&longs;t f, aufer&shy;<lb/>
tur pyramis, cuius centrum e; reliqu&aelig; magnitudinis, <lb/>
qu&aelig; ex &longs;olidis portionibus con&longs;tat, centrum grauitatis <lb/>
erit in linea ef protracta, &amp; in puncto q.</s>
          <s id="s.000623"> quod fieri <lb/>
non pote&longs;t: e&longs;t enim centrum grauitatis intra. </s>
          <s id="s.000624">Con&longs;tat <lb/>
igitur coni, uel coni portionis grauitatis centrum e&longs;&longs;e pun<lb/>
ctum e. </s>
          <s id="s.000625">qu&aelig; omnia demon&longs;trare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000626"><margin.target id="marg71"/>8 huius</s>
        </p>
        <p type="head">
          <s id="s.000627">THEOREMA XIX. PROPOSITIO XXIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000628">QVODLIBET fru&longs;tum &agrave; pyramide, qu&aelig; <lb/>
triangularem ba&longs;im habeat, ab&longs;ci&longs;&longs;um, diuiditur <lb/>
in tres pyramides proportionales, in ea proportio <lb/>
ne, qu&aelig; e&longs;t lateris maioris ba&longs;is ad latus minoris <lb/>
ip&longs;i re&longs;pondens.</s>
        </p>
        <p type="main">
          <s id="s.000629">Hoc demon&longs;trauit Leonardus Pi&longs;anus in libro, qui de&shy;<lb/>
praxi geometri&aelig; in&longs;cribitur. </s>
          <s id="s.000630">Sed quoniam is adhuc im&shy;<lb/>
pre&longs;&longs;us non e&longs;t, nos ip&longs;ius demon&longs;trationem breuiter <lb/>
per&longs;tringemus, rem ip&longs;am &longs;ecuti, non uerba. </s>
          <s id="s.000631">Sit fru&shy;<lb/>
&longs;tum pyramidis abcdef, cuius maior ba&longs;is triangulum <lb/>
abc, minor def: &amp; iunctis ae, cc, cd, per, line&shy;<lb/>
as ae, ec ducatur planum &longs;ecans fru&longs;tum: itemque per <lb/>
lineas ec, cd; &amp; per cd, da alia plana ducantur, qu&aelig; <lb/>
diuident fru&longs;tum in trcs pyramides abce, adce, defc. 



<pb xlink:href="023/01/068.jpg"/>Dico eas proportionales e&longs;&longs;e in proportione, qu&aelig; e&longs;t la&shy;<lb/>
teris ab adlatus de, ita ut earum maior &longs;it abce, me&shy;<lb/>
dia adce, &amp; minor defc. <!-- KEEP S--></s>
          <s id="s.000632">Quoniam enim line&aelig; de, <lb/>
ab &aelig;quidi&longs;tant; &amp; inter ip&longs;as &longs;unt triangula abe, ade; <lb/>
<arrow.to.target n="marg72"/><lb/>
<figure id="id.023.01.068.1.jpg" xlink:href="023/01/068/1.jpg"/><lb/>
erit triangulum abe <lb/>
ad triangulum abe, <lb/>
ut linea ab ad lineam <lb/>
de. </s>
          <s id="s.000633">ut autem triangu<lb/>
lum abe ad triangu&shy;<lb/>
<arrow.to.target n="marg73"/><lb/>
lum abe, ita pyramis <lb/>
abec ad pyramidem <lb/>
adec: habent enim <lb/>
altitudinem eandem, <lb/>
qu&aelig; e&longs;t&agrave; puncto cad <lb/>
planum, in quo qua&shy;<lb/>
<arrow.to.target n="marg74"/><lb/>
drilaterum abed. <!-- KEEP S--></s>
          <s id="s.000634">er&shy;<lb/>
go ut ab ad de, ita pyramis abec ad pyramidem adec. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000635">Rur&longs;us quoniam &aelig;quidi&longs;tantes &longs;unt ac, df; erit eadem <lb/>
<arrow.to.target n="marg75"/><lb/>
ratione pyramis adce ad pyramidem cdfe, ut ac ad <lb/>
df. </s>
          <s id="s.000636">Sed ut ac ad df, ita ab ad de, quoniam triangula <lb/>
abc, def &longs;imilia &longs;unt, ex nona huius. </s>
          <s id="s.000637">quare ut pyramis <lb/>
abce ad pyramidem abce, ita pyramis adce ad ip&longs;am<lb/>
defc. <!-- REMOVE S-->fru&longs;tum igitur abcdef diuiditur in tres pyramides <lb/>
proportionales in ea proportione, qu&aelig; e&longs;t lateris ab ad de <lb/>
latus, &amp; earum maior e&longs;t cabe, media adce, &amp; minor <lb/>
defc. <!-- REMOVE S-->quod demon&longs;trare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000638"><margin.target id="marg72"/>1. &longs;exti.</s>
        </p>
        <p type="margin">
          <s id="s.000639"><margin.target id="marg73"/>5. duodeci <lb/>
mi.</s>
        </p>
        <p type="margin">
          <s id="s.000640"><margin.target id="marg74"/>11. quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000641"><margin.target id="marg75"/>4 &longs;exti.</s>
        </p>
        <p type="head">
          <s id="s.000642">PROBLEMA V. PROPOSITIO XXIIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000643">QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>
uel coni portionis, plano ba&longs;i &aelig;quidi&longs;tanti ita &longs;e&shy;<lb/>
care, ut &longs;ectio &longs;it proportionalis inter maiorem, <lb/>
&amp; minorem ba&longs;im.</s>
        </p>
        <pb pagenum="31" xlink:href="023/01/069.jpg"/>
        <p type="main">
          <s id="s.000644">SIT fru&longs;tum pyramidis ae, cuius maior ba&longs;is triangu&shy;<lb/>
lum abc, minor def: &amp; oporteat ip&longs;um plano, quod ba&longs;i <lb/>
&aelig;quidi&longs;tet, ita &longs;ecare, ut &longs;ectio &longs;it proportionalis inter <expan abbr="tri&atilde;">trian</expan> <lb/>
gula abc, def. </s>
          <s id="s.000645">Inueniatur inter lineas ab, de media pro&shy;<lb/>
portionalis, qu&aelig; &longs;it bg: &amp; &agrave; puncto g erigatur gh &aelig;quidi&shy;<lb/>
&longs;tans be, <expan abbr="&longs;ecansq;">&longs;ecansque</expan> ad in h: deinde per h ducatur planum <lb/>
ba&longs;ibus &aelig;quidi&longs;tans, cuius &longs;ectio &longs;it triangulum hkl. <!-- KEEP S--></s>
          <s id="s.000646">Dico <lb/>
triangulum hKl proportionale e&longs;&longs;e inter triangula abc, <lb/>
<figure id="id.023.01.069.1.jpg" xlink:href="023/01/069/1.jpg"/><lb/>
def, hoc e&longs;t triangulum abc ad <lb/>
triangulum hKl eandem habere <lb/>
proportionem, quam <expan abbr="tri&atilde;gulum">triangulum</expan> <lb/>
hKl ad ip&longs;um def. </s>
          <s id="s.000647"><expan abbr="Quoni&atilde;">Quoniam</expan> enim <lb/>
<arrow.to.target n="marg76"/><lb/>
line&aelig; ab, hK &aelig;quidi&longs;tantium pla<lb/>
norum &longs;ectiones inter &longs;e &aelig;quidi&shy;<lb/>
&longs;tant: atque &aelig;quidi&longs;tant bk, gh: <lb/>
<arrow.to.target n="marg77"/><lb/>
linea hk ip&longs;i gb e&longs;t &aelig;qualis: &amp; pro<lb/>
pterea proportionalis inter ab, <lb/>
de. </s>
          <s id="s.000648">quare ut ab ad hK, ita e&longs;t hk<lb/>
ad de. </s>
          <s id="s.000649">fiat ut hk ad de, ita de <lb/>
ad aliam lineam, in qua &longs;it m. </s>
          <s id="s.000650">erit <lb/>
ex &aelig;quali ut ab ad de, ita hk ad <lb/>
<arrow.to.target n="marg78"/><lb/>
m. </s>
          <s id="s.000651">Et quoniam triangula abc, <lb/>
hKl, def &longs;imilia &longs;unt; <expan abbr="triangul&utilde;">triangulum</expan> <lb/>
<arrow.to.target n="marg79"/><lb/>
abc ad triangulum hkl e&longs;t, ut li&shy;<lb/>
nea ab ad lineam de: <expan abbr="triangul&utilde;">triangulum</expan> <lb/>
<arrow.to.target n="marg80"/><lb/>
autem hkl ad ip&longs;um def e&longs;t, ut hk ad m. </s>
          <s id="s.000652">ergo triangulum <lb/>
abc ad triangulum hkl eandem proportionem habet, <lb/>
quam triangulum hKl ad ip&longs;um def. </s>
          <s id="s.000653">Eodem modo in a&shy;<lb/>
liis fru&longs;tis pyramidis idem demon&longs;trabitur.</s>
        </p>
        <p type="margin">
          <s id="s.000654"><margin.target id="marg76"/>16. unde<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000655"><margin.target id="marg77"/>34. primi</s>
        </p>
        <p type="margin">
          <s id="s.000656"><margin.target id="marg78"/>9. huius <lb/>
corol.</s>
        </p>
        <p type="margin">
          <s id="s.000657"><margin.target id="marg79"/>20. &longs;exti</s>
        </p>
        <p type="margin">
          <s id="s.000658"><margin.target id="marg80"/>11. quinti</s>
        </p>
        <p type="main">
          <s id="s.000659">Sit fru&longs;tum coni, uel coni portionis ad: &amp; &longs;ecetur plano <lb/>
per axem, cuius &longs;ectio &longs;it abcd, ita ut maior ip&longs;ius ba&longs;is &longs;it <lb/>
circulus, uel ellip&longs;is circa diametrum ab; minor circa cd. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000660">Rur&longs;us inter lineas ab, cd inueniatur proportionalis be: <lb/>
&amp; ab e ducta ef &aelig;quidi&longs;tante bd, qu&aelig; lineam ca in f &longs;ecet, 



<pb xlink:href="023/01/070.jpg"/>per f planum ba&longs;ibus &aelig;quidi&longs;tans ducatur, ut &longs;it &longs;ectio cir<lb/>
culus, uel ellip&longs;is circa diametrum fg. <!-- KEEP S--></s>
          <s id="s.000661">Dico &longs;ectionem ab <lb/>
ad &longs;ectionem fg eandem proportionem habere, quam fg <lb/>
ad ip&longs;am cd. <!-- KEEP S--></s>
          <s id="s.000662">Simili enim ratione, qua &longs;upra, demon&longs;trabi&shy;<lb/>
tur quadratum ab ad quadratum fg ita e&longs;&longs;e, ut <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>
<arrow.to.target n="marg81"/><lb/>
fg ad cd quadratum. </s>
          <s id="s.000663">Sed circuli inter &longs;e eandem propor&shy;<lb/>
tionem habent, quam diametrorum quadrata. </s>
          <s id="s.000664">ellip&longs;es au&shy;<lb/>
tem circa ab, fg, cd, qu&aelig; &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="c&otilde;-mentariis">com&shy;<lb/>
mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>
&amp; &longs;ph&aelig;roidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadra<lb/>ta diametrorum, qu&aelig; eiu&longs;dem rationis &longs;unt, ex corollario&shy; <lb/>
<figure id="id.023.01.070.1.jpg" xlink:href="023/01/070/1.jpg"/><lb/>
&longs;eptim&aelig; propo&longs;itionis eiu&longs;dem li&shy;<lb/>
bri. </s>
          <s id="s.000665">ellip&longs;es enim nunc appello ip&shy;<lb/>
&longs;a &longs;pacia ellip&longs;ibus contenta. </s>
          <s id="s.000666">ergo <lb/>
circulus, uel ellip&longs;is ab ad <expan abbr="circul&utilde;">circulum</expan>, <lb/>
uel ellip&longs;im fg eam proportionem <lb/>
habet, quam circulus, uel ellip&longs;is <lb/>
fg ad circulum uel ellip&longs;im cd. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000667">quod quidem faciendum propo&shy;<lb/>
&longs;uimus.</s>
        </p>
        <p type="margin">
          <s id="s.000668"><margin.target id="marg81"/>2. duode<lb/>
cimi</s>
        </p>
        <p type="head">
          <s id="s.000669">THEOREMA XX. PROPOSITIO XXV.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000670">QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>
uel coni portionis ad pyramidem, uel conum, uel <lb/>
coni portionem, cuius ba&longs;is eadem e&longs;t, &amp; &aelig;qualis <lb/>
altitudo, eandem <expan abbr="proportion&etilde;">proportionem</expan> habet, quam utr&aelig; <lb/>
que ba&longs;es, maior, &amp; minor &longs;imul &longs;umpt&aelig; vn&agrave; <expan abbr="c&utilde;">cum</expan> <lb/>ea, qu&aelig; inter ip&longs;as &longs;it proportionalis, ad ba&longs;im ma<lb/>
iorem.</s>
        </p>
        <pb pagenum="32" xlink:href="023/01/071.jpg"/>
        <p type="main">
          <s id="s.000671">SIT <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> pyramidis, uel coni, uel coni portionis ad, <lb/>
cuius maior ba&longs;is ab, minor cd. <!-- KEEP S--></s>
          <s id="s.000672">&amp; &longs;ecetur altero plano <lb/>
ba&longs;i &aelig;quidi&longs;tante, ita ut &longs;ectio ef &longs;it proportionalis inter <lb/>
ba&longs;es ab, cd. <!-- KEEP S--></s>
          <s id="s.000673">con&longs;tituatur <expan abbr="aut&etilde;">autem</expan> pyramis, uel conus, uel co&shy;<lb/>
ni portio agb, cuius ba&longs;is &longs;it eadem, qu&aelig; ba&longs;is maior fru&shy;<lb/>
<figure id="id.023.01.071.1.jpg" xlink:href="023/01/071/1.jpg"/><lb/>
&longs;ti, &amp; altitudo &aelig;qualis. </s>
          <s id="s.000674">Di&shy;<lb/>
co fru&longs;tum ad ad pyrami&shy;<lb/>
dem, uel conum, uel coni <lb/>
portionem agb eandem <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> habere, <expan abbr="qu&atilde;">quam</expan> <lb/>
utr&aelig;que ba&longs;es, ab, cd un&agrave; <lb/>
cum ef ad ba&longs;im ab. </s>
          <s id="s.000675">e&longs;t <lb/>
enim fru&longs;tum ad &aelig;quale <lb/>
pyramidi, uel cono, uel co&shy;<lb/>
ni portioni, cuius ba&longs;is ex <lb/>
tribus ba&longs;ibus ab, ef, cd <lb/>
con&longs;tat; &amp; altitudo ip&longs;ius <lb/>
altitudini e&longs;t &aelig;qualis: quod mox o&longs;tendemus. </s>
          <s id="s.000676">Sed pyrami<lb/>
<figure id="id.023.01.071.2.jpg" xlink:href="023/01/071/2.jpg"/><lb/>
des, coni, uel coni <expan abbr="porti&otilde;es">portiones</expan>, <lb/>
qu&aelig; &longs;unt &aelig;quali altitudine, <lb/>
<expan abbr="e&atilde;dem">eandem</expan> inter &longs;e, quam ba&longs;es, <lb/>
proportionem habent, &longs;icu&shy;<lb/>
ti demon&longs;tratum e&longs;t, partim <lb/>
<arrow.to.target n="marg82"/><lb/>
ab Euclide in duodecimo li&shy;<lb/>
bro elementorum, partim &agrave; <lb/>
nobis in <expan abbr="c&otilde;mentariis">commentariis</expan> in un&shy;<lb/>
decimam <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> Ar&shy;<lb/>
chimedis de conoidibus, &amp; <lb/>
&longs;ph&aelig;roidibus. </s>
          <s id="s.000677">quare pyra&shy;<lb/>
mis, uel conus, uel coni por&shy;<lb/>
tio, cuius ba&longs;is e&longs;t tribus illis <lb/>
ba&longs;ibus &aelig;qualis ad agb eam <lb/>
habet proportionem, quam <lb/>
ba&longs;es ab, ef, cd ad ab ba&longs;im. </s>
          <s id="s.000678">Fru&longs;tum igitur ad ad agb 



