<?xml version="1.0"?>
<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></translator>
    <lang>la</lang>
    <cvs_file>comma_centr_023_la_1565.xml</cvs_file>
    <cvs_version></cvs_version>
    <locator>023.xml</locator>
  </info>
  <text>
    <pb xlink:href="023/01/001.jpg"></pb>
    <front>
      <section>
        <p type="head">
          <s id="s.000001">FEDERICI <lb></lb>COMMANDINI <lb></lb>VRBINATIS</s>
          <s id="s.000002">LIBER DE CENTRO <lb></lb>GRAVITATIS <lb></lb>SOLIDORVM.</s>
        </p>
        <figure id="id.023.01.001.1.jpg" xlink:href="023/01/001/1.jpg"></figure>
        <p type="head">
          <s id="s.000003">CVM PRIVILEGIO IN ANNOS X.</s>
        </p>
        <p type="head">
          <s id="s.000004">BONONIAE,</s>
        </p>
        <p type="head">
          <s id="s.000005">Ex Officina Alexandri Benacii.</s>
        </p>
        <p type="head">
          <s id="s.000006">MDLXV.</s>
        </p>
        <pb xlink:href="023/01/002.jpg"></pb>
        <pb xlink:href="023/01/003.jpg"></pb>
      </section>
      <section>
        <p type="head">
          <s id="s.000007">ALEXANDRO FARNESIO <lb></lb>
CARDINALI AMPLISSIMO. <lb></lb>
ET OPTIMO.</s>
        </p>
        <p type="main">
          <s id="s.000008">Cvm multæ res in mathematicis <lb></lb>
diſciplinis nequaquam ſatis ad­<lb></lb>
huc explicatæ ſint, tum perdif­<lb></lb>
ficilis, &amp; perobſcura quæſtio <lb></lb>
eſt de centro grauitatis corpo­<lb></lb>
rum ſolidorum; quæ, &amp; ad co­<lb></lb>
gnoſcendum pulcherrima eſt, <lb></lb>
&amp; ad multa, quæ à mathematicis proponuntur, præ­<lb></lb>
clare intelligenda maximum affert adiumentum. </s>
          <s id="s.000009">de <lb></lb>
qua neminem ex mathematicis, neque noſtra, neque <lb></lb>
patrum noſtrorum memoria ſcriptum reliquiſſe ſci­<lb></lb>
mus. </s>
          <s id="s.000010">&amp; quamuis in earum monumentis literarum <expan abbr="nõ">non</expan> <lb></lb>
nulla reperiantur, ex quibus in hanc ſententiam addu<lb></lb>
ci poſſumus, vt exiſtimemus hanc rem ab <expan abbr="ijſdẽ">ijſdem</expan> vber­<lb></lb>
rime tractatam eſſe; tamen neſcio quo fato adhuc <lb></lb>
in eiuſmodi librorum ignoratione verſamur. </s>
          <s id="s.000011">Archi­<lb></lb>
medes quidem <expan abbr="mathematicorũ">mathematicorum</expan> princeps in libello, <lb></lb>
cuius inſcriptio eſt, <foreign lang="grc">κέντρα βάρων ἐπιπέδων</foreign>, de centro pla­<lb></lb>
norum copioſiſsime, atque acutiſsime conſcripſit: &amp; <lb></lb>
in eo explicando <expan abbr="ſummã">ſummam</expan> ingenii, &amp; ſcientiæ <expan abbr="gloriã">gloriam</expan> eſt <lb></lb>
<expan abbr="cõſecutus">conſecutus</expan>. </s>
          <s id="s.000012">Sed de cognitione <expan abbr="cẽtri">centri</expan> grauitatis <expan abbr="corporũ">corporum</expan> <lb></lb>
<expan abbr="ſolidorũ">ſolidorum</expan> nulla in eius libris litera inuenitur. </s>
          <s id="s.000013">non mul<lb></lb>
tos abhinc annos MARCELLVS II. PONT. MAX.



<pb xlink:href="023/01/004.jpg"></pb>cum adhuc Cardinalis eſſet, mihi, quæ ſua erat hu­<lb></lb>
manitas, libros eiuſdem Archimedis de ijs, quæ ve­<lb></lb>
huntur in aqua, latine redditos dono dedit. </s>
          <s id="s.000014">hos cum <lb></lb>ego, ut aliorum ſtudia incitarem, <expan abbr="emendãdos">emendandos</expan>, &amp; <expan abbr="cõ-mentariis">com­<lb></lb>
mentariis</expan> illuſtrandos ſuſcepiſſem, animaduerti dubi <lb></lb>
tari non poſſe, quin Archimedes vel de hac materia <lb></lb>
ſcripſiſſet, vel aliorum mathematicorum ſcripta per­<lb></lb>
legiſſet. </s>
          <s id="s.000015">nam in iis tum alia nonnulla, tum maxime <lb></lb>
illam propoſitionem, ut euidentem, &amp; aliàs proba­<lb></lb>
tam aſſumit, <expan abbr="Centrũ">Centrum</expan> grauitatis in portionibus conoi<lb></lb>
dis rectanguli axem ita diuidere, vt pars, quæ ad verti<lb></lb>
cem terminatur, alterius partis, quæ ad baſim dupla <lb></lb>
ſit. </s>
          <s id="s.000016">Verum hæc ad eam partem mathematicarum <lb></lb>
diſciplinarum præcipue refertur, in qua de centro <lb></lb>
grauitatis corporum ſolidorum tractatur. </s>
          <s id="s.000017">non eſt au<lb></lb>
tem conſentaneum Archimedem illum admirabilem <lb></lb>
virum hanc propoſitionem ſibi argumentis con­<lb></lb>
firmandam exiſtimaturum non fuiſſe, niſi eam vel <lb></lb>
aliis in locis probauiſſet, vel ab aliis probatam eſſe <lb></lb>
comperiſſet. </s>
          <s id="s.000018">quamobrem nequid in iis libris intel­<lb></lb>
ligendis deſiderari poſſet, ſtatui hanc etiam partem <lb></lb>
vel à veteribus prætermiſſam, vel tractatam quidem, <lb></lb>
ſed in tenebris iacentem, non intactam relinquere; <lb></lb>
atque ex aſsidua mathematicorum, præſertim Archi­<lb></lb>
medis lectione, quæ mihi in mentem venerunt, ea in <lb></lb>
medium afferre; ut centri grauitatis corporum ſoli­<lb></lb>
dorum, ſi non perfectam,  at certe aliquam noti-



<pb xlink:href="023/01/005.jpg"></pb>tiam haberemus. </s>
          <s id="s.000019">Quem meum laborem <expan abbr="nõ">non</expan> mathe­<lb></lb>
maticis ſolum, verum iis etiam, qui naturæ obſcuri­<lb></lb>
tate delectantur, <expan abbr="nõ">non</expan> iniucundam fore ſperaui: multa <lb></lb>
enim <foreign lang="grc">προβλήματα</foreign> cognitione digniſsima, quæ ad <expan abbr="vtrã-que">vtran­<lb></lb>
que</expan> ſcientiam attinent, ſeſe legentibus obtuliſſent.</s>
          <lb></lb>
          <s id="s.000020">neque id vlli mirandum videri debet. </s>
          <s id="s.000021">vt enim in cor­<lb></lb>poribus noſtris omnia membra, ex quibus certa quæ<lb></lb>
dam officia naſcuntur, diuino quodam ordine inter <lb></lb>
ſe implicata, &amp; colligata ſunt: in <expan abbr="iisq́">iisque</expan>; admirabilis il­<lb></lb>
la conſpiratio, quam <foreign lang="grc">σύμπνοιαν</foreign> græci vocant, eluceſcit, <lb></lb>
ita tres illæ Philoſophiæ (ut Ariſtotelis verbo vtar) <lb></lb>
quæ veritatem ſolam propoſitam habent, licet qui­<lb></lb>
buſdam quaſi finibus ſuis regantur: tamen <expan abbr="earũ">earum</expan> vna­<lb></lb>
quæque per ſe ipſam quodammodo imperfecta eſt: <lb></lb>
neque altera ſine alterius auxilio plene comprehen­<lb></lb>
di poteſt. </s>
          <s id="s.000022">complures præterea mathematicorum no­<lb></lb>
di ante hac explicatu difficillimi nullo negotio expe<lb></lb>
diti eſſent: atque (ut vno verbo complectar) niſi <lb></lb>
mea valde amo, tractationem hanc meam ſtudioſis <lb></lb>
non mediocrem vtilitatem, &amp; magnam volupta­<lb></lb>
tem allaturam eſſe mihi perſuaſi. </s>
          <s id="s.000023">cum autem ad hoc <lb></lb>
ſcribendum aggreſſus eſsem, allatus eſt ad me liber <lb></lb>
Franciſci Maurolici Meſſanenſis, in quo vir ille do­<lb></lb>
ctiſsimus, &amp; in iis diſciplinis exercitatiſsimus af­<lb></lb>
firmabat ſe de centro grauitatis corporum ſolido­<lb></lb>
rum conſcripſiſſe. </s>
          <s id="s.000024">cum hoc intellexiſſem, ſuſtinui <lb></lb>
me pauliſper: tacitus que expectaui, dum opus cla-



<pb xlink:href="023/01/006.jpg"></pb>risſimi uiri, quem ſemper honoris cauſſa nomino, <lb></lb>
in lucem proferretur: mihi enim exploratisſimum <lb></lb>
erat: Franciſcum Maurolicum multo doctius, &amp; <lb></lb>
exquiſitius hoc diſciplinarum genus ſcriptis ſuis tra<lb></lb>
diturum. </s>
          <s id="s.000025">ſed cum id tardius fieret, hoc eſt, ut ego <lb></lb>
interpretor, diligentius, mihi diutius hac ſcriptione <lb></lb>
non ſuperſedendum eſſe duxi, præſertim cum iam li­<lb></lb>
bri Archimedis de iis, quæ uehuntur in aqua, opera <lb></lb>
mea illuſtrati typis <expan abbr="excudẽdi">excudendi</expan> eſſent. </s>
          <s id="s.000026">nec me alia cauſ<lb></lb>
ſa impuliſſet, ut de centro grauitatis corporum ſoli­<lb></lb>
dorum ſcriberem, niſi ut hac etiam ratione lux eis <lb></lb>
quàm maxime fieri poſſet afferretur. </s>
          <s id="s.000027"><expan abbr="atq;">atque</expan> id eò mihi <lb></lb>
faciendum exiſtimaui, quòd in ſpem ueniebam fore, <lb></lb>
ut cum ego ex omnibus mathematicis primus, hanc <lb></lb>
materiam explicandam ſuſcepiſſem; ſi quid errati for<lb></lb>
te à me commiſſum eſſet, boni uiri potius id meæ de <lb></lb>
ſtudioſis hominibus bene <expan abbr="merẽdi">merendi</expan> cupiditati, quàm <lb></lb>
arrogantiæ aſcriberent. </s>
          <s id="s.000028">reſtabat ut conſiderarem, cui <lb></lb>
potisſimum ex principibus uiris contemplationem <lb></lb>
hanc, nunc primum memoriæ, ac literis proditam de <lb></lb>
dicarem. </s>
          <s id="s.000029">harum mearum cogitationum ſumma fa­<lb></lb>
cta, exiſtimaui nemini conuenientius de centro graui <lb></lb>
tatis corporum opus dicari oportere, quàm ALE­<lb></lb>
XANDRO FARNESIO grauisſimo, ac prudentisſi­<lb></lb>
mo Cardinali, quo in uiro ſumma fortuna ſemper <expan abbr="cũ">cum</expan> <lb></lb>
ſumma uirtute certauit. </s>
          <s id="s.000030">quid enim maxime in te ad­<lb></lb>
mirati debeant homines, obſcurum eſt; uſum ne re-



<pb xlink:href="023/01/007.jpg"></pb>rum, qui pueritiæ tempus extremum principium ha<lb></lb>
buiſti, &amp; <expan abbr="imperiorũ">imperiorum</expan>, &amp; ad Reges, &amp; Imperatores ho­<lb></lb>
norificentiſsimarum legationum; an excellentiam <lb></lb>
in omni genere literarum, qui vix <expan abbr="adoleſcẽtulus">adoleſcentulus</expan>, quæ <lb></lb>
homines iam confirmata ætate ſummo ſtudio, <expan abbr="diu-turnisq́">diu­<lb></lb>
turnisque</expan>; laboribus didicerunt, ſcientia, &amp; cognitione <lb></lb>
comprehendiſti: an conſilium, &amp; ſapientiam in re­<lb></lb>
gendis, &amp; <expan abbr="gubernãdis">gubernandis</expan> Ciuitatibus, cuius grauiſsimæ <lb></lb>
ſententiæ in ſanctiſsimo Reip. Chriſtianæ conſilio di­<lb></lb>
ctæ, potius diuina oracula, quàm ſententiæ habitæ <lb></lb>
ſunt, &amp; habentur. </s>
          <s id="s.000031">prætermitto liberalitatem, &amp; mu­<lb></lb>
nificentiam tuam, quam in ſtudioſiſsimo quoque ho<lb></lb>
neſtando quotidie magis oſtendis, ne videar auribus <lb></lb>
tuis potius, quàm veritati ſeruire. </s>
          <s id="s.000032">quamuis à te in tot <lb></lb>
præclaros viros tanta beneficia collata ſunt, &amp; <expan abbr="confe-rũtur">confe­<lb></lb>
runtur</expan>, vt omnibus teſtatum ſit, nihil tibi eſſe charius, <lb></lb>
nihil iucundius, quàm eximia tua liberalitate homi­<lb></lb>
nes ad amplexandam virtutem, licet currentes incita­<lb></lb>
re. </s>
          <s id="s.000033">nihil dico de ceteris virtutibus tuis, quæ tantæ <lb></lb>
ſunt, quantæ ne cogitatione quidem comprehendi <lb></lb>
poſſunt. </s>
          <s id="s.000034">Quamobrem hac præcipue de cauſſa te hu­<lb></lb>
ius meæ lucubrationis patronum eſſe volui, quam ea, <lb></lb>
qua ſoles, humanitate accipies. </s>
          <s id="s.000035">te enim ſemper ob <lb></lb>
diuinas virtutes tuas colui, &amp; obſeruaui: <expan abbr="nihilq́">nihilque</expan>; mi­<lb></lb>
hi fuit optatius; quàm tibi perſpectum eſſe meum <lb></lb>
erga te animum; <expan abbr="ſingularemq́">ſingularemque</expan>; obſeruantiam. </s>
          <s id="s.000036">cœ­<lb></lb>
lum igitur digito attingam, ſi poſt grauiſsimas oc-



<pb xlink:href="023/01/008.jpg"></pb>cupationes tuas legendo Federici tui libro aliquid <lb></lb>
impertiri temporis non grauaberis: <expan abbr="cumq́">cumque</expan>; in iis, qui <lb></lb>
tibi ſemper addicti erunt, numerare. </s>
          <s id="s.000037">Vale.</s>
        </p>
        <p type="main">
          <s id="s.000038">Federicus Commandinus.</s>
        </p>
      </section>
      <section>
        <pb pagenum="1" xlink:href="023/01/009.jpg"></pb>
        <p type="head">
          <s id="s.000039">FEDERICI COMMANDINI <lb></lb>
VRBINATIS LIBER DE CENTRO <lb></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"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000042"><margin.target id="marg1"></margin.target>1</s>
        </p>
        <p type="main">
          <s id="s.000043">Centrvm grauitatis, Pappus <lb></lb>
Alexandrinus in octauo ma­<lb></lb>
thematicarum collectionum <lb></lb>
libro ita diffiniuit.</s>
        </p>
        <p type="main">
          <s id="s.000044"><foreign lang="grc">λέγομεν δέ κέντρον βάρους ἑκάστου σώ<lb></lb>
ματος ἐ̂ιναι σημε̂ιον τι κείμενον ἐντός, άφ&#039; <lb></lb>
ὅυ κατ&#039; έποίνιαν ὰρτνθέν τό βάρος ν̔μερε̂ι<lb></lb>
φερόμενον, καὶ φυλάσσει τήν ἐξ ἀρχῆς θέ­<lb></lb>
σιν, ὀυ μὴ περιτρεπόμενον ἐν τῆ φορᾶ</foreign>. hoc eſt,</s>
        </p>
        <p type="main">
          <s id="s.000045">Dicimus autem centrum grauitatis uniuſcu­<lb></lb>
iuſque corporis punctum quoddam intra poſi­<lb></lb>
tum, à quo ſi graue appenſum mente concipia­<lb></lb>
tur, dum fertur quieſcit; &amp; ſeruat eam, quam in <lb></lb>
principio habebat poſitionem: neque in ipſa la­<lb></lb>
tione circumuertitur.</s>
        </p>
        <p type="main">
          <s id="s.000046">Poſſumus etiam hoc modo diffinire.</s>
        </p>
        <p type="main">
          <s id="s.000047">Centrum grauitatis uniuſcuiuſque ſolidæ figu<lb></lb>
ræ eſt punctum illud intra poſitum, circa quod <lb></lb>
undique partes æqualium momentorum conſi­<lb></lb>
ſtunt. </s>
          <s id="s.000048">ſi enim per tale centrum ducatur planum <lb></lb>
figuram quomodocunque ſecans ſemper in par­



<pb xlink:href="023/01/010.jpg"></pb>tes æqueponderantes ipſam diuidet.</s>
        </p>
        <p type="main">
          <s id="s.000049"><arrow.to.target n="marg2"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000050"><margin.target id="marg2"></margin.target>2</s>
        </p>
        <p type="main">
          <s id="s.000051">Priſmatis, cylindri, &amp; portionis cylindri axem <lb></lb>
appello rectam lineam, quæ oppoſitorum plano­<lb></lb>
rum centra grauitatis coniungit.</s>
        </p>
        <p type="main">
          <s id="s.000052"><arrow.to.target n="marg3"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000053"><margin.target id="marg3"></margin.target>3</s>
        </p>
        <p type="main">
          <s id="s.000054">Pyramidis, coni, &amp; portionis coni axem dico li <lb></lb>
neam, quæ à uertice ad centrum grauitatis baſis <lb></lb>
perducitur.</s>
        </p>
        <p type="main">
          <s id="s.000055"><arrow.to.target n="marg4"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000056"><margin.target id="marg4"></margin.target>4</s>
        </p>
        <p type="main">
          <s id="s.000057">Si pyramis, conus, portio coni, uel conoidis ſe­<lb></lb>
cetur plano baſi æquidiſtante, pars, quæ eſt ad ba­<lb></lb>
ſim, fruſtum pyramidis, coni, portionis coni, uel <lb></lb>
conoidis dicetur; quorum plana æquidiſtantia, <lb></lb>
quæ opponuntur ſimilia ſunt, &amp; inæqualia: axes <lb></lb>
uero ſunt axium figurarum partes, quæ in ipſis <lb></lb>
comprehenduntur.</s>
        </p>
        <p type="head">
          <s id="s.000058">PETITIONES.</s>
        </p>
        <p type="main">
          <s id="s.000059"><arrow.to.target n="marg5"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000060"><margin.target id="marg5"></margin.target>1</s>
        </p>
        <p type="main">
          <s id="s.000061">Solidarum figurarum ſimilium centra grauita­<lb></lb>
tis ſimiliter ſunt poſita.</s>
        </p>
        <p type="main">
          <s id="s.000062"><arrow.to.target n="marg6"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000063"><margin.target id="marg6"></margin.target>2</s>
        </p>
        <p type="main">
          <s id="s.000064">Solidis figuris ſimilibus, &amp; æqualibus inter ſe <lb></lb>
aptatis, centra quoque grauitatis ipſarum inter ſe <lb></lb>
aptata erunt.</s>
        </p>
        <p type="head">
          <s id="s.000065">THEOREMA I. PROPOSITIO I.</s>
        </p>
        <p type="main">
          <s id="s.000066">Omnis figuræ rectilineæ in circulo deſcriptæ, <lb></lb>
quæ æqualibus lateribus, &amp; angulis contine­<lb></lb>




<pb pagenum="2" xlink:href="023/01/011.jpg"></pb>tur, centrum grauitatis eſt idem, quod circuli cen<lb></lb>
trum.</s>
        </p>
        <p type="main">
          <s id="s.000067">Sit primo triangulum æquilaterum abc in circulo de­<lb></lb>
ſcriptum: &amp; diuiſa ac bifariam in d, ducatur bd. </s>
          <s id="s.000068">erit in li­<lb></lb>
nea bd centrum grauitatis <expan abbr="triãguli">trianguli</expan> abc, ex tertia decima <lb></lb>
primi libri Archimedis de centro grauitatis planorum. </s>
          <s id="s.000069">Et <lb></lb>
<figure id="id.023.01.011.1.jpg" xlink:href="023/01/011/1.jpg"></figure><lb></lb>
quoniam linea ab eſt æqualis <lb></lb>
lineæ bc; &amp; ad ipſi dc; <expan abbr="eſtq́">eſtque</expan>; <lb></lb>
bd utrique communis: trian­<arrow.to.target n="marg7"></arrow.to.target><lb></lb>
gulum abd æquale erit trian<lb></lb>
gulo cbd: &amp; anguli angulis æ­<lb></lb>
quales, qui æqualibus lateri­<lb></lb>
<arrow.to.target n="marg8"></arrow.to.target><lb></lb>
bus ſubtenduntur. </s>
          <s id="s.000071">ergo angu<lb></lb>
li ad d <expan abbr="utriq;">utrique</expan> recti ſunt. </s>
          <s id="s.000072">quòd <lb></lb>
cum linea bd ſecet ae bifa­<lb></lb>
<arrow.to.target n="marg9"></arrow.to.target><lb></lb>
riam, &amp; ad angulos rectos; in <lb></lb>
ipſa bd eſt centrum circuli. </s>
          <s id="s.000073"><lb></lb>
quare in eadem bd linea erit <lb></lb>
centrum grauitatis trianguli, &amp; circuli centrum. </s>
          <s id="s.000074">Similiter <lb></lb>
diuiſa ab bifariam in e, &amp; ducta ce, oſtendetur in ipſa <expan abbr="utrũ">utrum</expan> <lb></lb>
que centrum contineri. </s>
          <s id="s.000075">ergo ea erunt in puncto, in quo li­<lb></lb>
neæ bd, ce conueniunt. </s>
          <s id="s.000076">trianguli igitur abc centrum gra<lb></lb>
uitatis eſt idem, quod circuli centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000077"><margin.target id="marg7"></margin.target>8. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000078"><margin.target id="marg8"></margin.target>13. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000079"><margin.target id="marg9"></margin.target>corol. pri<lb></lb>
mæ tertii</s>
        </p>
        <figure id="id.023.01.011.2.jpg" xlink:href="023/01/011/2.jpg"></figure>
        <p type="main">
          <s id="s.000080">Sit quadratum abcd in cir­<lb></lb>
culo deſcriptum: &amp; ducantur <lb></lb>
ac, bd, quæ conueniant in e. </s>
          <s id="s.000081">er­<lb></lb>
go punctum e eſt centrum gra<lb></lb>
uitatis quadrati, ex decima eiuſ <lb></lb>
dem libri Archimedis. </s>
          <s id="s.000082">Sed cum <lb></lb>
omnes anguli ad abcd recti <lb></lb>
<arrow.to.target n="marg10"></arrow.to.target><lb></lb>
ſint; erit abc ſemicirculus: <lb></lb>
<expan abbr="itemq́">itemque</expan>; bcd: &amp; propterea li­<lb></lb>
neæ ac, bd diametri circuli: 



<pb xlink:href="023/01/012.jpg"></pb>quæ quidem in centro conueniunt. </s>
          <s id="s.000083">idem igitur eſt centrum <lb></lb>
grauitatis quadrati, &amp; circuli centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000084"><margin.target id="marg10"></margin.target>31. tertii.</s>
        </p>
        <p type="main">
          <s id="s.000085">Sit pentagonum æquilaterum, &amp; æquiangulum in circu­<lb></lb>
<figure id="id.023.01.012.1.jpg" xlink:href="023/01/012/1.jpg"></figure><lb></lb>
lo deſcriptum abcd e. </s>
          <s id="s.000086">&amp; iun­<lb></lb>
cta bd, <expan abbr="bifariamq́">bifariamque</expan>; in f diuiſa, <lb></lb>
ducatur cf, &amp; producatur ad <lb></lb>
circuli circumferentiam in g; <lb></lb>
quæ lineam ae in h ſecet: de­<lb></lb>
inde iungantur ac, cc. </s>
          <s id="s.000087">Eodem <lb></lb>
modo, quo ſupra demonſtra­<lb></lb>
bimus angulum bcf æqualem <lb></lb>
eſſe. </s>
          <s id="s.000088">angulo dcf; &amp; angulos <lb></lb>
ad f utroſque rectos: &amp; idcir­<lb></lb>co lineam cfg per circuli cen<lb></lb>
trum tranſire. </s>
          <s id="s.000089">Quoniam igi­<lb></lb>
tur latera cb, ba, &amp; cd, de æqualia ſunt; &amp; æquales anguli <lb></lb>
<arrow.to.target n="marg11"></arrow.to.target><lb></lb>
cba, cde: erit baſis ca baſi: ce, &amp; angulus bca angulo <lb></lb>
dce æqualis. </s>
          <s id="s.000090">ergo &amp; reliquus ach, reliquo ech. </s>
          <s id="s.000091">eſt au­<lb></lb>
tem ch utrique triangulo ach, ech communis. </s>
          <s id="s.000092">quare <lb></lb>
baſis ah æqualis eſt baſi hc: &amp; anguli, qui ad h recti: <expan abbr="ſuntq́">ſuntque</expan>; <lb></lb>
<arrow.to.target n="marg12"></arrow.to.target><lb></lb>
recti, qui ad f. </s>
          <s id="s.000093">ergo lineæ ae, bd inter ſe ſe æquidiſtant. </s>
          <lb></lb>
          <s id="s.000094">Itaque cum trapezij abde latera bd, ae æquidiſtantia à li<lb></lb>
nea fh bifariam diuidantur; centrum grauitatis ipſius erit <lb></lb>
<arrow.to.target n="marg13"></arrow.to.target><lb></lb>
in linea fh, ex ultima eiuſdem libri Archimedis. </s>
          <s id="s.000095">Sed trian­<lb></lb>
guli bcd centrum grauitatis eſt in linea cf. </s>
          <s id="s.000096">ergo in eadem <lb></lb>
linea ch eſt centrum grauitatis trapezij abde, &amp; trian­<lb></lb>
guli bcd: hoc eſt pentagoni ipſius centrum: &amp; centrum <lb></lb>
circuli. </s>
          <s id="s.000097">Rurſus ſi iuncta ad, <expan abbr="bifariamq́">bifariamque</expan>; ſecta in k, duca­<lb></lb>
tur ekl: demonſtrabimus in ipſa utrumque centrum in <lb></lb>
eſſe. </s>
          <s id="s.000098">Sequitur ergo, ut punctum, in quo lineæ cg, el con­<lb></lb>
ueniunt, idem ſit centrum circuli, &amp; centrum grauitatis <lb></lb>
pentagoni.</s>
        </p>
        <p type="margin">
          <s id="s.000099"><margin.target id="marg11"></margin.target>4. Primi.</s>
        </p>
        <p type="margin">
          <s id="s.000100"><margin.target id="marg12"></margin.target>28. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000101"><margin.target id="marg13"></margin.target>13. Archi­<lb></lb>
medis.</s>
        </p>
        <p type="main">
          <s id="s.000102">Sit hexagonum abcdef æquilaterum, &amp; æquiangulum <lb></lb>
in circulo deſignatum: <expan abbr="iunganturq́">iunganturque</expan>; bd, ae: &amp; bifariam ſe­



<pb pagenum="3" xlink:href="023/01/013.jpg"></pb>cta bd in g puncto, ducatur cg; &amp; protrahatur ad circuli <lb></lb>
uſque circumferentiam; quæ ſecet ae in h. </s>
          <s id="s.000103">Similiter conclu<lb></lb>
demus cg per centrum circuli tranſire: &amp; bifariam ſecate <lb></lb>
lineam ae; <expan abbr="itemq́">itemque</expan>; lineas bd, ae inter ſe æquidiſtantes eſſe. <lb></lb>
</s>
          <s id="s.000104">Cum igitur cg per centrum circuli tranſeat; &amp; ad <expan abbr="punctũ">punctum</expan> <lb></lb>
f perueniat neceſſe eſt: quòd cdef ſit dimidium circumfe<lb></lb>
<figure id="id.023.01.013.1.jpg" xlink:href="023/01/013/1.jpg"></figure><lb></lb>
<arrow.to.target n="marg14"></arrow.to.target><lb></lb>
rentiæ circuli. </s>
          <s id="s.000105">Quare in eadem <lb></lb>
diametro cf erunt centra gra<lb></lb>
uitatis triangulorum bcd, <lb></lb>
afe, &amp; quadrilateri abde, ex <lb></lb>
quibus conſtat hexagonum ab <lb></lb>
cdef. </s>
          <s id="s.000106">perſpicuum eſt igitur in <lb></lb>
ipſa cf eſſe circuli centrum, &amp; <lb></lb>
centrum grauitatis hexagoni. <lb></lb>
</s>
          <s id="s.000107">Rurſus ducta altera diametro <lb></lb>
ad, eiſdem rationibus oſtende­<lb></lb>
mus in ipſa utrumque <expan abbr="cẽtrum">centrum</expan> <lb></lb>
ineſſe. </s>
          <s id="s.000108">Centrum ergo grauita­<lb></lb>
tis hexagoni, &amp; centrum circuli idem erit.</s>
        </p>
        <p type="margin">
          <s id="s.000109"><margin.target id="marg14"></margin.target>13 Archi<lb></lb>
medis.</s>
          <lb></lb>
          <s id="s.000110">9. <expan abbr="eiusdetilde;">eiusdem</expan> <lb></lb>
m</s>
        </p>
        <p type="main">
          <s id="s.000111">Sit heptagonum abcdefg æquilaterum atque æquian<lb></lb>
<figure id="id.023.01.013.2.jpg" xlink:href="023/01/013/2.jpg"></figure><lb></lb>
gulum in circulo deſcriptum: <lb></lb>
&amp; iungantur ce, bf, ag: di­<lb></lb>
uiſa autem ce bifariam in <expan abbr="pũtco">pun<lb></lb>
cto</expan> h: &amp; iuncta dh produca­<lb></lb>
tur in k. </s>
          <s id="s.000112">non aliter demon­<lb></lb>
ſtrabimus in linea dk eſſe cen<lb></lb>
trum circuli, &amp; centrum gra­<lb></lb>
uitatis trianguli cde, &amp; tra­<lb></lb>
peziorum bcef, abfg, hoc <lb></lb>
eſt centrum totius heptago­<lb></lb>
ni: &amp; rurſus eadem centra in <lb></lb>
alia diametro cl ſimiliter du­<lb></lb>
cta contineri. </s>
          <s id="s.000113">Quare &amp; centrum grauitatis heptagoni, &amp; <lb></lb>
centrum circuli in idem punctum conueniunt. </s>
          <s id="s.000114">Eodem mo



<pb xlink:href="023/01/014.jpg"></pb>do in reliquis figuris æquilateris, &amp; æquiangulis, quæ in cir­<lb></lb>
culo deſcribuntur, probabimus <expan abbr="cẽtrum">centrum</expan> grauitatis earum, <lb></lb>
&amp; centrum circuli idem eſſe. </s>
          <s id="s.000115">quod quidem demonſtrare <lb></lb>
oportebat.</s>
        </p>
        <p type="main">
          <s id="s.000116">Ex quibus apparet cuiuslibet figuræ rectilineæ <lb></lb>
in circulo plane deſcriptæ centrum grauitatis <expan abbr="idẽ">idem</expan> <lb></lb>
eſſe, quod &amp; circuli centrum.<lb></lb>
<arrow.to.target n="marg15"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000117"><margin.target id="marg15"></margin.target><foreign lang="grc">γνωρίμως</foreign></s>
        </p>
        <p type="main">
          <s id="s.000118">Figuram in circulo plane deſcriptam appella­<lb></lb>
mus, cuiuſmodi eſt ea, quæ in duodecimo elemen<lb></lb>
torum libro, propoſitione ſecunda deſcribitur. <lb></lb>
</s>
          <s id="s.000119">ex æqualibus enim lateribus, &amp; angulis conſtare <lb></lb>
perſpicuum eſt.</s>
        </p>
        <p type="head">
          <s id="s.000120">THEOREMA II, PROPOSITIO II.</s>
        </p>
        <p type="main">
          <s id="s.000121">Omnis figuræ rectilineæ in ellipſi plane deſcri­<lb></lb>
ptæ centrum grauitatis eſt idem, quod ellipſis <lb></lb>
centrum.</s>
        </p>
        <p type="main">
          <s id="s.000122">Quo modo figura rectilinea in ellipſi plane deſcribatur, <lb></lb>
docuimus in commentarijs in quintam propoſitionem li­<lb></lb>
bri Archimedis de conoidibus, &amp; ſphæroidibus.</s>
        </p>
        <p type="main">
          <s id="s.000123">Sit ellipſis abcd, cuius maior axis ac, minor bd: <expan abbr="iun-ganturq́">iun­<lb></lb>
ganturque</expan>; ab, bc, cd, da: &amp; bifariam diuidantur in pun­<lb></lb>
ctis efgh. </s>
          <s id="s.000124">à centro autem, quod ſit k ductæ lineæ ke, kf, <lb></lb>
kg, kh uſque ad ſectionem in puncta lmno protrahan­<lb></lb>
tur: &amp; iungantur lm, mn, no, ol, ita ut ac ſecet li­<lb></lb>
neas lo, mn, in z<foreign lang="grc">φ</foreign> punctis; &amp; bd ſecet lm, on in <foreign lang="grc">χψ.</foreign><lb></lb>
erunt lk, kn linea una, <expan abbr="itemq́ue">itemque</expan> linea una ipſæ mk, ko: <lb></lb>
&amp; lineæ ba, cd æquidiſtabunt lineæ mo: &amp; bc, ad ipſi <lb></lb>
ln. </s>
          <s id="s.000125">rurſus lo, mn axi bd æquidiſtabunt: &amp; lm, 



<pb pagenum="4" xlink:href="023/01/015.jpg"></pb>on ipſi ac. </s>
          <s id="s.000126">Quoniam enim triangulorum abk, adk, latus <lb></lb>
bk eſt æquale lateri kd, &amp; ak utrique commune; <expan abbr="anguliq́">angulique</expan>; <lb></lb>
<arrow.to.target n="marg16"></arrow.to.target><lb></lb>
ad k recti. </s>
          <s id="s.000127">baſis ab baſi ad; &amp; reliqui anguli reliquis an­<lb></lb>
gulis æquales erunt. </s>
          <s id="s.000128">eadem quoque ratione oſtendetur bc <lb></lb>
<figure id="id.023.01.015.1.jpg" xlink:href="023/01/015/1.jpg"></figure><lb></lb>
æqualis cd; &amp; ab ipſi <lb></lb>
bc. quare omnes ab, <lb></lb>
bc, cd, da ſunt æqua­<lb></lb>
les. </s>
          <s id="s.000129">&amp; quoniam anguli <lb></lb>
ad a æquales ſunt angu<lb></lb>
lis ad c; erunt anguli b <lb></lb>
ac, acd coalterni inter <lb></lb>
ſe æquales; <expan abbr="itemq́">itemque</expan>; dac, <lb></lb>
acb. </s>
          <s id="s.000130">ergo cd ipſi ba; <lb></lb>
&amp; ad ipſi bc æquidi­<lb></lb>
ſtat. </s>
          <s id="s.000131">At uero cum lineæ <lb></lb>
ab, cd inter ſe æquidi­<lb></lb>
ſtantes bifariam ſecen­<lb></lb>tur in punctis eg; erit li<lb></lb>
nea lekgn diameter ſe<lb></lb>
ctionis, &amp; linea una, ex <lb></lb>
demonſtratis in uigeſi­<lb></lb>
maoctaua ſecundi coni <lb></lb>
corum. </s>
          <s id="s.000132">Et eadem ratione linea una mfkho. </s>
          <s id="s.000133">Sunt <expan abbr="autẽ">autem</expan> ad, <lb></lb>
bc inter ſe ſe æquales, &amp; æquidiſtantes. </s>
          <s id="s.000134">quare &amp; earum di­<lb></lb>
<arrow.to.target n="marg17"></arrow.to.target><lb></lb>
midiæ ah, bf; <expan abbr="itemq́">itemque</expan>; hd, fe; &amp; quæ ipſas coniungunt rectæ <lb></lb>
lineæ æquales, &amp; æquidiſtantes erunt. </s>
          <s id="s.000135"><expan abbr="æquidiſtãt">æquidiſtant</expan> igitur ba, <lb></lb>
cd diametro mo: &amp; pariter ad, bc ipſi ln æquidiſtare o­<lb></lb>
ſtendemus. </s>
          <s id="s.000136">Si igitur <expan abbr="manẽte">manente</expan> diametro ac intelligatur abc <lb></lb>
portio ellipſis ad portionem adc moueri, cum primum b <lb></lb>
applicuerit ad d, <expan abbr="cõgruet">congruet</expan> tota portio toti portioni, <expan abbr="lineaq́">lineaque</expan>; <lb></lb>
ba lineæ ad; &amp; bc ipſi cd congruet: punctum uero e ca­<lb></lb>
det in h; f in g: &amp; linea ke in lineam kh: &amp; kf in kg. </s>
          <s id="s.000137">qua <lb></lb>
re &amp; el in ho, et fm in gn. </s>
          <s id="s.000138">At ipſa lz in zo; et m<foreign lang="grc">φ</foreign> in <foreign lang="grc">φ</foreign>n <lb></lb>
cadet. </s>
          <s id="s.000139">congruet igitur triangulum lkz triangulo okz: et 



