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<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >
<archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info>
<author>Commandino, Federico</author>
<title>Liber de centro gravitatis solidorum</title>
<date>1565</date>
<place>Bologna</place>
<translator/>
<lang>la</lang>
<cvs_file>comma_centr_01_la_1565</cvs_file>
<cvs_version/>
<locator>023.xml</locator>
</info><text>
<pb xlink:href="023/01/001.jpg"/><front><section><p type="head">
<s>FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</s>
<s>LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM.<!-- KEEP S--></s></p><figure id="id.023.01.001.1.jpg" xlink:href="023/01/001/1.jpg"/><p type="head">
<s>CVM PRIVILEGIO IN ANNOS X.<!-- KEEP S--></s></p><p type="head">
<s>BONONIAE,<!-- KEEP S--></s></p><p type="head">
<s>Ex Officina Alexandri Benacii.<!-- KEEP S--></s></p><p type="head">
<s>MDLXV.<!-- KEEP S--></s></p><pb xlink:href="023/01/002.jpg"/><pb xlink:href="023/01/003.jpg"/></section>
<section><p type="head">
<s>ALEXANDRO FARNESIO <lb/>
CARDINALI AMPLISSIMO. <lb/>
ET OPTIMO.</s></p><p type="main">
<s>Cvm multæ res in mathematicis <lb/>
di&longs;ciplinis nequaquam &longs;atis ad­<lb/>
huc explicatæ &longs;int, tum perdif­<lb/>
ficilis, & perob&longs;cura quæ&longs;tio <lb/>
e&longs;t de centro grauitatis corpo­<lb/>
rum &longs;olidorum; quæ, & ad co­<lb/>
gno&longs;cendum pulcherrima e&longs;t, <lb/>
& ad multa, quæ à mathematicis proponuntur, præ­<lb/>
clare intelligenda maximum affert adiumentum. </s>
<s>de <lb/>
qua neminem ex mathematicis, neque no&longs;tra, neque <lb/>
patrum no&longs;trorum memoria &longs;criptum reliqui&longs;&longs;e &longs;ci­<lb/>
mus. </s>
<s>& quamuis in earum monumentis literarum <expan abbr="nõ">non</expan> <lb/>
nulla reperiantur, ex quibus in hanc &longs;ententiam addu <lb/>
ci po&longs;&longs;umus, vt exi&longs;timemus hanc rem ab <expan abbr="ij&longs;d&etilde;">ij&longs;dem</expan> vber­<lb/>
rime tractatam e&longs;&longs;e; tamen ne&longs;cio quo fato adhuc <lb/>
in eiu&longs;modi librorum ignoratione ver&longs;amur. </s>
<s>Archi­<lb/>
medes quidem <expan abbr="mathematicorũ">mathematicorum</expan> princeps in libello, <lb/>
cuius in&longs;criptio e&longs;t, <foreign lang="greek">ke/ttraba/rwg ipipe/dwg,</foreign> de centro pla­<lb/>
norum copio&longs;i&longs;sime, atque acuti&longs;sime con&longs;crip&longs;it: & <lb/>
in eo explicando <expan abbr="&longs;ummã">&longs;ummam</expan> ingenii, & &longs;cientiæ <expan abbr="gloriã">gloriam</expan> e&longs;t <lb/>
<expan abbr="cõ&longs;ecutus">con&longs;ecutus</expan>. </s><s>Sed de cognitione <expan abbr="c&etilde;tri">centri</expan> grauitatis <expan abbr="corporũ">corporum</expan> <lb/>
<expan abbr="&longs;olidorũ">&longs;olidorum</expan> nulla in eius libris litera inuenitur. </s><s>non mul <lb/>
tos abhinc annos MARCELLVS II. PONT. MAX.<pb xlink:href="023/01/004.jpg"/>cum adhuc Cardinalis e&longs;&longs;et, mihi, quæ &longs;ua erat hu­<lb/>
manitas, libros eiu&longs;dem Archimedis de ijs, quæ ve­<lb/>
huntur in aqua, latine redditos dono dedit. </s><s>hos cum <lb/>
ego, ut aliorum &longs;tudia incitarem, <expan abbr="emendãdos">emendandos</expan>, & <expan abbr="cõ-mentariis">com­<lb/>
mentariis</expan> illu&longs;trandos &longs;u&longs;cepi&longs;&longs;em, animaduerti dubi <lb/>
tari non po&longs;&longs;e, quin Archimedes vel de hac materia <lb/>
&longs;crip&longs;i&longs;&longs;et, vel aliorum mathematicorum &longs;cripta per­<lb/>
legi&longs;&longs;et. </s><s>nam in iis tum alia nonnulla, tum maxime <lb/>
illam propo&longs;itionem, ut euidentem, & aliàs proba­<lb/>
tam a&longs;&longs;umit, <expan abbr="Centrũ">Centrum</expan> grauitatis in portionibus conoi <lb/>
dis rectanguli axem ita diuidere, vt pars, quæ ad verti <lb/>
cem terminatur, alterius partis, quæ ad ba&longs;im dupla <lb/>
&longs;it. </s><s>Verum hæc ad cam partem mathematicarum <lb/>
di&longs;ciplinarum præcipue refertur, in qua de centro <lb/>
grauitatis corporum &longs;olidorum tractatur. </s><s>non e&longs;t au <lb/>
tem con&longs;entaneum Archimedem illum admirabilem <lb/>
virum hanc propo&longs;itionem &longs;ibi argumentis con­<lb/>
firmandam exi&longs;timaturum non fui&longs;&longs;e, ni&longs;i eam vel <lb/>
aliis in locis probaui&longs;&longs;et, vel ab aliis probatam e&longs;&longs;e <lb/>
comperi&longs;&longs;et. </s><s>quamobrem nequid in iis libris intel­<lb/>
ligendis de&longs;iderari po&longs;&longs;et, &longs;tatui hanc etiam partem <lb/>
vel à veteribus prætermi&longs;&longs;am, vel tractatam quidem, <lb/>
&longs;ed in tenebris iacentem, non intactam relinquere; <lb/>
atque ex a&longs;sidua mathematicorum, præ&longs;ertim Archi­<lb/>
medis lectione, quæ mihi in mentem venerunt, ea in <lb/>
medium afferre; ut centri grauitatis corporum &longs;oli­<lb/>
dorum, &longs;i non perfectam, at certe aliquam noti-<pb xlink:href="023/01/005.jpg"/>tiam haberemus. </s><s>Quem meum laborem <expan abbr="nõ">non</expan> mathe­<lb/>
maticis &longs;olum, verum iis etiam, qui naturæ ob&longs;curi­<lb/>
tate delectantur, <expan abbr="nõ">non</expan> iniucundam fore &longs;peraui:multa <lb/>
enim <foreign lang="greek">problh/mata</foreign> cognitione digni&longs;sima, quæ ad <expan abbr="vtrã-que">vtran­<lb/>
que</expan> &longs;cientiam attinent, &longs;e&longs;e legentibus obtuli&longs;&longs;ent.</s> <lb/>
<s>neque id vlli mirandum videri debet. </s><s>vt enim in cor­<lb/>
poribus no&longs;tris omnia membra, ex quibus certa quæ <lb/>
dam officia na&longs;cuntur, diuino quodam ordine inter <lb/>
&longs;e implicata, & colligata &longs;unt: in <expan abbr="iis&qacute;">iisque</expan>; admirabilis il­<lb/>
la con&longs;piratio, quam <foreign lang="greek">tn/mpnoixn</foreign> græci vocant, eluce&longs;cit, <lb/>
ita tres illæ Philo&longs;ophiæ (ut Ari&longs;totelis verbo vtar) <lb/>
quæ veritatem &longs;olam propo&longs;itam habent, licet qui­<lb/>
bu&longs;dam qua&longs;i finibus &longs;uis regantur: tamen <expan abbr="earũ">earum</expan> vna­<lb/>
quæque per &longs;e ip&longs;am quodammodo imperfecta e&longs;t: <lb/>
neque altera &longs;ine alterius auxilio plene comprehen­<lb/>
di pote&longs;t. </s><s>complures præterea mathematicorum no­<lb/>
di ante hac explicatu difficillimi nullo negotio expe <lb/>
diti e&longs;&longs;ent: atque (ut vno verbo complectar) ni&longs;i <lb/>
mea valde amo, tractationem hanc meam &longs;tudio&longs;is <lb/>
non mediocrem vtilitatem, & magnam volupta­<lb/>
tem allaturam e&longs;&longs;e mihi per&longs;ua&longs;i. </s><s>cum autem ad hoc <lb/>
&longs;cribendum aggre&longs;&longs;us e&longs;sem, allatus e&longs;t ad me liber <lb/>
Franci&longs;ci Maurolici Me&longs;&longs;anen&longs;is, in quo vir ille do­<lb/>
cti&longs;simus, & in iis di&longs;ciplinis exercitati&longs;simus af­<lb/>
firmabat &longs;e de centro grauitatis corporum &longs;olido­<lb/>
rum con&longs;crip&longs;i&longs;&longs;e. </s><s>cum hoc intellexi&longs;&longs;em, &longs;u&longs;tinui <lb/>
me pauli&longs;per: tacitus que expectaui, dum opus cla-<pb xlink:href="023/01/006.jpg"/>ris&longs;imi uiri, quem &longs;emper honoris cau&longs;&longs;a nomino, <lb/>
in lucem proferretur: mihi enim exploratis&longs;imum <lb/>
erat: Franci&longs;cum Maurolicum multo doctius, & <lb/>
exqui&longs;itius hoc di&longs;ciplinarum genus &longs;criptis &longs;uis tra <lb/>
diturum. </s><s>&longs;ed cum id tardius fieret, hoc e&longs;t, ut ego <lb/>
interpretor, diligentius, mihi diutius hac &longs;criptione <lb/>
non &longs;uper&longs;edendum e&longs;&longs;e duxi, præ&longs;ertim cum iam li­<lb/>
bri Archimedis de iis, quæ uehuntur in aqua, opera <lb/>
mea illu&longs;trati typis <expan abbr="excud&etilde;di">excudendi</expan> e&longs;&longs;ent. </s><s>nec me alia cau&longs; <lb/>
&longs;a impuli&longs;&longs;et, ut de centro grauitatis corporum &longs;oli­<lb/>
dorum &longs;criberem, ni&longs;i ut hac etiam ratione lux eis <lb/>
quàm maxime fieri po&longs;&longs;et afferretur. </s><s><expan abbr="atq;">atque</expan> id eò mihi <lb/>
faciendum exi&longs;timaui, quòd in &longs;pem ueniebam fore, <lb/>
ut cum ego ex omnibus mathematicis primus, hanc <lb/>
materiam explicandam &longs;u&longs;cepi&longs;&longs;em; &longs;i quid errati for <lb/>
te à me commi&longs;&longs;um e&longs;&longs;et, boni uiri potius id meæ de <lb/>
&longs;tudio&longs;is hominibus bene <expan abbr="mer&etilde;di">merendi</expan> cupiditati, quàm <lb/>
arrogantiæ a&longs;criberent. </s><s>re&longs;tabat ut con&longs;iderarem, cui <lb/>
potis&longs;imum ex principibus uiris contemplationem <lb/>
hanc, nunc primum memoriæ, ac literis proditam de <lb/>
dicarem. </s><s>harum mearum cogitationum &longs;umma fa­<lb/>
cta, exi&longs;timaui nemini conuenientius de centro graui <lb/>
tatis corporum opus dicari oportere, quàm ALE­<lb/>
XANDRO FARNESIO grauis&longs;imo, ac prudentis&longs;i­<lb/>
mo Cardinali, quo in uiro &longs;umma fortuna &longs;emper <expan abbr="cũ">cum</expan> <lb/>
&longs;umma uirtute certauit. </s><s>quid enim maxime in te ad­<lb/>
mirati debeant homines, ob&longs;curum e&longs;t; u&longs;um'ne re-<pb xlink:href="023/01/007.jpg"/>rum, qui pueritiæ tempus extremum principium ha<lb/>
bui&longs;ti, & <expan abbr="imperiorũ">imperiorum</expan>, & ad Reges, & Imperatores ho­<lb/>
norificenti&longs;simarum legationum; an excellentiam <lb/>
in omni genere literarum, qui vix <expan abbr="adole&longs;c&etilde;tulus">adole&longs;centulus</expan>, quæ <lb/>
homines iam confirmata ætate &longs;ummo &longs;tudio, <expan abbr="diu-turnis&qacute;">diu­<lb/>
turnisque</expan>; laboribus didicerunt, &longs;cientia, & cognitione <lb/>
comprehendi&longs;ti: an con&longs;ilium, & &longs;apientiam in re­<lb/>
gendis, & <expan abbr="gubernãdis">gubernandis</expan> Ciuitatibus, cuius graui&longs;simæ <lb/>
&longs;ententiæ in &longs;ancti&longs;simo Reip. <!-- REMOVE S-->Chr&longs;tianæ con&longs;ilio di­<lb/>
ctæ, potius diuina oracula, quàm &longs;ententiæ habitæ <lb/>
&longs;unt, & habentur. </s>
<s>prætermitto liberalitatem, & mu­<lb/>
nificentiam tuam, quam in &longs;tudio&longs;i&longs;simo quoque ho <lb/>
ne&longs;tando quotidie magis o&longs;tendis, ne videar auribus <lb/>
tuis potius, quàm veritati &longs;eruire. </s><s>quamuis à te in tot <lb/>
præclaros viros tanta beneficia collata &longs;unt, & <expan abbr="confe-rũtur">confe­<lb/>
runtur</expan>, vt omnibus te&longs;tatum &longs;it, nihil tibi e&longs;&longs;e charius, <lb/>
nihil iucundius, quàm eximia tua liberalitate homi­<lb/>
nes ad amplexandam virtutem, licet currentes incita­<lb/>
re. </s><s>nihil dico de ceteris virtutibus tuis, quæ tantæ <lb/>
&longs;unt, quantæ ne cogitatione quidem comprehendi <lb/>
po&longs;&longs;unt. </s><s>Quamobrem hac præcipue de cau&longs;&longs;a te hu­<lb/>
ius meæ lucubrationis patronum e&longs;&longs;e volui, quam ea, <lb/>
qua &longs;oles, humanitate accipies. </s><s>te enim &longs;emper ob <lb/>
diuinas virtutes tuas colui, & ob&longs;eruaui: <expan abbr="nihil&qacute;">nihilque</expan>; mi­<lb/>
hi fuit optatius; quàm tibi per&longs;pectum e&longs;&longs;e meum <lb/>
erga te animum; <expan abbr="&longs;ingularem&qacute;">&longs;ingularemque</expan>; ob&longs;eruantiam. </s><s>cœ­<lb/>
lum igitur digito attingam, &longs;i po&longs;t graui&longs;simas oc-<pb xlink:href="023/01/008.jpg"/>cupationes tuas legendo Federici tui libro aliquid <lb/>
impertiri temporis non grauaberis: <expan abbr="cum&qacute;">cumque</expan>; in iis, qui <lb/>
tibi &longs;emper addicti erunt, numerare. </s><s>Vale.<!-- KEEP S--></s></p><p type="main">
<s>Federicus Commandinus.</s></p></section><section><pb xlink:href="023/01/009.jpg" pagenum="1"/><p type="head">
<s>FEDERICI COMMANDINI <lb/>
VRBINATIS LIBER DE CENTRO <lb/>
GRAVITATIS SOLIDORVM.</s></p></section></front><body><chap><p type="head">
<s>DIFFINITIONES.</s></p><p type="main">
<s><arrow.to.target n="marg1"/></s></p><p type="margin">
<s><margin.target id="marg1"/>1</s></p><p type="main">
<s>Centrvm grauitatis, Pappus <lb/>
Alexandrinus in octauo ma­<lb/>
thematicarum collectionum <lb/>
libro ita diffiniuit.</s></p><p type="main">
<s><foreign lang="greek">le/gomen de/ ke/ntron ba/rous e(/ka/stou sw/<lb/>
matos e)=inai shme=ion ti kei/menon e)nto/s, a/f'<lb/>
o(/u kat' e/poi/nian a\rtnqe/n to/ ba/ros n(mere=i<lb/>
fero/menon, kai\ fula/ssei th/n e)c a)rxh=s qe/­<lb/>
sin, o)u mh\ peritrepo/menon e)n th= fora=</foreign>. hoc e&longs;t,</s></p><p type="main">
<s>Dicimus autem centrum grauitatis uniu&longs;cu­<lb/>
iu&longs;que corporis punctum quoddam intra po&longs;i­<lb/>
tum, à quo &longs;i graue appen&longs;um mente concipia­<lb/>
tur, dum fertur quie&longs;cit; & &longs;eruat eam, quam in <lb/>
principio habebat po&longs;itionem: neque in ip&longs;a la­<lb/>
tione circumuertitur.</s></p><p type="main">
<s>Po&longs;&longs;umus etiam hoc modo diffinire.</s></p><p type="main">
<s>Centrum grauitatis uniu&longs;cuiu&longs;que &longs;olidæ figu <lb/>
ræ e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>
undique partes æqualium momentorum con&longs;i­<lb/>
&longs;tunt. </s><s>&longs;i enim per tale centrum ducatur planum <lb/>
figuram quomodocunque &longs;ecans &longs;emper in par­<pb xlink:href="023/01/010.jpg"/>tes æqueponderantes ip&longs;am diuidet.</s></p><p type="main">
<s><arrow.to.target n="marg2"/></s></p><p type="margin">
<s><margin.target id="marg2"/>2</s></p><p type="main">
<s>Pri&longs;matis, cylindri, & portionis cylindri axem <lb/>
appello rectam lineam, quæ oppo&longs;itorum plano­<lb/>
rum centra grauitatis coniungit.</s></p><p type="main">
<s><arrow.to.target n="marg3"/></s></p><p type="margin">
<s><margin.target id="marg3"/>3</s></p><p type="main">
<s>Pyramidis, coni, & portionis coni axem dico li <lb/>
neam, quæ à uertice ad centrum grauitatis ba&longs;is <lb/>
perducitur.</s></p><p type="main">
<s><arrow.to.target n="marg4"/></s></p><p type="margin">
<s><margin.target id="marg4"/>4</s></p><p type="main">
<s>Si pyramis, conus, portio coni, uel conoidis &longs;e­<lb/>
cetur plano ba&longs;i æquidi&longs;tante, pars, quæ e&longs;t ad ba­<lb/>
&longs;im, fru&longs;tum pyramidis, coni, portionis coni, uel <lb/>
conoidis dicetur; quorum plana æquidi&longs;tantia, <lb/>
quæ opponuntur &longs;imilia &longs;unt, & inæqualia: axes <lb/>
uero &longs;unt axium figurarum partes, quæ in ip&longs;is <lb/>
comprehenduntur.</s></p><p type="head">
<s>PETITIONES.<!-- KEEP S--></s></p><p type="main">
<s><arrow.to.target n="marg5"/></s></p><p type="margin">
<s><margin.target id="marg5"/>1</s></p><p type="main">
<s>Solidarum figurarum fimilium centra grauita­<lb/>
tis &longs;imiliter &longs;unt po&longs;ita.</s></p><p type="main">
<s><arrow.to.target n="marg6"/></s></p><p type="margin">
<s><margin.target id="marg6"/>2</s></p><p type="main">
<s>Solidis figuris &longs;imilibus, & æqualibus inter &longs;e <lb/>
aptatis, centra quoque grauitatis ip&longs;arum inter &longs;e <lb/>
aptata erunt.</s></p><p type="head">
<s>THEOREMA I. PROPOSITIO I.<!-- KEEP S--></s></p><p type="main">
<s>Omnis figuræ rectilineæ in circulo de&longs;criptæ, <lb/>
quæ æqualibus lateribus, & angulis contine­<lb/>
<pb xlink:href="023/01/011.jpg" pagenum="2"/>tur, centrum grauitatis e&longs;t idem, quod circuli cen<lb/>
trum.</s></p><p type="main">
<s>Sit primo triangulum æquilaterum abc in circulo de­<lb/>
&longs;criptum: & diui&longs;a ac bifariam in d, ducatur bd. <!-- KEEP S--></s><s>erit in li­<lb/>
nea bd centrum grauitatis <expan abbr="triãguli">trianguli</expan> abc, ex tertia decima <lb/>
primi libri Archimedis de centro grauitatis planorum. </s><s>Et <lb/>
<figure id="id.023.01.011.1.jpg" xlink:href="023/01/011/1.jpg"/><lb/>
quoniam linea ab e&longs;t æqualis <lb/>
lineæ bc; & ad ip&longs;i dc; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <lb/>
bd utrique communis: trian­</s></p><p type="main">
<s><arrow.to.target n="marg7"/><lb/>
gulum abd æquale erit trian <lb/>
gulo cbd: & anguli angulis æ­<lb/>
quales, qui æqualibus lateri­<lb/>
<arrow.to.target n="marg8"/><lb/>
bus &longs;ubtenduntur. </s><s>ergo angu <lb/>
li ad d <expan abbr="utriq;">utrique</expan> recti &longs;unt. </s><s>quòd <lb/>
cum linea bd &longs;ecet ae bifa­<lb/>
<arrow.to.target n="marg9"/><lb/>
riam, & ad angulos rectos; in <lb/>
ip&longs;a bd e&longs;t centrum circuli. </s><s><lb/>
quare in eadem bd linea erit <lb/>
centrum grauitatis trianguli, & circuli centrum. </s><s>Similiter <lb/>
diui&longs;a ab bifariam in e, & ducta ce, o&longs;tendetur in ip&longs;a <expan abbr="utrũ">utrum</expan> <lb/>
que centrum contineri. </s><s>ergo ea erunt in puncto, in quo li­<lb/>
neæ bd, ce conueniunt. </s><s>trianguli igitur abc centrum gra<lb/>
uitatis e&longs;t idem, quod circuli centrum.</s></p><p type="margin">
<s><margin.target id="marg7"/>8. primi.</s></p><p type="margin">
<s><margin.target id="marg8"/>13. primi.</s></p><p type="margin">
<s><margin.target id="marg9"/>corol. pri<lb/>
mæ tertii</s></p><figure id="id.023.01.011.2.jpg" xlink:href="023/01/011/2.jpg"/><p type="main">
<s>Sit quadratum abcd in cir­<lb/>
culo de&longs;criptum: & ducantur <lb/>
ac, bd, quæ conueniant in e. </s><s>er­<lb/>
go punctum e e&longs;t centrum gra<lb/>
uitatis quadrati, ex decima eiu&longs; <lb/>
dem libri Archimedis. <!-- KEEP S--></s><s>Sed cum <lb/>
omnes anguli ad abcd recti <lb/>
<arrow.to.target n="marg10"/><lb/>
&longs;int; erit abc &longs;emicirculus: <lb/>
<expan abbr="item&qacute;">itemque</expan>; bcd: & propterea li­<lb/>
neæ ac, bd diametri circuli: <pb xlink:href="023/01/012.jpg"/>quæ quidem in centro conueniunt. </s><s>idem igitur e&longs;t centrum <lb/>
grauitatis quadrati, & circuli centrum.</s></p><p type="margin">
<s><margin.target id="marg10"/>31. tertii.</s></p><p type="main">
<s>Sit pentagonum æquilaterum, & æquiangulum in circu­<lb/>
<figure id="id.023.01.012.1.jpg" xlink:href="023/01/012/1.jpg"/><lb/>
lo de&longs;criptum abcd e. </s><s>& iun­<lb/>
cta bd, <expan abbr="bifariam&qacute;">bifariamque</expan>; in f diui&longs;a, <lb/>
ducatur cf, & producatur ad <lb/>
circuli circumferentiam in g; <lb/>
quæ lineam ae in h &longs;ecet: de­<lb/>
inde iungantur ac, cc. <!-- KEEP S--></s><s>Eodem <lb/>
modo, quo &longs;upra demon&longs;tra­<lb/>
bimus angulum bcf æqualem <lb/>
e&longs;&longs;e. </s><s>angulo dcf; & angulos <lb/>
ad f utro&longs;que rectos: & idcir­<lb/>
co lineam cfg per circuli cen <lb/>
trum tran&longs;ire. </s><s>Quoniam igi­<lb/>
tur latera cb, ba, & cd, de æqualia &longs;unt; & æquales anguli <lb/>
<arrow.to.target n="marg11"/><lb/>
cba, cde: erit ba&longs;is ca ba&longs;i: ce, & angulus bca angulo <lb/>
dce æqualis. </s><s>ergo & reliquus ach, reliquo ech. </s><s>e&longs;t au­<lb/>
tem ch utrique triangulo ach, ech communis. </s><s>quare <lb/>
ba&longs;is ah æqualis e&longs;t ba&longs;i hc: & anguli, qui ad h recti: <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; <lb/>
<arrow.to.target n="marg12"/><lb/>
recti, qui ad f. </s><s>ergo lineæ ae, bd inter &longs;e &longs;e æquidi&longs;tant. </s><lb/>
<s>Itaque cum trapezij abde latera bd, ae æquidi&longs;tantia à li <lb/>
nea fh bifariam diuidantur; centrum grauitatis ip&longs;ius erit <lb/>
<arrow.to.target n="marg13"/><lb/>
in linea fh, ex ultima eiu&longs;dem libri Archimedis. <!-- KEEP S--></s><s>Sed trian­<lb/>
guli bcd centrum grauitatis e&longs;t in linea cf. </s><s>ergo in eadem <lb/>
linea ch e&longs;t centrum grauitatis trapezij abde, & trian­<lb/>
guli bcd: hoc e&longs;t pentagoni ip&longs;ius centrum: & centrum <lb/>
circuli. </s><s>Rur&longs;us &longs;i iuncta ad, <expan abbr="bifariam&qacute;">bifariamque</expan>; &longs;ecta in k, duca­<lb/>
tur ekl: demon&longs;trabimus in ip&longs;a utrumque centrum in <lb/>
e&longs;&longs;e. </s><s>Sequitur ergo, ut punctum, in quo lineæ cg, el con­<lb/>
ueniunt, idem &longs;it centrum circuli, & centrum grauitatis <lb/>
pentagoni.</s></p><p type="margin">
<s><margin.target id="marg11"/>4. Primi.<!-- KEEP S--></s></p><p type="margin">
<s><margin.target id="marg12"/>28. primi.</s></p><p type="margin">
<s><margin.target id="marg13"/>13. Archi­<lb/>
medis.<!-- KEEP S--></s></p><p type="main">
<s>Sit hexagonum abcdef æquilaterum, & æquiangulum <lb/>
in circulo de&longs;ignatum: <expan abbr="iungantur&qacute;">iunganturque</expan>; bd, ae: & bifariam &longs;e­<pb xlink:href="023/01/013.jpg" pagenum="3"/>cta bd in g puncto, ducatur cg; & protrahatur ad circuli <lb/>
u&longs;que circumferentiam; quæ &longs;ecet ae in h. </s><s>Similiter conclu <lb/>
demus cg per centrum circuli tran&longs;ire: & bifariam &longs;ecate <lb/>
lineam ae; <expan abbr="item&qacute;">itemque</expan>; lineas bd, ae inter &longs;e æquidi&longs;tantes e&longs;&longs;e. <lb/>
</s><s>Cum igitur cg per centrum circuli tran&longs;eat; & ad <expan abbr="punctũ">punctum</expan> <lb/>
f perueniat nece&longs;&longs;e e&longs;t: quòd cdef &longs;it dimidium circumfe <lb/>
<figure id="id.023.01.013.1.jpg" xlink:href="023/01/013/1.jpg"/><lb/>
<arrow.to.target n="marg14"/><lb/>
rentiæ circuli. </s><s>Quare in eadem <lb/>
diametro cf erunt centra gra<lb/>
uitatis triangulorum bcd, <lb/>
afe, & quadrilateri abde, ex <lb/>
quibus con&longs;tat hexagonum ab <lb/>
cdef. </s><s>per&longs;picuum e&longs;t igitur in <lb/>
ip&longs;a cf e&longs;&longs;e circuli centrum, & <lb/>
centrum grauitatis hexagoni. <lb/>
</s><s>Rur&longs;us ducta altera diametro <lb/>
ad, ei&longs;dem rationibus o&longs;tende­<lb/>
mus in ip&longs;a utrumque <expan abbr="c&etilde;trum">centrum</expan> <lb/>
ine&longs;&longs;e. </s><s>Centrum ergo grauita­<lb/>
tis hexagoni, & centrum circuli idem erit.</s></p><p type="margin">
<s><margin.target id="marg14"/>13 Archi<lb/>
medis.</s><lb/>
<s>9. <expan abbr="eiusdetilde;">eiusdem</expan> <lb/>
m</s></p><p type="main">
<s>Sit heptagonum abcdefg æquilaterum atque æquian<lb/>
<figure id="id.023.01.013.2.jpg" xlink:href="023/01/013/2.jpg"/><lb/>
gulum in circulo de&longs;criptum: <lb/>
& iungantur ce, bf, ag: di­<lb/>
ui&longs;a autem ce bifariam in <expan abbr="pũ">pun</expan> <lb/>
cto h: & iuncta dh produca­<lb/>
tur in k. </s><s>non aliter demon­<lb/>
&longs;trabimus in linea dk e&longs;&longs;e cen <lb/>
trum circuli, & centrum gra­<lb/>
uitatis trianguli cde, & tra­<lb/>
peziorum bcef, abfg, hoc <lb/>
e&longs;t centrum totius heptago­<lb/>
ni: & rur&longs;us eadem centra in <lb/>
alia diametro cl &longs;imiliter du­<lb/>
cta contineri. </s><s>Quare & centrum grauitatis heptagoni, & <lb/>
centrum circuli in idem punctum conueniunt. </s><s>Eodem mo<pb xlink:href="023/01/014.jpg"/>do in reliquis figuris æquilateris, & æquiangulis, quæ in cir­<lb/>
culo de&longs;cribuntur, probabimus <expan abbr="c&etilde;trum">centrum</expan> grauitatis earum, <lb/>
& centrum circuli idem e&longs;&longs;e. </s><s>quod quidem demon&longs;trare <lb/>
oportebat.</s></p><p type="main">
<s>Ex quibus apparet cuiuslibet figuræ rectilineæ <lb/>
in circulo plane de&longs;criptæ centrum grauitatis <expan abbr="id&etilde;">idem</expan> <lb/>
e&longs;&longs;e, quod & circuli centrum.<lb/>
<arrow.to.target n="marg15"/></s></p><p type="margin">
<s><margin.target id="marg15"/><foreign lang="greek">gnwri/mws</foreign></s></p><p type="main">
<s>Figuram in circulo plane de&longs;criptam appella­<lb/>
mus, cuiu&longs;modi e&longs;t ea, quæ in duodecimo elemen <lb/>
torum libro, propo&longs;itione &longs;ecunda de&longs;cribitur. <lb/>
</s><s>ex æqualibus enim lateribus, & angulis con&longs;tare <lb/>
per&longs;picuum e&longs;t.</s></p><p type="head">
<s>THEOREMA II, PROPOSITIO II.<!-- KEEP S--></s></p><p type="main">
<s>Omnis figuræ rectilineæ in ellip&longs;i plane de&longs;cri­<lb/>
ptæ centrum grauitatis e&longs;t idem, quod ellip&longs;is <lb/>
centrum.</s></p><p type="main">
<s>Quo modo figura rectilinea in ellip&longs;i plane de&longs;cribatur, <lb/>
docuimus in commentarijs in quintam propo&longs;itionem li­<lb/>
bri Archimedis de conoidibus, & &longs;phæroidibus.</s></p><p type="main">
<s>Sit ellip&longs;is abcd, cuius maior axis ac, minor bd: <expan abbr="iun-gantur&qacute;">iun­<lb/>
ganturque</expan>; ab, bc, cd, da: & bifariam diuidantur in pun­<lb/>
ctis efgh. </s><s>à centro autem, quod &longs;it k ductæ lineæ ke, kf, <lb/>
kg, kh u&longs;que ad &longs;ectionem in puncta lmno protrahan­<lb/>
tur: & iungantur lm, mn, no, ol, ita ut ac &longs;ecet li­<lb/>
neas lo, mn, in z<foreign lang="greek">f</foreign> punctis; & bd &longs;ecet lm, on in <foreign lang="greek">xy.</foreign><lb/>
erunt lk, kn linea una, <expan abbr="item&qacute;ue">itemque</expan> linea unaip&longs;æ mk, ko: <lb/>
& lineæ ba, cd æquidi&longs;tabunt lineæ mo: & bc, ad ip&longs;i <lb/>
ln. </s><s>rur&longs;us lo, mn axi bd æquidi&longs;tabunt: & lm, <pb xlink:href="023/01/015.jpg" pagenum="4"/>on ip&longs;i ac. <!-- KEEP S--></s><s>Quoniam enim triangulorum abk, adk, latus <lb/>
bk e&longs;t æquale lateri kd, & ak utrique commune; <expan abbr="anguli&qacute;">angulique</expan>; <lb/>
<arrow.to.target n="marg16"/><lb/>
ad k recti. </s><s>ba&longs;is ab ba&longs;i ad; & reliqui anguli reliquis an­<lb/>
gulis æquales erunt. </s><s>eadem quoqueratione o&longs;tendetur bc <lb/>
<figure id="id.023.01.015.1.jpg" xlink:href="023/01/015/1.jpg"/><lb/>
æqualis cd; & ab ip&longs;i <lb/>
bc. <!-- REMOVE S-->quare omnes ab, <lb/>
bc, cd, da &longs;unt æqua­<lb/>
les. </s>
<s>& quoniam anguli <lb/>
ad a æquales &longs;unt angu <lb/>
lis ad c; erunt anguli b <lb/>
ac, acd coalterni inter <lb/>
&longs;e æquales; <expan abbr="item&qacute;">itemque</expan>; dac, <lb/>
acb. </s><s>ergo cd ip&longs;i ba; <lb/>
& ad ip&longs;i bc æquidi­<lb/>
&longs;tat. </s><s>At uero cum lineæ <lb/>
ab, cd inter &longs;e æquidi­<lb/>
&longs;tantes bifariam &longs;ecen­<lb/>
tur in punctis eg; erit li <lb/>
nea lekgn diameter &longs;e <lb/>
ctionis, & linea una, ex <lb/>
demon&longs;tratis in uige&longs;i­<lb/>
maoctaua &longs;ecundi coni <lb/>
corum. </s><s>Et eadem ratione linea una mfkho. </s><s>Sunt <expan abbr="aut&etilde;">autem</expan> ad, <lb/>
bc inter &longs;e &longs;e æquales, & æquidi&longs;tantes. </s><s>quare & earum di­<lb/>
<arrow.to.target n="marg17"/><lb/>
midiæ ah, bf; <expan abbr="item&qacute;">itemque</expan>; hd, fe; & quæ ip&longs;as coniungunt rectæ <lb/>
lineæ æquales, & æquidi&longs;tantes erunt. </s><s><expan abbr="æquidi&longs;tãt">æquidi&longs;tant</expan> igitur ba, <lb/>
cd diametro mo: & pariter ad, bc ip&longs;i ln æquidi&longs;tare o­<lb/>
&longs;tendemus. </s><s>Si igitur <expan abbr="man&etilde;te">manente</expan> diametro ac intelligatur abc <lb/>
portio ellip&longs;is ad portionem adc moueri, cum primum b <lb/>
applicuerit ad d, <expan abbr="cõgruet">congruet</expan> tota portio toti portioni, <expan abbr="linea&qacute;">lineaque</expan>; <lb/>
ba lineæ ad; & bc ip&longs;i cd congruet: punctum uero e ca­<lb/>
det in h; f in g: & linea ke in lineam kh: & kf in kg. <!-- KEEP S--></s><s>qua <lb/>
re & el in ho, et fm in gn. </s><s>At ip&longs;a lz in zo; et m<foreign lang="greek">f</foreign> in <foreign lang="greek">f</foreign>n <lb/>
cadet. </s><s>congruet igitur triangulum lkz triangulo okz: et <pb xlink:href="023/01/016.jpg"/>triangulum mk<foreign lang="greek">f</foreign> triangulo nk<foreign lang="greek">f.</foreign> ergo anguli lzk, ozk, <lb/>
m <foreign lang="greek">f</foreign> k, n<foreign lang="greek">f</foreign>k æquales &longs;unt, ac recti. </s><s>quòd cum etram recti <lb/>
<arrow.to.target n="marg18"/><lb/>
&longs;int, qui ad k; æquidi&longs;tabunt lineæ lo, mn axi bd. <!-- KEEP S--></s><s>& ita <lb/>
demon&longs;trabuntur lm, on ip&longs;i ac æquidi&longs;tare. </s><s>Rurfus &longs;i <lb/>
iungantur al, lb, bm, mc, cn, nd, do, oa: & bifariam di <lb/>
uidantur: à centro autem k ad diui&longs;iones ductæ lineæ pro­<lb/>
trahantur u&longs;que ad &longs;ectionem in puncta pqrstuxy: & po <lb/>
&longs;tremo py, qx, ru, st, qr, ps, yt, xu coniungantur. </s><s>Simili­<lb/>
<figure id="id.023.01.016.1.jpg" xlink:href="023/01/016/1.jpg"/><lb/>
ter o&longs;tendemus lineas <lb/>
py, qx, ru, st axi bd æ­<lb/>
quidi&longs;tantes e&longs;&longs;e: & qr, <lb/>
ps, yt, xu æquidi&longs;tan­<lb/>
tes ip&longs;i ac. <!-- KEEP S--></s><s>Itaque dico <lb/>
harum figurarum in el­<lb/>
lip&longs;i de&longs;criptarum cen­<lb/>
trum grauitatis e&longs;&longs;e <expan abbr="pũ-ctum">pun­<lb/>
ctum</expan> k, idem quod & el <lb/>
lip&longs;is centrum. </s><s>quadri­<lb/>
lateri enim abcd cen­<lb/>
trum e&longs;t k, ex decima e­<lb/>
iu&longs;dem libri Archime­<lb/>
dis, quippe <expan abbr="cũ">cum</expan> in eo om <lb/>
nes diametri <expan abbr="cõueniãt">conueniant</expan>. </s><lb/>
<s>Sed in figura albmcn <lb/>
<arrow.to.target n="marg19"/><lb/>
do, quoniam trianguli <lb/>
alb centrum grauitatis <lb/>
<arrow.to.target n="marg20"/><lb/>
e&longs;t in linea le: <expan abbr="trapezij&qacute;">trapezijque</expan>; abmo centrum in linea ek: trape <lb/>
zijomcd in kg: & trianguli cnd in ip&longs;a gn: erit magnitu <lb/>
dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen <lb/>
trum grauitatis in linea ln: & ob eandem cau&longs;&longs;am in linea <lb/>
om. </s><s>e&longs;t enim trianguli aod centrum in linea oh: trapezij <lb/>
alnd in hk: trapezij lbcn in kf: & trianguli bmc in fm. </s><lb/>
<s>cum ergo figuræ albmcndo centrum grauitatis &longs;it in li­<lb/>
nea ln, & in linea om; erit centrum ip&longs;ius punctum k, in <pb xlink:href="023/01/017.jpg" pagenum="5"/>quo &longs;cilicet ln, om conueniunt. </s><s>Po&longs;tremo in figura <lb/>
aplqbrmsctnudxoy centrum grauitatis trian <lb/>
guli pay, & trapezii ploy e&longs;t in linea az: trapeziorum <lb/>
uero lqxo, qbdx centrum e&longs;t in linea zk: & <expan abbr="trapeziorũ">trapeziorum</expan> <lb/>
brud, rmnu in k<foreign lang="greek">f:</foreign> & denique trapezii mstn; & triangu <lb/>
li sct in <foreign lang="greek">f</foreign>c. </s><s>quare magnitudinis ex his compo&longs;itæ <expan abbr="centrũ">centrum</expan> <lb/>
in linea ac con&longs;i&longs;tit. </s><s>Rur&longs;us trianguli qbr, & trapezii ql<lb/>
mr centrum e&longs;t in linea b<foreign lang="greek">x.</foreign> trapeziorum lpsm, pacs, <lb/>
aytc, yont in linea <foreign lang="greek">xf:</foreign> <expan abbr="trapeziiq;">trapeziique</expan> oxun, & trianguli <lb/>
xdu centrum in <foreign lang="greek">y</foreign>d. <!-- KEEP S--></s><s>totius ergo magnitudinis centrum <lb/>
e&longs;t in linea bd. <!-- KEEP S--></s><s>ex quo &longs;equitur, centrum grauitatis figuræ <lb/>
aplqbrmsctnudxoy e&longs;&longs;e <expan abbr="punctũ">punctum</expan> K, lineis &longs;cilicet ac, <lb/>
bd commune, quæ omnia demon&longs;trare oportebat.</s></p><p type="margin">
<s><margin.target id="marg16"/>8. primi</s></p><p type="margin">
<s><margin.target id="marg17"/>33. primi</s></p><p type="margin">
<s><margin.target id="marg18"/>28. primi.</s></p><p type="margin">
<s><margin.target id="marg19"/>13. Archi<lb/>
medis.</s></p><p type="margin">
<s><margin.target id="marg20"/>Vltima.<!-- KEEP S--></s></p><p type="head">
<s>THEOREMA III. PROPOSITIO III.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet portio­<lb/>
nis circuli, & ellip&longs;is, <lb/>
quæ dimidia non &longs;it <lb/>
maior, centrum graui <lb/>
tatis in portionis dia­<lb/>
metro con&longs;i&longs;tit.</s></p><figure id="id.023.01.017.1.jpg" xlink:href="023/01/017/1.jpg"/><p type="main">
<s>HOC eodem pror&longs;us <lb/>
modo demon&longs;trabitur, <lb/>
quo in libro de centro gra<lb/>
uitatis planorum ab Ar­<lb/>
chimede <expan abbr="demon&longs;tratũ">demon&longs;tratum</expan> e&longs;t, <lb/>
in portione <expan abbr="cõtenta">contenta</expan> recta <lb/>
linea, & rectanguli coni &longs;e <lb/>
ctione grauitatis <expan abbr="c&etilde;trum">centrum</expan> <lb/>
e&longs;&longs;e in diametro portio­<lb/>
nis. </s><s>Et ita demon&longs;trari po <lb/>
<pb xlink:href="023/01/018.jpg"/>te&longs;t in portione, quæ recta linea & obtu&longs;ianguli coni &longs;e­<lb/>
ctione, &longs;eu hyperbola continetur.</s></p><p type="head">
<s>THEOREMA IIII. PROPOSITIO IIII.<!-- KEEP S--></s></p><p type="main">
<s>IN circulo & ellip&longs;i idem e&longs;t figuræ & graui­<lb/>
tatis centrum.</s></p><p type="main">
<s>SIT circulus, uel ellip&longs;is, cuius centrum a. </s><s>Dico a gra­<lb/>
uitatis quoque centrum e&longs;&longs;e. </s><s>Si enim fieri pote&longs;t, &longs;it b cen­<lb/>
trum grauitatis: & iuncta ab extra figuram in c produca <lb/>
tur: quam uero proportionem habet linea ca ad ab, ha­<lb/>
beat circulus a ad alium circulum, in quo d; uel ellip&longs;is ad <lb/>
aliam ellip&longs;im: & in circulo, uel ellip&longs;i figura rectilinea pla­<lb/>
ne de&longs;cribatur adco, ut tandem relinquantur portiones <lb/>
quædam minores circulo, uel ellip&longs;i d; quæ figura &longs;it abcefg <lb/>
hklmn. </s><s>Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>
elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con­<lb/>
<figure id="id.023.01.018.1.jpg" xlink:href="023/01/018/1.jpg"/><lb/>
&longs;tat; at in ellip&longs;i nos demon&longs;tra­<lb/>
uimus in commentariis in quin­<lb/>
tam propo&longs;itionem Archimedis <lb/>
de conoidibus, & &longs;phæroidibus. </s><lb/>
<s>erit igitur a centrum grauitatis <lb/>
ip&longs;ius figuræ, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;ten</expan> <lb/>
dimus. </s><s>Itaque quoniam circulus <lb/>
a ad circulum d, uel ellip&longs;is a ad <lb/>
ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>
habet, quam linea ca ad ab: <lb/>
portiones uero &longs;unt minores cir <lb/>
<arrow.to.target n="marg21"/><lb/>
culo uel ellip&longs;i d: habebit circu­<lb/>
lus, uel ellip&longs;is ad portiones ma­<lb/>
iorem proportionem, quàm ca <lb/>
<arrow.to.target n="marg22"/><lb/>
ad ab: & diuidendo figura recti­<lb/>
linea abcefghklmn ad portiones <pb xlink:href="023/01/019.jpg" pagenum="6"/><figure id="id.023.01.019.1.jpg" xlink:href="023/01/019/1.jpg"/><lb/>
habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam cb ad ba. </s><s>fiat ob ad ba, <lb/>
ut figura rectilinea ad portio­<lb/>
nes. </s><s>cum igitur à circulo, uel el­<lb/>
lip&longs;i, cuius grauitatis centrum <lb/>
e&longs;t b, auferatur figura rectilinea <lb/>
efghklmn, cuius centrum a; <lb/>
reliquæ magnitudinis ex portio <lb/>
<arrow.to.target n="marg23"/><lb/>
nibus compo&longs;itæ centrum graui <lb/>
tatis erit in linea ab producta, <lb/>
& in puncto o, extra figuram po <lb/>
&longs;ito. </s><s>quod quidem fieri nullo mo <lb/>
do po&longs;&longs;e per&longs;picuum e&longs;t. </s><s>&longs;equi­<lb/>
tur ergo, ut circuli & ellip&longs;is cen <lb/>
trum grauitatis &longs;it punctum a, <lb/>
idem quod figuræ centrum.</s></p><p type="margin">
<s><margin.target id="marg21"/>8. quinti</s></p><p type="margin">
<s><margin.target id="marg22"/>19. quinti <lb/>
apud <expan abbr="Cã">Cam</expan> <lb/>
panum.</s></p><p type="margin">
<s><margin.target id="marg23"/>8. Archi­<lb/>
medis.</s></p><p type="head">
<s>ALITER.<!-- KEEP S--></s></p><p type="main">
<s>Sit circulus, uel ellip&longs;is abcd, <lb/>
cuius diameter db, & centrum e: <expan abbr="ducaturq;">ducaturque</expan> per e recta li<lb/>
nea ac, &longs;ecans ip&longs;am db ad rectos angulos. </s><s>erunt adc, <lb/>
abc circuli, uel ellip&longs;is dimidiæ portiones. </s><s>Itaque quo­<lb/>
<figure id="id.023.01.019.2.jpg" xlink:href="023/01/019/2.jpg"/><lb/>
niam por <lb/>
<expan abbr="tiõis">tionis</expan> adc <lb/>
<expan abbr="c&etilde;trũ">centrum</expan> gra­<lb/>
uitatis e&longs;t <lb/>
in diame­<lb/>
tro de: & <lb/>
portionis <lb/>
abc cen­<lb/>
trum e&longs;t <expan abbr="ĩ">im</expan> <lb/>
ip&longs;a eb: to <lb/>
tius circu <lb/>
li, uel ellip&longs;is grauitatis centrum erit in diametro db. </s><lb/>
<s>Sit autem portionis adc <expan abbr="c&etilde;trum">centrum</expan> grauitatis f: & &longs;umatur <pb xlink:href="023/01/020.jpg"/>in linea eb <expan abbr="punctũ">punctum</expan> g, ita ut fit ge æqualis ef. </s><s>erit g por­<lb/>
tionis abc centrum. </s><s>nam &longs;i hæ portiones, quæ æquales <lb/>
& &longs;imiles &longs;unt, inter &longs;e &longs;e aptentur, ita ut be cadat in de, <lb/>
& punctum b in d cadet, & g in f: figuris autem æquali­<lb/>
bus, & &longs;imilibus inter &longs;e aptatis, centra quoque grauitatis <lb/>
ip&longs;arum inter &longs;e aptata erunt, ex quinta petitione Archi­<lb/>
medis in libro de centro grauitatis planorum. </s><s>Quare cum <lb/>
portionis adc centrum grauitatis &longs;it f: & portionis <lb/>
abc centrum g: magnitudinis; quæ ex utri&longs;que efficitur: <lb/>
hoc e&longs;t circuli uel ellip&longs;is grauitatis centrum in medio li­<lb/>
neæ fg, quod e&longs;t e, con&longs;i&longs;tet, ex quarta propo&longs;itione eiu&longs;­<lb/>
dem libri Archimedis. <!-- KEEP S--></s><s>ergo circuli, uel ellip&longs;is centrum <lb/>
grauitatis e&longs;t idem, quod figuræ centrum. </s><s>atque illud e&longs;t, <lb/>
quod demon&longs;trare oportebat.</s></p><p type="main">
<s>Ex quibus &longs;equitur portionis circuli, uel ellip­<lb/>
&longs;is, quæ dimidia maior &longs;it, centrum grauitatis in <lb/>
diametro quoque ip&longs;ius con&longs;i&longs;tere.</s></p><figure id="id.023.01.020.1.jpg" xlink:href="023/01/020/1.jpg"/><p type="main">
<s>Sit enim maior portio abc, cu<emph type="italics"/>i<emph.end type="italics"/>us diameter bd, & com­<lb/>
pleatur circulus, uel ellip&longs;is, ut portio reliqua fit aec, dia<pb xlink:href="023/01/021.jpg" pagenum="7"/>metrum habens ed. <!-- KEEP S--></s><s>Quoniam igitur circuli uel ellip&longs;is <lb/>
aecb grauitatis centrum e&longs;t in diametro be, & portio­<lb/>
nis aec centrum in linea ed: reliquæ portionis, uidelicet <lb/>
abc centrum grauitatis in ip&longs;a bd con&longs;i&longs;tat nece&longs;&longs;e e&longs;t, ex <lb/>
octaua propo&longs;itione eiu&longs;dem.</s></p><p type="head">
<s>THEOREMA V. PROPOSITIO V.<!-- KEEP S--></s></p><p type="main">
<s>SI pri&longs;ma &longs;ecetur plano oppo&longs;itis planis æqui <lb/>
di&longs;tante, &longs;ectio erit figura æqualis & &longs;imilis ei, <lb/>
quæ e&longs;t oppo&longs;itorum planorum, centrum graui <lb/>
tatis in axe habens.</s></p><p type="main">
<s>Sit pri&longs;ma, in quo plana oppo&longs;ita &longs;int triangula abc, <lb/>
def; axis gh: & &longs;ecetur plano iam dictis planis <expan abbr="æquidi&longs;tã">æquidi&longs;tan</expan> <lb/>
te; quod faciat &longs;ectionem klm; & axi in <expan abbr="pũcto">puncto</expan> n occurrat. </s><lb/>
<s>Dico klm triangulum æquale e&longs;&longs;e, & fimile triangulis abc <lb/>
def; atque eius grauitatis centrum e&longs;&longs;e punctum n. </s><s>Quo­<lb/>
<figure id="id.023.01.021.1.jpg" xlink:href="023/01/021/1.jpg"/><lb/>
niam enim plana abc <lb/>
Klm æquidi&longs;tantia <expan abbr="&longs;ecã">&longs;ecan</expan> <lb/>
<arrow.to.target n="marg24"/><lb/>
tur a plano ae; rectæ li­<lb/>
neæ ab, Kl, quæ &longs;unt ip <lb/>
&longs;orum <expan abbr="cõmunes">communes</expan> &longs;ectio­<lb/>
nes inter &longs;e &longs;e æquidi­<lb/>
&longs;tant. </s><s>Sed æquidi&longs;tant <lb/>
ad, be; cum ae &longs;it para <lb/>
lelogrammum, ex pri&longs;­<lb/>
matis diffinitione. </s><s>ergo <lb/>
& al <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb/>
erit; & propterea linea <lb/>
<arrow.to.target n="marg25"/><lb/>
kl, ip&longs;i ab æqualis. </s><s>Si­<lb/>
militer demon&longs;trabitur <lb/>
lm æquidi&longs;tans, & æqua <lb/>
lis bc; & mk ip&longs;i ca.</s><pb xlink:href="023/01/022.jpg"/>
<s>Itaque quoniam duæ lineæ Kl, lm &longs;e &longs;e tangentes, duabus <lb/>
lineis &longs;e &longs;e tangentibus ab, bc æquidi&longs;tant; nec &longs;unt in e o­<lb/>
dem plano: angulus klm æqualis e&longs;t angulo abc: & ita an <lb/>
<arrow.to.target n="marg26"/><lb/>
gulus lmk, angulo bca, & mk lip&longs;i cab æqualis probabi<lb/>
tur. </s><s>triangulum ergo klm e&longs;t æquale, & &longs;imile triang ulo <lb/>
abc. quare & triangulo def. </s>
<s>Ducatur linea cgo, & per ip <lb/>
&longs;am, & per cf ducatur planum &longs;ecans pri&longs;ma; cuius & paral <lb/>
lelogrammi ae communis &longs;ectio &longs;it opq.</s><s> tran&longs;ibit linea <lb/>
fq per h, & mp per n. </s><s>nam cum plana æquidi&longs;tantia &longs;ecen <lb/>
tur à plano cq, communes eorum &longs;ectiones cgo, mp, fq <lb/>
&longs;ibi ip&longs;is æquidi&longs;tabunt. </s><s>Sed & æquidi&longs;tant ab, kl, de. </s><s>an­<lb/>
<arrow.to.target n="marg27"/><lb/>
guli ergo aoc, kpm, dqf inter &longs;e æquales &longs;unt: & &longs;unt <lb/>
æquales qui ad puncta akd con&longs;tituuntur. </s><s>quare & reliqui <lb/>
reliquis æquales; & triangula aco, Kmp, dfq inter &longs;e &longs;imi <lb/>
<arrow.to.target n="marg28"/><lb/>
lia erunt. </s><s>Vt igitur ca ad ao, ita fd ad dq: & permutando <lb/>
ut ca ad fd, ita ao ad dq.</s><s>e&longs;t autem ca æqualis fd. <!-- KEEP S--></s><s>ergo & <lb/>
ao ip&longs;i dq.</s><s> eadem quoque ratione & ao ip&longs;i Kp æqualis <lb/>
demon&longs;trabitur. </s><s>Itaque &longs;i triangula, abc, def æqualia & <lb/>
<figure id="id.023.01.022.1.jpg" xlink:href="023/01/022/1.jpg"/><lb/>
&longs;imilia inter &longs;e <expan abbr="apt&etilde;tur">aptentur</expan>, <lb/>
cadet linea fq in lineam <lb/>
<arrow.to.target n="marg29"/><lb/>
cgo. </s><s>Sed & <expan abbr="centrũ">centrum</expan> gra<lb/>
uitatis h in g <expan abbr="centrũ">centrum</expan> ca­<lb/>
det. </s><s><expan abbr="trã&longs;ibit">tran&longs;ibit</expan> igitur linea <lb/>
fq per h: & planum per <lb/>
co & cf <expan abbr="ductũ">ductum</expan> per <expan abbr="ax&etilde;">axem</expan> <lb/>
gh ducetur: <expan abbr="idcircoq;">idcircoque</expan> li <lb/>
neam mp <expan abbr="etiã">etiam</expan> per n <expan abbr="trã">tram</expan> <lb/>
&longs;ire nece&longs;&longs;e erit. </s><s>Quo­<lb/>
niam ergo fh, cg æqua­<lb/>
les &longs;unt, & <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan>: <lb/>
<expan abbr="itemq;">itemque</expan> hq, go; rectæ li­<lb/>
neæ, quæ ip&longs;as <expan abbr="cõnectũt">connectunt</expan> <lb/>
cmf, gnh, opq æqua­<lb/>
les æquidi&longs;tantes <expan abbr="erũt">erunt</expan>.</s>
<pb xlink:href="023/01/023.jpg" pagenum="8"/><s>æquidi&longs;tant autem cgo, mnp. </s><s>ergo <expan abbr="parallelogrãma">parallelogramma</expan> &longs;unt <lb/>
on, gm, & linea mn æqualis cg; & np ip&longs;i go. </s><s>aptatis igi­<lb/>
tur klm, abc <expan abbr="triãgulis">triangulis</expan>, quæ æqualia & &longs;imilia <expan abbr="sũt">sunt</expan>; linea mp <lb/>
in co, & punctum n in g cadet. </s><s>Quòd <expan abbr="cũ">cum</expan> g &longs;it centrum gra­<lb/>
uitatis trianguli abc, & n trianguli klm grauitatis cen­<lb/>
trum erit id, quod demon&longs;trandum relinquebatur. </s><s>Simili <lb/>
ratione idem contingere demon&longs;trabimus in aliis pri&longs;ma­<lb/>
tibus, &longs;iue quadrilatera, &longs;iue plurilatera habeant plana, <lb/>
quæ opponuntur.</s></p><p type="margin">
<s><margin.target id="marg24"/>16. unde­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg25"/>34. primi</s></p><p type="margin">
<s><margin.target id="marg26"/>10. unde <lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg27"/>10. unde­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg28"/>4. &longs;exti</s></p><p type="margin">
<s><margin.target id="marg29"/>per 5. pe­<lb/>
titionem <lb/>
Archime<lb/>
dis.</s></p><p type="head">
<s>COROLLARIVM.<!-- KEEP S--></s></p><p type="main">
<s>Ex iam demon&longs;tratis per&longs;picue apparet, cuius <lb/>
libet pri&longs;matis axem, parallelogrammorum lateri <lb/>
bus, quæ ab oppo&longs;itis planis <expan abbr="ducũtur">ducuntur</expan> æquidi&longs;tare.</s></p><p type="head">
<s>THEOREMA VI. PROPOSITIO VI.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet pri&longs;matis centrum grauitatis e&longs;t in <lb/>
plano, quod oppo&longs;itis planis æquidi&longs;tans, reli­<lb/>
quorum planorum latera bifariam diuidit.</s></p><p type="main">
<s>Sit pri&longs;ma, in quo plana, quæ opponuntur &longs;int trian­<lb/>
gula ace, bdf: & parallelogrammorum latera ab, cd, <lb/>
ef bifariam <expan abbr="diuidãtur">diuidantur</expan> in punctis ghk: per diui&longs;iones au­<lb/>
<arrow.to.target n="marg30"/><lb/>
tem planum ducatur; cuius &longs;ectio figura gh K. <!-- KEEP S--></s><s>erit linea <lb/>
gh æquidi&longs;tans lineis ac, bd & hk ip&longs;is ce, df. </s><s>quare ex <lb/>
decimaquinta undecimi elementorum, planum illud pla <lb/>
nis ace, bdf æquidi&longs;tabit, & faciet &longs;ectionem figu­<lb/>
<arrow.to.target n="marg31"/><lb/>
ram ip&longs;is æqualem, & &longs;imilem, ut proxime demon&longs;tra­<lb/>
uimus. </s><s>Dico centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>
ghk. </s><s>Si enim fieri pote&longs;t, &longs;it eius centrum l: & ducatur <lb/>
lm u&longs;que ad planum ghk, quæ ip&longs;i ab æquidi&longs;tet. </s><pb xlink:href="023/01/024.jpg"/>
<s><arrow.to.target n="marg32"/>ergo linea ag continenter in duas partes æquales diui­<lb/>
&longs;a, relinquetur <expan abbr="tãdem">tandem</expan> pars aliqua ng, quæ minor erit lm. </s><lb/>
<s>Vtraque uero linearum ag, gb diuidatur in partes æqua­<lb/>
les ip&longs;i ng: & per puncta diui&longs;ionum plana oppo&longs;itis pla­<lb/>
<arrow.to.target n="marg33"/><lb/>
nis æquidi&longs;tantia ducantur. </s><s>erunt &longs;ectiones figuræ æqua­<lb/>
les, ac &longs;imiles ip&longs;is ace, bdf: & totum pri&longs;ma diui&longs;um erit <lb/>
in pri&longs;mata æqualia, & &longs;imilia: quæ cum inter &longs;e <expan abbr="congruãt">congruant</expan>; <lb/>
& grauitatis centra &longs;ibi ip&longs;is congruentia, <expan abbr="re&longs;pondentiaq;">re&longs;pondentiaque</expan> <lb/>
<figure id="id.023.01.024.1.jpg" xlink:href="023/01/024/1.jpg"/><lb/>
habebunt. </s><s><expan abbr="Itaq:">Itaque</expan> <lb/>
&longs;unt magnitudi­<lb/>
nes <expan abbr="quædã">quædam</expan> æqua­<lb/>
les ip&longs;i nh, & nu­<lb/>
mero pares, qua­<lb/>
rum centra gra­<lb/>
uitatis in <expan abbr="ead&etilde;re">eaderre</expan> <lb/>
cta linea con&longs;ti­<lb/>
tuuntur: duæ ue­<lb/>
ro mediæ æqua­<lb/>
les &longs;unt: & quæ ex <lb/>
utraque parte i­<lb/>
p&longs;arum &longs;imili-­<lb/>
ter æquales: & æ­<lb/>
quales rectæ li­<lb/>
neæ, quæ inter <lb/>
grauitatis centra <lb/>
interiiciuntur. </s><lb/>
<s>quare ex corolla­<lb/>
rio quintæ pro­<lb/>
po&longs;itionis primi <lb/>
libri Archimedis <lb/>
de centro graui­<lb/>
tatis planorum; magnitudinis ex his omnibus compo&longs;itæ <lb/>
centrum grauitatis e&longs;t in medio lineæ, quæ magnitudi­<lb/>
num mediarum centra coniungit. </s><s>at qui non ita res ha­<pb xlink:href="023/01/025.jpg" pagenum="9"/>bet, &longs;i quidem l extra medias magnitudines po&longs;itum e&longs;t. </s><lb/>
<s>Con&longs;tat igitur centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>
<figure id="id.023.01.025.1.jpg" xlink:href="023/01/025/1.jpg"/><lb/>
ghk, quod nos demon&longs;trandum propo&longs;uimus. </s><s>At &longs;i op­<lb/>
po&longs;ita plana in pri&longs;mate &longs;int quadrilatera, uel plurilatera, <lb/>
eadem erit in omnibus demon&longs;tratio.</s></p><p type="margin">
<s><margin.target id="marg30"/>33. primi</s></p><p type="margin">
<s><margin.target id="marg31"/>5. huius</s></p><p type="margin">
<s><margin.target id="marg32"/>1. decimi</s></p><p type="margin">
<s><margin.target id="marg33"/>5 huius</s></p><p type="head">
<s>THEOREMA VII. PROPOSITIO VII.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet cylindri, & cuiuslibet cylindri por <lb/>
tionis centrum grauitatis e&longs;t in plano, quod ba&longs;i­<lb/>
bus æquidi&longs;tans, parallelogrammi per axem late­<lb/>
ra bifariam &longs;ecat.</s></p><pb xlink:href="023/01/026.jpg"/><p type="main">
<s>SIT cylindrus, uel cylindri portio ac: & plano per a­<lb/>
xem ducto &longs;ecetur; cuius fectio &longs;it parallelogrammum ab <lb/>
c d: & bifariam diui&longs;is ad, bc parallelogrammi lateribus, <lb/>
per diui&longs;ionum puncta ef planum ba&longs;i æquidi&longs;tans duca­<lb/>
tur; quod faciet &longs;ectionem, in cylindro quidem circulum <lb/>
æqualem iis, qui &longs;unt in ba&longs;ibus, ut demon&longs;trauit Serenus <lb/>
in libro cylindricorum, propo&longs;itione quinta: in cylindr <lb/>
uero portione ellip&longs;im æqualem, & &longs;imilem eis, quæ &longs;unt <lb/>
<figure id="id.023.01.026.1.jpg" xlink:href="023/01/026/1.jpg"/><lb/>
in oppo&longs;itis planis, quod nos <lb/>
demon&longs;trauimus in commen <lb/>
tariis in librum Archimedis <lb/>
de conoidibus, & &longs;phæroidi­<lb/>
bus. </s><s>Dico centrum grauita­<lb/>
tis cylindri, uel cylindri por­<lb/>
tionis e&longs;&longs;e in plano ef. </s><s>Si <expan abbr="enĩ">enim</expan> <lb/>
fieri pote&longs;t, fit centrum g: & <lb/>
ducatur gh ip&longs;i ad æquidi­<lb/>
&longs;tans, u&longs;que ad ef planum. </s><lb/>
<s>Itaque linea ae continenter <lb/>
diui&longs;a bifariam, erit tandem <lb/>
pars aliqua ip&longs;ius ke, minor <lb/>
gh. </s><s>Diuidantur ergo lineæ <lb/>
ae, ed in partes æquales ip&longs;i <lb/>
ke: & per diui&longs;iones plana ba <lb/>
&longs;ibus æquidi&longs;tantia <expan abbr="ducãtur">ducantur</expan>. </s><lb/>
<s>erunt iam &longs;ectiones, figuræ æ­<lb/>
quales, & &longs;imiles eis, quæ &longs;unt <lb/>
in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: & cy <lb/>
lindri portio in portiones æquales, & &longs;imiles ip&longs;i kf. </s><s>reli­<lb/>
qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s></p>
<pb xlink:href="023/01/027.jpg" pagenum="10"/><figure id="id.023.01.027.1.jpg" xlink:href="023/01/027/1.jpg"/><p type="head">
<s>THEOREMA VIII. PROPOSITIO VIII.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet pri&longs;matis, & cuiuslibet cylindri, uel <lb/>
cylindri portionis grauitatis centrum in medio <lb/>
ip&longs;ius axis con&longs;i&longs;tit.</s></p><p type="main">
<s>Sit primum af pri&longs;ma æquidi&longs;tantibus planis <expan abbr="contentũ">contentum</expan>, <lb/>
quod &longs;olidum parallelepipedum appellatur: & oppo&longs;ito­<lb/>
rum planorum cf, ah, da, fg latera bifariam diuidantur in <lb/>
punctis klmnopqrstux: & per diui&longs;iones ducantur <lb/>
plana kn, or, sx. </s><s>communes autem eorum planorum &longs;e­<lb/>
ctiones &longs;int lineæ yz, <foreign lang="greek">qf, xy:</foreign> quæ in puncto <foreign lang="greek">w</foreign> <expan abbr="conueniãt">conueniant</expan>. </s><lb/>
<s>erit ex decima eiu&longs;dem libri Archimedis parallelogrammi <lb/>
cf centrum grauitatis punctum y; parallelogrammi ah <pb xlink:href="023/01/028.jpg"/>centrum z: parallelogrammi ad, <foreign lang="greek">q:</foreign> parallelogrammi fg, <foreign lang="greek">f:</foreign><lb/>
<figure id="id.023.01.028.1.jpg" xlink:href="023/01/028/1.jpg"/><lb/>
parallelogrammi dh, <foreign lang="greek">x:</foreign> & <lb/>
parallelogrammi cg <expan abbr="centrũ">centrum</expan> <lb/>
<foreign lang="greek">y:</foreign> atque erit <foreign lang="greek">w</foreign> punctum me <lb/>
dium uniu&longs;cuiu&longs;que axis, ui <lb/>
delicet eius lineæ quæ oppo <lb/>
&longs;itorum <expan abbr="planorũ">planorum</expan> centra con <lb/>
iungit. </s><s>Dico <foreign lang="greek">w</foreign> centrum e&longs;&longs;e <lb/>
grauitatis ip&longs;ius &longs;olidi. </s><s>e&longs;t <lb/>
<arrow.to.target n="marg34"/><lb/>
enim, ut demon&longs;trauimus, <lb/>
&longs;olidi af centrum grauitatis <lb/>
in plano Kn; quod oppo&longs;i­<lb/>
tis planis ad, gf æquidi&longs;tans <lb/>
reliquorum planorum late­<lb/>
ra bifariam diuidit: & &longs;imili <lb/>
ratione idem centrum e&longs;t in plano or, æquidi&longs;tante planis <lb/>
ae, bf oppo&longs;itis. </s><s>ergo in communi ip&longs;orum &longs;ectione: ui­<lb/>
delicet in linea yz. </s><s>Sed e&longs;t etiam in plano tu, quod <expan abbr="quid&etilde;">quidem</expan> <lb/>
yz &longs;ecatin <foreign lang="greek">w.</foreign> Con&longs;tatigitur centrum grauitatis &longs;olidi e&longs;&longs;e <lb/>
punctum <foreign lang="greek">w,</foreign> medium &longs;cilicet axium, hoc e&longs;t linearum, quæ <lb/>
planorum oppo&longs;itorum centra coniungunt.</s></p><p type="margin">
<s><margin.target id="marg34"/>6 huius</s></p><p type="main">
<s>Sit aliud prima af; & in eo plana, quæ opponuntur, tri­<lb/>
angula abc, def: <expan abbr="diui&longs;isq;">diui&longs;isque</expan> bifariam parallelogrammorum <lb/>
lateribus ad, be, cf in punctis ghk, per diui&longs;iones <expan abbr="planũ">planum</expan> <lb/>
ducatur, quod oppo&longs;itis planis æquidi&longs;tans faciet <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> <lb/>
triangulum ghx æquale, & &longs;imile ip&longs;is abc, def. </s><s>Rur&longs;us <lb/>
diuidatur ab bifariam in l: & iuncta cl per ip&longs;am, & per <lb/>
cKf planum ducatur pri&longs;ma &longs;ecans, cuius, & <expan abbr="parallelogrã">parallelogram</expan> <lb/>
mi ae communis &longs;ectio &longs;it lmn. </s><s>diuidet punctum m li­<lb/>
neam gh bifariam; & ita n diuidet lineam de: quoniam <lb/>
<arrow.to.target n="marg35"/><lb/>
triangula acl, gkm, dfn æqualia &longs;unt, & &longs;imilia, ut &longs;upra <lb/>
demon&longs;trauimus. </s><s>Iam ex iis, quæ tradita &longs;unt, con&longs;tat cen <lb/>
trum grauitatis pri&longs;matis in plano ghk contineri. </s><s>Dico <lb/>
ip&longs;um e&longs;&longs;e in linea km. </s><s>Si enim fieri pote&longs;t, &longs;it o centrum; <pb xlink:href="023/01/029.jpg" pagenum="11"/>& per o ducatur op ad km ip&longs;i hg æquidi&longs;tans. </s><s>Itaque li <lb/>
nea hm <expan abbr="bifariã">bifariam</expan> u&longs;que cò diuidatur, quoad reliqua &longs;it pars <lb/>
quædam qm, minor op. </s><s>deinde hm, mg diuidantur in <lb/>
partes æquales ip&longs;i mq: & per diui&longs;iones lineæ ip&longs;i mK <lb/>
æquidi&longs;tantes ducantur. </s><s>puncta uero, in quibus hæ trian­<lb/>
gulorum latera &longs;ecant, coniungantur ductis lineis rs, tu, <lb/>
<figure id="id.023.01.029.1.jpg" xlink:href="023/01/029/1.jpg"/><lb/>
xy; quæ ba&longs;i gh æquidi&longs;tabunt. </s><s>Quoniam enim lineæ gz, <lb/>
h<foreign lang="greek">a</foreign> &longs;unt æquales: <expan abbr="itemq;">itemque</expan> æquales gm, mh: ut mg ad gz, <lb/>