<pb xlink:href="023/01/072.jpg"/>pyramidem, uel conum, uel coni portionem eandem pro&shy;<lb/>
portionem habet, quam ba&longs;es ab, cd un&agrave; cum ef ad ba&shy;<lb/>
&longs;im ab. </s>
          <s id="s.000679">quod demon&longs;trare uolebamus.</s>
        </p>
        <p type="margin">
          <s id="s.000680"><margin.target id="marg82"/>6. 11. duo<lb/>
decimi</s>
        </p>
        <p type="main">
          <s id="s.000681">Fru&longs;tum uero ad &aelig;quale e&longs;&longs;e pyramidi, uel co<lb/>
no, uel coni portioni, cuius ba&longs;is con&longs;tat ex ba&longs;i&shy;<lb/>
bus ab, cd, ef, &amp; altitudo fru&longs;ti altitudini e&longs;t &aelig;&shy;<lb/>
qualis, hoc modo o&longs;tendemus.</s>
        </p>
        <p type="main">
          <s id="s.000682">Sit fru&longs;tum pyramidis abcdef, cuius maior ba&longs;is trian&shy;<lb/>
gulum abc; minor def: &amp; &longs;ecetur plano ba&longs;ibus &aelig;quidi&shy;<lb/>
&longs;tante, quod &longs;ectionem faciat triangulum ghk inter trian&shy;<lb/>
gula abc, def proportionale. </s>
          <s id="s.000683">Iam ex iis, qu&aelig; demon&longs;trata <lb/>
&longs;unt in 23. huius, patet fru&longs;tum abcdef diuidi in tres pyra<lb/>
mides proportionales; &amp; earum maiorem e&longs;&longs;e <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>
abcd <expan abbr="minor&etilde;">minorem</expan> uero defb. </s>
          <s id="s.000684">ergo pyramis &agrave; triangulo ghk <lb/>
con&longs;tituta, qu&aelig; altitudinem habeat fru&longs;ti altitudini &aelig;qua&shy;<lb/>
lem, proportionalis e&longs;t inter pyramides abcd, defb: &amp; <lb/>
idcirco fru&longs;tum abcdef tribus dictis pyramidibus &aelig;qua <lb/>
<figure id="id.023.01.072.1.jpg" xlink:href="023/01/072/1.jpg"/><lb/>
le erit. </s>
          <s id="s.000685">Itaque &longs;i intelligatur alia pyra&shy;<lb/>
mis &aelig;que alta, qu&aelig; ba&longs;im habeat ex tri<lb/>
bus ba&longs;ibus abc, def, ghk con&longs;tan&shy;<lb/>
tem; per&longs;picuum e&longs;t ip&longs;am ei&longs;dem py&shy;<lb/>
ramidibus, &amp; propterea ip&longs;i fru&longs;to &aelig;&shy;<lb/>
qualem e&longs;&longs;e.</s>
        </p>
        <p type="main">
          <s id="s.000686">Rur&longs;us &longs;it fru&longs;tum pyramidis ag, cu<lb/>
ius maior ba&longs;is quadrilaterum abcd, <lb/>
minor efgh: &amp; &longs;ecetur plano ba&longs;i&shy;<lb/>
bus &aelig;quidi&longs;tante, ita ut fiat &longs;ectio qua&shy;<lb/>
drilaterum Klmn, quod &longs;it proportio <lb/>
nale inter quadrilatera abcd, efgh. </s>
          <s id="s.000687">Dico pyramidem, <lb/>
cuius ba&longs;is &longs;it &aelig;qualis tribus quadrilateris abcd, klmn, <lb/>
efgh, &amp; altitudo &aelig;qualis altitudini fru&longs;ti, ip&longs;i fru&longs;to ag <lb/>
&aelig;qualem e&longs;&longs;e. </s>
          <s id="s.000688">Ducatur enim planum per lineas fb, hd, 



<pb pagenum="33" xlink:href="023/01/073.jpg"/>quod diuidat fru&longs;tum in duo fru&longs;ta triangulares ba&longs;es ha&shy;<lb/>
bentia, uidelicet in fru&longs;tum abdefh, &amp; in <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> bcdfgh. </s>
          <lb/>
          <s id="s.000689">erit triangulum kln proportionale inter triangula abd, <lb/>
efh: &amp; triangulum lmn proportionale inter bcd, fgh. </s>
          <lb/>
          <s id="s.000690">&longs;ed pyramis &aelig;que alta, cuius ba&longs;is con&longs;tat ex tribus trian&shy;<lb/>
<figure id="id.023.01.073.1.jpg" xlink:href="023/01/073/1.jpg"/><lb/>
gulis abd, klz, efh, demon&longs;trata <lb/>
e&longs;t fru&longs;to abdcfh &aelig;qualis. </s>
          <s id="s.000691">&amp; &longs;i&shy;<lb/>
militer pyramis, cuius ba&longs;is con&shy;<lb/>
&longs;tat ex triangulis bcd, lmn, fgh <lb/>
&aelig;qualis fru&longs;to bcdfgh: compo&shy;<lb/>
nuntur autem tria quadrilatera a <lb/>
bcd, klmn, efgh &egrave; &longs;ex triangu&shy;<lb/>
lis iam dictis. </s>
          <s id="s.000692">pyramis igitur ba&shy;<lb/>
&longs;im habens &aelig;qualem tribus qua&shy;<lb/>
drilateris, &amp; altitudinem eandem <lb/>
ip&longs;i fru&longs;to ag e&longs;t &aelig;qualis. </s>
          <s id="s.000693">Eodem <lb/>
modo illud <expan abbr="dem&otilde;&longs;trabitur">demon&longs;trabitur</expan> in aliis <lb/>
eiu&longs;modi fru&longs;tis.</s>
        </p>
        <p type="main">
          <s id="s.000694">Sit fru&longs;tum coni, uel coni portionis ad; cuius maior ba&shy;<lb/>
&longs;is circulus, uel ellip&longs;is circa diametrum ab; minor circa <lb/>
c d: &amp; &longs;ecetur plano, quod ba&longs;ibus &aelig;quidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> &longs;e&shy;<lb/>
ctionem circulum, uel ellip&longs;im circa diametrum ef, ita ut <lb/>
inter circulos, uel ellip&longs;es ab, cd &longs;it proportionalis. </s>
          <s id="s.000695">Dico <lb/>
conum, uel coni portionem, cuius ba&longs;is e&longs;t &aelig;qualis tribus <lb/>
circulis, uel tribus ellip&longs;ibus ab, ef, cd; &amp; altitudo eadem, <lb/>
qu&aelig; fru&longs;ti ad, ip&longs;i fru&longs;to &aelig;qualem e&longs;&longs;e. </s>
          <s id="s.000696">producatur enim <lb/>
fru&longs;ti &longs;uperficies quou&longs;que coeat in unum punctum, quod <lb/>
&longs;it g: &amp; coni, uel coni portionis agb axis &longs;it gh, occurrens <lb/>
planis ab, ef, cd in punctis hkl: circa circulum uero de&shy;<lb/>
&longs;cribatur quadratum mnop, &amp; circa ellip&longs;im <expan abbr="rectangul&utilde;">rectangulum</expan> <lb/>
mnop, quod ex ip&longs;ius diametris con&longs;tat: <expan abbr="iunctisq;">iunctisque</expan> gm, <lb/>
g n, go, gp, ex eodem uertice intelligatur pyramis ba&longs;im <lb/>
habens dictum quadratum, uel rectangulum: &amp; plana in <lb/>
quibus &longs;unt circuli, uel ellip&longs;es ef, cd u&longs;que ad eius latera 



<pb xlink:href="023/01/074.jpg"/>producantur. </s>
          <s id="s.000697">Quoniam igitur pyramis &longs;ecatur planis ba&longs;i <lb/>
<arrow.to.target n="marg83"/><lb/>
&aelig;quidi&longs;tantibus, &longs;ectiones &longs;imiles erunt: atque erunt qua&shy;<lb/>
drata, uel rectangula circa circulos, uel ellip&longs;es de&longs;cripta, <lb/>
quemadmodum &amp; in ip&longs;a ba&longs;i. </s>
          <s id="s.000698">Sed cum circuli inter &longs;e <expan abbr="e&atilde;">eam</expan> <lb/>
<arrow.to.target n="marg84"/><lb/>
proportionem habeant, quam diametrorum quadrata: <lb/>
<expan abbr="itemq;">itemque</expan> ellip&longs;es eam quam rectangula ex ip&longs;arum diametris <lb/>
<arrow.to.target n="marg85"/><lb/>
con&longs;tantia: &amp; &longs;it circulus, uel ellip&longs;is circa diametrum ef <lb/>
<figure id="id.023.01.074.1.jpg" xlink:href="023/01/074/1.jpg"/><lb/>
proportionalis inter circulos, uel ellip&longs;es ab, cd; erit re&shy;<lb/>
ctangulum ef etiam inter rectangula ab, cd proportio&shy;<lb/>
nale: per rectangulum enim nunc breuitatis cau&longs;a <expan abbr="eti&atilde;">etiam</expan> ip&shy;<lb/>
&longs;um quadratum intelligemus. </s>
          <s id="s.000699">quare ex iis, qu&aelig; proxime <lb/>
dicta &longs;unt, pyramis ba&longs;im habens &aelig;qualem dictis rectangu<lb/>
lis, &amp; altitudinem eandem, quam fru&longs;tum ad, ip&longs;i fru&longs;to &agrave; <lb/>
pyramide ab&longs;ci&longs;&longs;o &aelig;qualis probabitur. </s>
          <s id="s.000700">ut autem rectangu<lb/>
lum cd ad <expan abbr="rectangul&utilde;">rectangulum</expan> ef, ita circulus, uel ellip&longs;is cd ad ef <lb/>
circulum, uel ellip&longs;im: <expan abbr="componendoq;">componendoque</expan> ut rectangula cd, <lb/>
e f, ad ef rectangulum, ita circuli, uel ellip&longs;es ed, ef, ad ef: <lb/>
&amp; ut rectangulum ef ad rectangulum ab, ita circulus, uel <lb/>
ellip&longs;is ef ad ab circulum, uel ellip&longs;im. </s>
          <s id="s.000701">ergo ex &aelig;quali, &amp; <lb/>
componendo, ut <expan abbr="rect&atilde;gula">rectangula</expan> cd, ef, ab ad ip&longs;um ab, ita cir&shy;



<pb pagenum="34" xlink:href="023/01/075.jpg"/>culi, uel ellip&longs;es cd, ef ab ad circulum, uel ellip&longs;im ab. </s>
          <s id="s.000702">In&shy;<lb/>
telligatur pyramis q ba&longs;im habens &aelig;qualem tribus rectan <lb/>
gulis ab, ef, cd; &amp; altitudinem <expan abbr="e&atilde;dem">eandem</expan>, quam fru&longs;tum ad. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000703">intelligatur etiam conus, uel coni portio q, eadem altitudi<lb/>
ne, cuius ba&longs;is &longs;it tribus circulis, uel tribus ellip&longs;ibus ab, <lb/>
ef, cd &aelig;qualis. </s>
          <s id="s.000704">po&longs;tremo intelligatur pyramis alb, cuius. </s>
          <lb/>
          <s id="s.000705">ba&longs;is &longs;it rectangulum mnop, &amp; altitudo eadem, qu&aelig; fru&shy;<lb/>
&longs;ti: <expan abbr="itemq,">itemque</expan> intelligatur conus, uel coni portio alb, cuius <lb/>
ba&longs;is circulus, uel ellip&longs;is circa diametrum ab, &amp; eadem al<lb/>
<arrow.to.target n="marg86"/><lb/>
titudo. </s>
          <s id="s.000706">ut igitur rectangula ab, ef, cd ad rectangulum ab, <lb/>
ita pyramis q ad pyramidem alb; &amp; ut circuli, uel ellip&shy;<lb/>
&longs;es ab, ef, cd ad ab circulum, uel ellip&longs;im, ita conus, uel co<lb/>
ni portio q ad conum, uel coni portionem alb. </s>
          <s id="s.000707">conus <lb/>
igitur, uel coni portio q ad conum, uel coni portionem <lb/>
alb e&longs;t, ut pyramis q ad pyramidem alb. </s>
          <s id="s.000708">&longs;ed pyramis <lb/>
alb ad pyramidem agb e&longs;t, ut altitudo ad altitudinem, ex <lb/>
20. huius: &amp; ita e&longs;t conus, uel coni portio alb ad conum, <lb/>
uel coni portionem agb ex 14. duodecimi elementorum, <lb/>
&amp; ex iis, qu&aelig; nos demon&longs;trauimus in commentariis in un&shy;<lb/>
decimam de conoidibus, &amp; &longs;ph&aelig;roidibus, propo&longs;itione <lb/>
quarta. </s>
          <s id="s.000709">pyramis autem agb ad pyramidem cgd propor&shy;<lb/>
tionem habet compo&longs;itam ex proportione ba&longs;ium &amp; pro <lb/>
portione altitudinum, ex uige&longs;ima prima huius: &amp; &longs;imili&shy;<lb/>
ter conus, uel coni portio agb ad conum, uel coni portio&shy;<lb/>
nem cgd proportionem habet <expan abbr="compo&longs;it&atilde;">compo&longs;itam</expan> ex ei&longs;dem pro&shy;<lb/>
portionibus, per ea, qu&aelig; in dictis commentariis demon&shy;<lb/>
&longs;trauimus, propo&longs;itione quinta, &amp; &longs;exta: altitudo enim in<lb/>
utri&longs;que eadem e&longs;t, &amp; ba&longs;es inter &longs;e &longs;e eandem habent pro&shy;<lb/>
portionem. </s>
          <s id="s.000710">ergo ut pyramis agb ad pyramidem cgd, ita <lb/>
e&longs;t conus, uel coni portio agb ad agd conum, uel coni <lb/>
portionem: &amp; per <expan abbr="conuer&longs;ion&etilde;">conuer&longs;ionem</expan> rationis, ut pyramis agb <lb/>
ad <expan abbr="&longs;ru&longs;t&utilde;">fru&longs;tum</expan> &agrave; pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio <lb/>
agb ad fru&longs;tum ad. <!-- KEEP S--></s>
          <s id="s.000711">ex &aelig;quali igitur, ut pyramis q ad fru&shy;<lb/>
&longs;tum &agrave; pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio q ad 