<pb xlink:href="023/01/016.jpg"></pb>triangulum mk<foreign lang="grc">φ</foreign> triangulo nk<foreign lang="grc">φ.</foreign> ergo anguli lzk, ozk, <lb></lb>
m <foreign lang="grc">φ</foreign> k, n<foreign lang="grc">φ</foreign>k æquales ſunt, ac recti. </s>
          <s id="s.000140">quòd cum etiam recti <lb></lb>
<arrow.to.target n="marg18"></arrow.to.target><lb></lb>
ſint, qui ad k; æquidiſtabunt lineæ lo, mn axi bd. </s>
          <s id="s.000141">&amp; ita <lb></lb>
demonſtrabuntur lm, on ipſi ac æquidiſtare. </s>
          <s id="s.000142">Rurſus ſi <lb></lb>iungantur al, lb, bm, mc, cn, nd, do, oa: &amp; bifariam di<lb></lb>
uidantur: à centro autem k ad diuiſiones ductæ lineæ pro­<lb></lb>
trahantur uſque ad ſectionem in puncta pqrstuxy: &amp; po <lb></lb>
ſtremo py, qx, ru, st, qr, ps, yt, xu coniungantur. </s>
          <s id="s.000143">Simili­<lb></lb>
<figure id="id.023.01.016.1.jpg" xlink:href="023/01/016/1.jpg"></figure><lb></lb>
ter oſtendemus lineas <lb></lb>
py, qx, ru, st axi bd æ­<lb></lb>
quidiſtantes eſſe: &amp; qr, <lb></lb>
ps, yt, xu æquidiſtan­<lb></lb>
tes ipſi ac. </s>
          <s id="s.000144">Itaque dico <lb></lb>
harum figurarum in el­<lb></lb>
lipſi deſcriptarum cen­<lb></lb>
trum grauitatis eſſe <expan abbr="pũ-ctum">pun­<lb></lb>
ctum</expan> k, idem quod &amp; el<lb></lb>
lipſis centrum. </s>
          <s id="s.000145">quadri­<lb></lb>
lateri enim abcd cen­<lb></lb>
trum eſt k, ex decima e­<lb></lb>
iuſdem libri Archime­<lb></lb>
dis, quippe <expan abbr="cũ">cum</expan> in eo om<lb></lb>
nes diametri <expan abbr="cõueniãt">conueniant</expan>. </s>
          <lb></lb>
          <s id="s.000146">Sed in figura albmcn <lb></lb>
<arrow.to.target n="marg19"></arrow.to.target><lb></lb>
do, quoniam trianguli <lb></lb>
alb centrum grauitatis <lb></lb>
<arrow.to.target n="marg20"></arrow.to.target><lb></lb>
eſt in linea le: <expan abbr="trapezijq́">trapezijque</expan>; abmo centrum in linea ek: trape<lb></lb>
zij omcd in kg: &amp; trianguli cnd in ipſa gn: erit magnitu<lb></lb>
dinis ex his omnibus conſtantis, uidelicet totius figuræ cen <lb></lb>
trum grauitatis in linea ln: &amp; ob eandem cauſſam in linea <lb></lb>
om. </s>
          <s id="s.000147">eſt enim trianguli aod centrum in linea oh: trapezij <lb></lb>
alnd in hk: trapezij lbcn in kf: &amp; trianguli bmc in fm. </s>
          <lb></lb>
          <s id="s.000148">cum ergo figuræ albmcndo centrum grauitatis ſit in li­<lb></lb>
nea ln, &amp; in linea om; erit centrum ipſius punctum k, in 



<pb pagenum="5" xlink:href="023/01/017.jpg"></pb>quo ſcilicet ln, om conueniunt. </s>
          <s id="s.000149">Poſtremo in figura <lb></lb>
aplqbrmsctnudxoy centrum grauitatis trian<lb></lb>
guli pay, &amp; trapezii ploy eſt in linea az: trapeziorum <lb></lb>
uero lqxo, qbdx centrum eſt in linea zk: &amp; <expan abbr="trapeziorũ">trapeziorum</expan> <lb></lb>
brud, rmnu in k<foreign lang="grc">φ·</foreign> &amp; denique trapezii mstn; &amp; triangu<lb></lb>
li sct in <foreign lang="grc">φ</foreign>c. </s>
          <s id="s.000150">quare magnitudinis ex his compoſitæ <expan abbr="centrũ">centrum</expan> <lb></lb>
in linea ac conſiſtit. </s>
          <s id="s.000151">Rurſus trianguli qbr, &amp; trapezii ql<lb></lb>
mr centrum eſt in linea b<foreign lang="grc">χ.</foreign> trapeziorum lpsm, pacs, <lb></lb>
aytc, yont in linea <foreign lang="grc">χφ·</foreign> <expan abbr="trapeziiq;">trapeziique</expan> oxun, &amp; trianguli <lb></lb>
xdu centrum in <foreign lang="grc">ψ</foreign>d. </s>
          <s id="s.000152">totius ergo magnitudinis centrum <lb></lb>
eſt in linea bd. </s>
          <s id="s.000153">ex quo ſequitur, centrum grauitatis figuræ <lb></lb>
aplqbrmsctnudxoy eſſe <expan abbr="punctũ">punctum</expan> K, lineis ſcilicet ac, <lb></lb>
bd commune, quæ omnia demonſtrare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000154"><margin.target id="marg16"></margin.target>8. primi</s>
        </p>
        <p type="margin">
          <s id="s.000155"><margin.target id="marg17"></margin.target>33.  primi</s>
        </p>
        <p type="margin">
          <s id="s.000156"><margin.target id="marg18"></margin.target>28. primi.</s>
        </p>
        <p type="margin">
          <s id="s.000157"><margin.target id="marg19"></margin.target>13. Archi<lb></lb>
medis.</s>
        </p>
        <p type="margin">
          <s id="s.000158"><margin.target id="marg20"></margin.target>Vltima.</s>
        </p>
        <p type="head">
          <s id="s.000159">THEOREMA III. PROPOSITIO III.</s>
        </p>
        <p type="main">
          <s id="s.000160">Cuiuslibet portio­<lb></lb>
nis circuli, &amp; ellipſis, <lb></lb>
quæ dimidia non ſit <lb></lb>
maior, centrum graui <lb></lb>
tatis in portionis dia­<lb></lb>
metro conſiſtit.</s>
        </p>
        <figure id="id.023.01.017.1.jpg" xlink:href="023/01/017/1.jpg"></figure>
        <p type="main">
          <s id="s.000161">HOC eodem prorſus <lb></lb>
modo demonſtrabitur, <lb></lb>
quo in libro de centro gra<lb></lb>
uitatis planorum ab Ar­<lb></lb>
chimede <expan abbr="demonſtratũ">demonſtratum</expan> eſt, <lb></lb>
in portione <expan abbr="cõtenta">contenta</expan> recta <lb></lb>
linea, &amp; rectanguli coni ſe<lb></lb>
ctione grauitatis <expan abbr="cẽtrum">centrum</expan> <lb></lb>
eſſe in diametro portio­<lb></lb>
nis. </s>
          <s id="s.000162">Et ita demonſtrari po<lb></lb>




<pb xlink:href="023/01/018.jpg"></pb>teſt in portione, quæ recta linea &amp; obtuſianguli coni ſe­<lb></lb>
ctione, ſeu hyperbola continetur.</s>
        </p>
        <p type="head">
          <s id="s.000163">THEOREMA IIII. PROPOSITIO IIII.</s>
        </p>
        <p type="main">
          <s id="s.000164">IN circulo &amp; ellipſi idem eſt figuræ &amp; graui­<lb></lb>
tatis centrum.</s>
        </p>
        <p type="main">
          <s id="s.000165">SIT circulus, uel ellipſis, cuius centrum a. </s>
          <s id="s.000166">Dico a gra­<lb></lb>
uitatis quoque centrum eſſe. </s>
          <s id="s.000167">Si enim fieri poteſt, ſit b cen­<lb></lb>
trum grauitatis: &amp; iuncta ab extra figuram in c produca<lb></lb>
tur: quam uero proportionem habet linea ca ad ab, ha­<lb></lb>
beat circulus a ad alium circulum, in quo d; uel ellipſis ad <lb></lb>
aliam ellipſim: &amp; in circulo, uel ellipſi figura rectilinea pla­<lb></lb>
ne deſcribatur adco, ut tandem relinquantur portiones <lb></lb>
quædam minores circulo, uel ellipſi d; quæ figura ſit abcefg<lb></lb>
hklmn. </s>
          <s id="s.000168">Illud uero in circulo fieri poſſe ex duodecimo <lb></lb>
elementorum libro, propoſitione ſecunda manifeſte con­<lb></lb>
<figure id="id.023.01.018.1.jpg" xlink:href="023/01/018/1.jpg"></figure><lb></lb>
ſtat; at in ellipſi nos demonſtra­<lb></lb>
uimus in commentariis in quin­<lb></lb>
tam propoſitionem Archimedis <lb></lb>
de conoidibus, &amp; ſphæroidibus. </s>
          <lb></lb>
          <s id="s.000169">erit igitur a centrum grauitatis <lb></lb>
ipſius figuræ, quod proxime <expan abbr="oſtẽ">oſten</expan><lb></lb>
dimus. </s>
          <s id="s.000170">Itaque quoniam circulus <lb></lb>
a ad circulum d, uel ellipſis a ad <lb></lb>
ellipſim d eandem <expan abbr="proportionẽ">proportionem</expan> <lb></lb>
habet, quam linea ca ad ab: <lb></lb>
portiones uero ſunt minores cir<lb></lb>
<arrow.to.target n="marg21"></arrow.to.target><lb></lb>
culo uel ellipſi d: habebit circu­<lb></lb>
lus, uel ellipſis ad portiones ma­<lb></lb>
iorem proportionem, quàm ca <lb></lb>
<arrow.to.target n="marg22"></arrow.to.target><lb></lb>
ad ab: &amp; diuidendo figura recti­<lb></lb>
linea abcefghklmn ad portiones 



<pb pagenum="6" xlink:href="023/01/019.jpg"></pb><figure id="id.023.01.019.1.jpg" xlink:href="023/01/019/1.jpg"></figure><lb></lb>
habebit maiorem <expan abbr="proportionẽ">proportionem</expan>, <lb></lb>
quam cb ad ba. </s>
          <s id="s.000171">fiat ob ad ba, <lb></lb>
ut figura rectilinea ad portio­<lb></lb>
nes. </s>
          <s id="s.000172">cum igitur à circulo, uel el­<lb></lb>
lipſi, cuius grauitatis centrum <lb></lb>
eſt b, auferatur figura rectilinea <lb></lb>
efghklmn, cuius centrum a; <lb></lb>
reliquæ magnitudinis ex portio<lb></lb>
<arrow.to.target n="marg23"></arrow.to.target><lb></lb>
nibus compoſitæ centrum graui<lb></lb>
tatis erit in linea ab producta, <lb></lb>
&amp; in puncto o, extra figuram po<lb></lb>
ſito. </s>
          <s id="s.000173">quod quidem fieri nullo mo<lb></lb>
do poſſe perſpicuum eſt. </s>
          <s id="s.000174">ſequi­<lb></lb>
tur ergo, ut circuli &amp; ellipſis cen<lb></lb>
trum grauitatis ſit punctum a, <lb></lb>
idem quod figuræ centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000175"><margin.target id="marg21"></margin.target>8. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000176"><margin.target id="marg22"></margin.target>19. quinti <lb></lb>
apud <expan abbr="Cãpanum">Cam<lb></lb>
panum</expan> .</s>
        </p>
        <p type="margin">
          <s id="s.000177"><margin.target id="marg23"></margin.target>8. Archi­<lb></lb>
medis.</s>
        </p>
        <p type="head">
          <s id="s.000178">ALITER.</s>
        </p>
        <p type="main">
          <s id="s.000179">Sit circulus, uel ellipſis abcd, <lb></lb>
cuius diameter db, &amp; centrum e: <expan abbr="ducaturq;">ducaturque</expan> per e recta li<lb></lb>
nea ac, ſecans ipſam db ad rectos angulos. </s>
          <s id="s.000180">erunt adc, <lb></lb>
abc circuli, uel ellipſis dimidiæ portiones. </s>
          <s id="s.000181">Itaque quo­<lb></lb>
<figure id="id.023.01.019.2.jpg" xlink:href="023/01/019/2.jpg"></figure><lb></lb>
niam por<lb></lb>
<expan abbr="tiõis">tionis</expan> adc <lb></lb>
<expan abbr="cẽtrũ">centrum</expan> gra­<lb></lb>
uitatis eſt <lb></lb>
in diame­<lb></lb>
tro de: &amp; <lb></lb>
portionis <lb></lb>
abc cen­<lb></lb>
trum eſt <expan abbr="ĩ">im</expan> <lb></lb>
ipſa eb: to<lb></lb>
tius circu<lb></lb>
li, uel ellipſis grauitatis centrum erit in diametro db. </s>
          <lb></lb>
          <s id="s.000182">Sit autem portionis adc <expan abbr="cẽtrum">centrum</expan> grauitatis f: &amp; ſumatur 



<pb xlink:href="023/01/020.jpg"></pb>in linea eb <expan abbr="punctũ">punctum</expan> g, ita ut fit ge æqualis ef. </s>
          <s id="s.000183">erit g por­<lb></lb>
tionis abc centrum. </s>
          <s id="s.000184">nam ſi hæ portiones, quæ æquales <lb></lb>
&amp; ſimiles ſunt, inter ſe ſe aptentur, ita ut be cadat in de, <lb></lb>
&amp; punctum b in d cadet, &amp; g in f: figuris autem æquali­<lb></lb>
bus, &amp; ſimilibus inter ſe aptatis, centra quoque grauitatis <lb></lb>
ipſarum inter ſe aptata erunt, ex quinta petitione Archi­<lb></lb>
medis in libro de centro grauitatis planorum. </s>
          <s id="s.000185">Quare cum <lb></lb>
portionis adc centrum grauitatis ſit f: &amp; portionis <lb></lb>
abc centrum g: magnitudinis; quæ ex utriſque efficitur: <lb></lb>
hoc eſt circuli uel ellipſis grauitatis centrum in medio li­<lb></lb>
neæ fg, quod eſt e, conſiſtet, ex quarta propoſitione eiuſ­<lb></lb>
dem libri Archimedis. </s>
          <s id="s.000186">ergo circuli, uel ellipſis centrum <lb></lb>
grauitatis eſt idem, quod figuræ centrum. </s>
          <s id="s.000187">atque illud eſt, <lb></lb>
quod demonſtrare oportebat.</s>
        </p>
        <p type="main">
          <s id="s.000188">Ex quibus ſequitur portionis circuli, uel ellip­<lb></lb>
ſis, quæ dimidia maior ſit, centrum grauitatis in <lb></lb>
diametro quoque ipſius conſiſtere.</s>
        </p>
        <figure id="id.023.01.020.1.jpg" xlink:href="023/01/020/1.jpg"></figure>
        <p type="main">
          <s id="s.000189">Sit enim maior portio abc, cu<emph type="italics"></emph>i<emph.end type="italics"></emph.end>us diameter bd, &amp; com­<lb></lb>
pleatur circulus, uel ellipſis, ut portio reliqua fit aec, dia



<pb pagenum="7" xlink:href="023/01/021.jpg"></pb>metrum habens ed. </s>
          <s id="s.000190">Quoniam igitur circuli uel ellipſis <lb></lb>
aecb grauitatis centrum eſt in diametro be, &amp; portio­<lb></lb>
nis aec centrum in linea ed: reliquæ portionis, uidelicet <lb></lb>
abc centrum grauitatis in ipſa bd conſiſtat neceſſe eſt, ex <lb></lb>
octaua propoſitione eiuſdem.</s>
        </p>
        <p type="head">
          <s id="s.000191">THEOREMA V. PROPOSITIO V.</s>
        </p>
        <p type="main">
          <s id="s.000192">SI priſma ſecetur plano oppoſitis planis æqui <lb></lb>
diſtante, ſectio erit figura æqualis &amp; ſimilis ei, <lb></lb>
quæ eſt oppoſitorum planorum, centrum graui<lb></lb>
tatis in axe habens.</s>
        </p>
        <p type="main">
          <s id="s.000193">Sit priſma, in quo plana oppoſita ſint triangula abc, <lb></lb>
def; axis gh: &amp; ſecetur plano iam dictis planis <expan abbr="æquidiſtã">æquidiſtan</expan><lb></lb>
te; quod faciat ſectionem klm; &amp; axi in <expan abbr="pũcto">puncto</expan> n occurrat. </s>
          <lb></lb>
          <s id="s.000194">Dico klm triangulum æquale eſſe, &amp; ſimile triangulis abc <lb></lb>
def; atque eius grauitatis centrum eſſe punctum n. </s>
          <s id="s.000195">Quo­<lb></lb>
<figure id="id.023.01.021.1.jpg" xlink:href="023/01/021/1.jpg"></figure><lb></lb>
niam enim plana abc <lb></lb>
Klm æquidiſtantia <expan abbr="ſecã">ſecan</expan><lb></lb>
<arrow.to.target n="marg24"></arrow.to.target><lb></lb>
tur a plano ae; rectæ li­<lb></lb>
neæ ab, Kl, quæ ſunt ip <lb></lb>
ſorum <expan abbr="cõmunes">communes</expan> ſectio­<lb></lb>
nes inter ſe ſe æquidi­<lb></lb>
ſtant. </s>
          <s id="s.000196">Sed æquidiſtant <lb></lb>
ad, be; cum ae ſit para<lb></lb>
lelogrammum, ex priſ­<lb></lb>
matis diffinitione. </s>
          <s id="s.000197">ergo <lb></lb>
&amp; al <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb></lb>
erit; &amp; propterea linea <lb></lb>
<arrow.to.target n="marg25"></arrow.to.target><lb></lb>
kl, ipſi ab æqualis. </s>
          <s id="s.000198">Si­<lb></lb>
militer demonſtrabitur <lb></lb>
lm æquidiſtans, &amp; æqua <lb></lb>
lis bc; &amp; mk ipſi ca.</s>
          <pb xlink:href="023/01/022.jpg"></pb>
          <s id="s.000199">Itaque quoniam duæ lineæ Kl, lm ſe ſe tangentes, duabus <lb></lb>
lineis ſe ſe tangentibus ab, bc æquidiſtant; nec ſunt in e o­<lb></lb>
dem plano: angulus klm æqualis eſt angulo abc: &amp; ita an<lb></lb>
<arrow.to.target n="marg26"></arrow.to.target><lb></lb>
gulus lmk, angulo bca, &amp; mkl ipſi cab æqualis probabi<lb></lb>
tur. </s>
          <s id="s.000200">triangulum ergo klm eſt æquale, &amp; ſimile triangulo <lb></lb>
abc. quare &amp; triangulo def. </s>
          <s id="s.000201">Ducatur linea cgo, &amp; per ip<lb></lb>
ſam, &amp; per cf ducatur planum ſecans priſma; cuius &amp; paral<lb></lb>
lelogrammi ae communis ſectio ſit opq.</s>
          <s id="s.000202"> tranſibit linea <lb></lb>
fq per h, &amp; mp per n. </s>
          <s id="s.000203">nam cum plana æquidiſtantia ſecen <lb></lb>
tur à plano cq, communes eorum ſectiones cgo, mp, fq <lb></lb>
ſibi ipſis æquidiſtabunt. </s>
          <s id="s.000204">Sed &amp; æquidiſtant ab, kl, de. </s>
          <s id="s.000205">an­<lb></lb>
<arrow.to.target n="marg27"></arrow.to.target><lb></lb>
guli ergo aoc, kpm, dqf inter ſe æquales ſunt: &amp; ſunt <lb></lb>
æquales qui ad puncta akd conſtituuntur. </s>
          <s id="s.000206">quare &amp; reliqui <lb></lb>
reliquis æquales; &amp; triangula aco, Kmp, dfq inter ſe ſimi <lb></lb>
<arrow.to.target n="marg28"></arrow.to.target><lb></lb>
lia erunt. </s>
          <s id="s.000207">Vt igitur ca ad ao, ita fd ad dq: &amp; permutando <lb></lb>
ut ca ad fd, ita ao ad dq.</s>
          <s id="s.000208">eſt autem ca æqualis fd. </s>
          <s id="s.000209">ergo &amp; <lb></lb>
ao ipſi dq.</s>
          <s id="s.000210"> eadem quoque ratione &amp; ao ipſi Kp æqualis <lb></lb>
demonſtrabitur. </s>
          <s id="s.000211">Itaque ſi triangula, abc, def æqualia &amp; <lb></lb>
<figure id="id.023.01.022.1.jpg" xlink:href="023/01/022/1.jpg"></figure><lb></lb>
ſimilia inter ſe <expan abbr="aptẽtur">aptentur</expan>, <lb></lb>
cadet linea fq in lineam <lb></lb>
<arrow.to.target n="marg29"></arrow.to.target><lb></lb>
cgo. </s>
          <s id="s.000212">Sed &amp; <expan abbr="centrũ">centrum</expan> gra<lb></lb>
uitatis h in g <expan abbr="centrũ">centrum</expan> ca­<lb></lb>
det. </s>
          <s id="s.000213"><expan abbr="trãſibit">tranſibit</expan> igitur linea <lb></lb>
fq per h: &amp; planum per <lb></lb>
co &amp; cf <expan abbr="ductũ">ductum</expan> per <expan abbr="axẽ">axem</expan> <lb></lb>
gh ducetur: <expan abbr="idcircoq;">idcircoque</expan> li <lb></lb>
neam mp <expan abbr="etiã">etiam</expan> per n <expan abbr="trã">tran</expan><lb></lb>
ſire neceſſe erit. </s>
          <s id="s.000214">Quo­<lb></lb>
niam ergo fh, cg æqua­<lb></lb>
les ſunt, &amp; <expan abbr="æquidiſtãtes">æquidiſtantes</expan>: <lb></lb>
<expan abbr="itemq;">itemque</expan> hq, go; rectæ li­<lb></lb>
neæ, quæ ipſas <expan abbr="cõnectũt">connectunt</expan> <lb></lb>
cmf, gnh, opq æqua­<lb></lb>
les æquidiſtantes <expan abbr="erũt">erunt</expan>.</s>
          <pb pagenum="8" xlink:href="023/01/023.jpg"></pb>
          <s id="s.000215">æquidiſtant autem cgo, mnp. </s>
          <s id="s.000216">ergo <expan abbr="parallelogrãma">parallelogramma</expan> ſunt <lb></lb>
on, gm, &amp; linea mn æqualis cg; &amp; np ipſi go. </s>
          <s id="s.000217">aptatis igi­<lb></lb>
tur klm, abc <expan abbr="triãgulis">triangulis</expan>, quæ æqualia &amp; ſimilia <expan abbr="sũt">sunt</expan>; linea mp <lb></lb>
in co, &amp; punctum n in g cadet. </s>
          <s id="s.000218">Quòd <expan abbr="cũ">cum</expan> g ſit centrum gra­<lb></lb>
uitatis trianguli abc, &amp; n trianguli klm grauitatis cen­<lb></lb>
trum erit id, quod demonſtrandum relinquebatur. </s>
          <s id="s.000219">Simili <lb></lb>
ratione idem contingere demonſtrabimus in aliis priſma­<lb></lb>
tibus, ſiue quadrilatera, ſiue plurilatera habeant plana, <lb></lb>
quæ opponuntur.</s>
        </p>
        <p type="margin">
          <s id="s.000220"><margin.target id="marg24"></margin.target>16. unde­<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000221"><margin.target id="marg25"></margin.target>34. primi</s>
        </p>
        <p type="margin">
          <s id="s.000222"><margin.target id="marg26"></margin.target>10. unde<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000223"><margin.target id="marg27"></margin.target>10. unde­<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000224"><margin.target id="marg28"></margin.target>4. ſexti</s>
        </p>
        <p type="margin">
          <s id="s.000225"><margin.target id="marg29"></margin.target>per 5. pe­<lb></lb>
titionem <lb></lb>
Archime<lb></lb>
dis.</s>
        </p>
        <p type="head">
          <s id="s.000226">COROLLARIVM.</s>
        </p>
        <p type="main">
          <s id="s.000227">Ex iam demonſtratis perſpicue apparet, cuius <lb></lb>
libet priſmatis axem, parallelogrammorum lateri<lb></lb>
bus, quæ ab oppoſitis planis <expan abbr="ducũtur">ducuntur</expan> æquidiſtare.</s>
        </p>
        <p type="head">
          <s id="s.000228">THEOREMA VI. PROPOSITIO VI.</s>
        </p>
        <p type="main">
          <s id="s.000229">Cuiuslibet priſmatis centrum grauitatis eſt in <lb></lb>
plano, quod oppoſitis planis æquidiſtans, reli­<lb></lb>
quorum planorum latera bifariam diuidit.</s>
        </p>
        <p type="main">
          <s id="s.000230">Sit priſma, in quo plana, quæ opponuntur ſint trian­<lb></lb>
gula ace, bdf: &amp; parallelogrammorum latera ab, cd, <lb></lb>
ef bifariam <expan abbr="diuidãtur">diuidantur</expan> in punctis ghk: per diuiſiones au­<lb></lb>
<arrow.to.target n="marg30"></arrow.to.target><lb></lb>
tem planum ducatur; cuius ſectio figura ghK. </s>
          <s id="s.000231">erit linea <lb></lb>
gh æquidiſtans lineis ac, bd &amp; hk ipſis ce, df. </s>
          <s id="s.000232">quare ex <lb></lb>
decimaquinta undecimi elementorum, planum illud pla<lb></lb>
nis ace, bdf æquidiſtabit, &amp; faciet ſectionem figu­<lb></lb>
<arrow.to.target n="marg31"></arrow.to.target><lb></lb>
ram ipſis æqualem, &amp; ſimilem, ut proxime demonſtra­<lb></lb>
uimus. </s>
          <s id="s.000233">Dico centrum grauitatis priſmatis eſſe in plano <lb></lb>
ghk. </s>
          <s id="s.000234">Si enim fieri poteſt, ſit eius centrum l: &amp; ducatur <lb></lb>
lm uſque ad planum ghk, quæ ipſi ab æquidiſtet. </s>
          <pb xlink:href="023/01/024.jpg"></pb>
          <s id="s.000235"><arrow.to.target n="marg32"></arrow.to.target>ergo linea ag continenter in duas partes æquales diui­<lb></lb>
ſa, relinquetur <expan abbr="tãdem">tandem</expan> pars aliqua ng, quæ minor erit lm. </s>
          <lb></lb>
          <s id="s.000236">Vtraque uero linearum ag, gb diuidatur in partes æqua­<lb></lb>
les ipſi ng: &amp; per puncta diuiſionum plana oppoſitis pla­<lb></lb>
<arrow.to.target n="marg33"></arrow.to.target><lb></lb>
nis æquidiſtantia ducantur. </s>
          <s id="s.000237">erunt ſectiones figuræ æqua­<lb></lb>
les, ac ſimiles ipſis ace, bdf: &amp; totum priſma diuiſum erit <lb></lb>
in priſmata æqualia, &amp; ſimilia: quæ cum inter ſe <expan abbr="congruãt">congruant</expan>; <lb></lb>
&amp; grauitatis centra ſibi ipſis congruentia, <expan abbr="reſpondentiaq;">reſpondentiaque</expan> <lb></lb>
<figure id="id.023.01.024.1.jpg" xlink:href="023/01/024/1.jpg"></figure><lb></lb>
habebunt. </s>
          <s id="s.000238"><expan abbr="Itaq:">Itaque</expan> <lb></lb>
ſunt magnitudi­<lb></lb>
nes <expan abbr="quædã">quædam</expan> æqua­<lb></lb>
les ipſi nh, &amp; nu­<lb></lb>
mero pares, qua­<lb></lb>
rum centra gra­<lb></lb>
uitatis in <expan abbr="eadẽre">eadem</expan> re<lb></lb>
cta linea conſti­<lb></lb>
tuuntur: duæ ue­<lb></lb>
ro mediæ æqua­<lb></lb>
les ſunt: &amp; quæ ex <lb></lb>
utraque parte i­<lb></lb>
pſarum ſimili­<lb></lb>
ter æquales: &amp; æ­<lb></lb>
quales rectæ li­<lb></lb>
neæ, quæ inter <lb></lb>
grauitatis centra <lb></lb>
interiiciuntur. </s>
          <lb></lb>
          <s id="s.000239">quare ex corolla­<lb></lb>
rio quintæ pro­<lb></lb>
poſitionis primi <lb></lb>
libri Archimedis <lb></lb>
de centro graui­<lb></lb>
tatis planorum; magnitudinis ex his omnibus compoſitæ <lb></lb>
centrum grauitatis eſt in medio lineæ, quæ magnitudi­<lb></lb>
num mediarum centra coniungit. </s>
          <s id="s.000240">at qui non ita res ha­



<pb pagenum="9" xlink:href="023/01/025.jpg"></pb>bet, ſi quidem l extra medias magnitudines poſitum eſt. </s>
          <lb></lb>
          <s id="s.000241">Conſtat igitur centrum grauitatis priſmatis eſſe in plano <lb></lb>
<figure id="id.023.01.025.1.jpg" xlink:href="023/01/025/1.jpg"></figure><lb></lb>
ghk, quod nos demonſtrandum propoſuimus. </s>
          <s id="s.000242">At ſi op­<lb></lb>
poſita plana in priſmate ſint quadrilatera, uel plurilatera, <lb></lb>
eadem erit in omnibus demonſtratio.</s>
        </p>
        <p type="margin">
          <s id="s.000243"><margin.target id="marg30"></margin.target>33. primi</s>
        </p>
        <p type="margin">
          <s id="s.000244"><margin.target id="marg31"></margin.target>5. huius</s>
        </p>
        <p type="margin">
          <s id="s.000245"><margin.target id="marg32"></margin.target>1. decimi</s>
        </p>
        <p type="margin">
          <s id="s.000246"><margin.target id="marg33"></margin.target>5 huius</s>
        </p>
        <p type="head">
          <s id="s.000247">THEOREMA VII. PROPOSITIO VII.</s>
        </p>
        <p type="main">
          <s id="s.000248">Cuiuslibet cylindri, &amp; cuiuslibet cylindri por<lb></lb>
tionis centrum grauitatis eſt in plano, quod baſi­<lb></lb>
bus æquidiſtans, parallelogrammi per axem late­<lb></lb>
ra bifariam ſecat.</s>
        </p>
        <pb xlink:href="023/01/026.jpg"></pb>
        <p type="main">
          <s id="s.000249">SIT cylindrus, uel cylindri portio ac: &amp; plano per a­<lb></lb>
xem ducto ſecetur; cuius ſectio ſit parallelogrammum ab<lb></lb>
cd: &amp; bifariam diuiſis ad, bc parallelogrammi lateribus, <lb></lb>
per diuiſionum puncta ef planum baſi æquidiſtans duca­<lb></lb>
tur; quod faciet ſectionem, in cylindro quidem circulum <lb></lb>
æqualem iis, qui ſunt in baſibus, ut demonſtrauit Serenus <lb></lb>
in libro cylindricorum, propoſitione quinta: in cylindri <lb></lb>
uero portione ellipſim æqualem, &amp; ſimilem eis, quæ ſunt <lb></lb>
<figure id="id.023.01.026.1.jpg" xlink:href="023/01/026/1.jpg"></figure><lb></lb>
in oppoſitis planis, quod nos <lb></lb>demonſtrauimus in commen<lb></lb>
tariis in librum Archimedis <lb></lb>
de conoidibus, &amp; ſphæroidi­<lb></lb>
bus. </s>
          <s id="s.000250">Dico centrum grauita­<lb></lb>
tis cylindri, uel cylindri por­<lb></lb>
tionis eſſe in plano ef. </s>
          <s id="s.000251">Si <expan abbr="enĩ">enim</expan> <lb></lb>
fieri poteſt, fit centrum g: &amp; <lb></lb>
ducatur gh ipſi ad æquidi­<lb></lb>
ſtans, uſque ad ef planum. </s>
          <lb></lb>
          <s id="s.000252">Itaque linea ae continenter <lb></lb>
diuiſa bifariam, erit tandem <lb></lb>
pars aliqua ipſius ke, minor <lb></lb>
gh. </s>
          <s id="s.000253">Diuidantur ergo lineæ <lb></lb>
ae, ed in partes æquales ipſi <lb></lb>
ke: &amp; per diuiſiones plana ba<lb></lb>
ſibus æquidiſtantia <expan abbr="ducãtur">ducantur</expan>. </s>
          <lb></lb>
          <s id="s.000254">erunt iam ſectiones, figuræ æ­<lb></lb>
quales, &amp; ſimiles eis, quæ ſunt <lb></lb>
in baſibus: atque erit cylindrus in cylindros diuiſus: &amp; cy<lb></lb>
lindri portio in portiones æquales, &amp; ſimiles ipſi kf. </s>
          <s id="s.000255">reli­<lb></lb>
qua ſimiliter, ut ſuperius in priſmate concludentur.</s>
        </p>
        <pb pagenum="10" xlink:href="023/01/027.jpg"></pb>
        <figure id="id.023.01.027.1.jpg" xlink:href="023/01/027/1.jpg"></figure>
        <p type="head">
          <s id="s.000256">THEOREMA VIII. PROPOSITIO VIII.</s>
        </p>
        <p type="main">
          <s id="s.000257">Cuiuslibet priſmatis, &amp; cuiuslibet cylindri, uel <lb></lb>
cylindri portionis grauitatis centrum in medio <lb></lb>
ipſius axis conſiſtit.</s>
        </p>
        <p type="main">
          <s id="s.000258">Sit primum af priſma æquidiſtantibus planis <expan abbr="contentũ">contentum</expan>, <lb></lb>
quod ſolidum parallelepipedum appellatur: &amp; oppoſito­<lb></lb>
rum planorum cf, ah, da, fg latera bifariam diuidantur in <lb></lb>
punctis klmnopqrstux: &amp; per diuiſiones ducantur <lb></lb>
plana kn, or, sx. </s>
          <s id="s.000259">communes autem eorum planorum ſe­<lb></lb>
ctiones ſint lineæ yz, <foreign lang="grc">θφ, χψ·</foreign> quæ in puncto <foreign lang="grc">ω</foreign> <expan abbr="conueniãt">conueniant</expan>. </s>
          <lb></lb>
          <s id="s.000260">erit ex decima eiuſdem libri Archimedis parallelogrammi <lb></lb>
cf centrum grauitatis punctum y; parallelogrammi ah 



<pb xlink:href="023/01/028.jpg"></pb>centrum z: parallelogrammi ad, <foreign lang="grc">θ·</foreign> parallelogrammi fg, <foreign lang="grc">φ·</foreign><lb></lb>
<figure id="id.023.01.028.1.jpg" xlink:href="023/01/028/1.jpg"></figure><lb></lb>
parallelogrammi dh, <foreign lang="grc">χ·</foreign> &amp; <lb></lb>
parallelogrammi cg <expan abbr="centrũ">centrum</expan> <lb></lb>
<foreign lang="grc">ψ·</foreign> atque erit <foreign lang="grc">ω</foreign> punctum me <lb></lb>
dium uniuſcuiuſque axis, ui <lb></lb>
delicet eius lineæ quæ oppo <lb></lb>
ſitorum <expan abbr="planorũ">planorum</expan> centra con <lb></lb>
iungit. </s>
          <s id="s.000261">Dico <foreign lang="grc">ω</foreign> centrum eſſe <lb></lb>
grauitatis ipſius ſolidi. </s>
          <s id="s.000262">eſt <lb></lb>
<arrow.to.target n="marg34"></arrow.to.target><lb></lb>
enim, ut demonſtrauimus, <lb></lb>
ſolidi af centrum grauitatis <lb></lb>
in plano Kn; quod oppoſi­<lb></lb>
tis planis ad, gf æquidiſtans <lb></lb>
reliquorum planorum late­<lb></lb>
ra bifariam diuidit: &amp; ſimili <lb></lb>
ratione idem centrum eſt in plano or, æquidiſtante planis <lb></lb>
ae, bf oppoſitis. </s>
          <s id="s.000263">ergo in communi ipſorum ſectione: ui­<lb></lb>
delicet in linea yz. </s>
          <s id="s.000264">Sed eſt etiam in plano tu, quod <expan abbr="quidẽ">quidem</expan> <lb></lb>
yz ſecatin <foreign lang="grc">ω.</foreign> Conſtat igitur centrum grauitatis ſolidi eſſe <lb></lb>
punctum <foreign lang="grc">ω,</foreign> medium ſcilicet axium, hoc eſt linearum, quæ <lb></lb>
planorum oppoſitorum centra coniungunt.</s>
        </p>
        <p type="margin">
          <s id="s.000265"><margin.target id="marg34"></margin.target>6 huius</s>
        </p>
        <p type="main">
          <s id="s.000266">Sit aliud prima af; &amp; in eo plana, quæ opponuntur, tri­<lb></lb>
angula abc, def: <expan abbr="diuiſisq;">diuiſisque</expan> bifariam parallelogrammorum <lb></lb>
lateribus ad, be, cf in punctis ghk, per diuiſiones <expan abbr="planũ">planum</expan> <lb></lb>
ducatur, quod oppoſitis planis æquidiſtans faciet <expan abbr="ſectionẽ">ſectionem</expan> <lb></lb>
triangulum ghx æquale, &amp; ſimile ipſis abc, def. </s>
          <s id="s.000267">Rurſus <lb></lb>
diuidatur ab bifariam in l: &amp; iuncta cl per ipſam, &amp; per <lb></lb>
cKf planum ducatur priſma ſecans, cuius, &amp; <expan abbr="parallelogrã">parallelogram</expan><lb></lb>
mi ae communis ſectio ſit lmn. </s>
          <s id="s.000268">diuidet punctum m li­<lb></lb>
neam gh bifariam; &amp; ita n diuidet lineam de: quoniam <lb></lb>
<arrow.to.target n="marg35"></arrow.to.target><lb></lb>
triangula acl, gkm, dfn æqualia ſunt, &amp; ſimilia, ut ſupra <lb></lb>
demonſtrauimus. </s>
          <s id="s.000269">Iam ex iis, quæ tradita ſunt, conſtat cen<lb></lb>
trum grauitatis priſmatis in plano ghk contineri. </s>
          <s id="s.000270">Dico <lb></lb>
ipſum eſſe in linea km. </s>
          <s id="s.000271">Si enim fieri poteſt, ſit o centrum; 