ita erit mh, ad h<foreign lang="greek">a:</foreign> & diuidendo, ut mz ad zg, ita m<foreign lang="greek">a</foreign> ad <lb/>
<arrow.to.target n="marg36"/><lb/>
<foreign lang="greek">a</foreign>h. </s><s>Sed ut mz ad zg, ita kr ad rg: & ut m<foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign>h, ita ks <lb/>
ad sh. </s><s>quare ut kr ad rg, ita ks ad sh. </s><s>æquidi&longs;tant igitur <lb/>
<arrow.to.target n="marg37"/><lb/>
inter &longs;e &longs;e rs, gh. </s><s>eadem quoque ratione demon&longs;trabimus <pb xlink:href="023/01/030.jpg"/>tu, xy ip&longs;i gh æquidi&longs;tare. </s><s>Et quoniam triangula, quæ <lb/>
fiunt à lineis Ky, yu, us, sh æqualia &longs;unt inter &longs;e, & &longs;imilia <lb/>
<arrow.to.target n="marg38"/><lb/>
triangulo Kmh: habebit triangulum Kmh ad <expan abbr="triangulũ">triangulum</expan> <lb/>
K<foreign lang="greek">d</foreign>y duplam proportionem eius, quæ e&longs;t lineæ kh ad Ky. </s><lb/>
<s>&longs;ed Kh po&longs;ita e&longs;t quadrupla ip&longs;ius ky. </s><s>ergo triangulum <lb/>
kmh ad triangulum K<foreign lang="greek">d</foreign>y <expan abbr="eãdem">eandem</expan> proportionem habebit, <lb/>
quam &longs;exdecim ad <expan abbr="unũ">unum</expan>: & ad quatuor triangula k<foreign lang="greek">d</foreign>y, yu, <lb/>
us, s<foreign lang="greek">a</foreign>h habebit eandem, quam &longs;exdecim ad quatuor, hoc <lb/>
e&longs;t quam hK ad ky: & &longs;imiliter eandem habere demon&longs;tra <lb/>
<figure id="id.023.01.030.1.jpg" xlink:href="023/01/030/1.jpg"/><lb/>
bitur trian­<lb/>
gulum kmg <lb/>
ad quatuor <lb/>
<expan abbr="triãgula">triangula</expan> K<foreign lang="greek">d</foreign><lb/>
x, x<foreign lang="greek">g</foreign>t, t<foreign lang="greek">b</foreign>r, <lb/>
<arrow.to.target n="marg39"/><lb/>
rzg. <!-- KEEP S--></s><s>quare <lb/>
totum trian <lb/>
gulum Kgh <lb/>
ad omnia tri <lb/>
angula gzr, <lb/>
r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign><lb/>
K, K<foreign lang="greek">d</foreign>y, yu, <lb/>
us, s<foreign lang="greek">a</foreign>h ita <lb/>
erit, ut hk ad <lb/>
ky, hoc e&longs;t <lb/>
ut hm ad m<lb/>
q. </s><s>Si igitur in <lb/>
triangulis abc, def de&longs;cribantur figuræ &longs;imiles ei, quæ de­<lb/>
&longs;cripta e&longs;t in ghK triangulo: & per lineas &longs;ibi re&longs;ponden­<lb/>
tes plana ducantur: totum pri&longs;ma af diui&longs;um erit in tria <lb/>
&longs;olida parallelepipeda y<foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz, quorum ba&longs;es &longs;unt æqua <lb/>
les & &longs;imiles ip&longs;is parallelogrammis y <foreign lang="greek">g,</foreign>u<foreign lang="greek">b,</foreign> sz: & in octo <lb/>
pri&longs;mata gzr, r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign><lb/>
K, k<foreign lang="greek">d</foreign>y, yu, us, s<foreign lang="greek">a</foreign>h: quorum <lb/>
item ba&longs;es æquales, & &longs;imiles &longs;unt dictis triangulis; altitu­<lb/>
do autem in omnibus, totius pri&longs;matis altitudini æqualis.<pb xlink:href="023/01/031.jpg" pagenum="12"/>Itaque &longs;olidi parallelepipedi y<foreign lang="greek">g</foreign> centrum grauitatis e&longs;t in <lb/>
linea <foreign lang="greek">de:</foreign> &longs;olidi u<foreign lang="greek">b</foreign> centrum e&longs;t in linea <foreign lang="greek">eh:</foreign> & &longs;olidi sz in li<lb/>
nea <foreign lang="greek">h</foreign>m, quæ quidem lineæ axes &longs;unt, cum planorum oppo <lb/>
&longs;itorum centra coniungant. </s><s>ergo magnitudinis ex his &longs;oli <lb/>
dis compo&longs;itæ centrum grauitatis e&longs;t in linea <foreign lang="greek">d</foreign>m, quod &longs;it <lb/>
<foreign lang="greek">q</foreign>; & iuncta <foreign lang="greek">q</foreign>o producatur: à puncto autem h ducatur h<foreign lang="greek">a</foreign><lb/>
ip&longs;i mk æquidi&longs;tans, quæ cum <foreign lang="greek">q</foreign>o in <foreign lang="greek">m</foreign> conueniat. </s><s>triangu <lb/>
lum igitur ghk ad omnia triangula gzr, <foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign>k, <lb/>
k<foreign lang="greek">d</foreign>y, yu, us, s<foreign lang="greek">a</foreign>h eandem habet proportionem, quam hm <lb/>
ad mq; hoc e&longs;t, quam <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql:</foreign> nam &longs;i hm, <foreign lang="greek">mq</foreign> produci in <lb/>
telligantur, quou&longs;que coeant; erit ob linearum qy, mk æ­<lb/>
quidi&longs;tantiam, ut hq ad qm, ita <foreign lang="greek">ml</foreign> ad ad <foreign lang="greek">lq:</foreign> & componen <lb/>
do, ut hm ad mq, ita <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql.</foreign></s><s> linea uero <foreign lang="greek">q</foreign>o maior e&longs;t, <lb/>
<arrow.to.target n="marg40"/><lb/>
quàm <foreign lang="greek">ql:</foreign> habebit igitur <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql</foreign> maiorem proportio­<lb/>
nem, quàm ad <foreign lang="greek">q</foreign>o. </s><s>quare triangulum etiam ghk ad omnia <lb/>
iam dicta triangula maiorem <expan abbr="proportion&etilde;">proportionem</expan> habebit, quàm <lb/>
<foreign lang="greek">mq</foreign> ad <foreign lang="greek">q</foreign>o. </s><s>&longs;ed ut <expan abbr="triangulũ">triangulum</expan> ghk ad omnia triangula, ita <expan abbr="to-tũ">to­<lb/>
tum</expan> pri&longs;ma afad omnia pri&longs;mata gzr, r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">dk, kd</foreign> y, <lb/>
yu, us, s<foreign lang="greek">a</foreign>h: quoniam enim &longs;olida parallelepipeda æque al <lb/>
ta, eandem inter &longs;e proportionem habent, quam ba&longs;es; ut <lb/>
ex trige&longs;ima&longs;ecunda undecimi elementorum con&longs;tat. </s><s>&longs;unt <lb/>
<arrow.to.target n="marg41"/><lb/>
autem &longs;olida parallelepipeda pri&longs;matum triangulares ba­<lb/>
<arrow.to.target n="marg42"/><lb/>
&longs;es habentium dupla: &longs;equitur, ut etiam huiu&longs;modi pri&longs;­<lb/>
mata inter &longs;e &longs;int, &longs;icut eorum ba&longs;es. </s><s>ergo totum pri&longs;ma ad <lb/>
omnia pri&longs;mata maiorem proportionem habet, quam <foreign lang="greek">mq</foreign><lb/>
<arrow.to.target n="marg43"/><lb/>
ad <foreign lang="greek">q</foreign>o: & diuidendo &longs;olida parallelepipeda y<foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz ad o­<lb/>
mnia pri&longs;mata proportionem habent maiorem, quàm <foreign lang="greek">m</foreign>o <lb/>
ad o<foreign lang="greek">q</foreign>. </s><s>fiat <foreign lang="greek">n</foreign>o ad o<foreign lang="greek">q,</foreign> ut &longs;olida parallelepipeda y <foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz ad <lb/>
omnia pri&longs;mata. </s><s>Itaque cum à pri&longs;mate af, cuius <expan abbr="c&etilde;trum">centrum</expan> <lb/>
grauitatis e&longs;t o, auferatur magnitudo ex &longs;olidis parallelepi <lb/>
pedis y <foreign lang="greek">g,</foreign>u<foreign lang="greek">b,</foreign>sz con&longs;tans: atque ip&longs;ius grauitatis centrum <lb/>
&longs;it <foreign lang="greek">q:</foreign> reliquæ magnitudinis, quæ ex omnibus pri&longs;matibus <lb/>
con&longs;tat, grauitatis centrum erit in linea <foreign lang="greek">q</foreign> o producta: & <lb/>
in puncto <foreign lang="greek">v</foreign>, ex octava propo&longs;itione eiusdem libri Archi­<pb xlink:href="023/01/032.jpg"/>medis. </s><s>ergo punctum <foreign lang="greek">n</foreign> extra pri&longs;ma af po&longs;itum, <expan abbr="centrũ">centrum</expan> <lb/>
erit magnitudinis <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> ex omnibus pri&longs;matibus gzr, <lb/>
r <foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign>k, k<foreign lang="greek">d</foreign> y, yu, us, s<foreign lang="greek">a</foreign>h, quod fieri nullo modo po<lb/>
te&longs;t. </s><s>e&longs;t enim ex diffinitione centrum grauitatis &longs;olidæ figu <lb/>
ræ intra ip&longs;am po&longs;itum, non extra. </s><s>quare relinquitur, ut <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis pri&longs;matis &longs;it in linea Km. </s><s>Rur&longs;us bc bifa­<lb/>
riam in diuidatur: & ducta a<foreign lang="greek">x,</foreign> per ip&longs;am, & per lineam <lb/>
agd plan um ducatur; quod pri&longs;ma &longs;ecet: <expan abbr="faciatq;">faciatque</expan> in paral <lb/>
lelogrammo bf &longs;ectionem <foreign lang="greek">x p</foreign> diuidet punctum <foreign lang="greek">p</foreign> lineam <lb/>
quoque cf bifariam: & erit plani eius, & trianguli ghK <lb/>
communis &longs;ectio gu; quòd <expan abbr="pũctum">punctum</expan> u in medio lineæ hK <lb/>
<figure id="id.023.01.032.1.jpg" xlink:href="023/01/032/1.jpg"/><lb/>
po&longs;itum &longs;it. </s><s>Similiter demon&longs;trabimus centrum grauita­<lb/>
tis pri&longs;matis in ip&longs;a gu ine&longs;&longs;e. </s><s>&longs;it autem planorum cfnl, <lb/>
ad<foreign lang="greek">px</foreign> communis &longs;ectio linea <foreign lang="greek">rst;</foreign> quæ quidem pri&longs;matis <lb/>
axis erit, cum tran&longs;eat per centra grauitatis triangulorum <lb/>
abc, ghk def, ex quartadecima eiu&longs;dem. </s><s>ergo centrum <lb/>
grauitatis pri&longs;matis af e&longs;t punctum <foreign lang="greek">s,</foreign> centrum &longs;cilicet <pb xlink:href="023/01/033.jpg" pagenum="13"/>trianguli ghK, & ip&longs;ius <foreign lang="greek">rt</foreign> axis medium.</s></p><p type="margin">
<s><margin.target id="marg35"/>5.huius</s></p><p type="margin">
<s><margin.target id="marg36"/>2. &longs;exti.<lb/>
12 quinti.</s></p><p type="margin">
<s><margin.target id="marg37"/>2. &longs;exti.</s></p><p type="margin">
<s><margin.target id="marg38"/>
19. &longs;exti</s></p><p type="margin">
<s><margin.target id="marg39"/>2. uel 12. <lb/>
quinti.</s></p><p type="margin">
<s><margin.target id="marg40"/>8. quinti.<!-- KEEP S--></s></p><p type="margin">
<s><margin.target id="marg41"/>28. unde<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg42"/>15. quinti</s></p><p type="margin">
<s><margin.target id="marg43"/>19. quinti<lb/>
apud <expan abbr="Cã">Cam</expan><lb/>
panum</s></p><p type="main">
<s>Sit pri&longs;ma ag, cuius oppo&longs;ita plana &longs;int quadrilatera <lb/>
abcd, efgh: <expan abbr="&longs;ecenturq;">&longs;ecenturque</expan> ac, bf, cg, dh bifariam: & per di­<lb/>
ui&longs;iones planum ducatur; quod &longs;ectionem faciat quadrila­<lb/>
terum Klmn. </s><s>Deinde iuncta ac per lineas ac, ae ducatur <lb/>
planum <expan abbr="&longs;ecãs">&longs;ecans</expan> pri&longs;ma, quod ip&longs;um diuidet in duo pri&longs;mata <lb/>
triangulares ba&longs;es habentia abcefg, adcehg. <!-- KEEP S--></s><s>Sint <expan abbr="aut&etilde;">autem</expan> <lb/>
<figure id="id.023.01.033.1.jpg" xlink:href="023/01/033/1.jpg"/><lb/>
triangulorum abc, efg gra­<lb/>
uitatis centra op: & triangu­<lb/>
lorum adc, ehg centra qr: <lb/>
<expan abbr="iunganturq;">iunganturque</expan> op, qr; quæ pla­<lb/>
no klmn occurrant in pun­<lb/>
ctis st. </s><s>erit ex iis, quæ demon <lb/>
&longs;trauimus, punctum s grauita <lb/>
tis centrum trianguli klm; & <lb/>
ip&longs;ius pri&longs;matis abcefg: pun <lb/>
ctum uero t centrum grauita <lb/>
tis trianguli Knm, & pri&longs;ma­<lb/>
tis adc, ehg. <!-- KEEP S--></s><s>iunctis igitur <lb/>
oq, pr, st, erit in linea oq <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis quadrilateri <lb/>
abcd, quod &longs;it u: & in linea <lb/>
pr <expan abbr="c&etilde;trum">centrum</expan> quadrilateri efgh <lb/>
&longs;it autem x. </s><s>denique iungatur <lb/>
u x, quæ &longs;ecet lineam &longs; t in y. </s><s>&longs;e<lb/>
cabit enim cum &longs;int in eodem <lb/>
<arrow.to.target n="marg44"/><lb/>
plano: <expan abbr="atq;">atque</expan> erit y grauitatis centrum quadrilateri Klmn. </s><lb/>
<s>Dico idem punctum y centrum quoque gra uitatis e&longs;&longs;e to­<lb/>
tius pri&longs;matis. </s><s>Quoniam enim quadrilateri klmn graui­<lb/>
tatis centrum e&longs;t y: linea sy ad yt ean dem proportionem <lb/>
habebit, quam triangulum knm ad triangulum klm, ex 8 <lb/>
Archimedis de centro grauitatis planorum. </s><s>Vt autem <expan abbr="triã">trian</expan> <lb/>
gulum knm ad ip&longs;um klm, hoc e&longs;t ut triangulum adc ad <lb/>
triangulum abc, æqualia enim &longs;unt, ita pri&longs;ma adcehg <pb xlink:href="023/01/034.jpg"/>ad pri&longs;ma abcefg. <!-- KEEP S--></s><s>quare linea sy ad yt eandem propor­<lb/>
tionem habet, quam pri&longs;ma adcehg ad pri&longs;ma abcefg. <!-- KEEP S--></s><lb/>
<s>Sed pri&longs;matis abcefg centrum grauitatis e&longs;t s: & pri&longs;ma­<lb/>
tis adcehg centrum t. </s><s>magnitudinis igitur ex his compo <lb/>
&longs;itæ hoc e&longs;t totius pri&longs;matis ag centrum grauitatis e&longs;t pun <lb/>
ctum y; medium &longs;cilicet axis ux, qui oppo&longs;itorum plano­<lb/>
rum centra coniungit.</s></p><p type="margin">
<s><margin.target id="marg44"/>5. huius/></s></p><p type="main">
<s>Rur&longs;us &longs;it pri&longs;ma ba&longs;im habens pentagonum abcde: <lb/>
& quod ei opponitur &longs;it fghKl: &longs;ec<expan abbr="enturq;">enturque</expan> af, bg, ch, <lb/>
dk, el bifariam: & per diui&longs;iones ducto plano, &longs;ectio &longs;it <expan abbr="p&etilde;">pen</expan><lb/>
<expan abbr="tagonũ">tagonum</expan> mnopq. deinde iuncta eb per lineas le, eb aliud <lb/>
<figure id="id.023.01.034.1.jpg" xlink:href="023/01/034/1.jpg"/><lb/>
planum ducatur, <expan abbr="diuid&etilde;s">diuidens</expan> pri&longs;<lb/>
ma ak in duo pri&longs;mata; in pri&longs; <lb/>
ma &longs;cilicet al, cuius plana op­<lb/>
po&longs;ita &longs;int triangula ab abcefg l: <lb/>
& in prima bk cuius plana op <lb/>
po&longs;ita &longs;int quadrilatera bcde <lb/>
ghkl. <!-- KEEP S--></s><s>Sint autem triangulo­<lb/>
rum abe, fgl centra grauita <lb/>
tis puncta r &longs;: & bcde, ghkl <lb/>
quadrilaterorum centra tu: <lb/>
<expan abbr="iunganturq;">iunganturque</expan> rs, tu occurren­<lb/>
tes plano mnopq in punctis <lb/>
xy. </s><s>& itidem <expan abbr="iungãtur">iungantur</expan> rt, &longs;u, <lb/>
xy. </s><s>erit in linca rt <expan abbr="c&etilde;trum">centrum</expan> gra<lb/>
uitatis pentagoni abcde; <lb/>
quod &longs;it z: & in linea &longs;u cen­<lb/>
trum pentagoni fghkl :&longs;it au <lb/>
tem <foreign lang="greek">x:</foreign> & ducatur z<foreign lang="greek">x,</foreign> quæ di­<lb/>
cto plano in <foreign lang="greek">y</foreign> occurrat. </s><s><expan abbr="Itaq;">Itaque</expan> <lb/>
punctum x e&longs;t centrum graui <lb/>
tatis trianguli mnq, ac pri&longs;­<lb/>
matis al: & y grauitatis centrum quadrilateri nopq, ac <lb/>
pri&longs;matis bk. </s><s>quare y centrum erit pentagoni mnopq. </s><s> & <pb xlink:href="023/01/035.jpg" pagenum="14"/>&longs;imiliter demon&longs;trabitur totius pri&longs;matis aK grauitatis ef <lb/>
&longs;e centrum. </s><s>Simili ratione & in aliis pri&longs;matibus illud <lb/>
idem facile demon&longs;trabitur. </s><s>Quo autem pacto in omni <lb/>
figura rectilinea centrum grauitatis inueniatur, docuimus <lb/>
in commentariis in &longs;extam propo&longs;itionem Archimedis de <lb/>
quadratura parabolæ.</s></p><p type="main">
<s>Sit cylindrus, uel cylindri portio ce cuius axis ab: &longs;ece­<lb/>
turq, plano per axem ducto; quod &longs;ectionem faciat paral­<lb/>
lelogrammum cdef: & diui&longs;is cf, de bifariam in punctis <lb/>
<figure id="id.023.01.035.1.jpg" xlink:href="023/01/035/1.jpg"/><lb/>
gh, per ea ducatur planum ba&longs;i æquidi&longs;tans. </s><s>erit &longs;ectio gh <lb/>
circulus, uel ellip&longs;is, centrum habens in axe; quod &longs;it K at­<lb/>
<arrow.to.target n="marg45"/><lb/>
que erunt ex iis, quæ demon&longs;trauimus, centra grauitatis <lb/>
planorum oppo&longs;itorum puncta ab: & plani gh ip&longs;um k in <lb/>
quo quidem plano e&longs;t centrum grauitatis cylindri, uel cy­<lb/>
lindri portionis. </s><s>Dico punctum K cylindri quoque, uel cy <lb/>
lindri portionis grauitatis centrum e&longs;&longs;e. </s><s>Si enim fieri po­<lb/>
te&longs;t, &longs;it l centrum: <expan abbr="ducaturq;">ducaturque</expan> kl, & extra figuram in m pro­<lb/>
ducatur. </s><s>quam ucro proportionem habet linea mK ad kl <pb xlink:href="023/01/036.jpg"/>habeat circulus, uel ellip&longs;is gh ad aliud &longs;pacium, in quo u: <lb/>
& in cit culo, uel ellip&longs;i plane de&longs;cribatur rectilinea figura, <lb/>
ita ut <expan abbr="tãdem">tandem</expan> <expan abbr="relinquãtur">relinquantur</expan> portiones minores &longs;pacio u, quæ <lb/>
&longs;it opgqrsht: <expan abbr="de&longs;criptaq;">de&longs;criptaque</expan> &longs;imili figura in oppo&longs;itis pla­<lb/>
nis cd, fe, per lineas &longs;ibi ip&longs;is re&longs;pondentes plana <expan abbr="ducãtur">ducantur</expan>. </s><lb/>
<s>Itaque cylindrus, uel cylindri portio diuiditur in pri&longs;ma, <lb/>
cuius quidem ba&longs;is e&longs;t figura rectilinea iam dicta, centrum <lb/>
que grauitatis punctum K: & in multa &longs;olida, qaæ pro ba&longs;i <lb/>
bus habent relictas portiones, quas nos &longs;olidas portiones <lb/>
appellabimus. </s><s>cum igitur portiones &longs;int minores &longs;pacio <lb/>
u, circulus, uel ellip&longs;is gh ad portiones maiorem propor­<lb/>
tionem habebit, quàm linea mk ad Kl. <!-- KEEP S--></s><s>fiat nk ad Kl, ut <lb/>
circulus uel ellip&longs;is gh ad ip&longs;as portiones. </s><s>Sed ut circulus <lb/>
uel ellip&longs;is gh ad figuram rectilineam in ip&longs;a de&longs;cri­<lb/>
ptam, ita e&longs;t cylindrus uel cylindri portio ce ad pri&longs;ma, <lb/>
quod rectilineam figuram pro ba&longs;i habet, & altitudinem <lb/>
æqualem; id, quod infra demon&longs;trabitur. </s><s>crgo per conuer <lb/>
&longs;ionem rationis, ut circulus, uel ellip&longs;is gh ad portioncs re <lb/>
lictas, ita cylindrus, uel cylindri portio ce ad &longs;olidas por­<lb/>
tiones, quate cylindrus uel cylindri portio ad &longs;olidas por­<lb/>
tiones eandem proportionem habet, quam linea nk ad k <lb/>
& diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o­<lb/>
lidas portiones eandem proportionem habet, quam nl ad <lb/>
lk & quoniam a cylindro uel cylindri portione, cuius gra­<lb/>
uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili­<lb/>
neam <expan abbr="figurã">figuram</expan>, cuius <expan abbr="centrũ">centrum</expan> grauitatis e&longs;t K: re&longs;iduæ magnitu <lb/>
dinis ex &longs;olidis portionibus <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> grauitatis <expan abbr="c&etilde;trũ">centrum</expan> erit <lb/>
in linea kl protracta, & in puncto n; quod e&longs;t <expan abbr="ab&longs;urdũ">ab&longs;urdum</expan>. </s><s>relin <lb/>
quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>
tionis &longs;it <expan abbr="punctũ">punctum</expan> k. </s><s>quæ omnia <expan abbr="demon&longs;trãda">demon&longs;tranda</expan> propo&longs;uimus.</s></p><p type="margin">
<s><margin.target id="marg45"/>4. huius</s></p><p type="main">
<s>At uero cylindrum, uel cylindri <expan abbr="portion&etilde;">portionem</expan> ce <lb/>
ad pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura in &longs;pa­<lb/>
cio gh de&longs;cripta, & altitudo æqualis; eandem ha­<pb xlink:href="023/01/037.jpg" pagenum="15"/>bere proportionem, quam &longs;pacium gh ad <expan abbr="dictã">dictam</expan> <lb/>
figuram, hoc modo demon&longs;trabimus.</s></p><p type="main">
<s>Intelligatur circulus, uel ellip&longs;is x æqualis figuræ rectili­<lb/>
neæ in gh &longs;pacio de&longs;criptæ. </s><s>& ab x con&longs;tituatur conus, uel <lb/>
<figure id="id.023.01.037.1.jpg" xlink:href="023/01/037/1.jpg"/><lb/>
coni portio, <expan abbr="altitudin&etilde;">altitudinem</expan> habens <expan abbr="eand&etilde;">eandem</expan>, <expan abbr="quã">quam</expan> cylindrus uel cy <lb/>
lindri portio ce. </s><s>Sit deinde rectilinea figura, in qua y eade, <lb/>
quæ in &longs;pacio gh de&longs;cripta e&longs;t: & ab hac pyramis æquealta <lb/>
con&longs;tituatur. </s><s>Dico <expan abbr="conũ">conum</expan> uel coni portione x pyramidi y <expan abbr="æ-qual&etilde;">æ­<lb/>
qualem</expan> e&longs;&longs;e. </s><s>ni&longs;i enim &longs;it æqualis, uel maior, uel minor crit.</s></p><p type="main">
<s>Sit primum maior, et exuperet &longs;olido z. </s><s>Itaque in circu <lb/>
lo, uel ellip&longs;i x de&longs;cribatur figura rectilinea; & in ea pyra­<lb/>
mis eandem, quam conus, uel coni portio altitudinem ha­<lb/>
bens, ita ut portiones relictæ minores &longs;int &longs;olido a, quem­<lb/>
admodum docetur in duodecimo libro elementorum pro <lb/>
po&longs;itione undecima. </s><s>erit pyramis x adhuc pyramide y ma <lb/>
ior. </s><s>& quoniam piramides æque altæ inter &longs;e &longs;unt, &longs;icuti ba <lb/>
<arrow.to.target n="marg46"/><lb/>
&longs;es; pyramis x ad piramidem y eandem proportionem ha­<lb/>
bet, quàm figura rectilinea x ad figuram y. </s><s>Sed figura recti <pb xlink:href="023/01/038.jpg"/><figure id="id.023.01.038.1.jpg" xlink:href="023/01/038/1.jpg"/><lb/>
linea x cum &longs;it minor circulo, uel ellip&longs;i, e&longs;t etiam minor fi­<lb/>
gura rectilinea y. </s><s>ergo pyramis x pyramide y minor erit. </s><lb/>
<s>Sed & maior; quod ficri <expan abbr="nõ">non</expan> pote&longs;t. </s><s>At &longs;i conus, uel coni por <lb/>
tio x ponatur minor pyramide y: &longs;it alter conus æque al­<lb/>
tus, uel altera coni portio X ip&longs;i pyramidi y æqualis. </s><s>crit <lb/>
eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x, <lb/>
quorum exce&longs;&longs;us &longs;it &longs;pacium <foreign lang="greek">w.</foreign> Si igitur in circulo, uel eili­<lb/>
p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relictæ <lb/>
&longs;int <foreign lang="greek">w</foreign> &longs;pacio minores, ciu&longs;modi figura adhuc maior erit cir <lb/>
culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. </s><s>& pyramis in <lb/>
ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi­<lb/>
nor pyramide y. </s><s>e&longs;t ergo ut X figura rectilinea ad figuram <lb/>
rectilineam y, ita pyramis X ad pyramidem y. </s><s>quare cum <lb/>
figura rectilinea X &longs;it maior figura y: erit & pyramis X py­<lb/>
ramide y maior. </s><s>&longs;ed erat minor; quod rur&longs;us fieri non po­<lb/>
te&longs;t. </s><s>non e&longs;t igitur conus, uel coni portio x neque maior, <lb/>
neque minor pyramide y. </s><s>ergo ip&longs;i nece&longs;&longs;ario e&longs;t æqualis. </s><lb/>
<s>Itaque quoniam ut conus ad conum, uel coni portio ad co <pb xlink:href="023/01/039.jpg" pagenum="16"/><figure id="id.023.01.039.1.jpg" xlink:href="023/01/039/1.jpg"/><lb/>
ni portionem, ita e&longs;t cylindrus ad cylindrum, uel cylin­<lb/>
dri portio ad cylindri portionem: & ut pyramis ad pyra­<lb/>
midem, ita pri&longs;ma ad pri&longs;ma, cum eadem &longs;it ba&longs;is, & æqua <lb/>
lis altitudo; erit cylindrus uel cylindri portio x pri&longs;ma­<lb/>
ti y æqualis. </s><s><expan abbr="e&longs;tq;">e&longs;tque</expan> ut &longs;pacium gh ad &longs;pacium x, ita cylin­<lb/>
drus, uel cylindri portio ce ad cylindrum, uel cylindri por­<lb/>
tionem x. </s><s>Con&longs;tat igitur cylindrum uel cylindri <expan abbr="portion&etilde;">portionem</expan> <lb/>
c e, ad pri&longs;ma y, quippe cuius ba&longs;is e&longs;t figura rectilinea in <lb/>
<arrow.to.target n="marg47"/><lb/>
&longs;pacio gh de&longs;cripta, eandem proportionem habere, quam <lb/>
&longs;pacium gh habet ad &longs;pacium x, hoc e&longs;t ad dictam figuram. </s><lb/>
<s>quod demon&longs;trandum fuerat.</s></p><p type="margin">
<s><margin.target id="marg46"/>6. duode<lb/>
cimi.</s></p><p type="margin">
<s><margin.target id="marg47"/>7. quinti</s></p><p type="head">
<s>THEOREMA IX. PROPOSITIO IX.<!-- KEEP S--></s></p><p type="main">
<s>Si pyramis &longs;ecetur plano ba&longs;i æquidi&longs;tante; &longs;e­<lb/>
ctio erit figura &longs;imilis ei, quæ e&longs;t ba&longs;is, centrum <lb/>
grauitatis in axe habens.</s></p><pb xlink:href="023/01/040.jpg"/><p type="main">
<s>SIT pyramis, cuius ba&longs;is triangulum abc; axis dc: & <lb/>
&longs;ecetur plano ba&longs;i æquidi&longs;tante; quod <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> faciat fgh; <lb/>
<expan abbr="occurratq;">occurratque</expan> axi in puncto k. Dico fgh triangulum e&longs;&longs;e, ip&longs;i <lb/>
abc &longs;imile; cuius grauitatis centrum e&longs;t K. <expan abbr="Quoniã">Quoniam</expan> enim <lb/>
duo plana æquidi&longs;tantia abc, fgh &longs;ecantur à plano abd; <lb/>
communes eorum &longs;ectiones ab, fg æquidi&longs;tantes erunt: & <lb/>
eadem ratione æquidi&longs;tantes ip&longs;æ bc, gh: & ca, hf. </s><s>Quòd <lb/>
cum duæ lineæ fg, gh, duabus ab, bc æquidi&longs;tent, nec <lb/>
&longs;int in eodem plano; angulus ad g æqualis e&longs;t angulo ad <lb/>
b. </s><s>& &longs;imiliter angulus ad h angulo ad c: <expan abbr="angulusq;">angulusque</expan> ad fci, <lb/>
qui ad a e&longs;t æqualis. </s><s>triangulum igitur fgh &longs;imile e&longs;t tri­<lb/>
angulo abc. <!-- KEEP S--></s><s>Atuero punctum k centrum e&longs;&longs;e grauita­<lb/>
tis trianguli fgh hoc modo o&longs;tendemus. </s><s>Ducantur pla­<lb/>
na per axem, & per lineas da, db, dc: erunt communes &longs;e­<lb/>
ctiones fK, ae æquidi&longs;tantes: <expan abbr="pariterq;">pariterque</expan> kg, eb; & kh, ec: <lb/>
quare angulus kfh angulo eac; & angulus kfg ip&longs;i eab <lb/>
<figure id="id.023.01.040.1.jpg" xlink:href="023/01/040/1.jpg"/><lb/>
e&longs;t æqualis. </s><s>Eadem ratione <lb/>
anguli ad g angulis ad b: & <lb/>
anguli ad h iis, qui ad c æ­<lb/>
quales erunt. </s><s>ergo puncta <lb/>
eK in triangulis abc, fgh <lb/>
&longs;imiliter &longs;unt po&longs;ita, per &longs;e­<lb/>
xtam po&longs;itionem Archime­<lb/>
dis in libro de centro graui­<lb/>
tatis planorum. </s><s>Sed cum e <lb/>
&longs;it centrum grauitatis trian <lb/>
guli abc, erit ex undecima <lb/>
propo&longs;itione eiu&longs;dem libri, <lb/>
& K trianguli fgh grauita <lb/>
tis centrum. </s><s>id quod demon&longs;trare oportebat. </s><s>Non aliter <lb/>
in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra­<lb/>
bitur.</s></p><pb xlink:href="023/01/041.jpg" pagenum="17"/><p type="head">
<s>PROBLEMA I. PROPOSITIO X.<!-- KEEP S--></s></p><p type="main">
<s>DATA qualibet pyramide, fieri pote&longs;t, ut fi­<lb/>
gura &longs;olida in ip&longs;a in &longs;cribatur, & altera <expan abbr="circũ&longs;cri-batur">circum&longs;cri­<lb/>
batur</expan> ex pri&longs;matibus æqualem altitudinem <expan abbr="ha-b&etilde;tibus">ha­<lb/>
bentibus</expan>, ita ut circum&longs;cripta in&longs;criptam excedat <lb/>
magnitudine, quæ minor &longs;it <expan abbr="quacũque">quacunque</expan> &longs;olida ma <lb/>
gnitudine propo&longs;ita.</s></p><figure id="id.023.01.041.1.jpg" xlink:href="023/01/041/1.jpg"/><p type="main">
<s>Sit pyramis, cuius ba&longs;is <lb/>
<expan abbr="triangulũ">triangulum</expan> abc; axis de. </s><lb/>
<s><expan abbr="Sitq;">Sitque</expan> pri&longs;ma, quod <expan abbr="eand&etilde;">eandem</expan> <lb/>
ba&longs;im habeat, & axem eun <lb/>
dem. </s><s>Itaque hoc pri&longs;ma­<lb/>
te continenter &longs;ecto bifa­<lb/>
riam, plano ba&longs;i <expan abbr="æquidi&longs;tã">æquidi&longs;tan</expan><lb/>
te, relinquetur <expan abbr="tãdem">tandem</expan> pri&longs; <lb/>
ma quoddam minus pro­<lb/>
po&longs;ita magnitudine: quod <lb/>
quidem ba&longs;im eandem ha <lb/>
beat, quam pyramis, & a­<lb/>
xem ef. </s><s>diuidatur de in <lb/>
partes æquales ip&longs;i ef in <lb/>
punctis ghklmn: & per <lb/>
diui&longs;iones plana <expan abbr="ducãtur">ducantur</expan>: <lb/>
quæ ba&longs;ibus æquidi&longs;tent, <lb/>
erunt &longs;ectiones, triangula <lb/>
ip&longs;i abc &longs;imilia, ut proxi­<lb/>
me o&longs;tendimus. </s><s>ab uno <lb/>
quoque <expan abbr="aut&etilde;">autem</expan> horum trian <lb/>
gulorum duo pri&longs;mata <expan abbr="cõ">com</expan> <lb/>
&longs;truantur; unum quidem <lb/>
ad partes e; alterum ad <pb xlink:href="023/01/042.jpg"/>partes d. <!-- KEEP S--></s><s>in pyramide igitur in&longs;cripta erit quædam figura, <lb/>
ex pri&longs;matibus æqualem altitudinem habentibus <expan abbr="cõ&longs;tans">con&longs;tans</expan>, <lb/>
ad partes e: & altera circum&longs;cripta ad partes d. <!-- KEEP S--></s><s>Sed unum­<lb/>
quodque eorum pri&longs;matum, quæ in figura in&longs;cripta conti­<lb/>
nentur, æquale e&longs;t pri&longs;mati, quod ab eodem fit triangulo in <lb/>
figura circum&longs;cripta: nam pri&longs;ma pq pri&longs;mati po e&longs;t æ­<lb/>
quale; pri&longs;ma st æquale pri&longs;mati sr; pri&longs;ma xy pri&longs;mati <lb/>
xu; pri&longs;ma <foreign lang="greek">hq</foreign> pri&longs;mati <foreign lang="greek">h</foreign>z; pri&longs;ma <foreign lang="greek">mn</foreign> pri&longs;mati <foreign lang="greek">ml;</foreign> pri&longs;­<lb/>
ma <foreign lang="greek">rs</foreign> pri&longs;mati <foreign lang="greek">rp;</foreign> & pri&longs;ma <foreign lang="greek">fx</foreign> pri&longs;mati <foreign lang="greek">ft</foreign> æquale. </s><s>re­<lb/>
linquitur ergo, ut circum&longs;cripta figura exuperet <expan abbr="in&longs;criptã">in&longs;criptam</expan> <lb/>
pri&longs;mate, quod ba&longs;im habet abc triangulum, & axem ef. </s><lb/>
<s>Illud uero minus e&longs;t &longs;olida magnitudine propo&longs;ita. </s><s><expan abbr="Ead&etilde;">Eadem</expan> <lb/>
ratione in&longs;cribetur, & circum&longs;cribetur &longs;olida figura in py­<lb/>
ramide, quæ quadrilateram, uel <expan abbr="plurilaterã">plurilateram</expan> ba&longs;im habeat.</s></p><p type="head">
<s>PROBLEMA II. PROPOSITIO XI.<!-- KEEP S--></s></p><p type="main">
<s>DATO cono, fieri pote&longs;t, ut figura &longs;olida in­<lb/>
&longs;cribatur, & altera circum&longs;cribatur ex cylindris <lb/>
æqualem habentibus altitudinem, ita ut circum­<lb/>
&longs;cripta &longs;uperet in&longs;criptam, magnitudine, quæ &longs;o­<lb/>
lida magnitudine propo&longs;ita &longs;it minor.</s></p><p type="main">
<s>SIT conus, cuius axis bd: & &longs;ecetur plano per axem <lb/>
ducto, 'ut&longs;ectio &longs;it triangulum abc: <expan abbr="intelligaturq;">intelligaturque</expan> cylin­<lb/>
drus, qui ba&longs;im eandem, & eundem axem habeat. </s><s>Hocigi­<lb/>
tur cylindro continenter bifariam &longs;ecto, relinquetur cylin <lb/>
drus minor &longs;olida magnitudine propo&longs;ita. </s><s>Sit autem is cy <lb/>
lindrus, qui ba&longs;im habet circulum circa diametrum ac, & <lb/>
axem de. </s><s>Itaque diuidatur bd in partes æquales ip&longs;ide <lb/>
in punctis fghKlm: & per ea ducantur plana conum &longs;e­<lb/>
cantia; quæ ba&longs;i æquidi&longs;tent. </s><s>erunt &longs;ectiones circuli, cen­<lb/>
tra in axi habentes, ut in primo libro conicorum, propo&longs;i-<pb xlink:href="023/01/043.jpg" pagenum="18"/>tione quarta Apollonius demon&longs;trauit. </s><s>Si igitur à &longs;ingu­<lb/>
lis horum circulorum, duo cylindri fiant; unus quidem ad <lb/>
ba&longs;is partes; alter ad partes uerticis: in&longs;cripta erit in co­<lb/>
no &longs;olida quædam figura, & altera circum&longs;cripta ex cylin­<lb/>
dris æqualem altitudinem habentibus con&longs;tans; quorum <lb/>
<figure id="id.023.01.043.1.jpg" xlink:href="023/01/043/1.jpg"/><lb/>
unu&longs;qui&longs;que, qui in <lb/>
figura in&longs;cripta con­<lb/>
tinetur æqualis e&longs;t ei, <lb/>
qui ab eodem fit cir­<lb/>
culo in figura <expan abbr="circũ-&longs;cripta">circum­<lb/>
&longs;cripta</expan>. </s><s>Itaque cylin <lb/>
drus op æqualis e&longs;t <lb/>
cylindro on; cylin­<lb/>
drus rs <expan abbr="cylĩdro">cylindro</expan> rq.</s> <lb/><s>
cylindrus ux cylin­<lb/>
dro ut e&longs;t æqualis; <lb/>
& alii aliis &longs;imiliter. </s><lb/>
<s>quare con&longs;tat <expan abbr="circũ-&longs;criptam">circum­<lb/>
&longs;criptam</expan> figuram &longs;u­<lb/>
perare in&longs;criptam cy <lb/>
lindro, cuius ba&longs;is e&longs;t <lb/>
circulus circa diametrum ac, & axis de. </s><s>atque hic e&longs;t mi­<lb/>
nor &longs;olida magnitudine propo&longs;ita.</s></p><p type="head">
<s>PROBLEMA III. PROPOSITIO XII.<!-- KEEP S--></s></p><p type="main">
<s>DATA coni portione, pote&longs;t &longs;olida quædam <lb/>
figura in&longs;cribi, & altera circum&longs;cribi ex cylindri <lb/>
portionibus æqualem altitudinem habentibus; <lb/>
ita ut circum&longs;cripta in&longs;criptam exuperet, magni <lb/>
tudine, quæ minor fit &longs;olida magnitudine pro­<lb/>
po&longs;ita.</s></p><pb xlink:href="023/01/044.jpg"/><p type="main">
<s>Figuram ciu&longs;modi, & in&longs;cribemus, & <expan abbr="circũ&longs;cribemus">circum&longs;cribemus</expan>, ita <lb/>
ut in cono dictum e&longs;t.</s></p><figure id="id.023.01.044.1.jpg" xlink:href="023/01/044/1.jpg"/><p type="head">
<s>PROBLEMA IIII. PROPOSITIO XIII.<!-- KEEP S--></s></p><p type="main">
<s>DATA &longs;phæræ portione, quæ dimidia &longs;phæ­<lb/>
ra maior non &longs;it, pote&longs;t &longs;olida quædam portio in­<lb/>
&longs;cribi & altera circum&longs;cribi ex cylindris æqualem <lb/>
altitudinem habentibus, ita ut circum&longs;cripta in­<lb/>
&longs;criptam excedat magnitudine, quæ &longs;olida ma­<lb/>
gnitudine propo&longs;ita &longs;it minor.</s></p><p type="main">
<s>HOC etiam codem pror&longs;us modo &longs;iet: atque ut ab <lb/>
Archimedc traditum e&longs;t in conoidum, & &longs;phæroidum por <lb/>
tionibus, propo&longs;itione ulge&longs;imaprima libri de conoidi­<lb/>
bus, & &longs;phæroidibus.</s></p><pb xlink:href="023/01/045.jpg" pagenum="19"/><figure id="id.023.01.045.1.jpg" xlink:href="023/01/045/1.jpg"/><p type="head">
<s>THEOREMA X. PROPOSITIO XIIII.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet pyramidis, & cuiuslibet coni, uel <lb/>
coni portionis, centrum grauitatis in axe <expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>.</s></p><p type="main">
<s>SIT pyramis, cuius ba&longs;is triangulum abc: & axis de. </s><lb/>
<s>Dico in linea de ip&longs;ius grauitatis centrum ine&longs;&longs;e. </s><s>Si enim <lb/>
fieri pote&longs;t, &longs;it centrum f: & ab f ducatur ad ba&longs;im pyrami <lb/>
dis linea fg, axi æquidi&longs;tans: <expan abbr="iunctaq;">iunctaque</expan> eg ad latera trian­<lb/>
guli abc producatur in h. </s><s>quam uero proportionem ha­<lb/>
bet linea he ad eg, habeat pyramis ad aliud &longs;olidum, in <lb/>
quo K: <expan abbr="in&longs;cribaturq;">in&longs;cribaturque</expan> in pyramide &longs;olida figura, & altera cir <lb/>
cum&longs;cribatur ex pri&longs;matibus æqualem habentibus altitu­<lb/>
dinem, ita ut circum&longs;cripta in&longs;criptam exuperet magnitu­<lb/>
dine, quæ &longs;olido k &longs;it minor. </s><s>Et quoniam in pyramide pla<lb/>
num ba&longs;i æquidi&longs;tans ductum &longs;ectionem facit figuram &longs;i­<lb/>
milem ei, quæ e&longs;t ba&longs;is; <expan abbr="centrumq;">centrumque</expan> grauitatis in axe haben <lb/>
tem: erit pri&longs;matis st grauitatis <expan abbr="centrũ">centrum</expan> in linea rq; <lb/>
matis ux centrum in linea qp, pri&longs;matis yz in linea po; <lb/>
pri&longs;matis <foreign lang="greek">hq</foreign> in linea on; pri&longs;matis <foreign lang="greek">lm</foreign> in linea nm; pri&longs;­<lb/>
matis <foreign lang="greek">np</foreign> in ml; & denique pri&longs;matis <foreign lang="greek">rs</foreign> in le. </s><s>quare to­<pb xlink:href="023/01/046.jpg"/>tius figuræ in&longs;criptæ centrum grauitatis e&longs;t in linea re: <lb/>
<figure id="id.023.01.046.1.jpg" xlink:href="023/01/046/1.jpg"/>quod &longs;it <foreign lang="greek">t</foreign>: <expan abbr="iũ">iun</expan>­<lb/>
ctaque <foreign lang="greek">t</foreign>f, & <lb/>
producta, à <lb/>
puncto h du­<lb/>
catur linea a­<lb/>
xi pyramidis <lb/>
æquidi&longs;tans, <lb/>
quæ <expan abbr="cũ">cum</expan> linea <lb/>
<foreign lang="greek">t</foreign>f conueniat <lb/>
in <foreign lang="greek">f</foreign>.</s><s>habebit <lb/>
<foreign lang="greek">ft</foreign> ad <foreign lang="greek">t</foreign>f ean­<lb/>
dem propor­<lb/>
tionem, <expan abbr="quã">quam</expan> <lb/>
he ad eg. <lb/>
</s></p><p>
<s>Quoniam igi<lb/>
tur exce&longs;&longs;us, <lb/>
quo <expan abbr="circũ">circum</expan>&longs;cri<lb/>
pta figura in­<lb/>
&longs;criptam &longs;upe<lb/>
rat, minor e&longs;t <lb/>
&longs;olido <foreign lang="greek">x</foreign>; py­<lb/>
ramis ad eun­<lb/>
<expan abbr="d&etilde;">dem</expan> <expan abbr="exce&longs;&longs;ũ">exce&longs;&longs;um</expan> ma<lb/>
ioré propor­<lb/>
tioné habet, <lb/>
quàm ad K &longs;o<lb/>
lidum: uideli<lb/>
cet maiorem, <lb/>
quàm linea h<lb/>
e ad eg; hoc <lb/>
e&longs;t quàm <foreign lang="greek">ft</foreign> <lb/>
ad <foreign lang="greek">t</foreign>f: & propterea multo maiorem habet ad partem ex­<lb/>
ce&longs;&longs;us, quæ intra pyrimidem comprehenditur. </s><s>Itaque ha­<pb xlink:href="023/01/047.jpg" pagenum="20"/>beat eam, quam <foreign lang="greek">xt</foreign> ad <foreign lang="greek">t</foreign>f erit diuidendo ut <foreign lang="greek">x</foreign>f ad f<foreign lang="greek">t</foreign>, ita fi<lb/>
gura &longs;olida in&longs;cripta ad partem exce&longs;&longs;us, quæ e&longs;t intra pyra<lb/>
midem. </s><s>Cum ergo à pyramide, cuius grauitatis <expan abbr="ceũtrum">centrum</expan> e&longs;t <lb/>
punctum f, &longs;olida figura in&longs;cripta auferatur, cuius <expan abbr="centrũtrum">centrum</expan> <lb/>
<foreign lang="greek">t</foreign>: reliqua magnitudinis con&longs;tantis ex parte exce&longs;&longs;us, quæ <lb/>
e&longs;t intra pyramidem, centrum grauitatis erit in linea <foreign lang="greek">t</foreign>f <lb/>
producta, & in puncto <foreign lang="greek">x</foreign>. </s><s>quod fieri non pote&longs;t. </s><s>Sequitur <lb/>
igitur, ut centrum grauitatis pyramidis in linea de; hoc <lb/>
e&longs;t in eius axe con&longs;i&longs;tat.</s></p><p>
<s>Sit conus, uel coni portio, cuius axis bd: & &longs;ecetur plano <lb/>
per axem, ut &longs;ectio &longs;it triangulum abc. </s> <s>Dico centrum gra<lb/>
uitatis ip&longs;ius e&longs;&longs;e in linea bd. </s><s>Sit enim, &longs;i fieri pote&longs;t, <expan abbr="centrũ">centrum</expan> <lb/>
<figure id="id.023.01.047.1.jpg" xlink:href="023/01/047/1.jpg"/>
e: <expan abbr="perq;">perque</expan> e ducatur ef axi æquidi&longs;tans: & quam propor­<lb/>
tionem habet cd ad df, habeat conus, uel coni portio ad <lb/>
&longs;olidum g. </s><s>in&longs;cribatur ergo in cono, uel coni portione &longs;oli<pb xlink:href="023/01/048.jpg"/>da figura, & altera circum&longs;cribatur ex cylindris, uel cylin­<lb/>
dri portionibus, &longs;icuti dictum e&longs;t, ita ut exce&longs;&longs;us, quo figu­<lb/>
ra circum&longs;cripta in&longs;criptam &longs;uperat, &longs;it &longs;olido g minor. </s><lb/>
<s>Itaque centrum grauitatis cylindri, uel cylindri portionit <lb/>
qr e&longs;t in linea po; cylindri, uel cylindri portionis st cen­<lb/>
trum in linea on; centrum ux in linea nm; yz in mb; <foreign lang="greek">nq</foreign><lb/>
in lk; <foreign lang="greek">l m</foreign> in kh; & denique <foreign lang="greek">v p</foreign> centrum in hd. <!-- KEEP S--></s><s>ergo figu­<lb/>
<figure id="id.023.01.048.1.jpg" xlink:href="023/01/048/1.jpg"/><lb/>
ræ in&longs;criptæ centrum e&longs;t in linea pd. <!-- KEEP S--></s><s>Sit autem <foreign lang="greek">r</foreign>: & iun­<lb/>
cta <foreign lang="greek">r</foreign> e protendatur, ut cum linea, quæ à <expan abbr="pũcto">puncto</expan> c ducta &longs;ue­<lb/>
rit axi æquidi&longs;tans, conueniat in <foreign lang="greek">s.</foreign> erit <foreign lang="greek">s r</foreign> ad <foreign lang="greek">r</foreign>e, ut cd <lb/>
ad df: & conus, &longs;eu coni portio ad exce&longs;&longs;um, quo circum­<lb/>
&longs;cripta figura in&longs;criptam &longs;uperat, habebit maiorem pro­<lb/>
portionem, quàm <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. </s><s>ergo ad partem exce&longs;&longs;us, quæ <lb/>
intra ip&longs;ius &longs;uperficiem comprehenditur, multo maiorem <lb/>
proportionem habebit. </s><s>habeat eam, quam <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. </s><s>erit <pb xlink:href="023/01/049.jpg" pagenum="21"/>diuidendo figura &longs;olida in&longs;cripta ad dictam exce&longs;&longs;us par­<lb/>
tem, ut <foreign lang="greek">te</foreign> ad c<foreign lang="greek">p.</foreign> & quoniam à cono, &longs;cu coni portione, <lb/>
cuius grauitatis centrum e&longs;t e, aufertur figura in&longs;cripta, <lb/>
cuius centrum <foreign lang="greek">r:</foreign> re&longs;iduæ magnitudinis compo&longs;itæ cx par <lb/>
te exce&longs;&longs;us, quæ intra coni, uel coni portionis &longs;uperficiem <lb/>
continetur, centrum grauitatis erit in linea e protracta, <lb/>
atque in puncto t. </s><s>quod c&longs;t ab&longs;urdum. </s><s><expan abbr="cõ&longs;tat">con&longs;tat</expan> ergo <expan abbr="centrũ">centrum</expan> <lb/>
grauitatis coni, uel coni portionis, e&longs;&longs;e in axe bd: quod de <lb/>
mon&longs;trandum propo&longs;uimus.</s></p><p type="head">
<s>THEOREMA XI. PROPOSITIO XV.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet portionis &longs;phæræ uel &longs;phæroidis, <lb/>
quæ dimidia maior non &longs;it: <expan abbr="item&qacute;;">itemque</expan> cuiuslibet por <lb/>
tionis conoidis, uel ab&longs;ci&longs;&longs;æ plano ad axem recto, <lb/>
uel non recto, centrum grauitatis in axe con­<lb/>
&longs;i&longs;tit.</s></p><p type="main">
<s>Demon&longs;tratio &longs;imilis erit ei, quam &longs;upra in cono, uel co <lb/>
ni portione attulimus, ne toties eadem fru&longs;tra iterentur.</s></p><figure id="id.023.01.049.1.jpg" xlink:href="023/01/049/1.jpg"/><pb xlink:href="023/01/050.jpg"/><p type="head">
<s>THEOREMA XII. PROPOSITIO XVI.<!-- KEEP S--></s></p><p type="main">
<s>In &longs;phæra, & &longs;phæroidc idem e&longs;t grauitatis, & <lb/>
figuræ centrum.</s></p><p type="main">
<s>Secetur &longs;phæra, uel &longs;phæroides plano per axem ducto; <lb/>
quod &longs;ectionem faciat circulum, uel ellip&longs;im abcd, cuius <lb/>
diameter, & &longs;phæræ, uel &longs;phæroidis axis db; & centrum e. </s><lb/>
<s>Dico e grauitatis etiam centrum e&longs;&longs;e. </s><s>&longs;ecetur enim altero <lb/>
plano per e, ad planum &longs;ecans recto, cuius &longs;ectio &longs;it circu­<lb/>
lus circa diametrum ac. </s><s>erunt adc, abc dimidiæ portio­<lb/>
nes &longs;phæræ, uel &longs;phæroidis. </s><s>& quoniam portionis adc gra<lb/>
uitatis centrum e&longs;i in linea d, & centrum portionis abc in <lb/>
ip&longs;a be; totius &longs;phæræ, uel &longs;phæroidis grauitatis centrum <lb/>
in axe db con&longs;i&longs;iet, Quòd &longs;i portionis adc centrum graui <lb/>
tatis ponatur e&longs;&longs;e f & fiat ip&longs;i fe æqualis eg. <!-- REMOVE S--><expan abbr="punctũ">punctum</expan> g por <lb/>
<figure id="id.023.01.050.1.jpg" xlink:href="023/01/050/1.jpg"/><lb/>
<arrow.to.target n="marg48"/><lb/>
tionis abc centrum erit. </s>
<s>&longs;olidis enim figuris &longs;imilibus & <lb/>
æqualibus inter &longs;e aptatis, & centra grauitatis ip&longs;arum in­<lb/>
<arrow.to.target n="marg49"/><lb/>
ter se aptentur nece&longs;&longs;e e&longs;t. </s>
<s>ex quo fit, ut magnitudinis, quæ <lb/>
ex utilique <expan abbr="cõ&longs;lat">con&longs;tat</expan>, hoc e&longs;t ip&longs;ius &longs;phæræ, uel &longs;phæroidis gra<lb/>
uitatis centrum &longs;it in medio lineæ fg uidelicet in e. <!-- KEEP S--></s><s>Sphæ­<lb/>
ræ igitur, uel &longs;phæroidis grauitatis centrum e&longs;t idem, quod <lb/>
centrum figuræ.</s></p><pb xlink:href="023/01/051.jpg" pagenum="22"/><p type="margin">
<s><margin.target id="marg48"/>per 2. pe­<lb/>
titionem</s></p><p type="margin">
<s><margin.target id="marg49"/>4 Archi­<lb/>
medis.</s></p><p type="main">
<s>Ex demon&longs;tratis per&longs;picue apparet, portioni <lb/>
&longs;phæræ uel &longs;phæroidis, quæ dimidia maior e&longs;t, <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis in axe con&longs;i&longs;tere.</s></p><figure id="id.023.01.051.1.jpg" xlink:href="023/01/051/1.jpg"/><p type="main">
<s>Data enim <lb/>
qualibet maio<lb/>
ri <expan abbr="portiõe">portione</expan>, quo <lb/>
<expan abbr="niã">niam</expan> totius &longs;phæ <lb/>
ræ, uel &longs;phæroi <lb/>
dis grauitatis <lb/>
centrum e&longs;t in <lb/>
axe; e&longs;t autem <lb/>
& in axe cen­<lb/>
trum portio­<lb/>
nis minoris: <lb/>
reliquæ portionis uidelicet maioris centrum in axe nece&longs;­<lb/>
&longs;ario con&longs;i&longs;tet.</s></p><p type="head">
<s>THEOREMA XIII. PROPOSITIO XVII.<!-- KEEP S--></s></p><figure id="id.023.01.051.2.jpg" xlink:href="023/01/051/2.jpg"/><p type="main">
<s>Cuiuslibet pyramidis <expan abbr="triã">trian</expan> <lb/>
gularem ba&longs;im <expan abbr="hab&etilde;tis">habentis</expan> gra<lb/>
uitatis centrum e&longs;t in pun­<lb/>
cto, in quo ip&longs;ius axes con­<lb/>
ueniunt.</s></p><p type="main">
<s>Sit pyramis, cuius ba&longs;is trian <lb/>
gulum abc, axis de: <expan abbr="&longs;itq;">&longs;itque</expan> trian <lb/>
guli bdc grauitatis centrum f: <lb/>
& iungatur a &longs;. </s><s>crit & af axis eiu&longs; <lb/>
dem pyramidis ex tertia diffini­<lb/>
tione huius. </s><s>Itaque quoniam centrum grauitatis e&longs;t in <lb/>
axe de; e&longs;t autem & in axe af; &qgrave;uod proxime demon&longs;traui <pb xlink:href="023/01/052.jpg"/>mus: erit utique grauitatis centrum pyramidis punctum <lb/>
g. <!-- REMOVE S-->in quo &longs;cilicet ip&longs;i axes conueniunt.</s>
</p><p type="head">
<s>THEOREMA XIIII. PROPOSITIO XVIII.<!-- KEEP S--></s></p><p type="main">
<s>SI &longs;olidum parallelepipedum &longs;ecetur plano <lb/>
ba&longs;ibus æquidi&longs;tante; erit &longs;olidum ad &longs;olidum, <lb/>
&longs;icut altitudo ad altitudinem, uel &longs;icut axis ad <lb/>
axem.</s></p><figure id="id.023.01.052.1.jpg" xlink:href="023/01/052/1.jpg"/><p type="main">
<s>Sit &longs;olidum parallelepipe <lb/>
dum abcdefgh, cuius axis <lb/>
kl: <expan abbr="&longs;eceturq;">&longs;eceturque</expan> plano ba&longs;ibus <lb/>
æquidi&longs;tante, quod faciat <lb/>
&longs;ectionem mnop; & axi in <lb/>
puncto q occurrat. </s><s>Dico <lb/>
&longs;olidum gm ad &longs;olidum mc <lb/>
eam proportionem habere, <lb/>
quam altitudo &longs;olidi gm ha­<lb/>
bet ad &longs;olidi mc altitudi­<lb/>
nem; uel quam axis kq ad <lb/>
axem ql. <!-- KEEP S--></s><s>Si enim axis Kl ad <lb/>
ba&longs;is planum &longs;it perpendicu <lb/>
<figure id="id.023.01.052.2.jpg" xlink:href="023/01/052/2.jpg"/><lb/>
laris, & linea gc, quæ ex quin <lb/>
ta huius ip&longs;i kl æquidi&longs;tat, <lb/>
perpendicularis erit ad <expan abbr="id&etilde;">idem</expan> <lb/>
planum, & &longs;olidi altitudi­<lb/>
<arrow.to.target n="marg50"/><lb/>
nem dimetietur. </s><s>Itaque &longs;o­<lb/>
lidum gm ad &longs;olidum mc <lb/>
eam proportionem habet, <lb/>
quam parallelogramm<expan abbr="ũ">um</expan> gn <lb/>
ad parallelogrammum nc, <lb/>
<arrow.to.target n="marg51"/><lb/>
hoc e&longs;t quam linea go, quæ <pb xlink:href="023/01/053.jpg" pagenum="23"/>e&longs;t &longs;olidi gm altitudo ad oe altitudinem &longs;olidi mc, uel <expan abbr="quã">quam</expan> <lb/>
axis kq ad ql axem. </s><s>Si uero axis kl non &longs;it perpendicularis <lb/>
ad planum ba&longs;is; ducatur a puncto k ad idem planum per <lb/>
pendicularis kr, <expan abbr="occurr&etilde;s">occurrens</expan> plano mnop in s. </s><s>&longs;imiliter <expan abbr="de-mõ&longs;trabimus">de­<lb/>
mon&longs;trabimus</expan> &longs;olidum gm ad <expan abbr="&longs;olidũ">&longs;olidum</expan> mc ita e&longs;&longs;e, ut axis kq <lb/>
ad axem ql. <!-- KEEP S--></s><s>Sed ut Kq ad ql, ita ks altitudo ad altitudi­<lb/>
<arrow.to.target n="marg52"/><lb/>
nem sr; nam lineæ Kl, Kr à planis æquidi&longs;tantibus in ea&longs;­<lb/>
dem proportiones &longs;ecantur. </s><s>ergo &longs;olidum gm ad &longs;olidum <lb/>
mc <expan abbr="eand&etilde;">eandem</expan> proportionem habet, quam altitudo ad <expan abbr="altitudin&etilde;">altitu<lb/>
dinem</expan>, uel quam axis ad axem. </s><s>quod <expan abbr="demõ&longs;trare">demon&longs;trare</expan> oportebat.</s></p><p type="margin">
<s><margin.target id="marg50"/>25 undeci<lb/>
mi.</s></p><p type="margin">
<s><margin.target id="marg51"/><expan abbr="&longs;extĩ">&longs;extim</expan>.</s></p><p type="margin">
<s><margin.target id="marg52"/>17. unde­<lb/>
cimi</s></p><p type="head">
<s>THEOREMA XV. PROPOSITIO XIX.<!-- KEEP S--></s></p><p type="main">
<s>Solida parallelepipeda in eadem ba&longs;i, uel in <lb/>
æqualibus ba&longs;ibus con&longs;tituta eam inter &longs;e propor<lb/>
tionem habent, quam altitudines: & &longs;i axes ip&longs;o­<lb/>
rum cum ba&longs;ibus æquales angulos contineant, <lb/>
eam quoque, quam axes proportionem <expan abbr="habebũt">habebunt</expan>.</s></p><p type="main">
<s>Sint &longs;olida parallelepipeda in <expan abbr="ead&etilde;">eadem</expan> ba&longs;i <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> abcd, <lb/>
abef: & &longs;it &longs;olidi abcd altitudo minor: producatur au­<lb/>
tem planum cd adeo, ut &longs;olidum abef &longs;ecet; cuius &longs;ectio <lb/>
<figure id="id.023.01.053.1.jpg" xlink:href="023/01/053/1.jpg"/><lb/>
<arrow.to.target n="marg53"/><lb/>
&longs;it gh. </s><s><expan abbr="erũt">erunt</expan> &longs;oli <lb/>
da abcd, abgh <lb/>
in eadem ba&longs;i, <lb/>
& æquali altitu<lb/>
dine inter &longs;e æ­<lb/>
qualia. </s><s><expan abbr="Quoniã">Quoniam</expan> <lb/>
igitur &longs;olidum <lb/>
abef &longs;ecatur <lb/>
plano ba&longs;ibus <lb/>
<expan abbr="æquidi&longs;tãte">æquidi&longs;tante</expan>, erit <lb/>
<arrow.to.target n="marg54"/><lb/>
&longs;olidum ghef <lb/>
adip&longs;um abgh <pb xlink:href="023/01/054.jpg"/>ut altitudo ad altitudinem: & componendo conuertendo <lb/>
<arrow.to.target n="marg55"/><lb/>
que &longs;olidum abgh, hoc e&longs;t &longs;olidum abcd ip&longs;i æquale, ad­<lb/>
&longs;olidum abef, ut altitudo &longs;olidi abcd ad &longs;olidi abef al­<lb/>
titudinem.</s></p><p type="margin">
<s><margin.target id="marg53"/>29. unde­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg54"/>18. huius</s></p><p type="margin">
<s><margin.target id="marg55"/>7. quinti.</s></p><p type="main">
<s>Sint &longs;olida parallelopipeda ab, cd in æqualibus ba&longs;ibus <lb/>
con&longs;tituta: <expan abbr="&longs;itq;">&longs;itque</expan> be altitudo &longs;olidi ab: & &longs;olidi cd altitudo <lb/>
d f; quæ quidem maior &longs;it, quàm be. </s><s>Dico &longs;olidum ab ad <lb/>
&longs;olidum cd eandem habere proportionem, quam be ad <lb/>
d f. </s><s>ab&longs;cindatur enim à linea df æqualis ip&longs;i be, quæ &longs;it gf: <lb/>
& per g ducatur planum &longs;ecans &longs;olidum cd; quod ba&longs;ibus <lb/>
æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> hK. </s><s>erunt &longs;olida ab, ck æque <lb/>
<arrow.to.target n="marg56"/><lb/>
<figure id="id.023.01.054.1.jpg" xlink:href="023/01/054/1.jpg"/><lb/>
alta inter <lb/>
&longs;e æqualia <lb/>
<expan abbr="cũ">cum</expan> æqua­<lb/>
les ba&longs;es <lb/>
habeant. </s><lb/>
<s><arrow.to.target n="marg57"/><lb/>
Sed <expan abbr="&longs;olidũ">&longs;olidum</expan> <lb/>
hd ad &longs;oli <lb/>
dum cK <lb/>
e&longs;t, ut alti <lb/>
tudo dg <lb/>
ad gf <expan abbr="alti­tudin&etilde;">alti­<lb/>
tudinem</expan>; &longs;e <lb/>
catur enim &longs;olidum cd plano ba&longs;i <lb/>
<figure id="id.023.01.054.2.jpg" xlink:href="023/01/054/2.jpg"/><lb/>
bus æquidi&longs;tante: & rur&longs;us <expan abbr="cõpo-nende">compo­<lb/>
nende</expan>, <expan abbr="conuertendoq;">conuertendoque</expan> <expan abbr="&longs;olidũ">&longs;olidum</expan> ck <lb/>
<arrow.to.target n="marg58"/><lb/>
ad &longs;olidum cd, ut gf ad fd. <!-- KEEP S--></s><s>ergo <lb/>
&longs;olidum ab, quod e&longs;t æquale ip&longs;i <lb/>
ck ad &longs;olidum cd eam proportio <lb/>
nem habet, quam altitudo gf, hoc <lb/>
e&longs;t be ad df altitudinem.</s></p><p type="margin">
<s><margin.target id="marg56"/>31. unde <lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg57"/>18. huius</s></p><p type="margin">
<s><margin.target id="marg58"/>7. quinti.</s></p><p type="main">
<s>Sint deinde &longs;olida parallelepipe <lb/>
da ab, ac in eadem ba&longs;i; quorum <lb/>
axes de, &longs; e cum ip&longs;a æquales angu <pb xlink:href="023/01/055.jpg" pagenum="24"/>los contineant. </s><s>Dico &longs;olidum ab ad &longs;olidum ace idem ha <lb/>
bere proportionem, quam axis de ad axem ef. </s><s>Si enim <lb/>
axes in eadem recta linea fuerint con&longs;tituti, hæc duo &longs;oli­<lb/>
da, in unum, atque idem &longs;olidum conuenient. </s><s>quare ex <lb/>
iis, quæ proxime tradita &longs;unt, habebit &longs;olidum ab ad &longs;o­<lb/>
lidum ac eandem proportionem, quam axis de ad ef <lb/>
axem. </s><s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>
tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, dg, fh: <lb/>
& jungantur eg, eh. </s><s>Quoniam igitur axes cum ba&longs;ibus <lb/>
æquales angulos continent, erit deg angulus æqualis an­<lb/>
<figure id="id.023.01.055.1.jpg" xlink:href="023/01/055/1.jpg"/><lb/>
gulo feh: & &longs;unt <lb/>
anguli ad gh re­<lb/>
cti, quare & re­<lb/>
liquus edg æqua <lb/>
lis erit reliquo <lb/>
efh: & triangu­<lb/>
lum deg <expan abbr="triãgu-lo">triangu­<lb/>
lo</expan> feh &longs;imile. </s><s>er­<lb/>
go gd ad de e&longs;t', <lb/>
ut hf ad e: & per <lb/>
mutando gd ad <lb/>
hf, ut de ad cf. </s><lb/>
<figure id="id.023.01.055.2.jpg" xlink:href="023/01/055/2.jpg"/><lb/>
<s>Sed &longs;olidum ab <lb/>
ad &longs;olidum ac <lb/>
eandem propor­<lb/>
tionem habet, <lb/>
quam dg altitu­<lb/>
do ad <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>
fh. </s><s>ergo & <expan abbr="ean-d&etilde;">ean­<lb/>
dem</expan> habebit, <expan abbr="quã">quam</expan> <lb/>
axis de ad ef <expan abbr="ax&etilde;">axem</expan></s></p><p type="main">
<s>Po&longs;tremo &longs;int <lb/>
&longs;olidi paral le pi <lb/>
peda ab, cd in <pb xlink:href="023/01/056.jpg"/>æqualibus ba&longs;ibus, quorum axes cum ba&longs;ibus æquales an <lb/>
gulos faciant. </s><s>Dico &longs;olidum ab ad <expan abbr="&longs;olidũ">&longs;olidum</expan> cd ita e&longs;&longs;e, ut axis <lb/>
ef ad axem gh: nam &longs;i axes ad planum ba&longs;is recti &longs;int, il­<lb/>
lud per&longs;picue con&longs;tat: quoniam eadem linea, & axem & &longs;oli <lb/>
di altitudinem determinabit. </s><s>Si uero &longs;int inclinati, à pun­<lb/>
ctis eg ad &longs;ubiectum planum perpendiculares ducantur <lb/>
ek, gl: & iungantur fk, hl. <!-- KEEP S--></s><s>rur&longs;us quoniam axes cum ba <lb/>