<pb xlink:href="023/01/076.jpg"/>fru&longs;tum ad. <!-- KEEP S--></s>
          <s id="s.000712">Sed pyramis q &aelig;qualis e&longs;t fru&longs;to &agrave; pyramide <lb/>
ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. </s>
          <s id="s.000713">ergo &amp; conus, uel coni por&shy;<lb/>
tio q, cuius ba&longs;is ex tribus circulis, uel ellip&longs;ibus ab, ef, cd <lb/>
con&longs;tat, &amp; altitudo eadem, qu&aelig; fru&longs;ti: ip&longs;i fru&longs;to ad e&longs;t &aelig;&shy;<lb/>
qualis. </s>
          <s id="s.000714">atque illud e&longs;t, quod demon&longs;trare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000715"><margin.target id="marg83"/>9 huius</s>
        </p>
        <p type="margin">
          <s id="s.000716"><margin.target id="marg84"/>2. duode&shy;<lb/>cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000717"><margin.target id="marg85"/>7. de co&shy;<lb/>
noidibus <lb/>
&amp; &longs;ph&aelig;&shy;<lb/>
roidibus</s>
        </p>
        <p type="margin">
          <s id="s.000718"><margin.target id="marg86"/>6. II. duo <lb/>
decimi</s>
        </p>
        <p type="head">
          <s id="s.000719">THEOREMA XXI. PROPOSITIO XXVI.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000720">CVIVSLIBET fru&longs;ti &agrave; pyramide, uel cono, <lb/>
uel coni portione ab&longs;cis&longs;i, centrum grauitatis e&longs;t <lb/>
in axe, ita ut eo primum in duas portiones diui&shy;<lb/>
&longs;o, portio &longs;uperior, qu&aelig; minorem ba&longs;im attingit <lb/>
ad portionem reliquam eam habeat proportio&shy;<lb/>
nem, quam duplum lateris, uel diametri maioris <lb/>
ba&longs;is, vn&agrave; cum latere, uel diametro minoris, ip&longs;i <lb/>
re&longs;pondente, habet ad duplum lateris, uel diame&shy;<lb/>
tri minoris ba&longs;is vn&agrave; <expan abbr="c&utilde;">cum</expan> latere, uel diametro ma&shy;<lb/>
ioris: deinde &agrave; puncto diui&longs;ionis quarta parte &longs;u<lb/>
perioris portionis in ip&longs;a &longs;umpta: &amp; rur&longs;us ab in&shy;<lb/>
ferioris portionis termino, qui e&longs;t ad ba&longs;im maio<lb/>
rem, &longs;umpta quarta parte totius axis: centrum &longs;it <lb/>
in linea, qu&aelig; his finibus continetur, atque in eo li<lb/><lb/>
tem propinquiorem minori ba&longs;i, <expan abbr="e&atilde;dem">eandem</expan> propor&shy;<lb/>
tionem habeat, quam fru&longs;tum ad <expan abbr="pyramid&etilde;">pyramidem</expan>, uel <lb/>
conum, uel coni portionem, cuius ba&longs;is &longs;it ea&shy;<lb/>
dem, qu&aelig; ba&longs;is maior, &amp; altitudo fru&longs;ti altitudini <lb/>
&aelig;qualis.</s>
        </p>
        <pb pagenum="35" xlink:href="023/01/077.jpg"/>
        <p type="main">
          <s id="s.000721">Sit fru&longs;tum ae a pyramide, qu&aelig; triangularem ba&longs;im ha&shy;<lb/>
beat ab&longs;ci&longs;&longs;um: cuius maior ba&longs;is triangulum abc, minor <lb/>
def; &amp; axis gh. </s>
          <s id="s.000722">ducto autem plano per axem &amp; per <expan abbr="line&atilde;">lineam</expan> <lb/>
da, quod &longs;ectionem faciat dakl quadrilaterum; puncta <lb/>
Kl lineas bc, ef bifariam &longs;ecabunt. </s>
          <s id="s.000723">nam cum gh &longs;it axis <lb/>
fru&longs;ti: erit h centrum grauitatis trianguli abc: &amp; g <lb/>
<figure id="id.023.01.077.1.jpg" xlink:href="023/01/077/1.jpg"/><lb/>
<arrow.to.target n="marg87"/><lb/>
centrum trianguli def: cen&shy;<lb/>
trum uero cuiuslibet triangu<lb/>
li e&longs;t in recta linea, qu&aelig; ab an&shy;<lb/>
gulo ip&longs;ius ad <expan abbr="dimidi&atilde;">dimidiam</expan> ba&longs;im <lb/>
ducitur ex decimatertia primi <lb/>
libri Archimedis de <expan abbr="c&etilde;tro">centro</expan> gra<lb/>
<arrow.to.target n="marg88"/><lb/>
uitatis planorum. </s>
          <s id="s.000724">quare <expan abbr="cen-tr&utilde;">cen&shy;<lb/>
trum</expan> grauitatis trapezii bcfe <lb/>
e&longs;t in linea kl, quod &longs;it m: &amp; &agrave; <lb/>
puncto m ad axem ducta mn <lb/>
ip&longs;i ak, uel dl &aelig;quidi&longs;tante; <lb/>
erit axis gh diui&longs;us in portio&shy;<lb/>
nes gn, nh, quas diximus: ean <lb/>
dem enim proportionem ha&shy;<lb/>
bet gn ad nh, <expan abbr="qu&atilde;">quam</expan> lm ad mk. </s>
          <lb/>
          <s id="s.000725">At lm ad mK habet eam, <expan abbr="qu&atilde;">quam</expan> <lb/>
duplum lateris maioris ba&longs;is <lb/>
bc una cum latere minoris ef <lb/>
ad duplum lateris ef un&agrave; cum <lb/>
latere bc, ex ultima eiu&longs;dem <lb/>
libri Archimedis. <!-- KEEP S--></s>
          <s id="s.000726">Itaque &agrave; li&shy;<lb/>
nea ng ab&longs;cindatur, quarta <lb/>
pars, qu&aelig; fit np: &amp; ab axe hg ab&longs;cindatur itidem <lb/>
quarta pars ho: &amp; quam proportionem habet fru&longs;tum ad <lb/>
pyramidem, cuius maior ba&longs;is e&longs;t triangulum abc, &amp; alti&shy;<lb/>
tudo ip&longs;i &aelig;qualis; habeat op ad pq.</s>
          <s id="s.000727"> Dico centrum graui&shy;<lb/>
tatis fru&longs;ti e&longs;&longs;e in linea po, &amp; in puncto q.</s>
          <s id="s.000728"> namque ip&longs;um <lb/>
e&longs;&longs;e in linea gh manife&longs;te con&longs;tat. </s>
          <s id="s.000729">protractis enim fru&longs;ti pla<pb xlink:href="023/01/078.jpg"/>nis, quou&longs;que in unum punctum r conueniant; erit pyra&shy;<lb/>
midis abcr, &amp; pyramidis defr grauitatis centrum in li&shy;<lb/>
nea rh. </s>
          <s id="s.000730">ergo &amp; reliqu&aelig; magnitudinis, uidelicet fru&longs;ti cen&shy;<lb/>
trum in eadem linea nece&longs;&longs;ario comperietur. </s>
          <s id="s.000731">Iungantur <lb/>
db, dc, dh, dm: &amp; per lineas db, dc ducto altero plano <lb/>
intelligatur fru&longs;tum in duas pyramides diui&longs;um: in pyra&shy;<lb/>
midem quidem, cuius ba&longs;is e&longs;t triangulum abc, uertex d: <lb/>
&amp; in eam, cuius idem uertex, &amp; ba&longs;is trapezium bcfe. </s>
          <s id="s.000732">erit <lb/>
igitur pyramidis abcd axis dh, &amp; pyramidis bcfed axis <lb/>
d m: atque erunt tres axes gh, dh, dm in eodem plano <lb/>
daKl.</s>
          <s id="s.000733"> ducatur pr&aelig;terea per o linea &longs;t ip&longs;i aK <expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan>, <lb/>
qu&aelig; lineam dh in u &longs;ecet: per p uero ducatur xy &aelig;quidi&shy;<lb/>
<figure id="id.023.01.078.1.jpg" xlink:href="023/01/078/1.jpg"/><lb/>
&longs;tans eidem, &longs;ecansque dm in <lb/>
z: &amp; iungatur zu, qu&aelig; &longs;ecet <lb/>
gh in <foreign lang="greek">f.</foreign> tran&longs;ibit ea per q: &amp; <lb/>
erunt <foreign lang="greek">f</foreign>q unum, atque idem <lb/>
punctum; ut inferius appare&shy;<lb/>
bit. </s>
          <s id="s.000734">Quoniam igitur linea uo <lb/>
<arrow.to.target n="marg89"/><lb/>
&aelig;quidi&longs;tat ip&longs;i dg, erit du ad <lb/>
uh, ut go ad oh. </s>
          <s id="s.000735">Sed go tri&shy;<lb/>
pla e&longs;t oh. </s>
          <s id="s.000736">quare &amp; du ip&longs;ius <lb/>
uh e&longs;t tripla: &amp; ideo pyrami&shy;<lb/>
dis abcd centrum grauitatis <lb/>
erit punctum u. </s>
          <s id="s.000737">Rur&longs;us quo&shy;<lb/>
niam zy ip&longs;i dl &aelig;quidi&longs;tat, dz <lb/>
ad zm e&longs;t, ut ly ad ym: e&longs;tque <lb/>
ly ad ym, ut gp ad pn. </s>
          <s id="s.000738">ergo <lb/>
dz ad zm e&longs;t, ut gp ad pn. </s>
          <lb/>
          <s id="s.000739">Qu&ograve;d cum gp &longs;it tripla pn; <lb/>
erit etiam dz ip&longs;ius zm tri&shy;<lb/>
pla. </s>
          <s id="s.000740">atque ob eandem cau&longs;&shy;<lb/>
&longs;am punctum z e&longs;t <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>
uitatis pyramidis bcfed. </s>
          <s id="s.000741">iun<lb/>
cta igitur zu, in ea erit <expan abbr="c&etilde;trum">centrum</expan> 



<pb pagenum="36" xlink:href="023/01/079.jpg"/>grauitatis magnitudinis, qu&aelig; ex utri&longs;que pyramidibus <expan abbr="c&otilde;">con</expan><lb/>
&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s>
          <s id="s.000742">Sed fru&longs;ti centrum e&longs;t etiam in a&shy;<lb/>
xe gh. </s>
          <s id="s.000743">ergo in puncto <foreign lang="greek">f,</foreign> in quo line&aelig; zu, gh conueniunt. </s>
          <lb/>
          <s id="s.000744"><arrow.to.target n="marg90"/><lb/>
Itaque u<foreign lang="greek">f</foreign> ad <foreign lang="greek">f</foreign>z eam proportionem habet, quam pyramis <lb/>
bcfed ad pyramidem abcd. <!-- KEEP S--></s>
          <s id="s.000745">&amp; componendo uz ad z<foreign lang="greek">f</foreign><lb/>
eam habet, quam fru&longs;tum ad pyramidem abcd. <!-- KEEP S--></s>
          <s id="s.000746">Vt uero <lb/>
uz ad z<foreign lang="greek">f</foreign>, ita op ad p<foreign lang="greek">f</foreign> ob &longs;imilitudinem triangulorum, <lb/>
uo<foreign lang="greek">f</foreign>, zp<foreign lang="greek">f.</foreign> quare op ad p<foreign lang="greek">f</foreign> e&longs;t ut fru&longs;tum ad pyramidem <lb/>
abcd. <!-- KEEP S--></s>
          <s id="s.000747">&longs;ed ita erat op ad pq.</s>
          <s id="s.000748"> &aelig;quales igitur &longs;unt p<foreign lang="greek">f</foreign>, pq: &amp;<lb/>
<arrow.to.target n="marg91"/><lb/>
q<foreign lang="greek">f</foreign> unum atque idem punctum. </s>
          <s id="s.000749">ex quibus &longs;equitur lineam. </s>
          <lb/>
          <s id="s.000750">zu &longs;ecare op in q: &amp; propterea <expan abbr="p&utilde;ctum">punctum</expan> q ip&longs;ius fru&longs;ti gra&shy;<lb/>
uitatis centrum e&longs;&longs;e.</s>
        </p>
        <p type="margin">
          <s id="s.000751"><margin.target id="marg87"/>3. diffi. </s>
          <s id="s.000752">hu<lb/>
ius.</s>
        </p>
        <p type="margin">
          <s id="s.000753"><margin.target id="marg88"/>Vltima <expan abbr="e-iu&longs;d&etilde;">e&shy;<lb/>
iu&longs;dem</expan> libri <lb/>
Archime&shy;<lb/>
dis.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000754"><margin.target id="marg89"/>2. &longs;exti.</s>
        </p>
        <p type="margin">
          <s id="s.000755"><margin.target id="marg90"/>8. primi <lb/>
libri Ar&shy;<lb/>
chimedis <lb/>
de <expan abbr="c&etilde;tro">centro</expan> <lb/>
grauta&shy;<lb/>
tis plano <lb/>
rum</s>
        </p>
        <p type="margin">
          <s id="s.000756"><margin.target id="marg91"/>7. quinti.</s>
        </p>
        <p type="main">
          <s id="s.000757">Sit fru&longs;tum ag &agrave; pyramide, qu&aelig; quadrangularem ba&longs;im <lb/>
habeat ab&longs;ci&longs;&longs;um, cuius maior ba&longs;is abcd, minor efgh, <lb/>
&amp; axis kl. <!-- REMOVE S-->diuidatur autem <expan abbr="prim&utilde;">primum</expan> kl, ita ut quam propor&shy;<lb/>
tionem habet duplum lateris ab un&agrave; cum latere ef ad du <lb/>
plum lateris ef un&agrave; cum ab; habeat km ad ml. <!-- KEEP S--></s>
          <s id="s.000758">deinde &agrave; <lb/>
<expan abbr="p&utilde;cto">puncto</expan> m ad k &longs;umatur quarta pars ip&longs;ius mk qu&aelig; &longs;it mn. </s>
          <lb/>
          <s id="s.000759">&amp; rur&longs;us ab l &longs;umatur quarta pars totius axis lk, qu&aelig; &longs;it <lb/>
lo. </s>
          <s id="s.000760">po&longs;tremo fiat on ad np, ut fru&longs;tum ag ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
cuius ba&longs;is &longs;it eadem, qu&aelig; fru&longs;ti, &amp; altitudo &aelig;qualis. </s>
          <s id="s.000761">Dico <lb/>
punctum p fru&longs;ti ag grauitatis centrum e&longs;&longs;e. </s>
          <s id="s.000762">ducantur <lb/>
enim ac, eg: &amp; intelligantur duo fru&longs;ta triangulares ba&shy;<lb/>
&longs;es habentia, quorum alterum lf ex ba&longs;ibus abc, efg <expan abbr="c&otilde;-&longs;tet">con&shy;<lb/>
&longs;tet</expan>; alterum lh ex ba&longs;ibus acd, egh. </s>
          <s id="s.000763"><expan abbr="Sitq;">Sitque</expan> fru&longs;ti lf axis <lb/>
qr; in quo grauitatis centrum s: fru&longs;ti uero lh axis tu, &amp; <lb/>
x grauitatis centrum: deinde iungantur ur, tq, xs. </s>
          <s id="s.000764">tran&longs;i&shy;<lb/>
bit ur per l: quoniam l e&longs;t centrum grauitatis quadran&shy;<lb/>
guli abcd: &amp; puncta ru grauitatis centra triangulorum <lb/>
abc, acd; in qu&aelig; quadrangulum ip&longs;um diuiditur. </s>
          <s id="s.000765">eadem <lb/>
quoque ratione tq per punctum k tran&longs;ibit. </s>
          <s id="s.000766">At uero pro<lb/>
portiones, ex quibus fru&longs;torum grauitatis centra inquiri&shy;<lb/>
mus, e&aelig;dem &longs;unt in toto fru&longs;to ag, &amp; in fru&longs;tis lf, lh. </s>
          <s id="s.000767">Sunt <lb/>
enim per octauam huius quadrilatera abcd, efgh &longs;imilia: 