<pb pagenum="11" xlink:href="023/01/029.jpg"></pb>&amp; per o ducatur op ad km ipſi hg æquidiſtans. </s>
          <s id="s.000272">Itaque li<lb></lb>
nea hm <expan abbr="bifariã">bifariam</expan> uſque eò diuidatur, quoad reliqua ſit pars <lb></lb>
quædam qm, minor op. </s>
          <s id="s.000273">deinde hm, mg diuidantur in <lb></lb>
partes æquales ipſi mq: &amp; per diuiſiones lineæ ipſi mK <lb></lb>
æquidiſtantes ducantur. </s>
          <s id="s.000274">puncta uero, in quibus hæ trian­<lb></lb>
gulorum latera ſecant, coniungantur ductis lineis rs, tu, <lb></lb>
<figure id="id.023.01.029.1.jpg" xlink:href="023/01/029/1.jpg"></figure><lb></lb>
xy; quæ baſi gh æquidiſtabunt. </s>
          <s id="s.000275">Quoniam enim lineæ gz, <lb></lb>
h<foreign lang="grc">α</foreign> ſunt æquales: <expan abbr="itemq;">itemque</expan> æquales gm, mh: ut mg ad gz, <lb></lb>
ita erit mh, ad h<foreign lang="grc">α·</foreign> &amp; diuidendo, ut mz ad zg, ita m<foreign lang="grc">α</foreign> ad <lb></lb>
<arrow.to.target n="marg36"></arrow.to.target><lb></lb>
<foreign lang="grc">α</foreign>h. </s>
          <s id="s.000276">Sed ut mz ad zg, ita kr ad rg: &amp; ut m<foreign lang="grc">α</foreign> ad <foreign lang="grc">α</foreign>h, ita ks <lb></lb>
ad sh. </s>
          <s id="s.000277">quare ut kr ad rg, ita ks ad sh. </s>
          <s id="s.000278">æquidiſtant igitur <lb></lb>
<arrow.to.target n="marg37"></arrow.to.target><lb></lb>
inter ſe ſe rs, gh. </s>
          <s id="s.000279">eadem quoque ratione demonſtrabimus 



<pb xlink:href="023/01/030.jpg"></pb>tu, xy ipſi gh æquidiſtare. </s>
          <s id="s.000280">Et quoniam triangula, quæ <lb></lb>
fiunt à lineis Ky, yu, us, sh æqualia ſunt inter ſe, &amp; ſimilia <lb></lb>
<arrow.to.target n="marg38"></arrow.to.target><lb></lb>
triangulo Kmh: habebit triangulum Kmh ad <expan abbr="triangulũ">triangulum</expan> <lb></lb>
K<foreign lang="grc">δ</foreign>y duplam proportionem eius, quæ eſt lineæ kh ad Ky. </s>
          <lb></lb>
          <s id="s.000281">ſed Kh poſita eſt quadrupla ipſius ky. </s>
          <s id="s.000282">ergo triangulum <lb></lb>
kmh ad triangulum K<foreign lang="grc">δ</foreign>y <expan abbr="eãdem">eandem</expan> proportionem habebit, <lb></lb>
quam ſexdecim ad <expan abbr="unũ">unum</expan>: &amp; ad quatuor triangula k<foreign lang="grc">δ</foreign>y, yu, <lb></lb>
us, s<foreign lang="grc">α</foreign>h habebit eandem, quam ſexdecim ad quatuor, hoc <lb></lb>eſt quam hK ad ky: &amp; ſimiliter eandem habere demonſtra<lb></lb>
<figure id="id.023.01.030.1.jpg" xlink:href="023/01/030/1.jpg"></figure><lb></lb>
bitur trian­<lb></lb>
gulum kmg <lb></lb>
ad quatuor <lb></lb>
<expan abbr="triãgula">triangula</expan> K<foreign lang="grc">δ</foreign><lb></lb>
x, x<foreign lang="grc">γ</foreign>t, t<foreign lang="grc">β</foreign>r, <lb></lb>
<arrow.to.target n="marg39"></arrow.to.target><lb></lb>
rzg. </s>
          <s id="s.000283">quare <lb></lb>totum trian<lb></lb>
gulum Kgh <lb></lb>
ad omnia tri <lb></lb>
angula gzr, <lb></lb>
r<foreign lang="grc">β</foreign>t, t<foreign lang="grc">γ</foreign>x, x<foreign lang="grc">δ</foreign><lb></lb>
K, K<foreign lang="grc">δ</foreign>y, yu, <lb></lb>
us, s<foreign lang="grc">α</foreign>h ita <lb></lb>
erit, ut hk ad <lb></lb>
ky, hoc eſt <lb></lb>
ut hm ad m<lb></lb>
q. </s>
          <s id="s.000284">Si igitur in <lb></lb>
triangulis abc, def deſcribantur figuræ ſimiles ei, quæ de­<lb></lb>
ſcripta eſt in ghK triangulo: &amp; per lineas ſibi reſponden­<lb></lb>
tes plana ducantur: totum priſma af diuiſum erit in tria <lb></lb>
ſolida parallelepipeda y<foreign lang="grc">γ,</foreign> u<foreign lang="grc">β,</foreign> sz, quorum baſes ſunt æqua <lb></lb>
les &amp; ſimiles ipſis parallelogrammis y <foreign lang="grc">γ,</foreign>u<foreign lang="grc">β,</foreign> sz: &amp; in octo <lb></lb>
priſmata gzr, r<foreign lang="grc">β</foreign>t, t<foreign lang="grc">γ</foreign>x, x<foreign lang="grc">δ</foreign><lb></lb>
K, k<foreign lang="grc">δ</foreign>y, yu, us, s<foreign lang="grc">α</foreign>h: quorum <lb></lb>
item baſes æquales, &amp; ſimiles ſunt dictis triangulis; altitu­<lb></lb>
do autem in omnibus, totius priſmatis altitudini æqualis.



<pb pagenum="12" xlink:href="023/01/031.jpg"></pb>Itaque ſolidi parallelepipedi y<foreign lang="grc">γ</foreign> centrum grauitatis eſt in <lb></lb>
linea <foreign lang="grc">δε·</foreign> ſolidi u<foreign lang="grc">β</foreign> centrum eſt in linea <foreign lang="grc">εη·</foreign> &amp; ſolidi sz in li<lb></lb>
nea <foreign lang="grc">η</foreign>m, quæ quidem lineæ axes ſunt, cum planorum oppo<lb></lb>
ſitorum centra coniungant. </s>
          <s id="s.000285">ergo magnitudinis ex his ſoli <lb></lb>
dis compoſitæ centrum grauitatis eſt in linea <foreign lang="grc">δ</foreign>m, quod ſit <lb></lb>
<foreign lang="grc">θ</foreign>; &amp; iuncta <foreign lang="grc">θ</foreign>o producatur: à puncto autem h ducatur h<foreign lang="grc">α</foreign><lb></lb>
ipſi mk æquidiſtans, quæ cum <foreign lang="grc">θ</foreign>o in <foreign lang="grc">μ</foreign> conueniat. </s>
          <s id="s.000286">triangu<lb></lb>
lum igitur ghk ad omnia triangula gzr,  <foreign lang="grc">β</foreign>t, t<foreign lang="grc">γ</foreign>x, x<foreign lang="grc">δ</foreign>k, <lb></lb>
k<foreign lang="grc">δ</foreign>y, yu, us, s<foreign lang="grc">α</foreign>h eandem habet proportionem, quam hm <lb></lb>
ad mq ; hoc eſt, quam <foreign lang="grc">μθ</foreign> ad <foreign lang="grc">θλ·</foreign> nam ſi hm, <foreign lang="grc">μθ</foreign> produci in<lb></lb>
telligantur, quouſque coeant; erit ob linearum qy, mk æ­<lb></lb>
quidiſtantiam, ut hq ad qm, ita <foreign lang="grc">μλ</foreign> ad ad <foreign lang="grc">λθ·</foreign> &amp; componen <lb></lb>
do, ut hm ad mq, ita <foreign lang="grc">μθ</foreign> ad <foreign lang="grc">θλ.</foreign></s>
          <s id="s.000287"> linea uero <foreign lang="grc">θ</foreign>o maior eſt, <lb></lb>
<arrow.to.target n="marg40"></arrow.to.target><lb></lb>
quàm <foreign lang="grc">θλ·</foreign> habebit igitur <foreign lang="grc">μθ</foreign> ad <foreign lang="grc">θλ</foreign> maiorem proportio­<lb></lb>
nem, quàm ad <foreign lang="grc">θ</foreign>o. </s>
          <s id="s.000288">quare triangulum etiam ghk ad omnia <lb></lb>
iam dicta triangula maiorem <expan abbr="proportionẽ">proportionem</expan> habebit, quàm <lb></lb>
<foreign lang="grc">μθ</foreign> ad <foreign lang="grc">θ</foreign>o. </s>
          <s id="s.000289">ſed ut <expan abbr="triangulũ">triangulum</expan> ghk ad omnia triangula, ita <expan abbr="to-tũ">to­<lb></lb>
tum</expan> priſma afad omnia priſmata gzr, r<foreign lang="grc">β</foreign>t, t<foreign lang="grc">γ</foreign>x, x<foreign lang="grc">δκ, κδ</foreign> y, <lb></lb>
yu, us, s<foreign lang="grc">α</foreign>h: quoniam enim ſolida parallelepipeda æque al<lb></lb>
ta, eandem inter ſe proportionem habent, quam baſes; ut <lb></lb>
ex trigeſimaſecunda undecimi elementorum conſtat. </s>
          <s id="s.000290">ſunt <lb></lb>
<arrow.to.target n="marg41"></arrow.to.target><lb></lb>
autem ſolida parallelepipeda priſmatum triangulares ba­<lb></lb>
<arrow.to.target n="marg42"></arrow.to.target><lb></lb>
ſes habentium dupla: ſequitur, ut etiam huiuſmodi priſ­<lb></lb>
mata inter ſe ſint, ſicut eorum baſes. </s>
          <s id="s.000291">ergo totum priſma ad <lb></lb>
omnia priſmata maiorem proportionem habet, quam <foreign lang="grc">μθ</foreign><lb></lb>
<arrow.to.target n="marg43"></arrow.to.target><lb></lb>
ad <foreign lang="grc">θ</foreign>o: &amp; diuidendo ſolida parallelepipeda y<foreign lang="grc">γ,</foreign> u<foreign lang="grc">β,</foreign> sz ad o­<lb></lb>
mnia priſmata proportionem habent maiorem, quàm <foreign lang="grc">μ</foreign>o <lb></lb>
ad o<foreign lang="grc">θ</foreign>. </s>
          <s id="s.000292">fiat <foreign lang="grc">ν</foreign>o ad o<foreign lang="grc">θ,</foreign> ut ſolida parallelepipeda y <foreign lang="grc">γ,</foreign> u<foreign lang="grc">β,</foreign> sz ad <lb></lb>
omnia priſmata. </s>
          <s id="s.000293">Itaque cum à priſmate af, cuius <expan abbr="cẽtrum">centrum</expan> <lb></lb>
grauitatis eſt o, auferatur magnitudo ex ſolidis parallelepi<lb></lb>
pedis y <foreign lang="grc">γ,</foreign>u<foreign lang="grc">β,</foreign>sz conſtans: atque ipſius grauitatis centrum <lb></lb>
ſit <foreign lang="grc">θ·</foreign> reliquæ magnitudinis, quæ ex omnibus priſmatibus <lb></lb>
conſtat, grauitatis centrum erit in linea <foreign lang="grc">θ</foreign> o producta: &amp; <lb></lb>
in puncto <foreign lang="grc">f</foreign>, ex octava propoſitione eiusdem libri Archi­



<pb xlink:href="023/01/032.jpg"></pb>medis. </s>
          <s id="s.000294">ergo punctum <foreign lang="grc">ν</foreign> extra priſma af poſitum, <expan abbr="centrũ">centrum</expan> <lb></lb>
erit magnitudinis <expan abbr="cõpoſitæ">compoſitæ</expan> ex omnibus priſmatibus gzr, <lb></lb>
r <foreign lang="grc">β</foreign>t, t<foreign lang="grc">γ</foreign>x, x<foreign lang="grc">δ</foreign>k, k<foreign lang="grc">δ</foreign> y, yu, us, s<foreign lang="grc">α</foreign>h, quod fieri nullo modo po<lb></lb>
teſt. </s>
          <s id="s.000295">eſt enim ex diffinitione centrum grauitatis ſolidæ figu<lb></lb>
ræ intra ipſam poſitum, non extra. </s>
          <s id="s.000296">quare relinquitur, ut <expan abbr="cẽtrum">cen<lb></lb>
trum</expan> grauitatis priſmatis ſit in linea Km. </s>
          <s id="s.000297">Rurſus bc bifa­<lb></lb>
riam in diuidatur: &amp; ducta a<foreign lang="grc">χ,</foreign> per ipſam, &amp; per lineam <lb></lb>agd planum ducatur; quod priſma ſecet: <expan abbr="faciatq;">faciatque</expan> in paral<lb></lb>
lelogrammo bf ſectionem <foreign lang="grc">χ π</foreign> diuidet punctum <foreign lang="grc">π</foreign> lineam <lb></lb>
quoque cf bifariam: &amp; erit plani eius, &amp; trianguli ghK <lb></lb>
communis ſectio gu; quòd <expan abbr="pũctum">punctum</expan> u in medio lineæ hK <lb></lb>
<figure id="id.023.01.032.1.jpg" xlink:href="023/01/032/1.jpg"></figure><lb></lb>
poſitum ſit. </s>
          <s id="s.000298">Similiter demonſtrabimus centrum grauita­<lb></lb>
tis priſmatis in ipſa gu ineſſe. </s>
          <s id="s.000299">ſit autem planorum cfnl, <lb></lb>
ad<foreign lang="grc">πχ</foreign> communis ſectio linea <foreign lang="grc">ρστ;</foreign> quæ quidem priſmatis <lb></lb>
axis erit, cum tranſeat per centra grauitatis triangulorum <lb></lb>
abc, ghk def, ex quartadecima eiuſdem. </s>
          <s id="s.000300">ergo centrum <lb></lb>
grauitatis priſmatis af eſt punctum <foreign lang="grc">ς,</foreign> centrum ſcilicet 



<pb pagenum="13" xlink:href="023/01/033.jpg"></pb>trianguli ghK, &amp; ipſius <foreign lang="grc">ρτ</foreign> axis medium.</s>
        </p>
        <p type="margin">
          <s id="s.000301"><margin.target id="marg35"></margin.target>5.huius</s>
        </p>
        <p type="margin">
          <s id="s.000302"><margin.target id="marg36"></margin.target>2. ſexti.<lb></lb>
 12 quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000303"><margin.target id="marg37"></margin.target>2. ſexti.</s>
        </p>
        <p type="margin">
          <s id="s.000304"><margin.target id="marg38"></margin.target>
19. ſexti</s>
        </p>
        <p type="margin">
          <s id="s.000305"><margin.target id="marg39"></margin.target>2. uel 12. <lb></lb>
quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000306"><margin.target id="marg40"></margin.target>8. quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000307"><margin.target id="marg41"></margin.target>28. unde<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000308"><margin.target id="marg42"></margin.target>15. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000309"><margin.target id="marg43"></margin.target>19. quinti<lb></lb>
apud <expan abbr="Cãpanum">Cam<lb></lb>
panum</expan></s>
        </p>
        <p type="main">
          <s id="s.000310">Sit priſma ag, cuius oppoſita plana ſint quadrilatera <lb></lb>
abcd, efgh: <expan abbr="ſecenturq;">ſecenturque</expan> ac, bf, cg, dh bifariam: &amp; per di­<lb></lb>
uiſiones planum ducatur; quod ſectionem faciat quadrila­<lb></lb>
terum Klmn. </s>
          <s id="s.000311">Deinde iuncta ac per lineas ac, ae ducatur <lb></lb>
planum <expan abbr="ſecãs">ſecans</expan> priſma, quod ipſum diuidet in duo priſmata <lb></lb>
triangulares baſes habentia abcefg, adcehg. </s>
          <s id="s.000312">Sint <expan abbr="autẽ">autem</expan> <lb></lb>
<figure id="id.023.01.033.1.jpg" xlink:href="023/01/033/1.jpg"></figure><lb></lb>
triangulorum abc, efg gra­<lb></lb>
uitatis centra op: &amp; triangu­<lb></lb>
lorum adc, ehg centra qr: <lb></lb>
<expan abbr="iunganturq;">iunganturque</expan> op, qr; quæ pla­<lb></lb>
no klmn occurrant in pun­<lb></lb>
ctis st. </s>
          <s id="s.000313">erit ex iis, quæ demon<lb></lb>
ſtrauimus, punctum s grauita<lb></lb>
tis centrum trianguli klm; &amp; <lb></lb>
ipſius priſmatis abcefg: pun<lb></lb>
ctum uero t centrum grauita <lb></lb>
tis trianguli Knm, &amp; priſma­<lb></lb>
tis adc, ehg. </s>
          <s id="s.000314">iunctis igitur <lb></lb>
oq, pr, st, erit in linea oq <expan abbr="cẽ">cen</expan> <lb></lb>
trum grauitatis quadrilateri <lb></lb>
abcd, quod ſit u: &amp; in linea <lb></lb>
pr <expan abbr="cẽtrum">centrum</expan> quadrilateri efgh <lb></lb>
ſit autem x. </s>
          <s id="s.000315">denique iungatur <lb></lb>
u x, quæ ſecet lineam ſ t in y. </s>
          <s id="s.000316">ſe<lb></lb>
cabit enim cum ſint in eodem <lb></lb>
<arrow.to.target n="marg44"></arrow.to.target><lb></lb>
plano: <expan abbr="atq;">atque</expan> erit y grauitatis centrum quadrilateri Klmn. </s>
          <lb></lb>
          <s id="s.000317">Dico idem punctum y centrum quoque gra uitatis eſſe to­<lb></lb>
tius priſmatis. </s>
          <s id="s.000318">Quoniam enim quadrilateri klmn graui­<lb></lb>
tatis centrum eſt y: linea sy ad yt ean dem proportionem <lb></lb>
habebit, quam triangulum knm ad triangulum klm, ex 8 <lb></lb>
Archimedis de centro grauitatis planorum. </s>
          <s id="s.000319">Vt autem <expan abbr="triã">trian</expan><lb></lb>
gulum knm ad ipſum klm, hoc eſt ut triangulum adc ad <lb></lb>
triangulum abc, æqualia enim ſunt, ita priſma adcehg 



<pb xlink:href="023/01/034.jpg"></pb>ad priſma abcefg. </s>
          <s id="s.000320">quare linea sy ad yt eandem propor­<lb></lb>
tionem habet, quam priſma adcehg ad priſma abcefg. </s>
          <lb></lb>
          <s id="s.000321">Sed priſmatis abcefg centrum grauitatis eſt s: &amp; priſma­<lb></lb>
tis adcehg centrum t. </s>
          <s id="s.000322">magnitudinis igitur ex his compo<lb></lb>
ſitæ hoc eſt totius priſmatis ag centrum grauitatis eſt pun<lb></lb>
ctum y; medium ſcilicet axis ux, qui oppoſitorum plano­<lb></lb>
rum centra coniungit.</s>
        </p>
        <p type="margin">
          <s id="s.000323"><margin.target id="marg44"></margin.target>5. huius/&gt;</s>
        </p>
        <p type="main">
          <s id="s.000324">Rurſus ſit priſma baſim habens pentagonum abcde: <lb></lb>
&amp; quod ei opponitur ſit fghKl: ſec<expan abbr="enturq;">enturque</expan> af, bg, ch, <lb></lb>
dk, el bifariam: &amp; per diuiſiones ducto plano, ſectio ſit <expan abbr="pẽ">pen</expan><lb></lb>
<expan abbr="tagonũ">tagonum</expan> mnopq. deinde iuncta eb per lineas le, eb aliud <lb></lb>
<figure id="id.023.01.034.1.jpg" xlink:href="023/01/034/1.jpg"></figure><lb></lb>
planum ducatur, <expan abbr="diuidẽs">diuidens</expan> priſ<lb></lb>
ma ak in duo priſmata; in priſ<lb></lb>
ma ſcilicet al, cuius plana op­<lb></lb>
poſita ſint triangula abe fgl: <lb></lb>
&amp; in prima bk cuius plana op<lb></lb>
poſita ſint quadrilatera bcde <lb></lb>
ghkl. </s>
          <s id="s.000325">Sint autem triangulo­<lb></lb>rum abe, fgl centra grauita<lb></lb>
tis puncta r ſ: &amp; bcde, ghkl <lb></lb>
quadrilaterorum centra tu: <lb></lb>
<expan abbr="iunganturq;">iunganturque</expan> rs, tu occurren­<lb></lb>
tes plano mnopq in punctis <lb></lb>
xy. </s>
          <s id="s.000326">&amp; itidem <expan abbr="iungãtur">iungantur</expan> rt, ſu, <lb></lb>
xy. </s>
          <s id="s.000327">erit in linea rt <expan abbr="cẽtrum">centrum</expan> gra<lb></lb>
uitatis pentagoni abcde; <lb></lb>
quod ſit z: &amp; in linea ſu cen­<lb></lb>
trum pentagoni fghkl :ſit au <lb></lb>
tem <foreign lang="grc">χ·</foreign> &amp; ducatur z<foreign lang="grc">χ,</foreign> quæ di­<lb></lb>
cto plano in <foreign lang="grc">ψ</foreign> occurrat. </s>
          <s id="s.000328"><expan abbr="Itaq;">Itaque</expan> <lb></lb>
punctum x eſt centrum graui <lb></lb>
tatis trianguli mnq, ac priſ­<lb></lb>
matis al: &amp; y grauitatis centrum quadrilateri nopq, ac <lb></lb>
priſmatis 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"></pb>ſimiliter demonſtrabitur totius priſmatis aK grauitatis ef <lb></lb>
ſe centrum. </s>
          <s id="s.000331">Simili ratione &amp; in aliis priſmatibus illud <lb></lb>
idem facile demonſtrabitur. </s>
          <s id="s.000332">Quo autem pacto in omni <lb></lb>
figura rectilinea centrum grauitatis inueniatur, docuimus <lb></lb>
in commentariis in ſextam propoſitionem Archimedis de <lb></lb>
quadratura parabolæ.</s>
        </p>
        <p type="main">
          <s id="s.000333">Sit cylindrus, uel cylindri portio ce cuius axis ab: ſece­<lb></lb>
<expan abbr="turq,">turque</expan> plano per axem ducto; quod ſectionem faciat paral­<lb></lb>
lelogrammum cdef: &amp; diuiſis cf, de bifariam in punctis <lb></lb>
<figure id="id.023.01.035.1.jpg" xlink:href="023/01/035/1.jpg"></figure><lb></lb>
gh, per ea ducatur planum baſi æquidiſtans. </s>
          <s id="s.000334">erit ſectio gh <lb></lb>
circulus, uel ellipſis, centrum habens in axe; quod ſit K at­<lb></lb>
<arrow.to.target n="marg45"></arrow.to.target><lb></lb>
que erunt ex iis, quæ demonſtrauimus, centra grauitatis <lb></lb>
planorum oppoſitorum puncta ab: &amp; plani gh ipſum k in <lb></lb>
quo quidem plano eſt centrum grauitatis cylindri, uel cy­<lb></lb>
lindri portionis. </s>
          <s id="s.000335">Dico punctum K cylindri quoque, uel cy<lb></lb>
lindri portionis grauitatis centrum eſſe. </s>
          <s id="s.000336">Si enim fieri po­<lb></lb>
teſt, ſit l centrum: <expan abbr="ducaturq;">ducaturque</expan> kl, &amp; extra figuram in m pro­<lb></lb>
ducatur. </s>
          <s id="s.000337">quam ucro proportionem habet linea mK ad kl 



<pb xlink:href="023/01/036.jpg"></pb>habeat circulus, uel ellipſis gh ad aliud ſpacium, in quo u: <lb></lb>
&amp; in cit culo, uel ellipſi plane deſcribatur rectilinea figura, <lb></lb>
ita ut <expan abbr="tãdem">tandem</expan> <expan abbr="relinquãtur">relinquantur</expan> portiones minores ſpacio u, quæ <lb></lb>
ſit opgqrsht: <expan abbr="deſcriptaq;">deſcriptaque</expan> ſimili figura in oppoſitis pla­<lb></lb>
nis cd, fe, per lineas ſibi ipſis reſpondentes plana <expan abbr="ducãtur">ducantur</expan>. </s>
          <lb></lb>
          <s id="s.000338">Itaque cylindrus, uel cylindri portio diuiditur in priſma, <lb></lb>
cuius quidem baſis eſt figura rectilinea iam dicta, centrum <lb></lb>que grauitatis punctum K: &amp; in multa ſolida, quæ pro baſi<lb></lb>
bus habent relictas portiones, quas nos ſolidas portiones <lb></lb>
appellabimus. </s>
          <s id="s.000339">cum igitur portiones ſint minores ſpacio <lb></lb>
u, circulus, uel ellipſis gh ad portiones maiorem propor­<lb></lb>
tionem habebit, quàm linea mk ad Kl. </s>
          <s id="s.000340">fiat nk ad Kl, ut <lb></lb>
circulus uel ellipſis gh ad ipſas portiones. </s>
          <s id="s.000341">Sed ut circulus <lb></lb>
uel ellipſis gh ad figuram rectilineam in ipſa deſcri­<lb></lb>
ptam, ita eſt cylindrus uel cylindri portio ce ad priſma, <lb></lb>
quod rectilineam figuram pro baſi habet, &amp; altitudinem <lb></lb>
æqualem; id, quod infra demonſtrabitur. </s>
          <s id="s.000342">crgo per conuer<lb></lb>
ſionem rationis, ut circulus, uel ellipſis gh ad portiones re<lb></lb>
lictas, ita cylindrus, uel cylindri portio ce ad ſolidas por­<lb></lb>
tiones, quate cylindrus uel cylindri portio ad ſolidas por­<lb></lb>
tiones eandem proportionem habet, quam linea nk ad k <lb></lb>
&amp; diuidendo priſma, cuius baſis eſt rectilinea figura ad ſo­<lb></lb>
lidas portiones eandem proportionem habet, quam nl ad <lb></lb>
lk &amp; quoniam a cylindro uel cylindri portione, cuius gra­<lb></lb>
uitatis centrum eſt l, aufertur priſma baſim habens rectili­<lb></lb>
neam <expan abbr="figurã">figuram</expan>, cuius <expan abbr="centrũ">centrum</expan> grauitatis eſt K: reſiduæ magnitu<lb></lb>
dinis ex ſolidis portionibus <expan abbr="cõpoſitæ">compoſitæ</expan> grauitatis <expan abbr="cẽtrũ">centrum</expan> erit <lb></lb>
in linea kl protracta, &amp; in puncto n; quod eſt <expan abbr="abſurdũ">abſurdum</expan>. </s>
          <s id="s.000343">relin<lb></lb>
quitur ergo, ut <expan abbr="cẽtrum">centrum</expan> grauitatis cylindri; uel cylindri por<lb></lb>
tionis ſit <expan abbr="punctũ">punctum</expan> k. </s>
          <s id="s.000344">quæ omnia <expan abbr="demonſtrãda">demonſtranda</expan> propoſuimus.</s>
        </p>
        <p type="margin">
          <s id="s.000345"><margin.target id="marg45"></margin.target>4. huius</s>
        </p>
        <p type="main">
          <s id="s.000346">At uero cylindrum, uel cylindri <expan abbr="portionẽ">portionem</expan> ce <lb></lb>
ad priſma, cuius baſis eſt rectilinea figura in ſpa­<lb></lb>
cio gh deſcripta, &amp; altitudo æqualis; eandem ha­



<pb pagenum="15" xlink:href="023/01/037.jpg"></pb>bere proportionem, quam ſpacium gh ad <expan abbr="dictã">dictam</expan> <lb></lb>
figuram, hoc modo demonſtrabimus.</s>
        </p>
        <p type="main">
          <s id="s.000347">Intelligatur circulus, uel ellipſis x æqualis figuræ rectili­<lb></lb>
neæ in gh ſpacio deſcriptæ. </s>
          <s id="s.000348">&amp; ab x conſtituatur conus, uel <lb></lb>
<figure id="id.023.01.037.1.jpg" xlink:href="023/01/037/1.jpg"></figure><lb></lb>
coni portio, <expan abbr="altitudinẽ">altitudinem</expan> habens <expan abbr="eandẽ">eandem</expan>, <expan abbr="quã">quam</expan> cylindrus uel cy<lb></lb>
lindri portio ce. </s>
          <s id="s.000349">Sit deinde rectilinea figura, in qua y <expan abbr="eadẽ">eadem</expan>, <lb></lb>
quæ in ſpacio gh deſcripta eſt: &amp; ab hac pyramis æquealta <lb></lb>
conſtituatur. </s>
          <s id="s.000350">Dico <expan abbr="conũ">conum</expan> uel coni portione x pyramidi y <expan abbr="æ-qualẽ">æ­<lb></lb>
qualem</expan> eſſe. </s>
          <s id="s.000351">niſi enim ſit æqualis, uel maior, uel minor erit.</s>
        </p>
        <p type="main">
          <s id="s.000352">Sit primum maior, et exuperet ſolido z. </s>
          <s id="s.000353">Itaque in circu<lb></lb>
lo, uel ellipſi x deſcribatur figura rectilinea; &amp; in ea pyra­<lb></lb>
mis eandem, quam conus, uel coni portio altitudinem ha­<lb></lb>
bens, ita ut portiones relictæ minores ſint ſolido a, quem­<lb></lb>
admodum docetur in duodecimo libro elementorum pro<lb></lb>
poſitione undecima. </s>
          <s id="s.000354">erit pyramis x adhuc pyramide y ma<lb></lb>
ior. </s>
          <s id="s.000355">&amp; quoniam piramides æque altæ inter ſe ſunt, ſicuti ba<lb></lb>
<arrow.to.target n="marg46"></arrow.to.target><lb></lb>
ſes; pyramis x ad piramidem y eandem proportionem ha­<lb></lb>
bet, quàm figura rectilinea x ad figuram y. </s>
          <s id="s.000356">Sed figura recti 



<pb xlink:href="023/01/038.jpg"></pb><figure id="id.023.01.038.1.jpg" xlink:href="023/01/038/1.jpg"></figure><lb></lb>
linea x cum ſit minor circulo, uel ellipſi, eſt etiam minor fi­<lb></lb>
gura rectilinea y. </s>
          <s id="s.000357">ergo pyramis x pyramide y minor erit. </s>
          <lb></lb>
          <s id="s.000358">Sed &amp; maior; quod fieri <expan abbr="nõ">non</expan> poteſt. </s>
          <s id="s.000359">At ſi conus, uel coni por<lb></lb>
tio x ponatur minor pyramide y: ſit alter conus æque al­<lb></lb>
tus, uel altera coni portio X ipſi pyramidi y æqualis. </s>
          <s id="s.000360">erit <lb></lb>
eius baſis circulus, uel ellipſis maior circulo, uel ellipſi x, <lb></lb>
quorum exceſſus ſit ſpacium <foreign lang="grc">ω.</foreign> Si igitur in circulo, uel eili­<lb></lb>
pſi X figura rectilinea deſcribatur, ita ut portiones relictæ <lb></lb>
ſint <foreign lang="grc">ω</foreign> ſpacio minores, ciuſmodi figura adhuc maior erit cir <lb></lb>
culo, uel ellipſi x, hoc eſt figura rectilinea y. </s>
          <s id="s.000361">&amp; pyramis in <lb></lb>
ca conſtituta minor cono, uel coni portione X, hoc eſt mi­<lb></lb>
nor pyramide y. </s>
          <s id="s.000362">eſt ergo ut X figura rectilinea ad figuram <lb></lb>
rectilineam y, ita pyramis X ad pyramidem y. </s>
          <s id="s.000363">quare cum <lb></lb>
figura rectilinea X ſit maior figura y: erit &amp; pyramis X py­<lb></lb>
ramide y maior. </s>
          <s id="s.000364">ſed erat minor; quod rurſus fieri non po­<lb></lb>
teſt. </s>
          <s id="s.000365">non eſt igitur conus, uel coni portio x neque maior, <lb></lb>
neque minor pyramide y. </s>
          <s id="s.000366">ergo ipſi neceſſario eſt æqualis. </s>
          <lb></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"></pb><figure id="id.023.01.039.1.jpg" xlink:href="023/01/039/1.jpg"></figure><lb></lb>
ni portionem, ita eſt cylindrus ad cylindrum, uel cylin­<lb></lb>
dri portio ad cylindri portionem: &amp; ut pyramis ad pyra­<lb></lb>
midem, ita priſma ad priſma, cum eadem ſit baſis, &amp; æqua <lb></lb>
lis altitudo; erit cylindrus uel cylindri portio x priſma­<lb></lb>
ti y æqualis. </s>
          <s id="s.000368"><expan abbr="eſtq;">eſtque</expan> ut ſpacium gh ad ſpacium x, ita cylin­<lb></lb>
drus, uel cylindri portio ce ad cylindrum, uel cylindri por­<lb></lb>
tionem x. </s>
          <s id="s.000369">Conſtat igitur cylindrum uel cylindri <expan abbr="portionẽ">portionem</expan> <lb></lb>
c e, ad priſma y, quippe cuius baſis eſt figura rectilinea in <lb></lb>
<arrow.to.target n="marg47"></arrow.to.target><lb></lb>
ſpacio gh deſcripta, eandem proportionem habere, quam <lb></lb>
ſpacium gh habet ad ſpacium x, hoc eſt ad dictam figuram. </s>
          <lb></lb>
          <s id="s.000370">quod demonſtrandum fuerat.</s>
        </p>
        <p type="margin">
          <s id="s.000371"><margin.target id="marg46"></margin.target>6. duode<lb></lb>
cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000372"><margin.target id="marg47"></margin.target>7. quinti</s>
        </p>
        <p type="head">
          <s id="s.000373">THEOREMA IX. PROPOSITIO IX.</s>
        </p>
        <p type="main">
          <s id="s.000374">Si pyramis ſecetur plano baſi æquidiſtante; ſe­<lb></lb>
ctio erit figura ſimilis ei, quæ eſt baſis, centrum <lb></lb>
grauitatis in axe habens.</s>
        </p>
        <pb xlink:href="023/01/040.jpg"></pb>
        <p type="main">
          <s id="s.000375">SIT pyramis, cuius baſis triangulum abc; axis dc: &amp; <lb></lb>
ſecetur plano baſi æquidiſtante; quod <expan abbr="ſectionẽ">ſectionem</expan> faciat fgh; <lb></lb>
<expan abbr="occurratq;">occurratque</expan> axi in puncto k. Dico fgh triangulum eſſe, ipſi <lb></lb>
abc ſimile; cuius grauitatis centrum eſt K. <expan abbr="Quoniã">Quoniam</expan> enim <lb></lb>
duo plana æquidiſtantia abc, fgh ſecantur à plano abd; <lb></lb>
communes eorum ſectiones ab, fg æquidiſtantes erunt: &amp; <lb></lb>
eadem ratione æquidiſtantes ipſæ bc, gh: &amp; ca, hf. </s>
          <s id="s.000376">Quòd <lb></lb>
cum duæ lineæ fg, gh, duabus ab, bc æquidiſtent, nec <lb></lb>
ſint in eodem plano; angulus ad g æqualis eſt angulo ad <lb></lb>
b. </s>
          <s id="s.000377">&amp; ſimiliter angulus ad h angulo ad c: <expan abbr="angulusq;">angulusque</expan> ad fci, <lb></lb>
qui ad a eſt æqualis. </s>
          <s id="s.000378">triangulum igitur fgh ſimile eſt tri­<lb></lb>
angulo abc. </s>
          <s id="s.000379">Atuero punctum k centrum eſſe grauita­<lb></lb>
tis trianguli fgh hoc modo oſtendemus. </s>
          <s id="s.000380">Ducantur pla­<lb></lb>
na per axem, &amp; per lineas da, db, dc: erunt communes ſe­<lb></lb>
ctiones fK, ae æquidiſtantes: <expan abbr="pariterq;">pariterque</expan> kg, eb; &amp; kh, ec: <lb></lb>
quare angulus kfh angulo eac; &amp; angulus kfg ipſi eab <lb></lb>
<figure id="id.023.01.040.1.jpg" xlink:href="023/01/040/1.jpg"></figure><lb></lb>
eſt æqualis. </s>
          <s id="s.000381">Eadem ratione <lb></lb>
anguli ad g angulis ad b: &amp; <lb></lb>
anguli ad h iis, qui ad c æ­<lb></lb>
quales erunt. </s>
          <s id="s.000382">ergo puncta <lb></lb>
eK in triangulis abc, fgh <lb></lb>
ſimiliter ſunt poſita, per ſe­<lb></lb>
xtam poſitionem Archime­<lb></lb>
dis in libro de centro graui­<lb></lb>
tatis planorum. </s>
          <s id="s.000383">Sed cum e <lb></lb>
ſit centrum grauitatis trian<lb></lb>
guli abc, erit ex undecima <lb></lb>
propoſitione eiuſdem libri, <lb></lb>
&amp; K trianguli fgh grauita<lb></lb>
tis centrum. </s>
          <s id="s.000384">id quod demonſtrare oportebat. </s>
          <s id="s.000385">Non aliter <lb></lb>
in ceteris pyramidibus, quod propoſitum eſt demonſtra­<lb></lb>
bitur.</s>
        </p>
        <pb pagenum="17" xlink:href="023/01/041.jpg"></pb>
        <p type="head">
          <s id="s.000386">PROBLEMA I. PROPOSITIO X.</s>
        </p>
        <p type="main">
          <s id="s.000387">DATA qualibet pyramide, fieri poteſt, ut fi­<lb></lb>
gura ſolida in ipſa in ſcribatur, &amp; altera <expan abbr="circũſcri-batur">circumſcri­<lb></lb>
batur</expan> ex priſmatibus æqualem altitudinem <expan abbr="ha-bẽtibus">ha­<lb></lb>
bentibus</expan>, ita ut circumſcripta inſcriptam excedat <lb></lb>
magnitudine, quæ minor ſit <expan abbr="quacũque">quacunque</expan> ſolida ma<lb></lb>
gnitudine propoſita.</s>
        </p>
        <figure id="id.023.01.041.1.jpg" xlink:href="023/01/041/1.jpg"></figure>
        <p type="main">
          <s id="s.000388">Sit pyramis, cuius baſis <lb></lb>
<expan abbr="triangulũ">triangulum</expan> abc; axis de. </s>
          <lb></lb>
          <s id="s.000389"><expan abbr="Sitq;">Sitque</expan> priſma, quod <expan abbr="eandẽ">eandem</expan> <lb></lb>
baſim habeat, &amp; axem eun<lb></lb>
dem. </s>
          <s id="s.000390">Itaque hoc priſma­<lb></lb>
te continenter ſecto bifa­<lb></lb>
riam, plano baſi <expan abbr="æquidiſtã">æquidiſtan</expan><lb></lb>
te,  relinquetur <expan abbr="tãdem">tandem</expan> priſ<lb></lb>
ma quoddam minus pro­<lb></lb>
poſita magnitudine: quod <lb></lb>
quidem baſim eandem ha<lb></lb>
beat, quam pyramis, &amp; a­<lb></lb>
xem ef. </s>
          <s id="s.000391">diuidatur de in <lb></lb>
partes æquales ipſi ef in <lb></lb>
punctis ghklmn: &amp; per <lb></lb>
diuiſiones plana <expan abbr="ducãtur">ducantur</expan>: <lb></lb>
quæ baſibus æquidiſtent, <lb></lb>
erunt ſectiones, triangula <lb></lb>
ipſi abc ſimilia, ut proxi­<lb></lb>
me oſtendimus. </s>
          <s id="s.000392">ab uno <lb></lb>
quoque <expan abbr="autẽ">autem</expan> horum trian<lb></lb>
gulorum duo priſmata <expan abbr="cõ">con</expan><lb></lb>
ſtruantur; unum quidem <lb></lb>
ad partes e; alterum ad 