&longs;ibus æquales faciunt angulos, eodem modo demon&longs;trabi <lb/>
tur, triangulum efK triangulo ghl &longs;imile e&longs;&longs;e: & ek ad gl, <lb/>
ut ef ad gh. </s><s>Solidum autem ab ad &longs;olidum cd e&longs;t, ut <lb/>
eK ad gl. <!-- KEEP S--></s><s>ergo & ut axis ef ad axem gh. </s><s>quæ omnia de <lb/>
mon&longs;trare oportebat.</s></p><p type="main">
<s>Ex iis quæ demon&longs;trata &longs;unt, facile con&longs;tare <lb/>
pote&longs;t, pri&longs;mata omnia & pyramides, quæ trian­<lb/>
gulares ba&longs;es habent, &longs;iue in ei&longs;dem, &longs;iue in æqua <lb/>
<arrow.to.target n="marg59"/><lb/>
libus ba&longs;ibus con&longs;tituantur, eandem proportio­<lb/>
nem habere, quam altitudines: & &longs;i axes cum ba <lb/>
&longs;ibus æquales angulos contineant, &longs;imiliter ean­<lb/>
dem, quam axes, habere proportionem: &longs;unt <lb/>
<arrow.to.target n="marg60"/><lb/>
enim &longs;olida parallelepipeda pri&longs;matum triangula <lb/>
<arrow.to.target n="marg61"/><lb/>
res ba&longs;es <expan abbr="habentiũ">habentium</expan> dupla; & pyramidum &longs;extupla.</s></p><p type="margin">
<s><margin.target id="marg59"/>15. quinti</s></p><p type="margin">
<s><margin.target id="marg60"/>28. unde­<lb/>
cimi.</s></p><p type="margin">
<s><margin.target id="marg61"/>7. duode­<lb/>
cimi.</s></p><p type="head">
<s>THEOREMA XVI. PROPOSITIO XX.<!-- KEEP S--></s></p><p type="main">
<s>Pri&longs;mata omnia & pyramides, quæ in ei&longs;dem, <lb/>
uel æqualibus ba&longs;ibus con&longs;tituuntur, eam inter <lb/>
&longs;e proportionem habent, quam altitudines: & &longs;i <lb/>
axes cum ba&longs;ibus faciant angulos æquales, eam <lb/>
etiam, quam axes habent proportionem.</s></p><pb xlink:href="023/01/057.jpg" pagenum="25"/><p type="main">
<s>Sint duo pri&longs;mata ae, af, quorum eadem ba&longs;is quadri­<lb/>
latera abcd: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae altitudo eg; & pri&longs;matis <lb/>
af altitudo fh. </s><s>Dico pri&longs;ma ae ad pri&longs;ma af eam habere <lb/>
proportionem, quam eg ad fh. </s><s>iungatur enim ac: & in <lb/>
unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum <lb/>
<figure id="id.023.01.057.1.jpg" xlink:href="023/01/057/1.jpg"/><lb/>
ba&longs;es &longs;int triangu <lb/>
la abc, acd. <!-- KEEP S--></s><s>habe <lb/>
bunt duo pri&longs;ma­<lb/>
te in eadem ba&longs;i <lb/>
abc con&longs;tituta, <lb/>
proportionem <expan abbr="eã">eam</expan> <lb/>
dem, quam ip&longs;o­<lb/>
rum altitudines e <lb/>
g, fh, ex iam de­<lb/>
mon&longs;tratis. </s><s>& &longs;i­<lb/>
militer alia duo, <lb/>
quæ &longs;unt in ba&longs;i a <lb/>
<arrow.to.target n="marg62"/><lb/>
c d. <!-- KEEP S--></s><s>quare totum pri&longs;ma ae ad pri&longs;ma af eandem propor<lb/>
tionem habebit, quam altitudo eg ad fh altitudinem. </s><lb/>
<s>Quòd cum pri&longs;mata &longs;int pyramidum tripla, & ip&longs;æ pyrami <lb/>
des, quarum eadem e&longs;t ba&longs;is quadrilatera, & altitudo pri&longs;­<lb/>
matum altitudini æqualis, eam inter &longs;e proportionem ha­<lb/>
bebunt, quam altitudines.</s></p><p type="margin">
<s><margin.target id="marg62"/>12. quinti</s></p><p type="main">
<s>Si uero pri&longs;mata ba&longs;es æquales habeant, <expan abbr="nõ">non</expan> ea&longs;dem, &longs;int <lb/>
duo eiu&longs;modi pri&longs;mata ae, fl: & &longs;it ba&longs;is pri&longs;matis ae qua <lb/>
drilaterum abcd; & pri&longs;matis fl quadrilaterum fghk. </s><lb/>
<s>Dico pri&longs;ma ae ad pri&longs;ma fl ita e&longs;&longs;e, ut altitudo illius ad <lb/>
huius altitudinem. </s><s>nam &longs;i altitudo &longs;it eadem, <expan abbr="intelligãtur">intelligantur</expan> <lb/>
<arrow.to.target n="marg63"/><lb/>
duæ pyramides abcde, fghkl. <!-- KEEP S--></s><s>quæ <expan abbr="ĩtcr&longs;e">intcr&longs;e</expan> æquales <expan abbr="erũt">erunt</expan>, <lb/>
cum æquales ba&longs;es, & altitudinem eandem habeant. </s><s>quare <lb/>
<arrow.to.target n="marg64"/><lb/>
& pri&longs;mata ae, fl, quæ &longs;unt <expan abbr="harũ">harum</expan> pyramidum tripla, æqua­<lb/>
lia &longs;int nece&longs;&longs;e e&longs;t. </s><s>ex quibus per&longs;picue con&longs;tat <expan abbr="propo&longs;itũ">propo&longs;itum</expan>. </s><lb/>
<s>Si uero altitudo pri&longs;matis fl &longs;it maior, à pri&longs;mate fl ab­<lb/>
&longs;cindatur pri&longs;ma fm, quod æque altum &longs;it, <expan abbr="atq;">atque</expan> ip&longs;um ae. <pb xlink:href="023/01/058.jpg"/><figure id="id.023.01.058.1.jpg" xlink:href="023/01/058/1.jpg"/><lb/>
erunt eædem ra­<lb/>
tione pri&longs;mata a <lb/>
e, fm inter &longs;e æ­<lb/>
qualia. </s><s>quare &longs;i­<lb/>
militer demon­<lb/>
&longs;trabitur pri&longs;ma <lb/>
fm ad pri&longs;ma fl <lb/>
eandem habere <lb/>
proportionem, <lb/>
quam pri&longs;matis <lb/>
fm altitudo ad <lb/>
altitudinem ip­<lb/>
&longs;ius fl. <!-- KEEP S--></s><s>ergo & pri&longs;ma ae ad pri&longs;ma fl eandem propor­<lb/>
tionem habebit, quam altitudo ad altitudinem. </s><s>&longs;equitur <lb/>
igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;tituũ">con&longs;tituun</expan> <lb/>
tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>
habeant.</s></p><p type="margin">
<s><margin.target id="marg63"/>6. duode<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg64"/>25. quinti</s></p><figure id="id.023.01.058.2.jpg" xlink:href="023/01/058/2.jpg"/><p type="main">
<s>Sint deinde pri&longs;mata ae, af in eadem ba&longs;i abcd; <expan abbr="quorũ">quorum</expan> <lb/>
axes cum ba&longs;ibus æquales angulos contineant: & &longs;it pri&longs;­<pb xlink:href="023/01/059.jpg" pagenum="26"/>matis ae axis gh; & pri&longs;matis af axis lh. </s><s>Dico pri&longs;ma <lb/>
ae ad pri&longs;ma af eam proportionem habere, quam gh ad <lb/>
h l. <!-- REMOVE S-->ducantur à punctis gl perpendiculares ad ba&longs;is pla­<lb/>
<figure id="id.023.01.059.1.jpg" xlink:href="023/01/059/1.jpg"/><lb/>
num gK, lm: & iungantur kh, <lb/>
h m. </s>
<s>Itaque quoniam anguli gh <lb/>
k, lhm &longs;unt æquales, &longs;imiliter ut <lb/>
&longs;upra demon&longs;trabimus, triangu­<lb/>
la ghK, lhm &longs;imilia e&longs;&longs;e; & ut g <lb/>
K ad lm, ita gh ad hl. <!-- REMOVE S-->habet au <lb/>
tem pri&longs;ma ae ad pri&longs;ma af ean <lb/>
dem proportionem, quam altitu<lb/>
do gK ad altitudinem lm, &longs;icuti <lb/>
demon&longs;tratum e&longs;t. </s>
<s>ergo & ean­<lb/>
dem habebit, quam gh, ad hl. <!-- REMOVE S-->py <lb/>
ramis igitur abcdg ad pyrami­<lb/>
dem abcdl eandem proportio­<lb/>
nem habebit, quam axis gh ad hl axem.</s>
</p><figure id="id.023.01.059.2.jpg" xlink:href="023/01/059/2.jpg"/><p type="main">
<s>Denique &longs;int pri&longs;mata ae, ko in æqualibus ba&longs;ibus ab <lb/>
cd, klmn con&longs;tituta; quorum axes cum ba&longs;ibus æquales <lb/>
faciant angulos: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae axis fg, & altitudo fh: <lb/>
pri&longs;matis autem ko axis pq, & altitudo pr. </s><s>Dico pri&longs;ma <lb/>
ae ad pri&longs;ma ko ita e&longs;&longs;e, ut fg ad pq. </s><s>iunctis enim gh, <pb xlink:href="023/01/060.jpg"/>qr, eodem, quo &longs;upra, modo o&longs;tendemns fg ad pq, ut fh <lb/>
ad pr. </s><s>&longs;ed pri&longs;ma ae ad ip&longs;um ko e&longs;t, ut fh ad pr. </s><s>ergo <lb/>
& ut fg axis ad axem pq.</s><s> ex quibus &longs;it, ut pyramis abcdf <lb/>
<figure id="id.023.01.060.1.jpg" xlink:href="023/01/060/1.jpg"/><lb/>
ad <expan abbr="pyrami-d&etilde;">pyrami­<lb/>
dem</expan> klmnp <lb/>
eandem ha <lb/>
beat pro ­<lb/>
portion&etilde;, <lb/>
<expan abbr="quã">quam</expan> axis ad <lb/>
<expan abbr="ax&etilde;">axem</expan>. </s><s>quod <lb/>
<expan abbr="demon&longs;trã">demon&longs;tran</expan> <lb/>
<expan abbr="dũ">dum</expan> &longs;uerat.</s></p><p type="main">
<s>Simili ra<lb/>
tione in a­<lb/>
liis pri&longs;ma­<lb/>
tibus & py <lb/>
ramidibus eadem demon&longs;trabuntur.</s></p><p type="head">
<s>THEOREMA XVII. PROPOSITIO XXI.<!-- KEEP S--></s></p><p type="main">
<s>Pri&longs;mata omnia, & pyramides inter &longs;e propor<lb/>
tionem habent compo&longs;itam ex proportione ba­<lb/>
&longs;ium, & proportione altitudinum.</s></p><p type="main">
<s>Sint duo pri&longs;mata ae, gm: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae ba&longs;is qua <lb/>
drilaterum abcd, & altitudo ef: pri&longs;matis uero gm ba­<lb/>
&longs;is quadrilaterum ghKl, & altitudo mn. </s><s>Dico pri&longs;ma ae <lb/>
ad pri&longs;ma gm proportionem habere compo&longs;itam ex pro <lb/>
portione ba&longs;is abcd ad ba&longs;im ghkl, & ex proportione <lb/>
altitudinis ef, ad altitudinem mn.</s></p><p type="main">
<s>Sint enim primum ef, mn æquales: & ut ba&longs;is abcd <lb/>
ad ba&longs;im ghkl, ita fiat linea, in qua o ad lineam, in qua p: <lb/>
ut autem ef ad mn, ita linea p ad lineam q.</s><s> erunt lineæ <lb/>
pq inter &longs;e æquales. </s><s>Itaque pri&longs;ma ae ad pri&longs;ma gm <expan abbr="eã">eam</expan> <pb xlink:href="023/01/061.jpg" pagenum="27"/>proportionem habet, quam ba&longs;is abcd ad ba&longs;im ghkl: <lb/>
&longs;i enim intelligantur duæ pyramides abcde, ghklm, ha­<lb/>
bebunt hæ inter &longs;e proportionem eandem, quam ip&longs;ar um <lb/>
ba&longs;es ex &longs;exta duodecimi elementorum. </s><s>Sed ut ba&longs;is abcd <lb/>
ad ghKl ba&longs;im, ita linea o ad lineam p; hoc e&longs;t ad lineam q <lb/>
ei æqualem. </s><s>ergo pri&longs;ma ae ad pri&longs;ma gm e&longs;t, ut linea o <lb/>
ad lineam q.</s><s> proportio autem o ad q copo&longs;ita e&longs;t ex pro­<lb/>
portione o ad p, & ex proportione p ad q.</s><s> quare pri&longs;ma <lb/>
ae ad pri&longs;ma gm, & idcirco pyramis abcde, ad pyrami­<lb/>
dem ghKlm proportionem habet ex ei&longs;dem proportio­<lb/>
nibus compo&longs;itam, uidelicet ex proportione ba&longs;is abcd <lb/>
ad ba&longs;im ghKl, & ex proportione altitudinis ef ad mn al <lb/>
titudinem. </s><s>Quòd &longs;i lineæ ef, mn inæquales ponantur, &longs;it <lb/>
ef minor: & ut ef ad mn, ita fiat linea p ad lineam u: de <lb/>
<figure id="id.023.01.061.1.jpg" xlink:href="023/01/061/1.jpg"/><lb/>
inde ab ip&longs;a mn ab&longs;cindatur rn æqualis ef: & per r duca­<lb/>
tur planum, quod oppo&longs;itis planis æquidi&longs;tans faciat &longs;e­<lb/>
ctionem st. </s><s>erit pri&longs;ma ae, ad pri&longs;ma gt, ut ba&longs;is abcd <lb/>
ad ba&longs;im ghkl; hoc e&longs;t ut o ad p: ut autem pri&longs;ma gt ad <lb/>
<arrow.to.target n="marg65"/><lb/>
pri&longs;ma gm, ita altitudo rn; hoc e&longs;t ef ad altitudine mn; <lb/>
uidelicet linea p ad lineam u. </s><s>ergo ex æquali pri&longs;ma ae ad <lb/>
pri&longs;ma gm e&longs;t, ut linea o ad ip&longs;am u. </s><s>Sed proportio o ad <lb/>
u <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> e&longs;t ex proportione o ad p, quæ e&longs;t ba&longs;is abcd <lb/>
ad ba&longs;im ghkl; & ex proportione p ad u, quæ e&longs;t altitudi­<lb/>
nis ef ad altitudinem mn. </s><s>pri&longs;ma igitur ae ad pri&longs;ma gm <pb xlink:href="023/01/062.jpg"/>compo&longs;itam proportionem habet ex proportione <expan abbr="ba&longs;iũ">ba&longs;ium</expan>, <lb/>
& proportione altitudinum. </s><s>Quare & pyramis, cuius ba­<lb/>
&longs;is e&longs;t quadrilaterum abcd, & altitudo ef ad pyramidem, <lb/>
<figure id="id.023.01.062.1.jpg" xlink:href="023/01/062/1.jpg"/><lb/>
cuius ba&longs;is quadrilaterum ghKl, & altitudo mn, compo&longs;i <lb/>
tam habet proportionem ex proportione ba&longs;ium abcd, <lb/>
ghkl, & ex proportione altitudinum ef, mn. </s><s>quod qui­<lb/>
dem demon&longs;tra&longs;&longs;e oportebat.</s></p><p type="margin">
<s><margin.target id="marg65"/>20. huius</s></p><p type="main">
<s>Ex iam demon&longs;tratis per&longs;picuum e&longs;t, pri&longs;ma <lb/>
ta omnia, & pyramides, in quibus axes cum ba&longs;i­<lb/>
bus æquales angulos continent, proportionem <lb/>
habere compo&longs;itam ex ba&longs;ium proportione, & <lb/>
proportione axium. </s><s>demon&longs;ttatum e&longs;t enim, a­<lb/>
xes inter &longs;e eandem proportionem habere, quam <lb/>
ip&longs;æ altitudines.</s></p><p type="head">
<s>THEOREMA XVIII. PROPOSITIO XXII.<!-- KEEP S--></s></p><p type="main">
<s>CVIVSLIBEt pyramidis, & cuiuslibet coni, <pb xlink:href="023/01/063.jpg" pagenum="28"/>uel coni portionis axis à centro grauitatis ita diui <lb/>
ditur, ut pars, quæ terminatur ad uerticem reli­<lb/>
quæ partis, quæ ad ba&longs;im, &longs;it tripla.</s></p><p type="main">
<s>Sit pyramis, cuius ba&longs;is triangulum abc; axis de; & gra<lb/>
uitatis centrum K. <!-- KEEP S--></s><s>Dico lineam dk ip&longs;ius Ke triplam e&longs;&longs;e. </s><lb/>
<s>trianguli enim bdc centrum grauitatis &longs;it punctum f; <expan abbr="triã">triam</expan> <lb/>
guli adc <expan abbr="centrũ">centrum</expan> g; & trianguli adb &longs;it h: & iungantur af, <lb/>
b g, ch. </s><s>Quoniam igitur <expan abbr="centrũ">centrum</expan> grauitatis pyramidis in axe <lb/>
<arrow.to.target n="marg66"/><lb/>
<expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>: <expan abbr="&longs;untq;">&longs;untque</expan> de, af, bg, ch <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> pyramidis axes: conue <lb/>
nient omnes in <expan abbr="id&etilde;">idem</expan> <expan abbr="punctũ">punctum</expan> k, quod e&longs;t grauitatis centrum. </s><lb/>
<s>Itaque animo concipiamus hanc pyramidem diui&longs;am in <lb/>
quatuor pyramides, quarum ba&longs;es &longs;int ip&longs;a pyramidis <lb/>
<arrow.to.target n="marg67"/><lb/>
<figure id="id.023.01.063.1.jpg" xlink:href="023/01/063/1.jpg"/><lb/>
triangula; & <emph type="ul"/>axis<emph.end type="ul"/> pun­<lb/>
ctum k quæ quidem py­<lb/>
ramides inter &longs;e æquales <lb/>
&longs;unt, ut <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan>. </s><lb/>
<s>Ducatur <expan abbr="enĩ">enim</expan> per lineas <lb/>
dc, de planum <expan abbr="&longs;ecãs">&longs;ecans</expan>, ut <lb/>
&longs;it ip&longs;ius, & ba&longs;is abc <expan abbr="cõ">com</expan> <lb/>
munis &longs;ectio recta linea <lb/>
cel: <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> uero & <expan abbr="triã-guli">trian­<lb/>
guli</expan> adb &longs;it linea dhl. <!-- REMOVE S-->erit linea al æqualis ip&longs;i <lb/>
lb: nam centrum graui­<lb/>
tatis trianguli con&longs;i&longs;tit <lb/>
in linea, quæ ab angulo <lb/>
ad dimidiam ba&longs;im per­<lb/>
ducitur, ex tertia deci­<lb/>
ma Archimedis. <!-- KEEP S--></s><lb/>
<s>quare <lb/>
<arrow.to.target n="marg68"/><lb/>
triangulum acl æquale <lb/>
e&longs;t triangulo bcl: & propterea pyramis, cuius ba&longs;is trian­<lb/>
gulum acl, uertex d, e&longs;t æqualis pyramidi, cuius ba&longs;is bcl <lb/>
<arrow.to.target n="marg69"/><lb/>
triangulum, & idem uertex. </s><s>pyramides enim, quæ ab <expan abbr="eod&etilde;">eodem</expan> <pb xlink:href="023/01/064.jpg"/>&longs;unt uertice, eandem proportionem habent, quam <expan abbr="ip&longs;arũ">ip&longs;arum</expan> <lb/>
ba&longs;es. </s><s>eadem ratione pyramis aclk pyramidi bclk & py <lb/>
ramis adlk ip&longs;i bdlk pyramidi æqualis erit. </s><s>Itaque &longs;i a py <lb/>
ramide acld auferantur pyramides aclk, adlk: & à pyra <lb/>
mide bcld <expan abbr="auferãtur">auferantur</expan> pyramides bclk dblK: quæ relin­<lb/>
quuntur erunt æqualia. </s><s>æqualis igitur e&longs;t pyramis acdk<lb/>
pyramidi bcdK. <!-- KEEP S--></s><s>Rur&longs;us &longs;i per lineas ad, de ducatur pla­<lb/>
num quod pyramidem &longs;ccet: <expan abbr="&longs;itq;">&longs;itque</expan> eius & ba&longs;is communis <lb/>
&longs;ectio aem: &longs;imiliter o&longs;tendetur pyramis abdK æqualis <lb/>
pyramidi acdk. </s><s>ducto denique alio plano per lineas ca, <lb/>
af: ut eius, & trianguli cdb communis &longs;ectio &longs;it cfn, py­<lb/>
ramis abck pyramidi acdk æqualis demon&longs;trabitur. </s><s><expan abbr="cũ">cum</expan> <lb/>
ergo tres pyramides bcdk, abdk, abck uni, & eidem py <lb/>
ramidi acdk &longs;int æquales, omnes inter &longs;e &longs;e æquales <expan abbr="erũt">erunt</expan>. </s><lb/>
<s>Sed ut pyramis abcd ad pyramidem abck ita de axis ad <lb/>
axem ke, ex uige&longs;ima propo&longs;itione huius: &longs;unt enim hæ <lb/>
pyramides in eadem ba&longs;i, & axes cum ba&longs;ibus æquales con <lb/>
tinent angulos, quòd in eadem recta linea con&longs;tituantur. </s><lb/>
<s>quare diuidendo, ut tres pyramides acdk, bcdK, abdK <lb/>
ad pyramidem abcK, ita dk ad Ke. </s><s>con&longs;tat igitur lineam <lb/>
dK ip&longs;ius Ke triplam e&longs;&longs;e. </s><s>&longs;ed & ak tripla e&longs;t Kf: itemque <lb/>
bK ip&longs;ius kg: & ck ip&longs;ius kl tripla. </s><s>quod eodem modo <lb/>
demon&longs;trabimus.</s></p><p type="margin">
<s><margin.target id="marg66"/>17 huius</s></p><p type="margin">
<s><margin.target id="marg67"/><emph type="italics"/>ucrfex<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg68"/>1. sexti.</s></p><p type="margin">
<s><margin.target id="marg69"/>5. duode­<lb/>
cimi.</s></p><p type="main">
<s>Sit pyramis, cuius ba&longs;is quadrilaterum abcd; axis ef: <lb/>
& diuidatur ef in g, ita ut eg ip&longs;ius gf &longs;it tripla. </s><s>Dico cen­<lb/>
trum grauitatis pyramidis e&longs;&longs;e punctum g. <!-- REMOVE S-->ducatur enim <lb/>
linea bd diuidens ba&longs;im in duo triangula abd, bcd: ex <lb/>
quibus <expan abbr="intelligãtur">intelligantur</expan> <expan abbr="cõ&longs;titui">con&longs;titui</expan> duæ pyramides abde, bcde: <lb/>
&longs;itque pyramidis abde axis eh; & pyramidis bcde axis <lb/>
eK: & iungatur hK, quæ per f tran&longs;ibit: e&longs;t enim in ip&longs;a hK <lb/>
centrum grauitatis magnitudinis compo&longs;itæ ex triangulis <lb/>
abd, bcd, hoc e&longs;t ip&longs;ius quadrilateri. </s>
<s>Itaque centrum gra<lb/>
uitatis pyramidis abde &longs;it punctum l: & pyramidis bcde <lb/>
<arrow.to.target n="marg70"/><lb/>
&longs;it m. </s><s>ducta igitur lm ip&longs;i hm lineæ æquidi&longs;tabit. </s><s>nam el ad <pb xlink:href="023/01/065.jpg" pagenum="29"/>lh eandem habet proportionem, quam em ad mk, uideli­<lb/>
cet triplam. </s><s>quare linea lm ip&longs;am ef &longs;ecabit in puncto g: <lb/>
etenim eg ad gf e&longs;t, ut el ad lh. </s><s>præterea quoniam hk, lm <lb/>
æquidi&longs;tant, erunt triangula hef, leg &longs;imilia: <expan abbr="itemq;">itemque</expan> inter <lb/>
&longs;e &longs;imilia fek gem: & ut ef ad eg, ita hf ad lg: & ita fK ad <lb/>
gm. </s><s>ergo ut hf ad lg, ita fk ad gm: & permutando ut hf <lb/>
ad fK, ita lg ad gm. </s><s>&longs;ed cum h &longs;it centrum trianguli abd; <lb/>
& k <expan abbr="triãguli">trianguli</expan> bcd <expan abbr="punctũ">punctum</expan> uero f totius quadrilateri abcd <lb/>
centrum: erit ex 8. Archimedis de centro grauitatis plano <lb/>
rum hf ad fk ut triangulum bcd ad triangulum abd: ut, <lb/>
autem bcd triangulum ad triangulum abd, ita pyramis <lb/>
<figure id="id.023.01.065.1.jpg" xlink:href="023/01/065/1.jpg"/><lb/>
bcde ad pyramidem abde. </s><s>ergo <lb/>
linea lg ad gm erit, ut pyramis <lb/>
bcde ad <expan abbr="pyramid&etilde;">pyramidem</expan> abde. </s><s>ex quo <lb/>
&longs;equitur, ut totius pyramidis <lb/>
abcde punctum g &longs;it grauitatis <lb/>
centrum. </s><s>Rur&longs;us &longs;it pyramis ba­<lb/>
&longs;im habens pentagonum abcde: <lb/>
& axem fg: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="pũ">pun</expan> <lb/>
cto h, ita ut fh ad hg triplam habe <lb/>
at proportionem. </s><s>Dico h grauita­<lb/>
tis <expan abbr="centrũ">centrum</expan> e&longs;&longs;e pyramidis abcdef. </s><lb/>
<s>iungatur enim eb: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>
pyramis, cuius uertex f, & ba&longs;is <lb/>
triangulum abe: & alia pyramis <lb/>
intelligatur eundem uerticem ha­<lb/>
bens, & ba&longs;im bcde <expan abbr="quadrilaterũ">quadrilaterum</expan>: <lb/>
&longs;it autem pyramidis abef axis fk<lb/>
& grauitatis centrum l: & pyrami <lb/>
dis bcdef axis fm, & centrum gra <lb/>
uitatis n:<expan abbr="iunganturq;">iunganturque</expan> km, ln; <lb/>
quæ per puncta gh tran&longs;ibunt. </s><lb/>
<s>Rur&longs;us eodemmodo, quo &longs;up ra, <lb/>
demon&longs;trabimus lineas Kgm, lhn &longs;ibi ip&longs;is æquidi&longs;tare: <pb xlink:href="023/01/066.jpg"/>& denique punctum h pyramidis abcdef grauitatis e&longs;&longs;e <lb/>
centrum, & ita in aliis.</s></p><p type="margin">
<s><margin.target id="marg70"/>2. fexti.</s></p><p type="main">
<s>Sit conus, uel coni portio axem habens bd: &longs;eceturque <lb/>
plano per axem, quod &longs;ectionem faciat triangulum abc: <lb/>
& bd axis diuidatur in c, ita ut be ip&longs;ius ed &longs;it tripla. </s><lb/>
<s>Dico punctum e coni, uel coni portionis, grauitatis <lb/>
e&longs;&longs;e centrum. </s><s>Si enim fieri pote&longs;t, &longs;it centrum f: & pro­<lb/>
ducatur ef extra figuram in g. <!-- KEEP S--></s><s>quam uero proportionem <lb/>
habet ge ad ef, habeat ba&longs;is coni, uelconi portionis, hoc <lb/>
e&longs;t circulus, uel ellip&longs;is circa diametrum ac ad aliud &longs;pa­<lb/>
cium, in quo h. </s><s>Itaque in circulo, uel ellip&longs;i plane de&longs;cri­<lb/>
batur rectilinea figura axlmcnop, ita ut quæ <expan abbr="relinquũ-tur">relinquun­<lb/>
tur</expan> portiones &longs;int minores &longs;pacio h: & intelligatur pyra­<lb/>
mis ba&longs;im habens rectilineam figuram aKlmcnop, & <lb/>
axem bd; cuius quidem grauitatis centrum erit punctum <lb/>
e, utiam demon&longs;trauimus. </s><s>Et quoniam portiones &longs;unt <lb/>
minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma­<lb/>
<figure id="id.023.01.066.1.jpg" xlink:href="023/01/066/1.jpg"/><lb/>
iorem proportionem habet, quam ge ad ef. </s><s>&longs;ed ut circu­<lb/>
lus, uel ellip&longs;is ad figuram rectilineam &longs;ibi in&longs;criptam, ita <lb/>
conus, uel coni portio ad pyramidem, quæ figuram rectili­<lb/>
neam pro ba&longs;i habet; & altitudinem æqualem: etenim &longs;u­<pb xlink:href="023/01/067.jpg" pagenum="30"/><arrow.to.target n="marg71"/><lb/>
pra demon&longs;tratum e&longs;t, ita e&longs;&longs;e cylindrum, uel cylindri por­<lb/>
tionem ad pri&longs;ma, cuius ba&longs;is rectilinea figura, & æqua­<lb/>
lis altitudo. </s><s>ergo per conuer&longs;ionem rationis, ut circulus, <lb/>
uel ellip&longs;is ad portiones, ita conus, uel coni portio ad por­<lb/>
tiones &longs;olidas. </s><s>quare conus uel coni portio ad portiones <lb/>
&longs;olidas maiorem habet proportionem, quam ge ad ef: & <lb/>
diuidendo, pyramis ad portiones &longs;olidas maiorem pro­<lb/>
portionem habet, quam gf ad fe. </s><s>fiat igitur qf ad fe <lb/>
ut pyramis ad dictas portiones. </s><s>Itaque quoniam a cono <lb/>
uel coni portione, cuius grauitatis centrum e&longs;t f, aufer­<lb/>
tur pyramis, cuius centrum e; reliquæ magnitudinis, <lb/>
quæ ex &longs;olidis portionibus con&longs;tat, centrum grauitatis <lb/>
erit in linea ef protracta, & in puncto q.</s><s> quod fieri <lb/>
non pote&longs;t: e&longs;t enim centrum grauitatis intra. </s><s>Con&longs;tat <lb/>
igitur coni, uel coni portionis grauitatis centrum e&longs;&longs;e pun <lb/>
ctum e. </s><s>quæ omnia demon&longs;trare oportebat.</s></p><p type="margin">
<s><margin.target id="marg71"/>8 huius</s></p><p type="head">
<s>THEOREMA XIX. PROPOSITIO XXIII.<!-- KEEP S--></s></p><p type="main">
<s>QVODLIBET fru&longs;tum à pyramide, quæ <lb/>
triangularem ba&longs;im habeat, ab&longs;ci&longs;&longs;um, diuiditur <lb/>
in tres pyramides proportionales, in ea proportio <lb/>
ne, quæ e&longs;t lateris maioris ba&longs;is ad latus minoris <lb/>
ip&longs;i re&longs;pondens.</s></p><p type="main">
<s>Hoc demon&longs;trauit Leonardus Pi&longs;anus in libro, qui de­<lb/>
praxi geometriæ in&longs;cribitur. </s><s>Sed quoniam is adhuc im­<lb/>
pre&longs;&longs;us non e&longs;t, nos ip&longs;ius demon&longs;trationem breuiter <lb/>
per&longs;tringemus, rem ip&longs;am &longs;ecuti, non uerba. </s><s>Sit fru­<lb/>
&longs;tum pyramidis abcdef, cuius maior ba&longs;is triangulum <lb/>
abc, minor def: & iunctis ae, cc, cd, per, line­<lb/>
as ae, ec ducatur planum &longs;ecans fru&longs;tum: itemque per <lb/>
lineas ec, cd; & per cd, da alia plana ducantur, quæ <lb/>
diuident fru&longs;tum in trcs pyramides abce, adce, defc. <pb xlink:href="023/01/068.jpg"/>Dico eas proportionales e&longs;&longs;e in proportione, quæ e&longs;t la­<lb/>
teris ab adlatus de, itaut earum maior &longs;it abce, me­<lb/>
dia adce, & minor defc. <!-- KEEP S--></s><s>Quoniam enim lineæ de, <lb/>
ab æquidi&longs;tant; & inter ip&longs;as &longs;unt triangula abe, ade; <lb/>
<arrow.to.target n="marg72"/><lb/>
<figure id="id.023.01.068.1.jpg" xlink:href="023/01/068/1.jpg"/><lb/>
erit triangulum abe <lb/>
ad triangulum abe, <lb/>
utlinea ab ad lineam <lb/>
de. </s><s>ut autem triangu <lb/>
lum abe ad triangu­<lb/>
<arrow.to.target n="marg73"/><lb/>
lum abe, ita pyramis <lb/>
abec ad pyramidem <lb/>
adec: habent enim <lb/>
altitudinem eandem, <lb/>
quæ e&longs;tà puncto cad <lb/>
planum, in quo qua­<lb/>
<arrow.to.target n="marg74"/><lb/>
drilaterum abed. <!-- KEEP S--></s><s>er­<lb/>
go ut ab ad de, ita pyramis abec ad pyramidem adec. <!-- KEEP S--></s><lb/>
<s>Rur&longs;us quoniam æquidi&longs;tantes &longs;unt ac, df; erit eadem <lb/>
<arrow.to.target n="marg75"/><lb/>
ratione pyramis adce ad pyramidem cdfe, ut ac ad <lb/>
df. </s><s>Sed ut ac ad df, ita ab ad de, quoniam triangula <lb/>
abc, def &longs;imilia &longs;unt, ex nona huius. </s><s>quare ut pyramis <lb/>
abce ad pyramidem abce, ita pyramis adce ad ip&longs;am<lb/>
defc. <!-- REMOVE S-->fru&longs;tum igitur abcdef diuiditur in tres pyramides <lb/>
proportionales in ea proportione, quæ e&longs;t lateris ab ad de <lb/>
latus, & earum maior e&longs;t cabe, media adce, & minor <lb/>
defc. <!-- REMOVE S-->quod demon&longs;trare oportebat.</s>
</p><p type="margin">
<s><margin.target id="marg72"/>1. &longs;exti.</s></p><p type="margin">
<s><margin.target id="marg73"/>5. duodeci <lb/>
mi.</s></p><p type="margin">
<s><margin.target id="marg74"/>11. quinti.</s></p><p type="margin">
<s><margin.target id="marg75"/>4 &longs;exti.</s></p><p type="head">
<s>PROBLEMA V. PROPOSITIO XXIIII.<!-- KEEP S--></s></p><p type="main">
<s>QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>
uel coni portionis, plano ba&longs;i æquidi&longs;tanti ita &longs;e­<lb/>
care, ut &longs;ectio &longs;it proportionalis inter maiorem, <lb/>
& minorem ba&longs;im.</s></p><pb xlink:href="023/01/069.jpg" pagenum="31"/><p type="main">
<s>SIT fru&longs;tum pyramidis ae, cuius maior ba&longs;is triangu­<lb/>
lum abc, minor def: & oporteat ip&longs;um plano, quod ba&longs;i <lb/>
æquidi&longs;tet, ita &longs;ecare, ut &longs;ectio &longs;it proportionalis inter <expan abbr="triã">trian</expan> <lb/>
gula abc, def. </s><s>Inueniatur inter lineas ab, de media pro­<lb/>
portionalis, quæ &longs;it bg: & à puncto g erigatur gh æquidi­<lb/>
&longs;tans be, <expan abbr="&longs;ecansq;">&longs;ecansque</expan> ad in h: deinde per h ducatur planum <lb/>
ba&longs;ibus æquidi&longs;tans, cuius &longs;ectio &longs;it triangulum hkl. <!-- KEEP S--></s><s>Dico <lb/>
triangulum hKl proportionale e&longs;&longs;e inter triangula abc, <lb/>
<figure id="id.023.01.069.1.jpg" xlink:href="023/01/069/1.jpg"/><lb/>
def, hoc e&longs;t triangulum abc ad <lb/>
triangulum hKl eandem habere <lb/>
proportionem, quam <expan abbr="triãgulum">triangulum</expan> <lb/>
hKl ad ip&longs;um def. </s><s><expan abbr="Quoniã">Quoniam</expan> enim <lb/>
<arrow.to.target n="marg76"/><lb/>
lineæ ab, hK æquidi&longs;tantium pla <lb/>
norum &longs;ectiones inter &longs;e æquidi­<lb/>
&longs;tant: atque æquidi&longs;tant bk, gh: <lb/>
<arrow.to.target n="marg77"/><lb/>
linea hk ip&longs;i gb e&longs;t æqualis: & pro <lb/>
pterea proportionalis inter ab, <lb/>
de. </s><s>quare ut ab ad hK, ita e&longs;t hk<lb/>
ad de. </s><s>fiat ut hk ad de, ita de <lb/>
ad aliam lineam, in qua &longs;it m. </s><s>erit <lb/>
ex æquali ut ab ad de, ita hk ad <lb/>
<arrow.to.target n="marg78"/><lb/>
m. </s><s>Et quoniam triangula abc, <lb/>
hKl, def &longs;imilia &longs;unt; <expan abbr="triangulũ">triangulum</expan> <lb/>
<arrow.to.target n="marg79"/><lb/>
abc ad triangulum hkl e&longs;t, ut li­<lb/>
nea ab ad lineam de: <expan abbr="triangulũ">triangulum</expan> <lb/>
<arrow.to.target n="marg80"/><lb/>
autem hkl ad ip&longs;um def e&longs;t, ut hk ad m. </s><s>ergo triangulum <lb/>
abc ad triangulum hkl eandem proportionem habet, <lb/>
quam triangulum hKl ad ip&longs;um def. </s><s>Eodem modo in a­<lb/>
liis fru&longs;tis pyramidis idem demon&longs;trabitur.</s></p><p type="margin">
<s><margin.target id="marg76"/>16. unnde<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg77"/>34. primi</s></p><p type="margin">
<s><margin.target id="marg78"/>9. huius <lb/>
corol.</s></p><p type="margin">
<s><margin.target id="marg79"/>20. &longs;exti</s></p><p type="margin">
<s><margin.target id="marg80"/>11. quinti</s></p><p type="main">
<s>Sit fru&longs;tum coni, uel coni portionis ad: & &longs;ecetur plano <lb/>
per axem, cuius &longs;ectio &longs;it abcd, ita ut maior ip&longs;ius ba&longs;is &longs;it <lb/>
circulus, uel ellip&longs;is circa diametrum ab; minor circa cd. <!-- KEEP S--></s><lb/>
<s>Rur&longs;us inter lineas ab, cd inueniatur proportionalis be: <lb/>
& ab e ducta ef æquidi&longs;tante bd, quæ lineam ca in f &longs;ecet, <pb xlink:href="023/01/070.jpg"/>per f planum ba&longs;ibus æquidi&longs;tans ducatur, ut &longs;it &longs;ectio cir <lb/>
culus, uel ellip&longs;is circa diametrum fg. <!-- KEEP S--></s><s>Dico &longs;ectionem ab <lb/>
ad &longs;ectionem fg eandem proportionem habere, quam fg <lb/>
ad ip&longs;am cd. <!-- KEEP S--></s><s>Simili enim ratione, qua &longs;upra, demon&longs;trabi­<lb/>
tur quadratum ab ad quadratum fg ita e&longs;&longs;e, ut <expan abbr="quadratũ">quadratum</expan> <lb/>
<arrow.to.target n="marg81"/><lb/>
fg ad cd quadratum. </s><s>Sed circuli inter &longs;e eandem propor­<lb/>
tionem habent, quam diametrorum quadrata. </s><s>ellip&longs;es au­<lb/>
tem circa ab, fg, cd, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="cõ-mentariis">con­<lb/>
mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>
& &longs;phæroidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>
ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio­<lb/>
<figure id="id.023.01.070.1.jpg" xlink:href="023/01/070/1.jpg"/><lb/>
&longs;eptimæ propo&longs;itionis eiu&longs;dem li­<lb/>
bri. </s><s>ellip&longs;es enim nunc appello ip­<lb/>
&longs;a &longs;pacia ellip&longs;ibus contenta. </s><s>ergo <lb/>
circulus, uel ellip&longs;is ab ad <expan abbr="circulũ">circulum</expan>, <lb/>
uel ellip&longs;im fg eam proportionem <lb/>
habet, quam circulus, uel ellip&longs;is <lb/>
fg ad circulum uel ellip&longs;im cd. <!-- KEEP S--></s><lb/>
<s>quod quidem faciendum propo­<lb/>
&longs;uimus.</s></p><p type="margin">
<s><margin.target id="marg81"/>2. duode<lb/>
cimi</s></p><p type="head">
<s>THEOREMA XX. PROPOSITIO XXV.<!-- KEEP S--></s></p><p type="main">
<s>QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>
uel coni portionis ad pyramidem, uel conum, uel <lb/>
coni portionem, cuius ba&longs;is eadem e&longs;t, & æqualis <lb/>
altitudo, eandem <expan abbr="proportion&etilde;">proportionem</expan> habet, quam utræ <lb/>
que ba&longs;es, maior, & minor &longs;imul &longs;umptæ vnà <expan abbr="cũ">cum</expan> <lb/>
ca, quæ inter ip&longs;as &longs;it proportionalis, ad ba&longs;im ma <lb/>
iorem.</s></p><pb xlink:href="023/01/071.jpg" pagenum="32"/><p type="main">
<s>SIT <expan abbr="fru&longs;tũ">fru&longs;tum</expan> pyramidis, uel coni, uel coni portionis ad, <lb/>
cuius maior ba&longs;is ab, minor cd. <!-- KEEP S--></s><s>& &longs;ecetur altero plano <lb/>
ba&longs;i æquidi&longs;tante, ita ut &longs;ectio ef &longs;it proportionalis inter <lb/>
ba&longs;es ab, cd. <!-- KEEP S--></s><s>con&longs;tituatur <expan abbr="aut&etilde;">autem</expan> pyramis, uel conus, uel co­<lb/>
ni portio agb, cuius ba&longs;is &longs;it eadem, quæ ba&longs;is maior fru­<lb/>
<figure id="id.023.01.071.1.jpg" xlink:href="023/01/071/1.jpg"/><lb/>
&longs;ti, & altitudo æqualis. </s><s>Di­<lb/>
co fru&longs;tum ad ad pyrami­<lb/>
dem, uel conum, uel coni <lb/>
portionem agb eandem <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> habere, <expan abbr="quã">quam</expan> <lb/>
utræque ba&longs;es, ab, cd unà <lb/>
cum ef ad ba&longs;im ab. </s><s>e&longs;t <lb/>
enim fru&longs;tum ad æquale <lb/>
pyramidi, uel cono, uel co­<lb/>
ni portioni, cuius ba&longs;is ex <lb/>
tribus ba&longs;ibus ab, ef, cd <lb/>
con&longs;tat; & altitudo ip&longs;ius <lb/>
altitudini e&longs;t æqualis: quod mox o&longs;tendemus. </s><s>Sed pyrami <lb/>
<figure id="id.023.01.071.2.jpg" xlink:href="023/01/071/2.jpg"/><lb/>
des, coni, uel coni <expan abbr="portiões">portiones</expan>, <lb/>
quæ &longs;unt æquali altitudine, <lb/>
<expan abbr="eãdem">eandem</expan> inter &longs;e, quam ba&longs;es, <lb/>
proportionem habent, &longs;icu­<lb/>
ti demon&longs;tratum e&longs;t, partim <lb/>
<arrow.to.target n="marg82"/><lb/>
ab Euclide in duodecimo li­<lb/>
bro elementorum, partim à <lb/>
nobis in <expan abbr="cõmentariis">commentariis</expan> in un­<lb/>
decimam <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> Ar­<lb/>
chimedis de conoidibus, & <lb/>
&longs;phæroidibus. </s><s>quare pyra­<lb/>
mis, uel conus, uel coni por­<lb/>
tio, cuius ba&longs;is e&longs;t tribus illis <lb/>
ba&longs;ibus æqualis ad agb eam <lb/>
habet proportionem, quam <lb/>
ba&longs;es ab, ef, cd ad ab ba&longs;im. </s><s>Fru&longs;tum igitur ad ad agb <pb xlink:href="023/01/072.jpg"/>pyramidem, uel conum, uel coni portionem eandem pro­<lb/>
portionem habet, quam ba&longs;es ab, cd unà cum ef ad ba­<lb/>
&longs;im ab. </s><s>quod demon&longs;trare uolebamus.</s></p><p type="margin">
<s><margin.target id="marg82"/>6. 11. duo<lb/>
decimi</s></p><p type="main">
<s>Fru&longs;tum uero ad æquale e&longs;&longs;e pyramidi, uel co <lb/>
no, uel coni portioni, cuius ba&longs;is con&longs;tat ex ba&longs;i­<lb/>
bus ab, cd, ef, & altitudo fru&longs;ti altitudini e&longs;t æ­<lb/>
qualis, hoc modo o&longs;tendemus.</s></p><p type="main">
<s>Sit fru&longs;tum pyramidis abcdef, cuius maior ba&longs;is trian­<lb/>
gulum abc; minor def: & &longs;ecetur plano ba&longs;ibus æquidi­<lb/>
&longs;tante, quod &longs;ectionem faciat triangulum ghk inter trian­<lb/>
gula abc, def proportionale. </s><s>Iam ex iis, quæ demon&longs;trata <lb/>
&longs;unt in 23. huius, patet fru&longs;tum abcdef diuidi in tres pyra <lb/>
mides proportionales; & earum maiorem e&longs;&longs;e <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>
abcd <expan abbr="minor&etilde;">minorem</expan> uero defb. </s><s>ergo pyramis à triangulo ghk <lb/>
con&longs;tituta, quæ altitudinem habeat fru&longs;ti altitudini æqua­<lb/>
lem, proportionalis e&longs;t inter pyramides abcd, defb: & <lb/>
idcirco fru&longs;tum abcdef tribus dictis pyramidibus æqua <lb/>
<figure id="id.023.01.072.1.jpg" xlink:href="023/01/072/1.jpg"/><lb/>
le erit. </s><s>Itaque &longs;i intelligatur alia pyra­<lb/>
mis æque alta, quæ ba&longs;im habeat ex tri <lb/>
bus ba&longs;ibus abc, def, ghk con&longs;tan­<lb/>
tem; per&longs;picuum e&longs;t ip&longs;am ei&longs;dem py­<lb/>
ramidibus, & propterea ip&longs;i fru&longs;to æ­<lb/>
qualem e&longs;&longs;e.</s></p><p type="main">
<s>Rur&longs;us &longs;it fru&longs;tum pyramidis ag, cu <lb/>
ius maior ba&longs;is quadrilaterum abcd, <lb/>
minor efgh: & &longs;ecetur plano ba&longs;i­<lb/>
bus æquidi&longs;tante, ita ut fiat &longs;ectio qua­<lb/>
drilaterum Klmn, quod &longs;it proportio <lb/>
nale inter quadrilatera abcd, efgh. </s><s>Dico pyramidem, <lb/>
cuius ba&longs;is &longs;it æqualis tribus quadrilateris abcd, klmn, <lb/>
efgh, & altitudo æqualis altitudini fru&longs;ti, ip&longs;i fru&longs;to ag <lb/>
æqualem e&longs;&longs;e. </s><s>Ducatur enim planum per lineas fb, hd, <pb xlink:href="023/01/073.jpg" pagenum="33"/>quod diuidat fru&longs;tum in duo fru&longs;ta triangulares ba&longs;es ha­<lb/>
bentia, uidelicet in fru&longs;tum abdefh, & in <expan abbr="fru&longs;tũ">fru&longs;tum</expan> bcdfgh. </s><lb/>
<s>erit triangulum kln proportionale inter triangula abd, <lb/>
efh: & triangulum lmn proportionale inter bcd, fgh. </s><lb/>
<s>&longs;ed pyramis æque alta, cuius ba&longs;is con&longs;tat ex tribus trian­<lb/>
<figure id="id.023.01.073.1.jpg" xlink:href="023/01/073/1.jpg"/><lb/>
gulis abd, klz, efh, demon&longs;trata <lb/>
e&longs;t fru&longs;to abdcfh æqualis. </s><s>& &longs;i­<lb/>
militer pyramis, cuius ba&longs;is con­<lb/>
&longs;tat ex triangulis bcd, lmn, fgh <lb/>
æqualis fru&longs;to bcdfgh: compo­<lb/>
nuntur autem tria quadrilatera a <lb/>
bcd, klmn, efgh è &longs;ex triangu­<lb/>
lis iam dictis. </s><s>pyramis igitur ba­<lb/>
&longs;im habens æqualem tribus qua­<lb/>
drilateris, & altitudinem eandem <lb/>
ip&longs;i fru&longs;to ag e&longs;t æqualis. </s><s>Eodem <lb/>
modo illud <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan> in aliis <lb/>
eiu&longs;modi fru&longs;tis.</s></p><p type="main">
<s>Sit fru&longs;tum coni, uel coni portionis ad; cuius maior ba­<lb/>
&longs;is circulus, uel ellip&longs;is circa diametrum ab; minor circa <lb/>
c d: & &longs;ecetur plano, quod ba&longs;ibus æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> &longs;e­<lb/>
ctionem circulum, uel ellip&longs;im circa diametrum ef, ita ut <lb/>
inter circulos, uel ellip&longs;es ab, cd &longs;it proportionalis. </s><s>Dico <lb/>
conum, uel coni portionem, cuius ba&longs;is e&longs;t æqualis tribus <lb/>
circulis, uel tribus ellip&longs;ibus ab, ef, cd; & altitudo eadem, <lb/>
quæ fru&longs;ti ad, ip&longs;i fru&longs;to æqualem e&longs;&longs;e. </s><s>producatur enim <lb/>
fru&longs;ti &longs;uperficies quou&longs;que coeat in unum punctum, quod <lb/>
&longs;it g: & coni, uel coni portionis agb axis &longs;it gh, occurrens <lb/>
planis ab, ef, cd in punctis hkl: circa circulum uero de­<lb/>
&longs;cribatur quadratum mnop, & circa ellip&longs;im <expan abbr="rectangulũ">rectangulum</expan> <lb/>
mnop, quod ex ip&longs;ius diametris con&longs;tat: <expan abbr="iunctisq;">iunctisque</expan> gm, <lb/>
g n, go, gp, ex eodem uertice intelligatur pyramis ba&longs;im <lb/>
habens dictum quadratum, uel rectangulum: & plana in <lb/>
quibus &longs;unt circuli, uel ellip&longs;es ef, cd u&longs;que ad eius latera <pb xlink:href="023/01/074.jpg"/>producantur. </s><s>Quoniam igitur pyramis &longs;ecatur planis ba&longs;i <lb/>
<arrow.to.target n="marg83"/><lb/>
æquidi&longs;tantibus, &longs;ectiones &longs;imiles erunt: atque erunt qua­<lb/>
drata, uel rectangula circa circulos, uel ellip&longs;es de&longs;cripta, <lb/>
quemadmodum & in ip&longs;a ba&longs;i. </s><s>Sed cum circuli inter &longs;e <expan abbr="eã">eam</expan> <lb/>
<arrow.to.target n="marg84"/><lb/>
proportionem habeant, quam diametrorum quadrata: <lb/>
<expan abbr="itemq;">itemque</expan> ellip&longs;es eam quam rectangula ex ip&longs;arum diametris <lb/>
<arrow.to.target n="marg85"/><lb/>
con&longs;tantia: & &longs;it circulus, uel ellip&longs;is circa diametrum ef <lb/>
<figure id="id.023.01.074.1.jpg" xlink:href="023/01/074/1.jpg"/><lb/>
proportionalis inter circulos, uel ellip&longs;es ab, cd; erit re­<lb/>
ctangulum ef etiam inter rectangula ab, cd proportio­<lb/>
nale: per rectangulum enim nunc breuitatis cau&longs;a <expan abbr="etiã">etiam</expan> ip­<lb/>
&longs;um quadratum intelligemus. </s><s>quare ex iis, quæ proxime <lb/>
dicta &longs;unt, pyramis ba&longs;im habens æqualem dictis rectangu <lb/>
lis, & altitudinem eandem, quam fru&longs;tum ad, ip&longs;i fru&longs;to à <lb/>
pyramide ab&longs;ci&longs;&longs;o æqualis probabitur. </s><s>ut autem rectangu <lb/>
lum cd ad <expan abbr="rectangulũ">rectangulum</expan> ef, ita circulus, uel ellip&longs;is cd ad ef <lb/>
circulum, uel ellip&longs;im: <expan abbr="componendoq;">componendoque</expan> ut rectangula cd, <lb/>
e f, ad ef rectangulum, ita circuli, uel ellip&longs;es ed, ef, ad ef: <lb/>
& ut rectangulum ef ad rectangulum ab, ita circulus, uel <lb/>
ellip&longs;is ef ad ab circulum, uel ellip&longs;im. </s><s>ergo ex æquali, & <lb/>
componendo, ut <expan abbr="rectãgula">rectangula</expan> cd, ef, ab ad ip&longs;um ab, ita cir­<pb xlink:href="023/01/075.jpg" pagenum="34"/>culi, uel ellip&longs;es cd, ef ab ad circulum, uel ellip&longs;im ab. </s><s>In­<lb/>
telligatur pyramis q ba&longs;im habens æqualem tribus rectan <lb/>
gulis ab, ef, cd; & altitudinem <expan abbr="eãdem">eandem</expan>, quam fru&longs;tum ad. <!-- KEEP S--></s><lb/>
<s>intelligatur etiam conus, uel coni portio q, eadem altitudi<lb/>
ne, cuius ba&longs;is &longs;it tribus circulis, uel tribus ellip&longs;ibus ab, <lb/>
ef, cd æqualis. </s><s>po&longs;tremo intelligatur pyramis alb, cuius. </s><lb/>
<s>ba&longs;is &longs;it rectangulum mnop, & altitudo eadem, quæ fru­<lb/>
&longs;ti: <expan abbr="itemq,">itemque</expan> intelligatur conus, uel coni portio alb, cuius <lb/>
ba&longs;is circulus, uel ellip&longs;is circa diametrum ab, & eadem al <lb/>
<arrow.to.target n="marg86"/><lb/>
titudo. </s><s>ut igitur rectangula ab, ef, cd ad rectangulum ab, <lb/>
ita pyramis q ad pyramidem alb; & ut circuli, uel ellip­<lb/>
&longs;es ab, ef, cd ad ab circulum, uel ellip&longs;im, ita conus, uel co<lb/>
ni portio q ad conum, uel coni portionem alb. </s><s>conus <lb/>
igitur, uel coni portio q ad conum, uel coni portionem <lb/>
alb e&longs;t, ut pyramis q ad pyramidem alb. </s><s>&longs;ed pyramis <lb/>
alb ad pyramidem agb e&longs;t, ut altitudo ad altitudinem, ex <lb/>
20. huius: & ita e&longs;t conus, uel coni portio alb ad conum, <lb/>
uel coni portionem agb ex 14. duodecimi elementorum, <lb/>
& ex iis, quæ nos demon&longs;trauimus in commentariis in un­<lb/>
decimam de conoidibus, & &longs;phæroidibus, propo&longs;itione <lb/>
quarta. </s><s>pyramis autem agb ad pyramidem cgd propor­<lb/>
tionem habet compo&longs;itam ex proportione ba&longs;ium & pro <lb/>
portione altitudinum, ex uige&longs;ima prima huius: & &longs;imili­<lb/>
ter conus, uel coni portio agb ad conum, uel coni portio­<lb/>
nem cgd proportionem habet <expan abbr="compo&longs;itã">compo&longs;itam</expan> ex ei&longs;dem pro­<lb/>
portionibus, per ea, quæ in dictis commentariis demon­<lb/>
&longs;trauimus, propo&longs;itione quinta, & &longs;exta: altitudo enim in<lb/>
utri&longs;que eadem e&longs;t, & ba&longs;es inter &longs;e &longs;e eandem habent pro­<lb/>
portionem. </s><s>ergo ut pyramis agb ad pyramidem cgd, ita <lb/>
e&longs;t conus, uel coni portio agb ad agd conum, uel coni <lb/>
portionem: & per <expan abbr="conuer&longs;ion&etilde;">conuer&longs;ionem</expan> rationis, ut pyramis agb <lb/>
ad <expan abbr="&longs;ru&longs;tũ">fru&longs;tum</expan> à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio <lb/>
agb ad fru&longs;tum ad. <!-- KEEP S--></s><s>ex æquali igitur, ut pyramis q ad fru­<lb/>
&longs;tum à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio q ad <pb xlink:href="023/01/076.jpg"/>fru&longs;tum ad. <!-- KEEP S--></s><s>Sed pyramis q æqualis e&longs;t fru&longs;to à pyramide <lb/>
ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. </s><s>ergo & conus, uel coni por­<lb/>
tio q, cuius ba&longs;is ex tribus circulis, uel ellip&longs;ibus ab, ef, cd <lb/>
con&longs;tat, & altitudo eadem, quæ fru&longs;ti: ip&longs;i fru&longs;to ad e&longs;t æ­<lb/>
qualis. </s><s>atque illud e&longs;t, quod demon&longs;trare oportebat.</s></p><p type="margin">
<s><margin.target id="marg83"/>9 huius</s></p><p type="margin">
<s><margin.target id="marg84"/>2. duode­<lb/>
cnni.</s></p><p type="margin">
<s><margin.target id="marg85"/>7. de co­<lb/>
noidibus <lb/>
& &longs;phæ­<lb/>
roidibus</s></p><p type="margin">
<s><margin.target id="marg86"/>6. II. duo <lb/>
decimi</s></p><p type="head">
<s>THEOREMA XXI. PROPOSITIO XXVI.<!-- KEEP S--></s></p><p type="main">
<s>CVIVSLIBET fru&longs;ti à pyramide, uel cono, <lb/>
uel coni portione ab&longs;cis&longs;i, centrum grauitatis e&longs;t <lb/>
in axe, ita ut eo primum in duas portiones diui­<lb/>
&longs;o, portio &longs;uperior, quæ minorem ba&longs;im attingit <lb/>
ad portionem reliquam eam habeat proportio­<lb/>
nem, quam duplum lateris, uel diametri maioris <lb/>
ba&longs;is, vnà cum latere, uel diametro minoris, ip&longs;i <lb/>
re&longs;pondente, habet ad duplum lateris, uel diame­<lb/>
tri minoris ba&longs;is vnà <expan abbr="cũ">cum</expan> latere, uel diametro ma­<lb/>
ioris: deinde à puncto diui&longs;ionis quarta parte &longs;u <lb/>
perioris portionis in ip&longs;a &longs;umpta: & rur&longs;us ab in­<lb/>
ferioris portionis termino, qui e&longs;t ad ba&longs;im maio<lb/>
rem, &longs;umpta quarta parte totius axis: centrum &longs;it <lb/>
in linea, quæ his finibus continetur, atque in eo li <lb/>
neæ puncto, quo &longs;ic diuiditur, ut tota linea ad par <lb/>
tem propinquiorem minori ba&longs;i, <expan abbr="eãdem">eandem</expan> propor­<lb/>
tionem habeat, quam fru&longs;tum ad <expan abbr="pyramid&etilde;">pyramidem</expan>, uel <lb/>
conum, uel coni portionem, cuius ba&longs;is &longs;it ea­<lb/>
dem, quæ ba&longs;is maior, & altitudo fru&longs;ti altitudini <lb/>
æqualis.</s></p><pb xlink:href="023/01/077.jpg" pagenum="35"/><p type="main">
<s>Sit fru&longs;tum ae a pyramide, quæ triangularem ba&longs;im ha­<lb/>
beat ab&longs;ci&longs;&longs;um: cuius maior ba&longs;is triangulum abc, minor <lb/>
def; & axis gh. </s><s>ducto autem plano per axem & per <expan abbr="lineã">lineam</expan> <lb/>
da, quod &longs;ectionem faciat dakl quadrilaterum; puncta <lb/>
Kl lineas bc, ef bifariam &longs;ecabunt. </s><s>nam cum gh &longs;it axis <lb/>
fru&longs;ti: erit h centrum grauitatis trianguli abc: & g <lb/>
<figure id="id.023.01.077.1.jpg" xlink:href="023/01/077/1.jpg"/><lb/>
<arrow.to.target n="marg87"/><lb/>
centrum trianguli def: cen­<lb/>
trum uero cuiuslibet triangu <lb/>
li e&longs;t in recta linea, quæ ab an­<lb/>
gulo ip&longs;ius ad <expan abbr="dimidiã">dimidiam</expan> ba&longs;im <lb/>
ducitur ex decimatertia primi <lb/>
libri Archimedis de <expan abbr="c&etilde;tro">centro</expan> gra <lb/>
<arrow.to.target n="marg88"/><lb/>
uitatis planorum. </s><s>quare <expan abbr="cen-trũ">cen­<lb/>
trum</expan> grauitatis trapezii bcfe <lb/>
e&longs;t in linea kl, quod &longs;it m: & à <lb/>
puncto m ad axem ducta mn <lb/>
ip&longs;i ak, uel dl æquidi&longs;tante; <lb/>
erit axis gh diui&longs;us in portio­<lb/>
nes gn, nh, quas diximus: ean <lb/>
dem enim proportionem ha­<lb/>
bet gn ad nh, <expan abbr="quã">quam</expan> lm ad mk. </s><lb/>
<s>At lm ad mK habet eam, <expan abbr="quã">quam</expan> <lb/>
duplum lateris maioris ba&longs;is <lb/>
bc una cum latere minoris ef <lb/>
ad duplum lateris ef unà cum <lb/>
latere bc, ex ultima eiu&longs;dem <lb/>
libri Archimedis. <!-- KEEP S--></s><s>Itaque à li­<lb/>
nea ng ab&longs;cindatur, quarta <lb/>
pars, quæ fit np: & ab axe hg ab&longs;cindatur itidem <lb/>
quarta pars ho: & quam proportionem habet fru&longs;tum ad <lb/>
pyramidem, cuius maior ba&longs;is e&longs;t triangulum abc, & alti­<lb/>
tudo ip&longs;i æqualis; habeat op ad pq.</s><s> Dico centrum graui­<lb/>
tatis fru&longs;ti e&longs;&longs;e in linea po, & in puncto q.</s><s> namque ip&longs;um <lb/>
e&longs;&longs;e in linea gh manife&longs;te con&longs;tat. </s><s>protractis enim fru&longs;ti pla <pb xlink:href="023/01/078.jpg"/>nis, quou&longs;que in unum punctum r conueniant; erit pyra­<lb/>
midis abcr, & pyramidis defr grauitatis centrum in li­<lb/>
nca rh. </s><s>ergo & reliquæ magnitudinis, uidelicet fru&longs;ti cen­<lb/>
trum in eadem linea nece&longs;&longs;ario comperietur. </s><s>Iungantur <lb/>
db, dc, dh, dm: & per lineas db, dc ducto altero plano <lb/>
intelligatur fru&longs;tum in duas pyramides diui&longs;um: in pyra­<lb/>
midem quidem, cuius ba&longs;is e&longs;t triangulum abc, uertex d: <lb/>
& in eam, cuius idem uertex, & ba&longs;is trapezium bcfe. </s><s>erit <lb/>
igitur pyramidis abcd axis dh, & pyramidis bcfed axis <lb/>
d m: atque erunt tres axes gh, dh, dm in eodem plano <lb/>
daKl.</s><s> ducatur præterea per o linea &longs;t ip&longs;i aK <expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan>, <lb/>
quæ lineam dh in u &longs;ecet: per p uero ducatur xy æquidi­<lb/>
<figure id="id.023.01.078.1.jpg" xlink:href="023/01/078/1.jpg"/><lb/>
&longs;tans eidem, &longs;ecansque dm in <lb/>
z: & iungatur zu, quæ &longs;ecet <lb/>
gh in <foreign lang="greek">f.</foreign> tran&longs;ibit ea per q: & <lb/>
erunt <foreign lang="greek">f</foreign>q unum, atque idem <lb/>
punctum; ut inferius appare­<lb/>
bit. </s>
<s>Quoniam igitur linea uo <lb/>
<arrow.to.target n="marg89"/><lb/>
æquidi&longs;tat ip&longs;i dg, erit du ad <lb/>
uh, ut go ad oh. </s><s>Sed go tri­<lb/>
pla e&longs;t oh. </s><s>quare & du ip&longs;ius <lb/>
uh e&longs;t tripla: & ideo pyrami­<lb/>
dis abcd centrum grauitatis <lb/>
erit punctum u. </s><s>Rur&longs;us quo­<lb/>
niam zy ip&longs;i dl æquidi&longs;tat, dz <lb/>
ad zm e&longs;t, ut ly ad ym: e&longs;tque <lb/>
ly ad ym, ut gp ad pn. </s><s>ergo <lb/>
dz ad zm e&longs;t, ut gp ad pn. </s><lb/>
<s>Quòd cum gp &longs;it tripla pn; <lb/>
erit etiam dz ip&longs;ius zm tri­<lb/>
pla. </s><s>atque ob eandem cau&longs;­<lb/>
&longs;am punctuniz e&longs;t <expan abbr="centrũ">centrum</expan> gra­<lb/>
uitatis pyramidis bcfed. <!-- KEEP S--></s><s>iun <lb/>
cta igitur zu, in ea erit <expan abbr="c&etilde;trum">centrum</expan> <pb xlink:href="023/01/079.jpg" pagenum="36"/>grauitatis magnitudinis, quæ ex utri&longs;que pyramidibus <expan abbr="cõ">con</expan> <lb/>
&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s><s>Sed fru&longs;ti centrum e&longs;t etiam in a­<lb/>
xe gh. </s><s>ergo in puncto <foreign lang="greek">f,</foreign> in quo lineæ zu, gh conueniunt. </s><lb/>
<s><arrow.to.target n="marg90"/><lb/>
Itaque u<foreign lang="greek">f</foreign> ad <foreign lang="greek">f</foreign>z eam proportionem habet, quam pyramis <lb/>
bcfed ad pyramidem abcd. <!-- KEEP S--></s><s>& componendo uz ad z<foreign lang="greek">f</foreign><lb/>
eam habet, quam fru&longs;tum ad pyramidem abcd. <!-- KEEP S--></s><s>Vt uero <lb/>
uz ad z<foreign lang="greek">f</foreign>, ita op ad p<foreign lang="greek">f</foreign> ob &longs;imilitudinem triangulorum, <lb/>
uo<foreign lang="greek">f</foreign>, zp<foreign lang="greek">f.</foreign> quare op ad p<foreign lang="greek">f</foreign> e&longs;t ut fru&longs;tum ad pyramidem <lb/>
abcd. <!-- KEEP S--></s><s>&longs;ed ita erat op ad pq.</s><s> æquales igitur &longs;unt p<foreign lang="greek">f</foreign>, pq: &<lb/>
<arrow.to.target n="marg91"/><lb/>
q<foreign lang="greek">f</foreign> unum atque idem punctum. </s><s>ex quibus &longs;equitur lineam. </s><lb/>
<s>z u &longs;ecare op in q: & propterea <expan abbr="pũctum">punctum</expan> q ip&longs;ius fru&longs;ti gra­<lb/>
uitatis centrum e&longs;&longs;e.</s></p><p type="margin">
<s><margin.target id="marg87"/>3. diffi. </s><s>hu <lb/>
ius.</s></p><p type="margin">
<s><margin.target id="marg88"/>Vltima <expan abbr="e-iu&longs;d&etilde;">e­<lb/>
iu&longs;dem</expan> libri <lb/>
Archime­<lb/>
dis.<!-- KEEP S--></s></p><p type="margin">
<s><margin.target id="marg89"/>2. &longs;exti.</s></p><p type="margin">
<s><margin.target id="marg90"/>8. primi <lb/>
libri Ar­<lb/>
chimedis <lb/>
de <expan abbr="c&etilde;tro">centro</expan> <lb/>
grauta­<lb/>
tis plano <lb/>
rum</s></p><p type="margin">
<s><margin.target id="marg91"/>7. quinti.</s></p><p type="main">
<s>Sit fru&longs;tum ag à pyramide, quæ quadrangularem ba&longs;im <lb/>
habeat ab&longs;ci&longs;&longs;um, cuius maior ba&longs;is abcd, minor efgh, <lb/>
& axis kl. <!-- REMOVE S-->diuidatur autem <expan abbr="primũ">primum</expan> kl, ita ut quam propor­<lb/>
tionem habet duplum lateris ab unà cum latere ef ad du <lb/>
plum lateris ef unà cum ab; habeat km ad ml. <!-- KEEP S--></s>
<s>deinde à <lb/>
<expan abbr="pũcto">puncto</expan> m ad k &longs;umatur quarta pars ip&longs;ius mk quæ &longs;it mn. </s><lb/>
<s>& rur&longs;us ab l &longs;umatur quarta pars totius axis lk, quæ &longs;it <lb/>
lo. </s><s>po&longs;tremo fiat on ad np, ut fru&longs;tum ag ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
cuius ba&longs;is &longs;it eadem, quæ fru&longs;ti, & altitudo æqualis. </s><s>Dico <lb/>
punctum p fru&longs;ti ag grauitatis centrum e&longs;&longs;e. </s><s>ducantur <lb/>
enim ac, eg: & intelligantur duo fru&longs;ta triangulares ba­<lb/>
&longs;es habentia, quorum alterum lf ex ba&longs;ibus abc, efg <expan abbr="cõ-&longs;tet">con­<lb/>
&longs;tet</expan>; alterum lh ex ba&longs;ibus acd, egh. </s><s><expan abbr="Sitq;">Sitque</expan> fru&longs;ti lf axis <lb/>
qr; in quo grauitatis centrum s: fru&longs;ti uero lh axis tu, & <lb/>
x grauitatis centrum: deinde iungantur ur, tq, xs. </s><s>tran&longs;i­<lb/>
bit ur per l: quoniam l e&longs;t centrum grauitatis quadran­<lb/>
guli abcd: & puncta ru grauitatis centra triangulorum <lb/>
abc, acd; in quæ quadrangulum ip&longs;um diuiditur. </s><s>eadem <lb/>
quoque ratione tq per punctum k tran&longs;ibit. </s><s>At uero pro <lb/>
portiones, ex quibus fru&longs;torum grauitatis centra inquiri­<lb/>
mus, eædem &longs;unt in toto fru&longs;to ag, & in fru&longs;tis lf, lh. </s><s>Sunt <lb/>
enim per octauam huius quadrilatera abcd, efgh &longs;imilia: <pb xlink:href="023/01/080.jpg"/><expan abbr="itemq;">itemque</expan> &longs;imilia triangula abc, efg: & acd, egh. </s><s><expan abbr="idcir-coq;">idcir­<lb/>
coque</expan> latera &longs;ibi ip&longs;is re&longs;pondentia eandem inter &longs;e&longs;e pro­<lb/>
portionem &longs;eruant. </s><s>Vt igitur duplum lateris ab unà <lb/>
cum latere ef ad duplum lateris ef unà cum ab, ita e&longs;t <lb/>
<figure id="id.023.01.080.1.jpg" xlink:href="023/01/080/1.jpg"/><lb/>
duplum ad late­<lb/>
ris una cum late­<lb/>
re eh ad duplum <lb/>
eh unà cum ad: <lb/>
& ita in aliis. </s><lb/>
<s>Rur&longs;us fru&longs;tum <lb/>
ag ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
cuius eadem e&longs;t <lb/>
ba&longs;is, & æqualis <lb/>