<pb xlink:href="023/01/080.jpg"/><expan abbr="itemq;">itemque</expan> &longs;imilia triangula abc, efg: &amp; acd, egh. </s>
          <s id="s.000768"><expan abbr="idcir-coq;">idcir&shy;<lb/>
coque</expan> latera &longs;ibi ip&longs;is re&longs;pondentia eandem inter &longs;e&longs;e pro&shy;<lb/>
portionem &longs;eruant. </s>
          <s id="s.000769">Vt igitur duplum lateris ab un&agrave; <lb/>
cum latere ef ad duplum lateris ef un&agrave; cum ab, ita e&longs;t <lb/>
<figure id="id.023.01.080.1.jpg" xlink:href="023/01/080/1.jpg"/><lb/>
duplum ad late&shy;<lb/>
ris una cum late&shy;<lb/>
re eh ad duplum <lb/>
eh un&agrave; cum ad: <lb/>
&amp; ita in aliis. </s>
          <lb/>
          <s id="s.000770">Rur&longs;us fru&longs;tum <lb/>
ag ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
cuius eadem e&longs;t <lb/>
ba&longs;is, &amp; &aelig;qualis <lb/>
altitudo eandem <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> ha<lb/>
bet, quam <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> <lb/>
lf ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
qu&aelig; e&longs;t <expan abbr="ead&etilde;">eadem</expan> ba&shy;<lb/>
&longs;i, &amp; &aelig;quali alti&shy;<lb/>
tudine: &amp; &longs;imili&shy;<lb/>
ter quam lh fru&shy;<lb/>
&longs;tum ad pyrami&shy;<lb/>
dem, qu&aelig; ex <expan abbr="ea-d&etilde;">ea&shy;<lb/>
dem</expan> ba&longs;i, &amp; &aelig;quali <lb/>
altitudine con&shy;<lb/>
&longs;tat. </s>
          <s id="s.000771">nam &longs;i inter <lb/>
ip&longs;as ba&longs;es me&shy;<lb/>
di&aelig; proportio&shy;<lb/>
nales con&longs;tituan<lb/>
tur, tres ba&longs;es &longs;imul &longs;umpt&aelig; ad maiorem ba&longs;im in om&shy;<lb/>
nibus eodem modo &longs;e habebunt. </s>
          <s id="s.000772">Vnde fit, ut axes Kl, <lb/>
qr, tu &agrave; punctis psx in eandem proportionem &longs;ecen&shy;<lb/>
<arrow.to.target n="marg92"/><lb/>tur. </s>
          <s id="s.000773">ergo linea xs per p tran&longs;ibit: &amp; line&aelig; ru, sx, qt in&shy;<lb/>
ter &longs;e &aelig;quidi&longs;tantes erunt. </s>
          <s id="s.000774">Itaque cum fru&longs;ti ag latera pro&shy;<pb pagenum="37" xlink:href="023/01/081.jpg"/>ducta &longs;uerint, ita ut in unum punctum y coeant, erunt <expan abbr="tri&atilde;">trian</expan><lb/>
gula uyl, xyp, tyk inter &longs;e &longs;imilia: &amp; &longs;imilia etiam triangu<lb/>
la lyr, pys, kyq quare ut in 19 huius, demon&longs;trabitur <lb/>
xp, ad ps: <expan abbr="itemq;">itemque</expan> tk  ad kq eandem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam ul ad lr. </s>
          <s id="s.000775">Sed ut ul ad lr, ita e&longs;t triangulum abc ad <lb/>
triangulum acd: &amp; ut tk ad Kq, ita triangulum efg ad <lb/>
triangulum egh. </s>
          <s id="s.000776">Vt autem triangulum abc ad triangu&shy;<lb/>
lum acd, ita pyramis abcy ad pyramidem acdy. </s>
          <s id="s.000777">&amp; ut <lb/>
triangulum efg ad triangulum egh, ita pyramis efgy <lb/>
ad pyramidem eghy; ergo ut pyramis abcy ad <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>
<arrow.to.target n="marg93"/><lb/>
a cdy, ita pyramis efgy ad pyramidem eghy. </s>
          <s id="s.000778">reliquum <lb/>
igitur <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> lf ad reliquum <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan> lh e&longs;t ut pyramis abcy <lb/>
ad pyramidem acdy, hoc e&longs;t ut ul ad r, &amp; ut xp ad ps. </s>
          <lb/>
          <s id="s.000779">Qu&ograve;d cum fru&longs;ti lf centrum grauitatis &longs;its: &amp; fru&longs;ti lh &longs;it <lb/>
<arrow.to.target n="marg94"/><lb/>
centrum x: con&longs;tat punctum p totius fru&longs;ti ag grauitatis <lb/>
e&longs;&longs;e centrum. </s>
          <s id="s.000780">Eodem modo fiet demon&longs;tratio etiam in <lb/>
aliis pyramidibus.</s>
        </p>
        <p type="margin">
          <s id="s.000781"><margin.target id="marg92"/>a. </s>
          <s id="s.000782">&longs;exti.</s>
        </p>
        <p type="margin">
          <s id="s.000783"><margin.target id="marg93"/>19. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000784"><margin.target id="marg94"/>8. Archi&shy;<lb/>
medis.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000785">Sit fru&longs;tum ad &agrave; cono, uel coni portione ab&longs;ci&longs;&longs;um, eu&shy;<lb/>
ius maior ba&longs;is circulus, uel ellip&longs;is circa diametrum ab; <lb/>
minor circa diametrum cd: &amp; axis ef. </s>
          <s id="s.000786">diuidatur <expan abbr="aut&etilde;">autem</expan> ef <lb/>
in g, ita ut eg ad gf eandem proportionem habeat, quam <lb/>
duplum diametri ab un&agrave; cum diametro ed ad duplum cd <lb/>
un&agrave; cum ab. </s>
          <s id="s.000787"><expan abbr="Sitq;">Sitque</expan> gh quarta pars line&aelig; ge: &amp; &longs;it &longs; K item <lb/>
quarta pars totius fe axis. </s>
          <s id="s.000788">Rur&longs;us quam proportionem <lb/>
habet fru&longs;tum ad ad conum, uel coni portionem, in <expan abbr="ead&etilde;">eadem</expan> <lb/>
ba&longs;i, &amp; &aelig;quali altitudine, habeat linea Kh ad hl. <!-- KEEP S--></s>
          <s id="s.000789">Dico pun&shy;<lb/>
ctum l fru&longs;ti ad grauitatis centrum e&longs;&longs;e. </s>
          <s id="s.000790">Si enim fieri po&shy;<lb/>
te&longs;t, &longs;it m centrum: <expan abbr="producaturq;">producaturque</expan> lm extra fru&longs;tum in n: <lb/>
&amp; ut nl ad lm, ita fiat circulus, uel ellip&longs;is circa <expan abbr="diametr&utilde;">diametrum</expan> <lb/>
ab ad aliud &longs;pacium, in quo &longs;it o. </s>
          <s id="s.000791">Itaque in circulo, uel <lb/>
ellip&longs;i circa diametrum ab rectilinea figura plane de&longs;cri&shy;<lb/>
batur, ita ut qu&aelig; relinquuntur portiones &longs;int o &longs;pacio mi&shy;<lb/>
nores: &amp; intelligatur pyramis apb, ba&longs;im habens rectili&shy;<lb/>
neam figuram in circulo, uel ellip&longs;i ab de&longs;criptam: &agrave; qua 



<pb xlink:href="023/01/082.jpg"/>fru&longs;tum pyramidis &longs;it ab&longs;ci&longs;&longs;um. </s>
          <s id="s.000792">erit ex iis qu&aelig; proxime <lb/>
tradidimus, fru&longs;ti pyramidis ad centrum grauitatis l. <!-- KEEP S--></s>
          <s id="s.000793">Quo<lb/>
niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir <lb/>
<figure id="id.023.01.082.1.jpg" xlink:href="023/01/082/1.jpg"/><lb/>
culus, uel ellip&longs;is ab ad <lb/>
portiones dictas <expan abbr="maior&etilde;">maiorem</expan> <lb/>
proportionem, qu&agrave;m nl <lb/>
ad lm. </s>
          <s id="s.000794">&longs;ed ut circulus, uel <lb/>
ellip&longs;is ab ad portiones, <lb/>
ita apb conus, uel coni <lb/>
portio ad &longs;olidas portio&shy;<lb/>
nes, id quod &longs;upra demon <lb/>
&longs;tratum e&longs;t: &amp; ut circulus <lb/>
<arrow.to.target n="marg95"/><lb/>
uel ellip&longs;is cd ad portio&shy;<lb/>
nes, qu&aelig; ip &longs;i in&longs;unt, ita co<lb/>
nus, uel coni portio cpd <lb/>
ad &longs;olidas ip&longs;ius portio&shy;<lb/>
nes. </s>
          <s id="s.000795">Qu&ograve;d cum figur&aelig; in <lb/>
circulis, uel ellip&longs;ibus ab <lb/>
cd de&longs;cript&aelig; &longs;imiles &longs;int, <lb/>
erit proportio circuli, uel <lb/>
ellip&longs;is ab ad &longs;uas portio <lb/>
nes, <expan abbr="ead&etilde;">eadem</expan>, qu&aelig; circuli uel <lb/>
ellip&longs;is cd ad &longs;uas. </s>
          <s id="s.000796">ergo <lb/>
conus, uel coni portio ap<lb/>
b ad portiones &longs;olidas <expan abbr="e&atilde;-dem">ean&shy;<lb/>
dem</expan> habet <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam conus, uel coni por<lb/>
tio cpd ad &longs;olidas ip&longs;ius <lb/>
<arrow.to.target n="marg96"/><lb/>
portiones. </s>
          <s id="s.000797">reliquum igi&shy;<lb/>
tur coni, uel coni portionis <expan abbr="fru&longs;t&utilde;">fru&longs;tum</expan>, &longs;cilicet ad ad reliquas <lb/>
portiones &longs;olidas in ip&longs;o contentas eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>
habet, quam conus, uel coni portio apb ad &longs;olidas portio<lb/>nes: hoc e&longs;t eandem, quam circulus, uel ellip&longs;is ab ad por<lb/>
tiones planas. </s>
          <s id="s.000798">quare fru&longs;tum coni, uel coni portionis ad 



<pb pagenum="38" xlink:href="023/01/083.jpg"/>ad portiones &longs;olidas maiorem habet <expan abbr="proportion&etilde;">proportionem</expan>, qu&agrave;m <lb/>
nl ad lm: &amp; diuidendo fru&longs;tum pyramidis ad dictas por&shy;<lb/>
tiones maiorem proportionem habet, qu&agrave;m nm ad ml. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000799">fiat igitur ut fru&longs;tum pyramidis ad portiones, ita qm ad <lb/>
m l. <!-- KEEP S--></s>
          <s id="s.000800">Itaque quoniam &agrave; fru&longs;to coni, uel coni portionis ad, <lb/>
cuius grauitatis centrum e&longs;t m, aufertur fru&longs;tum pyrami&shy;<lb/>
dis habens centrum l; erit reliqu&aelig; magnitudinis, qu&aelig; ex <lb/>
portionibus &longs;olidis con&longs;tat; grauitatis <expan abbr="c&etilde;trum">centrum</expan> in linea lm <lb/>
producta, atque in puncto q, extra figuram po&longs;ito: quod <lb/>
fieri nullo modo pote&longs;t. </s>
          <s id="s.000801">relinquitur ergo, ut punctum l &longs;it <lb/>
fru&longs;ti ad grauitatis centrum. </s>
          <s id="s.000802">quz omnia demon&longs;tranda <lb/>
proponebantur.</s>
        </p>
        <p type="margin">
          <s id="s.000803"><margin.target id="marg95"/>22. huius</s>
        </p>
        <p type="margin">
          <s id="s.000804"><margin.target id="marg96"/>19. qu&iacute;nti</s>
        </p>
        <p type="head">
          <s id="s.000805">THEOREMA XXII. PROPOSITIO XXVII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000806">OMNIVM &longs;olidorum in &longs;ph&aelig;ra de&longs;cripto&shy;<lb/>
rum, qu&aelig; &aelig;qualibus, &amp; &longs;imilibus ba&longs;ibus conti&shy;<lb/>
nentur, centrum grauitatis e&longs;t idem, quod &longs;ph&aelig;&shy;<lb/>
r&aelig; centrum.</s>
        </p>
        <p type="main">
          <s id="s.000807">Solida eiu&longs;modi corpora regularia appellare &longs;olent, de <lb/>
quibus agitur in tribus ultimis libris elementorum: &longs;unt <lb/>
autem numero quinque, tetrahedrum, uel pyramis, hexa&shy;<lb/>
hedrum, uel cubus, octahedrum, dodecahedrum, &amp; ico&longs;a&shy;<lb/>
hedrum.</s>
        </p>
        <p type="main">
          <s id="s.000808">Sit primo abcd pyramis <expan abbr="&itilde;">im</expan> &longs;ph&aelig;ra de&longs;cripta, cuius &longs;ph&aelig;<lb/>
r&aelig; centrum &longs;it e. </s>
          <s id="s.000809">Dico e pyramidis abcd grauitatis e&longs;&longs;e <lb/>
centrum. </s>
          <s id="s.000810">Si enim iuncta dc producatur ad ba&longs;im abc in <lb/>
f; ex iis, qu&aelig; demon&longs;trauit Campanus in quartodecimo li<lb/>
bro elementorum, propo&longs;itione decima quinta, &amp; decima <lb/>
&longs;eptima, erit f centrum circuli circa triangulum abc de&shy;<lb/>
&longs;cripti: atque erit ef &longs;exta pars ip&longs;ius &longs;ph&aelig;r&aelig; axis. </s>
          <s id="s.000811">quare <lb/>
ex prima huius con&longs;tat trianguli abc grauitatis centrum <lb/>
e&longs;&longs;e punctum f: &amp; idcirco lineam df e&longs;&longs;e pyramidis axem. 



<pb xlink:href="023/01/084.jpg"/><figure id="id.023.01.084.1.jpg" xlink:href="023/01/084/1.jpg"/><lb/>
At cum ef &longs;it &longs;exta pars axis <lb/>
&longs;ph&aelig;r&aelig;, erit d  tripla ef. </s>
          <s id="s.000812">ergo <lb/>
punctum e e&longs;t grauitatis cen&shy;<lb/>
trum ip&longs;ius pyramidis: quod <lb/>
in uige&longs;ima &longs;ecunda huius de&shy;<lb/>
mon&longs;tratum &longs;uit. </s>
          <s id="s.000813">Sed e e&longs;t cen<lb/>
trum &longs;ph&aelig;r&aelig;. </s>
          <s id="s.000814">Sequitur igitur, <lb/>
ut centrum grauitatis pyrami&shy;<lb/>
dis in &longs;ph&aelig;ra de&longs;cript&aelig; idem <lb/>
&longs;it, quod ip&longs;ius &longs;ph&aelig;r&aelig; cen&shy;<lb/>
trum.</s>
        </p>
        <p type="main">
          <s id="s.000815">Sit cubus in &longs;ph&aelig;ra de&longs;criptus ab, &amp; oppo&longs;itorum pla&shy;<lb/>
norum lateribus bifariam diui&longs;is, per puncta diui&longs;ionum <lb/>
plana ducantur, ut communis ip&longs;orum &longs;ectio &longs;it recta li&shy;<lb/>
nea cd. <!-- KEEP S--></s>
          <s id="s.000816">Itaque &longs;i ducatur ab, &longs;olidi &longs;cilicet diameter, line&aelig; <lb/>
ab, cd ex trige&longs;imanona undecimi &longs;e&longs;e bifariam &longs;ecabunt. </s>
          <lb/>
          <s id="s.000817"><figure id="id.023.01.084.2.jpg" xlink:href="023/01/084/2.jpg"/><lb/>
&longs;ecent autem in puncto e. </s>
          <s id="s.000818">erit, <lb/>
e <expan abbr="centr&utilde;">centrum</expan> grauitatis &longs;olidi ab, <lb/>
id quod demon&longs;tratum e&longs;t in <lb/>
octaua huius. </s>
          <s id="s.000819">Sed quoniam ab <lb/>
e&longs;t &longs;ph&aelig;r&aelig; diametro &aelig;qualis, <lb/>
ut in decima quinta propo&longs;i&shy;<lb/>
tione tertii decimi libri <expan abbr="elem&etilde;">elemen</expan><lb/>
torum o&longs;tenditur: punctum e <lb/>
&longs;ph&aelig;r&aelig; quoque centrum erit. </s>
          <lb/>
          <s id="s.000820">Cubi igitur in &longs;ph&aelig;ra de&longs;cri&shy;<lb/>
pti grauitatis centrum idem <lb/>
e&longs;t, quod centrum ip&longs;ius &longs;ph&aelig;r&aelig;.</s>
        </p>
        <p type="main">
          <s id="s.000821">Sit octahedrum abcdef, in &longs;ph&aelig;ra de&longs;criptum, cuius <lb/>
&longs;ph&aelig;r&aelig; centrum &longs;it g. </s>
          <s id="s.000822">Dico punctum g ip&longs;ius octahedri <lb/>
grauitatis centrum e&longs;&longs;e. </s>
          <s id="s.000823">Con&longs;tat enim ex iis, qu&aelig; demon&shy;<lb/>
&longs;trata &longs;unt &agrave; Campano in quinto decimo libro elemento&shy;<lb/>
rum, propo&longs;itione &longs;extadecima eiu&longs;modi &longs;olidum diuidi <lb/>
in duas pyramides &aelig;quales, &amp; &longs;imiles; uidelicet in pyrami&shy;