<pb xlink:href="023/01/042.jpg"></pb>partes d. </s>
          <s id="s.000393">in pyramide igitur inſcripta erit quædam figura, <lb></lb>
ex priſmatibus æqualem altitudinem habentibus <expan abbr="cõſtans">conſtans</expan>, <lb></lb>
ad partes e: &amp; altera circumſcripta ad partes d. </s>
          <s id="s.000394">Sed unum­<lb></lb>
quodque eorum priſmatum, quæ in figura inſcripta conti­<lb></lb>
nentur, æquale eſt priſmati, quod ab eodem fit triangulo in <lb></lb>
figura circumſcripta: nam priſma pq priſmati po eſt æ­<lb></lb>
quale; priſma st æquale priſmati sr; priſma xy priſmati <lb></lb>
xu; priſma <foreign lang="grc">ηθ</foreign> priſmati <foreign lang="grc">η</foreign>z; priſma <foreign lang="grc">μν</foreign> priſmati <foreign lang="grc">μλ;</foreign> priſ­<lb></lb>
ma <foreign lang="grc">ρσ</foreign> priſmati <foreign lang="grc">ρπ;</foreign> &amp; priſma <foreign lang="grc">φχ</foreign> priſmati <foreign lang="grc">φτ</foreign> æquale. </s>
          <s id="s.000395">re­<lb></lb>
linquitur ergo, ut circumſcripta figura exuperet <expan abbr="inſcriptã">inſcriptam</expan> <lb></lb>
priſmate, quod baſim habet abc triangulum, &amp; axem ef. </s>
          <lb></lb>
          <s id="s.000396">Illud uero minus eſt ſolida magnitudine propoſita. </s>
          <s id="s.000397"><expan abbr="Eadẽ">Eadem</expan> <lb></lb>
ratione inſcribetur, &amp; circumſcribetur ſolida figura in py­<lb></lb>
ramide, quæ quadrilateram, uel <expan abbr="plurilaterã">plurilateram</expan> baſim habeat.</s>
        </p>
        <p type="head">
          <s id="s.000398">PROBLEMA II. PROPOSITIO XI.</s>
        </p>
        <p type="main">
          <s id="s.000399">DATO cono, fieri poteſt, ut figura ſolida in­<lb></lb>
ſcribatur, &amp; altera circumſcribatur ex cylindris <lb></lb>
æqualem habentibus altitudinem, ita ut circum­<lb></lb>
ſcripta ſuperet inſcriptam, magnitudine, quæ ſo­<lb></lb>
lida magnitudine propoſita ſit minor.</s>
        </p>
        <p type="main">
          <s id="s.000400">SIT conus, cuius axis bd: &amp; ſecetur plano per axem <lb></lb>ducto, &#039;ut ſectio ſit triangulum abc: <expan abbr="intelligaturq;">intelligaturque</expan> cylin­<lb></lb>
drus, qui baſim eandem, &amp; eundem axem habeat. </s>
          <s id="s.000401">Hoc igi­<lb></lb>
tur cylindro continenter bifariam ſecto, relinquetur cylin<lb></lb>
drus minor ſolida magnitudine propoſita. </s>
          <s id="s.000402">Sit autem is cy<lb></lb>
lindrus, qui baſim habet circulum circa diametrum ac, &amp; <lb></lb>
axem de. </s>
          <s id="s.000403">Itaque diuidatur bd in partes æquales ipſi de <lb></lb>
in punctis fghKlm: &amp; per ea ducantur plana conum ſe­<lb></lb>
cantia; quæ baſi æquidiſtent. </s>
          <s id="s.000404">erunt ſectiones circuli, cen­<lb></lb>
tra in axi habentes, ut in primo libro conicorum, propoſi-



<pb pagenum="18" xlink:href="023/01/043.jpg"></pb>tione quarta Apollonius demonſtrauit. </s>
          <s id="s.000405">Si igitur à ſingu­<lb></lb>
lis horum circulorum, duo cylindri fiant; unus quidem ad <lb></lb>
baſis partes; alter ad partes uerticis: inſcripta erit in co­<lb></lb>
no ſolida quædam figura, &amp; altera circumſcripta ex cylin­<lb></lb>
dris æqualem altitudinem habentibus conſtans; quorum <lb></lb>
<figure id="id.023.01.043.1.jpg" xlink:href="023/01/043/1.jpg"></figure><lb></lb>
unuſquiſque, qui in <lb></lb>
figura inſcripta con­<lb></lb>
tinetur æqualis eſt ei, <lb></lb>
qui ab eodem fit cir­<lb></lb>
culo in figura <expan abbr="circũ-ſcripta">circum­<lb></lb>
ſcripta</expan>. </s>
          <s id="s.000406">Itaque cylin<lb></lb>
drus op æqualis eſt <lb></lb>
cylindro on; cylin­<lb></lb>
drus rs <expan abbr="cylĩdro">cylindro</expan> rq.</s>
          <lb></lb>
          <s id="s.000407">
cylindrus ux cylin­<lb></lb>
dro ut eſt æqualis; <lb></lb>
&amp; alii aliis ſimiliter. </s>
          <lb></lb>
          <s id="s.000408">quare conſtat <expan abbr="circũ-ſcriptam">circum­<lb></lb>
ſcriptam</expan> figuram ſu­<lb></lb>
perare inſcriptam cy<lb></lb>
lindro, cuius baſis eſt <lb></lb>
circulus circa diametrum ac, &amp; axis de. </s>
          <s id="s.000409">atque hic eſt mi­<lb></lb>
nor ſolida magnitudine propoſita.</s>
        </p>
        <p type="head">
          <s id="s.000410">PROBLEMA III. PROPOSITIO XII.</s>
        </p>
        <p type="main">
          <s id="s.000411">DATA coni portione, poteſt ſolida quædam <lb></lb>figura inſcribi, &amp; altera circumſcribi ex cylindri <lb></lb>
portionibus æqualem altitudinem habentibus; <lb></lb>
ita ut circumſcripta inſcriptam exuperet, magni <lb></lb>
tudine, quæ minor fit ſolida magnitudine pro­<lb></lb>
poſita.</s>
        </p>
        <pb xlink:href="023/01/044.jpg"></pb>
        <p type="main">
          <s id="s.000412">Figuram cuiuſmodi, &amp; inſcribemus, &amp; <expan abbr="circũſcribemus">circumſcribemus</expan>, ita <lb></lb>
ut in cono dictum eſt.</s>
        </p>
        <figure id="id.023.01.044.1.jpg" xlink:href="023/01/044/1.jpg"></figure>
        <p type="head">
          <s id="s.000413">PROBLEMA IIII. PROPOSITIO XIII.</s>
        </p>
        <p type="main">
          <s id="s.000414">DATA ſphæræ portione, quæ dimidia ſphæ­<lb></lb>
ra maior non ſit, poteſt ſolida quædam portio in­<lb></lb>
ſcribi &amp; altera circumſcribi ex cylindris æqualem <lb></lb>
altitudinem habentibus, ita ut circumſcripta in­<lb></lb>
ſcriptam excedat magnitudine, quæ ſolida ma­<lb></lb>
gnitudine propoſita ſit minor.</s>
        </p>
        <p type="main">
          <s id="s.000415">HOC etiam eodem prorſus modo ſiet: atque ut ab <lb></lb>Archimede traditum eſt in conoidum, &amp; ſphæroidum por<lb></lb>
tionibus, propoſitione uigeſimaprima libri de conoidi­<lb></lb>
bus, &amp; ſphæroidibus.</s>
        </p>
        <pb pagenum="19" xlink:href="023/01/045.jpg"></pb>
        <figure id="id.023.01.045.1.jpg" xlink:href="023/01/045/1.jpg"></figure>
        <p type="head">
          <s id="s.000416">THEOREMA X. PROPOSITIO XIIII.</s>
        </p>
        <p type="main">
          <s id="s.000417">Cuiuslibet pyramidis, &amp; cuiuslibet coni, uel <lb></lb>
coni portionis, centrum grauitatis in axe <expan abbr="cõſiſtit">conſiſtit</expan>.</s>
        </p>
        <p type="main">
          <s id="s.000418">SIT pyramis, cuius baſis triangulum abc: &amp; axis de. </s>
          <lb></lb>
          <s id="s.000419">Dico in linea de ipſius grauitatis centrum ineſſe. </s>
          <s id="s.000420">Si enim <lb></lb>
fieri poteſt, ſit centrum f: &amp; ab f ducatur ad baſim pyrami<lb></lb>
dis linea fg, axi æquidiſtans: <expan abbr="iunctaq;">iunctaque</expan> eg ad latera trian­<lb></lb>
guli abc producatur in h. </s>
          <s id="s.000421">quam uero proportionem ha­<lb></lb>
bet linea he ad eg, habeat pyramis ad aliud ſolidum, in <lb></lb>
quo K: <expan abbr="inſcribaturq;">inſcribaturque</expan> in pyramide ſolida figura, &amp; altera cir<lb></lb>
cumſcribatur ex priſmatibus æqualem habentibus altitu­<lb></lb>
dinem, ita ut circumſcripta inſcriptam exuperet magnitu­<lb></lb>
dine, quæ ſolido k ſit minor. </s>
          <s id="s.000422">Et quoniam in pyramide pla<lb></lb>
num baſi æquidiſtans ductum ſectionem facit figuram ſi­<lb></lb>
milem ei, quæ eſt baſis; <expan abbr="centrumq;">centrumque</expan> grauitatis in axe haben<lb></lb>
tem: erit priſmatis st grauitatis <expan abbr="centrũ">centrum</expan> in linea rq ; <lb></lb>
matis ux centrum in linea qp, priſmatis yz in linea po; <lb></lb>
priſmatis <foreign lang="grc">ηθ</foreign> in linea on; priſmatis <foreign lang="grc">λμ</foreign> in linea  nm; priſ­<lb></lb>
matis <foreign lang="grc">νπ</foreign> in ml; &amp; denique priſmatis <foreign lang="grc">ρσ</foreign> in le. </s>
          <s id="s.000423">quare to­



<pb xlink:href="023/01/046.jpg"></pb>tius figuræ inſcriptæ centrum grauitatis eſt in linea re: <lb></lb>
<figure id="id.023.01.046.1.jpg" xlink:href="023/01/046/1.jpg"></figure>quod ſit <foreign lang="grc">τ</foreign>: <expan abbr="iũ">iun</expan>­<lb></lb>
ctaque <foreign lang="grc">τ</foreign>f, &amp; <lb></lb>
producta, à <lb></lb>
puncto h du­<lb></lb>
catur linea a­<lb></lb>
xi pyramidis <lb></lb>
æquidiſtans, <lb></lb>
quæ <expan abbr="cũ">cum</expan> linea <lb></lb>
<foreign lang="grc">τ</foreign>f conueniat <lb></lb>
in <foreign lang="grc">φ</foreign>.</s>
          <s id="s.000424">habebit <lb></lb>
<foreign lang="grc">φτ</foreign> ad <foreign lang="grc">τ</foreign>f ean­<lb></lb>
dem propor­<lb></lb>
tionem, <expan abbr="quã">quam</expan> <lb></lb>
he ad eg. <lb></lb>
</s>
        </p>
        <p>
          <s id="s.000425">Quoniam igi<lb></lb>
tur exceſſus, <lb></lb>
quo <expan abbr="circũ">circum</expan>ſcri<lb></lb>
pta figura in­<lb></lb>
ſcriptam ſupe<lb></lb>
rat, minor eſt <lb></lb>
ſolido <foreign lang="grc">χ</foreign>; py­<lb></lb>
ramis ad eun­<lb></lb>
<expan abbr="dẽ">dem</expan> <expan abbr="exceſſũ">exceſſum</expan> ma<lb></lb>
ioré propor­<lb></lb>
tioné  habet, <lb></lb>
quàm ad K ſo<lb></lb>
lidum: uideli<lb></lb>
cet maiorem, <lb></lb>
quàm linea h<lb></lb>
e ad eg; hoc <lb></lb>
eſt quàm <foreign lang="grc">φτ</foreign> <lb></lb>
ad <foreign lang="grc">τ</foreign>f: &amp; propterea multo maiorem habet ad partem ex­<lb></lb>
ceſſus, quæ intra pyrimidem comprehenditur. </s>
          <s id="s.000426">Itaque ha­



<pb pagenum="20" xlink:href="023/01/047.jpg"></pb>beat eam, quam <foreign lang="grc">χτ</foreign> ad <foreign lang="grc">τ</foreign>f erit diuidendo ut <foreign lang="grc">χ</foreign>f ad f<foreign lang="grc">τ</foreign>, ita fi<lb></lb>
gura ſolida inſcripta ad partem exceſſus, quæ eſt intra pyra<lb></lb>
midem. </s>
          <s id="s.000427">Cum ergo à pyramide, cuius grauitatis <expan abbr="ceũtrum">centrum</expan> eſt <lb></lb>
punctum f, ſolida figura inſcripta auferatur, cuius <expan abbr="centrũtrum">centrum</expan> <lb></lb>
<foreign lang="grc">τ</foreign>: reliqua magnitudinis conſtantis ex parte exceſſus, quæ <lb></lb>
eſt intra pyramidem, centrum grauitatis erit in linea <foreign lang="grc">τ</foreign>f <lb></lb>
producta, &amp; in puncto <foreign lang="grc">χ</foreign>. </s>
          <s id="s.000428">quod fieri non poteſt. </s>
          <s id="s.000429">Sequitur <lb></lb>
igitur, ut centrum grauitatis pyramidis in linea de; hoc <lb></lb>
eſt in eius axe conſiſtat.</s>
        </p>
        <p>
          <s id="s.000430">Sit conus, uel coni portio, cuius axis bd: &amp; ſecetur plano <lb></lb>
per axem, ut ſectio ſit triangulum abc. </s>
          <s id="s.000431">Dico centrum gra<lb></lb>
uitatis ipſius eſſe in linea bd. </s>
          <s id="s.000432">Sit enim, ſi fieri poteſt, <expan abbr="centrũ">centrum</expan> <lb></lb>
<figure id="id.023.01.047.1.jpg" xlink:href="023/01/047/1.jpg"></figure>
e: <expan abbr="perq;">perque</expan> e ducatur ef axi æquidiſtans: &amp; quam propor­<lb></lb>
tionem habet cd ad df, habeat conus, uel coni portio ad <lb></lb>
ſolidum g. </s>
          <s id="s.000433">inſcribatur ergo in cono, uel coni portione ſoli



<pb xlink:href="023/01/048.jpg"></pb>da figura, &amp; altera circumſcribatur ex cylindris, uel cylin­<lb></lb>
dri portionibus, ſicuti dictum eſt, ita ut exceſſus, quo figu­<lb></lb>
ra circumſcripta inſcriptam ſuperat, ſit ſolido g minor. </s>
          <lb></lb>
          <s id="s.000434">Itaque centrum grauitatis cylindri, uel cylindri portionis <lb></lb>
qr eſt in linea po; cylindri, uel cylindri portionis st cen­<lb></lb>
trum in linea on; centrum ux in linea nm; yz in mb; <foreign lang="grc">νθ</foreign><lb></lb>
in lk; <foreign lang="grc">λμ</foreign> in kh; &amp; denique <foreign lang="grc">fπ</foreign> centrum in hd. </s>
          <s id="s.000435">ergo figu­<lb></lb>
<figure id="id.023.01.048.1.jpg" xlink:href="023/01/048/1.jpg"></figure><lb></lb>
ræ inſcriptæ centrum eſt in linea pd. </s>
          <s id="s.000436">Sit autem <foreign lang="grc">ρ</foreign>: &amp; iun­<lb></lb>
cta <foreign lang="grc">ρ</foreign>e protendatur, ut cum linea, quæ à <expan abbr="pũcto">puncto</expan> c ducta ſue­<lb></lb>
rit axi æquidiſtans, conueniat in <foreign lang="grc">ς.</foreign> erit <foreign lang="grc">ς ρ</foreign> ad <foreign lang="grc">ρ</foreign>e, ut cd <lb></lb>
ad df: &amp; conus, ſeu coni portio ad exceſſum, quo circum­<lb></lb>
ſcripta figura inſcriptam ſuperat, habebit maiorem pro­<lb></lb>
portionem, quàm <foreign lang="grc">τρ</foreign> ad <foreign lang="grc">ρ</foreign>e. </s>
          <s id="s.000437">ergo ad partem exceſſus, quæ <lb></lb>
intra ipſius ſuperficiem comprehenditur, multo maiorem <lb></lb>
proportionem habebit. </s>
          <s id="s.000438">habeat eam, quam <foreign lang="grc">τρ</foreign> ad <foreign lang="grc">ρ</foreign>e. </s>
          <s id="s.000439">erit 



<pb pagenum="21" xlink:href="023/01/049.jpg"></pb>diuidendo figura ſolida inſcripta ad dictam exceſſus par­<lb></lb>
tem, ut <foreign lang="grc">τε</foreign> ad c<foreign lang="grc">π.</foreign> &amp; quoniam à cono, ſeu coni portione, <lb></lb>
cuius grauitatis centrum eſt e, aufertur figura inſcripta, <lb></lb>
cuius centrum <foreign lang="grc">ρ·</foreign> reſiduæ magnitudinis compoſitæ cx par <lb></lb>
te exceſſus, quæ intra coni, uel coni portionis ſuperficiem <lb></lb>
continetur, centrum grauitatis erit in linea e protracta, <lb></lb>
atque in puncto t. </s>
          <s id="s.000440">quod eſt abſurdum. </s>
          <s id="s.000441"><expan abbr="cõſtat">conſtat</expan> ergo <expan abbr="centrũ">centrum</expan> <lb></lb>
grauitatis coni, uel coni portionis, eſſe in axe bd: quod de <lb></lb>
monſtrandum propoſuimus.</s>
        </p>
        <p type="head">
          <s id="s.000442">THEOREMA XI. PROPOSITIO XV.</s>
        </p>
        <p type="main">
          <s id="s.000443">Cuiuslibet portionis ſphæræ uel ſphæroidis, <lb></lb>
quæ dimidia maior non ſit: <expan abbr="itemq́;">itemque</expan> cuiuslibet por<lb></lb>
tionis conoidis, uel abſciſſæ plano ad axem recto, <lb></lb>
uel non recto, centrum grauitatis in axe con­<lb></lb>
ſiſtit.</s>
        </p>
        <p type="main">
          <s id="s.000444">Demonſtratio ſimilis erit ei, quam ſupra in cono, uel co<lb></lb>
ni portione attulimus, ne toties eadem fruſtra iterentur.</s>
        </p>
        <figure id="id.023.01.049.1.jpg" xlink:href="023/01/049/1.jpg"></figure>
        <pb xlink:href="023/01/050.jpg"></pb>
        <p type="head">
          <s id="s.000445">THEOREMA XII. PROPOSITIO XVI.</s>
        </p>
        <p type="main">
          <s id="s.000446">In ſphæra, &amp; ſphæroide idem eſt grauitatis, &amp; <lb></lb>
figuræ centrum.</s>
        </p>
        <p type="main">
          <s id="s.000447">Secetur ſphæra, uel ſphæroides plano per axem ducto; <lb></lb>
quod ſectionem faciat circulum, uel ellipſim abcd, cuius <lb></lb>
diameter, &amp; ſphæræ, uel ſphæroidis axis db; &amp; centrum e. </s>
          <lb></lb>
          <s id="s.000448">Dico e grauitatis etiam centrum eſſe. </s>
          <s id="s.000449">ſecetur enim altero <lb></lb>
plano per e, ad planum ſecans recto, cuius ſectio ſit circu­<lb></lb>
lus circa diametrum ac. </s>
          <s id="s.000450">erunt adc, abc dimidiæ portio­<lb></lb>
nes ſphæræ, uel ſphæroidis. </s>
          <s id="s.000451">&amp; quoniam portionis adc gra<lb></lb>
uitatis centrum eſi in linea d, &amp; centrum portionis abc in <lb></lb>
ipſa be; totius ſphæræ, uel ſphæroidis grauitatis centrum <lb></lb>in axe db conſiſtet, Quòd ſi portionis adc centrum graui <lb></lb>
tatis ponatur eſſe f &amp; fiat ipſi fe æqualis eg: <expan abbr="punctũ">punctum</expan> g por<lb></lb>
<figure id="id.023.01.050.1.jpg" xlink:href="023/01/050/1.jpg"></figure><lb></lb>
<arrow.to.target n="marg48"></arrow.to.target><lb></lb>
tionis abc centrum erit. </s>
          <s id="s.000452">ſolidis enim figuris ſimilibus &amp; <lb></lb>
æqualibus inter ſe aptatis, &amp; centra grauitatis ipſarum in­<lb></lb>
<arrow.to.target n="marg49"></arrow.to.target><lb></lb>
ter se aptentur neceſſe eſt. </s>
          <s id="s.000453">ex quo fit, ut magnitudinis, quæ <lb></lb>
ex utilique <expan abbr="cõſlat">conſtat</expan>, hoc eſt ipſius ſphæræ, uel ſphæroidis gra<lb></lb>
uitatis centrum ſit in medio lineæ fg uidelicet in e. </s>
          <s id="s.000454">Sphæ­<lb></lb>
ræ igitur, uel ſphæroidis grauitatis centrum eſt idem, quod <lb></lb>
centrum figuræ.</s>
        </p>
        <pb pagenum="22" xlink:href="023/01/051.jpg"></pb>
        <p type="margin">
          <s id="s.000455"><margin.target id="marg48"></margin.target>per 2. pe­<lb></lb>
titionem</s>
        </p>
        <p type="margin">
          <s id="s.000456"><margin.target id="marg49"></margin.target>4 Archi­<lb></lb>
medis.</s>
        </p>
        <p type="main">
          <s id="s.000457">Ex demonſtratis perſpicue apparet, portioni <lb></lb>
ſphæræ uel ſphæroidis, quæ dimidia maior eſt, <expan abbr="cẽ">cen</expan><lb></lb>
trum grauitatis in axe conſiſtere.</s>
        </p>
        <figure id="id.023.01.051.1.jpg" xlink:href="023/01/051/1.jpg"></figure>
        <p type="main">
          <s id="s.000458">Data enim <lb></lb>
qualibet maio<lb></lb>
ri <expan abbr="portiõe">portione</expan>, quo <lb></lb>
<expan abbr="niã">niam</expan> totius ſphæ<lb></lb>
ræ, uel ſphæroi<lb></lb>
dis grauitatis <lb></lb>
centrum eſt in <lb></lb>
axe; eſt autem <lb></lb>
&amp; in axe cen­<lb></lb>
trum portio­<lb></lb>
nis minoris: <lb></lb>
reliquæ portionis uidelicet maioris centrum in axe neceſ­<lb></lb>
ſario conſiſtet.</s>
        </p>
        <p type="head">
          <s id="s.000459">THEOREMA XIII. PROPOSITIO XVII.</s>
        </p>
        <figure id="id.023.01.051.2.jpg" xlink:href="023/01/051/2.jpg"></figure>
        <p type="main">
          <s id="s.000460">Cuiuslibet pyramidis <expan abbr="triãgularem">trian<lb></lb>
gularem</expan> baſim <expan abbr="habẽtis">habentis</expan> gra<lb></lb>
uitatis centrum eſt in pun­<lb></lb>
cto, in quo ipſius axes con­<lb></lb>
ueniunt.</s>
        </p>
        <p type="main">
          <s id="s.000461">Sit pyramis, cuius baſis trian<lb></lb>
gulum abc, axis de: <expan abbr="ſitq;">ſitque</expan> trian<lb></lb>
guli bdc grauitatis centrum f: <lb></lb>
&amp; iungatur a ſ. </s>
          <s id="s.000462">erit &amp; af axis eiuſ<lb></lb>
dem pyramidis ex tertia diffini­<lb></lb>
tione huius. </s>
          <s id="s.000463">Itaque quoniam centrum grauitatis eſt in <lb></lb>
axe de; eſt autem &amp; in axe af; q̀uod proxime demonſtraui 



<pb xlink:href="023/01/052.jpg"></pb>mus: erit utique grauitatis centrum pyramidis punctum <lb></lb>
g. in quo ſcilicet ipſi axes conueniunt.</s>
        </p>
        <p type="head">
          <s id="s.000464">THEOREMA XIIII. PROPOSITIO XVIII.</s>
        </p>
        <p type="main">
          <s id="s.000465">SI ſolidum parallelepipedum ſecetur plano <lb></lb>
baſibus æquidiſtante; erit ſolidum ad ſolidum, <lb></lb>
ſicut altitudo ad altitudinem, uel ſicut axis ad <lb></lb>
axem.</s>
        </p>
        <figure id="id.023.01.052.1.jpg" xlink:href="023/01/052/1.jpg"></figure>
        <p type="main">
          <s id="s.000466">Sit ſolidum parallelepipe<lb></lb>
dum abcdefgh, cuius axis <lb></lb>
kl: <expan abbr="ſeceturq;">ſeceturque</expan> plano baſibus <lb></lb>
æquidiſtante, quod faciat <lb></lb>
ſectionem mnop; &amp; axi in <lb></lb>
puncto q occurrat. </s>
          <s id="s.000467">Dico <lb></lb>
ſolidum gm ad ſolidum mc <lb></lb>
eam proportionem habere, <lb></lb>
quam altitudo ſolidi gm ha­<lb></lb>
bet ad ſolidi mc altitudi­<lb></lb>
nem; uel quam axis kq ad <lb></lb>
axem ql. </s>
          <s id="s.000468">Si enim axis Kl ad <lb></lb>
baſis planum ſit perpendicu<lb></lb>
<figure id="id.023.01.052.2.jpg" xlink:href="023/01/052/2.jpg"></figure><lb></lb>
laris, &amp; linea gc, quæ ex quin<lb></lb>
ta huius ipſi kl æquidiſtat, <lb></lb>
perpendicularis erit ad <expan abbr="idẽ">idem</expan> <lb></lb>
planum, &amp; ſolidi altitudi­<lb></lb>
<arrow.to.target n="marg50"></arrow.to.target><lb></lb>
nem dimetietur. </s>
          <s id="s.000469">Itaque ſo­<lb></lb>
lidum gm ad ſolidum mc <lb></lb>
eam proportionem habet, <lb></lb>
quam parallelogramm<expan abbr="ũ">um</expan> gn <lb></lb>
ad parallelogrammum nc, <lb></lb>
<arrow.to.target n="marg51"></arrow.to.target><lb></lb>
hoc eſt quam linea go, quæ 



<pb pagenum="23" xlink:href="023/01/053.jpg"></pb>eſt ſolidi gm altitudo ad oe altitudinem ſolidi mc, uel <expan abbr="quã">quam</expan> <lb></lb>
axis kq ad ql axem. </s>
          <s id="s.000470">Si uero axis kl non ſit perpendicularis <lb></lb>
ad planum baſis; ducatur a puncto k ad idem planum per<lb></lb>
pendicularis kr, <expan abbr="occurrẽs">occurrens</expan> plano mnop in s. </s>
          <s id="s.000471">ſimiliter <expan abbr="de-mõſtrabimus">de­<lb></lb>
monſtrabimus</expan> ſolidum gm ad <expan abbr="ſolidũ">ſolidum</expan> mc ita eſſe, ut axis kq <lb></lb>
ad axem ql. </s>
          <s id="s.000472">Sed ut Kq ad ql, ita ks altitudo ad altitudi­<lb></lb>
<arrow.to.target n="marg52"></arrow.to.target><lb></lb>
nem sr; nam lineæ Kl, Kr à planis æquidiſtantibus in eaſ­<lb></lb>
dem proportiones ſecantur. </s>
          <s id="s.000473">ergo ſolidum gm ad ſolidum <lb></lb>
mc <expan abbr="eandẽ">eandem</expan> proportionem habet, quam altitudo ad <expan abbr="altitudinẽ">altitu<lb></lb>
dinem</expan>, uel quam axis ad axem. </s>
          <s id="s.000474">quod <expan abbr="demõſtrare">demonſtrare</expan> oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000475"><margin.target id="marg50"></margin.target>25 undeci<lb></lb>
mi.</s>
        </p>
        <p type="margin">
          <s id="s.000476"><margin.target id="marg51"></margin.target><expan abbr="ſextĩ">ſextim</expan>.</s>
        </p>
        <p type="margin">
          <s id="s.000477"><margin.target id="marg52"></margin.target>17. unde­<lb></lb>
cimi</s>
        </p>
        <p type="head">
          <s id="s.000478">THEOREMA XV. PROPOSITIO XIX.</s>
        </p>
        <p type="main">
          <s id="s.000479">Solida parallelepipeda in eadem baſi, uel in <lb></lb>
æqualibus baſibus conſtituta eam inter ſe propor<lb></lb>
tionem habent, quam altitudines: &amp; ſi axes ipſo­<lb></lb>
rum cum baſibus æquales angulos contineant, <lb></lb>
eam quoque, quam axes proportionem <expan abbr="habebũt">habebunt</expan>.</s>
        </p>
        <p type="main">
          <s id="s.000480">Sint ſolida parallelepipeda in <expan abbr="eadẽ">eadem</expan> baſi <expan abbr="cõſtituta">conſtituta</expan> abcd, <lb></lb>
abef: &amp; ſit ſolidi abcd altitudo minor: producatur au­<lb></lb>
tem planum cd adeo, ut ſolidum abef ſecet; cuius ſectio <lb></lb>
<figure id="id.023.01.053.1.jpg" xlink:href="023/01/053/1.jpg"></figure><lb></lb>
<arrow.to.target n="marg53"></arrow.to.target><lb></lb>
ſit gh. </s>
          <s id="s.000481"><expan abbr="erũt">erunt</expan> ſoli <lb></lb>
da abcd, abgh <lb></lb>
in eadem baſi, <lb></lb>
&amp; æquali altitu<lb></lb>
dine inter ſe æ­<lb></lb>
qualia. </s>
          <s id="s.000482"><expan abbr="Quoniã">Quoniam</expan> <lb></lb>
igitur ſolidum <lb></lb>
abef ſecatur <lb></lb>
plano baſibus <lb></lb>
<expan abbr="æquidiſtãte">æquidiſtante</expan>, erit <lb></lb>
<arrow.to.target n="marg54"></arrow.to.target><lb></lb>
ſolidum ghef <lb></lb>
adipſum abgh 



<pb xlink:href="023/01/054.jpg"></pb>ut altitudo ad altitudinem: &amp; componendo conuertendo <lb></lb>
<arrow.to.target n="marg55"></arrow.to.target><lb></lb>
que ſolidum abgh, hoc eſt ſolidum abcd ipſi æquale, ad <lb></lb>
ſolidum abef, ut altitudo ſolidi abcd ad ſolidi abef al­<lb></lb>
titudinem.</s>
        </p>
        <p type="margin">
          <s id="s.000483"><margin.target id="marg53"></margin.target>29. unde­<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000484"><margin.target id="marg54"></margin.target>18. huius</s>
        </p>
        <p type="margin">
          <s id="s.000485"><margin.target id="marg55"></margin.target>7. quinti.</s>
        </p>
        <p type="main">
          <s id="s.000486">Sint ſolida parallelopipeda ab, cd in æqualibus baſibus <lb></lb>
conſtituta: <expan abbr="ſitq;">ſitque</expan> be altitudo ſolidi ab: &amp; ſolidi cd altitudo <lb></lb>
d f; quæ quidem maior ſit, quàm be. </s>
          <s id="s.000487">Dico ſolidum ab ad <lb></lb>
ſolidum cd eandem habere proportionem, quam be ad <lb></lb>
d f. </s>
          <s id="s.000488">abſcindatur enim à linea df æqualis ipſi be, quæ ſit gf: <lb></lb>
&amp; per g ducatur planum ſecans ſolidum cd; quod baſibus <lb></lb>
æquidiſtet, <expan abbr="faciatq;">faciatque</expan> <expan abbr="ſectionẽ">ſectionem</expan> hK. </s>
          <s id="s.000489">erunt ſolida ab, ck æque <lb></lb>
<arrow.to.target n="marg56"></arrow.to.target><lb></lb>
<figure id="id.023.01.054.1.jpg" xlink:href="023/01/054/1.jpg"></figure><lb></lb>
alta inter <lb></lb>
ſe æqualia <lb></lb>
<expan abbr="cũ">cum</expan> æqua­<lb></lb>
les baſes <lb></lb>
habeant. </s>
          <lb></lb>
          <s id="s.000490"><arrow.to.target n="marg57"></arrow.to.target><lb></lb>
Sed <expan abbr="ſolidũ">ſolidum</expan> <lb></lb>
hd ad ſoli <lb></lb>
dum cK <lb></lb>
eſt, ut alti<lb></lb>
tudo dg <lb></lb>
ad gf <expan abbr="alti­tudinẽ">alti­<lb></lb>
tudinem</expan>; ſe<lb></lb>
catur enim ſolidum cd plano baſi<lb></lb>
<figure id="id.023.01.054.2.jpg" xlink:href="023/01/054/2.jpg"></figure><lb></lb>
bus æquidiſtante: &amp; rurſus <expan abbr="cõpo-nende">compo­<lb></lb>
nende</expan>, <expan abbr="conuertendoq;">conuertendoque</expan> <expan abbr="ſolidũ">ſolidum</expan> ck <lb></lb>
<arrow.to.target n="marg58"></arrow.to.target><lb></lb>
ad ſolidum cd, ut gf ad fd. </s>
          <s id="s.000491">ergo <lb></lb>
ſolidum ab, quod eſt æquale ipſi <lb></lb>
ck ad ſolidum cd eam proportio <lb></lb>
nem habet, quam altitudo gf, hoc <lb></lb>
eſt be ad df altitudinem.</s>
        </p>
        <p type="margin">
          <s id="s.000492"><margin.target id="marg56"></margin.target>31. unde<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000493"><margin.target id="marg57"></margin.target>18. huius</s>
        </p>
        <p type="margin">
          <s id="s.000494"><margin.target id="marg58"></margin.target>7. quinti.</s>
        </p>
        <p type="main">
          <s id="s.000495">Sint deinde ſolida parallelepipe<lb></lb>
da ab, ac in eadem baſi; quorum <lb></lb>axes de, ſ e cum ipſa æquales angu<pb pagenum="24" xlink:href="023/01/055.jpg"></pb>los contineant. </s>
          <s id="s.000496">Dico ſolidum ab ad ſolidum ace idem ha<lb></lb>
bere proportionem, quam axis de ad axem ef. </s>
          <s id="s.000497">Si enim <lb></lb>
axes in eadem recta linea fuerint conſtituti, hæc duo ſoli­<lb></lb>
da, in unum, atque idem ſolidum conuenient. </s>
          <s id="s.000498">quare ex <lb></lb>
iis, quæ proxime tradita ſunt, habebit ſolidum ab ad ſo­<lb></lb>
lidum ac eandem proportionem, quam axis de ad ef <lb></lb>
axem. </s>
          <s id="s.000499">Si uero axes non ſint in eadem recta linea, demittan<lb></lb>
tur a punctis d, ſ perpendiculares ad baſis planum, dg, fh: <lb></lb>
&amp; jungantur eg, eh. </s>
          <s id="s.000500">Quoniam igitur axes cum baſibus <lb></lb>
æquales angulos continent, erit deg angulus æqualis an­<lb></lb>
<figure id="id.023.01.055.1.jpg" xlink:href="023/01/055/1.jpg"></figure><lb></lb>
gulo feh: &amp; ſunt <lb></lb>
anguli ad gh re­<lb></lb>
cti, quare &amp; re­<lb></lb>
liquus edg æqua<lb></lb>
lis erit reliquo <lb></lb>
efh: &amp; triangu­<lb></lb>
lum deg <expan abbr="triãgu-lo">triangu­<lb></lb>
lo</expan>  feh ſimile. </s>
          <s id="s.000501">er­<lb></lb>
go gd ad de eſt, <lb></lb>
ut hf ad e: &amp; per <lb></lb>
mutando gd ad <lb></lb>
hf, ut de ad cf. </s>
          <lb></lb>
          <figure id="id.023.01.055.2.jpg" xlink:href="023/01/055/2.jpg"></figure>
          <lb></lb>
          <s id="s.000502">Sed ſolidum ab <lb></lb>
ad ſolidum ac <lb></lb>
eandem propor­<lb></lb>
tionem habet, <lb></lb>
quam dg altitu­<lb></lb>
do ad <expan abbr="altitudinẽ">altitudinem</expan> <lb></lb>
fh. </s>
          <s id="s.000503">ergo &amp; <expan abbr="ean-dẽ">ean­<lb></lb>
dem</expan> habebit, <expan abbr="quã">quam</expan> <lb></lb>
axis de ad ef <expan abbr="axẽ">axem</expan></s>
        </p>
        <p type="main">
          <s id="s.000504">Poſtremo ſint <lb></lb>
ſolidi parallepi<lb></lb>
peda ab, cd in 