altitudo eandem <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> ha <lb/>
bet, quam <expan abbr="fru&longs;tũ">fru&longs;tum</expan> <lb/>
lf ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
quæ e&longs;t <expan abbr="ead&etilde;">eadem</expan> ba­<lb/>
&longs;i, & æquali alti­<lb/>
tudine: & &longs;imili­<lb/>
ter quam lh fru­<lb/>
&longs;tum ad pyrami­<lb/>
dem, quæ ex <expan abbr="ea-d&etilde;">ea­<lb/>
dem</expan> ba&longs;i, & æquali <lb/>
altitudine con­<lb/>
&longs;tat. </s><s>nam &longs;i inter <lb/>
ip&longs;as ba&longs;es me­<lb/>
diæ proportio­<lb/>
nales con&longs;tituan <lb/>
tur, tres ba&longs;es &longs;imul &longs;umptæ ad maiorem ba&longs;im in om­<lb/>
nibus codem modo &longs;e habebunt. </s><s>Vnde fit, ut axes Kl, <lb/>
qr, tu à punctis psx in eandem proportionem &longs;ecen­<lb/>
<arrow.to.target n="marg92"/><lb/>
tur. </s><s>ergo linea xs per p tran&longs;ibit: & lineæ ru, sx, qt in­<lb/>
ter &longs;e æquidi&longs;tantes erunt. </s><s>Itaque cum fru&longs;ti ag latera pro­<pb xlink:href="023/01/081.jpg" pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y coeant, erunt <expan abbr="triã">trian</expan><lb/>
gula uyl, xyp, tyk inter &longs;e &longs;imilia: & &longs;imilia etiam triangu <lb/>
la lyr, pys, kyq quare ut in 19 huius, demon&longs;trabitur <lb/>
xp, ad ps: <expan abbr="itemq;">itemque</expan> tk ad kq eandem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam ul ad lr. </s><s>Sed ut ul ad lr, ita e&longs;t triangulum abc ad <lb/>
triangulum acd: & ut tk ad Kq, ita triangulum efg ad <lb/>
triangulum egh. </s><s>Vt autem triangulum abc ad triangu­<lb/>
lum acd, ita pyramis abcy ad pyramidem acdy. </s><s>& ut <lb/>
triangulum efg ad triangulum egh, ita pyramis efgy <lb/>
ad pyramidem eghy; ergo ut pyramis abcy ad <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>
<arrow.to.target n="marg93"/><lb/>
a cdy, ita pyramis efgy ad pyramidem eghy. </s><s>reliquum <lb/>
igitur <expan abbr="fru&longs;tũ">fru&longs;tum</expan> lf ad reliquum <expan abbr="fru&longs;tũ">fru&longs;tum</expan> lh e&longs;t ut pyramis abcy <lb/>
ad pyramidem acdy, hoc e&longs;t ut ul ad r, & ut xp ad ps. </s><lb/>
<s>Quòd cum fru&longs;ti lf centrum grauitatis &longs;its: & fru&longs;ti lh &longs;it <lb/>
<arrow.to.target n="marg94"/><lb/>
centrum x: con&longs;tat punctum p totius fru&longs;ti ag grauitatis <lb/>
e&longs;&longs;e centrum. </s><s>Eodem modo fiet demon&longs;tratio etiam in <lb/>
aliis pyramidibus.</s></p><p type="margin">
<s><margin.target id="marg92"/>a. </s><s>&longs;exti.</s></p><p type="margin">
<s><margin.target id="marg93"/>19. quinti</s></p><p type="margin">
<s><margin.target id="marg94"/>8. Archi­<lb/>
medis.<!-- KEEP S--></s></p><p type="main">
<s>Sit fru&longs;tum ad à cono, uel coni portione ab&longs;ci&longs;&longs;um, eu­<lb/>
ius maior ba&longs;is circulus, uel ellip&longs;is circa diametrum ab; <lb/>
minor circa diametrum cd: & axis ef. </s><s>diuidatur <expan abbr="aut&etilde;">autem</expan> ef <lb/>
in g, ita ut eg ad gf eandem proportionem habeat, quam <lb/>
duplum diametri ab unà cum diametro ed ad duplum cd <lb/>
unà cum ab. </s><s><expan abbr="Sitq;">Sitque</expan> gh quarta pars lineæ ge: & &longs;it &longs; K item <lb/>
quarta pars totius fe axis. </s><s>Rur&longs;us quam proportionem <lb/>
habet fru&longs;tum ad ad conum, uel coni portionem, in <expan abbr="ead&etilde;">eadem</expan> <lb/>
ba&longs;i, & æquali altitudine, habeat linea Kh ad hl. <!-- KEEP S--></s><s>Dico pun­<lb/>
ctum l fru&longs;ti ad grauitatis centrum e&longs;&longs;e. </s><s>Si enini fieri po­<lb/>
te&longs;t, &longs;it m centrum: <expan abbr="producaturq;">producaturque</expan> lm extra fru&longs;tum in n: <lb/>
& ut nl ad lm, ita fiat circulus, uel ellip&longs;is circa <expan abbr="diametrũ">diametrum</expan> <lb/>
ab ad aliud &longs;pacium, in quo &longs;it o. </s><s>Itaque in circulo, uel <lb/>
ellip&longs;i circa diametrum ab rectilinea figura plane de&longs;cri­<lb/>
batur, ita ut quæ relinquuntur portiones &longs;int o &longs;pacio mi­<lb/>
nores: & intelligatur pyramis apb, ba&longs;im habens rectili­<lb/>
neam figuram in circulo, uel ellip&longs;i ab de&longs;criptam: à qua <pb xlink:href="023/01/082.jpg"/>fru&longs;tum pyramidis &longs;it ab&longs;ci&longs;&longs;um. </s><s>erit ex iis quæ proxime <lb/>
tradidimus, fru&longs;ti pyramidis ad centrum grauitatis l. <!-- KEEP S--></s><s>Quo <lb/>
niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir <lb/>
<figure id="id.023.01.082.1.jpg" xlink:href="023/01/082/1.jpg"/><lb/>
culus, uel ellip&longs;is ab ad <lb/>
portiones dictas <expan abbr="maior&etilde;">maiorem</expan> <lb/>
proportionem, quàm nl <lb/>
ad lm. </s><s>&longs;ed ut circulus, uel <lb/>
ellip&longs;is ab ad portiones, <lb/>
ita apb conus, uel coni <lb/>
portio ad &longs;olidas portio­<lb/>
nes, id quod &longs;upra demon <lb/>
&longs;tratum e&longs;t: & ut circulus <lb/>
<arrow.to.target n="marg95"/><lb/>
uel ellip&longs;is cd ad portio­<lb/>
nes, quæ ip &longs;i in&longs;unt, ita co <lb/>
nus, uel coni portio cpd <lb/>
ad &longs;olidas ip&longs;ius portio­<lb/>
nes. </s><s>Quòd cum figuræ in <lb/>
circulis, uel ellip&longs;ibus ab <lb/>
cd de&longs;criptæ &longs;imiles &longs;int, <lb/>
erit proportio circuli, uel <lb/>
ellip&longs;is ab ad &longs;uas portio <lb/>
nes, <expan abbr="ead&etilde;">eadem</expan>, quæ circuli uel <lb/>
ellip&longs;is cd ad &longs;uas. </s><s>ergo <lb/>
conus, uel coni portio ap <lb/>
b ad portiones &longs;olidas <expan abbr="eã-dem">ean­<lb/>
dem</expan> habet <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam conus, uel coni por <lb/>
tio cpd ad &longs;olidas ip&longs;ius <lb/>
<arrow.to.target n="marg96"/><lb/>
portiones. </s><s>reliquum igi­<lb/>
tur coni, uel coni portionis <expan abbr="fru&longs;tũ">fru&longs;tum</expan>, &longs;cilicet ad ad reliquas <lb/>
portiones &longs;olidas in ip&longs;o contentas eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>
habet, quam conus, uel coni portio apb ad &longs;olidas portio <lb/>
nes: hoc e&longs;t eandem, quam circulus, uel ellip&longs;is ab ad por <lb/>
tiones planas. </s><s>quare fru&longs;tum coni, uel coni portionis ad <pb xlink:href="023/01/083.jpg" pagenum="38"/>ad portiones &longs;olidas maiorem habet <expan abbr="proportion&etilde;">proportionem</expan>, quàm <lb/>
nl ad lm: & diuidendo fru&longs;tum pyramidis ad dictas por­<lb/>
tiones maiorem proportionem habet, quàm nm ad ml. <!-- KEEP S--></s><lb/>
<s>fiat igitur ut fru&longs;tum pyramidis ad portiones, ita qm ad <lb/>
m l. <!-- KEEP S--></s><s>Itaque quoniam à fru&longs;to coni, uel coni portionis ad, <lb/>
cuius grauitatis centrum e&longs;t m, aufertur fru&longs;tum pyrami­<lb/>
dis habens centrum l; erit reliquæ magnitudinis, quæ ex <lb/>
portionibus &longs;olidis con&longs;tat; grauitatis <expan abbr="c&etilde;trum">centrum</expan> in linea lm <lb/>
producta, atque in puncto q, extra figuram po&longs;ito: quod <lb/>
fieri nullo modo pote&longs;t. </s><s>relinquitur ergo, ut punctum l &longs;it <lb/>
fru&longs;ti ad grauitatis centrum. </s><s>quz omnia demon&longs;tranda <lb/>
proponebantur.</s></p><p type="margin">
<s><margin.target id="marg95"/>22. huius</s></p><p type="margin">
<s><margin.target id="marg96"/>19. quínti</s></p><p type="head">
<s>THEOREMA XXII. PROPOSITIO XXVII.<!-- KEEP S--></s></p><p type="main">
<s>OMNIVM &longs;olidorum in &longs;phæra de&longs;cripto­<lb/>
rum, quæ æqualibus, & &longs;imilibus ba&longs;ibus conti­<lb/>
nentur, centrum grauitatis e&longs;t idem, quod &longs;phæ­<lb/>
ræ centrum.</s></p><p type="main">
<s>Solida eiu&longs;modi corpora regularia appellare &longs;olent, de <lb/>
quibus agitur in tribus ultimis libris elementorum: &longs;unt <lb/>
autem numero quinque, tetrahedrum, uel pyramis, hexa­<lb/>
hedrum, uel cubus, octahedrum, dodecahedrum, & ico&longs;a­<lb/>
hedrum.</s></p><p type="main">
<s>Sit primo abcd pyramis <expan abbr="ĩ">im</expan> &longs;phæra de&longs;cripta, cuius &longs;phæ <lb/>
ræ centrum &longs;it e. </s><s>Dico e pyramidis abcd grauitatis e&longs;&longs;e <lb/>
centrum. </s><s>Si enim iuncta dc producatur ad ba&longs;im abc in <lb/>
f; ex iis, quæ demon&longs;trauit Campanus in quartodecimo li <lb/>
bro elementorum, propo&longs;itione decima quinta, & decima <lb/>
feptima, erit f centrum circuli circa triangulum abc de­<lb/>
fcripti: atque erit ef &longs;exta pars ip&longs;ius &longs;phæræ axis. </s><s>quare <lb/>
ex prima huius con&longs;tat trianguli abc grauitatis centrum <lb/>
e&longs;&longs;e punctum f: & idcirco lineam df e&longs;&longs;e pyramidis axem. <pb xlink:href="023/01/084.jpg"/><figure id="id.023.01.084.1.jpg" xlink:href="023/01/084/1.jpg"/><lb/>
At cum ef &longs;it &longs;exta pars axis <lb/>
&longs;phæræ, erit d tripla ef. </s><s>ergo <lb/>
punctum e e&longs;t grauitatis cen­<lb/>
trum ip&longs;ius pyramidis: quod <lb/>
in uige&longs;ima &longs;ecunda huius de­<lb/>
mon&longs;tratum &longs;uit. </s><s>Sed e e&longs;t cen <lb/>
trum &longs;phæræ. </s><s>Sequitur igitur, <lb/>
ut centrum grauitatis pyrami­<lb/>
dis in &longs;phæra de&longs;criptæ idem <lb/>
&longs;it, quod ip&longs;ius &longs;phæræ cen­<lb/>
trum.</s></p><p type="main">
<s>Sit cubus in &longs;phæra de&longs;criptus ab, & oppo&longs;itorum pla­<lb/>
norum lateribus bifariam diui&longs;is, per puncta diui&longs;ionum <lb/>
plana ducantur, ut communis ip&longs;orum &longs;ectio &longs;it rectali­<lb/>
nea cd. <!-- KEEP S--></s><s>Itaque &longs;i ducatur ab, &longs;olidi &longs;cilicet diameter, lineæ <lb/>
ab, cd ex trige&longs;imanona undecimi&longs;e&longs;e bifariam &longs;ecabunt. </s><lb/>
<s><figure id="id.023.01.084.2.jpg" xlink:href="023/01/084/2.jpg"/><lb/>
&longs;ecent autem in puncto e. </s><s>erit, <lb/>
e <expan abbr="centrũ">centrum</expan> grauitatis &longs;olidi ab, <lb/>
id quod demon&longs;tratum e&longs;t in <lb/>
octaua huius. </s><s>Sed quoniam ab <lb/>
e&longs;t &longs;phæræ diametro æqualis, <lb/>
ut in decima quinta propo&longs;i­<lb/>
tione tertii decimilibri <expan abbr="elem&etilde;">elemen</expan> <lb/>
torum o&longs;tenditur: punctum e <lb/>
&longs;phæræ quoque centrum erit. </s><lb/>
<s>Cubi igitur in &longs;phæra de&longs;cri­<lb/>
pti grauitatis centrum idem <lb/>
e&longs;t, quod centrum ip&longs;ius &longs;phæræ.</s></p><p type="main">
<s>Sit octahedrum abcdef, in &longs;phæra de&longs;criptum, cuius <lb/>
&longs;phæræ centrum &longs;it g. </s><s>Dico punctum g ip&longs;ius octahedri <lb/>
grauitatis centrum e&longs;&longs;e. </s><s>Con&longs;tat enim ex iis, quæ demon­<lb/>
&longs;trata &longs;unt à Campano in quinto decimo libro elemento­<lb/>
rum, propo&longs;itione &longs;extadecima eiu&longs;modi &longs;olidum diuidi <lb/>
in duas pyramides æquales, & &longs;imiles; uidelicet in pyrami­<pb xlink:href="023/01/085.jpg" pagenum="39"/>dem, cuius ba&longs;is e&longs;t quadratum abcd, & altitudo eg: & <lb/>
in pyramidem, cuius <expan abbr="ead&etilde;">eadem</expan> ba&longs;is, <expan abbr="altitudoq;">altitudoque</expan> fg; ut &longs;int eg, <lb/>
gf &longs;emidiametri &longs;phæræ, & linea una. </s><s><expan abbr="Cũ">Cum</expan> igitur g &longs;it &longs;phæ­<lb/>
ræ centrum, erit etiam centrum circuli, qui circa <expan abbr="quadratũ">quadratum</expan> <lb/>
abcd de&longs;cribitur: & propterea eiu&longs;dem quadrati grauita <lb/>
tis centrum: quod in prima propo&longs;itione huius demon­<lb/>
&longs;tratum e&longs;t. </s><s>quare pyramidis abcde axis erit eg: & pyra <lb/>
midis abcdf axis fg. <!-- KEEP S--></s><s>Itaque &longs;it h centrum grauitatis py­<lb/>
ramidis abcde, & pyramidis abcdf centrum &longs;it <emph type="italics"/>K:<emph.end type="italics"/> per­<lb/>
&longs;picuum e&longs;t ex uige&longs;ima &longs;ecunda propo&longs;itione huius, <expan abbr="lineã">lineam</expan> <lb/>
<figure id="id.023.01.085.1.jpg" xlink:href="023/01/085/1.jpg"/><lb/>
ch triplam e&longs;&longs;e hg: <expan abbr="cõ">com</expan> <lb/>
<expan abbr="ponendoq;">ponendoque</expan> eg ip&longs;ius g <lb/>
h quadruplam. </s><s>& <expan abbr="ead&etilde;">eadem</expan> <lb/>
ratione fg <expan abbr="quadruplã">quadruplam</expan> <lb/>
ip&longs;ius gk quod cum e <lb/>
g, gf &longs;int æquales, & h <lb/>
g, g <emph type="italics"/>K<emph.end type="italics"/> nece&longs;&longs;ario æqua­<lb/>
les erunt. </s><s>ergo ex quar <lb/>
ta propo&longs;itione primi <lb/>
libri Archimedis de <expan abbr="c&etilde;-tro">cen­<lb/>
tro</expan> grauitatis <expan abbr="planorũ">planorum</expan>, <lb/>
totius octahedri, quod <lb/>
ex dictis pyramidibus <lb/>
con&longs;tat, centrum graui <lb/>
tatis erit punctum g idem, quod ip&longs;ius &longs;phæræ centrum.</s></p><p type="main">
<s>Sit ico&longs;ahedrum ad de&longs;criptum in &longs;phæra, cuius <expan abbr="centrũ">centrum</expan> <lb/>
&longs;it g. <!-- KEEP S--></s><s>Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. </s><s>Si <lb/>
enim ab angulo a per g ducatur recta linea u&longs;que ad &longs;phæ <lb/>
ræ &longs;uperficiem; con&longs;tat ex &longs;exta decima propo&longs;itione libri <lb/>
tertii decimi elementorum, cadere eam in angulum ip&longs;i a <lb/>
oppo&longs;itum. </s><s>cadat in d: <expan abbr="&longs;itq;">&longs;itque</expan> una aliqua ba&longs;is ico&longs;ahedri tri­<lb/>
angulum abc: & iunctæ bg, producantur, & cadant in <lb/>
angulos ef, ip&longs;is bc oppo&longs;itos. </s><s>Itaque per triangula <lb/>
abc, def ducantur plana &longs;phæram &longs;ecantia.</s><s> erunt hæ &longs;e-<pb xlink:href="023/01/086.jpg"/>ctiones circuli ex prima propo&longs;itione &longs;phæricorum Theo <lb/>
do&longs;ii: unus quidem circa triangulum abc de&longs;criptus: al­<lb/>
ter uero circa def: & quoniam triangula abc, def æqua­<lb/>
lia &longs;unt, & &longs;imilia; erunt ex prima, & &longs;ecunda propo&longs;itione <lb/>
duodecimi libri elementorum, circuli quoque inter &longs;e &longs;e <lb/>
æquales. </s><s>po&longs;tremo a centro g ad circulum abc perpendi <lb/>
cularis ducatur gh; & alia perpendicularis ducatur ad cir <lb/>
culum def, quæ &longs;it gk; & iungantur ah, dk per&longs;picuum <lb/>
e&longs;t ex corollario primæ &longs;phæricorum Theodo&longs;ii, punctum <lb/>
h centrum e&longs;&longs;e circuli abc, & k centrum circuli def. </s><s>Quo <lb/>
niam igitur triangulorum gah, gdK latus ag e&longs;t æquale la <lb/>
teri gd; &longs;unt enim à centro &longs;phæræ ad &longs;uperficiem: atque <lb/>
e&longs;t ah æquale dk: & ex &longs;exta propo&longs;itione libri primi &longs;phæ <lb/>
ricorum Theodo&longs;ii gh ip&longs;i gK: triangulum gah æquale <lb/>
erit, & &longs;imile gdk triangulo: & angulus agh æqualis an­<lb/>
<arrow.to.target n="marg97"/><lb/>
gulo dg <emph type="italics"/>K.<emph.end type="italics"/> &longs;ed anguli agh, hgd &longs;unt æquales duobus re­<lb/>
ctis. </s><s>ergo & ip&longs;i hgd, dgk duobus rectis æquales erunt. </s><lb/>
<s><arrow.to.target n="marg98"/><lb/>
& idcirco hg, g <emph type="italics"/>K<emph.end type="italics"/> una, atque eadem erit linea. </s><s>cum autem <lb/>
<figure id="id.023.01.086.1.jpg" xlink:href="023/01/086/1.jpg"/><lb/>
h &longs;it <expan abbr="centrũ">centrum</expan> circuli, & tri­<lb/>
anguli abc grauitatis cen <lb/>
<expan abbr="trũ">trum</expan> probabitur ex iis, quæ <lb/>
in prima propo&longs;itione hu <lb/>
ius tradita &longs;unt. </s><s>quare gh <lb/>
erit pyramidis abcg axis. </s><lb/>
<s>& ob eandem cau&longs;&longs;am gk <lb/>
axis pyramidis defg. <!-- KEEP S--></s><s>lta­<lb/>
que centrum grauitatls py <lb/>
ramidis abcg &longs;it <expan abbr="pũctum">punctum</expan> <lb/>
l, & pyramidis defg &longs;it m. </s><lb/>
<s>Similiter ut &longs;upra demon­<lb/>
&longs;trabimus mg, gl inter &longs;e æquales e&longs;&longs;e, & punctum g graui <lb/>
tatis centrum magnitudinis, quæ ex utri&longs;que pyramidibus <lb/>
con&longs;tat. </s><s>eodem modo demon&longs;trabitur, quarumcunque <lb/>
duarum pyramidum, quæ opponuntur, grauitatis <expan abbr="centrũ">centrum</expan> <pb xlink:href="023/01/087.jpg" pagenum="40"/>e&longs;&longs;e punctum g. <!-- KEEP S--></s><s>Sequitur ergo ut ico&longs;ahedri centrum gra<lb/>
uitatis &longs;it idem, quod ip&longs;ius &longs;phæræ centrum.</s></p><p type="margin">
<s><margin.target id="marg97"/>13. primi</s></p><p type="margin">
<s><margin.target id="marg98"/>14. primi</s></p><p type="main">
<s>Sit dodecahedrum af in &longs;phæra de&longs;ignatum, &longs;itque &longs;phæ <lb/>
ræ centrum m. </s><s>Dico m centrum e&longs;&longs;e grauitatis ip&longs;ius do­<lb/>
decahedri. </s><s>Sit enim pentagonum abcde una ex duode­<lb/>
cim ba&longs;ibus &longs;olidi af: & iuncta am producatur ad &longs;phæræ <lb/>
&longs;uperficiem. </s><s>cadet in angulum ip&longs;i a oppo&longs;itum; quod col­<lb/>
ligitur ex decima &longs;eptima propo&longs;itione tertiidecimi libri <lb/>
elementorum. </s><s>cadat in f. </s><s>at &longs;i ab aliis angulis bcde per <expan abbr="c&etilde;">cen</expan> <lb/>
trum itidem lineæ ducantur ad &longs;uperficiem &longs;phæræ in pun <lb/>
cta ghkl; cadent hæ in alios angulos ba&longs;is, quæ ip&longs;i abcd <lb/>
ba&longs;i opponitur. </s><s>tran&longs;eant ergo per pentagona abcde, <lb/>
fghKl plana &longs;phæram &longs;ecantia, quæ facient &longs;ectiones cir­<lb/>
culos æquales inter &longs;e &longs;e: po&longs;tea ducantur ex centro &longs;phæræ <lb/>
<figure id="id.023.01.087.1.jpg" xlink:href="023/01/087/1.jpg"/><lb/>
m perpendiculares ad pla­<lb/>
na dictorum <expan abbr="circulorũ">circulorum</expan>; ad <lb/>
circulum quidem abcde <lb/>
perpendicularis mn: & ad <lb/>
circulum fghKl ip&longs;a mo, <lb/>
<arrow.to.target n="marg99"/><lb/>
erunt puncta no <expan abbr="circulorũ">circulorum</expan> <lb/>
centra: & lineæ mn, mo in <lb/>
ter &longs;e æquales: quòd circu­<lb/>
<arrow.to.target n="marg100"/><lb/>
li æquales &longs;int. </s><s>Eodem mo <lb/>
do, quo &longs;upra, demon&longs;trabi <lb/>
mus lineas mn, mo in <expan abbr="unã">unam</expan> <lb/>
atque eandem lineam con­<lb/>
uenire. </s><s>ergo cum puncta no &longs;int centra circulorum, con­<lb/>
&longs;tat ex prima huius & <expan abbr="pentagonorũ">pentagonorum</expan> grauitatis e&longs;&longs;e centra: <lb/>
<expan abbr="idcircoq;">idcircoque</expan> mn, mo pyramidum abcdem, fghklm axes. </s><lb/>
<s>ponatur abcdem pyramidis grauitatis centrum p: & py <lb/>
ramidis fghklm ip&longs;um q centrum. </s><s>erunt pm, mq æqua­<lb/>
les, & punctum m grauitatis centrum magnitudinis, quæ <lb/>
ex ip&longs;is pyramidibus con&longs;tat. </s><s><expan abbr="eod&etilde;">eodem</expan> modo probabitur qua­<lb/>
rumlibet pyramidum, quæ è regione opponuntur, <expan abbr="centrũ">centrum</expan> <pb xlink:href="023/01/088.jpg"/>grauitatis e&longs;&longs;e punctum m. </s><s>patet igitur totius dodecahe­<lb/>
dri, centrum grauitatis <expan abbr="id&etilde;">idem</expan> e&longs;&longs;e, quod & &longs;phæræ ip&longs;um com <lb/>
prehendentis centrum. </s><s>quæ quidem omnia demon&longs;tra&longs;&longs;e <lb/>
oportebat.</s></p><p type="margin">
<s><margin.target id="marg99"/>corol. </s><s>pri<lb/>
mæ &longs;phæ<lb/>
ricorum <lb/>
Theod.<!-- REMOVE S--><margin.target id="marg100"/>6. primi <lb/>
phærico <lb/>
rum.</s></p><p type="head">
<s>PROBLEMA VI. PROPOSITIO XXVIII.<!-- KEEP S--></s></p><p type="main">
<s>DATA qualibet portione conoidis rectangu <lb/>
li, ab&longs;ci&longs;&longs;a plano ad axem recto, uel non recto; fie­<lb/>
ri pote&longs;t, ut portio &longs;olida in&longs;cribatur, uel circum­<lb/>
&longs;cribatur ex cylindris, uel cylindri portionibus, <lb/>
æqualem habentibus altitudinem, ita ut recta li­<lb/>
nea, quæ inter centrum grauitatis portionis, & <lb/>
figuræ in&longs;criptæ, uel circum&longs;criptæ interiicitur, <lb/>
&longs;it minor qualibet recta linea propo&longs;ita.</s></p><p type="main">
<s>Sit portio conoidis rectanguli abc, cuius axis bd, <expan abbr="gra-uitatisq;">gra­<lb/>
uitatisque</expan> centrum e: & &longs;it g recta linea propo&longs;ita. </s><s>quam ue <lb/>
ro proportionem habet linea be ad lineam g, eandem ha­<lb/>
beat portio conoidis ad &longs;olidum h: & circum&longs;cribatur por <lb/>
tioni figura, &longs;icuti dictum e&longs;t, ita ut portiones reliquæ &longs;int <lb/>
&longs;olido h minores: cuius quidem figuræ centrum grauitatis <lb/>
&longs;it punctum k. </s><s>Dico <expan abbr="lineã">lineam</expan> ke minorem e&longs;&longs;e linea g propo­<lb/>
&longs;ita. </s><s>ni&longs;i enim &longs;it minor, uel æqualis, uel maior erit. </s><s>& quo­<lb/>
niam figura circum&longs;cripta ad reliquas portiones maiorem <lb/>
<arrow.to.target n="marg101"/><lb/>
proportionem habet, quàm portio conoidis ad &longs;olidum h; <lb/>
hoc e&longs;t maiorem, quàm bc ad g: & be ad g non minorem <lb/>
habet proportionem, quàm ad ke, propterea quod ke non <lb/>
ponitur minor ip&longs;a g: habebit figura circum&longs;cripta ad por <lb/>
tiones reliquas maiorem proportionem quàm be ad ek: <lb/>
<arrow.to.target n="marg102"/><lb/>
& diuidendo portio conoidis ad reliquas portiones habe­<lb/>
bit maiorem, quàm bk ad Ke. </s><s>quare &longs;i fiat ut portio co­<pb xlink:href="023/01/089.jpg" pagenum="41"/>noidis ad portiones reliquas, ita alia linea, quæ &longs;it lk ad <lb/>
ke: erit lk maior, quam bk: & ideo punctum l extra por­<lb/>
<figure id="id.023.01.089.1.jpg" xlink:href="023/01/089/1.jpg"/><lb/>
tionem cadet. </s><s><expan abbr="Quoniã">Quoniam</expan> <lb/>
igitur à figura circum­<lb/>
&longs;cripta, cuius grauitatis <lb/>
centrum e&longs;t k, aufertur <lb/>
portio conoidis, cuius <lb/>
centrum e. </s><s><expan abbr="habetq;">habetque</expan> lK <lb/>
ad Ke eam proportio­<lb/>
nem, quam portio co­<lb/>
noidis ad reliquas por­<lb/>
tiones; erit punctum l <lb/>
extra portionem <expan abbr="cad&etilde;s">cadens</expan>, <lb/>
centrum magnitudinis <lb/>
ex reliquis portionibus compo&longs;itæ. </s><s>illud autem fieri nullo <lb/>
modo pote&longs;t. </s><s>quare con&longs;tat lineam ke ip&longs;a g linea propo&longs;i <lb/>
ta minorem e&longs;&longs;e.</s></p><p type="margin">
<s><margin.target id="marg101"/>8. quínti.</s></p><p type="margin">
<s><margin.target id="marg102"/>29. quínti <lb/>
ex tradi­<lb/>
tione <expan abbr="Cã-l">Can­<lb/>
l</expan> ani.</s></p><p type="main">
<s>Rur&longs;us in&longs;cribatur portioni figura, uidelicet cylindrus <lb/>
<figure id="id.023.01.089.2.jpg" xlink:href="023/01/089/2.jpg"/><lb/>
mn, ut &longs;it ip&longs;ius altitudo <lb/>
æqualis dimidio axis bd: <lb/>
& quam proportionem <lb/>
habet be ad g, habeat mn <lb/>
cylindrus ad &longs;olidum o. </s><lb/>
<s>in&longs;cribatur deinde eidem <lb/>
alia figura, ita ut portio­<lb/>
nes reliquæ &longs;int &longs;olido o <lb/>
minores: & centrum gra<lb/>
uitatis figuræ &longs;it p. </s><s>Dico <lb/>
lineam pe ip&longs;a g <expan abbr="minor&etilde;">minorem</expan> <lb/>
e&longs;&longs;e. </s><s>&longs;i enim non &longs;it mi­<lb/>
nor, codem, quo &longs;upra modo demon&longs;trabimus figuram in <lb/>
&longs;criptam ad reliquas portiones maiorem proportionem <lb/>
habere, quàm be ad ep. </s><s>& &longs;i fiat alia linea le ad ep, ut e&longs;t <lb/>
figura in&longs;cripta ad reliquas portiones, <expan abbr="pũctum">punctum</expan> l extra por <pb xlink:href="023/01/090.jpg"/>tionem cadet: Itaque cum à portione conoidis, cuius gra­<lb/>
uitatis centrum e auferatur in&longs;cripta figura, centrum ha­<lb/>
bens p: & &longs;it le ad ep, ut figura in&longs;cripta ad portiones reli <lb/>
quas: erit magnitudinis, quæ ex reliquis portionibus con <lb/>
&longs;tat, centrum grauitatis punctum l, extra portionem ca­<lb/>
dens. </s><s>quod fieri nequit. </s><s>ergo linea pe minor e&longs;tip&longs;a g li­<lb/>
nea propo&longs;ita.</s></p><p type="main">
<s>Ex quibus per&longs;picuum e&longs;t centrum grauitatis <lb/>
figuræ in&longs;criptæ, & circum&longs;criptæ eo magis acce <lb/>
dere ad portionis centrum, quo pluribus cylin­<lb/>
dris, uel cylindri portionibus con&longs;tet: <expan abbr="fiat&qacute;">fiatque</expan>; figu <lb/>
rain&longs;cripta maior, & circum&longs;cripta minor. </s><s>& <lb/>
quanquam continenter ad portionis <expan abbr="centrũ">centrum</expan> pro­<lb/>
pius admoueatur: nunquam tamen ad ip&longs;um per <lb/>
ueniet. </s><s>&longs;equeretur enim figuram in&longs;criptam, <expan abbr="nõ">non</expan> <lb/>
&longs;olum portioni, &longs;ed etiam circum&longs;criptæ figuræ <lb/>
æqualem e&longs;&longs;e. </s><s>quod e&longs;t ab&longs;urdum.</s></p><p type="head">
<s>THEOREMA XXIII. PROPOSITIO XXIX.<!-- KEEP S--></s></p><p type="main">
<s>CVIVSLIBET portionis conoidis rectangu­<lb/>
li axis à <expan abbr="c&etilde;tro">centro</expan> grauitatis ita diuiditur, ut pars quæ <lb/>
terminatur ad uerticem, reliquæ partis, quæ ad ba <lb/>
&longs;im &longs;it dupla.</s></p><p type="main">
<s>SIT portio conoidis rectanguli uel ab&longs;ci&longs;&longs;a plano ad <lb/>
axem recto, uel non recto: & &longs;ecta ip&longs;a altero plano per <expan abbr="ax&etilde;">axem</expan><lb/>
&longs;it &longs;uperficici &longs;ectio abc rectanguli coni &longs;ectio, uel parabo <lb/>
le; plani ab&longs;cindentis portionem &longs;ectio &longs;it recta linea ac: <lb/>
axis portionis, & &longs;ectionis diameter bd. </s><s>Sumatur autem <lb/>
in linea bd punctum e, ita ut be &longs;it ip&longs;ius ed dupla. </s><s>Dico <pb xlink:href="023/01/091.jpg" pagenum="42"/><figure id="id.023.01.091.1.jpg" xlink:href="023/01/091/1.jpg"/><lb/>
e portionis ab <lb/>
c grauitatis e&longs;&longs;e <lb/>
centrum. </s><s>Diui­<lb/>
datur enim bd <lb/>
bifariam in m: <lb/>
& rur&longs;us dm, m <lb/>
b bifariam diui­<lb/>
dantur in pun­<lb/>
ctis n, o: <expan abbr="in&longs;cri-baturq;">in&longs;cri­<lb/>
baturque</expan> portio­<lb/>
ni figura &longs;olida, <lb/>
& altera circum <lb/>
&longs;cribatur ex cy­<lb/>
lindris æqualem <lb/>
altitudinem ha­<lb/>
bentibus, ut &longs;u­<lb/>
perius <expan abbr="dictũ">dictum</expan> e&longs;t'. </s><lb/>
<s>Sit autem pri­<lb/>
mum figura in­<lb/>
&longs;cripta <expan abbr="cylĩdrus">cylindrus</expan> <lb/>
f g: & <expan abbr="circũ&longs;cri">circum&longs;cri</expan>
­<lb/> ex cylindris <lb/>
ah, Kl con&longs;tet. </s><lb/>
<s><arrow.to.target n="marg103"/><lb/>
punctum n erit <lb/>
centrum graui­<lb/>
tatis figuræ in­<lb/>
&longs;criptæ, <expan abbr="mediũ">medium</expan> <lb/>
&longs;cilicet ip&longs;ius d <lb/>
m axis: <expan abbr="atq;">atque</expan> <expan abbr="id&etilde;">idem</expan> <lb/>
erit centrum cy <lb/>
lindri ah: & cy­<lb/>
lindri kl <expan abbr="centrũ">centrum</expan> <lb/>
o, axis bm me­<lb/>
dium. </s><s>quare &longs;i li <pb xlink:href="023/01/092.jpg"/><figure id="id.023.01.092.1.jpg" xlink:href="023/01/092/1.jpg"/><lb/>
neam on ita di <lb/>
ui&longs;erimus in p, <lb/>
ut <expan abbr="quã">quam</expan> <expan abbr="propor-tion&etilde;">propor­<lb/>
tionem</expan> habet cy­<lb/>
lindrus ah ad <lb/>
cylindrum kl, <lb/>
habeat linea op <lb/>
<arrow.to.target n="marg104"/><lb/>
ad pn: centrum <lb/>
grauitatis toti­<lb/>
us figuræ <expan abbr="circũ-&longs;criptæ">circum­<lb/>
&longs;criptæ</expan> erit pun <lb/>
<arrow.to.target n="marg105"/><lb/>
ctum p. </s><s>Sed cy­<lb/>
lindri, qui &longs;unt <lb/>
æquali altitudi­<lb/>
ne, eandem in­<lb/>
ter &longs;e &longs;e, quam <lb/>
ba&longs;es propor— <lb/>
tionem habent: <lb/>
<expan abbr="e&longs;tq;">e&longs;tque</expan> ut linea db <lb/>
ad bm, ita <expan abbr="qua-dratũ">qua­<lb/>
dratum</expan> lineæ ad <lb/>
ad <expan abbr="quadratũ">quadratum</expan> ip­<lb/>
&longs;ius Km, ex uige <lb/>
&longs;ima primi libri <lb/>
<arrow.to.target n="marg106"/><lb/>
<expan abbr="conicorũ">conicorum</expan> & ita <lb/>
quadratum ac <lb/>
ad <expan abbr="quadratũ">quadratum</expan> K <lb/>
<arrow.to.target n="marg107"/><lb/>
g: hoc e&longs;t circu­<lb/>
lus circa diame<lb/>
trum ac ad cir­<lb/>
culum circa dia <lb/>
metrum kg. <!-- KEEP S--></s><s>du <lb/>
pla e&longs;t autem li­<lb/>
nea db lineæ <pb xlink:href="023/01/093.jpg" pagenum="43"/>bm. </s><s>ergo circulus ac circuli kg: & idcirco cylindrus <lb/>
ah cylindri k. </s><s>l duplus erit. </s><s>quare & linea op dupla <lb/>
ip&longs;ius pn. </s><s>Deinde in&longs;cripta & circum&longs;cripta portioni <lb/>
alia figura, ita ut in&longs;cripta con&longs;tituatur ex tribus cylin­<lb/>
dris qr, sg, tu: circum&longs;cripta uero ex quatuor ax, yz, <lb/>
K<foreign lang="greek">v, ql:</foreign> diuidantur bo, om, mn, nd bifariam in punctis <lb/>
<foreign lang="greek">mnpr.</foreign> Itaque cylindri <foreign lang="greek">ql</foreign> centrum grauitatis e&longs;t punctum <lb/>
<foreign lang="greek">m:</foreign> & cylindri k<foreign lang="greek">h</foreign> centrum <foreign lang="greek">n.</foreign> ergo &longs;i linea <foreign lang="greek">mg</foreign> diuidatur in <foreign lang="greek">s,</foreign><lb/>
ita ut <foreign lang="greek">ms</foreign> ad <foreign lang="greek">sg</foreign> <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="eã">eam</expan> habeat, quam cylindrus K<foreign lang="greek">h</foreign><lb/>
ad cylindrum <foreign lang="greek">ql,</foreign> uidelicet quam quadratum knr ad qua­<lb/>
<arrow.to.target n="marg108"/><lb/>
dratum <foreign lang="greek">q</foreign>o, hoc e&longs;t, quam linea mb ad bo: erit <foreign lang="greek">s</foreign> centrum <lb/>
magnitudinis compo&longs;itæ ex cylindris <foreign lang="greek">kg, ql.</foreign> & cum linea <lb/>
mb &longs;it dupla bo, erit & <foreign lang="greek">ms</foreign> ip&longs;ius <foreign lang="greek">sn</foreign> dupla. </s><s>præterea quo­<lb/>
niam cylindri yz centrum grauitatis e&longs;t <foreign lang="greek">p,</foreign> linea <foreign lang="greek">sp</foreign> ita diui <lb/>
&longs;ain <foreign lang="greek">t,</foreign> ut <foreign lang="greek">st</foreign> ad <foreign lang="greek">tp</foreign> eam habeat proportionem, quam cylin <lb/>
drus yz ad duos cylindros K<foreign lang="greek">n, ql:</foreign> crit <foreign lang="greek">t</foreign> centrum magnitu <lb/>
dinis, quæ ex dictis tribus cylindris con&longs;tat. </s><s>cylindrus <expan abbr="au-t&etilde;">au­<lb/>
tem</expan> yz ad cylindrum <foreign lang="greek">ql</foreign> e&longs;t, utlinea nb ad bo, hoc e&longs;t ut 3 <lb/>
ad 1: & ad cylindrum k<foreign lang="greek">h</foreign>, ut nb ad bm, uidelicet ut 3 ad 2. <!-- KEEP S--></s><lb/>
<s>quare yz <expan abbr="cylĩdrus">cylindrus</expan> duobus cylindris k<foreign lang="greek">n, ql</foreign> æqualis erit. </s><s>& <lb/>
propterea linea <foreign lang="greek">st</foreign> æqualis ip&longs;i <foreign lang="greek">tp.</foreign> denique cylindri ax <lb/>
centrum grauitatis e&longs;t punctum <foreign lang="greek">r.</foreign> & cum <foreign lang="greek">tr</foreign> diui&longs;a fuerit <lb/>
in <expan abbr="eã">eam</expan> proportionem, quam habet cylindrus ax ad tres cy­<lb/>
lindros yz, k<foreign lang="greek">n, ql:</foreign> erit in eo puncto centrum grauitatis <lb/>
totius figuræ <expan abbr="circũ&longs;criptæ">circum&longs;criptæ</expan>. </s><s>Sed cylindrus ax ad ip&longs;um yz <lb/>
e&longs;t ut linea db ad bn: hoc e&longs;t ut 4 ad 3: & duo cylindri k<foreign lang="greek">h<lb/>
ql</foreign> cylindro y &longs;unt æquales. </s><s>cylindrus igitur ax ad tres <lb/>
iam dictos cylindros e&longs;t ut 2 ad 3. Sed <expan abbr="quoniã">quoniam</expan> <foreign lang="greek">m s</foreign> e&longs;t dua­<lb/>
rum partium, & <foreign lang="greek">s g</foreign> unius, qualium <foreign lang="greek">m p</foreign> e&longs;t &longs;ex; erit <foreign lang="greek">s p</foreign> par­<lb/>
tium quatuor: <expan abbr="proptereaq;">proptereaque</expan> <foreign lang="greek">tp</foreign> duarum, & <foreign lang="greek">np,</foreign> hoc e&longs;t <foreign lang="greek">pr</foreign><lb/>
trium. </s><s>quare &longs;equitur ut punctum <foreign lang="greek">p</foreign> totius figuræ circum <lb/>
&longs;criptæ &longs;it centrum. </s><s>Itaque fiat <foreign lang="greek">nu</foreign> ad <foreign lang="greek">up,</foreign> ut <foreign lang="greek">ms</foreign> ad <foreign lang="greek">sg.</foreign> & <foreign lang="greek">ur</foreign><lb/>
bifariam diuidatur in <foreign lang="greek">f.</foreign> Similiter ut in circum&longs;cripta figu <lb/>
ra o&longs;tendetur centrum magnitudinis compo&longs;itæ ex cylin-<pb xlink:href="023/01/094.jpg"/><figure id="id.023.01.094.1.jpg" xlink:href="023/01/094/1.jpg"/><lb/>
dris sg, tu e&longs;&longs;e <lb/>
punctum <foreign lang="greek">u:</foreign> & <lb/>
totius figuræ in <lb/>
&longs;criptæ, quæ <expan abbr="cõ-&longs;tat">con­<lb/>
&longs;tat</expan> ex cylindris <lb/>
qr, &longs; g, tu e&longs;&longs;e <foreign lang="greek">f</foreign><lb/>
centrum. </s><s>Sunt <lb/>
enim hi cylindri <lb/>
æquales & &longs;imi­<lb/>
les cylindris yz, <lb/>
K<foreign lang="greek">h, ql,</foreign> figuræ <lb/>
circum&longs;criptæ. </s><lb/>
<s><expan abbr="Quoniã">Quoniam</expan> igitur <lb/>
ut be ad ed, ita <lb/>
e&longs;t op ad pn; <lb/>
<expan abbr="utraq;">utraque</expan> enim u­<lb/>
triu&longs;que e&longs;t du­<lb/>
pla: erit compo <lb/>
nendo, ut bd ad <lb/>
de, ita on ad n <lb/>
p; & permutan <lb/>
do, ut bd ad o<lb/>
n, ita de ad np. </s><lb/>
<s>Sed bd dupla <lb/>
e&longs;t on. </s><s>ergo & <lb/>
ed ip&longs;ius np du <lb/>
pla erit. </s><s>quòd &longs;i <lb/>
ed bifariam di­<lb/>
uidatur <expan abbr="ĩ">im</expan> <foreign lang="greek">x,</foreign> erit <lb/>
<foreign lang="greek">x</foreign> d, uel e <foreign lang="greek">x</foreign> æ­<lb/>
qualis np: & <lb/>
&longs;ublata en, quæ <lb/>
e&longs;t <expan abbr="cõmunis">communis</expan> u­<lb/>
trique e <foreign lang="greek">x,</foreign> pn, <pb xlink:href="023/01/095.jpg" pagenum="44"/>relinquetur pe ip&longs;i n<foreign lang="greek">x</foreign> æqualis. </s><s>cum autem be &longs;it dupla <lb/>
ed, & op dupla pn, hoc e&longs;t ip&longs;ius e <foreign lang="greek">x,</foreign> & reliquum, uideli­<lb/>
<arrow.to.target n="marg109"/><lb/>
cet bo unà cum pe ip&longs;ius reliqui <foreign lang="greek">x</foreign> d duplum erit. </s><s>e&longs;tque <lb/>
bo dupla <foreign lang="greek">r</foreign> d. <!-- KEEP S--></s><s>ergo pe, hoc e&longs;t n<foreign lang="greek">x</foreign> ip&longs;ius <foreign lang="greek">xr</foreign> dupla. </s><s>&longs;ed dn <lb/>
dupla e&longs;t n<foreign lang="greek">r.</foreign> reliqua igitur d<foreign lang="greek">x</foreign> dupla reliquæ <foreign lang="greek">x</foreign> n. </s><s>&longs;unt au­<lb/>
tem d<foreign lang="greek">x,</foreign> pn inter &longs;e æquales: <expan abbr="itemq;">itemque</expan> æquales <foreign lang="greek">x</foreign> n, pe. </s><s>qua­<lb/>
re con&longs;tat np ip&longs;ius pe duplam e&longs;&longs;e. </s><s>& idcirco pe ip&longs;i en <lb/>
æqualem. </s><s>Rur&longs;us cum &longs;it <foreign lang="greek">mn</foreign> dupla o<foreign lang="greek">n,</foreign> & <foreign lang="greek">m s</foreign> dupla <foreign lang="greek">s g;</foreign> erit <lb/>
etiam reliqua <foreign lang="greek">ns</foreign> reliquæ <foreign lang="greek">s</foreign> o dupla. </s><s>Eadem quoque ratione <lb/>
<expan abbr="cõcludetur">concludetur</expan> <foreign lang="greek">p u</foreign> dupla <foreign lang="greek">u</foreign> m. </s><s>ergo ut <foreign lang="greek">ns</foreign> ad <foreign lang="greek">s</foreign> o, ita <foreign lang="greek">pu</foreign> ad <foreign lang="greek">u</foreign> m: <lb/>
<expan abbr="componendoq;">componendoque</expan>, & permutando, ut <foreign lang="greek">n</foreign>o ad <foreign lang="greek">p</foreign>m, ita o<foreign lang="greek">s</foreign> ad <lb/>
m<foreign lang="greek">u:</foreign> & &longs;unt æquales <foreign lang="greek">n</foreign>o, <foreign lang="greek">p</foreign>m. </s><s>quare & o<foreign lang="greek">s,</foreign> m<foreign lang="greek">u</foreign> æquales. </s><s>præ <lb/>
terea <foreign lang="greek">sp</foreign> dupla e&longs;t <foreign lang="greek">pt,</foreign> & <foreign lang="greek">np</foreign> ip&longs;ius <foreign lang="greek">p</foreign>m. </s><s>reliquaigitur <foreign lang="greek">sn</foreign> re <lb/>
liquæ m<foreign lang="greek">t</foreign> dupla. </s><s>atque erat <foreign lang="greek">ns</foreign> dupla <foreign lang="greek">s</foreign>o. </s><s>ergo m<foreign lang="greek">t, s</foreign>o æ­<lb/>
quales &longs;unt: & ita æquales m<foreign lang="greek">u,</foreign> n<foreign lang="greek">f.</foreign> at o<foreign lang="greek">s,</foreign> e&longs;t æqualis <lb/>
m<foreign lang="greek">u.</foreign> Sequitur igitur, ut omnes o<foreign lang="greek">s,</foreign> m<foreign lang="greek">t,</foreign> m<foreign lang="greek">u,</foreign> n<foreign lang="greek">f</foreign> in­<lb/>
ter &longs;e &longs;int æquales. </s><s>Sed ut <foreign lang="greek">rp</foreign> ad <foreign lang="greek">pt,</foreign> hoc e&longs;t ut 3 ad 2, ita nd <lb/>
ad d<foreign lang="greek">x:</foreign> <expan abbr="permutãdoq;">permutandoque</expan> ut <foreign lang="greek">rp</foreign> ad nd, ita <foreign lang="greek">pt</foreign> ad d<foreign lang="greek">x.</foreign> & <expan abbr="&longs;ũt">&longs;unt</expan> æqua <lb/>
les <foreign lang="greek">rp,</foreign> nd. <!-- KEEP S--></s><s>ergo d<foreign lang="greek">x,</foreign> hoc e&longs;t np, & <foreign lang="greek">pt</foreign> æquales. </s><s>Sed etiam æ­<lb/>
quales n<foreign lang="greek">p, p</foreign>m. </s><s>reliqua igitur <foreign lang="greek">p</foreign> preliquæ m<foreign lang="greek">t,</foreign> hoc e&longs;t ip&longs;i <lb/>
n<foreign lang="greek">f</foreign> æqualis erit. </s><s>quare dempta p<foreign lang="greek">p</foreign> ex pe, & <foreign lang="greek">f</foreign>n dempta ex <lb/>
ne, relinquitur pe æqualis e<foreign lang="greek">f.</foreign> Itaque <foreign lang="greek">p, f</foreign> centra <expan abbr="figurarũ">figurarum</expan> <lb/>
&longs;ecundo loco de&longs;criptarum a primis centris pn æquali in­<lb/>
teruallo recedunt. </s><s>quòd &longs;i rur&longs;us aliæ figuræ de&longs;cribantur, <lb/>
codem modo demon&longs;trabimus earum centra æqualiter ab <lb/>
his recedere, & ad portionis conoidis centrum propius ad <lb/>
moueri. </s><s>Ex quibus con&longs;tat lineam <foreign lang="greek">pf</foreign> à centro grauitatis <lb/>
portionis diuidi in partes æquales. </s><s>Si enim fieri pote&longs;t, non <lb/>
&longs;it centrum in puncto e, quod e&longs;t lineæ <foreign lang="greek">pf</foreign> medium: &longs;ed in <lb/>
<foreign lang="greek">y:</foreign> & ip&longs;i <foreign lang="greek">py</foreign> æqualis fiat <foreign lang="greek">fw.</foreign> Cum igitur in portione &longs;olida <lb/>
quædam figura in&longs;cribi pos&longs;it, ita ut linea, quæ inter cen­<lb/>
trum grauitatis portionis, & in&longs;criptæ figuræ interiicitur, <lb/>
qualibet linea propo&longs;ita &longs;it minor, quod proxime demon­<lb/>
&longs;trauimus: perueniet tandem <foreign lang="greek">f</foreign> centrum in&longs;criptæ figuræ <pb xlink:href="023/01/096.jpg"/><figure id="id.023.01.096.1.jpg" xlink:href="023/01/096/1.jpg"/><pb xlink:href="023/01/097.jpg" pagenum="45"/>ad punctum <foreign lang="greek">w.</foreign> Sed quoniam <foreign lang="greek">p</foreign> circum&longs;cripta itidem alia <lb/>
figura æquali interuallo ad portionis centrum accedit, ubi <lb/>
primum <foreign lang="greek">f</foreign> applicuerit &longs;e ad <foreign lang="greek">w,</foreign> & <foreign lang="greek">p</foreign> ad <expan abbr="punctũ">punctum</expan> <foreign lang="greek">y,</foreign> hoc e&longs;t ad <lb/>
portionis centrum &longs;e applicabit. </s><s>quod fieri nullo modo <lb/>
po&longs;&longs;e per&longs;picuum e&longs;t. </s><s>non aliter idem ab&longs;urdum &longs;equetur, <lb/>
fi ponamus centrum portionis recedere à medio ad par­<lb/>
tes <foreign lang="greek">w;</foreign> e&longs;&longs;et enim aliquando centrum figuræ in&longs;criptæ idem <lb/>
quod portionis <expan abbr="centrũ">centrum</expan>. </s><s>ergo punctum e centrum erit gra<lb/>
uitatis portionis abc. quod demon&longs;trare oportebat.</s>
</p><p type="margin">
<s><margin.target id="marg103"/>7. huius</s></p><p type="margin">
<s><margin.target id="marg104"/>8. primi <lb/>
libri Ar­<lb/>
chimedis</s></p><p type="margin">
<s><margin.target id="marg105"/>11. duo­<lb/>
decimi.</s></p><p type="margin">
<s><margin.target id="marg106"/>15. quinti</s></p><p type="margin">
<s><margin.target id="marg107"/>2. duode­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg108"/>20. primi <lb/>
<expan abbr="conicorũ">conicorum</expan></s></p><p type="margin">
<s><margin.target id="marg109"/>19.<lb/>
quinti</s></p><p type="main">
<s>Quod autem &longs;upra <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t in portione conoi­<lb/>
dis recta per figuras, quæ ex cylindris æqualem altitudi­<lb/>
dinem habentibus con&longs;tant, idem &longs;imiliter demon&longs;trabi­<lb/>
mus per figuras ex cylindri portionibus con&longs;tantes in ea <lb/>
portione, quæ plano non ad axem recto ab&longs;cinditur. </s><s>ut <lb/>
enim tradidimus in commentariis in undecimam propo&longs;i <lb/>
tionem libri Archimedis de conoidibus & &longs;phæroidibus. </s><lb/>
<s>portiones cylindri, quæ æquali &longs;unt altitudine eam inter &longs;e <lb/>
&longs;e proportionem habent, quam ip&longs;arum ba&longs;es: ba&longs;es <expan abbr="aut&etilde;">autem</expan> <lb/>
<arrow.to.target n="marg110"/><lb/>
quæ &longs;unt ellip&longs;es &longs;imiles eandem proportionem habere, <lb/>
quam quadrata diametrorum eiu&longs;dem rationis, ex corol­<lb/>
lario &longs;eptimæ propo&longs;itionis libri de conoidibus, & &longs;phæ­<lb/>
roidibus, manife&longs;te apparet.</s></p><p type="margin">
<s><margin.target id="marg110"/>corol. 15<lb/>
de conoi­<lb/>
dibus & <lb/>
&longs;phæroi­<lb/>
dibus.</s></p><p type="head">
<s>THEOREMA XXIIII. PROPOSITIO XXX.<!-- KEEP S--></s></p><p type="main">
<s>Si à portione conoidis rectanguli alia portio <lb/>
ab&longs;cindatur, plano ba&longs;i æquidi&longs;tante; habebit <lb/>
portio tota ad eam, quæ ab&longs;ci&longs;&longs;a e&longs;t, duplam pro <lb/>
portion em eius, quæ e&longs;t ba&longs;is maioris portionis <lb/>
ad ba&longs;i m minoris, uel quæ axis maioris ad axem <lb/>
minoris.</s></p><pb xlink:href="023/01/098.jpg"/><p type="main">
<s>ABSCINDATVR à portione conoidis rectanguli <lb/>
abc alia portio ebf, plano ba&longs;i æquidi&longs;tante: & eadem <lb/>
portio &longs;ecetur alio plano per axem; ut &longs;uperficiei &longs;ectio &longs;it <lb/>
parabole abc: <expan abbr="planorũ">planorum</expan> portiones ab&longs;cindentium rectæ <lb/>
lincæ ac, ef: axis autem portionis, & &longs;ectionis diameter <lb/>
bd; quam linea ef in puncto g &longs;ecet. </s><s>Dico portionem co­<lb/>
noidis abc ad portionem ebf duplam proportionem ha­<lb/>
bere eius, quæ e&longs;t ba&longs;is ac ad ba&longs;im ef; uel axis db ad bg<lb/>
axem. </s><s>Intelligantur enim duo coni, &longs;eu coni portiones <lb/>
abc, ebf, <expan abbr="eãdem">eandem</expan> ba&longs;im, quam portiones conoidis, & æqua <lb/>
lem habentes altitudinem. </s><s>& quoniam abc portio conoi <lb/>
dis &longs;e&longs;quialtera e&longs;t coni, &longs;eu portionis coni abc; & portio <lb/>
ebf coni &longs;eu portionis coni bf e&longs;t &longs;e&longs;quialtera, quod de­<lb/>
<figure id="id.023.01.098.1.jpg" xlink:href="023/01/098/1.jpg"/><lb/>
mon&longs;trauit Archimedes in propo&longs;itionibus 23, & 24 libri <lb/>
de conoidibus, & &longs;phæroidibus: erit conoidis portio ad <lb/>
conoidis portionem, ut conus ad conum, uel ut coni por­<lb/>
tio ad coni portionem. </s><s>Sed conus, nel coni portio abc ad <lb/>
conum, uel coni portionem ebf compo&longs;itam proportio­<lb/>
nem habet ex proportione ba&longs;is ac ad ba&longs;im ef, & ex pro­<lb/>
portione altitudinis coni, uel coni portionis abc ad alti­<lb/>
tudinem ip&longs;ius ebf, ut nos demon&longs;trauimus in com men­<lb/>
tariis in undecimam propo&longs;itionem eiu&longs;dem libri Archi­<lb/>
medis: altitudo autem ad altitudinem c&longs;t, ut axis ad axem. </s><lb/>
<s>quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue <pb xlink:href="023/01/099.jpg" pagenum="46"/>ro ita demon&longs;trabitur. </s><s>Ducatur à puncto b ad planum ba­<lb/>
&longs;is ac perpendicularis linea bh, quæ ip&longs;am ef in K &longs;ecet. </s><lb/>
<s>erit bh altitudo coni, uel coni portionis abc: & bK altitu<lb/>
<arrow.to.target n="marg111"/><lb/>
do efg. </s><s>Quod cum lineæ ac, ef inter &longs;e æquidi&longs;tent, &longs;unt <lb/>
enim planorum æquidi&longs;tantium &longs;ectiones: habebit db ad <lb/>
<arrow.to.target n="marg112"/><lb/>
bg proportionem eandem, quam hb ad bk quare por­<lb/>
tio conoidis abc ad portionem efg proportionem habet <lb/>
compo&longs;itam ex proportione ba&longs;is ac ad ba&longs;im ef; & ex <lb/>
<arrow.to.target n="marg113"/><lb/>
proportione db axis ad axem bg. <!-- KEEP S--></s><s>Sed circulus, uel <lb/>
ellip&longs;is circa diametrum ac ad circulum, uel ellip&longs;im <lb/>
<arrow.to.target n="marg114"/><lb/>
circa ef, e&longs;t ut quadratum ac ad quadratum ef; hoc e&longs;t ut <lb/>
<expan abbr="quadratũ">quadratum</expan> ad ad <expan abbr="quadratũ">quadratum</expan> eg. <!-- REMOVE S-->& quadratum ad ad quadra <lb/>
tum eg e&longs;t, ut linea db ad lineam bg. <!-- KEEP S--></s>
<s>circulus igitur, uel el <lb/>
<arrow.to.target n="marg115"/><lb/>
lip&longs;is circa diametrum ac ad <expan abbr="circulũ">circulum</expan>, uel ellip&longs;im circa ef, <lb/>
<arrow.to.target n="marg116"/><lb/>
hoc e&longs;t ba&longs;is ad ba&longs;im eandem proportionem habet, <expan abbr="quã">quam</expan> <lb/>
db axis ad axem bg. <!-- KEEP S--></s><s>ex quibus &longs;equitur portionem abc <lb/>
ad portionem ebf habere proportionem duplam eius, <lb/>
quæ e&longs;t ba&longs;is ac ad ba&longs;im ef: uel axis db ad bg axem. </s><s>quod <lb/>
demon&longs;trandum proponebatur.</s></p><p type="margin">
<s><margin.target id="marg111"/>16. unde­<lb/>
cimi.</s></p><p type="margin">
<s><margin.target id="marg112"/>4 sexti.</s></p><p type="margin">
<s><margin.target id="marg113"/>2. duode<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg114"/>7. de co­<lb/>
noidibus <lb/>
& &longs;phæ­<lb/>
roidibus</s></p><p type="margin">
<s><margin.target id="marg115"/>15. quinti. </s><s>quinti</s></p><p type="margin">
<s><margin.target id="marg116"/>20. primi <lb/>
<expan abbr="conicorũ">conicorum</expan></s></p><p type="head">
<s>THEOREMA XXV. PROPOSITIO XXXI.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet fru&longs;ti à portione rectanguli conoi <lb/>
dis ab&longs;cis&longs;i, centrum grauitatis e&longs;t in axe, ita ut <lb/>
demptis primum à quadrato, quod fit ex diame­<lb/>
tro maioris ba&longs;is, tertia ip&longs;ius parte, & duabus <lb/>
tertiis quadrati, quod fit ex diametro ba&longs;is mino­<lb/>
ris: deinde à tertia parte quadrati maioris ba&longs;is <lb/>
rur&longs;us dempta portione, ad quam reliquum qua <lb/>
drati ba&longs;is maioris unà cum dicta portione <expan abbr="duplã">duplam</expan> <lb/>
proportionem habeat eius, quæ e&longs;t quadrati ma­<pb xlink:href="023/01/100.jpg"/>ioris ba&longs;is ad quadratum minoris: centrum &longs;it in <lb/>
eo axis puncto, quo ita diuiditur ut pars, quæ mi<lb/>
norem ba&longs;im attingit ad alteram partem eandem <lb/>
proportionem habeat, quam dempto quadrato <lb/>
minoris ba&longs;is à duabus tertiis quadrati maioris, <lb/>
habet id, quod reliquum e&longs;t unà cum portione à <lb/>
tertia quadrati maioris parte dempta, ad <expan abbr="reliquã">reliquam</expan> <lb/>
eiu&longs;dem tertiæ portionem.</s></p><p type="main">
<s>SIT fru&longs;tum à portione rectanguli conoidis ab&longs;ci&longs;&longs;um <lb/>
abcd, cuius maior ba&longs;is circulus, uel ellip&longs;is circa diame­<lb/>
trum bc, minor circa diametrum ad; & axis ef. </s><s>de&longs;criba­<lb/>
tur autem portio conoidis, à quo illud ab&longs;ci&longs;&longs;um e&longs;t, & pla­<lb/>
<figure id="id.023.01.100.1.jpg" xlink:href="023/01/100/1.jpg"/><lb/>
no per axem ducto &longs;ecetur; ut &longs;uperficiei &longs;ectio &longs;it parabo­<lb/>
le bgc, cuius diameter, & axis portionis gf: deinde gf diui <lb/>
datur in puncto h, ita ut gh &longs;it dupla hf: & rur&longs;us ge in ean <lb/>
dem proportionem diuidatur: <expan abbr="&longs;itq;">&longs;itque</expan> gk ip&longs;ius ke dupla. </s><s><expan abbr="Iã">Iam</expan> <lb/>
ex iis, quæ proxime demon&longs;trauimus, con&longs;tat centrum gra<lb/>
uitatis portionis bgc e&longs;&longs;e h punctum: & portionis agc <lb/>
punctum k. </s><s>&longs;umpto igitur infra h puncto l, ita ut kh ad hl <pb xlink:href="023/01/101.jpg" pagenum="47"/>eam proportionem habeat, quam abcd fru&longs;tum ad por­<lb/>
tionem agd; erit punctum l eius fru&longs;ti grauitatis <expan abbr="c&etilde;trum">centrum</expan>: <lb/>
<expan abbr="habebitq;">habebitque</expan> componendo Kl ad lh proportionem eandem, <lb/>
<arrow.to.target n="marg117"/><lb/>
quam portio conoidis bgc ad agd portionem. </s><s><expan abbr="Itaq;">Itaque</expan> quo <lb/>
niam quadratum bf ad quadratum ae, hoc e&longs;t quadratum <lb/>
bc ad quadratum ad e&longs;t, ut linea fg ad ge: erunt duæ ter­<lb/>
tiæ quadrati bc ad duas tertias quadrati ad, ut hg ad gk: <lb/>
& &longs;i à duabus tertiis quadrati bc demptæ fuerint duæ ter­<lb/>
tiæ quadrati ad: erit <expan abbr="diuid&etilde;do">diuidendo</expan> id, quod relinquitur ad duas <lb/>
tertias quadrati ad, ut hk ad kg. <!-- KEEP S--></s><s>Rur&longs;us duæ tertiæ quadra <lb/>
ti ad ad duas tertias quadrati bc &longs;unt, ut kg ad gh: & duæ <lb/>
tertiæ quadrati bc ad <expan abbr="tertiã">tertiam</expan> <expan abbr="part&etilde;">partem</expan> ip&longs;ius, ut gh ad hf. </s><s>ergo <lb/>
ex æquali id, quod relinquitur ex duabus tertiis quadrati <lb/>
bc, demptis ab ip&longs;is quadrati ad duabus tertiis, ad <expan abbr="tertiã">tertiam</expan> <lb/>
partem quadrati bc, ut kh ad hf: & ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>
tertiæ partis, ad quam unà cum ip&longs;a portione, duplam pro <lb/>
portionem habeat eius, quæ e&longs;t quadrati bc ad <expan abbr="quadratũ">quadratum</expan> <lb/>
ad, ut Kl ad lh. </s><s>habet enim Kl ad lh eandem proportio­<lb/>
nem, quam conoidis portio bgc ad portionem agd: por­<lb/>
tio autem bgc ad portionem agd duplam proportionem <lb/>
habet eius, quæ e&longs;t ba&longs;is bc ad ba&longs;im ad: hoc e&longs;t quadrati <lb/>
<arrow.to.target n="marg118"/><lb/>
bc ad quadratum ad; ut proxime demon&longs;tratum e&longs;t. </s><s>quare <lb/>
dempto ad quadrato à duabus tertiis quadrati bc, erit id, <lb/>
quod relinquitur unà cum dicta portione tertiæ partis ad <lb/>
reliquam eiu&longs;dem portionem, ut el ad lf. </s><s>Cum igitur cen­<lb/>
trum grauitatis fru&longs;ti abcd &longs;it l, à quo axis ef in eam, <expan abbr="quã">quam</expan> <lb/>
diximus, proportionem diuidatur; con&longs;tat <expan abbr="uerũ">uerum</expan> e&longs;&longs;e illud, <lb/>
quod demon&longs;trandum propo&longs;uimus.</s></p><p type="margin">
<s><margin.target id="marg117"/>20. 1. coni<lb/>
corum.</s></p><p type="margin">
<s><margin.target id="marg118"/>30 huius</s></p><p type="head">
<s>FINIS LIBRI DE CENTRO<!-- REMOVE S-->GRAVITATIS SOLIDORVM.<!-- KEEP S--></s></p><p type="main">
<s>Impre&longs;&longs;. <!-- REMOVE S-->Bononiæ cum licentia Superiorum, </s>
</p> </chap> </body> <back/> </text></archimedes>
| |
| | |
| | <?xml version="1.0"?>
<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >
<archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info>
<author>Commandino, Federico</author>
<title>Liber de centro gravitatis solidorum</title>
<date>1565</date>
<place>Bologna</place>
<translator/>
<lang>la</lang>
<cvs_file>comma_centr_01_la_1565</cvs_file>
<cvs_version/>
<locator>023.xml</locator>
</info><text>
<pb xlink:href="023/01/001.jpg"/><front><section><p type="head">
<s>FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</s>
<s>LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORVM.<!-- KEEP S--></s></p><figure id="id.023.01.001.1.jpg" xlink:href="023/01/001/1.jpg"/><p type="head">
<s>CVM PRIVILEGIO IN ANNOS X.<!-- KEEP S--></s></p><p type="head">
<s>BONONIAE,<!-- KEEP S--></s></p><p type="head">
<s>Ex Officina Alexandri Benacii.<!-- KEEP S--></s></p><p type="head">
<s>MDLXV.<!-- KEEP S--></s></p>
<pb xlink:href="023/01/002.jpg"/>
<pb xlink:href="023/01/003.jpg"/></section>
<section><p type="head">
<s>ALEXANDRO FARNESIO <lb/>
CARDINALI AMPLISSIMO. <lb/>
ET OPTIMO.</s></p><p type="main">
<s>Cvm multæ res in mathematicis <lb/>
di&longs;ciplinis nequaquam &longs;atis ad­<lb/>
huc explicatæ &longs;int, tum perdif­<lb/>
ficilis, & perob&longs;cura quæ&longs;tio <lb/>
e&longs;t de centro grauitatis corpo­<lb/>
rum &longs;olidorum; quæ, & ad co­<lb/>
gno&longs;cendum pulcherrima e&longs;t, <lb/>
& ad multa, quæ à mathematicis proponuntur, præ­<lb/>
clare intelligenda maximum affert adiumentum. </s>
<s>de <lb/>
qua neminem ex mathematicis, neque no&longs;tra, neque <lb/>
patrum no&longs;trorum memoria &longs;criptum reliqui&longs;&longs;e &longs;ci­<lb/>
mus. </s>
<s>& quamuis in earum monumentis literarum <expan abbr="nõ">non</expan> <lb/>
nulla reperiantur, ex quibus in hanc &longs;ententiam addu <lb/>
ci po&longs;&longs;umus, vt exi&longs;timemus hanc rem ab <expan abbr="ij&longs;d&etilde;">ij&longs;dem</expan> vber­<lb/>
rime tractatam e&longs;&longs;e; tamen ne&longs;cio quo fato adhuc <lb/>
in eiu&longs;modi librorum ignoratione ver&longs;amur. </s>
<s>Archi­<lb/>
medes quidem <expan abbr="mathematicorũ">mathematicorum</expan> princeps in libello, <lb/>
cuius in&longs;criptio e&longs;t, <foreign lang="greek">ke/ttraba/rwg ipipe/dwg,</foreign> de centro pla­<lb/>
norum copio&longs;i&longs;sime, atque acuti&longs;sime con&longs;crip&longs;it: & <lb/>
in eo explicando <expan abbr="&longs;ummã">&longs;ummam</expan> ingenii, & &longs;cientiæ <expan abbr="gloriã">gloriam</expan> e&longs;t <lb/>
<expan abbr="cõ&longs;ecutus">con&longs;ecutus</expan>. </s><s>Sed de cognitione <expan abbr="c&etilde;tri">centri</expan> grauitatis <expan abbr="corporũ">corporum</expan> <lb/>
<expan abbr="&longs;olidorũ">&longs;olidorum</expan> nulla in eius libris litera inuenitur. </s><s>non mul <lb/>
tos abhinc annos MARCELLVS II. PONT. MAX.