<pb pagenum="39" xlink:href="023/01/085.jpg"/>dem, cuius ba&longs;is e&longs;t quadratum abcd, &amp; altitudo eg: &amp; <lb/>
in pyramidem, cuius <expan abbr="ead&etilde;">eadem</expan> ba&longs;is, <expan abbr="altitudoq;">altitudoque</expan> fg; ut &longs;int eg, <lb/>
gf &longs;emidiametri &longs;ph&aelig;r&aelig;, &amp; linea una. </s>
          <s id="s.000824"><expan abbr="C&utilde;">Cum</expan> igitur g &longs;it &longs;ph&aelig;&shy;<lb/>
r&aelig; centrum, erit etiam centrum circuli, qui circa <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>
abcd de&longs;cribitur: &amp; propterea eiu&longs;dem quadrati grauita<lb/>
tis centrum: quod in prima propo&longs;itione huius demon&shy;<lb/>
&longs;tratum e&longs;t. </s>
          <s id="s.000825">quare pyramidis abcde axis erit eg: &amp; pyra<lb/>
midis abcdf axis fg. <!-- KEEP S--></s>
          <s id="s.000826">Itaque &longs;it h centrum grauitatis py&shy;<lb/>
ramidis abcde, &amp; pyramidis abcdf centrum &longs;it <emph type="italics"/>K:<emph.end type="italics"/> per&shy;<lb/>
&longs;picuum e&longs;t ex uige&longs;ima &longs;ecunda propo&longs;itione huius, <expan abbr="line&atilde;">lineam</expan> <lb/>
<figure id="id.023.01.085.1.jpg" xlink:href="023/01/085/1.jpg"/><lb/>
ch triplam e&longs;&longs;e hg: <expan abbr="c&otilde;">com</expan><lb/>
<expan abbr="ponendoq;">ponendoque</expan> eg ip&longs;ius g <lb/>
h quadruplam. </s>
          <s id="s.000827">&amp; <expan abbr="ead&etilde;">eadem</expan> <lb/>
ratione fg <expan abbr="quadrupl&atilde;">quadruplam</expan> <lb/>
ip&longs;ius gk quod cum e <lb/>
g, gf &longs;int &aelig;quales, &amp; h <lb/>
g, g <emph type="italics"/>K<emph.end type="italics"/> nece&longs;&longs;ario &aelig;qua&shy;<lb/>
les erunt. </s>
          <s id="s.000828">ergo ex quar<lb/>
ta propo&longs;itione primi <lb/>
libri Archimedis de <expan abbr="c&etilde;-tro">cen&shy;<lb/>
tro</expan> grauitatis <expan abbr="planor&utilde;">planorum</expan>, <lb/>
totius octahedri, quod <lb/>
ex dictis pyramidibus <lb/>
con&longs;tat, centrum graui <lb/>
tatis erit punctum g idem, quod ip&longs;ius &longs;ph&aelig;r&aelig; centrum.</s>
        </p>
        <p type="main">
          <s id="s.000829">Sit ico&longs;ahedrum ad de&longs;criptum in &longs;ph&aelig;ra, cuius <expan abbr="centr&utilde;">centrum</expan> <lb/>
&longs;it g. <!-- KEEP S--></s>
          <s id="s.000830">Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. </s>
          <s id="s.000831">Si <lb/>
enim ab angulo a per g ducatur recta linea u&longs;que ad &longs;ph&aelig;<lb/>
r&aelig; &longs;uperficiem; con&longs;tat ex &longs;exta decima propo&longs;itione libri <lb/>
tertii decimi elementorum, cadere eam in angulum ip&longs;i a <lb/>
oppo&longs;itum. </s>
          <s id="s.000832">cadat in d: <expan abbr="&longs;itq;">&longs;itque</expan> una aliqua ba&longs;is ico&longs;ahedri tri&shy;<lb/>
angulum abc: &amp; iunct&aelig; bg, producantur, &amp; cadant in <lb/>
angulos ef, ip&longs;is bc oppo&longs;itos. </s>
          <s id="s.000833">Itaque per triangula <lb/>
abc, def ducantur plana &longs;ph&aelig;ram &longs;ecantia.</s>
          <s id="s.000834"> erunt h&aelig; &longs;e-



<pb xlink:href="023/01/086.jpg"/>ctiones circuli ex prima propo&longs;itione &longs;ph&aelig;ricorum Theo<lb/>
do&longs;ii: unus quidem circa triangulum abc de&longs;criptus: al&shy;<lb/>
ter uero circa def: &amp; quoniam triangula abc, def &aelig;qua&shy;<lb/>
lia &longs;unt, &amp; &longs;imilia; erunt ex prima, &amp; &longs;ecunda propo&longs;itione <lb/>
duodecimi libri elementorum, circuli quoque inter &longs;e &longs;e <lb/>
&aelig;quales. </s>
          <s id="s.000835">po&longs;tremo a centro g ad circulum abc perpendi<lb/>
cularis ducatur gh; &amp; alia perpendicularis ducatur ad cir<lb/>
culum def, qu&aelig; &longs;it gk; &amp; iungantur ah, dk per&longs;picuum <lb/>
e&longs;t ex corollario prim&aelig; &longs;ph&aelig;ricorum Theodo&longs;ii, punctum <lb/>
h centrum e&longs;&longs;e circuli abc, &amp; k centrum circuli def. </s>
          <s id="s.000836">Quo<lb/>
niam igitur triangulorum gah, gdK latus ag e&longs;t &aelig;quale la<lb/>
teri gd; &longs;unt enim &agrave; centro &longs;ph&aelig;r&aelig; ad &longs;uperficiem: atque <lb/>
e&longs;t ah &aelig;quale dk: &amp; ex &longs;exta propo&longs;itione libri primi &longs;ph&aelig;<lb/>
ricorum Theodo&longs;ii gh ip&longs;i gK: triangulum gah &aelig;quale <lb/>
erit, &amp; &longs;imile gdk triangulo: &amp; angulus agh &aelig;qualis an&shy;<lb/>
<arrow.to.target n="marg97"/><lb/>
gulo dg <emph type="italics"/>K.<emph.end type="italics"/> &longs;ed anguli agh, hgd &longs;unt &aelig;quales duobus re&shy;<lb/>
ctis. </s>
          <s id="s.000837">ergo &amp; ip&longs;i hgd, dgk duobus rectis &aelig;quales erunt. </s>
          <lb/>
          <s id="s.000838"><arrow.to.target n="marg98"/><lb/>
&amp; idcirco hg, g <emph type="italics"/>K<emph.end type="italics"/> una, atque eadem erit linea. </s>
          <s id="s.000839">cum autem <lb/>
<figure id="id.023.01.086.1.jpg" xlink:href="023/01/086/1.jpg"/><lb/>
h &longs;it <expan abbr="centr&utilde;">centrum</expan> circuli, &amp; tri&shy;<lb/>
anguli abc grauitatis cen<lb/>
<expan abbr="tr&utilde;">trum</expan> probabitur ex iis, qu&aelig; <lb/>
in prima propo&longs;itione hu<lb/>
ius tradita &longs;unt. </s>
          <s id="s.000840">quare gh <lb/>
erit pyramidis abcg axis. </s>
          <lb/>
          <s id="s.000841">&amp; ob eandem cau&longs;&longs;am gk <lb/>
axis pyramidis defg. <!-- KEEP S--></s>
          <s id="s.000842">lta&shy;<lb/>
que centrum grauitatls py<lb/>
ramidis abcg &longs;it <expan abbr="p&utilde;ctum">punctum</expan> <lb/>
l, &amp; pyramidis defg &longs;it m. </s>
          <lb/>
          <s id="s.000843">Similiter ut &longs;upra demon&shy;<lb/>
&longs;trabimus mg, gl inter &longs;e &aelig;quales e&longs;&longs;e, &amp; punctum g graui <lb/>
tatis centrum magnitudinis, qu&aelig; ex utri&longs;que pyramidibus <lb/>
con&longs;tat. </s>
          <s id="s.000844">eodem modo demon&longs;trabitur, quarumcunque <lb/>
duarum pyramidum, qu&aelig; opponuntur, grauitatis <expan abbr="centr&utilde;">centrum</expan> 



<pb pagenum="40" xlink:href="023/01/087.jpg"/>e&longs;&longs;e punctum g. <!-- KEEP S--></s>
          <s id="s.000845">Sequitur ergo ut ico&longs;ahedri centrum gra<lb/>
uitatis &longs;it idem, quod ip&longs;ius &longs;ph&aelig;r&aelig; centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000846"><margin.target id="marg97"/>13. primi</s>
        </p>
        <p type="margin">
          <s id="s.000847"><margin.target id="marg98"/>14. primi</s>
        </p>
        <p type="main">
          <s id="s.000848">Sit dodecahedrum af in &longs;ph&aelig;ra de&longs;ignatum, &longs;itque &longs;ph&aelig;<lb/>
r&aelig; centrum m. </s>
          <s id="s.000849">Dico m centrum e&longs;&longs;e grauitatis ip&longs;ius do&shy;<lb/>
decahedri. </s>
          <s id="s.000850">Sit enim pentagonum abcde una ex duode&shy;<lb/>
cim ba&longs;ibus &longs;olidi af: &amp; iuncta am producatur ad &longs;ph&aelig;r&aelig; <lb/>
&longs;uperficiem. </s>
          <s id="s.000851">cadet in angulum ip&longs;i a oppo&longs;itum; quod col&shy;<lb/>
ligitur ex decima &longs;eptima propo&longs;itione tertiidecimi libri <lb/>
elementorum. </s>
          <s id="s.000852">cadat in f. </s>
          <s id="s.000853">at &longs;i ab aliis angulis bcde per <expan abbr="c&etilde;">cen</expan><lb/>
trum itidem line&aelig; ducantur ad &longs;uperficiem &longs;ph&aelig;r&aelig; in pun<lb/>
cta ghkl; cadent h&aelig; in alios angulos ba&longs;is, qu&aelig; ip&longs;i abcd <lb/>
ba&longs;i opponitur. </s>
          <s id="s.000854">tran&longs;eant ergo per pentagona abcde, <lb/>
fghKl plana &longs;ph&aelig;ram &longs;ecantia, qu&aelig; facient &longs;ectiones cir&shy;<lb/>
culos &aelig;quales inter &longs;e &longs;e: po&longs;tea ducantur ex centro &longs;ph&aelig;r&aelig; <lb/>
<figure id="id.023.01.087.1.jpg" xlink:href="023/01/087/1.jpg"/><lb/>
m perpendiculares ad pla&shy;<lb/>
na dictorum <expan abbr="circulor&utilde;">circulorum</expan>; ad <lb/>
circulum quidem abcde <lb/>
perpendicularis mn: &amp; ad <lb/>
circulum fghKl ip&longs;a mo, <lb/>
<arrow.to.target n="marg99"/><lb/>
erunt puncta no <expan abbr="circulor&utilde;">circulorum</expan> <lb/>
centra: &amp; line&aelig; mn, mo in<lb/>
ter &longs;e &aelig;quales: qu&ograve;d circu&shy;<lb/>
<arrow.to.target n="marg100"/><lb/>
li &aelig;quales &longs;int. </s>
          <s id="s.000855">Eodem mo<lb/>
do, quo &longs;upra, demon&longs;trabi<lb/>
mus lineas mn, mo in <expan abbr="un&atilde;">unam</expan> <lb/>
atque eandem lineam con&shy;<lb/>
uenire. </s>
          <s id="s.000856">ergo cum puncta no &longs;int centra circulorum, con&shy;<lb/>
&longs;tat ex prima huius &amp; <expan abbr="pentagonor&utilde;">pentagonorum</expan> grauitatis e&longs;&longs;e centra: <lb/>
<expan abbr="idcircoq;">idcircoque</expan>  mn, mo pyramidum abcdem, fghklm axes. </s>
          <lb/>
          <s id="s.000857">ponatur abcdem pyramidis grauitatis centrum p: &amp; py<lb/>
ramidis fghklm ip&longs;um q centrum. </s>
          <s id="s.000858">erunt pm, mq &aelig;qua&shy;<lb/>
les, &amp; punctum m grauitatis centrum magnitudinis, qu&aelig; <lb/>
ex ip&longs;is pyramidibus con&longs;tat. </s>
          <s id="s.000859"><expan abbr="eod&etilde;">eodem</expan> modo probabitur qua&shy;<lb/>
rumlibet pyramidum, qu&aelig; &egrave; regione opponuntur, <expan abbr="centr&utilde;">centrum</expan> 



<pb xlink:href="023/01/088.jpg"/>grauitatis e&longs;&longs;e punctum m. </s>
          <s id="s.000860">patet igitur totius dodecahe&shy;<lb/>
dri, centrum grauitatis <expan abbr="id&etilde;">idem</expan> e&longs;&longs;e, quod &amp; &longs;ph&aelig;r&aelig; ip&longs;um com<lb/>
prehendentis centrum. </s>
          <s id="s.000861">qu&aelig; quidem omnia demon&longs;tra&longs;&longs;e <lb/>
oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000862"><margin.target id="marg99"/>corol. </s>
          <s id="s.000863">pri<lb/>
m&aelig; &longs;ph&aelig;<lb/>
ricorum <lb/>
Theod.<!-- REMOVE S--><margin.target id="marg100"/>6. primi <lb/>sph&aelig;rico<lb/>
rum.</s>
        </p>
        <p type="head">
          <s id="s.000864">PROBLEMA VI. PROPOSITIO XXVIII.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000865">DATA qualibet portione conoidis rectangu<lb/>
li, ab&longs;ci&longs;&longs;a plano ad axem recto, uel non recto; fie&shy;<lb/>
ri pote&longs;t, ut portio &longs;olida in&longs;cribatur, uel circum&shy;<lb/>
&longs;cribatur ex cylindris, uel cylindri portionibus, <lb/>
&aelig;qualem habentibus altitudinem, ita ut recta li&shy;<lb/>
nea, qu&aelig; inter centrum grauitatis portionis, &amp; <lb/>
figur&aelig; in&longs;cript&aelig;, uel circum&longs;cript&aelig; interiicitur, <lb/>
&longs;it minor qualibet recta linea propo&longs;ita.</s>
        </p>
        <p type="main">
          <s id="s.000866">Sit portio conoidis rectanguli abc, cuius axis bd, <expan abbr="gra-uitatisq;">gra&shy;<lb/>
uitatisque</expan> centrum e: &amp; &longs;it g recta linea propo&longs;ita. </s>
          <s id="s.000867">quam ue<lb/>
ro proportionem habet linea be ad lineam g, eandem ha&shy;<lb/>
beat portio conoidis ad &longs;olidum h: &amp; circum&longs;cribatur por<lb/>
tioni figura, &longs;icuti dictum e&longs;t, ita ut portiones reliqu&aelig; &longs;int <lb/>
&longs;olido h minores: cuius quidem figur&aelig; centrum grauitatis <lb/>
&longs;it punctum k. </s>
          <s id="s.000868">Dico <expan abbr="line&atilde;">lineam</expan> ke minorem e&longs;&longs;e linea g propo&shy;<lb/>
&longs;ita. </s>
          <s id="s.000869">ni&longs;i enim &longs;it minor, uel &aelig;qualis, uel maior erit. </s>
          <s id="s.000870">&amp; quo&shy;<lb/>
niam figura circum&longs;cripta ad reliquas portiones maiorem <lb/>
<arrow.to.target n="marg101"/><lb/>
proportionem habet, qu&agrave;m portio conoidis ad &longs;olidum h; <lb/>
hoc e&longs;t maiorem, qu&agrave;m bc ad g: &amp; be ad g non minorem <lb/>
habet proportionem, qu&agrave;m ad ke, propterea quod ke non <lb/>
ponitur minor ip&longs;a g: habebit figura circum&longs;cripta ad por<lb/>
tiones reliquas maiorem proportionem qu&agrave;m be ad ek: <lb/>
<arrow.to.target n="marg102"/><lb/>
&amp; diuidendo portio conoidis ad reliquas portiones habe&shy;<lb/>
bit maiorem, qu&agrave;m bk ad Ke. </s>
          <s id="s.000871">quare &longs;i fiat ut portio co&shy;