<pb xlink:href="023/01/056.jpg"></pb>æqualibus baſibus, quorum axes cum baſibus æquales an<lb></lb>
gulos faciant. </s>
          <s id="s.000505">Dico ſolidum ab ad <expan abbr="ſolidũ">ſolidum</expan> cd ita eſſe, ut axis <lb></lb>
ef ad axem gh: nam ſi axes ad planum baſis recti ſint, il­<lb></lb>
lud perſpicue conſtat: quoniam eadem linea, &amp; axem &amp; ſoli<lb></lb>
di altitudinem determinabit. </s>
          <s id="s.000506">Si uero ſint inclinati, à pun­<lb></lb>
ctis eg ad ſubiectum planum perpendiculares ducantur <lb></lb>
ek, gl: &amp; iungantur fk, hl. </s>
          <s id="s.000507">rurſus quoniam axes cum ba<lb></lb>
ſibus æquales faciunt angulos, eodem modo demonſtrabi<lb></lb>
tur, triangulum efK triangulo ghl ſimile eſſe: &amp; ek ad gl, <lb></lb>
ut ef ad gh. </s>
          <s id="s.000508">Solidum autem ab ad ſolidum cd eſt, ut <lb></lb>
eK ad gl. </s>
          <s id="s.000509">ergo &amp; ut axis ef ad axem gh. </s>
          <s id="s.000510">quæ omnia de<lb></lb>
monſtrare oportebat.</s>
        </p>
        <p type="main">
          <s id="s.000511">Ex iis quæ demonſtrata ſunt, facile conſtare <lb></lb>
poteſt, priſmata omnia &amp; pyramides, quæ trian­<lb></lb>gulares baſes habent, ſiue in eiſdem, ſiue in æqua<lb></lb>
<arrow.to.target n="marg59"></arrow.to.target><lb></lb>
libus baſibus conſtituantur, eandem proportio­<lb></lb>
nem habere, quam altitudines: &amp; ſi axes cum ba<lb></lb>
ſibus æquales angulos contineant, ſimiliter ean­<lb></lb>
dem, quam axes, habere proportionem: ſunt <lb></lb>
<arrow.to.target n="marg60"></arrow.to.target><lb></lb>
enim ſolida parallelepipeda priſmatum triangula<lb></lb>
<arrow.to.target n="marg61"></arrow.to.target><lb></lb>
res baſes <expan abbr="habentiũ">habentium</expan> dupla; &amp; pyramidum ſextupla.</s>
        </p>
        <p type="margin">
          <s id="s.000512"><margin.target id="marg59"></margin.target>15. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000513"><margin.target id="marg60"></margin.target>28. unde­<lb></lb>
cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000514"><margin.target id="marg61"></margin.target>7. duode­<lb></lb>
cimi.</s>
        </p>
        <p type="head">
          <s id="s.000515">THEOREMA XVI. PROPOSITIO XX.</s>
        </p>
        <p type="main">
          <s id="s.000516">Priſmata omnia &amp; pyramides, quæ in eiſdem, <lb></lb>
uel æqualibus baſibus conſtituuntur, eam inter <lb></lb>
ſe proportionem habent, quam altitudines: &amp; ſi <lb></lb>
axes cum baſibus faciant angulos æquales, eam <lb></lb>
etiam, quam axes habent proportionem.</s>
        </p>
        <pb pagenum="25" xlink:href="023/01/057.jpg"></pb>
        <p type="main">
          <s id="s.000517">Sint duo priſmata ae, af, quorum eadem baſis quadri­<lb></lb>
latera abcd: <expan abbr="ſitq;">ſitque</expan> priſmatis ae altitudo eg; &amp; priſmatis <lb></lb>
af altitudo fh. </s>
          <s id="s.000518">Dico priſma ae ad priſma af eam habere <lb></lb>
proportionem, quam eg ad fh. </s>
          <s id="s.000519">iungatur enim ac: &amp; in <lb></lb>
unoquoque priſmate duo priſmata intelligantur, quorum <lb></lb>
<figure id="id.023.01.057.1.jpg" xlink:href="023/01/057/1.jpg"></figure><lb></lb>
baſes ſint triangu<lb></lb>
la abc, acd. </s>
          <s id="s.000520">habe <lb></lb>
bunt duo priſma­<lb></lb>
te in eadem baſi <lb></lb>
abc conſtituta, <lb></lb>
proportionem <expan abbr="eã">eam</expan> <lb></lb>
dem, quam ipſo­<lb></lb>
rum altitudines e <lb></lb>
g, fh, ex iam de­<lb></lb>
monſtratis. </s>
          <s id="s.000521">&amp; ſi­<lb></lb>
militer alia duo, <lb></lb>
quæ ſunt in baſi a <lb></lb>
<arrow.to.target n="marg62"></arrow.to.target><lb></lb>
c d. </s>
          <s id="s.000522">quare totum priſma ae ad priſma af eandem propor<lb></lb>
tionem habebit, quam altitudo eg ad fh altitudinem. </s>
          <lb></lb>
          <s id="s.000523">Quòd cum priſmata ſint pyramidum tripla, &amp; ipſæ pyrami<lb></lb>
des, quarum eadem eſt baſis quadrilatera, &amp; altitudo priſ­<lb></lb>
matum altitudini æqualis, eam inter ſe proportionem ha­<lb></lb>
bebunt, quam altitudines.</s>
        </p>
        <p type="margin">
          <s id="s.000524"><margin.target id="marg62"></margin.target>12. quinti</s>
        </p>
        <p type="main">
          <s id="s.000525">Si uero priſmata baſes æquales habeant, <expan abbr="nõ">non</expan> eaſdem, ſint <lb></lb>duo eiuſmodi priſmata ae, fl: &amp; ſit baſis priſmatis ae qua<lb></lb>
drilaterum abcd; &amp; priſmatis fl quadrilaterum fghk. </s>
          <lb></lb>
          <s id="s.000526">Dico priſma ae ad priſma fl ita eſſe, ut altitudo illius ad <lb></lb>
huius altitudinem. </s>
          <s id="s.000527">nam ſi altitudo ſit eadem, <expan abbr="intelligãtur">intelligantur</expan> <lb></lb>
<arrow.to.target n="marg63"></arrow.to.target><lb></lb>
duæ pyramides abcde, fghkl. </s>
          <s id="s.000528">quæ <expan abbr="ĩtcrſe">interſe</expan> æquales <expan abbr="erũt">erunt</expan>, <lb></lb>
cum æquales baſes, &amp; altitudinem eandem habeant. </s>
          <s id="s.000529">quare <lb></lb>
<arrow.to.target n="marg64"></arrow.to.target><lb></lb>
&amp; priſmata ae, fl, quæ ſunt <expan abbr="harũ">harum</expan> pyramidum tripla, æqua­<lb></lb>
lia ſint neceſſe eſt. </s>
          <s id="s.000530">ex quibus perſpicue conſtat <expan abbr="propoſitũ">propoſitum</expan>. </s>
          <lb></lb>
          <s id="s.000531">Si uero altitudo priſmatis fl ſit maior, à priſmate fl ab­<lb></lb>
ſcindatur priſma fm, quod æque altum ſit, <expan abbr="atq;">atque</expan> ipſum ae. 



<pb xlink:href="023/01/058.jpg"></pb><figure id="id.023.01.058.1.jpg" xlink:href="023/01/058/1.jpg"></figure><lb></lb>
erunt eædem ra­<lb></lb>
tione priſmata a <lb></lb>
e, fm inter ſe æ­<lb></lb>
qualia. </s>
          <s id="s.000532">quare ſi­<lb></lb>
militer demon­<lb></lb>
ſtrabitur priſma <lb></lb>
fm ad priſma fl <lb></lb>
eandem habere <lb></lb>
proportionem, <lb></lb>
quam priſmatis <lb></lb>
fm altitudo ad <lb></lb>
altitudinem ip­<lb></lb>
ſius fl. </s>
          <s id="s.000533">ergo &amp; priſma ae ad priſma fl eandem propor­<lb></lb>
tionem habebit, quam altitudo ad altitudinem. </s>
          <s id="s.000534">ſequitur <lb></lb>
igitur ut &amp; pyramides, quæ in æqualibus baſibus <expan abbr="conſtituũ">conſtituun</expan><lb></lb>
tur, eandem inter ſe ſe, quam altitudines, proportionem <lb></lb>
habeant.</s>
        </p>
        <p type="margin">
          <s id="s.000535"><margin.target id="marg63"></margin.target>6. duode<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000536"><margin.target id="marg64"></margin.target>25. quinti</s>
        </p>
        <figure id="id.023.01.058.2.jpg" xlink:href="023/01/058/2.jpg"></figure>
        <p type="main">
          <s id="s.000537">Sint deinde priſmata ae, af in eadem baſi abcd; <expan abbr="quorũ">quorum</expan> <lb></lb>
axes cum baſibus æquales angulos contineant: &amp; ſit priſ­



<pb pagenum="26" xlink:href="023/01/059.jpg"></pb>matis ae axis gh; &amp; priſmatis af axis lh. </s>
          <s id="s.000538">Dico priſma <lb></lb>
ae ad priſma af eam proportionem habere, quam gh ad <lb></lb>
h l. ducantur à punctis gl perpendiculares ad baſis pla­<lb></lb>
<figure id="id.023.01.059.1.jpg" xlink:href="023/01/059/1.jpg"></figure><lb></lb>
num gK, lm: &amp; iungantur kh, <lb></lb>
h m. </s>
          <s id="s.000539">Itaque quoniam anguli gh <lb></lb>
k, lhm ſunt æquales, ſimiliter ut <lb></lb>
ſupra demonſtrabimus, triangu­<lb></lb>
la ghK, lhm ſimilia eſſe; &amp; ut g <lb></lb>
K ad lm, ita gh ad hl. </s>
          <s id="s.000540">habet au<lb></lb>
tem priſma ae ad priſma af ean <lb></lb>
dem proportionem, quam altitu<lb></lb>
do gK ad altitudinem lm, ſicuti <lb></lb>
demonſtratum eſt. </s>
          <s id="s.000541">ergo &amp; ean­<lb></lb>
dem habebit, quam gh, ad hl. py<lb></lb>
ramis igitur abcdg ad pyrami­<lb></lb>
dem abcdl eandem proportio­<lb></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"></figure>
        <p type="main">
          <s id="s.000542">Denique ſint priſmata ae, ko in æqualibus baſibus ab <lb></lb>
cd, klmn conſtituta; quorum axes cum baſibus æquales <lb></lb>
faciant angulos: <expan abbr="ſitq;">ſitque</expan> priſmatis ae axis fg, &amp; altitudo fh: <lb></lb>
priſmatis autem ko axis pq, &amp; altitudo pr. </s>
          <s id="s.000543">Dico priſma <lb></lb>
ae ad priſma ko ita eſſe, ut fg ad pq. </s>
          <s id="s.000544">iunctis enim gh, 



<pb xlink:href="023/01/060.jpg"></pb>qr, eodem, quo ſupra, modo oſtendemus fg ad pq, ut fh <lb></lb>
ad pr. </s>
          <s id="s.000545">ſed priſma ae ad ipſum ko eſt, ut fh ad pr. </s>
          <s id="s.000546">ergo <lb></lb>
&amp; ut fg axis ad axem pq.</s>
          <s id="s.000547"> ex quibus ſit, ut pyramis abcdf <lb></lb>
<figure id="id.023.01.060.1.jpg" xlink:href="023/01/060/1.jpg"></figure><lb></lb>
ad <expan abbr="pyrami-dẽ">pyrami­<lb></lb>
dem</expan> klmnp <lb></lb>
eandem ha<lb></lb>
beat pro ­<lb></lb>
portionẽ, <lb></lb>
<expan abbr="quã">quam</expan> axis ad <lb></lb>
<expan abbr="axẽ">axem</expan>. </s>
          <s id="s.000548">quod <lb></lb>
<expan abbr="demonſtrã">demonſtran</expan> <lb></lb>
<expan abbr="dũ">dum</expan> ſuerat.</s>
        </p>
        <p type="main">
          <s id="s.000549">Simili ra<lb></lb>
tione in a­<lb></lb>
liis priſma­<lb></lb>
tibus &amp; py<lb></lb>
ramidibus eadem demonſtrabuntur.</s>
        </p>
        <p type="head">
          <s id="s.000550">THEOREMA XVII. PROPOSITIO XXI.</s>
        </p>
        <p type="main">
          <s id="s.000551">Priſmata omnia, &amp; pyramides inter ſe propor<lb></lb>
tionem habent compoſitam ex proportione ba­<lb></lb>
ſium, &amp; proportione altitudinum.</s>
        </p>
        <p type="main">
          <s id="s.000552">Sint duo priſmata ae, gm: <expan abbr="ſitq;">ſitque</expan> priſmatis ae baſis qua<lb></lb>
drilaterum abcd, &amp; altitudo ef: priſmatis uero gm ba­<lb></lb>
ſis quadrilaterum ghKl, &amp; altitudo mn. </s>
          <s id="s.000553">Dico priſma ae <lb></lb>
ad priſma gm proportionem habere compoſitam ex pro<lb></lb>
portione baſis abcd ad baſim ghkl, &amp; ex proportione <lb></lb>
altitudinis ef, ad altitudinem mn.</s>
        </p>
        <p type="main">
          <s id="s.000554">Sint enim primum ef, mn æquales: &amp; ut baſis abcd <lb></lb>
ad baſim ghkl, ita fiat linea, in qua o ad lineam, in qua p: <lb></lb>
ut autem ef ad mn, ita linea p ad lineam q.</s>
          <s id="s.000555"> erunt lineæ <lb></lb>
pq inter ſe æquales. </s>
          <s id="s.000556">Itaque priſma ae ad priſma gm <expan abbr="eã">eam</expan> 



<pb pagenum="27" xlink:href="023/01/061.jpg"></pb>proportionem habet, quam baſis abcd ad baſim ghkl: <lb></lb>
ſi enim intelligantur duæ pyramides abcde, ghklm, ha­<lb></lb>
bebunt hæ inter ſe proportionem eandem, quam ipſarum <lb></lb>
baſes ex ſexta duodecimi elementorum. </s>
          <s id="s.000557">Sed ut baſis abcd <lb></lb>
ad ghKl baſim, ita linea o ad lineam p; hoc eſt ad lineam q <lb></lb>
ei æqualem. </s>
          <s id="s.000558">ergo priſma ae ad priſma gm eſt, ut linea o <lb></lb>
ad lineam q.</s>
          <s id="s.000559"> proportio autem o ad q copoſita eſt ex pro­<lb></lb>
portione o ad p, &amp; ex proportione p ad q.</s>
          <s id="s.000560"> quare priſma <lb></lb>
ae ad priſma gm, &amp; idcirco pyramis abcde, ad pyrami­<lb></lb>
dem ghKlm proportionem habet ex eiſdem proportio­<lb></lb>
nibus compoſitam, uidelicet ex proportione baſis abcd <lb></lb>
ad baſim ghKl, &amp; ex proportione altitudinis ef ad mn al<lb></lb>
titudinem. </s>
          <s id="s.000561">Quòd ſi lineæ ef, mn inæquales ponantur, ſit <lb></lb>
ef minor: &amp; ut ef ad mn, ita fiat linea p ad lineam u: de <lb></lb>
<figure id="id.023.01.061.1.jpg" xlink:href="023/01/061/1.jpg"></figure><lb></lb>
inde ab ipſa mn abſcindatur rn æqualis ef: &amp; per r duca­<lb></lb>
tur planum, quod oppoſitis planis æquidiſtans faciat ſe­<lb></lb>
ctionem st. </s>
          <s id="s.000562">erit priſma ae, ad priſma gt, ut baſis abcd <lb></lb>
ad baſim ghkl; hoc eſt ut o ad p: ut autem priſma gt ad <lb></lb>
<arrow.to.target n="marg65"></arrow.to.target><lb></lb>
priſma gm, ita altitudo rn; hoc eſt ef ad altitudine mn; <lb></lb>
uidelicet linea p ad lineam u. </s>
          <s id="s.000563">ergo ex æquali priſma ae ad <lb></lb>
priſma gm eſt, ut linea o ad ipſam u. </s>
          <s id="s.000564">Sed proportio o ad <lb></lb>
u <expan abbr="cõpoſita">compoſita</expan> eſt ex proportione o ad p, quæ eſt baſis abcd <lb></lb>
ad baſim ghkl; &amp; ex proportione p ad u, quæ eſt altitudi­<lb></lb>
nis ef ad altitudinem mn. </s>
          <s id="s.000565">priſma igitur ae ad priſma gm 



<pb xlink:href="023/01/062.jpg"></pb>compoſitam proportionem habet ex proportione <expan abbr="baſiũ">baſium</expan>, <lb></lb>
&amp; proportione altitudinum. </s>
          <s id="s.000566">Quare &amp; pyramis, cuius ba­<lb></lb>
ſis eſt quadrilaterum abcd, &amp; altitudo ef ad pyramidem, <lb></lb>
<figure id="id.023.01.062.1.jpg" xlink:href="023/01/062/1.jpg"></figure><lb></lb>
cuius baſis quadrilaterum ghKl, &amp; altitudo mn, compoſi<lb></lb>
tam habet proportionem ex proportione baſium abcd, <lb></lb>
ghkl, &amp; ex proportione altitudinum ef, mn. </s>
          <s id="s.000567">quod qui­<lb></lb>
dem demonſtraſſe oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000568"><margin.target id="marg65"></margin.target>20. huius</s>
        </p>
        <p type="main">
          <s id="s.000569">Ex iam demonſtratis perſpicuum eſt, priſma <lb></lb>
ta omnia, &amp; pyramides, in quibus axes cum baſi­<lb></lb>
bus æquales angulos continent, proportionem <lb></lb>
habere compoſitam ex baſium proportione, &amp; <lb></lb>
proportione axium. </s>
          <s id="s.000570">demonſtratum eſt enim, a­<lb></lb>
xes inter ſe eandem proportionem habere, quam <lb></lb>
ipſæ altitudines.</s>
        </p>
        <p type="head">
          <s id="s.000571">THEOREMA XVIII. PROPOSITIO XXII.</s>
        </p>
        <p type="main">
          <s id="s.000572">CVIVSLIBEt pyramidis, &amp; cuiuslibet coni, 



<pb pagenum="28" xlink:href="023/01/063.jpg"></pb>uel coni portionis axis à centro grauitatis ita diui <lb></lb>
ditur, ut pars, quæ terminatur ad uerticem reli­<lb></lb>
quæ partis, quæ ad baſim, ſit tripla.</s>
        </p>
        <p type="main">
          <s id="s.000573">Sit pyramis, cuius baſis triangulum abc; axis de; &amp; gra<lb></lb>
uitatis centrum K. </s>
          <s id="s.000574">Dico lineam dk ipſius Ke triplam eſſe. </s>
          <lb></lb>
          <s id="s.000575">trianguli enim bdc centrum grauitatis ſit punctum f; <expan abbr="triã">trian</expan><lb></lb>
guli adc <expan abbr="centrũ">centrum</expan> g; &amp; trianguli adb ſit h: &amp; iungantur af, <lb></lb>
b g, ch. </s>
          <s id="s.000576">Quoniam igitur <expan abbr="centrũ">centrum</expan> grauitatis pyramidis in axe <lb></lb>
<arrow.to.target n="marg66"></arrow.to.target><lb></lb>
<expan abbr="cõſiſtit">conſiſtit</expan>: <expan abbr="ſuntq;">ſuntque</expan> de, af, bg, ch <expan abbr="eiuſdẽ">eiuſdem</expan> pyramidis axes: conue<lb></lb>
nient omnes in <expan abbr="idẽ">idem</expan> <expan abbr="punctũ">punctum</expan> k, quod eſt grauitatis centrum. </s>
          <lb></lb>
          <s id="s.000577">Itaque animo concipiamus hanc pyramidem diuiſam in <lb></lb>
quatuor pyramides, quarum baſes ſint ipſa pyramidis <lb></lb>
<arrow.to.target n="marg67"></arrow.to.target><lb></lb>
<figure id="id.023.01.063.1.jpg" xlink:href="023/01/063/1.jpg"></figure><lb></lb>
triangula; &amp; <emph type="ul"></emph>axis<emph.end type="ul"></emph.end> pun­<lb></lb>
ctum k quæ quidem py­<lb></lb>
ramides inter ſe æquales <lb></lb>
ſunt, ut <expan abbr="demõſtrabitur">demonſtrabitur</expan>. </s>
          <lb></lb>
          <s id="s.000578">Ducatur <expan abbr="enĩ">enim</expan> per lineas <lb></lb>
dc, de planum <expan abbr="ſecãs">ſecans</expan>, ut <lb></lb>
ſit ipſius, &amp; baſis abc <expan abbr="cõ">com</expan><lb></lb>
munis ſectio recta linea <lb></lb>
cel: <expan abbr="eiuſdẽ">eiuſdem</expan> uero &amp; <expan abbr="triã-guli">trian­<lb></lb>
guli</expan> adb ſit linea dhl. erit linea al æqualis ipſi <lb></lb>
lb: nam centrum graui­<lb></lb>
tatis trianguli conſiſtit <lb></lb>
in linea, quæ ab angulo <lb></lb>
ad dimidiam baſim per­<lb></lb>
ducitur, ex tertia deci­<lb></lb>
ma Archimedis. </s>
          <lb></lb>
          <s id="s.000579">quare <lb></lb>
<arrow.to.target n="marg68"></arrow.to.target><lb></lb>
triangulum acl æquale <lb></lb>
eſt triangulo bcl: &amp; propterea pyramis, cuius baſis trian­<lb></lb>
gulum acl, uertex d, eſt æqualis pyramidi, cuius baſis bcl <lb></lb>
<arrow.to.target n="marg69"></arrow.to.target><lb></lb>
triangulum, &amp; idem uertex. </s>
          <s id="s.000580">pyramides enim, quæ ab <expan abbr="eodẽ">eodem</expan> 



<pb xlink:href="023/01/064.jpg"></pb>ſunt uertice, eandem proportionem habent, quam <expan abbr="ipſarũ">ipſarum</expan> <lb></lb>
baſes. </s>
          <s id="s.000581">eadem ratione pyramis aclk pyramidi bclk &amp; py<lb></lb>
ramis adlk ipſi bdlk pyramidi æqualis erit. </s>
          <s id="s.000582">Itaque ſi a py<lb></lb>ramide acld auferantur pyramides aclk, adlk: &amp; à pyra<lb></lb>
mide bcld <expan abbr="auferãtur">auferantur</expan> pyramides bclk dblK: quæ relin­<lb></lb>
quuntur erunt æqualia. </s>
          <s id="s.000583">æqualis igitur eſt pyramis acdk <lb></lb>
pyramidi bcdK. </s>
          <s id="s.000584">Rurſus ſi per lineas ad, de ducatur pla­<lb></lb>
num quod pyramidem ſccet: <expan abbr="ſitq;">ſitque</expan> eius &amp; baſis communis <lb></lb>
ſectio aem: ſimiliter oſtendetur pyramis abdK æqualis <lb></lb>
pyramidi acdk. </s>
          <s id="s.000585">ducto denique alio plano per lineas ca, <lb></lb>
af: ut eius, &amp; trianguli cdb communis ſectio ſit cfn, py­<lb></lb>
ramis abck pyramidi acdk æqualis demonſtrabitur. </s>
          <s id="s.000586"><expan abbr="cũ">cum</expan> <lb></lb>
ergo tres pyramides bcdk, abdk, abck uni, &amp; eidem py<lb></lb>
ramidi acdk ſint æquales, omnes inter ſe ſe æquales <expan abbr="erũt">erunt</expan>. </s>
          <lb></lb>
          <s id="s.000587">Sed ut pyramis abcd ad pyramidem abck ita de axis ad <lb></lb>
axem ke, ex uigeſima propoſitione huius: ſunt enim hæ <lb></lb>
pyramides in eadem baſi, &amp; axes cum baſibus æquales con<lb></lb>
tinent angulos, quòd in eadem recta linea conſtituantur. </s>
          <lb></lb>
          <s id="s.000588">quare diuidendo, ut tres pyramides acdk, bcdK, abdK <lb></lb>
ad pyramidem abcK, ita dk ad Ke. </s>
          <s id="s.000589">conſtat igitur lineam <lb></lb>
dK ipſius Ke triplam eſſe. </s>
          <s id="s.000590">ſed &amp; ak tripla eſt Kf: itemque <lb></lb>
bK ipſius kg: &amp; ck ipſius kl tripla. </s>
          <s id="s.000591">quod eodem modo <lb></lb>
demonſtrabimus.</s>
        </p>
        <p type="margin">
          <s id="s.000592"><margin.target id="marg66"></margin.target>17 huius</s>
        </p>
        <p type="margin">
          <s id="s.000593"><margin.target id="marg67"></margin.target><emph type="italics"></emph>ucrfex<emph.end type="italics"></emph.end></s>
        </p>
        <p type="margin">
          <s id="s.000594"><margin.target id="marg68"></margin.target>1. sexti.</s>
        </p>
        <p type="margin">
          <s id="s.000595"><margin.target id="marg69"></margin.target>5. duode­<lb></lb>
cimi.</s>
        </p>
        <p type="main">
          <s id="s.000596">Sit pyramis, cuius baſis quadrilaterum abcd; axis ef: <lb></lb>
&amp; diuidatur ef in g, ita ut eg ipſius gf ſit tripla. </s>
          <s id="s.000597">Dico cen­<lb></lb>
trum grauitatis pyramidis eſſe punctum g. ducatur enim <lb></lb>
linea bd diuidens baſim in duo triangula abd, bcd: ex <lb></lb>
quibus <expan abbr="intelligãtur">intelligantur</expan> <expan abbr="cõſtitui">conſtitui</expan> duæ pyramides abde, bcde: <lb></lb>
ſitque pyramidis abde axis eh; &amp; pyramidis bcde axis <lb></lb>
eK: &amp; iungatur hK, quæ per f tranſibit: eſt enim in ipſa hK <lb></lb>
centrum grauitatis magnitudinis compoſitæ ex triangulis <lb></lb>
abd, bcd, hoc eſt ipſius quadrilateri. </s>
          <s id="s.000598">Itaque centrum gra<lb></lb>
uitatis pyramidis abde ſit punctum l: &amp; pyramidis bcde <lb></lb>
<arrow.to.target n="marg70"></arrow.to.target><lb></lb>
ſit m. </s>
          <s id="s.000599">ducta igitur lm ipſi hm lineæ æquidiſtabit. </s>
          <s id="s.000600">nam el ad 



<pb pagenum="29" xlink:href="023/01/065.jpg"></pb>lh eandem habet proportionem, quam em ad mk, uideli­<lb></lb>
cet triplam. </s>
          <s id="s.000601">quare linea lm ipſam ef ſecabit in puncto g: <lb></lb>
etenim eg ad gf eſt, ut el ad lh. </s>
          <s id="s.000602">præterea quoniam hk, lm <lb></lb>
æquidiſtant, erunt triangula hef, leg ſimilia: <expan abbr="itemq;">itemque</expan> inter <lb></lb>
ſe ſimilia fek gem: &amp; ut ef ad eg, ita hf ad lg: &amp; ita fK ad <lb></lb>
gm. </s>
          <s id="s.000603">ergo ut hf ad lg, ita fk ad gm: &amp; permutando ut hf <lb></lb>
ad fK, ita lg ad gm. </s>
          <s id="s.000604">ſed cum h ſit centrum trianguli abd; <lb></lb>
&amp; k <expan abbr="triãguli">trianguli</expan> bcd <expan abbr="punctũ">punctum</expan> uero f totius quadrilateri abcd <lb></lb>
centrum: erit ex 8. Archimedis de centro grauitatis plano<lb></lb>
rum hf ad fk ut triangulum bcd ad triangulum abd: ut, <lb></lb>
autem bcd triangulum ad triangulum abd, ita pyramis <lb></lb>
<figure id="id.023.01.065.1.jpg" xlink:href="023/01/065/1.jpg"></figure><lb></lb>
bcde ad pyramidem abde. </s>
          <s id="s.000605">ergo <lb></lb>
linea lg ad gm erit, ut pyramis <lb></lb>
bcde ad <expan abbr="pyramidẽ">pyramidem</expan> abde. </s>
          <s id="s.000606">ex quo <lb></lb>
ſequitur, ut totius pyramidis <lb></lb>
abcde punctum g ſit grauitatis <lb></lb>
centrum. </s>
          <s id="s.000607">Rurſus ſit pyramis ba­<lb></lb>
ſim habens pentagonum abcde: <lb></lb>
&amp; axem fg: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="pũ">pun</expan><lb></lb>
cto h, ita ut fh ad hg triplam habe<lb></lb>
at proportionem. </s>
          <s id="s.000608">Dico h grauita­<lb></lb>
tis <expan abbr="centrũ">centrum</expan> eſſe pyramidis abcdef. </s>
          <lb></lb>
          <s id="s.000609">iungatur enim eb: <expan abbr="intelligaturq;">intelligaturque</expan> <lb></lb>
pyramis, cuius uertex f, &amp; baſis <lb></lb>
triangulum abe: &amp; alia pyramis <lb></lb>
intelligatur eundem uerticem ha­<lb></lb>
bens, &amp; baſim bcde <expan abbr="quadrilaterũ">quadrilaterum</expan>: <lb></lb>
ſit autem pyramidis abef axis fk<lb></lb>
&amp; grauitatis centrum l: &amp; pyrami<lb></lb>
dis bcdef axis fm, &amp; centrum gra <lb></lb>
uitatis n:<expan abbr="iunganturq;">iunganturque</expan> km, ln; <lb></lb>
quæ per puncta gh tranſibunt. </s>
          <lb></lb>
          <s id="s.000610">Rurſus eodem modo, quo ſup ra, <lb></lb>
demonſtrabimus lineas Kgm, lhn ſibi ipſis æquidiſtare: 



<pb xlink:href="023/01/066.jpg"></pb>&amp; denique punctum h pyramidis abcdef grauitatis eſſe <lb></lb>
centrum, &amp; ita in aliis.</s>
        </p>
        <p type="margin">
          <s id="s.000611"><margin.target id="marg70"></margin.target>2. fexti.</s>
        </p>
        <p type="main">
          <s id="s.000612">Sit conus, uel coni portio axem habens bd: ſeceturque <lb></lb>
plano per axem, quod ſectionem faciat triangulum abc: <lb></lb>
&amp; bd axis diuidatur in c, ita ut be ipſius ed ſit tripla. </s>
          <lb></lb>
          <s id="s.000613">Dico punctum e coni, uel coni portionis, grauitatis <lb></lb>
eſſe centrum. </s>
          <s id="s.000614">Si enim fieri poteſt, ſit centrum f: &amp; pro­<lb></lb>
ducatur ef extra figuram in g. </s>
          <s id="s.000615">quam uero proportionem <lb></lb>
habet ge ad ef, habeat baſis coni, uelconi portionis, hoc <lb></lb>
eſt circulus, uel ellipſis circa diametrum ac ad aliud ſpa­<lb></lb>
cium, in quo h. </s>
          <s id="s.000616">Itaque in circulo, uel ellipſi plane deſcri­<lb></lb>
batur rectilinea figura axlmcnop, ita ut quæ <expan abbr="relinquũ-tur">relinquun­<lb></lb>
tur</expan> portiones ſint minores ſpacio h: &amp; intelligatur pyra­<lb></lb>
mis baſim habens rectilineam figuram aKlmcnop, &amp; <lb></lb>
axem bd; cuius quidem grauitatis centrum erit punctum <lb></lb>
e, ut iam demonſtrauimus. </s>
          <s id="s.000617">Et quoniam portiones ſunt <lb></lb>
minores ſpacio h, circulus, uel ellipſis ad portiones ma­<lb></lb>
<figure id="id.023.01.066.1.jpg" xlink:href="023/01/066/1.jpg"></figure><lb></lb>
iorem proportionem habet, quam ge ad ef. </s>
          <s id="s.000618">ſed ut circu­<lb></lb>
lus, uel ellipſis ad figuram rectilineam ſibi inſcriptam, ita <lb></lb>
conus, uel coni portio ad pyramidem, quæ figuram rectili­<lb></lb>
neam pro baſi habet; &amp; altitudinem æqualem: etenim ſu­



<pb pagenum="30" xlink:href="023/01/067.jpg"></pb><arrow.to.target n="marg71"></arrow.to.target><lb></lb>
pra demonſtratum eſt, ita eſſe cylindrum, uel cylindri por­<lb></lb>
tionem ad priſma, cuius baſis rectilinea figura, &amp; æqua­<lb></lb>
lis altitudo. </s>
          <s id="s.000619">ergo per conuerſionem rationis, ut circulus, <lb></lb>
uel ellipſis ad portiones, ita conus, uel coni portio ad por­<lb></lb>
tiones ſolidas. </s>
          <s id="s.000620">quare conus uel coni portio ad portiones <lb></lb>
ſolidas maiorem habet proportionem, quam ge ad ef: &amp; <lb></lb>
diuidendo, pyramis ad portiones ſolidas maiorem pro­<lb></lb>
portionem habet, quam gf ad fe. </s>
          <s id="s.000621">fiat igitur qf ad fe <lb></lb>
ut pyramis ad dictas portiones. </s>
          <s id="s.000622">Itaque quoniam a cono <lb></lb>
uel coni portione, cuius grauitatis centrum eſt f, aufer­<lb></lb>
tur pyramis, cuius centrum e; reliquæ magnitudinis, <lb></lb>
quæ ex ſolidis portionibus conſtat, centrum grauitatis <lb></lb>
erit in linea ef protracta, &amp; in puncto q.</s>
          <s id="s.000623"> quod fieri <lb></lb>
non poteſt: eſt enim centrum grauitatis intra. </s>
          <s id="s.000624">Conſtat <lb></lb>
igitur coni, uel coni portionis grauitatis centrum eſſe pun<lb></lb>
ctum e. </s>
          <s id="s.000625">quæ omnia demonſtrare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000626"><margin.target id="marg71"></margin.target>8 huius</s>
        </p>
        <p type="head">
          <s id="s.000627">THEOREMA XIX. PROPOSITIO XXIII.</s>
        </p>
        <p type="main">
          <s id="s.000628">QVODLIBET fruſtum à pyramide, quæ <lb></lb>
triangularem baſim habeat, abſciſſum, diuiditur <lb></lb>
in tres pyramides proportionales, in ea proportio <lb></lb>
ne, quæ eſt lateris maioris baſis ad latus minoris <lb></lb>
ipſi reſpondens.</s>
        </p>
        <p type="main">
          <s id="s.000629">Hoc demonſtrauit Leonardus Piſanus in libro, qui de­<lb></lb>
praxi geometriæ inſcribitur. </s>
          <s id="s.000630">Sed quoniam is adhuc im­<lb></lb>
preſſus non eſt, nos ipſius demonſtrationem breuiter <lb></lb>
perſtringemus, rem ipſam ſecuti, non uerba. </s>
          <s id="s.000631">Sit fru­<lb></lb>
ſtum pyramidis abcdef, cuius maior baſis triangulum <lb></lb>
abc, minor def: &amp; iunctis ae, cc, cd, per, line­<lb></lb>
as ae, ec ducatur planum ſecans fruſtum: itemque per <lb></lb>
lineas ec, cd; &amp; per cd, da alia plana ducantur, quæ <lb></lb>
diuident fruſtum in trcs pyramides abce, adce, defc. 