<pb xlink:href="023/01/004.jpg"/>cum adhuc Cardinalis e&longs;&longs;et, mihi, quæ &longs;ua erat hu­<lb/>
manitas, libros eiu&longs;dem Archimedis de ijs, quæ ve­<lb/>
huntur in aqua, latine redditos dono dedit. </s><s>hos cum <lb/>
ego, ut aliorum &longs;tudia incitarem, <expan abbr="emendãdos">emendandos</expan>, & <expan abbr="cõ-mentariis">com­<lb/>
mentariis</expan> illu&longs;trandos &longs;u&longs;cepi&longs;&longs;em, animaduerti dubi <lb/>
tari non po&longs;&longs;e, quin Archimedes vel de hac materia <lb/>
&longs;crip&longs;i&longs;&longs;et, vel aliorum mathematicorum &longs;cripta per­<lb/>
legi&longs;&longs;et. </s><s>nam in iis tum alia nonnulla, tum maxime <lb/>
illam propo&longs;itionem, ut euidentem, & aliàs proba­<lb/>
tam a&longs;&longs;umit, <expan abbr="Centrũ">Centrum</expan> grauitatis in portionibus conoi <lb/>
dis rectanguli axem ita diuidere, vt pars, quæ ad verti <lb/>
cem terminatur, alterius partis, quæ ad ba&longs;im dupla <lb/>
&longs;it. </s><s>Verum hæc ad cam partem mathematicarum <lb/>
di&longs;ciplinarum præcipue refertur, in qua de centro <lb/>
grauitatis corporum &longs;olidorum tractatur. </s><s>non e&longs;t au <lb/>
tem con&longs;entaneum Archimedem illum admirabilem <lb/>
virum hanc propo&longs;itionem &longs;ibi argumentis con­<lb/>
firmandam exi&longs;timaturum non fui&longs;&longs;e, ni&longs;i eam vel <lb/>
aliis in locis probaui&longs;&longs;et, vel ab aliis probatam e&longs;&longs;e <lb/>
comperi&longs;&longs;et. </s><s>quamobrem nequid in iis libris intel­<lb/>
ligendis de&longs;iderari po&longs;&longs;et, &longs;tatui hanc etiam partem <lb/>
vel à veteribus prætermi&longs;&longs;am, vel tractatam quidem, <lb/>
&longs;ed in tenebris iacentem, non intactam relinquere; <lb/>
atque ex a&longs;sidua mathematicorum, præ&longs;ertim Archi­<lb/>
medis lectione, quæ mihi in mentem venerunt, ea in <lb/>
medium afferre; ut centri grauitatis corporum &longs;oli­<lb/>
dorum, &longs;i non perfectam, at certe aliquam noti-
<pb xlink:href="023/01/005.jpg"/>tiam haberemus. </s><s>Quem meum laborem <expan abbr="nõ">non</expan> mathe­<lb/>
maticis &longs;olum, verum iis etiam, qui naturæ ob&longs;curi­<lb/>
tate delectantur, <expan abbr="nõ">non</expan> iniucundam fore &longs;peraui: multa <lb/>
enim <foreign lang="greek">problh/mata</foreign> cognitione digni&longs;sima, quæ ad <expan abbr="vtrã-que">vtran­<lb/>
que</expan> &longs;cientiam attinent, &longs;e&longs;e legentibus obtuli&longs;&longs;ent.</s> <lb/>
<s>neque id vlli mirandum videri debet. </s><s>vt enim in cor­<lb/>
poribus no&longs;tris omnia membra, ex quibus certa quæ <lb/>
dam officia na&longs;cuntur, diuino quodam ordine inter <lb/>
&longs;e implicata, & colligata &longs;unt: in <expan abbr="iis&qacute;">iisque</expan>; admirabilis il­<lb/>
la con&longs;piratio, quam <foreign lang="greek">tn/mpnoixn</foreign> græci vocant, eluce&longs;cit, <lb/>
ita tres illæ Philo&longs;ophiæ (ut Ari&longs;totelis verbo vtar) <lb/>
quæ veritatem &longs;olam propo&longs;itam habent, licet qui­<lb/>
bu&longs;dam qua&longs;i finibus &longs;uis regantur: tamen <expan abbr="earũ">earum</expan> vna­<lb/>
quæque per &longs;e ip&longs;am quodammodo imperfecta e&longs;t: <lb/>
neque altera &longs;ine alterius auxilio plene comprehen­<lb/>
di pote&longs;t. </s><s>complures præterea mathematicorum no­<lb/>
di ante hac explicatu difficillimi nullo negotio expe <lb/>
diti e&longs;&longs;ent: atque (ut vno verbo complectar) ni&longs;i <lb/>
mea valde amo, tractationem hanc meam &longs;tudio&longs;is <lb/>
non mediocrem vtilitatem, & magnam volupta­<lb/>
tem allaturam e&longs;&longs;e mihi per&longs;ua&longs;i. </s><s>cum autem ad hoc <lb/>
&longs;cribendum aggre&longs;&longs;us e&longs;sem, allatus e&longs;t ad me liber <lb/>
Franci&longs;ci Maurolici Me&longs;&longs;anen&longs;is, in quo vir ille do­<lb/>
cti&longs;simus, & in iis di&longs;ciplinis exercitati&longs;simus af­<lb/>
firmabat &longs;e de centro grauitatis corporum &longs;olido­<lb/>
rum con&longs;crip&longs;i&longs;&longs;e. </s><s>cum hoc intellexi&longs;&longs;em, &longs;u&longs;tinui <lb/>
me pauli&longs;per: tacitus que expectaui, dum opus cla-
<pb xlink:href="023/01/006.jpg"/>ris&longs;imi uiri, quem &longs;emper honoris cau&longs;&longs;a nomino, <lb/>
in lucem proferretur: mihi enim exploratis&longs;imum <lb/>
erat: Franci&longs;cum Maurolicum multo doctius, & <lb/>
exqui&longs;itius hoc di&longs;ciplinarum genus &longs;criptis &longs;uis tra <lb/>
diturum. </s><s>&longs;ed cum id tardius fieret, hoc e&longs;t, ut ego <lb/>
interpretor, diligentius, mihi diutius hac &longs;criptione <lb/>
non &longs;uper&longs;edendum e&longs;&longs;e duxi, præ&longs;ertim cum iam li­<lb/>
bri Archimedis de iis, quæ uehuntur in aqua, opera <lb/>
mea illu&longs;trati typis <expan abbr="excud&etilde;di">excudendi</expan> e&longs;&longs;ent. </s><s>nec me alia cau&longs; <lb/>
&longs;a impuli&longs;&longs;et, ut de centro grauitatis corporum &longs;oli­<lb/>
dorum &longs;criberem, ni&longs;i ut hac etiam ratione lux eis <lb/>
quàm maxime fieri po&longs;&longs;et afferretur. </s><s><expan abbr="atq;">atque</expan> id eò mihi <lb/>
faciendum exi&longs;timaui, quòd in &longs;pem ueniebam fore, <lb/>
ut cum ego ex omnibus mathematicis primus, hanc <lb/>
materiam explicandam &longs;u&longs;cepi&longs;&longs;em; &longs;i quid errati for <lb/>
te à me commi&longs;&longs;um e&longs;&longs;et, boni uiri potius id meæ de <lb/>
&longs;tudio&longs;is hominibus bene <expan abbr="mer&etilde;di">merendi</expan> cupiditati, quàm <lb/>
arrogantiæ a&longs;criberent. </s><s>re&longs;tabat ut con&longs;iderarem, cui <lb/>
potis&longs;imum ex principibus uiris contemplationem <lb/>
hanc, nunc primum memoriæ, ac literis proditam de <lb/>
dicarem. </s><s>harum mearum cogitationum &longs;umma fa­<lb/>
cta, exi&longs;timaui nemini conuenientius de centro graui <lb/>
tatis corporum opus dicari oportere, quàm ALE­<lb/>
XANDRO FARNESIO grauis&longs;imo, ac prudentis&longs;i­<lb/>
mo Cardinali, quo in uiro &longs;umma fortuna &longs;emper <expan abbr="cũ">cum</expan> <lb/>
&longs;umma uirtute certauit. </s><s>quid enim maxime in te ad­<lb/>
mirati debeant homines, ob&longs;curum e&longs;t; u&longs;um'ne re-
<pb xlink:href="023/01/007.jpg"/>rum, qui pueritiæ tempus extremum principium ha<lb/>
bui&longs;ti, & <expan abbr="imperiorũ">imperiorum</expan>, & ad Reges, & Imperatores ho­<lb/>
norificenti&longs;simarum legationum; an excellentiam <lb/>
in omni genere literarum, qui vix <expan abbr="adole&longs;c&etilde;tulus">adole&longs;centulus</expan>, quæ <lb/>
homines iam confirmata ætate &longs;ummo &longs;tudio, <expan abbr="diu-turnis&qacute;">diu­<lb/>
turnisque</expan>; laboribus didicerunt, &longs;cientia, & cognitione <lb/>
comprehendi&longs;ti: an con&longs;ilium, & &longs;apientiam in re­<lb/>
gendis, & <expan abbr="gubernãdis">gubernandis</expan> Ciuitatibus, cuius graui&longs;simæ <lb/>
&longs;ententiæ in &longs;ancti&longs;simo Reip. <!-- REMOVE S-->Chri&longs;tianæ con&longs;ilio di­<lb/>
ctæ, potius diuina oracula, quàm &longs;ententiæ habitæ <lb/>
&longs;unt, & habentur. </s>
<s>prætermitto liberalitatem, & mu­<lb/>
nificentiam tuam, quam in &longs;tudio&longs;i&longs;simo quoque ho <lb/>
ne&longs;tando quotidie magis o&longs;tendis, ne videar auribus <lb/>
tuis potius, quàm veritati &longs;eruire. </s><s>quamuis à te in tot <lb/>
præclaros viros tanta beneficia collata &longs;unt, & <expan abbr="confe-rũtur">confe­<lb/>
runtur</expan>, vt omnibus te&longs;tatum &longs;it, nihil tibi e&longs;&longs;e charius, <lb/>
nihil iucundius, quàm eximia tua liberalitate homi­<lb/>
nes ad amplexandam virtutem, licet currentes incita­<lb/>
re. </s><s>nihil dico de ceteris virtutibus tuis, quæ tantæ <lb/>
&longs;unt, quantæ ne cogitatione quidem comprehendi <lb/>
po&longs;&longs;unt. </s><s>Quamobrem hac præcipue de cau&longs;&longs;a te hu­<lb/>
ius meæ lucubrationis patronum e&longs;&longs;e volui, quam ea, <lb/>
qua &longs;oles, humanitate accipies. </s><s>te enim &longs;emper ob <lb/>
diuinas virtutes tuas colui, & ob&longs;eruaui: <expan abbr="nihil&qacute;">nihilque</expan>; mi­<lb/>
hi fuit optatius; quàm tibi per&longs;pectum e&longs;&longs;e meum <lb/>
erga te animum; <expan abbr="&longs;ingularem&qacute;">&longs;ingularemque</expan>; ob&longs;eruantiam. </s><s>cœ­<lb/>
lum igitur digito attingam, &longs;i po&longs;t graui&longs;simas oc-
<pb xlink:href="023/01/008.jpg"/>cupationes tuas legendo Federici tui libro aliquid <lb/>
impertiri temporis non grauaberis: <expan abbr="cum&qacute;">cumque</expan>; in iis, qui <lb/>
tibi &longs;emper addicti erunt, numerare. </s><s>Vale.<!-- KEEP S--></s></p><p type="main">
<s>Federicus Commandinus.</s></p></section><section>
<pb xlink:href="023/01/009.jpg" pagenum="1"/><p type="head">
<s>FEDERICI COMMANDINI <lb/>
VRBINATIS LIBER DE CENTRO <lb/>
GRAVITATIS SOLIDORVM.</s></p></section></front><body><chap><p type="head">
<s>DIFFINITIONES.</s></p><p type="main">
<s><arrow.to.target n="marg1"/></s></p><p type="margin">
<s><margin.target id="marg1"/>1</s></p><p type="main">
<s>Centrvm grauitatis, Pappus <lb/>
Alexandrinus in octauo ma­<lb/>
thematicarum collectionum <lb/>
libro ita diffiniuit.</s></p><p type="main">
<s><foreign lang="greek">le/gomen de/ ke/ntron ba/rous e(/ka/stou sw/<lb/>
matos e)=inai shme=ion ti kei/menon e)nto/s, a/f'<lb/>
o(/u kat' e/poi/nian a\rtnqe/n to/ ba/ros n(mere=i<lb/>
fero/menon, kai\ fula/ssei th/n e)c a)rxh=s qe/­<lb/>
sin, o)u mh\ peritrepo/menon e)n th= fora=</foreign>. hoc e&longs;t,</s></p><p type="main">
<s>Dicimus autem centrum grauitatis uniu&longs;cu­<lb/>
iu&longs;que corporis punctum quoddam intra po&longs;i­<lb/>
tum, à quo &longs;i graue appen&longs;um mente concipia­<lb/>
tur, dum fertur quie&longs;cit; & &longs;eruat eam, quam in <lb/>
principio habebat po&longs;itionem: neque in ip&longs;a la­<lb/>
tione circumuertitur.</s></p><p type="main">
<s>Po&longs;&longs;umus etiam hoc modo diffinire.</s></p><p type="main">
<s>Centrum grauitatis uniu&longs;cuiu&longs;que &longs;olidæ figu <lb/>
ræ e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>
undique partes æqualium momentorum con&longs;i­<lb/>
&longs;tunt. </s><s>&longs;i enim per tale centrum ducatur planum <lb/>
figuram quomodocunque &longs;ecans &longs;emper in par­
<pb xlink:href="023/01/010.jpg"/>tes æqueponderantes ip&longs;am diuidet.</s></p><p type="main">
<s><arrow.to.target n="marg2"/></s></p><p type="margin">
<s><margin.target id="marg2"/>2</s></p><p type="main">
<s>Pri&longs;matis, cylindri, & portionis cylindri axem <lb/>
appello rectam lineam, quæ oppo&longs;itorum plano­<lb/>
rum centra grauitatis coniungit.</s></p><p type="main">
<s><arrow.to.target n="marg3"/></s></p><p type="margin">
<s><margin.target id="marg3"/>3</s></p><p type="main">
<s>Pyramidis, coni, & portionis coni axem dico li <lb/>
neam, quæ à uertice ad centrum grauitatis ba&longs;is <lb/>
perducitur.</s></p><p type="main">
<s><arrow.to.target n="marg4"/></s></p><p type="margin">
<s><margin.target id="marg4"/>4</s></p><p type="main">
<s>Si pyramis, conus, portio coni, uel conoidis &longs;e­<lb/>
cetur plano ba&longs;i æquidi&longs;tante, pars, quæ e&longs;t ad ba­<lb/>
&longs;im, fru&longs;tum pyramidis, coni, portionis coni, uel <lb/>
conoidis dicetur; quorum plana æquidi&longs;tantia, <lb/>
quæ opponuntur &longs;imilia &longs;unt, & inæqualia: axes <lb/>
uero &longs;unt axium figurarum partes, quæ in ip&longs;is <lb/>
comprehenduntur.</s></p><p type="head">
<s>PETITIONES.<!-- KEEP S--></s></p><p type="main">
<s><arrow.to.target n="marg5"/></s></p><p type="margin">
<s><margin.target id="marg5"/>1</s></p><p type="main">
<s>Solidarum figurarum &longs;imilium centra grauita­<lb/>
tis &longs;imiliter &longs;unt po&longs;ita.</s></p><p type="main">
<s><arrow.to.target n="marg6"/></s></p><p type="margin">
<s><margin.target id="marg6"/>2</s></p><p type="main">
<s>Solidis figuris &longs;imilibus, & æqualibus inter &longs;e <lb/>
aptatis, centra quoque grauitatis ip&longs;arum inter &longs;e <lb/>
aptata erunt.</s></p><p type="head">
<s>THEOREMA I. PROPOSITIO I.<!-- KEEP S--></s></p><p type="main">
<s>Omnis figuræ rectilineæ in circulo de&longs;criptæ, <lb/>
quæ æqualibus lateribus, & angulis contine­<lb/>
<pb xlink:href="023/01/011.jpg" pagenum="2"/>tur, centrum grauitatis e&longs;t idem, quod circuli cen<lb/>
trum.</s></p><p type="main">
<s>Sit primo triangulum æquilaterum abc in circulo de­<lb/>
&longs;criptum: & diui&longs;a ac bifariam in d, ducatur bd. <!-- KEEP S--></s><s>erit in li­<lb/>
nea bd centrum grauitatis <expan abbr="triãguli">trianguli</expan> abc, ex tertia decima <lb/>
primi libri Archimedis de centro grauitatis planorum. </s><s>Et <lb/>
<figure id="id.023.01.011.1.jpg" xlink:href="023/01/011/1.jpg"/><lb/>
quoniam linea ab e&longs;t æqualis <lb/>
lineæ bc; & ad ip&longs;i dc; <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; <lb/>
bd utrique communis: trian­</s></p><p type="main">
<s><arrow.to.target n="marg7"/><lb/>
gulum abd æquale erit trian <lb/>
gulo cbd: & anguli angulis æ­<lb/>
quales, qui æqualibus lateri­<lb/>
<arrow.to.target n="marg8"/><lb/>
bus &longs;ubtenduntur. </s><s>ergo angu <lb/>
li ad d <expan abbr="utriq;">utrique</expan> recti &longs;unt. </s><s>quòd <lb/>
cum linea bd &longs;ecet ae bifa­<lb/>
<arrow.to.target n="marg9"/><lb/>
riam, & ad angulos rectos; in <lb/>
ip&longs;a bd e&longs;t centrum circuli. </s><s><lb/>
quare in eadem bd linea erit <lb/>
centrum grauitatis trianguli, & circuli centrum. </s><s>Similiter <lb/>
diui&longs;a ab bifariam in e, & ducta ce, o&longs;tendetur in ip&longs;a <expan abbr="utrũ">utrum</expan> <lb/>
que centrum contineri. </s><s>ergo ea erunt in puncto, in quo li­<lb/>
neæ bd, ce conueniunt. </s><s>trianguli igitur abc centrum gra<lb/>
uitatis e&longs;t idem, quod circuli centrum.</s></p><p type="margin">
<s><margin.target id="marg7"/>8. primi.</s></p><p type="margin">
<s><margin.target id="marg8"/>13. primi.</s></p><p type="margin">
<s><margin.target id="marg9"/>corol. pri<lb/>
mæ tertii</s></p><figure id="id.023.01.011.2.jpg" xlink:href="023/01/011/2.jpg"/><p type="main">
<s>Sit quadratum abcd in cir­<lb/>
culo de&longs;criptum: & ducantur <lb/>
ac, bd, quæ conueniant in e. </s><s>er­<lb/>
go punctum e e&longs;t centrum gra<lb/>
uitatis quadrati, ex decima eiu&longs; <lb/>
dem libri Archimedis. <!-- KEEP S--></s><s>Sed cum <lb/>
omnes anguli ad abcd recti <lb/>
<arrow.to.target n="marg10"/><lb/>
&longs;int; erit abc &longs;emicirculus: <lb/>
<expan abbr="item&qacute;">itemque</expan>; bcd: & propterea li­<lb/>
neæ ac, bd diametri circuli:
<pb xlink:href="023/01/012.jpg"/>quæ quidem in centro conueniunt. </s><s>idem igitur e&longs;t centrum <lb/>
grauitatis quadrati, & circuli centrum.</s></p><p type="margin">
<s><margin.target id="marg10"/>31. tertii.</s></p><p type="main">
<s>Sit pentagonum æquilaterum, & æquiangulum in circu­<lb/>
<figure id="id.023.01.012.1.jpg" xlink:href="023/01/012/1.jpg"/><lb/>
lo de&longs;criptum abcd e. </s><s>& iun­<lb/>
cta bd, <expan abbr="bifariam&qacute;">bifariamque</expan>; in f diui&longs;a, <lb/>
ducatur cf, & producatur ad <lb/>
circuli circumferentiam in g; <lb/>
quæ lineam ae in h &longs;ecet: de­<lb/>
inde iungantur ac, cc. <!-- KEEP S--></s><s>Eodem <lb/>
modo, quo &longs;upra demon&longs;tra­<lb/>
bimus angulum bcf æqualem <lb/>
e&longs;&longs;e. </s><s>angulo dcf; & angulos <lb/>
ad f utro&longs;que rectos: & idcir­<lb/>
co lineam cfg per circuli cen <lb/>
trum tran&longs;ire. </s><s>Quoniam igi­<lb/>
tur latera cb, ba, & cd, de æqualia &longs;unt; & æquales anguli <lb/>
<arrow.to.target n="marg11"/><lb/>
cba, cde: erit ba&longs;is ca ba&longs;i: ce, & angulus bca angulo <lb/>
dce æqualis. </s><s>ergo & reliquus ach, reliquo ech. </s><s>e&longs;t au­<lb/>
tem ch utrique triangulo ach, ech communis. </s><s>quare <lb/>
ba&longs;is ah æqualis e&longs;t ba&longs;i hc: & anguli, qui ad h recti: <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; <lb/>
<arrow.to.target n="marg12"/><lb/>
recti, qui ad f. </s><s>ergo lineæ ae, bd inter &longs;e &longs;e æquidi&longs;tant. </s><lb/>
<s>Itaque cum trapezij abde latera bd, ae æquidi&longs;tantia à li <lb/>
nea fh bifariam diuidantur; centrum grauitatis ip&longs;ius erit <lb/>
<arrow.to.target n="marg13"/><lb/>
in linea fh, ex ultima eiu&longs;dem libri Archimedis. <!-- KEEP S--></s><s>Sed trian­<lb/>
guli bcd centrum grauitatis e&longs;t in linea cf. </s><s>ergo in eadem <lb/>
linea ch e&longs;t centrum grauitatis trapezij abde, & trian­<lb/>
guli bcd: hoc e&longs;t pentagoni ip&longs;ius centrum: & centrum <lb/>
circuli. </s><s>Rur&longs;us &longs;i iuncta ad, <expan abbr="bifariam&qacute;">bifariamque</expan>; &longs;ecta in k, duca­<lb/>
tur ekl: demon&longs;trabimus in ip&longs;a utrumque centrum in <lb/>
e&longs;&longs;e. </s><s>Sequitur ergo, ut punctum, in quo lineæ cg, el con­<lb/>
ueniunt, idem &longs;it centrum circuli, & centrum grauitatis <lb/>
pentagoni.</s></p><p type="margin">
<s><margin.target id="marg11"/>4. Primi.<!-- KEEP S--></s></p><p type="margin">
<s><margin.target id="marg12"/>28. primi.</s></p><p type="margin">
<s><margin.target id="marg13"/>13. Archi­<lb/>
medis.<!-- KEEP S--></s></p><p type="main">
<s>Sit hexagonum abcdef æquilaterum, & æquiangulum <lb/>
in circulo de&longs;ignatum: <expan abbr="iungantur&qacute;">iunganturque</expan>; bd, ae: & bifariam &longs;e­
<pb xlink:href="023/01/013.jpg" pagenum="3"/>cta bd in g puncto, ducatur cg; & protrahatur ad circuli <lb/>
u&longs;que circumferentiam; quæ &longs;ecet ae in h. </s><s>Similiter conclu <lb/>
demus cg per centrum circuli tran&longs;ire: & bifariam &longs;ecate <lb/>
lineam ae; <expan abbr="item&qacute;">itemque</expan>; lineas bd, ae inter &longs;e æquidi&longs;tantes e&longs;&longs;e. <lb/>
</s><s>Cum igitur cg per centrum circuli tran&longs;eat; & ad <expan abbr="punctũ">punctum</expan> <lb/>
f perueniat nece&longs;&longs;e e&longs;t: quòd cdef &longs;it dimidium circumfe <lb/>
<figure id="id.023.01.013.1.jpg" xlink:href="023/01/013/1.jpg"/><lb/>
<arrow.to.target n="marg14"/><lb/>
rentiæ circuli. </s><s>Quare in eadem <lb/>
diametro cf erunt centra gra<lb/>
uitatis triangulorum bcd, <lb/>
afe, & quadrilateri abde, ex <lb/>
quibus con&longs;tat hexagonum ab <lb/>
cdef. </s><s>per&longs;picuum e&longs;t igitur in <lb/>
ip&longs;a cf e&longs;&longs;e circuli centrum, & <lb/>
centrum grauitatis hexagoni. <lb/>
</s><s>Rur&longs;us ducta altera diametro <lb/>
ad, ei&longs;dem rationibus o&longs;tende­<lb/>
mus in ip&longs;a utrumque <expan abbr="c&etilde;trum">centrum</expan> <lb/>
ine&longs;&longs;e. </s><s>Centrum ergo grauita­<lb/>
tis hexagoni, & centrum circuli idem erit.</s></p><p type="margin">
<s><margin.target id="marg14"/>13 Archi<lb/>
medis.</s><lb/>
<s>9. <expan abbr="eiusdetilde;">eiusdem</expan> <lb/>
m</s></p><p type="main">
<s>Sit heptagonum abcdefg æquilaterum atque æquian<lb/>
<figure id="id.023.01.013.2.jpg" xlink:href="023/01/013/2.jpg"/><lb/>
gulum in circulo de&longs;criptum: <lb/>
& iungantur ce, bf, ag: di­<lb/>
ui&longs;a autem ce bifariam in <expan abbr="pũ">pun</expan> <lb/>
cto h: & iuncta dh produca­<lb/>
tur in k. </s><s>non aliter demon­<lb/>
&longs;trabimus in linea dk e&longs;&longs;e cen <lb/>
trum circuli, & centrum gra­<lb/>
uitatis trianguli cde, & tra­<lb/>
peziorum bcef, abfg, hoc <lb/>
e&longs;t centrum totius heptago­<lb/>
ni: & rur&longs;us eadem centra in <lb/>
alia diametro cl &longs;imiliter du­<lb/>
cta contineri. </s><s>Quare & centrum grauitatis heptagoni, & <lb/>
centrum circuli in idem punctum conueniunt. </s><s>Eodem mo
<pb xlink:href="023/01/014.jpg"/>do in reliquis figuris æquilateris, & æquiangulis, quæ in cir­<lb/>
culo de&longs;cribuntur, probabimus <expan abbr="c&etilde;trum">centrum</expan> grauitatis earum, <lb/>
& centrum circuli idem e&longs;&longs;e. </s><s>quod quidem demon&longs;trare <lb/>
oportebat.</s></p><p type="main">
<s>Ex quibus apparet cuiuslibet figuræ rectilineæ <lb/>
in circulo plane de&longs;criptæ centrum grauitatis <expan abbr="id&etilde;">idem</expan> <lb/>
e&longs;&longs;e, quod & circuli centrum.<lb/>
<arrow.to.target n="marg15"/></s></p><p type="margin">
<s><margin.target id="marg15"/><foreign lang="greek">gnwri/mws</foreign></s></p><p type="main">
<s>Figuram in circulo plane de&longs;criptam appella­<lb/>
mus, cuiu&longs;modi e&longs;t ea, quæ in duodecimo elemen <lb/>
torum libro, propo&longs;itione &longs;ecunda de&longs;cribitur. <lb/>
</s><s>ex æqualibus enim lateribus, & angulis con&longs;tare <lb/>
per&longs;picuum e&longs;t.</s></p><p type="head">
<s>THEOREMA II, PROPOSITIO II.<!-- KEEP S--></s></p><p type="main">
<s>Omnis figuræ rectilineæ in ellip&longs;i plane de&longs;cri­<lb/>
ptæ centrum grauitatis e&longs;t idem, quod ellip&longs;is <lb/>
centrum.</s></p><p type="main">
<s>Quo modo figura rectilinea in ellip&longs;i plane de&longs;cribatur, <lb/>
docuimus in commentarijs in quintam propo&longs;itionem li­<lb/>
bri Archimedis de conoidibus, & &longs;phæroidibus.</s></p><p type="main">
<s>Sit ellip&longs;is abcd, cuius maior axis ac, minor bd: <expan abbr="iun-gantur&qacute;">iun­<lb/>
ganturque</expan>; ab, bc, cd, da: & bifariam diuidantur in pun­<lb/>
ctis efgh. </s><s>à centro autem, quod &longs;it k ductæ lineæ ke, kf, <lb/>
kg, kh u&longs;que ad &longs;ectionem in puncta lmno protrahan­<lb/>
tur: & iungantur lm, mn, no, ol, ita ut ac &longs;ecet li­<lb/>
neas lo, mn, in z<foreign lang="greek">f</foreign> punctis; & bd &longs;ecet lm, on in <foreign lang="greek">xy.</foreign><lb/>
erunt lk, kn linea una, <expan abbr="item&qacute;ue">itemque</expan> linea unaip&longs;æ mk, ko: <lb/>
& lineæ ba, cd æquidi&longs;tabunt lineæ mo: & bc, ad ip&longs;i <lb/>
ln. </s><s>rur&longs;us lo, mn axi bd æquidi&longs;tabunt: & lm,
<pb xlink:href="023/01/015.jpg" pagenum="4"/>on ip&longs;i ac. <!-- KEEP S--></s><s>Quoniam enim triangulorum abk, adk, latus <lb/>
bk e&longs;t æquale lateri kd, & ak utrique commune; <expan abbr="anguli&qacute;">angulique</expan>; <lb/>
<arrow.to.target n="marg16"/><lb/>
ad k recti. </s><s>ba&longs;is ab ba&longs;i ad; & reliqui anguli reliquis an­<lb/>
gulis æquales erunt. </s><s>eadem quoque ratione o&longs;tendetur bc <lb/>
<figure id="id.023.01.015.1.jpg" xlink:href="023/01/015/1.jpg"/><lb/>
æqualis cd; & ab ip&longs;i <lb/>
bc. <!-- REMOVE S-->quare omnes ab, <lb/>
bc, cd, da &longs;unt æqua­<lb/>
les. </s>
<s>& quoniam anguli <lb/>
ad a æquales &longs;unt angu <lb/>
lis ad c; erunt anguli b <lb/>
ac, acd coalterni inter <lb/>
&longs;e æquales; <expan abbr="item&qacute;">itemque</expan>; dac, <lb/>
acb. </s><s>ergo cd ip&longs;i ba; <lb/>
& ad ip&longs;i bc æquidi­<lb/>
&longs;tat. </s><s>At uero cum lineæ <lb/>
ab, cd inter &longs;e æquidi­<lb/>
&longs;tantes bifariam &longs;ecen­<lb/>
tur in punctis eg; erit li <lb/>
nea lekgn diameter &longs;e <lb/>
ctionis, & linea una, ex <lb/>
demon&longs;tratis in uige&longs;i­<lb/>
maoctaua &longs;ecundi coni <lb/>
corum. </s><s>Et eadem ratione linea una mfkho. </s><s>Sunt <expan abbr="aut&etilde;">autem</expan> ad, <lb/>
bc inter &longs;e &longs;e æquales, & æquidi&longs;tantes. </s><s>quare & earum di­<lb/>
<arrow.to.target n="marg17"/><lb/>
midiæ ah, bf; <expan abbr="item&qacute;">itemque</expan>; hd, fe; & quæ ip&longs;as coniungunt rectæ <lb/>
lineæ æquales, & æquidi&longs;tantes erunt. </s><s><expan abbr="æquidi&longs;tãt">æquidi&longs;tant</expan> igitur ba, <lb/>
cd diametro mo: & pariter ad, bc ip&longs;i ln æquidi&longs;tare o­<lb/>
&longs;tendemus. </s><s>Si igitur <expan abbr="man&etilde;te">manente</expan> diametro ac intelligatur abc <lb/>
portio ellip&longs;is ad portionem adc moueri, cum primum b <lb/>
applicuerit ad d, <expan abbr="cõgruet">congruet</expan> tota portio toti portioni, <expan abbr="linea&qacute;">lineaque</expan>; <lb/>
ba lineæ ad; & bc ip&longs;i cd congruet: punctum uero e ca­<lb/>
det in h; f in g: & linea ke in lineam kh: & kf in kg. <!-- KEEP S--></s><s>qua <lb/>
re & el in ho, et fm in gn. </s><s>At ip&longs;a lz in zo; et m<foreign lang="greek">f</foreign> in <foreign lang="greek">f</foreign>n <lb/>
cadet. </s><s>congruet igitur triangulum lkz triangulo okz: et
<pb xlink:href="023/01/016.jpg"/>triangulum mk<foreign lang="greek">f</foreign> triangulo nk<foreign lang="greek">f.</foreign> ergo anguli lzk, ozk, <lb/>
m <foreign lang="greek">f</foreign> k, n<foreign lang="greek">f</foreign>k æquales &longs;unt, ac recti. </s><s>quòd cum etram recti <lb/>
<arrow.to.target n="marg18"/><lb/>
&longs;int, qui ad k; æquidi&longs;tabunt lineæ lo, mn axi bd. <!-- KEEP S--></s><s>& ita <lb/>
demon&longs;trabuntur lm, on ip&longs;i ac æquidi&longs;tare. </s><s>Rur&longs;us &longs;i <lb/>
iungantur al, lb, bm, mc, cn, nd, do, oa: & bifariam di <lb/>
uidantur: à centro autem k ad diui&longs;iones ductæ lineæ pro­<lb/>
trahantur u&longs;que ad &longs;ectionem in puncta pqrstuxy: & po <lb/>
&longs;tremo py, qx, ru, st, qr, ps, yt, xu coniungantur. </s><s>Simili­<lb/>
<figure id="id.023.01.016.1.jpg" xlink:href="023/01/016/1.jpg"/><lb/>
ter o&longs;tendemus lineas <lb/>
py, qx, ru, st axi bd æ­<lb/>
quidi&longs;tantes e&longs;&longs;e: & qr, <lb/>
ps, yt, xu æquidi&longs;tan­<lb/>
tes ip&longs;i ac. <!-- KEEP S--></s><s>Itaque dico <lb/>
harum figurarum in el­<lb/>
lip&longs;i de&longs;criptarum cen­<lb/>
trum grauitatis e&longs;&longs;e <expan abbr="pũ-ctum">pun­<lb/>
ctum</expan> k, idem quod & el<lb/>
lip&longs;is centrum. </s><s>quadri­<lb/>
lateri enim abcd cen­<lb/>
trum e&longs;t k, ex decima e­<lb/>
iu&longs;dem libri Archime­<lb/>
dis, quippe <expan abbr="cũ">cum</expan> in eo om<lb/>
nes diametri <expan abbr="cõueniãt">conueniant</expan>. </s><lb/>
<s>Sed in figura albmcn <lb/>
<arrow.to.target n="marg19"/><lb/>
do, quoniam trianguli <lb/>
alb centrum grauitatis <lb/>
<arrow.to.target n="marg20"/><lb/>
e&longs;t in linea le: <expan abbr="trapezij&qacute;">trapezijque</expan>; abmo centrum in linea ek: trape<lb/>
zij omcd in kg: & trianguli cnd in ip&longs;a gn: erit magnitu<lb/>
dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen <lb/>
trum grauitatis in linea ln: & ob eandem cau&longs;&longs;am in linea <lb/>
om. </s><s>e&longs;t enim trianguli aod centrum in linea oh: trapezij <lb/>
alnd in hk: trapezij lbcn in kf: & trianguli bmc in fm. </s><lb/>
<s>cum ergo figuræ albmcndo centrum grauitatis &longs;it in li­<lb/>
nea ln, & in linea om; erit centrum ip&longs;ius punctum k, in
<pb xlink:href="023/01/017.jpg" pagenum="5"/>quo &longs;cilicet ln, om conueniunt. </s><s>Po&longs;tremo in figura <lb/>
aplqbrmsctnudxoy centrum grauitatis trian<lb/>
guli pay, & trapezii ploy e&longs;t in linea az: trapeziorum <lb/>
uero lqxo, qbdx centrum e&longs;t in linea zk: & <expan abbr="trapeziorũ">trapeziorum</expan> <lb/>
brud, rmnu in k<foreign lang="greek">f:</foreign> & denique trapezii mstn; & triangu<lb/>
li sct in <foreign lang="greek">f</foreign>c. </s><s>quare magnitudinis ex his compo&longs;itæ <expan abbr="centrũ">centrum</expan> <lb/>
in linea ac con&longs;i&longs;tit. </s><s>Rur&longs;us trianguli qbr, & trapezii ql<lb/>
mr centrum e&longs;t in linea b<foreign lang="greek">x.</foreign> trapeziorum lpsm, pacs, <lb/>
aytc, yont in linea <foreign lang="greek">xf:</foreign> <expan abbr="trapeziiq;">trapeziique</expan> oxun, & trianguli <lb/>
xdu centrum in <foreign lang="greek">y</foreign>d. <!-- KEEP S--></s><s>totius ergo magnitudinis centrum <lb/>
e&longs;t in linea bd. <!-- KEEP S--></s><s>ex quo &longs;equitur, centrum grauitatis figuræ <lb/>
aplqbrmsctnudxoy e&longs;&longs;e <expan abbr="punctũ">punctum</expan> K, lineis &longs;cilicet ac, <lb/>
bd commune, quæ omnia demon&longs;trare oportebat.</s></p><p type="margin">
<s><margin.target id="marg16"/>8. primi</s></p><p type="margin">
<s><margin.target id="marg17"/>33. primi</s></p><p type="margin">
<s><margin.target id="marg18"/>28. primi.</s></p><p type="margin">
<s><margin.target id="marg19"/>13. Archi<lb/>
medis.</s></p><p type="margin">
<s><margin.target id="marg20"/>Vltima.<!-- KEEP S--></s></p><p type="head">
<s>THEOREMA III. PROPOSITIO III.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet portio­<lb/>
nis circuli, & ellip&longs;is, <lb/>
quæ dimidia non &longs;it <lb/>
maior, centrum graui <lb/>
tatis in portionis dia­<lb/>
metro con&longs;i&longs;tit.</s></p><figure id="id.023.01.017.1.jpg" xlink:href="023/01/017/1.jpg"/><p type="main">
<s>HOC eodem pror&longs;us <lb/>
modo demon&longs;trabitur, <lb/>
quo in libro de centro gra<lb/>
uitatis planorum ab Ar­<lb/>
chimede <expan abbr="demon&longs;tratũ">demon&longs;tratum</expan> e&longs;t, <lb/>
in portione <expan abbr="cõtenta">contenta</expan> recta <lb/>
linea, & rectanguli coni &longs;e<lb/>
ctione grauitatis <expan abbr="c&etilde;trum">centrum</expan> <lb/>
e&longs;&longs;e in diametro portio­<lb/>
nis. </s><s>Et ita demon&longs;trari po<lb/>
<pb xlink:href="023/01/018.jpg"/>te&longs;t in portione, quæ recta linea & obtu&longs;ianguli coni &longs;e­<lb/>
ctione, &longs;eu hyperbola continetur.</s></p><p type="head">
<s>THEOREMA IIII. PROPOSITIO IIII.<!-- KEEP S--></s></p><p type="main">
<s>IN circulo & ellip&longs;i idem e&longs;t figuræ & graui­<lb/>
tatis centrum.</s></p><p type="main">
<s>SIT circulus, uel ellip&longs;is, cuius centrum a. </s><s>Dico a gra­<lb/>
uitatis quoque centrum e&longs;&longs;e. </s><s>Si enim fieri pote&longs;t, &longs;it b cen­<lb/>
trum grauitatis: & iuncta ab extra figuram in c produca<lb/>
tur: quam uero proportionem habet linea ca ad ab, ha­<lb/>
beat circulus a ad alium circulum, in quo d; uel ellip&longs;is ad <lb/>
aliam ellip&longs;im: & in circulo, uel ellip&longs;i figura rectilinea pla­<lb/>
ne de&longs;cribatur adco, ut tandem relinquantur portiones <lb/>
quædam minores circulo, uel ellip&longs;i d; quæ figura &longs;it abcefg<lb/>
hklmn. </s><s>Illud uero in circulo fieri po&longs;&longs;e ex duodecimo <lb/>
elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con­<lb/>
<figure id="id.023.01.018.1.jpg" xlink:href="023/01/018/1.jpg"/><lb/>
&longs;tat; at in ellip&longs;i nos demon&longs;tra­<lb/>
uimus in commentariis in quin­<lb/>
tam propo&longs;itionem Archimedis <lb/>
de conoidibus, & &longs;phæroidibus. </s><lb/>
<s>erit igitur a centrum grauitatis <lb/>
ip&longs;ius figuræ, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;ten</expan><lb/>
dimus. </s><s>Itaque quoniam circulus <lb/>
a ad circulum d, uel ellip&longs;is a ad <lb/>
ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>
habet, quam linea ca ad ab: <lb/>
portiones uero &longs;unt minores cir<lb/>
<arrow.to.target n="marg21"/><lb/>
culo uel ellip&longs;i d: habebit circu­<lb/>
lus, uel ellip&longs;is ad portiones ma­<lb/>
iorem proportionem, quàm ca <lb/>
<arrow.to.target n="marg22"/><lb/>
ad ab: & diuidendo figura recti­<lb/>
linea abcefghklmn ad portiones
<pb xlink:href="023/01/019.jpg" pagenum="6"/><figure id="id.023.01.019.1.jpg" xlink:href="023/01/019/1.jpg"/><lb/>
habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam cb ad ba. </s><s>fiat ob ad ba, <lb/>
ut figura rectilinea ad portio­<lb/>
nes. </s><s>cum igitur à circulo, uel el­<lb/>
lip&longs;i, cuius grauitatis centrum <lb/>
e&longs;t b, auferatur figura rectilinea <lb/>
efghklmn, cuius centrum a; <lb/>
reliquæ magnitudinis ex portio<lb/>
<arrow.to.target n="marg23"/><lb/>
nibus compo&longs;itæ centrum graui<lb/>
tatis erit in linea ab producta, <lb/>
& in puncto o, extra figuram po<lb/>
&longs;ito. </s><s>quod quidem fieri nullo mo<lb/>
do po&longs;&longs;e per&longs;picuum e&longs;t. </s><s>&longs;equi­<lb/>
tur ergo, ut circuli & ellip&longs;is cen<lb/>
trum grauitatis &longs;it punctum a, <lb/>
idem quod figuræ centrum.</s></p><p type="margin">
<s><margin.target id="marg21"/>8. quinti</s></p><p type="margin">
<s><margin.target id="marg22"/>19. quinti <lb/>
apud <expan abbr="Cã">Cam</expan> <lb/>
panum.</s></p><p type="margin">
<s><margin.target id="marg23"/>8. Archi­<lb/>
medis.</s></p><p type="head">
<s>ALITER.<!-- KEEP S--></s></p><p type="main">
<s>Sit circulus, uel ellip&longs;is abcd, <lb/>
cuius diameter db, & centrum e: <expan abbr="ducaturq;">ducaturque</expan> per e recta li<lb/>
nea ac, &longs;ecans ip&longs;am db ad rectos angulos. </s><s>erunt adc, <lb/>
abc circuli, uel ellip&longs;is dimidiæ portiones. </s><s>Itaque quo­<lb/>
<figure id="id.023.01.019.2.jpg" xlink:href="023/01/019/2.jpg"/><lb/>
niam por<lb/>
<expan abbr="tiõis">tionis</expan> adc <lb/>
<expan abbr="c&etilde;trũ">centrum</expan> gra­<lb/>
uitatis e&longs;t <lb/>
in diame­<lb/>
tro de: & <lb/>
portionis <lb/>
abc cen­<lb/>
trum e&longs;t <expan abbr="ĩ">im</expan> <lb/>
ip&longs;a eb: to <lb/>
tius circu <lb/>
li, uel ellip&longs;is grauitatis centrum erit in diametro db. </s><lb/>
<s>Sit autem portionis adc <expan abbr="c&etilde;trum">centrum</expan> grauitatis f: & &longs;umatur
<pb xlink:href="023/01/020.jpg"/>in linea eb <expan abbr="punctũ">punctum</expan> g, ita ut fit ge æqualis ef. </s><s>erit g por­<lb/>
tionis abc centrum. </s><s>nam &longs;i hæ portiones, quæ æquales <lb/>
& &longs;imiles &longs;unt, inter &longs;e &longs;e aptentur, ita ut be cadat in de, <lb/>
& punctum b in d cadet, & g in f: figuris autem æquali­<lb/>
bus, & &longs;imilibus inter &longs;e aptatis, centra quoque grauitatis <lb/>
ip&longs;arum inter &longs;e aptata erunt, ex quinta petitione Archi­<lb/>
medis in libro de centro grauitatis planorum. </s><s>Quare cum <lb/>
portionis adc centrum grauitatis &longs;it f: & portionis <lb/>
abc centrum g: magnitudinis; quæ ex utri&longs;que efficitur: <lb/>
hoc e&longs;t circuli uel ellip&longs;is grauitatis centrum in medio li­<lb/>
neæ fg, quod e&longs;t e, con&longs;i&longs;tet, ex quarta propo&longs;itione eiu&longs;­<lb/>
dem libri Archimedis. <!-- KEEP S--></s><s>ergo circuli, uel ellip&longs;is centrum <lb/>
grauitatis e&longs;t idem, quod figuræ centrum. </s><s>atque illud e&longs;t, <lb/>
quod demon&longs;trare oportebat.</s></p><p type="main">
<s>Ex quibus &longs;equitur portionis circuli, uel ellip­<lb/>
&longs;is, quæ dimidia maior &longs;it, centrum grauitatis in <lb/>
diametro quoque ip&longs;ius con&longs;i&longs;tere.</s></p><figure id="id.023.01.020.1.jpg" xlink:href="023/01/020/1.jpg"/><p type="main">
<s>Sit enim maior portio abc, cu<emph type="italics"/>i<emph.end type="italics"/>us diameter bd, & com­<lb/>
pleatur circulus, uel ellip&longs;is, ut portio reliqua fit aec, dia
<pb xlink:href="023/01/021.jpg" pagenum="7"/>metrum habens ed. <!-- KEEP S--></s><s>Quoniam igitur circuli uel ellip&longs;is <lb/>
aecb grauitatis centrum e&longs;t in diametro be, & portio­<lb/>
nis aec centrum in linea ed: reliquæ portionis, uidelicet <lb/>
abc centrum grauitatis in ip&longs;a bd con&longs;i&longs;tat nece&longs;&longs;e e&longs;t, ex <lb/>
octaua propo&longs;itione eiu&longs;dem.</s></p><p type="head">
<s>THEOREMA V. PROPOSITIO V.<!-- KEEP S--></s></p><p type="main">
<s>SI pri&longs;ma &longs;ecetur plano oppo&longs;itis planis æqui <lb/>
di&longs;tante, &longs;ectio erit figura æqualis & &longs;imilis ei, <lb/>
quæ e&longs;t oppo&longs;itorum planorum, centrum graui <lb/>
tatis in axe habens.</s></p><p type="main">
<s>Sit pri&longs;ma, in quo plana oppo&longs;ita &longs;int triangula abc, <lb/>
def; axis gh: & &longs;ecetur plano iam dictis planis <expan abbr="æquidi&longs;tã">æquidi&longs;tan</expan> <lb/>
te; quod faciat &longs;ectionem klm; & axi in <expan abbr="pũcto">puncto</expan> n occurrat. </s><lb/>
<s>Dico klm triangulum æquale e&longs;&longs;e, & fimile triangulis abc <lb/>
def; atque eius grauitatis centrum e&longs;&longs;e punctum n. </s><s>Quo­<lb/>
<figure id="id.023.01.021.1.jpg" xlink:href="023/01/021/1.jpg"/><lb/>
niam enim plana abc <lb/>
Klm æquidi&longs;tantia <expan abbr="&longs;ecã">&longs;ecan</expan><lb/>
<arrow.to.target n="marg24"/><lb/>
tur a plano ae; rectæ li­<lb/>
neæ ab, Kl, quæ &longs;unt ip <lb/>
&longs;orum <expan abbr="cõmunes">communes</expan> &longs;ectio­<lb/>
nes inter &longs;e &longs;e æquidi­<lb/>
&longs;tant. </s><s>Sed æquidi&longs;tant <lb/>
ad, be; cum ae &longs;it para<lb/>
lelogrammum, ex pri&longs;­<lb/>
matis diffinitione. </s><s>ergo <lb/>
& al <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb/>
erit; & propterea linea <lb/>
<arrow.to.target n="marg25"/><lb/>
kl, ip&longs;i ab æqualis. </s><s>Si­<lb/>
militer demon&longs;trabitur <lb/>
lm æquidi&longs;tans, & æqua <lb/>
lis bc; & mk ip&longs;i ca.</s>
<pb xlink:href="023/01/022.jpg"/>
<s>Itaque quoniam duæ lineæ Kl, lm &longs;e &longs;e tangentes, duabus <lb/>
lineis &longs;e &longs;e tangentibus ab, bc æquidi&longs;tant; nec &longs;unt in e o­<lb/>
dem plano: angulus klm æqualis e&longs;t angulo abc: & ita an<lb/>
<arrow.to.target n="marg26"/><lb/>
gulus lmk, angulo bca, & mkl ip&longs;i cab æqualis probabi<lb/>
tur. </s><s>triangulum ergo klm e&longs;t æquale, & &longs;imile triangulo <lb/>
abc. quare & triangulo def. </s>
<s>Ducatur linea cgo, & per ip<lb/>
&longs;am, & per cf ducatur planum &longs;ecans pri&longs;ma; cuius & paral<lb/>
lelogrammi ae communis &longs;ectio &longs;it opq.</s><s> tran&longs;ibit linea <lb/>
fq per h, & mp per n. </s><s>nam cum plana æquidi&longs;tantia &longs;ecen <lb/>
tur à plano cq, communes eorum &longs;ectiones cgo, mp, fq <lb/>
&longs;ibi ip&longs;is æquidi&longs;tabunt. </s><s>Sed & æquidi&longs;tant ab, kl, de. </s><s>an­<lb/>
<arrow.to.target n="marg27"/><lb/>
guli ergo aoc, kpm, dqf inter &longs;e æquales &longs;unt: & &longs;unt <lb/>
æquales qui ad puncta akd con&longs;tituuntur. </s><s>quare & reliqui <lb/>
reliquis æquales; & triangula aco, Kmp, dfq inter &longs;e &longs;imi <lb/>
<arrow.to.target n="marg28"/><lb/>
lia erunt. </s><s>Vt igitur ca ad ao, ita fd ad dq: & permutando <lb/>
ut ca ad fd, ita ao ad dq.</s><s>e&longs;t autem ca æqualis fd. <!-- KEEP S--></s><s>ergo & <lb/>
ao ip&longs;i dq.</s><s> eadem quoque ratione & ao ip&longs;i Kp æqualis <lb/>
demon&longs;trabitur. </s><s>Itaque &longs;i triangula, abc, def æqualia & <lb/>
<figure id="id.023.01.022.1.jpg" xlink:href="023/01/022/1.jpg"/><lb/>
&longs;imilia inter &longs;e <expan abbr="apt&etilde;tur">aptentur</expan>, <lb/>
cadet linea fq in lineam <lb/>
<arrow.to.target n="marg29"/><lb/>
cgo. </s><s>Sed & <expan abbr="centrũ">centrum</expan> gra<lb/>
uitatis h in g <expan abbr="centrũ">centrum</expan> ca­<lb/>
det. </s><s><expan abbr="trã&longs;ibit">tran&longs;ibit</expan> igitur linea <lb/>
fq per h: & planum per <lb/>
co & cf <expan abbr="ductũ">ductum</expan> per <expan abbr="ax&etilde;">axem</expan> <lb/>
gh ducetur: <expan abbr="idcircoq;">idcircoque</expan> li <lb/>
neam mp <expan abbr="etiã">etiam</expan> per n <expan abbr="trã">tran</expan> <lb/>
&longs;ire nece&longs;&longs;e erit. </s><s>Quo­<lb/>
niam ergo fh, cg æqua­<lb/>
les &longs;unt, & <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan>: <lb/>
<expan abbr="itemq;">itemque</expan> hq, go; rectæ li­<lb/>
neæ, quæ ip&longs;as <expan abbr="cõnectũt">connectunt</expan> <lb/>
cmf, gnh, opq æqua­<lb/>
les æquidi&longs;tantes <expan abbr="erũt">erunt</expan>.</s>
<pb xlink:href="023/01/023.jpg" pagenum="8"/><s>æquidi&longs;tant autem cgo, mnp. </s><s>ergo <expan abbr="parallelogrãma">parallelogramma</expan> &longs;unt <lb/>
on, gm, & linea mn æqualis cg; & np ip&longs;i go. </s><s>aptatis igi­<lb/>
tur klm, abc <expan abbr="triãgulis">triangulis</expan>, quæ æqualia & &longs;imilia <expan abbr="sũt">sunt</expan>; linea mp <lb/>
in co, & punctum n in g cadet. </s><s>Quòd <expan abbr="cũ">cum</expan> g &longs;it centrum gra­<lb/>
uitatis trianguli abc, & n trianguli klm grauitatis cen­<lb/>
trum erit id, quod demon&longs;trandum relinquebatur. </s><s>Simili <lb/>
ratione idem contingere demon&longs;trabimus in aliis pri&longs;ma­<lb/>
tibus, &longs;iue quadrilatera, &longs;iue plurilatera habeant plana, <lb/>
quæ opponuntur.</s></p><p type="margin">
<s><margin.target id="marg24"/>16. unde­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg25"/>34. primi</s></p><p type="margin">
<s><margin.target id="marg26"/>10. unde <lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg27"/>10. unde­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg28"/>4. &longs;exti</s></p><p type="margin">
<s><margin.target id="marg29"/>per 5. pe­<lb/>
titionem <lb/>
Archime<lb/>
dis.</s></p><p type="head">
<s>COROLLARIVM.<!-- KEEP S--></s></p><p type="main">
<s>Ex iam demon&longs;tratis per&longs;picue apparet, cuius <lb/>
libet pri&longs;matis axem, parallelogrammorum lateri<lb/>
bus, quæ ab oppo&longs;itis planis <expan abbr="ducũtur">ducuntur</expan> æquidi&longs;tare.</s></p><p type="head">
<s>THEOREMA VI. PROPOSITIO VI.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet pri&longs;matis centrum grauitatis e&longs;t in <lb/>
plano, quod oppo&longs;itis planis æquidi&longs;tans, reli­<lb/>
quorum planorum latera bifariam diuidit.</s></p><p type="main">
<s>Sit pri&longs;ma, in quo plana, quæ opponuntur &longs;int trian­<lb/>
gula ace, bdf: & parallelogrammorum latera ab, cd, <lb/>
ef bifariam <expan abbr="diuidãtur">diuidantur</expan> in punctis ghk: per diui&longs;iones au­<lb/>
<arrow.to.target n="marg30"/><lb/>
tem planum ducatur; cuius &longs;ectio figura ghK. <!-- KEEP S--></s><s>erit linea <lb/>
gh æquidi&longs;tans lineis ac, bd & hk ip&longs;is ce, df. </s><s>quare ex <lb/>
decimaquinta undecimi elementorum, planum illud pla <lb/>
nis ace, bdf æquidi&longs;tabit, & faciet &longs;ectionem figu­<lb/>
<arrow.to.target n="marg31"/><lb/>
ram ip&longs;is æqualem, & &longs;imilem, ut proxime demon&longs;tra­<lb/>
uimus. </s><s>Dico centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>
ghk. </s><s>Si enim fieri pote&longs;t, &longs;it eius centrum l: & ducatur <lb/>
lm u&longs;que ad planum ghk, quæ ip&longs;i ab æquidi&longs;tet. </s>
<pb xlink:href="023/01/024.jpg"/>
<s><arrow.to.target n="marg32"/>ergo linea ag continenter in duas partes æquales diui­<lb/>
&longs;a, relinquetur <expan abbr="tãdem">tandem</expan> pars aliqua ng, quæ minor erit lm. </s><lb/>
<s>Vtraque uero linearum ag, gb diuidatur in partes æqua­<lb/>
les ip&longs;i ng: & per puncta diui&longs;ionum plana oppo&longs;itis pla­<lb/>
<arrow.to.target n="marg33"/><lb/>
nis æquidi&longs;tantia ducantur. </s><s>erunt &longs;ectiones figuræ æqua­<lb/>
les, ac &longs;imiles ip&longs;is ace, bdf: & totum pri&longs;ma diui&longs;um erit <lb/>
in pri&longs;mata æqualia, & &longs;imilia: quæ cum inter &longs;e <expan abbr="congruãt">congruant</expan>; <lb/>
& grauitatis centra &longs;ibi ip&longs;is congruentia, <expan abbr="re&longs;pondentiaq;">re&longs;pondentiaque</expan> <lb/>
<figure id="id.023.01.024.1.jpg" xlink:href="023/01/024/1.jpg"/><lb/>
habebunt. </s><s><expan abbr="Itaq:">Itaque</expan> <lb/>
&longs;unt magnitudi­<lb/>
nes <expan abbr="quædã">quædam</expan> æqua­<lb/>
les ip&longs;i nh, & nu­<lb/>
mero pares, qua­<lb/>
rum centra gra­<lb/>
uitatis in <expan abbr="ead&etilde;re">eadem</expan> re<lb/>
cta linea con&longs;ti­<lb/>
tuuntur: duæ ue­<lb/>
ro mediæ æqua­<lb/>
les &longs;unt: & quæ ex <lb/>
utraque parte i­<lb/>
p&longs;arum &longs;imili­<lb/>
ter æquales: & æ­<lb/>
quales rectæ li­<lb/>
neæ, quæ inter <lb/>
grauitatis centra <lb/>
interiiciuntur. </s><lb/>
<s>quare ex corolla­<lb/>
rio quintæ pro­<lb/>
po&longs;itionis primi <lb/>
libri Archimedis <lb/>
de centro graui­<lb/>
tatis planorum; magnitudinis ex his omnibus compo&longs;itæ <lb/>
centrum grauitatis e&longs;t in medio lineæ, quæ magnitudi­<lb/>
num mediarum centra coniungit. </s><s>at qui non ita res ha­
<pb xlink:href="023/01/025.jpg" pagenum="9"/>bet, &longs;i quidem l extra medias magnitudines po&longs;itum e&longs;t. </s><lb/>
<s>Con&longs;tat igitur centrum grauitatis pri&longs;matis e&longs;&longs;e in plano <lb/>
<figure id="id.023.01.025.1.jpg" xlink:href="023/01/025/1.jpg"/><lb/>
ghk, quod nos demon&longs;trandum propo&longs;uimus. </s><s>At &longs;i op­<lb/>
po&longs;ita plana in pri&longs;mate &longs;int quadrilatera, uel plurilatera, <lb/>
eadem erit in omnibus demon&longs;tratio.</s></p><p type="margin">
<s><margin.target id="marg30"/>33. primi</s></p><p type="margin">
<s><margin.target id="marg31"/>5. huius</s></p><p type="margin">
<s><margin.target id="marg32"/>1. decimi</s></p><p type="margin">
<s><margin.target id="marg33"/>5 huius</s></p><p type="head">
<s>THEOREMA VII. PROPOSITIO VII.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet cylindri, & cuiuslibet cylindri por<lb/>
tionis centrum grauitatis e&longs;t in plano, quod ba&longs;i­<lb/>
bus æquidi&longs;tans, parallelogrammi per axem late­<lb/>
ra bifariam &longs;ecat.</s></p>
<pb xlink:href="023/01/026.jpg"/><p type="main">
<s>SIT cylindrus, uel cylindri portio ac: & plano per a­<lb/>
xem ducto &longs;ecetur; cuius &longs;ectio &longs;it parallelogrammum ab<lb/>
cd: & bifariam diui&longs;is ad, bc parallelogrammi lateribus, <lb/>
per diui&longs;ionum puncta ef planum ba&longs;i æquidi&longs;tans duca­<lb/>
tur; quod faciet &longs;ectionem, in cylindro quidem circulum <lb/>
æqualem iis, qui &longs;unt in ba&longs;ibus, ut demon&longs;trauit Serenus <lb/>
in libro cylindricorum, propo&longs;itione quinta: in cylindri <lb/>
uero portione ellip&longs;im æqualem, & &longs;imilem eis, quæ &longs;unt <lb/>
<figure id="id.023.01.026.1.jpg" xlink:href="023/01/026/1.jpg"/><lb/>
in oppo&longs;itis planis, quod nos <lb/>
demon&longs;trauimus in commen <lb/>
tariis in librum Archimedis <lb/>
de conoidibus, & &longs;phæroidi­<lb/>
bus. </s><s>Dico centrum grauita­<lb/>
tis cylindri, uel cylindri por­<lb/>
tionis e&longs;&longs;e in plano ef. </s><s>Si <expan abbr="enĩ">enim</expan> <lb/>
fieri pote&longs;t, fit centrum g: & <lb/>
ducatur gh ip&longs;i ad æquidi­<lb/>
&longs;tans, u&longs;que ad ef planum. </s><lb/>
<s>Itaque linea ae continenter <lb/>
diui&longs;a bifariam, erit tandem <lb/>
pars aliqua ip&longs;ius ke, minor <lb/>
gh. </s><s>Diuidantur ergo lineæ <lb/>
ae, ed in partes æquales ip&longs;i <lb/>
ke: & per diui&longs;iones plana ba<lb/>
&longs;ibus æquidi&longs;tantia <expan abbr="ducãtur">ducantur</expan>. </s><lb/>
<s>erunt iam &longs;ectiones, figuræ æ­<lb/>
quales, & &longs;imiles eis, quæ &longs;unt <lb/>
in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: & cy<lb/>
lindri portio in portiones æquales, & &longs;imiles ip&longs;i kf. </s><s>reli­<lb/>
qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.</s></p>
<pb xlink:href="023/01/027.jpg" pagenum="10"/><figure id="id.023.01.027.1.jpg" xlink:href="023/01/027/1.jpg"/><p type="head">
<s>THEOREMA VIII. PROPOSITIO VIII.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet pri&longs;matis, & cuiuslibet cylindri, uel <lb/>
cylindri portionis grauitatis centrum in medio <lb/>
ip&longs;ius axis con&longs;i&longs;tit.</s></p><p type="main">
<s>Sit primum af pri&longs;ma æquidi&longs;tantibus planis <expan abbr="contentũ">contentum</expan>, <lb/>
quod &longs;olidum parallelepipedum appellatur: & oppo&longs;ito­<lb/>
rum planorum cf, ah, da, fg latera bifariam diuidantur in <lb/>
punctis klmnopqrstux: & per diui&longs;iones ducantur <lb/>
plana kn, or, sx. </s><s>communes autem eorum planorum &longs;e­<lb/>
ctiones &longs;int lineæ yz, <foreign lang="greek">qf, xy:</foreign> quæ in puncto <foreign lang="greek">w</foreign> <expan abbr="conueniãt">conueniant</expan>. </s><lb/>
<s>erit ex decima eiu&longs;dem libri Archimedis parallelogrammi <lb/>
cf centrum grauitatis punctum y; parallelogrammi ah
<pb xlink:href="023/01/028.jpg"/>centrum z: parallelogrammi ad, <foreign lang="greek">q:</foreign> parallelogrammi fg, <foreign lang="greek">f:</foreign><lb/>
<figure id="id.023.01.028.1.jpg" xlink:href="023/01/028/1.jpg"/><lb/>
parallelogrammi dh, <foreign lang="greek">x:</foreign> & <lb/>
parallelogrammi cg <expan abbr="centrũ">centrum</expan> <lb/>
<foreign lang="greek">y:</foreign> atque erit <foreign lang="greek">w</foreign> punctum me <lb/>
dium uniu&longs;cuiu&longs;que axis, ui <lb/>
delicet eius lineæ quæ oppo <lb/>
&longs;itorum <expan abbr="planorũ">planorum</expan> centra con <lb/>
iungit. </s><s>Dico <foreign lang="greek">w</foreign> centrum e&longs;&longs;e <lb/>
grauitatis ip&longs;ius &longs;olidi. </s><s>e&longs;t <lb/>
<arrow.to.target n="marg34"/><lb/>
enim, ut demon&longs;trauimus, <lb/>
&longs;olidi af centrum grauitatis <lb/>
in plano Kn; quod oppo&longs;i­<lb/>
tis planis ad, gf æquidi&longs;tans <lb/>
reliquorum planorum late­<lb/>
ra bifariam diuidit: & &longs;imili <lb/>
ratione idem centrum e&longs;t in plano or, æquidi&longs;tante planis <lb/>
ae, bf oppo&longs;itis. </s><s>ergo in communi ip&longs;orum &longs;ectione: ui­<lb/>
delicet in linea yz. </s><s>Sed e&longs;t etiam in plano tu, quod <expan abbr="quid&etilde;">quidem</expan> <lb/>
yz &longs;ecatin <foreign lang="greek">w.</foreign> Con&longs;tat igitur centrum grauitatis &longs;olidi e&longs;&longs;e <lb/>
punctum <foreign lang="greek">w,</foreign> medium &longs;cilicet axium, hoc e&longs;t linearum, quæ <lb/>
planorum oppo&longs;itorum centra coniungunt.</s></p><p type="margin">
<s><margin.target id="marg34"/>6 huius</s></p><p type="main">
<s>Sit aliud prima af; & in eo plana, quæ opponuntur, tri­<lb/>
angula abc, def: <expan abbr="diui&longs;isq;">diui&longs;isque</expan> bifariam parallelogrammorum <lb/>
lateribus ad, be, cf in punctis ghk, per diui&longs;iones <expan abbr="planũ">planum</expan> <lb/>
ducatur, quod oppo&longs;itis planis æquidi&longs;tans faciet <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> <lb/>
triangulum ghx æquale, & &longs;imile ip&longs;is abc, def. </s><s>Rur&longs;us <lb/>
diuidatur ab bifariam in l: & iuncta cl per ip&longs;am, & per <lb/>
cKf planum ducatur pri&longs;ma &longs;ecans, cuius, & <expan abbr="parallelogrã">parallelogram</expan><lb/>
mi ae communis &longs;ectio &longs;it lmn. </s><s>diuidet punctum m li­<lb/>
neam gh bifariam; & ita n diuidet lineam de: quoniam <lb/>
<arrow.to.target n="marg35"/><lb/>
triangula acl, gkm, dfn æqualia &longs;unt, & &longs;imilia, ut &longs;upra <lb/>
demon&longs;trauimus. </s><s>Iam ex iis, quæ tradita &longs;unt, con&longs;tat cen<lb/>
trum grauitatis pri&longs;matis in plano ghk contineri. </s><s>Dico <lb/>
ip&longs;um e&longs;&longs;e in linea km. </s><s>Si enim fieri pote&longs;t, &longs;it o centrum;
<pb xlink:href="023/01/029.jpg" pagenum="11"/>& per o ducatur op ad km ip&longs;i hg æquidi&longs;tans. </s><s>Itaque li <lb/>
nea hm <expan abbr="bifariã">bifariam</expan> u&longs;que eò diuidatur, quoad reliqua &longs;it pars <lb/>
quædam qm, minor op. </s><s>deinde hm, mg diuidantur in <lb/>
partes æquales ip&longs;i mq: & per diui&longs;iones lineæ ip&longs;i mK <lb/>
æquidi&longs;tantes ducantur. </s><s>puncta uero, in quibus hæ trian­<lb/>
gulorum latera &longs;ecant, coniungantur ductis lineis rs, tu, <lb/>
<figure id="id.023.01.029.1.jpg" xlink:href="023/01/029/1.jpg"/><lb/>
xy; quæ ba&longs;i gh æquidi&longs;tabunt. </s><s>Quoniam enim lineæ gz, <lb/>
h<foreign lang="greek">a</foreign> &longs;unt æquales: <expan abbr="itemq;">itemque</expan> æquales gm, mh: ut mg ad gz, <lb/>
ita erit mh, ad h<foreign lang="greek">a:</foreign> & diuidendo, ut mz ad zg, ita m<foreign lang="greek">a</foreign> ad <lb/>
<arrow.to.target n="marg36"/><lb/>
<foreign lang="greek">a</foreign>h. </s><s>Sed ut mz ad zg, ita kr ad rg: & ut m<foreign lang="greek">a</foreign> ad <foreign lang="greek">a</foreign>h, ita ks <lb/>
ad sh. </s><s>quare ut kr ad rg, ita ks ad sh. </s><s>æquidi&longs;tant igitur <lb/>
<arrow.to.target n="marg37"/><lb/>
inter &longs;e &longs;e rs, gh. </s><s>eadem quoque ratione demon&longs;trabimus
<pb xlink:href="023/01/030.jpg"/>tu, xy ip&longs;i gh æquidi&longs;tare. </s><s>Et quoniam triangula, quæ <lb/>
fiunt à lineis Ky, yu, us, sh æqualia &longs;unt inter &longs;e, & &longs;imilia <lb/>
<arrow.to.target n="marg38"/><lb/>
triangulo Kmh: habebit triangulum Kmh ad <expan abbr="triangulũ">triangulum</expan> <lb/>
K<foreign lang="greek">d</foreign>y duplam proportionem eius, quæ e&longs;t lineæ kh ad Ky. </s><lb/>
<s>&longs;ed Kh po&longs;ita e&longs;t quadrupla ip&longs;ius ky. </s><s>ergo triangulum <lb/>
kmh ad triangulum K<foreign lang="greek">d</foreign>y <expan abbr="eãdem">eandem</expan> proportionem habebit, <lb/>
quam &longs;exdecim ad <expan abbr="unũ">unum</expan>: & ad quatuor triangula k<foreign lang="greek">d</foreign>y, yu, <lb/>
us, s<foreign lang="greek">a</foreign>h habebit eandem, quam &longs;exdecim ad quatuor, hoc <lb/>
e&longs;t quam hK ad ky: & &longs;imiliter eandem habere demon&longs;tra <lb/>
<figure id="id.023.01.030.1.jpg" xlink:href="023/01/030/1.jpg"/><lb/>
bitur trian­<lb/>
gulum kmg <lb/>
ad quatuor <lb/>
<expan abbr="triãgula">triangula</expan> K<foreign lang="greek">d</foreign><lb/>
x, x<foreign lang="greek">g</foreign>t, t<foreign lang="greek">b</foreign>r, <lb/>
<arrow.to.target n="marg39"/><lb/>
rzg. <!-- KEEP S--></s><s>quare <lb/>
totum trian <lb/>
gulum Kgh <lb/>
ad omnia tri <lb/>
angula gzr, <lb/>
r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign><lb/>
K, K<foreign lang="greek">d</foreign>y, yu, <lb/>
us, s<foreign lang="greek">a</foreign>h ita <lb/>
erit, ut hk ad <lb/>
ky, hoc e&longs;t <lb/>
ut hm ad m<lb/>
q. </s><s>Si igitur in <lb/>
triangulis abc, def de&longs;cribantur figuræ &longs;imiles ei, quæ de­<lb/>
&longs;cripta e&longs;t in ghK triangulo: & per lineas &longs;ibi re&longs;ponden­<lb/>
tes plana ducantur: totum pri&longs;ma af diui&longs;um erit in tria <lb/>
&longs;olida parallelepipeda y<foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz, quorum ba&longs;es &longs;unt æqua <lb/>
les & &longs;imiles ip&longs;is parallelogrammis y <foreign lang="greek">g,</foreign>u<foreign lang="greek">b,</foreign> sz: & in octo <lb/>
pri&longs;mata gzr, r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign><lb/>
K, k<foreign lang="greek">d</foreign>y, yu, us, s<foreign lang="greek">a</foreign>h: quorum <lb/>
item ba&longs;es æquales, & &longs;imiles &longs;unt dictis triangulis; altitu­<lb/>
do autem in omnibus, totius pri&longs;matis altitudini æqualis.