<pb pagenum="41" xlink:href="023/01/089.jpg"/>noidis ad portiones reliquas, ita alia linea, qu&aelig; &longs;it lk ad <lb/>
ke: erit lk maior, quam bk: &amp; ideo punctum l extra por&shy;<lb/>
<figure id="id.023.01.089.1.jpg" xlink:href="023/01/089/1.jpg"/><lb/>
tionem cadet. </s>
          <s id="s.000872"><expan abbr="Quoni&atilde;">Quoniam</expan> <lb/>
igitur &agrave; figura circum&shy;<lb/>
&longs;cripta, cuius grauitatis <lb/>
centrum e&longs;t k, aufertur <lb/>
portio conoidis, cuius <lb/>
centrum e. </s>
          <s id="s.000873"><expan abbr="habetq;">habetque</expan> lK <lb/>
ad Ke eam proportio&shy;<lb/>
nem, quam portio co&shy;<lb/>
noidis ad reliquas por&shy;<lb/>
tiones; erit punctum l <lb/>
extra portionem <expan abbr="cad&etilde;s">cadens</expan>, <lb/>
centrum magnitudinis <lb/>
ex reliquis portionibus compo&longs;it&aelig;. </s>
          <s id="s.000874">illud autem fieri nullo <lb/>
modo pote&longs;t. </s>
          <s id="s.000875">quare con&longs;tat lineam ke ip&longs;a g linea propo&longs;i<lb/>
ta minorem e&longs;&longs;e.</s>
        </p>
        <p type="margin">
          <s id="s.000876"><margin.target id="marg101"/>8. qu&iacute;nti.</s>
        </p>
        <p type="margin">
          <s id="s.000877"><margin.target id="marg102"/>29. qu&iacute;nti <lb/>
ex tradi&shy;<lb/>
tione <expan abbr="C&atilde;-pani">Cam&shy;<lb/>pani
</expan> .</s>
        </p>
        <p type="main">
          <s id="s.000878">Rur&longs;us in&longs;cribatur portioni figura, uidelicet cylindrus <lb/>
<figure id="id.023.01.089.2.jpg" xlink:href="023/01/089/2.jpg"/><lb/>
mn, ut &longs;it ip&longs;ius altitudo <lb/>
&aelig;qualis dimidio axis bd: <lb/>
&amp; quam proportionem <lb/>
habet be ad g, habeat mn <lb/>
cylindrus ad &longs;olidum o. </s>
          <lb/>
          <s id="s.000879">in&longs;cribatur deinde eidem <lb/>
alia figura, ita ut portio&shy;<lb/>
nes reliqu&aelig; &longs;int &longs;olido o <lb/>
minores: &amp; centrum gra<lb/>
uitatis figur&aelig; &longs;it p. </s>
          <s id="s.000880">Dico <lb/>
lineam pe ip&longs;a g <expan abbr="minor&etilde;">minorem</expan> <lb/>
e&longs;&longs;e. </s>
          <s id="s.000881">&longs;i enim non &longs;it mi&shy;<lb/>
nor, eodem, quo &longs;upra modo demon&longs;trabimus figuram in <lb/>
&longs;criptam ad reliquas portiones maiorem proportionem <lb/>
habere, qu&agrave;m be ad ep. </s>
          <s id="s.000882">&amp; &longs;i fiat alia linea le ad ep, ut e&longs;t <lb/>
figura in&longs;cripta ad reliquas portiones, <expan abbr="p&utilde;ctum">punctum</expan> l extra por



<pb xlink:href="023/01/090.jpg"/>tionem cadet: Itaque cum &agrave; portione conoidis, cuius gra&shy;<lb/>
uitatis centrum e auferatur in&longs;cripta figura, centrum ha&shy;<lb/>
bens p: &amp; &longs;it le ad ep, ut figura in&longs;cripta ad portiones reli<lb/>
quas: erit magnitudinis, qu&aelig; ex reliquis portionibus con<lb/>
&longs;tat, centrum grauitatis punctum l, extra portionem ca&shy;<lb/>
dens. </s>
          <s id="s.000883">quod fieri nequit. </s>
          <s id="s.000884">ergo linea pe minor e&longs;t ip&longs;a g li&shy;<lb/>
nea propo&longs;ita.</s>
        </p>
        <p type="main">
          <s id="s.000885">Ex quibus per&longs;picuum e&longs;t centrum grauitatis <lb/>
figur&aelig; in&longs;cript&aelig;, &amp; circum&longs;cript&aelig; eo magis acce<lb/>
dere ad portionis centrum, quo pluribus cylin&shy;<lb/>
dris, uel cylindri portionibus con&longs;tet: <expan abbr="fiat&qacute;">fiatque</expan>; figu<lb/>
ra in&longs;cripta maior, &amp; circum&longs;cripta minor. </s>
          <s id="s.000886">&amp; <lb/>
quanquam continenter ad portionis <expan abbr="centr&utilde;">centrum</expan> pro&shy;<lb/>
pius admoueatur: nunquam tamen ad ip&longs;um per <lb/>
ueniet. </s>
          <s id="s.000887">&longs;equeretur enim figuram in&longs;criptam, <expan abbr="n&otilde;">non</expan> <lb/>
&longs;olum portioni, &longs;ed etiam circum&longs;cript&aelig; figur&aelig; <lb/>
&aelig;qualem e&longs;&longs;e. </s>
          <s id="s.000888">quod e&longs;t ab&longs;urdum.</s>
        </p>
        <p type="head">
          <s id="s.000889">THEOREMA XXIII. PROPOSITIO XXIX.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000890">CVIVSLIBET portionis conoidis rectangu&shy;<lb/>
li axis &agrave; <expan abbr="c&etilde;tro">centro</expan> grauitatis ita diuiditur, ut pars qu&aelig; <lb/>
terminatur ad uerticem, reliqu&aelig; partis, qu&aelig; ad ba <lb/>
&longs;im &longs;it dupla.</s>
        </p>
        <p type="main">
          <s id="s.000891">SIT portio conoidis rectanguli uel ab&longs;ci&longs;&longs;a plano ad <lb/>
axem recto, uel non recto: &amp; &longs;ecta ip&longs;a altero plano per <expan abbr="ax&etilde;">axem</expan><lb/>
 &longs;it &longs;uperficiei &longs;ectio abc rectanguli coni &longs;ectio, uel parabo <lb/>
le; plani ab&longs;cindentis portionem &longs;ectio &longs;it recta linea ac: <lb/>
axis portionis, &amp; &longs;ectionis diameter bd. </s>
          <s id="s.000892">Sumatur autem <lb/>
in linea bd punctum e, ita ut be &longs;it ip&longs;ius ed dupla. </s>
          <s id="s.000893">Dico 



<pb pagenum="42" xlink:href="023/01/091.jpg"/><figure id="id.023.01.091.1.jpg" xlink:href="023/01/091/1.jpg"/><lb/>
e portionis ab <lb/>
c grauitatis e&longs;&longs;e <lb/>
centrum. </s>
          <s id="s.000894">Diui&shy;<lb/>
datur enim bd <lb/>
bifariam in m: <lb/>
&amp; rur&longs;us dm, m<lb/>
b bifariam diui&shy;<lb/>
dantur in pun&shy;<lb/>
ctis n, o: <expan abbr="in&longs;cri-baturq;">in&longs;cri&shy;<lb/>
baturque</expan> portio&shy;<lb/>
ni figura &longs;olida, <lb/>
&amp; altera circum <lb/>&longs;cribatur ex cy<lb/>
lindris &aelig;qualem <lb/>
altitudinem ha&shy;<lb/>
bentibus, ut &longs;u&shy;<lb/>
perius <expan abbr="dict&utilde;">dictum</expan> e&longs;t. </s>
          <lb/>
          <s id="s.000895">Sit autem pri&shy;<lb/>
mum figura in&shy;<lb/>
&longs;cripta <expan abbr="cyl&itilde;drus">cylindrus</expan> <lb/>
fg: &amp; <expan abbr="circ&utilde;&longs;cri">circum&longs;cri</expan>
&shy;<lb/>
pta ex cylindris <lb/>
ah, Kl con&longs;tet. </s>
          <lb/>
          <s id="s.000896"><arrow.to.target n="marg103"/><lb/>
punctum n erit <lb/>
centrum graui&shy;<lb/>
tatis figur&aelig; in&shy;<lb/>
&longs;cript&aelig;, <expan abbr="medi&utilde;">medium</expan> <lb/>
&longs;cilicet ip&longs;ius d <lb/>
m axis: <expan abbr="atq;">atque</expan> <expan abbr="id&etilde;">idem</expan> <lb/>
erit centrum cy<lb/>
lindri ah: &amp; cy&shy;<lb/>
lindri kl <expan abbr="centr&utilde;">centrum</expan> <lb/>
o, axis bm me&shy;<lb/>
dium. </s>
          <s id="s.000897">quare &longs;i li



<pb xlink:href="023/01/092.jpg"/><figure id="id.023.01.092.1.jpg" xlink:href="023/01/092/1.jpg"/><lb/>
neam on ita di <lb/>
ui&longs;erimus in p, <lb/>
ut <expan abbr="qu&atilde;">quam</expan> <expan abbr="propor-tion&etilde;">propor&shy;<lb/>
tionem</expan> habet cy&shy;<lb/>
lindrus ah ad <lb/>
cylindrum kl, <lb/>
habeat linea op <lb/>
<arrow.to.target n="marg104"/><lb/>
ad pn: centrum <lb/>
grauitatis toti&shy;<lb/>
us figur&aelig; <expan abbr="circ&utilde;-&longs;cript&aelig;">circum&shy;<lb/>
&longs;cript&aelig;</expan> erit pun<lb/>
<arrow.to.target n="marg105"/><lb/>
ctum p. </s>
          <s id="s.000898">Sed cy&shy;<lb/>
lindri, qui &longs;unt <lb/>
&aelig;quali altitudi&shy;<lb/>
ne, eandem in&shy;<lb/>
ter &longs;e &longs;e, quam <lb/>ba&longs;es propor-<lb/>
tionem habent: <lb/>
<expan abbr="e&longs;tq;">e&longs;tque</expan> ut linea db <lb/>
ad bm, ita <expan abbr="qua-drat&utilde;">qua&shy;<lb/>
dratum</expan> line&aelig; ad <lb/>
ad <expan abbr="quadrat&utilde;">quadratum</expan> ip&shy;<lb/>
&longs;ius Km, ex uige <lb/>
&longs;ima primi libri <lb/>
<arrow.to.target n="marg106"/><lb/>
<expan abbr="conicor&utilde;">conicorum</expan> &amp; ita <lb/>
quadratum ac <lb/>
ad <expan abbr="quadrat&utilde;">quadratum</expan> K <lb/>
<arrow.to.target n="marg107"/><lb/>
g: hoc e&longs;t circu&shy;<lb/>
lus circa diame<lb/>
trum ac ad cir&shy;<lb/>
culum circa dia<lb/>
metrum kg. <!-- KEEP S--></s>
          <s id="s.000899">du<lb/>
pla e&longs;t autem li&shy;<lb/>
nea db line&aelig; 



<pb pagenum="43" xlink:href="023/01/093.jpg"/>bm. </s>
          <s id="s.000900">ergo circulus ac circuli kg: &amp; idcirco cylindrus <lb/>
ah cylindri k. </s>
          <s id="s.000901">l duplus erit. </s>
          <s id="s.000902">quare &amp; linea op dupla <lb/>
ip&longs;ius pn. </s>
          <s id="s.000903">Deinde in&longs;cripta &amp; circum&longs;cripta portioni <lb/>
alia figura, ita ut in&longs;cripta con&longs;tituatur ex tribus cylin&shy;<lb/>
dris qr, sg, tu: circum&longs;cripta uero ex quatuor ax, yz, <lb/>
K<foreign lang="greek">v, ql:</foreign> diuidantur bo, om, mn, nd bifariam in punctis <lb/>
<foreign lang="greek">mnpr.</foreign> Itaque cylindri <foreign lang="greek">ql</foreign> centrum grauitatis e&longs;t punctum <lb/>
<foreign lang="greek">m:</foreign> &amp; cylindri k<foreign lang="greek">h</foreign> centrum <foreign lang="greek">n.</foreign> ergo &longs;i linea <foreign lang="greek">mg</foreign> diuidatur in <foreign lang="greek">s,</foreign><lb/>
ita ut <foreign lang="greek">ms</foreign> ad <foreign lang="greek">sg</foreign> <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="e&atilde;">eam</expan> habeat, quam cylindrus K<foreign lang="greek">h</foreign><lb/>
ad cylindrum <foreign lang="greek">ql,</foreign> uidelicet quam quadratum knr ad qua&shy;<lb/>
<arrow.to.target n="marg108"/><lb/>
dratum <foreign lang="greek">q</foreign>o, hoc e&longs;t, quam linea mb ad bo: erit <foreign lang="greek">s</foreign> centrum <lb/>
magnitudinis compo&longs;it&aelig; ex cylindris <foreign lang="greek">kg, ql.</foreign> &amp; cum linea <lb/>
mb &longs;it dupla bo, erit &amp; <foreign lang="greek">ms</foreign> ip&longs;ius <foreign lang="greek">sn</foreign> dupla. </s>
          <s id="s.000904">pr&aelig;terea quo&shy;<lb/>
niam cylindri yz centrum grauitatis e&longs;t <foreign lang="greek">p,</foreign> linea <foreign lang="greek">sp</foreign> ita diui<lb/>
&longs;a in <foreign lang="greek">t,</foreign> ut <foreign lang="greek">st</foreign> ad <foreign lang="greek">tp</foreign> eam habeat proportionem, quam cylin<lb/>
drus yz ad duos cylindros K<foreign lang="greek">n, ql:</foreign> erit <foreign lang="greek">t</foreign> centrum magnitu<lb/>
dinis, qu&aelig; ex dictis tribus cylindris con&longs;tat. </s>
          <s id="s.000905">cylindrus <expan abbr="au-t&etilde;">au&shy;<lb/>
tem</expan> yz ad cylindrum <foreign lang="greek">ql</foreign> e&longs;t, ut linea nb ad bo, hoc e&longs;t ut 3 <lb/>
ad 1: &amp; ad cylindrum k<foreign lang="greek">h</foreign>, ut nb ad bm, uidelicet ut 3 ad 2. <!-- KEEP S--></s>
          <lb/>
          <s id="s.000906">quare yz <expan abbr="cyl&itilde;drus">cylindrus</expan> duobus cylindris k<foreign lang="greek">n, ql</foreign> &aelig;qualis erit. </s>
          <s id="s.000907">&amp; <lb/>
propterea linea <foreign lang="greek">st</foreign> &aelig;qualis ip&longs;i <foreign lang="greek">tp.</foreign> denique cylindri ax <lb/>
centrum grauitatis e&longs;t punctum <foreign lang="greek">r.</foreign> &amp; cum <foreign lang="greek">tr</foreign> diui&longs;a fuerit <lb/>
in <expan abbr="e&atilde;">eam</expan> proportionem, quam habet cylindrus ax ad tres cy&shy;<lb/>
lindros yz, k<foreign lang="greek">n, ql:</foreign> erit in eo puncto centrum grauitatis <lb/>
totius figur&aelig; <expan abbr="circ&utilde;&longs;cript&aelig;">circum&longs;cript&aelig;</expan>. </s>
          <s id="s.000908">Sed cylindrus ax ad ip&longs;um yz <lb/>
e&longs;t ut linea db ad bn: hoc e&longs;t ut 4 ad 3: &amp; duo cylindri k<foreign lang="greek">h<lb/>
ql</foreign> cylindro y  &longs;unt &aelig;quales. </s>
          <s id="s.000909">cylindrus igitur ax ad tres <lb/>
iam dictos cylindros e&longs;t ut 2 ad 3. Sed <expan abbr="quoni&atilde;">quoniam</expan> <foreign lang="greek">m s</foreign> e&longs;t dua&shy;<lb/>
rum partium, &amp; <foreign lang="greek">s g</foreign> unius, qualium <foreign lang="greek">m p</foreign> e&longs;t &longs;ex; erit <foreign lang="greek">s p</foreign> par&shy;<lb/>
tium quatuor: <expan abbr="proptereaq;">proptereaque</expan> <foreign lang="greek">tp</foreign> duarum, &amp; <foreign lang="greek">np,</foreign> hoc e&longs;t <foreign lang="greek">pr</foreign><lb/>
trium. </s>
          <s id="s.000910">quare &longs;equitur ut punctum <foreign lang="greek">p</foreign> totius figur&aelig; circum <lb/>
&longs;cript&aelig; &longs;it centrum. </s>
          <s id="s.000911">Itaque fiat <foreign lang="greek">nu</foreign> ad <foreign lang="greek">up,</foreign> ut <foreign lang="greek">ms</foreign> ad <foreign lang="greek">sg.</foreign> &amp; <foreign lang="greek">ur</foreign><lb/>
bifariam diuidatur in <foreign lang="greek">f.</foreign> Similiter ut in circum&longs;cripta figu<lb/>
ra o&longs;tendetur centrum magnitudinis compo&longs;it&aelig; ex cylin-