<pb xlink:href="023/01/068.jpg"></pb>Dico eas proportionales eſſe in proportione, quæ eſt la­<lb></lb>
teris ab adlatus de, ita ut earum maior ſit abce, me­<lb></lb>
dia adce, &amp; minor defc. </s>
          <s id="s.000632">Quoniam enim lineæ de, <lb></lb>
ab æquidiſtant; &amp; inter ipſas ſunt triangula abe, ade; <lb></lb>
<arrow.to.target n="marg72"></arrow.to.target><lb></lb>
<figure id="id.023.01.068.1.jpg" xlink:href="023/01/068/1.jpg"></figure><lb></lb>
erit triangulum abe <lb></lb>
ad triangulum abe, <lb></lb>
ut linea ab ad lineam <lb></lb>
de. </s>
          <s id="s.000633">ut autem triangu<lb></lb>
lum abe ad triangu­<lb></lb>
<arrow.to.target n="marg73"></arrow.to.target><lb></lb>
lum abe, ita pyramis <lb></lb>
abec ad pyramidem <lb></lb>
adec: habent enim <lb></lb>
altitudinem eandem, <lb></lb>
quæ eſtà puncto cad <lb></lb>
planum, in quo qua­<lb></lb>
<arrow.to.target n="marg74"></arrow.to.target><lb></lb>
drilaterum abed. </s>
          <s id="s.000634">er­<lb></lb>
go ut ab ad de, ita pyramis abec ad pyramidem adec. </s>
          <lb></lb>
          <s id="s.000635">Rurſus quoniam æquidiſtantes ſunt ac, df; erit eadem <lb></lb>
<arrow.to.target n="marg75"></arrow.to.target><lb></lb>
ratione pyramis adce ad pyramidem cdfe, ut ac ad <lb></lb>
df. </s>
          <s id="s.000636">Sed ut ac ad df, ita ab ad de, quoniam triangula <lb></lb>
abc, def ſimilia ſunt, ex nona huius. </s>
          <s id="s.000637">quare ut pyramis <lb></lb>
abce ad pyramidem abce, ita pyramis adce ad ipſam<lb></lb>
defc. fruſtum igitur abcdef diuiditur in tres pyramides <lb></lb>
proportionales in ea proportione, quæ eſt lateris ab ad de <lb></lb>
latus, &amp; earum maior eſt cabe, media adce, &amp; minor <lb></lb>
defc. quod demonſtrare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000638"><margin.target id="marg72"></margin.target>1. ſexti.</s>
        </p>
        <p type="margin">
          <s id="s.000639"><margin.target id="marg73"></margin.target>5. duodeci <lb></lb>
mi.</s>
        </p>
        <p type="margin">
          <s id="s.000640"><margin.target id="marg74"></margin.target>11. quinti.</s>
        </p>
        <p type="margin">
          <s id="s.000641"><margin.target id="marg75"></margin.target>4 ſexti.</s>
        </p>
        <p type="head">
          <s id="s.000642">PROBLEMA V. PROPOSITIO XXIIII.</s>
        </p>
        <p type="main">
          <s id="s.000643">QVODLIBET fruſtum pyramidis, uel coni, <lb></lb>
uel coni portionis, plano baſi æquidiſtanti ita ſe­<lb></lb>
care, ut ſectio ſit proportionalis inter maiorem, <lb></lb>
&amp; minorem baſim.</s>
        </p>
        <pb pagenum="31" xlink:href="023/01/069.jpg"></pb>
        <p type="main">
          <s id="s.000644">SIT fruſtum pyramidis ae, cuius maior baſis triangu­<lb></lb>
lum abc, minor def: &amp; oporteat ipſum plano, quod baſi <lb></lb>
æquidiſtet, ita ſecare, ut ſectio ſit proportionalis inter <expan abbr="triã">trian</expan> <lb></lb>
gula abc, def. </s>
          <s id="s.000645">Inueniatur inter lineas ab, de media pro­<lb></lb>
portionalis, quæ ſit bg: &amp; à puncto g erigatur gh æquidi­<lb></lb>
ſtans be, <expan abbr="ſecansq;">ſecansque</expan> ad in h: deinde per h ducatur planum <lb></lb>
baſibus æquidiſtans, cuius ſectio ſit triangulum hkl. </s>
          <s id="s.000646">Dico <lb></lb>
triangulum hKl proportionale eſſe inter triangula abc, <lb></lb>
<figure id="id.023.01.069.1.jpg" xlink:href="023/01/069/1.jpg"></figure><lb></lb>
def, hoc eſt triangulum abc ad <lb></lb>
triangulum hKl eandem habere <lb></lb>
proportionem, quam <expan abbr="triãgulum">triangulum</expan> <lb></lb>
hKl ad ipſum def. </s>
          <s id="s.000647"><expan abbr="Quoniã">Quoniam</expan> enim <lb></lb>
<arrow.to.target n="marg76"></arrow.to.target><lb></lb>
lineæ ab, hK æquidiſtantium pla<lb></lb>
norum ſectiones inter ſe æquidi­<lb></lb>
ſtant: atque æquidiſtant bk, gh: <lb></lb>
<arrow.to.target n="marg77"></arrow.to.target><lb></lb>
linea hk ipſi gb eſt æqualis: &amp; pro<lb></lb>
pterea proportionalis inter ab, <lb></lb>
de. </s>
          <s id="s.000648">quare ut ab ad hK, ita eſt hk<lb></lb>
ad de. </s>
          <s id="s.000649">fiat ut hk ad de, ita de <lb></lb>
ad aliam lineam, in qua ſit m. </s>
          <s id="s.000650">erit <lb></lb>
ex æquali ut ab ad de, ita hk ad <lb></lb>
<arrow.to.target n="marg78"></arrow.to.target><lb></lb>
m. </s>
          <s id="s.000651">Et quoniam triangula abc, <lb></lb>
hKl, def ſimilia ſunt; <expan abbr="triangulũ">triangulum</expan> <lb></lb>
<arrow.to.target n="marg79"></arrow.to.target><lb></lb>
abc ad triangulum hkl eſt, ut li­<lb></lb>
nea ab ad lineam de: <expan abbr="triangulũ">triangulum</expan> <lb></lb>
<arrow.to.target n="marg80"></arrow.to.target><lb></lb>
autem hkl ad ipſum def eſt, ut hk ad m. </s>
          <s id="s.000652">ergo triangulum <lb></lb>
abc ad triangulum hkl eandem proportionem habet, <lb></lb>
quam triangulum hKl ad ipſum def. </s>
          <s id="s.000653">Eodem modo in a­<lb></lb>
liis fruſtis pyramidis idem demonſtrabitur.</s>
        </p>
        <p type="margin">
          <s id="s.000654"><margin.target id="marg76"></margin.target>16. unde<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000655"><margin.target id="marg77"></margin.target>34. primi</s>
        </p>
        <p type="margin">
          <s id="s.000656"><margin.target id="marg78"></margin.target>9. huius <lb></lb>
corol.</s>
        </p>
        <p type="margin">
          <s id="s.000657"><margin.target id="marg79"></margin.target>20. ſexti</s>
        </p>
        <p type="margin">
          <s id="s.000658"><margin.target id="marg80"></margin.target>11. quinti</s>
        </p>
        <p type="main">
          <s id="s.000659">Sit fruſtum coni, uel coni portionis ad: &amp; ſecetur plano <lb></lb>
per axem, cuius ſectio ſit abcd, ita ut maior ipſius baſis ſit <lb></lb>
circulus, uel ellipſis circa diametrum ab; minor circa cd. </s>
          <lb></lb>
          <s id="s.000660">Rurſus inter lineas ab, cd inueniatur proportionalis be: <lb></lb>
&amp; ab e ducta ef æquidiſtante bd, quæ lineam ca in f ſecet, 



<pb xlink:href="023/01/070.jpg"></pb>per f planum baſibus æquidiſtans ducatur, ut ſit ſectio cir<lb></lb>
culus, uel ellipſis circa diametrum fg. </s>
          <s id="s.000661">Dico ſectionem ab <lb></lb>
ad ſectionem fg eandem proportionem habere, quam fg <lb></lb>
ad ipſam cd. </s>
          <s id="s.000662">Simili enim ratione, qua ſupra, demonſtrabi­<lb></lb>
tur quadratum ab ad quadratum fg ita eſſe, ut <expan abbr="quadratũ">quadratum</expan> <lb></lb>
<arrow.to.target n="marg81"></arrow.to.target><lb></lb>
fg ad cd quadratum. </s>
          <s id="s.000663">Sed circuli inter ſe eandem propor­<lb></lb>
tionem habent, quam diametrorum quadrata. </s>
          <s id="s.000664">ellipſes au­<lb></lb>
tem circa ab, fg, cd, quæ ſimiles ſunt, ut oſtendimus in <expan abbr="cõ-mentariis">com­<lb></lb>
mentariis</expan> in principium libri Archimedis de conoidibus, <lb></lb>
&amp; ſphæroidibus, eam <expan abbr="habẽt">habent</expan> proportionem, quam quadra<lb></lb>ta diametrorum, quæ eiuſdem rationis ſunt, ex corollario­ <lb></lb>
<figure id="id.023.01.070.1.jpg" xlink:href="023/01/070/1.jpg"></figure><lb></lb>
ſeptimæ propoſitionis eiuſdem li­<lb></lb>
bri. </s>
          <s id="s.000665">ellipſes enim nunc appello ip­<lb></lb>
ſa ſpacia ellipſibus contenta. </s>
          <s id="s.000666">ergo <lb></lb>
circulus, uel ellipſis ab ad <expan abbr="circulũ">circulum</expan>, <lb></lb>
uel ellipſim fg eam proportionem <lb></lb>
habet, quam circulus, uel ellipſis <lb></lb>
fg ad circulum uel ellipſim cd. </s>
          <lb></lb>
          <s id="s.000667">quod quidem faciendum propo­<lb></lb>
ſuimus.</s>
        </p>
        <p type="margin">
          <s id="s.000668"><margin.target id="marg81"></margin.target>2. duode<lb></lb>
cimi</s>
        </p>
        <p type="head">
          <s id="s.000669">THEOREMA XX. PROPOSITIO XXV.</s>
        </p>
        <p type="main">
          <s id="s.000670">QVODLIBET fruſtum pyramidis, uel coni, <lb></lb>
uel coni portionis ad pyramidem, uel conum, uel <lb></lb>
coni portionem, cuius baſis eadem eſt, &amp; æqualis <lb></lb>
altitudo, eandem <expan abbr="proportionẽ">proportionem</expan> habet, quam utræ <lb></lb>
que baſes, maior, &amp; minor ſimul ſumptæ vnà <expan abbr="cũ">cum</expan> <lb></lb>ea, quæ inter ipſas ſit proportionalis, ad baſim ma<lb></lb>
iorem.</s>
        </p>
        <pb pagenum="32" xlink:href="023/01/071.jpg"></pb>
        <p type="main">
          <s id="s.000671">SIT <expan abbr="fruſtũ">fruſtum</expan> pyramidis, uel coni, uel coni portionis ad, <lb></lb>
cuius maior baſis ab, minor cd. </s>
          <s id="s.000672">&amp; ſecetur altero plano <lb></lb>
baſi æquidiſtante, ita ut ſectio ef ſit proportionalis inter <lb></lb>
baſes ab, cd. </s>
          <s id="s.000673">conſtituatur <expan abbr="autẽ">autem</expan> pyramis, uel conus, uel co­<lb></lb>
ni portio agb, cuius baſis ſit eadem, quæ baſis maior fru­<lb></lb>
<figure id="id.023.01.071.1.jpg" xlink:href="023/01/071/1.jpg"></figure><lb></lb>
ſti, &amp; altitudo æqualis. </s>
          <s id="s.000674">Di­<lb></lb>
co fruſtum ad ad pyrami­<lb></lb>
dem, uel conum, uel coni <lb></lb>
portionem agb eandem <lb></lb>
<expan abbr="proportionẽ">proportionem</expan> habere, <expan abbr="quã">quam</expan> <lb></lb>
utræque baſes, ab, cd unà <lb></lb>
cum ef ad baſim ab. </s>
          <s id="s.000675">eſt <lb></lb>
enim fruſtum ad æquale <lb></lb>
pyramidi, uel cono, uel co­<lb></lb>
ni portioni, cuius baſis ex <lb></lb>
tribus baſibus ab, ef, cd <lb></lb>
conſtat; &amp; altitudo ipſius <lb></lb>
altitudini eſt æqualis: quod mox oſtendemus. </s>
          <s id="s.000676">Sed pyrami<lb></lb>
<figure id="id.023.01.071.2.jpg" xlink:href="023/01/071/2.jpg"></figure><lb></lb>
des, coni, uel coni <expan abbr="portiões">portiones</expan>, <lb></lb>
quæ ſunt æquali altitudine, <lb></lb>
<expan abbr="eãdem">eandem</expan> inter ſe, quam baſes, <lb></lb>
proportionem habent, ſicu­<lb></lb>
ti demonſtratum eſt, partim <lb></lb>
<arrow.to.target n="marg82"></arrow.to.target><lb></lb>
ab Euclide in duodecimo li­<lb></lb>
bro elementorum, partim à <lb></lb>
nobis in <expan abbr="cõmentariis">commentariis</expan> in un­<lb></lb>
decimam <expan abbr="propoſitionẽ">propoſitionem</expan> Ar­<lb></lb>
chimedis de conoidibus, &amp; <lb></lb>
ſphæroidibus. </s>
          <s id="s.000677">quare pyra­<lb></lb>
mis, uel conus, uel coni por­<lb></lb>
tio, cuius baſis eſt tribus illis <lb></lb>
baſibus æqualis ad agb eam <lb></lb>
habet proportionem, quam <lb></lb>
baſes ab, ef, cd ad ab baſim. </s>
          <s id="s.000678">Fruſtum igitur ad ad agb 



<pb xlink:href="023/01/072.jpg"></pb>pyramidem, uel conum, uel coni portionem eandem pro­<lb></lb>
portionem habet, quam baſes ab, cd unà cum ef ad ba­<lb></lb>
ſim ab. </s>
          <s id="s.000679">quod demonſtrare uolebamus.</s>
        </p>
        <p type="margin">
          <s id="s.000680"><margin.target id="marg82"></margin.target>6. 11. duo<lb></lb>
decimi</s>
        </p>
        <p type="main">
          <s id="s.000681">Fruſtum uero ad æquale eſſe pyramidi, uel co<lb></lb>
no, uel coni portioni, cuius baſis conſtat ex baſi­<lb></lb>
bus ab, cd, ef, &amp; altitudo fruſti altitudini eſt æ­<lb></lb>
qualis, hoc modo oſtendemus.</s>
        </p>
        <p type="main">
          <s id="s.000682">Sit fruſtum pyramidis abcdef, cuius maior baſis trian­<lb></lb>
gulum abc; minor def: &amp; ſecetur plano baſibus æquidi­<lb></lb>
ſtante, quod ſectionem faciat triangulum ghk inter trian­<lb></lb>
gula abc, def proportionale. </s>
          <s id="s.000683">Iam ex iis, quæ demonſtrata <lb></lb>
ſunt in 23. huius, patet fruſtum abcdef diuidi in tres pyra<lb></lb>
mides proportionales; &amp; earum maiorem eſſe <expan abbr="pyramidẽ">pyramidem</expan> <lb></lb>
abcd <expan abbr="minorẽ">minorem</expan> uero defb. </s>
          <s id="s.000684">ergo pyramis à triangulo ghk <lb></lb>
conſtituta, quæ altitudinem habeat fruſti altitudini æqua­<lb></lb>
lem, proportionalis eſt inter pyramides abcd, defb: &amp; <lb></lb>
idcirco fruſtum abcdef tribus dictis pyramidibus æqua <lb></lb>
<figure id="id.023.01.072.1.jpg" xlink:href="023/01/072/1.jpg"></figure><lb></lb>
le erit. </s>
          <s id="s.000685">Itaque ſi intelligatur alia pyra­<lb></lb>
mis æque alta, quæ baſim habeat ex tri<lb></lb>
bus baſibus abc, def, ghk conſtan­<lb></lb>
tem; perſpicuum eſt ipſam eiſdem py­<lb></lb>
ramidibus, &amp; propterea ipſi fruſto æ­<lb></lb>
qualem eſſe.</s>
        </p>
        <p type="main">
          <s id="s.000686">Rurſus ſit fruſtum pyramidis ag, cu<lb></lb>
ius maior baſis quadrilaterum abcd, <lb></lb>
minor efgh: &amp; ſecetur plano baſi­<lb></lb>
bus æquidiſtante, ita ut fiat ſectio qua­<lb></lb>
drilaterum Klmn, quod ſit proportio <lb></lb>
nale inter quadrilatera abcd, efgh. </s>
          <s id="s.000687">Dico pyramidem, <lb></lb>
cuius baſis ſit æqualis tribus quadrilateris abcd, klmn, <lb></lb>
efgh, &amp; altitudo æqualis altitudini fruſti, ipſi fruſto ag <lb></lb>
æqualem eſſe. </s>
          <s id="s.000688">Ducatur enim planum per lineas fb, hd, 



<pb pagenum="33" xlink:href="023/01/073.jpg"></pb>quod diuidat fruſtum in duo fruſta triangulares baſes ha­<lb></lb>
bentia, uidelicet in fruſtum abdefh, &amp; in <expan abbr="fruſtũ">fruſtum</expan> bcdfgh. </s>
          <lb></lb>
          <s id="s.000689">erit triangulum kln proportionale inter triangula abd, <lb></lb>
efh: &amp; triangulum lmn proportionale inter bcd, fgh. </s>
          <lb></lb>
          <s id="s.000690">ſed pyramis æque alta, cuius baſis conſtat ex tribus trian­<lb></lb>
<figure id="id.023.01.073.1.jpg" xlink:href="023/01/073/1.jpg"></figure><lb></lb>
gulis abd, klz, efh, demonſtrata <lb></lb>
eſt fruſto abdcfh æqualis. </s>
          <s id="s.000691">&amp; ſi­<lb></lb>
militer pyramis, cuius baſis con­<lb></lb>
ſtat ex triangulis bcd, lmn, fgh <lb></lb>
æqualis fruſto bcdfgh: compo­<lb></lb>
nuntur autem tria quadrilatera a <lb></lb>
bcd, klmn, efgh è ſex triangu­<lb></lb>
lis iam dictis. </s>
          <s id="s.000692">pyramis igitur ba­<lb></lb>
ſim habens æqualem tribus qua­<lb></lb>
drilateris, &amp; altitudinem eandem <lb></lb>
ipſi fruſto ag eſt æqualis. </s>
          <s id="s.000693">Eodem <lb></lb>
modo illud <expan abbr="demõſtrabitur">demonſtrabitur</expan> in aliis <lb></lb>
eiuſmodi fruſtis.</s>
        </p>
        <p type="main">
          <s id="s.000694">Sit fruſtum coni, uel coni portionis ad; cuius maior ba­<lb></lb>
ſis circulus, uel ellipſis circa diametrum ab; minor circa <lb></lb>
c d: &amp; ſecetur plano, quod baſibus æquidiſtet, <expan abbr="faciatq;">faciatque</expan> ſe­<lb></lb>
ctionem circulum, uel ellipſim circa diametrum ef, ita ut <lb></lb>
inter circulos, uel ellipſes ab, cd ſit proportionalis. </s>
          <s id="s.000695">Dico <lb></lb>
conum, uel coni portionem, cuius baſis eſt æqualis tribus <lb></lb>
circulis, uel tribus ellipſibus ab, ef, cd; &amp; altitudo eadem, <lb></lb>
quæ fruſti ad, ipſi fruſto æqualem eſſe. </s>
          <s id="s.000696">producatur enim <lb></lb>
fruſti ſuperficies quouſque coeat in unum punctum, quod <lb></lb>
ſit g: &amp; coni, uel coni portionis agb axis ſit gh, occurrens <lb></lb>
planis ab, ef, cd in punctis hkl: circa circulum uero de­<lb></lb>
ſcribatur quadratum mnop, &amp; circa ellipſim <expan abbr="rectangulũ">rectangulum</expan> <lb></lb>
mnop, quod ex ipſius diametris conſtat: <expan abbr="iunctisq;">iunctisque</expan> gm, <lb></lb>
g n, go, gp, ex eodem uertice intelligatur pyramis baſim <lb></lb>
habens dictum quadratum, uel rectangulum: &amp; plana in <lb></lb>
quibus ſunt circuli, uel ellipſes ef, cd uſque ad eius latera 



<pb xlink:href="023/01/074.jpg"></pb>producantur. </s>
          <s id="s.000697">Quoniam igitur pyramis ſecatur planis baſi <lb></lb>
<arrow.to.target n="marg83"></arrow.to.target><lb></lb>
æquidiſtantibus, ſectiones ſimiles erunt: atque erunt qua­<lb></lb>
drata, uel rectangula circa circulos, uel ellipſes deſcripta, <lb></lb>
quemadmodum &amp; in ipſa baſi. </s>
          <s id="s.000698">Sed cum circuli inter ſe <expan abbr="eã">eam</expan> <lb></lb>
<arrow.to.target n="marg84"></arrow.to.target><lb></lb>
proportionem habeant, quam diametrorum quadrata: <lb></lb>
<expan abbr="itemq;">itemque</expan> ellipſes eam quam rectangula ex ipſarum diametris <lb></lb>
<arrow.to.target n="marg85"></arrow.to.target><lb></lb>
conſtantia: &amp; ſit circulus, uel ellipſis circa diametrum ef <lb></lb>
<figure id="id.023.01.074.1.jpg" xlink:href="023/01/074/1.jpg"></figure><lb></lb>
proportionalis inter circulos, uel ellipſes ab, cd; erit re­<lb></lb>
ctangulum ef etiam inter rectangula ab, cd proportio­<lb></lb>
nale: per rectangulum enim nunc breuitatis cauſa <expan abbr="etiã">etiam</expan> ip­<lb></lb>
ſum quadratum intelligemus. </s>
          <s id="s.000699">quare ex iis, quæ proxime <lb></lb>
dicta ſunt, pyramis baſim habens æqualem dictis rectangu<lb></lb>
lis, &amp; altitudinem eandem, quam fruſtum ad, ipſi fruſto à <lb></lb>
pyramide abſciſſo æqualis probabitur. </s>
          <s id="s.000700">ut autem rectangu<lb></lb>
lum cd ad <expan abbr="rectangulũ">rectangulum</expan> ef, ita circulus, uel ellipſis cd ad ef <lb></lb>
circulum, uel ellipſim: <expan abbr="componendoq;">componendoque</expan> ut rectangula cd, <lb></lb>
e f, ad ef rectangulum, ita circuli, uel ellipſes ed, ef, ad ef: <lb></lb>
&amp; ut rectangulum ef ad rectangulum ab, ita circulus, uel <lb></lb>
ellipſis ef ad ab circulum, uel ellipſim. </s>
          <s id="s.000701">ergo ex æquali, &amp; <lb></lb>
componendo, ut <expan abbr="rectãgula">rectangula</expan> cd, ef, ab ad ipſum ab, ita cir­



<pb pagenum="34" xlink:href="023/01/075.jpg"></pb>culi, uel ellipſes cd, ef ab ad circulum, uel ellipſim ab. </s>
          <s id="s.000702">In­<lb></lb>
telligatur pyramis q baſim habens æqualem tribus rectan <lb></lb>
gulis ab, ef, cd; &amp; altitudinem <expan abbr="eãdem">eandem</expan>, quam fruſtum ad. </s>
          <lb></lb>
          <s id="s.000703">intelligatur etiam conus, uel coni portio q, eadem altitudi<lb></lb>
ne, cuius baſis ſit tribus circulis, uel tribus ellipſibus ab, <lb></lb>
ef, cd æqualis. </s>
          <s id="s.000704">poſtremo intelligatur pyramis alb, cuius. </s>
          <lb></lb>
          <s id="s.000705">baſis ſit rectangulum mnop, &amp; altitudo eadem, quæ fru­<lb></lb>
ſti: <expan abbr="itemq,">itemque</expan> intelligatur conus, uel coni portio alb, cuius <lb></lb>
baſis circulus, uel ellipſis circa diametrum ab, &amp; eadem al<lb></lb>
<arrow.to.target n="marg86"></arrow.to.target><lb></lb>
titudo. </s>
          <s id="s.000706">ut igitur rectangula ab, ef, cd ad rectangulum ab, <lb></lb>
ita pyramis q ad pyramidem alb; &amp; ut circuli, uel ellip­<lb></lb>
ſes ab, ef, cd ad ab circulum, uel ellipſim, ita conus, uel co<lb></lb>
ni portio q ad conum, uel coni portionem alb. </s>
          <s id="s.000707">conus <lb></lb>
igitur, uel coni portio q ad conum, uel coni portionem <lb></lb>
alb eſt, ut pyramis q ad pyramidem alb. </s>
          <s id="s.000708">ſed pyramis <lb></lb>
alb ad pyramidem agb eſt, ut altitudo ad altitudinem, ex <lb></lb>
20. huius: &amp; ita eſt conus, uel coni portio alb ad conum, <lb></lb>
uel coni portionem agb ex 14. duodecimi elementorum, <lb></lb>
&amp; ex iis, quæ nos demonſtrauimus in commentariis in un­<lb></lb>
decimam de conoidibus, &amp; ſphæroidibus, propoſitione <lb></lb>
quarta. </s>
          <s id="s.000709">pyramis autem agb ad pyramidem cgd propor­<lb></lb>
tionem habet compoſitam ex proportione baſium &amp; pro <lb></lb>
portione altitudinum, ex uigeſima prima huius: &amp; ſimili­<lb></lb>
ter conus, uel coni portio agb ad conum, uel coni portio­<lb></lb>
nem cgd proportionem habet <expan abbr="compoſitã">compoſitam</expan> ex eiſdem pro­<lb></lb>
portionibus, per ea, quæ in dictis commentariis demon­<lb></lb>
ſtrauimus, propoſitione quinta, &amp; ſexta: altitudo enim in<lb></lb>
utriſque eadem eſt, &amp; baſes inter ſe ſe eandem habent pro­<lb></lb>
portionem. </s>
          <s id="s.000710">ergo ut pyramis agb ad pyramidem cgd, ita <lb></lb>
eſt conus, uel coni portio agb ad agd conum, uel coni <lb></lb>
portionem: &amp; per <expan abbr="conuerſionẽ">conuerſionem</expan> rationis, ut pyramis agb <lb></lb>
ad <expan abbr="ſruſtũ">fruſtum</expan> à pyramide abſciſſum, ita conus uel coni portio <lb></lb>
agb ad fruſtum ad. </s>
          <s id="s.000711">ex æquali igitur, ut pyramis q ad fru­<lb></lb>
ſtum à pyramide abſciſſum, ita conus uel coni portio q ad 



<pb xlink:href="023/01/076.jpg"></pb>fruſtum ad. </s>
          <s id="s.000712">Sed pyramis q æqualis eſt fruſto à pyramide <lb></lb>
abſciſſo, ut demonſtrauimus. </s>
          <s id="s.000713">ergo &amp; conus, uel coni por­<lb></lb>
tio q, cuius baſis ex tribus circulis, uel ellipſibus ab, ef, cd <lb></lb>
conſtat, &amp; altitudo eadem, quæ fruſti: ipſi fruſto ad eſt æ­<lb></lb>
qualis. </s>
          <s id="s.000714">atque illud eſt, quod demonſtrare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000715"><margin.target id="marg83"></margin.target>9 huius</s>
        </p>
        <p type="margin">
          <s id="s.000716"><margin.target id="marg84"></margin.target>2. duode­<lb></lb>cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000717"><margin.target id="marg85"></margin.target>7. de co­<lb></lb>
noidibus <lb></lb>
&amp; ſphæ­<lb></lb>
roidibus</s>
        </p>
        <p type="margin">
          <s id="s.000718"><margin.target id="marg86"></margin.target>6. II. duo <lb></lb>
decimi</s>
        </p>
        <p type="head">
          <s id="s.000719">THEOREMA XXI. PROPOSITIO XXVI.</s>
        </p>
        <p type="main">
          <s id="s.000720">CVIVSLIBET fruſti à pyramide, uel cono, <lb></lb>
uel coni portione abſcisſi, centrum grauitatis eſt <lb></lb>
in axe, ita ut eo primum in duas portiones diui­<lb></lb>
ſo, portio ſuperior, quæ minorem baſim attingit <lb></lb>
ad portionem reliquam eam habeat proportio­<lb></lb>
nem, quam duplum lateris, uel diametri maioris <lb></lb>
baſis, vnà cum latere, uel diametro minoris, ipſi <lb></lb>
reſpondente, habet ad duplum lateris, uel diame­<lb></lb>
tri minoris baſis vnà <expan abbr="cũ">cum</expan> latere, uel diametro ma­<lb></lb>
ioris: deinde à puncto diuiſionis quarta parte ſu<lb></lb>
perioris portionis in ipſa ſumpta: &amp; rurſus ab in­<lb></lb>
ferioris portionis termino, qui eſt ad baſim maio<lb></lb>
rem, ſumpta quarta parte totius axis: centrum ſit <lb></lb>
in linea, quæ his finibus continetur, atque in eo li<lb></lb><lb></lb>
tem propinquiorem minori baſi, <expan abbr="eãdem">eandem</expan> propor­<lb></lb>
tionem habeat, quam fruſtum ad <expan abbr="pyramidẽ">pyramidem</expan>, uel <lb></lb>
conum, uel coni portionem, cuius baſis ſit ea­<lb></lb>
dem, quæ baſis maior, &amp; altitudo fruſti altitudini <lb></lb>
æqualis.</s>
        </p>
        <pb pagenum="35" xlink:href="023/01/077.jpg"></pb>
        <p type="main">
          <s id="s.000721">Sit fruſtum ae a pyramide, quæ triangularem baſim ha­<lb></lb>
beat abſciſſum: cuius maior baſis triangulum abc, minor <lb></lb>
def; &amp; axis gh. </s>
          <s id="s.000722">ducto autem plano per axem &amp; per <expan abbr="lineã">lineam</expan> <lb></lb>
da, quod ſectionem faciat dakl quadrilaterum; puncta <lb></lb>
Kl lineas bc, ef bifariam ſecabunt. </s>
          <s id="s.000723">nam cum gh ſit axis <lb></lb>
fruſti: erit h centrum grauitatis trianguli abc: &amp; g <lb></lb>
<figure id="id.023.01.077.1.jpg" xlink:href="023/01/077/1.jpg"></figure><lb></lb>
<arrow.to.target n="marg87"></arrow.to.target><lb></lb>
centrum trianguli def: cen­<lb></lb>
trum uero cuiuslibet triangu<lb></lb>
li eſt in recta linea, quæ ab an­<lb></lb>
gulo ipſius ad <expan abbr="dimidiã">dimidiam</expan> baſim <lb></lb>
ducitur ex decimatertia primi <lb></lb>
libri Archimedis de <expan abbr="cẽtro">centro</expan> gra<lb></lb>
<arrow.to.target n="marg88"></arrow.to.target><lb></lb>
uitatis planorum. </s>
          <s id="s.000724">quare <expan abbr="cen-trũ">cen­<lb></lb>
trum</expan> grauitatis trapezii bcfe <lb></lb>
eſt in linea kl, quod ſit m: &amp; à <lb></lb>
puncto m ad axem ducta mn <lb></lb>
ipſi ak, uel dl æquidiſtante; <lb></lb>
erit axis gh diuiſus in portio­<lb></lb>
nes gn, nh, quas diximus: ean <lb></lb>
dem enim proportionem ha­<lb></lb>
bet gn ad nh, <expan abbr="quã">quam</expan> lm ad mk. </s>
          <lb></lb>
          <s id="s.000725">At lm ad mK habet eam, <expan abbr="quã">quam</expan> <lb></lb>
duplum lateris maioris baſis <lb></lb>
bc una cum latere minoris ef <lb></lb>
ad duplum lateris ef unà cum <lb></lb>
latere bc, ex ultima eiuſdem <lb></lb>
libri Archimedis. </s>
          <s id="s.000726">Itaque à li­<lb></lb>
nea ng abſcindatur, quarta <lb></lb>
pars, quæ fit np: &amp; ab axe hg abſcindatur itidem <lb></lb>
quarta pars ho: &amp; quam proportionem habet fruſtum ad <lb></lb>
pyramidem, cuius maior baſis eſt triangulum abc, &amp; alti­<lb></lb>
tudo ipſi æqualis; habeat op ad pq.</s>
          <s id="s.000727"> Dico centrum graui­<lb></lb>
tatis fruſti eſſe in linea po, &amp; in puncto q.</s>
          <s id="s.000728"> namque ipſum <lb></lb>
eſſe in linea gh manifeſte conſtat. </s>
          <s id="s.000729">protractis enim fruſti pla<pb xlink:href="023/01/078.jpg"></pb>nis, quouſque in unum punctum r conueniant; erit pyra­<lb></lb>
midis abcr, &amp; pyramidis defr grauitatis centrum in li­<lb></lb>
nea rh. </s>
          <s id="s.000730">ergo &amp; reliquæ magnitudinis, uidelicet fruſti cen­<lb></lb>
trum in eadem linea neceſſario comperietur. </s>
          <s id="s.000731">Iungantur <lb></lb>
db, dc, dh, dm: &amp; per lineas db, dc ducto altero plano <lb></lb>
intelligatur fruſtum in duas pyramides diuiſum: in pyra­<lb></lb>
midem quidem, cuius baſis eſt triangulum abc, uertex d: <lb></lb>
&amp; in eam, cuius idem uertex, &amp; baſis trapezium bcfe. </s>
          <s id="s.000732">erit <lb></lb>
igitur pyramidis abcd axis dh, &amp; pyramidis bcfed axis <lb></lb>
d m: atque erunt tres axes gh, dh, dm in eodem plano <lb></lb>
daKl.</s>
          <s id="s.000733"> ducatur præterea per o linea ſt ipſi aK <expan abbr="æquidiſtãs">æquidiſtans</expan>, <lb></lb>
quæ lineam dh in u ſecet: per p uero ducatur xy æquidi­<lb></lb>
<figure id="id.023.01.078.1.jpg" xlink:href="023/01/078/1.jpg"></figure><lb></lb>
ſtans eidem, ſecansque dm in <lb></lb>
z: &amp; iungatur zu, quæ ſecet <lb></lb>
gh in <foreign lang="grc">φ.</foreign> tranſibit ea per q: &amp; <lb></lb>
erunt <foreign lang="grc">φ</foreign>q unum, atque idem <lb></lb>
punctum; ut inferius appare­<lb></lb>
bit. </s>
          <s id="s.000734">Quoniam igitur linea uo <lb></lb>
<arrow.to.target n="marg89"></arrow.to.target><lb></lb>
æquidiſtat ipſi dg, erit du ad <lb></lb>
uh, ut go ad oh. </s>
          <s id="s.000735">Sed go tri­<lb></lb>
pla eſt oh. </s>
          <s id="s.000736">quare &amp; du ipſius <lb></lb>
uh eſt tripla: &amp; ideo pyrami­<lb></lb>
dis abcd centrum grauitatis <lb></lb>
erit punctum u. </s>
          <s id="s.000737">Rurſus quo­<lb></lb>
niam zy ipſi dl æquidiſtat, dz <lb></lb>
ad zm eſt, ut ly ad ym: eſtque <lb></lb>
ly ad ym, ut gp ad pn. </s>
          <s id="s.000738">ergo <lb></lb>
dz ad zm eſt, ut gp ad pn. </s>
          <lb></lb>
          <s id="s.000739">Quòd cum gp ſit tripla pn; <lb></lb>
erit etiam dz ipſius zm tri­<lb></lb>
pla. </s>
          <s id="s.000740">atque ob eandem cauſ­<lb></lb>
ſam punctum z eſt <expan abbr="centrũ">centrum</expan> gra­<lb></lb>
uitatis pyramidis bcfed. </s>
          <s id="s.000741">iun<lb></lb>
cta igitur zu, in ea erit <expan abbr="cẽtrum">centrum</expan> 



<pb pagenum="36" xlink:href="023/01/079.jpg"></pb>grauitatis magnitudinis, quæ ex utriſque pyramidibus <expan abbr="cõ">con</expan><lb></lb>
ſtat; hoc eſt ipſius fruſti. </s>
          <s id="s.000742">Sed fruſti centrum eſt etiam in a­<lb></lb>
xe gh. </s>
          <s id="s.000743">ergo in puncto <foreign lang="grc">φ,</foreign> in quo lineæ zu, gh conueniunt. </s>
          <lb></lb>
          <s id="s.000744"><arrow.to.target n="marg90"></arrow.to.target><lb></lb>
Itaque u<foreign lang="grc">φ</foreign> ad <foreign lang="grc">φ</foreign>z eam proportionem habet, quam pyramis <lb></lb>
bcfed ad pyramidem abcd. </s>
          <s id="s.000745">&amp; componendo uz ad z<foreign lang="grc">φ</foreign><lb></lb>
eam habet, quam fruſtum ad pyramidem abcd. </s>
          <s id="s.000746">Vt uero <lb></lb>
uz ad z<foreign lang="grc">φ</foreign>, ita op ad p<foreign lang="grc">φ</foreign> ob ſimilitudinem triangulorum, <lb></lb>
uo<foreign lang="grc">φ</foreign>, zp<foreign lang="grc">φ.</foreign> quare op ad p<foreign lang="grc">φ</foreign> eſt ut fruſtum ad pyramidem <lb></lb>
abcd. </s>
          <s id="s.000747">ſed ita erat op ad pq.</s>
          <s id="s.000748"> æquales igitur ſunt p<foreign lang="grc">φ</foreign>, pq: &amp;<lb></lb>
<arrow.to.target n="marg91"></arrow.to.target><lb></lb>
q<foreign lang="grc">φ</foreign> unum atque idem punctum. </s>
          <s id="s.000749">ex quibus ſequitur lineam. </s>
          <lb></lb>
          <s id="s.000750">zu ſecare op in q: &amp; propterea <expan abbr="pũctum">punctum</expan> q ipſius fruſti gra­<lb></lb>
uitatis centrum eſſe.</s>
        </p>
        <p type="margin">
          <s id="s.000751"><margin.target id="marg87"></margin.target>3. diffi. </s>
          <s id="s.000752">hu<lb></lb>
ius.</s>
        </p>
        <p type="margin">
          <s id="s.000753"><margin.target id="marg88"></margin.target>Vltima <expan abbr="e-iuſdẽ">e­<lb></lb>
iuſdem</expan> libri <lb></lb>
Archime­<lb></lb>
dis.</s>
        </p>
        <p type="margin">
          <s id="s.000754"><margin.target id="marg89"></margin.target>2. ſexti.</s>
        </p>
        <p type="margin">
          <s id="s.000755"><margin.target id="marg90"></margin.target>8. primi <lb></lb>
libri Ar­<lb></lb>
chimedis <lb></lb>
de <expan abbr="cẽtro">centro</expan> <lb></lb>
grauta­<lb></lb>
tis plano <lb></lb>
rum</s>
        </p>
        <p type="margin">
          <s id="s.000756"><margin.target id="marg91"></margin.target>7. quinti.</s>
        </p>
        <p type="main">
          <s id="s.000757">Sit fruſtum ag à pyramide, quæ quadrangularem baſim <lb></lb>
habeat abſciſſum, cuius maior baſis abcd, minor efgh, <lb></lb>
&amp; axis kl. diuidatur autem <expan abbr="primũ">primum</expan> kl, ita ut quam propor­<lb></lb>
tionem habet duplum lateris ab unà cum latere ef ad du <lb></lb>
plum lateris ef unà cum ab; habeat km ad ml. </s>
          <s id="s.000758">deinde à <lb></lb>
<expan abbr="pũcto">puncto</expan> m ad k ſumatur quarta pars ipſius mk quæ ſit mn. </s>
          <lb></lb>
          <s id="s.000759">&amp; rurſus ab l ſumatur quarta pars totius axis lk, quæ ſit <lb></lb>
lo. </s>
          <s id="s.000760">poſtremo fiat on ad np, ut fruſtum ag ad <expan abbr="pyramidẽ">pyramidem</expan>, <lb></lb>
cuius baſis ſit eadem, quæ fruſti, &amp; altitudo æqualis. </s>
          <s id="s.000761">Dico <lb></lb>
punctum p fruſti ag grauitatis centrum eſſe. </s>
          <s id="s.000762">ducantur <lb></lb>
enim ac, eg: &amp; intelligantur duo fruſta triangulares ba­<lb></lb>
ſes habentia, quorum alterum lf ex baſibus abc, efg <expan abbr="cõ-ſtet">con­<lb></lb>
ſtet</expan>; alterum lh ex baſibus acd, egh. </s>
          <s id="s.000763"><expan abbr="Sitq;">Sitque</expan> fruſti lf axis <lb></lb>
qr; in quo grauitatis centrum s: fruſti uero lh axis tu, &amp; <lb></lb>
x grauitatis centrum: deinde iungantur ur, tq, xs. </s>
          <s id="s.000764">tranſi­<lb></lb>
bit ur per l: quoniam l eſt centrum grauitatis quadran­<lb></lb>
guli abcd: &amp; puncta ru grauitatis centra triangulorum <lb></lb>
abc, acd; in quæ quadrangulum ipſum diuiditur. </s>
          <s id="s.000765">eadem <lb></lb>
quoque ratione tq per punctum k tranſibit. </s>
          <s id="s.000766">At uero pro<lb></lb>
portiones, ex quibus fruſtorum grauitatis centra inquiri­<lb></lb>
mus, eædem ſunt in toto fruſto ag, &amp; in fruſtis lf, lh. </s>
          <s id="s.000767">Sunt <lb></lb>
enim per octauam huius quadrilatera abcd, efgh ſimilia: 