<pb xlink:href="023/01/031.jpg" pagenum="12"/>Itaque &longs;olidi parallelepipedi y<foreign lang="greek">g</foreign> centrum grauitatis e&longs;t in <lb/>
linea <foreign lang="greek">de:</foreign> &longs;olidi u<foreign lang="greek">b</foreign> centrum e&longs;t in linea <foreign lang="greek">eh:</foreign> & &longs;olidi sz in li<lb/>
nea <foreign lang="greek">h</foreign>m, quæ quidem lineæ axes &longs;unt, cum planorum oppo<lb/>
&longs;itorum centra coniungant. </s><s>ergo magnitudinis ex his &longs;oli <lb/>
dis compo&longs;itæ centrum grauitatis e&longs;t in linea <foreign lang="greek">d</foreign>m, quod &longs;it <lb/>
<foreign lang="greek">q</foreign>; & iuncta <foreign lang="greek">q</foreign>o producatur: à puncto autem h ducatur h<foreign lang="greek">a</foreign><lb/>
ip&longs;i mk æquidi&longs;tans, quæ cum <foreign lang="greek">q</foreign>o in <foreign lang="greek">m</foreign> conueniat. </s><s>triangu <lb/>
lum igitur ghk ad omnia triangula gzr, <foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign>k, <lb/>
k<foreign lang="greek">d</foreign>y, yu, us, s<foreign lang="greek">a</foreign>h eandem habet proportionem, quam hm <lb/>
ad mq; hoc e&longs;t, quam <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql:</foreign> nam &longs;i hm, <foreign lang="greek">mq</foreign> produci in<lb/>
telligantur, quou&longs;que coeant; erit ob linearum qy, mk æ­<lb/>
quidi&longs;tantiam, ut hq ad qm, ita <foreign lang="greek">ml</foreign> ad ad <foreign lang="greek">lq:</foreign> & componen <lb/>
do, ut hm ad mq, ita <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql.</foreign></s><s> linea uero <foreign lang="greek">q</foreign>o maior e&longs;t, <lb/>
<arrow.to.target n="marg40"/><lb/>
quàm <foreign lang="greek">ql:</foreign> habebit igitur <foreign lang="greek">mq</foreign> ad <foreign lang="greek">ql</foreign> maiorem proportio­<lb/>
nem, quàm ad <foreign lang="greek">q</foreign>o. </s><s>quare triangulum etiam ghk ad omnia <lb/>
iam dicta triangula maiorem <expan abbr="proportion&etilde;">proportionem</expan> habebit, quàm <lb/>
<foreign lang="greek">mq</foreign> ad <foreign lang="greek">q</foreign>o. </s><s>&longs;ed ut <expan abbr="triangulũ">triangulum</expan> ghk ad omnia triangula, ita <expan abbr="to-tũ">to­<lb/>
tum</expan> pri&longs;ma afad omnia pri&longs;mata gzr, r<foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">dk, kd</foreign> y, <lb/>
yu, us, s<foreign lang="greek">a</foreign>h: quoniam enim &longs;olida parallelepipeda æque al <lb/>
ta, eandem inter &longs;e proportionem habent, quam ba&longs;es; ut <lb/>
ex trige&longs;ima&longs;ecunda undecimi elementorum con&longs;tat. </s><s>&longs;unt <lb/>
<arrow.to.target n="marg41"/><lb/>
autem &longs;olida parallelepipeda pri&longs;matum triangulares ba­<lb/>
<arrow.to.target n="marg42"/><lb/>
&longs;es habentium dupla: &longs;equitur, ut etiam huiu&longs;modi pri&longs;­<lb/>
mata inter &longs;e &longs;int, &longs;icut eorum ba&longs;es. </s><s>ergo totum pri&longs;ma ad <lb/>
omnia pri&longs;mata maiorem proportionem habet, quam <foreign lang="greek">mq</foreign><lb/>
<arrow.to.target n="marg43"/><lb/>
ad <foreign lang="greek">q</foreign>o: & diuidendo &longs;olida parallelepipeda y<foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz ad o­<lb/>
mnia pri&longs;mata proportionem habent maiorem, quàm <foreign lang="greek">m</foreign>o <lb/>
ad o<foreign lang="greek">q</foreign>. </s><s>fiat <foreign lang="greek">n</foreign>o ad o<foreign lang="greek">q,</foreign> ut &longs;olida parallelepipeda y <foreign lang="greek">g,</foreign> u<foreign lang="greek">b,</foreign> sz ad <lb/>
omnia pri&longs;mata. </s><s>Itaque cum à pri&longs;mate af, cuius <expan abbr="c&etilde;trum">centrum</expan> <lb/>
grauitatis e&longs;t o, auferatur magnitudo ex &longs;olidis parallelepi<lb/>
pedis y <foreign lang="greek">g,</foreign>u<foreign lang="greek">b,</foreign>sz con&longs;tans: atque ip&longs;ius grauitatis centrum <lb/>
&longs;it <foreign lang="greek">q:</foreign> reliquæ magnitudinis, quæ ex omnibus pri&longs;matibus <lb/>
con&longs;tat, grauitatis centrum erit in linea <foreign lang="greek">q</foreign> o producta: & <lb/>
in puncto <foreign lang="greek">v</foreign>, ex octava propo&longs;itione eiusdem libri Archi­
<pb xlink:href="023/01/032.jpg"/>medis. </s><s>ergo punctum <foreign lang="greek">n</foreign> extra pri&longs;ma af po&longs;itum, <expan abbr="centrũ">centrum</expan> <lb/>
erit magnitudinis <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> ex omnibus pri&longs;matibus gzr, <lb/>
r <foreign lang="greek">b</foreign>t, t<foreign lang="greek">g</foreign>x, x<foreign lang="greek">d</foreign>k, k<foreign lang="greek">d</foreign> y, yu, us, s<foreign lang="greek">a</foreign>h, quod fieri nullo modo po<lb/>
te&longs;t. </s><s>e&longs;t enim ex diffinitione centrum grauitatis &longs;olidæ figu<lb/>
ræ intra ip&longs;am po&longs;itum, non extra. </s><s>quare relinquitur, ut <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis pri&longs;matis &longs;it in linea Km. </s><s>Rur&longs;us bc bifa­<lb/>
riam in diuidatur: & ducta a<foreign lang="greek">x,</foreign> per ip&longs;am, & per lineam <lb/>
agd plan um ducatur; quod pri&longs;ma &longs;ecet: <expan abbr="faciatq;">faciatque</expan> in paral<lb/>
lelogrammo bf &longs;ectionem <foreign lang="greek">x p</foreign> diuidet punctum <foreign lang="greek">p</foreign> lineam <lb/>
quoque cf bifariam: & erit plani eius, & trianguli ghK <lb/>
communis &longs;ectio gu; quòd <expan abbr="pũctum">punctum</expan> u in medio lineæ hK <lb/>
<figure id="id.023.01.032.1.jpg" xlink:href="023/01/032/1.jpg"/><lb/>
po&longs;itum &longs;it. </s><s>Similiter demon&longs;trabimus centrum grauita­<lb/>
tis pri&longs;matis in ip&longs;a gu ine&longs;&longs;e. </s><s>&longs;it autem planorum cfnl, <lb/>
ad<foreign lang="greek">px</foreign> communis &longs;ectio linea <foreign lang="greek">rst;</foreign> quæ quidem pri&longs;matis <lb/>
axis erit, cum tran&longs;eat per centra grauitatis triangulorum <lb/>
abc, ghk def, ex quartadecima eiu&longs;dem. </s><s>ergo centrum <lb/>
grauitatis pri&longs;matis af e&longs;t punctum <foreign lang="greek">s,</foreign> centrum &longs;cilicet
<pb xlink:href="023/01/033.jpg" pagenum="13"/>trianguli ghK, & ip&longs;ius <foreign lang="greek">rt</foreign> axis medium.</s></p><p type="margin">
<s><margin.target id="marg35"/>5.huius</s></p><p type="margin">
<s><margin.target id="marg36"/>2. &longs;exti.<lb/>
12 quinti.</s></p><p type="margin">
<s><margin.target id="marg37"/>2. &longs;exti.</s></p><p type="margin">
<s><margin.target id="marg38"/>
19. &longs;exti</s></p><p type="margin">
<s><margin.target id="marg39"/>2. uel 12. <lb/>
quinti.</s></p><p type="margin">
<s><margin.target id="marg40"/>8. quinti.<!-- KEEP S--></s></p><p type="margin">
<s><margin.target id="marg41"/>28. unde<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg42"/>15. quinti</s></p><p type="margin">
<s><margin.target id="marg43"/>19. quinti<lb/>
apud <expan abbr="Cã">Cam</expan><lb/>
panum</s></p><p type="main">
<s>Sit pri&longs;ma ag, cuius oppo&longs;ita plana &longs;int quadrilatera <lb/>
abcd, efgh: <expan abbr="&longs;ecenturq;">&longs;ecenturque</expan> ac, bf, cg, dh bifariam: & per di­<lb/>
ui&longs;iones planum ducatur; quod &longs;ectionem faciat quadrila­<lb/>
terum Klmn. </s><s>Deinde iuncta ac per lineas ac, ae ducatur <lb/>
planum <expan abbr="&longs;ecãs">&longs;ecans</expan> pri&longs;ma, quod ip&longs;um diuidet in duo pri&longs;mata <lb/>
triangulares ba&longs;es habentia abcefg, adcehg. </s><s>Sint <expan abbr="aut&etilde;">autem</expan> <lb/>
<figure id="id.023.01.033.1.jpg" xlink:href="023/01/033/1.jpg"/><lb/>
triangulorum abc, efg gra­<lb/>
uitatis centra op: & triangu­<lb/>
lorum adc, ehg centra qr: <lb/>
<expan abbr="iunganturq;">iunganturque</expan> op, qr; quæ pla­<lb/>
no klmn occurrant in pun­<lb/>
ctis st. </s><s>erit ex iis, quæ demon<lb/>
&longs;trauimus, punctum s grauita<lb/>
tis centrum trianguli klm; & <lb/>
ip&longs;ius pri&longs;matis abcefg: pun<lb/>
ctum uero t centrum grauita <lb/>
tis trianguli Knm, & pri&longs;ma­<lb/>
tis adc, ehg. <!-- KEEP S--></s><s>iunctis igitur <lb/>
oq, pr, st, erit in linea oq <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis quadrilateri <lb/>
abcd, quod &longs;it u: & in linea <lb/>
pr <expan abbr="c&etilde;trum">centrum</expan> quadrilateri efgh <lb/>
&longs;it autem x. </s><s>denique iungatur <lb/>
u x, quæ &longs;ecet lineam &longs; t in y. </s><s>&longs;e<lb/>
cabit enim cum &longs;int in eodem <lb/>
<arrow.to.target n="marg44"/><lb/>
plano: <expan abbr="atq;">atque</expan> erit y grauitatis centrum quadrilateri Klmn. </s><lb/>
<s>Dico idem punctum y centrum quoque gra uitatis e&longs;&longs;e to­<lb/>
tius pri&longs;matis. </s><s>Quoniam enim quadrilateri klmn graui­<lb/>
tatis centrum e&longs;t y: linea sy ad yt ean dem proportionem <lb/>
habebit, quam triangulum knm ad triangulum klm, ex 8 <lb/>
Archimedis de centro grauitatis planorum. </s><s>Vt autem <expan abbr="triã">trian</expan><lb/>
gulum knm ad ip&longs;um klm, hoc e&longs;t ut triangulum adc ad <lb/>
triangulum abc, æqualia enim &longs;unt, ita pri&longs;ma adcehg
<pb xlink:href="023/01/034.jpg"/>ad pri&longs;ma abcefg. <!-- KEEP S--></s><s>quare linea sy ad yt eandem propor­<lb/>
tionem habet, quam pri&longs;ma adcehg ad pri&longs;ma abcefg. <!-- KEEP S--></s><lb/>
<s>Sed pri&longs;matis abcefg centrum grauitatis e&longs;t s: & pri&longs;ma­<lb/>
tis adcehg centrum t. </s><s>magnitudinis igitur ex his compo<lb/>
&longs;itæ hoc e&longs;t totius pri&longs;matis ag centrum grauitatis e&longs;t pun<lb/>
ctum y; medium &longs;cilicet axis ux, qui oppo&longs;itorum plano­<lb/>
rum centra coniungit.</s></p><p type="margin">
<s><margin.target id="marg44"/>5. huius/></s></p><p type="main">
<s>Rur&longs;us &longs;it pri&longs;ma ba&longs;im habens pentagonum abcde: <lb/>
& quod ei opponitur &longs;it fghKl: &longs;ec<expan abbr="enturq;">enturque</expan> af, bg, ch, <lb/>
dk, el bifariam: & per diui&longs;iones ducto plano, &longs;ectio &longs;it <expan abbr="p&etilde;">pen</expan><lb/>
<expan abbr="tagonũ">tagonum</expan> mnopq. deinde iuncta eb per lineas le, eb aliud <lb/>
<figure id="id.023.01.034.1.jpg" xlink:href="023/01/034/1.jpg"/><lb/>
planum ducatur, <expan abbr="diuid&etilde;s">diuidens</expan> pri&longs;<lb/>
ma ak in duo pri&longs;mata; in pri&longs;<lb/>
ma &longs;cilicet al, cuius plana op­<lb/>
po&longs;ita &longs;int triangula abe fgl: <lb/>
& in prima bk cuius plana op<lb/>
po&longs;ita &longs;int quadrilatera bcde <lb/>
ghkl. <!-- KEEP S--></s><s>Sint autem triangulo­<lb/>
rum abe, fgl centra grauita <lb/>
tis puncta r &longs;: & bcde, ghkl <lb/>
quadrilaterorum centra tu: <lb/>
<expan abbr="iunganturq;">iunganturque</expan> rs, tu occurren­<lb/>
tes plano mnopq in punctis <lb/>
xy. </s><s>& itidem <expan abbr="iungãtur">iungantur</expan> rt, &longs;u, <lb/>
xy. </s><s>erit in linca rt <expan abbr="c&etilde;trum">centrum</expan> gra<lb/>
uitatis pentagoni abcde; <lb/>
quod &longs;it z: & in linea &longs;u cen­<lb/>
trum pentagoni fghkl :&longs;it au <lb/>
tem <foreign lang="greek">x:</foreign> & ducatur z<foreign lang="greek">x,</foreign> quæ di­<lb/>
cto plano in <foreign lang="greek">y</foreign> occurrat. </s><s><expan abbr="Itaq;">Itaque</expan> <lb/>
punctum x e&longs;t centrum graui <lb/>
tatis trianguli mnq, ac pri&longs;­<lb/>
matis al: & y grauitatis centrum quadrilateri nopq, ac <lb/>
pri&longs;matis bk. </s><s>quare y centrum erit pentagoni mnopq. </s><s> &
<pb xlink:href="023/01/035.jpg" pagenum="14"/>&longs;imiliter demon&longs;trabitur totius pri&longs;matis aK grauitatis ef <lb/>
&longs;e centrum. </s><s>Simili ratione & in aliis pri&longs;matibus illud <lb/>
idem facile demon&longs;trabitur. </s><s>Quo autem pacto in omni <lb/>
figura rectilinea centrum grauitatis inueniatur, docuimus <lb/>
in commentariis in &longs;extam propo&longs;itionem Archimedis de <lb/>
quadratura parabolæ.</s></p><p type="main">
<s>Sit cylindrus, uel cylindri portio ce cuius axis ab: &longs;ece­<lb/>
<expan abbr="turq,">turque</expan> plano per axem ducto; quod &longs;ectionem faciat paral­<lb/>
lelogrammum cdef: & diui&longs;is cf, de bifariam in punctis <lb/>
<figure id="id.023.01.035.1.jpg" xlink:href="023/01/035/1.jpg"/><lb/>
gh, per ea ducatur planum ba&longs;i æquidi&longs;tans. </s><s>erit &longs;ectio gh <lb/>
circulus, uel ellip&longs;is, centrum habens in axe; quod &longs;it K at­<lb/>
<arrow.to.target n="marg45"/><lb/>
que erunt ex iis, quæ demon&longs;trauimus, centra grauitatis <lb/>
planorum oppo&longs;itorum puncta ab: & plani gh ip&longs;um k in <lb/>
quo quidem plano e&longs;t centrum grauitatis cylindri, uel cy­<lb/>
lindri portionis. </s><s>Dico punctum K cylindri quoque, uel cy<lb/>
lindri portionis grauitatis centrum e&longs;&longs;e. </s><s>Si enim fieri po­<lb/>
te&longs;t, &longs;it l centrum: <expan abbr="ducaturq;">ducaturque</expan> kl, & extra figuram in m pro­<lb/>
ducatur. </s><s>quam ucro proportionem habet linea mK ad kl
<pb xlink:href="023/01/036.jpg"/>habeat circulus, uel ellip&longs;is gh ad aliud &longs;pacium, in quo u: <lb/>
& in cit culo, uel ellip&longs;i plane de&longs;cribatur rectilinea figura, <lb/>
ita ut <expan abbr="tãdem">tandem</expan> <expan abbr="relinquãtur">relinquantur</expan> portiones minores &longs;pacio u, quæ <lb/>
&longs;it opgqrsht: <expan abbr="de&longs;criptaq;">de&longs;criptaque</expan> &longs;imili figura in oppo&longs;itis pla­<lb/>
nis cd, fe, per lineas &longs;ibi ip&longs;is re&longs;pondentes plana <expan abbr="ducãtur">ducantur</expan>. </s><lb/>
<s>Itaque cylindrus, uel cylindri portio diuiditur in pri&longs;ma, <lb/>
cuius quidem ba&longs;is e&longs;t figura rectilinea iam dicta, centrum <lb/>
que grauitatis punctum K: & in multa &longs;olida, qaæ pro ba&longs;i <lb/>
bus habent relictas portiones, quas nos &longs;olidas portiones <lb/>
appellabimus. </s><s>cum igitur portiones &longs;int minores &longs;pacio <lb/>
u, circulus, uel ellip&longs;is gh ad portiones maiorem propor­<lb/>
tionem habebit, quàm linea mk ad Kl. <!-- KEEP S--></s><s>fiat nk ad Kl, ut <lb/>
circulus uel ellip&longs;is gh ad ip&longs;as portiones. </s><s>Sed ut circulus <lb/>
uel ellip&longs;is gh ad figuram rectilineam in ip&longs;a de&longs;cri­<lb/>
ptam, ita e&longs;t cylindrus uel cylindri portio ce ad pri&longs;ma, <lb/>
quod rectilineam figuram pro ba&longs;i habet, & altitudinem <lb/>
æqualem; id, quod infra demon&longs;trabitur. </s><s>crgo per conuer<lb/>
&longs;ionem rationis, ut circulus, uel ellip&longs;is gh ad portiones re<lb/>
lictas, ita cylindrus, uel cylindri portio ce ad &longs;olidas por­<lb/>
tiones, quate cylindrus uel cylindri portio ad &longs;olidas por­<lb/>
tiones eandem proportionem habet, quam linea nk ad k <lb/>
& diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o­<lb/>
lidas portiones eandem proportionem habet, quam nl ad <lb/>
lk & quoniam a cylindro uel cylindri portione, cuius gra­<lb/>
uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili­<lb/>
neam <expan abbr="figurã">figuram</expan>, cuius <expan abbr="centrũ">centrum</expan> grauitatis e&longs;t K: re&longs;iduæ magnitu<lb/>
dinis ex &longs;olidis portionibus <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan> grauitatis <expan abbr="c&etilde;trũ">centrum</expan> erit <lb/>
in linea kl protracta, & in puncto n; quod e&longs;t <expan abbr="ab&longs;urdũ">ab&longs;urdum</expan>. </s><s>relin<lb/>
quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>
tionis &longs;it <expan abbr="punctũ">punctum</expan> k. </s><s>quæ omnia <expan abbr="demon&longs;trãda">demon&longs;tranda</expan> propo&longs;uimus.</s></p><p type="margin">
<s><margin.target id="marg45"/>4. huius</s></p><p type="main">
<s>At uero cylindrum, uel cylindri <expan abbr="portion&etilde;">portionem</expan> ce <lb/>
ad pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura in &longs;pa­<lb/>
cio gh de&longs;cripta, & altitudo æqualis; eandem ha­
<pb xlink:href="023/01/037.jpg" pagenum="15"/>bere proportionem, quam &longs;pacium gh ad <expan abbr="dictã">dictam</expan> <lb/>
figuram, hoc modo demon&longs;trabimus.</s></p><p type="main">
<s>Intelligatur circulus, uel ellip&longs;is x æqualis figuræ rectili­<lb/>
neæ in gh &longs;pacio de&longs;criptæ. </s><s>& ab x con&longs;tituatur conus, uel <lb/>
<figure id="id.023.01.037.1.jpg" xlink:href="023/01/037/1.jpg"/><lb/>
coni portio, <expan abbr="altitudin&etilde;">altitudinem</expan> habens <expan abbr="eand&etilde;">eandem</expan>, <expan abbr="quã">quam</expan> cylindrus uel cy<lb/>
lindri portio ce. </s><s>Sit deinde rectilinea figura, in qua y <expan abbr="ead&etilde;">eadem</expan>, <lb/>
quæ in &longs;pacio gh de&longs;cripta e&longs;t: & ab hac pyramis æquealta <lb/>
con&longs;tituatur. </s><s>Dico <expan abbr="conũ">conum</expan> uel coni portione x pyramidi y <expan abbr="æ-qual&etilde;">æ­<lb/>
qualem</expan> e&longs;&longs;e. </s><s>ni&longs;i enim &longs;it æqualis, uel maior, uel minor erit.</s></p><p type="main">
<s>Sit primum maior, et exuperet &longs;olido z. </s><s>Itaque in circu<lb/>
lo, uel ellip&longs;i x de&longs;cribatur figura rectilinea; & in ea pyra­<lb/>
mis eandem, quam conus, uel coni portio altitudinem ha­<lb/>
bens, ita ut portiones relictæ minores &longs;int &longs;olido a, quem­<lb/>
admodum docetur in duodecimo libro elementorum pro<lb/>
po&longs;itione undecima. </s><s>erit pyramis x adhuc pyramide y ma<lb/>
ior. </s><s>& quoniam piramides æque altæ inter &longs;e &longs;unt, &longs;icuti ba<lb/>
<arrow.to.target n="marg46"/><lb/>
&longs;es; pyramis x ad piramidem y eandem proportionem ha­<lb/>
bet, quàm figura rectilinea x ad figuram y. </s><s>Sed figura recti
<pb xlink:href="023/01/038.jpg"/><figure id="id.023.01.038.1.jpg" xlink:href="023/01/038/1.jpg"/><lb/>
linea x cum &longs;it minor circulo, uel ellip&longs;i, e&longs;t etiam minor fi­<lb/>
gura rectilinea y. </s><s>ergo pyramis x pyramide y minor erit. </s><lb/>
<s>Sed & maior; quod fieri <expan abbr="nõ">non</expan> pote&longs;t. </s><s>At &longs;i conus, uel coni por<lb/>
tio x ponatur minor pyramide y: &longs;it alter conus æque al­<lb/>
tus, uel altera coni portio X ip&longs;i pyramidi y æqualis. </s><s>erit <lb/>
eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x, <lb/>
quorum exce&longs;&longs;us &longs;it &longs;pacium <foreign lang="greek">w.</foreign> Si igitur in circulo, uel eili­<lb/>
p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relictæ <lb/>
&longs;int <foreign lang="greek">w</foreign> &longs;pacio minores, ciu&longs;modi figura adhuc maior erit cir <lb/>
culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. </s><s>& pyramis in <lb/>
ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi­<lb/>
nor pyramide y. </s><s>e&longs;t ergo ut X figura rectilinea ad figuram <lb/>
rectilineam y, ita pyramis X ad pyramidem y. </s><s>quare cum <lb/>
figura rectilinea X &longs;it maior figura y: erit & pyramis X py­<lb/>
ramide y maior. </s><s>&longs;ed erat minor; quod rur&longs;us fieri non po­<lb/>
te&longs;t. </s><s>non e&longs;t igitur conus, uel coni portio x neque maior, <lb/>
neque minor pyramide y. </s><s>ergo ip&longs;i nece&longs;&longs;ario e&longs;t æqualis. </s><lb/>
<s>Itaque quoniam ut conus ad conum, uel coni portio ad co
<pb xlink:href="023/01/039.jpg" pagenum="16"/><figure id="id.023.01.039.1.jpg" xlink:href="023/01/039/1.jpg"/><lb/>
ni portionem, ita e&longs;t cylindrus ad cylindrum, uel cylin­<lb/>
dri portio ad cylindri portionem: & ut pyramis ad pyra­<lb/>
midem, ita pri&longs;ma ad pri&longs;ma, cum eadem &longs;it ba&longs;is, & æqua <lb/>
lis altitudo; erit cylindrus uel cylindri portio x pri&longs;ma­<lb/>
ti y æqualis. </s><s><expan abbr="e&longs;tq;">e&longs;tque</expan> ut &longs;pacium gh ad &longs;pacium x, ita cylin­<lb/>
drus, uel cylindri portio ce ad cylindrum, uel cylindri por­<lb/>
tionem x. </s><s>Con&longs;tat igitur cylindrum uel cylindri <expan abbr="portion&etilde;">portionem</expan> <lb/>
c e, ad pri&longs;ma y, quippe cuius ba&longs;is e&longs;t figura rectilinea in <lb/>
<arrow.to.target n="marg47"/><lb/>
&longs;pacio gh de&longs;cripta, eandem proportionem habere, quam <lb/>
&longs;pacium gh habet ad &longs;pacium x, hoc e&longs;t ad dictam figuram. </s><lb/>
<s>quod demon&longs;trandum fuerat.</s></p><p type="margin">
<s><margin.target id="marg46"/>6. duode<lb/>
cimi.</s></p><p type="margin">
<s><margin.target id="marg47"/>7. quinti</s></p><p type="head">
<s>THEOREMA IX. PROPOSITIO IX.<!-- KEEP S--></s></p><p type="main">
<s>Si pyramis &longs;ecetur plano ba&longs;i æquidi&longs;tante; &longs;e­<lb/>
ctio erit figura &longs;imilis ei, quæ e&longs;t ba&longs;is, centrum <lb/>
grauitatis in axe habens.</s></p>
<pb xlink:href="023/01/040.jpg"/><p type="main">
<s>SIT pyramis, cuius ba&longs;is triangulum abc; axis dc: & <lb/>
&longs;ecetur plano ba&longs;i æquidi&longs;tante; quod <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> faciat fgh; <lb/>
<expan abbr="occurratq;">occurratque</expan> axi in puncto k. Dico fgh triangulum e&longs;&longs;e, ip&longs;i <lb/>
abc &longs;imile; cuius grauitatis centrum e&longs;t K. <expan abbr="Quoniã">Quoniam</expan> enim <lb/>
duo plana æquidi&longs;tantia abc, fgh &longs;ecantur à plano abd; <lb/>
communes eorum &longs;ectiones ab, fg æquidi&longs;tantes erunt: & <lb/>
eadem ratione æquidi&longs;tantes ip&longs;æ bc, gh: & ca, hf. </s><s>Quòd <lb/>
cum duæ lineæ fg, gh, duabus ab, bc æquidi&longs;tent, nec <lb/>
&longs;int in eodem plano; angulus ad g æqualis e&longs;t angulo ad <lb/>
b. </s><s>& &longs;imiliter angulus ad h angulo ad c: <expan abbr="angulusq;">angulusque</expan> ad fci, <lb/>
qui ad a e&longs;t æqualis. </s><s>triangulum igitur fgh &longs;imile e&longs;t tri­<lb/>
angulo abc. <!-- KEEP S--></s><s>Atuero punctum k centrum e&longs;&longs;e grauita­<lb/>
tis trianguli fgh hoc modo o&longs;tendemus. </s><s>Ducantur pla­<lb/>
na per axem, & per lineas da, db, dc: erunt communes &longs;e­<lb/>
ctiones fK, ae æquidi&longs;tantes: <expan abbr="pariterq;">pariterque</expan> kg, eb; & kh, ec: <lb/>
quare angulus kfh angulo eac; & angulus kfg ip&longs;i eab <lb/>
<figure id="id.023.01.040.1.jpg" xlink:href="023/01/040/1.jpg"/><lb/>
e&longs;t æqualis. </s><s>Eadem ratione <lb/>
anguli ad g angulis ad b: & <lb/>
anguli ad h iis, qui ad c æ­<lb/>
quales erunt. </s><s>ergo puncta <lb/>
eK in triangulis abc, fgh <lb/>
&longs;imiliter &longs;unt po&longs;ita, per &longs;e­<lb/>
xtam po&longs;itionem Archime­<lb/>
dis in libro de centro graui­<lb/>
tatis planorum. </s><s>Sed cum e <lb/>
&longs;it centrum grauitatis trian<lb/>
guli abc, erit ex undecima <lb/>
propo&longs;itione eiu&longs;dem libri, <lb/>
& K trianguli fgh grauita<lb/>
tis centrum. </s><s>id quod demon&longs;trare oportebat. </s><s>Non aliter <lb/>
in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra­<lb/>
bitur.</s></p>
<pb xlink:href="023/01/041.jpg" pagenum="17"/><p type="head">
<s>PROBLEMA I. PROPOSITIO X.<!-- KEEP S--></s></p><p type="main">
<s>DATA qualibet pyramide, fieri pote&longs;t, ut fi­<lb/>
gura &longs;olida in ip&longs;a in &longs;cribatur, & altera <expan abbr="circũ&longs;cri-batur">circum&longs;cri­<lb/>
batur</expan> ex pri&longs;matibus æqualem altitudinem <expan abbr="ha-b&etilde;tibus">ha­<lb/>
bentibus</expan>, ita ut circum&longs;cripta in&longs;criptam excedat <lb/>
magnitudine, quæ minor &longs;it <expan abbr="quacũque">quacunque</expan> &longs;olida ma<lb/>
gnitudine propo&longs;ita.</s></p><figure id="id.023.01.041.1.jpg" xlink:href="023/01/041/1.jpg"/><p type="main">
<s>Sit pyramis, cuius ba&longs;is <lb/>
<expan abbr="triangulũ">triangulum</expan> abc; axis de. </s><lb/>
<s><expan abbr="Sitq;">Sitque</expan> pri&longs;ma, quod <expan abbr="eand&etilde;">eandem</expan> <lb/>
ba&longs;im habeat, & axem eun<lb/>
dem. </s><s>Itaque hoc pri&longs;ma­<lb/>
te continenter &longs;ecto bifa­<lb/>
riam, plano ba&longs;i <expan abbr="æquidi&longs;tã">æquidi&longs;tan</expan><lb/>
te, relinquetur <expan abbr="tãdem">tandem</expan> pri&longs;<lb/>
ma quoddam minus pro­<lb/>
po&longs;ita magnitudine: quod <lb/>
quidem ba&longs;im eandem ha<lb/>
beat, quam pyramis, & a­<lb/>
xem ef. </s><s>diuidatur de in <lb/>
partes æquales ip&longs;i ef in <lb/>
punctis ghklmn: & per <lb/>
diui&longs;iones plana <expan abbr="ducãtur">ducantur</expan>: <lb/>
quæ ba&longs;ibus æquidi&longs;tent, <lb/>
erunt &longs;ectiones, triangula <lb/>
ip&longs;i abc &longs;imilia, ut proxi­<lb/>
me o&longs;tendimus. </s><s>ab uno <lb/>
quoque <expan abbr="aut&etilde;">autem</expan> horum trian<lb/>
gulorum duo pri&longs;mata <expan abbr="cõ">con</expan><lb/>
&longs;truantur; unum quidem <lb/>
ad partes e; alterum ad
<pb xlink:href="023/01/042.jpg"/>partes d. <!-- KEEP S--></s><s>in pyramide igitur in&longs;cripta erit quædam figura, <lb/>
ex pri&longs;matibus æqualem altitudinem habentibus <expan abbr="cõ&longs;tans">con&longs;tans</expan>, <lb/>
ad partes e: & altera circum&longs;cripta ad partes d. <!-- KEEP S--></s><s>Sed unum­<lb/>
quodque eorum pri&longs;matum, quæ in figura in&longs;cripta conti­<lb/>
nentur, æquale e&longs;t pri&longs;mati, quod ab eodem fit triangulo in <lb/>
figura circum&longs;cripta: nam pri&longs;ma pq pri&longs;mati po e&longs;t æ­<lb/>
quale; pri&longs;ma st æquale pri&longs;mati sr; pri&longs;ma xy pri&longs;mati <lb/>
xu; pri&longs;ma <foreign lang="greek">hq</foreign> pri&longs;mati <foreign lang="greek">h</foreign>z; pri&longs;ma <foreign lang="greek">mn</foreign> pri&longs;mati <foreign lang="greek">ml;</foreign> pri&longs;­<lb/>
ma <foreign lang="greek">rs</foreign> pri&longs;mati <foreign lang="greek">rp;</foreign> & pri&longs;ma <foreign lang="greek">fx</foreign> pri&longs;mati <foreign lang="greek">ft</foreign> æquale. </s><s>re­<lb/>
linquitur ergo, ut circum&longs;cripta figura exuperet <expan abbr="in&longs;criptã">in&longs;criptam</expan> <lb/>
pri&longs;mate, quod ba&longs;im habet abc triangulum, & axem ef. </s><lb/>
<s>Illud uero minus e&longs;t &longs;olida magnitudine propo&longs;ita. </s><s><expan abbr="Ead&etilde;">Eadem</expan> <lb/>
ratione in&longs;cribetur, & circum&longs;cribetur &longs;olida figura in py­<lb/>
ramide, quæ quadrilateram, uel <expan abbr="plurilaterã">plurilateram</expan> ba&longs;im habeat.</s></p><p type="head">
<s>PROBLEMA II. PROPOSITIO XI.<!-- KEEP S--></s></p><p type="main">
<s>DATO cono, fieri pote&longs;t, ut figura &longs;olida in­<lb/>
&longs;cribatur, & altera circum&longs;cribatur ex cylindris <lb/>
æqualem habentibus altitudinem, ita ut circum­<lb/>
&longs;cripta &longs;uperet in&longs;criptam, magnitudine, quæ &longs;o­<lb/>
lida magnitudine propo&longs;ita &longs;it minor.</s></p><p type="main">
<s>SIT conus, cuius axis bd: & &longs;ecetur plano per axem <lb/>
ducto, 'ut&longs;ectio &longs;it triangulum abc: <expan abbr="intelligaturq;">intelligaturque</expan> cylin­<lb/>
drus, qui ba&longs;im eandem, & eundem axem habeat. </s><s>Hoc igi­<lb/>
tur cylindro continenter bifariam &longs;ecto, relinquetur cylin<lb/>
drus minor &longs;olida magnitudine propo&longs;ita. </s><s>Sit autem is cy<lb/>
lindrus, qui ba&longs;im habet circulum circa diametrum ac, & <lb/>
axem de. </s><s>Itaque diuidatur bd in partes æquales ip&longs;i de <lb/>
in punctis fghKlm: & per ea ducantur plana conum &longs;e­<lb/>
cantia; quæ ba&longs;i æquidi&longs;tent. </s><s>erunt &longs;ectiones circuli, cen­<lb/>
tra in axi habentes, ut in primo libro conicorum, propo&longs;i-
<pb xlink:href="023/01/043.jpg" pagenum="18"/>tione quarta Apollonius demon&longs;trauit. </s><s>Si igitur à &longs;ingu­<lb/>
lis horum circulorum, duo cylindri fiant; unus quidem ad <lb/>
ba&longs;is partes; alter ad partes uerticis: in&longs;cripta erit in co­<lb/>
no &longs;olida quædam figura, & altera circum&longs;cripta ex cylin­<lb/>
dris æqualem altitudinem habentibus con&longs;tans; quorum <lb/>
<figure id="id.023.01.043.1.jpg" xlink:href="023/01/043/1.jpg"/><lb/>
unu&longs;qui&longs;que, qui in <lb/>
figura in&longs;cripta con­<lb/>
tinetur æqualis e&longs;t ei, <lb/>
qui ab eodem fit cir­<lb/>
culo in figura <expan abbr="circũ-&longs;cripta">circum­<lb/>
&longs;cripta</expan>. </s><s>Itaque cylin<lb/>
drus op æqualis e&longs;t <lb/>
cylindro on; cylin­<lb/>
drus rs <expan abbr="cylĩdro">cylindro</expan> rq.</s> <lb/><s>
cylindrus ux cylin­<lb/>
dro ut e&longs;t æqualis; <lb/>
& alii aliis &longs;imiliter. </s><lb/>
<s>quare con&longs;tat <expan abbr="circũ-&longs;criptam">circum­<lb/>
&longs;criptam</expan> figuram &longs;u­<lb/>
perare in&longs;criptam cy<lb/>
lindro, cuius ba&longs;is e&longs;t <lb/>
circulus circa diametrum ac, & axis de. </s><s>atque hic e&longs;t mi­<lb/>
nor &longs;olida magnitudine propo&longs;ita.</s></p><p type="head">
<s>PROBLEMA III. PROPOSITIO XII.<!-- KEEP S--></s></p><p type="main">
<s>DATA coni portione, pote&longs;t &longs;olida quædam <lb/>
figura in&longs;cribi, & altera circum&longs;cribi ex cylindri <lb/>
portionibus æqualem altitudinem habentibus; <lb/>
ita ut circum&longs;cripta in&longs;criptam exuperet, magni <lb/>
tudine, quæ minor fit &longs;olida magnitudine pro­<lb/>
po&longs;ita.</s></p>
<pb xlink:href="023/01/044.jpg"/><p type="main">
<s>Figuram ciu&longs;modi, & in&longs;cribemus, & <expan abbr="circũ&longs;cribemus">circum&longs;cribemus</expan>, ita <lb/>
ut in cono dictum e&longs;t.</s></p><figure id="id.023.01.044.1.jpg" xlink:href="023/01/044/1.jpg"/><p type="head">
<s>PROBLEMA IIII. PROPOSITIO XIII.<!-- KEEP S--></s></p><p type="main">
<s>DATA &longs;phæræ portione, quæ dimidia &longs;phæ­<lb/>
ra maior non &longs;it, pote&longs;t &longs;olida quædam portio in­<lb/>
&longs;cribi & altera circum&longs;cribi ex cylindris æqualem <lb/>
altitudinem habentibus, ita ut circum&longs;cripta in­<lb/>
&longs;criptam excedat magnitudine, quæ &longs;olida ma­<lb/>
gnitudine propo&longs;ita &longs;it minor.</s></p><p type="main">
<s>HOC etiam codem pror&longs;us modo &longs;iet: atque ut ab <lb/>
Archimedc traditum e&longs;t in conoidum, & &longs;phæroidum por<lb/>
tionibus, propo&longs;itione uige&longs;imaprima libri de conoidi­<lb/>
bus, & &longs;phæroidibus.</s></p>
<pb xlink:href="023/01/045.jpg" pagenum="19"/><figure id="id.023.01.045.1.jpg" xlink:href="023/01/045/1.jpg"/><p type="head">
<s>THEOREMA X. PROPOSITIO XIIII.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet pyramidis, & cuiuslibet coni, uel <lb/>
coni portionis, centrum grauitatis in axe <expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>.</s></p><p type="main">
<s>SIT pyramis, cuius ba&longs;is triangulum abc: & axis de. </s><lb/>
<s>Dico in linea de ip&longs;ius grauitatis centrum ine&longs;&longs;e. </s><s>Si enim <lb/>
fieri pote&longs;t, &longs;it centrum f: & ab f ducatur ad ba&longs;im pyrami<lb/>
dis linea fg, axi æquidi&longs;tans: <expan abbr="iunctaq;">iunctaque</expan> eg ad latera trian­<lb/>
guli abc producatur in h. </s><s>quam uero proportionem ha­<lb/>
bet linea he ad eg, habeat pyramis ad aliud &longs;olidum, in <lb/>
quo K: <expan abbr="in&longs;cribaturq;">in&longs;cribaturque</expan> in pyramide &longs;olida figura, & altera cir<lb/>
cum&longs;cribatur ex pri&longs;matibus æqualem habentibus altitu­<lb/>
dinem, ita ut circum&longs;cripta in&longs;criptam exuperet magnitu­<lb/>
dine, quæ &longs;olido k &longs;it minor. </s><s>Et quoniam in pyramide pla<lb/>
num ba&longs;i æquidi&longs;tans ductum &longs;ectionem facit figuram &longs;i­<lb/>
milem ei, quæ e&longs;t ba&longs;is; <expan abbr="centrumq;">centrumque</expan> grauitatis in axe haben<lb/>
tem: erit pri&longs;matis st grauitatis <expan abbr="centrũ">centrum</expan> in linea rq; <lb/>
matis ux centrum in linea qp, pri&longs;matis yz in linea po; <lb/>
pri&longs;matis <foreign lang="greek">hq</foreign> in linea on; pri&longs;matis <foreign lang="greek">lm</foreign> in linea nm; pri&longs;­<lb/>
matis <foreign lang="greek">np</foreign> in ml; & denique pri&longs;matis <foreign lang="greek">rs</foreign> in le. </s><s>quare to­
<pb xlink:href="023/01/046.jpg"/>tius figuræ in&longs;criptæ centrum grauitatis e&longs;t in linea re: <lb/>
<figure id="id.023.01.046.1.jpg" xlink:href="023/01/046/1.jpg"/>quod &longs;it <foreign lang="greek">t</foreign>: <expan abbr="iũ">iun</expan>­<lb/>
ctaque <foreign lang="greek">t</foreign>f, & <lb/>
producta, à <lb/>
puncto h du­<lb/>
catur linea a­<lb/>
xi pyramidis <lb/>
æquidi&longs;tans, <lb/>
quæ <expan abbr="cũ">cum</expan> linea <lb/>
<foreign lang="greek">t</foreign>f conueniat <lb/>
in <foreign lang="greek">f</foreign>.</s><s>habebit <lb/>
<foreign lang="greek">ft</foreign> ad <foreign lang="greek">t</foreign>f ean­<lb/>
dem propor­<lb/>
tionem, <expan abbr="quã">quam</expan> <lb/>
he ad eg. <lb/>
</s></p><p>
<s>Quoniam igi<lb/>
tur exce&longs;&longs;us, <lb/>
quo <expan abbr="circũ">circum</expan>&longs;cri<lb/>
pta figura in­<lb/>
&longs;criptam &longs;upe<lb/>
rat, minor e&longs;t <lb/>
&longs;olido <foreign lang="greek">x</foreign>; py­<lb/>
ramis ad eun­<lb/>
<expan abbr="d&etilde;">dem</expan> <expan abbr="exce&longs;&longs;ũ">exce&longs;&longs;um</expan> ma<lb/>
ioré propor­<lb/>
tioné habet, <lb/>
quàm ad K &longs;o<lb/>
lidum: uideli<lb/>
cet maiorem, <lb/>
quàm linea h<lb/>
e ad eg; hoc <lb/>
e&longs;t quàm <foreign lang="greek">ft</foreign> <lb/>
ad <foreign lang="greek">t</foreign>f: & propterea multo maiorem habet ad partem ex­<lb/>
ce&longs;&longs;us, quæ intra pyrimidem comprehenditur. </s><s>Itaque ha­
<pb xlink:href="023/01/047.jpg" pagenum="20"/>beat eam, quam <foreign lang="greek">xt</foreign> ad <foreign lang="greek">t</foreign>f erit diuidendo ut <foreign lang="greek">x</foreign>f ad f<foreign lang="greek">t</foreign>, ita fi<lb/>
gura &longs;olida in&longs;cripta ad partem exce&longs;&longs;us, quæ e&longs;t intra pyra<lb/>
midem. </s><s>Cum ergo à pyramide, cuius grauitatis <expan abbr="ceũtrum">centrum</expan> e&longs;t <lb/>
punctum f, &longs;olida figura in&longs;cripta auferatur, cuius <expan abbr="centrũtrum">centrum</expan> <lb/>
<foreign lang="greek">t</foreign>: reliqua magnitudinis con&longs;tantis ex parte exce&longs;&longs;us, quæ <lb/>
e&longs;t intra pyramidem, centrum grauitatis erit in linea <foreign lang="greek">t</foreign>f <lb/>
producta, & in puncto <foreign lang="greek">x</foreign>. </s><s>quod fieri non pote&longs;t. </s><s>Sequitur <lb/>
igitur, ut centrum grauitatis pyramidis in linea de; hoc <lb/>
e&longs;t in eius axe con&longs;i&longs;tat.</s></p><p>
<s>Sit conus, uel coni portio, cuius axis bd: & &longs;ecetur plano <lb/>
per axem, ut &longs;ectio &longs;it triangulum abc. </s> <s>Dico centrum gra<lb/>
uitatis ip&longs;ius e&longs;&longs;e in linea bd. </s><s>Sit enim, &longs;i fieri pote&longs;t, <expan abbr="centrũ">centrum</expan> <lb/>
<figure id="id.023.01.047.1.jpg" xlink:href="023/01/047/1.jpg"/>
e: <expan abbr="perq;">perque</expan> e ducatur ef axi æquidi&longs;tans: & quam propor­<lb/>
tionem habet cd ad df, habeat conus, uel coni portio ad <lb/>
&longs;olidum g. </s><s>in&longs;cribatur ergo in cono, uel coni portione &longs;oli
<pb xlink:href="023/01/048.jpg"/>da figura, & altera circum&longs;cribatur ex cylindris, uel cylin­<lb/>
dri portionibus, &longs;icuti dictum e&longs;t, ita ut exce&longs;&longs;us, quo figu­<lb/>
ra circum&longs;cripta in&longs;criptam &longs;uperat, &longs;it &longs;olido g minor. </s><lb/>
<s>Itaque centrum grauitatis cylindri, uel cylindri portionit <lb/>
qr e&longs;t in linea po; cylindri, uel cylindri portionis st cen­<lb/>
trum in linea on; centrum ux in linea nm; yz in mb; <foreign lang="greek">nq</foreign><lb/>
in lk; <foreign lang="greek">l m</foreign> in kh; & denique <foreign lang="greek">v p</foreign> centrum in hd. <!-- KEEP S--></s><s>ergo figu­<lb/>
<figure id="id.023.01.048.1.jpg" xlink:href="023/01/048/1.jpg"/><lb/>
ræ in&longs;criptæ centrum e&longs;t in linea pd. <!-- KEEP S--></s><s>Sit autem <foreign lang="greek">r</foreign>: & iun­<lb/>
cta <foreign lang="greek">r</foreign> e protendatur, ut cum linea, quæ à <expan abbr="pũcto">puncto</expan> c ducta &longs;ue­<lb/>
rit axi æquidi&longs;tans, conueniat in <foreign lang="greek">s.</foreign> erit <foreign lang="greek">s r</foreign> ad <foreign lang="greek">r</foreign>e, ut cd <lb/>
ad df: & conus, &longs;eu coni portio ad exce&longs;&longs;um, quo circum­<lb/>
&longs;cripta figura in&longs;criptam &longs;uperat, habebit maiorem pro­<lb/>
portionem, quàm <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. </s><s>ergo ad partem exce&longs;&longs;us, quæ <lb/>
intra ip&longs;ius &longs;uperficiem comprehenditur, multo maiorem <lb/>
proportionem habebit. </s><s>habeat eam, quam <foreign lang="greek">t r</foreign> ad <foreign lang="greek">r</foreign> e. </s><s>erit
<pb xlink:href="023/01/049.jpg" pagenum="21"/>diuidendo figura &longs;olida in&longs;cripta ad dictam exce&longs;&longs;us par­<lb/>
tem, ut <foreign lang="greek">te</foreign> ad c<foreign lang="greek">p.</foreign> & quoniam à cono, &longs;cu coni portione, <lb/>
cuius grauitatis centrum e&longs;t e, aufertur figura in&longs;cripta, <lb/>
cuius centrum <foreign lang="greek">r:</foreign> re&longs;iduæ magnitudinis compo&longs;itæ cx par <lb/>
te exce&longs;&longs;us, quæ intra coni, uel coni portionis &longs;uperficiem <lb/>
continetur, centrum grauitatis erit in linea e protracta, <lb/>
atque in puncto t. </s><s>quod c&longs;t ab&longs;urdum. </s><s><expan abbr="cõ&longs;tat">con&longs;tat</expan> ergo <expan abbr="centrũ">centrum</expan> <lb/>
grauitatis coni, uel coni portionis, e&longs;&longs;e in axe bd: quod de <lb/>
mon&longs;trandum propo&longs;uimus.</s></p><p type="head">
<s>THEOREMA XI. PROPOSITIO XV.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet portionis &longs;phæræ uel &longs;phæroidis, <lb/>
quæ dimidia maior non &longs;it: <expan abbr="item&qacute;;">itemque</expan> cuiuslibet por <lb/>
tionis conoidis, uel ab&longs;ci&longs;&longs;æ plano ad axem recto, <lb/>
uel non recto, centrum grauitatis in axe con­<lb/>
&longs;i&longs;tit.</s></p><p type="main">
<s>Demon&longs;tratio &longs;imilis erit ei, quam &longs;upra in cono, uel co <lb/>
ni portione attulimus, ne toties eadem fru&longs;tra iterentur.</s></p><figure id="id.023.01.049.1.jpg" xlink:href="023/01/049/1.jpg"/>
<pb xlink:href="023/01/050.jpg"/><p type="head">
<s>THEOREMA XII. PROPOSITIO XVI.<!-- KEEP S--></s></p><p type="main">
<s>In &longs;phæra, & &longs;phæroidc idem e&longs;t grauitatis, & <lb/>
figuræ centrum.</s></p><p type="main">
<s>Secetur &longs;phæra, uel &longs;phæroides plano per axem ducto; <lb/>
quod &longs;ectionem faciat circulum, uel ellip&longs;im abcd, cuius <lb/>
diameter, & &longs;phæræ, uel &longs;phæroidis axis db; & centrum e. </s><lb/>
<s>Dico e grauitatis etiam centrum e&longs;&longs;e. </s><s>&longs;ecetur enim altero <lb/>
plano per e, ad planum &longs;ecans recto, cuius &longs;ectio &longs;it circu­<lb/>
lus circa diametrum ac. </s><s>erunt adc, abc dimidiæ portio­<lb/>
nes &longs;phæræ, uel &longs;phæroidis. </s><s>& quoniam portionis adc gra<lb/>
uitatis centrum e&longs;i in linea d, & centrum portionis abc in <lb/>
ip&longs;a be; totius &longs;phæræ, uel &longs;phæroidis grauitatis centrum <lb/>
in axe db con&longs;i&longs;iet, Quòd &longs;i portionis adc centrum graui <lb/>
tatis ponatur e&longs;&longs;e f & fiat ip&longs;i fe æqualis eg. <!-- REMOVE S--><expan abbr="punctũ">punctum</expan> g por <lb/>
<figure id="id.023.01.050.1.jpg" xlink:href="023/01/050/1.jpg"/><lb/>
<arrow.to.target n="marg48"/><lb/>
tionis abc centrum erit. </s>
<s>&longs;olidis enim figuris &longs;imilibus & <lb/>
æqualibus inter &longs;e aptatis, & centra grauitatis ip&longs;arum in­<lb/>
<arrow.to.target n="marg49"/><lb/>
ter se aptentur nece&longs;&longs;e e&longs;t. </s>
<s>ex quo fit, ut magnitudinis, quæ <lb/>
ex utilique <expan abbr="cõ&longs;lat">con&longs;tat</expan>, hoc e&longs;t ip&longs;ius &longs;phæræ, uel &longs;phæroidis gra<lb/>
uitatis centrum &longs;it in medio lineæ fg uidelicet in e. <!-- KEEP S--></s><s>Sphæ­<lb/>
ræ igitur, uel &longs;phæroidis grauitatis centrum e&longs;t idem, quod <lb/>
centrum figuræ.</s></p>
<pb xlink:href="023/01/051.jpg" pagenum="22"/><p type="margin">
<s><margin.target id="marg48"/>per 2. pe­<lb/>
titionem</s></p><p type="margin">
<s><margin.target id="marg49"/>4 Archi­<lb/>
medis.</s></p><p type="main">
<s>Ex demon&longs;tratis per&longs;picue apparet, portioni <lb/>
&longs;phæræ uel &longs;phæroidis, quæ dimidia maior e&longs;t, <expan abbr="c&etilde;">cen</expan> <lb/>
trum grauitatis in axe con&longs;i&longs;tere.</s></p><figure id="id.023.01.051.1.jpg" xlink:href="023/01/051/1.jpg"/><p type="main">
<s>Data enim <lb/>
qualibet maio<lb/>
ri <expan abbr="portiõe">portione</expan>, quo <lb/>
<expan abbr="niã">niam</expan> totius &longs;phæ <lb/>
ræ, uel &longs;phæroi <lb/>
dis grauitatis <lb/>
centrum e&longs;t in <lb/>
axe; e&longs;t autem <lb/>
& in axe cen­<lb/>
trum portio­<lb/>
nis minoris: <lb/>
reliquæ portionis uidelicet maioris centrum in axe nece&longs;­<lb/>
&longs;ario con&longs;i&longs;tet.</s></p><p type="head">
<s>THEOREMA XIII. PROPOSITIO XVII.<!-- KEEP S--></s></p><figure id="id.023.01.051.2.jpg" xlink:href="023/01/051/2.jpg"/><p type="main">
<s>Cuiuslibet pyramidis <expan abbr="triã">trian</expan> <lb/>
gularem ba&longs;im <expan abbr="hab&etilde;tis">habentis</expan> gra<lb/>
uitatis centrum e&longs;t in pun­<lb/>
cto, in quo ip&longs;ius axes con­<lb/>
ueniunt.</s></p><p type="main">
<s>Sit pyramis, cuius ba&longs;is trian <lb/>
gulum abc, axis de: <expan abbr="&longs;itq;">&longs;itque</expan> trian <lb/>
guli bdc grauitatis centrum f: <lb/>
& iungatur a &longs;. </s><s>erit & af axis eiu&longs; <lb/>
dem pyramidis ex tertia diffini­<lb/>
tione huius. </s><s>Itaque quoniam centrum grauitatis e&longs;t in <lb/>
axe de; e&longs;t autem & in axe af; &qgrave;uod proxime demon&longs;traui
<pb xlink:href="023/01/052.jpg"/>mus: erit utique grauitatis centrum pyramidis punctum <lb/>
g. <!-- REMOVE S-->in quo &longs;cilicet ip&longs;i axes conueniunt.</s>
</p><p type="head">
<s>THEOREMA XIIII. PROPOSITIO XVIII.<!-- KEEP S--></s></p><p type="main">
<s>SI &longs;olidum parallelepipedum &longs;ecetur plano <lb/>
ba&longs;ibus æquidi&longs;tante; erit &longs;olidum ad &longs;olidum, <lb/>
&longs;icut altitudo ad altitudinem, uel &longs;icut axis ad <lb/>
axem.</s></p><figure id="id.023.01.052.1.jpg" xlink:href="023/01/052/1.jpg"/><p type="main">
<s>Sit &longs;olidum parallelepipe <lb/>
dum abcdefgh, cuius axis <lb/>
kl: <expan abbr="&longs;eceturq;">&longs;eceturque</expan> plano ba&longs;ibus <lb/>
æquidi&longs;tante, quod faciat <lb/>
&longs;ectionem mnop; & axi in <lb/>
puncto q occurrat. </s><s>Dico <lb/>
&longs;olidum gm ad &longs;olidum mc <lb/>
eam proportionem habere, <lb/>
quam altitudo &longs;olidi gm ha­<lb/>
bet ad &longs;olidi mc altitudi­<lb/>
nem; uel quam axis kq ad <lb/>
axem ql. <!-- KEEP S--></s><s>Si enim axis Kl ad <lb/>
ba&longs;is planum &longs;it perpendicu <lb/>
<figure id="id.023.01.052.2.jpg" xlink:href="023/01/052/2.jpg"/><lb/>
laris, & linea gc, quæ ex quin <lb/>
ta huius ip&longs;i kl æquidi&longs;tat, <lb/>
perpendicularis erit ad <expan abbr="id&etilde;">idem</expan> <lb/>
planum, & &longs;olidi altitudi­<lb/>
<arrow.to.target n="marg50"/><lb/>
nem dimetietur. </s><s>Itaque &longs;o­<lb/>
lidum gm ad &longs;olidum mc <lb/>
eam proportionem habet, <lb/>
quam parallelogramm<expan abbr="ũ">um</expan> gn <lb/>
ad parallelogrammum nc, <lb/>
<arrow.to.target n="marg51"/><lb/>
hoc e&longs;t quam linea go, quæ
<pb xlink:href="023/01/053.jpg" pagenum="23"/>e&longs;t &longs;olidi gm altitudo ad oe altitudinem &longs;olidi mc, uel <expan abbr="quã">quam</expan> <lb/>
axis kq ad ql axem. </s><s>Si uero axis kl non &longs;it perpendicularis <lb/>
ad planum ba&longs;is; ducatur a puncto k ad idem planum per <lb/>
pendicularis kr, <expan abbr="occurr&etilde;s">occurrens</expan> plano mnop in s. </s><s>&longs;imiliter <expan abbr="de-mõ&longs;trabimus">de­<lb/>
mon&longs;trabimus</expan> &longs;olidum gm ad <expan abbr="&longs;olidũ">&longs;olidum</expan> mc ita e&longs;&longs;e, ut axis kq <lb/>
ad axem ql. <!-- KEEP S--></s><s>Sed ut Kq ad ql, ita ks altitudo ad altitudi­<lb/>
<arrow.to.target n="marg52"/><lb/>
nem sr; nam lineæ Kl, Kr à planis æquidi&longs;tantibus in ea&longs;­<lb/>
dem proportiones &longs;ecantur. </s><s>ergo &longs;olidum gm ad &longs;olidum <lb/>
mc <expan abbr="eand&etilde;">eandem</expan> proportionem habet, quam altitudo ad <expan abbr="altitudin&etilde;">altitu<lb/>
dinem</expan>, uel quam axis ad axem. </s><s>quod <expan abbr="demõ&longs;trare">demon&longs;trare</expan> oportebat.</s></p><p type="margin">
<s><margin.target id="marg50"/>25 undeci<lb/>
mi.</s></p><p type="margin">
<s><margin.target id="marg51"/><expan abbr="&longs;extĩ">&longs;extim</expan>.</s></p><p type="margin">
<s><margin.target id="marg52"/>17. unde­<lb/>
cimi</s></p><p type="head">
<s>THEOREMA XV. PROPOSITIO XIX.<!-- KEEP S--></s></p><p type="main">
<s>Solida parallelepipeda in eadem ba&longs;i, uel in <lb/>
æqualibus ba&longs;ibus con&longs;tituta eam inter &longs;e propor<lb/>
tionem habent, quam altitudines: & &longs;i axes ip&longs;o­<lb/>
rum cum ba&longs;ibus æquales angulos contineant, <lb/>
eam quoque, quam axes proportionem <expan abbr="habebũt">habebunt</expan>.</s></p><p type="main">
<s>Sint &longs;olida parallelepipeda in <expan abbr="ead&etilde;">eadem</expan> ba&longs;i <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> abcd, <lb/>
abef: & &longs;it &longs;olidi abcd altitudo minor: producatur au­<lb/>
tem planum cd adeo, ut &longs;olidum abef &longs;ecet; cuius &longs;ectio <lb/>
<figure id="id.023.01.053.1.jpg" xlink:href="023/01/053/1.jpg"/><lb/>
<arrow.to.target n="marg53"/><lb/>
&longs;it gh. </s><s><expan abbr="erũt">erunt</expan> &longs;oli <lb/>
da abcd, abgh <lb/>
in eadem ba&longs;i, <lb/>
& æquali altitu<lb/>
dine inter &longs;e æ­<lb/>
qualia. </s><s><expan abbr="Quoniã">Quoniam</expan> <lb/>
igitur &longs;olidum <lb/>
abef &longs;ecatur <lb/>
plano ba&longs;ibus <lb/>
<expan abbr="æquidi&longs;tãte">æquidi&longs;tante</expan>, erit <lb/>
<arrow.to.target n="marg54"/><lb/>
&longs;olidum ghef <lb/>
adip&longs;um abgh
<pb xlink:href="023/01/054.jpg"/>ut altitudo ad altitudinem: & componendo conuertendo <lb/>
<arrow.to.target n="marg55"/><lb/>
que &longs;olidum abgh, hoc e&longs;t &longs;olidum abcd ip&longs;i æquale, ad­<lb/>
&longs;olidum abef, ut altitudo &longs;olidi abcd ad &longs;olidi abef al­<lb/>
titudinem.</s></p><p type="margin">
<s><margin.target id="marg53"/>29. unde­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg54"/>18. huius</s></p><p type="margin">
<s><margin.target id="marg55"/>7. quinti.</s></p><p type="main">
<s>Sint &longs;olida parallelopipeda ab, cd in æqualibus ba&longs;ibus <lb/>
con&longs;tituta: <expan abbr="&longs;itq;">&longs;itque</expan> be altitudo &longs;olidi ab: & &longs;olidi cd altitudo <lb/>
d f; quæ quidem maior &longs;it, quàm be. </s><s>Dico &longs;olidum ab ad <lb/>
&longs;olidum cd eandem habere proportionem, quam be ad <lb/>
d f. </s><s>ab&longs;cindatur enim à linea df æqualis ip&longs;i be, quæ &longs;it gf: <lb/>
& per g ducatur planum &longs;ecans &longs;olidum cd; quod ba&longs;ibus <lb/>
æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> hK. </s><s>erunt &longs;olida ab, ck æque <lb/>
<arrow.to.target n="marg56"/><lb/>
<figure id="id.023.01.054.1.jpg" xlink:href="023/01/054/1.jpg"/><lb/>
alta inter <lb/>
&longs;e æqualia <lb/>
<expan abbr="cũ">cum</expan> æqua­<lb/>
les ba&longs;es <lb/>
habeant. </s><lb/>
<s><arrow.to.target n="marg57"/><lb/>
Sed <expan abbr="&longs;olidũ">&longs;olidum</expan> <lb/>
hd ad &longs;oli <lb/>
dum cK <lb/>
e&longs;t, ut alti <lb/>
tudo dg <lb/>
ad gf <expan abbr="alti­tudin&etilde;">alti­<lb/>
tudinem</expan>; &longs;e <lb/>
catur enim &longs;olidum cd plano ba&longs;i <lb/>
<figure id="id.023.01.054.2.jpg" xlink:href="023/01/054/2.jpg"/><lb/>
bus æquidi&longs;tante: & rur&longs;us <expan abbr="cõpo-nende">compo­<lb/>
nende</expan>, <expan abbr="conuertendoq;">conuertendoque</expan> <expan abbr="&longs;olidũ">&longs;olidum</expan> ck <lb/>
<arrow.to.target n="marg58"/><lb/>
ad &longs;olidum cd, ut gf ad fd. <!-- KEEP S--></s><s>ergo <lb/>
&longs;olidum ab, quod e&longs;t æquale ip&longs;i <lb/>
ck ad &longs;olidum cd eam proportio <lb/>
nem habet, quam altitudo gf, hoc <lb/>
e&longs;t be ad df altitudinem.</s></p><p type="margin">
<s><margin.target id="marg56"/>31. unde <lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg57"/>18. huius</s></p><p type="margin">
<s><margin.target id="marg58"/>7. quinti.</s></p><p type="main">
<s>Sint deinde &longs;olida parallelepipe <lb/>
da ab, ac in eadem ba&longs;i; quorum <lb/>
axes de, &longs; e cum ip&longs;a æquales angu
<pb xlink:href="023/01/055.jpg" pagenum="24"/>los contineant. </s><s>Dico &longs;olidum ab ad &longs;olidum ace idem ha <lb/>
bere proportionem, quam axis de ad axem ef. </s><s>Si enim <lb/>
axes in eadem recta linea fuerint con&longs;tituti, hæc duo &longs;oli­<lb/>
da, in unum, atque idem &longs;olidum conuenient. </s><s>quare ex <lb/>
iis, quæ proxime tradita &longs;unt, habebit &longs;olidum ab ad &longs;o­<lb/>
lidum ac eandem proportionem, quam axis de ad ef <lb/>
axem. </s><s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>
tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, dg, fh: <lb/>
& jungantur eg, eh. </s><s>Quoniam igitur axes cum ba&longs;ibus <lb/>
æquales angulos continent, erit deg angulus æqualis an­<lb/>
<figure id="id.023.01.055.1.jpg" xlink:href="023/01/055/1.jpg"/><lb/>
gulo feh: & &longs;unt <lb/>
anguli ad gh re­<lb/>
cti, quare & re­<lb/>
liquus edg æqua <lb/>
lis erit reliquo <lb/>
efh: & triangu­<lb/>
lum deg <expan abbr="triãgu-lo">triangu­<lb/>
lo</expan> feh &longs;imile. </s><s>er­<lb/>
go gd ad de e&longs;t', <lb/>
ut hf ad e: & per <lb/>
mutando gd ad <lb/>
hf, ut de ad cf. </s><lb/>
<figure id="id.023.01.055.2.jpg" xlink:href="023/01/055/2.jpg"/><lb/>
<s>Sed &longs;olidum ab <lb/>
ad &longs;olidum ac <lb/>
eandem propor­<lb/>
tionem habet, <lb/>
quam dg altitu­<lb/>
do ad <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>
fh. </s><s>ergo & <expan abbr="ean-d&etilde;">ean­<lb/>
dem</expan> habebit, <expan abbr="quã">quam</expan> <lb/>
axis de ad ef <expan abbr="ax&etilde;">axem</expan></s></p><p type="main">
<s>Po&longs;tremo &longs;int <lb/>
&longs;olidi paral le pi <lb/>
peda ab, cd in
<pb xlink:href="023/01/056.jpg"/>æqualibus ba&longs;ibus, quorum axes cum ba&longs;ibus æquales an <lb/>
gulos faciant. </s><s>Dico &longs;olidum ab ad <expan abbr="&longs;olidũ">&longs;olidum</expan> cd ita e&longs;&longs;e, ut axis <lb/>
ef ad axem gh: nam &longs;i axes ad planum ba&longs;is recti &longs;int, il­<lb/>
lud per&longs;picue con&longs;tat: quoniam eadem linea, & axem & &longs;oli <lb/>
di altitudinem determinabit. </s><s>Si uero &longs;int inclinati, à pun­<lb/>
ctis eg ad &longs;ubiectum planum perpendiculares ducantur <lb/>
ek, gl: & iungantur fk, hl. <!-- KEEP S--></s><s>rur&longs;us quoniam axes cum ba <lb/>
&longs;ibus æquales faciunt angulos, eodem modo demon&longs;trabi <lb/>
tur, triangulum efK triangulo ghl &longs;imile e&longs;&longs;e: & ek ad gl, <lb/>
ut ef ad gh. </s><s>Solidum autem ab ad &longs;olidum cd e&longs;t, ut <lb/>
eK ad gl. <!-- KEEP S--></s><s>ergo & ut axis ef ad axem gh. </s><s>quæ omnia de <lb/>
mon&longs;trare oportebat.</s></p><p type="main">
<s>Ex iis quæ demon&longs;trata &longs;unt, facile con&longs;tare <lb/>
pote&longs;t, pri&longs;mata omnia & pyramides, quæ trian­<lb/>
gulares ba&longs;es habent, &longs;iue in ei&longs;dem, &longs;iue in æqua <lb/>
<arrow.to.target n="marg59"/><lb/>
libus ba&longs;ibus con&longs;tituantur, eandem proportio­<lb/>
nem habere, quam altitudines: & &longs;i axes cum ba <lb/>
&longs;ibus æquales angulos contineant, &longs;imiliter ean­<lb/>
dem, quam axes, habere proportionem: &longs;unt <lb/>
<arrow.to.target n="marg60"/><lb/>
enim &longs;olida parallelepipeda pri&longs;matum triangula <lb/>
<arrow.to.target n="marg61"/><lb/>
res ba&longs;es <expan abbr="habentiũ">habentium</expan> dupla; & pyramidum &longs;extupla.</s></p><p type="margin">
<s><margin.target id="marg59"/>15. quinti</s></p><p type="margin">
<s><margin.target id="marg60"/>28. unde­<lb/>
cimi.</s></p><p type="margin">
<s><margin.target id="marg61"/>7. duode­<lb/>
cimi.</s></p><p type="head">
<s>THEOREMA XVI. PROPOSITIO XX.<!-- KEEP S--></s></p><p type="main">
<s>Pri&longs;mata omnia & pyramides, quæ in ei&longs;dem, <lb/>
uel æqualibus ba&longs;ibus con&longs;tituuntur, eam inter <lb/>
&longs;e proportionem habent, quam altitudines: & &longs;i <lb/>
axes cum ba&longs;ibus faciant angulos æquales, eam <lb/>
etiam, quam axes habent proportionem.</s></p>
<pb xlink:href="023/01/057.jpg" pagenum="25"/><p type="main">
<s>Sint duo pri&longs;mata ae, af, quorum eadem ba&longs;is quadri­<lb/>
latera abcd: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae altitudo eg; & pri&longs;matis <lb/>
af altitudo fh. </s><s>Dico pri&longs;ma ae ad pri&longs;ma af eam habere <lb/>
proportionem, quam eg ad fh. </s><s>iungatur enim ac: & in <lb/>
unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum <lb/>
<figure id="id.023.01.057.1.jpg" xlink:href="023/01/057/1.jpg"/><lb/>
ba&longs;es &longs;int triangu <lb/>
la abc, acd. <!-- KEEP S--></s><s>habe <lb/>
bunt duo pri&longs;ma­<lb/>
te in eadem ba&longs;i <lb/>
abc con&longs;tituta, <lb/>
proportionem <expan abbr="eã">eam</expan> <lb/>
dem, quam ip&longs;o­<lb/>
rum altitudines e <lb/>
g, fh, ex iam de­<lb/>
mon&longs;tratis. </s><s>& &longs;i­<lb/>
militer alia duo, <lb/>
quæ &longs;unt in ba&longs;i a <lb/>
<arrow.to.target n="marg62"/><lb/>
c d. <!-- KEEP S--></s><s>quare totum pri&longs;ma ae ad pri&longs;ma af eandem propor<lb/>
tionem habebit, quam altitudo eg ad fh altitudinem. </s><lb/>
<s>Quòd cum pri&longs;mata &longs;int pyramidum tripla, & ip&longs;æ pyrami <lb/>
des, quarum eadem e&longs;t ba&longs;is quadrilatera, & altitudo pri&longs;­<lb/>
matum altitudini æqualis, eam inter &longs;e proportionem ha­<lb/>
bebunt, quam altitudines.</s></p><p type="margin">
<s><margin.target id="marg62"/>12. quinti</s></p><p type="main">
<s>Si uero pri&longs;mata ba&longs;es æquales habeant, <expan abbr="nõ">non</expan> ea&longs;dem, &longs;int <lb/>
duo eiu&longs;modi pri&longs;mata ae, fl: & &longs;it ba&longs;is pri&longs;matis ae qua <lb/>
drilaterum abcd; & pri&longs;matis fl quadrilaterum fghk. </s><lb/>
<s>Dico pri&longs;ma ae ad pri&longs;ma fl ita e&longs;&longs;e, ut altitudo illius ad <lb/>
huius altitudinem. </s><s>nam &longs;i altitudo &longs;it eadem, <expan abbr="intelligãtur">intelligantur</expan> <lb/>
<arrow.to.target n="marg63"/><lb/>
duæ pyramides abcde, fghkl. <!-- KEEP S--></s><s>quæ <expan abbr="ĩtcr&longs;e">intcr&longs;e</expan> æquales <expan abbr="erũt">erunt</expan>, <lb/>
cum æquales ba&longs;es, & altitudinem eandem habeant. </s><s>quare <lb/>
<arrow.to.target n="marg64"/><lb/>
& pri&longs;mata ae, fl, quæ &longs;unt <expan abbr="harũ">harum</expan> pyramidum tripla, æqua­<lb/>
lia &longs;int nece&longs;&longs;e e&longs;t. </s><s>ex quibus per&longs;picue con&longs;tat <expan abbr="propo&longs;itũ">propo&longs;itum</expan>. </s><lb/>
<s>Si uero altitudo pri&longs;matis fl &longs;it maior, à pri&longs;mate fl ab­<lb/>
&longs;cindatur pri&longs;ma fm, quod æque altum &longs;it, <expan abbr="atq;">atque</expan> ip&longs;um ae.