<pb xlink:href="023/01/094.jpg"/><figure id="id.023.01.094.1.jpg" xlink:href="023/01/094/1.jpg"/><lb/>
dris sg, tu e&longs;&longs;e <lb/>
punctum <foreign lang="greek">u:</foreign> &amp; <lb/>
totius figur&aelig; in <lb/>
&longs;cript&aelig;, qu&aelig; <expan abbr="c&otilde;-&longs;tat">con&shy;<lb/>
&longs;tat</expan> ex cylindris <lb/>
qr, &longs; g, tu e&longs;&longs;e <foreign lang="greek">f</foreign><lb/>
centrum. </s>
          <s id="s.000912">Sunt <lb/>
enim hi cylindri <lb/>
&aelig;quales &amp; &longs;imi&shy;<lb/>
les cylindris yz, <lb/>
K<foreign lang="greek">h, ql,</foreign> figur&aelig; <lb/>
circum&longs;cript&aelig;. </s>
          <lb/>
          <s id="s.000913"><expan abbr="Quoni&atilde;">Quoniam</expan> igitur <lb/>
ut be ad ed, ita <lb/>
e&longs;t op ad pn; <lb/>
<expan abbr="utraq;">utraque</expan> enim u&shy;<lb/>
triu&longs;que e&longs;t du&shy;<lb/>
pla: erit compo<lb/>
nendo, ut bd ad <lb/>
de, ita on ad n <lb/>
p; &amp; permutan <lb/>
do, ut bd ad o<lb/>
n, ita de ad np. </s>
          <lb/>
          <s id="s.000914">Sed bd dupla <lb/>
e&longs;t on. </s>
          <s id="s.000915">ergo &amp; <lb/>
ed ip&longs;ius np du<lb/>
pla erit. </s>
          <s id="s.000916">qu&ograve;d &longs;i <lb/>
ed bifariam di&shy;<lb/>
uidatur <expan abbr="&itilde;">im</expan> <foreign lang="greek">x,</foreign> erit <lb/>
<foreign lang="greek">x</foreign> d, uel e <foreign lang="greek">x</foreign> &aelig;&shy;<lb/>
qualis np: &amp; <lb/>
&longs;ublata en, qu&aelig; <lb/>
e&longs;t <expan abbr="c&otilde;munis">communis</expan> u&shy;<lb/>
trique e <foreign lang="greek">x,</foreign> pn, 



<pb pagenum="44" xlink:href="023/01/095.jpg"/>relinquetur pe ip&longs;i n<foreign lang="greek">x</foreign> &aelig;qualis. </s>
          <s id="s.000917">cum autem be &longs;it dupla <lb/>
ed, &amp; op dupla pn, hoc e&longs;t ip&longs;ius e <foreign lang="greek">x,</foreign> &amp; reliquum, uideli&shy;<lb/>
<arrow.to.target n="marg109"/><lb/>
cet bo un&agrave; cum pe ip&longs;ius reliqui <foreign lang="greek">x</foreign> d duplum erit. </s>
          <s id="s.000918">e&longs;tque <lb/>
bo dupla <foreign lang="greek">r</foreign> d. <!-- KEEP S--></s>
          <s id="s.000919">ergo pe, hoc e&longs;t n<foreign lang="greek">x</foreign> ip&longs;ius <foreign lang="greek">xr</foreign> dupla. </s>
          <s id="s.000920">&longs;ed dn <lb/>
dupla e&longs;t n<foreign lang="greek">r.</foreign> reliqua igitur d<foreign lang="greek">x</foreign> dupla reliqu&aelig; <foreign lang="greek">x</foreign> n. </s>
          <s id="s.000921">&longs;unt au&shy;<lb/>
tem d<foreign lang="greek">x,</foreign> pn inter &longs;e &aelig;quales: <expan abbr="itemq;">itemque</expan> &aelig;quales <foreign lang="greek">x</foreign> n, pe. </s>
          <s id="s.000922">qua&shy;<lb/>
re con&longs;tat np ip&longs;ius pe duplam e&longs;&longs;e. </s>
          <s id="s.000923">&amp; idcirco pe ip&longs;i en <lb/>
&aelig;qualem. </s>
          <s id="s.000924">Rur&longs;us cum &longs;it <foreign lang="greek">mn</foreign> dupla o<foreign lang="greek">n,</foreign> &amp; <foreign lang="greek">m s</foreign> dupla <foreign lang="greek">s g;</foreign> erit <lb/>
etiam reliqua <foreign lang="greek">ns</foreign> reliqu&aelig; <foreign lang="greek">s</foreign> o dupla. </s>
          <s id="s.000925">Eadem quoque ratione <lb/>
<expan abbr="c&otilde;cludetur">concludetur</expan> <foreign lang="greek">p u</foreign> dupla <foreign lang="greek">u</foreign> m. </s>
          <s id="s.000926">ergo ut <foreign lang="greek">ns</foreign> ad <foreign lang="greek">s</foreign> o, ita <foreign lang="greek">pu</foreign> ad <foreign lang="greek">u</foreign> m: <lb/>
<expan abbr="componendoq;">componendoque</expan>, &amp; permutando, ut <foreign lang="greek">n</foreign>o ad <foreign lang="greek">p</foreign>m, ita o<foreign lang="greek">s</foreign> ad <lb/>
m<foreign lang="greek">u:</foreign> &amp; &longs;unt &aelig;quales <foreign lang="greek">n</foreign>o, <foreign lang="greek">p</foreign>m. </s>
          <s id="s.000927">quare &amp; o<foreign lang="greek">s,</foreign> m<foreign lang="greek">u</foreign> &aelig;quales. </s>
          <s id="s.000928">pr&aelig;<lb/>
terea <foreign lang="greek">sp</foreign> dupla e&longs;t <foreign lang="greek">pt,</foreign> &amp; <foreign lang="greek">np</foreign> ip&longs;ius <foreign lang="greek">p</foreign>m. </s>
          <s id="s.000929">reliqua igitur <foreign lang="greek">sn</foreign> re<lb/>
liqu&aelig; m<foreign lang="greek">t</foreign> dupla. </s>
          <s id="s.000930">atque erat <foreign lang="greek">ns</foreign> dupla <foreign lang="greek">s</foreign>o. </s>
          <s id="s.000931">ergo m<foreign lang="greek">t, s</foreign>o &aelig;&shy;<lb/>
quales &longs;unt: &amp; ita &aelig;quales m<foreign lang="greek">u,</foreign> n<foreign lang="greek">f.</foreign> at o<foreign lang="greek">s,</foreign> e&longs;t &aelig;qualis <lb/>
m<foreign lang="greek">u.</foreign> Sequitur igitur, ut omnes o<foreign lang="greek">s,</foreign> m<foreign lang="greek">t,</foreign> m<foreign lang="greek">u,</foreign> n<foreign lang="greek">f</foreign> in&shy;<lb/>
ter &longs;e &longs;int &aelig;quales. </s>
          <s id="s.000932">Sed ut <foreign lang="greek">rp</foreign> ad <foreign lang="greek">pt,</foreign> hoc e&longs;t ut 3 ad 2, ita nd <lb/>
ad d<foreign lang="greek">x:</foreign> <expan abbr="permut&atilde;doq;">permutandoque</expan> ut <foreign lang="greek">rp</foreign> ad nd, ita <foreign lang="greek">pt</foreign> ad d<foreign lang="greek">x.</foreign> &amp; <expan abbr="&longs;&utilde;t">&longs;unt</expan> &aelig;qua<lb/>
les <foreign lang="greek">rp,</foreign> nd. <!-- KEEP S--></s>
          <s id="s.000933">ergo d<foreign lang="greek">x,</foreign> hoc e&longs;t np, &amp; <foreign lang="greek">pt</foreign> &aelig;quales. </s>
          <s id="s.000934">Sed etiam &aelig;&shy;<lb/>
quales n<foreign lang="greek">p, p</foreign>m. </s>
          <s id="s.000935">reliqua igitur <foreign lang="greek">p</foreign>p reliqu&aelig; m<foreign lang="greek">t,</foreign> hoc e&longs;t ip&longs;i <lb/>
n<foreign lang="greek">f</foreign> &aelig;qualis erit. </s>
          <s id="s.000936">quare dempta p<foreign lang="greek">p</foreign> ex pe, &amp; <foreign lang="greek">f</foreign>n dempta ex <lb/>
ne, relinquitur pe &aelig;qualis e<foreign lang="greek">f.</foreign> Itaque <foreign lang="greek">p, f</foreign> centra <expan abbr="figurar&utilde;">figurarum</expan> <lb/>
&longs;ecundo loco de&longs;criptarum a primis centris pn &aelig;quali in&shy;<lb/>
teruallo recedunt. </s>
          <s id="s.000937">qu&ograve;d &longs;i rur&longs;us ali&aelig; figur&aelig; de&longs;cribantur, <lb/>
eodem modo demon&longs;trabimus earum centra &aelig;qualiter ab <lb/>
his recedere, &amp; ad portionis conoidis centrum propius ad <lb/>
moueri. </s>
          <s id="s.000938">Ex quibus con&longs;tat lineam <foreign lang="greek">pf</foreign> &agrave; centro grauitatis <lb/>
portionis diuidi in partes &aelig;quales. </s>
          <s id="s.000939">Si enim fieri pote&longs;t, non <lb/>
&longs;it centrum in puncto e, quod e&longs;t line&aelig; <foreign lang="greek">pf</foreign> medium: &longs;ed in <lb/>
<foreign lang="greek">y:</foreign> &amp; ip&longs;i <foreign lang="greek">py</foreign> &aelig;qualis fiat <foreign lang="greek">fw.</foreign> Cum igitur in portione &longs;olida <lb/>
qu&aelig;dam figura in&longs;cribi pos&longs;it, ita ut linea, qu&aelig; inter cen&shy;<lb/>
trum grauitatis portionis, &amp; in&longs;cript&aelig; figur&aelig; interiicitur, <lb/>
qualibet linea propo&longs;ita &longs;it minor, quod proxime demon&shy;<lb/>
&longs;trauimus: perueniet tandem <foreign lang="greek">f</foreign> centrum in&longs;cript&aelig; figur&aelig; 



<pb xlink:href="023/01/096.jpg"/><figure id="id.023.01.096.1.jpg" xlink:href="023/01/096/1.jpg"/>