<pb xlink:href="023/01/080.jpg"></pb><expan abbr="itemq;">itemque</expan> ſimilia triangula abc, efg: &amp; acd, egh. </s>
          <s id="s.000768"><expan abbr="idcir-coq;">idcir­<lb></lb>
coque</expan> latera ſibi ipſis reſpondentia eandem inter ſeſe pro­<lb></lb>
portionem ſeruant. </s>
          <s id="s.000769">Vt igitur duplum lateris ab unà <lb></lb>
cum latere ef ad duplum lateris ef unà cum ab, ita eſt <lb></lb>
<figure id="id.023.01.080.1.jpg" xlink:href="023/01/080/1.jpg"></figure><lb></lb>
duplum ad late­<lb></lb>
ris una cum late­<lb></lb>
re eh ad duplum <lb></lb>
eh unà cum ad: <lb></lb>
&amp; ita in aliis. </s>
          <lb></lb>
          <s id="s.000770">Rurſus fruſtum <lb></lb>
ag ad <expan abbr="pyramidẽ">pyramidem</expan>, <lb></lb>
cuius eadem eſt <lb></lb>
baſis, &amp; æqualis <lb></lb>
altitudo eandem <lb></lb>
<expan abbr="proportionẽ">proportionem</expan> ha<lb></lb>
bet, quam <expan abbr="fruſtũ">fruſtum</expan> <lb></lb>
lf ad <expan abbr="pyramidẽ">pyramidem</expan>, <lb></lb>
quæ eſt <expan abbr="eadẽ">eadem</expan> ba­<lb></lb>
ſi, &amp; æquali alti­<lb></lb>
tudine: &amp; ſimili­<lb></lb>
ter quam lh fru­<lb></lb>
ſtum ad pyrami­<lb></lb>
dem, quæ ex <expan abbr="ea-dẽ">ea­<lb></lb>
dem</expan> baſi, &amp; æquali <lb></lb>
altitudine con­<lb></lb>
ſtat. </s>
          <s id="s.000771">nam ſi inter <lb></lb>
ipſas baſes me­<lb></lb>
diæ proportio­<lb></lb>
nales conſtituan<lb></lb>
tur, tres baſes ſimul ſumptæ ad maiorem baſim in om­<lb></lb>
nibus eodem modo ſe habebunt. </s>
          <s id="s.000772">Vnde fit, ut axes Kl, <lb></lb>
qr, tu à punctis psx in eandem proportionem ſecen­<lb></lb>
<arrow.to.target n="marg92"></arrow.to.target><lb></lb>tur. </s>
          <s id="s.000773">ergo linea xs per p tranſibit: &amp; lineæ ru, sx, qt in­<lb></lb>
ter ſe æquidiſtantes erunt. </s>
          <s id="s.000774">Itaque cum fruſti ag latera pro­<pb pagenum="37" xlink:href="023/01/081.jpg"></pb>ducta ſuerint, ita ut in unum punctum y coeant, erunt <expan abbr="triã">trian</expan><lb></lb>
gula uyl, xyp, tyk inter ſe ſimilia: &amp; ſimilia etiam triangu<lb></lb>
la lyr, pys, kyq quare ut in 19 huius, demonſtrabitur <lb></lb>
xp, ad ps: <expan abbr="itemq;">itemque</expan> tk  ad kq eandem habere <expan abbr="proportionẽ">proportionem</expan>, <lb></lb>
quam ul ad lr. </s>
          <s id="s.000775">Sed ut ul ad lr, ita eſt triangulum abc ad <lb></lb>
triangulum acd: &amp; ut tk ad Kq, ita triangulum efg ad <lb></lb>
triangulum egh. </s>
          <s id="s.000776">Vt autem triangulum abc ad triangu­<lb></lb>
lum acd, ita pyramis abcy ad pyramidem acdy. </s>
          <s id="s.000777">&amp; ut <lb></lb>
triangulum efg ad triangulum egh, ita pyramis efgy <lb></lb>
ad pyramidem eghy; ergo ut pyramis abcy ad <expan abbr="pyramidẽ">pyramidem</expan> <lb></lb>
<arrow.to.target n="marg93"></arrow.to.target><lb></lb>
a cdy, ita pyramis efgy ad pyramidem eghy. </s>
          <s id="s.000778">reliquum <lb></lb>
igitur <expan abbr="fruſtũ">fruſtum</expan> lf ad reliquum <expan abbr="fruſtũ">fruſtum</expan> lh eſt ut pyramis abcy <lb></lb>
ad pyramidem acdy, hoc eſt ut ul ad r, &amp; ut xp ad ps. </s>
          <lb></lb>
          <s id="s.000779">Quòd cum fruſti lf centrum grauitatis ſits: &amp; fruſti lh ſit <lb></lb>
<arrow.to.target n="marg94"></arrow.to.target><lb></lb>
centrum x: conſtat punctum p totius fruſti ag grauitatis <lb></lb>
eſſe centrum. </s>
          <s id="s.000780">Eodem modo fiet demonſtratio etiam in <lb></lb>
aliis pyramidibus.</s>
        </p>
        <p type="margin">
          <s id="s.000781"><margin.target id="marg92"></margin.target>a. </s>
          <s id="s.000782">ſexti.</s>
        </p>
        <p type="margin">
          <s id="s.000783"><margin.target id="marg93"></margin.target>19. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000784"><margin.target id="marg94"></margin.target>8. Archi­<lb></lb>
medis.</s>
        </p>
        <p type="main">
          <s id="s.000785">Sit fruſtum ad à cono, uel coni portione abſciſſum, eu­<lb></lb>
ius maior baſis circulus, uel ellipſis circa diametrum ab; <lb></lb>
minor circa diametrum cd: &amp; axis ef. </s>
          <s id="s.000786">diuidatur <expan abbr="autẽ">autem</expan> ef <lb></lb>
in g, ita ut eg ad gf eandem proportionem habeat, quam <lb></lb>
duplum diametri ab unà cum diametro ed ad duplum cd <lb></lb>
unà cum ab. </s>
          <s id="s.000787"><expan abbr="Sitq;">Sitque</expan> gh quarta pars lineæ ge: &amp; ſit ſ K item <lb></lb>
quarta pars totius fe axis. </s>
          <s id="s.000788">Rurſus quam proportionem <lb></lb>
habet fruſtum ad ad conum, uel coni portionem, in <expan abbr="eadẽ">eadem</expan> <lb></lb>
baſi, &amp; æquali altitudine, habeat linea Kh ad hl. </s>
          <s id="s.000789">Dico pun­<lb></lb>
ctum l fruſti ad grauitatis centrum eſſe. </s>
          <s id="s.000790">Si enim fieri po­<lb></lb>
teſt, ſit m centrum: <expan abbr="producaturq;">producaturque</expan> lm extra fruſtum in n: <lb></lb>
&amp; ut nl ad lm, ita fiat circulus, uel ellipſis circa <expan abbr="diametrũ">diametrum</expan> <lb></lb>
ab ad aliud ſpacium, in quo ſit o. </s>
          <s id="s.000791">Itaque in circulo, uel <lb></lb>
ellipſi circa diametrum ab rectilinea figura plane deſcri­<lb></lb>
batur, ita ut quæ relinquuntur portiones ſint o ſpacio mi­<lb></lb>
nores: &amp; intelligatur pyramis apb, baſim habens rectili­<lb></lb>
neam figuram in circulo, uel ellipſi ab deſcriptam: à qua 



<pb xlink:href="023/01/082.jpg"></pb>fruſtum pyramidis ſit abſciſſum. </s>
          <s id="s.000792">erit ex iis quæ proxime <lb></lb>
tradidimus, fruſti pyramidis ad centrum grauitatis l. </s>
          <s id="s.000793">Quo<lb></lb>
niam igitur portiones ſpacio o minores ſunt; habebit cir <lb></lb>
<figure id="id.023.01.082.1.jpg" xlink:href="023/01/082/1.jpg"></figure><lb></lb>
culus, uel ellipſis ab ad <lb></lb>
portiones dictas <expan abbr="maiorẽ">maiorem</expan> <lb></lb>
proportionem, quàm nl <lb></lb>
ad lm. </s>
          <s id="s.000794">ſed ut circulus, uel <lb></lb>
ellipſis ab ad portiones, <lb></lb>
ita apb conus, uel coni <lb></lb>
portio ad ſolidas portio­<lb></lb>
nes, id quod ſupra demon <lb></lb>
ſtratum eſt: &amp; ut circulus <lb></lb>
<arrow.to.target n="marg95"></arrow.to.target><lb></lb>
uel ellipſis cd ad portio­<lb></lb>
nes, quæ ip ſi inſunt, ita co<lb></lb>
nus, uel coni portio cpd <lb></lb>
ad ſolidas ipſius portio­<lb></lb>
nes. </s>
          <s id="s.000795">Quòd cum figuræ in <lb></lb>
circulis, uel ellipſibus ab <lb></lb>
cd deſcriptæ ſimiles ſint, <lb></lb>
erit proportio circuli, uel <lb></lb>
ellipſis ab ad ſuas portio <lb></lb>
nes, <expan abbr="eadẽ">eadem</expan>, quæ circuli uel <lb></lb>
ellipſis cd ad ſuas. </s>
          <s id="s.000796">ergo <lb></lb>
conus, uel coni portio ap<lb></lb>
b ad portiones ſolidas <expan abbr="eã-dem">ean­<lb></lb>
dem</expan> habet <expan abbr="proportionẽ">proportionem</expan>, <lb></lb>
quam conus, uel coni por<lb></lb>
tio cpd ad ſolidas ipſius <lb></lb>
<arrow.to.target n="marg96"></arrow.to.target><lb></lb>
portiones. </s>
          <s id="s.000797">reliquum igi­<lb></lb>
tur coni, uel coni portionis <expan abbr="fruſtũ">fruſtum</expan>, ſcilicet ad ad reliquas <lb></lb>
portiones ſolidas in ipſo contentas eandem <expan abbr="proportionẽ">proportionem</expan> <lb></lb>
habet, quam conus, uel coni portio apb ad ſolidas portio<lb></lb>nes: hoc eſt eandem, quam circulus, uel ellipſis ab ad por<lb></lb>
tiones planas. </s>
          <s id="s.000798">quare fruſtum coni, uel coni portionis ad 



<pb pagenum="38" xlink:href="023/01/083.jpg"></pb>ad portiones ſolidas maiorem habet <expan abbr="proportionẽ">proportionem</expan>, quàm <lb></lb>
nl ad lm: &amp; diuidendo fruſtum pyramidis ad dictas por­<lb></lb>
tiones maiorem proportionem habet, quàm nm ad ml. </s>
          <lb></lb>
          <s id="s.000799">fiat igitur ut fruſtum pyramidis ad portiones, ita qm ad <lb></lb>
m l. </s>
          <s id="s.000800">Itaque quoniam à fruſto coni, uel coni portionis ad, <lb></lb>
cuius grauitatis centrum eſt m, aufertur fruſtum pyrami­<lb></lb>
dis habens centrum l; erit reliquæ magnitudinis, quæ ex <lb></lb>
portionibus ſolidis conſtat; grauitatis <expan abbr="cẽtrum">centrum</expan> in linea lm <lb></lb>
producta, atque in puncto q, extra figuram poſito: quod <lb></lb>
fieri nullo modo poteſt. </s>
          <s id="s.000801">relinquitur ergo, ut punctum l ſit <lb></lb>
fruſti ad grauitatis centrum. </s>
          <s id="s.000802">quz omnia demonſtranda <lb></lb>
proponebantur.</s>
        </p>
        <p type="margin">
          <s id="s.000803"><margin.target id="marg95"></margin.target>22. huius</s>
        </p>
        <p type="margin">
          <s id="s.000804"><margin.target id="marg96"></margin.target>19. quínti</s>
        </p>
        <p type="head">
          <s id="s.000805">THEOREMA XXII. PROPOSITIO XXVII.</s>
        </p>
        <p type="main">
          <s id="s.000806">OMNIVM ſolidorum in ſphæra deſcripto­<lb></lb>
rum, quæ æqualibus, &amp; ſimilibus baſibus conti­<lb></lb>
nentur, centrum grauitatis eſt idem, quod ſphæ­<lb></lb>
ræ centrum.</s>
        </p>
        <p type="main">
          <s id="s.000807">Solida eiuſmodi corpora regularia appellare ſolent, de <lb></lb>
quibus agitur in tribus ultimis libris elementorum: ſunt <lb></lb>
autem numero quinque, tetrahedrum, uel pyramis, hexa­<lb></lb>
hedrum, uel cubus, octahedrum, dodecahedrum, &amp; icoſa­<lb></lb>
hedrum.</s>
        </p>
        <p type="main">
          <s id="s.000808">Sit primo abcd pyramis <expan abbr="ĩ">im</expan> ſphæra deſcripta, cuius ſphæ<lb></lb>
ræ centrum ſit e. </s>
          <s id="s.000809">Dico e pyramidis abcd grauitatis eſſe <lb></lb>
centrum. </s>
          <s id="s.000810">Si enim iuncta dc producatur ad baſim abc in <lb></lb>
f; ex iis, quæ demonſtrauit Campanus in quartodecimo li<lb></lb>
bro elementorum, propoſitione decima quinta, &amp; decima <lb></lb>
ſeptima, erit f centrum circuli circa triangulum abc de­<lb></lb>
ſcripti: atque erit ef ſexta pars ipſius ſphæræ axis. </s>
          <s id="s.000811">quare <lb></lb>
ex prima huius conſtat trianguli abc grauitatis centrum <lb></lb>
eſſe punctum f: &amp; idcirco lineam df eſſe pyramidis axem. 



<pb xlink:href="023/01/084.jpg"></pb><figure id="id.023.01.084.1.jpg" xlink:href="023/01/084/1.jpg"></figure><lb></lb>
At cum ef ſit ſexta pars axis <lb></lb>
ſphæræ, erit d  tripla ef. </s>
          <s id="s.000812">ergo <lb></lb>
punctum e eſt grauitatis cen­<lb></lb>
trum ipſius pyramidis: quod <lb></lb>
in uigeſima ſecunda huius de­<lb></lb>
monſtratum ſuit. </s>
          <s id="s.000813">Sed e eſt cen<lb></lb>
trum ſphæræ. </s>
          <s id="s.000814">Sequitur igitur, <lb></lb>
ut centrum grauitatis pyrami­<lb></lb>
dis in ſphæra deſcriptæ idem <lb></lb>
ſit, quod ipſius ſphæræ cen­<lb></lb>
trum.</s>
        </p>
        <p type="main">
          <s id="s.000815">Sit cubus in ſphæra deſcriptus ab, &amp; oppoſitorum pla­<lb></lb>
norum lateribus bifariam diuiſis, per puncta diuiſionum <lb></lb>
plana ducantur, ut communis ipſorum ſectio ſit recta li­<lb></lb>
nea cd. </s>
          <s id="s.000816">Itaque ſi ducatur ab, ſolidi ſcilicet diameter, lineæ <lb></lb>
ab, cd ex trigeſimanona undecimi ſeſe bifariam ſecabunt. </s>
          <lb></lb>
          <s id="s.000817"><figure id="id.023.01.084.2.jpg" xlink:href="023/01/084/2.jpg"></figure><lb></lb>
ſecent autem in puncto e. </s>
          <s id="s.000818">erit, <lb></lb>
e <expan abbr="centrũ">centrum</expan> grauitatis ſolidi ab, <lb></lb>
id quod demonſtratum eſt in <lb></lb>
octaua huius. </s>
          <s id="s.000819">Sed quoniam ab <lb></lb>
eſt ſphæræ diametro æqualis, <lb></lb>
ut in decima quinta propoſi­<lb></lb>
tione tertii decimi libri <expan abbr="elemẽ">elemen</expan><lb></lb>
torum oſtenditur: punctum e <lb></lb>
ſphæræ quoque centrum erit. </s>
          <lb></lb>
          <s id="s.000820">Cubi igitur in ſphæra deſcri­<lb></lb>
pti grauitatis centrum idem <lb></lb>
eſt, quod centrum ipſius ſphæræ.</s>
        </p>
        <p type="main">
          <s id="s.000821">Sit octahedrum abcdef, in ſphæra deſcriptum, cuius <lb></lb>
ſphæræ centrum ſit g. </s>
          <s id="s.000822">Dico punctum g ipſius octahedri <lb></lb>
grauitatis centrum eſſe. </s>
          <s id="s.000823">Conſtat enim ex iis, quæ demon­<lb></lb>
ſtrata ſunt à Campano in quinto decimo libro elemento­<lb></lb>
rum, propoſitione ſextadecima eiuſmodi ſolidum diuidi <lb></lb>
in duas pyramides æquales, &amp; ſimiles; uidelicet in pyrami­



<pb pagenum="39" xlink:href="023/01/085.jpg"></pb>dem, cuius baſis eſt quadratum abcd, &amp; altitudo eg: &amp; <lb></lb>
in pyramidem, cuius <expan abbr="eadẽ">eadem</expan> baſis, <expan abbr="altitudoq;">altitudoque</expan> fg; ut ſint eg, <lb></lb>
gf ſemidiametri ſphæræ, &amp; linea una. </s>
          <s id="s.000824"><expan abbr="Cũ">Cum</expan> igitur g ſit ſphæ­<lb></lb>
ræ centrum, erit etiam centrum circuli, qui circa <expan abbr="quadratũ">quadratum</expan> <lb></lb>
abcd deſcribitur: &amp; propterea eiuſdem quadrati grauita<lb></lb>
tis centrum: quod in prima propoſitione huius demon­<lb></lb>
ſtratum eſt. </s>
          <s id="s.000825">quare pyramidis abcde axis erit eg: &amp; pyra<lb></lb>
midis abcdf axis fg. </s>
          <s id="s.000826">Itaque ſit h centrum grauitatis py­<lb></lb>
ramidis abcde, &amp; pyramidis abcdf centrum ſit <emph type="italics"></emph>K:<emph.end type="italics"></emph.end> per­<lb></lb>
ſpicuum eſt ex uigeſima ſecunda propoſitione huius, <expan abbr="lineã">lineam</expan> <lb></lb>
<figure id="id.023.01.085.1.jpg" xlink:href="023/01/085/1.jpg"></figure><lb></lb>
ch triplam eſſe hg: <expan abbr="cõ">com</expan><lb></lb>
<expan abbr="ponendoq;">ponendoque</expan> eg ipſius g <lb></lb>
h quadruplam. </s>
          <s id="s.000827">&amp; <expan abbr="eadẽ">eadem</expan> <lb></lb>
ratione fg <expan abbr="quadruplã">quadruplam</expan> <lb></lb>
ipſius gk quod cum e <lb></lb>
g, gf ſint æquales, &amp; h <lb></lb>
g, g <emph type="italics"></emph>K<emph.end type="italics"></emph.end> neceſſario æqua­<lb></lb>
les erunt. </s>
          <s id="s.000828">ergo ex quar<lb></lb>
ta propoſitione primi <lb></lb>
libri Archimedis de <expan abbr="cẽ-tro">cen­<lb></lb>
tro</expan> grauitatis <expan abbr="planorũ">planorum</expan>, <lb></lb>
totius octahedri, quod <lb></lb>
ex dictis pyramidibus <lb></lb>
conſtat, centrum graui <lb></lb>
tatis erit punctum g idem, quod ipſius ſphæræ centrum.</s>
        </p>
        <p type="main">
          <s id="s.000829">Sit icoſahedrum ad deſcriptum in ſphæra, cuius <expan abbr="centrũ">centrum</expan> <lb></lb>
ſit g. </s>
          <s id="s.000830">Dico g ipſius icoſahedri grauitatis eſſe centrum. </s>
          <s id="s.000831">Si <lb></lb>
enim ab angulo a per g ducatur recta linea uſque ad ſphæ<lb></lb>
ræ ſuperficiem; conſtat ex ſexta decima propoſitione libri <lb></lb>
tertii decimi elementorum, cadere eam in angulum ipſi a <lb></lb>
oppoſitum. </s>
          <s id="s.000832">cadat in d: <expan abbr="ſitq;">ſitque</expan> una aliqua baſis icoſahedri tri­<lb></lb>
angulum abc: &amp; iunctæ bg, producantur, &amp; cadant in <lb></lb>
angulos ef, ipſis bc oppoſitos. </s>
          <s id="s.000833">Itaque per triangula <lb></lb>
abc, def ducantur plana ſphæram ſecantia.</s>
          <s id="s.000834"> erunt hæ ſe-



<pb xlink:href="023/01/086.jpg"></pb>ctiones circuli ex prima propoſitione ſphæricorum Theo<lb></lb>
doſii: unus quidem circa triangulum abc deſcriptus: al­<lb></lb>
ter uero circa def: &amp; quoniam triangula abc, def æqua­<lb></lb>
lia ſunt, &amp; ſimilia; erunt ex prima, &amp; ſecunda propoſitione <lb></lb>
duodecimi libri elementorum, circuli quoque inter ſe ſe <lb></lb>
æquales. </s>
          <s id="s.000835">poſtremo a centro g ad circulum abc perpendi<lb></lb>
cularis ducatur gh; &amp; alia perpendicularis ducatur ad cir<lb></lb>
culum def, quæ ſit gk; &amp; iungantur ah, dk perſpicuum <lb></lb>
eſt ex corollario primæ ſphæricorum Theodoſii, punctum <lb></lb>
h centrum eſſe circuli abc, &amp; k centrum circuli def. </s>
          <s id="s.000836">Quo<lb></lb>
niam igitur triangulorum gah, gdK latus ag eſt æquale la<lb></lb>
teri gd; ſunt enim à centro ſphæræ ad ſuperficiem: atque <lb></lb>
eſt ah æquale dk: &amp; ex ſexta propoſitione libri primi ſphæ<lb></lb>
ricorum Theodoſii gh ipſi gK: triangulum gah æquale <lb></lb>
erit, &amp; ſimile gdk triangulo: &amp; angulus agh æqualis an­<lb></lb>
<arrow.to.target n="marg97"></arrow.to.target><lb></lb>
gulo dg <emph type="italics"></emph>K.<emph.end type="italics"></emph.end> ſed anguli agh, hgd ſunt æquales duobus re­<lb></lb>
ctis. </s>
          <s id="s.000837">ergo &amp; ipſi hgd, dgk duobus rectis æquales erunt. </s>
          <lb></lb>
          <s id="s.000838"><arrow.to.target n="marg98"></arrow.to.target><lb></lb>
&amp; idcirco hg, g <emph type="italics"></emph>K<emph.end type="italics"></emph.end> una, atque eadem erit linea. </s>
          <s id="s.000839">cum autem <lb></lb>
<figure id="id.023.01.086.1.jpg" xlink:href="023/01/086/1.jpg"></figure><lb></lb>
h ſit <expan abbr="centrũ">centrum</expan> circuli, &amp; tri­<lb></lb>
anguli abc grauitatis cen<lb></lb>
<expan abbr="trũ">trum</expan> probabitur ex iis, quæ <lb></lb>
in prima propoſitione hu<lb></lb>
ius tradita ſunt. </s>
          <s id="s.000840">quare gh <lb></lb>
erit pyramidis abcg axis. </s>
          <lb></lb>
          <s id="s.000841">&amp; ob eandem cauſſam gk <lb></lb>
axis pyramidis defg. </s>
          <s id="s.000842">lta­<lb></lb>
que centrum grauitatls py<lb></lb>
ramidis abcg ſit <expan abbr="pũctum">punctum</expan> <lb></lb>
l, &amp; pyramidis defg ſit m. </s>
          <lb></lb>
          <s id="s.000843">Similiter ut ſupra demon­<lb></lb>
ſtrabimus mg, gl inter ſe æquales eſſe, &amp; punctum g graui <lb></lb>
tatis centrum magnitudinis, quæ ex utriſque pyramidibus <lb></lb>
conſtat. </s>
          <s id="s.000844">eodem modo demonſtrabitur, quarumcunque <lb></lb>
duarum pyramidum, quæ opponuntur, grauitatis <expan abbr="centrũ">centrum</expan> 



<pb pagenum="40" xlink:href="023/01/087.jpg"></pb>eſſe punctum g. </s>
          <s id="s.000845">Sequitur ergo ut icoſahedri centrum gra<lb></lb>
uitatis ſit idem, quod ipſius ſphæræ centrum.</s>
        </p>
        <p type="margin">
          <s id="s.000846"><margin.target id="marg97"></margin.target>13. primi</s>
        </p>
        <p type="margin">
          <s id="s.000847"><margin.target id="marg98"></margin.target>14. primi</s>
        </p>
        <p type="main">
          <s id="s.000848">Sit dodecahedrum af in ſphæra deſignatum, ſitque ſphæ<lb></lb>
ræ centrum m. </s>
          <s id="s.000849">Dico m centrum eſſe grauitatis ipſius do­<lb></lb>
decahedri. </s>
          <s id="s.000850">Sit enim pentagonum abcde una ex duode­<lb></lb>
cim baſibus ſolidi af: &amp; iuncta am producatur ad ſphæræ <lb></lb>
ſuperficiem. </s>
          <s id="s.000851">cadet in angulum ipſi a oppoſitum; quod col­<lb></lb>
ligitur ex decima ſeptima propoſitione tertiidecimi libri <lb></lb>
elementorum. </s>
          <s id="s.000852">cadat in f. </s>
          <s id="s.000853">at ſi ab aliis angulis bcde per <expan abbr="cẽ">cen</expan><lb></lb>
trum itidem lineæ ducantur ad ſuperficiem ſphæræ in pun<lb></lb>
cta ghkl; cadent hæ in alios angulos baſis, quæ ipſi abcd <lb></lb>
baſi opponitur. </s>
          <s id="s.000854">tranſeant ergo per pentagona abcde, <lb></lb>
fghKl plana ſphæram ſecantia, quæ facient ſectiones cir­<lb></lb>
culos æquales inter ſe ſe: poſtea ducantur ex centro ſphæræ <lb></lb>
<figure id="id.023.01.087.1.jpg" xlink:href="023/01/087/1.jpg"></figure><lb></lb>
m perpendiculares ad pla­<lb></lb>
na dictorum <expan abbr="circulorũ">circulorum</expan>; ad <lb></lb>
circulum quidem abcde <lb></lb>
perpendicularis mn: &amp; ad <lb></lb>
circulum fghKl ipſa mo, <lb></lb>
<arrow.to.target n="marg99"></arrow.to.target><lb></lb>
erunt puncta no <expan abbr="circulorũ">circulorum</expan> <lb></lb>
centra: &amp; lineæ mn, mo in<lb></lb>
ter ſe æquales: quòd circu­<lb></lb>
<arrow.to.target n="marg100"></arrow.to.target><lb></lb>
li æquales ſint. </s>
          <s id="s.000855">Eodem mo<lb></lb>
do, quo ſupra, demonſtrabi<lb></lb>
mus lineas mn, mo in <expan abbr="unã">unam</expan> <lb></lb>
atque eandem lineam con­<lb></lb>
uenire. </s>
          <s id="s.000856">ergo cum puncta no ſint centra circulorum, con­<lb></lb>
ſtat ex prima huius &amp; <expan abbr="pentagonorũ">pentagonorum</expan> grauitatis eſſe centra: <lb></lb>
<expan abbr="idcircoq;">idcircoque</expan>  mn, mo pyramidum abcdem, fghklm axes. </s>
          <lb></lb>
          <s id="s.000857">ponatur abcdem pyramidis grauitatis centrum p: &amp; py<lb></lb>
ramidis fghklm ipſum q centrum. </s>
          <s id="s.000858">erunt pm, mq æqua­<lb></lb>
les, &amp; punctum m grauitatis centrum magnitudinis, quæ <lb></lb>
ex ipſis pyramidibus conſtat. </s>
          <s id="s.000859"><expan abbr="eodẽ">eodem</expan> modo probabitur qua­<lb></lb>
rumlibet pyramidum, quæ è regione opponuntur, <expan abbr="centrũ">centrum</expan> 



<pb xlink:href="023/01/088.jpg"></pb>grauitatis eſſe punctum m. </s>
          <s id="s.000860">patet igitur totius dodecahe­<lb></lb>
dri, centrum grauitatis <expan abbr="idẽ">idem</expan> eſſe, quod &amp; ſphæræ ipſum com<lb></lb>
prehendentis centrum. </s>
          <s id="s.000861">quæ quidem omnia demonſtraſſe <lb></lb>
oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000862"><margin.target id="marg99"></margin.target>corol. </s>
          <s id="s.000863">pri<lb></lb>
mæ ſphæ<lb></lb>
ricorum <lb></lb>
Theod.<margin.target id="marg100"></margin.target>6. primi <lb></lb>sphærico<lb></lb>
rum.</s>
        </p>
        <p type="head">
          <s id="s.000864">PROBLEMA VI. PROPOSITIO XXVIII.</s>
        </p>
        <p type="main">
          <s id="s.000865">DATA qualibet portione conoidis rectangu<lb></lb>
li, abſciſſa plano ad axem recto, uel non recto; fie­<lb></lb>
ri poteſt, ut portio ſolida inſcribatur, uel circum­<lb></lb>
ſcribatur ex cylindris, uel cylindri portionibus, <lb></lb>
æqualem habentibus altitudinem, ita ut recta li­<lb></lb>
nea, quæ inter centrum grauitatis portionis, &amp; <lb></lb>
figuræ inſcriptæ, uel circumſcriptæ interiicitur, <lb></lb>
ſit minor qualibet recta linea propoſita.</s>
        </p>
        <p type="main">
          <s id="s.000866">Sit portio conoidis rectanguli abc, cuius axis bd, <expan abbr="gra-uitatisq;">gra­<lb></lb>
uitatisque</expan> centrum e: &amp; ſit g recta linea propoſita. </s>
          <s id="s.000867">quam ue<lb></lb>
ro proportionem habet linea be ad lineam g, eandem ha­<lb></lb>
beat portio conoidis ad ſolidum h: &amp; circumſcribatur por<lb></lb>
tioni figura, ſicuti dictum eſt, ita ut portiones reliquæ ſint <lb></lb>
ſolido h minores: cuius quidem figuræ centrum grauitatis <lb></lb>
ſit punctum k. </s>
          <s id="s.000868">Dico <expan abbr="lineã">lineam</expan> ke minorem eſſe linea g propo­<lb></lb>
ſita. </s>
          <s id="s.000869">niſi enim ſit minor, uel æqualis, uel maior erit. </s>
          <s id="s.000870">&amp; quo­<lb></lb>
niam figura circumſcripta ad reliquas portiones maiorem <lb></lb>
<arrow.to.target n="marg101"></arrow.to.target><lb></lb>
proportionem habet, quàm portio conoidis ad ſolidum h; <lb></lb>
hoc eſt maiorem, quàm bc ad g: &amp; be ad g non minorem <lb></lb>
habet proportionem, quàm ad ke, propterea quod ke non <lb></lb>
ponitur minor ipſa g: habebit figura circumſcripta ad por<lb></lb>
tiones reliquas maiorem proportionem quàm be ad ek: <lb></lb>
<arrow.to.target n="marg102"></arrow.to.target><lb></lb>
&amp; diuidendo portio conoidis ad reliquas portiones habe­<lb></lb>
bit maiorem, quàm bk ad Ke. </s>
          <s id="s.000871">quare ſi fiat ut portio co­



<pb pagenum="41" xlink:href="023/01/089.jpg"></pb>noidis ad portiones reliquas, ita alia linea, quæ ſit lk ad <lb></lb>
ke: erit lk maior, quam bk: &amp; ideo punctum l extra por­<lb></lb>
<figure id="id.023.01.089.1.jpg" xlink:href="023/01/089/1.jpg"></figure><lb></lb>
tionem cadet. </s>
          <s id="s.000872"><expan abbr="Quoniã">Quoniam</expan> <lb></lb>
igitur à figura circum­<lb></lb>
ſcripta, cuius grauitatis <lb></lb>
centrum eſt k, aufertur <lb></lb>
portio conoidis, cuius <lb></lb>
centrum e. </s>
          <s id="s.000873"><expan abbr="habetq;">habetque</expan> lK <lb></lb>
ad Ke eam proportio­<lb></lb>
nem, quam portio co­<lb></lb>
noidis ad reliquas por­<lb></lb>
tiones; erit punctum l <lb></lb>
extra portionem <expan abbr="cadẽs">cadens</expan>, <lb></lb>
centrum magnitudinis <lb></lb>
ex reliquis portionibus compoſitæ. </s>
          <s id="s.000874">illud autem fieri nullo <lb></lb>
modo poteſt. </s>
          <s id="s.000875">quare conſtat lineam ke ipſa g linea propoſi<lb></lb>
ta minorem eſſe.</s>
        </p>
        <p type="margin">
          <s id="s.000876"><margin.target id="marg101"></margin.target>8. quínti.</s>
        </p>
        <p type="margin">
          <s id="s.000877"><margin.target id="marg102"></margin.target>29. quínti <lb></lb>
ex tradi­<lb></lb>
tione <expan abbr="Cã-pani">Cam­<lb></lb>pani
</expan> .</s>
        </p>
        <p type="main">
          <s id="s.000878">Rurſus inſcribatur portioni figura, uidelicet cylindrus <lb></lb>
<figure id="id.023.01.089.2.jpg" xlink:href="023/01/089/2.jpg"></figure><lb></lb>
mn, ut ſit ipſius altitudo <lb></lb>
æqualis dimidio axis bd: <lb></lb>
&amp; quam proportionem <lb></lb>
habet be ad g, habeat mn <lb></lb>
cylindrus ad ſolidum o. </s>
          <lb></lb>
          <s id="s.000879">inſcribatur deinde eidem <lb></lb>
alia figura, ita ut portio­<lb></lb>
nes reliquæ ſint ſolido o <lb></lb>
minores: &amp; centrum gra<lb></lb>
uitatis figuræ ſit p. </s>
          <s id="s.000880">Dico <lb></lb>
lineam pe ipſa g <expan abbr="minorẽ">minorem</expan> <lb></lb>
eſſe. </s>
          <s id="s.000881">ſi enim non ſit mi­<lb></lb>
nor, eodem, quo ſupra modo demonſtrabimus figuram in <lb></lb>
ſcriptam ad reliquas portiones maiorem proportionem <lb></lb>
habere, quàm be ad ep. </s>
          <s id="s.000882">&amp; ſi fiat alia linea le ad ep, ut eſt <lb></lb>
figura inſcripta ad reliquas portiones, <expan abbr="pũctum">punctum</expan> l extra por



<pb xlink:href="023/01/090.jpg"></pb>tionem cadet: Itaque cum à portione conoidis, cuius gra­<lb></lb>
uitatis centrum e auferatur inſcripta figura, centrum ha­<lb></lb>
bens p: &amp; ſit le ad ep, ut figura inſcripta ad portiones reli<lb></lb>
quas: erit magnitudinis, quæ ex reliquis portionibus con<lb></lb>
ſtat, centrum grauitatis punctum l, extra portionem ca­<lb></lb>
dens. </s>
          <s id="s.000883">quod fieri nequit. </s>
          <s id="s.000884">ergo linea pe minor eſt ipſa g li­<lb></lb>
nea propoſita.</s>
        </p>
        <p type="main">
          <s id="s.000885">Ex quibus perſpicuum eſt centrum grauitatis <lb></lb>
figuræ inſcriptæ, &amp; circumſcriptæ eo magis acce<lb></lb>
dere ad portionis centrum, quo pluribus cylin­<lb></lb>
dris, uel cylindri portionibus conſtet: <expan abbr="fiatq́">fiatque</expan>; figu<lb></lb>
ra inſcripta maior, &amp; circumſcripta minor. </s>
          <s id="s.000886">&amp; <lb></lb>
quanquam continenter ad portionis <expan abbr="centrũ">centrum</expan> pro­<lb></lb>
pius admoueatur: nunquam tamen ad ipſum per <lb></lb>
ueniet. </s>
          <s id="s.000887">ſequeretur enim figuram inſcriptam, <expan abbr="nõ">non</expan> <lb></lb>
ſolum portioni, ſed etiam circumſcriptæ figuræ <lb></lb>
æqualem eſſe. </s>
          <s id="s.000888">quod eſt abſurdum.</s>
        </p>
        <p type="head">
          <s id="s.000889">THEOREMA XXIII. PROPOSITIO XXIX.</s>
        </p>
        <p type="main">
          <s id="s.000890">CVIVSLIBET portionis conoidis rectangu­<lb></lb>
li axis à <expan abbr="cẽtro">centro</expan> grauitatis ita diuiditur, ut pars quæ <lb></lb>
terminatur ad uerticem, reliquæ partis, quæ ad ba <lb></lb>
ſim ſit dupla.</s>
        </p>
        <p type="main">
          <s id="s.000891">SIT portio conoidis rectanguli uel abſciſſa plano ad <lb></lb>
axem recto, uel non recto: &amp; ſecta ipſa altero plano per <expan abbr="axẽ">axem</expan><lb></lb>
 ſit ſuperficiei ſectio abc rectanguli coni ſectio, uel parabo <lb></lb>
le; plani abſcindentis portionem ſectio ſit recta linea ac: <lb></lb>
axis portionis, &amp; ſectionis diameter bd. </s>
          <s id="s.000892">Sumatur autem <lb></lb>
in linea bd punctum e, ita ut be ſit ipſius ed dupla. </s>
          <s id="s.000893">Dico 