<pb xlink:href="023/01/058.jpg"/><figure id="id.023.01.058.1.jpg" xlink:href="023/01/058/1.jpg"/><lb/>
erunt eædem ra­<lb/>
tione pri&longs;mata a <lb/>
e, fm inter &longs;e æ­<lb/>
qualia. </s><s>quare &longs;i­<lb/>
militer demon­<lb/>
&longs;trabitur pri&longs;ma <lb/>
fm ad pri&longs;ma fl <lb/>
eandem habere <lb/>
proportionem, <lb/>
quam pri&longs;matis <lb/>
fm altitudo ad <lb/>
altitudinem ip­<lb/>
&longs;ius fl. <!-- KEEP S--></s><s>ergo & pri&longs;ma ae ad pri&longs;ma fl eandem propor­<lb/>
tionem habebit, quam altitudo ad altitudinem. </s><s>&longs;equitur <lb/>
igitur ut & pyramides, quæ in æqualibus ba&longs;ibus <expan abbr="con&longs;tituũ">con&longs;tituun</expan> <lb/>
tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>
habeant.</s></p><p type="margin">
<s><margin.target id="marg63"/>6. duode<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg64"/>25. quinti</s></p><figure id="id.023.01.058.2.jpg" xlink:href="023/01/058/2.jpg"/><p type="main">
<s>Sint deinde pri&longs;mata ae, af in eadem ba&longs;i abcd; <expan abbr="quorũ">quorum</expan> <lb/>
axes cum ba&longs;ibus æquales angulos contineant: & &longs;it pri&longs;­
<pb xlink:href="023/01/059.jpg" pagenum="26"/>matis ae axis gh; & pri&longs;matis af axis lh. </s><s>Dico pri&longs;ma <lb/>
ae ad pri&longs;ma af eam proportionem habere, quam gh ad <lb/>
h l. <!-- REMOVE S-->ducantur à punctis gl perpendiculares ad ba&longs;is pla­<lb/>
<figure id="id.023.01.059.1.jpg" xlink:href="023/01/059/1.jpg"/><lb/>
num gK, lm: & iungantur kh, <lb/>
h m. </s>
<s>Itaque quoniam anguli gh <lb/>
k, lhm &longs;unt æquales, &longs;imiliter ut <lb/>
&longs;upra demon&longs;trabimus, triangu­<lb/>
la ghK, lhm &longs;imilia e&longs;&longs;e; & ut g <lb/>
K ad lm, ita gh ad hl. <!-- REMOVE S-->habet au <lb/>
tem pri&longs;ma ae ad pri&longs;ma af ean <lb/>
dem proportionem, quam altitu<lb/>
do gK ad altitudinem lm, &longs;icuti <lb/>
demon&longs;tratum e&longs;t. </s>
<s>ergo & ean­<lb/>
dem habebit, quam gh, ad hl. <!-- REMOVE S-->py <lb/>
ramis igitur abcdg ad pyrami­<lb/>
dem abcdl eandem proportio­<lb/>
nem habebit, quam axis gh ad hl axem.</s>
</p><figure id="id.023.01.059.2.jpg" xlink:href="023/01/059/2.jpg"/><p type="main">
<s>Denique &longs;int pri&longs;mata ae, ko in æqualibus ba&longs;ibus ab <lb/>
cd, klmn con&longs;tituta; quorum axes cum ba&longs;ibus æquales <lb/>
faciant angulos: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae axis fg, & altitudo fh: <lb/>
pri&longs;matis autem ko axis pq, & altitudo pr. </s><s>Dico pri&longs;ma <lb/>
ae ad pri&longs;ma ko ita e&longs;&longs;e, ut fg ad pq. </s><s>iunctis enim gh,
<pb xlink:href="023/01/060.jpg"/>qr, eodem, quo &longs;upra, modo o&longs;tendemns fg ad pq, ut fh <lb/>
ad pr. </s><s>&longs;ed pri&longs;ma ae ad ip&longs;um ko e&longs;t, ut fh ad pr. </s><s>ergo <lb/>
& ut fg axis ad axem pq.</s><s> ex quibus &longs;it, ut pyramis abcdf <lb/>
<figure id="id.023.01.060.1.jpg" xlink:href="023/01/060/1.jpg"/><lb/>
ad <expan abbr="pyrami-d&etilde;">pyrami­<lb/>
dem</expan> klmnp <lb/>
eandem ha <lb/>
beat pro ­<lb/>
portion&etilde;, <lb/>
<expan abbr="quã">quam</expan> axis ad <lb/>
<expan abbr="ax&etilde;">axem</expan>. </s><s>quod <lb/>
<expan abbr="demon&longs;trã">demon&longs;tran</expan> <lb/>
<expan abbr="dũ">dum</expan> &longs;uerat.</s></p><p type="main">
<s>Simili ra<lb/>
tione in a­<lb/>
liis pri&longs;ma­<lb/>
tibus & py <lb/>
ramidibus eadem demon&longs;trabuntur.</s></p><p type="head">
<s>THEOREMA XVII. PROPOSITIO XXI.<!-- KEEP S--></s></p><p type="main">
<s>Pri&longs;mata omnia, & pyramides inter &longs;e propor<lb/>
tionem habent compo&longs;itam ex proportione ba­<lb/>
&longs;ium, & proportione altitudinum.</s></p><p type="main">
<s>Sint duo pri&longs;mata ae, gm: <expan abbr="&longs;itq;">&longs;itque</expan> pri&longs;matis ae ba&longs;is qua <lb/>
drilaterum abcd, & altitudo ef: pri&longs;matis uero gm ba­<lb/>
&longs;is quadrilaterum ghKl, & altitudo mn. </s><s>Dico pri&longs;ma ae <lb/>
ad pri&longs;ma gm proportionem habere compo&longs;itam ex pro <lb/>
portione ba&longs;is abcd ad ba&longs;im ghkl, & ex proportione <lb/>
altitudinis ef, ad altitudinem mn.</s></p><p type="main">
<s>Sint enim primum ef, mn æquales: & ut ba&longs;is abcd <lb/>
ad ba&longs;im ghkl, ita fiat linea, in qua o ad lineam, in qua p: <lb/>
ut autem ef ad mn, ita linea p ad lineam q.</s><s> erunt lineæ <lb/>
pq inter &longs;e æquales. </s><s>Itaque pri&longs;ma ae ad pri&longs;ma gm <expan abbr="eã">eam</expan>
<pb xlink:href="023/01/061.jpg" pagenum="27"/>proportionem habet, quam ba&longs;is abcd ad ba&longs;im ghkl: <lb/>
&longs;i enim intelligantur duæ pyramides abcde, ghklm, ha­<lb/>
bebunt hæ inter &longs;e proportionem eandem, quam ip&longs;ar um <lb/>
ba&longs;es ex &longs;exta duodecimi elementorum. </s><s>Sed ut ba&longs;is abcd <lb/>
ad ghKl ba&longs;im, ita linea o ad lineam p; hoc e&longs;t ad lineam q <lb/>
ei æqualem. </s><s>ergo pri&longs;ma ae ad pri&longs;ma gm e&longs;t, ut linea o <lb/>
ad lineam q.</s><s> proportio autem o ad q copo&longs;ita e&longs;t ex pro­<lb/>
portione o ad p, & ex proportione p ad q.</s><s> quare pri&longs;ma <lb/>
ae ad pri&longs;ma gm, & idcirco pyramis abcde, ad pyrami­<lb/>
dem ghKlm proportionem habet ex ei&longs;dem proportio­<lb/>
nibus compo&longs;itam, uidelicet ex proportione ba&longs;is abcd <lb/>
ad ba&longs;im ghKl, & ex proportione altitudinis ef ad mn al <lb/>
titudinem. </s><s>Quòd &longs;i lineæ ef, mn inæquales ponantur, &longs;it <lb/>
ef minor: & ut ef ad mn, ita fiat linea p ad lineam u: de <lb/>
<figure id="id.023.01.061.1.jpg" xlink:href="023/01/061/1.jpg"/><lb/>
inde ab ip&longs;a mn ab&longs;cindatur rn æqualis ef: & per r duca­<lb/>
tur planum, quod oppo&longs;itis planis æquidi&longs;tans faciat &longs;e­<lb/>
ctionem st. </s><s>erit pri&longs;ma ae, ad pri&longs;ma gt, ut ba&longs;is abcd <lb/>
ad ba&longs;im ghkl; hoc e&longs;t ut o ad p: ut autem pri&longs;ma gt ad <lb/>
<arrow.to.target n="marg65"/><lb/>
pri&longs;ma gm, ita altitudo rn; hoc e&longs;t ef ad altitudine mn; <lb/>
uidelicet linea p ad lineam u. </s><s>ergo ex æquali pri&longs;ma ae ad <lb/>
pri&longs;ma gm e&longs;t, ut linea o ad ip&longs;am u. </s><s>Sed proportio o ad <lb/>
u <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> e&longs;t ex proportione o ad p, quæ e&longs;t ba&longs;is abcd <lb/>
ad ba&longs;im ghkl; & ex proportione p ad u, quæ e&longs;t altitudi­<lb/>
nis ef ad altitudinem mn. </s><s>pri&longs;ma igitur ae ad pri&longs;ma gm
<pb xlink:href="023/01/062.jpg"/>compo&longs;itam proportionem habet ex proportione <expan abbr="ba&longs;iũ">ba&longs;ium</expan>, <lb/>
& proportione altitudinum. </s><s>Quare & pyramis, cuius ba­<lb/>
&longs;is e&longs;t quadrilaterum abcd, & altitudo ef ad pyramidem, <lb/>
<figure id="id.023.01.062.1.jpg" xlink:href="023/01/062/1.jpg"/><lb/>
cuius ba&longs;is quadrilaterum ghKl, & altitudo mn, compo&longs;i <lb/>
tam habet proportionem ex proportione ba&longs;ium abcd, <lb/>
ghkl, & ex proportione altitudinum ef, mn. </s><s>quod qui­<lb/>
dem demon&longs;tra&longs;&longs;e oportebat.</s></p><p type="margin">
<s><margin.target id="marg65"/>20. huius</s></p><p type="main">
<s>Ex iam demon&longs;tratis per&longs;picuum e&longs;t, pri&longs;ma <lb/>
ta omnia, & pyramides, in quibus axes cum ba&longs;i­<lb/>
bus æquales angulos continent, proportionem <lb/>
habere compo&longs;itam ex ba&longs;ium proportione, & <lb/>
proportione axium. </s><s>demon&longs;ttatum e&longs;t enim, a­<lb/>
xes inter &longs;e eandem proportionem habere, quam <lb/>
ip&longs;æ altitudines.</s></p><p type="head">
<s>THEOREMA XVIII. PROPOSITIO XXII.<!-- KEEP S--></s></p><p type="main">
<s>CVIVSLIBEt pyramidis, & cuiuslibet coni,
<pb xlink:href="023/01/063.jpg" pagenum="28"/>uel coni portionis axis à centro grauitatis ita diui <lb/>
ditur, ut pars, quæ terminatur ad uerticem reli­<lb/>
quæ partis, quæ ad ba&longs;im, &longs;it tripla.</s></p><p type="main">
<s>Sit pyramis, cuius ba&longs;is triangulum abc; axis de; & gra<lb/>
uitatis centrum K. <!-- KEEP S--></s><s>Dico lineam dk ip&longs;ius Ke triplam e&longs;&longs;e. </s><lb/>
<s>trianguli enim bdc centrum grauitatis &longs;it punctum f; <expan abbr="triã">triam</expan> <lb/>
guli adc <expan abbr="centrũ">centrum</expan> g; & trianguli adb &longs;it h: & iungantur af, <lb/>
b g, ch. </s><s>Quoniam igitur <expan abbr="centrũ">centrum</expan> grauitatis pyramidis in axe <lb/>
<arrow.to.target n="marg66"/><lb/>
<expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>: <expan abbr="&longs;untq;">&longs;untque</expan> de, af, bg, ch <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> pyramidis axes: conue <lb/>
nient omnes in <expan abbr="id&etilde;">idem</expan> <expan abbr="punctũ">punctum</expan> k, quod e&longs;t grauitatis centrum. </s><lb/>
<s>Itaque animo concipiamus hanc pyramidem diui&longs;am in <lb/>
quatuor pyramides, quarum ba&longs;es &longs;int ip&longs;a pyramidis <lb/>
<arrow.to.target n="marg67"/><lb/>
<figure id="id.023.01.063.1.jpg" xlink:href="023/01/063/1.jpg"/><lb/>
triangula; & <emph type="ul"/>axis<emph.end type="ul"/> pun­<lb/>
ctum k quæ quidem py­<lb/>
ramides inter &longs;e æquales <lb/>
&longs;unt, ut <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan>. </s><lb/>
<s>Ducatur <expan abbr="enĩ">enim</expan> per lineas <lb/>
dc, de planum <expan abbr="&longs;ecãs">&longs;ecans</expan>, ut <lb/>
&longs;it ip&longs;ius, & ba&longs;is abc <expan abbr="cõ">com</expan> <lb/>
munis &longs;ectio recta linea <lb/>
cel: <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> uero & <expan abbr="triã-guli">trian­<lb/>
guli</expan> adb &longs;it linea dhl. <!-- REMOVE S-->erit linea al æqualis ip&longs;i <lb/>
lb: nam centrum graui­<lb/>
tatis trianguli con&longs;i&longs;tit <lb/>
in linea, quæ ab angulo <lb/>
ad dimidiam ba&longs;im per­<lb/>
ducitur, ex tertia deci­<lb/>
ma Archimedis. <!-- KEEP S--></s><lb/>
<s>quare <lb/>
<arrow.to.target n="marg68"/><lb/>
triangulum acl æquale <lb/>
e&longs;t triangulo bcl: & propterea pyramis, cuius ba&longs;is trian­<lb/>
gulum acl, uertex d, e&longs;t æqualis pyramidi, cuius ba&longs;is bcl <lb/>
<arrow.to.target n="marg69"/><lb/>
triangulum, & idem uertex. </s><s>pyramides enim, quæ ab <expan abbr="eod&etilde;">eodem</expan>
<pb xlink:href="023/01/064.jpg"/>&longs;unt uertice, eandem proportionem habent, quam <expan abbr="ip&longs;arũ">ip&longs;arum</expan> <lb/>
ba&longs;es. </s><s>eadem ratione pyramis aclk pyramidi bclk & py <lb/>
ramis adlk ip&longs;i bdlk pyramidi æqualis erit. </s><s>Itaque &longs;i a py <lb/>
ramide acld auferantur pyramides aclk, adlk: & à pyra <lb/>
mide bcld <expan abbr="auferãtur">auferantur</expan> pyramides bclk dblK: quæ relin­<lb/>
quuntur erunt æqualia. </s><s>æqualis igitur e&longs;t pyramis acdk<lb/>
pyramidi bcdK. <!-- KEEP S--></s><s>Rur&longs;us &longs;i per lineas ad, de ducatur pla­<lb/>
num quod pyramidem &longs;ccet: <expan abbr="&longs;itq;">&longs;itque</expan> eius & ba&longs;is communis <lb/>
&longs;ectio aem: &longs;imiliter o&longs;tendetur pyramis abdK æqualis <lb/>
pyramidi acdk. </s><s>ducto denique alio plano per lineas ca, <lb/>
af: ut eius, & trianguli cdb communis &longs;ectio &longs;it cfn, py­<lb/>
ramis abck pyramidi acdk æqualis demon&longs;trabitur. </s><s><expan abbr="cũ">cum</expan> <lb/>
ergo tres pyramides bcdk, abdk, abck uni, & eidem py <lb/>
ramidi acdk &longs;int æquales, omnes inter &longs;e &longs;e æquales <expan abbr="erũt">erunt</expan>. </s><lb/>
<s>Sed ut pyramis abcd ad pyramidem abck ita de axis ad <lb/>
axem ke, ex uige&longs;ima propo&longs;itione huius: &longs;unt enim hæ <lb/>
pyramides in eadem ba&longs;i, & axes cum ba&longs;ibus æquales con <lb/>
tinent angulos, quòd in eadem recta linea con&longs;tituantur. </s><lb/>
<s>quare diuidendo, ut tres pyramides acdk, bcdK, abdK <lb/>
ad pyramidem abcK, ita dk ad Ke. </s><s>con&longs;tat igitur lineam <lb/>
dK ip&longs;ius Ke triplam e&longs;&longs;e. </s><s>&longs;ed & ak tripla e&longs;t Kf: itemque <lb/>
bK ip&longs;ius kg: & ck ip&longs;ius kl tripla. </s><s>quod eodem modo <lb/>
demon&longs;trabimus.</s></p><p type="margin">
<s><margin.target id="marg66"/>17 huius</s></p><p type="margin">
<s><margin.target id="marg67"/><emph type="italics"/>ucrfex<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg68"/>1. sexti.</s></p><p type="margin">
<s><margin.target id="marg69"/>5. duode­<lb/>
cimi.</s></p><p type="main">
<s>Sit pyramis, cuius ba&longs;is quadrilaterum abcd; axis ef: <lb/>
& diuidatur ef in g, ita ut eg ip&longs;ius gf &longs;it tripla. </s><s>Dico cen­<lb/>
trum grauitatis pyramidis e&longs;&longs;e punctum g. <!-- REMOVE S-->ducatur enim <lb/>
linea bd diuidens ba&longs;im in duo triangula abd, bcd: ex <lb/>
quibus <expan abbr="intelligãtur">intelligantur</expan> <expan abbr="cõ&longs;titui">con&longs;titui</expan> duæ pyramides abde, bcde: <lb/>
&longs;itque pyramidis abde axis eh; & pyramidis bcde axis <lb/>
eK: & iungatur hK, quæ per f tran&longs;ibit: e&longs;t enim in ip&longs;a hK <lb/>
centrum grauitatis magnitudinis compo&longs;itæ ex triangulis <lb/>
abd, bcd, hoc e&longs;t ip&longs;ius quadrilateri. </s>
<s>Itaque centrum gra<lb/>
uitatis pyramidis abde &longs;it punctum l: & pyramidis bcde <lb/>
<arrow.to.target n="marg70"/><lb/>
&longs;it m. </s><s>ducta igitur lm ip&longs;i hm lineæ æquidi&longs;tabit. </s><s>nam el ad
<pb xlink:href="023/01/065.jpg" pagenum="29"/>lh eandem habet proportionem, quam em ad mk, uideli­<lb/>
cet triplam. </s><s>quare linea lm ip&longs;am ef &longs;ecabit in puncto g: <lb/>
etenim eg ad gf e&longs;t, ut el ad lh. </s><s>præterea quoniam hk, lm <lb/>
æquidi&longs;tant, erunt triangula hef, leg &longs;imilia: <expan abbr="itemq;">itemque</expan> inter <lb/>
&longs;e &longs;imilia fek gem: & ut ef ad eg, ita hf ad lg: & ita fK ad <lb/>
gm. </s><s>ergo ut hf ad lg, ita fk ad gm: & permutando ut hf <lb/>
ad fK, ita lg ad gm. </s><s>&longs;ed cum h &longs;it centrum trianguli abd; <lb/>
& k <expan abbr="triãguli">trianguli</expan> bcd <expan abbr="punctũ">punctum</expan> uero f totius quadrilateri abcd <lb/>
centrum: erit ex 8. Archimedis de centro grauitatis plano <lb/>
rum hf ad fk ut triangulum bcd ad triangulum abd: ut, <lb/>
autem bcd triangulum ad triangulum abd, ita pyramis <lb/>
<figure id="id.023.01.065.1.jpg" xlink:href="023/01/065/1.jpg"/><lb/>
bcde ad pyramidem abde. </s><s>ergo <lb/>
linea lg ad gm erit, ut pyramis <lb/>
bcde ad <expan abbr="pyramid&etilde;">pyramidem</expan> abde. </s><s>ex quo <lb/>
&longs;equitur, ut totius pyramidis <lb/>
abcde punctum g &longs;it grauitatis <lb/>
centrum. </s><s>Rur&longs;us &longs;it pyramis ba­<lb/>
&longs;im habens pentagonum abcde: <lb/>
& axem fg: <expan abbr="diuidaturq;">diuidaturque</expan> axis in <expan abbr="pũ">pun</expan> <lb/>
cto h, ita ut fh ad hg triplam habe <lb/>
at proportionem. </s><s>Dico h grauita­<lb/>
tis <expan abbr="centrũ">centrum</expan> e&longs;&longs;e pyramidis abcdef. </s><lb/>
<s>iungatur enim eb: <expan abbr="intelligaturq;">intelligaturque</expan> <lb/>
pyramis, cuius uertex f, & ba&longs;is <lb/>
triangulum abe: & alia pyramis <lb/>
intelligatur eundem uerticem ha­<lb/>
bens, & ba&longs;im bcde <expan abbr="quadrilaterũ">quadrilaterum</expan>: <lb/>
&longs;it autem pyramidis abef axis fk<lb/>
& grauitatis centrum l: & pyrami <lb/>
dis bcdef axis fm, & centrum gra <lb/>
uitatis n:<expan abbr="iunganturq;">iunganturque</expan> km, ln; <lb/>
quæ per puncta gh tran&longs;ibunt. </s><lb/>
<s>Rur&longs;us eodemmodo, quo &longs;up ra, <lb/>
demon&longs;trabimus lineas Kgm, lhn &longs;ibi ip&longs;is æquidi&longs;tare:
<pb xlink:href="023/01/066.jpg"/>& denique punctum h pyramidis abcdef grauitatis e&longs;&longs;e <lb/>
centrum, & ita in aliis.</s></p><p type="margin">
<s><margin.target id="marg70"/>2. fexti.</s></p><p type="main">
<s>Sit conus, uel coni portio axem habens bd: &longs;eceturque <lb/>
plano per axem, quod &longs;ectionem faciat triangulum abc: <lb/>
& bd axis diuidatur in c, ita ut be ip&longs;ius ed &longs;it tripla. </s><lb/>
<s>Dico punctum e coni, uel coni portionis, grauitatis <lb/>
e&longs;&longs;e centrum. </s><s>Si enim fieri pote&longs;t, &longs;it centrum f: & pro­<lb/>
ducatur ef extra figuram in g. <!-- KEEP S--></s><s>quam uero proportionem <lb/>
habet ge ad ef, habeat ba&longs;is coni, uelconi portionis, hoc <lb/>
e&longs;t circulus, uel ellip&longs;is circa diametrum ac ad aliud &longs;pa­<lb/>
cium, in quo h. </s><s>Itaque in circulo, uel ellip&longs;i plane de&longs;cri­<lb/>
batur rectilinea figura axlmcnop, ita ut quæ <expan abbr="relinquũ-tur">relinquun­<lb/>
tur</expan> portiones &longs;int minores &longs;pacio h: & intelligatur pyra­<lb/>
mis ba&longs;im habens rectilineam figuram aKlmcnop, & <lb/>
axem bd; cuius quidem grauitatis centrum erit punctum <lb/>
e, utiam demon&longs;trauimus. </s><s>Et quoniam portiones &longs;unt <lb/>
minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma­<lb/>
<figure id="id.023.01.066.1.jpg" xlink:href="023/01/066/1.jpg"/><lb/>
iorem proportionem habet, quam ge ad ef. </s><s>&longs;ed ut circu­<lb/>
lus, uel ellip&longs;is ad figuram rectilineam &longs;ibi in&longs;criptam, ita <lb/>
conus, uel coni portio ad pyramidem, quæ figuram rectili­<lb/>
neam pro ba&longs;i habet; & altitudinem æqualem: etenim &longs;u­
<pb xlink:href="023/01/067.jpg" pagenum="30"/><arrow.to.target n="marg71"/><lb/>
pra demon&longs;tratum e&longs;t, ita e&longs;&longs;e cylindrum, uel cylindri por­<lb/>
tionem ad pri&longs;ma, cuius ba&longs;is rectilinea figura, & æqua­<lb/>
lis altitudo. </s><s>ergo per conuer&longs;ionem rationis, ut circulus, <lb/>
uel ellip&longs;is ad portiones, ita conus, uel coni portio ad por­<lb/>
tiones &longs;olidas. </s><s>quare conus uel coni portio ad portiones <lb/>
&longs;olidas maiorem habet proportionem, quam ge ad ef: & <lb/>
diuidendo, pyramis ad portiones &longs;olidas maiorem pro­<lb/>
portionem habet, quam gf ad fe. </s><s>fiat igitur qf ad fe <lb/>
ut pyramis ad dictas portiones. </s><s>Itaque quoniam a cono <lb/>
uel coni portione, cuius grauitatis centrum e&longs;t f, aufer­<lb/>
tur pyramis, cuius centrum e; reliquæ magnitudinis, <lb/>
quæ ex &longs;olidis portionibus con&longs;tat, centrum grauitatis <lb/>
erit in linea ef protracta, & in puncto q.</s><s> quod fieri <lb/>
non pote&longs;t: e&longs;t enim centrum grauitatis intra. </s><s>Con&longs;tat <lb/>
igitur coni, uel coni portionis grauitatis centrum e&longs;&longs;e pun <lb/>
ctum e. </s><s>quæ omnia demon&longs;trare oportebat.</s></p><p type="margin">
<s><margin.target id="marg71"/>8 huius</s></p><p type="head">
<s>THEOREMA XIX. PROPOSITIO XXIII.<!-- KEEP S--></s></p><p type="main">
<s>QVODLIBET fru&longs;tum à pyramide, quæ <lb/>
triangularem ba&longs;im habeat, ab&longs;ci&longs;&longs;um, diuiditur <lb/>
in tres pyramides proportionales, in ea proportio <lb/>
ne, quæ e&longs;t lateris maioris ba&longs;is ad latus minoris <lb/>
ip&longs;i re&longs;pondens.</s></p><p type="main">
<s>Hoc demon&longs;trauit Leonardus Pi&longs;anus in libro, qui de­<lb/>
praxi geometriæ in&longs;cribitur. </s><s>Sed quoniam is adhuc im­<lb/>
pre&longs;&longs;us non e&longs;t, nos ip&longs;ius demon&longs;trationem breuiter <lb/>
per&longs;tringemus, rem ip&longs;am &longs;ecuti, non uerba. </s><s>Sit fru­<lb/>
&longs;tum pyramidis abcdef, cuius maior ba&longs;is triangulum <lb/>
abc, minor def: & iunctis ae, cc, cd, per, line­<lb/>
as ae, ec ducatur planum &longs;ecans fru&longs;tum: itemque per <lb/>
lineas ec, cd; & per cd, da alia plana ducantur, quæ <lb/>
diuident fru&longs;tum in trcs pyramides abce, adce, defc.
<pb xlink:href="023/01/068.jpg"/>Dico eas proportionales e&longs;&longs;e in proportione, quæ e&longs;t la­<lb/>
teris ab adlatus de, itaut earum maior &longs;it abce, me­<lb/>
dia adce, & minor defc. <!-- KEEP S--></s><s>Quoniam enim lineæ de, <lb/>
ab æquidi&longs;tant; & inter ip&longs;as &longs;unt triangula abe, ade; <lb/>
<arrow.to.target n="marg72"/><lb/>
<figure id="id.023.01.068.1.jpg" xlink:href="023/01/068/1.jpg"/><lb/>
erit triangulum abe <lb/>
ad triangulum abe, <lb/>
utlinea ab ad lineam <lb/>
de. </s><s>ut autem triangu <lb/>
lum abe ad triangu­<lb/>
<arrow.to.target n="marg73"/><lb/>
lum abe, ita pyramis <lb/>
abec ad pyramidem <lb/>
adec: habent enim <lb/>
altitudinem eandem, <lb/>
quæ e&longs;tà puncto cad <lb/>
planum, in quo qua­<lb/>
<arrow.to.target n="marg74"/><lb/>
drilaterum abed. <!-- KEEP S--></s><s>er­<lb/>
go ut ab ad de, ita pyramis abec ad pyramidem adec. <!-- KEEP S--></s><lb/>
<s>Rur&longs;us quoniam æquidi&longs;tantes &longs;unt ac, df; erit eadem <lb/>
<arrow.to.target n="marg75"/><lb/>
ratione pyramis adce ad pyramidem cdfe, ut ac ad <lb/>
df. </s><s>Sed ut ac ad df, ita ab ad de, quoniam triangula <lb/>
abc, def &longs;imilia &longs;unt, ex nona huius. </s><s>quare ut pyramis <lb/>
abce ad pyramidem abce, ita pyramis adce ad ip&longs;am<lb/>
defc. <!-- REMOVE S-->fru&longs;tum igitur abcdef diuiditur in tres pyramides <lb/>
proportionales in ea proportione, quæ e&longs;t lateris ab ad de <lb/>
latus, & earum maior e&longs;t cabe, media adce, & minor <lb/>
defc. <!-- REMOVE S-->quod demon&longs;trare oportebat.</s>
</p><p type="margin">
<s><margin.target id="marg72"/>1. &longs;exti.</s></p><p type="margin">
<s><margin.target id="marg73"/>5. duodeci <lb/>
mi.</s></p><p type="margin">
<s><margin.target id="marg74"/>11. quinti.</s></p><p type="margin">
<s><margin.target id="marg75"/>4 &longs;exti.</s></p><p type="head">
<s>PROBLEMA V. PROPOSITIO XXIIII.<!-- KEEP S--></s></p><p type="main">
<s>QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>
uel coni portionis, plano ba&longs;i æquidi&longs;tanti ita &longs;e­<lb/>
care, ut &longs;ectio &longs;it proportionalis inter maiorem, <lb/>
& minorem ba&longs;im.</s></p>
<pb xlink:href="023/01/069.jpg" pagenum="31"/><p type="main">
<s>SIT fru&longs;tum pyramidis ae, cuius maior ba&longs;is triangu­<lb/>
lum abc, minor def: & oporteat ip&longs;um plano, quod ba&longs;i <lb/>
æquidi&longs;tet, ita &longs;ecare, ut &longs;ectio &longs;it proportionalis inter <expan abbr="triã">trian</expan> <lb/>
gula abc, def. </s><s>Inueniatur inter lineas ab, de media pro­<lb/>
portionalis, quæ &longs;it bg: & à puncto g erigatur gh æquidi­<lb/>
&longs;tans be, <expan abbr="&longs;ecansq;">&longs;ecansque</expan> ad in h: deinde per h ducatur planum <lb/>
ba&longs;ibus æquidi&longs;tans, cuius &longs;ectio &longs;it triangulum hkl. <!-- KEEP S--></s><s>Dico <lb/>
triangulum hKl proportionale e&longs;&longs;e inter triangula abc, <lb/>
<figure id="id.023.01.069.1.jpg" xlink:href="023/01/069/1.jpg"/><lb/>
def, hoc e&longs;t triangulum abc ad <lb/>
triangulum hKl eandem habere <lb/>
proportionem, quam <expan abbr="triãgulum">triangulum</expan> <lb/>
hKl ad ip&longs;um def. </s><s><expan abbr="Quoniã">Quoniam</expan> enim <lb/>
<arrow.to.target n="marg76"/><lb/>
lineæ ab, hK æquidi&longs;tantium pla <lb/>
norum &longs;ectiones inter &longs;e æquidi­<lb/>
&longs;tant: atque æquidi&longs;tant bk, gh: <lb/>
<arrow.to.target n="marg77"/><lb/>
linea hk ip&longs;i gb e&longs;t æqualis: & pro <lb/>
pterea proportionalis inter ab, <lb/>
de. </s><s>quare ut ab ad hK, ita e&longs;t hk<lb/>
ad de. </s><s>fiat ut hk ad de, ita de <lb/>
ad aliam lineam, in qua &longs;it m. </s><s>erit <lb/>
ex æquali ut ab ad de, ita hk ad <lb/>
<arrow.to.target n="marg78"/><lb/>
m. </s><s>Et quoniam triangula abc, <lb/>
hKl, def &longs;imilia &longs;unt; <expan abbr="triangulũ">triangulum</expan> <lb/>
<arrow.to.target n="marg79"/><lb/>
abc ad triangulum hkl e&longs;t, ut li­<lb/>
nea ab ad lineam de: <expan abbr="triangulũ">triangulum</expan> <lb/>
<arrow.to.target n="marg80"/><lb/>
autem hkl ad ip&longs;um def e&longs;t, ut hk ad m. </s><s>ergo triangulum <lb/>
abc ad triangulum hkl eandem proportionem habet, <lb/>
quam triangulum hKl ad ip&longs;um def. </s><s>Eodem modo in a­<lb/>
liis fru&longs;tis pyramidis idem demon&longs;trabitur.</s></p><p type="margin">
<s><margin.target id="marg76"/>16. unnde<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg77"/>34. primi</s></p><p type="margin">
<s><margin.target id="marg78"/>9. huius <lb/>
corol.</s></p><p type="margin">
<s><margin.target id="marg79"/>20. &longs;exti</s></p><p type="margin">
<s><margin.target id="marg80"/>11. quinti</s></p><p type="main">
<s>Sit fru&longs;tum coni, uel coni portionis ad: & &longs;ecetur plano <lb/>
per axem, cuius &longs;ectio &longs;it abcd, ita ut maior ip&longs;ius ba&longs;is &longs;it <lb/>
circulus, uel ellip&longs;is circa diametrum ab; minor circa cd. <!-- KEEP S--></s><lb/>
<s>Rur&longs;us inter lineas ab, cd inueniatur proportionalis be: <lb/>
& ab e ducta ef æquidi&longs;tante bd, quæ lineam ca in f &longs;ecet,
<pb xlink:href="023/01/070.jpg"/>per f planum ba&longs;ibus æquidi&longs;tans ducatur, ut &longs;it &longs;ectio cir <lb/>
culus, uel ellip&longs;is circa diametrum fg. <!-- KEEP S--></s><s>Dico &longs;ectionem ab <lb/>
ad &longs;ectionem fg eandem proportionem habere, quam fg <lb/>
ad ip&longs;am cd. <!-- KEEP S--></s><s>Simili enim ratione, qua &longs;upra, demon&longs;trabi­<lb/>
tur quadratum ab ad quadratum fg ita e&longs;&longs;e, ut <expan abbr="quadratũ">quadratum</expan> <lb/>
<arrow.to.target n="marg81"/><lb/>
fg ad cd quadratum. </s><s>Sed circuli inter &longs;e eandem propor­<lb/>
tionem habent, quam diametrorum quadrata. </s><s>ellip&longs;es au­<lb/>
tem circa ab, fg, cd, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="cõ-mentariis">con­<lb/>
mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>
& &longs;phæroidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>
ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio­<lb/>
<figure id="id.023.01.070.1.jpg" xlink:href="023/01/070/1.jpg"/><lb/>
&longs;eptimæ propo&longs;itionis eiu&longs;dem li­<lb/>
bri. </s><s>ellip&longs;es enim nunc appello ip­<lb/>
&longs;a &longs;pacia ellip&longs;ibus contenta. </s><s>ergo <lb/>
circulus, uel ellip&longs;is ab ad <expan abbr="circulũ">circulum</expan>, <lb/>
uel ellip&longs;im fg eam proportionem <lb/>
habet, quam circulus, uel ellip&longs;is <lb/>
fg ad circulum uel ellip&longs;im cd. <!-- KEEP S--></s><lb/>
<s>quod quidem faciendum propo­<lb/>
&longs;uimus.</s></p><p type="margin">
<s><margin.target id="marg81"/>2. duode<lb/>
cimi</s></p><p type="head">
<s>THEOREMA XX. PROPOSITIO XXV.<!-- KEEP S--></s></p><p type="main">
<s>QVODLIBET fru&longs;tum pyramidis, uel coni, <lb/>
uel coni portionis ad pyramidem, uel conum, uel <lb/>
coni portionem, cuius ba&longs;is eadem e&longs;t, & æqualis <lb/>
altitudo, eandem <expan abbr="proportion&etilde;">proportionem</expan> habet, quam utræ <lb/>
que ba&longs;es, maior, & minor &longs;imul &longs;umptæ vnà <expan abbr="cũ">cum</expan> <lb/>
ca, quæ inter ip&longs;as &longs;it proportionalis, ad ba&longs;im ma <lb/>
iorem.</s></p>
<pb xlink:href="023/01/071.jpg" pagenum="32"/><p type="main">
<s>SIT <expan abbr="fru&longs;tũ">fru&longs;tum</expan> pyramidis, uel coni, uel coni portionis ad, <lb/>
cuius maior ba&longs;is ab, minor cd. <!-- KEEP S--></s><s>& &longs;ecetur altero plano <lb/>
ba&longs;i æquidi&longs;tante, ita ut &longs;ectio ef &longs;it proportionalis inter <lb/>
ba&longs;es ab, cd. <!-- KEEP S--></s><s>con&longs;tituatur <expan abbr="aut&etilde;">autem</expan> pyramis, uel conus, uel co­<lb/>
ni portio agb, cuius ba&longs;is &longs;it eadem, quæ ba&longs;is maior fru­<lb/>
<figure id="id.023.01.071.1.jpg" xlink:href="023/01/071/1.jpg"/><lb/>
&longs;ti, & altitudo æqualis. </s><s>Di­<lb/>
co fru&longs;tum ad ad pyrami­<lb/>
dem, uel conum, uel coni <lb/>
portionem agb eandem <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> habere, <expan abbr="quã">quam</expan> <lb/>
utræque ba&longs;es, ab, cd unà <lb/>
cum ef ad ba&longs;im ab. </s><s>e&longs;t <lb/>
enim fru&longs;tum ad æquale <lb/>
pyramidi, uel cono, uel co­<lb/>
ni portioni, cuius ba&longs;is ex <lb/>
tribus ba&longs;ibus ab, ef, cd <lb/>
con&longs;tat; & altitudo ip&longs;ius <lb/>
altitudini e&longs;t æqualis: quod mox o&longs;tendemus. </s><s>Sed pyrami <lb/>
<figure id="id.023.01.071.2.jpg" xlink:href="023/01/071/2.jpg"/><lb/>
des, coni, uel coni <expan abbr="portiões">portiones</expan>, <lb/>
quæ &longs;unt æquali altitudine, <lb/>
<expan abbr="eãdem">eandem</expan> inter &longs;e, quam ba&longs;es, <lb/>
proportionem habent, &longs;icu­<lb/>
ti demon&longs;tratum e&longs;t, partim <lb/>
<arrow.to.target n="marg82"/><lb/>
ab Euclide in duodecimo li­<lb/>
bro elementorum, partim à <lb/>
nobis in <expan abbr="cõmentariis">commentariis</expan> in un­<lb/>
decimam <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> Ar­<lb/>
chimedis de conoidibus, & <lb/>
&longs;phæroidibus. </s><s>quare pyra­<lb/>
mis, uel conus, uel coni por­<lb/>
tio, cuius ba&longs;is e&longs;t tribus illis <lb/>
ba&longs;ibus æqualis ad agb eam <lb/>
habet proportionem, quam <lb/>
ba&longs;es ab, ef, cd ad ab ba&longs;im. </s><s>Fru&longs;tum igitur ad ad agb
<pb xlink:href="023/01/072.jpg"/>pyramidem, uel conum, uel coni portionem eandem pro­<lb/>
portionem habet, quam ba&longs;es ab, cd unà cum ef ad ba­<lb/>
&longs;im ab. </s><s>quod demon&longs;trare uolebamus.</s></p><p type="margin">
<s><margin.target id="marg82"/>6. 11. duo<lb/>
decimi</s></p><p type="main">
<s>Fru&longs;tum uero ad æquale e&longs;&longs;e pyramidi, uel co <lb/>
no, uel coni portioni, cuius ba&longs;is con&longs;tat ex ba&longs;i­<lb/>
bus ab, cd, ef, & altitudo fru&longs;ti altitudini e&longs;t æ­<lb/>
qualis, hoc modo o&longs;tendemus.</s></p><p type="main">
<s>Sit fru&longs;tum pyramidis abcdef, cuius maior ba&longs;is trian­<lb/>
gulum abc; minor def: & &longs;ecetur plano ba&longs;ibus æquidi­<lb/>
&longs;tante, quod &longs;ectionem faciat triangulum ghk inter trian­<lb/>
gula abc, def proportionale. </s><s>Iam ex iis, quæ demon&longs;trata <lb/>
&longs;unt in 23. huius, patet fru&longs;tum abcdef diuidi in tres pyra <lb/>
mides proportionales; & earum maiorem e&longs;&longs;e <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>
abcd <expan abbr="minor&etilde;">minorem</expan> uero defb. </s><s>ergo pyramis à triangulo ghk <lb/>
con&longs;tituta, quæ altitudinem habeat fru&longs;ti altitudini æqua­<lb/>
lem, proportionalis e&longs;t inter pyramides abcd, defb: & <lb/>
idcirco fru&longs;tum abcdef tribus dictis pyramidibus æqua <lb/>
<figure id="id.023.01.072.1.jpg" xlink:href="023/01/072/1.jpg"/><lb/>
le erit. </s><s>Itaque &longs;i intelligatur alia pyra­<lb/>
mis æque alta, quæ ba&longs;im habeat ex tri <lb/>
bus ba&longs;ibus abc, def, ghk con&longs;tan­<lb/>
tem; per&longs;picuum e&longs;t ip&longs;am ei&longs;dem py­<lb/>
ramidibus, & propterea ip&longs;i fru&longs;to æ­<lb/>
qualem e&longs;&longs;e.</s></p><p type="main">
<s>Rur&longs;us &longs;it fru&longs;tum pyramidis ag, cu <lb/>
ius maior ba&longs;is quadrilaterum abcd, <lb/>
minor efgh: & &longs;ecetur plano ba&longs;i­<lb/>
bus æquidi&longs;tante, ita ut fiat &longs;ectio qua­<lb/>
drilaterum Klmn, quod &longs;it proportio <lb/>
nale inter quadrilatera abcd, efgh. </s><s>Dico pyramidem, <lb/>
cuius ba&longs;is &longs;it æqualis tribus quadrilateris abcd, klmn, <lb/>
efgh, & altitudo æqualis altitudini fru&longs;ti, ip&longs;i fru&longs;to ag <lb/>
æqualem e&longs;&longs;e. </s><s>Ducatur enim planum per lineas fb, hd,
<pb xlink:href="023/01/073.jpg" pagenum="33"/>quod diuidat fru&longs;tum in duo fru&longs;ta triangulares ba&longs;es ha­<lb/>
bentia, uidelicet in fru&longs;tum abdefh, & in <expan abbr="fru&longs;tũ">fru&longs;tum</expan> bcdfgh. </s><lb/>
<s>erit triangulum kln proportionale inter triangula abd, <lb/>
efh: & triangulum lmn proportionale inter bcd, fgh. </s><lb/>
<s>&longs;ed pyramis æque alta, cuius ba&longs;is con&longs;tat ex tribus trian­<lb/>
<figure id="id.023.01.073.1.jpg" xlink:href="023/01/073/1.jpg"/><lb/>
gulis abd, klz, efh, demon&longs;trata <lb/>
e&longs;t fru&longs;to abdcfh æqualis. </s><s>& &longs;i­<lb/>
militer pyramis, cuius ba&longs;is con­<lb/>
&longs;tat ex triangulis bcd, lmn, fgh <lb/>
æqualis fru&longs;to bcdfgh: compo­<lb/>
nuntur autem tria quadrilatera a <lb/>
bcd, klmn, efgh è &longs;ex triangu­<lb/>
lis iam dictis. </s><s>pyramis igitur ba­<lb/>
&longs;im habens æqualem tribus qua­<lb/>
drilateris, & altitudinem eandem <lb/>
ip&longs;i fru&longs;to ag e&longs;t æqualis. </s><s>Eodem <lb/>
modo illud <expan abbr="demõ&longs;trabitur">demon&longs;trabitur</expan> in aliis <lb/>
eiu&longs;modi fru&longs;tis.</s></p><p type="main">
<s>Sit fru&longs;tum coni, uel coni portionis ad; cuius maior ba­<lb/>
&longs;is circulus, uel ellip&longs;is circa diametrum ab; minor circa <lb/>
c d: & &longs;ecetur plano, quod ba&longs;ibus æquidi&longs;tet, <expan abbr="faciatq;">faciatque</expan> &longs;e­<lb/>
ctionem circulum, uel ellip&longs;im circa diametrum ef, ita ut <lb/>
inter circulos, uel ellip&longs;es ab, cd &longs;it proportionalis. </s><s>Dico <lb/>
conum, uel coni portionem, cuius ba&longs;is e&longs;t æqualis tribus <lb/>
circulis, uel tribus ellip&longs;ibus ab, ef, cd; & altitudo eadem, <lb/>
quæ fru&longs;ti ad, ip&longs;i fru&longs;to æqualem e&longs;&longs;e. </s><s>producatur enim <lb/>
fru&longs;ti &longs;uperficies quou&longs;que coeat in unum punctum, quod <lb/>
&longs;it g: & coni, uel coni portionis agb axis &longs;it gh, occurrens <lb/>
planis ab, ef, cd in punctis hkl: circa circulum uero de­<lb/>
&longs;cribatur quadratum mnop, & circa ellip&longs;im <expan abbr="rectangulũ">rectangulum</expan> <lb/>
mnop, quod ex ip&longs;ius diametris con&longs;tat: <expan abbr="iunctisq;">iunctisque</expan> gm, <lb/>
g n, go, gp, ex eodem uertice intelligatur pyramis ba&longs;im <lb/>
habens dictum quadratum, uel rectangulum: & plana in <lb/>
quibus &longs;unt circuli, uel ellip&longs;es ef, cd u&longs;que ad eius latera
<pb xlink:href="023/01/074.jpg"/>producantur. </s><s>Quoniam igitur pyramis &longs;ecatur planis ba&longs;i <lb/>
<arrow.to.target n="marg83"/><lb/>
æquidi&longs;tantibus, &longs;ectiones &longs;imiles erunt: atque erunt qua­<lb/>
drata, uel rectangula circa circulos, uel ellip&longs;es de&longs;cripta, <lb/>
quemadmodum & in ip&longs;a ba&longs;i. </s><s>Sed cum circuli inter &longs;e <expan abbr="eã">eam</expan> <lb/>
<arrow.to.target n="marg84"/><lb/>
proportionem habeant, quam diametrorum quadrata: <lb/>
<expan abbr="itemq;">itemque</expan> ellip&longs;es eam quam rectangula ex ip&longs;arum diametris <lb/>
<arrow.to.target n="marg85"/><lb/>
con&longs;tantia: & &longs;it circulus, uel ellip&longs;is circa diametrum ef <lb/>
<figure id="id.023.01.074.1.jpg" xlink:href="023/01/074/1.jpg"/><lb/>
proportionalis inter circulos, uel ellip&longs;es ab, cd; erit re­<lb/>
ctangulum ef etiam inter rectangula ab, cd proportio­<lb/>
nale: per rectangulum enim nunc breuitatis cau&longs;a <expan abbr="etiã">etiam</expan> ip­<lb/>
&longs;um quadratum intelligemus. </s><s>quare ex iis, quæ proxime <lb/>
dicta &longs;unt, pyramis ba&longs;im habens æqualem dictis rectangu <lb/>
lis, & altitudinem eandem, quam fru&longs;tum ad, ip&longs;i fru&longs;to à <lb/>
pyramide ab&longs;ci&longs;&longs;o æqualis probabitur. </s><s>ut autem rectangu <lb/>
lum cd ad <expan abbr="rectangulũ">rectangulum</expan> ef, ita circulus, uel ellip&longs;is cd ad ef <lb/>
circulum, uel ellip&longs;im: <expan abbr="componendoq;">componendoque</expan> ut rectangula cd, <lb/>
e f, ad ef rectangulum, ita circuli, uel ellip&longs;es ed, ef, ad ef: <lb/>
& ut rectangulum ef ad rectangulum ab, ita circulus, uel <lb/>
ellip&longs;is ef ad ab circulum, uel ellip&longs;im. </s><s>ergo ex æquali, & <lb/>
componendo, ut <expan abbr="rectãgula">rectangula</expan> cd, ef, ab ad ip&longs;um ab, ita cir­
<pb xlink:href="023/01/075.jpg" pagenum="34"/>culi, uel ellip&longs;es cd, ef ab ad circulum, uel ellip&longs;im ab. </s><s>In­<lb/>
telligatur pyramis q ba&longs;im habens æqualem tribus rectan <lb/>
gulis ab, ef, cd; & altitudinem <expan abbr="eãdem">eandem</expan>, quam fru&longs;tum ad. <!-- KEEP S--></s><lb/>
<s>intelligatur etiam conus, uel coni portio q, eadem altitudi<lb/>
ne, cuius ba&longs;is &longs;it tribus circulis, uel tribus ellip&longs;ibus ab, <lb/>
ef, cd æqualis. </s><s>po&longs;tremo intelligatur pyramis alb, cuius. </s><lb/>
<s>ba&longs;is &longs;it rectangulum mnop, & altitudo eadem, quæ fru­<lb/>
&longs;ti: <expan abbr="itemq,">itemque</expan> intelligatur conus, uel coni portio alb, cuius <lb/>
ba&longs;is circulus, uel ellip&longs;is circa diametrum ab, & eadem al <lb/>
<arrow.to.target n="marg86"/><lb/>
titudo. </s><s>ut igitur rectangula ab, ef, cd ad rectangulum ab, <lb/>
ita pyramis q ad pyramidem alb; & ut circuli, uel ellip­<lb/>
&longs;es ab, ef, cd ad ab circulum, uel ellip&longs;im, ita conus, uel co<lb/>
ni portio q ad conum, uel coni portionem alb. </s><s>conus <lb/>
igitur, uel coni portio q ad conum, uel coni portionem <lb/>
alb e&longs;t, ut pyramis q ad pyramidem alb. </s><s>&longs;ed pyramis <lb/>
alb ad pyramidem agb e&longs;t, ut altitudo ad altitudinem, ex <lb/>
20. huius: & ita e&longs;t conus, uel coni portio alb ad conum, <lb/>
uel coni portionem agb ex 14. duodecimi elementorum, <lb/>
& ex iis, quæ nos demon&longs;trauimus in commentariis in un­<lb/>
decimam de conoidibus, & &longs;phæroidibus, propo&longs;itione <lb/>
quarta. </s><s>pyramis autem agb ad pyramidem cgd propor­<lb/>
tionem habet compo&longs;itam ex proportione ba&longs;ium & pro <lb/>
portione altitudinum, ex uige&longs;ima prima huius: & &longs;imili­<lb/>
ter conus, uel coni portio agb ad conum, uel coni portio­<lb/>
nem cgd proportionem habet <expan abbr="compo&longs;itã">compo&longs;itam</expan> ex ei&longs;dem pro­<lb/>
portionibus, per ea, quæ in dictis commentariis demon­<lb/>
&longs;trauimus, propo&longs;itione quinta, & &longs;exta: altitudo enim in<lb/>
utri&longs;que eadem e&longs;t, & ba&longs;es inter &longs;e &longs;e eandem habent pro­<lb/>
portionem. </s><s>ergo ut pyramis agb ad pyramidem cgd, ita <lb/>
e&longs;t conus, uel coni portio agb ad agd conum, uel coni <lb/>
portionem: & per <expan abbr="conuer&longs;ion&etilde;">conuer&longs;ionem</expan> rationis, ut pyramis agb <lb/>
ad <expan abbr="&longs;ru&longs;tũ">fru&longs;tum</expan> à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio <lb/>
agb ad fru&longs;tum ad. <!-- KEEP S--></s><s>ex æquali igitur, ut pyramis q ad fru­<lb/>
&longs;tum à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio q ad
<pb xlink:href="023/01/076.jpg"/>fru&longs;tum ad. <!-- KEEP S--></s><s>Sed pyramis q æqualis e&longs;t fru&longs;to à pyramide <lb/>
ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. </s><s>ergo & conus, uel coni por­<lb/>
tio q, cuius ba&longs;is ex tribus circulis, uel ellip&longs;ibus ab, ef, cd <lb/>
con&longs;tat, & altitudo eadem, quæ fru&longs;ti: ip&longs;i fru&longs;to ad e&longs;t æ­<lb/>
qualis. </s><s>atque illud e&longs;t, quod demon&longs;trare oportebat.</s></p><p type="margin">
<s><margin.target id="marg83"/>9 huius</s></p><p type="margin">
<s><margin.target id="marg84"/>2. duode­<lb/>
cnni.</s></p><p type="margin">
<s><margin.target id="marg85"/>7. de co­<lb/>
noidibus <lb/>
& &longs;phæ­<lb/>
roidibus</s></p><p type="margin">
<s><margin.target id="marg86"/>6. II. duo <lb/>
decimi</s></p><p type="head">
<s>THEOREMA XXI. PROPOSITIO XXVI.<!-- KEEP S--></s></p><p type="main">
<s>CVIVSLIBET fru&longs;ti à pyramide, uel cono, <lb/>
uel coni portione ab&longs;cis&longs;i, centrum grauitatis e&longs;t <lb/>
in axe, ita ut eo primum in duas portiones diui­<lb/>
&longs;o, portio &longs;uperior, quæ minorem ba&longs;im attingit <lb/>
ad portionem reliquam eam habeat proportio­<lb/>
nem, quam duplum lateris, uel diametri maioris <lb/>
ba&longs;is, vnà cum latere, uel diametro minoris, ip&longs;i <lb/>
re&longs;pondente, habet ad duplum lateris, uel diame­<lb/>
tri minoris ba&longs;is vnà <expan abbr="cũ">cum</expan> latere, uel diametro ma­<lb/>
ioris: deinde à puncto diui&longs;ionis quarta parte &longs;u <lb/>
perioris portionis in ip&longs;a &longs;umpta: & rur&longs;us ab in­<lb/>
ferioris portionis termino, qui e&longs;t ad ba&longs;im maio<lb/>
rem, &longs;umpta quarta parte totius axis: centrum &longs;it <lb/>
in linea, quæ his finibus continetur, atque in eo li <lb/>
neæ puncto, quo &longs;ic diuiditur, ut tota linea ad par <lb/>
tem propinquiorem minori ba&longs;i, <expan abbr="eãdem">eandem</expan> propor­<lb/>
tionem habeat, quam fru&longs;tum ad <expan abbr="pyramid&etilde;">pyramidem</expan>, uel <lb/>
conum, uel coni portionem, cuius ba&longs;is &longs;it ea­<lb/>
dem, quæ ba&longs;is maior, & altitudo fru&longs;ti altitudini <lb/>
æqualis.</s></p>
<pb xlink:href="023/01/077.jpg" pagenum="35"/><p type="main">
<s>Sit fru&longs;tum ae a pyramide, quæ triangularem ba&longs;im ha­<lb/>
beat ab&longs;ci&longs;&longs;um: cuius maior ba&longs;is triangulum abc, minor <lb/>
def; & axis gh. </s><s>ducto autem plano per axem & per <expan abbr="lineã">lineam</expan> <lb/>
da, quod &longs;ectionem faciat dakl quadrilaterum; puncta <lb/>
Kl lineas bc, ef bifariam &longs;ecabunt. </s><s>nam cum gh &longs;it axis <lb/>
fru&longs;ti: erit h centrum grauitatis trianguli abc: & g <lb/>
<figure id="id.023.01.077.1.jpg" xlink:href="023/01/077/1.jpg"/><lb/>
<arrow.to.target n="marg87"/><lb/>
centrum trianguli def: cen­<lb/>
trum uero cuiuslibet triangu <lb/>
li e&longs;t in recta linea, quæ ab an­<lb/>
gulo ip&longs;ius ad <expan abbr="dimidiã">dimidiam</expan> ba&longs;im <lb/>
ducitur ex decimatertia primi <lb/>
libri Archimedis de <expan abbr="c&etilde;tro">centro</expan> gra <lb/>
<arrow.to.target n="marg88"/><lb/>
uitatis planorum. </s><s>quare <expan abbr="cen-trũ">cen­<lb/>
trum</expan> grauitatis trapezii bcfe <lb/>
e&longs;t in linea kl, quod &longs;it m: & à <lb/>
puncto m ad axem ducta mn <lb/>
ip&longs;i ak, uel dl æquidi&longs;tante; <lb/>
erit axis gh diui&longs;us in portio­<lb/>
nes gn, nh, quas diximus: ean <lb/>
dem enim proportionem ha­<lb/>
bet gn ad nh, <expan abbr="quã">quam</expan> lm ad mk. </s><lb/>
<s>At lm ad mK habet eam, <expan abbr="quã">quam</expan> <lb/>
duplum lateris maioris ba&longs;is <lb/>
bc una cum latere minoris ef <lb/>
ad duplum lateris ef unà cum <lb/>
latere bc, ex ultima eiu&longs;dem <lb/>
libri Archimedis. <!-- KEEP S--></s><s>Itaque à li­<lb/>
nea ng ab&longs;cindatur, quarta <lb/>
pars, quæ fit np: & ab axe hg ab&longs;cindatur itidem <lb/>
quarta pars ho: & quam proportionem habet fru&longs;tum ad <lb/>
pyramidem, cuius maior ba&longs;is e&longs;t triangulum abc, & alti­<lb/>
tudo ip&longs;i æqualis; habeat op ad pq.</s><s> Dico centrum graui­<lb/>
tatis fru&longs;ti e&longs;&longs;e in linea po, & in puncto q.</s><s> namque ip&longs;um <lb/>
e&longs;&longs;e in linea gh manife&longs;te con&longs;tat. </s><s>protractis enim fru&longs;ti pla
<pb xlink:href="023/01/078.jpg"/>nis, quou&longs;que in unum punctum r conueniant; erit pyra­<lb/>
midis abcr, & pyramidis defr grauitatis centrum in li­<lb/>
nca rh. </s><s>ergo & reliquæ magnitudinis, uidelicet fru&longs;ti cen­<lb/>
trum in eadem linea nece&longs;&longs;ario comperietur. </s><s>Iungantur <lb/>
db, dc, dh, dm: & per lineas db, dc ducto altero plano <lb/>
intelligatur fru&longs;tum in duas pyramides diui&longs;um: in pyra­<lb/>
midem quidem, cuius ba&longs;is e&longs;t triangulum abc, uertex d: <lb/>
& in eam, cuius idem uertex, & ba&longs;is trapezium bcfe. </s><s>erit <lb/>
igitur pyramidis abcd axis dh, & pyramidis bcfed axis <lb/>
d m: atque erunt tres axes gh, dh, dm in eodem plano <lb/>
daKl.</s><s> ducatur præterea per o linea &longs;t ip&longs;i aK <expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan>, <lb/>
quæ lineam dh in u &longs;ecet: per p uero ducatur xy æquidi­<lb/>
<figure id="id.023.01.078.1.jpg" xlink:href="023/01/078/1.jpg"/><lb/>
&longs;tans eidem, &longs;ecansque dm in <lb/>
z: & iungatur zu, quæ &longs;ecet <lb/>
gh in <foreign lang="greek">f.</foreign> tran&longs;ibit ea per q: & <lb/>
erunt <foreign lang="greek">f</foreign>q unum, atque idem <lb/>
punctum; ut inferius appare­<lb/>
bit. </s>
<s>Quoniam igitur linea uo <lb/>
<arrow.to.target n="marg89"/><lb/>
æquidi&longs;tat ip&longs;i dg, erit du ad <lb/>
uh, ut go ad oh. </s><s>Sed go tri­<lb/>
pla e&longs;t oh. </s><s>quare & du ip&longs;ius <lb/>
uh e&longs;t tripla: & ideo pyrami­<lb/>
dis abcd centrum grauitatis <lb/>
erit punctum u. </s><s>Rur&longs;us quo­<lb/>
niam zy ip&longs;i dl æquidi&longs;tat, dz <lb/>
ad zm e&longs;t, ut ly ad ym: e&longs;tque <lb/>
ly ad ym, ut gp ad pn. </s><s>ergo <lb/>
dz ad zm e&longs;t, ut gp ad pn. </s><lb/>
<s>Quòd cum gp &longs;it tripla pn; <lb/>
erit etiam dz ip&longs;ius zm tri­<lb/>
pla. </s><s>atque ob eandem cau&longs;­<lb/>
&longs;am punctuniz e&longs;t <expan abbr="centrũ">centrum</expan> gra­<lb/>
uitatis pyramidis bcfed. <!-- KEEP S--></s><s>iun <lb/>
cta igitur zu, in ea erit <expan abbr="c&etilde;trum">centrum</expan>
<pb xlink:href="023/01/079.jpg" pagenum="36"/>grauitatis magnitudinis, quæ ex utri&longs;que pyramidibus <expan abbr="cõ">con</expan> <lb/>
&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s><s>Sed fru&longs;ti centrum e&longs;t etiam in a­<lb/>
xe gh. </s><s>ergo in puncto <foreign lang="greek">f,</foreign> in quo lineæ zu, gh conueniunt. </s><lb/>
<s><arrow.to.target n="marg90"/><lb/>
Itaque u<foreign lang="greek">f</foreign> ad <foreign lang="greek">f</foreign>z eam proportionem habet, quam pyramis <lb/>
bcfed ad pyramidem abcd. <!-- KEEP S--></s><s>& componendo uz ad z<foreign lang="greek">f</foreign><lb/>
eam habet, quam fru&longs;tum ad pyramidem abcd. <!-- KEEP S--></s><s>Vt uero <lb/>
uz ad z<foreign lang="greek">f</foreign>, ita op ad p<foreign lang="greek">f</foreign> ob &longs;imilitudinem triangulorum, <lb/>
uo<foreign lang="greek">f</foreign>, zp<foreign lang="greek">f.</foreign> quare op ad p<foreign lang="greek">f</foreign> e&longs;t ut fru&longs;tum ad pyramidem <lb/>
abcd. <!-- KEEP S--></s><s>&longs;ed ita erat op ad pq.</s><s> æquales igitur &longs;unt p<foreign lang="greek">f</foreign>, pq: &<lb/>
<arrow.to.target n="marg91"/><lb/>
q<foreign lang="greek">f</foreign> unum atque idem punctum. </s><s>ex quibus &longs;equitur lineam. </s><lb/>
<s>z u &longs;ecare op in q: & propterea <expan abbr="pũctum">punctum</expan> q ip&longs;ius fru&longs;ti gra­<lb/>
uitatis centrum e&longs;&longs;e.</s></p><p type="margin">
<s><margin.target id="marg87"/>3. diffi. </s><s>hu <lb/>
ius.</s></p><p type="margin">
<s><margin.target id="marg88"/>Vltima <expan abbr="e-iu&longs;d&etilde;">e­<lb/>
iu&longs;dem</expan> libri <lb/>
Archime­<lb/>
dis.<!-- KEEP S--></s></p><p type="margin">
<s><margin.target id="marg89"/>2. &longs;exti.</s></p><p type="margin">
<s><margin.target id="marg90"/>8. primi <lb/>
libri Ar­<lb/>
chimedis <lb/>
de <expan abbr="c&etilde;tro">centro</expan> <lb/>
grauta­<lb/>
tis plano <lb/>
rum</s></p><p type="margin">
<s><margin.target id="marg91"/>7. quinti.</s></p><p type="main">
<s>Sit fru&longs;tum ag à pyramide, quæ quadrangularem ba&longs;im <lb/>
habeat ab&longs;ci&longs;&longs;um, cuius maior ba&longs;is abcd, minor efgh, <lb/>
& axis kl. <!-- REMOVE S-->diuidatur autem <expan abbr="primũ">primum</expan> kl, ita ut quam propor­<lb/>
tionem habet duplum lateris ab unà cum latere ef ad du <lb/>
plum lateris ef unà cum ab; habeat km ad ml. <!-- KEEP S--></s>
<s>deinde à <lb/>
<expan abbr="pũcto">puncto</expan> m ad k &longs;umatur quarta pars ip&longs;ius mk quæ &longs;it mn. </s><lb/>
<s>& rur&longs;us ab l &longs;umatur quarta pars totius axis lk, quæ &longs;it <lb/>
lo. </s><s>po&longs;tremo fiat on ad np, ut fru&longs;tum ag ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
cuius ba&longs;is &longs;it eadem, quæ fru&longs;ti, & altitudo æqualis. </s><s>Dico <lb/>
punctum p fru&longs;ti ag grauitatis centrum e&longs;&longs;e. </s><s>ducantur <lb/>
enim ac, eg: & intelligantur duo fru&longs;ta triangulares ba­<lb/>
&longs;es habentia, quorum alterum lf ex ba&longs;ibus abc, efg <expan abbr="cõ-&longs;tet">con­<lb/>
&longs;tet</expan>; alterum lh ex ba&longs;ibus acd, egh. </s><s><expan abbr="Sitq;">Sitque</expan> fru&longs;ti lf axis <lb/>
qr; in quo grauitatis centrum s: fru&longs;ti uero lh axis tu, & <lb/>
x grauitatis centrum: deinde iungantur ur, tq, xs. </s><s>tran&longs;i­<lb/>
bit ur per l: quoniam l e&longs;t centrum grauitatis quadran­<lb/>
guli abcd: & puncta ru grauitatis centra triangulorum <lb/>
abc, acd; in quæ quadrangulum ip&longs;um diuiditur. </s><s>eadem <lb/>
quoque ratione tq per punctum k tran&longs;ibit. </s><s>At uero pro <lb/>
portiones, ex quibus fru&longs;torum grauitatis centra inquiri­<lb/>
mus, eædem &longs;unt in toto fru&longs;to ag, & in fru&longs;tis lf, lh. </s><s>Sunt <lb/>
enim per octauam huius quadrilatera abcd, efgh &longs;imilia:
<pb xlink:href="023/01/080.jpg"/><expan abbr="itemq;">itemque</expan> &longs;imilia triangula abc, efg: & acd, egh. </s><s><expan abbr="idcir-coq;">idcir­<lb/>
coque</expan> latera &longs;ibi ip&longs;is re&longs;pondentia eandem inter &longs;e&longs;e pro­<lb/>
portionem &longs;eruant. </s><s>Vt igitur duplum lateris ab unà <lb/>
cum latere ef ad duplum lateris ef unà cum ab, ita e&longs;t <lb/>
<figure id="id.023.01.080.1.jpg" xlink:href="023/01/080/1.jpg"/><lb/>
duplum ad late­<lb/>
ris una cum late­<lb/>
re eh ad duplum <lb/>
eh unà cum ad: <lb/>
& ita in aliis. </s><lb/>
<s>Rur&longs;us fru&longs;tum <lb/>
ag ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
cuius eadem e&longs;t <lb/>
ba&longs;is, & æqualis <lb/>
altitudo eandem <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> ha <lb/>
bet, quam <expan abbr="fru&longs;tũ">fru&longs;tum</expan> <lb/>
lf ad <expan abbr="pyramid&etilde;">pyramidem</expan>, <lb/>
quæ e&longs;t <expan abbr="ead&etilde;">eadem</expan> ba­<lb/>
&longs;i, & æquali alti­<lb/>
tudine: & &longs;imili­<lb/>
ter quam lh fru­<lb/>
&longs;tum ad pyrami­<lb/>
dem, quæ ex <expan abbr="ea-d&etilde;">ea­<lb/>
dem</expan> ba&longs;i, & æquali <lb/>
altitudine con­<lb/>
&longs;tat. </s><s>nam &longs;i inter <lb/>
ip&longs;as ba&longs;es me­<lb/>
diæ proportio­<lb/>
nales con&longs;tituan <lb/>
tur, tres ba&longs;es &longs;imul &longs;umptæ ad maiorem ba&longs;im in om­<lb/>
nibus codem modo &longs;e habebunt. </s><s>Vnde fit, ut axes Kl, <lb/>
qr, tu à punctis psx in eandem proportionem &longs;ecen­<lb/>
<arrow.to.target n="marg92"/><lb/>
tur. </s><s>ergo linea xs per p tran&longs;ibit: & lineæ ru, sx, qt in­<lb/>
ter &longs;e æquidi&longs;tantes erunt. </s><s>Itaque cum fru&longs;ti ag latera pro­
<pb xlink:href="023/01/081.jpg" pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y coeant, erunt <expan abbr="triã">trian</expan><lb/>
gula uyl, xyp, tyk inter &longs;e &longs;imilia: & &longs;imilia etiam triangu <lb/>
la lyr, pys, kyq quare ut in 19 huius, demon&longs;trabitur <lb/>
xp, ad ps: <expan abbr="itemq;">itemque</expan> tk ad kq eandem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam ul ad lr. </s><s>Sed ut ul ad lr, ita e&longs;t triangulum abc ad <lb/>
triangulum acd: & ut tk ad Kq, ita triangulum efg ad <lb/>
triangulum egh. </s><s>Vt autem triangulum abc ad triangu­<lb/>
lum acd, ita pyramis abcy ad pyramidem acdy. </s><s>& ut <lb/>
triangulum efg ad triangulum egh, ita pyramis efgy <lb/>
ad pyramidem eghy; ergo ut pyramis abcy ad <expan abbr="pyramid&etilde;">pyramidem</expan> <lb/>
<arrow.to.target n="marg93"/><lb/>
a cdy, ita pyramis efgy ad pyramidem eghy. </s><s>reliquum <lb/>
igitur <expan abbr="fru&longs;tũ">fru&longs;tum</expan> lf ad reliquum <expan abbr="fru&longs;tũ">fru&longs;tum</expan> lh e&longs;t ut pyramis abcy <lb/>
ad pyramidem acdy, hoc e&longs;t ut ul ad r, & ut xp ad ps. </s><lb/>
<s>Quòd cum fru&longs;ti lf centrum grauitatis &longs;its: & fru&longs;ti lh &longs;it <lb/>
<arrow.to.target n="marg94"/><lb/>
centrum x: con&longs;tat punctum p totius fru&longs;ti ag grauitatis <lb/>
e&longs;&longs;e centrum. </s><s>Eodem modo fiet demon&longs;tratio etiam in <lb/>
aliis pyramidibus.</s></p><p type="margin">
<s><margin.target id="marg92"/>a. </s><s>&longs;exti.</s></p><p type="margin">
<s><margin.target id="marg93"/>19. quinti</s></p><p type="margin">
<s><margin.target id="marg94"/>8. Archi­<lb/>
medis.<!-- KEEP S--></s></p><p type="main">
<s>Sit fru&longs;tum ad à cono, uel coni portione ab&longs;ci&longs;&longs;um, eu­<lb/>
ius maior ba&longs;is circulus, uel ellip&longs;is circa diametrum ab; <lb/>
minor circa diametrum cd: & axis ef. </s><s>diuidatur <expan abbr="aut&etilde;">autem</expan> ef <lb/>
in g, ita ut eg ad gf eandem proportionem habeat, quam <lb/>
duplum diametri ab unà cum diametro ed ad duplum cd <lb/>
unà cum ab. </s><s><expan abbr="Sitq;">Sitque</expan> gh quarta pars lineæ ge: & &longs;it &longs; K item <lb/>
quarta pars totius fe axis. </s><s>Rur&longs;us quam proportionem <lb/>
habet fru&longs;tum ad ad conum, uel coni portionem, in <expan abbr="ead&etilde;">eadem</expan> <lb/>
ba&longs;i, & æquali altitudine, habeat linea Kh ad hl. <!-- KEEP S--></s><s>Dico pun­<lb/>
ctum l fru&longs;ti ad grauitatis centrum e&longs;&longs;e. </s><s>Si enini fieri po­<lb/>
te&longs;t, &longs;it m centrum: <expan abbr="producaturq;">producaturque</expan> lm extra fru&longs;tum in n: <lb/>
& ut nl ad lm, ita fiat circulus, uel ellip&longs;is circa <expan abbr="diametrũ">diametrum</expan> <lb/>
ab ad aliud &longs;pacium, in quo &longs;it o. </s><s>Itaque in circulo, uel <lb/>
ellip&longs;i circa diametrum ab rectilinea figura plane de&longs;cri­<lb/>
batur, ita ut quæ relinquuntur portiones &longs;int o &longs;pacio mi­<lb/>
nores: & intelligatur pyramis apb, ba&longs;im habens rectili­<lb/>
neam figuram in circulo, uel ellip&longs;i ab de&longs;criptam: à qua
<pb xlink:href="023/01/082.jpg"/>fru&longs;tum pyramidis &longs;it ab&longs;ci&longs;&longs;um. </s><s>erit ex iis quæ proxime <lb/>
tradidimus, fru&longs;ti pyramidis ad centrum grauitatis l. <!-- KEEP S--></s><s>Quo <lb/>
niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir <lb/>
<figure id="id.023.01.082.1.jpg" xlink:href="023/01/082/1.jpg"/><lb/>
culus, uel ellip&longs;is ab ad <lb/>
portiones dictas <expan abbr="maior&etilde;">maiorem</expan> <lb/>
proportionem, quàm nl <lb/>
ad lm. </s><s>&longs;ed ut circulus, uel <lb/>
ellip&longs;is ab ad portiones, <lb/>
ita apb conus, uel coni <lb/>
portio ad &longs;olidas portio­<lb/>
nes, id quod &longs;upra demon <lb/>
&longs;tratum e&longs;t: & ut circulus <lb/>
<arrow.to.target n="marg95"/><lb/>
uel ellip&longs;is cd ad portio­<lb/>
nes, quæ ip &longs;i in&longs;unt, ita co <lb/>
nus, uel coni portio cpd <lb/>
ad &longs;olidas ip&longs;ius portio­<lb/>
nes. </s><s>Quòd cum figuræ in <lb/>
circulis, uel ellip&longs;ibus ab <lb/>
cd de&longs;criptæ &longs;imiles &longs;int, <lb/>
erit proportio circuli, uel <lb/>
ellip&longs;is ab ad &longs;uas portio <lb/>
nes, <expan abbr="ead&etilde;">eadem</expan>, quæ circuli uel <lb/>
ellip&longs;is cd ad &longs;uas. </s><s>ergo <lb/>
conus, uel coni portio ap <lb/>
b ad portiones &longs;olidas <expan abbr="eã-dem">ean­<lb/>
dem</expan> habet <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>
quam conus, uel coni por <lb/>
tio cpd ad &longs;olidas ip&longs;ius <lb/>
<arrow.to.target n="marg96"/><lb/>
portiones. </s><s>reliquum igi­<lb/>
tur coni, uel coni portionis <expan abbr="fru&longs;tũ">fru&longs;tum</expan>, &longs;cilicet ad ad reliquas <lb/>
portiones &longs;olidas in ip&longs;o contentas eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>
habet, quam conus, uel coni portio apb ad &longs;olidas portio <lb/>
nes: hoc e&longs;t eandem, quam circulus, uel ellip&longs;is ab ad por <lb/>
tiones planas. </s><s>quare fru&longs;tum coni, uel coni portionis ad
<pb xlink:href="023/01/083.jpg" pagenum="38"/>ad portiones &longs;olidas maiorem habet <expan abbr="proportion&etilde;">proportionem</expan>, quàm <lb/>
nl ad lm: & diuidendo fru&longs;tum pyramidis ad dictas por­<lb/>
tiones maiorem proportionem habet, quàm nm ad ml. <!-- KEEP S--></s><lb/>
<s>fiat igitur ut fru&longs;tum pyramidis ad portiones, ita qm ad <lb/>
m l. <!-- KEEP S--></s><s>Itaque quoniam à fru&longs;to coni, uel coni portionis ad, <lb/>
cuius grauitatis centrum e&longs;t m, aufertur fru&longs;tum pyrami­<lb/>
dis habens centrum l; erit reliquæ magnitudinis, quæ ex <lb/>
portionibus &longs;olidis con&longs;tat; grauitatis <expan abbr="c&etilde;trum">centrum</expan> in linea lm <lb/>
producta, atque in puncto q, extra figuram po&longs;ito: quod <lb/>
fieri nullo modo pote&longs;t. </s><s>relinquitur ergo, ut punctum l &longs;it <lb/>
fru&longs;ti ad grauitatis centrum. </s><s>quz omnia demon&longs;tranda <lb/>
proponebantur.</s></p><p type="margin">
<s><margin.target id="marg95"/>22. huius</s></p><p type="margin">
<s><margin.target id="marg96"/>19. quínti</s></p><p type="head">
<s>THEOREMA XXII. PROPOSITIO XXVII.<!-- KEEP S--></s></p><p type="main">
<s>OMNIVM &longs;olidorum in &longs;phæra de&longs;cripto­<lb/>
rum, quæ æqualibus, & &longs;imilibus ba&longs;ibus conti­<lb/>
nentur, centrum grauitatis e&longs;t idem, quod &longs;phæ­<lb/>
ræ centrum.</s></p><p type="main">
<s>Solida eiu&longs;modi corpora regularia appellare &longs;olent, de <lb/>
quibus agitur in tribus ultimis libris elementorum: &longs;unt <lb/>
autem numero quinque, tetrahedrum, uel pyramis, hexa­<lb/>
hedrum, uel cubus, octahedrum, dodecahedrum, & ico&longs;a­<lb/>
hedrum.</s></p><p type="main">
<s>Sit primo abcd pyramis <expan abbr="ĩ">im</expan> &longs;phæra de&longs;cripta, cuius &longs;phæ <lb/>
ræ centrum &longs;it e. </s><s>Dico e pyramidis abcd grauitatis e&longs;&longs;e <lb/>
centrum. </s><s>Si enim iuncta dc producatur ad ba&longs;im abc in <lb/>
f; ex iis, quæ demon&longs;trauit Campanus in quartodecimo li <lb/>
bro elementorum, propo&longs;itione decima quinta, & decima <lb/>
feptima, erit f centrum circuli circa triangulum abc de­<lb/>
fcripti: atque erit ef &longs;exta pars ip&longs;ius &longs;phæræ axis. </s><s>quare <lb/>
ex prima huius con&longs;tat trianguli abc grauitatis centrum <lb/>
e&longs;&longs;e punctum f: & idcirco lineam df e&longs;&longs;e pyramidis axem.