<pb pagenum="45" xlink:href="023/01/097.jpg"/>ad punctum <foreign lang="greek">w.</foreign> Sed quoniam <foreign lang="greek">p</foreign> circum&longs;cripta itidem alia <lb/>
figura &aelig;quali interuallo ad portionis centrum accedit, ubi <lb/>
primum <foreign lang="greek">f</foreign> applicuerit &longs;e ad <foreign lang="greek">w,</foreign> &amp; <foreign lang="greek">p</foreign> ad <expan abbr="punct&utilde;">punctum</expan> <foreign lang="greek">y,</foreign> hoc e&longs;t ad <lb/>
portionis centrum &longs;e applicabit. </s>
          <s id="s.000940">quod fieri nullo modo <lb/>
po&longs;&longs;e per&longs;picuum e&longs;t. </s>
          <s id="s.000941">non aliter idem ab&longs;urdum &longs;equetur, <lb/>
fi ponamus centrum portionis recedere &agrave; medio ad par&shy;<lb/>
tes <foreign lang="greek">w;</foreign> e&longs;&longs;et enim aliquando centrum figur&aelig; in&longs;cript&aelig; idem <lb/>
quod portionis <expan abbr="centr&utilde;">centrum</expan>. </s>
          <s id="s.000942">ergo punctum e centrum erit gra<lb/>
uitatis portionis abc. quod demon&longs;trare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000943"><margin.target id="marg103"/>7. huius</s>
        </p>
        <p type="margin">
          <s id="s.000944"><margin.target id="marg104"/>8. primi <lb/>
libri Ar&shy;<lb/>
chimedis</s>
        </p>
        <p type="margin">
          <s id="s.000945"><margin.target id="marg105"/>11. duo&shy;<lb/>
decimi.</s>
        </p>
        <p type="margin">
          <s id="s.000946"><margin.target id="marg106"/>15. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000947"><margin.target id="marg107"/>2. duode&shy;<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000948"><margin.target id="marg108"/>20. primi <lb/>
<expan abbr="conicor&utilde;">conicorum</expan></s>
        </p>
        <p type="margin">
          <s id="s.000949"><margin.target id="marg109"/>19.<lb/>
quinti</s>
        </p>
        <p type="main">
          <s id="s.000950">Quod autem &longs;upra <expan abbr="dem&otilde;&longs;tratum">demon&longs;tratum</expan> e&longs;t in portione conoi&shy;<lb/>
dis recta per figuras, qu&aelig; ex cylindris &aelig;qualem altitudi&shy;<lb/>
dinem habentibus con&longs;tant, idem &longs;imiliter demon&longs;trabi&shy;<lb/>
mus per figuras ex cylindri portionibus con&longs;tantes in ea <lb/>
portione, qu&aelig; plano non ad axem recto ab&longs;cinditur. </s>
          <s id="s.000951">ut <lb/>
enim tradidimus in commentariis in undecimam propo&longs;i<lb/>
tionem libri Archimedis de conoidibus &amp; &longs;ph&aelig;roidibus. </s>
          <lb/>
          <s id="s.000952">portiones cylindri, qu&aelig; &aelig;quali &longs;unt altitudine eam inter &longs;e <lb/>
&longs;e proportionem habent, quam ip&longs;arum ba&longs;es: ba&longs;es <expan abbr="aut&etilde;">autem</expan> <lb/>
<arrow.to.target n="marg110"/><lb/>
qu&aelig; &longs;unt ellip&longs;es &longs;imiles eandem proportionem habere, <lb/>
quam quadrata diametrorum eiu&longs;dem rationis, ex corol&shy;<lb/>
lario &longs;eptim&aelig; propo&longs;itionis libri de conoidibus, &amp; &longs;ph&aelig;&shy;<lb/>
roidibus, manife&longs;te apparet.</s>
        </p>
        <p type="margin">
          <s id="s.000953"><margin.target id="marg110"/>corol. 15<lb/>
de conoi&shy;<lb/>
dibus &amp; <lb/>
&longs;ph&aelig;roi&shy;<lb/>
dibus.</s>
        </p>
        <p type="head">
          <s id="s.000954">THEOREMA XXIIII. PROPOSITIO XXX.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000955">Si &agrave; portione conoidis rectanguli alia portio <lb/>
ab&longs;cindatur, plano ba&longs;i &aelig;quidi&longs;tante; habebit <lb/>
portio tota ad eam, qu&aelig; ab&longs;ci&longs;&longs;a e&longs;t, duplam pro <lb/>
portion em eius, qu&aelig; e&longs;t ba&longs;is maioris portionis <lb/>
ad ba&longs;i m minoris, uel qu&aelig; axis maioris ad axem <lb/>
minoris.</s>
        </p>
        <pb xlink:href="023/01/098.jpg"/>
        <p type="main">
          <s id="s.000956">ABSCINDATVR &agrave; portione conoidis rectanguli <lb/>
abc alia portio ebf, plano ba&longs;i &aelig;quidi&longs;tante: &amp; eadem <lb/>
portio &longs;ecetur alio plano per axem; ut &longs;uperficiei &longs;ectio &longs;it <lb/>
parabole abc: <expan abbr="planor&utilde;">planorum</expan> portiones ab&longs;cindentium rect&aelig; <lb/>
linc&aelig; ac, ef: axis autem portionis, &amp; &longs;ectionis diameter <lb/>
bd; quam linea ef in puncto g &longs;ecet. </s>
          <s id="s.000957">Dico portionem co&shy;<lb/>
noidis abc ad portionem ebf duplam proportionem ha&shy;<lb/>
bere eius, qu&aelig; e&longs;t ba&longs;is ac ad ba&longs;im ef; uel axis db ad bg<lb/>
axem. </s>
          <s id="s.000958">Intelligantur enim duo coni, &longs;eu coni portiones <lb/>
abc, ebf, <expan abbr="e&atilde;dem">eandem</expan> ba&longs;im, quam portiones conoidis, &amp; &aelig;qua <lb/>
lem habentes altitudinem. </s>
          <s id="s.000959">&amp; quoniam abc portio conoi <lb/>
dis &longs;e&longs;quialtera e&longs;t coni, &longs;eu portionis coni abc; &amp; portio <lb/>
ebf coni &longs;eu portionis coni  bf e&longs;t &longs;e&longs;quialtera, quod de&shy;<lb/>
<figure id="id.023.01.098.1.jpg" xlink:href="023/01/098/1.jpg"/><lb/>
mon&longs;trauit Archimedes in propo&longs;itionibus 23, &amp; 24 libri <lb/>
de conoidibus, &amp; &longs;ph&aelig;roidibus: erit conoidis portio ad <lb/>
conoidis portionem, ut conus ad conum, uel ut coni por&shy;<lb/>
tio ad coni portionem. </s>
          <s id="s.000960">Sed conus, nel coni portio abc ad <lb/>
conum, uel coni portionem ebf compo&longs;itam proportio&shy;<lb/>
nem habet ex proportione ba&longs;is ac ad ba&longs;im ef, &amp; ex pro&shy;<lb/>
portione altitudinis coni, uel coni portionis abc ad alti&shy;<lb/>
tudinem ip&longs;ius ebf, ut nos demon&longs;trauimus in commen&shy;<lb/>
tariis in undecimam propo&longs;itionem eiu&longs;dem libri Archi&shy;<lb/>
medis: altitudo autem ad altitudinem c&longs;t, ut axis ad axem. </s>
          <lb/>
          <s id="s.000961">quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue 



<pb pagenum="46" xlink:href="023/01/099.jpg"/>ro ita demon&longs;trabitur. </s>
          <s id="s.000962">Ducatur &agrave; puncto b ad planum ba&shy;<lb/>
&longs;is ac perpendicularis linea bh, qu&aelig; ip&longs;am ef in K &longs;ecet. </s>
          <lb/>
          <s id="s.000963">erit bh altitudo coni, uel coni portionis abc: &amp; bK altitu<lb/>
<arrow.to.target n="marg111"/><lb/>
do efg. </s>
          <s id="s.000964">Quod cum line&aelig; ac, ef inter &longs;e &aelig;quidi&longs;tent, &longs;unt <lb/>
enim planorum &aelig;quidi&longs;tantium &longs;ectiones: habebit db ad <lb/>
<arrow.to.target n="marg112"/><lb/>
bg proportionem eandem, quam hb ad bk quare por&shy;<lb/>
tio conoidis abc ad portionem efg proportionem habet <lb/>
compo&longs;itam ex proportione ba&longs;is ac ad ba&longs;im ef; &amp; ex <lb/>
<arrow.to.target n="marg113"/><lb/>
proportione db axis ad axem bg. <!-- KEEP S--></s>
          <s id="s.000965">Sed circulus, uel <lb/>
ellip&longs;is circa diametrum ac ad circulum, uel ellip&longs;im <lb/>
<arrow.to.target n="marg114"/><lb/>
circa ef, e&longs;t ut quadratum ac ad quadratum ef; hoc e&longs;t ut <lb/>
<expan abbr="quadrat&utilde;">quadratum</expan> ad ad <expan abbr="quadrat&utilde;">quadratum</expan> eg. <!-- REMOVE S-->&amp; quadratum ad ad quadra<lb/>
tum eg e&longs;t, ut linea db ad lineam bg. <!-- KEEP S--></s>
          <s id="s.000966">circulus igitur, uel el<lb/>
<arrow.to.target n="marg115"/><lb/>
lip&longs;is circa diametrum ac ad <expan abbr="circul&utilde;">circulum</expan>, uel ellip&longs;im circa ef, <lb/>
<arrow.to.target n="marg116"/><lb/>
hoc e&longs;t ba&longs;is ad ba&longs;im eandem proportionem habet, <expan abbr="qu&atilde;">quam</expan> <lb/>
db axis ad axem bg. <!-- KEEP S--></s>
          <s id="s.000967">ex quibus &longs;equitur portionem abc <lb/>
ad portionem ebf habere proportionem duplam eius, <lb/>
qu&aelig; e&longs;t ba&longs;is ac ad ba&longs;im ef: uel axis db ad bg axem. </s>
          <s id="s.000968">quod <lb/>
demon&longs;trandum proponebatur.</s>
        </p>
        <p type="margin">
          <s id="s.000969"><margin.target id="marg111"/>16. unde&shy;<lb/>
cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000970"><margin.target id="marg112"/>4 sexti.</s>
        </p>
        <p type="margin">
          <s id="s.000971"><margin.target id="marg113"/>2. duode<lb/>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000972"><margin.target id="marg114"/>7. de co&shy;<lb/>
noidibus <lb/>
&amp; &longs;ph&aelig;&shy;<lb/>
roidibus</s>
        </p>
        <p type="margin">
          <s id="s.000973"><margin.target id="marg115"/>15. quinti. </s>
          <s id="s.000974">quinti</s>
        </p>
        <p type="margin">
          <s id="s.000975"><margin.target id="marg116"/>20. primi <lb/>
<expan abbr="conicor&utilde;">conicorum</expan></s>
        </p>
        <p type="head">
          <s id="s.000976">THEOREMA XXV. PROPOSITIO XXXI.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000977">Cuiuslibet fru&longs;ti &agrave; portione rectanguli conoi<lb/>
dis ab&longs;cis&longs;i, centrum grauitatis e&longs;t in axe, ita ut <lb/>
demptis primum &agrave; quadrato, quod fit ex diame&shy;<lb/>
tro maioris ba&longs;is, tertia ip&longs;ius parte, &amp; duabus <lb/>
tertiis quadrati, quod fit ex diametro ba&longs;is mino&shy;<lb/>
ris: deinde &agrave; tertia parte quadrati maioris ba&longs;is <lb/>
rur&longs;us dempta portione, ad quam reliquum qua<lb/>
drati ba&longs;is maioris un&agrave; cum dicta portione <expan abbr="dupl&atilde;">duplam</expan> <lb/>
proportionem habeat eius, qu&aelig; e&longs;t quadrati ma&shy;



<pb xlink:href="023/01/100.jpg"/>ioris ba&longs;is ad quadratum minoris: centrum &longs;it in <lb/>
eo axis puncto, quo ita diuiditur ut pars, qu&aelig; mi<lb/>
norem ba&longs;im attingit ad alteram partem eandem <lb/>
proportionem habeat, quam dempto quadrato <lb/>
minoris ba&longs;is &agrave; duabus tertiis quadrati maioris, <lb/>
habet id, quod reliquum e&longs;t un&agrave; cum portione &agrave; <lb/>
tertia quadrati maioris parte dempta, ad <expan abbr="reliqu&atilde;">reliquam</expan> <lb/>
eiu&longs;dem terti&aelig; portionem.</s>
        </p>
        <p type="main">
          <s id="s.000978">SIT fru&longs;tum &agrave; portione rectanguli conoidis ab&longs;ci&longs;&longs;um <lb/>
abcd, cuius maior ba&longs;is circulus, uel ellip&longs;is circa diame&shy;<lb/>
trum bc, minor circa diametrum ad; &amp; axis ef. </s>
          <s id="s.000979">de&longs;criba&shy;<lb/>
tur autem portio conoidis, &agrave; quo illud ab&longs;ci&longs;&longs;um e&longs;t, &amp; pla&shy;<lb/>
<figure id="id.023.01.100.1.jpg" xlink:href="023/01/100/1.jpg"/><lb/>
no per axem ducto &longs;ecetur; ut &longs;uperficiei &longs;ectio &longs;it parabo&shy;<lb/>
le bgc, cuius diameter, &amp; axis portionis gf: deinde gf diui<lb/>
datur in puncto h, ita ut gh &longs;it dupla hf: &amp; rur&longs;us ge in ean <lb/>
dem proportionem diuidatur: <expan abbr="&longs;itq;">&longs;itque</expan> gk ip&longs;ius ke dupla. </s>
          <s id="s.000980"><expan abbr="I&atilde;">Iam</expan> <lb/>
ex iis, qu&aelig; proxime demon&longs;trauimus, con&longs;tat centrum gra<lb/>
uitatis portionis bgc e&longs;&longs;e h punctum: &amp; portionis agc <lb/>
punctum k. </s>
          <s id="s.000981">&longs;umpto igitur infra h puncto l, ita ut kh ad hl 



<pb pagenum="47" xlink:href="023/01/101.jpg"/>eam proportionem habeat, quam abcd fru&longs;tum ad por&shy;<lb/>
tionem agd; erit punctum l eius fru&longs;ti grauitatis <expan abbr="c&etilde;trum">centrum</expan>: <lb/>
<expan abbr="habebitq;">habebitque</expan> componendo Kl ad lh proportionem eandem, <lb/>
<arrow.to.target n="marg117"/><lb/>
quam portio conoidis bgc ad agd portionem. </s>
          <s id="s.000982"><expan abbr="Itaq;">Itaque</expan> quo <lb/>
niam quadratum bf ad quadratum ae, hoc e&longs;t quadratum <lb/>
bc ad quadratum ad e&longs;t, ut linea fg ad ge: erunt du&aelig; ter&shy;<lb/>
ti&aelig; quadrati bc ad duas tertias quadrati ad, ut hg ad gk: <lb/>
&amp; &longs;i &agrave; duabus tertiis quadrati bc dempt&aelig; fuerint du&aelig; ter&shy;<lb/>
ti&aelig; quadrati ad: erit <expan abbr="diuid&etilde;do">diuidendo</expan> id, quod relinquitur ad duas <lb/>
tertias quadrati ad, ut hk ad kg. <!-- KEEP S--></s>
          <s id="s.000983">Rur&longs;us du&aelig; terti&aelig; quadra<lb/>
ti ad ad duas tertias quadrati bc &longs;unt, ut kg ad gh: &amp; du&aelig; <lb/>
terti&aelig; quadrati bc ad <expan abbr="terti&atilde;">tertiam</expan> <expan abbr="part&etilde;">partem</expan> ip&longs;ius, ut gh ad hf. </s>
          <s id="s.000984">ergo <lb/>
ex &aelig;quali id, quod relinquitur ex duabus tertiis quadrati <lb/>
bc, demptis ab ip&longs;is quadrati ad duabus tertiis, ad <expan abbr="terti&atilde;">tertiam</expan> <lb/>
partem quadrati bc, ut kh ad hf: &amp; ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>
terti&aelig; partis, ad quam un&agrave; cum ip&longs;a portione, duplam pro<lb/>
portionem habeat eius, qu&aelig; e&longs;t quadrati bc ad <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>
ad, ut Kl ad lh. </s>
          <s id="s.000985">habet enim Kl ad lh eandem proportio&shy;<lb/>
nem, quam conoidis portio bgc ad portionem agd: por&shy;<lb/>
tio autem bgc ad portionem agd duplam proportionem <lb/>
habet eius, qu&aelig; e&longs;t ba&longs;is bc ad ba&longs;im ad: hoc e&longs;t quadrati <lb/>
<arrow.to.target n="marg118"/><lb/>
bc ad quadratum ad; ut proxime demon&longs;tratum e&longs;t. </s>
          <s id="s.000986">quare <lb/>
dempto ad quadrato &agrave; duabus tertiis quadrati bc, erit id, <lb/>
quod relinquitur un&agrave; cum dicta portione terti&aelig; partis ad <lb/>
reliquam eiu&longs;dem portionem, ut el ad lf. </s>
          <s id="s.000987">Cum igitur cen&shy;<lb/>
trum grauitatis fru&longs;ti abcd &longs;it l, &agrave; quo axis ef in eam, <expan abbr="qu&atilde;">quam</expan> <lb/>
diximus, proportionem diuidatur; con&longs;tat <expan abbr="uer&utilde;">uerum</expan> e&longs;&longs;e illud, <lb/>
quod demon&longs;trandum propo&longs;uimus.</s>
        </p>
        <p type="margin">
          <s id="s.000988"><margin.target id="marg117"/>20. 1. coni<lb/>
corum.</s>
        </p>
        <p type="margin">
          <s id="s.000989"><margin.target id="marg118"/>30 huius</s>
        </p>
        <p type="head">
          <s id="s.000990">FINIS LIBRI DE CENTRO<!-- REMOVE S-->GRAVITATIS SOLIDORVM.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000991">Impre&longs;&longs;. <!-- REMOVE S-->Bononi&aelig; cum licentia Superiorum, </s>
        </p>
      </chap>
    </body>
    <back/>
  </text>
</archimedes>