<pb pagenum="42" xlink:href="023/01/091.jpg"></pb><figure id="id.023.01.091.1.jpg" xlink:href="023/01/091/1.jpg"></figure><lb></lb>
e portionis ab <lb></lb>
c grauitatis eſſe <lb></lb>
centrum. </s>
          <s id="s.000894">Diui­<lb></lb>
datur enim bd <lb></lb>
bifariam in m: <lb></lb>
&amp; rurſus dm, m<lb></lb>
b bifariam diui­<lb></lb>
dantur in pun­<lb></lb>
ctis n, o: <expan abbr="inſcri-baturq;">inſcri­<lb></lb>
baturque</expan> portio­<lb></lb>
ni figura ſolida, <lb></lb>
&amp; altera circum <lb></lb>ſcribatur ex cy<lb></lb>
lindris æqualem <lb></lb>
altitudinem ha­<lb></lb>
bentibus, ut ſu­<lb></lb>
perius <expan abbr="dictũ">dictum</expan> eſt. </s>
          <lb></lb>
          <s id="s.000895">Sit autem pri­<lb></lb>
mum figura in­<lb></lb>
ſcripta <expan abbr="cylĩdrus">cylindrus</expan> <lb></lb>
fg: &amp; <expan abbr="circũſcri">circumſcri</expan>
­<lb></lb>
pta ex cylindris <lb></lb>
ah, Kl conſtet. </s>
          <lb></lb>
          <s id="s.000896"><arrow.to.target n="marg103"></arrow.to.target><lb></lb>
punctum n erit <lb></lb>
centrum graui­<lb></lb>
tatis figuræ in­<lb></lb>
ſcriptæ, <expan abbr="mediũ">medium</expan> <lb></lb>
ſcilicet ipſius d <lb></lb>
m axis: <expan abbr="atq;">atque</expan> <expan abbr="idẽ">idem</expan> <lb></lb>
erit centrum cy<lb></lb>
lindri ah: &amp; cy­<lb></lb>
lindri kl <expan abbr="centrũ">centrum</expan> <lb></lb>
o, axis bm me­<lb></lb>
dium. </s>
          <s id="s.000897">quare ſi li



<pb xlink:href="023/01/092.jpg"></pb><figure id="id.023.01.092.1.jpg" xlink:href="023/01/092/1.jpg"></figure><lb></lb>
neam on ita di <lb></lb>
uiſerimus in p, <lb></lb>
ut <expan abbr="quã">quam</expan> <expan abbr="propor-tionẽ">propor­<lb></lb>
tionem</expan> habet cy­<lb></lb>
lindrus ah ad <lb></lb>
cylindrum kl, <lb></lb>
habeat linea op <lb></lb>
<arrow.to.target n="marg104"></arrow.to.target><lb></lb>
ad pn: centrum <lb></lb>
grauitatis toti­<lb></lb>
us figuræ <expan abbr="circũ-ſcriptæ">circum­<lb></lb>
ſcriptæ</expan> erit pun<lb></lb>
<arrow.to.target n="marg105"></arrow.to.target><lb></lb>
ctum p. </s>
          <s id="s.000898">Sed cy­<lb></lb>
lindri, qui ſunt <lb></lb>
æquali altitudi­<lb></lb>
ne, eandem in­<lb></lb>
ter ſe ſe, quam <lb></lb>baſes propor-<lb></lb>
tionem habent: <lb></lb>
<expan abbr="eſtq;">eſtque</expan> ut linea db <lb></lb>
ad bm, ita <expan abbr="qua-dratũ">qua­<lb></lb>
dratum</expan> lineæ ad <lb></lb>
ad <expan abbr="quadratũ">quadratum</expan> ip­<lb></lb>
ſius Km, ex uige <lb></lb>
ſima primi libri <lb></lb>
<arrow.to.target n="marg106"></arrow.to.target><lb></lb>
<expan abbr="conicorũ">conicorum</expan> &amp; ita <lb></lb>
quadratum ac <lb></lb>
ad <expan abbr="quadratũ">quadratum</expan> K <lb></lb>
<arrow.to.target n="marg107"></arrow.to.target><lb></lb>
g: hoc eſt circu­<lb></lb>
lus circa diame<lb></lb>
trum ac ad cir­<lb></lb>
culum circa dia<lb></lb>
metrum kg. </s>
          <s id="s.000899">du<lb></lb>
pla eſt autem li­<lb></lb>
nea db lineæ 



<pb pagenum="43" xlink:href="023/01/093.jpg"></pb>bm. </s>
          <s id="s.000900">ergo circulus ac circuli kg: &amp; idcirco cylindrus <lb></lb>
ah cylindri k. </s>
          <s id="s.000901">l duplus erit. </s>
          <s id="s.000902">quare &amp; linea op dupla <lb></lb>
ipſius pn. </s>
          <s id="s.000903">Deinde inſcripta &amp; circumſcripta portioni <lb></lb>
alia figura, ita ut inſcripta conſtituatur ex tribus cylin­<lb></lb>
dris qr, sg, tu: circumſcripta uero ex quatuor ax, yz, <lb></lb>
K<foreign lang="grc">f, θλ·</foreign> diuidantur bo, om, mn, nd bifariam in punctis <lb></lb>
<foreign lang="grc">μνπρ.</foreign> Itaque cylindri <foreign lang="grc">θλ</foreign> centrum grauitatis eſt punctum <lb></lb>
<foreign lang="grc">μ·</foreign> &amp; cylindri k<foreign lang="grc">η</foreign> centrum <foreign lang="grc">ν.</foreign> ergo ſi linea <foreign lang="grc">μγ</foreign> diuidatur in <foreign lang="grc">ς,</foreign><lb></lb>
ita ut <foreign lang="grc">μσ</foreign> ad <foreign lang="grc">σγ</foreign> <expan abbr="proportionẽ">proportionem</expan> <expan abbr="eã">eam</expan> habeat, quam cylindrus K<foreign lang="grc">η</foreign><lb></lb>
ad cylindrum <foreign lang="grc">θλ,</foreign> uidelicet quam quadratum knr ad qua­<lb></lb>
<arrow.to.target n="marg108"></arrow.to.target><lb></lb>
dratum <foreign lang="grc">θ</foreign>o, hoc eſt, quam linea mb ad bo: erit <foreign lang="grc">σ</foreign> centrum <lb></lb>
magnitudinis compoſitæ ex cylindris <foreign lang="grc">κγ, θλ.</foreign> &amp; cum linea <lb></lb>
mb ſit dupla bo, erit &amp; <foreign lang="grc">μσ</foreign> ipſius <foreign lang="grc">σν</foreign> dupla. </s>
          <s id="s.000904">præterea quo­<lb></lb>
niam cylindri yz centrum grauitatis eſt <foreign lang="grc">π,</foreign> linea <foreign lang="grc">σπ</foreign> ita diui<lb></lb>
ſa in <foreign lang="grc">τ,</foreign> ut <foreign lang="grc">στ</foreign> ad <foreign lang="grc">τπ</foreign> eam habeat proportionem, quam cylin<lb></lb>
drus yz ad duos cylindros K<foreign lang="grc">ν, θλ·</foreign> erit <foreign lang="grc">τ</foreign> centrum magnitu<lb></lb>
dinis, quæ ex dictis tribus cylindris conſtat. </s>
          <s id="s.000905">cylindrus <expan abbr="au-tẽ">au­<lb></lb>
tem</expan> yz ad cylindrum <foreign lang="grc">θλ</foreign> eſt, ut linea nb ad bo, hoc eſt ut 3 <lb></lb>
ad 1: &amp; ad cylindrum k<foreign lang="grc">η</foreign>, ut nb ad bm, uidelicet ut 3 ad 2. </s>
          <lb></lb>
          <s id="s.000906">quare yz <expan abbr="cylĩdrus">cylindrus</expan> duobus cylindris k<foreign lang="grc">ν, θλ</foreign> æqualis erit. </s>
          <s id="s.000907">&amp; <lb></lb>
propterea linea <foreign lang="grc">στ</foreign> æqualis ipſi <foreign lang="grc">τπ.</foreign> denique cylindri ax <lb></lb>
centrum grauitatis eſt punctum <foreign lang="grc">ρ.</foreign> &amp; cum <foreign lang="grc">τρ</foreign> diuiſa fuerit <lb></lb>
in <expan abbr="eã">eam</expan> proportionem, quam habet cylindrus ax ad tres cy­<lb></lb>
lindros yz, k<foreign lang="grc">ν, θλ·</foreign> erit in eo puncto centrum grauitatis <lb></lb>
totius figuræ <expan abbr="circũſcriptæ">circumſcriptæ</expan>. </s>
          <s id="s.000908">Sed cylindrus ax ad ipſum yz <lb></lb>
eſt ut linea db ad bn: hoc eſt ut 4 ad 3: &amp; duo cylindri k<foreign lang="grc">η<lb></lb>
θλ</foreign> cylindro y  ſunt æquales. </s>
          <s id="s.000909">cylindrus igitur ax ad tres <lb></lb>
iam dictos cylindros eſt ut 2 ad 3. Sed <expan abbr="quoniã">quoniam</expan> <foreign lang="grc">μ σ</foreign> eſt dua­<lb></lb>
rum partium, &amp; <foreign lang="grc">ς γ</foreign> unius, qualium <foreign lang="grc">μ π</foreign> eſt ſex; erit <foreign lang="grc">ς π</foreign> par­<lb></lb>
tium quatuor: <expan abbr="proptereaq;">proptereaque</expan> <foreign lang="grc">τπ</foreign> duarum, &amp; <foreign lang="grc">νπ,</foreign> hoc eſt <foreign lang="grc">πρ</foreign><lb></lb>
trium. </s>
          <s id="s.000910">quare ſequitur ut punctum <foreign lang="grc">π</foreign> totius figuræ circum <lb></lb>
ſcriptæ ſit centrum. </s>
          <s id="s.000911">Itaque fiat <foreign lang="grc">νυ</foreign> ad <foreign lang="grc">υπ,</foreign> ut <foreign lang="grc">μσ</foreign> ad <foreign lang="grc">σγ.</foreign> &amp; <foreign lang="grc">υρ</foreign><lb></lb>
bifariam diuidatur in <foreign lang="grc">φ.</foreign> Similiter ut in circumſcripta figu<lb></lb>
ra oſtendetur centrum magnitudinis compoſitæ ex cylin-



<pb xlink:href="023/01/094.jpg"></pb><figure id="id.023.01.094.1.jpg" xlink:href="023/01/094/1.jpg"></figure><lb></lb>
dris sg, tu eſſe <lb></lb>
punctum <foreign lang="grc">υ·</foreign> &amp; <lb></lb>
totius figuræ in <lb></lb>
ſcriptæ, quæ <expan abbr="cõ-ſtat">con­<lb></lb>
ſtat</expan> ex cylindris <lb></lb>
qr, ſ g, tu eſſe <foreign lang="grc">φ</foreign><lb></lb>
centrum. </s>
          <s id="s.000912">Sunt <lb></lb>
enim hi cylindri <lb></lb>
æquales &amp; ſimi­<lb></lb>
les cylindris yz, <lb></lb>
K<foreign lang="grc">η, θλ,</foreign> figuræ <lb></lb>
circumſcriptæ. </s>
          <lb></lb>
          <s id="s.000913"><expan abbr="Quoniã">Quoniam</expan> igitur <lb></lb>
ut be ad ed, ita <lb></lb>
eſt op ad pn; <lb></lb>
<expan abbr="utraq;">utraque</expan> enim u­<lb></lb>
triuſque eſt du­<lb></lb>
pla: erit compo<lb></lb>
nendo, ut bd ad <lb></lb>
de, ita on ad n <lb></lb>
p; &amp; permutan <lb></lb>
do, ut bd ad o<lb></lb>
n, ita de ad np. </s>
          <lb></lb>
          <s id="s.000914">Sed bd dupla <lb></lb>
eſt on. </s>
          <s id="s.000915">ergo &amp; <lb></lb>
ed ipſius np du<lb></lb>
pla erit. </s>
          <s id="s.000916">quòd ſi <lb></lb>
ed bifariam di­<lb></lb>
uidatur <expan abbr="ĩ">im</expan> <foreign lang="grc">χ,</foreign> erit <lb></lb>
<foreign lang="grc">χ</foreign> d, uel e <foreign lang="grc">χ</foreign> æ­<lb></lb>
qualis np: &amp; <lb></lb>
ſublata en, quæ <lb></lb>
eſt <expan abbr="cõmunis">communis</expan> u­<lb></lb>
trique e <foreign lang="grc">χ,</foreign> pn, 



<pb pagenum="44" xlink:href="023/01/095.jpg"></pb>relinquetur pe ipſi n<foreign lang="grc">χ</foreign> æqualis. </s>
          <s id="s.000917">cum autem be ſit dupla <lb></lb>
ed, &amp; op dupla pn, hoc eſt ipſius e <foreign lang="grc">χ,</foreign> &amp; reliquum, uideli­<lb></lb>
<arrow.to.target n="marg109"></arrow.to.target><lb></lb>
cet bo unà cum pe ipſius reliqui <foreign lang="grc">χ</foreign> d duplum erit. </s>
          <s id="s.000918">eſtque <lb></lb>
bo dupla <foreign lang="grc">ρ</foreign> d. </s>
          <s id="s.000919">ergo pe, hoc eſt n<foreign lang="grc">χ</foreign> ipſius <foreign lang="grc">χρ</foreign> dupla. </s>
          <s id="s.000920">ſed dn <lb></lb>
dupla eſt n<foreign lang="grc">ρ.</foreign> reliqua igitur d<foreign lang="grc">χ</foreign> dupla reliquæ <foreign lang="grc">χ</foreign> n. </s>
          <s id="s.000921">ſunt au­<lb></lb>
tem d<foreign lang="grc">χ,</foreign> pn inter ſe æquales: <expan abbr="itemq;">itemque</expan> æquales <foreign lang="grc">χ</foreign> n, pe. </s>
          <s id="s.000922">qua­<lb></lb>
re conſtat np ipſius pe duplam eſſe. </s>
          <s id="s.000923">&amp; idcirco pe ipſi en <lb></lb>
æqualem. </s>
          <s id="s.000924">Rurſus cum ſit <foreign lang="grc">μν</foreign> dupla o<foreign lang="grc">ν,</foreign> &amp; <foreign lang="grc">μ σ</foreign> dupla <foreign lang="grc">ς γ;</foreign> erit <lb></lb>
etiam reliqua <foreign lang="grc">νσ</foreign> reliquæ <foreign lang="grc">σ</foreign> o dupla. </s>
          <s id="s.000925">Eadem quoque ratione <lb></lb>
<expan abbr="cõcludetur">concludetur</expan> <foreign lang="grc">π υ</foreign> dupla <foreign lang="grc">υ</foreign> m. </s>
          <s id="s.000926">ergo ut <foreign lang="grc">νσ</foreign> ad <foreign lang="grc">σ</foreign> o, ita <foreign lang="grc">πυ</foreign> ad <foreign lang="grc">υ</foreign> m: <lb></lb>
<expan abbr="componendoq;">componendoque</expan>, &amp; permutando, ut <foreign lang="grc">ν</foreign>o ad <foreign lang="grc">π</foreign>m, ita o<foreign lang="grc">σ</foreign> ad <lb></lb>
m<foreign lang="grc">υ·</foreign> &amp; ſunt æquales <foreign lang="grc">ν</foreign>o, <foreign lang="grc">π</foreign>m. </s>
          <s id="s.000927">quare &amp; o<foreign lang="grc">ς,</foreign> m<foreign lang="grc">υ</foreign> æquales. </s>
          <s id="s.000928">præ<lb></lb>
terea <foreign lang="grc">σπ</foreign> dupla eſt <foreign lang="grc">πτ,</foreign> &amp; <foreign lang="grc">νπ</foreign> ipſius <foreign lang="grc">π</foreign>m. </s>
          <s id="s.000929">reliqua igitur <foreign lang="grc">σν</foreign> re<lb></lb>
liquæ m<foreign lang="grc">τ</foreign> dupla. </s>
          <s id="s.000930">atque erat <foreign lang="grc">νσ</foreign> dupla <foreign lang="grc">σ</foreign>o. </s>
          <s id="s.000931">ergo m<foreign lang="grc">τ, σ</foreign>o æ­<lb></lb>
quales ſunt: &amp; ita æquales m<foreign lang="grc">υ,</foreign> n<foreign lang="grc">φ.</foreign> at o<foreign lang="grc">ς,</foreign> eſt æqualis <lb></lb>
m<foreign lang="grc">υ.</foreign> Sequitur igitur, ut omnes o<foreign lang="grc">ς,</foreign> m<foreign lang="grc">τ,</foreign> m<foreign lang="grc">υ,</foreign> n<foreign lang="grc">φ</foreign> in­<lb></lb>
ter ſe ſint æquales. </s>
          <s id="s.000932">Sed ut <foreign lang="grc">ρπ</foreign> ad <foreign lang="grc">πτ,</foreign> hoc eſt ut 3 ad 2, ita nd <lb></lb>
ad d<foreign lang="grc">χ·</foreign> <expan abbr="permutãdoq;">permutandoque</expan> ut <foreign lang="grc">ρπ</foreign> ad nd, ita <foreign lang="grc">πτ</foreign> ad d<foreign lang="grc">χ.</foreign> &amp; <expan abbr="ſũt">ſunt</expan> æqua<lb></lb>
les <foreign lang="grc">ρπ,</foreign> nd. </s>
          <s id="s.000933">ergo d<foreign lang="grc">χ,</foreign> hoc eſt np, &amp; <foreign lang="grc">πτ</foreign> æquales. </s>
          <s id="s.000934">Sed etiam æ­<lb></lb>
quales n<foreign lang="grc">π, π</foreign>m. </s>
          <s id="s.000935">reliqua igitur <foreign lang="grc">π</foreign>p reliquæ m<foreign lang="grc">τ,</foreign> hoc eſt ipſi <lb></lb>
n<foreign lang="grc">φ</foreign> æqualis erit. </s>
          <s id="s.000936">quare dempta p<foreign lang="grc">π</foreign> ex pe, &amp; <foreign lang="grc">φ</foreign>n dempta ex <lb></lb>
ne, relinquitur pe æqualis e<foreign lang="grc">φ.</foreign> Itaque <foreign lang="grc">π, φ</foreign> centra <expan abbr="figurarũ">figurarum</expan> <lb></lb>
ſecundo loco deſcriptarum a primis centris pn æquali in­<lb></lb>
teruallo recedunt. </s>
          <s id="s.000937">quòd ſi rurſus aliæ figuræ deſcribantur, <lb></lb>
eodem modo demonſtrabimus earum centra æqualiter ab <lb></lb>
his recedere, &amp; ad portionis conoidis centrum propius ad <lb></lb>
moueri. </s>
          <s id="s.000938">Ex quibus conſtat lineam <foreign lang="grc">πφ</foreign> à centro grauitatis <lb></lb>
portionis diuidi in partes æquales. </s>
          <s id="s.000939">Si enim fieri poteſt, non <lb></lb>
ſit centrum in puncto e, quod eſt lineæ <foreign lang="grc">πφ</foreign> medium: ſed in <lb></lb>
<foreign lang="grc">ψ·</foreign> &amp; ipſi <foreign lang="grc">πψ</foreign> æqualis fiat <foreign lang="grc">φω.</foreign> Cum igitur in portione ſolida <lb></lb>
quædam figura inſcribi posſit, ita ut linea, quæ inter cen­<lb></lb>
trum grauitatis portionis, &amp; inſcriptæ figuræ interiicitur, <lb></lb>
qualibet linea propoſita ſit minor, quod proxime demon­<lb></lb>
ſtrauimus: perueniet tandem <foreign lang="grc">φ</foreign> centrum inſcriptæ figuræ 



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



<pb pagenum="45" xlink:href="023/01/097.jpg"></pb>ad punctum <foreign lang="grc">ω.</foreign> Sed quoniam <foreign lang="grc">π</foreign> circumſcripta itidem alia <lb></lb>
figura æquali interuallo ad portionis centrum accedit, ubi <lb></lb>
primum <foreign lang="grc">φ</foreign> applicuerit ſe ad <foreign lang="grc">ω,</foreign> &amp; <foreign lang="grc">π</foreign> ad <expan abbr="punctũ">punctum</expan> <foreign lang="grc">ψ,</foreign> hoc eſt ad <lb></lb>
portionis centrum ſe applicabit. </s>
          <s id="s.000940">quod fieri nullo modo <lb></lb>
poſſe perſpicuum eſt. </s>
          <s id="s.000941">non aliter idem abſurdum ſequetur, <lb></lb>
fi ponamus centrum portionis recedere à medio ad par­<lb></lb>
tes <foreign lang="grc">ω;</foreign> eſſet enim aliquando centrum figuræ inſcriptæ idem <lb></lb>
quod portionis <expan abbr="centrũ">centrum</expan>. </s>
          <s id="s.000942">ergo punctum e centrum erit gra<lb></lb>
uitatis portionis abc. quod demonſtrare oportebat.</s>
        </p>
        <p type="margin">
          <s id="s.000943"><margin.target id="marg103"></margin.target>7. huius</s>
        </p>
        <p type="margin">
          <s id="s.000944"><margin.target id="marg104"></margin.target>8. primi <lb></lb>
libri Ar­<lb></lb>
chimedis</s>
        </p>
        <p type="margin">
          <s id="s.000945"><margin.target id="marg105"></margin.target>11. duo­<lb></lb>
decimi.</s>
        </p>
        <p type="margin">
          <s id="s.000946"><margin.target id="marg106"></margin.target>15. quinti</s>
        </p>
        <p type="margin">
          <s id="s.000947"><margin.target id="marg107"></margin.target>2. duode­<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000948"><margin.target id="marg108"></margin.target>20. primi <lb></lb>
<expan abbr="conicorũ">conicorum</expan></s>
        </p>
        <p type="margin">
          <s id="s.000949"><margin.target id="marg109"></margin.target>19.<lb></lb>
quinti</s>
        </p>
        <p type="main">
          <s id="s.000950">Quod autem ſupra <expan abbr="demõſtratum">demonſtratum</expan> eſt in portione conoi­<lb></lb>
dis recta per figuras, quæ ex cylindris æqualem altitudi­<lb></lb>
dinem habentibus conſtant, idem ſimiliter demonſtrabi­<lb></lb>
mus per figuras ex cylindri portionibus conſtantes in ea <lb></lb>
portione, quæ plano non ad axem recto abſcinditur. </s>
          <s id="s.000951">ut <lb></lb>
enim tradidimus in commentariis in undecimam propoſi<lb></lb>
tionem libri Archimedis de conoidibus &amp; ſphæroidibus. </s>
          <lb></lb>
          <s id="s.000952">portiones cylindri, quæ æquali ſunt altitudine eam inter ſe <lb></lb>
ſe proportionem habent, quam ipſarum baſes: baſes <expan abbr="autẽ">autem</expan> <lb></lb>
<arrow.to.target n="marg110"></arrow.to.target><lb></lb>
quæ ſunt ellipſes ſimiles eandem proportionem habere, <lb></lb>
quam quadrata diametrorum eiuſdem rationis, ex corol­<lb></lb>
lario ſeptimæ propoſitionis libri de conoidibus, &amp; ſphæ­<lb></lb>
roidibus, manifeſte apparet.</s>
        </p>
        <p type="margin">
          <s id="s.000953"><margin.target id="marg110"></margin.target>corol. 15<lb></lb>
de conoi­<lb></lb>
dibus &amp; <lb></lb>
ſphæroi­<lb></lb>
dibus.</s>
        </p>
        <p type="head">
          <s id="s.000954">THEOREMA XXIIII. PROPOSITIO XXX.</s>
        </p>
        <p type="main">
          <s id="s.000955">Si à portione conoidis rectanguli alia portio <lb></lb>
abſcindatur, plano baſi æquidiſtante; habebit <lb></lb>
portio tota ad eam, quæ abſciſſa eſt, duplam pro <lb></lb>
portion em eius, quæ eſt baſis maioris portionis <lb></lb>
ad baſi m minoris, uel quæ axis maioris ad axem <lb></lb>
minoris.</s>
        </p>
        <pb xlink:href="023/01/098.jpg"></pb>
        <p type="main">
          <s id="s.000956">ABSCINDATVR à portione conoidis rectanguli <lb></lb>
abc alia portio ebf, plano baſi æquidiſtante: &amp; eadem <lb></lb>
portio ſecetur alio plano per axem; ut ſuperficiei ſectio ſit <lb></lb>
parabole abc: <expan abbr="planorũ">planorum</expan> portiones abſcindentium rectæ <lb></lb>
lincæ ac, ef: axis autem portionis, &amp; ſectionis diameter <lb></lb>
bd; quam linea ef in puncto g ſecet. </s>
          <s id="s.000957">Dico portionem co­<lb></lb>
noidis abc ad portionem ebf duplam proportionem ha­<lb></lb>
bere eius, quæ eſt baſis ac ad baſim ef; uel axis db ad bg<lb></lb>
axem. </s>
          <s id="s.000958">Intelligantur enim duo coni, ſeu coni portiones <lb></lb>
abc, ebf, <expan abbr="eãdem">eandem</expan> baſim, quam portiones conoidis, &amp; æqua <lb></lb>
lem habentes altitudinem. </s>
          <s id="s.000959">&amp; quoniam abc portio conoi <lb></lb>
dis ſeſquialtera eſt coni, ſeu portionis coni abc; &amp; portio <lb></lb>
ebf coni ſeu portionis coni  bf eſt ſeſquialtera, quod de­<lb></lb>
<figure id="id.023.01.098.1.jpg" xlink:href="023/01/098/1.jpg"></figure><lb></lb>
monſtrauit Archimedes in propoſitionibus 23, &amp; 24 libri <lb></lb>
de conoidibus, &amp; ſphæroidibus: erit conoidis portio ad <lb></lb>
conoidis portionem, ut conus ad conum, uel ut coni por­<lb></lb>
tio ad coni portionem. </s>
          <s id="s.000960">Sed conus, nel coni portio abc ad <lb></lb>
conum, uel coni portionem ebf compoſitam proportio­<lb></lb>
nem habet ex proportione baſis ac ad baſim ef, &amp; ex pro­<lb></lb>
portione altitudinis coni, uel coni portionis abc ad alti­<lb></lb>
tudinem ipſius ebf, ut nos demonſtrauimus in commen­<lb></lb>
tariis in undecimam propoſitionem eiuſdem libri Archi­<lb></lb>
medis: altitudo autem ad altitudinem cſt, ut axis ad axem. </s>
          <lb></lb>
          <s id="s.000961">quod quidem in conis rectis perſpicuum eſt, in ſcalenis ue 



<pb pagenum="46" xlink:href="023/01/099.jpg"></pb>ro ita demonſtrabitur. </s>
          <s id="s.000962">Ducatur à puncto b ad planum ba­<lb></lb>
ſis ac perpendicularis linea bh, quæ ipſam ef in K ſecet. </s>
          <lb></lb>
          <s id="s.000963">erit bh altitudo coni, uel coni portionis abc: &amp; bK altitu<lb></lb>
<arrow.to.target n="marg111"></arrow.to.target><lb></lb>
do efg. </s>
          <s id="s.000964">Quod cum lineæ ac, ef inter ſe æquidiſtent, ſunt <lb></lb>
enim planorum æquidiſtantium ſectiones: habebit db ad <lb></lb>
<arrow.to.target n="marg112"></arrow.to.target><lb></lb>
bg proportionem eandem, quam hb ad bk quare por­<lb></lb>
tio conoidis abc ad portionem efg proportionem habet <lb></lb>
compoſitam ex proportione baſis ac ad baſim ef; &amp; ex <lb></lb>
<arrow.to.target n="marg113"></arrow.to.target><lb></lb>
proportione db axis ad axem bg. </s>
          <s id="s.000965">Sed circulus, uel <lb></lb>
ellipſis circa diametrum ac ad circulum, uel ellipſim <lb></lb>
<arrow.to.target n="marg114"></arrow.to.target><lb></lb>
circa ef, eſt ut quadratum ac ad quadratum ef; hoc eſt ut <lb></lb>
<expan abbr="quadratũ">quadratum</expan> ad ad <expan abbr="quadratũ">quadratum</expan> eg. &amp; quadratum ad ad quadra<lb></lb>
tum eg eſt, ut linea db ad lineam bg. </s>
          <s id="s.000966">circulus igitur, uel el<lb></lb>
<arrow.to.target n="marg115"></arrow.to.target><lb></lb>
lipſis circa diametrum ac ad <expan abbr="circulũ">circulum</expan>, uel ellipſim circa ef, <lb></lb>
<arrow.to.target n="marg116"></arrow.to.target><lb></lb>
hoc eſt baſis ad baſim eandem proportionem habet, <expan abbr="quã">quam</expan> <lb></lb>
db axis ad axem bg. </s>
          <s id="s.000967">ex quibus ſequitur portionem abc <lb></lb>
ad portionem ebf habere proportionem duplam eius, <lb></lb>
quæ eſt baſis ac ad baſim ef: uel axis db ad bg axem. </s>
          <s id="s.000968">quod <lb></lb>
demonſtrandum proponebatur.</s>
        </p>
        <p type="margin">
          <s id="s.000969"><margin.target id="marg111"></margin.target>16. unde­<lb></lb>
cimi.</s>
        </p>
        <p type="margin">
          <s id="s.000970"><margin.target id="marg112"></margin.target>4 sexti.</s>
        </p>
        <p type="margin">
          <s id="s.000971"><margin.target id="marg113"></margin.target>2. duode<lb></lb>
cimi</s>
        </p>
        <p type="margin">
          <s id="s.000972"><margin.target id="marg114"></margin.target>7. de co­<lb></lb>
noidibus <lb></lb>
&amp; ſphæ­<lb></lb>
roidibus</s>
        </p>
        <p type="margin">
          <s id="s.000973"><margin.target id="marg115"></margin.target>15. quinti. </s>
          <s id="s.000974">quinti</s>
        </p>
        <p type="margin">
          <s id="s.000975"><margin.target id="marg116"></margin.target>20. primi <lb></lb>
<expan abbr="conicorũ">conicorum</expan></s>
        </p>
        <p type="head">
          <s id="s.000976">THEOREMA XXV. PROPOSITIO XXXI.</s>
        </p>
        <p type="main">
          <s id="s.000977">Cuiuslibet fruſti à portione rectanguli conoi<lb></lb>
dis abſcisſi, centrum grauitatis eſt in axe, ita ut <lb></lb>
demptis primum à quadrato, quod fit ex diame­<lb></lb>
tro maioris baſis, tertia ipſius parte, &amp; duabus <lb></lb>
tertiis quadrati, quod fit ex diametro baſis mino­<lb></lb>
ris: deinde à tertia parte quadrati maioris baſis <lb></lb>
rurſus dempta portione, ad quam reliquum qua<lb></lb>
drati baſis maioris unà cum dicta portione <expan abbr="duplã">duplam</expan> <lb></lb>
proportionem habeat eius, quæ eſt quadrati ma­



<pb xlink:href="023/01/100.jpg"></pb>ioris baſis ad quadratum minoris: centrum ſit in <lb></lb>
eo axis puncto, quo ita diuiditur ut pars, quæ mi<lb></lb>
norem baſim attingit ad alteram partem eandem <lb></lb>
proportionem habeat, quam dempto quadrato <lb></lb>
minoris baſis à duabus tertiis quadrati maioris, <lb></lb>
habet id, quod reliquum eſt unà cum portione à <lb></lb>
tertia quadrati maioris parte dempta, ad <expan abbr="reliquã">reliquam</expan> <lb></lb>
eiuſdem tertiæ portionem.</s>
        </p>
        <p type="main">
          <s id="s.000978">SIT fruſtum à portione rectanguli conoidis abſciſſum <lb></lb>
abcd, cuius maior baſis circulus, uel ellipſis circa diame­<lb></lb>
trum bc, minor circa diametrum ad; &amp; axis ef. </s>
          <s id="s.000979">deſcriba­<lb></lb>
tur autem portio conoidis, à quo illud abſciſſum eſt, &amp; pla­<lb></lb>
<figure id="id.023.01.100.1.jpg" xlink:href="023/01/100/1.jpg"></figure><lb></lb>
no per axem ducto ſecetur; ut ſuperficiei ſectio ſit parabo­<lb></lb>
le bgc, cuius diameter, &amp; axis portionis gf: deinde gf diui<lb></lb>
datur in puncto h, ita ut gh ſit dupla hf: &amp; rurſus ge in ean <lb></lb>
dem proportionem diuidatur: <expan abbr="ſitq;">ſitque</expan> gk ipſius ke dupla. </s>
          <s id="s.000980"><expan abbr="Iã">Iam</expan> <lb></lb>
ex iis, quæ proxime demonſtrauimus, conſtat centrum gra<lb></lb>
uitatis portionis bgc eſſe h punctum: &amp; portionis agc <lb></lb>
punctum k. </s>
          <s id="s.000981">ſumpto igitur infra h puncto l, ita ut kh ad hl 



<pb pagenum="47" xlink:href="023/01/101.jpg"></pb>eam proportionem habeat, quam abcd fruſtum ad por­<lb></lb>
tionem agd; erit punctum l eius fruſti grauitatis <expan abbr="cẽtrum">centrum</expan>: <lb></lb>
<expan abbr="habebitq;">habebitque</expan> componendo Kl ad lh proportionem eandem, <lb></lb>
<arrow.to.target n="marg117"></arrow.to.target><lb></lb>
quam portio conoidis bgc ad agd portionem. </s>
          <s id="s.000982"><expan abbr="Itaq;">Itaque</expan> quo <lb></lb>
niam quadratum bf ad quadratum ae, hoc eſt quadratum <lb></lb>
bc ad quadratum ad eſt, ut linea fg ad ge: erunt duæ ter­<lb></lb>
tiæ quadrati bc ad duas tertias quadrati ad, ut hg ad gk: <lb></lb>
&amp; ſi à duabus tertiis quadrati bc demptæ fuerint duæ ter­<lb></lb>
tiæ quadrati ad: erit <expan abbr="diuidẽdo">diuidendo</expan> id, quod relinquitur ad duas <lb></lb>
tertias quadrati ad, ut hk ad kg. </s>
          <s id="s.000983">Rurſus duæ tertiæ quadra<lb></lb>
ti ad ad duas tertias quadrati bc ſunt, ut kg ad gh: &amp; duæ <lb></lb>
tertiæ quadrati bc ad <expan abbr="tertiã">tertiam</expan> <expan abbr="partẽ">partem</expan> ipſius, ut gh ad hf. </s>
          <s id="s.000984">ergo <lb></lb>
ex æquali id, quod relinquitur ex duabus tertiis quadrati <lb></lb>
bc, demptis ab ipſis quadrati ad duabus tertiis, ad <expan abbr="tertiã">tertiam</expan> <lb></lb>
partem quadrati bc, ut kh ad hf: &amp; ad portionem <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb>
tertiæ partis, ad quam unà cum ipſa portione, duplam pro<lb></lb>
portionem habeat eius, quæ eſt quadrati bc ad <expan abbr="quadratũ">quadratum</expan> <lb></lb>
ad, ut Kl ad lh. </s>
          <s id="s.000985">habet enim Kl ad lh eandem proportio­<lb></lb>
nem, quam conoidis portio bgc ad portionem agd: por­<lb></lb>
tio autem bgc ad portionem agd duplam proportionem <lb></lb>
habet eius, quæ eſt baſis bc ad baſim ad: hoc eſt quadrati <lb></lb>
<arrow.to.target n="marg118"></arrow.to.target><lb></lb>
bc ad quadratum ad; ut proxime demonſtratum eſt. </s>
          <s id="s.000986">quare <lb></lb>
dempto ad quadrato à duabus tertiis quadrati bc, erit id, <lb></lb>
quod relinquitur unà cum dicta portione tertiæ partis ad <lb></lb>
reliquam eiuſdem portionem, ut el ad lf. </s>
          <s id="s.000987">Cum igitur cen­<lb></lb>
trum grauitatis fruſti abcd ſit l, à quo axis ef in eam, <expan abbr="quã">quam</expan> <lb></lb>
diximus, proportionem diuidatur; conſtat <expan abbr="uerũ">uerum</expan> eſſe illud, <lb></lb>
quod demonſtrandum propoſuimus.</s>
        </p>
        <p type="margin">
          <s id="s.000988"><margin.target id="marg117"></margin.target>20. 1. coni<lb></lb>
corum.</s>
        </p>
        <p type="margin">
          <s id="s.000989"><margin.target id="marg118"></margin.target>30 huius</s>
        </p>
        <p type="head">
          <s id="s.000990">FINIS LIBRI DE CENTROGRAVITATIS SOLIDORVM.</s>
        </p>
        <p type="main">
          <s id="s.000991">Impreſſ. Bononiæ cum licentia Superiorum, </s>
        </p>
      </chap>
    </body>
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</archimedes>