<pb xlink:href="023/01/084.jpg"/><figure id="id.023.01.084.1.jpg" xlink:href="023/01/084/1.jpg"/><lb/>
At cum ef &longs;it &longs;exta pars axis <lb/>
&longs;phæræ, erit d tripla ef. </s><s>ergo <lb/>
punctum e e&longs;t grauitatis cen­<lb/>
trum ip&longs;ius pyramidis: quod <lb/>
in uige&longs;ima &longs;ecunda huius de­<lb/>
mon&longs;tratum &longs;uit. </s><s>Sed e e&longs;t cen <lb/>
trum &longs;phæræ. </s><s>Sequitur igitur, <lb/>
ut centrum grauitatis pyrami­<lb/>
dis in &longs;phæra de&longs;criptæ idem <lb/>
&longs;it, quod ip&longs;ius &longs;phæræ cen­<lb/>
trum.</s></p><p type="main">
<s>Sit cubus in &longs;phæra de&longs;criptus ab, & oppo&longs;itorum pla­<lb/>
norum lateribus bifariam diui&longs;is, per puncta diui&longs;ionum <lb/>
plana ducantur, ut communis ip&longs;orum &longs;ectio &longs;it rectali­<lb/>
nea cd. <!-- KEEP S--></s><s>Itaque &longs;i ducatur ab, &longs;olidi &longs;cilicet diameter, lineæ <lb/>
ab, cd ex trige&longs;imanona undecimi&longs;e&longs;e bifariam &longs;ecabunt. </s><lb/>
<s><figure id="id.023.01.084.2.jpg" xlink:href="023/01/084/2.jpg"/><lb/>
&longs;ecent autem in puncto e. </s><s>erit, <lb/>
e <expan abbr="centrũ">centrum</expan> grauitatis &longs;olidi ab, <lb/>
id quod demon&longs;tratum e&longs;t in <lb/>
octaua huius. </s><s>Sed quoniam ab <lb/>
e&longs;t &longs;phæræ diametro æqualis, <lb/>
ut in decima quinta propo&longs;i­<lb/>
tione tertii decimilibri <expan abbr="elem&etilde;">elemen</expan> <lb/>
torum o&longs;tenditur: punctum e <lb/>
&longs;phæræ quoque centrum erit. </s><lb/>
<s>Cubi igitur in &longs;phæra de&longs;cri­<lb/>
pti grauitatis centrum idem <lb/>
e&longs;t, quod centrum ip&longs;ius &longs;phæræ.</s></p><p type="main">
<s>Sit octahedrum abcdef, in &longs;phæra de&longs;criptum, cuius <lb/>
&longs;phæræ centrum &longs;it g. </s><s>Dico punctum g ip&longs;ius octahedri <lb/>
grauitatis centrum e&longs;&longs;e. </s><s>Con&longs;tat enim ex iis, quæ demon­<lb/>
&longs;trata &longs;unt à Campano in quinto decimo libro elemento­<lb/>
rum, propo&longs;itione &longs;extadecima eiu&longs;modi &longs;olidum diuidi <lb/>
in duas pyramides æquales, & &longs;imiles; uidelicet in pyrami­
<pb xlink:href="023/01/085.jpg" pagenum="39"/>dem, cuius ba&longs;is e&longs;t quadratum abcd, & altitudo eg: & <lb/>
in pyramidem, cuius <expan abbr="ead&etilde;">eadem</expan> ba&longs;is, <expan abbr="altitudoq;">altitudoque</expan> fg; ut &longs;int eg, <lb/>
gf &longs;emidiametri &longs;phæræ, & linea una. </s><s><expan abbr="Cũ">Cum</expan> igitur g &longs;it &longs;phæ­<lb/>
ræ centrum, erit etiam centrum circuli, qui circa <expan abbr="quadratũ">quadratum</expan> <lb/>
abcd de&longs;cribitur: & propterea eiu&longs;dem quadrati grauita <lb/>
tis centrum: quod in prima propo&longs;itione huius demon­<lb/>
&longs;tratum e&longs;t. </s><s>quare pyramidis abcde axis erit eg: & pyra <lb/>
midis abcdf axis fg. <!-- KEEP S--></s><s>Itaque &longs;it h centrum grauitatis py­<lb/>
ramidis abcde, & pyramidis abcdf centrum &longs;it <emph type="italics"/>K:<emph.end type="italics"/> per­<lb/>
&longs;picuum e&longs;t ex uige&longs;ima &longs;ecunda propo&longs;itione huius, <expan abbr="lineã">lineam</expan> <lb/>
<figure id="id.023.01.085.1.jpg" xlink:href="023/01/085/1.jpg"/><lb/>
ch triplam e&longs;&longs;e hg: <expan abbr="cõ">com</expan> <lb/>
<expan abbr="ponendoq;">ponendoque</expan> eg ip&longs;ius g <lb/>
h quadruplam. </s><s>& <expan abbr="ead&etilde;">eadem</expan> <lb/>
ratione fg <expan abbr="quadruplã">quadruplam</expan> <lb/>
ip&longs;ius gk quod cum e <lb/>
g, gf &longs;int æquales, & h <lb/>
g, g <emph type="italics"/>K<emph.end type="italics"/> nece&longs;&longs;ario æqua­<lb/>
les erunt. </s><s>ergo ex quar <lb/>
ta propo&longs;itione primi <lb/>
libri Archimedis de <expan abbr="c&etilde;-tro">cen­<lb/>
tro</expan> grauitatis <expan abbr="planorũ">planorum</expan>, <lb/>
totius octahedri, quod <lb/>
ex dictis pyramidibus <lb/>
con&longs;tat, centrum graui <lb/>
tatis erit punctum g idem, quod ip&longs;ius &longs;phæræ centrum.</s></p><p type="main">
<s>Sit ico&longs;ahedrum ad de&longs;criptum in &longs;phæra, cuius <expan abbr="centrũ">centrum</expan> <lb/>
&longs;it g. <!-- KEEP S--></s><s>Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. </s><s>Si <lb/>
enim ab angulo a per g ducatur recta linea u&longs;que ad &longs;phæ <lb/>
ræ &longs;uperficiem; con&longs;tat ex &longs;exta decima propo&longs;itione libri <lb/>
tertii decimi elementorum, cadere eam in angulum ip&longs;i a <lb/>
oppo&longs;itum. </s><s>cadat in d: <expan abbr="&longs;itq;">&longs;itque</expan> una aliqua ba&longs;is ico&longs;ahedri tri­<lb/>
angulum abc: & iunctæ bg, producantur, & cadant in <lb/>
angulos ef, ip&longs;is bc oppo&longs;itos. </s><s>Itaque per triangula <lb/>
abc, def ducantur plana &longs;phæram &longs;ecantia.</s><s> erunt hæ &longs;e-
<pb xlink:href="023/01/086.jpg"/>ctiones circuli ex prima propo&longs;itione &longs;phæricorum Theo <lb/>
do&longs;ii: unus quidem circa triangulum abc de&longs;criptus: al­<lb/>
ter uero circa def: & quoniam triangula abc, def æqua­<lb/>
lia &longs;unt, & &longs;imilia; erunt ex prima, & &longs;ecunda propo&longs;itione <lb/>
duodecimi libri elementorum, circuli quoque inter &longs;e &longs;e <lb/>
æquales. </s><s>po&longs;tremo a centro g ad circulum abc perpendi <lb/>
cularis ducatur gh; & alia perpendicularis ducatur ad cir <lb/>
culum def, quæ &longs;it gk; & iungantur ah, dk per&longs;picuum <lb/>
e&longs;t ex corollario primæ &longs;phæricorum Theodo&longs;ii, punctum <lb/>
h centrum e&longs;&longs;e circuli abc, & k centrum circuli def. </s><s>Quo <lb/>
niam igitur triangulorum gah, gdK latus ag e&longs;t æquale la <lb/>
teri gd; &longs;unt enim à centro &longs;phæræ ad &longs;uperficiem: atque <lb/>
e&longs;t ah æquale dk: & ex &longs;exta propo&longs;itione libri primi &longs;phæ <lb/>
ricorum Theodo&longs;ii gh ip&longs;i gK: triangulum gah æquale <lb/>
erit, & &longs;imile gdk triangulo: & angulus agh æqualis an­<lb/>
<arrow.to.target n="marg97"/><lb/>
gulo dg <emph type="italics"/>K.<emph.end type="italics"/> &longs;ed anguli agh, hgd &longs;unt æquales duobus re­<lb/>
ctis. </s><s>ergo & ip&longs;i hgd, dgk duobus rectis æquales erunt. </s><lb/>
<s><arrow.to.target n="marg98"/><lb/>
& idcirco hg, g <emph type="italics"/>K<emph.end type="italics"/> una, atque eadem erit linea. </s><s>cum autem <lb/>
<figure id="id.023.01.086.1.jpg" xlink:href="023/01/086/1.jpg"/><lb/>
h &longs;it <expan abbr="centrũ">centrum</expan> circuli, & tri­<lb/>
anguli abc grauitatis cen <lb/>
<expan abbr="trũ">trum</expan> probabitur ex iis, quæ <lb/>
in prima propo&longs;itione hu <lb/>
ius tradita &longs;unt. </s><s>quare gh <lb/>
erit pyramidis abcg axis. </s><lb/>
<s>& ob eandem cau&longs;&longs;am gk <lb/>
axis pyramidis defg. <!-- KEEP S--></s><s>lta­<lb/>
que centrum grauitatls py <lb/>
ramidis abcg &longs;it <expan abbr="pũctum">punctum</expan> <lb/>
l, & pyramidis defg &longs;it m. </s><lb/>
<s>Similiter ut &longs;upra demon­<lb/>
&longs;trabimus mg, gl inter &longs;e æquales e&longs;&longs;e, & punctum g graui <lb/>
tatis centrum magnitudinis, quæ ex utri&longs;que pyramidibus <lb/>
con&longs;tat. </s><s>eodem modo demon&longs;trabitur, quarumcunque <lb/>
duarum pyramidum, quæ opponuntur, grauitatis <expan abbr="centrũ">centrum</expan>
<pb xlink:href="023/01/087.jpg" pagenum="40"/>e&longs;&longs;e punctum g. <!-- KEEP S--></s><s>Sequitur ergo ut ico&longs;ahedri centrum gra<lb/>
uitatis &longs;it idem, quod ip&longs;ius &longs;phæræ centrum.</s></p><p type="margin">
<s><margin.target id="marg97"/>13. primi</s></p><p type="margin">
<s><margin.target id="marg98"/>14. primi</s></p><p type="main">
<s>Sit dodecahedrum af in &longs;phæra de&longs;ignatum, &longs;itque &longs;phæ <lb/>
ræ centrum m. </s><s>Dico m centrum e&longs;&longs;e grauitatis ip&longs;ius do­<lb/>
decahedri. </s><s>Sit enim pentagonum abcde una ex duode­<lb/>
cim ba&longs;ibus &longs;olidi af: & iuncta am producatur ad &longs;phæræ <lb/>
&longs;uperficiem. </s><s>cadet in angulum ip&longs;i a oppo&longs;itum; quod col­<lb/>
ligitur ex decima &longs;eptima propo&longs;itione tertiidecimi libri <lb/>
elementorum. </s><s>cadat in f. </s><s>at &longs;i ab aliis angulis bcde per <expan abbr="c&etilde;">cen</expan> <lb/>
trum itidem lineæ ducantur ad &longs;uperficiem &longs;phæræ in pun <lb/>
cta ghkl; cadent hæ in alios angulos ba&longs;is, quæ ip&longs;i abcd <lb/>
ba&longs;i opponitur. </s><s>tran&longs;eant ergo per pentagona abcde, <lb/>
fghKl plana &longs;phæram &longs;ecantia, quæ facient &longs;ectiones cir­<lb/>
culos æquales inter &longs;e &longs;e: po&longs;tea ducantur ex centro &longs;phæræ <lb/>
<figure id="id.023.01.087.1.jpg" xlink:href="023/01/087/1.jpg"/><lb/>
m perpendiculares ad pla­<lb/>
na dictorum <expan abbr="circulorũ">circulorum</expan>; ad <lb/>
circulum quidem abcde <lb/>
perpendicularis mn: & ad <lb/>
circulum fghKl ip&longs;a mo, <lb/>
<arrow.to.target n="marg99"/><lb/>
erunt puncta no <expan abbr="circulorũ">circulorum</expan> <lb/>
centra: & lineæ mn, mo in <lb/>
ter &longs;e æquales: quòd circu­<lb/>
<arrow.to.target n="marg100"/><lb/>
li æquales &longs;int. </s><s>Eodem mo <lb/>
do, quo &longs;upra, demon&longs;trabi <lb/>
mus lineas mn, mo in <expan abbr="unã">unam</expan> <lb/>
atque eandem lineam con­<lb/>
uenire. </s><s>ergo cum puncta no &longs;int centra circulorum, con­<lb/>
&longs;tat ex prima huius & <expan abbr="pentagonorũ">pentagonorum</expan> grauitatis e&longs;&longs;e centra: <lb/>
<expan abbr="idcircoq;">idcircoque</expan> mn, mo pyramidum abcdem, fghklm axes. </s><lb/>
<s>ponatur abcdem pyramidis grauitatis centrum p: & py <lb/>
ramidis fghklm ip&longs;um q centrum. </s><s>erunt pm, mq æqua­<lb/>
les, & punctum m grauitatis centrum magnitudinis, quæ <lb/>
ex ip&longs;is pyramidibus con&longs;tat. </s><s><expan abbr="eod&etilde;">eodem</expan> modo probabitur qua­<lb/>
rumlibet pyramidum, quæ è regione opponuntur, <expan abbr="centrũ">centrum</expan>
<pb xlink:href="023/01/088.jpg"/>grauitatis e&longs;&longs;e punctum m. </s><s>patet igitur totius dodecahe­<lb/>
dri, centrum grauitatis <expan abbr="id&etilde;">idem</expan> e&longs;&longs;e, quod & &longs;phæræ ip&longs;um com <lb/>
prehendentis centrum. </s><s>quæ quidem omnia demon&longs;tra&longs;&longs;e <lb/>
oportebat.</s></p><p type="margin">
<s><margin.target id="marg99"/>corol. </s><s>pri<lb/>
mæ &longs;phæ<lb/>
ricorum <lb/>
Theod.<!-- REMOVE S--><margin.target id="marg100"/>6. primi <lb/>
phærico <lb/>
rum.</s></p><p type="head">
<s>PROBLEMA VI. PROPOSITIO XXVIII.<!-- KEEP S--></s></p><p type="main">
<s>DATA qualibet portione conoidis rectangu <lb/>
li, ab&longs;ci&longs;&longs;a plano ad axem recto, uel non recto; fie­<lb/>
ri pote&longs;t, ut portio &longs;olida in&longs;cribatur, uel circum­<lb/>
&longs;cribatur ex cylindris, uel cylindri portionibus, <lb/>
æqualem habentibus altitudinem, ita ut recta li­<lb/>
nea, quæ inter centrum grauitatis portionis, & <lb/>
figuræ in&longs;criptæ, uel circum&longs;criptæ interiicitur, <lb/>
&longs;it minor qualibet recta linea propo&longs;ita.</s></p><p type="main">
<s>Sit portio conoidis rectanguli abc, cuius axis bd, <expan abbr="gra-uitatisq;">gra­<lb/>
uitatisque</expan> centrum e: & &longs;it g recta linea propo&longs;ita. </s><s>quam ue <lb/>
ro proportionem habet linea be ad lineam g, eandem ha­<lb/>
beat portio conoidis ad &longs;olidum h: & circum&longs;cribatur por <lb/>
tioni figura, &longs;icuti dictum e&longs;t, ita ut portiones reliquæ &longs;int <lb/>
&longs;olido h minores: cuius quidem figuræ centrum grauitatis <lb/>
&longs;it punctum k. </s><s>Dico <expan abbr="lineã">lineam</expan> ke minorem e&longs;&longs;e linea g propo­<lb/>
&longs;ita. </s><s>ni&longs;i enim &longs;it minor, uel æqualis, uel maior erit. </s><s>& quo­<lb/>
niam figura circum&longs;cripta ad reliquas portiones maiorem <lb/>
<arrow.to.target n="marg101"/><lb/>
proportionem habet, quàm portio conoidis ad &longs;olidum h; <lb/>
hoc e&longs;t maiorem, quàm bc ad g: & be ad g non minorem <lb/>
habet proportionem, quàm ad ke, propterea quod ke non <lb/>
ponitur minor ip&longs;a g: habebit figura circum&longs;cripta ad por <lb/>
tiones reliquas maiorem proportionem quàm be ad ek: <lb/>
<arrow.to.target n="marg102"/><lb/>
& diuidendo portio conoidis ad reliquas portiones habe­<lb/>
bit maiorem, quàm bk ad Ke. </s><s>quare &longs;i fiat ut portio co­
<pb xlink:href="023/01/089.jpg" pagenum="41"/>noidis ad portiones reliquas, ita alia linea, quæ &longs;it lk ad <lb/>
ke: erit lk maior, quam bk: & ideo punctum l extra por­<lb/>
<figure id="id.023.01.089.1.jpg" xlink:href="023/01/089/1.jpg"/><lb/>
tionem cadet. </s><s><expan abbr="Quoniã">Quoniam</expan> <lb/>
igitur à figura circum­<lb/>
&longs;cripta, cuius grauitatis <lb/>
centrum e&longs;t k, aufertur <lb/>
portio conoidis, cuius <lb/>
centrum e. </s><s><expan abbr="habetq;">habetque</expan> lK <lb/>
ad Ke eam proportio­<lb/>
nem, quam portio co­<lb/>
noidis ad reliquas por­<lb/>
tiones; erit punctum l <lb/>
extra portionem <expan abbr="cad&etilde;s">cadens</expan>, <lb/>
centrum magnitudinis <lb/>
ex reliquis portionibus compo&longs;itæ. </s><s>illud autem fieri nullo <lb/>
modo pote&longs;t. </s><s>quare con&longs;tat lineam ke ip&longs;a g linea propo&longs;i <lb/>
ta minorem e&longs;&longs;e.</s></p><p type="margin">
<s><margin.target id="marg101"/>8. quínti.</s></p><p type="margin">
<s><margin.target id="marg102"/>29. quínti <lb/>
ex tradi­<lb/>
tione <expan abbr="Cã-l">Can­<lb/>
l</expan> ani.</s></p><p type="main">
<s>Rur&longs;us in&longs;cribatur portioni figura, uidelicet cylindrus <lb/>
<figure id="id.023.01.089.2.jpg" xlink:href="023/01/089/2.jpg"/><lb/>
mn, ut &longs;it ip&longs;ius altitudo <lb/>
æqualis dimidio axis bd: <lb/>
& quam proportionem <lb/>
habet be ad g, habeat mn <lb/>
cylindrus ad &longs;olidum o. </s><lb/>
<s>in&longs;cribatur deinde eidem <lb/>
alia figura, ita ut portio­<lb/>
nes reliquæ &longs;int &longs;olido o <lb/>
minores: & centrum gra<lb/>
uitatis figuræ &longs;it p. </s><s>Dico <lb/>
lineam pe ip&longs;a g <expan abbr="minor&etilde;">minorem</expan> <lb/>
e&longs;&longs;e. </s><s>&longs;i enim non &longs;it mi­<lb/>
nor, codem, quo &longs;upra modo demon&longs;trabimus figuram in <lb/>
&longs;criptam ad reliquas portiones maiorem proportionem <lb/>
habere, quàm be ad ep. </s><s>& &longs;i fiat alia linea le ad ep, ut e&longs;t <lb/>
figura in&longs;cripta ad reliquas portiones, <expan abbr="pũctum">punctum</expan> l extra por
<pb xlink:href="023/01/090.jpg"/>tionem cadet: Itaque cum à portione conoidis, cuius gra­<lb/>
uitatis centrum e auferatur in&longs;cripta figura, centrum ha­<lb/>
bens p: & &longs;it le ad ep, ut figura in&longs;cripta ad portiones reli <lb/>
quas: erit magnitudinis, quæ ex reliquis portionibus con <lb/>
&longs;tat, centrum grauitatis punctum l, extra portionem ca­<lb/>
dens. </s><s>quod fieri nequit. </s><s>ergo linea pe minor e&longs;tip&longs;a g li­<lb/>
nea propo&longs;ita.</s></p><p type="main">
<s>Ex quibus per&longs;picuum e&longs;t centrum grauitatis <lb/>
figuræ in&longs;criptæ, & circum&longs;criptæ eo magis acce <lb/>
dere ad portionis centrum, quo pluribus cylin­<lb/>
dris, uel cylindri portionibus con&longs;tet: <expan abbr="fiat&qacute;">fiatque</expan>; figu <lb/>
rain&longs;cripta maior, & circum&longs;cripta minor. </s><s>& <lb/>
quanquam continenter ad portionis <expan abbr="centrũ">centrum</expan> pro­<lb/>
pius admoueatur: nunquam tamen ad ip&longs;um per <lb/>
ueniet. </s><s>&longs;equeretur enim figuram in&longs;criptam, <expan abbr="nõ">non</expan> <lb/>
&longs;olum portioni, &longs;ed etiam circum&longs;criptæ figuræ <lb/>
æqualem e&longs;&longs;e. </s><s>quod e&longs;t ab&longs;urdum.</s></p><p type="head">
<s>THEOREMA XXIII. PROPOSITIO XXIX.<!-- KEEP S--></s></p><p type="main">
<s>CVIVSLIBET portionis conoidis rectangu­<lb/>
li axis à <expan abbr="c&etilde;tro">centro</expan> grauitatis ita diuiditur, ut pars quæ <lb/>
terminatur ad uerticem, reliquæ partis, quæ ad ba <lb/>
&longs;im &longs;it dupla.</s></p><p type="main">
<s>SIT portio conoidis rectanguli uel ab&longs;ci&longs;&longs;a plano ad <lb/>
axem recto, uel non recto: & &longs;ecta ip&longs;a altero plano per <expan abbr="ax&etilde;">axem</expan><lb/>
&longs;it &longs;uperficici &longs;ectio abc rectanguli coni &longs;ectio, uel parabo <lb/>
le; plani ab&longs;cindentis portionem &longs;ectio &longs;it recta linea ac: <lb/>
axis portionis, & &longs;ectionis diameter bd. </s><s>Sumatur autem <lb/>
in linea bd punctum e, ita ut be &longs;it ip&longs;ius ed dupla. </s><s>Dico
<pb xlink:href="023/01/091.jpg" pagenum="42"/><figure id="id.023.01.091.1.jpg" xlink:href="023/01/091/1.jpg"/><lb/>
e portionis ab <lb/>
c grauitatis e&longs;&longs;e <lb/>
centrum. </s><s>Diui­<lb/>
datur enim bd <lb/>
bifariam in m: <lb/>
& rur&longs;us dm, m <lb/>
b bifariam diui­<lb/>
dantur in pun­<lb/>
ctis n, o: <expan abbr="in&longs;cri-baturq;">in&longs;cri­<lb/>
baturque</expan> portio­<lb/>
ni figura &longs;olida, <lb/>
& altera circum <lb/>
&longs;cribatur ex cy­<lb/>
lindris æqualem <lb/>
altitudinem ha­<lb/>
bentibus, ut &longs;u­<lb/>
perius <expan abbr="dictũ">dictum</expan> e&longs;t'. </s><lb/>
<s>Sit autem pri­<lb/>
mum figura in­<lb/>
&longs;cripta <expan abbr="cylĩdrus">cylindrus</expan> <lb/>
f g: & <expan abbr="circũ&longs;cri">circum&longs;cri</expan>
­<lb/> ex cylindris <lb/>
ah, Kl con&longs;tet. </s><lb/>
<s><arrow.to.target n="marg103"/><lb/>
punctum n erit <lb/>
centrum graui­<lb/>
tatis figuræ in­<lb/>
&longs;criptæ, <expan abbr="mediũ">medium</expan> <lb/>
&longs;cilicet ip&longs;ius d <lb/>
m axis: <expan abbr="atq;">atque</expan> <expan abbr="id&etilde;">idem</expan> <lb/>
erit centrum cy <lb/>
lindri ah: & cy­<lb/>
lindri kl <expan abbr="centrũ">centrum</expan> <lb/>
o, axis bm me­<lb/>
dium. </s><s>quare &longs;i li
<pb xlink:href="023/01/092.jpg"/><figure id="id.023.01.092.1.jpg" xlink:href="023/01/092/1.jpg"/><lb/>
neam on ita di <lb/>
ui&longs;erimus in p, <lb/>
ut <expan abbr="quã">quam</expan> <expan abbr="propor-tion&etilde;">propor­<lb/>
tionem</expan> habet cy­<lb/>
lindrus ah ad <lb/>
cylindrum kl, <lb/>
habeat linea op <lb/>
<arrow.to.target n="marg104"/><lb/>
ad pn: centrum <lb/>
grauitatis toti­<lb/>
us figuræ <expan abbr="circũ-&longs;criptæ">circum­<lb/>
&longs;criptæ</expan> erit pun <lb/>
<arrow.to.target n="marg105"/><lb/>
ctum p. </s><s>Sed cy­<lb/>
lindri, qui &longs;unt <lb/>
æquali altitudi­<lb/>
ne, eandem in­<lb/>
ter &longs;e &longs;e, quam <lb/>
ba&longs;es propor— <lb/>
tionem habent: <lb/>
<expan abbr="e&longs;tq;">e&longs;tque</expan> ut linea db <lb/>
ad bm, ita <expan abbr="qua-dratũ">qua­<lb/>
dratum</expan> lineæ ad <lb/>
ad <expan abbr="quadratũ">quadratum</expan> ip­<lb/>
&longs;ius Km, ex uige <lb/>
&longs;ima primi libri <lb/>
<arrow.to.target n="marg106"/><lb/>
<expan abbr="conicorũ">conicorum</expan> & ita <lb/>
quadratum ac <lb/>
ad <expan abbr="quadratũ">quadratum</expan> K <lb/>
<arrow.to.target n="marg107"/><lb/>
g: hoc e&longs;t circu­<lb/>
lus circa diame<lb/>
trum ac ad cir­<lb/>
culum circa dia <lb/>
metrum kg. <!-- KEEP S--></s><s>du <lb/>
pla e&longs;t autem li­<lb/>
nea db lineæ
<pb xlink:href="023/01/093.jpg" pagenum="43"/>bm. </s><s>ergo circulus ac circuli kg: & idcirco cylindrus <lb/>
ah cylindri k. </s><s>l duplus erit. </s><s>quare & linea op dupla <lb/>
ip&longs;ius pn. </s><s>Deinde in&longs;cripta & circum&longs;cripta portioni <lb/>
alia figura, ita ut in&longs;cripta con&longs;tituatur ex tribus cylin­<lb/>
dris qr, sg, tu: circum&longs;cripta uero ex quatuor ax, yz, <lb/>
K<foreign lang="greek">v, ql:</foreign> diuidantur bo, om, mn, nd bifariam in punctis <lb/>
<foreign lang="greek">mnpr.</foreign> Itaque cylindri <foreign lang="greek">ql</foreign> centrum grauitatis e&longs;t punctum <lb/>
<foreign lang="greek">m:</foreign> & cylindri k<foreign lang="greek">h</foreign> centrum <foreign lang="greek">n.</foreign> ergo &longs;i linea <foreign lang="greek">mg</foreign> diuidatur in <foreign lang="greek">s,</foreign><lb/>
ita ut <foreign lang="greek">ms</foreign> ad <foreign lang="greek">sg</foreign> <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="eã">eam</expan> habeat, quam cylindrus K<foreign lang="greek">h</foreign><lb/>
ad cylindrum <foreign lang="greek">ql,</foreign> uidelicet quam quadratum knr ad qua­<lb/>
<arrow.to.target n="marg108"/><lb/>
dratum <foreign lang="greek">q</foreign>o, hoc e&longs;t, quam linea mb ad bo: erit <foreign lang="greek">s</foreign> centrum <lb/>
magnitudinis compo&longs;itæ ex cylindris <foreign lang="greek">kg, ql.</foreign> & cum linea <lb/>
mb &longs;it dupla bo, erit & <foreign lang="greek">ms</foreign> ip&longs;ius <foreign lang="greek">sn</foreign> dupla. </s><s>præterea quo­<lb/>
niam cylindri yz centrum grauitatis e&longs;t <foreign lang="greek">p,</foreign> linea <foreign lang="greek">sp</foreign> ita diui <lb/>
&longs;ain <foreign lang="greek">t,</foreign> ut <foreign lang="greek">st</foreign> ad <foreign lang="greek">tp</foreign> eam habeat proportionem, quam cylin <lb/>
drus yz ad duos cylindros K<foreign lang="greek">n, ql:</foreign> erit <foreign lang="greek">t</foreign> centrum magnitu <lb/>
dinis, quæ ex dictis tribus cylindris con&longs;tat. </s><s>cylindrus <expan abbr="au-t&etilde;">au­<lb/>
tem</expan> yz ad cylindrum <foreign lang="greek">ql</foreign> e&longs;t, utlinea nb ad bo, hoc e&longs;t ut 3 <lb/>
ad 1: & ad cylindrum k<foreign lang="greek">h</foreign>, ut nb ad bm, uidelicet ut 3 ad 2. <!-- KEEP S--></s><lb/>
<s>quare yz <expan abbr="cylĩdrus">cylindrus</expan> duobus cylindris k<foreign lang="greek">n, ql</foreign> æqualis erit. </s><s>& <lb/>
propterea linea <foreign lang="greek">st</foreign> æqualis ip&longs;i <foreign lang="greek">tp.</foreign> denique cylindri ax <lb/>
centrum grauitatis e&longs;t punctum <foreign lang="greek">r.</foreign> & cum <foreign lang="greek">tr</foreign> diui&longs;a fuerit <lb/>
in <expan abbr="eã">eam</expan> proportionem, quam habet cylindrus ax ad tres cy­<lb/>
lindros yz, k<foreign lang="greek">n, ql:</foreign> erit in eo puncto centrum grauitatis <lb/>
totius figuræ <expan abbr="circũ&longs;criptæ">circum&longs;criptæ</expan>. </s><s>Sed cylindrus ax ad ip&longs;um yz <lb/>
e&longs;t ut linea db ad bn: hoc e&longs;t ut 4 ad 3: & duo cylindri k<foreign lang="greek">h<lb/>
ql</foreign> cylindro y &longs;unt æquales. </s><s>cylindrus igitur ax ad tres <lb/>
iam dictos cylindros e&longs;t ut 2 ad 3. Sed <expan abbr="quoniã">quoniam</expan> <foreign lang="greek">m s</foreign> e&longs;t dua­<lb/>
rum partium, & <foreign lang="greek">s g</foreign> unius, qualium <foreign lang="greek">m p</foreign> e&longs;t &longs;ex; erit <foreign lang="greek">s p</foreign> par­<lb/>
tium quatuor: <expan abbr="proptereaq;">proptereaque</expan> <foreign lang="greek">tp</foreign> duarum, & <foreign lang="greek">np,</foreign> hoc e&longs;t <foreign lang="greek">pr</foreign><lb/>
trium. </s><s>quare &longs;equitur ut punctum <foreign lang="greek">p</foreign> totius figuræ circum <lb/>
&longs;criptæ &longs;it centrum. </s><s>Itaque fiat <foreign lang="greek">nu</foreign> ad <foreign lang="greek">up,</foreign> ut <foreign lang="greek">ms</foreign> ad <foreign lang="greek">sg.</foreign> & <foreign lang="greek">ur</foreign><lb/>
bifariam diuidatur in <foreign lang="greek">f.</foreign> Similiter ut in circum&longs;cripta figu <lb/>
ra o&longs;tendetur centrum magnitudinis compo&longs;itæ ex cylin-
<pb xlink:href="023/01/094.jpg"/><figure id="id.023.01.094.1.jpg" xlink:href="023/01/094/1.jpg"/><lb/>
dris sg, tu e&longs;&longs;e <lb/>
punctum <foreign lang="greek">u:</foreign> & <lb/>
totius figuræ in <lb/>
&longs;criptæ, quæ <expan abbr="cõ-&longs;tat">con­<lb/>
&longs;tat</expan> ex cylindris <lb/>
qr, &longs; g, tu e&longs;&longs;e <foreign lang="greek">f</foreign><lb/>
centrum. </s><s>Sunt <lb/>
enim hi cylindri <lb/>
æquales & &longs;imi­<lb/>
les cylindris yz, <lb/>
K<foreign lang="greek">h, ql,</foreign> figuræ <lb/>
circum&longs;criptæ. </s><lb/>
<s><expan abbr="Quoniã">Quoniam</expan> igitur <lb/>
ut be ad ed, ita <lb/>
e&longs;t op ad pn; <lb/>
<expan abbr="utraq;">utraque</expan> enim u­<lb/>
triu&longs;que e&longs;t du­<lb/>
pla: erit compo <lb/>
nendo, ut bd ad <lb/>
de, ita on ad n <lb/>
p; & permutan <lb/>
do, ut bd ad o<lb/>
n, ita de ad np. </s><lb/>
<s>Sed bd dupla <lb/>
e&longs;t on. </s><s>ergo & <lb/>
ed ip&longs;ius np du <lb/>
pla erit. </s><s>quòd &longs;i <lb/>
ed bifariam di­<lb/>
uidatur <expan abbr="ĩ">im</expan> <foreign lang="greek">x,</foreign> erit <lb/>
<foreign lang="greek">x</foreign> d, uel e <foreign lang="greek">x</foreign> æ­<lb/>
qualis np: & <lb/>
&longs;ublata en, quæ <lb/>
e&longs;t <expan abbr="cõmunis">communis</expan> u­<lb/>
trique e <foreign lang="greek">x,</foreign> pn,
<pb xlink:href="023/01/095.jpg" pagenum="44"/>relinquetur pe ip&longs;i n<foreign lang="greek">x</foreign> æqualis. </s><s>cum autem be &longs;it dupla <lb/>
ed, & op dupla pn, hoc e&longs;t ip&longs;ius e <foreign lang="greek">x,</foreign> & reliquum, uideli­<lb/>
<arrow.to.target n="marg109"/><lb/>
cet bo unà cum pe ip&longs;ius reliqui <foreign lang="greek">x</foreign> d duplum erit. </s><s>e&longs;tque <lb/>
bo dupla <foreign lang="greek">r</foreign> d. <!-- KEEP S--></s><s>ergo pe, hoc e&longs;t n<foreign lang="greek">x</foreign> ip&longs;ius <foreign lang="greek">xr</foreign> dupla. </s><s>&longs;ed dn <lb/>
dupla e&longs;t n<foreign lang="greek">r.</foreign> reliqua igitur d<foreign lang="greek">x</foreign> dupla reliquæ <foreign lang="greek">x</foreign> n. </s><s>&longs;unt au­<lb/>
tem d<foreign lang="greek">x,</foreign> pn inter &longs;e æquales: <expan abbr="itemq;">itemque</expan> æquales <foreign lang="greek">x</foreign> n, pe. </s><s>qua­<lb/>
re con&longs;tat np ip&longs;ius pe duplam e&longs;&longs;e. </s><s>& idcirco pe ip&longs;i en <lb/>
æqualem. </s><s>Rur&longs;us cum &longs;it <foreign lang="greek">mn</foreign> dupla o<foreign lang="greek">n,</foreign> & <foreign lang="greek">m s</foreign> dupla <foreign lang="greek">s g;</foreign> erit <lb/>
etiam reliqua <foreign lang="greek">ns</foreign> reliquæ <foreign lang="greek">s</foreign> o dupla. </s><s>Eadem quoque ratione <lb/>
<expan abbr="cõcludetur">concludetur</expan> <foreign lang="greek">p u</foreign> dupla <foreign lang="greek">u</foreign> m. </s><s>ergo ut <foreign lang="greek">ns</foreign> ad <foreign lang="greek">s</foreign> o, ita <foreign lang="greek">pu</foreign> ad <foreign lang="greek">u</foreign> m: <lb/>
<expan abbr="componendoq;">componendoque</expan>, & permutando, ut <foreign lang="greek">n</foreign>o ad <foreign lang="greek">p</foreign>m, ita o<foreign lang="greek">s</foreign> ad <lb/>
m<foreign lang="greek">u:</foreign> & &longs;unt æquales <foreign lang="greek">n</foreign>o, <foreign lang="greek">p</foreign>m. </s><s>quare & o<foreign lang="greek">s,</foreign> m<foreign lang="greek">u</foreign> æquales. </s><s>præ <lb/>
terea <foreign lang="greek">sp</foreign> dupla e&longs;t <foreign lang="greek">pt,</foreign> & <foreign lang="greek">np</foreign> ip&longs;ius <foreign lang="greek">p</foreign>m. </s><s>reliquaigitur <foreign lang="greek">sn</foreign> re <lb/>
liquæ m<foreign lang="greek">t</foreign> dupla. </s><s>atque erat <foreign lang="greek">ns</foreign> dupla <foreign lang="greek">s</foreign>o. </s><s>ergo m<foreign lang="greek">t, s</foreign>o æ­<lb/>
quales &longs;unt: & ita æquales m<foreign lang="greek">u,</foreign> n<foreign lang="greek">f.</foreign> at o<foreign lang="greek">s,</foreign> e&longs;t æqualis <lb/>
m<foreign lang="greek">u.</foreign> Sequitur igitur, ut omnes o<foreign lang="greek">s,</foreign> m<foreign lang="greek">t,</foreign> m<foreign lang="greek">u,</foreign> n<foreign lang="greek">f</foreign> in­<lb/>
ter &longs;e &longs;int æquales. </s><s>Sed ut <foreign lang="greek">rp</foreign> ad <foreign lang="greek">pt,</foreign> hoc e&longs;t ut 3 ad 2, ita nd <lb/>
ad d<foreign lang="greek">x:</foreign> <expan abbr="permutãdoq;">permutandoque</expan> ut <foreign lang="greek">rp</foreign> ad nd, ita <foreign lang="greek">pt</foreign> ad d<foreign lang="greek">x.</foreign> & <expan abbr="&longs;ũt">&longs;unt</expan> æqua <lb/>
les <foreign lang="greek">rp,</foreign> nd. <!-- KEEP S--></s><s>ergo d<foreign lang="greek">x,</foreign> hoc e&longs;t np, & <foreign lang="greek">pt</foreign> æquales. </s><s>Sed etiam æ­<lb/>
quales n<foreign lang="greek">p, p</foreign>m. </s><s>reliqua igitur <foreign lang="greek">p</foreign> preliquæ m<foreign lang="greek">t,</foreign> hoc e&longs;t ip&longs;i <lb/>
n<foreign lang="greek">f</foreign> æqualis erit. </s><s>quare dempta p<foreign lang="greek">p</foreign> ex pe, & <foreign lang="greek">f</foreign>n dempta ex <lb/>
ne, relinquitur pe æqualis e<foreign lang="greek">f.</foreign> Itaque <foreign lang="greek">p, f</foreign> centra <expan abbr="figurarũ">figurarum</expan> <lb/>
&longs;ecundo loco de&longs;criptarum a primis centris pn æquali in­<lb/>
teruallo recedunt. </s><s>quòd &longs;i rur&longs;us aliæ figuræ de&longs;cribantur, <lb/>
codem modo demon&longs;trabimus earum centra æqualiter ab <lb/>
his recedere, & ad portionis conoidis centrum propius ad <lb/>
moueri. </s><s>Ex quibus con&longs;tat lineam <foreign lang="greek">pf</foreign> à centro grauitatis <lb/>
portionis diuidi in partes æquales. </s><s>Si enim fieri pote&longs;t, non <lb/>
&longs;it centrum in puncto e, quod e&longs;t lineæ <foreign lang="greek">pf</foreign> medium: &longs;ed in <lb/>
<foreign lang="greek">y:</foreign> & ip&longs;i <foreign lang="greek">py</foreign> æqualis fiat <foreign lang="greek">fw.</foreign> Cum igitur in portione &longs;olida <lb/>
quædam figura in&longs;cribi pos&longs;it, ita ut linea, quæ inter cen­<lb/>
trum grauitatis portionis, & in&longs;criptæ figuræ interiicitur, <lb/>
qualibet linea propo&longs;ita &longs;it minor, quod proxime demon­<lb/>
&longs;trauimus: perueniet tandem <foreign lang="greek">f</foreign> centrum in&longs;criptæ figuræ
<pb xlink:href="023/01/096.jpg"/><figure id="id.023.01.096.1.jpg" xlink:href="023/01/096/1.jpg"/>
<pb xlink:href="023/01/097.jpg" pagenum="45"/>ad punctum <foreign lang="greek">w.</foreign> Sed quoniam <foreign lang="greek">p</foreign> circum&longs;cripta itidem alia <lb/>
figura æquali interuallo ad portionis centrum accedit, ubi <lb/>
primum <foreign lang="greek">f</foreign> applicuerit &longs;e ad <foreign lang="greek">w,</foreign> & <foreign lang="greek">p</foreign> ad <expan abbr="punctũ">punctum</expan> <foreign lang="greek">y,</foreign> hoc e&longs;t ad <lb/>
portionis centrum &longs;e applicabit. </s><s>quod fieri nullo modo <lb/>
po&longs;&longs;e per&longs;picuum e&longs;t. </s><s>non aliter idem ab&longs;urdum &longs;equetur, <lb/>
fi ponamus centrum portionis recedere à medio ad par­<lb/>
tes <foreign lang="greek">w;</foreign> e&longs;&longs;et enim aliquando centrum figuræ in&longs;criptæ idem <lb/>
quod portionis <expan abbr="centrũ">centrum</expan>. </s><s>ergo punctum e centrum erit gra<lb/>
uitatis portionis abc. quod demon&longs;trare oportebat.</s>
</p><p type="margin">
<s><margin.target id="marg103"/>7. huius</s></p><p type="margin">
<s><margin.target id="marg104"/>8. primi <lb/>
libri Ar­<lb/>
chimedis</s></p><p type="margin">
<s><margin.target id="marg105"/>11. duo­<lb/>
decimi.</s></p><p type="margin">
<s><margin.target id="marg106"/>15. quinti</s></p><p type="margin">
<s><margin.target id="marg107"/>2. duode­<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg108"/>20. primi <lb/>
<expan abbr="conicorũ">conicorum</expan></s></p><p type="margin">
<s><margin.target id="marg109"/>19.<lb/>
quinti</s></p><p type="main">
<s>Quod autem &longs;upra <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t in portione conoi­<lb/>
dis recta per figuras, quæ ex cylindris æqualem altitudi­<lb/>
dinem habentibus con&longs;tant, idem &longs;imiliter demon&longs;trabi­<lb/>
mus per figuras ex cylindri portionibus con&longs;tantes in ea <lb/>
portione, quæ plano non ad axem recto ab&longs;cinditur. </s><s>ut <lb/>
enim tradidimus in commentariis in undecimam propo&longs;i <lb/>
tionem libri Archimedis de conoidibus & &longs;phæroidibus. </s><lb/>
<s>portiones cylindri, quæ æquali &longs;unt altitudine eam inter &longs;e <lb/>
&longs;e proportionem habent, quam ip&longs;arum ba&longs;es: ba&longs;es <expan abbr="aut&etilde;">autem</expan> <lb/>
<arrow.to.target n="marg110"/><lb/>
quæ &longs;unt ellip&longs;es &longs;imiles eandem proportionem habere, <lb/>
quam quadrata diametrorum eiu&longs;dem rationis, ex corol­<lb/>
lario &longs;eptimæ propo&longs;itionis libri de conoidibus, & &longs;phæ­<lb/>
roidibus, manife&longs;te apparet.</s></p><p type="margin">
<s><margin.target id="marg110"/>corol. 15<lb/>
de conoi­<lb/>
dibus & <lb/>
&longs;phæroi­<lb/>
dibus.</s></p><p type="head">
<s>THEOREMA XXIIII. PROPOSITIO XXX.<!-- KEEP S--></s></p><p type="main">
<s>Si à portione conoidis rectanguli alia portio <lb/>
ab&longs;cindatur, plano ba&longs;i æquidi&longs;tante; habebit <lb/>
portio tota ad eam, quæ ab&longs;ci&longs;&longs;a e&longs;t, duplam pro <lb/>
portion em eius, quæ e&longs;t ba&longs;is maioris portionis <lb/>
ad ba&longs;i m minoris, uel quæ axis maioris ad axem <lb/>
minoris.</s></p>
<pb xlink:href="023/01/098.jpg"/><p type="main">
<s>ABSCINDATVR à portione conoidis rectanguli <lb/>
abc alia portio ebf, plano ba&longs;i æquidi&longs;tante: & eadem <lb/>
portio &longs;ecetur alio plano per axem; ut &longs;uperficiei &longs;ectio &longs;it <lb/>
parabole abc: <expan abbr="planorũ">planorum</expan> portiones ab&longs;cindentium rectæ <lb/>
lincæ ac, ef: axis autem portionis, & &longs;ectionis diameter <lb/>
bd; quam linea ef in puncto g &longs;ecet. </s><s>Dico portionem co­<lb/>
noidis abc ad portionem ebf duplam proportionem ha­<lb/>
bere eius, quæ e&longs;t ba&longs;is ac ad ba&longs;im ef; uel axis db ad bg<lb/>
axem. </s><s>Intelligantur enim duo coni, &longs;eu coni portiones <lb/>
abc, ebf, <expan abbr="eãdem">eandem</expan> ba&longs;im, quam portiones conoidis, & æqua <lb/>
lem habentes altitudinem. </s><s>& quoniam abc portio conoi <lb/>
dis &longs;e&longs;quialtera e&longs;t coni, &longs;eu portionis coni abc; & portio <lb/>
ebf coni &longs;eu portionis coni bf e&longs;t &longs;e&longs;quialtera, quod de­<lb/>
<figure id="id.023.01.098.1.jpg" xlink:href="023/01/098/1.jpg"/><lb/>
mon&longs;trauit Archimedes in propo&longs;itionibus 23, & 24 libri <lb/>
de conoidibus, & &longs;phæroidibus: erit conoidis portio ad <lb/>
conoidis portionem, ut conus ad conum, uel ut coni por­<lb/>
tio ad coni portionem. </s><s>Sed conus, nel coni portio abc ad <lb/>
conum, uel coni portionem ebf compo&longs;itam proportio­<lb/>
nem habet ex proportione ba&longs;is ac ad ba&longs;im ef, & ex pro­<lb/>
portione altitudinis coni, uel coni portionis abc ad alti­<lb/>
tudinem ip&longs;ius ebf, ut nos demon&longs;trauimus in com men­<lb/>
tariis in undecimam propo&longs;itionem eiu&longs;dem libri Archi­<lb/>
medis: altitudo autem ad altitudinem c&longs;t, ut axis ad axem. </s><lb/>
<s>quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue
<pb xlink:href="023/01/099.jpg" pagenum="46"/>ro ita demon&longs;trabitur. </s><s>Ducatur à puncto b ad planum ba­<lb/>
&longs;is ac perpendicularis linea bh, quæ ip&longs;am ef in K &longs;ecet. </s><lb/>
<s>erit bh altitudo coni, uel coni portionis abc: & bK altitu<lb/>
<arrow.to.target n="marg111"/><lb/>
do efg. </s><s>Quod cum lineæ ac, ef inter &longs;e æquidi&longs;tent, &longs;unt <lb/>
enim planorum æquidi&longs;tantium &longs;ectiones: habebit db ad <lb/>
<arrow.to.target n="marg112"/><lb/>
bg proportionem eandem, quam hb ad bk quare por­<lb/>
tio conoidis abc ad portionem efg proportionem habet <lb/>
compo&longs;itam ex proportione ba&longs;is ac ad ba&longs;im ef; & ex <lb/>
<arrow.to.target n="marg113"/><lb/>
proportione db axis ad axem bg. <!-- KEEP S--></s><s>Sed circulus, uel <lb/>
ellip&longs;is circa diametrum ac ad circulum, uel ellip&longs;im <lb/>
<arrow.to.target n="marg114"/><lb/>
circa ef, e&longs;t ut quadratum ac ad quadratum ef; hoc e&longs;t ut <lb/>
<expan abbr="quadratũ">quadratum</expan> ad ad <expan abbr="quadratũ">quadratum</expan> eg. <!-- REMOVE S-->& quadratum ad ad quadra <lb/>
tum eg e&longs;t, ut linea db ad lineam bg. <!-- KEEP S--></s>
<s>circulus igitur, uel el <lb/>
<arrow.to.target n="marg115"/><lb/>
lip&longs;is circa diametrum ac ad <expan abbr="circulũ">circulum</expan>, uel ellip&longs;im circa ef, <lb/>
<arrow.to.target n="marg116"/><lb/>
hoc e&longs;t ba&longs;is ad ba&longs;im eandem proportionem habet, <expan abbr="quã">quam</expan> <lb/>
db axis ad axem bg. <!-- KEEP S--></s><s>ex quibus &longs;equitur portionem abc <lb/>
ad portionem ebf habere proportionem duplam eius, <lb/>
quæ e&longs;t ba&longs;is ac ad ba&longs;im ef: uel axis db ad bg axem. </s><s>quod <lb/>
demon&longs;trandum proponebatur.</s></p><p type="margin">
<s><margin.target id="marg111"/>16. unde­<lb/>
cimi.</s></p><p type="margin">
<s><margin.target id="marg112"/>4 sexti.</s></p><p type="margin">
<s><margin.target id="marg113"/>2. duode<lb/>
cimi</s></p><p type="margin">
<s><margin.target id="marg114"/>7. de co­<lb/>
noidibus <lb/>
& &longs;phæ­<lb/>
roidibus</s></p><p type="margin">
<s><margin.target id="marg115"/>15. quinti. </s><s>quinti</s></p><p type="margin">
<s><margin.target id="marg116"/>20. primi <lb/>
<expan abbr="conicorũ">conicorum</expan></s></p><p type="head">
<s>THEOREMA XXV. PROPOSITIO XXXI.<!-- KEEP S--></s></p><p type="main">
<s>Cuiuslibet fru&longs;ti à portione rectanguli conoi <lb/>
dis ab&longs;cis&longs;i, centrum grauitatis e&longs;t in axe, ita ut <lb/>
demptis primum à quadrato, quod fit ex diame­<lb/>
tro maioris ba&longs;is, tertia ip&longs;ius parte, & duabus <lb/>
tertiis quadrati, quod fit ex diametro ba&longs;is mino­<lb/>
ris: deinde à tertia parte quadrati maioris ba&longs;is <lb/>
rur&longs;us dempta portione, ad quam reliquum qua <lb/>
drati ba&longs;is maioris unà cum dicta portione <expan abbr="duplã">duplam</expan> <lb/>
proportionem habeat eius, quæ e&longs;t quadrati ma­
<pb xlink:href="023/01/100.jpg"/>ioris ba&longs;is ad quadratum minoris: centrum &longs;it in <lb/>
eo axis puncto, quo ita diuiditur ut pars, quæ mi<lb/>
norem ba&longs;im attingit ad alteram partem eandem <lb/>
proportionem habeat, quam dempto quadrato <lb/>
minoris ba&longs;is à duabus tertiis quadrati maioris, <lb/>
habet id, quod reliquum e&longs;t unà cum portione à <lb/>
tertia quadrati maioris parte dempta, ad <expan abbr="reliquã">reliquam</expan> <lb/>
eiu&longs;dem tertiæ portionem.</s></p><p type="main">
<s>SIT fru&longs;tum à portione rectanguli conoidis ab&longs;ci&longs;&longs;um <lb/>
abcd, cuius maior ba&longs;is circulus, uel ellip&longs;is circa diame­<lb/>
trum bc, minor circa diametrum ad; & axis ef. </s><s>de&longs;criba­<lb/>
tur autem portio conoidis, à quo illud ab&longs;ci&longs;&longs;um e&longs;t, & pla­<lb/>
<figure id="id.023.01.100.1.jpg" xlink:href="023/01/100/1.jpg"/><lb/>
no per axem ducto &longs;ecetur; ut &longs;uperficiei &longs;ectio &longs;it parabo­<lb/>
le bgc, cuius diameter, & axis portionis gf: deinde gf diui <lb/>
datur in puncto h, ita ut gh &longs;it dupla hf: & rur&longs;us ge in ean <lb/>
dem proportionem diuidatur: <expan abbr="&longs;itq;">&longs;itque</expan> gk ip&longs;ius ke dupla. </s><s><expan abbr="Iã">Iam</expan> <lb/>
ex iis, quæ proxime demon&longs;trauimus, con&longs;tat centrum gra<lb/>
uitatis portionis bgc e&longs;&longs;e h punctum: & portionis agc <lb/>
punctum k. </s><s>&longs;umpto igitur infra h puncto l, ita ut kh ad hl
<pb xlink:href="023/01/101.jpg" pagenum="47"/>eam proportionem habeat, quam abcd fru&longs;tum ad por­<lb/>
tionem agd; erit punctum l eius fru&longs;ti grauitatis <expan abbr="c&etilde;trum">centrum</expan>: <lb/>
<expan abbr="habebitq;">habebitque</expan> componendo Kl ad lh proportionem eandem, <lb/>
<arrow.to.target n="marg117"/><lb/>
quam portio conoidis bgc ad agd portionem. </s><s><expan abbr="Itaq;">Itaque</expan> quo <lb/>
niam quadratum bf ad quadratum ae, hoc e&longs;t quadratum <lb/>
bc ad quadratum ad e&longs;t, ut linea fg ad ge: erunt duæ ter­<lb/>
tiæ quadrati bc ad duas tertias quadrati ad, ut hg ad gk: <lb/>
& &longs;i à duabus tertiis quadrati bc demptæ fuerint duæ ter­<lb/>
tiæ quadrati ad: erit <expan abbr="diuid&etilde;do">diuidendo</expan> id, quod relinquitur ad duas <lb/>
tertias quadrati ad, ut hk ad kg. <!-- KEEP S--></s><s>Rur&longs;us duæ tertiæ quadra <lb/>
ti ad ad duas tertias quadrati bc &longs;unt, ut kg ad gh: & duæ <lb/>
tertiæ quadrati bc ad <expan abbr="tertiã">tertiam</expan> <expan abbr="part&etilde;">partem</expan> ip&longs;ius, ut gh ad hf. </s><s>ergo <lb/>
ex æquali id, quod relinquitur ex duabus tertiis quadrati <lb/>
bc, demptis ab ip&longs;is quadrati ad duabus tertiis, ad <expan abbr="tertiã">tertiam</expan> <lb/>
partem quadrati bc, ut kh ad hf: & ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>
tertiæ partis, ad quam unà cum ip&longs;a portione, duplam pro <lb/>
portionem habeat eius, quæ e&longs;t quadrati bc ad <expan abbr="quadratũ">quadratum</expan> <lb/>
ad, ut Kl ad lh. </s><s>habet enim Kl ad lh eandem proportio­<lb/>
nem, quam conoidis portio bgc ad portionem agd: por­<lb/>
tio autem bgc ad portionem agd duplam proportionem <lb/>
habet eius, quæ e&longs;t ba&longs;is bc ad ba&longs;im ad: hoc e&longs;t quadrati <lb/>
<arrow.to.target n="marg118"/><lb/>
bc ad quadratum ad; ut proxime demon&longs;tratum e&longs;t. </s><s>quare <lb/>
dempto ad quadrato à duabus tertiis quadrati bc, erit id, <lb/>
quod relinquitur unà cum dicta portione tertiæ partis ad <lb/>
reliquam eiu&longs;dem portionem, ut el ad lf. </s><s>Cum igitur cen­<lb/>
trum grauitatis fru&longs;ti abcd &longs;it l, à quo axis ef in eam, <expan abbr="quã">quam</expan> <lb/>
diximus, proportionem diuidatur; con&longs;tat <expan abbr="uerũ">uerum</expan> e&longs;&longs;e illud, <lb/>
quod demon&longs;trandum propo&longs;uimus.</s></p><p type="margin">
<s><margin.target id="marg117"/>20. 1. coni<lb/>
corum.</s></p><p type="margin">
<s><margin.target id="marg118"/>30 huius</s></p><p type="head">
<s>FINIS LIBRI DE CENTRO<!-- REMOVE S-->GRAVITATIS SOLIDORVM.<!-- KEEP S--></s></p><p type="main">
<s>Impre&longs;&longs;. <!-- REMOVE S-->Bononiæ cum licentia Superiorum, </s>
</p> </chap> </body> <back/> </text></archimedes>
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